Page 119 - Neural Network Modeling and Identification of Dynamical Systems

108 3. NEURAL NETWORK BLACK BOX APPROACH TO THE MODELING AND CONTROL OF DYNAMICAL SYSTEMS

outputs from these target ones, i.e., minimize the
error E(w). Thus, there arises the need to solve

the inverse problem of dynamics for some plant.
If the plant model is a traditional nonlinear sys-
tem of ODEs, then the solution of this problem
is complicated to obtain. An alternative option is
to use as a plant model some ANN, for which, as
a rule, the solution of the inverse problem does
not cause serious difficulties.
Thus, the neural network approach to the so-
lution of the problem in question requires the
use of two ANNs: one as the neurocontroller
and the other as the plant model. FIGURE 3.5 The neural network model of the short-
period longitudinal motion of the aircraft. V z , q are the val-
So, the first thing we need to be able to do
ues of the aircraft state variables at time t i ; δ e is the value of
to solve the problem of adjusting the dynamic the deviation angle of the stabilizer at time t i ; V z , q are
properties of the plant in the way suggested the increments of the values of the aircraft state variables at
above is to approximate the source system of dif- time t i + t (From [99], used with permission from Moscow
Aviation Institute).
ferential equations (3.13) (or, concerning the par-
ticular problem in question, the system (3.23)).
We can consider this problem as an ordinary the deflection angle of stabilizer δ e for the time
task of identifying the mathematical model of moment t i . Values of the state variables V z and
the plant [59,69] for the case when the values of q go to one group of neurons, and the value of
the outputs (state variables) of the plant are not the control variable δ e goes to another group of
obtained as a result of measurements but with neurons of the first hidden layer, which is the
the help of a numerical solution of the corre- preprocessing layer of the input signals. The re-
sponding system of differential equations. sults of this preprocessing are applied to all four
The approach consisting in the use of ANNs neurons of the second hidden layer. At the out-
to approximate a mathematical model of a plant put of the ANN, the values of V z and q are
(a mathematical model of aircraft motion, in par- increments of the values of the aircraft state vari-
ticular) is becoming increasingly widespread [4, ables at the time moment t i + t. The neurons of
31,34,38,70–72]. the ANN hidden layers in Fig. 3.5 have activa-
The structure of such models, the acquisition tion functions of the Gaussian type, the output
of data for their training, as well as the learn- layer neurons are linear activation functions.
ing algorithms, were considered in Chapter 2 for The model of the short-period aircraft mo-
both feedforward and recurrent networks. tion (3.23) contains the deflection angle of the
For the case of a plant of the form (3.23), all-turn stabilizer δ e as the control variable. In
i.e., for the aircraft performing the longitudinal the model (3.23), the character of the process of
short-period motion, a neural network approxi- forming the value δ e is not taken into account.
mating the motion model (3.23), after some com- However, such a process, determined by the dy-
putational experiments, has the form shown in namic properties of a controlled stabilizer (ele-
Fig. 3.5. vator) actuator, can have a significant effect on
The ANN inputs in Fig. 3.5 are two state vari- the dynamic properties of the controlled system
ables: the vertical velocity V z and the angular being created.
velocity of the pitch q in the body-fixed coordi- The dynamics of the stabilizer actuator in this
nate system at time t i , and the control variable is problem is described by the following differen-