Perturbation Theory of Transfer Function Matrices
Vanni Noferini, Lauri Nyman, Javier J. Pérez, María C. Quintana · SIAM Journal on Matrix Analysis and Applications · 2023
Abstract. Zeros of rational transfer function matrices [Formula: see text] are the eigenvalues of associated polynomial system matrices [Formula: see text] under minimality conditions. In this paper, we define a structured condition number for a simple eigenvalue [Formula: see text] of a (locally) minimal polynomial system matrix [Formula: see text], which in turn is a simple zero [Formula: see text] of its transfer function matrix [Formula: see text]. Since any rational matrix can be written as the transfer function of a polynomial system matrix, our analysis yields a structured perturbation theory for simple zeros of rational matrices [Formula: see text]. To capture all the zeros of [Formula: see text], regardless of whether they are poles, we consider the notion of root vectors. As corollaries of the main results, we pay particular attention to the special case of [Formula: see text] being not a pole of [Formula: see text] since in this case the results get simpler and can be useful in practice. We also compare our structured condition number with Tisseur’s unstructured condition number for eigenvalues of matrix polynomials and show that the latter can be unboundedly larger. Finally, we corroborate our analysis by numerical experiments.