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自抗扰控制框架下的摩擦力振动分析

ANALYSIS OF FRICTION INDUCED VIBRATION UNDER THE ACTIVE DISTURBANCE REJECTION CONTROL FRAMEWORK

  • 摘要: 自抗扰控制(active disturbance rejection control, ADRC)是一种具有两自由度控制结构的工程化方法, 由于其能够直观有效地处理多种扰动, 近些年来在许多机电系统上得到了成功应用. 当采用ADRC对带有摩擦力的机电系统进行调节时, 可能会产生极限环振动. 目前, 还没有ADRC框架下摩擦力振动精确分析的相关工作. 因此, 本文采用非线性动力学系统的分析工具对这一问题进行研究. 首先, 考虑两种典型摩擦力模型, 静态切换模型和动态LuGre 模型, 对一类二阶运动系统设计不同阶次的ADRC, 得到控制器的等效形式, 并揭示出与比例积分微分(proportional-integral-derivative, PID)控制之间的联系. 然后, 采用打靶法结合拟弧长延拓方法求解系统中的极限环, 并根据Floquet理论判断极限环的稳定性、可能出现的分岔以及分岔类型. 此外, 通过雅克比矩阵和近似数值方法对系统平衡点集的局部稳定性进行了分析. 最后, 通过数值计算研究了摩擦力模型和参数、ADRC阶次和参数对极限环和平衡点集的影响. 计算结果表明, 决定摩擦力Stribeck效应负斜率的参数\beta作用较大. 当\beta>1时, 两种摩擦力模型下的闭环系统呈现出相同的特性, 极限环会出现环面折叠分岔(cyclic fold bifurcation, CFB)且平衡点集是局部稳定的. 然而当\beta<1时, 两种闭环系统呈现出完全不同的特性. 此外, 不同阶次的ADRC在极限环的存在性和稳定性、平衡点集的稳定性上面的结论是相同的, 而低阶次的ADRC能够更好地解决摩擦力补偿和稳定鲁棒性之间的矛盾问题. 这些结论对实际现象的理解、ADRC阶次的选择以及参数整定提供了一定指导.

     

    Abstract: Active disturbance rejection control(ADRC) is a practical control method with a two-degree-of-freedom structure. Due to its capability of handling multifarious disturbances in a straightforward and effective manner, ADRC has been successfully applied to many mechanical systems. However, the limit cycle vibration may be induced when employing the ADRC for mechanical systems with friction. At present, there is no precise analysis work about the friction induced vibration under the ADRC framework. Therefore, this paper investigates this problem by using the analysis tools of nonlinear dynamic systems. First, two representative friction models, static switch model and dynamic LuGre model, respectively, are considered, and active disturbance rejection controllers of different orders are designed for a class of second-order motion systems. Equivalent forms of the controllers are obtained and their relationships with the proportional-integral-derivative(PID) controller are revealed. Then, the limit cycle is calculated by using the shooting method combined with the pseudo arc-length continuation approach. Based on the Floquet theory, the stability, occurrence and type of bifurcation of the limit cycle can be determined. In addition, the local stability of the equilibrium points is analyzed based on the Jacobian matrix and approximate numerical method. Finally, the effects of the model and parameter of friction, the order and parameters of the ADRC on the limit cycle are investigated by numerical calculations. As shown by the calculation results, the parameter \beta, which determines the negative slope of the Stribeck effect, has a significant effect. When \beta>1, closed-loop systems with these two friction models have the same characteristics. Cyclic fold bifurcation(CFB) of the limit cycle occurs and the set of equilibrium points is locally stable. However, characteristics of these two closed-loop systems are totally different when \beta<1. As for the ADRC order, it is found that the order does not affect the conclusions in terms of the existence and stability of the limit cycle, and the stability of the set of equilibrium points. Moreover, low-order ADRC has superior performance in tackling the conflict between the friction compensation and stability robustness. These results can provide some guidelines on the understanding of practical phenomena, selection of the ADRC order, and parameter tuning.

     

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