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 Type ess requirement wgc td, tr, tp, ts w 0dB w -90 PM=70-Mp -180

Overall Loop shaping strategy Overall gain K to select wgc By shifting gain plot up and down PD or lead to add more PM @ wgc Both PD and lead has positive phase PI or lag to increase low freq gain But both has negative phase, deteriorates PM Low pass filter to reduce high freq gain For noise attenuation Also has negative effect on PM

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; 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);

After design Before design

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

Alternative use of lead Use Lead and Proportional together To fix overshoot and ess No speed requirement Select K so that KG(s) meet ess req. Find wgc and PM, also find PMd Determine phi_max, and alpha 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%

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

n=[1]; d=[1 5 0 0]; 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

Lag Controller Design

Desired effect: low freq gain boost for improved steady state tracking z/p=40 Desired effect: low freq gain boost for improved steady state tracking z/p=20 z/p=10 z=wgcd/5 wgcd z/p=5 Kill PM by 10 to 12 deg

Desired effect: low freq gain boost for improved steady state tracking z/p=40 Desired effect: low freq gain boost for improved steady state tracking z/p=20 z/p=10 z=wgcd/10 wgcd z/p=5 Kill PM by 5 to 7 deg

z/p=40 z/p=20 Want these: DC gain boosting z/p=10 z/p=5 z=wgcd/20 wgcd Don’t want these: PM reduction! Kill PM by 2 to 3 deg

Lag and lead-lag Design Steps From plant, draw Bode plot From specs => PMd and wgcd If there is speed or BW req,  wgcd, In this case, if PM not enough, design PD or lead Otherwise, choose wgcd to have PM>PMd Find K to enforce wgcd: Find Kp,v,a-have with K and C above Find Kp,v,a-des from ess specs zlag/plag = Kp,v,a-des/Kp,v,a-have Let zlag= wgcd/10~20, depending on PM room Compute plag

Lag design example Plant transfer function is given by: Desired design specifications are: Step response overshoot <= 10% Steady state tracking error to ramp input is <=0.01 Note: no speed or BW requirement, so just choose wgcd to have enough PM

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; PMd = 70 – Mp + 3; semilogx(V(1:2), [PMd-180 PMd-180],':r'); %get desired w_gc x=ginput(1); w_gcd = x(1); K = 1/abs(evalfr(tf(n,d),j*w_gcd)); Kva = K*n(end)/d(end-1); ess=0.01; Kvd=1/ess; z = w_gcd/5; p = z/(Kvd/Kva); 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;

Designed for 63 Lag killed 12

Overshoot too much

n=[1]; d=[1/5/50 1/5+1/50 1 0]; figure(1); clf; margin(n,d); hold on; grid; V=axis; Mp = 10; PMd = 70 – Mp + 3; semilogx(V(1:2), [PMd-180 PMd-180],':r'); x=ginput(1); w_gcd = x(1); %get desired w_gc K = 1/abs(polyval(n,j*w_gcd)/polyval(d,j*w_gcd)); Kva = K*n(end)/d(end-1); ess=0.01; Kvd=1/ess; z = w_gcd/10; p = z/(Kvd/Kva); ngc = conv(n, K*[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;

clear all; n=[1]; d=[1/5/50 1/5+1/50 1 0]; figure(1); clf; margin(n,d); grid; V=axis; hold on; Mp = 10; PMd = 70 – Mp + 7; semilogx(V(1:2), [PMd-180 PMd-180],':r'); x=ginput(1); w_gcd = x(1); %get desired w_gc K = 1/abs(polyval(n,j*w_gcd)/polyval(d,j*w_gcd)); Kva = K*n(end)/d(end-1); ess=0.01; Kvd=1/ess; z = w_gcd/10; p = z/(Kvd/Kva); ngc = conv(n, K*[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;

Lead-Lag design example Plant transfer function is given by: Desired design specifications are: Step response overshoot <= 16% Step response rise time <= 2 sec Steady state tracking error to unit acceleration input is <=1 Note: we have all three types of specs: speed, relative stability, and tracking

Strategy First do a lead design to fix speed and overshoot requirement Then do a lag design to fix the ess.

n=[1]; d=[1 5 0 0]; figure(1); clf; margin(n,d); grid; hold on; Mp=16; PMd = 70 – Mp + 1; tr = 2; wn = 1.8/tr; w_gcd = wn*0.8; PM = pi+angle(evalfr(tf(n,d),j*w_gcd)); 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(evalfr(tf(n,d),j*w_gcd))); ngc = conv(n, K*[1 zlead]); dgc = conv(d, [1 plead]); figure(1); margin(ngc,dgc); [ncl,dcl]=feedback(ngc,dgc,1,1); figure(2); step(ncl,dcl); grid; figure(3); margin(ncl*1.414,dcl); grid;

Need to reduce by Mp by 10% So increase PMd by 10 deg

Overshoot is too large. Plus, we know the lag controller will further deteriorate Mp. So, redesign for better Mp.

About 12% overshoot. So, let’s go ahead with lag design.

Kaa = ngc(end)/dgc(end-2); ess=1; Kad=1/ess; zlag = w_gcd/20; plag = zlag/(Kad/Kaa); ngcc = conv(ngc, [1 zlag]); dgcc = conv(dgc, [1 plag]); figure(1); margin(ngcc,dgcc); [ncl,dcl]=feedback(ngcc,dgcc,1,1); figure(4); step(ncl,dcl); grid; figure(5); margin(ncl*1.414,dcl); grid; We don’t have too much room to spare for Mp, so choose 20 so that the lag controller only kills about 3 degrees of PM, which will add about 3 percentage points to Mp.

Lead-Lag design HW Desired design specifications are: C(s) G(s) Kv = 20 PMd=60

figure(3); margin(100, [1 6 5 0]); grid; hold; Low freq Gain will Give Kv=20 w=1 2 3 4

Problem asks for a lag lead controller Suppose lead can make a phi-max of about 60 degrees (1+sin(62*pi/180))/(1-sin(62*pi/180)) =16.1 (1+sin(63*pi/180))/(1-sin(63*pi/180))=17.3 Take alpha =16, rt_alp = 4. Lag will reduce PM a little, so take wgc at a frequency where phase is just a couple deg above -180 So take wgc =2

Then z_lead =2/4; p_lead=2*4; Make z_lag to be 4 times below z_lead To keep K*z_lead*z_lag/p_lag/p_lead unchanged, use p_lag = z_lag/4^2 T=1/2;a=4;margin(100*conv([T*a^2 1],[T*a 1]), conv(conv([T/a 1],[T*a^4 1]),[1 6 5 0]))

But max phase bump is at a freq quite bit lower than wgc Both specs are met! But max phase bump is at a freq quite bit lower than wgc So move all zeros and poles up by a factor of 2: T=1/4;a=4;margin(100*conv([T*a^2 1],[T*a 1]), conv(conv([T/a 1],[T*a^4 1]),[1 6 5 0]))

All specs are still met. The corner frequencies are shifted to higher freq. Phase trough and peak shifted to high freq. Phase looks better. Wgc actually became lower. So, which one is better? Green or red?