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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.
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Mr and BW are widely used
Closed-loop phase resp. rarely used
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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
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Closed-loop BW to wn ratio
BW≈wn z
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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
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Prototype 2nd order system closed-loop frequency response
Mr vs z
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Mr in dB
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Percentage Overshoot in closed-loop step response
z > 0.5 is good z
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Mr < 15% is good, >40% dB not tolerable
Percentage Overshoot in closed-loop step response Mr < 15% is good, >40% dB not tolerable Mr
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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
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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
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Open-loop wgc to wn ratio
wgc≈0.7wn z
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In the range of good zeta,
PM is about 100*z z=0.1 0.2 0.3 0.4 w/wn
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Phase Margin PM = 100z z
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PM+Mp =70 line Percentage Overshoot in closed-loop step response Phase Margin in degrees
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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 % PM+Mp ≈70
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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
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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
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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
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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);
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Lead Controller Design
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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!
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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
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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;
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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);
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After design Before design
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Closed-loop Bode plot by:
margin(ncl*1.414,dcl); Magnitude plot shifted up 3dB So, gc is BW
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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;
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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;
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