Stabilisation of matrix polynomials

Abstract

A state feedback is proposed to analyse the stability of a matrix polynomial in closed loop. First, it is shown that a matrix polynomial is stable if and only if a state space realisation of a ladder form of certain transfer matrix is stable. Following the ideas of the Routh–Hurwitz stability procedure for scalar polynomials, certain continued-fraction expansions of polynomial matrices are carrying out by unimodular matrices to achieve the Euclid’s division algorithm which leads to an extension of the well-known Routh–Hurwitz stability criteria but this time in terms of matrix coefficients. After that, stability of the closed-loop matrix polynomial is guaranteed based on a Corollary of a Lyapunov Theorem. The sufficient stability conditions are: (i) The matrices of one column of the presented array must be symmetric and positive definite and (ii) the matrices of the cascade realisation must satisfy a commutative condition. These stability conditions are also necessary for matrix polynomial of second order. The results are illustrated through examples.

Keywords:

• matrix polynomials
• stabilisation
• ladder form
• state feedback
• Routh–Hurwitz criterion

Acknowledgements

The author would like to thank the reviewers for their valuable comments.

Disclosure statement

No potential conflict of interest was reported by the author.