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21.8: Exercises

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  • Exercise \(\PageIndex{21.1}\)

    In decentralized control, the plant is assumed to be diagonal and controllers are de- signed independently for each diagonal element. If however, the real process is not completely decou- pled, the interactions between these separate subsystems can drive the system to instability.

    Consider the \(2 \times 2\) plant

    P_{11} & P_{12} \\
    P_{21} & P_{22}

    Assume that \(P_{12}\) and \(P_{21}\) are stable and relatively small in comparison to the diagonal elements, and only a bound on their frequency response is available. Suppose a controller \(K=\operatorname{diag}\left(K_{1}, K_{2}\right)\) is designed to stabilize the system \(P_{0}=\operatorname{diag}\left(P_{11}, P_{22}\right)\).

    1. Set-up the problem as a stability robustness problem, i.e., put the problem in the \(M - \Delta\) form.
    2. Derive a non-conservative condition (necessary and sufficient) that guarantees the stability robustness of the above system. Assume the off-diagonal elements are perturbed independently. Reduce the result to the simplest form (an answer like \(\mu(M)<1\) is not acceptable; this problem has an exact solution which is computable).
    3. How does your answer change if the off-diagonal elements are perturbed simultaneously with the same \(\Delta\).

    Exercise \(\PageIndex{21.2}\)

    Consider the rank 1 \(\mu\) problem. Suppose \(\Delta_{0}\), contains only real perturbations. Compute the exact expression of \(\mu(M)\).

    Exercise \(\PageIndex{21.3}\)

    Consider the set of plants characterized by the following sets of numerators and denominators of the transfer function:

    \[N(s)=N_{0}(s)+N_{\delta}(s) \delta, \quad D(s)=D_{0}(s)+D_{\delta}(s) \delta\nonumber\]

    Where both \(N_{0}\) and \(D_{0}\) are polynomials in \(s, \delta \in \mathbb{R}^{n}\), and \(N_{\delta}, D_{\delta}\) are polynomial row vectors. The set of all plants is then given by:

    \[\Omega=\left\{\frac{N(s)}{D(s)}\left|\delta \in \mathbb{R}^{n},\right| \delta_{i} \mid \leq \gamma\right\}\nonumber\]

    Let \(K\) be a controller that stabilizes \(N_{0}/D_{0}\) . Compute the exact stability margin; i.e., compute the largest \(\gamma\) such that the system is stable.

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