Page 24 - Neural Network Modeling and Identification of Dynamical Systems

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

1.1.2.2 Systems With Uncertainties the part of the components of the vector ξ,which
Systems of class DS whose properties are de- takes values from some domain . The “uncon-
terministic represent a somewhat rare particu- trollability” of an external perturbation means
lar case of dynamical systems. Much more of- that for the system VS there is no complete a pri-
ten, dynamical systems contain uncertainties of ori information about the characteristics of this
some kind from those listed in Section 1.1.1.3.In disturbance. In this case, corresponding compo-
particular, these uncertainties may be due to in- nents of the vector ξ can take random or fuzzy
complete and inaccurate knowledge of specific values.
properties of the object. For example, for aircraft,
1.1.2.3 Controllable Systems
we often have a situation where its aerodynamic
As noted above, the dynamical system S in-
characteristics are known not precisely and not
teracts with the environment E, i.e., perceives
wholly (in particular, there are no data on them
the impact of the environment and responds to
for a part of the flight regimes).
The system VS, 10 as well as the system DS,re- it accordingly. The response of the system to en-
mains uncontrollable and reacts to influences of vironmental influences can be either passive or
active.
the environment VS E, but it contains the un-
Passive interaction is, for example, the move-
certainties of the parametric and/or functional
ment of a stone or the flight of an artillery shell
type associated with the system or the environ-
or an uncontrolled rocket under the influence
ment (see Section 1.1.1.3). Thus, we can define
of gravitational and aerodynamic forces. Exactly
the system VS as follows:
this kind of interaction is realized by the systems
VS = X VS ,T VS , VS , DS of the form (1.3)and VS of the form (1.4).
In the case of active interaction, the system,
X VS ⊆ X,T VS ⊆ T, VS = VS (x,ξ,t), (1.4) influenced by the environment, generates and
x ∈X VS , ξ ∈ , t ∈ T VS ⊆ T. implements the response of the system to this
action; for example, it deflects the aircraft con-
The notations for the system (1.4) are similar to trol surfaces to compensate for the disturbance.
those introduced above for the system (1.3). The This means that if the dynamical system is able
difference between these systems lies in the fact to actively interact with the environment, then it
that the rule VS includes the uncertainty fac- is a controllable system.
tors ξ = (ξ 1 ,...,ξ q ) ∈ .AswasshowninSec- A controllable dynamical system CS 11 ac-
tion 1.1.1.3, usually systems of the class VS not tively responds to the effects of the environment
only contain uncertainties due to the features of and is capable of compensating the perturba-
the system and the available information about tions arising in the interaction CS E within
them, but they also interact with the environ- certain limits. We describe this system as fol-
ment E that contains uncertainties itself (an ex- lows:
ample of such environment is the turbulent at- CS CS CS
CS = X ,T , , ,
mosphere in which an aircraft is flying). There- CS CS CS CS
fore, in this case, the state of the dynamical sys- X ⊆ X, T ⊆ T, = (x,u,ξ,t), (1.5)
tem depends not only on the current state x(t i ) of x ∈ X CS ,u ∈ U CS , ξ ∈ , t ∈ T CS ⊆ T.
the system and time, but also on the value of the
external uncontrolled disturbances described by As in the case of the VS system of the form
(1.4), the notations for the system (1.5)are sim-
10 VS is the abbreviation for the Vague System, i.e., for a sys-
tem containing uncertainties of some kind. 11 CS is the abbreviation for the Controllable System.