On the estimation of the domain of attraction for uncertain polynomial systems via LMIs

Estimating the domain of attraction (DA) of equilibrium points is a problem of fundamental importance in systems engineering. Several approaches have been proposed for the case of known polynomial systems allowing one to find the largest estimate of the DA (LEDA) for a given Lyapunov function (LF). However, the problem of estimating the robust DA (RDA), that is the DA guaranteed for all possible uncertainties in an uncertain system, it is still an unsolved problem. In this paper, some methods are proposed for dealing with such a problem in the case of systems depending polynomially in the state and in the uncertainty which is supposed to belong to a polytope. Specifically, the issue of estimating the robust LEDA (RLEDA), that is the intersection of all LEDAs in the uncertain system, is considered for common and parameter-dependent LFs, providing constant and parameter-dependent lower bounds through LMI optimizations. In order to obtain easy descriptions of the RLEDA in the case of parameter-dependent LFs, an LMI method for computing approximations with simple shape is presented.