Optimal switching of nonlinear systems with free mode sequence

Approximate dynamic programming is utilized in this study to develop solutions for optimal switching problems. The order of the active subsystems and the number of switches are free and an online solution is developed that produces optimal performance. Motivated by the development in the adaptive critics literature, the proposed method uses a critic and as many actors as the number of subsystems in order to conduct optimal scheduling in real-time based on the instantaneous states of the system. It is observed that once the neural networks are trained offline based on the proposed algorithm, they provide solutions for a range of initial conditions and different final times. Finally, the theoretical results are analyzed with numerical results from a fourth order switching system.