發(fā)布時間：2018-06-11 22:14

  本文選題：分數(shù)階系統(tǒng) + 始前過程　； 參考：《中國科學技術大學》2017年碩士論文

【摘要】：在現(xiàn)代科學技術的眾多領域當中,系統(tǒng)與控制理論的研究極大地提高了人們的生活水平和生產效率,而分數(shù)階微積分的引入,無疑又為相關研究注入了嶄新的活力。相較于整數(shù)階的情況而言,一方面對于分數(shù)階系統(tǒng)理論的探索為人們理解自然并創(chuàng)造價值提供了全新的思路,另一方面分數(shù)階系統(tǒng)本身所具有的特殊性也為相關研究的開展帶來了重重困難,其中初始條件問題就是最為典型的代表之一。對于分數(shù)階系統(tǒng)初始條件問題的研究雖然極具挑戰(zhàn),但也是分數(shù)階系統(tǒng)科學領域所不得不面對的重點和難點,是分數(shù)階系統(tǒng)研究從理論走向應用的重要基礎和必要前提。因此,本文將著重探究分數(shù)階系統(tǒng)的初始條件問題。首先,本文通過引入分數(shù)階系統(tǒng)所特有的越軌現(xiàn)象,明確了分數(shù)階系統(tǒng)初始條件問題的復雜性和重要性。進一步通過從無窮維特性和長記憶特性角度揭示越軌現(xiàn)象發(fā)生的內在本質,給出了 Riemann-Liouville定義和Caputo定義下分數(shù)階系統(tǒng)偽狀態(tài)空間模型和無窮維真實狀態(tài)空間模型之間的關系,同時引入了始前過程和初始化函數(shù)的概念,為后續(xù)的研究提供了理論基礎。其次,本文對非零初始條件下的分數(shù)階數(shù)值實現(xiàn)進行了研究。對于分數(shù)階微分的數(shù)值實現(xiàn),給出了不同定義下分數(shù)階微積分的一般計算方法,實現(xiàn)了時間最優(yōu)意義下的分數(shù)階跟蹤微分器設計,同時考慮非零初始條件,明確了始前過程對于分數(shù)階微分計算的影響。對于分數(shù)階系統(tǒng)響應的求解,給出了適用于一般分數(shù)階系統(tǒng)響應求解的數(shù)值方法,并針對非零初始條件的情況,提出了具體的系統(tǒng)響應數(shù)值實現(xiàn)方案。此外,考慮到分數(shù)階系統(tǒng)的有理逼近為在整數(shù)階框架下研究分數(shù)階系統(tǒng)問題提供了依據(jù),本文從頻域辨識的角度出發(fā),運用矢量擬合的方法,實現(xiàn)了從分數(shù)階積分算子到一般分數(shù)階系統(tǒng)的有理逼近,并提出了 一種低階模型的直接逼近方法。同時考慮非零初始條件,分別針對Riemann-Liouville定義和Caputo定義,提出了逼近模型真實初始狀態(tài)的分配策略,在保證系統(tǒng)頻域和時域特性在逼近前后相似性的同時,也保持了系統(tǒng)初始條件的一致性。最后,本文研究了始前過程未知的分數(shù)階系統(tǒng)非零初始條件估計問題。從無窮維特性的角度出發(fā),針對分數(shù)階系統(tǒng)的真實初始狀態(tài),提出了一種基于最小二乘的估計方法,實現(xiàn)了對系統(tǒng)輸出的在線跟蹤,并借助于整數(shù)階狀態(tài)觀測器的概念,完成了分數(shù)階系統(tǒng)真實初始狀態(tài)觀測器的設計。另外,從長記憶特性的角度出發(fā),本文還給出了初始化函數(shù)的擬合方法,實現(xiàn)了對分數(shù)階系統(tǒng)始前過程的估計。
[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.
【學位授予單位】：中國科學技術大學
【學位級別】：碩士
【學位授予年份】：2017
【分類號】：O231

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本文編號：2006885