Catálogo de publicaciones - libros

We consider nonlinear systems with polynomial vector fields and pose two classes of system theoretic problems that may be solved by sum of squares programming. The first is disturbance analysis using three different norms to bound the reachable set. The second is the synthesis of a polynomial state feedback controller to enlarge the provable region of attraction. We also outline a variant of the state feedback synthesis for handling systems with input saturation. Both classes of problems are demonstrated using two-state nonlinear systems.

Recent advances in semidefinite programming along with use of the sum of squares decomposition to check nonnegativity have paved the way for efficient and algorithmic analysis of systems with polynomial vector fields. In this paper we present a systematic methodology for analyzing the more general class of non-polynomial vector fields, by recasting them into rational vector fields. The sum of squares decomposition techniques can then be applied in conjunction with an extension of the Lyapunov stability theorem to investigate the stability and other properties of the recasted systems, from which properties of the original, non-polynomial systems can be inferred. This will be illustrated by some examples from the mechanical and chemical engineering domains.

Recent improvements of semi-definite programming solvers and developments on polynomial optimization have resulted in a large increase of the research activity on the application of the so-called sum-of-squares (SOS) technique in control. In this approach non-convex polynomial optimization programs are approximated by a family of convex problems that are relaxations of the original program [4, 22]. These relaxations are based on decompositions of certain polynomials into a sum of squares. Using a theorem of Putinar [28] it can be shown (under suitable constraint qualifications) that the optimal values of these relaxed problems converge to the optimal value of the original problem. These relaxation schemes have recently been applied to various nonconvex problems in control such as Lyapunov stability of nonlinear dynamic systems [25, 5] and robust stability analysis [15].

A general controller design technique is proposed for scalar linear systems, based on properties of positive polynomial matrices. The order of the controller is fixed from the outset, independently of the order of the plant and weighting functions. A sufficient LMI condition is used to overcome nonconvexity of the original design problem. The key design step, as well as the whole degrees of freedom are in the choice of a central polynomial, or desired closed-loop characteristic polynomial.

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).

We study here the static state-feedback stabilization of linear finite dimensional systems depending polynomially upon a finite set of real, bounded, parameters. These parameters are a priori unknown, but available in realtime for control. In consequence, it is natural to allow possible dependence of the gain with respect to the parameters ().

We consider the problem of minimizing a form on the standard simplex [equivalently, the problem of minimizing an even form on the unit sphere]. Two converging hierarchies of approximations for this problem can be constructed, that are based, respectively, on results by Schmüdgen-Putinar and by Pólya about representations of positive polynomials in terms of sums of squares. We show that the two approaches yield, in fact, the same approximations.

Consider an ideal generated by polynomials . Let us call := {, ... , } a and let a term ordering of monomials with < ... < be given.

Given a semialgebraic set determined by a finite set of polynomial inequalities { ≥ 0, ..., ≥ 0} , we want to characterize a polynomial f which is positive (or non-negative) on K in terms of sums of squares and the polynomials g i used to describe . Such a representation of f is an immediate witness to the positivity condition. Theorems about the existence of such representations also have various applications, notably in problems of optimizing polynomial functions on semialgebraic sets.

Stability of two-dimensional polynomials arises in fields as diverse as 2D digital signal and image processing ([11], [7], [18]), time-delay systems [14], repettitive, or multipass, processes [24], and target tracking in radar systems. For this reason, there have been a large number of stability criteria for 2D polynomials, which have been surveyed and discussed in a number of papers ([13], [22], [9], [4], [5], [21]). In achieving the maximal efficiency of 2D stability tests, the reduction of algebraic complexity offered by the stability criteria in [26] has been useful. Apart from some minor conditions, the criteria convert stability testing of a 2D polynomial to testing of only two 1D polynomials, one for stability and the other for positivity.