Stability of matrix polynomials†

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Abstract
The paper considers the following question : Given a square, non-singular polynomial matrix C(s)how do we check, without evaluating the determinant, whether all the zeros of det C(s) are in the open left-half plane ? The approach used to answer this question is to derive from c(s) a rational transfer function matrix which is lossless positive real (l.p.r.) if arid only if det C(s)is Hurwitz. The l.p.r. property is easily checked using the coefficients of the rational function only. The construction of the l.p.r. function requires solution of a polynomial matrix equation, and the later part of the paper discusses both existence questions and solution procedures ; if no solution exists to the matrix equation then det C(s)is non-Hurwitz The connection is also illustrated between the l.p.r. stability test and that of Shieh and Sacheti (1976). Prospects for development of the theory are discussed.

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