Page 209 - Neural Network Modeling and Identification of Dynamical Systems

200 6. NEURAL NETWORK SEMIEMPIRICAL MODELING OF AIRCRAFT MOTION

and moments acting on the aircraft, i.e., The recovery object from the experimental
data with this approach is the set of partial
X = C x ¯qS; Y = C y ¯qS; Z = C z ¯qS; derivatives C z α , C z δ e , ..., C m α , ....
(6.2)
L =C l ¯qSb; M = C m ¯qS ¯c; N = C n ¯qSb. In some cases, when the assumption of small
changes in the parameters is not satisfied, we
The dimensionless coefficients of aerody- leave the terms of the second-order expansion in
namic forces and moments are nonlinear func- the representations for the forces and moments.
tions of several variables; for example, in the In other words, we introduce in these cases non-
typical case linearities into expressions for the coefficients of
forces and moments.
In comparison with the traditional approach,
C x = C x (α,β,δ e ,q,M 0 );
which is based on the linearization of the rela-
C y = C y (α,β,δ r ,δ a ,p,r,M 0 );
tionships for aerodynamic forces and moments,
C z = C z (α,β,δ e ,q,M 0 ); in the semiempirical approach to the modeling
(6.3)
C l = C l (α,β,δ r ,δ a ,p,r,M 0 ); and identification of dynamical systems, we re-
C m = C m (α,β,δ e ,q,M 0 ); store the nonlinear functions C x , C y , C z , C l , C m ,
C n = C n (α,β,δ r ,δ a ,p,r,M 0 ). C n (6.3) as whole objects in the entire range of
changing the values of their arguments. In con-
The problem of identification in this case is trast, in the traditional approach, the derivatives
that it is required to restore the unknown depen- C z α , C z δ e , ..., C m α , ...are restored.
dencies for the available experimental data for
C x , C y , C z , C l , C m , C n .
The system of equations of motion for the air- 6.2 SEMIEMPIRICAL MODELING OF
craft (6.1) can be substantially simplified if we LONGITUDINAL
accept the assumption of small changes in all pa- SHORT-PERIOD MOTION FOR A
rameters relative to some initial (reference) mo- MANEUVERABLE AIRCRAFT
tion. If this condition is satisfied, instead of the
source nonlinear motion model, its variant can In this chapter, we demonstrate that semiem-
be used, linearized with respect to some refer- pirical ANN models (gray box models) pro-
ence motion.
vide highly efficient solutions to applied prob-
With this approach, which is traditional for
lems. We use in this section as the first example
flight simulation problems [7–12], the depen-
the modeling of longitudinal short-period (rota-
dencies for the aerodynamic forces and the mo-
tional) motion of a maneuverable aircraft. These
ments acting on the aircraft in flight are repre- models are based on the traditional theoretical
sented as Taylor series in powers of the incre-
model of aircraft motion in the form of a sys-
ments of the parameters relative to the reference
tem of ODEs. The semiempirical ANN model
flight regime, with terms in the expansions of or- designed in this particular example includes two
der not higher than the first; for example, for the
black box module elements. These elements de-
coefficient of normal force C z = C z (α,δ e ),
scribe the dependence of the normal force and
pitch moment on the state variables (angle of at-
∂C z ∂C z
+ α + tack, pitch angular velocity, and a deflection an-
C z = C z 0 δ e
∂α ∂δ e
gle of the controlled stabilizer) which is initially
δ e . (6.4)
= C z 0 + C z α α + C z δ e unknown and meant to be extracted from avail-