An approach to the continuous‐time stability criterion of polynomial matrices via orthogonal polynomial matrices

Abstract The stability of a scalar polynomial can be tested by the Routh‐Hurwitz method, but a stability criterion for polynomial matrices is not known yet. A square polynomial matrix D(s) is said to be (continuous‐time) stable when all the zeros of det D(s) have negative real parts. In this paper, orthogonal polynomial matrices (Rj(s); j = 0, 1,…; associated with an inner product are considered in studying the stability of D(s). A recursive algorithm to produce {Rj(s)} is presented which elucidates the relation between the stability and the polynomial matrix structure of D(s). When D(s) is a scalar polynomial the algorithm is shown to be equivalent to the Routh‐Hurwitz test. The results are applied to the construction of the Schwartz‐form realizations and the Routh approximations of multi‐variable linear systems.