Design of robust active queue management controllers for a class of TCP communication networks

Abstract

This paper describes the design of active queue management (AQM) controllers for a class of TCP communication networks. In TCP/IP networks, the packet-dropping probability function is considered as a control input. Therefore, a TCP AQM controller was modeled as a time-delayed system with a saturated input. The objective of the work described here was to design robust controllers capable of achieving the desired queue size and guaranteeing asymptotic stability of the operating point. To achieve this aim, we have proposed two control strategies, namely a static state feedback controller and an observer-based controller. By applying the Lyapunov–Krasovskii functional approach and the linear matrix inequality technique, control laws and delay-independent stability criteria for the AQM controllers were derived. The performance of the two control schemes was evaluated in various network scenarios via a series of numerical simulations. The simulation results confirm that the proposed schemes outperform other AQM schemes.

Full text

Page 1

Proceedings of 2005 CACS Automatic Control Conference
Tainan, Taiwan, Nov 18-19, 2005
Design of Robust Active Queue Management Controllers for a
Class of TCPCommunication Networks
Chang-Kuo Chen, Yung-Ching Hung, Teh-Lu Liao
Department of Engineering Science, National Cheng Kung University
Tainan, 701 Taiwan, e-mail: tlliao@mail.ncku.edu.tw
Abstract
This paper is concerned with the design of active
queue management (AQM) controllers for a class of
TCP communicationnetworks.
networks, the packet drop probability is limited
between 0 and 1. Therefore we model TCP/AQM as a
time-delay system with a saturating input. The object
of this paper is to design controllers to achieve the
desired queue sign and guarantee asymptotic stability
of the operating point. To achieve this aim, we
propose two control strategies: static state feedback
controller and observer-based controller for the
time-delay system with input saturation. Based on the
Lyapunov-Krasovskii functional approach and LMI
method, the control laws and their delay-independent
stability criteria are derived. The proposed control
laws are then validated and tested on different network
scenarios through numerical simulations in both
Matlab and Network Simulator-2.
In theTCP/IP
Key Word:AQM, TCP, Lyapunov-Krasovskii,
LMI.
1.Introduction
In the past few years, networks have become an
essential part of many science and engineering
applications. Due toa
applications cannot be exactly recognized in the
networking environment Therefore, traffic congestion
is oneofthe major
experienced by millions of user in the current Internet.
The congestion control of AQM is used for
networkmeasurecongestion
congestion control (ECN), which is developed by
Internet Engineering Task Force (IEFT). Active queue
management is one of the key congestion control
schemes, which has been developed to reduce the
packet drop and network utility in advance of
prevention. The RED (Random Early Detection) [1]
algorithm, the earliest well known AQM scheme,
eliminates the flow synchronization problem and
attenuates the traffic load by domination of average
queue length. However, RED algorithm leads to
oscillated and unstable with the parameter sensitivity
and system responsiveness. Random Exponential
Marking (REM) [2] is another AQM scheme, which
can achieve both high utilization and rare packet loss
through decuple price function form system behavior.
Recently, control theories have been widely
applied to model and analyze the network congestion
numberofusersand
communicationproblems
levelonexplicit
control. Based on theory of the stochastic equations, a
fluid based model of TCP and RED dynamics was
developed in [3]. This model was described the
evolution of the average characteristics variables on
the network, such as the average TCP windows size
and the average queue length. Meanwhile, it was
shown that the captures indeed the qualitative
behavior of TCP traffic flows. Based on the TCP
model, several congestion control schemes using
control theory have been proposed to improve the
performance of communication networks. Adaptive
REDis describedby [4]
parameters to reduce RED’s parameter sensitivity. In
[5],aPI(proportional-integral)
developed for new AQM approach through linear
system analysis and implemented by difference
equation. However, those methods can reduce the
sensitivity problem of RED, but in more realistic
networks, the RTT (run-trip time) variation of
congestion phenomena and other effects may cause
the traffic load become unstable, such as the presence
of responsive sources variations different points in the
network, rerouting, additional unresponsive source
traffic.
The work in [6] converted the inherently
nonlinear model into a linear delay system, through
the technique of linearization and subsequently
applied thecorresponding
approach. Theback-stepping
filter-based approach has been applied to design
adaptive both state feedback and output feedback
controllers [7]. Also, the study in [8] designed an
AQM controllerwith variable structure control
scheme, and validated its performance via different
simulation environments.
On the other hand, the saturating actuator
problemsareencountered
engineering systems frequently, their existence always
be the source of instability, like an integral wind up or
limit cycle [9-13]. In case of the control system
designed tobestable
saturation, we cannot guarantee the stability of this
closed-loop system when the controller applied to the
saturating input. Although, several control schemes of
linear delay system with input saturation have be
developed, but rare control schemes can be applied to
congestion control due to the Internet is a large scale
complex system.
Inspired by theabove
developments, we analyze and design AQM controller
using time-delay and saturation model. We present
that someadaptive
controllerwas
controller withLMI
andtechnique
inmany practical
withoutconsideringthe
congestion control

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Proceedings of 2005 CACS Automatic Control Conference
Tainan, Taiwan, Nov 18-19, 2005
both state feedback and output feedback controllers
for AQM supporting TCP via LMI technique. It is
shown that the proposed design has good asymptotic
as well as stability robustness with respect to RTTs
and the number of the active TCP sessions.
The rest of this paper is organized as follows.
The TCP Model and Design objective is discuss in
Section 2. Section 3 presents the AQM stabilizing
controllerdesignbased
observer-basedcontrolschemes.
results are presented in Section 4 to validate our
design. Finally, we present our conclusions in Section
5.
state feedback
The
and
simulation
2.TCP Model and Design Objective
For the TCP networks, a nonlinear fluid-flow
dynamic model was developed by using the theory of
stochastic analysis. This model relates the average
value of characteristic variables on the network such
as the average TCP window size and the average
queue length, and is described by coupled nonlinear
differential that can accurately captures the dynamics
of TCP. Under some assumptions, this nonlinear
dynamic was simplified s follows:
))((
))((
))((

2
)(
)(
C
1
)(
tRtp
T
C
tRtq
tRtwtw
T
tq
tw

pp






(1.a)



























0if,)(
)(
C
)(
, 0max
0if),(
)(
C
)(
)(
qtw
T
tq
tN
C
qtw
T
tq
tN
C
tq

p
p
(1.b)
where w is the average TCP window size (packets),
qthe average queue length (packets),
propagation delay (second), R0is the transmission
round-trip time ( R q
CTp), C is the link capacity
(packets/s), N denotes the number of TCP sessions
and
10
 p
is the probability of a packet being
marked, which is considered as the control input to
reduce the sending rate and maintain the bottleneck
queue. All variables are assumed non-negative.
Equation (1.a) describes both the additive increase and
multiplicative decrease (AIMD) congestion control
algorithm with window size evolution of TCP flow.
Equation (1.b) represents the dynamics of queue size
which will be accumulated when the sending packets
exceed the link capacity. This model considers
homogeneous multiple TCP sessions, one single
bottleneck link, and delay feedback configuration.
Detailed justification of this model can be found in
[14-17].
For the design and analysis of AQM scheme, we
p
T
denotes the
assume
operation point
and
)( 
tq 
0
)(
RtR

and
w
Nt
)
N
p

by letting
)(
and obtain the
w 
,,(
000
q
0)(

t
0
, thereby implying that the operation point
),,(
000
pqw
satisfies
2
0
2
0

pw
and
N
CR
w
0
0
.
Further, we linearize the system (1) about the
operating point
),,(
000
pqw
of the time-delay argument
nonlinear model can be rewritten by the following
linear model
()()(
1
tCx ty

with the system matrices as follows:



1
RR
and ignore the dependence
t 
, and then the
0 R
)()(
)
0
Bu RtxAtAx tx


(2)













00
2
0
2
0
1
N
CRCR
N
A
;









00
1
2
0
2
0
1
CRCR
N
A
;









0
2
2
2
0
N
CR
B
;
 1 0

C
(3)
where





)(
)(
)(
tq
tw
tx

, u(t)  p(t) , w(t)  w(t)w0,
 q(t)  q(t)q0, and  p(t)  p(t) p0 denote the
perturbations of state variables and control input
around the operating point (w0, q0, p0), respectively.
Based on the time-delayed linear system (2) with
a saturation input and the measured queue length
 q , the object of this paper is to design a controller
to achieve asymptotic stability of the desired operating
point (w0, q0, p0).
3.AQM Stabilizing Controller Design
Due to the behavior of the TCP networks, the
packet drop probability is limited between 0 and 1 and
there exists a delay in the state. In this section we
consider two control schemes. One is a state feedback
control scheme in which both window size and queue
length are available. Another one is an output
feedback design relying only on link buffer queue
lengthmeasurement.Consider
time-delay system with saturating input


] 0,[),()(

tttx
where
x(t)  
u(t)   ,
the state, the control input, and output of the system
(in this paper, we assume the size of queue is the
system output). The time delay   
and
C
areconstant
dimensions,and
(t)  
vector-valued initial function. The main assumption
about the system (4) is that the pairs (A,B) and
thefollowing




)()(
))((sat)()()(
1

tCxty
tuBtAtAxtx 
(4)
n,
y(t)  
represent
, A, A1, B
appropriate
continuous
matrices
n
of
theis

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Proceedings of 2005 CACS Automatic Control Conference
Tainan, Taiwan, Nov 18-19, 2005
(A,C) is stabilizable and observable, respectively,
and
A is an asymptotically stable matrix. The
saturation actuator is described by the nonlinearity
,
))((sat

The saturation term can be written as
sat(u(t))  (u(t))u(t)
where
(
))((

and satisfies
0)(
)(0
u
)(
, 0
),(


t

u



ut
utu
tu
u
tu
m
mm
(5)
0)(if
)(0
u
if
)(if
, 0
, 1
, )
t


t

u



ut
utuuu
tu
m
mm

(6)
0 allfor, 1))((0

ttu

(7)
Thus, based on (5)-(7), the system (4) can be rewritten
equivalently as follows:


] 0,[),()(

tttx




)()(
)())(()()()(
1

t Cxty
tutuBtAtAxtx 
(8)
A. State Feedback Controller Design
It is possible to design a state feedback
controller for system (8) under the assumption that all
states are available. We implement a memoryless the
state feedback controller in the form of
u(t) Kx(t)
where K is the gain matrix such that the resulting
closed-loopsystem is
Substituting (9) into (8) yields the closed-loop system,
which is written as follows:
)()()(
1

Kt KxBA
In order to guarantee the asymptotic stability of the
system (10) for any time delay, we apply the
Lyapunov-Krasovskii functional approach and LMI
method to design the suitable feedback gain matrix K.
The controller gain and delay-independent stability
criterion are given by the following theorem.
(9)
asymptotically stable.
)()())) (((
)(

))
x
((
1





x

tAt
t KxtuBtxAt Axtx 
(10)
Theorem 1: Consider the system (8). Then, the system
is stabilized by the controller (9) for any constant
delay 
if there exist symmetric positive definite
matrices P and Q satisfying the following matrix
inequality:
A
TP PAQ
A1
PA1
Q
TP




 0

(11)
and the state feedback gain in then given by
K  B
TP
(12)
Proof:Consider the Lyapunov functional candidate
V  x x
t
The time derivative of the Lyapunov function (13)
T(t)Px(t)
T(s)Qx(s)ds
t
(13)
along the trajectory of system (10) is given by
)()()()
P
)(
t
(2
)()
t
(2

)
t
())( (
)()(
)()()()


t
(

)(

P
)(
1



K
V



A

x

Qx

QxxKxPB
, we obtain

PA
tx

tx PAtxtxQ PAt
ttx
tQxtxtx

PtxtPxtx

V
TT
TTT
T
TTT

(14)
Let
B
T
T






(15)








A





)(
)

(
)()(
)()(
)(2) )((
)()()()(2
)()(2)()(
1
1
1
1





tx
tx
QP
PAQPA
T
PA
txtx
tQxtx
txPAxQPAPAtx
tQxtxtBxPBBtx
tx PAtxtxQPAx
T
TT
T
TTT
TTT
TT
The negative inequality of (11) leads to the
negativity of the derivative V
Therefore, from the Lyapunov-Krasovskii Stability
Theorem,itconcludes
stability of the system (10).
along with (10).
the uniformasymptotic
B. Observer-Based Controller Design
Since the Internet is a large scale complex
system, the state variable window size can’t be
measured locally. It is more suitable to deploy an
observed-basedcontroller
measurement. We therefore propose the following
output feedback controller, where q is considered as
the measured output, and the window size w is an
unmeasured state variable.
Consider the observer-based controller in the
following form
( ˆ
)( ˆ
)( ˆ
1
tytyL


basedontheoutput
)( ˆ
x
)(
)( ˆ
x
)( ˆ
y
)) ( ˆ
)((
)()) (()
tKtu
tCt
tutuBtxA

txA

tx








(16)
where L is the observer gain matrix. Let us define
the state error e(t}  x(t) ? x(t), then the dynamics of
state error is given by
)( ˆ
)()(


AteLC A
For this given controller, the closed-loop system
(16)-(17) can be written as
)( ˆ
)( ˆ
)( ˆ
1

BtxAtxAtx


)()()(
)) ( ˆ

)((
)( ˆ
1
xA
)( ˆ
x
)()(
LC
1
1




x


te
ttx
ttAtxAtAx
tx

tx

te

(17)
)()(
previous
)(
))
A
( ˆ

)((
)( ˆ
1
xA
)( ˆ
x
)(

t
)(
LC
)( ˆ

x
)()(
)()( ˆ
x
))( ˆ
x
(
1
1

section,

t



A

x



teeLC
the
t
ttAtxAtAx
txtxte
teLCtKtK

(18)
As
Lyapunov-Krasovskii functional approach and LMI
method to design the suitable observer gain matrix L
inweapplythe

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Proceedings of 2005 CACS Automatic Control Conference
Tainan, Taiwan, Nov 18-19, 2005
and feedback gain matrix K. The delay-independent
stability criterion is stated by the following theorem.
Theorem 2: Consider the system (18). Then, the
system is stabilized by the observer-based controller
(16) for any constant delay  if the observer gain L
is chosen such that AL (ALC) is stable and there
exist symmetric positive definite matrices P and Q
satisfying the following matrix inequality:
0
00
00P
0H)(
0PM
LC
1
1
12
11

















QQA
PA
QA
ALC
T
T
T
(19)
where
M
1
And the state feedback gain in then given by
PPAAP
T

and
QQAQAH
LL

T
2
.
1 -
PBK
T

(20)
Proof:
follows.
Consider a Lyaponuv functional candidate as
dsseQ
)
sesx
( ˆ
Psx
teQ
)
tetx
( ˆ
Ptx
ˆ

time
tVtVt
(
VtV
T
t
t
T
TT
))(())(
ˆ (
)(())
)()

()()(
321



(21)
Taking
trajectories of system (18) yields
)( ˆ
)(
ˆ
1
BAtx

thederivativeof
V
alongthe
)()( ˆ
2
x
)( ˆ
1
x
)
LC
)( ˆ
2
x
)( ˆ )
PLC
)))(

(()(
ˆ
x
)
t
( ˆ
x
))))
u

(

()((
ˆ
)(
ˆ
x
)
t
( ˆ
u
tettPAt
xKtB

APt
tPK
tPtxtx

Ptx

V
T
TT
TT







(22)
)()(2
)()()()())( (
)()(()(
t
1
)
2


RA


tete
teLC ARteteRAe
te

RteteRte

V
T
TTT
TT

(23)
)()(
)()()( ˆ
x
)(

ˆ
x
( ˆ

)
t
(
ˆ
x
3



z

tQz
tzQtztPttxPtV
T
TTT

(24)
Then, by using (22)-(24), we have
()()(
21

APtx
)()(
)( ˆ
x
)(
ˆ
x
)()(2
e
)())(( (

)
t
(
e
)
t
( ˆ
2
e
)( ˆ

)( ˆ
2
x
)( ˆ )
xP
)( (
ˆ
x
)
t
()()()(
t
)( ˆ
e
)
Q
(
t
ˆ
x
)( ˆ
e
)(
ˆ
x
)()

(
t
2
P
)(
t
)(
x
)(
)

())((
)()( ˆ
2
t
)
R
( ˆ

A
)( ˆ
2
e
)( ˆ )
PLCe
)))
x
((()(
ˆ
)( ˆ
x
))))
u
(()((
ˆ
x
)()


tx
1
1
1
1
3



















teQt
tPtQA
teQQAQAtetPLCetx
txPAt PAPA
tetQte
xtPt
teRAtete LCRte
eLCAt
tt PAtx
tKtB
txPKtuBAt
tVtVtVtV
T
TT
T
L
T
L
T
TT
TT
TT
TT
TT
T
TT

(25)
By
z
obtain a more compact form of

letting
(
ˆ
tx
P

B
(
K
)
T

ˆ
x
and
(
e
assuming
,
 ) )())(

tttet
TTTTT
we
)(
00
0
Q
0
0)(
0
)(
1
1
12
11
tz
QA
PPA
QA HPLC
PAPLC H
tzV
T
T
T
T















(26)
where
H1 A
Letting P  P
TP PA P
and
H2 AL
TQQALQ
.
1, then the (26) can be rewritten as
ALC M

)(
00
00
0
0
)(
1
1
12
11
tz
QQA
PAP
QA H(LC)

P
tzV
T
T
T
T


















(27)
where

z
The negative inequality of (19) leads to the negativity
of the derivative V along with (18). Therefore, from
theLyapunov-Krasovskii
concludes the uniform asymptotic stability of the
system (18).
 )()(
ˆ
x
)()(
ˆ
x
)(

tetPtetPt
TTTTT
.
StabilityTheorem,it
4.Simulation Results
The proposed static state
controller and observer-based AQM controller for the
time-delay system with input saturation are verified
the performance and effectiveness through numerical
simulations in both Matlab and Network Simulator-2.
First, we consider N homogeneous TCP connections
shareing a bottleneck link with capacity
(packet/sec)wherethe
packets,
N 100 and
R
results in Matalb are depicted in Figure 1 and Figure 2.
From these results, we can conclude that the desired
queue size can be achieved and the TCP network is
stabilized to the operating point by proposed AQM
controllers.
feedback AQM
1250

C
desired
2 . 0
queue
.. The simulation
qd150
0
Figure1. System response using the static state
feedback controller

Page 5

Proceedings of 2005 CACS Automatic Control Conference
Tainan, Taiwan, Nov 18-19, 2005
Figure 2. System response using the
observer-based controller
Meanwhile, we also compare with other different
AQM schemes, like RED, REM and PI schemes. The
dumbbell network topology [10] is adopted in our
experiment which depicted in Figure 3. In the
following simulations, 150 TCP-Reno connections are
used as the transport protocol and packet field ECN is
enabled at time
0

t
. At time
connections stop transmitting data, and at time
they resume transmitting again. The queue evolutions
of different AQM scheme are depicted in Figures 4-7.
Note that REM and PI are not robust with respect to
the variation of load. Even though, RED is not
sensitive, but it lead to degradation of link utilization
due to excess oscillation. However, the proposed static
feedback controller is robust against the variation of
connections.
40

t
, 50 of TCP
70

t
Figure 2. Dumbbell network topology
Figure 4. Queue evolution using static feedback
controller
Figure 5. Queue evolution using RED controller
Figure 6. Queue evolution using REM controller
Figure 7. Queue evolution using PI controller
5.Conclusion
This paper has modeled the TCP/AQM as a
time-delay system with a saturating input. In order to
achieve asymptotic stability of the desired operating
point of the TCP network, we have proposed both
static state feedback controller and observer-based
controller for the time-delay system. Based on the
Lyapunov-Krasovskii functional approach and LMI
method, the congestion control laws and their
delay-independent stability criteria have been derived.
Numerical simulations in both Matlab and Network
Simulator-2 have validated and tested the performance