Feedback Solution to Optimal Switching Problems with Switching Cost

The problem of optimal switching between nonlinear autonomous subsystems is investigated in this study where the objective is not only bringing the states to close to the desired point, but also adjusting the switching pattern, in the sense of penalizing switching occurrences and assigning different preferences to utilization of different modes. The mode sequence is unspecified and a switching cost term is used in the cost function for penalizing each switching. It is shown that once a switching cost is incorporated, the optimal cost-to-go function depends on the already active subsystem, i.e., the subsystem which was engaged in the previous time step. Afterwards, an approximate dynamic programming based method is developed which provides an approximation of the optimal solution to the problem in a feedback form and for different initial conditions. Finally, the performance of the method is analyzed through numerical examples.