Entropy formulation of optimal and adaptive control

The use of entropy as the common measure to evaluate the different levels of intelligent machines is reported. At the execution level, the design of the desirable control can be expressed by the uncertainty of selecting the optimal control that minimizes a given performance index. By choosing a density function over the set of admissible controls to minimize the differential control entropy, it can be shown that the optimal control problem is equivalent to the problem of minimization of the assigned entropy function with respect to the association control. The adaptive control problem can be analyzed by considering the same entropy over extended space that includes the uncertain parameters. It is shown that the optimal entropy is decomposed into three terms: the optimal control term with given parameters, the parameter identification term, and the equivocation term which accounts for the active transition of dual control. The equivocation when calculated can serve as a measure of optimality of the adaptive control algorithms that involve only distinct identification and optimal control algorithms. An upper bound can be used instead, when the equivocation is hard to calculate. An example illustrates the method