An asymmetric Lyapunov function for linear systems with asymmetric actuator saturation

Summary This paper proposes an asymmetric Lyapunov function approach to the estimation of the domain of attraction and the domain with a guaranteed regional gain 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 2 m regions. In each region, the linear system with asymmetrically saturated inputs can be expressed as a linear system with symmetric dead zone. A quadratic function of the augmented state vector containing the system state and the symmetric dead‐zone function 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, 2 generalized asymmetric Lyapunov functions are proposed that lead to reduced conservativeness. A set of conditions are established under which the level sets of these asymmetric Lyapunov functions are contractively invariant and are thus estimates of the domain of attraction. Another set of conditions are derived under which the level sets are subsets of the domain with a guaranteed regional gain. 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 and of the domain with a guaranteed regional gain. Simulation results demonstrate the effectiveness of the proposed approach.