Control Methods for Data Flow in Communication Networks

DISSERTATION

Presented in Partial Fulﬁllment of the Requirements for the Degree Doctor of Philosophy in the

Graduate School of The Ohio State University

Peng Yan, M.S.

*****

The Ohio State University

2003

Dissertation Committee: Approved by

Professor Hitay Ozbay¨ , Adviser Professor Kevin M. Passino Adviser ¨ ¨ Professor Umit Ozg¨uner Department of Electrical Engineering c Copyright by

Peng Yan

2003 ABSTRACT

In this dissertation, we investigate various control methods for data ﬂow in communica-

tion networks. First, we develop a rate-based ﬂow controller for self-similar network trafﬁc, ½ which consists of a robust À control block and an adaptive LMMSE capacity predictor.

The controller guarantees robust stability against time-varying time delay uncertainties and improves the transient response by predicting the self-similar cross traffic. Window-based congestion control methods are also explored for TCP traffic on IP networks. We propose a variable structure approach in Active Queue Management (AQM) support Explicit Con- gestion Notification (ECN). By analyzing the robustness and performance of the control scheme for the nonlinear TCP/AQM model, we show that the proposed design has good performance and robustness with respect to the uncertainties of the round-trip time (RTT)

and the number of active TCP sessions, which are central to the notion of AQM. Alterna- ½

tively, we design robust À AQM controllers for the linearized TCP/AQM model, with ½

the presence of uncertain time delays. The À performance is analyzed and a switching

control scheme is introduced to improve the system performance. Motivated by the Lin- ½ ear Parameter Varying (LPV) nature of the linearized TCP/AQM model, a switching À

control method is further investigated for LPV systems where we provide some stability

conditions in terms of the dwell time and the average dwell time.

ii To my wife, Yang Sun

iii ACKNOWLEDGMENTS

I would like to thank my adviser, Professor Hitay Ozbay,¨ for his guidance, support, and encouragement throughout the course of my research at The Ohio State University. I also thank Professor Kevin Passino and Professor Umit¨ Ozg¨¨ uner for serving on my committee and for their review of this thesis.

I would like to thank Dr. Yuan Gao for his collaboration on the topic of the Variable

Structure AQM in Chapter 3 and for providing me with the ns-2 code in the packet-level simulations of Chapter 3. I also thank Dr. Pierre-Franc¸ois Quet for various technical discussions.

A special thanks goes to the entire Control group, professors and students, for creating a wonderful academic environment that encourages learning and pursuit of research. I would like to express my gratitude to all my friends for their help and encouragement.

Last but not least, I wish to specially thank my family for their love, support and encouragement throughout my studies.

Finally I would like to acknowledge that the ﬁnancial support for this work came form

NSF grants Nos. ANI-9806660, ANI-0073725, SBC/Ameritech Faculty Research Grant, and Air Force Research Laboratory under agreement No. F33615-01-2-3154.

iv VITA

January 21, 1975 ...... Born - P. R. China

1997 ...... B.S. Southeast University, Nanjing, P. R. China 1999 ...... M.S. Southeast University, Nanjing, P. R. China 1999-present ...... Graduate Research Associate, The Ohio State University.

PUBLICATIONS

Research Publications

Peng Yan, Yuan Gao, and Hitay Ozbay¨ Variable Structure Control in Active Queue Man- agement for TCP with ECN. Proceedings of the 8th IEEE Symposium on Computers and Communications, Antalya, TURKEY, July 2003.

¨ ½ Peng Yan and Hitay Ozbay À Performance Analysis of Robust Controllers Designed for AQM. Proceedings of the American Control Conference, Denver, Colorado, USA, June 2003.

¨ ½ Peng Yan and Hitay Ozbay On the À -Based Controllers for Automatic Steering of Vehi- cles with Actuator Delays. Proceedings of the 6th ASME Biennial Conference on Engieer- ing Systems Design and Analysis, Istanbul, Turkey, July 2002.

Peng Yan and Hitay Ozbay¨ Flow Controller Design and Performance Analysis for Self- Similar Network Trafﬁc. Proceedings of the 39th Annual Allerton Conference on Commu- nication, Control, and Computing, Monticello, Illinois, USA, October 2001.

v FIELDS OF STUDY

Major Field: Electrical Engineering

vi TABLE OF CONTENTS

Page

Abstract ...... ii

Dedication ...... iii

Acknowledgments ...... iv

Vita...... v

List of Figures ...... x

List of Tables ...... xiii

Chapters:

1. Introduction ...... 1

1.1 Congestion Control in Communication Networks ...... 2 1.1.1 Rate-Based Control Methods for ATM Traffic ...... 3 1.1.2 Window-Based Congestion Control Methods for TCP Traffic . . 4 1.1.3 Active Queue Management ...... 5 1.1.4 Stability of Distributed Congestion Control ...... 7 1.2 Problem Definition and Motivation ...... 8 1.3 Contributions of the Dissertation ...... 9 1.4 Structure of the Dissertation ...... 11

2. Flow Controller Design and Performance Analysis for Self-Similar Network Trafﬁc...... 12

2.1 Overview ...... 12 2.2 Preliminary deﬁnitions ...... 14 2.3 Mathematical Model ...... 15

vii 2.4 Short-Term Prediction of Self-Similar Trafﬁc ...... 19 2.5 Simulation Results of the LMMSE Predictor ...... 20 2.6 System Level Simulations ...... 25 2.7 Concluding Remarks ...... 29

3. A Variable Structure Control Approach to Active Queue Management for TCP with ECN ...... 30

3.1 Overview ...... 30 3.2 Variable Structure Control in AQM ...... 32 3.2.1 Nonlinear TCP dynamics ...... 32 3.2.2 VS based AQM with ECN ...... 34 3.3 Robustness and Performance Analysis ...... 36 3.4 Related Work ...... 42 3.5 Packet-Level Simulations ...... 47 3.5.1 Simulation Conﬁguration ...... 47 3.5.2 The Scenario of Single Bottleneck Topology ...... 48 3.5.3 The Scenario of Multiple Bottleneck Topology ...... 55

3.6 Concluding Remarks ...... 58 ½

4. Robust Controller Design for AQM and À -Performance Analysis ...... 61

4.1 Introduction ...... 61

4.2 Mathematical Model of TCP/AQM ...... 62 ½

4.3 À Controller Design for AQM ...... 64

4.4 Multiplicative Uncertainty Bound ...... 66 ½

4.5 À -Performance Analysis ...... 69 4.6 Simulations ...... 72 4.6.1 The Case of a Single Controller ...... 73 4.6.2 The Case of Switching Control ...... 75

4.7 Conclusions ...... 76 ½

5. On Switching À Controllers for a Class of LPV Systems ...... 79

5.1 Introduction ...... 80 5.2 Problem Deﬁnition ...... 81 5.3 Main Results ...... 84 5.4 Numerical Example ...... 92 5.5 Concluding Remarks ...... 99

viii 6. Conclusions ...... 100

6.1 Summary of Results ...... 100 6.2 Future Work ...... 102

Bibliography ...... 104

ix LIST OF FIGURES

Figure Page

1.1 The ﬂow control model ...... 4

1.2 Data ﬂow in communication networks ...... 8

2.1 LMMSE based feedback control system ...... 16

2.2 One-step prediction for different values of ...... 21

2.3 Error for different prediction steps ...... 22

2.4 Prediction Error and relative error variance in one-step prediction ...... 23

2.5 Discretized model ...... 25

¼ 2.6 System responses for ¼ time scale: the middle one denotes the comparison controller and the bottom one denotes the LMMSE based scheme . 26

2.7 Performance indexes as a function of ...... 28 2.8 System response for ½ time scale: The middle one denotes the comparison controller and the bottom one denotes the LMMSE one ...... 29

3.1 Aggregated dynamics of TCP and VS based AQM ...... 34

3.2 System responses using the VS controller ...... 42

3.3 Phase portrait of the closed loop system ...... 43

3.4 Dumbbell network topology for ns simulations ...... 48

x 3.5 The network topology with multiple bottleneck links ...... 49

3.6 Instantaneous queue size using VS control ...... 50

3.7 System responses for DropTail, RED, PI and REM ...... 51

3.8 Queue evolution using RED, PI, REM and VS control ...... 52

3.9 Average queue length w.r.t. the number of TCP ﬂows ...... 53

3.10 Link utilization w.r.t. the number of TCP ﬂows ...... 54

3.11 Packet loss ratio w.r.t. the number of TCP ﬂows ...... 55

3.12 Comparison of PI and the VS controller ...... 56

3.13 Performance in the presence of short-lived TCP ﬂows ...... 57

3.14 Queue evolution using RED, PI, REM and VS control in the scenario with a much smaller propagation delay ...... 58

3.15 Queue evolution using RED, PI, REM and VS control in the scenario of a larger propagation delay ...... 59

3.16 Evolution of Queue ¿ (in packet) using RED, PI, REM and VS control . . . 60

3.17 Evolution of Queue (in packet) using RED, PI, REM and VS control . . . 60

¢ ¢ ¢ ¾

4.1 Partition of by ½ and ...... 69

4.2 À performance with respect to ...... 71

4.3 Performance cost w.r.t. and ...... 72

4.4 Performance cost w.r.t. and ...... 73

4.5 Performance cost w.r.t. and ...... 74

½ ¾ ¼

4.6 System responses of ¼ , and at ...... 75

¡ ¼¿

½ ¼ 4.7 System responses of ¼ and at ...... 76

·¡ ¼

¾ ¼

4.8 System responses of ¼ and at ...... 77

4.9 A single controller ¼ ...... 77

¾ 4.10 Switching control between ½ and ...... 78

5.1 The switching control system ...... 82

5.2 LPV plant and the controller ...... 83

5.3 Switching logic ...... 89

5.4 Block diagram ...... 93

¾ 5.5 Weights for uncertainties with respect to ½ and ...... 95

5.6 Robustness test ...... 96 ½

5.7 The case of a single À controller ...... 97 ½

5.8 The switching À control method ...... 98

xii LIST OF TABLES

Table Page 2.1 One-step prediction at a larger time scale (½ )...... 24

2.2 Comparison of performance index ...... 28

5.1 Parameters , , and ...... 95 É

xiii CHAPTER 1

Introduction

Communication networks have seen explosive growth in their size, diversity and application domains in the past decades. Particularly, the Internet becomes a global network connecting tens of millions of hosts and users with different transmission links (optical

fiber, satellite link, etc.), different traffic sources (data, voice, video etc.), and different nodes (e.g. endhosts, switches, routers). The huge scale and the heterogeneousness of today’s networks give rise to complexity and difficulty in network management and control.

Quite different from the telephone network which uses circuit-switching and can provide a guaranteed bandwidth for each connection, the modern communication networks use packet-switching and the available bandwidth is competed by statistical multiplexing [76].

Correspondingly, the service provided to sources in high-speed communication networks is called best-effort service which provides unreliable Quality-of-Service (QoS). In fact, the best-effort trafﬁc in ATM (Asynchronous Transfer Mode) networks may be guaranteed a minimum data rate which is known as Available Bit Rate (ABR) service [2], while in the

Internet, there are no guarantees. Many network applications such as FTP (File Transfer

Protocol) and HTTP (Hypertext Transfer Protocol) require reliable end-to-end data transmission and others (e.g. Voice over IP and on-line multimedia services) require guaranteed

QoS. Thus how to regulate the data ﬂow to avoid congestions at bottleneck nodes, thereby

1 achieving better network performance has been a challenging problem over the last twenty

years [49].

1.1 Congestion Control in Communication Networks

The transmission of data ﬂow in communication networks is regulated by network protocols such as TCP (Transmission Control Protocol) and UDP (User Datagram Protocol).

For the purpose of routing and reliable data transmission, the network architecture is capable of fragmenting data into “packets” which can be of ﬁxed (the 53-byte cell in ATM networks) or variable size (the TCP packet) depending on the network. At the network layer, the delivery of data packets is governed by some routing mechanisms such as the datagram routing method and the virtual circuit routing method, with the objective of max- imizing the throughput and minimizing the packet delay. On top of the network layer, the transport layer is introduced where transmission protocols are performed to deal with congestion avoidance, packet retransmission and buffer management.

Traffic congestion on today’s high speed networks is one of the major communication problems experienced by millions of users. It occurs at some transmission links when the packet arrival rate exceeds the packet outgoing rate (available link capacity). The accu- mulation of packets in the congested buffers give rise to queuing delay, packet dropping, retransmission, and even traffic collapse. Since the landmark work of Jacobson [35] in the late 1980s’, significant research efforts have been made on effective control mechanisms for the data flow at the level of the transport layer, in the hope of congestion avoidance, better utilization of bandwidth, and better fairness [49, 76]. Congestion control methods can be classified as window-based algorithms and rate-based algorithms. The widely deployed

TCP protocol is the most important example of the window-based approach [78], where the

2 sending rate of each endhost is governed by a sliding window with adjustable window size

corresponding to the congestion indication such as packet loss or packet marking. In contrast, the ATM-ABR trafﬁc represents the rate-based approach, where a control cell is generated by the sender periodically so that the switches can explicitly suggest the sending rate and feed it back to the ABR sources [2].

1.1.1 Rate-Based Control Methods for ATM Trafﬁc

From the viewpoint of control systems, the data ﬂow control problem for ATM-ABR

trafﬁc can be modeled as the feedback system depicted in Figure 1.1, where source nodes are feeding the bottleneck node with a ﬂow controller assigning rates for each source node.

A fundamental aspect of this ﬂow control problem is the presence of time delays, which consist of forward delays and backward delays. This problem has been studied widely in the framework of feedback control theory (see e.g., [4, 51, 58, 59, 66, 73, 93] and references therein). Under the assumption of known and constant time delays, [51] provided a Smith-predictor based ﬂow controller and validated its implementation in ATM congestion control. A similar model was considered from the perspective of game theory in [73],

where individual source utility was optimized based on explicit rate feedback. A more ½

recent result of Quet et al in [66] designed À controllers for ATM ﬂow control using ½ inﬁnite dimensional À optimization, which allows for time varying and uncertain time

delays. Another challenging aspect in designing ﬂow controllers for ATM-ABR trafﬁc is

the existence of cross trafﬁc, which results in the complex stochastic behavior of the avail-

´µ

able capacity ( ÓÙØ in Figure 1.1.(b)); [65] extended the result of [66] with an additional control block using the estimation of the available capacity.

3 Sources Time Delays

1 r 1 c (t) Bottleneck Node out r2 2 Cout queue Controller integrator r n r (t) rb (t) − q(t) q + s RTTs + d K τ n − Bottleneck Node

assigned rates Flow Controller desired queue

(a) Flow Control Problem (b) Feedback Control Model

Figure 1.1: The ﬂow control model

1.1.2 Window-Based Congestion Control Methods for TCP Trafﬁc

Contrary to the data ﬂow control problem of ATM-ABR trafﬁc, congestion control of

TCP traffic is end-to-end and stateless [20, 78]. In fact, the routers of IP networks are not capable of recording per-flow state hence can’t admit a central flow controller assigning explicit rate for each endhost. Meanwhile, the signaling of congestion for TCP traffic is based on the receiving of ACK (Acknowledgment) or NACK (Negative Acknowledgment) generated by the destinations (not the routers). In essence, congestion control for TCP traf-

fic consists of distributed end-host control algorithms and queue management algorithms at routers. This separation of TCP congestion control increases the scalability of the Internet, while adding more difficulties to modeling and control of the TCP traffic. The Tahoe version and the Reno version of TCP are the predominate TCP implementations. The control scheme of these protocols has the AIMD (Additive Increase and Multiplicative Decrease) behavior when the sources enter the congestion avoidance phase, where the sliding window increases its size by one per RTT (round trip time-delay) while cutting its size by half on

detecting a packet loss. Besides the control schemes at end-hosts, the routers have FIFO

4 (First-In-First-Out) buffers managed by DropTail policy, which drops an arrival to a full

buffer. The current TCP is called implicit feedback control because the sliding window

size of each end-host is adjusted based on ACKs or NACKs implicitly related to the packet

loss at routers.

The information of congestion in the current TCP deployment is typically generated

through packet loss. An improved TCP protocol, TCP Vegas, was proposed in [12] us-

ing queuing delay as congestion indication, which has been shown to be more accurate

and more responsive in congestion detection. Alternatively, an Explicit Congestion No-

tiﬁcation (ECN) mechanism was introduced in [67] which enables congestion indication

using packet marking instead of packet loss. Various implementations of TCP control have

helped alleviate severe congestions observed earlier in the Internet [35]. But further analy-

sis shows that many problems still remain:

(1) Fairness: The long TCP ﬂows (elephants) and short TCP ﬂows (mice) can not share

the bandwidth with equal opportunity; non-TCP ﬂows such as UDP may grab all the

available bandwidth when competing with TCP trafﬁc.

(2) Oscillations: The packet-loss based congestion signaling tends to keep the buffer oc-

cupancy high and oscillating.

(3) Global Synchronization: The bursty TCP trafﬁc, together with the heavily occupied

buffer makes a loss event at routers involving many packet loss, thereby leading to

synchronization among TCP ﬂows sharing a congested link [74].

1.1.3 Active Queue Management

The earliest efforts on TCP trafﬁc analysis and control are based on heuristic methods and empirical study. In the last few years, however, analytical models have been brought

5 into TCP congestion control, among which the feedback control based models are receiv-

ing wide-spread investigations. At the same time, the AQM (Active Queue Management)

technology was proposed to improve the Internet performance [20, 22, 23]. From control

systems point of view, the AQM algorithms are indeed feedback controllers implemented

at routers for buffer management and TCP trafﬁc regulation.

A benchmark result in AQM is the Random Early Detection (RED) scheme [24], which

has been shown to prevent trafﬁc synchronization, remove higher loss bias against bursty

traffic and reduce RTT [20,53]. Unfortunately, the performance of RED is very sensitive to link’s traffic load and its parameter setting [13]. The drawbacks of RED prompted research on its modifications and alternatives, such as BLUE [21], flow RED [47] (FRED), balanced

RED [5] (BRED) and stabilized RED [57] (SRED). Although the above control schemes outperform RED in the aspects of fairness and throughput, they are essentially based on heuristics without systematic analysis.

Notably, more and more researchers consider the congestion control problem from the perspective of complex systems and use classical feedback control approaches to analyze and control the TCP data flow [8, 10, 34, 41, 49, 50, 62, 64]. We refer to [39, 50, 54] for the details of the nonlinear dynamical models for TCP/AQM. It has been shown in [54] that a fluid-based deterministic model can be used to capture the main dynamics of TCP traffic. In [32], a control theoretic analysis was given for RED based on the linearized

model provided in [54]. Later, [33] developed a PI controller for AQM using linear system ½ analysis, and [64] investigated similar problem using À optimization which allows for the uncertainty of RTT and other parameter uncertainties. Alternatively, the congestion control system can be viewed as a convex optimization problem for a certain aggregate utility function, where TCP/AQM can be interpreted as carrying out a gradient algorithm to

6 maximize aggregate source utility [40,48,49]. Some representative control schemes in this

category are REM (Random Exponential Marking) [8], AVQ (Adaptive Virtual Queue) [41]

and the scalable control scheme in [50]. Note that the control mechanism proposed in [50]

comprises not only a dynamic AQM at the router side, but a static endhost control law as

well. Furthermore, it has been shown in [62] that a dynamic lead-lag compensator can be

introduced to replace the static endhost source control for better performance.

1.1.4 Stability of Distributed Congestion Control

Besides the surge of interests in designing congestion controllers for best-effort service in high speed networks, significant research efforts have been made to study the stability of the global networks in the presence of distributed control mechanisms and heterogeneous time delays [10, 16, 37, 40, 42, 52, 61, 84]. In [10], the transient and steady state behaviors of TCP flows were investigated in the framework of hybrid systems, while [61] considered the deployment of REM for TCP congestion control and demonstrated the global asymptotic stability of the equilibrium for the delay free systems. A more general objective of congestion control is to maximize the sums of utilities based on the network usage. In [38], a local stability condition was provided for a single congestion controller with delayed feedback. A related approach has been developed in [16] where global asymptotic stability and semiglobal exponential stability are established. We should also mention that [37] investigated the local stability of the feedback system with distributed time delays under the assumption that these delays are common to all users within classes while they can be different from class to class. Massoulié [52] further extended the results of [37] by exploring the stability conditions for congestion control in the presence of heterogeneous feedback delays.

7 shared link of capacity co

ingress cout network egress edge router cloud edge router

cross traffic cross traffic

Figure 1.2: Data ﬂow in communication networks

1.2 Problem Deﬁnition and Motivation

Given the complexity of the network topology and the heterogeneousness of the network components, a system level description of the data ﬂow control problem is challenging. From the viewpoint of dynamical systems, we are particularly interested in the congestion control of the network model depicted in Figure 1.2, where we have a number of endhost users sharing the congested bottleneck link with the disturbance of cross trafﬁc.

We address and develop control solutions for window-based and rate-based congestion control problems in the level of the transport layer. Note that the rate-based control algorithms for ATM networks operate only on long-lived aggregate ﬂows deployed in the core carriers of the Internet, instead of the end-to-end TCP trafﬁc. Therefore, the explicit rate-based control scheme in Chapter 2 is discussed in the framework of private edge networks [28]

(e.g. the part in Figure 1.2 with dotted lines). On the other hand, the window-based control schemes in Chapter 3 and 4 are considered in the framework of active queue management for TCP data ﬂow.

Earliest efforts in congestion control are focused on congestion avoidance and TCP enhancement. The control theoretic approach offers a new look into this problem for the

8 purpose of eliminating buffer overﬂow, reducing packet loss and increasing bandwidth uti-

lization. Some fundamental aspects of the network congestion control problem are:

(1) The network trafﬁc has the bursty behavior on a wide range of time scales [14, 46],

which reﬂects the self-similarity or Long-Range-Dependence (LRD) of the data traf-

ﬁc. The traditional approaches, such as Poisson process, ARMA model or Markovian

representation, can not fully characterize the stochastic behaviors of the self-similar

trafﬁc.

(2) The ﬂuid based dynamical models addressing TCP/AQM are nonlinear with time de-

lays. To further complicate the situation, these models have parameter uncertainties

such as RTT, number of connections, and available bandwidth.

(3) The dynamics of TCP/AQM exhibit a parameter varying nature, where the operating

range of the parameters could be too large for a single controller.

1.3 Contributions of the Dissertation

We investigate in this thesis various control methods for data ﬂow in communication networks. Speciﬁcally, the major contributions include the following four parts:

Rate-based ﬂow controller for self-similar network trafﬁc Recent studies of high-

resolution trafﬁc measurement discovered the self-similarity in both LAN and WAN trafﬁc.

We introduce a two-degree of freedom rate based ﬂow controller, which includes a robust ½

À control block and an LMMSE based adaptive capacity predictor. The former part de-

signed in [66] can guarantee the robust stability against time-varying time delay uncertain-

ties and the latter improves the transient response by predicting the self-similar cross-trafﬁc.

9 The LMMSE predictor is validated by data traces generated from packet-level simulations.

Performance analysis is provided for the closed loop system at different time scales.

Variable structure control in AQM for TCP trafﬁc supporting ECN We develop a variable structure based control scheme in Active Queue Management (AQM) supporting explicit congestion notiﬁcation (ECN). By analyzing the robustness and performance of the control scheme for the nonlinear TCP/AQM model, we show that the proposed design has good performance and robustness with respect to the uncertainties of the round-trip time (RTT) and the number of active TCP sessions, which are central to the notion of

AQM. Implementation issues are discussed and ns packet-level simulations are provided to validate the design and compare its performance to other peer schemes’ in different scenarios. The results show that the proposed design signiﬁcantly outperforms the peer

AQM schemes in terms of packet loss ratio, throughput and buffer ﬂuctuation. ½

Robust controller for AQM and À -performance analysis Linearization of the non-

linear TCP/AQM dynamics results in Linear Parameter Varying (LPV) systems with time ½

delays. Earlier in [64] robust AQM controllers were designed using the À optimization ½ method for inﬁnite dimensional systems. Here we investigate the À -performance with respect to the uncertainty bound of RTT, and further introduce controller switching to im-

prove the system performance, which is validated by simulations. ½

Switching À controllers for LPV systems with possible applications to AQM de-

sign Motivated by the LPV nature of the linearized TCP/AQM model, we propose a switch- ½ ing À control method for the purpose of better system performance and larger robustness with respect to the scheduling parameter (e.g. RTT for TCP/AQM dynamics). A funda-

mental aspect of switching control is the stability in the presence of switching sequences. ½

To this end, we consider the stability conditions on switching À controllers for a class

10 of LPV systems, where the candidate controllers are selected from a given controller set

according to the switching rules based on the scheduling variable. Sufﬁcient conditions are

provided to guarantee the stability of the switching LPV systems in terms of the dwell time

and the average dwell time.

1.4 Structure of the Dissertation

The structure of this thesis is as follows. In Chapter 2, we develop an explicit rate- based ﬂow controller for communication networks exhibiting self-similarity, which is based on the work published in [90]. The nonlinear TCP/AQM dynamics are investigated in

Chapter 3, where we present a variable structure AQM control scheme for TCP trafﬁc

supporting ECN. The results of Chapter can be found in [89] and [88]. In Chapter 4, robust ½ controllers for AQM are developed and À -performance analysis are provided, which motivate us to perform controller switching for better system performance [91]. The LPV nature of the linearized TCP/AQM model prompts the research in Chapter 5 on switching

control methods for possible applications in AQM, where stability analysis is provided for ½ switching À controllers with respect to LPV systems. Chapter 5 is based on the results published in [92]. This dissertation concludes in Chapter 6 with a summary of research contributions as well as further discussions on future work.

11 CHAPTER 2

Flow Controller Design and Performance Analysis for Self-Similar Network Trafﬁc

Recent studies of high-resolution trafﬁc measurement discovered the self-similarity in

both LAN and WAN traffic. In this chapter, we introduce a two-degree of freedom rate ½ based flow controller, which includes a robust À control block and an LMMSE based adaptive capacity predictor. The former part can guarantee the robust stability against time- varying time delay uncertainties and the latter improves the transient response by predicting the self-similar cross-traffic. Implementation issues are discussed and performance analysis is provided to validate our design. We also investigate the prediction and control in larger time scale which is more applicable for the real network environment.

2.1 Overview

Congestion control for trafﬁc management is one of the main concerns in the design of modern high speed communication networks. For best-effort services, ﬂow control mechanisms are implemented to ensure a good Quality of Service (QoS) by means of regulating the data rate generated by the sources [11] [36].

In ATM ABR (Available Bit Rate) services, the rate-based ﬂow control has been chosen by the ATM forum [2], which allows for the intermediate nodes to adjust a data rate to

12 ABR sources based on network feedback; this problem can be viewed as a feedback control

problem where the queue length is used as explicit feedback [3] [4] [51]. In [4], for exam-

ple, the problem was formulated as an LQG stochastic control problem with action delays,

and the information of available rate as well as the queue length was used to compute the

sending rate for each ABR source. In [3], however, the authors studied the problem in a

game-theoretic context where each user minimized its own performance objective.

A challenging aspect of ﬂow control is the presence of round trip time delays (RTT) which consist of forward delays and backward delays. The problem is further complicated

due to the fact that these time delays are usually time-varying and uncertain because of ½ data buffering at the bottleneck nodes. For such time delay systems, the À based robust control techniques were discussed in [58] [59] [66], where the robustness against delay

uncertainties, weighted fairness, as well as tracking performance were considered in the ½ framework of À optimization for inﬁnite dimensional systems. Internally robust implementation of this controller was discussed in [59], and the case of multiple time-delays at

different channels was discussed in [66]. ½

In a more recent result [65], the authors investigated a two-degree of freedom À ﬂow controller design, which used capacity information as well as queue length as feedback.

This controller, as demonstrated by the simulations of [65], had faster transient response

with respect to time-varying available bandwidth. The available bandwidth was assumed Ä

to be generated by applying an unknown ¾ signal to a known low-pass ﬁlter. Thus they ½ designed an À based capacity predictor, because it was the most natural for this kind of disturbance.

However, the available bandwidth is actually a random variable determined by the network cross trafﬁc. Recent statistical analysis of high-resolution trafﬁc measurement have

revealed the self-similar behaviors of network trafﬁc [69] [86]. Thus the available capacity

´µ ´µ Ö Ó××

¼ also exhibits LRD (Long-Range-Dependence), which is not in the form

´µ Ö Ó×× considered in [65]. Here ¼ is the full link capacity and is the capacity taken out by the cross trafﬁc. In this work, we assume the LRD cross trafﬁc and design an LMMSE

(Linear Minimum Mean Square Error) predictor for available bandwidth estimation. It is ½ used with the À controller designed for robustness against the time-varying time delay

uncertainty [66].

We also consider a more realistic case where control and prediction are in a larger time

scale. In fact, for ATM networks, once every cells the sources send a control cell which can be used by switches to convey feedback. And for TCP/IP networks, each source updates its sending window size once every RTT when an ACK is received [35]. For this scenario, implementation issues are discussed, and the performance is analyzed in the sense of the multifractal nature of cross trafﬁc as well as the robustness of our controller. All the above results are demonstrated by MATLAB simulations, where the cross trafﬁc comes from the data traces generated by packet-level simulations using ns-2.

2.2 Preliminary deﬁnitions

The measurements of LAN or WAN trafﬁc have revealed the presence of self-similarity

or fractal behavior. The most broadly studied fractal model is the fractal Brownian motion

´ µ ´´ · ½µ¡µ ´ ¡µ (fBm), whose increment process is called fractional

Gaussian noise (fGn).

Heavy-Tailedness: A random variable is called heavy-tailed if

½ ¼ ¾ as , where . A simple heavy-tailed distribution is the Pareto distribution

whose probability density function is given by:

« « ½

´µ

(2.1)

where is the shape parameter and is the location parameter. Its distribution func-

´µ½ ´µ

tion obeys . Note that smaller means heavier tail of the distribution.

In network models, we are interested in choosing ½ , [63].

Self-Similarity: Consider a wide-sense stationary (W.S.S.) time series Ø . The

´Òµ

aggregated series is obtained by averaging Ø over non-overlapping blocks of length Ø

and replacing each block by its mean:

· ¡¡¡ ·

ØÒ Ò·½ ØÒ

´Òµ

(2.2)

´Òµ ´Òµ

´ µ ´ µ

Denote the auto-covariance of Ø and by and respectively. We say that

Ø is self-similar if the following conditions hold:

´Òµ

´ µ ´ µ ¡ Ò

(2.3)

½ ¼ ½

for where . That is, the correlation structure is preserved at different

´ µ

time scales and decays according to a power-law. The Hurst parameter is denoted

½ ¾ ½¾ ½ by , where . The Hurst parameter is an indicator of the degree

of self-similarity in the sense that larger means stronger LRD.

Long Range Dependence: Ø is said to exhibit LRD in that its auto-covariance decays

´ µ½

hyperbolically with , which is implied by (2.3). One example of an LRD

¼ process is fGn.

2.3 Mathematical Model

The congestion control problem to be considered here is for queue management and bandwidth allocation at the bottleneck node, where the output capacity is determined by

15 Bottleneck Node c(t)=c 0−c cross(t) Integrator

r (t) rb(t) − q(t) Source s Forward Time Delay + Controller

^ cτ (t) Predictor r(t) Zero− ^ + Backward Order r(t) K 2 Time Delay Hold + − Robust e q(t) + u(t) Controller q d(t) K 1

Figure 2.1: LMMSE based feedback control system

self-similar cross trafﬁc. For simplicity, consider the single bottleneck node network with

one source, the model of which can be depicted in Figure 2.1.

The dynamics of the system can be modeled as:

´µ ´µ ´µ ´ µ ´µ

(2.4)

´µ ´µ

where is the queue length at time , is the data rate received at the bottleneck

´µ ¼

node, and is the ﬂow rate assigned by the controller. The round-trip delay (RTT),

´µ

is deﬁned as:

´µ ´µ· ´µ

(2.5)

´µ · Æ ´µ

where is the forward time delay from the source node to the bottleneck

Æ ´µ

node where is the nominal forward delay and is the time-varying forward time

´µ · Æ ´µ

delay uncertainty. Similarly, is the backward time delay from the

bottleneck node (controller output) to the source node where is the nominal backward

Æ ´µ delay and is the time-varying backward time delay uncertainty.

Thus, the total nominal time-delay can be deﬁned as , and the total time

Æ ´µÆ ´µ·Æ ´µ delay uncertainty is .

Our two-degree of freedom ﬂow controller consists of a stochastic predictor and a ro-

bust control block, together with a zero-order hold (ZOH). In digital implementation, the

´µ

ZOH holds for -step, which means a larger time scale prediction and control at the

½ ×

scale × , where is the sampling period. In the reasoning of this section, we set

´µ ´µ

for simplicity. Thus we have . This leads to:

´µ ´ µ ´µ ´ µ ´´µ ´ µµ

(2.6)

´µ

The cross trafﬁc Ö Ó×× is assumed to be stochastically self-similar. Thus the output

´µ ´µ Ö Ó××

capacity ¼ also exhibits self-similarity. We introduce the LMMSE

´ · µ ´µ

based predictor, the output of which is a causal estimate of , namely ,or

´µ

for simplicity. Note that by omitting the time delay uncertainties, a good predictor

´µ

can act as a compensator by which the impact of can be canceled. In other words,

´µ ´µ ´ µ should be small enough. Details of the predictor design and

validation are discussed in the next section.

´µ´ µ In this sense, we would like to have . Substitute with its nominal value

, and consider the frequency domain expression, we will have:

× ×

´µ ´µ ´µ ´µ Õ

½ (2.7)

where

´µ ´µ ´µ

Õ (2.8)

´µ ´µ

where is the desired queue length and is its frequency domain expression. The objectives in the design of robust control block ½ are:

Æ ´µ

¯ Robust stability against the time-varying time delay uncertainty

¯ ¾ Queue management, e.g. minimizing Õ

Combining the robust stability and nominal performance conditions (e.g. tracking, weighted ½

fairness, and transient response), we come up with an inﬁnite dimensional À optimization

problem as follows:

Minimize , such that ½ is stabilizing and

Û Û

´½ · µ

× ¼ ½

Û Û

(2.9)

Û Û

´½ · µ

½ ¼ ½

´µ ×

where ¼ , and are chosen to meet desired robustness and performance ×

respectively. See [58] [59] for details.

The optimal controller ½ is computed in [66]:

´µ

½ (2.10)

½· ´µ

Æ Æ where are determined by the upper bound of delay uncertainty , the upper bound

on the derivative of delay uncertainty and the upper bound on the derivative of forward

delay uncertainty , as well as the nominal time delay . We refer to [66] for details. An

internally robust digital implementation of the controller (2.10) includes a PI term which is

´µ

cascaded with a feedback block containing an FIR (Finite Impulse Response) ﬁlter .

× The length of the FIR ﬁlter is × , where is the sampling period, see [66] for the

calculation of FIR coefﬁcients.

¾ In our setting, robust controller ½ and predictor are combined together to form a

ﬂow controller which has access to queue information as well as capacity information.

18 2.4 Short-Term Prediction of Self-Similar Trafﬁc

In the digital implementation of the ﬂow controller of Figure 2.1, we should provide

× a × -step prediction of the available bandwidth, where is the sampling period of the discretized system. The positive correlations of LRD trafﬁc implies the possibility of nontrivial short-term trafﬁc prediction. Since the derivation of an optimal nonlinear predictor will depend on higher order moment functions that may not be available, we will consider designing causal LMMSE predictors. In fact, It has been demonstrated in [26] that the causal linear predictors have good prediction results for self-similar signals.

First, let’s consider one step prediction. Denote the available capacity observed at time

´ µ ´ µ ¼ ½

by . We have a random sequence by time . Suppose we will

´ µ´ ·½µ

be using a sliding window of length , , to generate a one-step

´ µ ´ ·½µ

prediction, namely ½ , which will be the predicted value of . In order to have

Ð ½

´Æ µ

¾ ´ µ

better estimation, we further assume . Let Ð be the mean value

´ µ´ µ ´ µ ·½ ¡¡¡

and denote Ð . Then we have

´ µ ´ µ· ´ µ

Ð ½

½ (2.11)

´ µ ´ ·½µ

where ½ is the one-step prediction of zero-mean W.S.S. random sequence

´ µ ·½ ¡¡¡

given , which can be written as:

Ð ½

´Æ µ

´ µ ´ ·½·µ

½ (2.12)

´Æ µ

¼ ¡¡¡ ½

where are optimal parameters of the LMMSE predictor, which satisfy

the orthogonality condition, [77],

Ð ½

´Æ µ

´ ·½µ ´ ·½·µ ´ µ ·½

(2.13)

´Æ µ ´Æ µ ´Æ µ

´Æµ

¡¡¡

We deﬁne the column vector and the row vector

¼ ½

Ð ½

´ ·½µ´ ·½µ ´ ·½µ´ ·¾µ ¡¡¡ ´ ·½µ´ µ

½ ¡¡¡ ½

ÜÜ ÜÜ

ÜÜ (2.14)

and Toeplitz covariance matrix ÜÜ , where

´ µ ´ ·½·µ´ ·½· µ

ÜÜ ÜÜ (2.15)

From (2.13), we have:

´Æµ ½

½ (2.16) ÜÜ

This prediction is non-biased and the LMMSE is given by

¢ £

´´ ·½µ ´ µµ

ÑÒ ½

¢ £

½ Ì

´´ µµ

½ (2.17)

ÜÜ ½

From (2.11), (2.12) and (2.16), we can determine the algorithm for one-step prediction.

Note that the covariance ÜÜ can be estimated by the time average under

the assumption that the sequence is ergodic in correlation, i.e.

· · ·

ÜÜ (2.18)

where we choose ¾ to guarantee accuracy.

Similarly, the multiple-step prediction can be easily formulated by substituting ½ with

· ½ · ¾ ¡¡¡

ÜÜ ÜÜ ÜÜ × in the above derivation, where is the prediction steps.

2.5 Simulation Results of the LMMSE Predictor

In the following, the performance of the LMMSE predictor is evaluated for self-similar network trafﬁc prediction by simulations using data traces generated by ns-2. We set the

20 network topology to be a single bottleneck node with 50 UDP ﬂows sharing the link. Each

ﬂow, at the application layer, obeys a Pareto distribution with shape parameter . The link

¼¼ capacity is supposed to be 4 Mbps and we sample the aggregated trafﬁc with × .

It has been shown that the heavy-tailed distributions of objects (e.g. files) at the application layer will result in the self-similarity of the aggregated traffic [63]. Thus this simulation can examine whether the LMMSE predictor designed here is qualified for self-similar traffic

prediction.

½¼ ½¼¼

For one step prediction, we set and deﬁne the relative error to be

Ü Ü

½¾¼ ½ ½

. By Choosing , we have the results shown in Figure 2.2. The Ü

error for each case is around ±, which indicates good match between the trafﬁc data and our predictions.

Traffic Trace 2.5 Predicted Data Flow α=1.20 2

in Mbps 1.5

1 0 10 20 30 40 50 60 70 80 90 100 k 2 α=1.45

1.5 in Mbps

1 0 10 20 30 40 50 60 70 80 90 100 k 2 α=1.95

1.5 in Mbps

1 0 10 20 30 40 50 60 70 80 90 100 k

Figure 2.2: One-step prediction for different values of

Remark 2.1 It has been demonstrated in [56] that it makes little difference whether we

´ ½µ ´¼µ

know on or for short-term prediction of fBm trafﬁc. What is more, our simulations show that larger and cannot improve the performance very much, although

they increase complexity.

Intuitively, for multiple-step prediction, the error might be larger with more prediction

steps, which can be veriﬁed by Figure 2.3, where we plot the errors for up to ¾¼ prediction steps for each case.

13 α =1.20 α =1.45 α =1.95 12

10 Error (%) 9

6 0 2 4 6 8 10 12 14 16 18 20 Prediction Steps

Figure 2.3: Error for different prediction steps

From Figure 2.3, we can also ﬁnd that for smaller , which means heavier distribu-

tion in the tail, the errors become smaller for both one-step and multiple-step predictions.

In order to validate this claim, we introduce the relative error variance (RV)

µ ´µ (see [56] for details), which is another index for evaluating the predictor. For

½¼ ½¾¼ ½ ½¼ ½ respectively, we plot the errors and the relative error variances as depicted in Figure 2.4.

0.58

0.56

0.54

0.52

Error (RV) 0.5

0.48

0.46 1 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2 Shape Parameter α

7.8

7.6

7.4

7.2

7 Error (%) 6.8

6.6

6.4 1 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2 Shape Parameter α

Figure 2.4: Prediction Error and relative error variance in one-step prediction

¿ «

Remark 2.2 Theoretically, the Hurst Parameter can be calculated by , which ¾

comes from the ON/OFF model in the idealized case corresponding to a fGn process [15, 86]. In reality, can still be linearly related to , although its slope changes case by

case [63, 81].

The results in [56] show that the relative error variance decays to zero when ap-

proaches ½, that means the LMMSE based predictor works better for the case of stronger

LRD. In other words, our prediction is more reliable for smaller shape parameter , which has been demonstrated in Figure 2.3 and 2.4. Note that estimation errors are not monotone

with respect to the shape parameter because our data traces came from the aggregation of heavy-tailed distributions, not idealized fGn.

23 Meanwhile, the self-similarity implies nontrivial correlation structure at larger time

scales, with which we can realize larger time scale predictions. For the above data traces,

´ µ

we change the time scale to × , which means is aggregated over samples and

´Òµ

´ µ

replaced by its mean , as (2.2) shows. So all prediction at time scale × will be

´Òµ ·

´ µ ¾ ¼ based on this new sequence , . Following the above simulations, we obtained similar results, which again veriﬁed the self-similar nature of network trafﬁc as

well as the feasibility of the LMMSE predictor for different time scale predictions. We only

list the one step prediction error for the scenario of ½ time scale (which corresponds

¾¼ to ) for different values of in Table 2.1. Other results are omitted because they

are quite similar to the above ﬁgures. Recall from the above discussion that for between

¾ ½ ± ¼¼ ½ and the relative error was about for time scale, these results are comparable to the ones shown in Table 2.1. But for the same time interval of prediction, the larger time scale prediction needs fewer prediction steps compared with that at smaller

time scale. We will make use of this fact in the system simulations in Section 2.6.

½¼ ½¾¼ ½ ½¼ ½

¼± ± ½± ¾± ½±

Prediction error Table 2.1: One-step prediction at a larger time scale (½ )

Remark 2.3 Note that the LMMSE predictor is an adaptive estimator since the predictor updates its parameters at every sampling time as described by (2.12), (2.16) and (2.18).

Moreover this scheme can be easily extended for a multi-sources network topology, since

½ the predictor block ¾ is designed independently of the control block and that the

design of the block ½ for a multi-source network topology with uncertain time-varying

time delays has been considered in [66].

2.6 System Level Simulations

The control scheme combining the LMMSE predictor ¾ and the robust control block

½ is applied to the discretized model depicted in Figure 2.5, where the sampling time is

× × ¾

× and the time delay is equal to with ; predicts the trafﬁc at time

scale × which results from the trafﬁc aggregation obeying (2.2). Correspondingly the

Zero Order Hold holds the signal for steps which gives the output ÄÅÅË of the overall

controller sampled at × . Note that corresponds to the case where every signal is

sampled at the value × .

ÄÅ Å Ë

Figure 2.5: Discretized model

The simulations of our control scheme is compared with those of the two-degree of ½

freedom À controller of [65] which we call comparison controllers for short.

¼¼ ½ ½ ´ µ ¼ ¾ ¼

We assume × , , , with in our

´ µ

ﬁrst setting. The cross trafﬁc ÖÓ×× is generated by using the similar network topology

½¼ as mentioned in Section 2.5 by ns-2, where the shape parameter is . The LMMSE system responses are depicted in Figure 2.6.

c(t) in Mbps 20

0 0 20 40 60 80 100 120 140 160 180 200 time t in second 150 Queue Length Desired Queue 100

in Mb 50

0 0 20 40 60 80 100 120 140 160 180 200 time t in second 150 Queue Length Desired Queue 100

in Mb 50

0 0 20 40 60 80 100 120 140 160 180 200

time t in second

¼ Figure 2.6: System responses for ¼ time scale: the middle one denotes the comparison controller and the bottom one denotes the LMMSE based scheme

We can see that the LMMSE control scheme and the comparison controller have similar

performance, with nearly the same overshoot, settling time, and oscillations. In fact, by

¾!

Æ ´ µ ¼×Ò´ µ

adding time-varying time delay uncertainty, say × , we can still get ¼ the same conclusion, which implies that the two kinds of controllers even have similar robustness.

26 However, the LMMSE predictor is designed directly for self-similar network trafﬁc, that works equally well in the larger time scale, with the extra advantage of reducing the prediction steps which has been discussed in Section 2.5. Thus we would like to implement it with the robust control block at larger time scales, that is more applicable for the real network environment, where control actions are updated once every RTT.

Before such implementation, we need to investigate the performance of the robust con-

trol block ½ at different time scales (i.e. holding the control output for steps in our

simulations). In this case we quantify the performance of the control system in two ways:

¯ ÑÒ ´µ ´µ ´µ ±

× Ø

Settling time × , deﬁned as ,

Ì"Ì

¯ ´ ´ µ ´ µµ

× ×

Cost index × .

½ ½ ¼ ´ µ ¼ ¾

For , we choose holding steps from to and set for

, by which we omit the impact of the cross trafﬁc. The responses of ½ alone and

the comparison controller are examined with respect to different values of : performance

indexes, × and are shown in Figure 2.7.

¾¼

As shown in Fig.2.7, ½ preserves good performance even for , i.e. for the case

½ ¼¼ where the assigned rates (controller output) is updated every while × .

The comparison controller, however, does not possess this property. Larger holding time deteriorates the performance of the comparison controller signiﬁcantly. Note that we did not plot the cost index curve for the comparison controller in Fig.2.7, because it diverges

very rapidly.

Based on the above simulations at different time scales for ½ and the multi-scale properties of the LMMSE predictor, we consider applying our control scheme at a larger time

27 200 K1,without uncertainty (h=1) K1, with uncertainty (h=1) 150 Comparison Controller (h=1)

100

50 Settling time in second

0 0 5 10 15 20 25 30 35 40

0.95

K1, without uncertainty (h=1) 0.9 K1, with uncertainty (h=1)

0.85 Cost Index 0.8

0.75 0 5 10 15 20 25 30 35 40 n (holding steps)

Figure 2.7: Performance indexes as a function of

scale, where the control output is updated once every second. We adopt the same parame-

¾¼ ter setting as in the ﬁrst scenario, except that in this case we have . The responses for the LMMSE controller and the comparison controller are shown in Fig.2.8, where the

performance improvement of the LMMSE control scheme is apparent.

¼¼ ½ Furthermore, we list the performance index for time scale and , for both control schemes in Table 2.2. In conclusion, the controller which uses the LMMSE predictor shows good performance with respect to self-similar trafﬁc and time delay uncertainties, both in smaller time scales and in larger ones.

time scale 0.05 sec time scale 1 sec

½ ½·Æ ´ µ ½ ½·Æ ´ µ LMMSE controller 0.6765 0.6869 0.7398 0.7928 Comparison controller 0.6305 0.6520 1.9751 1.9818

Table 2.2: Comparison of performance index

28 60

c(t) in Mbps 20

0 0 20 40 60 80 100 120 140 160 180 200 time t in second 200 Queue Length 150 Desired Queue

100 in Mb 50

0 0 20 40 60 80 100 120 140 160 180 200 time t in second 150 Queue Length Desired Queue 100

in Mb 50

0 0 20 40 60 80 100 120 140 160 180 200

time t in second Figure 2.8: System response for ½ time scale: The middle one denotes the comparison controller and the bottom one denotes the LMMSE one

2.7 Concluding Remarks ½

In this chapter, an LMMSE based predictor is added to the robust À controller designed in earlier works [59] [66]. The two-degree of freedom flow controller has better transient response for self-similar cross traffic and good robustness against time delay uncertainty. Notably, the flow controller developed here preserves good performance in the scenario of larger time scale prediction and control, which is more applicable for real networks. An interesting extension of this work is to consider prediction and control at

different time scales or multiple time scales, by which we could make better use of the ½ multifractal properties of network trafﬁc and the robustness of the À controller.

29 CHAPTER 3

A Variable Structure Control Approach to Active Queue Management for TCP with ECN

It has been shown that the TCP connections through the congested routers can be modeled as a feedback dynamic system. In this chapter, we design a variable structure (VS) based control scheme in Active Queue Management (AQM) supporting explicit congestion notification (ECN). By analyzing the robustness and performance of the control scheme for the nonlinear TCP/AQM model, we show that the proposed design has good performance and robustness with respect to the uncertainties of the round-trip time (RTT) and the number of active TCP sessions, which are central to the notion of AQM. Implementation issues are discussed and ns packet-level simulations [1] are provided to validate the design and compare its performance to other peer schemes’ in different scenarios. The results show that the proposed design significantly outperforms the peer AQM schemes in terms of packet loss ratio, throughput and buffer fluctuation.

3.1 Overview

The TCP connections through the congested routers can be modeled as a feedback dynamic system, where control theory based approaches can be used to analyze the network behavior, tune AQM’s parameter settings, and design new AQM schemes. We refer to

30 [39,50,54] for the details of the nonlinear dynamic models for the TCP’s AIMD (Additive

Increase and Multiplicative Decrease) behavior and the AQM’s congestion feedback. It is believed that the control system based analysis offers new insight into the AQM design. In

[32], a control theoretic analysis was given for RED, which provided a more systematic and in-depth study on RED parameter tuning; and [33] developed a PI controller as a new AQM scheme using linear system analysis. On the other hand, the AQM congestion control can be viewed as a convex optimization problem for the dynamical system [39,41,50,62], where steady state properties and equilibriums are investigated. Meanwhile Explicit Congestion

Notiﬁcation (ECN) [67] has been proposed, where packet marking is used as congestion indication. We refer to the AVQ algorithm [41] for an ECN based AQM design. Taking the stability and performance indices into consideration, the above control theoretic AQM schemes achieve better performance in terms of high link utilization and low packet loss ratio.

Due to the challenging nature of nonlinearity in the TCP dynamics, most of the current results on the AQM analysis and design are based on linearized models which are valid only in the neighborhood of the equilibrium points [32, 33, 41, 68]. To further complicate the situation, the TCP/AQM dynamics have time varying round-trip times (RTT) and uncertainties with respect to the number of active TCP sessions through the congested AQM router, which requires more robustness for the designed schemes. In this chapter, we introduce a robust variable structure based AQM scheme designed directly for the nonlinear

TCP model, which to the best of our knowledge has not yet been employed to TCP/AQM system analysis and design. The motivations behind this work are (i) VS sliding mode control is robust and powerful for nonlinear systems [82], thus being well-suited for an

AQM scheme; (ii) DropTail, a widely deployed AQM scheme, can be treated as a VS con-

´µ ½ ´µ ´µ ¼

troller in the sense that its dropping policy is when Ñ#Ü , and

´µ ´µ ´µ Ñ#Ü when Ñ#Ü , where is the dropping probability, and are the occupied buffer and the maximum buffer length respectively. Note that the undesirable behavior of

DropTail (e.g. heavy queue ﬂuctuation and high dropping rate), from the VS control perspective, is due to the fact that its binomial control doesn’t ﬁt TCP dynamics to enforce a stable sliding mode.

In this study, we present guidelines to design VS controller for AQM supporting ECN.

By analyzing the robustness and performance of the proposed AQM scheme, we show that the proposed design has good asymptotic properties as well as stability robustness with respect to RTTs and the number of the active TCP sessions through the router, which are central to the notion of AQM. Meanwhile, we also discuss the implementation issues and compare our design against related works. Our VS based AQM scheme is shown via ns packet-level simulations to be a robust controller that performs better than a number of well-known AQM schemes under different scenarios.

3.2 Variable Structure Control in AQM

3.2.1 Nonlinear TCP dynamics

In [39], a nonlinear dynamic model for TCP congestion control was derived, where

the network topology was assumed to be a single bottleneck with homogeneous TCP sources that share the bottleneck link and have roughly the same RTTs, but don’t necessarily transverse the same path. For TCP with ECN, the AIMD behavior in congestion avoidance

phase can be modeled as follows: each positive acknowledgment increases the value of congestion window cwnd by ½ while each congestion indication (ECN) reduces the

½ Ô

Û Ò

cwnd by half, thus the expected change in congestion window is given by ,

Û Ò ¾ where is the marking probability. Aggregating the TCP ﬂows through one congested

router, we have

Ö ´Øµ

Å Å

´µ ´ · µ´µ

¾ ¾

´Øµ ´Øµ ¾Å

(3.1)

´µ ´µ ¼

and

´µ

´µ ·

Ô (3.2)

´µ ½ ´µ ´µ

where ¼ is the marking probability, is the round trip time delay, is

´µ

the instantaneous queue length on the router, is the incoming trafﬁc rate per unit time,

¼ is the number of the active TCP sessions, Ô is the propagation delay and is the

link capacity. Note that (3.1) is a simpliﬁed model without considering the time delays

´µ ´µ in and , and also we assume full link capacity is available. The impact of this

simpliﬁcation will be discussed in Section 3.5.

´µ

To simplify the solution, instead of using (3.2), we assume to be any continuous

´µ

function independent on with

Ñ#Ü

· ´µ

Ô ¼ ½

Ô (3.3)

where Ñ#Ü is the buffer length of the router. Note that (3.3) captures its time varying nature

´µ ´µ and decouples from . We further assume the number of active TCP connections to

be uncertain, obeying

¼ (3.4) which is more reasonable in practice.

Recall that most of the current AQM schemes [32, 33, 41, 68] are based on the linearized model, which are valid only in the neighborhood of the operating points. Mean- while, some important parts of TCP are even not included in the nonlinear model (3.1):

33 (i) non-responsive UDP ﬂows are not modeled; (ii) the impact of short-lived connections

(the so called web mice), such as Telnet and HTTP. Taking the nonlinearity and the unmod- eled uncertainty into consideration, we believe that variable structure sliding model control would be an ideal methodology for a robust AQM.

3.2.2 VS based AQM with ECN

The nonlinear TCP dynamics and the VS based AQM can be modeled as a feedback control system depicted in Figure 3.1, where the VS controller uses the queue and trafﬁc

incoming rate information to generate marking rate as congestion indication.

Õ Õ Ô

Ê Ê

Ö Ö ·

¾Å

Ô · Ô

Ë Ô

ÎË × ÉÅ

Figure 3.1: Aggregated dynamics of TCP and VS based AQM

´µ

½ Õ ¾ Õ Let Õ and denote , , where is the desired queue length.

We have

´µ ´µ

Õ ¼ ¾ (3.5)

and the plant (3.1) can be described as

½ ¾

Ü · µ

´ (3.6)

Å Å

¾ ¼

´ · µ´µ

¾ ¾ ¾

´Øµ ´Øµ ¾Å

Note that (3.6) is an afﬁne nonlinear system in the form of

´ µ· ´ µ´µ

where

´ µ

´Ü · µ

¾ ¼

´ · µ

´Øµ ¾Å Thus the equivalent control method (ECM) [82,83] can be used to construct the VS control

law. We select the sliding mode surface as

´ µ · ¼

½ ¾ Ë Ë (3.7) which corresponds to a linear combination of the queue length error and the error between incoming trafﬁc rate and link capacity.

The corresponding existence condition [83] for the sliding mode is

·´ µ

(3.8)

where . Observe

´ · µ

¾ ¼

· ¼ ´ µ

´µ ¾

and

´ µ ¼

´ µ

½ ¾

It is straightforward to check that (3.8) is satisﬁed. The typical sliding mode controller

( [83]) is given by

´ µ ´ µ ½ ´ µ ´ µ

and Õ (3.9)

where

¾ Ë ¾

´Øµ

´ µ

Õ (3.10)

´ µ ¾

Unfortunately, controller (3.9) is not practically feasible because the negative part of

´µ ¼ ´µ ½ in (3.9) is out of the bound of . In what follows, we would like to

improve the sliding mode controller by introducing a feedback term. Deﬁne

´ · µ

¾ ¼

· ´ µ

¾ (3.11)

Ì ·Ì

Å ·Å

¼ ½

´µ

where and are the nominal values of and respectively.

¾ ¾

According to the sliding mode existence condition and the robustness criteria, we construct

the VS controller as following

½ ¾

·´ ´ µ · Æ µ ´ ´ µµ

´µ

´ µ

´ µ ¼ ´ µ if

(3.12)

´ µ ´ µ ¼

½ Æ¼

where , are constants and

½ ¾

´ µ ·´ · Æ µ

´ µ

¾ ¼

¾ ½

· Æ µ ´ ´ µ

Ë (3.13)

´ µ

¾ ¼ ¾

Stability and robustness with respect to the proposed VS controller will be discussed in

Æ the next section, where we also provide guidelines of choosing the parameters Ë , and .

3.3 Robustness and Performance Analysis

In order to guarantee the robust stability of the closed loop system, the existence condition for the sliding mode should be satisﬁed, which is given by the following theorem.

Theorem 3.1 The VS controller (3.12) robustly stabilizes the nonlinear system (3.6) for all

´µ ½

and obeying (3.3) and (3.4) respectively, if and

½ ¼

Æ ´ · · µ

µ´ (3.14)

· ·

¼ ½ ¼

Proof. First, notice that

¾ ´ · µ

¾ ¼

´ µ · ·

¾ ¼

¾ (3.15)

´µ ¾ ´µ

´ µ ·

¾ ¼

Similarly, we have ¾ . Thus

´µ´ µ

´ µ

¾ ¾

Å Å

´ µ

¾Å

´ · µ

¾ ¼

¾ ¾

´ µ´ µ

¾ ¾

Å Å

µ ´

¾Å

¾ ¾

´µ ´ µ

· ´ µ·´ µ

(3.16)

which is straightforward from (3.3), (3.4), (3.14) and (3.15).

´ µ Recall (3.7), the time derivative of along the trajectory of (3.6) under the control

(3.12) is given as

´ µ´ µ ´ µ ·

¾ Ë ¾

´µ

· ´ µ

Ë ¾ ¾

´µ

´µ´ µ

´µ · Æ µ ´ ´ µµ · ´ ¾

Ë (3.17)

´ µ ´ µ ¾ Note that and ¾ are positive. Invoke (3.14), (3.15) and (3.16), we have:

´ µ ¼

(i) If ,

´ µ ´ µ

½ ¾ ¾

´µ ·

Ë ¾ Ë ¾

´ µ

½ Ë

´ µ´´ ½µ ´µ · Æ µ

¾ ¾

´ µ ´ µ

½ ¾ ¾

´µ · ´ µ

Ë ¾ Ë ¾ ¾

´ µ

½ ¾

µ ´½

Ë ¾

¾ ·

¾ ¼

· ´ µ´ µ

´ µ

´µ´ µ

´½ µ ¼ ¾

Ë (3.18)

¾ ·

¾ ¼

´ µ ¼

(ii) If ,

´ µ ´ µ

¾ ½ ¾

· ´µ ·

Ë ¾ Ë ¾

´ µ

½ Ë

´µ · Æ µ · ´ µ´´ ½µ

¾ ¾

´ µ ´ µ

½ ¾ ¾

· ´ µ · ´µ ·

¾ Ë ¾ Ë ¾

´ µ

½ ¾

½µ ´

Ë ¾

¾ ·

¾ ¼

· ´ µ´ µ

´ µ

´µ´ µ

´ ½µ ¼ ¾

Ë (3.19)

¾ ·

¾ ¼

ÐÑ ´ µ ´ µ ¼

¼ which indicates Ë . According to the existence condition of the sliding mode [83], the proposed VS controller is robustly stable and the state trajectory of the feedback system converges to the sliding mode. Thus the proof is complete.

Inserting the equivalent controller (3.10) into system (3.6), we obtain the motion on the

´ µ¼

sliding manifold ,

Ë (3.20)

´µ

Thus the instantaneous queue length exponentially converges to the desired queue

length with decay rate once the trajectory reaches the sliding mode surface.

´µ

Remark 3.1 Note that Ë is the decay rate of . For the purpose of fast convergence,

we would like to pick up Ë as large as possible. On the other hand, a larger results in a

´µ

larger magnitude of the marking probability (from (3.12) and (3.13)), which increases

¼ ´µ ½

Ë ½

the risk of saturation ( ). Practically, we choose Ë so that the two terms

½¼ ½¼¼ ½¼¼¼

Ë ¼ and ¾ in (3.7) are balanced (e.g. is feasible when and ).

Another important aspect of the system performance is how fast the closed loop system

reaches the sliding mode surface. In the following, we will also investigate the reaching

condition and the corresponding reaching time for the states to reach the sliding manifold

from any initial point.

´µ Theorem 3.2 For the nominal system with and in (3.6), the states of the closed loop system implementing the VS controller (3.12) can reach the sliding manifold

from any initial point, and the reaching time is no larger than

´¼µ · ´¼µ

Ñ#Ü ½ ¾

Ë (3.21)

´µ

Proof. With and , the nominal system is given as follows:

½ ¾

´ · µ

¾ ¼

µ´µ ´ ·

¾ (3.22)

¾ ¾

Introduce the Lyapunov functional as

´ µ ´ µ

(3.23) ¾

i.e.

´ µ ¾ (3.24)

´ µ Find the time derivative of along the trajectories of system (3.22) with control

(3.12):

´ µ´ µµ ´ ·

¾ Ë ¾

´ µ´ ´µ · Æ µ ´ µ

Ë ¾ ¾ Ë ¾

´ µ´ ´µ · Æ µ·

¾ Ë ¾ Ë ¾

´Æ ´ µ· µ

¾ Ë ¾ Ë ¾

(3.25)

Recalling (3.24) yields

Ô Ô

¾Æ

(3.26)

Since the Lyapunov function decays at a ﬁnite rate, it vanishes and sliding mode occurs

after a ﬁnite time interval. Thus the reaching condition is fulﬁlled.

´ µ Note that the solution to the differential inequality in (3.26) is non-negative and

is bounded by

´ µ ´ · µ ´¼µ ¼

¼ (3.27)

From (3.24) and (3.27), we come up with an estimate of the reaching time Ö #

´¼µ · ´¼µ

Ë ½ ¾ Ñ#Ü

Ö # (3.28)

Æ and complete the proof.

With a view toward implementation of the proposed VS based AQM scheme (3.12), we

´µ ½ have to take the saturation constraint ¼ into consideration, which results in the

following modiﬁcation:

´µ©´´µµ

Ö #Ð (3.29)

where

¼ ¼

µ ¼ ½

©´ if

½ ½ if Such modiﬁcation decreases the control magnitude, which may introduce chattering behavior and even system divergence. Note that the small ﬂuctuation of the queue, caused by the chattering of the state trajectory around the sliding manifold, is acceptable in network buffer management.

Remark 3.2 Theoretically, the VS controller in the form of (3.12) can achieve robust sta-

´µ bility regarding any uncertainty bounds of and by choosing the control magnitude large enough, which may result in heavy saturation and performance deterioration. This is

a trade-off with respect to the system robustness and performance.

The above algorithm can be validated using the following example:

Example 1. Consider homogeneous TCP connections sharing a link with capacity

½¾¼ ¼ ½¾¼ ¼

¼ packet/sec , where we choose randomly distributed in with

½¾¼

, and the RTT is time varying obeying

¼½ ´µ¼¾·¼¼ ×Ò´ µ ¼¾ ¼

sec ½ sec

½¼¼

¼¾ ½¼ ½¼¼

Thus and . We further assume the desired queue packet.

Recalling (3.14) gives

½ ¼

´ · · µ´ µ¼¾

· ·

¼ ½ ¼

½¼ ½ Æ ¼¿

Based on Theorem 3.1, we choose Ë , and . Inserting these parameters

into (3.12), we come up with the VS based AQM controller. The MATLAB simulations

are depicted in Figure 3.2 and Figure 3.3, where the nominal system with and

´µ is also simulated as comparison.

1corresponds to a 10Mb/s link with average packet size 1000 Bytes

41 200

150

100

q(t) (packet) real sys. −50 nominal sys. −100 0 5 10 15 20 25 30 35 40 time (sec)

1000

−1000 S(x,t)

−2000 real sys. nominal sys. −3000 0 5 10 15 20 25 30 35 40 time (sec)

Figure 3.2: System responses using the VS controller

3.4 Related Work

Current queue management schemes, such as DropTail and RED, have been observed to have some drawbacks such as link underutilization, heavy queue ﬂuctuation and global synchronization. Several improved AQM schemes have been proposed to remedy these drawbacks based on more systematic analysis of the TCP dynamics [5,8,34,41,47,57]. In what follows, we provide a brief review and comparison of some representative TCP/AQM algorithms in terms of their motivations, performance objectives and methodologies.

1. Modiﬁcations and alternatives of RED. To remedy RED’s drawbacks (e.g.parameter sensitivity), several approaches have been proposed in the recent years, e.g. FRED [47], balanced RED (BRED) [5] and stabilized RED (SRED) [57]. FRED and BRED share the

same objective of improving per-ﬂow fairness by monitoring the per-ﬂow queue length

and tuning the dropping rate correspondingly. Meanwhile, BRED can be considered

42 1500 nominal sys. real sys.

1000

500 2 x

sliding manifold −500 S=0

−1000 −250 −200 −150 −100 −50 0 50 x 1

Figure 3.3: Phase portrait of the closed loop system

as an extension of FRED in the sense that BRED has a reﬁned dropping policy by dividing the space of into four regions with four dropping probabilities in each of the regions.

Note that both of them use only the queue length as the congestion index to calculate drop-

´µ ping probability , which is believed to be difﬁcult to achieve high link utilization and low packet loss ratio simultaneously [8]. On the other hand, SRED was proposed to alleviate the heavy ﬂuctuation of the instantaneous queue caused by RED. Using the estimate

of the number of active ﬂows , SRED modiﬁed the dropping probability as an increasing

function of . Thus SRED can adaptively adjust its dropping probability with respect to

TCP ﬂow number. Although the above modiﬁcations outperform RED in the aspects of fairness and throughput, they are essentially heuristic without systematic analysis.

2. AQM schemes based on equilibrium structure and utility optimization. The congestion control system can be considered as a convex optimization problem for a certain

43 aggregate utility function, where TCP/AQM can be interpreted as carrying out a gradient

algorithm to maximize aggregate source utility. Schemes in this category (e.g. REM [8] and

AVQ [41]) aim at achieving both high utilization and low packet loss ratio by introducing

the price function as congestion index.

´ µ In REM (Random Exponential Marking), a price function is deﬁned based on the

queuing information and the incoming rate

´ · ½µ ÑÜ´¼´ µ· ´ ´ ´ µ µ·´ µ µµ

Ö (3.30)

´ µ ´ µ where and are constants, and are the queue length and the incoming rate

respectively, and is the link capacity. Correspondingly, REM uses an adaptive marking

´ µ

probability to regulate the queue length

&´ µ

´ ·½µ ½

(3.31)

½ ´ ´ µ µ·´ µ where is a constant. Note that the term Ö in (3.30) is in fact

the weighted mismatch of the queue and the rate, which is similar to the sliding manifold

´ µ in our design.

The AVQ (Adaptive Virtual Queue) is another adaptive AQM scheme, whose marking

probability is determined by the virtual queue capacity updated according to

´ ´ µµ (3.32)

where is the desired link utilization. Based on the linearized TCP dynamics, AVQ cal-

´ µ culates marking probability using only the incoming rate , so it is primarily based on

incoming trafﬁc rate to provide early congestion feedback.

´µ

Scalable control scheme proposed in [50] uses link’s price Ð as congestion index

Ô ´Øµ

½ ½ ´µ and marks packets with probability . The link updates its price Ð using

´µ

the aggregate input rate Ð according to:

Ð Ð

´µ ¼

if Ð

´µ

(3.33)

Ð Ð

ÑÜ¼ ´µ¼

if Ð

in which Ð is the that is strictly less than the real link capacity. The source

´µ

will set its sending rate as an exponential function with respect to the aggregate price ,

« Õ ´Øµ

´µ Ñ#Ü( i.e. . To utilized this scheme, the current TCP’s congestion control

and avoidance scheme has to be changed which is not an easy task.

3. AQM schemes based on feedback control theory. The dynamic models of TCP

make it possible to design AQM in the literature of feedback control theory. In [32], RED

was considered as a AQM controller whose parameters can be analyzed based on a lin-

earized TCP model. An extension to this work is PI AQM [34], whose marking probability

is updated based on the queue length as

´ ·½µ ´ µ·´ ´ ·½µ µ ´ ´ µ µ Ö

Ö (3.34) where and are constants.

It has been shown in [34] that the PI AQM scheme can outperform RED in terms of system response and steady state error. On the other hand, the PI controller has some inherent limitations: 1) the linearization inevitably introduces model error; 2) it is based on the frequency domain analysis which is invalid for time-varying systems (TCP dynamics are essentially time varying); 3) although gain-phase margin can be analyzed for the PI controller, it can not incorporate robustness directly in the design. However, our VS based

AQM is capable of performing robust control for the time-varying nonlinear TCP dynamics due to the nature of sliding mode control.

45 It is worth mentioning that DropTail can be viewed as a variable structure controller in

the sense that its dropping probability is given as

½ ´µ ¼

if Ñ#Ü

´µ

(3.35)

¼ ´µ ¼

if Ñ#Ü which is similar to the typical structure of sliding mode control. From the point of view of

VS control, the reason why DropTail performs poorly is that it could not enforce a stable sliding manifold.

The idea of sliding mode control was also discussed in [68], where a sliding mode AQM algorithm (which is called SMVS) was proposed for the linearized delay-free system. The

dropping rate of SMVS is calculated by

´ ´µ µ ´ ´µ µ¼

´µ

(3.36)

´ ´µ µ¼ ´ ´µ µ if

where

´ ´µ µ· ´ ´µ µ ¼

(3.37)

is the sliding manifold and , are constants. It has been shown in [68] that SMVS has

better robustness and performance than PI AQM. However, the SMVS controller has a

´µ ½

fundamental mistake. Taking the constraint ¼ into consideration, (3.36) is

´µ ½ ÑÜ

valid only in the range of . Using the recommended values

in the paper, we ﬁnd that SMVS will always work at the saturation states

¼ ½ ´µ ½¼ ½¼ of or , except when , indicating that it is degraded to a DropTail scheme most of the time. Furthermore, the theoretical analysis in [68] is invalid for the implementation of SMVS ((10) in [68]), so that the stability of the AQM algorithm can not be guaranteed.

Compared with the above schemes, the VS based AQM proposed in this chapter is the only one that is directly designed for the time-varying nonlinear TCP dynamics, which has

46 good robustness and fast response inherited from sliding mode control. More comprehen-

sive comparison will be given in Section 3.5 using ns-2 simulations.

3.5 Packet-Level Simulations

To validate the performance and the robustness of the proposed VS AQM, we implement it in ns-2 and conduct a packet-level simulation study in different scenarios. Some representative AQM schemes, namely, DropTail, RED [24], REM [8] and PI [34], are also simulated for the purpose of comparison. Note that the sliding mode AQM (SMVS) in [68] is not considered for comparison because the simulations in [68] are questionable due to the severe problem discussed in the previous section.

3.5.1 Simulation Conﬁguration

The dynamic behaviors of the above AQM schemes are simulated under a variety of network topologies and trafﬁc sources. In particular, we consider the dumbbell network

topology depicted in Figure 3.4, where TCP connections share a single bottleneck link.

We assume that the TCP sources always have data to send. The links between the TCP

½¼ ¼

sources and the router ½ are Mbps links with a ms propagation delay, which are

the same as those between the TCP sinks and the outer ¾ . Router is connected to

½¼ ¾¼

¾ through a Mbps ms delay link. The maximum buffer size of each router is set to

¿¼¼ packets (of size 1000 bytes). Meanwhile, we also consider the network topology with

multiple bottleneck links (Figure 3.5), where the maximum buffer of each router is ¾¼¼

packets, the bandwidth and the propagation delay of each link are indicated in Figure 3.5

and each sender-receiver pair has Ö Ó×× TCP connections as cross trafﬁc. In both scenarios,

TCP-Reno is used as the transport agent.

ÌÈ ËÓÙÖ× ÌÈ ËÒ×

½ ½

ÓØØÐÒ Ä Ò

ÊÓÙØÖ ÊÓÙØÖ

Ê Ê

½ ¾

½¼ÅÔ×¸¾¼Ñ×

Å Ë ½¼ÅÔ×¸¼Ñ× ½¼ÅÔ×¸¼Ñ× Å

Figure 3.4: Dumbbell network topology for ns simulations

½¼¼ ¼¾ ½¼

The parameters used in VS based AQM (3.12) are: , , × ,

½ Æ ¼¿

and (see example 1 for details). In PI controller (3.34), we use the suggested

½¾¾ ¢ ½¼ ½½ ¢ ½¼

parameter values and given in [33]. The desired

½¼ queue length is set to packets for VS control and PI control. The parameters

of RED are set as recommended in http://www.aciri.org/ﬂoyd/REDparameters.txt.For

¼½ ¼¼¼½ ½¼¼½ REM (3.30) and (3.31), the parameters are set as , and , which

are recommended in [8]. In the following simulations, ECN is enabled for VS AQM, PI,

´µ REM and RED respectively, where corresponds to the marking rate, and packet loss is observed only when the buffer overﬂows.

3.5.2 The Scenario of Single Bottleneck Topology

Performance Comparison of Different AQM Schemes: In this experiment, we choose

½¼¼ in Figure 3.4, which corresponds to 100 greedy FTP ﬂows sharing the bottleneck link. The system response using the VS controller is depicted in Figure 3.6, where

48 Cross−traffic Cross−traffic Senders Senders Senders Receivers

10Mbps, 45ms 10Mbps, 45ms 10Mbps, 25ms 10Mbps, 25ms

10Mbps, 10ms 10Mbps, 10ms 10Mbps, 10ms 2Mbps, 10ms 10Mbps, 10ms RRR R RR Queue 1 Queue 2 Queue 3 Queue 4 Queue 5

10Mbps, 45ms 10Mbps, 25ms 10Mbps, 45ms 10Mbps, 25ms

Cross−traffic Cross−traffic Receivers Receivers

Figure 3.5: The network topology with multiple bottleneck links

the performance shows fast response and the stabilized queue size. Meanwhile, we repeat the same experiment using DropTail, RED, PI and REM respectively, and depict their instantaneous queues in Figure 3.7.

Note that the heavy oscillation of DropTail (Figure 3.7 (a)) coincides with our analysis in Section 3.4 from the point of view of sliding mode control. As compared to the AQM schemes shown in Figure 3.7, we clearly see that the VS AQM outperforms other schemes in terms of system stability and performance, which implies higher link utilization, lower packet loss ratio and smaller queue ﬂuctuation. Although the PI controller is also a control theory based design which could regulate the queue to the desired value, its transient response is sluggish, which deteriorates its performance (Figure 3.7 (c)).

Performance Under Dynamic Trafﬁc Changes: In this scenario, we provide some

time-varying dynamics and investigate the performance of the VS controller and other rep-

¼ ¼

resentative schemes. We use 150 TCP connections at time . At time ,50of

¼ the TCP connections stop transmitting data, and at time they resume transmitting

49 300

250

200

150

Queue size (packet) 100

0 0 10 20 30 40 50 60 70 80 90 100 Time (second)

Figure 3.6: Instantaneous queue size using VS control

again. The queue evolution is depicted in Figure 3.8. Note that PI (Figure 3.8 (b)) and

REM (Figure 3.8 (c)) are not very robust with respect to such connection number variation, which result in heavy queue ﬂuctuation during 40-70 second. Although RED (Figure 3.8

(a)) is not very sensitive in this scenario, it tends to over-mark the incoming trafﬁc so that the link utilization is degraded. As is evident from (Figure 3.8 (d)), the VS controller is very robust against the variation of connections and keeps very good response even in the presence of such variations.

Robustness w.r.t. Number of TCP Connections: The performance and robustness of the VS scheme are explored with respect to different TCP loads. We conduct simulations

with the same setting as in Experiment 1, except that the number of connections varies ½¼¼ from 50 to 250. Figure 3.9 plots the average queue length (from ¾¼ second to second)

50 (a) DropTail (b) RED 300 300

250 250

200 200

150 150

100 100

Queue size (packet) 50 50 Queue size (packet) 0 0 0 20 40 60 80 100 0 20 40 60 80 100 Time (second) Time (second) (c) PI (d) REM 300 300

250 250

200 200

150 150

100 100

Queue size (packet) 50 Queue size (packet) 50

0 0 0 20 40 60 80 100 0 20 40 60 80 100 Time (second) Time (second)

Figure 3.7: System responses for DropTail, RED, PI and REM

for different AQM schemes. Correspondingly the link utilization and the packet loss ratio are depicted in Figure 3.10 and Figure 3.11 respectively.

It is observed that the VS controller can robustly stabilize the queue length around ½¼¼ packets. The average queue lengths of REM and RED vary slightly with respect to the ﬂow number, while the average queue of PI blows up when the ﬂow number increases, which is due to the inherent nature of PID control that system response is highly dependent on the system parameters.

From Figure 3.10, we clearly see that the VS and PI controller have better link utilization than REM and RED, and the link utilization of REM and RED are sensitive to the variation of the ﬂow numbers. Note that the packet loss ratio of PI increases with respect to higher loads (Figure 3.11), which is not desirable for congestion control. REM has lower

51 (a) RED (b) PI 300 300

250 250

200 200

150 150

100 100

Queue size (packet) 50 Queue size (packet) 50

0 0 0 20 40 60 80 100 0 20 40 60 80 100 Time (second) Time (second) (c) REM (d) VS control 300 300

250 250

200 200

150 150

100 100

Queue size (packet) 50 Queue size (packet) 50

0 0 0 20 40 60 80 100 0 20 40 60 80 100 Time (second) Time (second)

Figure 3.8: Queue evolution using RED, PI, REM and VS control

drop rate compared with PI, but its link utilization is not that good. Notice that in Figure

3.11, the drop rate for RED is not plotted due to the fact that RED has a much higher drop ½¿± rate in the range between ± and in this simulation. Compared with RED, REM and

PI, the VS controller has much better performance in terms of robustly stabilized queue length, high link utilization and low packet loss ratio (in this experiment, the packet loss

ratio for VS AQM is always ¼).

The above analysis shows that the control theory based AQM designs, say, PI and VS

AQM, have better link utilization by stabilizing the queue length to the desired values.

Note that the system response (response time and overshoot) of PI is deteriorated by higher loads. Thus the high link utilization of PI is in the expense of high packet drop rate when the number of TCP ﬂows increases. On the other hand, the VS controller keeps fast response,

52 300 VS control PI REM 250 RED

200

150

100

Average queue size (packet) 50

0 50 100 150 200 250 Number of connections

Figure 3.9: Average queue length w.r.t. the number of TCP ﬂows

stability robustness with respect to a large range of variation of TCP ﬂow numbers, which

is well-suited as an AQM scheme.

Comparison with PI AQM under Higher TCP Loads: It has been shown in [33]

that the PI controller has robustness with respect to the number of connections, although a

larger number of connections results in a slower system response. In this experiment, we ¿¼¼ set the ﬂow number to and compare the system response of PI and the VS AQM.

As shown in Figure 3.12, the VS controller has very good transient response and very low overshoot. On the other hand, the PI controller exhibits much slower response and larger

overshoot, which implies higher packet loss ratio and larger .

Performance in the Presence of Short-lived TCP Flows: In this experiment, we investigate the system performance in the presence of short-lived TCP ﬂows. The single

bottleneck topology we considered is depicted in Figure 3.4, where ½¾¼ greedy FTP ﬂows

53 1

0.995

0.99

0.985

Link utilization 0.98

VS control PI 0.975 REM RED

0.97 50 100 150 200 250 Number of connections

Figure 3.10: Link utilization w.r.t. the number of TCP ﬂows

and ¼ short-lived TCP flows share the link. Each short-lived TCP flow is configured to

randomly turn ON/OFF in each s period. As shown in Figure 3.13, VS AQM has good robustness against the disturbance of such short-lived TCP ﬂows, and can signiﬁcantly improve the system performance.

Robustness w.r.t. RTT: As discussed in Section 3.3, the VS controller has very good

robustness against the uncertainty of , which is essential as an AQM scheme. In this experiment, we change the propagation delays in the network topology (Figure 3.4) and

evaluate the robustness of the VS controller. First, we set the propagation delay between

½¼ ¾ ¾ router ½ and to ms and the delays between the routers and the end hosts ms, which

corresponds to a much smaller than the nominal one. The regulated queues are depicted in Figure 3.14 where the VS controller is still capable of robustly stabilizing the

54 4 VS control PI 3.5 REM

2.5

1.5

Packet loss ratio (%) 1

0.5

0 50 100 150 200 250 Number of connections

Figure 3.11: Packet loss ratio w.r.t. the number of TCP ﬂows

queue length. Note that for this scenario, RED (Figure 3.14 (a)) also exhibits good performance.

Meanwhile, we also consider the scenario with a much larger , where we set the

½¾¼ ¾

propagation delay between ½ and to ms and the delays between the routers and the

½¼¼ end hosts ¾¼ms. We repeat the simulation with the ﬂow number and obtain the system responses depicted in Figure 3.15. As we can observe, the VS controller contin- ues to exhibit good performance in the sense of queue stability and fast response, which outperforms other peer schemes.

3.5.3 The Scenario of Multiple Bottleneck Topology

Using the multiple bottleneck network topology depicted in Figure 3.5, we study the

behaviors of different AQM schemes in the presence of cross trafﬁc. We set ¿¼¼ TCP

55 300 VS AQM PI AQM

250

200

150

Queue size (packet) 100

0 0 20 40 60 80 100 120 140 160 180 200 Time (second)

Figure 3.12: Comparison of PI and the VS controller

connections with sender at the left hand side and receivers at the right hand side, with

¼ ¼

TCP ﬂows ( Ö Ó×× ) for each cross trafﬁc sender-receiver pair. The instantaneous

queues of Queue ¿ for different AQMs are depicted in Figure 3.16, and those of Queue

¾ ¿

are plotted in Figure 3.17. Note that Queue , and exhibit similar trends. Queue ½ and Queue are almost empty, indicating that these two links are not bottleneck links.

Similar results to Figure 3.16 and Figure 3.17 can be obtained under different TCP loads and different cross trafﬁc loads.

Once again, the VS controller shows much better performance than other AQM methods. In fact, the performance of RED in this experiment is sensitive to the network config- urations (e.g. TCP loads, cross traffic and propagation delays), which affirms coincidence to [13]. On the other hand, REM tends to mark too many packets and keeps too small a queue size, so that the link utilization is lower than other AQMs. Meanwhile, PI AQM

56 300

150 RED

0 0 10 20 30 40 50 60 70 80 90 100 300

150 PI

0 0 10 20 30 40 50 60 70 80 90 100 300

150 REM

0 0 10 20 30 40 50 60 70 80 90 100 300

150 VSC

0 0 10 20 30 40 50 60 70 80 90 100 Time (second)

Figure 3.13: Performance in the presence of short-lived TCP ﬂows

suffers a sluggish transient behavior and makes the queue full in its transient period, which

results in buffer overﬂow and packet losses.

ÐÑ ´µ

½ Remark 3.3 Note that the VS controller has a steady state error ( Ø )of

about ¼ packets in the above scenarios. In fact, the VS control is designed for the delay- free nonlinear system (3.6) and the corresponding sliding manifold (3.7) is also delay-free.

A more accurate model of the real network trafﬁc is the delayed version of (3.6)

´ µ ´ µ ´µ ´ µ

´µ ´ · µ´ µ

¾ ¾

´µ ´µ ´µ ´µ ¾

´µ ´µ

¼ (3.38)

´ µ ¼ which was proposed in [39]. The presence of time delays results in in the

system steady state, which causes the steady state error of the queue size.

57 (a) RED (b) PI 300 300

250 250

200 200

150 150

100 100

Queue size (packet) 50 Queue size (packet) 50

0 0 0 20 40 60 80 100 0 20 40 60 80 100 Time (second) Time (second) (c) REM (d) VS control 300 300

250 250

200 200

150 150

100 100 Queue size (packet)

Queue size (packet) 50 50

0 0 0 20 40 60 80 100 0 20 40 60 80 100 Time (second) Time (second)

Figure 3.14: Queue evolution using RED, PI, REM and VS control in the scenario with a much smaller propagation delay

3.6 Concluding Remarks

In this chapter, we developed a variable structure based AQM control scheme supporting ECN. We presented guidelines for designing the robust VS sliding mode controller directly for the nonlinear TCP dynamics. The robustness with respect to the RTTs and the number of the active TCP sessions was analyzed, and the asymptotic properties of the closed loop system was discussed. It was shown that the VS controller, from control theoretic point of view, has many desirable properties such as good robustness and fast system response. We also provided packet-level simulations using ns-2 in different scenarios to validate our results. The simulation experiments showed that the proposed AQM scheme performs better than a number of well-known AQM schemes in terms of packet loss ratio,

58 (a) RED (b) PI 300 300

250 250

200 200

150 150

100 100

Queue size (packet) 50 Queue size (packet) 50

0 0 0 20 40 60 80 100 0 20 40 60 80 100 Time (second) Time (second) (c) REM (d) VS control 300 300

250 250

200 200

150 150

100 100

Queue size (packet) 50 Queue size (packet) 50

0 0 0 20 40 60 80 100 0 20 40 60 80 100 Time (second) Time (second)

Figure 3.15: Queue evolution using RED, PI, REM and VS control in the scenario of a larger propagation delay

link utilization and queue ﬂuctuation. A challenging extension of this work is to consider

VS control for the TCP model in the presence of time delays as shown in (3.38).

59 200

100 RED

0 0 10 20 30 40 50 60 70 80 90 100 200

100 PI

0 0 10 20 30 40 50 60 70 80 90 100 200

100 REM

0 0 10 20 30 40 50 60 70 80 90 100 200

100 VSC

0 0 10 20 30 40 50 60 70 80 90 100 Time (second)

Figure 3.16: Evolution of Queue ¿ (in packet) using RED, PI, REM and VS control

200

100 RED

0 0 10 20 30 40 50 60 70 80 90 100 200

100 PI

0 0 10 20 30 40 50 60 70 80 90 100 200

100 REM

0 0 10 20 30 40 50 60 70 80 90 100 200

100 VSC

0 0 10 20 30 40 50 60 70 80 90 100 Time (second)

Figure 3.17: Evolution of Queue (in packet) using RED, PI, REM and VS control

CHAPTER 4 ½

Robust Controller Design for AQM and À -Performance Analysis

It has recently become clear that the dynamics of TCP/AQM can be described as a ½ nonlinear feedback system. In this chapter, we design À robust controllers for AQM

based on the linearized TCP model with time delays. For the linear system model exhibiting ½

LPV nature, we investigate the À -performance with respect to the uncertainty bound of

RTT (round trip time). Robust controller switching is further introduced to improve the system performance. Simulations are also provided to validate our design.

4.1 Introduction

Active Queue Management has recently been proposed in [20] to support the end-to-end congestion control for TCP trafﬁc regulation on the Internet. For the purpose of alleviating congestion for IP networks and providing some notion of quality of service (QoS), the

AQM schemes are designed to improve the Internet applications. Earliest efforts on AQM

(e.g. RED in [24]) are essentially heuristic without systematic analysis. The dynamic models of TCP ( [39, 54]) make it possible to design AQM in the literature of feedback control theory. We refer to [49] for a general review of Internet congestion control.

In [54], a TCP/AQM model was derived using delay differential equations. They further provided a control theoretic analysis for RED where the parameters of RED can be

61 tuned as an AQM controller [32]. In [33], a Proportional-Integral controller was developed

based on the linearized model of [54]. Their controller could ensure robust stability of the

closed loop system in the sense of gain-phase margin of the PI AQM [33, 34]. A chal-

lenging nature in the design of AQM is the presence of a time delay which is called RTT ½

(round trip time), and the time delays are usually time varying and uncertain. In [64], À

optimization method was proposed for AQM controller design, which allows for parameter

uncertainties of RTT, the number of TCP connections and available link capacity. In a sim- ½ ilar fashion, we develop in this chapter robust AQM controllers based on the À control techniques for SISO inﬁnite dimensional systems [25, 80]. However, the model we con-

sidered here is a LPV system with RTT being the scheduling parameter. We also analyze ½ the À performance for the robust controllers with respect to the uncertainty bound of the

scheduling parameter RTT. Our results show that a smaller operating range of RTT results ½

in better À performance of the AQM controller, which indicates that switching control

among a set of robust controllers designed at selected smaller operating ranges can have ½

better performance than a single À controller for the whole range.

4.2 Mathematical Model of TCP/AQM

In [54], a nonlinear dynamic model for TCP congestion control was derived, where

the network topology was assumed to be a single bottleneck with homogeneous TCP

ﬂows sharing the link. The congestion avoidance phase of TCP can be modeled as AIMD

(additive-increase and multiplicative-decrease), where each positive ACK increases the

´µ ´µ TCP window size by one per RTT and a congestion indication reduces by

half. Aggregating TCP ﬂows through one congested router results in the following TCP

dynamics [34, 54]:

´ ´µµ ½ ´µ

´µ ´ ´µµ

´µ ¾ ´ ´µµ

´µ

´µ ´µ ´µ

(4.1)

´µ

´µ ¼ ´µ ½ ´µ where is the RTT, is the marking probability, is the queue length at

the router, and is the link capacity. Note

´µ

´µ ·

´µ

where Ô is the propagation delay and is the queuing delay.

´µ ´µ ¼

Assume and , the operating point of (4.1) is deﬁned by

· Ô

¼ (4.2)

¼ (4.3)

¼ (4.4)

Æ Æ ¼ Let ¼ and , the linearization of (4.1) results in the following LPV

time delay system, [34],

´ µ×

Æ´µ ´ µ

´µ

(4.5)

Æ´µ ´ ´ µ · ½µ´ ´ µ ·½µ

½ ¾

where

¿ ¿

´ µ

(4.6)

´ µ

½ (4.7)

´ µ

¾ (4.8)

´ µ (4.9)

´µ ¾ ·

Ô Ñ#Ü Ñ#Ü

and Ô is the scheduling parameter of (4.5) where is the

Ä ´ µ ´µ

buffer size. Note that we employ to describe the LPV dynamic

¼ ¼

equations in Laplace domain at ﬁxed parameter values. ½

4.3 À Controller Design for AQM

Consider the nominal system

´ µ×

´ µ

´µ ´µ

¼ (4.10)

´ ´ µ · ½µ´ ´ µ ·½µ

½ ¼ ¾ ¼

´µ

¼ ¼ where ¼ is the nominal RTT. We would like to design a robust AQM controller

for the nominal plant (4.10) so that

´µ ´µ ¾ ¢ ¡ ·¡

¼ ¼

(i) ¼ robustly stabilizes for ;

(ii) The closed loop nominal system has good tracking of the desired queue length ¼

which is a step-like signal.

Notice that the plant (4.5) can be written as

´µ ´µ´½ · ¡ ´µµ

(4.11)

¡ ´µ

where is the multiplicative plant uncertainty.

´ (¡ µ ¼

It can be shown that an uncertainty bound satisfying

´ (¡ µ

¡ ´µ ´µ ¾ Ê

×) ×)

(4.12) ¾

is (see details of the derivation in Section 4.4)

´ (¡ µ

´µ · ·

(4.13)

¡ where , and are defined in (4.29). Note that once ¼ and are fixed, these coefficients are fixed.

Combining the robust stability and the nominal tracking performance condition, we ½

come up with a two block inﬁnite dimensional À optimization problem as follows:

´µ ´µ ¼

Minimize , such that robust controller ¼ is stabilizing and

Û Û

´µ ´µ

½ ¼

Û Û

(¡ µ

´ (4.14)

Û Û

´µ ´µ

¾ ½

where

´µ ´½ · ´µ ´µµ

¼ ¼ ¼

´µ ½ ´µ ´µ ´µ´½ · ´µ ´µµ

¼ ¼ ¼ ¼ ¼ ¼

´µ½

and ½ is for good tracking of step-like reference inputs.

By applying the formulae given in [80] and [25], the optimal solution to (4.14) can be

determined as follows [64]:

´ ´ µ · ½µ´ ´ µ ·½µ ½

½ ¼ ¾ ¼

´µ

¼ (4.15)

´ µ ½·´µ· ´µ ¼

where

´µ

(4.16)

´µ

and is a ﬁnite impulse response (FIR) ﬁlter with time domain response

µ ´ · µ #Ó×´

½ Ø

·´ · µ×Ò´ µ ´µ ´ µ

for ¼ (4.17)

¼ otherwise

where

¾ ¾

¾ ¾ ¾ ¾ ¾

´ ¾µ

·¾ (4.18)

¾ ¾ ¾ ¾

with the unique positive root of

¾ ¾ ¾ ¾ ¾ ¾ ¾ ¾ ¾

¾ ´¾ µ · ´ µ

¿ ¾ ¾ ¾

· ´ µ ¼

(4.19)

¾ ¾

The optimal À performance cost is determined as the largest root of

´ µ×

½ (4.20)

´ · µ´ · · µ ½

Note that an internally robust digital implementation of the À AQM controller (4.15)

includes a second-order term which is cascaded with a feedback block containing an FIR

´µ ´ µ

× × ﬁlter . The length of the FIR ﬁlter is ¼ , where is the sampling period.

4.4 Multiplicative Uncertainty Bound

Recall (4.5) and (4.10), we have

´µ ´µ

×)

¬ ¬

´ µ× ´ µ×

¬ ¬

´ µ ´ µ

¬ ¬

´ ´ µ ·½µ´ ´ µ ·½µ ´ ´ µ · ½µ´ ´ µ ·½µ

½ ¾ ½ ¼ ¾ ¼

×)

¬ ¬

¡×

¬ ¬

´ µ ´ µ

¬ ¬

´ ´ µ ·½µ´ ´ µ ·½µ ´ ´ µ · ½µ´ ´ µ ·½µ

½ ¾ ½ ¼ ¾ ¼

×)

¬ ¬

¡×

¬ ¬

´ µ ´ µ · ´ µ ´ µ

¬ ¬

´ ´ µ · ½µ´ ´ µ ·½µ

½ ¾

×)

¬ ¬

Ã ´ µ´Ì ´ µ×·½µ´Ì ´ µ×·½µ

¼ ½ ¾

¬ ¬

´ µ

´Ì ´ µ×·½µ´Ì ´ µ×·½µ ¬ ¬

½ ¼ ¾ ¼

¬ ¬

´ ´ µ · ½µ´ ´ µ ·½µ

½ ¾

×)

¬ ¬

¡×

¬ ¬

· ´ µ

¬ ¬

´Ì ´ µ×·½µ´Ì ´ µ×·½µ

½ ¾

¬ ¬

´ ´ µ · ½µ´ ´ µ ·½µ

½ ¾

×)

¬ ¬

´ ´ µ ´ µ ´ µ ´ µµ ·´¡ ·¡ µ

½ ¾ ½ ¼ ¾ ¼ ½ ¾

¬ ¬

· ´ µ

¼ (4.21)

¬ ¬

´µ

×)

where

´µ ´ ´ µ · ½µ´ ´ µ · ½µ´ ´ µ · ½µ´ ´ µ ·½µ

½ ¾ ½ ¼ ¾ ¼

¡ ´ µ ´ µ ¡ ´ µ ´ µ ¡ ´ µ ´ µ ¡ ´ µ

¼ ½ ½ ½ ¼ ¾ ¾

¼ , , and

´ µ ¼ ¾ .

Note that

¬ ¬

¡×

¬ ¬

×)

and

¬ ¬

´ ´ µ · ½µ´ ´ µ ·½µ

½ ¾

¬ ¬

ÑÜ´ µ

½ ¾

¬ ¬

×)

where

ÑÒ ´ µ ¾ ·

½ Ô Ô Ñ#Ü Ô

ÑÒ ´ µ ¾ ·

¾ Ô Ô Ñ#Ü

which are straightforward from (4.7) and (4.8). Thus

¬ ¬

¡×

¬ ¬

¬ (4.22)

´Ì ´ µ×·½µ´Ì ´ µ×·½µ

½ ¾

¬ ¬

ÑÜ´ µ

½ ¾

×)

Recall

¡ ´ µ ´ µ ´ µ ´ µ

½¾ ½ ¾ ½ ¼ ¾ ¼

´ ´ µ·¡ µ´ ´ µ·¡ µ ´ µ ´ µ

½ ¼ ½ ¾ ¼ ¾ ½ ¼ ¾ ¼

¡ ¡ · ´ µ¡ · ´ µ¡

¾ ½ ¼ ¾ ¾ ¼ ½ ½ (4.23)

We have

¬ ¬

¡ ·´¡ ·¡ µ

½¾ ½ ¾

¬ ¬

´µ

×)

¬ ¬

¡ · ´¡ ·¡ µ

½¾ ½ ¾

¬ ¬

´µ

×)

¬ ¬

¡ ¡ · ´ µ¡ · ´ µ¡ ¡ ·¡

¬ ¬

½ ¾ ½ ¼ ¾ ¾ ¼ ½ ½ ¾

¬ ¬

´Ì ´ µ×·½µ´Ì ´ µ×·½µ´Ì ´ µ×·½µ´Ì ´ µ×·½µ

½ ¾ ½ ¼ ¾ ¼

¬ ¬

ÑÜ´ ´ µ ´ µµ

½ ¼ ¾ ¼

×)

¡ ¡ ¡ ¡ ¡ · ¡

½ ¾ ½ ¾ ½ ¾

· · ·

´ µ ´ µ ´ µ ´ µ ÑÜ´ ´ µ ´ µµ

½ ¼ ¾ ¼ ½ ¼ ¾ ¼ ½ ¼ ¾ ¼

Invoking (4.21) and (4.22), we have

´µ ´µ

¼ ×)

¡ ¡ ¡

½ ¾

· ¡ · ´ µ´ · ´ µ

´ µ ´ µ

µ ÑÜ´

½ ¼ ¾ ¼

¾ ½

¡ ¡ ¡ · ¡

½ ¾ ½ ¾

· µ

· (4.24)

´ µ ´ µ ÑÜ´ ´ µ ´ µµ

½ ¼ ¾ ¼ ½ ¼ ¾ ¼

Deﬁning

´ · µ

Ô Ñ#Ü

ÑÜ ´ µ ¾ ·

Ô Ô Ñ#Ü

and assuming

´ µ ´ µ

Ì (4.25)

the additive uncertainty (4.24) can be rewritten as

´µ ´µ ¡

¼ ×)

´ (¡ µ

´ µ

¼ Ì Ì

½ ¾

· ´¡ µ ·´

´ µ ´ µ

µ ÑÜ´

½ ¼ ¾ ¼

¾ ½

´ µ ´ µ´ · µ ´ µ

¼ Ì ¼ Ì Ì ¼ Ì

¾ ½ ¾ ½

· · µ¡ ·

´ µ ´ µ ÑÜ´ ´ µ ´ µµ

½ ¼ ¾ ¼ ½ ¼ ¾ ¼

(4.26)

¡ ´µ

With (4.11) and (4.26), the multiplicative uncertainty can be bounded by

´ (¡ µ

¡ ´µ ¡ ´µ ´µ

×) ¼ ×) ×)

(¡ µ

´ (4.27)

¼ ¾

where

´ (¡ µ

µ · ·

´ (4.28) ¾

with

´ (¡ µ

´ µ

¡ ´ ´ µ· ´ µµ

½ ¼ ¾ ¼

´ (¡ µ

´ µ

¡ ´ µ ´ µ

½ ¼ ¾ ¼

´ (¡ µ

(4.29)

´ µ

¼ ½

4.5 À -Performance Analysis

À ´µ

As shown in Section 4.3, the AQM controller (4.15) is designed for

¾ ¢ ¡ ·¡

and allows for . In this section, we would like to investigate ½ the À -performance for the corresponding closed loop system, which indicates the system

robustness and system response.

¢ ¢

½ ¾

´×µ ´×µ

½ ¾

¾ ¾

¡ ·¡

½ ½ ½ ¾ ¾ ¾

·¡

½ ½

¡ ·¡

¼ ¼ ¼

´×µ

¢ ¢ ¢ ¾

Figure 4.1: Partition of by ½ and

À ´µ ´µ

Deﬁne the -performance of controller ¼ with respect to as follows:

Û Û

´µ ´µ

Û Û

´ µ

(¡ µ

´ (4.30)

Û Û

´µ ´µ ´µ ´µ

¼ ¼

¾ ½

¾ ¢ ¡ ·¡ ¼

for any ¼ , where

´µ ´½ · ´µ ´µµ ¼

(4.31)

´ (¡ µ

´µ ´µ

here the term ¼ can be seen as a bound on the additive plant uncer- ¾ tainty.

Furthermore, we deﬁne

×ÙÔ ´ µ

(4.32)

¾¢

´µ ´µ

which corresponds to the worst system response of controller ¼ for plant with

¾ ¡ ·¡ ¼

¼ . Notice that a smaller means better performance of the robust

controller within the operating range ¢.

Particularly, we are interested in the scenario depicted in Figure 4.1, where ¢ is equally

¢ ¡ ·¡ ¢ ¡ ·¡ ¡

½ ½ ½ ½ ¾ ¾ ¾ ¾ ¾ ½

partitioned by ½ and , with

¡ ¾ ¢ ½ ¾ À ´µ

.For , , we design controller obeying (4.15) with the

´µ ´µ

nominal plant . Similar to (4.30) and (4.32), we have

Û Û

´µ ´µ

Û Û

´ µ

(¡ µ

´ (4.33)

Û Û

´µ ´µ ´µ ´µ

¾ ¢ ½ ¾

for any , and

×ÙÔ ´ µ ½ ¾

(4.34)

¾¢

´µ ´½ · ´µ ´µµ

where is deﬁned similarly to (4.31). ½

In what follows, we provide numerical analysis of the À performance with respect to ½¼

the operating ranges and corresponding controllers shown in Figure 4.1. Assume ,

¼¼ ¡ ¼¾ ¼ À ´ µ ½ ¾ ´ µ

, , and ¼ , the performance and can be

¼ numerically obtained from (4.30) and (4.33). As depicted in Figure 4.2, it is straightforward

to have

¡ ¡

½ ¾

ÑÜ´ µ ¾ ½¼

½ ¾

which means that the partition of Figure 4.1 can improve system performance in the sense ½

of smaller À performance cost. In fact, it is a general trend that

¡ ¡

½ ¾

ÑÜ´

µ (4.35)

½ ¾

which can be further veriﬁed by Figure 4.3, Figure 4.4, and Figure 4.5, where is chosen

¾¼¼ ¼¼ ¼¼ from ½¼¼ to , from to .

120

100 Θ

80 Θ Θ 1 2

60 γ (θ) C performance 0 ∞ H 40

γ (θ) C 20 1 γ (θ) C 2

0 0.3 0.35 0.4 0.45 0.5 0.55 0.6 0.65 0.7

½ Figure 4.2: À performance with respect to

Based on the observation of better performance obtained by the partition shown in ½

Figure 4.1, it is natural to consider switching robust control among a set of À controllers,

71 1400

1200

1000

800 0 ∆θ C γ 600

400

200

0 200 180 600 TCP 160flow number 550 140 500 120 450 Link capacity C

N 100 400

Figure 4.3: Performance cost w.r.t. and

each of which is designed for a smaller operating range. We provide in Section 4.6 the

simulation results of switching control between two robust controllers.

4.6 Simulations

The closed loop system with the determined controllers is implemented in MATLAB to ½

validate the controller design as well as the À performance analyzed in previous sections.

½¼ ¼¼

We assume the TCP ﬂow number , the link capacity packets/sec. The

¼¿ ½¼¼ ¼

propagation delay Ô is set to be sec and the desired queue size is packets.

¼ ¼

Therefore, the nominal RTT is sec ( ¼ ), which is straightforward from (4.2).

72 26

25.5 1 1 ∆θ C γ 25

24.5

24 200 180 600 TCP flow160 number 550 140 500 120 450

N 100 400 Link capacity C

Figure 4.4: Performance cost w.r.t. and

4.6.1 The Case of a Single Controller

¡ ¼¾ ´µ ¡ ¡ ¼½ ´µ ´µ

½ ¾ ½ ¾ We use in the design of ¼ and in and .

The following three scenarios are considered:

¯ ´µ ´µ ¼

Assuming the plant is the nominal one, i.e. , we implement controller

´µ ´µ ´µ

½ ¾

¼ as well as and . It is shown in Figure 4.6 that the three controllers

can stabilize the queue length because the nominal value ¼ is within the operating

¢ ¢ ¢ ´µ

¾ ¼

range of , ½ , and . Note that the system response of is better than the

other two due to the fact that it achieves the optimal performance at ¼ .

¯ ¡ ¼¿

¼ ½ ¾

Assuming ¼ , we implement controller and ( is not eligible

½ in this scenario). As depicted in Figure 4.7, ¼ and can robustly stabilize the

73 7.2

6.8 2 2 ∆θ C γ 6.6

6.4 200 180 600 TCP 160flow number 550 140 500 120 450

N 100 400 Link capacity C

Figure 4.5: Performance cost w.r.t. and

queue length. Observe that the system response of ½ is better because it has much ½

smaller À performance cost, which has been shown in Section 4.5.

¯ ·¡ ¼

Similarly, we choose ¼ and repeat the simulation for controller

¾ ½

¼ and ( is not eligible). As depicted in Figure 4.8, the two controllers can ½

robustly stabilize the queue length and their system responses coincide with the À

performance analysis given previously.

The above simulations show that the proposed robust AQM controllers have good per-

formance and robustness in the presence of parameter uncertainties. Meanwhile, the system ½ responses also afﬁrm a good coincidence with the À performance analysis in Section 4.5.

74 160

140

120

100

60 Queue length q(t) (packet) 40

With C 20 0 With C 1 With C 2 0 0 10 20 30 40 50 60 70 80 90 100

time t (second)

½ ¾ ¼ Figure 4.6: System responses of ¼ , and at

4.6.2 The Case of Switching Control

Motivated by the analysis in Section 4.5, we perform control switching in this exper-

iment. We assume the same simulation conﬁguration as Section 4.6.1 and investigate the

closed loop system performance in the presence of switching between controller ½

´µ ¾ ¢

and ¾ for a slow time varying signal . For the purpose of comparison, we also

À provide the system response with a single controller ¼ . As depicted in Figure 4.9 and

Figure 4.10, the switching control method has better transient behavior in terms of smaller

overshoot, faster convergence and less oscillations. Note that the large oscillations around

¼ on both plots are due to the fact that is not assumed to be time varying in the proposed design. Instead, we assume it is piece-wise constant but uncertain in the derivation of the system unertainty bound (see Section 4.3 and 4.4 for details).

75 120

100

40 Queue length q(t) (packet)

20 With C 1 With C 0 0 0 10 20 30 40 50 60 70 80 90 100

time t (second)

¡ ¼¿

½ ¼ Figure 4.7: System responses of ¼ and at

4.7 Conclusions

We provided in this chapter the guidelines of designing robust controllers for AQM,

½ ½ À where the À techniques for inﬁnite dimensional systems were implemented. The -

performance was numerically analyzed with respect to the bound of the scheduling param-

½ À eter . It was shown that smaller uncertainty bound could result in better -performance of the corresponding closed loop systems. Accordingly, we proposed switching control between two robust controllers which outperforms a single controller. Simulations were conducted to validate the design and analysis.

76 200

180

160

140

120

100

60 Queue length q(t) (packet)

20 With C 2 With C 0 0 0 10 20 30 40 50 60 70 80 90 100

time t (second)

·¡ ¼

¾ ¼ Figure 4.8: System responses of ¼ and at

0.8

0.7

0.6

0.5 (sec) θ 0.4

0.3

0.2 0 20 40 60 80 100 120 140 160

300 q 0 250 q(t) 200

150

100 q(t) (packet) 50

0 0 20 40 60 80 100 120 140 160

time t (sec)

Figure 4.9: A single controller ¼

77 0.8

0.4 (sec) θ

0 0 20 40 60 80 100 120 140 160 300 q 0 200 q(t)

100 q(t) (packet) 0 0 20 40 60 80 100 120 140 160 3

2 Controller C Controller C 1 2 1 Controllers

0 0 20 40 60 80 100 120 140 160

time t (sec)

¾ Figure 4.10: Switching control between ½ and

CHAPTER 5 ½

On Switching À Controllers for a Class of LPV Systems

It has been shown in Chapter 4 that linearization of the nonlinear TCP/AQM model results in a class of LPV (Linear Parameter Varying) systems scheduled along a measurable parameter trajectory. For the purpose of better system performance and larger robustness with respect to the scheduling parameter (e.g. RTT for TCP/AQM dynamics), switching control can be introduced where the switching among candidate controllers is determined by a high-level switching logic or supervisor. Apparently this control method gives rise to the hybrid behavior of the closed loop system. A fundamental aspect of switching control is

the stability in the presence of switching sequences. To this end, we consider the stability ½ conditions on switching À controllers for a class of LPV systems in this chapter. The candidate controllers are selected from a given controller set according to the switching rules based on the scheduling variable. We provide sufﬁcient conditions to guarantee the stability of the switching LPV systems in terms of the dwell time and the average dwell time. Our results are illustrated with an example, where switching between two robust controllers is performed for an LPV system.

5.1 Introduction ½

This chapter addresses a switching À control strategy for a class of linear parameter varying (LPV) systems scheduled along a measurable parameter trajectory. LPV systems are ubiquitous in chemical processes, robotics systems, automotive systems and many man- ufacturing processes. Particularly, the dynamics of TCP/AQM exhibit parameter varying nature with scheduling parameter being RTT. The analysis and control of LPV systems has been studied widely [6, 7, 44, 60, 72, 79, 85, 87]. A systematic gain scheduling method was developed in [6,7] based on LMI (Linear Matrix Inequality) algorithms; [87] provided sufﬁcient conditions for the stability of LPV systems with parameter-varying time delays, where gain scheduled controller was designed based on LMIs. Fast gain scheduling was considered in [44], where derivative information on the scheduling variable was utilized in a new control law. In a very recent publication [85], an improved stability analysis for

LPV systems was given and the robust gain-scheduled controller was constructed in terms of LMIs. We refer to [70] for a general review on gain scheduling methods.

An alternative method is switching control where a family of controllers are designed at different operating points and the system performs controller switching based on the switching logic (see [29] and references therein). As stated in [9], a challenging point of switching control is its hybrid nature of the continuous and discrete-valued signals. Sta- bility analysis and the design methodology have been investigated recently in the literature of hybrid dynamical systems [30, 45, 55, 71, 75]. For LTI systems, [75] provided sufficient conditions on the stability of the switching control systems based on Filippov solutions to discontinuous differential equations and Lyapunov functionals; [55] proposed a dwell-time based switching control, where a sufficiently large dwell-time can guarantee the system stability. A more flexible result was obtained in [30], where the average dwell-time was

80 introduced for switching control. Besides stability analysis, a number of results have been

published on related topics, such as optimal control [71] and tracking problem [31].

Due to the time-varying and the hybrid natures of the switching LPV systems, it is

challenging to explore the stability conditions and switching schemes similarly to those for

LTI systems. Theoretical and practical results have been presented in recent publications

[9, 45, 70]. In particular, [9] analyzed the bounded amplitude performance and derived the ½

conditions related to dwell time, and [45] proposed switching À controllers for nonlinear

systems which exhibits LPV nature after linearization. In this chapter, we discuss the

½ ½ À switching À control methodology for a class of LPV systems, where each candidate

controller guarantees robust property at the selected operating condition and the switching

rules are developed to cover a large operating range. By constructing Lyapunov functionals

for time-varying systems as [17, 43], this work extends the stability results of [30, 55] to

LPV systems.

5.2 Problem Deﬁnition

The general structure of the switching control scheme that we consider in this chapter is

Ò Ò

Û Ù

¾ Ê ¾ Ê

depicted in Figure 5.1, where Ô is the exogenous input, is the control in-

Ò Ò

Þ Ý

¾ Ê ¾ Ê

put, Ô is the regulated output and is the measured output. The LPV system

´µ ´µ ¾ Ê

depends on a parameter , where is assumed to be continuously differentiable

¾ ¢ ¢

and where is a compact set.

´µ ´µ

Under further assumptions of and , the stability and performance ½ analysis of LPV system (1) can be formulated in terms of LMIs, where gain scheduling À

controllers can be derived based on convex optimization using LMIs [6, 7, 60, 70, 85]. In ½

the present work, we propose to construct a family of À controllers designed at selected

Ã ´×µ

ËÛ Ø Ò

ÄÓ

Ã ´×µ

Û Þ

Ô Ô

ËÝ×ØÑ

ÄÈÎ

Ù Ý

È ´ µ

Ã ´×µ Ð

Figure 5.1: The switching control system

½ ¾ operating points , , and perform controller switching for the above LPV system, which allows for more freedom in the controller design and has the advantage of

simplicity.

Ã ´µ ½ ¾

The candidate controllers are chosen from a controller set ,

´µ À ¢

where is an LTI controller designed at . Consider an operating range ,

¾ ¢ ´ ¡ µ ¡

, the LPV system in Figure 5.1 can be represented as , where is the

time varying portion of the LPV system, is the LTI portion with nominal value and

Ù denotes the upper LFT (Linear Fractional Transformation). The closed loop system is

depicted in Figure 5.2, where is the nominal transfer function at a speciﬁed :

¾ ¿

´ µ ´ µ ´ µ

½ ¾

´µ ½ ¾

´ µ ´ µ ´ µ

½½ ½¾

½ (5.1)

´ µ ´ µ ´ µ

¾ ¾½ ¾¾

À ´µ

and an optimization problem is deﬁned as ﬁnding for the LTI plant such that

¾ ¢

(i). The closed loop system is asymptotically stable for ;

Ò' ×ÙÔ ´µ

(ii). satisﬁes (i) for the smallest possible , where

Û ¼

Ö Ö

Ô Ô

LTV portion ∆ θi

wr zr wp LTI portion zp u y Gθi LPV plant LTI controller Ki

Mθi

Figure 5.2: LPV plant and the controller

¡ Ä

¾ Ö

Denote ( to be the induced norm and let to be the transfer function from

½ ¡

½ (¾

to Ö . A sufﬁcient condition on robust stability satisfying (i) is and

½, which can be obtained by applying small gain analysis [6, 70, 94]. The above treatment

½ ½ À results in the À controller design for the LTI system, where standard optimization

methods can be employed [18]. The state space expression of each candidate controller

´µ

is given by

Ã Ã

´µ ½ ¾

(5.2)

Ã Ã

´µ ¡ ½

(¾

Note that robustly stabilizes the LPV system for , which can be guaran-

teed by properly choosing , and , such that

¾ ¢ ´µ

(5.3)

¢ ´µ Accordingly, the robust bound can be determined for each candidate controller .

In order to cover a large operating range ¢, we need to develop stable switching

schemes over Ã . Obviously, a necessary condition for stable switching is:

¢ ¢

(5.4)

5.3 Main Results

Applying the switching rules over Ã and invoking (5.1) and (5.2), we obtain the closed

¾ ´ µ ½ ¾

loop A-matrix Ð , where

½ ½

· ´ µ ´ µ

¾ Ã ¾¾ Ã ¾ ¾ Ã ¾¾ Ã

´ µ

½ ½

´ µ · ´ µ

Ã ¾¾ Ã ¾ Ã Ã ¾¾ Ã ¾¾ Ã

For switching LTI systems, it has been shown in [55] that a sufﬁciently large dwell time

can guarantee stability; and [30] provided a more ﬂexible and stronger result based on

the average dwell time. We claim that similar results can be obtained for switching LPV

systems.

Consider the following switching LPV system:

´µ ´ ´µµ ´µ ¼

Õ (5.5)

½ ¾ ´µ

where is a piecewise constant signal taken values on the set , i.e.

· Ø

¾ ¾ µ ¾ ¼

·½

, , for , where , , is the switching time instant. Here

¾ ´ ´µµ ¾´µ ¾ ¢ , which is a family of parameter varying matrices.

We further assume that:

¼ ¾ ¢ ´ µ

H1. There is a , such that for any , the eigenvalues of have real parts

¾ ¾

no greater than , ;

¼ ´ ´µµ ¾

H2. , , ;

- -

/- ´ µ

¼ ¾

H3. , , ;

. . /

where ¡ denotes the (pointwise in time) Euclidean norm of a time-varying vector and

the corresponding induced norm on matrices.

The dwell time based switching rule set is denoted by . , where is a constant

¾ ´µ

such that for any . , the distance between any consecutive discontinuities of ,

¾ ¼

·½ .

, , is larger than [30, 55]. Clearly

½ . ¾ . ½ . ¾ . (5.6)

A sufﬁcient condition on the minimum dwell time to guarantee the stable switching can now be given using Lyapunov stability analysis (a similar result is obtained in [45], using the same technique, for switched gain scheduling controllers in uncertain nonlinear

systems).

¼ ¼

Theorem 5.1 Assume (H1-H3). Then there exist ﬁnite constants , . , such that

the switching LPV system (5.5) is stable in the sense of Lyapunov for any switching rule

¾ ´µ

. if .

Proof. First we notice that

´ ´µµ ´ ´µµ · ¾

(5.7)

is Hurwitz, which is straightforward from (H1). Let

- ´ ´Øµµ0 - ´ ´Øµµ0

´µ ¾

(5.8) ¼ 85

´µ

Note that is well deﬁned, continuously differentiable, and is the unique positive-

deﬁnite solution of

´ ´µµ ´µ· ´µ ´ ´µµ

(5.9)

i.e.

´ ´µµ ´µ· ´µ ´ ´µµ ¾ ´µ

(5.10)

Deﬁne a family of Lyapunov functions

Î ´ ´µµ ´µ ´µ ´µ ¾

(5.11)

for the following LPV systems respectively

´µ ´ ´µµ ´µ ¾

(5.12)

¼ ¾

Recall that there exist positive constants , , depending only on and

, such that

¾ ¾

´µ ´ ´µµ ´µ ¼

(5.13)

We refer to [17, 43] for details.

µ ´µ ¾ ¾

·½

Consider an arbitrary switching interval , where , for

·½

. Using the quadratic form of as shown in (5.11), a straightforward calculation

´ ´µµ

gives the time derivative of along the trajectory of (5.12)

Ì Ì Ì

´ ´µµ ´µ ´µ ¾ ´µ ´µ ´µ· ´µ ´µ ´µ ¾ µ

·½

(5.14)

Note that differentiating (5.8) with respect to gives

- ´ ´Øµµ0 Ì - ´ ´Øµµ0

´ ´µµ ´µ· ´µ ´ ´µµ ´µ

(5.15)

where

´ ´µµ ´µ ¼ ´ ´µµ

(5.16) 86

Invoking (H3) and Lemma 3 of [43] we have

´ ´µµ ´µ

´µ ´µ

(5.17)

where is a constant depending only on , and .

É - .

Deﬁne

½· ¾

ÑÒ

(5.18)

´µ ¼ ´µ

Since , we can pick up such that . Thus

´ ´µµ ´½ · ¾ µ ´µ

´µ ¾ µ

·½ (5.19)

where

´½ · ¾ µ ¼

(5.20) É

Recall (5.13) and (5.19), we have

´ ´µµ ´µ

(5.21)

´ ´µµ ´µ

Thus

´Ø Ø µ

¾Å

¾ µ ´µ ´ µ

·½

(5.22)

Choosing the minimum dwell time as follows:

¾ ÐÒ

ÑÜ

. (5.23)

¾ we claim that any switching rule . is stable in the sense of Lyapunov.

¾ ¼

·½ . In fact, recall the deﬁnition of the dwell time . ,wehave , .

Thus

´Ø Ø µ

¾Å

´ µ ÐÑ ´µ ÐÑ ´ µ

·½

Ø Ø Ø Ø

·½ ·½

´Ø Ø µ

·½

¾Å

´ µ

¾Å

´ µ ´ µ

´ µ ¼ ½ ¾ ´ µ ¼

Thus we have a decreasing sequence with upper bound

´¼µ , which indicates the stability of the overall switching system. Thus the proof is

complete.

Note that for switching LTI systems, we can set

ÑÒ Ñ#Ü

(5.24)

Ñ#Ü

where ÑÒ denotes the smallest singular value of and the largest singular

value of . ½

The dwell time condition in Theorem 5.1 can be applied to the switching À control problem discussed in Section 5.2. As depicted in Figure 5.3, two possible switching schemes [45] are (a) critical-point switching, (b) hysteresis switching. For the critical-point

switching, the stability of the closed-loop system cannot be guaranteed. In fact, in the worst

´µ ·½ case where oscillates within a neighborhood of ( , fast switching or chattering will

happen, which may violate the dwell time requirement. The following corollary addresses

½ Ã a sufﬁcient condition for the hysteresis switching scheme over À controller set .

Corollary 5.1 For the hysteresis switching over robust controller set Ã with operating ¢

range obeying (5.4), a sufﬁcient condition to guarantee Lyapunov stability is

(·½

´µ ÑÒÑÒ

(5.25)

. 88 Κ Ki+1

Ki (a) c i, i+1

θ Κ Ki+1

Ki d i, i+1 (b) ∆i ∆i+1 − − + + θi θi+1 θi θi+1 θ

t θ(t)

Figure 5.3: Switching logic

¢ ¢

·½ ·½

where ( is the hysteresis interval as shown in Figure 5.3.

´µ ´µ

·½

Proof. For simplicity, we consider only two neighboring controllers, i.e. and

µ ¾ ¼

·½ ·½

in switching time interval , . As discussed in Theorem 3.1,

. should be satisﬁed to guarantee stability of the switching system, which requires the

´µ .

currently working controller to hold on at least . In the worst case of switching

´µ ¾

·½ (·½

where oscillates around the center of the interval ( , with amplitude , the

´µ

·½ .

condition ( is sufﬁcient to guarantee stable switching. Taking all the

´µ possible controllers into consideration and invoking (5.3) and , we come up with

(5.25) and complete the proof.

89 Note that Theorem 5.1, as well as Corollary 5.1, is in fact conservative in the sense that the minimum dwell time should be satisﬁed, which does not allow for fast switching. In the following, we present another result based on the average dwell time for switching LPV systems, which can guarantee exponential stability of switching LPV systems in the more

general sense. £

Similar to [30], we deﬁne the average dwell time and the corresponding switching

£ ·

¼ ´ µ ¾ ¼ ¼

rule set #Ú as follows. For , let denote the number

´ µ

of discontinuities (switching number) of a switching signal in the time interval ;

#Ú is deﬁned as the set of all switching rules, , that satisfy:

´ µ ·

¼ (5.26)

where is called the average dwell time and ¼ the chatter bound. Obviously

£ £

#Ú

. .

In the rest of this section, a sufﬁcient condition on the exponential stability is given in term of the average dwell time, which is an extension of theorem 1 of [30] to the switching

LPV systems.

Theorem 5.2 Deﬁne as

ÑÒ

(5.27)

¾ ´¼ µ

where and can be obtained from (5.20) and (5.13) respectively. For ,

there exists such that the switching LPV system (5.5) is exponentially stable with

¼ ¼

decay rate no slower than for all the switching rules over #Ú , where is . any ﬁnite chatter bound.

¡¡¡ ¼

¼ ½ ¾ ¼

´Ø(Ø µ

Proof. Given time interval , where , denote Æ to be

´ µ

the switching time instants of in ¼ . Let

ÑÜ

(5.28)

Recall (5.22), we have

´Ø Ø µ

½ ¼

¾Å

´ µ ÐÑ ´µ ´ µ

½ ¼

Ø Ø

2´Ø Ø µ

½ ¼

´ µ

¼ (5.29)

¼ ´ µ ½

Iterating the above inequality from to ¼ yields

2´Ø Ø µ

Æ ´ØØ µ Æ ´ØØ µ ½

¼ ¼

´ µ ´ µ ¡¡¡

Æ ´Ø(Ø µ Æ ´Ø(Ø µ ½

¼ ¼

Æ ´Ø(Ø µ 2´Ø Ø µ

¼ ¼

Æ ´ØØ µ

´ µ

¼ (5.30)

Based on (5.22), (5.30),

Æ ´Ø(Ø µ·½ 2´Ø Ø µ

¼ ¼

´µ ´ µ

2´Ø Ø µ·´Æ ´Ø(Ø µ·½µ ÐÒ 3

¼ ¼

´ µ

¼ (5.31)

¾ ´¼ µ

For any , we deﬁne

ÐÒ

¼ and (5.32)

ÐÒ

¼ ¼

where is a constant. Based on the deﬁnition of #Ú , we come up with

´ µ ·

¼ ¼

which is equivalent to

´ µ· ´ µÐÒ ´ µ

¼ ¼ ¼ (5.33)

Thus

2´Ø Ø µ·Æ ´Ø(Ø µÐÒ3 2´Ø Ø µ

¼ ¼ ¼

´µ ´ µ ´ µ ¼ ¼ (5.34)

We conclude from (5.34) that the switching LPV system (5.5) is exponentially stable

for all switching rules over #Ú with decay rate no slower than . .

Recall (5.32), (5.27) and (5.28), we have

ÐÒ ÑÜ

ÐÒ

. (5.35)

ÑÒ

¾Å

Thus the average dwell time derived in Theorem 5.2 is larger than the minimum dwell

time . in Theorem 5.1. However, the former doesn’t require any minimum dwell time for

switching, which could allow for some fast switchings.

´µ

Note that we assume is a scalar function of time in this paper. For the scenario

´µ ¾ Ê being a vector, similar results can be easily obtained without further complica- tion.

5.4 Numerical Example ½

In this section, we apply the above switching À control method to the following LPV

Ä ´ µ ´µ

system shown in Figure 5.4. We employ to describe the LPV

¼ ¼

dynamic equations in Laplace domain at ﬁxed parameter values, by which the LPV plant

can be written as:

´½ µ´½ · µ

´µ

(5.36)

´½ · µ´ ·¾ ´ µ · µ´½ µ

¼ ¼

¼½ ½¼ ½ ´ µ ¼¼ ·¼¼ ´µ ´ · µ ¼ where , ¼ , , , and is periodical

obeying

¿ ¿

´µ´¿ · ¾ ×Ò´ µµ´Í´µ Í´ µµ · Í´ µ

Í´ µ · ´¿ · ¾ ×Ò´ µµÍ´ µ

¿ ¢ ½¼ Í´µ ¾ ¢ ½

where and is the unit step function. Thus and

´ µ ¾ ¼½ ¼

¼ .

Ï ¯

· · ·

Ã Ï È

Þ ½

Figure 5.4: Block diagram

×ÙÔ

We would like to design À controllers to stabilize the system and minimize ,

Û ¼

where the regulated output and the exogenous input are deﬁned as:

½ ½

¾ ¾

Note that ¾ is a ﬁctitious noise that we added so that the rank conditions of standard four

À ¾

block design can be satisﬁed [18]. The weighting functions ½ and are chosen as

´ · ½¼¼µ´ ·µ ¾ ¾

½ and respectively. ½

We consider switching À control scheme as discussed in the above sections, where we

À ¾¾

¾ ½

design two controllers ½ and at the selected operating points and

½ ¾

¾ respectively, and employ controller switching between and . The operating

· ·

¢ ½ ¿ ¢ ¾

½ ¾

range is chosen as ½ for controller , and

½ ½ ¾ ¾

½ ¾ for ¾ . The two candidate controllers and can be constructed using standard

93 ½

À optimization methods [18, 94]:

Ã Ã

½ ½

´µ

Ã Ã

½ ½

¿ ¾

·½ ¢ ½¼ ·½ ¢ ½¼ ·½½½ ¢ ½¼ · ¢ ½¼

¿ ¾

· · ½ ½¼ ·¾¿ ¢ ½¼ ·¾ ¢ ½¼ ·½¼

and

Ã Ã

¾ ¾

´µ

Ã Ã

¾ ¾

¿ ¾

½ ·½ ¢ ½¼ ·½ ¢ ½¼ ·½½¿ ¢ ½¼ · ¢ ½¼

¿ ¾

·¿ · ½½ ·¿¼ ¢ ½¼ ·¿¼ ¢ ½¼ ·½¼½

The following analysis shows that ½ and can robustly stabilize the LPV system

¢ ¢ ¾ within the operating range ½ and respectively.

Deﬁne

´µ ´µ ´µ ½ ¾

and assume

´µ ½ ¾

(5.37)

A sufﬁcient condition to guarantee robust stability is given by [94]:

´½ · µ ½ ½ ¾

(5.38)

As depicted in Figure 5.5, (5.37) can be satisﬁed by choosing

´ ·¾µ

´µ

´ · µ ´ · µ´ · µ´ · ½¾µ

¿¼´ ·¾µ

´µ

´ · µ ´ · µ´ · µ´ · ½¾µ

¾ Figure 5.6 shows that the robust stability condition (5.38) is satisﬁed for ½ and

respectively.

94 0 10 W1 e

−5 10 magnitude

−10 10 −1 0 1 2 10 10 10 10 frequency (rad/sec) 0 10 W2 e

−5 10 magnitude

−10 10 −1 0 1 2 10 10 10 10

frequency (rad/sec)

Figure 5.5: Weights for uncertainties with respect to ½ and

¢ ¢

¾ ½ ¾

Thus ½ and can robustly stabilize the LPV system with respect to and .

´ µ ¾ ¢ ´ µ ¾ ¢

½ ¾ ¾ Consider the closed-loop A-matrices ½ and , we can numerically

obtain the following parameters listed in Table 5.1:

É ½

0.8 1.0 261.59 3028.2 ¾

1.4 1.0 261.49 1190.9

Table 5.1: Parameters , , and É

95 ∞ H bound test 0 10 at θ=θ1 at θ=θ2

−1 10 magnitude −2 10

−3 10 −1 0 1 2 10 10 10 10 frequency (rad/sec)

Figure 5.6: Robustness test

¢ ½¼

Recall (5.18), we have . Also notice that

¢ ½¼ ´µ

½¼¼¼

½ ¢ ½¼ ¼½ ¾ ¾

Choosing and invoking (5.20), we have ½ and . Fur-

½¼¾ ¢ ½¼ ½ ¢ ½¼

thermore, we can pick such that , which is straightforward

´µ ½

from (5.27), (5.28) and (5.32). Thus, the switching scheme for belongs to #Ú , .

which is due to the fact that there are only 2 switchings per period (Figure 5.8). Based

on Theorem 3.2, we conclude that the switching LPV system with ½ and are stable. ½

The closed loop system with the determined switching À control scheme is simulated

using MATLAB. For the purpose of comparison, we also provide an À controller

½ ¾

designed at for the LPV system, by which the performance of a single

À ´µ ×Ò´¾¼¼¼µ · ½

controller can be simulated. The disturbance ½ is set to be

×Ò´¾¿¼¼¼µ · Æ ´µ ´µ ¼

, where Æ is a Gaussian distributed signal of mean and variance

¾ ¼ .

5 1

0 Disturbance n −5 0 5 10 15 20 25 30 35 40 45 19 X 10 1 2

Regulated output z −2 0 5 10 15 20 25 30 35 40 45 6

4 (t) θ 2

0 0 5 10 15 20 25 30 35 40 45

Time in 103 seconds ½

Figure 5.7: The case of a single À controller

First, we give the simulation result for the case of single À controller (for compar-

ison purposes) in Figure 5.7, where the divergence of the output signal is observed because

itself can not robustly stabilize the LPV system for the whole operating range . The ½

simulation results of the switching À control method are depicted in Figure 5.8. Note that

the system remains stable and the magnitude of the regulated output ½ is much smaller than

¾ ¢ the magnitude of the disturbance ½ for all .

97 4 K (s) 2 1 2 2

0 K (s) 1

Disturbance n −2 1 Controller switching

−4 0 20 40 0 20 40

0.2 6

1 5 0.1 4

0 (t) 3 θ 2 −0.1

Regulated output z 1

−0.2 0 0 20 40 0 20 40

Time in 103 seconds Time in 103 seconds ½

Figure 5.8: The switching À control method

Note that for the proposed switching control scheme, Theorem 5.1 is not valid. In fact,

the minimum dwell time . to guarantee stability in Theorem 5.1 is given by

¾ ÐÒ

½ ¢ ½¼ ÑÜ

ÑÒ .

½(¾

¼¼¼

where ÑÒ is the minimum distance between two consecutive switchings in our design, which is depicted in Figure 5.8. Meanwhile, Corollary 5.1 also turns out to

be too conservative for this design due to the fact that

½(¾

¾ ¢ ½¼ ÑÜ ´µ

½¼¼¼ ½¼ .

98 which violates (5.25). The analysis of this numerical example afﬁrms a good coincidence

with the discussion of Section 5.3. It suggests that Theorem 5.2 is a less conservative result

allowing faster switching.

5.5 Concluding Remarks ½

Switching À controllers are proposed for a class of LPV systems with slow parameter

variations. Controller robustness is combined with the switching policy, which results in ½

the hysteresis switching over a set of À controllers designed at selected operating points.

The stability analysis is provided in terms of the dwell time and the average dwell time. The ½

proposed switching À control method is illustrated by a numerical example, where the ½ comparison between the single À controller and our design is also given. The switching control methods considered in this chapter have possible applications in designing AQM with respect to TCP trafﬁc exhibiting a parameter varying nature.

99 CHAPTER 6

Conclusions

In this chapter, we provide a brief summary of this dissertation and suggest several research areas for future work.

6.1 Summary of Results

In this dissertation we have investigated various control methods for data ﬂow in communication networks.

Recent studies of high-resolution trafﬁc measurement discovered the self-similarity in

both LAN and WAN traffic. In Chapter 2, we introduced a two-degree of freedom rate ½ based flow controller, which includes a robust À control block and an LMMSE based adaptive capacity predictor. The former part can guarantee the robust stability against time- varying time delay uncertainties and the latter improves the transient response by predicting the self-similar cross-traffic. The LMMSE predictor was validated by data traces generated from packet-level simulations. Performance analysis was provided for the closed loop system at different time scales.

In Chapter 3, we developed a variable structure based control scheme in Active Queue

Management (AQM) supporting explicit congestion notiﬁcation (ECN). By analyzing the robustness and performance of the control scheme for the nonlinear TCP/AQM model, we

100 showed that the proposed design has good performance and robustness with respect to the

uncertainties of the round-trip time (RTT) and the number of active TCP sessions, which are central to the notion of AQM. Implementation issues were discussed and ns packet-level simulations were provided to validate the design and compare its performance to other peer schemes’ in different scenarios. The results showed that the proposed design signiﬁcantly outperforms the peer AQM schemes in terms of packet loss ratio, throughput and buffer

ﬂuctuation.

Linearization of the nonlinear TCP/AQM dynamics results in Linear Parameter Varying

(LPV) systems with time delays. In Chapter 4 we designed robust AQM controllers using

½ ½ À the À optimization method for inﬁnite dimensional systems and investigated the - performance with respect to the uncertainty bound of RTT. Robust controller switching was further introduced to improve the system performance, which was validated by simulations.

Finally, stability of switching control systems was addressed in Chapter 5. Motivated ½ by the LPV nature of the linearized TCP/AQM model, we proposed a switching À control method for the purpose of better system performance and larger robustness with respect to the scheduling parameter (e.g. RTT for TCP/AQM dynamics). A fundamental aspect of

switching control is the stability in the presence of switching sequences. To this end, we ½ considered the stability conditions on switching À controllers for a class of LPV systems, where the candidate controllers were selected from a given controller set according to the switching rules based on the scheduling variable. Sufﬁcient conditions were provided to guarantee the stability of the switching LPV systems in terms of the dwell time and the average dwell time.

101 6.2 Future Work

Analysis and control of the data ﬂow in communication networks prompts research in many fundamental areas. Speciﬁcally, we identify, from the perspective of control systems, some challenging topics for further investigation.

(1) The explicit rate-based ﬂow controller designed in Chapter 2 has been shown to have

good performance at different time scales. An interesting extension is to consider

prediction and control at multiple time scales to make better use of the multifractal

properties of network trafﬁc.

(2) The nonlinear TCP/AQM dynamics can be described by (3.38), which has time vary-

ing uncertain time delays. The variable structure AQM controller in Chapter 3 is

considered for delay free nonlinear systems. A challenging extension is to explore

sliding mode control for TCP trafﬁc obeying (3.38) in the presence of time delays.

(3) We assume a single bottleneck link for the network topology in the AQM designs of

this dissertation. The primary objective of congestion control is to ﬁnd robust control

protocols in the decentralized, scalable way for arbitrary network topology. Also,

we would be interested in exploring the real implementations of the control schemes

proposed in this thesis (e.g. building the control protocols into FreeBSD/Linux ker-

nels).

(4) The switching control scheme considered in Chapter 5 allows for only slow parame-

ter variations, which is indicated by Theorem 5.1 and Theorem 5.2. A further exten-

sion of this work would be switching control for LPV systems with fast parameter

variations.

102 (5) Most of the current results on stability analysis for switching control systems are

limited to delay free systems due to the inﬁnite dimensional nature of time delay

systems. Although the stability results of Chapter 5 can be used for time delay sys-

tems by Pad´e approximation, direct methods on switching time delay systems are

theoretically fundamental. Given the delay dependent stability conditions [19], we

are interested in exploring piecewise Lyapunov-Razumikhin functionals [27] for time

delay system in the existence of switching sequences, in the hope that a similar min-

imum dwell time as that of Chapter 5 can be provided to guarantee stability in the

sense of Lyapunov.

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