Page 41 - Neural Network Modeling and Identification of Dynamical Systems

1.3 A GENERAL APPROACH TO DYNAMICAL SYSTEM MODELING 29
ordered triple of the following form: • uncertainties caused by changes of the prop-
erties of the object due to failures of its equip-
S = U,P,Y , (1.27) ment and structural damage.
Analysis of the problems associated with the
where U is the input to the simulated/con- development of complex systems operating un-
trolled object; P is the simulated/controlled ob- der uncertainty conditions leads to the conclu-
ject (plant); Y is the object’s response to the input sion that we need to adopt an idea of adaptabil-
effects. ity. As shown above, models of simulated ob-
Inputs U include initial conditions, controls, jects play a crucial role in the process of develop-
and uncontrolled external disturbances for the ment for adaptive systems. These models are in-
object P. A simulated object P might be, in par- tended primarily for the use in onboard systems
ticular, an aircraft of some type. Outputs Y of the (for example, in aircraft control systems) in real
dynamical system S represent the observed reac- or even advanced time, which imposes some re-
tions of the object P to the input actions U. quirements for them, namely increased accuracy
Based on these definitions, we can state three and high computational speed and adaptability.
Traditional models (in the form of ODEs
main classes of problems related to dynamical
systems as follows: or DAEs) do not meet all of the requirements
regarding accuracy and computational speed.
1. U,P,Y is the analysis problem for the be- They do not satisfy the requirement of adapt-
havior of the dynamical system (find Y,given ability at all. The ways of meeting these require-
Uand P); ments can be as follows. Accuracy and com-
2. U,P,Y is the synthesis problem for the dy- putational speed of the model can be ensured
namical system control (find U, given P and by maximizing the use of knowledge and data
Y); about the simulated object. The adaptability of
3. U,P,Y is the identification problem for the the model can be interpreted as the ability to
dynamical system (find P, given U and Y). rapidly restore its adequacy to the simulated ob-
ject by means of structural and/or parametric
Here problem 1 belongs to the class of direct adjustments.
problems, while problems 2 and 3 belong to the The process of development for the dynam-
class of inverse problems of system dynamics. ical system model is reduced to the solution of
In this case, problem 3 is actually the problem four main problems. Namely, we need to find:
of designing a model of a dynamical system,
1) a set of variables that describe the simulated
whereas problems 1 and 2 are solved using this
object;
model. 2) a class (family) of models, which includes
Another important question that we need to
the required (desired) model;
address for the considered dynamical systems is
3) tools for selecting a particular model from a
the problem of the uncertainties present in these
given class (the criterion of the adequacy and
systems as well as their environment. the search algorithm for this model);
As already noted in Section 1.1.1.2, uncertain- 4) a representative set of experimental data re-
ties may be divided into the following types: quired to design and evaluate the model.
• uncertainties caused by uncontrolled distur- Problem 1, that is, the formation of a set of
bances acting on the object; variables that describe the simulated object, is
• insufficient knowledge of the simulated object the subject of a separate study aimed at un-
and its environment; derstanding how the meaningful interpretation