A Unifying Passivity Framework for Network Flow Control - RPI

A Unifying Passivity Framework for Network Flow Control John T. Wen and Murat Arcak

∗

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.

1

Introduction

Network flow is governed by the interconnection between the information sources and communication links through the routing matrix, R, as shown in Figure 1. Packets from each source (with rate xi ) are routed through the links (with the aggregate link rate y` ). Each link has a fixed capacity c` , and based on its congestion and queue size, a link price, p` , 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 relationship: q = RT p

y = Rx,

(1)

where x ∈ RN is the source rate, y ∈ RL is the aggregate rate, p ∈ RL is the link price, and q ∈ RN is the aggregate price. In this paper, we assume that there is no delay in the loop and the link capacity c is a constant vector. The flow control problem aims to find decentralized source and link control algorithms (x as a function of q, and p as a function of y) to achieve the following objectives: • Utilization: Maximize throughput by keeping y` near c` . • Fairness: All sources have “equitable” shares of capacity. ∗

The authors are with the Department of Electrical, Computer, and Systems Engineering, Rensselaer Polytechnic Institute, Troy, NY 12180. Emails: [email protected], [email protected]

1

x ∈ RN

y∈R L

R

N sources

L links

Source . control .

link . control .

diagonal

q∈R

diagonal

RT

N

p∈R L

Figure 1: Network Flow Control Model • Stability: All signals converge to desired equilibrium values. • Robustness: Maintain stability and performance under model variation, including time delays, unmodeled flows (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), x∗ , y ∗ , p∗ , and q ∗ . 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 the utilization function Ui (xi ) for the sources while complying with capacity constraints in the links; that is max x≥0

N X

Ui (xi )

subject to

i=1

Rx ≤ c, |{z}

(2)

y

where Ui is a strictly concave function. By using the Lagrange multiplier, p, the inequality constraint can be folded into the optimization problem: min max L(x, p) = min max p≥0 x≥0

p≥0 x≥0

       

N X

Ui (xi ) −

i=1

L X

p` (y` − c` )

`=1





       

   X X L L X  N   p` c`  . = min max  Ui (xi ) − p` y`  + p≥0  x≥0   `=1  i=1   `=1     |P{z }       qx i i i

2

If U is differentiable, the first order condition for the maximization problem is qi = Ui0 (xi ).

(3)

The condition for the Lagrange multiplier p is (

p`

=0 ≥0

if y` < c` if y` = c` .

(4)

As shown in Appendix A, if each Ui is strictly concave and R is of full row rank, (3)-(4) is sufficient to determine the unique equilibrium. The utility function Ui (xi ) for each source determines the 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: • Decentralization: xi can only depend on qi , and p` can only depend on y` . • No routing information: The routing matrix R 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, 4, 5], only a few can ensure global stability subject to the structural and information constraints described above. 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, 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 perturbation in [8] and for the single-bottleneck case (i.e., only one congested link is considered) in [9], and in the delay free case, [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 H with state x, input u and output y, is said to be passive if there exists a continuously differentiable ‘storage function’ V (x) ≥ 0 such that V˙ ≤ −W (x) + uT y

(5)

for some function W (x) ≥ 0. If W (x) 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 [12], where V (x) corresponds to an energy function. Passivity is a useful tool for nonlinear stability analysis [13, 14] and control design [15, 16]. Indeed, the celebrated Passivity Theorem states that, if two passive systems H1 and H2 with positive definite and radially unbounded storage functions V1 (x1 ) 3

and V2 (x2 ), respectively, are interconnected as in Figure 2, then the equilibrium of the interconnection is stable in the sense of Lyapunov. This is because the sum of the storage functions, V (x) = V1 (x1 ) + V2 (x2 ), can now be used as a Lyapunov function, and satisfies V˙ ≤ −W1 (x1 ) − W2 (x2 ) + uT1 y1 + uT2 y2 = −W1 (x1 ) − W2 (x2 )

(6)

from the interconnection conditions u1 = −y2 and y1 = u2 . 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].

u1

H1

y1

− y2

H2

u2

Figure 2: Feedback interconnection of two passive systems. 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 [19, 7] can be augmented with a class of dynamic systems motivated by the Zames-Falb 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 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 2 discusses the passivity interpretation of the primal controller in [1] and the generalization to passive source control and multiplier-based link control. Section 3 presents the passivity framework for the dual controller in [7] and 4

the generalization to passive link control and multiplier-based source control. A combined primal/dual algorithm is also shown in this section. Simulation results based on a foursource/three-link network are shown in Section 4. Conclusion and future work are given in Section 5. Throughout the paper, we shall make the following assumptions: 1. The routing matrix R is of full row rank (to ensure the uniqueness of the optimal solution given by (3)–(4); see Appendix A). 2. The optimal source rates, x∗i , i = 1, . . . , N , are all positive (to ensure positive projection does not change the optimal solution; see Appendix C). Notation: Given U : RN → R, we will also use U 0 (x) to denote the column vector 

2



∂U ∂x

T



and U 00 (x) to denote the square matrix ∂∂xU2 . Given a function f : R+ → R, its positive projection is defined as: (

(f (x))+ x

:=

f (x) 0

if x > 0, or x = 0 and f (x) ≥ 0 if x = 0 and f (x) < 0 .

When the first condition holds, we say that the projection is inactive; otherwise, the projection is active. If x and f (x) are vectors, then (f (x))x is interpreted in the component-wise sense. 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.

2

Primal Flow Control Algorithms

In [1], a flow control law based on the primal approach to the optimization problem (2) is proposed (see Figure 3). Given the utility function for each source, Ui (xi ), the source update law is given by x˙ = K(U 0 (x) − q)+ (7) x where K = diag {ki }, ki > 0, U 0 (x) ∈ RN with ith component Ui0 (xi ). The link control is given by p = h(y) (8) where h(y) ∈ RL , with `th component h` (y` ), is a penalty function that enforces the link capacity constraint, y` ≤ c` . The equilibrium can be computed by setting x˙ = 0 in (7): q ∗ = U 0 (x∗ ),

p∗ = h(y ∗ ),

(9)

which approximately satisfies the desired condition (3)-(4). Appendix B discusses the uniqueness of the equilibrium and its relationship to the optimal solution in (3)-(4). We assume that each penalty function is monotonically non-decreasing, such as the following used in [1]: h` (y) = (y` − c` + )+ /2 . 5

(10)

The assumption x∗i > 0 for all i ensures that the positive projection in (7) does not change the equilibrium of (7)-(8), as shown in Lemma 1 in Appendix C. This fact will also be useful in the stability proof.

+

-p

-q

RT

x

y

R

x& = K (U ′( x) − q ) +x

p

y

h

Figure 3: Primal Flow Controller based on [1]

2.1

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 2.2. We express the interconnected system in Figure 3 in an equivalent form in Figure 4 based on the deviation from the equilibrium condition, and rewrite the link update as h(y − y ∗ ) := h(y) − h(y ∗ ).

(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]. -(p-p*)

+

-(q-q*)

RT

x-x*

x& = K (U ′( x) − q ) +x

.

y-y*

R

y

s IL

p-p*

y-y*

h

.

s-1

IL

y

Figure 4: Equivalent Representation of Primal Flow Controller Proposition 1 Consider the feedback interconnection shown in Figure 4. The forward system from −(p − p∗ ) to y, ˙ and the return system from y˙ to p − p∗ are both passive. Furthermore, the equilibrium x = x∗ is globally asymptotically stable for the state space X := {xi ≥ 0, i = 1, . . . , N }. 6

Proof: For the forward system, we let V1 (x − x∗ ) =

X

(−(Ui (xi ) − Ui (x∗i )) + qi∗ (xi − x∗i )) ,

(12)

i

where V1 (0) = 0. The gradient of V1 with respect to x − x∗ is ∇V1 = −U 0 (x) + q ∗ which, when set to zero, has the unique solution at x = x∗ . Because the Hessian of V1 is ∇2 V1 = −U 00 (x) > 0, V1 is a positive definite function. Now we use V1 as a storage function to prove passivity of the forward system. The derivative of V1 along the solution of (7) is V˙ 1 = (−U 0 (x) + q ∗ )T x. ˙ Adding and subtracting q to q ∗ , and substituting (7) and q = RT p, we get V˙ 1 = (−U 0 (x) + q)T x˙ − (q − q ∗ )T x˙ = (−U 0 (x) + q)T K(U 0 (x) − q)+ − (p − p∗ )T y. ˙ Because the first term is negative semidefinite, the forward system from −(p − p∗ ) to y˙ is passive. Now consider the return system, and let ∗

V2 (y − y ) =

X Z `

y`

y`∗

!

(h` (σ) −

h` (y`∗ ))

dσ ,

where V2 (0) = 0, ∇V2 (0) = (h(y) − h(y ∗ ))|y=y∗ = 0, and ∇2 V2 = h0 (y) ≥ 0, so V2 is a non-negative definite function. The return system from y˙ to p − p∗ is passive since V˙ 2 = (h(y) − h(y ∗ ))T y˙ = (p − p∗ )T y. ˙ We can now use V = V1 + V2 as a Lyapunov function and obtain V˙ ≤ (−U 0 (x) + q)T K(U 0 (x) − q)+ x. It follows from LaSalle’s Invariance principle [14] that x(t) converges to the largest invariant set contained in n

o

0 Ω = x : (U 0 (x) − q)T K(U 0 (x) − q)+ x = 0 = {x : Ui (xi ) = qi or xi = 0, 1 ≤ i ≤ N } .

Because x = x∗ is the unique equilibrium of (7)-(8) (see Appendix C), we conclude that Ω = {x∗ }; that is, x∗ is globally asymptotically stable.

7

2.2

An 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 Figure 5(a), where g(x − x∗ ) := −U 0 (x) + U 0 (x∗ ).

(13)

Note that since U is assumed to be diagonal and concave, g is also diagonal with each diagonal entry consisting of a memoryless first-third quadrant nonlinearity: (x−x∗ )T g(x−x∗ ) = (x−x∗ )T (−U 0 (x)+U 0 (x∗ )) = −(x−x∗ )T U 00 (ξ)(x−x∗ ) ≥ 0,

for some ξ.

The forward system in Figure 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 extend this approach, the gain K in the forward system may be replaced by a strictly positive real (SPR) transfer function (D + C(sI − A)−1 B) [24, 25], as shown in Figure 5(b). In this case, the forward system remains strictly passive, and the modified source control law would therefore still be passive.

(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 by an SPR transfer function D + C(sI − A)−1 B. Figure 5: Extension of Source Rate Update in Primal Flow Controller The above argument does not take into account the non-negativity restriction on x. The following theorem shows that for certain class of U (x), the source rates would remain positive and the overall system is globally asymptotically stable. 8

Theorem 1 Consider the source rate control law for source i, i = 1, . . . , N , given by ξ˙i = Ai ξi + Bi (Ui0 (xi ) − qi ), x˙ i = Ci ξi + Di (Ui0 (xi ) − qi ),

ξi ∈ Rni

(14) (15)

where (Ai , Bi , Ci , Di ) is SPR, the link update given by (8), and the routing connection given by (1). If Ui (xi ) → −∞ as xi → 0 for i = 1, . . . , N , then the equilibrium of the interconnected system is globally asymptotically stable for the state space X+ × Rn1 × . . . RnN , where X+ := {x : xi > 0, i = 1, . . . , N }. Proof: From [24, 25], (Ai , Bi , Ci , Di ) is SPR if and only if there exist symmetric positive definite matrices Pi and Li , and matrices Qi , Wi , such that Pi Ai + ATi Pi = −Li − QTi Qi BiT Pi − Ci = WiT Qi WiT Wi = 2Di .

(16) (17) (18)

For the forward system, let V1 (x − x∗ , ξ) =

X 1 i

( ξiT Pi ξi − (Ui (xi ) − Ui (x∗i )) + qi∗ (xi − x∗i )). 2

(19)

Since Ui is strictly concave, V1 is a positive definite function. The derivative of V1 along the solution is V˙ 1 =

X 1 i

[ ξiT (Pi Ai + ATi Pi )ξi + ξiT Pi Bi (Ui0 (xi ) − qi ) + (−Ui0 (xi ) + qi∗ )T x˙ i ]. 2

Adding and subtracting qi to qi∗ , and using (15) and (16)–(18), we get V˙ 1 =

Xh 1 i

2

ξiT (Pi Ai + ATi Pi )ξi + (Ci ξi + WiT Qi ξi )(Ui0 (xi ) − qi ) i

−(Ui0 (xi ) − qi )(Ci ξi + Di (Ui0 (xi ) − qi )) − (qi − qi∗ )x˙ i i Xh 1 1 1 2 = − ξiT Lξi − kQi ξi k2 + (Qi ξi )T Wi (Ui0 (xi ) − qi ) − kWi (Ui0 (xi ) − qi )k − (qi − qi∗ )x˙ i 2 2 2 i i Xh 1 1 2 = − ξiT Lξi − kQi ξi − Wi (Ui0 (xi ) − qi )k − (qi − qi∗ )x˙ i 2 2 i ≤

Xh

i

−λmin (Li ) kξi k2 − (qi − qi∗ )x˙ i ,

i

where λmin (Li ) denotes the minimum eigenvalue of Li . Because the first term is negative semidefinite, the forward system from −(q − q ∗ ) to x˙ is passive. Now consider the return system, and let V2 (y − y ∗ ) =

X Z `

y`

y`∗

9

!

(h(σ) − h(y`∗ )) dσ ,

Since h(y` ) − h(y`∗ ) is first-third quadrant, V2 is a non-negative definite function. The return system from x˙ to q − q ∗ is passive since V˙ 2 = (h(y) − h(y ∗ ))T y˙ = (p − p∗ )T y˙ = (q − q ∗ )T x. ˙ We can now use V = V1 + V2 as a Lyapunov function and obtain V˙ ≤ −

X

λmin (Li ) kξi k2 .

(20)

i

Since Ui (xi ) → −∞ as xi → 0, the level sets of V are confined in X+ × Rn1 × . . . RnN . Thus, X+ is positive-invariant. It follows from (20) and LaSalle’s Invariance principle that (x(t), ξ(t)) converges to the largest invariant set contained in Ω = {(x, ξ) : ξ = 0} . On Ω, ξ˙i = 0 for all i, therefore, from (14), Ui0 (xi ) = qi . Because x = x∗ is the unique equilibrium of (7)-(8), we conclude that Ω = {(x∗ , 0)}; that is, (x∗ , 0) is globally asymptotically stable. The assumption Ui (x) → −∞ as x → 0 ensures that X+ is positive-invariant, i.e., if x is initially in X+ , it will remain in X+ for all t ≥ 0. For commonly used utility functions such as Ui (xi ) = ai log(xi ) (proportional fair, also TCP Vegas) and Ui (xi ) = −ai /xi (Variant of TCP Reno) [5], this assumption is satisfied. However, for Ui (xi ) = ai tan−1 (bxi ) (TCP Reno), the assumption does not hold. In this case, we can prove global asymptotic stability by imposing positive projections on (14)-(15): + ξ˙i = (Ai ξi + Bi (Ui0 (xi ) − qi ))ξi

x˙ i = (Ci ξi +

Di (Ui0 (xi )

−

+ qi ))xi

(21) ,

(22)

and restrict (Ai , Bi , Ci , Di ) to 

−ai1 ..



Ai =   Ci =

h

0

0

. −aini

ci1 . . . ci,ni

i

   





bi1  .   Bi =   ..  bi,ni

Di = di

(23)

where aij , bij , cij , and di are positive constants.

2.3

An Alternate Class of Source Rate Control Laws

The passivity argument in Section 2.1 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. 10

Proposition 2 Consider the feedback interconnection shown in Figure 4. The system from −(p − p∗ ) to y − y ∗ is strictly passive. Proof: Consider V1 (x − x∗ ) =

1 X (xi − x∗i )2 . 2 i ki

(24)

The derivative along the solution is V˙ 1 =

X

(xi − x∗i )(Ui0 (xi ) − qi )+ xi .

i

We observe that ∗ 0 (xi − x∗i )(Ui0 (xi ) − qi )+ xi ≤ (xi − xi )(Ui (xi ) − qi ).

This follows because, if the projection is inactive then both sides of the inequality are equal, and if the projection is active, xi = 0 and Ui0 (xi ) − qi < 0, then the left hand side is zero, and the right hand side is non-negative. By adding and subtracting q ∗ from q, we have V˙ 1 ≤

X

(xi − x∗i )(Ui0 (xi ) − qi∗ +qi∗ − qi )

i

|{z}

Ui0 (x∗i )

= (x − x∗ )T (U 0 (x) − U 0 (x∗ )) − (y − y ∗ )T (p − p∗ ).

(25)

The last equality follows from X

(xi − x∗i )(qi∗ − qi ) = (x − x∗ )T (q ∗ − q) = (x − x∗ )T RT (p∗ − p) = −(y − y ∗ )(p − p∗ ).

i

Since V1 is positive definite and each Ui is strictly concave, the system from −(p − p∗ ) to y − y ∗ is strictly passive. Remark: In [1], the gain ki depends on xi , ki = di /U 0 (xi ). For this case, the storage function (24) needs to be modified to 1 X Z xi (σ − x∗i )Ui0 (σ) V1 (x − x ) = dσ. 2 i x∗i di ∗

(26)

Then V˙ 1 =

X

(xi − x∗i )(Ui0 (xi ) − qi ),

i

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, [26, 20] showed that a monotone firstthird 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: 11

Definition 1 The class of proper transfer function H(s) is called inverse–ZF (Zames-Falb) if H(s) is strictly stable and 1/H(s) = (m0 − Z(s) + ηs) where m0 , η are positive constants and the impulse response of Z(s), z(t), satisfies z(t) > 0,

Z

∞

0

z(t) < m0 .

Examples of inverse-ZF functions include first-order lag filters, 1/(a0 s + a1 ) and (as + 1)/(s + a), 0 ≤ a < 1. 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 Figure6(a), it is equivalent to the same modification applied to the error system as in Figure 6(b). p

p

h

H(s) p

p

H(s)

h

y

H(s)y

h

y

H(s)

y

h

(a) Link controllers augmented with inverse-ZF functions. p-p*

y-y*

h y-y*

p-p*

h

H(s)

H(s) p-p*

y-y*

H(s) y-y*

p-p*

p-p*

h

H(s)

p-p*

p-p*

h1

h1

y-y*

y-y* H(s)

H(s) p-p*

y-y*

H(s)y-y* h1

h hin1 the error H(s) (b) Link controllers augmented with inverse-ZF functions, represented coordinate. Figure 6: Extension of Link Price Update in Primal Flow Controller

Proposition 3 If the source update law is given by (7) and the update law for the `th link is replaced by p` = h` (H` (s)y` ), p` = H` (s)h` (y` )

or

(27) (28)

where H` (s) is inverse-ZF and H` (0) = 1, then the equilibrium of the closed loop system is the same as in (9) and is globally asymptotically stable. 12

Proof: At the equilibrium, H` (s)y`∗ = H` (0)y`∗ = y`∗ which implies p∗` = h(y`∗ ). Therefore, the equilibrium is unchanged. Due to the linearity of H(s), we can represent the return system as the cascade of the inverse-ZF function H(s) and the nonlinearity h defined in (11), driven by (y − y ∗ ) (see Fig. 6). Since h is first-third quadrant and monotone, it follows from the property of inverse-ZF functions that the return system is passive, i.e., T

Z 0

(p − p∗ )T (y − y ∗ ) dt ≥ −γ(xH (0))

(29)

where xH is the internal state of H(s) and γ(·) is a nonnegative function (see [26, 20] for proofs of this property). We now show the closed loop stability by using V1 in Proposition 2. The derivative of V1 along the solution again satisfies (25). Integrating both sides and denoting x˜ := x − x∗ , we get V1 (˜ x(T )) − V1 (˜ x(0)) ≤ −

Z

T

W (˜ x(t)) dt −

Z

0

T

(p − p∗ )T (y − y ∗ ) dt

(30)

0

where W (˜ x(t)) := −(x − x∗ )T (U 0 (x) − U 0 (x∗ )) is positive definite as in (25). Substituting (29) in (30), we obtain V1 (˜ x(T )) ≤ V1 (˜ x(0)) + γ(xH (0)) −

Z

T

0

W (˜ x(t))dt ≤ V1 (˜ x(0)) + γ(xH (0))

(31)

which proves boundedness of x˜. To prove x˜ → 0, we note from (31) that Z 0

T

W (˜ x(t))dt ≤ V1 (˜ x(0)) + γ(xH (0)).

Then, using boundedness of x, ˙ we can apply Barbalat’s Lemma [17] to conclude that x˜ → 0 asymptotically. Since H(s) is stable, its internal states also converge to the corresponding equilibrium values. In (27), p is non-negative since the output of h is non-negative. 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 p cannot be realized. If p is transmitted explicitly (which would require more communication bandwidth), then negative values of p may be acceptable.

3

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, b, is explicitly modeled, and the rate of change of the link price, p, ˙ is determined from the combination of queue size and link rate: b˙ = (y − c)+ b p˙ = Γ(Λb + y − c)+ p 13

(32) (33)

where Γ = diag {γ` }, γ` > 0, Λ = diag {λ` }, λ` > 0. The source update is directly given by the primal solution (3): −1 x = U 0 (q) (34) where the ith component of U 0 −1 (q) is Ui0 −1 (qi ). The closed loop system is shown in Figure 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 below, which applies the passivity theory to extend the link controller to a broader class.

+

-q

x

−U ′−1

p

q

y b& = ( y − c)

RT

+ b

R

p& = Γ( y − c + Λb) +p

x

Figure 7: Dual Flow Controller based on [7]

3.1

An Extended Class of Link Price Control Laws

. * yl -Section cl l -pl As in 2.2, we now pextend theplsecond order link update algorithm in (32)-(33) to a (s+λ γl larger class ofl ) passive dynamic systems. As a motivation, ignoring the positive projection, s s s (32)-(33) can be regarded as a first order PR transfer function from y` − c` to p˙` (see Figure 8(a)). The result below shows that this transfer function can be generalized to a class of positive real (PR) transfer functions (see Figure 8(b)), and, under positive projection, the γl closed loop system remains globally asymptotically stable. yl − cl

pl − pl* s + λl s

γl s

p& l

yl - cl

s

.

D+C(sI-A)-1B

pl

γl

(a) Link controller in [7], from y` − c` to p˙` .

(b) PR extension of the link controller in [7]

Figure 8: Extension of Link Rate Update in Dual Flow Controller Theorem 2 Consider the link update law η˙ = (Aη + B(y − c))+ η p˙ = (Cη + D(y − c))+ p 14

(35) (36)

where (A, B, C, D) are block diagonal, partitioned according to each link, with the `th subsystem given by 

−a`1 ...



A` =   C` =

h



0

  

−a`,n`

0

c`1 . . . c`,n`

i





b`1  .   B` =   ..  b`,n`

D` = d`

(37)

where a`m is non-negative, and b`m , c`m and d` are positive constants. Then the equilibrium (η, p) = (0, p∗ ), with p∗ 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 definite function for the `th link: V2` =

( X c`m m

2b`m

)

+ (c` − y`∗ )p` .

2 η`m

The derivative along the solution is (after adding and subtracting y` from y`∗ ): V˙ 2` =

X c`m m

b`m

!+

η`m (−a`m η`m + b`m (y` −

c` ))+ η`m +(c` −y` )

X

+(y` −y`∗ )p˙` .

c`m η`m + d` (y` − c` )

m

p`

(38) First observe that a`m c`m 2 c`m η`m (−a`m η`m + b`m (y` − c` ))+ η + c`m η`m (y` − c` ), η`m = − b`m b`m `m

(39)

which is immediate, if the projection is inactive. If the projection is active, i.e., η`m = 0, then both sides of the equality are zero. Next we claim that !+

(c` − y` )

X

c`m η`m + d` (y` − c` )

m

≤ (c` − y` )

X

p`

c`m η`m .

(40)

m

If the projection is inactive, the inequality holds since d` (y` − c` )2 ≥ 0. If the projection is active, then X c`m η`m + d` (y` − c` ) < 0 m

which implies y` − c` < 0 and therefore the right hand side is non-negative. Since the left hand side is zero, it follows that (40) is true. Substituting (39) and (40) in (38), we obtain V˙ 2` ≤

( X m

)

a`m c`m 2 η + (y` − y`∗ )p˙` . − b`m `m

Summing V2` over all links and using the condition (1), it follows that the return system from (x − x∗ ) to q˙ is passive. 15

Now consider the forward system, and let ∗

V1 (q − q ) =

XZ i

qi



qi∗

−1



x∗i − Ui0 (σ) dσ.

where V1 (0) = 0, ∇V1 (0) = (x∗ − U 0 −1 (q ∗ )) = 0, and ∇2 V1 = −(U 00 (x))−1 > 0, so V1 is a positive definite function. The forward system from −q˙ to x − x∗ is passive since V˙ 1 =

X

−1



x∗i − Ui0 (qi ) q˙i =

X

(x∗i − xi ) q˙i .

i

i

We now use V = V1 + V2 as a Lyapunov function, and obtain V˙ ≤ −W (η, p) :=

 X X `



c`m η`m (y` − c` ) + (c` − y` )

m

X

!+  

c`m η`m + d` (y` − c` )

m

p`



where W (η, p) is non-negative as shown before, and vanishes on the set {(η, p) : for all ` = 1, . . . , L, either y` = c` , or, y` < c` and p` = η`m = 0, m = 1, . . . , n` } . 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. It thus 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) with n` = 1, a`,1 = 0, b`,1 = 1, c`,1 = γ` λ` , d`,1 = γ` . Our extended class also encompasses the first order link control law used in [19, 10] p˙ = Γ(y − c)+ (41) p by setting n` = 0, c`,1 = 0, d`,1 = γ` . 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 non-zero. A non-zero 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 size is desired, then at least one of the a`,m should be set to zero to emulate the queue dynamics.

3.2

An Alternate Class of Link Price Control Laws

In Section 2.3, we showed that the first order primal source control is passive from −(q − q ∗ ) to x − x∗ , and this property allows the inverse-ZF functions to be introduced into the link control. In this section, we present the dual result where the link controller is passive from (x − x∗ ) to q − q ∗ . Proposition 4 Consider the link control law (41). The return system from y − y ∗ to p − p∗ is passive. 1

Due to the discontinuity of the differential equations (35)–(36), the standard LaSalle Principle is not applicable. Instead, we use a variant for differential inclusions, proved in [27]. Our function W (η, p) above is indeeded lower-semi-continuous (in the sense of real-valued functions) as assumed in [27].

16

Proof: Consider the following positive definite function: 1 X −1 γ (p` − p∗` )2 . 2 ` `

V2 (p − p∗ ) =

(42)

After taking the derivative of V2 along the solution and substituting in (41), we get V˙ 2 (p − p∗ ) =

X

(p` − p∗` )(y` − c` )+ p` .

(43)

`

Note that ∗ (p` − p∗` )(y` − c` )+ p` ≤ (p` − p` )(y` − c` ),

(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 (−p∗` (·) ≥ 0). |{z} ≤0

Next, we claim that (p` − p∗` )(y` − c` ) ≤ (p` − p∗` )(y` − y`∗ ),

(45)

which is immediate if y`∗ = c` . If y`∗ < c` , then p∗` = 0 and (p` − p∗` )(y` − c` ) = (p` − p∗` )(y` − y`∗ ) + (p` − p∗` )(y`∗ − c` ) = (p` − p∗` )(y` − y`∗ ) + p` (y`∗ − c` ) ≤ (p` − p∗` )(y` − y`∗ ) where the last inequality follows from p` ≥ 0 and y`∗ < c` . By using (44)–(45), (43) becomes V˙ 2 (p − p∗ ) =

X

(p` − p∗` )(y` − y`∗ ).

(46)

`

By integrating both sides, it follows that the system from y − y ∗ to p − p∗ is passive. By slightly modifying the storage function, we can show that the passivity property holds for the adaptive virtual queue link controller in [28, 8]. Note that in [8], a nonlinear γ` is needed to show stability; here we only require γ` to be a positive constant. Proposition 5 Consider the following link control law: η˙ ` = γ` (y` − c` )+ η` p` = f` (y` , η` )

(47) (48)

where γ` > 0, f` is a non-negative function that is strictly increasing in both variables, and f` (y` , 0) = 0 for all y` . Then the map from y` − y`∗ to p` − p∗` (where p∗` := f` (y`∗ , η`∗ )) is passive. Proof: Consider the storage function for the `th link: V` (η` − η`∗ ) =

Z

η`

η`∗

(f` (y`∗ , z) − f` (y`∗ , η`∗ )) dz. 17

(49)

Since f` is strictly increasing with respect to η` , V` is a positive definite function. Take the derivative of V` along the solution, we have ∗ ∗ ∗ V˙ ` = γ` (y` − c` )+ η` (f` (y` , η` ) − f` (y` , η` )).

(50)

We first observe that ∗ ∗ ∗ ∗ ∗ ∗ γ` (y` − c` )+ η` (f` (y` , η` ) − f` (y` , η` )) ≤ γ` (y` − c` )(f` (y` , η` ) − f` (y` , η` )).

This is immediate if the projection is inactive. If the projection is active, the left hand side is zero, and the right hand is −γ` (y` − c` )f` (y`∗ , η`∗ ) which is non-negative. Next, we show γ` (y` − c` )(f` (y`∗ , η` ) − f` (y`∗ , η`∗ )) ≤ γ` (y` − y`∗ )(f` (y`∗ , η` ) − f` (y`∗ , η`∗ )). If y`∗ = c` , then both sides are identical. If y`∗ < c` , then η`∗ = 0. Since f` (y`∗ , 0) = 0 and f` is non-negative, the inequality follows immediately. By adding and subtracting f` (y` , η` ) to the right hand side of (50), we have V˙ ` ≤ γ` (y` − y`∗ )(−(f` (y` , η` ) − f` (y`∗ , η` )) + (f` (y` , η` ) − f` (y`∗ , η`∗ ))). Since f is strictly increasing with respect to the first variable, (y` − y`∗ ) has the same sign as f` (y` , η` ) − f` (y`∗ , η` ). It follows V˙ ` ≤ γ` (y` − y`∗ )(f` (y` , η` ) − f (y`∗ , η`∗ )). |

{z p`

}

|

{z p∗`

}

Since V` is positive definite, the passivity from y` − y`∗ to p` − p∗` follows by integrating both sides of the inequality. The above passivity property does not hold for the second order controller in [7]. This is because, as shown in Figure 8(a), if we ignore the positive projection, the transfer function from y` − c` to p` − p∗` 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 xi = Hi (s)U 0 i −1 (qi ) xi = U 0 i −1 (Hi (s)qi )

or

(51) (52)

where Hi (s) is inverse-ZF and Hi (0) = 1, then the equilibrium of the closed loop system is the same as in (3)–(4) and is globally asymptotically stable.

18

3.3

Combined Primal/Dual Algorithms

In Section 2.3, we showed passivity of the first order source rate controllers from −(p − p∗ ) to (y − y ∗ ), and in Section 3.2, we showed passivity of the first order link price controller from -p consequence -q x y (y−y ∗ ) to (p−p∗ ). As a direct of the . passivity analysis, we can combine these two + + ξ=(Aasymptotic classes of controllers together to R achieve stability T Sξ-BSx) ξ .global R as in Figure 9. Among + x=(C ξ+D (U’(x)-q)) the congestion control laws in the literatureS which guarantee global asymptotic stability, x S either the source rate update or the link price update is static. When both are dynamic, . y + stability has only been shown usingpsingularη=(A perturbation Lη -BL p) η in [8] and for the single-bottleneck .p = (C η+D + case in [9, 10]. With our passivity approach, extension from single to multiple bottlenecks is p L L(y - c)) T straightforward because pre-multiplication by R and post-multiplication by R as in Figure 9 does not change the passivity property of the forward system. -p

+

-q

RT

x

y

x& = K (U ′( x) − q) +x

R

p& = Γ( y − c)+p

y

p

Figure 9: Primal/Dual Congestion Control

4

Simulation Results

To illustrate the new classes of stable flow controllers presented in this paper, we consider a simple 4-source/3-link example shown in Figure 10. The corresponding routing matrix is 



1 0 1 0  R= 1 1 1 0  . 1 1 0 1 We assume that all source utility functions are Ui (xi ) = log(xi ) and the link capacities are all 1. For this choice of Ui , the assumption in Theorem 1 is satisfied. The solution to the optimality condition (3)-(4) is x∗ =

h

0.25 0.25 0.5 0.5

q∗ =

h

4 4 2 2

y∗ =

h

0.75 1 1

p∗ =

h

0 2 2

19

iT

iT iT

.

iT

The initial source rate is set to x(0) =

h

0.4 0.6 0.8 1.0

iT

.

The initial link price is set to zero.

Figure 10: A Network Example

4.1

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 PM , (53) Tmax = ωgc where Tmax is the maximum delay allowed in the feedback loop without destabiizing the system, P M is the phase margin, and ωgc 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 below (A1) Flow control in [1] with ki = 0.1 and  = 0.1. (B1) Flow control in Theorem 1: with Ai = −20, Bi = 1, Ci = −1.9, Di = 0.1, i.e., , and  = 0.1. Di + Ci (sI − Ai )−1 Bi = (0.1s+0.1) (s+20) We will introduce time delay in the price feedback channel, therefore, the corresponding loop gain of the linearized system (about the source rate x) is GS (s) = −Rh0 (Rx)(I − s−1 W (s)U 00 (x))−1 s−1 W (s)RT .

(54)

where W (s) = diag{ki } for controller A1, and W (s) = diag{Di + Ci (sI − Ai )−1 Bi } 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 x∗ ) for the two controllers are shown in Figure 11. Controller A1 has a larger bandwidth, ωgc = 21.9rad/sec, and smaller phase margin, P M = 108.2◦ , than controller B1, ωgc = 8.4rad/sec,P M = 155.5◦ . The corresponding predicted delay stability margin for A1 is 0.086sec and 0.322sec 20

for Controller B1. This is indeed verified in simulation as shown in Figure 12 when 0.25sec delay is introduced in the price 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.

Figure 11: Loop Gain Bode Plot Comparison: Flow Controller in [1] vs. Controller based on Theorem 1

4.2

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 with the second-order link controller in [7]. Consider the following controllers: (A2) Flow control in [7], with γ` = 2 and λ` = 2. (B2) Flow control in Theorem 2, with A` = −1, B` = 1, C` = 2, D` = 2, i.e., D` + C` (sI − . A` )−1 B` = 2(s+2) (s+1) Note that the only difference between (A2) and (B2) is that A` is zero in (A2) and negative in (B2). With a constant delay of 1 second 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 Figure 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 x) is GL (s) = −RU 00 (x)−1 RT s−1 (DL + CL (sI − AL )−1 BL ).

21

(55)

Figure 12: Source Rate Comparison under 2 sec Delay: Flow Controller in [1] vs. Controller based on Theorem 1 Figure 14 shows the Bode plot comparison of the (1,1) element of the loop gain transfer matrix with x = x∗ (other channels are similar). The phase margin corresponding to Controller B2 is much larger than the phase margin of Controller A2 (62.4◦ vs. 4.5◦ ), and both controllers have comparable bandwidth (0.59rad/sec for B2 vs. 0.79rad/sec for A2). The predicted delay robustness margine based on the linearized system is 1.85sec for B2 and only 0.1sec for A2. The enhanced delay robustness is indeed verified by the simulation result.

5

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, 19, 10, 8] 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.

22

link rate comparison 2 A2 B2 1.8

1.6

1.4

y

1.2

1

0.8

0.6

0.4

0.2

0

50

100

150 time

200

250

300

Figure 13: Link Rate Comparison under 1 sec Delay: Flow Controller in [7] vs. Controller based on Theorem 2

Bode Diagram

Bode Diagram

Gm = Inf, Pm = 4.5274 deg (at 0.79182 rad/sec)

Gm = Inf, Pm = 62.441 deg (at 0.59237 rad/sec)

40

20

0

0

Magnitude (dB)

Magnitude (dB)

20

−20 −40

−40

−60

−60 −80

−80

−90

−90

Phase (deg)

Phase (deg)

−20

−135

−180 −1 10

0

1

10

10

−135

−180 −1 10

2

10

Frequency (rad/sec)

0

1

10

10

2

10

Frequency (rad/sec)

Controller (A2)

Controller (B2)

Figure 14: Bode Plot Comparison of Linearized System: Flow Controller in [7] vs. Controller based on Theorem 2

23

Acknowledgment The authors would like to thank helpful discussion of this research with Shivkumar Kalyanaraman, Ariel Loncarich, and Xingzhe Fan. We also thank Andy Teel for pointing to reference [27]. This research is supported in part by the RPI Office of Research through an Exploratory Seed Grant.

24

Appendix A

Uniqueness of Optimization Solution

Proposition A.1 If U : RN → RN satisfies U 00 (x) < −δI,

δ > 0,

(56)

and the routing matrix R is of full row rank, then the optimality condition given by (1), (3), (4) has a unique solution. Proof: Suppose that at the optimal solution, links αk , k = 1, . . . , K, are uncongested, i.e., yα∗ k < cαk , and the other links, βj , j = 1, . . . , L − K, are congested, i.e., yβ∗j = cβj . Define S ∈ RK×L and S˜ ∈ RL×(L−K) as: (

Sk,` = (

S˜`,j =

1 ` = αk 0 otherwise

k = 1, . . . K, ` = 1, . . . , L

1 ` = βj 0 otherwise

` = 1, . . . , L, j = 1, . . . L − K.

By construction, S S˜ = 0. From (4), prices for the uncongested links are zero, therefore, ˜ p∗ = Sη

(57)

where η contains the prices of the congested links. The congested links operate at capacity, so S˜T y = S˜T c. Substituting (1) for y and (3) for x, we obtain −1 ˜ = S˜T c. S˜T RU 0 (RT Sη)

(58)

From (56), U 00 is a negative definite matrix. Therefore, the gradient of the right hand side ˜ which is a negative definite matrix. By the inverse funcwith respect to η is S˜T RU 00 −1 RT S, tion theorem, there is at least one solution of η. Suppose that there are multiple solutions. Then, by the mean value theorem, S˜T RU 00 −1 RT S˜ must be singular which contradicts the strict concavity assumption on U . Therefore, the solution η is unique. It follows from (57), (1), and (3), that p∗ , q ∗ , x∗ , and y ∗ are also unique. Note that the full row rank condition on R is only sufficient; a unique equilibrium may exist even when the "condition is violated. For example, consider a single source and two-link # 1 network with R = . A unique optimum exists as long as the two links do not have 1 identical capacities.

25

B

Approximate Solution to the Optimization Problem by using Link Penalty Function

We first show that the approximate solution to the optimization problem is unique. Proposition A.2 If U : RN → RN satisfies (56) and h is monotonically non-decreasing, then (1), (3), and (8) has a unique solution. Proof: Combining (1), (3), and (8), the source rate x solves F (x) := U 0 (x) − RT h(Rx) = 0. Since U satisfies (56) and h is monotonically non-decreasing, F 0 (x) = U 00 (x) − RT h0 (Rx)R < δI. It follows from the inverse function theorem that F (x) = 0 has a unique solution. Note that the full row rank condition on R is no longer required since p is uniquely determined by x. We now address the issue of closeness of the approximate solution to the optimal solution when R is of full row rank. Assume that the penalty function h(y) has the following property: (

h` (y` )

=0 >0

if y` ≤ c` −  if y` > c` −  .

(59)

Suppose that at the optimal solution, links αk , k = 1, . . . , K, satisfy yα∗ k ≤ cαk − , and the ˜ where η contains other links, βj , j = 1, . . . , L − K, satisfy yβ∗j > cβj − . From (59), p = Sη the prices of the congested links. Using (1), (3), and (8), we have −1 ˜ = S˜T h−1 (Sη). ˜ S˜T RU 0 (RT Sη)

(60)

˜ > 0. Express h in terms of deviation from the capacity: Note that h is invertible since Sη ˆ − c + ) := h(y). h(y Then ˆ −1 (Sη) ˜ =h ˜ + c − . h−1 (Sη) Substituting back to (60), we have ˆ −1 (Sη) ˜ + S˜T RU 0 −1 (RT Sη) ˜ = S˜T (c − ). −S˜T h ˆ −1 → 0 as  → 0, then this equation becomes the condition for the optimal solution, (58), If h i.e., the approximate optimal solution converges to the optimal solution.

26

C

Uniqueness of Equilibrium under Positive Projection

Lemma 1 Consider the system x˙ = f (x)

(61)

where f is continuously differentiable and f (x) = 0 has a unique fixed point x∗ , with strictly positive entries x∗i > 0, i = 1, · · · , n. If, for all xi ≥ 0, ∂f ≤ 0, ∂x

(62)

x˙ = [f (x)]+ x.

(63)

then x∗ is also a unique equilibrium for

Proof: We note that x∗ is an equilibrium for (63), because (61) and (63) coincide when xi > 0, ∀i = 1, · · · , n. To show that it is unique we assume that there exists another equilibrium xˆ∗ , and proceed with a contradiction argument. Substituting f (x∗ ) = 0 in the mean value theorem ∗

∗

f (ˆ x ) − f (x ) =

Z 0

1



∂f (ˆ x∗ − x∗ )dζ. ∂x x=x∗ +ζ(ˆx∗ −x∗ )

(64)

Multiplying both sides from the left by (ˆ x∗ − x∗ )T , and using (62), we obtain (ˆ x∗ − x∗ )T f (ˆ x∗ ) ≤ 0.

(65)

x∗i ) < 0 and x∗i ) = 0, or fi (ˆ Because [f (ˆ x∗ )]+ x = 0 by assumption, we note that either fi (ˆ ∗ xˆi = 0. This means that ∗

∗ T

∗

(ˆ x − x ) f (ˆ x )=

n X

−x∗i fi (ˆ x∗i ) ,

(66)

i=1

where fi (ˆ x∗i ) is either zero or strictly negative. Because x∗i > 0, and at least one of the fi (ˆ x∗i )’s is strictly negative (if all were zero, xˆ∗ would have been another equilibrium for (61)), we conclude that (66) is strictly positive, thus contradicting (65).

27

References [1] F. Kelly, A. Maulloo, and D. Tan. Rate control in communication networks: shadow prices, proportional fairness and stability. Journal of the Operational Research Society, 49:237–252, 1998. [2] S.H. Low and D.E. Lapsley. Optimization flow control – I: basic algorithm and convergence. IEEE/ACM Transaction on Networking, 7(6):861–874, 1999. [3] R. Jain. Congestion control and traffic management in ATM networks: Recent advances and a survey. Computer Networks and ISDN Systems, 28(13):1723–1738, February 1995. [4] R. Srikant. Control of communication networks. In Tariq Samad, editor, Perspectives in Control Engineering: Technologies, Applications, New Directions, pages 462–488. IEEE Press, 2000. [5] S.H. Low, F. Paganini, and J.C. Doyle. Internet congestion control. IEEE Control Systems Magazine, 22(1):28–43, 2002. [6] S. Athuraliya, S.H. Low, V.H. Li, and Q. Yin. REM: active queue management. IEEE Network, 15(3):48–53, 2001. [7] F. Paganini. A global stability result in network flow control. Systems and Control Letters, 46:165–172, 2002. [8] S. Kunniyur and R. Srikant. A time-scale decomposition approach to adaptive ECN marking. IEEE Transactions on Automatic Control, 47(6):884–894, June 2002. [9] E. Altman, T. Ba¸sar, and R. Srikant. Robust rate control for ABR sources. In Proceedings of IEEE INFOCOM 1998, pages 166–173, San Francisco, CA, 1998. [10] C.V. Hollot and Y. Chait. Nonlinear stability analysis for a class of TCP/AQM networks. In Proceedings of the IEEE Conference on Decison and Control, pages 2309–2314, Orlando, FL, December 2001. [11] B.D.O. Anderson and S. Vongpanitlerd. Network Analysis and Synthesis – A Modern Systems Theory Approach. Prentice–Hall, New Jersey, 1973. [12] R.J. Benhabib, R.P. Iwens, and R.L. Jackson. Stability of distributed control for large flexible structures using positivity concepts. In AIAA Guidance and Control Conference, Paper No. 79–1780, Boulder, Co., August 1979. [13] G. Zames. On the input/output stability of time–varying nonlinear feedback systems, part I: Conditions derived using concepts of loop gain, conicity, and positivity, part II: Conditions involving circles in the frequency plane and sector nonlinearities. IEEE Transactions on Automatic Control, 11(3):465–476, 1966. [14] J.C. Willems. Dissipative dynamical systems, part I: General theory, part II: Linear systems with quadratic supply rate. Arch. Rational Mech. Anal., 45:321–393, 1972. 28

[15] P.V. Kokotovi´c and H.J. Sussmann. A positive real condition for global stabilization of nonlinear systems. Systems and Control Letters, 19:177–185, 1989. [16] R. Sepulchre, M. Jankovi´c, and P. Kokotovi´c. Constructive Nonlinear Control. Springer, New York, 1997. [17] H.K. Khalil. Nolinear Systems. Prentice-Hall, third edition, 2002. [18] A. J. van der Schaft. L2 -gain and Passivity Techniques in Nonlinear Control. SpringerVerlag, New York and Berlin, second edition, 2000. [19] F. Paganini, J.C. Doyle, and S.H. Low. Scalable laws for stable network congestion control. In Proceedings of 2001 Conference on Decision and Control, pages 185–190, Orlando, FL, December 2001. [20] J.L. Willems. Stability Theory of Dynamical Systems. Wiley, New-York, 1970. [21] G.F. Franklin, J.D. Powell, and A. Emami-Naeini. Feedback Control of Dynamic Systems. Addison-Wesley, third edition, 1994. [22] A.F. Filippov. Differential equations with discontinuous righthand sides. Kluwer Academic Publishers, Netherlands, 1988. [23] C.A. Desoer and M. Vidyasagar. Feedback Systems: Input–Output Properties. Academic Press, New York, 1975. [24] P. Ioannou and G. Tao. Frequency domain conditions for strictly positive real functions. IEEE Trans. on Automatic Control, 32(10):53–54, Oct 1987. [25] J.T. Wen. Time domain and frequency domain conditions for strict positive realness. IEEE Trans. on Automatic Control, 33(10):988–992, Oct 1988. [26] G. Zames and P. L. Falb. Stability conditions for systems with monotone and sloperestricted nonlinearities. SIAM J. Control, 6:89–109, 1968. [27] E.P. Ryan. An integral invariance principle for differential inclusions with applications in adaptive control. SIAM Journal on Control and Optimization, 36(3):960–980, 1998. [28] S. Kunniyur and R. Srikant. End-to-end congestion control: Utility functions, random losses and ecn marks. In Proceedings of INFOCOM 2000, Tel Aviv, Israel, March 2000.

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