Minimizing control energy in a class of bounded-controllinear-quadratic regulator problems

Abstract

Minimal-control-energy strategies are substantiated and illus- trated for linear-quadratic problems with penalized endpoints and no state- trajectory cost, when bounds in control values are imposed. The optimal solu- tion for a given process with restricted controls, starting at a known initial state, is shown to coincide with the saturated solution to the unrestricted problem that has the same coe¢ cients but starts at a generally difeerent initial state. This result reduces the searching span for the solution: from the in?nite-dimensional set of admissible control trajectories, to the finite-dimensional Euclidean space of initial conditions. An efcient on-line scheme is proposed here to approxi- mate (eventually to ?nd) the optimal control strategy, based on the detection of the appropriate initial state while avoiding as much as possible the gener- ation and evaluation of state and control trajectories. Numerical (including Model Predictive Control) simulations are provided, compared, and checked against the analytical solution to 'the cheapest stop of a train' problem in its pure-upper-bounded brake, flexible-endpoint setting.