On a property of optimal controllers with boundedness constraints

This paper deals with a theory for optimal control systems designed to operate a plant of known characteristics. It is assumed that error-free measurement of the system input and the system behavior is permitted, but only limited changes in the system characteristics can be effected by the control variables at the designer´s disposal. It is shown that within this limitation for a wide class of inputs and systems, and for a certain class of measures of the system performance, an optimal system behaves as a relay or switched system during the transient period, and as a continuous system during periods in which the input is produced identically. A procedure is described for determining the switching times during the transient period in terms of the permissible measurements. The result is the design of the optimal controller. Typically, its realization requires analog computation of the switching function and digital switching of the control variables. The design of a second-order regulator system, in which the control variable is the gain in the feedback path, is obtained. Marked improvement in the system performance is noted.