非线性电路与系统的脉冲建模及其稳定性
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摘要
非线性电路与系统是非线性科学的一个分支,近年来受到来自各个领域的越来越多的关注。本论文主要内容涉及非线性电路与系统的脉冲建模和稳定性的研究。其中包括:1)关于蔡氏混沌电路的脉冲建模及其控制的研究;2)关于脉冲控制洛伦兹混沌系统族的稳定性研究;3)关于鲁里叶系统和时变时滞非线性系统的脉冲建模及其控制的研究;4)关于模糊混沌系统与脉冲时滞模糊系统的稳定性研究;5)关于不确定性时滞系统的脉冲建模及其控制的研究。
本文的主要创新之处可概括如下:
1.关于蔡氏混沌电路的脉冲建模及其控制的研究
蔡氏电路、蔡氏振荡器和时滞蔡氏振荡器是典型的非线性电路,是第一个真正能够用物理手段实现的混沌系统。通过脉冲的引入及其控制器的设计,研究了蔡氏振荡器的脉冲模型在李雅普诺夫意义下的全局指数稳定性。根据这种脉冲控制思想,研究了时滞蔡氏振荡器的脉冲模型在李雅普诺夫意义下的稳定、渐进稳定和全局指数稳定。
2.关于脉冲控制洛伦兹混沌系统族的稳定性研究
Lorenz系统、Chen系统和L(u|¨)系统都是三参数形式的混沌系统,组成洛伦兹混沌系统家族,可用单参数的统一混沌系统来建模。在本文中,我们提出了洛伦兹混沌系统族的脉冲模型,并进一步利用Lyapunov函数法和不等式工具研究了由第一个混沌模型——洛伦兹系统拓广和延伸的一大类相联系统的渐进稳定,得出了稳定性对时变脉冲间距的依赖关系。
3.关于鲁里叶系统和时变时滞非线性系统的脉冲建模及其控制的研究
根据脉冲控制系统的结构,把脉冲控制的思想推广到一般的非线性反馈控制系统——鲁里叶系统和时变时滞非线性系统。构造适当的李雅普诺夫函数并利用比较原理、不等式定理和线性矩阵不等式(LMI)工具,对鲁里叶系统和时变时滞非线性系统的脉冲控制问题进行了研究,得到了一系列的较易验证的渐进稳定和全局指数稳定准则,并推出了渐进稳定控制的脉冲间距和时变时滞的上界估计。
4.关于模糊混沌系统与脉冲时滞模糊系统的稳定性研究
把脉冲和时滞引入到模糊系统中是一种全新的思想。根据中心平均模糊(center-average defuzzifier)方法和平行分配补偿(PDC)设计技术,时滞效应的引入,得到了一种新的基于当前状态和时滞状态反馈的模糊控制器的设计方法。利用Lyapunov-Krasovskii泛函和线性矩阵不等式,研究了这类模糊混沌系统的渐进稳定。此外,还讨论了非线性系统的模糊和脉冲的混合建模,并利用Lyapunov直接法和Schur补对这类脉冲时滞模糊系统的全局指数稳定性问题进行了研究。
5.关于不确定性时滞系统的脉冲建模及其控制的研究
将脉冲引入到不确定性时滞系统中,建立起基于脉冲模型和时滞效应的不确定混合系统,并进一步利用李雅普诺夫直接法和比较原理研究了其鲁棒渐进稳定性,得出了鲁棒稳定对不确定性参数、脉冲间距和时滞的依赖关系。这种脉冲建模及其控制技术,也可推广应用在不确定性时变时滞非线性系统。
Recently, nonlinear circuits and systems as a branch of nonlinear science attract more and more attentions from various fields of science and engineering. In this dissertation, we perform impulsive modeling and stability study on nonlinear circuits and systems. The main contents of this dissertation include: 1) Impulsive modeling of Chua's chaotic circuit and its control; 2) Stability analysis and design of impulsive control Lorenz chaotic systems family; 3) Impulsive modeling and control of Lurie systems and the nonlinear systems with time-varying delays; 4) Stability of fuzzy chaotic systems and impulsive fuzzy systems with time-delays; 5) Impulsive modeling of a class of uncertain systems with time-delays and its control.
The main originality in this paper can be summarized as follows:
1. The study of impulsive modeling of Chua's chaotic circuit and its control
It's well known that Chua's circuit, Chua's oscillator and time-delayed Chua's oscillator are typical nonlinear circuits and the first chaotic systems which can be implemented by physical devices. Introducing impulse and design of its controller, we study globally exponential stability of the impulsive model of Chua's oscillator. According to the same idea of impulsive control, we still study stability, asymptotical stability and globally exponential stability of the impulsive model of time-delayed Chua's oscillator.
2. Stability study of impulsive control Lorenz chaotic systems family
Lorenz chaotic systems family consisting of Lorenz chaotic systems, Chen chaotic systems and Lüchaotic systems, whose mathematical models are presented by three parameters, can be modeled as a united chaotic system with one parameter. In this thesis, we formulate the impulsive model of Lorenz chaotic systems family, and further investigate asmptotical stability of a class of interrelation systems extended by the first chaotic model of Lorenz system by using the Lyapunov function method and the tool of inequalities. We derive explicit relationship between the stability of impulsive control Lorenz chaotic systems family and time-varying impulse interval.
3. Study on impulsive modeling and control of Lurie systems and the nonlinear systems with time-varying delays
With the structure of impulsive control system, we extend the idea of impulsive control to the general nonlinear control systems with feedback named as Lurie systems and the nonlinear systems with time-varying delays. By constructing the very Lyapunov function and using comparison principle, inequality theorem and linear matrix inequalities (LMIs), we investigate impulsive control of Lurie systems and the nonlinear systems with time-varying delays, and obtain some easy-verified asymptotically stable and globally exponentially stable criteria, further derive the estimate the upper bounds of impulse intervals and time-varying delays for asymptotically stable control.
4. Stability study of fuzzy chaotic systems and impulsive fuzzy systems with time-delays
New Conceptual idea is to introduce impulse and time delay to fuzzy systems. According to the method of center-average defuzzifier and design of parallel distributed compensation (PDC), we introduce time-delayed effect to design a new fuzzy controller based on current state feedback and time-delayed state feedback. By using Lyapunov-Krasovskii functionals and LMIs, we investigate asmptotical stability of this kind of fuzzy chaotic systems. Furthermore, we consider fuzzy and impulsive hybrid modeling of the nonlinear systems, and study globally exponential stability of this kind of impulsive fuzzy system with time delays by using Lyapunov direct method and Schur complement.
5. Impulsive modeling of a class of uncertain systems with time-delays and its control
By introducing impulses to the uncertain systems with time delays, we establish the uncertain hybrid systems based on impulsive model and time-delayed effect, futher study these robust asmptotical stability by the use of Lyapunov direct method and comparison principle, and derive explicit relationship between robust stability and uncertain parameter, impulse interval and time delay. We still can spread the art of impulsive modeling and its control to apply the uncertain nonlinear systems with time-varying delays extensively.
本文的主要创新之处可概括如下:
1.关于蔡氏混沌电路的脉冲建模及其控制的研究
蔡氏电路、蔡氏振荡器和时滞蔡氏振荡器是典型的非线性电路,是第一个真正能够用物理手段实现的混沌系统。通过脉冲的引入及其控制器的设计,研究了蔡氏振荡器的脉冲模型在李雅普诺夫意义下的全局指数稳定性。根据这种脉冲控制思想,研究了时滞蔡氏振荡器的脉冲模型在李雅普诺夫意义下的稳定、渐进稳定和全局指数稳定。
2.关于脉冲控制洛伦兹混沌系统族的稳定性研究
Lorenz系统、Chen系统和L(u|¨)系统都是三参数形式的混沌系统,组成洛伦兹混沌系统家族,可用单参数的统一混沌系统来建模。在本文中,我们提出了洛伦兹混沌系统族的脉冲模型,并进一步利用Lyapunov函数法和不等式工具研究了由第一个混沌模型——洛伦兹系统拓广和延伸的一大类相联系统的渐进稳定,得出了稳定性对时变脉冲间距的依赖关系。
3.关于鲁里叶系统和时变时滞非线性系统的脉冲建模及其控制的研究
根据脉冲控制系统的结构,把脉冲控制的思想推广到一般的非线性反馈控制系统——鲁里叶系统和时变时滞非线性系统。构造适当的李雅普诺夫函数并利用比较原理、不等式定理和线性矩阵不等式(LMI)工具,对鲁里叶系统和时变时滞非线性系统的脉冲控制问题进行了研究,得到了一系列的较易验证的渐进稳定和全局指数稳定准则,并推出了渐进稳定控制的脉冲间距和时变时滞的上界估计。
4.关于模糊混沌系统与脉冲时滞模糊系统的稳定性研究
把脉冲和时滞引入到模糊系统中是一种全新的思想。根据中心平均模糊(center-average defuzzifier)方法和平行分配补偿(PDC)设计技术,时滞效应的引入,得到了一种新的基于当前状态和时滞状态反馈的模糊控制器的设计方法。利用Lyapunov-Krasovskii泛函和线性矩阵不等式,研究了这类模糊混沌系统的渐进稳定。此外,还讨论了非线性系统的模糊和脉冲的混合建模,并利用Lyapunov直接法和Schur补对这类脉冲时滞模糊系统的全局指数稳定性问题进行了研究。
5.关于不确定性时滞系统的脉冲建模及其控制的研究
将脉冲引入到不确定性时滞系统中,建立起基于脉冲模型和时滞效应的不确定混合系统,并进一步利用李雅普诺夫直接法和比较原理研究了其鲁棒渐进稳定性,得出了鲁棒稳定对不确定性参数、脉冲间距和时滞的依赖关系。这种脉冲建模及其控制技术,也可推广应用在不确定性时变时滞非线性系统。
Recently, nonlinear circuits and systems as a branch of nonlinear science attract more and more attentions from various fields of science and engineering. In this dissertation, we perform impulsive modeling and stability study on nonlinear circuits and systems. The main contents of this dissertation include: 1) Impulsive modeling of Chua's chaotic circuit and its control; 2) Stability analysis and design of impulsive control Lorenz chaotic systems family; 3) Impulsive modeling and control of Lurie systems and the nonlinear systems with time-varying delays; 4) Stability of fuzzy chaotic systems and impulsive fuzzy systems with time-delays; 5) Impulsive modeling of a class of uncertain systems with time-delays and its control.
The main originality in this paper can be summarized as follows:
1. The study of impulsive modeling of Chua's chaotic circuit and its control
It's well known that Chua's circuit, Chua's oscillator and time-delayed Chua's oscillator are typical nonlinear circuits and the first chaotic systems which can be implemented by physical devices. Introducing impulse and design of its controller, we study globally exponential stability of the impulsive model of Chua's oscillator. According to the same idea of impulsive control, we still study stability, asymptotical stability and globally exponential stability of the impulsive model of time-delayed Chua's oscillator.
2. Stability study of impulsive control Lorenz chaotic systems family
Lorenz chaotic systems family consisting of Lorenz chaotic systems, Chen chaotic systems and Lüchaotic systems, whose mathematical models are presented by three parameters, can be modeled as a united chaotic system with one parameter. In this thesis, we formulate the impulsive model of Lorenz chaotic systems family, and further investigate asmptotical stability of a class of interrelation systems extended by the first chaotic model of Lorenz system by using the Lyapunov function method and the tool of inequalities. We derive explicit relationship between the stability of impulsive control Lorenz chaotic systems family and time-varying impulse interval.
3. Study on impulsive modeling and control of Lurie systems and the nonlinear systems with time-varying delays
With the structure of impulsive control system, we extend the idea of impulsive control to the general nonlinear control systems with feedback named as Lurie systems and the nonlinear systems with time-varying delays. By constructing the very Lyapunov function and using comparison principle, inequality theorem and linear matrix inequalities (LMIs), we investigate impulsive control of Lurie systems and the nonlinear systems with time-varying delays, and obtain some easy-verified asymptotically stable and globally exponentially stable criteria, further derive the estimate the upper bounds of impulse intervals and time-varying delays for asymptotically stable control.
4. Stability study of fuzzy chaotic systems and impulsive fuzzy systems with time-delays
New Conceptual idea is to introduce impulse and time delay to fuzzy systems. According to the method of center-average defuzzifier and design of parallel distributed compensation (PDC), we introduce time-delayed effect to design a new fuzzy controller based on current state feedback and time-delayed state feedback. By using Lyapunov-Krasovskii functionals and LMIs, we investigate asmptotical stability of this kind of fuzzy chaotic systems. Furthermore, we consider fuzzy and impulsive hybrid modeling of the nonlinear systems, and study globally exponential stability of this kind of impulsive fuzzy system with time delays by using Lyapunov direct method and Schur complement.
5. Impulsive modeling of a class of uncertain systems with time-delays and its control
By introducing impulses to the uncertain systems with time delays, we establish the uncertain hybrid systems based on impulsive model and time-delayed effect, futher study these robust asmptotical stability by the use of Lyapunov direct method and comparison principle, and derive explicit relationship between robust stability and uncertain parameter, impulse interval and time delay. We still can spread the art of impulsive modeling and its control to apply the uncertain nonlinear systems with time-varying delays extensively.
引文
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