基于变步长LMS方法的磁悬浮飞轮振动抑制

王雨楠，刘昆; Wang Yunan; Liu Kun

引 en

基于变步长LMS方法的磁悬浮飞轮振动抑制*

王雨楠
机构：
中山大学航空航天学院,广州 510275
×
，刘昆
机构：
中山大学航空航天学院,广州 510275
×

中山大学航空航天学院,广州 510275；

VIBRATION SUPPRESSION OF MAGNETIC LEVITATION FLYWHEEL BASED ON VARIABLE STEP SIZE LMS METHOD*

Wang Yunan
Affiliation：
School of Aeronautics and Astronautics ,Sun Yat-sen University,Guangzhou 510275 ,China
×
，Liu Kun
Affiliation：
School of Aeronautics and Astronautics ,Sun Yat-sen University,Guangzhou 510275 ,China
×

School of Aeronautics and Astronautics ,Sun Yat-sen University,Guangzhou 510275 ,China；

通讯作者:

刘昆,E-mail:liukun6@mail.sysu.edu.cn

中图分类号:TH133.3

文献标识码:A

文章编号:1672-6553-2022-20(3)-077-06

DOI:10.6052/1672-6553-2021-046

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参考文献
出版信息

参考文献 1

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目录contents

摘要 Abstract
关键词 Keywords
1 磁悬浮轴承系统建模
1.1 磁轴承-转子系统动力学建模
1.2 振动源建模
1.3 参数辨识
2 基于自适应LMS的主动振动控制
2.1 标准LMS算法
2.2 变步长LMS算法
3 仿真分析
4 结论
参考文献

摘要

针对磁悬浮飞轮转子位移传感器扰动引起的多频谐波振动,提出一种基于最小均方差(LMS)的变步长算法实现谐波振动的有效抑制.建立模型对磁悬浮轴承转子的位移刚度系数k_h和电流刚度系数k_i进行辨识,以位移传感器扰动引起的八倍频噪声为抑制目标,在自适应LMS算法抑制电流扰动基础上,改进一种随转子位移信号频率变化而变化的变步长因子实现自动平衡,利用广义根轨迹分析了引入补偿后系统的闭环稳定性.仿真结果表明所提出的方法能够有效实现对八倍频振动的抑制.

Abstract

To suppress multi-frequency harmonic vibration caused by displacement sensor runout in a magnetic suspension flywheel system, an adaptive variable step-size least mean square (LMS) algorithm is proposed. The model is established to identify the displacement stiffness coefficient k_h and current stiffness coefficient k_i of the magnetic bearing rotor. The octave frequency noise caused by the displacement sensor runout is taken as the suppression target. On the basis of the adaptive LMS algorithm, a modified variable step-size factor with changing frequency is improved to achieve automatic balancing. The stability of close loop system is analyzed by generalized root locus. Simulation results show that the proposed method can effectively suppress octave frequency disturbance.

关键词

主动磁悬浮轴承； LMS算法；位移传感器扰动；振动控制

Keywords

active magnetic bearing ； variable step-size LMS algorithm ； displacement sensor runout ； vibration control

引言
磁悬浮轴承相比传统轴承有许多优点,如不存在机械接触、无磨损、低功耗,在极端温度和较高速度下运行能力较好^[1].因此电磁轴承已经广泛应用于高速透平机械等工业领域以及磁悬浮飞轮等航天领域.电磁轴承有两个主要的振动源,转子质量不平衡和传感器噪声.质量不平衡是由于实际制造的材料与加工工艺限制,主轴的惯性轴与几何轴线不重合,转子高速旋转时会产生不平衡振动^[2].传感器扰动是由于传感器表面不均匀及其他非线性因素,导致在传感器测量环节引入干扰,传感器扰动主要导致谐波干扰.
对于电磁轴承的振动抑制主要有两种:一种为不平衡补偿,以“零转子位移”为目的.另一种为自动平衡,实现力或电流的最小补偿^[3].国内外学者对此作了许多研究,Herzog等^[4]提出一种具有自由极点位置的广义陷波滤波器,可以处理不同转速下的同频不平衡信号,但是不能补偿位移刚度力引起的振动.Shi等^[5]提出一种基于LMS自适应滤波算法的自适应前馈法,实现转子最小位移补偿.Liu等^[6]提出一种开环轴承力补偿方法来抑制不平衡振动,采用自适应陷波器识别位置信号中的同频位移,但是并没有考虑功率放大器的低通特性.这些方法主要针对同频振动,针对传感器谐波噪声引起的扰动问题,Setiawan等^[7]提出了一种自适应的传感器扰动补偿算法.它保证了转子几何中心的渐近稳定,并利用持续激励在线前馈消除扰动.Xu等^[8]提出了一种基于重复控制器的谐波振动力消除方法,实验结果表明该方法能有效地消除谐波电流.Cui等^[9]提出利用改进的重复控制器抑制主被动磁悬浮控制力矩陀螺中的谐波电流,试验证明了该方法具有较好的抑制性能.张激扬等^[10]基于一种级联相移陷波器对磁悬浮飞轮全转速范围内的传感器谐波扰动进行了主动抑制.Zhou等^[11]提出针对奇次谐波的改进重复控制器,消除PWM逆变器跟踪误差.
本文对磁轴承-转子系统的位移刚度系数k_h和电流刚度系数k_i进行辨识,提出利用自适应LMS算法实现对传感器扰动引起的八倍频振动进行抑制,在此基础上改进一种随转子位移信号频率变化而变化的变步长因子实现自动平衡,利用广义根轨迹分析插入自适应LMS算法后系统的稳定性,并通过仿真分析验证了该方法的振动抑制效果.
1 磁悬浮轴承系统建模
1.1 磁轴承-转子系统动力学建模
磁轴承-转子系统考虑细长转子,转子两端分别安装一个径向磁轴承,分别产生沿转子径向的电磁控制力.图1是转子受电磁力作用的示意图.
图1 转子受力分析
Fig.1 Free-body diagram of the rotor
由牛顿第二定律和动量矩定理可得到转子动力学方程:

\{\begin{matrix} m {\ddot{x}}_{g} = F_{A X} + F_{B X} \\ m {\ddot{y}}_{g} = F_{A Y} + F_{B Y} \\ J_{r} \ddot{α} + J_{p} Ω \dot{β} = - F_{A X} l_{a} + F_{B X} l_{b} \\ J_{r} \ddot{β} - J_{p} Ω \dot{α} = F_{A Y} l_{a} - F_{B Y} l_{b} \end{matrix}

(1)

式中,F_AX、F_BX为转子沿x方向的电磁力,F_AY、F_BY为转子沿y方向的电磁力,α、β分别为转子绕x、y轴旋转的角位移,Ω为转速,l为转子长度,m为转子质量.对磁轴承的磁力线性化可得:

F = k_{i} i + k_{h} x

(2)

式中,k_i为电流刚度系数,k_h为位移刚度系数.
1.2 振动源建模
磁悬浮飞轮悬浮控制系统中存在多个干扰源,不平衡干扰力是在转子动力学环节引入的振动干扰.受到加工工艺和安装误差等因素影响,传感器测量环节和磁力产生环节也会引入干扰.传感器各环节引入的噪声可以等效为测量面的不规则,可以称之为传感器扰动(Sensor Runout),如图2所示.
图2 传感器扰动示意图
Fig.2 Sensor runout diagram
在转子旋转时测量面的不规则会给悬浮控制系统引入周期干扰,这些干扰可以用转子同频和倍频的傅里叶级数建模.传感器扰动在传感器坐标下表示为^[12]:

X_{s d} = [\begin{matrix} x_{s d a x} \\ x_{s d b x} \\ x_{s d a y} \\ x_{s d b y} \end{matrix}] = [\begin{matrix} \sum_{i = 1}^{n} x_{a i} c o s (i Ω t + θ_{a i}) \\ \sum_{i = 1}^{n} x_{b i} c o s (i Ω t + θ_{b i}) \\ \sum_{i = 1}^{n} x_{a i} s i n (i Ω t + θ_{a i}) \\ \sum_{i = 1}^{n} x_{b i} s i n (i Ω t + θ_{b i}) \end{matrix}]

(3)

式中,i为谐波次数, $i = 1,2, \dots n; x_{a i} 、 x_{b i}$ 为磁轴承A、B两端传感器输出信号的第i次谐波的幅值,θ_bi分别为相应的初始相位.本文主要考虑传感器扰动的八次谐波.
1.3 参数辨识
对位移刚度系数k_hx和电流刚度系数k_ix进行辨识.当转子水平放置在试验台上时,动力学方程为:

\{\begin{matrix} m \ddot{x} (t) = k_{h x} x (t) + k_{i x} i_{x} (t) \\ m \ddot{y} (t) = k_{h y} y (t) + k_{i y} i_{y} (t) \end{matrix}

(4)

当转子稳定悬浮时,认为:

\{\begin{matrix} E [\ddot{x} (t)] = 0 \\ E [\ddot{y} (t)] = 0 \end{matrix}

(5)

对式(4)两边取均值

\{\begin{matrix} 0 = k_{h x} E [x_{h} (t)] + k_{i x} E [i_{h x} (t)] \\ 0 = k_{h y} E [y_{h} (t)] + k_{i y} E [i_{h y} (t)] \end{matrix}

(6)

根据式(6)可得:

\{\begin{matrix} \frac{k_{h x}}{k_{i x}} = - \frac{E [i_{h x} (t)]}{E [x_{h} (t)]} \\ \frac{k_{h y}}{k_{i y}} = - \frac{E [i_{h y} (t)]}{E [y_{h} (t)]} \end{matrix}

(7)

对于X通道,测量得到 $E [i_{h x} (t)]$ 和 $E [x_{h} (t)]$ 后,可以计算得到电流刚度系数和位移刚度系数的比值,Y通道与之相同.当转子垂直安装时,动力学方程:

\{\begin{matrix} m \ddot{x} (t) = k_{h x} x (t) + k_{i x} i_{x} (t) + m g c o s θ \\ m \ddot{y} (t) = k_{h y} y (t) + k_{i y} i_{y} (t) + m g s i n θ \end{matrix}

(8)

当转子稳定悬浮时,认为:

\{\begin{matrix} E [\ddot{x} (t)] = 0 \\ E [\ddot{y} (t)] = 0 \end{matrix}

(9)

对式(8)两边取均值

\begin{matrix} 0 = k_{h x} E [x_{v} (t)] + k_{i x} E [i_{v x} (t)] + m g c o s θ \\ 0 = k_{h y} E [y_{v} (t)] + k_{i y} E [i_{v y} (t)] + m g s i n θ \end{matrix}

(10)

图3 垂直悬浮时转子受力示意图
Fig.3 Force diagram of the rotor in vertical suspension
由于存在控制作用,偏置位移产生的位移刚度力与水平放置时相同,而控制电流中额外包含了克服重力而多出的部分.由式(10)减去式(6)得:

\{\begin{matrix} 0 = k_{h x} E [x_{v} (t) - x_{h} (t)] + k_{i x} E [i_{v x} (t) - i_{h x} (t)] + m g c o s θ \\ 0 = k_{h y} E [y_{v} (t) - y_{h} (t)] + k_{i y} E [i_{v y} (t) - i_{h y} (t)] + m g s i n θ \end{matrix}

(11)

受PID控制器中积分项作用,给定位移偏置指令后,最终稳态的位移偏差为0.令给定的偏置位移指令相等,有:

\{\begin{matrix} 0 = k_{i x} (E [i_{v x} (t)] - E [i_{h x} (t)]) + m g c o s θ \\ 0 = k_{i y} (E [i_{v y} (t)] - E [i_{h y} (t)]) + m g s i n θ \end{matrix}

(12)

可得

\{\begin{matrix} k_{i x} = - \frac{m g c o s θ}{E [i_{v x} (t)] - E [i_{h x} (t)]} \\ k_{i y} = - \frac{m g s i n θ}{E [i_{v y} (t)] - E [i_{h y} (t)]} \end{matrix}

(13)

测量得到 $[i_{v x} (t)]$ 、 $E [i_{h x} (t)]$ 、m和θ,可以计算得到k_ix.为了提高精度,可以多点测量求均值,根据式(7)和式(13)可以得到k_hx.
2 基于自适应LMS的主动振动控制
2.1 标准LMS算法
图4给出了加入LMS自适应算法的磁轴承-转子系统悬浮控制结构框图.可以看出LMS滤波器加在了悬浮控制器之前的位置.图5则给出了LMS算法结构图.
图4 加入LMS自适应算法的磁轴承结构图
Fig.4 Structure diagram of magnetic bearing based on LMS adaptive feedforward
图5 LMS算法结构图
Fig.5 Structure of LMS algorithm
定义 $X (t) = {[s i n (ω_{0} t) c o s (ω_{0} t)]}^{T}$ 为输入信号向量, $W (t) = {[w_{L 1} (t) w_{L 2} (t)]}^{T}$ 为权向量,d(t) 为需要滤除的信号,Y(t) 为算法输入信号,e(t) 为误差信号,ω₀为滤波角频率,μ为步长.由图5可以得到LMS算法的时域方程为:

\{\begin{matrix} e (t) = d (t) - Y (t) = d (t) - X^{T} (t) W (t) \\ Y (t) = Y_{1} (t) w_{L 1} (t) + Y_{2} (t) w_{L 2} (t) \\ W_{k + 1} (t) = W_{k} (t) + μ e (t) X (t) \end{matrix}

(14)

由文献^[13]可知从d(t)到 e(t)的闭环传递函数为:

H (z) = \frac{z^{2} - 2 z c o s ω_{0} + 1}{z^{2} - 2 (1 - μ) z c o s ω_{0} + 1 - 2 μ}

(15)

式中,ω₀=2πf₀T,T为系统采样周期.为保证收敛,步长要满足条件:

0 < μ < \frac{1}{λ_{m a x}}

(16)

该函数零点为 $z_{0} = e^{\pm j ω_{0}}$ ,当d(t) 的频率f等于算法输入信号频率f₀时,有H(z)=0,可以实现对d(t) 的滤波.
2.2 变步长LMS算法
固定步长LMS算法只能对某些固定转动频率具有好的补偿效果,对于不同的旋转频率,需要选择合适的步长因子,才能有效地滤除振动干扰.文献^[14]分析了几种变步长LMS算法的滤波效果,提出一种变步长改进算法,步长因子为

μ_{v} = \frac{f_{r} μ_{1}}{1 + γ f_{r} μ_{1} e (k)}

(17)

式中,f_r为实时监测到的转子转动频率, μ₁为一个很小的正参数,γ为加权系数,一般在20-30之间.但是式子中有两个调节参数,对于高速旋转的转子,需要调节的参数越多,在试验中的不利因素越多,因此文献^[15]在此基础上提出改进的变步长LMS算法,步长因子为:

μ_{v 1} = \frac{f μ_{1}^{'}}{1 - | e (k) |}

(18)

$μ_{1}^{'}$ 是一个很小的值,只有一个需要调节的参数.
3 仿真分析
搭建SIMULINK模型对上述算法进行仿真.表1给出了仿真中用到的系统参数.
表1 模型参数
Table1 Parameter of the model
采用PID控制器,其传递函数的具体形式为:

G_{c} = k_{P} + k_{I} \frac{1}{s} + k_{D} \frac{N s}{s + N}

(19)

式中,k_P是比例系数, k_I是积分系数, k_D 是微分系数,N表示微分器的深度.功放模型简化为一阶惯性环节,即:

G_{w} (s) = \frac{k_{w}}{τ s + 1}

(20)

式中,k_w为功放增益,τ为功放时间常数.
转子系统传递函数为:

P (s) = \frac{k_{i}}{m s^{2} - k_{h}}

(21)

式中,m为转子质量,k_i 为电流刚度系数,k_h为位移刚度系数.位移传感器用比例增益表示:

G_{s} (s) = k_{s}

(22)

绘出加入基于LMS陷波器的磁悬浮轴承系统转速根轨迹的主导零极点如图6所示,从图中可知系统在低转速时存在不稳定极点.可以考虑在低频时不加补偿,转速达到一定值后再加入补偿.
图6 根轨迹图
Fig.6 Diagram of root locus
考虑正弦信号x_sd=Acos(8Ωt+θ).可以看出,该信号的频率是转速的8倍,将其作为传感器扰动,加入系统后进行仿真.在仿真开始前3s内,不启动LMS算法,3s时启动LMS算法,仿真结果如图7和图8所示,仿真中的转速Ω=2000rpm.
图7 电流收敛过程时间关系曲线
Fig.7 The simulation results of the coil currents
图8 位移收敛过程时间关系曲线
Fig.8 The simulation results of the displace ment signal
从仿真结果可以看出,对于传感器扰动引起的八倍频扰动,加入LMS反馈补偿后,转子的位移和电流都迅速收敛到极低值地扰动进行了抑制.
4 结论
本文对磁悬浮转子存在传感器谐波噪声情况下的振动控制问题进行了研究,通过引入自适应LMS反馈补偿,减小了电流和位移信号中的八倍频扰动,提出了一种改进型的随转速变化的变步长因子,能够实时对变转速的转子实现振动抑制,通过仿真研究验证了该方法具有较好的抑制效果.但是本文只针对八倍频扰动进行了研究,对于其他倍频扰动的同步抑制还有待研究.
参考文献
- [1] Zheng S Q,Han B C.Investigations of an integrated angular velocity measurement and attitude control system for spacecraft using magnetically suspended double-gimbal CMGs.Advances in Space Research,2013,51(12):2216~2228
- [2] Schweitzer G,Maslen H.Magnetic bearings:theory,design,and application to rotating machinery.New York:Springer,2009
- [3] Tang L,Chen Y Q.Model development and adaptive imbalance vibration control of magnetic suspended system.Acta Astronautica,2009,65(9):1506~1514
- [4] Herzog R,Buhter P,Gahler C,et al.Unbalance compensation using generalized notch filters in the multivariable feedback of magnetic bearings.IEEE Transactions on Control Systems Technology,1996,4(5):580~586
- [5] Shi J,Zmood R,Qin L.The direct method for adaptive feed-forward vibration control in magnetic bearing systems.In:Proceedings of the seventh International Conference on Control,Automation,Robotics and Vision,Singapore,2002:675~680
- [6] 刘彬,房建成,刘刚,等.磁悬浮飞轮不平衡振动控制方法与试验研究.机械工程学报,2010,46(12):188~194;(Liu B,Fang J C,Liu G,et al.Unbalance vibration control and experiment research of magnetically suspended flywheels.Journal of Mechanical Engineering,2010,46(12):188~194(in Chinese))
- [7] Setiawan J D,Mukherjee R,Maslen E H.Synchronous sensor runout and unbalance compensation in active magnetic bearings using bias current excitation.Journal of Dynamic Systems Measurement & Control,2002,124(1):14~24
- [8] Xu X B,Fang J C,Liu G,et al.Model development and harmonic current reduction in active magnetic bearing systems with rotor imbalance and sensor runout.Journal of Vibration and Control,2015,21(13):2520~2535
- [9] Cui P L,Li S,Zhao G Z,et al.Suppression of harmonic current in active-passive magnetically suspended using improved repetitive controller.IEEE/ASME Transactions on Mechatronics,2016,21(4):2132~2141
- [10] 张激扬,陈宗基,刘虎.磁悬浮飞轮位移传感器谐波扰动的主动抑制.宇航学报,2015,11:1289~1295;(Zhang J Y,Chen Z J,Liu H.Active control of multi-frequency vibration caused by displacement sensor runout in magnetic suspension flywheel.Journal of Astronautics.2015,11:1289~1295(in Chinese))
- [11] Zhou K,Low K S,Wang D,et al.Zero-phase odd-harmonic repetitive controller for a single-phase pwminverter.IEEE Transactions on Power Electronics,2006,21(1):193~201
- [12] 魏静波.磁悬浮微框架动量轮控制技术研究[博士学位论文].长沙:国防科学技术大学,2016;(Wei J B.Study on control technologies of magnetically suspended gimballing momentum wheel [Ph.D Thesis].Changsha:National University of Defense Technology,2016(in Chinese))
- [13] 沈福民.自适应信号处理.西安:西安电子科技大学出版社,2001;(Shen F M.Adaptive signal processing.Xi′an:Xi′an University of Electronic Science and Technology Press,2001(in Chinese))
- [14] Chen S L,Lin S Y,Toh C S.Adaptive unbalance compensation for a three-pole active magnetic bearing system.IEEE Transactions on Industrial Electronics,2020,67(3):2097~2106
- [15] 陈琪,刘刚,郑世强.基于自适应变步长最小均方差算法的磁悬浮电动机自动平衡方法.机械工程学报,2015,51(15):119~127;(Chen Q,Liu G,Zhen S Q.Automatic balancing for magnetically suspended high-speed motor based on adaptive variable step-size LMS algorithm.Journal of Mechanical Engineering,2015,51(15):119~127(in Chinese))

基本信息

中图分类号: TH133.3
文献标识码: A
DOI: 10.6052/1672-6553-2021-046
文章编号: 1672-6553-2022-20(3)-077-06

基金信息

深圳基础研究(重点项目)资助项目(JCYJ20200109142205924)；

引用信息

稿件历史

收稿日期: 2021-04-13

参考文献

[1] Zheng S Q,Han B C.Investigations of an integrated angular velocity measurement and attitude control system for spacecraft using magnetically suspended double-gimbal CMGs.Advances in Space Research,2013,51(12):2216~2228
[2] Schweitzer G,Maslen H.Magnetic bearings:theory,design,and application to rotating machinery.New York:Springer,2009
[3] Tang L,Chen Y Q.Model development and adaptive imbalance vibration control of magnetic suspended system.Acta Astronautica,2009,65(9):1506~1514
[4] Herzog R,Buhter P,Gahler C,et al.Unbalance compensation using generalized notch filters in the multivariable feedback of magnetic bearings.IEEE Transactions on Control Systems Technology,1996,4(5):580~586
[5] Shi J,Zmood R,Qin L.The direct method for adaptive feed-forward vibration control in magnetic bearing systems.In:Proceedings of the seventh International Conference on Control,Automation,Robotics and Vision,Singapore,2002:675~680
[6] 刘彬,房建成,刘刚,等.磁悬浮飞轮不平衡振动控制方法与试验研究.机械工程学报,2010,46(12):188~194;(Liu B,Fang J C,Liu G,et al.Unbalance vibration control and experiment research of magnetically suspended flywheels.Journal of Mechanical Engineering,2010,46(12):188~194(in Chinese))
[7] Setiawan J D,Mukherjee R,Maslen E H.Synchronous sensor runout and unbalance compensation in active magnetic bearings using bias current excitation.Journal of Dynamic Systems Measurement & Control,2002,124(1):14~24
[8] Xu X B,Fang J C,Liu G,et al.Model development and harmonic current reduction in active magnetic bearing systems with rotor imbalance and sensor runout.Journal of Vibration and Control,2015,21(13):2520~2535
[9] Cui P L,Li S,Zhao G Z,et al.Suppression of harmonic current in active-passive magnetically suspended using improved repetitive controller.IEEE/ASME Transactions on Mechatronics,2016,21(4):2132~2141
[10] 张激扬,陈宗基,刘虎.磁悬浮飞轮位移传感器谐波扰动的主动抑制.宇航学报,2015,11:1289~1295;(Zhang J Y,Chen Z J,Liu H.Active control of multi-frequency vibration caused by displacement sensor runout in magnetic suspension flywheel.Journal of Astronautics.2015,11:1289~1295(in Chinese))
[11] Zhou K,Low K S,Wang D,et al.Zero-phase odd-harmonic repetitive controller for a single-phase pwminverter.IEEE Transactions on Power Electronics,2006,21(1):193~201
[12] 魏静波.磁悬浮微框架动量轮控制技术研究[博士学位论文].长沙:国防科学技术大学,2016;(Wei J B.Study on control technologies of magnetically suspended gimballing momentum wheel [Ph.D Thesis].Changsha:National University of Defense Technology,2016(in Chinese))
[13] 沈福民.自适应信号处理.西安:西安电子科技大学出版社,2001;(Shen F M.Adaptive signal processing.Xi′an:Xi′an University of Electronic Science and Technology Press,2001(in Chinese))
[14] Chen S L,Lin S Y,Toh C S.Adaptive unbalance compensation for a three-pole active magnetic bearing system.IEEE Transactions on Industrial Electronics,2020,67(3):2097~2106
[15] 陈琪,刘刚,郑世强.基于自适应变步长最小均方差算法的磁悬浮电动机自动平衡方法.机械工程学报,2015,51(15):119~127;(Chen Q,Liu G,Zhen S Q.Automatic balancing for magnetically suspended high-speed motor based on adaptive variable step-size LMS algorithm.Journal of Mechanical Engineering,2015,51(15):119~127(in Chinese))

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基于变步长LMS方法的磁悬浮飞轮振动抑制*

VIBRATION SUPPRESSION OF MAGNETIC LEVITATION FLYWHEEL BASED ON VARIABLE STEP SIZE LMS METHOD*

摘要

Abstract

关键词

Keywords

1 磁悬浮轴承系统建模

1.1 磁轴承-转子系统动力学建模

(1)

(2)

1.2 振动源建模

(3)

1.3 参数辨识

(4)

(5)

(6)

(7)

(8)

(9)

(10)

(11)

(12)

(13)

2 基于自适应LMS的主动振动控制

2.1 标准LMS算法

(14)

(15)

(16)

2.2 变步长LMS算法

(17)

(18)

3 仿真分析

(19)

(20)

(21)

(22)

4 结论

参考文献

基本信息

基金信息

引用信息

稿件历史

参考文献

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