Optimal saturated feedback laws for LQR problems with bounded controls

VICENTE COSTANZA; PABLO S. RIVADENEIRA

Computational and Applied Mathematics

SP Birkhäuser Verlag Basel

Año: 2013 vol. 32 p. 355 - 371

0101-8205

Optimal and suboptimal strategies are substantiated and illustrated for linearquadraticproblems with penalized endpoints, when bounds in control values are imposed.The optimal solution for a given process with restricted controls, starting at a known initialstate, is shown to coincide with the saturated solution to some unrestricted problem thathas the same coefficients, except for the final penalization matrix S, and starts at a generallydifferent initial state. This result reduces the searching span for the solution: from the infinitedimensionalset of admissible control trajectories to the finite-dimensional space of symmetricpositive semi-definite symmetric matrices ?S and initial states ?x0. An efficient scheme is alsoproposed to approximate (and eventually to find) the optimal feedback strategy on-line, basedon the updating of ?S at successive sampling times tk , and on the possibility to generate thecorresponding Riccati matrix P(t, T, ?S) for tk < t ≤ tk+1 from auxiliary matrices stored inmemory. Numerical simulations are provided, compared, and checked against the analyticalsolutions of two classical case-studies.