Use of dynamic programming for the analytical solution of input-constrained LQR problems

This paper addresses the input-constrained LQR problem, i.e., the problem of finding the optimal control law for a linear system such that a quadratic cost functional over a horizon of length N is minimised subject to the satisfaction of input constraints. A global solution (i.e., valid in the entire state space) for this problem, and for arbitrary horizon TV, is derived analytically by using dynamic programming. The scalar input case is considered in this paper. Solutions to this problem (and to more general problems: state constraints, multiple inputs) have been reported recently in the literature; for example, approaches that use the geometric structure of the underlying quadratic programming problem and approaches that use multi-parametric quadratic programming properties. The solution by dynamic programming proposed in the present paper coincides with the ones obtained by the aforementioned approaches. However, being derived using a different approach that exploits the dynamic nature of the optimal control problem to obtain an analytical solution, the present result complements the previous methods and reveals additional insights into the intrinsic structure of the optimal solution.