162 ieee transactions on automatic control, vol. 49, no....

162 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 49, NO. 2, FEBRUARY 2004

A Unifying Passivity Framework forNetwork Flow Control

John T. Wen, Fellow, IEEE, and Murat Arcak, Member, IEEE

Abstract—Network flow control regulates the traffic betweensources and links based on congestion, and plays a critical role inensuring satisfactory performance. In recent studies, global sta-bility has been shown for several flow control schemes. By usinga passivity approach, this paper presents a unifying frameworkwhich encompasses these stability results as special cases. In ad-dition, the new approach significantly expands the current classesof stable flow controllers by augmenting the source and link up-date laws with passive dynamic systems. This generality offers thepossibility of optimizing the controllers, for example, to improverobustness in stability and performance with respect to time de-lays, unmodeled flows, and capacity variation.

Index Terms—Network congestion control, network flow con-trol, network optimization, nonlinear control, passivity.

I. INTRODUCTION

NETWORK flow is governed by the interconnection be-tween the information sources and communication links

through the routing matrix, , as shown in Fig. 1. Packets fromeach source (with rate ) are routed through the links (withthe aggregate link rate ). Each link has a fixed capacity ,and based on its congestion and queue size, a link price, , iscomputed. The link price information is then sent back to eachsource to regulate the traffic back into the links. Since the linksonly feed back the price information to the sources that utilizethem, we have the following relationship:

(1)

where is the source rate, is the aggregate rate,is the link price, and is the aggregate price. In

this paper, we assume that there is no delay in the loop and thelink capacity is a constant vector.

The flow control problem aims to find decentralized sourceand link control algorithms ( as a function of , and as afunction of ) to achieve the following objectives.

• Utilization: Maximize throughput by keeping near .• Fairness: All sources have “equitable” shares of capacity.• Stability: All signals converge to desired equilibrium

values.• Robustness: Maintain stability and performance under

model variation, including time delays, unmodeled flows

Manuscript received August 15, 2002; revised October 14, 2003. Recom-mended by Associate Editor H. Zhang. This work was supported in part by theRPI Office of Research through an Exploratory Seed Grant.

The authors are with the Department of Electrical, Computer, and SystemsEngineering, Rensselaer Polytechnic Institute, Troy, NY 12180 USA (e-mail:[email protected]; [email protected]).

Digital Object Identifier 10.1109/TAC.2003.822858

Fig. 1. Network flow control model.

(flows that do not conform to this framework), andcapacity variation.

A common approach to flow control is to decompose theproblem into a static optimization problem and a dynamic sta-bilization problem [1], [2]. The static optimization incorporatesfairness, capacity constraint, and utilization, and its solutionprovides the desired steady state operating point (equilibriumof the closed loop system), , , , and . The source rateand link price update laws are then designed to guarantee sta-bility and robustness of the equilibrium.

The static optimization problem is to maximize the sum ofthe utilization function for the sources while complyingwith capacity constraints in the links; that is

subject to (2)

where is a strictly concave function. By using the Lagrangemultiplier, , the inequality constraint can be folded into theoptimization problem

If is differentiable, the first order condition for the maximiza-tion problem is

(3)

0018-9286/04$20.00 © 2004 IEEE

WEN AND ARCAK: UNIFYING PASSIVITY FRAMEWORK FOR NETWORK FLOW CONTROL 163

The condition for the Lagrange multiplier is

ifif .

(4)

As shown in Appendix I, if each is strictly concave andis of full row rank, (3) and (4) are sufficient to determine theunique equilibrium.

The utility function for each source determines theequilibrium, and consequently the steady state fairness and uti-lization. The objective of source and link update laws is now todrive the actual source rates and link prices to their respectiveequilibrium values. The constraints in this problem are as fol-lows.

• Decentralization: can only depend on , and canonly depend on .

• No routing information: The routing matrix is unknownto the sources and the links.

• No coordination among sources and links: The sourcesdo not have knowledge of the objective functions of othersources, and the links do not have information of the ca-pacities of other links. Therefore, the equilibrium value isunknown.

Among many congestion control methods surveyed in[3]–[5], only a few can ensure global stability subject to thestructural and information constraints described before. Moti-vated by the gradient update for the optimization problem (2),a “primal” algorithm is proposed in [1] which consists of a firstorder source update law and a static link penalty function tokeep the aggregate rate below its capacity. A “dual” algorithmis also proposed in [1] where the static source update is theoptimality condition (3) and the link uses a first order dynamicsof the price update. In [6] and [7], the link update is replacedby second order dynamics. When both source and link updatesare dynamic in the so-called primal/dual algorithm, stabilityhas only been shown using singular perturbations in [8] andfor the single-bottleneck case (i.e., only one congested link isconsidered) in [9] and [10].

In this paper, we develop a unifying framework for stable net-work flow control by using a passivity approach. This frame-work includes the primal and dual control schemes in [1], [7] asspecial cases and extends them to broader classes of flow con-trol laws. A system with state , input and output , is saidto be passive if there exists a continuously differentiable storagefunction such that

(5)

for some function . If is positive definite,then the system is strictly passive. The notion of passivity ismotivated by physical systems that conserve or dissipate en-ergy, such as passive circuits [11] and mechanical structures[12], where corresponds to an energy function. Passivityis a useful tool for nonlinear stability analysis [13], [14] andcontrol design [15], [16]. Indeed, the celebrated Passivity The-orem states that, if two passive systems and with pos-itive definite and radially unbounded storage functionsand , respectively, are interconnected as in Fig. 2, thenthe equilibrium of the interconnection is stable in the sense of

Fig. 2. Feedback interconnection of two passive systems.

Lyapunov. This is because the sum of the storage functions,, can now be used as a Lyapunov func-

tion, and satisfies

(6)

from the interconnection conditions and .A detailed treatment of passivity and its applications in stabilityanalysis is given in [17], and furthered with feedback designapplications in [16] and [18].

By using the passivity approach, we show in this paper thatthe first-order source controller in the primal approach in [1],and the first or second-order link controllers in the dual approachin [1], [7], [19] can be replaced by dynamic systems with pre-scribed passivity properties. In addition, the static link update in[1] and the static source update in [7], [19] can be augmentedwith a class of dynamic systems motivated by the Zames–Falb(ZF) multiplier [20]. The dynamic source update in the primalcontroller [1] can also be combined with the dynamic link up-date in the dual controller [7] to obtain a dynamic-source/dy-namic-link stabilizing control law. As an example, the singlebottleneck congestion controller in [10] becomes a special case.

In addition to unifying the existing stabilizing controllers inthe literature, our result also offers the potential for additionaloptimization for robustness in stability and performance withrespect to time delay, unmodeled flows, and capacity variation.Time delay adds phase lag and compromises stability in a feed-back system. From the linear systems point of view, to enhancerobustness with respect to the delay, the controller should addphase lead or increase gain roll-off or do both (subject to thegain-phase relationship imposed by the Bode integral formula[21]). The controllers in this paper can provide additional phaselead or gain roll-off which may be exploited for robustnessenhancement. Another consideration is network variation.The model considered in this paper is highly idealized; realnetworks have constantly changing topology and capacity.For the actual deployment of congestion control, averaging isused to smooth out the variation in various signals. Within ourpassivity framework, the smoothing filter can be designed andoptimized without compromising stability.

This paper is organized as follows. Section II discusses thepassivity interpretation of the primal controller in [1] and thegeneralization to passive source control and multiplier-basedlink control. Section III presents the passivity framework forthe dual controller in [7] and the generalization to passivelink control and multiplier-based source control. A combined


Fig. 3. Primal flow controller based on [1].

primal/dual algorithm is also shown in this section. Simulationresults based on a four-source/three-link network are shown inSection IV. Conclusions are given in Section V.

Throughout the paper, we shall make the following assump-tions.

1) The routing matrix is of full row rank [to ensure theuniqueness of the optimal solution given by (3) and (4);see Appendix I].

2) The optimal source rates, , , are all pos-itive (to ensure that positive projection does not changethe optimal solution; see Appendix III).

A. Notation

Given , we will also use to denotethe column vector and to denote the squarematrix .

Given a function , its positive projection isdefined as

if , or andif and .

When the first condition holds, we say that the projection is in-active; otherwise, the projection is active. If and are vec-tors, then is interpreted in the component-wise sense.We will use projection functions to ensure nonnegative valuesfor physical quantities, such as source rates, computed by thecontrollers. The issues of existence and uniqueness of solutionsresulting from the discontinuity of these functions is not em-phasized in this paper. Existence is guaranteed for a differen-tial inclusion that encompasses the projection discontinuity, asin [22]. Uniqueness is not necessary in our results because ourLyapunov estimates apply to all solutions.

II. PRIMAL FLOW CONTROL ALGORITHMS

In [1], a flow control law based on the primal approach tothe optimization problem (2) is proposed (see Fig. 3). Given theutility function for each source, , the source update lawis given by

(7)

where , , with th compo-nent . The link control is given by

(8)

Fig. 4. Equivalent representation of primal flow controller.

where , with th component , is a penaltyfunction that enforces the link capacity constraint, . Theequilibrium can be computed by setting in (7)

(9)

which approximately satisfies the desired condition (3)–(4). Ap-pendix II discusses the uniqueness of the equilibrium and itsrelationship to the optimal solution in (3)–(4). We assume thateach penalty function is monotonically nondecreasing, such asthe following used in [1]:

(10)

The assumption for all ensures that the positiveprojection in (7) does not change the equilibrium of (7)–(8), asshown in Lemma 1 in Appendix III. This fact will also be usefulin the stability proof.

A. Passivity-Based Stability Analysis

We now present a stability argument by using passivity anal-ysis. The advantage of this approach is that it leads to an ex-tended class of stabilizing control laws to be discussed in Sec-tion II-B.

We express the interconnected system in Fig. 3 in an equiva-lent form in Fig. 4 based on the deviation from the equilibriumcondition, and rewrite the link update as

(11)

The following result shows that both the forward and returnsystems are passive. Adding the two storage functions, we re-cover the Lyapunov function used in [1].

Proposition 1: Consider the feedback interconnection shownin Fig. 4. The forward system from to , and thereturn system from to are both passive. Furthermore,the equilibrium is globally asymptotically stable for thestate space .

Proof: For the forward system, we let

(12)

where . The gradient of with respect to is

which, when set to zero, has the unique solution at .Because the Hessian of is

is a positive–definite function.


Now, we use as a storage function to prove passivity of theforward system. The derivative of along the solution of (7) is

Adding and subtracting to , and substituting (7) and, we get

Because the first term is negative semidefinite, the forwardsystem from to is passive.

Now, consider the return system, and let

where , , and, so is a nonnegative definite function.

The return system from to is passive since

We can now use as a Lyapunov function and obtain

The right-hand side is negative definite because, as we prove inAppendix III, only when . We thusconclude that is globally asymptotically stable.

B. Extended Class of Source Rate Control Laws

The advantage of the passivity perspective is that it allows usto consider a broader class of stabilizing source control laws. Anatural generalization is to replace the first order source update(7) by a more general class of passive systems. To motivate thisextension, we express the source control law (7) as a feedbackinterconnection as shown in Fig. 5(a), where

(13)

Note that since is assumed to be diagonal and concave, isalso diagonal with each diagonal entry consisting of a memory-less first–third quadrant nonlinearity

for some

The forward system in Fig. 5(a) is strictly passive. The feed-back system is passive since it is diagonal with diagonal entriesconsisting of an integrator in series with a first–third quadrantnonlinearity [23]. Since the feedback interconnection betweena strictly passive system and a passive system is itself passive[23], it follows that the source control law (7) is passive. Toextend this approach, the gain in the forward system maybe replaced by a strictly positive real (SPR) transfer function

(a)

(b)

Fig. 5. Extension of source rate update in primal flow controller. (a) Sourcecontrol law (7) expressed as a feedback interconnection of a strictly passiveforward system and a passive feedback system. (b) Extension of source controllaw (7) by replacing K with an SPR transfer function D +C(sI �A) B.

( ) [24], [25], as shown in Fig. 5(b). In thiscase, the forward system remains strictly passive, and the mod-ified source control law would, therefore, still be passive.

The previous argument does not take into account the non-negativity restriction on . The following theorem shows thatfor certain class of , the source rates would remain posi-tive and the overall system is globally asymptotically stable.

Theorem 1: Consider the source rate control law for source, , given by

(14)

(15)

where ( ) is SPR, the link update given by (8), andthe routing connection given by (1). If asfor , then the equilibrium of the interconnectedsystem is globally asymptotically stable for the state space

, where .Proof: From [24] and [25], ( ) is SPR if and

only if there exist symmetric positive–definite matrices and, and matrices , , such that

(16)

(17)

(18)

For the forward system, let

(19)Since is strictly concave, is a positive–definite function.The derivative of along the solution is


Adding and subtracting to , and using (15) and (16)–(18),we get the equation shown at the bottom of the page, where

denotes the minimum eigenvalue of . Because thefirst term is negative semidefinite, the forward system from

to is passive.Now, consider the return system, and let

Since is first–third quadrant, is a nonnega-tive–definite function. The return system from to ispassive since

We can now use as a Lyapunov function and obtain

(20)

Since as , the level sets of are confinedin . Thus, is positive invariant.

It follows from (20) and LaSalle’s invariance principle that( ) converges to the largest invariant set contained in

On , for all , therefore, from (14), . Be-cause is the unique equilibrium of (7)–(8), we concludethat ; that is, is globally asymptoticallystable.

The assumption as ensures thatis positive-invariant, i.e., if is initially in , it will remainin for all . For commonly used utility functions suchas (proportional fair, also TCP Vegas) and

(cariant of TCP Reno) [5], this assumptionis satisfied. However, for (TCP Reno),the assumption does not hold. In this case, we can prove globalasymptotic stability by imposing positive projections on (14)and (15)

(21)

(22)

and restrict ( ) to

. . ....

(23)

where , , , and are positive constants.

C. Alternate Class of Source Rate Control Laws

The passivity argument in Section II-A uses the fact that theforward system cascaded with the derivative block and the re-turn system pre-multiplied by the integral block are both pas-sive. In this section, we use an alternative passivity argument toshow that the forward system is passive without the addition ofthe derivative block. This will then allow us to use dynamic linkcontrollers and still achieve global asymptotic stability.

Proposition 2: Consider the feedback interconnection shownin Fig. 4. The system from to is strictly passive.

Proof: Consider

(24)

The derivative along the solution is

We observe that

This follows because, if the projection is inactive then both sidesof the inequality are equal, and if the projection is active,and , then the left-hand side is zero, and theright-hand side is nonnegative.

By adding and subtracting from , we have

(25)


The last equality follows from:

Since is positive definite and each is strictly concave, thesystem from to is strictly passive.

Remark: In [1], the gain depends on ,. For this case, the storage function (24) needs to be

modified to

(26)

Then

and the rest of proof follows as before.Proposition 2 can be used to extend the static link controller

(8) to a dynamical controller. In the context of passivity-basedstability analysis, [20], [26] showed that a monotone first-thirdquadrant nonlinearity cascaded with a class of transfer functionsis passive. To use the result in link control, we first define thefollowing class of transfer functions.

Definition 1: The class of proper transfer function iscalled inverse ZF if is strictly stable and

where , are positive constants and the impulse response of, , satisfies

Examples of inverse-ZF functions include first-order lag filters,and , . We now ex-

tend the static link update in Theorem 1 to the penalty functioncascaded with an inverse-ZF function. The main idea is to showthat when the link penalty function is modified by inverse-ZFfunctions as in Fig. 6(a), it is equivalent to the same modifica-tion applied to the error system as in Fig. 6(b).

Proposition 3: If the source update law is given by (7) andthe update law for the th link is replaced by

or (27)

(28)

where is inverse-ZF and , then the equilibriumof the closed-loop system is the same as in (9) and is globallyasymptotically stable.

Proof: At the equilibrium,which implies . Therefore, the equilibrium is un-changed. Due to the linearity of , we can represent the re-turn system as the cascade of the inverse-ZF function andthe nonlinearity defined in (11), driven by ( ) (see Fig. 6).Since is first–third quadrant and monotone, it follows from the

(a)

(b)

Fig. 6. Extension of link price update in primal flow controller. (a) Link con-trollers augmented with inverse-ZF functions. (b) Link controllers augmentedwith inverse-ZF functions, represented in the error coordinate.

property of inverse-ZF functions that the return system is pas-sive, i.e.,

(29)

where is the internal state of and is a nonnega-tive function (see [20] and [26] for proofs of this property). Wenow show the closed loop stability by using in Proposition 2.The derivative of along the solution again satisfies (25). In-tegrating both sides and denoting , we get

(30)where

is positive definite as in (25). Substituting (29) in (30), we obtain

(31)

which proves boundedness of . To prove , we note from(31) that

Then, using boundedness of , we can apply Barbalat’s Lemma[17] to conclude that asymptotically. Since isstable, its internal states also converge to the correspondingequilibrium values.

In (27), is nonnegative since the output of is nonnegative.However, in (28), this is no longer guaranteed. If the link pricefeedback is implemented in an averaged sense through packetmarking or packet drop, then negative cannot be realized. Ifis transmitted explicitly (which would require more communi-cation bandwidth), then negative values of may be acceptable.


Fig. 7. Dual flow controller based on [7].

(a)

(b)

Fig. 8. Extension of link rate update in dual flow controller.

III. DUAL FLOW CONTROL ALGORITHMS

The flow control algorithm in [1] is based on the primaloptimization in (2) and the link rate constraint is enforced byusing the penalty function (8). Though this approach guaranteesglobal stability of the source rate, it does not take the linkqueue dynamics explicitly into account. In [7], a dual approachis proposed where the queue size, , is explicitly modeled, andthe rate of change of the link price, , is determined from thecombination of queue size and link rate

(32)

(33)

where , , , . Thesource update is directly given by the primal solution (3)

(34)

where the th component of is . The closedloop system is shown in Fig. 7. Global asymptotic stability of thesystem is shown in [7] by using a Lyapunov function argument.We can recover this result as a special case of Theorem 2, whichapplies the passivity theory to extend the link controller to abroader class.

A. Extended Class of Link Price Control Laws

As in Section II-B, we now extend the second-order link up-date algorithm in (32)–(33) to a larger class of passive dynamicsystems. As a motivation, ignoring the positive projection,(32)–(33) can be regarded as a first-order positive real (PR)transfer function from to [see Fig. 8(a)]. The resultbelow shows that this transfer function can be generalized toa class of PR transfer functions [see Fig. 8(b)], and, underpositive projection, the closed-loop system remains globallyasymptotically stable.

Theorem 2: Consider the link update law

(35)

(36)

where ( ) are block diagonal, partitioned according toeach link, with the th subsystem given by

. . ....

(37)

where is nonnegative, and , and are positive con-stants. Then, the equilibrium , with satisfying(4), of the interconnected system described by (34), (35)–(36),and (1) is globally asymptotically stable.

Proof: We first show that the modified return system(35)–(36) is passive. Consider the following positive–definitefunction for the th link:

The derivative along the solution is (after adding and subtractingfrom )

(38)

First, observe that

(39)

which is immediate, if the projection is inactive. If the projectionis active, i.e., , then both sides of the equality are zero.

Next, we claim that

(40)If the projection is inactive, the inequality holds since

. If the projection is active, then

which implies and therefore the right-hand side isnonnegative. Since the left-hand side is zero, it follows that (40)is true.

Substituting (39) and (40) in (38), we obtain

Summing over all links and using the condition (1), it fol-lows that the return system from ( ) to is passive.


Now, consider the forward system, and let

where , , and, so is a positive–definite func-

tion. The forward system from to is passive since

We now use as a Lyapunov function, and obtain

where is nonnegative as shown before, and vanishes onthe set

However, arguments similar to those in [7, p. 170] show thatthe only trajectory that can remain identically in this set is theunique equilibrium. Thus, it follows from a LaSalle-type invari-ance argument1 that this equilibrium is globally asymptoticallystable.

The link controller (32)–(33) is a special case of (35)–(36)with , , , , . Ourextended class also encompasses the first-order link control lawused in [10] and [19]

(41)

by setting , , .The steady-state queue size in (32)–(33) and (41) is zero.

Since the queue dynamics is not directly modeled in our ex-tended controller (35)–(36), the steady state queue size is in gen-eral nonzero. A nonzero queue may be desirable to ensure thefull utilization of the link. However, the queue should not be toolarge in order to avoid excessive queuing delay. If zero queuesize is desired, then at least one of the should be set to zeroto emulate the queue dynamics.

B. Alternate Class of Link Price Control Laws

In Section II-C, we showed that the first order primal sourcecontrol is passive from to , and this propertyallows the inverse-ZF functions to be introduced into the linkcontrol. In this section, we present the dual result where the linkcontroller is passive from ( ) to .

Proposition 4: Consider the link control law (41). The returnsystem from to is passive.

Proof: Consider the following positive–definite function:

(42)

1Due to the discontinuity of the differential (35)–(36), the standard LaSallePrinciple is not applicable. Instead, we use a variant for differential inclusions,proved in [27]. Our functionW (�; p) shown previously is indeeded lower semi-continuous (in the sense of real-valued functions) as assumed in [27].

After taking the derivative of along the solution and substi-tuting in (41), we get

(43)

Note that

(44)

which is immediate if the projection is inactive. If the projectionis active, the left-hand side is zero and the right-hand side isnonnegative ( ).

Next, we claim that

(45)

which is immediate if . If , then . Sinceis nonnegative, the inequality follows immediately.By using (44) and (45), (43) becomes

(46)

By integrating both sides, it follows that the system fromto is passive.

By slightly modifying the storage function, we can show thatthe passivity property holds for the adaptive virtual queue linkcontroller in [8] and [28]. Note that in [8], a nonlinear isneeded to show stability; here we only require to be a positiveconstant.

Proposition 5: Consider the following link control law:

(47)

(48)

where , is a nonnegative function that is strictly in-creasing in both variables, and for all . Then,the map from to (where ) ispassive.

Proof: Consider the storage function for the th link

(49)

Since is strictly increasing with respect to , is a posi-tive–definite function.

Take the derivative of along the solution, we have

(50)

We first observe that

This is immediate if the projection is inactive. If the projec-tion is active, the left-hand side is zero, and the right hand is

which is nonnegative.Next, we show


Fig. 9. Primal/dual congestion control.

If , then both sides are identical. If , then. Since and is nonnegative, the inequality

follows immediately.By adding and subtracting to the right-hand side of

(50), we have

Since is strictly increasing with respect to the first variable,( ) has the same sign as . It followsthat

Since is positive definite, the passivity from tofollows by integrating both sides of the inequality.

The previous passivity property does not hold for the secondorder controller in [7]. This is because, as shown in Fig. 8(a),if we ignore the positive projection, the transfer function from

to has two poles at the origin, which cannot bePR.

We can also make the same modification as in Proposition 3for the source control by using the inverse-ZF functions.

Proposition 6: If the link update law is (41) and the sourceupdate law is

(51)

(52)

where is inverse-ZF and , then the equilibriumof the closed loop system is the same as in (3)–(4) and is globallyasymptotically stable.

C. Combined Primal/Dual Algorithms

In Section II-C, we showed passivity of the first order sourcerate controllers from to ( ), and in Section III-B,we showed passivity of the first-order link price controller from( ) to ( ). As a direct consequence of the passivityanalysis, we can combine these two classes of controllers to-gether to achieve global asymptotic stability as in Fig. 9. Amongthe congestion control laws in the literature which guaranteeglobal asymptotic stability, either the source rate update or thelink price update is static. When both are dynamic, stability hasonly been shown using singular perturbations in [8] and for thesingle-bottleneck case in [9], [10]. With our passivity approach,extension from single to multiple bottlenecks is straightforwardbecause pre-multiplication by and post-multiplication byas in Fig. 9 does not change the passivity property of the for-ward system.

Fig. 10. Network example.

IV. SIMULATION RESULTS

To illustrate the new classes of stable flow controllers pre-sented in this paper, we consider a simple four-source/three-linkexample shown in Fig. 10. The corresponding routing matrix is

We assume that all source utility functions areand the link capacities are all 1. For this choice of , the as-sumption in Theorem 1 is satisfied. The solution to the opti-mality condition (3)–(4) is

The initial source rate is set to

The initial link price is set to zero.

A. Delay Robustness of Primal Controller in Theorem 1

We first illustrate the effect of adding an SPR source con-troller as in Theorem 1. From linear system theory, the delay ro-bustness for a single-input–single-output (SISO) system is givenby

(53)

where is the maximum delay allowed in the feedback loopwithout destabiizing the system, is the phase margin, and

is the gain crossover frequency. Therefore, to enhance delayrobustness, one needs to increase the phase margin and/or de-crease the bandwidth. This may be accomplished by introducingan SPR filter in the source controller, as in (14)–(15). To illus-trate the result, consider the controller that follows.

A1) Flow control in [1] with and .B1) Flow control in Theorem 1: with , ,

, , i.e.,, and .


Fig. 11. Loop gain Bode plot comparison: Flow controller in [1] versuscontroller based on Theorem 1.

Fig. 12. Source rate comparison under 2-s delay. Flow controller in [1] versuscontroller based on Theorem 1.

We will introduce time delay in the price feedback channel,therefore, the corresponding loop gain of the linearized system(about the source rate ) is

(54)

where for controller A1), andfor controller B1). To show the

connection with SISO concept of phase margin and bandwidth,we only consider delay in the price feedback of link 3 (so thefirst two loops are closed with unity gains). The SISO loop gain(based on the linearization about ) for the two controllersare shown in Fig. 11. Controller A1) has a larger bandwidth,

, and smaller phase margin, ,than controller B1, , . The corre-sponding predicted delay stability margin for A1) is 0.086 s and0.322 s for Controller B1). This is indeed verified in simulationas shown in Fig. 12 when 0.25-s delay is introduced in the price

Fig. 13. Link rate comparison under 1-s delay. Flow controller in [7] versuscontroller based on Theorem 2.

feedback of link 3. Persistent oscillatory response is present inController A1), while a stable response is obtained using Con-troller B1). However, the smaller bandwidth of B1) also resultsin a lower rate of convergence.

B. Delay Robustness of Dual Controller in Theorem 2

We next show the effect of our extended dual controller(based on Theorem 2) on delay robustness by comparing withthe second-order link controller in [7]. Consider the followingcontrollers.

A2) Flow control in [7], with and .B2) Flow control in Theorem 2, with , ,

, , i.e.,.

Note that the only difference between A2) and B2) is that iszero in A2) and negative in B2). With a constant delay of 1 sintroduced in all source to link channels, a persistent oscillationappears with A2) while a stable response is maintained with B2),as shown in the link rate comparison in Fig. 13. This differencein robustness can again be explained by the phase margin andgain crossover frequency comparison of the linearized system.The loop gain of the linearized system (about the source rate )is

(55)

Fig. 14 shows the Bode plot comparison of the (1,1) elementof the loop gain transfer matrix with (other channelsare similar). The phase margin corresponding to Controller B2)is much larger than the phase margin of Controller A2) (62.4versus 4.5 ), and both controllers have comparable bandwidth[0.59 rad/s for B2) versus 0.79 rad/s for A2)]. The predicteddelay robustness margine based on the linearized system is1.85 s for B2) and only 0.1 s for A2). The enhanced delayrobustness is indeed verified by the simulation result.


Fig. 14. Bode plot comparison of linearized system. Flow controller in [7]versus controller based on Theorem 2.

V. CONCLUSION

This paper has addressed source and link control which canguarantee global asymptotic stability of the desired networkequilibrium. By using the passivity approach in nonlinear anal-ysis, we have developed a unifying framework for stabilizingsource and link control laws which encompass existing algo-rithms in [1], [7], [8], [10], and [19] as special cases. In ourapproach, we first interpret the existing results in a passivityperspective and then augment the source and link control lawswith suitable passive systems. The global asymptotic stabilityof the closed loop system then follows from the passivity-basedstability analysis. In addition to unifying and extending the ex-isting results, we have shown that the added design freedom inflow control can be exploited to improve the disturbance rejec-tion property and robustness with respect to time delays. The ro-bustness enhancement potential is illustrated through linearizedanalysis and simulation. A significant research direction is to de-velop systematic procedures for the selection of design param-eters to improve robustness and performance. We are currentlyalso studying the delay problem in a nonlinear framework andinvestigating stability and robustness under discrete time imple-mentations.

APPENDIX IUNIQUENESS OF OPTIMIZATION SOLUTION

Proposition A.1: If satisfies

(56)

and the routing matrix is of full row rank, then the optimalitycondition given by (1), (3), and (4) has a unique solution.

Proof: Suppose that at the optimal solution, links ,, are uncongested, i.e., , and the other links,

, , are congested, i.e., . Define

and as

otherwise

otherwise

By construction, . From (4), prices for the uncongestedlinks are zero, therefore

(57)

where contains the prices of the congested links. The con-gested links operate at capacity, so

Substituting (1) for and (3) for , we obtain

(58)

From (56), is a negative–definite matrix. Therefore, the gra-dient of the right-hand side with respect to is ,which is a negative definite matrix. By the inverse functiontheorem, there is at least one solution of . Suppose that thereare multiple solutions. Then, by the mean value theorem,

must be singular which contradicts the strictconcavity assumption on . Therefore, the solution is unique.It follows from (57), (1), and (3), that , , , and arealso unique.

Note that the full-row rank condition on is only sufficient;a unique equilibrium may exist even when the condition is vi-olated. For example, consider a single source and two-link net-

work with . A unique optimum exists as long as the

two links do not have identical capacities.

APPENDIX IIAPPROXIMATE SOLUTION TO THE OPTIMIZATION PROBLEM BY

USING LINK PENALTY FUNCTION

We first show that the approximate solution to the optimiza-tion problem is unique.

Proposition A.2: If satisfies (56) and ismonotonically nondecreasing, then (1), (3), and (8) has a uniquesolution.


Proof: Combining (1), (3), and (8), the source ratesolves

Since satisfies (56) and is monotonically nondecreasing

It follows from the inverse function theorem that hasa unique solution.

Note that the full-row rank condition on is no longer re-quired since is uniquely determined by .

We now address the issue of closeness of the approximatesolution to the optimal solution when is of full-row rank. As-sume that the penalty function has the following property:

ifif .

(59)

Suppose that at the optimal solution, links , ,satisfy , and the other links, , ,satisfy . From (59), where contains theprices of the congested links. Using (1), (3), and (8), we have

(60)

Note that is invertible since . Express in terms ofdeviation from the capacity

Then

Substituting back to (60), we have

If as , then this equation becomes the conditionfor the optimal solution, (58), i.e., the approximate optimal so-lution converges to the optimal solution.

APPENDIX IIIUNIQUENESS OF EQUILIBRIUM UNDER POSITIVE PROJECTION

Lemma 1: Consider the system

(61)

where is continuously differentiable and has aunique fixed point , with strictly positive entries ,

. If, for all

(62)

then is also a unique equilibrium for

(63)

Proof: We note that is an equilibrium for (63), because(61) and (63) coincide when , . To show

that it is unique we assume that there exists another equilib-rium , and proceed with a contradiction argument. Substi-tuting in the mean value theorem

(64)

Multiplying both sides from the left by , and using(62), we obtain

(65)

Because by assumption, we note that either, or and . This means that

(66)

where is either zero or strictly negative. Because ,and at least one of the ’s is strictly negative [if all werezero, would have been another equilibrium for (61)], we con-clude that (66) is strictly positive, thus contradicting (65).

ACKNOWLEDGMENT

The authors would like to thank the helpful discussions onthis research with S. Kalyanaraman, A. Loncarich, and X. Fan.They also thank A. Teel for pointing to [27].

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John Ting-Yung Wen (S’79–M’81–SM’93–F’01)received the B.Eng. degree from McGill University,Montreal, QC, Canada, in 1979, the M.S. degreefrom the University of Illinois, Urbana, in 1981, andthe Ph.D. degree from Rensselaer Polytechnic Insti-tute, Troy, NY, in 1985, all in electrical engineering.

From 1981 to 1983, he was a System Engineer inFisher Control Company, Marshalltown, IA, wherehe worked on the coordination control of a pulp andpaper plant. From 1985 to 1988, he was a Memberof the Technical Staff at the Jet Propulsion Labora-

tory, Pasadena, CA, where he worked on the modeling and control of large flex-ible structures and multiple-robot coordination. Since 1988, he has been withthe Department of Electrical, Computer, and Systems Engineering with a jointappointment in the Department of Mechanical, Aerospace, and Nuclear Engi-neering at Rensselaer Polytechnic Institute, where he is currently a Professor.His current research interests are in the area of precision motion control, dis-tributed control, and modeling and control of mechanical and material systems.

Murat Arcak (S’97–M’00) was born in Istanbul,Turkey, in 1973. He received the B.S. degree inelectrical and electronics engineering from theBogazici University, Istanbul, in 1996, and theM.S. and Ph.D. degrees in electrical and computerengineering from the University of California, SantaBarbara, in 1997 and 2000, under the direction ofPetar Kokotovic.

In 2001, he joined the Rensselaer PolytechnicInstitute, Troy, NY, as an Assistant Professor ofElectrical, Computer, and Systems Engineering. His

research interests are in nonlinear control theory and applications.Dr. Arcak is a Member of SIAM and an Associate Editor on the Conference

Editorial Board of IEEE Control Systems Society. He was a finalist for the Stu-dent Best Paper Award at the 2000 American Control Conference, and receiveda National Science Foundation CAREER Award in 2003.

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