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

作者：Ait Rami Mustapha; Henrion Didier^*

来源：Automatica, 2011, 47(7): 1455-1460.

DOI：10.1016/j.automatica.2011.02.026

• Stability
• Positive polynomials
• LMI
• Toeplitz matrices

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.

• 出版日期2011-7