【摘要】：近些年,隨著計算機技術(shù)和交叉學(xué)科的發(fā)展,分?jǐn)?shù)階微積分在理論和應(yīng)用方面取得巨大進(jìn)步。分?jǐn)?shù)階微積分作為整數(shù)階微積分在任意階次上的推廣,具有非局限性特性,非常適合描述那些具有記憶和遺傳特性的材料和過程,被廣泛用于科技和工程領(lǐng)域,例如多孔材料中的流體流動、反常擴散、粘彈性材料中的聲波傳播、自相似結(jié)構(gòu)的動力學(xué)、信號處理、金融理論、保密通信和生物系統(tǒng)電導(dǎo)等。分?jǐn)?shù)階微積分已經(jīng)成為當(dāng)下的研究熱點領(lǐng)域。本文主要研究了一種改進(jìn)的預(yù)估-校正算法、分?jǐn)?shù)階單時滯復(fù)Lorenz系統(tǒng)的動力學(xué)分析及其自時滯同步及分?jǐn)?shù)階多時滯非線性系統(tǒng)的穩(wěn)定性控制等問題。本文的主要內(nèi)容和創(chuàng)新之處如下:1.改進(jìn)的預(yù)估-校正算法通過對預(yù)估-校正算法的預(yù)估部分作出改進(jìn),提出了一種改進(jìn)的預(yù)估-校正算法。該算法通過提高預(yù)估部分的計算精度,來提高整體算法的計算精度。此外,利用MATLAB數(shù)值仿真工具分析了三個數(shù)值仿真實例,通過與預(yù)估-校正算法比較,體現(xiàn)了改進(jìn)的預(yù)估-校正算法在計算精度上的優(yōu)越性。2.分?jǐn)?shù)階單時滯復(fù)Lorenz系統(tǒng)的動力學(xué)行為及其自時滯同步通過相圖法和最大Lyapunov指數(shù)法,分析了分?jǐn)?shù)階單時滯復(fù)Lorenz系統(tǒng)的動力學(xué)行為。在這部分,我們通過固定系統(tǒng)階數(shù),增加時滯項系數(shù),發(fā)現(xiàn)分?jǐn)?shù)階單時滯復(fù)Lorenz系統(tǒng)具有豐富的動力學(xué)行為。此外通過構(gòu)造反饋控制器,實現(xiàn)了分?jǐn)?shù)階單時滯復(fù)Lorenz系統(tǒng)的自時滯同步。利用MATLAB數(shù)值仿真工具做了數(shù)值仿真,驗證了所得結(jié)論的有效性和可行性。3.分?jǐn)?shù)階多時滯非線性系統(tǒng)的穩(wěn)定性控制基于分?jǐn)?shù)階Lyapunov直接方法和分?jǐn)?shù)階時滯非線性系統(tǒng)的穩(wěn)定性理論,提出了一種通過構(gòu)造反饋控制器實現(xiàn)分?jǐn)?shù)階多時滯非線性系統(tǒng)穩(wěn)定性控制的方法。不同于已有的工作,該方法中,反饋控制器為線性反饋控制器,且只涉及系統(tǒng)當(dāng)前狀態(tài)變量,與時滯項無關(guān),結(jié)構(gòu)簡單,易于工程實現(xiàn)。最后將該方法應(yīng)用于三個典型的分?jǐn)?shù)階多時滯非線性受控系統(tǒng),通過MATLAB數(shù)值仿真工具,驗證了所得結(jié)果的有效性與可行性。
[Abstract]:In recent years, with the development of computer technology and interdiscipline, fractional calculus has made great progress in theory and application. Fractional calculus, as a generalization of integral order calculus at any order, has the characteristics of no limitation and is very suitable for describing materials and processes with memory and genetic properties, and is widely used in science, technology and engineering. For example, fluid flow in porous materials, anomalous diffusion, acoustic propagation in viscoelastic materials, dynamics of self-similar structures, signal processing, financial theory, secure communications and biological system conductance. Fractional calculus has become a hot research field. In this paper, an improved predictor-correction algorithm, dynamic analysis of fractional single-delay complex Lorenz systems and stability control of self-delay synchronization and fractional multi-delay nonlinear systems are studied. The main contents and innovations of this paper are as follows: 1. An improved predictor-correction algorithm is proposed by improving the prediction part of the predictor-correction algorithm. The algorithm improves the accuracy of the whole algorithm by improving the accuracy of the prediction part. In addition, three numerical simulation examples are analyzed by using the MATLAB numerical simulation tool. By comparing with the predictor-correction algorithm, the superiority of the improved predictor-correction algorithm in the calculation accuracy is demonstrated. The dynamic behavior of fractional single-delay complex Lorenz system and its self-delay synchronization by phase diagram method and maximum Lyapunov exponent method are analyzed. The dynamic behavior of fractional single-delay complex Lorenz system is analyzed. In this part, we find that fractional single-delay complex Lorenz systems have rich dynamic behavior by increasing the coefficients of delay terms by the fixed order of the system. In addition, a feedback controller is constructed to realize the self-delay synchronization of fractional single-delay complex Lorenz systems. The effectiveness and feasibility of the conclusions are verified by using the MATLAB numerical simulation tool. The stability control of fractional multi-delay nonlinear systems is based on the fractional Lyapunov direct method and the stability theory of fractional time-delay nonlinear systems. In this paper, a feedback controller is proposed to control the stability of fractional multi-delay nonlinear systems. In this method, the feedback controller is a linear feedback controller, which only involves the current state variables of the system, is independent of the time-delay term, and is simple in structure and easy to be implemented in engineering. Finally, the method is applied to three typical fractional multi-delay nonlinear controlled systems. The validity and feasibility of the obtained results are verified by MATLAB numerical simulation tool.
【學(xué)位授予單位】：重慶郵電大學(xué)
【學(xué)位級別】：碩士
【學(xué)位授予年份】：2016
【分類號】：TP13

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本文編號：2197074