Kaplan, Sargent and Goodman 
wy py. SAS (EL OP SL A PERM (2) 
where the weighting factors w are chosen to make each sum non- 
dimensional and of the same order of magnitude. Thus, the solution 
of (1) is sought which is in best agreement with the measurements 
in a least square sense. The parameter vector, a , is suppressed 
in (1) by considering its components to be additional state variables 
subject to the equation 
a= 0 (3) 
The number n is thereby increased to include the additional state 
variables and the extended c vector includes the unknown parameter 
vector in addition to the state variable initial conditions. 
The parameters of the system are determined in the following 
way : the initial vector is estimated and (1) is integrated. The es- 
timated initial vector is denoted by c”* and the resulting solution of 
(1) by Y*. The deviation can then be calculated and its value denoted 
by e*. Assuming the initial vector to be changed by an increment 
6c, this would cause the solution vector to be changed by an incre- 
ment 6Y and the deviation by an increment 6e . From (2) itis 
seen that 
M 
M, _ 
i= wi, by ai ly, (t_) sm bY, (tat... (4) 
igay = il 
The equations which the incremental solution vector safisfies are 
called the equations of differential corrections and are obtained by 
expanding (1) in a Taylor series and retaining only linear terms : 
n 
ad 
bY, (t) = ) 08; 8, (5) 
ait tay 
: j 
1636 
