Controllability and stability radii for companion form systems (Q1111496)

The paper unites and extends earlier work on robustness measures for the controllability, stabilizability and stability of finite-dimensional linear state-space systems [cf., \textit{R. Eising}, Syst. Control Lett. 4, 263-264 (1984; Zbl 0542.93007); \textit{C. F. Van Loan}, Contemp. Math. 47, 465-478 (1985; Zbl 0576.15011); \textit{D. Hinrichsen} and \textit{A. J. Pritchard}, Syst. Control Lett. 7, 1-10 (1986; Zbl 0631.93064)]. By deriving simple formulas for the singular values of companion form matrices, the authors are able to bound these robustness measures for systems which are in controllable canonical form. As a result it is shown for example, that almost all high-order controllable canonical forms are nearly uncontrollable. This supports earlier empirical observations that high-order companion system models have certain undesirable numerical properties.