Characterizing the solution set of polynomial systems in terms of homogeneous forms: an LMI approach

This paper considers the problem of determining the solution set of polynomial systems, a well-known problem in control system analysis and design. A novel approach is developed as a viable 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 suitable symmetric matrix. Such a matrix is obtained via the solution of a 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 solution set from the kernel of the obtained matrix. For polynomial systems of degree m in n variables, a basic procedure is available if the kernel dimension does not exceed m+1, while an extended procedure can be applied if the kernel dimension is less than n(m−1)+2. Finally, some application examples are illustrated to show the features of the approach and to make a brief comparison with polynomial resultant techniques.