Stability, Multivariate polynomial, Bernstein expansion, Companion matrix.
In this study, the problem of robust asymptotic stability of n by n polynomial matrix family, in both continuous-time and discrete-time cases, is considered. It is shown that in the continuous case the problem can be reduced to positivity of two specially constructed multivariable polynomials, whereas in the discrete-time case it is required three polynomials. A number of examples are given, where the Bernstein expansion method and sufficient conditions from [L.H. Keel and S.P. Bhattacharya. Robust stability via sign-definite decomposition. IEEE Transactions on Automatic Control, 56(1):140–145, 2011.] are applied to test positivity of the obtained multivariable polynomials. Sufficient conditions for matrix polytopes and one interesting negative result for companion matrices are also considered.
Buyukkoroglu, Taner; Celebi, Gokhan; and Dzhafarov, Vakif.
"On the Robust Stability of Polynomial Matrix Families",
Electronic Journal of Linear Algebra,
Volume 30, pp. 905-915.