Page 28 - Neural Network Modeling and Identification of Dynamical Systems

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

According to the level of potential capabilities • the complex K UE = DS,UE , which includes
DS
(in order of increasing capabilities), these classes an uncontrollable deterministic dynamical
of systems are arranged as follows: system DS, interacting with the environment
UE that contains uncertainty factors (an ex-
DS ⊂ VS ⊂ CS ⊂ AS ⊂ IS.
ample is an uncontrolled missile moving in a
Similarly, the hierarchy of environmental classes turbulent atmosphere);
can be structured as follows: • the complex K SE = CS,SE , which includes
CS
a controllable deterministic dynamical system
SE ⊂ UE ⊂ RE ⊂ AE ⊂ IE. CS that regularly interacts with the determin-
istic environment SE (an example is an air-
As noted above, the system S should be con-
sidered in interaction with the environment E. craft performing controlled movement in a
quiet atmosphere);
Symbolically, we will denote this assertion as
• the complex K UE = AS,UE , which includes
follows: AS
an adaptive dynamical system AS, interact-
System-Complex K = System-Object S ing with the environment UE that contains
uncertainty factors (an example is an aircraft
+ System-Environment E,
that operates in an environment with uncer-
K = S E, tainties 21 while being able to quickly adapt to
them).
that is, at the most general level, we consider the
system-complex K, which consists of two inter-
acting systems, namely, the system-object S and 1.1.5 Formalization of the Dynamical
the system-environment E, in which the system- System Concept
object operates, 20 so we have We now introduce the formalized concept of
the system S in the form in which it will be used
K = S,E,T,
, S E. (1.8)
later. In the general case, in this description we
Here
is the law of the interaction between S have to define the following elements related to
and E in time T . S:
The specific form of the complex (1.8) is deter- 1) the set of variables (with the range of their
mined by the way its constituent parts S, E, T ,
admissible values) describing S and the con-
are defined. For example, the following variants ditions in which S operates;
are possible: 2) the set of variables (with the range of their
• the complex K SE = DS,SE , which includes admissible values) describing the factors af-
DS
an uncontrollable deterministic dynamical fecting the states of the system S;
system DS; this system regularly interacts 3) thetimeinwhich S is running;
with the deterministic environment SE (an 4) law of the functioning of S, that is, a set of
example is an object moving in the gravita- rules, 22 according to which the collection of
tional field of a celestial body that does not variables describing S changes with time.
have an atmosphere);
21 The uncertainties that might arise in the problems of mod-
20 In the following text, for brevity, we will simply refer to eling the behavior of dynamical systems are diverse (see, for
System-Complex as the “complex”, to System-Object as the example, [22–27]).
“system” (dynamical system) or “object” (“plant”), and to 22 In the theory of dynamical systems [7,9], this set of rules is
System-Environment as just the “environment.” often also referred to as the evolution law of the system S.