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.

Mr and BW are widely used Closed-loop phase resp. rarely used

BW Prototype 2nd order system closed-loop frequency response No resonance for z <= 0.7 Mr=0.3dB for z=0.6 Mr=1.2dB for z=0.5 Mr=2.6dB for z=0.4 For small zeta, resonance freq is about wn BW ranges from 0.5 wn to 1.5 wn For good z range, BW is 0.8~1.25 wn So take BW ≈ wn z=0.1 0.2 0.3 -3dB BW w/wn 0.63 1.58 0.79 1.26

Closed-loop BW to wn ratio BW≈wn z

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 Since we design for z >=0.5, Mr and wr are of less value No resonance for z <= 0.7 Mr<0.5 dB for z=0.6 Mr=1.2 dB for z=0.5 Mr=2.6 dB for z=0.4 w=wn w/wn

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

Mr in dB

Percentage Overshoot in closed-loop step response z > 0.5 is good z

Mr < 15% is good, >40% dB not tolerable Percentage Overshoot in closed-loop step response Mr < 15% is good, >40% dB not tolerable Mr

Mr < 1 dB is good, >3 dB not tolerable Percentage Overshoot in closed-loop step response Mr < 1 dB is good, >3 dB not tolerable Mr in dB

0.2 z=0.1 0.3 0.4 wgc In the range of good zeta, wgc is about 0.7 times wn w/wn

Open-loop wgc to wn ratio wgc≈0.7wn z

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

Phase Margin PM = 100z z

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

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 0.4 0.5 0.6 0.7 PM 44 53 61 67 deg ≈100z Mp 25 16 10 5 % PM+Mp ≈70

Desired Bode plot shape High low freq gain for steady state tracking Low high freq gain for noise attenuation Sufficient PM near wgc for stability wgc w 0dB w -90 -180

Controller design with Bode C(s) Gp(s) From specs: => desired Bode shape of Gol(s) Make Bode plot of Gp(s) Add C(s) to change Bode shape Get closed loop system Run step response, or sinusoidal response

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

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; %overshoot in percentage PMd = 70-Mp + 3; semilogx(V(1:2), [PMd-180 PMd-180],':r'); %get desired w_gc x=ginput(1); w_gcd = x(1); KP = 1/abs(evalfr(tf(n,d),j*w_gcd)); figure(2); margin(KP*n,d); figure(3); mystep(KP*n, d+KP*n);

Lead Controller Design

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!

Lead Design From specs => PMd and wgcd From plant, draw Bode plot Find PMhave = 180 + angle(G(jwgcd) DPM = PMd - PMhave + a few degrees Choose a=plead/zlead so that fmax = DPM and it happens at wgcd

Lead design example Plant transfer function is given by: n=[50000]; d=[1 60 500 0]; Desired design specifications are: Step response overshoot <= 16% Closed-loop system BW>=20;

n=[50000]; d=[1 60 500 0]; 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 + 3; 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);

After design Before design

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

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 * 100 + 10; 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;

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 * 100 + 10; 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;