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A Unifying Passivity Framework for Network Flow Control John T. Wen, Fellow, IEEE, and Murat Arcak, Member, IEEE
Abstract—Network flow control regulates the traffic between sources and links based on congestion, and plays a critical role in ensuring satisfactory performance. In recent studies, global stability has been shown for several flow control schemes. By using a passivity approach, this paper presents a unifying framework which encompasses these stability results as special cases. In addition, the new approach significantly expands the current classes of stable flow controllers by augmenting the source and link update laws with passive dynamic systems. This generality offers the possibility of optimizing the controllers, for example, to improve robustness in stability and performance with respect to time delays, unmodeled flows, and capacity variation. Index Terms—Network congestion control, network flow control, network optimization, nonlinear control, passivity.
I. INTRODUCTION
N
ETWORK flow is governed by the interconnection between the information sources and communication links through the routing matrix, , as shown in Fig. 1. Packets from each source (with rate ) are routed through the links (with the aggregate link rate ). Each link has a fixed capacity , and based on its congestion and queue size, a link price, , is computed. The link price information is then sent back to each source to regulate the traffic back into the links. Since the links only feed back the price information to the sources that utilize them, we have the following relationship:
Fig. 1. Network flow control model.
(flows that do not conform to this framework), and capacity variation. A common approach to flow control is to decompose the problem into a static optimization problem and a dynamic stabilization problem [1], [2]. The static optimization incorporates fairness, capacity constraint, and utilization, and its solution provides the desired steady state operating point (equilibrium of the closed loop system), , , , and . The source rate and link price update laws are then designed to guarantee stability and robustness of the equilibrium. The static optimization problem is to maximize the sum of for the sources while complying the utilization function with capacity constraints in the links; that is
(1) subject to
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 the link capacity is a constant vector. The flow control problem aims to find decentralized source and link control algorithms ( as a function of , and as a function 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. Recommended by Associate Editor H. Zhang. This work was supported in part by the RPI Office of Research through an Exploratory Seed Grant. The authors are with the Department of Electrical, Computer, and Systems Engineering, Rensselaer Polytechnic Institute, Troy, NY 12180 USA (e-mail:
[email protected];
[email protected]). Digital Object Identifier 10.1109/TAC.2003.822858
(2)
where is a strictly concave function. By using the Lagrange multiplier, , the inequality constraint can be folded into the optimization problem
If is differentiable, the first order condition for the maximization problem is
0018-9286/04$20.00 © 2004 IEEE
(3)
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The condition for the Lagrange multiplier if if
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is .
(4)
is strictly concave and As shown in Appendix I, if each is of full row rank, (3) and (4) are sufficient to determine the unique equilibrium. for each source determines the The utility function equilibrium, and consequently the steady state fairness and utilization. The objective of source and link update laws is now to drive the actual source rates and link prices to their respective equilibrium values. The constraints in this problem are as follows. can only depend on , and can • Decentralization: only depend on . • No routing information: The routing matrix is unknown to the sources and the links. • No coordination among sources and links: The sources do not have knowledge of the objective functions of other sources, and the links do not have information of the capacities of other links. Therefore, the equilibrium value is unknown. Among many congestion control methods surveyed in [3]–[5], only a few can ensure global stability subject to the structural and information constraints described before. Motivated by the gradient update for the optimization problem (2), a “primal” algorithm is proposed in [1] which consists of a first order source update law and a static link penalty function to keep the aggregate rate below its capacity. A “dual” algorithm is also proposed in [1] where the static source update is the optimality condition (3) and the link uses a first order dynamics of the price update. In [6] and [7], the link update is replaced by second order dynamics. When both source and link updates are dynamic in the so-called primal/dual algorithm, stability has only been shown using singular perturbations in [8] and for the single-bottleneck case (i.e., only one congested link is considered) in [9] and [10]. In this paper, we develop a unifying framework for stable network flow control by using a passivity approach. This framework includes the primal and dual control schemes in [1], [7] as special cases and extends them to broader classes of flow control laws. A system with state , input and output , is said to be passive if there exists a continuously differentiable storage such that function (5) for some function . If is positive definite, then the system is strictly passive. The notion of passivity is motivated by physical systems that conserve or dissipate energy, such as passive circuits [11] and mechanical structures corresponds to an energy function. Passivity [12], where is a useful tool for nonlinear stability analysis [13], [14] and control design [15], [16]. Indeed, the celebrated Passivity Theand with posorem states that, if two passive systems itive definite and radially unbounded storage functions and , respectively, are interconnected as in Fig. 2, then the 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 function, and satisfies
(6) and . from the interconnection conditions A detailed treatment of passivity and its applications in stability analysis is given in [17], and furthered with feedback design applications in [16] and [18]. By using the passivity approach, we show in this paper that the first-order source controller in the primal approach in [1], and the first or second-order link controllers in the dual approach in [1], [7], [19] can be replaced by dynamic systems with prescribed passivity properties. In addition, the static link update in [1] and the static source update in [7], [19] can be augmented with a class of dynamic systems motivated by the Zames–Falb (ZF) multiplier [20]. The dynamic source update in the primal controller [1] can also be combined with the dynamic link update in the dual controller [7] to obtain a dynamic-source/dynamic-link stabilizing control law. As an example, the single bottleneck congestion controller in [10] becomes a special case. In addition to unifying the existing stabilizing controllers in the literature, our result also offers the potential for additional optimization for robustness in stability and performance with respect to time delay, unmodeled flows, and capacity variation. Time delay adds phase lag and compromises stability in a feedback system. From the linear systems point of view, to enhance robustness with respect to the delay, the controller should add phase lead or increase gain roll-off or do both (subject to the gain-phase relationship imposed by the Bode integral formula [21]). The controllers in this paper can provide additional phase lead or gain roll-off which may be exploited for robustness enhancement. Another consideration is network variation. The model considered in this paper is highly idealized; real networks have constantly changing topology and capacity. For the actual deployment of congestion control, averaging is used to smooth out the variation in various signals. Within our passivity framework, the smoothing filter can be designed and optimized without compromising stability. This paper is organized as follows. Section II discusses the passivity interpretation of the primal controller in [1] and the generalization to passive source control and multiplier-based link control. Section III presents the passivity framework for the dual controller in [7] and the generalization to passive link control and multiplier-based source control. A combined
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Fig. 4.
Primal flow controller based on [1].
primal/dual algorithm is also shown in this section. Simulation results based on a four-source/three-link network are shown in Section IV. Conclusions are given in Section V. Throughout the paper, we shall make the following assumptions. 1) The routing matrix is of full row rank [to ensure the uniqueness of the optimal solution given by (3) and (4); see Appendix I]. , are all pos2) The optimal source rates, , itive (to ensure that positive projection does not change the optimal solution; see Appendix III). A. Notation Given the column vector . matrix Given a function defined as
, we will also use to denote and to denote the square , its positive projection is
if if
, or and
and .
When the first condition holds, we say that the projection is inare vecactive; otherwise, the projection is active. If and is interpreted in the component-wise sense. tors, then We will use projection functions to ensure nonnegative values for physical quantities, such as source rates, computed by the controllers. The issues of existence and uniqueness of solutions resulting from the discontinuity of these functions is not emphasized in this paper. Existence is guaranteed for a differential inclusion that encompasses the projection discontinuity, as in [22]. Uniqueness is not necessary in our results because our Lyapunov estimates apply to all solutions. II. PRIMAL FLOW CONTROL ALGORITHMS In [1], a flow control law based on the primal approach to the optimization problem (2) is proposed (see Fig. 3). Given the , the source update law utility function for each source, is given by
Equivalent representation of primal flow controller.
where , with th component , is a penalty function that enforces the link capacity constraint, . The in (7) equilibrium can be computed by setting (9) which approximately satisfies the desired condition (3)–(4). Appendix II discusses the uniqueness of the equilibrium and its relationship to the optimal solution in (3)–(4). We assume that each penalty function is monotonically nondecreasing, such as the following used in [1]: (10) for all ensures that the positive The assumption projection in (7) does not change the equilibrium of (7)–(8), as shown in Lemma 1 in Appendix III. This fact will also be useful in the stability proof. A. Passivity-Based Stability Analysis We now present a stability argument by using passivity analysis. The advantage of this approach is that it leads to an extended class of stabilizing control laws to be discussed in Section II-B. We express the interconnected system in Fig. 3 in an equivalent form in Fig. 4 based on the deviation from the equilibrium condition, and rewrite the link update as (11) The following result shows that both the forward and return systems are passive. Adding the two storage functions, we recover the Lyapunov function used in [1]. Proposition 1: Consider the feedback interconnection shown to , and the in Fig. 4. The forward system from return system from to are both passive. Furthermore, is globally asymptotically stable for the the equilibrium . state space Proof: For the forward system, we let (12) where
. The gradient of
with respect to
is
(7) where nent
, , . The link control is given by
with th compo-
(8)
which, when set to zero, has the unique solution at Because the Hessian of is
is a positive–definite function.
.
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Now, we use as a storage function to prove passivity of the forward system. The derivative of along the solution of (7) is
Adding and subtracting , we get
to
, and substituting (7) and (a)
Because the first term is negative semidefinite, the forward to is passive. system from Now, consider the return system, and let (b)
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 in only when . We thus Appendix III, conclude that is globally asymptotically stable.
Fig. 5. Extension of source rate update in primal flow controller. (a) Source control law (7) expressed as a feedback interconnection of a strictly passive forward system and a passive feedback system. (b) Extension of source control law (7) by replacing K with an SPR transfer function D + C (sI A) B .
0
) [24], [25], as shown in Fig. 5(b). In this ( case, the forward system remains strictly passive, and the modified source control law would, therefore, still be passive. The previous argument does not take into account the nonnegativity restriction on . The following theorem shows that , the source rates would remain posifor certain class of tive and the overall system is globally asymptotically stable. Theorem 1: Consider the source rate control law for source , , given by (14) (15)
B. Extended Class of Source Rate Control Laws The advantage of the passivity perspective is that it allows us to consider a broader class of stabilizing source control laws. A natural generalization is to replace the first order source update (7) by a more general class of passive systems. To motivate this extension, we express the source control law (7) as a feedback interconnection as shown in Fig. 5(a), where (13)
where ( ) is SPR, the link update given by (8), and as the routing connection given by (1). If for , then the equilibrium of the interconnected system is globally asymptotically stable for the state space , where . ) is SPR if and Proof: From [24] and [25], ( only if there exist symmetric positive–definite matrices and , and matrices , , such that (16) (17)
Note that since is assumed to be diagonal and concave, is also diagonal with each diagonal entry consisting of a memoryless first–third quadrant nonlinearity
(18) For the forward system, let
for some The forward system in Fig. 5(a) is strictly passive. The feedback system is passive since it is diagonal with diagonal entries consisting of an integrator in series with a first–third quadrant nonlinearity [23]. Since the feedback interconnection between a strictly passive system and a passive system is itself passive [23], it follows that the source control law (7) is passive. To in the forward system may extend this approach, the gain be replaced by a strictly positive real (SPR) transfer function
(19) is strictly concave, is a positive–definite function. Since along the solution is The derivative of
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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 the first term is negative semidefinite, the forward system from to is passive. Now, consider the return system, and let
..
We can now use
as a Lyapunov function and obtain (20)
.. .
.
(23) where
Since is first–third quadrant, is a nonnegais tive–definite function. The return system from to passive since
) to
and restrict (
,
,
, and
are positive constants.
C. Alternate Class of Source Rate Control Laws The passivity argument in Section II-A uses the fact that the forward system cascaded with the derivative block and the return system pre-multiplied by the integral block are both passive. In this section, we use an alternative passivity argument to show that the forward system is passive without the addition of the derivative block. This will then allow us to use dynamic link controllers and still achieve global asymptotic stability. Proposition 2: Consider the feedback interconnection shown to is strictly passive. in Fig. 4. The system from Proof: Consider
Since as , the level sets of are confined . Thus, is positive invariant. in It follows from (20) and LaSalle’s invariance principle that ) converges to the largest invariant set contained in (
(24) The derivative along the solution is
On , for all , therefore, from (14), . Beis the unique equilibrium of (7)–(8), we conclude cause that ; that is, is globally asymptotically stable. The assumption as ensures that is positive-invariant, i.e., if is initially in , it will remain for all . For commonly used utility functions such in (proportional fair, also TCP Vegas) and as (cariant of TCP Reno) [5], this assumption is satisfied. However, for (TCP Reno), the assumption does not hold. In this case, we can prove global asymptotic stability by imposing positive projections on (14) and (15)
We observe that
This follows because, if the projection is inactive then both sides of the inequality are equal, and if the projection is active, and , then the left-hand side is zero, and the right-hand side is nonnegative. By adding and subtracting from , we have
(21) (22)
(25)
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The last equality follows from:
Since is positive definite and each is strictly concave, the to is strictly passive. system from Remark: In [1], the gain depends on , . For this case, the storage function (24) needs to be modified to
(a)
(26) (b)
Then
Fig. 6. Extension of link price update in primal flow controller. (a) Link controllers augmented with inverse-ZF functions. (b) Link controllers augmented with inverse-ZF functions, represented in the error coordinate.
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-based stability analysis, [20], [26] showed that a monotone first-third quadrant nonlinearity cascaded with a class of transfer functions is passive. To use the result in link control, we first define the following class of transfer functions. is Definition 1: The class of proper transfer function called 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 extend the static link update in Theorem 1 to the penalty function cascaded with an inverse-ZF function. The main idea is to show that when the link penalty function is modified by inverse-ZF functions as in Fig. 6(a), it is equivalent to the same modification applied to the error system as in Fig. 6(b). Proposition 3: If the source update law is given by (7) and the update law for the th link is replaced by or
property of inverse-ZF functions that the return system is passive, i.e., (29) is the internal state of and is a nonnegawhere tive function (see [20] and [26] for proofs of this property). We now show the closed loop stability by using in Proposition 2. along the solution again satisfies (25). InThe derivative of , we get tegrating both sides and denoting
(30) where
is positive definite as in (25). Substituting (29) in (30), we obtain
(31) which proves boundedness of . To prove (31) that
, we note from
(27) (28)
where is inverse-ZF and , then the equilibrium of the closed-loop system is the same as in (9) and is globally asymptotically stable. Proof: At the equilibrium, which implies . Therefore, the equilibrium is un, we can represent the rechanged. Due to the linearity of turn system as the cascade of the inverse-ZF function and ) (see Fig. 6). the nonlinearity defined in (11), driven by ( Since is first–third quadrant and monotone, it follows from the
Then, using boundedness of , we can apply Barbalat’s Lemma asymptotically. Since is [17] to conclude that stable, its internal states also converge to the corresponding equilibrium values. In (27), is nonnegative since the output of is nonnegative. However, in (28), this is no longer guaranteed. If the link price feedback is implemented in an averaged sense through packet marking or packet drop, then negative cannot be realized. If is transmitted explicitly (which would require more communication bandwidth), then negative values of may be acceptable.
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Theorem 2: Consider the link update law (35) (36) ) are block diagonal, partitioned according to where ( each link, with the th subsystem given by Fig. 7. Dual flow controller based on [7].
..
.
.. . (37)
is nonnegative, and , and are positive conwhere , with satisfying stants. Then, the equilibrium (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–definite function for the th link:
(a)
(b) Fig. 8. Extension of link rate update in dual flow controller.
The derivative along the solution is (after adding and subtracting from )
III. DUAL FLOW CONTROL ALGORITHMS The flow control algorithm in [1] is based on the primal optimization in (2) and the link rate constraint is enforced by using the penalty function (8). Though this approach guarantees global stability of the source rate, it does not take the link queue dynamics explicitly into account. In [7], a dual approach is proposed where the queue size, , is explicitly modeled, and the rate of change of the link price, , is determined from the combination of queue size and link rate
(38) First, observe that
(32) (33)
(39)
where , , , . The source update is directly given by the primal solution (3)
which is immediate, if the projection is inactive. If the projection , then both sides of the equality are zero. is active, i.e., Next, we claim that
(34) is . The closed where the th component of loop system is shown in Fig. 7. Global asymptotic stability of the system is shown in [7] by using a Lyapunov function argument. We can recover this result as a special case of Theorem 2, which applies the passivity theory to extend the link controller to a broader class. A. Extended Class of Link Price Control Laws As in Section II-B, we now extend the second-order link update algorithm in (32)–(33) to a larger class of passive dynamic systems. As a motivation, ignoring the positive projection, (32)–(33) can be regarded as a first-order positive real (PR) to [see Fig. 8(a)]. The result transfer function from below shows that this transfer function can be generalized to a class of PR transfer functions [see Fig. 8(b)], and, under positive projection, the closed-loop system remains globally asymptotically stable.
(40) If the projection is inactive, the inequality holds since . If the projection is active, then
which implies and therefore the right-hand side is nonnegative. 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) to is passive. lows that the return system from (
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Now, consider the forward system, and let
After taking the derivative of tuting in (41), we get
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along the solution and substi(43)
where
,
, so tion. The forward system from
We now use
, and is a positive–definite functo is passive since
as a Lyapunov function, and obtain
Note that (44) which is immediate if the projection is inactive. If the projection is active, the left-hand side is zero and the right-hand side is ). nonnegative ( Next, we claim that (45)
where the set
is nonnegative as shown before, and vanishes on
. If , then . Since which is immediate if is nonnegative, the inequality follows immediately. By using (44) and (45), (43) becomes (46)
However, arguments similar to those in [7, p. 170] show that the only trajectory that can remain identically in this set is the unique equilibrium. Thus, it follows from a LaSalle-type invariance argument1 that this equilibrium is globally asymptotically stable. The link controller (32)–(33) is a special case of (35)–(36) , , , , . Our with extended class also encompasses the first-order link control law used in [10] and [19]
By integrating both sides, it follows that the system from to is passive. By slightly modifying the storage function, we can show that the passivity property holds for the adaptive virtual queue link is controller in [8] and [28]. Note that in [8], a nonlinear needed to show stability; here we only require to be a positive constant. Proposition 5: Consider the following link control law: (47) (48)
(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 extended controller (35)–(36), the steady state queue size is in general nonzero. A nonzero queue may be desirable to ensure the full utilization of the link. However, the queue should not be too large in order to avoid excessive queuing delay. If zero queue should be set to zero size is desired, then at least one of the to emulate the queue dynamics. B. Alternate Class of Link Price Control Laws In Section II-C, we showed that the first order primal source to , and this property control is passive from allows the inverse-ZF functions to be introduced into the link control. In this section, we present the dual result where the link ) to . controller is passive from ( Proposition 4: Consider the link control law (41). The return to is passive. system from Proof: Consider the following positive–definite function: (42) 1Due to the discontinuity of the differential (35)–(36), the standard LaSalle Principle is not applicable. Instead, we use a variant for differential inclusions, proved in [27]. Our function W (; p) shown previously is indeeded lower semicontinuous (in the sense of real-valued functions) as assumed in [27].
, is a nonnegative function that is strictly inwhere for all . Then, creasing in both variables, and to (where ) is the map from passive. Proof: Consider the storage function for the th link (49) is strictly increasing with respect to , is a posiSince 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 projection is active, the left-hand side is zero, and the right hand is which is nonnegative. Next, we show
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Fig. 9. Primal/dual congestion control. Fig. 10.
If , then both sides are identical. If , then . Since and is nonnegative, the inequality follows immediately. to the right-hand side of By adding and subtracting (50), we have
Since ( that
Network example.
IV. SIMULATION RESULTS To illustrate the new classes of stable flow controllers presented in this paper, we consider a simple four-source/three-link example shown in Fig. 10. The corresponding routing matrix is
is strictly increasing with respect to the first variable, ) has the same sign as . It follows
Since is positive definite, the passivity from to follows by integrating both sides of the inequality. The previous passivity property does not hold for the second order 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 be PR. We can also make the same modification as in Proposition 3 for the source control by using the inverse-ZF functions. Proposition 6: If the link update law is (41) and the source update law is (51) (52) where is inverse-ZF and , then the equilibrium of the closed loop system is the same as in (3)–(4) and is globally asymptotically stable. C. Combined Primal/Dual Algorithms In Section II-C, we showed passivity of the first order source to ( ), and in Section III-B, rate controllers from we showed passivity of the first-order link price controller from ) to ( ). As a direct consequence of the passivity ( analysis, we can combine these two classes of controllers together to achieve global asymptotic stability as in Fig. 9. Among the congestion control laws in the literature which guarantee global asymptotic stability, either the source rate update or the link price update is static. When both are dynamic, stability has only been shown using singular perturbations in [8] and for the single-bottleneck case in [9], [10]. With our passivity approach, extension from single to multiple bottlenecks is straightforward and post-multiplication by because pre-multiplication by as in Fig. 9 does not change the passivity property of the forward system.
We assume that all source utility functions are and the link capacities are all 1. For this choice of , the assumption in Theorem 1 is satisfied. The solution to the optimality 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 controller as in Theorem 1. From linear system theory, the delay robustness for a single-input–single-output (SISO) system is given by (53) where is the maximum delay allowed in the feedback loop is the phase margin, and without destabiizing the system, is the gain crossover frequency. Therefore, to enhance delay robustness, one needs to increase the phase margin and/or decrease the bandwidth. This may be accomplished by introducing an SPR filter in the source controller, as in (14)–(15). To illustrate the result, consider the controller that follows. and . A1) Flow control in [1] with , , B1) Flow control in Theorem 1: with , , i.e., , and .
WEN AND ARCAK: UNIFYING PASSIVITY FRAMEWORK FOR NETWORK FLOW CONTROL
Fig. 11. Loop gain Bode plot comparison: Flow controller in [1] versus controller based on Theorem 1.
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Fig. 13. Link rate comparison under 1-s delay. Flow controller in [7] versus controller based on Theorem 2.
feedback of link 3. Persistent oscillatory response is present in Controller A1), while a stable response is obtained using Controller B1). However, the smaller bandwidth of B1) also results in a lower rate of convergence. B. Delay Robustness of Dual Controller in Theorem 2
Fig. 12. Source rate comparison under 2-s delay. Flow controller in [1] versus controller 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
We next show the effect of our extended dual controller (based on Theorem 2) on delay robustness by comparing with the second-order link controller in [7]. Consider the following controllers. 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 is zero in A2) and negative in B2). With a constant delay of 1 s introduced in all source to link channels, a persistent oscillation appears with A2) while a stable response is maintained with B2), as shown in the link rate comparison in Fig. 13. This difference in robustness can again be explained by the phase margin and gain crossover frequency comparison of the linearized system. The loop gain of the linearized system (about the source rate ) is
(54) for controller A1), and for 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 the first two loops are closed with unity gains). The SISO loop gain (based on the linearization about ) for the two controllers are shown in Fig. 11. Controller A1) has a larger bandwidth, , and smaller phase margin, , than controller B1, , . The corresponding predicted delay stability margin for A1) is 0.086 s and 0.322 s for Controller B1). This is indeed verified in simulation as shown in Fig. 12 when 0.25-s delay is introduced in the price where
(55) Fig. 14 shows the Bode plot comparison of the (1,1) element of the loop gain transfer matrix with (other channels are similar). The phase margin corresponding to Controller B2) is much larger than the phase margin of Controller A2) (62.4 versus 4.5 ), and both controllers have comparable bandwidth [0.59 rad/s for B2) versus 0.79 rad/s for A2)]. The predicted delay robustness margine based on the linearized system is 1.85 s for B2) and only 0.1 s for A2). The enhanced delay robustness is indeed verified by the simulation result.
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APPENDIX I UNIQUENESS OF OPTIMIZATION SOLUTION Proposition A.1: If
satisfies (56)
and the routing matrix is of full row rank, then the optimality condition 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 uncongested links are zero, therefore (57) where contains the prices of the congested links. The congested links operate at capacity, so
Substituting (1) for
and (3) for , we obtain (58)
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 can guarantee global asymptotic stability of the desired network equilibrium. By using the passivity approach in nonlinear analysis, we have developed a unifying framework for stabilizing source and link control laws which encompass existing algorithms in [1], [7], [8], [10], and [19] as special cases. In our approach, we first interpret the existing results in a passivity perspective and then augment the source and link control laws with suitable passive systems. The global asymptotic stability of the closed loop system then follows from the passivity-based stability analysis. In addition to unifying and extending the existing results, we have shown that the added design freedom in flow control can be exploited to improve the disturbance rejection property and robustness with respect to time delays. The robustness enhancement potential is illustrated through linearized analysis and simulation. A significant research direction is to develop systematic procedures for the selection of design parameters to improve robustness and performance. We are currently also studying the delay problem in a nonlinear framework and investigating stability and robustness under discrete time implementations.
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 function theorem, there is at least one solution of . Suppose that there are multiple solutions. Then, by the mean value theorem, must be singular which contradicts the strict concavity assumption on . Therefore, the solution is unique. are It follows from (57), (1), and (3), that , , , and also unique. Note that the full-row rank condition on is only sufficient; a unique equilibrium may exist even when the condition is violated. For example, consider a single source and two-link network with
. A unique optimum exists as long as the
two links do not have identical capacities. APPENDIX II APPROXIMATE SOLUTION TO THE OPTIMIZATION PROBLEM BY USING LINK PENALTY FUNCTION We first show that the approximate solution to the optimization problem is unique. satisfies (56) and is Proposition A.2: If monotonically nondecreasing, then (1), (3), and (8) has a unique solution.
WEN AND ARCAK: UNIFYING PASSIVITY FRAMEWORK FOR NETWORK FLOW CONTROL
Proof: Combining (1), (3), and (8), the source rate solves
Since
satisfies (56) and
173
that it is unique we assume that there exists another equilibrium , and proceed with a contradiction argument. Substiin the mean value theorem tuting (64)
is monotonically nondecreasing
It follows from the inverse function theorem that has a unique solution. Note that the full-row rank condition on is no longer required since is uniquely determined by . We now address the issue of closeness of the approximate solution to the optimal solution when is of full-row rank. Ashas the following property: sume that the penalty function if if
(60) . Express
, and using (65)
Because , or
by assumption, we note that either and . This means that (66)
(59)
.
, Suppose that at the optimal solution, links , , and the other links, , , satisfy . From (59), where contains the satisfy prices of the congested links. Using (1), (3), and (8), we have
Note that is invertible since deviation from the capacity
Multiplying both sides from the left by (62), we obtain
in terms of
is either zero or strictly negative. Because , where ’s is strictly negative [if all were and at least one of the zero, would have been another equilibrium for (61)], we conclude that (66) is strictly positive, thus contradicting (65). ACKNOWLEDGMENT The authors would like to thank the helpful discussions on this research with S. Kalyanaraman, A. Loncarich, and X. Fan. They also thank A. Teel for pointing to [27]. REFERENCES
Then
Substituting back to (60), we have
If as , then this equation becomes the condition for the optimal solution, (58), i.e., the approximate optimal solution converges to the optimal solution. APPENDIX III UNIQUENESS OF EQUILIBRIUM UNDER POSITIVE PROJECTION Lemma 1: Consider the system (61) where is continuously differentiable and unique fixed point , with strictly positive entries . If, for all
has a ,
(62) then
is also a unique equilibrium for (63)
Proof: We note that is an equilibrium for (63), because , . To show (61) and (63) coincide when
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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. degree from the University of Illinois, Urbana, in 1981, and the Ph.D. degree from Rensselaer Polytechnic Institute, Troy, NY, in 1985, all in electrical engineering. From 1981 to 1983, he was a System Engineer in Fisher Control Company, Marshalltown, IA, where he worked on the coordination control of a pulp and paper plant. From 1985 to 1988, he was a Member of the Technical Staff at the Jet Propulsion Laboratory, Pasadena, CA, where he worked on the modeling and control of large flexible structures and multiple-robot coordination. Since 1988, he has been with the Department of Electrical, Computer, and Systems Engineering with a joint appointment in the Department of Mechanical, Aerospace, and Nuclear Engineering at Rensselaer Polytechnic Institute, where he is currently a Professor. His current research interests are in the area of precision motion control, distributed 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 in electrical and electronics engineering from the Bogazici University, Istanbul, in 1996, and the M.S. and Ph.D. degrees in electrical and computer engineering from the University of California, Santa Barbara, in 1997 and 2000, under the direction of Petar Kokotovic´ . In 2001, he joined the Rensselaer Polytechnic Institute, Troy, NY, as an Assistant Professor of Electrical, 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 Student Best Paper Award at the 2000 American Control Conference, and received a National Science Foundation CAREER Award in 2003.
