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52
Stability criteria for switched and hybrid systems
 SIAM Review
, 2007
"... The study of the stability properties of switched and hybrid systems gives rise to a number of interesting and challenging mathematical problems. The objective of this paper is to outline some of these problems, to review progress made in solving these problems in a number of diverse communities, an ..."
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Cited by 114 (8 self)
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The study of the stability properties of switched and hybrid systems gives rise to a number of interesting and challenging mathematical problems. The objective of this paper is to outline some of these problems, to review progress made in solving these problems in a number of diverse communities, and to review some problems that remain open. An important contribution of our work is to bring together material from several areas of research and to present results in a unified manner. We begin our review by relating the stability problem for switched linear systems and a class of linear differential inclusions. Closely related to the concept of stability are the notions of exponential growth rates and converse Lyapunov theorems, both of which are discussed in detail. In particular, results on common quadratic Lyapunov functions and piecewise linear Lyapunov functions are presented, as they represent constructive methods for proving stability, and also represent problems in which significant progress has been made. We also comment on the inherent difficulty of determining stability of switched systems in general which is exemplified by NPhardness and undecidability results. We then proceed by considering the stability of switched systems in which there are constraints on the switching rules, through both dwell time requirements and state dependent switching laws. Also in this case the theory of Lyapunov functions and the existence of converse theorems is reviewed. We briefly comment on the classical Lur’e problem and on the theory of stability radii, both of which contain many of the features of switched systems and are rich sources of practical results on the topic. Finally we present a list of questions and open problems which provide motivation for continued research in this area.
A Unifying Passivity Framework for Network Flow Control
 IEEE Transactions on Automatic Control
, 2002
"... Network flow control regulates the traffic between sources and links based on congestion, and plays a critical role in ensuring satisfactory performance. In recent studies, global stability has been shown for several flow control schemes. By using a passivity approach, this paper presents a unifying ..."
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Cited by 80 (14 self)
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Network flow control regulates the traffic between sources and links based on congestion, and plays a critical role in ensuring satisfactory performance. In recent studies, global stability has been shown for several flow control schemes. By using a passivity approach, this paper presents a unifying framework which encompasses these stability results as special cases. In addition, the new approach significantly expands the current classes of stable flow controllers by augmenting the source and link update laws with passive dynamic systems. This generality offers the possibility of optimizing the controllers, for example, to improve robustness and performance with respect to time delay, unmodeled flows, and capacity variation.
Global Stability of Congestion Controllers for the Internet
 IEEE Transactions on Automatic Control
, 2002
"... We consider a single link accessed by a single source which responds to congestion signals from the network. The design of controllers for such sources in the presence of feedback delay has received much attention recently. Here we present conditions for the global, asymptotic stability and semi ..."
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Cited by 76 (6 self)
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We consider a single link accessed by a single source which responds to congestion signals from the network. The design of controllers for such sources in the presence of feedback delay has received much attention recently. Here we present conditions for the global, asymptotic stability and semiglobal exponential stability of congestion controllers which are natural extensions of earlier linearized analysis of such systems. Our result on exponential stability provides the missing link in the proof of how one obtains a single deterministic congestion control equation from a system with many congestioncontrolled sources and random disturbances. Using numerical examples, we compare the conditions on the congestioncontrol parameters obtained using local and global stability analysis.
A positive systems model of TCPlike congestion control: Asymptotic results
 IEEE/ACM Transactions on Networking
, 2004
"... In this paper we study communication networks that employ droptail queueing and AdditiveIncrease MultiplicativeDecrease (AIMD) congestion control algorithms. We show that the theory of nonnegative matrices may be employed to model such networks. In particular, we show that important network p ..."
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Cited by 64 (10 self)
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In this paper we study communication networks that employ droptail queueing and AdditiveIncrease MultiplicativeDecrease (AIMD) congestion control algorithms. We show that the theory of nonnegative matrices may be employed to model such networks. In particular, we show that important network properties such as: (i) fairness; (ii) rate of convergence; and (iii) throughput; can be characterised by certain nonnegative matrices that arise in the study of AIMD networks. We demonstrate that these results can be used to develop tools for analysing the behaviour of AIMD communication networks. The accuracy of the models is demonstrated by means of several NSstudies.
Global Stability of Internet Congestion Controllers with Heterogeneous Delays
, 2004
"... In this paper, we study the problem of designing globally stable, scalable congestion control algorithms for the Internet. Prior work has primarily used linear stability as the criterion for such a design. Global stability has been studied only for single node, single source problems. Here, we obtai ..."
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Cited by 57 (1 self)
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In this paper, we study the problem of designing globally stable, scalable congestion control algorithms for the Internet. Prior work has primarily used linear stability as the criterion for such a design. Global stability has been studied only for single node, single source problems. Here, we obtain conditions for a general topology network accessed by sources with heterogeneous delays. We obtain a sufficient condition for global stability in terms of the increase/decrease parameters of the congestion control algorithm and the price functions used at the links.
Global stability with timedelay in network congestion control
 in Proc. IEEE Conf. Decision Control, 2002
"... This paper concerns the global stability of recently proposed laws for network congestion control. In earlier work it was shown that such laws were able to maintain local dynamic stability on networks of arbitrary delay, capacity and topology. In this paper we give conditions under which the stabili ..."
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Cited by 38 (3 self)
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This paper concerns the global stability of recently proposed laws for network congestion control. In earlier work it was shown that such laws were able to maintain local dynamic stability on networks of arbitrary delay, capacity and topology. In this paper we give conditions under which the stability can be shown to be global, focusing on the case of a single bottleneck network. The results include: a general boundedness result; a theorem on global asymptotic stability for the case of a single source, or many homogeneous ones, under quite sharp conditions; and a somewhat weaker asymptotic stability theorem for the heterogeneous source case. 1
RateBased versus QueueBased Models of Congestion Control
 in Proceedings of ACM SIGMETRICS
, 2004
"... Mathematical models of congestion control capture the congestion indication mechanism at the router in two different ways: ratebased models, where the queuelength at the router does not explicitly appear in the model, and queuebased models, where the queue length at the router is explicitly a p ..."
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Cited by 30 (3 self)
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Mathematical models of congestion control capture the congestion indication mechanism at the router in two different ways: ratebased models, where the queuelength at the router does not explicitly appear in the model, and queuebased models, where the queue length at the router is explicitly a part of the model. Even though most congestion indication mechanisms use the queue length to compute the packet marking or dropping probability to indicate congestion, we argue that, depending upon the choice of the parameters of the AQM scheme, one would obtain a ratebased model or a rateandqueuebased model as the deterministic limit of a stochastic system with a large number of users.
Remote Stabilization via TimeVarying Communication Network Delays: Application to TCP Networks
 in Proc. of the IEEE Conference on Control Applications
, 2004
"... In this paper we investigate the problem of remote stabilization via communication networks. This problem arises when the control law is remotely implemented. The exchange of data between the controller and the system is done via a data communication network described by a deterministic model. This ..."
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Cited by 17 (11 self)
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In this paper we investigate the problem of remote stabilization via communication networks. This problem arises when the control law is remotely implemented. The exchange of data between the controller and the system is done via a data communication network described by a deterministic model. This leads to the problem of stabilizing an openloop unstable system with timevarying delay. Assuming a known general model for the timedelay dynamics, we develop a timevarying horizon predictor, which is used as a basis to build a stabilizing control law that includes explicitly the delay dynamics. This control law is then applied to the case of TCP networks. Experimental results are presented.
lp stability and delay robustness of network flow control
 in Proceedings of the IEEE Conference on Decision and Control, Maui
, 2003
"... This paper studies robustness of Kelly’s source and link control laws in [1] with respect to disturbances and timedelays. This problem is of practical importance because of unmodeled flows, and propagation and queueing delays, which are ubiquitous in networks. We first show Lpstability, for p ∈ [1 ..."
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Cited by 10 (3 self)
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This paper studies robustness of Kelly’s source and link control laws in [1] with respect to disturbances and timedelays. This problem is of practical importance because of unmodeled flows, and propagation and queueing delays, which are ubiquitous in networks. We first show Lpstability, for p ∈ [1, ∞], with respect to additive disturbances. We pursue L∞stability within the inputtostate stability (ISS) framework of Sontag [2], which makes explicit the vanishing effect of initial conditions. Next, using this ISS property and a loop transformation, we prove that global asymptotic stability is preserved for sufficiently small timedelays in forward and return channels. For larger delays, we achieve global asymptotic stability by scaling down the control gains as in Paganini et al. [3]. 1