Stochastic control of linear systems with structural changes with applications to economic systems

A very important class of problems deals with the control of systems whose structure can undergo changes. The simplest class of changes are the ones that affect only the system parameters and not the rest of the basic structure of the system like order. In this paper we examine the class of linear, discrete time, multiple-input multiple-output systems, with additive system disturbances and exact output measurements. This class of models is also appropriate for describing some economic processes. The assumption will be that the system parameters can change according to a known low. The objective is to find the sequence of controls that will minimize a known cost of the output and the control sequence. We have assumed a quadratic cost. If the noise statistics and the initial conditions are assumed known, then the problem is an adaptive control one. We use dynamic programming to solve the resulting dual adaptive structure. We use an approximation to the optimal cost-to-go that renders the solution quite tractable. The approximation results in an actively adaptive law because it takes into account future measurements. The computational overhead required of this approximation is not too large, a property that renders the control law quite feasible. Some thought is given to what happens when some of the noise statistics are assumed not known with certainty, a case that is most appropriate in models of econcmic processes.