Oded Yaniv Auth. Quantitative Feedback Design of Linear and Nonlinear Control Systems
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QU NTIT TIVE FEEDB FEED B CK DESIGN LINE E R ND NONLIN NONLINE E R O LIN CONT CO NTRO ROL L SYSTEM SYSTEMS S
THE KLUWER INTERN
ND
TION
L SERIES IN ENGINEERING
OMPUTER SCIENCE
QUANTITATIVE FEEDBACK DESIGN QUANTITATIVE OF LINEAR AND NONLINEAR CONTROL SYSTEMS
ODEDYANIV Faculty
o
Engineering, Tel Aviv University, Israel
oreword by
Isaac Horowitz
~
pringer Science Business Media, LLC
Library
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Publicatiorll D a t a iJt Publicatior C o n g r e ss C a taloging iJt
Y aniv, O d e d , ear control c ontrol ar and nonlin and nonlinear linear feedback back design d esign Df line Q u a ntitative ntitative feed itz. Horowitz. by Isaac Horow Yan iv ; forew ord by Isaac ms O d e d Yaniv system syste g and and ineering onal s e d e s in e n g ineerin i nternational K l u w e r internati cm . ciencee c o m puter sscienc
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rences. s. raphical ical refe reference ludes bibliDg bibliDgraph I n c ludes k) -63 3 31-7 eBoo eBook) 78-1-4757-6 89 - 5 ISB N 9 78-1-4757 ISBN -1-4419-5089 ISBN 97 8 -1-4419-50 75 7-6331-7 1 007/978-1-4 007/978-1-475 DOI 10. 1 0.1 1
feedb ck
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Copyright
rol systems. sy stems. control cont
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edia N N e w York York siness M edia ringerr Science+Bu Scienc e+Business 1999 y S p ringe © 1999
19 99 Pub1ishers ishers in in 199 Ac ademic Pub1 b y Kluw er Academic y pub1ished pub1is hed by Orig inally Originall 1 999 1st st edition edi tion 1999 hard cover 1 ftcover er reprint reprin t of t he hardcover S o ftcov st o r e d in a n m a y be rep rodu ced, sto publicatio lication N o p a r t o f th i s pub ica1, photo mechanica1, transmitted mitted in in an y form o r y any m eans, mechan sys tem or trans retrieva l system ssion o f th e p ermission î o r w ritien permi se, without wit hout ih e p r îo rding,, or otherwi ot herwise, pying, g, reco rding c o pyin reserved. ved. AI AIII rights reser
publisher publ isher,, Springer Spring er Science+B Sc ience+Busine usiness ss Media, Med ia, LLC. LL C.
paper rinted d o n a cid-fre c id-freee paper P rinte
NOTE TO RE
DER
The software software ffor or all all the examples in the form available at t t p : / / w w w e n g t a u a c i l / ~ y a n i v
o
Matlab ™ scri script pt files is
OR Michal nd Yotam
ontents
xv
Foreword
xvii
Preface Acknow ledgments
xix
Abbreviations Notation and Symbols
xxi
1
1
Basic Components of Feedback Controlled Controlled Systems
1 1
2.
Why Embed a Plant in in a Feedback Feedba ck System?
2
3.
The Design Process of Feedback Control Systems
4.
Book Outline
INTRODUCTION
Part I 2
LINE
9
R SYSTEMS
BASICS O F SISO FEEDBACK CONTROLLED SYSTEMS 1 Introduction 2 Basic Frequency Domain Characteristi Characteristics cs 21 Relative Stabi Stability lity,, Cross-Over Cros s-Over Frequency Frequen cy and Bandw idth Conditionally Stable Systems Conditionally High Frequency Gain Stability Analysis Using Nichols Charts 241 Continuous Systems Discrete Time Systems 2.4.2 243 Some Remarks About the Nyquist Plot Closed Loop Specifications 31 t-Domain Specification Specification 32 w Domain Specification 33 Translation of Specifications From t-Domain to 22 23 24
3
Domain 331
4
15
15 15 16 17 18
20 21
22 25 25 26 27 w
Model Ba sed Technique 3.3.2 Krishnan Krish nan and Cruickshanks Technique Performance Limitations of NMP or Unstable Systems 41 Stable Plants 411 Extension to Several RHP Zeros and/or Delay
29 30 33
34 35 37
x
QU NTIT TIVE FEEDB CK DESIGN
4.1.2 4.2
5 6
7. 8 3
Comparison to Loop Transmissions Which Violate Assumption 2 1 Unstable Plants 4.2.1 Unstable Plants With a Single RHP Pole 4.2.2 An Example and Limitations 4.2.3 Extension to Several RHP Poles
Loop Shaping Summary Exercises Notes and References
SYNTHESIS OF LTI CONTROLLERS FOR MISO LTI PLANTS 1 Introduction 2 One DOF System 21 Sensitivity Reduction Problem 2.2 Bound Calculations 2. 2.3 3 Control Effort Problem Proble m 2.4 Examples 3
4 5 6
7 8
Tw Two F Systems 3. 3.1 1 o DOBound Calculations 3.1.1 3.1 .1 Bound Calculat Calculations ions With The Aid of Plant Templates 3.1.2 Bound Calculations Using a Closed Form Algorithm An Example 3.2 Extension to NMP Plants Extension to Sampled Data Systems Summary Exercises Notes and References
4. SYNTHESIS SYNTH ESIS OF LTI LTI CONTROLLERS FOR MIMO L LTI TI PLANTS 1 Synthesis of One DOF Feedback Systems 11 2 x 2 Plants and Disturbances at th thee Pla Plant nt s Inputs 1.2 2 x 2 Plants and Disturbance Distur bancess at the Plant Pla nt s Outputs 1.3 Distur bances at the Plants Pla nts Inputs n x n Plants and Disturbances 1.4 Plant s Outputs n x n Plants and Disturbances at the Plants 1. 1.5 5 Design Improvements Improvem ents by Iteration 1.6 Shortcuts in Low Frequency Bound Calculations 2 Synthesis of Two DOF Fee Feedback dback Systems 2 x 2 Plants 21 2.2 n x n Plants Model Matching Matc hing Specifications 2.3 Shortcuts in Low Frequency Bound Calculations 2.4 Synthesis Synthe sis for Margins at the Plants Plant s Outputs 3. 2 x 2 Plants and Diagonal Controllers 3.1 n x n Plants and Diagonal Controllers 3.2
40 42 43 46 47 49 55
56 57
59 59 59 61
62 64 65 73 75 75 78 8
81
85 88 88
90 93 94 95 113 113 125 125 134 135 137 137 139 148 157 157 158 158 160 163 170
Contents 4 5 6
7.
Synthe sis for Margins at the Plants Synthesis Plan ts Inputs Synthesis o Non-diagonal Controllers for n x m Plants Plan ts Synthesis Synthe sis for Minimum Phase Diagonal Elements 6. 6.1 1 2 x 2 Plants Plant s 6.2 n x n Plants Plant s Synthesis Synthes is for the General Control Problem Using LFT Notation
xi 180 186 194 196 201 20 1 205 205
7. 7.1 1 7.2 7.3
8
9 10 11
12. 13. 14. 15. 16. Part II 5
Some Special Cases 206 Statement o the Problem 208 Development o the Design Equations Equation s 209 7.3.1 Special Cases 210 7.4 A Design Procedure For The Stated Problems Proble ms 211 211 Sensitivity Reduction Limitations and Tradeoffs in NMP Feedbac Feedback k Systems 212 81 SISO Plants 214 8.2 MIMO Plants 217 8.3 8.3 Sensitivity Reduction Limitations For a Single Row o S 217 8.4 Sensitivity Reduction Limitations For Several Rows o 222 8.5 Necessary Conditions 225 A Design Example 8.6 227 Exercises 230 Appendix A 237 Appendix B 238 Appendix C 239 AppendixD 239 Appendix E 239 Summary 240 Notes and References 241 NONLINEAR NONLINEA R SYSTEMS
SYNTHESIS OF LTI CONTROLLERS FOR NONLINEAR SISO PLANTS 247 1 Synthesis for Tracking Specifications 247 11 The Schauder Technique Technique 250 1.2 G u i ~ e l i ~ e s for the Choice o PN y dN y and Theoretical Limitatlons 253 1.2.1 1.2 .1 How to choose choo se the pairs PN y dN y 253 1.2.2 Some Theoretical Limitations 256 13 The Homotopic Invariance Technique 258 1.4 An Example Exa mple 261 26 1 1.4.1 Implementation o the Design Procedure A Four-Step Four-S tep Process Proce ss 261 261 2 Synthesis for Zeroing the Plant Output 264 2.1 2.1 The Schauder Scha uder Technique 267
xu
QUANTITATIVE
2.2
3
4 5
FEEDB
CK DESIGN
for the Choice o PN y dN y and Theoretical LImItatIOns 2.2.1 How to choose cho ose the pairs PN y dN y 2.2.2 Some Theoretical Limitations 2.3 The Homotopic Homot opic Invariance Technique Techniqu e 2.4 An Example 2.4.1 Implementation o the Design Procedure Procedur e a Two Step Process Appendix A Summary Notes and References G . u i ~ e l i
1 e s
270 270 74
275 276 276 278 280 280
6
SYNTHESIS OF LTV CONTROLLERS FOR NONLINEAR SISO PLANTS 281 1 Statement o the Problem Pro blem 281 28 1 2 The Design Procedure 283 21 Guidelines for the Choice o PN y dN y and the Time Slices 284 3 An Example 286 3.1 Statement o the Problem Prob lem 286 3.2 Single Time Slice Design 288 3.3 The Two Two Consecutive Consecutiv e Time Slices Design Desig n 290 3.3.1 3.3. 1 Design Desig n on the First Time Slice 290 3.3.2 Design on the Second Time Slice 292 3.4 Comparisons and Discussion 293 293 4 Summary 294
7
SYNTHESIS OF LTI CONTROLLERS FOR NONLINEAR MIMO PLANTS 295 Synthesis for Tracking Specifications 295 1 11 The Schauder Schaude r Technique 299 1.2 Guidelines Guidel ines for Choosing Choosin g o PN y dN y and Remarks 305 1.3 1.3 The Homotopic Homoto pic Invariance Technique Techniqu e 308 1.4 An Example Examp le 310 2 Synthesis for Zeroing the Plant Outputs 314 21 The Schauder Schaud er Technique 317 2.2 Guidelines for Choosing PN y dN y and Remarks 319 2.3 The Homotopic Homotop ic Invariance Technique 322 2.4 An example exam ple 323 3 Synthesis o the MIMO LTI Problem 326 3.1 3.1 2 x 2 plants plan ts 328 3.1. 3.1.1 1 A design Procedure With With Less Stringent Stringen t High Frequency Conditions 331 3.1.2 A Third Design Procedure 332 32
n x.1n Plants 3.2.1 3.2 A Design Procedure With With Less Stringent High Frequency Conditions 3.2.2 A Third Design Procedure
332 334 336
ontents
4
5
6 8
Rational TF Approximations From Input Output Data Dat a 41 SISO Systems 42 MIMO systems Summary Notes and References LTV
lll
337 338 339 341 341
SYNTHESIS OF CONTROLLERS FOR NONLINEAR MIMO 343 PLANTS 1 Statement State ment of the Problem 343 2 The Design Procedure 344 21 Guidelines for the Choice of PN y dN y and Time Slices 346 3 An Example 347 4 Notes 354 Index 367
oreword
This book is a very welcome valuable addition addition to the Feedba Fee dback ck Control literature in general and especially to to its practical Engineeri Engi neering ng part. There Ther e are so few books devoted devoted to to genuine practical Feedback Control design. design. There Ther e is a great need for many more such such books in order to hopefully hopeful ly overcome over come the huge gap between much o Engineering Control Academia and Industry i only for appreciation appreciation of the tremendous tremendous power o the simplest kind o feedback compensation compensa tion when it is properly understood and quantitatively formulated. It is also very important that such such books books be written written by active active designers whose elbows have been deep in detailed designs and their practical implementation. e are satiated with superficial review books. t is natural that such a variety of research-oriented practical designers will present a variety o approaches insights - each his his own own specia speciall techniques techniques and short cuts. This is a boon to the genuine student stud ent and design engineer engineer enhanci enhancing ng his perspective perspect ive and design desig n tools. There are very few self-contained books dealing with the theory and practice of Quantitative Feedback T Theo heory. ry. This book is one o the only two that are available. Control engineers will find find in this book many-detailed examples which can be easily applied to many industrial applications from Robotics to Flight control to Ecology. This book presents feedback synthesis for single-input single-output and multi-input multi-output linear time invariant invariant and and nonlinear plants based on the QFT method. t includes design details and graphs which do not appear in the literature. literature. These will enable engineers and researchers researchers understand QFT QF T in great depth. Many examples ar aree presented to help the serious reader to to understand and apply the extremely powerful design techniques. Professor Profe ssor Oded Yani Yaniv v has extensive highly successful practical pract ical design ex perience. He has solved practical extremely challenging challengi ng problems probl ems which were given up as hopeless. In additi addition on he is a leading researcher in this highly im portantt but academia - mutilated subject portan subject.. The Control community communit y is is indebted
XVI
QU NTIT TIVE FEEDB CK DESIGN
to him for this sorely needed book this gift
t s
hoped that they will take advantage of
Isaac Horowitz Dec 1998
reface
This book presents practical techniques for designing CONTROL systems for linear and nonlinear plants based on Quantitative Feedback Theory (QFT), (QFT), whose origins can be b e traced back bac k to I Horowitz (1959). (1959). The QFT QF T s a frequency domain method based on two observations: i) feedback s needed to achieve a desired plant output response n the presence of plant uncertainty and/or unknown disturbances; and U) a controller which produces less control effort s preferred, that is, a controller whose b ndwidth is smaller is preferable. The QFT s qu ntit tive n the sense that it synthesizes a controller for the exact amount o plant uncertainty, disturbances and required specifications. One additional very attractive property o the QFT technique s the fact that narrow bandwidth controllers controllers result as compared to designs which assume special structures for plant uncertainties, disturbances and specifications such as norm-bounded uncertainties etc. This book is about feedback feedback synthesis for single-input single-output si ngle-output (SISO), multi-input single-output (MISO) and mUlti-input multi-output (MIMO) linear time invariant invariant and nonlinear plants, plants, by the QFT method. method. It includes design details which do not appear in the literature, and thus can help engineers and researchers understand QFT n greater depth. depth. Many examples example s are presented, so that the reader can study them and acquire the experience and a nd understanding o the design techniques involved involved.. In particular each MI MIMO MO technique is first first explained for the simple two-input two-output case, and then extended to the general multi-input multi-output case. Local linearization is most often used to design feedback around nonlinear plants.. The QFT techniques are suitable for feedback design around the loc lly plants line rized model because the design takes into account the exact uncertainty without inducing over-desig over-design. n. However this technique may not be suitable for systems whose operating point changes rapi rapidly dly.. One design method which overcomes this difficulty and is presented present ed in this book is the glob l line riz tion approach which s base based d on functional aanalysis nalysis the theory. ory. The reader rea der who is
X V lll
QU NTIT TIVE FEEDB
CK DESIGN
unfamiliar with this branch of mathematics may skip the proofs, since a good understanding of Linear Systems Theory s really all that is required in order to understand the material presented in this book. This book bo ok can be used aass a text in in any course on control system sys tem design at both the graduate and undergraduate levels. levels. The prerequisites are a course in clas sical linear systems, minimal understanding o f sampled data systems and the z-transf z-tr ansform orm only for section 5.), and an introductory course in in c1assica c1as sicallin llinear ear control theory theory.. Som Somee functional analysis s needed for the nonlinear techniques but these can be explained and applied without understanding the underlying mathematical mathema tical the theory ory.. Engineers are highly highly encouraged to use this book to ap ply QFT to industrial applications; the techniques are not only quantitative but also offer the following two important properties for practical feedback design tradeoffs among various various considerations such as: as: solution com i) insight into tradeoffs plexity, amount o f uncertainty, specifications, amount of scheduling, sampling rate, and cost o f feedback; and ii) no need for a model in state-space form or any other form, since the QFT techniques can be applied to the measured plant responses at a dense set o f frequencies. Property ii) enables us to extend QFT nonlinearr plants plants.. Researchers n control systems are encouraged to to many nonlinea use this book n order to gain insight into synthesis of feedback systems based on the QFT methodology, compare their design results with that o f QFT, and assess and improve their design results where applicable. All o f the examples where implemented using a t l a b ™ version 5.3, the script file filess can be found at the following W Web eb si site: te: http://www http://www.eng.tau.a .eng.tau.ac.il/yaniv c.il/yaniv.. ODED Y
NIV
cknowledgments
is a pleasure for me to acknowledge several individuals who have helped m e thro through ughout out the years. The first first is Prof. Isaac Horowitz Horowi tz who w was as my Ph.D supervisor and as such taught me QFT Prof. C.H. Houpis hosted me sev sev t
eral times at the Air-force institute of technology (AFIT), Dayton Ohio, and encouraged me to develop QFT software and the extension o the Golubev identificatio identi fication n algorithm algorith m to MIMO systems. Prof. Yossi Yossi Chait and I have been working together for the last 10 years and along with Craig Borghesani we developed the QFT toolbox. Dr Yah Yahali ali Theodor has shared share d his extensive prac tical experience with me in the form o many long discussions and I ve enjoyed numerous stimulating discussions with my colleague Dr Per-Olof Gutmann. Dr Marcel Sidi and I cooperated on the topic o bandwidth limitations o non minimum-phase minimum -phase SISO systems. systems. Special appreciation is due to Igor Igo r Chipovet sky for preparation o many o the LTI MIMO examples, to Ronen Boneh for preparation o the SISO LTV example and to Haim Weiss from Rafael, Israel, for his contribution to section 1.3. To Arik Dickman whose excellent insight into industrial control problems he shared with me during our cooperative years in the control industry. My fi final nal word word o acknowledgment goes to Aron Pila o IMI (Israel (Israel Military Military Industries Industries - Advanced Systems Division) who did an outstanding job o reviewing the manuscript, improving its presentation and generating some o the plots and figures. figures. His contributio contr ibution n is deeply appreciated.
bbreviations Notation and Symbols
bbreviations
DOF - Degr Degreeee-Of-F Of-Free reedom dom LHP - Left-H Left-Half-Pl alf-Plane ane LFT - Linear Fractional Transfo Transformatio rmation n LTII LT - Linear-Time-Invaria Linear-Time-Invariant nt LTV LT V - Linear-Time- Var Varyin ying g MIMO - Mult Multi-Inp i-Input-M ut-Multiulti-Outpu Outputt MTF - Mat Matrix rix Tran Transfer sfer Func Function tion - Min Minimu imum-P m-Phase hase MP NL NMP QFT RHP SISO SIMO MISO TF
ZOH
- NonNon-Line Linear ar - Non-M Non-Minimum inimum-Phase -Phase - Quantit Quantitative ative Feedback Theory - Righ Right-Half t-Half-Plan -Planee - SingJe-Inp SingJe-Input-Single ut-Single-Output -Output - Single Single-Inpu -Input-Mu t-Multi-Ou lti-Output tput - Mult Multi-Inp i-Input-Si ut-Singlengle-Outpu Outputt - Transf Transfer er Functio Function n - ZeroZero-Orde Order-Hol r-Hold d
Notation
• Capital or lower case non-bold italic italic letters letters denote SIS SISO O TF s, for example P , G F ,g , f • Signals and their Laplac Laplacee transfor transforms ms are denoted by the same iitalic talic JJet ette ter. r. In order to avoid confusion, s , j w or t are added as arguments whenever necessary, for example r , r t), r s), r jw). • The capital letter P , in all its forms, is reserved for the plant: P ,
Pk
P,
Pk> PN , y , P N,y·
• Fo Forr LTI plants, in order orde r to avoid confu confusion, sion, s , j w
or t
are added as
arguments whenever necessary, for example P s), P(jw), P s), P jw). • Bold capital letters stand for matrices; the same non-bold lower lower-case -case letters = [mij] stand for its entries:
XXll
QUANTITATIVE FEEDBAC FEEDBACK K D ESIGN
• Bold Bol d lower-case letters letters stand stand for vectors; vectors; the same non-bold lower-case lower- case letters stand st and for its entries: entries: v = [Vi] v = [VI , Vn] v = [VI , Vn]T. • MIMO plant inverses inverses p I have entries: p I =
[nij].
• Sets are denoted by parenthesis { }, for example { P} is a set o f plants P . • Matrix Matri x and vector inequalities, for example A element inequality.
B
means element by
• The notation PN,y P N,y , means that the plant, P and y N and y).
P), is a function o f N
• The notation dN,y dN,y), means that the disturbance, d d), is a function o f N a n d y N and y).
• diag(gl ... , gn denotes a square n x n matrix or MTF) whose diagonal i i element is gi and all other elements are zeros.
ymbols
H2
- The Hardy space o f all complex valued functions F s) which are analytic in the open right half plane and which satisfy the condition
[ s u P ~ > a Hoo
L2
~
jw dw]
< 00
- The Hardy space o f all complex-valued functions F(s) which are
analytic and bounded in the open right half plane, R e s > O Bounded means that there is a real number b such that I F(s) < b, R e s > O The least such bound b is the Hoo norm o f F denoted by F 11 . Equivalently F 11 = sup{1 F(s) I R e s > o}. By the maximum Modulus Theorem, the open right half plane can be replaced by the imaginary axis and thus F 11 = sUPw{1 F(jw) I w E R}. - The Hilbert space of all measurable functions f : R+ - R with the property that oo f t) 2 dt < 0 0 , and where the inner product is defined as the square o f the norm, that is < f, f >= oo f(t) 2 dt, while . signifies the Euclidean norm.
~ x n _ The
L}
I ~ o o F
set of k x n matrices whose elements are in L 2 .
space - The Hilbert space
of all measurable functions
f : R+
f(t) 0 < t < T { property that h E L2 where h = 0 t >-T -
-
R with the .
for all fillIte T .
XXlll
consisting o f real-rational functions. F RHoo i f and only if F is proper
o
and Ui t)
=
-Uo i f Ui t) < -Uo;
• the sensor noise resp response onse at the plant output may be too large; • if the plant has a pure delay and/or NMP and/or the controller is digital, the controller cannot b bee too strong theoretical limitations discussed in section 4.); • due to unknown high ffrequency requency plant dynamics and/or high frequency problematic dynamics for example hig highly hly under-da under-damped mped resonances), the
I
The next equation is valid under the conditions that G
-+
00
implies L
I
jw)
-+
01 and
[I LJ- 1 P G H -+ 1 2For SISO controllers we say that G dj w ) is stronger or larger) than G2 jW) if ILdjw 1 > 1 and the amplitude of h jw) is greater than the amplitude of G 2 jw). If the controller is MIMO. we can qualitatively say that that a stronger larger) controller is one for which sensitivity reduction is better.
4
QUANTI QUANTITATIVE TATIVE FEEDBACK DESIG N
controlle r must be small enough abov controller abovee certain frequenc frequencies, ies, thus limiting the controller strength as in case of NMP plants; plants; • stronger strong er controllers fforce orce the actuators producing the plant input to move faster and with larger amplitudes, which may decrease the life-time o f the actuators and may also produce larger friction with the surroundings. Du e to these drawbacks we shall always seek a controlle Due contr ollerr and filter) filter) which satisfy the desired specifications but produce small plant inputs u. The plant input is called the control effort. 3
THE DESIGN PROCESS O
FEEDBACK CONTROL
SYSTEMS
Any feedback controlled system must include the plant to be controlled, as well as its sensors a controller and the actuators which drive the plant. A schematic description of a typical system and its signals is depicted in Fig. 1.2.
External
External
command
disturbances
Controller
ensor
nOlse
-
Uact
-
Actuator
'
-
'
Plant
Measu ed data
Sensor
igure 1.2.
u sys
output t o b e controlled
-
A simplified feedback system
While undertaking the design of a feedback system, the control engineer should consider many issues, even before purchasing a single component. These include a variety of topics such as: choice o f the system structure and components, modelling the plant, choice of sensors and actuators including their location, mechanical design, hardware and software issues, control design and simulation, simula tion, and making adequate provisions hardware and and software) for testing. A recomm recommended ended feedback control design process is described describe d schemat ically in the flow chart of Fig. 1.3 followed by a detailed explanation o f each of its blocks.
Introduction
5
Block description o system
relax
improve
Simulation
Fabrication A feedback control design process, see see also Fried Friedland land 1996)
Figure 1 3
The first phase is to perfonn a block description o the system which should include: l
Block diagram
2
Characterization
3
Definition
o
o
the system. o
its subsystems.
the physical variables
to
be controlled.
The second phase consists o a determination o the performance object objectives ives the system and should deal with the following:
o
QU NTIT TIVE FEEDB CK
ESIGN
1
A statement o f the goals regulation, or both both..
2
A decision as to the closed loop specifications, which may include distur bance rejection specs, tracking accuracy, system type, rise times, settling times, step response overshoots, minimum mini mum damping ratios o f the closed loop
of
the feedback system such as tracking (servo),
poles, minimum mini mum gain and phase margin margins, s, ma ximum limit cycle amplitudes, etc. 3. A summary summa ry o f any outstanding non technical item itemss such as pricing, cus custome tomerr demands, cost/performance ratio, weight, reliability, etc. 4. In many applications, the mechanical design o f the plant should e as rigid as possible possib le and guided by the cont control rol eng engineer. ineer. A basic rule o f thumb is that the plantt s lowes plan lowestt resonance frequency should be at least twice the OdB crossing frequency of the open-loop gain (determined by the specs) specs).. Para Parameter meterss such as friction and backlash should be minimized if possible. t is good practice not to let the control engineer solve problems that could e solved by the mechanical designer, since the control based solution will usually e at the expense o f other equally important criteri criteria. a. The performance objectives should therefore include damping ratio and resonance limitations o f the system. 5
It is possible that a relaxation o f the performance specification specificationss will e possible as a result o f simulation studies (including Monte-Carlo simulations).
In the third phase the control hardware is specified in conformity with the following requirements, which may depend on the choice o f the system con figuration, see first item in the control design phase: 1
Sensorr sele Senso selection ction - inert inertial ial un unit itss such as gy gyros, ros, non-inert non-inertial ial types such as tachometers, resolvers, cameras, etc., all o f which should e chosen in accordance with the environment in which the plant must operate and the required performance o f the closed loop system.
2. Choice o f sensor location location - for mechanical system systems, s, in genera general, l, at the most rigid point o f the plant to avoid additional highly underdamped complex poles and zeros, due to a non-collocated configuration. 3
Selection o f the required actuators, including their precise location within the syste system m - cho choic icee o f actuators should usually consider the moments (in case o f mechanical systems) that could be developed (these depend on the disturbances, comma commands nds and bandwidth), the required valve sizes and types in the cas casee o f process control, the supply voltages voltages and currents and the power powe r dissipation that is converted to heat and affects the temperature conditions o f the entire system system..
Introduction 4
7
digitally interfaced sensors are used, the number o f bits should be chosen according accord ing to the required dynamic range and accura accuracy. cy. Com Compone ponents nts such as A I D s and D I A s should be chosen so that the required dynamic range and accuracy are not degraded. f
5
fa
digital con controller troller is imple implemente mented d with a fi fixe xed d poin pointt processor, a minimal word length should be chosen such that sufficient accuracy is maintained, unless a floating point processor is used.
6
The
7
Analog components should be chosen with sufficient bandwidth and mini
processing speed of a CPU CP U usual usually ly DSP) should be sufficien sufficiently tly fas fastt so that all required computations are performed within the minimal sam pling interval interval or at the maximum sampling frequency. Sa Sampl mpling ing frequency between 5-10 times the 3 d B crossing frequency o f the closed loop gain response, or small enough to enable digital filtering o f high frequencies and/or high modes is usually a good selection.
mal offset and distortion especially if the ffirs irstt in the forward chain loop).
of
the
8
Care should be taken, to include test points, status reporting and the capa bility to check different blocks in the control system and modify internal parameters from the outside.
9
is possible that a change in the choice as a result o f the simulation phase. t
The fourth phase involves the development of the system which should include
of
of
control hardware will be needed
a truth model and a desi design gn model
1
Detailed mathematical modelling o f the plant, plant, including nonlinear elements, to be used for evaluating evaluating the feedb feedback ack system performanc performancee by simulations. In many applications the model will depend on experiments.
2
Choice o f a simplified model o f the plant for control design, most likely linear time invariant or linear time invariant with modelling uncerta uncertainties. inties. It is very important that the design model be comprehensive enough to include all o f the dominant effects to allow for a reliable design, while remaining simple enough for the design procedure to proceed smoothly.
3
Modelling o f the chosen sensors and actuators including their noise spectra, drifts and nonlinearities.
4
Estimation o f the exogenous disturbance models, spectra or signals which act on the system.
8
QUANTITATIVE
5. Estimation
of
FEEDB
the degree
CK DESIGN
of
modelling uncertainty in the plant, sensors and
actuators. The fifth phase is the control design that is, design algorithm, and should consist of: 1. Choice
of
of
the feedback control
of
the feedback configuration in the form a one, two, or higher degrees o f freedom structure, such as a cascade structure for exam example ple aan n outer position loop aided by an inner velocity loop for improved closed loop damping), which is superior, in many applications for instance in flight control), to the standard one or two degrees o f freedom scheme.
2. Digital compon component ent structures direc direct, t, parallel, etc.) should be considered consid ered
to minimize limit cycles, computation nois noise, e, etc. and increase robustness due to coefficient quantization quantization). ). 3.
computer controlled systems the determination rates must be made. or
4. Design
o f the
required sampling required
the controllers and feed-forward signals. signals. This may include: partitioning of the system into subsystems by decoupling, it is reco recommen mmended ded designing each subsystem separately with the simplest possible controllers, if the system can be decoupled adequately; choice o f a hierarchical structure loops within loops); use of the separation principle, if possible or needed, to design full state feedback low and an observer; using scheduling, nonlinear and adaptation algorithm; etc. of
an d iteration iteration on the the previous The sixth si xth and fina finall phase is to perform simulations and phases. This is a key key tool tool for evalu evaluating ating system perform performance ance before equipme equipment nt purchase) and should include: 1
Simulation
of
the complete linear and nonlinear closed loop system.
2. Base Based d on the simulation resu results: lts: 1) improve the design, design, and if necessary add scheduling and/or adaptation and/or nonlinear control element; ii) decide upon the required modifications in order to evaluate tradeoffs amongst the
hardwar e components and the chosen performa hardware performance nce objectives bandwidth, closed loop performance, expected amplitudes o f possible limit l imit cycles, etc). When a system already exists at the hardware level, the control engineer may be called upon to perform the following testing tasks: 1. Identify the plant, sensors and actutators, which may include pure delays,
linear time invariant invaria nt as models models whichfriction, may depend on th the e input lev level), el), non linear elements such dead-zone, saturation, etc. 2. Modify the estimated model to fit the measured model.
Introduction
3
Redesign the controller and simulate the overall system to assess the ex pected closed loop performance. f the controller includes a nonlinear algorithm, this is the opportunity to improve its parameters.
4
Test the final integrated hardware and software system.
Note that it might be b e impossible or too expensive expensive to perform experiments experi ments with all o the actual hardware. hardware. For Fo r example, in the design process o a missile system, the missile seeker is placed on a multi axis table which is driven by full six-degree-of-fre six-deg ree-of-freedom edom aerodynamic simulations. In this way way it is possible to increase the confidence level o the system s simulation and save expensive firings. While performing all o the above-mentioned above-mentioned steps might seem complicated and require much muc h knowhow, knowhow, in most applications previous previous knowledge knowl edge o similar systems or older versions of the same system provides answers to the major items listed above. above. For more about the challenges o the process o feedback control system design and an example, see Friedland (1996). 4
OOK OUTLIN OUTLINE E
This book deals with synthesis synthesis of o f feedback for uncertain plants by the quan titative feedback theory (QFT). The QFT design technique emphasizes that feedback is necessary to achieve desired closed loop specifications (in the form o tolerances), in spite of o f plant plant uncertainty and/or unknown disturbances, while trying to minim minimize ize the control effor effort. t. The plants can be single-input single-output (SISO), mUlti-input single-output (MISO), multi-input multi output (MIMO), linear time invariant (LTI) or nonlinear; the synthesized con trollers are LTI or linear time varying (LTV); the plant can be given by a model (or a set o models i it is uncertain) or by its measured transfer function (TF) at a dense enough set of frequencies. The synthesis techni techniques ques presented pres ented are suitable for continuous conti nuous as well as as discrete controllers and plants. The Th e plant plan t can include pure delay. Chapter 2: presents the basic properties o SISO feedback systems without considering consideri ng robustness issues issues.. Firs Firstt the notion o gain margin, phase margin, bandwidth bandw idth and cross-over cross-ov er frequencies are discussed. Then it is explained why it is so important to decrease the controller bandwidth and in this regard the high-frequency-gain is defined. A controller is designed to comply with the desired closed loop specifica tions, and time domain as well as frequency domain specifications are defined. Although there is no one to one translation from the time domain specifications to the frequency domain specifications, two efficient algorithms are proposed. The benefits o feedback for for non-minimum-phase (NMP) open loop systems are limited in the sense that their bandwidth has an upper limit, the same is true
10
QU NTIT TIVE FEEDB CK
ESIGN
the open loop is is unstable but the bandwidth has a lower limit. Equations Equat ions and graphs which whi ch relates these limitations as a function o margins and cross-over frequencies are given. i
Finally a section is devoted to helping the reader gain skills in loop shaping via examples and exercises. Chapter 3: details the QFT design technique for SISO and MISO systems. The important features o this approach are: i) it is robust to the exact amount o plant uncertainty; uncertainty; ii) it tailors the closed loop precisely to the specifications which are given at each frequency; and iii) the technique is graphically based therefore allowing for insight into tradeoffs amongst design parameters such as complexity scheduling amount o uncertainty sampling samplin g time margins and bandwidth. bandwid th. As a result the method allows the design designer er to come up with low bandwidth bandwidt h designs. The design technique is developed separately for one and two degree-of freedom systems and detailed algorithms algorithms for computing bounds are given. given. Then it is extended to to sampled-data syste systems ms and NMP p plants lants and a detailed explanation o the NMP phenomenon is provided. Chapter 4: extends the QFT technique to the design o controllers for MIMO fixed or highly uncertain plants in order to achieve desired closed loop spec ifications. The Th e basic idea is to break the design process down into a series o stages. stage s. Eac Each h stage o this sequential process is a simplified SISO or MISO feedback problem which can be solved by the techniques presented in Chap ter 2 A solution to the original problem is then simply a combination o the solutions obtained at each stage stage.. The types o closed loop specifications considered here are given in the frequency domain and include include:: disturbance d isturbance rejection reference tracking gain and phase margins margins control eeffort ffort and design design for diagonal minimum-phase minimum-p hase elements o the tracking MTF. MTF. The plants considered consider ed can be stable unstable unstabl e NMP NM P NMP and unstable or given by measured data. The benefits o NMP plants are restricted; these restrictions are discussed with reference both to SISO and MIMO systems. systems. Finally it is shown how to square plants with more inputs than outputs such that the splitting o the control action will be within specified tolerances. Section 8 is optional and was included as a theoretical complement to the less theoretically oriented sections of chapter 4 I t was felt that its inclusion would supply an appropriate mathematical backdrop for researchers interested in NMP systems and could provide a convenient point o departure for further research into this subject. Chapter 5: presents the QFT technique for the design
o
LTI controller and
ntroduction
11
feed-forward command, for uncertain nonlinear plants, in order to achieve closed clos ed loop specifications on a finite or infinite time interval. Two types o problems are considered: (i) Tracking with given initial conditions; and ii) zeroing the plant output for non non-zero -zero initial conditions. The addition o disturbance attenuation to each o these problems is a trivial extension. The proposed techniques are based on the Schauder mapping theorem de veloped by Horowitz and the Homotopic invariance technique suggested by Barnard, which is a natural extension o Horowitz s method and is the pre ferred technique. Chapter 6: In Chapter 5 we show how to design an LTV controller for an uncer tain nonlinear plant, based on the assumption that the nonlinear nonli near plant pla nt equation, y = N u where y is the output and u is the input, can be replaced by an LTI uncertain plant, PN,y, with disturbance, dN,y, in the form y = PN,yU dN,y. Essential conditions for a successful application o the technique are that the uncertainty on PN,y be limited such that there exists an LTI controller which stabilizes PN,y and that dN,y be small small enough. enough. These conditions restrict the sets nonlinear plants to which the the proposed design process can be applied. applied. Thus, qualitatively we can say that i the nonlinear plants deviate too much from LTI plants, the existence o a solution is questionable. Clearly, the deviation o a nonlinear plant pl ant from an LTI plant is much smaller on a finite time interval than for all t O Hence the idea is to update the chosen LTI plant and disturbance set {PN,y, dN,y} along the system trajectory, that is, for different time inter vals va ls.. The result will be a piecewise LTI controll controller, er, the structure o which may not be fixe fixed d on the different time intervals. intervals. This chapter presents this technique. o
Chapter 7: proposes a synthesis technique for the design o a MIMO LTI controller and feed-forward, for an uncertain MIMO nonlinear plant, in order to achieve closed loop specifications, on finite or infinite time intervals. The basic idea is to integrate the following three techniques: i) The MIMO MI MO design design technique for nonlinear systems suggested by Horowitz and Breiner (1981) which is an extension o the nonlinear technique for SISO systems presented in Chapter 5; U) the MIMO technique for LTI plants presented in Chapte Chapterr 4; 4; and iii) the nonlinear SISO technique based on Homotopic invariance presented in Chapter 5 Tw Two o types types of o f problems are consid considered: ered: tracking for given zero initial conditions; and zeroing the plant output output for non-zero initial conditions. The addition o disturbance attenuation to each of these these problems is also considere considered. d. Chapter 8: In Chapter 7 we saw how to design an LTV controller for an uncer tain nonlinear MIMO plant, based on the assumption that the nonlinear plant equation, y = N u where y is the output and u is the input, can be replaced by an uncertain set o LTI plants, P N,y, with disturbances, dN,y, in the form
12
QU NTIT TIVE FEEDBACK DESIGN
= P N,yU
Essential conditions for a successful application o f the technique are that the uncertainty o f the plant P N,y, be limited such that there exists an LTI controller which stabilizes it and that the disturbance d N , y , be small enough. These conditions restrict the ssets ets of nonlinear plants to which the prop proposed osed design proce process ss can be applied. applied. Thus qualitatively we can say y
dN,y
i f the nonlinear plant deviates that deviates too the much from ano fLTI LaTInonlinear plant theplant existence of a solution solu tion is is questionab questionable. le. Clearly deviation from an LTI plant s much smaller on a finite time interval than for infinite time intervals. Hence the idea o f updating the chosen LTI plant and disturbance set { N ,y , d N , y } along the system system tra trajecto jectory ry and updating the the controller control ler and it itss initial conditions accordingly is appropriate. appropriate. This results iin n a piecew piecewise ise L LTI TI controller or linear time varying controller.
LIN E
R SYS SYSTEM TEMS S
Chapter 2 BASICS OF SISO FEEDBACK CONTROLLED SYSTEMS
1
INTRODUCTION
In this chapter we present the basic properties o f single-input single-output feedback systems without considering robustness robustness issues. issues. First the notion o f gain margin, phase margin, bandwidth and cross-over frequencies are defined and discus discussed. sed. The Then n it is expla explained ined why it is so important to decrease the controller bandwidth and in this regard the high-frequency-gain is defined. A controller is designed in order to comply with desired closed loop spec ifications, and time domain as well as frequency domain specifications are defined. defin ed. Although th there ere is no one-to-one translation from time domain to the frequency domain specifications, two algorithms are proposed which attempt to bridge the gap. The benefits o f feedback for non-minimum-phase open loop systems are limited in the sense that their bandwidth has aan n uppe upperr lim limit. it. On the other hand, if the open loop iis s unstable the bandwidth bandwidt h has a lower lim limit. it. Equat Equations ions and graphs which relate these limitations as a function o f margins and cross-over frequencies are given. Finally a section is devoted to helping the reader gain skill via examples and exercises.
2
in
loop shaping,
BASIC FREQUENCY DOMAIN CHARACTERISTICS
The feedback system considered here is described in Fig. 2.1 2.1.. An experi enced control engine e ngineer er gains excellent insight into the sy syste stem m s beha behavior vior by P G H and filter F. The looking at the Bode or Nichols plot o f the TF s L T F L s) is called the open loop or loop-transmission and the T F from any command input, say r to any output, say 5
y
is called the closed loop
O. Yaniv, Quantitative Feedback Design of Linear and Nonlinear Control Systems © Springer Science+Business Media New York 1999
TF
from
16
QUANTITATIVE QUANTITATIVE FEEDBACK DESIGN
r
-
-
F
e
G
...
u
Y
p
I
H Figure 2.1.
A SISO 2 degree-of-freedom feedback system
r to y. Several key parameters which govern the system s behavior are now defined and explained.
2.1
RELATIVE STABILITY STABILITY CROSS-OVE CROS S-OVER R FREQUENCY AND BANDWIDTH
An example o f a Nichols plot o f an open loop is shown in in Fig. 2.2. It includes 5 parameters which characterize the open loop system and have a strong impact on closed loop system behavior: 1
Cross-over frequency or phase-margin frequency, where the open loop is OdB, IL jwcj» = OdB.
wcj>:
is the frequency
1
2
Phase-margin, ¢: is the phase above -180° ¢ = argL jwcj»
3
Gain-marginfrequency, WM: is the frequency wherearg L jwM)
4
Gain-margin, M : is the distance in d B of L jw M from the
o f the
open loop at the cross-over frequency 180.
Nyquist plot OdB,-180° on the Nichols chart), M 5
=
1
=
-180°.
point on the
- 2 0 log IL jwM 1.
Bandwidth, Wb: several definitions of bandwidth appear in the literature, all almost coincide, we choose here the frequency where the closed loop · d amp1Itu e
S
L ( jjWb) Wb) - 3dB 1I+PG
I --
- 3dB .
In general, the relative stability gain-margin and phase-margin, are figures o f merit for two important aspects o f the closed loop system: i) how much uncertainty uncertaint y the plant can can tolerate tolerate and and remain closed loop loop stable - for example i f the gain-margin is lOdB, the system can tolerate a lOdB increase in gain without losing closed loop stability; and ii) it puts an upper bound on the amplitude o f the closed loop TF from sensor noise to the plant output, and an upper bound on the sensitivi sensitivity. ty. Having large large gain-margins and phase-margins, phase-margins , implies suppression o f the closed loop resonances at frequencies around the bandwidth frequency Wb, but a serious drawback arises in the form o f an
Basics o f SISO Feedback Controlled Systems
Nichols Chart
30
17
o dB
.0.25 dB 0.5 dB
.. .1: dB
3dB
-30 -360
-270
-180
Open-L Ope n-Loop oop Phase Phase deg) deg)
-90
o
Figure 2.2. Definition of gain-margin M, phase-margin cp cross-over frequenc frequency y w ¢ bandwidth frequency W b , and gain-margin frequency W M , for the TF L s
increase of sensor noise response at the plant input This is exemplified by the following example: example : Figs. 2.3-2.4 depict depic t the Nichols and Bode plots o f 3 loop transmissions, L1 L2 and L 3 with margins o f (130°, 14dB), (140°, 12dB) and (150°, 6.5dB) respectively, and all having the same cross-over frequency. From Fig. 2.4 the sensor noise at the plant input of L is much higher than that o f L2 which is much higher than that of L 3 . The noise, whose spectrum is located in the high frequency range, is amplified by L to the order o f 6dB more than it is amplified by L2 and by 16dB more than it is amplified by L 3 . The above definitions don t hold if there are several cross-over frequencies and/or several gain-margin frequencies, in which case the above figures of merit may be misleading. Examples are shown in Fig. 2.5. 22
CONDITIONALLY STA STABLE BLE SYSTE SYSTEMS MS
Open-loop TF s may have have positiv positivee and neg negative ative gain gain-margins. -margins. Any practical open loop system has a positive gain-margin which means that when the loop is closed the system loses stability if the open loop gain increases too much. f
18
QU NTIT TIVE
FEEDB
CK DESIGN
30 20 10
0 aJ
Cl
-10 -20 -30 -40 -360
-270
-180
deg
-90
o
Figure 2.3. OpenOpen-loop loop Nichols plots for £1 £2 and £3 having almost the same low frequency characteristics and same cross-over frequency, but different margins
an open loop system s ystem has a neg negative ative gain-marg gain-margin, in, then the system in closed loop loses stability if the open loop gain decreases too much, and such systems are called conditionally stable systems. The main practical precaution with such systems is to guarantee that during operation, the plant input will not saturate, becaus e such systems can easily lose stab because stabilit ility. y. The simplest mathe mathematic matical al explanation o f this phenomenon uses describing functions and is based on the assumption that saturation is equivalent to an effective decrease in open loop gain Gelb and V Vander ander Ve Velde lde,, 197 1970). 0). Anothe Anotherr explanati explanation on is based on the circle criterion Friedlan Friedland d 1 1996) 996) which guarantees stability in the case o f saturation if I ~ L
j w ) /
l
L
j w ) ) 1
<
1 for all frequencies, a condition which never
occurs for conditionally stable systems systems.. Examples o f such systems are shown in Fig. 2.6. All ar aree open loop un unstable stable except the upper-rightmost upper-right most graph which is open loop stable.
23
HIGH FREQUENCY GAIN
As mentioned in Chapter 1 one o f the factors which limits system band width and thus lim limits its system performance) is the sensor noise at the plant input. For example, let us look at three controllers for the plant 1 / s2 all possessing
Basics o f SISO Feedback Controlled Systems 6
0
~
~
~
~
19
~
~
150
200
:::J40
+ -.. T
(J
20 100 rad/sec
50
T
-..
(J
10 O
L
10 1
~
~
~
log m)
~
~
~
~
~
~
102
Figure 2.4. TF s from sensor noise to the plant input for the loop transmission upper plot in arithmetic scale, lower in log scale
of
Fig. 2.3,
almost the same low frequency open loop behavior and the same cross-over frequency but having different high frequency responses as shown in Fig. 2.7. Plots o f the TF s from the sensor sens or noise to the plant pl ant input, G / 1 are L), 80 given in Fig. 2.8. Clearly if the sensor noise spectrum is concentrated+above rad/sec, the controller G 3 is superior to controllers G 1 , G 2 , and G 2 is superior to G 1 G amplifies high frequency noise signals by 7dB more than G 2 and by 20dB more than G 3 ). Thus an important figure o f merit for comparing two controllers controll ers is: is: how much is the sensor noise amplified in the high frequency range by one of the designs as compared to the other. W e shall then say that the high frequency gain of one design is larger than the other by xdB s, i f its sensor noise amplification amplification at high frequencie frequenciess is larger than the other ot her design by xdB s. This topic was dealt with by by Helton and Merino (1998) (199 8) pp. 23-24 with respect to the controller roll-off and trade-offs between bandwidth and other performance measures. Example: Given the plant P = 1 0 / s and the two controllers such that their open loop TF s are L and L3 of Figs. 2.7-2.8. 2.7-2.8. Their closed loop performances
20
QUANTITATIVE QUANTITA TIVE FEEDBA CK
ESIGN
Nichols plot a)
20 III
0
0
-20
-360
20
m
-270
-180
0
-90
b)
o
0
-20
-360
-270
-180
-90 deg
o
90
180
Figure 2.5. (a) an open loop with 3 cross-over frequencies, marked x ; (b) an open loop with two cross-over frequencies, marked x
are expected to be the same because L l L3 up to the cross-over frequency. Now suppose that the sensor noise is a white noise colored by passing it through a bandpass filter [40,3001 1 rad/sec. Fig. 2.9 describes the plant s input signal due to that noise, the upper plot for L l and the lower one for L 3 Clearly the RMS o f the upper plot is at least twice that o f the lower plot. Moreover i f the plant input saturates at ±30 the design whose open loop is L l might lead to instability whereas the design whose open loop is L3 can handle disturbances and/or tracking signals whose response at the plant input is up to ±1O ±20 for the noise and ±1O for the control signal). 24
STABILITY ANAL ANALYSI YSIS S USING NICHOL NICHOLS S CHARTS
We now develop the graphical stability criterion using the Nichols chart rather than the standard complex plane techniques techniques (Bode or Nyquist plots), the lThe symbol [x y defines a bandpass filter with lower and upper cutoff frequencies of x and y rad/sec respectively.
Basics oIS/SO Feedback Controlled Systems 8
O
0
.
270
~
~
o
90
180
o
2
.
~
~
0
~
~
270
~
~
180
~
~
~
90
~
o
~
10
o
0
~
360
20 o
~
20·
20
360
0
~
o
20
4
~
40
o
o
~
60
20
0
0
21
O
0
20
0 10 20
40
360
270
180
deg
90
o
~ g 6 0
270
180
deg
90
o
Figure 2.6 2.6.. Nichols plots of conditionally stable stable open loop loopss TF s: All o f the open loop systems are unstable except for the upper-rightmost plot which is open loop stable
motivation being that the use of Nichols charts is an attractive pictorial way to design design and represent feedback feedback systems. systems. The development presented here is based on the work
2.4. 2. 4.1 1
of
Cohen et al
1994).
CONT CONTINUOU INUOUS S SYSTEMS
Given a feedback system whose open loop, L 8), is strictly proper, define the standard Nyquist contour, r , with positive clock-wise direction, as the semi-circle whose radius is R with jw-axis indentations added as necessary to account for the imaginary poles of L 8). I t is assumed that R is large enough to include all the unstable poles o f L 8). The net total mUltiplicity o f these poles is denoted by n The Nyquist plot is the image o f L 8) in the complex plane due to r The Nyquist stability criterion for a feedback system is as follows:
2.1 The feedback system whose open loop is L 8) is stable i and only i he Nyquist plot o f L 8) does not intersect the 1 point and encircle it n times in the counter-clockwise direction. THEOREM
22
QUANTITATIVE QUANTI TATIVE FEEDBACK DESIGN
30 20 10 0 O 0
-10 -20
-30
-4 ~ 60
-270
-180
deg
-90
0
Figure 2.7. 2.7.
Different open loops with almost the same lo low w frequency characterist charac teristics ics and same cross-over frequency frequency
Vidyasagar et al 1988) presented the following simplification to the Nyquist criterion. Letting Ro b e the ray - 0 0 - 1 ] , a crossing occurs when the Nyquist plot o f L s intersects Ro The crossing crossin g is said to be positive if the intersection is upward and negative otherwise, see Fig. 2.10.
The feedb feedback ack system whose op open en loop is L s) is stable, i and only i he Nyquist plot o f L s) does not intersect the - 1 poin pointt and the net sum o f its crossing is equal to n. THEOREM
2.2
Note that in Theorem 2.2 the ray Ro can b e replaced by any curve connecting the point -1, 0 and the point at 0 0 under a suitable definition o f crossing orientation.
2.4.2
DIS
RETE TIME
SYSTEMS
sy stem with open loop denoted denote d Let us now consider a sampled data feedback system by L z), which is assumed to b e strictly proper while 1 L z) is proper. The standard Nyquist contour, r , encloses the unstable region o f the complex z plane and is therefore the space outside the unit circle with indentations on the
23
Basics o f SISO Feedback Controlled Systems
25 20 ~
-
5
:::::: 10
(9
L L
25
50
100 m
75
125
150
175
200
-
:::::: 10 (9
-5 O
L
~
10 1
~
~
~
~
~
~
~
~
~
10 2
log m)
TF s from sensor noise noise to the plant input. upper plot in arithmetic scale. lower in
Figure 2.8. log scale
unit circle as required, to account for poles o f L z) on the unit circle. Let Zi and Pi denote the zeros and poles o f 1 L z) inside the unit circle, and Zoo Po the zeros and poles o f 1 outsideothe unit circle. respectively. respectively. From Fr om the principle thez)number o f the argument. L f encirclements o f the -1 point made by the Nyquist plot o f L z) is N where N = - Z i - Pd Now, since L z) is strictly proper, the numerator and denominator o f 1 L z) have the same polynomial order, say m and therefore
hence = Zo - Po Stability is guaranteed if and only i f Zo = 0 and hence if and only if = - Po The Nyquist stability criterion for systems with open loop TF s L z), where n is the total number o f poles o f L z) outside the discrete time Nyquist contour (including multiplicity), is the following: TH
OR
M
2.3 Thefeedback system whose open loop is L z) is stable ifand
only i f the Nyquist plot o f L z) does not intersect tthe he n times in the counter-clockwise direction.
1
point and encircle it
QUANTITATIVE FEEDBACK DESIGN
4
8
0
,
-
-
-
~
-
-
-
-
- ,
-
-
-
-
~
-
-
-
-
~
-
- -
-
-
-
-
-
~
-
-
-
-
~
- -
-
-
~
60 l
0
c
40
20
0
@ 20
0. 40
60 8 0 ~
~ 0.5
o
8
0
,
-
-
-
-
-
-
-
-
-
,
~
~
~
~
~ 1.5
1 tim -
-
-
-
-
-
-
-
-
~
-
-
-
-
-
-
-
-
~
-
~
-
-
-
-
-
~
-
~
-
2
~
60 l 0
_c C u
40
20
0
20
0. 40
60 - 8 0 ~
o
-
-
-
-
-
-
-
- ~ 0.5
-
-
-
-
-
-
-
~ 1
-
-
-
-
-
-
-
~ - 1.5
-
-
-
-
-
-
~
2
tim
Figure 2.9 2.9.. Plant input signal for different controllers with the sa same me closed loop bandwidth bandwidth,, upper plot for L 1 lower one for L3
Complex plane
Nichols chart
mag
d
Real
~ - - - - - - -
- - - - - - - - ~ -
0
deg _36 2
10 The notion Figure is the ray Ro
o
0
180
0
crossing in [dB deg] right) and [real imag] left), the thick arrow
Basics o f SISO Feedback Controlled Systems
25
With the above definitions, the simplification in Vidyasagar et al. (1988) can be readily extended to the sampled data case. case. The crossing noti notion on is the same as for the continuous case. THEOREM
feedba ck system whose open loop is L z), is stable i and 2.4 The feedback
only i he Nyquist plot o f L z ) does not intersect the
1
point and the net sum
o f its crossings is equal to n.
243
SOME REMARKS ABOUT THE NYQUIST PLOT
It has become customary in control system design to use only half o f the Nyquist plot (only the positive imaginary part o f the Nyquist co contour) ntour).. Because of conjugate symmetry, a crossing at some So, implies another crossing at its complex conjugate pair o f the same ssign ign.. That iis, s, each crossi crossing ng o f the half Nyquist plot should be counted twice. twice. Using half o f the Nyquist plot, non integer crossings may occur which become an integer when counted twice. Take for example L(s) = 1O/(s - I), L O) crosses the ray Ro 1/2 time but when counted twice the crossings become an integer, l Let us assume that that L s) does not have poles on the imaginary axis, 1 L(s) doesn t have any RHP zeros, and wq denotes the smallest cross-over frequency which is larger than all the crossing frequencies on the ray Ro, then: •
the number o f RHP poles, n, is odd, arg L(jwq,) must be larger than -180°, and L O) must lie on the ray Ro, otherwise the number o f crossings cannot be b e an od odd d numbe number. r. f
• I f the number ofRHP poles, n , is even and IL O)I > OdB, then arg L(jwq,) must be larger than -180° and L O) must not lie on the ray R o. • I f L s) is strictly proper and I 0) I > OdB, then the gain sign is positive, that is, in the following representation ao > 0
(s)
=
amsm sn
am-IS m
bn_ls n -
I
... ao ... bo
provided that there is no RHP pole zero cancellation in L(s) (Yaniv (1988) appendix B). Open loop TF s, L(s), such that 1 L(s) does not have any RHP zeros, have the following ver very y important propert property: y: The phase o f the cross-over cross -over frequ frequency, ency, arg L(jwq,), is negative. A large large class for which this is true is developed in section 8. This class includes any open loop, L s ), such that IL 0) I > 1, L s ) is strictly proper and L(s) has a unique cross-over frequency. 3
CLOSED LOOP SPECIFICATIONS
Consi der the feedback system shown in Fig. 2.11. Consider 2.11. The TF P s) belongs to a set {P} of plants with uncertainties, and the TF s G(s) and F(s) denote
QUANTITATIVE QUANTITAT IVE FEEDBAC K DESIGN
26
the controller and prefilter to be synthesized in order to meet robust stability and closed close d loop specifications. specifications. Closed loop loop specifications specifications o f the system in
d n
r
F
G
p
filter
controller
plant
y
sensor Figure 2.11.
A SISO feedback system:
T
n
d and
Ud
are inputs,
U
and y outputs
Fig. 2.11 are typically described in the time-domain and/or frequency domain, abbreviated as t domain and w domain respe respectivel ctively. y. The t-domain t-domai n specifica tions relate to the system response at any desired output due to a given input, while the w-domain specifications are the frequency domain analogue to the t-domain specs. In the next section section these specification specification types are described and a nd a quantitative translation translat ion from the t-domain to the w-domain is given. given. The ability to translate specifications from one domain to the other is very important since all o f the design techniques presented here are in the w-domain, while a true evaluation o f a feedback system s performance is done in the t-domain. 3.1
T DOMAIN SPECIFICATION
Clo sed loop specifications on the system in Closed in Fig. 2.11 2.11 are typically described descri bed in terms o f the plant input and output signals. signals. These Thes e should b e bounded such that the system syst em will function within its working region and will give the desired desire d time response. ome examples follow: follow: plant output should be kept close to zero (a regulation system), the specifications can be o f the following form form:: for given given non-zero non-zer o initial conditions or given disturbances, the plant output is bounded by a given time function (see Fig. 2.12a,b).
1
The
2
The
plant output should follow a given desired output (a servo system) where, due to plant uncertainty, the desired output is bounded between upper and lower time functions (see Fig. 2.12c).
Basics o f SISO Feedback Feedback Controlled Systems
27
The plant input, for a given sensor noise and/or given disturbance and/or given tracking command, is bounded by a given time function (often in order to avoid saturation, see Fig. 2.12d).
3
1 .
~
1
5
0.5
::J
a.
c
0.. -0.5
region
a.-0.5
o
2
1
-1
3
1 . 5 .
c)
-
0
1
2
20
~
-
~ 1
:::J
a.
o
d)
S.region
0
C
5
3
10
c
::J
a.
S
co
1
~ 0
0
o
co
b)
05
::J
0
o
~
co
0..- 1
o
1
time
2
3
2
0
L
o
1
~
time
2
~
~
3
Figure 2.12.
Examples of t-domain specifications: (a), (b) on the plant output for a given set of initial conditions; (c) on the plant output for a step command; and (d) on the plant input
The classical t-domain specifications such as step response rise-time, settling time, maximum overshoot coefficient errors (D azzo and Houpis 1988) are special cases o f the above above exa examples. mples. The maxi maximum mum overshoot and coefficien coefficientt errors are in fact w-domain specifications (D azzo and Houpis 1988). 3.2 3. 2
w DOMAIN SPECIFIC
TION
Closed loop specifications o f the system in Fig. 2.11 are typically described in terms o f inequalities on the system s TF s from some inputs to some outputs, these are (using the notation L = PGH : 1
Plant output disturbance rejection Sensitivity): for any P E {P } the T F from the disturbance at the plant output to the plant output is bounded by
~
=
+~ U w
< os w).
QUANTITATIVE QUANTIT ATIVE FEEDBACK DESIGN
28
2. Plan Plantt input disturbance rejecti rejection: on: for an any y P E {P} the T F from the distur
bance at the plant input to the plant output is bounded by
iL 1Ud
=
I P(jw
I< J
L(Jw)
1
P
w).
P E {P} the distance the given optimal TF, Fm(jw), is bounded by
3. Mode Modell mat matching: ching: for any
o f the T F from
r
to y from
y I IPGF(jw) . I I :;: - Fm = 1 L(jw) - Fm(}w) < Jm w). 4
Tracking: for any P
E
{P} the amplitude o f the T F from
r
to y is bounded
by
PGF jw) I 3 ) L jw):S; w.
) w :s; I 1 5. Nois Noisee rejection: for any
P E {P} the
T F from the sensor output to the
plant output is bounded by
6. Cont Control rol effort: fo forr any P input is bounded by
E
{P} the TF from the sensor output to the plant
For specifications 1 - 4 the following hold H = 1 for simplicity) 1imIG jw)l-+oo
11
jw)
limIG jw)l-+oo
I1
P(jw) 1= L(jw)
1imIG jw)l-+oo
I
LF jw)
I=
L(jw)
1
-
0
Fml
=
0
F=Fm
LF jw)
1imIG jw)l-+oo
maxp jw) l+L jw)
- - - ~ -
.
, - - ; - , - = 1 . LF jw)
m m p j w ) l+L jw)
Therefore as the amplitude o f the controller increases, increases, the plant output respon response se to disturbances decreases and the sensitivity o f the plant output to tracking
Basics o f SISO Feedback Controlled Systems
29
comman ds decreases. We shall then say commands say that the specifications are tighter. Tighter specifications therefore require a larger IG jw) I which in tum implies a larger open loop L jw), and hence a larger cross-over cross-o ver freque frequency. ncy. Tighter Tighte r specifications thus imply a larger bandwidth. Now let us consider specifications 5 and 6. A larger IG jw) I increases the plant output response to sensor noise, therefore suggesting larger 6n w) input and 6and c w) on a wider bandwidth. Qualitatively this can be explained as follows: The TF considered in specification 5 is L / 1 L) which is in general a low-pass filter operating on the sensor noise, with bandwidth o f Wb L jWb) ; ; -3dB . A larger IG jw)1 causes an increase o f the bandwidth, W b o f the low-pass TF from the noise to the plant output, thus increasing the noise response at both the plant output and input. input. The same explanation holds for specification 6 because beca use G / 1 L = f ~ and P is not a design parameter. Hence there is tradeoff between specifications 1 4 and 5, 6. A good feedback design is is one wherein wherein spec specific ificatio ations ns 1 - 4 are are wel welll satis satisfied fied without ma making king specifications o f the type 5 6 too stringent. of
relative stability and phase-margins areshown special specifica tionThe 1 and 5 where 6s andgain can be bycases simple arithmetic n are constants. It can that for s = , the gain and phase-margins are 20 log { - - } [dB], 2 s i n - 1 -I
and for 6n
=,
{ ~ } 2
[deg]
the gain and phase-margins are 20 log {
1 } [dB], 2 s in- 1
{
~ } [deg]
The disturbance rejection specification 1 with constant 6s bounds the dis tance o f the open loop L s) evaluated on L jw) from I o n the Nyquist plot OdB, - 180° on the Nichols chart) thus placing an upper bound on the ampli fication o f the plant output response to disturbances at the plant output, in the frequency domain. As such it is a more attractive specification than the relative gain and phase-margins (which theoretically place the above upper bound only at two discrete frequencies: wrj and w M . The same is true for specification 5 with a constant 6n . It puts an upper bound on the TF from the noise no ise to the plant output and, as such, eliminates closed loop under-damped poles in the system s significant frequency rang range. e. 3.3
TRANSLATION OF SPECIFICATIONS FROM T DOMAIN TO w DOMAIN
There is no one-to-one translation o f the proposed propos ed t-domain to the w-domain specifications. specificat ions. Fro From m a practical point o f view, however, very good translations do exist, two o f which are described now. From our experience these techniques techniques
30
QUANTITATIVE QUANTIT ATIVE FEEDBACK DESIGN
give very good results, but if the designer is not satisfied he/she can iterate, for example by simply adding gain in the low frequencies (with appropriate modifications of the high frequencies to satisfy the specified margins), until the desired results are achieved. It is also recommended relaxing the w-domain specifications above a certain Wh,
frequency, to reducing satisf satisfy y only gaineffort and phase-margin This is very attractive for control with negligiblespecifications. impact on closed loop performance. Iterations on Wh may be needed, but it is very easy to carry out and evaluate the tradeoffs amongst Wh, control effort and t-domain specifications. 331
MODEL BASED BASED TECHNIQUE
This method is based on an assumed plant and controller model structure (number (num ber of poles and zeros of the plant and and the controller are known). known). Thus for a given input, the model structure on which the specifications are imposed can be calculated. calculated. The idea idea is to search for for the parameters paramet ers of the assumed plant and controller models, and then use the maximum or minimum of the amplitude of the resulting TF on the j w axis as the w-domain specification. Example 1: Suppose that the plant is a simple integrator 1 s, embedded in a feedback structure to achieve the following closed loop t-domain specifica tions:: The tions Th e plant output for a step step disturbance at the plant output should be bounded by the two curves in Fig. 2.l3a. Let us assume that the controller structure is a simple pole, that is G = k / 8 + a . The first step is to calculate the plant output model structure 1
1
Y(8)=I+PG8
1
1
s
1 _k_-;
82
s(s+a)
a
+ as + k ·
The second step is to search for the a s and k s which satisfy the t-domain specifications. This set of outputs is denoted by {y s)} in the frequency domain. The third step is to calculate the w-domain specification, in this case the sensitivity specification, by
I1 +1/L(s) Is = j w = 8
max y E { y (j w )}
I
(j w
I.
The results o f the above calculation, for a E [1,8] and k E [2,12] are given was designed to satisfy the the w-domain specification specificat ion in Fig. 2.13c. The system was using the technique described in Chapter 3, and the resulting controller is G = 6/ s + 2.7). The design was checked by simulations and the results are presented in Fig. 2.l3 2. l3c, c,d. d. Clearly the original t-domain and the translated w-domain specifications ar aree sa satisf tisfied. ied. But more importantly, the t-domain specifications are barely satisfied, thus implying that the translation to the wdomain is not conservative.
Basics o f SISO Feedback Controlled Systems
t-specs 1
.
2
.
-
-
-
-
-
~
-
~
-
-
-
-
-
-
-
6
,
o
-
~
:J
c..
D
....J
.
~
~
OJ-specs ~
~
~
~
31
~
~
(c),
-6
-12
+ ::: -18 .--
:J
o
-24 -
0
.
4
-
-
-
1
-
~
-
-
~
-
2
-
-
-30
3
10 1 OJ-spec OJ-s pecs+s s+sii mulatio mula tion n --)
t-specs+simulation - - )
6
1 . 2 . - - - - - ~ - - - - - - - - - - - - - ,
0 ~
:J c..
D
....J
:J
o
,
+
- --.
.
-
0
.
4
o
-
-
-
-
~
1
-
-
~
-
sec
-
.
2
.
.
.
.
J
-12
:::-18 .--
-.-
;,..;.
-6
3
10°
log OJ)
Figure 2.13.
Examples o f t-domain specifications: (a) the the output for a step disturban disturbance ce at th thee plant output must be between the curves; (b) t-domain simulations (dashed) for a step disturbance at the plant output; (c) w-domain specs on the sensitivity TF; (d) the designed sensitivity TF (dashed)
Example 2: Suppose that the plant is a simple integrator, 1/8 embedded in a feedback structure to achieve the the following following closed loop t-domain t-doma in specifications: The plant output for a step disturbance at the plant input should be bounded by the two curves in Fig. 2.14a. I t is assumed that the controller includes an integrator and a lead element, that is, G = k 8 + a) / 8 8 + b). The first step is to calculate the plant output model structure, which is
P
1
Y 8) = 1 + PG ;
1+
1
1
8+b
s+a 8 2 s s+b
The second step is to search for the a s, b s and k s for which the t-domain specificatio specif ications ns are satisfied. This set o f outputs is denoted by {y 8 )} in the frequency domain. The third step is to calculate the w domain specification, in
3
QUANTITAT QUAN TITATIVE IVE FEEDBACK DES IGN
this case the sensitivity specification 1 82
1 + L 8)
.
s=Jw
=
max
YE{y jw)}
ly jw)l.
The results o f the above calculation, for E [0.5,2], b E [2.5,9] and k E [3.5,14], are given in Fig. 2.14c. 2.14c. The system was designed to satisfy the w-domain specification, using the technique described in Chapter 3 and the resulting controller is G = 7.48 + 6.6)/ 8 2 + 4.38). The design was checked checke d by simulations and the results results are presented in Fig. 2.14c,d. 2.14c,d. Clearly the original t-domain and the translated w-domain specifications are satisfied.
igure 2.14. Examples of t-domain and and w-domain w-domain specifications: specifications: a) tthe he output for a step disturbance must be between the curves; curves; b) t-domain simulations dashed dashed)) for a step disturbance
at the plan plantt input; c) w-domain w-domai n specs on the sensitivity sensitivit y TF; d) th thee desig designed ned sensiti sensitivity vity TF dashed)
Basics o f SISO Feedback Controlled Controll ed Systems
3.3.2
33
KRISHNAN AND CRUICKSHANKS CRUICKSHANKS TECHNIQUE
Let us assume that the t-domain specifications are
o
the form 2.1)
where y t) is the closed loop signal and m t), v t) are specified time functions. That is, y t) should not deviate from m t) by more than v t). condition than this is the following:
fat
Iy t) - m t)12
;
fat v 2 t),
Vt
:
0,
A weaker
2.2)
which means that instead o the upper bound on 1y t) - m t) 1 being v t), the energy o the signal on the time interval [0, t is bounded by the energy o the signal v t) over the same inte interval rval for any t). Krishnan and Cruickshanks suggested using this weaker condition for which the sufficient condition in the w-domain corresponding to inequality 2.2) is Krishnan and Cruickshanks 1977)
Iy jw) - m jw) 1
;
Iv jw) I·
2.3)
The main drawback o this technique is that a weaker condition is used, but in general, it is a reasonable alternative to the original specification. Example 3: Consider the system described in Fig. 2.1 and the closed loop t-domain specification on the plant output y t)
a t) ::; y t) ::; b t) where a t), b t) are given in in Fig. 2.1Sa. 2.1Sa. Clearly Clear ly this specific specification ation is equivalen equivalentt to
Iy t) - m t)1 ::; v t); m t) =
a b -2-
v t)
b-a
= -2-
shown also in Fig. 2.1Sa. 2.1Sa. The w-domain specifications will then be
I PGFr P G - m jw)
I1
;
Iv jw)1 ,
which, for the choice o prefilter such that F r sensitivity specification
m jw)
I I+L
I :::;lv
m s ), is the w-domain
jw)I,L=PG.
The amplitudes o m jw)w and v jw)w are shown in Fig. 2.1Sc in place o m jw) and v jw), due to conveni convenience ence in in plotting plotti ng the graphs. As is argued in
QUANTITATIVE QUANTITATIV E FEEDBACK DESIG N
34
Chapter 3 it is important to relax the specifications at high-frequencies in order to save bandwidth. The dashed curve in Fig. 2.15c is the relaxed Iv jw)wl specs used for design. design. Time domain simulations simulati ons for G = 35/ 8 + 17.5) are shown in Fig. 2.15b; clearly the t-domain specifications are satisfied except near t = 0 Fig. 2.15d), 2.15d), the reason being the relaxation o f the specification
Iv jw)1 at higher frequencies. w-specs
t-specs
o o 0
-6 -12 -18
3 1 2 4 t-specs+simulation - -
5 t-specs+simulation - 0.1
1
0.08 >
0.06 0.04 0.02
1
2
sec
3
4
5
00
0.1
0.2 sec
0.3
0.4
Figure 2.15. Examples of t-domain specifications, a) th thee plant outpu outputt ffor or a step tracking command should be between the curves a t),b t) m t) = a 1 b and v = a ~ b ; b) t-domain simulations simul ations dashed) for all plant uncertainty where F r = m s); c) w-domain sensitivity specifications, the dashed curve is the true vwl used for design; d) t-domain simula simulation tion near t=O
4
PERFORMAN PERFO RMANCE CE LIMITATI LIMITATIONS ONS OF NMP OR UNSTABLE SYSTEMS
NMP plants are those whose model includes one or more RHP zeros and/or pure delay sampling inherently inherently includes includes d dela elay, y, therefore feedback systems which include sampling are also NMP). A well known example is the inverted pendulum on on a cart Kailath 1980 for which it is easy to explain the NMP
Basics o f SISO Feedback Controlled Systems
35
phenomenon: I f he/she wants to move the tip o f the pendulum to the right, the cart first has to move to the left in order to let the tip fall to the right and then the cart is is driven to th thee rig right. ht. The tip will the then n move slightly to the left and then to the righ right. t. It will therefore take more time to bring the tip to to a desired location on the right as compared compar ed to a plant which does not have to move to the left first. The open loop crossover frequency, w ¢ o f an NMP system has an upper bound, hence the amplitude o f the loop transmission at frequencies below the cross-over frequency is also bounded. 2 The reason is due to the Bode phase-amplitude phase-am plitude rela relationships. tionships. In mini minimum mum phase systems, one can shape the open loop with lead-lags, lag-leads et etc. c. such that any desired cross-over frequency can be achiev achieved. ed. Fo Forr NMP plan plants, ts, howeve however, r, the loop transmission cross-over frequency is bounded (see section 8. for further explanations and formulae), and the loop gain at low frequencies is bounded. or example, if the cross-over frequency of a system which behaves like 1/ s2 in open loop at low frequencies, frequencies, is 10 ra rad/sec, d/sec, iitt s loop transmissi transmission on g gain ain at w = 1 cannot exceed 40dB Therefo Therefore re the benefits o f feedback for NMP plants are limited in the sense that not any closed loop specifications can be achieved using an LTI controller. contro ller. These limitations depend on the following parameters: the RHP pole and zero locations, gain-margin, phase-margin, and the open loop delay. A quantitative discussion now fol follows lows.. 41
STABLE PLANTS PLAN TS
Any stable open loop TF L s), can be uniquely factored into L s) = LM S)A s) where LM S) is minimum-phase (stable and has no RHP zero) and A s) an all-pass T (stable and IA jw)1 = 1 for all w), for example LM(S)
L s) = s
s-
A(s)
~
A
1
2) s
1
l
2) s
3) s
s
3
s
s
1.
Based on Bode Bod e s relationship between the amplitude and phase o f a minimum phase T (Bode 1945, Horowitz 1963) i f the phase o f a minimum-phase stable T over a large frequency range is fixed at, say ¢[deg], then within that fre quency range it can be approximated by the T k/ s ¢ 90 (it is equivalent to asking that the Bode amplitude plot have a fixed slope over a large frequency range).. Adding this approximation to the practical criterion that the bandwi range) bandwidth dth o f the controller should be bounded, we make the following assumption which is satisfied to a very large extent in realistic feedback systems and will be
2True in general, but at a ssingle ingle frequency we can achieve achieve,, theoret theoretically, ically, any o pen loop gain using oscillators as a part o f the controller of the form S
n
QUANTITATIVE FEEDBAC K DESIGN
36
verified numerically in the sequel verified sequel see also Sidi 1997) and Horowitz and Sidi 1972)): ASSUMPTION
frequency,
W M,
2.1 Betwe Between en the cross-over frequency frequency,, W¢ and the gain-margin the minimum-phase TF, LAds), can be approximated by
Under assumption 2.1, if L(s) includes a single RHP zero at a, then L s) L M s ) ~ and therefore argL jw)
2 t a n - 1 w/a); w¢:::; w:::;
a1f -
=
WM,
which yields for phase-margin ¢ cross-over frequency w¢ and gain-margin frequency W M def
W a¢ =
Wa M
W¢ -a
def W M
l-a)1f-¢} tan { 2
2.4)
{ l - a2 ) 1 f } .
2.5)
tan
a
Fr From om equatio equations ns 2.4,2.5) an and d assumption 2.1, 2.1, the the gain-margin, denoted by M , is
M =
Q [ ] Wa M
_
wa¢
-
t
an t
~ 2]
an
Q
Q
2.6)
r
2
Also from equations 2.4,2.5) and equation 2.6) the relations betwe between en the gain margin, phase-margin, cross-over frequency and a can be calculated and are shown in Fig. 2.16. Example: Given a plant with a single RHP zero at a
=
3 for example
p(s)= 3-s s 8+s
then for phase-margin 40° and gain-margin M a
=
0.61, wa¢
=
0.275, w¢
=
a
=
lOdE see Fig. 2.16),
x wa¢ = 3 x 0.275 = 0.825
and by equation 2.6) WM
=
loglO M 2a
x w¢ = 2.6 x 0.875 = 2.8.
Basics o f SISO Feedback Feedback Controlled Systems
90
37
- O/a-
0.9
.5 80
;
.55
70
.6
Q)
::2 ~ 6 0 Ol
.65
ct
E Q) /)
50
.7
ct
..c 0
40
.75
30
.8
20 0
5 Figure 2.16 2.16..
15 Gain margin[dB] 10
20
Phase-margin vs vs.. gain-margin for different a and
25
w1> a
Using loop shaping trying to maximize w jJ for the specification - 3 . 5 d B (which preserve the same gain and phase-margins) gives
L s)
2.8s 2 s2
values
L jw) I
>
5.5 P s) 7.8s 3.5s 0.67
whose TF is is shown in Fig. 2.17. 2.17. It' It'ss cross-over frequency is 0.93 rad/sec which is greater than the estimated value based on assumption 2.1 by 12 . Note that the price for this small increase is a 150 increase in wM and hence a very large increase in noise amplification at the plant input and in the minimum required sampling rate in case of digital implementation. 4.1.1
EXTENSION TO SEVERAL RHP ZEROS AND/OR DELAY
zeros: os: NMP plants with only real RHP zer
A reasonab reasonable le estimation for the the relationship between gain-margin, phase-margin and cross-over frequency can be achieved by replacing the RHP zeros by an equivalent single RH RHP P zero whose phase is the first order approximation of the original RHP zeros. A simple formula is: for the RHP zeros located at Z l , Z n , the zero Z which replaces
38
QUANTITATIVE QUANTI TATIVE FEEDBACK DESIGN
20 15
10
5
1
. ·0.70.5
+
0
o '0
-5 -10 -15
-2E - 60 Figure 2.17 2.17..
L jw)1
>
-220
-180
-140
deg
Loop transmiss transmission ion of L ( s) for which
w¢
-100
-60
is close to maximum subject to
-3.5dB
them is found from the following low frequency first order approximation arg
I-8/Z1
...
1+8/Z1
which gives
1
:::::;
Z
I
8/zn
:::::; arg
1 + 8 / zn
1 -+
I 8/z
1+8/z
1 '+ - .
Zl
2.7)
Zn
This approximation is valid in the frequency range where all of the parameters involved can be replaced by the linear relationship tan This result can :::::; ~ be demonstrated demonst rated with the following open loop TF s which obey assumption 2.1. L
8)
L2 8) Ld 8)
k 3- 8
82 8 3 k (6-8)(6-8) 82 k
(8+6)(8+6)
2- e
8a
sT
,
=
2/3.
Basics o f SISO Feedback Controlled Systems
39
The equivalent NMP zero, zo of the RHP zeros of L2 and the Pade approx imation of the delay TF L d , is at 3. These TF s are shown in Fig. 2.18 with gain-margins around lOdE and phase-margins of approximately 40°. All three TF's are almost the same even up to W M , the gain-margin of L2 is only IdE less than that o f L1 and the gain-margin of Ld is only 1.5dE less. In order to get the same gain-margin for all three TF's, the cross-over frequency o f L2 and Ld should be about 0.7 rad/sec which is a deviation o f less than 12 from that of L
1
20 15
ill=0;2
10
5 [()
0
0 -5
-10
-15
-28 - 70
-240
-210
-180 deg
-150
-120
-90
Figure 2.18.
Comparison of three NMP TF's whose approximation is the same, L1 includes an all-pass TF with a RHP zero at 3 L2 includes an all-pass TF with two RHP zeros at 6 Ld includes a pure delay of 2 / 3
NMP plants with a single highly under damped RHP zeros : sumption 2.1, L 8) takes on a very simple form
L 8) =
8
82 a
2
82
-
~ W 8 ~ W 8
Under as as
w2.
w2
As its damping factor tends to zero, its Nichols plot converges to the simple 3 straight line structure o f Fig. 2.19. Hence, the relation relat ion betwe between en the gain-
40
QUANTITATIVE QUANTITATIVE FEEDBACK DESIGN
margin M phase-margin cp W and cross-over frequency w.:
. .... : ~ ~ ... . ':: .
0>;·
: '3dB
.
'
- 5 ..
···c· :
'6
dB
.
,
. : ,.
-10'
,
,
..
.
:
:-12
dB
.
..;
-15:: -20::: . -25
~ g 6 ~
-330 -300 -270 -240 -21 0 -180 -150 -120 -90 - 6 0
deg
-30:
o
igure 2.22. Definition of upper and lower gain-margins, ML, MH, phase-margin, cp cross over frequenc frequency, y, wrp and gain-margin frequency, W M
4. 4.2. 2.1 1
UNS UNSTAB TABLE LE PLANTS WI WITH TH A SIN SINGLE GLE RH RHP P PO POLE LE
The simplest unstable open loop transmission with a single RHP pole and finite gain-margin will be used for our discussion, it is o f the form
L s) =
k
s j a - 1 s2
+
w;
~ w n s
+
~
.
(2.8)
Without loss o f generality we can normalize the pole such that a = 1 (if a I 1 the following equations and results are true where Wn is replaced aWn, W M by aWM, and w< > by aw
2.11) - w ~ 2 2 ~ w n w e / > ) 2 / w ~ . Note that from stability considerations M L > 1 M L > OdE , hence the following inequality must hold: ML =
V
~ V ( w ~
2.12) The phase angle at the upper gain-margin frequency, WM, tion 2.10), argL jwM) = - 7 r : -7r
=
-7r
+ tan
1
WM - tan
1
{
IS,
from equa
2 2~ W n W 2M }
Wn-WM
which results in 2.13) Substituting Substit uting equation 2.13) into equation 2.9) give givess M H in arithmetic units) 1
WnML
MH
-
~
2 ~ ( 1
2.14)
2 ~ w n )
Note that from stability considerations M H < 1 in arithmetic units, hence the following inequality must hold 2.15) Equat ion 2.14) and equation 2.11) show the intimate connection between Equation gain-margins M L M H and the cross-over frequency we/> The next important entity is the phase-margin ¢ From equation 2.l0) it satisfies: -7r
We >
1
tan
-1
We >
- tan¢ we >
tan¢
- tan - 1
{
2 ~ w n w e / > } 2
2
Wn -We/>
2.16) 2.17)
By use o f equations 2.11,2.14,2.17) we can get a set of graphs relating the phase-margin ¢, gain-margins ML, MH and we > for different ~ s . First we differentiate equation 2.10) with respect tto o W in order to obtain the maximum
Basics o f SISO Feedback Controlled Systems
argument o f L jw) at W = order equation:
The result is the solution
wr/J
of
45
the following
th
Having found wr/J as a function of W n we then find the phase-margin from equation 2.16), and M L and M H from equation 2.11) and equation 2.14), = 0.5, where respectively. respec tively. These relations are shown in Fig. 2.23 for the maximum phase-margin for a given normalized frequency wn a and the minimum wr J a that can be achie achieved ved ffor or each case are depicted. Fig. 2.24
1
~
~
~
a) ~ ~
~
~
~
~
3
~ 0
7
6 . 5
..
4
3 2
~
~
~
~
~
~
~
0>2 6 ~ 2 4
>8 c5
~
2.8
9
~
~
c)
·rof
~
~
~
~
~
·
n
~
·
~
~
~
-2.2 2 1.8
g 1.6
1.4 1. 2
~
1
o 102030405060708090
L
L
~
~
•... oj fa: ~
~
.n
~
~
~
~
~
2 3 4 5 6 7 8 9 10 d)
b)
15
g 10
en 0
5·
00
102030405060708090 ro
2.23.. Figure 2.23 vs. W n for €
3
fa n
Cros Cross-ove s-overr frequency = 0.5
4
5
6 ro
w¢
and phase-margin
rjJ
7
8
9
10
fa n
upper) and and gain margins lower)
shows the resulting open loop transmissions on the Nichols chart for wn a = 3.65,5.3,8.3, 14.5, 33 and 100 with phase-margins o f 30 40 50 60 70 and 80 deg, respectively. Fig. 2.25 is the same as Fig. 2.24 but with = 1.0. Clearly its phase-margin is lower than for the = 0.5 case, but its gain-margin is higher.
QUANTITATI QUANT ITATIVE VE FEEDBACK DESIGN
6
25
20
· ~ = 0
5
15
10 5 0
III 0
.9
5 10 15 20
-2 ~
-225
70
Figure 2.24. Wn
Nichols plot
of
-180
deg
130
equation (2.S) for maximum phase-margin,
values
4.2.2 4.2. 2
-150
AN EX EXAMP AMPLE LE
AND
110
90
e= 0.5 and several
LIMITATIONS
In many practical feedback control problems, we are interested in reducing as much as possible, in order to minimize the sensor s noise amplification at the plant input (see section 3 also Horowitz (1963) and Horowitz and Sidi (1972)). (1972 )). The results o f Fig. 2.23 can be used to find the constraints posed by the open open loop RHP pole. pole. For instan instance, ce, suppose we need a phase-margin o f P = 40°, then from Fig. 2.23c, wn/a = 5.3 is the smallest value that can be used, for which (from the same graph) wcp a = 1.8, M L = 5.8dB and M 7.2dB from Fig. Fig. 2.23d. These results are confirmed b by y the Nichols pl plot ot of Fig. 2.24 on which are shown some L jw) s corresponding to equation (2.8). Practical open loop transfer functions will take more complicated complicate d forms than assumed ass umed here, i.e., will contain more poles and zeros than equation equat ion (2.8). How ever, optimal shaping of the open loop TF (in the sense o f lowest bandwidth) will have the same basic margin characteristics shown for the simplified struc ture o f equation (2.8) and plotted in Fig. 2.24, for which the results o f Fig. 2.23 hold exactly.
Wn
Basics o f SISO Feedback Controlled Systems
47
25 20 ~ = 1
15
10
5 ID
0
0
-5 -10 -15 -20
-2 ~ 70 Figure 2.25. Wn
-180
-225
-150 -130 -110
deg
Nichols plot o f equation 2.8) for maximum phase-ma phase-margin, rgin,
-90
= 1 and several
values
4.2.3
EXTENSION T O SEVERAL R H P PO LES
only real R H P poles poles:: A reasonable estimation for the relation between gain-margin, phase-margin and cross-over frequency can be achieved by replacing the RHP poles by a single single equivalent RHP pole, whose w hose phase is the first fir st orde orderr approxi approximation mation to the original RHP RH P poles at high frequencies. This is because the high frequency region region dominates the bandwidth equations. A simple formula is: for the RHP poles located at PI, Pn the pole, P which replaces them is the first order approximation to Plants with
arg
1 PI s 1 Pn/ ... 1 - PI s 1 - Pn/ s
arg
1 1-
p/ s p/ s
_...0:.....:...,.-
which gives
P
PI
...
Pn·
2.18)
The reason for choosing this approximation is that the frequency range in which all the parameters involved is the range where w Pi and the linear
«
48
QUANTITATIVE FEEDBA FEEDBACK CK D ESIGN
approximation tan the following following T F s
is applicable applicable.. This approximat approximation ion is illustrated by
8
8
4
8 - 4 (8 2 /21.2 2 + 8/21.2 + 1)(8 + 4)' 8
(8+1)(8+3)
(8 - 1)(8 - 3) (8 2 /21.2 2 + 8/21.2 + 1)(8 + 4)'
has the structure o f equation (2.8) with a maximum phase-margin o f 40° and has the same structure where the all-pass ~ ~ was replaced by its equivalent
L L
all-pass i ~ ~ i H ~ ~ ~ l Both transfer functions are shown in Fig. 2.26. Clearly the phase-margins, cross-over frequency and upper gain-margin o f L are very close to that o f L 1 the low frequency gain-margin o f L is 1 5dB lower than that o f L which is about 12 o f the sum o f margins M ML.
15
10 L
5 0 (()
0
-5 -10 -15
2B - 70
-225 Figure 2.26 2.26..
-135
-180
deg
Nichols plot of
Ll
and L
-90
Basics o f SISO Feedback Controlled Systems
Plants with highly under damped RHP poles poles:: is o f the form:
assumption ption 2. 2.1 1 Under assum
49
L 8)
As its damping damp ing factor, tends to zero, its Nichols plot p lot converges to its minimum phase form at frequencies larger than than W n , and can thus be treated as a minimum minim um phase T F in that frequency range range.. S
LOOP SHAPING shaping ing is the skill to to generate a controller, G Loop shap
8), such that an open loop
satisfies ies certain certain specification specifications. s. Thes e specifications specifications can be o f T F L ( 8) = G P satisf
many kinds, the most important being that L 8) satisfies the Nyquist stability criteria. crite ria. Others include one or more o f the following: following: gain-margin, phase margin, cross-over frequency, bandwidth, gain-margin frequencies, coefficient error gain a t w = 0 and number o f integr integrators ators in the open loop) and satisfaction o f bounds requirements. This section is is devoted devoted to helping the reader reade r tto o master the skills o f loop shaping via examples. The difficulties in loop shaping arise because o f the amplitude and phase
relations as given by the Hilbert transform and the Bode integrals Horowitz 1963), especially when the plant, P is NMP and/or includes a pure delay and/or is open loop unstable. unstable. This is because closed loop stability stability,, alon along g with the other specifications, decreases the freedom in choosing the controller and may even impose contradictory requirements, in which case a controller cannot be created and the specifications specifications should be relaxed. relaxed. softwaree package use d by the designe r The QFT Matlab ™ toolbox is a CAD softwar to append and iterate) iterate) basic TF s to the controller and plot the specifications and the open loop response on the screen. screen. Thes e basic TF s are: l
Simpl e gain: gain: k.
2. Simple pole or simple zero: - - L
s+p
s +p
3. Simpl e lead or lag: lag: ~
~ .
4. Second order pole or zero:
5. Notch:
s 2
2 ~ l W S
p
w2 s 2
2 ~ w s
w 2
,
s 2
2 ~ w s
w2
w 2
w 2
s2+ 26w s+ w 2 .
k, is to shift L(jw) up by kdB i f k > OdB or down w 2 p2) if k < OdB. A simple pole located at p shifts L(jw) by - 1 0 log(l d B and by - tan 1 w j p) deg. The shift in [dB, deg 1o f a second order zero
The effect o f a simple gain,
and pole with different damping factors, is shown in Fig. 2.27. The shift in
50
QU NTIT TIVE
FEEDBACK DESIGN
28 24 20 16 12
8 4 [ l
'0
0 4 8 12 16 20 24
28 180 150 120
90
60
30
0
deg
30
60
90
120 150 180
Nichols plot of the complex pole 8 2 /1 6 2 + 21;-;168 + 1 - 1 appears on the left hand side of the graph and the complex zero 8 2 /1 6 2 + 2 0 1 6 8 + 1) appears on the right hand side of the graph Figure 2.27.
2.28. It can be shown, shown, by [dB deg of a lead/lag element is shown in Fig. 2.28. differentiation, that the maximum minimum) phase of a lead lag) element eleme nt ~ t
appears at w = VOJj and is
¢ = 90 ° - 2 t a n
1
Fib
2.19)
The shift in [dB deg of a Notch for different damping factors is shown in Fig. 2.29. Example 1: Given the plant P = lO s see L o n Fig. 2.30); it is required to shape a controller so that the overall loop transmission will have a cross-over frequency at w< > = 10, phase-margin of 45° and gain-margin o f lOdB. Design steps: 1 Add a lag element whose maximum lag phase is 45° at w = 10 to achieve the desired phase-margin atw
= 10) 10).. The lag element elem ent is G 1 =
i ~ :
~
where
Basics o f SISO Feedback Controlled Systems
51
20 18 16 14 12 108 6 4 2 0
n
0
-2 -4 -6 -8 -10 -12 -14 -16 -18 -20
-75
-45
-60
Figure 2.28.
Nichols plot
-30
of
the lead element ~ ~ : ~ ~
45
30
15
-15
and lag element ~ ; :
60
75
for different alb
values, b = 16
JCJj = 10 and equation 2.19). The Th e TF kl G 1 P is modified by the gain kl = 7 8dB so that L2 ( j 10) = OdE shown in Fig. 2.30, with the label L2) a
= 24, b = 4 are calculated from
2
Add a zero such that the phase o f L 2 will be about - 9 0 0 with 4dB less than the desired gain-margin, i.e., -14dB, in order to allow for a complex pole at the third step). step). The element elem ent is G 2 = 1 + 8/45 and the TF l G 1 G 2 P is shown in Fig. 2.30 with the label L 3 .
3
The frequency where L is - 4 d E below the desired gain-margin is about 30 rad/se rad/sec. c. Adding a complex pole with with a damping factor o f 0.5, will shift L j30) to have a phase of - 1 8 0 0 . Iterations are required on the parameters, including the gain to to achieve tthe he desired margins. The result is the TF L 4 in Fig. 2.30 where:
G
=
2
1
8/24
2 1+
8/ 4
1
8/45
82/302 + 0.6/308 +
l
L4
=
10 4 ;.
QUANTITATIVE QUANTITAT IVE FEEDBA CK DESIGN
52
20 18 16 14 12 10 a:l 0
8 6 4 2
0 -2 -4 -75
Figure 2.29.
- 60
-45
-30
0
-15
deg
15
Nichols plot of the Notch filter, s s 2 / / \ 6 : 2 : 2 ~ g : : : 1 1
30
45
60
75
, or different damping factor,
Example 2: Given the plant P = 101 s ; it is required to shape a controller so that the overall loop transmission will have a cross-over frequency w< > = 10, phase-margin o f 45° and gain-margin o f 10dB. Design steps: 1. Add a lead element whose maximum lead phase is 45° at w
= 10 to
achieve the desired phase-margin at w = 10 10). ). The ele element ment is G l = 1 sl a)/ l sib where a = 4, b = 24 are calculated from O j = 10 and equation 2.1 2.19). 9). The TF k l G I P is modified by the gain k l = 12.3dB and the resulting L 2 j10) = Od is shown in Fig. 2.31. 2. Add a zero such that the phase
will be about -1 1 0 ° with 4dB less than the desired gain margin, i.e., -1 4 d B , in order to allow for a complex pole at the third third step step). ). The eleme element nt iiss G 2 = 1 s 150 and the TF kl G 1G 2 P of L
is shown in Fig. 2.31 labeled as L 3 . 3. The frequency where L3 is - 4 d B below the desired gain-margin is about
30 rad/sec.
Adding a complex pole with a dampi damping ng factor
of
0.5, will
Basics o f SISO Feedback Controlled Systems
20
L
10
2
4
5 'C
.
m=
15
m
53
7
10
0 -5 -10
-15 -28
- 70
Figure 2.30.
-225
-180
deg
-135
-45
-90
Nichols plot for lOdE gain-margin and 45° 45° phase-margin, the plant is 1 0 / s
shift L 3 j30) to have a phase o f -180°. Iterations are required on the parameters, including the gain to achieve the desired mar margins. gins. The result is the TF L4 in Fig. 2.31
G - [12 3dBJ 1 + 8/4 .
1 + 8/50
1 + 8 / 2 4 8 2/4 0 2 + 0 . 8 / 4 0 8 + 1
L
G =
1 4 82
.
Example 3: Fig. 2.32 describes a set o f bounds, for the nominal plant
P
S(S lO). It is required to shape G 8), such that L jw) = PG jw) will be
above the curves marked B 2), B 4), B 8) and B 20) at w = 2,4,8,20, respectively, and outside the closed curved at all frequencie frequencies. s. The first step is to adjust the nominal plant gain to satisfy the low frequency bounds, marked a in Fig. 2.32. The bounds are satisfied excep exceptt around w = 80, therefore the second step is to add a lead element whose maximum lead is around w = 80 see loop marked b . Although the bounds are satisfied by loop b we want to add another pole to the controller to achieve nice roll-off characteristics at high frequencies, and this is done as follows: Add a zero so that about 4dB below the bottom part of the bounds at hi high gh frequenc frequencies, ies, - 28dB the phase of
5
QU NTIT TIVE FEEDB CK DESI GN
25 20 15 10 5
o -5 -10 -15
-20 -270
-225
Figure 2.31.
-180
Nicho Nichols ls plot of design steps and 45° phase-margin, the plant is 10/8
deg
£2, £3
-135
and final design
-90
£4
for
lOdE
-45
gain-margin
the loop transmission is around -120°, see curve c ; then add a complex pole with damping = 0.5 and natural frequency that will touch the bounds, see loop-transmission d . The design process described in example 3 is suitable for a large set o f practical examples and proceeds along the following lines: 1
Star t with a simple gain to satisfy Start satisfy low low frequency frequency specifications known in QFT as bounds).
2
Add lead and/or lag elements to satis satisfy fy the low frequency bounds and at the same time decrease the controller amplification.
3
Add lead elements
4
Iterate on the controller parameters to decrease the controller bandwidth by
to
satisfy the high frequency bounds.
trying to decrease its high frequency gain. 5
Add as many excess poles over zeros as needed, by adding a zeroes) then a far enough pole s) or a complex pole s) pair.
Basics o S1S Feedback Controlled Systems
55
40 30 20
o 10
40
80 150
20
250 400 600
30
270
Figure 2.32.
-225
-180
135
deg
90
45
o
Loop-sha Loop -shaping ping steps a,b,c and fin final al desi design gn d for example 3
The final step can involve either iterations on the control parameters or the ad dition of many lead and lag elements followed by a model reduction procedure - the QFT QFT Matlab toolbox Borghesani et al 1994) is well suited for this process. Finally Final ly it is advisable to do the exercises at the end of this chapter. QFT toolbox, one can gain good W e have no doubt that using the Matlab skill in loop shaping within a day or so. 6
SUMM
RY
In this chapter chapte r we reviewed reviewed the basic classical parameters par ameters which characterize cha racterize a feedback system system - margins, margins, bandwidth bandwidth and and cross-over freque frequency ncy.. It was was emphasized that these parameters can serve as excellent figures figures of merit for the robustness and performance performa nce of a feedback system. system. The existence existe nce of several cross-over frequencies however, invalidates what was stated above. A controller is designed in order to comply with desired closed loop speci fications, in the time as well as frequency domain. Although there is no one to one translation from time to frequency domain specifications, two algorithms were proposed which possesses the following two important features: i) rere-
6
QUANTITATIVE QUANTITATIVE FEEDBACK DESIGN
duction o f sensitivity or complem complementary entary sensitivity therefore can be solved using the design technique o f Chapter 3), and ii) usefulness as a very good first order guess for a proposed translation with a minimal number o f iterations to achieve a solution fo forr the t-domain sp specs. ecs. The Krishnan and Cruicks Cruickshanks hanks translation technique is accurate but on energy type t-domain specifications. t can be used for tthe he t-domain specifications suggested her heree which are practical) i f they do not oscillate too rapidly in the time domain. The existenc existencee o f RHP zeros in a plant model, limits the achievable cross cross-over -over frequency o f the open loop, thus limiting the benefits o f feedback with respect to closed loop sensi sensitivi tivity. ty. Simple equations and design graphs were prese presented nted in order to find find the limitati limitations ons and tradeoffs between the cross-over frequency, gain-margin and phase-margin as a function o f the RHP zeros, in the region around the cross-over frequency where the slope of the Bode plot was given. The existence o f an unstable pole pol e has aan n opposi opposite te effect, it puts a lowe lowerr bou bound nd on the cross-over freque frequency ncy.. Equations and desig design n graphs were presented in order to find find a good estimation to the limitations and tradeo tradeoffs ffs bet between ween the cross cross-over -over frequency, upper and lower gain-margins and phase-margin as a function o f the frequency, RHP poles, based on a simplified structure o f its loop transmission. The last section is devoted to helping the reader gain skill in loop shaping, via examples and exercises.
7.
EXER
ISES
2.1 Given the plant P 8)
=
Estimate the maximum cross over frequency and margin frequency for minimum gain-margin 12dB and phase margin 45°. EXERCISE
1 6 6 ~ S
(3-s)(5-s) E · I P 8) = S1 (16+s)(20+s) . E X E R C I S E2. 2 G Iven the pant stImate the maxImum cross over frequency and margin frequency for minimum gain-margin 12dB and phase margin 45°.
EXERCISE
2.3 Given the plant
EXERCISE
2.4 Repeat exercise 2.3 with the plant
shape a controller such that the phase o f L(jw) is in the interval [-145 -155]deg and where the open loop is in the interval [30, -30]dB, and cross-over frequency at 10 rad/sec. Hint: start with a pole at 1 then choose a lead element whose maximum lead is at a higher frequency, then a lag, etc., iterate at each step i f needed. zero at
1
EXERCISE
1/8
1/82.
Hint: start with a
and continue as in example 2.3.
2.5 Repeat exercise 2.3 for the phase intervals [-145 -140]
[-140 -130] and [-130 -125].
Given the plant 1 8 - 1 , shape a controller with the smallest possible cross-over frequency you can achieve such that the closed loop is E X E R C I S E 2.6
Basics o f SIS 0 Feedback Controlled Systems
57
stable, the gain at w = 0 is 20dB and the closed loop satisfies the sensitivity specs LI-- 1 s 3dB + LI
E X E R C I S E 2.7
Repeat exercise 2.3 for the plant e
sT /
8 - 1). What Wh at is the
maximum T for which you can satisfy the spec? compare your results with the prediction you can get from Fig. 2.16 using Pade approximation for the delay. 8
NOTES
ND REFERENCE REFERENCES S
Time domain specifications and characteristics o f closed loop systems can be found in many classical text books, for example D Azzo and Houpis 1988) and Skoges Skogestad tad and Postlethwaite Postlethwa ite 1996). Design Desig n methods for t-domai t-domain n sspecs pecs can be found in Horowitz Horowit z 1992) and Pritchard Pritch ard and Wigdorowitz Wigdorow itz 1997). While frequency domain specifications o f the sensitivity or o r tracking type are sufficient to satisfy time domain specifications o f the form a m S y m) S bm where y m) denotes the mth derivative o f the the pla plant nt o output utput - see Horowitz Horowitz 1978), in practice one should tradeoff between time domain specifications and controller contro ller bandwidth. Iterations, based on the Krishnan and an d Cruickshanks technique, are therefore highly recommended. Sidi 1976) and and Horowitz and and Sidi 1978), presented presente d an optimal optim al robust con troller synthesis technique for an uncertain NMP plant with required closed loop performance. performance. Their synthe synthesis sis method provides provides the designer with insight into the tradeoffs between closed loop performance and bandwidth, and also defines an implicit criterion for determining whether or not a solution exists. Sidi 1980) deve developed loped a criterion criterion to to estimate the the maximum bandwidth o f a sampled data plant for a given gain and phase-margin by assuming ideal Bode open loop transmission characteristics and using asymptotic approximations. Horowit Horowitz zos.and Liaushowed 1984)how extended this technique stable severa severall RHP RH P zer zeros. They to aachieve chieve large loop to gain overplants several severawith l frequency ranges - eve even n though there wi will ll always always be some frequency frequency ranges which are determined by the RHP zeros, in which the loop transmission must be less than OdB This was proven proven by Francis and Zames 1984) and by Freudenberg and Looz Loozee 1985), who showed that for NMP NM P plants, small sensitivity in one frequency range results in large sensit sensitivity ivity in an adjacent adjace nt ran range. ge. Freude Freudenberg nberg and Looze Looz e 1985, 1987) 1987) developed several constraints on the closed clos ed loop sen sitivity o f NMP and/or unstable plants in the form o f weighted integrals o f the sensitivity on a log scale for all frequencies or on a frequency range where the open loop is much less than 1 Middleton 1991) used their results to provide a bandwidth bandw idth limitation on NMP and/or unstable plants. Sidi 1997) used a dif ferent approach based on the Bode relationships to obtain graphical results on bandwidth limitations for plants containing containing a single single RH RHP P zero or one unstable pole.. A v pole very ery good discussion on performance limitations limitati ons due to RHP poles
8
QU NTIT TIVE
FEEDB
CK DESIGN
and/or zeros, based on the ideal Bode characteristics, appears in Maciejowski 1990). he fir first st paper which presented an efficient techni technique que for loop sshapi haping ng was by Geraa and Horowit Ger Horowitzz 1980 1980). ). Automatic lo loop op shaping software was develo developed ped by Bailey e t al 1992). 1992 ). Chait 1997) presented an algorithm using convex optimization which optimizes only the zeros o f the contro controller. ller. Tho Thomps mpson on and Nwokah Nwok ah 1994) developed an algorithm for shaping minimu minimum-gain m-gain controllers.
Chapter SYNTHESIS OF LTIPLANTS
1
LTI
CONTROLLERS FOR MISO
INTRODUCTION
In this chapte chapterr the engineering engine ering tool for feedback design desig n o single-input single output and multi-input single-output systems, known as QFT, is introduced. i)
o
The important features this approach are: it is robust to the exact amount o plant uncertainty; ii) it tailors the closed loop precisely to the specifications which are given at each frequency; frequency; and iii) the technique is graphically based, therefore allowing for insight into tradeoffs amongst design parameters such s complexity, scheduling, amount o uncertainty, sampling time, margins and bandwidth. As a result, the method allows the designer to come up with low bandwidth designs. The design method is developed separately for one and two degree-of freedom systems, and the algorithms for computing computi ng bounds are given. t is then extended to sampled-data systems and NMP plants, and a detailed explanation o the NMP phenomenon is provided.
2
ONE DOF SYSTEM
The feedback system o interest is depicted schematically in Fig. 3.1 and described by the equations:
y
Pu
u
-GHy
Pdd
Gn.
The problem discussed in the sequel is how to design a controller, G s , such that for a given set o plants, {P} the closed loop is stable and certain w-domain specifications specifi cations are satisfied. All
o
the w-domain specifications for one DOF 59
O. Yaniv, Quantitative Feedback Design of Linear and Nonlinear Control Systems
© Springer Science+Business Media New York 1999
60
QUANTITATIVE FEEDBACK DESIGN
d ,.
n
-
-
-
Pd
-
u
G
Y,.
P
I
0
Figure 3.1 3.1..
A single OOF Single feedback system
systems can be reduced to the following inequality on the controller G s :
I
A jw)
B jw)G jw) 1 +PHG jw)
I < 6 ( ) VP -
w,
E {P}
,
where A, B , C and 6 may depend on the plant P, as well as on w. All o f the wdomain specifications, o w), are absolute values and therefore only functions o f w i.e. phase is not o f importa importance nce as as far as the specifi specifications cations go). We distinguish between the following two major problem categories: 1
Sensitivity reduction - the specifications specifications aare: re:
P Note that G jw) = 2
00
is
G j w ) I
6 s(w).
always a solution.
Control effort minimization - the specifications specifications take take the form form::
I1
G jw) P HG jw)
I
~
oc(w)
Note that G jw) = 0 is always a solution to the above inequality. The sensitivity reduction problem tends
to
increase the loop gain while the
control effort problem puts an upper limit on the control controller ler output assuming c < P HI- 1 . All o f the w-domain specifications o f section 3.2 for the one DOF DO F problem can be reduced to one of the above above two two categories. Two examples now follow.
Synthesis o f LTI Controller Controllerss fo forr MISO LTI Plants
61
Example 1: Fo Forr a given d s) and Pd s) we require the plant output y o f Fig. 3.1, to be bounded by e( w). This problem can be recast in the form o f a sensitivity reduction problem as follows:
I<
e w) = 6 w I1 + PHG(jw) - IPdd jw) I - s ) 1
Example 2: For a given n s) in Fig. 3.1, we require the plant input to be bound bounded ed by e w). This may be written as a control effort problem:
I< I 1 + G(jw). PHG Jw) 21
e
~ )
In Jw)1
== 6u w)
SENSITIVITY REDUCTION PRO LEM
Consider the system shown in Fig. 3.1, where P is an LTI plant. The problem we face is how to design a controller, G s , such that, for a given set plants, {P}, the closed loop is stable and the open loop sensitivity satisfies specifications o f the form:
of
I1 + G P1H (JW
I ;
6s w), i P E {P}.
(3.1)
Note that s w) and H jw) may depend on the plant, P. The solution to inequality (3.1), for a given frequency w, plant P jw), and sensor TF H jw), is a circle in the complex plane. Using the notation L s) = P G H, then the exterior o f the circle centered at [ - 1 0] with radius 8 s w 1 1 is the allowed region for which inequality (3.1) is valid for L jw) (see Fig. 3.2), while the interior area isn t. Similarly, Similarly, G jw) defin defines es anoth another er circle in the complex plane, outside o f which inequality (3.1) will will be satisfied satisfied (see Fig. 3.2). The design process will then be: l . Calculate circles at each frequency for all plants P E {P}, denoting the
curve of the union o f all the circles at a given w by B w). In QFT QF T parlance, B w) is the bound on G s) at frequency w.
2
Find (shape) a TF G s) such that at any frequency w, G jw) lies outside the bound B w), in conformity with inequality (3.1), and the closed loop is stable for all P E {P};
The resulting G s) will then be a solution, in so far as the closed loop is stable and satisfies the the specifications. From Fr om simple geometry, it can be shown that the center o f the forbidden circle, for the parameter L jw) = G P H jw), is located at [ - 1 0 ] and its radius radiu s is 8 ; 1 w). For the parameter G jw) the center cente r iiss shifted to I I P H jw) and the radius is accordingly, 6;1 liP H(jw) I Calculation o f
62
QUANTITATIVE FEEDBACK DESIGN
these bounds with the aid o f the corresponding circles is not recommended howeve how ever. r. The preferred procedure also the one used by tthe he QFT Matlab™ toolbox, Borghesani et al. 1994)) will now be presented.
SCw /1 PHUw) I
-l/PH Uw
on GUw
on L=PGHUw)
3.2.. Figure 3.2
Circle on the left with radius of 8;1 w) and centered at [ - 1
OJ
defines the allowed
region for L j w ) = G P H j w ) . The circle on the right with radius I P ~ ~ ; W I 1 / P H jw) defines the allowed region for G jw)
22
centered at
BOUND CALCULATIO CALCULATIONS NS
With the polar pol ar form ofG jw), at a given frequency, w that is, G jw) = ge j ¢ and PH jw) =
pej J,
inequality 3.1) reduces to:
+ p~j ¢+O) I
;
o w).
The squared magnitudes magnitudes o f both sides o f the inequality become:
Rearranging terms gives us the quadratic form: 3.2)
The unknown parameter in the above equation is the complex number ge j ¢ , which is a circle for all ¢ E [0 27fJ see also Thompson Thomp son 1995)). Th Thee curve o f the union o f all these circles is the bound w). n practice it is not possible
Synthesis o f LT LTll Controller Controllerss for MISO LTl Plants Plants
63
to calculate these bounds bound s for all wand infi infinite nite numbers o f plants, therefore the proposed bound calculation algorithm is:
[0,00) into a finite set
1. Discretize the frequencies
n=
2. Discret Discretize ize the uncertain plant p lant set into a finit finitee set {P}
{WI, ... , w m }.
= {PI, ... , Pn }.
3. Discretize the controller phase ¢ into an ordered set, for example = {O - 5 . . , -360}.
4. Choo se a single frequency
wEn
5. Choose a single single phase ¢ E . 6. Choose a single plant Pi from {P}. 3.2). 7. Compute 9max(P (jw), ¢ and 9min(Pi (jW), ¢) using inequality 3.2). 8. Repeat step 7 for all Pi (jw) s and denote
9max(¢,w) 9min ¢, w steps 5 - 8 for all phases phases in in . 9. Repeat steps 10. Repeat steps steps 4 - 9 for al alll frequenci frequencies es in For a given w E n and phase
n.
¢ E the interval from which w hich IG jw)1 is allowed
to take its values is:
IG jw)1
:s: 9min
or
> IG jw)1 > IG jw)1
9max, i f 9min(¢,W)
9max, i f 9min(¢,W)
>
:s:
0 0,
and its curve over all ¢ E 1 that is, the region exterior to this circle conforms to inequality (3.3) while the interior area doesn does n t. However i f ouIPH(jw) I < 1 inequality (3.3) is true within the above above circle s interior interior.. The design process will then be: 1. Calculate circles corresponding to inequality inequality (3.3) at each fre
quency and for all P E {P}. Intersect all o f the allowed regions o f all the calculated circles for a given w, denoting the resulting curve by (w).
2. Find (shape) a TF,
G(s), such that at any frequency w, G(jw) lies inside
its bound B(w) in conformity with inequality (3.3) and the closed loop is stable for all P E {P}.
G (s) will then be said to be a solution to the control effort problem. From simple geometric considerations, it can be shown that the center o f the circle which solves the inequality:
11 LI 1) it to inequality (3.3) for PHG is the same circle with a = o/IPHI; and for the parameter G, since G = it is the circle whose center, denoted by Cg , and radius, denoted by R g , are
lH
Cg w)
=
a2 1 _ a 2 p H(jw)' Rg(w)
=
a a2
PH(jw)l·
Calculation o f these bounds with the aid o f the corresponding circles is not recommende recom mended d howeve however. r. The preferred procedure was presented in section 2.2. Example: For the plant 1 / s, the closed loop plant input at w = 1 is required to be less than 1.1 and less than 0.14 (arithmetic units) at w = 5. The curves (bounds) on Lo(jw) = GPo(jw) for Po = l / s are shown in Fig. 3. 3.4. 4. For w = 1 it is the closed curve s interior which is the forbidden region, while for w = 5 the area below the open curve is the allowed region. 24
EXAMPLES
Example 1: Given an uncertain plant which includes an integrator, a simple pole at - 1 , and uncertain gain in the interval [1,5] {P} -
k
- s (s + 1 )
k
E
[
]
1,5,
QUANTITATIV QUANT ITATIVE E FEEDBACK DESIG DESIGN N
30 25
20 15 10 III 0
;
forbidden· region· .
: for 0
5
=
1
Inside· CirCle
o -10
-15 -20 -360
-315
-225
-270
-180
deg
-135
-90
-45
o
=
effort;; for w 1 the interior of the closed curve is the forbidden igure 3.4. Bounds on control effort region for Lo jw). while for w 5 the area below the open curve is the allowed region for Lo jw). In the the complex plane. w with ith real and imaginary arithmetic scales. both bot h curves are circles
=
the requirement is to design a controller such that for any plant in {P}: (i) the closed loop is stable; U) at frequencies [1,2,4] rad/sec, the sensitivity is less than [0.05,0.2,0.8] arithmetic units, respectively; and iii) at all frequencies, L jw)I- < 3.5dB are satisfied. gain and phase-margins o f the form The bounds and two designs are shown in Fig. 3.5 and Fig. 3.6 where the nominal plant is 1/(82 + 8). The designed controller in Fig. 3.5 is
_ 2 1 + 8/30)(1 + 8/6) G l - 8 1 + 8/20)(1 + 8/90)(1 + 8/220)' and is composed o f only real poles and zeros, while the second controller design, which appears in Fig. 3.6, and is 78 2 G = 28 1 + 8/27)(1 + 8/6) 2 (1+8/14) 82+788+782
Synthesis o LTI Controllers Controllers fo forr MISO LTI Plants
67
consists o f a complex pole pair with a damping ratio o f 0.5. The second design is better in the sense that the high frequency sensor noise at the plant input is reduced, in compa comparis rison on to the fi first rst design. A compar comparison, ison, in the form o f a Bode plot o f the two TF s from the sensor output to the plant input, is depicted in Fig. 3.7. Note that the loop loop shaping algori algorithm thm tries to position positi on the open loop response at each frequenc frequency, y, as close as possible to its corresponding bounding curve - in the allowed region, region, which quite naturally imposes a smaller band band width solution (Horowitz (1978) appendix 1 . Since at high frequencies the
30 20 10 0 co
C
-10
-20
-30
48 - 60
-270
Figure
-180
-90
deg
35
Bounds and
L
o
= G1Po
benefits o f feedback are negligible, any specifications at high frequencies will result in a large bandwidth with negligible closed loop performance improve ment. men t. From the bounds bounds at w 4 we can expect that at higher frequencies, the open loop gain will be less than OdB and therefore the specifications chosen at frequencies w > 4 are only of the gain and phase margin type. Example 2: The former example presents us with a good good opportunity to demon strate the effect o f uncertainty on the controller s high frequency gain. The de sign G2 is compared to a design for the same example but with less uncertainty
68
QUANTITATIVE
FEEDBACK FEEDBAC K DESIGN
10
0 [ l 0
-10
-20
-30
100
-4 ~ 60
-270
-180
-90
deg
Figure
Bounds
36
and
L
= G
o
o
(the gain is in the interval [1,2]). The controller G 3 turns out to be: 39 2
G _ 28 1 + 8/6)(1 + 8/18.6)
82+398+392
1+8/14
Fig. 3.8 shows the bounds o f both uncertain plants. plants. The low frequency bou bounds nds are the same, but the high frequency bounds for the expanded uncertainty case are d B larger than than for the lo low w uncertainty uncert ainty plant. As a result, the high frequency controller gain, for the low uncertainty plant, (also the controller amplitude amplitu de and sensor noise amplification at the the plant input in high frequencies) is about lOdB less than corresponding gain for th thee plant with large uncertainty Example 3: The simplest model mISSIon
of
a loaded DC motor with flexible trans
IS:
z Tm ikt
-
Bl } l
Tdl
- Bmem -
Tdn
Synthesis o f LTI Control Controllers lers for MISO LTI Plan Plants ts 2
5
0
~
~
~
~
~
~
~
~
~
350
400
~
9
~
200 ~ 1 5
50
100
150
200
250 0)
300
450
500
50 aJ45 E. ~ 4 - '
35
)
30 1 10
10g(0))
102
Figure 3.7. Comparison of the two TF s from the sensor output to the plant input for the two designs G 1 and G 2 upper plot in arithmetic scale lower is a Bode plot
It
Bmll n Bm
-
Bl cml
The symbols have the following meaning: motor current k t - the motor e m J constant Bm - the viscous friction on the motor axis Jm - the motor inertia Tm - the electrical moment m - the motor angle n - the gear gea r ratio Cml - the transmission spring factor dml - the transmission damping factor Tl - the moment transferred to the load i
the
l
Jl l Tdl
-
the the - the - the -
viscous friction on the load load inertia load angle disturbance moment on the load.
iJmln
- iJ l )
dml·
QUANTITATIVE QUANTITAT IVE FEEDBACK DESIG N
70
30 20 10
o III 0
10
20
30
40 360
-270
180
90
deg
o
Figure 3.B. Comparison of two designs: Lb for the expanded uncertainty case (rigid bounds), La for the low uncertainty case (rigid bounds in low frequencies and dashed bounds at high
frequencies)
Bl and B m can be neglected, the TF s from current command, i, to the motor angle, Om and load angle, It, are: If
Om
kt JI s2
:mL J[ s
£mL) J[
+ n:ry;;; + n:ry;;; emf) J m Jls2 s2 + 1:mL ~ s + £mL J[ J[ (3.4)
For the plant described above for Om/i, we assume the following numerical values: dmL/ Jl = 2 cmL/ Jl = 100, dml/ J m = 0.4, cml/ J m = 20, n = 1, kt/ Jm = 1. It is assumed that the uncertainty only acts on the load, Jt, which can be up to 5 times its nominal nomin al value. value. A contr controller oller is required suc such h that (i) the closed loop is stable; ii) at frequencies between [1 2] rad/sec, the sensitivity is less than 20, -7]dB respectively; and iii) at all frequencies gain and phase-margins o f the form 1 11 L < 3.5dB are satisfied.
Synthesis o f LTI Controllers for MISO LTI Plants
71
The bounds and design are shown in Fig. 3.9. Note the change cha nge in the the bounds boun ds near the resonance frequency, as a result of large plant uncertainty near the resonance frequenc frequency. y. The resulting controller is: 95 2
G = 22 1 + 8/3) 1 + 8/76) 1 8 /1 5
1
30
82 1508 952
81)
20 10
O
'0
l
0
-10
-20
.,
4
7.5 10.5 12
,
........
,
18
3B 70
30
-180
-225
Figure 3.9,
deg
-135
-90
Bounds and open loop for Example 3
Example 4: The aim o f this example is to demonstrate how QFT handles un certain under-damped plants. plants. The plant was was first first suggested by Wie and Byun 1987). I t can represent a model of o f a loaded loaded DC motor whose sensor is located on the load itself its elf see equation equati on 3.4) for very very small d m1
{P} --
k1 C
8 2 s2
+ 0.2y
ks
l
-
+ 2k
l k E 0.5,2.
A controller design is required such that, for any plant in {P}: i) the closed loop is stable; ii) at all frequencies gain and phase-margins o f the form +
72
QU NTIT TIVE FEEDB CK DESIGN
L jw)I- < d B are satisfied; and iii) the cross-over frequency is the highest possible but less less than the the reson resonance ance frequ frequency ency.. The motivation for using specification iii) is to limit the control effort, because it increases rapidly for cross-over frequencies for which the phase o f the plant is much less than 1800 • The bounding curves and corresponding design are shown in Fig. Fig. 3.10 where the nominal plant is chosen to be k = 0.5. Each sub-plot contains a single bound along with the the nominal loop response and a single frequen frequency. cy. Since Sinc e the open loop almost touches its bounding curve below the cross-over frequency, it is considered to be an an efficie efficient nt desig design. n. Trying to increase the cross-over cross-ove r frequency will increase the bandwidth tremen tremendously dously (see exercise 3.6). The controller is:
G
0.14
=
82 /0.01 1 8/5)(1
. . . .
..
. .
0.16/0.018 1 8/0.6)(1 8/0.04)
.
30 20 10
.
=0.3
;0:6
0
10 20
-270
-180
-270
90
-180
90
30 0
~
10
-270
-180
0
~ 8
90
:
10 20
20
-360
-360 30 20 10
20 10
30
30
-270
-180
90
30
-270
-180
90
90
-270
-180
90
90
-270
-180
90
-270
-180
90
-270
-180
-270
-180
-360
30 20 10
0
10 20
-270
-180
90
-270
-180
90
30
-360
Bounds and loop transmission for L1 Figure 3 10 G1PO - each sub-plot represents the loop transmission and bounds at a single frequency (denoted = by ow = frequency on the bounding curve)
Synthesis o f LT Contr Controlle ollers rs for fo r MISO LT Plants
3
TWO
73
OF SYSTEMS
The feedback system o f interest is depicted schematically in Fig. 3.11, and described mathematically by the equations: y
Pu
u
-GHy+GFr.
Our concern is how to design a controller, G s , and prefilter, F s , such that
-
-
-
u
-
G
P
-
H
Figure 3 11
-
Y,
A two DOF feedback system
for a given set o f plants, {P}, the closed loop is stable and certain w-domain specifications specifi cations are satisfied. The w-domain specifications specifica tions are o f the following two types: i) model matching - for a given plant set, {P}, and a T Fm(s)
. I L(jw) - Fm(Jw)
PGF jw)
I1
m W), 6m 6
E
V
{P};
3.5)
(ii) tracking - for a given plant set, {P},
a(w)
PGF jw)
I1
L(jw)
I
{3(w),
V
E
{Pl·
3.6)
The model matching specification limits the distance o f Fm s , the desired TF from the tracking T by 8m w), while the tracking specification places upper and lower limits, {3(w) and a(w) respectively, on the amplitude o f the tracking TF.
The model matching problem can be reduced to the sensitivity problem discussed in section 2 I f we allow F == H Fm , then inequality 3.5) becomes:
I
Fm(jw) I 1 L(jw)
8m (w), 8m
V
E
{Pl·
The tracking specification divides itself between the task o f the controller, G, and that of the prefilter, F . The job o f the controller is to reduce the difference
QUANTITATIVE QUANTIT ATIVE FEEDBACK DESIG N
74
between the plant s minimum and maxi mum outpu outputs, ts, Y m i n and Y m a x (on a log scale), due to plant uncertainty, while at the same time ensuring closed loop stability. The principal task of F , on the other hand, is to shift the Bode plots of P G / l PG) (on a log scale), so that they are within the specification boundaries. This is shown pictorially in Fig. 3.12. The effect o f F and G will be explained in the sequel.
-2 -4
-6 -8 -10
-12 -14 -16
-18 2
0
~
~
~
~
~
~
~
~
~
~
~
~
~
~
10 1
10°
~
~
10 2
log m )
3.12. Differ Different ent effects of C and F on the closed loop TF: The response of IPC / 1 igure 3.12. PC)I is situated between betwe en the the upper two solid lines, due to plant uncertainty. The requ required ired specs for the closed loop response are the two two dotted line lines. s. The filter F shapes the upper two solid lines so that they fall within the specs (dashed lines).
The TF, F
s , is a solution to inequality (3.6)
if and only if it satisfies the
inequalities: loga w)
10g,8 w)
min log
PE{P}
PE{P} max
log
Il Il
G
P G
~ ~ ) )
LJw
~ ~ ) )
LJw
I S log IF jw)l, I
log IF jw)l·
and
(3.7)
It should shoul d be noted that for a given given amplitude ampl itude function IA w) I there always exists a stable minimum-phase TF, F s), such that IA w)1 = IF jw) 1 (Horowitz
Synthesis o f LTI Controllers for MISO LTI Plants
75
1963). This in tum implies the existence o f a minimum-phase solution, F s), 1963). if the following inequality holds max log
PE{P}
1 PG jw) 1 1 + L jw)
1 PG jw) 1 < log 3 w) . 1 + L jw) a w)
min log
PE{P}
(3.8)
Notice that inequality (3.8) depends only on G s) and the plant uncertainty. The job, therefore, o f the controller, is to reduce the closed loop uncertainty the filter does not contribute to this end. Th Thee prefilter, F simply positions the TF s PG/ 1 + L), for all P E {P}, to lie within the specified curves, but doesn't change the uncertainty o f PGF/ 1 + PGF) on a log scale 31
BOUN BOUND D CALCULATIO CALCULATIONS NS
Two additional algorithms for calculating Two calcul ating bounds on G s) are now presented. The first is based on the notion o f 'templates' and its major advantage is the excellent insight it provides into the relationships between plant uncertainty, closed loop specifications, specifications, bandwidth, bandwidth, scheduling and controller controll er structure. The second o f the two is a closed form algorithm, which is recommended when the discrete number o f plants in {P} is small. From here on we shall shall assume, without loss o f generality, that H 1. 311
BOUND CALCULA CALCULATIONS TIONS WITH THE AID OF PLANT TEMPLATES
A TF, P s), evaluated at a given frequency, w, is a complex number, the set o f all complex numbers for a given set o f TF's, {P} evaluated at a given frequency w, that is Tw = {P jw)
IP
E {P} }
is termed the Template o f { P } at w. The region labeled 'A' on the Nichols chart o f Fig. 3.13, describes the template o f {P} = k / s s + a) for k E [1,5J and a E [1,2J at the frequency o f w = 1 rad/s rad/sec. ec. The closed loop values values for this template, that is the intersection of o f the extremities o f the template with the P G / 1 + PG) curves on the Nichols chart, for all P E T w , range from - 6 d B to 3dB. Hence for G = 1 (template A), the closed loop uncertainty is:
2010g max /1 PE{P}
P ~ ~ ~ ) + JW
20 log min / PE{P}
P
j ~ ~ ) /= 3 -
1 + P JW
(-6) = 9dB.
The template for {PG} bears the same shape as the template {P}, but is arg
located 20two logexamples, it and forGG jw) degrees to labeled the right. right.BFig. IG jw)1 above includes the template and3.13 for = [16dB,00J G = [19dB,500] labeled C . By simply sliding the template around on the Nichols chart, a readout o f the amount of uncertainty which the closed loop
76
QUANTITATIVE QUANTITATI VE FEEDBACK DESIGN DESIG N
Nichols Chart 40 OdB .
30
. ...... · · · · ·
~ : ~ ~ d B
20 10 .....
.:
..
: . .. : .... .
,
;::;:
•. : .• i·,
-12 dB
, .. .· :
··: ... 20 dB
:::
- ~ g 6 ~ -330 -300 -270 -240 -210 -180 -150 -120 -90 -60· - 30 [deg]
•• 40 dB
o
Templates for w = 1 rad/sec consists of all complex numbers within the closed curve 'A'; 'B' is the same template but shifted by 16dB; 'c' is shifted by 19dB and 50°. The symbol x on the symbol the bottom right right hand comer of all of the templates denotes the nominal plant Figure 3.13.
l ~ ; g t j L
can tolerate, tolerate, may be easily obtained. obtained. For a given given phase o f G jw),
I
there always exists a minimum amplitude o f G jw) above which the closed loop uncertainty o f ; d j w s less than than a desired value. The bound calculat calc ulation ion procedure for a set o f frequencies a , /3 will then be as follows:
w
n
and given specifications in the form
1. Pick a frequency w E n .
2. Plot the curve o f the template
Tw o f
{P} on a Nichols chart (for an efficient
algorithm see Ballance and Hughes (199 (1996)). 6)). 3. Arbitrarily choose a single plant o j w from {P}, designate it as the nominal plant and mark its location on the template. 4. Discretize the controller phase ¢ in steps of, for example .. ,-360}. < > = {O -5
5. Choose a single phase ¢ E
.
77
Synthesis o f LT Controllers Controllers fo forr MISO LT Plants
6. Shift the template horizontally by po, and then vertically by zdB (open
loop gain) without changing the templa te s attitude, until the point where wh ere the template s closed loop uncertainty uncertainty meets the specification, specification, i. i.e. e. the point where
PG(jw) 2010 g Pmax E{P} 1 I PG(jw)
I
-
I
201 g Pm .P } 1 PG(jw) PG(jw) E {m
I
f3 w)
20 log a(w)
Mark the location o f the nominal point o f the template on the Nichols chart (shifted by cpo and zdB). The significance significance o f this point is that i f arg G (jw) = cpo and IG (jw) I > zdB (open loop), the closed loop uncertainty at w is satisfied, that is, 2010
g
max
PE{P}
PG(jw) I I 1 + PG(jw) <
7. Repeat step 5 for all
cp E
201
.
og ~ { ~ }
I PG(jw) I 1
PG(jw)
f3 w)
20 log a w) .
1> then connect conn ect all the marked marke d points.
This curve is the bound on PoG at w; and it means that i f G (s) is designed such that PoG(jw) is located above it, then the closed loop specifications are satisfied.
8. Repeat step 2 - 7 for all frequencies in f2.
As long as f3(w)/a(w) is close enough to 1 then for each phase angle cp E [0, 21T 1 there exists a finite amplitude o f G (jw) above which the closed loop uncertainty is sati satisfie sfied. d. The bound will then then be a curve which crosses the Nichols chart from right cp = 0) to left cp = -360 . A bound o f this kind is called an open bound and the frequencies for which the bounds are open are called low frequency freque ncy bounds. frequencies, the specifications specifications are only gain and phase margins, that At high frequencies, IS
I1
PG I PG
;
6n W)
or
11
1
PGI
;
6s W),
for some 6n w w)) > 1 and/or 6s W) > 1. The same technique described above can be used to find bounds, but for each phase angle there exist two values o f g l and g2 gl < g2 such that the margin specifications are satisfied IG(jw)l, if IG (jw) I S g l or if IG (jw) I :: g2. The resulting bounding curve is closed, unless the uncertainty is too wide in phase. A bound bou nd o f this kind is called a closed bound and the frequencies for which the bounds are closed and larger
QUANTITATIVE FEEDBACK DESIGN
78
than the low frequency bounds are called high frequency bounds. Example: Given the uncertain plant {P} the specifications are 1
=a
<
=
s ( s ~ a )
for k E [1,5] and a E [1,2]
I1PGF PGjw) jw) I
f
= 1.12, for w = 1
PG jw) I I 1 + PG jw)
2
= 6dB,
for w
;
o
The open bound, labeled B l in Fig. 3.14, is the bound for w = 1 rad/sec. The plant at w = 1 is the lower one with a 1 at its cente center. r. By moving the temp templat latee vertically (increasing the open loop gain by 17 dB), its closed loop uncertainty is reduced to 1dB (1.12 in arithmetic units). units). The nominal plant Po = S ( S ~ 2 ) marked on the template by an x , impinges upon the bounding curve B l The same is true when the template is shifted by 18.7 d B and 50°, (see the plant in the top right comer with a 1 at it itss ce center). nter). The closed bound curve B 5 , in Fig. 3.14, is the bound for w = 5 rad/se rad/sec. c. The template for w = 5 rad/sec is the lowest one on the graph with a 5 at its center. center. Increasing Incre asing its open loop gain by l l d B and shifting it by - 2 0 ° results in its closed loop uncertainty being less than 6dB (the template doesn t enter the 6dB region but touches it only) and the nominal plant, Po = s ( s ~ 2 ) marked on the template by an x , touches the lower portion o f the closed bounding curve (see the template marked 5 whose nominal point is at about -18dB). The same is true when the template is shifted vertically by 35dB (open loop gain) without changing its phase angle (see the uppermost template with a 5 at its its center on on the graph graph). ). Ther Therefor eforee for a controller with a phase angle o f - 20° at w = 5 rad/sec, there are two points on the closed bounding curve which means: IG j 5 I should be less than l l d B or greater tha than n 35dB in order to satisfy the specifications at w = 5. 312
BOUND CALCULATIONS CALCULATIONS USING A CLOSED FORM ALGORITHM
For a given frequency I l : } , ~ G
I
w
nd
two plants say Pi, P2, where I l : ~ ~ G
1
>
inequality (3.8) reduces to
I
1 1
PiG) P21 < f3 w) P2G) Pi - a w)
if its left side is greater than 1, otherwise it becomes:
I
1
P2G) Pi I < f3 w) .
(3.9)
PiG) P2 - a w)
1
Synthesis o f LTI Controllers for MISO LTI Plants
79
Nichols Chart
40 30 20
-
10
. :.-3 dB
en
0 : . . . . . ..
-10
. .. :
-20
-30
~ g 6 0
.
:
3 3 0
3 0 0
:
2 7 0
2 4 0
2 1
:
0-180-150-120 -90
[deg]
dB
;-6
dB
: ~
2 d B
.-20dB
40 dB
-60
- 30
o
Templates and Bounds: For w = 1 the template is the lowest structure with a 1 at its center, the other templates are of the same size and shape but shifted vertically (open loop gain increase or decrease) or horizontally (open loop phase added or subtracted) or both, The bounding curve BJ is an open bound, For w = 5 the template is the lowest one with a 5 at its center and its bounding curve B5 is a closed bound Figure 3,14,
Its solution is a circle in the complex plane, such that if G jw) is found in its interior, the inequality inequal ity does not hold. This can be seen lettin letting g G 0 0 so that inequality (3.9) tends to 1. This implies that the inequality is true, which further implies that G is not in th thee circle s interior interior.. The cente centerr and radius of this circle can easily be calculated by analogy to the procedure in section 2.3. Letting L
gives:
=
1
PI G) P
_ _\ < ~
\
P2 1 -
P
2
a
I+L-a-
whose solution, to the parameter L is the circle located at Z I = 2 / 1 - a2 , ] with radius TI = a II - a2 (see section 2 2.3) .3).. The solution to the parameter 1
to
1
80
QUANTITATIVE QUANTIT ATIVE FEEDBA FEEDBACK CK DESIGN
+ P i G will then be the circle located at
Pl/ P2 - 1) with radius r2 = rilPl/ P2 - Similarly the solution to the parameter Pi G will be the circle located at Z3 = Z - 1 with radius r3 = r2 and the solution to the parameter param eter G will be the circle locat located ed at Z4 = Z3 / Pi with radius r 4 = r3/ IPi I which is is the solution to inequality 3.9) for the para paramete meterr G. An algo algorithm rithm
1
Z
=
Zi
to find bounds on G(jw) which solves inequality 3.7) will therefore be as follows: 1. Discretize the uncertain plant set into a finite set {P}
= {Pi, . . . , Pn } .
2. Choose two plants Pi, Pj from {Pl.
3. Comp Compute ute the circle which solv solves es ineq inequality uality 3.9 3.9)) for Pi
=
Pi and P2 = Pj .
4. Rep Repea eatt step 3 for all pairs o f plants, Pi, Pj in {P}. 5. The curve made up o f the union o f all circles computed in step 4 is the bound B(w). 32
AN EXAMPLE
Plant set and speci specificati fication on
{P} =
k
88+a
k
)
E
[1,10], a E [1,10].
is required to design a controller, G(8), and prefilter, F(8), such that i) the open-loop response o f L(8) will satisfy the margin specifications t
L jw) jw)
I1 + L
I
< 3dB,
Vw
0
and (ii) the amplitude o f the closed loop T F from r to y will be bounded between the following f3 w) and a w) functions w
I 1 w)[dB] I a w)[dB]
I 0.5 I I 0 I I - 0 .5 I
1
0 1
I 2 I I -1 I I 3 I
4 4 8
I 8 I I 7 I I -14 I
Controller desi design gn - The cho chosen sen set of frequencies, n is the set for which the tracking specifications are given and includes the high frequencies 15, 30, 60, 100 and 200. 200. The bounds are shown in Fig. 3.15 for the nominal plant S ( S ~ l ) which also includes the shaped loop transmission for the controller G _
52 1
+ 8/70)(1 + 8/6)
-
1
+ 8/0.3)(8 2 /170 2 + 8/170 + 1 )
Synthesis o f LTf Controllers for MISO LTf Plants
81
50 40
30 20 10 c 0
0 -10
I
I
I
I
15 30
-20
60
-30 -4£
120 180
- 60
-315
-270
-225
Figure 3.15.
-180
deg
-135
-90
-45
0
Bounds and open loop
Prefilter Prefi lter desi design gn - The amplitud amplitudee of the T F l : ~ g t j ~ )
was simulated for 100
plants for each integer in the range o f k 8 and a 8). The maximum and minimum mini mum values at the tracking sp specification ecification frequencies frequencie s are shown in Fig. 3.16, rigid line, which also include the specifications and maximum and minimum closed loop o f amplitude, 1 1 : c : J j ~ ) I using the prefilter 82
= 1
11.28
8/2.8) 1
64)/64 8/8)(1
8/35)
clearly the specifications are satisfied. 4
EXTENSION TO NMP PL NTS
The NMP N MP plant ty types pes addressed here are plan plants ts with pure delay and/or RHP zeros emb embedd edded ed in a feedback structure Fig. 3.1 3.1 or Fig. 3.11 3.11). ). He Hence nce their loop transmission, L = PG, is bandwidth limited, that is, there exists an w such that L jw) = OdE where w decreases as the RHP zeros decrease or the delay gets larger see Chapter 2). 2). The design techniq technique ue described in the
previous sections is also suitable for NMP plants, but difficulties may arise
82
QUANTITATIVE QUANTITA TIVE FEEDBACK
ESIGN
4 2 0 -2
D
"'0
- :; = := = :: : - - -
PGI··
1.+PGI
.j.. ,
-4
....... \,.
,
-6
,
,.
,:
.
,:' Ir.,
o-
.
"
,
-8
.......
-10
. ... .
.........
,.
"
•
,
: ......
,
.
+;
.
,
. :
: ...... .
: IPGFI
: " : Ifl-PGI
" , :: \
-12
.
:
:
-14
log w) Prefilt Prefilter er design: Rigid lines are maximum and minimu minimum m o f I PG/ l + PG I. dashed lines are maximum and minimum of IPGF / l + PG I. the + signs are the specs 3 w) and a w) respectively Figure 3.16.
because o f the bandwidth limitations, which are: (i) for stable plants the con flicting requirements o f large bandwidth for tight ti ght closed loop specific specifications ations and the maximum bandwidth bandwi dth limitations due to RHP zeros and delays; delays; and ii) for unstable plants the conflicting requirements o f large bandwidth for tight ti ght closed loop specifications, specifications, the minimum bandwidth limitations limit ations due to the RHP poles, and the maximum bandwidth limitations due to RHP zeros and delays. Plants which includes both RHP poles and zeros (or delays) may have serious closed loop limitations, limitations, especially ga gain in and phase-margin limitations. For Fo r a quantitative discussion see Horowitz (1979). A quantitative argument, based on the Hilbert transform, transform, is now described in order to reinforce a pictorial explanation o f the NMP closed loop performance limitation phenomenon. phenomenon. Letting L M s) denote a minimum-phase and stable TF, the Bode (Horowitz 1963) relating its phase, argequation: LM jW)), to its amplitude, logintegrals ILM jW)I, are given by the principle integral
-
ln oo dlog LM jW) W
)d
h
-
7r
d
0
U
U
U,
were
Synthesis o f LTI Contr Controllers ollers for MISO LTI Plants
u
= l o g ~
Wo
W u)
= log
coth
M2
= log
83
+
w wo. W - Wo
The weighting function W u is infinite at wo, very large within the vicinity o f W o and decreases with Iw - W o I, that is, it is close to the 8 w - wo) function. Thus, on the average for a large enough frequency interval we can conclude that: tha t: the phase o f LM jW) (which is negative) is larger (closer to zero) i f the slope o f the amplitude o f LM jW) in log scale (which is negative) is larger (closer to to zzero ero). ). Moreove Moreoverr if its phase is positive (negative) then its amplitude is an increasing (decreasing) function of w. This observation is true on the average over large enough frequency intervals, but as the phase o f LM jW) tends to zero, it becomes tru truee almost p pointw ointwise. ise. The proo prooff and conditions under which this is true follow. Let us assume that arg L M jw) ::; 0 for all frequencies except for those inside the interval [WI, W2J wherearg LM jW) > O Using the Hilbert transform
-- ~ ° O Iog IL M (JWo . )1 )17r 0
2 2 W, wargLfvdjw)d Wo - w
it can easily be shown that
which is negative because the sign of argLM jw) is positive on [Wl,W2J and negative on the intervals (O,wd, and W2,OO). This is depicted in Fig. 3.17. Thus as arg L M jw) t 0 in the interval [WI, W2], IL M jW2) I IL M jwd 11 . The observations above will help us to understand the NMP phenomenon o f the following example. exam ple. Fig. 3.18 shows a loo loop p transmi transmission, ssion, Lo = PoG, for th thee NMP plant {P} =
k 20 - s s
20
+s
k E [1,3J
with margin specifications o f the form 1/11 not only
Lo = 20 - s ~ G s + 20 s
+ L jw)1 < 3.5.
Fig. 3.18 shows
I it is assumed that
M is continuous on the interval [Wi, W ]
84
QUANTITATIVE QUANTIT ATIVE FEEDB ACK DESIG N
41(jw)
41(jw) dB
I
dB
t
-180
(a)
t
o
I
-180
\
(b)
figures es labele labeled d (a) (a) and b) show two trajectories of a minimum phase plant on Figure 3.17. The figur a Nichols chart. The trajectory of figure (a) is not feasible mathematically, while that o f figure (b) is. Note that WI :::; W2
o 0
30
30
20
20
10
10
o
o 10
10
20
20
~ B 7 0
2 2 5
1 8 0
1 3 5 9 0 deg
45
0
~ B 7 0
2 2 5
1 8 0
1 3 5 9 0 deg
45
0
TF's 's L jw) and their minimum phase parts LM: the low frequency Figure 3.18. Two NMP TF gain of the right figure is 1 5dE more than the one on the left but the cost of control in terms o f amplitude is increasing, for example, G jw) has increased by at least lOdE at w > 300
Synthesis o f LTl Controllers for MISO LTl Plants
85
but also its minimum-phase part, L M = iG s), for two designs. Clearly as one tries to increase the open-loop bandwidth the following happens: i) the gain margin frequency increase increasess in this exam example ple from 70 to 300), ii) the interval o f frequencies close to the margin frequencies increase, and iii) the phase o f the minimum phase part o f the loop-transmission approaches zero, and therefore cannot be decreased in order to achieve more gain margin margin.. The result is bandwidth limitation. In the example above the high frequency bounds are one closed curve, because o f the choice o f the nominal plant as an NMP system. I f the nominal plant will be chosen as minimum-phase, the high frequency bounds will move to the right. right. The bandw bandwidth idth will then be limit limited ed b by y the ability to sha shape pe the loop such that it will will encircle the bounds from the rig right. ht. Otherwise Other wise the bounds will be encircled from the left which means, by the Nyquist stability criterion, that the closed loop, with open loop L s), is unstable. This enci encircle rclement ment must take place when the phase of L M jw) is negati negative ve and whenever L M jw) passes near the bounds. Othe Otherwis rwise, e, by the discuss discussion ion above, the ampl amplitu itude de o f L M jw) will be an increasing function the right.
of wan d
therefore will not encircle the bounds from
EXTENSION TO S MPLED DATA SYSTEMS
5
The feedback system dealt with is depicted schematically in Fig. 3.20 where T denotes the sampling time o f the AID converter ws = 2n I T is the sampling frequency) and signifies a sampled signal such that x* t)
=
L x kT)r5 t -
kT).
k=O
The Laplace transform
of
this sequence is
X* 8) =
L x kT)e-
skT ,
k=O
while 280)
if
X 8) has more poles than zeros A strom and Witten mark 1996) p.
X* 8)
1 = -
T
LX80
j k w s ).
The D/A converter is a ZOH. Mathematically the system is described by the equations Pzoh z) denotes the T from the ZOH input to the output o f P where
z
=
esT)
Y
Pzohu
+d
- G z ) H z)y* + n
u
- F z)r*) ,
QUANTITATIVE FEEDBAC FEEDBACK K DESIG N
86
30
20
10
0.5 -10
1 2 4
8 15 -20
30 70
200
200 -30 -270
-225
-180
-135
deg
o
-45
-90
Figure 3.19.
An NMP plant, the nominal open loop, L M s), is minimum phase, therefore the bounds move with frequency to the rig right. ht. Near w = 30, the phase of L jw) must be negative so that its amplitude will be a decreasing function of w, and thus the bounds will be encircled by L jw) from the right, which means that the Nyquist stability criterion is satisfied
~
r
F z)
d
D/A G Z)
u
ZOH
u
p
H z)
Figure 3 20
Two DOF sampled data control system
y
Synthesis o f LTI Controll Controllers ers for MISO LTI Plants
87
the sampled output y* and sampled input u* are d* - P;ohG z)n*
y* s)
u* s)
+ P;ohG z)F z)r*
P;ohG z)H z)
1
-G z)H z)d* - G z)n* z)n*
=
G z)F z)r*
P;ohG z)H z)
1
Now let us look at the following closed loop specifications: 1
Sensitivity Reduction Pro Proble blem: m: for all P E {P} the T F from disturbance at the plant output to the plant output is bounded by
z) IY* d* z)
=
1
1
P;ohG z)H z) z=e j w
< os w). -
2. Model matching problem (given Fm z)): for all P E {P} the distance o f the T F from the command input r* to the plant output from the optimal TF,
Fm z), is bounded by Y: z) - Fm z)1 I r z)
=
I P ; o ~ G z ) F z ) - Fm z) PzohG z)H z) z=e j w 1
::; om w).
3. Tracking problem: for all P E {P} the amplitude o f the T F from the command input input r* to the plant output is bounded by
a w) < IY* z) = P;ohG z)F z) - r* z) 1 + P;ohG z)H z) z=e j w 4. Control effort problem: for all P the plant input is bounded by
u* z) I = I In* z) 1
E
< 3 w). -
{P} the T F from the sensor output to
G z) I P;ohG z)H z) z=e j w
< oc w). -
All o f these specifications are equivalent to those discussed for continuous systems. Therefore There fore bounds for each frequency, frequency, w, are calculated exactly as for the continuous continuo us case. case. The only only difference is that the controller, G z), is discrete, but in practice it is the same as shaping a continuous contin uous controller. In the M atlab™ QFT toolbox (Borghesani e t al. (1994)) the designer shapes discrete TF s based on his skills to shape continuous TF s. The idea is the following: The designer decides to to add a pole, zero, etc. to the loop-transmission, loop-transmis sion, the software picks a discrete approximation o f the chosen chos en pole, zero, etc. and displays a Nichols plot o f the discrete design using the transformation z = esT. When Whe n the designer is satisfied with the the shaping (which (whi ch is a pictorial pictori al criteria), the discrete controller is already available without the need to use transformations
o f any kind from the continuous to the discrete domain.
88 6
QUANTITATIV QUANTI TATIVE E FEEDBACK DESIGN DESI GN
SUMM RY
In Chapter 1 it was shown that the main reasons for using feedback are plantt uncertaint plan uncertainty y and unknown distu disturbances. rbances. In thi thiss chapte chapterr the QFT design technique is presented as a solution to the parameter uncertainty and disturbance attenua attenuation problems. The resulting open open loop transmission function is very narrow tion in bandwidth. Reasonable closed loop time domain specifications in the form o f upper and lower limits are given and converted to specs in the frequency domain in the form o f an upper bound on the sensitivity (or complementary sensitivity) o f the transfer function at each freq frequenc uency. y. The QFT technique desc described ribed here is suitable for these types o f specs, and moreover m oreover iitt s solution exactly satisfies these specs without over-design. The QFT technique suits all kind o f LTI plants: stable, unstable unstable,, NMP, NMP, NMP and unstable, sampled data, which include pure delay, and even measured frequency domain data (no model). NMP plants deserve special treatment because their feedback properties are limiting. This limitation appears in the form o f an upper bound on the cross over frequency (or closed loo loop p bandwidth). A qu quantitative antitative proc procedure edure to assess this limitation was presented. 7
EXERCISES
Given the open loop = P G whose uncertainty is 5 < IL I < < 0, find the maximum and minimum values o f the 50 and - 3 6 0 < arg closed loop amplitude of L / 1 L . Repeat for the controllers 2G and lOG and compare by how much the uncertainty o f the amplitude o f L / 1 L was reduced relative to G (on a d B scale). EXERCI SE 3.1
EXERCI SE 3.2
o f IL/ 1
Repeat exercise 3.1 for the closed loop TF 1 11
L)[. What are the conclusions?
L[ instead
Given the plant k / s 1 where k E [1,5]. Plo Plott the plant step response for k = 1 2 3 5; then use the controller G = 10 and plot the closed loop response for the same plant cases. Comp Compare are the closed loop uncertai uncertainty. nty. E X E R C I S E 3.3
E X E R C I S E 3.4
rad/sec and W =
5 6
Given the plant P
eE [0.2,0.7]. [0.2, 0.7].
=
S
2
2 ~ k + 2 sW Wn n
where k
=
10,
Wn
=
10
Plot the plant templates using a [dB deg scale for 15 rad/sec. Repeat for ~ = 0.5 and W E [8,12] rad/sec.
Given the plant P = ~ e s T where T E [0,0.1], k E [1,3]. Plot the plant templates using [dB deg scale for W = 0.1,0.2, 1 5 rad/sec. EXERCI SE 3.5
Repeat example (4) in section 2.4 for cross-over frequencies 0.4, 0.6, 1 and 2 rad/sec. Comp Compare are the Bode plots o f the contr controllers ollers - what ar aree E X E R C I S E 3.6
the conclusions?
Synthesis o f LTI Controlle Controllers rs for fo r MISO LTI Plants EXERCISE
=
3.7 Given the plant P
Design a controller with margins
I I I EXERCISE
I
w
I
i3 w) n w)
I
1 1.1 0.9
I I I
2 1.1 0.8
where k E [2,10] a E [1 4].
s ( s ~ a )
LI
1
I
3
I I
< 6dB and tracking specs
I
1 0.6
I
5
I
0.7 0.2
I
I I
I
10
I
0.7 0.1
I
3.8 Repeat exercise 3.7 where the plant has a delay T
find the
maximum T for which you can shape a controller. EXERCISE
. the plant P 3.9 GIVen
=
2
2 ~
89
kw 2 w
n
s
] where k E [ 2,6,
+ 2 n
Wn
E
[5 9] rad/sec ande E [.5 .9]. Design a controller with margins 11 LI-1 < 6dB and tracking specs 2 1 I w I ,i3 w) I 1.1 I 1.1 I 1
In EXERCISE
w)
and tracking specs
1
3
1
5
1
I 0.7 I
1
I 0.9 I 0.8 I 0.6 I 0.2 I
3.10 Given the plant P
5rad sec and e E
1
kw 2
=
s 2+2€w nn s + Wn2
where k E [2 6]
[.1 1]. Design a controller with margins w
I
i3 w)
In
w)
< 6dB
I 0.9 I 0.8 I 0.6 I 0.2 I
3.11 3. 11 Give Given n the plant P = [1,10]. Design a controller with margins
s l;s/p)
LI
where k E [2 6] and p E 1 < 6dB and tracking specs
I ,i3 w)
11 12 13 15 1 I 1.1 I 1.1 I 1 I 0.7 I
In
I 0.9 I 0.8 I 0.6 I 0.2 I
w)
1
=
11 12 13 15 1 I 1.1 I 1.1 I 1 I 0.7 I
EXERCISE
w
LI
W
Repeat exercise 3.10 by designing two controllers, one with two poles more than zeros and one with one pole more than zeros (do not use damping factor lower than 0.5). Compare the controllers using a Bode plot. plo t. Which one of the controllers would you prefer to use and under what conditions? EXERCISE 3.12
EXERCISE 3.13
Given the plant P
=
( s _ i ) O ( ~ + l )
can you stabilize it using a
PI
controller, or a PI controller?
9
8
QUANTITATIVE FEEDBACK
NOTES
DESIGN
ND REFERENCES
The QFT single-loop feedback design technique presented here is based on the work o f Horowitz and Sidi (1972 (1972). ). The technique matches sensitivity as well as robustness specifications specifications for the exact exa ct amount o f uncertainty, and its criterion for a good design is the high frequency gain originated by Bode transmission). nsmission). Base Based d on the above 1945) ideal Bode characteristic of a loop tra attractive properties o f the QFT design technique, Yaniv et al (1990) proposed an adaptive algorithm to modify the controller to reduce plant uncertainty without reducing the closed loop performance. An algorithm to identify the reduced plant uncertainty was published by Gutman (1988). A partial extension of the QFT technique to a single-input multi-output specification was published by Breiner (1981). (1981). An extension to allow allow for time varying plant gain was presented by Wang at al (1990), whose technique is based on adding a quantity of unstructured uncertainty, the amount o f which is dictated by the circle criteri criterion. on. Theod Theodor or aand nd Shaked (1994) combine combined d QFT and H methods, to design design robust H controllers with almost no over-design. Plants which include uncertain highly under-damped complex poles are well suited to the the QFT techniqu technique. e. Som Somee insight insight into such such systems may be obtained from the articles by Kidron and Yaniv (1996) and Yaniv and Horowitz (1990a). There The re exist several CAD packages tto o compute bounds and help in loop shap ing.. Th ing Thee fir first st paper which presented an efficient efficient technique for loop shaping was published by Gera and Horowitz 198 1980, 0, and since then then many CAD C AD software packages appeared, amongst am ongst whi which ch are are in included cluded:: the QFT atlab ™ Tool box, the U.S. Air-force QFT package developed by Houpis and Sating (1997), and Gutman Gutma n (199 (1996). 6). The software developed by by Bailey et al (1992) includes automatic automat ic loop sshaping haping.. Chait (1997) presented an algorithm using convex optimization which optimizes only the zeros o f the control controller. ler. Gutier Gutierrez rez and Rabins (1993) developed a computer loop shaping algorithm for controllers for minimum-phase minimum- phase systems to sati satisfy sfy QFT type type bou bounds. nds. Efficient algorithms for calculating bounds were developed by Rodrigues et al 1995, and Yong dong and Jayasuriya (1994), who presented a very efficient algorithm for the calculation o f bounds for general interval plants. The QFT extension to NMP plants plants was discussed in several several papers. Sidi (1976) and Horowitz and Sidi (1978) presented an optimal robust synthe sis technique to design a feedback controller for an uncertain NMP plant to achieve a given closed loop performance. performance. Their Thei r synthesis technique provide providess the designer with insight into the tradeoffs between closed loop performance and bandwidth, and also defines n implicit criterion for determining whether or not a solution exists. exists. Sidi (1980) developed a criterion to estimat estimatee the max imum bandwidth of a sampled plant for give given n gai gain n and phase-margins. He assumed an open loop transmission of the ideal Bode characteristics form and
used asymptotic approximations. Horowitz and Liau (1984) extended this this tech-
Synthesis o LTI Control Controllers lers or MISO LTI Plants
9
nique to stable plants with several several RHP RH P ze zeros. ros. They showed how to achieve a large loop transmission in several frequency ranges, although there will always be some frequency ranges which are determined by the RHP RH P zeros, in which the the loop transmission must be less tha than n OdE This well known fact was proven by Francis and Zames Zame s 1984) and by Freudenberg and Looze Loo ze 1985), who showed that for NMP plants, a small sensitivity in one frequency range forces a large sensitivity in the adjoining range. For a rigorous extension o the QFT technique to sampled data systems see Ya Yani niv v and Chait 1993), whose technique includes algorithms to achieve closed loop specs on the sampled plant output, as well as on the continuous plant output. output. Many papers include sections on existence theorems, but the first was by Horowitz 1979) appendix 1 and Yani Yaniv v 1991), Yani Yaniv v and Sch Schwar wartz tz 1991), Jayasuriya and Zhao 1993,1994) who also included mixed uncertainty in their existence proofs.
Chapter 4 SYNTHESIS OF L T LTIPL NTS
CONTROLLERS CONTRO LLERS FO FOR R MIMO
The objective objective o f this chapter is to present quantitative design techniques for synthesizing a controller and a prefilter for a MIM MIMO O plant plan t either known or highly uncertain), in order to achiev achievee desired closed loop specifications. The of
is
basic toquential break the design down into seriesfeed stages. Each stage oidea f thi thiss se sequential process is aprocess simplified SIS SISO O or aMISO feedback back pr problem oblem which is supported by the QFT Matlab™ Toolb Toolbox. ox. A solution tto o the original problem is then simply a combination o f the solutions obtained at each stage. The types o f closed loop specifications considered here are given in the frequency domain and include: include: disturbance rejection, reference referen ce tracking, gain and phase margins, control effort and design for diagonal minimum-phase elements o f the tracking MTF. The plants considered can be stable, unstable, NMP, NM P, NMP and unstable, or given by measured data. data. The benefits o f NMP plants are restricted; these restrictions are discussed with reference both to SISO and MIMO MI MO syste systems. ms. A hidden assumption made in many practical systems is that a plant, which may be piecewise LTI, locally linearized around a set of operating points, LTV or nonlinear, can be represented by a set of LTI plants such that the solution to this set is appli applicable cable to th thee original plant. We shall thus talk abou aboutt sets o f uncertain plants. In order to concentrate on practical plants, the following assumptions are made throughout this chapter: • Any plant P and any controller, G or W G is strictly proper; • Any plant is o f full rank, that is, if is square then its inverse and if is non-square then then there exists a weighting MTF MTF,, W P W ) - l exists; 9
exists, such that 1
O. Yaniv, Quantitative Feedback Design of Linear and Nonlinear Control Systems © Springer Science+Business Media New York 1999
94
QUANTITAT QUAN TITATIVE IVE FEEDBACK DESIGN DES IGN
• Whenev Whenever er a S1SO S1SO controller controll er is designed for a plant set, {P} no RHP pole zero cancellation takes place between any member o f the set, {P} and the controller. 1
SYNTHESIS OF ONE DOF FEEDBACK SYSTEMS SYSTE MS
The feedback system o f interest to us is depicted schematically in Fig. 4.1. Mathematically it is described by the equations y
Pu+Pdd
u
-GHy.
d
-
G
U
Pd
-
H
Figure 4 1
P
-
-
Y
-
A single DOF MIMO feedback system
The problem under consideration is how to design the controller, G , such that for a given set o f plants {P} i) the closed loop is stable and its sensitivity to plant members in {P} remains within the given tolerances, and U) for a given set o f disturbances {d}, the plant output continues to be bounded by the given specifications. The set {d} can be all signals whose spectral densities are bounded by the spectral density o f a given signal, i.e., if d = [d 1 , d2 hen d 1 can be any signal whose spectral density is bounded by 1 / w 2 and d 2 any signal whose spectral density is bounded by 10/ 100 w2 . The set {d} can also be a finite set o f given signals for example the two signals [1/ s 1 / s and [ 1 / s 10 / s 10) V. Note that anyone o f the elements in d can be correlated to the plant P E {Pl. Disturbances introduced into into the plant s input P d = P), and at the plant s output P d = I ) are treated se separa parately. tely. A statement stat ement o f the problem for the general case now follows:
V
PROBLEM
4.1 Consider the system shown in Fig. 4.1 where P is an n x n LTI
plant pla nt belonging tto o a set {P}, {d, P d} is a set o f disturbances and disturbance
Synthesis o f LTf Controllers For MIMO LTf Plants
95
MT F s resp MTF respecti ectively vely,, e w) is a specification vector, and W h is afrequency such that the specification vector vect or iiss applicable for all W ::::; Who Design the controller, G, such thatfor all P E { P } • the syst system em is stable; an and d • for all pairs d, P d E {d P d}, th thee plant output y = [Yl ... Yn] by ydjw I ; edw ; k = 1 ... n, w::::; Who
is bounde bounded d
4.1)
4.1 The case where the specifications apply to the MTF from d to y is a sp speci ecial al ccase. ase. Here we use, use, for each { P d} the set o f n disturbances d = 1 s) (an impulse vector where the impulse is introduced into the ith input channel and zeros are applied to all the other input channels) applied to each o f the input channe channels. ls. Th Thee specifica specification tion vector for each channel, i w), will be th thee upper boun bound d on the TF s fr from om input chann channel el i to all outputs. REMARK
e w) may depend on the plant P and/or the couple d P d, which may in turn depend on the plant P E {P}. R E M A R K 4.2
R E M A R K 4.3
t high frequencies the benefits o f feedback are negligible.
Specifications at high frequencies will result in large bandwidth with very small closed loop perf performa ormance nce impr improvemen ovement. t. Choosing th thee lowes lowestt possi possible ble W h is thus recommended.
The design procedur proceduree is first developed for 2 X 2 plants and then for n X n plants. Conditions for a successful application o f the design process and some comments on NMP plants follow the development. 1.1
2 X 2 PLANTS AND DISTURBANCES
T THE
PLANT S INPU INPUTS TS
The feedback system o f interest to us is schematically depicted in Fig. 4.2. The question under consideration is how to design the diagonal controller
such that it simultaneously stabilizes a given set of plants {P} and decreases the plant output due to disturbance to a specifie specified d lleve evel. l. Put more formally our problem is the following: P R O B L E M 4.2 Consider the system shown in Fig. 4.2 where P is a 2 X 2 LTI plant belonging to a set {P} d a disturbance belonging to a given set {d}, e w) a specification vector, and W h a freque frequency ncy such that the specificat specifications ions are
applicable for all W
::::; Who
Design the controller, G suchthatforallP E {P}
QUANTITATIVE FEEDBACK DESIGN
9
d
Figure 4.2.
u
-
G
-
p
Y
-
A MIMO feedback system with disturbances distur bances at the input to the plant plan t
• the system is stable and • for all d E {d} the the pla plant nt output y
=
[Y1, Y f is bounded by 4.2)
Development o f the Design Process:
In Fig. 4.2
I (P - 1 Using the notation explicitly
1
I
Y PG)y
(4.3)
Pd
G)y
=
PG ) - 1Pd,
4.4)
d.
=
[1I ij] and G
diag g1, g2), equation 4.4) is 4.5)
MUltiplying both sides of the equation on the left by the matrix
[ 1
7I 1l
91
0 1
1
gives [
where
~
(4.6)
g
2
6
11'2111'12
11'22
= 11 22-
11'11
4.7)
g
Synthesis o f LTI Controllers For MIMO LTI Plants
Fr From om equatio equations ns 4.5,4.6), we solve for
Y
and Y2 respectively:
d 1 - 7r12Y2
Y
7ru
Y2
+9
7r2 I ( 7r
=
9
7 r ~
4.8)
+ 91 d1 d 1 + d2 + 92
4.9)
The design process is is based on equations 4.8,4.9) which aid us in transforming the problem into the design o f two sequential SIS SISO O systems, as follows follows:: from equation 4.8) and specification 4.2) 4.2),, 9 s) should be designed such that
dl-7r12Y21::; e1 w); 7rll + 9 v d = [d 1 , d2 f E {d}, P E {P}, w ::;
I
who
4.10)
A large enough 9 jw) I satisfie satisfiess equation 4.10) if an upper bound on IY2 jw) I exists and is known for purposes of practical computati computation. on. We then deliberat deliberately ely choose its maximum value, assuming that Y2 satisfies specification 4.2) 4.2).. Thus, 9 is designed to satisfy:
IId
+ l7rd e2 7ru + 9
11
jw)
I::;
e1 w);
v d = [d 1 , d2 ] T E {d}, P E {P}, w ::;
who
4.11)
A TF 9 which is a solution to the following single DOF SISO feedback problem also solves inequality 4.11) 4.11).. Consider the system shown in Fig. 4.3, where plant. Design 9 such that for all P E {P} PROBLEM 4 3
1/7rU
is a 5150
• the system is stable stable;; and
VE
• for all d = [d 1, d 2
{d}, th thee plant output Y is bounded by
IY1 jW)1 ::; e1 w); w::; Wh where Idl ::; Id 11+ l7rde2(w).
Note that the maximum of Idl, maxdl (ld11+ 17r12Ie2 W)), is not necessarily one o f the entries o f the members in the set d E {d}. From Fro m equatio equation n 4.9) 4.9),, and specification 4.2), 92 should be designed to satisfy the inequality
4.12)
98
QUANTITATIVE FEEDBA CK
ESIGN
d
-
gl
Figure 4. 4.3. 3.
-
-
The SISO feedback system
to
./1
1
n
be solved in the design
of
g
A T F g which is a solution tto o the following single DOF SISO SIS O feedback probl pr oblem em also solves inequality 4.12). 4.12). PROBLEM 4 4
plant Design
g
Consider the system shown in Fig 4.4, where such that for all P E {P}
is a SISO
1 / 1 f ~ 2
• the system syste m is stable; and an d • for all d = [d 1 d2
V E {d}, the the pla plant nt output Y2 is bounded by
d H
-
Figure 4.4 4.4..
1
g2
The SISO feedback system
Y
2
n 22
to
be solved in the design
of
g2
As will be shown in section 8., a necessary condition for the existence o f a solution to problem 4.4 is that the plant, 1 / 1 f ~ 2 should be minimum-phase or its RHP RH P zeros ar aree sufficiently sufficiently distant from the origin. origin. The RHP zeros o f the plant 1 / 1 f ~ 2 are the poles o f its inverse 1 f ~ 2 By equation 4.7) these poles include the zeros o f 1 f l l + l if 1f121f21 is not identically equal to zero. I f 1f121f21 = 0, the zeros o f 1 f l l l are poles o f the MTF of the system in Fig. 4.2 from d to y so that if they are in the RHP, the system will be unstable. g2 must not be in the RHP, otherwise the system Similarly the zeros of 1f22
will be unstable
s
well. Hence the conclusion:
Synthesis o f LTf Controllers For MIMO LTf Plants
99
gl should be designed suc such h thatJor all P E {P} gl 7r1l does not have RHP R HP zzer eros os,, or 1 +gl /7r11 shoul should d not have RH RHP P zzero eros, s, and gl/7r11 does not have any RH RHP P pole-zero cancellations that is the system in Fig. 4.3 is stable). f 7r217r12 is not identically equal to zero and other conditions to be developed in the sequel are satisfied, 1 g d r l l may have RHP zeros i f they CONCLUSION 4.1
are far enough from the origin to render problem 4.4 solvable.
Summary
of
the Design Procedure
A two-stage sequential procedure: procedure: At the the first first stage, stage, a single DOF SISO feedback problem is defined whose solution is gl and at the second stage, a single DOF SISO feedback problem is defined whose solution is g2. G = diag gl,g2) will be a solution to problem 4.2. A t the first stage we solve problem 4.3, and at the second stage problem 4.4. R E M A R K 4.4
By conclusion 4.1, gl, design designed ed at th thee first stage stage,, can be chosen g d r l l will have RHP zeros sufficiently distant from the origin,
such that 1 is 7r11) of 1 / that the solution problem may cause gl / in comparison to be unstable. for usingglsuch to a The reason a g1 is 4.3 to decrease its 1bandwidth stabilizing g1. But this is possible only i he bandwidth o f he loop transmission designed at the second stage is much smaller than the loop transmission band width o f he fir first st stage. This is because RHP R HP zero zeross of1 gl/7r11 appear as RHP - the plant o f the second zeros o f 1 secon d sstag tage, e, an and d so i they are far removed from the ori origin, gin, they contribute very llittle ittle phase lag at the the crosscross-over over fre frequen quency cy o f the loop transmission o f the 2nd 2n d sta stage. ge. How far from the origin such RHP zeros should be depends on the existence o f a solution to problem 4.4. The proposed design technique was developed for gl to be designed first first and
then used in defining the second single DOF S1S0 feedback problem in order to design g2. But this is not essential, since one can design g2 first and then gl. Problems 4.3 and 4.4 will then be changed in accordance with the following lemma:
Let A be an n x n matrix which represents row or column permutations, that is one element in each row is 1 and all the others are 0 so that det A) = ±1. Suppose G is a solution to problem 4.1 when P is replaced by P - 1 P d by P d and the specification e(w) by Ae(w); then A - l G A is a solution to problem 4.1. L E M M A 4.1
Proof: The lemma le mma conditions mean that for for all all P E {P} d E {d} (I
P
-lG)-1
P dd
A(1 Ay
where the notation
E
means
P E
- l G A ) - l p dd
Ae(w),
(4.13)
A y satisfies specification (4.1) when e(w) is
replaced by Ae(w).
Now by lemma 4.13, (section 2.1), A can be cancelled
100
QUANTITATIVE FEEDB CK DESIGN
out from both sides of the P E {P},d E {d} 1
+P
relation relatio n in equation equati on 4.13) which gives: for all
- l G A ) - l p dd
Y E e w),
which in tum implies that A - l G A solves problem 4.1.D Example: for 2 x 2 systems
that is, 91 and 92 change position. Hence in in order to design the second ro row w first, we simply apply the design process to problem 4.2, replacing {P} by the set { P -I} the set {d, P d} by {d, P d} and e w) by Ae w). A solution to the original problem will then be G = dia9 92, 91). Specifications and Stability Note that whenever we refer to to specification specification 4.2), 4.2), we are actually referring to inequality 4.2) 4.2).. Lemma Lem ma 4.2 below proves that if conditions on the set {P} and the controller, G , are such that closed loop stability is guaranteed then specification specificati on 4.2) is satisfied. Note that in general, the pol poles es o f a closed loop system are common to all the TF s from any any input to any out output put Bode 1945, Horowitz 1963). 1963). This being the case, stability is guaranteed if the last design stage is successfully executed. executed. The stability stability proofs provided here are unexceptional. LE MMA
4.2 Suppose G = dia9 91,92) stabilizes the system defined by
Fig. 4.2 for all P E { P } and inequalities 4.11,4.12) are true true,, then the closed loop specification 4.2) is satisfied.
Proof: We have to show that specificati specification on 4.2) is satisfied for k = 1,2. Inequality Inequali ty 4.12) guarantees that Y2 which is the closed loop response to disturbance d satisfies specification specificati on 4.2) for = 2, that is,
Now by inequality inequality 4.11), the the closed loop response to disturbance, d, satisfies specification 4.2) for = 1 if IY2 jw)1 S e2 w), which is true from the first part o f the proof. Thus
We shall now state several conditions which guarantee closed loop stability,
as defined by Vidyasagar 1985, p. 99) 99),, who offers offers a frequency domain d omain criterion
101
Synthesis o f LTI Controllers Controllers For MIMO LTI Plants
to ensure internal stability i f the state-space representation o f P and G is minimal (Vidyasagar 1985, p. 102).
4.1 We shall say that the system in Fig 4.5 is stable if i) det I P G ) is not identically equal to zero; and (ii) the the follo following wing MTF is stable DEFINITION
H =
G+ I + PG)-l
GP)-l + + GP)-l ] I - P I + GP)-lG - P I + GP)-l ] (I + G P ) - l G (I + G P ) - l - I + PG)-l P ] (I + P G ) - l G I + PG)-l I - G I + PG)-lp .
[ (I [ [
PG)-l
-P I
(I
4.14) 4.15) 4.16)
REMARK 4.5 Fo Forr physical sys systems tems l i m s oo P G = 0, hence we shall assume that det I + PG) is not identically equal to zero
REMARK 4.6 The MTF H comes from the relationship between inputs and outputs as depicted in Fig. 4.5, that is [
] = H [
].
d2 dl
-
+
ul
YI
G
.
Figure 4.5 4.5..
u2
+
P
-
,Y2
A feedback system
To show that equations (4.14,4.16) are identical, use the following identity (Kailath 1980, p. 656), which is true for the non-singular matrices A and C , of dimensions m x m and n x n respectively:
For A
A
+ BCD)-l
=
I, B
=
= A-I - A -lB DA -l B
P, D
=
G , and C
+ PG)-l (I + G P ) - l (I
=
+ C - 1)-lDA -1.
I we find that
+ GP)-lG, I + PG)-lp.
=
I - P I
(4.17)
=
I - G
(4.18)
By pre-multip pre-multiplying lying equation (4.17) on the right by P and equation (4.18) on
the right by G , we have:
(I
+ PG)-lp
P I
+ GP)-l,
(4.19)
(I
+ GP)-lG
G I
+ PG)-l.
(4.20)
102
QUANTITATIVE FEEDBACK DESIGN
The stability definition, definition 4.1, states that the four MTF s o f Fig. 4.5 from d l and d 2 to Ul and U2 are stable. I t can be shown that this is equivalent to stability o f the four MTF s from d l and d 2 to Yl and Y2. LEMMA 4.3
Consider the feedback system defined in Fig 4.5. Suppose that
G P - l is stable; then the system is stable. (i) G is stable, and (ii) P ( I Note that by equation (4.19), the lemma condition ii) can be replaced by the stability of ( I PG -lp.
Proof: By equations (4.17-4.20) H ~ f
[ H11 H2l
H12]
H22
= [
H 12 G G ( I - H12G)
1-
1
H12 ] G H 12
hence the stability conditions o f definition 4.1 4.1 are satisfied. 0 We are now now ready to state conditions which guarantee that the design process proces s leads to a stable solution solution.. Th Thee ffirs irstt lem lemma ma states that for mini minimummum-phase phase plants and stable controllers, stability is guaranteed if the SISO system designed at each stage is stable, i.e., 1 g d H 1 1 as well as 1 2 H ~ 2 do not have RHP zeros. Consider the system shown in Fig 4.2. Suppose that for all 1 = [Hij] is stable, (ii) 1 + g d H 1 1 and 1 + g2/H§2 do not P E {P} (i) no t have RHP zeros, and (iii) G is stable; then the system is stable. LEMMA 4.4
Proof: From equation (4.6) the T F from d to Y is:
(4.21)
which is a multiplication o f two stable MTF s. Thus by lemma 4.3 the system is stable. 0 I f a MIMO plant is derived from differential equations o f the form D y =
where D and N are matrices o f polynomials in s sk represents the kth derivative), and u and y a r e the plant inputs and outputs respectively, then p - l = Adj N)D/det N). It is thus expected that the elements o f P - l will have the same poles or o r at least the same RHP poles, as the RHP zeros o f det N). I f this is indeed the case, then the system is termed NMP (Cheng and Desoer Nu
1980). Note that each of H11, H12 H2 and H22 can have RHP poles. However, 1980). as will be seen in the following lemma, certain combinations o f these elements
must be stable or not possess RHP zeros.
Synthesis o LTI Controllers For MIMO LTI Plants
103
Consider the system shown in Fig 4.2. Suppose that or all P E {P } (i) 7r1d7r11 and 7r21/7r11 are stable, (ii) 1 91/7r11 and 1 d 7 r ~ do not have RHP zero zeros, s, and (iii) G is stable; then the system is stable. LEMMA 4.5
Proof: By i ) and ii)
is stable. stable. Simila Similarly rly 7r21/ 7r11 stable. Hence from equat equation ion (4.21), 9 d is stable. I + P G ) - l p is stable; thus by lemma 4.3 the system is stable.D The next lemma states conditions on minimum-phase plants such that sta bility is guaranteed if the controller designed at the second stage produces a stable closed loop, whereas, the system designed at the first stage need not be stable. Consider the system shown in Fig 4.2. Suppose that for all 1 is stable, (ii) 1 2 / 7 r ~ 2 does not have RHP zeros and 9 2 / 7 r ~ 2 does not have any RHP pole-zero cancellations, (iii) G is stable, and (iv) 7r127r21 is not identically equal to zero and its RHP zeros are not zeros o 7r11 91; then the system is stable. LEMMA 4.6
P E {P} (i)
Proof: By straightforward calculation
(I
PG)-lp =
p-l
G)-l
=
[ 7r22 7r11
-
-
92
7r 2
7r 12
]
7rll 91 , (4.22) 92) - 7r127r21
9d(7r22
r ~ 2 ) which, by (ii) and the fact that 9 2 / 7 r ~ 2 with it s 22 element being 1/ 92 must not have any RHP pole-zero cancellations, is stable. The denominator o f equation (4.22) is
7r11
9d(7r22
92) - 7r127r21
=
7r11
d
7 r ~ 2
92)
Thus a RHP zero o f the denominator o f equation (4.22) must be a RHP zero o f 7r11 91 since 7 r ~ 2 92) does doesn n t possess possess anyRHP zeros. This implies that aatt a RHP zero o f 7r11 91, the term 7r127r21 = 0, which by hypothesis isn t true. Hence the denominator o f equation (4.22) does not have RHP RH P zeros. By (i) and stable. ble. Use o f lemma 4.3 completes the proof. 0 (iii) all the other elements are sta The next lemma combines lemma 4.5 and lemma 4.6 to cover the situation where the plant may be NMP and the SISO system designed at the first stage
is unstable.
1 4
QUANTITATIVE FEEDBACK DESIGN
4.2. Suppose that for all P E {P} (i) 7r12/7rll, 7r21/7rU and 7r22/7rU are stable, (ii) 1 + 2 / 7 r ~ 2 does not have RHP zeros and 9 2 / 7 r ~ 2 does not have any RHP pole-zero cancellations, (iii) G is stable, and (iv) 7r127r21 is not identically equal to zero and its RHP zeros are not zeros o f 7ru + 91; then the system is stable. LEMMA
4.7 Consider the system shown in Fig
1 is stable, we can use the same argument as in lemma 4.6 Proof: When to show that the denominator o f equation (4.22) does not have RHP ze 1 is not stable the proof is as follows: As was ros. However Howe ver when shown in lemma 4.6, any RHP zero of the denominator o f equation (4.22), (7rU + 91)(7r22 + 92) - 7r127r21, must be a RHP zero o f (7ru + 91). We now show by contradiction that any RHP zero of 7rU + 9t) must be a RHP zero o f (7rU + 9t ) 7r22 + 92) Let us assume that it is not true, hence 7r22 + 92) must have the same RHP pole in order to cancel this particular RHP zero o f (7rU + 9t)· However since 7r22 + 92) has an unstable pole, 7r22 must possess the same unstable pole and hence from (i) this unstable pole also belongs to
9t) From iii) we know that 91 and 92 are both stable and hence 7rU is endowed with the same unstable pole as 7ru. This implies that 7rU + 91) must have the same unstable pole as 7r22 + 92) However this contradicts the assumption that the unstable pole o f 7r22 + 92) was cancelled by the RHP zero o f 7r11 + 91) since 7rU + 9 d must then possess the same RHP pole pol e and zero. Thus any RHP zero o f the denominator o f equation (4.22) is also a RHP zero o f (7rll + d 7r22 + 92) Based on iv) and the same argument as in lemma 4.6, we then conclude that the denominator of equation (4.22) does not have RHP zeros. The 22 element of equation (4.22) can be shown to be 1/(92 + r ~ 2 ) From (ii) and the fact that 9 2 7 r ~ 2 must not have any RHP pole-zero cancel lations, we conclude that the 22 element is stable. Now since the 22 element o f equation (4.22) is stable, this implies that the RHP poles o f (7rU + d are cancelled by the RHP poles o f (( 7ru + 91)( 7r22 + 92) - 7r127r12) (which are the RHP zeros o f [(7rll + 9t}(7r22 + 92) - 7r12 7r127r 7r12 12t1 t1). ). We ve already shown that the RHP poles o f 7r11 + 91) are the same as the RHP poles o f 7r11. Furthermore from (i) we notice that 7ru possesses all the RHP poles o f 7r12, 7r21 and 7f22 , and thus we conclude that if the RHP zeros of [ 7ru + 91)( 7r22 + 92) - 7r127r12]-1 cancel the RHP poles o f 7rll, they also cancel the RHP poles of 7r12, 7r21 and 7r22. Hence all o f the elements of equation (4.22) are stabl stable. e. Using llemma emma 4.3 completes the proof. D 7ru.
In the degenerate case, when 7r12 0 or 7r21 0, the system may not be stabilizable by an LTI diagonal controller. For example: Le Lett G = dia9 91 , 92) and
P = [
1
1
then
Synthesis o f LTf Controllers For MIMO LTf Plants
I
,1
[
PG -l
105
which is an unstable MTF. Thus by definition 4.1 the closed loop system is unstable.
4.1 Solve problem 4.2 for the follo following wing unce uncertain rtain plant distu disturrbance se sett and close closed d loop spec specific ificatio ations. ns. EX
MPLE
The uncertain plant {P} includes all the 2 x 2 plants
P =
S
k12
[kll
]
k22
k21
where the k j are uncorrelated and can take any values in the ranges kll k22 E [2,4],
E [1.0,1.8].
k12, k21
The disturbance set { d = [d 1 s), d2 s) ]T} includes all the elements such that
Id1 jw)1 Id 2 jw)1
< 11fjwl < Ilfjwl
and
Id 2 1=
0
and
Id
o.
=
4.23)
or
4.24)
The clos closed ed loop specific specifications ations ar are: e: For al alll P E {P} • the system is stable; and • for all d E {d} the plant output, y
= [Yl Y2V is bounded by
IYk jW)1 ::; ek w); Vw ::; 3 where
el
w) and
e2
k = 1,2,
w) are given for w = 1,2,3 in the following table: I w [rad sec]
1
-26
I -30
3
-32
Implementation of the design process A two-stage procedure): 1. First, the SISO SI SO proble problem m 4.3 iiss solved. Its plant, 1 / 1 f u , is calculated from 1
=
S
kllkn -
k12k21
The selected nominal case is which
[kn k21
kll
- k 12 kll
]
=
= k n = 2 and
1f12]
[1fll
1fn
1f21
k12
=
k21
= 1.8, for
1
0.38
s
s
QUANTITATIVE QUANTIT ATIVE FEEDBAC K DESIGN
106
The low low frequency bou bounds nds at W = 1,2,3 are calculated to satisfy the disturbance rejection inequality 4.11), 4.11), which for the disturbance set se t defined by inequalities 4.23,4.24) 4.23,4.24) is
<
Ill/ jw)1 + 1?T12Ie2 w) I ::; e1 w)j ?Tn + 91
<
1?T12 e2 W )
?Tn + 91
I::; e1 w)j
\i
\i
E {P}
E {P}
4.25)
The chosen stability margin for 1 91/?T11)-1 is 4dB which means that the following inequality should be satisfied at all frequencies: 11
91/?T11 I = 11 + 0.3891/81;::: 0.6
=
-4dB, 8 = jw.
4.26)
The finite number of plants on which calculations were performed are all uncorrelated maximum and minimum values o f k . The calculated bounds for inequalit inequalities ies 4.25,4.26) and the nominal loop L1 in Fig. 4.6, where the controller is
91 =
=
91/?T11 are shown
47.3 1 + 8/8) 1 8/49) 47.3 1 + 8/1.8) 1 + 8/20.5) 1 + 8/120 + 8 2/120 2 .
4.27)
Clearly L d j w ) satisfies the bounds for w = 1,2,3 at each frequency L1 jw) lies above its bound) and at all frequencies the margin bounds are satisfied at each frequency frequency it lie liess outside o f its margin bound). 2. The problem to be solved is problem 4.4, with plant 1 2
?T22
T
2 -
1 1f11
+91
and nominal case:
1 ? T ~
0.38 0.38 1 8/18.2) 8/18 .2) 1 0.88/6 0.88 /6 + 82/ 6 2) 1 + 1.18/119 - 8 1 + 8/11.5) 1 + 0.98/19 + 82/ 19 2) 1 + 1.18/115
82/119 2) 82/115 2) .
The low low frequency bounds at W = 1,2,3) are calculated to satisfy the disturbance rejection inequality 4.12 4.12), ), which for the the d disturbance isturbance set defined by inequalities 4.23,4.24) 4.23,4.24) is:
<
1 ? T 2 1 / ~ ? T 1 1 ?T22
9 d / 8 1::;
92
e2 w)j
\i
E
{P}, 8
jw
<
I
//
?T22
8
+ g2
I ;
e2 W);
\i
E {P}
8 =
4.28)
jw.
Synthesis o f LTI Controllers
For
MIMO L TI Plants
107
30 2 0 .... . 8 1
10 0 O 0
10 - 10
20 - 20
- 30
- 40 -270
-240
-210
Figure 4.6.
-180
-150
-120
-90
-60
-3 0
o
L1 jw) and its bounds for the nominal case
The stability margin for 1 2 / 7 f ~ 2 - 1 is 4dB which means that the following inequality should be satisfied in all frequencies: (4.29) The calculated bounds for inequalities (4.28,4.29) and the nominal loop L = 9 2 7 f ~ 2 are shown in Fig. 4.7, where the controller is:
92
=
1
21.7(1 8/13)(1 8/40) 8/3)(1 8/26.5)(1 8/88.5 82 /88.5 2 )
(4.30)
Clearly, L 2 jw) satisfies the bounds for w = 1 2 3 (at each frequency L jw) lies above its bound) and satisfies the margin bounds at all frequencies (at each frequency it lies outside o f its margin bound). Verification by simulation: Frequency domain simulations for a step disturbance from each one o f the plant inputs to the plant outputs, for all uncorrel uncorrelated ated maximum and minimum values o f the k j s o f the plant uncertainty, are shown
in Fig. 4 4.8. .8. The asterisk asterisks, s, *, correspond to the disturbance rejection specifications el w) and e2 w). In Fig. 4.7 the loop transmission L2 jW) touches
lOS
QUANTITATIVE FEEDBACK DESIGN
30 2
~
~
~
__
______________
____
10
o -10
-20
-30 -4 0 -270
-240
-210
-180
-150
-90
-120
deg
-6 0
-3 0
0
L2 (jw) and its bounds for the nominal case
Figure 4.7.
the bounds at w = 1,2,3 , so that no over-design should b e expected at these frequencies. This is shown in the simulation o f Fig. 4.S, where the asterisks touch the maximum plant output for output Y2 due to a disturbance d 2 . The disturb ance rejection specifica disturbance specifications tions ar aree satisfied satisfied for all all disturbances. A time domain simulation for the same input and uncertainty is shown in Fig. 4.9.
Solve problem 4.2 for the following uncertain plant, disturbance set, and closed loop specifications. EX
M P L E 4.2
The uncertain plant
{P} is: p =
S
kll
k21
k12] k22
where the k i j are uncorrelated and can take any values in the ranges kll E [2,8]; The disturbance set
k22 E [2,3];
k12, k21 E [0.5,1.5].
{ d = [dd s , d2 s )V} include includess all the elements such that
Id jw)1
<
jjwl
and
1
=
0,
or
4.31)
2
Iljjwl
Id 2 jw)1
and
1
4.32)
= 0.
Synthesis o f LTI
2
0
~
ontroll ontrollers ers Fo Forr MIMO LTI Plants
~
2
0
1 9
~
~
40 D
0
40
5
0
L
·
10°
~
·
~ Y
~
~
~
ue to d
~
8
~
0
10 1
~
=
~
~
~
~
~
10°
10 1
2 0 ~
~
D
0
log oo) Figure 4.8. Frequency Frequenc y domain simulation to validate closed close d loop specifications: the specs e 1 w) and e2 w) respectively
are the
The closed loop loop specifications specifications are: are: For all P E {P} • the system is stable; and • for all d E {d} the plant output, y is bounded by
where
e1
w) and
e2
w) are given for
I W [Tad/sec I I
el e2
[dB)
I
=
1 2 3 in the following table:
1
I
2
I
3
-26
I
-30
I
-32
Implementation of the design process A two-s two-stage tage procedure): The designer
has to decide which of the controller elements to design first, 91 or 92. Both options will be tried and compared. Note that the the plant for 91 designed first
QUANTITATIVE FEEDBACK DESIGN
110
Y due to d2
due to d
Y
0.25
0.25
0.2 .
0.2
0.15
0.15
01
01
0.05
0.05
0
0
-0.05
-0.05
0
0.2
0.4
0.6
0.8
1
0
0.2
Y2 due to d
0.8
1
0.8
1
0.25
0.2 .
0.2
0.15
0.15
01
01
0.05
0.05
0
0
0
0.6
Y2 due to d2
0.25
-0.05
0.4
0.2
0.4
0.6
0.8
1
-0.05
0
0.2
0.4
sec
Figure
49
Time domain simulations for example
0.6
sec
41
IS
7r11
with an uncertainty ratio of 7.9/0.8 = 9.875, while the plant for first is (use lem lemma ma 4.1 4.1 where A = [0,1; 1,0])
1
kllk22 - kl2k2d/kll
92
designed
[0.9,2.9]
7r22
with an uncertainty ratio of 2.9/0.9 = 3.22, which is 33% o f the uncertainty of 91 desi designed gned first. first. The design process is carried out exactly as in example 4.1. Design results for g designed first followed by gl: Using lemma lem ma 4.1 4.1 with wit h A = [0,1; 1,0], the problem is modified to a problem in which 9 2 is designed first.
The solution will then be G = diag g2,91). Using the notation p l = [7rij] and G = dia9 gl,92), the first loop to design is 9 2 / 7 r 2 2 ' The design and its
Synthesis o f LTI Controllers For MIMO LTI Plants
111
bounds are shown in Fig. 4.10, where the nominal case is
and the controller 92 is 29(1 1
92 =
3/2)(1
3/8.5)(1 3/18.5)(1
3/36) 3/80
32/ 8 0 2) .
The nominal second loop 9l/nf1 and its bounds are shown in Fig. 4.11, where 40
30 2
~
~
~
10
o
10
-10
-20 -30 -40 -270
-240
-210
-180
-150
deg
l jw) and its bounds,
Figure 4.10.
-9 0
-120
92
-60
designed first'
and the controller 91 is 29(1 gl =
1
8/2)(1
3/15.5)(1 3/45) 8/31)(1 8/111 82/1112)
-30
0
The solution to the original problem is G = diag g2, g d
QUANTITATIVE QUANTIT ATIVE FEEDBAC K DESIGN
112
40
30
]J
'0
B
o -10
-20
10 20 50
-30
70
-40 -270
100 150 -240
-210
Figure 4.11.
-180
-150
-90
-120
deg
L2 jW) and its bounds,
92
-60
-30
0
designed first
Design results for 9 designed first followed by 92: The nominal loops and bounds are given in Fig. 4.12 and Fig. Fig. 4.13. 4.13. The controller G = dia9 91 , 92) is 9
92
=
37.5(1 + 8/10.5)(1 + 8/60) 1 + 8/1.7)(1 + 8/26.5)(1 + 1.18/150 + 82 /150 2 ) 37.5(1 + 8/10)(1 + 8/47.5) 1
+ 8/1.4)(1 + 8/30)(1 + 8/69 + 82 /6 /69 92 )
Comparison: A Bode plot o f the two designs is given in Fig. 4.14. I t shows that at high frequencies, 9 and 92 of the first design are lower by 6dB and higher by 3dB, respectively, as compared to 9 and 92 of the second design (compare ~ to and ~ to respectively). The preferred prefer red solution depends on many factors, such as sensor noise at the outputs Yl and Y and the nonlinearities of the two actuators. Verification by simulation: Frequency domain simulations simulati ons
of
the output re
sponses due to a step disturbance introduced into each one of the plant inputs, for all uncorrelated maximum and minimum values of the k i s o f the plant
Synthesis o f LT Control Controllers lers Fo Forr MI MIMO MO LT Plants
113
40 30
B
20
10 Ol
0
'0
10
20
-10 -20 -30 -48
- 70 -240 -210
Figure 4.12.
-150
-90
-120
deg
L1 (jw) and its bounds,
91
-60
-30
0
designed first
uncertainty, are shown in Figs. 4.15-4.16. The asteri asterisks sks corres correspond pond to the disturbance rejection specifications e1 w) and e2 w) respectively, which are satisfied for all d disturbances. isturbances. Time d domain omain simulations for step disturbances introduced into the plant inputs are shown in Figs. 4.17 -4.18 . Up to plot ting accuracy, both designs meet the specs and produce similar time-domain responses. 1.2
2 X 2 PLANTS AND DISTURBANCES
T THE
PLANT S OUTPUTS
The feedback system under consideration is schematically described in Fig. 4.19. The problem o f intere interest st iiss h how ow to design the controller controller,, G = diag g1 , g2), such that it simultaneously stabilizes a given set o f plants {P} and decr decrease easess the plant output due to disturbances, disturban ces, to a specified lev level. el. The problem may be formally stated s follows: PR O
LEM
4.5 Consid Consider er th thee system shown in Fig. 4.19, where P is a 2 x 2
Tl
plant pla nt which belongs to a set {P}, d a disturbance belonging to a given set { d}, e( w) a specification vector, and Wh is a frequency such that the specification
QUANTITATIVE
4
FEEDB
CK DESIGN
4
3 2
o 10
20
30
40 270
240
180
210
Figure 4.13. 2
5
r
-
~
-
-
-
-
-
-
~
-
-
-
120
eg
jw) and its bounds,
L -
150
~
-
-
~
-
-
~
~
~
~
90
60
30
91 designed first ~
-
-
-
-
-
-
-
-
~
-
-
-
-
20
Figure 4.14 4.14..
A Bode plot comparison
o
the controllers
o
both designs for example 4.2
-
.
Synthesis o f LTf Controllers For M MO LTf Plants Y
~
due
t
Y1 due
d1
-20
-20
-25
-30
3 0
-40
-35
-50
-40
10°
Y2 due to d 1
101
10°
101
10°
t
l S
d2
Y2 due to d 2
101
-20 -30 ~
4 0
-50
-60
10°
log CD) Figure 4. 5.
log
CD)
101
Bode plots o f closed loop g2 designed first)
vector is applicable for all W
::; who
P E {P}
Design the controller G, such that for all
• the syste system m is stable; and • for all d E { } the plant output y
=
[Yl Y2jT is bounded by
4.33)
Development o f the Design Design Process: In Fig. 4.19 Y
I
PG)y
I d
PG)-ld,
p-l
P-1d.
G )y
4.34)
QUANTITATIVE QUANTI TATIVE FEEDBACK DESIGN
6
10°
due to d,
Y
Y2 due to d 2
101
20
20
*
30
~
10°
10 1
25
*
4 0 50
60
10°
log ro)
log ro) Figure 4.16 4.16..
Bode plots of closed loop 91 designed first)
Using the notation p l = [7fij] and G = diag gl, g2), equation 4.34) may be explicitly written as:
[
+ gl
7f~~ ] [
7f
MUltiplying both sides
of
] =
[ ~ ~
~ ~ ]
[
] .
4.35)
the equation on the left by the matrix
gives 4.36) where
Synthesis o f LTI Controllers For MIMO LTI Plants
o
0.25 Y
0.5
0.75
o
1
0.25
0.5
0.75
117
1
Y2 due to d 2
due to d 1
0.2 0.1
o o
0.25
0.5
0.75
sec Figure 4. J 7.
0.25
o
1
0.5
0.75
1
sec
Time domain simulations 92 design designed ed first) for example 4.2
4.37) From Fro m equations 4.35,4.36) 4.35,4.36) for Yl and Y2 respectively 7 rll d
Yl
7r 2d2 -
gl
7 rll
Y2
=
7 r ~
d
r ~
7 r ~ 2
g2
7 r 1 2 Y2
d
4.38) 4.39)
The design process is is based on equations 4.38,4.39), which wh ich aid us iin n transform ing the problem into the design o f two sequential SISO systems, as follows: from equation 4.38) and specificati specification on 4.33) for k = 1 g l should be designed such that IYll
=
l7r
d1
7r 2d2 -
7 rll
gl
7 r 1 2 Y2 1 S
el w)j
d =
1 d2 ]
E
{d}
P E {P}
w:S
Who
4.40)
118
QUANTITATIVE FEEDB
o
0.25
0.5
CK DESIGN
o
0.75
0.25
0.75
Y2 due to d 2
due to d 1
Y
0.5
01
01
o
0.25
0.5
0.75
o
1
sec
4.18.. Figure 4.18
0.25
0.5
0.75
1
se
Time domain simulations g1 designed first) first) fo forr example 4.2
d
-
-
Figure 4.19 4.19..
U
G
P
-
y
A MIMO feedback system with disturba disturbances nces at the plan plantt s outpu outputt
A large enough Ig1 jw) satisfies equation (4.40) 1
i
an upper bound on IY2 jw)
1
exists and is known for purposes o practical computation. We then deliberat deliberately ely choose its maximum value, assuming that Y satisfi satisfies es specification ((4.33). 4.33). Thus
Synthesis o f LTl Controllers For MIMO LTl Plants
119
91 is designed to satisfy
d l + 7r12d21 + 17r12ie2{W) 17r ------' -'--'-------'--------'-'--- '-'---'--'----'--'- < 7r
d=
+
-
9
[dl, d 2 f E {d}
P E {P}
el
)
W
j
w:S;
(4.41)
Who
A TF 91 which is a solution to the following single DOF 5 I 5 0 feedback problem is also a solution to inequality (4.41). Conside r the Consider the system shown sho wn in Fig 4.20 where 1/7r11 is a SISO plant. Design 91 such that for all P E {P} P R O B LE M 4 6
• the system syste m is stable; and • for all d = [dl, d 2 jT E {d}, the plant output y is bounded by
IYl jW)1 s; e1{w)j w:S; where Idl s; l7r l 1 d 1
(4.42)
Wh
7r 2l e 2{W).
7r 2d21
d
-
-
Figure 4 20
gl
-
-
1
Yl
7tll
The SISO feedback system to be solved in the design o f g1
Note thatthe thatth e maximum o f 17r11dl +7r12d21 + 17r121e2 w) over d is not necessarily am em b ero fd = [d l d2 jT in {d} From equation (4.39) and specification (4.33), 92 should be designed to satisfy the inequality:
A TF 92 which is a solution to the following singl singlee DOF 5 I 5 0 feedbac feedback k problem is a solution to equation (4.43). Consider the system shown show n in Fig 4.21, where plant. Design 92 such that for all P E {P} PROBLEM 4 7
1
7 r ~ 2
is a SISO
• the system syste m is stable; and
120
QUANTITATIVE QUANTITA TIVE FEEDBACK DESIGN
• for all d E {d}, the plant output Y2 is bounded b y
d
-
g2
4.21.. Figure 4.21
,
1 2 1t22
-
...Y 2
The SISO feedback system to be solved in the design of 92
As will be shown in section 8., a necessary condition for the existence o f a solution to problem 4.7 is that the plant, 1 / 1 f ~ 2 should be minimum-phase or that its RHP zeros be sufficient sufficiently ly distant from the origin. origin. The RHP zeros of the plant are the poles of its inverse, 1 f ~ 2 . By equation 4.37) these poles include includ e the zeros of 1f11 gl if 1f121f21 is not identically equal to zero. I f 1f121f21 = 0, the zeros of 1 gl /1f11 are poles of the MTF of the system in Fig. 4.19 from d to y so that if they are in the RHP the system will be unstable; similarly the zeros o f 1f22 g2 must not be in the RHP otherwise the system will be unstable as well well.. Hence the conclusion: conclusion: CONCLUSION
4.2
gl should be designed such that for all
P E {P},
gl
1f11
does not have RHP zeros. That is 1 gI/1fl1 does not have RHP zeros and gI/ 1f11 does not have any RHP pole-zero cancellations, equivalently the system in Fig. 4.20 is stable. f 1f211f12 is not identically equal to zero and other conditions to be developed in the sequel are satisfied, 1 gI/ 1f11 may have RHP zeros i they are far enough from the origin to render problem 4.7 solvable.
Summary o the Design Procedure
A two-stage sequential procedure: At the first first stage, stage, a single DOF DO F SISO SIS O feedback problem is defined whose solution is gl and at the second stage, a single DOF SISO feedback problem is defined whose solution is g2. G = Proble m 4.6 must be solved at diag gl, g2) will be a solution to problem 4.5. Problem the first stage and problem 4.7 at the second stage.
same as remark 4.4) B y conclusion 4.2, gl, designed a t the first stage, can be chosen such that 1 gI/ 1f11 will have RHP zeros sufficiently REM
RK
4.7
Synthesis o f LTl Controllers For MIMO LTl Plants
121
distant from the origi origin, n, that is, the solution gl o f problem 4.3 may cause 1/(1 gl/,rru) to be unstable. The reason for using such a gl is to decrease its bandwidth in comparison to a stabilizing gl. But this is possible only i the bandwidth o f the loop transmission designed at the second stage is much smaller small er than than the loop transmission transmission bandwidth bandw idth o f he first stage. stage. This is because bec ause g1/7r11 appear as RHP zeros o f 1 7 r ~ 2 - the plant o f he second hey are far removed from the origin origin,, they contribute very liule phase lag a t the cross-over frequency frequenc y o f he loop transmission o f he 2nd stage. How far from the origin such RHP zeros should be, depends on the existence o f a solution to problem 4.7. RHP zeros of1 stage, and so i
In the proposed design technique, gl is designed first and is then used to define the second single DOF SISO feedback problem in order to design g2. But this is not essential: one can design g2 first and then gl. Problems 4.6 and 4.7 will then be changed in accordance with lemma 4.1 and the instructions following the lemma.
Specifications and Stability Note that whenever we refer to specification (4.33), we are actually referring to inequality (4.33). (4.33). The conditions on the set set {P} and the controller contr oller G which guarantee guarant ee closed loop stability are the the same as for distu disturbance rbance at a t the plant s input, because stability depends only on the plant and the controller and not on the exact point at which the input is introduced. The following lemma provides the conditions under which specification (4.33) is satisfied.
4.8 Suppose G = diag(gl g2) stabilizes the system defined by Fig. 4.19 for all P E {P} and inequalities (4.41,4.43) are true true,, then the n the close d loop specification specification (4.33) is satisfied. L E MMA
Proof: We have to show that specification (4.33) is satisfied for k = 1 2. Inequality (4.43) guarantees that Y2, which is the closed loop response to disturbance d satisfies specification (4.33) for k = 2, that is
Now by inequality (4.41), the closed loop response to disturbance d satisfies specification (4.33) for k = 1, i f Y2(jW) ; e2 w), which is true from the first part o f the proof. Therefore
which means that specification (4.33) is also satisfied for k
=
I.D
Solve problem 4.5 for the following uncertain plant, distur bance set, and closed loop specifications. E X A MP L E 4.3
122
QUANTITATIVE FEEDBACK DESIGN
The uncertain plant {P} is:
P
[kl1
=
k12]
k21
S
k22
where the k i/s are uncorrelated and can take any values in the ranges kl1,k22 E [2,4];
k12,k21 E [1.0,1.8].
[dd s), d2 (s) V} includes all elements such that
The disturbance set { d =
Idl jW)1
S
Id2 jw)1
S
+51 15/ jw + 5 1 15/(jw
and Id 21= 0,
4.44)
or
and Id1 = O.
4.45)
The clo closed sed loop specifications are are:: For aall ll P E {P}
• the system syste m is stable; and • for all d E {d} the plant output, y is bounded by
IYk jW)1 :S edw); Vw :S 3, k = 1,2, where el (w) and e2 (w) are given for w = 1,2, 3 in the following table
I I
W
[Tad sec] I
el , e2
1
I
I
2
I
3
[dB] I - 2 6 I - 2 0 I - 1 4 I
Implementation o f the design process A two-stage procedure): 1. The SISO problem 4.6 is solved and its plant, 1 / 7f11, is calculated by
P -1 =
S
kl1k22 - k12k21
[k k21
- k12 ] kl1
= [7fll 7f12].
=
1.8,
7f21
7f22
The chosen nominal case is
kll
=
k22
=
k12
2
k21
=
for which the nominal plant for the first stage is
1
(kllk22 - k12k2d/k22
0.38
s
s
th e dis = 1,2,3 are calculated to satisfy the turbance rejection inequality 4.41). 4.41). These inequalities inequalities,, for the disturbance disturbanc e set defined by inequalities inequalitie s 4.44,4.45) are
The low frequency bounds at w
Yl1
<
157f11/ S
w) < el w) + 5 1++9117f12ie2 17fll I -
VP E {P} s
=
jw.
IYll
<
157f12/ S
+ 5 1 + 17f12Ie2 w) < el(w);
17f11
+ 911
-
VP E {Pl.
4.46)
Synthesis o f LTI Contro Controllers llers For For MIMO LTI Plants
123
gl/,Tr11 -l is arbitrarily chosen to be 4dB The stability margin for 1 which means that the following inequality should be satisfied at all frequen CIes: 11
gl/7I 11 I =
11
0.6 = - 4 d B 8 = jw.
0.38gl/81
4.47)
The calculated bounds bounds for for inequalities 4.46,4.47) and the nominal lo loop op L jw) = l /71 11 are shown in Fig. 4.22. The controller is 30
Bl
20
10 0 OJ 0
-1 0
-2 0
-3 0
-40 -2 7 0
-2 4 0
Figure 4.22.
l = (1
-2 1 0
-180
-1 5 0
-1 2 0
deg
-90
-60
-30
0
L1 jw) and its bounds for the nominal case of example 4.3
68 1 8/12) 1 8/3) 1 8/20.5) 1
8/47.5) 8/145 82 /145 2 )
4.48)
Clearly L jw) satisfies the bounds for w = 1,2,3 at each frequency L jw) lies above its bound) and the margin bounds for all frequ frequenci encies es at each frequency L jw) lies outside o f its margin bound). 2. The problem to be solved is problem 4.7, with plant:
1 71 2
22
1 -
71
22
-
El..2.2£2.L • 7rll
91
QUANTITATI QUANT ITATIVE VE FEEDBACK D ESIGN
24
The low frequency bounds at w = 1,2,3 are calculated to satisfy the disturbance distu rbance rejection inequality 4.43), which for the disturban disturbance ce set defined by inequalities 4.44,4.45) are the following two inequalities
<
IY2
157f~~/ 8 7f22
<
IY2
5 7 f ~
/ 8 2
7f22
5) 92
5) 92
I
e2 w);
;
E {P}, 8 = jw.
\l
)
::; e2 w ; \ l
E
{P}, 8 = jw.
4.49)
The stability margin for 1 2 / 7 f ~ 2 - 1 is arbitrarily chosen to be 4dB, which means m eans that the following inequality should be satisfied at all frequencies 4.50) Th e calculated bounds for inequalities 4.49,4.50) and the nominal loop The L2 = 9 2 7 f ~ 2 are shown in Fig. Fig. 4.23. The controller contro ller is is::
30
8
o ......
> . . a . ~ o
.j.
-10
10 20 50 70 100 150
-20 -30
.
-40 -270
-240
Figure 4.23.
-210
-180
-150
.
-120
deg
, ...
,
-90
-60
-30
L2 jw) and its bounds for the nominal case of example 4.3
o
92 =
1
56.6 1 8/40 8/4.8) 1 8/178 82/178 2)
4.51)
Synthesis o f LTI Control Controllers lers For MIM MIMO O LTI Plants
125
Clearly, L2 jW) satisfies the bounds for w = 1 2 3 (at each frequency L jw) lies above its bound) and satisfies the margin bounds for all frequencies (at each frequency L jw) lies outside o f its margin bound). Verification by simulation: Frequency domain simulations for all uncorrelated maximum and minimum values o f the k i j s are shown in Fig. 4.24. The disturbance model o f 5 / (3+5) is added to each channel chann el o f the plant s output, and the output is plotted as a function of frequency in response to the disturbances. The asterisks, , correspond to the disturbance rejection specifications l w) and e2 w), which are clearly satisfi satisfied ed for all all disturbances. Tim Timee doma domain in Y1
due to d 1
0 -10
-20
~
3 0
-40 -50 -60
10°
Y2
due to d 1
10°
10 1
0
Y2 due to d 2
10 1
0 -10
-20
~
-20
3 0
~
3 0
-40
-40
-50
-50
-60
10°
10g 0))
10
1
-60
10°
10 1
10g 0))
Figure 4.24 4.24..
Frequency domain simulations to validate close closed d loop specifications: the the specs e w) and e2 w) respectively
simulations for the same disturbances, d 1 = [5/ (3+5), O]T and d 2 5)]T, at the plant output are shown in Fig. 4.25. 1.3
n
X
n PLANTS AND DISTURBANCES
* are
= [0,5/ (3+
T THE
PLANTS INPUTS
The feedback system o f interest is depicted schematically in Fig. 4.26. The
126
QUANTITATIVE FEEDB
CK DESIGN
Y1 due to d 2
due to d 1
Y
1
1
D
-0.5
-1
0
01
-1
0.2 0.3 0.4 0.5 0.6
0
01
Y2 due to d 1
0.2 0.3 0.4 0.5 0.6 Y2 due to d2
1 0.5 0 -0.5 -1
0
01
-1
0.2 0.3 0.4 0.5 0.6 sec
Figure 4.25.
0
01
0.2 0.3 0.4 0.5 0.6 sec
Time domain simulation fo forr example 4.3
d
G
Figure 4.26.
-
+
u
p
y
A MIM MIMO O feedback system with disturbances at the plant s inputs
problem under consideration is how to design the controll controller er G = diag gl, ... , gn such that it stabilizes a given set o f plants {P} and decreases the plant s out puts, given disturbance inputs, to a specified lleve evel. l. Forma Formally lly the probl problem em may
be stated as follows:
Synthesis o f LTI Contr Controllers ollers Fo Forr MIM O LTI Plants
127
Consider the system shown in Fig. 4.26 where P is an n x n LTI plant belonging to a set {P}, {d} a set o f disturbance signals, e w) a specifi specification cation vec vecto tor, r, an and d W h a frequency such that the specifications are Who Desig applicable for all W Design n the contro controller ller,, G, such that for all P E { P } P R O B L E M 4.8
• the system is stable; and • for all d E {d} the elements o f y
I
ydjw
=
[Y1 ... , YnV ar aree bounded bou nded by
ek w);
= 1, ... .. . , n,
w
4.52)
Who
esign Process:
evelopment o the In Fig. 4.26
P-1
Y
4.53)
G) y
4.54)
1 = [7rlj] d = [di, ... d ; V and G = dia9(91, ... Using the notation ... ,9n) ,9 n) the superscript 1 is used to denote the original plant inverse for the first sequentiall design stage), equation 4.54) is explicitly sequentia
7r
1
91
1 [ 7r21
..
1
7r 12
1 7r22
92
...
,
1
4.55)
1
7rn1
7rn2
The Gaussi Gaussian an elimination algorithm Gant Gantmac macher her 1960 1960)) iiss now applie applied d to equatio n 4.55). The firs equation firstt step is to zero the elements in the first column o f the left matrix, except exce pt for the 11th element. This is done b by y multi multiplying plying bot both h sides on the left by the matrix 0
1
G2
7r}1+ 9
=
7r}1+ 9
0
1
def
1
0
which gives
[ ,1
9
7r12 1 2 7r22
92
1
7r13 1 2 7r23
0
7 r ~ 1 9
7 r ;1 9
1 0
:1
, 4.56) 2
2
7r n 2
7r n3
128
QUANTITATIVE FEEDBACK DESIGN
where 1
1 7r i l 7r 1j 1 7r11 91
_
07 1. Z)
d1
1 d1 7ri1 1 . 7ri1 91'
_
Z
2
-
Z-
=
i
.. n
n
j=I
2, ... , n
Note also that
[
1 7r1l 2 7r21
1
2 7r22
1 7r13 2 7r23
2 7rn2
7r n 3
7r 12
2 7rn1
[
9
1
7r I n
2 7r2n
2
2
7r n n
1 7r12 2 7r22
1
and
1 7r13 2 7r23
92
2 7rn2
1
7r I n
2 7r2n
7r n 23
7r n 2n
9n
1
The second Gaussian elimination step involves multiplication o f both sides o f
equation (4.56) on the left by the matrix
1 0
G3 =
0 0
0 1
_7r 2
7r 2 2 + 9 2
_7r 2
0 0
0
1
0
0
7 r ~ 2
0
-7rJ2
0
7 r ~ 2
1 def
0
7r 2 2 + 92
-7 r 2
0
0 0
r 22 + 9 2
1
0
0
0 1 92 92
0 0
0 0
1
0
0 0
0
1
92
which gives 1
7 rll
0 0
91
1
1
7r 12
2 7r22
0
92
2 7r23 3 r33
7rln
2 7r2n 3 7r3n
93
3
3
0
7r n 3
7r n n
9n
where
3 rij
7r Z 2. _
2 2 7ri27r2j 2 7r22 92
; i
=
3 ... , n
[: I j = l
n
1 1 2 d2
3 dn
d3 Z
d2
_
Z
2 d2 7ri2 2 . 2 7r22 92
= 3, ... , n
i
Synthesis o f LT LTff Controllers Controllers
129
MIMO LTf Plants
For
Note also that
G3
G 2P - 1
G 3 G 2 P-l
G
1 7r11 2 7r21 3 r31
1 7r12 2 7r22 3 7r32
1 7r13 2 7r23 3 7r33
3 7rn1
3 7rn2
3 7r n 3
1 7ru
91
1 7r12 2 7r22
1
7rln
2 7r2n 3 7r 7r3n 3n
,
and
3
7r n n
0 0
0
1 7r13 2 7r23 3 7r33
0
0
3 7r n 3
92
1 7r1n 2 7r2n 3 7r3n
93
r ~ n
n
Applying the Gaussian elimination algorithm recursively for k - 1 iterations yields 1 7r11
0
91
1 7r12 2 7r22
2 7r2k
92
0
k 7rkk
0
k 7rnk
where
7rt,
k
1 7r1n 2 7r2n
Y1 Y2
1 1 d d22
k 7rkn
Yk
dkk
n
Yn
dnk
k
7r n n
(4.57)
f and G k are defined recursively by 1
,
k. _
HZ)
k k i 7r 7r k k kj
7rkk
d kZ
_
k
k dk 7rik k k
7r k k
,2= k
.
9k
i
=
k
1,
n .] =1
n
458)
(4.59)
1, ... , n.
and (denoting m = k - 1).
Gk 1
0
0
1
0
0
0
1
0
0
1
0
0
_7r mm 7r;:;::m gm
0
0
-7rtm gm
1
1
0
(4.60)
0
1
7 r ~ l 2 J
7r;:;::m gm
7 r ~ m
0
1
gm
130
QUANTITATIVE FEEDBACK DESIGN
Note also that 1 7r11 2 7r21
G k ···
G 2P - 1
7rk1 k
k
7rn1 G 2 P 1
Gk· 1
0 0
0
7rkk
0
0
7rnk
The
91
k
7rn2
1 7r1n 2 7r2n
7r k3 k
,
7rkn k
k
and
k
r nn
7r n 3
1
2 7r2k
1 7r1n 2 7r2n
k
k
7r lk
92
9k
k
4.61)
7rkn
k
9n
r nn
following follo wing identity is also important import ant to be used later): later):
LEMMA
4 9 91
2 - 7r 21
Gk
7rk2 k
1 7rl3 2 7r23
G)
1 7r12 2 7r22
7rll
1 7r12 2 7r22
G 2G
=
k
- 7r k1
k
-7 rn 1
0
0 0
92
0 0
k
9k
- 7r k2
k
-7 rn 2
7 r ~ 3
0
k
=
2 ... n .
9n
Proof: Trivial from equation equati on 4.61) and the identity
following lemma and remark show that it is always possible to apply the Gaussian elimination algorithm to equation 4.55). 4.55). The
Necessary and sufficient conditions for applying the Gaussian elimination algorithm to equation 4.55) are i) 7 r ~ k 9k is not identically equal to zero for k = 1 ... , n o r ii) the diagonal minors o f G k . . . G 2 P - 1 G) formed from rows and columns 1 ... , k denote denoted d by Mk), are no nott the zero TF for = 1 ... , n, that is Mk =I O LEMMA
Proof:
4 10
The
first condit con dition ion is trivial. To show ii) use Gantmacher 1960 1960 equation
14), p. 26):
k=n
= T
Mk
k
9k).
7 r ~ k
l
Synthesis o f LTI Controller Controllerss For MIM MIMO O LTI Plants
Thus i) implies ii), and k 7rkk
gk
=
131
ii) is true it implie impliess i) bec becaus ausee
if
lkfk
i-
l k fk - 1
0; k
=
1 ... , n; lkfo
=
1 0
, g n are designed such that 1 4.8 Since g l k 7 r ~ k is not iden tically zero zero most likely ev even en stable stable), ), the suff sufficient icient conditio conditions ns Jor application o j the Gaussian elimination algorithm are satisfied. REMARK
From Fro m equation 4.57) for k Y1
=
1 ... , n
d11
-
,\ ,n
rfl
d22
Y2
g1
2 L..Ju=3 7r2uYu
dm
g2
,\ ,n
=
1;
k
=
2;
m
L..Ju=m+1 7r m u Yu
m -
dn
r;;{m
_----'-'-n_,
Yn
k
,\ ,n
-
7 r ~ 2
Ym
1
L..Ju=2 7rluYu
k=m;
gm
4.62)
k=n
gn
r ~ n
The design process is based o on n equations 4.62), which transforms the proce procedure dure into n sequential designs o f SISO systems as follows: From equation 4.62) for k = 1 and specific specification ation 4.52) for Y1 g1 should be designed to satisfy IYll
=
dl
I
-
1 L..Ju-2 7rluYu 1 7r11 g1 ,\ ,n
A large enough
Ig1
I
: :;
e
{d},, P E {P} {P},, w ::; w); lid E {d}
Who
4.63)
jw) satis satisfies fies equation 4.63), provided that a bound on 1
exists. But such an upper bound iiss not known. We shall there fore deliberately choose the maximum value, assuming that Y2 , Y n satisfy specification 4.52). Thus g l is designed to satisfy
Y2
, Yn
Idil
~ = 2 1 7 r f l l e u w )
7r l l
gl
:S
el(w);
lid E {d}, P E {P}, w:S
Who
4.64)
A T F g l whic which h iiss a solutio solution n tto o the follow following ing si single ngle DOF SISO feedback problem satisfies equation 4.64). Consider the system shown in Fig. 4.27, where LTl plant. Design g l suc such h thatJor all P E {P}
PROBLEM 4 9
• the syste system m is stable; and
1 / 7r l 1
is a S1S0
• Jor all d
E
{d} the the plant output Y l is bounded by
132
QUANTITATIVE FEEDBACK DESIGN
d
1
l
-
) 1
1
7tll
Figure 4.2 4.27. 7.
where Idl ::; Idll
The SISO feedba feedback ck structure in the design of 9
: ~ = 2 1 7 T l u l e u { w
and d = [dl, ... , ~ f ·
The same line o f reasoning is repeated in the design o f 92, ... , 9 n that is, 9m originates from the use o f equation 4.62) for k = m and is designed such that : ~ = m + 1
I d ~ 1
1 7 T ~ u l e u { w
9ml
1 7 T ~ m
d E {d}
P
E {P}
4.65)
W ::; Who
Note that when m = n , the term L : ~ = m + l l 7 T ~ u l e u ( w ) drops out because o f equation 4.62), the design equation for 9n will therefore be
A T F which is a solution to the following single DOF SISO feedback problem satisfies inequality 4.65). 4.65).
Consider the system shown in Fig. 4.28, where SISO LTI plant. Design 9m such that for all P E { P} P R O B L E M 4 10
l
7 T ~ m
is a
• the system is stable; and • fo forr al alll d E {d}, the plant output Ym is bounded by IYm jw)1 ::; em{w ; w::;
where Idl ::; m = n, and
I d ~ 1 ~
: ~ = m + l l 7 T ~ u l e u { w )
Wh
when m
is defined by equation 4.59).
<
n, Idl ::;
I d ~ 1
when
Note that at the mth stage, where 9m is to be designed, 91, ... , 9m-1 are known and the upper bound assumption is made only on Ym+1, ... , Yn. Thus at the last nth) stage no such assumption is needed. As is is shown in sectio section n 8., a necessar nece ssary y
condit ion for the existence o f a solution to problem 4.10 is that its condition its plant, 1 T ~ m should be minimum-phase or that its RHP zeros are sufficiently distant from
Synthesis o f LTl Controllers For
d gm
-
, +
-
1 m
M fM O LTl
Plants
33
.}m
nmm
Figure 4.28.
The SISO feedback system used in the design of gm
the origin. The RHP zeros o f the plant are the poles o f its inverse, 7r;;{m which by equation 4.58) include the zeros of 7rZ k + gk) where k = m - 1. Hence the following conclusion may be drawn: CONCLUSION
4.3 gk should should,, in gener general, al, be des design igned ed such that 7rZk
+ gk does
not have RHP zeros for all P E {Pl. That is 1 + gk/7rZk does not have RHP zeros and 9k/7rZk does not have any RHP pole-zero cancellations. f it has RHP zeros they should be far enough from the origin to make problem 4.10 solvable for all the subsequent design stages. Summary o f the Design Procedure Procedure::
This is an n-stage procedure. At stage k, a single DOF SISO feedback prob problem lem is defined whose solution is gk. G = diag(gl ... , gn) will then be a solution to problem 4.8. 4.8. The single DOF SISO feedback problem to be solv solved ed at the the kth stage is problem 4.10, for m = k. and
Specifications Stability: The first first lem lemma ma pr proves oves that specification 4.52) is satisf satisfied. ied. Condit Conditions ions on the set {P} and the controller, G guaranteeing closed loop stability are then given. 4.11 Suppose that G = diag gl, ... , gn) stabilizes the system defined by Fig. 4.26 for all P E { } and that inequality 4.65) is true for m = 1 ... , n, then the closed loop specification 4.52) is satisfied. L E MMA
Proof: By induction from th thee nth stage to the 1st stage stage:: Fro From m inequal inequality ity 4.65), when m = n , and equation 4.62) with k = n, specification 4.52) for Yn is satisfied.. Now assume that specification 4.52) is satisfied for Yn, ... , Ym+1, satisfied then by equation 4.62) for k = m and inequality 4.65), the specificati specification on for Ym is satisfied. D
We shall now now present a condition guarante guaranteeing eing closed loop stabi stability. lity. It is valid for minimum minimum-phas -phasee plants and a stable controllers. The NMP case is
QUANTITATIVE QUANTITAT IVE FEEDBACK DESIGN
34
treated in section 8.. 8.. The test for closed loop stability is is due to Vidyasagar (1985) (see also definition 4.1). Consider the system shown in Fig. 4.26. Suppose that for all P E {P}, (i) p 1 is stable, (ii) 1 k 1 f ~ k does not have RHP zeros and LEMMA 4.12
does not have any RHP pole-zero cancellati cancellations, ons, an and d (iii) G is stable; then the system is stable.
9 k
1 f ~ k
Proof: Using equation (4.58) and equation (4.59) it can be show shown n by induction that ft and ~ are stable for all i j and k. Thus in equation (4.57) for k = n, y is stable because: i) the left matrix is upper diagonal with stable elements, hence its inverse is stable, and ii) the right matrix is stable. A special spec ial case arises when the disturbance d is an anyo yone ne of the the colum columns ns of the identity MTF I , for which by equation (4.53) y = (I hence (I 14
P G -1p,
PG -1 P is stable. Use of lemma 4.3 completes the proof. 0
n X n PLANTS AND DISTURBANCES
T
PLANTS OUTPUTS
THE
The feedback system under consideration is schematically described in Fig. 4.29. The problem of interest is how to design the controller G =
d f
u
Figure 4.29.
G
y
-
P
-
A MIMO feedback system with distu disturbance rbance
at
the plants outputs
diag g1 , ... , gn) such that it stabilizes a given set of plants {P}, and decreases the plants plan ts outputs for given disturbance inputs, inputs, to a specified specified level. level. Formally Formal ly the problem may be stated as PROBLEM
4.11 Consider the system shown in Fig. 4.29, where P is an n x n
to aWset {P} {d} a set o f disturbance signals, e w) LTI plant which a specif specificatio ication n vec vbelongs ecto tor, r, a and nd h a frequency such that the specifications are
applicable for all W
Design the controller, G, such that for all P E { P }
Who
• the system is stable; and
Synthesis o f LT Controllers
• for all d E {d} the elements o f y = [Y1
,
or
YnV
MIMO LT Plants
135
are bounded by
Development o f the Design Process:
In Fig. 4.29
(I
Y
P-1
PG)-ld,
(4.66)
1 d.
G)y
(4.67)
Equation (4.67) is the same as equation (4.54), where the disturbance d being replaced by P - 1 d. Thus the same design technique is applicable appli cable w with ith the distur bance d replaced by p 1 d. At design stage k the right side o f equation (4.57) for disturbances at the plants outputs, outputs, may be written by by equation equati on (4.61) as as:: 1 7f11 2 7f21
=
k
G 2
1d
=
k 7fk1 k
7fn1
15
1 7f 2 2 7f22
1 7f 3 2 7f 23
k 7fk2
k 7f k3
fkn
k
dk
k
k 7fnk
k 7fn n
dn
7fn2
1
7f1n
2
7f2n
d1 d2
DESIGN DESI GN IMPROVEMENTS BY ITER TION
When W < W h the low frequency bounds, are calculated from inequal ity (4.65), which is of the form: ~ = m
I d ~ 1 1
m
7fm m
1
1 7 f ~ u l e u 9m
I
I n
I d ~ 1
7f n n
<
w )
9n
-
I
;
em
)
w
£
or m
<
n
en(w), for m = n .
(4.68)
The reason for using this inequality is that u for u = m 1, ... , n is not known and we deliberately choose the worst case, on the assumption that u satisfies the closed loop specifications, that is, IYu jw)1 ; eu w). Thus it is expected that the bounds calcula calculated ted with inequality (4.68) will form a larger domain than is actually needed. The result will be a sol solution ution whose bandwidth band width may be (i (in n general will be) larger than required. e shall then say that the solution has an over-design. Another source of over-design arises from the fact that a solution for any specification for which some o f the eu (w)'s are less than the original e u w) , (at some frequencies), frequencies), can produce a smaller small er bandwi bandwidth dth solution. Why
this results results in over-design can be explained as follows: follows: Le Lett s assume ass ume that
9m
a solution to inequality (4.68). If we now allow for a decrease in all of the eu (w)'s on the left hand side of inequality (4.68), while maintaining the same
is
136 em
QUANTITATIVE FEEDBACK DESIGN
w) on the right hand side, then a smaller gm may be used and we can still
achieve the conditions of the inequal inequality, ity, thus resulting in a narrower ban bandwidt dwidth h solution. However if we allo allow w em w) on the right hand side o f inequality (4.68) to decrease, and don't allow for a decrease in any o f the eu(w) s on the left hand side, we must then expect an increase in the amplitude o f gm and hence a wider bandwidth bandwidt h solution. solution. Hence we arr arrive ive at a tradeoff situation where if we decrease the eu (w) s on the left hand side and em w) on the right hand side, we run into conflicting demands on gm and thus on the bandwidth which can result in over-design over-design.. Hence we propose the following following iterative process, which overcomes the above limitations and iiss based on the assumption tha thatt a controller contr oller which solves the problem is already available (the controller designed at the first iteration stage): 1. Design G .
2. Simulate in the frequency domain to find the true performance of the system
related to the specifications. 3. Decide which o f the g/s to improve, say gk. 4. Calculate the bounds on g b assuming that gk is to be designed at the
last stage, when gl, ... , gk-1, gk+1, ... , gn are known from stage 1 without violating the specifications. 5. Redesign gk.
6. Go back to stage 2 where the know known n controller control ler parameters are those o f
the new controller. The inequalities with which to calculate the bounds at stage 4 can be cast into the following bilinear form form::
la1
i
gk 3i gk 6k
<
-
e(w) 1
,
i = 1 ... ,n,
where ai(jw), 3i jW) and 6djw) are derived as follows: Let
Gk
diag(0, ... ,0,9k,0, .. .... ,0), ,0) ,
Ck
1 + P G - G k ),
Pk e
Plk, P2k, ... , Pmk f
(plant transfer functions) and
[0 ... ,0,1,0, ... , 0 ] ) , 1 only i n position k.
(4.69)
Then,, using Kailat Then Kailath h 1980, 1980, pp. 655-656, 655- 656,
(I
PG)-l
Synthesis o f LTI Controllers Controllers Fo Forr MIMO LTI Plants
137
(4.70) Hence, for example, for disturbance vec vector, tor, d , introduc introduced ed at the plants outputs outputs,, i and /3i, which appear in equation (4.69) form the following vectors:
al, Note that the scalar
16
anf k
= Ad, /31,
,/3nf=Bd.
jw) does not depend on the output channel i.
SHORTCUTS IN LOW LOW FREQUENCY BOUND CALCULATIONS
As explained in section 1.5, the algorithm for low frequency bound calcu
lations suffers suffers from ove over-design. r-design. The following approxima approximation tion reduces this over-design and decreases the computational computati onal effort involv involved. ed. Th Thee approxi mation matio n used for disturbances at the plant plantss inputs or outputs is based on the assumption that the controller gain, at low frequencies, is large enough such that the following approximation is applicable
(I
PG -lpd
(I
G-1d,
PG -ld
and
G-1p-1d.
Thus for disturbance disturbancess at the plants inputs inputs,, the bounds on k are calculated by (4.71) and for disturbances at th thee plants outputs th thee bounds on k are calculated by
. 19k Jw)1
I Lek jPkjdj I jw) ;
Vd E {d} P E {P} w
Who
(4.72)
These formulae considerab considerably ly reduce the computational effort for low frequenc frequency y bound calculations, because the bounds on k jw) are calculated for a single case for which the right sides o f equation equationss (4.71,4 (4.71,4.72) .72) are the max maxima ima over all {P} and {d}. 2
SYNTHESIS O F TWO DOF FEEDBACK SYSTEMS The feedback system o f interest to us
is
depicted schematically in Fig. 4.30
and described mathematically by the equations y
=
u
138
QUANTITATIVE QUANTITA TIVE FEEDBACK DESIGN
u
=
G Fr - Hy).
(4.73)
PGH)-lpGF.
(4.74)
Hence the MTF from r to y is
T = (I
r
-
F
e - ~
-
G
u
H
Figure 4.30.
P
-
-
Two DOF MIMO feedback structure
Note that H denotes the sensor MTF and/or low-pass and/or notch filte filter. r. For Fo r simplicity and without loss of generality, we assume that H = I . The problem under consideration is how to design a controller, G , and a prefilter, F which simultaneously stabilize a given set o f plants {P} and satisfy closed loop sensitivity specifications. Formally the problem may be stated as: PROBLEM
4.12 Consider the system shown sho wn in Fig. 4.30, where P is an n x n
LTI plan plantt which belongs to a g given iven set {P},
B = [bij w)]
and A =
[aij w)] are specification n x n matrices, and h is afrequen afreq uency cy suc such h that the specificat specifications ions are applicable for all W ; ho Denote the MTF from r to y by T = [tij]. Design Des ign a contr controller oller,, G and a prefilter, F to satisfy the following specifications
for all
P E {P}
• stability: the system is stable; and • clo closed sed loop specific specifications: ations:
4.75)
The closed loop specifications stipulate that the Bode plot o f each element o f the system s MTF, tij jW), must lie between the two specified Bode plots, and w) bij bij w). Hence when w)aij is closer to aij the W).t i s sensitivity to plant uncertainty is smaller
REMARK
4.9 The specification matrices A and B may depend on the plant
P E {Pl.
Synthesis o f LTl Controllers For MIMO LTl Plan Plants ts R E M A R K 4.10
139
t high frequencies the benefits o f feedback are negligible.
High frequency specifications will result in large bandwidth with very little closed clo sed loop perfo performance rmance improve improvement. ment. It is thus recommended recomm ended that tthe he lowest possible Wh be chosen.
The design process is first developed for 2 x 2 and then for x plants. It is similar to the procedure for disturbance rejection developed in section 1 1 for 2 x 2 plants and in section 1 3 for n x n plants plants.. Closed loop specification specificationss o f the model matching type type instead o f equation 4.75)) are treated later, later, and and some hints for shortcuts in low frequency bound calculations are then given.
21
2 X 2 PLANTS
The system under consideration is as depicted in Fig. 4.3 4.31. 1. The problem in
r
F
-
e
,
-
Figure 4.31.
-
G
u
-
Y,
P
Two DOF 2 x 2 systems
question is how to design the controller, G = diag g1 , g2), and prefilter, F such that the sensitivity o f the MTF from command input r to output y satisfies given specifications. Forma Formally lly the probl problem em may be stated as follows: PROBLEM
4.13 Consider the system shown in Fig. 4.31, where P is a 2 x 2
LTI pla plant nt which belongs to a giv given en set {P}
B = [bij W)]
and A
=
[aij w)] are specification 2 x 2 matrices, and Wh is af afreq requen uency cy such that the specifica specifications tions are applicable for all W ::; ho Denote the MTF from r to y by T = [tij]. Design Des ign a contro controller, ller, G and aprefilterF to satisfy thefollowingforall P E { P } • stability: the system is stabl stable; e; and • clo closed sed loop specifica specification: tion:
4.76)
Development o f the Design Process:
In Fig. 4.31
T
I
PG)-1PGF,
4.77)
I
PGF
PG)T
p-l
G)T
4.78)
GF.
14
QUANTITATIVE FEEDBACK DESIGN DESIG N
Using the notation p l explicitly [ 71 11 7T2
gl
7T 2 71 22
g2
=
[7TiJl
] [tll t2
and G
=
diag gl,g2), equation
t12] = [91
0
t22
0 ] g2
[111
2
h2]
22
4.78) is
4.79)
MUltiplying both sides on the left by the matrix
gives
us:
4.80) where
From equations 4.79,4.80) 4.79,4.80) tll
gIi l l - 7T 2t2 71 11
gl
gIi12 - 7T 2t22 gl Tll
4.82)
g2hl - 7 T ~ d l l 7 T ~
g2
g2h2 - 7 T ~ d I 7 T ~
g2
4.83)
The design process is based on on equations 4.82,4.83) which transform transfo rm the the problem into the design o two sequential MISO systems, as follows: From equation 4.82) and specifica specification tion 4.76), 4.76), gl should be designed such that
Synthesis o f LTI Controllers Controllers Fo Forr MIMO LTI Plants
4
For a large enough Igl jw) I there exist 111 and h 2 which satisfy equation (4.84), if upper bounds on It21 jW)1 and It22 jW)1 exist, and are known. But these are not known, we shall then deliberately choose these upper bounds, assuming that t21, t22 meet specification (4.76). Thus gl, 111 and 112 are designed to satisfy all W) ~
Ilgdlll ±
I
b
w); VP E {P}
W
Wh,
a12 w) ~
IIglh 21±
I
b 2 W); VP E {P}
W
Who
l7fdb 2 7fll + gl l7fdb 22 7fll + gl
4.85)
TF s g1. 111 and h 2 which are a solution to the following two DOF MISO feedback problem also solve inequality (4.85).
l/7fll
4 14 Consider the system shown in Fig. 4.32, where SISO plant. Design gl, 111 and h 2 such that Jar all P E { } PROBLEM
is a
• the system is stable; and • in Fig.
4 32a,for r(t)
= o t) and Id 11
IYl jw)1
all
and in Fig.
4
32b, Jar r(t) a12
bll ; w
= o t) and
1Y1 jw)1
l7fdb 2 Wh;
Id 21
17f121b22
b 2 ; W
Who
4.86)
From equation (4.83), and specification (4.76), g2 should be designed to satisfy the inequalities:
(4.87) TF s g2, h i and 122, which are a solution to the following two DOF MISO feedback problem, also solve inequality (4.87). 4 15 Consider the system shown in Fig. 4.33, where SISO plant. Design g2, 121 and 122 such that Jar all P E {P} PROBLEM
• the system is stable; and
1
7 f ~ 2
is a
• in Fig.
4
33a, Jar r(t) = o t) and dI = a21
IY2 jW)1
7 f ~ d l l
b2 ; w
Wh;
142
QUANTITATIVE QUANTITATI VE FEEDBACK DESIGN DESIG N
d1
+
r
III
1
gl
Yl
7t l l
a)
+
r
1 b)
Figure 4.32.
The SISO feedback system to be solved in the design
and in Fig 4.33b,for r t) = 6 t) and d§ =
of
g l 111 and 12
7 r ~ d 1 2
As is shown in section 8., a necessary condition for the existence o f a solution to problem 4.15 is that its plant, 1 / 7 r ~ 2 should be minimum-phase or that its RHP zeros are sufficientl sufficiently y distant from the orig origin. in. The RHP zeros o f the plant are the poles o f its inverse, 7 r ~ 2 which by equation equati on 4.81) include the zeros o f 7r11 + 9 d if 7r127r21 is not identically equal to zero zero.. Moreover, if 7r127r21 = 0, the zeros o f 7r11 + 91 are poles o f the MTF o f the system in Fig. 4.31 from r to y so that the system is unstable. Hence the following conclusion, conc lusion, which wh ich is is the same as conclusion 4.1, may be drawn: should be designed such that,for all P E {P} 7r11 + 91 does not have RHP zeros, o r 1 + 9 1 7r11 should not have RHP zeros and 9 1 7r11 CONCLUSION
4.4
91
does not have any RHP pole-z pole-zero ero cancellations. The system syste m in in Fig. Fig. 4.32 will then be stable. I f 7r217r12 is not identically equal to zero and other conditions conditions
to be developed in the sequel are satisfied, 1 + 91 7r11 may have RHP zeros i they are far enough from the origin to render problem 4.15 solvable.
Synthesis o f LTl Contro Controllers llers For MIMO LTl Plants
143
1
r
2 11:22
a)
1
r
Y2
2 11:22
b)
4.33. 3. Figure 4.3
The SISO feedback system
to
be solved
to
design 92, 121 and i
Summary o the Design De sign Procedure
A two-s two-stage tage sequential seque ntial procedure: At the fi first rst stage a two two DOF MISO feedback feedback problem is defined whose solution is 91 III and h2. At the second stage a two DOF MISO feedback problem is defined whose solution is 92,121 and 122 G = dia9 91,92) and F [ ij] solve problem 4.l3. Problem 4.14 solved at the first stage and=problem 4.15 at the second stage.
is
to be
4.11 By conclusion 4.4, 91, desi designed gned at the first sta stage, ge, can be chosen such that 1 91/ 7rll will have RHP zeros sufficiently distant from the origin, that is the solution 91 o f problem 4.14 may cause 1/ 1 9I/7r1l) to be unstable. The reason fo forr using such a 91 is to decrease its bandwidth in comparison with a stabilizing 91. But this is possible only i f he bandwidth o f the loop transmission designed at the second stage is much smaller than the loop transmission bandwidth o f the fir first st sstage. tage. This is because RHP zeros o f 1 91/7rll appear as RHP zeros o f the plant o f the second stage, nd so if REMARK
they are f r remov removed ed from the o origin, rigin, they contribute vvery ery little phase pha se lag at the the cross-over crossover frequency o f the loop transmission o f the 2nd 2n d stage. stage. How f r from
the origin such RHP zeros should be depends on the existence o f a solution to problem 4.15 4.15..
144
QUANTITATIVE QUANTITAT IVE FEEDBACK DESIGN
In the proposed design technique, g1 h 2 and 121 are designed firs firstt and are then used to define the two DOF MISO feedback problem involved in designing 92, 121 and 122. This is not essential; one can design g2 i2l and 122 first and then go on to find g1 ill and h 2 . Problems 4.14 and 4.15 will then change in accordance with the next lemma.
Let D be an n x n matrix which repre represents sents row or column per per-mutations, that is one element in each row is 1 and all the others are 0 so that det D) = ± 1. Suppose that G and F are a solution to problem 4.13 when P is replaced by D P D 1 and the specification A and B by D A D 1 and D B D 1 respectively; then D -1 G D and D -1 F D are a solution to problem 4.13. LEMMA
4 13
The following lemma is needed in order to prove lemma 4
l3:
Given two n x n matrices, A = [aij] and B = bij] with real elements, and a matrix D which represents row or column permutations, that is one element in each row is 1 and all the others are 0 so that det D) = LEMMA
4 14
± 1. Suppose that A
::; B th thee matrix inequality means elem element ent by element elem ent inequality), then D A ::; DB, A D ::; B D and D A D - 1 ::; D B D - 1.
Proof: Any j element o f D A and of D B is the same element for example, the k l element) o f A and o f B . Hence A::; Band D A ::; D B include the same n 2 inequalities. inequal ities. The proof for A D and B D is the ssame. ame. The proo prooff for D A D - 1 and D B D - 1 is a trivial result of the above argument and o f the fact that D 1 has the same properties as D , that is D 1 D T and det D) = d e t D - . 0 Proof o f lemma I
D I
4 l3:
The MTF from r to y is
DPD-1G)-lDPD-1GF
=
P D 1G D ) - 1 P D - 1G D D - 1F D D 1
D T D 1,
where T is the MTF o f the system in problem 4 l3 for which D 1 G D and D -1 F D replace G and F respectively, and D T D -1 satisfies specification 4.76) for which A and B are replaced by D A D 1 and D B D 1. Thus by lemma 4.14, T satisf satisfies ies specification 4. 4.76 76). ).0 0 Example: for 2 x 2 systems =}
D-1 [ g1
=}
D-1 [
ill 121
Tha t is, g1 and g2 change position, ill and 122 change position, That position, and 121 and h 2 change cha nge position. Henc Hencee in order to design g2 2 and 122 first, simply apply
Synthesis o f LTI Control Controllers lers Fo Forr MIMO LTI Plants
145
the design process to problem 4.13, replacing { P } by the set { D P D 1 } and A and B in specification specific ation 4.76) by D D 1 and D B D 1 respectively. The resulting controller contr oller and prefilter will will be G and F respectively and the controller and prefilter o f the original problem will be D 1 G D and D 1 F D Specifications and Stability Stability
The first first lemma b below elow proves that specification 4.76) is satisfied. The conditions on the set {P} and the controller, G guaranteeing closed loop stability are the same as for the disturbance rejection problems see section 1.1). 1.1). This is because the same SISO TFs 1/nt1 and I / n ~ 2 are stabilized at each design stage. Suppose that G = diag g1,g2) stabilizes the system as depicted in Fig. 4.31 for all P E { P } and inequalities 4.85,4.87) are true, then the closed loop specification 4.76) is satisfied. L E M M A 4.15
Proof: W e have to show that specification 4.76) is satisfied for = 1,2. Inequality 4.87) guarantees that t2 and t22 satisfy specification 4.76) for k = 2 Now by inequali inequality ty 4.85), tl1 and t 2 satisfy specification 4.76) for k = 1 if t2 and t22 satisfy specificati specification on 4.76) for k = 2, which is true from the first part o f the proof. 0 EXAMPLE
4.4 Solve problem 4.13 for the following uncertain plant and
close d loop specif specification ications. s.
The uncertain plant {P}
P where the
ij
is:
=
+
s s1 1
[ kk2 l1
k12] k22
are un correlated correl ated and can have any values in the range rangess k l1
, k22
E [2,4], k12,
k2
E [-1.8 1.8].
The closed loop specifications are: are: For al alll P E {P} • the system
is
stable; and
• the MTF T from r
to
y of Fig. 4.31
is
bounded by 4.88)
where
ij
w) and bi j w) are given in the following table:
146
QUANTITATIVE QUANTITAT IVE FEEDBAC K DESIGN
Iw
I bll
[r [rad ad// sec] sec]
I all
b
a22
I b12, b21 I a12, a21
0.5
l 1
0.9
0.02
0
1.0
l 1
0.8
0.05
0
2.0
1.0
0.7
0.10
0
4.0
0.7
0.4
0.16
0
Carryi ng out the design proces Carrying process: s: The closed loop specifications call for the smallest possible off-diagonal response a12 = n = 0); we shall then choose the non-diagonal elements o f the prefilter as, h i = 0 and h2 = 0 note that this may not be the best choice). first. st. The low freque frequency ncy bound boundss at 1. The MISO problem 4.14 is solved fir w = 0.5,1,2,4 are calculated to satisfy inequality 4.85) which for specifi cation catio n 4.88) is a l l w)
<
±
119dlll
l1fdb21I ::; bll w); \lP
1fll
<
1
12 b22
1fll
91
I
91
;
b12 W); \lP
E
E
{Pl.
{Pl.
4.89)
1 91/1fll)-1 is 4dB which means that the following inequality should be satisfied at all frequencies:
The chosen stability margin for
91/1f11
I
:
0.6
=
-4dB
8 =
jw.
4.90)
Bounds on L l jw) = 91/ 1fll, such that there exists exis ts a filter 111 which solves inequality 4.89) aand nd satisfi satisfies es the margin inequality 4.90), are shown iin n Fig. 4.34, where the nominal case is
and the controller is 64 1 91 =
1
8/1.3) 1
8/12) 1
8/61.5)
8/258
82 /258 2 .
The filter 111, which, along with 91, satisfies inequali inequality ty 4.89), is 1
111 = 1
8/2.25·
2. The MISO problem problem 4.15 is now solve solved. d. The low frequency boun bounds ds are calculated to satisfy ine inequality quality 4.87), that is, is, the bounds on 92 are
Synthesis o f LT LTll Controllers Controllers For MIMO
Tl Plants
147
40 30
m
o
'C
10 20
30 40 50 270
240
210
Figure 4.34.
150
180
L
120
deg
90
60
30
o
jw) and its bounds for the nominal case
calculated such that
<
2 7r22
7 r ~ d l
I7r222
92
92122 92
I -< b w)·, I l/ij jw)1 >
bij w)
-ltij jW)
+2 aij W)
-/ij jw)1
_ bij w) - aij W) _ 2
-
.. ( )
a Z)
W .
The approximation used in section 1.6 can be applied to equation (4.112) in order ord er tto o decre decrease ase the computational effo effort. rt. The Th e inequalities for low frequency bound calculations can b e shown to be be:: Fo Forr all P E {P} and W s; Wh IG-1p-1FI
bij w) - aij w) 2
IG-1p-1FI
bij W),
if
aij =
Letting X = P-1F eij = b ij - aij)/2 i f aij the bounds on gi jw) are calculated to satisfy:
==
·f
1
-L
a Z)
0
I
O.
0 and eij
= bij
if
aij
=
0;
Xij jW)
I gi jW)
I s;
.
eij w); J = 1 ... , n, \iP E {P}.
160
QUANTITATIVE QUANTI TATIVE FEEDBACK DESIGN
SYNTHESIS FOR MARGINS AT THE PLANTS
3
OUTPUTS The synthesis techniques for feedback design o f MIMO systems described
in the previous sections ensures: ensures: robustness, tracking and disturbance rejection specifications over a range o f frequencies from d.c to Who Amplification o f noise and other external inputs inputs at the plants inputs and/or outputs may occ occur ur at frequencies W > W h and should be eliminated. Increasing the rang rangee o f frequencies beyond Wh is an unacceptable solution because i) the bandwidth o f the solution may increase, and/or ii) amplification of external inputs will appear at higher frequencies. The synthesis techniques previously described where n-stage sequential routines, where, at each stage, bounds were calculated and a controller (and also prefilters for for tracking problems) was designed to satis satisfy fy the bounds. Since a bound is the border o f a region in the complex domain, any design which meets the specifications over a more extensive region region than originally stipulated or required, will still satisfy the original spe specificatio cifications. ns. The probl problem em unde underr consideration is how to extend any bound such that the designed controller will still guarantee that the amplification o f external signals at the plants inputs and/or outputs will will be eliminat eliminated ed aatt aall ll frequenc frequencies. ies. In SISO systems this amplification problem s solved by the use o f sufficient gain and phase margins; for a qualitative discussion based on second order systems see D Azzo and Houpis (1988, chapter 9). 9). The concept o f phase and gain margins for SISO systems s extended to MIMO systems because o f its attractive features. Margin ofaS1S0 system: LetL s) den denote ote a TF,forexample the loop transmission o f a S1S0 system. We shall say that the margin o f L s ) is m w) i f DE FINIT ION 4.2
L jw)I- 1
=
m w).
Note that a necessary condition for a finite bandwidth loop transmission, L s , is that exists W h such that m w) > 1 for all W Who The well known gain and phase margins o f SIS SISO O systems are are related tto o definiti definition on 4.2 as fo follows llows:: A phase margin o f a degrees means that at the frequency W such that IL jw) I = 1
m w)
=
2 - 2 co s h ) = 2 i n ~
and a gain margin of20 lo log( g(y) y) means that at the frequency for which arg L jw) -180°
=
m w)
=
y/ y - 1 .
4.3 The margin o f a MIMO system system:: Consider th thee system shown in Fig 4.41. Let Lk S) denote the loop transmission from the kth input o f the DEFINITION
Synthesis o f LTI Controllers For MIMO LTI Plants
161
controller, G to its itself elf (see for example Fig 4.42 for a 2 x 2 system, where L 1 (s) is the TF from input a to output b . The loop transmission, L k (s , is determined determi ned as follows: close all o f he loops except for the kth loop, where the loop is opened at the kth input to the controller, G, with input point a and Fig 4.42 output poin pointt output b respectively respe ctively (see the loop transmission, Lk s),from point b (se to einput point and a . calculate We shall say that the margin o f
the MIMO systemfor channel k is mk w) i Lk jw)I-
e
r
-
-
= mk u
G
Figure 4.41.
w).
-
y
P
A feedback system syst em
b
.:Jf.::...
-
Figure 4.42.
G
P
u
-y
-
Margin definition for L1 (s) for a 2 x 2 system
4.14 The system in Fig 4.41 can be looked upon as a SISO LTl feedback system for command rk in channel k ofr, o r disturbance d k at output k o f P , when 9k is the controller and the output is Yk (the kth output ofy). The loop transmission for this SISO feedback system is L k and (for a diagona diagonall controller) it satisfies (see Fig 4.43 : REM
RK
Yk
rk Yk
Lk
1
Lk
1
dk
1
Lk
Hence desirable margin properties o f SISO feedback systems, which, for ininstance, result in the elimination o f closed loop resonance resonances, s, can be impo imposed sed o on n
162
QUANTITAT QUAN TITATIVE IVE FEEDBACK D ESIGN
MIM O feedback system MIMO systemss for the TF s from rk and/or dk to output Yk by use o f the appropriate mk w),
Sub-system Pk dashed) Yl
P Yk-1
L Uk
l
Yk l
A Yn
N ,,
Figure 4.43.
System used in
the
T
, Yk
definition of L k L k = gkPk
The feedback system under scrutiny is the MIMO system o f Fig. 4.41 and described mathematically by the following equations:
+ PG
e
I
u
I + Gp -lGr, I + PG -lPGr.
y
is
-lr,
The problem being considered is how to design a controller, G , such that for a given set of plants, {P}, the given margin specifications will be satisfied for all P E {Pl. Formally the problem is stated as: Consider the system shown in Fig. 4.41, where P is an x LTI plant belonging to a set {P}; Design the controller, G, such that for all P E { P} P R O B L E M 4.20
• G is a solution o f a given problem o f the form o f problem 4.1; and • the foll followin owing g margin specifications are satis satisfied fied::
4.113)
Synthesis o f LTI Controllers For MIMO LTI Plants
163
4.15 mk w) may depend on the plant P E {P} and, as in SISO feedback systems, should be greater than 1 at highfrequencies in order to allow REMARK
for a finite bandwidth solution.
The design procedure is first developed for 2 x 2 and then for n x n plants. 31
2 X 2 PLANTS AND DIAGONAL CONTROLLERS
The feedback system o f interest t us is depicted schematically in Fig. 4.44. The problem proble m under consideratio consideration n is how how to compu compute te bounds on the controll controller, er,
+ e
r
G
u
p
y
+ Figure 4.44.
2 x 2 feedback feedback syste system m
G = diag(g1 g2), such that the given margin specifications will be satisfied for a given set o f plants, {P} During the design procedure these bounds will intersect the bounds generated to satisfy other specifications, so that the solution will also satisfy the margin specifications. For Formal mally ly the proble problem m is as stated below: PROBLEM
4.21 Consider the system shown in Fig. 4.44 where P is a 2 x 2
LTI plant which belongs to a set {Pl Show how to calculate bounds on the elements o f the controller, G, such that for all P E {P} the following margin specifications are satisfied:
+ Lk 4.16
jw I- 1
mk w);
W
: Wmb
k = 1,2.
4.114)
is a transition transition frequency between low and high freq frequenuencies. cie s. It depends on the other speci specifications fications im imposed posed on the clo closed sed loop system. A good go od approximation approximation can be fou found nd after several desig design n ite iterati rations, ons, but for reasonable values o f m (greater than 1 an and d clos closed ed loop specifications specifications,, the margin bounds at low frequencies ar aree not no t th thee dominant bounds bounds;; hence Wmb = 0 is a satisfa sat isfacto ctory ry choice. choice. Thus there is no need to carry out design iterations to find Wmb· REMARK
Wmb
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