Desired Bode plot shape dB  gc High low freq gain for steady state tracking Low high freq gain for noise attenuation Sufficient PM near  gc for stability   Low freq ess, type High freq Noise immu Want high gain Want low gain Mid freq Speed, BW Want sufficient Phase margin Use low pass filters Use PI or lag control Use lead or PD control PM+Mp=70 Mr, Mp

Overall Loop shaping strategy Determine mid freq requirements –Speed/bandwidth   gc –Overshoot/resonance  PM d Use PD or lead to achieve PM  gc Use overall gain K to enforce  gc PI or lag to improve steady state tracking –Use PI if type increase neede –Use lag if ess needs to be reduced Use low pass filter to reduce high freq gain

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 = (0.65~0.8)*  n and K P = 1/|G(j  gcd )|

PD Controller

PD control design

PD control design Variation Restricted to using K P = 1 Meet M p requirement Find  gc and PM Find PM d Let  = PMd – PM + (a few degrees) Compute T D = tan(  )/w gcd K P = 1; K D =K P T D

Lead Controller Design

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

Lead Design From specs => PM d and  gcd From plant, draw Bode plot Find PM have = angle(G(j  gcd )  PM = PM d - PM have + a few degrees Choose  =p lead /z lead so that  max =  PM and it happens at  gcd

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;

n=[50000]; d=[ ]; G=tf(n,d); figure(1); margin(G); Mp_d = 16/100; PMd = 70 – Mp_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);

Before design After design

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

Alternative use of lead Use Lead and Proportional together –To fix overshoot and ess –No speed requirement 1.Select K so that KG(s) meet ess req. 2.Find wgc and PM, also find PMd 3.Determine phi_max, and alpha 4.Place phi_max a little higher than wgc

Alternative lead design example Plant transfer function is given by: n=[50]; d=[1/50 1 0]; Desired design specifications are: –Step response overshoot <= 20% –Steady state tracking error for ramp input <= 1/200; –No speed requirements –No settling concern –No bandwidth requirement

n=[50]; d=[1/5 1 0]; figure(1); clf; margin(n,d); grid; hold on; Mp = 20; PMd = 70 – Mp + 7; 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); %phimax located higher p=w_gcd*alpha^.75; %sqrt(alpha); %than wgc 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;

Lead design tuning example C(s)G(s) Design specifications: rise time <=2 sec overshoot <16%

4. Go back and take wgcd = 0.6*wn so that tr is not too small Desired tr < 2 sec We had tr = 1.14 in the previous 4 designs

n=[1]; d=[ ]; G=tf(n,d); Mp_d = 16; %in percentage PMd = 70 - Mp_d + 4; %use Mp + PM =70 tr_d = 2; wnd = 1.8/tr_d; w_gcd = 0.6*wnd; 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]); [ncl,dcl]=feedback(ngc,dgc,1,1); stepchar(ncl,dcl); grid figure(2); margin(G); hold on; margin(ngc,dgc); hold off; grid