View source for Other Controllers

{|border="1" width=100%
! width=20% | Visual Cue
! width=60% | 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: 

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    PI = (s+1)/s
    evans(PI*System)

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    sgrid(zeta, wn)  
    Kpi = -1/real(horner(PI*System, [1 %i]*locate(1)))
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We use the same values of zeta and omega_n from the proportional controller. 

We now check the performance of the PI controller. 
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    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

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    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
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From theory we know, <read from slide> 

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    zeta = sqrt((log(OS))^2/((log(OS))^2 + %pi^2))
    wn = 4/(zeta*Ts)

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Therefore, we design our PD controller: 

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    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)))
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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: 
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    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: 

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    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
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We add a zero at origin to the PD controller. 


We use the same values of zeta and omega_n from the PD system. 


Now we check the performance of the PID 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: 
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    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"])
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END
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