Continuous representation and control of hybrid systems

This article introduces a continuous representation for controlled model and state jump hybrid systems. Such a representation is valuable in that it manages to cope with the difficulties related to discontinuities, by replacing the study of these systems by that of continuous ones, which is simple. The studied hybrid systems are first introduced, together with their associated continuous systems. Some localized and global convergences of the states of continuous systems to those of hybrid systems are proved. Then, the continuous representation is used within the optimal control framework. Free end-time and trajectory pursuit hybrid problems are studied, as well as their respective continuous versions. Convergences of solutions between continuous and hybrid problems are established, which shows that the resolution of a problem ensuing from the representation enables one to obtain an approximation of the solution of the initial hybrid problem. Finally, the formalism of calculus of variations is used to obtain the expressions of the gradients of cost functions associated with a trajectory pursuit problem in hybrid and continuous cases. Theoretical and technical simplifications, brought by the representation, are highlighted, as well as its practical interest shown by means of a numerical application