Robust schur stability with polynomial parameters

We propose a stability test for discrete-time systems whose coefficients depend polynomially on some bounded parameters. The test is a particular form of Positivstellensatz, appeals to sum-of-squares polynomials and can be implemented as a semidefinite programming problem. Although implementable only in relaxed form, due to the necessity of limiting the degrees of the polynomial variables involved, the experiments show a good accuracy with degrees smaller than for other tests.

I. Robust Stability Problem

To check the stability of a system whose coefficients depend polynomially on some bounded parameters is a difficult problem in robust stability analysis. Let $$A(z,{\mmb q})=\sum_{n=0}^{N}a_{n}({\mmb q})z^{n}\eqno{\hbox{(1)}}$$be the denominator of a discrete-time transfer function. The coefficients are -variate polynomials in the parameters ; each parameter , , is bounded by some constants and, without any loss of generality, we can consider ; thus, if the parameters are real, then and if the parameters are complex, then , where is the unit disk. The robust stability problem consists of deciding if the polynomial (1) is Schur, i.e., has no roots inside the unit circle, for any .