Page 110 - Neural Network Modeling and Identification of Dynamical Systems

3.3 ANN-BASED MODELING AND IDENTIFICATION OF DYNAMICAL SYSTEMS 99
• state variables are observable, hence they can selves are dynamical systems, which determines
be interpreted as outputs of the system, and the validity of this approach. However, dynamic
the problem is reduced to the one previously networks are very difficult to learn. For this rea-
considered for the input–output representa- son, it is advisable, in situations where possible,
tion case; the ideal model will be a feedfor- to use feedforward networks that are simpler in
ward neural network, which can be used as a terms of their learning processes.
one-step-ahead predictor; Feedforward networks can be used both in
• state variables are not observable, and there- tasks of modeling dynamical systems and for
fore an ideal model cannot be constructed; In control of such systems in two situations.
this case, we should use the input–output rep- In the first one, we solve the problem of
resentation (with some loss of generality of modeling some uncontrolled dynamical sys-
the model), or build some recurrent model, al- tem, which implements the trajectory depend-
though it will not be optimal in this situation. ing only on the initial conditions (and possibly
disturbances acting on the system). For a sin-
The last type of the noise influence on the sim-
ulated system is the case when the noise simul- gle variant of the initial conditions, the solution
of the problem will be a trajectory described by
taneously affects both the outputs and the states
some function which is nonlinear in the general
of the dynamical system. This assumption leads
case. As is well known [14,25–29], the feedfor-
to the following model:
ward networks have universal approximating
x(k) = ϕ(x(k − 1),u(k − 1),ξ 1 (k − 1)), properties, i.e., the task of describing the be-
(3.12) havior of a dynamical system, in this case, is
y(k) = ψ(x(k),ξ 2 (k)).
reduced to the formation of appropriate archi-
Similar to the previous case, two situations are tecture and the training of a feedforward neural
possible: network.
In real-world problems, the case of a single
• if the state variables are observable, they can variant of the initial conditions is not typical.
be interpreted as outputs of the dynamical Usually, there is a range of relevant initial condi-
system, and the problem is reduced to the one tions for the dynamical system under consider-
previously considered for the input–output ation. In this case, we can enter the parametriza-
representation case; tion of the trajectories implemented by the sys-
• if the state variables are not observable, the tem, where the parameters are the initial condi-
ideal model should include both states and tions. The simplest variant is to cover the range
the observed output of the system. of reasonable values of the initial conditions
with a finite set of their “typical values” and con-
struct a bundle of trajectories corresponding to
3.3 ANN-BASED MODELING AND these initial conditions. In this case, we form this
IDENTIFICATION OF bundle in such a way that the distance between
DYNAMICAL SYSTEMS trajectories does not exceed a specific predeter-
mined threshold value. Then, with the appear-
3.3.1 Feedforward Neural Networks for ance of initial conditions that do not coincide
Modeling of Dynamical Systems with any of the available sets, we take from this
set the value closest to the one presented. This
The most natural approach to implementing approach is conceptually close to the one used
models of dynamical systems is the use of re- in Chapter 5 to form a set of reference trajec-
current neural networks. Such networks them- tories in the task of obtaining training data for