分数阶非线性时滞系统的稳定性理论及控制研究
摘要
自1983年Mandelbort指出自然界及许多科学技术领域中存在大量的分数维以来,分数阶微积分在各个领域得到了广泛研究,分数阶微积分较整数阶微积分具有诸多突出优势迅速成为当前的研究热点；在实际系统中,不可避免存在非线性特征,是非线性系统。与此同时,一般控制系统由于反馈延时、信号传输延时等都具有时滞。而时滞的存在往往是系统的稳定性和性能变差的原因,时滞的存在可能导致通信网拥塞、数据丢包,可能导致交通系统中车辆流动不畅、交通拥堵。如何判定分数阶时滞非线性系统的稳定性与控制至今未得到有效解决,该课题的科学意义不仅在于发展分数阶时滞非线性系统稳定性理论实现分数阶时滞系统控制,推动分数阶控制理论和控制方法的应用,更有助于丰富非线性系统控制理论,推动控制理论和控制工程的发展。
本论文提出了分数阶无时滞与含时滞非线性系统稳定性理论；提出了矩阵配置的控制器设计方法。实现了分数阶无时滞与含时滞非线性系统的控制与同步及系统的未知参数辨识。
本论文的主要研究内容和创新点包括以下内容：
1.分数阶非线性系统稳定性与控制研究。基于整数阶非线性系统稳定性理论与分数阶线性系统稳定性理论,从系统Jacobian矩阵特征值的角度出发提出了分数阶非线性系统稳定性理论。与此同时,将整数阶Lyapunov稳定性理论拓展到分数阶系统中,提出构造正定函数V并对函数V求整数阶导数的稳定性分析理论。该稳定性理论克服了分数阶Lyapunov稳定性理论中对函数V的复杂的分数阶求导、稳定性分析困难,且系统参数发生摄动时稳定性分析失效这一难题。在此基础上,研究无时滞分数阶系统的脉冲控制理论,建立了脉冲强度与脉冲间隔的一般关系,实现了分数阶系统的脉冲控制。最后研究了分数阶系统的间歇控制稳定性理论,建立了间歇控制一般稳定性条件,实现了分数阶系统间歇控制。数值仿真结果证实所提理论的正确有效性。
2.分数阶时滞系统稳定性理论与控制研究。基于本文提出的分数阶非线性系统稳定性理论和整数阶时滞系统稳定性理论,研究了如何建立正定函数V的扩展函数并对该扩展函数求整数阶导数而不是分数阶导数的分数阶系统Lyapunov-krasovskii理论,建立了扩展函数的一般构造方法。利用该理论,整数阶时滞系统的大部分控制方法可拓展到分数阶时滞系统。且从定理的形式可以看出,当系统系数和微分阶次在一定范围内摄动时该理论仍然适用。基于特征值稳定性分析理论,结合不等式利用反证法等建立了分数阶时滞系统极限发散和收敛速率计算方法。根据分数阶时滞非线性系统的极限发散和极限收敛速率,建立了分数阶离散时滞非线性系统间歇控制稳定性理论,实现了离散时滞分数阶系统间歇控制方法。最后,仿真实例显示其结果的有效性。
3.基于矩阵配置法的分数阶非线性系统控制器设计。尽管非线性系统的控制与同步有大量报道,但如何简单有效设计控制器实现系统控制鲜见介绍。针对这一情况提出矩阵配置控制器设计法。针对无时滞参数己知、参数未知的分数阶系统提出了如何利用矩阵配置控制器设计方法设计控制器实现系统控制和未知参数辨识。基于本文提出的分数阶时滞系统稳定性理论,利用时滞分离方法,提出了含离散时滞的分数阶非线性系统矩阵配置控制器设计方法,实现了离散时滞分数阶非线性系统控制。进一步将这一方法拓展到了整数阶分布式时滞系统,实现了整数阶分布式时滞系统控制。这些工作表明该控制器设计方法具有较好的通用性。
最后,总结全文并提出了进一步研究的方向。
In view of the fractional calculus's outstanding advantage than integer or-der's, since Mandelbort point out that there are vast fractional phenomena in the nature and the domain of science and technology, the fractional calculus has been widely studied involving all fields, and became the research hotspot. Nonlinear always exists in real system, nonlinear is an essential attribute of real system. Meanwhile, owing to feedback delay and siginal transmission delay, time-delay is commonly encountered. Time-delay is frequently a source of instability poor performance, such as network congestion, data packet dropout, poor flow the ve-hicles, traffic jams, etc. However, how to predicate the stability on the fractional nonlinear system with delay have not been solved effectively, let alone realize control. So the scientific significance of the subject lies not only in developing the stability theorem and realizing control about the fractional nonlinear system, promoting the application of the control theorem and control method, but also in enriching the nonlinear system control theorem, pushing the development of control theory and control engineer.
Firstly, the stability theories of the fractional nonlinear system with or with-out delay are proposed. Then, the Matrix configuration controller design methods are presented. Finally, control and synchronization of fractional nonlinear sys-tems with and without delay, as well as identification of unknown parameters of the systems are realized.
The main achievements and innovations of this dissertation are as follows:
1. Stability and control study of fractional nonlinear system. Based on stability theorems of integer order nonlinear systems and fractional linear systems, the stability theorem and control of fractional nonlinear system with-out delay are studied from Jacobian matrix eigenvalues point of view. In the meantime, by expanding integer order Lyapunov stability theory to fractional systems, a new stability theorem on fractional nonlinear system is proposed via constructing positive definite function V and taking integer derivatives of V. The superiority of the proposed theorem is reflected in not only avoiding difficultly taking fractional derivatives of V, but also avoiding losing stable efficacy while parameters perturbing. And on this basis we study impulsive control theorem on fractional nonlinear system, establish the general relationship between pulse strength and pulse period, and realize fractional impulsive control. Then we conduct intermittent control theorem study, create general intermittent stability condition, realize fractional intermittent control. Finally, simulation examples are given to illustrate the effectiveness of the developed theoretical results.
2. stability and control study of fractional nonlinear system with time-delay. Based on the proposed stability theorem of the fractional nonlinear system and the stability theorem of integer time-delay nonlinear system, we s-tudy fractional Lyapunov-krasovskii theory by building positive definite function V and taking integer derivatives of V instead of taking fractional derivatives, and propose general constructing method of expanding function. Using this theory, most control method of integer delayed system can be expanded to fractional delayed system. Based on the eigenvalue stability theory, combined inequali-ty and reduction to absurdum, the computing methods of limit divergence and convergence are presented. Subsequently, intermittent control theorem on frac-tional nonlinear system with discrete time-delay is set up and the intermittent control on delayed fractional nonlinear system is realized. Finally, numerical ex-amples are provided to demonstrate the effectiveness and the applicability of the proposed method.
3. Controller design of fractional nonlinear system via matrix con-figuration method. Although there are a large number of reports on nonlinear control and synchronization, but how to simply design controller is seldom report-ed so far. Stimulated by this, the matrix configuration controller design method is put forward. For non-delay fractional nonlinear system with or without unknown parameters, controller can be designed by matrix configuration method to realize system control and parameter identification. Based on the proposed stability theorem on the delayed fractional nonlinear system, by separating time-delay terms, the matrix configuration controller design method of fractional nonlinear system with discrete time-delay is presented also, and for delayed fractional non-linear system with or without unknown parameters, controller can be designed by matrix configuration method to realize system control and parameter identi-fication. We as well as expand this method to the integer order nonlinear system with distributed delay and achieve control. All the works show that the matrix configuration controller design method has good versatility and effectiveness.
Finally, the main results of the dissertation are concluded, and the issues of future investigation are proposed.
本论文提出了分数阶无时滞与含时滞非线性系统稳定性理论；提出了矩阵配置的控制器设计方法。实现了分数阶无时滞与含时滞非线性系统的控制与同步及系统的未知参数辨识。
本论文的主要研究内容和创新点包括以下内容：
1.分数阶非线性系统稳定性与控制研究。基于整数阶非线性系统稳定性理论与分数阶线性系统稳定性理论,从系统Jacobian矩阵特征值的角度出发提出了分数阶非线性系统稳定性理论。与此同时,将整数阶Lyapunov稳定性理论拓展到分数阶系统中,提出构造正定函数V并对函数V求整数阶导数的稳定性分析理论。该稳定性理论克服了分数阶Lyapunov稳定性理论中对函数V的复杂的分数阶求导、稳定性分析困难,且系统参数发生摄动时稳定性分析失效这一难题。在此基础上,研究无时滞分数阶系统的脉冲控制理论,建立了脉冲强度与脉冲间隔的一般关系,实现了分数阶系统的脉冲控制。最后研究了分数阶系统的间歇控制稳定性理论,建立了间歇控制一般稳定性条件,实现了分数阶系统间歇控制。数值仿真结果证实所提理论的正确有效性。
2.分数阶时滞系统稳定性理论与控制研究。基于本文提出的分数阶非线性系统稳定性理论和整数阶时滞系统稳定性理论,研究了如何建立正定函数V的扩展函数并对该扩展函数求整数阶导数而不是分数阶导数的分数阶系统Lyapunov-krasovskii理论,建立了扩展函数的一般构造方法。利用该理论,整数阶时滞系统的大部分控制方法可拓展到分数阶时滞系统。且从定理的形式可以看出,当系统系数和微分阶次在一定范围内摄动时该理论仍然适用。基于特征值稳定性分析理论,结合不等式利用反证法等建立了分数阶时滞系统极限发散和收敛速率计算方法。根据分数阶时滞非线性系统的极限发散和极限收敛速率,建立了分数阶离散时滞非线性系统间歇控制稳定性理论,实现了离散时滞分数阶系统间歇控制方法。最后,仿真实例显示其结果的有效性。
3.基于矩阵配置法的分数阶非线性系统控制器设计。尽管非线性系统的控制与同步有大量报道,但如何简单有效设计控制器实现系统控制鲜见介绍。针对这一情况提出矩阵配置控制器设计法。针对无时滞参数己知、参数未知的分数阶系统提出了如何利用矩阵配置控制器设计方法设计控制器实现系统控制和未知参数辨识。基于本文提出的分数阶时滞系统稳定性理论,利用时滞分离方法,提出了含离散时滞的分数阶非线性系统矩阵配置控制器设计方法,实现了离散时滞分数阶非线性系统控制。进一步将这一方法拓展到了整数阶分布式时滞系统,实现了整数阶分布式时滞系统控制。这些工作表明该控制器设计方法具有较好的通用性。
最后,总结全文并提出了进一步研究的方向。
In view of the fractional calculus's outstanding advantage than integer or-der's, since Mandelbort point out that there are vast fractional phenomena in the nature and the domain of science and technology, the fractional calculus has been widely studied involving all fields, and became the research hotspot. Nonlinear always exists in real system, nonlinear is an essential attribute of real system. Meanwhile, owing to feedback delay and siginal transmission delay, time-delay is commonly encountered. Time-delay is frequently a source of instability poor performance, such as network congestion, data packet dropout, poor flow the ve-hicles, traffic jams, etc. However, how to predicate the stability on the fractional nonlinear system with delay have not been solved effectively, let alone realize control. So the scientific significance of the subject lies not only in developing the stability theorem and realizing control about the fractional nonlinear system, promoting the application of the control theorem and control method, but also in enriching the nonlinear system control theorem, pushing the development of control theory and control engineer.
Firstly, the stability theories of the fractional nonlinear system with or with-out delay are proposed. Then, the Matrix configuration controller design methods are presented. Finally, control and synchronization of fractional nonlinear sys-tems with and without delay, as well as identification of unknown parameters of the systems are realized.
The main achievements and innovations of this dissertation are as follows:
1. Stability and control study of fractional nonlinear system. Based on stability theorems of integer order nonlinear systems and fractional linear systems, the stability theorem and control of fractional nonlinear system with-out delay are studied from Jacobian matrix eigenvalues point of view. In the meantime, by expanding integer order Lyapunov stability theory to fractional systems, a new stability theorem on fractional nonlinear system is proposed via constructing positive definite function V and taking integer derivatives of V. The superiority of the proposed theorem is reflected in not only avoiding difficultly taking fractional derivatives of V, but also avoiding losing stable efficacy while parameters perturbing. And on this basis we study impulsive control theorem on fractional nonlinear system, establish the general relationship between pulse strength and pulse period, and realize fractional impulsive control. Then we conduct intermittent control theorem study, create general intermittent stability condition, realize fractional intermittent control. Finally, simulation examples are given to illustrate the effectiveness of the developed theoretical results.
2. stability and control study of fractional nonlinear system with time-delay. Based on the proposed stability theorem of the fractional nonlinear system and the stability theorem of integer time-delay nonlinear system, we s-tudy fractional Lyapunov-krasovskii theory by building positive definite function V and taking integer derivatives of V instead of taking fractional derivatives, and propose general constructing method of expanding function. Using this theory, most control method of integer delayed system can be expanded to fractional delayed system. Based on the eigenvalue stability theory, combined inequali-ty and reduction to absurdum, the computing methods of limit divergence and convergence are presented. Subsequently, intermittent control theorem on frac-tional nonlinear system with discrete time-delay is set up and the intermittent control on delayed fractional nonlinear system is realized. Finally, numerical ex-amples are provided to demonstrate the effectiveness and the applicability of the proposed method.
3. Controller design of fractional nonlinear system via matrix con-figuration method. Although there are a large number of reports on nonlinear control and synchronization, but how to simply design controller is seldom report-ed so far. Stimulated by this, the matrix configuration controller design method is put forward. For non-delay fractional nonlinear system with or without unknown parameters, controller can be designed by matrix configuration method to realize system control and parameter identification. Based on the proposed stability theorem on the delayed fractional nonlinear system, by separating time-delay terms, the matrix configuration controller design method of fractional nonlinear system with discrete time-delay is presented also, and for delayed fractional non-linear system with or without unknown parameters, controller can be designed by matrix configuration method to realize system control and parameter identi-fication. We as well as expand this method to the integer order nonlinear system with distributed delay and achieve control. All the works show that the matrix configuration controller design method has good versatility and effectiveness.
Finally, the main results of the dissertation are concluded, and the issues of future investigation are proposed.
引文
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