Hopf bifurcation control of congestion control model in a wireless access network
Introduction
Nowadays, with an increasing number of Internet applications, congestion control in a wireless access network becomes more and more important [1], [2], [3], [4], [5]. However, because of the limited spectrum and the serious damage of the wireless medium which suffer from time-varying fading, shadow, interference, etc, congestion control for a wireless network is much more challenging than that for wired networks [3]. Thus, in order to study the congestion control of a wireless access network, many simplified models have been established and studied recently [2], [3], [4], [5]. These simplified models are very useful to discuss the congestion control algorithms and their control effects in a wireless access network [5].
Additionally, as we know, one of the most important factors on the performance of congestion control is its stability, and a lot of research has been devoted to its analysis [2], [3], [4], [5]. A stable state of the algorithm can guarantee the good performance of congestion control system. While the congestion control algorithm loses its stability, it causes some nonlinear dynamical behaviors such as periodic oscillatory behavior, chaos and bifurcation. However, these complex dynamic behaviors are usually harmful to the system as it means the system has been changed from a stable state to an unstable one [6], [7]. Hence, much work has been used to delay or even avoid this kind of behaviors.
In this work, we focus on the problem of bifurcation control for a wireless access network model. Bifurcation control generally refers to the problem of modifying the bifurcation characteristics so that achieving some desirable dynamical behaviors [8], for example, postponing the onset of the bifurcation, stabilizing an unstable bifurcating solution and changing the critical points of an existing bifurcation [6]. Recently, a lot of methods are studied and investigated to control Hopf bifurcation. For instance, an impulsive control method was proposed for controlling bifurcations in the Internet congestion control system [4]. In [6], [8], the authors proposed a dynamic delayed feedback control method is utilized for stabilizing unstable fixed points near Hopf bifurcation. Later, a dynamic state-feedback control law incorporating a washout filter in [9] and parameters delay feedback control in [10] were also used in controlling the Hopf bifurcation. In [11], [12], [13], [14], [15], a hybrid control strategy using both state feedback and parameter perturbation was applied to control the Hopf bifurcation. Moreover, there have been many other ideas and methods of bifurcation control [16], [17], [18], [20], [21]. These methods can also be applied to bifurcation control problem for a wireless access network model.
In this paper, we will apply a hybrid control strategy in [11], [12], [13], [14], [15] to a wireless access network model, the control strategy has two advantages: firstly, the hybrid control strategy combines state feedback with parameter perturbation for realizing bifurcation control; secondly, the strategy can realize bifurcation control without changing the equilibrium point of the system, thus the properties of the original system can be retained completely. Therefore, by using the hybrid control strategy, we only modify the congestion control method in the source side of the wireless access network, while keeping the congestion control of the router unchanged in the network. This makes the control strategy easy to be applied in reality. At the same time, we extend the stable range in parameter so that the controlled system can maintain its stable dynamical behaviors in a large range of parameter value variation.
The remainder of this paper is organized as follows. In Section 2, we introduce a hybrid control strategy to a wireless access network model, and then study the existence of the Hopf bifurcation of this system. In Section 3, we apply a method based on the center manifold theorem and the normal form theory to study the direction of the Hopf bifurcation and the stability of the bifurcating periodic solutions. To verify the theoretic analysis, numerical simulations are given in Section 4. Finally, Section 5 concludes with some discussions.
Section snippets
Hopf bifurcation analysis of uncontrolled system
In this section, we consider a simplified version of the TCP/AQM wireless network mode. The uncontrolled system can be modeled by the following delay differential equations [5]:where is the average of the TCP window size (packets), denotes the average queue length (packets), τ is the round-trip time (second), denotes the packet loss probability in the down link transmission, denotes the packet
Direction and stability of the Hopf bifurcation for controlled system
In this section, we will study the direction of the Hopf bifurcation and the stability of the bifurcating periodic solutions by the center manifold and the normal form theories.
Applying Taylor expansion to the system (3) at the equilibrium point, we havewhere
Numerical examples
In this section, we present numerical results to verify the analytical predictions obtained in the previous section, using the hybrid control to control the Hopf bifurcation in a wireless access network.
For a consistent comparison, we choose the system parameters as used in [5], with , , C=1000 (packets/s), , and .
Firstly, we choose , so the system is the uncontrolled model. For the uncontrolled model, we know that when , we get , and
Conclusions and discussion
In this paper, the problem of Hopf bifurcation control for a wireless access network model with time delays has been studied. To control the Hopf bifurcation, a hybrid control strategy has been proposed. By selecting control parameters appropriately, this method can effectively postpone the onset of the Hopf bifurcation so that achieving some desirable dynamical behaviors. Furthermore, we also applied a method based on the center manifold theorem and normal form theory to study the properties
Acknowledgments
This paper is supported by NSFC-Guangdong Joint Fund, People's Republic of China (No. U1201255), the National Natural Science Foundation of China, People's Republic of China (Nos. 61201227 and 61172127), the Natural Science Foundation of Anhui (No. 1208085MF93) and 211 Innovation Team of Anhui University, People's Republic of China (Nos. KJTD007A and KJTD001B).
Dawei Ding works as an associate professor in Department of Electronics and Information Engineering of Anhui University, Hefei, China. He graduated from the Radio technology and received the master׳s degree in Circuits and Systems from Anhui University, in 1997 and 2004, respectively. In 2008, he received the Ph.D. degree in Circuits and Systems. He has held and participated in a number of national natural science fund research. Main research direction includes communications networks, the
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2021, NeurocomputingCitation Excerpt :In the past, research on dynamic characteristics of congestion control systems has produced a lot of results. The stability and the Hopf bifurcation problems of system with time delays are the most representative studies [11–16]. For example, Ding et al. established an Internet congestion control dual model in [11,16], they studied the Hopf bifurcation caused by time delay and obtained the bifurcation periodic solution of the system by the disturbance method.
Complex dynamic behaviors of a congestion control system under a novel PD<sup> [Formula presented]</sup> control law: Stability, bifurcation and periodic oscillations
2019, Chaos, Solitons and FractalsCitation Excerpt :Furthermore, the state feedback control has been proposed to control unstable steady states or periodic orbits [25,26]. Moreover, the hybrid controller has been advanced to achieve some desirable dynamical performance with changing the feedback gain [27,28]. Under these control strategies, the stability range of congestion control systems has been extended and bifurcation mechanism of the controlled system can be moderately altered.
Congestion tracking control for uncertain TCP/AQM network based on integral backstepping
2019, ISA TransactionsCitation Excerpt :Nowadays, communication networks play an important role in contemporary society, however, the traffic congestion has become a main problem in current network control. During the past three decades, the attention on network congestion control has been increasing [1–3], and a major method is called active queue management (AQM) algorithm. Among them, an earliest AQM method, called random early detection (RED), was proposed by [4].
Dawei Ding works as an associate professor in Department of Electronics and Information Engineering of Anhui University, Hefei, China. He graduated from the Radio technology and received the master׳s degree in Circuits and Systems from Anhui University, in 1997 and 2004, respectively. In 2008, he received the Ph.D. degree in Circuits and Systems. He has held and participated in a number of national natural science fund research. Main research direction includes communications networks, the nonlinear circuit network, the network congestion control, nonlinear dynamics and chaos, bifurcation, etc. He has published a number of papers in international journals and conferences, and he was repeatedly invited to be a high level of foreign journals to review.
Xuemei Qin was born in Anhui Province, China, in 1989. She is currently a graduate student with Signal and Information Processing, Department of Electronics and Information Engineering, Anhui University, Hefei, China. Her research interests include the network congestion control, nonlinear dynamics and chaos, bifurcation, etc.
Tingting Wu was born in Anhui Province, China, in 1989. She is currently a graduate student with Signal and Information Processing, Department of Electronics and Information Engineering, Anhui University, Hefei, China. Her research interests include the network congestion control, nonlinear dynamics and chaos, bifurcation, etc.
Nian Wang is currently a deputy dean in the Department of Electronics and Information Engineering of Anhui University. He graduated from the Radio Technology, Circuits and Systems and Computer Application Technology from Anhui University, in 1986, 1998 and 2005, respectively. He has presided over many meetings and participated in a number of scientific research projects. Main research interests include internet congestion, computer vision and image processing, etc.
Dong Liang is a Ph.D., professor and doctoral supervisor. He acts as the dean in the Department of Electronics and Information Engineering and the director in the ministry of education power quality engineering research center, Anhui University, Hefei, China. In the past five years, he hosted the national natural science foundation, national 973 program, national 863 plan, science and technology research of Anhui province, high technology industrialization of Anhui province, information industry development fund of Anhui province and so on more than 30 scientific research project, and he participated in more than 10 kinds of projects. His current research directions include computer vision and image processing and power quality detection and control, etc.