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Robust stability analysis of state space models with respect to real parametric uncertainty is a widely studied challenging problem. In this paper, a quite general uncertainty model is considered, which allows one to consider polynomial nonlinearities in the uncertain parameters. A class of parameter-dependent Lyapunov functions is used to establish stability of a matrix depending polynomially on a vector of parameters constrained in a polytope. Such class, denoted as Homogeneous Polynomially Parameter-Dependent Quadratic Lyapunov Functions (HPD-QLFs), contains quadratic Lyapunov functions whose dependence on the parameters is expressed as a polynomial homogeneous form. Its use is motivated by the property that the considered matricial uncertainty set is stable if and only there exists a HPD-QLF. The paper shows that a sufficient condition for the existence of a HPD-QLF can be derived in terms of Linear Matrix Inequalities (LMIs). [ABSTRACT FROM AUTHOR]

This paper considers the problem of determining the minimum euclidean distance of a point from a polynomial surface in Rn. It is well known that this problem is in general non-convex. The main purpose of the paper is to investigate to what extent Linear Matrix Inequality (LMI) techniques can be exploited for solving this problem. The first result of the paper shows that a lower bound to the global minimum can be achieved via the solution of a one-parameter family of LMIs. Each LMI problem consists in the minimization of the maximum eigen value of a symmetric matrix. It is also pointed out that for some classes of problems the solution of a single LMI provides the lower bound. The second result concerns the tightness of the bound. It is shown that optimality of the lower bound can be readily checked via the solution of a system of linear equations. In addition, it is pointed out that lower bound tightness is strictly related to some properties concerning real homogeneous forms. Finally, an application example is developed throughout the paper to show the features of the approach. [ABSTRACT FROM AUTHOR]