Proceedings of the 42nd IEEE Conference on Decision and Control Maui, Hawaii USA, December 2003
ThM03-5
Lp Stability and Delay Robustness of Network Flow Control 1 Xingzhe Fan2 , Murat Arcak, and John T. Wen Department of Electrical, Computer, and Systems Engineering Rensselaer Polytechnic Institute Troy, NY 12180 Emails:
[email protected],
[email protected],
[email protected] b source i to link j, and τij is the backward delay from the link j to source i.
Abstract This paper studies robustness of Kelly’s source and link control laws in [1] with respect to disturbances and time-delays. This problem is of practical importance because of unmodeled flows, and propagation and queueing delays, which are ubiquitous in networks. We first show Lp -stability, for p ∈ [1, ∞], with respect to additive disturbances. We pursue L∞ -stability within the input-to-state stability (ISS) framework of Sontag [2], which makes explicit the vanishing effect of initial conditions. Next, using this ISS property and a loop transformation, we prove that global asymptotic stability is preserved for sufficiently small time-delays in forward and return channels. For larger delays, we achieve global asymptotic stability by scaling down the control gains as in Paganini et al. [3].
Figure 1: Network Flow Control Model A common approach to the flow control problem is to decompose it into a static optimization problem and a dynamic stabilization problem. The static optimization problem computes the desired equilibrium condition by maximizing the sum of the source utility functions Ui (xi ), while complying with capacity constraints in the links; that is, Rx ≤ c, where c is a vector of link capacities cl , l = 1, · · · , L. The dynamic problem is to design the source rate update law based on the aggregate price, and the link price update law based on the aggregate rate, to guarantee stability of the equilibrium. Recently, this dynamic problem has been explored in several papers, including [1, 3, 5, 6], which present source and link control laws and provide stability proofs. However, robustness of these control laws against disturbances and time-delay, which is crucial for practical implementation, has received less attention.
1 Introduction Network flows are modelled as the interconnection of information sources and communication links through the routing matrices as shown in Fig. 1 ( [1,4,5]). Packets from each source i (with sending rate xi ) are routed through the links with the aggregate link rate y = Rf x, where Rf is the forward routing matrix. 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 the sources with the aggregate source price, q = RbT p, where Rb is the return routing matrix. If the delays are ignored, then Rf = Rb =: R since the links only feed back price information to the sources that utilize them. In the presence of delays, Rf (respectively, Rb ) is obtained f
from R by multiplying its entries with e−τij s (respecb f is the forward delay from the tively, e−τij s ), where τij
Robustness against time delay has mostly been studied for linearized models. In [3], the authors consider first-order link updates and static sources, and propose scalable controllers, which reduce the control gains to preserve local stability in the presence of large delays. In [7–9], a similar linearization-based analysis is pursued for first-order dynamic source- and linkcontrollers. The Lyapunov-Razumikhin theory is used
1 This research is supported in part by the RPI Office of Research through an Exploratory Seed Grant. 2 Corresponding author. Electrical, Computer and Systems Engineering Department, Rensselaer Polytechnic Institute, Troy, NY 12180-3590, USA. Tel.: +1-518-276-8205; fax: +1-518-2766261.
0-7803-7924-1/03/$17.00 ©2003 IEEE
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rium. Likewise, y˜ = y − y ∗ , p˜ = p − p∗ , and q˜ = q − q ∗ . Given a function f (z), its positive projection is defined as:
for nonlinear models in [10, 11]; however, the analysis is restricted to a single source and a single bottleneck, and the non-negativity constraint on the source rate is not taken into account. A global stability proof is given in [12] for the scalable controller in [3]. However, the analysis is, again, restricted to a single link. For multiple links, global convergence is proved for a discretetime algorithm in [4], using an optimization approach. As we further discuss in Remark 4 in Section 4, the study in [4] has common features with our results for static source update laws. It does not, however, discuss dynamic source laws. To the best of our knowledge, robustness with respect to disturbances (caused by e.g., modelling errors, unmodelled flows, and uncooperative users) has not been addressed in the literature.
+ ∆
(f ½ (z))z = f (z) if z > 0, or z = 0 and f (z) ≥ 0, 0 if z = 0 and f (z) < 0. +
If z and f (z) are vectors, then (f (z))z is interpreted in + the component-wise sense. When (f (z))z = 0, we say + that the projection is active. When (f (z))z = f (z), the projection is inactive. We denote by kzk the vector norm of z, and by kzkLp the Lp -norm of z(t), p ∈ [1, ∞]. For d ∈ L∞ , we define kdka = lim sup kd (t)k . t→∞
In this paper, we study robustness of Kelly’s primal and dual algorithms [1] against such disturbances and time-delays. The two algorithms provide natural generalizations to large-scale networks of simple additive increase/multiplicative decrease schemes, and was shown to be stable about a system optimum characterized by a proportional fairness criterion [1]. By using the Lyapunov functions motivated by the passivity analysis of network flow control in [13], we show that both primal and dual algorithms are robust to additive Lp disturbances. In particular, L∞ -stability, pursued here in the ISS framework [2], is instrumental for our proof of delay robustness. Using a loop transformation, we first represent the delayed model as a feedback interconnection of the nominal delay-free model, and a perturbation block, the ISS-gain of which depends on the amount of delay. Next, using the ISS Small-Gain Theorem [14], we prove global asymptotic stability (GAS) for sufficiently small delays. For larger delays, we achieve GAS by scaling down the controller gains as in [3]. All of these stability results are global, and are applicable to multiple sources and links.
A system z˙ = f (z, u) is said to be input-to-state stable (ISS) if there exist class-K functions 1 γ0 (·) and γ(·) n such that, for any input u(·) ∈ Lm ∞ and z0 ∈ R , the response z(t) from the initial state z(0) = z0 satisfies ¡ ¢ kzkL∞ ≤ γ0 (kz0 k) + γ kukL∞ kzka ≤ γ (kuka ) . Kelly’s primal algorithm consists of a first order source update law, and a static penalty function for the link to keep the aggregate rate below its capacity: x˙ = K(U 0 (x) − q)+ x,
p = h(y).
(1)
Here, K = diag {κi } is a diagonal matrix of the source controller gains κi > 0, i = 1, · · · , N , and U 0 (x) ∈ RN is a vector whose ith component is the derivative Ui0 (xi ) of the utility function Ui (xi ). Likewise, h(y) ∈ RL consists of the penalty functions h` (y` ), which are designed to enforce the link capacity constraints y` ≤ c` , ` = 1, · · · , L, i.e., to keep the aggregate rate yl below its capacity cl . The dual algorithm consists of a static source update and a first order dynamic price update:
This paper is organized as follows: Section 2 gives the notation and definitions used in the paper, and overviews Kelly’s primal and dual control laws. Section 3 proves their robustness to additive Lp disturbances. Section 4 proves GAS for sufficiently small delays, and, proceeds with the scaled redesign for large delays. Conclusions are given in Section 5. Due to the space restriction, proofs of all theorems are omitted. Please refer to our technical report [15].
x = U 0−1 (q),
p˙ = Γ(y − c)+ p,
(2)
where Γ = diag {γ` } is a diagonal matrix of link controller gains γ` > 0, ` = 1, · · · , L. From (2), the unique equilibrium for the dual control law is obtained from the equations qi∗ = Ui0 (x∗i ), i = 1, . . . , N ½ = 0 if y`∗ ≤ c` ∗ , ` = 1, . . . , L, p` ≥ 0 if y`∗ = c`
2 Preliminaries
(3) (4)
which, as shown in [1], correspond to the solution of the optimization problem
In this section, we introduce the notation and definitions used in the paper, and overview Kelly’s primal and dual control laws.
max xi ≥0
We let x denote the vector of sending rates xi of each source i, and x ˜ denote x − x∗ , where x∗ is the equilib-
N X i=1
Ui (xi ) subject to
Rx ≤ c, |{z} y
1 A function γ(·) is defined to be class-K if it is continuous, zero at zero, and strictly increasing.
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in which p` ’s play the role of Lagrange multipliers for the capacity constraints.
where = κ−1 η2 kRk β1 = √ (ζ kd2 k + kd1 k) 2κ κ = max{κi }, κ = min{κi },
α1
For the primal control law, the equilibrium obtained from qi∗
= Ui0 (x∗i ), i = 1, . . . , N
(5)
p∗`
= h` (y`∗ ), ` = 1, . . . , L,
(6)
i
i
(12) (13) (14)
and q and p are complementary indices, that is,
approximates the optimality condition (3)-(4) with the help of the penalty functions h` (y` ).
q−1 + p−1 = 1.
(15)
When p = ∞, the system is ISS with °√ ° √ k˜ x(t)k≤κ−1 ° x ˜(0)T K −1 x ˜(0)°e−α1 t + κα2
3 Robustness to Disturbances
1
In this section, we prove Lp and input-to-state stability of Kelly’s primal and dual algorithms with respect to additive disturbances. These disturbances could be from non-congestion-related flows such as flows that do not confirm to the optimization framework in the wireline network or noise on wireless links on the route. We start with Kelly’s primal algorithm, and assume that disturbances act on the inputs and outputs of the links, as depicted in Fig. 2. This assumption is not restrictive because we can use loop transformations to represent other disturbances as in Fig. 2. In Theorem 2 below, we use the Lyapunov function derived in [13], and obtain an explicit bound on k˜ xkLp and k˜ pkLp . When p = ∞, we prove that the system is ISS.
kβ1 kL∞ .
(16)
Remark 1: The main assumption in Theorem 1 is that the utility functions satisfy (7), which is an incremental sector condition on Ui 0 (·). However, as shown in the proof [15], this assumption can be relaxed as 2
(ξ − x∗i ) (Ui0 (ξ) − Ui0 (x∗i )) ≤ −η2 (ξ − x∗i ) , ∀ξ ≥ 0, (17) which only requires that U 0 i (·) be a sector nonlinearity about the base-point x∗ . We have used the more restrictive incremental sector condition (7) in Theorem 1, because it does not depend on x∗ , and is easier to verify. Finally, we wish to emphasize that, even when this assumption does not hold globally, regional stability results can be derived using the same proof technique as in Theorem 1. 2 We next study Kelly’s dual algorithm. To simplify the derivations in the next section, this time we assume that disturbances act on the inputs and outputs of the sources as in Figure 3. Again, other disturbances can be transformed to appear in this form.
Figure 2: Kelly’s primal flow control algorithm with disturbances d1 and d2 .
Theorem 2 Consider the feedback interconnection shown in Fig. 3:
Theorem 1 Consider the feedback interconnection shown in Fig. 2, where Ui ’s satisfy, for all xi ≥ 0, i = 1, · · · , N , Ui 00 (xi ) ≤ −η2 , (7)
q = R T p − d1 ,
−η1 ≤ Ui 00 (xi ) ≤ −η2 ,
for all yl ≥ 0, and ` = 1, . . . , L, (8)
p = h(y)
x = U 0−1 (q),
(9)
where K = diag {κ1 , . . . , κN }, guarantee °q ° ° 1 ° −p −1 T −1 ° k˜ xkLp ≤ κ (α1 p) ˜(0) K x ˜(0)° ° x ° √ −1 1 + 2κ (α1 q)− q kβ1 kLp (10) k˜ pkLp
≤ ζ kRk k˜ xkLp + ζ kd2 kLp
(19)
for some constants η1 > η2 > 0. If d1 and d2 are both Lp , p ∈ [1, ∞), then the source and link controllers
where ζ is a positive constant. If d1 and d2 are Lp , p ∈ [0, ∞), then the source and link controllers x˙ = K(U 0 (x) − q)+ x,
(18)
Assume that R is of full row rank and that the utility functions Ui (·) satisfy, for all xi ≥ 0, i = 1, · · · , N ,
and link penalty functions h` (·) satisfy ζ > h0` (yl ) ≥ 0
y = R(x + d2 ).
p˙ = Γ(y − c)+ p,
where Γ = diag {γ1 , · · · , γL }, guarantee: q 1 k˜ pkLp ≤ γ −1 (α2 p)− p p˜(0)T Γ−1 p˜(0) √ 1 + 2γ −1 (α2 q)− q kβ2 kLp k˜ xkLp ≤ η2−1 kd1 kLp + η2−1 kRk k˜ pkLp
(11)
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(20)
(21) (22)
where α2
=
β2
=
γ
=
(η1 γ)−1 σ(R)2 kRk p (η2−1 kd1 k + kd2 k) 2γ min{γ` }, `
γ = max{γ` } `
(23) (24) (25) Figure 4: Kelly’s Primal flow control algorithm in the
and q and p are complementary indices as in (15). When p = ∞, the system is ISS with √ q 2 −1 −α2 t T −1 p˜(0) Γ p˜(0)e + k˜ p (t)k ≤ γ kβ2 kL∞ . γα2 (26)
case of time-delay.
2 Figure 5: Equivalent system of Kelly’s primal flow control algorithm after loop-transformation.
From Theorem 1, it is not difficult to show that the ISS gain of the feedback path from d2 to p˜ is r 1 −1 2 α kRk ζ 2 + ζ. (29) g1 = κ3 1
Figure 3: Kelly’s dual flow control algorithm with disturbances d1 and d2 .
Remark 2: From the proof in [15], it is not difficult to show that the incremental sector assumption (19) can be relaxed as: ³ ´ −1 −1 2 (θ − qi∗ ) U 0 i (θ) − U 0 i (qi∗ ) ≤ −η1−1 (θ − qi∗ ) ,
In Theorem 3 below, we show that the feedforward path from p˜ to d2 has gain √ ¢ ¡ g2 = 2LN τ¯ kRk κ (30) ¯ η1 η2−1 κκ−1 + 1 ,
¯ ¯ ¯ 0 −1 ¯ −1 ¯U i (θ + δ) − U 0 i (θ)¯ ≤ η2−1 |δ|
τ¯ := max{τij },
where
(27) (28)
ij
(31)
and prove global asymptotic stability from the smallgain condition g1 g2 < 1. (32)
for all θ ≥ 0 and all δ ∈ R.
Because g2 is smaller when τ¯ gets smaller, we conclude that the small-gain condition (32) is satisfied for small delays and, thus, global asymptotic stability is preserved:
4 Robustness to Time Delay In this section, we analyze robustness of Kelly’s primal and dual control laws to time-delay. This robustness property is crucial because of queuing and propagation delays in networks. We first study Kelly’s primal algorithm. To simplify the derivations, we combine the feedforward and feedback delays in one block in the feedforward path in Figure 4.
Theorem 3 Consider the feedback interconnection in Figure 4, and suppose that the penalty functions hi (·), i = 1, · · · , L, are such that (8) is satisfied with ζ > 0, and the utility functions Ui (xi ), i = 1, · · · , N are such that (19) is satisfied with η1 > η2 > 0. If the delay τ¯ in (31) is small enough that (32) is satisfied, then the source and link controllers (9) guarantee global asymptotic stability.
To transform the delay robustness problem to the framework of Section III, we add and subtract the term R to the feedforward routing matrix R (e−sτli ) in Figure 4 and represent it as in Figure 5 below, where the inner loop represents the nominal system without delay, and the outer loop is the perturbation due to delay.
Remark 3: If all controller gains ki , i = 1, · · · , N are same, condition (32) will become k¯ τ < G,
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(33)
where G is a constant which depends upon global parameters. Note that similar results could be found in [5], where these conditions are derived based on linearization, or in [5], where the condition is just for single source and single link.
that the utility functions Ui (xi ), i = 1, · · · , N are such that (19) is satisfied for with η1 > η2 > 0. If
Remark 3: Theorem 3 relies on the small-gain condition (32), which holds if the delay τ¯ is sufficiently small. For larger delays that violate (32), we can scale the controller gain K by λ > 0, and rewrite (32) as ¶ µ √ κ ¯ ζkRk2 √ 3 κ(η1 κ−1 κ ¯ η2−1 +1)kRk
