Page 23 - Neural Network Modeling and Identification of Dynamical Systems

1.1 THE DYNAMICAL SYSTEM AS AN OBJECT OF STUDY 11
in principle. Therefore, it seems appropriate for in the same way. The properties of such systems
aircraft control to interpret possible sharp and are either constant or vary according to some
unpredictable changes in the dynamical prop- preserved relationship. Examples of such sys-
erties of aircraft due to failures and damage as tems are an unguided missile with a detachable
another class of uncertainties. Compensation of booster and an uncontrolled aircraft with a mass
the impact of these factors on the aircraft behav- that varies due to fuel consumption.
ior is the responsibility of the adaptation mech- We assume that the system DS is a triple
anisms. These mechanisms must ensure fault- DS DS DS
tolerant and damage-tolerant control of the air- DS = X , ,T ,
craft, i.e., control that can adapt to a change in X DS ⊆ X,T DS ⊆ T, (1.3)
the dynamics of the controlled object generated DS DS DS DS
by failures or damage. The result of such adap- = (x,t),x ∈ X ,t ∈ T .
tation should be the restoration of the aircraft
Here X is the phase space (state space) of the
stability and controllability to a level acceptable system (1.3), the elements (“points”) x ∈ X of
8
from the flight safety point of view. which are the possible phase states (phase vec-
T
tor) of the given system, x = (x 1 ,...,x n ) , x 1 ∈
1.1.2 Classes of Dynamical Systems X 1 ,...,x n ∈ X n , X = X 1 × ... × X n ; x 1 ,...,x n are
the state variables (phase coordinates) of the sys-
In this section, we form a hierarchy of system- tem DS; X DS ⊆ X is the range of admissible val-
objects as a sequence of definitions of their ues of phase states of the system (1.3); T is the
classes, ordered by their level of capabilities. set of all possible time instants (“world time”)
The system-objects, considered below, differ endowed with a structure of linear order (that is,
from each other in their properties and, as a con- ordered by the relation ), T DS ⊆ T is the set of
sequence, the level of potential capabilities. instants of the system operation time (“system
We use the following features to differentiate time”); DS = DS (x,t), x ∈ X DS , t ∈ T DS ,isa
systems of various classes: rule that allows us to determine the state of the
system (1.3) at each time instant t ∈ T DS ,given
• the presence/absence of uncertainties in the its states at all previous time instants. Systems
9
system that affect the properties of the system;
of the class DS are the subject of the study of the
• the presence/absence of the possibility for the
modern theory of dynamical systems (see, for
system to control its behavior as a way of ac-
example, [7,20]).
tively responding to changes in the current
and/or predicted situation;
8 The terminology adopted here goes back to the traditions
• the presence/absence of the possibility for the
of mechanics, the theory of dynamical systems, and also the
system to adapt to changes in the properties
theory of control and controlled systems. This seems quite
of the object and/or environment; logical, because of the fundamental role played by the con-
• the presence/absence of goal-setting capabil- cept of a dynamical system for the class of problems we
ities in the system. consider.
9 This means, generally speaking, that the rule must have
S
1.1.2.1 Deterministic Systems infinite memory in order to store all the states previously at-
tained by the system S. Most often, based on the specifics of
At the lowest level of the hierarchy of sys- the problem being solved, it can be argued that this require-
tems S lie the deterministic dynamical systems ment is excessive and it is sufficient to use the state prehis-
7
DS, i.e., those that respond to the same actions tory of finite length; in a number of cases it can be assumed
S
that all future states of the system S for t> t i , t,t i ∈ T ,are
determined only by its state x(t i ) at the given current time
7 DS is the abbreviation for the Deterministic System. instant t i ∈ T S (and, of course, by the rule ).
S