【摘要】：分?jǐn)?shù)階系統(tǒng)是由微分階次為任意實(shí)數(shù)甚至為任意復(fù)數(shù)的微分方程所描述的動(dòng)力學(xué)系統(tǒng)。能更準(zhǔn)確、更有效地描述實(shí)際系統(tǒng)中出現(xiàn)的復(fù)雜現(xiàn)象,現(xiàn)已被廣泛應(yīng)用于諸多高精尖領(lǐng)域�；煦缦到y(tǒng)是一類(lèi)特殊的非線性系統(tǒng)。自二十世紀(jì)初學(xué)者們提出混沌系統(tǒng)以來(lái),由于它具有的隨機(jī)性和對(duì)初始條件的極端敏感性等特點(diǎn),混沌系統(tǒng)的研究一直處于前沿課題。對(duì)于特殊的無(wú)平衡點(diǎn)的分?jǐn)?shù)階混沌系統(tǒng)來(lái)說(shuō),常規(guī)判定不能證明其混沌特性。對(duì)于這樣的系統(tǒng),本文的研究將著重以下幾個(gè)方面:首先,研究了分?jǐn)?shù)階微積分的基本定義、性質(zhì)以及它的穩(wěn)定性理論。結(jié)合自然界和社會(huì)科學(xué)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(duì)一個(gè)新的無(wú)平衡點(diǎn)分?jǐn)?shù)階混沌系統(tǒng)進(jìn)行了分析,利用分?jǐn)?shù)階微分變換方法,得到了它的解序列。研究了無(wú)平衡點(diǎn)分?jǐn)?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(yīng)控制方法,使系統(tǒng)的狀態(tài)可以追蹤期望軌跡。最后通過(guò)數(shù)值仿真,驗(yàn)證了該控制方法的有效性。其次,研究了一類(lèi)參數(shù)不確定的無(wú)平衡點(diǎn)分?jǐn)?shù)階混沌系統(tǒng)的同步問(wèn)題。根據(jù)自適應(yīng)控制理論,將自適應(yīng)控制法引入到混沌同步中。把它應(yīng)用到一類(lèi)新的無(wú)平衡點(diǎn)分?jǐn)?shù)階混沌系統(tǒng)中,設(shè)計(jì)了性能可靠的同步控制器和參數(shù)自適應(yīng)律,實(shí)現(xiàn)了上述提到的無(wú)平衡點(diǎn)分?jǐn)?shù)階混沌系統(tǒng)與其自結(jié)構(gòu)的同步。同時(shí)運(yùn)用主動(dòng)控制方法構(gòu)造主動(dòng)控制器對(duì)系統(tǒng)進(jìn)行了同步研究。把誤差系統(tǒng)中無(wú)益的非線性項(xiàng)通過(guò)設(shè)計(jì)控制器進(jìn)行抵消,提出了一類(lèi)簡(jiǎn)化分?jǐn)?shù)階混沌系統(tǒng)的同步控制器設(shè)計(jì)方法。與上述所研究的自適應(yīng)法進(jìn)行數(shù)值仿真對(duì)比,既驗(yàn)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(guò)對(duì)比,可以得出改進(jìn)的自適應(yīng)控制法具有更快的控制速度和更高的控制效率,而主動(dòng)控制方法具有更簡(jiǎn)單的計(jì)算過(guò)程。最后針對(duì)無(wú)平衡點(diǎn)的分?jǐn)?shù)階混沌系統(tǒng),令其為驅(qū)動(dòng)系統(tǒng),令一個(gè)新的四維分?jǐn)?shù)階混沌系統(tǒng)為響應(yīng)系統(tǒng),采用自適應(yīng)控制方法進(jìn)行異結(jié)構(gòu)同步研究。設(shè)計(jì)了合適的同步控制器和可以很好辨識(shí)參數(shù)的自適應(yīng)律,實(shí)現(xiàn)了無(wú)平衡點(diǎn)分?jǐn)?shù)階混沌系統(tǒng)與新的四維分?jǐn)?shù)階混沌系統(tǒng)之間的異結(jié)構(gòu)同步,并進(jìn)行仿真,驗(yàn)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(xué)位授予單位】：東北石油大學(xué)
【學(xué)位級(jí)別】：碩士
【學(xué)位授予年份】：2017
【分類(lèi)號(hào)】：O415.5;TP273
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本文編號(hào)：2462706