A Multiobjective Cumulant Control Problem

The H2/H∞ control problem is well known in the control community. It mixes the results of two powerful control techniques; to balance two objectives: minimizing the H2 norm of the system, while constraining the system’s H∞ norm. In the presence of random noise, this is akin to solving a Nash game with the players’ objectives to minimize the mean of their costs. In this chapter, recent trends in minimizing further cumulants will be analyzed, in particular one in which wishes to minimize the variance and other cumulants of a cost, while constraining the system’s H∞ norm. This problem formulation will begin for a class of nonlinear systems with nonquadratic costs. Sufficient conditions for a Nash equilibrium for a two player game in which the control wishes to minimize the variance of its costs and the disturbance wishes to minimize the mean of its cost are found. The case of linear systems and quadratic costs is applied and equilibrium solutions are determined. Further cumulants are also examined. The results of the control formulation are applied to a problem in structural control, namely, the third generation structural benchmark for tall buildings subject to high winds.