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21.3: The Structured Singular Value

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    For an unstructured perturbation, the supremum of the maximum singular value of \(M\) (i.e. \(\|M\|_{\infty}\)) provides a clean and numerically tractable method for evaluating robust stability. Recall that, for the standard \(M-\Delta\) loop, the system fails to be robustly stable if there exists an admissible \(\Delta\) such that \((I - M(\Delta\)) is singular. What distinguishes the current situation from the unstructured case is that we have placed constraints on the set \(\Delta_{0}\). Given this more limited set of admissible perturbations, we desire a measure of robust stability similar to \(\|M\|_{\infty}\). This can be derived from the structured singular value \(\mu(M)\).

    Definition: Word

    The structured singular value of a complex matrix M with respect to a class of perturbations \(\Delta_{0}\) is given by

    \[\mu(M) \triangleq \frac{1}{\inf \left\{\sigma_{\max }(\Delta) \mid \operatorname{det}(I-M \Delta)=0\right\}}, \quad \Delta \in \Delta_{0}\label{21.4}\]

    If \(\operatorname{det}(I-M \Delta) \neq 0\) for all \(\Delta \in \Delta_{0}\), then \(\mu(M)=0\).

    Theorem \(\PageIndex{21.1}\)

    The \(M-\Delta\) System is stable for all \(\Delta \in \Delta_{0} \text { with }\|\Delta\|_{\infty}<1\) if and only if

    \[\sup _{\omega} \mu(M(j \omega)) \leq 1\nonumber\]

    Proof

    Immediate, from the definition. Clearly, if \(\mu \leq 1\), then the norm of the smallest allowable destabilizing perturbation \(\Delta\) must by definition be greater than 1.

    This page titled 21.3: The Structured Singular Value was last modified on Fri, 05 Mar 2021 08:19:37 GMT and is shared under a CC BY-NC-SA 4.0 license and was authored, remixed, and/or curated by Mohammed Dahleh, Munther A. Dahleh, and George Verghese (MIT OpenCourseWare) via source content that was edited to the style and standards of the LibreTexts platform.