Types of classical controllers Proportional control Needed to make a specific point on RL to be closed-loop system dominant pole Proportional plus derivative control (PD control) Needed to “bend” R.L. into the desired region Lead control Similar to PD, but without the high frequency noise problem; max angle contribution limited to < 75 deg Proportional plus Integral Control (PI control) Needed to “eliminate” a non-zero steady state tracking error Lag control Needed to reduce a non-zero steady state error, no type increase PID control When both PD and PI are needed, PID = PD * PI Lead-Lag control When both lead and lag are needed, lead-lag = lead * lag

Proportional control design Draw R.L. for given plant Draw desired region for poles from specs Pick a point on R.L. and in desired region Use ginput to get point and convert to complex # Compute Obtain closed-loop TF Obtain step response and compute specs Decide if modification is needed nump=…; denp= …; sysp=tf(nump, denp); rlocus(sysp); [x,y]=ginput(1); pd=x+j*y; Gpd=evalfr(sysp,pd); K=1/Gpd; sysc = K; syscl = feedback(sysc*sysp,1); use your program from several weeks ago to do all these

PD controller design Design steps: From specs, draw desired region for pole. Pick from region, not on RL Compute Select Select: [x,y]=ginput(1); pd=x+j*y; Gpd=evalfr(sysp,pd) phi=pi - angle(Gpd) z=abs(real(pd))+abs(imag(pd)/tan(phi)); Kd=1/abs(pd+z)/abs(Gpd); Kp=z*Kd;

From specs draw region for desired c.l. poles Select pd from region Approximation to PD Same usefulness as PD Lead Control: Draw R.L. for G From specs draw region for desired c.l. poles Select pd from region Let Pick –z somewhere below pd on –Re axis Let Select [x,y]=ginput(1); pd=x+j*y; Gpd=evalfr(sysp,pd) phi=pi - angle(Gpd) [x,y]=ginput(1); z=abs(x); phi1=angle(pd+z); phi2=phi1-phi; p=abs(real(pd))+abs(imag(pd)/tan(phi2)); K=abs((pd+p)/(pd+z)/Gpd); sysc=tf(K*[1 z],[1 p]); Hold on; rlocus(sysc*sysp);

Alternative Lead Control Draw R.L. for G From specs draw region for desired c.l. poles Select pd from region Let Select phipd=angle(pd); phi1=(phipd+phi)/2; phi2=phi1-phi;

Lag Design steps Draw R.L. for G(s). From specs, draw desired pole region Select pd on R.L. & in region Get With that K, compute error constant (Kpa, Kva, Kaa) from KG(s) From specs, compute Kpd, Kvd, Kad sysol = sysc*sysp; [nol, dol]=tfdata(sysol,'v'); dn0=dol(dol~=0); Kact=nol(end)/dn0(end); Kdes = 1/ess;

If K#a > K#d , done else: pick Re-compute Closed-loop simulation & tuning as necessary z=-real(pd)/…; p=z*Kact/Kdes/(1+…); 0.05 or 0.1

PI Design steps First design: design PD for G(s)/s Second design: Draw R.L. for G(s) From specs, draw desired region Pick pd on R.L. & in region i. Choose ii. Choose Simulate & tune

Alternative PI design Since PI = PD/s, Can first multiply system by 1/s Then design using PD, lead, lag The overall controller is the controller you designed divided by s

Example: Want: Solution: C(s)

Draw R.L. Pick pd on R.L. & in Region pick pd = – 0.35 + j0.5 Since there is one in G(s)

New root locus: RL going north-east,  reduce K will increase s and and increase z

Use original K=0.86 instead of 0.914; ess should be OK Mp reduced by not enough.

Change that divide # from 5 to 15. ans = yss=1; ess=0; tr=2.78; td=2.78; ts=10.6; tp=6.94; Mp=15.8

Lag control can improve ess, but cannot eliminate ess Use PI control to eliminate ess PI :

Overall controller design R(s) E(s) C(s) U(s) Gp(s) Y(s) Draw R.L. for G(s), hold graph Draw desired region for closed-loop poles based on desired specs If R.L. goes through region, pick pd on R.L. and in region. Go to step 7.

Pick pd in region (near corner but inside region for safety margin) Compute angle deficiency: a. PD control, choose zpd such that then

b. Lead control: choose zlead, plead such that You can select zlead & compute plead. Or you can use the “bisection” method to compute z and p. Then

Compute overall gain: If there is no steady-state error requirement, go to 14. With K from 7, evaluate error constant that you already have:

The 0, 1, 2 should match p, v, a This is for lag control. For PI:

Compute desired error const. from specs: For PI : set K*a = K*d & solve for zpi For lag : pick zlag & let

Re-compute K Get closed-loop T.F. Do step response analysis. If not satisfactory, go back to 3 and redesign.

If we have both PI and PD we have PID control:

If we have both Lead and Lag, we have lead-lag control:

Control System Implementation R(s) E(s) C(s) U(s) Gp(s) Y(s) disturbance input Reference Command output error Controller control Actuator Plant + _ plant input Sensor noise

PC-in-the-loop Control disturbance input Reference Command PC I/O output D/A power Amp Actuator Plant Signal Conditioner and amplifier A/D Sensor All control algorithms implemented in PC (could be Matlab Real-Time) Needs data acquisition system, including A/D and D/A Needs power amplifier

m-Controller based control disturbance input Reference Command m-Controller I/O output power Amp Actuator Plant Signal Conditioner and amplifier Sensor Very similar architecture to PC-in-the-loop control All control algorithms implemented in m-controller m-controller has its own A/D and D/A, but make sure resolution is OK Still needs power amplifier, because m-controller outputs weak signal

Power electronic based control disturbance input Reference Command Difference amplifier output Op Amp circuit Actuator Plant Sensor Analog operation, fast Inexpensive All algorithms in circuit hardware No sampling and aliasing issues