Stability analysis for a class of nonlinear switched systems using variational principle

This paper presents a new theorem for stability of all possible trajectories starting at a given initial condition of a class of nonlinear switched systems with an arbitrary switching law. By utilizing the idea of variational principle, it is shown that a nonlinear switched system is asymptotically stable if and only if a critical trajectory is asymptotically convergent to the origin. In this regard, a necessary and sufficient condition is obtained. In order to investigate the stability property, the critical trajectory, characterized as a solution that maximizes a cost function, should be calculated. To this end, an affine control system, instead of the original switched system, and optimal control techniques are utilized. Afterwards, to calculate the critical trajectory, the initial condition x(0) is assumed given, and an algorithm is introduced for stability of all possible trajectories starting at a given initial condition of n-dimensional nonlinear switched systems under arbitrary switching with several operating modes. Simulation results reveal the efficiency of proposed technique.