Page 154 - Neural Network Modeling and Identification of Dynamical Systems

144 4. NEURAL NETWORK BLACK BOX MODELING OF AIRCRAFT CONTROLLED MOTION

In this model, ω rm = 1, ζ rm = 0.9, the vector when the network learns only this particular
of the state variables x =[n x a rm , ˙n x a rm ], r is the segment, forgetting about all the others.
reference signal. 2. Learning the network on large segments al-
The neurocontroller is configured to mini- ways leads to a bad local minimum.
mize the error y rm − y, i.e., to approximate the 3. Learning the network on medium-sized seg-
behavior of the reference model with the re- ments also leads to bad local minimum; how-
sponse of the plant model coupled with the con- ever, the rotation of these segments allows
troller. For a good ANN model, this means min- circumventing this problem to some extent.
imizing the “real” error y rm −y to a certain level.
For these reasons, it is necessary to use
Although the neurocontroller is static, it
medium-sized segments, to perform training
works as part of a dynamical system, so we need
to configure it as a part of the whole recurrent with three to seven epochs for each of them, and
network. This configurable network consists of to loop over the segments several times, and fi-
two subnets (the neurocontroller itself and the nally to consolidate the segments to improve the
training performance. As a result, the learning
closed-loop object model), closed by the external
process of the ANN becomes very computation-
feedback loop. During the configuration, the pa-
ally intensive (up to several hours, depending
rameters of the model subnet do not change, i.e.,
the ANN model only serves to close the external on the implementation details).
feedback loop and represent the entire system in According to the previous considerations, it
a neural network form (to estimate the sensitivi- is advantageous to use the sequential training
ties of the outputs of the controlled object to the mode mentioned in the last section to train the
parameters of the neurocontroller). neurocontroller in the batch mode (i.e., for its
In the batch mode, such a network can be pretraining); the only difference is that we need
trained using the same Levenberg–Marquardt to use dynamic backpropagation to compute the
method. However, it requires the computation Jacobian.
of dynamic derivatives; hence to compute the In this case, the Kalman filter acts as the “sta-
Jacobian, we have to either apply the backprop- pler” of the individual segments into one data
agation through time or the real-time recurrent array. Moreover, the segments can be chosen to
learning method. A recurrent form of the net- be small (30–100 points, which saves consider-
work presents additional difficulties in the pro- able computational time), so long as the dynam-
cess of ANN learning: the larger the sample, ics of the controlled object is reflected on this
the higher the chance that the learning process interval. Although in general sequential meth-
will get stuck in some of the local minima. Such ods achieve lower accuracy, it is more important
chances increase with the length of the sample to circumvent the problem of local minima and
with catastrophic speed. Therefore, we divide to decrease the training time.
the entire sample into segments. Thus, the procedure of the neurocontroller
To configure parameters uniquely, we require configuration is as follows:
the closed-loop network with the controller to
1. Set the initial conditions on the reference tra-
start from the reference trajectory on each seg- jectory. Usually, a first few points of the seg-
ment, since the neurocontroller cannot affect the ment are assigned to the initial conditions.
initial conditions.
2. Simulation of the coupled network on this
Thus it is necessary to consider the following
segment (prediction of the behavior of the
factors:
controlled object with the current parameters
1. Learning the network on small segments (less of the neural controller), estimation of the er-
than 500–1000 points) leads to the situation ror of the reference model tracking, compu-