An approach for solving general switched linear quadratic optimal control problems

This paper successfully addresses an important class of hybrid optimal control problems of practical significance. It provides a viable general approach to hybrid optimal control based on nonlinear optimization and it shows that when this approach is applied to linear quadratic problems it leads to computationally attractive algorithms. Unlike conventional optimal control problems, optimal control problems for switched systems require the solutions of not only optimal continuous inputs but also optimal switching sequences. Many practical problems only involve optimization where the number of switchings and the sequence of active subsystems are given. This is stage 1 of the two stage optimization method proposed by the authors in previous papers. In order to solve stage 1 problems using efficient nonlinear optimization techniques, the derivatives of the optimal cost with respect to the switching instants need to be known. In this paper, we focus on and solve a special class of optimal control problems, namely, general switched linear quadratic problems. The approach first transcribes a stage 1 problem into an equivalent problem parameterized by the switching instants and then obtains the derivative values based on the solution of an initial value ordinary differential equation formed by the general Riccati equation and its differentiations. Examples illustrate the results.