几类分数阶系统的稳定性分析与镇定控制器设计

发布时间：2018-08-14 19:28

【摘要】：分数阶微积分是整数阶微积分的延伸与拓展,其发展几乎与整数阶微积分的发展同步。分数阶微积分在越来越多的领域中都发挥着极其重要的作用。与整数阶模型相比,分数阶模型能够更加准确地描述自然现象,更好地模拟自然界的物理现象和动态过程。随着分数阶微积分理论在不同的科学领域里出现,对其理论或应用价值的研究都显得尤为迫切。因此,对分数阶微分方程和系统进行深入研究具有广泛的理论意义与实际应用价值。有关分数阶微分方程和系统的研究引起了国内外学者的广泛关注并逐渐成为一个热点问题。本文针对几类分数阶系统的稳定性分析、镇定控制器设计问题和两类分数阶微分方程边值问题解的存在性进行了研究,给出了几类分数阶系统一些新的稳定性判据、镇定控制器的设计方法和分数阶微分方程边值问题解存在的若干充分条件,并分别用仿真例子验证了所得到结果的有效性。主要研究内容如下：1.基于Caputo分数阶导数已有的基本性质,给出了Caputo分数阶导数的一些新性质。这些新性质可以帮助寻找一个给定的分数阶系统的二次Lyapunov函数。2.研究了几类分数阶系统的稳定性和镇定性。首先,利用分数阶Lyapunov函数方法,研究了分数阶线性系统的稳定性,并给出了分数阶线性受控系统的状态反馈控制器设计。其次,利用分数阶Razumikhin定理,研究了分数阶线性时滞系统的稳定性,并给出了分数阶线性时滞受控系统的状态反馈控制器设计。最后,利用分数阶Lyapunov函数方法,研究了分数阶非线性系统的稳定性,并利用Backstepping设计方法,给出了一类分数阶非线性三角系统的状态反馈控制器设计。3.研究了分数阶非线性三角系统的反馈镇定控制器设计问题。通过引入适当的状态变换,将分数阶非线性三角系统的反馈镇定控制器设计问题转化为待定参数的选取问题。利用静态增益控制设计方法和分数阶Lyapunov函数方法,分别给出了分数阶非线性下、上三角系统的状态反馈和输出反馈控制器设计。4.研究了分数阶非线性时滞三角系统的反馈镇定控制器设计问题。通过引入适当的状态变换,将分数阶非线性时滞三角系统的反馈镇定控制器设计问题转化为待定参数的选取问题。利用静态增益控制设计方法和分数阶Razumikhin定理,分别设计了分数阶非线性时滞下、上三角系统的状态反馈和输出反馈控制器。5.研究了两类分数阶微分方程边值问题解的存在性。利用上下解方法、Shauder不动点定理和Leggett-Williams不动点定理,建立了一类分数阶微分方程边值问题至少存在一个或三个正解的几个充分条件。利用Banach代数上的Dhage不动点定理,给出了一类混合分数阶微分方程边值问题存在一个解的充分条件。
[Abstract]:Fractional calculus is an extension and extension of integral order calculus, and its development almost keeps pace with the development of integer order calculus. Fractional calculus plays an important role in more and more fields. Compared with the integer order model, the fractional order model can describe the natural phenomena more accurately and simulate the physical phenomena and dynamic processes better. With the emergence of fractional calculus theory in different fields of science, it is urgent to study its theory or application value. Therefore, the study of fractional differential equations and systems has a wide range of theoretical significance and practical application value. The study of fractional differential equations and systems has attracted the attention of scholars at home and abroad and has gradually become a hot issue. In this paper, the existence of solutions to the stability analysis, stabilization controller design problem and boundary value problem of two kinds of fractional differential equations are studied, and some new stability criteria are given. The design method of stabilizing controller and some sufficient conditions for the existence of solutions to the boundary value problem of fractional differential equations are discussed. Simulation examples are given to verify the validity of the obtained results. The main research contents are as follows: 1. Based on the basic properties of Caputo fractional derivative, some new properties of Caputo fractional derivative are given. These new properties can help to find the quadratic Lyapunov function of a given fractional system. The stability and stability of several fractional order systems are studied. Firstly, the stability of fractional linear systems is studied by using the fractional Lyapunov function method, and the state feedback controller design for fractional linear controlled systems is given. Secondly, the stability of fractional linear time-delay systems is studied by using fractional Razumikhin theorem, and the state feedback controller design for fractional linear time-delay controlled systems is given. Finally, the stability of fractional nonlinear systems is studied by using the fractional Lyapunov function method, and the state feedback controller design of a class of fractional nonlinear triangular systems is given by using the Backstepping design method. The design of feedback stabilization controllers for fractional nonlinear triangular systems is studied. By introducing appropriate state transformation, the design problem of feedback stabilization controller for fractional nonlinear triangular systems is transformed into the selection of undetermined parameters. Using the static gain control design method and the fractional Lyapunov function method, the state feedback and output feedback controller design of the upper triangular system under fractional order nonlinearity are given respectively. 4. In this paper, the design of feedback stabilization controller for fractional nonlinear delay-triangular systems is studied. By introducing appropriate state transformation, the problem of feedback stabilization controller design for fractional nonlinear delay-triangular systems is transformed into the selection of undetermined parameters. Using the static gain control design method and fractional order Razumikhin theorem, the state feedback and output feedback controllers for upper triangular systems with fractional nonlinear delay are designed respectively. The existence of solutions for two kinds of boundary value problems for fractional differential equations is studied. By using the upper and lower solution method and the Leggett-Williams fixed point theorem, some sufficient conditions for the existence of at least one or three positive solutions for a class of fractional differential equation boundary value problems are established. By using the Dhage fixed point theorem on Banach algebra, a sufficient condition for the existence of a solution to the boundary value problem for a class of mixed fractional differential equations is given.
【学位授予单位】：山东大学
【学位级别】：博士
【学位授予年份】：2016
【分类号】：TP13

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