Design a model predictive controller for the process Gp(s) = e^−6s/10s + 1 Gv = Gm = 1 Select a value of N based on 95% completion of the step response and Δt = 2. Simulate the closed-loop response for a set-point change using the following design parameters: (a) M = 1 P = 7 R = 0 (b) M = 1 P = 5

Question

Design a model predictive controller for the process

[latex]G_{p}(s)=\frac{e^{-6 s}}{10 s+1} \quad G_{v}=G_{m}=1[/latex]

Select a value of [latex]N[/latex] based on [latex]95 \%[/latex] completion of the step response and [latex]\Delta t=2[/latex]. Simulate the closed-loop response for a set-point change using the following design parameters:

[latex]\begin{array}{lll} ext { (a) } M=1 & P=7 & R = 0 \end{array}[/latex]

[latex]\begin{array}{lll} ext { (b) } M=1 & P=5 & R = 0 \end{array}[/latex]

[latex]\begin{array}{lll} ext { (c) } M=4 & P=30 & R=0\end{array}[/latex]

Accepted Answer

The open-loop unit step response of [latex]G_{p}(s)[/latex] is

[latex]y(t)= L ^{-1}\left(\frac{e^{-6 s}}{10 s+1} \frac{1}{s}\right)= L ^{-1}\left(e^{-6 s}\left(\frac{1}{s}-\frac{10}{10 s+1}\right)\right)=S(t-6)\left[1-e^{-(t-6) / 10}\right][/latex]

By trial and error, [latex]y(34)<0.95, y(36)>0.95[/latex].

Therefore [latex]N \Delta t=36[/latex] or [latex]N=18[/latex].

The coefficients [latex]\left\{S_{i}\right\}[/latex] are obtained from the expression for [latex]y(t)[/latex] and the predictive controller is obtained following the procedure of Example 20.5. The closed-loop responses for a unit set-point change are shown below for the three sets of controller design parameters.

Q. 20.9

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