In analyzing systems and their control, one must find a way to 
represent the unpredictable disturbances. Such disturbances cannot be 
modeled by analytic functions since the value of an analytic function at 
any point is predictable from its value on an arbitrary short interval. 
One answer to modeling these disturbances is to describe them as stochastic 
sEoeaeses.” The theory of such processes was developed to model the 
fluctuation observed in physical systems. Wiener processes or the 
Brownian motion process are of particular interest to the stochastic 
control problem; many of the disturbances that affect a control system 
can be modeled by processes generated from Wiener processes. A Wiener 
process is a stochastic process in which the statistical properties over 
the interval (t, ttt) are the same as those over the interval (s, s+t); 
moreover, the behavior of the process is independent over time intervals 
which do not overlap, and there is no trend in the behavior. 
Once the stochastic disturbances have been introduced into the 
control theory, the problem is no longer deterministic. The state 
variables and control variables are no longer predictable but must be 
described by their statistical properties. Kalman and Bue provide a 
solution to the stochastic control problem for nonstationary linear 
systems. Their solution consists of using an optimal filter to estimate 
from the observed system performance the state of the system in terms of 
the conditional mean; the estimated state is fed back to the control 
signal through linear feedback. The linear feedback is determined by 
solving a deterministic control problem; the filter depends on the 
disturbances and on the system dynamics, but it is independent of the 
cost. Although the nonlinear stochastic control problem or its equiva- 
lent, the nonlinear filter problem, has not been solved, some headway 
has been made by Bucy and Jeaeping this lecture considers only the 
linear problem. 
Aslerem, K. J., "Introduction to Stochastic Control Theory," Academic 
Press, Inc., New York (1970). 
alle, R. E. and R. S. Bucy, "New Results in Linear Filtering and 
Prediction Theory," Journal of Basic Engineering Series D, American 
Society of Mechanical Engineers, Vol. 83, pp. 95--108 (1961). 
“ney. R. S. and P. D. Joseph, "Filtering for Stochastic Processes with 
Applications to Guidance," Interscience Publishers, Inc., New York (1968). 
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