Consider a continuous time plant with nominal transfer function Go(s) given by Go(s) = 2/(s + 1)(s + 2) (13.7.5) Assume that this plant has to be digitally controlled with sampling period ∆ = 0.2[s] in such a way that the plant output tracks a periodic reference, r[k], given by

Question

Consider a continuous time plant with nominal transfer function [latex]G_{o}(s)[/latex] given by

[latex]G_{o}(s) = \frac{2}{(s + 1)(s + 2)}[/latex] (13.7.5)

Assume that this plant has to be digitally controlled with sampling period ∆ = 0.2[s] in such a way that the plant output tracks a periodic reference, r[k], given by

[latex]r\left[k\right] = \sum\limits_{i=0}^{\infty }{r_{T}\left[k - 10i\right] } \Longleftrightarrow R_{q}(z) = R_{Tq}(z)\frac{z^{10}}{z^{10} - 1}[/latex] (13.7.6)

where {[latex]r_{T}[/latex] [k]} = {0.0; 0.1; 0.25; 0.6; 0.3; 0.2; −0.1; −0.3; −0.4; 0.0} and³ [latex]R_{Tq}(z) = Z [r_{T} [k]][/latex].
Synthesize the digital control which achieves zero steady state at-sample errors.

Accepted Answer

From (13.7.6) we observe that the reference generating polynomial, [latex]Γ_{qr}[/latex](z), is given by [latex]z^{10}[/latex] − 1. Thus the IMP leads to the following controller structure

[latex]C_{q}(z)=\frac{P_{q}(z)}{L_{q}(z)}=\frac{P_{q}(z)}{\bar{L}_{q}(z)\Gamma _{qr}(z) }[/latex] (13.7.7)

We then apply pole assignment with the closed loop polynomial chosen as [latex]A_{clq}(z) = z^{12}(z − 0.2)[/latex]. The solution of the Diophantine equation yields

[latex]P_{q}(z)=13.0z^{11}+11.8z^{10}-24.0z^{9}+19.7z^{8}-16.1z^{7}+13.2z^{6}-10.8z^{5}+8.8z^{4}-7.3z^{3}+36.4z^{2}-48.8z+17.6[/latex] (13.7.8)

[latex]L_{q}(z)=(z^{10}-1)(z+0.86)[/latex] (13.7.9)

Figure 13.9 shows the performance of the resulting digital control loop.
In Figure 13.9 we verify that,after a transient period,the plant output, y(t) follows exactly the periodic reference, at the sampling instants. Notice the dangers of a purely at-sample analysis and design. The intersample behavior of this example can be predicted with the techniques for hybrid systems which will be presented in Chapter 14.

Question 13.10: Consider a continuous time plant with nominal transfer funct...

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