New Interpretations and Justifications for Worst Case Min-Max Design of Linear Control Systems

A method of designing a feedback controller for a linear dynamic system disturbed by additive white noise treats the noise as a maximizing (vindictive) opponent and minimaximizes a quadratic performance criterion which is the difference between the original criterion, for the system performance, and a quadratic function of the maximizing player (noise). This procedure, though well known, is viewed with some suspicion as it appears to be a very pessimistic design method in that it assumes that the noise will “conspire” to oppose the controller in the most perverse way. We show, however, in this paper that the controller obtained in this manner is the same as that obtained by minimizing the expected value of an exponential function of the given quadratic performance criterion. When looked at from this viewpoint the “worst case design” does not appear to be too pessimistic since the exponential criterion is rather appealing. Of theoretical interest is the fact that existence of a solution to the zero sum differential game (which arises in the worst case formulation) implies and is implied by existence of the expected value of the exponential performance criterion.