Stabilisation of matrix polynomials |
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Authors: | R Galindo |
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Affiliation: | 1. Department of Electrical and Mechanical Engineering, Autonomous University of Nuevo Leon, Nuevo Leon, Mexicorgalindo@gama.fime.uanl.mx |
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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. |
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Keywords: | matrix polynomials stabilisation ladder form state feedback Routh–Hurwitz criterion |
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