An approach to optimal control of switched systems with internally forced switchings

Provides an approach to optimal control of switched systems with internally forced switchings (IFS). Many practical problems only involve optimization where the number of switchings and the sequence of active subsystems are specified. This is the stage 1 optimization we study in this paper. In previous papers, we proposed an approach for optimal control problems for switched systems with externally forced switchings (EFS). Here we extend such an approach to problems with IFS. The approach first transcribes a stage 1 problem into an equivalent problem and then the values of the derivatives of the optimal cost with respect to the switching instants are obtained based on the solution of a two point boundary value differential algebraic equation formed by the state, costate, stationarity, boundary equations and the equations for the state and the costate at the switching instants, along with their differentiations. With the knowledge of the derivatives, nonlinear optimization methods can be applied to find the implicitly-generated optimal switching instants along with the corresponding continuous input.