非线性动力系统的时滞反馈分岔控制研究
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摘要
分岔控制和时滞动力学作为非线性科学中的前沿研究课题,极具挑战性,是目前非线性研究的新热点。分岔控制的目的是对给定的非线性动力系统设计一个控制器,用来改变系统的分岔特性,从而消除系统中有害的动力学行为,使之产生人们需要的动力学行为。本文在全面分析和总结非线性动力系统分岔控制研究现状的基础上,基于非线性控制理论、分岔理论、时滞动力学等非线性科学的现代分析方法和理论,设计了时滞反馈控制器,对非线性微分动力系统分岔控制的基础理论和应用进行了系统和深入的研究,工作具有较大的理论意义和应用价值。研究内容如下:
第一章对非线性控制理论、分岔控制和时滞反馈控制的研究方法、现状和进展进行了综述,介绍了本文的研究目的、研究内容和创新点。
第二章介绍动力学研究的一些基本概念,简述发生鞍结分岔、跨临界分岔、叉形分岔的充分必要条件,以及这三种静态分岔相互转换的条件;介绍分岔控制器设计及分析的主要方法以及时滞动力学的一些分析方法。
第三章设计含有线性时滞位移和时滞阻尼的时滞反馈控制器,对含有平方和立方非线性项的强迫Duffing振动系统在主共振和亚超谐共振时的分岔进行控制,找到了系统产生鞍结分岔的临界条件,消除了系统的鞍结分岔,同时降低了系统的振幅,得到系统稳态响应的振幅与控制参数之间的关系,给出设计该类时滞反馈控制器的思路。
第四章用含有两个时滞量的线性时滞反馈控制器对参数激励的van der Pol-Duffing系统的主参数共振进行控制。通过对平均方程和分岔响应方程分析,得到时滞参数对分岔响应的影响,进而提出控制策略,设计时滞控制器,对系统分岔进行控制。
研究轴向周期激励作用下梁的后屈曲动力学分岔行为,设计线性时滞阻尼控制器,对临界力进行了有效地控制,并有效地消除超临界分岔,改变亚临界分岔的位置。
第五章对线性和非线性时滞反馈控制器联合作用下的分岔控制进行了研究。以van der Pol-Duffing系统为例,分析了静平衡情况时的Hopf分岔,得到了时滞参数与Hopf分岔产生条件的关系;对于存在稳态周期响应的情况,推导出了时滞参数与稳定极限环幅值的关系,从而达到通过时滞反馈对稳定极限环幅值的控制。
第六章对受移动载荷作用的非线性梁进行研究,设计一类反馈控制器,分析
As a leading subject and new interests of nonlinear researches, bifurcation control and time delay dynamics come up with great challenge. It aims at designing a controller to modify the bifurcation properties of a given nonlinear system, and achieving some desirable behaviors. Through a complete summary and examination of the history and the actuality of the bifurcation control research, a systematic investigation into the fundamental theory and application of the bifurcation control of a differential nonlinear system has been studied based on the nonlinear vibration theory, the nonlinear dynamics theory, the bifurcation theory and the time delay dynamics theory, and the time delay feedback controller has been designed. The studies have more profound theoretical significances and important engineering application values, which contribute to the development and application of bifurcation control. The main contents in this paper can be stated as follows:Chapter one outlines some study methods, nonlinear theory and recent advances on bifurcation control first, and introduces the aim, contents and innovations of the study in this paper.Chapter two introduces some basic conceptions on dynamics. The necessary and sufficient conditions of genesis for three elementary static bifurcations, including saddle-node bifurcation, transcritical bifurcation and pitchfork bifurcation, and transfer conditions between each other are introduced. Then some design and analysis methods on bifurcation controller and time delay dynamics are presented.In chapter three, a controller with two linear time delays-----displacement andvelocity, is designed to the forced Duffing system with quadratic and cubic nonlinearities, Studies cover primary resonance, subresonance and superresonance. The critical conditions of bifurcation are obtained based on the study. The saddle-node bifurcation is eliminated as well as the resonance amplitude is reduced; the relationship between steady response amplitude and controller parameters is obtained and design method of this kind time delay controller is provided.In chapter four, a linear controller with time delay displacement and time delay velocity is designed to van der Pol-Duffing system, and principle parametric resonance is studied. Using the perturbation method, the average equations and bifurcation equation are obtained. Studying the obtained equations, the effects of main parameters to the bifurcation are found, so the control strategy is advanced. The
desirable controller is designed and the bifurcation of this system is controlled well.As an engineering case, a post-bulked dynamics bifurcation of a beam subjected to harmonic axial excitation is studied. Using the time delay velocity feedback controller, the critical force control is achieved. The study indicates this controller can effectively eliminate super-critical bifurcation or change the position of sub-critical bifurcation.In the chapter five, van der Pol-Duffing system with the controller including linear time delay items and nonlinear items at the same time is investigated. Hopf bifurcation for the static steady case is studied and time delay parameters' effects on the condition of this bifurcation are obtained. For the steady periodic resonance case, the relationship between limit cycle amplitude and time delay parameters is obtained, so the aim of amplitude control of limit cycle with time delays is achieved.In the chapter six, a nonlinear beam under moving loads is investigated. The dynamics differential equation with time delays is obtained and the dynamics behaviors of this system are analyzed. Particularly, a kind of nonlinear time delay controller is introduced, which can obtain the results that cannot be obtained with the linear time delay controller.In this paper, some innovative thinking is that using bifurcation control theory investigates the nonlinear dynamical systems, which enriches the nonlinear dynamics theory and expands the nonlinear theory. The incorporates are as follows:1. The time delay feedback control is used to investigate nonlinear dynamic systems control, especially for the bifurcation control. Some kind of time delay feedback controllers are designed and studied. The systems' dynamic behaviors are optimized and some control strategies for the various systems are obtained.2. Time delay controllers with multi time delays are new exposure in the interrelated field. In this paper, controller with two time delays, including displacement and velocity, is investigated. Using the designed controllers, the Duffing system with quadratic and cubic nonlinearities and van der Pol-Duffing system with parametric motivation are investigated. The results indicate these controllers can be well used to bifurcation control of primary resonance.3. Using linear time delay velocity controller, the flexible post-bulking beam under parametric excitation and harmonic motivation is investigated. The aims, including controlling critical force, eliminating super-critical bifurcation and changing the bifurcation position of sub-critical bifurcation, are achieved.4. A nonlinear beam under moving loads is studied. The dynamics differential
equation with time delays is obtained and the dynamics behaviors of this system are analyzed. A nonlinear time delay controller is obtained, which can content the bifurcation aims and meantime make the response amplitude invariably. This result is meaning for practical engineering.Through the present research work, the bifurcation control theory is enriched and developed. The studied have put forward the theoretical foundations and approaches for designing time delay feedback controllers for optimizing the dynamics behaviors. The bifurcation control is a new researching field that needs not only more deeper and wider theoretical studies but practical applications. Due to the widely existence of time delays and the superiority of time delay control, it can be anticipated that new results publications on this subject will be appeared, more and more researchers will pursue further in this stimulating and promising direction of the new research.
第一章对非线性控制理论、分岔控制和时滞反馈控制的研究方法、现状和进展进行了综述,介绍了本文的研究目的、研究内容和创新点。
第二章介绍动力学研究的一些基本概念,简述发生鞍结分岔、跨临界分岔、叉形分岔的充分必要条件,以及这三种静态分岔相互转换的条件;介绍分岔控制器设计及分析的主要方法以及时滞动力学的一些分析方法。
第三章设计含有线性时滞位移和时滞阻尼的时滞反馈控制器,对含有平方和立方非线性项的强迫Duffing振动系统在主共振和亚超谐共振时的分岔进行控制,找到了系统产生鞍结分岔的临界条件,消除了系统的鞍结分岔,同时降低了系统的振幅,得到系统稳态响应的振幅与控制参数之间的关系,给出设计该类时滞反馈控制器的思路。
第四章用含有两个时滞量的线性时滞反馈控制器对参数激励的van der Pol-Duffing系统的主参数共振进行控制。通过对平均方程和分岔响应方程分析,得到时滞参数对分岔响应的影响,进而提出控制策略,设计时滞控制器,对系统分岔进行控制。
研究轴向周期激励作用下梁的后屈曲动力学分岔行为,设计线性时滞阻尼控制器,对临界力进行了有效地控制,并有效地消除超临界分岔,改变亚临界分岔的位置。
第五章对线性和非线性时滞反馈控制器联合作用下的分岔控制进行了研究。以van der Pol-Duffing系统为例,分析了静平衡情况时的Hopf分岔,得到了时滞参数与Hopf分岔产生条件的关系;对于存在稳态周期响应的情况,推导出了时滞参数与稳定极限环幅值的关系,从而达到通过时滞反馈对稳定极限环幅值的控制。
第六章对受移动载荷作用的非线性梁进行研究,设计一类反馈控制器,分析
As a leading subject and new interests of nonlinear researches, bifurcation control and time delay dynamics come up with great challenge. It aims at designing a controller to modify the bifurcation properties of a given nonlinear system, and achieving some desirable behaviors. Through a complete summary and examination of the history and the actuality of the bifurcation control research, a systematic investigation into the fundamental theory and application of the bifurcation control of a differential nonlinear system has been studied based on the nonlinear vibration theory, the nonlinear dynamics theory, the bifurcation theory and the time delay dynamics theory, and the time delay feedback controller has been designed. The studies have more profound theoretical significances and important engineering application values, which contribute to the development and application of bifurcation control. The main contents in this paper can be stated as follows:Chapter one outlines some study methods, nonlinear theory and recent advances on bifurcation control first, and introduces the aim, contents and innovations of the study in this paper.Chapter two introduces some basic conceptions on dynamics. The necessary and sufficient conditions of genesis for three elementary static bifurcations, including saddle-node bifurcation, transcritical bifurcation and pitchfork bifurcation, and transfer conditions between each other are introduced. Then some design and analysis methods on bifurcation controller and time delay dynamics are presented.In chapter three, a controller with two linear time delays-----displacement andvelocity, is designed to the forced Duffing system with quadratic and cubic nonlinearities, Studies cover primary resonance, subresonance and superresonance. The critical conditions of bifurcation are obtained based on the study. The saddle-node bifurcation is eliminated as well as the resonance amplitude is reduced; the relationship between steady response amplitude and controller parameters is obtained and design method of this kind time delay controller is provided.In chapter four, a linear controller with time delay displacement and time delay velocity is designed to van der Pol-Duffing system, and principle parametric resonance is studied. Using the perturbation method, the average equations and bifurcation equation are obtained. Studying the obtained equations, the effects of main parameters to the bifurcation are found, so the control strategy is advanced. The
desirable controller is designed and the bifurcation of this system is controlled well.As an engineering case, a post-bulked dynamics bifurcation of a beam subjected to harmonic axial excitation is studied. Using the time delay velocity feedback controller, the critical force control is achieved. The study indicates this controller can effectively eliminate super-critical bifurcation or change the position of sub-critical bifurcation.In the chapter five, van der Pol-Duffing system with the controller including linear time delay items and nonlinear items at the same time is investigated. Hopf bifurcation for the static steady case is studied and time delay parameters' effects on the condition of this bifurcation are obtained. For the steady periodic resonance case, the relationship between limit cycle amplitude and time delay parameters is obtained, so the aim of amplitude control of limit cycle with time delays is achieved.In the chapter six, a nonlinear beam under moving loads is investigated. The dynamics differential equation with time delays is obtained and the dynamics behaviors of this system are analyzed. Particularly, a kind of nonlinear time delay controller is introduced, which can obtain the results that cannot be obtained with the linear time delay controller.In this paper, some innovative thinking is that using bifurcation control theory investigates the nonlinear dynamical systems, which enriches the nonlinear dynamics theory and expands the nonlinear theory. The incorporates are as follows:1. The time delay feedback control is used to investigate nonlinear dynamic systems control, especially for the bifurcation control. Some kind of time delay feedback controllers are designed and studied. The systems' dynamic behaviors are optimized and some control strategies for the various systems are obtained.2. Time delay controllers with multi time delays are new exposure in the interrelated field. In this paper, controller with two time delays, including displacement and velocity, is investigated. Using the designed controllers, the Duffing system with quadratic and cubic nonlinearities and van der Pol-Duffing system with parametric motivation are investigated. The results indicate these controllers can be well used to bifurcation control of primary resonance.3. Using linear time delay velocity controller, the flexible post-bulking beam under parametric excitation and harmonic motivation is investigated. The aims, including controlling critical force, eliminating super-critical bifurcation and changing the bifurcation position of sub-critical bifurcation, are achieved.4. A nonlinear beam under moving loads is studied. The dynamics differential
equation with time delays is obtained and the dynamics behaviors of this system are analyzed. A nonlinear time delay controller is obtained, which can content the bifurcation aims and meantime make the response amplitude invariably. This result is meaning for practical engineering.Through the present research work, the bifurcation control theory is enriched and developed. The studied have put forward the theoretical foundations and approaches for designing time delay feedback controllers for optimizing the dynamics behaviors. The bifurcation control is a new researching field that needs not only more deeper and wider theoretical studies but practical applications. Due to the widely existence of time delays and the superiority of time delay control, it can be anticipated that new results publications on this subject will be appeared, more and more researchers will pursue further in this stimulating and promising direction of the new research.
引文
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