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20.9: Exercises

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  • Exercise \(\PageIndex{20.1}\)

    Consider a plant described by the transfer function matrix

    \frac{\alpha}{s-1} & \frac{1}{s-1} \\
    \frac{2 s-1}{s(s-1)} & \frac{1}{s-1}

    where \(\alpha\) is a real but uncertain parameter, confined to the range [0:5 , 1:5]. We wish to design a feedback compensator \(K(s)\) for robust stability of a standard servo loop around the plant.

    (a) We would like to find a value of \(\alpha\), say \(\tilde{\alpha}\), and a scalar, stable, proper rational \(W(s)\) such that the set of possible plants \(P_{\alpha}(s)\) is contained within the "uncertainty set"

    \[P_{\tilde{\alpha}}(s)[I+W(s) \Delta(s)]\nonumber\]

    where \(\Delta(s)\) ranges over the set of stable, proper rational matrices with \(\|\Delta\|_{\infty} \leq 1\). Try and find (no assurances that this is possible!) a suitable \(\tilde{\alpha}\) and \(W(s)\), perhaps by keeping in mind that what we really want to do is guarantee

    \[\sigma_{\max }\left\{P_{\tilde{\alpha}}^{-1}(j \omega)\left[P_{\alpha}(j \omega)-P_{\tilde{\alpha}}(j \omega)\right]\right\} \leq|W(j \omega)|\nonumber\]

    What specific choice of \(\Delta(s)\) yields the plant \(P_{1}(s)\) (i.e. the plant with

    (b) Repeat part (a), but now working with the uncertainty set

    \[P_{\tilde{\alpha}}(s)\left[I+W_{1}(s) \Delta(s) W_{2}(s)\right]\nonumber\]

    where \(W_{1}(s)\) and \(W_{2}(s)\) are column and row vectors respectively, and \(\Delta(s)\) is scalar. Plot the upper bound on

    \[\sigma_{\max }\left\{P_{\tilde{\alpha}}^{-1}(j \omega)\left[P_{\alpha}(j \omega)-P_{\tilde{\alpha}}(j \omega)\right]\right\}\nonumber\]

    that you obtain in this case.

    (c) For each of the cases above, write down a suffcient condition for robust stability of the closed-loop system, stated in terms of a norm condition involving the nominal complementary sensitivity function \(T=\left(I+K P_{\tilde{\alpha}}\right)^{-1} K P_{\tilde{\alpha}}\) and \(W-\) or, in part (b), \(W_{1}\) and \(W_{2}\).

    Exercise \(\PageIndex{20.2}\)

    It turns out that the small gain theorem holds for nonlinear systems as well. Consider a feedback configuration with a stable system \(M\) in the forward loop and a stable, unknown perturbation in the feedback loop. Assume that the configuration is well-posed. Verify that the closed loop system is stable if \(\|M\|\|\Delta\|<1\). Here the norm is the gain of the system over any p-norm. (This result is also true for both DT and CT systems; the same proof holds).

    Exercise \(\PageIndex{20.3}\)

    The design of a controller should take into consideration quantization effects. Let us assume that the only variable in the closed loop which is subject to quantization is the output of the plant. Two very simple schemes are proposed:

    Quantization in the closed loop

    Figure 20.8: Quantization in the Closed Loop

    Screen Shot 2020-11-13 at 10.30.11 PM.png

    Figure 20.9: Quantization Modeled as Bounded Noise

    1. Assume that the output is passed through a quantization operator \(Q\) defined as:

    \[Q(x)=a\left\lfloor\frac{|x|}{.5+a}\right\rfloor \operatorname{sgn}(x), \quad a>0\nonumber\]

    where \([r]\) denotes the largest integer smaller than \(r\). The output of this operator feeds into the controller as in Figure 20.8. Derive a suffcient condition that guarantees stability in the presence of \(Q\).

    2. Assume that the input of the controller is corrupted with an unknown but bounded signal, with a small bound as in Figure 20.9. Argue that the controller should be designed so that it does not amplify this disturbance at its input.

    Compare the two schemes, i.e., do they yield the same result? Is there a difference?

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