Page 217 - Neural Network Modeling and Identification of Dynamical Systems

208 6. NEURAL NETWORK SEMIEMPIRICAL MODELING OF AIRCRAFT MOTION

gle of the stabilizer δ e and the angle of attack α: box modules that represent the normal and lat-
=−0.5. In a similar way, we can eral force coefficients, as well as the pitch, yaw,
C L α = 0.5, C m α
compute derivatives for any other combinations and roll moment coefficients, each of which de-
of the values of the arguments for the functions pends nonlinearly on several parameters of the
C L and C m . aircraft motion. These five dependencies need
Based on these results, we can conclude that to be extracted (restored) from available experi-
the semiempirical neural network modeling ap- mental data for the observed variables of the dy-
proach, which combines domain-specific knowl- namical system, i.e., we need to solve the identi-
edge and experience with computational math- fication problem for the aerodynamic character-
ematics methods, is a powerful and promising istics of the aircraft.
tool potentially suitable for solving complicated The proposed approach to the identification
problems of describing and analyzing the con- of aerodynamic characteristics of an aircraft dif-
trolled motion of aircraft. Comparison of the re- fers substantially from the traditionally accepted
sults obtained using the semiempirical approach way of solving such problems. Namely, the tra-
with the traditional (black box) ANN modeling ditional approach [7–11,23–29] relies on the use
(NARX-type models) approach shows the defi- of a linearized model of the disturbed motion
nite advantages of semiempirical models. of an aircraft. In this case, the dependencies
for the aerodynamic forces and moments act-
ing on the aircraft are represented in the form of
6.3 SEMIEMPIRICAL MODELING OF the Taylor series expansion, truncated after the
AIRCRAFT THREE-AXIS first-order terms (in rare cases after the second-
ROTATIONAL MOTION order terms). In such a case, we reduce the so-
lution of the identification problem to the recon-
In the previous section, we have demon- struction of the coefficients of the Taylor expan-
strated the effectiveness of the semiempirical sion using the experimental data. In this expan-
approach to ANN modeling of dynamical sys- sion, the dominant terms are the partial deriva-
tems by applying it to the problem of longitudi- tives of the dimensionless coefficients of aerody-
nal angular motion of the maneuverable aircraft. namic forces and moments concerning the vari-
This task is a relatively simple one, due to its ous parameters of the aircraft motion (C z α , C y β ,
low dimensionality and, more importantly, due C m α , C m q , etc.). In contrast, the semiempirical ap-
to the use of single-channel control (pitch chan- proach implements the reconstruction of the re-
nel, a single control surface is used, namely an lations for the coefficients of the forces C x , C y ,
all-movable stabilizer). In this section, we solve C z and the moments C l , C n , C m as whole non-
a much more complicated problem. We will de- linear dependencies from the corresponding ar-
sign the ANN model of three-axis rotational mo- guments. We perform this reconstruction with-
tion (with three simultaneously used controls: out resorting to a Taylor series expansion for the
stabilizer, rudder, and ailerons) and perform the aerodynamic coefficients. That is, the functions
identification for five of the six unknown aero- C x , C y , C z , C l , C n , C m themselves are estimated,
dynamic coefficients. and not the coefficients of their series expan-
As in the previous case, the theoretical model sion. We represent each of these dependencies
for the problem being solved is the correspond- as a separate ANN module embedded into the
ing traditional model of aircraft motion, which semiempirical model. If the derivatives C z α , C y β ,
contains some uncertainty factors. To eliminate C m α , C m q ,etc.are required forthe solution of
the existing uncertainties, we form the semiem- some problems, for example, for the analysis of
pirical ANN model, which includes five black the stability characteristics and controllability of