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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.
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Pick pd in region (leave some safety flex)
Compute angle deficiency: a. PD control, choose zpd such that then
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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
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Compute overall gain: If there is no steady-state error requirement, go to 14. With K from 7, evaluate error constant. You already have:
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The 0, 1, 2 should match p, v, a This is for lag control. For PI:
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Compute desired error const. from specs:
For PI : set K*a = K*d & solve for zi For lag : pick zlag & let
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Re-compute K Get closed-loop T.F. Do step response analysis. If not satisfactory, go back to 3 and redesign.
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If we have both PI and PD we have PID control:
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Lead-lag design example
Too much overshoot, too slow & ess to ramp is too large.
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Draw R.L. for G(s) & the desired region
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Clearly R.L. does not pass through desired region. need PD or lead.
Let’s do lead. Pick pd in region
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Now choose zlead & plead.
Could use bisection. Let’s pick zlead to cancel plant pole s + 0.5
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Use our formula to get plead
Now compute K : Now evaluate error constant Kva
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Should re-compute K, but let’s skip:
do step response.
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Op-amp controller circuit:
Proportional:
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Integral:
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Derivative control:
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PD controller:
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PI controller:
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PID controller:
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Lead or lag controller:
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If R1C1 > R2C2 then z < p This is lead controller
If R1C1 < R2C2 then z > p This is lag controller
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Lead-lag controller:
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