Schur stability of polytopic systems through positivity analysis of matrix-valued polynomials

This paper is concerned with robust stability of uncertain discrete-time linear systems. The matrix defining the linear system (system matrix) is assumed to depend affinely on a set of time-invariant unknown parameters lying on a known polytope. Robust stability is investigated by checking whether a certain integer power K of the uncertain system matrix has spectral norm less than one. This peculiar stability test is shown to be equivalent to the positivity of a homogeneous symmetric matrix polynomial with known coefficients and degree indexed by K. A unique feature is that no extra variables need to be added to the problems being solved. Numerical experiments reveal that the value of K needed to test robust stability is mostly independent of the system dimension but grows sharply as the eigenvalues of the uncertain system approach the unit circle. By identifying the proposed stability test with a particular choice of a parameter-dependent Lyapunov function, extra variables can be introduced that can help mitigate such convergence problems for systems of small dimension.