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

TCPcommunicationnetworks.

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.

IntheTCP/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. Duetoa

applications cannot be exactly recognized in the

networking environment Therefore, traffic congestion

isoneofthemajor

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],a PI(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

appliedthecorresponding

approach.The back-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

designedtobestable

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 some adaptive

controllerwas

controllerwithLMI

andtechnique

inmanypractical

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

CTp), 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

tCxty

with the system matrices as follows:

1

RR

and ignore the dependence

t

, and then the

0 R

)()(

)

0

BuRtxAtAxtx

(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

;

10

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

Page 3

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)

0allfor, 1))((0

ttu

(7)

Thus, based on (5)-(7), the system (4) can be rewritten

equivalently as follows:

] 0,[),()(

tttx

)()(

)( ))(()()()(

1

tCxty

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-loop system is

Substituting (9) into (8) yields the closed-loop system,

which is written as follows:

)()()(

1

KtKxBA

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)

asymptoticallystable.

)()()))(((

)(

))

x

((

1

x

tAt

tKxtuBtxAtAxtx

(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 PAQ

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 xx

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

txPAtxtxQPAx

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).

theuniformasymptotic

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

)()(

AteLCA

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

Page 4

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 (ALC) 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

teLCARteteRAe

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

teQQA QAtetPLCe tx

txPA t PAPA

tetQte

xtPt

teRA teteLC Rte

eLC At

ttPA tx

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

TQQALQ

.

1, then the (26) can be rewritten as

ALCM

)(

00

00

0

0

)(

1

1

12

11

tz

QQA

PAP

QAH(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

qd150

0

Figure1. System response using the static state

feedback controller

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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