Page 36 - Neural Network Modeling and Identification of Dynamical Systems

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

In practice, we can not always meet these re- possible values of the characteristics of the air-
quirements. Also, in the process of operation, craft), and ξ k (which define the possible values
the parameters and characteristics of the object of the parameters of the atmosphere and atmo-
and environment can vary considerably. In these spheric influences), respectively, i.e.,
cases, traditional methods often give unsatisfac-
tory results. w i ∈ W i , W i =[w i min ,w max ], i = 1,2,...,p,
i
For this reason, there is a need to build con- v j ∈ V j , V j =[v min ,v max ], j = 1,2,...,q,
trol systems that do not require a complete a j min j max
priori knowledge of the control object and the ξ ∈ k , k =[ξ k ,ξ k ], k = 1,2,...,r.
conditions for its operation. Such a system must (1.24)
be able to adjust to the changing object prop-
erties and environmental conditions. These re- A particular combination of the parameters
quirements are met by the adaptive systems w i , v j , ξ k generates a tuple ω s of length p +q +r,
[40–54], in which the current available informa- i.e.,
tion is used not only to generate a control action
ω s = w 1 ,...,w p ,v 1 ,...,v q ,ξ 1 ,...,ξ r ,
(as in conventional nonadaptive systems), but
also for changing (adjusting) the control algo- s = 1,2,...,p · q · r. (1.25)
rithm.
We usually distinguish two main classes of All possible combinations of values of the un-
adaptive systems [41,44]: certainty factors characterizing control problems
for some dynamical system 24 form a collection
• self-tuning systems, in which the structure of of tuples ω s as the Cartesian product of the
the control algorithm does not change during sets W, V , , i.e.,
operation, but only its parameters change;
• self-organizing systems, in which the struc- = W × V × , ω s ∈ , (1.26)
ture of the control algorithm changes during
operation. or as a subset of ⊂ , if not all tuples ω s are
valid. For other types of dynamical systems, we
Incomplete knowledge of the parameters and
characteristics of the control object and environ- define the uncertainty factors and their possible
combinations in a similar way.
ment in which it operates is typical for adaptive Having in mind this definition of the operat-
systems. We treat this incomplete knowledge as ing conditions for the considered dynamical sys-
additional uncertainty factors and include them tem, we can formulate the problem of adaptive
in the corresponding class . control as follows: the controller will be adaptive
For example, we can define the uncertainty
in the class , if after a finite time T a , called the
factors associated with the aircraft control prob-
adaptation time, it will ensure the fulfillment of
lem through the following three sets of parame-
the control goal.
ters:
W = W 1 × W 2 × ... × W p ,
1.2.3 Basic Structural Variants of
V = V 1 × V 2 × ... × V q , (1.23) Adaptive Systems
= 1 × 2 × ... × r ,
As already noted, the control system is con-
sidered adaptive if the current information on
where W i , V j , k are the ranges of the values
of w i (which define the possible values of the
parameters of the aircraft), v j (which define the 24 For example, a flight control problem for aircraft.