A hierarchy of LMI inner approximations of the set of stable polynomials

Exploiting spectral properties of symmetric banded Toeplitz matrices, we describe simple sufficient conditions for the positivity of a trigonometric polynomial formulated as linear matrix inequalities (LMIs) in the coefficients. As an application of these results, we derive a hierarchy of convex LMI inner approximations (affine sections of the cone of positive definite matrices of size m ) of the nonconvex set of Schur stable polynomials of given degree n < m . It is shown that when m tends to infinity the hierarchy converges to a lifted LMI approximation (projection of an LMI set defined in a lifted space of dimension quadratic in n ) already studied in the technical literature. An application to robust controller design is described.