分数阶系统的初始条件问题研究

发布时间：2018-06-11 22:14

本文选题：分数阶系统 + 始前过程　；参考：《中国科学技术大学》2017年硕士论文

【摘要】：在现代科学技术的众多领域当中,系统与控制理论的研究极大地提高了人们的生活水平和生产效率,而分数阶微积分的引入,无疑又为相关研究注入了崭新的活力。相较于整数阶的情况而言,一方面对于分数阶系统理论的探索为人们理解自然并创造价值提供了全新的思路,另一方面分数阶系统本身所具有的特殊性也为相关研究的开展带来了重重困难,其中初始条件问题就是最为典型的代表之一。对于分数阶系统初始条件问题的研究虽然极具挑战,但也是分数阶系统科学领域所不得不面对的重点和难点,是分数阶系统研究从理论走向应用的重要基础和必要前提。因此,本文将着重探究分数阶系统的初始条件问题。首先,本文通过引入分数阶系统所特有的越轨现象,明确了分数阶系统初始条件问题的复杂性和重要性。进一步通过从无穷维特性和长记忆特性角度揭示越轨现象发生的内在本质,给出了 Riemann-Liouville定义和Caputo定义下分数阶系统伪状态空间模型和无穷维真实状态空间模型之间的关系,同时引入了始前过程和初始化函数的概念,为后续的研究提供了理论基础。其次,本文对非零初始条件下的分数阶数值实现进行了研究。对于分数阶微分的数值实现,给出了不同定义下分数阶微积分的一般计算方法,实现了时间最优意义下的分数阶跟踪微分器设计,同时考虑非零初始条件,明确了始前过程对于分数阶微分计算的影响。对于分数阶系统响应的求解,给出了适用于一般分数阶系统响应求解的数值方法,并针对非零初始条件的情况,提出了具体的系统响应数值实现方案。此外,考虑到分数阶系统的有理逼近为在整数阶框架下研究分数阶系统问题提供了依据,本文从频域辨识的角度出发,运用矢量拟合的方法,实现了从分数阶积分算子到一般分数阶系统的有理逼近,并提出了一种低阶模型的直接逼近方法。同时考虑非零初始条件,分别针对Riemann-Liouville定义和Caputo定义,提出了逼近模型真实初始状态的分配策略,在保证系统频域和时域特性在逼近前后相似性的同时,也保持了系统初始条件的一致性。最后,本文研究了始前过程未知的分数阶系统非零初始条件估计问题。从无穷维特性的角度出发,针对分数阶系统的真实初始状态,提出了一种基于最小二乘的估计方法,实现了对系统输出的在线跟踪,并借助于整数阶状态观测器的概念,完成了分数阶系统真实初始状态观测器的设计。另外,从长记忆特性的角度出发,本文还给出了初始化函数的拟合方法,实现了对分数阶系统始前过程的估计。
[Abstract]:In many fields of modern science and technology, the study of system and control theory has greatly improved people's living standard and production efficiency, and the introduction of fractional calculus has undoubtedly injected new vitality into related research. Compared with the integer order, on the one hand, the exploration of fractional order system theory provides a new way for people to understand nature and create value. On the other hand, the particularity of fractional order system also brings many difficulties for the related research, among which the initial condition is one of the most typical representatives. Although the research on the initial conditions of fractional systems is very challenging, it is also a key and difficult point in the field of fractional system science. It is an important foundation and necessary prerequisite for the research of fractional order systems from theory to application. Therefore, this paper will focus on the initial conditions of fractional systems. Firstly, the complexity and importance of the initial condition problem of fractional order system are clarified by introducing the characteristic deviant phenomenon of fractional order system. By revealing the intrinsic nature of deviant phenomena from the point of view of infinite dimension characteristic and long memory characteristic, the relationship between the pseudo-state space model of fractional order system and the infinite dimensional real state space model under the definition of Riemann-Liouville and Caputo is given. At the same time, the concepts of prestart process and initialization function are introduced, which provide a theoretical basis for further research. Secondly, the fractional numerical realization under non-zero initial condition is studied in this paper. For the numerical realization of fractional differential, the general calculation method of fractional calculus under different definitions is given, and the fractional order tracking differentiator is designed in the sense of optimal time, and the non-zero initial condition is considered at the same time. The effect of the process on fractional differential calculation is clarified. For solving the response of fractional order system, a numerical method suitable for solving the response of general fractional order system is presented, and a numerical realization scheme of system response is proposed for the case of non-zero initial conditions. In addition, considering that the rational approximation of fractional order system provides the basis for the study of fractional order system under the frame of integer order, this paper uses vector fitting method from the view of frequency domain identification. The rational approximation from fractional integral operator to general fractional order system is realized, and a direct approximation method for low order model is proposed. At the same time, considering the non-zero initial condition, for Riemann-Liouville definition and Caputo definition, the assignment strategy for the real initial state of the approximation model is proposed, which ensures the similarity between the frequency domain and time domain characteristics before and after approximation. The consistency of the initial conditions of the system is also maintained. Finally, the problem of nonzero initial condition estimation for fractional order systems with unknown processes before the beginning is studied. From the point of view of infinite dimension property, an estimation method based on least square is proposed for the real initial state of fractional order system. The on-line tracking of system output is realized, and the concept of integer order state observer is used. The design of the real initial state observer for fractional order system is completed. In addition, from the point of view of long memory characteristics, this paper presents a fitting method of initialization function, which realizes the estimation of the process before the beginning of fractional order system.
【学位授予单位】：中国科学技术大学
【学位级别】：硕士
【学位授予年份】：2017
【分类号】：O231

【参考文献】