A block diagram for internal model control, a control technique that is considered in Chapter 12, is shown in Fig. E11.7. Transfer function Gp denotes the process model, while Gp denotes the actual process transfer function. It has been assumed that Gv = Gm = 1 for simplicity. Derive closed-loop

Question

A block diagram for internal model control, a control technique that is considered in Chapter 12 , is shown in Fig. E11.7. Transfer function [latex]\widetilde{G}_{p}[/latex] denotes the process model, while [latex]G_{p}[/latex] denotes the actual process transfer function. It has been assumed that [latex]G_{v}=G_{m}=1[/latex] for simplicity. Derive closed-loop transfer functions for both the servo and regulator problems.

Accepted Answer

Using block diagram algebra

[latex]\begin{aligned}&Y=G_{d} D+G_{p} U\quad\quad\quad (1) \&U=G_{c}\left[Y_{s p}-\left(Y-\widetilde{G}_{p} U ight) ight] \quad\quad\quad (2)\end{aligned}[/latex]

From (2), [latex]\quad U=\frac{G_{c} Y_{s p}-G_{c} Y}{1-G_{c} \widetilde{G}_{p}}[/latex]

Substituting for [latex]U[/latex] in Eq. 1

[latex]\left[1+G_{c}\left(G_{p}- ilde{G}_{p} ight) ight] Y=G_{d}\left(1-G_{c} \widetilde{G}_{p} ight) D+G_{p} G_{c} Y_{s p}[/latex]

Therefore,

[latex]\frac{Y}{Y_{s p}}=\frac{G_{p} G_{c}}{1+G_{c}\left(G_{p}-\widetilde{G}_{p} ight)}[/latex]

and

[latex]\frac{Y}{D}=\frac{G_{d}\left(1-G_{c} ilde{G}_{p} ight)}{1+G_{c}\left(G_{p}- ilde{G}_{p} ight)}[/latex]

Question 11.7: A block diagram for internal model control, a control techni...

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