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 Want high gain Use PI or lag control wgc w High freq 0dB Low freq Mid freq Use low pass filters Use lead or PD control Want low gain w Want sufficient Phase margin -90 -180 PM+Mp=100

Bode-based Control design C(s) Gp(s) Use series controller Specs: => desired Bode shape of C(s)Gp(s) Start with Bode plot of Gp(s) Use poles/zero of C(s) to change Bode shape

Overall Loop shaping strategy Determine mid freq requirements Speed/bandwidth  wgc Overshoot/resonance  PMd Use PD or lead to achieve PMd@ wgc Use overall gain K to enforce wgc 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 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 = (0.65~0.8)*wn and KP = 1/|G(jwgcd)|

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

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 from KG(s), find PMd from specs Let phi_max = PMd – PM +5~7, find alpha Place phi_max a little higher than wgc

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 Let zlag/plag = Kp,v,a-des/Kp,v,a-have Let zlag= wgcd/10~20, depending on PM room Compute plag

PD control design

PD control design Variation Restricted to using KP = 1 Meet Mp requirement Find wgc and PM Find PMd Let f = PMd – PM + (a few degrees) Compute TD = tan(f)/wgcd KP = 1; KD=KPTD

Basic PI 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: Let KP = K And KI = Kwgcd/10~20, depending on extra PM room to spare Need to increase type to make a nonzero ess to be zero. But no requirement on ess after type increase.

PI Design with ess specs From plant, draw Bode plot From specs => Kv,a-des, PMd and wgcd For required ess, Kv,a-des =1/ess With C(s)=1/s, compute Kv,a-have 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: Let KP = K, KIdes= Kv,a-des/Kv,a-have If KIdes <= Kwgcd/5~20, done, let KI = KIdes Else, increase wgcd and go back to previous step Need to increase type by 1 to make a nonzero ess to be zero, and after type increase, there is further requirement on ess.

PI Design with PD Design Steps From required ess, Kv,a-des =1/ess With C(s)=1/s, compute Kv,a-have Let KI = Kv,a-des/Kv,a-have Multiply G(s) by KI/s Do a PD design for KIG(s)/s, with DC gain=1: Find wgc and PM Find PMd Let f = PMd – PM + (a few degrees) Compute TD = tan(f)/wgcd KP = KI*TD

Alternative PI Design Steps For required ess, Kv,a-des =1/ess With C(s)=1/s, compute Kv,a-have Let KI = Kv,a-des/Kv,a-have Rewrite char eq: (KP + KI/s)G(s) + 1=0 KP*n/d + KI*n/d/s +1 = 0 KP *n*s + KI*n+d*s =0, KP*n*s/(KI*n+d*s) + 1 =0 So do a KP design for n*s/(KI*n+d*s), with KI above Draw Bode plot for n*s/(KI*n+d*s) Select max PM frequency Compute KP to make that frequency wgc

Two d.o.f. Control design C(s) Gp(s) C1(s) Gp(s) C2(s) + r + e C1(s) Gp(s) y + _ _ C2(s) C1(s) = C(s) – C2(s)

Loop transfer function is still C(s)Gp(s) TF from d to y: Gp(s)/(1+C(s)Gp(s)) TF from r to y: C1(s)Gp(s)/(1+C1(s)Gp(s) +C2(s)Gp(s))=C1(s)Gp(s)/(1+C(s)Gp(s)) Design C(s) to achieve desired loop shape Split C(s) into C1 and C2 Dual loop implementation to individually control disturbance TF and reference TF

Dual loop control to increase type Recall: type with respect to r = #integrators in TF from e to where r enters This = #integrator in C1 + #integrator in inner loop Therefore, choose C2 so that #integrators in inner loop is more than #integrators in Gp(s) Overall loop TF not affected Input/output TF Poles not affected Input/output TF Zeros affected Steady state tracking improved

Inner loop TF Not easy to see in this general form But simply pick C2 to cancel one or two lowest order terms in Gp See example

y _ C2(s) C2 = -2s Implement as: y _ -2

Example Design problem from text book But our solution is much more meaningful and much easier Plant TF Gp(s) = 10/s(s+1) Specs: Mp <19%, but >2% Ts <1 sec Ess to step, ramp, acc all = 0

s=tf('s'); Gp=10/s/(s+1); figure; margin(Gp); grid; ts = 1; %specification t=linspace(0,3*ts,301); Gcl=Gp/(1+Gp); figure; step(Gcl,t); grid; %specification for Mp is 2% to 19% %can use the mid point as initial target Mp=10; zeta=0.6; %for Mp=10% sigma=4/ts; %no tolerance band is given, so use 2% wn=sigma/zeta; wgcd=0.7*wn; Gwgc = evalfr(Gp, j*wgcd); PM=angle(Gwgc)/pi*180 + 180;

PMd = 70 - Mp + 10; %add 10 deg because we need PI later DPM = PMd - PM; %phase margin deficiency z_PD = wgcd /tan(DPM*pi/180); %PD control K = 1/abs(evalfr((s+z_PD)*Gp,j*wgcd)); C=K*(s+z_PD); figure; margin(C*Gp); grid; %Bode with PD figure; step(C*Gp/(1+C*Gp),t); grid; %now add PI, place zero of PI at wgcd/10 %Mp target of 10% together with 10 deg extra PMd %can tolerate a little more phase delay from PI. z_PI = wgcd/10; C=C*(s+z_PI)/s; %multiply PI and PD figure; margin(C*Gp); grid;

Plant Bode plot

Original step response

Bode plot after PD

Step response with initial PD

Step response after PD Ts is two large, need to increase wgc by 1.5X

wgc=0.7*wn*1.6; z_PI = wgc/20;

Wgc increased from 4.47 to 7.47 After PD

Ts is now about right After PD

After PID

After PID Ts is OK Mp is OK

PD +0.1s y PI _ _ -0.1