Outline. Classical Control. Lecture 5

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1 Outline

2 Outline Outline 1 What is 2

3 Outline What is Why use? Sketching a 1 What is Why use? Sketching a 2 Gain Controller Lead Compensation Lag Compensation

4 What is Properties of a General System Why use? Sketching a + D(s) G(s) Transfer Functions Open loop: D(s)G(s) Closed loop: D(s)G(s) 1+D(s)G(s) Sensitivity: 1 1+D(s)G(s)

5 What is The Goal of Why use? Sketching a What do we want? Show how changes in a systems feedback characteristics and other parameters influence the pole locations Study the effects of additional poles and zeros when designing dynamic compensation

6 What is Why use? Sketching a Effects of Additional Poles and Zeros

7 What is S-plane Design in MATLAB Why use? Sketching a sgrid(linspace(0,1,11),linspace(0,9,10), new )

8 What is Why use? Sketching a is the set of values for s for which the following equation holds for some real value of K 1 + KL(s) = 0 which is the characteristic equation for the closed loop system D(s)G(s) 1 + D(s)G(s) = KL(s) 1 + KL(s) The roots of the characteristic equation is thus the poles of the closed loop system

9 What is Why use? Sketching a Generally the root locus can be formulated as: 1 + KL(s) = K b(s) a(s) = 0 a(s) + Kb(s) = 0 L(s) = 1 K

10 What is Why use? Sketching a : Example 1 (FC pp. 234) System G(s) = 1 s(s + 1) D(s) = K D(s)G(s) = 1 + K s(s + 1) = 1 + KL(s) 1 L(s) = s(s + 1) MATLAB sys=tf(1,[1 1 0]); rlocus(sys) axis([ ])

11 What is : Example 1 Why use? Sketching a Imaginary Axis System: sys Gain: Pole: Damping: 1 Overshoot (%): 0 Frequency (rad/sec): System: sys Gain: Pole: i Damping: 0.66 Overshoot (%): 6.32 Frequency (rad/sec): System: sys Gain: Pole: i Damping: Overshoot (%): 8.05 Frequency (rad/sec): Real Axis

12 What is Why use? Sketching a : Example 2 (FC pp. 235) Plant G(s) = D(s) = D(s)G(s) = s(s + c) L(s) = b(s) a(s) = 1 s(s + c) = 0 = s2 + cs + 1 = a(s) + cb(s) s s MATLAB sys=tf([1 0],[1 0 1]); rlocus(sys) axis([ ])

13 What is : Example 2 Why use? Sketching a Imaginary Axis System: sys Gain: 2.31 Pole: 1.73 Damping: 1 Overshoot (%): 0 Frequency (rad/sec): System: sys Gain: 2.09 Pole: Damping: 1 Overshoot (%): 0 Frequency (rad/sec): System: sys Gain: 1.19 Pole: i Damping: Overshoot (%): 9.8 Frequency (rad/sec): Real Axis

14 What is Why use? Sketching a Sketching a (FC 3rd ed. pp. 260)

15 What is : Example 3 Why use? Sketching a Plant (D(s) = K) G(s) = s + 1 s 2 (s + 4) = L(s) a(s) = s 3 + 4s 2 b(s) = s + 1 b da ds adb = 0 ds s = 0, 1.75 ± j MATLAB sys=tf([1 1],[ ]); rlocus(sys) axis([ ])

16 What is : Example 3 Why use? Sketching a Imaginary Axis Real Axis

17 What is : Example 4 Why use? Sketching a Plant (D(s) = K) G(s) = s + 1 s 2 (s + 9) = L(s) a(s) = s 3 + 9s 2 b(s) = s + 1 b da ds adb ds = 0 MATLAB sys=tf([1 1],[ ]); rlocus(sys) axis([ ]) s = 0, 3, 3

18 What is : Example 4 Why use? Sketching a Imaginary Axis Real Axis

19 What is : Example 5 Why use? Sketching a Plant (D(s) = K) G(s) = s + 1 s 2 (s + 12) = L(s) a(s) = s s 2 b(s) = s + 1 b da ds adb = 0 ds s = 0, 2.3, 5.2 MATLAB sys=tf([1 1],[ ]); rlocus(sys) axis([ ])

20 What is : Example 5 Why use? Sketching a Imaginary Axis Real Axis

21 What is : Example 6 Why use? Sketching a Plant (D(s) = K) G(s) = 1 s(s + 2)[(s + 1) 2 + 4] = L(s) a(s) = s 4 + 4s 3 + 9s s b(s) = 1 b da ds adb = 0 ds s = 1, 1 ± 1.2j MATLAB sys=tf(1,[ ]); rlocus(sys) axis([ ])

22 What is : Example 6 Why use? Sketching a Imaginary Axis Real Axis

23 Outline What is Gain Controller Lead Compensation Lag Compensation 1 What is Why use? Sketching a 2 Gain Controller Lead Compensation Lag Compensation

24 What is Gain Controller Lead Compensation Lag Compensation Objective: Select a particular value of K that will meet the specifications for static and dynamic response Magnitude Condition: 1 + KG(s) = 0 K = 1 G(s)

25 What is Gain Controller Lead Compensation Lag Compensation

26 Gain Controller? What is Gain Controller Lead Compensation Lag Compensation + K G(s) MATLAB sys=tf(1,[10 1 0]); rlocus(sys) axis([ ]) grid

27 Gain Controller? What is Gain Controller Lead Compensation Lag Compensation Imaginary Axis Real Axis

28 What is Dynamic Compensation Gain Controller Lead Compensation Lag Compensation Objective: If satisfactory process dynamics cannot be obtained by a gain adjustment alone then some modification or compensation of the dynamics is needed 1 + KD(s)G(s) = 0 Lead and Lag Compensation: Lead Compensation acts mainly to lower rise time and decrease the transient overshoot: D(s) = s+z s+p,z < p Lag Compensation acts mainly to improve the steady-state accuracy: D(s) = s+z s+p,z > p + KD(s) G(s)

29 Lead Compensation What is Gain Controller Lead Compensation Lag Compensation Approximates PD control Moves the locus to the left and typically improves the system damping Zero and pole selection: The zero is placed in the neighborhood of the closed-loop ω n as determined by rise-time or settling-time requirements The pole is located at a distance 3 to 20 times the value of the zero location

30 What is Lead Compensation: Example Gain Controller Lead Compensation Lag Compensation System K G(s) = s(s + 1) D 1 (s) = s + 2 D 2 (s) = s + 2 s + 20 D 3 (s) = s + 2 s + 10 MATLAB sys=tf(1,[1 1 0]); D1=tf([1 2],[1]); D2=tf([1 2],[1 20]); D3=tf([1 2],[1 10]); rlocus(sys,d1*sys,d2*sys,d3*sys) axis([ ])

31 What is Lead Compensation: Example Gain Controller Lead Compensation Lag Compensation System: untitled3 Gain: 84.8 Pole: i Damping: Overshoot (%): 16 Frequency (rad/sec): Imaginary Axis Real Axis

32 What is Gain Controller Lead Compensation Lag Compensation Lead Compensation: Design Example Design the closed loop response for the following system to have a damping ratio ζ > 0.5 and a natural frequency ω n > 7 rad sec. System K G(s) = s(s + 1) D 1 (s) = s + 2 s + 10 MATLAB sys=tf(1,[1 1 0]); D1=tf([1 2],[1 10]); D2=tf([1 6],[1 30]); rlocus(d1*sys,d2*sys) axis([ ]) D 2 (s) = s + 6 s + 30

33 What is Gain Controller Lead Compensation Lag Compensation Lead Compensation: Design Example Imaginary Axis System: untitled2 Gain: Pole: i Damping: Overshoot (%): 16 System: untitled1 Frequency (rad/sec): 23.1 Gain: 60.2 Pole: i Damping: Overshoot (%): 8.98 Frequency 10 (rad/sec): Real Axis

34 What is Lead Compensation: Antenna Gain Controller Lead Compensation Lag Compensation Step 1 Select the lead compensator as D(s) = s+0.9 s+10 sysdg=tf(1,[10 1 0])*tf([1 0.9],[1 10]); Step 2 Draw the root locus to determine K rlocus(sysdg)

35 What is Lead Compensation: Antenna Gain Controller Lead Compensation Lag Compensation Step 3 Construct the closed-loop system and check whether the requirements are satisfied or not step(feedback(k*sysdg,1)) Step 4 If the requirements are not satisfied, iterate the design procedure

36 Lag Compensation What is Gain Controller Lead Compensation Lag Compensation Approximates PI control Creates a zero and a pole: Effect of the zeros are to move the locus to the left Effect of the poles are to press the locus towards the right Zero and pole selection: Select a pole near s = 0 Select the zero nearby such that the dipole does not significantly interfere with the response The lag compensation usually degrades stability

37 What is Lag Compensation: Example Gain Controller Lead Compensation Lag Compensation System K G(s) = s(s + 1) D 1 (s) = s + 2 s + 20 D 2 (s) = s s MATLAB sys=tf(1,[1 1 0]); D1=tf([1 2],[1 20]); D2=tf([1 0.1],[1 0.01]); rlocus(d1*d2*sys) axis([ ])

38 What is Lag Compensation: Example Gain Controller Lead Compensation Lag Compensation Imaginary Axis Real Axis

39 What is Lag Compensation: Example Gain Controller Lead Compensation Lag Compensation axis([ ]) Imaginary Axis Real Axis

40 What is Book: Feedback Control 1 Exercise 5.2 in FC pp Consider a DC motor control using a PI controller, Where the motor and PI controller are modeled as G(s) = K (τs + 1), D(s) = K p(t i s + 1) T i s with parameters K = 30, τ = 0.35, T i = Use the root locus method to determine the largest vaule of K p such that ζ = Try to use the root locus method to design a lead compensator for the examplified antenna system.

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