An LMI-based approach for characterizing the solution set of polynomial systems

Considers the problem of solving certain classes of polynomial systems. This is a well known problem in control system analysis and design. A novel approach is developed as a possible alternative to the commonly employed algebraic geometry and homotopy methods. The first result of the paper shows that the solution set of the polynomial system belongs to the kernel of a symmetric matrix. Such a matrix is obtained via the solution of a suitable linear matrix inequality (LMI) involving the maximization of the minimum eigenvalue of an affine family of symmetric matrices. The second result concerns the computation of the solutions from the kernel of the obtained matrix. In particular, it is shown that the solutions can be recovered quite easily if the dimension of the kernel is smaller than the degree of the polynomial system. Finally, some application examples are illustrated to show the features of the approach and to make a brief comparison with the algebraic geometry techniques.