On the estimation of the equilibrium points of uncertain nonlinear systems

Abstract

The analysis and control of nonlinear systems often require information about the location of their equilibrium points. This paper addresses the problem of estimating the set of equilibrium points of uncertain nonlinear systems, in particular, systems whose dynamics are described by a nonlinear function of the state depending polynomially on an uncertainty vector constrained in a polytope. It is shown that estimates of this set can be obtained by solving LMI problems, which are built through sum of squares techniques by introducing worst-case truncations of the nonlinearities and by exploiting homogeneity of equivalent representations. In particular, the computation of estimates with fixed shape and the problem of establishing their tightness is firstly considered. Then, the paper shows how this methodology can be used to address the computation of the minimum volume estimate and the construction of the smallest convex estimate. Examples with random and real systems illustrate the proposed methodology. © 2011 John Wiley & Sons, Ltd.