一类弱奇性积分微分模型的稳定性分析及其在分数阶等效电路中的应用

发布时间：2018-02-23 13:42

本文关键词： 积分微分方程分数阶系统稳定性等效电路模型 Lyapunov方法复路径积分法　出处：《山东大学》2017年硕士论文　论文类型：学位论文

【摘要】：积分微分方程是一类有效的建模工具,对积分微分模型性质的研究可以增进对系统的深刻了解,并可通过分析积分微分方程的稳定性来指导系统的预测和控制等。但由于积分微分方程种类繁杂、形式差异大,且相对缺少现成的分析方法或实现技术,从而在一定程度上限制了其在系统科学领域的应用。因此,积分微分模型稳定性的分析以及模型的数字与模拟实现,将为复杂系统建模、控制性能优化和故障诊断等提供可靠保障,具有重要的现实意义。本文针对几类典型的积分微分方程进行了详细的描述并给出了若干典型的积分微分方程的解法。其中,将具有弱奇性核函数的类型归于分数阶(非整数阶)积分微分方程。与整数阶模型相比,分数阶模型具有更丰富的可调参数,能够更精确地描述模型,并对揭示自然界中普遍存在的分数阶现象至关重要。因此,针对这种带有弱奇性核函数的积分微分方程的研究具有一定的前沿性和应用价值。稳定是控制系统的基本要求之一,也是保证系统正常工作的基本条件。然而,稳定性研究是积分微分方程课题中的一个难点,即使是线性时不变系统的稳定性分析仍然存在一定的难度,特别是Lyapunov方法这一积分微分方程稳定性研究中十分重要的方法,对于分数阶积分微分方程要找到其Lyapunov函数比较困难,尤其是非线性方程。本文首先给出了整数阶情况下的Lyapunov分析方法,并且分别分析了一维和多维两种不同情况,之后又给出了分数阶Caputo形式情况下的Lyapunov分析方法,为分数阶积分微分系统的稳定性分析提供了理论支持。在分数阶电路中,用积分微分方程对其进行建模不仅可以得到电路中变量的时域解,还可以利用Lyapunov方法分析电路的稳定性问题,以及进行时频域分析,这对分数阶等效电路模型的研究十分重要。然而,由于探索Lyapunov函数是一个艰巨的任务,即使是在线性时不变的分数阶微分系统中,利用Lyapunov方法来研究分数阶等效电路模型的稳定性问题也是十分困难的。因此,寻找一种简单有效的分数阶电路的稳定性分析方法显得尤为重要。本文引入了分数阶稳定性分析方法,并将其运用于生物医学中的分数阶等效电路模型,并且提出了一种路径积分方法来分析分数阶电路的稳定性问题。如果直接用逆拉氏变换法求解系统解,那么系统的稳定性问题无法确定。如果用Lyapunov方法来分析系统的稳定性,那么将无法得到精确解,并且探索Lyapunov函数也是一个艰巨的过程。而本文提出的复路径积分方法不仅可以得到系统解,还可以判定解的稳定性问题。因此,将这种方法运用于电路分析及电路的建模和辨识中具有显著的作用。大量数值方针及其分析验证了上述结果的准确性和实用性。
[Abstract]:Integro-differential equation is a kind of effective modeling tool. The study of the properties of integro-differential model can enhance the deep understanding of the system. The stability of the integrodifferential equation can be used to guide the prediction and control of the system. Therefore, the analysis of the stability of integral-differential model and the realization of digital and simulation of the model will model the complex system. The optimization of control performance and fault diagnosis are of great practical significance. In this paper, several typical integro-differential equations are described in detail and the solutions of some typical integro-differential equations are given. The types of kernel functions with weak singularity are classified into fractional (non-integer) integro-differential equations. Compared with the integer order model, the fractional order model has more adjustable parameters and can describe the model more accurately. Therefore, the study of integro-differential equation with weak singularity kernel function has some vanguard and application value. Stability is one of the basic requirements of control system. It is also the basic condition to ensure the normal operation of the system. However, the study of stability is a difficult problem in the subject of integro-differential equation, even the stability analysis of linear time-invariant system still has some difficulty. In particular, the Lyapunov method is a very important method in the study of the stability of integrodifferential equations. It is difficult to find its Lyapunov function for fractional integrodifferential equations. In this paper, we first give the Lyapunov analysis method in the case of integer order, and analyze two different cases of one or more dimensions, and then give the Lyapunov analysis method in fractional order Caputo form. It provides theoretical support for the stability analysis of fractional integro-differential systems. In fractional order circuits, not only the time-domain solution of variables in the circuit can be obtained by modeling it with integro-differential equations, but also the time-domain solution of the variables in the circuit can be obtained. The Lyapunov method can also be used to analyze the stability of circuits and time-frequency domain analysis, which is very important for the study of fractional equivalent circuit models. However, it is a difficult task to explore Lyapunov functions. Even in linear time-invariant fractional differential systems, it is very difficult to study the stability of fractional equivalent circuit models by using Lyapunov method. It is very important to find a simple and effective method for the stability analysis of fractional order circuits. In this paper, a fractional order stability analysis method is introduced and applied to the fractional equivalent circuit model in biomedicine. A path integral method is proposed to analyze the stability of fractional circuits. If the inverse Laplace transform is directly used to solve the stability problem of the system, the stability problem of the system cannot be determined. If the stability of the system is analyzed by the Lyapunov method, Therefore, the complex path integral method proposed in this paper can not only obtain the solution of the system, but also determine the stability of the solution. The application of this method to circuit analysis and circuit modeling and identification has a significant effect. A large number of numerical guidelines and their analysis verify the accuracy and practicability of the above results.
【学位授予单位】：山东大学
【学位级别】：硕士
【学位授予年份】：2017
【分类号】：O175;O231

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