Dynamites of Naval Craft - System Identification 
unknown parameters ina. The equation given in Equation (34) is 
easily absorbed into the total system representation in establishing 
the g-matrix, and the remaining equations for P readily follow. 
The only problem resulting from the introduction of the additional 
elements is the increase in the total number of equations to be solved 
which increases the computational complexities. Illustrations of the 
results obtained by application of the equations presented above are 
given in the following sections of this paper. 
APPLICATIONS OF SEQUENTIAL ESTIMATION TECHNIQUE 
The digital computer program established for solution of 
the coupled estimator and gain equations is structured to carry out the 
solution of the equations as a group of coupled first order ordinary 
differential equations. These equations are nonlinear and time-varying, 
in general, and the technique for solution is based upon use ofa 
Runge-Kutta fourth order integration scheme. The integration step 
size is not arbitrary, since there is a maximum basic frequency of the 
vehicle motion modes for any particular craft, and hence sampling of 
any data would have to be made at twice the rate of the largest frequen- 
cy manifested in the system response in accordance with the requi- 
rements of the sampling theorem [23] . The integration time step 
can be less than this amount, and the values to be used would depend 
upon requirements of stability, total time of solution, bandwidth pro- 
perties, etc. All of the computer operations for a general equation 
system are carried out in matrix form, as indicated by the represen- 
tation of the equations given in the preceding section, and various 
subroutines to make use of matrix manipulations are employed, which 
are standard procedures associated with digital computer operations, 
The results of application of these equations for various illustrative, 
cases and for different naval craft are provided below. 
In order to illustrate the capabilities of the basic method of 
analysis, a series of computational experiments were carried out on 
simple systems with known parameter values. The first problem 
considered is that of a second order system, where it is required to 
find the two unknown parameters, and the second problem is to obtain 
one of the coefficients in a third order system. The test cases are 
selected with no external forcing function, thereby being transient 
response trajectory data as input information. These particular pro- 
blems are similarto test problems solved using analog computers in 
[14] , with noisy forcing functions there, and serve to validate the 
digital computer program and procedures. 
The first problem is represented by the differential equa- 
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