A Generalized Piecewise Quadratic Lyapunov Function Approach to Estimating the Domain of Attraction of a Saturated System∗

This paper revisits the problem of estimating the domain of attraction of a saturated system with an algebraic loop. A piecewise quadratic Lyapunov function of an augmented state vector composing of the system state and the saturated input has been proposed for use in estimating the domain of attraction for a saturated system. Considering the relationship between the system states and the saturation function, we propose in this paper a generalized piecewise quadratic Lyapunov function, which results from adding a term that characterizes the regional sector condition of the saturated input to the piecewise quadratic Lypunov function. The matrix associated with the generalized piecewise quadratic Lyapunov function is not required to be positive definite, and thus a set of less conservative stability conditions are established, from which a larger estimate of the domain of attraction can be obtained. Simulation results indicate that the effectiveness and superiority of the proposed method.