Other Controllers
| Visual Cue | Narration |
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In order to eliminate the steady state error, we add an integrator to the proportional controller That is to say, we add a pole at origin. In order to have the root locus pass through the same point as before so as to maintain the same transient performance, we add a zero near the origin: | |
PI = (s+1)/s evans(PI*System) |
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sgrid(zeta, wn) Kpi = -1/real(horner(PI*System, [1 %i]*locate(1))) |
We use the same values of zeta and omega_n from the proportional controller. We now check the performance of the PI controller. |
PISystem = Kpi*PI*System/. 1
yPI = csim('step', t, PISystem);
plot(t, yPI)
plot(t, ones(t), 'r'), // Compare with step
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We can also compare this response with the Proportional controller |
plot(t, [ones(t); y; yPI; yProp]) |
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Now, let us try to design a PD controller that has better transient performance than the plain Proportional controller. Let us aim for the following parameters: Overshoot of 0.05 and Settling time of 0.5 seconds. | |
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Show first slide of PD controller: OS = 0.05 Ts = 0.5 |
From theory we know, <read from slide> |
zeta = sqrt((log(OS))^2/((log(OS))^2 + %pi^2)) wn = 4/(zeta*Ts) |
Therefore, we design our PD controller: |
PD = s + 20 evans(PD*System) sgrid(zeta, wn) // Zoom first, then execute next line: Kpd = -1/real(horner(PD*System, [1 %i]*locate(1))) |
You might need to go through a few trials to find the location of the pole required so that the modified root locus passes through the point of intersection of the curves corresponding to zeta and omega_n. We do not wish to waste our time here- let me tell you that -20 is an appropriate location. We check the performance of the controller: |
PDSystem = Kpd*PD*System/. 1
yPD = csim('step', t, PDSystem);
plot(t, yPD)
plot(t, ones(t), 'r'), // Compare with step
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We note that the transient performance has improved significantly, but the steady state performance has degraded somewhat. |
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Let us now try to design a PID controller by removing the steady state error from the PD controlled system, in a manner similar to how we designed a PI controller: | |
PID = (s + 20)*(s+1)/s evans(PID*System) sgrid(zeta, wn) Kpid = -1/real(horner(PID*System, [1 %i]*locate(1))) PIDSystem = Kpid*PID*System/. 1 |
We add a zero at origin to the PD controller.
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yPID = csim('step', t, PIDSystem);
plot(t, yPID)
plot(t, ones(t), 'r')
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We note the significantly improved steady state as well as transient response. |
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As a final step, let us compare the performance of all the controllers we have studied so far: | |
plot(t, [ones(t); y; yFeedback; yProp; yPI; yPD; yPID]) legend(["Step"; "Open Loop TF"; "Closed Loop TF"; "Proportional Controller"; "PI Controller"; "PD Controller"; "PID Controller"]) |
END |
