Robust Power Control of Optical Networks with Time-Delays. Nemanja Stefanovic

Robust Power Control of Optical Networks with Time-Delays by Nemanja Stefanovic A thesis submitted in conformity with the requirements for the degree of Doctor of Philosophy Graduate Department of Electrical and Computer Engineering University of Toronto Copyright c 2010 by Nemanja Stefanovic

Abstract Robust Power Control of Optical Networks with Time-Delays Nemanja Stefanovic Doctor of Philosophy Graduate Department of Electrical and Computer Engineering University of Toronto 2010 We study the stability of power control algorithms applied to optical networks in the presence of both time-delays and uncertainties. The objective of power control algorithms acting on optical networks is to ensure each signal channel attains an optimal optical signal-to-noise ratio (OSNR) value such that transmission errors are minimized. The inputs to the optical network are the transmitter powers and the outputs of the optical network are the OSNR values. The primal control algorithms adjust the channel powers at the transmitters using the channel OSNR values as feedbacks to attain OSNR optimality. We also present the dual control algorithm located at the links which transmits a channel price as an additional feedback to the primal control algorithms. Together, these are called primal-dual control algorithms. We present robust OSNR models for optical networks with multiple time-delays. Specifically, we consider additive system uncertainties, input multiplicative uncertainties on the signal powers, and transmitter noise uncertainties, all within a norm-bounded uncertainty framework. We analyze and modify both central cost based algorithms and game-theoretic based algorithms, with an emphasis on the latter, to ensure the stability of the closed-loop system. We apply time-delay stability analyses to exploit the structures of the closed-loop systems for each type of control algorithm. These techniques include frequency analyses, Lyapunov-Razumikhin techniques, and Lyapunov-Krasovskii techniques. Due to nonlinearities in the closed-loop system models, and their time-scale ii

separated dynamics, we apply singular perturbation theory modified to handle either Lyapunov-Razumikhin theory or Lyapunov-Krasovskii theory. Singular perturbation theory, modified for time-delays, allows us to decouple complicated closed-loop systems into two simpler subsystems, one on a slow time-scale, and the other on a fast time-scale. We develop stability conditions for primal algorithms applied to arbitrary networks with delays. We also develop stability conditions for primal-dual algorithms applied to singlelinks, single-sink networks, two channel networks, and multi-link networks with both time-delays and uncertainties. The main results are presented as either LMI conditions and algebraic criteria. Simulations verify the stability of the closed-loop systems in the presence of time-delays. In addition, the simulations show the stabilization of perturbed systems at the expense of transient convergence time. iii

Dedication I dedicate this work to my wife and family. iv

Acknowledgements I give special thanks to my supervisor, Prof. Lacra Pavel, for her guidance over the past six years. v

Contents 1 Introduction 1 1.1 Optical Network Overview.......................... 2 1.2 Literature Review............................... 6 1.2.1 Optical Network OSNR Model and Control Algorithms...... 7 1.2.2 Time-Delay in Communication Networks.............. 9 1.2.3 Uncertainty and Time-Delay in Communication Networks.... 13 1.3 Problem Statement.............................. 15 1.4 Thesis Organization.............................. 16 2 Review of Preliminary Theory 18 2.1 Time-Delay Stability Theory......................... 18 2.1.1 Definitions............................... 19 2.1.2 Razumikhin Stability Theory.................... 22 2.1.3 Krasovskii Stability Theory..................... 25 2.2 Robust Theory with Multiple Time-delays................. 27 3 Game-Theoretic and Central Cost Algorithms 31 3.1 Review of OSNR Models in Optical Networks............... 31 3.2 Central Cost Based Control Algorithm................... 38 3.3 Game-Theoretic Based Control Algorithm................. 43 3.4 Link Algorithm................................ 47 vi

3.5 Multi-link Framework............................. 48 4 Primal Control of Optical Networks with Time-Delays 55 4.1 OSNR Models with Time-Delays...................... 56 4.2 Closed-Loop Game-Theoretic System with Time-Delays.......... 58 4.3 Stability Analysis of Game-Theoretic Algorithms............. 61 4.4 Stability Analysis of Central Cost Algorithms............... 70 4.5 Discussion................................... 73 4.6 Simulations.................................. 76 4.6.1 Simulation Parameters........................ 76 4.6.2 Game-theoretic Based Control.................... 77 4.6.3 Central Cost Based Control..................... 80 5 Primal-Dual Control of Optical Links with Time-Delay 86 5.1 Singular Perturbation Theory........................ 87 5.2 Continuous-Time Closed-Loop System................... 90 5.3 Razumikhin Stability Analysis........................ 96 5.4 Krasovskii Stability Analysis......................... 101 5.5 Scalar Analysis................................ 109 5.6 Discussion................................... 112 5.7 Simulations.................................. 114 5.7.1 Simulation Parameters........................ 114 5.7.2 Razumikhin Based Time-Delay Bound............... 115 5.7.3 Krasovskii Based Time-Delay Bound................ 117 6 Robust Primal-Dual Network Control with Time-delays 120 6.1 OSNR Models with Time-Delays and Uncertainties............ 121 6.2 Single-Link Analysis............................. 123 6.3 Single-Sink Network Analysis........................ 135 vii

6.4 Arbitrary Two Channel Network Topology................. 141 6.5 General Multi-link Framework........................ 149 6.6 Discussion................................... 150 6.7 Simulations.................................. 153 6.7.1 Single-Link Simulations....................... 153 6.7.2 Single-Sink Simulations........................ 154 7 Conclusions and Future Work 159 7.1 Summary of Main Results.......................... 159 7.2 Conclusions.................................. 160 7.3 Future Work.................................. 162 A Linear Algebra 165 B Robust Time-Delay Framework 173 B.1 Single-Link Framework............................ 173 B.2 Multi-link Network Framework....................... 175 C Ancillary Theory Proofs 178 D Pricing Strategies 202 D.1 Review of Proportional Pricing Strategy.................. 202 D.2 Proportional Pricing for Primal-Dual Control............... 203 E List of Symbols 206 E.1 Acronyms................................... 206 E.2 Symbols.................................... 206 Bibliography 214 viii

List of Tables 4.1 Primal Algorithm Simulation Parameters.................. 78 5.1 Primal-Dual Simulation Parameters..................... 115 ix

List of Figures 1.1 An Example of a Global Optical Network [1]............... 2 1.2 A Segment of An Optical Network with All Components Labeled.... 3 1.3 Physical diagram of the system signals over a network cloud........ 4 1.4 Block diagram of the controllers acting on the optical network....... 5 3.1 Mesh Optical Network............................ 32 3.2 Optical spans in a link segment........................ 33 3.3 Optical path within a network........................ 33 3.4 Block diagram for the OSNR model (3.16).................. 38 3.5 Central cost based control algorithm applied to the optical network.... 39 3.6 Discrete-time Control Algorithm (3.24) on the OSNR system (3.16).... 41 3.7 Discrete-time Control Algorithm (3.38) on the OSNR system (3.16)... 46 3.8 Block diagram of the primal-dual algorithm acting on the optical network. 47 3.9 Discrete-time Control Algorithm (3.38) and Link Algorithm (3.41) applied to OSNR system (3.11)............................ 49 3.10 Three link example where the convexity condition (3.44) does not hold.. 51 3.11 Single sink network configuration such that the convexity condition (3.44) is preserved................................... 52 3.12 Two channel network configuration such that the convexity condition (3.44) is preserved................................... 53 3.13 Multi-link network broken up into single-link stages............. 54 x

4.1 Example general multi-link network..................... 56 4.2 Block diagram for the time-delayed OSNR model (4.1)........... 58 4.3 Single-link configuration........................... 58 4.4 Continuous-time closed-loop block diagram of algorithm (4.4) applied to the OSNR model (4.1)............................. 61 4.5 Unity feedback system with loop transfer function, L(s).......... 63 4.6 No Time-Delay, ρ i = 0.1.......................... 78 4.7 Time-Delay 10ms, ρ i = 0.1......................... 79 4.8 Time-Delay 10ms, ρ i = 0.95......................... 79 4.9 Conservative Case, Time-Delay 10ms, ρ i = 1............... 80 4.10 2 Channel multi-link configuration...................... 80 4.11 2 Channel multi-link configuration, ρ i = 0.1................ 81 4.12 2 Channel multi-link configuration, ρ i = 0.4................ 81 4.13 2 Channel multi-link configuration, ρ i = 0.63............... 82 4.14 2 Channel multi-link configuration, ρ i = 0.65............... 82 4.15 Simulation: 8 channels, single-link, 10ms delay, ρ i = 1, maximal OSNR target..................................... 83 4.16 Simulation: 8 channels, single-link, 250ms delay, ρ i = 1, maximal OSNR target..................................... 84 4.17 Simulation: 8 channels, single-link, 10ms delay, µ = 1, 23dB median target 84 4.18 Simulation: 2 channels, 4 links, 10ms delay per link, ρ i = 1, maximal OSNR target................................. 85 5.1 Continuous-time Control Algorithm (5.8) and Link Algorithm (5.12) applied to OSNR system (4.2) with time-delay................. 93 5.2 µ decoupled with ρ i = 0.1.......................... 116 5.3 µ = 0.25 µ with ρ i = 0.1......................... 116 5.4 µ = 0.25 µ with ρ i = 0.35......................... 117 xi

5.5 The case ρ i = 0.1 with 10 optical spans and a time-delay of 15ms.... 118 5.6 The case ρ i = 0.1 with 10 optical spans and a time-delay of 20ms.... 119 6.1 OSNR Model (6.4) with time-delays and uncertainties.......... 123 6.2 Multi-link, single-sink network with time-delays.............. 136 6.3 Two channel configuration with decoupled OSNR outputs......... 143 6.4 Two channel configuration with coupled OSNR output........... 143 6.5 The base case ˆq = 0, no uncertainty in the system with nominal control laws (3.38) and (3.41)............................. 154 6.6 The case ˆq = 1.823 10 4 for the unstable perturbed system with nominal control laws (3.38) and (3.41)......................... 155 6.7 The case ˆq = 1.823 10 4 for the perturbed system with robust compensation with a i increased by a factor of 2.9.................. 155 6.8 No uncertainty in the system without robust compensation and ρ = 0.01. 157 6.9 Perturbed system without robust compensation and ρ = 0.01....... 157 6.10 Perturbed system with robust compensation and ρ = 0.002........ 158 C.1 Nyquist Plot of (C.38)............................ 191 C.2 Lemma 4.3.2: Solution range for (C.47)................... 192 C.3 Lemma 4.3.2:Nyquist Shape Exploit..................... 193 xii

Chapter 1 Introduction Our dependence on communication networks to distribute information is growing at an exponential rate. A multitude of communication networks has evolved to provide a variety of services to end users. These communication networks include radio networks, satellite networks, and optical networks to name a few. Over the past decade, system control theory has been utilized to optimize the performance of these networks where heuristics were previously used. However, current communication control schemes ignore the effects of propagation delay. Time-delays may not just impact the performance efficiency of network control algorithms, but they may destabilize the networks. As communication networks grow in geographical size and complexity, the effects of timedelays become more pronounced and can not be ignored. An extreme example of timedelays in communication networks is the developing interplanetary Internet system where time-delays manifest themselves in minutes. The work proposed herein remedies the impact of time-delay in optical power control algorithms. We first present a high-level overview of optical communication networks. We then present a survey of current research in communication networks with time-delays and uncertainties. Next, we outline our research and formulate the problem statement. This is followed by a section that briefly summarizes the content of the remaining chapters. 1

Chapter 1. Introduction 2 Figure 1.1: An Example of a Global Optical Network [1] 1.1 Optical Network Overview Optical networks form the backbone for ultra-long haul communication in the Internet. In fact, over 85 percent of all global communications traverse optical fiber [7], including cable and telephone signals. Optical networks are global in size and and traverse both oceans and continents. The study of time-delays is justified since trans-oceanic cables and continental cables span thousands of kilometers. Thus, propagation delays are nonnegligible. In addition, mesh optical networks are distributed over large surface areas. The path that a signal traverses may not be direct thereby adding additional propagation delays. Figure 1.1 depicts an example of a global optical mesh configuration. Optical networks are comprised of a set of fiber optic links that connect nodes which facilitate the adding, dropping and rerouting of signal channels dynamically. Lasers transmit the signals into the optical network. The signals (channels) are multiplexed into a single beam of light using wavelength division multiplexing (WDM) before being transmitted through the network. Each channel is assigned a unique wavelength in the

Chapter 1. Introduction 3 Figure 1.2: A Segment of An Optical Network with All Components Labeled beam of light. The signals are demultiplexed at their destination points (Rx) to be processed outside the optical domain. The device that multiplexes/demultiplexes the channels is called the optical line terminal (OLT). The individual powers of the channels can be adjusted at the signal sources (Tx). Optical amplifiers (OAs) are used to amplify the channel powers every 80 to 100 kilometers along an optical link. Channels can be rerouted dynamically through different links within the network using optical crossconnects (OXCs). Figure 1.2 shows a segment of an optical network with all of the major components labeled. The transmission rate of error-free data in optical communication networks depends on the optical signal-to-noise ratio (OSNR) of the signal channels at reception, Rx. As the signal power in a channel increases, the OSNR value of the channel increases, and the bit error rate decreases. Thus, we are interested in optimizing all of the OSNR values in an optical network. The OSNR optimization problem is complicated because there are trade-offs in the attainable OSNR values for all channels. As one channel increases its signal power, thereby increasing its OSNR, it may adversely affect the OSNR values of the other channels in the network. This occurs due to cross-gain modulation [2] in the optical amplifiers. Optical amplifiers amplify multiple channels simultaneously according to a fixed gain profile. As such, if one channel increases its power, it requires more power from

Chapter 1. Introduction 4 Rx Optical Network Rx Tx Tx Tx Figure 1.3: Physical diagram of the system signals over a network cloud. the optical amplifier, which results in less power available for the remaining channels. From a systems level perspective, we view optical networks as follows. The inputs to the network are the signal powers at the sources, Tx. The outputs are the OSNR values at Rx, which are measurable in real-time. Proposed power control schemes utilize the OSNR values as feedbacks that are sent via dedicated channels. This information is used by control algorithms to adjust the source input powers as a function of OSNR. The control algorithms operate in real-time to achieve optimal OSNR values for the network. Figure 1.3 depicts the signal transmissions from Tx by solid lines, and the OSNR feedbacks from Rx as dashed lines. The signals (channels) may be geographically distributed across the network. Each transmitter for each signal has its own control algorithm at Tx. Figure 1.4 illustrates the closed-loop block diagram. In the work proposed herein, we modify existing control algorithms to ensure stability in the presence of both time-delays and uncertainties. Time-delays are not negligible in optical networks due to their global size. The timedelays in optical networks are the result of propagation delays in the signals traversing the network. The signals in optical networks are carried by pulses of light. The light is guided

Chapter 1. Introduction 5 Figure 1.4: Block diagram of the controllers acting on the optical network. through the optical fibers by the process of total internal reflection [42]. The signals travel through the silica in the fiber at a velocity of approximately 200, 000km/s. Thus, for every 100km span of fiber, the round trip signal propagation delay is approximately 1ms. The impact of the time-delays on the stability of the control algorithms depends on the rate at which the algorithms update the control signals. The faster the algorithm update rate, and hence the better the network performance, the more significant are the impact of time-delays in destabilizing the system. Thus, for high-performance optical networks, it is essential to understand the impact of time-delays on the control of optical networks. For example, packet-switched optical networks, which are currently experimental [42], require fast algorithm update rates. The following example illustrates the need to understand the effects of time-delays on optical networks in practice. Consider an optical network, with transmitter hardware and control software implemented to ensure power optimization. The channel powers at Tx are adjusted based on OSNR feedbacks every 100ms, which is reasonable by today s standards. We assume that there exist round-trip time-delays of 50ms, which is the case for a network covering a distance of 5000km. The network engineers believe that they

Chapter 1. Introduction 6 may improve the quality of service and robustness of the network by implementing significantly faster channel update times of 10ms. This requires purchasing more expensive transmitter hardware. However, for the transmitter to update every 10ms, the OSNR feedback signal is delayed by five times this update period. This delay is not an issue at slow update times, but these faster update times do not ensure system stability. If the system is no longer stable as a result of the delay, the channel update time-period must be increased which diminishes the benefit of the new hardware. In addition to time-delays, it is crucial to account for system uncertainties. In general, optical networks have system uncertainties due to both slow and fast-varying parameter changes over time. The fast-varying parameter changes are due to the inherent noise introduced by OAs called amplified spontaneous emission (ASE). The network gains change as a result of ASE being dependent on the channel powers. The slow parameter changes occur due to parameter drift in the components of the network over long periods of time. An additive uncertainty applied to the system gains accounts for such perturbations. Additive uncertainty also effectively models the changes in the transmitter noise over long periods of time. Furthermore, the optical fiber lengths may change due to environmental effects or system reconfigurations. In either case, these unmodeled changes result in inaccurate time-delays and signal power values. Input multiplicative uncertainty applied to the OSNR model compensates for this. Thus, the aforementioned uncertainties, combined with time-delays, need to be accounted for in modeling optical networks. 1.2 Literature Review In this section, we present a high-level overview of relevant technical results for optical network power control, time-delay effects in communication networks, and uncertainties coupled with time-delays in communications. The first subsection summarizes current

Chapter 1. Introduction 7 research in power control in optical networks. The second subsection broadly surveys all relevant work in general communication networks with time-delays. The last subsection presents the work relevant to both time-delays and uncertainties in communications. 1.2.1 Optical Network OSNR Model and Control Algorithms This section surveys current results in OSNR optical network modeling and their power control algorithms. The material focuses primarily on game-theoretic and central cost based solutions to the OSNR optimization problem. In addition, introductory references to optical networks are provided. Time-delays and uncertainties are not present in the following works. In [38], Pavel proposes a game-theoretic solution to the OSNR optimization problem in optical communication networks. She derives the OSNR model that is used to describe arbitrary optical networks. Note that the OSNR model does not include timedelays or uncertainties. We apply this OSNR model to the work in this thesis. The author then applies a game-theoretic framework to the OSNR optimization problem and proves that a Nash equilibrium exists under certain prerequisite conditions. This gametheoretic framework is also used for our work on time-delays and system uncertainties. Based on the game-theoretic framework, a discrete-time updating algorithm is derived which ensures the channel powers converge to the Nash equilibrium point in real-time. Furthermore, we use this control algorithm to study the effects of time-delays on the stability of the closed-loop system. The above results are extended in [40] and [36] as follows. First, [40] extends [38] to decouple the Nash game into a low level Nash game with no coupled constraints between channels and a high level optimization problem for pricing parameters. This approach results in a dual control law to complement the primal control law developed in [38]. The dual law measures the total channel power and computes a channel price to feed back to the source algorithms. We study the stability of the primal-dual control algorithms, i.e. both the primal channel algorithms and the dual

Chapter 1. Introduction 8 link algorithms applied in the presence of time-delays and uncertainties. In addition, [36] extends [38] to incorporate a link capacity upper bound for the single-link and single source case. Finally, [40] generalizes the network topology to allow intermediate source adjustment in the network to give more control over OSNR tuning. This control scheme generalizes the network topology presented in [38], but requires more control over the optical network. This work provides a framework to address multiple Rx points in a network where the OSNR is read and the information is fed back to the network sources. As such, this configuration results in a more intricate network of feedbacks to the channels so that each channel experiences different backward propagation delays from multiple Rx locations. A multi-link framework is presented in [37] that generalizes [38, 40]. The work breaks up arbitrary network configurations into cascades of link level stages. On each link stage, the primal-dual algorithms of [38, 40] are modified to optimize OSNR degradation rather than total channel OSNR. Here, OSNR degradation is the measure of the difference between the reciprocals of the OSNR values at the beginning and the end of a link. The sum of the cascaded OSNR degradation reflects the total OSNR degradation of an optical channel along its optical path. One key requirement in the multi-link framework is that the channel powers are adjustable at intermediary nodes, i.e. optical cross connects [42], in the network. This requires significant control over the network nodes. We incorporate time-delays and uncertainties into this multi-link framework. Another result by Pavel, [39], provides a solution to the OSNR optimization problem using convex optimization methods. It also provides a discrete-time update algorithm for the sources to ensure convergence to the equilibrium value, along with the same OSNR model provided in [38]. In contrast to the game theoretic model of [38], the work in [39] strives to achieve pre-specified OSNR targets. Both [38] and [39] are unique in the literature in dealing with convergence of OSNR algorithms in optical communication networks. We study the effects of time-delays on the central-cost based control algorithms

Chapter 1. Introduction 9 and compare them with the game-theoretic based results. The above mentioned works rely on a preliminary understanding of the principles of optical communications. The textbook [42] gives a broad introductory background on optical communication networks, as well as their components. Other books, such as [2], provide a much more in depth and detailed study of the physics involved with optical communication. The material in [2] also provides the physical modeling necessary to understand the development of the OSNR model from [38] and [39]. 1.2.2 Time-Delay in Communication Networks There are numerous works related to the study of time-delay effects in communication networks. We present a broad survey of time-delay analysis techniques including frequency domain analyses, Lyapunov-based analyses, and various other approaches. The following works apply to time-delayed systems without uncertainties, which represents the single largest portion of work in this thesis. The works [34, 55, 56, 57, 35] present congestion control algorithms based on frequency domain stability analyses with time-delays. A congestion control scheme for communication networks with arbitrary network scaling and time-delays is presented. Beginning with [34], Paganini first outlines a local control law for a system with loop transfer function comprised of exponentials multiplied by integrators. The significance of this paper lies in the control tools derived therein that apply to this system structure. The loop transfer function is very similar to that derived from [38] and [39] when time-delays are incorporated into the model. Thus, the results of [34] provide the ideal starting point to incorporate time-delays into the OSNR model for optical communication networks [38, 39]. However, the utility function presented in the control problem in [34] is decoupled, which is simpler than the coupled utility function in the OSNR model from [38]. The work in [35] provides important proofs that are not included in [34]. The paper [35] is an excellent consolidation of the work done in [34, 55, 56, 57] with added simulations.

Chapter 1. Introduction 10 The main body of work by Paganini is extended in [55, 56, 57] by utilizing Lyapunov stability theory in addition to frequency domain techniques. Global stability for network congestion control is addressed in [55] and [56]. A primal-dual approach is used, but the results are restricted to single source and single-link configurations. This is an appropriate first step, but the results do not depict a typical real-world communication network. In contrast to [34], which utilizes classical control theory[43], the proof of global convergence is based on Lyapunov-Razumikhin methods and singular perturbation theory. These methods are relevant to this thesis since we also apply singular perturbation theory coupled with time-delays. The paper [57] extends the work done in [56] to redefine the boundedness of trajectories in the single source and single-link case. The work produces a nonlinear primal-dual congestion control law, but the utility function is still decoupled and the network is restricted to the single source single-link case. Both of these qualities are too simplistic to apply to optical networks since they possess coupled utilities and multi-source, multi-link configurations. The stability analysis of a class of time-delayed nonlinear system is studied in [29]. The inspiration to study the particular class of system derives from congestion control problems for the Internet. A Lyapunov-Krasovskii approach is taken to produce computable stability criteria. This work relates to this thesis since we study nonlinear, time-delayed systems and we apply a Lyapunov-Krasovskii stability analysis as well. The paper also serves as a technical reference for nonlinear time-delayed stability analyses in general. A passivity framework for network flow control is developed in [58] that also utilizes a frequency domain analysis. This work is significant in that it generalizes the works of [21, 34] as well as others. Both the primal and dual control laws are augmented and global stability for the no delay case is demonstrated using nonlinear Lyapunov based proofs. However, [58] only tests robustness with respect to time-delay. The work utilizes classical SISO frequency domain techniques to ensure network stability with respect to

Chapter 1. Introduction 11 time-delays. There is still an upper limit to the time-delay before network instability occurs. Furthermore, the work assumes a decoupled utility function, which does not fit the OSNR network model. A local stability condition with time-delay is derived using a frequency domain analysis for the linearized system in [26]. This paper provides an excellent synopsis of past research and its context with other research directions. A more general primal-dual control law is introduced to accommodate TCP-AQM. Both local and global stability without delay is shown. In addition, local delay dependent stability conditions are proposed. An interesting result from this work is that local stability of the primal-dual controller depends solely on the link adaptation speed, and not the source rate adaptation speed. It is also worth mentioning [53] utilizes a frequency domain analysis to study the stability of a system with time-delays. It gives conditions for local stability of TCPlike congestion control algorithms. The result relies on linearization and the Nyquist criterion. It is significant in that it shows parallels to [34], with slight generalizations to the delay models. Other works by Vinnicombe include analyzing stochastically modeled communication systems, as in [25], but the work in this thesis does not consider stochastic modeling. The work [4] utilizes Lyapunov stability theory. It extends the results of [3] to consider fixed heterogeneous delays in the network. Global stability is proved under a mild symmetric condition using Lyapunov stability theory. Functional Lyapunov theory is utilized to prove the stability results. In addition, a series of works study singularly perturbed systems with time-delays. The paper [10] studies the stability of linear, singularly perturbed systems with state delays. It shows that if the origins of the reduced and boundary-layer systems are exponentially stable, then the origin of the full singularly perturbed system is also exponentially stable. This thesis studies nonlinear singularly perturbed systems with time-delays as well. The work [12] decomposes a linear singularly perturbed system with small time-

Chapter 1. Introduction 12 delays into a slow system of ordinary differential equations and a fast functional equation. It generalizes previous work and also presents reduced order models and stability criteria. The decomposition technique in the slow and fast time-scales is a critical tool that we apply in our analysis. The effects of small delays on linear, singularly-perturbed systems are also studied in [13]. Furthermore, [51] considers linear singularly perturbed systems with small state delay in the context of the infinite horizon H state-feedback control problem. The condition for the existence of a solution for this problem is presented which is independent of the singular perturbation parameter. The work [17] presents a composite feedback control that stabilizes a non-standard linear singularly perturbed system with a small time-delay. Another paper, [54], also studies non-standard singularly perturbed systems. In general, the aforementioned papers present some specific results that combine singular perturbation theory with time-delays. However, there is no body of work to encompass singularly perturbed systems with time-delays in general forms. Although the above papers do not directly relate to communication networks, the topic of singular perturbation with time-delays is extremely relevant to the results of this thesis. The pioneering paper in congestion control is [21], which utilizes discrete-time maps. The paper introduces a primal and a dual controller. The stability proof is based on Lyapunov techniques and discrete-time maps with time-delays. Although we do not directly apply the theory presented in [21], it serves as the basis for most modern congestion control theory. Another result that relies on the discrete-time map to study the stability of network control algorithms is [41]. In [41], global stability conditions are derived for rate control and arbitrary network delays. The work utilizes the well established theory from [21] to show that under certain weak conditions and certain network delay configurations, the stability of the discrete-time map of delay differential equations is sufficient to show stability. Furthermore, [41] highlights that if a system is stable with homogeneous feedback delays from the network, but with heterogeneous feed-forward delays from the users,

Chapter 1. Introduction 13 then the system is also stable with appropriate initial conditions. The result is significant in that it shows system stability is sensitive to user utility functions and resource price functions, but not on the various delay values throughout the network. The presented works utilize the techniques of frequency domain analysis, Lyapunov functional analysis and discrete-time maps in the study of stability in time-delayed systems. The basics of the aforementioned techniques are described in [18]. This book provides a comprehensive tutorial in general time-delay systems. Also, [30] presents the stability and stabilization theory for linear time-delayed systems based on an eigenvalue approach. Furthermore, the books [43, 15, 22] provide good reference notes for the control theory that is common to the works mentioned in this subsection. Specifically, [22] gives a good mathematical background necessary to follow the stability proofs. In addition, [5] is also a good reference for linear algebra. 1.2.3 Uncertainty and Time-Delay in Communication Networks There are few results in literature for the robust stability of optical networks with timedelays and uncertainties. In [63], classic control techniques are applied to the linear optical signal-noise difference model to separately study additive uncertainty and timedelays. The work does not include link dynamics or pricing. In this thesis, we study the combined uncertain, time-delayed system for the nonlinear OSNR based primal-dual control algorithms (see [38, 40]). In [61], the application of a converse Lyapunov Theorem to time-delayed systems is relevant here. In [16], polytopic uncertainty is applied to genetic regulatory networks with time-varying delays. The work uses Lyapunov-Krasovskii theory and linear matrix inequality (LMI) techniques. We apply Lyapunov-Krasovskii techniques with norm bounded uncertainty. Polytopic uncertainty is also studied in [19]. An implicit model transformation is presented in [24] to solve the guaranteed cost control problem with memoryless state feedback controllers. The result applies to a linear system with time-delays. The uncertainty is norm bounded. The work of [20] studies the

Chapter 1. Introduction 14 delay-dependent robust stability problem for linear uncertain systems using a Lyapunov- Krasovskii approach. The framework presented therein allows for interval time-varying delay, which may extend the theory presented herein. In [33], a norm bounded uncertainty framework is applied to a class of time-delay systems with actuator saturation. The work considers both input uncertainty and state uncertainty, as we do herein. Thus, these results serve as a good aid for some of the technical challenges presented in the thesis. A simpler Lyapunov-Razumikhin approach is utilized, which we study herein as well. In [59], input uncertainty is used to deal with parametric uncertainties for time-delay systems. Again, the input uncertainty models present interesting parallels to our upcoming research, with the exception that we focus on structured uncertainties. In [11], an output-feedback guaranteed cost controller for uncertain time-varying time-delay systems is developed. Lyapunov-Krasovskii techniques are used, which are relevant to this thesis. Also, [50] applies a Lyapunov-Razumikhin approach to derive sufficient conditions for robust time-delay stability. A study of structured uncertainties for linear systems is found in [9]. This is useful for further modeling plant uncertainties as needed by a designer. Finally, [32] takes a robust control approach to the stability of time-delayed systems. The aforementioned research provides the starting point for studying the effects of both uncertainty and time-delays in optical communication networks. In [3] a control algorithm that is robust to time-delays is outlined. The paper defines a mathematical framework for congestion control in general topology networks. The approach uses game theory to define the source transmission rates as users in a noncooperative game. The dynamics of the system are characterized by gradient algorithms. Although the paper does briefly discuss network delays, it does so from a robustness perspective, that does not explicitly analyze the effects of large propagation delay. The authors further acknowledge a future research direction where large propagation delays should be analyzed in the model. In addition, the utility function is uncoupled in the source rate, which does not capture the coupling effects in optical communication net-

Chapter 1. Introduction 15 works. Finally, in [14], the state feedback H control problem is studied for singularly perturbed systems with time-delays and norm-bounded uncertainties. This paper is relevant to this thesis since we also apply norm-bounded uncertainties to a singularly perturbed system 1.3 Problem Statement The problem is formulated as follows: For a given set of time-delays and norm bounded uncertainties in an optical network, find the conditions on the control parameters such that asymptotic stability is ensured for the OSNR values of the closed-loop system. Our primary interest is to ensure the stability of optical networks in the presence of time-delays due to signal propagation. Our secondary objective is to ensure network stability in the presence of both time-delays and uncertainties. We account for the timedelays and uncertainties in the system by deriving a new OSNR model which extends from [38]. The OSNR model is valid for multiple time-delays. We apply additive and input multiplicative uncertainties to the OSNR model via a norm-bounded uncertainty framework. The stability analysis progresses in stages. First, a multi-link, multiple time-delay stability analysis without uncertainties is presented for the primal control based closedloop system. This system is linear, so a frequency techniques are used to solve the problem. Next, the problem is solved for the primal-dual control algorithm applied to the single-link configuration. The closed loop system is nonlinear with a single roundtrip time-delay. This analysis also does not take uncertainties into account. Singular perturbation theory modified for Lyapunov-Razumikhin and Lyapunov-Krasovskii based techniques are used to derive algebraic stability conditions. Finally, the primal-dual control algorithm is applied to arbitrary multi-link network configurations. The stability

Chapter 1. Introduction 16 analysis is extended to include uncertainties and multiple time-delays. 1.4 Thesis Organization The remaining chapters of the thesis are organized as follows: Chapter 2: We present the background on stability of time-delayed systems, and robust theory with time-delays. We present a section dedicated to the stability of time-delayed systems, which is the main focus of this thesis. We break up the time-delay theory into two sections: Lyapunov-Razumikhin theory and Lyapunov- Krasovskii theory. We then introduce a section on robust theory coupled with multiple time-delays. Chapter 3: We review the existing OSNR model from [38] and the power control algorithms [38, 40, 39] for optical networks. We begin by presenting the central cost based primal control algorithm. The central cost based algorithm is only studied in Chapter 4. The game-theoretic based primal control algorithm is presented next. The link algorithm, which is complementary to the game-theoretic based control algorithm, is then presented. The primal-dual algorithm is studied in Chapter 5 for the single-link time-delayed OSNR model. The primal-dual algorithm is also studied in chapter 6 for the multi-link uncertain OSNR model with time-delays. Chapter 4: We extend the multi-link OSNR model to include arbitrary timedelays. We study both the central cost based and game-theoretic based control algorithms applied to this OSNR model. We first derive the closed-loop system with time-delays for the game-theoretic control algorithms. We then derive stability conditions for the closed-loop system as functions of the control gains. We apply the linear time-delay results from [35] to derive the main theorem. Next, we apply a similar procedure to derive the stability conditions for the central cost based

Chapter 1. Introduction 17 control algorithms. Finally, we validate our results through simulations. Chapter 5: We study the primal-dual control algorithms applied to a single-link network configuration with time-delays. We first present a section dedicated to singular perturbation theory [22], which is used in the proofs of the main theorems for the primal-dual analyses. The main results utilize singular perturbation theory modified to handle time-delays. We apply both Lyapunov-Razumikhin theory and Lyapunov-Krasovskii theory to obtain upper bounds on the time-delays for closed-loop stability. The implicit model transformation of the Lyapunov- Krasovskii theory is more general than the explicit model transformation of the Lyapunov-Razumikhin theory. We then compare and contrast the results of both approaches by performing a scalar analysis and discussing the structural differences in the time-delay bounds. Finally, we validate our results through simulations. Chapter 6: We extend the time-delayed OSNR model from Chapter 3 to account for additive and input-multiplicative uncertainties. We study the primal-dual control algorithms applied to arbitrary multi-link network configurations. We incorporate additive uncertainties applied to the system gains and the transmitter noise. We also apply input multiplicative uncertainties to the signal powers. We break-up the analysis into the single-link, single-sink, two channel, and multi-link network cases. The main results are presented in the form of LMIs. The work utilizes singular perturbation theory modified to handle Lyapunov-Krasovskii time-delay theory. We then compare and contrast the results of each section. Our results are validated by simulations. Chapter 7: We present our concluding remarks, where we summarize our results from a high-level perspective. We follow up with a section on future work, where we address potential improvements and future research directions.

Chapter 2 Review of Preliminary Theory This section reviews the background theory necessary to study the stability of optical networks with both time-delays and uncertainties. Incorporating time-delays and uncertainties into the network model is important for a practical implementation of the control theory. We first present a section on time-delay theory, which is the main focus of this thesis. We outline both the Lyapunov-Razumikhin and Lyapunov-Krasovskii time-delay theorems [18] in detail here. We also review robust theory augmented to include timedelays [18]. Appendix A reviews the linear algebra used in the following sections and throughout the thesis. 2.1 Time-Delay Stability Theory In this section, we present the theory needed to study the stability of time-delayed systems. We first define the continuous norm, the retarded functional differential equation, and the definitions of stability from [18]. We then present the Lyapunov-Razumikhin time-delay theorems [18], including all derivations. These theorems do not rely on functionals, but typical Lyapunov functions, and as such they produce simpler, more conservative stability criteria. This is followed by a section on Lyapunov-Krasovskii timedelay theory [18], which is more general than Lyapunov-Razumikhin theory. Lyapunov- 18

Chapter 2. Review of Preliminary Theory 19 Krasovskii theory relies on functional equations which give less conservative stability criteria at the expense of both analytical and computational complexity. 2.1.1 Definitions We define C([ τ, 0], R n ) as the set of all continuous functions mapping [ τ, 0] to R n. Let C = C([ τ, 0], R n ). The general form of a retarded functional differential equation [6] is defined as dx dt = f(t, x t) (2.1) where x t C is the function mapping [t τ, t] to R n, and f : R C R n. The following norm is used in the definition of stability for time-delayed systems as well as the proofs of the main results herein. Definition 2.1.1. The continuous norm is defined as x t c = max τ θ 0 x(t + θ) 2 where x t C is the function mapping [t τ, t] to R n. Note that x(t) 2 x t c. We now define stability for (2.1). Definition 2.1.2. For the system (2.1), the origin is: stable if for any t 0 R and any ˆǫ > 0, there exists a ˆδ = ˆδ(t 0, ˆǫ) > 0 such that x t0 c < ˆδ implies x(t) < ˆǫ for t t 0. asymptotically stable if it is stable and for any t 0 R and any ˆǫ > 0, there exists a ˆδ a = ˆδ a (t 0, ˆǫ) > 0 such that x t0 c < ˆδ a implies lim t 0 x(t) = 0. uniformly stable if it is stable and ˆδ(t 0, ˆǫ) can be chosen independently of t 0. uniformly asymptotically stable if it is uniformly stable and there exists a ˆδ a > 0 such that for any η a > 0, there exists a T = T(ˆδ a, η a ) such that x t0 c < ˆδ implies x(t) < η a for t t 0 + T and t 0 R.

Chapter 2. Review of Preliminary Theory 20 We define C h as the space of continuous functions, φ, mapping [ τ, 0] to R n such that φ c < h, where h is a positive real number. Thus, C h is the open ball of C of radius h. We use the following definition of exponential stability from [61]. Definition 2.1.3. The origin of (2.1) is exponentially stable if there exist positive real numbers h, a, and b such that for every initial condition φ C h the solution x t (φ) of (2.1) exists for all t 0 and furthermore satisfies x t (φ) c ae bt φ c The above results of this section apply for general retarded functional differential equations of the form (2.1). We next define the stability criteria from [18] for linear time-delay systems. Consider the system of the form, ẋ(t) = a 0 x(t) + a 1 x(t τ) (2.2) where x(t), τ, a 0, a 1 R. The initial condition for (2.2) is defined as the function φ : [ τ, 0] R such that x(t) = φ(t) t [ τ, 0] (2.3) The Laplace transform of (2.2) with initial conditions φ exists, and may be written as follows where X(s) = 1 0 } {φ(0) + a 1 e s(v+τ) φ(v)dv (s) τ (2.4) (s) = s a 0 a 1 e τs (2.5) is the characteristic quasipolynomial of (2.2). Let Φ(t) denote the inverse Laplace transform of 1/ (s). We call Φ(t) the fundamental solution of (2.2). Notice that with the initial condition φ(0) = 1, and φ(v) = 0 for all v [ τ, 0), then Φ(t) is the solution of (2.2). The growth of the general solution of (2.2) is related to the exponential growth of

Chapter 2. Review of Preliminary Theory 21 the fundamental solution Φ(t), which is determined by the poles of the system (2.4), i.e., the solutions of the characteristic equation (s) = 0 (2.6) In general, (2.6) has an infinite number of solutions. Denote by Re[s] the real component of the complex value s. The following proposition from [18] presents the theory for the stability of (2.2). Proposition 2.1.1. For any α R, there are a finite number of poles with real parts greater than α. Let s i for i = 1, 2,... be the poles of the system (2.2), i.e. the solutions of (2.6), and let α 0 = max i Re[s i ] (2.7) Then for any α > α 0, there exists a L > 0 such that the solution of (2.2) with the initial condition (2.3) satisfies the inequality x(t) Le αt φ c (2.8) By Proposition 2.1.1, for the solution of (2.2) to approach zero as t for any initial condition, it is sufficient that the poles of the system have negative real parts. This stability result is similar to the stability result for systems without time-delay. In fact, this condition is both necessary and sufficient [18]. As in [18], the above discussion for (2.2) may be generalized to linear systems of the following form: K ẋ(t) = A 0 x(t) + A k x(t τ k ), τ k 0 (2.9) k=1 where A 0, A k R n n, and τ k R for k = 1,..., K. The stability of (2.9) is determined by the characteristic quasipolynomial p(s; e τ 1s,..., e τ Ks ) = det ( (s)) (2.10)

Chapter 2. Review of Preliminary Theory 22 where K (s) = si A 0 A k e τ ks We say (2.9) is asymptotically stable if and only if p(s; e τ1s,..., e τks ) has no root in the closed RHP, or C +. This definition of stability is stated formally as follows[18]. k=1 Definition 2.1.4. The system (2.9) is said to be asymptotically stable if and only if its characteristic quasipolynomial (2.10) satisfies p(s; e τ 1s,..., e τ Ks ) 0, s C + (2.11) The following section presents the Lyapunov-Razumikhin stability theorems which provide alternative stability criteria for linear time-delay systems. 2.1.2 Razumikhin Stability Theory We present the Lyapunov-Razumikhin theorem and a set of stability theorems for linear systems with time-delays from [18]. We use the Lyapunov-Razumikhin theorem because it produces simpler stability criteria than the Lyapunov-Krasovskii theorem. The Lyapunov-Razumikhin theorem does not depend on Lyapunov functionals, but on the traditional Lyapunov functions used in systems without delays. Consider the following linear time-invariant system of the form dx(t) dt = A 0 x(t) + A 1 x(t τ) (2.12) where A 0 and A 1 are (n n) real matrices. The following theorem, known as the restricted Lyapunov-Razumikhin theorem [18](Proposition 5.1, pg.149) gives conditions for asymptotic stability. Lemma A.0.2 defines a quadratic function which is used in the Theorem.

Chapter 2. Review of Preliminary Theory 23 Theorem 2.1.1. The time-delay system (2.12) with maximum time-delay τ is asymptotically stable if there exists a quadratic Lyapunov function V : R n R such that for some ψ 1 > 0, ψ 2 > 0 it satisfies ψ 1 x 2 2 V (x) ψ 2 x 2 2 where the time derivative along the system trajectory dv (x) dt satisfies dv (x) dt ψ 1 x(t) 2 2 (2.13) if p > 1, t, ξ [ τ, 0] such that V (x(t + ξ)) pv (x(t)) Notice that Theorem 2.1.1 closely resembles the Lyapunov stability theorem [22]. The first condition is the same as in the Lyapunov stability theorem. The second condition requires that the time derivative of the Lyapunov function be non-positive which is also similar to the Lyapunov stability theorem. The third requirement is an extra condition that only appears for time-delayed systems. We explain this extra condition as follows. The negation of the third condition of Theorem 2.1.1 is p > 1, t, ξ [ τ, 0] V (x(t + ξ)) > pv (x(t)) (2.14) Thus, for some p > 1, at every instant of time, t, the value V (x(t)) is less than a past instant. Intuitively, this means the state trajectory is approaching its equilibrium point, and hence, the system is stable. Thus, we only need to ensure the first two conditions of Theorem 2.1.1 are satisfied if the third condition holds. If the third condition does not hold, then the system is implicitly stable. An interesting note is that the Lyapunov function and its time derivative are compared to the square of the norm of the state, which implies exponential stability.

Chapter 2. Review of Preliminary Theory 24 Theorem 2.1.1 presents a general set of stability conditions that do not exploit the structure of (2.12), i.e., A 0 and A 1 do not appear in the stability criteria. In addition, it is not obvious how to pick a Lyapunov function to satisfy the stability criteria in Theorem 2.1.1. We show that in the foregoing work, the Lyapunov function V = x T Px may be used. The remainder of this section is dedicated to presenting the delay-independent and delay-dependent stability criteria specific to the structure of the linear system (2.12). We first outline the time-delay independent stability conditions. The time-delay independent stability conditions provide the simplest and most conservative stability conditions. They are presented as Proposition 5.3 in [18] and provided below as Proposition 2.1.2. The proof is found in Appendix C. Proposition 2.1.2. The system (2.12) is asymptotically stable if there exists a scalar ˆα > 0 and a real symmetric matrix P such that PA 0 + A T 0 P + ˆαP PA 1 A T 1 P ˆαP < 0 (2.15) If A 0 = 0 then Proposition 2.1.2 can not be directly applied since this does not produces a feasible solution. Thus, more general time-delay dependent conditions need to be applied. Time-delay dependent stability conditions produce less conservative stability criteria. The following theorem presents a set of time-delay dependent conditions to ensure asymptotic stability of (2.12). This is Corollary 5.8 in [18] with the proof presented in Appendix C. Theorem 2.1.2. The system (2.12) is asymptotically stable if there exists a real symmetric matrix P and real scalars ˆα 0 > 0, ˆα 1 > 0 such that M PA 1 A 0 PA 2 1 A T 0 AT 1 P ˆα 0P 0 < 0 (2.16) (A 2 1) T P 0 ˆα 1 P

Chapter 2. Review of Preliminary Theory 25 where M = 1 τ [P(A 0 + A 1 ) + (A 0 + A 1 ) T P] + (ˆα 0 + ˆα 1 )P (2.17) 2.1.3 Krasovskii Stability Theory The following Lyapunov-Krasovskii stability theory is less conservative than the Lyapunov- Razumikhin theory presented in the previous section. The trade-off in using Lyapunov- Krasovskii theory is in its analytical and computational complexity. This theory depends on Lyapunov functionals, which are structurally more complicated than traditional Lyapunov functions. The following theorems provide an alternative and more general timedelay analysis to those in Section 2.1.2. We state the following theorem from [18] (Proposition 5.2, pg. 150), also known as the restricted Lyapunov-Krasovskii theorem, which gives the conditions for asymptotic stability of time-delay systems based on Definition 2.1.2. The theorem is stated without proof. Theorem 2.1.3. A time-delay system of the form (2.12) is asymptotically stable if there exists a Lyapunov-Krasovskii functional V : C R such that for some ψ 1 > 0 and ψ 2 > 0, it satisfies ψ 1 x(t) 2 2 V (x t) ψ 2 x t 2 c where the continuous norm x t c is defined in Definition 2.1.1, and the functional satisfies dv (x t ) dt ψ 1 x(t) 2 2 dv (xt) dt We note that the functional dv (xt) dt is obtained as the derivative of V (x t ) with respect to time t along the system trajectory and based on the chain-rule [27]. It can also be

Chapter 2. Review of Preliminary Theory 26 evaluated as dv (x t ) dt = V (x t ) dx t dt where V (x t ) or dv dx t denotes the Fréchet derivative of V with respect to its argument. In the finite dimensional case, or the no-delay case, V (x t ) is equal to the gradient of V with respect to its argument. Alternatively, the Dini derivative may be applied instead of the Fréchet derivative. Both derivatives are generalizations of the directional derivative. Theorem 2.1.3 presents a set of conditions to ensure the stability of a linear system of the form (2.12) without explicitly accounting for A 0 and A 1. The next theorem derives from (2.1.3) and provides explicit, computable stability conditions. It is more general result than Theorem 2.1.2. The following derivation proceeds as in [18](Section 5.5.2, pp.173-175). We show that (2.12) may be rewritten by applying an implicit model transformation. The proof of the following theorem is found in Appendix C. Theorem 2.1.4. The system (2.12) is asymptotically stable if there exist real matrices X T = X, Y and P = P T such that N PA 1 Y A T 0 Y T A T 1 P Y T S A T 1 Y T Y A 0 Y A 1 1 X τ < 0 (2.18) where N = PA 0 + A 0 P + S + τx + Y + Y T (2.19) Remark: If we select Y = PA 1, and apply the following variable transformation S = τs 1 X = 1 τ [P(A 0 + A 1 ) + (A 0 + A 1 ) T P] S 0 S 1 where we also select S j = ˆα j P for j = 0, 1, then we recover the result of Theorem 2.1.2. Finally, from [23](Lemma 33.1) and [61], the following converse Lyapunov Theorem is used in Krasovskii-based proofs.

Chapter 2. Review of Preliminary Theory 27 Theorem 2.1.5. Let the system of the form (2.1), where f is continuous and Lipschitz on bounded sets, and the initial conditions are in C h, be exponentially stable. Then, there exists a continuous functional W(φ) defined on C h a i = 1, 2, 3, 4 such that the following conditions hold φ, ξ Ch: a and positive constants C i for C 1 φ c W(φ) C 2 φ c (2.20) dw(φ) dt C 3 φ c (2.21) W(φ) W(ξ) C 4 φ ξ c (2.22) 2.2 Robust Theory with Multiple Time-delays In this section, we extend the results of Section 2.1 to account for multiple time-delays and norm-bounded uncertainties. This theory is necessary for multi-link optical networks since multiple inputs and outputs produce multiple unique time-delays. We first present the theory associated with a single time-delay system with uncertainties. We then present the extended theory for multiple time-delays. The theory presented in this section comes from [18]. The framework for transforming a linear system with time-delays and uncertainties into the standard norm-bounded uncertainty structure is found in Appendix B. We present the uncertain system with a single time-delay as dx dt = A 0(t)x(t) + A 1 (t)x(t τ) (2.23) where A 0 R n n and A 1 R n n are uncertain matrices defined in a compact set Ω (A 0 (t), A 1 (t)) Ω t 0 and τ > 0 is a known constant time-delay.

Chapter 2. Review of Preliminary Theory 28 We rewrite (2.23) into a distributed delay system form to derive delay-dependent stability criteria. We use the following transformation, x(t τ) = x(t) 0 τ dx (t + θ)dθ dt where dx (t + θ) is the derivative of x evaluated at time t + θ. Substitute (2.23) into the dt above equation to obtain x(t τ) = x(t) 0 We then substitute (2.24) into (2.23) to get τ [A 0 (t + θ)x(t + θ) + A 1 (t + θ)x(t τ + θ)]dθ (2.24) dx(t) dt dx(t) dt = (A 0 (t) + A 1 (t))x(t) A 1 (t) = (A 0 (t) + A 1 (t))x(t) + + τ 2τ 0 τ 0 τ [ A 1 (t)a 1 (t + θ + τ)x(t + θ)]dθ [A 0 (t + θ)x(t + θ) + A 1 (t + θ)x(t τ + θ)]dθ [ A 1 (t)a 0 (t + θ)x(t + θ)]dθ Thus, the system (2.23) may be rewritten into the distributed delay form dx(t) dt = Ā0(t)x(t) + 0 2τ Ā(t, θ)x(t + θ)dθ (2.25) where Ā 0 = A 0 (t) + A 1 (t) A 1 (t)a 0 (t + θ) for τ < θ 0 Ā(t, θ) = A 1 (t)a 1 (t + θ) for 2τ < θ τ (2.26) We first present an intermediary result from [18] to prove the upcoming propositions. Proposition 2.2.1. The distributed delay system described by (2.25) is asymptotically stable if there exists a symmetric matrix P > 0, a symmetric matrix function R(θ, Ā0(t), Ā(t, θ)) with the constraint 0 τ R(θ, Ā0(t), Ā(t, θ))dθ = 0

Chapter 2. Review of Preliminary Theory 29 and symmetric matrix function S(θ) such that M(θ) PĀ(t, θ) Ā T (t, θ)p for all (Ā0(t), Ā(t, θ)) Ω where S(θ) < 0 τ θ 0 (2.27) M(θ) = 1 τ (PĀ0(t) + ĀT 0 (t)p) + S(θ) + R(θ, Ā0(t), Ā(t, θ)) (2.28) The proof is found in Appendix C. The following proposition provides stability results for (2.23) using Proposition 2.2.1. We state the result as an LMI. This result is an extension of Theorem 2.1.2 since it also accounts for uncertainties in the system. Its proof is found in Appendix C. Proposition 2.2.2. The system described by (2.23) is asymptotically stable if there exist symmetric matrices P > 0, S 0 > 0 and S 1 > 0 such that M PA 1 (t)a 0θ PA 1 (t)a 1θ A T 0θ AT 1 (t)p S 0 0 A T 1θ AT 1 (t)p 0 S 1 < 0 (2.29) where A jθ = A j (t + θ) for j = 0, 1, and for all (A 0 (t), A 1 (t)) Ω and (A 0θ, A 1θ ) Ω where M = 1 τ [P(A 0(t) + A 1 (t)) + (A 0 (t) + A 1 (t)) T P] + S 0 + S 1 (2.30) Finally, we present the delay-dependent stability results from [18] for the following uncertain system with multiple time-delays dx(t) dt = K A i (t)x(t τ i ) (2.31) i=0

Chapter 2. Review of Preliminary Theory 30 where 0 = τ 0 < τ 1 < < τ K = τ, and ω Ω for all t 0, for Ω a compact set, and ω = ( A 0 (t) A 1 (t)... A K (t) ) (2.32) As above, we transform (2.31) into a distributed delay form by using x(t τ i ) = x(t) which we substitute into (2.31) and replace dx dt system dx(t) dt = K A i (t)x(t) i=0 K i=1 K j=0 0 0 τ i dx dt (t + θ)dθ with (2.31) to obtain the distributed delay τ i A i (t)a j (t + θ)x(t + θ τ j )dθ (2.33) We state the stability criteria for (2.31) as the following proposition from [18]. Its proof is found in Appendix C. Denote by diag(v i ) the diagonal matrix with elements v i Proposition 2.2.3. The system with multiple time-delays described by (2.31) is asymptotically stable if there exist symmetric matrices P > 0, S ij > 0 for i = 1, 2,..., K; j = 0, 1, 2..., K such that M PA 1 (t)ω θ PA 1 (t)ω θ... PA K (t)ω θ (PA 1 (t)ω θ ) T 1 τ 1 S 1 0... 0 (PA 2 (t)ω θ ) T 1 0 τ 2 S 2... 0....... (PA 1 (t)ω θ ) T 0 0... is satisfied for all ω Ω and ω θ Ω where 1 τ K S K > 0 S i = diag( S i0 S i1... S ik ) and ( ) ω = A 0 (t) A 1 (t)... A K (t) ( ) ω θ = A 0θ A 1θ... A Kθ

Chapter 3 Game-Theoretic and Central Cost Algorithms In this chapter, we review the OSNR model for multi-link optical networks and its control algorithms from [38] and [39]. These control algorithms adjust the channel powers at the transmitters to ensure system-wide OSNR optimization. We begin by presenting the OSNR model for arbitrary multi-link optical networks from [38]. Next, we introduce the central cost and game-theoretic based control algorithms without time-delays and uncertainties as presented in [38] and [39]. We present the derivations for both control algorithms. We also present the multi-link framework of [37] without time-delays and uncertainties. This framework decomposes multi-link networks into cascades of singlelinks to help study the stability of the full networks. 3.1 Review of OSNR Models in Optical Networks We begin by reviewing the OSNR model in [38]. The OSNR model is valid for arbitrary network configurations. Consider an optical network that is defined by a set of optical links, L = {1,..., L}, that connect to optical nodes. The optical nodes allow channels in the network to be added, dropped, or rerouted by optical cross-connects (OXCs). Figure 31

Chapter 3. Game-Theoretic and Central Cost Algorithms 32 Figure 3.1: Mesh Optical Network 3.1 depicts a mesh optical network. A link, l, is composed of a series of N l optical spans that include one optical amplifier (OA) and a length of fiber, eg. 100km, per span. Figure 3.2 depicts a series of optical spans within a link. A set of channels, M = {1,..., n}, (intensity modulated wavelengths) are multiplexed and transmitted across the network. We denote by M l the set of channels transmitted over link l, l L. Also, we denote by R i, i M, the set of links from source (Tx) to destination (Rx), that channel i uses in its optical path. We denote by u i, s i, and n i, the optical input power for channel i at Tx, the output signal at Rx, and the output noise at Rx, respectively. We develop the functional dependence of s i and n i later in the section. Figure 3.3 depicts an optical path for a channel along with the signals u 1, s 1, and n 1. The optical signal-to-noise ratio (OSNR) for a channel, i M, is defined as Later in the section, we express s i and n i as functions of u i. OSNR i = s i n i (3.1) The following provides the framework for modeling OSNR in a general optical network. The k th optical span in link l is composed of an OA with channel dependent gain, G l,k,i, for channel i, and an optical fiber with wavelength independent loss coefficient, L l,k (see Figure 3.2). The OA introduces amplified spontaneous emission (ASE) noise power, denoted by ASE l,k,i. The optical span transmission which represents the total

Chapter 3. Game-Theoretic and Central Cost Algorithms 33 Figure 3.2: Optical spans in a link segment. Figure 3.3: Optical path within a network.

Chapter 3. Game-Theoretic and Central Cost Algorithms 34 power transmitted for the i th channel, in the k th span on the l th link, is given by h l,k,i = G l,k,i L l,k, k = 1,..., N l (3.2) We define the l th link transmission as N l T l,i = h l,q,i, l = 1,..., L (3.3) q=1 which is the total power transmitted for channel i across all spans in link l. Assumption 1. ASE noise power is small compared with the signal power and it does not contribute to gain saturation[2]. ASE power is self-generated in the k th amplifier, on the l th link, for the i th channel. ASE is modeled as: ASE l,k,i = 2n sp [G l,k,i 1]hν i B 1 (3.4) where G l,k,i is the amplifier gain in the wavelength λ i, n sp > 1 is the excess noise factor, h is the Planck constant, B 1 is the optical bandwidth, and ν i is the optical frequency corresponding to wavelength λ i. Assumption 2. All spans in a link have equal length. Assumption 3. All the amplifiers in a link have the same spectral shape and are operated in automatic power control (APC) mode, with the same total power target P 0,l. Thus, G l,k,i is reduced to G l,k,i = G l,i k = 1,..., N l. The operation of the OAs in APC mode is common in application. This ensures a common power output for each optical span and it compensates for variations in fiberspan loss across a link [2]. Mathematically, the total power target for the k th span in the l th link is expressed as p l,k,i = P 0,l (3.5) i where p l,k,i is the output signal power of the i th channel, on the l th link at the k th optical span.

Chapter 3. Game-Theoretic and Central Cost Algorithms 35 We now present the following two OSNR models from [39]. The first OSNR model is valid when the optical amplifiers operate in gain control mode. In this case, the OSNR of the i th channel only depends on the i th channel power. Thus, this OSNR model is decoupled from the remaining channel powers, which results in a simpler OSNR model. The second OSNR model builds on top of the first OSNR model, and it describes the case where the amplifiers are in automatic power control mode. Automatic power control mode requires the same total power to be launched into each span of a link to limit nonlinear effects. Optical amplifiers in the power control mode produce a coupled OSNR model, where each OSNR value is dependent on multiple channel powers. The second OSNR model is more realistic, since the power control mode is used in practice. The following lemma from [39] describes the OSNR model for optical networks where the OAs are in gain control mode. The proof is in [39]. Lemma 3.1.1. Under Assumption 1, the OSNR for the i th channel along a path R i is OSNR i = u i n 0,i + Nl 1 ASE l,k,i (3.6) l R i k=1 l 1 q 1 T q,i H l,k,i where H l,k,i = k q=1 h l,q,i is the intermediary transmission for the l th link and n 0,i is the noise optical power at transmitter (Tx) for the i th channel. Proof. The full proof for Lemma 3.1.1 is found in [39]. For simplicity, we outline the proof for the single-link case. Thus, in the foregoing proof, we drop the l subscript in all of the variables. Assume there is only one link in the optical network. Let p k,i and v k,i be the signal and noise powers, respectively, for channel i at the output of the k th span in the single-link. We write p k,i = h k,i p k 1,i v k,i = h k,i v k 1,i + ASE k,i (3.7)

Chapter 3. Game-Theoretic and Central Cost Algorithms 36 where ASE k,i is defined in (3.4). Let p 0,i = u i and v 0,i = n 0,i be the link input power and the noise input power, respectively. We cascade the equations (3.7) after a series of k optical spans p k,i v k,i = u i k q=1 h q,i k = n 0,i h q,i + N ASE r,i q=1 r=1 q=r+1 k h q,i (3.8) where N is the number of optical spans in the link, and h q,i is defined in (3.2). The last term of (3.8) represents each ASE q,i term, the ASE for the q th span, being transmitted across the remaining spans along the link up to the k th span. Thus, we write the following relationships between the inputs and the outputs of a link s i = T i u i n i = T i n 0,i + N r=1 where T i is defined in (3.3). By (3.1), we obtain T i H r,i ASE r,i (3.9) OSNR i = s i n i = u i n 0,i + N ASE r,i (3.10) r=1 H r,i where T i cancels out. Notice that (3.6) simplifies to (3.10) if we only have one link and we remove the l subscripts from the variables. The OSNR model (3.6) is static. In other words, there are no dynamics within (3.1). However, the dynamic control algorithms introduced in the following sections of the chapter do have dynamics. These dynamics in the control algorithms are the dynamics that are present in the upcoming closed-loop system models. Next, we present the two OSNR models that more typically describe optical networks than the model presented in Lemma 3.1.1. Specifically, the operation of OAs in APC mode as in Assumption 3 accurately describes conventional optical networks. In this case, the signal channels are coupled in the OSNR outputs. We first present the simpler

Chapter 3. Game-Theoretic and Central Cost Algorithms 37 single-link model. The proof for the single-link is simpler than the proof for the multi-link OSNR model. This is later followed by the presentation of the multi-link model. The OSNR model for an optical link is presented as the following lemma. Note that we drop the l subscript for the variables in the single-link case. Lemma 3.1.2. Under Assumptions 1-3, the OSNR for channel i M at the output of a single-link is OSNR i = u i n 0,i + j M Γ i,ju j (3.11) where we define Γ as the (n n) link matrix with elements Γ i,j, given by Γ i,j = N G k j G k k=1 i ASE k,i P 0 i, j M (3.12) Proof. The proof follows similarly to [38], and it relies on Lemma 3.1.1. We rewrite (3.5) as H k,j u j = P 0 (3.13) j M Then, by Assumption 3, we decompose G k,i, the channel dependent gain, as G k,i = G i κ k where κ k is the variable optical attenuator loss, which is adjusted to achieve the total output power P 0. Thus, from the definition of H r,i in Lemma 3.1.1 where we drop the l subscript and use r instead of k, we obtain H r,i = G r i r κ q (3.14) where κ q = κ q L q and L q is the wavelength independent loss coefficient. Substitute (3.14) into (3.13) to get r κ q = q=1 q=1 P 0 j M Gr j u j (3.15) which is wavelength independent. By substituting (3.14) and (3.15) into (3.10), and substituting k back for the r index, we immediately obtain (3.11).

Chapter 3. Game-Theoretic and Central Cost Algorithms 38 Figure 3.4: Block diagram for the OSNR model (3.16). The following lemma from [38] describes the OSNR model for arbitrary multi-link optical networks. The lemma serves as the basis for much of the foregoing analysis in the thesis. The proof is found in [38] and is similar to the proof of Lemma 3.1.2. As such, it is omitted here. Lemma 3.1.3. Under Assumptions 1-3, the OSNR for the i th channel is given as OSNR i = u i n 0,i + j M Γ i,ju j (3.16) where Γ i,j, elements of the full (n n) system matrix Γ, are defined as Γ i,j = N l G k l,j G k l R i R j k=1 l,i ( l 1 q=1 T q,j T q,i ) ASE l,k,i P 0,l (3.17) Figure 3.4 depicts the block diagram for the OSNR model (3.16). Note that n 0 is a noise vector with elements n 0,i. 3.2 Central Cost Based Control Algorithm Using Lemma 3.1.3, a central cost optimization problem is defined in [39] subject to the constraints that each channel has to achieve a minimum OSNR target,ˆγ i, i.e., OSNR i ˆγ i. Recall from Section 3.1 that OSNR is a function of the channel powers, u i. Thus, the OSNR targets are attained by adjusting the channel powers at the transmitters, Tx, in

Chapter 3. Game-Theoretic and Central Cost Algorithms 39 Figure 3.5: Central cost based control algorithm applied to the optical network. real time. The objective of the central cost approach is to minimize the sum of the input powers of all of the network channels. The central cost problem assumes that all network information is available for the control design. The problem and solution are presented in [39] without time-delays. Figure 3.5 depicts the central cost based control algorithm acting on an optical network. The optimization problem is stated as [39] min i M u i (3.18) subject to Define the matrix vector form of (3.19) as u i n 0,i + j M Γ i,ju j ˆγ i i M (3.19) u i 0 i (3.20) u ˆΓu + ˆn 0 (3.21) where ˆΓ = ˆγ Γ, ˆn 0 = ˆγ n 0, ˆγ = ˆγ 1 0... 0 ˆγ m

Chapter 3. Game-Theoretic and Central Cost Algorithms 40 We call ˆΓ the weighted transmission matrix. The vectors u and n 0 are the input signal powers and the input noise powers, respectively. We restate the solution to the optimization problem from [39] as follows. By Perron s Theorem [31], a solution for (3.20) and (3.19) is ensured if ρ(ˆγ) < 1, where ρ denotes the spectral radius of the nonnegative matrix ˆΓ. If ρ(ˆγ) < 1, then (I ˆΓ) 1 exists. This fact guarantees that the solution to the central cost optimization problem (3.18)-(3.20) is unique. Furthermore, [60] shows that the input powers (3.20) that satisfy (3.19) with equality also minimize the sum of the total input powers (3.18). Thus, if ρ(ˆγ) < 1, the unique and optimal solution, u, to the central cost problem (3.18)-(3.20) is the solution of u = ˆΓu + ˆn 0 (3.22) The solution, u, of (3.22) produces a static, centralized value. We present the discretetime distributed and iterative control algorithm developed in [39] based on (3.22). Consider the structure of (3.22) to motivate the design of a control algorithm. For channel i, take the i th row of (3.22) and substitute (3.16) into the RHS, to obtain u i (k) u i (k + 1) = ˆγ i OSNR i (k) (3.23) where k is the iterative time step. The time period between the k and k + 1 steps is a constant that depends on the speed of the algorithm. For example, the algorithm may be updated every T 0 milliseconds, i.e. T 0 milliseconds elapses between time k and k + 1. In other words, the units of the time index k are normalized to the time period T 0. We have imposed the LHS of (3.23) as the next iteration of the channel power, u i (k + 1). Next, we include a tunable adjustment parameter, ρ 0 > 0, which acts as a weight between u i (k) and the RHS of (3.23) to get u i (k) u i (k + 1) = (1 ρ 0 )u i (k) + ρ 0ˆγ i OSNR i (k) (3.24) Figure 3.6 depicts the control algorithm (3.24) acting on the OSNR system (3.16). Notice that (3.24) is decentralized in its form since each channel, u i, only depends on the

Chapter 3. Game-Theoretic and Central Cost Algorithms 41 Figure 3.6: Discrete-time Control Algorithm (3.24) on the OSNR system (3.16). OSNR at the output of channel i. However, due to the problem set-up, and the stability conditions that are presented next, the convergence of (3.24) is only ensured if we have complete knowledge of the system matrix for the entire optical network. Thus, (3.24) is a centralized control algorithm. When (3.24) is applied at the sources of the optical network, the following theorem, restated from [39], ensures convergence to the optimal solution. Theorem 3.2.1. Assume that an optimal solution exists, i.e., ρ(ˆγ) < 1. Then algorithm (3.24) applied to the OSNR model (3.16) is globally stable and converges to the optimal solution, u, if 0 < ρ 0 < 2 1 + ρ(ˆγ) (3.25) Proof. The proof follows as in [39]. We substitute (3.16) into (3.24) to obtain u i (k + 1) = (1 ρ 0 )u i (k) + ρ 0ˆγ i (n 0,i + j Γ i,j u j (k)) In matrix form, this becomes u(k + 1) = (1 ρ 0 )u(k) + ρ 0 (ˆΓu(k) + ˆn 0 ) u(k + 1) = (I ρ 0 (I ˆΓ))u(k) + ρ 0ˆn 0 u(k + 1) = B 0 u(k) + ρ 0ˆn 0 (3.26)

Chapter 3. Game-Theoretic and Central Cost Algorithms 42 where B 0 = I ρ 0 (I ˆΓ). Define e(k) = u(k) u (3.27) We next prove that e(k) 0 as k which ensures the closed-loop system is asymptotically stable. Substitute (3.27) into (3.26) to get e(k + 1) + Iu = B 0 (e(k) + u ) + ρ 0ˆn 0 and using (3.22) we obtain e(k + 1) = B 0 e(k) (3.28) The solution to (3.28) is e(k) = B0e(0) k Thus, e(k) 0 if and only if the eigenvalues of B 0 are inside the unit circle. We relate the eigenvalues of B 0 to the eigenvalues of ˆΓ. Let (λ i (ˆΓ), w) be an eigenvalue/eigenvector pair of ˆΓ. Thus, we have ˆΓw = λ i (ˆΓ)w which we use in the following B 0 w = (I ρ 0 (I ˆΓ))w = (1 ρ 0 (1 λ i (ˆΓ)))w = λ i (B 0 )w (3.29) where λ i (B 0 ) = 1 ρ 0 (1 λ i (ˆΓ)) is an eigenvalue for B 0 as a function of an eigenvalue of ˆΓ. We then write λ i (B 0 ) = 1 ρ 0 + ρ 0 λ i (ˆΓ) 1 ρ 0 + ρ 0 λ i (B 0 ) 1 ρ 0 + ρ 0 max i λ i (ˆΓ) = 1 ρ 0 + ρ 0 ρ(ˆγ) (3.30)

Chapter 3. Game-Theoretic and Central Cost Algorithms 43 where we used the triangle inequality and ρ(ˆγ) is the spectral radius of ˆΓ. Thus, λ i (B 0 ) < 1 if the RHS of (3.30) is less than 1, or 1 ρ 0 + ρ 0 ρ(ˆγ) < 1 Thus, as long as (3.25) holds, then e(k) 0 as k. Despite having a convergent control algorithm for the OSNR optimization problem, the practical implementation of such a control law would have to take time-delays into account. 3.3 Game-Theoretic Based Control Algorithm A non-cooperative game between channels is defined in [38] using Lemma 3.1.3 without time-delays. We present the work from [38] here. Recall from Lemma 3.1.3 that as the channel power u i increases, thereby increasing the OSNR value of the i th channel, it also has the effect of decreasing the OSNR values of the remaining channels. As such, it is not obvious how to dynamically adjust the channel input powers to optimize the OSNR values in the optical network given these trade-offs in OSNR between the channels. A game-theoretic framework can be used to develop control algorithms for optical networks. In such a framework, the maximization of the utility function for a channel i is directly related to the maximization of its OSNR value. Let the vector u i represent the vector with elements u i, the channel powers, where the i th entry is deleted, i.e. u i = [ u 1... u i 1 u i+1... u n ] T. We define the following cost function for each player i [38], J i (u i, u i ) = α i u i β i U i (u i, u i ) i (3.31) with ( ) OSNR i U i = ln 1 + a i 1 Γ i,i OSNR i (3.32)

Chapter 3. Game-Theoretic and Central Cost Algorithms 44 where α i is the channel cost, β i is the utility weight, a i is a tunable channel dependent design parameter. The utility function, U i, is what the i th channel is trying to maximize. The utility function U i is a continuously differentiable function in u i, monotone increasing and strictly concave in u i. Also, u i = 0 is not a solution to the minimization of the cost function J i, i.e., J i (0) > J i (u i ) u i 0. We define the Nash equilibrium as follows. Definition 3.3.1. Consider n-players in a game. Each player minimizes a coupled cost function, J i, such that u i [0, u max ], where u max is the upper bound for all u i. The Nash equilibrium solution is defined by the vector u if J i (u ) inf ui [0, u max]j i (u i, u i) i. By Definition 3.3.1, a game achieves the Nash equilibrium when no player can improve its cost function by unilaterally changing its action as long as the other players maintain their own actions. The following theorem from [38] ensures (3.31) has a unique Nash equilibrium. Theorem 3.3.1. The n-player game problem with cost functions (3.31) admits a unique Nash equilibrium, u, if we satisfy the internal condition a i > j i Γ i,j i (3.33) Proof. We begin by substituting (3.16) into (3.31), to obtain ( J i (u i, u i ) = α i u i β i ln 1 + a i u i n 0,i + j i Γ i,ju j ) i (3.34) The unique Nash equilibrium, u, is obtained from (3.34) by setting J i u i = 0. Note we may set J i u i = 0 due to the properties of the utility function, U i, defined in (3.32), which results in a strictly convex cost function, J i, in (3.31). Thus, we obtain a i u i + j i Γ i,j u j = a iβ i α i n 0,i i (3.35)

Chapter 3. Game-Theoretic and Central Cost Algorithms 45 We rewrite (3.35) in the equivalent matrix form Γu = b (3.36) where b is a column vector with elements a iβ i α i n 0,i and the elements of Γ are defined as a i, i = j Γ i,j = (3.37) Γ i,j, i j Thus, if Γ is invertible, a unique Nash equilibrium exists for (3.31). Note that all of the entries of Γ defined in (3.37) are positive. Hence, if (3.33) holds, then Γ is diagonally dominant, which means Γ is also invertible, and (3.31) admits a unique Nash equilibrium. We now present the channel algorithm from [38] that ensures the convergence of the OSNR values of the system (3.16) to the Nash equilibrium, u, in (3.35). The channel algorithm is based on the structure of (3.35) but with the substitution of (3.16), u i (k + 1) = β i 1 ( ) 1 µ i a i OSNR i (k) Γ i,i u i (k) (3.38) where k is the iterative time step and µ i is the channel price. The units of the time index k are normalized to the time period T 0, i.e. T 0 milliseconds elapses between time k and k + 1. We also call the channel algorithm the primal algorithm. The term primal originates from congestion control problems in technical literature [21, 35]. In addition, the term dual is also used to describe the link algorithm, which is introduced in the following section. For systems in which there is both a channel and link algorithm present, we refer to them as primal-dual algorithms. Without the link algorithm, which is introduced in the following section, we have µ i = α i for all i. Figure 3.7 depicts the control algorithm (3.38) acting on the OSNR system (3.16). Notice that we have included the µ vector as an input to the control algorithm (3.38) although we treat it as a constant for now.

Chapter 3. Game-Theoretic and Central Cost Algorithms 46 Figure 3.7: Discrete-time Control Algorithm (3.38) on the OSNR system (3.16) We now state the following theorem along with its proof from [38]. Theorem 3.3.2. The channel algorithm (3.38) applied to (3.16), for µ i = α i constant, admits a unique Nash equilibrium, u, if (3.33) is satisfied. Proof. We define e i (k) = u i (k) u i. Thus, we obtain the following using (3.35) and (3.16), e i (k + 1) = u i (k + 1) u ( i = β i 1 n 0,i + Γ i,j u j (k) u i α i a i j i = 1 Γ i,j e j (k) (3.39) a i We use (3.39), and the triangle inequality to obtain, ( ) 1 e(k + 1) = max i e i (k + 1) max i Γ i,j e j (k) a i j i We next use e j (k) e(k) for all j, to get ( 1 e(k + 1) max i a i Finally, by (3.33), we have j i Γ i,j ) j i ) e(k) e(k + 1) < e(k), k = 0, 1, 2,... (3.40)

Chapter 3. Game-Theoretic and Central Cost Algorithms 47 Figure 3.8: Block diagram of the primal-dual algorithm acting on the optical network. which implies e(k) 0 as k increases, and hence, u(k) u. 3.4 Link Algorithm In [40], a link algorithm is proposed that computes a channel price, µ, in real-time such that total power constraints are satisfied. The link algorithm is located at the OSNR output locations, away from the channel sources where the channel algorithms are located. The inputs to the link algorithm are the total channel powers and its output is a scalar channel price µ. The total channel powers are measurable in real-time. The link algorithm complements the game-theoretic channel algorithm from Section 3.3. Figure 3.8 shows both control algorithms acting on the optical network. The channel price, µ, is computed every K iterations of the control algorithm, (3.38), where K is a large number, e.g., K = 100. Let k denote the slow time index of the link algorithm with respect to the fast time-scale, k, of the channel algorithm. The two time indexes are related by k = K k. The time period between the k and k + 1 steps is a