*109489*
Yunicheva N.R., Yunicheva R.R
Institute of Informatics and
Control Problems, Kazakhstan
Al-Farabi Kazakh National
University
The formal type of solving
parametrical synthesis task of
inexact data objects control
In
article [1] it has been shown, that a task of parametrical synthesis of control
by objects with inexact data it is reduced to resolvability of interval system
of the linear algebraic equations. On the other hand, well-known, that the
finding of the decision of such interval systems is NP – difficult task.
The
review of references under the applied interval analysis reveals existing types
of solving of similar systems.
The
concept of "solving" of interval system of inclusions demands special
specification since interval uncertainty of the data of system can be treated
doubly, according to dual understanding of intervals [2].
In the
first case, the interval 
is set of all
material numbers from 
up to 
, and in the second - to contain in itself even one value
between 
and 
.
From
the point of view of mathematics, this distinction is expressed by the use of
quantifiers of universality 
and quantifier of
existence 
: in the first case enters the name 
, and in the second 
.
As to
parameters of system of the linear interval equations 
about which the belonging to some intervals basic distinction
between two types of interval uncertainty is shown as distinction between
parameters which can change within the limits of the intervals specified by it
as investigation of external unpredictable indignations and parameters with
which we can vary within the limits of the set intervals at the will is known
only i.e. to operate them.
In the
interval analysis there are following various definitions of concepts of
solving of interval system of the algebraic interval equations [2]:
The
united solution set ,
, (1)
which is formed by
decisions of all systems 
with 
and 
.
The task of
construction of set of a kind (2) can be named a task of identification.
Tolerlable
solution set
, (2)
which is formed by
all such vectors 
, that product 
gets in 
for anyone 
.
The
task of construction of set of a kind (2) refers to as a linear tolerance task.
That fact is curious, that the initial motivation of introduction and studying äîïóñêîâûõ decisions has come from a task of designing of
the elevating crane and a task of calculation of interbranch economic balance
at the inexact data [2].
Controllable
solution set
,
(3)
educated by such
vectors 
, that for anyone desirable 
it is possible to
pick up corresponding 
satisfying 
.
The
task of construction of set of a kind (3) is a task of control. Controllable
solution set for the first time have been entered by Shary [2].
Formal
decisions (them still name "algebraic") for the first time have been
entered in [3]. The description and calculation of formal decisions of interval
linear systems rather difficult process, however dot formal decisions are
deprived similar difficulties.
In
article the formal type of the decision of the above-stated interval systems is
considered. The computing algorithm is developed.
As it has been
marked, the task in view is shown to resolvability of system of linear interval
algebraic inclusions:
(4)
where 
- the interval matrix made of elements of a matrix 
and a vector 
of object of
management; 
- the interval vector
made of factors at the corresponding degree 
of a characteristic
polynom of object of management; 
- the interval vector
made of factors at the corresponding degree 
of a desirable
characteristic polynom of the closed control system [4].
The
interval vector 
refers to as the formal decision of system 
if his substitution in this system and performance of all
operations by rules of interval mathematics results in true equality.
Further
we shall allocate the dot formal decision which satisfies to the following
equation:
(5)
Let's
prove, that the vector is the dot formal decision of system 
in only case when it satisfies to system of the following
kind:
, (6)
where 
, 
medial matrixes; 
, 
, (7)
where 
always a non-negative
matrix of radiuses 
; 
always a non-negative
vector of radiuses 
.
Under
the offer [5] system (2) is equivalent to below-mentioned system.
,
(8)
That
attracts (6), (7). However, on the other hand, from system (6), (7) follows (8)
and therefore, follows (5).
According
to the theorem from [5] vector is the dot formal decision of system 
in only case when it is as äîïóñêîâûì, and its controlled decision. If the matrix 
is nonexceptional the
system 
has the dot formal
decision in only case when these data
satisfy to the following system
(9)
and in this case
the vector 
is unique such
decision.
References
1. Yunicheva N.R.
Questions of the analysis and synthesis of control systems by objects in uncertainty conditions. Almaty, Printing house
«Ñlassics». 2011. –95p.
2. Shary S.P. Solving the linear interval tolerance problem // Math.Comput.
Simulation. 1995. V.39.–P.53-85.
3. Jolen L., Kiefer Ì., Didri Î. Walter E.. The applied interval analysis. M.: Institute of computer
researches. 2007. – 467p.
4. Khlebalin N.A.
Modal Control of Plants with Uncertain Interval Parameters, in: Proc. Intern.
Workshop «Control System Syntesis: Theory and Application», Novosibirsk, 1991.
-P. 168-173.
5. Fiedler M., Nedoma J, Ramik J., Rohn J., Zimmerman K. Linear
optimization problems with inexact data M.: Institute of computer researches.
2008. – 288p.