Saturating Servomechanism to Follow a Random Process†

Abstract

Abstract : Solutions to the problem of obtaining the optimal bounded input to a servomechanism which must follow a discrete-time random process are given in the case of a first order system. If the random process is a finite Markov chain, solutions can be obtained for any servomechanism with finite state vector, with each component having only a finite range. The cost function is any function of the state and of the control, which is restricted to a finite set of values. The generalizations of these problems can be done in various directions. Keeping the assumptions of a discrete-time random process x(t) and the form of the cost functional, which both ake sense in engineering applic tions, a first extension would be o consider servom c anisms of order greater than one, in the cases of independent and of M rkov dependent random variables. As a further extension, the cost functional could include a function of the control itself, instead of depending only on the error. In the finite case, when the inputs and the servomechanism take values in finite sets only, the problem is theoretically solved by application of the minimization procedure in a product space of sufficiently high dimensions. However, this b comes rapidly numerically unworkable as the number of variabl s increases. How to reduce the dimensionality of the problem is then worth a careful investigation. (Author)