基于Frenkel-Kontorova模型的控制系统设计与稳定性分析
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摘要
Frenkel-Kontorova模型是一类非线性耦合动态系统,有着广泛的应用价值与理论指导意义。结合目前FK模型广泛的实际应用,阐明了研究FK模型的重要意义。对国内外关于FK模型的研究现状和本文的研究问题进行了分析与总结。分别从定量和定性两种角度,对非线性系统的研究方法进行了总结。给出了本文需要用到的理论工具。
从实际应用背景和制约条件出发,提出控制问题并针对该问题设计输出反馈控制律。在所设计的输出反馈控制律作用下,分目标速度等于零与不等于零两种情况,分别分析了系统的平衡点的稳定性和状态有界的充分条件,并讨论了在摩擦系数足够大这一特殊情况下系统的状态。并通过Matlab仿真说明本章所得结论的有效性和正确性。
针对Morse型Frenkel-Kontorova模型的缺陷进行改进以确保粒子序列的不变性。在原设计的输出状态反馈控制律作用下,分别利用庞加莱影射和李雅普诺夫直接方法对改进后的Morse型Frenkel-Kontorova模型进行研究。分析了状态混沌与系统摩擦系数及目标速度之间的关系,并分目标速度等于零、不等于零两种情况,分别讨论了平衡点的稳定性、状态有界的充分条件以及运动状态轨迹的估计等问题。并通过数值算例验证了所得结果的有效性和正确性。
利用李雅普诺夫直接方法和鲁棒控制的思想,设计了非线性输出反馈控制律,使得系统的平均速度精确收敛于目标值。分别给出了在该非线性输出反馈控制律的作用下,改进后的Morse型Frenkel-Kontorova模型中粒子运动状态有界与经典Frenkel-Kontorova模型中粒子运动状态同步的充分条件。通过仿真实例,将所设计的非线性输出反馈控制律与以往的进行比较以说明本控制律的优势,并分别验证在该非线性输出反馈控制律的作用下,改进后的Morse型Frenkel-Kontorova模型中粒子运动状态有界与经典Frenkel-Kontorova模型中粒子运动状态同步的充分条件。
Frenkel-Kontorova model is a class of nonlinear coupled dynamical system and has broadened applications. With the broadened applications of Frenkel-Kontorova model, the importance of research for this model is stated. Current research of Frenkel-Kontorova model and the problem studied here are summarized and analyzed. Different methods for researching nonlinear dynamical systems are summarized, including both numerical and analytical methods. Theory tools used in this research are given.
According to the background of applications and constraints, a control problem is proposed and an output feedback control law is designed to solve this control problem. Under the designed output feedback control law, the dynamical system is divided into two cases; targeted velocity is zero and non-zero, to study the stability of equilibrium and sufficient conditions for boundedness of states, respectively. Moreover, for the special case that damping coefficient is sufficiently large, the states of the researched dynamical system are studied. Simulations are given to show the effectiveness and validity of the derived results with Matlab.
To conquer a drawback of Morse-type Frenkel-Kontorova model, a modification for this model is proposed, which assures the sequence of particles in the model is unchanged. Under the designed output feedback control law, dynamical features of the modified model are studied using Poincare mapping and Lyapunov methods, respectively. The relationship between friction coefficient and range of chaos in targeted velocity is revealed. And the modified dynamical system is divided into two cases, which are zero targeted velocity and non-zero targeted velocity, to study the stability of equilibrium, sufficient conditions for the boundedness of states and the estimation of states. Computer simulation shows the validity of derived results.
For Frenkel-Kontorova model, a continuous nonlinear output feedback control law is designed, using the idea of robustness control and direct Lyapunov method, such that average velocity tends to targeted value precisely. Under the continuous nonlinear output feedback control law, the sufficient conditions for bounded states of modified Morse-type Frenkel-Kontorova model and synchronization of classical Frenkel-Kontorova model are studied, respectively. Numerical examples show the effectiveness and advantage of the designed continuous nonlinear control law and derived sufficient conditions for bounded states of modified Morse-type Frenkel-Kontorova model and synchronization of classical Frenkel-Kontorova model.
从实际应用背景和制约条件出发,提出控制问题并针对该问题设计输出反馈控制律。在所设计的输出反馈控制律作用下,分目标速度等于零与不等于零两种情况,分别分析了系统的平衡点的稳定性和状态有界的充分条件,并讨论了在摩擦系数足够大这一特殊情况下系统的状态。并通过Matlab仿真说明本章所得结论的有效性和正确性。
针对Morse型Frenkel-Kontorova模型的缺陷进行改进以确保粒子序列的不变性。在原设计的输出状态反馈控制律作用下,分别利用庞加莱影射和李雅普诺夫直接方法对改进后的Morse型Frenkel-Kontorova模型进行研究。分析了状态混沌与系统摩擦系数及目标速度之间的关系,并分目标速度等于零、不等于零两种情况,分别讨论了平衡点的稳定性、状态有界的充分条件以及运动状态轨迹的估计等问题。并通过数值算例验证了所得结果的有效性和正确性。
利用李雅普诺夫直接方法和鲁棒控制的思想,设计了非线性输出反馈控制律,使得系统的平均速度精确收敛于目标值。分别给出了在该非线性输出反馈控制律的作用下,改进后的Morse型Frenkel-Kontorova模型中粒子运动状态有界与经典Frenkel-Kontorova模型中粒子运动状态同步的充分条件。通过仿真实例,将所设计的非线性输出反馈控制律与以往的进行比较以说明本控制律的优势,并分别验证在该非线性输出反馈控制律的作用下,改进后的Morse型Frenkel-Kontorova模型中粒子运动状态有界与经典Frenkel-Kontorova模型中粒子运动状态同步的充分条件。
Frenkel-Kontorova model is a class of nonlinear coupled dynamical system and has broadened applications. With the broadened applications of Frenkel-Kontorova model, the importance of research for this model is stated. Current research of Frenkel-Kontorova model and the problem studied here are summarized and analyzed. Different methods for researching nonlinear dynamical systems are summarized, including both numerical and analytical methods. Theory tools used in this research are given.
According to the background of applications and constraints, a control problem is proposed and an output feedback control law is designed to solve this control problem. Under the designed output feedback control law, the dynamical system is divided into two cases; targeted velocity is zero and non-zero, to study the stability of equilibrium and sufficient conditions for boundedness of states, respectively. Moreover, for the special case that damping coefficient is sufficiently large, the states of the researched dynamical system are studied. Simulations are given to show the effectiveness and validity of the derived results with Matlab.
To conquer a drawback of Morse-type Frenkel-Kontorova model, a modification for this model is proposed, which assures the sequence of particles in the model is unchanged. Under the designed output feedback control law, dynamical features of the modified model are studied using Poincare mapping and Lyapunov methods, respectively. The relationship between friction coefficient and range of chaos in targeted velocity is revealed. And the modified dynamical system is divided into two cases, which are zero targeted velocity and non-zero targeted velocity, to study the stability of equilibrium, sufficient conditions for the boundedness of states and the estimation of states. Computer simulation shows the validity of derived results.
For Frenkel-Kontorova model, a continuous nonlinear output feedback control law is designed, using the idea of robustness control and direct Lyapunov method, such that average velocity tends to targeted value precisely. Under the continuous nonlinear output feedback control law, the sufficient conditions for bounded states of modified Morse-type Frenkel-Kontorova model and synchronization of classical Frenkel-Kontorova model are studied, respectively. Numerical examples show the effectiveness and advantage of the designed continuous nonlinear control law and derived sufficient conditions for bounded states of modified Morse-type Frenkel-Kontorova model and synchronization of classical Frenkel-Kontorova model.
引文
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