State space measures of robustness of pole locations for structured and unstructured perturbations

Abstract

Measures of pole location robustness for linear feedback systems are derived from a state space model of the system. The robustness tests ensure that the eigenvalues of the perturbed systems matrix A + E remain in a desired region D of the complex plane containing the eigenvalues of the nominal system matrix A. The region D may be any open set of the complex plane whatsoever. The results are expressed in terms of induced matrix norms and apply to structured perturbations of the form E = BΔC, where B and C define the structure of E, and may be nonsquare matrices. Rank one perturbations E of minimal norm and with the given structure that will cause A + E to have an eigenvalue outside of D are constructed for the cases when the matrix norm is induced by the vector 1-norm or the vector ∞-norm. The advantages of having robustness measures for several matrix norms that can be computed are illustrated with a simple example that demonstrates how the conservatism of single tests can be reduced using several tests (i.e. several matrix norms). A method for computing numerically the robustness measures for particular norms is presented. It can be used to compute, with a guaranteed degree of accuracy, the maximum of the norm of the frequency response of a system.