Optimal Dynamic Lyapunov Function and The Largest Estimation of Domain of Attraction

摘要

Estimation of the domain of attraction is one of the major difficulties in the analysis of any nonlinear control system. Exact expression of a candidate Lyapunov function is indispensable for the estimation. The optimal Lyapunov function which helps to estimate the domain of attraction exactly is acquired by solving a partial differential equation that is not solvable easily for all of the systems. In this paper, an algebraic equation is proposed instead of the partial differential equation to acquire optimal Lyapunov function. The key tool for this purpose is dynamic Lyapunov function which makes it possible to have an analytic expression of a family of candidate Lyapunov functions that are parametric and functional. The parameters and functions would be selected such that the optimal Lyapunov function is achieved. As the second contribution of this paper, the problem of optimality in the sense of the largest elliptical estimation of region of attraction would be followed up. Using a linear matrix inequality technique and a special criterion it is shown that dynamic Lyapunov function can leads to a larger elliptical estimation than that of the conventional Lyapunov function.