Stabilization of a Limit Cycle for Discrete-Time Switched Nonlinear Systems

This letter studies global exponential stabilisation of a limit cycle of interest for discrete-time switched nonlinear systems, in which the subsystems may have different equilibria. As a first step, a set of candidate limit cycles is determined according to a criterion related to the steady-state behavior of the system trajectories. Afterwards, a state-dependent switching function, based on sufficient conditions derived from a time-periodic Lyapunov function, is proposed to ensure global exponential stability of the limit cycle and a guaranteed performance level for the overall system. A class of polynomial switched systems is used to illustrate the main results. For this class, new LMI conditions are obtained that ensure local exponential stability of the limit cycle, inside a polyhedral set given by the designer. An ellipsoidal set of maximum volume is determined such that any trajectory starting inside it does not leave the polyhedron. The main features of this methodology are illustrated by academic examples.