On the characterization of the solution set of polynomial systems via LMI techniques

Abstract

The problem addressed in this paper is the computation of the solution set for systems of polynomial equations, a key issue in several system analysis and control problems. A new approach is presented, which represents a possible alternative to well-known techniques, based on algebraic geometry and homotopy methods. The basic idea is to characterize the solution set in terms of the kernel of a symmetric matrix, associated to a suitable quadratic homogeneous form. This matrix is obtained via a Linear Matrix Inequality (LMI) optimization problem. The actual computation of the solution set can be performed quite easily, provided that the dimension of the kernel does not exceed a prescribed value. It is shown that this value turns out to be quite large, so that the proposed procedure can be applied to a fairly wide variety of polynomial systems.