Controlling a linear dynamic system according to asymmetric preferences

In this paper an attempt is made to introduce an asymmetric loss function in the context of a dynamic decision problem, where the target variables and instruments are related linearly, and where uncertainty is introduced by means of additive disturbances. Because of the particular form of the loss function an optimal linear feedback rule and an analytical expression for the minimal expected loss can only be formulated in the case of as many instruments as target variables. In the case of more targets than instruments an open-loop policy has to be adopted. Numerical experiments with a small artificial model, in which this open-loop policy is compared with the closed-loop policies resulting from two quadratic approximations are presented. Both in the case of a deterministic as well as a stochastic model the open-loop policy turns out to be better than or at least as well as the other approaches.