On settling time function and stability of vector relay systems

The paper addresses the problem of discovering the necessary and sufficient stability conditions of vector relay systems and studying the settling time function of this system. It shows that this function is piecewise linear in some cases and the linearity domains of this function are some convex cones. This fact helps to prove that the existence of the piecewise linear Lyapunov function for such vector relay system is a criterion of its stability. Some theoretical results are supported by numerical example.