互联网拥塞控制系统的非线性稳定性研究
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摘要
随着网络规模和服务质量的不断升级,互联网拥塞控制的重要性将更加突出。在通信时延、时变的网络拓扑等客观因素的影响下,互联网拥塞控制系统具有明显的非线性特征,在一定条件下将失去稳定性而呈现出分岔、混沌等动力学行为特征。此时路由器队列长度以及用户窗口大小将大幅振荡,最终导致网络效率急速下降甚至拥塞崩溃。本论文围绕这一主题,首先研究了时滞系统的混沌控制与同步等非线性问题,然后采用模型改进、信号处理等方法,研究推迟分岔和增大稳定区的有效方法。本论文的主要工作体现在以下几个方面。
(1)研究了同结构和异结构(超)混沌系统的同步。首先以Chen系统和n涡卷Chua系统为同步对象,分别在参数确定和不确定两种情况下,研究了相同和不同类型三维混沌系统的同步策略,推导了相应的同步控制器和参数自适应更新率;然后推导了高维混沌系统全状态混合投影同步的广义公式,以四维Chen系统和Lü系统为例,得到超混沌系统全状态混合投影同步的控制器公式;最后,应用Lyapunov稳定性理论和计算机仿真结果,证明所推导的同步控制器的作用效果。
(2)研究了时滞多智能体网络的同步一致性。首先根据已有多智能体网络模型在稳定区、响应速度等方面存在的问题,提出了一种改进措施。改进模型时引进比例系数k来增强同步控制的灵活度,同时在通信接收端引入相同的时滞τ以兼顾运算复杂度。然后应用频域分析法和几何法推导出改进模型的同步条件,给出了应用该同步条件和网络拓扑确定时滞τ取值范围的具体方法。最后,通过计算机仿真以及跟已有模型的比较,证明了新模型在同步控制方面的优势。
(3)研究了互联网拥塞控制中TCP/AQM时滞对偶模型的非线性稳定性。首先论述拥塞控制对偶模型的概念、分类和特点,通过线性化分析和计算机仿真,表明已有模型存在稳定区小、易发生分岔和混沌等问题,在此基础上提出了一种改进措施。然后以通信时延为分岔参数重点研究了新模型的分岔条件。分别采用摄动法、中心流形定理和正规形法理论法研究了新模型的分岔方向和稳定性。两种分析方法的比较表明,在小参数存在的基础上,摄动法具有求解思路简明、运算量小等优点。最后,尝试推导改进模型的同伦近似解并与数值解进行比较,从运算复杂度和精确度等方面证明了同伦分析法在拥塞控制系统求解中的有效性。
(4)研究了互联网拥塞控制中TCP/AQM流体流模型的非线性稳定性。首先论述流体流模型的演变和简化过程,分析了几种简化策略的基本思想和共同存在的问题,从提高模型稳定性的角度出发提出了一种改进措施。然后从线性近似和非线性两个层面分析了改进模型的稳定性,其中非线性分析基于摄动理论,通过计算得到的Floquet指数的正负号判断分岔周期解的稳定性。相关的研究结果概括为两个定理,并通过计算机仿真进行验证。
(5)研究了互联网拥塞控制中TCP/RED频闪模型的分岔控制和非线性稳定性。首先回顾了TCP/RED频闪模型的基本特点。以模型的窗口加权因子w为分岔参数,研究由倍周期分岔引起的不稳定性。为扩大模型的稳定工作区,我们将滑动平均滤波器引入频闪模型。引入滑动平均滤波器后,系统的分岔起点得到明显延缓,还可以通过调节滤波器的尺度来灵活控制频闪模型的稳定区。最后分析了滑动平均滤波器类型对系统运算复杂度的影响,相关的研究结论都通过了仿真验证。
With the development of Internet’s scale and quality of service (QoS), its congestion control will be more important than before. Impacted by communication delay and time-variant network topology, Internet congestion control system shows obvious nonlinear characteristics such as bifurcation and chaos, which means that the router’s queue length and the user’window size will oscillate radically, and network resource cannot be used effectively. Around such a problem, after studying on the chaos and synchronization of nonlinear and time-delayed systems, we use signal processing and modifying models methods to study how to put off the bifurcation outset or how to enlarge the stable range. The main contents and contributions of this dissertation are as follows.
(1) Synchronizations between same or different chaotic systems are studied. First, taking Chen’s system and Chua’s system as instances, adaptive synchronizations of 3-dimension chaotic systems with uncertain or unknown parameters are investigated, where adaptive controllers and parameter update laws are designed with Lyapunov’s stability theorem. Next, general formula for full state hybrid projective synchronization (FSHPS) is derived. Based on the proposed formula, a concrete synchronization controller about 4-dimension chaotic systems, i.e. Chen’s hyper-chaotic system and Lü’s hyper-chaotic system is given. Finally, synchronization results are proved by numerical simulations. Studies show that two same or different chaotic systems can be synchronized asymptotically by the proposed controller.
(2) Synchronization and consensus of time-delayed multi-agent network models are investigated. First, several prior networks’synchronization characteristics with or without time delay are studied. Then, in the aspect of enlarging models’stable domain, an improved model is presented, whose main characteristics lie in two aspects: One is that a proportional coefficient k is introduced to raise the model’s flexibility, the other is that time delay is introduced into each communication receiver. Finally, by geometrical method and frequency-domain method, the synchronization condition is derived, how to choose time delay is also provided and verification is come out by numerical simulations. It is found that synchronization domain of the novel model is larger than those of some prior models and it can be regulated.
(3) Nonlinear stability of time-delayed dual model of Internet congestion control system is studied. Firstly, some notions and typical examples of the time-delayed dual model are reviewed. It's found that these models’stable domains are narrow and prone to enter into bifurcated or chaotic states. Then an improved model is proposed correspondingly, and the Hopf bifurcation phenomenon is studied by setting time delay as the bifurcation parameter. Both perturbation method and center manifest theorem and normal form theory are used to complete the nonlinear stability analysis, including the bifurcation direction, the stability and the change of the periodic solution. In addition, we try to use homotopy analysis method (HAM) to calculate bifurcation solution of the improved model, and the calculation’s complexity is compared with some traditional methods. In the final, the theoretical conclusions above are verified by numerical experiments.
(4) Nonlinear stability of a fluid flow model in Internet congestion control is studied. First, the simplification and evolvement of fluid flow models are analyzed. Then a modified fluid flow model is presented in order to enlarge its stable range. Finally the linear and nonlinear stability analyses are finished, respectively. Its nonlinear stability analysis is based on perturbation method, and the conclusion can be described by its Floquet exponent’s sign. Fluid flow model’s research is summarized as two theorems which are verified by numerical simulations.
(5) Bifurcation control and nonlinear stability of TCP/RED stroboscopic model in Internet congestion control are studied. First basic characteristics of TCP/RED stroboscopic model are reviewed. By numerical simulation, it is shown that TCP/RED stroboscopic model is unstable when the parameter w passes through some critical bifurcation value, and the critical bifurcation value is too small to satisfy the general request of Internet congestion control. Then, in order to enlarge the model’s stable domain, a moving average filter is added to the TCP/RED stroboscopic mode. Studies show that the moving average filter can be used to put off the bifurcation’s outset, and the model’s stable domain can be regulated by choosing different filter sizes. In addition, the computation complexity after being added to the filter is analyzed and the conclusion in point is verified by numerical simulations.
(1)研究了同结构和异结构(超)混沌系统的同步。首先以Chen系统和n涡卷Chua系统为同步对象,分别在参数确定和不确定两种情况下,研究了相同和不同类型三维混沌系统的同步策略,推导了相应的同步控制器和参数自适应更新率;然后推导了高维混沌系统全状态混合投影同步的广义公式,以四维Chen系统和Lü系统为例,得到超混沌系统全状态混合投影同步的控制器公式;最后,应用Lyapunov稳定性理论和计算机仿真结果,证明所推导的同步控制器的作用效果。
(2)研究了时滞多智能体网络的同步一致性。首先根据已有多智能体网络模型在稳定区、响应速度等方面存在的问题,提出了一种改进措施。改进模型时引进比例系数k来增强同步控制的灵活度,同时在通信接收端引入相同的时滞τ以兼顾运算复杂度。然后应用频域分析法和几何法推导出改进模型的同步条件,给出了应用该同步条件和网络拓扑确定时滞τ取值范围的具体方法。最后,通过计算机仿真以及跟已有模型的比较,证明了新模型在同步控制方面的优势。
(3)研究了互联网拥塞控制中TCP/AQM时滞对偶模型的非线性稳定性。首先论述拥塞控制对偶模型的概念、分类和特点,通过线性化分析和计算机仿真,表明已有模型存在稳定区小、易发生分岔和混沌等问题,在此基础上提出了一种改进措施。然后以通信时延为分岔参数重点研究了新模型的分岔条件。分别采用摄动法、中心流形定理和正规形法理论法研究了新模型的分岔方向和稳定性。两种分析方法的比较表明,在小参数存在的基础上,摄动法具有求解思路简明、运算量小等优点。最后,尝试推导改进模型的同伦近似解并与数值解进行比较,从运算复杂度和精确度等方面证明了同伦分析法在拥塞控制系统求解中的有效性。
(4)研究了互联网拥塞控制中TCP/AQM流体流模型的非线性稳定性。首先论述流体流模型的演变和简化过程,分析了几种简化策略的基本思想和共同存在的问题,从提高模型稳定性的角度出发提出了一种改进措施。然后从线性近似和非线性两个层面分析了改进模型的稳定性,其中非线性分析基于摄动理论,通过计算得到的Floquet指数的正负号判断分岔周期解的稳定性。相关的研究结果概括为两个定理,并通过计算机仿真进行验证。
(5)研究了互联网拥塞控制中TCP/RED频闪模型的分岔控制和非线性稳定性。首先回顾了TCP/RED频闪模型的基本特点。以模型的窗口加权因子w为分岔参数,研究由倍周期分岔引起的不稳定性。为扩大模型的稳定工作区,我们将滑动平均滤波器引入频闪模型。引入滑动平均滤波器后,系统的分岔起点得到明显延缓,还可以通过调节滤波器的尺度来灵活控制频闪模型的稳定区。最后分析了滑动平均滤波器类型对系统运算复杂度的影响,相关的研究结论都通过了仿真验证。
With the development of Internet’s scale and quality of service (QoS), its congestion control will be more important than before. Impacted by communication delay and time-variant network topology, Internet congestion control system shows obvious nonlinear characteristics such as bifurcation and chaos, which means that the router’s queue length and the user’window size will oscillate radically, and network resource cannot be used effectively. Around such a problem, after studying on the chaos and synchronization of nonlinear and time-delayed systems, we use signal processing and modifying models methods to study how to put off the bifurcation outset or how to enlarge the stable range. The main contents and contributions of this dissertation are as follows.
(1) Synchronizations between same or different chaotic systems are studied. First, taking Chen’s system and Chua’s system as instances, adaptive synchronizations of 3-dimension chaotic systems with uncertain or unknown parameters are investigated, where adaptive controllers and parameter update laws are designed with Lyapunov’s stability theorem. Next, general formula for full state hybrid projective synchronization (FSHPS) is derived. Based on the proposed formula, a concrete synchronization controller about 4-dimension chaotic systems, i.e. Chen’s hyper-chaotic system and Lü’s hyper-chaotic system is given. Finally, synchronization results are proved by numerical simulations. Studies show that two same or different chaotic systems can be synchronized asymptotically by the proposed controller.
(2) Synchronization and consensus of time-delayed multi-agent network models are investigated. First, several prior networks’synchronization characteristics with or without time delay are studied. Then, in the aspect of enlarging models’stable domain, an improved model is presented, whose main characteristics lie in two aspects: One is that a proportional coefficient k is introduced to raise the model’s flexibility, the other is that time delay is introduced into each communication receiver. Finally, by geometrical method and frequency-domain method, the synchronization condition is derived, how to choose time delay is also provided and verification is come out by numerical simulations. It is found that synchronization domain of the novel model is larger than those of some prior models and it can be regulated.
(3) Nonlinear stability of time-delayed dual model of Internet congestion control system is studied. Firstly, some notions and typical examples of the time-delayed dual model are reviewed. It's found that these models’stable domains are narrow and prone to enter into bifurcated or chaotic states. Then an improved model is proposed correspondingly, and the Hopf bifurcation phenomenon is studied by setting time delay as the bifurcation parameter. Both perturbation method and center manifest theorem and normal form theory are used to complete the nonlinear stability analysis, including the bifurcation direction, the stability and the change of the periodic solution. In addition, we try to use homotopy analysis method (HAM) to calculate bifurcation solution of the improved model, and the calculation’s complexity is compared with some traditional methods. In the final, the theoretical conclusions above are verified by numerical experiments.
(4) Nonlinear stability of a fluid flow model in Internet congestion control is studied. First, the simplification and evolvement of fluid flow models are analyzed. Then a modified fluid flow model is presented in order to enlarge its stable range. Finally the linear and nonlinear stability analyses are finished, respectively. Its nonlinear stability analysis is based on perturbation method, and the conclusion can be described by its Floquet exponent’s sign. Fluid flow model’s research is summarized as two theorems which are verified by numerical simulations.
(5) Bifurcation control and nonlinear stability of TCP/RED stroboscopic model in Internet congestion control are studied. First basic characteristics of TCP/RED stroboscopic model are reviewed. By numerical simulation, it is shown that TCP/RED stroboscopic model is unstable when the parameter w passes through some critical bifurcation value, and the critical bifurcation value is too small to satisfy the general request of Internet congestion control. Then, in order to enlarge the model’s stable domain, a moving average filter is added to the TCP/RED stroboscopic mode. Studies show that the moving average filter can be used to put off the bifurcation’s outset, and the model’s stable domain can be regulated by choosing different filter sizes. In addition, the computation complexity after being added to the filter is analyzed and the conclusion in point is verified by numerical simulations.
引文
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