Bi-Cumulant Games: A Generalization of H∞ and H2/H∞ Control

Cost cumulants have been used more frequently in the recent literature and have demonstrated their usefulness in controlling the vibrations of buildings that are subject to high winds and seismic disturbances. This paper extends the use of cumulants into stochastic game theory. The development is for a class of nonlinear systems with non-quadratic costs, over a finite time horizon. Sufficient and necessary conditions for Nash and minimax equilibrium solutions are given; and, for the case of linear systems with quadratic costs, the equilibrium solutions are determined. In the process, coupled Riccati equations are found. It is shown that these bi-cumulant games constitute a generalization of H∞ and H2/H∞. Moreover, the H∞ generalization is applied to the control of a four-story building under a seismic disturbance; and the results are compared with those of linear quadratic gaussian, minimum cost variance, and H∞ control.