A second-order process with transfer function
$G(s)= \frac{-5e^{-4s}}{(3s+1)(5s+1)};\quad G_{m}= 1$
is controlled with a digital IMC controller (Δt= 0.2). There is no modeling error. An IMC filter time constant of zero is used (F(z)=1). A unit step change in load occurs at time t=10, with a known disturbance transfer function
$G_{d} = \frac{1}{7s+1}$
Compute the IMC closed-loop response.

Question

A second-order process with transfer function
[latex] G(s)= \frac{-5e^{-4s}}{(3s+1)(5s+1)};\quad G_{m}= 1 [/latex]
is controlled with a digital IMC controller (Δt= 0.2). There is no modeling error. An IMC filter time constant of zero is used (F(z)=1). A unit step change in load occurs at time t=10, with a known disturbance transfer function
[latex] G_{d} = \frac{1}{7s+1} [/latex]
Compute the IMC closed-loop response.

Accepted Answer

Using MATLAB c2d function with sampling time Δt=0.2,
[latex] G(z)= \frac{(−0.006434z^{−1} −0.00621z^{−2})}{1−1.896z^{−1} +0.8988z^{−2}}z^{-20} [/latex]
Since a zero is at −0.9652, it should be included in [latex] \widetilde{G}_{+}(z) [/latex].
Therefore,
[latex] \widetilde{G}_{+}(z) = \frac{(−0.006434z^{−1} −0.00621z^{−2})}{(−0.006434−0.00621)}z^{-20} [/latex]
[latex] = 0.5089z^{−21} +0.4911z^{−22} [/latex]
[latex] \widetilde{G}_{-}(z) = \frac{ (−0.006434−0.00621)}{1−1.896z^{−1} +0.8988z^{−2}} [/latex]

For the filter F(z)=1, analogous to Equation
[latex] G^{*}_{c} = \frac{1}{\widetilde{G}_{-} }f [/latex],

the IMC controller is
[latex] G_{c*}(z)= \widetilde{G}^{-1}_{-}(z)F(z)= \frac{1−1.896z^{−1} +0.8988z^{−2}}{(−0.006434−0.00621) } [/latex]
By using Simulink-MATLAB, the IMC response for a unit step in disturbance at t=10 is shown in Figure.

Question 17.8: A second-order process with transfer function G(s) = -5e^-4s...

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