Robust Stability Analysis of Smith Predictor-based Congestion Control Algorithms for Computer Networks ✩ Luca De Cicco a,∗ , Saverio Mascolo a , Silviu-Iulian Niculescu b a Politecnico di Bari, Dipartimento di Elettrotecnica ed Elettronica, Via Orabona, 4, Bari, Italy b Laboratoire des Signaux et Syst` emes (L2S), CNRS-SUPELEC, 3 rue Joliot-Curie 91192 Gif-sur-Yvette cedex, France Abstract Congestion control is a fundamental building block in packet switching networks such as the Internet due to the fact that commu- nication resources are shared. It has been shown that the plant dynamics is essentially made of an integrator plus time delay and that a proportional controller plus a Smith predictor defines a simple and effective controller. It has been also shown that the today running TCP congestion control can be modelled using a Smith predictor plus a proportional controller. Due to the importance of this control structure in the field of data network congestion control, we analyze the robust stability of the closed-loop system in the face of delay uncertainties that in data networks are present due to queuing. In particular, by applying a geometric approach, we derive a bound on the proportional controller gain which is necessary and sufficient to guarantee the closed-loop stability for a given bound on the delay uncertainty. Keywords: Time-delay systems, robust stability, congestion control, Smith predictor 1. Introduction Time delays are often present in feedback control systems due to reasons such as the transport of material or information. From the control theoretic point of view, it is well-accepted that, quite often, an increase of the time delay may lead to instabil- ity of the closed-loop system and to performance degradation as well (see, e.g. Niculescu (2001), Niculescu and Michiels (2007), Zhong (2006) and the references therein). In such cases, the design of a finite-dimensional controller, such as the clas- sical PID, is very challenging since the closed-loop system has an infinite number of characteristic roots and the resulting con- troller could provide an unacceptable sluggish closed-loop dy- namics (Astrom and Hagglund, 1995). The Smith principle (Smith, 1959) is a classical ap- proach which is often employed to design effective infinite- dimensional controllers for time delay systems using an appro- priate transformation of the control scheme which takes the de- lay out of the loop (see, for instance, Niculescu and Michiels (2007) for further details). Such a controller design proved its interest in various applications covering congestion mech- anisms in communication networks (Mascolo, 1999), motion synchronization in various network configurations (Cheong et al., 2009a) or collaborative simulations in ring-like networks (Cheong et al., 2009b). It is known that, by assuming the exact knowledge of both the plant model and time delay, controllers ✩ This paper is the extended version of a paper that has been presented at the 8th IFAC Workshop on Time-Delay Systems, Sinaia, Romania, September 2009. ∗ Corresponding author. Tel.: +39 080 5963851; Fax: +39 080 5963410 Email addresses: ldecicco@gmail.com (Luca De Cicco), mascolo@poliba.it (Saverio Mascolo), Silviu.Niculescu@lss.supelec.fr (Silviu-Iulian Niculescu) designed using a Smith predictor are very effective in counter- acting the effect of time delays. It is worth to mention that such a method is less effective in the presence of modelling errors in the delay terms. Therefore, robustness issues of the Smith predictor with respect to uncertainties in the knowledge of the time delay have been extensively studied since 1980 see, for instance, Palmor (1980), Yamanaka and Shimemura (1987), Gudin and Mirkin (2007). Indeed, the Internet represents a challenging case study in the context of time delay systems due to the presence of delays that are caused by the propagation of the information, which is sent in form of data packets, from a source to a destination through a series of communication links and router queues. A cornerstone component of the Internet protocol stack is the end-to-end congestion control which has been implemented in the Transmission Control Protocol (TCP) by Van Jacobson in order to avoid congestion and preserve network stability (Ja- cobson, 1988). Basically, the goal of a network congestion control algo- rithm is to adequately throttle the input rate at each source of a connection so that router buffer overflows are avoided. The congestion control proposed by V. Jacobson, and introduced in the TCP, proposes to address this problem implementing a dis- tributed end-to-end algorithm that closes the loop at the end- point of the connection and does not require any explicit feed- back from the routers. The only (implicit) feedback signal con- sidered by TCP is represented by packet loss events that are in- terpreted by the congestion control algorithm as a synonymous of network congestion. In the last decade, the issue of modelling the TCP congestion control algorithm has gained a great deal of attention in the sci- entific community. Indeed, fluid flow models play an important Preprint submitted to Automatica January 19, 2011