What differential equation model represents the parallel PID controller with a derivative filter? (Hint: Find a common denominator for the transfer function first.)

(a) Repeat for the series PID controller with a derivative filter.

(b) Simulate the time response of each controller for a unit step change in $e(t)$ and the following parameter values:

$K_{c}=2, \quad \tau_{I}=3 \min , \quad \tau_{D}=5 \min , \quad \alpha=0.1, \quad M=1$

Question

What differential equation model represents the parallel PID controller with a derivative filter? (Hint: Find a common denominator for the transfer function first.)

(a) Repeat for the series PID controller with a derivative filter.

(b) Simulate the time response of each controller for a unit step change in [latex]e(t)[/latex] and the following parameter values:

[latex]K_{c}=2, \quad au_{I}=3 \min , \quad au_{D}=5 \min , \quad \alpha=0.1, \quad M=1[/latex]

Accepted Answer

a)

[latex]\begin{aligned}\frac{P^{\prime}(s)}{E(s)} &=K_{c}\left(1+\frac{1}{ au_{I} s}+\frac{ au_{D} s}{\alpha au_{D} s+1} ight) \&=K_{c} \frac{\left( au_{I} s\left(\alpha au_{D} s+1 ight)+\alpha au_{D} s+1+ au_{D} s au_{I} s ight)}{ au_{I} s\left(\alpha au_{D} s+1 ight)} \&=K_{c}\left[\frac{1+\left( au_{I}+\alpha au_{D} ight) s+(1+\alpha) au_{I} au_{D} s^{2}}{ au_{I} s\left(\alpha au_{D} s+1 ight)} ight]\end{aligned}[/latex]

Cross- multiplying

[latex]\left(\alpha au_{I} au_{D} s^{2}+ au_{I} s ight) P^{\prime}(s)=K_{c}\left(1+\left( au_{I}+\alpha au_{D} ight) s+(1+\alpha) au_{I} au_{D} s^{2} ight) E(s)[/latex]

Taking inverse Laplace transforms gives,

[latex]\alpha au_{I} au_{D} \frac{d^{2} p^{\prime}(t)}{d t^{2}}+ au_{I} \frac{d p^{\prime}(t)}{d t}=K_{c}\left(e(t)+\left( au_{I}+\alpha au_{D} ight) \frac{d e(t)}{d t}+(1+\alpha) au_{I} au_{D} \frac{d^{2} e(t)}{d t^{2}} ight)[/latex]

b) [latex]\quad \frac{P^{\prime}(s)}{E(s)}=K_{c}\left(\frac{ au_{I} s+1}{ au_{I} s} ight)\left(\frac{ au_{D} s}{\alpha au_{D} s+1} ight)[/latex]

Cross-multiplying

[latex] au_{I} s^{2}\left(\alpha au_{D} s+1 ight) P^{\prime}(s)=K_{c}\left(\left( au_{I} s+1 ight)\left( au_{D} s+1 ight) ight) E(s)[/latex]

[latex]\alpha au_{I} au_{D} \frac{d^{2} p^{\prime}(t)}{d t^{2}}+ au_{I} \frac{d p^{\prime}(t)}{d t}=K_{c}\left(e(t)+\left( au_{I}+ au_{D} ight) \frac{d e(t)}{d t}+ au_{I} au_{D} \frac{d^{2} e(t)}{d t^{2}} ight)[/latex]

c) The simulation is performed for the following parameter values:

[latex]K_{c}=2, \quad au_{I}=3, \quad au_{D}=0.5 \quad, \quad \alpha=0.1 \quad, \quad M=1[/latex]

The Simulink-MATLAB results are shown in Figure S8.10.:

Question 8.10: What differential equation model represents the parallel PID...

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