四类分数阶时滞非线性系统稳定性研究
发布时间：2018-08-22 12:15
【摘要】：近些年,随着计算机技术和交叉学科的发展,分数阶微积分在理论和应用方面取得巨大进步。分数阶微积分作为整数阶微积分在任意阶次上的推广,具有非局限性特性,非常适合描述那些具有记忆和遗传特性的材料和过程,被广泛用于科技和工程领域,例如多孔材料中的流体流动、反常扩散、粘弹性材料中的声波传播、自相似结构的动力学、信号处理、金融理论、保密通信和生物系统电导等。分数阶微积分已经成为当下的研究热点领域。本文主要研究了一种改进的预估-校正算法、分数阶单时滞复Lorenz系统的动力学分析及其自时滞同步及分数阶多时滞非线性系统的稳定性控制等问题。本文的主要内容和创新之处如下:1.改进的预估-校正算法通过对预估-校正算法的预估部分作出改进,提出了一种改进的预估-校正算法。该算法通过提高预估部分的计算精度,来提高整体算法的计算精度。此外,利用MATLAB数值仿真工具分析了三个数值仿真实例,通过与预估-校正算法比较,体现了改进的预估-校正算法在计算精度上的优越性。2.分数阶单时滞复Lorenz系统的动力学行为及其自时滞同步通过相图法和最大Lyapunov指数法,分析了分数阶单时滞复Lorenz系统的动力学行为。在这部分,我们通过固定系统阶数,增加时滞项系数,发现分数阶单时滞复Lorenz系统具有丰富的动力学行为。此外通过构造反馈控制器,实现了分数阶单时滞复Lorenz系统的自时滞同步。利用MATLAB数值仿真工具做了数值仿真,验证了所得结论的有效性和可行性。3.分数阶多时滞非线性系统的稳定性控制基于分数阶Lyapunov直接方法和分数阶时滞非线性系统的稳定性理论,提出了一种通过构造反馈控制器实现分数阶多时滞非线性系统稳定性控制的方法。不同于已有的工作,该方法中,反馈控制器为线性反馈控制器,且只涉及系统当前状态变量,与时滞项无关,结构简单,易于工程实现。最后将该方法应用于三个典型的分数阶多时滞非线性受控系统,通过MATLAB数值仿真工具,验证了所得结果的有效性与可行性。
[Abstract]:In recent years, with the development of computer technology and interdiscipline, fractional calculus has made great progress in theory and application. Fractional calculus, as a generalization of integral order calculus at any order, has the characteristics of no limitation and is very suitable for describing materials and processes with memory and genetic properties, and is widely used in science, technology and engineering. For example, fluid flow in porous materials, anomalous diffusion, acoustic propagation in viscoelastic materials, dynamics of self-similar structures, signal processing, financial theory, secure communications and biological system conductance. Fractional calculus has become a hot research field. In this paper, an improved predictor-correction algorithm, dynamic analysis of fractional single-delay complex Lorenz systems and stability control of self-delay synchronization and fractional multi-delay nonlinear systems are studied. The main contents and innovations of this paper are as follows: 1. An improved predictor-correction algorithm is proposed by improving the prediction part of the predictor-correction algorithm. The algorithm improves the accuracy of the whole algorithm by improving the accuracy of the prediction part. In addition, three numerical simulation examples are analyzed by using the MATLAB numerical simulation tool. By comparing with the predictor-correction algorithm, the superiority of the improved predictor-correction algorithm in the calculation accuracy is demonstrated. The dynamic behavior of fractional single-delay complex Lorenz system and its self-delay synchronization by phase diagram method and maximum Lyapunov exponent method are analyzed. The dynamic behavior of fractional single-delay complex Lorenz system is analyzed. In this part, we find that fractional single-delay complex Lorenz systems have rich dynamic behavior by increasing the coefficients of delay terms by the fixed order of the system. In addition, a feedback controller is constructed to realize the self-delay synchronization of fractional single-delay complex Lorenz systems. The effectiveness and feasibility of the conclusions are verified by using the MATLAB numerical simulation tool. The stability control of fractional multi-delay nonlinear systems is based on the fractional Lyapunov direct method and the stability theory of fractional time-delay nonlinear systems. In this paper, a feedback controller is proposed to control the stability of fractional multi-delay nonlinear systems. In this method, the feedback controller is a linear feedback controller, which only involves the current state variables of the system, is independent of the time-delay term, and is simple in structure and easy to be implemented in engineering. Finally, the method is applied to three typical fractional multi-delay nonlinear controlled systems. The validity and feasibility of the obtained results are verified by MATLAB numerical simulation tool.
【学位授予单位】：重庆邮电大学
【学位级别】：硕士
【学位授予年份】：2016
【分类号】：TP13
本文编号：2197074
[Abstract]:In recent years, with the development of computer technology and interdiscipline, fractional calculus has made great progress in theory and application. Fractional calculus, as a generalization of integral order calculus at any order, has the characteristics of no limitation and is very suitable for describing materials and processes with memory and genetic properties, and is widely used in science, technology and engineering. For example, fluid flow in porous materials, anomalous diffusion, acoustic propagation in viscoelastic materials, dynamics of self-similar structures, signal processing, financial theory, secure communications and biological system conductance. Fractional calculus has become a hot research field. In this paper, an improved predictor-correction algorithm, dynamic analysis of fractional single-delay complex Lorenz systems and stability control of self-delay synchronization and fractional multi-delay nonlinear systems are studied. The main contents and innovations of this paper are as follows: 1. An improved predictor-correction algorithm is proposed by improving the prediction part of the predictor-correction algorithm. The algorithm improves the accuracy of the whole algorithm by improving the accuracy of the prediction part. In addition, three numerical simulation examples are analyzed by using the MATLAB numerical simulation tool. By comparing with the predictor-correction algorithm, the superiority of the improved predictor-correction algorithm in the calculation accuracy is demonstrated. The dynamic behavior of fractional single-delay complex Lorenz system and its self-delay synchronization by phase diagram method and maximum Lyapunov exponent method are analyzed. The dynamic behavior of fractional single-delay complex Lorenz system is analyzed. In this part, we find that fractional single-delay complex Lorenz systems have rich dynamic behavior by increasing the coefficients of delay terms by the fixed order of the system. In addition, a feedback controller is constructed to realize the self-delay synchronization of fractional single-delay complex Lorenz systems. The effectiveness and feasibility of the conclusions are verified by using the MATLAB numerical simulation tool. The stability control of fractional multi-delay nonlinear systems is based on the fractional Lyapunov direct method and the stability theory of fractional time-delay nonlinear systems. In this paper, a feedback controller is proposed to control the stability of fractional multi-delay nonlinear systems. In this method, the feedback controller is a linear feedback controller, which only involves the current state variables of the system, is independent of the time-delay term, and is simple in structure and easy to be implemented in engineering. Finally, the method is applied to three typical fractional multi-delay nonlinear controlled systems. The validity and feasibility of the obtained results are verified by MATLAB numerical simulation tool.
【学位授予单位】：重庆邮电大学
【学位级别】：硕士
【学位授予年份】：2016
【分类号】：TP13
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