Takagi-Sugeno Fuzzy Control of Hopf Bifurcation in
the Internet Congestion Control System
F. Abdous*, **, A. Ranjbar N. **, R. Ghaderi**, S.H. Hosseinnia**
* Young Researchers Club
* *Noushirvani University of Technology, Faculty of Electrical and Computer Engineering,
P.O. Box: 47135-484, Babol, Iran ([email protected])

Abstract: The main objective of this paper is to show the effectiveness of the proposed fuzzy controller
in the congestion control of the internet. The chaotic behaviour of the congestion is primarily shown when
a network with a single link in connection with a single source is considered. The gain variation as a
critical parameter produces the Hopf bifurcation. As soon as instability occurs, the effective range of gain
parameter, and therefore the system optimum performance is restricted. The chaos and the Hopf
bifurcation are controlled through two controllers of Feedback Linearization and TDFC schemes. The
performance is shown promoted when a fuzzy controller has stabilized the bifurcation.
Keywords: Hopf bifurcation, Fuzzy controller, Feedback linearization, Time-Delayed Feedback Control,
Chaos

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1. INTRODUCTION
Due to increasing use of the internet, congestion as a
challenging problem occurs. Drop-tail approach as an
ordinary management skill will be used in routers. It means,
when the capacity becomes full, there is no room available
and incoming data packet will have no fortune to get a
desired service. The main draw back of this scheme is the
higher servicing time delay and also the need for
synchronism (Hashem, 1989).The Active Queuing
Management (AQM) has been proposed to handle the
congestion problem in the routers. The effectiveness of
Random Early Detection method (RED) has also been
investigated (Floyd and Jacobson, 1993). Based upon RED
some other optimum technique e.g. Random Early Marking
(REM) (Lapsley and Low, 1999), is also suggested.
Continuous and discrete models have been proposed to
interpret the congestion in the internet network (Deb and
Srikant, 2002; Hollot et. al., 2001; Hespanha et. al. 2001;
Veres and Boda, 2000). A model with a single link which is
connected by a single source will be used to represent the
congestion problem (Deb and Srikant, 2002; Hespanha et. al.,
2001).
The usual models are either nonlinear or linearized in a small
interval e.g. at the operating point. Nonetheless, nonlinear
systems produce some effects such as chaos, which makes
the analyze more complicated. An internet network may have
such chaotic behaviour (Veres and Boda 2000). When a
change in the gain of an internet network is of interest – a
single source along with a single link– bifurcation
phenomenon may be occurred (Li et. al., 2004). It was
primarily assumed that the controllability of chaotic systems
would be failed. This view was changed when Grebogi,
Yorke, and Ott (Ott et. al., 1990) showed otherwise.
Thereafter, several attempts were made to develop such other
control techniques. These include Occasional Proportional
Feedback (OPF) (Hunt, 1991) and Time-Delay Feedback
Control (TDFC) (Pyragas, 1991).
In (Wang, 2002) RED was used to control an internet
congestion problem. In this, RED parameter behaves as a
chaotic parameter. The work was developed in (Chen et. al.,
2003) together with applying Time-Delay Feedback to
control the internet congestion. A combination of methods in
(Wang, 2002) and (Chen et. al., 2003) was proposed in (Chen
et. al., 2004) to gain their advantages. Meanwhile some of
nonlinear control techniques such as sliding mode, robust and
fuzzy control may cope with the nonlinearities in the chaotic
dynamics. Furthermore, a control approach based on solving
nonlinear differential equation can also be used (HoseinNia
et. al., 2008a, b). Useful techniques based on TDFC and
feedback Linearization (FL) methods were presented to
control the congestion in the network (Abdous et. al., 2008a
,b).
Fuzzy control approach as a novel contribution, which was
widely used in nonlinear systems, will be gained here to
control the Hopf bifurcation. This phenomenon occurs when
the gain of the system as a critical parameter varies, in a wide
range. The proposed fuzzy controller provides the stability as
well as fast transient response. The outcome of the proposed
technique will be compared with TDFC and F.L. to signify
the performance.
The paper is organized as follows: A network model is
primarily introduced in section II. In section III, Hopf
bifurcation and the cause of occurrence will be briefly
introduced. The proposed fuzzy controller will be described
in section IV. Section V is devoted to simulate and to show
the performance and the significance of the method.
Ultimately, the work will be concluded in section VI.
2. MODEL OF CONGESTION IN THE INTERNET
NETWORK
(2)
x* p ( x* )  w
This is achieved, assuming the stationary situation for the
states and therefore, no delay in the system. The appropriate
Eigen value will be obtained when the following equation is
substituted in (1) as:
l  k[ p( x* )  x* p( x* )]
Consider an internet network with a single link and single
source. The appropriate representing model is often described
(Deb and Strikant, 2002; Hespanha et. al., 2001) as:
(1)
dx(t )
 k[w  x(t  D) p( x(t  D))]
dt
3.2 The Hopf Bifurcation in the Internet Congestion Control
The following equation evaluates the equilibrium points
i.e. x * :
Where, x(t) stands for the instantaneous speed of sending
information from the reference unit in time t. A positive gain
parameter and the set point are denoted by k and w,
respectively. D also represents total delay time in sending and
receiving data. The lost probability is shown by a
nonnegative and ascending function, p(.). This shows the
probability of loosing the sent information packets. It will
shortly be shown that this system becomes unstable when the
gain parameter k, varies in some interval. An initial and more
important objective is to maintain the stability.
3. CRITICAL PARAMETER AND HOPF BIFURACATION
IN THE INTERNET CONGESTION CONTROL
3.1 Bifurcation Theory
The dependency study of the equilibrium point to the
parameters of the system is called the Bifurcation theory
(Hilborn, 2000). In this theory, variation of the Eigen values
with respect to the parameter changes is usually plotted.
Accordingly, a specific value of parameter (critical
parameter) where the sign of Eigen value varies- spots the
chaos occurrence. On the other hand, equating the Eigen
value to zero, obtains the critical value. This behaviour is
schematically shown in Figure 1. As shown in this graph, 
as a critical parameter, shows the existence of chaos.
Therefore, chaos appears when  crosses the lower threshold
i.e.   1 . In this case, where the Eigen value becomes zero, a
stable limit cycle will be produced.
l
(3)
It was shown (Li et. al., 2004) that the following equation
yields the real part of Eigen value:
k* 

(4)
2 D( p( x )  x p( x ) )
*
*
*
Consequently, the Hopf bifurcation will be occurred in the
equilibrium point x * when k takes k * value.
4. APPLYING FUZZY TECHNIQUE TO CONTROL THE
INTERNET CONGESTION
The first step to design a controller is to define the goal. A set
point can be an achievable aim, which implicitly needs the
stability of system. The same goal may be defined for the
congestion in the internet. This means the congestion will be
controlled if the error tends to zero, of course when the gain
k, will be greater than k*. Meaningfully, a primary goal is to
design a fuzzy controller to reduce the steady error. Assume
w is a set point of system in (1). Therefore the error is as
follows:
e=x-w
(5)
where x denotes the state. The error is an input to the
controller which will be assumed a fuzzy controller, here.
Membership function N, Z and P will be defined as in Figure
2.
1.2
N
1
P
Z
0.8
0.6
0.4
0.2
l0
1
l>0

l<0
0
-2
-1.5
-1
-0.5
0
0.5
1
1.5
2
Fig. 2: The input membership function
Fig1. A schematic diagram of dependency of Eigen value l ,
and chaos parameter 
Due to ease of use, a Takagi-Sugeno fuzzy system will be
used for designation. Consider the following rules:
IF (e is P) then ( u=a1e+b1 )
IF (e is N) then ( u=a 2 e+b 2 )
The variables a 1 , a 2 , b1 ,and b2 are chosen as the following for
the simulation:
Table 1. Control Parameter Value
a1
4
b1
0.8
a2
3
b2
0.7
These values are obtained by a trial and error algorithm,
whereas can be assigned by such a genetic algorithm.
5. SIMULATION RESULTS
(20)
PAQM (t ) part has the duty to mange the queue. Meanwhile,
Pc (t ) controls the chaos. The latter will be generated by a
fuzzy mechanism.
Simultaneously, a REM is selected as the queuing
management (Lapsley and Low, 1999) Accordingly, the
probability function i.e. p(.) is substituted by an equation,
which is as follows:
(23)
PREM (q (t ))  1  exp( q (t ))
Where, q(t) is the instantaneous queue length in time t and 
is a constant integer parameter. Using the Brownian motion
approximation (Kelly, 2000), the probability function p(t),
will be substituted by the following equation:
PREM ( x) 
 2 x
 2 x  2(C  x)
(24)
x* .PREM ( x* )  1
x*
)  1  x*  3.2170
(30)
20  3x*
This evaluates the critical value of the gain parameter, k that
is as follows:
k *  1.7231
(31)
In order to spot the performance of the proposed controller,
two different situations of with and without controller are
simulated, choosing various values of the gain.
In a second step, it is chosen as k=1.5, 2.2 and 3.3. It is worth
noticing that these values are selected from either sides of the
critical value as stated in (31). The corresponding simulation
results are shown in Figure 3 - 8. In the first place, the
outcome of fuzzy controller is compared with Feedback
linearization (F.L.) (Abdous et. al., 2008a) and the Timedelayed feedback control (TDFC) (Abdous et. al., 2008b)
controllers and the case with no control in action.
The time response and the phase portrait are shown in Figure
3 and 4 respectively, caused by a change in the gain, i.e.
k=1.5. As it can be seen, if the gain value is less than the
critical one, the system is still stable. However, the fuzzy
controller provides quick response together with less
maximum peak and no steady state error. These will be
confirmed in the phase portrait in Figure 4, as well.
x
(26)
20  3x
According to (20) and (26), p (t ) will be obtained, which is
as follows:
PREM ( x) 
x (t )
 Pc (t )
(27)
20  3x (t )
To set a regulation problem, w is considered as unit.
Consequently, the system in (1) will be stated as:
p (t ) 
Time Responce For k=1.5
5
without control
with TDFC
with F. L.
with Fuzzy control
4
(25)
3
x
In the mean time, the transfer capacity and total delay along
with the length D are assumed as 1 Mbps and 40 ms (Li et.
al., 2004), respectively. Let us consider the total delay as a
unit for the packet size of 1000 Bytes. This immediately
evaluates the capacity of the data transfer line as 5 packets/
time unit. Therefore, PREM (x ) will be obtained as:
(29)
Considering (26) and (29) yields the equilibrium point as:
Where,  2 stands for the congestion deviation in the data
package whilst C denotes the data transmission capacity in
packets/time unit. Similar to (Deb and Strikant, 2002), let
 is stated as:
 2  0.5
(28)
A free running stage for the system ignores the roll of the
controller. It means the control signal p(t) only
includes PREM (t ) . In the first step, to compute the critical
value of k it is needed to find the equilibrium point.
According to (2), it is necessary to consider:
x* (
The appropriate tool to adjust the queue length in system (1)
is p(t). This must control the queue and the required service.
When a chaos takes effect, the control must overcome this
phenomenon. To meet the requirements and to control the
nonlinear chaos, p(t) is considered by two partitions as:
p (t )  PAQM (t )  Pc (t )
x  k[1  x(t  1) p( x(t  1))]
2
1
0
0
10
20
30
40
Time(s)
50
60
70
80
Figure 3. The time response of the system in (28) for k=1.5
in four different situations of no control, TDFC,
Feedback Linearization (F.L.) and Fuzzy control
The roll of the controllers become more significant when the
higher value is assigned to the gain variable e.g. k=3.3. The
appropriate phase portrait is also plotted in Figure 8. The
performance of the fuzzy controller is shown when this
controller stabilizes the chaos in the system. The transient
response is provided satisfactory in Figure 7.
x (t-1)
x (t-1)
Phase Plan without controller Phase plan with TDFC controller
4
4
3
2
4
2.5
2
1
2
3
2
1
0
4
19
1
time response without controller
x 10
0
x
3
x (t-1)
2
2
1
2
3
4
x
x
Phase plan with F.L. controller Phase plan with Fuzzy controller
3
3
x (t-1)
1
3
1
2
x
3
4
-1
x
-2
Figure 4. The phase portrait in different situations of: No
controller (up-Left), TDFC (up-Right), F.L. (Down-Left)
and Fuzzy control (Down-Right)
20
40
60
80
time response with control
10
TDFC
F. L.
Fuzzy control
5
x
The quality of the system when the gain crosses the critical
value, is also shown in Figure 5 – 8. It is easily seen that,
system is unstable when the controller is not in place. The
occurrence of chaos can also be partially evident in the phase
portrait in Figure 6 and 8. The significance of the Fuzzy
controller can be seen when this controller stabilizes the
system. Therefore, the chaos is completely vanished.
0
0
0
20
40
Time(s)
60
80
Figure 7. The time response for much higher gain value
Time Responce For k=2.2
7
without control
TDFC
F. L.
Fuzzy control
4
3
-1
-2
-10
2
1
Phase plan with TDFC controller
10
-5
0
x
5
0
-5
-2
5
0
18
2
x
4
6
x 10
Phase plan with F. L. controller Phase plan with Fuzzy controller
5
6
0
20
40
Time(s)
60
80
x(t-1)
0
Figure 5. The controllers performace when the critical value
is assigned to the gain i.e. k=2.2.
4
x(t-1)
x
0
x(t-1)
5
19
Phase
x 10Plan without controller
1
x(t-1)
6
3
2
2
4
x
6
4
2
0
2
4
6
x
x (t-1)
x (t-1)
Phase Plan without controller Phase plan with TDFC controller
6
4
2
4
0
2
2
4
2
4
6
x
x
Phase plan with F. L. controllerPhase plan with Fuzzy controller
4
4
x (t-1)
x (t-1)
0
3
2
2
3
x
4
2
0
1
2
3
4
x
Figure 6. Occurance of chaos and the performance of the
controllers
Figure 8. Phase portrait of the output for higher value of the
gain, e.g. k=3.3, No controller (up-Left), TDFC (upRight), F.L. (Down-Left) and Fuzzy control (DownRight)
The instantaneous behaviour of p(t) in two different
situations of no controller in action and under control is
shown in Figure 9– 11.
Time Response of Signal Control
p
0.5
0
0
10
20
30
40
50
60
70
80
0
10
20
30
40
50
60
70
80
0
10
20
30
40
50
60
70
80
p
40
20
0
p
2
0
-2
p
2
0
-2
0
10
20
30
40
Time(s)
50
60
70
80
Figure 9. Instantaneous behaviour of p (t ) in four different
cases of no controller (1), TDFC (2) , F.L. (3) and Fuzzy
controllers (4), considering k=1.5.
Time Response of Signal Control
p
1
0
-1
0
10
20
30
40
50
60
70
80
0
10
20
30
40
50
60
70
80
0
10
20
30
40
50
60
70
80
0
10
20
30
40
Time(s)
50
60
70
80
p
40
20
0
p
5
0
-5
p
5
0
-5
Figure 10. Instantaneous behavior of p (t ) in four different
cases of no controller (1), TDFC (2) , F.L. (3) and Fuzzy
controllers (4), considering k=2.2.
Time Response of Signal Control
p
2
0
-2
0
10
20
30
40
50
0
10
20
30
40
50
0
10
20
30
40
50
0
10
20
30
40
50
p
40
20
0
p
2
0
-2
p
2
0
-2
Time(s)
Figure11. Instantaneous behavior of p (t ) in four different
cases of no controller (1), TDFC (2) , F.L. (3) and Fuzzy
controllers (4), considering k=3.3.
6. CONCLUSION
A fuzzy controller is used to control the Hopf bifurcation due
to the congestion in the internet. The performance of the
proposed method is compared with Feedback Linearization
and TDFC. In addition of the stabilization, the steady state
error tends to zero. Meanwhile the speed of the transient time
is other advantage of the proposed controller. The simulation
result signifies the performance of the fuzzy controller over
two Feedback Linearization and TDFC schemes.
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