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.

3. 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=1dB for z=0.6
Mr=3dB for z=0.5
Mr=7dB for z=0.4
w/wn

4. 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

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

6. Important relationships
Prototype wn, open-loop wgc, closed-loop BW are all very close to each other
When there is visible resonance peak, it is located near or just below wn,
This happens when z <= 0.6
When z >= 0.7, no resonance
z determines phase margin and Mp: z
PM deg ≈100z
Mp %

7. Important relationships
wgc determines wn and bandwidth
As wgc ↑, ts, td, tr, tp, etc ↓
Low frequency gain determines steady state tracking:
L.F. magnitude plot slope/(-20dB/dec) = type
L.F. asymptotic line evaluated at w = 1: the value gives Kp, Kv, or Ka, depending on type
High frequency gain determines noise immunity

8. Desired Bode plot shape

9. 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

12. 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);

15. n=[1]; d=[1/5/50 1/5+1/50 1 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));
Kv = Kp*n(1)/d(3); ess=0.01; Kvd=1/ess;
z = w_gcd/5; p = z/(Kvd/Kv);
ngc = conv(n, Kp*[1 z]); dgc = conv(d, [1 p]);
figure(1); hold on; margin(ngc,dgc);
[ncl,dcl]=feedback(ngc,dgc,1,1);
figure(2); step(ncl,dcl); grid;
figure(3); margin(ncl*1.414,dcl); grid;

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

25. PD Control C(s)=KP + KDs = KP(1+TDs) For fixing wgcd and PMd
Compute wgcd from tr, td, etc
Compute PMd from Mp
Compute f = PMd –
Compute TD = tan(f)/wgcd
KP = 1/sqrt(1+Td2wgcd2)/abs(G(jwgcd))
KD=KPTD

26. Want: maximum overshoot <= 10% rise time <= 0.3 sec
Example
C(s)
G(s)
Want: maximum overshoot <= 10%
rise time <= 0.3 sec

29. 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

31. Less than spec

32. 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

35. 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);

38. 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);