![](bg4.png)
corresponding certainty indexes, a task consisting in
adequate design of the examined control system that
aims at making the system stable, i.e. stabilization of
the system, can be considered. The task of
designing is a parametric problem, i.e. parameters in
the set form of a control algorithm of an uncertain
system should be determined. Stabilization of an
uncertain system consists in maximization of the
certainty index that considered system is stable.
However, bearing in mind the fact that usually, as a
result of a stability analysis, the value of the
certainty index is not obtained directly but only its
estimation (6) is obtained, the task of stabilization
can be formulated as a selection of controller
parameters that maximize the certainty index:
-
w
v determined from the sufficient condition,
-
w
v subject to constraint
α
≤
g
v , for a given
10 <<
or
β
≤−
wg
vv , for a given
10 <<
,
-
g
v subject to constraint
γ
≥
w
v , for a given
10 <<
or
λ
≤−
wg
vv , for a given 10 <<
.
The task of stabilization for the first of above-
mentioned approaches consists in determination of a
set of controller parameters
c
D that are
maximizing
w
v :
}maxarg:{
,
*
*
*
*
w
ba
c
v
b
a
b
a
D =
⎥
⎦
⎤
⎢
⎣
⎡
⎥
⎦
⎤
⎢
⎣
⎡
= .
For the considered system, the result of this task is
the set
}]*...**[:{
)()1()0(*
*
*
zw
Tk
c
Dzzzz
b
a
D ∈=
⎥
⎦
⎤
⎢
⎣
⎡
= ,
where
)(maxarg
)(
,
)*(
)(
i
iz
z
i
zhz
i
= , ( ki ,0∈ ).
The value of the certainty index
w
v for controller
parameters selected from the
c
D , is equal 1.
For example, for 50
*
=z ,
15=g
and
25=e
:
,50
)2(2
1
,50
2
1
:{
)1()0()0(
)1(
)0(
≥
+
−
≤
⎥
⎦
⎤
⎢
⎣
⎡
=
aaa
a
a
D
c
}.0,02.0,043,0
)1()0()1()0()1()0(
<><+>+ aaaaaa
By selection of controller parameters from
c
D , for
example 03.0
)0(
=a , 018.0
)1(
−=a the certainty
index that the system is stable 1=
s
v is obtained.
5 FINAL REMARKS
In this paper, the stability analysis and stabilization
of the congestion control system in the ATM
network with application of uncertain variables has
been presented. The specific results in a form of
estimation of the certainty indexes that considered
control system is stable has been obtained. The
selection of controller parameters, which assures
stabilization of the examined system, has been
considered. Uncertain variables can be utilized also
for the quality analysis of control in the considered
control system.
This work was supported by Polish State Committee
for Scientific Research under the grant no. 4 T11C 001 22.
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