THE KALMAN-BUCY FILTER 
The solution of the optimal control problem for a linear stochastic 
system is given by the separation theorem. It consists of an optimal 
filter for estimating the state of the system from the observed data and 
a linear feedback of the estimated state of the system; see Figure 7. 
The linear feedback is the same as the feedback that would be obtained 
if there were no stochastic perturbation of the system. This section 
will develop the explicit computational schemes for solving the filter- 
ing problem. 
Suppose we have the stochastic process described in the previous 
section 
dx = A x dt +d w(t) Gr BD) 
x(0) =e (5.2) 
where w(t) is a Wiener process and c is a Gaussion zero mean n-vector. 
In an actual case in which the process is realized, it is important to 
know the state of the system. It is, however, not always possible to 
measure x directly; instead, a set of quantities z(t) dependent on x are 
measured. Assume that the dependence of z on x is linear and is given 
by 
dz = H x dt + dv Ges) 
where the perturbation v is a Wiener process independent of x. 
The filter problem can be formulated as follows. Assume that a 
realization of the output z has been observed over the interval 
0 <T< t. Determine the best estimate of the value of the state vector 
x at time t. It is assumed here that the admissible estimates of x are 
linear functionals F(z) of the observed output z. The criterion 
5)3) 
