Parameterizing PI Congestion Controllers Ahmad T. Al-Hammouri, Vincenzo Liberatore, Michael S. Branicky Stephen M. Phillips Department of Electrical Engineering and Computer Science Department of Electrical Engineering Case Western Reserve University Arizona State University Cleveland, Ohio 44106 USA Tempe, Arizona 85287 USA E-mail: { ata5, vl, mb } @case.edu E-mail: stephen.phillips@asu.edu URL: http://vincenzo.liberatore.org/NetBots/ that the previously proposed PIP controller [2] can be unstable Abstract — This paper describes a method for finding the stability regions of the PI and PIP controllers for TCP AQM. The in the presence of delays, even for the control parameters given method is applied on several representative examples, showing in the literature. that stable controllers can exhibit widely different performance. AQM is one of the most mature areas in network control, but Thus, the results highlight the importance of optimizing the previous work has neglected the investigation of the stability design of PI AQM. Furthermore, the paper shows that the region. The original RED controller has been analyzed in previously proposed PIP controller can be unstable in the presence of delays even for the control parameters given in the control-theoretical terms, and shown to be outperformed by PI literature. [3]. The PI controller is a natural choice due to its robustness and its ability to eliminate the steady-state error. The original I. I NTRODUCTION PI AQM gives a single pair of the proportional gain k p and the Control-theoretical methods lead to stable, effective, and ro- integral gain k i that guarantees the stability of the closed-loop bust congestion control, but the limits of control performance system as a function of the network parameters [3]. However, are still largely unknown. Congestion control regulates the rate there are other ( k p , k i ) pairs that stabilize the closed-loop at which traffic sources inject packets into a network to ensure system and result in better performance. The PIP controller high bandwidth utilization while avoiding network congestion. is a variant of PI [2]. Although PIP is stable in the absence of End-point congestion control can be helped by Active Queue time delays, we show in this paper that PIP becomes unstable Management (AQM), whereby intermediate routers mark or with time delays even in the exact scenarios considered by drop packets prior to the inception of congestion. AQM has previous work. been extensively addressed by control-theoretical methods (see This paper is organized as follows. In Section II, we for example [1] and the references therein). However, it is still introduce the linearized TCP-AQM model with PI and PIP unclear whether existing AQM controllers achieve “optimal” controllers, and we present the method we used to obtain performance. In particular, previous work lacks a complete the complete stabilizing region. In Section III, we compute characterization of the stability region, a definition of network- the complete set, S R , of stabilizing PI parameters. Simulations relevant control performance, and the design of provably that stress the importance of extracting a complete stabilizing optimal AQM controllers. region are presented in Section IV. Directions for future work are given in Section V, and conclusions in Section VI. A long-term goal in congestion control is to understand the limits of control performance. In this paper, we describe II. B ACKGROUND the stability region of the Proportional-Integral (PI) AQM A. Linearized TCP Model with the PI Controller controllers. The stability region describes the set of feasible A fluid-based linearized model for TCP congestion control, design points. Stable designs can be subsequently considered delays, and queues is expressed by the transfer function [4]: within an optimization framework. Therefore, the character- ization of a stability region is the first essential step toward B ( s + α )( s + β ) e − sd , P ( s ) = (1) the design of optimal AQM controllers. The derivation of a stability region is involved due to time delays that arise from where d is the round-trip delay (seconds), α = 2 N / ( d 2 C ) , the non-negligible network latencies between sources, sinks, β = 1 / d , B = C 2 / ( 2 N ) , C is the bottleneck link capacity and routers. The paper exploits recent results on PI control (packets/second), and N is the number of TCP flows traversing theory for time-delay systems to obtain the PI stability region. the link. The introduction of PI AQM results in the feedback We find that the controller performance varies significantly control shown in Fig. 1 [3], where q ( s ) is the Laplace trans- across the stability region and, in particular, there are stable form of the instantaneous queue length q ( t ) , q 0 is the desired controllers that have significantly better performance than queue length around which the controller should stabilize q ( t ) , previously proposed ones. Since stable PI controllers differed and widely in performance, the results support the importance of s = k p s + k i G ( s ; k p , k i ) = k p + k i finding optimal PI controllers. Furthermore, the paper shows s
q + q(s) 0 The stability region S R is the complete set of points ( k p , k i ) G(s) P(s) − for which the closed-loop system in Fig. 1 is stable for all delays L between 0 and d . The stability region S R can be expressed as S R = S 1 \ S L [6, p. 249], where • S 1 = S 0 \ S N . The closed-loop system of TCP-AQM linearized model P ( s ) , with Fig. 1. the PI controller, G ( s ) . • S 0 is the set of k p and k i values that stabilize the delay- free system P 0 ( s ) . • S N is the set of k p and k i values such that G ( s ; k p , k i ) P 0 ( s ) is the Laplace transform of the PI controller. The controller is an improper transfer function. Formally, S N is G ( s ; k p , k i ) will be denoted simply as G ( s ) when the pro- portional gain k p and the integral gain k i are clear from the context. � � S N = ( k p , k i ) : lim s → ∞ | G ( s ; k p , k i ) P 0 ( s ) | ≥ 1 . (2) B. The PIP Controller To enhance the speed of response of the PI controller, a position feedback compensation technique was proposed as an • S L is the set of ( k p , k i ) values such that G ( s ; k p , k i ) P ( s ) inner loop to the system in Fig. 1 [2]. The new arrangement, has a minimal destabilizing delay that is less than or equal which is called a PIP controller, is shown in Fig. 2. to d . Formally, S L is It can be shown with some mathematical manipulation that the characteristic equation of the closed-loop transfer function ∈ S N : ∃ L ∈ [ 0 , d ] , ω ∈ R s.t. with PIP control is equal to that with PI control, except that = { ( k p , k i ) / S L the proportional gain, k p , of the PI controller is increased by G ( j ω ; k p , k i ) P 0 ( j ω ) e − jL ω = − 1 } . (3) the value of k h . Therefore, the stability region in the ( k p , k i ) plane for the system with the PIP scheme is simply the same stability region as with a PI controller shifted to the left by To compute S R , fi rst defi ne the projection of the stability k h . Therefore, the stability analysis can be restricted, without region S R on the line k p = ˆ k p as: loss of generality, to the PI system in Fig. 1. The original PIP stability argument disregards the delay term e − sd . This simplifi cation is a signifi cant weakness be- k p = { ( k p , k i ) ∈ S R : k p = ˆ k p } , S R , ˆ cause delays in feedback loops are known to reduce stability margins drastically. The experiments in Section IV will con- fi rm this. so that the stability region can be calculated for each value of the proportional gain ˆ k p : C. Analysis of Time-Delay Systems Given a network topology with specifi c C and N , our goal is to determine all values of the ( k p , k i ) gains so that [ S R = k p . (4) S R , ˆ the feedback closed-loop system in Fig. 1 is stable for all ˆ k p values of delay less than d . Delays in the feedback loop are captured in (1) in the exponential term e − sd , which in turn greatly complicates the stability analysis beyond the To compute S R , ˆ k p , defi ne the projections traditional textbook techniques of Control Theory [5]. Previous work sidestepped the problem through assumptions and by constraining the PI gains [3]. An alternative approach is to { ( k p , k i ) ∈ S 1 : k p = ˆ = k p } , S 1 , ˆ exploit recent results on time-delay systems [6]. This section k p reviews one such recent method for time-delay PI control, and { ( k p , k i ) ∈ S N : k p = ˆ = k p } , S N , ˆ k p the rest of the paper will apply this method for the stability { ( k p , k i ) ∈ S L : k p = ˆ = k p } . S L , ˆ analysis of TCP AQM. k p q + q(s) + 0 k p = S 1 , ˆ k p \ S L , ˆ Then, S R , ˆ k p . It remains to compute S L , ˆ k p by G(s) P(s) − − evaluating the condition in (3) that G ( j ω ; k p , k i ) P 0 ( j ω ) e − jL ω = − 1. The set S L , ˆ k p can be further decomposed and computed k h as: Fig. 2. The PIP controller: an inner loop with constant gain is introduced = S + k p ∪ S − k p , S L , ˆ to provide position feedback compensation. k p L , ˆ L , ˆ
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