The Structure of Optimum Control Systems

A large class of optimum control problems can be formulated as the variational problem of minimizing a known functional subject to isoperimetric and nonholonomic constraints. The (vector) Euler equation for this problem leads directly to the structure of the optimum controller, which turns out to comprise a dynamic portion which is the adjoint of the plant to be controlled and instantaneous nonlinear elements determined by the performance functional and input constraints. Continuous measurement of the state of the plant results in the elimination of the dynamic portion, and the entire optimum controller is instantaneous. An example is given which illustrates the complete design of regulators for a simple plant with constraints on either amplitude or energy of the actuating signal which minimize response time or integrated square error.