Elsevier

Automatica

Volume 40, Issue 8, August 2004, Pages 1361-1370
Automatica

Dynamics and stability in optical communication networks: a system theory framework

https://doi.org/10.1016/j.automatica.2004.03.014Get rights and content

Abstract

This paper addresses the problem of dynamics analysis in optical networks from a system control perspective. A general framework for finding the transfer matrix representation of an optical network is developed, based on linear fractional transformations. Under the natural assumption of equal time-delay for all channels in a link, the network transfer matrix is simplified such that channel cross-coupling is evidenced. The optical network stability problem is then reformulated as a robust stability problem and stability conditions are developed by applying μ-analysis.

Introduction

In the context of evolution from optical point-to-point communication links towards reconfigurable optical networks, dynamic aspects in optical networks have started to be considered very recently, as in Bala and Brackett (1996), Mukherjee (2000) and Barnes (2002). Reconfigurable optical networks operate in a dynamic environment, with existing channels being continuously in-service while network reconfiguration (channel add/drop) is being performed. Such optical networks are in fact large-scale, time-delay dynamical systems, with channel add/drop acting as a disturbance for the existent channels. This paper addresses the problem of dynamic response analysis in optical networks from a system and control theory perspective.

In optical communications, information is transmitted by modulating the optical power of light pulses of a specific frequency (wavelength). Multichannel optical systems are realized by wavelength division multiplexing (WDM), which consists of several sources multiplexed in the wavelength domain. Optical signals, corresponding to multiple channels at different wavelengths, are transmitted together on a single optical fiber. Each channel's information is recovered at the receiver after demultiplexing and conversion from optical to electrical domain. An essential parameter in determining the performance of an optical communication system is the optical power per channel. Optical signals experience power loss during propagation through an optical fiber, which is compensated by using optical amplifiers every few tens of km.

The need to study dynamics and dynamic effects in optical networks is motivated by two important facts. Firstly, the presence of optical amplifiers which have active control, together with their operation in reconfiguration scenarios introduces a time-dependent component. The most widely used multichannel optical amplifier is the Erbium-doped fiber amplifier (EDFA), firstly studied from a dynamical point of view in Sun, Zyskind, and Srivastava (1997). Due to their slow gain-dynamics, EDFAs do not introduce any channel cross-talk (cross-coupling), at the high bit-rates used in optical communications. EDFAs react only to variation in average optical signal power. However in dynamically reconfigurable networks, channel add/drop and hence changes in optical signal power, occur on time-scales comparable with the EDFA time-constant. This requires consideration of dynamics as well as transient control in optical amplifiers, see for example Sun et al. (1997), Srivastava, Sun, Zyskind, and Sulhoff (1997) and Pavel (2003). Other devices such as dynamic equalizing filters (see Ford & Walker, 1998), are also used to adjust optical power profile in wavelength domain. This enables all channels to reach the receiver with equal optical power, and hence with equal optical signal-to-noise ratio. Effectively, an optical link is a multivariable dynamical system composed from cascaded dynamical optical elements.

Secondly, for reconfigurable optical networks, mesh type configurations can be realized, not only link configurations. Since channel routes can be changed dynamically via channel add/drop or switching, closed cycles can be formed, (see Bala & Brackett, 1996), and time-delay effects need to be considered. Network reconfiguration, failures or protection switching can cause abrupt changes in optical power across some wavelengths, which can be transferred to other wavelengths. Due to dynamic cross-coupling effects as well as due to propagation time-delay, for ring-type configurations, bursts of power fluctuations occur, resulting in possible network instability. This was shown recently in simulation and experimental studies, as in Yoo, Xin, and Garrett (1998), Kim, Bae, Joon Ahn, and Park (2000), Pavel (2002). The resulting dynamic behavior of such a network is complex and, to the best of our knowledge, no analytical network dynamics studies exist.

In conventional switched communication networks, dynamics is an important aspect of the data-layer, in terms of traffic congestion, control and stability, (see Altman, Basar, & Srikant, 1999; Alpcan & Basar, 2002; Wang & Paganini, 2002). In these networks, channels are characterized by static loss/gain only. Given the inherent dynamics of an optical link, optical network dynamics is an important aspect even at the physical layer. The analytical study of optical network dynamics is the problem considered in this paper. For tractability, linearized models are assumed.

The paper is organized as follows. In Section 2, we formulate the problem for a generic optical network in ring-type configuration. In Section 3, we find the network transfer matrix, in terms of a modified Redheffer star-product with propagation delay feedback. We specialize these results for transient response analysis under channel add/drop. In Section 4, we address the network stability by reformulating it as a robust stability problem and resorting to μ-analysis. Simulation results are shown in Section 5, with conclusions being given in Section 6.

Section snippets

Problem formulation

The following general notations are defined. Assume as in general network theory, (see Anderson, 1973), that the optical network is characterized by a set of nodes, a set of optical links L={1,…,L} connecting the nodes, and a set of channels, M={1,…,m}. Optical add-drop multiplexers (OADM) that are used to separate and recombine wavelength channels, constitute network nodes. The input to each optical link is either from optical sources (Tx) or from other optical links, via the OADM (switching)

Optical network transfer matrix

In this section we find a closed form of the optical network transfer matrix. We start by finding optical span and optical link transfer matrices. We show how the distributed multichannel time-delay can be lumped on a link-by-link basis. The network transfer matrix is then determined by applying linear fractional techniques. Next we show that network dynamics depends explicitly on time-delay along the optical path and on channel group cross-coupling across the network.

Consider the general

Stability analysis framework

In addition to being useful for analyzing optical network dynamic response, Theorem 3 can be used to address the optical network stability analysis. This problem will be considered in this section, by reformulating it as a robust stability problem and solving it via μ-analysis.

An underlying basic assumption is that all optical links are stable systems, so that from (23) in Theorem 3 it follows that the internal stability of the overall optical network is equivalent to stability of ΨD, (23),ΨD

Simulation results

Consider a network configuration as in Fig. 1, with twelve optical dynamic amplifier spans, each span of approximately 70km, corresponding to about τ=0.33ms delay. The network has 80 wavelengths grouped in 10 sub-bands propagating across a total 12×70km loop, resulting in a total nominal delay of about τ0=4ms. Each dynamic optical amplifier is composed of an optical amplifier (EDFA) and a DGE, which is modeled as a wavelength decoupled system. Each EDFA is modeled as a (10×10) MIMO transfer

Conclusions

In this paper, we have developed an analytical approach for studying dynamic behavior in optical networks. To our knowledge, this represents the first theoretical study of dynamic response in optical networks, from a control theory perspective. Such a problem is extremely complex as the optical network is a large-scale multivariable, interconnected system, with distributed time-delays. This work is relevant for reconfigurable optical networks where closed cycles, affected by time-delay, can be

Lacra Pavel has received the M. Eng. degree in Automatic Control from the Technical University Iasi, Romania in 1989 and the Ph.D. degree in Electrical Engineering (Control Theory) from Queen's University, Kingston, Canada in 1996. Between 1997 and 1998 she was with the National Research Council in Ottawa, as an NSERC Postdoctoral Fellow, working on noise and vibration control for flexible structures. From 1998 to 2002 she worked in the optical communication industry as a Senior Control

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Lacra Pavel has received the M. Eng. degree in Automatic Control from the Technical University Iasi, Romania in 1989 and the Ph.D. degree in Electrical Engineering (Control Theory) from Queen's University, Kingston, Canada in 1996. Between 1997 and 1998 she was with the National Research Council in Ottawa, as an NSERC Postdoctoral Fellow, working on noise and vibration control for flexible structures. From 1998 to 2002 she worked in the optical communication industry as a Senior Control Engineer. In 2002 she joined University of Toronto, where she is currently an Assistant Professor cross-appointed to Systems Control and Optical Communications. Her research interests are in the general area of control theory with applications to optical communications networks, control of networks, robust and H-infinity optimal control, nonlinear control systems.

This paper was not presented at any IFAC meeting. This paper was recommended for publication in revised form by Associate Editor Hitay Ozbay under the direction of Editor Tamer Başar. This work was supported by the Natural Sciences and Engineering Research Council of Canada.

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