Where is w=0+, where is w=0-, where is w=+inf, where is w=-inf, what is the system type, what is the relative order of the TF, how should you complete the nyquist plot, what are P/N/Z values as in the nyquist criterion, is the closed-loop system stable, what the is the phase margin, by how much can the gain be varied without affecting stability? how many gain cross-over points and how many phase cross-over points are there?

G(s) Open vs Closed Loop Frequency Response And Frequency Domain Specifications C(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.

 = No resonance for  <= 0.7 Mr=1dB for  =0.6 Mr=3dB for  =0.5 Mr=7dB for  =0.4 For small zeta, resonance freq is about  n BW ranges from 0.5wn to 1.5  n For good z range, BW is 0.8 to 1.1  n So take BW =  n Prototype 2 nd order system closed-loop frequency response  n

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

 n z=  gc In the range of good zeta,  gc is about 0.65 times to 0.8 times  n

 n  = In the range of good zeta, PM is about 100* 

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

 gc determines  n and bandwidth –As  gc ↑, 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  = 1: the value gives Kp, Kv, or Ka, depending on type High frequency gain determines noise immunity Important relationships

Desired Bode plot shape

Proportional controller design Obtain open loop Bode plot Convert design specs into Bode plot req. Select K P based on requirements: –For improving ess: K P = K p,v,a,des / K p,v,a,act –For fixing Mp: select  gcd to be the freq at which PM is sufficient, and K P = 1/|G(j  gcd )| –For fixing speed: from td, tr, tp, or ts requirement, find out  n, let  gcd =  n and choose K P 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 = 10/100; 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);

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;

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

C(s)G p (s)