Readings: FC: p : lead compensation. 9/9/2011 Classical Control 1

Size: px

Start display at page:

Download "Readings: FC: p : lead compensation. 9/9/2011 Classical Control 1"

Cody Willis
6 years ago
Views:

1 MM0 Frequency Response Design Readings: FC: p : lead compensation 9/9/20 Classical Control

2 What Have We Talked about in MM9? Control design based on Bode plot Stability margins (Gain margin and phase margin) Transient performance Steady-state performance Nyquist Diagram What s Nyquist diagram? What we can gain from Nyquist diagram Matlab functions: bode(), margin(), nyquist() 9/9/20 Classical Control 2

3 Nyquist Criterion for Stability (MM9) The Nyquist criterion states that: P = the number of open-loop (unstable) poles of G(s)H(s) N = the number of times the Nyquist diagram encircles clockwise encirclements of - count as positive encirclements counter-clockwise (or anti-clockwise) encirclements of - count as negative encirclements Z = the number of right half-plane (positive, real) poles of the closed-loop system The important equation: Z = P + N 9/9/20 Classical Control 3

4 Goals for this lecture (MM0) An illustrative example Frequency response analysis Frequency response design Lead and lag compensators What s a lead/lag compensator? Their frequency features A systematical procedure for lead compensator design A practical design example Beam and Ball Control 9/9/20 Classical Control 4

5 An Illustrative Example: Antenna Position Control Control system design for a satellite tracking antenna (one-dimentional) Design specifications: Overshoot to a step input less than 6%; Settling time to 2% to be less than 0 sec.; Tracking error to a ramp input of slope 0.0rad/sec to be less than 0.0rad; Sampling time to give at at least 0 samples in a rise time. 9/9/20 Classical Control 5

6 9/9/20 Classical Control 6 System model: Transfer function: T c T d B J... B t T t u J B a a s s s U s c ) ( ) ( 0.,, ) ( ) ( ) ( Example: Mathematical Modeling

7 Example: Open-Loop Analysis Transfer function: ( s) B Tc ( t), a 0., u( t) U( s) s s( ) J B a Open-loop properties Step response Impulse response impulse(tf(,[0 0])); figure; step(tf(,[0 0]),0) Stability? 9/9/20 Classical Control 7

8 Example: Possible Control Structure K G(s) KD(s) G(s) G(s) D(s) G(s) KD(s) D2(s) Different control structures Cascade controller: Gain, dynamic compensator? Feedback controller: Gain, dynamic compensator? Single or multiple loops?... 9/9/20 Classical Control 8

9 Example: Just Gain Controller? sys=tf(,[0 0]), step(feedback(00*sys,)) K G(s) sys=tf(,[0 0]), s^2 + s rlocus(sys) 9/9/20 Classical Control 9

10 syscl=feedback(0.*sys,); bode(syscl) ; figure, margin(0.*sys) Closed-loop transfer function s^2 + s /9/20 Classical Control 0

11 Dynamic Compensation Objective: If a satisfactory process dynamics can not be obtained by a gain adjustment alone, some modification or compensation of the process dynamics is needed +KD(s)G(s)=0 Controller KD(s) Plant G(s) 9/9/20 Classical Control

12 Lead and Lag Compensators Lead compensation: acts mainly to lower rise time and decrease the transient overshoot: D(s)=(s+z)/(s+p) with z < p Lag compensation: acts mainly to improve the steadystate accuracy: D(s)=(s+z)/(s+p) with z > p Compensationis typically placed in series with the plant in the feedforward path Controller KD(s) Plant G(s) 9/9/20 Classical Control 2

13 Frequency Properties of Lead and Lag 9/9/20 Classical Control 3

14 Lead and PD Controllers (I) PD compensation: D(s)=K(T D s+) Increasing the phase margin Amplify the high frequency noise Lead compensation: D(s)=K(Ts+)/(Ts+), < 9/9/20 Classical Control 4

15 Lead and PD Controllers (II) Lead compensation: D(s)=K(Ts+)/(Ts+), < Lead compensator is a high-pass filter (app.pd control) It is used whenever substantial improvement in damping is required The maximum phase contribution is Example: sysd=tf([0 ],[ ]) bode(sysd) max T sin sin max max 9/9/20 Classical Control 5

16 Example: PD Controller for Antenna Control (I) Design the low frequency gain K with respect to the steadystate error specification: Tracking error to a ramp input of slope 0.0rad/sec to be less than 0.0rad Ramp Input (R(s) = /s^2): (MM5) Antenna system case... K= 9/9/20 Classical Control 6

17 Example: PD Controller for Antenna Control (II) PD controller: D(s)=0s+ sysd=tf(,[ ])*tf([0.0.0],); syscl2=feedback(sysd,); step(syscl2) Check whether this controller is ok? 9/9/20 Classical Control 7

18 Example: Intuitive Lead Design Design the low frequency gain K with respect to the steadystate error specification Antenna system case... K= Lead controller: D(s)=(0s+)/(s+) sysd=tf(,[ ])*tf([0.0.0],[ ]); syscl3=feedback(sysd,); step(syscl3) 9/9/20 Classical Control 8

19 Example: Comparison of PD and Lead Design bode(syscl2,syscl3); grid 9/9/20 Classical Control 9

20 Goals for this lecture (MM0) An illustrative example Frequency response analysis Frequency response design Lead and lag compensators What s a lead/lag compensator? Their frequency features A systematical procedure for lead compensator design 9/9/20 Classical Control 20

21 Lead Compensator Design Procedure (I) Step : Design the low frequency gain K with respect to the staedy-state error specification Antenna system case... K= Step Input (R(s) = /s): Ramp Input (R(s) = /s^2): Parabolic Input (R(s) = /s^3): 9/9/20 Classical Control 2

22 Lead Compensator Design Procedure (II) Step 2: Determine the needed phase lead Original system PM: sys=tf(,[0 0]), margin(sys)... PM=8 at Expected PM: Expected overshoot limit (6%) Dampling ratio ξ 0.5 Expected PM 00*0.5=50 degree Directly needed phase lead: 50-8=32 degree Expected phase lead: 32+ (7~0) degree 9/9/20 Classical Control 22

23 Lead Compensator Design Procedure (III) Lead compensation: D(s)=K(Ts+)/(Ts+), < Step 3: Determine coefficient α sin sin max max Step 4: Determine coefficient T T max ( n / 2) sin sin max T sin sin max max 9/9/20 Classical Control 23

24 Lead Compensator Design Procedure (IV) Step 5: Draw the compensated frequency response, check PM sysd=tf([4.622 ],[.037 ]); sysc=sys*sysd; margin(sysc); step(feedback(sysc,)) 9/9/20 Classical Control 24

25 Why this Lead Compensator doesn t work? (I) Check the poles & zeros of the closed-loop... syscl=feedback(sysc,); pzmap(syscl); Compare with a standard 2nd-order system... Sys2=tf(0.3099, conv([ i], [ i])) ; step(feedback(sysc,),sys2) 9/9/20 Classical Control 25

26 Lead Compensator Design Procedure (V) Step 6: Iterate on the design until all specifications are met sysd=tf([ ],[.09 ]) ; sysc=sys*sysd; margin(sysc); syscl=feedback(sysc,); step(syscl) sysd=tf([9 ],[ ]); syscl=feedback(sys*sysd,); step(syscl) T D ( s ) max sin sin max max s s ( n /.5) sin sin /9/20 Classical Control 26

27 9/9/20 Classical Control 27

28 System: A ball rolls along the track of a beam that is pivoted at some position. Objective: To steadily place the ball at any given position along the track Strategy: To control the track angle through the control of a servo motor 4. What s B&B System? 9/9/20 Classical Control 28

29 4.2 Why focus on B&B System? The ball and beam apparatus demonstrates the control problems associated with unstable systems. An example of such a system is a missile during launch; active control is required to prevent the missile going unstable and toppling over. 9/9/20 Classical Control 29

30 4.4 AUE Beam and Ball System Implement at least one control method 5-0% overshoot and 3-6 second settling time The potentiometer and axle a hybrid stepping motor TI MSP430 fix-point 9/9/20 Classical Control 30

31 4.4. Modelling the AUE B&B System r B R 2 p 2 p Lagrangian modelling technique Block diagram of stepping motor and load r s m g s J 2 ball m s 2 R 9/9/20 Classical Control 3

32 4.4.2 Analysis of the AUE B&B From the nyquist plot, it can be Observed that the system is unstable since - is encircled clockwise by the nyquist plot System The system is unstable if it is exposed to a step input. From this can it be concluded that the system needs some kind of controller. 9/9/20 Classical Control 32

33 4.4.3 Control Strategy for the B&B System Cascade control Master loop (outer loop) Slave loop (inner loop) 9/9/20 Classical Control 33

4.4.4 Control Design for Slave Loop The block control contains the P-controller, Kp2. The slave loop must be faster enough (e.g., 0 times faster) comparing with the master loop kp 2 4.

34 4.4.4 Control Design for Slave Loop The block control contains the P-controller, Kp2. The slave loop must be faster enough (e.g., 0 times faster) comparing with the master loop kp s Through bode plot it can be seen that the system has a cutoff frequency at appr. 5 rad/sec The system is settled in approximately 0.3 sec 9/9/20 Classical Control 34

35 4.4.5 Control Design for Master Loop Kp( s ) D s s the system has a phase margin of 47.5 deg at a frequency of.38 Hz 9/9/20 Classical Control 35

36 4.4.6 Simulation Tests 9/9/20 Classical Control 36

37 4.4.7 Real Test Videos 9/9/20 Classical Control 37

38 Exercise Could you repeat the antenna design using. Continuous lead compensation; 2. Emulation method for digital control; Such that the design specifications: Overshoot to a step input less than 5%; Settling time to % to be less than 4 sec.; Tracking error to a ramp input of slope 0.0rad/sec to be less than 0.0rad; Sampling time to give at at least 0 samples in a rise time. (Write your analysis and program on a paper!) 9/9/20 Classical Control 38

Outline. Digital Control. Lecture 3

Outline. Digital Control. Lecture 3 Outline Outline Outline 1 ler Design 2 What have we talked about in MM2? Sampling rate selection Equivalents between continuous & digital Systems Outline ler Design Emulation Method for 1 ler Design