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

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

2010-06-30

arxiv.org/abs/1006.5876v1

Mathematics

Optimization and Control

Scientific paper

Exploiting spectral properties of symmetric banded Toeplitz matrices, we describe simple sufficient conditions for positivity of a trigonometric polynomial formulated as linear matrix inequalities (LMI) 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.

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