Optimal Saturated Feedback Laws for LQR Problems with Bounded Controls

Abstract

Optimal and suboptimal strategies are substantiated and illustrated for linear-quadratic problems with penalized endpoints, when bounds in control values are imposed. The optimal solution for a given process with restricted controls, starting at a known initial state, is shown to coincide with the saturated solution to some unrestricted problem that has the same coefficients, except for the final penalization matrix S, and starts at a generally different initial state. This result reduces the searching span for the solution: from the infinite-dimensional set of admissible control trajectories to the finite-dimensional space of symmetric positive semi-definite symmetric matrices Ŝ and initial states x̂ 0. An efficient scheme is also proposed to approximate (and eventually to find) the optimal feedback strategy on-line, based on the updating of Ŝ at successive sampling times tk, and on the possibility to generate the corresponding Riccati matrix P(t,T,Ŝ ) for tk<t≤tk+1 from auxiliary matrices stored in memory. Numerical simulations are provided, compared, and checked against the analytical solutions of two classical case-studies.