Consider an FOPTD model: G̃(s) = 100 / 100s + 1 e^−s This model is lag-dominant because θ/τ = 0.01. Specify three PI controllers using the following methods: (a) IMC (τc = 1) (b) IMC (τc = 2) and the integrator approximation method (Eq. 12-33) (c) IMC (τc = 1) and SIMC (Eq. 12-34) Evaluate the three

Question

Consider an FOPTD model:
[latex] ilde{G}(s)=\frac{100}{100s+1}e^{-s} [/latex]
This model is lag-dominant because [latex] heta/ au = 0.01[/latex]. Specify three PI controllers using the following methods:
(a) IMC ([latex] au_{c} = 1[/latex])
(b) IMC ([latex] au_{c} = 2[/latex]) and the integrator approximation method (Equation [latex]G(s)=\frac{K^{\ast }e^{- heta s}}{s} [/latex])
(c) IMC ([latex] au_{c} = 1[/latex]) and SIMC (Equation [latex] au_{I}=\min \left\{ au,4( au_{c}+ heta) ight\} [/latex])

Evaluate the three controllers by comparing their performance for unit step changes in both set point and disturbance. Assume that the model is perfect and that [latex]G_{d}(s) = G(s)[/latex].

Accepted Answer

The PI controller settings are:

Controller	K_c	[latex] au_{I}[/latex]
(a) IMC	0.5	100
(b) Integrator approximation	0.57	5
(c) SIMC	0.5	8

The simulation results in Figure on the right indicate that the IMC controller provides an excellent set-point response. The two special controllers provide significantly improved disturbance rejection with much smaller settling times.
Thus, although the standard IMC tuning rules produce very sluggish disturbance responses for FOPTD models with small [latex] heta / au[/latex] ratios, simple remedies are available as demonstrated in this example. Also, the set-point overshoots can be reduced without affecting the disturbance responses by adjusting the set-point weighting coefficient, as considered in Section 12.4.

Question 12.4: Consider an FOPTD model: G̃(s) = 100 / 100s + 1 e^−s This mo...

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