A process model for a liquid storage system is given by Chien and Fruehauf (1990): G̃(s) = Ke^−7.4s/s Use Table 12.1 to calculate PI and PID controller settings for K = 0.2 and τc = 8. Repeat for τc = 15 and do the following:(a) Compare the four controllers for unit step changes in the set point and

Question

A process model for a liquid storage system is given by Chien and Fruehauf (1990):
[latex] ilde{G}(s)=\frac{Ke^{-7.4s}}{s} [/latex]
Use Table below to calculate PI and PID controller settings for K = 0.2 and [latex] au_{c} = 8[/latex]. Repeat for [latex] au_{c} = 15[/latex] and do the following:
(a) Compare the four controllers for unit step changes in the set point and disturbance, assuming that [latex]G_{d} = ilde{G}[/latex].
(b) In order to characterize the robustness of each controller of part (a), determine [latex]K_{\max}[/latex], the largest value of K that results in a stable closed-loop system for each controller.

Accepted Answer

(a) For this integrating process, [latex] ilde{G}_{+}=e^{- heta s}[/latex], and thus C = −θ in Equation
[latex]f=\frac{(2 au_{c}-C)s+1}{( au_{c}s+1)^{2}} \quad where \quad C=\frac{d ilde{G}_{+}}{ds}\mid _{s=0} [/latex].
The IMC controller settings for controllers M and N in Table above are

	K_c	τ_I	τ_D
PI (τ_c = 8)	0.493	23.4	–
PI (τ_c = 15)	0.373	37.4	–
PID (τ_c= 8)	0.857	23.4	3.12
PID (τ_c = 15)	0.535	37.4	3.33

The closed-loop responses in Figure on the right are more sluggish and less oscillatory for [latex] au_{c} = 15[/latex] than they are for [latex] au_{c} = 8[/latex]. Also, for [latex] au_{c} = 15[/latex] the overshoot is smaller for the set-point change, and the maximum deviation is larger after the disturbance. The PID controller provides a better disturbance response than the PI controller with a smaller maximum deviation. In addition, the PID controller has a very short settling time for [latex] au_{c} = 8[/latex], which gives it the best performance of the four controllers considered.

(b) The numerical value of [latex]K_{\max} [/latex] can be obtained from a stability analysis using the Direct Substitution Method of Chapter 11.
The numerical results shown in the following table indicate that K can increase significantly from its nominal value of 0.2 before the closed-loop system becomes unstable. Thus, these IMC controllers are quite robust and become even more so as [latex] au_{c}[/latex] increases. The approximate values of [latex]K_{\max} [/latex] were obtained by using the time-delay approximation in Equation
[latex]e^{- heta s}\cong 1- heta s[/latex].

		K_max
Controller	τ_c	Approximate	Exact
PI	8	0.274	0.356
PI	15	0.363	0.515
PID	8	0.376	0.277
PID	15	0.561	0.425

Question 12.3: A process model for a liquid storage system is given by Chie...

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