Time optimisation problem for switched stochastic systems with multi-switching times

In this study, a time optimisation problem is solved for a class of linear time-invariant switched stochastic systems with multi-switching times using the calculus of variations. The objective of authors´ study is to minimise a cost functional defined on the system state, where the sequence of active subsystems is pre-specified and the switching times are the only control variables. The gradient of the cost functional with respect to the switching times is derived, which has an especially simple form and can be directly used in gradient descent algorithms to locate the optimal switching instants. The proposed method is applied on a two-tank system, which is a practical example of switched systems. Combining with another numerical example with three switching times case, the viability of the proposed method is further demonstrated.