Introduction Many interesting control system design and analysis problems can be recast as systems of inequalities for multivariate polynomials in real variables. In particular, for linear time-invariant systems, important control issues such as robust stability and robust performance can be reduced to such systems. Typically, the variables in the (multivariate) polynomials come from plant (controlled system) and compensator (controller) parameters. In this chapter, we describe a method for solving such systems of inequalities. By solving we mean that we end up with a collection of axis-parallel boxes in the parameter space whose union provides an inner approximation of the solution set, i.e., the polynomial inequalities are fulfilled for each parameter vector taken from such a box. This method is based on the expansion of a multivariate polynomial into Bernstein polynomials. It provides an alternative to symbolic methods like quantifier elimination whose application to control

We survey some recent applications of Bernstein expansion to robust stability, viz. checking robust Hurwitz and Schur stability of polynomials with polynomial parameter dependency by testing determinantal criteria and by inspection of the value set. Then we show how Bernstein expansion can be used to solve systems of strict polynomial inequalities.

. The paper considers the robust Schur stability verification of polynomials with coefficients depending polynomially on parameters varying in given intervals. A new algorithm is presented which relies on the expansion of a multivariate polynomial into Bernstein polynomials and is based on the decomposition of the family of polynomials into its symmetric and antisymmetric parts. It is shown how the inspection of both polynomial families on the upper half of the unit circle can be reduced to the analysis of two related polynomial families on the real interval [\Gamma1; 1]. Then the Bernstein expansion can be applied in order to check whether both polynomial families have a zero in this interval in common. 1. Introduction Bernstein expansion has proved to be a well established and important tool for solving robust stability problems such as checking a polynomial with polynomial parameter dependency for stability. Stability regions covered include the open left half of the complex plane ...