Analytical Solution of Input Constrained Reference Tracking Problems by Dynamic Programming

This paper is concerned with the explicit solution to constrained receding-horizon reference tracking control problems. The goal of this work is, for any scalar reference trajectory, to find the optimal control law for SISO linear systems such that a quadratic cost functional is minimised over a horizon of length N, subject to the satisfaction of input constraints, and under the assumption that the reference is known over the entire horizon. A global solution (i.e., valid in the entire data-space) for this problem, and for arbitrary horizon N, is derived analytically by using dynamic programming. The optimal solution is given by a piece-wise affine function of the data (the initial state of the system and the reference sequence), and the data-space is partitioned into a number of polyhedral regions, inside each of which a unique affine function is applied. From the dynamic programming solution, a clear relationship is exposed between input-constrained reference tracking problems and state estimation problems in the presence of constrained disturbances.