On convergence properties of piecewise affine systems

In this paper convergence properties of piecewise affine (PWA) systems are studied. In general, a system is called convergent if all its solutions converge to some bounded globally asymptotically stable steady-state solution. The notions of exponential, uniform and quadratic convergence are introduced and studied. It is shown that for non-linear systems with discontinuous right-hand sides, quadratic convergence, i.e., convergence with a quadratic Lyapunov function, implies exponential convergence. For PWA systems with continuous right-hand sides it is shown that quadratic convergence is equivalent to the existence of a common quadratic Lyapunov function for the linear parts of the system dynamics in every mode. For discontinuous bimodal PWA systems it is proved that quadratic convergence is equivalent to the requirements that the system has some special structure and that certain passivity-like condition is satisfied. For a general multimodal PWA system these conditions become sufficient for quadratic convergence. An example illustrating the application of the obtained results to a mechanical system with a one-sided restoring characteristic, which is equivalent to an electric circuit with a switching capacitor, is provided. The obtained results facilitate bifurcation analysis of PWA systems excited by periodic inputs, substantiate numerical methods for computing the corresponding periodic responses and help in controller design for PWA systems.