On the estimation of the domain of attraction for linear systems with asymmetric actuator saturation via asymmetric Lyapunov functions

This paper proposes an asymmetric Lyapunov function approach to the estimation of the domain of attraction for a linear system subject to asymmetric actuator saturation. Depending on the sign of each of the m inputs, the input space is divided into 2m regions. In each region, the linear system with asymmetrically saturated inputs can be expressed as a linear system with symmetric deadzones. A quadratic function of the augmented state vector containing the system states and the symmetric deadzone functions is constructed for each region. From these quadratic functions, an asymmetric Lyapunov function is composed. Furthermore, based on the special properties of the intersections between regions, a generalized asymmetric Lyapunov function is proposed that leads to reduced conservativeness. Conditions are established under which the level sets of both of these asymmetric Lyapunov functions are contractively invariant. Based on these conditions, LMI-based optimization problems are formulated and solved to obtain the largest level sets as the estimates of the domain of attraction. Simulation results demonstrate the effectiveness of the proposed approach.