Dynamtes of Naval Craft - System Identtficatton 
(the asterisk means that the coefficients are calculated using solu- 
tion Y™*). Equation (5) is now integrated n times; the jth time 
the integration is performed the initial conditions are that 6 Y; 
(0) = 1 and all the other 6Yj (0)'s vanish. This special solution is 
denoted by 6Yij , and the general solution can then be written, by 
superposition, ‘as 
n 
= ¥ oi Gt 
bY, > bes 8 i; | ) (6) 
jel 
This incremental solution vector is used to express 46Y, , OMgiry 
. interms of 6c, and upon substituting into (4) and interchanging 
the order of summation the variation of the deviation becomes 
where the repeated suffix implies summation on j from 1 to n, 
The variation of the deviation has thus been expressed directly in 
terms of the variation of each of the initial conditions. In order for 
¢€ to be minimum de must vanish for an arbitrary variation in the 
initial conditions. This means that if U; is defined to be 
Waa Saicing, ano y, (t_) “bn éY,, (ti bp Be ns (8) 
igayresioal 
thenthe error will be minimized with respect to the ae whenever. 
Car si Oy ape botaion (9) 
In general, using the estimated vector c* and the resulting 
solution vector y*, the values of Uy will not vanish. Denoting the 
value of Uj , as calculated in this way by u; , the objective is 
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