## Date of Award

Spring 1969

## Document Type

Dissertation - Restricted

## Degree Name

Doctor of Philosophy (PhD)

## Department

Electrical and Computer Engineering

## Abstract

Since 1827 when Robert Brown, a biologist, first discovered "Brownian Motion," the analysis of stochastic systems has been considered by many investigators. They have studied conditional probability functions, moments and sample characteristics of these systems. More recently the stability and optimal control of these systems has been treated using time domain techniques such as dynamic programming and stochastic Liapunov functions. The purpose of the present investigation is to make use of time domain techniques in developing new engineering design and analysis methods for the class of stochastic systems modeled by Ito's stochastic differential equation. In the study of this class of systems a special system, called the random gain system, will be used extensively. The random gain system is very similar to the Lurie system. If the nonlinear element in a Lurie system is replaced by a time-varying random gain characterized as Gaussian white noise with known first and second moments, then the system is called a random gain system. The stability of this system is studied using stochastic Liapunov theorems developed by Kushner for asymptotic stability in the large with probability one (A.S.I.L.-W.P.1). A frequency domain criterion is developed such that the unforced system is stable (A.S.I.L.-W.P.l) and the forced system is L2-bounded input - bounded output stable with probability one. A frequency criterion is also derived for the discrete or sample-data version of the random gain system such that it is stable (A,S,I.L.-W,P.l). Using the technique of vector Liapunov functions, a method is developed to study the stability (A,S,I,L.-W.P.l) of a certain class of large complex stochastic systems. The stability of the complex system is determined from the stability of its subsystems and the interconnections. This method is somewhat conservative; however its value lies in the ease with which is applied to large complex systems as is illustrated by the seventh order example. The random gain system is also optimized for finite and infinite time intervals. The optimal control for a quadratic cost functional is determined from the solution of a matrix differential equation, similar to the matrix Riccati equation for the deterministic regulator problem. or the infinite time optimization problem, it is shown that optimal control is determined from a steady state solution of this matrix differential equation. Sufficient conditions are given for the existence and stability (A.S,I.L,-W.P.l) of this optimal control law. In general, the results presented here provide new and simple methods for design and analysis of closed loop systems subject to random time-varying gain fluctuations.