三维和四维非线性系统Hopf分岔反馈控制
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摘要
作为一个新出现的前沿研究课题,高维系统的分岔控制极具挑战性。低维系统的分岔控制研究已经取得了一些成果,而高维系统的分岔控制比低维系统的难度要大得多。延迟和消除分岔现象,避免不良后果,或有目的地创建或强化有益的分岔现象,使其为人们所利用,是分岔控制理论研究的主要内容。本文主要研究典型的三维和四维非线性系统的Hopf分岔控制问题,分析系统的分岔特性,提出控制方法,设计分岔控制器,使系统产生所期望的动力学行为。
本文首先综述了非线性控制理论、分岔控制、Hopf分岔控制以及混沌控制等的研究现状与发展状况。然后介绍了非线性动力学研究的一些基本概念和几种分岔控制方法,阐述了Hopf分岔定义与判据、Hopf分岔理论、以及Hopf分岔周期解的近似求解方法,同时给出了极限环稳定性指标与极限环幅值近似解析解的计算。
作为Hopf分岔控制研究的一个重要方面,本文着重研究了三维和四维非线性系统的极限环幅值控制问题,根据中心流形定理和规范型降维理论,提出了方便有效的极限环幅值控制方法。
对于三维系统,研究了作用于标准型下系统的普适的多项式控制器设计以及相应的极限环稳定性指标的计算,建立了受控系统的极限环幅值与控制增益之间的近似解析关系。以Chen系统和平滑Chua系统为示例,验证了所提出的幅值控制方法的简便与有效性。不同于对Chen系统和Chua系统的幅值控制,对于同样是三维的Langford系统,其极限环的幅值控制更加直接而有效,便于实现控制增益的优化设计,而且,对于给定的幅值控制目标,可直接针对原系统进行控制器设计。此外还设计了一些特殊的控制器,在这些控制器作用下,能够写出受控系统极限环幅值与控制增益之间的精确解析关系。
对于四维Qi系统的极限环幅值控制,采用的是washout filter非线性反馈控制器,受控系统的维数达到了五维,成功建立了极限环幅控近似解析关系,给出了反映在不同的控制增益下受控系统极限环的幅值控制结果的幅值曲线。对于所得到的近似解析关系,本文均与数值模拟结果进行了对比验证,验证结果说明:在Hopf分岔邻域,这些近似解的误差很小。
本文分别对Langford系统的一次Hopf分岔、二次Hopf分岔及其混沌运动的控制进行了研究,分别得到控制增益与分岔参数之间的解析关系。采用线性状态反馈控制器,转移了系统的一次Hopf分岔点;采用非线性状态反馈控制器,在不改变一次Hopf分岔点的前提下,转移了系统的二次Hopf分岔点;通过对通向混沌运动的准周期道路的控制来控制混沌运动,实现了对该系统混沌运动的延迟控制。
研究了四维Qi系统Hopf分岔的反控制问题。在线性与非线性状态反馈控制共同作用下,在该系统原为鞍点的零平衡点上,创建了稳定的Hopf分岔行为。针对不同的分岔参数,设计了不同的控制方案。
本文的研究工作丰富了非线性控制理论的研究内容,发展了分岔控制的研究方法,为高维系统分岔控制理论的建立作出了贡献,具有较大的理论意义和实际应用价值。
As an emerging leading research field, bifurcation control of high dimensional systems has become more and more challenging. Some fruits have been achieved in low dimensional systems and bifurcation control of high dimensional systems is quite difficult than that in low ones. It is the main topics in bifurcation control to either delay (or eliminate) bifurcations for avoiding bad affects or creat (or enhance) beneficial bifurcation behaviors on purpose of utilizing them. This paper focuss on studying the Hopf bifurcation control of the typical 3D and 4D nonlinear systems, e.g. analyzing the bifurcation characteristics, offering approaches of control, designing bifurcation contollers and thereby achieving some desirable dynamical behaviors.
Firstly, the recent advances about the nonlinear control theory, the bifurcation control, the Hopf bifurcation control and the chaos control are summarized in this paper. Secondly, some basic concepts about nonlinear dynamics and several bifurcation control methods are introduced, the definition, the criterion and the theory of Hopf bifurcation as well as the approach of the periodic approximate solutions are illustrated and simutanously the formulae for stability index and analytical amplitude approximations of limit cycle are presented.
As an important aspect in Hopf bifurcation control, the amplitude control of limit cycle is stressly investigated in 3D and 4D nonlinear systems. A convenient approach to achieve an effective amplitude control is proposed from the center manifold theory and normal form reduction.
For a 3D system, formulae for a general polynomial controller applied to the normal form of a system and for the relvant stability index are derived. Thereby the amplitude approximation in terms of control gains of the controlled system is solved. The Chen system and smooth Chua's equation are used to illustrate the application of the amplitude control technology given above and show that the control technology is convenient and valid. The Langford system, which is the same as the Chen and Chua's system in being 3D system but different from them in amplitude control application, can be controlled in amplitude of limit cycle directly and effectively and the control gains can be optimized conveniently. The controller employed in the original Langford system can be directly designed for a desirable amplitude target and the exact solutions for amplitudes of limit cycles in systems under some special controllers can be obtained.
For modifying amplitude of limit cycle in the 4D Qi system, the washout filter controller is applied in the feedback nonlinear control and adds the dimension of Qi system to SD, the analytical amplitude approximation in terms of control gains is well established and the curves showing that result of amplitude control are given with different choices of control gains. The analytical amplitude approximations are all compared with the numerical simulations that it is varified that the errors are small enough in vicinity of Hopf bifurcation.
The first and secondary Hopf bifurcation and the chaos motion of the Langford system are controlled and the relationships between bifurcation parameter and control gains are obtained. The first Hopf bifurcation points are shifted under a state feedback linear control while the secondary Hopf bifurcation value is changed preserving one of the first Hopf bifurcation points under a nonlinear control. The chaos motion is delayed by the way of controlling the quasi-periodic bifurcation route to chaos.
Anti-control of Hopf bifurcation of the 4D Qi system is investigated. The Hopf bifurcation behavior from the zero equilibrium, which is a saddle point, is created under the combination control of linear and nonlinear feedback. With respect to various bifurcation parameters, different approaches of anti-control are advanced.
The research work enriches the nonlinear control theory, develops the technique of bifurcation control, contributes to the establishment of bifurcation control theory of high dimensional systems and has great theoretical meanings and pratical values.
本文首先综述了非线性控制理论、分岔控制、Hopf分岔控制以及混沌控制等的研究现状与发展状况。然后介绍了非线性动力学研究的一些基本概念和几种分岔控制方法,阐述了Hopf分岔定义与判据、Hopf分岔理论、以及Hopf分岔周期解的近似求解方法,同时给出了极限环稳定性指标与极限环幅值近似解析解的计算。
作为Hopf分岔控制研究的一个重要方面,本文着重研究了三维和四维非线性系统的极限环幅值控制问题,根据中心流形定理和规范型降维理论,提出了方便有效的极限环幅值控制方法。
对于三维系统,研究了作用于标准型下系统的普适的多项式控制器设计以及相应的极限环稳定性指标的计算,建立了受控系统的极限环幅值与控制增益之间的近似解析关系。以Chen系统和平滑Chua系统为示例,验证了所提出的幅值控制方法的简便与有效性。不同于对Chen系统和Chua系统的幅值控制,对于同样是三维的Langford系统,其极限环的幅值控制更加直接而有效,便于实现控制增益的优化设计,而且,对于给定的幅值控制目标,可直接针对原系统进行控制器设计。此外还设计了一些特殊的控制器,在这些控制器作用下,能够写出受控系统极限环幅值与控制增益之间的精确解析关系。
对于四维Qi系统的极限环幅值控制,采用的是washout filter非线性反馈控制器,受控系统的维数达到了五维,成功建立了极限环幅控近似解析关系,给出了反映在不同的控制增益下受控系统极限环的幅值控制结果的幅值曲线。对于所得到的近似解析关系,本文均与数值模拟结果进行了对比验证,验证结果说明:在Hopf分岔邻域,这些近似解的误差很小。
本文分别对Langford系统的一次Hopf分岔、二次Hopf分岔及其混沌运动的控制进行了研究,分别得到控制增益与分岔参数之间的解析关系。采用线性状态反馈控制器,转移了系统的一次Hopf分岔点;采用非线性状态反馈控制器,在不改变一次Hopf分岔点的前提下,转移了系统的二次Hopf分岔点;通过对通向混沌运动的准周期道路的控制来控制混沌运动,实现了对该系统混沌运动的延迟控制。
研究了四维Qi系统Hopf分岔的反控制问题。在线性与非线性状态反馈控制共同作用下,在该系统原为鞍点的零平衡点上,创建了稳定的Hopf分岔行为。针对不同的分岔参数,设计了不同的控制方案。
本文的研究工作丰富了非线性控制理论的研究内容,发展了分岔控制的研究方法,为高维系统分岔控制理论的建立作出了贡献,具有较大的理论意义和实际应用价值。
As an emerging leading research field, bifurcation control of high dimensional systems has become more and more challenging. Some fruits have been achieved in low dimensional systems and bifurcation control of high dimensional systems is quite difficult than that in low ones. It is the main topics in bifurcation control to either delay (or eliminate) bifurcations for avoiding bad affects or creat (or enhance) beneficial bifurcation behaviors on purpose of utilizing them. This paper focuss on studying the Hopf bifurcation control of the typical 3D and 4D nonlinear systems, e.g. analyzing the bifurcation characteristics, offering approaches of control, designing bifurcation contollers and thereby achieving some desirable dynamical behaviors.
Firstly, the recent advances about the nonlinear control theory, the bifurcation control, the Hopf bifurcation control and the chaos control are summarized in this paper. Secondly, some basic concepts about nonlinear dynamics and several bifurcation control methods are introduced, the definition, the criterion and the theory of Hopf bifurcation as well as the approach of the periodic approximate solutions are illustrated and simutanously the formulae for stability index and analytical amplitude approximations of limit cycle are presented.
As an important aspect in Hopf bifurcation control, the amplitude control of limit cycle is stressly investigated in 3D and 4D nonlinear systems. A convenient approach to achieve an effective amplitude control is proposed from the center manifold theory and normal form reduction.
For a 3D system, formulae for a general polynomial controller applied to the normal form of a system and for the relvant stability index are derived. Thereby the amplitude approximation in terms of control gains of the controlled system is solved. The Chen system and smooth Chua's equation are used to illustrate the application of the amplitude control technology given above and show that the control technology is convenient and valid. The Langford system, which is the same as the Chen and Chua's system in being 3D system but different from them in amplitude control application, can be controlled in amplitude of limit cycle directly and effectively and the control gains can be optimized conveniently. The controller employed in the original Langford system can be directly designed for a desirable amplitude target and the exact solutions for amplitudes of limit cycles in systems under some special controllers can be obtained.
For modifying amplitude of limit cycle in the 4D Qi system, the washout filter controller is applied in the feedback nonlinear control and adds the dimension of Qi system to SD, the analytical amplitude approximation in terms of control gains is well established and the curves showing that result of amplitude control are given with different choices of control gains. The analytical amplitude approximations are all compared with the numerical simulations that it is varified that the errors are small enough in vicinity of Hopf bifurcation.
The first and secondary Hopf bifurcation and the chaos motion of the Langford system are controlled and the relationships between bifurcation parameter and control gains are obtained. The first Hopf bifurcation points are shifted under a state feedback linear control while the secondary Hopf bifurcation value is changed preserving one of the first Hopf bifurcation points under a nonlinear control. The chaos motion is delayed by the way of controlling the quasi-periodic bifurcation route to chaos.
Anti-control of Hopf bifurcation of the 4D Qi system is investigated. The Hopf bifurcation behavior from the zero equilibrium, which is a saddle point, is created under the combination control of linear and nonlinear feedback. With respect to various bifurcation parameters, different approaches of anti-control are advanced.
The research work enriches the nonlinear control theory, develops the technique of bifurcation control, contributes to the establishment of bifurcation control theory of high dimensional systems and has great theoretical meanings and pratical values.
引文
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