RDA in Uncertain Polynomial Systems

Abstract

This chapter addresses the problem of estimating the RDA of common equilibrium points of uncertain polynomial systems by using polynomial LFs and SOS programming. In particular, the chapter considers dynamical systems whose dynamics is described by a polynomial function of the state with coefficients depending polynomially on a time-invariant uncertain vector constrained over a semialgebraic set. It is shown that a condition for establishing whether a sublevel set of a common LF is an estimate of the RDA can be obtained in terms of an LMI feasibility test by exploiting SOS parameter-dependent polynomials and their representation through the SMR. Then, a GEVP is proposed for computing a candidate of the largest of such estimates, and a sufficient and necessary condition for establishing the tightness of the candidate is provided. This methodology is hence extended to the case of polynomial LFs with coefficients depending polynomially on the uncertainty. For this case, the computation of parameter-dependent estimates is discussed. Moreover, a strategy is proposed for estimating the intersection of such estimates in order to provide a common estimate that is independent on the uncertainty. Lastly, the chapter addresses the computation of optimal estimates of the RDA by using variable LFs, the problem of establishing robust global asymptotical stability of the common equilibrium point, and the design of polynomial static output controllers for enlarging the RDA.