Robust Control System Design Advanced State Space Techniques 2ed PDF
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Robust Control S y s t e m Design d v an c ed S t a t e Space Techniques Sec ond Edition Edition Revised Revised and Expanded
Chia Chi Tsui DeVly Znstitute o Technology Long Island City New York U.S.A.
Copyright 2004 by Marcel Dekker, Inc. All Rights Reserved.
Although great care has been taken to provide accurate and current information, ne neit ither her th thee au auth thor or(s (s)) no norr the the pu publ blis ishe her, r, no norr an anyo yone ne else else asso associ ciat ated ed wi with th this this publication, shall be liable for any loss, damage, or liability directly or indirectly caused or alleged to be caused by this book. The material contained herein is not intended to provide specific advice or recommendations for any specific situation. Trademark Tradem ark notice notice:: Produc Productt or corpor corporate ate names names may be tradem trademark arkss or regist registere ered d trademarks and are used only for identification and explanation without intent to infringe. Library of Congress Cataloging-in-Publication Data A catalog record for this book is available from the Library of Congress. ISBN: 0-8247-4869-7
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Copyright 2004 by Marcel Dekker, Inc. All Rights Reserved.
CONT ROL ENGINEERING ENGINEERING A Series of Reference Books and Textbooks
Editors NEIL MUNRO PH.D. D.Sc. Professor Applied C ontrol Engineering Engineering University of M anchester Institute Institute of Science and Technology Manchester, United Kingdom FR NK L.
LEWIS PH.D.
Moncrief-O Donnell Endow ed Chair and Associate D irector irector of Research Automation Robotics Research Institute Institute University of Texas, Arlington
1. Nonlinear Control Contr ol o off Elec Electri tric c Machi Machinery nery Darren M. Dawson, Jun Hu, 2 3
4. 5
and Timothy C . Burg Computational Intelli Intelligence gence n Control Engine Engineering ering Robert E . King Quantitative Feedback Theory: Fundamentals and Applications Constantine H. H oupis and S teven J. Rasm ussen Self-Learning Control of Finite Mark Markov ov Chains A S. Poznya k, K. Najim, and E. Gomez-Ramirez Robust Control and Filtering for Time-Delay Systems Magdi S. Mahmoud
6. Classical Enright Feedback Control: With MATLAB Boris J. Lurie and P aul J. 7 Optimal Control of Singularly Perturbed Linear Systems and Applications: High-Accuracy Techniques Zoran Gajic and Myo-Taeg Lim 8 Engineering System Dynamics: A Unified Graph-Centered Approach Forbes T. Brown 9. Advanced Process Identification and Control Enso lkonen and Kaddour Najim 10. Modern Control Engineering P. N. Paraskevopoulos 11. Sliding Mode Control in Engineering edited by Wilfrid Perruquetti and Jean Pierre Barbot 12. Actuator Saturation Control edited by by Vikram Kapila and Karolos M. Grigoriadis
Copyright 2004 by Marcel Dekker, Inc. All Rights Reserved.
13. Nonlinear Control Systems
DonlagiC, Donla giC, Sejid T esnjak
Zoran Vukic, Ljubomir KuljaCa, Dali
14. Linear Control System Analysis and Design Desig n with MATLAB: Fifth Edition Revised and Expanded John J. D’Azzo, Constanfine H . Houpis, and
Stuart N . Sheldon
15. Robot Manipulator Manipula tor Control: Theory and Practice Second Edition Revised and Expanded Frank L. Lewis, Darren M. Dawson, and Chaouki T . Abdallah 16. Robust Control System Design: Design: Advanced State Space Techniques Second Edition Edition Revised and Expanded Chia-Chi Tsui Additional Volumes in Preparation
Copyright 2004 by Marcel Dekker, Inc. All Rights Reserved.
To Susan and James and Shane
Copyright 2004 by Marcel Dekker, Inc. All Rights Reserved.
Series Introduction
Many textbooks have been written on control engineering, describing new techniques for controlling systems, or new and better ways of mathematically all y for formul mulati ating ng exi existi sting ng met methods hods to sol solve ve the ever ever-in -incre creasi asing ng com comple plex x problems faced by practicing engineers. However, few of these books fully address the applications aspects of control engineering. It is the intention of this new series to redress this situation. The series will stress applications issues, and not just the mathematics
of control engineering. It will provide texts that present not only both new and an d we well ll-e -est stab abli lish shed ed te tech chni niqu ques es,, bu butt also also de deta tail iled ed exam exampl ples es of th thee application of these methods to the solution of real-world problems. The auth au thor orss wi will ll be dr draw awn n fr from om bot both h th thee aca academ demic ic wo worl rld d and th thee rele relevan vantt applications sectors.
Copyright 2004 by Marcel Dekker, Inc. All Rights Reserved.
There are already many exciting examples of the applic application ation of control tech techniq niques ues in th thee es esta tabl blis ished hed fiel fields ds of el elect ectri rical cal,, me mech chani anical cal (incl (includi uding ng ae aero rosp space ace), ), and che chemi mica call eng engin ineer eerin ing. g. We hav havee onl only y to look look aroun around d in to toda day’s y’s hi highl ghly y aut autom omate ated d so soci ciet ety y to se seee the the use of ad advan vanced ced ro robot botic icss techniques in the manufacturing industries; the use of automated control and navigation systems in air and surface transport systems; the increasing us usee of in inte tell llig igent ent con contr trol ol sy syst stem emss in th thee ma many ny arti artifa facts cts ava avail ilabl ablee to th thee domes do mesti ticc con consu sume merr ma mark rket et;; and the the re reli liabl ablee suppl supply y of wat water er,, ga gas, s, and electrical power to the domestic consumer and to industry. However, there are cur curren rently tly man many y chal challen lenging ging pro proble blems ms tha thatt cou could ld bene benefit fit fro from m wid wider er exposure to the applicability of control methodologies, and the systematic systems-oriented basis inherent in the application of control techniques. This Th is se seri ries es pres present entss boo books ks th that at dr draw aw on exp exper erti tise se from from bot both h th thee academic world and the applications domains, and will be useful not only as ac acad adem emic ical ally ly re reco comm mmen ende ded d co cour urse se te text xtss bu butt al also so as ha hand ndbo book okss fo forr practi pra ctitio tioner nerss in man many y appl applica icatio tions ns doma domains ins.. Robust Robust Contr Control ol Syst Systems ems is another outstanding entry in Dekker’s Control Engineering series.
Copyright 2004 by Marcel Dekker, Inc. All Rights Reserved.
Preface
This seco This second nd edi editi tion on of of Robust Robust Cont Control rol Syst System em Desi Design gn intro introduc duces es a new de desi sign gn ap appr proa oach ch to mo mode dern rn cont contro roll syst system ems. s. Th This is de desi sign gn ap appr proa oach ch
guarantees, for the first time, the full realization of robustness properties of generalized state feedback control for most open-loop system conditions. State and generalized state feedback control can achieve feedback system performance and robustness far more effectively than other basic forms of control. Performance and robustness (versus model uncertainty and control distur dis turbanc bance) e) are mut mutuall ually y cont contrad radicto ictory, ry, yet the they y are the key pro proper perties ties required by practical control systems. Hence, this design approach not only enriches the existing modern control system design theory, but also makes possible its wide application.
Copyright 2004 by Marcel Dekker, Inc. All Rights Reserved.
Modern (or state space) control theory was developed in the 1960s. The th The theo eory ry ha hass ev evol olve ved d su such ch th that at th thee stat statee fe feed edba back ck co cont ntro roll an and d it itss impleme imp lementi nting ng obse observe rverr are des design igned ed separately (follow (following ing the soso-cal called led sepa separa rati tion on prin princi ciple ple [W [Wil il,, 199 1995] 5]). ). Wi With th th this is exi exist stin ing g desi design gn app appro roach ach,, although the direct state feedback system can be designed to have good performance and robustness, almost all the actual corresponding observer feed feedbac back k sy syst stem emss ha have ve ent entir irel ely y diff differ eren entt robus robustn tnes ess. s. In th thee new desi design gn approach presented here, the state feedback control and its implementing observer observ er are designed designed together together.. More explicitly, the state feedback control is designed based on the results of its implementing observer. The resulting state feedback control is the generalized the generalized state feedback control [Tsui, [Tsui, 1999b]. This Th is fu funda ndamen menta tall lly y new ap appr proa oach ch gua guara rant ntees ees—f —for or all all op openen-lo loop op syste ystems ms wi witth mo morre ou outp tput utss tha han n inp nput utss or wi with th at lea easst on onee sta tabl blee tr tran ansm smis issi sion on zer zero— o—the the sa same me loop loop tr tran ansf sfer er funct function ion and ther therefo efore re th thee samee rob sam robust ustness ness of the obs observe erverr feed feedback back sys system tem and the cor corres respond ponding ing direct state feedback system. Most open-loop systems satisfy either of these two conditions. For all other open-loop systems, this approach guarantees that the difference between the loop transfer functions of the above two feedback systems be kept minimal in a simple least-square sense. Modern and classical control theories are the two major components of control systems theory. Compared with classical control theory, modern control theory can describe a single system’s performance and robustness more accurately, but it lacks a clear concept of feedback system robustness, such as the loop transfer function of classical control theory. By fully using the concept of loop transfer functio functions, ns, the approa approach ch exploi exploits ts the advanta advantages ges of both classi classical cal and modern control theorie theories. s. This approac approach h guarant guarantees ees the robust rob ustnes nesss and loo loop p tra transf nsfer er fun functi ction on of clas classic sical al cont control rol the theory, ory, whi while le desig des igni ning ng this this loop loop tr trans ansfe ferr fu funct nction ion mu much ch mo more re ef effe fect ctive ively ly (t (tho hough ugh indirectly) using modern control design techniques. Thus it achieves both good robustness and performance for feedback control systems.
If the first edition of this book emphasized the first of the above two advantages (i.e., the true realization of robustness properties of feedback control), then this second edition highlights the second of the above two adva ad vant ntag ages es—t —the he fa farr mo more re ef effe fect ctiv ivee de desi sign gn of high high pe perf rfor orma manc ncee an and d robustness feedback control itself. A useful control theory should provide general and effective guidance on co com mplic plicat ated ed co cont ntro roll sy syst stem em de desi sign gn.. To achi achiev evee this, his, the the de desi sign gn formulation must fully address both performance and robustness. It must also exploit fully the existing design freedom and apply a general, simple, and explicit design procedure. The approach presented here truly satisfies th thes esee requ requir irem emen ents ts.. Sinc Sincee th this is bo book ok conc concen entr trat ates es on th this is ne new w de desi sign gn approach and its relevant analysis, other analytical control theory results are
Copyright 2004 by Marcel Dekker, Inc. All Rights Reserved.
presented presen ted wit with h an emph emphasi asiss on the their ir phys physical ical meanin meanings, gs, ins instea tead d of the their ir detailed mathematical derivations and proofs. The following list shows several of the book’s most important results. With the exception of the third item, these results are not presented in any other books: 1.
The fir first st gen genera erall dynam dynamic ic out output put fe feedba edback ck comp compens ensator ator tthat hat can impleme imp lement nt sta state te or gene general ralized ized sta state te fee feedbac dback k cont control rol,, and its design procedure. The feedback system of this compensator is the fir first st gene genera rall fe feed edba back ck sy syst stem em that that ha hass th thee sa same me robu robust stne ness ss prope propert rtie iess of its its corr corresp espon ondi ding ng dire direct ct stat statee fe feed edbac back k syst system em (Chapters 3 3 to to 6) 6)..
2.
A syste systemat matic, ic, sim simple ple,, and exp explic licit it eigen eigenvalu valuee ass assign ignmen mentt pro proceceduree usi dur using ng sta static tic out output put fee feedbac dback k cont control rol or gene general ralize ized d sta state te fe feed edbac back k con contr trol ol (Sec (Secti tion on 8.1) 8.1).. Thi Thiss pr proce ocedur duree ena enabl bles es th thee systematic eigenvector assignment procedures of this book, and is general to most open-loop system conditions if based on the generalized state feedback control of this book. Eige Eigenv nvec ecto torr ass assig ignm nmen entt proc proced edur ures es th that at ca can n fully ully use use the the fr free eedom dom of th this is as assi sign gnmen ment. t. Bo Both th num numer erica icall algo algori rith thms ms an and d analytical procedures are presented (Section 8.2). A ge gene nera rall fa fail ilur uree de dete tect ctio ion, n, isol isolat atio ion, n, an and d ac acco comm mmod odat atio ion n comp compen ensa sato torr th that at is ca capa pabl blee of cons consid ider erin ing g syst system em mo mode dell uncerta unce rtaint inty y and mea measur suremen ementt noi noise, se, and its sys system temati aticc des design ign 10).. procedure (Chapter 10) procedure Th Thee simpl simples estt po poss ssib ible le fo form rmul ulat atio ion, n, and a trul truly y syst system emat atic ic an and d general procedure, procedure, of minim minimal al order obser observer ver design (Chapter 7) 7).. So Solu luti tion on of th thee matr matrix ix equ equat atio ion n TA FT FT = LC LC [matrix pair
3.
4.
5. 6.
7.
8.
(A, C ) is observable and eigenvalues of matrix F are are arbitrarily assigned]. This solution is general and has all eigenvalues of of F and all rows of T T completely decoupled (F (F is in Jordan form). This Th is so solu luti tion on un uniq iquel uely y enabl enables es th thee full full use of th thee rema remaini ining ng fr free eedo dom m of th this is ma matr trix ix equa equati tion on,, wh whic ich h is fu fund ndam amen enta tall lly y impo im port rtan antt in mo most st of th thee ba basi sicc de desi sign gn prob proble lems ms of mo mode dern rn control theory (Chapters theory (Chapters 5 5 to to 8, 8, 10) 10).. The bas basic ic desi design gn conc concept ept of gen genera eratin ting g a state fee feedbac dback k contr control ol signal sig nal wit without hout est estima imatin ting g all sta state te var variab iables les,, and the gene general ral-ization of this design concept from function observers only to all feedback compensators (Chapters (Chapters 3 3 to to 10) 10).. Th Thee co comp mpllet etee un uniific ficat atio ion n of tw two o ex exis isti ting ng ba basi sicc fee eedb dbac ack k structures of modern control theory—the zero input gain state
Copyright 2004 by Marcel Dekker Inc All Rights Reserved
9.
10.
11.. 11
12.
observer obser ver fe feedb edbac ack k st stru ruct ctur uree an and d th thee stati staticc ou outp tput ut fe feedb edbac ack k structure (Section 6.3). A mo more re ge gene nera rall lly y ac accu cura rate te robu robust st stab stabil ilit ity y me meas asur uree th that at is expressed in terms of the sensitivities of each system pole. This analy ana lyti tical cal me meas asure ure can be used used to gui guide de syst system emat atic ic fe feed edbac back k system design (Sections 2.2.2 and 8.2). Com Compar pariso ison n of comput computati ational onal comp complexi lexity ty and the theref refore ore trac trackkability (ability to adjust the original design formulation based on the final and numerical design results) of all feedback control design techniques (Section 9.3). Em Emph phas asis is on th thee dist distin inct ct ad adva vant ntag ages es of high high pe perf rfor orma manc nce/ e/ ro robu bust stnes nesss con contr trol ol des desig ign n usin using g ei eige genst nstru ruct ctur uree assi assignm gnmen entt te techn chniq iques ues ove overr th thee te tech chniq niques ues fo forr th thee dire direct ct des desig ign n of loop loop transfer functions (Chapters (Chapters 2, 2, 3, 3, 8, 8, 9) 9).. Th Thee conce concept pt of ad adapt aptiv ivee con contr trol ol and its its app appli lica cati tion on in fa fail ilur uree accommodation and control (Section 10.2).
The first five of the above results are actual design results. The last seven are new theoretical results and concepts that have enabled the establishment of the first five results. In other words, the main new result (result 1, the full realiz rea lizati ation on of rob robust ustnes nesss prop propert erties ies of sta state/ te/gene general ralized ized sta state te fee feedbac dback k control) is enabled by some significant and fundamental developments (such as results 6 to 8), and is validated by the distinct effectiveness of state/ generalized state feedback control (results 2 to 3 and 9 to 11). This book also addresses the computational reliability of its analysis and design algorithms. This is because practical control problems usually
require a large amount of computation, and unreliable computation can yield totally unreliable results. Every effort has been made to use reliable computational methods in design algorithms, such as the computation of Hessenberg form (instead of the canonical form) and of orthogonal matrix operation (instead of elementary matrix operation). As a result, the computation required in this book is slightly more complicated, but the more reliable results thus obtained make the effort wort wo rthw hwhi hile le.. It sh shou ould ld be no note ted d th that at th thee comp comput utat atio ion n of po poly lynom nomia ials ls req equi uire red d by the clas classi sica call co cont ntro roll the heor ory y is usua usuall lly y un unre rellia iabl ble. e. Th Thee develop deve lopment ment of comp computa utatio tional nal sof softwa tware re has als also o eas eased ed con consid siderab erably ly the complexity of computation. Each design procedure is presented in algorithm form, and each step of these algorithms can be implemented directly by the existing computational software. This book will be useful to control system designers and researchers. Although a solid background in basic linear algebra is required, it requires remarka rem arkably bly les lesss mat mathem hemati atical cal sop sophis histica ticatio tion n tha than n oth other er book bookss sim simila ilarr in
Copyright 2004 by Marcel Dekker, Inc. All Rights Reserved.
scope. This book can also be used as a textbook for students who have had a firstt cour firs course se (pr (prefer eferabl ably y inc includ luding ing sta state te space space the theory ory)) in cont control rol sys system tems. s. Multi-input and multi-output systems are discussed throughout. However, reader rea derss wil willl find tha thatt the res result ultss have been subs substan tantia tially lly sim simpli plified fied to be quite easily understandable, and that the results have been well unified with the single-input and single-output system results. In addition, this book is comprehensive and self-contained, with every topic introduced at the most basic level. Thus it could also be used by honor program students with background in signals and systems only. An ov over ervi view ew of eac each h cha chapt pter er fo foll llows ows.. Cha Chapt pter er 1 introduces introduces basic Chapter pter 2 analyzes analyzes the per perfor forman mance ce and system sys tem mode models ls and pro proper pertie ties. s. Cha sensitivity of a single overall system. Chapter 3 3 describes describes the critical role of loop transfer functions on the sensitivity of feedback systems, including the observer feedback systems. Chapter 4 proposes the new design approach and anal analyzes yzes its adva advanta ntages ges.. Cha Chapter pter 5 prese present ntss the the so solu luti tion on of a ba basi sicc matrix equation. This solution is used throughout the remaining chapters (except Chapter (except Chapter 9 9). ). Chapter Chapter 6 presents 6 presents the design of the dynamic part of the observer such that for any state feedback control signal generated by this observer, the loop transfer function of this control is also fully realized. Chapter 7 presents 7 presents the design of the function observer, which generates an arbitr arb itrari arily ly giv given en sta state te fee feedbac dback k cont control rol signal signal,, wit with h min minimi imized zed obse observe rverr order. Chapter 8 presents the eigenvalue/vector assignment control design order. methods. Chapter methods. Chapter 9 9 introduces introduces the linear linear quadratic optimal control design 8 and and 9 9 will will determine the output part of methods. Both designs of Chapters Chapters 8
the observer of Chapter Chapter 6 6,, as well as the ‘‘target’’ closed-loop system loop tr tran ansf sfer er funct functio ion. n. Co Comp mpari ariso son n of var vario ious us des desig igns ns revea reveals ls tw two o dist distin inct ct advantages of eigenstructure assignment design. Chapter 10 10 deals with the design of a general failure detection, isolation, and (adaptive) accommodation compensator that is capable of considering system model uncertainty and meas measurem urement ent noi noise. se. Thi Thiss com compens pensato atorr has the com compati patible ble str structu ucture re of—and of— and can be imp impleme lemente nted d in coor coordin dinati ation on wit with—t h—the he norm normal al (fr (free ee of major failure) robust control compensator of this book. There is a set of simple exercises simple exercises at at the end of each chapter. To ma make ke the the bo book ok se self lf-c -con ontai taine ned, d, Ap Appen pendi dix x A provi provides des a simpl simplee introduction to the relevant mathematical background material. Appendix B lists the mathematical models of eight real-world systems for synthesized design practi practice. ce. I would like to thank everyone who helped me, especially during my student years. I also thank my former student Reza Shahriar, who assisted with some of the computer graphics. Chia-Chi Tsui
Copyright 2004 by Marcel Dekker, Inc. All Rights Reserved.
Contents
Series Introduction Preface 1.
System Sys tem Math Mathemat ematical ical Model Modelss and Basic Basic Prope Properti rties es 1.1 1.1 1.2 1.3 1.4 1. 4
Two Ki Two Kind ndss of Mat Mathem hemat atica icall Mo Model delss Eigens Eig enstru tructu cture re Deco Decompo mposit sition ion of a Sta State te Spac Spacee Model Model System Sys tem Ord Order, er, Cont Control rollab labili ility, ty, and Obs Observ ervabi abilit lity y Syst Sy stem em Po Pole less an and d Ze Zero ross Exercises
Copyright 2004 by Marcel Dekker, Inc. All Rights Reserved.
2
SingleSin gle-Sys System tem Per Perfor formanc mancee and Sens Sensiti itivit vity y 2.1 2.1 2.2 2. 2
3
4
System Syst em Pe Perf rfor orma manc ncee Syst Sy stem em Sen Sensi siti tivi vity ty an and d Ro Robus bustne tness ss Conclusion Exercises
Feedb Fe edbac ack k Sys Syste tem m Sen Sensi siti tivi vity ty 3.1 3. 1
Sensit Sens itiv ivit ity y and and Lo Loop op Tr Trans ansfer fer Fu Func ncti tion on of Feedback Systems
3.2 3. 2
Se Sens nsit itiv ivit ity y of Fee Feedba dback ck Sys Syste tems ms of of Mode Modern rn Cont Contro roll Theory Summary
A New New Fe Feedb edback ack Co Cont ntro roll Des Desig ign n Appr Approac oach h 4.1 4. 1
4.2 4. 2
Basicc Des Basi Desig ign n Conc Concept ept of Ob Obse serv rver ers— s—Di Direc rectt Generation of State Feedback Control Signal Without Explicit System States Perf Pe rfor orman mance ce of Ob Obse serv rver er Fee Feedb dback ack Sys Syste tems ms— —
4.3 4.3 4.4 4. 4
5
Solut So lutio ion n of Mat Matri rix x Eq Equat uatio ion n TA FT FT = = LC 5.1 5.2 5. 2
6
Separation Property Thee Cur Th Curre rent nt St Stat atee of of LTR LTR Ob Obse serv rver er De Desi sign gn A New New De Desi sign gn Ap Appr proa oach ch an and d New New Fe Feed edba back ck Structure—A Dynamic Output Feedback Compensator that Generates State/ Generalized State Feedback Control Signal Exercises
Computati Comput ation on of a Sys System’ tem’ss Obs Observa ervable ble Hes Hessenb senberg erg Form Solv So lvin ing g Ma Matr trix ix Eq Equa uati tion on TA FT FT = = LC Exercises
Observe Obs erverr (Dynam (Dynamic ic Part) Part) Des Design ign for for Robus Robustne tness ss Real Realiza ization tion 6.1 6.1 6.2 6. 2 6.3 6. 3 6.4
Soluti Solu tion on of Ma Matr trix ix Eq Equa uati tion on TB = 0 Anal An alys ysis is an and d Exam Exampl ples es of Thi Thiss Des Desig ign n Solu Soluti tion on Compl Co mplet etee Uni Unific ficati ation on of Tw Two o Exis Existi ting ng Ba Basi sicc Modern Control System Structures Observe Obs erverr Ord Order er Adj Adjust ustmen mentt to Tra Tradeo deoff ff Bet Between ween Performance and Robustness Exercises
Copyright 2004 by Marcel Dekker, Inc. All Rights Reserved.
7
Obse Ob serv rver er De Desi sign gn fo forr Min Minim imiz ized ed Ord Order er 7.1 7.1 7.2 7. 2 7.3 7. 3
8
Design Des ign of of Feedb Feedback ack Contr Control— ol—Eig Eigens enstru tructu cture re Assi Assignm gnment ent 8.1 8.1 8.2 8. 2
9
Design Desi gn Fo Form rmul ulat atio ion n Desi De sign gn Al Algor gorit ithm hm an and d It Itss An Anal alys ysis is Exam Ex ampl ples es and Si Signi gnifica ficanc ncee of of Thi Thiss De Desi sign gn Exercises
Select Sele ctio ion n and and Plac Placem ement ent of Fee Feedba dback ck Sys Syste tem m Pole Poless Eigen Ei genvec vector tor As Assi sign gnmen mentt Summary Exercises
Design Des ign of of Feedb Feedback ack Contr Control— ol—Quad Quadrat ratic ic Opti Optimal mal Cont Control rol 9.1 9.1 9.2 9. 2 9.3 9. 3
Design Desi gn of Di Dire rect ct St Stat atee Feed Feedbac back k Cont Contro roll Desi De sign gn of Ge Gener neral aliz ized ed Sta State te Fe Feedb edbac ack k Cont Contro roll Compa Co mpari riso son n and and Conc Conclu lusi sion on of Fee Feedba dback ck Con Contr trol ol
Designs Exercises 10
Design of Fai Design Failur luree Detec Detectio tion, n, Iso Isolat lation ion,, and and Accommodation Compensators 10.1 10.1 10.2 10. 2 10.3 10. 3
Failuree Det Failur Detecti ection on and Iso Isolat lation ion Adapti Ada ptive ve State State Feedb Feedback ack Cont Control rol for for Failu Failure re Accommodation The Tre Treatm atment ent of Model Model Unc Uncert ertain ainty ty and and Measurement Noise Exercises
Appendix A: Relevant Linear Algebra and Numerical Linear Algebra Appendix B: Design Projects and Problems References
Copyright 2004 by Marcel Dekker, Inc. All Rights Reserved.
1 System Mathematical Models and Basic Properties
Unlike other Unlike other eng engine ineerin ering g specia specialit lities ies wh whose ose sub subjec jectt of study study is a spe specifi cificc engineering system such as an engine system or an airborne system, control systems theory studies only a general mathematical model of engineering systems. This chapter introduces two basic mathematical models and some basic system properties revealed by these models. There are four sections in this chapter. Sectio Sec tion n 1.1 introd introduce ucess the state state space space mo model del and tra transf nsfer er fun functi ction on model of linear time-invariant multi-input and multi-output systems, and the basic relationship between these two models. Sectio Sec tion n 1.2 describ describes es the eigens eigenstru tructur cturee decom decompos positi ition on of the sta state te space model, where the dynamic matrix of this model is in Jordan form.
Copyright 2004 by Marcel De Dekker, kker, Inc. All Rights Reserved.
Section 1.3 introd Section introduce ucess two basic basic system system proper propertie ties—c s—cont ontrol rollab labilit ility y and observability. Section 1.4 introduces two basic system parameters—system poles and zeros. These properties and parameters can be simply and clearly described based on the eigenstructure decomposition of the state space model.
1.1
TWO KIN KINDS DS O OF F MA MATHE THEMAT MATICA ICAL L MODE MODELS LS
This book studies only the linear time-invariant systems, which have also been the main subject of control systems theory. A linear time-invariant system can be represented by two kinds of mathematical models—the state space model and the transfer function model. The control theory based on thee stat th statee sp spac acee mo mode dell is ca call lled ed the the ‘‘st ‘‘stat atee spac spacee co cont ntro roll th theo eory ry’’ ’’ or the the
modern mode rn cont contro roll theo theory ry,, and and the the cont contro roll th theo eory ry ba base sed d on th thee tr tran ansf sfer er function model is called the ‘‘classical control theory.’’ We will first introduce the state space model and its derivation. A state space model is formed by a set of first-order linear differential equati equ ations ons wit with h consta constant nt coe coeffic fficien ients ts (1. (1.1a) 1a) and a set of lin linear ear equ equati ations ons (1.1b) x_ ðtÞ ¼ A xðtÞ þ BuðtÞ yðtÞ ¼ C xðtÞ þ DuðtÞ
ð1:1aÞ ð1:1bÞ
where 0
xðtÞ ¼ ½x1 ðtÞ; . . . ; xn ðtÞ is the system state vector (the prime symbol stands for transpose) xi ðtÞ; i ¼ ¼ 1 ; . . . ; n are the system state variables 0 uðtÞ ¼ ½u1 ðtÞ; . . . ; u p ðtÞ is the system input 0 y1 ðtÞ; . . . ; ym ðtÞ is the system output yðtÞ ¼ ½ y
and th thee sy syst stem em matr matric icees ðA; B; C ; DÞ are are real real,, con onst stan ant, t, an and d wit ith h dimensions n6n; n6 p; m6n, and m6 p, respe respectivel ctively. y. In the above model, Eq. (1.1a) is called the ‘‘dynamic equation,’’ which describes the ‘‘dynamic part’’ of the system and how the initial system state xð0Þ and system input u ðtÞ will determine the system state x ðtÞ. Hence matrix A is called the ‘‘dynamic matrix’’ of the system. Equation (1.1b) describes how the system state xðtÞ and system input uðtÞ will insta instantly ntly deter determine mine system output yðtÞ. This is the ‘‘output part’’ of the system and is static (memoryless) as compared with the dynamic part of the system.
Copyright 2004 by Marcel De Dekker, kker, Inc. All Rights Reserved.
From the definition of (1.1), parameters p and m represent the number of sy syst stem em in inpu puts ts an and d outp output uts, s, resp respec ecti tive vely. ly. If p > 1, then we call the corresponding system ‘‘multi-input.’’ If m > 1, then we call the corresponding system ‘‘multi-output.’’ A multi-input or multi-output system is also called a ‘‘MIMO system.’’ On the other hand, a system is called ‘‘SISO’’ if it is both single-input and single-output. In (1.1), the physical meaning of system state xðtÞ is used to describe completely comp letely the energy distribution distribution of the system at time t , espec especially ially at t ¼ 0 (initial time of system operation). For exampl example, e, in electr electrical ical circui circuitt system systemss with with lin linear ear tim time-in e-invar varian iantt circui cir cuitt elemen elements ts (in (indu ducto ctors, rs, res resist istors ors,, and capaci capacitors tors), ), the sys system tem state state is formed by all independent capacitor voltages and inductor currents. Thus its initial condition x ð0Þ can completely describe the initial electrical charge and initial magnetic flux stored in the circuit system.
Another example is in linear motion mechanical systems with linear time-invariant elements (springs, dampers, and masses), in which the system state is formed by all independent mass velocities and spring forces. Thus its initial state xð0Þ completely describes the initial dynamic energy and initial potential energy stored in the mechanical system. Because of this reason, the number ðnÞ of system states also indicates the number of the system’s independent energy storage devices.
Example 1.1 The fo The foll llow owing ing el elec ectri trica call ci circ rcui uitt syste system m is a line linear ar ti time me-i -inv nvar arian iantt sy syst stem em (Fig. 1.1). Letting v1 ðtÞ and v2 ðtÞ be the node voltages of the circuit, and letting the capacitor voltage and inductor current be the two system states x 1 ðtÞ and x2 ðtÞ, respectively, we have v1 ðtÞ ¼ x 1 ðtÞ and v2 ðtÞ ¼ x 1 ðtÞ R2 x2 ðtÞ
ð1:2Þ
Figure 1.1 A linear ti time-inv me-invariant ariant ci circuit rcuit sys system. tem.
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In other words, all node voltages and branch currents can be expressed in terms of system states and inputs. Thus the system’s output part (1.1b) can be dire direct ctly ly de deri rive ved. d. For For exam exampl ple, e, if the the ou outp tput ut yðtÞ is de desi sign gnat ated ed as 0 ½v1 ðtÞ; v2 ðtÞ , then from (1.2),
ð Þ
v1 t 1 0 ¼ yðtÞ ¼ v2 ðtÞ 1 R2
ð Þ
x1 t þ 0 4 C xðtÞ þ 0uðtÞ x2 ðtÞ ¼
The dynamic equation of this circuit system can also be derived by standard circuit analysis. Applying Kirchoff’s current law at each node of the circuit, we have
v1 ðtÞ ½v1 ðtÞ v2 ðtÞ i ðtÞ ¼ C v_1 ðtÞ þ þ R2 R1 ½ v2 ðtÞ v1 ðtÞ ½ v2 ðtÞ dt þ 0 ¼ R2 L
ð1:3aÞ
R
ð1:3bÞ
Substituting (1.2) into (1.3) and after simple manipulation [including taking derivatives on both sides of (1.3b)], we can have the form of (1.1a)
1 1 1 x_ 1 ðtÞ ¼ ðCR1 Þ x1 ðtÞ þ C x2 ðtÞ þ C i ðtÞ R2 1 Þx2 ðtÞ x_ 2 ðtÞ ¼ x1 ðtÞ þ ð
L
L
Thus comparing (1.1a), the system matrices are A ¼
=ð
1 CR1 Þ 1=L
1=C R2 =L
B ¼
= 1 C 0
Example 1.2 The following linear motion mechanical system is a linear time-invariant system (Fig. system (Fig. 1.2). Letting v 1 ðtÞ and v2 ðtÞ be the node velocities in the system, and letting thee ma th mass ss velo veloci city ty and and spri spring ng forc forcee be the the sy syst stem em st stat ates es x1 ðtÞ and x2 ðtÞ, respectively, then v1 ðtÞ ¼ x 1 ðtÞ and v2 ðtÞ ¼ x 1 ðtÞ D21 x2 ðtÞ
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Figure 1.2 A linear tim time-invar e-invariant iant mech mechanical anical sys system. tem.
ð1:4Þ
In other words, all velocities and forces within this mechanical system can be expressed in terms of the system states and the applied input force. The system’s output output part (1.1b) can thus be direc directly tly derived. For example, if 0 the system output yðtÞ is designated as ½v1 ðtÞ; v2 ðtÞ , then from (1.4),
ð Þ
v2 t 1 0 ¼ yðtÞ ¼ 1 v2 ðtÞ 1 D 2
ð Þ
x1 t 4 C xðtÞ þ 0uðtÞ x2 ðtÞ ¼
The dynamic equation of this mechanical system can also be derived using standard dynamic analysis. Balancing the forces at each node of this system, we have f ðtÞ ¼ M v_1 ðtÞ þ D1 v1 ðtÞ þ D2 ½v1 ðtÞ v2 ðtÞ
R ½ ðÞ
0 ¼ D 2 ½v2 ðtÞ v1 ðtÞ þ K v2 t dt
ð1:5aÞ ð1:5bÞ
Substituting (1.4) into (1.5) and after simple manipulation [including taking derivatives on both sides of (1.5b)], we can have the form of (1.1a)
1 1 D1 x2 ðtÞ þ f ðtÞ x1 ðtÞ þ M M M
x_ 1 ðtÞ ¼
x_ 2 ðtÞ ¼ Kx 1 ðtÞ þ
K x2 ðtÞ D2
Comparing (1.1a), the system matrices of this system are
A ¼
D1 =M 1=M ; K K =D2
B ¼
= 1 M 0
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In the above two examples, the forms and derivations of state space models are very similar to each other. We call different physical systems that aree si ar simi milar lar in te term rmss of math mathem emat atica icall mode models ls ‘‘an ‘‘anal alog ogs.’ s.’’’ This This pr prop oper erty ty enables the simulation of the behavior of one physical system (such as a mechanical system) by comparison with an analog but different physical syst sy stem em (suc (such h as a ci circ rcui uitt syst system em), ), or by th thee nume numeri rica call so solu luti tion on of the the mathematical model of that system. We call the former ‘‘analog simulation’’ and the latter ‘‘digital simulation.’’ The use of analogs can be extended to a wide range of linear time-
invariant physical systems, such as rotational mechanical systems, thermodyna dy nami micc syste systems ms,, an and d fluid fluid dyna dynami micc syste systems ms.. Th Ther eref efor ore, e, alth althou ough gh th thee mathematical models and the control theory which is based on these models are abstract, they can have very general applications. A linear time-invariant system can have another kind of mathematical model, called the transfer function model, which can be derived from its corresponding state space model. Taking the Laplace transforms on both sides of (1.1), 1
X ðsÞ ¼ ðsI AÞ
1
AÞ xð0Þ þ ð sI
BU ðsÞ
ð1:6aÞ ð1:6bÞ
Y ðsÞ ¼ CX ðsÞ þ DU ðsÞ
where X ðsÞ, U ðsÞ, and Y ðsÞ are the Laplace transforms of xðtÞ, uðtÞ, and yðtÞ, respectively, and I stands for an n-dimensional identity matrix such that sIX ðsÞ ¼ sX ðsÞ. Substituting (1.6a) into (1.6b), we have 1
Y ðsÞ ¼ C ðsI AÞ
1
xð0Þ þ ½C ðsI AÞ
Zero input response Y zi ðsÞ
B þ DU ðsÞ
ð1:6cÞ
Zero state response Y zs zs ðsÞ
From superposition principle of linear systems, Eqs. (1.6a) and (1.6c) each have two terms or two contributing factors. The first term is due to the system’s initial state xð0Þ only and the second is due to system input U ðsÞ only on ly.. For For ex exam ampl ple, e, in (1 (1.6 .6c) c),, the the syst system em outp output ut (als (also o ca call lled ed th thee sy syst stem em ‘‘response’’) Y ðsÞ eequ qual alss the the first first term term if the the syst system em inpu inputt is ze zero ro.. We ther th erefo efore re de defin finee the the first first te term rm of (1 (1.6c .6c)) as ‘‘ze ‘‘zero ro inpu inputt resp respon onse se Y zi zi ðsÞ.’’ Similarly, Y ðsÞ equals the second term of (1.6c) if system initial state is zero, and it is therefore defined as the ‘‘zero state response Y zzss ðsÞ.’’ The form of (1.6) is guaranteed by the linearity property of the state space model (1.1) and of the Laplace transform operator.
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The system’s transfer function model G ðsÞ is defined from the system’s zero state response as Y zs zs ðsÞ ¼ G ðsÞU ðsÞ
ð1:7Þ
Therefore from (1.6c), 1
GðsÞ ¼ C ðsI AÞ
B þ D
ð1:8Þ
The de The defin finit itio ion n of G s show showss th that at it refle reflects cts on only ly th thee rela relatio tions nshi hip p ð ðÞsÞ and output Y ðsÞ. This relationship (1.7, 1.8) between the system input U is derived by combining and simplifying a more detailed system structure (1.6a,b), which involves explicitly system state X ðsÞ and which is derived directly from the state space model (1.1). In addition, the transfer function mode mo dell do does es not not refle reflect ct di dire rect ctly ly an and d expl explic icit itly ly th thee syst system em’s ’s ze zero ro inpu inputt response, which is as important as zero state response.
Example 1.3 Consider the following RC circuit circuit system (a) and mechanical system (b) with a mass M and and a frictional force D (Fig. 1.3): 1.3): Balancing the currents of (a) and the forces of (b), we have i ðtÞ ¼ C v_ðtÞ þ
½vðtÞ 0 R
and f ðtÞ ¼ M v_ðtÞ þ D½vðtÞ 0
Figure 1.3 First-o First-order rder circ circuit uit and mechan mechanical ical systems. systems.
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l; BÞ equal ð1=RC ; 1=C Þ and Comparing (1.1a), the system matrices ðA4 ¼ ðD=M ; 1=M Þ for the above two systems, respectively. Taki king ng Lap apla lacce tran transf sfo orms rms on the these tw two o eq equ uat atio ions ns and af afte terr manipulation, we have the form of (1.6a) or (1.6c) as
V ðsÞ ¼
1
vð0Þ þ
B
U ðsÞ
s l
s l
where V ðsÞ and U ðsÞ are the Laplace transforms of vðtÞ and system input signal ½i ðtÞ or f ðtÞ, respectivel respectively. y. Letting U ðsÞ ¼ F =s (or step function) and taking the inverse Laplace transforms on the above equation, we have, for t50,
vðtÞ ¼ l1 fV ðsÞg ¼ e lt vð0Þ þ F
B
l
½1 elt
4vzi ðtÞ þ vzs ðtÞ ¼
In each of the above expressions of V ðsÞ and vðtÞ, the two terms are zero input response and zero state response, respectively. The two terms of vðtÞ have the waveforms shown in Fig. 1.4. The first waveform of Fig. 1.4 shows that the zero input response starts at its initial condition and then decays exponentially to zero with a time constant j1=lj. In other words, the response decays to 36.8% of its initial value at t ¼ j1=lj. This waveform has very clear physical meaning. In the circuit system (a), this waveform shows (when the input current is zero) how the capacitor charge ½¼ Cv ðtÞ is discharged to zero through the resistor R with current vðtÞ=R, and and wit ith h a ti time me cons consta tant nt RC . In ot oth her words, rds, the the larg larger er th thee capacitor or resistor, the slower the discharge process. In the mechanical system this waveform with zeroforce input force how athe momentum ð¼ Mv ð(b), tÞÞ slows Dv ðtÞ, with to zero byshows the frictional time constant M /D. In other words, the larger the mass and the smaller the friction D, the longer the time for the velocity to slow to 36.8 % of its initial value. The second waveform of Fig. 1.4 shows that the zero state response starts at zero and then reaches exponentially to its steady state level, which is specified by the input level F . This process also has a time constant j1=lj, which means that the response reaches 1 36:8% ¼ 63:2% of its final value at t ¼ j1=lj. This waveform also has very clear physical meaning. In the circuit system (a), this waveform shows how the capacitor is charged from zero until vðtÞ ¼ ðB=lÞF ¼ RF , by a constant current source F uðtÞ: The final
Copyright 2004 by Marcel De Dekker, kker, Inc. All Rights Reserved.
S y s t e m M a t h e m a t i c a l M o
d e l s a n d B a s i c P r o p e r t i e s
Figure 1.4 Waveforms Waveforms of zero input response response and zero state response response of a first-order first-order system.
Copyright 2004 by Marcel Dekk Dekker, er, Inc. All Rights Reserved.
value of vðtÞ equals the supply side voltage, which means that the capacitor is fully charged. This charging process has a time constant RC , which means the larger the capacitor or the resistor, the slower the charging process. In the mechanical system (b), this waveform shows how the mass is accelerated from zero to ðB=lÞF ¼ F =D by a constant force F uðtÞ. This acceleration process has a time constant M /D, which implies that the larger the mass or the higher the final velocity F /D, which is implied by a lower D, the longer the time for the mass to accelerate to 63.2 % of its final velocity. This example shows a very fitting analogy between the two systems, and the solution of their common mathematical model. This example also show sh owss th thee im impo porta rtanc ncee of the the in init itia iall stat statee of the the sy syste stem m (i (ini niti tial al ca capa paci cito torr chargee and initial mass velocity, charg velocity, respectively) respectively) and its effects on the system— the zero input response (discharging and de-acceleration, respectively). The definition (1.7)–(1.8) of transfer function model GðsÞ implies that GðsÞ cannot in general describe explicitly and directly the system’s zero input response, especially when the system has many state variables, inputs, and outpu ou tputs ts.. Beca Becaus usee tran transi sien entt resp respon onse se is defin defined ed as th thee co comp mple lete te syst system em response before reaching steady state and is therefore closely related to the system’s zero input response, the inherent feature of the transfer function model will inevitably jeopardize the understanding of the system’s transient re resp spon onse se,, wh whos osee quic quickn knes esss an and d smoo smooth thne ness ss is a ma majo jorr pa part rt of sy syst stem em performance, as will be defined in the next chapter. In Example 1.3, the concept of time constant is used as a measure of transient response and is closely related to zero input response. In both the state space model (1.1) and the transfer function model (1.8), the system matrix D reflects only an independent and static relation between system inputs and outputs. This relation can be easily measured and cancelled in the analysis and design. For this reason, we will assume
D ¼ 0 in the rest of this book. Using this assumption, the transfer function model of (1.8) now becomes 1
GðsÞ ¼ C ðsI AÞ
B
ð1:9Þ
Figure 1.5 Partit Partitioned ioned bloc block k diagram repr representat esentation ion of a system’ system’s s transfer function model.
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Finally, the transfer function model (1.9) can be represented by the block diagram in Fig. in Fig. 1.5, which 1.5, which is a series connection of two blocks.
1.2
EIG EIGENS ENSTRU TRUCTU CTURE RE DECOM DECOMPOS POSITI ITION ON OF A STATE STATE SPACE MODEL
To gain gain a si simp mple lerr yet yet deep deeper er unde unders rsta tand ndin ing g of sy syst stem em st stru ruct ctur uree an and d properties, we partition the system dynamic matrix
A ¼ V LV 1
2 6 44 ¼
j V 1 :
j
1
... :
2 3 j 75664 j
V q
L1 .
.
.
Lq
32 775664
-T 1 . . .
-T q -
3 775
ð1:10aÞ
4 ¼ T LT called ed a ‘‘ ‘‘Jo Jord rdan an fo form rm ma matr trix ix,’ ,’’’ wh whos osee where L ¼ diagfL1 ; . . . ; Lq g is call diagonal matrix blocks Li (i ¼ ¼ 1 ; . . . ; q, called ‘‘Jordan blocks’’) are formed ¼ 1 ; . . . ; nÞ of matrix A according to the following by the eigenvalues ðli ; i ¼ rules: Li ¼ li ; if li is real and distinct Li ¼
si oi
oi si
;
if the corresponding li and li þ1 are s + j o
i a complex conjugate pair i Li ¼ diagfLi ; j ; j ¼ 1 ; . . . ; qi g, if the corresponding li repeats ni times,
and the ni ; j dimensional matrix
2 666 ¼ 6 64
Li ; j
li 1 .
li
.
. .
. .
3 777 775
(blank entries are all 0’s)
1 li
ð1:10bÞ
and is called “bidiagonal form matrix,” where
ni ;1 þ þ ni ;qi ¼ n i
Finally, the sum of dimensions of all Jordan blocks Li ði ¼ ¼ 1 ; . . . ; qÞ equals n. When matrix A is in (1.10), the corresponding state space model is said to be in ‘‘Jordan canonical form.’’ Any real square matrix (and any dynamic matrix) A can have the eigenstructure decomposition such as (1.10).
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Because Becau se (1.10) (1.10) implie impliess AV V L ¼ 0, we ca call ll ma matr trix ix V the ‘‘r ‘‘righ ightt ei eige genv nvec ecto torr matr matrix ix’’ ’’ of matr matrix ix A, and call each column of matrix V ; vi ði ¼ ¼ 1 ; . . . ; nÞ, the ‘‘right eigenvector’’ of matrix A correspondi corresponding ng to li . Similarly, because TA LT ¼ 0, we call matrix T the the ‘‘left eigenvector
matrix’’ of matrix A and call each row of matrix T ; ti ði ¼ ¼ 1 ; . . . ; nÞ, the ‘‘left eigenvector’’ of matrix A corresponding to li . All but the first eigenvectors corresponding to the Jordan block (1.10b) are derived based on each other and are called the ‘‘generalized eigenvectors.’’ From (1.10),
ðsI AÞ1 ¼ ½V ðsI LÞV 1 1 ¼ V ðsI LÞ1 V 1 Therefore, from (1.9) and the inverse matrix rules, 1
GðsÞ ¼ CV ðsI LÞ
V 1 B
CV adjðsI LÞV 1 B ¼ detðsI LÞ CV adjðsI LÞV 1 B ¼ ðs l1 Þ . . . ðs ln Þ
ð1:11aÞ ð1:11bÞ
ð1:11cÞ
where adjðÞ and detðÞ stand for the adjoint and the determinant of the corresponding matrix, respectively. From (1.11c), transfer function GðsÞ is a rational polynomial matrix. It has an n-th order order den denomi ominat nator or pol polyno ynomial mial wh whose ose n root rootss eq equa uall th thee n
eig igeenval nvalu ues of the the syste ystem m dyna dynami micc matri atrix x, an and d which hich is call calleed th thee ‘‘characteristic polynomial’’ of the system. Comparing (1.11a) with (1.9), a new system matrix triple ðL; V 1 B; CV Þ has has the the sa same me tran transf sfer er fu func nctio tion n as th that at of syst system em matr matrix ix triple ðA; B; C Þ, provided that A ¼ V LV 1 . We call these two state space models mode ls and their corresponding corresponding systems ‘‘simi ‘‘similar’’ lar’’ to each other and call the tr tran ansfo sforma rmati tion on be betw twee een n the the two two simi simila larr st stat atee spac spacee mo mode dels ls ‘‘sim ‘‘simil ilari arity ty transformation.’’ This property can be extended to any system matrix triple ðQ1 AQ; Q1 B; CQÞ for a nonsingular Q. The physical meaning of similar systems can be interpreted as follows. Let xðtÞ and xðtÞ be the state vectors of state space models ðA; B; C Þ and ðQ1 AQ; Q1 B; CQÞ, respectively. Then from (1.1), x_ ðtÞ ¼ Q 1 AQxðtÞ þ Q1 BuðtÞ yðtÞ ¼ CQ xðtÞ þ DuðtÞ
ð1:12aÞ ð1:12bÞ
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It is clear from (1.1a) and (1.12a) that xðtÞ ¼ Q xðtÞ
or
xðtÞ ¼ Q 1 xðtÞ
ð1:13Þ
From De From Defin finit itio ions ns A.3– A.3–A. A.4 4 of Ap Appe pend ndix ix A, (1.1 (1.13) 3) impli implies es th that at the the on only ly difference between the state space models (1.1) and (1.12) is that the state vectors are based on different basis vector matrices ( I and Q, respectively). Simila Sim ilarit rity y tra transf nsform ormati ation on,, especi especiall ally y wh when en the sta state te space space model model is transformed to ‘‘Jordan canonical form’’ where the dynamic matrix is in Jordan form, is a very effective and very frequently used scheme which can substantially simplify the understanding of the system, as will be shown in the rest of this chapter.
1.3
SY SYST STEM EM ORDE ORDER, R, C CONT ONTROL ROLLAB LABILI ILITY, TY, AND OBSERVABILITY
Definition 1.1 The order order n of a syst system em equa equals ls the the orde orderr of th thee sy syst stem em’s ’s char charac acte teri rist stic ic polynomial. It is clear from (1.11c) that system order also equals the number of states of the system. Lett us di Le disc scus usss the the situ situat atio ion n of th thee ex exis iste tenc ncee of co comm mmon on fa fact ctor orss
between the transfer function s numerator and denominator polynomials. Because this denominator polynomial is defined as the system’s characteristic polynomial, and because common factors can cancel out each other, the above situation implies that the corresponding system order is reducible. We call ca ll this this ki kind nd of syst system em ‘‘red ‘‘reduc ucib ible. le.’’ ’’ Othe Otherw rwis isee the the sy syst stem em is sa said id to be ‘‘irreducible.’’ The situation of reducible systems can be more explicitly described by thei th eirr co corr rresp espon ondi ding ng stat statee spac spacee mo mode dels ls.. De Defin finit itio ion n 1.1 1.1 im impli plies es th that at in re redu duci cible ble sy syst stem ems, s, so some me of the the syst system em st stat ates es are are no nott invo involv lved ed wi with th th thee syst sy stem em’s ’s inpu inputt and and outp output ut rela relatio tion n GðsÞ. In ot othe herr wo word rds, s, in redu reduci cibl blee systems, some of the system states either cannot be influenced by any of the system inputs, or cannot influence any of the system outputs. We will define these two situations separately in the following.
Definition 1.2 If there is at least one system state which cannot be influenced by any of the system inputs, then the system is uncontrollable; otherwise the system is
Copyright 2004 by Marcel De Dekker, kker, Inc. All Rights Reserved.
controllable. Among many existing criteria of controllability, perhaps the simplest is that a system is controllable if and only if there exists no constant l such that the rank of matrix ½lI A : B is less than n.
Definition 1.3 If there is at least one system state which cannot influence any of the system outputs, then the system is unobservable; otherwise the system is observable. Among many existing criteria of observability, perhaps the simplest is that a system is observable if and only if there exists no constant l such that the rank of matrix ½lI 0 A0 : C 0 is less than n. Because Becau se the the rank rank of ma matr trix ix lI A always always equa equals ls n if l is not an eigenvalue of A, the above criteria can be checked only for the n values of l which equal the eigenvalues of matrix A. It is clear that an irreducible system must be both controllable and observable. Any uncontrollable or unobservable system is also reducible. Up to this this poin point, t, we can can see see a comm common on an and d dist distin inct ct phen phenom omeno enon n of line linear ar syste systems ms—d —dua uali lity ty.. For For exam exampl ple, e, in li line near ar sy syst stem ems, s, cu curr rren entt and and voltage, force and velocity, charge and flux, dynamic ener erg gy and potent po tential ial energy energy,, capaci capacitan tance ce and induct inductanc ance, e, mas masss and spr spring ing are dua duall pair pa irs. s. In line linear ar al alge gebr bra a and and line linear ar cont contro roll th theo eory ry wh whic ich h de desc scrib ribee line linear ar
sy syste stems ms, ,puts, ma matr ix co colu lumn mns sllabili and anility d tyrows roand ws,, observ righ right t an and dlity left le eige eigenv nvec tors, rs, ineach puts ts and out output s,trix and con contro trollab obs ervabi abilit y ftare als also o ecto dua dual l toinpu eac h other. The phenomenon of duality can not only help us understand linear systems comprehensively, but also help us solve some specific analysis and desi de sign gn pr prob oble lems ms.. For For exam example ple,, the the dete determ rmin inati ation on of wheth whether er a sy syst stem em ðA; BÞ is controllable can be replaced by the determination of whether a system ðA ¼ A 0 ; C ¼ B 0 Þ is observable instead. Becau Be cause se mat matrix rix ½lI Q1 AQ : Q 1 B ¼ Q 1 ½ðlI AÞQ : B has has th thee same rank as that of matrix ½lI A : B , similarity transformation will not chan ch ange ge the the co contr ntrol olla labil bilit ity y prop proper erty ty of the the or orig igin inal al syst system em.. Simil Similar arit ity y tr tran ansfo sforma rmati tion on chan change gess only only th thee basi basiss ve vect ctor or ma matr trix ix of st stat atee vect vectors ors of the system’s state space model and therefore cannot change the system’s basic properties such as controllability. From duality, similarity transformation cannot change the observability of the system either. It is therefore valid to determine a system’s controllability and observability conditions after similarity transformation. The foll follow owin ing g thre threee exam exampl ples es show show th thee rela relati tive ve sim imp pli lici city ty of determining controllability and observability when the system matrices are
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in special forms (especially the Jordan canonical form), which can be derived from any system matrices by similarity transformation.
Example 1.4 Determine whether the system
02 Þ ¼ @4
ðA; B; C
1 0 0
0 0 2 0 0 3
32 3 5; 4 5; ½ -b1 -b2 -b3 -
c1 : c 2 : c 3
1 A
is controllable and observable. From Definition 1.2, it is clear that if any row of matrix B equals zero, say bi ¼ 0 ði ¼ ¼ 1 ; 2; 3Þ, then there exist a constant l ¼ i such such that the i -th -th row of matrix ½lI A : B equals zero. Only when every row of matrix B is non onze zero ro,, th then en th thee rank rank of mat atri rix x ½lI A : B equals n, for l ¼ i ði ¼ ¼ 1 ; 2; 3Þ ¼ all eigenvalues of matrix A. Thus the necessary and sufficient condition for this system to be controllable is that every row of matrix B is nonzero.
Similarly (from duality), the necessary and sufficient condition for this system to be observable is that every column of matrix C is is nonzero. From (1.9), the transfer function of this system is 1
GðsÞ ¼ C ðsI AÞ
¼
B
c 1 ðs þ 2Þðs þ 3Þb1 þ c2 ðs þ 1Þðs þ 3Þb2 þ c3 ðs þ 1Þðs þ 2Þb3 ðs þ 1Þðs þ 2Þðs þ 3Þ
It is clear that if any bi or ci equals zero ði ¼ ¼ 1 ; 2; 3Þ, then there will be common factors between the numerator and denominator polynomials of GðsÞ. Howeve However, r, the reduci reducible ble transfe transferr fun functi ction on GðsÞ cannot cannot indica indicate te the converse: whether a row of matrix B or a column of matrix C is zero, or whether the system is uncontrollable or unobservable or both. In this sense, the information provided by the transfer function model is less complete and explicit than the state space model. Cont Co ntro roll llab abil ilit ity y and and obse observ rvab abil ilit ity y co cond ndit itio ions ns can can also also be cl clea earl rly y revealed from the system’s block diagram. 1.6 shows clearly that any system state x i ðtÞ is influenced by the Figure 1.6 shows input uðtÞ if and only if the corresponding bi =0 ði ¼ ¼ 1 ; 2; 3Þ, and that any xi ðtÞ influences output yðtÞ if and only if ci =0.
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Figure 1.6 Block di diagram agram of the sys system tem from Exam Example ple 1.4.
Example 1.5 Example 1.4 is a Jordan canonical formed system with distinct and real eigenv eig envalu alues. es. Th Thee presen presentt exa exampl mplee studie studiess the sam samee sys system tem wit with h multip multiple le eigenvalues [see (1.10b)]. Let
02 Þ¼ @4
ðA; B; C
l 1 0 0 l 1
0
0 l
32 3 ; ; ½ 54 5 -0-0-b3 -
c1 : 0 : 0
1 A
It is clear that the rank of matrix ½lI A : B equals n if and only if b3 =0, and the rank of matrix ½lI 0 A0 : C 0 equals n if and only if c1 =0. In examining the block diagram of this system (Fig. system (Fig. 1.7), it 1.7), it is clear that b3 and c 1 are the only links between the system states and the system’s inputs and outputs, respectively. Because all system states are on a single path in Fig. 1.7, it is of interest to observe that any system state is observable if and only if all gains on that path and on the right side of this state are nonzero. In the dual sense, any state is controllable if and only if all gains on that path and on the left side of this state are nonzero. This property can help
Figure 1.7 Block dia diagram gram of the sys system tem from Exam Example ple 1.5.
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one to extract the controllable or observable part of the system from the rest of the system (see Sec. 5.1). Exa xam mples les 1.4 and 1. 1.5 5 show show that that a sy syst steem’s m’s co cont ntro roll llab abil ilit ity y and observability properties can be easily checked based on Jordan canonical forms of the system’s state space model. Unfortunately, the computational prob pr oble lem m of th thee simi similar larit ity y tran transf sfor orma mati tion on to Jo Jord rdan an ca cano noni nica call fo form rm is difficult and is usually very sensitive to the initial data variation. On the other hand, the form of state space model of Example 1.5 is a special case of a so-called Hessenberg form, which can be easily and reliably computed and which can also be used to determine system controllability and observability (see Sec. 5.1). In the next two examples, we will study a second special case of the Hessenberg form state space model.
Example 1.6 The observable canonical form state space model:
02
32
3
1
a1 1 0 a2 0 1
BBB666 Þ¼ B@64
ðA; B; C
. . .
.. .
. . .
an1 0 an 0
.. .
0 0 1 0
777 666 ; 75 64
-b1 -b2 . . .
-bn1 -bn -
777 ; ½ 75
CCC ; CA
c1 ; 0; . . . 0
ð1:14Þ
This is a single-output (although it can be a multiple input) system. The above system matrices are said to be in the ‘‘observable canonical form.’’ In addition, the system matrix A of (1.14) is called a ‘‘companion form’’ or ‘‘canonical form’’ matrix. Let us examine the block diagram of this system. Figure 1.8 shows that all system states can influence system output c 1 =0, but if any of the 1’s of matrix A becomes 0, (observable) if and only if c then all system states left of this 1 on the main path (with all system states) of Fig. 1.8 will become unobservable. It has been proven that any singleoutput (n-th order) obse observable rvable system is similar similar to (1.14) (1.14) [Lue [Luenberg nberger, er, 1967 1967;; Chen, 1984]. wher eree th thee sy syst stem em From Fr om du dual alit ity, y, if a syst system em mo mode dell is ðA0 ; C 0 ; B0 Þ, wh matrix triple ð A; B; C Þ is from (1.14), then this system model is said to be in ‘‘controllable canonical form’’ and is controllable if and only if c1 =0. Any single-input controllable system is similar to this ðA0 ; C 0 ; B0 Þ: Controllable and observable canonical form state space models share an im impo port rtan antt prop proper erty ty in thei theirr corr corres espo pond nding ing tr tran ansf sfer er fu func ncti tion on GðsÞ.
Copyright 2004 by Marcel De Dekker, kker, Inc. All Rights Reserved.
Figure 1.8 Block diagram of a single-output system in observable canonical form.
Substituting (1.14) into (1.9), we have 1
GðsÞ ¼ C ðsI AÞ
B
c b sn1 b sn2 b s b n n1 2 1 þ ¼ sn1þ ð a n1þþ a sn2þþ þ an1 þ s þ aÞn 1s 2 N ðsÞ 4 ¼ DðsÞ
ð1:15Þ
In other word words, s, the unknown parameters parameters of the canonica canonicall state space model fully match the unknown parameters of the corresponding transfer function. In addition, the n unknown parameters of the companion form matrix A fully match the n unknown coefficients of its characteristic polynomial D ðsÞ, which wh ich furthe furtherr ful fully ly det determ ermine iness all n eige eigenv nval alue uess of th thee ma matr trix ix.. Fo Forr th this is reason, we also call all (either Jordan, controllable, or observable) canonical form state space model the ‘‘minimal parameter’’ model. Thee com Th comput putati ation on of simila similarit rity y transf transform ormati ation on fro from m a gen genera erall sta state te spacee model to canonica spac canonicall forms (1.14) and (1.10) (1.10) impli implies es the compres compression sion of system sys tem dynami dynamicc mat matrix rix parame parameter terss fro from m gen genera erall n6n to only n. In this sense, the computation of (1.14) and (1.10) can be equally difficult [Laub, 1985]. In this single-output system, the corresponding transfer function has the denominator DðsÞ as a scalar polynomial, and the numerator N ðsÞ as a polynomial row vector. In the next example, we will extend this result into
Copyright 2004 by Marcel De Dekker, kker, Inc. All Rights Reserved.
multi-output systems whose transfer function has both its denominator D ðsÞ and numerator N ðsÞ as a polynomial matrix.
Example 1.7 A multi-output observable system in canonical form:
A
2 66 ¼ 6 664
A1 I 2 0
.. .
. . .
A.2 0. I 3 . .
. .
Av1 0
0
0
I v
3 77 777 5
B
2 666 ¼ 6 64
B1 B2 :
Bv1 Bv
3 777 775
ð1:16Þ
Av 0 C ¼ ¼ ½ I 1 0
.. .
.. .
0
0
¼ 1 ; . . . ; v have have dime dimension nsionss mi 1 xmi where the where the ma matr trix ix bl bloc ocks ks I i i and i ¼ ðm0 ¼ m Þ and equal an mi 1 dimens dimension ional al ide identi ntity ty mat matrix rix wit with h mi 1 mi columns eliminated. Here m1 þ þ mv ¼ n . For example, for m i 1 ¼ 3, the corresponding I i i matrix blocks can be:
2 4
1 0 0
0 1 0
32 32 32 32 32 3 5 4 5; 4 5; 4 5; 4 5; 4 5;
0 0 ; 1
1 0 0
0 1 0
1 0 0
0 0 1
0 1 0
0 0 1
1 0 0
0 1 0
and
23 45 0 0 1
Without loss of generality (by assuming that all system outputs are be an and let let I 1 be line linear arly ly inde indepe pend nden entt [Che [Chen, n, 1984 1984]), ]), we let let m1 ¼ m and m-dimensional identity matrix. These m columns will disappear gradually I i subsequent at matrices I ii , this column and its at
to I 1 ði ¼ ¼ 2 ;
. . .
j -th Once the -th column disappears correspondingÞ.row will disappear at subsequent can ther theref efor oree dist distin ingu guis ish h an and d as assi sign gn a co cons nsta tant nt matrices I ii þ1 ; . . . . We can parameter v j ¼ i ; j ¼ 1 ; . . . ; m. From Example 1.6, the disappearance of the j -th -th column also implies that the j -th -th output is no longer influenced by any more system states. It is ap appa pare rent nt that that the the la larg rges estt valu valuee of v j equals v becaus becausee all m (1.16) 6).. It is also also pro rov ven th tha at any column col umnss disapp disappear ear at mat matrix rix I vþ1 in (1.1 observable system is similar to (1.16) [Luenberger, 1967; Chen, 1984], which is called the ‘‘block-observable canonical form’’ (see Sec. 5.1 also).
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To match all unknown parameters of (1.16) directly to all unknown parameters of the corresponding transfer function GðsÞ ¼ D 1 ðsÞN ðsÞ
ð1:17aÞ
as was done in Example 1.6, where DðsÞ and N ðsÞ are m6m and m6 p di dime mens nsio iona nall po poly lyno nomi mial al matr matric ices es,, resp respec ecti tive vely, ly, we need need to perf perfor orm m th thee following two preliminary and simple operations. zero rows into each matrix block Ai and Fir irsst, fil filll m mi zer Bi ði ¼ ¼ 1 ; . . . ; vÞ. The rows will be filled at the positions corresponding to all missing columns of matrix block I i an and its preceding I j ’s ð j ¼ i 1; . . . ; 1Þ. For exam exampl plee, if m ¼ 3 and I i i take takess th thee ab abov ovee seve seven n different forms, then the zero rows shall be filled at the third, the second, the
first,, th first thee se seco cond nd an and d thir third, d, the the first first an and d th third ird,, and and th thee fir first st an and d seco second nd positions of the second to the seventh matrix, respectively. At the end of this opera op erati tion on,, al alll matri matrix x bl bloc ocks ks Ai and Bi will will become become m6m and m6 p ¼ 1 ; . . . ; vÞ, respectively. dimensional matrix blocks Ai and Bi ði ¼ Second Sec ond,, form form matric matrices es ½I : A1 : A2 : . . . : Av and ½B1 : B 2 : . . . : Bv and then circular shift (shift in zeros) each row (say, the j -th -th row) of these two matric rices to the right by mðv v j Þ or pðv v j Þ positions, respectively, j ¼ 1 ; . . . ; m. We deno denote te the the two two resu result ltin ing g ma matr tric ices es of th this is ~1 : A ~2 : . . . : A ~v and ½B ~1 : B ~2 : . . . : B ~v , respectively. step as ½I ~0 : A Finally, in (1.17a), ~ sv1 þ þ A ~ ~ DðsÞ ¼ I ~0 sv þ A 1 v1 s þ Av
ð1:17bÞ
and ~ ~ ~ sv2 þ þ B N ðsÞ ¼ B~1 sv1 þ B v1 s þ Bv 2
ð1:17cÞ
It can be verified that the above (1.17) equals the GðsÞ of (1.9), which is computed from ðA; B; C Þ of (1.16) [Tsui and Chen, 1983a]. (See Exercise 1.3 to 1.6 for the numerical examples of this result.) The above two steps do not change, add or eliminate any parameter of ðA; B; C Þ of (1.16). Therefore, these two steps, which have not appeared explicitly before, enable the direct match between the parameters of state spac sp acee mode modell (1 (1.1 .16) 6) an and d the the pa para rame mete ters rs of the the tr tran ansfe sferr fu func nctio tion n mode modell (1.17a). (1.17 a). A sign significant ificant aspec aspectt of this direct direct para parametric metric match is that it enables enables thee fin th findi ding ng of th thee corr corres espo pond ndin ing g stat statee spac spacee mo mode dell (1.1 (1.16) 6) fr from om a give given n transfer function model (1.17). This problem is called ‘‘realization.’’
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Comparing the forms of (1.14) and (1.16), the former is truly a special case of the latter when m ¼ 1. Therefore, the novel operation of (1.17) is the direct generalization of realization problem, from the SISO case ðm ¼ 1 Þ to the MIMO case ðm > 1 Þ. Because the realization from (1.17) to (1.16) is easy, a transfer function model-b mod el-base ased d design design method method can easily easily find its corres correspon pondin ding g met method hod in state space theory. On the other hand, the computation of (1.16) from a gene ge nera rall st stat atee sp spac acee mode modell is very very di diffi fficu cult lt (s (see ee the the pr prev evio ious us ex exam ampl ple) e).. Therefore it is difficult to find the corresponding design method in classical control theory. This is another important reflection of the advantage of state space control theory over classical control theory. This book discusses only controllable and observable systems.
1. 1.4 4
SY SYST STEM EM PO POLE LES SA AND ND ZERO ZEROS S
Definition 1.4 A system pole is a constant l such that G ðs ¼ lÞ ¼ ?. From (1.11), a syste system m pole is a root of the characteristic polynomial of the system G ðsÞ and is also an eigenvalue of the dynamic matrix of the system. Thus the number of poles of an irreducible system is n.
Definition 1.5 In SISO systems, a system zero is a finite constant z such that G ðs ¼ z Þ ¼ 0. From (1.11), a system zero is a root of the numerato torr polynomial ial CV adj adjðsI LÞV 1 B of GðsÞ, of an irreducible system. In MIMO systems, CV adj adjðsI LÞV 1 B is not a scalar. Therefore, the definition of system zeros is more complicated. From Rosenbrock [1973], we define any finite constant z such that Gðs ¼ z Þ ¼ 0 as ‘‘blocking zero.’’ A system with blocking zero z has zero response to input u0 ezt for any u0. We also define any finite constant z such that the rank of Gðs ¼ z Þ is m and p ) as ‘‘transmission zero.’’ Thus less than min fm; pg (the minimum of m a system with transmission zero z and with more outputs than inputs ðm > p Þ has at least one constant vector u 0 such that G ðs ¼ z Þu0 ¼ 0. In other words, such su ch a sy syst stem em ha hass ze zero ro resp respon onse se to inpu inputt u0 ezt , wh wher eree u0 must must sat satisf isfy y Gðs ¼ z Þu0 ¼ 0. Therefore, blocking zero is a special case of transmission zero. There is no difference between blocking zeros and transmission zeros in SISO systems.
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There is a clear and simple relationship between system transmission zeros and the system’s state space model ðA; B; C ; DÞ [Chen, 1984]. Because
I
zI A
0
1
I
C ðzI AÞ
B
C
D
zI A
¼
0
B Gðs ¼ z Þ
hence
rank½S 4 rank ¼
zI A C
B
D
¼ rank
rank
zI A
0
B Gðs ¼ z Þ rank
½zI A þ ¼ n þ minfm; pg
½Gðs z Þ
ð1:18Þ
In other words, transmission zero z must make the rank of matrix S (which (which is formed by state space model parameters) less than n þ minfm; pg. This relation is based on the assumption of irreducible systems so that z cannot be a system pole and so that rank ½zI A is guaranteed to be n.
Example 1.8 Let the transfer function of a system with three outputs and two inputs be
2 ð Þ¼4
G s
0 2
sðs þ 1Þ=ðs þ 1Þ sðs þ 1Þðs þ 2Þ=ðs4 þ 2Þ
3 ð þ Þ=ð þ Þ ð þ Þð þ Þ=ð þ þ Þ 5 s
2
1 s 1 2 s 1 s 2 s 2s 3 ðs þ 1Þðs þ 2Þ=ðs2 þ 2s þ 2Þ
Fro rom m De Defin finit itio ion n 1. 1.5 5, th this is syst systeem has a blo block ckin ing g zero ero 1 and two transmission zeros 1 and 0, but 2 is not a transmission zero. when of a system has a zeros different number of inputsThis and example outputs ð pshows =mÞ, that its number transmission is usually much less than its number of system poles. However, when a system has the same number of inputs and outputs ðm ¼ p Þ, its number of transmission zeros is usually n m. In addition, if such a system (with m ¼ p ) has matrix product CB nonsi nonsingu ngular lar,, then then its numbe numberr of transm transmiss ission ion zer zeros os is alw always ays n m. These properties have been proved based on the determinant of matrix S of (1.18) [Davison and Wang, 1974]. An interesting property of transmission zeros is as follows. Suppose there are r transmission zeros of system ð A; B; C Þ, then for any nonsingular matrix K which which app approa roache chess infi infinit nity, y, among among the n eigenva eigenvalue luess of matrix matrix
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A BKC , r of them will approach each of the transmission zeros and n r of them will approach infinity [Davison, 1978]. Anot An othe herr in inte tere rest stin ing g prop proper erty ty of tran transm smiss issio ion n ze zero ross is that that when when a system GðsÞ is con connec nected ted with with a dynami dynamicc fee feedba dback ck com compen pensato satorr sys system tem H ðsÞ, the set of transmission zeros of the overall feedback system equals the union of the transmission zeros of GðsÞ and the poles of H ðsÞ [Patel, 1978]. In addition, we will assign all stable transmission zeros of GðsÞ as the poles of its corresponding dynamic feedback compensator H ðsÞ (in (in Chap. 5). Chap. 5). Hence the accurate computation of transmission zeros of a given system is important.
There are several methods of computing transmission zeros of a given system [Davison and Wang, 1974; Davison, 1976, 1978; Kouvaritakis and MacFarlane, 1976; MacFarlane and Karcaniar, 1976; Sinswat et al., 1976]. The following is a brief description of the so-called QZ method method [Laub and Moore Mo ore,, 197 1978]. 8]. Thi Thiss method method comput computes es all fini finite te genera generaliz lized ed eigenv eigenvalu alues es z such that there exists an n þ p dimensional vector w satisfying S w ¼ 0
ð1:19Þ
where matrix S is is already defined in (1.18). Equation (1.19) is valid for the case m 5 p. The transpose (or the dual) of (1.19) can be used for the case m 4 p. The advantage of this method arises from the existence of a numerically stable algorithm [Moler and Stewart, 1973] for computing the generalized eigenvalues [Laub and Moore, 1978]. We have have briefl riefly y disc iscuss ussed the the prope ropert rtie iess of sy syst steem zero eros. The prop pr oper ertie tiess of sy syst stem em po pole less wi will ll be di disc scus usse sed d in th thee ne next xt chap chapte ter, r, wh whic ich h show sh owss tha that th thee syst systeem pol olees are are the the most imp importa ortant nt pa para ram met eteers in determining a system’s performance.
EXERCISES Fig. 1.9: time-inva e-invariant riant cir circuit cuit system system shown in in Fig. 1.1 For a linear tim (a (a))
Let Let th thee curr curren ents ts of the two res resis isto tors rs be the two out outpu puts ts of th this is syste sy stem, m, resp respec ectiv tivel ely. y. Find Find the the st stat atee spac spacee mo mode dell (1.1) (1.1) of th this is system. (b) Der Derive ive the tran transfe sferr func function tion m mode odell (1.9) (1.9) of this sy syste stem. m. (c) Plo Plott the linear linear motion motion mechan mechanica icall syste system m which which is ana analog logous ous to this th is ci circ rcui uitt syst system em.. Indi Indica cate te all all sign signal alss an and d el elem emen ents ts of th this is mechanical system in terms of the corresponding circuit system signals and elements.
Copyright 2004 by Marcel De Dekker, kker, Inc. All Rights Reserved.
Figure 1.9 A linear ti time-inv me-invariant ariant c circuit ircuit sy system. stem.
1.2 Let a contro controllable llable can canonic onical al form state space space model be the dual from Example 1.6.
A ¼ A 0
B ¼ C 0 ¼ ½c1 ; 0 . . . 0
C ¼ ¼ B 0 ¼ ½b01 :
0
0 0
. . . : b n
(a (a)) (b)) (b
Plot Plot th thee blo block ck dia diagr gram am si simi milar lar tto o Fig. 1.8 1.8.. ¼ 0 is the necessary and sufficient condition for the Prov Provee that that c1 6 system ðA; B; C Þ to be controllable. (c) Pro Prove ve that that the transf transfer er ffunc unctio tions ns of of ðA; B; C Þ is the transpose of that from Example 1.6. 1.3 Let a two-out two-output put observable observable canonical canonical form system system state space mode modell be
A
2 6 ¼ 4
2
3
1
4
5
0
6
7
0
3 75
2 3 6 7 ¼ 4 5 8
B
9
C ¼ ¼
10
1
0
0
0
1
0
(a)
From From the des descri cripti ption on from from Example Example 1.7 (o (orr Defin Definiti ition on 5.1 5.1), ), find ¼ 1 ; 2Þ. the observability indices vi ði ¼ (b) Fo Follo llowin wing g the two-step two-step proced procedure ure from from Exam Example ple 1.7, 1.7, deriv derivee the polynomial matrix fraction description of the transfer function of 1
this system GðsÞ ¼ D
ðsÞN ðsÞ.
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(c (c))
Fi Find nd the the poles poles and and zeros zeros (if (if any any)) of thi thiss syste system. m.
Answer : v1 ¼ 2 ; v2 ¼ 1 ; DðsÞ ¼ N ðsÞ ¼
8s þ 10
s2 2s 6 3s 7
4
s 5
9 1.4 Repe Repeat at 1.3 for tthe he system system
A
2 6 ¼ 4
a
b 0
c
d 1
e
f 0
3 75
2 3 6 75 ¼ 4 ð Þ¼ B
N ðsÞ ¼
g
h
i
j
k
l
s
Answer : v1 ¼ 1 ; v2 ¼ 2 ; D s
g
C ¼ ¼
1
0
0
0
1
0
b e s2 ds f
a
cs
h
is k
js þ l
1.5 Repe Repeat at 1.3 for tthe he system system
A
2 6 64 ¼ 6
a
b 1
0
c
d 0
1
e
f 0
0
g
h 0
0
3 777 5
Answer : v1 ¼ v 2 ¼ 2 ; DðsÞ ¼ N ðsÞ ¼ is þ k js þ l
B
2 6 64 ¼ 6
i j k l
3 777 5
C ¼ ¼
1
0
0
0
0
1
0
0
s2 as e bs f
cs g
s2 ds h
Repeat at 1.3 for tthe he system system 1.6 Repe
A
2 6 64 ¼ 6
a
b 1
0
c
d 0
0
e
f 0
1
g
h 0
0
3 777 5
B
2 6 64 ¼ 6
i j k l
3 777 5
C ¼ ¼
1
0
0
0
0
1
0
0
Answer : v1 ¼ 3 ; v2 ¼ 1
Copyright 2004 by Marcel De Dekker, kker, Inc. All Rights Reserved.
s3 as2 es g bs2 fs h
ð Þ¼
D s
N ðsÞ ¼
c is2 þ ks þ l j
s d
two o syste system m dyna dynamic mic matrices matrices be 1.7 Let tw
A1
2 ¼ 4
1 1 0 0 1 1 0 0 2
35
A2
2 ¼ 4
1 0 0 0 1 1 0 0 2
35
Compute the Jordan form decomposition (1.10) of the two matrices. 1.8 Verify N ðsÞ [in GðsÞ ¼ D 1 ðsÞN ðsÞ] from Examples 6.1 and 6.3, according to the two-step procedure from Example 1.7.
Copyright 2004 by Marcel De Dekker, kker, Inc. All Rights Reserved.
2 Single-System Performance and Sensitivity
High system performance and low sensitivity are the two required properties of control systems. Low sensitivity is defined with respect to the system’s mathem mat hemati atical cal mod model el uncer uncertain tainty ty and termin terminal al distur disturban bance, ce, and is cal called led ‘‘robustness.’’ Unfortunately, high performance and robustness are usually contradictor dic tory y to eac each h other— other—hig higher her per perform formanc ancee sys system temss usu usually ally hav havee hig higher her sensitivity sensit ivity and worse worse robustness robustness properties. properties. Yet both high perfo performanc rmancee and high robustness are essential to most practical engineering systems. Usually, only high-performance systems have serious robustness problems and only such systems controlling. be considered as re relia liabi bili lity ty,,are is worthy al also so esse esof senti ntial al in most moRobustness, st pr prac acti tica call which ca case ses. s. can Th Ther eref efor ore, e, bo both th
Copyright 2004 by Marcel De Dekker, kker, Inc. All Rights Reserved.
perfor perf orma manc ncee an and d sens sensiti itivi vity ty prop proper ertie tiess mu must st be st stud udied ied.. Th This is ch chap apte terr consis con sists ts of two section sectionss which which study study system system per perfor forman mance ce and sen sensit sitivi ivity ty
properties, respectively. Section 2.1 studies some system properties such as system stability, quickness, and smoothness of system transient response, which are most impo im porta rtant nt (a (and nd most most di diffi fficu cult lt to achi achiev eve) e) in sy syste stem m pe perf rfor orma manc nce. e. Th This is se sect ctio ion n expl explai ains ns how how thes thesee prop proper ertie tiess ar aree mo most st dire direct ctly ly an and d ex expl plici icitl tly y determined by the system poles. Sect Se ctio ion n 2.2 2.2 stud studie iess the the pr prop oper erty ty of syst system em sens sensit itiv ivit ity y via via a no nove vell perspective of the sensitivities of system poles. A basic result of numerical linear algebra is that the sensitivity of an eigenvalue is determined by its corresponding left and right eigenvectors.
2.1
SYSTE SYSTEM M PE PERF RFORM ORMANC ANCE E
The reason that systems control theory has concentrated mainly on linear time-invariant systems is that only the mathematical models of this kind of syste sy stems ms can can ha have ve gene genera rall an and d expl explic icit it solu solutio tions ns.. Fu Furth rther ermo more re,, on only ly th thee gene ge nera rall an and d expl explic icit it unde unders rsta tand ndin ing g of the the syst system em ca can n be us used ed to gu guid idee genera gen erally lly,, system systematic aticall ally, y, and effect effective ively ly the compli complicat cated ed con contro troll system system design. The analytical solution of the state space model (1.1a) is, for t > 0, t
ð Þ ¼ e
x t
At
Z ð Þþ
x 0
eAðttÞ Bu t d t
ð Þ
ð2:1Þ
0
ðÞ
ð Þð
Þ
where x 0 and u t 04t4t are are gi give ven n sy syst stem em initi initial al st stat atee an and d sy syst stem em input, respectively. One way of deriving this result is by taking the inverse Laplace transform on (1.6a). We call (2.1) the ‘‘complete system response’’ of system state x t . Substituting (1.10) into (2.1) and using the Cayley–Hamilton theorem
ðÞ
t
Z ð Þþ ð Þ ðÞ¼ X ! Z X ¼ ð Þþ
x t
V eLt V 1 x 0
V eL t
t
V 1 Bu t d t
ðÞ
ð2:2Þ
0
t
q
V i i e
¼
i 1
Lit
q
V i i eLi ðttÞ T i i Bu t d t
T i i x 0
0
¼
i 1
ðÞ
ð2:3Þ
Therefore, eLit i
ð ¼ ¼ 1; . . . ; qÞ are the only time function terms related to the
Copyright 2004 by Marcel De Dekker, kker, Inc. All Rights Reserved.
system in the system response of (2.1)–(2.3). In other words, the eigenvalues
ðLÞ of system dynamic matrix A (or the system poles) are parameters which
most directly and explicitly determine the system response. Let us analyze all possible waveforms of the function eLit based on the definitions (1.10) of Jordan blocks Li . eLit
lit
¼e ; if L ¼ l e cos ot e ¼ e sinððotÞÞ e i
i
st
eLit
st
st
st
s sin ot if L ¼ o cosððotÞÞ
i
o
tan
s
ð2:4aÞ ð2:4bÞ
The linear combinations of the elements of this matrix can be simplified as: aest cos ot
ð Þ þ be
st
ð Þ ¼ ða þ b Þ = e
sin ot
2
2 1 2 st
cos ot
1
b a
where a and b are real numbers.
eLit
266 ¼6 4
1 t t2 =2 0 1 t 0 0 1 .. . 0 0 ...
. . . tn1 = n 1 ! . . . tn2 = n 2 ! . . . tn3 = n 3 ! elit .. . ... 01
ð Þ ð Þ
377 75
ð2:4cÞ
if Li is an n-dimensional bidiagonal matrix of (1.10b). Figure 2.1 plot Figure plotss al alll di diff ffer erent ent wave wavefo form rmss of (2.4 (2.4). ). In th thee fig figur ure, e, an eigenvalue (or a pole) is indicated by a symbol ‘‘x’’ and its coordinative position, and the corresponding waveform of this eigenvalue is plotted near that position. We can derive the following important conclusions directly from Fig. 2.1.
Definition 2.1 A system is asymptotically stable if and only if for any initial state x(0) the system’s zero-input response eAt x 0 converges to zero.
ðÞ
Conclusion 2.1 From Fig. 2.1, a system is asymptotically stable if and only if every system pole (or dynamic matrix eigenvalue) has a negative real part. We will refer to ‘‘asymptotic stable’’ as ‘‘stable’’ in the rest of this book.
Copyright 2004 by Marcel De Dekker, kker, Inc. All Rights Reserved.
Figure 2.1 Possib Figure Possible le system system pole poles s and wavefo waveforms rms of their corre correspon sponding ding system response.
Definition 2.2 The system response (2.1) of an asymptotically stable system always reaches a steady state, which is called ‘‘steady state response’’ and which is often the desi de sire red d st stat atee of resp respon onse se.. The The syste system m resp respon onse se (2.1 (2.1)) be befo fore re reac reachin hing g it itss steady ste ady state state is called called ‘‘tran ‘‘transie sient nt respo response nse.’’ .’’ The Theref refore ore,, the fas faster ter and the smoother the transient response, the better (higher) the performance of the system.
Conclusion 2.2 From (2.1), the transient response is mainly determined by the term eLit . Some conclusions about system performance can be drawn from Fig. 2.1. (a (a))
The The more more neg negat ativ ivee the the real real p par artt s of the system poles, especially the poles with least negative s, the faster the corresponding term
Copyright 2004 by Marcel De Dekker, kker, Inc. All Rights Reserved.
(b)) (b
est co conv nver erge gess to zero zero,, an and d ther theref efor oree th thee high higher er th thee sy syst stem em performance. Fo Forr comp comple lex x conju conjuga gate te syst system em pole poles, s, th thee larg larger er th thee im imagi agina nary ry part o of the system poles, the higher the oscillation frequency o of th thee corr corres espo pond nding ing tran transi sien entt resp respon onse se,, an and d th thee fast faster er th that at response reaches its first zero. However, the oscillatory feature of re resp spon onse se is ge gene nera rall lly y unde undesi sira rabl blee rega regard rding ing th thee smoo smooth thne ness ss
requirement (see Definition 2.2). Mu Mult ltip iple le pole poless gene genera rall lly y caus causee slow slower er and and ro roug ughe herr tr tran ansi sien entt response.
(c) (c)
We define stability, and the fastness and smoothness of the system transient response, as the main measures of system performance. Conclusions 2.1 and 2.2 indicat atee that the syste tem m poles det eteermine system performance most directly, accurately, and comprehensively. For the first-orde first-orderr system exampl examples es from Exam Example ple 1.3, the systems are stable because their only pole l is negative. Furthermore, the more negative the l, the smaller the time constant 1=l , and the faster the zero-input FB=l , response and zero-state response reach zero and steady state respec res pective tively. ly. Fur Furthe thermo rmore, re, the firs first-o t-orde rderr sys system temss do not hav havee mul multip tiple le eigenvalues. Hence their responses are smooth. In cla classic ssical al contro controll theory theory,, the sys system tem perfor performan mance ce is mea measur sured ed by band ba ndw wid idth th (B (BW) W).. Assu Assume me a seco second nd-o -ord rder er SI SISO SO sy syst stem em has has co comp mple lex x conjugate poles s+ j o0 :
ð¼
j j
Þ
o2n
ð Þ ¼ ½s ðs þ j o Þ½s ðs j o Þ
G s
0 2
0
o ¼ s þ ð 2sÞs þ ðs þ o Þ o 4¼ s þ 2zo s þ o n
2
2 0
2
ð2:5aÞ
2 n
2
n
2 n
where
on
¼ ðs þ o Þ = 2
2 1 2 0
and
¼ os ð0 < z < 1Þ
z
n
j ð Þj
ð2:5bÞ
The magnit magnitude ude of fre frequ quenc ency y respon response se G j o of th this is syst system em (als (also o called an ‘‘underdamped system’’) is shown in Fig. in Fig. 2.2.
Copyright 2004 by Marcel De Dekker, kker, Inc. All Rights Reserved.
Figure 2.2 Frequency response of an underda underdamped mped system system..
Figure 2.2 shows that as frequency o increases from 0 to infinity, the function G j o starts at 1 and eventually decays to 0. The bandwidth is defined defi ned as the fre freque quency ncy o at which which G j o 1 = 2 &0:707 707.. Fig Figure ure 2.2 shows that [Chen, 1993]
j ð Þj
BW&1:6on
p j ð Þj ¼
! 0:6o
n
¼ 0:1 ! 1
when z
ffiffi
ð2:6Þ j j
j j
In other words, BW is proportional with respect to on , or s and o0 . Therefore from Conclusion 2.2, the wider the bandwidth, the higher the performance (generally) of the system. However, relation (2.6) is based on a rather strict assumption (2.5) of the system, and the indication of BW is indirectly derived from Conclusion 2.2. The bandwidth, although it is simpler to measure, is generally far less accurate than the system poles in indicating the system performance. If this tradeoff in accuracy was formerly necessary because of the lack of effective computati comp utational onal means, means, the developme development nt of computer-a computer-aided ided desig design n (CAD) capability has obviated this necessity.
Copyright 2004 by Marcel De Dekker, kker, Inc. All Rights Reserved.
Example 2.1 Zero-Input Zero-Input Respon Response se of two Third-Ord Third-Order er Syste Systems ms Let the dynamic matrices of two systems be
2 3 ¼ 4 5
A1
1 2
2 3
0 0
0
1
2
2 3 ¼ 4 5
A2
and
The two matrices have the same eigenvalues Jordan forms.
¼ V L T
A1
1
1
2 ¼ 64
1
1
0
0
1
1=2
0
1
1=2
1
11 ; L12
¼ V diagfL 1
¼ V L T
A2
2
2
1 1
0 1=2
0 0
1
1=2
1
2 ¼ 64
2
¼ V diagfL
0 1
0 0
2
1
2
1; 1, and 2, but different
32 32 7564 7564 1
1
0
1
0
0
0
1
0
2
2
0
0
0
2
2
1
1
1 0
0 1
0 0
1 2
0 2
0 0
0
0
2
2
1
1
gT
1
375264 375264
21 ; L22 ; L23
2
1 0
gT
2
3 75 375
From (2.4),
eL1 t
2 ¼ 4
t
e 0 0
tet et
3 5 0 0
e
eL2 t
and an
2t
0 e
2 ¼ 4
eA1 t x 0
eA2 t x 0
4 6 2 ð Þ¼
ðÞ
et
þ 6te 2e 6te 4e 6te þ 7e t
t
t
t
t
et
2t
3 75
2
and
2et 4et þ 7e2t
375
3 5
0 0 0 et 0 0 0 e 2t
ð Þ ¼ ½ 1
From (2.1)–(2.2), for a common initial state x 0 input response for the state x t is
2 ð Þ ¼ 64
1
The waveforms of these two functions are shown in Fig. 2.3.
0
3 , the zero-
Copyright 2004 by Marcel De Dekker, kker, Inc. All Rights Reserved.
Figure 2.3 Wav Wavefor eforms ms of stat state e zero zero-inp -input ut respons responses es of two system systems s with
same poles.
Copyright 2004 by Marcel De Dekker, kker, Inc. All Rights Reserved.
The second waveform is remarkably better than the first in terms of both fastness and smoothness. This is caused by the only difference between the two systems: there is a generalized eigenvector (for eigenvalue 1) for thee fir th first st dy dyna nami micc matr matrix ix and and none none for for the the seco second nd dy dyna nami micc ma matr trix ix.. Th This is difference cannot be reflected by the transfer function model G s . From Fr om (2.2 (2.2), ), the the di diff ffer eren ence ce in the the fu func ncti tion on eLt inev inevita itabl bly y ma make kess a
ðÞ
difference in the system’s zero-state response. Hence the state space model can also describe the zero-state response (the transient part) more explicitly than the transfer function model, even though the transfer function model is defined from the system’s zero-state response only.
Example 2.2 Zero-State Zero-State Respon Response se of a Third Third-Order -Order Sys System tem Let
32 3 02 5 4 5 Þ ¼ @4 1 3 0
ðA; B; C
3 1 0
0 0 ; 2
0 5=3 1 ; 0 1
0 5
1 1
1A
Because matrix A is already in Jordan form, we apply (2.4a, b) directly and get
2 ¼ 4 t
et sinð3tÞ et cosð3tÞ
3 ð Þ ð Þ 5 ð ð Þ ¼ ðÞ 2 þ ð p p ffi ffiffiffi ffiffi Þ Z ð Þ ¼ ð Þ ¼ 4 þ ð Þ e
eAt
cos 3t e t sin 3t 0
0
0 0 e 2t
The hen n from from (2.1 (2.1), ), fo forr a uni nitt ste step in inp put u t response of x t is t
eA
x t
t t
B d t
0
ðÞ
3=10 1=10
Þ
1 ; t 5 0 , th thee ze zero ro-s -sta tate te
1= 10 e cosð3t 198 Þ 1= 10 e t cosð3t 108 Þ 1=2 ð1=2Þe2t t
3 5 ð Þ 4 ½ ð Þ
y1 t The waveform of x t and the corres correspon pondin ding g sys system tem output output y t ¼ y 0 y2 t C x t are shown in Fig. in Fig. 2.4. The waveforms all start at zero, which conforms to the assumptions of zero initial state and of finite power input signal. The waveforms of states
ð Þ ¼ ð Þ
ðÞ
ðÞ
¼
x1 t and x2 t oscillate with period 2p=o 2 p=3&2 before reaching their respective steady states 0.3 and 0.1. This feature conforms with Conclusion 2.2 (Part B).
Copyright 2004 by Marcel De Dekker, kker, Inc. All Rights Reserved.
Figure 2.4 The zerozero-stat state e response of system stat state e and of system outp output, ut, due to unit step input.
ðÞ
The above result on steady state of system output y t can also be directly derived from the system’s transfer function model.
Copyright 2004 by Marcel De Dekker, kker, Inc. All Rights Reserved.
Figure 2.4 (Continued)
From (1.9),
ð Þ ¼ C ðsI AÞ B g ðsÞ 4¼ g ðsÞ ¼ ðs þ 2s þ1 10Þðs þ 2Þ 1
G s
1
2
2
s
2
6s2
þ 7s þ 20 þ 17s þ 20
Copyright 2004 by Marcel De Dekker, kker, Inc. All Rights Reserved.
ðÞ
ð Þ¼ ð Þ ð Þ
G s U s From (1.7), y t equals the inverse Laplace transform of Y s G s =s (for unit step input). In addition, from the final value theorem of the Lap Laplac lacee transf transform orm,, the consta constant nt ste steady ady sta state te of y t can be de deri rive ved d directly as
¼ ðÞ
ðÞ
y t
li)m sY ðsÞ ¼ ½ 1 1 0 ð ) ?Þ ¼ lim s
0
2.4.. This derivation shows that the This result is in accordance with Fig. 2.4 clas cl assi sica call co cont ntro roll theo theory ry,, whic which h conc concen entr trat ates es on sy syst stem em inpu input/ t/ou outp tput ut relatio rel ations ns esp especi ecially ally at ste steady ady state, state, is easier easier than than the sta state te space space contro controll theory for deriving steady state response. However, in measuring the transient part of this input/output relation, g 1 s and g 2 s (3.815 and 9.21, respectively) are incorrect the bandwidths of g because y1 t and y2 t reach their steady state at about the same time. In addition, the waveform of y1 t is noticeably smoother than that of y2 t in
ðÞ
ðÞ ðÞ
ðÞ
Fig. 2.4. Overall, based on the actual step responses y1 t and y2 t , system g1 s is cert certai ain nly muc much more pref prefer erab able le than than sy syst stem em g2 s , yet the corresponding BW1 is two and a half times narrower than BW2 .
ðÞ
2.2
ðÞ
ðÞ
ð Þ ð Þ ðÞ
SYSTE SYSTEM M SE SENSI NSITIV TIVITY ITY AND R ROBU OBUST STNES NESS S
Whereas the previous section showed the critical importance of system poles (eigenvalu (eige nvalues es of system system dynamic dynamic matrix) matrix) on syste system m perfo performanc rmance, e, this section is based on a basic result of numerical linear algebra that the sensitivity of eigenvalues is determined by their corresponding eigenvectors. Numerical linear algebra, which has not been commonly used in the exi xist stin ing g tex textb tbo ooks on con control trol syste ystems ms,, is a br bra anc nch h of st stud udy y which ich concentrates on the sensitivity of linear algebraic computation with respect to the initial data variation and computational round-off errors [Fox, 1964]. Beca Be caus usee line linear ar al alge gebr bra a is th thee ba basi sicc ma math them emat atic ical al to tool ol in line linear ar co cont ntro roll systems theory, the results of numerical linear algebra can be used directly in analyzing linear system sensitivities. Some basic results of numerical linear algebra have been introduced in Appendix A. Let us first define the norm A of a matrix
k k
a11 A
¼
264
a
.
a m1
1n
.
a
mn
375
The norm of a matrix can provide a scalar measure to the magnitude of the matrix.
Copyright 2004 by Marcel De Dekker, kker, Inc. All Rights Reserved.
Before establishing the matrix norm, it is necessary to establish the 0 norm x of a vector x x1 ; . . . ; xn , where the vector elements xi i 1; . . . ; n can can be comp comple lex x numb number ers. s. Li Like ke th thee abso absolu lute te valu valuee of a scal scalar ar variab var iable, le, the vec vector tor norm norm x must must hav havee the fol follow lowing ing thr three ee pro proper perties ties [Chen, 1984]:
k k Þ 1. 2. 3.
¼ ½ k k
ð ¼ ¼
kxk50 and kxk ¼ 0 if and only if x ¼ 0 kaxk4jajkxk, where a is a scalar kx þ yk4kxk þ kyk, where y is also an n-dimensional vector
The third property is also called ‘‘triangular inequality.’’
Definition 2.3
k k 1. kxk ¼ jx j þ þ jx j = = 2. kxk ¼ ðjx j þ þ jx j Þ ¼ ðx* xÞ (‘ (‘‘*’ ‘*’’’ st stan ands ds fo forr tr tran ansspose and complex conjugate operation) 3. kxk ¼ max jx j In most cases only the norm kxk is being used. Therefore kxk is the default of vector norm kxk in this book unless specified otherwise. The vector norm x is defined as follows: 1
2
1
1
i
?
n
2
n
2 1 2
1 2
i
2
2
Vector norms Vector norms have have the fol follow lowing ing com commo mon n and imp import ortant ant proper property ty (Cauchy–Schwartz inequality) [Chen, 1984]:
jx*yj ¼ jy* xj4kxkkyk k k
ð2:7Þ
The matrix norm A , where the entries of matrix A can be complex numbers, must also have the following four properties:
kAk50 and kAk ¼ 0 if and only if A ¼ 0 kaAk ¼ jajkAk, where a is a scalar 3. k kAAx þk B4kk4Akk kAxkkþ kBk, where B is a matrix of same dimensionð2:8Þ 4.
1. 2.
Based on the above properties, especially (2.8), there can be three different definitions of matrix norm A according to the three different vector norms of Definition 2.3, respectively [Chen, 1984].
k k
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Definition 2.4 1.
kAk ¼ max 1
A
2.
j
j
j
j þ þ ja
a1 j
mj
max eigenvalue of A* A max singular value of A ain max i ai 1
2
1=2
k k ¼¼ ffð ð g ÞÞ g ð2:9Þ 3. kAk ¼ fj j þ þ j jg kAk ; which is also called Unless specified otherwise, kAk is the default of k the ‘‘spectrum norm.’’ There is another commonly used matrix norm kAk , which is called ?
2
F
the ‘‘Frobenius norm’’ and is defined as follows: 1=2
4:
2
ja j
kAk ¼
ij
F
¼ ½TraceðA* AÞ =
1 2
ð2:10Þ
X ! i ; j j
wheree the wher the matr matrix ix oper operat ator or ‘‘Tr ‘‘Trac ace’ e’’’ stan stands ds for for th thee sum sum of all all diag diagon onal al elements. Based on the singular value decomposition of a matrix with m 5n (see Appendix A, Sec. A.3), A
¼ U SV * ¼ U S0
1
V *
* 1 diag 5s singular values of A : s i i 1 ; . . . ; n ; U U where and s1 5s2 5 n 50. Then from (2.9–2.10), S
ð ¼ ¼
¼ f
kAk ¼ ½TraceðS*SÞ = ¼ ðs þ þ s Þ =
Þg
¼ ¼
* I ; V V
¼ I , ¼
1 2
F
2 1
2 1 2 n
p ffiffi ¼ p ffi k k n A
4 n s1
¼ kAk
5s1
2
2
ð2:11aÞ ð2:11bÞ
Equation (2.11) is useful in estimating the matrix spectrum norm.
Definition 2.5 Condition number of a computational problem: Let A be data and f A be the result of a computational problem f A . Let DA be the variation of data A a and nd D f be be the corresponding variation of
ð Þ
ð Þ
Copyright 2004 by Marcel De Dekker, kker, Inc. All Rights Reserved.
ð Þ f ðA þ DAÞ ¼ f ðAÞ þ D f
result f A due to DA such that
Then The n the condit condition ion number number k f of the com comput putati ationa onall pro proble blem m f A is defined by the following inequality:
ðÞ
ð Þ
kD f k 4 kð f ÞkDAk ð2:12Þ kAk k f k Therefore, kð f Þ is the relative sensitivity of problem f with respect to the relative relat ive varia variation tion of data A . A small kð f Þ implies low sensitivity of problem f , which is then called a ‘‘well-conditioned problem.’’ On the other hand, a large kð f Þ implies high sensitivity of the problem f , which is then called an ‘‘ill-conditioned problem’’ [Wilkinson, 1965].
Example Exa mple 2.3
[Wi [Wilki lkinso nson, n, 1965; 1965; Tsui, Tsui, 1983b 1983b]]
Let the computational problem be the computation of solution x of a set of linear equations Ax b, where A and b are given data. Let Db be the variation of b (no variation of A). Then A x Dx b Db implies that
¼
ð þ Þ ¼
ð þ Þ
Dx
k k¼k
A
1
Db 4 A
Thus from (2.8),
kAkkxk5kbk; kDx
1
Db
k k kk k kDbk
A k k kk k bk kxk 4 A
1
From Definition 2.5, this inequality implies that the condition number of this problem is A A1 . Supp Su ppos osee in th thee same same prob proble lem m th that at DA is the the va vari riat atio ion n of A (no b implies (assuming DADx is variation of b). Then A DA x Dx very small):
k kk k ð þ Þð þ Þ ¼
k
k
¼ A ðDAxÞ Thuss from Thu from (2.8), (2.8), kDxk=kxk4kAkkA kkDAk=kAk. Fr From om De Defin finiti ition on 2.5, 2.5, 1
Dx
1
Copyright 2004 by Marcel De Dekker, kker, Inc. All Rights Reserved.
this inequality again implies that the condition number of this problem is A A 1 . Because of the result of Example 2.3, we define the condition number of a matrix A as
k kk k
ð Þ ¼ kAkkA k 1
k A
ð2:13Þ
In the following we will first analyze the sensitivity of the eigenvalues of system dynamic matrix, and then use this result to analyze the sensitivity of system stability property.
2.2.1 2.2. 1
The Sens Sensitivit itivity y of Eigenval Eigenvalues ues ((Robu Robust st Perform Performance) ance)
Robust performance is defined as the low sensitivity of system performance with respect to system model uncertainty and terminal disturbance. Because Sec. 2.1 indicated that the eigenvalues of system dynamic matrix (or system pole po les) s) mo most st di dire rectl ctly y and and expl explici icitl tly y dete determ rmin inee sy syst stem em pe perf rfor orma manc nce, e, it is obvious that the sensitivities of these eigenvalues most directly determine a system’s robust performance. From (1.10), V 1 AV L, where matrix L is a Jordan form matrix with all eigenvalues of matrix A. Therefore, if A becomes A DA, then
¼ ¼
V 1 A
þ
ð þ DAÞV ¼ ¼ L þ V DAV D¼ L þ DL
DL
1
1
D
D
ð2:14Þ
D
k k4kV kk kkV kk Ak ¼ kðV Þk Þ k Ak
ð2:15aÞ Ineq In equa uali lity ty (2.1 (2.15a 5a)) in indi dica cate tess th that at the the co cond ndit itio ion n numb number er kðV Þ of eigenvector matrix V can decide the magnitude of k kDLk. However, DL is
not necessarily in Jordan form, and hence may not accurately indicate the actual variation of the eigenvalues. Base Ba sed d on (2 (2.1 .14) 4),, a resu result lt us usin ing g k V to indi indica cate te the the vari variati ation on of eigenvalues was derived by Wilkinson (1965):
ð Þ
min i
fjl ljg D¼ min fjDl jg4kðV Þk ÞkDAk i
ð ¼ ¼
ð2:15bÞ
i
i
Þ
ð þ Þ
where li i 1 ; . . . ; n and l are eigenvalues of matrices A and A DA , respectively. Because the left-hand side of (2.15b) takes the minimum of the difference Dli between the eigenvalues of A and A DA, the upper bound on the right-hand side of (2.15b) does not apply to other Dli ’s. To summarize, from (2.15), it is still reasonable to use the condition numb nu mber er of eige eigenv nvec ecto torr ma matr trix ix V of matr matrix ix A, k V , to measure the
þ
ð Þ
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ð Þ sðLÞ D kðV Þ ¼ kV kk kkV k
ð Þ
sensitivity of all eigenvalues L of matrix A , s L . In other words, we define 1
ð2:16Þ
¼
ð Þ k k
even though s L is not an accurate measure of the variation (sensitivity) of each individual eigenvalue. The advantage of this measure is that it is valid for large DA [Wilkinson, 1965]. In or orde derr to obta obtain in a more more ac accu cura rate te meas measur uree of th thee sens sensit itiv ivit ity y of individual eigenvalues, first-order perturbation analysis is applied and the following result is obtained under the assumption of small DA [Wilkinson, 1965]:
k k
Theorem 2.1 Let li ; vi , and ti be the i --th th eigenvalue, right and left eigenvectors of matrix A, respectively i 1 ; . . . ; n . Let li Dli be the i -th -th eigenvalue of matrix A DA i 1 ; . . . ; n . Then for small enough DA ,
ð ¼ ¼
Þ
þ
þ ð ¼ ¼ Þ jDl j4kt kkv kkDAk D¼ sðl ÞkDAk; i
i
i
i
k k i ¼ ¼ 1; . . . ; n
ð2:17Þ
Proof
¼
Let DA dB , where d is is a positive yet small enough scalar variable, and B is -th an n6n dimensio dimensional nal matrix matrix.. Let li d and vi d i 1 ; . . . ; n be the i -th
ðÞ
ð ÞÞðð ¼ ¼
Þ
eigenvalue and eigenvector of matrix A
þ dB, respectively. Then
ðA þ dBÞv ðd Þ ¼ l ðd Þv ðd Þ i
ð2:18Þ
i
i
¼ ¼
Without loss of generality, we assume i 1. From the perturbation theory, 2
ð2:19aÞ
ð Þ ¼ l þ k d þ þ k d þ
l1 d
1
1
2
and 2
ð Þ ¼ v þ ðl v þ þ l v Þd þ þ ðl v þ þ l v Þd þ ð2:19bÞ ¼ 2; . . . ; n; j ¼ ¼ 1; 2; . . .Þ are constants. For small enough d where k and l ði ¼ v1 d
1
2 21 1 2
n1 n
2 22 2 2
n2 n
ij ij
j
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(or DA), (2.19) can be simplified as k1 d D l1
l1 d &l1
and
¼
2:20a
Dl1
þ
ðÞ
þ
ðÞ
þ ðl v þ þ l v Þd
v1 d &v1
Þ
ð2:20bÞ
n1 n
2 21 1 2
ð 2
¼ l v and d 5 1, we have ½ðl l Þl v þ þ ðl l Þl v þ Bv d ¼ ¼ k v d ð2:21Þ Multiplying t ðt v ¼ d Þ on the left of both sides of (2.21), we have t Bv ¼ k Substituting (2.20) into (2.18) and from Avi 2
1
1
1
2 21 1 2
1
i j
n
1
n1 n
i i
1
1 1
ij
1
From (2.20a) and (2.8),
jDl j ¼ jt Bv d j4kt kkv kkdBk ¼ kt kkv kkDAk 1
1
1
1
1
1
1
The derivation after (2.18) is valid for other eigenvalues and eigenvectors. Hence the proof. This Th is theo theore rem m show showss cl clea early rly that that the the sens sensit itivi ivity ty of an eige eigenv nval alue ue is determined by its corresponding left and right eigenvectors. From now on, we will use the notation s li to represent the sensitivity of li , even though
ð Þ
ð Þ
s li is not the condition number of li as defined in (2.13).
Example 2.4 Consider the following two matrices: n 1
266 6 ¼ 6 66 4
A1
0
0 n 1 1 .. . 0 n 2 .. .. . . .. .
0
0
0
... 0 . . ...
1 ..
..
.
.
0
2
1
0 ... 0
1
377 77 77 5
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and
n
n
0
266 ¼ 66 64
0 n 1 n .. . 0 n 2 .. .. . . .. .
26 66 ¼ 6 ¼ 66 4
1 0
x
0 .. . .. .
0
0
0
A2
0
0
... 0 . . ...
0 n ..
.
..
.
0
2 n
0 ... 0
1
377 77 75
f 1; . . . ; 1g.
Clearly, the two matrices have the same set of eigenvalues n; n The right and left eigenvector matrices are:
V
and
2
1
2
x3 x4 . . . x2 x3 . . .
ð1Þ x ð1Þ x 1 x . . . ð1Þ x
.
n 2
n 3
2
..
n 1 n
..
1 0 ... 0
3
.
.. .
x
2
1
n 1 n 2
37 77 77 75
1 x2 x3 . . . xn
66 6 ¼ 6 ¼ 64
T
1 x2 . . . xn1 .. .. . 0 1 . .. .. .. . . .
0 0 .. . .. .
1 x2 0 ... 0
0
1
77 77 75
where
¼ 2; . . . ; n; ðx ¼ 1Þ; or ¼ x =ði 1Þ ¼ 1=ði 1Þ!; i ¼ x ¼ 1 ; x ¼ 1 =2!; x ¼ 1 =3!; . . . ; x ¼ 1 =ðn 1Þ!; ¼ 2; . . . ; n; ðx ¼ 1Þ; for A : x ¼ nx =ði 1Þ ¼ n =ði 1Þ!; i ¼ or x ¼ n ; x ¼ n =2!; . . . ; x ¼ n =ðn 1Þ!: ¼ 1; . . . ; nÞ are much greater The eigenvector matrix parameters ðx ; i ¼ for A1 : xi
2
1
i 1
2
3
i
i 1
4
n
i 1
2
1
2
3
n 1
n
i
Copyright 2004 by Marcel De Dekker, kker, Inc. All Rights Reserved.
for A2 than for A1 . From (2.17), the sensitivity of the eigenvalues of A2 is much higher than that of A1 . For example,
ð ¼ nÞ ¼ kt kkv k ¼ kv kkt k ¼ s ðl Þ ¼ ð1 þ x þ þ x Þ = ð1Þ ð1 þ ð1=2Þ þ þ 1=ðn 1Þ! Þ = &1 ðfor A Þ ð2:22aÞ ¼ ð1 þ ðn=2Þ þ þ ½n =ðn 1Þ! Þ = ðfor A Þ &n =ðn 1Þ!
s l1
1
1
n
2 1 2 n
2 2
2
8>< >:
s l n=2
ð ¼
n
n
2 1 2
2
1
2 1 2
n 1
n 1
n =2
2
t
v n=2
n=2
v
s l
t n=2 1 1=2
n=2
1
n 2
1
Þ ¼¼ kð1 þ kk þ xk ¼ k ðÞ Þþð1kk þ ð Þþþ xk ¼Þ ð= ð = Þþ Þ ð = Þþ = ð 1Þð1Þ ¼ 1 ðfor A Þ & ð 2:22bÞ = ðn =ðn=2Þ!Þ ðfor A Þ
(
ð Þ
2 n 2
1
1 2 2 n 2
1
n 2
2
2
The values of s li are much greater for A2 than for A1 . For A1 , all s li values are close to 1 i 1 ; . . . ; n . Thus Thus ev ever ery y eige eigenv nvalu aluee of A1 is almost the least possibly sensitive to the parameter variation of A1 , and the computati comp utations ons of these eigenvalues eigenvalues are there therefore fore all well conditio conditioned. ned. On the
ð Þ
ð ¼ ¼
Þ
l
6
6
i for A2 equals 5.2, 275, and 2 :155 s respectively, i for other the20 1086 for 1 and n 5,hand, 10, and and equals 6 :9446106 and 10 12 i n=2 and n 10 and 20, respectively. Thus the eigenvalues (especially l n=2 ) of A2 are ve Therefo fore re th thee very ry sens sensit itiv ivee to the the para parame mete terr va vari riat atio ion n of A2 . There computations of the eigenvalues of A2 are ill conditioned. The difference between matrices A1 and A2 is at the upper diagonal line. From Example 1.5 and (2.4c), the upper diagonal elements of A1 and A2 are the coupling links between the eigenvalues of A1 and A2 . Therefore the weaker these coupling links, the smaller the norm of each row of matrix compute ted d from from al alll colu column mnss of ma matr trix ix V , an and d th thee low lower th thee T V 1 compu sensitivity of each eigenvalue.
¼
¼
ð Þ
¼ ¼ ¼ ¼
Þ
ð¼
From another point of view, the weaker the coupling links, the weaker thee effe th effect ct of the the matri atrix x par aram amet eteer var variati iatio on on th thee corres rrespo pond ndin ing g eigenvalues (see Gerschgorin’s theorem [Wilkinson, 1965]). An even more direct inspection of the original matrices A1 and A2 shows that the smaller these upper diagonal elements, the closer the matrices to Jordan form, and therefore the lower the sensitivity of their Jordan forms to the variation of thes th esee two two ma matr tric ices es.. Th This is ob obse serv rvat atio ion n is no nott gene genera rall lly y va vali lid d fo forr ot othe herr matrices.
Copyright 2004 by Marcel De Dekker, kker, Inc. All Rights Reserved.
To summ summar ariz ize, e, this this ex exam ampl plee show showss th that at deco decoup upli ling ng is ex extr trem emel ely y ef effe fect ctive ive in lowe loweri ring ng ei eige genv nval alue ue sens sensiti itivi vity ty.. It is co comm mmon on sens sensee th that at if a system relies more heavily on more system components in order to run, then that system has higher sensitivity with respect to these system components. Although the coupled systems usually have higher performance. The theoretical analysis on the sensitivity of eigenvalues of A1 and A2 can be shown by the following example of DA [Wilkinso [Wilkinson, n, 1965; Chen Chen,, 1984]. Let
DA
¼ dB
Then
2 6 ¼ 64
0 ... 0 .. .. . . 0 0 ... 0 d 0 . . . 0 0 .. .
3 7 75
½ ðA þ DAÞ ¼ ðl nÞðl 2Þðl 1Þ þ d ð ð1Þ
det lI
1
n 1
and
½ ðA þ DAÞ ¼ ðl nÞðl 2Þðl 1Þ þ d ð ð n Þ
det lI
2
n 1
Hence the constant coefficient of characteristic polynomial is affected
by the above data variation d (or DA), and this effect is much more serious for A2 than for A1. A root locus plot (with respect to d ) in Chen [1984] demonstrates the sensitivity of the eigenvalues of A2 vs. d . Read Re ader erss can can also also refe referr to Wilk Wilkin inso son n [196 [1965] 5] fo forr mo more re th theo eore reti tica call discussions discu ssions on A2. However, the comparison of A1 and A2 in this book offers a clearer explanation for understanding the eigenvalue sensitivity of this example.
2.2.2
The S Sensit ensitivity ivity o off System System Stabili Stability ty (Robust (Robust Stabil Stability) ity)
Stability is the foremost system property. Therefore the sensitivity of this property (called ‘‘robust stability’’) with respect to system model uncertainty is also critically important. Consequently, a generally accurate quantitative measure of this sensitivity is also essential to guide robust stability analysis and design. From Conclusion 2.1, the most basic and direct criterion of system stability is that every dynamic matrix eigenvalue has a negative real part. Henc He ncee th thee se sens nsit itivi ivity ty of th thes esee ei eige genv nval alue uess wi with th resp respec ectt to sy syste stem m mo mode dell uncertainty (or dynamic matrix variation) should be the most direct and
Copyright 2004 by Marcel De Dekker, kker, Inc. All Rights Reserved.
critic crit ical al fact factor or in me meas asur urin ing g the the sens sensit itiv ivit ity y of syst system em st stab abil ilit ity y (r (rob obus ustt stability). Lett us comp Le compar aree the the Rout Routh– h–Hu Hurw rwit itzz crit criter erio ion n of sy syst stem em st stab abil ility ity,, where wh ere the sys system tem cha charac racteri teristi sticc pol polyno ynomial mial mus mustt be first first com comput puted. ed. The sen ensi siti tivi vity ty of this this step step of compu omputa tati tio on can be as high igh as th thee direc irectt comput com putati ation on of eigenv eigenvalu alues es (se (seee Wilkin Wilkinson son,, 196 1965 5 and Exa Exampl mples es 1.6 and 2.4). The Routh–Hurwitz criterion requires additional determination based on the the ch char arac acte teri risti sticc po poly lyno nomi mial al coef coeffic ficie ient ntss an and d on the the ba basi sicc st stab abili ility ty criter cri terion ion of Co Concl nclusi usion on 2.1. 2.1. This This ind indire irectn ctness ess wil willl inevit inevitabl ably y red reduce uce the accu ac cura racy cy of bo both th th thee stab stabili ility ty dete determ rmin inati ation on an and d th thee me meas asur uree of ro robu bust st stability. Let us compare another stability criterion, the Nyquist criterion. This crit cr iter erio ion n also also requ requir ires es tw two o gene genera rall st step eps. s. Th Thee fir first st st step ep plot plotss sy syst stem em frequency response G j o o 0 ?? . The second step applies the Nyquist stability criterion, which is based on the basic criterion of Conclusion 2.1 and on the Cauchy integral theorem, on the plot of step one. Both steps are indirect with respect to Conclusion 2.1 and will cause inaccuracy in each step. Stability is an internal system system property about the convergence of a time domain response, while the Nyquist criterion determines this property based on the information of system’s input/output terminal relation in frequency
ð Þ ð ¼
Þ
domain. Because of this fundamental reason, the Nyquist criterion is very difficult to apply to multivariable systems [Rosenbrock, 1974; Hung et al., 1979; Postlethwaite et al., 1982; Doyle et al., 1992], and its corresponding robust stability measures (gain margin and phase margin) are not generally accurate [Vidyasagar, 1984]. In this book, the result of sensitivity of system poles of Sec. 2.2.1 is used us ed to meas measur uree robu robust st stab stabil ilit ity. y. Comp Compar ared ed to th thee ab abov ovee tw two o ro robu bust st stability measures of classical control theory, this measure has not only the appa ppare ren nt adva advant nta age of gener eneral al accu accura raccy, but also lso an anot oth her crit critic ical al adva ad vant ntag age— e—th thee ab abil ilit ity y to acco accomm mmod odat atee po pole le as assi sign gnme ment nt and and th thus us to guarantee performance. The analysis in Sec. 2.1 shows that system poles can most mo st directl directly y and explic explicitl itly y det determ ermine ine the corres correspon pondin ding g sys system tem per perfor for-mance.
As stare atedtheintwo the contradictory beginnin ing g oyet f tcritical his chaproperties pte pter, perfoof rmaapractical nce and robustness engineering system. Therefore, it would be very impractical to concentrate on only only one one of thes thesee tw two o prop proper erti ties es [suc [such h as po pole le assi assign gnme ment nt only only or 1 (see Sec. 3.1) only]. The main purpose of sensitivity function I L s this book is to introduce a new design approach which can really and fully consider both properties. There are three existing robust stability measures using the sensitivity of system poles. In this book they are called M 1, M 2, and M 3. Among the
½ ð Þ
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three measures, M 1 and M 2 were developed in the mid 1980s [Kautsky et al., 19 1985 85;kowic ; wicz Qiuz an and d Siv David av idso son n, 8], 1986 19and 86;; Juan Ju g setde avelo l. l.,,lope 1ped 98 986 Dic cekman km an, 19 1987 87;s; M ang Lewko Lew and Sivan, an, 1988], 198 wa deve d6;inDi th the ea earl rly y, 19 1990 90s 3 was [Tsui, 1990, 1994a]. We will analyze and compare the general accuracy and the optimization feasibility of these three measures. Let us first introduce these three measures.
¼ min< fsðA j oI Þg; ðs equals the smallest singular valueÞ ð2:23Þ M ¼ s ðLÞ jRefl gj; ðjRefl gj4 4jRefl gjÞ ð2:24Þ M ¼ min fsðl Þ jRefl Þjg ð2:25Þ
M 1
04o ?
1
2
3
14i 4n
n
i
1
1
n
i
ð f g
Þ
where all eigenvalues are assumed stable Re li < 0 ; Vi . In addition, we assu as sume me al alll ei eige genv nvalu alues es are are al alre read ady y arbi arbitr trar arily ily assig assigne ned d fo forr gu guar aran ante teed ed performance. We wi will ll anal analyz yzee thes thesee th thre reee meas measur ures es in th thee fo foll llow owin ing. g. Al Alll th thre reee
measur meas ures es ar aree de defin fined ed such such that that th thee more more ro robu bust stly ly st stab able le the the sy syst stem em,, th thee greater the value of its robust stability measure. Because s indicates the smallest possible norm of matrix variation for a ma matr trix ix to beco become me sing singula ularr (see (see Theo Theorem rem A. A.8) 8),, M 1 equals equals the sma smalle llest st poss po ssib ible le ma matr trix ix vari variat atio ion n norm norm fo forr the the dy dyna nami micc ma matr trix ix A to have an unstab uns table le and pure pure imagin imaginary ary eigenv eigenvalu aluee j o. Th There erefor foree M 1 sho should be a generally accurate robust stability measure. The main drawback of M 1 seems to be its difficulty to design. For example, it is very difficult to design a matrix K such such that the M 1 of matrix A BK is is maximized, where matrices A; B are given and the eigenvalues A BK are of A are also prespecified to guarantee the desired performance. In the existing analysis about maximizing M 1, the only simple and analytical result Re ln if s ln is at its is that M 1 will be at its maximum possible value minimal value 1 [Lewkowicz and Sivan, 1988]. Unfortunately, this is
ð
Þ
ð¼ j f gjÞ ð Þ
impossible to achieve in most cases. In the measure M 2, the term Re ln is obviously the shortest distance Fig. 2.1. M 2 equals between the unstable region and the eigenvalues li on on Fig. this distance divided (or weighted) by the sensitivity of all eigenvalue matrix L. The lower the sensitivity s L , the greater M 2. In other words, M 2 may be considered as the weighted distance for l to become unstable, or as the likelihood margin for ln to become unstable. There exist several general and systematic numerical algorithms which 1 or M is maximized, can compute matrix K such that the value of s L 2
ð¼ Þ
j f gj
ð Þ
n
ð Þ
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with arbitrarily assigned eigenvalues in matrix A
BK [Kautsky [Kautsky et al., 1985;
MATLAB, 1990]. However, M 2 seems to be less accurate in measuring the likelihood margin for l to become unstable, because s L is not an accurate measure of the sensitivity of ln [see the discussion of (2.14)]. In the the defin definit itio ion n of me meas asur uree M 3 , th thee like likeli liho hood od ma marg rgin inss fo forr every eigenvalue to become unstable are considered. Here the likelihood margin forr each fo each li equals Re li div divid ided ed by it itss co corr rres espo pond ndin ing g sens sensit itiv ivit ity y 1 s li ; i 1 ; . . . ; n. In practice, the algorithms for maximizing M 2 or s L 1 can also be used to maximize M , after adding a weighting factor k V 3 1 Re li on each column vi of matrix V ; i 1 ; . . . ; n. Based on the above analysis and some basic principles, there are two obvious reasons that M 3 is generally more accurate than M 1 and M 2 . First, M 1 and M 2 consider only the likelihood margin for ln to become a ny eigenvalue can cause system instability unstable, while the instability of any (Concl (Co nclusi usion on 2.1 2.1). ). There Therefore fore M 3 meas measur ures es th thee ro robu bust st st stab abil ilit ity y mo more re completely and more rigorously than M 1 and M 2 . genera rall lly y no nott an ac accu cura rate te meas measur uree of Second Sec ond,, the s L of M 2 is gene
ð Þ
n
ð Þ ¼ ¼ ¼ ð Þ Þ j f gj
j f gj
ð ð Þ
¼ ¼
ð Þ
individual individ ual eig eigenv envalu aluee sensiti sensitivit vity y and is obv obviou iously sly not as acc accura urate te as the sensitivity s li of li itself in measuring the sensitivity of li ; Vi (including i n ). Hence M 2 is too conservative compared to M 3 . This is reflected in the follo fo llowi wing ng lowe lowerr bo boun und d of M 3 , even even th thou ough gh M 3 more more com comple pletel tely y and rigorously reflects the instability likelihood of all eigenvalues.
ð Þ
¼ ¼
ð Þ 4¼ kV kk kkV k > kv kkt k D¼ sðl Þ51; i ¼ ¼ 1; . . . ; n ;M ¼ s ðLÞ jRefl gj4M 4jRefl gj 1
,s L
1
2
i
i
3
n
i
n
ð2:26Þ ð2:27Þ
It has been proved that M 1 shares the same upper and lower bounds with M 3 [Kautsky et al., 1985; Lewkowicz and Sivan, 1988]. From (2.26–2.27), if the overall eigenvalue sensitivity s L k V is at the lowest possible value 1 , then all three measures M i i i 1 ; 2; 3 will re reac ach h th thei eirr co comm mmon on hi high ghes estt po poss ssib ible le valu valuee Re ln . However, it is
ð Þ¼ ð Þ ð ¼ ¼ Þ
ð¼ Þ
impossible to make s L 1 for most cases. In those cases, a lower s L does not necessaril necessarily y imply a higher higher M 1 or M 3 [Lewkowicz and Sivan, 1988]. Furthermore, in those cases, (2.27) implies that M 1 and M 3 have higher resolution and therefore higher accuracy than M 2 .
j f gj
ð Þ¼
ð Þ
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Example Exa mple 2.5
[Lewko [Lewkowicz wicz and Sivan, Sivan, 1988; 1988; Tsui, Tsui, 1994 1994a] a]
Let
2 3 5 ¼ 4 3 4:5 0
A1
0 2 0
0 0 1
and
2 3 5 ¼ 4 3 1:5 3
A2
0 2 0
0 0 1
The two matrice matricess have have same same eig eigenv envalu alues es but dif differ ferent ent eig eigenv envect ectors ors.. Hence the eigenvalue sensitivity as well as the robust stability are different for these two matrices. The eigenstructure decomposition of these two matrices are
¼ V L T 0:217
A1
1
2
1
1
0
0
32
3
0
0
32
4:61
0
0
3 ðk k ¼ t
Þ
4 :61
1
¼ 64
0:976 0
1
0
0
1
0
7564
0
2
0
4:5
1
0
0
1
0
0
1
7564
t2
4 :61
t3
1
75 ðk k ¼ Þ ðk k ¼ Þ
and A2
¼ V L T 2
2
2 ¼ 64
2
0:4264
0
0
0:6396
1
0
0:6396
0
1
32 32 7564 7564 3
0
0
2:345
0
0
0
2
0
1:5
1
0
0
0
1
1:5
0
1
3 ðk k ¼ 75 ðk k ¼ ðk k ¼ t1
t2
t3
2:345
Þ 1:803Þ 1:803Þ
In the above result, the norm of every right eigenvector in V matrix matrix equals one. Thus from (2.17), the eigenvalue sensitivity s li equals the norm of the correspond corre sponding ing left eigenvecto eigenvectorr ti , which has been listed along with the corresponding vector above.
ð Þ
k k
i calcula culated ted in TableBased 2.1. on this result, the values of M i 1 ; 2; 3 are cal Thee in Th insp spec ecti tion on of the the tw two o matr matric ices es sh show owss th that at un unli like ke in A2 , the ln 1. 1 in A1 is completely decoupled and thus has sensitivity s 1 This feature is reflected by M 1 , which reaches its maximal value for A1 and is considered by M 3 also. Also, unlike A2 , A1 has a large element (4.5) which A 1 as causes higher sensitivity of other two adjacent eigenvalues 2; 3 of A well as a higher value of ss L . This feature is reflected by a smaller value of M 2 for A1 and only M 3 can and is con consid sidere ered by M 3 also. Th There erefor fore, e, on comprehensively reflect these two conflicting features about robust stability.
ð ¼ ¼
Þ
ð Þ ¼
ð¼ Þ
ð Þ
ð Þ
Copyright 2004 by Marcel De Dekker, kker, Inc. All Rights Reserved.
Stability ility Measurem Measurements ents of Two Dynamic Mat Matrices rices Table 2.1 Robust Stab A1
M 1 M 2 2
¼ s ðLÞ1j1 1j s ð ð1Þ1 s ð ð2Þ1 s ð ð3Þ
M 3 3
1 0.1097 1 0.2169 0.2169 1 2 2 0 :4338
ð Þ j j ¼ ð
s
A2
0.691 0.2014 0.5546 0.5546 0.4264 1 1 1 0 :5546
ð Þ j j ¼ ð
s
eigenv envalu aluee 2 has the for ma matr trix ix A1 , eig From th From thee defin definit itio ion n of M 3 , for shortest likelihood margin (0.4338) to instability and therefore is most likely to beco become me unst unstab able le,, even even th thou ough gh the the ei eige genv nvalu aluee ln 1 is closest to unstable region. Thus for matrix A1 , M 3 has considered accurately the low consid sidere ered d the hig high h has no not, t, an and d M 3 has con sensiti sen sitivit vity y of its ln while M 2 has sensitivity of other eigenvalues while M 1 has not. Overall, matrices A 1 and A 2 are quite similar—with one element 4.5 in A being divided into two elements (1.5 and 3) in A . Hence a reasonable 2 1 robust stability measure should not differ too much for these two matrices. We notice that this is the case for M 3 but not for M 1 or M 2 . This example shows quite convincingly that M 3 is considerably more accurate than M 1 and M 2 . V V 1 may not Although maximizing M 2 or minimizing s L improve robust stability as directly as maximizing M 3 , it also implies in a simple, scalar, and unified sense the improvement of other system aspects such as the lowering of feedback control gain K and the smoothing of transient response (see Chap. 8 and Kautsky, 1985). Both aspects are very importa imp ortant, nt, esp especi eciall ally y when when the dynami dynamicc matrix matrix eig eigenv envalu alues es are alr alread eady y
¼
ð Þð¼ k kk kk
kÞ
k k
assigned. We have mentioned that the numerical algorithms used to minimize s L can also be used to maximize M 3 . In addition, there is an analytical method for improving M 3 . This method is based on the possibility of simple decoupling of the feedback system eigenstructure into p blocks ( p number of system inputs). The decoupling is extremely effective in improving the system sys tem’s ’s robust robustnes ness. s. For exa exampl mple, e, the eig eigenv envalu aluee 1 of Exam ampl plee 2.5 2.5 is and th thu us ha hass th thee low lowest est pos ossi sibl blee comp co mple lete tely ly deco decoup uple led d in matr matrix ix A1 and se sens nsit itiv ivit ity. y. Exam Exampl plee 2.4 2.4 also also sh show owss conv convin inci cing ngly ly th thee st stro rong ng ef effe fect ct of coupling on eigenvalue sensitivity.
ð Þ
¼
Copyright 2004 by Marcel De Dekker, kker, Inc. All Rights Reserved.
CONCLUSION State space control theory provides distinctly general, accurate, and clear analysis on linear time-invariant systems, especially their performance and sensitivity properties. Only this kind of analysis and understanding can be used us ed to gu guid idee gene genera rall lly y and and effe effect ctiv ivel ely y the the de desi sign gn of comp comple lex x co cont ntro roll systems. This is the reason that linear time-invariant system control results form the basis of the study of other systems such as nonlinear, distributive, and time-varying systems, even though most practical systems belong to the latter category. This is also the reason that the development of state space control theory has always been significant and useful. For example, because of the lack of accurate measure of system performance and robustness, the direct
design of loop transfer function has not been generally effective (see also the end of Secs. 3.1 and 9.3). Starting with the next chapter, we will see that there are basic, practical, and significant design problems which can only now be solved satisfactorily using state space techniques.
EXERCISES 2.1 Let the dy dynamic namic matrice matricess of two systems systems be
2 3 ¼ 4 5 1 0 0
A1
1 1 0
0 1 2
and
2 3 5 ¼ 4 1 0 0
A2
0 1 0
0 1 2
(a)
Based Based on the eigens eigenstru tructu cture re decompo decomposit sition ion of Exer Exercis cisee probl problem em 1.7 and based on (2.4), derive the time function eAi t ; i 1 ; 2. 1 i 1 ; 2 . (b) Deriv ivee eAi t using eAi t l1 sI Ai 0 (c) (c) De Deri rive ve zero zero-i -inp nput ut resp respon onse se eAi t xi 0 with xi 0 1 2 3 i 1 ; 2 . Plot and compare the wave forms of these two responses.
¼
ð ¼ ¼ Þ
¼ ¼
fð g gð ¼ ¼ Þ ðÞ ð Þ¼½
2.2 Repe Repeat at 2.1 for the tw two o matric matrices es from Exa Example mple 2.5 2.5.. Consider sider the system 2.3 Con
A1 ; B
¼ ½ 0
1
1 0
and
¼ ¼
C
1 0
0 1
1 1
where matrix A1 is similar to that of 2.1 above.
Copyright 2004 by Marcel De Dekker, kker, Inc. All Rights Reserved.
(a)
Using Using (2. (2.1) 1) and the the result result of 2.1, 2.1, deri derive ve the ze zeroro-sta state te res respon ponse se of the two outputs of this system for unit step input. 1 Y zzss s G s U s G s =s derive y t , (b) Usin (b) Using g (1.7 (1.7), ), y t l where G s is derived based on (1.9). (c) Co Compu mpute te the the ban bandwi dwidth dth for for the the two elem element entss of G s . (d) Plo Plott and co compa mpare re th thee wavefo waveforms rms of th thee two o outp utputs uts..
ðÞ
ðÞ¼
f ð Þ¼ ð Þ ð Þ¼ ð Þ g ðÞ
ðÞ
2.4 Anal Analyze yze the robust stab stability ility of the two systems from Ex Example ample 2.1. Notice that for eigenvalue l within a Jordan block larger than 161, the corresponding sensitivity s l should be modified from (2.17)
ðÞ
[Golu [G olub b an and d Wilk Wilkin inso son, n, 1976 1976]. ]. A simp simple le me meth thod od is to add add to toge gethe therr
sensitivities (2.17) of all eigenvalues within a same Jordan block. For example, in matrix A1 , suppose the first two left and right eigenvectors are t1 ; t2 ; v1 ; v2 and correspond to a multiple eigenvalue l1 1 in a 262 Jordan block, then
ð¼ Þ p p sðl Þ ¼ kt kkv k þ kt kkv kð¼ ð1Þð 3 Þ þ ð 8 Þð 1= 2ÞÞ 1
1
1
2
ffiffi ffiffi p ffiffi ffi ffi
2
2.5 Repe Repeat at 2.4 for the two dyna dynamic mic matr matrices ices from 2.1. 2.1. 2.6 Veri Verify fy the expre expression ssion (2. (2.22) 22) from Example Example 2.4. 2.7 Veri Verify fy the concl conclusion usion from from (2.26) to (2.27). (2.27). 2.8 Re Repe peat at Ex Exer erci cise sess 2. 2.1 1 and and 2. 2.4 4 for for th thee fo foll llow owing ing dy dyna nami micc ma matr tric ices es.. Compare the results.
2 3 2 3 2 3 64 75 64 75 64 75 2 3 2 3 64 75 64 75 3
0
0
3
0
0
3
0
0
3
2
0 ;
3
2
0 ;
2
2
0 ;
1:5
0
1
1:5
0
1
2:5
0
1
3
0
0
3
0
0
2:5
2
0 ;
2
2
0
2
0
1
2:5
0
1
Copyright 2004 by Marcel De Dekker, kker, Inc. All Rights Reserved.
3
Feedback System Sensitivity
The feedback system discussed in this book consists of two basic subsystem compo com ponen nents— ts—an an ‘‘open ‘‘open-lo -loop op system system,’’ ,’’ which which con contai tains ns the giv given en ‘‘plan ‘‘plantt system,’’ and a feedback controller system, called a ‘‘compensator.’’ Hence thee anal th analys ysis is of su such ch fe feed edba back ck sy syst stem emss is diffe differe rent nt fr from om th that at of a sing single le system. Of the the tw two o crit critic ical al prop proper erti ties es of perf perfor orma manc ncee an and d low low sens sensiti itivi vity ty (robustness) of feedback systems, sensitivity has been less clearly analyzed in state space control theory. It is analyzed in this chapter, which is divided into two sections. Sect Se ctio ion n 3. 3.1 1 hi high ghli ligh ghts ts a conc concep eptt in clas classi sica call cont contro roll theo theory ry ab abou outt feedback system sensitivity—the decisive role of loop transfer function in
Copyright 2004 by Marcel De Dekker, kker, Inc. All Rights Reserved.
the se the sens nsit itiv ivit ity y of feed feedbac back k syst system ems. s. This This co conc ncep eptt wi will ll gu guid idee th thee de desi sign gn throughout this book, even though the focus remains on state space models of the systems. Sect Se ction ion 3. 3.2 2 anal analyz yzes es the the sens sensit itiv ivit ity y pr prop oper erti ties es of th thre reee ba basi sicc an and d existin exi sting g fee feedba dback ck contro controll str struct ucture uress of state state space space contro controll the theory ory—di —direc rectt state feedback, static output feedback, and observer feedback. The emphasis is on the observer feedback structure, which is more commonly used than other two structures. A key design requirement on the robustness property
of this structure, called loop transfer recovery (LTR), is introduced.
3.1
SENSI SENSITIV TIVITY ITY A AND ND LOO LOOP P TRANS TRANSFER FER FUNC FUNCTIO TION N OF FEEDBACK SYSTEMS
The basic feedback control structure studied by control systems theory is shown in Fig. 3.1. In this system structure, there is a feedback path from the plant system output Y ðsÞ to input U ðsÞ through a general feedback controller system, called ‘‘compensator’’ H ðsÞ. Here RðsÞ and DðsÞ are Laplace transforms of an exte extern rnal al re refe feren rence ce sign signal al rðtÞ and and an inpu inputt distu disturb rban ance ce sign signal al dðtÞ, respectively. The plant system, which is subject to control, is either GðsÞ itself or a compon com ponent ent sys system tem of GðsÞ and and with with output output Y ðsÞ. In this book, we will generally treat the plant system as G ðsÞ. Hence the controller to be designed is H ðsÞ. The structure of Fig. 3.1 is very basic. For more complicated control system configurations, the analysis and design is usually carried out block by block and module by module, with each block (or module) structured like Fig. 3.1. Because Becau se input U ðsÞ can co cont ntro roll th thee be beha havi vior or of outp output ut Y ðsÞ, such input is called ‘‘control input signal.’’ Because the control signal usually
Figure 3.1 The basic stru structure cture of feedback feedback (closed (closed-loop -loop)) systems.
Copyright 2004 by Marcel De Dekker, kker, Inc. All Rights Reserved.
re requ quire iress a larg largee amou amount nt of po powe wer, r, th ther eree wi will ll very very li like kely ly be dist distur urba banc ncee asso as soci ciat ated ed with with the the ge gene nera ratio tion n of U ðsÞ. This This dist distur urba banc ncee is co comm mmon only ly treate tre ated d in the sys system tem’s ’s mat mathem hemati atical cal mod model el of of Fig Fig.. 3. 3.1 1 as an add additi itiona onall signal DðsÞ. Thee pu Th purp rpos osee and and requ requir irem emen entt of cont contro roll syste systems ms is ge gene nera rall lly y th thee control of plant system output (or response) Y ðsÞ so that it can quickly reach and stabilize to its desired state, such as the desired vehicle and engine
speed, the desired radar and airborne system angle, the desired robot arm position, the desired container pressure and temperature, etc. The desired system output state is usually specified by the reference signal RðsÞ. Hence how well the how the sy syst steem outp utput reac reach hes it itss desi desire red d sta tate te det eteerm rmin ines es th thee performance of the system. The final steady state of system response is relatively easy to analyze (using the final value theorem for example) and relatively easy to satisfy via feedback control design. Hence the transient response properties (such as the convergent speed) are critical factors to system performance and are the main challenges of feedback control system design. The most basic feature of the feedback control system structure of Fig. 3.1 is that the control signal U ðsÞ, which controls signal Y ðsÞ, is itself contro con trolle lled d bas based ed on Y ðsÞ. This feedback of Y ðsÞ to U ðsÞ creates a loop which starts and ends at U ðsÞ, and whose transfer function called ‘‘loop transfer function’’ is LðsÞ ¼ H ðsÞGðsÞ
ð3:1Þ
We therefore call the feedback system a ‘‘closed-loop system.’’ On the other hand ha nd,, a sy syst stem em with withou outt fe feed edba back ck [o [orr H ðsÞ ¼ 0] is call called ed an ‘‘op ‘‘open en-lo -loop op system.’’ Figure 3.2 shows a block diagram where the control signal U ðsÞ is not influenced by its control object Y ðsÞ. The loop transfer function of this system is LðsÞ ¼ 0
ð3:2Þ
Figure 3.2 The str structur ucture e of open-l open-loop oop sys systems. tems.
Copyright 2004 by Marcel De Dekker, kker, Inc. All Rights Reserved.
A main main di diffe ffere renc ncee be betw twee een n feed feedba back ck co cont ntro roll an and d co cont ntro roll wi with thou outt feedback concerns the sensitivity to the plant system mathematical model uncertainty, defined as DGðsÞ, and to the control input disturbance DðsÞ. This section shows that this difference is determined almost solely by the loop transfer function LðsÞ, which is created by the feedback configuration
itself. To si simp mpli lify fy th thee de desc scrip riptio tion n of this this co conc ncep ept, t, only only SI SISO SO sy syst stem emss ar aree studied in this section. However, this basic and simple concept is general to MIMO systems as well.
3.1.1 3.1 .1
Sensi Sensitiv tivity ity to Sy Syste stem m Model Model Unc Uncerta ertaint inty y
In most most practic practical al sit situat uation ions, s, the given given mat mathem hemati atical cal mod model el (ei (eithe therr sta state te space or transfer function) of the plant system is inaccurate. This is because the practical physical system is usually nonlinear, and its parameters are usuall usu ally y dis distrib tributi utive ve and are dif difficu ficult lt to mea measur suree acc accura uratel tely. y. Even Even for an initially accurate model, the actual plant system will inevitably experience wear-out wear -out and accid accidental ental damage, both of which which can make the mathemati mathematical cal model of the plant system inaccurate. To sum summar marize ize,, there there is an inevit inevitabl ablee dif differ ferenc encee bet betwee ween n the act actual ual plant pla nt sys system tem and its mat mathem hematic atical al model model GðsÞ. Th This is diff differ eren ence ce is calle called d ‘‘model uncertainty’’ and is defined as DGðsÞ. Therefore, it is essential that thee co th cont ntro roll sy syst stem ems, s, whic which h are are desi design gned ed ba base sed d on th thee give given n av avai aila labl blee mathematical model GðsÞ, have low sensitivity to DGðsÞ. In single-variable systems, the transfer function from RðsÞ to Y ðsÞ of control systems of Figs Figs 3.1 3.1 and and 3.2 3.2 are, respectively GðsÞ T c ðsÞ ¼ 1 þ H ðsÞGðsÞ
ð3:3aÞ
and To ðsÞ
¼ G ðsÞ
ð3:3bÞ
Let DT ðsÞ be the uncertainty of overall control system T ðsÞ caused by DGðsÞ. We will use relative plant system model uncertainty DGðsÞ=GðsÞ and relativ rel ativee contro controll system system unc uncert ertain ainty ty DT ðsÞ=T ðsÞ to meas measur uree th thee ov over eral alll control system sensitivity vs. plant system model uncertainty.
Copyright 2004 by Marcel De Dekker, kker, Inc. All Rights Reserved.
Definition 3.1 The sensitivity of a control system T ðsÞ to DGðsÞ is defined as
Þj ¼
sðT
ð Þ= ð Þ ð Þ= ð Þ
DT
G
s T s DG s G s
ð3:4aÞ
For small enough DGðsÞ and DT ðsÞ, sðT ÞjG &
qT ðsÞ G
ð Þ ð Þ
s qGðsÞ T s
ð3:4bÞ
Equation (3.4b) is the general formula for determining sðT ÞjG . Substituting (3.3a) and (3.3b) into (3.4b), we have sðT c
Þj ¼ þ
G
1
1 1 ¼ H ðsÞGðsÞ 1 LðsÞ
ð3:5aÞ
and sðT o ÞjG ¼ 1
ð3:5bÞ
A comparison of (3.5a) and (3.5b) shows clearly that the sensitivity to the plant system model uncertainty of a closed-loop system can be much lower than that of the open-loop system. The difference is determined solely by loop transfer function LðsÞ.
Exampl Exa mple e 3.1
Sensi Sensitiv tivity ity to th the e Unc Uncert ertain ainty ty of Some Some Individual Plant System Parameters
Let GðsÞ ¼
K s þ l
and H ðsÞ ¼ 1
Copyright 2004 by Marcel De Dekker, kker, Inc. All Rights Reserved.
Then from (3.3),
T c ðsÞ ¼
K s þ l þ K
and T o ðsÞ ¼
K l
s þ Thus from (3.4b),
sðT c
ð Þ þ þ ¼ Þj ¼ =ð þ þ Þ ð Þ ð þ þ Þ ¼ ð Þ ðÞ ¼ ¼ Þj ¼ ð Þ ð Þ ð þÞ =ð þ Þ Þj ¼ ¼ ð Þ ð þ þ Þ =ð þ þ Þ =ð þ Þ ¼ ð Þ qT c
K s qK T c s
K
1
sðT o
sðT c
s K qK T o s
qT c
l
s
qT o
s
ql
l
s
2
K s
K 1 s l K s l
l
K
1
K
s
l
ð Þ ¼ Þj ¼ ðÞ ð þ ¼ ð þ Þ l
K
l
K
K
l
2
K s
l
K
s l L s
1
sðT o
s
l
T c s
ql
l
and
K K
l
1 L s
qT o
K
s
l
K
l
T o s
s
l
2 l Þ K =ðs þ l
Þ
Therefore, the sensitivity to either plant system parameter K or or l of a closedloop system equals that of an open-loop system divided by 1 LðsÞ. For open-loop systems, at s ¼ 0, this sensitivity equals 1 ¼ 100 %, which is quite high.
3.1.2 3.1 .2
Sensi Sensitiv tivity ity to Co Contr ntrol ol Inp Input ut Dis Distur turban bance ce
As introduced in the beginning of this section, disturbance DðsÞ associated wit ith h the the ge gene nera rati tion on of la larg rgee powe powerr cont contro roll inpu inputt U ðsÞ is seri seriou ouss an and d
Copyright 2004 by Marcel De Dekker, kker, Inc. All Rights Reserved.
inevitable. Therefore, a practical control system must have low sensitivity to
DðsÞ.
In pra practic ctice, e, the contro controller ller which which actual actually ly genera generates tes and ass assert ertss the control is usually called the ‘‘actuator.’’ From the superposition principle of linear systems, in the presence of ¼ 0 Þ, the disturbance DðsÞ ð6 the clos closed ed-lo -loop op sy syst stem em (Fig. Fig. 3.1 3.1)) and open open-loop -loop (Fig. 3.2 3.2)) responses are system (Fig. system Y c ðsÞ ¼ GðsÞ RðsÞ þ GðsÞ DðsÞ 1 LðsÞ 1 LðsÞ
ð3:6aÞ
Y o ðsÞ ¼ G ðsÞRðsÞ þ GðsÞDðsÞ
ð3:6bÞ
and
respectively. If amo mong ng the the resp respec ecti tiv ve two two te term rmss of (3 (3.6 .6a, a, b) th thee fir firsst te term rm is the desired control system response which follows RðsÞ and which is the respon res ponse se when when DðsÞ ¼ 0, th then en the the seco second nd te term rm is th thee de devi via ati tion on fr from om the desired response and is the sole effect of disturbance D ðsÞ. Therefore, the gain ga in (m (mag agni nitu tude de of tran transf sfer er fu func ncti tion on)) of this this seco second nd te term rm repr repres esen ents ts the sensitivity of the corresponding system to D ðsÞ. The higher the gain, the higher the sensitivity to DðsÞ.
Definition 3.2 A system’s sensitivity to its control input disturbance is represented by its gain from this disturbance to its output. Similar to the conclusions from Subsection 3.1.1, a comparison of the second terms of (3.6a) and (3.6b) shows clearly that the sensitivity to control input disturbance of closed-loop systems can be much lower than that of open-loop systems. The difference is an additional denominator 1 LðsÞ, which is determined solely by loop transfer function LðsÞ.
Exampl Exa mple e 3.2
Sensi Sensitiv tivity ity to Out Output put Me Measu asurem rement ent No Noise ise
It is important to measure the sensitivity to output measurement noise. In practical feedback control systems, besides the undesirable effect of control input disturbance, there is another common and undesirable effect, caused by output output measuremen measurementt noise. noise. This noise is represent represented ed in the mathemat mathematical ical model as an additional signal N ðsÞ to Y ðsÞ, and in the block diagram of
Copyright 2004 by Marcel De Dekker, kker, Inc. All Rights Reserved.
Figure 3.3 Feedba Feedback ck contro controll system with outp output ut measure measurement ment noise noise..
Fig. 3.3, which shows a feedback control system with output measurement noise. In many practical analog systems, especially nonelectrical systems, the signal Y ðsÞ, su such ch as velo velocit city, y, an angl gle, e, pres pressu sure re,, and and te temp mper erat atur ure, e, is ve very ry difficult diffi cult to measure measure accur accurately. ately. In addition, addition, the imple implementat mentation ion of feedb feedback ack control often requires that the measured analog signal be transformed to a different analog signal such as an electrical signal. The device that performs this operation is called a ‘‘transducer.’’ Such operations can also introduce err rro or. Beca ecaus usee th thee prese resenc ncee of outp outpu ut me meas asu urem remen entt nois noisee is alm lmo ost inevitable, a feedback system must have low sensitivity to such noise. The purpose of measuring the feedback system output Y ðsÞ is to help generate a desirable control U ðsÞ, so the undesirable effect of system output measurement noise is reflected mainly in its effect on U ðsÞ. Applying Mason’s formula to the system in Fig. 3.3, when RðsÞ ¼ 0,
U ðsÞ ¼
H ðsÞ H ðsÞ N ðsÞ ¼ N ðsÞ 1 LðsÞ 1 þ H ðsÞGðsÞ
ð3:7Þ
N ðsÞ on U ðsÞimplies This the effect of N . Similarlower to Definition 3.2, lower magnitude of theistransfer function of (3.7) sensitivity against N ðsÞ. It is clear from (3.7) that the sensitivity to N ðsÞ is very much related to the loop loop transf transfer er functi function on LðsÞ. For For exam exampl ple, e, fr from om (3 (3.7 .7), ), in open open-l -loo oop p syst sy stem emss which hich have have no feed feedba back ck ½H ðsÞ ¼ L ðsÞ ¼ 0 an and in whic ich h the Y ðsÞ does not affect the system, the sensitivity to the output measurement of Y measurement noise N ðsÞ is zero. Substituting (3.7) into Y ðsÞ ¼ G ðsÞU ðsÞ,
Y ðsÞ ¼
LðsÞ GðsÞH ðsÞ N ðsÞ N ðsÞ ¼ 1 LðsÞ 1 LðsÞ
This is the effect of N ðsÞ on system output Y ðsÞ.
Copyright 2004 by Marcel De Dekker, kker, Inc. All Rights Reserved.
ð3:8Þ
In the analysis of feedback system sensitivity to plant system model uncertainty and control input disturbance, it seems that the higher the loop gain jLðsÞj, the lower the sensitivity. However, Example 3.2 shows that a high jLðsÞj or or a large jH ðsÞj does does no nott lowe lowerr th thee sens sensit itiv ivit ity y to outp output ut measurement noise at all. In fact, it is equally undesirable to indiscriminately increase the loop gain jLðsÞj because of the following three reasons. 1. 2.
3.
A hi high gh loop gain gain is likely likely to caus causee fe feed edba back ck system system ins instab tabil ilit ity, y, from root locus results. This is especially true for plant systems either with pole-zero excess exceeding two or with unstable zeros. A high high lo loop op gain gain j LðsÞj can generally reduce system performance. From (3.3a) and the definition of bandwidth of Sec. 2.1, a higher jLðsÞj often implies a lower overall feedback system gain jT c ðsÞj and therefore a narrower bandwidth. A high high lo loop op ga gain in or a hig high h contr control olle lerr gain gain j H ðsÞj is more difficult to im imple pleme ment nt in prac practi tice ce.. A sy syst stem em with with high higher er ga gain in ge gene nera rally lly cons co nsu ume mess mo more re con ontr tro ol en ener erg gy an and d is mo more re li like kely ly to infl inflic ictt disturbance and cause failure.
Because of the above three reasons, the loop gain jLð j oÞj is shaped only at certain frequency bands. For MIMO systems, the loop gain is represented by the largest singular value of the p6 p dimensional matrix Lð j oÞ [Doyle and Stein, 1981; Zhou et al., 1995]. However, as described in Sec. 2.1, bandwidth is far less direct and far less generally accurate in reflecting system performance. Subsections 2.2.2 and 3.2.1 (at the end) also indicated that robust stability is far less generally accurately measured by the loop transfer function based gain margins and phase margins. In addition, the loop-shaping operation, though it is already very complicated, is lessdesign refined than state space design methods terms of how fully the available freedom is utilized. For example, it in seems that only on ly th thee gai ain n (bu (but not tthe he phas phasee angl angle) e) of loop loop tr tran ansf sfer er func functi tion on is considered by this operation. To summarize, the critical factor of feedback system sensitivity is the system loop transfer function itself, but not the high gain or only the gain, of this loop transfer function.
3.2
SE SENSI NSITIV TIVITY ITY OF OF FEEDBA FEEDBACK CK SYSTE SYSTEMS MS OF MODE MODERN RN CONTROL THEORY
Section 3.1 described the critical importance of loop transfer function for feedback system sensitivity. The same concept will be used to analyze the sensitivity of three existing and basic feedback control structures of state
Copyright 2004 by Marcel De Dekker, kker, Inc. All Rights Reserved.
space control theory. These three structures are state feedback, static output feedback, and observer feedback. Of the three structures, observer feedback system structure is much more commonly used than the other two. Because loop transfer function is determined by the internal feedback syste sy stem m stru struct ctur ure, e, from from now now on we will will let let th thee ex exte tern rnal al sy syst stem em refe refere renc ncee signal rðtÞ ¼ 0. In addition, we will assume that the plant system GðsÞ is irreducible.
3.2.1 3.2 .1
State State F Fee eedba dback ck Cont Control rol Sy Syste stems ms
The state feedback control systems (or direct state feedback systems) have a control signal uðtÞ of uðtÞ ¼ K xðtÞ
ð3:9Þ
where xðtÞ is th thee syste system m stat statee vect vector or,, and and K , which which is calle called d th thee ‘‘st ‘‘stat atee fe feed edba back ck ga gain in’’ ’’ or ‘‘st ‘‘stat atee fe feed edbac back k cont contro roll law, law,’’ ’’ is cons consta tant nt.. The The bloc block k diagram of this feedback control structure is shown in Fig. 3.4. It is clear from Fig. 3.4 that the loop transfer function of this system is 1
LðsÞ ¼ K ðsI AÞ
B D LKx ðsÞ ¼
ð3:10Þ
Substituting (3.9) into (1.1a), the dynamic equation of this feedback system becomes x_ ðtÞ ¼ ðA BK ÞxðtÞ þ BrðtÞ
ð3:11Þ
Hence matrix Asystem, theitsdynamic matrix thepoles corresponding direct BK is state feedback and eigenvalues areofthe of that feedback system. From Sec. 1.1, system state provides the most explicit and detailed information about that system. Therefore state feedback control, if designed
Figure 3.4 Direct state ffeedbac eedback k system systems. s.
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properly, should be most effective in improving system performance and robustness properties, even though this design is not aimed at shaping the loop transfer function LKx ðsÞ directly.
Theorem 3.1 For any con contro trolla llable ble plant plant system system,, the dir direct ect state state feedba feedback ck contro controll can assign arbitrary eigenvalues to matrix A BK , and the direct state feedback feedback system remains controllable.
Proof Any controllable system is similar to its corresponding block-controllable cano ca noni nica call form form,, whic which h is th thee dual dual vers versio ion n ðA0 ; C 0 Þ of its cor corresp respond onding ing block-observable canonical form of ðA; C Þ of (1.16). 0
The form of (1.16) implies that there exists a matrix K such that all unknown parameters of matrix A K 0 C can be arbitrarily assigned, and that A K 0 C remains remains to be in observable canonical form for any K 0 . Hence thee ei th eige genv nval alue uess of matr matrix ix A K 0 C can be ar arbi bitr trar arily ily as assi sign gned ed an and d th thee system ðA K 0 C ; C Þ remains to be observable for any K 0 . From the duality phenomenon, the above conclusions imply that the can be arbitrarily assigned, and that system eigenvalues of matrix A0 C 0 K can ðA0 C 0 K ; C 0 Þ remains to be controllable for any K . However, Howeve r, matrix matrix A BK in gene genera rall ca cann nnot ot pres preser erve ve th thee bloc blockkobse ob serv rvab able le cano canoni nica call form form of the the orig origin inal al ma matr trix ix A. He Henc ncee dire direct ct st stat atee feedback system cannot preserve the observability property of the original open-loop plant system ðA; B; C Þ. In addition, eigenvectors can also be assigned if p > 1, thus achieving robustness (see Sec. 2.2). The explicit design algorithms of state feedback control for eigenvalue/vector assignment will be introduced in Chap. 8. Beside Bes idess the abi abilit lity y to assign assign arbitr arbitrary ary pol poles es and the cor corres respon pondin ding g eigenvectors to the feedback system, state feedback control can also realize a so ca call lleed ‘‘ ‘‘li lin near ear quad uadrati raticc opti tima mall cont contrrol,’ ol,’’’ whose hose de desi sign gn wil illl be introduced in in Chap. 9. It has been proved that the loop transfer function LKx ðsÞ of such control systems satisfies the ‘‘Kalman inequality’’ such that
½I LKx ð j oÞ* R½I LKx ð j oÞ5R Vo
ð3:12aÞ
where R is symmetrical positive definite ðR ¼ R 0 > 0 Þ [Kalman, 1960].
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Based on (3.12a), it has been proved that for R ¼ rI ðr > 0 Þ, si ½I LKx ð j oÞ51 Vo
ð3:12bÞ
where si ði ¼ ¼ 1 ; . . . ; pÞ is the i -th -th singular value of the matrix. From (3.12b), the values of gain margin and phase margin of the feedback system are 1=2 ! ? and 560 , respectively [Lehtomati et al., 1981]. The SISO case of the above result can be shown in Fig. 3.5. 3.5. The shaded area of Fig. 3.5 indicates all possible values of LKx ð j oÞ that satisfy (3.12b). It is clear that the margin between these values and the 1 point is at least 1=2 to ? in magnitude, and 608 in phase angle. Since according to the Nyquist stability criterion, the number of encirclements of the 1 point determines determines feedbac feedback k system stabilit stability, y, this resul resultt implie impliess a good robust stability of quadratic optimal feedback systems. Notice that at this good robust stability, no large gain (distance to the origin) of LKx ð j oÞ is required at all. However, as will be introduced at the beginning of Chap. Chap. 9, the 9, the linear quad adra rati ticc opt ptim imal al con contro trol sy sysstem tems can be fo form rmu ulate lated d to have ave poor robustness (such as the minimum time problem). Yet the gain margin and phase margin indicate good robustness for all such systems. This is another proof that the gain margins and phase margins are not generally accurate measures of system robustness (see Subsection 2.2.2). The main drawback of direct state feedback control is that it cannot be generally implemented. In most practical plant systems, only the terminal inputs and outputs of the system are directly measurable; not the entire set of internal system states. In other words, the available information about most mo st pra practic ctical al system systemss can canno nott be as comple complete te and explicit explicit as for sys system tem
Fi Figu gure re 3. 3.5 5 Lo Loop op tran transf sfer er freq freque uenc ncy y re resp spon onse se of sin singl glee-in inpu putt qu quad adra ratic tic
optimal control systems.
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states. There states. Therefore, fore, direct state feed feedback back control should be considere considered d only as an ideal and theoretical form of control.
3.2.2 3.2 .2
Static Static Ou Outpu tputt Fee Feedba dback ck Contr Control ol Sy Syste stems ms
In static output feedback control systems, the control signal uðtÞ is uðtÞ ¼ K y y yðtÞ ¼ K y y C xðtÞ
ð3:13Þ
where yðtÞ ¼ C xðtÞ is a system output that is directly measurable and K yy is constant. The block diagram of this feedback system is shown in Fig. 3.6. The loop transfer function and the dynamic matrix of this feedback system are, respectively 1
LðsÞ ¼ K y y C ðsI AÞ
B
ð3:14Þ
and A BK y C , which are very similar to that of the direct state feedback system. The only difference is that the constant gain on xðtÞ is K yy C instead instead of K , wh wher eree C is a gi give ven n syst system em ma matr trix ix.. He Henc ncee st stat atic ic outp output ut fe feed edba back ck implements a constrained state feedback control with constraint K ¼ K yy C
ð3:15Þ
In other words, K must must be a linear combination of the rows of given matrix C , or K 0 [ RðC 0 Þ4 range space of C 0 (see Subsection A.1.2). Because the ¼ dimension of this space is m , which is usually smaller than n , this constraint can be serious.
Example 3.3 In a second-ord rdeer SISO system tem ðn ¼ 2 ; p ¼ m ¼ 1 Þ, if C is eit ithe herr [1 0] or [0 1], then from (3.15) the state feedback control law K ¼ ½k1 k2
Figure 3.6 Static o output utput ffeedbac eedback k system systems. s.
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realized by the static output feedback must have either k2 ¼ 0 or k1 ¼ 0, respectively. This situation generally implies a reduction of the effectiveness of the control from dimension 2 to dimension 1. If m ¼ n and if C C is nonsingular, then (3.15) is no longer a constraint, and static output feedback becomes direct state feedback in the sense that 1 xðtÞ ¼ C 1 yðtÞ and K y y ¼ KC . Therefore, direct state feedback control can be considered a special case of static output feedback control when C is nonsingular, and static output feedback control may be called ‘‘generalized state feedback control,’’ as is done in this book. Thee ad Th adva vant ntag agee of stati staticc outp output ut fe feed edba back ck cont contro roll is it itss ge gene nera ralit lity y because yðtÞ is directly measurable. Besides, its corresponding loop transfer 1 function is guaranteed to be K ðsI AÞ B of (3.14) for whatever K ¼ K yy C of of (3.15). This property is not shared by many other feedback systems (such as observer feedback systems). Finally, from the same argument of Theo Th eore rem m 3.1 3.1 and and it itss proo proof, f, stat static ic ou outp tput ut fe feed edba back ck co cont ntro roll pr pres eser erve vess cont co ntro roll llab abil ility ity and and obse observ rvab abili ility ty prop proper ertie tiess of th thee or orig igin inal al op open en-l -loo oop p system. Thee main Th main draw drawba back ck of stati staticc outp output ut fe feed edba back ck co cont ntro roll is that that it is usua us uall lly y too too we weak ak comp compar ared ed with with di dire rect ct st stat atee fe feed edba back ck co cont ntro rol. l. Th This is is because m is usua usually lly much much sm smal alle lerr than than n in pr prac actic tice, e, wh whic ich h ma make kess th thee const onstrrai ain nt (3 (3.1 .15 5) of stat static ic outp output ut fe feed edba back ck cont contro roll too too sev severe. ere. For exam ex ampl ple, e, on only ly when when m is la larg rgee en enou ough gh (as (as co comp mpar ared ed to n) such such tha that m þ p > n , can arbitra arbitrary ry eigenv eigenvalu alues es be assign assigned ed to the fee feedba dback ck system system dynamic matrix A BK y y C [Kimura, 1975]. Example 3.3 is another such example. As a result, the design of static output feedback control is far from satis sa tisfa fact ctor ory y [S [Syr yrmo moss et al al., ., 1994 1994]. ]. In th this is bo book ok,, st stat atic ic outp output ut fe feed edba back ck contro con troll design design algori algorithm thmss for either either pole pole ass assign ignmen mentt (Algor (Algorith ithm m 8.1 8.1)) or quadratic optimal control (Algorithm 9.2) are presented in Chaps 8 8 and 9, respectively.
3.2.3 3.2. 3
Obse Observer rver Feedback Feedback Systems— Systems—Loop Loop Transfer Transfer Recovery Recovery
An observer feedback system does not require the direct observation of all system states, and implements a generalized state feedback control which is
much stronger than the normal static output feedback Therefore observer feedback control structure overcomes the main control. drawbacks of both direct state feedback control structure and static output feedback control stru struct ctur ure; e; it is the the mo most st co comm mmon only ly used used cont contro roll st stru ructu cture re in st stat atee sp spac acee control theory.
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An observer is itself a linear linear time-invari time-invariant ant dynamic system, which has the general state space model z_ ðtÞ ¼ F zðtÞ þ LyðtÞ þ TBuðtÞ
ð3:16aÞ ð3:16bÞ
K xðtÞ ¼ K z zðtÞ K y yðtÞ
where zðtÞ is the state vector of observer system, B comes from the plant syst sy stem em st stat atee sp spac acee mode modell ðA; B; C Þ; and and ot othe herr ob obse serv rver er pa para rame mete ters rs ðF ; T ; L; K z ; K y Þ are free to be designed. This observer definition is more general than the existing ones. It is defined from the most basic and general observer function that it has yðtÞ and uðtÞ as its inputs and K xðtÞ as its output. The many distinct advantages of this general definition will be made obvious in the rest of this book. Let us first analyze the conditions for an observer of (3.16) to generate a desired state feedback control signal K xðtÞ. Becaus Bec ausee bot both h xðtÞ and yðtÞ ¼ C xðtÞ are are tim time-v e-vary arying ing sig signal nals, s, and because K and C are are cons consta tant nts, s, it is obvio obvious us th that at to gene genera rate te K xðtÞ in (3.16b), the observer state zðtÞ must converge to T xðtÞ for a constant T . This is the foremost important requirement of observer design.
Theorem 3.2 The necessary and sufficient condition for observer state zðtÞ to converge to T xðtÞ for a constant T , or for observer output to converge to K xðtÞ for a constant K , and for any zð0Þ and any xð0Þ, is T A F T ¼ L C
ð3:17Þ
where all eigenvalues of matrix F must must be stable.
Proof [Luenberger, 1971] From (1.1a), _ ðtÞ ¼ T AxðtÞ þ T BuðtÞ T _x
ð3:18Þ
Subtracting (3.18) from (3.16a), we have _ ðtÞ ¼ F zðtÞ þ LC xðtÞ T AxðtÞ z_ ðtÞ T _x
ð3:19Þ
¼ F zðtÞ FT xðtÞ þ FT xðtÞ þ LC xðtÞ T AxðtÞ ¼ F ½zðtÞ T xðtÞ
ð3:20Þ
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if and only if (3.17) holds. Because the solution of (3.20) is zðtÞ T xðtÞ ¼ eF t ½zð0Þ T xð0Þ
zeigenvalues xðtÞ for T stable. converges ges for any any zð0Þ and and an any y xð0Þ if if and only if all ðtÞ conver of F to are are This proof also shows that it is necessary to let the observer gain to uðtÞ be defined as TB in (3.16a), in order for (3.19) to hold.
After zðtÞ ¼ T xðtÞ is satisfied, replacing this zðtÞ into the output part of observer (3.16b) yields
K ¼ K Z Z T þ K y C ¼ ½K Z Z : K y y
T 4 K C C ¼
ð3:21Þ
Therefore (3.17) (with stable F ) and (3.21) together form the necessary and sufficient conditions for observer (3.16) to generate a desired state feedback K xðtÞ. The above introduction of (3.17) and (3.21) shows clearly that the two conditions have naturally and completely separate physical meanings. More expl ex plic icit itly ly,, (3.1 (3.17) 7) dete determ rmin ines es the the dyna dynami micc pa part rt of ob obse serv rver er ðF ; T ; LÞ exclusively and guarantees the observer state zðtÞ ) T xðtÞ exclusively, while (3.21)) presu (3.21 presumes mes that zðtÞ ) T xðtÞ is already satisfie satisfied d and det determ ermine iness the outp ou tput ut part part of obse observ rver er (K or K ¼ K C ) exclusively. This This ba basi sicc de desi sign gn concept has not been applied before (except for a very narrow application of function observer design) and will be emphasized throughout the rest of this book. There are many design algorithms that can satisfy (3.17) and (3.21) for arbitrary arbi trary (stable) (stable) eigen eigenvalue valuess of F and arbitrary K (assuming (assuming obse observable rvable sys yste tems ms). ). Howe oweve ver, r, th this is book ook wil illl pres presen entt only one su such ch algo lgorith rithm m (Alg (A lgor orit ithm hm 7. 7.1) 1),, whic which h has has an additional featur featuree of min minimiz imized ed ob obser server ver orde or der. r. This This is be beca caus usee (3 (3.17 .17)) and and (3 (3.2 .21) 1) have have not address addressed ed the critica criticall robustness property of the observer feedback systems. This property will be analyzed in the rest of this chapter.
As st stat ated ed in the the begi beginn nnin ing g of th this is subs subsec ecti tion on,, ob obse serv rver er fe feed edba back ck systems have been the most commonly used control structure in state space control theory. Because an observer can generate the state feedback control signal K xðtÞ ¼ KC xðtÞ [if (3.17) holds] and because the union of observer pole po less and and ei eige genv nval alue uess of A BK ¼ A BKC form formss th thee enti entirre set set of observer feedback system poles [if (3.17) holds, see Theorem 4.1], it has been presumed that observer feedback systems have the same ideal robustness
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proper prop erti ties es as th thos osee of the the di dire rect ct stat statee fe feed edba back ck sy syst stem em co corr rres espo pond ndin ing g K ¼ KC . Howeve How ever, r, in pra practic cticee sin since ce the 196 1960s, 0s, bad rob robust ustnes nesss pro proper pertie tiess of observer feedback systems have commonly been experienced, even though the observ observer er imp implem lement entss a state state feedba feedback ck con contro troll who whose se cor corres respon pondin ding g direct state feedback system is supposed to have ideal robustness properties (see Subsection 3.2.1). Because robustness with respect to model uncertainty and control disturbance is critically important for most practical engineering syst sy stem emss (s (see ee Se Sec. c. 3.1) 3.1),, stat statee spac spacee cont contro roll th theo eory ry ha hass no nott fo foun und d ma many ny successful practical applications since the 1960s. At the same time, the application of the polynomial matrix and the rati ra tion onal al poly polyno nomia miall matri matrix x has has exte extend nded ed cl clas assic sical al co cont ntro roll th theo eory ry into into MIMO systems [Rosenbrock, 1974; Wolovich, 1974; Kaileth, 1980; Chen, 1984 19 84;; Vidy Vidyas asag agar ar,, 1985 1985]. ]. Usin Using g the the co conc ncep eptt of loop loop tr tran ansf sfer er fu func nctio tions ns,, clas classi sica call cont contro roll theo theory ry clar clarifi ifies es be bett tter er th than an mo mode dern rn co cont ntro roll th theo eory ry th thee analysis of feedback system robustness properties (see Sec. 3.1). Furthermore, matrix singular values which can simply and accurately represent the matrix matri x norm (such as loop transf transfer er function matrix norm or loop gain) have become practically computable by computers (see Sec. A.3). As a result, clas cl assi sica call cont contro roll theo theory ry,, espe especi cial ally ly in te term rmss of it itss robu robust st de desi sign gn,, ha hass witnessed witn essed significant significant developme development nt duri during ng the past two decades [Doyle et al., 1992]. For example, the H ? problem, which may be briefly formulated as 1 minfmax fk½I Lð j oÞ k? gg o
ðsee Definition 2:4Þ
has received much attention [Zames, 1981; Francis, 1987; Doyle et al., 1989; Kwakernaak, 1993; Zhou et al., 1995]. Until the end of 1970s, there was a consensus of understanding on the cause of the problems of bad robustness observer feedback systems. This understanding was based solely on the perspective of loop transfer functions [Doyle, 1978]. We will describe this understanding in the following. The feedback system of the general observer (3.16) can be depicted as
in Fig. 3.7, which shows that an observer can be considered a feedback compensator H ðsÞ with input yðtÞ and output uðtÞ, where U ðsÞ ¼ H ðsÞY ðsÞ 1 1 1 ¼ ½I þ K Z Z ðsI F Þ TB ½K y þ K Z Z ðsI F Þ LY ðsÞ
ð3:22Þ
It should be noticed from (3.16) that the transfer function from signal y ðtÞ to
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Figure 3.7 Block dia diagram gram of genera generall observe observerr feedback syste systems. ms.
K xðtÞ is 1
H Kx ðsÞ ¼ ½K y þ K Z ðsI F Þ
L
ð3:23Þ
which is different from H ðsÞ of (3.22). The difference is caused solely by the feedback of signa signall uðtÞ to the observer. If this feedback, which is defined 1 by its path gain TB and its loop gain K Z Z ðsI F Þ TB, equals zero, then H ðsÞ ¼ H Kx Kx ðsÞ.
Theorem 3.3 The loop transfer function at the break point K xðtÞ of Fig. 3.7, LKx ðsÞ, equals that of the corresponding direct state feedback system (3.10), or 1
LKx ðsÞ ¼ K ðsI AÞ
B
ð3:24Þ
Proof [Tsui, 1988a] From Fig. 3.7, 1
LKx ðsÞ ¼ H Kx Kx ðsÞGðsÞ K Z Z ðsI F Þ
TB
ð3:25aÞ
by (3.23) 1 ¼ K y GðsÞ K Z Z ðsI F Þ ½LGðsÞ þ TB
ð3:25bÞ
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by (1.9) 1 ¼ ½K y C þ þ K Z þ sT TAÞðsI AÞ1 B Z ðsI F Þ ðLC þ
by (3.17) 1 ¼ ½K y C þ þ K Z ðsI AÞ1 B Z ðsI F Þ ðsI F ÞT ð
by (3.21)
¼ K ðsI AÞ1 B Fi Figu gure re 3. 3.7 7 also also show showss that that K xðtÞ is only nly an inte intern rnal al sign signal al of compensator H ðsÞ, whil whilee uðtÞ is the real anal analog og co cont ntro roll sign signal al that that is attributed to the plant system GðsÞ and which is where the disturbance is introduced (see Subsection 3.1.2). Therefore, the loop transfer function LðsÞ, which really determines the sensitivity properties of the observer feedback system, should be the one at break point uðtÞ [Doyle, 1978]. From Fig. 3.7, LðsÞ ¼ H ðsÞGðsÞ
by (3.22)
¼ ½I þ K Z F Þ1 TB1 ½K y þ K Z F Þ1 LGðsÞ Z ðsI Z ðsI
ð3:26Þ
Because LðsÞ ¼ AÞ1 B and because loop transfer func6 L Kx ðsÞ ¼ K ðsI tion tio n plays plays a cri critic tical al role role in the fee feedba dback ck system system sensit sensitivit ivity, y, the obs observ erver er feed feedba back ck syst system em ha hass di diff ffer eren entt ro robu bust stne ness ss pr prop oper erti ties es fr from om th that at of the the corresponding direct state feedback system [Doyle, 1978].
Example 3.4 In order to further understand the difference between the two loop transfer functions of (3.25) and (3.26), we will analyze two more system diagrams of observer feedback systems. The first diagram (Fig. 3.8) is called a ‘‘signal flow diagram.’’ For simplicity of presentation, we may assume the path branch with gain K y ¼ 0 and ignore this path branch. Then Fig. 3.8 shows that at node 1 uðtÞ there is only one loop path. The loop with gain K Z Z ðsI F Þ TB is attached to this single loop path. In contrast, at node K xðtÞ, there are two
Copyright 2004 by Marcel De Dekker, kker, Inc. All Rights Reserved.
Figure 3.8 Sign Signal al flow diagra diagram m of observer feedback feedback systems. systems.
loop paths. The loop with gain K Z F Þ1 TB is an independent loop Z ðsI path between the two. The second block diagram diagram (Fig. 3.9) 3.9) is also common in literature. In this equivalent block diagram of observer feedback systems, 1
H y ðsÞ ¼ ½K y þ K Z Z ðsI F Þ
L ¼ H Kx Kx ðsÞ of ð3:23Þ
and 1
H u ðsÞ ¼ K Z Z ðsI F Þ
TB
ð3:27Þ
We should reach the same conclusion from Figs 3.8 and 3.9 on the loop transfer functions at nodes uðtÞ and K xðtÞ. They are 1
LðsÞ ¼ ½I H u ðsÞ
H y ðsÞGðsÞ
1 1 1 ¼ ½I þ K Z Z ðsI F Þ TB ½K y þ K Z Z ðsI F Þ LGðsÞ
ð3:26Þ
Figure 3.9 An equiva equivalent lent block diag diagram ram of observe observerr feedback sys systems. tems.
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and LKx ðsÞ ¼ H y y ðsÞGðsÞ þ H u ðsÞ
ð3:25aÞ
Z ðsI F Þ1 ½LGðsÞ þ TB ¼ K y GðsÞ K Z
ð3:25bÞ
respectively.
Theorem 3.4 The necessary and sufficient condition for observer feedback system loop transfer function LðsÞ to be the same as that of the corresponding direct state feedback system LKx ðsÞ is Z ðsI F Þ1 TB ¼ 0 Vs H u ðsÞ ¼ K Z
ð3:28aÞ
For freely designed state feedback gain K [or K Z Z of (3.21)], (3.28a) becomes 1
H u ðsÞ ¼ K Z Z ðsI F Þ
TB ¼ 0 Vs and K Z Z
ð3:28bÞ
The necessary and sufficient condition for (3.28b) is TB ¼ 0
ð3:29Þ
Proof 3.7, Example Figure 3.7, Example 3.4, 3.4, and and the comparison between (3.25) and (3.26) all
indicate clearly that the difference between LKx ðsÞ and LðsÞ is caused solely by the the feed feedba back ck lo loop op [wit [with h gain gain H u ðsÞ]. Ther Theref efor ore, e, th thee ne nece cess ssar ary y an and d sufficient condition for LðsÞ ¼ L Kx ðsÞ is H u ðsÞ ¼ 0 [or (3.28a)]. 1 Because ðsI F Þ should be nonsingular Vs and K Z Z should be freely designed, TB ¼ 0 is obvi obviou ously sly the the nece necess ssar ary y an and d su suffi ffici cien entt co cond ndit itio ion n fo forr (3.28b). 3.1 1 and 3.9, this this th theo eore rem m indi indica cate tess th that at only only th thee Comparing Figs Figs 3. system structure of Fig. 3.1, which does not have the feedback from input uðtÞ and which is therefore called the ‘‘output feedback compensator’’ (see Sec. 4.4), can guarantee the same loop transfer function of the direct state feedback system. In papers [Doyle and Stein, 1979, 1981] subsequent to Doyle [1978], the authors imposed the problem of making LðsÞ ¼ L Kx ðsÞ, which is called
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‘‘lo ‘‘loop op tr tran ansf sfer er reco recove very ry’’ ’’ (L (LTR TR). ). This This pr prob oble lem m is cl clea early rly an ad addi diti tion onal al requirement of observer design—the observer is required not only to realize a de desi sire red d st stat atee fe feed edba back ck cont contro roll sign signal, al, but but also also to ha have ve LðsÞ ¼ L Kx ðsÞ. Mathematically speaking, from Theorems 3.2–3.4, the observer is required to satisfy not only (3.17) and (3.21), but also (3.29) (if the state feedback control is freely designed). Thee LT Th LTR R requ require ireme ment nt can can el elim imin inate ate th thee basi basicc ca caus usee of sens sensit itiv ivit ity y problems of observer feedback systems and is therefore of great practical importance to the entire state space control theory. Unfortunately, for almost all given given plant systems, it is impossible to have ha ve an obse observ rver er that that can can gene genera rate te th thee ar arbi bitra trari rily ly give given n st stat atee fe feed edba back ck signal K xðtÞ whil whilee sati satisf sfyi ying ng (3.2 (3.28a 8a)) or (3.2 (3.29) 9) (s (see ee Sec. Sec. 4.3) 4.3).. Fo Forr th this is reason, this book proposes a new and systematic design approach which is general for all plant systems. This new approach can design an observer thatt gen tha genera erates tes a con constr strain ained ed state state feedba feedback ck signal signal K xðtÞ ¼ KC xðtÞ (K is comp co mple lete tely ly free freely ly de desi sign gned ed)) that that sati satisfi sfies es (3.2 (3.29) 9) exac exactl tly y fo forr most most plan plantt systems (see Sec. 4.4) and that satisfies (3.29) in a least-square sense for all other plant systems. Alth Al thou ough gh a stat statee ob obse serv rver er th that at ca can n ge gene nera rate te the the arbitrarily given K xðtÞ cannot satisfy (3.28a) or (3.29) for almost all plant systems, such an obse serrve verr is req required ired by al alll oth other LTR de desi sign gn me meth tho ods. ds. At th thee oth ther er every pos possib sible le K xðtÞ that extr ex trem eme, e, th thee stud study y of every that ca can n be ge gene nera rate ted d by an obser ob server ver [sa [satis tisfyin fying g (3.28a (3.28a), ), (3.17) (3.17),, and (3.21) (3.21)]] and tha thatt can sta stabili bilize ze the matrix A BK has been reported [Saberi, 1991]. Obviously, the K (or K Z Z ) that is constrained on (3.28a), (3.17), (3.21), and stable A BK is only a theoretical formulation (or reformulation). The K under this formulation
that (K Z Z is not free) cannot be systematically designed, in contrast to the K that is co cons nstra train ined ed only only on K ¼ KC (K or K Z free ee)) of ou ourr de desi sign gn (s (see ee Z are fr Subsection 3.2.2, the paragraph at the end of Sec. 4.2, and the corresponding technical argument in Sec. 4.4).
SUMMARY Loop transfer function is a critical factor which determines the feedback syste sy stem m se sens nsiti itivi vity ty,, and and requ require ireme ment nt (3 (3.2 .29) 9) is ne nece cess ssar ary y an and d su suffi fficie cient nt to preserve observer feedback system loop transfer function from that of its corresponding direct state feedback system, for either arbitrarily given or freely [but with constraint (3.21)] designed state feedback. State feedback control, either unconstrained or constrained by (3.21), is the general and the basic form of control of state space control theory, and is by far the best among all existing basic forms of control.
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The observer (3.16) is the main feedback compensator structure of state space control th theeory, but it is required red to satisfy (3.17) and nonsingular C [or [or (3.21) for all K ] in most of the literature. Observers with additional requirement (3.28a) or (3.29) in the existing literature are very se seve vere rely ly limi limite ted. d. This This bo book ok in intr trod oduc uces es a fu fund ndam amen enta tall lly y ne new w ob obse serv rver er desi de sign gn appr approa oach ch wh whic ich h can can sati satisf sfy y (3 (3.1 .17) 7),, (3.2 (3.21) 1),, an and d (3.2 (3.29) 9) mu much ch mo more re generally.
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4 A New Feedback Control Design Approach
Chapter 3 analyzed 3 analyzed the observer design requirements, which can be outlined as follows. To guarantee observer state zðtÞ ) T xðtÞ, we require TA FT ¼ LC ðF is stableÞ
ð4:1Þ
To gua guaran rantee tee the gen genera eratio tion n of signal signal K xðtÞ, we requir requiree [as [assum suming ing zðtÞ ) T xðtÞ]
K ¼ ½K Z y Z : K y
T D K C C ¼
ð4:2Þ
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Finally, to realize the same robustness properties of the state feedback control which can be designed systematically, we require (Theorem 3.3 and 3.4) TB ¼ 0
ð4:3Þ
The real challenge is how to generally and systematically satisfy these three thr ee req requir uireme ements. nts. A fun fundam dament entall ally y new des design ign app approa roach ch of sat satisf isfyin ying g these three requirements is proposed in this chapter, which is divided into four sections. Section 4.1 points out a basic and general observer design concept that (4.1) should be satisfied separately and before satisfying (4.2) for arbitrary K (or nonsingular C ). ). In most existing observer design and in all existing LTR obse serv rver er desi design gn,, only only sta state ob obse serv rveers ar aree design signed ed whic ich h im imp ply th thee simultaneo simul taneous us satis satisfactio faction n of (4.1) and nonsingular C . This This ba basi sicc co conc ncep eptt implies imp lies the genera generatio tion n of K xðtÞ directly directly from zðtÞ ½) T xðtÞ and yðtÞ ½¼ 1 0 0 0 C xðtÞ instead of from the explicit xðtÞ ¼ C ½zðtÞ : y ðtÞ . This concept is used throughout the rest of this book. Sect Se ctio ion n 4.2 4.2 anal analyz yzes es the the pole poless (or (or pe perf rfor orma manc nce) e) of th thee obse observ rver er feedback system. It proves a revised version of the ‘‘separation property’’ that (4.1) alone (not nonsingular C ) is the sufficient condition for observer fe feed edba back ck syst system em pole poless bein being g comp compos osed ed of the the eige eigenv nval alue uess of F and
A BKC .
Sect Se ction ion 4. 4.3 3 revi review ewss th thee curr curren entt st stat atee of exis existi ting ng resu results lts of LTR. LTR. It points out that while state observers can be designed generally, the LTR state observers are very severely limited. Section 4.4 summarizes the conclusions of the first three sections and propos pro poses es a fundam fundament entally ally new design design app approa roach ch which which sat satisfi isfies es (4. (4.1) 1) and (4.3) first (not nonsingular C ), ), and which satisfies (4.1)–(4.3) much more generally, simply, and systematically. The only tradeoff of this new design approach is that its state feedback K xðtÞ can be constrained on (4.2) because C may may not always be nonsingular. This tradeoff is obviously necessary and worthwhile in light of the severe drawbacks of the results of Sec. 4.3.
4.1
BASIC BASIC DESIGN DESIGN CONCE CONCEPT PT OF OBSE OBSERVE RVERS RS—DI —DIRE RECT CT GENERATION OF STATE FEEDBACK CONTROL SIGNAL WITHOUT EXPLICIT SYSTEM STATES
We will use the design examples of three basic observers to explain that satisfying (4.1) first and then (4.2) keeps with the basic physical meanings of these two requirements. Because (4.1) alone implies that zðtÞ ) T xðtÞ, this
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separation also implies the direct generation of K xðtÞ in (4.2) from zðtÞ ½) T xðtÞ and yðtÞ ½¼ C xðtÞ.
Example Exa mple 4.1 4.1
Full-Or Full-Order der Id Ident entity ity St State ate O Obse bserve rvers rs
Let T ¼ I and F ¼ A LC in the the obse observ rver er part part (3 (3.1 .16a 6a). ). Th Then en (4 (4.1 .1)) is obviously satisfied and (3.16a) becomes z_ ðtÞ ¼ ðA LC ÞzðtÞ þ LyðtÞ þ BuðtÞ
¼ A zðtÞ þ BuðtÞ þ L½yðtÞ C zðtÞ
ð4:4Þ ð4:5Þ
Subtracting (1.1a) from (4.4), z_ ðtÞ x_ ðtÞ ¼ ðA LC ÞÞ½½zðtÞ xðtÞ ¼ F ½zðtÞ xðtÞ
Therefore, zðtÞ ) x ðtÞ if F is stable. Thus we have repeated the proof of Theorem 3.2. In the above argument, (4.2) is not involved and (4.1) alone completely determines the observer (4.4)–(4.5), which generates xðtÞ. Only after zðtÞ ) T xðtÞ is generated do we multiply z ðtÞ by K [or [or let ½ K : K 4 K ¼ ½K : 0 in
Z
y
¼
(3.16b) and (4.2)] in order to generate the desired state feedback K x(t). Because parameter T has n rows, this observer has n states, and it is T is therefore called ‘‘full order.’’ In addition, if T is not an identity matrix but is nonsingular, then x ðtÞ does not equal z ðtÞ but equals T 1 zðtÞ. We define any observer of (3.16) that estimates x ðtÞ as a ‘‘state observer.’’ We therefore call the observer with T ¼ I an ‘‘identity observer’’ and consider it a special case of full-order state observers. It is obvious that TB cannot be 0 for a nonsingular T . Therefore, a full-order state observer cannot satisfy LTR (4.3). The observer structure of (4.5) is also the structure of Kalman filters [Ander [An derson son,, 1979; 1979; Balakr Balakrish ishnan nan,, 1984], 1984], where where L is th thee filte filterr ga gain in.. The Kalman filter can therefore be considered a special case of full-order identity state observer. The full-order identity state observer feedback system has the block diagram shown in Fig. in Fig. 4.1.
Example Exa mple 4.2
Red Reduce uced-Or d-Order der Sta State te Obse Observe rvers rs
Contrary to full-order state observers, the order of a reduced-order state observer obse rver equal equalss n m and yðtÞ is used sed in (3 (3.1 .16 6b) ðK y ¼ 6 0 Þ. Thus the parameter T of of this observer has only n m rows and cannot be square.
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Figure 4.1 Full-order id identity entity state observer fe feedback edback system system..
As in th thee desi design gn of Ex Exam ampl plee 4. 4.1, 1, (4 (4.1 .1), ), an and d zðtÞ ) T xðtÞ must must be satis sa tisfie fied d first first (see (see Theo Theore rem m 3. 3.2) 2).. Only Only th then en,, to ge gene nera rate te xðtÞ ¼ I xðtÞ in (3.16b) or to satisfy 0
0 0
0
0 0 I xðtÞ ¼ ½K Z y ½zðtÞ : yðtÞ ¼ K ½T : C xðtÞ Z : K y
ð4:6Þ
0
1
0 C 0 T must be nonsingular and K ¼ C . Therefore, in this : matrix C 4 ½ ¼ design des ign,, the req requir uireme ement nt (4. (4.2) 2) ðI ¼ KC Þ again again com comes es after (4.1 (4.1)) an and d is separated from (4.1).
The reason that this observer can have order lower than n comes from the utilization of the information of yðtÞ ¼ C xðtÞ in (3.16b), (4.2), and (4.6). Mathematically speaking, with the addition of m m rows of matrix C in in matrix C , the number of rows of T T can can be reduced from n to n m in order to make matrix C squa square re and and nons nonsin ingu gular lar.. The The redu reduce cedd-ord order er st stat atee ob obse serv rver er feedback system can be depicted as shown in Fig. 4.2.
Figure 4.2 Reduced-orde Reduced-orderr state obse observer rver feed feedback back system.
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In the formulation (3.16) of observers, the signal K xðtÞ is estimated, with K being a general matrix. Therefore the state observer that estimates xðtÞ ¼ I xðtÞ is a special case of the observers of (3.16) in the sense that the general matrix K of the latter becomes a special identity matrix I of the former. Examples 4.1 and 4.2 also show that because matrix I has rank n, 0 matrix C (which (which equal equalss T in ful full-o l-orde rderr sta state te obs observ ervers ers and ½T 0 : C 0 in re redu duce cedd-or orde derr stat statee ob obse serv rver ers) s) mu must st be a no nons nsin ingu gula larr sq squa uare re ma matr trix ix.. Therefore the number of rows of matrix T or or the order of these two types of state observers must be n and n m, respectivel respectively. y. However, Howe ver, after zðtÞ ) T xðtÞ is sati satisfi sfied ed by (4.1 (4.1), ), th thee de desi sire red d st stat atee feedback K xðtÞ can be gene genera rated ted directly from zðtÞ ¼ T xðtÞ and yðtÞ ¼ C xðtÞ without explicit information on x ðtÞ. From a linear algebraic point of 1 view, Eq. (4.2) ðK ¼ KC Þ can be solved without the computation of C . More important, for p 5 n, which is generally true, a very wide range of desirable K can be satisfied by (4.2) without a nonsingular C , as long as 0 K 0 [ RðC Þ, even though a nonsingular C [or the estimation of xðtÞ] is still
required VK . This This basic basic und unders erstan tandin ding g offers offers the fol follow lowing ing two possib possible le significant improvements of observer design. The first is observer order reduction, because the observer order equals T in the number of rows of T in matrix C . We will call the observer that estimates the desired state feedback K xðtÞ and with minimal order the ‘‘minimal order observer.’’ The design results of this observer will be reviewed in Example 4.3, and the first systematic and general design algorithm of this observer [Tsui, 1985] is presented in Chap. in Chap. 7. Thee se Th seco cond nd,, and and even even mo more re sign signifi ifica cant nt,, impr improv ovem emen entt is that that no nott requiring C to be nonsingular implies that the entire remaining observer design freedom after (4.1) can be fully used to satisfy (4.3), or to realize the robustness properties of the state feedback control that the observer is trying to realize. This is the key concept behind the new design approach, which is formally proposed in Sec. 4.4 [Tsui, 1987b]. The exact and analytical solution 5 and 6. of (4.1) and (4.3) [Tsui, 1992, 1993b] will be described in Chaps 5 It sh shou ould ld be emph emphas asize ized d that that the the single purp purpos osee of an ob obse serv rver er in almost all control system applications is to realize a state feedback control K xðtÞ but not to estimat estimatee exp explic licit it plant plant sys system tem sta state te xðtÞ. When xðtÞ is estimated by a state observer, it is multiplied immediately by K (see Figs (see Figs 4.1 and 4.2).
Definition 4.1 The obser The observe verr (3 (3.1 .16) 6) that that gene genera rates tes the the de desi sire red d K xðtÞ directly directly [without [without generating gene rating expli explicitly citly xðtÞ] is ca call lled ed the the ‘‘fu ‘‘func nctio tion n obse observ rver. er.’’ ’’ Ob Obvio vious usly ly,,
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only fu only func ncti tion on ob obse serv rver erss ca can n have have mini minima mall or orde ders rs that that ar aree lowe lowerr th than an n m.
Example Exa mple 4.3
Ove Overvi rview ew of Min Minima imall Orde Orderr Functio Function n Observer Design
Order reduction has been an important problem in control systems theory [Kung, 1981 981] and high observer order has been a major cause of impracticality of state space control theory. Based on the analysis of this section, the only difference between the minimal order observer and the other observers is at Eq. (4.2):
K ¼ ½K Z y Z : K y
T ¼ K C C
in which the least possible number of rows of matrix T is is sought in design computation. To do this computation generally and systematically, every row of matrix T in (4.2) must be completely decoupled from each other and an d mu must st corr corres espo pond nd to only only on onee ei eige genv nvalu aluee of ma matri trix x F (or (or on only ly on onee observer pole). In addition, the complete freedom of T must also be fully used use d in this this design design comput computati ation. on. Bec Becaus ausee T must must satis satisfy fy (4.1 (4.1)) first first,, th thee freedom of T to to be used in (4.2) can be considered the remaining freedom of (4.1). Although there have been many attempts at minimal order observer design des ign [Go [Gopin pinath ath,, 1971; 1971; Fortm Fortmann ann and Wil Willia liamso mson, n, 197 1972; 2; Gupta Gupta et al., al., 1981 19 81;; Va Van n Loan Loan,, 1984 1984;; Fowe Fowell ll et al., al., 19 1986 86], ], which hich ha have ve be been en cl clea earl rly y documented in O’Reilly [1983], the above solution matrix T of (4.1) has not been derived [Tsui, 1993a]. As a result, it has been necessary to solve (4.1) and (4.2) together and it has not been possible to solve (4.2) separately and therefore systematically [Tsui, 1993a]. As a result, the general and systematic minimal order observer design problem has been considered a difficult and unsolved [Kaileth, 1980, Chen, 1984, p. 371]. T has The problem above solution matrix p. has527; been derived by Tsui [1985]. Thus the minimal order observer design has been really and uniquely simplified to the solving of only (4.2), which is only a set of linear equations. A general and systematic algorithm of minimal order observer design [or the solving of (4.2) for minimal number of rows of T ] is proposed in Tsui [1985] and is introduced as Algorithm 7.1 in Chap. in Chap. 7 7 of of this book. Minima Min imall order order functi function on observ observer er is the only existin existing g obse observer rver that genera gen erates tes the des desire ired d K xðtÞ signal signal direct directly, ly, wit withou houtt the exp explic licit it xðtÞ [or, satisfying (4.1), without a nonsingular C ], ], and it is the only application of
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this basicand design concept. This example shows thatathis observer can be generally systematically designed, based only on desirable solution of (4.1) [Tsui, 1985]. The above three examples of existing basic observer design demonst stra rate te th that at sati satisf sfyi ying ng (4.1 (4.1)) first first with withou outt a no nons nsin ingu gula larr C in (4 (4.2 .2)) [or [or 1 0 0 0 generating K xðtÞ directly, directly, without without generating generating xðtÞ ¼ C ½zðtÞ : yðtÞ ] fits the original physical meanings of these two conditions and is in keeping with the existing basic observer design procedures. This Th is de desi sign gn co conc ncep eptt enab enable less the the el elim imin inat atio ion n of th thee diffi difficu cult lt an and d unnecessary requirement of complete state estimation or the requirement that C be no nons nsin ingu gula lar, r, an and d thus thus enab enable less th thee po poss ssib ibil ility ity of sign signific ifican antt imp im prove roveme ment ntss on obser bserv ver desig esign n (on (one of whic hich is observ server er ord rder er reduction).
Exampl Exam plee 4.3 4.3 al also so demo demons nstr trat ates es that that this this basi basicc co conc ncep eptt ha hass be been en obscured by the fact that almost all observer results involve state observers only, and and by th thee previ revio ous unsu unsucc cces essf sfu ul att ttem empt ptss at th thee ge gene nerral and systematic design of minimal order function observers.
4.2
PER PERFORM FORMANCE ANCE OF OBSE OBSERVER RVER FEEDBACK FEEDBACK SYSTEMS SYSTEMS— — SEPARATION PROPERTY
In the previous section, we discussed the design concept of satisfying (4.1) separately without satisfying a nonsingular matrix C 4 ½T 0 : C 0 0 . ¼ In this section, we will prove that (4.1) alone ( not with a nonsingular C ) guarantees that the observer feedback system poles be the eigenvalues of F and A BKC . Thus (4.1) alone also guarantees explicitly and to a certain degree the observer feedback system’s performance (see Sec. 2.1). This is an essential validation of the new design approach of this book, which seeks the satisfaction of (4.1) and (4.3) first, without a nonsingular matrix C .
Theorem 4.1 (Separation property) If (4.1) is satisfied, then the poles of the feedback system that is formed by the plant system (1.1) and the general observer (3.16) are composed of the eigenvalues of matrices F of of (3.16) and A BK of (1.1) and (4.2).
Proof [Tsui, 1993b] Substituting (3.16b) into the plant system input uðtÞ and then substituting this uðtÞ and yðtÞ ¼ C xðtÞ into the dynamic part of plant system (1.1a) and
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observer (3.16a), the dynamic equation of the observer feedback system is
ð Þ
A BK y C BK Z x_ t xðtÞ xðtÞ Z 4 Ac ¼ LC TBK y C F TBK Z zðtÞ ¼ z_ ðtÞ zðtÞ Z
Multiplying Q1 ¼
I 0 T I
and
I 0
ð4:7Þ
Q ¼
T
I
on the left and right side of Ac , respectively, we have
A BK y C BK Z Z T BK Z Z Ac ¼ Q 1 Ac Q ¼ TA þ FT þ LC F
if (4.1)
¼
A BKC BK Z Z F 0
ð4:8Þ
A BKC and F will A c of The eigenvalues of A and of F will constitute all eigenvalues of A (4.8), which has the same eigenvalues of Ac .
In the the norm normal al and and exis existin ting g stat statee sp spac acee de desi sign gn pr prac acti tice ce,, eith either er th thee eigenvalues of F and A BKC are are assigned without considering the overall feedback system poles, or the overall system is designed without considering the poles of its feedback compensator. The separation property guarantees the overall observer feedback system poles once the eigenvalues of F and A BKC are assigned. Therefore from Sec. 2.1, it guarantees explicitly the over ov eral alll obse observ rver er feed feedba back ck syst system em pe perf rfor orma manc ncee to th thee degr degree ee of th thos osee assigned poles. It also guarantees the poles and the stability of observer (3.16) from the stability of the overall observer feedback system, in case the design is carried out from the perspective of the overall feedback systems. The separation property is thus extremely important and has appeared in almost all state space control literature, such as O’Reilly [1983]. Howeve How ever, r, the gen genera erall observ observer er (3. (3.16) 16) for formul mulati ation on (wi (with th genera generalize lized d dynamic part and generalized state feedback output) has not really been
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extended to the existing literature. More important, the property that (4.1) alone (not a nonsingular C ) is the sufficient condition of Theorem 4.1 has not really been clarified in the existing literature either [Tsui, 1993b]. Because in the original version of Theorem 4.1 the parameter KC is replaced by an arbitrary K , it has been customary to assign the eigenvalues of F in (4.1) and the eigenvalues of A BK (3.11) completely separately. Hence the name ‘‘separation property.’’ Howe Ho weve ver, r, as wi will ll be desc descri ribe bed d in the the ne next xt sect sectio ion, n, fo forr mo most st plan plantt systems, an arbitrarily designed state feedback K xðtÞ cannot be implemented by an observer with a nonsingular C and with with exact LTR [or (4.3)]. The new design approach of this book fundamentally changes this traditional
design desi gn pr prac acti ticce by desi design gnin ing g the the stat statee fe feed edba back ck gain ain K based on 0 K ¼ K ½T 0 : C 0 , where observer parameter T satisfies (4.1) and (4.3). This new design approach is validated partly by the above revised separation proper pro perty, ty, which which shows shows that that (4.1) (4.1) alo alone ne is the suffici sufficient ent conditi condition on of this this 0 property, while the addition of constraint K ¼ K ½T 0 : C 0 generalizes this 0 property from K to K ¼ K ½T 0 : C 0 . Finally, for the sake of theoretical integrity, we shall point out that the cond co ndit itio ion n (4 (4.1 .1)) is not not a nece necess ssar ary y cond condit itio ion n of Theo Theore rem m 4.1 4.1 fo forr ev ever ery y possible combination of (C ; K ; F ; T ; L). This point can be simply proved by the following special example.
Example 4.4 Let a matrix Ac and its characteristic polynomial be
jsI Ac
ð Þ j¼ ¼ sI
A
TA s
BKC BK z
FT
a
b
c
LC
. . .
b
s
1
. .
a
c
sI F
:
1
s
f
where parameters ða; b; c; f Þ are scalars. Then
Þj þ Þj
jsI Ac j ¼ ðs f ÞjsI ðA BKC
¼ ðs f ÞjsI ðA BKC ÞÞjj
s
c
a
þ b
c
b
c
s a c
ð4:9Þ
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The equality of (4.9) (or separation separation property) property) holds even if parameter parameter c 6 ¼ 0, or even if (4.1) is not satisfied. In any practical design, the parameters of ðC ; K ; F ; T ; LÞ have to be designed to satisfy (4.1), (4.3), and a satisfactory A BKC , but not to fit the special case of Example 4.4. Thus this argument on the necessity of (4.1) to Theorem 4.1 is totally meaningless in practice. This situation is very similar to the argument of Saberi et al. [1991] that (4.3) is not necessary for exact LTR and for every possible combination of parameters ðK ; F ; T Þ. This is because the parameters of ðK ; F ; T Þ have
to be designed to satisfy (4.1) and a satisfactory A BKC , but not to fit those special cases that satisfy (3.28a) but not (4.3).
4.3
THE CURR CURRENT ENT S STAT TATE E OF LTR LTR OB OBSE SERVE RVER R DESIGN DESIGN
As discussed in Sec. 4.1, besides observer order reduction, a much more importa imp ortant nt observ observer er des design ign imp improv roveme ement nt is the sig signifi nifican cantly tly more more gen genera erall and an d syste systema matic tic ro robu bust stne ness ss pres preser erva vatio tion n (or (or LT LTR) R) of obse observ rver er fe feed edba back ck systems. From Theorems 3.3 and 3.4, the requirement of LTR [or (4.3)] can el elimi imina nate te the the ba basi sicc caus causee of sens sensiti itivi vity ty pr prob oble lems ms of ob obse serv rver er fe feed edba back ck syste sy stems ms an and d is ther theref efore ore of grea greatt prac practic tical al im impo port rtan ance ce.. As a resu result lt,, th this is prob pr oble lem m ha hass rece receiv ived ed much much atte attent ntio ion n sinc sincee it itss prop propos osit itio ion n [Sog [Sogaa aard rd-Andersen, 1986; Stein and Athans, 1987; Dorato, 1987; Tsui, 1987b; Moore and Tay, 1989; Saberi and Sannuti, 1990; Liu and Anderson, 1990; Niemann et al., 1991; Saeki, 1992; Tsui, 1992, 1993b; Saberi et al., 1993; Tsui, 1996a, b; Tsui, 1998b]. However, the mere proposition and formulation of a problem does not imply that the problem is solved, and experience shows that the latter can be much more difficult than the former. Even the derivation of some initial solutions of a problem does not imply the problem is solved satisfactorily, and experience also shows that the latter can be much more difficult than the former. Furthermore, only the theoretical problem with a really satisfactory solution can have real practical value. This Th is se sect ctio ion n show showss that that al alll othe otherr ex exis istin ting g LT LTR R obse observ rver erss ar aree st stat atee observers. While a state observer without the LTR requirement (4.3) can be generally designed, the state observer with (4.3), which is called the ‘‘exact LTR state observer,’’ is very severely limited. It has been proved that to have an exact LTR state observer or to satisfy (4.1) and (4.3) with arbitrarily given K [or [or to satisfy (4.1) and (4.3) with wit h a non nonsin singul gular ar C ]],, the the pl plan antt syst system em mu must st sati satisf sfy y ei eith ther er one one of th thee following two restrictions [Kudva et al., 1980]. These two restrictions are
Copyright 2004 by Marcel De Dekker, kker, Inc. All Rights Reserved.
originally derived for the existence of the ‘‘unknown input observers.’’ An unkn un know own n inpu inputt obse observ rver er is a stat statee obse observ rver er wi with th zero zero ga gain in to the the plan plantt system’s unknown input [Wang et al., 1975]. Hence it is equivalent to an exac ex actt LT LTR R st stat atee obse observ rver er,, if we co cons nsid ider er th thee plan plantt sy syst stem em ga gain in to th thee unknown input signal as matrix B. The first restriction is that the plant system must have n m stable transmission zeros. This is extremely restrictive because most systems with m 6 ¼ p do do no nott have have that that many many tran transm smiss issio ion n ze zero ross in th thee fir first st plac placee (s (see ee
Example 1.8 and Davison and Wang, 1974). Thee se Th seco cond nd is a set set of thre threee rest restri rict ctio ions ns:: (1 (1)) mi mini nimu mumm-ph phas asee (all (all transmission zeros are stable), (2) rank ðCBÞ ¼ p , and (3) m5 p. This again is very hard hard to requ extr ex trem emel ely y re rest stri rict ctiv ivee be beca caus usee it is very requir iree all existing transmission zeros be stable (see Exercises 4.2 and 4.6), and rank ðCBÞ ¼ p is also not satisfied by many practical systems such as airborne systems. The above two restrictions can be related by the following property of tr tran ansm smis issi sion on zero zeross [D [Dav avis ison on and and Wang Wang,, 19 1974 74]; ]; na name mely ly,, th that at almo almost st all all systems with m ¼ p have n m transmission zeros, and that all systems with m ¼ p and with rank ðCBÞ ¼ p have n m transmission zeros. Therefore the second restriction is a little more general than the first restriction because it admits some additional plant systems with m > p . For plant systems not satisfying the above two restrictions, if they are minimum-phase, then there is an asymptotic LTR state observer for these systems, while there exist no other unknown input observer results for these syst sy stem emss be beca caus usee the the ab abov ovee two two rest restri rict ctio ions ns ar aree ne nece cess ssary ary co cond ndit ition ionss of unknown input observers. Asymptotic LTR state observers have been widely documented [Doyle and an d St Stei ein, n, 19 1979 79;; Stei Stein n and and Atha Athans ns,, 1987 1987;; Dora Dorato to,, 1987 1987;; Mo Moor oree and and Ta Tay, y, 19 1989 89;; Saberi and Sannuti, 1990; Niemann et al., 1991; Saberi et al., 1993] and have been considered the main result of LTR because minimum-phase restriction is less strict than the above two restrictions for exact LTR state observers. There are mainly two design approaches for asymptotic LTR state observers. The first is valid for minimal-phase systems only, and is to asymptotically increase the plant system input noise level when designing the Kalman filter [Doyle and Stein, 1979] or to asymptotically increase the time scale of state observer poles [Saberi and Sannuti, 1990]. Unfortunately, this approach inevitably and asymptotically increases the observer gain L. As discussed in Sec. 3.1 and Shaked and Soroka, (1985); Tahk and Speyer, (1987); and Fu, (1990), the large gain L is even more harmful to system sensitivity properties than not having LTR at all. Thee se Th seco cond nd appr approa oach ch is to co comp mput utee a loop loop tr tran ansf sfer er fu func nctio tion n LðsÞ whose difference to the target loop transfer function LKx ðsÞ has an H ? norm
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bound over frequency [Moore and Tay, 1989]. Unfortunately, this bound is itself generally unpredictable. For example, in the actual design it is ever increased until a numerical solution of a bounded value Riccati equation exists—it does not converge to a lower level at all [Weng and Shi, 1998]. Even more critically, at the frequency o of this bound, no consideration is made and no bound exists for the phase angle of Lð j oÞ LKx ð j oÞ. To summarize, the existing exact LTR state observer is too restrictive,
while the existing asymptotic LTR state observers are far from satisfactory. The main reason for these unsat satisfactory ory LTR result ltss is the re requ quir irem emen entt of stat statee esti estima mati tion on or the the requ requir irem emen entt of im impl plem emen enti ting ng arbitrarily give given n state state fe feed edba back ck cont contro rol. l. Ma Math them emat atic ical ally ly sp spea eaki king ng,, C nonsingular is a difficult yet unnecessary additional requirement [in addition to necessary conditions (4.1) and (4.3)] to satisfy. For example, most of the existing LTR results involve Kalman filters. The Kalman filter design freedom is used almost completely for minimum variance state estimation [Anderson and Moore, 1979; Balakrishnan, 1984] and not for LTR. The only remaining design freedom of Kalman filters for LTR is a scalar plant system input noise level q [Doyle and Stein, 1979]. As q is increased asymptotically for achieving LTR, the Kalman filter poles must approach each of the plant system transmission zeros and negative infinity at Butterworth pattern [Anderson and Moore, 1979]. This is the reason that the Kal Kalman man filterfilter-bas based ed exa exact ct LTR LTR observ observer er req requir uires es n m stable stable plan plantt system transmission zeros [Stein and Athans, 1987; Friedland, 1989], and is the reason that the asymptotic LTR state observer requires that the plant system be minimum-phase [Doyle and Stein, 1979, 1981].
Example Exa mple 4.5
The U Unsa nsatis tisfac factory tory S Stat tate e of the E Exis xisting ting Asymptotic LTR Result
Let the given plant system be
0 3 2 ðA; B; C Þ ¼ ; ;½0 1 4 1
1
and GðsÞ ¼
s þ 2 ðs þ 1Þðs þ 3Þ
which has n m ¼ 2 1 stable transmission zero 2.
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Let us design an observer which can implement a quadratic optimal state feedback K ¼ ½30 50
whose corresponding loop transfer function (3.10) is 1
LKx ðsÞ ¼ K ðsI AÞ
B ¼
ð10s þ 50Þ ðs þ 1Þðs þ 3Þ
This example was raised by Doyle and Stein [1979], which provided two types of observer results: 1.
A full-or full-order der iden identity tity stat statee observ observer er wit with h pol poles es 7+ j 2: 2:
ðF ¼ A LC ; T ; L; K Z Z ; K y Þ ¼
0 53 50 ; I ; ; K ; 0 1 14 10
whose corresponding loop transfer function is computed as 1
1
½0 þ K ðsI F Þ1 LGðsÞ s þ 2 100ð10s þ 26Þ 6 ¼ 2 s þ 24s 797 ðs þ 1Þðs þ 3Þ
LðsÞ ¼ ½1 þ K ðsI F Þ
2.
B
and is very different from LKx ðsÞ. A Kalma Kalman n fil filte terr with with as asym ympt ptot otic ic LT LTR R ðq ¼ 100 Þ:
ðF ¼ A LC ; T ; L; K Z Z ; K y Þ 0 206 7 1 102:4
¼
I 203 7 98:4
: ; ;
K 0
: ; ;
whose corresponding loop transfer function is similarly computed as s þ 2 ð1191s þ 5403Þ LðsÞ ¼ 2 6 s þ 112:4s þ 49:7 ðs þ 1Þðs þ 3Þ
This is already the best LTR result of Doyle and Stein [1979]. It is achieved by a hi high gh-i -inp nput ut nois noisee leve levell q ¼ 100 100 and and th thee as asso socia ciate ted d larg largee filte filterr ga gain in ðk Lk ¼ 226 :2Þ, which is extremely undesirable. The poles of this filter are
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around 2 and 100. Nonetheless, L ð j oÞ is still very different from L Kx ð j oÞ at o < 10 (see Fig. 3 of Doyle and Stein, 1979). The simple and exact LTR result is derived as [Tsui, 1988b]
ðF ; T ; L; K Z Z ; K y Þ ¼ ð2; ½1 2; 1; 30; 10Þ It can be verified that the corresponding L ðsÞ ¼ L Kx ðsÞ, which is guaranteed by TB ¼ 0 (see Theorem 3.4). The observer gain L ¼ 1 while the output gain of this observer ½K Z Z : K y y is less than K . It should be noted that there is no explicit state estimation in this design. The or orig igin inal al exam exampl plee of Doyl Doylee and Ste tein in [197 [1979] 9] used sed th thee dual controllable canonical form ð A0 ; C 0 ; B0 Þ to represent the same plant system. The corre correspon sponding ding state feedback feedback K is ½ 50 10 , and the cor corres respon pondin ding g gains L for for th thee abov abovee two two ob obse serv rver erss of Doyl Doylee an and d St Stei ein n [197 [1979] 9] were were ½ 30 50 and ½ 6:9 84:6 , resp respec ectiv tivel ely. y. No None neth thel eles ess, s, all all co comp mpat atib ible le transfer functions and loop transfer functions of Example 4.5 and Doyle and Stein [1979] remain the same. This Th is exam exampl plee show showss that that the the asym asympt ptot otic ic LT LTR R resu result lt is fa farr fr from om satisfa sat isfacto ctory. ry. It also also shows shows the eff effect ective ivenes nesss of the des design ign con concep ceptt of not explicitly estimating plant system states. The plant system of Example 4.5 has n m stable transmission zeros and therefore satisfies the first of the above two sets of restrictions for exact LTR. The real adva advantage ntage of the new design design appr approach oach (of not requiring requiring state es estim timat atio ion) n) of this this bo book ok is for for sy syste stems ms not sati satisf sfyin ying g th thes esee tw two o sets sets of restrictions. Several such examples will be illustrated in Sec. 6.2, after the explicit algorithms of (4.1) and (4.3) are described. A reason that only state observers [satisfying (4.1) and nonsingular C together] are involved in the existing LTR results concerns the difficulty in deriving a really satisfactory solution of (4.1), as was true in the minimal order observer design (see Example 4.3). To summarize, the solving of (4.1) and (4.3) (4.3) [but not nonsingular C ] is not a retre etreat at in into to a simp simple lerr desi design gn app pprroa oach ch no norr an av avoi oid dan ance ce of arbi ar bitra trary ry stat statee fe feed edba back ck impl implem emen enta tati tion on,, bu butt a ne nece cess ssar ary y an and d diffi difficu cult lt step to eliminate the very unsatisfactory state of the existing LTR results, and an d a no nove vell step step wh whic ich h is enab enable led d only by a te tech chni nica call br brea eakt kthr hrou ough gh in thee so th solu luti tion on of (4 (4.1 .1)) [the [the rem remai ain nin ing g fr free eed dom of (4 (4.1 .1)) is full fully y us useed to satisfy satisf y (4.3)]. (4.3)].
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4.4
A NE NEW W DESIG DESIGN N APP APPRO ROACH ACH AND AND N NEW EW F FEE EEDBA DBACK CK STRUCTURE—A DYNAMIC OUTPUT FEEDBACK
COMPENSATOR THAT GENERATES STATE/ GENERALIZED STATE FEEDBACK CONTROL SIGNAL The conclusions of the first three sections of this chapter can be listed as follows.
Conclusion 4.1 Equation (4.1) is a necessary and sufficient condition for an observer (3.16) to generate a signal K xðtÞ for a constant K , where xðtÞ is the plant system state vector (Theorem 3.2).
Conclusion 4.2 Equation Equati on (4. (4.1) 1) is also also the suf suffici ficient ent conditi condition on for the ob obser server ver fee feedba dback ck system poles to be composed of the eigenvalues of F and and of A BK , where K xðtÞ is the state feedback generated by the observer (3.16) (Theorem 4.1). This theorem guarantees the observer feedback system performance.
Conclusion 4.3 0
For a free For freely ly desi design gned ed stat statee feed feedba back ck K xðtÞ (K ¼ KC ; C ¼ ½T 0 : C 0 is determined and K is completely completely free), the necessary necessary and suffic sufficient ient condition condition for the observer feedback system to realize the robustness properties of this K xðtÞ is (4.3) (or TB ¼ 0, Theorem 3.4).
Conclusion 4.4 To satisfy (4.1), (4.3), and a nonsingular C , or to have an exact LTR state observer, the plant system either must have n m stable transmission zeros or satisfy (1) minimum-phase, (2) rank ðCBÞ ¼ p , and (3) m5 p [Kudva et al., 1980]. Most practical plant systems do not satisfy these restrictions. The other existing asymptotic LTR state observer is far from satisfactory either, mainly because of its asymptotic large gain. Becausee of this Becaus this conclu conclusion sion,, even even though though the ide ideally ally and sep separa aratel tely y designed state feedback can always be implemented by a state observer, its ideal robustness property is lost in the actual observer feedback system in most cases. This is intolerable because robustness is a key property of most
Copyright 2004 by Marcel De Dekker, kker, Inc. All Rights Reserved.
engineering systems. Conversely, even though a state observer has generated
the desi the desire red d stat statee fe feed edba back ck cont contro roll sign signal al (e (eve ven n op opti tima mally lly in a mi mini nima mall variance sense), the purpose of this state observer is also lost because it has failed to realize the critical robustness properties of the same state feedback control in a deterministic sense. The reason for this state of existing results of Conclusion 4.4 can be interp int erpret reted ed as follow follows. s. Becaus Becausee the sta state te feedb feedback ack con contro troll K is des design igned ed separately from from the the stat statee ob obse serv rver ers, s, the the st stat atee ob obser serve vers rs ar aree ex expe pect cted ed to implement arbitrarily given state feedback. This is proven to be too much of a requirement if the LTR requirement (4.3) is added. Let us analyze the above situation from another different perspective. Thee di Th dire rect ct (a (and nd id idea eal) l) stat statee fe feed edbac back k is desi design gned ed ba base sed d on th thee dy dyna namic mic matrix A BK or the information of the plant system’s input dynamic part ðA; BÞ only, and and is sepa separa rate ted d comp comple lete tely ly fr from om th thee kn know owle ledg dgee of plan plantt system’s output observation (with key parameter C ) and the knowledge of the observer (with key parameter T ) which actually realizes and implements it. Therefore such design cannot be considered mature and is not based on comp co mple lete te info inform rmat atio ion. n. This This im imma matu turit rity y is refle reflect cted ed by th thee fact fact th that at the the result res ulting ing sta state te feedba feedback ck con contro troll and its robu robust stne ness ss pr prop oper erty ty ca cann nnot ot be actually realized in most cases if the states are not all directly measurable, even though such a state feedback control is itself ideal and superb (see Subsection 3.2.1). Based on the above conclusions and analysis, this book proposes a fund fu ndam amen enta tall lly y ne new w desi design gn ap appr proa oach ch.. In th this is ne new w appr approa oach ch,, th thee st stat atee fe feed edba back ck cont contro roll is desi design gned ed base based d on the the fe feed edba back ck syst system em dy dyna nami micc 0 A BK ½T 0 : C 0 , whic which h co comp mpri rise sess th thee info inform rmati ation on of matrix A BKC 4 ¼ nott only no only th thee pl plan antt syst system em’s ’s in inpu putt dy dyna nami micc part part ðA; BÞ; but but also also ot othe herr plant pla nt sys system tem parame parameter ter C and and obser observer ver par parame ameter ter T . The new state feedba fee dback ck con contro troll is guaran guarantee teed d of ob obser server ver imp implem lement entati ation on,, sep separa aratio tion n property, proper ty, and robust robustnes nesss rea realiz lizatio ation n for sig signifi nifican cantly tly more more gen genera erall cas cases. es. Thus th this is ne new w app approa roach is mat atu ure and and is divid ivided ed nat atu ura rall lly y into into the the follow fol lowing ing two major major ste steps. ps. The first step determines the observer dynamic part (3.16a) by solving (4.1) and using the remaining freedom of (4.1) to best satisfy (4.3). Thus the resulting observer is able to generate a state feedback signal K xðtÞ with a constant K ¼ KC (see (see Conc Conclu lusi sion on 4.1) 4.1) and, and, for for wh what atev ever er this this K , the feedback system poles of this observer are guaranteed to be the eigenvalues of A BK and F (see Conclusion 4.2). In addition, every effort has been made to realize the robustness property of this state feedback control (see Conclusion 4.3). and d 6. The design algorithms of this step are described in Chaps 5 an Condition (4.1) is satisfied first in Chap. 5 and for all plant systems, and
Copyright 2004 by Marcel De Dekker, kker, Inc. All Rights Reserved.
(4.3) is then best satisfied in Chap. in Chap. 6. 6. It It is proved in Sec. 6.2 that for all plant systems either with at least one stable transmission zero or with m > p , the exact solution of (4.1) and (4.3) can be computed, with the rank of matrix C also maximized by the available remaining freedom of (4.1) and (4.3). This is significantly more general than the existing exact LTR state observers, and is general for most plant systems (see Exercises 4.3 and 4.7). For all other plant systems, the least square solution of (4.3) can be computed, without large gain. The second step fully determines the output part of the observer (3.16b) by designing the dynamic matrix A BKC , where K is the completely free parameter of (3.16b). The loop transfer function L Kx ðsÞ is indirectly (though much more effectively) determined by this design (see Chaps 2, 2, 3, 3, 8 an and d 9). The explicit design algorithms are described in Chaps 8 and 9. It should be noted that the design of A BKC is exactly compatible math ma them emat atic ical ally ly wi with th th thee stat static ic outp output ut fe feed edba back ck de desi sign gn A BK y C of Subs Su bsec ectio tion n 3. 3.2. 2.2. 2. The The on only ly di diff ffer eren ence ce is th that at ra rank nk ðC Þ ¼ m while while rank rank ðC Þ ¼ r þ m5m, where r is the number of rows of T , or the order of the observer of the first step. In addition, because the rank of C of the first step can be between n and m, this new design approach unifies completely the exact LTR state observ obs erver, er, whi which ch corres correspon ponds ds to ran rank k ðC Þ ¼ maximum n, an and d th thee st stat atic ic output feedback, which corresponds to rank ðC Þ ¼ m ¼ minimum of rank ðC Þ. This unification will be discussed in Sec. 6.3. In this sense, we also call thee feed th feedba back ck co cont ntro roll whic which h is imple impleme ment nted ed by th this is new new ob obse serv rver er as th thee ‘‘generalized state feedback control.’’ Becaus Bec ausee (4. (4.3) 3) ðTB ¼ 0 Þ is sati satisfi sfied ed in the the first rst st steep of this this de desi sign gn approach, the corresponding observer of (3.16) will have the following state space model z_ ðtÞ ¼ F zðtÞ þ LyðtÞ
K xðtÞ ¼ K Z Z zðtÞ K y yðtÞ
ð4:10aÞ ð4:10bÞ
Because this observer (which is also called ‘‘feedback compensator’’) is not involved with the plant system input u ðtÞ, we call it the ‘‘output feedback compensator.’’ In addition, compared to static output feedback systems of Sec. Se c. 3. 3.2. 2.2, 2, th this is comp compen ensa sato torr ha hass an addi additi tion onal al dy dyna namic mic part part with with st stat atee zðtÞ ) T xðtÞ, and the control signal produced by this compensator has an additi add itiona onall ter term m K Z Z T xðtÞ, which is provided by the above Z zðtÞ ) K Z additi add dyn dynami amic c edba part. par t.ckTherefo The remple this thi sstcompen com sator com comple pletely unifies uni fies th the e itiona stat stonal atic icl ou outp tput ut fe feed back as refore its its simp si lest ca case sepensato an and d isr ca call lled ed ately ‘‘dy ‘‘dyna nami mic c output feedback compensator.’’
Copyright 2004 by Marcel De Dekker, kker, Inc. All Rights Reserved.
The fe The feed edba back ck sy syste stem m of th this is comp compen ensa sator tor is depi depict cted ed in th thee bloc block k diagram in Fig. in Fig. 4.3. Finally, let us clarify three technical arguments concerning this new design approach. First, the ideal compensator does not universally exist in practice. The significant advantage of this new design approach in very significantly more general robustness realization (see Exercises 4.2, 4.3, 4.6, and 4.7) certainly has therefore state criticism feedbackthat control K xðtits KC xðtÞ (ifconstrained n ). This Þ ¼price—the rank ð C Þ 0, see Conclusion 6.1 .1). ). Base ased on the the assu assump mpti tion on and resu esult of 4.2 4.2, calcu alcula late te the the probability of r > 0 for n m ¼ 1 to 8, and compare this probability with the probability of minimum-phase Pnm of Part (b) of 4.2. Answer: Pðr > 0 Þ ¼ 1 P0 ¼ 0.5, 0.75, 0.875, 0.9375, 0.9688, 0.9844, 0.9922, 0.9964: The probability Pðr > 0 Þ of this new design approach is almost 100 % as soon as n m is > 3, and is very significantly greater than Pnm (probability of minimum-phase, one of the necessary conditions of the existing LTR results). 4.4 The suffic sufficient ient condition condition for the generalize generalized d state feed feedback back contr control ol of this book to assign arbitrarily given poles and some eigenvectors is r þ m þ p > n , or r > n m p [see (6.19) or Step 2 of Algorithm 8.1]. Based on the assumption and result of 4.2, calculate the probability of r > n m p ð¼ 100 % if n m p < 0Þ.
Copyright 2004 by Marcel De Dekker, kker, Inc. All Rights Reserved.
Answer: n ¼
m ¼ p ¼
3 %
4
5
6
7
8
9
10
%
%
%
%
%
%
%
2
100
75
50
3 4
100 100
1 10 00 1 10 00
1 10 00 1 10 00
31.25 87.5 1 10 00
18.75
1 10 0.94
6.25
3.52
68.75 100
5 50 0 93.75
34.38 8 81 1.25
2 22 2.66 6 65 5.63
Compa ompare red d to the the po popu pula larr stat static ic out utp put fe feeedbac dback k con ontr tro ol, th thee prob pr obab abili ility ty to ac achi hiev evee th this is arbi arbitra trary ry po pole le as assi sign gnme ment nt an and d pa part rtia iall eigenvector assignment is 0% in the above table if the number is not 100%. Thus the improvement of our generalized state feedback control from the static output feedback control is very significant. 4.5 The suffic sufficient ient condition condition for the generalize generalized d state feedback contr control ol of this th is bo book ok to assi assign gn arbi arbitr trar ary y po pole less an and d to gu guar aran ante teee st stab abil ilit ity y is r þ mÞ p > n or r > n = p m [see (6.18), Adjustment 2 of Sec. 8.1.4, ðand Wang, 1996]. Based on the assumption and result of 4.2, calculate
the probability of r > n = p m ð¼ 100% if n=p m < 0Þ. Answer: n ¼
m ¼ p ¼
2 3
3 %
100 100
4
5
6
7
8
9
10
11
12
%
%
%
%
%
%
%
%
%
75 100
88 100
69 100
81 100
66 100
77 98
63 99
75 99þ 98
The probability is very high as soon as m is increased higher than 2, and decreases very slowly so that no substantial decrease can be shown in the above table. For example, it can be calculated that when n ¼ 16, thee prob th probab abil ilit ity y is stil stilll 98% for m ¼ 3, an and d th tha at when n ¼ 26 th thee probability is still 99.7% for m ¼ 4. This result indicates that the great majority of the open loop systems can be guaranteed of arbitrary pole assignment and stabilization by our generalized state feedback control. Compa Co mpared red to the popul popular ar sta static tic output output fee feedba dback ck con contro trol, l, the probability to achieve this arbitrary pole assignment is 0% in the above table if the number is not 100%. Thus Thus th thee im impr prov ovem emen entt of ou ourr genera gen eraliz lized ed sta state te feedba feedback ck con contro troll fro from m the sta static tic output output fee feedba dback ck
control is very significant. Repeat at 4.2 by changin changing g p to 3/4. This new p implies that each plant 4.6 Repe system transmission zero is three times more likely to be stable than to be unstable. This p is significantly more favorable than the half-andz
z
z
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p of 4.2 to 4.5, even though that old p is still reasonable (see 4.2). half p Therefore we may assume that most of the practical values of p would fall between these two values of p . z
z
z
z
(a) Answer: Pr ¼ ðassume pz ¼ 3 =4Þ n m ¼
1
2
3
4
5
6
7
8
1/42 6/42 9/42
1/43 9/43 27/43 27/43
1/44 12/44 54/44 108/44 81/44
1/45 15/45 90/45 270/45 405/45 243/45
1/46 18/46 135/46 540/46 1,215/46 1,458/46 729/46
1/47 21/47 189/47 945/47 2,835/47 5,103/47 5,103/47 2,187/47
1/48 24/48 252/48 1,512/48 5,670/48 13,608/48 20,412/48 17,496/48 6,561/48
r
0 1 2 3 4 5 6 7 8
1/4 3/4
(b)
Ba Based sed on the the result result of Part (a (a), ), find the the prob probabi abilit lity y of mini minimum mum-phase ðr ¼ n mÞ for n m ¼ 1 to 8. Answer: Pnm ¼ 0 :75; 0:56; 0:42; 0:32; 0:24; 0:18; 0:13; 0:1. Although much higher than the corresponding probabilities of Exercise Exer cise 4.2, the P nm is still lower than 1/2 as soon as n m > 2, and is decreasing as n m increases. 4.7 One of the suffi sufficie cient nt conditio conditions ns of the new desi design gn approach approach of thi thiss book is at least one stable transmission zero ( r > 0, see Conclusion 6.1). Based on the assumption and result of 4.6, calculate the probability of r > 0 for n m ¼ 1 to 8, and and com compa pare re th this is pr pro obab abil ilit ity y wit ith h th thee probability of minimum-phase P of Part b of 4.6. Answer: Pðr > 0 Þ ¼ 1 P0 ¼ 0 :75; 0:9375; 0:9844; 0:9964; . . . The probability Pðr > 0Þ of this new design approach is almost 100 % as soon as n m >1, an and d is very very sign signifi ifica cant ntly ly grea greate terr th that at P (probability of minimum-phase, one of the necessary conditions of the existing LTR designs). sufficient ient condition condition for the generalize generalized d state feed feedback back contr control ol of 4.8 The suffic n
m
n
m
this given poles eigenvectors is r þ mbook þ p >ton, assign or r > arbitrarily n m p (see (6.19) or and Stepsome 2 of Algorithm 8.1). Based on the assumption and result of 4.6, calculate the probability of r > n m p ð¼ 100% if n m p < 0Þ:
Copyright 2004 by Marcel De Dekker, kker, Inc. All Rights Reserved.
n ¼ m ¼ p ¼
3
4
5
6
7
8
9
10
%
%
%
%
%
%
%
%
2 3
100 100
94 1 10 00
84 1 10 00
74 98
63 97
53 9 90 0
44 83
37 7 76 6
4
100
100
100
100
100
99.6
98
96
Compare to the popula larr static output feedback control, the prob pr obab abili ility ty to ac achi hiev evee th this is arbi arbitra trary ry po pole le as assi sign gnme ment nt an and d pa part rtia iall eigenvector assignment is 0% in the above table if the number is not 100%. Thus the improvement of our generalized state feedback control from the static output feedback control is very significant, is much more significant than that of Exercise 4.4, and makes this very effective and difficult design goal (see Chaps (see Chaps 8 an 8 and d 9) achievable 9) achievable to a very large port po rtio ion n of pr prac acti tica call syst system ems. s. This This ta tabl blee of data data shou should ld be mo most st relevant amoand ng robustness all tables,system to thdesign. e practical and current high performance 4.9 Th Thee suf suffici ficient ent condit condition ion for the genera generaliz lized ed sta state te fee feedb dback ack con contro troll of this this book book to assig assign n ar arbi bitr trar ary y pole poless an and d to guar guaran ante teee st stab abili ility ty is ðr þ mÞ p > n or r > n/ p m [see (6.18), Adjustment 2 of Sec. 8.1.4, and Wang, 1996]. Based on the assumption and result of 4.6, calculate the probability of r > n/p m ð¼ 100% in n/p m < 0Þ. Answer: n ¼
3
4
5
6
7
8
9
10
11
12 12
m ¼ p ¼
%
%
%
%
%
%
%
%
%
%
94 100
98 100
95 100
98 100
96 100
2 3
100 100
99 97 99þ 99þ 99þ 99þ
The probability is almost all 100 %, and does not decrease as n increases. Thiss result Thi result indica indicates tes tha thatt arb arbitr itrary ary pole pole assign assignmen mentt and sta stabil biliza izatio tion n are virtually guaranteed by our generalized state feedback control. Compare to the popula larr static output feedback control, the probab pro babili ility ty to achiev achievee thi thiss arb arbitr itrary ary pole pole assign assignmen mentt is 0% in the above
table if the number is not 100 %. Thus the improvement of our generalized st stat atee fe feed edba back ck cont contro roll from from the the stati staticc ou outp tput ut fe feed edba back ck co cont ntro roll is ve very ry significant and much more significant than the improvement of Exercise 4.5 (which is based on a less favorable assumption of 4.2).
Copyright 2004 by Marcel De Dekker, kker, Inc. All Rights Reserved.
5 Solution of Matrix Equation TA FT ¼ LC
Chapter 4 proposed the new design approach of satisfying (4.1) and (4.3) first fir st,, an and d ex expl plain ained ed th thee nece necess ssit ity y an and d adva advant ntag ages es of th this is appr approa oach ch.. The The prob pr oble lem m of so solv lvin ing g (4 (4.1 .1)) and and (4.3 (4.3)) was first first rais raised ed in Tsui Tsui [198 [1987b 7b]. ]. It Itss
satisfactory solution appeared in Tsui [1992], much delayed from its first verbal presentation at the 1990 American Control Conference. This solution has made this new design approach 2000]. in Chaps 5 and The design algorithms of (4.1) possible and (4.3)[Tsui, are presented in Chaps 5 6, respectively. Chapter 5 has two sections. Sect Se ctio ion n 5.1 5.1 in intr trod oduc uces es the the algo algori rith thm m fo forr co comp mput utin ing g th thee bloc blockkobse ob serv rvab able le Hess Hessen enbe berg rg fo form rm of the the plan plantt sy syst stem em’s ’s st stat atee sp spac acee mo mode del. l. Althou Alt hough gh this this comput computati ation on is unnece unnecessa ssary ry for the ana analyti lytical cal sol soluti ution on of
Copyright 2004 by Marcel De Dekker, kker, Inc. All Rights Reserved.
(4.1), it significantly improves the numerical computation of this solution, and naturally separates the observable part from the unobservable part of the plant system. Sect Se ction ion 5. 5.2 2 pr pres esen ents ts th thee solu solutio tion n of (4.1 (4.1). ). It de demo mons nstr trat ates es also also th thee analytical and computational advantages of this solution over other existing solutions of (4.1).
5.1 5.1.1 5.1 .1
COMPUT COMPUTATI ATION ON OF A S SYS YSTEM TEM’S ’S OBSE OBSERVAB RVABLE LE HESSENBERG FORM Single Single-Ou -Outpu tputt Syste Systems ms
The Hessenberg form matrix is defined as follows:
A ¼
2 66 66 4
x * 0 x x * .. : . : : x ...
... 0 0 : .. . :
0 x * ... x
3 77 77 5
ð5:1Þ
where the elements ‘‘x’’ are arbitrary and the elements ‘‘*’’ are nonzero. The matr ma trix ix of (5.1) (5.1) is al also so call called ed th thee ‘‘ ‘‘lo lowe werr He Hess ssen enbe berg rg fo form rm’’ ’’ matr matrix ix.. Th Thee transpose of the matrix form (5.1) is called the ‘‘upper Hessenberg form.’’ The Hessenberg form is the simplest possible matrix form which can be computed from a general matrix by orthogonal matrix operation without iteration. For example the Schur triangular form, which differs from the Hessenberg form by having all ‘‘*’’ entries of (5.1) equal 0, is computed by iterative methods (QR method).
In the established computational algorithms of some basic numerical linear algebra problems, whether in the QR method of computing matrix eigenstructure decomposition [Wilkinson, 1965] and singular value decomposition [Golub and Reinsch, 1970], or in the computation of solution of the Sylvester equation [Golub et al., 1979] and the Riccati equation (Laub, 1979, Sec. 8.1), the computation of the Hessenberg form has always been the first step [Laub and Linnemann, 1986]. As the first step of the design algorithm for solving (4.1), a special form of system matrix ðA; C Þ called ‘‘observable Hessenberg form,’’ in which matrix A is in the lower Hessenberg form of (5.1), is also computed [Van Dooren et al., 1978; Van Dooren, 1981]. The
Copyright 2004 by Marcel De Dekker, kker, Inc. All Rights Reserved.
Solution of Matrix Equation TA FT ¼ LC
single-output case of this form is * 0
2 4
CH . . . :: H 0 AH
3 5
¼
266 66 64
. x. . .*. . x x : : x
... 0
. 0. . .. .. .. . 0. . * 0 : .. .. . . 0
x * ... x x
377 77 75
ð5:2Þ
where matrix H is an unitary similarity transformation matrix ðH 0 H ¼ I Þ which transforms the plant system matrix pair ðA; C Þ into the form of (5.2). The matrix H and its result (5.2) can be computed by the following algorithm.
Algorith Algo rithm m 5.1
Step 1: Step St ep 2 2:: Step St ep 3:
Computa Computatio tion n of Singl Single-O e-Outpu utputt Obse Observa rvable ble Hessenberg Form System Matrix
Let j ¼ 1 ; H ¼ I ; c1 ¼ C , and A1 ¼ A . Co Comp mput utee th thee unit unitary ary m mat atrix rix H j such that c j H j ¼ ½ c j ; 0 . . . 0 (see Appendix A, Sec. 2). Comp Comput utee
H j 0 A j H j ¼
2 66 4
a jj : ... x
...
c j þ1
. . . :
: A j þ1
3 77 ) 5
n j
ð5:3Þ
:
Step St ep 4:
Upda Update te ma matri trix x
1 I . j . . . .: . . 0. . H ¼ H 0 : H j :
26 4
37 5
where I j 1 is an identity matrix with dimension j 1.
Copyright 2004 by Marcel De Dekker, kker, Inc. All Rights Reserved.
Step 5: If c j þ1 of (5.3) equals 0, then go to Step 7. Step 6: Let j ¼ j þ þ 1 (so that c j þ1 and A j þ1 of (5.3) become c j and A j , respective resp ectively). ly). If j ¼ n then go to Step 7; otherwise return to Step 2. Step St ep 7: The The fina finall res resul ultt is is
2 64
CH . . . ::
H 0 AH
2 66 6 3 66 75 66 64 2 66 6 64 ¼
c1 0 . . . : . . . 0 : 0 . . . :0 . . . : . . . : . . . : . . . : . . . : :
:
a11 c2 0 . . . : 0 : .. .. : a22 . : : . .. . . : .. 0 : ..
:
: x
: :
c j : : a jj : 0 . . . :0
.
. . . : . . . : . . . : . . . : . . . : : . . . . . . :
X
C o : 0
...
:
4 Ao : 0 ¼
...
...
...
3 77 7 75
g j
: Ao
3 77 77 7 77 75 ð5:4Þ
X : Ao gn j
where the matrix rix pair ðAo ; C o Þ is in the observable Hessenberg form of (5.2) and is separated from the unobservable part of the system Ao . The dimension of this
observable part is j but but will be replaced by n elsewhere in this book because only observable systems are being considered. (see Exercise 5.4).
5.1.2 5.1 .2
Multip Multiple le Outpu Outputt System Systems s
In multi-output systems of m outputs, C is a matrix of m rows and is no longer a row vector. The corresponding observable Hessenberg form in this
Copyright 2004 by Marcel De Dekker, kker, Inc. All Rights Reserved.
case is the so-called block-observable Hessenberg form, as in:
gm0
C 1 0 . . . : . . . : . . . : 0
2 4
CH . . . :: H 0 AH
266 66 3 6 5 66 66 4 ¼
. . . : . . . : . . . : . . . : . . . : . . . : A11 C 2 0 . . . : . . . : 0 .. : A22 C 3 . : .. .. .. .. . : . . : . .. .. .. . . : . 0 .. .. . : . C n
An1 . . .
Ann
m1 m2 . . . . . . . . . . . . . . . : mn
377 77 77 77 7 5
gm1 gm2 :
ð5:5Þ
:
gmv1 gmv
where the Aij is an mi 6m j dimensional arbitrary matrix block, and C j ð j ¼ 1; . . . ; nÞ is is an m j 1 6m j dimen dimensi sion onal al ‘‘ ‘‘lo lowe werr-ec eche helo lon n ma matr trix ix’’ ’’ ðm0 ¼ m 5m1 5m2 5 5mn > 0 Þ. We will use the following example to illustrate the lower-echelon-form matrix.
Example 5.1 All lower-echelon-form matrices with three rows are in the following seven different forms:
2 4
32 32 32 32 32 32 3 54 54 54 54 54 54 5
* 0 0 x * 0 ; x x *
* 0 x * ; x x
* 0 x 0 ; x *
0 0 * 0 ; x *
* x ; x
0 * ; x
0 0 *
where ‘‘x’’ entries are arbitrary and ‘‘*’’ entries are nonzero. From Sec. A.2 of Appendix A, there exists a unitary matrix H j such that C j j H j ¼ ½ C j ; 0 . . . 0 for any matrix C j j with m j 1 rows, where C j j is an m j 1 6m j dimensional lower-echelon matrix. From Example 5.1, all m j columns of C j are linearly independent of each other, and so are the m j rows (those with a ‘‘*’’ element) of C C j j . Each of the other other m j 1 m j rows rows (tho (those se with withou outt a ‘‘* ‘‘*’’ ’’ el elem emen ent) t) ca can n alwa always ys be expressed as a linear combination of the linearly independent rows which are above this linearly dependent row in C j (see Sec. A.2 of Appendix A).
Copyright 2004 by Marcel De Dekker, kker, Inc. All Rights Reserved.
Example 5.2 In the last six of the seven matrices of Example 5.1, the linearly dependent rows are, respectively, the 3rd, the 2nd, the 1st, the 3rd and 2nd, the 3rd and 1st, and the 2nd and 1st of the corresponding matrix. All three rows of the first matrix are linearly independent of each other. Forr ea Fo each ch of th thee la last st six six of th thee seve seven n ma matr tric ices es C j ð j ¼ 2 ; . . . ; 7Þ of Exam Ex ampl plee 5. 5.1, 1, ther theree ex exis ists ts at leas leastt on onee row row ve vect ctor or d j such tha thatt d j C j ¼ 0. For example, d j ¼ ½ x x 1 ; ½ x 1 0 ; ½ 1 0 0 ; ½ x 0 1 or ½ x 1 0 ; ½ 0 x 1 or ½ 1 0 0 , and ½ 0 1 0 or ½ 1 0 0 , for j ¼ 2 ; . . . ; 7, respectively. It is clear that in these d j vectors, the position of elem el emen entt ‘‘ ‘‘1’ 1’’’ al alwa ways ys corr corres espo pond ndss to the the li line near arly ly de depe pend nden entt ro row w of the the corresponding C j , while all ‘‘x’’ el elem emen ents ts ar aree th thee line linear ar co comb mbin inat atio ion n coef co effic ficie ient ntss for for that that li line near arly ly depe depend nden entt ro row. w. It is also also cl clea earr th that at th thes esee coeffic coe fficien ients ts cor corres respon pond d only only to the linear linearly ly indepe independe ndent nt row rowss wh which ich are above that linearly dependent row. Without loss of generality, we assume m1 ¼ m , so that all m system outputs are linearly independent [Chen, 1984]. In other words, each row of matrix C corresponds to a linearly independent output. As in the singleoutput case, during the computation of the block-observable Hessenberg C j becomes 0 or becomes linearly dependent on its previous form, if a row of C rows of C j , then the corresponding output is no longer influenced by more syste sy stem m stat states es.. Thus Thus this this row/ row/co colu lumn mn wi will ll disa disapp ppea earr at the the su subs bseq eque uent nt C i ði > j Þ blocks (or no longer appear at the observable part of the system). With this adaptation, Algorithm 5.1 can be generalized to the following
Algorithm 5.2 for multi-output case.
Algorit Algo rithm hm 5. 5.2 2
Computa Computation tion o off Blo Block-O ck-Obse bserva rvable ble Hessenberg Form
Step 1: Step St ep 2 2::
Let j ¼ 1 ; H ¼ I ; C 1 ¼ C ; A1 ¼ A ; m0 ¼ m , and n0 ¼ 0. Comp Comput utee a un unit itar ary y ma matr trix ix H j such that C j H j ¼ ½C j ; 0 . . . 0, where C j is an m j 1 6m j dimensional lower echelon matrix.
Step St ep 3:
Comp Comput utee
2 66 4
H j 0 A j H j ¼
A jj : C j þ1 ...
. . . :
...
X : A j þ1 :
m j
3 77 5
gm j ð5:6Þ
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Step St ep 4:
Upda Update te ma matri trix x j
26 64
H ¼ H
I ...
: 0 ... ...
0 : H j :
37 75
where I j is an identity matrix with dimension n j 1 . Step 5: n j ¼ n j 1 þ m j . If n j ¼ n or if C j þ1 ¼ 0, then let n ¼ j and go to Step 7. þ 1 (so that the C j þ1 and A j þ1 of (5.6) become C j Step 6: Let j ¼ j þ and A j , respectively), and return to Step 2. Step St ep 7: The The fina finall resu result lt is is
2 64
CH . . . :: 0
H AH
2 66 66 3 6 75 6 ¼
C 1 0 . . .
... 0
:
. . . : . . . : . . . : . . . : . . . : : .. A11 C 2 . : : .. .. . : : . : A22 .. .. .. . . : . 0 : . . : .. .. C n :
0 ... 0
3 77 77 7 7
gm
¼
66 64 2 66 64
..
An1
Ann : 0 . . . 0
.
. . . : . . . : . . . : . . . : . . . : . . .
X C o : 0 ...
:
Ao : 0 . . . : . . .
...
X : Ao
. . . :
:
: Ao
3 77 75
77 75 ð5:7Þ
is alre alread ady y in th thee bloc blockkwheree th wher thee matr matrix ix pa pair ir ðAo ; C o Þ is observable Hessenberg form (5.5) and is already separated from the unobservable part A o of the system. The dimension of Ao is n j ¼ m 1 þ þ mv (replaced by n in the rest of this book).
Copyright 2004 by Marcel De Dekker, kker, Inc. All Rights Reserved.
It is clear that Algorithm 5.2 is a generalized version of Algorithm 5.1, when the parameter m is generalized from 1 to m ð51Þ. The main computation of this algorithm is of at this Stepalgorithm 3. According to Sec. A.2 of 3 the order =3. of computation (based on Step 3) Appendix is about 4nA,
Definition 5.1 From the description of the block-observable Hessenberg form (5.5), each row of matrix C of (5.5) corresponds to one of system outputs and is linked to one more system state if that row is linearly independent in the next matrix block C j þ1 of (5.5). Therefore the number of system states which influence the i -th -th system output equals the number of matrix blocks C j in which the i -th -th row of C C is is linearly independent. We define this number as the ¼ 1 ; . . . ; m. i -th -th observability index ni ; i ¼ that i -th -th row bec becom omes es linear linearly ly dep depend endent ent It is clea lear th tha at ni ¼ j if that in ma matr trix ix bl bloc ock k C j þ1 , and that n1 þ þ n m ¼ n . It is also clear that maxfni g ¼ n of of Step 5, and that all observability indices can be determined by Algorithm 5.2. Another set of parameters m j ; j ¼ 1 ; . . . ; n of (5.5) can also be used to ind indic icat atee th thee observ servab abil ilit ity y in ind dex. ex. From rom the the descr escrip ipti tion on of (5.5 (5.5)) and and Definition 5.1, m j indicates the number of observability indices which are
5 j .
Example 5.3 Let the block-observable Hessenberg form of a four-output and ninth-order system be C 1 0
266 64 C ¼ A
A11 A21 A31 A41
C 2 A22 A32 A42
0
0
0 0 C 3 0 A33 C 4 A43 A44
377 75
Copyright 2004 by Marcel De Dekker, kker, Inc. All Rights Reserved.
Solution of Matrix Equation TA FT ¼ LC
¼
2 66 66 66 66 6 66 66 66 66 4
* 0
0
0 : 0
0
0
0
0
x þ 0
0 : 0
0
0
0
0
x x
x & 0 : 0 x x # : 0
0 0
0 0
0 0
0 0
x
x
x
x : * 0
0 :
0
0
x
x
x
x : x þ 0 :
0
0
x
x
x
x : x
x
0
0
x
x
x
x : x
x #: 0
0
...
...
...
x
x
x
x : x
x
x : 0 : 0
x
x
x
x : x
x
x : þ : 0
x
x
x
x : x
x
x : x : 0
0 :
. . . . . . : . . . : . . .
...
...
. . . :
::
m0 ¼ m ¼ 4
) )
m1 ¼ 4
m2 ¼ 3
:: x x x x : x x x : x : þ gm3 ¼ 1 . . . . . . . . . . . . . . . : . . . : . . . . . . : :: x x x x : x x x : x : x gm4 ¼ 1 ...
. . . . . . : . . . : . . .
. . . :
3 77 77 77 77 7 77) 77 77 77 5
ð5:8aÞ
From Definition 5.1, corresponding to the four system outputs which are represented by the nonzero elements with symbols ‘‘*,’’ ‘‘þ,’’ ‘‘&,’’ and ‘‘#,’’ ‘‘# ,’’ respec respectiv tively ely,, the observ observabi abilit lity y indice indicess are n1 ¼ 2 ; n2 ¼ 4 ; n3 ¼ 1, and n4 ¼ 2. These indices also equal the number of appearances of the corresponding symbols in (5.8a). We can verify that tha at n ¼ n2 ¼ 4. We n1 þ n 2 þ n 3 þ n 4 ¼ m 1 þ m2 þ m3 þ m4 ¼ n ¼ 9, and th can also verify that m j equals the number of observability indices which are greater than or equal to j ð j ¼ 1 ; . . . ; n ¼ 4 Þ. In the literature Chen [1984], the block-observable Hessenberg form (5.8a) can be further transformed to the block-observable canonical form (1.16) by elementary similarity transformation:
CE E 1 AE
2 6 66 64 ¼
I 1 0
0
0
A1 I 2 0
0
A2 0 I 3 0 A3 0
0 I 4
A4 0
0
0
3 77 77 5
Copyright 2004 by Marcel De Dekker, kker, Inc. All Rights Reserved.
1
¼
266 66 66 66 6 66 66 66 66 6
0
0
0 : 0
0
0
0
0
x 1 0 x x 1
0 : 0 0 : 0
0 0
0 0
0 0
0 0
0
0
0
x
x
x 1 : 0
0
x
x
x
x : 1
0
0 : 0
0
x
x
x
x : 0
1
0 : 0
0
x
x
x
x : 0
0
0 : 0
0
x
x
x
x : 0
0
1 : 0
0
...
...
...
. . . . . . : . . . : . . .
. . . :
::
x
x
x
x : 0
0
0 : 0 : 0
x
x
x
x : 0
0
0 : 1 : 0
x
x
x
x : 0
0
0 : 0 : 0
...
...
...
x
x
x
. . . . . . : . . . : . . .
x : 0
0
. . . :
::
0 : 0 : 1
377 77 77 77 7 77 77 77 77 7
ð5:8bÞ
...
...
...
x
x
x
. . . . . . : . . . : . . .
x : 0
0
. . . :
::
0 : 0 : 0
4
5
where matrix E represents represents elementary matrix operations [Chen, 1984] and is usually not a unitary matrix. The comparison of (5.8a) and (5.8b) shows that the block-observable canonical form is a special case of the block-observable Hessenberg form, in C j j blocks of the sense that in (5.8b), all nonzero elements (those symbols) of C (5.8a) become 1, and all other arbitrary ‘‘x’’ elements of (5.8a) except those in the left m1 columns become 0. Although the parameters of a block-observable canonical form system matr ma trix ix can can be subs substi titu tuted ted di dire rect ctly ly in into to th thee po poly lyno nomi mial al ma matr trix ix fr frac acti tion on descriptio descr iption n of its correspon corresponding ding transfer transfer funct function ion GðsÞ ¼ D 1 ðsÞN ðsÞ (see Exam Ex ampl plee 1.7), 1.7), th this is uniq unique ue ad adva vant ntag agee is offse offsett by th thee un unre reli liab abil ilit ity y of it itss computati comp utation on (matr (matrix ix E of (5 (5.8 .8b) b) is usua usuall lly y ill ill co cond ndit itio ione ned d [W [Wil ilki kins nson on,, 1965]). For this reason, the actual design algorithms of this book are based only on the observable Hessenberg form.
Copyright 2004 by Marcel De Dekker, kker, Inc. All Rights Reserved.
5.2
SO SOLVIN LVING G MAT MATRIX RIX EQU EQUATI ATION ON TA FT ¼ LC
The co The comp mput utat atio iona nall al algo gori rith thm m for for the the solu soluti tion on of matr matrix ix eq equa uatio tion n (4.1) (4.1) ðTA FT ¼ LC Þ is pres presen ente ted d in this this sect sectio ion. n. He Here re the the n6n and m6n dimensiona dimen sionall system matri matrices ces ðA; C Þ are are give given n an and d ar aree ob obse serv rvab able. le. The The number of rows of solution ðF ; T ; LÞ is presumed to be n m, although this numb nu mber er is free freely ly adju adjust stab able le beca becaus usee each each ro row w of this this solu soluti tion on wi will ll be completely decoupled. To simplify the computation of solution of (4.1), we have computed block-obse block -observable rvable Hessenberg Hessenberg form ðH 0 AH ; CH Þ in Al Algo gori rith thm m 5.2. 5.2. Su Subbstituting ð H 0 AH ; CH Þ into (4.1), we have T ðH 0 AH Þ FT ¼ L ðCH Þ, which implies that the solution matrix T of of this equation must be postmultiplied by 0
0
H , in order toÞ.be recovered to the solution ð TH Þ, which corresponds to the original ðA; C ¼ 1 ; . . . ; n mÞ of matrix F of Mathematically, the eigenvalues ðli ; i ¼ (4.1) (4. 1) can be arbitr arbitraril arily y giv given. en. We wil will, l, howeve however, r, sel select ect the these se eig eigenv envalu alues es based on the following analytical understandings. First, these eigenvalues must have negative and sufficiently negative re real al part partss in or orde derr to achi achiev evee obse observ rver er st stab abil ilit ity y an and d su suffi ffici cien entl tly y fa fast st
convergence of observer output to K xðtÞ (Theorem 3.2). Seco Se cond nd,, th thee magn magnit itud udee of thes thesee ei eige genv nval alue uess ca cann nnot ot be to too o larg largee because it would cause large observer gain L (see Secs. 3.1 and 4.3). Third, each plant system stable transmission zero must be matched by one of th one thee eige eigenv nval alu ues of F . This is th thee ne nece cess ssar ary y co con nditi ditio on fo forr th thee corresponding rows of T to to be linearly independent if TB=0 (see Sec. 6.2). Finally, all n m eigenvalues of F are the transmission zeros of the corresponding observer feedback system [Patel, 1978] and should be selected with the properties of transmission zeros in mind (see Sec. 1.4). There have been some other suggesti tio ons for the selection of eigenvalues of F , but they are unsatisfactory. For example, the suggestion that th at the the ei eige genv nvalu alues es of F othe otherr th than an thos thosee which hich ma matc tche hed d th thee st stab able le transmission zeros be negative infinity with Butterworth pattern, is criticized by Sogaard-Andersen [1987]. The other suggestion that all eigenvalues of F be clustered around the plant system stable transmission zeros causes near singul sin gular ar mat matrix rix C ¼ ¼ ½T 0 : C 0 0 , and and ther theref efor oree larg largee an and d un unsa sati tisf sfac acto tory ry observer output gain K in (4.2) [Tsui, 1988b]. Hence the eigenvalues of F should be selected by following the preceding four guidelines. Once the eigenvalues of matrix F are are selected, the matrix F is is required in our algorithm to be a Jordan form matrix, with all multiple eigenvalues form fo rmin ing g a si sing ngle le Jord Jordan an bl bloc ock k [see [see (1 (1.1 .10) 0)]. ]. Henc Hencee th thee ma matr trix ix F is fu fully lly dete de term rmin ined ed.. In addi additi tion on,, each each ro row w or ea each ch bloc block k of ro rows ws of solu soluti tion on ðF ; T ; LÞ corresponding to a Jordan block of F is decoupled and can be
Copyright 2004 by Marcel De Dekker, kker, Inc. All Rights Reserved.
separa sep arately tely comput computed. ed. Theref Therefore ore,, our algorit algorithm hm tre treats ats the follow following ing two cases of Jordan block size (¼ 1 or > 1), separately.
5.2.1 5.2 .1
Eigens Eigenstru tructu cture re Ca Case se A
For dis For distin tinct ct and real real eigenv eigenvalue alue li ði ¼ ¼ 1 ; . . . ; n mÞ of F , (4 (4.1 .1)) can can be partitioned as ti A l i ti ¼ 1i C ;
i ¼ ¼ 1 ; . . . ; n m
ð5:9Þ
where ti and li are the i -th -th row of matrix T and L correspondin corresponding g to li , respectively. Based on the observable Hessenberg form (5.5) where
C ¼½ C 1 0 . . . 0 m
Eq. (5.9) can be partitioned as the left m columns
ti ðA l i I Þ
I m 0
¼ 1i C
I m 0
¼ 1i C 1
ð5:10aÞ
and the right n m columns
ti ðA l i I Þ
0 I nm
¼ 1i C
0 I nm
¼ 0
ð5:10bÞ
Because C 1 of (5.5) is of full-column rank and 1i is free, (5.10a) can always be satisfied by 1i for whatever ti . Therefore, the problem of (5.9) is simplified to the solving of ti of (5.10b) only, which has only n m columns instead of the n columns of (5.9). From Fr om ob obse serv rvab abili ility ty crit criter eria ia an and d the the fo form rm of ma matr trix ix C , th thee matr matrix ix product on the left-hand side of (5.10b) must have m linearly dependent
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rows. Furthermore, this matrix
2 6 66 64
ðA li I Þ
0 I nm
¼
C 2
A22 l i I : : : An2
0 . . . : . . . : 0 .. . : C 3 .. .. . : . .. . 0 C v ... Ann li I
3 77 77 5
ð5:11Þ
A maintains the same form of that of the right n m columns of matrix A, if A
is in block-observable Hessenberg form (5.5). From the definition of this form, the C j matrices ð j ¼ 2 ; . . . ; vÞ and the matrix of (5.11) are in lower echelon form. In other words, the n m linearly independent rows of matrix (5 (5.1 .11) 1) are are cl clea earl rly y in indi dica cate ted d as th thee row rows co corr rres espo pond ndin ing g to the the no nonz nzer ero o elements of matrices C j ð j ¼ 2 ; . . . ; vÞ. Each of the rest of m rows of matrix
(5.11) can always be expressed as a linear combination of its previous and line linear arly ly inde indepe pend nden entt ro rows ws in th that at matr matrix ix.. Th Thus us we have have th thee fo foll llow owin ing g conclusion.
Conclusion 5.1 The soluti solution on ti of Eq Eq.. (5.1 (5.10b 0b)) has has m basis basis vecto vectors rs dij ð j ¼ 1 ; . . . ; mÞ. If ðA; C Þ is already in block-observable Hessenberg form, then each of these m basis vectors can correspond to one of the m linearly dependent rows of matrix (5.11), each can be formed by the linear combination coefficients of the preceding and linearly independent rows of this linearly dependent row, and each can be computed by back substitution.
For For a Singl Singlee-Ou Outpu tputt Case Case ðm ¼ 1 Þ
Exam Ex ample ple 5 5.4 .4 From (5.2),
2 6 66 64
ðA li I Þ
0 I n1
¼
* 0 . . . : . . . : 0 .. * x . : . . .. .. : : .. : . 0 * : x . . . . . . : . . . : x
3 77 77 5
Copyright 2004 by Marcel De Dekker, kker, Inc. All Rights Reserved.
which hasThe onlysolution one ð m ¼ 1 Þ linearlyhas dependent (the row and without a ‘‘*’’ only onerow basis vector is unique, element). ti therefore and can be computed by back substitution.
Exam Ex ample ple 5.5 5.5
For For a Multi Multi-Ou -Outpu tputt Case Case
In Example 5.3 ð m ¼ 4 Þ, for each li , the corresponding solution ti of (5.10b) has m ð¼ 4 Þ basis vectors as di 1 ¼
½x di 2 ¼ ½ x di 3 ¼ ½ x di 4 ¼ ½ x
x 0 x : 1
0
0 : 0 : 0
x 0 x : 0 x 0 : x : 1 x 1
0 : 0
0
0 : 0 : 0
x 0 x : 0 x 1 : 0 : 0
Each of the above vectors dij has a ‘‘1’’ element, whose position corresponds to where the j -th row becomes linearly dependent in (5.8a) ð j ¼ 1 ; . . . ; m ¼ 4 Þ. The ‘‘x’’ elem elemen ents ts of dij ar aree th thee line linear ar co comb mbin inat atio ion n coeffic coe fficien ients ts of the linear linearly ly indepe independe ndent nt and pre preced ceding ing row rowss on tha thatt j -th -th linearly dependent row. Because each dij vector satisfies (5.10b), the actual solution ti of (5.10b) can be an arbitrary linear combination of the dij ’s. At the same position of each ‘‘1’’ element of dij , the elements of other three basis vectors are all 0. Therefore the four basis vectors are linearly independent of each other. From Fr om Co Conc nclu lusi sion on 5.1, 5.1, fo forr mu mult ltip iple le ð4mÞ and and real real ei eige genv nval alue uess ðsay; li ; i ¼ ¼ 1 ; . . . ; mÞ, it is possible to assign their corresponding rows of T as ti ¼ dii ði ¼ ¼ 1 ; . . . ; mÞ. This This wa way, y, thes thesee mu mult ltip iple le ei eige genv nval alue uess be beco come me equivalent of the distinct eigenvalues in the sense that their corresponding Jordan block in F becomes becomes diagfli ; i ¼ ¼ 1 ; . . . ; mg. However, by making this assig ign nment, there is certainly no more freedom left for soluti tio ons ¼ 1 ; . . . ; mÞ, and hence this possible solution is not recommended for ti ði ¼ solving (4.1) and (4.3) [but is recommended for solving the dual of (4.1) in eigenstructure assignment problems of Chap. Chap. 8]. Replacing the block-observable Hessenberg form (5.8a) by its special case, the block-observable canonical form (5.8b), the four basis vectors of ti of (5.10b) are di 1 ¼
½ li 0 3 di 2 ¼ ½ 0 li di 3 ¼ ½ 0 0 di 4 ¼ ½ 0 0
0 0
0 : 1
0
0 : 0 : 0
0 : 0 li 2 0 : li : 1 0 : 0
0
0 : 0 : 0
0 li : 0
0
1 : 0 : 0
1
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These four vectors not only are linearly independent of each other, but also have additional algebraic properties as follows.
Conclusion 5.2 From the above example, for a fixed parameter j ð j ¼ 1 ; . . . ; mÞ, any set of v j of the n dij vectors are linearly independent, because these vectors form a matrix which equals a v j dimensional Vandermonde matrix added with n v j zero columns. This conclusion is valid for block-observable Hessenberg form-based vectors too, because (5.8a) and (5.8b) are similar to each other. This conclusion can also be extended to multiple eigenvalue and generalized eigenvector cases. See the more rigorous proof in Theorem 8.1.
It is also obvious from the above example that for a fixed parameter j ð j ¼ 1 ; . . . ; mÞ, any v j 1 of the n dij vectors are also linearly independent of matrix C of of (5.5).
5.2.2 5.2 .2
Eigen Eigenstr struct ucture ure Ca Case se B
For complex conjugate or multiple eigenvalues of F , the results of Case A can be generalized. Letting li and li þ1 be a+ jb, and their corresponding Jordan block be F ii ¼
a b
b a
as in (1.10), the corresponding Eqs of (5.9), (5.10a), and (5.10b) become
ti
ti þ1
A 0
1i
¼
ti þ1
I m
ti
1i þ1
I m
ti
F i
C
¼
1i þ1
0 I nm
¼ 0
0
ti
ti þ1
A
0 I nm
ti þ1
ti
F i i
ti þ1
ð5:12Þ
1i
ti þ1
and
ti
A F i i
C 1
ð5:13aÞ
ð5:13bÞ
respectively. Becaus Bec ausee in (5.13a (5.13a)) C 1 is of full full-c -colu olumn mn ra rank nk an and d 1i and 1i þ1 are completely free, we need only to solve (5.13b) for ti and ti þ1 . (5.13b) can be
Copyright 2004 by Marcel De Dekker, kker, Inc. All Rights Reserved.
written as a set of linear equations:
02 BB66 @4 2 64 A
½ti : ti þ1
0
I nm ......
0
a
0
I nm ...... 0
:
: ...... : : :
: ...... :
3 77 5 31 75CA 0
. . . . . . :
A
b
0
I nm
ð5:13cÞ
0
I nm .........
0
¼ 0
b
I nm
a
:
I nm
where the two matrices in the bracket have dimension 2n62ðn mÞ, and can be expressed as
1 0
0 0 6A I nm 1
and
F i 0 6
0 Inm
respectively, where the operator ‘‘6’’ stands for ‘‘Kronecker product.’’ It is not difficult to verify that like the matrix (5.10b) of Case A, the who hole le ma matr trix ix in the the brac bracke kett of (5.1 (5.13c 3c)) has has 2m linearly linearly depe dependent ndent rows rows.. Ther Th eref efor oree th thee solu soluti tion on ½ti : ti þ1 of of (5.13c) has 2m basi basiss ve vect ctor orss ½dij : di þ1; j ð j ¼ 1 ; . . . ; 2mÞ.
Example Exa mple 5.6
Sing Single-O le-Outp utput ut Cas Case e ðm ¼ 1 Þ
Let matrix A of (5.2) be
2 4
x * 0 x x * x x x
3 5
ðn ¼ 3 Þ
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then the matrix of (5.13c) will be
2 66 66
* 0 x * x x ... ... 0 0 b 0 0 b
: 0 : b : 0 ... ... : * : x : x
0 0 b . . . : 0 * x
3 77 77
4
5
Clearly, this matrix has 2m ð¼ 2 Þ linearly dependent rows which do not have ‘‘*’’ elements. Therefore the solution ½ti : ti þ1 has 2m ð¼ 2 Þ basis vectors of the following forms:
½ di 1 :
di þ1;1
¼ ½x x 1 : x x 0
and d
d
½
i 2
i þ1;2
:
¼ ½x x 0 : x x 1
where the position of element ‘‘1’’ corresponds to one of the two linearly dependent rows, and ‘‘x’’ elements are the linear combination coefficients of all lin linear early ly indepe independe ndent nt rows. rows. These These two bas basis is vec vector torss can be comput computed ed se sepa para ratel tely, y, eith either er by modi modifie fied d back back subs substit titut utio ion n me meth thod od or by Gi Give vens ns’’ rotational method [Tsui, 1986a]. For a multiple of q eigenvalues li and and th thei eirr co corr rresp espon ondi ding ng q-dimensional Jordan block
F i 0 ¼
2 66 66 64
li 1
0 li : 0 : :
0 ...
0 . . . : . . . : 0 .. . : 1 .. ... ... . : .. .. .. . 0 . . .. . li 1 . . . . . . : 0 li
3 77 77 75
where ‘‘0 ’’ stands for transpose, its corresponding (5.12) and (5.13a,b,c) are,
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respectively:
2 64 2 6
t1
: tq t1
:
3 75 3 7
2 64
A F i i
A
I m
t1
: tq
3 2 3 75 64 75 2 3 2 3 67 6 7 11
¼
:
C
ð5:14Þ
1q
ti
F i i :
I m
11
¼
:
C 1
ð5:15aÞ
0
tq
426 4
t1
: tq
0
tq
1q
537 4 26 5 37 4 5 5 45 A
t1
0
F i i :
I nm
tq
0
¼ 0
I nm
ð5:15bÞ
and
½ t1
:
...
:
0
0
tq
I q 6A
0
I nm
F i 6
I nm
¼ 0
ð5:15cÞ
where ti ði ¼ corresponding to ¼ 1 ; . . . ; qÞ are the q rows of solution matrix T corresponding the Jordan block F i i . Because C 1 is of fu full ll-c -col olum umn n rank rank an and d 1i ði ¼ ¼ 1 ; . . . ; qÞ are free in ¼ 1 ; . . . ; qÞ. (5.15a), we need to solve (5.15b,c) only for ti ði ¼ It is not difficult to verify that like (5.13c), the whole matrix in the brac br acke kett of (5 (5.1 .15c 5c)) has has qm linea linearl rly y depe depend nden entt ro rows ws.. Thus Thus th thee so solu lutio tion n ½ t1 : . . . : tq of of (5.15c) has qm ba basis vectors ½ d1 j : . . . : dqj ; j ¼ 1 ; . . . ; qm. Because of the simplicity of bidiagonal form of the Jordan block F i i , (5.15b,c) can be expressed as
t j ðA l i I Þ
0 I nm
¼ t j 1
0
I nm
; j ¼ 1 ; . . . ; q; t0 ¼ 0
ð5:15dÞ
Equation (5.15d) shows that all t j vectors except t1 are computed based on its its pr prev evio ious us vect vector or t j 1 . These hese vect vecto ors are ca call lleed ‘‘ ‘‘ge gene nera rali lizzed’’ ed’’ or ‘‘defe ‘‘d efecti ctive. ve.’’ ’’ Be Becau cause se the vector vectorss t j are also also th thee le left ft ei eige genv nvec ecto tors rs [s [see ee (1.1 (1 .10 0)] )],, we al also so call call t j ð j ¼ 2 ; . . . ; qÞ of (5.15d (5.15d)) ‘‘g ‘‘gene eneral ralize ized/d d/defe efectiv ctivee eigenvectors’’ [Golub and Wilkinson, 1976b]. Thee abov Th abovee ca case sess of A and B can can be su summ mmar ariz ized ed in th thee fo foll llow owing ing algorithm for solving (4.1) [Tsui, 1987a].
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Algorith Algo rithm m 5.3
Computa Computatio tion n of Sol Solutio ution n of Matrix Matrix Equa Equation tion TA FT ¼ LC
Step Ste p1 1::
Ba Based sed on each each eeige igenva nvalue lue of F (say, (say, li , which is distinct real, complex conjugate, or multiple [of q]), compute the m, 2m, and qm basis vectors of the corresponding ti ; ½ti : ti þ1 , and ½ ti : . . . : ti þq1 , acco accord rdin ing g to (5 (5.1 .10b 0b), ), (5 (5.1 .13c 3c), ), and and
(5.15c), respectively. Step 2: The row (or rows) ti ; ½ti : ti þ1 , and ½ ti :
Steep 3 St 3::
...
:
ti þq1
equal an, 2arbitrary linear of their corresponding m m, and qm ði ¼ ¼ 1 ; . . . ; nÞ. set of m basiscombination vectors, respectively There are a total of nm free linear combination coefficients. Afte fter a all ll ti rows and the corresponding matrix T are fully dete de termi rmine ned d in St Step ep 2, sati satisf sfy y th thee left left m columns columns of TA FT ¼ LC [or (5.10a), (5.13a), and (5.15a)] by solving
ðTA FT Þ
I m 0
¼ LC 1
ð5:16Þ
The so The solu luti tion on L is uniq unique ue be beca caus usee C 1 has m linearly independent columns.
Conclusion 5.3 The above Algorithm 5.3 computes ðF ; T ; LÞ which satisfies (4.1). It is clear that the first two steps of the algorithm satisfy the right n m columns of (4.1), and Step 3 satisfies the left m columns of (4.1). This solution does not assume any restrictions and is therefore completely general. The complete re rema main inin ing g free freedo dom m of (4 (4.1 .1)) is also also ex expr pres esse sed d ex expl plic icit itly ly (a (ass th thee li line near ar combination coefficients) in Step 2. Let us an Let anal alyz yzee the the comp comput utat atio iona nall reli reliab abili ility ty an and d ef effic ficie ienc ncy y of th this is algorithm. Beca Be caus usee th thee in init itia iall step step of an algo algori rith thm m affe affect ctss the the co comp mput utat atio ion n reliability of that algorithm most, and because most of the computation of Algorithm 5.3 concerns Step 1, the analysis will concentrate on this step only. This step can be carried out by back substitution (see Sec. A.2 of Appendix A), which is itself numerically stable [Wilkinson, 1965]. However, this operation requires repeated divisions by those ‘‘*’’ nonzero elements of
Copyright 2004 by Marcel De Dekker, kker, Inc. All Rights Reserved.
the Hessenberg form matrix (5.5). Therefore this step can be ill conditioned if these nonzero elements are not large enough in magnitude. According to the Householder method (see Sec. A.2 of Appendix A), these nonzero elements (computed at Step 2 of Algorithm 5.2) equal the norm no rm of the the co corr rres espo pond ndin ing g row row ve vect ctor or.. Th This is st step ep also also co cons nsis ists ts of the the
determ dete rmin inat atio ion n of whet whethe herr that that no norm rm is ze zero ro or no nonz nzer ero. o. Th Ther eref efor ore, e, to improve the condition of Step 1 of Algorithm 5.3, it is plausible to admit only the large enough vector norms as nonzero. From the description of Example 1.5, each of these nonzero elements is the only link between one of the pl the plan antt syst system em stat states es to syst system em outp output ut.. Thus Thus ad admi mitt ttin ing g only only the the larg largee enough elements as nonzero implies admitting only the strongly observable states as observable states. However, reducing the dimension of a system’s observable part also implies the reduction of system information. This tradeoff of accuracy and solution magnitude is studied in depth in Lawson and Hanson [1974], Golub et al. [1976a]. [1976a]. To best handle this tradeoff, the singu singular lar value deco decomposit mposition ion (SVD) method can be used to replace the Householder method in Step 2 of Algorithm 5.2 [Van Dooren, 1981; Patel, 1981]. However, the SVD method cannot determine at that step which row among the m j 1 rows of matrix C j is li line near arly ly depe depend nden entt or in inde depe pend nden ent, t, an and d th thus us ca cann nnot ot de dete term rmin inee th thee observability index, which is the analytic information about multi-output system and is as important as the system order of single-output systems. In addition, the SVD method cannot result in echelon form matrix C j —the form fo rm whic which h made made the the simp simple le back back subs substi titu tuti tion on oper operat atio ion n of Step Step 1 of Algorithm 5.3 possible. The distinct advantage of the computational efficiency of Step 1 of Algo Al gori rith thm m 5.3 5.3 is th that at this this co comp mput utat atio ion n ca can n be ca carr rrie ied d ou outt in complete parallel . This advantage is uniquely enabled by the distinct feature that all basis vectors dij are completely decoupled for all j ð j ¼ 1 ; . . . ; mÞ and for all i ði ¼ ¼ 1 ; . . . ; nÞ as long as the l i ’s are in different Jordan blocks of F F . In other words, the computation of dij does not depend on the information of other dij ’s. Only the dij ’s corresponding to the same Jordan block and the same j aree co ar coup uple led d [see [see (5 (5.1 .13c 3c)) an and d (5 (5.1 .15c 5c)] )].. In ad addi diti tion on,, th thee ba back ck su subs bsti titu tutio tion n operation is itself very simple and efficient (see Sec. A.2 of Appendix A). The basic reason for the good computational properties of Algorithm 5.3 is the Jordan form of matrix F . It should be noticed that simplicity and decoupling are the fundamental features and advantages of eigenstructure decompos deco mposition. ition. This is the reason reason that the eigen eigenstruct structure ure decompositi decomposition on (or Jordan form) is computed from a given matrix in the first place. In the part pa rtic icul ular ar pr prob oble lem m of so solv lvin ing g (4 (4.1) .1),, the the ei eige genv nval alue uess ar aree given and and ar aree unnecessary to be computed. Therefore it is certainly plausible to set matrix F in in Jordan form—the form that is much sought after in other problems.
Copyright 2004 by Marcel De Dekker, kker, Inc. All Rights Reserved.
Conclusion 5.4 The co The comp mput utati ation on of Algo Algori rithm thm 5. 5.3 3 is very very relia reliabl blee an and d ve very ry ef effic ficien ient, t, as compared with other algorithms for solving (4.1).
The much more important advantage of the solution of Algorithm 5.3 concerns its analytical aspects. Equation (4.1) is not only the most fundamental equation of observer feedback compensator (3.16) design (see Chaps (see Chaps 3 3 and and 4) 4),, but also the most fundamental equation of state/generalized state feedback design. The dual of (4.1) is ; AV V L ¼ B K
ð5:17Þ
which implies which implies A BK ¼ V LV 1 , where K ¼ K K V 1 is the state feedback control gain, and V and L are the right eigenvector matrix and the Jordan forrm matr fo matrix ix of the the stat statee fe feeedb dbac ack k sys syste tem m dy dyna nami micc ma matr trix ix A BK , respectively. Because of these reasons, if Lyapunov/Sylvester equations AV VA0 ¼ B = AV V L ¼ B
ð5:18Þ
are considered fundamental in system analysis, and if the algebraic Riccati equatio equ ation n is con consid sidere ered d fundam fundament ental al in quadra quadratic tic optima optimall con contro troll sys system tem design, then Eqs (4.1) and (5.17) should be considered the most fundamental equations in state space control system design. However, the really general solution of (4.1), with really fully usable remain rem aining ing freedo freedom m and with with fully fully decoup decoupled led pro prope pertie rties, s, was not not der derive ived d until 1985 [Tsui, 1987a, 1993a]. For example, the solution of the Sylvester equa eq uatio tion n (5 (5.18 .18)) has has gene genera rally lly been been us used ed as th thee su subs bsti titu tute te of th thee ge gene nera rall at solution soluti on of (5.17) (5.17) [Tsui, [Tsui, 1986 1986c]. c]. Because Because (5.18) lack lackss the free parameter parameter K its right-hand side as compared with (5.17) [or lacks parameter L of (4.1)], it cannot be simplified to the form of (5.10b), (5.13c), or (5.15c). Thus the existence of solution of (5.18) is questionable when A and a nd L share common eige eigenv nval alue uess [G [Gan antm tmac ache her, r, 19 1959 59;; Ch Chen en,, 19 1984 84;; Fr Frie iedla dland nd,, 19 1986 86]. ]. Su Such ch a solution is certainly not a general solution of (4.1) or (5.17). From Conclusion 5.3, the general solution of (4.1) or (5.17) has been derive der ived, d, wit with h explic explicitl itly y and ful fully ly expres expressed sed rem remain aining ing fre freedo edom m and wit with h completely decoupled rows corresponding to the different Jordan blocks of F . Such a solution to such a fundamental equation of design will certainly have impact oncontrol state space control system design and onin the practical valuegreat of state space theory. In fact, as will be shown the rest of
Copyright 2004 by Marcel De Dekker, kker, Inc. All Rights Reserved.
this book, this solution has uniquely enabled the dynamic output feedback compensator design [Tsui, 1992, 1993b] (Sec. 6.1), the systematic minimal
orderr ob orde obse serv rver er desi design gn [Tsu [Tsui, i, 1985 1985]] (Chap Chap.. 7) 7),, the sys system temati aticc eig eigenv envalu aluee assign ass ignmen mentt [Ts [Tsui, ui, 1999a] 1999a] (Sec. (Sec. 8.1) 8.1) and eigenv eigenvecto ectorr ass assign ignmen mentt [Ka [Kauts utsky ky et al., 1985; Tsui, 1986a] (Sec. 8.2), and the robust failure detector design [Tsui, 1989] (Sec. 10.1). 5.1 outlines outlines the sequential relationship of these design results. Figure 5.1
EXERCISES 5.1
5.2 5.3
5.4
Re Repe peat at th thee comp comput utat atio ion n of simi simila lari rity ty tr tran ansf sfor orma mati tion on to bloc blockkobservable Hessenberg form of Example 6.2, according to Algorithm 5.2 (also Algorithm A.1 for QR decomposition). Repe Repeat at 5.1 for Exa Example mple 8.7 (d (dual ual version). version). Repe Repeat at the comput computation ation of satis satisfying fying (4.1 (4.1)) for Example Exampless 6.1, 6.2, 6.3, 7.3 and 8.1, 8.2, 8.3, 8.4 (Step 1), according to Algorithm 5.3 (first two steps mainly). Verify (4.1) for these results. 0 0 0 Par Partit tition ioning ing the state of syste system m (5.7) as ½xo ðtÞ : xo ðtÞ , the system’s bloc lock dia iagr gra am can can be depic epicte ted d as in Fig. Fig. 5. 5.2 2 whic which h sh show owss that that
Figure 5.1 Sequ Sequence ence of desi design gn algori algorithm thms s of this boo book. k.
Copyright 2004 by Marcel De Dekker, kker, Inc. All Rights Reserved.
Figure 5.2 Blo Figure Block ck dia diagra gram m of sys system tems s wit with h ob obser servab vable le and uno unobse bserva rvable ble parts.
ðAo ; Bo ; C o Þ is observable le,, while the other part of the syste tem m ðAo ; Bo ; 0Þ is not. not. Repe Repeat at this this proo prooff fo forr it itss du dual al case case (c (con ontro troll llab able le 5.5
Hessenberg form). Co Comp mput utee the the solu soluti tion on ðT 4 ½t t ; LÞ whic which h sa sati tisfi sfiees th thee ma matr trix ix ¼ 1 2 equation (4.1) ðTA FT ¼ LC Þ, where [Chen, 1993] A ¼
0
1
0 1
C ¼ ¼ ½ 1
0
and
F ¼ 4 and 1; respectively Answer : For F ¼ 4 : T ½ 1
0
3 ¼ 0 ) T ¼ ½ 3t2 t2 ðarbitrary
t2 6 ¼ 0 Þ; then; L ¼ T ½ 4 For F ¼ 1 : T ½ 1
0
0 ¼ 12t2 :
0
0 ¼ 0 ) T ¼ ½ 0 t2 ðarbitrary
6 0 Þ; then; L ¼ T ½ 1 t2 ¼
0 0 ¼ 0 :
Copyright 2004 by Marcel De Dekker, kker, Inc. All Rights Reserved.
6 Observer (Dynamic Part) Design for Robustness Realization
Step 2 of Al Step Algo gori rith thm m 5. 5.3 3 reve reveal aled ed the the rema remain inin ing g fr free eedo dom m of (4.1 (4.1). ). Th This is freedom will be fully used for the various design applications listed at the end of Chap. Chap. 5. 5. This chapter describes the first of such applications—observer design forr th fo thee guar guaran ante teed ed full full real realiz izati ation on of th thee ro robu bust stne ness ss pr prop oper erti ties es of st state ate feedback control. Failure to realize the robustness properties of this control is perhaps the drawback that has limited the practical applications of state space control theory. This chapter freedom will demonstrate for the first time, with the full use of the remaining of (4.1), this drawback canthat be effectively overcome for most open loop system conditions.
Copyright 2004 by Marcel De Dekker, kker, Inc. All Rights Reserved.
The design of this chapter will fully determine the dynamic part of the observer, which can also be considered as a feedback compensator. This chapter consists of four sections. Sect Se ction ion 6. 6.1 1 pres presen ents ts the the desi design gn algo algori rith thm m that that uses uses the the rema remaini ining ng freedom of (4.1) to best satisfy equation TB ¼ 0 (4.3). As described in Chap. in Chap. 3, TB ¼ 0 is the key requirement of realizing the robustness properties of state feedback control [Tsui, 2000]. 6.2 analyzes the generality the above solutionexamples. of (4.1) and (4.3), Section and illustrates this design algorithmofwith six numerical Sect Se ction ion 6. 6.3 3 demo demons nstr trat ates es a theo theore reti tica call sign signifi ifican cance ce of th this is desi design gn algorithm—the complete unification of exact LTR state observer feedback system and the static output feedback system. Sect Se ctio ion n 6.4 6.4 desc descri ribe bess th thee adju adjust stme ment nt of ob obse serv rver er or orde der, r, whic which h is completely adjustable under the design algorithm of this book. The higher observer order implies a less constrained and therefore a more powerful generalized state feedback control, while the lower observer order implies an easier realization of robustness properties of this control.
6.1
SOLUTIO SOLUTION N OF MATRIX MATRIX EQU EQUATI ATION ON TB ¼ 0
Let us first summarize the results at Step 2 of Algorithm 5.3. For distinct and real eigenvalue li ,
ti ¼ ci Di
ð6:1aÞ
For complex conjugate li and li þ1 ,
½ti : t i þ1 ¼ ½ci : c i þ1 ½Di : D i þ1
ð6:1bÞ
þ q 1Þ, For multiple of q eigenvalues l j ð j ¼ i ; . . . ; i þ ½ti :
. . . : ti þq1
¼ ½ci : . . . : ci þq1 ½Di :
. . . : D i þq1
ð6:1cÞ
¼ 1 ; . . . ; n mÞ are n and m, The dimensions of each row vector ti and ci ði ¼ respectively.
Copyright 2004 by Marcel De Dekker, kker, Inc. All Rights Reserved.
Algor Al gorith ithm m 6. 6.1 1 Step Ste p 1:
Solv Solve e TB ¼ 0 ð4:3Þ½Tsui ; 1992; 1993b
Substi Substitut tutee (6.1) (6.1) iinto nto (4.3), (4.3), we we have have ci ½Di B ¼ 0
½ci : c i þ1 ½Di B : D i þ1 B ¼ 0
ð6:2aÞ ð6:2bÞ
and
½ci :
Step St ep 2:
. . . : ci þq1 ½Di B : . . . : D i þq1 B
¼ 0
ð6:2cÞ
respectively. ¼ 1 ; . . . ; n mÞ of (6.2). Comp Comput utee th thee solu soluti tion on ci ði ¼
Equati Equa tion on (6.2 (6.2)) is only only a set set of line linear ar eq equa uati tion onss (s (see ee Appe Append ndix ix A) A).. None No neth thele eless ss,, ther theree are are two two spec specia iall case casess of (6.2 (6.2)) wh whic ich h will will be tr trea eate ted d separately in the following. To simplify the description, only the distinct and real eigenvalue case (6.2a) will be described.
Case A If the exact nonzero solution of (6.2a) does not exist (this usually happens when m < p þ 1), then compute the least square solution of (6.2a): ci ¼ u0m
ð6:3Þ
where um is th thee m-th -th colu column mn of matr matrix ix U , and where here U SV 0 ¼ D i B is the singul singular ar value value decomp decomposi ositio tion n of D B (with (with nonze nonzero ro singul singular ar val values ues i ¼ 1 ; . . . ; m) of (A si > 0 ; i ¼ (A.2 .21 1). The corre or resp spo ond ndin ing g rig right ht-h -han and d side side of (6.2a) will be sm vm , where vm is the m-th row of matrix V 0 of (A.21) (see Example A.6). Because the solution of case A implies that TB=0, the corresponding obse ob serv rver er (3 (3.1 .16) 6) ca cann nnot ot be cons consid ider ered ed as a dy dyna nami micc ou outp tput ut fe feed edba back ck compensator (4.10), even though this observer approximates the dynamic output feedback compensator requirement ðTB ¼ 0 Þ in least-square sense.
Case B If th thee ex exact act solu solutio tion n of (6 (6.2 .2a) a) is not not uniq unique ue (t (thi hiss us usua ually lly ha happ ppen enss wh when en m > p þ 1) 1),, then then the the rema remain inin ing g free freedo dom m of (6.2 (6.2a) a) [and [and (4 (4.1 .1)] )] ex exis ists ts.. Th This is fr free eedo dom m wi will ll be fu fully lly used used to ma maxi ximi mize ze the the an angl gles es be betw twee een n th thee ro rows ws of
Copyright 2004 by Marcel De Dekker, kker, Inc. All Rights Reserved.
0
matrix C ¼ ½T 0 : C 0 by the following three substeps. The purpose of this opera op erati tion on is to be best st stre streng ngth then en th thee stat statee an and d ge gene nera rali lize zed d st stat atee fe feed edba back ck control K xðtÞ ¼ KC xðtÞ which is eventually implemented by this observer. Step 2a:
Compu mpute all m p possi possible ble and and line linear arly ly inde indepe pend nden entt solutions cij of (6.2a) such that cij ½Di B ¼ 0 ; j ¼ 1 ; . . . ; m p
Step St ep 2b: 2b:
ð6:4Þ
Co Comp mput utee matr matrix ix
Di ¼
Step 2c:
2 64
ci 1 Di . . .
ci ;m p Di
3 75
ð6:5Þ
¼ Compute the m p dime dimens nsio iona nall ro row w vect vector or ci ði ¼ 1; . . . ; n mÞ such that the angles between the rows
ti
0
I nm
¼ c i Di
0
I nm
ð6:6Þ
are maximized (as close to +90 as possible). The explicit algorithms of Substep 2c will be described as Algorithms Becaus usee of th thee sp speci ecial al fo form rm of 8.2 and 8.3 in Chap. Chap. 8. Beca matrix C in in (5.5), Substep 2c implies the maximization of 0 the angles between the rows of matrix ½T 0 : C 0 . In addi ad diti tion on,, maxi maximi mizi zing ng row row ve vect ctor or an angl gles es also also im impl plie iess maximizing the row rank of the same matrix. The second maximization is much easier than the first (only nonzero angles between the vectors are required), and is guaranteed to be achieved by Algorithms 8.2 and 8.3 even though the first maximization may not be. It is obvious that TB ¼ 0 is satisfied or best satisfied by Algorithm 6.1 [after (4.1) is satisfied].
6.2
ANALYS ANALYSIS IS A AND ND E EXAM XAMPLE PLES S OF THIS THIS DESIGN DESIGN SOLUTION
Design Desi gn algo algori rith thm m 5.3 5.3 [f [for or (4 (4.1 .1)] )] and and desi design gn algo algori rith thm m 6.1 6.1 [f [for or (4 (4.3 .3)] )] completely determine the dynamic part of observer feedback compensator
Copyright 2004 by Marcel De Dekker, kker, Inc. All Rights Reserved.
and essentially define the new design approach of this book. This design is analyzed theoretically and is illustrated by six numerical examples in this section.
Conclusion 6.1 A sufficient condition to satisfy (4.1) and (4.3) exactly is m > p . It is obvious that Algorithm 5.3 satisfies (4.1) for all observable plant systems ðA; B; C Þ, while (4.3) [or (6.2)] can always be satisfied by the remaining freedom of (4.1) (the ci parameters) if m > p . Another sufficient condition to satisfy (4.1) and (4.3) exactly is that the plant system has at least one stable transmission zero. This is because from the property of transmission zeros (say zi ) described after Definition 1.5, there exists at least one vector (say, ½ti : l i ) such that : i ½t li S ¼
: i ½t l i
A zi I
C
: :
B
0 ¼ 0
ð6:7Þ
F (see m 6 (see the beginning of if m > p . Because zi is matched by an eigenvalue l i of F Sec. 5.2), the comparison between (4.1) and the left n columns of (6.7) and the comparison between (4.3) and the right p columns of (6.7) indicate that t i and li of (6.7) are the i -th -th row of T and L of (4.1) and (4.3), respectively. In othe ot herr wo word rds, s, (4 (4.1 .1)) and and (4.3 (4.3)) are are auto automa mati tica call lly y sa sati tisfi sfied ed to toge geth ther er if zi is matched by li . It should be noticed that the number of rows of solution ðF ; T ; LÞ of (4.1) is freely adjustable and can be as low as one. Therefore the existence of z at least one stable transmission zero i implies the existence of solution of (4.1) and (4.3). A su suffi ffici cien entt cond condit itio ion n for for m 6 > p is also also a su suffi fficie cient nt co cond ndit itio ion n fo forr m > p , because the former case is more difficult (has less output meas me asur urem emen entt in info form rmati ation on but but more more co cont ntro rols ls to real realiz ize) e) than than th thee latte latterr case, as proved by the first part of this conclusion. Definition 1.5 also implies that the existence of stable transmission zeros is also a necessary condition to satisfy (4.1) and (4.3) exactly if m 6 > p .
Conclusion 6.2 It is obvious that Algorithms 5.3 and 6.1 fully used the entire design freedom of ob obse serv rver er dyna dynami micc part part ðF ; T ; LÞ (aft (after er th thee eige eigenv nval alue uess of F are
determined) to satisfy (4.1) and (4.3) and to maximize the angles between
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0
¼ ½T 0 : C 0 [se the rows of matrix C ¼ [seee Co Concl nclusi usion on 5.3 and Alg Algori orithm thm 6.1 (Case B)]. Conclusion 6.3 If the plant system ðA; B; C Þ either has n m stable transmission zeros or satisfies satisfi es (1) min minimu imum-p m-phas hase; e; (2) rankðCBÞ ¼ p ; and (3) m5 p, then the 0 0 0 resulting matrix C ¼ ½T : C of Algorithms 5.3 and 6.1 is nonsingular. In othe ot herr wo word rds, s, Algo Algori rith thms ms 5.3 5.3 an and d 6.1 6.1 will will resu result lt in an ex exac actt LT LTR R st stat atee observer if the plant system satisfies the above conditions.
Proof The proof is divided into two parts, A and B. Part A: The plant system has n m stable transmission zeros From Conclusion 6.1, for general plant system ðA; B; C Þ with m 6 > p , there exists an additional linearly independent row of solution ðF ; T ; LÞ of (4.1) and (4.3) if and only if there exists an additional plant system stable T corresponding transmission zero. From Conclusion 5.2, the n m rows of T corresponding to the n m stab stable le tran transm smis issi sion on ze zero ross ca can n alwa always ys be made made li line near arly ly independent of each other and of the rows of matrix C . Thus the necessary and sufficient condition for the plant system GðsÞ with m 6 > p , to have an 0 exact solution of (4.1), (4.3) and a nonsingular matrix C ¼ ½T 0 : C 0 , is that GðsÞ has n m stable transmission zeros. Similar to the last argument of Conclusion 6.1, the sufficient condition for m 6 > p is also a sufficient condition for m > p . Part B: The Part The plan plantt sy syste stem m satis satisfie fiess ð1Þ minimum-phase, (2) rankðCBÞ ¼ p , and (3) m5 p First, Firs t, because because m ¼ p and rankðCBÞ ¼ p guarantee n m plant system transmission zeros [Davison and Wang, 1974], the additional condition of minimum-phase guarantees n m stable plant system transmission zeros. Thus the proof of Part A of this conclusion can be used to prove Part B for the case of m ¼ p . m > p of Part B, the proof is indirect via the proof that For the case of m thee ab th abov ovee th thre reee cond condit itio ions ns are are suffi suffici cien entt cond condit itio ions ns fo forr the the ex exis iste tence nce of unknown input observers or exact LTR state observers which satisfy (4.1), (4.3), and rank ðC Þ ¼ n (see Sec. 4.3). Because Conclusion 6.2 shows that Algo Al gori rith thms ms 5. 5.3 3 and and 6. 6.1 1 fully used used the rem remain aining ing obs observ erver er dyn dynami amicc par partt
design freedom to satisfy (4.1), (4.3) and maximized rank of C after the F are F and eigenvalues of F are assigned, and because the eigenvalues of F and the poles
Copyright 2004 by Marcel De Dekker, kker, Inc. All Rights Reserved.
of unknown input observers are similarly assigned, matrix C of of Algorithms 5.3 and 6.1 will have the maximum rank n and will be nonsingular if the unknown input observer exists. There have been a number of such proofs in the literature [Kudva et al., 1980; Hou and Muller, 1992; Syrmos, 1993b]. It seems that the proof in Hou and Muller [1992] is most complete and explicit. This proof is presented in the following, with minor revision. Let a nonsingular matrix Q ¼ ½B : B
ð6:8Þ
where B is an arbitrary matrix which makes Q nonsingular. Then make a similarity transformation on the plant system ðA; B; C Þ: 0
0 0
xðtÞ ¼ Q 1 xðtÞ4½x1 ðtÞ : x 2 ðtÞ
ð6:9Þ
¼
and
ðQ1 AQ; Q1 B; CQÞ4 ¼
A11 A21
: :
A12 I p CB B ; ½CB : C ; A22 0
ð6:10Þ
From (6.9) and (6.10), x_ 2 ðtÞ ¼ A 21 x1 ðtÞ þ A22 x2 ðtÞ
ð6:11aÞ
CB Bx2 ðtÞ yðtÞ ¼ CB x1 ðtÞ þ C
ð6:11bÞ
Because m > p and and Rank Rank ðCBÞ ¼ p , all columns of CB are line linear arly ly independent. Hence we can set a nonsingular matrix P ¼ ½CB : CB
where CB is an arbi arbitr trar ary y ma matr trix ix whi hich ch ma make kess ma matr trix ix P nonsingular. Multiplying P1 on the left-hand side of (6.11b) we have P1 yðtÞ 4 ¼
P1 I yðtÞ ¼ p P2 0
: :
P1 CB P2 CB
x1 ðtÞ x2 ðtÞ
ð6:12Þ
From the first p rows of (6.12), CB Bx2 ðtÞ x1 ðtÞ ¼ P 1 ½yðtÞ C
ð6:13Þ
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Substituting (6.13) and (6.12) into (6.11), we have the following system of order n p x_ 2 ðtÞ ¼ ðA22 A21 P1 CB Þx2 ðtÞ þ A21 P1 yðtÞ
ð6:14aÞ
4A~x2 ðtÞ þ B~yðtÞ ¼
~x ðtÞ yðtÞ4P2 yðtÞ ¼ P 2 CBx2 ðtÞ4C 2
¼
ð6:14bÞ
¼
Because system (6.14) does not involve the original plant system input uðtÞ, its corresponding state observer is an unknown input observer. In addition, if x2 ðtÞ is estimated by this observer, then from (6.13) x1 ðtÞ can also be estimated. Thus the sufficient condition for the existence of unknown input observer of the original plant system is equivalent to the detectability of system (6.14), plus the Rank ðCBÞ ¼ p and m > p conditions which made the system formulation (6.14) possible. Because for the system (6.10),
Rank
2 64
sI p A11
A21 CB
A12 sI n p A22 CB
¼ p þ Rank
:
I p
:
0
:
0
CB
CB
2 64 "
0 P1
sI n p
A~
CB
A21
0
I p
~ C
0
~ sI n p A ~ C
A21 sI n p A22 P1 CB
I n p 0
¼ 2 p þ Rank
ð6:15aÞ
A21 sI n p A22
¼ p þ Rank
¼ p þ Rank
3 75
#
3 75
CB
gn p
"
I p
I n p 0
#!
g p gm p
ð6:15bÞ
A comparison of (6.15a) and (6.15b) shows that the transmission zeros of syst sy stem em (6 (6.1 .10) 0) eq equa uall the the pole poless of unob unobse serv rvab able le part part of sy syst stem em (6.1 (6.14) 4).. Therefore, the necessary and sufficient condition for system (6.14) to be detectable is that all transmission zeros of plant system (6.10) are stable [or that (6.10) is minimum-phase]. Thus the proof.
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Conclusion 6.4 The cond The condit itio ions ns that that a pl plan antt sy syste stem m is mi mini nimu mumm-ph phas asee an and d th that at RankðCBÞ ¼ p are necessary for the plant system to have exact LTR state observer.
Proof The proo The prooff of Co Conc nclu lusi sion on 6. 6.3 3 show showss the the co cond ndit itio ion n that that all all plan plantt sy syste stem m transmission zeros are stable (minimum-phase) is a necessary condition for the existence of unknown input observers. For matrix C ¼ ¼ ½T 0 : C 0 0 be nonsingular, CB must have full-column rank if TB ¼ 0 (see Example A.7). The co The cond ndit itio ions ns of Conc Conclu lusi sion on 6.3 6.3 an and d 6.4 6.4 ar aree su summ mmar ariz ized ed in the the following Tabl Tablee 6. 6.1, 1, whi which ch show showss th that at the the cond condit itio ion n of n m stable transmission zeros is stronger than the condition of minimum-phase and rankðCBÞ ¼ p . Th Thee two two cond condit itio ions ns are are eq equi uiva vale lent nt (b (bot oth h ne nece cess ssar ary y an and d sufficient) for the case m ¼ p , but the former is not a necessary condition, while the latter is if m > p ; and the latter is not a sufficient condition, while the former is if m < p . Thus between the two conditions themselves, the form fo rmer er is a su suffi ffici cien entt cond condit itio ion n of th thee latt latter er,, whil whilee the the latt latter er is on only ly a necessary condition (but not a sufficient condition) of the former. Hence the former condition is even more strict than the latter. This result conforms with the existing properties about transmission zeros [Davison and Wang, 1974]. Table 6.1 also shows that in any case the condition of minimum-phase and RankðCBÞ ¼ p is a necessary condition for the existence of exact LTR state observers. It is difficult to require that all existing existing transmission zeros be
Table 6.1 Nece Necessary ssary and Suffi Sufficien cientt Conditi Conditions ons for the Existen Existence ce of a Dynamic Output Feedback Compensator Which Implements Arbitrarily Given State Feedback Control Conditions
m < p
m ¼ p
m > p
Has n m stable
Necessary and
Necessary and
transmission zeros Minimum-phase and CB full fullcolumn rank
sufficient
sufficient
Necessary
Sufficient
Necessary and sufficient
Necessary and sufficient
Copyright 2004 by Marcel De Dekker, kker, Inc. All Rights Reserved.
st stab able le (s (see ee Exer Exerci cise sess 4.2 4.2 an and d 4.6) 4.6).. In SI SISO SO sy syst stem ems, s, rankðCBÞ ¼ p (or CB 6 ¼ 0) im impl plie iess the the exis existe tenc ncee of n m zer zeros os.. In MI MIMO MO sy syste stems ms,, this this condition is also closely related to the number of zeros [see Davison and Wang, 1974] and is unsatisfied by many practical systems such as airborne systems. Thus the existing result of LTR is very severely limited and is in fact invalid for for most plant systems. From Conclusion 6.1, the new design approach of Algorithms 5.3 and 6.1 requires either the existence of at least one stable transmission zero or m > p . Because almost all plants with m ¼ p have n m transmission zeros [Davison and Wang, 1974], m ¼ p can also be the sufficient condition of (4.1 (4 .1)) and and (4.3 (4.3)) for for most ost case ases (see (see Exerc ercises ises 4.3 4.3 an and d 4.7 4.7). Thus th thee restric res trictio tions ns of min minimu imum-ph m-phase ase and rank rankðCBÞ ¼ p of Co Conc nclu lusi sion on 6.3 6.3 ar aree almost completely elim elimin inat ated ed.. Thus hus our new new desi design gn is va vali lid d fo forr most practical systems. It is also common to have m > p because it is generally much easier to add measurements (or m) to a system than to add controls significant gene generaliza ralization tion of th (or p) to a syst system em.. Thi hiss significant thee criti critica call ro robu bust st control design is possible because the new design approach of this book avoids avo ids the realiz realizatio ation n of separa separatel tely y design designed ed and arb arbitr itrari arily ly giv given en sta state te feedback control.
Example 6.1 This is an example of four plant systems which share a common system matrix pair ðA; C Þ
A ¼
266 66 66 4
x
x
x
:
x x
x x
x x
:
:
...
...
...
x x x
x x x
x x x
...
...
...
x
x
x
...
:
:
: ...
:
1 0 0 ...
0 0 0 ...
0
0 1 0
0 0 1
...
...
0 0 0
0 0 0
...
...
0
0
:
:
: ...
:
:
: ...
:
0 0 0 ...
1 0 0 ...
0
377 77 77 5
and
C ¼ ¼
2 4
1 0 0 x 1 0 x x 1
: : :
0 0 0
0 0 0
0 0 0
: : :
0 0 0
3 5
ð6:16Þ
where ‘‘x’’’s are arbitrary elements. Thus this example is very general.
Copyright 2004 by Marcel De Dekker, kker, Inc. All Rights Reserved.
The matrix pair ðA; C Þ of (6.16) is in observable canonical form (1.16) or (5.8b). The four plant systems are distinguished by their respective B matrices:
B1 ¼
26 66 66 64
1 1 1
0 1 0
.. .:
1 1 1
2
1 1 2
.. .:
.. .:
.. .:
2
37 77 77 75
B2 ¼
26 66 66 64
0 0 1
0 1 0
...
...
1 2 3
0 2 1
...
...
1
1
37 77 77 75
B3 ¼
26 66 66 64
1 1 1
0 0 0
...
...:
1 1 1
1 2 1
...
...:
2 2
37 77 77 75
and
B4 ¼
2 66 66 6 64
1 1 1
0 1 0
.. .:
1 2 2
2
1 1 2
.. .:
.. .:
.. .:
1
3 77 77 7 75
Using the method of Example 1.7 we can derive directly the polynomial matr ma trix ix frac fracti tion on desc descrrip ipti tio on of th thee corr orresp espond ndin ing g tr tran ansf sfeer fu func ncti tion on GðsÞ ¼ D 1 ðsÞN ðsÞ, where polynomial matrix DðsÞ is common for all four systems, and the four different N ðsÞ polynomial matrices are:
N 1 ðsÞ ¼
N 2 ðsÞ ¼
2 642 64
ðs þ 1Þ ðs 2Þ ðs þ 1Þ ðs þ 1Þ ðs þ 1Þ 2
: :
:
:
ðs þ 3Þ
:
:
1
ðs þ 2Þ
1
ðs 2Þ ðs 1Þ
3 75
2
3 75
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N 3 ðsÞ ¼
2 4
ðs þ 1Þ ðs 2Þ ðs þ 1Þ ðs þ 1Þ
:
ðs 2Þ
: :
2 1
3 5
and
N 4 ðsÞ ¼
2 4
ðs 2Þ ðs þ 1Þ ðs 2Þ ðs 2Þ
: :
ðs 1Þ ðs 1Þ
:
2
3 5
The fo The four ur N ðsÞ matr matric ices es reve reveal al that that whil whilee all all fo four ur sy syst stem emss ha have ve m ¼ 3 > 2 ¼ p , only the first and the third syste tem ms have one stable transm tra nsmiss ission ion zer zero o 1, and only the fourth system has an unstable transmission zero 2. Thus the dynamic matrix F of of the four corresponding corresponding dynamic dynamic outpu outputt feedback compensators can be commonly set as F ¼ diag f1; 2; 3; 4g
which whic h incl includ udes es al alll poss possib ible le stab stable le tran transm smiss issio ion n ze zero ross of the the fo four ur plan plantt systems. Because Step 1 of Algorithm 5.3 is based on matrices ð A; C ; F Þ which aree comm ar common on for for th thee four four pl plan antt syst system ems, s, th thee resu result lt of th this is st step ep is also also common for the four systems. The following four basis vector matrices for the four eigenvalues of F are are computed according to (5.10b):
D1 ¼
2 64
1
0
0
:
0 1
0
:
1 0 0
0
1
0
:
1
:
0
3 75
0 D2 ¼
D3 ¼
2 264 64
4
0 1
:
0
0
2 0 0
0
0
:
0 2
0
:
0
1
0
0 2
:
0
0
9
0
0
:
0 3
0
:
0
1
0 3
:
0
0
0
1
:
0
:
1
0
:
0
1
:
0
3 0 0
:
1
0
:
0
1
:
0
3 375 75
Copyright 2004 by Marcel De Dekker, kker, Inc. All Rights Reserved.
and 16 D4 ¼
24
0
0
:
0 4 0 0 0 4
: :
4 0 0
0 0
1 0
0 1
:
1
: :
0 0
35
The result of Step 2 of Algorithm 5.3 (or Algorithm 6.1) is computed according to (6.2a):
T 1 ¼
¼
T 2 ¼
¼
2 6 642 66 4 2 6 642 66 4
½0 1 1D1 ½1 4=5 16=5D2 ½1 5=6 25=6D3
3 7 75
½1 6=7 36=7D4 0 1 1 4 8=5 32=5 9 15=6 75=6 16 24=7 144=7 ½0 1 1D1 ½1 1 1D2 ½1 1 0D3 ½0 1 2D4 0 1 1 0 1 4 2 2 2 1 9 3 0 3 1
3 7 75
0
1
1
0
2 3 4
4=5
16=5
1
5=6
25=6
1
6=7
36=7
1
1 0 1 1 0
1
3 77 5
3 77 5
T 3 ¼
¼
2 66 4 26 64
0 4 8 ½0 1 2D1
½1 0 ½1 0 ½1 0
4D2 5D3 6D4
0
1
2
0
3 77 5
0 1 2 0 4 0 8 2
1 2 0 4
0 1
9
0 15 3
0
5
1
16
0 24 4
0
6
1
37 75
Copyright 2004 by Marcel De Dekker, kker, Inc. All Rights Reserved.
and
½2 1 T 4 ¼
¼
266 4 2 66 4
1D1
½1 1=5 6=5D2 ½1 0 2D3 ½1 1=7 20=7D4 2 1 1 4 2=5 12=5 9 0 6 16 4=7 80=7
377 5
2 1 1 2 1=5 6=5 3 0 2 4 1=7 20=7
2 1 1 1
3 77 5
It can be easily verified that the above four matrices satisfy (4.1) [the right n m ð¼ 4 Þ columns] and (4.3) ðT i i Bi ¼ 0 ; i ¼ ¼ 1 ; . . . ; 4Þ. Because the left m ð¼ 3 Þ columns of (4.1) can always be satisfied by matrix L as shown in (5.16), we consider the above four matrices the exact solution of (4.1) and (4 (4.3 .3). ). Th Thee matr matrix ix trip triple less ðF ; T i i ; Li Þ ði ¼ ¼ 1 ; . . . ; 4Þ full fully y de dete term rmin inee th thee dynamic part (4.10a) of the four dynamic output feedback compensators. This result conforms with Conclusion 6.1 (the first part). Let us now analyze the design of output part (4.10b) of these four dynami dyn amicc out output put feedba feedback ck co compe mpensa nsator tors. s. Be Becau cause se the mat matric rices es C i i ¼ ½T i 0 : 0 C 0 ði ¼ ¼ 1 ; 2Þ are no nons nsin ingu gula lar, r, th thee first first two two co comp mpen ensa sator torss ca can n ge gene nera rate te ¼ 1 ; 2Þ. This result arbitrary and ideal state feedback control K i i ¼ K i C i i ði ¼ conf co nfor orms ms wi with th Conc Conclu lusi sion on 6.3 6.3 and and Tabl Tablee 6.1. 6.1. On th thee ot othe herr ha hand nd,, th thee ¼ 3 ; 4Þ have rank 6 and are singular. Hence only matrices C ii ¼ ½T i 0 : C 0 0 ði ¼ ¼ 3 ; 4Þ can be implemented constrained state feedback control K i i ¼ K i C i i ði ¼ by the last two compensators. This result again conforms to Conclusion 6.4
and Table 6.1 because because the third plant system has rankðCBÞ ¼ 1 < 2 ¼ p and the fourth plant system has a nonminimum-phase zero (2). For the third and fourth plant systems, there exists no other general and systematic design method which can fully use the design freedom to achieve feedback system performance and robustness. However, Algorithms 5.3 and 6.1 have systematically and generally designed the dynamic part of the dynamic output feedback compensator for these two plant systems as follows. ¼ 3 ; 4Þ, we can select six out of the Because rank ðC i i Þ ¼ 6 < 7 ¼ n ; ði ¼ seven rows of C C i i to form a new C i i ði ¼ ¼ 3 ; 4Þ so that Rank ðC i i Þ still equals 6. Suppose we select the first three rows of matrix T i i and all three rows of ¼ 3 ; 4Þ. Then the new dynamic part of the matrix C to form form C i i ði ¼ corresponding dynamic output feedback compensator would be ðF i ; T i i ; Li Þ, ¼ 3 ; 4Þ, and which is formed by the first three rows of original ðF i i ; T i i ; Li Þ ði ¼
Copyright 2004 by Marcel De Dekker, kker, Inc. All Rights Reserved.
the state feedback control gain implemented by these two compensators is
0
0
K 3 ¼ K 3 ½T 3 : C 0 ¼ K 3
26 66 64
0 1 2 0 4 0 8 2 9 0 15 3
1 2 0 4 0 5
0 1 1
1 0 x 1 x x
0 0 0
0 0 0
0 0 1
0 0 0
0 0 0
37 77 75
and 0
K 4 ¼ K 4 ½T 4 : C 0 0 2 1
¼ K 4
2 66 66 66 4
4 9 1
1 2 1 1 2 2=5 12=5 2 1=5 6=5 1 0 6 3 0 2 1 0
0
0
0
0
0
x 1
0
0
0
0
0
1
0
0
0
0
x
x
3 77 77 77 5
respectively. These two state feedback controls are equivalent to a static output feedback control with six independent outputs. Therefore, they are
not mu not much ch weak weaker er th than an the the id idea eall stat statee fe feed edba back ck co cont ntro roll an and d ar aree mu much ch strong str onger er than than the ordina ordinary ry sta static tic output output feedba feedback ck con contro trol, l, whi which ch cor correrespon sp onds ds to only only thre threee outpu tputs. ts. For exam exampl plee, th theese tw two o co cont ntro rols ls can can arbitr arb itrari arily ly ass assign ign the eigenv eigenvalu alues es of the corre correspo spondi nding ng fee feedba dback ck sys system tem matrix A BK i C i i ði ¼ ¼ 3 ; 4Þ becau because se 6 þ p ¼ 6 þ 2 ¼ 8 > 7 ¼ n [Kimura, 1975], 197 5], while while the ord ordina inary ry static static output output feedba feedback ck con contro troll can cannot not bec becaus ausee 3 þ p ¼ 3 þ 2 ¼ 5 < 7 ¼ n . More important, all four compensators guarantee that the feedback system poles be the union of f1; 2; 3g and and the eig eigenv envalu alues es of A BK ii ði ¼ ¼ 1 ; . . . ; 4Þ (Theorem 4.1), and guarantee that the feedback system 1 ¼ 1 ; . . . ; 4Þ (Theorem 3.4). loop transfer function equals K i i ðsI AÞ Bði ¼ Thiss result Thi result certai certainly nly canno cannott be achiev achieved ed sys system tematic aticall ally y by oth other er exi existin sting g design desig n methods methods for the third and fourth plant systems, systems, espec especially ially the fourth fourth,, which is nonminimum-phase.
Copyright 2004 by Marcel De Dekker, kker, Inc. All Rights Reserved.
Example Exa mple 6.2
Let
A
The C Case ase When When E Eige igenva nvalues lues o off F ar are e Comple Complex x Conjugate
:
C
¼
:
2 66 66 64
B
0
1:0048 0:0068 0:1704 18:178
7:7779
0:8914
:
39:611
10:784
0
:
0
1
0
0
0
:
0
0
0
0
0
:
1
1
0
0
0
:
0
0
1
0
0
:
0
3 77 77 75
This is the state space model of a combustion engine system [Liubakka, 1987]. Its four states are manifold pressure, engine rotation speed, manifold pressure (previous rotation), and throttle position, respectively. Its control input is the throttle position (next rotation) and its two output measurements are manifold pressure and engine rotation speed, respectively. Apply the operation of Steps 2 and 3 of Algorithm 5.2 ð j ¼ 2 Þ, where the operator matrix
2
3
H ¼
64
1 0 0 0
0 0 0 1 0 0 0 0:0093735 0:999956 0 0:99996 0:0093735
75
is dete determ rmin ined ed by the the el elem emen ents ts ½ 0:1704 18:178 of ma matr trix ix A. The resulting block-observable Hessenberg form system matrices are 0
H AH CH
: :
0
H B 0
Copyright 2004 by Marcel De Dekker kker Inc All Rights Reserved Design for Robustness Realization
¼
2 66 66 64
1:0048 0:0068
7:7779
18:1788
0:8914 0:1011
0:00937
0
:
10:7835
:
39:6111
0
1 0:0093735
0 0
0 0
:
1
0 0
:
1
0
0
0
:
0
0
1
0
0
:
0
3 77 77 75
Because this system does not have any stable transmission zeros, we arbitrarily select matrix 1 1 F ¼ 1 1
with eigenvalues 1+ j . Substituting matrices H 0 AH and F into into (5.13b) of Step 1, Algorithm 5.3, we have
0
½D1 : D2 I 2 6 H AH 0:05501
0
I 2
0
0
F
6
1
0
I 2
0
:
0:05501
0
0
0
¼
266 64
0:0005157 0:092734 0 1
:
0:0005157 0:092734 0 0 0:05501 0 1 0 0:0005157 0:092734 0 0
0:05501
0
0
0
:
0:0005157
0:092734
0
0
:
2 66 66 66 4
18:1788
0
:
0
0
0:1011 10:7835
:
0
0
1
0
:
1
0
0
1
:
0
1
6
0
0
:
18:1788
0
0
:
0:1011 10:7835
1
0
:
1
0
1
:
0
1
0
0
3 77 77 77 5
377 75
¼ 0
Copyright 2004 by Marcel De Dekker, kker, Inc. All Rights Reserved.
Substituting ½D1 : D2 in (6.2b) of Algorithm 6.1, we have
2 66 64
3:179 0:029801
: :
2:179 0:02043
½0:0093735 1 0 0 c
2:179 0:02043 D1 B
:
:
3:179 0:029801 D2 B ¼ 0
3 77 75
¼ 0
Thus the result of Step 2 of Algorithm 5.3 is
T ¼
0 0:092734 0:0093735 cD1 ¼ cD2 0 0:92734 0
1 0
This matrix corresponds to system matrix ðH 0 AH ; H 0 B; CH Þ. Hence it must be adjusted to correspond to the original system matrix ðA; B; C Þ (see the beginning of Sec. 5.2):
T ¼ TH 0 ¼
0 0:92734
1
0
0 0:92734
0
0
Substituting this T into into (5.16) of Step 3, Algorithm 5.3, we have
L ¼ ðTA FT Þ
I 2 1:721276 0:082663 ¼ 0 0:721276 0:26813
It can be verified that ðF ; T ; LÞ sati satisfi sfies es (4 (4.1 .1)) an and d (4.3 (4.3), ), bu butt th thee ma matr trix ix 0 0 0 C ¼ ¼ ½T : C is singular. This is because the system has a nonminimumphasee ze phas zero ro (0.4 (0.458 589) 9).. None Noneth thel eles ess, s, matr matrix ix C has has one mo more re li lin nea earl rly y indepe ind epende ndent nt row tha than n the origin original al matrix matrix C . Hence with the guaranteed robustness realization [by (4.3)], the compensator F ; T ; LÞ of (4.10) realizes a stronger state feedback control KC xðtÞ than K yy C xðtÞ of the ordinary static output feedback control. In addition, Example 7.3 of of Chap. 7 provides a numerical example abou ab outt th thee mu mult ltip iple le ei eige genv nval alue ue case case of Algo Algori rith thm m 5.3. 5.3. Th Thus us co comp mple lete te eigenvalue cases have been shown by numerical examples in this book.
Copyright 2004 by Marcel De Dekker, kker, Inc. All Rights Reserved.
Example Exa mple 6.3
A Cas Case e wit with h App Approxi roxima mate te Sol Solutio ution n of (4.1) (4.1) and and (4.3)
Let the system matrix be
ðA; B; C Þ ¼
02 B@64
x x x x
x x x x
1 0 0 0
32 75 64
0 1 ; 0 0
1 3 1 2 2 6 1 2
3 75 ;
1 0
0 1
0 0
0 0
1 CA
where ‘‘x’’’s are arbitrary entries. Because this system has the same number of inpu inputs ts an and d outp output utss ðm ¼ p Þ and satis satisfie fiess ra rank nk ðCBÞ ¼ p ¼ 2, it has n m ¼ 4 2 ¼ 2 transmission zeros. Because this system is in observable canonical form, using the procedure of Examples 1.7, matrix N ðsÞ of the poly po lyno nomia miall matri matrix x frac fracti tion on desc descrip ripti tion on of th thee co corr rres espo pond ndin ing g tr tran ansfe sferr function GðsÞ ¼ D 1 ðsÞN ðsÞ can be directly derived as
s þ 2 N ðsÞ ¼ s 1
3ðs þ 2Þ 2ðs 1Þ
Thus this system has two ð¼ n mÞ transmission zeros (2 and 1) and is nonminimum-phase. Let us set F ¼ diag f2; 1g, where 2 matches the stable transmis ðA; B; C Þ and 1 is arbitrarily chosen. Solving (5.10b), we have sion zero of ð D1 ¼
2
0 0 2
1 0
0 1
and
D2 ¼
1
0 0 1
1 0
0 1
Substituting this result into (6.2a), c1 D1 B ¼ c1
c2 D2 B ¼ c2
0 0 ¼ 0 3 6
and
1 3 ¼ 0 2 4
ð6:17Þ
imply that c1 ¼ ½ x 0 ðx=0Þ, while the exact solution of c2 does not exist. This is because the corresponding transmission zero ð2Þ of c1 is matched F , while the transmission zero (1) corresponding to c 2 is by the eigenvalue of F
Copyright 2004 by Marcel De Dekker, kker, Inc. All Rights Reserved.
not. This example shows that having n m stable transmission zeros is a mampl necessary plant system with have exact solution ¼ p Þe toalso to (4 (4.1 .1), ), condition (4.3 (4.3)) and and for nons noa nsin ingu gula lar r C . T his ðex exam ple ls o co conf nfo orms rm s wit ith h Conclusion 6.1 (the second part). To minimize c2 D2 B of (6.17) in a least-square sense, we use (6.3) of Algorithm 6.1 such that c2 ¼ u02 ¼ ½ 0:8174
0:576 0
Here u2 is the normalized right eigenvector of matrix ½D2 B½D2 B and its smallest eigenvalue s22 ¼ 0 :13393. In other words, s2 ¼ 0 :366 is the smallest 2 B and u2 is the second column of unitary matrix eDdeco singular of matrix U of th theevalue si sing ngul ular ar valu value de comp mpos ositi ition on of D2 B. It can can be verifi erified ed th that at kc2 D2 Bk ¼ s2 which is the least-square residual of (6.17) (see Example A.6). The above result provides us with two possible feedback compensators, whose dynamic part (3.16a) will be, respectively,
ðF 1 ; T 1 Þ ¼ ð2; ½2c1 : c1 Þ
and
ðF 2 ; T 2 Þ ¼ 2
0 ; 2c1 0 1 0:8174 0:576
: :
c1 0:8174 0:576
Of th thes esee two two poss possib ible le co comp mpen ensa sato tors rs,, th thee fir first st is a dy dyna namic mic outp output ut feedback compensator (4.10) because it satisfies TB ¼ 0, while the second does not satisfy TB ¼ 0 and hence is an observer (3.16) only. Therefore, the first fir st comp compen ensa sato torr gu guar aran ante tees es that that the the fe feed edba back ck sy syst stem em loop loop tr tran ansf sfer er 0 1 function equals K 1 ½T 10 : C 0 ðsI AÞ B for whatever K 1 , while the second compensator does not (for its corresponding freely designed K 2 ) (Theorem 3.4), even though the least-square gain TB of (6.17) is used in this observer. On th thee ot othe herr hand hand,, the the first first co comp ensa sato torr ca can n im impl plem emen only ly a 0 entt on 0 mpen constrained state feedback K 1 ½T 10 : C 0 because Rank ½T 10 : C 0 ¼ 3 < 4 ¼ n , even ev en th thou ough gh arbi arbitr trar ary y ei eige genv nval alue uess ca can n st stil illl be as assi sign gned ed to th thee ma matr trix ix 0 A BK 1 ½T 10 : C 0 be beca caus usee 3 þ p ¼ 3 þ 2 ¼ 5 > 4 ¼ n , while the second 0 compen com pensat sator or can imp implem lement ent arbitra arbitrary ry sta state te fee feedb dback ack K 2 ½T 20 : C 0 because 0 the matrix ½T 20 : C 0 is nonsingular. We recall for nonminimum-phase plant systems, such as the one in this example, that there is no other design method which can systematically and analytically derive as strong a result as these two compensators.
Copyright 2004 by Marcel De Dekker, kker, Inc. All Rights Reserved.
6.3
CO COMPL MPLETE ETE U UNIF NIFICA ICATIO TION N OF TWO TWO EXISTI EXISTING NG BASIC BASIC
MODERN CONTROL SYSTEM STRUCTURES Besides general robustness realization, the new design approach of this book has another major theoretical significance. That is the complete unification of two existing basic control structures of modern control theory. These two basic structures are the exact LTR state observer feedback system and the static output feedback system. State observer and static output feedback have been the main control structures of modern control theory for years, butt no atte bu attemp mptt has has be been en made made to unif unify y th thes esee seem seemin ingl gly y ve very ry diff differ eren entt structures. Thee ne Th new w co cont ntro roll stru structu cture re de desi sign gned ed in th this is ch chap apte ter— r—th thee dy dyna nami micc output out put fee feedba dback ck contro controlle llerr whi which ch can implem implement ent sta state/ te/gen genera eralize lized d sta state te fe feed edba back ck cont contro rol, l, can can comp comple lete tely ly unif unify y th thee ab abov ovee tw two o exis existin ting g co cont ntro roll structures as its two extreme cases. This unification can be shown in Fig. 6.1 6.1,, and the properties of these three structures can be summarized in Table in Table 6.2 6.2.. Table 6.2 shows clearly that the new control structure of this book [structure (b)] completely unifies in all aspects the existing two basic control
structures of (a) and (c). The common feature which makes this unification uniquely possible is the realization of state feedback control [K xðtÞ; K is a 1 constant] and its robustness properties ðLðsÞ ¼ K ðsI AÞ BÞ. Tabl Ta blee 6. 6.2 2 show showss th that at co cont ntro roll stru struct ctur uree (a) (a) ex exer erci cise sess th thee st stro rong nges estt control but is least generally designed, while control structure (c) exercises the weakest control but is general to all plant systems. The table also shows that control structure (b) completely unifies these two extreme properties. A direct consequence of this unification is that the design of the output part pa rt of dy dyna nami micc outp output ut feed feedba back ck comp compen ensa sato torr K ¼ KC is dire direct ctly ly
Fi Figu gure re 6. 6.1 1 Th Thre ree e mo mode dern rn cont contro roll stru struct ctur ures es ca capa pabl ble e of re real aliz izin ing g st stat ate/ e/ generalized state feedback control and their their robustness properties.
Copyright 2004 by Marcel De Dekker, kker, Inc. All Rights Reserved.
Table 6.2 Thre Three e Contro Controll Systems of Moder Modern n Control Theo Theory ry Control structure
(a)
(b)
Controller order r
n m
n m5r 50
½T 0 : C 0 0
½T 0 : C 0 0
C
n
n 5r þ m5m
m
0
Matrix C ¼ ½T 0 : C 0 Rank ðC Þ ¼ q ¼ r þ m State feedback gain K ¼ KC
Dynamic matrix Loop transfer function
Arbitrary K (C nonsingular)
A BK
Arbitrary to severely constrained K ¼ KC A B KC
(c)
0
Severely constrained K ¼ K yy C A BK yy C
K ðsI AÞ1 B KC ðsI AÞ1 B K yy C ðsI AÞ1 B
Generality (conditions on plant system)
m stable n transmission zeros or minimum phase, rank ðCB Þ ¼ p , and m5p
At least one stable transmission zero or m > p
None
q ¼ n ) and the existing compatible with the existing state feedback design (if q static output feedback design (if q < n ). This design will be described in 8 and Chaps 8 and 9.
6.4
OBSER OBSERVER VER O ORDE RDER R ADJ ADJUST USTME MENT NT TO TRAD TRADEO EOFF FF BETWEEN PERFORMANCE AND ROBUSTNESS [Tsui, 1999c]
One of the main and unique features of the observers based on the result of Algorithm 6.3, is that the observer order r is completely flexible. On the cont co ntra rary ry,, th thee ex exis isti ting ng ob obse serv rver er orde orders rs ar aree fix fixed ed.. Fo Forr ex exam ampl ple, e, the the st stat atee observer orders are fixed to be either n or n m, and the order of a static output feedback controller is 0. Also because of this unique feature, our observer compensator can comp co mple lete tely ly unif unify y exac exactt LTR LTR stat statee obse observ rver er an and d st stati aticc ou outp tput ut fe feed edba back ck control, as described clearly in Sec. 6.3.
Copyright 2004 by Marcel De Dekker, kker, Inc. All Rights Reserved.
The reason behind unique isled. that thes dynamic part ofbled, our observer observ er compen com pensat sator or isthis comple com pletel tely yfeature decoup decoupled . Thi This is fur furthe ther r ena enable d, uniquely, by the Jordan form of matrix F in (4.1) and in Algorithm 5.3, and by the design concept that a nonsingular matrix C in (4.2) is unnecessary (see Chap. (see 4). Chap. 4).
Example 6.4 Example 6.1 (the third and the fourth compensators) and Example 6.3 (the first compensator) all show that when matrix F is is in Jordan form and when a nonsingular matrix C is no longer required, the compensator order can be freely adjusted. More specifically, the third and the fourth compensators of Example 6.1 have order r ¼ 3 while n m ¼ 4, and the first compensator of Example 6.3 has order r ¼ 1 while n m ¼ 2.
This se This sect ctio ion n deal dealss with with the the ac actu tual al dete determ rmin inati ation on of this this ob obse serv rver er compensator order r . Our determination is based on the following two basic and clear understandings. Thee fir Th first st un unde ders rsta tand ndin ing g is ba base sed d on the the fo form rmul ulat atio ion n (4.2 (4.2)) of our our control K ¼ KC , where ere C is formed by the rows of matric rices C of (1.1b) (1. 1b) and T of (4 (4.1 .1)) and and (4.3 (4.3). ). Equa Equati tion on (4 (4.2 .2)) is a co cons nstr trai aint nt on the the state sta te fee feedba dback ck gain gain K (see (see Su Subs bsec ectio tion n 3.2. 3.2.2) 2).. Th Ther eref efor ore, e, the the high higher er the the observer obse rver order r (whi (which ch equa equals ls the the row row rank rank of ma matr trix ix T )),, the the high higher er thee ro th row w ran ank k ðr þ mÞ of matr matrix ix C , the the les less th thee const onstra rain intt on K (see Appe Ap pend ndix ix A. A.1) 1),, and the the more powerfu erfull th thee co corr rres espo pon ndin ding co con ntr trol ol K xðtÞ. The second understanding is based on Eq. (4.3) ð TB ¼ 0 Þ, which is the key condition for realizing the loop transfer function/robustness properties of our control. Because B is given, the smaller the row rank r of matrix T , the easier to satisfy (4.3) (see Appendix A.1). In addition to these two simple and basic understandings, our observer order determination is further based on the following two obvious system design principles. The first system design principle is that the system must be stable. Therefore, based on the first of the above two basic understandings, the order r has to be high enough so that the corresponding matrix A BKC is stabilizable. Stabilization, which only requires all eigenvalues of matrix A BKC be in the stable region rather than in exact locations, is substantially easier than arbitrary eigenvalue assignment of matrix A BKC (see Subsection 8.1.4). Now because rank ðC Þ6 p > n is generically sufficient for arbitrary
Copyright 2004 by Marcel De Dekker, kker, Inc. All Rights Reserved.
eigenvalue assignment of A BKC [Wang, 1996], this condition should be sufficient for the stabilization of A A BKC . Therefore, we should have a high enough observer order r such that n p
ðr þ mÞ6 p > n or r > m
ð6:18Þ
This should be the lower bound of observer order r. The second system design principle is that the effectiveness (especially TB =0. Therefore, the robustness property) of control K xðtÞ is totally lost if TB based on the second of the above two basic understandings, the observer order r should be low enough so that TB can be sufficiently minimized. Based on Conclusion 6.1 and the second column of Table Table 6.2, if the
open-loo openloop p sys system tem ðA; B; C Þ has has eith either er m > p or or at least one stable transmission zero, than TB ¼ 0 can be fully satisfied [in addition to (4.1)]. Then from the first of the above two basic understandings, we should have thee hi th high ghes estt po poss ssib ible le obse observ rver er orde orderr r, say r0 , whil whilee keep keepin ing g TB ¼ 0 satisfied.
Definition 6.1 0
0
0 0
¼ maximal possible rank ðC 4 ¼ ½T : C Þ m where matrix T satisfies satisfies Let r 4 (4.1) and (4.3).
From Conclusion 6.3 and its proof, r0 equals the number of stable transmission zeros of system ðA; B; C Þ if m 4 p. m > p ? It differs from system to What is the value of rr 0 for the cases of m system, depends on parameters such as rank( CB) and the numbers of system sta tab ble and and unsta nstabl blee tra transm nsmissi issio on ze zero ross (e (eve ven n th tho ough ugh su such ch sys yste tems ms generi gen erical cally ly do no nott have have tra transm nsmiss ission ion zer zeros os [Da [Davis vison on and Wan Wang, g, 197 1974]) 4]),, and ranges between 0 and n m. There is no simple and general formula for r0 directly from the parameters of system ð A; B; C Þ. Fortunately, Case B of Algo Al gori rith thm m 6. 6.1 1 guar guaran antee teess th thee simp simple le an and d dire direct ct comp computa utati tion on of r0 , as convincingly argued by Conclusion 6.2. There is another way to compute the value of r0 and it computes r0 before the computation of the solution of (4.1) and (4.3). This computation is base based d on a spec specia iall simi simila lari rity ty tran transf sfor orma mati tion on on th thee sy syst stem em ðA; B; C Þ called cal led the ‘‘s ‘‘spec pecial ial coordi coordinat natee bas basis is (s.o.b (s.o.b.)’ .)’’’ [Saber [Saberii et al., al., 199 1993]. 3]. In the s.o.b., the system is decoupled into five parts with five dimensions such as the number of system stable transmission zeros and the number of system unstable transmission zeros, etc. The value of r0 can be determined easily from these five dimensions, because the state observers of some of these
Copyright 2004 by Marcel De Dekker, kker, Inc. All Rights Reserved.
decoupled system parts of s.o.b. satisfy automatically TB ¼ 0. However, it is obvious and it is accepted that the computation of this s.o.b. itself is very difficu dif ficult lt and ill conditi condition oned ed [Chu, [Chu, 200 2000], 0], even even tho though ugh num numeri erical cally ly mor moree stable algorithm of computing this s.o.b. is presented in Chu [2000]. In ad addi diti tion on to the the il illl cond condit itio ion n of the the comp comput utat atio ion n of s.o. s.o.b. b.,, th thee corr co rres espo pond ndin ing g state state obse observ rver er of s.o. s.o.b. b. has has or orde derr fix fixed ed at r0 an and d is not adjustable at all. Then what if this r0 cannot satisfy (6.18) or (6.19) (if a higher design requirement is imposed), or what if this r0 is too high to be re real aliz ized ed?? Thes Thesee prob proble lems ms cann cannot ot be even even disc discus usse sed d ba base sed d on the the st stat atee observers since the state observer order is fixed. 0
r guaranteed. of (6.1 (6.18) 8),, then thBecause en (4 (4.1 .1), ), (4.1) (4 (4.3 .3))implies an and d th the e st stab abil iliza izatio tion n of system ma matri trix x If r 5 A BKC are that the feedback
KC poless are pole are co comp mpos osed ed of the the ei eige genv nval alue uess of matr matric ices es F and A B BKC (The (T heor orem em 4. 4.1) 1),, and and (4 (4.3 .3)) impl implie iess an outp output ut fe feed edba back ck co comp mpen ensa sato torr [see [see (4 (4.1 .10) 0)], ], a solu soluti tion on to the the stro strong ng stab stabil iliza izati tion on pr prob oble lem m is auto automa mati tica call lly y deri de rive ved d by our our de desi sign gn.. The The stro strong ng stab stabil iliz izat atio ion n pr prob oble lem m is defin defined ed as stabilizing the feedback system [say matrix A BKC ] by a stable output feedback compensator [Youla et al., 1974 and Vidyasagar, 1985]. In pra practic cticee a contro controll system system design design tha thatt req requir uires es adv advanc anced ed contro controll theory usually deserve both high performance and robustness, in addition to st stab abil ility ity only only.. Ther Theref efor oree th thee cont contro roll KC xðtÞ shou should ld be ab able le to as asssign ign
arbitrary arbitr ary eigenv eigenvalu alues es and at lea least st some some eig eigenv envect ectors ors.. Fo Fortu rtunat nately ely,, suc such h desi de sign gn al algo gorit rithm hm is pres presen ente ted d in Su Subs bsec ecti tion on 8.1. 8.1.3, 3, an and d is ex exec ecut utab able le if rankðC Þ þ p > n . Therefore in such designs, it is required that at least
ðr þ mÞ þ p > n or r > n p m
ð6:19Þ
is satisfied. It is prov proven en main mainly ly by the the exer exerci cise sess of Ch Chap ap.. 4, an and d pa part rtia iall lly y by Exercises 8.6 and 8.7, that (6.19) can be satisfied by most open-loop systems, and that (6.18) can be satisfied by a great majority of the open-loop systems. Comp Co mpar arin ing g the the stati staticc outp output ut feed feedba back ck co cont ntro rols ls wh wher eree r ¼ 0 (see Table 6.2 and Subsection 3.2.2), (6.18) and (6.19) cannot be satisfied as soon as m6 p4n and m þ p4n, respectively (see for Example 6.3 and Exercises 6.7 and 8.6). In case the desired value of rr of (6.19) or even (6.18) is higher than the value of r0 (which guarantees TB ¼ 0), the remaining r r0 rows of T [or their corre correspon sponding ding ci vectors of (6.1)] should be computed to make the 0 C 0 0 T full row rank instead of making TB (or : corresponding matrix C 4 ½ ¼ ¼ r 0 þ 1 to rÞ ¼ 0. Nonetheless, these r r0 rows of T T should should still be ci Di B; i ¼ selected out of the n m r0 rows of T T and and should still be computed, so that
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¼ r 0 þ 1 to r) has th the pr the prod oduc uctt TB (or kci Di Bk; i ¼ thee sma mall lles estt poss ossible ible magnitude.
EXERCISES 6.1 Ve Verif rify y the com comput putati ation on of Algori Algorithm thm 6.1 to sat satisf isfy y (4. (4.3) 3) for the fou fourr systems of Example 6.1. Verify fy the comput computation ation of Algorithm Algorithm 6.2 to satisfy (4. (4.3) 3) for Exampl Examplee 6.2 Veri
6.2. 6.3 Veri Verify fy the comput computation ation of Algorithm Algorithm 6.3 to satisfy (4. (4.3) 3) for Exampl Examplee 6.3. ¼ 1 ; . . . ; 4Þ are all the same as 6.4 Suppose matrices A , C , and D i ðand li ; i ¼ that th at of Ex Exam ampl plee 7.3. 7.3. Le Lett the the matr matrix ix B be gen genera erally lly giv given. en. Repeat Repeat Algorithm 6.1. (a)
Let c1 ¼ ½1; c1 ; c2 . Compute c1 such that c1 D1 B ¼ 0. Answer :
½ c1 c2 ¼ ½ ½ 2 0 1 1 0 0 1 B6
(b)
1 1 1
1
1
0
0
0
0
0
1
0
0
0
1
B
Let c2 ¼ ½c1 ; 1; c2 . Compute c2 such that c2 D2 B ¼ 0. Answer :
½ c1 c2 ¼ ½ ½ 1 2 1 1 1 0 0 B6 1
(c)
1 1 1
0 1 1
0 0
0 0
0 1
1 B 0
Let c3 ¼ ½c1 ; c2 ; 1. Compute c3 such that c3 D3 B ¼ 0. Answer :
½ c1 c2 ¼ ½ 2 2 2 0 0 1 0 B6 1
2 2 1 1 0 0 1 B 3 3 1 1 1 0 0
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(d)
Let c4 ¼ ½1; c1 ; c2 . Compute c4 such that c4 D4 B ¼ 0. Answer :
½ c1 c2 ¼ ½ 6 1 2 2 0 0 1 B6
3
0 1
1 1
1
1
1
0
0
0
0
1
0
1
B
Of course the ci vectors do not need to (and some times cannot) be fixed at the above forms. 6.5 Change matrix B of the system of Example 6.1 to
1 0
1 0
1 0
: :
3 1 1 1
2
:
1
:
2 2
0
so th tha at Rank ankðCBÞ ¼ 1 ¼ p 1 and the system has one unstable transmission zero 1. What is the value r0 of this system? Answer: r0 ¼ 2 ¼ n m 2. 6.6 Change matrix B of the system of Example 6.1 to
1
1
1
:
0
0
0
:
3 1 1 1
0
0
:
2
:
2
0
so th that at Ran RankðCBÞ ¼ 1 ¼ p 1 and the system has two unstable transmission zeros 1 and 2. What is the value r0 of this system? Answer: r0 ¼ 1 ¼ n m 3. 6.7 Repe Repeat at Exam Example ple 6.3 for for a similar similar system
ðA; B; C Þ ¼
02 BB66 @4
x
x 1
0
x
x 0
1
x
x 0
0
x
x 0
0
32 77 66 54 ;
1
2
1
3
3
6
1 3
3 77 5 ;
1
0
0
0
0
1
0
0
1 CC A
Instead of having 2 and 1 as transmission zeros of Example 6.3, this new system has 3 and 1 as transmission zeros. 6.8 In si sing ngle le-in -inpu putt and and sing single le-o -out utpu putt syste systems ms GðsÞ ¼ D 1 ðsÞN ðsÞ, the cond co ndit itio ion n ra rank nkðCBÞ ¼ p (or CB 6 ¼ 0) im impl plie iess N ðsÞ has has or orde derr n m ¼ n 1. Thus Thus the the SISO SISO syst system emss ha have ve ge gene neri rica call lly y n 1 ze zero ross
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[Davis [Dav ison on and and Wang Wang,, 1974 1974]. ]. Us Usin ing g th thee resu result lt of Exam Exampl plee 1.7 1.7 an and d Exercises 1.3 to 1.6, repeat the above analysis on how the condition rankðCBÞ ¼ p will imply to the number of MIMO system transmission zeros.
Copyright 2004 by Marcel De Dekker, kker, Inc. All Rights Reserved.
7 Observer Design for Minimized Order
As stated at the beginning of Chap. Chap. 6, 6, Step 2 of Algorithm 5.3 revealed the remaining freedom of (4.1). The first application of using this freedom is to realize the robustness properties of state feedback control, and is presented in Chap. 6. The second application of using this freedom is to minimize the observer order, and is presented in this chapter. The objectives of these two applications are very different. Like the failure to realize the robustness properties of state feedback cont co ntro rol, l, hi high gh ob obse serv rver er orde orderr ha hass al also so been been a ma majo jorr dr draw awba back ck th that at has has limited the practical application of state space control theory. Lower order observers not only are much easier to realize, but also have generally much smoo sm ooth ther er co corr rres espo pond ndin ing g resp respon onse se.. Like Like in Ch Chap ap.. 6, th this is ch chap apte terr wi will ll
Copyright 2004 by Marcel De Dekker, kker, Inc. All Rights Reserved.
demonstrate that with the full use of the remaining freedom of (4.1), this drawback can be effectively overcome. Howe Ho weve ver, r, unli unlike ke th thee desi design gn of Chap Chap.. 6 wh which ich de dete term rmin ines es on only ly the the dyna dy nami micc part part of th thee obse observ rver er and and whic which h resu results lts in an ou outp tput ut fe feed edba back ck
compensator (4.10), the design of this chapter will completely determine the whole observer which cannot qualify as an output feedback compensator. The design of this chapter is also based on the unique feature of the solution of (4.1) of Algorithm 5.3, that the rows of this solution ðF ; T ; LÞ are completely decoupled. Thus the number of rows of this solution can be dete de term rmin ined ed fr free eely ly.. Fr From om the the ob obse serv rver er de defin finit itio ion n of (3.1 (3.16) 6),, th this is nu numb mber er equals the observer order r (see also Sec. 6.4 for the determination of r, but for a purpose totally different from a minimized r). Section 7.1 describes the design formulation of this problem, which is claimed in Sec. 7.3 to be far simpler and the simplest possible general design formulation of this problem. Sect Se ctio ion n 7.2 7.2 pres presen ents ts the the simp simple le and and sy syst stem emat atic ic desi design gn algo algori rith thm m (Algorithm 7.1) based on this formulation, and analyzes the general upper and lower bounds of r which is computed by this algorithm. Section 7.3 proves that the general observer order bounds of Sec. 7.2 are far lower than the existing ones, are the lowest possible general bounds, and are lower enough to be practically significant even at the computer age. Seve Se vera rall exam exampl ples es are are pres presen ente ted d to de demo mons nstr trat atee th this is sign signifi ifica canc ncee an and d Algorithm 7.1. 7.1
DES DESIGN IGN F FORM ORMULA ULATIO TION N [Ts [Tsui, ui, 1 198 985, 5, 19 1993a 93a]]
As described in Example 4.3, minimal order observer design fully uses the remaining freedom of (4.1) to satisfy (4.2) [but not (4.3)] with arbitrarily given K , wi with th arbi arbitr trar aril ily y give given n obse observ rver er pole poless fo forr gu guar aran ante teed ed ra rate te of observation convergence, and with a minimal value of r. As reviewed in Example 4.3, minimal order observer design has been attempted for years since 1970 [Gopinath, 1971; Fortmann and Williamson, 1972; Kaileth, 1980, p. 527; Gupta et al., 1981; O’Reilly, 1983; Chen, 1984, p. 371; 371; Van Van Door Dooren en,, 1984 1984;; Fowe Fowell ll et al., al., 19 1986 86]. ]. Bu Butt none none has has us used ed th thee solution of (4.1) of Algorithm 5.3. This solution is uniquely decoupled and shows completely and explicitly the remaining freedom of (4.1) (see Sec. 5.2 and the beginning of Chaps Chaps 6 6 and and 7) 7).. Thus only based on this solution of (4.1), can the minimal order observer design problem be simplified to the solving of (4.2) only and therefore really systematically. As reviewed in Subsection 3.2.3, only Eq. (4.2) reveals the difference between different types of observers, suchofasDefinition the state observers Examples 4.1 and 4.2 vs. the function observers 4.1, andof such as the
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¼ 0 Þ. The following strictly proper type ðK y ¼ 0 Þ vs. the proper type ðK y 6 design formulation (7.1c) and the corresponding design algorithm (Algorithm 7.1) are for proper type observers. However they can be very easily
adapted to solve the strictly proper type observer problems. Based on the block-observable Hessenberg form of ðA; C Þ, Eq. (4.2), like (4.1), can be partitioned into its left m columns:
K
I m
¼ ½K Z Z : K y y
T I m C
0
K y C 1
ð7:1aÞ
0
0
and its right n m columns:
K
I m
þ
¼ K Z Z T
0 0 ¼ K Z 4 K 4 K Z Z T Z T I nm ¼ I nm ¼
ð7:1bÞ
Because rank ð C 1 Þ ¼ m and K y is completely free in (7.1a), only (7.1b) need to be satisfied. To simplify the problem, problem, we assum assumee that all observer observer poles are distinct distinct and real. Substituting the result of (6.1a) of Algorithm 5.3 (Step 1) into (7.1b), we have
2 664
¼ K Z K Z T ¼K Z Z
c1
..
. cr
m ... m
32 775664
1 D .. . r D
3 775
ð7:1cÞ
n m
; T ; D , are the right n m columns of K K ; T ; D ði ¼ ¼ 1 ; . . . ; rÞ of (4.2) where K i i and (6.1a), respectively, and r equals the number of rows of matrix T or the corresponding minimal observer order. Thee un Th unkn know own n solu soluti tion on of (7 (7.1 .1c) c) is K Z ¼ 1 ; . . . ; rÞ, wh wher eree Z and ci ði ¼ parameter K Z Z represents the design freedom of observer output part while ¼ 1; . . . ; rÞ represen represents ts the remaining freedom of obse observer rver dynamic part. ci ði ¼ The parameters ci can also be considered the remaining freedom of (4.1), or the freedom of observer eigenvector assignment because F is is in Jordan form. Hence the observer design freedom is fully used in (7.1c). i of (7.1c) are completely decoupled In addition, the given row blocks D for all i becaus becausee they they are bas basis is vector vector matrice matricess of observ observer er eig eigenv envect ectors ors..
Hence unlike any other existing minimal order observer design formulations, (7.1c) is truly very similar to a set of linear equations.
Copyright 2004 by Marcel De Dekker, kker, Inc. All Rights Reserved.
As a result, for the first time, (7.1c) can be solved systematically by
matrix triangularization operations from the right side of the given matrix of (7.1c), and by back substitution (see Appendix A, Sec. A.2).
7. 7.2 2
DESI DESIGN GN ALGOR ALGORIT ITHM HM AN AND D ITS AN ANAL ALYS YSIS IS
Thee follow Th following ing desig design n algori algorithm thm sol solves ves (7. (7.1c) 1c) by mat matrix rix tri triang angula ulariz rizati ation on operations the right side of each the given equation of (7.1c), and by back substitutionfrom operation following triangularization (see Sec. A.2).
Algori Alg orithm thm 7.1
Des Design ign of Min Minima imall Orde Orderr Obse Observe rvers rs [Ts [Tsui, ui, 1985] 1985]
Triang Triangular ularize ize the the foll followi owing ng matr matrix ix S until until it becomes
Step Ste p 1: 1 D
2 3 6 7 4 64 75 2 666 666 666 666 ¼ 666 666 666 664
SH
.. .
¼
H
Dnm K *
0
..
.
*
...
...
...
0
: ...
...
...
...
...
...
...
...
...
...
r þ1 H D 1
nm H D
X
(7.2)
m
X ...
r1 m rows
m
.. .
9> >>= þ >> >;
r1 1Þm 1 to r1 m rows
X
x ... x
4 S
: :
X
...
3 777 ð 777 777 777 777 g 777 777 g 775
0 ... 0
/the q1 -th row 11
¼
Step St ep 2:
is a The The fo form rm of S of of (7.2) indicates that the q1 -th row of K i ði ¼ li lin near ear com omb bin ina ati tio on of the the rows ows of D ¼ 1 ; . . . ; r1 Þ, or i H ði ¼ ¼ 1 ; . . . ; r1 Þ. Co 11 ¼ Sci D Compu mpute te ci by bac back k sub substi stitutu-
Copyright 2004 by Marcel De Dekker, kker, Inc. All Rights Reserved.
tion. Also set the q1 -th row of matrix K Z Z as ½1::1 : 0 . . . 0 with r1 ‘‘1’’s. Triang Triangula ulariz rizee the follow following ing matr matrix ix S 1 until it becomes:
Step Ste p 3:
S 1 H 1
2 6 66 46 666 664 2 66 666 666 6 ¼6 666 666 666 6 64
1 H c1 D .. .
r1 H cr1 D
¼
3 7 77 777 777 5
r1þ1 H H 1 D .. .
nm H D
KH K * ..
0
.
: :
X
*
...
...
...
r1
0 ...
...
...
...
...
...
X ...
...
...
... r1þr2þ1 HH 1 D
0 ... 0
/the q2 -t -th h ro row w 12
q2
X
4 S 1 ¼
Step St ep 4:
q 1
is a The The form form of S 1 of (7.3) indicates that the q2 -th row of K i ði ¼ linear combination of the rows ci D ¼ 1 ; . . . ; r1 Þ and the i ði ¼ ¼ r 1 þ 1; . . . ; r1 þ r2 Þ, or rows of D r1þr2
r1
i HH 1 Þ þ k2i ðci D
12 ¼
X i ¼1
i HH 1 Þ ci ðD
X
i ¼r1þ1
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(7.3)
m
X x ... x
r2 m rows
.. .
nm HH 1 D .. .
1Þm þ 1
m
.. .
r2
to r1
: ...
3 77 þ ð þ 777 777 777 777g 777 777g 75ð 6¼ Þ
¼ r 1 þ 1; . . . ; r1 þ r2 Þ, and then set the k and ci ði ¼ Compute ½ k21 ; . . . ; k2;r1 4 ¼ 2 q2 -th row of K Z Z as ½ k2 : 1 . . . 1 : 0 . . . 0
ð7:4aÞ
r1 r2 is Steps 3 and 4 are repeated until each of the p rows of K is expressed as i ði ¼ a linear combination of the rows of D ¼ 1 ; ; r1 þ :: þ r p 4 ¼ rÞ, where r is the observer order. Finally, parameters T and L are determined by Step 3 of Algorithm 5.3, and parameter K y is determined by (7.1a).
¼ 1 ; . . . ; pÞ, then the Without loss of generality, we assume qi ¼ i ði ¼ corresponding
2 6 ¼ 6 666 4
1...1 : k2
...
:
0
...
:
0
: 1. . . 1 : ...
:
0
:
: 1...1 : k3
K Z Z
0
0 :
.. kp
r1
r2
. : 1...1
r3
...
3 77 777 5
ð7:4bÞ
rp
It is obvious that observer order is tried and increased one by one starting from 0, in Algorithm 7.1. At any stage of this algorithm, if the i will be redeployed at the lower calculated ci ¼ 0, then the corresponding D . Therefore it is also obvious that part of matrix S to to express other rows of K ¼ 1 ; . . . ; rÞ is fully used. all remaining freedom of (4.1) ðci ; i ¼ Based on Conclusion 5.2 and the general assumption that v1 5v2 5 5vm ;
and that
r1 5r2 5 5r p
ð7:5Þ
it is proven that [Tsui, 1986b] in Algorithm 7.1 ri 4vi 1;
i ¼ ¼ 1 ; . . . ; p
ð7:6aÞ
Thus r ¼ ðr1 þ þ r p Þ4ðv1 1Þ þ þ ðv p 1Þ
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ð7:6bÞ
It is also proven that [Tsui, 1986b] in this algorithm r4n m
ð7:7Þ
If parameter K y is predetermined to be 0, then Eq. (7.1) becomes D1
c1
..
266 4
K ¼ K Z Z T ¼ K Z Z
. cr
377266 54
m ... m
.. .
Dr
377 5
ð7:8Þ
n
Because the only difference between (7.8) and (7.1c) is that the former has m additional columns, Algorithm 7.1 can be used directly to design this type of minimal order observers, and (7.6a,b) and (7.7) can be replaced by ri 4vi ;
i ¼ ¼ 1 ; . . . ; p
r ¼ ðr1 þ þ r p Þ4v1 þ þ v p
ð7:9aÞ ð7:9bÞ
and
r4n
ð7:10Þ
respectively. Now we have the complete formula for the general lower and upper bounds of orders of minimal order observers. Table le 7.1 show showss th that at the the orde orderr of a fu func ncti tion on ob obse serv rver er whic which h ca can n Tab implem imp lement ent arb arbitra itrary ry sta state te feedba feedback ck con contro troll var varies ies bet betwe ween en its lower lower and upper bounds. Unlike state observer orders, the actual value r of this order depends on the actual values of K and T (Di’s) in either (7.8) (if K y ¼ 0) or (7.1) (if K y =0).
Table 7.1 Lower an and d Upper Bou Bounds nds for Orders Orders of Minim Minimal al Order Observers with Arbitrarily Given Poles Stateo obser observers vers Observer type p ¼ n ; K ¼ I K yy ¼ 0 K yy =0
r ¼ n r ¼ n m
Function observers (p 4n ; K arbitrary, and v 1 5 5v m ) 14r 4minfn ; v 1 þ þ v p g 04r 4minfn m; ðv 1 1Þ þ þ ðv pp 1Þg
Copyright 2004 by Marcel De Dekker, kker, Inc. All Rights Reserved.
7.3
EXA EXAMP MPLES LES A AND ND SIG SIGNIF NIFICA ICANCE NCE OF T THIS HIS DE DESIGN SIGN [[Tsu Tsui, i, 1998a]
Example 7.1 In the single-output case ð m ¼ 1 Þ, the basis vector matrices D i of Algorithm 5.3 become row vectors ti ði ¼ ¼ 1 ; . . . ; rÞ. Hence the corresponding (7.8) and (7.1) become
2 664
K ¼ K Z Z
t1
.. . tr
n
3 775
and
2 664
K ¼ K Z Z
t1
.. . tr
3 775 0
I nm
n m
respectively. Thus the upper bound of function observer order of this case is n and n m, respectively. The lower bound remains 1 and 0 respectively,
because K cannot cannot be 0 while K can. The single output (SO) system is a special case of the multiple output ðMO; m51Þ system in the sense of m ¼ 1. No Noti tice ce that that fo forr such such sy syst stem emss n1 ¼ n , which makes the two terms of the upper bounds of r of Table 7.1 well unified with each other when m ¼ 1. Example 7.1 shows that the observer order bounds of this special case is well unified by the bounds of Table 7.1. The single input (SI) system is also a special case of the multiple input ðMI ; p51Þ system in the sense of p ¼ 1. In this special case K is a row vector. The upper bound of r is n1 and n1 1 for the two types of observers resp especti ectiv vely ely becau ecause se of (7.9 (7.9a a) and and (7.6 (7.6a a), resp espec ecti tiv vely. ely. Not otic icee tha that n1 ¼ n if m ¼ 1. This makes the two terms of upper bounds of r unified with each other for m ¼ p ¼ 1. As p increases from 1 [or the problem is changed to generate more signals of K xðtÞ], the upper bound of r should increase to n1 þ þ n p or ðn1 1Þ þ þ ðn p 1Þ but should not exceed the most difficult state observer case n or n m, respectively, for the two types of observers. Because the observability indices satisfy n1 þ þ nm ¼ n in Definition 5.1, the two terms of the upper bounds of rr are also perfectly unified as p is increased up to m. This unification is not achieved by other existing general upper bounds of r such as pn1 or pðn1 1Þ [Chen, 1984] because the ni ’s may not be all the same. For(7.1c) all SISO or MIMOalso cases, the lower of r isbe still 1 and K 0 of in K of (7.8) and respectively, because of bound (7.8) cannot 0 while (7.1c) can. The first case implies that K is is a linear combination of the rows of D1 (see Part (c) of Example 7.3 and Exercise 7.1 or see Part (d) of Exercise
Copyright 2004 by Marcel De Dekker, kker, Inc. All Rights Reserved.
7.1 for variation). The second case implies that the corresponding K is a linear combination of the rows of matrix C . To summarize, the lower and upper bounds of minimal order observer order of Table 7.1 are perfectly and uniquely unified from SISO cases to the MIMO cases. The derivation of (7.8) to (7.10) for the strictly proper observers ðK y ¼ 0 the derivation of (7.1) and (7.5)r of to (7.7) type observers Þ and ðK Tablefor 7.1the areproper also perfectly unified y =0Þ also show that the bounds of r for these two types of observers. Forr th Fo thee stat statee ob obse serv rver er case case when when ra rank nk ðK ¼ I Þ ¼ maximum n, the upper bounds of rr should reach the ultimate high levels n and n m for the two types of observers, respectively. For K ¼ I , the matrix T of of (7.8) and the 0 matrix ½ T 0 : C 0 of (7.1) should be square and nonsingular for the two types T ((r) should be n and n m for the of observers. Thus the number of rows of T two tw o ty type pess of obse observ rvers ers,, resp respec ectiv tivel ely. y. This This is sh show own n in Tabl Tablee 7.1. 7.1. Th Thus us Table 7.1 also unifies the state observer case and function observer case perfectly. From the perfect unification of SISO and MIMO systems, the perfect unifica uni ficatio tion n of strictl strictly y pro proper per and pro proper per type type obs observ ervers ers,, and the per perfect fect unification of state and function observers, all bounds of observer order of Table 7.1 should be the lowest possible. Any other bound that is lower than any of these bounds of Table 7.1 cannot be general because it cannot unify the special cases. Although the upper bounds of minimal function observer order is not as simple as that of the state observer order in Table 7.1, it often offers substantial order reduction in practice. The lower bounds (1 and 0) of r are the lowest possible and can be achieved by Algorithm 7.1 systematically whenever it applies (see Example 7.3 and Exercise 7.1). However, it is the upper bound that guarantees the significant order reduction from the state observer orders. Because the observability indices satisfy n1 þ þ nm ¼ n in Definition 5.1, the upper bound of rr of Table 7.1 is lower than the state observer order whenever m > p . In addition, this upper bound can be significantly lower than the state observer order in the situation that p 5 m 5 n and that the ni ’s are evenly valued. This situation is indeed common in practice because it is generally much easier to add measurements (or m) to a system than to add controls (of p) to a system.
Example 7.2 In a circuit system with 100 capacitors, 10 current or voltage meters, and 2
controlled contr olled current current or voltage voltage sour sources, ces, n 100 ; m 10 an and d p 2. 2. Gi Give ven n
Copyright 2004 by Marcel De Dekker, kker, Inc. All Rights Reserved.
that v1 ¼ ¼ v 10 ¼ 10 ðv1 þ þ v10 ¼ 100 ¼ n Þ, the the fu func nctio tion n ob obse serv rver er order of Algorithm 7.1 will not exceed v1 þ v2 ¼ 20 and ðv1 1Þ þ ðv2 1Þ ¼ 18, respectively (see Table 7.1 and Exercise 7.3). This is significantly lower than the state observer order. In addition, it is possible that the function observer order can be systematically designed to be even lower than its upper bound of 20 or 18. The improvement from a hund hu ndre redt dthh-ord order er comp compen ensa sator tor to a twen twenti tiet ethh-or orde derr on onee ca can n ha hard rdly ly be discounted, even by today’s computer numerical computation capability. The de The deve velo lopm pmen entt of comp comput uter er nu nume meric rical al co comp mput utati ation on capa capabi bili lity ty should only be a challenge, instead of a reason of abandonment, for such rese esearc arch ta task skss as min inim ima al orde orderr ob obse serv rveer desig esign n. For ex exam amp ple, le, the the develo dev elopme pment nt of high-s high-spee peed d comput computers ers has now made made pos possib sible le the digita digitall re real aliz izat atio ion n of a tw twen enti tiet ethh-or orde derr comp compen ensa sator tor of Exam Exampl plee 7.2. 7.2. In ot othe herr words, the significance of Example 7.2 is feasible because of the computer development. It should be noted that the result of Table 7.1 is analytical and general. Hence the 100-to-20-order reduction of Example 7.2 can easily be a 1000-to-200-order reduction (assuming n ¼ 1000 and v1 ¼ ¼ v 10 ¼ 100; other parameters of Example 7.2 remain unchanged). In addition, the unsuccessful past attempts of developing a simple, genera gen eral, l, and sys system temati aticc minima minimall order order ob obser server ver des design ign alg algori orithm thm sho should uld only be a challenge, instead of a reason of abandonment, for developing one.
Exam Ex ampl ple e 7.3 7.3
[Tsu [Tsui, i, 1985 1985]]
Let the block-observable Hessenberg form system matrices be
2 666 6 ¼ 66 66
1 2 0
...
A
0 0 3 ...
0
0
0 1
0 0
0 1 0
: : : ... ... 0 :
0 0
: :
1 1 0 ...
3
0 0
0 1 1 ...
0
0 : 0 : 1 : ... 1 :
...
0 0 0 ...
1
1 0 : 1 0 1 : 0
3 777 7 77 77
4
...
...
...
0
1
0
... ... : 0
...
...
... 0 :
1
...
2
5
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and
2 ¼ 4 ¼
C
1 1 1
0 1 0
0 : 0 0 : 0 1 : 0
0 0 0
0 : 0 0 : 0 0 : 0
3 5
From Definiti From Definition on 5.1, 5.1, v1 ¼ 3 ; v2 ¼ 2 ; v3 ¼ 2. Le Lett us desi design gn th thre reee mi mini nima mall order observers for the following three state feedbacks:
K 1
¼
K 2 ¼
3 2 2 : 1
2
1 : 0
0 1 : 1 0 2 : 1
1 0
0 : 0 1 : 1
2 2
3 3 2 : 1 2 0 : 0
and K 3 ¼
0 2 3 : 0 1 1 1 : 1
0 1
0 : 0 0 : 0
From Table 7.1, the observer order cannot exceed ðv1 1Þ þ ðv2 1Þ ¼ 3. But let us first set the n m ¼ 7 3 ¼ 4 possible eigenvalues of matrix F as as fl1 ; l2 ; l3 ; l4 g ¼ f1; 2; 3; 1g. In Step Step 1 of Al Algo gori rith thm m 5.3 5.3 we comp comput utee th thee ba basi siss ve vect ctor or ma matr tric ices es Di ði ¼ ¼ 1 ; 2; 3Þ from (5.10b) and D4 from (5.15d) (based on D1):
2 666 666 666 2 3 66 64 75 ¼ 666 66 6
2 0 1 : 1 1 1 : 0 0 0 :
...
D1 .. .
D4
...
...
1 1 0 1 2 1 1 1 1
...
...
...
2 2 1 3 3 1 2 2 2
...
...
...
... : : : ... : : : ...
0 1 0
...
...
0 1 0
0 1 0
...
...
... ... 0 : 1 0 : 0 1 : 0 ... ... ...
1 0
0 1 0
0 1 0 0 1 : 0
...
...
1
0 : 1 0 : 0 1 : 0
1 1 0
...
...
...
...
3 777 777 777 777 777 7 7
666 4
6 1 2 : 3 0 1 : 1 1 1 :
2 1 0
0 1 0
0 : 1 0 : 0 1 : 0
777
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We will apply Algorithm 7.1 to each of the three different K ’s. ’s. For simplicity of co comp mput utat atio ion, n, we use use el elem emen enta tary ry ma matr trix ix oper operat atio ion n (H ) instead of orth or thon onor orma mall ma matr trix ix oper operat atio ion n (H ) to tr tria ian ngu gula lari rizze th thee matri atrix x of th this is example. For K 1 Step 1:
2 32 666 7776 66 777664 ¼ 6 64 75 2 666 666 666 666 666 6 ¼6 666 666 666 666 666 4 1 D .. .
SH
4 D
---1 K
1 1
1
0
0 1
0
1
0
1
0
0
1
0
0
0
0
1
0 : 0 1 : 0
...
...
0
0
...
...
0
...
3 777 5 0 0
...
...
1
0
...
...
0
0
1
1
1
0
0
0
0
1
0
1
0
0
2
1
1
0
0
0
0
1
0
2
0
0 1
1
1
0
0
0
0
1
0
1
2
1
1
1
1 : 0
0
...
3 777 777 777 777 777 77 777 777 777 777 777 5
ð7:11Þ
/q1 ¼ 2 ; 11
Step 2: r1 ¼ 1 ; c1 ¼ ½ 0
1 H ¼ 11 ¼ ½ 1 0 ; c1 D
1
1
0
0 .
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Step 3:
S 1 H 1
1
1
0
0
0
0
0
1
1 0
1 0
0 1
0 0
1
0
0
2
1
1
0
0
0
0
1
0
2
1
1
1 1 1 0
0
0
2 66 666 6 ¼ 6 666 666 664
:
1
266 666 666 6 ¼6 666 666 6 64
3 772 777 777666 7774 777 75 0
1 1
0
0
0
1
0
0
0
0
1
0
0
0
0
1
0
0
0
0
1
1
0
0
0
0
0
1
0
1
1
0
2
...
...
...
...
... : :
1
1
1
1
x
x
x
x
Step 4: r2 ¼ 2 ; c2 ¼ ½ 1
0
1 ; c3 ¼ ½ 1
3 777 5
0
377 777 777 777 777 777 75
/ 12
0 , and k2 ¼ 2, so that
1 HH 1 Þ þ c 2 ðD 2 HH 1 Þ þ c 3 ðD 3 HH 1 Þ 12 ¼ k2 ðc1 D
Finally, the observer for K 1 is ðr ¼ r 1 þ r2 ¼ 3 Þ, and
2 4 ¼
1
F
0
0
0 2 0 0 0 3
35
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and
2 3 2 ¼ ¼ 4 5 4
T
c1 D1 c2 D2
1 1 1 2 2 1
c3 D3
2 2
1 0
1 0
0 0 1 1
1 1
0
0
From (7.4a),
K Z Z ¼
1 1 : 0
1 0
k2 :
2 : 1 1 : 0
¼
1 0
From (5.10a) and (7.1a), 0 4 2 7 0 0 5 2 1
2 ¼4
L ¼ ðTA FT Þ
I 3 C 11 0
3 5
and
K y ¼ ðK K Z Z T Þ
I 3 1 C 11 ¼ 0 0
0 1
0 0
For K 2
Step St ep 1 1::
Th Thee resu result lt is sim simil ilar ar to to that that of of K 1 in (7.11), except the part
2 H ¼ 1 K
0
1
0 /q1 ¼ 1 ; 11
1
3 5
1
2
Step 2: r1 ¼ 1 ; c1 ¼ ½ 1
0
1 1 H Þ ¼ 11 . 1 so that c1 ðD
0
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Step 3:
S 1 H 1
2 66 666 ¼ 6 666 664 2 66 6 66 ¼6 666 64
1
0
1
0
0
0
0
1
1
1
0
0
0
0
1
0
: :
1
0
1
1 2 1 0
0
3 772 77766 77764 777 5
0 1
0
0
1
0
0
0
0
1
0
0
0
0
1
0 1 0 0
0
0
0
1
1 0
1 1 0 1
0 0
...
1
...
...
...
:
x
x
x
x
1
2 1
1
Step 4: r2 ¼ 1 ; c2 ¼ ½ 1
2
3 777 5
3 77 777 777 775
/ 12
1 , and k2 ¼ 1 such that
1 HH 1 Þ þ c 2 ðD 2 HH 1 Þ 12 ¼ k2 ðc1 D
Finally, the minimal order observer for K 2 is ðr ¼ r 1 þ r2 ¼ 2 Þ, and
F ¼
1
0
0 2
T ¼
c1 D1 c
D
¼
2
0 1
1
0
1
1
2 4 3 2 2 1 1
2
K Z Z ¼
1 : 0 k2 :
r
1
1 : 0
¼
1
2
1 : 1
r 2
L ¼ ðTA FT Þ
I 3
0
C 11 ¼
2 2 1
3 16 10
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and
K y ¼ ðK K Z Z TÞ
I 3 1 C 11 ¼ 0 0
0 1
1 0
For K 3 K 3 is already a linear combination of rows of C , we Because the first row of K let r 1 ¼ 0 and let the linear combination coefficients be k 1 . Then, because the 3 equa 1 , we have se seco cond nd ro row w of K equals ls the the seco second nd row row of K have th thee fo foll llow owin ing g minimal order observer for K 3 : ð r ¼ r 1 þ r2 ¼ 0 þ ðr1 for K 1 Þ ¼ 1 Þ F ¼ 1; T ¼ the first r1 ð¼ 1 Þ rows of T for K 1
¼ ½ 1 1 1 1 1 0 0 1
0 Z K Z
¼ 1
K y
¼
0
2 0
3 0
¼
0
k1
0
0
L ¼ ½ 0 4 2
To summarize, the order of the three minimal order observers is 3, 2, and 1, respectively, which is systematically and generally determined by Algorithm 7. 7.1. 1. All All th thre reee or orde ders rs do not not exce exceed ed ðv1 1Þ þ ðv2 1Þ ¼ 3, which is the upper bound of Table 7.1. The minimal order observer design problem has been studied using classical control methods also. The most recent result can be found in Chen [1984] and Zhang [1990]. Although for years the upper bound of minimal order observer order from these methods has been minfn m; pðv1 1Þg [Chen, 1984], (see Exercise 7.4), the classical control methods differ much from Algorithm 7.1 in determining systematically and generally the lowest
possible observer order (see Example 7.3 and the argument between (7.1c) and an d Al Algo gori rith thm m 7. 7.1) 1).. The The di diffe ffere renc ncee appe appear arss at ho how w syst system emati atica call lly y th thee equation equa tion [such as (7.1c)] (7.1c)] is being being solved, solved, at how the observer dynamic dynamic part is decoupled, and at how fully the design freedom (such as the free parameters ci ) is being used. It seems that the classical control methods cannot match Algorithm 7.1 in the above three technical aspects. The fore foremo most st the theore oreti tica call sign signifi ificcan ance ce of Algo lgorith rithm m 7.1 7.1 is the the si simp mpli lific ficati ation on of the the desi design gn pr prob oble lem m into into a true set of line linear ar eq equa uatio tions ns (7.1c) or (7.8) with fully usable freedom. The general and lowest possible lower and upper bounds of minimal order observer order (Table 7.1) are also derived simply based on this set of linear equations. Thus it can be
Copyright 2004 by Marcel De Dekker, kker, Inc. All Rights Reserved.
clai claime med d with with co confi nfide denc ncee that that this this set set of line linear ar eq equa uati tion onss is alre alread ady y th thee simplest possible theoretical and general form of the minimal order observer design problem [Tsui, 1993a]. From Example 4.3 and Algorithm 7.1, this development is enabled solely by the development on the decoupled solution of (4.1) (Algorithm 5.3). Other state space minimal order observer design methods cannot reach this simple form because the result of Algorithm 5.3 has not been used [Van Dooren, 1984; Fowell et al., 1986]. The act ctu ual solv solvin ing g of this this set set of li lin near ear equ quat atio ion ns is te tech chni niccal. al. Although Algorithm 7.1 is general and systematic, guarantees the upper boun bo und d of ob obse serv rver er orde orderr of Ta Tabl blee 7.1 and and tr trie iess th thee ob obse serv rver er or orde derr on onee by one (startin ing g from 0), it stil illl has room for improvement. This ¼ 1 ; 2; . . . ; but algorithm operates on the Di matrices in the sequence of i ¼ does do es no nott tr try y different different sequences sequences among among the these se mat matrice rices, s, wh which ich may off offer er addi ad diti tion onal al ob obser serve verr orde orderr redu reduct ctio ion. n. For For ex exam ample ple,, if oper operat atin ing g in th thee 1 ; D2 ; D3 Þ, Al D seq sequen uence ce of Algo gori rith thm m 7.1 7.1 ca can n de dete tect ct th that at K is lin linear early ly ð dependent on the rows of D 1 and D2 ðr ¼ 2 Þ, but it is still possible that operating on a different sequence of ðD3 ; D2 ; D1 Þ the Algorithm 7.1 can detect det ect tha thatt K is li line near arly ly depe depend nden entt on th thee ro rows ws of D3 only ðr ¼ 1 Þ (see Exerci Exe rcise se 7.1 7.1,, Pa Part rt (d) (d)). ). In the the li lite tera ratu ture re,, th ther eree are are othe otherr repo report rtss of mi mini nimi mizi zing ng func functi tion on observer obse rver orde orderr by observer observer pole selection selection [Fortmann and Willi Williamson amson,, 1972; Whistle, 1985]. However, these design methods are much more complicated, while the additional observer order reduction offered by these methods is not generally significant. Finally, it should be emphasized again that the minimal order observer
design (Algorithm 7.1) uses up completely the remaining design freedom of (4.1) (or of the observer) and therefore cannot take care of the robustness of thee corr th corres espo pond ndin ing g obse observ rver er fe feed edba back ck syste system m [suc [such h as (4.3) (4.3)]. ]. He Henc ncee th this is design is useful only for the situation in which the plant system model and meas me asur urem emen ents ts ar aree ac accu cura rate te—a —and nd cont contin inue ue to be accu accura rate te—a —and nd th that at
di dist sturb urban ance ce an and d fa failu ilure re are are rela relati tive vely ly free free.. In ot othe herr wo word rds, s, the the mi mini nima mall order observer should be used when only performance (but not robustness) is required. Alth Al thou ough gh mi mini nima mall orde orderr obse observ rver er and and dy dyna nami micc outp output ut fe feed edba back ck compensator (capable of implementing state feedback control) differ from each ea ch othe otherr in desi design gn pr prio iori rity ty,, both both thei theirr de desi sign gnss ar aree pa part rt of St Step ep 2 of Algorithm 5.3 and both are in the similar form of sets of linear equations. Also, they both are successful and actual attempts of the basic observer desi de sign gn conc concep ept— t—imp imple leme ment nting ing stat statee fe feed edba back ck co cont ntro roll dire directl ctly y wi with thou outt explic exp licit it inf inform ormati ation on of system system sta states tes.. In additi addition, on, bot both h ord order er red reduct uction ion (whi (w hich ch is pa part rt of perf perfor orma manc nce) e) and and robu robust stne ness ss ar aree im impo port rtan antt sy syst stem em properties, even though the emphasis of this book is more on robustness
Copyright 2004 by Marcel De Dekker, kker, Inc. All Rights Reserved.
properties. Therefore, both results are useful and both may be used in some situations.
EXERCISES 7.1
Repe Repeat at Examp Example le 7.3 for a modifie modified d system of Exam Example ple 6.1:
2 6 ¼ 4
Let C
1
0
0 : 0
0
0
0
2
1
0 : 0
0
0
0
3
4
1 : 0
0
0
0
3 75
¼ 1 ; . . . ; 4Þ be the same as that of Example 6.1. and let matrices Di ði ¼ The system has parameters n ¼ 7 ; m ¼ 3 ; p ¼ 2 ; v1 ¼ 3, and v2 ¼ v 3 ¼ 2. (a)
K ¼
1 1
0
0
0 : 1
2
3
1
1 : 4
3 2
0
2½ 6 ¼ 4 ½
Answer : r ¼ 3 ; T
¼
2
3
2
3D1
2
0
0D2
½ 4 3 2D3 3 2 3 : 3 2 3 8
36 9 1 1 : 0
64
K Z Z ¼
0
0
2 : 1
3 75
3
0 2
3
0 : 4
0
6 : 12
3 2 4 19 11
K y ¼
63
75
3
29 5
(b)
1
K ¼
0
0 : 1
0 1
2
3
1
1 : 2 4 6 4 ½ 3 2 3D1
Answer : r ¼ 2 ; T ¼
½ 2 0 0D2 3 2 3 : 3 2 3 3 ¼ 8 0 0 : 4 0 0 2 1 1 17 10 3 K Z K y ¼ Z ¼ 2 1 17 15 5
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(c)
K ¼
11
7
6
1 : 0
0
0
2 4 : 6 2
0
4 6
Answer : r ¼ 1 ; F ¼ 1; T ¼ ½ 3 1 2 D1 ¼ ½ 3 1 2 : 3 K Z Z ¼
(d)
K ¼
0
2
3 10
K y ¼
2
3
1
0
0
0
1
1 3
6
0
0
0
=
Answer : r ¼ 1 ; F ¼ 3; T ¼ ½ 1 2 3 D3 ¼ ½ 9 6 9 : 3 2 K Z Z ¼ 7.2
= 1 3 0
K y ¼
0
0
0
3
2
1
3
2 3 : 1 2=3 1 : 0
1 2
3
1
Le Lett a sys system tem and its state state feed feedbac back k gain be give given n as
02 Þ ¼ @4
ðA; B; C ; K
0 1 0
32 3 5 4 5; ½
0 5 0 5 ; 1 2
1 0 0
0
0
1 ;
½ 0 5 3 8
Design a minimal order observer according to Algorithm 7.1.
1 A
(a) (b)
7.3
Let K y ¼ 0 and observer poles ¼ f5:25; 2:44; 4g. Answer: r ¼ 2 < n . ¼ 0, and observer poles ¼ f5=3; 10=3g. Let K y 6 Answer: r ¼ 1 < n m.
In Example Example 7.2, let n ¼ 1000; m ¼ 100 ; p ¼ 2 ; v1 ¼ ¼ v 100 ¼ 10, and K y ¼ 0. What are the state observer order and the upper bound of minimal observer order of Table 7.1? Answer: n ¼ 1000 and v1 þ v2 ¼ 20.
7.4
¼ 0. Let n ¼ 21 ; m ¼ 5 ; p ¼ 2 ; v1 ¼ 9 ; v2 ¼ ¼ v 5 ¼ 3, and K y 6 What are the state observer order, the upper bound pðv1 1Þ of the existing minimal observer order, and the upper bound of our minimal observer order of Table 7.1? Answer: n m ¼ 16 ; pðv1 1Þ ¼ 16 ; v1 þ v2 2 ¼ 10.
Copyright 2004 by Marcel De Dekker, kker, Inc. All Rights Reserved.
8 Design of Feedback Control— Eigenstructure Assignment
The new design approach of this book is divided into two major steps. The first concerns the dynamic part of the observer/compensator and is covered in Chap. in Chap. 6 (for 6 (for robustness realization). The second step, which is covered by 8 and 9 Chaps 8 and 9,, deals with the design of the output part of the compensator, or the design of the generalized state feedback control K C x t with a given C . Th This is de desi sign gn al also so full fully y dete determ rmin ines es the the fe feed edba back ck sy syste stem m loop loop tr tran ansf sfer er function because (4.3) is already guaranteed. Among the existing design results of this control, the eigenvalue and eigenv eig envect ector or ass assign ignmen mentt (calle (called d ‘‘eige ‘‘eigenstr nstruct ucture ure ass assign ignmen ment’’) t’’) and lin linear ear quadra qua dratic tic optima optimall contro controll are perhap perhapss mo most st commo commonly nly kno known wn,, and are capabl cap ablee of consid consideri ering ng effect effective ively ly both both per perfor forman mance ce and rob robust ustnes ness. s. In
ðÞ
Copyright 2004 by Marcel De Dekker kker Inc All Rights Reserved
partic part icul ular, ar, ac acco cord rdin ing g to the the anal analys ysis is of of Chap Chap.. 2, the eige eigenv nvalu alues es and and eigenv eig envecto ectors rs can determ determine ine system system perfor performan mance ce and rob robust ustnes nesss far more more directly explicitly thanperformance other indicators. Hence their assignment should improveand feedback system and robustness distinctly effectively. In thi thiss book, book, eigens eigenstru tructu cture re ass assign ignmen mentt design design met method hodss and lin linear ear 8 and and 9, quadratic optimal control design methods are introduced in Chaps in Chaps 8 respectively. The design of the generalized state feedback control K C x t is based on the single overall feedback system matrix A BK C . Therefore if the design of Chap. Chap. 6 is 6 is based mainly on the understanding of feedback systems of Chaps Chaps 3 3 and and 4, 4, then then the design of Chaps Chaps 8 8 and and 9 is based mainly on the analysis of the single overall system of Chap. of Chap. 2. According to Table 6.2, the generalized state feedback control K C x t
ðÞ
unifies the arbitrary state feedback control (or state feedback control) K x t (if rank C n ) and static output feedback control (if rank C C m ). Chaps 8 and 9 present present the design methods in these two categories. The Both Chaps Both 8 and 9 arbitrary state feedback control, which is a special case of the generalized state feedback control in the sense of C I , is presented first.
ð Þ ð ¼ ¼ Þ ¼
ð Þ ¼
¼ ¼
8.1 8.1.1 8.1 .1
SELEC SELECTIO TION N AND PLACEM PLACEMENT ENT OF F FEED EEDBAC BACK K SYSTEM SYSTEM POLES Eigenv Eigenvalu alue e ((Pol Pole) e) Selec Selectio tion n
Although system poles most directly determine system performance, there are no general, explicit and optimal rules for feedback system pole selection. Furthermore, there is no real optimal pole selection without trial and error. This is because plant systems are usually very different and complex, and al also so beca becaus usee th thee perf perfor orma manc ncee and and robu robust stne ness ss de desi sign gn requ requir irem emen ents ts are are contradictory to each other. Noneth Non ethele eless, ss, there there are sti still ll some some bas basic ic and gen genera erall und unders erstan tandin dings gs about the relationship between the system poles and the system performance and robustness. The following six general rules of pole selection are guided by these basic understandings (see Truxal, 1955 and Conclusion 2.2). (a)
The more more n nega egative tive the the rea reall part part of the po poles les,, the fas faster ter th thee speed speed with which the system reaches its steady state. (b) In regulato regulatorr problems problems,, it is often often rrequi equired red th that at the zero ffrequ requency ency response of the control system T s 0 be a finite constant. For example, if the unit-step response of a single-input and singleoutpu out putt system system y t is requ requir ired ed to appr approa oach ch 1 at st stea eady dy st stat atee t?? , then y t?? sY s?0 sT s?0 =s T s?0 1.
ð ¼ Þ
ð
Þ
ð
ðÞ
Þ¼ ð
Þ¼ ð
Þ ¼ ð
Þ¼
Copyright 2004 by Marcel De Dekker, kker, Inc. All Rights Reserved.
ð ¼ Þ ¼ ð ¼ Þ
ð Þ¼ ð Þ ð Þ
D s 0 in T s N s =D s . It is This implies that N s 0 well known that N s 0 equals the product of zeros of T T s and
is invariant under state feedback control [Patel, 1978]. Therefore D s 0 im thee rela th relati tion on N s 0 impo pose sess a con onst stra rain intt on D s 0 , whic which h equa equals ls th thee prod produc uctt of th thee pole poless of fe feed edba back ck system T s . (c) (c) Fr From om the the resu result ltss of root root lo locu cus, s, th thee fu furt rthe herr aw away ay th thee fe feed edba back ck system poles from the loop transfer function poles, the higher the loop lo op gain gain (o (orr feed feedba back ck cont contro roll ga gain in)) requ requir ired ed to plac placee th thes esee feedback system poles. The severe disadvantages of high feedback control gain are listed in Subsection 3.1.2. If rul rulee (a) is concer concerned ned mai mainly nly wit with h system system perfor performan mance, ce, then rules (b) and (c) are concerned mainly with robustness, and are constraints on rule (a). (d) If the eeige igenva nvalue luess of a matrix matrix differ differ too too much much in mag magnit nitude ude,, then then the difference between the largest and the smallest singular values of that matrix will also differ too much. This implies the bad cond co ndit itio ion n and and the the bad bad robu robust stne ness ss of th thee ei eige genv nvalu alues es,, of th that at matrix. (e (e)) Mu Mult ltip iple le ei eige genv nval alue uess can can caus causee de defe fect ctiv ivee ei eige genv nvect ector orss (5.1 (5.15d 5d), ), which are very sensitive sensitive to matrix para parameter meter variatio variation n (see Golub and an d Wilk Wilkin inso son, n, 1976 1976b) b) and and whic which h ge gene nera rall lly y resu result lt in roug rough h
ð ¼ Þ
ð ð¼ ¼Þ ¼Þ ð ¼ Þ
ðÞ
ðÞ
re resp spon onse sess (see (see Exam Exampl plee 2. 2.1 1 and and Fig. Fig. 2. 2.1 1). Therefore Therefore multiple multiple (f) (f)
poles, even clustered poles, should generally be avoided. For so some me opti tima mall cont contrrol syste ystems ms in th thee sen ense se of min minimal imal ‘‘Integral of time multiplied by absolute error (ITAE)’’ [Graham and Lathrop, 1953]:
Z ¼ ½ j ð Þ j ¼ ?
t y y t
J
1 dt
0
or in th thee sens sensee of mini minima mall ‘‘In ‘‘Inte tegra grall of quad quadra rati ticc erro errorr (I (ISE SE)’ )’’’ [Chang, 1961]:
Z ¼ ½ ð ð Þ Þ þ ð Þ ¼ ?
q y t
J
1
2
ru t
2
dt;
q??
0
the feedback system poles are required to have similar magnitude and evenly distributed phase angles between 90 and 90 . This result conforms with rules (d) and (e).
þ
Copyright 2004 by Marcel De Dekker, kker, Inc. All Rights Reserved.
These ese six six rule ruless are are conc concer ern ned more more wit ith h th thee ef effe fect ctiv iveeness ness an and d limita lim itatio tions ns of practi practical cal analog analog con contro troll system systems. s. In con contra trast, st, the sel select ection ion of fe feed edba back ck comp compen ensa sator tor pole poless (see (see the the be begi ginn nning ing of Sec. Sec. 5.2) 5.2) ar aree mo more re specifically and explicitly guided. The feedback compensators are usually digital and can therefore be made ideal and precise, while the analog systems cannot be made ideal and precise. To summ summar ariz ize, e, the the pole pole sele selecti ction on rule ruless ar aree ne neit ithe herr ex exha haus usti tive ve nor nor generally optimal. This should be a true and reasonable reflection of the reality of practical engineering systems, and should impose a challenge to control engineers.
8.1.2 8.1. 2
Eige Eigenval nvalue ue Assignm Assignment ent b by y State State Fe Feedb edback ack Contro Controll
The eigenvalue assignment design methods are presented in this subsection and in Subse Subsecti ction on 8.1.3, 8.1.3, for arbitr arbitrary ary sta state te fee feedba dback ck con contro troll K x t and gene ge nera rali lize zed d stat statee feed feedba back ck co cont ntro roll K C x t , resp respec ectiv tively ely.. These These desi design gn metho met hods ds have have the distin distinct ct pro proper perty ty that that the corres correspon pondin ding g eig eigenv envect ectors ors aree expr ar expres esse sed d in term termss of thei theirr corr corres espo pond ndin ing g ba basi siss ve vect ctor ors, s, an and d ca can n therefore be assigned by really systematic and effective numerical methods. These eigenvector assignment design methods will be presented in Sec. 8.2.
ðÞ
ðÞ
Let L be be th thee Jord Jordan an form form matr matrix ix th that at is fo form rmed ed by th thee sele select cted ed eige ei genv nval alue uess of Subs Subsec ecti tion on 8. 8.1. 1.1. 1. Then Then the the ei eige gens nstr truc uctu ture re as assi sign gnme ment nt problem can be formulated as (1.10):
ðA BK ÞV ¼ V L
ð8:1aÞ
or
AV V L
ð8:1bÞ
¼ BK ðK ¼ KV Þ
b b
Let matrix F of of Eq. (4.1) be the same Jordan form matrix (in transpose) as L, and be set to have dimension n, then this equation TA
FT ¼ ¼ LC
becomes the dual of (8.1b). In other words, we can take the transpose of 0 both sides of (8.1b) and then consider the resulting A0 ; B0 ; V 0 , and K as the matrices A, C , T , and L of Eq. (4.1), respectively. Therefore, we can use the dual version of Algorithm 5.3 to compute directly the solution V ; K of (8.1b). Incidentally, Algorithm 5.3 and its dual version were published formally in the same year in Tsui [1985] and Kautsky et al. [1985], respectively.
ð
b
bÞ
Copyright 2004 by Marcel De Dekker, kker, Inc. All Rights Reserved.
The only difference between these two design computations is that 1
V after K only is is co comp mput uted ed corresponds from from (8.1 (8.1b) b), , it mu st be feedback ad adju just sted ed dynamic to K K K because matrix corresponds to themust original matrix A BK . Th This is adju adjustm stmen entt is unne unnece cess ssar ary y in ob obse serv rver er de desi sign gn be beca caus usee th thee observer dynamic matrix in (3.16) is matrix F instead instead of matrix A LC . The dual version of Algorithm 5.3 is introduced in the following with some simplifications. Let the Li be an ni -dimensional Jordan block of L. Let
b
¼ ¼
b
4¼ ½v j . . . jv
V ii
i 1
and
ini
4¼ ½k j . . . jk
K i i
i 1
ini
b
be n6ni and p6ni dimens dimensio iona nal, l, and and be the the pa part rt of ma matri trice cess V and K corresponding to Li in (8.1b), respectively. Then (8.1b) can be partitioned as
ð8:2Þ where r is the number of Jordan blocks in L and n þ þ n ¼ n . V L ¼ BK ;
AV ii
ii
i
i i
¼ 1; . . . ; r ¼
i
1
r
Using the Kronecker product operator 6, Eq. (8.2) can be rewritten as
¼ 1; . . . ; r ½I 6A L 6I j I 6Bw ¼ 0; i ¼ ni
i
ni
i
ð8:3aÞ
where
¼ ½v0 : . . . : v0 : k0 : . . . : k0 0 For example, when n ¼ 1, (8.3) becomes wi
i 1
ini
i 1
ini
ð8:3bÞ
i
¼
½A l I : i
B
vi ki
0
ð8:4Þ ð þ Þ
Because the matrix of (8.3a) has dimension ni n6ni n p [see (5.13c)], and beca be caus usee co cont ntro roll llab abili ility ty crite criteri rion on impl implie iess th that at all all ro rows ws of th this is matr matrix ix ar aree linearly independent (see Definition 1.2), the vector wi of (8.3) has ni 6 p basis vectors and can be set as an arbitrary linear combination of these basis vectors. Naturally, the determination of this linear combination constitutes the assignment of eigenvectors V i i vi 1 : . . . : vini . For example, when ni 1, the matrix of (8.3) or (8.4) has dimension n6 n p . Henc Hencee ei eige genv nvec ecto torr vi of (8.4) can be an arbitr tra ary linear
¼
ð þ Þ
¼ ½
Copyright 2004 by Marcel De Dekker, kker, Inc. All Rights Reserved.
ð ¼ ¼ 1; . . . ; pÞ which also
combination of its p corresponding basis vectors dij j satisfy (8.4):
¼ ½d
i 1 : . . . : dip
vi
c 4¼ D c i
i i
ð8:5Þ
where ci is a p-dimensional free column vector. The vector ki of (8.4) will be the same linear combination (coefficient vector is ci ) of its own corresponding basis vectors. p 1 (single-input case) and n i 1, then the matrix of (8.3) or (8.4) If p has dimension n6 n 1 . Hence the solution vi and ki is unique (ci is a sc scal alar ar). ). Th This is impl implie iess that that in sing single le-i -inp nput ut ca case se,, th ther eree is no ei eige genv nvec ecto torr
¼
ð þ Þ
¼
assignment freedom, and the eigenvalues alone can uniquely determine the feedback system dynamic matrix. Equa uati tion on (8.5 (8.5)) is a uniq unique uely ly expli xplici citt an and d uniq nique uely ly de deco coup uple led d formul for mulati ation on of eig eigenv envecto ectorr assign assignmen ment. t. Onl Only y bas based ed on thi thiss for formul mulati ation, on, thee gene th genera rall an and d syst system emat atic ic desi design gn algo algori rith thms ms fo forr ro robu bust st ei eige genv nvec ecto torr assignment are developed in Kautsky et al. [1985]. These methods will be
introduced in Sec. 8.2. Equations (8.3) and (8.4) are the formulas for computing the basis vect ve ctors ors of eige eigenv nvec ecto torr matri matrix x V . Like Step 1 of Algo lgorithm 5.3, this computation can be carried out by direct back substitution if based on the block-controllable Hessenberg form
½A : B
26 ¼ 66 4
A11 A12 B2 A22 0 B3 . . .
0
A1m
...
...
...
...
:
:
... . . . . . .
...
:
:
B1 0 0
:
:
. . .
:
0
.. .
0 Bm Amm
ð ¼ ¼
Þ
ðð ¼ ¼ m þ m þ þ m ¼ n
Þ
:
37 775
ð8:6Þ
where matrix blocks B j j 1 ; . . . ; m are the upper echelon-form matrices, and m is the largest controllability index of the system A; B . As the dual of the observability index of Definition 5.1, there are p controllability indices m j j 1 ; . . . ; p of system A; B and
ð Þ
Þ
ð ð8:7Þ In addition, each basis vector of (8.5) d ð i ¼ ¼ 1; . . . ; n; j ¼ ¼ 1; . . . ; pÞ can 1
2
p
ij
be computed corresponding to one of the p inputs which is indicated by j . If th thee dij vecto vectors rs are are co comp mput uted ed this this wa way, y, then then fr from om th thee du dual al of Conclusion 5.2, for a fixed value of j , any set of m j of the n dij vectors are linearly independent of each other (see Example 8.6 and Theorem 8.1). This
Copyright 2004 by Marcel De Dekker, kker, Inc. All Rights Reserved.
analyt anal ytic ical al prop proper erty ty is very very usef useful ul in th thee an anal alyti ytica call ru rule less of ei eige genv nvec ecto torr assignment (Subsection 8.2.2). If matr matrix ix V is co comp mput uted ed ba base sed d on a simi simila lari rity ty tr tran ansf sfor orma mati tion on HAH 0 ; HB instead of the original A; B , [one example of HAH 0 ; HB is the blo blockck-con contro trolla llable ble Hessen Hessenber berg g form form (8. (8.6)] 6)],, then then the cor corres respon pondin ding g (8.1b) becomes
ð
Þ
ð
HAH 0 V V L
Þ
ðð
¼ HBK ^
Þ
ð8:8Þ
A comparison of (8.1b) and (8.8) indicates that the matrix V of of (8.8) should be adjusted to V H 0 V in in order to correspond to the original system matrix
¼ ¼
ðA; BÞAs. stated following (8.1b), after this adjustment of V , it will then be used to adjust the feedback gain matrix K ¼ ¼ K V . 8.1.3 8.1 .3
b
1
Eigen Eigenval value ue Ass Assign ignme ment nt by Gen Genera eraliz lized ed State State
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