Page 221 - Neural Network Modeling and Identification of Dynamical Systems

212 6. NEURAL NETWORK SEMIEMPIRICAL MODELING OF AIRCRAFT MOTION

The structural diagram of the semiempirical If we expand the initial theoretical model
model corresponding to the system (6.6)–(6.9) (6.6)–(6.9) by adding the equations for the trans-
is quite cumbersome and thus not shown here. lational motion of the aircraft as well as the
This diagram is conceptually similar to the one equations that describe the engine dynamics,
shown in Fig. 6.5; however, it includes a much it becomes possible to reconstruct all the six
larger number of elements and connections be- functions C x , C y , C z , C l , C m , C n by training the
tween them. Most of these elements correspond semiempirical ANN model. This problem is con-
to the additional terms in the initial theoret- ceptually similar, although the model training is
ical model and do not contain any unknown somewhat more time consuming due to the in-
tunable parameters. Also, the ANN model of creased dimensionality.
the system (6.6)–(6.9) contains five black box– As already noted, to ensure the adequacy of
type ANN modules that represent unknown de- the semiempirical ANN model being created,
pendencies for the coefficients of aerodynamic we require a representative (informative) train-
forces and moments (C y , C z , C l , C n , C m )tobere- ing set that describes the response of the sim-
constructed, as compared to only two modules ulated object to control signals from a given
(C z , C m ) for the system (6.5). range. These constraints on the values of con-
It is important to note that since we con- trol signals, in turn, lead to the constraints on
sider the problem of aircraft short-period angu- the values of the state variables that describe the
2
lar motion modeling, we can assume the alti- system. Adequacy of the designed model can
tude H and the airspeed V to be constant (these only be ensured within the corresponding do-
variables do not change significantly during the main of values for control and state variables,
transient time). This assumption allows us to re- which is formed by the constraints mentioned
duce the initial theoretical model by eliminating above.
the differential equations for translational mo- In the computational experiments, the con-
tion of an aircraft as well as equations that de- trol variables δ e act , δ r act , δ a act took values within
scribe the engine dynamics. However, this also intervals specified in Table 6.4 for both the train-
leads to the lack of possibility to efficiently con- ing phase (polyharmonic control signal) and the
trol the aircraft velocity using the engine thrust testing phase (random control signal). Corre-
or air brake deflection. Thus, we cannot obtain sponding intervals for the values of the state
a representative training set for the axial force variables p, q, r, φ, θ, ψ, α, β are also included in
coefficient C x using only the stabilizer, rudder, Table 6.4.
and ailerons deflections. In order to overcome In order to expand these ranges of values for
this problem, we first train the ANN module control and state variables up to the full opera-
for C x directly using the wind tunnel data [20], tional area of the simulated system, we have to
separately from the whole model. Then, we em- develop appropriate algorithms for model gen-
bed this ANN module into the semiempirical eration. One of the approaches to solving this
model and “freeze” its parameters (i.e., prohibit problem relies upon the incremental learning
their modification during the model training). methods for the ANN model [30,31]. Under this
Finally, we perform training for the semiempir- approach, initially only the core of the model is
ical model to simultaneously approximate the designed that provides required accuracy within
unknown functions C y , C z , C l , C m , C n . 1 some subspace of the operational area, and then

1 All the ANN modules, both for the functions C y , C z , C l , 2 That is, the model should generalize well enough to pro-
C m , C n and for the function C x , are represented by sigmoidal vide the required simulation accuracy for the whole range of
feedforward neural networks with one hidden layer. its operating modes.