Optimality with a finite number of trajectory penalties

Abstract

Abstract The state of a continuous system most naturally enters the performance index as a discrete sum if there is a finite set of observation times, or if there are only a few key times over which errors really matter. Such continuous-discrete problems are studied here for linear plants under a broad class of continuous metrics on control and summed metrics on trajectory error. The approach is through the dual optimization problem and alignment properties of the primal and dual solution. The dual problem, is often more attractive computationally than the primal, and is always finite-dimensional as opposed to the infinite-dimensional primal problem. Occasionally the dual problem has a straightforward analytical solution, as in the case of the quadratic continuous-discrete regulator. In any case, information on the structure of the primal control is available through the transformation even before the dual problem is solved. Three examples are detailed involving L1, L2 and L∞ measures on trajectory errors.