Page 31 - Neural Network Modeling and Identification of Dynamical Systems

1.1 THE DYNAMICAL SYSTEM AS AN OBJECT OF STUDY 19
outputs of the system S, respectively; F and G variable u(t). Similarly, the u (j) (t i ) designation
are the rules governing the evolution of the state or, in the abbreviated form, u (j) , denotes the in-
i
and output of the system S over time, respec- stantaneous value of the jth component of the
tively; X and Y are respectively the set of states vector variable u(t) at the instant of discrete time
and observable outputs of the system S.Ele- t i ∈[t 0 ,t f ]. The corresponding notation for the
ments of these sets will be denoted as u ∈ U, variables x ∈ X, y ∈ Y, ξ ∈ , ζ ∈ Z is introduced
ξ ∈ , ζ ∈ Z, x ∈ X, y ∈ Y, t ∈ T . similarly.
In (1.16), T denotes the time interval on which As noted above, our object of study is a con-
the system S is considered. The time within this trollable dynamical system, operating under un-
interval can be either continuous, that is, T ⊂ R, certainty conditions. We can divide these uncer-
or discrete. In the case of discrete time, the se- tainties into the following main types:
quence of time instants is given by the following
rule: • uncertainties generated by uncontrolled dis-
turbances acting on the object (for example,
T ={t 0 ,t 1 ,...,t i−1 ,t i ,t i+1 ,...,t N−1 ,t N },t N = t f , atmospheric turbulence, wind gusts);
• insufficient knowledge of the simulated object
t i+1 = t i + t, i = 0,1,...,N. and the environment in which it operates (for
(1.17)
example, insufficiently known or unknown
aerodynamic characteristics of the aircraft);
In the following text, unless otherwise speci-
• uncertainties caused by a change in the prop-
fied, a discrete time (1.17) will be used. This ap-
erties of the object due to equipment failures
proach seems quite natural for the range of prob- and structural damage (for example, combat
lems under consideration. Systems S are usually and/or operational structure damage and air-
described in continuous time by ordinary differ-
ential equations (ODEs) or differential-algebraic craft equipment failures that change the object
equations (DAEs). However, in order to numer- properties).
ically solve problems of the analysis, identifica- In order to describe the current (instanta-
tion, and control synthesis for such systems, we neous) state of the complex K, we introduce the
need to approximate them with finite difference concept of a situation that includes components
(discrete time) equations given by correspond- describing the state of both the system S and
ing recurrent schemes (Runge–Kutta, Adams, the environment E. The components describing
etc.). Thus, in the process of solving these prob- the system S will be called internal, while com-
lems, there is a mandatory transition from con- ponents describing the environment E will be
tinuous to discrete time. called external; hence
Similarly, the onboard implementation of
control systems for modern and advanced air- Situation = External-Situation
craft is also performed in the digital environ- + Internal-Situation
ment, so these systems will also operate in dis- λ(t i ) = λ int (t i ),λ ext (t i ) ,
crete time. λ(t i ) ∈ , λ int (t i ) ∈ int ,λ ext (t i ) ∈ ext .
The notation u(t) here and below denotes a
vector variable u as a function of time t ∈ T . Along with the concept of the situation, the
The designation u(t i ) denotes the instantaneous idea of situational awareness plays an important
value of this variable at the instant of discrete role. While the concept of the situation describes
time t i ∈[t 0 ,t f ]. We will also use the abbrevi- objective reality (object + environment), the con-
ated notation u i instead of u(t i ). The notation cept of situational awareness describes the de-
u (j) (t) denotes the jth component of the vector gree of awareness of the system S about this