(a) Using the process, sensor, and valve transfer functions in Exercise 11.21, find the ultimate controller gain $K_{c u}$ using a Bode plot. Using simulation, verify that values of $K_{c}>K_{c u}$ cause instability.

(b) Next fit a FOPTD model to $G$ and tune a PI controller for a set-point change. What is the gain margin for the controller?

Question

(a) Using the process, sensor, and valve transfer functions in Exercise 11.21, find the ultimate controller gain $K_{c u}$ using a Bode plot. Using simulation, verify that values of $K_{c}>K_{c u}$ cause instability.

(b) Next fit a FOPTD model to $G$ and tune a PI controller for a set-point change. What is the gain margin for the controller?

(b) Next fit a FOPTD model to [latex]G[/latex] and tune a PI controller for a set-point change. What is the gain margin for the controller?

Accepted Answer

(a)

[latex]G=G_{p} G_{v} G_{m}=\frac{2 e^{-1.5 s}}{(60 s+1)(5 s+1)} \frac{0.5 e^{-0.3 s}}{3 s+1} \frac{3 e^{-0.2 s}}{2 s+1}=\frac{3 e^{-2 s}}{(60 s+1)(5 s+1)(3 s+1)(2 s+1)}[/latex]

[latex]\omega_{ c }[/latex] occurs where [latex]\varphi=-180[/latex]

[latex]\begin{aligned}&\omega_{c}=0.152 \quad A R\left(\omega_{c} ight)=0.227 \&K_{a u}=\frac{1}{A R\left(\omega_{c} ight)}=4.41\end{aligned}[/latex]

Simulation results with different [latex]K c[/latex] are shown in Fig. S14.19b. [latex]K c>K c u[/latex], the system becomes unstable as expected.

(b)

Use Skogestad's half rule

[latex]\begin{aligned}& au=60+0.5 imes 5=62.5 \& heta=2.5+3+2+2=9.5\end{aligned}[/latex]

The approximated FOPTD model:

[latex]G=\frac{3 e^{-9.5 s}}{62.5 s+1}[/latex]

Using Table 12.3, [latex]K_{c}=0.586(9.5 / 62.5)^{-0.916} / 3=1.10[/latex];

[latex] au_{I}=\frac{62.5}{-0.165(9.5 / 62.5)+1.03}=62.19[/latex]

Then,

[latex]\begin{aligned}&G_{c}=1.10\left(1+\frac{1}{62.19 s} ight), G_{O L}=G G_{c} \&\omega_{c} ext { occurs where } \varphi=-180: \&\omega_{c}=0.153 \quad A R\left(\omega_{c} ight)=0.249 \&K_{c u}=\frac{1}{A R\left(\omega_{c} ight)}=4.02\end{aligned}[/latex]

Table 12.3 Guidelines for choosing [latex] au_{c}[/latex] for the IMC and SIMC tuning methods and an FOPTD model
Recommendations	Conditions	References
[latex] au_{c}=\left\{\begin{array}{c}0.5 heta \ heta \1.5 heta\end{array} ight.[/latex]	aggressive control default value smooth responses	Grimholt and Skogestad (2013)
[latex] au_{c}=3 heta[/latex]	robust control	Åström and Hägglund (2006, p. 187); McMillan (2015, p. 29)
[latex]\left.\begin{array}{l} au_{c}> au \ au_{c}=2-3 au\end{array} ight\}[/latex]	[latex]\left.\begin{array}{l} ext{default value} \ ext{robust control}\end{array} ight\}[/latex]	Blevins et al. (2013, p. 102)

Question 14.19: (a) Using the process, sensor, and valve transfer functions ...

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