Question 3 of 8: IMC Design for a First-Order-Plus-Dead-Time Process

Problem 3: IMC Design for a First-Order-Plus-Dead-Time Process (20%)

Question text not reproduced: the examination questions are © Engineers and Geoscientists BC. Open the official past paper (linked at the top of this page) to read the question, then follow the worked solution below.

Given. A FOPDT model and its IMC filter:

Find. (1) $G_c^{*}$, the classical equivalent $G_c$, and whether it is PID; (2) $\delta C(t)$ for a unit set-point step.

Approach. Factor $G$ into its invertible and non-invertible (dead-time) parts, invert only the invertible part and append the filter to get $G_c^{*}$, convert to the classical loop via $G_c=G_c^{*}/(1-G_c^{*}G)$, and use the perfect-model identity $C/R=G_c^{*}G$ for the response.

1. Factor the model. Split $G=G_-G_+$ into invertible $G_-=\dfrac{5}{10s+1}$ and all-pass/dead-time $G_+=e^{-2s}$ (a delay cannot be inverted causally).
2. (1) IMC controller. Invert $G_-$ and add a first-order filter $f=\dfrac{1}{\tau_Cs+1}$: $$\boxed{G_c^{*}=G_-^{-1}f=\frac{10s+1}{5}\cdot\frac{1}{20s+1}=\frac{10s+1}{5(20s+1)}.}$$
3. (1) Classical equivalent. Using $G_c^{*}G=\dfrac{10s+1}{5(20s+1)}\cdot\dfrac{5e^{-2s}}{10s+1}=\dfrac{e^{-2s}}{20s+1}$, $$G_c=\frac{G_c^{*}}{1-G_c^{*}G}=\frac{\dfrac{10s+1}{5(20s+1)}}{1-\dfrac{e^{-2s}}{20s+1}}=\boxed{\frac{10s+1}{5\,(20s+1-e^{-2s})}.}$$
4. Is $G_c$ PID? No. The $e^{-2s}$ in the denominator makes $G_c$ a transcendental (dead-time-compensating, Smith-predictor-like) controller, not a rational $K_c(1+1/\tau_Is+\tau_Ds)$. Only if the delay were replaced by a Padé rational approximation would $G_c$ reduce to (approximately) PID form.
5. (2) Perfect-model set-point response. With $\tilde G=G$ the IMC closed loop gives $\dfrac{C}{R}=G_c^{*}G=\dfrac{e^{-2s}}{20s+1}$. For a unit step $R=1/s$: $$\delta C(s)=\frac{e^{-2s}}{s(20s+1)}\ \Rightarrow\ \boxed{\delta C(t)=\Big(1-e^{-(t-2)/20}\Big)\,\mathbf 1(t\ge 2).}$$ The controlled variable stays at zero for the $2$ s of dead time, then rises as a first-order curve with time constant $\tau_C=20$ s toward $1$ — perfect (offset-free) set-point tracking, delayed only by the unavoidable dead time.