Dynamites of Naval Craft - System Identtfication 
partial differential equation that is solved by an expansion of the 
solution about the desired reference condition, which in the present 
case is the estimate x(T) (see [71). The result of this invariant 
imbedding approach is a sequential estimator, which is such that 
previous data points do not have to be repeated whenever new obser- 
vations are added, and hence the estimation operation can be carried 
out at a fast computational rate. 
The estimator equations for the scalar case are 
aT elas) it 2 P(T) h, (x, T) fe(r) 4 nT) (26) 
dP Ass be x a A a an 
a 2P(T)g*(x,T) + 2P - h(x, T) Ee ne, 7) P 
1 
2w (x, T) (27) 
where 
_ 9h(x, T) __9g(x, T) 
i Ss ay PTR OS ax ani 
The above results are somewhat similar to, and represent a genera- 
lization of the results of linear Kalman filtering [8] . The weighting 
function P(T) is found from a Riccati-type equation, and the two 
equations are solved when given the initial conditions. The initial 
value x(0) represents the best estimate of the system state at 
t = 0, which is based on available a priori information, and the 
initial value P(0) reflects the confidence in the initial value of x 
and the observed signal y(t). 
The estimator equations for the vector case are derived in 
[7] and are given below as 
= g(%,T) + 2P(T) H(%,T)Q | ¥(T) - ne, T) “2 
& = g(x, T) P + Pg-(x, Ty (30) 
+ 
2P ES | x(x) - h(x, of | a + k(x, t)v 1(z, T)k'(x, T) 
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