Kaplan, Sargent and Goodman 
No assumptions regarding the statistics of the unknown input functions 
or the measurements error is made. With measurements of the out- 
put y(t), for 0<t <T, itis required to estimate x(T) on the basis 
of minimizing with respect to x(t) (a nominal trajectory) the func- 
tional 
where w(x,t) is a positive weighting factor, and the errors e, and 
1 
e, are defined by 
y(t) - h(x, t) (24) 
a} 
= 
— 
ct 
— 
iH] 
a2 (f): 22° = Gt) (25) 
The least squares estimate of x(T), denoted as x(T), is obtained 
from minimizing the integral of the (weighted) mean square errors, 
where the error e, (t) represents the difference between a nominal 
trajectory and the assumed form of its equation representation. 
The minimization problem is then a problem in variational 
calculus, which leads to the associated Euler-Lagrange equations 
that contain an unknown Lagrange multiplier. The boundary conditions 
for this Lagrange multiplier are known at the ends of the interval, 
i.e. Oand T, but there is no information about the value of x(T), 
and hence the problem reduces to a two point boundary value problem 
(TPBVP) that yields the optimal estimate x(T). With the variable T 
now considered as a running time variable, the problem is treated as 
a family of problems with different final points, T, and the problem 
becomes one of sequential estimation, i.e. the TPBVP must be 
continuously solved for all values of T (the running time variable). 
The problem is solved by application of the concept of in- 
variant imbedding [12] , which is used to converta TPBVP into 
an initial value problem that can be easily solved. The missing 
"initial condition'' is represented in a general manner for different 
values of T, thereby establishing a family of problems. On the basis 
that neighboring processes (i.e. system responses) are related to 
each other, the missing condition is found by examining the relation- 
ships between such neighboring processes. The procedure leads toa 
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