A multiobjective cost cumulant control problem: the Nash game solution

Recently cost cumulant control has been effectively applied to vibration control problems. Similarly so have multiobjective control paradigms. In this paper the cost cumulant control idea is used to solve the case when the system has some structured uncertainty, with two players, the control and disturbance, having their own performance indices. The development at first is done for a class of nonlinear systems, non-quadratic cost and then is later applied to the linear, quadratic special case. The two performance indices will be linear combinations of cumulants, in which one shall be for the control and the other will constrain the input-output relationship of the disturbance to a output. The control solution is then applied to the First Generation Benchmark for a 3-story building under seismic excitation. The results will be compared with those of several different controllers.

I. Introduction

In recent times the cost cumulant (kCC) and the minimum cost variance (MCV) control problems have gained attention [7]–[12], [13]. These control methods generalize the approach of minimizing the mean of a cost function that is so prevalent in the area of control. They let the control minimize a linear combination of the cumulants. In the MCV problem this means minimizing a linear combination of the mean and the variance of a cost function, whereas the kCC goes beyond these two most well known cumulants. These methods have been applied successfully to vibration control problems, in particular the control of structures excited by winds and seismic disturbances. Also, there has been the application of multiobjective methods [5] to this same problem. The approach taken in this paper is to combine these two problems. The idea presented is to allow the control to minimize a linear combination of the first two cumulants while satisfying some constraint on the systems induced norm. In doing this a Nash game approach shall be taken in a manner similar to that of [6], [3]. The development will be carried out for a class of nonlinear systems with non-quadratic costs. It shall then be applied to the case when the system is linear and costs are quadratic. The Nash game shall involve two players, a control and a disturbance. Later the disturbance will be given as the result of some “structured” uncertainty inherent in the system. Lastly the control will be applied to the First Generation Structural Benchmark for buildings under seismic excitation.