Optimal Bounded Control of Linear Sampled-Data Systems With Quadratic Loss

A method is presented for the computation of optimal control for linear sampled-data systems when the control variable is a bounded scalar. It is shown that for this problem the optimal control is a piecewise linear function of the state and may be computed by piecewise iteration of suitable recurrence relations. The optimal control is presented in terms of the control coefficients (matrices) and the regions to which they apply. No solution other than computer storage is suggested for the synthesis of these controls. In the second section of the paper, it is shown that the method applies with trivial modification to the random-input, random-observation noise case. The optimal control law has the same form as the deterministic case with the conditional expectation used in the control law in place of the stale itself. A simple deterministic example computed on an IBM 1620 is presented. As might be expected, the computer capacity required for the problem is intermediate between the unbounded control case, where the control is linear, and more general problems.