Domain of attraction: estimates for non-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 developed for computing the Largest Estimate of the DA (LEDA) corresponding to a Lyapunov function in the case of polynomial systems. In the case of non-polynomial systems, the computation of the LEDA is still an open problem. In this paper, an LMI technique is proposed to deal with such a problem for a class of non-polynomial systems. The key point consists of using sum of squares relaxations for taking into account the worst-case remainders corresponding to truncated Taylor expansions of the non-polynomial terms. As shown by some examples, low degree remainders may be sufficient to obtain almost tight estimates.