Activity 03

Feedback and Control Systems Name:__________________________________ Date Performed: _________________________

Rating:__________________________ Date Submitted: __________________

Feedback and Control Systems STABILITY AND STEADY-STATE ERROR ANALYSIS AND DESIGN OF SYSTEMS Activity No. 3 I. ACTIVITY OBJECTIVES This activity aims to 1. demonstrate the use of computer aided tools in analyzing the stability and steady-state error of linear systems; and 2. equip the students with the skills and knowledge in designing systems with the aid of tools to achieve transient response and steady-state error requirements of systems while ensuring stability. II. INTENDED LEARNING OUTCOMES At the end of this activity, the students shall be able to 1. analyze the stability and steady-state error of dynamic systems described by transfer functions; 2. design component values of systems to meet steady-state error requirements while ensuring stability. III. BACKGROUND INFORMATION Stability is the most important requirement of any control system. If the system is unstable, it cannot be designed for transient response and steady-state error. It also poses threat to life and property, as instability can mean a motor that has uncontrollable speed, or too much heat produced by a heater. In the discussion, two definitions of stability are offered:  A system is stable if the natural response approaches zero as time approaches infinity.  A system is stable if every bounded input yields a bounded output (the bounded-input boundedoutput or BIBO requirement). It was also discussed that stability is also related to the location of the closed-loop poles. In the discussion, the following were concluded:  A system is stable if all of its closed-loop poles are in the left-half of the complex s-plane.  A system is marginally stable if it has poles of multiplicity one at the jω-axis.  A system is unstable if it has at least one pole on the right-half of the complex s-plane or has multiple poles on a single location at the jω-axis. To find how the poles are distributed on the complex s-plane, the Routh-Hurwitz criterion is being used, although tools such as MATLAB and LabVIEW can compute the exact location of closed-loop poles of a higher-ordered system.

Activity No. 3 – Stability and Steady-state Error Analysis and Design of Systems

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Feedback and Control Systems Steady-state error is the difference of the actual output to the desired output of the system. It can be evaluated using the closed-loop transfer function or an equivalent unity feedback system. In the discussion, the latter approach was preferred, since it also provides perspective on the static error constants which relates to the error of the system. In this activity, the analysis and design of systems related to stability and steady-state error using MATLAB and LabVIEW will be explored. IV. RESOURCES NEEDED To perform this activity, a computer workstation with MATLAB R2012a or higher and LabVIEW 8.6 or higher installed is required. For MATLAB, the control systems toolbox is required and for LabVIEW, the control design and simulation module. V. LEARNING ACTIVITIES Activity 3.1 – Stability via pole location 1. Use the Routh-Hurwitz criterion to determine the pole location distribution of the system whose configuration is shown below. Complete the table.

System

Closed-loop pole distribution (via Routh-Hurwitz criterion) Left-half plane Right-half plane -axis

Q1.1(a) What can be said about the stability of the system? _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ 2. MATLAB. The pole-zero map of the closed-loop transfer function can be plotted and from there, the number of poles on the left-half, right-half and the jω-axis of the complex s-plane. The command roots()computes the roots of a polynomial whose coefficients are written as a row matrix. If the polynomial has the form Activity No. 3 – Stability and Steady-state Error Analysis and Design of Systems

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Feedback and Control Systems P(s) = a s + a

s

+⋯+ a s +a

then the command is entered in the following manner >> roots([an an-1 ... a1 a0]) Q1.2(a) Using MATLAB, complete the table below. Sketch the pole-zero plot of the closed-loop transfer function. Indicate the number of poles, as well as the exact location of the poles under each region of the -plane.

System

Closed-loop pole distribution (via MATLAB) Left-half plane Right-half plane -axis

Pole-zero Map

Q1.2(b) Does the results returned by MATLAB agree with the results generated by the Routh table. Is the conclusion about the stability of system the same when the results generated by MATLAB were interpreted? _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ 3. LabVIEW. Build the act03-01.vi VI as shown below. The VI analyzes the stability of the system whose configuration is shown in the front panel.

Activity No. 3 – Stability and Steady-state Error Analysis and Design of Systems

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Feedback and Control Systems

Q1.3(a) Using the VI, complete the table below. Sketch the pole-zero plot of the closed-loop transfer function. Indicate the number of closed-loop poles, as well as the exact location of these poles under each region of the -plane.

System

Closed-loop pole distribution (via LabVIEW) Right-half plane Left-half plane -axis

Pole-zero Map

Activity No. 3 – Stability and Steady-state Error Analysis and Design of Systems

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Feedback and Control Systems Q1.3(b) Will you reach the same conclusions about the stability of the system when the LabVIEW virtual instrument is used? _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ Q1.3(c) Use MATLAB and LabVIEW to complete the table below. Indicate the number of closed-loop poles, as well as their exact location under each region of the complex -plane. Under “Remarks”, tell whether the system is stable, unstable or marginally stable. On separate sheets of paper, sketch the closed-loop pole-zero map of each of the systems. Verify the results using Routh table. System

T(s) =

Closed-loop pole distribution and location LHP RHP -axis

Remarks

34 s + 10s + 35s + 50s + 34

with

Activity No. 3 – Stability and Steady-state Error Analysis and Design of Systems

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Feedback and Control Systems Activity 3.2 – Analysis of steady-state error. 1. The steady-state error will be evaluated using the configuration below. Refer to the lecture on the formulas to be used in evaluating the static error constants and the error for step, ramp and parabolic test inputs.

Remember that the system must be tested first for stability before analyzing it for transient response or steady-state error. Thus, the techniques learned in Activity 3.1 can be applied first before proceeding. 2. MATLAB. To use MATLAB, the object representing G(s) must be converted first to a symbolic object. If G contains the transfer function object, use the following commands to convert G into a symbolic object Gsym. >> [num den] = tfdata(G); >> syms s >> Gsym = poly2sym(cell2mat(num),s)/poly2sym(cell2mat(den),s) Gsym is now a symbolic math object. The function limit()can now be used to evaluate the static error constants, which will be then used to evaluate the error for various test inputs. As an example, if Gsym is the symbolic object representing the open-loop transfer function of the unity feedback system as shown in step one of this sub-activity, then the static error constant K and the error due to the step (∞) are evaluated as input e >> Kp = limit(Gsym,0) >> estep = 1/(1+Kp) Q2.1(a) What does the following functions in MATLAB do: tfdata(), syms, poly2sym(), cell2mat(). Discuss the syntax and the required arguments of each function. _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ Q2.2(b) Use MATLAB to evaluate the static error constants and steady-state errors of the systems shown below. Complete the table. Verify the values obtained using manual calculations. For the last system, assume that the input and output are the same quantity. Activity No. 3 – Stability and Steady-state Error Analysis and Design of Systems

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Feedback and Control Systems System (check if stable)

Static error constants

Steady-state errors (∞)

(∞)

(∞)

3. LabVIEW. Build the act03-02.vi as shown below. The Array of Polynomial Coefficients to Formula String.vi can be obtained from your instructor or from this link: https://decibel.ni.com/content/docs/DOC-22590 if you have an available internet connection.

Activity No. 3 – Stability and Steady-state Error Analysis and Design of Systems

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Feedback and Control Systems

Q2.3(a) Use the VI above to evaluate the static error constants and steady-state errors of the systems shown below. Complete the table. For the last system, assume that the input and output are the same quantity. Static error constants Steady-state errors System (check if stable) (∞) (∞) (∞)

VI. CONCLUSIONS _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ Activity No. 3 – Stability and Steady-state Error Analysis and Design of Systems

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Feedback and Control Systems _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ VII. ASSESSMENT TASKS - MACHINE PROBLEMS 1. For the system shown below, do the following: (a) At K = 10, is the system stable? Find the system type, the static error constant and the error of the system at this gain, then plot and determine the time response parameters if possible. (b) Repeat part (a) at K = 10 . (c) Plot the value of the static error constant and the steady-state error as a function of the gain K for the range at which the system is stable. What conclusions can be drawn from the plot?

2. The open-loop transfer function of a swivel controller and plant for an industrial robot is given as ω (s) K G (s) = = V (s) (s + 10)(s + 4s + 10) where ω (s) is the Laplace transform of the robot’s angular swivel velocity and V (s) is the input voltage to the controller. Assume G (s) is the forward transfer function of a velocity control loop with an input transducer and a sensor, each represented by a constant gain of 3 (Schneider, 1992), do the following: (a) Plot the value of the error of the system as a function of the gain K at the range of K for which the system is stable. (b) Design the value of the gain K to minimize the steady-state error between the input commanded angular swivel velocity and the output actual angular swivel velocity. Show that the system is still stable at the design point. (c) For the chosen value of the gain at part (b), determine the system type, steady-state error and the transient response of the system. VIII. REFERENCES N. Nise. (2011). Control Systems Engineering 6th Edition. United States of America: John Wiley & Sons. R. Dorf& R. Bishop. (2011). Modern Control Systems 12th Edition. New Jersey: Prentice Hall. MathWorks (2009 June 26). How can I convert a transfer function object from the Control System Toolbox into a symbolic object for use with the Symbolic Math Toolbox?. From Activity No. 3 – Stability and Steady-state Error Analysis and Design of Systems

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Feedback and Control Systems http://www.mathworks.com/support/solutions/en/data/1-1BS2O/?product=SM&solution=1-1BS2O retrieved 2012 Nov 02. Ryan-C (2012 May 23). Convert Array of Polynomials to Formula String Input. Message posted to https://decibel.ni.com/content/docs/DOC-22590 IX. ASSESSMENT RUBRIC A. Assessment rubric for the activity’s intended learning outcomes (70%) INTENDED LEARNING OUTCOMES

1

2

3

Analyze the stability and steady-state error of dynamic systems described by transfer functions. (MP 1)

The student was not able to analyze the stability and steady-state error of the system.

The student was able to analyze the stability of the system but not the steady-state error, or was able to obtain the steady-state error but did not check for stability.

The student was able to analyze the stability and steady-state error of the system properly.

The student was not able to design the component values of the systems.

The student was able to The student was able to design correctly design component component values but does not values of the system that fully meet the steady-state meets stability and steady-state error requirements or that the error requirements and the design was not verified. design is correctly verified.

Design component values of systems to meet steady-state error requirements while ensuring stability. (MP 2)

Points

Total Score Mean Score = (Total Score /2) Percentage Score = (Total Score / 6) x 100% A = 70% of the Percentage Score

B. Assessment rubric for the conduct of laboratory experiments (30%) Performance Indicators

Conduct experiments in accordance with good and safe laboratory practice.

1

2

3

Points

Members follow good Members do not follow Members follow good and safe laboratory good and safe laboratory and safe laboratory practice most of the time practice in the conduct of practice at all times in the in the conduct of experiments. conduct of experiments. experiments.

Activity No. 3 – Stability and Steady-state Error Analysis and Design of Systems

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Feedback and Control Systems Performance Indicators

1

2

3

Operate equipment and instruments with ease

Members are unable to operate the equipment and instruments.

Members are able to operate equipment and instrument with supervision.

Members are able to operate the equipment and instruments with ease and with minimum supervision.

The group has complete data but has no analysis and valid conclusion.

The group has complete data, validates experimental values against theoretical values, and provides valid conclusion.

Analyze data, validate experimental values against theoretical values to determine possible experimental errors, and provide valid conclusions.

The group has incomplete data.

Points

Total Score Mean Score = (Total Score /3) Percentage Score = (Total Score / 9) x 100% B = 30% of the Percentage Score

Laboratory Rating A

Activity No. 3 – Stability and Steady-state Error Analysis and Design of Systems

B

Total

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