Analysis of linear systems with piecewise linear functions in the input

This paper revisits the problem of estimating the domain of attraction of linear systems with piecewise linear function in the input. We construct a generalized piecewise quadratic Lyapunov function of the augmented state vector consisting of the system states and the terms that characterize the regional sector conditions of the saturation functions embedded in the piecewise linear function. Conditions are established under which the level sets of the generalized piecewise quadratic Lyapunov function are contractively invariant. Based on these conditions, we formulate a bilinear matrix inequality optimization problem and solve it by using an iterative algorithm that is based on linear matrix inequalities (LMIs) to obtain the largest level set as an estimate of the domain of attraction. Simulation results indicate that the proposed approach indeed has the ability to obtain a significantly larger estimate of the domain of attraction than the existing methods.