Page 228 - Neural Network Modeling and Identification of Dynamical Systems

6.4 SEMIEMPIRICAL MODELING OF AIRCRAFT LONGITUDINAL TRANSLATIONAL AND ANGULAR MOTION 219
2
ment of inertia I y = 75673.6 kg·m ; center of then this situation is equivalent to assigning the
gravity is located at 5 percent of the mean aero- weight K to some typical example from this re-
dynamic chord; the time constant of the stabi- gion. Thus, the uneven distribution of examples
lizer actuator T φ = 0.025 sec; coefficient of rela- can lead to a model with high accuracy in some
tive damping of the stabilizer actuator ζ = 0.707. regions of input space and a much lower accu-
In these experiments we consider a range of alti- racy in the others. To avoid this, at the end of
tudes from 1000 mto 9000 m and Mach numbers the procedure for synthesizing the training set,
from 0.1 to 0.6. we assign weights to its elements. For each ele-
When solving problems of the type in ques- ment λ ∈ , we find the elements λ ∈ located
˜
tion, one of the most critical tasks is the genera- in its ε-neighborhood. Then, we assign a weight
tion of a representative set of data that presents to each example from Q, which is inversely pro-
the behavior of the simulated dynamical system portional to the number of neighbors found for
on a sufficiently wide range of values of the vari- this example.
ables describing the given object. This task is When implementing this algorithm on a com-
essential for obtaining a reliable model of such puter, one should choose an appropriate data
a system, but it has no simple solution. We can structure for the representation of the sets , ,
¯
¯
collect the required training data for the gener- and m , which would ensure the efficient oper-
ated ANN model using the specially organized ations of the nearest neighbor search, the search
test excitation signals applied to the simulated for neighbors in a given region, and the addition
dynamical system. of new items. An example of such structure is a
In this section, we propose an automatic pro- k-dimensional tree, implemented, for example,
cedure for synthesizing control actions that pro- in the FLANN library [37].
vide sufficiently dense coverage of the region of This algorithm was successfully applied to
change for the values of the variables describ- the generation of a training set for a semiem-
ing the dynamical system. Such technique as- pirical model of the longitudinal motion of a
sumes the availability of some initial theoretical maneuverable aircraft. The following ranges for
model of the dynamical system. This model may variables were considered:
have low accuracy, or for other reasons not sat-
isfy the requirements for final models. However, δ e act ∈[−25,25] deg,δ e ∈[−25,25] deg,
it can be used to synthesize control signals cor- δ th ∈[0,1],P c ∈[0,100]%,
responding to sufficiently diverse trajectories in θ ∈[−90,90] deg,q ∈[−100,100] deg/sec,
the state space.
V ∈[35,180] m/sec,α ∈[−20,90] deg.
Then, we apply the resulting set of control
actions to the simulated object, and the result- The effectiveness of this algorithm can be es-
ing trajectories are used to fill the training set. timated using the coverage diagrams [38], for
The test set is generated similarly. The descrip- the range of acceptable values of the variables
tion of this procedure is presented in Algo- and their derivatives that describe the simulated
rithm 1. object, using examples obtained when the test
In addition to the representative training set, signal is applied to the object. These diagrams
we use the weighting of individual examples make it possible to evaluate the representative-
from the training set to improve the general- ness (informativeness) of training sets obtained
ization error of the neural network model. This by applying various test excitations to the mod-
procedure is based on the following considera- eled object. The set will be better if it covers
tions: if the arguments of K examples from the the required range of values describing the be-
training set are located in a small neighborhood, havior of the object under consideration more