APPLICATION OF UNCERTAIN VARIABLES TO STABILITY
ANALYSIS AND STABILIZATION FOR ATM ABR
CONGESTION CONTROL SYSTEMS
Magdalena Turowska
Institute of Control and Systems Engineering, Wroclaw University of Technology,
Wyb. Wyspianskiego 27, 50-370 Wroclaw, Poland
Keywords: Uncertain variables, Congestion control system, Stability analysis, Stabilization
Abstract: The paper presents the application of uncertain variables to stabilization for ATM ABR congestion control
systems. The time-varying system with unknown parameters is considered. The unknown parameter is
assumed to be a value of uncertain variable described by the certainty distribution given by an expert. The
estimation of the certainty index that the system is stable is presented. The estimation consists in the
determination of the lower and upper bounds for the certainty index via the determination of sufficient and
necessary stability conditions. A specific stabilization problem is considered.
1 INTRODUCTION
A theory of uncertain variables and their
applications (Bubnicki, 2001a, 2001b, 2002b, 2004)
developed in the recent years has been successfully
applied to stability analysis for uncertain systems
(Bubnicki, 2000; 2002a). The uncertain variable
x
is defined by a set of values
X
and the certainty
distribution
)
~
()( xxvxh
x
== given by an expert,
where
v
denotes the certainty index that x is
approximately equal to
x
. For the given set
X
D
⊂
the certainty index that
x approximately belongs to
D
is defined as follows
)(max)
~
( xhDxv
x
Dx∈
=∈ .
Computer networks are often treated as control
systems, for which well-known methods of the
control theory and artificial intelligence are applied.
Taking into account a specific character of a
network as a system, the control may concern
various functions, for example – the prevention of
congestion (Altman, et al., 1999; Benhomed, et al.,
1993; Kolarov, et al., 1999). The purpose of this
paper is to demonstrate how uncertain variables can
be used for stability analysis and stabilization of the
selected congestion control system in the ATM
network, under assumption that in the description of
uncertainty concerning the time-varying system, the
parameters that are the values of uncertain variables
occur.
2 THE MODEL OF THE
CONTROL SYSTEM
One of the significant functions of the computer
network is prevention of congestions. In this paper,
the congestion control system proposed in (Kolarov,
et al., 1999) is considered, where the problem of
determination of the explicit rate for the ABR source
has been resolved as a classical control problem in a
closed-loop system (Figure 1). The control system is
a computer network. The model of the system has
the following form
,...
1
)(
1
)2()1(
1
)0(
1
nkn
k
n
nnnnnnnn
wul
ulululyy
++
++++=
−+
−++
(1)
where the input of the system
n
u is the common rate
of the ABR sources, the output
n
y is the number of
ABR cells in a buffer of a switch port,
n
w - the
available bandwidth to ABR sources, is treated as an
unmodelled disturbance. In the equation of the
523
Turowska M. (2004).
APPLICATION OF UNCERTAIN VARIABLES TO STABILITY ANALYSIS AND STABILIZATION FOR ATM ABR CONGESTION CONTROL SYSTEMS.
In Proceedings of the Sixth International Conference on Enterprise Information Systems, pages 523-526
DOI: 10.5220/0002624205230526
Copyright
c
 SciTePress

model, there appears the vector of the parameters,
Controller
n
u
n
w
*y
n
ε
n
y
+
+
-
ATM Network
Figure 1: Block diagram of the closed-loop system
Tk
nnnn
llll ]...[
)()1()0(
= , where
0
)(
≥
i
n
l
is the
number of connections with a round trip delay equal
to
i ki ,0∈ time slots and k means the largest,
measured round-trip delay. In this paper, it is
assumed that
n
l
is a vector of unknown time-
varying parameters. For determination of the
n
u
quantity, it is proposed to use the control algorithm
described by the equation
,...
)(
2
)2(
1
)1()0(
1
)1()0(
1
kn
k
nnn
nnnn
ubububub
aauu
−−−
−+
−−−−
−++=
εε
(2)
where the control error
nn
yy −=
*
ε is the input of a
controller,
*
y is a buffer set point,
T
aaa ][
)1()0(
=
and
Tk
bbbb ]...[
)()1()0(
= are the vectors of
controller parameters. The description of the above-
mentioned system in a state space, assuming the
state vector
T
nknknn
x ]...[
1
εεε
−−−
= , has a form
n
k
nnn
xxlbaCx
⎥
⎥
⎥
⎥
⎦
⎤
⎢
⎢
⎢
⎢
⎣
⎡
==
+
+
2321
1
...
...
0...000
0...100
0...010
),,(
αααα
MMMM
, (3)
)(
)()1(
1
k
k
n
bla −= α ,
)1(
)()1()1()()0(
2
−
−
−++=
k
kk
n
k
n
bblala α ,
)2()1()2()1()1()0(
3
−−−−
−++=
kkk
n
k
n
bblala α ,
…
2
)0(0)0(
2
−+=
+
bla
nk
α .
3 STABILITY ANALYSIS
Applying the methods proposed in (Bubnicki, 2000,
2002a), the stability analysis of the considered
congestion control system can be carried out, under
assumption of time-varying character of a control
system. This analysis consists in evaluation of the
certainty index that considered system is stable.
Let us assume that the uncertainty concerning
unknown parameters of the considered control
system, and thus the number of the connections with
the specified round-trip delay, is described by a set
of allowable values
)}(:{),(
)(
)(
)(
},...,1,0{
i
i
i
ki
l
zleLlzeD ≤≤∈=
∈
(4)
in a following way
),( zeDl
ln
n
∈ , (5)
where
Tk
eeee ]...[
)()1()0(
= is a vector of lower
bounds of parameters
n
l ,
)(i
e is the minimum
possible number of connections with a round-trip
delay equal to
i time slots,
)(i
z
is the upper limit,
which is an unknown parameter, the value of the
uncertain variable with the certainty distribution
)(
)(
,
i
iz
zh , ( ki ,0∈ ) given by an expert.
The problem of a stability assessment can be
formulated in the following way: for given certainty
distributions )(
)(
,
i
iz
zh , one should determine
estimation of the certainty index
s
v that considered
system is stable
gsw
vvv ≤≤ . (6)
This estimation can be achieved by determination of
the necessary condition
)(zG
and sufficient stability
condition )(zW and then by determination of
certainty indexes that conditions are satisfied
)(max zhv
z
Dz
w
zw
∈
= , )(max zhv
z
Dz
g
zg
∈
= ,
where )}(:{ zWZzD
zw
∈= , )}(:{ zGZzD
zg
∈= and
)}(...,),(),({min)(
2
)2(
2
2
1
1
+
+
=
k
k
zzzz
zhzhzhzh is
ICEIS 2004 - ARTIFICIAL INTELLIGENCE AND DECISION SUPPORT SYSTEMS
524

a joint certainty distribution.
In general
zgzw
DD ⊆ and
zwzg
DD − is called
“a grey zone” which is a result of an additional
uncertainty caused by the fact that:
)()( zGzW ≠
.
Consider a congestion control system, in which,
in respect of delays, only one kind of connections
exists, i.e. all connections have the same round-trip
delays (k = 0), the number of these connections
equals
n
l . The uncertainty concerning an unknown
parameter
n
l is described by the following formula
zle
n
n
≤≤ , (7)
where e (the minimum allowable number of
connections) is given, whereas z is an unknown
parameter, about which it is known that this
parameter is a value of an uncertain variable with a
certainty distribution )(zh
z
. The description of the
system in the state space has the following form
n
nn
n
x
lala
x
⎥
⎦
⎤
⎢
⎣
⎡
+−−−
=
+
21
10
)0()1(
1
. (8)
Applying stability conditions proposed in (Bubnicki,
2000, 2002a) from the sufficient condition we
obtained
},
2
1
,0,043,0
,
)2(2
1
2
1
:{
)0()1()1()0()1()0(
)1()0()0(
e
aaaaaa
aa
z
a
ZzD
zw
><<+>+
+
−
<<∈=
from the necessary condition
}.0),0
,
4
()0,
4
(:{
)1()0()1(
)1()0(
)1(
)0(
>+<
−
<∨><∈=
aaa
aa
za
a
zZzD
zg
For the given triangular certainty distribution
(
*
z
, g) described by the formula
⎪
⎪
⎪
⎪
⎩
⎪
⎪
⎪
⎪
⎨
⎧
+≤≤++−
≤≤−+−
=
otherwise,
for
for
0
,1
1
,1
1
)(
**
*
**
*
gzzz
g
z
z
g
zzgz
g
z
z
g
zh
z
the certainty index that the necessary stability
condition is satisfied
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎩
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎨
⎧
>+<
≤
−
≤−
+−
−
>+>
≤≤−
+−
>+
<≥
−
∧
>≥
=
otherwise, 0
for
for
for
,0,0
,
4
1
)(
4
,0,0
,
4
1
4
,0
),0,
4
(
)0,
4
(1
)1()0()1(
*
)1()0(
*
*
)1()0(
)1()0()1(
*
)0(
*
*
)0(
)1()0(
)1(*
)1()0(
)1(*
)0(
aaa
z
aa
gz
g
z
aag
aaa
z
a
gz
g
z
ag
aa
az
aa
az
a
v
g
and for
0
)1()0(
>+aa , 043
)1()0(
<+ aa ,
e
a
2
1
)0(
> ,
0
)1(
<a the certainty index that the necessary
condition is satisfied
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎩
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎨
⎧
+≤≤
++
−
<
+
−
<−
+−
+
−
≥
+
−
≤
=
otherwise. 0
for
for
for
,
2
1
1
1
,
)2(2
1
1
*
)
)1(
2
)0(
(2
1
,
)2(2
1
,
2
1
1
*
)0(
*
*
)0(
*
)1()0(
*
*
)1()0(
*
)0(
gz
a
z
g
z
ag
z
aa
gz
g
z
aag
z
aa
z
a
v
w
For example, for 35
*
=z ,
15=g
, ,02.0
)0(
=a
018.0
)1(
−=a , we obtained: 175.0 ≤≤
s
v .
4 SYSTEM STABILIZATION
Having determined the stability conditions and
APPLICATION OF UNCERTAIN VARIABLES TO STABILITY ANALYSIS AND STABILIZATION FOR ATM ABR
CONGESTION CONTROL SYSTEMS
525

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 <<
α
or β ≤−
wg
vv , for a given
10 <<
β
,
-
g
v subject to constraint γ ≥
w
v , for a given
10 <<
γ
or λ ≤−
wg
vv , for a given 10 <<
λ
.
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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