Kaplan, Sargent and Goodman 
by use of this method, are given in following sections of this paper. 
When considering the case of a craft in waves, the effect of 
continuous random forcing functions (due to the waves) is present. 
Therefore another technique different from the one used in the pre- 
vious work described above, which is based on transient responses 
with no ''noisy'' measurements or random forcing functions, must be 
used. The method proposed for application to this problem is based 
on developments in recent literature of modern control theory 
(maximum principle, two-points boundary value problems, invariant 
imbedding, and sequential estimation) which are described in [6] 
and [7] . The basic technique is applied to problems that are gener- 
ally nonlinear, with the possibility of measurement observation 
errors and with unknown random inputs. Using continuous time his- 
tories of the observed output measurements, the task is then to 
obtain optimal estimates of the state variables and also various 
parameters in the equations (such as coefficients and other unknown 
magnitude mathematical forms) by a procedure that is based on 
minimizing an integral of the sum of weighted squares of residual 
errors. The errors are the difference between the observed data and 
the actual desired system outputs (i.e. eliminating the measurement 
noise), and also the difference between the nominal trajectory of the 
system and the assumed form of the equation representation (i.e. 
eliminating the noisy input excitation and achieving a proper repre- 
sentation of the basic system dynamics). In this case, the unknown 
parameters are also added as additional variables in the complete 
dynamic representation. 
The equations that result for the estimates of the system 
state and also for the parameters provide an on-line filtering pro- 
cedure together with a sequential estimation technique, which does 
not require repeating all calculations after additional observations 
or measurements are made, as in classical estimation schemes. The 
resulting equations are of a form that is somewhat similar to that of 
the Kalman filter [8], but they are applicable to nonlinear systems. 
In addition the terms entering the equations are not dependent upon a 
knowledge of the statistical characteristics of the input disturbances 
or the measurement errors, thereby allowing consideration of 
vehicles in arbitrary seaway conditions and hence increasing the 
generality of the approach. 
The equations developed for this system identification pro- 
cedure use the continuous measurements of the actual system outputs 
as observed, and those signals are operated on and processed with 
the special estimator equations. As time evolves the combined fil- 
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