1. Open vs Closed Loop Frequency Response
And Frequency Domain Specifications
C(s)
G(s)
Goal: 1) Define typical “good” freq resp shape for closed-loop
2) Relate closed-loop freq response shape to step response shape
3) Relate closed-loop freq shape to open-loop freq resp shape
4) Design C(s) to make C(s)G(s) into “good” shape.

2. Prototype 2nd order system closed-loop frequency response
For small zeta,
resonance freq
is about wn
BW ranges from
0.5wn to 1.5 wn
For good z range,
BW is 0.8 to 1.1 wn
So take BW = wn
z=0.1
0.2
0.3
No resonance
for z <= 0.7
Mr=0.5dB for z=0.6
Mr=3dB for z=0.5
Mr=7dB for z=0.4
w/wn

3. Closed-loop BW to wn ratio
BW≈1.4wn z

4. Prototype 2nd order system closed-loop frequency response
z=0.1
When z <=0.5
 visible resonance peak near w=wn
When z >=0.6
no visible resonance peak
0.2
0.3
No resonance
for z <= 0.7
Mr<0.5 dB for z=0.6
Mr=1.2 dB for z=0.5
Mr=2.5 dB for z=0.4
w=wn
w/wn

5. Prototype 2nd order system closed-loop frequency response
Mr vs z

6. Percentage Overshoot in closed-loop step responsez

7. Percentage Overshoot in closed-loop step responseMr

8. Percentage Overshoot in closed-loop step response
Mr in dB

9. Phase Margin
PM = 100z z

10. PM+Mp =70 line
Percentage Overshoot in closed-loop step response
Phase Margin in degrees

11. 0.2
z=0.1
0.3
0.4
wgc
In the range of good zeta,
wgc is about 0.65 times to 0.8 times wn
w/wn

12. Open-loop wgc to wn ratio
wgc≈0.7wn z

13. In the range of good zeta,
PM is about 100*z
z=0.1
0.2
0.3
0.4
w/wn

14. Important relationships
Closed-loop BW are very close to wn
Open-loop gain cross over wgc ≈ (0.65~0.8)* wn,
When z <= 0.6, wr and wn are close
When z >= 0.7, no resonance
z determines phase margin and Mp: z
PM deg ≈100z
Mp %

15. Desired Bode plot shape

16. Proportional controller design
Obtain open loop Bode plot
Convert design specs into Bode plot req.
Select KP based on requirements:
For improving ess: KP = Kp,v,a,des / Kp,v,a,act
For fixing Mp: select wgcd to be the freq at which PM is sufficient, and KP = 1/|G(jwgcd)|
For fixing speed: from td, tr, tp, or ts requirement, find out wn, let wgcd = wn and choose KP as above

17. clear all;
n=[0 0 40]; d=[1 2 0];
figure(1); clf; margin(n,d);
%proportional control design:
figure(1); hold on; grid; V=axis;
Mp = 10/100;
zeta = sqrt((log(Mp))^2/(pi^2+(log(Mp))^2));
PMd = zeta * ;
semilogx(V(1:2), [PMd-180 PMd-180],':r');
%get desired w_gc
x=ginput(1); w_gcd = x(1);
KP = 1/abs(polyval(n,j*w_gcd)/polyval(d,j*w_gcd));
figure(2); margin(KP*n,d);
figure(3); stepchar(KP*n, d+KP*n);

18. PD Controller

19. KP/KD
20*log(KP)
Place wgcd here

21. figure(1); clf; margin(n,d); Mp = 10/100;
zeta = sqrt((log(Mp))^2/(pi^2+(log(Mp))^2));
PMd = zeta * ;
tr = 0.3; w_n=1.8/tr; w_gcd = w_n;
PM = angle(polyval(n,j*w_gcd)/polyval(d,j*w_gcd));
phi = PMd*pi/180-PM; Td = tan(phi)/w_gcd;
KP = 1/abs(polyval(n,j*w_gcd)/polyval(d,j*w_gcd));
KP = KP/sqrt(1+Td^2*w_gcd^2); KD=KP*Td;
ngc = conv(n, [KD KP]);
figure(2); margin(ngc,d);
figure(3); stepchar(ngc, d+ngc);
Could be a little less

22. Variation Compute TD = tan(f)/wgcd KP = 1; KD=KPTD
Restricted to using KP = 1
Meet Mp requirement
Find wgc and PM
Find PMd
Let f = PMd – PM + (a few degrees)
Compute TD = tan(f)/wgcd
KP = 1; KD=KPTD

23. n=[0 0 5]; d=[ ];
figure(1); clf; margin(n,d);
Mp = 10/100;
zeta = sqrt((log(Mp))^2/(pi^2+(log(Mp))^2));
PMd = zeta * ;
[GM,PM,wgc,wpc]=margin(n,d);
phi = (PMd-PM)*pi/180; Td = tan(phi)/wgc;
Kp=1; Kd=Kp*Td;
ngc = conv(n, [Kd Kp]);
figure(2); margin(ngc,d);
figure(3); stepchar(ngc, d+ngc);

29. Lead Controller Design

30. Goal: select z and p so that max phase lead is at desired wgc
plead
zlead
20log(Kzlead/plead)
Goal: select z and p so that max phase lead is at desired wgc
and max phase lead = PM defficiency!

32. Lead Design From specs => PMd and wgcd From plant, draw Bode plot
Find PMhave = angle(G(jwgcd)
DPM = PMd - PMhave + a few degrees
Choose a=plead/zlead so that fmax = DPM and it happens at wgcd

33. Lead design example Plant transfer function is given by:
n=[50000]; d=[ ];
Desired design specifications are:
Step response overshoot <= 16%
Closed-loop system BW>=20;

34. n=[50000]; d=[ ]; G=tf(n,d);
figure(1); margin(G);
Mp_d = 16/100;
zeta_d =0.5; % or calculate from Mp_d
PMd = 100*zeta_d ;
BW_d=20;
w_gcd = BW_d*0.7; Gwgc=evalfr(G, j*w_gcd);
PM = pi+angle(Gwgc);
phimax= PMd*pi/180-PM;
alpha=(1+sin(phimax))/(1-sin(phimax));
zlead= w_gcd/sqrt(alpha);
plead=w_gcd*sqrt(alpha);
K=sqrt(alpha)/abs(Gwgc);
ngc = conv(n, K*[1 zlead]);
dgc = conv(d, [1 plead]);
figure(1); hold on; margin(ngc,dgc); hold off;
[ncl,dcl]=feedback(ngc,dgc,1,1);
figure(2); step(ncl,dcl);

35. After design
Before design

36. Closed-loop Bode plot by:
margin(ncl*1.414,dcl);
Magnitude plot shifted up 3dB
So, gc is BW

38. n=[50]; d=[1/5 1 0];
figure(1); clf; margin(n,d); grid; hold on;
Mp = 20/100; zeta = sqrt((log(Mp))^2/(pi^2+(log(Mp))^2));
PMd = zeta * ;
ess2ramp= 1/200; Kvd=1/ess2ramp;
Kva = n(end)/d(end-1); Kzp = Kvd/Kva;
figure(2); margin(Kzp*n,d); grid;
[GM,PM,wpc,wgc]=margin(Kzp*n,d);
w_gcd=wgc; phimax = (PMd-PM)*pi/180;
alpha=(1+sin(phimax))/(1-sin(phimax));
z=w_gcd/sqrt(alpha);
p=w_gcd*sqrt(alpha);
ngc = conv(n, alpha*Kzp*[1 z]); dgc = conv(d, [1 p]);
figure(3); margin(tf(ngc,dgc)); grid;
[ncl,dcl]=feedback(ngc,dgc,1,1);
figure(4); step(ncl,dcl); grid;
figure(5); margin(ncl*1.414,dcl); grid;

45. n=[50]; d=[1/5 1 0];
figure(1); clf; margin(n,d); grid; hold on;
Mp = 20/100; zeta = sqrt((log(Mp))^2/(pi^2+(log(Mp))^2));
PMd = zeta * ;
ess2ramp= 1/200; Kvd=1/ess2ramp;
Kva = n(end)/d(end-1); Kzp = Kvd/Kva;
figure(2); margin(Kzp*n,d); grid;
[GM,PM,wpc,wgc]=margin(Kzp*n,d);
w_gcd=wgc; phimax = (PMd-PM)*pi/180;
alpha=(1+sin(phimax))/(1-sin(phimax));
z=w_gcd/alpha^.25; %sqrt(alpha);
p=w_gcd*alpha^.75; %sqrt(alpha);
ngc = conv(n, alpha*Kzp*[1 z]); dgc = conv(d, [1 p]);
figure(3); margin(tf(ngc,dgc)); grid;
[ncl,dcl]=feedback(ngc,dgc,1,1);
figure(4); step(ncl,dcl); grid;
figure(5); margin(ncl*1.414,dcl); grid;