(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?
(a)
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)}
\omega_{ c } occurs where \varphi=-180
\begin{aligned}&\omega_{c}=0.152 \quad A R\left(\omega_{c}\right)=0.227 \\&K_{a u}=\frac{1}{A R\left(\omega_{c}\right)}=4.41\end{aligned}
Simulation results with different K c are shown in Fig. S14.19b. K c>K c u, the system becomes unstable as expected.
(b)
Use Skogestad’s half rule
\begin{aligned}&\tau=60+0.5 \times 5=62.5 \\&\theta=2.5+3+2+2=9.5\end{aligned}
The approximated FOPTD model:
G=\frac{3 e^{-9.5 s}}{62.5 s+1}
Using Table 12.3, K_{c}=0.586(9.5 / 62.5)^{-0.916} / 3=1.10;
\tau_{I}=\frac{62.5}{-0.165(9.5 / 62.5)+1.03}=62.19
Then,
\begin{aligned}&G_{c}=1.10\left(1+\frac{1}{62.19 s}\right), G_{O L}=G G_{c} \\&\omega_{c} \text { occurs where } \varphi=-180: \\&\omega_{c}=0.153 \quad A R\left(\omega_{c}\right)=0.249 \\&K_{c u}=\frac{1}{A R\left(\omega_{c}\right)}=4.02\end{aligned}
| Table 12.3 Guidelines for choosing \tau_{c} for the IMC and SIMC tuning methods and an FOPTD model | ||
| Recommendations | Conditions | References |
| \tau_{c}=\left\{\begin{array}{c}0.5 \theta \\\theta \\1.5 \theta\end{array}\right. | aggressive control default value smooth responses | Grimholt and Skogestad (2013) |
| \tau_{c}=3 \theta | robust control | Åström and Hägglund (2006, p. 187); McMillan (2015, p. 29) |
| \left.\begin{array}{l}\tau_{c}>\tau \\\tau_{c}=2-3 \tau\end{array}\right\} | \left.\begin{array}{l}\text{default value} \\\text{robust control}\end{array}\right\} | Blevins et al. (2013, p. 102) |
