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# H-Infinity, Cumulants, and Games

thesis

posted on 2006-08-04, 00:00 authored by Ronald Walter DiersingCost cumulant control is a generalization of linear quadratic Gaussian (LQG) control. Instead of one cost cumulant being used, more cumulants are utilized. This has produced encouraging results on vibration problems. Other interesting areas are H2/H-Infinity control and H-Infinity control. With stochastic noise present, these techniques can be reformulated as games involving the mean of the players' cost. Through cost cumulants, this dissertation generalizes H2/H-Infinity and H-Infinity control. Results from the discrete time cumulant control theory are reviewed, with some novel applications. Then the discrete time minimum cost variance (MCV) problem is applied to the first generation earthquake structural benchmark. The discussion includes output feedback conditional cumulant control. A performance index consisting of a linear combination of conditional cumulants is used and a control law is found. With the aid of these results, a discrete time, cost variance game is given. A two player, Nash game and its generalization of H2/H-Infinity are discussed. Furthermore, a two-player, zero sum game is developed. To make these results more general, an N-player, cost variance, Nash game is given. Attention is also directed to the continuous time case. Both non-zero and zero sum games are developed. At first this is for the case of two cumulants, and then for k cumulants. Furthermore, an N player, k cumulant Nash game is presented. An H-Infinity connection is made and the results are applied to a four story building. The case of the control minimizing the k-th cumulant, with the disturbance minimizing the mean of its cost, is also examined. Generalization of H2/H-Infinity control is made and a new multi-objective cumulant (MCC) control problem emerges. MCC control is applied to various building problems. Then the output feedback problem is discussed. An MCC solution is determined for two cases, one in which the disturbance has full state information, and one where it does not. The results are applied to the first generation earthquake structural benchmark. Finally, a minimax game involving the mean of conditional cumulants is developed. The problem is one where the players both have partial state information. An equilibrium solution is determined.

## History

## Date Modified

2017-06-02## Defense Date

2006-07-28## Research Director(s)

Dr. Paulo Tabuada## Committee Members

Dr. Peter Bauer Dr. Paulo Tabuada Dr. Panos Antsaklis Dr. Michael K. Sain## Degree

- Doctor of Philosophy

## Degree Level

- Doctoral Dissertation

## Language

- English

## Alternate Identifier

etd-08042006-124801## Publisher

University of Notre Dame## Program Name

- Electrical Engineering

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