Page 210 - Neural Network Modeling and Identification of Dynamical Systems

6.2 SEMIEMPIRICAL MODELING OF LONGITUDINAL SHORT-PERIOD MOTION FOR A MANEUVERABLE AIRCRAFT 201
able experimental data for the observed state 0.02 sec for the partially observed state vec-
T
variables of the dynamical system. tor y(t) =[α(t);q(t)] , corrupted by additive
In this example, we consider the theoretical Gaussian noise with a mean square deviation
model of the angular longitudinal motion of the σ = 0.01.
aircraft that is traditional for aircraft flight dy- As already noted, one of the critical issues
namics [13–19]. This model is written as follows: arising in the development of empirical and
semiempirical ANN models is the problem of
¯ qS g
˙ α = q − C L (α,q,δ e ) + cosθ, the acquisition of a training set that adequately
mV V describes the behavior of the modeled system.
c
¯ qS ¯ (6.5) We get this training data set by developing an
˙ q = C m (α,q,δ e ),
I y appropriate test control signal for the simulated
2 object and evaluating the response of the object
T ¨ δ e =−2Tζ ˙ δ e − δ e + δ e act ,
to this signal.
where α is the angle of attack, deg; θ is angle of Let us analyze the influence of the type of
pitch, deg; q is the angular velocity of the pitch, test control signal on the representativeness of
deg/sec; δ e is the deflection angle of the con- the resulting set of experimental data used as
trolled stabilizer, deg; C L is the lift coefficient; a training set for the ANN model. We compare
C m is the pitch moment coefficient; m is mass the typical excitations (step, impulse, doublet,
of the aircraft, kg; V is the airspeed, m/sec; ¯q = and random signal) with the polyharmonic sig-
2
ρV /2 is the dynamic pressure, kg·m −1 sec −2 ; ρ nal designed specially. The comparison is based
3
is air density, kg/m ; g is the acceleration due on the simulation results for various test maneu-
2
2
to gravity, m/sec ; S is the wing area, m ; ¯c is vers, including a straight-line horizontal flight
the mean aerodynamic chord of the wing, m; I y with a constant speed and a flight with a mono-
is the moment of inertia of the aircraft relative tonically increasing angle of attack.
2
to the lateral axis, kg·m ; the dimensionless co- When solving problems of the considered
efficients C L and C m are nonlinear functions of type, one of the most critical tasks is the gener-
their arguments; T , ζ are the time constant and ation of an informative (representative) data set
the relative damping coefficient of the actuator; that characterizes the behavior of the simulated
is the command signal to the actuator dynamical system over the whole range of the
and δ e act
of the all-turn controllable stabilizer (limited to values of the variables describing the dynamical
±25 deg). In the model (6.5), the variables α, q, system and the derivatives (rates of change) of
δ e ,and ˙ δ e are the states of the controlled object, these quantities. As shown in Section 2.2.2,the
is the control. Here, g(H) and required training data for the generated ANN
the variable δ e act
ρ(H) are the variables describing the state of model can be obtained by application of the spe-
the environment (gravitational field and atmo- cially designed test excitations to the simulated
sphere, respectively), where H is the altitude of dynamical system. As evidenced by computa-
the flight; m, S, ¯, I z , T , ζ are constant parameters tional experiments, sufficient informativity of
c
of the simulated object, C L and C m are variable the training set for the considered problem is
parameters of the simulated object. provided only by the random and polyharmonic
As an example of a particular simulation signals (Fig. 6.1) of all the test signals listed in
object, a maneuverable F-16 aircraft was con- Section 2.2.2.
sidered, the source data for which were taken We can graphically present the effectiveness
from [20,21]. Computational experiments with of various types of test signals, mentioned above.
the model (6.5) were performed on the time in- For this purpose, we will use the coverage dia-
terval t ∈[0;20] sec with a sample step t = grams for the range of acceptable values of the