Feedback_control_for_router_management_and_TCPIP_network_stability-JdP.pdf

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IEEE TRANSACTIONS ON NETWORK AND SERVICE MANAGEMENT, VOL. 6, NO. 4, DECEMBER 2009 255

Feedback Control for Router Management andTCP/IP Network Stability

Yassine Ariba, Frdric Gouaisbaut, and Yann Labit

AbstractSeveral works have established links between con-gestion control in communication networks and feedback controltheory. In this paper, following this paradigm, the design ofan AQM (Active Queue Management) ensuring the stability ofthe congestion phenomenon at a router is proposed. To thisend, a modified fluid flow model of TCP (Transmission ControlProtocol) that takes into account all delays of the topology isintroduced. Then, appropriate tools from control theory are usedto address the stability issue and to cope with the time-varyingnature of the multiple delays. More precisely, the design of theAQM is formulated as a structured state feedback for multipletime delay systems through the quadratic separation framework.The objective of this mechanism is to ensure the regulation of

the queue size of the congested router as well as flow ratesto a prescribed level. Furthermore, the proposed methodologyallows to set arbitrarily the QoS (Quality of Service) of thecommunications following through the controlled router. Finally,a numerical example and some simulations support the exposedtheory.

Index TermsActive queue management, congestion control,control theory, multiple time delay system, stability, TCP networkmodel.

I. INTRODUCTION

C

ONGESTION control consists in regulating rates of

traffi

c sent into the network based on its state. The end-to-end congestion control mechanism TCP [1] is widely usedin the current Internet to prevent from congestion collapse.To this end, TCP algorithms use only implicit congestioninformations, such that delays or losses, at end-sources withoutany assistance from the network. As a matter of fact, therehas been a growing recognition that the network itself mustparticipate in congestion control and resource management[2], [3]. Regarding Internet routers, they use only a store andforward mechanism in which packets are stored in an outputlink buffer and are forwarded when the bandwidth of thislatter link is available. If the buffer becomes full, then extrapackets are dropped, thats the so-called drop tail mechanism.However, this latter is known to have many drawbacks suchas synchronization of sources, monopolization of resources byfew connections, large oscillations and saturation of the bufferunder persisting heavy traffics.

To prevent from this behavior, additional network algo-rithms need to be implemented not only to provide better

Manuscript received January 25, 2009; revised May 27, 2009 and August31, 2009. The associate editor coordinating the review of this paper andapproving it for publication was Q. Wang (corresponding guest editor).

Y. Ariba is with CNRS; LAAS; 7, avenue du Colonel Roche, F-31077Toulouse, France (e-mail: [email protected]).

F Gouaisbaut and Y. Labit are with the Universit de Toulouse, UPS, INSA,INP, ISAE ; LAAS ; F-31077 Toulouse, France.

Digital Object Identifier 10.1109/TNSM.2009.04.090405

congestion information to sources but also to control conges-tion effectively. Such mechanisms like Active Queue Manage-ment (AQM), executed by routers, detect congestion problemand inform sources (either implicitly or explicitly with themechanism of Explicit Congestion Notification ECN [4]). TheAQM principle consists in dropping (or marking when theECN option is enabled) some packets before the buffer sat-urates. Hence, following the Additive-Increase Multiplicative-

Decrease (AIMD) behavior, sources reduce their congestionwindow size avoiding then the full saturation of the router.Basically, an AQM drops/marks incoming packet with a given

probability related to a congestion index (such as queue lengthor delays) allowing then a control on the buffer occupation atrouters. The basic design issue is then How does the AQMhave to adjust its dropping probability to regulate the queuelength?.

Various mechanisms have been proposed in the litera-ture for the development of AQM such as Random EarlyDetection (RED) [5], Random Early Marking (REM) [6],Adaptive Virtual Queue (AVQ) [7] and many others [8]. Theirperformances have been evaluated [8] and empirical studieshave shown the effectiveness of these algorithms [3]. Then,significant researches have been devoted to the use of control

theory to develop more efficient AQM. Using dynamicalmodel developed by [9], a PI (Proportional Integral) [10]have been designed. In the same framework, other tools havebeen used to extend this preliminary work such as a PIDcontroller [8], [11] or robust control [12]. However, most ofthese papers do not take into account the delay and ensurethe stability in closed loop for all possible delays which couldbe very conservative in practice. Indeed, it should be moreconvenient to control the network congestion only for admis-sible delays, i.e. with limited and reasonable upperbounds.Furthermore, most of these works have been dedicated tothe stability analysis of networks composed of homogeneous

sources (see for example [13], [11], [14] and [15]). In thispaper, networks with heterogeneous sources are consideredintroducing then several delays. This case has already beeninvestigated in [16] and [17]. In [17], the construction of theAQM required to invoke the Generalized Nyquist Theoremand [16] provides a delay dependent state feedback involvingdelay compensations with a memory feedback control. Allthese latter methodologies are interesting in theory but hardlysuitable in practice. While these latter studies have considereda simplified model of TCP/AQM from [9], we use in thiscontribution a more accurate model presented in [18]. Indeed,contrary to [16] and [17], in this paper both forward andbackward delays are taken into account. Moreover, we do not

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256 IEEE TRANSACTIONS ON NETWORK AND SERVICE MANAGEMENT, VOL. 6, NO. 4, DECEMBER 2009

Fig. 1. Network topology.

make the usual assumption that delays are invariant. Then,congestion control of networks consisting in heterogeneousTCP sources is transformed into a stabilization problem for

multiple time-varying delays systems. Using the robust anal-ysis framework and especially quadratic separation approachdeveloped for time-delay systems by [19] and extended in[20], a stabilizing AQM is designed. The approach employedin this paper allows the formulation of the problem into matrixinequalities [21]. Using semi-definite programming solver, thislatter stability condition provides a simple and systematicmethod to design the parameters of the AQM. That is, nomathematical calculations are required, one just has to run thesystematic solver algorithm.

The paper is organized as follows. The second part presentsthe mathematical model of a network composed of a singlerouter and several heterogeneous sources supporting TCP.Section III is dedicated to the design of the AQM ensuring thestabilization of TCP. Section IV presents a numerical exampleand simulation results using NS-2. Finally, Section V con-cludes the paper. Appendix sections detail some mathematicalproofs and give the notations used in this paper.

II . NETWORK DYNAMICS

A. Fluid-flow model of TCP

In this paper, we consider a network consisting in a singlerouter and heterogeneous TCP sources. By heterogeneous,we mean that each source is linked to the router with differentpropagation delays (see Figure 1). Since the bottleneck isshared by flows, TCP applies the congestion avoidancealgorithm to cope with the network saturation [1].

Deterministic fluid-flow models have been widely used ([9],[18] and references therein) to describe congestion controland AQM schemes in IP networks. These models capturethe mean behavior of the TCP dynamic. While most of thestudies using control theory for network control considerthe model proposed by [9], we consider in this paper themodel introduced in [18] and described by (1). Contraryto the former, the model (1) does not neglect the forwardand backward delays. Furthermore, we will take into accountthe time-varying nature of network delays. The model and

Fig. 2. A single connection.

notations are as follow:

() =( )

( )(1 ())

1

()

( )

( )

()

2(),

() = +

(

)

( ),

=()

+ =

+

,

(1)where is the congestion window size of the source , isthe queue length of the buffer at the router, is the RoundTrip Time (RTT) perceived by the source . This latter quantitycan be decomposed as the sum of the forward and backwarddelays ( and

), standing for, respectively, the trip time

from the source to the router (the one way) and from therouter to the source via the receiver (the return). Note thatwe assume all RTT and backward delays are time-varying

(because of the queueing delay) but bounded: () and ()

= {1,...,}. , and areparameters related to the network configuration and represent,respectively, the link capacity, the propagation time of the pathtaken by the connection and the number of TCP sources. is the number of sessions established by the source .The first equation of (1) describes the AIMD behavior ofthe congestion window size of the transmission protocolapplied by the source . Roughly, the first part of the right-hand side represents the additive-increase, whereas the sec-ond follows the multiplicative decrease [18]. Expressed bythe second equation of (1), the length of the FIFO queue

integrates the difference between incoming traffi

c and thelink capacity. The RTTs, in the third equation, comprise thequeueing delay ()/ and the propagation delay. The signal() = (

) corresponds to the dropping probability of a

packet applied by the AQM. Besides, we also do not supposeany knowledge (like bounds) on the delay derivative whichcorresponds to the delay jitter.

Now, the question is What values should be assigned tothe dropping probability () in order to control flow rates,ensuring then the stability of the congestion ? In this paper,the dropping strategy, set by the AQM, will be designed withthe help of robust control tools. As usual in control theory,it is required to obtain some relevant informations about thenetwork status, ensuring then a feedback control structure.



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ARIBA et al.: FEEDBACK CONTROL FOR ROUTER MANAGEMENT AND TCP/IP NETWORK STABILITY 257

Fig. 3. Network control.

Because the congestion window cannot be measured, wepropose a second model (2) that considers flow rates ,expressed as () =

()()

. Hence, the dynamic of this newquantity is of the form

() =

(()

() )=

()() () ()

2

()=

() () ()

().

Based on the expressions of(), (), (), (1) and () =() , a new model of the TCP behavior is derived

() =( )

()2(1 (

)

( )()

2(

)

=1 (

)

() +

()

,

() = +=1

( ),

(2)

Remark 1 Model (2) allows us to use instead of which is more suitable to handle in practice as proposed in

AVQ [22]. Effectively, numerous works have developed tools

that enable flow rates measurements, especially in anomaly

detection framework (see for example [23]).

Remark 2 Our work focuses on the congestion control of

a single router with a fixed topology. and are thensupposed to be constant.

B. Linearization of the nonlinear model

The objective of this study is to regulate the queue length() as well as flow rates () {1,...,} around anarbitrary operating point. By definition, at the equilibriumpoint, () = 0, () = 0, and all variables are set to anequilibrium value (for instance, () = ( ) = 0 ). Letus define the set of equilibrium points of the nonlinear model(2) as

0 = + 0/,

() = 0

0 = ,() = 0 0 =

22+(00 )

2 .(3)

As expected, the equilibrium is set if the amount of dataarriving to the router equals its capacity (see the second

equation in (3)). Moreover, the dropping probability must beset according to the third equation to ensure the equilibriumof the sending rates at the stead state. We can observe thatthe more the source is far away (higher RTT 0 ), the higheris the dropping probability at the equilibrium.Model (2) can thus be approximated by the following linearmodel

1()...()

()

=

1()...()

()

+

1(

1 )...

( )

()

+

1( 1)

...(

)

,

(4)where = 0 , = 0 and = 0 arethe state and input variations around the equilibrium point (3).The matrices of equation (4) are defined by

=

1 0 0 1

0. . . 0

...0 0 0 0 0 0

, =

1 0 0

0. . . 0

0 0 0 0 0

,

=

11 . . . 1 0...

... 01 . . . 0

1 . . . 0

,

(5)with =

100

2

0

00

2 , = 2(10 )

30

, = 00

and

= 12

0

20

2. The linearization is detailed in Appendix B.

Note that the equilibrium point (3) can be arbitrarily chosenaccording to some desired specifications:

0, the queue length at the equilibrium in the firstequation of (3), can be chosen to set the queueing delayat a desired value.

0 , in the second equation of (3), can be chosen to setthe part of the bandwidth that source is allowed to use.So fixing the same value for all 0 , fairness is ensured.Besides, allocating more resources to a given source (greater 0 than others), ensures to user a better Qualityof Service (QoS).

III. STABILIZATION: DESIGN OF AN AQM

A. Preliminaries

Before the design of the AQM, we first present a priorresult on control theory that will be used in the sequel.Although it exists a large literature about the stabilization oftime delay systems [24], it is still an open problem whereconservative methods (BMI, relaxation algorithms, conserva-tive inequalities) are involved [25]. Actually, all results use theLyapunov method and several conservative inequalities (likeParks, Moons or Jensens inequalities) must be introducedto derive constructive conditions [26]. Then, for stabilizationpurpose, some techniques propose to cope with non-linearform via model transformations (like descriptor form or the



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Fig. 4. An interconnected system.

Leibniz-Newton formula) and additional decision variablescalled slack variables [24][27]. Nevertheless, the relevancyand the conservatism of such methods are difficult to quantifyand would lead to tedious calculations with many decisionvariables in the case of the system considered here. In thispaper, we propose another conservative method in an originalframework: the quadratic separation. This latter presents theadvantage to separate a system as the interconnection of alinear equation and a set of operators, allowing then to copewith several delays in an unified and simple way (with lessdecision variables). Besides, system (4) has a particular form:

involving different delays in the same delayed state

vector, because the inputs are subjected to different backward

delays, delayed subvector of the state are entailed in themodel (see system (10) in Section III-B).

Such particular structure cannot be efficiently addressed withclassical control tools and requires to find a particular Lya-punov functional while quadratic separation provides a con-venient method which decomposes the structure of the systemand does not require the system to be in a standard form.Theorem 1 states a stability condition for a closed-loop systemas illustrated in Figure 4 [28], [20]. Coming from the robustcontrol theory, this result provides a suitable framework for

the analysis of multiple time-varying delays systems.

Theorem 1 Given two possibly non-squared matrices , and an uncertain matrix belonging to a set . The inter-connected system represented on Figure 4 is stable for all

matrices if there exists a symmetric matrix =

satisfying both conditions[

][

]

> , (6)

2,

,

0 (7)

The matrices and define the system and matrix

contains operators that characterize the system. The key ideaconsists in embedding the delay operator in an uncertain setsatisfying the constraint (7). The vectors and describe thestate of the system whereas and are exogeneous signals.

B. State feedback control: the dropping strategy

As previously mentioned, an AQM computes the droppingprobability () in order to force source to reduce its sendingrate. Hence, an AQM acts as a controller and feedbacksthe congestion information (via loss rate) to sources basedon network measurements (see Figure 5). In [17], [18], thegeneralized Nyquist criterion is invoked to tune the parametersof RED and PI control laws which depend on the queuelength at the router. Both will be evaluated and compared

Fig. 5. Implementation of the AQM.

to our method in the Section IV. In [16], the predictiveapproach is used to compensate the delays and to stabilize

the communication protocol. However, this latter techniquepresents a high computational cost for the router requiring thecalculus of multiple integrals, multiplications and exponentialsof matrices. Besides, all these methods [16][18] (frequentialmethods, predictive approach) make necessarily the assump-tion that delays are invariant. Similarly to a structured statefeedback, we propose in this paper to compute () as alinear combination of the input rate perceived by the routerand the queue length:

() = 1( ) + 2() (8)

where 1 , 2 {1,...,} are tuning parameters of the

AQM. Applying this control law to each source yields in thefollowing input vector:

1()

...()

=

11 0. . .

0 1

1

1( 1 )

...

( )

+

21 0. . .

0 2

2

()...

()

.

(9)

This is the dropping strategy that must be applied by the AQMat the router (see Figure 5). Note that the dropping probability perceived by the source will be delayed ( ) becauseof the corresponding backward delay. We look for a structuredstate feedback (1 and 2 are diagonal matrices) rather thanfull state feedback for two reasons:

to avoid several extra signals associated to the differentdelay combinations in the state feedback: (

)

for , {1,...,}, introducing then many additionaldelays,

it provides a light computational cost (with less opera-

tions) compared to full matrices 1 and 2, reducingthen the processing time at router.



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Fig. 6. Splitting principle of a TCP connection.

Applying the structured state feedback control law (9) tomodel (4), a closed-loop system is obtained

1()...

()()

=

1()...

()()

+

1( 1 )

...(

)

()

+1 1( 1)

...( )

+ 2 ( 1)

...( )

.(10)

Equation (10) depicts the mean behavior of TCP flow ratesregulated by a structured state feedback type AQM around anequilibrium point.

Remark 3 Since the proposed AQM handles both quantities

and , it appears to be a mix of queue-based and rate-based [22] algorithm. Combining and processing the two

informations, we expect ensuring an efficient regulation but

at the expense of additional measures.

Remark 4 Although we have made a restrictive assumption

in Remark 2 for the mathematical tractability, this latter can

be motivated by using the splitting principle of the TCP

connections [29]. The key idea is to divide an end-to-end

TCP connection into multiple split TCP connections via some

TCP Proxy, as illustrated on Figure 6, in order to isolate

a critical part of the network towards the QoS (wireless

network, satellite communication, congestion phenomenon).

Hence, choosing strategically few TCP proxy around a bot-

tleneck being subject to severe congestion, we can recover the

considered topology (see Figure 7).

Based on the prior result exposed in Section III-A, we aimnow at designing the AQM parameters 1 and 2 throughthe stability analysis of system (10). To this end, followingthe general outline proposed in [19], [20], system (10) can betransformed into a interconnected system of the form of Figure4 choosing matrices , , and vectors , appropriately(see Appendix C). It yields to the following theorem.

Theorem 2 For given upperbounds on delays ,

and

( = {1,...,}), system (10) isasymptotically stable, if there exists positive definite matrices

(+1)(+1), positive diagonal matrices , = {1, 2, 3} and diagonal matrices 1, 2 such that the

Fig. 7. Splitting principle for congestion control.

following inequality holds:

11 12 23 11 1 12 2 13 3

> , (11)

where , 1, 2 and , = {1, 2, 3} (18) are defined as(15)-(19).

The proof is defered in Appendix C. The matrices 1 and 2are derived solving the inequality (11) in a systematic mannerwith a semi-definite optimization algorithm [21].

Remark 5 Remark that the inequality (11) is bilinear w.r.t.

the decision variables. Hence, the optimization problem is

non convex and difficult to handle. Nevertheless, the feasibility

problem can still be tested to provide a suboptimal solution

using a BMI solver [30] or a relaxation algorithm with an

appropriate semi-definite programming solver [25].

IV. NS-2 SIMULATIONS

In this section, we present several simulations that have beenperformed with the network simulator NS-2 (release 2.30).Throughout this part, we consider the numerical exampleillustrated on the Figure 8 where several sources send dataflows to their respective receiver through a bottleneck. Asuch topology induces a congestion phenomenon at the firstrouter. That is why, we propose to implement our AQM tocope with this issue regulating the queue length of the routerto a desired level 0 = 100 packets (while the maximalbuffer size is set to 400 packets). Propagation times areas illustrated on Figure 8. The link bandwidth is fixed to10, that is 2500 packets/s considering packet size of500 bytes. Hence, at the equilibrium the queueing delay isequal to 40. The three sources use TCP/Reno and haveestablished 10 connections generating long lived flows. Uponthese latter specifications, the equilibrium point (3) is derived:0 = 100, 10 = 150, 20 = 250, 30 = 350,10 = 20 = 30 = 83/ (rate for each connections of

the three sources) and 0 = 103 [9.508 3.444 1.760].



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0 10 20 30 40 50 60 70 80250

300

350

400

450

SF

0 10 20 30 40 50 60 70 80250

300

350

400

450

PI

0 10 20 30 40 50 60 70 80250

300

350

400

450

RED

0 10 20 30 40 50 60 70 80250

300

350

400

450

Time (s)

REM

Fig. 10. Evolution of the RTT of connections from source 3 (ms): theexpected value is 350.

Note that REM has the lowest queueing delay, however theobjective is to guarantee the queue length stability inducingthen a queueing delay equals to 40. If a lower queueingdelay is required, one just has to change the equilibriumpoint (reducing the desired queue length). Then, although REDperforms a good regulation on the queue length and the RTTat the steady state, its response time is very slow which makesRED inefficient against short-lived traffic perturbations [13].At last, the stability of the congestion phenomenon keeping astable queue length (and thus a stable queueing delay), allowsto control the RTT for all sources to a desired value with lowvariations (see Figure 10).

Figures 11, 12, 13 and 14 show the packet arrival ratesof each user when different AQM are implemented. The

prescribed equilibrium rate for each connection (according to(3)) that establishes fairness is 10 = 20 = 30 = 83/.As it can be seen on figures, only is able to keeparrival rates comparatively close to the equilibrium value.Table III presents statistics related to these figures and bearsout the accuracy and the control efficiency of the proposedSF as AQM. Futhermore, based on the Jains fairness index= (

)

2

(

2 )

[31] (the more the index is close to 1, the morethe distribution of the resources is fair), we observe that applies a fair strategy.

As mentioned earlier in Remark 6, setting the equi-librium point (3) appropriately, one can allocate arbitrarilydifferent amounts of resource to each source. Consider againexample of Figure 8, the objective is still to regulate the

TABLE IISOME STATISTICS ON THE QUEUE LENGTH FOR DIFFERENT AQM

DT RED REM PI SF() 317.6 103.8 94.8 99.4 102.3

..() 84 21.7 70.7 35.1 23.3

()127 41.5 37.9 39.8 40.9

()

33.6 8.7 28.3 14 9.3

0 10 20 30 40 50 60 70 800

50

100

150

200

Source

1

(pqts/s)

0 10 20 30 40 50 60 70 800

50

100

150

200

Source

2

(pqts/s)

0 10 20 30 40 50 60 70 800

50

100

150

200

Temps (s)

Source

3

(pqts/s)

Fig. 11. Our State feedback strategy: sending rates of sources, expectedrates for fairness 0 = 83/.

queue length at 0 = 100 but assigning different sharesof the link capacity: 10 = 50/, 20 = 80/ and30 = 120/. Hence, the dropping probability at theequilibrium (3) is 0 = 103 [25.9 3.73 0.84] and solvingcondition of Theorem 2 we obtain:

1 = 103diag

0.303, 0.198, 0.062

,

2 = 103diag

0.215, 0.057, 0.017

.

Simulation of this new configuration still show a stable queuesize around 100 (as previously) but users send data flowswith different prescribed rates (see Figure 15). At last, we havetested the effect of additional unexpected and non-responsivetraffic (see Figure 16). The bursty traffic consists in 1, 6 or10 sources that inject in the congested router UDP flows with

a rate of 0.5 between 80 and 100 sec. Although ourAQM is able to maintain a stable queue length in presenceof light unexpected traffic, it is not efficient anymore underheavy traffic. In that case, the QoS related to the queueingdelay is not guaranteed.

V. CONCLUSION

In this paper the design of an AQM for congestion controlof a single router has been presented. The considered topologyconsists in several TCP sources sending long-lived flowsthrough a router to their respective receivers. To supply theTCP congestion control mechanism, an AQM must be imple-mented at the router. Based on a modified mathematical modelof the protocol behavior, a such AQM has been developed with



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0 10 20 30 40 50 60 70 800

50

100

150

200

Source

1

(pqts/s)

0 10 20 30 40 50 60 70 800

50

100

150

200

Temps (s)

Source

3

(pqts/s)

0 10 20 30 40 50 60 70 800

50

100

150

200

Source

2

(pqts/s)

Fig. 12. RED strategy: sending rates of sources, expected rates for fairness0 = 83/.

0 10 20 30 40 50 60 70 800

50

100

150

200

Source

1

(pqts/s)

0 10 20 30 40 50 60 70 800

50

100

150

200

Source

2

(pqts/s)

0 10 20 30 40 50 60 70 800

50

100

150

200

Temps (s)

Source

3

(pqts/s)

Fig. 13. REM strategy: sending rates of sources, expected rates for fairness0 = 83/.

control theory tools. Indeed, in a time delay system frame-work, a control law has been proposed and then the stabilityanalysis of the feedback system has been ensured. That is, theregulation of flows and the queue size of at the router havebeen ensured. Moreover, the choice of the equilibrium pointallows us to fix arbitrarily the different shares of the routercapacity allocated to each user. At last, a numerical exampleand simulations have shown the effectiveness and the fairnessof the proposed methodology. Future works concern the designof resilient AQM to cope with implementation uncertaintiesand tuning errors. Indeed, numerical computations have a finiteprecision and the resulting effective control may be slightlydifferent from the one expected. A second critical issue tohandle is buffer control under bursty and non-responsive trafficlike UDP.

TABLE IIISOME STATISTICS ON ARRIVAL RATES FOR DI FFERENT AQM

RED REMusers

Mean (pkts/s)Stand. dev. (pkts/s)

1 2 3147 83 14861 36 372

1 2 3147 81 6056 34 23

Jains fairness index 0.9450 0.8703

PI SFusers

Mean (pkts/s)Stand. dev. (pkts/s)

1 2 3

146 88 10966 35 335

1 2 3

104 91 8841 30 40Jains fairness index 0.9579 0.9946

0 10 20 30 40 50 60 70 800

50

100

150

200

Source

1

(pqts/s)

0 10 20 30 40 50 60 70 800

50

100

150

200

Sou

rce

2

(pqts/s)

0 10 20 30 40 50 60 70 800

50

100

150

200

Temps (s)

Source

3

(pqts/s)

Fig. 14. PI strategy: sending rates of sources, expected rates for fairness0 = 83/.

APPENDIX ANOTATIONS:

For two symmetric matrices, and , > () meansthat is (semi-) positive definite. denotes thetranspose of. and denote respectively the identitymatrix of size and null matrix of size . If the contextallows it, the dimensions of these matrices are often omitted.We also define the set 2 [0, +) = 2 consisting of allmeasurable functions : + such that the following

norm 2

= (

0 (()()))1/2

< is bounded. Note

that this norm is related to the inner product defined as

, =+0

()(). Hence, the norm is also defined

as 22 = , . Finally, diag(, ) stands for the block

diagonal matrix: diag(, ) =

00

.

APPENDIX BLINEARIZATION OF THE FLUID-FLOW MODEL

In this appendix, we linearize the equations (2) around theequilibrium point defined in (3). First, let us recall the model



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0 10 20 30 40 50 60 70 800

50

100

150

200

Source

1

(pqts/s)

0 10 20 30 40 50 60 70 800

50

100

150

200

Source

2

(pqts/s)

0 10 20 30 40 50 60 70 800

50

100

150

200

Temps (s)

Source

3

(pqts/s)

Fig. 15. Our state feedback strategy with different QoS: sending rates ofsources, expected rates 10 = 50/, 20 = 80/ and 30 =120/

0 50 100 1500

100

200

300

400

Queuesize(pkts)

0 50 100 1500

100

200

300

400

Queuesize(pkts)

0 50 100 1500

100

200

300

400

Time (s)

Queuesize(pkts)

10 unexpected sources: 5Mbps

6 unexpected sources: 3Mbps

1 unexpected sources: 0.5Mbps

Fig. 16. Our state feedback strategy with unexpected traffic (UDP) between80 and 100 sec

and introduce functions and :

() =( )

()(()

+ )2

(1 ( )

( )()

2(

)

=1 ( )

() + () +

()()

+ ,

=((), ( ), (

), (), (

)),

() = +

=1( ),

=((

)).

Now, we evaluate the partials of and , w.r.t. theirarguments, at the equilibrium point

()

=

[0(1 0)

2020

+00

2+

0

1

0

]

=

[(1 0)

200+

002

],

( ) =(1 0)

020

00

2

= 0,

( )

= 00

,

()

=

[020

0

220+

0(1 0)2

030

],

= (1 0)2

30,

( )

= 0

020

20

2,

( )

=

with =

=1 0 . Hence, for small variations around theequilibrium, model (2) can be approximate by the followinglinear time-delay system:

() =

[(1 0)

200+

002

]()

00

=1

( )

(1 0)2

30()

[1

20+

202

](

),

() =

=1

( ),

where = 0 , = 0 and = 0represent the variations of , and , respectively, aroundthe operating point (3).

APPENDIX CPROOF OF THEOREM 2

In this appendix, we detail the proof of the main resultconcerning the stabilization of the closed-loop system (10).First, let us write this system as

[ ()() ] = [ ()() ] + [ 1()() ]

+ 12() + 2

( 1)...

( )

where

() =

1()...

()

, 1() =

1( 1 )

...

( )

,

2() = 1( 1)

...( )

.



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Secondly, we aim to transform this latter model as an inter-connection of the form of the Figure 4 in order to apply thestability criterion of Theorem 1. It is required foremost todefine operators that characterize the system:

:()

0

(),

:() ( ),

(1 ) :() =

().(14)

Operators and correspond to the integral operator anddelay operator, respectively. Note that traditional Laplacenotations (1 and ) cannot be used since delays are time-varying. The third operator is another delay-related operatorwhich will be used to provide rate-independent and delay-dependent stability condition. Hence, no information on thedelay variations will be required [32], [20] and maximalupperbounds on delays will be used [19]. Then, defining

the set of delay-related operators

1 =

(1 1

)

. . .

(1

)

,

2 =

(1 1). . .

(1 )

,

3 =

(1

1

). . .

(1

)

,

we can reconstruct the different signals that compose thesystem:

()

()

=

()

()

,

() 1() =1(),

() 2() =2(),

() ( 1)

.

..() ( )

=3 ()

.

..()

.

Finally, the model (10) is reworded in equivalent way as aninterconnected system with

()

() 1()() 2()

() ( 1)...

() ( )

=

1

2

3

()()

()

()

()...

()

and the linear equation (16) where

1 =[

1

], 2 =

1...1

,

= + + 11 + 22.

(15)

It can be shown [19], [20] that for a such interconnection the

following separator fulfills the second inequality (7)

=

11

2233

1

23

(17)where and for = {1, 2, 3} are positive definite matricesand

1 =diag21 ,...,2 ,2 =diag

21 ,...,

2

,

3 =diag

2

1 ,...,2

,

(18)

() ,

and () are maximal ad-

missible delays. Note that is invariant but may be uncertainand provides an upperbound. Invoking Theorem 1, theabove interconnection, and thus system (10), is asymptoticallystable if the second requirement (6) is satisfied with , and defined in (16) and (17). Indeed, the second one (7) beingverified (by construction of ), the first inequality provides

the criterion to be tested. Some basic calculations show thatthis latter condition can be developed as the following one

1 (11 + 22)1 2 332

where

=

+ 1

1 1

1

1 1 2 2

2 3

2 3

=1 11 +

1 21 +

2 32,

1 =[

1 11 11 12

],

2 =[

2 21 21 22

].

(19)At last, applying the Schur complement yields in the condition(11) of Theorem 2 which concludes the proof.

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[2] B. Braden, D. Clark, and J. Crowcroft, Recommendations on queuemanagement and congestion avoidance in the internet," RFC 2309, Apr.1998.



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112

(

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) =

1 1 2

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() ( 1 )...

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(16)

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[14] S. Manfredi, M. di Bernardo, and F. Garofalo, Robust output feedbackactive queue management control in TCP networks," in Proc. IEEEConf. Decision Control, Dec. 2004, pp. 1004-1009.

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[16] K. B. Kim, Design of feedback controls supporting TCP based on thestate space approach," in IEEE Trans. Automat. Control, vol. 51, no. 7,July 2006.

[17] H. Han, C. V. Hollot, Y. Chait, and V. Misra, TCP networks stabilizedby buffer-based AQMs," in Proc. IEEE INFOCOM, Mar. 2004, pp. 964-974.

[18] H. S. Low, F. Paganini, and J. Doyle, Internet Congestion Control,"IEEE Control Syst. Mag., Feb. 2002, vol. 22, pp. 28-43.[19] F. Gouaisbaut and D. Peaucelle, A note on stability of time delay

systems," in 5 IFAC Symp. Robust Control Design (ROCOND06),Toulouse, France, July 2006.

[20] Y. Ariba, F. Gouaisbaut, and D. Peaucelle, Stability analysis oftime-varying delay systems in quadratic separation framework," in

International Conf. Mathematical Problems Engineering, Aerospace

Sciences (ICNPAA08), Gene, Italie, June 2008. [Online]. Available:http://hal.archives-ouvertes.fr/hal-00357766/fr/.

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[22] D. Supratim and R. Srikant, Rate-based versus queue-based models ofcongestion control," IEEE Trans. Automat. Control, vol. 51, pp. 606-619, Apr. 2006.

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implicit linear transformation," Automatica, vol. 43, no. 5, pp. 795-804,2007.[29] I. Maki, G. Hasegawa, M. Murata, and T. Murase, Performance analysis

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[31] R. Jain, D. Chiu, and W. Hawe, A quantitative measure of fairnessand discrimination for resource allocation in shared computer systems,"Technical Report TR301, Digital Equipment Corp, 1984.

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Lett., vol. 55, pp. 1041-1053, Sept. 2006.

Yassine Ariba was born in Marrakech, Maroc,in November 13, 1983. He received the Masterdegree from the Paul Sabatier University, Toulouse,France, in July 2006. From October 2006 to Novem-ber 2009, he was a Ph.D. student at the Laboratoryfor Analysis and Architecture of Systems (LAAS)in Toulouse, France. In November 2009, he receivedthe Ph.D. degree from the Paul Sabatier University,Toulouse. He is currently assistant professor at theUniversity Paul Sabatier and researcher at LAAS inToulouse. His research activity concerns time delay

systems, robust analysis and network control.

Frdric Gouaisbaut was born in Rennes (France)in April 26, 1973. He received the "DiplmedIngnieur" (Engineers degree) from the EcoleCentrale de Lille, France, in September 1997 andthe "Diplme dEtudes Approfondies" (Masters De-gree) from the University of Science and Technologyof Lille, France, in September 1997. From October1998 to October 2001 he was a Ph.D. student atthe Laboratoire dAutomatique, Gnie Informatiqueet Signal (LAGIS) in Lille, France. He receivedthe "Diplme de Doctorat" (Ph.D. degree) from the

Ecole Centrale de Lille, France, in october 2001. Since October 2003, he isan associate professor at the Paul Sabatier University, Toulouse. His researchinterests include robust control, time delay systems, networked systems andsliding mode control.



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Yann Labit was born in Lorient, France, in 1976.He received the Masters Degree (Diplme dEtudesApprofondies) from the University of Science PaulSabatier in Toulouse, France, in June 1999. FromOctober 1999 to October 2001 he was a Ph.D. Stu-dent at the Laboratoire dAnalyse et dArchitecturedes Systmes of the Centre National de la RechercheScientifique (LAAS-CNRS), Toulouse, France. Hereceived the Ph.D. Degree from INSAs School,Toulouse, France, in October 2002. Since October

2002, he is associate professor at the Paul SabatierUniversity, Toulouse. Yann Labits research interests are monitoring andsecurity in networks, networked systems, time delay systems and robustcontrol.

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