【摘要】：分數(shù)階系統(tǒng)是由微分階次為任意實數(shù)甚至為任意復數(shù)的微分方程所描述的動力學系統(tǒng)。能更準確、更有效地描述實際系統(tǒng)中出現(xiàn)的復雜現(xiàn)象,現(xiàn)已被廣泛應用于諸多高精尖領域。混沌系統(tǒng)是一類特殊的非線性系統(tǒng)。自二十世紀初學者們提出混沌系統(tǒng)以來,由于它具有的隨機性和對初始條件的極端敏感性等特點,混沌系統(tǒng)的研究一直處于前沿課題。對于特殊的無平衡點的分數(shù)階混沌系統(tǒng)來說,常規(guī)判定不能證明其混沌特性。對于這樣的系統(tǒng),本文的研究將著重以下幾個方面:首先,研究了分數(shù)階微積分的基本定義、性質(zhì)以及它的穩(wěn)定性理論。結合自然界和社會科學中普遍存在的混沌現(xiàn)象,介紹了整數(shù)階混沌系統(tǒng)同步的基本方法。對一個新的無平衡點分數(shù)階混沌系統(tǒng)進行了分析,利用分數(shù)階微分變換方法,得到了它的解序列。研究了無平衡點分數(shù)階混沌系統(tǒng)的Kaplan-Yorke維數(shù)和耗散性,然后基于系統(tǒng)的離散映射,根據(jù)QR分解方法得到最大Lyapunov特征指數(shù),從而判斷系統(tǒng)是否保持混沌。最后給出了一種全狀態(tài)自適應控制方法,使系統(tǒng)的狀態(tài)可以追蹤期望軌跡。最后通過數(shù)值仿真,驗證了該控制方法的有效性。其次,研究了一類參數(shù)不確定的無平衡點分數(shù)階混沌系統(tǒng)的同步問題。根據(jù)自適應控制理論,將自適應控制法引入到混沌同步中。把它應用到一類新的無平衡點分數(shù)階混沌系統(tǒng)中,設計了性能可靠的同步控制器和參數(shù)自適應律,實現(xiàn)了上述提到的無平衡點分數(shù)階混沌系統(tǒng)與其自結構的同步。同時運用主動控制方法構造主動控制器對系統(tǒng)進行了同步研究。把誤差系統(tǒng)中無益的非線性項通過設計控制器進行抵消,提出了一類簡化分數(shù)階混沌系統(tǒng)的同步控制器設計方法。與上述所研究的自適應法進行數(shù)值仿真對比,既驗證了兩種同步方法的有效性也體現(xiàn)出了兩種方法所存在的差異。經(jīng)過對比,可以得出改進的自適應控制法具有更快的控制速度和更高的控制效率,而主動控制方法具有更簡單的計算過程。最后針對無平衡點的分數(shù)階混沌系統(tǒng),令其為驅(qū)動系統(tǒng),令一個新的四維分數(shù)階混沌系統(tǒng)為響應系統(tǒng),采用自適應控制方法進行異結構同步研究。設計了合適的同步控制器和可以很好辨識參數(shù)的自適應律,實現(xiàn)了無平衡點分數(shù)階混沌系統(tǒng)與新的四維分數(shù)階混沌系統(tǒng)之間的異結構同步,并進行仿真,驗證了該方法的有效性。
[Abstract]:Fractional-order systems are dynamical systems described by differential equations with differential orders of arbitrary real numbers or even arbitrary complex numbers. It can more accurately and effectively describe the complex phenomena in the actual system, and has been widely used in many high-precision fields. Chaotic system is a kind of special nonlinear system. Since chaos system was put forward by scholars in the early 20th century, because of its randomness and extreme sensitivity to initial conditions, the study of chaotic system has been in the forefront of the subject. For a special fractional chaotic system with no equilibrium point, the conventional decision can not prove its chaotic characteristics. For such a system, this paper will focus on the following aspects: firstly, the basic definition, properties and stability theory of fractional calculus are studied. Based on the common chaotic phenomena in nature and social science, this paper introduces the basic methods of synchronization of integer-order chaotic systems. In this paper, a new fractional chaotic system with no equilibrium point is analyzed, and its solution sequence is obtained by means of fractional differential transformation. The Kaplan-Yorke dimension and dissipation of fractional order chaotic systems without equilibrium point are studied. Then based on the discrete mapping of the system the maximum Lyapunov characteristic exponent is obtained according to the QR decomposition method so as to judge whether the system is chaotic or not. Finally, a full-state adaptive control method is presented to make the state of the system track the desired trajectory. Finally, the effectiveness of the control method is verified by numerical simulation. Secondly, the synchronization problem of a class of unequilibrium fractional chaotic systems with uncertain parameters is studied. According to the adaptive control theory, the adaptive control method is introduced into chaos synchronization. It is applied to a new class of fractional chaotic systems without equilibrium point, and the synchronization controller and parameter adaptive law with reliable performance are designed. The synchronization between the above mentioned fractional order chaotic system without equilibrium point and its self-structure is realized. At the same time, the active control method is used to construct the active controller to study the synchronization of the system. A synchronization controller design method for a class of simplified fractional chaotic systems is proposed by canceling the non-linear terms of the error system by designing a controller. Compared with the self-adaptive method mentioned above, the effectiveness of the two methods is verified and the difference between the two methods is demonstrated. Through comparison, it can be concluded that the improved adaptive control method has faster control speed and higher control efficiency, while the active control method has a simpler calculation process. Finally, for the fractional chaotic system without equilibrium point, it is a driving system, and a new four-dimensional fractional chaotic system is a response system. An adaptive control method is used to study the heterogeneous structure synchronization. A suitable synchronization controller and an adaptive law which can identify the parameters well are designed, and the heterogeneous synchronization between the unequilibrium fractional order chaotic system and the new four dimensional fractional chaotic system is realized, and the simulation is carried out. The validity of the method is verified.
【學位授予單位】：東北石油大學
【學位級別】：碩士
【學位授予年份】：2017
【分類號】：O415.5;TP273
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本文編號：2462706