Page 19 - Neural Network Modeling and Identification of Dynamical Systems

CHAPTER
1

The Modeling Problem for Controlled

Motion of Nonlinear Dynamical Systems

1.1 THE DYNAMICAL SYSTEM AS cluded in this system, that is, the environ-
∗ 1
AN OBJECT OF STUDY ment E .
Postulate 3. Both the system S and the envi-
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ronment E are sets of interacting systems
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Let us first formulate the concept of a dynam- ∗ ∗
of lower (as compared to S and E )hier-
ical system on an intuitive level, and then for- archical level which are, in turn, composed
malize it. of systems of even lower hierarchical level,
etc.
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Postulate 4. The system S and the environ-
1.1.1 The General Concept of a ∗
Dynamical System ment E interact with each other; in other
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words, S E . This interaction can be ei-
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1.1.1.1 Concept of the System, Basic ther statical, if the impact of the environ-
ment on the system and the response of the
Postulates
system to this effect are constant, or dynam-
The considerations presented below are based ical, if the impacts and/or reactions vary
2
on some postulates that are assumptions based in time. The classes of systems that cor-
on the ideas of general systems theory [1–5]. respond to these types of interaction are
Postulate 1. The World is a fragment of the usually referred to as statical and dynam-
Macrocosm, and this fragment is isolated
in some way. This World is the universe U 1 Generally speaking, the separation of the system-universe
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of our reasoning. U into the system S and the environment E is somewhat
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∗
arbitrary. The boundary between S and E does not only
Postulate 2. The World (system-universe) con- depend on the problem being solved (that is, what we want
sists of two systems: the system S ,which to get from the system), but can be even movable, i.e., it can
∗
∗
is the subject of our study, and the environ- vary during the lifetime of the system S .Anexample of this
kind of situation arises in the case of object adaptation (see
ment E , which includes all other elements. Section 1.2.1.3).
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In other words, we will assume that the 2 The interaction of the system and the environment is sym-
world is arranged like this: there is a spe- metric in the sense that one can speak of the effect of the
environment on the system with the response of the system
∗
cific system S , which is the subject of our to this effect, and the effect of the system on the environment
study, and there is everything else not in- with the response of the environment to this effect.
Neural Network Modeling and Identification of Dynamical Systems
https://doi.org/10.1016/B978-0-12-815254-6.00011-3 7 Copyright © 2019 Elsevier Inc. All rights reserved.