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

Pick pd in region (leave some safety flex) Compute angle deficiency: a. PD control, choose zpd such that then

b. Lead control: choose zlead, plead such that 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. 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 zi 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:

Lead-lag design example Too much overshoot, too slow & ess to ramp is too large.

Draw R.L. for G(s) & the desired region

Clearly R.L. does not pass through desired region. need PD or lead. Let’s do lead. Pick pd in region

Now choose zlead & plead. Could use bisection. Let’s pick zlead to cancel plant pole s + 0.5

Use our formula to get plead Now compute K : Now evaluate error constant Kva

Should re-compute K, but let’s skip: do step response.

Op-amp controller circuit: Proportional:

Integral:

Derivative control:

PD controller:

PI controller:

PID controller:

Lead or lag controller:

If R1C1 > R2C2 then z < p This is lead controller If R1C1 < R2C2 then z > p This is lag controller

Lead-lag controller: