Explicit solutions to optimal control problems for constrained continuous-time linear systems

An algorithmic framework is presented for the derivation of the explicit optimal control policy for continuous-time linear dynamic systems that involve constraints on the process inputs and outputs. The control actions are usually computed by regularly solving an on-line optimisation problem in the discrete-time space based on a set of measurements that specify the current process state. A way to derive the explicit optimal control law, thereby, eliminating the need for rigorous on-line computations has already been reported in the literature, but it is limited to discrete-time linear dynamic systems. The currently presented approach derives the optimal state-feedback control law off-line for a continuous-time dynamic plant representation. The control law is proved to be nonlinear piecewise differentiable with respect to the system state and does not require the repetitive solution of on-line optimisation problems. Hence, the on-line implementation is reduced to a sequence of function evaluations. The key advantages of the proposed algorithm are demonstrated via two illustrative examples.