problem is given by the separation ihaoreemn = The solution consists of 
an optimal filter for estimating the state of the system from the ob- 
served data and a linear feedback of the estimated state of the system; 
see Figure 7. 
OBSERVED 
DATA 
PROCESS 
OPTIMAL 
FILTER 
CONTROL SIGNAL 
LINEAR 
FEEDBACK 
ESTIMATED STATE 
Figure 7 -- Stochastic Control System 
The optimal filter is the Kalman-Bucy filter, which will be dis- 
cussed in detail in the next section; the linear feedback is the same as 
would be obtained if the state of the system could be measured exactly 
and if there were no randum disturbances in the system. Thus, the 
linear feedback can be determined by solving a deterministic problem. 
Because of time limitations, we will not prove but merely accept the 
separation theorem. 
One objection to the use of stochastic control theory is that the 
process to which the theory is applied may not be random but merely 
irregular. For instance, the traffic flow on the Washington Beltway may 
not be truely random but it is certainly highly irregular. If I need to 
reach Dullis Airport from DINSRDC by 1 pm, it might take me 45 to 50 
minutes; but to reach the airport at 6 pm, I would have to allow 2 
hours. The reason for this variation in lead time is that there will be 
bumper-—to-bumper traffic on the Beltway during the rush hour and any 
accident brings this traffic to a halt. It is not the microscopic but 
the macroscopic properties of the traffic flow that govern our lead time 
estimate. The traffic flow could be analyzed as a stochastic process; 
such a model would be acceptable provided it predicted the macroscopic 
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