Page 22 - Neural Network Modeling and Identification of Dynamical Systems

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

the environment operating time, T is the set of maximum possible values for the parameter λ,
all possible time instants, and E is the activity respectively.
E
mechanism of the environment E. We can define uncertainties of a functional
type in the form of parametric families of curves.
1.1.1.3 The Uncertainties in the For example, consider the following representa-
System-Complex tion:
When solving problems related to system-
complex K, it is necessary to take into account ϕ(x) = ϕ(x) nom + ϕ(x),
various uncertainty factors: ϕ(x) = w 0 + w 1 x + w 2 x ,
2
• uncertainties, caused by uncontrolled distur- w 0 ∈[w min ,w max ], w 1 ∈[w min ,w max ],
0 0 1 1
bances acting on the system-object S; min max
w 2 ∈[w ,w ],
• incomplete and inaccurate knowledge of the 2 2
properties and characteristics of the system-
where ϕ(x) is one of the aircraft characteristics
object S and the conditions in which it
C x , C y , C z , C l , C m , C n , T ; ϕ(x) nom is some nomi-
will operate (i.e., properties of the system-
nal value of the characteristic ϕ(x); ϕ(x) is the
environment E);
deviation of the actual characteristic ϕ(x) from
• uncertainties caused by a change in the prop-
its nominal value.
erties of the system-object S due to equipment
Uncertainties associated with environmental
failures and structural damage.
influences, i.e., air density ρ, atmospheric pres-
We can distinguish the following typical sure p, wind speed W, atmospheric turbulence,
classes of uncertainty factors: etc., are usually defined according to generally
accepted probabilistic models.
• uncertainties of parametric type associated
Let us explain the meaning of uncertainty
with variables describing the parameters of
caused by the change in the properties of the
the system-object S (for example, the mass m
system-object S due to equipment failures and
and the moments of inertia I x , I y , I z , I xz for the
damage in the structure, by examples related to
case when the system-object S is an aircraft);
aircraft.
• uncertainties of functional type related to the
During the flight of aircraft, various kinds
characteristics of the system-object S (for ex-
of abnormal (emergency) situations can occur,
ample, the coefficients of the aerodynamic
caused by failures of equipment and aircraft sys-
forces C x , C y , C z , the moment coefficients C l ,
tems or damage to the airframe and power plant
C m , C n , and thrust of the power plant T for the
of the aircraft. Some of these failures and kinds
case when the system-object S is an aircraft);
of damage have a direct impact on the dynami-
• uncertainties associated with the effects of
cal characteristics of aircraft as an object of mod-
system-environment E (for example, air den-
eling and control.
sity ρ; atmospheric pressure p; wind speed W;
For these reasons, there arises a need for an
atmospheric turbulence).
adjustment of the aircraft control algorithms that
Uncertainties of the parametric type are usu- would provide an opportunity to adapt to the
ally defined in the interval form as changed dynamical properties of the aircraft. In
this case it is extremely difficult, if possible at
λ ∈[λ min ,λ max ], all, to anticipate all possible failures and dam-
age as well as their combinations in advance. As
where λ is one of the parameters m, I x , I y , I z , for the damage to the airframe and power plant,
I xz ,etc.; λ min and λ max are the minimum and it is impossible to foresee all feasible options