21.3: The Structured Singular Value

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.