基于自抗扰控制的2.5 m望远镜主轴伺服控制系统

刘洋; 李洪文; 邓永停; 王帅; 费强; 杨晓霞; 曹玉岩

doi:10.37188/OPE.20223023.3058

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基于自抗扰控制的2.5 m望远镜主轴伺服控制系统

更新时间：2022-12-20

- 基于自抗扰控制的2.5 m望远镜主轴伺服控制系统
- Main axes servo control system for 2.5 m telescope based on active disturbance rejection control
- 光学精密工程
- 作者机构：
  
  1.中国科学院长春光学精密机械与物理研究所，吉林长春 130033
  2.中国科学院大学，北京 100049
  3.季华实验室，广东佛山 528000
- 作者简介：
  
  [ "刘洋(1987-)，男，吉林德惠人，博士研究生，助理研究员，2011年于西北工业大学获得学士学位，2014年于西北工业大学获得硕士学位，主要从事光电望远镜伺服控制方面的研究。E-mail: liuyang@ciomp.ac.cn" ]
  
  刘洋(1987-)，男，吉林德惠人，博士研究生，助理研究员，2011年于西北工业大学获得学士学位，2014年于西北工业大学获得硕士学位，主要从事光电望远镜伺服控制方面的研究。E-mail: liuyang@ciomp.ac.cn
  [ "邓永停(1987-)，男，山东潍坊人，博士，研究员，2010年于中国石油大学获得学士学位，2015年于中科院长春光机所获得博士学位，主要从事大型望远镜精密跟踪控制及电子学设计方面的研究。E-mail:dyt0612@163.com" ]
  
  邓永停(1987-)，男，山东潍坊人，博士，研究员，2010年于中国石油大学获得学士学位，2015年于中科院长春光机所获得博士学位，主要从事大型望远镜精密跟踪控制及电子学设计方面的研究。E-mail:dyt0612@163.com
- 基金信息：
  
  国家自然科学基金资助项目(11973041;12122304);中国科学院青年创新促进会会员项目(2019218);天文联合基金资助项目(U2031126)
- DOI：10.37188/OPE.20223023.3058 中图分类号： TP273;TH743
- 纸质出版日期：2022-12-10，
  
  收稿日期：2022-09-27，
  
  修回日期：2022-11-14，
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引用本文
刘洋,李洪文,邓永停等.基于自抗扰控制的2.5 m望远镜主轴伺服控制系统[J].光学精密工程,2022,30(23):3058-3069.

LIU Yang,LI Hongwen,DENG Yongting,et al.Main axes servo control system for 2.5 m telescope based on active disturbance rejection control[J].Optics and Precision Engineering,2022,30(23):3058-3069.
刘洋,李洪文,邓永停等.基于自抗扰控制的2.5 m望远镜主轴伺服控制系统[J].光学精密工程,2022,30(23):3058-3069. DOI： 10.37188/OPE.20223023.3058.

LIU Yang,LI Hongwen,DENG Yongting,et al.Main axes servo control system for 2.5 m telescope based on active disturbance rejection control[J].Optics and Precision Engineering,2022,30(23):3058-3069. DOI： 10.37188/OPE.20223023.3058.

基于自抗扰控制的2.5 m望远镜主轴伺服控制系统 Main axes servo control system for 2.5 m telescope based on active disturbance rejection control 1 2 first-author 刘洋(1987-)，男，吉林德惠人，博士研究生，助理研究员，2011年于西北工业大学获得学士学位，2014年于西北工业大学获得硕士学位，主要从事光电望远镜伺服控制方面的研究。E-mail: liuyang@ciomp.ac.cn http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37919917&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37919921&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37919919&type= 22.01333237 32.00399780 刘洋 (1987-)，男，吉林德惠人，博士研究生，助理研究员，2011年于西北工业大学获得学士学位，2014年于西北工业大学获得硕士学位，主要从事光电望远镜伺服控制方面的研究。E-mail: liuyang@ciomp.ac.cn liuyang@ciomp.ac.cn 1 1 corresp E-mail： dyt0612@163.com E-mail： dyt0612@163.com 邓永停(1987-)，男，山东潍坊人，博士，研究员，2010年于中国石油大学获得学士学位，2015年于中科院长春光机所获得博士学位，主要从事大型望远镜精密跟踪控制及电子学设计方面的研究。E-mail:dyt0612@163.com http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37919923&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37919925&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37919924&type= 22.01332855 32.00399780 邓永停 (1987-)，男，山东潍坊人，博士，研究员，2010年于中国石油大学获得学士学位，2015年于中科院长春光机所获得博士学位，主要从事大型望远镜精密跟踪控制及电子学设计方面的研究。E-mail: dyt0612@163.com dyt0612@163.com 1 3 1 1 2 中国科学院长春光学精密机械与物理研究所，吉林长春 130033 中国科学院长春光学精密机械与物理研究所，吉林长春 130033 Changchun Institute of Optics， Fine Mechanics and Physics， Chinese Academy of Sciences， Changchun 130033， China Changchun Institute of Optics， Fine Mechanics and Physics， Chinese Academy of Sciences， Changchun 130033， China 中国科学院大学，北京 100049 中国科学院大学，北京 100049 University of Chinese Academy of Sciences， Beijing 100049， China University of Chinese Academy of Sciences， Beijing 100049， China 季华实验室，广东佛山 528000 季华实验室，广东佛山 528000 Ji Hua Laboratory， Foshan 528000， China Ji Hua Laboratory， Foshan 528000， China 为了满足2.5 m大视场望远镜的跟踪要求，提出了望远镜主轴伺服三闭环控制设计方法。在整定好的电流环输入端注入正弦信号，扫频测试主轴伺服速度环的开环频率特性。根据谐振频率设计结构滤波器抑制机械谐振，从而达到更高的闭环带宽。最后，依据辨识得到的控制模型，设计速度环线性自抗扰控制器和位置环比例控制器。速度环基于线性扩张状态观测器估计扰动并补偿其影响，可以达到更高的低速跟踪精度。为解决快速搜索问题，根据设备允许的最大速度、加速度，基于离散最速跟踪微分器安排过渡过程。实验结果表明：线性自抗扰控制相比PI控制，2.5 m大视场望远镜进行1.24°视场角快速步进搜索的时间由1.6 s降低为1.0 s；以速度为2 （°）/s，加速度为1 （°）/s 2 作为等效正弦引导的误差均方根值由1.08″降低为0.60″；低速跟踪斜率为0.000 1 （°）/s的位置斜坡曲线时，位置跟踪稳态误差均方根值由0.015 8″降低为0.010 6″。线性自抗扰控制能够满足大视场角天文望远镜高效率快速搜索和低速精密跟踪的要求。 To meet the requirements of fast search and high precision tracking at ultra-low speed for a 2.5 m wide field telescope， the design method of the three closed-loop control system of position， velocity， and current of telescope main axes AC servo is introduced. Firstly， a sinusoidal signal is injected into the set current closed-loop reference input end， and the speed open-loop frequency characteristic curve of the main axes servo system for the telescope is measured using a frequency sweep. Then， the structural filter is designed according to the resonant frequency to suppress the mechanical resonance so as to achieve a higher closed-loop bandwidth. Finally， according to the identified control model， the speed loop linear active disturbance rejection （LADRC） controller and position loop proportional controller are designed. The velocity loop estimates the disturbance and compensates for its effects based on the linear extended state observer， which achieves higher low-speed tracking accuracy. To solve the fast search problem， the position loop arranges the transition process according to the maximum allowable velocity and acceleration of the device and based on the discrete maximum velocity tracking differentiator. The experimental results show that， compared with the PI control， LADRC reduces the fast step search time of 1.24° FOV for a 2.5 m wide field telescope from 1.6 s to 1.0 s. The equivalent sinusoidal guidance error RMS value decreases from 1.08″ to 0.60″ when tracking the sine curve with a velocity of 2 （°）/s and acceleration of 1 （°）/s 2 ， The steady-state error RMS of position decreases from 0.015 8″ to 0.010 6" when tracking the slope of position at 0.000 1 （°）/s. The experimental results indicate that LADRC can meet the requirements of high efficiency， fast search， and precise tracking at low speed for large field angle telescopes. 大视场望远镜线性自抗扰频率特性结构滤波器最速跟踪微分器 wide field telescope linear active disturbance rejection control frequency characteristics structural filters fastest tracking differentiator 1 引言 1 随着深空探测能力、空间分辨率需求的提高，地基望远镜的口径越来越大，对伺服系统驱动能力和跟踪精度的要求也越来越高［ 1 1 - 2 2 1-2 ］。在系统稳定的前提下，望远镜伺服系统最为关键的指标就是跟踪精度、响应速度和抗扰性能。提高系统带宽是优化上述指标最直接的方法，但系统带宽受到结构谐振频率限制。随着望远镜口径的增大，机械谐振频率降低，限制了控制带宽，同时随机风载、非线性摩擦等扰动增大，因此，大口径望远镜的高精度控制面临着挑战。在望远镜驱动方式上，相比于齿轮或者摩擦传动，直驱方式结构刚度更高，可显著提高谐振频率和锁定谐振频率，有利于增加系统带宽，从而提高响应速度和跟踪精度［ 3 3 ］。由于永磁同步电机（Permanent Magnetic Synchronous Motor, PMSM）功率密度高、低速平稳性好，随着矢量控制技术日趋成熟，地基大口径望远镜永磁同步电机直接驱动技术的应用越来越广泛，如日本的8.2 m subaru望远镜、西班牙的10.4 m GTC望远镜、欧洲南方天文台的4架8.2 m VLT望远镜，均采用永磁同步电机直驱［ 1 1 - 2 2 1-2 ］。在控制算法上，PID控制因不依赖模型，简单易用，在大型望远镜工程中应用最广泛，但是PID控制的动态响应能力和超调量存在不可调和的矛盾，在超低速高精度跟踪、抗扰性能方面有进一步提高的空间。因此，人们提出PID改进型来提高控制性能，如anti-windup内模PID ［ 4 4 ］、变结构PID ［ 5 5 ］，应用于大型望远镜速度环，实现了无超调响应，提高了稳态精度。在扰动抑制方面，LQG和H ∞ 控制在大型射电望远镜上有效克服了风载扰动，相比PID具有更好的控制性能［ 6 6 - 7 7 6-7 ］。但是，参数摄动时，LQG和H ∞ 控制的性能变化较大，鲁棒性不足［ 8 8 ］，H ∞ -GESO控制器却获得了明显的性能提升。文献［ 9 9 ］对近年来自抗扰控制在PMSM高性能伺服中的研究成果进行了总结，指出模型不确定性因素包括参数摄动和未建模动态，属于内部扰动；不确定载荷扰动属于外部扰动。在望远镜伺服系统中，内部扰动主要包括转动惯量摄动、齿槽转矩和摩擦扰动；外部扰动主要包括风载扰动。韩京清于1998年提出自抗扰控制器（Active Disturbance Rejection Control, ADRC）［ 10 10 ］，包括TD，NLSEF和ESO 3部分，不区分内扰和外扰，通过ESO估计内扰外扰之和，即总扰动，是一种新的抗扰控制思路。高志强对自抗扰思想和控制范式进行深入的剖析［ 11 11 ］，并引入线性化方法，得到线性自抗扰控制（Linear Active Disturbance Rejection Control, LADRC），用熟悉的带宽概念对LADRC参数进行整定［ 12 12 ］，极大地推动了ADRC的应用。文献［ 13 13 - 14 14 13-14 ］将LADRC应用于望远镜速度环，在与PI控制具有同等带宽的前提下，LADRC抑制了摩擦和齿槽力矩波动，在低速跟踪精度和中低频段的抗扰性能相比PI控制有较大提升。文献［ 15 15 ］提出位置-电流双闭环策略，采用位置速度复合二阶非线性ADRC控制，电流环采用一阶LADRC，设计实现相对复杂。文献［ 16 16 ］采用位置速度一体化设计，引入速度限幅的位置环二阶ADRC控制，实现了位置最速跟踪定位，但是缺少独立的速度控制模式。由于望远镜伺服系统通常需要工作在等速、定点和数引等工作模式，因此，本文首先辨识望远镜的频率特性，然后设计结构滤波器进行机械谐振抑制，采用位置、速度、电流三闭环串级控制结构，依据带宽法由内环向外环依次整定各环路的控制参数。由于望远镜伺服控制属于低速应用，电流环带宽较高，所以采用PI控制即可，速度环设计一阶LADRC，抑制扰动，实现低速高精度跟踪；位置环设计最速TD规划和比例控制器，实现最大速度、加速度下的快速搜索。 2 望远镜主轴伺服控制系统模型 1 2.1　永磁同步电机矢量控制数学模型 2 地基大口径望远镜伺服系统一般采用表面贴装式PMSM。采用 <math id="M1"><msub><mrow><mi>i</mi></mrow><mrow><mi>d</mi></mrow></msub><mo>=</mo><mn mathvariant="normal">0</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37920020&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37920017&type= 9.14400005 3.72533321 的矢量控制方式，永同步电机在 d - q 轴坐标系下的数学模型如下：电压方程： <math display="block" id="M2"><mfenced open="{" close="" separators="|"><mrow><mtable columnalign="left"><mtr><mtd><msub><mrow><mi>u</mi></mrow><mrow><mi>q</mi></mrow></msub><mo>=</mo><msub><mrow><mi>R</mi></mrow><mrow><mi 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http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37919933&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37919932&type= 34.96733475 7.87400007 ，式中： <math id="M5"><msub><mrow><mi>u</mi></mrow><mrow><mi>d</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37919938&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37919936&type= 3.21733332 3.72533321 ， <math id="M6"><msub><mrow><mi>u</mi></mrow><mrow><mi>q</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37919942&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37919940&type= 3.04800010 3.72533321 分别为 d ， q 轴电压； <math id="M7"><msub><mrow><mi>i</mi></mrow><mrow><mi>d</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37919946&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37919944&type= 2.45533323 3.72533321 ， <math id="M8"><msub><mrow><mi>i</mi></mrow><mrow><mi>q</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37919948&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37919947&type= 2.28600001 3.72533321 分别为 d ， q 轴电流； <math id="M9"><msub><mrow><mi>L</mi></mrow><mrow><mi>d</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37919950&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37919949&type= 3.80999994 3.72533321 ， <math id="M10"><msub><mrow><mi>L</mi></mrow><mrow><mi>q</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37919956&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37919953&type= 3.64066648 3.72533321 分别为 d ， q 轴电感，且 <math id="M11"><msub><mrow><mi>L</mi></mrow><mrow><mi>d</mi></mrow></msub><mo>=</mo><msub><mrow><mi>L</mi></mrow><mrow><mi>q</mi></mrow></msub><mo>=</mo><mi>L</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37919963&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37919960&type= 19.38866615 3.72533321 ； <math id="M12"><msub><mrow><mi>R</mi></mrow><mrow><mi mathvariant="normal">s</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37919970&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37919966&type= 3.72533321 3.72533321 为定子相电阻； <math id="M13"><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=37919976&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37919973&type= 2.87866688 4.48733330 为永磁体磁链； <math id="M14"><msub><mrow><mi>n</mi></mrow><mrow><mi mathvariant="normal">p</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37919980&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37919978&type= 3.04800010 3.72533321 为极对数； <math id="M15"><msub><mrow><mi>K</mi></mrow><mrow><mi mathvariant="normal">t</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37919988&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37919985&type= 3.64066648 3.72533321 为力矩系数； <math id="M16"><msub><mrow><mi>ω</mi></mrow><mrow><mi mathvariant="normal">e</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37919996&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37919991&type= 3.55599999 3.72533321 为转子旋转的电角速度， <math id="M17"><mi>Ω</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37920279&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37920275&type= 2.70933342 2.79399991 为转子旋转的机械角速度，且 <math id="M18"><msub><mrow><mi>ω</mi></mrow><mrow><mi mathvariant="normal">e</mi></mrow></msub><mo>=</mo><msub><mrow><mi>n</mi></mrow><mrow><mi mathvariant="normal">p</mi></mrow></msub><mi mathvariant="normal">Ω</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37920008&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37920005&type= 14.22399998 3.72533321 ； <math id="M19"><msub><mrow><mi>T</mi></mrow><mrow><mi mathvariant="normal">L</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37920014&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37920012&type= 4.06400013 3.72533321 为负载转矩； B 为黏滞摩擦系数。 2.2　电流环PI控制器设计 2 2.5 m级口径望远镜交流伺服系统属于低速应用，采用位置、速度、电流三闭环控制。内环电流环相对外环速度环、位置环带宽较高，通常采用简单易用的PI控制，即可满足快速无超调的要求。采用 <math id="M20"><msub><mrow><mi>i</mi></mrow><mrow><mi>d</mi></mrow></msub><mo>=</mo><mn mathvariant="normal">0</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37920020&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37920017&type= 9.14400005 3.72533321 的矢量控制，忽略反电动势影响，电流环开环传递函数为： <math display="block" id="M21"><msub><mrow><mi>G</mi></mrow><mrow><mi mathvariant="normal">c</mi><mi mathvariant="normal">o</mi></mrow></msub><mo stretchy="false">(</mo><mi>s</mi><mo stretchy="false">)</mo><mo>=</mo><msub><mrow><mi>K</mi></mrow><mrow><mi mathvariant="normal">c</mi><mi mathvariant="normal">p</mi></mrow></msub><mfenced separators="|"><mrow><mn mathvariant="normal">1</mn><mo>+</mo><mfrac><mrow><mn mathvariant="normal">1</mn></mrow><mrow><msub><mrow><mi>τ</mi></mrow><mrow><mi>i</mi></mrow></msub><mi>s</mi></mrow></mfrac></mrow></mfenced><mfenced separators="|"><mrow><mfrac><mrow><mn mathvariant="normal">1</mn></mrow><mrow><msub><mrow><mi>L</mi></mrow><mrow><mi>q</mi></mrow></msub><mi>s</mi><mo>+</mo><msub><mrow><mi>R</mi></mrow><mrow><mi mathvariant="normal">s</mi></mrow></msub></mrow></mfrac></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37920025&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37920023&type= 52.32400131 9.99066734 ，按照I型系统整定， <math id="M22"><msub><mrow><mi>τ</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>=</mo><msub><mrow><mi>τ</mi></mrow><mrow><mi>e</mi></mrow></msub><mo>=</mo><msub><mrow><mi>L</mi></mrow><mrow><mi>q</mi></mrow></msub><mo>/</mo><msub><mrow><mi>R</mi></mrow><mrow><mi mathvariant="normal">s</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37920031&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37920028&type= 23.70666695 4.14866638 ，电流环闭环传递函数为： <math display="block" id="M23"><msub><mrow><mi>G</mi></mrow><mrow><mi mathvariant="normal">c</mi><mi mathvariant="normal">c</mi></mrow></msub><mo stretchy="false">(</mo><mi>s</mi><mo stretchy="false">)</mo><mo>=</mo><mfrac><mrow><msub><mrow><mi>G</mi></mrow><mrow><mi mathvariant="normal">c</mi><mi mathvariant="normal">o</mi></mrow></msub><mo stretchy="false">(</mo><mi>s</mi><mo stretchy="false">)</mo></mrow><mrow><mn mathvariant="normal">1</mn><mo>+</mo><msub><mrow><mi>G</mi></mrow><mrow><mi mathvariant="normal">c</mi><mi mathvariant="normal">o</mi></mrow></msub><mo stretchy="false">(</mo><mi>s</mi><mo stretchy="false">)</mo></mrow></mfrac><mo>=</mo><mfrac><mrow><mn mathvariant="normal">1</mn></mrow><mrow><mo stretchy="false">(</mo><msub><mrow><mi>L</mi></mrow><mrow><mi>q</mi></mrow></msub><mo>/</mo><msub><mrow><mi>K</mi></mrow><mrow><mi mathvariant="normal">c</mi><mi mathvariant="normal">p</mi></mrow></msub><mo stretchy="false">)</mo><mi>s</mi><mo>+</mo><mn mathvariant="normal">1</mn></mrow></mfrac></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37920038&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37920035&type= 59.52066422 9.39799976 . 整定后电流环的时间常数 <math id="M24"><msub><mrow><mi>τ</mi></mrow><mrow><mi mathvariant="normal">c</mi></mrow></msub><mo>=</mo><msub><mrow><mi>L</mi></mrow><mrow><mi>q</mi></mrow></msub><mo>/</mo><msub><mrow><mi>K</mi></mrow><mrow><mi mathvariant="normal">c</mi><mi mathvariant="normal">p</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37920043&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37920040&type= 17.78000069 4.14866638 ，可见，通过调节 <math id="M25"><msub><mrow><mi>K</mi></mrow><mrow><mi mathvariant="normal">c</mi><mi mathvariant="normal">p</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37920049&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37920046&type= 4.91066647 3.72533321 可以设计合适的电流环闭环带宽。对于2.5 m级望远镜，电流环闭环带宽一般在100 Hz以上。 2.3　望远镜主轴伺服系统频率特性辨识 2 速度环开环频率特性测试原理框图如图1 图1 所示，电流环处于闭环状态，在电流参考输入端注入正弦信号，通过计算机同步记录转台的速度响应数据。该方法辨识的控制模型包括电流闭环控制器、驱动放大部分、电机部分和机械结构部分［ 17 17 ］。注入的正弦电流指令信号如下： <math display="block" id="M26"><msubsup><mrow><mi>i</mi></mrow><mrow><mi>q</mi></mrow><mrow><mi mathvariant="normal">*</mi></mrow></msubsup><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo>=</mo><msub><mrow><mi>I</mi></mrow><mrow><mi>q</mi></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><mi mathvariant="normal">π</mi><mi>f</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37920067&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37920064&type= 33.78200150 5.07999992 ，其中： <math id="M27"><mi>f</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo>=</mo><msub><mrow><mi>f</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><mo stretchy="false">(</mo><mn mathvariant="normal">1</mn><mo>+</mo><mi>c</mi><msup><mrow><mi>t</mi></mrow><mrow><mi>n</mi></mrow></msup><mo stretchy="false">)</mo><mi>t</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37920072&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37920069&type= 30.22599983 5.07999992 ， <math id="M28"><mi>c</mi><mo>=</mo><mfrac><mrow><msub><mrow><mi>f</mi></mrow><mrow><mi>T</mi></mrow></msub><mo>/</mo><msub><mrow><mi>f</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><mo>-</mo><mn mathvariant="normal">1</mn></mrow><mrow><mo stretchy="false">(</mo><mi>n</mi><mo>+</mo><mn mathvariant="normal">1</mn><mo stretchy="false">)</mo><msup><mrow><mi>T</mi></mrow><mrow><mi>n</mi></mrow></msup></mrow></mfrac></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37920077&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37920075&type= 22.94466591 8.97466660 。正弦扫描频率为 <math id="M29"><mo stretchy="false">[</mo><msub><mrow><mi>f</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><mo>,</mo><msub><mrow><mi>f</mi></mrow><mrow><mi>T</mi></mrow></msub><mo stretchy="false">]</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37920083&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37920080&type= 11.00666618 4.91066647 ，要覆盖低阶机械谐振频率范围，此处取［1，150］ Hz。扫描时间为Ｔ，取 T =40 s。 <math id="M30"><msub><mrow><mi>I</mi></mrow><mrow><mi>q</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37920089&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37920086&type= 2.45533323 3.72533321 为正弦电流信号的幅值，一般要足以驱动望远镜来回摆动， n 为多项式的阶次， n =3。在获得系统的开环速度响应数据后，对输入电流序列 <math id="M31"><msub><mrow><mi>U</mi></mrow><mrow><mi>i</mi></mrow></msub><mo stretchy="false">(</mo><mi>ω</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37920119&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37920116&type= 9.99066734 4.48733330 和输出速度序列 <math id="M32"><msub><mrow><mi>Y</mi></mrow><mrow><mi>i</mi></mrow></msub><mo stretchy="false">(</mo><mi>ω</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37920100&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37920098&type= 9.82133293 4.48733330 进行离散傅里叶变换，得到输入和输出序列的功率谱密度函数： <math display="block" id="M33"><msub><mrow><mi>P</mi></mrow><mrow><mi>u</mi><mi>y</mi></mrow></msub><mo stretchy="false">(</mo><mi>ω</mi><mo stretchy="false">)</mo><mo>=</mo><mfrac><mrow><mn mathvariant="normal">2</mn></mrow><mrow><mi>n</mi><mi>T</mi></mrow></mfrac><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>i</mi><mo>=</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi>n</mi></mrow></munderover></mstyle><mrow><msubsup><mrow><mi>U</mi></mrow><mrow><mi>i</mi></mrow><mrow><mi>*</mi></mrow></msubsup><mo stretchy="false">(</mo><mi>ω</mi><mo stretchy="false">)</mo></mrow><msub><mrow><mi>Y</mi></mrow><mrow><mi>i</mi></mrow></msub><mo stretchy="false">(</mo><mi>ω</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37920106&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37920103&type= 46.73600006 8.63599968 ，其中： <math id="M34"><msubsup><mrow><mi>U</mi></mrow><mrow><mi>i</mi></mrow><mrow><mi mathvariant="normal">*</mi></mrow></msubsup><mo stretchy="false">(</mo><mi>ω</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37920114&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37920109&type= 10.75266743 4.65666676 为 <math id="M35"><msub><mrow><mi>U</mi></mrow><mrow><mi>i</mi></mrow></msub><mo stretchy="false">(</mo><mi>ω</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37920119&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37920116&type= 9.99066734 4.48733330 的共轭， <math id="M36"><mi>T</mi><mo>=</mo><mi>n</mi><msub><mrow><mi>T</mi></mrow><mrow><mi mathvariant="normal">s</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37920125&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37920122&type= 12.78466606 3.72533321 ，表示每段数据总的采样时间， <math id="M37"><msub><mrow><mi>T</mi></mrow><mrow><mi mathvariant="normal">s</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37920129&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37920127&type= 3.55599999 3.72533321 表示数据采样周期。激励信号的功率谱密度函数表示为： <math display="block" id="M38"><msub><mrow><mi>P</mi></mrow><mrow><mi>u</mi><mi>u</mi></mrow></msub><mo stretchy="false">(</mo><mi>ω</mi><mo stretchy="false">)</mo><mo>=</mo><mfrac><mrow><mn mathvariant="normal">2</mn></mrow><mrow><mi>n</mi><mi>T</mi></mrow></mfrac><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>i</mi><mo>=</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi>n</mi></mrow></munderover></mstyle><mrow><mo stretchy="false">|</mo><msub><mrow><mi>U</mi></mrow><mrow><mi>i</mi></mrow></msub><mo stretchy="false">(</mo><mi>ω</mi><mo stretchy="false">)</mo></mrow><msup><mrow><mo stretchy="false">|</mo></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37920135&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37920132&type= 38.69266510 8.63599968 . 因此，系统的开环频率特性表示为： <math display="block" id="M39"><mover accent="true"><mi>G</mi><mo>^</mo></mover><mo stretchy="false">(</mo><mi>ω</mi><mo stretchy="false">)</mo><mo>=</mo><mfrac><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>u</mi><mi>y</mi></mrow></msub><mo stretchy="false">(</mo><mi>ω</mi><mo stretchy="false">)</mo></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>u</mi><mi>u</mi></mrow></msub><mo stretchy="false">(</mo><mi>ω</mi><mo stretchy="false">)</mo></mrow></mfrac></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37920139&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37920137&type= 24.89200020 8.97466660 . 根据上述方法，以俯仰轴为例，对2.5 m望远镜进行频率特性辨识，得到俯仰轴转台的开环频率特性曲线，如图2 图2 所示。系统的锁定谐振频率 <math id="M40"><msub><mrow><mi>ω</mi></mrow><mrow><mi mathvariant="normal">A</mi><mi mathvariant="normal">R</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37920144&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37920141&type= 5.58799982 3.72533321 为24.8 Hz，谐振频率 <math id="M41"><msub><mrow><mi>ω</mi></mrow><mrow><mi mathvariant="normal">R</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37920151&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37920148&type= 4.06400013 3.72533321 为27.0 Hz。 3 望远镜主轴伺服系统控制器设计 1 3.1　结构滤波器设计 2 速度回路的主要作用是减小机电时间滞后，提高系统的动态响应能力，减小扰动误差。为了减小低速干扰误差，应尽量增大系统的增益和带宽。速度回路的带宽主要受到系统机械谐振频率的限制。为了减小谐振频率对系统的影响，在速度环PI控制器输出串入结构滤波器，以增大系统闭环带宽，增强速度回路的动态跟踪性能，结构滤波器的表达式如下［ 17 17 - 18 18 17-18 ］： <math display="block" id="M42"><msub><mrow><mi>W</mi></mrow><mrow><mi mathvariant="normal">S</mi><mi mathvariant="normal">F</mi></mrow></msub><mo stretchy="false">(</mo><mi>s</mi><mo stretchy="false">)</mo><mo>=</mo><mfrac><mrow><msup><mrow><mi>s</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mo>+</mo><mn mathvariant="normal">2</mn><msub><mrow><mi>ζ</mi></mrow><mrow><mi mathvariant="normal">Z</mi></mrow></msub><msub><mrow><mi>ω</mi></mrow><mrow><mi mathvariant="normal">N</mi><mi mathvariant="normal">F</mi></mrow></msub><mi>s</mi><mo>+</mo><msubsup><mrow><mi>ω</mi></mrow><mrow><mi mathvariant="normal">N</mi><mi mathvariant="normal">F</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msubsup></mrow><mrow><msup><mrow><mi>s</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mo>+</mo><mn mathvariant="normal">2</mn><msub><mrow><mi>ζ</mi></mrow><mrow><mi mathvariant="normal">P</mi></mrow></msub><msub><mrow><mi>ω</mi></mrow><mrow><mi mathvariant="normal">N</mi><mi mathvariant="normal">F</mi></mrow></msub><mi>s</mi><mo>+</mo><msubsup><mrow><mi>ω</mi></mrow><mrow><mi mathvariant="normal">N</mi><mi mathvariant="normal">F</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msubsup></mrow></mfrac></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37920167&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37920164&type= 46.48199844 9.22866726 ，式中： <math id="M43"><msub><mrow><mi>ω</mi></mrow><mrow><mi mathvariant="normal">N</mi><mi mathvariant="normal">F</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37920174&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37920169&type= 5.41866684 3.72533321 为系统的谐振频率，称为陷波频率， <math id="M44"><msub><mrow><mi>ζ</mi></mrow><mrow><mi mathvariant="normal">Z</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37920178&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37920175&type= 2.87866688 4.57200003 ， <math id="M45"><msub><mrow><mi>ζ</mi></mrow><mrow><mi mathvariant="normal">P</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37920184&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37920183&type= 2.96333337 4.57200003 分别为结构滤波器的零点和极点阻尼系数。比值 <math id="M46"><msub><mrow><mi>ζ</mi></mrow><mrow><mi mathvariant="normal">Z</mi></mrow></msub><mo>/</mo><msub><mrow><mi>ζ</mi></mrow><mrow><mi mathvariant="normal">P</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37920194&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37920188&type= 7.87400007 4.82600021 决定谐振频率 <math id="M47"><msub><mrow><mi>ω</mi></mrow><mrow><mi mathvariant="normal">N</mi><mi mathvariant="normal">F</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37920199&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37920196&type= 5.41866684 3.72533321 处陷波深度，比值越小，凹陷越深。极点阻尼系数 <math id="M48"><msub><mrow><mi>ζ</mi></mrow><mrow><mi mathvariant="normal">P</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37920204&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37920201&type= 2.96333337 4.57200003 决定谐振频率 <math id="M49"><msub><mrow><mi>ω</mi></mrow><mrow><mi mathvariant="normal">N</mi><mi mathvariant="normal">F</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37920210&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37920207&type= 5.41866684 3.72533321 处陷波宽度，其值越大宽度越宽。由图2 图2 可知， <math id="M50"><msub><mrow><mi>ω</mi></mrow><mrow><mi mathvariant="normal">N</mi><mi mathvariant="normal">F</mi></mrow></msub><mo>=</mo><mn mathvariant="normal">27.0</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37920219&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37920213&type= 17.86466599 3.72533321 Hz，取 <math id="M51"><msub><mrow><mi>ζ</mi></mrow><mrow><mi mathvariant="normal">P</mi></mrow></msub><mo>=</mo><mn mathvariant="normal">0.6</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37920229&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37920224&type= 13.54666710 4.57200003 ， <math id="M52"><msub><mrow><mi>ζ</mi></mrow><mrow><mi mathvariant="normal">Z</mi></mrow></msub><mo>/</mo><msub><mrow><mi>ζ</mi></mrow><mrow><mi mathvariant="normal">P</mi></mrow></msub><mo>=</mo><mn mathvariant="normal">0.1</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37920237&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37920233&type= 18.37266731 4.82600021 ，俯仰轴加入与未加入结构滤波器的开环频率特性曲线如图3 图3 所示，加入结构滤波器后谐振频率27 Hz处机械谐振峰值得到 - 15 dB的衰减。同理，可在其他机械谐振频率处加入结构滤波器。 3.2　速度环线性自抗扰控制器设计 2 电流环整定完成后，忽略扰动力矩的影响，速度环被控对象的传递函数为： <math display="block" id="M53"><msub><mrow><mi>G</mi></mrow><mrow><mi mathvariant="normal">v</mi><mo>_</mo><mi mathvariant="normal">p</mi><mi mathvariant="normal">l</mi><mi mathvariant="normal">a</mi><mi mathvariant="normal">n</mi><mi mathvariant="normal">t</mi></mrow></msub><mo stretchy="false">(</mo><mi>s</mi><mo stretchy="false">)</mo><mo>=</mo><mfrac><mrow><mi mathvariant="normal">Ω</mi><mo stretchy="false">(</mo><mi>s</mi><mo stretchy="false">)</mo></mrow><mrow><msub><mrow><mi>I</mi></mrow><mrow><mi>q</mi></mrow></msub><mo stretchy="false">(</mo><mi>s</mi><mo stretchy="false">)</mo></mrow></mfrac><mo>=</mo><mfrac><mrow><msub><mrow><mi>K</mi></mrow><mrow><mi>t</mi></mrow></msub></mrow><mrow><mo stretchy="false">(</mo><mi>J</mi><mi>s</mi><mo>+</mo><mi>B</mi><mo stretchy="false">)</mo><mo stretchy="false">(</mo><msub><mrow><mi>τ</mi></mrow><mrow><mi mathvariant="normal">c</mi></mrow></msub><mi>s</mi><mo>+</mo><mn mathvariant="normal">1</mn><mo stretchy="false">)</mo></mrow></mfrac></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37920256&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37920253&type= 60.11333084 9.39799976 . 大口径望远镜的速度环带宽一般受到机械谐振频率的限制，速度环闭环带宽一般只有一阶锁定谐振频率的二分之一甚至三分之一。相比之下，电流环带宽较高，忽略其时间常数 <math id="M54"><msub><mrow><mi>τ</mi></mrow><mrow><mi mathvariant="normal">c</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37920262&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37920259&type= 2.62466669 3.72533321 ，则速度环可以简化为一阶系统，即： <math display="block" id="M55"><mover><mrow><mi>Ω</mi></mrow><mrow><mo>·</mo></mrow></mover><mo>=</mo><mfrac><mrow><msub><mrow><mi>K</mi></mrow><mrow><mi>t</mi></mrow></msub></mrow><mrow><mi>J</mi></mrow></mfrac><msub><mrow><mi>i</mi></mrow><mrow><mi>q</mi></mrow></msub><mo>-</mo><mfrac><mrow><mi>B</mi></mrow><mrow><mi>J</mi></mrow></mfrac><mi>Ω</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37920268&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37920265&type= 26.33133125 7.95866632 . 电流 <math id="M56"><msub><mrow><mi>i</mi></mrow><mrow><mi>q</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37920274&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37920271&type= 2.28600001 3.72533321 作为系统输入 u ，机械角速度 <math id="M57"><mi>Ω</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37920279&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37920275&type= 2.70933342 2.79399991 作为系统输出，并引入随机扰动 <math id="M58"><mi>w</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37920325&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37920322&type= 2.96333337 2.79399991 。系统参数记作 <math id="M59"><msub><mrow><mi>a</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><mo>=</mo><mfrac><mrow><mi>B</mi></mrow><mrow><mi>J</mi></mrow></mfrac></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37920289&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37920286&type= 11.09133339 7.95866632 ，系统控制增益记作 <math id="M60"><msub><mrow><mi>b</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><mo>=</mo><mfrac><mrow><msub><mrow><mi>K</mi></mrow><mrow><mi>t</mi></mrow></msub></mrow><mrow><mi>J</mi></mrow></mfrac></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37920295&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37920292&type= 12.19200039 7.95866632 ，则有： <math display="block" id="M61"><mover><mrow><mi>Ω</mi></mrow><mrow><mo>·</mo></mrow></mover><mo>=</mo><msub><mrow><mi>b</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><mi>u</mi><mo>-</mo><msub><mrow><mi>a</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><mi>Ω</mi><mo>+</mo><mi>w</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37920303&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37920301&type= 31.32666588 5.41866684 . 考虑到控制增益误差，引入控制器增益 b ，得到： <math display="block" id="M62"><mover><mrow><mi>Ω</mi></mrow><mrow><mo>·</mo></mrow></mover><mo>=</mo><mo stretchy="false">(</mo><msub><mrow><mi>b</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><mo>-</mo><mi>b</mi><mo stretchy="false">)</mo><mi>u</mi><mo>-</mo><msub><mrow><mi>a</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><mi mathvariant="normal">Ω</mi><mo>+</mo><mi>w</mi><mo>+</mo><mi>b</mi><mi>u</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37920311&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37920307&type= 49.44533539 5.41866684 . 系统总扰动记作 <math id="M63"><mi>f</mi><mo>=</mo><mo stretchy="false">(</mo><msub><mrow><mi>b</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><mo>-</mo><mi>b</mi><mo stretchy="false">)</mo><mi>u</mi><mo>-</mo><msub><mrow><mi>a</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><mi mathvariant="normal">Ω</mi><mo>+</mo><mi>w</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37920319&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37920315&type= 39.87799835 4.91066647 。望远镜主轴力矩扰动 <math id="M64"><mi>w</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37920325&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37920322&type= 2.96333337 2.79399991 包括摩擦、齿槽、风扰和随机扰动等，总扰动 <math id="M65"><mi>f</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37920348&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37920345&type= 2.45533323 3.64066648 包含系统内部扰动和外部扰动，包括已知和未知部分。系统模型可描述为： <math display="block" id="M66"><mover><mrow><mi>Ω</mi></mrow><mrow><mo>·</mo></mrow></mover><mo>=</mo><mi>f</mi><mo>+</mo><mi>b</mi><mi>u</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37920336&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37920334&type= 17.35666656 5.84200001 . 总扰动 <math id="M67"><mi>f</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37920348&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37920345&type= 2.45533323 3.64066648 不能直接测量，LADRC通过线性扩张状态观测器（Linear Extended State Observes， LESO）的扩张状态估计 <math id="M68"><mi>f</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37920348&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37920345&type= 2.45533323 3.64066648 ，并前馈补偿。速度环LADRC原理如图4 图4 所示。对于一阶系统设计二阶LESO，即： <math display="block" id="M69"><mfenced open="{" close="" separators="|"><mrow><mtable columnalign="left"><mtr><mtd><msub><mrow><mi>e</mi></mrow><mrow><mi>Ω</mi></mrow></msub><mo>=</mo><mi>Ω</mi><mo>-</mo><msub><mrow><mi>z</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub></mtd></mtr><mtr><mtd><mover><mrow><msub><mrow><mi>z</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub></mrow><mrow><mo>·</mo></mrow></mover><mo>=</mo><msub><mrow><mi>z</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo>+</mo><msub><mrow><mi>β</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><msub><mrow><mi>e</mi></mrow><mrow><mi>Ω</mi></mrow></msub><mo>+</mo><mi>b</mi><mi>u</mi></mtd></mtr><mtr><mtd><mover><mrow><msub><mrow><mi>z</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub></mrow><mrow><mo>·</mo></mrow></mover><mo>=</mo><msub><mrow><mi>β</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><msub><mrow><mi>e</mi></mrow><mrow><mi>Ω</mi></mrow></msub></mtd></mtr></mtable></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37920363&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37920358&type= 32.42733002 19.30400085 ，式中： <math id="M70"><msub><mrow><mi>z</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37920368&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37920364&type= 2.96333337 3.72533321 为速度信号 <math id="M71"><mi>Ω</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37920376&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37920373&type= 2.70933342 2.79399991 的估计值； <math id="M72"><msub><mrow><mi>e</mi></mrow><mrow><mi>Ω</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37920381&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37920379&type= 3.04800010 3.72533321 为速度环LESO的估计误差； <math id="M73"><msub><mrow><mi>z</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37920386&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37920383&type= 2.96333337 3.72533321 为扰动 <math id="M74"><mi>f</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37920392&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37920390&type= 2.45533323 3.64066648 的估计值； u 为系统控制量，实际系统控制量 u 为有限值，所以需要限幅； <math id="M75"><msub><mrow><mi>β</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37920398&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37920394&type= 3.38666677 4.57200003 和 <math id="M76"><msub><mrow><mi>β</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37920431&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37920427&type= 3.38666677 4.57200003 为观测器增益。通过速度环LESO角频率带宽 <math id="M77"><msub><mrow><mi>ω</mi></mrow><mrow><mi mathvariant="normal">v</mi><mi mathvariant="normal">o</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37920413&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37920409&type= 4.65666676 3.72533321 确定观测器增益 <math id="M78"><msub><mrow><mi>β</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37920422&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37920417&type= 3.38666677 4.57200003 和 <math id="M79"><msub><mrow><mi>β</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37920431&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37920427&type= 3.38666677 4.57200003 ，使LESO特征方程的特征值为 <math id="M80"><mo>-</mo><msub><mrow><mi>ω</mi></mrow><mrow><mi mathvariant="normal">v</mi><mi mathvariant="normal">o</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37920438&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37920434&type= 8.12800026 3.72533321 ，其公式如下［ 12 12 ］： <math display="block" id="M81"><mfenced open="{" close="" separators="|"><mrow><mtable columnalign="left"><mtr><mtd><msub><mrow><mi>β</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>=</mo><mn mathvariant="normal">2</mn><msub><mrow><mi>ω</mi></mrow><mrow><mi mathvariant="normal">v</mi><mi mathvariant="normal">o</mi></mrow></msub></mtd></mtr><mtr><mtd><msub><mrow><mi>β</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo>=</mo><msubsup><mrow><mi>ω</mi></mrow><mrow><mi mathvariant="normal">v</mi><mi mathvariant="normal">o</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msubsup></mtd></mtr></mtable></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37920450&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37920443&type= 16.67933464 10.15999985 . 误差反馈控制率设计为比例控制器，即： <math display="block" id="M82"><msub><mrow><mi>u</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><mo>=</mo><msub><mrow><mi>K</mi></mrow><mrow><mi mathvariant="normal">v</mi><mi mathvariant="normal">p</mi></mrow></msub><mo stretchy="false">(</mo><msup><mrow><mi>Ω</mi></mrow><mrow><mi mathvariant="normal">*</mi></mrow></msup><mo>-</mo><msub><mrow><mi>z</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37920458&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37920453&type= 28.10933495 4.65666676 ，式中： <math id="M83"><msup><mrow><mi>Ω</mi></mrow><mrow><mi mathvariant="normal">*</mi></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37920468&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37920464&type= 3.80999994 3.04800010 为机械角速度参考输入， <math id="M84"><msub><mrow><mi>K</mi></mrow><mrow><mi mathvariant="normal">v</mi><mi mathvariant="normal">p</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37920477&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37920473&type= 4.99533367 3.72533321 为速度环比例增益。扰动前馈补偿，得到实际控制量： <math display="block" id="M85"><mi>u</mi><mo>=</mo><mo stretchy="false">(</mo><msub><mrow><mi>u</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><mo>-</mo><msub><mrow><mi>z</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo stretchy="false">)</mo><mo>/</mo><mi>b</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37920485&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37920481&type= 24.46866608 4.48733330 . 选取合适的 <math id="M86"><msub><mrow><mi>β</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37920494&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37920490&type= 3.38666677 4.57200003 和 <math id="M87"><msub><mrow><mi>β</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37920502&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37920497&type= 3.38666677 4.57200003 ，使 <math id="M88"><msub><mrow><mi>z</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>→</mo><mi>Ω</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37920509&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37920505&type= 10.41399956 3.72533321 ， <math id="M89"><msub><mrow><mi>z</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo>→</mo><mi>f</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37920516&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37920512&type= 9.56733322 4.57200003 ，将式（19）~ 式（18）式（18）带入式（16）式（16），得到速度环闭环传递函数： <math display="block" id="M90"><msub><mrow><mi>G</mi></mrow><mrow><mi mathvariant="normal">v</mi><mi mathvariant="normal">c</mi></mrow></msub><mo>=</mo><mfrac><mrow><mi>Ω</mi><mo stretchy="false">(</mo><mi>s</mi><mo stretchy="false">)</mo></mrow><mrow><msup><mrow><mi>Ω</mi></mrow><mrow><mi mathvariant="normal">*</mi></mrow></msup><mo stretchy="false">(</mo><mi>s</mi><mo stretchy="false">)</mo></mrow></mfrac><mo>=</mo><mfrac><mrow><mn mathvariant="normal">1</mn></mrow><mrow><mo stretchy="false">(</mo><mn mathvariant="normal">1</mn><mo>/</mo><msub><mrow><mi>K</mi></mrow><mrow><mi mathvariant="normal">v</mi><mi mathvariant="normal">p</mi></mrow></msub><mo stretchy="false">)</mo><mi>s</mi><mo>+</mo><mn mathvariant="normal">1</mn></mrow></mfrac></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37920523&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37920519&type= 45.04266739 9.39799976 . 取 <math id="M91"><msub><mrow><mi>K</mi></mrow><mrow><mi mathvariant="normal">v</mi><mi mathvariant="normal">p</mi></mrow></msub><mo>=</mo><msub><mrow><mi>ω</mi></mrow><mrow><mi mathvariant="normal">v</mi><mi mathvariant="normal">c</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37920532&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37920528&type= 14.39333248 3.72533321 ，根据速度环所需角频率带宽 <math id="M92"><msub><mrow><mi>ω</mi></mrow><mrow><mi mathvariant="normal">v</mi><mi mathvariant="normal">c</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37920540&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37920536&type= 4.48733330 3.72533321 确定控制器增益 <math id="M93"><msub><mrow><mi>K</mi></mrow><mrow><mi mathvariant="normal">v</mi><mi mathvariant="normal">p</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37920550&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37920544&type= 4.99533367 3.72533321 。 3.3　位置环比例控制器与跟踪微分器设计 2 位置环控制原理如图5 图5 所示。速度环整定完成后，忽略较小的电流环时间常数，位置环被控对象传递函数为： <math display="block" id="M94"><msub><mrow><mi>G</mi></mrow><mrow><mi mathvariant="normal">p</mi><mo>_</mo><mi mathvariant="normal">p</mi><mi mathvariant="normal">l</mi><mi mathvariant="normal">a</mi><mi mathvariant="normal">n</mi><mi mathvariant="normal">t</mi></mrow></msub><mo stretchy="false">(</mo><mi>s</mi><mo stretchy="false">)</mo><mo>=</mo><mfrac><mrow><mi>θ</mi><mo stretchy="false">(</mo><mi>s</mi><mo stretchy="false">)</mo></mrow><mrow><msup><mrow><mi>Ω</mi></mrow><mrow><mi mathvariant="normal">*</mi></mrow></msup><mo stretchy="false">(</mo><mi>s</mi><mo stretchy="false">)</mo></mrow></mfrac><mo>=</mo><mfrac><mrow><mn mathvariant="normal">1</mn></mrow><mrow><mi>s</mi><mo stretchy="false">(</mo><mo stretchy="false">(</mo><mn mathvariant="normal">1</mn><mo>/</mo><msub><mrow><mi>K</mi></mrow><mrow><mi mathvariant="normal">v</mi><mi mathvariant="normal">p</mi></mrow></msub><mo stretchy="false">)</mo><mi>s</mi><mo>+</mo><mn mathvariant="normal">1</mn><mo stretchy="false">)</mo></mrow></mfrac></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37920555&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37920552&type= 58.41999817 9.39799976 . 位置环控制器设计为比例控制器，则位置环闭环传递函数为： <math display="block" id="M95"><msub><mrow><mi>G</mi></mrow><mrow><mi mathvariant="normal">p</mi><mi mathvariant="normal">c</mi></mrow></msub><mo stretchy="false">(</mo><mi>s</mi><mo stretchy="false">)</mo><mo>=</mo><mfrac><mrow><mi>θ</mi><mo stretchy="false">(</mo><mi>s</mi><mo stretchy="false">)</mo></mrow><mrow><msub><mrow><mi>θ</mi></mrow><mrow><mi mathvariant="normal">r</mi></mrow></msub><mo stretchy="false">(</mo><mi>s</mi><mo stretchy="false">)</mo></mrow></mfrac><mo>=</mo><mfrac><mrow><msub><mrow><mi>K</mi></mrow><mrow><mi mathvariant="normal">v</mi><mi mathvariant="normal">p</mi></mrow></msub><msub><mrow><mi>K</mi></mrow><mrow><mi mathvariant="normal">p</mi><mi mathvariant="normal">p</mi></mrow></msub></mrow><mrow><msup><mrow><mi>s</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mo>+</mo><msub><mrow><mi>K</mi></mrow><mrow><mi mathvariant="normal">v</mi><mi mathvariant="normal">p</mi></mrow></msub><mi>s</mi><mo>+</mo><msub><mrow><mi>K</mi></mrow><mrow><mi mathvariant="normal">v</mi><mi mathvariant="normal">p</mi></mrow></msub><msub><mrow><mi>K</mi></mrow><mrow><mi mathvariant="normal">p</mi><mi mathvariant="normal">p</mi></mrow></msub></mrow></mfrac></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37920575&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37920571&type= 57.48867035 9.31333351 . 选取位置环比例增益 <math id="M96"><msub><mrow><mi>K</mi></mrow><mrow><mi mathvariant="normal">p</mi><mi mathvariant="normal">p</mi></mrow></msub><mo>=</mo><mfrac><mrow><msub><mrow><mi>ω</mi></mrow><mrow><mi mathvariant="normal">v</mi><mi mathvariant="normal">c</mi></mrow></msub></mrow><mrow><mn mathvariant="normal">4</mn></mrow></mfrac></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37920583&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37920579&type= 15.07066536 7.87400007 ，则位置环传递函数为： <math display="block" id="M97"><msub><mrow><mi>G</mi></mrow><mrow><mi mathvariant="normal">p</mi><mi mathvariant="normal">c</mi></mrow></msub><mo stretchy="false">(</mo><mi>s</mi><mo stretchy="false">)</mo><mo>=</mo><mfrac><mrow><mi>θ</mi><mo stretchy="false">(</mo><mi>s</mi><mo stretchy="false">)</mo></mrow><mrow><msub><mrow><mi>θ</mi></mrow><mrow><mi mathvariant="normal">r</mi></mrow></msub><mo stretchy="false">(</mo><mi>s</mi><mo stretchy="false">)</mo></mrow></mfrac><mo>=</mo><mfrac><mrow><msup><mrow><mo stretchy="false">(</mo><mfrac><mrow><msub><mrow><mi>ω</mi></mrow><mrow><mi mathvariant="normal">v</mi><mi mathvariant="normal">c</mi></mrow></msub></mrow><mrow><mn mathvariant="normal">2</mn></mrow></mfrac><mo stretchy="false">)</mo></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup></mrow><mrow><msup><mrow><mo stretchy="false">(</mo><mi>s</mi><mo>+</mo><mfrac><mrow><msub><mrow><mi>ω</mi></mrow><mrow><mi mathvariant="normal">v</mi><mi mathvariant="normal">c</mi></mrow></msub></mrow><mrow><mn mathvariant="normal">2</mn></mrow></mfrac><mo stretchy="false">)</mo></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup></mrow></mfrac></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37920592&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37920588&type= 44.78866577 17.18733406 . 由此可见，位置环闭环角频率带宽 <math id="M98"><msub><mrow><mi>ω</mi></mrow><mrow><mi mathvariant="normal">p</mi><mi mathvariant="normal">c</mi></mrow></msub><mo>=</mo><mfrac><mrow><msub><mrow><mi>ω</mi></mrow><mrow><mi mathvariant="normal">v</mi><mi mathvariant="normal">c</mi></mrow></msub></mrow><mrow><mn mathvariant="normal">2</mn></mrow></mfrac></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37920599&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37920596&type= 14.56266594 7.87400007 。位置环采用跟踪微分器（Tracking-Differentiator， TD）安排过渡过程， <math id="M99"><msub><mrow><mi>θ</mi></mrow><mrow><mi mathvariant="normal">r</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37920606&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37920603&type= 2.62466669 3.72533321 和 <math id="M100"><msub><mrow><mi>Ω</mi></mrow><mrow><mi mathvariant="normal">r</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37920614&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37920610&type= 3.30200005 3.72533321 为根据最大加速度、最大速度规划的参考位置和速度，以抑制谐振和减小超调。当位置误差较大时，位置环比例控制器饱和，位置环开环，采用TD规划的速度指令进行速度闭环控制，当位置误差较小时，位置环闭环，速度指令为位置环比例控制器输出，线性区 <math id="M101"><mi>δ</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37920619&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37920617&type= 1.94733346 2.79399991 的选取应该保证速度尽量平缓切换，避免速度响应剧烈跳变。具体表达式如下： <math display="block" id="M102"><msup><mrow><mi>Ω</mi></mrow><mrow><mi mathvariant="normal">*</mi></mrow></msup><mo>=</mo><mfenced open="{" close="" separators="|"><mrow><mtable columnalign="left"><mtr><mtd><msub><mrow><mi mathvariant="normal">Ω</mi></mrow><mrow><mi mathvariant="normal">r</mi></mrow></msub><mo>,</mo><mtext> </mtext><mfenced open="|" close="|" separators="|"><mrow><msub><mrow><mi>e</mi></mrow><mrow><mi>θ</mi></mrow></msub></mrow></mfenced><mo>></mo><mi>δ</mi></mtd></mtr><mtr><mtd><msub><mrow><mi>K</mi></mrow><mrow><mi mathvariant="normal">p</mi><mi mathvariant="normal">p</mi></mrow></msub><msub><mrow><mi>e</mi></mrow><mrow><mi>θ</mi></mrow></msub><mo>,</mo><mtext> </mtext><mfenced open="|" close="|" separators="|"><mrow><msub><mrow><mi>e</mi></mrow><mrow><mi>θ</mi></mrow></msub></mrow></mfenced><mo>≤</mo><mi>δ</mi></mtd></mtr></mtable></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37920626&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37920623&type= 35.13666534 10.58333302 . 此处TD采用韩氏离散最速控制函数fhan ［ 15 15 ］。离散最速跟踪函数fhan公式如下： <math display="block" id="M103"><mtable columnalign="left"><mtr><mtd><mi mathvariant="normal">f</mi><mi mathvariant="normal">h</mi><mi mathvariant="normal">a</mi><mi mathvariant="normal">n</mi><mo stretchy="false">(</mo><msub><mrow><mi>x</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>x</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo>,</mo><mi>r</mi><mo>,</mo><mi>h</mi><mo stretchy="false">)</mo><mo>:</mo></mtd></mtr><mtr><mtd><mfenced open="{" close="" separators="|"><mrow><mtable columnalign="left"><mtr><mtd><mi>d</mi><mo>=</mo><mi>r</mi><mi>h</mi></mtd></mtr><mtr><mtd><msub><mrow><mi>d</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><mo>=</mo><mi>h</mi><mi>d</mi></mtd></mtr><mtr><mtd><mi>y</mi><mo>=</mo><msub><mrow><mi>x</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>+</mo><mi>h</mi><msub><mrow><mi>x</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub></mtd></mtr><mtr><mtd><msub><mrow><mi>a</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><mo>=</mo><mroot><mrow><msup><mrow><mi>d</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mo>+</mo><mn mathvariant="normal">8</mn><mi>r</mi><mfenced open="|" close="|" separators="|"><mrow><mi>y</mi></mrow></mfenced></mrow><mrow/></mroot></mtd></mtr><mtr><mtd><mi>a</mi><mo>=</mo><mfenced open="{" close="" separators="|"><mrow><mtable columnalign="left"><mtr><mtd><msub><mrow><mi>x</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo>+</mo><mfrac><mrow><mo stretchy="false">(</mo><msub><mrow><mi>a</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><mo>-</mo><mi>d</mi><mo stretchy="false">)</mo></mrow><mrow><mn mathvariant="normal">2</mn></mrow></mfrac><mi mathvariant="normal">s</mi><mi mathvariant="normal">i</mi><mi mathvariant="normal">g</mi><mi mathvariant="normal">n</mi><mo stretchy="false">(</mo><mi>y</mi><mo stretchy="false">)</mo><mo>,</mo><mtext> </mtext><mfenced open="|" close="|" separators="|"><mrow><mi>y</mi></mrow></mfenced><mo>></mo><msub><mrow><mi>d</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub></mtd></mtr><mtr><mtd><msub><mrow><mi>x</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo>+</mo><mfrac><mrow><mi>y</mi></mrow><mrow><mi>h</mi></mrow></mfrac><mo>,</mo><mtext> </mtext><mfenced open="|" close="|" separators="|"><mrow><mi>y</mi></mrow></mfenced><mo>≤</mo><msub><mrow><mi>d</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub></mtd></mtr></mtable></mrow></mfenced></mtd></mtr><mtr><mtd><mi mathvariant="normal">f</mi><mi mathvariant="normal">h</mi><mi mathvariant="normal">a</mi><mi mathvariant="normal">n</mi><mo>=</mo><mo>-</mo><mfenced open="{" close="" separators="|"><mrow><mtable columnalign="left"><mtr><mtd><mi>r</mi><mi mathvariant="normal">s</mi><mi mathvariant="normal">i</mi><mi mathvariant="normal">g</mi><mi mathvariant="normal">n</mi><mo stretchy="false">(</mo><mi>a</mi><mo stretchy="false">)</mo><mo>,</mo><mtext> </mtext><mfenced open="|" close="|" separators="|"><mrow><mi>a</mi></mrow></mfenced><mo>></mo><mi>d</mi></mtd></mtr><mtr><mtd><mi>r</mi><mfrac><mrow><mi>a</mi></mrow><mrow><mi>d</mi></mrow></mfrac><mo>,</mo><mtext> </mtext><mfenced open="|" close="|" separators="|"><mrow><mi>a</mi></mrow></mfenced><mo>≤</mo><mi>d</mi></mtd></mtr></mtable></mrow></mfenced></mtd></mtr></mtable></mrow></mfenced></mtd></mtr></mtable></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37920635&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37920629&type= 62.22999954 64.26200104 . 依据望远镜最大速度、最大加速度限幅，改进最速离散跟踪微分器，设计如下： <math display="block" id="M104"><mfenced open="{" close="" separators="|"><mrow><mtable columnalign="left"><mtr><mtd><mi>f</mi><mi>h</mi><mo>=</mo><mi mathvariant="normal">f</mi><mi mathvariant="normal">h</mi><mi mathvariant="normal">a</mi><mi mathvariant="normal">n</mi><mo stretchy="false">(</mo><msub><mrow><mi>x</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo stretchy="false">(</mo><mi>k</mi><mo stretchy="false">)</mo><mo>-</mo><mi>v</mi><mo stretchy="false">(</mo><mi>k</mi><mo stretchy="false">)</mo><mo>,</mo><msub><mrow><mi>x</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo stretchy="false">(</mo><mi>k</mi><mo stretchy="false">)</mo><mo>,</mo><mi>r</mi><mo>,</mo><msub><mrow><mi>h</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><mo stretchy="false">)</mo></mtd></mtr><mtr><mtd><msub><mrow><mi>x</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo stretchy="false">(</mo><mi>k</mi><mo>+</mo><mn mathvariant="normal">1</mn><mo stretchy="false">)</mo><mo>=</mo><msub><mrow><mi>x</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo stretchy="false">(</mo><mi>k</mi><mo stretchy="false">)</mo><mo>+</mo><mi>h</mi><msub><mrow><mi>x</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo stretchy="false">(</mo><mi>k</mi><mo stretchy="false">)</mo></mtd></mtr><mtr><mtd><msub><mrow><mi>x</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo stretchy="false">(</mo><mi>k</mi><mo>+</mo><mn mathvariant="normal">1</mn><mo stretchy="false">)</mo><mo>=</mo><msub><mrow><mi>x</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo stretchy="false">(</mo><mi>k</mi><mo stretchy="false">)</mo><mo>+</mo><mi>h</mi><mi>f</mi><mi>h</mi></mtd></mtr><mtr><mtd><msub><mrow><mi>x</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo stretchy="false">(</mo><mi>k</mi><mo>+</mo><mn mathvariant="normal">1</mn><mo stretchy="false">)</mo><mo>=</mo><mi mathvariant="normal">s</mi><mi mathvariant="normal">a</mi><mi mathvariant="normal">t</mi><mo stretchy="false">(</mo><msub><mrow><mi>x</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo stretchy="false">(</mo><mi>k</mi><mo>+</mo><mn mathvariant="normal">1</mn><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mtd></mtr><mtr><mtd><mi mathvariant="normal">s</mi><mi mathvariant="normal">a</mi><mi mathvariant="normal">t</mi><mo stretchy="false">(</mo><msub><mrow><mi>x</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo stretchy="false">)</mo><mo>=</mo><mi mathvariant="normal">s</mi><mi mathvariant="normal">i</mi><mi mathvariant="normal">g</mi><mi mathvariant="normal">n</mi><mo stretchy="false">(</mo><msub><mrow><mi>x</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo stretchy="false">)</mo><mi mathvariant="normal">m</mi><mi mathvariant="normal">i</mi><mi mathvariant="normal">n</mi><mfenced open="{" close="}" separators="|"><mrow><msub><mrow><mi>x</mi></mrow><mrow><mn mathvariant="normal">2</mn><mi mathvariant="normal">m</mi><mi mathvariant="normal">a</mi><mi mathvariant="normal">x</mi></mrow></msub><mo>,</mo><mfenced open="|" close="|" separators="|"><mrow><msub><mrow><mi>x</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub></mrow></mfenced></mrow></mfenced></mtd></mtr></mtable></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37920643&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37920639&type= 60.70600128 27.77066803 ，式中： <math id="M105"><mi>v</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37920747&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37920744&type= 1.86266661 2.79399991 为被跟随的输入信号； <math id="M106"><mi>r</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37920679&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37920676&type= 1.69333339 2.79399991 为速度因子， <math id="M107"><mi>r</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37920699&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37920696&type= 1.69333339 2.79399991 越大， <math id="M108"><msub><mrow><mi>x</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37920667&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37920664&type= 3.38666677 3.72533321 越快到达设定值 <math id="M109"><mi>v</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37920673&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37920670&type= 1.86266661 2.79399991 ， <math id="M110"><mi>r</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37920679&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37920676&type= 1.69333339 2.79399991 还决定TD的转折频率 <math id="M111"><msub><mrow><mi>ω</mi></mrow><mrow><mi mathvariant="normal">T</mi><mi mathvariant="normal">D</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37920687&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37920683&type= 5.58799982 3.72533321 ，一般 <math id="M112"><msub><mrow><mi>ω</mi></mrow><mrow><mi mathvariant="normal">T</mi><mi mathvariant="normal">D</mi></mrow></msub><mo>=</mo><mn mathvariant="normal">1.14</mn><mroot><mrow><mi>r</mi></mrow><mrow/></mroot></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37920692&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37920689&type= 23.45266724 4.31799984 ， <math id="M113"><mi>r</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37920699&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37920696&type= 1.69333339 2.79399991 一般取望远镜最大加速度，即 <math id="M114"><mi>r</mi><mo>=</mo><msub><mrow><mi>a</mi></mrow><mrow><mi mathvariant="normal">m</mi><mi mathvariant="normal">a</mi><mi mathvariant="normal">x</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37920705&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37920702&type= 11.76866722 3.72533321 ； <math id="M115"><msub><mrow><mi>x</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37920710&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37920708&type= 3.38666677 3.72533321 为 <math id="M116"><msub><mrow><mi>x</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37920713&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37920711&type= 3.38666677 3.72533321 的微分，可以作为 <math id="M117"><mi>v</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37920747&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37920744&type= 1.86266661 2.79399991 的近似微分； <math id="M118"><mi>h</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37920728&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37920726&type= 1.94733346 2.79399991 为积分步长， <math id="M119"><mi>h</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37920728&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37920726&type= 1.94733346 2.79399991 一般为采样周期； <math id="M120"><msub><mrow><mi>h</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37920735&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37920732&type= 3.04800010 3.72533321 为滤波因子，一般取 <math id="M121"><msub><mrow><mi>h</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><mo>=</mo><mi>m</mi><mi>h</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37920741&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37920738&type= 12.61533356 3.72533321 ， m 为大于1的正整数，可以滤除输入信号 <math id="M122"><mi>v</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37920747&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37920744&type= 1.86266661 2.79399991 的噪声， m 越大，滤波性能越好，但是相位滞后越多； <math id="M123"><mi mathvariant="normal">s</mi><mi mathvariant="normal">a</mi><mi mathvariant="normal">t</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37920755&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37920752&type= 4.31799984 2.79399991 为限幅函数，此处要跟随的信号为位置信号 <math id="M124"><msup><mrow><mi>θ</mi></mrow><mrow><mi mathvariant="normal">*</mi></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37920764&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37920759&type= 2.96333337 3.04800010 ； <math id="M125"><msub><mrow><mi>x</mi></mrow><mrow><mn mathvariant="normal">2</mn><mi mathvariant="normal">m</mi><mi mathvariant="normal">a</mi><mi mathvariant="normal">x</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37920773&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37920768&type= 6.77333355 3.72533321 为系统允许最大机械角速度值 <math id="M126"><msub><mrow><mi>Ω</mi></mrow><mrow><mi mathvariant="normal">m</mi><mi mathvariant="normal">a</mi><mi mathvariant="normal">x</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37920780&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37920776&type= 6.09600019 3.72533321 。 4 实验与结果分析 1 2.5 m望远镜转台及其伺服驱动控制器如图6 图6 所示。望远镜由永磁同步电机直接驱动方位和俯仰转台运动，采用32位绝对式光电编码器作为位置反馈元件，速度信号由位置信号差分及低通滤波得到。控制器采用DSP+FPGA 的架构设计，DSP作为主控制器完成位置环和速度环校正，FPGA作为协控制器完成电流环、故障保护以及脉宽调制。驱动器采用智能功率模块，最大驱动电压为1 200 Ｖ，最大驱动电流为75 Ａ，伺服系统采用直流开关电源供电，直流母线电压为60 Ｖ。伺服系统参数如表1 表1 所示。矢量控制电子换向基于七段式空间矢量脉宽调制（Space Vector Pulse Width Modulation， SVPWM）实现，电流环的采样频率为15 kHz，位置环的采样频率为1 000 Hz，速度信号经过位置差分和低通滤波得到。参数数值同步电感 <math id="M127"><mi>L</mi><mo>/</mo><mi mathvariant="normal">m</mi><mi mathvariant="normal">H</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37920799&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37920795&type= 8.63599968 2.79399991 23.75 定子相电阻 <math id="M128"><msub><mrow><mi>R</mi></mrow><mrow><mi mathvariant="normal">s</mi></mrow></msub><mo>/</mo><mi mathvariant="normal">Ω</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37920807&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37920804&type= 6.94266701 3.55599999 2.45 极对数 <math id="M129"><msub><mrow><mi>n</mi></mrow><mrow><mi mathvariant="normal">p</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37920814&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37920811&type= 2.62466669 3.21733332 45 转矩系数 <math id="M130"><msub><mrow><mi>K</mi></mrow><mrow><mi mathvariant="normal">t</mi></mrow></msub><mo>/</mo><mo stretchy="false">(</mo><mi mathvariant="normal">N</mi><mo>⋅</mo><mi mathvariant="normal">m</mi><mo>⋅</mo><msup><mrow><mi mathvariant="normal">A</mi></mrow><mrow><mo>-</mo><mn>1</mn></mrow></msup><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37920820&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37920817&type= 21.75933456 4.06400013 118 转动惯量 <math id="M131"><mi>J</mi><mo>/</mo><mo stretchy="false">(</mo><mi mathvariant="normal">k</mi><mi mathvariant="normal">g</mi><mo>⋅</mo><msup><mrow><mi mathvariant="normal">m</mi></mrow><mrow><mn>2</mn></mrow></msup><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37920828&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37920824&type= 14.64733315 3.55599999 7 100 粘滞摩擦系数 <math id="M132"><mi>B</mi><mo>/</mo><mo stretchy="false">(</mo><mi mathvariant="normal">N</mi><mo>⋅</mo><mi mathvariant="normal">m</mi><mo>⋅</mo><mi mathvariant="normal">s</mi><mo>⋅</mo><mi mathvariant="normal">r</mi><mi mathvariant="normal">a</mi><msup><mrow><mi mathvariant="normal">d</mi></mrow><mrow><mo>-</mo><mn>1</mn></mrow></msup><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37920838&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37920835&type= 25.31533241 3.21733332 30 母线电压 <math id="M133"><msub><mrow><mi>V</mi></mrow><mrow><mi mathvariant="normal">B</mi><mi mathvariant="normal">u</mi><mi mathvariant="normal">s</mi></mrow></msub><mo>/</mo><mi mathvariant="normal">V</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37920845&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37920842&type= 9.65200043 3.55599999 60 电流限幅 <math id="M134"><msub><mrow><mi>I</mi></mrow><mrow><mi mathvariant="normal">q</mi><mi mathvariant="normal">m</mi></mrow></msub><mo>/</mo><mi mathvariant="normal">A</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37920853&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37920849&type= 7.70466709 3.55599999 10 4.1　控制器设计带宽验证 2 交、直轴电流环带宽设计为150 Hz，根据系统参数，选取电流环PI控制器的积分时间常数 <math id="M135"><msub><mrow><mi>τ</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>=</mo><mn mathvariant="normal">0.009</mn><mtext> </mtext><mn mathvariant="normal">7</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37920869&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37920866&type= 19.21933365 3.80999994 ，比例系数 <math id="M136"><msub><mrow><mi>K</mi></mrow><mrow><mi mathvariant="normal">c</mi><mi mathvariant="normal">p</mi></mrow></msub><mo>=</mo><mn mathvariant="normal">22.38</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37920876&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37920873&type= 19.13466644 3.72533321 。电流环闭环后，得到速度环被控对象的传递函数为： <math display="block" id="M137"><msub><mrow><mi>G</mi></mrow><mrow><mi mathvariant="normal">v</mi><mo>_</mo><mi mathvariant="normal">p</mi><mi mathvariant="normal">l</mi><mi mathvariant="normal">a</mi><mi mathvariant="normal">n</mi><mi mathvariant="normal">t</mi></mrow></msub><mo stretchy="false">(</mo><mi>s</mi><mo stretchy="false">)</mo><mo>=</mo><mfrac><mrow><mn mathvariant="normal">118</mn></mrow><mrow><mo stretchy="false">(</mo><mn mathvariant="normal">7100</mn><mi>s</mi><mo>+</mo><mn mathvariant="normal">30</mn><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mn mathvariant="normal">0.001</mn><mi>s</mi><mo>+</mo><mn mathvariant="normal">1</mn><mo stretchy="false">)</mo></mrow></mfrac></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37920886&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37920880&type= 60.19800186 8.63599968 . 根据伺服系统参数，得到速度环LADRC增益 <math id="M138"><mi>b</mi><mo>=</mo><mn mathvariant="normal">0.016</mn><mtext> </mtext><mn mathvariant="normal">62</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37920892&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37920888&type= 20.65866661 2.87866688 ，俯仰轴一阶锁定谐振频率为24.8 Hz，速度环带宽设计为8 Hz，约为锁定谐振频率的三分之一，所以，速度环线性自抗扰控制器参数选取为 <math id="M139"><msub><mrow><mi>ω</mi></mrow><mrow><mi mathvariant="normal">v</mi><mi mathvariant="normal">o</mi></mrow></msub><mo>=</mo><msub><mrow><mi>ω</mi></mrow><mrow><mi mathvariant="normal">v</mi><mi mathvariant="normal">c</mi></mrow></msub><mo>=</mo><mn mathvariant="normal">50.24</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37920898&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37920896&type= 28.27866554 3.72533321 。实际速度闭环伯德图如图7 图7 所示，实测速度闭环带宽为7.9 Hz，与设计值基本吻合。加入结构滤波器后速度环的低频特性基本不变，高频噪声和机械谐振峰值得到有效抑制。速度环整定完成后，得到位置环被控对象传递函数为： <math display="block" id="M140"><msub><mrow><mi>G</mi></mrow><mrow><mi mathvariant="normal">p</mi><mo>_</mo><mi mathvariant="normal">p</mi><mi mathvariant="normal">l</mi><mi mathvariant="normal">a</mi><mi mathvariant="normal">n</mi><mi mathvariant="normal">t</mi></mrow></msub><mo stretchy="false">(</mo><mi>s</mi><mo stretchy="false">)</mo><mo>=</mo><mfrac><mrow><mn mathvariant="normal">1</mn></mrow><mrow><mi>s</mi><mo stretchy="false">(</mo><mn mathvariant="normal">0.019</mn><mtext> </mtext><mn mathvariant="normal">9</mn><mi>s</mi><mo>+</mo><mn mathvariant="normal">1</mn><mo stretchy="false">)</mo></mrow></mfrac></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37920918&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37920914&type= 43.85733414 8.63599968 . 位置环比例增益 <math id="M141"><msub><mrow><mi>K</mi></mrow><mrow><mi mathvariant="normal">p</mi><mi mathvariant="normal">p</mi></mrow></msub><mo>=</mo><mn mathvariant="normal">12.56</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37920924&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37920920&type= 19.30400085 3.72533321 ，设计的位置环带宽为4.0 Hz，实际位置环伯德图如图8 图8 所示，实测位置环闭环带宽为5.1 Hz，达到了设计值。 4.2　TD安排过渡过程验证 2 过渡过程参数如下：最大加速度 <math id="M142"><msub><mrow><mi>a</mi></mrow><mrow><mi mathvariant="normal">m</mi><mi mathvariant="normal">a</mi><mi mathvariant="normal">x</mi></mrow></msub><mo>=</mo><mn mathvariant="normal">7</mn><mtext> </mtext><mo stretchy="false">(</mo><mo>°</mo><mo stretchy="false">)</mo><mo>/</mo><msup><mrow><mi mathvariant="normal">s</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37920943&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37920938&type= 21.75933456 4.65666676 ，最大速度 <math id="M143"><msub><mrow><mi mathvariant="normal">Ω</mi></mrow><mrow><mi mathvariant="normal">m</mi><mi mathvariant="normal">a</mi><mi mathvariant="normal">x</mi></mrow></msub><mo>=</mo><mn mathvariant="normal">10</mn><mtext> </mtext><mo stretchy="false">(</mo><mo>°</mo><mo stretchy="false">)</mo><mo>/</mo><mi mathvariant="normal">s</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37920951&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37920947&type= 23.11400032 4.48733330 ，位置环采样频率为1 000 Hz，所以积分步长 <math id="M144"><mi>h</mi><mo>=</mo><mn mathvariant="normal">0.001</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37920959&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37920956&type= 16.17133331 2.79399991 ，取 <math id="M145"><msub><mrow><mi>h</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><mo>=</mo><mn mathvariant="normal">0.002</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37920966&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37920963&type= 17.35666656 3.72533321 ，线性区 <math id="M146"><mi>δ</mi><mo>=</mo><mn mathvariant="normal">0.005</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37920974&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37920970&type= 16.25600052 2.79399991 °，即位置误差 <math id="M147"><mfenced open="|" close="|" separators="|"><mrow><msub><mrow><mi>e</mi></mrow><mrow><mi>θ</mi></mrow></msub></mrow></mfenced><mo>≤</mo><mi>δ</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37920979&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37920977&type= 11.59933376 3.97933316 切入位置闭环，避免跟踪微分器误差带来位置环稳态误差。离散最速跟踪函数为 <math id="M148"><mi mathvariant="normal">f</mi><mi mathvariant="normal">h</mi><mi mathvariant="normal">a</mi><mi mathvariant="normal">n</mi><mo stretchy="false">(</mo><msub><mrow><mi>θ</mi></mrow><mrow><mi mathvariant="normal">r</mi></mrow></msub><mo>-</mo><msup><mrow><mi>θ</mi></mrow><mrow><mi mathvariant="normal">*</mi></mrow></msup><mo>,</mo><msub><mrow><mi>Ω</mi></mrow><mrow><mi mathvariant="normal">r</mi></mrow></msub><mo>,</mo><msub><mrow><mi>a</mi></mrow><mrow><mi mathvariant="normal">m</mi><mi mathvariant="normal">a</mi><mi mathvariant="normal">x</mi></mrow></msub><mo>,</mo><msub><mrow><mi>h</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37920988&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37920983&type= 37.93066406 4.65666676 。 1.24°视场角快速步进搜索时的位置、速度响应曲线如图9 图9 所示。从图中可以看出，TD规划速度曲线为三角形，先以最大加速度 <math id="M149"><msub><mrow><mi>a</mi></mrow><mrow><mi mathvariant="normal">m</mi><mi mathvariant="normal">a</mi><mi mathvariant="normal">x</mi></mrow></msub><mo>=</mo><mn mathvariant="normal">7</mn><mtext> </mtext><mo stretchy="false">(</mo><mo>°</mo><mo stretchy="false">)</mo><mo>/</mo><msup><mrow><mi mathvariant="normal">s</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37920995&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37920991&type= 21.59000015 4.65666676 加速运行，速度未达到最大速度就开始减速运行，调节时间为1.0 s。位置环进入线性区时虽然有很小的速度跳跃，但是对最终跟踪响应时间的影响很小。不采用TD规划过渡过程，最大加速度放大至10 （°）/s 2 ，减速过程速度分段切入线性区，此时1.24°阶跃时的位置、速度响应曲线如图10 图10 所示，速度切换过程有3次较大的跳跃，且调节时间为1.6 s。当进行60°大范围搜索时的位置、速度响应曲线如图11 图11 所示。TD规划速度曲线为梯形，先以最大加速度 <math id="M150"><msub><mrow><mi>a</mi></mrow><mrow><mi mathvariant="normal">m</mi><mi mathvariant="normal">a</mi><mi mathvariant="normal">x</mi></mrow></msub><mo>=</mo><mn mathvariant="normal">7</mn><mtext> </mtext><mo stretchy="false">(</mo><mo>°</mo><mo stretchy="false">)</mo><mo>/</mo><msup><mrow><mi mathvariant="normal">s</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37921022&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37921021&type= 21.75933456 4.65666676 加速运行，速度达到最大速度 <math id="M151"><msub><mrow><mi>Ω</mi></mrow><mrow><mi mathvariant="normal">m</mi><mi mathvariant="normal">a</mi><mi mathvariant="normal">x</mi></mrow></msub><mo>=</mo><mn mathvariant="normal">10</mn><mtext> </mtext><mo stretchy="false">(</mo><mo>°</mo><mo stretchy="false">)</mo><mo>/</mo><mi mathvariant="normal">s</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37921027&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37921025&type= 23.19866753 4.48733330 后匀速运行，然后减速运行，调节时间为7.6 s。不采用TD规划过渡过程，最大加速度放大至10 （°）/s 2 ，减速过程速度分段切入线性区，此时60°阶跃时的位置、速度响应曲线如图12 图12 所示，速度切换过程有5次较大的跳跃，且调节时间为8.6 s。可见，采用TD规划过渡过程，不论小范围阶跃还是大范围阶跃，速度切换都更平缓，且调节时间更短。 4.3　跟踪性能验证 2 为验证快速目标跟踪性能，以速度为2 （°）/s，加速度为1 （°）/s 2 作为等效正弦引导，即 <math id="M152"><msup><mrow><mi>θ</mi></mrow><mrow><mi mathvariant="normal">*</mi></mrow></msup><mo>=</mo><mn mathvariant="normal">4</mn><mi mathvariant="normal">s</mi><mi mathvariant="normal">i</mi><mi mathvariant="normal">n</mi><mo stretchy="false">(</mo><mn mathvariant="normal">0.5</mn><mi>t</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37921045&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37921042&type= 24.46866608 3.80999994 。为描述方便，位置环采用TD规划+P控制，速度环采用LADRC控制，电流环采用PI控制，本文记作LADRC控制；位置环分段限幅+P控制，速度环采用PI控制，电流环采用PI控制，记作PI控制。作为对比，速度环PI控制带宽与LADRC相当。LADRC和PI控制的位置、速度和误差曲线分别如图13 图13 ~ 图14 图14 所示。LADRC等效正弦引导位置误差峰值为2.62″，RMS值约为0.60″。PI控制等效正弦引导位置误差峰值为3.6″，RMS值约为1.08″。可见，LADRC比PI控制等效正弦跟踪误差更小，跟踪精度更高。为验证超低速跟踪性能，以斜率为0.000 1 （°）/s的位置曲线作为引导，即 <math id="M153"><msup><mrow><mi>θ</mi></mrow><mrow><mi mathvariant="normal">*</mi></mrow></msup><mo>=</mo><mn mathvariant="normal">0.000</mn><mtext> </mtext><mn mathvariant="normal">1</mn><mi>t</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37921069&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37921067&type= 21.08200073 3.13266683 。LADRC和PI控制的位置、速度及误差曲线分别如图15 图15 ~ 图16 图16 所示。由图可见，LADRC跟踪超低速斜坡位置引导信号的误差RMS值为0.010 6″，PI控制跟踪超低速斜坡位置引导信号的误差 RMS值为0.015 8″。LADRC的超低速跟踪误差RMS值更小，跟踪精度更高。 4.4　抗扰性能验证 2 2.5 m口径望远镜要求在15 m/s风速下工作，那么风阻转矩的影响不可忽略。俯仰轴垂直指向天顶，且风以垂直镜筒方向流动时，望远镜承受最大的风阻力矩。为简化分析，以静态风载的作用力公式进行计算，其表达式为： <math display="block" id="M154"><mi>F</mi><mo>=</mo><mi>P</mi><mi>A</mi><mo>=</mo><mfrac><mrow><mn mathvariant="normal">1</mn></mrow><mrow><mn mathvariant="normal">2</mn></mrow></mfrac><msub><mrow><mi>C</mi></mrow><mrow><mi mathvariant="normal">D</mi></mrow></msub><mi>ρ</mi><mi>A</mi><msup><mrow><mi>V</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37921084&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37921083&type= 34.54399872 7.87400007 ，其中： <math display="block" id="M155"><mi>P</mi><mo>=</mo><mfrac><mrow><mn mathvariant="normal">1</mn></mrow><mrow><mn mathvariant="normal">2</mn></mrow></mfrac><msub><mrow><mi>C</mi></mrow><mrow><mi mathvariant="normal">D</mi></mrow></msub><mi>ρ</mi><msup><mrow><mi>V</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37921089&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37921086&type= 21.59000015 7.87400007 ，式中： <math id="M156"><msub><mrow><mi>C</mi></mrow><mrow><mi mathvariant="normal">D</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37921093&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37921091&type= 4.23333359 3.72533321 为风阻系数，对于长圆柱体其值取0.8； <math id="M157"><mi>ρ</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37921097&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37921095&type= 2.11666679 3.64066648 为海拔高度处的空气密度； A 为结构迎风面积； V 为风速； P 为风压力。海拔高度为400 m，风速 V =15 m/s，空气密度 ρ =1.2 kg/m 3 ，则风压 <math id="M158"><mi>P</mi><mo>=</mo><mfrac><mrow><mn mathvariant="normal">1</mn></mrow><mrow><mn mathvariant="normal">2</mn></mrow></mfrac><msub><mrow><mi>C</mi></mrow><mrow><mi mathvariant="normal">D</mi></mrow></msub><mi>ρ</mi><msup><mrow><mi>V</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mo>=</mo><mn mathvariant="normal">108</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37921103&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37921099&type= 31.91933441 7.87400007 ，风压乘以作用面积、作用力臂，就是风阻产生的力矩。按照机械设计参数，俯仰轴承受风阻的受力面积及力臂如表2 表2 所示。其中，俯仰轴上半部分的定义为正，下半部分定义为负。结构名称受力面积/m 2 力臂/m 作用力矩/（N·m）探测器外罩 0.5 1.95 105 主镜遮光罩 4.29 1.05 486 主镜室支撑 -2.74 0.815 -241 综合以上数据，计算俯仰轴最大风阻力矩为 <math id="M159"><msub><mrow><mi>T</mi></mrow><mrow><mi mathvariant="normal">w</mi><mi mathvariant="normal">i</mi><mi mathvariant="normal">n</mi><mi mathvariant="normal">d</mi></mrow></msub><mo>=</mo><mn mathvariant="normal">351</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37921111&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37921109&type= 17.10266685 3.72533321 N·m。通常风载为低频扰动，频率集中在1 Hz以下。通过调节电永磁吸盘充磁电流从而调节摩擦吸附力，使加载摩擦转矩保持在351 N·m，模拟风载的加载和卸载。在0.1 s时给0.01 （°）/s速度阶跃指令，在0.5 s时充磁突加351 N·m负载，在1.5 s时退磁卸掉351 N·m负载，测试速度波动情况如图17 图17 所示。在速度环PI和LADRC控制的闭环带宽（7.9 Hz）相同的情况下，LADRC在突加、突减负载时，速度波动后可以迅速恢复到指令值，而PI控制在加载期间已不能稳定跟随输入指令。LADRC的LESO估计的负载扰动如图18 图18 所示，LESO可以很好地估计负载扰动，通过前馈补偿负载扰动的影响，从而具有更高的跟踪精度。由此可见，ADRC比PI控制具有更好的抗扰性能。 5 结论 1 本文根据2.5 m望远镜对快速搜索和超低速精密跟踪要求，提出望远镜伺服控制器的位置、速度、电流三闭环串级控制设计流程和方法，并设计了望远镜伺服系统频率特性辨识和结构滤波器。依据伺服系统一阶锁定谐振频率，介绍了带宽法设计速度环LADRC参数、位置环比例控制参数，使得控制器参数设计有据可循，避免了PI控制器参数的试凑调试。依据设备最大速度值、最大加速度值，设计离散最速跟踪微分器安排位置环过渡过程，相比其他指令平滑方法无需参数调试，提高了视场角步进搜索和大范围掉转的动态响应能力。实验结果表明：LADRC相比传统PI控制，具有更好的跟踪精度和抗扰性能。在2.5 m大视场望远镜做1.24°视场角快速步进搜索时，调节时间由1.6 s减小到1.0 s，速度更平缓；以速度为2 （°）/s，加速度为1 （°）/s 2 作等效正弦引导时，位置最大稳态误差由3.6″降低为2.62″，RMS值由1.08″降低为0.60″；超低速跟踪斜率为0.000 1 （°）/s的位置斜坡曲线时，位置跟踪稳态误差RMS值由0.015 8″降低为0.010 6″。综上可知，LADRC能够满足2.5 m大视场角天文望远镜高效率快速搜索和超低速精密跟踪要求，为大口径大视场望远镜主轴伺服控制系统设计提供了参考。参考文献邓永停，李洪文，王建立 . 大型望远镜交流伺服控制系统综述［J］. 中国光学， 2015 ， 8 （ 6 ）： 895 - 908 . 大型望远镜交流伺服控制系统综述 DENG Y T ， LI H W ， WANG J L . 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