21.4: Properties of the Structured Singular Value

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It is important to note that \(\mu\) is a function that depends on the perturbation class \(\Delta_{0}\) (sometimes, this function is denoted by \(\mu_{Delta_{0}}\) to indicate this dependence). The following are useful properties of such a function.

1. \(\mu(M) \geq 0\).

2. If \(\Delta_{0}=\{\lambda I \mid \lambda \in \mathbb{C}\}\), then \(\mu(M)=\rho(M)\), the spectral radius of \(M\) (which is equal to the magnitude of the eigenvalue of \(M\) with maximum magnitude).

3. If \(\Delta=\{\Delta_{0} \mid \Delta\) is an arbitrary complex matrix} then \(\mu=\sigma_{\max }(M)\), from which \(\sup _{\omega} \mu=\|M\|_{\infty}\).

Property 2 shows that the spectral radius function is a particular \(\mu\) function with respect to a perturbation class consisting of matrices of the form of scaled identity. Property 3 shows that the maximum singular value function is a particular \(\mu\) function with respect to a perturbation class consisting of arbitrary norm bounded perturbations (no structural constraints).

4. If \(\Delta=\left\{\operatorname{diag}\left(\Delta_{1}, \ldots, \Delta_{n}\right) \mid \Delta_{i}\right.\) complex}, then \(\mu(M)=\mu\left(D^{-1} M D\right)\) for any \(D = \operatorname{diag}\left(d_{1}, \ldots, d_{n}\right),\left|d_{i}\right| \geq 0\). The set of such scales is denoted \(\mathcal{D}\).

This can be seen by noting that \(\operatorname{det}(I-A B)=\operatorname{det}(I-B A)\), so that \(\operatorname{det}\left(I-D^{-1} M D \Delta\right)=\operatorname{det}(I-\left.M D \Delta D^{-1}\right)=\operatorname{det}(I-M \Delta)\). The last equality arises since the diagonal matrices \(\Delta)\) and \(D\) commute.

5. If \(\Delta_{0}=\operatorname{diag}\left(\Delta_{1}, \ldots, \Delta_{n}\right), \Delta_{i}\) complex, then \(\rho(M)<\mu(M)<\sigma_{\max }(M)\).

This property follows from the following observation: If \(\Delta_{01} \subset \Delta_{02}, \text { then } \mu_{1} \leq \mu_{2}\). It is clear that the class of perturbations consisting of scaled identity matrices is a subset of \(\Delta_{0}\) which is a subset of the class of all unstructured perturbations.

6. From 4 and 5 we have that \(\mu(M)=\mu\left(D^{-1} M D\right) \leq \inf _{D \in \mathcal{D}} \sigma_{\max }\left(D^{-1} M D\right)\)