Page 225 - Neural Network Modeling and Identification of Dynamical Systems

216 6. NEURAL NETWORK SEMIEMPIRICAL MODELING OF AIRCRAFT MOTION

acterize the total effect that the errors of these practical due to the difficulties encountered in
function approximations have on the accuracy obtaining the informative data set required to
of the trajectory predictions given by the model. find the axial force coefficient C x . For this reason,
These results can be regarded as completely sat- the overall problem of aerodynamic character-
isfactory. However, it is also of interest to an- istics identification has been divided into two
alyze how accurately the problem of the aero- subproblems. The first subproblem, considered
dynamic characteristics identification has been
in Section 6.3, is the problem of identification
solved.
of the five coefficients C y , C z , C l , C n , C m in the
To answer this question, we need to extract
case of three-axis rotational motion. The second
the ANN modules corresponding to the approx-
subproblem, considered in this section, amounts
imated functions C y , C z , C l , C n , C m and then
to the identification of the remaining axial force
compare the values they yield with the avail-
coefficient C x in the case of longitudinal motion
able experimental data [20]. Integral estimates of
(both translational and angular). In addition, in
the accuracy can be obtained, for example, with
the RMSE function. In the experiments above we [3,4], in order to reduce the computational com-
plexity of the problem being solved, the solution
=
have the following error estimates: RMSE C y
5.4257 · 10 −4 ,RMSE C z = 9.2759 · 10 −4 , RMSE C l = of the modeling and identification problem was
2.1496·10 −5 ,RMSE C m = 1.4952·10 −4 , RMSE C n = not carried out for the full range of possible val-
1.3873 · 10 −5 . The values of the reproduction er- ues of the state variables and the controls for
ror for the functions C y , C z , C l , C n , C m for each the dynamical system under consideration, but
time instant during the testing of the semiempir- only for its part (on the order of several percent
ical model are shown in Fig. 6.10. of the range of values of each of the variables).
In this section, we extract the dependencies for
the coefficients C x , C z , C m on a radically more
6.4 SEMIEMPIRICAL MODELING OF wide range of possible values of their arguments
LONGITUDINAL (for the list of these arguments, see (6.11)and
TRANSLATIONAL AND (6.12)).
ANGULAR MOTION FOR A The identification problem for the axial aero-
¯
dynamic force X as a nonlinear function of the
MANEUVERABLE AIRCRAFT
corresponding arguments is traditionally chal-
lenging to solve (see, for example, [32,33]). Sim-
In this section, we consider the problem of the ilarly, the problem of finding the aircraft engine
longitudinal motion modeling for a maneuver- thrust F T value is difficult [32,33]. We need this
able aircraft as well as the identification prob- value to extract X from the total force R X mea-
¯
lem for its aerodynamic characteristics, such as
sured during the flight experiment. The ANN
the coefficients of aerodynamic axial and nor-
mal forces, and the pitching moment. We solve modeling methods seem to be a promising tool
for the solution of this problem in the same
this problem in the same way as the problems
in the previous two sections, by using a class way as it was for the other aerodynamic char-
of semiempirical dynamic models that combine acteristics identification. This hypothesis is sup-
the possibilities of theoretical and neural net- ported by the theoretical results (see, for exam-
work modeling. ple, [34–36]), which show that an artificial neural
In Section 6.3, it was shown that the simul- network has the properties of a universal ap-
taneous reconstruction of the dependencies for proximator, i.e., it can represent any mapping of
all six aerodynamic forces and moments is im- an n-dimensional input into an m-dimensional