Geometric synthesis of a hybrid limit cycle for the stabilizing control of a class of nonlinear switched dynamical systems

This paper proposes a new constructive method for synthesizing a hybrid limit cycle for the stabilizing control of a class of switched dynamical systems in R2R2, switching between two discrete modes and without state discontinuity. For each mode, the system is continuous, linear or nonlinear. This method is based on a geometric approach. The first part of this paper demonstrates a necessary and sufficient condition of the existence and stability of a hybrid limit cycle consisting of a sequence of two operating modes in R2R2 which respects the technological constraints (minimum duration between two successive switchings, boundedness of the real valued state variables). It outlines the established method for reaching this hybrid limit cycle from an initial state, and then stabilizing it, taking into account the constraints on the continuous variables. This is then illustrated on a Buck electrical energy converter and a nonlinear switched system in R2R2. The second part of the paper proposes and demonstrates an extension to RnRn for a class of systems, which is then illustrated on a nonlinear switched system in R3R3.