On D-stable and D-semistable matrices and the structured singular value

It is shown that testing a matrix A ∈ R^nxn for discrete-time D-stability is equivalent to testing if the structured singular value (SSV or μ) of a matrix M ∈ R^2nx2n obtained from A, is less than unity. Testing for D-semistability (i.e. the property that the product AD has all eigenvalues in the closed left half plane) is shown to be equivalent to testing if the SSV of (M-I)^-1(M+I) is less than or equal to unity. The existence of the Fan-Tits-Doyle LMI-based upper bound for CL (1991) is shown to imply the existence of a diagonal solution to the discrete-time Lyapunov equation in A