Page 20 - Neural Network Modeling and Identification of Dynamical Systems

8 1. THE MODELING PROBLEM FOR CONTROLLED MOTION OF NONLINEAR DYNAMICAL SYSTEMS

ical systems. The overwhelming majority tion of an aircraft, then it is not necessary to in-
of practically existing classes of systems troduce into this model the variables associated
are dynamical, and they will be the subject with the aircraft lateral motion.
of our study. 3 Similarly, we introduce a partial representa-
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It should be noted that the universe U is a too tion E for the environment E , which interacts
general construction regarding the purposes of with the system S. We will denote by E ⊂ E the
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this book: both the system S and the environ- fact that E is a partial representation of the en-
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ment E in the interpretation, as they are given vironment E . This representation also depends
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above, are redundant. We need to reduce this significantly on the problem being solved. Be-
redundancy to decrease the level of complexity cause of this, the “radius of the environment” in
of the problem being solved. This can be done some problems can be of the order of meters (for
based on the following considerations. In the example, to take into account the effects of at-
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universe U, the system S is represented with all mospheric disturbances on the aircraft), and in
its inherent properties, while the environment others it can be of the order of hundreds of kilo-
E , as noted above, is the part of the universe meters (for example, tracking targets in systems
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U that did not enter the system S . For practi- such as AWACS).
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cal purposes, this composition of S and E is Accordingly, instead of the universe
clearly redundant.
We introduce a partial representation for the ∗ ∗
U = S ∪ E
system S , which we denote as S, and we denote
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the relation between the system and its partial we will consider the union of the “truncated”
representation as S ⊂ S . versions of the system S ⊂ S and the environ-
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When we talk about the “partial representa- ment E ⊂ E . We will call this combination the
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tion” S for S ,wemeanthat S takes into account “System-Complex” K:
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only a part of the properties inherent to the sys-
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tem S . In this case, only those properties of the System-Complex K = System-Object S
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system S are taken into account in S which are
significant for the problem being solved. Differ- + System-Environment E,
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ent problems solved for the same system S re- K = S,E , S E.
quire consideration of its various properties, i.e.,
each time only a part of the properties that S ∗ 1.1.1.2 Concepts of System-Object and
possesses is required. For example, if we solve System-Environment
the problem of analyzing the longitudinal mo-
The dynamical system S is a system whose
state varies with time under the influence of
3 In fact, statical systems in nature, most likely, simply do not some external and/or internal factors [1,6–19].
exist. Even those systems about which we are accustomed to The source of external factors is the environment
think of as statical (more precisely, not dynamical), for ex- E in which the dynamical system operates, and
ample, buildings and structures of various kinds, in fact, are
not. For example, television towers, skyscrapers, and other the source of internal factors is the set of features
high-rise buildings are characterized by the presence of os- that characterize the system as well as events oc-
cillations in their upper parts with a rather large amplitude curring in the system (for example, failures and
(due to wind effects). So the “statical system” is just a simpli- damage affecting the dynamic properties of the
fication of a dynamical system in cases where the “dynami- system).
cal” component can be neglected under the conditions of the
problem being solved. Consequently, without loss of gener- In general, the dynamical system S is not iso-
ality, we can only talk about dynamical systems. lated, but rather it operates in some environ-