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.