非线性动力系统的分岔控制研究
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摘要
分岔控制作为一个非线性科学中新出现的前沿研究课题,极具挑战性。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,从而去掉系统中有害的动力学行为,使之产生人们所需要的动力学行为。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,基于非线性动力学、非线性控制理论、分岔理论等非线性科学的现代分析方法,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工程应用价值,获得了较为丰硕的研究成果。主要研究内容和结论如下。
1.利用开环控制的方法,实现了平衡点分岔的控制。推导出一维非线性微分动力系统发生鞍结分岔、跨临界分岔和叉形分岔三种基本平衡点分岔的条件。然后利用开环控制来改变非线性系统的分岔参数,使之获得理想的平衡点分岔方程。通过状态反馈控制器的设计,可实现所希望的任意平衡点分岔,同时去掉所不需要的轨道分支。
2.设计了线性和非线性反馈控制器,实现了对带有平方和立方非线性项的强迫Duffing动力系统的分岔控制。当系统处于主共振和超谐共振状态时,设计了线性控制器,消除了系统的鞍结分岔;设计了非线性控制器来延迟系统鞍结分岔的出现;设计了线性和非线性项联合作用的控制器,可以适当的调整控制参数,使得系统不发生鞍结分岔,或延迟鞍结分岔的出现;同时,大大降低了系统响应的幅值。对线性控制器、非线性控制器、线性和非线性项联合作用的控制器进行了数值模拟分析,说明了控制器的设计是成功的、有效的。
3.对二阶非线性常微分参数激励模型进行了动力学分析,设计了速度立方项的状态控制器,对参数激励系统的2倍超谐共振进行了控制。通过对平均方程的频响曲线分析和分岔分析,检验了控制器的效率,系统的响应幅值大大降低,鞍结分岔被消除,系统的动力学行为得到了优化。同时,利用改进的LP法对强非线性含Duffing-van der Pol振子的参数激励系统在1/2阶次谐共振时进行了分岔分析,由奇异性理论和普适开折理论,获得了系统在不同参数情况下的转迁集和分岔图,为今后进一步对系统进行分岔控制研究打下了良好的基础。同时,设计了一个简单的单摆模型,通过适当的外部激励信号的作用,完全可以实现对参数激励系统的分岔控制。这说明线性或非线性控制器在工程实际中是可以设计出来的,是完全可以实现的。
4.设计了不同的含时间的非线性参数控制器,实现了对非线性动力系统的分
Bifurcation control as an emerging new research field has become more and more challenging. It aims at designing a controller to modify the bifurcation properties of a given nonlinear system, and achieving some desirable dynamical behaviors.Through a complete summary and examination of the history and the actuality of the bifurcation control research, in this paper a systematic investigation into the fundamental theory and application of the bifurcation control is made by using the nonlinear vibration control theory, the nonlinear dynamics theory, the bifurcation theory. The studies have more profound theoretical significances and important engineering application values, which contribute to the development and application of the bifurcation control. The main achievements and conclusions in this dissertation are obtained as follows:1. The strategy for controlling the equilibrium bifurcation is obtained by using the open-loop control approach. The conditions of three elementary static bifurcations as saddle-node, transcritical, and pitchfork types of bifurcations for a one-dimensional ordinary differential equation are formulated. The open loop control is used to adjust the bifurcation parameter in order to obtain a desired equilibrium bifurcation diagram. By adopting the state feedback control strategy, the required bifurcations is obtained and the unwanted branches are eliminated.2. The linear and nonlinear feedback controllers are designed to control the saddle-nodes bifurcation of the forced Duffing system with the quadratic and cubic nonlinearities. In the cases of primary and superharmonic resonances, the linear feedback controller is designed to eliminate saddle-node bifurcations which would occur in the uncontrolled system, while the nonlinear one is designed to delay the occurrence of saddle-node bifurcations. Accordingly, either a linear feedback, or a nonlinear one, or a synthesis of both is adequate for the purpose of bifurcation control. Moreover, an appropriate feedback can also decrease the amplitude of the steady state response. Through the numerical simulations, the results are qualitative agreement with these of the theoretical analysis.
3. The theoretical studies reveal that the designed cubic velocity feedback is affective for controlling the superharmonic resonance responses of a parametrically excited system. The amplitude of the response are reduced and the saddle-node bifurcations have are eliminated, which would take place in the resonance responses. By analyzing the bifurcation function associated with the corresponding frequency-response equation and the Jacobi matrix, the gain of the feedback control is determined. A parametrically excited oscillator with strong nonlinearity including van der Pol and Duffing type is studied for static bifurcations. The applicable range of the MLP method is extended to 1/2 subharmonic resonance systems and the bifurcation equation of a strongly nonlinear oscillator which is transformed into a small parameter system is determined by using the multiple scales method. On the basis of the singularity theory, the transition set and the bifurcation diagram in various regions of the parameter plane are analyzed. The parametrically excited pendulum with the linear and nonlinear feedback is found which validates that the controller can be designed in the engineering applications.4. The different nonlinear parametric feedback control including time is used to control the bifurcations in various nonlinear dynamical systems, such as the forced Duffing system, the Duffing-van der Pol system and the parametrically excited system. The designed controllers are testified theoretically to eliminate the saddle-node bifurcation successfully in the case of primary and superharmonic resonances and to reject the steady-state response in the case of subharmonic. The results of the numerical simulations show that the proposed feedback control method is quite effective to achieve the goal of bifurcation control.5. The coupled oscillators are studied, and the control law is obtained in the case of the one-to-one internal resonance. An approximate solution for the nonlinear differential equations is got by using the method of multiple scales. And the bifurcation analysis and the performance of the control strategy are investigated theoretically. By adjusting the control parameter, the high-amplitude periodic and chaotic motions are removed.In this paper, the innovative thinking is that the bifurcation control theory is used to investigate the nonlinear dynamical systems, which enriches the nonlinear dynamics theory and expands the nonlinear control theory. The creative things are as follows: The open-loop control approach is used to control the equilibrium bifurcation. The state feedback is extended to control
1.利用开环控制的方法,实现了平衡点分岔的控制。推导出一维非线性微分动力系统发生鞍结分岔、跨临界分岔和叉形分岔三种基本平衡点分岔的条件。然后利用开环控制来改变非线性系统的分岔参数,使之获得理想的平衡点分岔方程。通过状态反馈控制器的设计,可实现所希望的任意平衡点分岔,同时去掉所不需要的轨道分支。
2.设计了线性和非线性反馈控制器,实现了对带有平方和立方非线性项的强迫Duffing动力系统的分岔控制。当系统处于主共振和超谐共振状态时,设计了线性控制器,消除了系统的鞍结分岔;设计了非线性控制器来延迟系统鞍结分岔的出现;设计了线性和非线性项联合作用的控制器,可以适当的调整控制参数,使得系统不发生鞍结分岔,或延迟鞍结分岔的出现;同时,大大降低了系统响应的幅值。对线性控制器、非线性控制器、线性和非线性项联合作用的控制器进行了数值模拟分析,说明了控制器的设计是成功的、有效的。
3.对二阶非线性常微分参数激励模型进行了动力学分析,设计了速度立方项的状态控制器,对参数激励系统的2倍超谐共振进行了控制。通过对平均方程的频响曲线分析和分岔分析,检验了控制器的效率,系统的响应幅值大大降低,鞍结分岔被消除,系统的动力学行为得到了优化。同时,利用改进的LP法对强非线性含Duffing-van der Pol振子的参数激励系统在1/2阶次谐共振时进行了分岔分析,由奇异性理论和普适开折理论,获得了系统在不同参数情况下的转迁集和分岔图,为今后进一步对系统进行分岔控制研究打下了良好的基础。同时,设计了一个简单的单摆模型,通过适当的外部激励信号的作用,完全可以实现对参数激励系统的分岔控制。这说明线性或非线性控制器在工程实际中是可以设计出来的,是完全可以实现的。
4.设计了不同的含时间的非线性参数控制器,实现了对非线性动力系统的分
Bifurcation control as an emerging new research field has become more and more challenging. It aims at designing a controller to modify the bifurcation properties of a given nonlinear system, and achieving some desirable dynamical behaviors.Through a complete summary and examination of the history and the actuality of the bifurcation control research, in this paper a systematic investigation into the fundamental theory and application of the bifurcation control is made by using the nonlinear vibration control theory, the nonlinear dynamics theory, the bifurcation theory. The studies have more profound theoretical significances and important engineering application values, which contribute to the development and application of the bifurcation control. The main achievements and conclusions in this dissertation are obtained as follows:1. The strategy for controlling the equilibrium bifurcation is obtained by using the open-loop control approach. The conditions of three elementary static bifurcations as saddle-node, transcritical, and pitchfork types of bifurcations for a one-dimensional ordinary differential equation are formulated. The open loop control is used to adjust the bifurcation parameter in order to obtain a desired equilibrium bifurcation diagram. By adopting the state feedback control strategy, the required bifurcations is obtained and the unwanted branches are eliminated.2. The linear and nonlinear feedback controllers are designed to control the saddle-nodes bifurcation of the forced Duffing system with the quadratic and cubic nonlinearities. In the cases of primary and superharmonic resonances, the linear feedback controller is designed to eliminate saddle-node bifurcations which would occur in the uncontrolled system, while the nonlinear one is designed to delay the occurrence of saddle-node bifurcations. Accordingly, either a linear feedback, or a nonlinear one, or a synthesis of both is adequate for the purpose of bifurcation control. Moreover, an appropriate feedback can also decrease the amplitude of the steady state response. Through the numerical simulations, the results are qualitative agreement with these of the theoretical analysis.
3. The theoretical studies reveal that the designed cubic velocity feedback is affective for controlling the superharmonic resonance responses of a parametrically excited system. The amplitude of the response are reduced and the saddle-node bifurcations have are eliminated, which would take place in the resonance responses. By analyzing the bifurcation function associated with the corresponding frequency-response equation and the Jacobi matrix, the gain of the feedback control is determined. A parametrically excited oscillator with strong nonlinearity including van der Pol and Duffing type is studied for static bifurcations. The applicable range of the MLP method is extended to 1/2 subharmonic resonance systems and the bifurcation equation of a strongly nonlinear oscillator which is transformed into a small parameter system is determined by using the multiple scales method. On the basis of the singularity theory, the transition set and the bifurcation diagram in various regions of the parameter plane are analyzed. The parametrically excited pendulum with the linear and nonlinear feedback is found which validates that the controller can be designed in the engineering applications.4. The different nonlinear parametric feedback control including time is used to control the bifurcations in various nonlinear dynamical systems, such as the forced Duffing system, the Duffing-van der Pol system and the parametrically excited system. The designed controllers are testified theoretically to eliminate the saddle-node bifurcation successfully in the case of primary and superharmonic resonances and to reject the steady-state response in the case of subharmonic. The results of the numerical simulations show that the proposed feedback control method is quite effective to achieve the goal of bifurcation control.5. The coupled oscillators are studied, and the control law is obtained in the case of the one-to-one internal resonance. An approximate solution for the nonlinear differential equations is got by using the method of multiple scales. And the bifurcation analysis and the performance of the control strategy are investigated theoretically. By adjusting the control parameter, the high-amplitude periodic and chaotic motions are removed.In this paper, the innovative thinking is that the bifurcation control theory is used to investigate the nonlinear dynamical systems, which enriches the nonlinear dynamics theory and expands the nonlinear control theory. The creative things are as follows: The open-loop control approach is used to control the equilibrium bifurcation. The state feedback is extended to control
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