一类多自由度机械系统的时滞反馈镇定
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摘要
近二十年来,在计算机、传感器和驱动器等技术的带动下,主动控制技术得到了快速发展。然而,随着人们对控制速度和性能要求的不断提高,控制回路中不可避免的时滞成为制约主动控制技术发展和应用的一个重要因素。另一方面,时滞并非总是不利因素,若主动巧妙地利用时滞则可在某些情况下改善控制系统的性能。此外,非线性时滞系统具有丰富的动力学现象,在动力学分析的基础上设计控制器可获得一些独特的控制效果。所以,时滞系统的动力学与控制研究具有重要的科学意义和工程应用价值。
本文以一类具有多个不稳定平衡点的多自由度机械系统作为主要对象,研究时滞反馈镇定方法和相关的非线性动力学问题,进而深化对时滞系统动力学与控制的认识。论文的主要研究工作与学术贡献如下:
1.根据研究目的,设计并研制了一套可用于时滞动力学与控制研究的小车～二级摆控制实验系统。通过理论、数值和实验手段研究了数字滤波器对该系统稳定性和分叉的影响。实验发现,若滤波器设计指标设置过高会因引入过长的群时延量而导致系统平衡点失稳,发生Hopf分叉。在理论和数值分析中,将滤波器简化为纯滞后环节,得到连续的时滞动力系统,进而可分析其稳定性和平衡点失稳后的Hopf分叉。理论、数值和实验结果的一致性肯定了将数字滤波器等效为纯滞后环节的合理性。
2.对现有的几种基于状态变换的时滞线性二次型(LQ)控制方法做了简要的介绍,并在小车～二级摆控制实验中尝试使用时滞LQ控制方法避免滤波器群时延引起的闭环系统失稳。实验的成功证实了控制方法的有效性,并说明了将滤波器简化为时滞环节的合理性。
3.揭示了针对输入时滞系统的一种连续状态变换和一种离散状态变换之间的比例关系。在研究两者关系的过程中,提出了一种新的连续时滞LQ控制方法。在该方法设计的控制律中,反馈增益矩阵不随着时滞量的变化而改变,为控制含时变时滞的系统提供了可能。随后探讨了新方法在慢变输入时滞系统中的应用。
4.提出了一套通过位移和滞后位移(PDP)反馈来镇定多自由度线性无阻尼机械系统的方法。对于全驱动系统,先通过模态解耦和单自由度系统的稳定区域图完成对应每个模态自由度的控制器设计,然后按照反解耦过程重构物理空间的反馈控制器。对于欠驱动系统,提出一种两步控制策略:先设计位移反馈控制,将闭环系统极点配置到虚轴上;再基于非线性特征值灵敏度分析设计位移和滞后位移差分反馈控制,使闭环系统极点移动到左半复平面。数值仿真演示了PDP反馈控制器的设计过程并验证了其有效性。
Last two decades have witnessed a rapid development of active control of various mechanical systems, owing to the recent advances in computing, sensing and driving technology. However, with increasingly strict requirements for control speed and system performance, the unavoidable time delays in control loop have become a severe limitation to the development and application of active control. On the other hand, several studies have shown that the voluntary introduction of time delays can also benefit the control. Moreover, the nonlinear time-delay systems exhibit rich interesting dynamic behaviors, which can be employed to design controls to achieve some special kinds of motion. Therefore, dynamics and control of time-delay systems are of great scientific significance and practical value.
This dissertation mainly focuses on the stabilization of mechanical systems of multiple degrees of freedom with multiple unstable equilibrium points via delayed feedback and the conrresponding nonlinear dynamics, so as to deepen understanding of dynamics and control of time-delay systems. The contents and contribution of the dissertation are as follows.
1. According to the research objective, an experimental setup of a double pendulum on a cart is designed and established to study the dynamics and control of mechanical system with delayed feedback. Effects of digital filters on the stability and dynamic behaviors of the controlled double pendulum are studeied theoretically, numerically and experimentally. The experiment results show that an over demanding selection of the filter specifications will lead to the instability of the closed-loop system through a Hopf bifurcation, because the group delay of the filter exceeds a critical value. In the theoretical and numerical analysis, the filters are modeled into the dynamic equation of closed loop system as the components of pure time delays. This simplification yields a continuous time-delay system so that one can make not only the stability analysis, but also the Hopf bifurcation analysis. The experimental results show a good agreement with the theoretical and numerical results, and positively confirm the simplification of the digital filters as the components of pure time delays.
2. A brief review on three existing state-transformation based delayed linear quadratic (LQ) control methods is presented. A discrete delayed LQ controller is successfully applied to the stabilization experiment of the double pendulum system to avoid the instablility induced by the group delay of a digital filter. The experimental results indicate the effectiveness of the control methods and support the adequacy of simplifying the digital filter as a time delay component.
3. The relations between a continous transformation and a discrete transformation for the system with an input delay are revealed and inspired by which, a new continuous delayed LQ control is proposed. The new control method has a very interesting property that the feedback gain matrix keeps the same for different values of an input delay. Based on this property, the new delayed LQ control method, intended to deal with the dynamic system with a constant delay, can also be applied to the dynamic system with a slowly time-varying input delay.
4. A systematic approach to stabilizing a kind of linear undamped systems of multiple degrees of freedom by using both position and delayed position (PDP) feedbacks is proposed. For the fully-actuated system, the approach enables one to complete the design of controller directly through the use of modal decoupling and a stability chart. For the under-actuated system, the approach includes two steps. The first step is to move all the eigenvalues of the system on the imaginary axis of the complex plane by using a position feedback, and the second step is to drag all the eigenvalues of the system to the open left half of the complex plane through the use of a delayed position feedback, which can be determined on the basis of sensitivity analysis of eigenvalues. Two illustrative examples well demonstrate the design procedure of PDP feedback controllers and their efficacy.
本文以一类具有多个不稳定平衡点的多自由度机械系统作为主要对象,研究时滞反馈镇定方法和相关的非线性动力学问题,进而深化对时滞系统动力学与控制的认识。论文的主要研究工作与学术贡献如下:
1.根据研究目的,设计并研制了一套可用于时滞动力学与控制研究的小车～二级摆控制实验系统。通过理论、数值和实验手段研究了数字滤波器对该系统稳定性和分叉的影响。实验发现,若滤波器设计指标设置过高会因引入过长的群时延量而导致系统平衡点失稳,发生Hopf分叉。在理论和数值分析中,将滤波器简化为纯滞后环节,得到连续的时滞动力系统,进而可分析其稳定性和平衡点失稳后的Hopf分叉。理论、数值和实验结果的一致性肯定了将数字滤波器等效为纯滞后环节的合理性。
2.对现有的几种基于状态变换的时滞线性二次型(LQ)控制方法做了简要的介绍,并在小车～二级摆控制实验中尝试使用时滞LQ控制方法避免滤波器群时延引起的闭环系统失稳。实验的成功证实了控制方法的有效性,并说明了将滤波器简化为时滞环节的合理性。
3.揭示了针对输入时滞系统的一种连续状态变换和一种离散状态变换之间的比例关系。在研究两者关系的过程中,提出了一种新的连续时滞LQ控制方法。在该方法设计的控制律中,反馈增益矩阵不随着时滞量的变化而改变,为控制含时变时滞的系统提供了可能。随后探讨了新方法在慢变输入时滞系统中的应用。
4.提出了一套通过位移和滞后位移(PDP)反馈来镇定多自由度线性无阻尼机械系统的方法。对于全驱动系统,先通过模态解耦和单自由度系统的稳定区域图完成对应每个模态自由度的控制器设计,然后按照反解耦过程重构物理空间的反馈控制器。对于欠驱动系统,提出一种两步控制策略:先设计位移反馈控制,将闭环系统极点配置到虚轴上;再基于非线性特征值灵敏度分析设计位移和滞后位移差分反馈控制,使闭环系统极点移动到左半复平面。数值仿真演示了PDP反馈控制器的设计过程并验证了其有效性。
Last two decades have witnessed a rapid development of active control of various mechanical systems, owing to the recent advances in computing, sensing and driving technology. However, with increasingly strict requirements for control speed and system performance, the unavoidable time delays in control loop have become a severe limitation to the development and application of active control. On the other hand, several studies have shown that the voluntary introduction of time delays can also benefit the control. Moreover, the nonlinear time-delay systems exhibit rich interesting dynamic behaviors, which can be employed to design controls to achieve some special kinds of motion. Therefore, dynamics and control of time-delay systems are of great scientific significance and practical value.
This dissertation mainly focuses on the stabilization of mechanical systems of multiple degrees of freedom with multiple unstable equilibrium points via delayed feedback and the conrresponding nonlinear dynamics, so as to deepen understanding of dynamics and control of time-delay systems. The contents and contribution of the dissertation are as follows.
1. According to the research objective, an experimental setup of a double pendulum on a cart is designed and established to study the dynamics and control of mechanical system with delayed feedback. Effects of digital filters on the stability and dynamic behaviors of the controlled double pendulum are studeied theoretically, numerically and experimentally. The experiment results show that an over demanding selection of the filter specifications will lead to the instability of the closed-loop system through a Hopf bifurcation, because the group delay of the filter exceeds a critical value. In the theoretical and numerical analysis, the filters are modeled into the dynamic equation of closed loop system as the components of pure time delays. This simplification yields a continuous time-delay system so that one can make not only the stability analysis, but also the Hopf bifurcation analysis. The experimental results show a good agreement with the theoretical and numerical results, and positively confirm the simplification of the digital filters as the components of pure time delays.
2. A brief review on three existing state-transformation based delayed linear quadratic (LQ) control methods is presented. A discrete delayed LQ controller is successfully applied to the stabilization experiment of the double pendulum system to avoid the instablility induced by the group delay of a digital filter. The experimental results indicate the effectiveness of the control methods and support the adequacy of simplifying the digital filter as a time delay component.
3. The relations between a continous transformation and a discrete transformation for the system with an input delay are revealed and inspired by which, a new continuous delayed LQ control is proposed. The new control method has a very interesting property that the feedback gain matrix keeps the same for different values of an input delay. Based on this property, the new delayed LQ control method, intended to deal with the dynamic system with a constant delay, can also be applied to the dynamic system with a slowly time-varying input delay.
4. A systematic approach to stabilizing a kind of linear undamped systems of multiple degrees of freedom by using both position and delayed position (PDP) feedbacks is proposed. For the fully-actuated system, the approach enables one to complete the design of controller directly through the use of modal decoupling and a stability chart. For the under-actuated system, the approach includes two steps. The first step is to move all the eigenvalues of the system on the imaginary axis of the complex plane by using a position feedback, and the second step is to drag all the eigenvalues of the system to the open left half of the complex plane through the use of a delayed position feedback, which can be determined on the basis of sensitivity analysis of eigenvalues. Two illustrative examples well demonstrate the design procedure of PDP feedback controllers and their efficacy.
引文
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