An automated algorithm for stability analysis of hybrid dynamical systems

There are many hybrid dynamical systems encountered in nature and in engineering, that have a large number of subsystems and a large number of switching conditions for transitions between subsystems. Bifurcation analysis of such systems poses a problem, because the detection of periodic orbits and the computation of their Floquet multipliers become difficult in such systems. In this paper we propose an algorithm to solve this problem. It is based on the computation of the fundamental solution matrix over a complete period-where the orbit may contain transitions through a large number of subsystems. The fundamental solution matrix is composed of the exponential matrices for evolution through the subsystems (considered linear time invariant in this paper) and the saltation matrices for the transitions through switching conditions. This matrix is then used to compose a Newton-Raphson search algorithm to converge on the periodic orbit. The algorithm-which has no restriction of the complexity of the system-locates the periodic orbit (stable or unstable), and at the same time computes its Floquet multipliers. The program is written in a sufficiently general way, so that it can be applied to any hybrid dynamical system.