On the set of obtainable reference trajectories using minimum variance control (Q1106078)

This paper deals with the problem which target paths can be ultimately tracked by means of minimum variance control (MV), if the base system is described by a linear, finite-dimensional, time-varying difference equation. A target path which has this property is called admissible. It is shown that every MV-admissible target path satisfies a recurrence equation which corresponds to the base system. However, this intuitively very appealing result is not yet sufficient to conclude that a target path is admissible. An additional necessary and sufficient condition is derived. Loosely speaking this condition tells us that components of the system which show up in the part of the system that cannot be controlled, must be stabilized by the closed-loop system matrix. These two establishments are particularly interesting for economics. Therefore, the results are interpreted in a number of economic settings. In section 5, the effect of limited control possibilities on the admissibility property of a target path is analyzed. Two approaches are taken. First, control costs are introduced in the minimum variance cost criterion which are related with the MV-controller. By minimizing this extended cost criterion a new, more cautious, controller is obtained. If this controller is used to regulate the system, it turns out that every admissible target path takes explicitly account of the dynamics of the desired control variables. In the second approach it is assumed that every control variable may vary only within a prescribed interval. Consequently, the control scheme becomes nonlinear, which makes a nice characterization of the admissible target paths impossible. Moreover, the control scheme may result in that case into a bang-bang controller. So, the first approach seems to be preferable. Throughout the paper, results are interpreted for time-invariant systems and illustrated by means of a simulation study. Finally, the effect on the admissibility conditions is discussed if the base system is disturbed by white noise.