Page 21 - Neural Network Modeling and Identification of Dynamical Systems

1.1 THE DYNAMICAL SYSTEM AS AN OBJECT OF STUDY 9
ment E and interacts with this environment. 4 We assume that the system S is completely
The response of the system to the influence of defined if certain sets of variables (states, con-
5
the environment can be either passive, or ac- trols, disturbances, etc.) are given, as well as
tive. 6 time and a rule that determines the next state of
Dynamical systems, actively interacting with the system based on its current and/or previous
the environment, are controllable dynamical states. We will represent the system S as an or-
systems. In this case, the nature of the inter- dered triple
action of the system with the environment de-
S
S
S
pends on the properties of the system, as well as S = X , ,W ,
on the resources that it has. We can divide such S S S S
X ={X },i=1,...,N S ; ={ },j =1,...,N R ;
j
i
resources into two types: internal and external.
S S S S
If we take as an example the task of controlling W = T ,E ,T ⊆ T.
the movement of an aircraft, these resources will (1.1)
be as follows:
S
S
Here X ={X }, i = 1,...,N S ,isthe setof struc-
i
• internal resources are control laws, digital ter- tures of the system S (the term “structure” is
rain map, tools of inertial navigation, etc.; treated here in the general mathematical sense,
• external resources are systems of satellite nav- as a collection of sets with the collection of re-
igation (GLONASS and GPS), tools for remote S S
lations defined on these sets); ={ }, j =
control and radio navigation, etc. j
1,...,N R ,isthe setof rescripts of the system S
The less the system depends on external re- (the term “rescript” is used here as a generic
sources (information sources and control com- name for transformations of all kinds, i.e., map-
mands), the higher the degree of its autonomy. pings, algorithms, inference procedures, etc.);
S
S
S
The study of highly autonomous control sys- W = T ,E is the clock of the system, i.e.,
tems is becoming increasingly relevant at the the set of instants of the system operating time
present time due to the rapid development of (“system time”), T is the set of all possible time
unmanned vehicles of various classes, in partic- instants (“world time”), endowed with a struc-
ular unmanned aerial vehicles (UAVs) and un- ture of linear order (i.e., ordered by the relation
manned cars. ), and E S is the activity mechanism (“clock
generator”) of the system S.
4 In most of the applied problems, only part of this interac- The system-object S is not isolated; it interacts
tion is taken into account, namely the influence of the en- with the system-environment E, represented by
vironment E on the system S. For example, it can be the an ordered triple of the form
influence of the gravitational field and/or atmosphere on the
aircraft. However, in some cases it is also necessary to take
E
E
E
into account the second half of this interaction, that is, the E = Q , ,W ,
influence of S on E. In particular, this kind of interaction gen- E E E E
Q ={Q },i= 1,...,M E ; ={ },j =1,...,M R ;
erates a trail (wake turbulence) behind the aircraft. We must i j
take into account such factors to solve problems related, for W = T ,E ,T ⊆ T.
E
E
E
E
example, to formation flying of aircraft.
5 Examples of passive interaction with the environment are (1.2)
such tasks as the flight of an artillery shell or an unguided
E
E
rocket. Here Q ={Q }, i = 1,...,M E , is a collection
i
6 In case of active interaction, the system generates and im- of structures of the environment E; ={ },
E
E
plements some reaction to the impact of the environment j
according to some “control law”. For example, the elevator j = 1,...,M R ,isthe setof rescripts of the envi-
E
E
E
will be deflected as a reaction to a vertical gust of wind that ronment E; W = T ,E is the clock of the
has affected the aircraft. environment E, T E ⊆ T is the set of instants of