Speed fluctuation suppression of the gimbal system with harmonic drive

CHEN Xiangwen; LI Haitao

doi:10.37188/OPE.20223020.2457

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Speed fluctuation suppression of the gimbal system with harmonic drive

Micro/Nano Technology and Fine Mechanics | Updated：2022-10-27

- Speed fluctuation suppression of the gimbal system with harmonic drive
- Optics and Precision Engineering
- Affiliations：
  
  1.School of Instrumentation Science and Optoelectronics Engineering， Beihang University，Beijing 100191， China
  2.Ningbo Institute of Technology， Beihang University， Ningbo 315800， China
  3.Research Institute for Frontier Science， Beihang University， Beijing 100191， China
- Author bio：
- Funds：
- DOI：10.37188/OPE.20223020.2457 CLC： TM351
- Published：25 October 2022，
  
  Received：06 June 2022，
  
  Revised：22 July 2022，
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Cite this article
陈祥文,李海涛.带有谐波减速器的框架系统速度波动抑制[J].光学精密工程,2022,30(20):2457-2466.

CHEN Xiangwen,LI Haitao.Speed fluctuation suppression of the gimbal system with harmonic drive[J].Optics and Precision Engineering,2022,30(20):2457-2466.
陈祥文,李海涛.带有谐波减速器的框架系统速度波动抑制[J].光学精密工程,2022,30(20):2457-2466. DOI： 10.37188/OPE.20223020.2457.

CHEN Xiangwen,LI Haitao.Speed fluctuation suppression of the gimbal system with harmonic drive[J].Optics and Precision Engineering,2022,30(20):2457-2466. DOI： 10.37188/OPE.20223020.2457.

带有谐波减速器的框架系统速度波动抑制 Speed fluctuation suppression of the gimbal system with harmonic drive 1 2 first-author 陈祥文（1994-），男，山东临沂人，博士研究生，2017年于内蒙古大学获得学士学位，主要从事高精度永磁同步电机控制和磁悬浮控制力矩陀螺框架伺服系统控制方面的研究。E-mail： xiangwen@buaa.edu.cn http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=36182395&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=36182405&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=36182400&type= 22.01332855 32.00400543 陈祥文（1994-），男，山东临沂人，博士研究生，2017年于内蒙古大学获得学士学位，主要从事高精度永磁同步电机控制和磁悬浮控制力矩陀螺框架伺服系统控制方面的研究。E-mail： xiangwen@buaa.edu.cn xiangwen@buaa.edu.cn 2 3 corresp E-mail：haitao@buaa.edu.cn E-mail：haitao@buaa.edu.cn 李海涛（1979-），男，山东临清人，博士，副教授，博士生导师，2002年、2005年于山东大学分别获得学士、硕士学位，2009年于北京航空航天大学获得博士学位，主要从事磁悬浮惯性执行机构技术、磁悬浮控制力矩陀螺超低速框架伺服系统的高精度控制方面的研究。E-mail： haitao@buaa.edu.cn http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=36182411&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=36182424&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=36182413&type= 22.01332855 32.00399780 李海涛（1979-），男，山东临清人，博士，副教授，博士生导师，2002年、2005年于山东大学分别获得学士、硕士学位，2009年于北京航空航天大学获得博士学位，主要从事磁悬浮惯性执行机构技术、磁悬浮控制力矩陀螺超低速框架伺服系统的高精度控制方面的研究。E-mail： haitao@buaa.edu.cn haitao@buaa.edu.cn 北京航空航天大学仪器科学与光电工程学院，北京 100191 北京航空航天大学仪器科学与光电工程学院，北京 100191 School of Instrumentation Science and Optoelectronics Engineering， Beihang University，Beijing 100191， China School of Instrumentation Science and Optoelectronics Engineering， Beihang University，Beijing 100191， China 北京航空航天大学宁波创新研究院，浙江宁波 315800 北京航空航天大学宁波创新研究院，浙江宁波 315800 Ningbo Institute of Technology， Beihang University， Ningbo 315800， China Ningbo Institute of Technology， Beihang University， Ningbo 315800， China 北京航空航天大学前沿科学技术创新研究院，北京 100191 北京航空航天大学前沿科学技术创新研究院，北京 100191 Research Institute for Frontier Science， Beihang University， Beijing 100191， China Research Institute for Frontier Science， Beihang University， Beijing 100191， China 为了实现带有谐波减速器的间驱式框架系统高精度速度跟踪控制，提出了一种基于位置域迭代学习控制的前馈补偿方法。首先，分析了谐波减速器运动误差导致的负载端角速率波动的位置域周期特性，并在此基础上提出了一种位置域迭代学习前馈补偿方法。该方法通过对框架系统速度调节误差信号进行累加获取前馈补偿信号，通过补偿力矩来抑制转速波动。然后，对迭代学习算法进行了分析，得到了迭代学习的收敛条件和迭代终止条件。最后，利用仿真和试验对所提方法的有效性和可行性进行了验证。结果表明，应用所提的迭代学习前馈补偿方法可使框架系统负载端的转速波动量被抑制30%以上。所提方法实现简单，具有较好的速度波动抑制能力，有效提高了控制力矩陀螺输出力矩精度。 To realize high-precision speed tracking control of the gimbal system with harmonic drive， a feedforward compensation method based on the iterative learning control of the position domain is proposed. First， the periodicity characteristics of the load angular speed fluctuation in the position domain， which is caused by the kinematic error of the harmonic drive， are analyzed. According to the analysis results， an iteration compensation method of the position domain is proposed. This method obtains the feedforward compensation signal by accumulating the speed adjustment error signal. The compensation signal compensates the torque to suppress the speed fluctuation. Then， the convergence and iteration termination conditions are obtained by the analysis of the iterative learning algorithm. Finally， the effectiveness and feasibility of the proposed method were verified with simulations and experiments. The results show that the proposed iterative learning feedforward compensation method suppresses the load angular speed fluctuation of the gimbal system by more than 30%. The proposed method is simple to implement and has good velocity fluctuation suppression. Simultaneously， the output torque of the control moment gyro is effectively improved. 位置域迭代学习框架系统速度跟踪控制 position domain iterative learning gimbal system speed tracking control 1 引言 1 控制力矩陀螺以其输出力矩大、动态响应快、控制精度高等优点成为超敏捷卫星和大型航天器的理想执行机构［ 1 1 - 3 3 1-3 ］。控制力矩陀螺主要由高速转子系统和框架系统两部分组成。其中高速转子系统产生角动量 <math id="M1"><mi mathvariant="bold-italic">H</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=36182433&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=36182429&type= 3.47133350 2.79399991 ，框架系统通过提供转速 <math id="M2"><mi mathvariant="bold-italic">ω</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=36182443&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=36182436&type= 2.62466669 2.79399991 来改变角动量 <math id="M3"><mi mathvariant="bold-italic">H</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=36182460&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=36182456&type= 3.47133350 2.79399991 的方向，从而对外输出力矩，陀螺输出力矩为二者的矢量积。在控制力矩陀螺正常工作的状态下，高速转子系统处在稳速状态，为陀螺提供恒定的角动量 <math id="M4"><mi mathvariant="bold-italic">H</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=36182460&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=36182456&type= 3.47133350 2.79399991 。此时，陀螺输出力矩的精度取决于框架系统转速 <math id="M5"><mi mathvariant="bold-italic">ω</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=36182471&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=36182466&type= 2.62466669 2.79399991 的精度［ 4 4 ］。因此，实现框架系统输出转速的高精度控制对于提高控制力矩陀螺输出力矩精度具有十分重要的意义。在航天应用当中，为满足航天器对于体积和重量的要求，控制力矩陀螺需要在保证输出大力矩的同时，尽可能减小其自身尺寸。因此，需要在框架系统中增加减速机构以实现这一需求。相较于其它减速机构，谐波减速器具有体积小、传动比大、传动精度高、结构和安装简单等优势，成为控制力矩陀螺理想的减速机构。但是由于运动误差、迟滞特性和非线性摩擦等因素，谐波减速器的输出转矩表现出非线性特性，使得输出角速率存在较大波动［ 5 5 ］，而由于谐波减速器各组件的加工和装配误差导致的运动误差是框架系统产生速率波动的主要因素［ 6 6 ］。Miyazaki等人认为谐波减速器运动误差是电机端角位置偶数倍频的正弦叠加，呈现出位置域周期性［ 7 7 ］。为抑制运动误差造成的框架系统速率波动，文献［ 8 8 ］通过非线性微分跟踪器获取角加速度信号，实现了加速度反馈控制。Godler等人将加速度反馈与重复控制相结合以抑制负载端角速度的周期性波动［ 9 9 ］。但是重复控制需要一定的收敛时间，虽然在匀速状态下可以获得良好的控制效果，但当系统处于变速状态时，由于收敛时间的存在，其动态响应性能较差。为更好地抑制谐波减速器的非线性传输力矩，提高转速的控制精度，一般采取前馈控制对干扰进行抑制。相较于反馈控制，前馈控制将干扰量直接引入调节装置，按照干扰的作用对被控量进行调节，对于干扰的抑制要比反馈控制及时。在谐波减速器的相关研究中，目前常用的前馈控制方式有两类：（1）以非线性传输力矩建模为基础，基于模型的前馈；（2）将非线性传输力矩视为扰动，使用干扰观测器对其观测并在控制器中进行补偿。对于第一类方式，文献［ 10 10 ］提出了一种基于Preisach模型的建模方法，通过建立柔轮输出力矩与扭转角之间的关系，得到谐波减速器迟滞特性模型。党选举等人用非线性函数代替Prandtl-Ishlinskii模型框架下Play算子的线性部分，并在此基础上建立了神经网络迟滞模型，提高了建模精度［ 11 11 ］。针对非线性摩擦，有研究人员利用采样电流和估计的角加速度建立了库伦+粘滞+Stribeck摩擦模型，并设计了基于该模型的前馈补偿控制算法用以提高框架系统的速度控制精度［ 12 12 ］。文献［ 13 13 ］提出了一种基于拉格朗日方程的在线建模方法，通过对谐波减速器非线性传输转矩建模补偿以实现对负载端转速波动的抑制。但是建模过程中存在参数辨识困难，前期工作量大等问题。另一方面，由于谐波减速器的非线性传输力矩成因复杂，在建模过程中需要对一些因素进行取舍，难以实现精确建模，所以基于模型的控制方法的补偿精度受到一定限制。对于第二类利用干扰观测器观测非线性传输转矩并补偿的方式，文献［ 14 14 ］提出一种基于干扰观测器和自适应方法相结合的前馈补偿方法，用以补偿谐波减速器转矩波动导致的速度波动。在此基础上，王冲冲等人在内外回路中分别采用干扰观测器对非线性摩擦和周期性扰动进行观测和补偿［ 15 15 ］。但基于传统状态干扰观测器的补偿方法，其观测器增益矩阵选取困难，同时，传统的状态干扰观测器设计通常集中于保证控制器与观测器相结合之后的系统稳定性，很少关注扰动观测精度。为简化观测器增益参数选取，提高观测精度，文献［ 16 16 ］使用级联扩张状态观测器来替换传统的状态干扰观测器，但是带有谐波减速器的框架系统的动力学方程不满足积分串联形式，需要通过李导数对其进行坐标变换，增加了系统的计算复杂度。除上述两类方式之外，从跟踪性能的角度出发，迭代学习控制也是一种很有效的前馈控制方式［ 17 17 - 19 19 17-19 ］。迭代学习控制最初由日本学者有本卓在1984年首次提出，由于其结构设计简单、不依赖于被控对象的模型、对周期性扰动信号具有很好的抑制作用，因此得到了广泛的应用［ 20 20 ］。文献［ 21 21 ］通过对输入、输出信号的迭代学习来获取定位系统复杂动力学的等效逆模型，实现了音圈电机电磁驱动平台的高精度控制。Bolder等人使用迭代学习控制来调节参数化前馈控制器的自由参数，以实现系统的无静差跟踪［ 22 22 ］。同样是针对参数化前馈控制器设计，Dai等人通过基于逆模型的迭代学习控制方法在匀速状态下获取前馈补偿信号，以实现系统对参考信号的无静差跟踪，然后通过多项式对前馈补偿信号进行拟合，获取参数化的前馈控制器，从而达到在变参考状态下的无静差跟踪［ 23 23 ］。传统的迭代学习控制是在时间域内进行迭代，但是运动误差导致的速度波动是在电机端的位置域内呈周期性的，所以不能直接应用。为解决这一问题本文将位置域与迭代学习控制相结合，利用运动误差导致的速度波动在位置域内的相位不变性［ 24 24 ］，在位置域内对调节误差进行迭代学习在线建立前馈补偿表，实现了变速情况下的快速响应和精确补偿，提高了间驱式框架系统负载端的速度跟踪精度。 2 间驱式框架系统速率波动分析 1 带有谐波减速器的框架系统动力学方程可以表示为： <math display="block" id="M6"><mfenced open="{" close="" separators="|"><mrow><mtable columnalign="left"><mtr><mtd><mfrac><mrow><mi 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mathvariant="normal">l</mi></mrow></msub><mo>-</mo><mfrac><mrow><mn mathvariant="normal">1</mn></mrow><mrow><msub><mrow><mi>J</mi></mrow><mrow><mi mathvariant="normal">l</mi></mrow></msub></mrow></mfrac><msub><mrow><mi>T</mi></mrow><mrow><mi mathvariant="normal">d</mi></mrow></msub></mtd></mtr></mtable></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=36182480&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=36182476&type= 53.42466354 29.04066467 ，其中： <math id="M7"><msub><mrow><mi>ω</mi></mrow><mrow><mi mathvariant="normal">m</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=36182492&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=36182487&type= 4.23333359 3.72533321 和 <math id="M8"><msub><mrow><mi>ω</mi></mrow><mrow><mi mathvariant="normal">l</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=36182499&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=36182496&type= 3.21733332 3.72533321 分别是电机端和负载端的转速， <math id="M9"><msub><mrow><mi>J</mi></mrow><mrow><mi mathvariant="normal">m</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=36182508&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=36182503&type= 3.47133350 3.72533321 和 <math id="M10"><msub><mrow><mi>J</mi></mrow><mrow><mi mathvariant="normal">l</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=36182518&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=36182513&type= 2.45533323 3.72533321 分别是电机端和负载端的转动惯量， <math id="M11"><msub><mrow><mi>B</mi></mrow><mrow><mi mathvariant="normal">m</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=36182528&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=36182523&type= 4.31799984 3.72533321 和 <math id="M12"><msub><mrow><mi>B</mi></mrow><mrow><mi mathvariant="normal">l</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=36182540&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=36182535&type= 3.30200005 3.72533321 分别是电机端和负载端的阻尼系数， <math id="M13"><msub><mrow><mi>K</mi></mrow><mrow><mi mathvariant="normal">h</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=36182550&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=36182545&type= 4.06400013 3.72533321 是谐波减速器的扭转刚度， <math id="M14"><mi>N</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=36182558&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=36182555&type= 3.30200005 2.87866688 是谐波减速器的传动比； <math id="M15"><mi mathvariant="normal">Δ</mi><mi>θ</mi><mo>=</mo><mfrac><mrow><msub><mrow><mi>θ</mi></mrow><mrow><mi mathvariant="normal">m</mi></mrow></msub></mrow><mrow><mi>N</mi></mrow></mfrac><mo>-</mo><msub><mrow><mi>θ</mi></mrow><mrow><mi mathvariant="normal">l</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=36182567&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=36182563&type= 20.65866661 8.04333401 表示谐波减速器折算到负载端的扭转角， <math id="M16"><msub><mrow><mi>θ</mi></mrow><mrow><mi mathvariant="normal">m</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=36182577&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=36182572&type= 3.55599999 3.72533321 和 <math id="M17"><msub><mrow><mi>θ</mi></mrow><mrow><mi mathvariant="normal">l</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=36182585&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=36182580&type= 2.45533323 3.72533321 分别是电机端和负载端的角位置； <math id="M18"><msub><mrow><mi>T</mi></mrow><mrow><mi mathvariant="normal">e</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=36182592&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=36182588&type= 3.64066648 3.72533321 表示电磁力矩， <math id="M19"><msub><mrow><mi>T</mi></mrow><mrow><mi mathvariant="normal">d</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=36182600&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=36182596&type= 3.72533321 3.72533321 表示负载端的扰动力矩。间驱式框架控制系统采用速度环和电流环双闭环结构，负载端角速率和运动误差 <math id="M20"><msub><mrow><mi>θ</mi></mrow><mrow><mi mathvariant="normal">e</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=36182608&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=36182604&type= 2.87866688 3.72533321 之间的关系为： <math display="block" id="M21"><mtable><mtr><mtd><msub><mrow><mi>ω</mi></mrow><mrow><mi mathvariant="normal">l</mi></mrow></msub><mo>=</mo></mtd></mtr><mtr><mtd><mfrac><mrow><msub><mrow><mi>B</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><mi>L</mi><msup><mrow><mi>s</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mo>+</mo><msub><mrow><mi>B</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mi>s</mi><mo>+</mo><mi>N</mi><msub><mrow><mi>B</mi></mrow><mrow><mn mathvariant="normal">3</mn></mrow></msub><msub><mrow><mi>B</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo>+</mo><msub><mrow><mi>B</mi></mrow><mrow><mn mathvariant="normal">4</mn></mrow></msub><mo stretchy="false">(</mo><mi>N</mi><msub><mrow><mi>G</mi></mrow><mrow><mi>ω</mi></mrow></msub><mo stretchy="false">(</mo><mi>s</mi><mo stretchy="false">)</mo><mo>+</mo><msub><mrow><mi>B</mi></mrow><mrow><mi mathvariant="normal">l</mi></mrow></msub><mo stretchy="false">)</mo></mrow><mrow><msub><mrow><mi>a</mi></mrow><mrow><mn mathvariant="normal">4</mn></mrow></msub><msup><mrow><mi>s</mi></mrow><mrow><mn mathvariant="normal">4</mn></mrow></msup><mo>+</mo><msub><mrow><mi>a</mi></mrow><mrow><mn mathvariant="normal">3</mn></mrow></msub><msup><mrow><mi>s</mi></mrow><mrow><mn mathvariant="normal">3</mn></mrow></msup><mo>+</mo><msub><mrow><mi>a</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><msup><mrow><mi>s</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mo>+</mo><msub><mrow><mi>a</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mi>s</mi><mo>+</mo><msub><mrow><mi>a</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub></mrow></mfrac><msub><mrow><mi>θ</mi></mrow><mrow><mi mathvariant="normal">e</mi></mrow></msub></mtd></mtr></mtable></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=36182619&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=36182615&type= 70.86599731 13.54666710 ，其中： <math id="M22"><msub><mrow><mi>a</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><mo>=</mo><mi>N</mi><mfenced open="[" close="]" separators="|"><mrow><msub><mrow><mi>B</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><msub><mrow><mi>B</mi></mrow><mrow><mn mathvariant="normal">3</mn></mrow></msub><mo>+</mo><msub><mrow><mi>B</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><msub><mrow><mi>B</mi></mrow><mrow><mn mathvariant="normal">7</mn></mrow></msub><msub><mrow><mi>B</mi></mrow><mrow><mi mathvariant="normal">l</mi></mrow></msub></mrow></mfenced><mo>+</mo><msub><mrow><mi>K</mi></mrow><mrow><mi mathvariant="normal">h</mi></mrow></msub><msub><mrow><mi>G</mi></mrow><mrow><mi>ω</mi></mrow></msub><mo stretchy="false">(</mo><mi>s</mi><mo stretchy="false">)</mo><msub><mrow><mi>K</mi></mrow><mrow><mi mathvariant="normal">m</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=36182627&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=36182623&type= 62.39933777 4.06400013 <math id="M23"><msub><mrow><mi>G</mi></mrow><mrow><mi>i</mi></mrow></msub><mo stretchy="false">(</mo><mi>s</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=36182634&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=36182630&type= 8.72066593 4.48733330 ； <math id="M24"><msub><mrow><mi>a</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>=</mo><mi>N</mi><mo stretchy="false">[</mo><msub><mrow><mi>B</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><msub><mrow><mi>B</mi></mrow><mrow><mn mathvariant="normal">6</mn></mrow></msub><mo>+</mo><msub><mrow><mi>B</mi></mrow><mrow><mn mathvariant="normal">3</mn></mrow></msub><mi>L</mi><mo>+</mo><msub><mrow><mi>B</mi></mrow><mrow><mn mathvariant="normal">7</mn></mrow></msub><mo stretchy="false">(</mo><msub><mrow><mi>J</mi></mrow><mrow><mi mathvariant="normal">l</mi></mrow></msub><mi>R</mi><mo>+</mo><msub><mrow><mi>B</mi></mrow><mrow><mi mathvariant="normal">l</mi></mrow></msub><mi>L</mi><mo stretchy="false">)</mo><mo>+</mo><msub><mrow><mi>K</mi></mrow><mrow><mi mathvariant="normal">m</mi></mrow></msub><msub><mrow><mi>B</mi></mrow><mrow><mi mathvariant="normal">l</mi></mrow></msub><msub><mrow><mi>C</mi></mrow><mrow><mi mathvariant="normal">e</mi></mrow></msub><mo stretchy="false">]</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=36182641&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=36182636&type= 79.75599670 4.48733330 ； <math id="M25"><msub><mrow><mi>a</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo>=</mo><mi>N</mi><mo stretchy="false">[</mo><msub><mrow><mi>B</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><msub><mrow><mi>B</mi></mrow><mrow><mn mathvariant="normal">5</mn></mrow></msub><mo>+</mo><msub><mrow><mi>B</mi></mrow><mrow><mn mathvariant="normal">6</mn></mrow></msub><mi>L</mi><mo>+</mo><msub><mrow><mi>B</mi></mrow><mrow><mn mathvariant="normal">7</mn></mrow></msub><msub><mrow><mi>J</mi></mrow><mrow><mi mathvariant="normal">l</mi></mrow></msub><mi>L</mi><mo>+</mo><msub><mrow><mi>K</mi></mrow><mrow><mi mathvariant="normal">m</mi></mrow></msub><msub><mrow><mi>J</mi></mrow><mrow><mi mathvariant="normal">l</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=36182650&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=36182646&type= 59.18199921 4.48733330 <math id="M26"><msub><mrow><mi>C</mi></mrow><mrow><mi mathvariant="normal">e</mi></mrow></msub><mo stretchy="false">]</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=36182660&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=36182656&type= 5.24933338 4.48733330 ； <math id="M27"><msub><mrow><mi>a</mi></mrow><mrow><mn mathvariant="normal">3</mn></mrow></msub><mo>=</mo><mi>N</mi><mo stretchy="false">[</mo><msub><mrow><mi>J</mi></mrow><mrow><mi mathvariant="normal">m</mi></mrow></msub><msub><mrow><mi>J</mi></mrow><mrow><mi mathvariant="normal">l</mi></mrow></msub><msub><mrow><mi>B</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo>+</mo><msub><mrow><mi>B</mi></mrow><mrow><mn mathvariant="normal">5</mn></mrow></msub><mi>L</mi><mo stretchy="false">]</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=36182668&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=36182663&type= 37.25333405 4.48733330 ； <math id="M28"><msub><mrow><mi>a</mi></mrow><mrow><mn mathvariant="normal">4</mn></mrow></msub><mo>=</mo><mi>N</mi><msub><mrow><mi>J</mi></mrow><mrow><mi mathvariant="normal">m</mi></mrow></msub><msub><mrow><mi>J</mi></mrow><mrow><mi mathvariant="normal">l</mi></mrow></msub><mi>L</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=36182675&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=36182672&type= 19.55800056 3.80999994 ； <math id="M29"><msub><mrow><mi>B</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><mo>=</mo><mi>N</mi><msub><mrow><mi>K</mi></mrow><mrow><mi mathvariant="normal">h</mi></mrow></msub><msub><mrow><mi>J</mi></mrow><mrow><mi mathvariant="normal">m</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=36182684&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=36182678&type= 19.13466644 3.80999994 ； <math id="M30"><msub><mrow><mi>B</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>=</mo><msub><mrow><mi>B</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><msub><mrow><mi>B</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo>+</mo><mi>N</mi><msub><mrow><mi>B</mi></mrow><mrow><mn mathvariant="normal">3</mn></mrow></msub><mi>L</mi><mo>+</mo><msub><mrow><mi>B</mi></mrow><mrow><mn mathvariant="normal">4</mn></mrow></msub><msub><mrow><mi>J</mi></mrow><mrow><mi mathvariant="normal">l</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=36182692&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=36182687&type= 41.65599823 3.80999994 ； <math id="M31"><msub><mrow><mi>B</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo>=</mo><mi>R</mi><mo>+</mo><msub><mrow><mi>G</mi></mrow><mrow><mi>i</mi></mrow></msub><mo stretchy="false">(</mo><mi>s</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=36182698&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=36182694&type= 24.80733490 4.48733330 ； <math id="M32"><msub><mrow><mi>B</mi></mrow><mrow><mn mathvariant="normal">3</mn></mrow></msub><mo>=</mo><msub><mrow><mi>K</mi></mrow><mrow><mi mathvariant="normal">h</mi></mrow></msub><msub><mrow><mi>B</mi></mrow><mrow><mi mathvariant="normal">m</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=36182706&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=36182703&type= 17.18733406 3.72533321 ； <math id="M33"><msub><mrow><mi>B</mi></mrow><mrow><mn mathvariant="normal">4</mn></mrow></msub><mo>=</mo><msub><mrow><mi>K</mi></mrow><mrow><mi mathvariant="normal">h</mi></mrow></msub><msub><mrow><mi>C</mi></mrow><mrow><mi mathvariant="normal">e</mi></mrow></msub><msub><mrow><mi>K</mi></mrow><mrow><mi mathvariant="normal">m</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=36182714&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=36182709&type= 21.33600044 3.72533321 ； <math id="M34"><msub><mrow><mi>B</mi></mrow><mrow><mn mathvariant="normal">5</mn></mrow></msub><mo>=</mo><msub><mrow><mi>J</mi></mrow><mrow><mi mathvariant="normal">m</mi></mrow></msub><msub><mrow><mi>B</mi></mrow><mrow><mi mathvariant="normal">l</mi></mrow></msub><mo>+</mo><msub><mrow><mi>B</mi></mrow><mrow><mi mathvariant="normal">m</mi></mrow></msub><msub><mrow><mi>J</mi></mrow><mrow><mi mathvariant="normal">l</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=36182722&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=36182719&type= 27.26266861 3.80999994 ； <math id="M35"><msub><mrow><mi>B</mi></mrow><mrow><mn mathvariant="normal">6</mn></mrow></msub><mo>=</mo><msub><mrow><mi>B</mi></mrow><mrow><mi mathvariant="normal">m</mi></mrow></msub><msub><mrow><mi>B</mi></mrow><mrow><mi mathvariant="normal">l</mi></mrow></msub><mo>+</mo><msub><mrow><mi>K</mi></mrow><mrow><mi mathvariant="normal">h</mi></mrow></msub><msub><mrow><mi>J</mi></mrow><mrow><mi mathvariant="normal">m</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=36182730&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=36182726&type= 28.95599937 3.80999994 ； <math id="M36"><msub><mrow><mi>B</mi></mrow><mrow><mn mathvariant="normal">7</mn></mrow></msub><mo>=</mo><mfrac><mrow><mn mathvariant="normal">1</mn></mrow><mrow><msup><mrow><mi>N</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup></mrow></mfrac><msub><mrow><mi>K</mi></mrow><mrow><mi mathvariant="normal">h</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=36182737&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=36182734&type= 18.37266731 8.21266651 ； <math id="M37"><msub><mrow><mi>G</mi></mrow><mrow><mi>ω</mi></mrow></msub><mfenced separators="|"><mrow><mi>s</mi></mrow></mfenced><mo>=</mo><msubsup><mrow><mi>k</mi></mrow><mrow><mi>p</mi></mrow><mrow><mi>ω</mi></mrow></msubsup><mo>+</mo><mfrac><mrow><msubsup><mrow><mi>k</mi></mrow><mrow><mi>i</mi></mrow><mrow><mi>ω</mi></mrow></msubsup></mrow><mrow><mi>s</mi></mrow></mfrac></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=36182745&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=36182741&type= 26.16200066 8.21266651 和 <math id="M38"><msub><mrow><mi>G</mi></mrow><mrow><mi>i</mi></mrow></msub><mfenced separators="|"><mrow><mi>s</mi></mrow></mfenced><mo>=</mo><msubsup><mrow><mi>k</mi></mrow><mrow><mi>p</mi></mrow><mrow><mi>i</mi></mrow></msubsup><mo>+</mo><mfrac><mrow><msubsup><mrow><mi>k</mi></mrow><mrow><mi>i</mi></mrow><mrow><mi>i</mi></mrow></msubsup></mrow><mrow><mi>s</mi></mrow></mfrac></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=36182753&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=36182749&type= 24.12999916 8.21266651 分别是速度环和电流环PI（Proportional Integral）控制器的传递函数， <math id="M39"><msubsup><mrow><mi>k</mi></mrow><mrow><mi>p</mi></mrow><mrow><mi>ω</mi></mrow></msubsup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=36182763&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=36182758&type= 3.38666677 3.97933316 和 <math id="M40"><msubsup><mrow><mi>k</mi></mrow><mrow><mi>p</mi></mrow><mrow><mi>i</mi></mrow></msubsup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=36182772&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=36182768&type= 2.79399991 3.97933316 分别是速度环和电流环的比例项系数， <math id="M41"><msubsup><mrow><mi>k</mi></mrow><mrow><mi>i</mi></mrow><mrow><mi>ω</mi></mrow></msubsup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=36182781&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=36182777&type= 3.38666677 3.97933316 和 <math id="M42"><msubsup><mrow><mi>k</mi></mrow><mrow><mi>i</mi></mrow><mrow><mi>i</mi></mrow></msubsup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=36182789&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=36182785&type= 2.62466669 3.97933316 分别是速度环和电流环的积分项系数； <math id="M43"><msub><mrow><mi>G</mi></mrow><mrow><mi mathvariant="normal">m</mi></mrow></msub><mfenced separators="|"><mrow><mi>s</mi></mrow></mfenced><mo>=</mo><mfrac><mrow><mn mathvariant="normal">1</mn></mrow><mrow><mi>L</mi><mi>s</mi><mo>+</mo><mi>R</mi></mrow></mfrac></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=36182798&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=36182793&type= 25.99266624 8.04333401 ， <math id="M44"><msub><mrow><mi>G</mi></mrow><mrow><mi mathvariant="normal">m</mi><mi mathvariant="normal">l</mi></mrow></msub><mfenced separators="|"><mrow><mi>s</mi></mrow></mfenced><mo>=</mo><mfrac><mrow><mn mathvariant="normal">1</mn></mrow><mrow><msub><mrow><mi>J</mi></mrow><mrow><mi mathvariant="normal">m</mi></mrow></msub><mi>s</mi><mo>+</mo><msub><mrow><mi>B</mi></mrow><mrow><mi mathvariant="normal">m</mi></mrow></msub></mrow></mfrac></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=36182806&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=36182802&type= 29.37933159 8.38199997 和 <math id="M45"><msub><mrow><mi>G</mi></mrow><mrow><mi mathvariant="normal">l</mi></mrow></msub><mfenced separators="|"><mrow><mi>s</mi></mrow></mfenced><mo>=</mo><mfrac><mrow><mn mathvariant="normal">1</mn></mrow><mrow><msub><mrow><mi>J</mi></mrow><mrow><mi mathvariant="normal">l</mi></mrow></msub><mi>s</mi><mo>+</mo><msub><mrow><mi>B</mi></mrow><mrow><mi mathvariant="normal">l</mi></mrow></msub></mrow></mfrac></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=36182814&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=36182810&type= 25.73866653 8.38199997 分别是电机数学模型，电机端负载数学模型和负载端负载数学模型， <math id="M46"><mi>L</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=36182822&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=36182818&type= 2.45533323 2.79399991 为定子电感， R 是定子电阻， <math id="M47"><msub><mrow><mi>K</mi></mrow><mrow><mi mathvariant="normal">m</mi></mrow></msub><mo>=</mo><mn mathvariant="normal">1.5</mn><mi>p</mi><msub><mrow><mi>ψ</mi></mrow><mrow><mi mathvariant="normal">f</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=36182830&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=36182826&type= 19.81200027 4.57200003 是电机的力矩系数， <math id="M48"><msub><mrow><mi>ψ</mi></mrow><mrow><mi mathvariant="normal">f</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=36182840&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=36182836&type= 2.87866688 4.48733330 是电机的磁链系数； <math id="M49"><msub><mrow><mi>C</mi></mrow><mrow><mi mathvariant="normal">e</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=36182850&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=36182846&type= 3.64066648 3.72533321 是电机的反电势系数。文献［ 6 6 ］给出了谐波减速器运动误差的一般表现形式： <math display="block" id="M50"><mtable><mtr><mtd><msub><mrow><mi>θ</mi></mrow><mrow><mi mathvariant="normal">e</mi></mrow></msub><mo>=</mo><msub><mrow><mi>A</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mi mathvariant="normal">s</mi><mi mathvariant="normal">i</mi><mi mathvariant="normal">n</mi><mo stretchy="false">(</mo><mn mathvariant="normal">2</mn><msub><mrow><mi>θ</mi></mrow><mrow><mi mathvariant="normal">m</mi></mrow></msub><mo stretchy="false">)</mo><mo>+</mo><msub><mrow><mi>A</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mi mathvariant="normal">s</mi><mi mathvariant="normal">i</mi><mi mathvariant="normal">n</mi><mo stretchy="false">(</mo><mn mathvariant="normal">4</mn><msub><mrow><mi>θ</mi></mrow><mrow><mi mathvariant="normal">m</mi></mrow></msub><mo stretchy="false">)</mo><mo>+</mo></mtd></mtr><mtr><mtd><msub><mrow><mi>A</mi></mrow><mrow><mn mathvariant="normal">3</mn></mrow></msub><mi mathvariant="normal">s</mi><mi mathvariant="normal">i</mi><mi mathvariant="normal">n</mi><mo stretchy="false">(</mo><mn mathvariant="normal">6</mn><msub><mrow><mi>θ</mi></mrow><mrow><mi mathvariant="normal">m</mi></mrow></msub><mo stretchy="false">)</mo><mo>+</mo><mo>…</mo><mo>+</mo><msub><mrow><mi>A</mi></mrow><mrow><mi>n</mi></mrow></msub><mi mathvariant="normal">s</mi><mi mathvariant="normal">i</mi><mi mathvariant="normal">n</mi><mo stretchy="false">(</mo><mi>n</mi><msub><mrow><mi>θ</mi></mrow><mrow><mi mathvariant="normal">m</mi></mrow></msub><mo stretchy="false">)</mo></mtd></mtr></mtable></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=36182856&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=36182852&type= 52.57799911 9.82133293 ，其中： <math id="M51"><msub><mrow><mi>A</mi></mrow><mrow><mi>i</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=36182865&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=36182860&type= 3.80999994 3.72533321 （ i =1，2，3，…， n ）表示各倍频的正弦幅值。根据运动误差表达式（3）式（3）可以知道其引起的负载端角速率波动具有位置域周期性，将位置域频率 <math id="M52"><mover><mrow><mi>f</mi></mrow><mrow><mi>˜</mi></mrow></mover></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=36182874&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=36182870&type= 2.45533323 4.40266657 定义为： <math display="block" id="M53"><mover><mrow><mi>f</mi></mrow><mrow><mi>˜</mi></mrow></mover><mo>=</mo><mfrac><mrow><mn mathvariant="normal">1</mn></mrow><mrow><msub><mrow><mi>ω</mi></mrow><mrow><mi mathvariant="normal">m</mi></mrow></msub><mi>T</mi></mrow></mfrac><mo>=</mo><mfrac><mrow><mn mathvariant="normal">1</mn></mrow><mrow><msub><mrow><mi>ω</mi></mrow><mrow><mi mathvariant="normal">m</mi></mrow></msub></mrow></mfrac><mi>f</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=36182884&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=36182878&type= 27.26266861 8.29733276 ，其中， <math id="M54"><mi>f</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=36182892&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=36182887&type= 2.45533323 3.64066648 为时间域内的频率。对框架系统不同转速下的速率曲线进行FFT分析，得到运动误差在位置域中的频谱图，如图1 图1 所示。由图可知，运动误差在不同速度下都是电机端角位置的偶数倍频，而且集中在低频段，其主要频率分量为 <math id="M55"><msub><mrow><mover><mrow><mi>f</mi></mrow><mrow><mi>˜</mi></mrow></mover></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>=</mo><mfrac><mrow><mn mathvariant="normal">1</mn></mrow><mrow><mi>π</mi></mrow></mfrac><mtext> </mtext><mi mathvariant="normal">r</mi><mi mathvariant="normal">a</mi><msup><mrow><mi mathvariant="normal">d</mi></mrow><mrow><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=36182921&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=36182915&type= 19.55800056 7.87400007 ， <math id="M56"><msub><mrow><mover><mrow><mi>f</mi></mrow><mrow><mi>˜</mi></mrow></mover></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo>=</mo><mfrac><mrow><mn mathvariant="normal">2</mn></mrow><mrow><mi>π</mi></mrow></mfrac><mtext> </mtext><mi mathvariant="normal">r</mi><mi mathvariant="normal">a</mi><msup><mrow><mi mathvariant="normal">d</mi></mrow><mrow><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=36182931&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=36182926&type= 19.55800056 7.87400007 ， <math id="M57"><msub><mrow><mover><mrow><mi>f</mi></mrow><mrow><mi>˜</mi></mrow></mover></mrow><mrow><mn mathvariant="normal">3</mn></mrow></msub><mo>=</mo><mfrac><mrow><mn mathvariant="normal">3</mn></mrow><mrow><mi>π</mi></mrow></mfrac><mtext> </mtext><mi mathvariant="normal">r</mi><mi mathvariant="normal">a</mi><msup><mrow><mi mathvariant="normal">d</mi></mrow><mrow><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=36182942&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=36182937&type= 19.55800056 7.87400007 。 3 迭代学习前馈补偿控制 1 为抑制运动误差导致的负载端转速波动，在传统双闭环控制结构中引入基于位置域迭代学习控制的前馈补偿方法，通过对速度环调节误差进行迭代获取补偿表。传统的迭代学习控制所解决的周期性扰动问题，通常是指在时间域内的周期性，而谐波减速器运动误差导致的负载端速率波动在电机端的位置域内呈现周期性。针对此类问题，文献［ 25 25 ］提出一种“域转换”方法，通过该方法可以将位置域内的周期性扰动转换为时间域的周期性扰动。本文首先在位置域内建立了迭代学习控制算法用以抑制谐波减速器负载端角速率波动，然后通过“域转换”方法将其转换到时间域内讨论其收敛性问题。所提出的迭代学习控制方法框图如图2 图2 所示，其中 e j 为第 j 个位置周期内的速度调节误差。根据图2 图2 可以得到位置域中迭代学习控制表达式为： <math display="block" id="M58"><msub><mrow><mi>u</mi></mrow><mrow><mi>j</mi></mrow></msub><mfenced separators="|"><mrow><mover><mrow><mi>s</mi></mrow><mrow><mi>˜</mi></mrow></mover></mrow></mfenced><mo>=</mo><mi>Γ</mi><msub><mrow><mi>e</mi></mrow><mrow><mi>j</mi></mrow></msub><mfenced separators="|"><mrow><mover><mrow><mi>s</mi></mrow><mrow><mi>˜</mi></mrow></mover></mrow></mfenced><mo>+</mo><mi>Φ</mi><msub><mrow><mi>e</mi></mrow><mrow><mi>j</mi><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msub><mfenced separators="|"><mrow><mover><mrow><mi>s</mi></mrow><mrow><mi>˜</mi></mrow></mover></mrow></mfenced><mo>+</mo><mfenced separators="|"><mrow><mn mathvariant="normal">1</mn><mo>-</mo><mi>α</mi></mrow></mfenced><msub><mrow><mi>u</mi></mrow><mrow><mi>j</mi><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msub><mfenced separators="|"><mrow><mover><mrow><mi>s</mi></mrow><mrow><mi>˜</mi></mrow></mover></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=36182966&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=36182962&type= 67.90267181 4.06400013 ，其中： <math id="M59"><mover><mrow><mi>s</mi></mrow><mrow><mi>˜</mi></mrow></mover></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=36182976&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=36182971&type= 1.26999998 2.79399991 表示位置域中的拉普拉斯算子； <math id="M60"><mi>Γ</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=36182985&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=36182978&type= 2.70933342 2.79399991 表示反馈增益； <math id="M61"><mi>Φ</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=36182993&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=36182990&type= 2.87866688 2.79399991 是学习增益； <math id="M62"><mi>α</mi><mo>∈</mo><mo stretchy="false">(</mo><mn mathvariant="normal">0,1</mn><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=36183002&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=36182999&type= 14.13933277 3.55599999 为遗忘因子，用于抑制非周期性信号的影响，其大小与收敛速度负相关，为保证系统动态响应速度， <math id="M63"><mi>α</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=36183014&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=36183008&type= 1.94733346 2.79399991 应在条件允许范围内尽可能的小。该表达式是以位置域中的拉普拉斯算子为自变量得到的，相较于传统的时间域中的拉普拉斯算子，二者之间只是选取变量的表达形式有所不同，其所代表的实际物理意义是一样的。二者之间可以通过“域转换”原理进行变换，具体变换形式为： <math display="block" id="M64"><mi>f</mi><mfenced separators="|"><mrow><mi>t</mi></mrow></mfenced><mo>=</mo><mover><mrow><mi>f</mi></mrow><mrow><mi>˜</mi></mrow></mover><mfenced separators="|"><mrow><msub><mrow><mi>θ</mi></mrow><mrow><mi mathvariant="normal">m</mi></mrow></msub><mfenced separators="|"><mrow><mi>t</mi></mrow></mfenced></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=36183021&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=36183018&type= 26.07733345 4.74133301 ，其中： <math id="M65"><mi>f</mi><mfenced separators="|"><mrow><mi>t</mi></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=36183028&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=36183025&type= 7.53533268 4.06400013 是时间域中的函数， <math id="M66"><mover><mrow><mi>f</mi></mrow><mrow><mi>˜</mi></mrow></mover><mfenced separators="|"><mrow><msub><mrow><mi>θ</mi></mrow><mrow><mi mathvariant="normal">m</mi></mrow></msub><mfenced separators="|"><mrow><mi>t</mi></mrow></mfenced></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=36183043&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=36183040&type= 14.39333248 4.74133301 是位置域中的函数，对 <math id="M67"><mover><mrow><mi>f</mi></mrow><mrow><mi>˜</mi></mrow></mover><mfenced separators="|"><mrow><msub><mrow><mi>θ</mi></mrow><mrow><mi mathvariant="normal">m</mi></mrow></msub><mfenced separators="|"><mrow><mi>t</mi></mrow></mfenced></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=36183043&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=36183040&type= 14.39333248 4.74133301 求导可得： <math display="block" id="M68"><mfrac><mrow><mi mathvariant="normal">d</mi><mover><mrow><mi>f</mi></mrow><mrow><mi>˜</mi></mrow></mover><mfenced open="[" close="]" separators="|"><mrow><msub><mrow><mi>θ</mi></mrow><mrow><mi mathvariant="normal">m</mi></mrow></msub><mfenced separators="|"><mrow><mi>t</mi></mrow></mfenced></mrow></mfenced></mrow><mrow><mi mathvariant="normal">d</mi><msub><mrow><mi>θ</mi></mrow><mrow><mi mathvariant="normal">m</mi></mrow></msub><mfenced separators="|"><mrow><mi>t</mi></mrow></mfenced></mrow></mfrac><mo>=</mo><mfrac><mrow><mi mathvariant="normal">d</mi><mi>t</mi></mrow><mrow><mi mathvariant="normal">d</mi><msub><mrow><mi>θ</mi></mrow><mrow><mi mathvariant="normal">m</mi></mrow></msub><mfenced separators="|"><mrow><mi>t</mi></mrow></mfenced></mrow></mfrac><mo>⋅</mo><mfrac><mrow><mi mathvariant="normal">d</mi><mi>f</mi><mfenced separators="|"><mrow><mi>t</mi></mrow></mfenced></mrow><mrow><mi mathvariant="normal">d</mi><mi>t</mi></mrow></mfrac></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=36183049&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=36183046&type= 43.43399811 10.15999985 . 在框架系统中 <math id="M69"><msub><mrow><mi>ω</mi></mrow><mrow><mi mathvariant="normal">m</mi></mrow></msub><mfenced separators="|"><mrow><mi>t</mi></mrow></mfenced><mo>=</mo><mfrac><mrow><mi mathvariant="normal">d</mi><msub><mrow><mi>θ</mi></mrow><mrow><mi mathvariant="normal">m</mi></mrow></msub><mfenced separators="|"><mrow><mi>t</mi></mrow></mfenced></mrow><mrow><mi mathvariant="normal">d</mi><mi>t</mi></mrow></mfrac></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=36183056&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=36183051&type= 24.55333328 8.29733276 ，因此 <math id="M70"><mover><mrow><mi>s</mi></mrow><mrow><mi>˜</mi></mrow></mover></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=36183061&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=36183058&type= 1.26999998 2.79399991 和 <math id="M71"><mi>s</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=36183067&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=36183064&type= 1.26999998 2.79399991 之间的关系可以化简为： <math display="block" id="M72"><mover><mrow><mi>s</mi></mrow><mrow><mi>˜</mi></mrow></mover><mo>=</mo><mfrac><mrow><mn mathvariant="normal">1</mn></mrow><mrow><msub><mrow><mi>ω</mi></mrow><mrow><mi mathvariant="normal">m</mi></mrow></msub><mfenced separators="|"><mrow><mi>t</mi></mrow></mfenced></mrow></mfrac><mi>s</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=36183074&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=36183071&type= 17.78000069 9.05933285 . 根据（8），可以将（5）在时间域中表示为： <math display="block" id="M73"><msub><mrow><mi>u</mi></mrow><mrow><mi>j</mi></mrow></msub><mfenced separators="|"><mrow><mi>s</mi></mrow></mfenced><mo>=</mo><mi>Γ</mi><msub><mrow><mi>e</mi></mrow><mrow><mi>j</mi></mrow></msub><mfenced separators="|"><mrow><mi>s</mi></mrow></mfenced><mo>+</mo><mi>Φ</mi><msub><mrow><mi>e</mi></mrow><mrow><mi>j</mi><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msub><mfenced separators="|"><mrow><mi>s</mi></mrow></mfenced><mo>+</mo><mfenced separators="|"><mrow><mn mathvariant="normal">1</mn><mo>-</mo><mi>α</mi></mrow></mfenced><msub><mrow><mi>u</mi></mrow><mrow><mi>j</mi><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msub><mfenced separators="|"><mrow><mi>s</mi></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=36183083&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=36183078&type= 67.90267181 4.06400013 . 使用双线性变换法将（9）离散化为： <math display="block" id="M74"><msub><mrow><mi>u</mi></mrow><mrow><mi>j</mi></mrow></msub><mfenced separators="|"><mrow><mi>z</mi></mrow></mfenced><mo>=</mo><mi>Γ</mi><msub><mrow><mi>e</mi></mrow><mrow><mi>j</mi></mrow></msub><mfenced separators="|"><mrow><mi>z</mi></mrow></mfenced><mo>+</mo><mi>Φ</mi><msub><mrow><mi>e</mi></mrow><mrow><mi>j</mi><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msub><mfenced separators="|"><mrow><mi>z</mi></mrow></mfenced><mo>+</mo><mfenced separators="|"><mrow><mn mathvariant="normal">1</mn><mo>-</mo><mi>α</mi></mrow></mfenced><msub><mrow><mi>u</mi></mrow><mrow><mi>j</mi><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msub><mfenced separators="|"><mrow><mi>z</mi></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=36183092&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=36183088&type= 66.88666534 4.06400013 . 由于迭代学习控制律中不包含具体的拉普拉斯算子，因此（9）和（10）只需要更换自变量即可。结合迭代学习控制之后，完整的框架系统控制框图如图3 图3 所示，其中红框部分可以等效为被控对象 G p （ s ）： <math display="block" id="M75"><mtable><mtr><mtd><msub><mrow><mi>G</mi></mrow><mrow><mi>p</mi></mrow></msub><mfenced separators="|"><mrow><mi>s</mi></mrow></mfenced><mo>=</mo><mfrac><mrow><mn mathvariant="normal">1</mn></mrow><mrow><mi mathvariant="normal">Δ</mi></mrow></mfrac><mi>P</mi><mfenced separators="|"><mrow><mi>s</mi></mrow></mfenced><msub><mrow><mi mathvariant="normal">Δ</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>=</mo></mtd></mtr><mtr><mtd><mfrac><mrow><mi>N</mi><msub><mrow><mi>K</mi></mrow><mrow><mi mathvariant="normal">m</mi></mrow></msub><msub><mrow><mi>K</mi></mrow><mrow><mi mathvariant="normal">h</mi></mrow></msub><msub><mrow><mi>G</mi></mrow><mrow><mi>i</mi></mrow></msub><mfenced separators="|"><mrow><mi>s</mi></mrow></mfenced><msub><mrow><mi>G</mi></mrow><mrow><mi mathvariant="normal">m</mi></mrow></msub><mfenced separators="|"><mrow><mi>s</mi></mrow></mfenced><msub><mrow><mi>G</mi></mrow><mrow><mi mathvariant="normal">m</mi><mi mathvariant="normal">l</mi></mrow></msub><mfenced separators="|"><mrow><mi>s</mi></mrow></mfenced><msub><mrow><mi>G</mi></mrow><mrow><mi mathvariant="normal">l</mi></mrow></msub><mfenced separators="|"><mrow><mi>s</mi></mrow></mfenced></mrow><mrow><msup><mrow><mi>N</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mfenced open="[" close="]" separators="|"><mrow><msub><mrow><mi>Q</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mfenced separators="|"><mrow><mi>s</mi></mrow></mfenced><mo>+</mo><msub><mrow><mi>Q</mi></mrow><mrow><mn mathvariant="normal">3</mn></mrow></msub><mfenced separators="|"><mrow><mi>s</mi></mrow></mfenced></mrow></mfenced><msub><mrow><mi>Q</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mfenced separators="|"><mrow><mi>s</mi></mrow></mfenced><mo>+</mo><msub><mrow><mi>K</mi></mrow><mrow><mi mathvariant="normal">h</mi></mrow></msub><msub><mrow><mi>G</mi></mrow><mrow><mi mathvariant="normal">m</mi><mi mathvariant="normal">l</mi></mrow></msub><mfenced separators="|"><mrow><mi>s</mi></mrow></mfenced><msub><mrow><mi>Q</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mfenced separators="|"><mrow><mi>s</mi></mrow></mfenced></mrow></mfrac><mo>,</mo></mtd></mtr></mtable></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=36183098&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=36183096&type= 69.17266846 19.89666748 其中： <math id="M76"><msub><mrow><mi>Q</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mfenced separators="|"><mrow><mi>s</mi></mrow></mfenced><mo>=</mo><mn mathvariant="normal">1</mn><mo>+</mo><msub><mrow><mi>G</mi></mrow><mrow><mi>i</mi></mrow></msub><mfenced separators="|"><mrow><mi>s</mi></mrow></mfenced><msub><mrow><mi>G</mi></mrow><mrow><mi mathvariant="normal">m</mi></mrow></msub><mfenced separators="|"><mrow><mi>s</mi></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=36183107&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=36183102&type= 38.43866730 3.97933316 ， <math id="M77"><msub><mrow><mi>Q</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mfenced separators="|"><mrow><mi>s</mi></mrow></mfenced><mo>=</mo><mi>s</mi><mo>+</mo><msub><mrow><mi>K</mi></mrow><mrow><mi mathvariant="normal">h</mi></mrow></msub><msub><mrow><mi>G</mi></mrow><mrow><mi mathvariant="normal">l</mi></mrow></msub><mfenced separators="|"><mrow><mi>s</mi></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=36183112&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=36183108&type= 32.00400162 3.97933316 ， <math id="M78"><msub><mrow><mi>Q</mi></mrow><mrow><mn mathvariant="normal">3</mn></mrow></msub><mfenced separators="|"><mrow><mi>s</mi></mrow></mfenced><mo>=</mo><msub><mrow><mi>K</mi></mrow><mrow><mi mathvariant="normal">m</mi></mrow></msub><msub><mrow><mi>C</mi></mrow><mrow><mi mathvariant="normal">e</mi></mrow></msub><msub><mrow><mi>G</mi></mrow><mrow><mi mathvariant="normal">m</mi></mrow></msub><mfenced separators="|"><mrow><mi>s</mi></mrow></mfenced><msub><mrow><mi>G</mi></mrow><mrow><mi mathvariant="normal">m</mi><mi mathvariant="normal">l</mi></mrow></msub><mfenced separators="|"><mrow><mi>s</mi></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=36183118&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=36183115&type= 41.74066544 3.97933316 。使用双线性变换法，将 <math id="M79"><msub><mrow><mi>G</mi></mrow><mrow><mi>p</mi></mrow></msub><mfenced separators="|"><mrow><mi>s</mi></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=36183125&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=36183122&type= 8.72066593 4.06400013 和 <math id="M80"><msub><mrow><mi>G</mi></mrow><mrow><mi>ω</mi></mrow></msub><mfenced separators="|"><mrow><mi>s</mi></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=36183131&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=36183127&type= 9.05933285 3.97933316 离散化为： <math display="block" id="M81"><mtable><mtr><mtd><msub><mrow><mi>G</mi></mrow><mrow><mi>p</mi></mrow></msub><mfenced separators="|"><mrow><mi>z</mi></mrow></mfenced><mo>=</mo></mtd></mtr><mtr><mtd><mfrac><mrow><mi>N</mi><mi>T</mi><msub><mrow><mi>K</mi></mrow><mrow><mi mathvariant="normal">m</mi></mrow></msub><msub><mrow><mi>K</mi></mrow><mrow><mi mathvariant="normal">h</mi></mrow></msub><msub><mrow><mi>G</mi></mrow><mrow><mi>i</mi></mrow></msub><mfenced separators="|"><mrow><mi>z</mi></mrow></mfenced><msub><mrow><mi>G</mi></mrow><mrow><mi mathvariant="normal">m</mi></mrow></msub><mfenced separators="|"><mrow><mi>z</mi></mrow></mfenced><msub><mrow><mi>G</mi></mrow><mrow><mi mathvariant="normal">m</mi><mi mathvariant="normal">l</mi></mrow></msub><mfenced separators="|"><mrow><mi>z</mi></mrow></mfenced><msub><mrow><mi>G</mi></mrow><mrow><mi mathvariant="normal">l</mi></mrow></msub><mfenced separators="|"><mrow><mi>z</mi></mrow></mfenced><mfenced separators="|"><mrow><mi>z</mi><mo>+</mo><mn mathvariant="normal">1</mn></mrow></mfenced></mrow><mrow><msup><mrow><mi>N</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mfenced open="[" close="]" separators="|"><mrow><msub><mrow><mi>H</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mfenced separators="|"><mrow><mi>z</mi></mrow></mfenced><mo>+</mo><msub><mrow><mi>H</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mfenced separators="|"><mrow><mi>z</mi></mrow></mfenced></mrow></mfenced><msub><mrow><mi>H</mi></mrow><mrow><mn mathvariant="normal">3</mn></mrow></msub><mfenced separators="|"><mrow><mi>z</mi></mrow></mfenced><mo>+</mo><msub><mrow><mi>K</mi></mrow><mrow><mi mathvariant="normal">h</mi></mrow></msub><msub><mrow><mi>G</mi></mrow><mrow><mi mathvariant="normal">m</mi><mi mathvariant="normal">l</mi></mrow></msub><mfenced separators="|"><mrow><mi>z</mi></mrow></mfenced><msub><mrow><mi>H</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mfenced separators="|"><mrow><mi>z</mi></mrow></mfenced></mrow></mfrac><mo>,</mo></mtd></mtr></mtable></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=36183137&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=36183133&type= 74.25267029 16.17133331 <math display="block" id="M82"><msub><mrow><mi>G</mi></mrow><mrow><mi>ω</mi></mrow></msub><mfenced separators="|"><mrow><mi>z</mi></mrow></mfenced><mo>=</mo><msubsup><mrow><mi>k</mi></mrow><mrow><mi>p</mi></mrow><mrow><mi>ω</mi></mrow></msubsup><mo>+</mo><mfrac><mrow><mn mathvariant="normal">2</mn><msubsup><mrow><mi>k</mi></mrow><mrow><mi>i</mi></mrow><mrow><mi>ω</mi></mrow></msubsup><mfenced separators="|"><mrow><mi>z</mi><mo>-</mo><mn mathvariant="normal">1</mn></mrow></mfenced></mrow><mrow><mi>T</mi><mfenced separators="|"><mrow><mi>z</mi><mo>+</mo><mn mathvariant="normal">1</mn></mrow></mfenced></mrow></mfrac></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=36183140&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=36183138&type= 40.30133438 9.56733322 ，其中： <math id="M83"><msub><mrow><mi>G</mi></mrow><mrow><mi>i</mi></mrow></msub><mfenced separators="|"><mrow><mi>z</mi></mrow></mfenced><mo>=</mo><msubsup><mrow><mi>k</mi></mrow><mrow><mi>p</mi></mrow><mrow><mi>i</mi></mrow></msubsup><mo>+</mo><mfrac><mrow><mn mathvariant="normal">2</mn><msubsup><mrow><mi>k</mi></mrow><mrow><mi>i</mi></mrow><mrow><mi>i</mi></mrow></msubsup><mfenced separators="|"><mrow><mi>z</mi><mo>-</mo><mn mathvariant="normal">1</mn></mrow></mfenced></mrow><mrow><mi>T</mi><mfenced separators="|"><mrow><mi>z</mi><mo>+</mo><mn mathvariant="normal">1</mn></mrow></mfenced></mrow></mfrac></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=36183144&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=36183142&type= 38.26933289 9.56733322 ， <math display="block" id="M84"><msub><mrow><mi>G</mi></mrow><mrow><mi mathvariant="normal">m</mi></mrow></msub><mfenced separators="|"><mrow><mi>z</mi></mrow></mfenced><mo>=</mo><mfrac><mrow><mi>T</mi><mfenced separators="|"><mrow><mi>z</mi><mo>+</mo><mn mathvariant="normal">1</mn></mrow></mfenced></mrow><mrow><mn mathvariant="normal">2</mn><mi>L</mi><mfenced separators="|"><mrow><mi>z</mi><mo>-</mo><mn mathvariant="normal">1</mn></mrow></mfenced><mo>+</mo><mi>T</mi><mi>R</mi><mfenced separators="|"><mrow><mi>z</mi><mo>+</mo><mn mathvariant="normal">1</mn></mrow></mfenced></mrow></mfrac></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=36183148&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=36183146&type= 53.76333237 9.65200043 ， <math display="block" id="M85"><msub><mrow><mi>G</mi></mrow><mrow><mi mathvariant="normal">m</mi><mi mathvariant="normal">l</mi></mrow></msub><mfenced separators="|"><mrow><mi>z</mi></mrow></mfenced><mo>=</mo><mfrac><mrow><mi>T</mi><mfenced separators="|"><mrow><mi>z</mi><mo>+</mo><mn mathvariant="normal">1</mn></mrow></mfenced></mrow><mrow><mn mathvariant="normal">2</mn><msub><mrow><mi>J</mi></mrow><mrow><mi mathvariant="normal">m</mi></mrow></msub><mfenced separators="|"><mrow><mi>z</mi><mo>-</mo><mn mathvariant="normal">1</mn></mrow></mfenced><mo>+</mo><mi>T</mi><msub><mrow><mi>B</mi></mrow><mrow><mi mathvariant="normal">m</mi></mrow></msub><mfenced separators="|"><mrow><mi>z</mi><mo>+</mo><mn mathvariant="normal">1</mn></mrow></mfenced></mrow></mfrac></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=36183153&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=36183151&type= 56.13399887 9.65200043 ， <math display="block" id="M86"><msub><mrow><mi>G</mi></mrow><mrow><mi mathvariant="normal">l</mi></mrow></msub><mfenced separators="|"><mrow><mi>z</mi></mrow></mfenced><mo>=</mo><mfrac><mrow><mi>T</mi><mfenced separators="|"><mrow><mi>z</mi><mo>+</mo><mn mathvariant="normal">1</mn></mrow></mfenced></mrow><mrow><mn mathvariant="normal">2</mn><msub><mrow><mi>J</mi></mrow><mrow><mi mathvariant="normal">l</mi></mrow></msub><mfenced separators="|"><mrow><mi>z</mi><mo>-</mo><mn mathvariant="normal">1</mn></mrow></mfenced><mo>+</mo><mi>T</mi><msub><mrow><mi>B</mi></mrow><mrow><mi mathvariant="normal">l</mi></mrow></msub><mfenced separators="|"><mrow><mi>z</mi><mo>+</mo><mn mathvariant="normal">1</mn></mrow></mfenced></mrow></mfrac></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=36183157&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=36183155&type= 52.57799911 9.65200043 ， <math display="block" id="M87"><msub><mrow><mi>H</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mfenced separators="|"><mrow><mi>z</mi></mrow></mfenced><mo>=</mo><msub><mrow><mi>G</mi></mrow><mrow><mi>i</mi></mrow></msub><mfenced separators="|"><mrow><mi>z</mi></mrow></mfenced><msub><mrow><mi>G</mi></mrow><mrow><mi mathvariant="normal">m</mi></mrow></msub><mfenced separators="|"><mrow><mi>z</mi></mrow></mfenced><mo>+</mo><mn mathvariant="normal">1</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=36183161&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=36183159&type= 40.21666718 3.97933316 ， <math display="block" id="M88"><msub><mrow><mi>H</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mfenced separators="|"><mrow><mi>z</mi></mrow></mfenced><mo>=</mo><msub><mrow><mi>K</mi></mrow><mrow><mi mathvariant="normal">m</mi></mrow></msub><msub><mrow><mi>C</mi></mrow><mrow><mi mathvariant="normal">e</mi></mrow></msub><msub><mrow><mi>G</mi></mrow><mrow><mi mathvariant="normal">m</mi></mrow></msub><mfenced separators="|"><mrow><mi>z</mi></mrow></mfenced><msub><mrow><mi>G</mi></mrow><mrow><mi mathvariant="normal">m</mi><mi mathvariant="normal">l</mi></mrow></msub><mfenced separators="|"><mrow><mi>z</mi></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=36183166&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=36183163&type= 43.51866913 3.97933316 ， <math display="block" id="M89"><msub><mrow><mi>H</mi></mrow><mrow><mn mathvariant="normal">3</mn></mrow></msub><mfenced separators="|"><mrow><mi>z</mi></mrow></mfenced><mo>=</mo><mn mathvariant="normal">2</mn><mfenced separators="|"><mrow><mi>z</mi><mo>-</mo><mn mathvariant="normal">1</mn></mrow></mfenced><mo>+</mo><msub><mrow><mi>K</mi></mrow><mrow><mi mathvariant="normal">h</mi></mrow></msub><mi>T</mi><mfenced separators="|"><mrow><mi>z</mi><mo>+</mo><mn mathvariant="normal">1</mn></mrow></mfenced><msub><mrow><mi>G</mi></mrow><mrow><mi mathvariant="normal">l</mi></mrow></msub><mfenced separators="|"><mrow><mi>z</mi></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=36183172&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=36183169&type= 61.29866791 4.06400013 . 收敛性是迭代学习控制的主要指标之一，只有当迭代学习控制算法收敛，才能够确保控制系统的整体稳定性，根据图3 图3 可以得到此时系统的状态： <math display="block" id="M90"><mfenced open="{" close="" separators="|"><mrow><mtable columnalign="left"><mtr><mtd><msub><mrow><mi>u</mi></mrow><mrow><mi>s</mi><mo>,</mo><mi>j</mi></mrow></msub><mfenced separators="|"><mrow><mi>z</mi></mrow></mfenced><mo>=</mo><msub><mrow><mi>u</mi></mrow><mrow><mi>c</mi><mo>,</mo><mi>j</mi></mrow></msub><mfenced separators="|"><mrow><mi>z</mi></mrow></mfenced><mo>+</mo><msub><mrow><mi>u</mi></mrow><mrow><mi>j</mi></mrow></msub><mfenced separators="|"><mrow><mi>z</mi></mrow></mfenced></mtd></mtr><mtr><mtd><msub><mrow><mi>u</mi></mrow><mrow><mi>s</mi><mo>,</mo><mi>j</mi><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msub><mfenced separators="|"><mrow><mi>z</mi></mrow></mfenced><mo>=</mo><msub><mrow><mi>u</mi></mrow><mrow><mi>c</mi><mo>,</mo><mi>j</mi><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msub><mfenced separators="|"><mrow><mi>z</mi></mrow></mfenced><mo>+</mo><msub><mrow><mi>u</mi></mrow><mrow><mi>j</mi><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msub><mfenced separators="|"><mrow><mi>z</mi></mrow></mfenced></mtd></mtr><mtr><mtd><msub><mrow><mi>u</mi></mrow><mrow><mi>c</mi><mo>,</mo><mi>j</mi></mrow></msub><mfenced separators="|"><mrow><mi>z</mi></mrow></mfenced><mo>=</mo><msub><mrow><mi>G</mi></mrow><mrow><mi>ω</mi></mrow></msub><mfenced separators="|"><mrow><mi>z</mi></mrow></mfenced><msub><mrow><mi>e</mi></mrow><mrow><mi>j</mi></mrow></msub><mfenced separators="|"><mrow><mi>z</mi></mrow></mfenced></mtd></mtr><mtr><mtd><msub><mrow><mi>u</mi></mrow><mrow><mi>c</mi><mo>,</mo><mi>j</mi><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msub><mfenced separators="|"><mrow><mi>z</mi></mrow></mfenced><mo>=</mo><msub><mrow><mi>G</mi></mrow><mrow><mi>ω</mi></mrow></msub><mfenced separators="|"><mrow><mi>z</mi></mrow></mfenced><msub><mrow><mi>e</mi></mrow><mrow><mi>j</mi><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msub><mfenced separators="|"><mrow><mi>z</mi></mrow></mfenced></mtd></mtr><mtr><mtd><msub><mrow><mi>u</mi></mrow><mrow><mi>j</mi></mrow></msub><mfenced separators="|"><mrow><mi>z</mi></mrow></mfenced><mo>=</mo><mi>Γ</mi><msub><mrow><mi>e</mi></mrow><mrow><mi>j</mi></mrow></msub><mfenced separators="|"><mrow><mi>z</mi></mrow></mfenced><mo>+</mo><mi>Φ</mi><msub><mrow><mi>e</mi></mrow><mrow><mi>j</mi><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msub><mfenced separators="|"><mrow><mi>z</mi></mrow></mfenced><mo>+</mo><mfenced separators="|"><mrow><mn mathvariant="normal">1</mn><mo>-</mo><mi>α</mi></mrow></mfenced><msub><mrow><mi>u</mi></mrow><mrow><mi>j</mi><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msub><mfenced separators="|"><mrow><mi>z</mi></mrow></mfenced></mtd></mtr><mtr><mtd><msub><mrow><mi>ω</mi></mrow><mrow><mi>l</mi><mo>,</mo><mi>j</mi></mrow></msub><mfenced separators="|"><mrow><mi>z</mi></mrow></mfenced><mo>=</mo><msub><mrow><mi>G</mi></mrow><mrow><mi>p</mi></mrow></msub><mfenced separators="|"><mrow><mi>z</mi></mrow></mfenced><msub><mrow><mi>u</mi></mrow><mrow><mi>s</mi><mo>,</mo><mi>j</mi></mrow></msub><mfenced separators="|"><mrow><mi>z</mi></mrow></mfenced></mtd></mtr><mtr><mtd><msub><mrow><mi>ω</mi></mrow><mrow><mi>l</mi><mo>,</mo><mi>j</mi><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msub><mfenced separators="|"><mrow><mi>z</mi></mrow></mfenced><mo>=</mo><msub><mrow><mi>G</mi></mrow><mrow><mi>p</mi></mrow></msub><mfenced separators="|"><mrow><mi>z</mi></mrow></mfenced><msub><mrow><mi>u</mi></mrow><mrow><mi>s</mi><mo>,</mo><mi>j</mi><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msub><mfenced separators="|"><mrow><mi>z</mi></mrow></mfenced></mtd></mtr></mtable></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=36183185&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=36183183&type= 71.96665955 39.96266937 ，其中： <math id="M91"><msub><mrow><mi>u</mi></mrow><mrow><mi>j</mi></mrow></msub><mfenced separators="|"><mrow><mi>z</mi></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=36183189&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=36183187&type= 7.87400007 4.06400013 、 <math id="M92"><msub><mrow><mi>u</mi></mrow><mrow><mi>c</mi><mo>,</mo><mi>j</mi></mrow></msub><mfenced separators="|"><mrow><mi>z</mi></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=36183193&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=36183191&type= 9.65200043 4.06400013 和 <math id="M93"><msub><mrow><mi>u</mi></mrow><mrow><mi>s</mi><mo>,</mo><mi>j</mi></mrow></msub><mfenced separators="|"><mrow><mi>z</mi></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=36183197&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=36183195&type= 9.48266602 4.06400013 分别是第 j 个时刻迭代学习前馈的输出、速度环控制器的输出和 <math id="M94"><msub><mrow><mi>G</mi></mrow><mrow><mi>p</mi></mrow></msub><mfenced separators="|"><mrow><mi>z</mi></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=36183200&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=36183198&type= 9.22866726 4.06400013 的参考。根据（14），可以得到： <math display="block" id="M95"><mtable><mtr><mtd><msub><mrow><mi>e</mi></mrow><mrow><mi>j</mi><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msub><mfenced separators="|"><mrow><mi>z</mi></mrow></mfenced><mo>=</mo><msub><mrow><mi>ω</mi></mrow><mrow><mi mathvariant="normal">m</mi></mrow></msub><mfenced separators="|"><mrow><mi>z</mi></mrow></mfenced><mo>-</mo><mi>N</mi><msub><mrow><mi>ω</mi></mrow><mrow><mi mathvariant="normal">l</mi><mo>,</mo><mi>j</mi><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msub><mfenced separators="|"><mrow><mi>z</mi></mrow></mfenced><mo>=</mo></mtd></mtr><mtr><mtd><mfrac><mrow><msub><mrow><mi>ω</mi></mrow><mrow><mi mathvariant="normal">m</mi></mrow></msub><mfenced separators="|"><mrow><mi>z</mi></mrow></mfenced></mrow><mrow><mn mathvariant="normal">1</mn><mo>+</mo><mi>N</mi><msub><mrow><mi>G</mi></mrow><mrow><mi>ω</mi></mrow></msub><mfenced separators="|"><mrow><mi>z</mi></mrow></mfenced><msub><mrow><mi>G</mi></mrow><mrow><mi>p</mi></mrow></msub><mfenced separators="|"><mrow><mi>z</mi></mrow></mfenced></mrow></mfrac><mo>-</mo><mfrac><mrow><mi>N</mi><msub><mrow><mi>G</mi></mrow><mrow><mi>p</mi></mrow></msub><mfenced separators="|"><mrow><mi>z</mi></mrow></mfenced><msub><mrow><mi>u</mi></mrow><mrow><mi>j</mi><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msub><mfenced separators="|"><mrow><mi>z</mi></mrow></mfenced></mrow><mrow><mn mathvariant="normal">1</mn><mo>+</mo><mi>N</mi><msub><mrow><mi>G</mi></mrow><mrow><mi>ω</mi></mrow></msub><mfenced separators="|"><mrow><mi>z</mi></mrow></mfenced><msub><mrow><mi>G</mi></mrow><mrow><mi>p</mi></mrow></msub><mfenced separators="|"><mrow><mi>z</mi></mrow></mfenced></mrow></mfrac></mtd></mtr></mtable></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=36183204&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=36183202&type= 64.00800323 15.40933418 ，将（14）中第5式与（15）相结合，通过递推的方式可以得到第 j 个时刻的速度调节误差表达式： <math display="block" id="M96"><mtable><mtr><mtd><msub><mrow><mi>e</mi></mrow><mrow><mi>j</mi></mrow></msub><mfenced separators="|"><mrow><mi>z</mi></mrow></mfenced><mo>=</mo></mtd></mtr><mtr><mtd><mfrac><mrow><mfenced separators="|"><mrow><mn mathvariant="normal">1</mn><mo>-</mo><mi>α</mi></mrow></mfenced><mo>+</mo><mi>N</mi><msub><mrow><mi>G</mi></mrow><mrow><mi>p</mi></mrow></msub><mfenced separators="|"><mrow><mi>z</mi></mrow></mfenced><mfenced open="[" close="]" separators="|"><mrow><mfenced separators="|"><mrow><mn mathvariant="normal">1</mn><mo>-</mo><mi>α</mi></mrow></mfenced><msub><mrow><mi>G</mi></mrow><mrow><mi>ω</mi></mrow></msub><mfenced separators="|"><mrow><mi>z</mi></mrow></mfenced><mo>-</mo><mi>Φ</mi></mrow></mfenced></mrow><mrow><mn mathvariant="normal">1</mn><mo>+</mo><mi>N</mi><msub><mrow><mi>G</mi></mrow><mrow><mi>p</mi></mrow></msub><mfenced separators="|"><mrow><mi>z</mi></mrow></mfenced><mfenced open="[" close="]" separators="|"><mrow><msub><mrow><mi>G</mi></mrow><mrow><mi>ω</mi></mrow></msub><mfenced separators="|"><mrow><mi>z</mi></mrow></mfenced><mo>+</mo><mi>Γ</mi></mrow></mfenced></mrow></mfrac><msub><mrow><mi>e</mi></mrow><mrow><mi>j</mi><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msub><mfenced separators="|"><mrow><mi>z</mi></mrow></mfenced><mo>+</mo></mtd></mtr><mtr><mtd><mtext> </mtext><mfrac><mrow><mi>α</mi><msub><mrow><mi>ω</mi></mrow><mrow><mi mathvariant="normal">m</mi></mrow></msub><mfenced separators="|"><mrow><mi>z</mi></mrow></mfenced></mrow><mrow><mn mathvariant="normal">1</mn><mo>+</mo><mi>N</mi><msub><mrow><mi>G</mi></mrow><mrow><mi>p</mi></mrow></msub><mfenced separators="|"><mrow><mi>z</mi></mrow></mfenced><mfenced open="[" close="]" separators="|"><mrow><msub><mrow><mi>G</mi></mrow><mrow><mi>ω</mi></mrow></msub><mfenced separators="|"><mrow><mi>z</mi></mrow></mfenced><mo>+</mo><mi>Γ</mi></mrow></mfenced></mrow></mfrac><mo>.</mo><mtext> </mtext><mo stretchy="false">(</mo><mn mathvariant="normal">16</mn><mo stretchy="false">)</mo></mtd></mtr></mtable></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=36183206&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=36183205&type= 76.87733459 28.87133408 为了尽可能减少阻尼等与速度相关的扰动的影响，使得速率波动的位置域周期性更加明显，从而获取准确的补偿表，系统需要运行于匀速状态。因此（16）式的右侧第二项为一恒值常量，系统的收敛条件可以表示为： <math display="block" id="M97"><mfenced open="|" close="|" separators="|"><mrow><mfrac><mrow><mfenced separators="|"><mrow><mn mathvariant="normal">1</mn><mo>-</mo><mi>α</mi></mrow></mfenced><mo>+</mo><mi>N</mi><msub><mrow><mi>G</mi></mrow><mrow><mi>p</mi></mrow></msub><mfenced separators="|"><mrow><mi>z</mi></mrow></mfenced><mfenced open="[" close="]" separators="|"><mrow><mfenced separators="|"><mrow><mn mathvariant="normal">1</mn><mo>-</mo><mi>α</mi></mrow></mfenced><msub><mrow><mi>G</mi></mrow><mrow><mi>ω</mi></mrow></msub><mfenced separators="|"><mrow><mi>z</mi></mrow></mfenced><mo>-</mo><mi>Φ</mi></mrow></mfenced></mrow><mrow><mn mathvariant="normal">1</mn><mo>+</mo><mi>N</mi><msub><mrow><mi>G</mi></mrow><mrow><mi>p</mi></mrow></msub><mfenced separators="|"><mrow><mi>z</mi></mrow></mfenced><mfenced open="[" close="]" separators="|"><mrow><msub><mrow><mi>G</mi></mrow><mrow><mi>ω</mi></mrow></msub><mfenced separators="|"><mrow><mi>z</mi></mrow></mfenced><mo>+</mo><mi>Γ</mi></mrow></mfenced></mrow></mfrac></mrow></mfenced><mo>≤</mo><mi>ρ</mi><mo><</mo><mn mathvariant="normal">1</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=36183208&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=36183207&type= 74.08333588 12.02266598 . 在参数选取时，首先根据噪声抑制效果与迭代次数之间的关系确定遗忘因子，然后通过极点配置和收敛条件获取参数选取范围，最后通过仿真和试验效果确定参数取值。在实际系统中，补偿信号会在迭代过程中引入遗忘因子难以完全抑制的非周期性噪声。多次迭代后，当调节误差中周期性波动的幅值小于噪声的幅值时，继续迭代会放大噪声，使得补偿效果变差。为此，本文通过如下停止条件来确定迭代停止时刻。 <math display="block" id="M98"><msubsup><mrow><mi>ω</mi></mrow><mrow><mi>l</mi><mo>,</mo><mi>j</mi></mrow><mrow><mi mathvariant="normal">m</mi><mi mathvariant="normal">a</mi><mi mathvariant="normal">x</mi></mrow></msubsup><mo>=</mo><mi mathvariant="normal">m</mi><mi mathvariant="normal">i</mi><mi mathvariant="normal">n</mi><mfenced separators="|"><mrow><mfrac><mrow><mn mathvariant="normal">1</mn></mrow><mrow><mn mathvariant="normal">3</mn></mrow></mfrac><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>p</mi><mo>=</mo><mn mathvariant="normal">1</mn></mrow><mrow><mn mathvariant="normal">3</mn></mrow></munderover></mstyle><mrow><msubsup><mrow><mi>ω</mi></mrow><mrow><mi mathvariant="normal">l</mi><mo>,</mo><mi>j</mi><mo>-</mo><mi>p</mi></mrow><mrow><mi mathvariant="normal">m</mi><mi mathvariant="normal">a</mi><mi mathvariant="normal">x</mi></mrow></msubsup></mrow><mo>,</mo><msubsup><mrow><mi>ω</mi></mrow><mrow><mi mathvariant="normal">l</mi><mo>,</mo><mi>j</mi></mrow><mrow><mi mathvariant="normal">m</mi><mi mathvariant="normal">a</mi><mi mathvariant="normal">x</mi></mrow></msubsup><mo>,</mo><mfrac><mrow><mn mathvariant="normal">1</mn></mrow><mrow><mn mathvariant="normal">3</mn></mrow></mfrac><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>p</mi><mo>=</mo><mn mathvariant="normal">1</mn></mrow><mrow><mn mathvariant="normal">3</mn></mrow></munderover></mstyle><mrow><msubsup><mrow><mi>ω</mi></mrow><mrow><mi mathvariant="normal">l</mi><mo>,</mo><mi>j</mi><mo>+</mo><mi>p</mi></mrow><mrow><mi mathvariant="normal">m</mi><mi mathvariant="normal">a</mi><mi mathvariant="normal">x</mi></mrow></msubsup></mrow></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=36183210&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=36183209&type= 64.85466003 10.49866676 ，其中： <math id="M99"><msubsup><mrow><mi>ω</mi></mrow><mrow><mi mathvariant="normal">l</mi><mo>,</mo><mi>j</mi></mrow><mrow><mi mathvariant="normal">m</mi><mi mathvariant="normal">a</mi><mi mathvariant="normal">x</mi></mrow></msubsup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=36183212&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=36183211&type= 6.09600019 3.97933316 表示单个角位置周期内的负载端转速最大值，通过定角测时方法获得，该方法将电机端角位置等分为36个测速区间，计算每个区间内的平均速度，36个平均速度中的最大值即为 <math id="M100"><msubsup><mrow><mi>ω</mi></mrow><mrow><mi mathvariant="normal">l</mi><mo>,</mo><mi>j</mi></mrow><mrow><mi mathvariant="normal">m</mi><mi mathvariant="normal">a</mi><mi mathvariant="normal">x</mi></mrow></msubsup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=36183214&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=36183213&type= 6.09600019 3.97933316 。当条件（18）成立时，迭代学习控制算法停止迭代，前馈补偿表建立。前馈补偿量根据当前电机端角位置查表获取，不受参考指令变化的影响，确保了框架系统在变速条件下的动态性能。 4 仿真验证 1 为验证所提方法的有效性和可行性，利用Matlab/Simulink对该方法进行了仿真验证，仿真中框架系统的主要参数如表1 表1 所示。参数参考值参数参考值反馈增益 <math id="M101"><mi>Γ</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=36183216&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=36183215&type= 2.28600001 2.37066650 3 减速比 <math id="M102"><mi>N</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=36183218&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=36183217&type= 2.79399991 2.45533323 100 学习增益 <math id="M103"><mi>Φ</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=36183220&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=36183219&type= 2.45533323 2.37066650 11 极对数 <math id="M104"><mi>P</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=36183222&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=36183221&type= 2.20133328 2.37066650 8 遗忘因子 <math id="M105"><mi>α</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=36183225&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=36183223&type= 1.69333339 2.37066650 0.01 电感 <math id="M106"><mi>L</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=36183229&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=36183227&type= 2.11666679 2.37066650 0.017 15 H 速度环比例环节 <math id="M107"><msubsup><mrow><mi>K</mi></mrow><mrow><mi>p</mi></mrow><mrow><mi>ω</mi></mrow></msubsup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=36183233&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=36183231&type= 4.14866638 3.38666677 5 电机绕组电阻 <math id="M108"><mi>R</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=36183235&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=36183234&type= 2.37066650 2.37066650 3.2 Ω 速度环积分环节 <math id="M109"><msubsup><mrow><mi>K</mi></mrow><mrow><mi>i</mi></mrow><mrow><mi>ω</mi></mrow></msubsup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=36183237&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=36183236&type= 4.14866638 3.38666677 0.08 电机力矩系数 <math id="M110"><msub><mrow><mi>K</mi></mrow><mrow><mi mathvariant="normal">m</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=36183239&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=36183238&type= 3.97933316 3.21733332 0.6 电流环比例环节 <math id="M111"><msubsup><mrow><mi>K</mi></mrow><mrow><mi>p</mi></mrow><mrow><mi>i</mi></mrow></msubsup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=36183241&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=36183240&type= 3.47133350 3.38666677 7 电机转动惯量 <math id="M112"><msub><mrow><mi>J</mi></mrow><mrow><mi mathvariant="normal">m</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=36183243&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=36183242&type= 3.04800010 3.21733332 0.001 1 kg·m 2 电流环积分环节 <math id="M113"><msubsup><mrow><mi>K</mi></mrow><mrow><mi>i</mi></mrow><mrow><mi>i</mi></mrow></msubsup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=36183245&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=36183244&type= 3.47133350 3.38666677 0.03 负载转动惯量 <math id="M114"><msub><mrow><mi>J</mi></mrow><mrow><mi mathvariant="normal">l</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=36183247&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=36183246&type= 2.11666679 3.21733332 0.278 kg·m 2 谐波减速器扭转刚度 <math id="M115"><msub><mrow><mi>K</mi></mrow><mrow><mi mathvariant="normal">h</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=36183249&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=36183248&type= 3.47133350 3.21733332 3×10 4 N·m·rad -1 负载阻尼 <math id="M116"><msub><mrow><mi>B</mi></mrow><mrow><mi mathvariant="normal">l</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=36183251&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=36183250&type= 2.79399991 3.21733332 0.8 N·m·s·rad -1 电机阻尼 <math id="M117"><msub><mrow><mi>B</mi></mrow><mrow><mi mathvariant="normal">m</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=36183253&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=36183252&type= 3.72533321 3.21733332 0.01 N·m·s·rad -1 考虑到运动误差导致的负载端转速波动主要集中在低频段，仿真中施加的扰动模型如下： <math display="block" id="M118"><mtable><mtr><mtd><msub><mrow><mi>T</mi></mrow><mrow><mi mathvariant="normal">d</mi></mrow></msub><mo>=</mo><msub><mrow><mi>K</mi></mrow><mrow><mi mathvariant="normal">h</mi></mrow></msub><msub><mrow><mi>θ</mi></mrow><mrow><mi mathvariant="normal">e</mi></mrow></msub><mo>=</mo></mtd></mtr><mtr><mtd><msub><mrow><mi>K</mi></mrow><mrow><mi mathvariant="normal">h</mi></mrow></msub><mfenced open="[" close="" separators="|"><mrow><mn mathvariant="normal">0.002</mn><mtext> </mtext><mn mathvariant="normal">511</mn><mi mathvariant="normal">*</mi><mi mathvariant="normal">s</mi><mi mathvariant="normal">i</mi><mi mathvariant="normal">n</mi><mo stretchy="false">(</mo><mn mathvariant="normal">2</mn><msub><mrow><mi>θ</mi></mrow><mrow><mi mathvariant="normal">m</mi></mrow></msub><mo stretchy="false">)</mo></mrow></mfenced><mo>+</mo><mn mathvariant="normal">0.001</mn><mtext> </mtext><mn mathvariant="normal">584</mn><mi mathvariant="normal">*</mi></mtd></mtr><mtr><mtd><mi mathvariant="normal">s</mi><mi mathvariant="normal">i</mi><mi mathvariant="normal">n</mi><mo stretchy="false">(</mo><mn mathvariant="normal">4</mn><msub><mrow><mi>θ</mi></mrow><mrow><mi mathvariant="normal">m</mi></mrow></msub><mo stretchy="false">)</mo><mfenced open="" close="]" separators="|"><mrow><mo>+</mo><mn mathvariant="normal">0.000</mn><mtext> </mtext><mn mathvariant="normal">794</mn><mtext> </mtext><mn mathvariant="normal">3</mn><mi mathvariant="normal">*</mi><mi mathvariant="normal">s</mi><mi mathvariant="normal">i</mi><mi mathvariant="normal">n</mi><mo stretchy="false">(</mo><mn mathvariant="normal">6</mn><msub><mrow><mi>θ</mi></mrow><mrow><mi mathvariant="normal">m</mi></mrow></msub><mo stretchy="false">)</mo></mrow></mfenced><mo>.</mo></mtd></mtr></mtable></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=36183257&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=36183256&type= 59.68999863 15.66333294 图4 图4 给出了不同迭代次数下的补偿信号。通过该图可以发现，随着迭代次数的增加，补偿信号逐渐趋于稳定。这是由于迭代学习控制输出补偿信号是对同一角位置下的速度调节误差不断累积得到的，随着迭代次数增加，速度调节误差逐渐减小，补偿信号的变化逐渐趋于平缓。在系统中不存在非周期性噪声的情况下，随着迭代次数的增加，在补偿信号的作用下，负载端速率波动逐渐趋于零，不同迭代次数下的负载端速率波动如图5 图5 所示。当系统中存在非周期性噪声时，补偿效果与迭代次数之间不再是正相关，通过在仿真中增加高斯白噪声模拟系统噪声，补偿信号仿真结果如图6 图6 所示。通过图6 图6 中的结果可以发现，在噪声存在的情况下，补偿效果在第15次迭代时最佳，之后随迭代次数的增加补偿效果变差。这是由于随着迭代次数的增加，负载端转速中噪声的幅值逐渐增大并最终超过周期性的转速波动，使得噪声对补偿信号的作用远高于周期性扰动的作用。换言之，补偿信号中噪声的含量随着迭代次数的增加而升高，最终淹没周期性补偿信号。仿真结果表明，当系统中存在噪声时，为保证补偿信号的准确性，迭代的次数必然是有限的，而且有限次的迭代在一定程度上节省了系统资源，提高了系统的运行效率。 5 试验结果 1 本研究的试验平台是基于单框架磁悬浮控制力矩陀螺工程样机的试验系统，如图7 图7 所示。框架控制系统是基于DSP芯片TMS320F28377D实现的数字控制系统。永磁同步电机用于输入力矩，谐波减速器将输入力矩放大。负载端的角位置通过23位精度的双通道旋转变压器检测，并对该角位置进行后向差分获取转速用于速度闭环控制。电机端的角位置通过编码器来检测，用于电流坐标变换。系统的主要参数与仿真中的参数一致。为检验误差补偿表对速度波动的抑制效果，将在10°/s匀速条件下建立的补偿表加入到系统中，试验结果如图8 图8 所示。可以发现，加入补偿表之后，在各自转速条件下，速度波动被分别抑制掉了31.8%（5°/s）和40.9%（10°/s）。试验结果表明，所提方法获取的补偿表可以有效地补偿谐波减速器运动误差导致的转速波动，提高转速控制精度。图9 图9 给出了基于所提方法的正弦跟踪曲线及跟踪误差，正弦参考速度曲线为3sin（6π t ）（°）·s - 1 。从图中可以发现，补偿后的跟踪误差被抑制了30.5%（从4.07°/s下降到2.83°/s）。同时，跟踪误差中的二倍频（图9 图9 中红框内）被有效补偿。试验结果表明基于位置域迭代学习控制建立的补偿表在动态条件下依然具有良好的转速波动抑制效果，这是由于所提方法在补偿表建立完成之后，迭代停止，补偿值根据电机端角位置查表获取，即使参考指令变化也无需重新学习，极大地提高了系统的动态响应性能。另一方面，从图8 图8 和图9 图9 的补偿效果可以看出在噪声特性不变的情况下，所建立的误差补偿表具有一定的普适性。 6 结论 1 本文在分析了谐波减速器运动误差的基础上，提出了基于位置域迭代学习控制的前馈补偿算法，用于对谐波减速器运动误差造成的负载端角速率波动进行抑制。同时对位置域迭代学习控制算法进行了分析，给出了算法的收敛条件和迭代停止条件。最后，通过仿真和试验验证了所提方法的有效性和可行性。试验结果表明，在施加了基于位置域迭代学习控制方法建立的补偿表之后，框架系统负载端速度波动量被抑制了30 %以上，所提方法为进一步提高带有谐波减速器的框架系统的速度控制精度提供了良好的基础。参考文献 LIU G ， LIU Y F ， ZHENG S Q . 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