For the process in Exercise 14.23, the measurement is to be filtered using a noise filter with transfer function GF(s) = 1/(0.1s + 1). Would you expect this change to result in better or worse control system performance? Compare the ultimate gains and critical frequencies with and without the

Question

For the process in Exercise 14.23, the measurement is to be filtered using a noise filter with transfer function [latex]G_{F}(s)=1 /(0.1 s+1)[/latex]. Would you expect this change to result in better or worse control system performance? Compare the ultimate gains and critical frequencies with and without the filter. Justify your answer.

Accepted Answer

[latex]K_{c u}[/latex] and [latex]\omega_{c}[/latex] are obtained using Eqs. [latex]14-56[/latex] and 14-57. Including the filter [latex]G_{F}[/latex] into these equations gives

[latex]\begin{aligned}A R_{O L}\left(\omega_{c} ight) &=\left|G_{O L}\left(j \omega_{c} ight) ight|=1 (14-56)\\phi_{O L}\left(\omega_{c} ight) &=\angle G_{O L}\left(j \omega_{c} ight)=-180^{\circ} (14-57)\end{aligned}[/latex]

[latex]-180^{\circ}=0+\left[-0.2 \omega_{c}- an ^{-1}\left(\omega_{c} ight) ight]+\left[- an ^{-1}\left( au_{F} \omega_{c} ight) ight][/latex]

Solving,

[latex]\begin{array}{lll}\omega_{c}=8.443 & ext { for } & au_{F}=0 \\omega_{ c }=5.985 & ext { for } & au_{F}=0.1\end{array}[/latex]

Then, from Eq. 14-57,

[latex]1=\left(K_{c u} ight)\left(\frac{2}{\sqrt{\omega_{c}^{2}+1}} ight)\left(\frac{1}{\sqrt{ au_{F}^{2} \omega_{c}^{2}+1}} ight)[/latex]

Solving for [latex]K_{c u}[/latex] gives,

[latex]\begin{array}{lll}K_{c u}=4.251 & ext { for } & au_{F}=0 \K_{c u}=3.536 & ext { for } & au_{F}=0.1\end{array}[/latex]

Therefore,

[latex]\begin{array}{lll}\omega_{c} K_{c u}=35.9 & ext { for } & au_{F}=0 \\omega_{c} K_{c u}=21.2 & ext { for } & au_{F}=0.1\end{array}[/latex]

Since [latex]\omega_{c} K_{c u}[/latex] is lower for [latex] au_{F}=0.1[/latex], filtering the measurement results in worse control performance.

Question 14.26: For the process in Exercise 14.23, the measurement is to be ...

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