Robust Stability of Polytopic Systems via Homogeneous Polynomials

This paper considers the robust stability problem for state space models via homogeneous polynomially parameter-dependent Lyapunov functions. The suggested approach is based on a quadratic in the vector of parametric monomials matrix representation of the time derivative of the Lyapunov function. At each step, sufficiency is provided by a set of relaxed linear matrix inequalities and check for positive definiteness of a matrix, the size of which depends on the polynomial degree. Necessity is attained through a generalization of Polya’s Theorem to the case of polynomial matrix valued functions of arbitrary degree.