An Analytic Theory of Modeling for a Class of Minimal-Energy Control Systems (Disturbance-Free Case)

A quantitative theory is developed for modeling a class of optimal control systems. A mathematical representation—a model system—is fit to an actual system solely on the basis of the respective optimal performances of the two systems, where performance is defined by a generalized quadratic criterion of the minimum energy, minimal endpoint-error type. The plant to be controlled is assumed to be linear time-varying (at least in the small), and the model is taken to be linear, but constant-coefficient. Necessary and sufficient conditions are derived for achieving certain pertinent tasks of performance prediction and optimal control, wherein particular attention is paid to the accomplishment of these tasks by computer methods. It is found that the very structure of the plant representation may prohibit some model activities, e.g., if a certain inequality relation is not maintained between the respective dimensions of the state and control vectors. Finally, the given performance index is used to partition the universe of linear systems into equivalence classes, and the conditions are presented for two systems to be performance-equivalent. These are shown to be the necessary and sufficient conditions for the optimal control laws of nonidentical systems to be, in fact, interchangeable in the large.