Desired Bode plot shape

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1 Desired Bode plot shape 0dB Want high gain Use PI or lag control Low freq ess, type High low freq gain for steady state tracking Low high freq gain for noise attenuation Sufficient PM near ω gc for stability ω gc Mid freq Speed, BW Use lead or PD control High freq Noise immu Use low pass filters Want low gain ω Want sufficient Phase margin Mr, Mp PM+Mp=70 ω

2 Overall Loop shaping strategy Determine mid freq requirements Speed/bandwidth è ω gc Overshoot/resonance è PM d Use PD or lead to achieve PM ω gc Use overall gain K to enforce ω gc PI or lag to improve steady state tracking Use PI if type increase neede Use lag if ess needs to be reduced Use low pass filter to reduce high freq gain

3 PI or PID Design Steps From plant, draw Bode plot From specs => PM d and ω gcd If there is speed or BW req, è ω gcd, In this case, if PM not enough, design PD or lead Otherwise, choose ω gcd to have PM>PMd Find K to enforce ω gcd : Let K P = K ( ) CG( jω 1 gcd ) And K I = Kω gcd /5~20, depending on extra PM room to spare K I location affected by ess spec at other frequencies other than 0 K =

4 PI-PID Design with ess specs From plant, draw Bode plot From specs => K v,a-des, PM d and ω gcd For required ess, K v,a-des =1/ess With C(s)=1/s, compute K v,a-have If there is speed or BW req, è ω gcd, In this case, if PM not enough, design PD or lead Otherwise, choose ω gcd to have PM>PMd Find K to enforce ω gcd : ( ( ) CG j 1 = ω gcd ) Let K P = K, KI des = K v,a-des /K v,a-have If KI des <= Kω gcd /5~20, done, let K I = KI des Else, increase ω gcd and go back to previous step Need to increase type by 1 to make a nonzero ess to be zero, and after type increase, there is further requirement on ess. K

5 PI Design with PD Design Steps From required ess, K v,a-des =1/ess With C(s)=1/s, compute K v,a-have Let KI = K v,a-des /K v,a-have Multiply G(s) by KI/s Do a PD design for KIG(s)/s, with DC gain=1: Find ω gc and PM Find PM d Let φ = PMd PM + (a few degrees) Compute T D = tan(φ)/w gcd K P = KI*T D

6 Example Design problem from text book But our solution is much more meaningful and much easier Plant TF Gp(s) = 10/s(s+1) Specs: Mp <19%, but >2% Ts <1 sec Ess to step, ramp, acc all = 0

7 s=tf('s'); Gp=10/s/(s+1); figure; margin(gp); grid; ts = 1; %specification t=linspace(0,3*ts,301); Gcl=Gp/(1+Gp); figure; step(gcl,t); grid; %specification for Mp is 2% to 19% %can use the mid point as initial target Mp=10; zeta=0.6; %for Mp=10% sigma=4/ts; %no tolerance band is given, so use 2% wn=sigma/zeta; wgcd=0.7*wn; Gwgc = evalfr(gp, j*wgcd); PM=angle(Gwgc)/pi* ;

8 PMd = 70 - Mp + 10; %add 10 deg because we need PI later DPM = PMd - PM; %phase margin deficiency z_pd = wgcd /tan(dpm*pi/180); %PD control K = 1/abs(evalfr((s+z_PD)*Gp,j*wgcd)); C=K*(s+z_PD); figure; margin(c*gp); grid; %Bode with PD figure; step(c*gp/(1+c*gp),t); grid; %now add PI, place zero of PI at wgcd/10 %Mp target of 10% together with 10 deg extra PMd %can tolerate a little more phase delay from PI. z_pi = wgcd/10; C=C*(s+z_PI)/s; %multiply PI and PD figure; margin(c*gp); grid; figure; step(c*gp/(1+c*gp),t); grid;

9 Plant Bode plot Bode Diagram Gm = Inf db (at Inf rad/sec), Pm = 18 deg (at 3.08 rad/sec) 100 Magnitude (db) Phase (deg) Frequency (rad/sec)

10 Original step response 1.8 Step Response Amplitude Time (sec)

11 Magnitude (db) Bode plot after PD Bode Diagram Gm = Inf, Pm = 70 deg (at 4.67 rad/sec) Phase (deg) Frequency (rad/sec)

12 Step response with initial PD 1.4 Step Response Amplitude Time (sec)

13 Step response after PD 1.4 Step Response Amplitude Ts is two large, need to increase wgc by 1.5X Time (sec)

14 Bode Diagram Gm = -Inf db (at 0 rad/sec), Pm = 64.4 deg (at 4.69 rad/sec) 100 Magnitude (db) Phase (deg) Frequency (rad/sec)

15 wgc=0.7*wn*1.6; z_pi = wgc/20; K I =K z_pi

16 Magnitude (db) Wgc increased from 4.47 to 7.47 Bode Diagram Gm = Inf, Pm = 70 deg (at 7.47 rad/sec) After PD Phase (deg) Frequency (rad/sec)

17 1.4 Ts is now about right Step Response Amplitude After PD Time (sec)

18 Bode Diagram Gm = -Inf db (at 0 rad/sec), Pm = 67.2 deg (at 7.47 rad/sec) 100 Magnitude (db) Phase (deg) After PID Frequency (rad/sec)

19 1.4 Step Response Amplitude After PID Ts is OK Mp is OK Time (sec)

20 Design specs: use PID to achieve 1. Ess to ramp input = 0 2. As fast as possible 3. Overshoot <= 25% Plant is type 1, so need PI after PD. 25% Mp è PM >= 45, but with PI, need PMd 55 PD can contribute a maximum of about 75 So, select max wgc at phase=-200

21 s=tf('s'); Gp=1/s/(s+1)/(s+5); figure(2); margin(gp); grid; V=axis; Mp=25; PMd = 70 - Mp + 10; %+10 for PI later DPM=75; PM=PMd-DPM; hold on; plot([v(1) V(2)],[PM PM]-180,'r:'); [x, y]=ginput(1); wgcd=x; z_pd = wgcd/tan(dpm*pi/180); %PD control K = 1/abs(evalfr((s+z_PD)*Gp,j*wgcd)); C=K*(s+z_PD); margin(c*gp); grid; %Bode with PD z_pi = wgcd/10; C=C*(s+z_PI)/s; %multiply PI and PD margin(c*gp); grid; hold off; %Bode with PID t=linspace(0, 15/wgcd, 301); figure(3); step(c*gp/(1+c*gp),t); grid;

22 100 Bode Diagram Gm = -Inf db (at 0 rad/sec), Pm = 48.2 deg (at 3.68 rad/sec) 50 Magnitude (db) Phase (deg) Frequency (rad/sec)

23 1.4 Step Response Amplitude Time (sec)

24 Ziegler-Nichols rules for tuning PIDs K=K P (1+1/T I /s+t d s) K cr = critical gain for oscillations P cr =period of oscilaltions K P =0.6K cr T I =0.5 P cr T d =0.125P cr

25 Figure 8-8 Unit-step response curve of PID-controlled system designed by use of the Ziegler Nichols tuning rule (second method).

26 Figure 8-10 Unit-step response of the system shown in Figure 8 6 with PID controller after numerical tuning of PID parameters.

27 Observations: Can shift red bump toward high freq a bit to increase speed But at higher freq, phase is lower, so PD needs to contribute more DPM PM=48 can be reduced by about 3

28 s=tf('s'); Gp=1/s/(s+1)/(s+5); figure(2); margin(gp); grid; V=axis; Mp=25; PMd = 70 - Mp + 7; %+7 for PI later DPM=85; PM=PMd-DPM; hold on; plot([v(1) V(2)],[PM PM]-180,'r:'); [x, y]=ginput(1); wgcd=x; z_pd = wgcd/tan(dpm*pi/180); %PD control K = 1/abs(evalfr((s+z_PD)*Gp,j*wgcd)); C=K*(s+z_PD); margin(c*gp); grid; %Bode with PD z_pi = wgcd/10; C=C*(s+z_PI)/s; %multiply PI and PD margin(c*gp); grid; hold off; %Bode with PID t=linspace(0, 15/wgcd, 301); figure(3); step(c*gp/(1+c*gp),t); grid;

29 100 Bode Diagram Gm = -Inf db (at 0 rad/sec), Pm = 45.5 deg (at 5.02 rad/sec) 50 Magnitude (db) Phase (deg) Frequency (rad/sec)

30 1.4 Step Response Amplitude Time (sec)

31 Lag and lead-lag Design Steps From plant, draw Bode plot From specs => PM d and ω gcd If there is speed or BW req, è ω gcd, In this case, if PM not enough, design PD or lead Otherwise, choose ω gcd to have PM>PMd Find K to enforce ω gcd : Find K p,v,a-have with K and C above Find K p,v,a-des from ess specs z lag /p lag = K p,v,a-des /K p,v,a-have Let z lag = ω gcd /5~20, depending on PM room Compute p lag K ( ) CG( j 1 = ω gcd ) C = 1 s+ z C = s+ p lead lead

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