面向纳米栅格扫描的时滞压电微动平台耦合补偿

张志名; 闫鹏

doi:10.37188/OPE.20233111.1641

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面向纳米栅格扫描的时滞压电微动平台耦合补偿

微纳技术与精密机械 | 更新时间：2023-07-08

- 面向纳米栅格扫描的时滞压电微动平台耦合补偿
- Coupling compensation for nano raster scanning of piezoelectric nano-stages with delays
- 光学精密工程
- 作者机构：
  
  1.山东大学机械工程学院，山东济南 250061
  2.高效洁净机械制造教育部重点实验室，山东济南 250061
- 作者简介：
  
  [ "张志名（1993-），男，山东滕州人，博士研究生，2016年于山东大学获得学士学位，主要从事柔性机构学、超精密机电系统的建模与控制等方面的研究。E-mail： jimmyzhangsdu@126.com" ]
  
  张志名（1993-），男，山东滕州人，博士研究生，2016年于山东大学获得学士学位，主要从事柔性机构学、超精密机电系统的建模与控制等方面的研究。E-mail： jimmyzhangsdu@126.com
  [ "闫鹏（1975-），男，山东济南人，教授，博士生导师，1997年、1999年于东南大学分别获得学士、硕士学位，2003年于俄亥俄州立大学获得博士学位，主要从事微纳测量/控制与制造、超精密机电系统的设计与控制、微纳智能系统等方面的研究。E-mail： yanpeng@sdu.edu.cn" ]
  
  闫鹏（1975-），男，山东济南人，教授，博士生导师，1997年、1999年于东南大学分别获得学士、硕士学位，2003年于俄亥俄州立大学获得博士学位，主要从事微纳测量/控制与制造、超精密机电系统的设计与控制、微纳智能系统等方面的研究。E-mail： yanpeng@sdu.edu.cn
- 基金信息：
  
  山东省自然科学基金重大基础研究项目(ZR2019ZD08)
- DOI：10.37188/OPE.20233111.1641
  中图分类号： TP273;TH113
- 纸质出版日期：2023-06-10，
  
  收稿日期：2023-01-10，
  
  修回日期：2023-02-01，
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引用本文
张志名,闫鹏.面向纳米栅格扫描的时滞压电微动平台耦合补偿[J].光学精密工程,2023,31(11):1641-1651.

ZHANG Zhiming,YAN Peng.Coupling compensation for nano raster scanning of piezoelectric nano-stages with delays[J].Optics and Precision Engineering,2023,31(11):1641-1651.
张志名,闫鹏.面向纳米栅格扫描的时滞压电微动平台耦合补偿[J].光学精密工程,2023,31(11):1641-1651. DOI： 10.37188/OPE.20233111.1641.

ZHANG Zhiming,YAN Peng.Coupling compensation for nano raster scanning of piezoelectric nano-stages with delays[J].Optics and Precision Engineering,2023,31(11):1641-1651. DOI： 10.37188/OPE.20233111.1641.

面向纳米栅格扫描的时滞压电微动平台耦合补偿 Coupling compensation for nano raster scanning of piezoelectric nano-stages with delays 1 2 first-author corresp 张志名（1993-），男，山东滕州人，博士研究生，2016年于山东大学获得学士学位，主要从事柔性机构学、超精密机电系统的建模与控制等方面的研究。E-mail： jimmyzhangsdu@126.com http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45412435&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45412455&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45412443&type= 22.01332855 32.00400162 张志名（1993-），男，山东滕州人，博士研究生，2016年于山东大学获得学士学位，主要从事柔性机构学、超精密机电系统的建模与控制等方面的研究。E-mail： jimmyzhangsdu@126.com jimmyzhangsdu@126.com 1 2 corresp E-mail： yanpeng@sdu.edu.cn E-mail： yanpeng@sdu.edu.cn 闫鹏（1975-），男，山东济南人，教授，博士生导师，1997年、1999年于东南大学分别获得学士、硕士学位，2003年于俄亥俄州立大学获得博士学位，主要从事微纳测量/控制与制造、超精密机电系统的设计与控制、微纳智能系统等方面的研究。E-mail： yanpeng@sdu.edu.cn http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45412447&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45412459&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45412458&type= 22.01333237 32.00400162 闫鹏（1975-），男，山东济南人，教授，博士生导师，1997年、1999年于东南大学分别获得学士、硕士学位，2003年于俄亥俄州立大学获得博士学位，主要从事微纳测量/控制与制造、超精密机电系统的设计与控制、微纳智能系统等方面的研究。E-mail： yanpeng@sdu.edu.cn yanpeng@sdu.edu.cn 山东大学机械工程学院，山东济南 250061 山东大学机械工程学院，山东济南 250061 School of Mechanical Engineering， Shandong University， Jinan 250061， China School of Mechanical Engineering， Shandong University， Jinan 250061， China 高效洁净机械制造教育部重点实验室，山东济南 250061 高效洁净机械制造教育部重点实验室，山东济南 250061 Key Laboratory of High-efficiency and Clean Mechanical Manufacture of the Ministry of Education， Jinan 250061， China Key Laboratory of High-efficiency and Clean Mechanical Manufacture of the Ministry of Education， Jinan 250061， China 针对栅格纳米扫描特别是高速扫描应用中，带测量滞后的压电纳米平台轴间周期耦合补偿的问题，提出一种无需复杂逆模型求解的鲁棒周期干扰观测控制策略。建立压电平台的机电耦合多扰动模型用以描述平台系统的动力学行为。在此基础上，构建周期干扰观测控制结构。通过推导的性能优化函数和鲁棒稳定性条件，将最优控制器参数求解问题转化为延时系统的混合灵敏度优化问题，并通过无穷维 H ∞ 控制方法进行求解。最后，在压电纳米平台上进行综合的实验探究。结果表明，建立的动力学模型能够有效拟合平台的实验响应；相较于无干扰观测器的方法和传统的周期干扰观测控制方法，提出的鲁棒优化方法在基频的抗干扰性能分别有超过99%和50%的提升。提出的鲁棒周期干扰控制方法有优越的抗干扰和不确定补偿性能。 Aiming to compensate for the cross-periodic coupling of piezoelectric nano-stages with measurement delays in raster nano-scanning， particularly in high-speed scanning applications， this paper presents a robust periodic disturbance observer （PDOB） control strategy that does not require solving complex inverse models. First， an electromechanical multi-perturbation model is developed to describe the dynamic behavior of piezoelectric stage systems. Then， a PDOB control structure based on the model is constructed. The problem of optimal controller parameter solution is transformed into a mixed sensitivity optimization problem for time-delay systems with the formulated performance optimization function and robust stability condition， which is solved via the infinite-dimensional H ∞ control method. Finally， comprehensive experimental investigations are conducted on the piezoelectric nano-stage. The results indicate that the developed dynamic model can effectively fit the experimental response of the stage and that the proposed robust periodic disturbance control method has superior anti-disturbance and uncertainty compensation performance to the method without disturbance observer and the conventional PDOB control method， with improvements of > 99% and > 50%， respectively， in the anti-disturbance performance at the fundamental harmonic frequency. 压电微动平台耦合补偿动力学建模周期干扰观测器时滞 H ∞ 优化 piezoelectric nano-stage coupling compensation dynamic modeling periodic disturbance observer time delay H ∞ optimization 1 引言 1 并联压电微动平台的平面纳米扫描是原子力显微镜［ 1 1 - 2 2 1-2 ］、扫描探针显微镜［ 3 3 ］等显微成像高端装备的关键部件。与李萨如扫描、螺旋扫描等非栅格扫描模式相比，基于三角波的栅格平面扫描有着成像速度恒定、成像质量好的优点［ 4 4 ］，在工程中得到了广泛的应用。然而，在栅格扫描模式下，并联平台扫描轴对定位轴存在严重的轴间耦合，特别在高速扫描时，轴间耦合会导致实际栅格扫描轨迹的畸变现象［ 5 5 ］。因此，实现轴间耦合特别是定位轴周期耦合扰动补偿控制方法设计，是并联压电纳米平台高速精密扫描应用中的一项重要课题。面向并联压电微动平台的轴间周期耦合补偿控制方法可分为三类：反馈整形控制［ 6 6 ］、基于耦合模型的解耦控制［ 7 7 ］以及基于内模的抗干扰控制［ 8 8 ］。然而，现有方法中，反馈整形方法多以提高扰动抑制带宽的方式减缓轴间耦合效应，并没有真正实现解耦控制。基于耦合模型的解耦控制方法依赖于耦合动力学的建模精度，对复杂应用工况下的误差十分敏感。尽管传统的内模抗干扰控制方法可以有效补偿周期耦合带来的运动误差，但对系统逆模型精确建模的要求同样会导致稳定性差的问题，而且压电纳米平台动力学中复杂的非线性和高阶模态［ 9 9 ］，也给该方法的控制应用带来困难。此外，对于采用电容位移传感器进行位置测量与反馈的压电微动平台，传感器的模拟输出信号调理和A/D转换会导致系统动力学存在不可忽略的输出延时［ 10 10 ］，延时的无穷维和非因果逆特性，以及对稳定裕度的衰减使上述控制方法的设计和优化变得更加复杂。本文针对栅格纳米扫描特别是高速扫描应用中，带有输出延时的并联压电微动平台周期耦合扰动补偿问题，提出一种无需复杂逆模型求解的稳定周期干扰观测控制策略。首先，分析并联纳米平台机电耦合动力学行为，采用一种多扰动模型对平台系统的动力学进行描述。在得到模型结构的基础上，构建周期干扰观测控制结构。将控制器参数的求解转化为面向时滞系统的混合灵敏度优化问题，并通过无穷维 H ∞ 控制方法进行求解。最后，在压电微动平台上，实验验证了建立的模型和提出的稳定周期干扰观测控制方法的有效性。 2 原理 1 并联纳米平台存在无法避免的轴间交叉耦合现象，尽管采用对称机构设计能够有效减小轴间耦合［ 11 11 ］，但始终存在的加工/装配误差仍会导致平台沿轴运动的不对称性，从而造成轴间耦合，如图1 图1 （a）所示。在纳米栅格扫描任务中，定位轴单步位移小，相比之下，扫描轴多执行大运动范围的周期运动，故平台的轴间耦合以扫描轴对定位轴的寄生耦合为主导。如图1 图1 （b）所示，以 X -轴为扫描轴做三角波跟踪， Y -轴为定位轴做斜坡定位，以三角波上升段为扫描段， X -轴对 Y -轴的耦合使Y-轴的输出出现较大偏差，进而严重影响了平面扫描的精度。因此，探究面向栅格扫描的轴间耦合补偿控制方法，有着重要的工程意义和应用价值。本文针对栅格扫描中的定位轴，提出一种无需进行复杂逆模型求解的周期干扰观测控制策略，以有效补偿扫描轴带来的周期耦合扰动，实现高动态的纳米定位和高速精密扫描。首先对平台定位轴的主运动方向动力学进行建模，为控制设计提供模型基础。综合考虑压电驱动器电路、机构特性和传感器引入的输出延时，压电微位移平台系统的机电耦合动力学模型如图2 图2 所示。不同于已有的复杂Hammerstein模型推导［ 12 12 ］，本文将平台动力学描述为带有迟滞输入扰动、耦合输出扰动、不确定干扰等多扰动的时滞线性模型。对于压电驱动器电路部分，在电压 <math id="M1"><mi>v</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45412479&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45412478&type= 1.86266661 2.79399991 的输入下，电压放大器被建模为放大增益 <math id="M2"><msub><mrow><mi>K</mi></mrow><mrow><mi mathvariant="normal">a</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45412499&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45412481&type= 3.89466691 3.72533321 和等效电阻 <math id="M3"><msub><mrow><mi>R</mi></mrow><mrow><mi mathvariant="normal">a</mi></mrow></msub></math> 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<math id="M10"><msub><mrow><mi>C</mi></mrow><mrow><mi mathvariant="normal">p</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45412545&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45412536&type= 3.72533321 3.72533321 中的电荷为 <math id="M11"><msub><mrow><mi>q</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=45412539&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45412538&type= 3.80999994 4.57200003 。 <math id="M12"><msub><mrow><mi>q</mi></mrow><mrow><mi mathvariant="normal">p</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45412560&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45412542&type= 3.04800010 4.57200003 为陶瓷受到机械作用力由正压电效应转换的电荷。电压 <math id="M13"><msub><mrow><mi>v</mi></mrow><mrow><mi mathvariant="normal">p</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45412561&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45412552&type= 3.04800010 3.72533321 进行逆压电效应产生推力 <math id="M14"><msub><mrow><mi>F</mi></mrow><mrow><mi mathvariant="normal">i</mi><mi mathvariant="normal">n</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45412682&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45412680&type= 4.23333359 3.72533321 满足 <math id="M15"><msub><mrow><mi>F</mi></mrow><mrow><mi mathvariant="normal">i</mi><mi mathvariant="normal">n</mi></mrow></msub><mo>=</mo><msub><mrow><mi>T</mi></mrow><mrow><mi mathvariant="normal">p</mi></mrow></msub><msub><mrow><mi>v</mi></mrow><mrow><mi mathvariant="normal">p</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45412575&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45412573&type= 16.17133331 3.72533321 ［ 13 13 ］。压电驱动器的电路方程可以表示为： <math display="block" id="M16"><msub><mrow><mi>R</mi></mrow><mrow><mi mathvariant="normal">a</mi></mrow></msub><mover accent="true"><mi>q</mi><mo>˙</mo></mover><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo>+</mo><mi>H</mi><mo stretchy="false">(</mo><mi>q</mi><mo stretchy="false">)</mo><mo>+</mo><mfrac><mrow><mn mathvariant="normal">1</mn></mrow><mrow><msub><mrow><mi>T</mi></mrow><mrow><mi mathvariant="normal">p</mi></mrow></msub></mrow></mfrac><msub><mrow><mi>F</mi></mrow><mrow><mi mathvariant="normal">i</mi><mi mathvariant="normal">n</mi></mrow></msub><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo>=</mo><msub><mrow><mi>K</mi></mrow><mrow><mi mathvariant="normal">a</mi></mrow></msub><mi>v</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45412570&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45412569&type= 58.33533096 8.80533314 . 定义 <math id="M17"><msub><mrow><mi>y</mi></mrow><mrow><mi mathvariant="normal">i</mi><mi mathvariant="normal">n</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45412651&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45412649&type= 3.72533321 4.57200003 为机械端的输入位移。由于 <math id="M18"><msub><mrow><mi>v</mi></mrow><mrow><mi mathvariant="normal">p</mi></mrow></msub><mo>=</mo><msub><mrow><mi>q</mi></mrow><mrow><mi mathvariant="normal">c</mi><mi mathvariant="normal">p</mi></mrow></msub><msubsup><mrow><mi>C</mi></mrow><mrow><mi mathvariant="normal">p</mi></mrow><mrow><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msubsup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45412597&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45412594&type= 17.61066628 4.82600021 ， <math id="M19"><msub><mrow><mi>q</mi></mrow><mrow><mi mathvariant="normal">c</mi><mi mathvariant="normal">p</mi></mrow></msub><mo>=</mo><mi>q</mi><mo>-</mo><msub><mrow><mi>q</mi></mrow><mrow><mi mathvariant="normal">p</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45412581&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45412579&type= 18.28800011 4.57200003 及正压电效应 <math id="M20"><msub><mrow><mi>q</mi></mrow><mrow><mi mathvariant="normal">p</mi></mrow></msub><mo>=</mo><msub><mrow><mi>T</mi></mrow><mrow><mi mathvariant="normal">p</mi></mrow></msub><msub><mrow><mi>y</mi></mrow><mrow><mi mathvariant="normal">i</mi><mi mathvariant="normal">n</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45412586&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45412585&type= 15.66333294 4.57200003 ，总电荷 <math id="M21"><mi>q</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45412606&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45412603&type= 1.86266661 3.64066648 亦可表示为： <math display="block" id="M22"><mi>q</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo>=</mo><mfrac><mrow><msub><mrow><mi>C</mi></mrow><mrow><mi mathvariant="normal">p</mi></mrow></msub></mrow><mrow><msub><mrow><mi>T</mi></mrow><mrow><mi mathvariant="normal">p</mi></mrow></msub></mrow></mfrac><msub><mrow><mi>F</mi></mrow><mrow><mi mathvariant="normal">i</mi><mi mathvariant="normal">n</mi></mrow></msub><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo>+</mo><msub><mrow><mi>T</mi></mrow><mrow><mi mathvariant="normal">p</mi></mrow></msub><msub><mrow><mi>y</mi></mrow><mrow><mi mathvariant="normal">i</mi><mi mathvariant="normal">n</mi></mrow></msub><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45412619&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45412602&type= 43.09533310 8.80533314 . 将式（2）式（2）代入式（1）式（1）中可以得到压电平台主运动方向的综合动力学模型为： <math display="block" id="M23"><mtable><mtr><mtd><mfrac><mrow><msub><mrow><mi>C</mi></mrow><mrow><mi mathvariant="normal">p</mi></mrow></msub></mrow><mrow><msub><mrow><mi>T</mi></mrow><mrow><mi mathvariant="normal">p</mi></mrow></msub></mrow></mfrac><msub><mrow><mover accent="true"><mi>F</mi><mo>˙</mo></mover></mrow><mrow><mi mathvariant="normal">i</mi><mi mathvariant="normal">n</mi></mrow></msub><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo>+</mo><mfrac><mrow><mn mathvariant="normal">1</mn></mrow><mrow><msub><mrow><mi>R</mi></mrow><mrow><mi mathvariant="normal">a</mi></mrow></msub><msub><mrow><mi>T</mi></mrow><mrow><mi mathvariant="normal">p</mi></mrow></msub></mrow></mfrac><msub><mrow><mi>F</mi></mrow><mrow><mi mathvariant="normal">i</mi><mi mathvariant="normal">n</mi></mrow></msub><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo>+</mo><msub><mrow><mi>T</mi></mrow><mrow><mi mathvariant="normal">p</mi></mrow></msub><msub><mrow><mover accent="true"><mi>y</mi><mo>˙</mo></mover></mrow><mrow><mi mathvariant="normal">i</mi><mi mathvariant="normal">n</mi></mrow></msub><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo>=</mo></mtd></mtr><mtr><mtd><mfrac><mrow><msub><mrow><mi>K</mi></mrow><mrow><mi mathvariant="normal">a</mi></mrow></msub></mrow><mrow><msub><mrow><mi>R</mi></mrow><mrow><mi mathvariant="normal">a</mi></mrow></msub></mrow></mfrac><mfenced separators="|"><mrow><mi>v</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo>-</mo><mfrac><mrow><mn mathvariant="normal">1</mn></mrow><mrow><msub><mrow><mi>K</mi></mrow><mrow><mi mathvariant="normal">a</mi></mrow></msub></mrow></mfrac><mi>H</mi><mo stretchy="false">(</mo><mi>q</mi><mo stretchy="false">)</mo></mrow></mfenced><mo>.</mo></mtd></mtr></mtable></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45412623&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45412622&type= 58.50466156 19.72733307 式（3）式（3）表明，迟滞效应可以用满足有界条件 <math id="M24"><mo stretchy="false">|</mo><msub><mrow><mi>d</mi></mrow><mrow><mi>H</mi></mrow></msub><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo stretchy="false">|</mo><mo>≤</mo><mi mathvariant="normal">s</mi><mi mathvariant="normal">u</mi><msub><mrow><mi mathvariant="normal">p</mi></mrow><mrow><mi>t</mi><mo>≥</mo><mn mathvariant="normal">0</mn></mrow></msub><mo stretchy="false">|</mo><msub><mrow><mi>d</mi></mrow><mrow><mi>H</mi></mrow></msub><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=45412615&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45412613&type= 35.89866638 4.82600021 的输入扰动 <math id="M25"><msub><mrow><mi>d</mi></mrow><mrow><mi>H</mi></mrow></msub><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo>=</mo><mo>-</mo><msubsup><mrow><mi>K</mi></mrow><mrow><mi mathvariant="normal">a</mi></mrow><mrow><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msubsup><mi>H</mi><mo stretchy="false">(</mo><mi>q</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45412617&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45412626&type= 32.42733002 5.07999992 来描述。在机械部分中，定义机构输入-输出位移的放大倍数为 <math id="M26"><mi>λ</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45412635&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45412630&type= 1.77800000 2.79399991 ，即平台输入位移 <math id="M27"><msub><mrow><mi>y</mi></mrow><mrow><mi mathvariant="normal">i</mi><mi mathvariant="normal">n</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45412651&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45412649&type= 3.72533321 4.57200003 与输出位移 <math id="M28"><msub><mrow><mi>y</mi></mrow><mrow><mi mathvariant="normal">o</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45412654&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45412639&type= 3.21733332 4.57200003 的关系为： <math display="block" id="M29"><msub><mrow><mi>y</mi></mrow><mrow><mi mathvariant="normal">i</mi><mi mathvariant="normal">n</mi></mrow></msub><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo>=</mo><mfrac><mrow><mn mathvariant="normal">1</mn></mrow><mrow><mi>λ</mi></mrow></mfrac><msub><mrow><mi>y</mi></mrow><mrow><mi mathvariant="normal">o</mi></mrow></msub><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45412660&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45412657&type= 25.06133461 7.87400007 . 平台的机构动力学部分可以简化为一个集总的质量-弹簧-阻尼模型，如图2 图2 所示。其中， <math id="M30"><msub><mrow><mi>m</mi></mrow><mrow><mi>y</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45412662&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45412643&type= 3.97933316 3.72533321 ， <math id="M31"><msub><mrow><mi>k</mi></mrow><mrow><mi>y</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45412667&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45412664&type= 2.79399991 3.72533321 和 <math id="M32"><msub><mrow><mi>c</mi></mrow><mrow><mi>y</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45412670&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45412669&type= 2.62466669 3.72533321 分别为其等效的质量、刚度和阻尼系数。由牛顿第二定律可以得到输入力 <math id="M33"><msub><mrow><mi>F</mi></mrow><mrow><mi mathvariant="normal">i</mi><mi mathvariant="normal">n</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45412682&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45412680&type= 4.23333359 3.72533321 对平台输出位移 <math id="M34"><msub><mrow><mi>y</mi></mrow><mrow><mi mathvariant="normal">o</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45412685&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45412673&type= 3.21733332 4.57200003 的力学方程为： <math display="block" id="M35"><msub><mrow><mi>F</mi></mrow><mrow><mi mathvariant="normal">i</mi><mi mathvariant="normal">n</mi></mrow></msub><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo>=</mo><msub><mrow><mi>m</mi></mrow><mrow><mi>y</mi></mrow></msub><msub><mrow><mover accent="true"><mrow><mi>y</mi></mrow><mi>¨</mi></mover></mrow><mrow><mi mathvariant="normal">o</mi></mrow></msub><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo>+</mo><msub><mrow><mi>c</mi></mrow><mrow><mi>y</mi></mrow></msub><msub><mrow><mover accent="true"><mi>y</mi><mo>˙</mo></mover></mrow><mrow><mi mathvariant="normal">o</mi></mrow></msub><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo>+</mo><msub><mrow><mi>k</mi></mrow><mrow><mi>y</mi></mrow></msub><msub><mrow><mi>y</mi></mrow><mrow><mi mathvariant="normal">o</mi></mrow></msub><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45412688&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45412676&type= 58.16600037 4.91066647 . 令 <math id="M36"><mi>u</mi><mo>=</mo><mi>v</mi><mo>+</mo><msub><mrow><mi>d</mi></mrow><mrow><mi>H</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45412708&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45412691&type= 17.27199936 3.80999994 为控制输入。将式（4）式（4）和式（5）式（5）代入式（3）式（3），同时考虑因信号调理和A/D转换而导致的传感器延时 <math id="M37"><msub><mrow><mi>τ</mi></mrow><mrow><mi>n</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45412774&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45412758&type= 2.70933342 3.72533321 ，从控制输入 <math id="M38"><mi>u</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45414196&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45414189&type= 1.86266661 2.79399991 到测量输出 <math id="M39"><mi>y</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45414182&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45414180&type= 2.11666679 3.72533321 的名义动力学可由如下时滞线性模型描述： <math display="block" id="M40"><msub><mrow><mi>P</mi></mrow><mrow><mi>n</mi></mrow></msub><mo stretchy="false">(</mo><mi>s</mi><mo stretchy="false">)</mo><mo>=</mo><mfrac><mrow><msub><mrow><mi>K</mi></mrow><mrow><mi mathvariant="normal">p</mi></mrow></msub></mrow><mrow><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><msup><mrow><mi mathvariant="normal">e</mi></mrow><mrow><mo>-</mo><msub><mrow><mi>τ</mi></mrow><mrow><mi>n</mi></mrow></msub><mi>s</mi></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45412707&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45412705&type= 51.05400085 8.63599968 ，其中： <math id="M41"><msub><mrow><mi>K</mi></mrow><mrow><mi mathvariant="normal">p</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45412743&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45412740&type= 4.06400013 3.72533321 ， <math id="M42"><msub><mrow><mi>a</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45412734&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45412731&type= 3.04800010 3.72533321 ， <math id="M43"><msub><mrow><mi>a</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45412754&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45412737&type= 3.04800010 3.72533321 ， <math id="M44"><msub><mrow><mi>a</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45412769&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45412756&type= 3.04800010 3.72533321 和延时 <math id="M45"><msub><mrow><mi>τ</mi></mrow><mrow><mi>n</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45412774&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45412758&type= 2.70933342 3.72533321 为与系统机电特性相关的名义参数。在得到模型阶次的基础上，模型名义参数通过参数辨识方法确定。通过将未建模动力学部分视为有界乘不确定 <math id="M46"><mi>Δ</mi><mo stretchy="false">(</mo><mi>s</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45413044&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45413041&type= 7.28133297 3.55599999 ，压电纳米平台系统的实际被控模型可以表示为 <math id="M47"><mi>P</mi><mo stretchy="false">(</mo><mi>s</mi><mo stretchy="false">)</mo><mo>=</mo><msub><mrow><mi>P</mi></mrow><mrow><mi>n</mi></mrow></msub><mo stretchy="false">(</mo><mi>s</mi><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mn mathvariant="normal">1</mn><mo>+</mo><mi mathvariant="normal">Δ</mi><mo stretchy="false">(</mo><mi>s</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45412788&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45412784&type= 38.43866730 4.48733330 ，其中 <math id="M48"><msub><mrow><mi>P</mi></mrow><mrow><mi>n</mi></mrow></msub><mo stretchy="false">(</mo><mi>s</mi><mo stretchy="false">)</mo><mo>=</mo><msub><mrow><mover accent="true"><mi>P</mi><mo>¯</mo></mover></mrow><mrow><mi>n</mi></mrow></msub><mo stretchy="false">(</mo><mi>s</mi><mo stretchy="false">)</mo><msub><mrow><mi>m</mi></mrow><mrow><mi>n</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=45412793&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45412790&type= 31.91933441 4.65666676 ， <math id="M49"><msub><mrow><mover accent="true"><mi>P</mi><mo>¯</mo></mover></mrow><mrow><mi>n</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=45413272&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45413256&type= 8.89000034 4.65666676 是 <math id="M50"><msub><mrow><mi>P</mi></mrow><mrow><mi>n</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=45413814&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45413813&type= 8.89000034 4.48733330 的无延时部分， <math id="M51"><msub><mrow><mi>m</mi></mrow><mrow><mi>n</mi></mrow></msub><mo stretchy="false">(</mo><mi>s</mi><mo stretchy="false">)</mo><mo>=</mo><msup><mrow><mi mathvariant="normal">e</mi></mrow><mrow><mo>-</mo><msub><mrow><mi>τ</mi></mrow><mrow><mi>n</mi></mrow></msub><mi>s</mi></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45412809&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45412818&type= 19.64266586 4.48733330 。进一步考虑轴间交叉耦合导致的输出周期扰动 <math id="M52"><msub><mrow><mi>d</mi></mrow><mrow><mi mathvariant="normal">c</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45412958&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45412929&type= 3.21733332 3.72533321 ，最终得到的压电微动平台的多扰动模型可由图3 图3 表示。 3 稳定周期干扰观测控制 1 为实现定位轴周期耦合误差的有效补偿，在得到名义动力学模型的基础上，本文提出一种基于周期干扰观测器（Periodic Disturbance Observer， PDOB）的控制策略，如图4 图4 所示。其中， <math id="M53"><mi>K</mi><mo stretchy="false">(</mo><mi>s</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45412858&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45412839&type= 8.04333401 3.55599999 是反馈控制器， <math id="M54"><mi>r</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45412852&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45412850&type= 1.69333339 2.79399991 ， <math id="M55"><mi>e</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45412867&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45412864&type= 1.52400005 2.79399991 ， <math id="M56"><mi>c</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45412874&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45412870&type= 1.52400005 2.79399991 ， <math id="M57"><mi>d</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45414193&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45414184&type= 2.28600001 2.79399991 和 <math id="M58"><mover accent="true"><mi>d</mi><mo>^</mo></mover></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45412884&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45412880&type= 2.28600001 3.72533321 分别是参考输入、输出误差、反馈控制器输出、输入扰动（包括 <math id="M59"><msub><mrow><mi>d</mi></mrow><mrow><mi>H</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45412887&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45412895&type= 3.89466691 3.72533321 ）和扰动估计， <math id="M60"><mi>Q</mi><mo stretchy="false">(</mo><mi>s</mi><mo stretchy="false">)</mo><msub><mrow><mi>m</mi></mrow><mrow><mn mathvariant="normal">1</mn></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=45412907&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45412904&type= 17.52599907 4.48733330 为干扰观测器的 Q -滤波器（ Q -filter），设计 <math id="M61"><msub><mrow><mi>m</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo stretchy="false">(</mo><mi>s</mi><mo stretchy="false">)</mo><mo>=</mo><msup><mrow><mi mathvariant="normal">e</mi></mrow><mrow><mo>-</mo><mo stretchy="false">(</mo><mi>T</mi><mo>-</mo><msub><mrow><mi>τ</mi></mrow><mrow><mi>n</mi></mrow></msub><mo stretchy="false">)</mo><mi>s</mi></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45412901&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45412900&type= 25.06133461 4.57200003 ，其中 <math id="M62"><mi>T</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45413761&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45413759&type= 2.87866688 2.79399991 是待抑制扰动的周期， <math id="M63"><mi>Q</mi><mo stretchy="false">(</mo><mi>s</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45414171&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45414157&type= 7.78933382 3.55599999 为待求的单位低通滤波器。干扰观测控制设计的核心任务是设计和优化 Q -滤波器［ 14 14 ］。对于带有延时的系统，时滞的无穷维特性及对稳定裕度的衰减使 Q -滤波器的优化变得复杂。现有的无需求解系统逆模型的结果，多是在固定的滤波器结构（如二项式滤波器，Binomial Filter ［ 14 14 ］）下对控制参数进行试错调节，这进一步限制了这类高动态敏感型系统的性能。因此，探究面向延时系统的周期干扰观测器直接优化设计，是一项具有挑战性的工作。计算得到从 <math id="M64"><msub><mrow><mi>d</mi></mrow><mrow><mi mathvariant="normal">c</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45412958&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45412929&type= 3.21733332 3.72533321 到 <math id="M65"><mi>y</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45414182&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45414180&type= 2.11666679 3.72533321 的名义传递函数为： <math display="block" id="M66"><msub><mrow><mi>G</mi></mrow><mrow><mi>d</mi><mi>c</mi><mi>y</mi></mrow></msub><mo stretchy="false">(</mo><mi>s</mi><mo stretchy="false">)</mo><mo>=</mo><mfrac><mrow><mn mathvariant="normal">1</mn><mo>-</mo><mi>Q</mi><mo stretchy="false">(</mo><mi>s</mi><mo stretchy="false">)</mo><msub><mrow><mi>m</mi></mrow><mrow><mn mathvariant="normal">2</mn></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>P</mi></mrow><mrow><mi>n</mi></mrow></msub><mo stretchy="false">(</mo><mi>s</mi><mo stretchy="false">)</mo><mi>K</mi><mo stretchy="false">(</mo><mi>s</mi><mo stretchy="false">)</mo></mrow></mfrac></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45412978&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45412964&type= 40.55533600 8.97466660 ，其中 <math id="M67"><msub><mrow><mi>m</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo stretchy="false">(</mo><mi>s</mi><mo stretchy="false">)</mo><mo>=</mo><msup><mrow><mi mathvariant="normal">e</mi></mrow><mrow><mo>-</mo><mi>T</mi><mi>s</mi></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45412967&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45412982&type= 19.30400085 4.65666676 。轴间周期耦合的补偿可由分子项的内模单元 <math id="M68"><mn mathvariant="normal">1</mn><mo>-</mo><mi>Q</mi><mo stretchy="false">(</mo><mi>s</mi><mo stretchy="false">)</mo><msub><mrow><mi>m</mi></mrow><mrow><mn mathvariant="normal">2</mn></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=45412989&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45412986&type= 24.21466637 4.48733330 保证。同时，观察到从 <math id="M69"><mi>d</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45414193&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45414184&type= 2.28600001 2.79399991 到 <math id="M70"><mi>u</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45414196&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45414189&type= 1.86266661 2.79399991 的名义传递函数 <math id="M71"><msub><mrow><mi>G</mi></mrow><mrow><mi>d</mi><mi>u</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=45412996&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45412975&type= 10.24466610 4.48733330 与式（7）式（7）相等，故图4 图4 的周期干扰观测控制结构能够实现一致的输入干扰和输出干扰抑制性能，以及对迟滞输入扰动的有效抑制。 <math id="M72"><mi>Q</mi><mo stretchy="false">(</mo><mi>s</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45414171&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45414157&type= 7.78933382 3.55599999 的性能优化函数可以表述为： <math display="block" id="M73"><msub><mrow><mfenced open="‖" close="‖" separators="|"><mrow><msub><mrow><mi>W</mi></mrow><mrow><mi mathvariant="normal">S</mi></mrow></msub><mo stretchy="false">(</mo><mi>s</mi><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mn mathvariant="normal">1</mn><mo>-</mo><mi>Q</mi><mo stretchy="false">(</mo><mi>s</mi><mo stretchy="false">)</mo><msub><mrow><mi>m</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo stretchy="false">(</mo><mi>s</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow></mfenced></mrow><mrow><mi mathvariant="normal">∞</mi></mrow></msub><mo><</mo><mn mathvariant="normal">1</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45413013&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45413010&type= 50.29199982 5.16466665 ，其中， <math id="M74"><msub><mrow><mi>W</mi></mrow><mrow><mi mathvariant="normal">S</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=45413006&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45413003&type= 10.32933331 4.48733330 是性能权函数选择为： <math display="block" id="M75"><msub><mrow><mi>W</mi></mrow><mrow><mi mathvariant="normal">S</mi></mrow></msub><mo stretchy="false">(</mo><mi>s</mi><mo stretchy="false">)</mo><mo>=</mo><mfrac><mrow><msub><mrow><mi>K</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub></mrow><mrow><mi>s</mi></mrow></mfrac></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45413021&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45413023&type= 19.64266586 7.95866632 ，以保证 <math id="M76"><mi>Q</mi><mo stretchy="false">(</mo><mi>s</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45414171&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45414157&type= 7.78933382 3.55599999 的低通和零静差特性。图4 图4 所示的闭环系统对应于 <math id="M77"><mi>Δ</mi><mo stretchy="false">(</mo><mi>s</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45413044&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45413041&type= 7.28133297 3.55599999 的补灵敏度函数为： <math display="block" id="M78"><msub><mrow><mi>T</mi></mrow><mrow><mi mathvariant="normal">c</mi><mi mathvariant="normal">l</mi></mrow></msub><mo stretchy="false">(</mo><mi>s</mi><mo stretchy="false">)</mo><mo>=</mo><mfrac><mrow><msub><mrow><mi>L</mi></mrow><mrow><mi>n</mi></mrow></msub><mo stretchy="false">(</mo><mi>s</mi><mo stretchy="false">)</mo><mo>+</mo><mi>Q</mi><mo stretchy="false">(</mo><mi>s</mi><mo stretchy="false">)</mo><msub><mrow><mi>m</mi></mrow><mrow><mn mathvariant="normal">2</mn></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>L</mi></mrow><mrow><mi>n</mi></mrow></msub><mo stretchy="false">(</mo><mi>s</mi><mo stretchy="false">)</mo></mrow></mfrac></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45413029&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45413028&type= 45.29666519 8.97466660 ，其中 <math id="M79"><msub><mrow><mi>L</mi></mrow><mrow><mi>n</mi></mrow></msub><mo stretchy="false">(</mo><mi>s</mi><mo stretchy="false">)</mo><mo>=</mo><msub><mrow><mi>P</mi></mrow><mrow><mi>n</mi></mrow></msub><mo stretchy="false">(</mo><mi>s</mi><mo stretchy="false">)</mo><mi>K</mi><mo stretchy="false">(</mo><mi>s</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45413052&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45413049&type= 30.73399925 4.48733330 。根据小增益定理［ 15 15 ］，当满足： <math display="block" id="M80"><msub><mrow><mfenced open="‖" close="‖" separators="|"><mrow><msub><mrow><mi>W</mi></mrow><mrow><mi mathvariant="normal">Δ</mi></mrow></msub><mo stretchy="false">(</mo><mi>s</mi><mo stretchy="false">)</mo><msub><mrow><mi>T</mi></mrow><mrow><mi mathvariant="normal">c</mi><mi mathvariant="normal">l</mi></mrow></msub><mo stretchy="false">(</mo><mi>s</mi><mo stretchy="false">)</mo></mrow></mfenced></mrow><mrow><mi mathvariant="normal">∞</mi></mrow></msub><mo><</mo><mn mathvariant="normal">1</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45413056&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45413054&type= 32.34266663 5.16466665 ，此时闭环控制系统稳定，其中， <math id="M81"><msub><mrow><mi>W</mi></mrow><mrow><mi mathvariant="normal">Δ</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=45413897&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45413893&type= 10.49866676 4.48733330 为不确定的界满足 <math id="M82"><mfenced open="|" close="|" separators="|"><mrow><msub><mrow><mi>W</mi></mrow><mrow><mi mathvariant="normal">Δ</mi></mrow></msub><mo stretchy="false">(</mo><mi>j</mi><mi>ω</mi><mo stretchy="false">)</mo></mrow></mfenced><mo>≥</mo><mfenced open="|" close="|" separators="|"><mrow><mi mathvariant="normal">Δ</mi><mo stretchy="false">(</mo><mi>j</mi><mi>ω</mi><mo stretchy="false">)</mo></mrow></mfenced><mo>,</mo><mtext> </mtext><mo>∀</mo><mi>ω</mi><mo>∈</mo><msup><mrow><mi mathvariant="bold">R</mi></mrow><mrow><mo>+</mo></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45413062&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45413037&type= 47.92133331 4.74133301 。已知 <math id="M83"><mo>∀</mo><mi>ω</mi><mo>∈</mo><msup><mrow><mi mathvariant="bold">R</mi></mrow><mrow><mo>+</mo></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45413067&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45413069&type= 12.95400047 3.38666677 时， <math display="block" id="M84"><mtable columnalign="left"><mtr><mtd><mfenced open="|" close="|" separators="|"><mrow><msub><mrow><mi>L</mi></mrow><mrow><mi>n</mi></mrow></msub><mfenced separators="|"><mrow><mi mathvariant="normal">j</mi><mi>ω</mi></mrow></mfenced><mo>+</mo><mi>Q</mi><mo stretchy="false">(</mo><mi mathvariant="normal">j</mi><mi>ω</mi><mo stretchy="false">)</mo><msub><mrow><mi>m</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo stretchy="false">(</mo><mi mathvariant="normal">j</mi><mi>ω</mi><mo stretchy="false">)</mo></mrow></mfenced><mo>≤</mo></mtd></mtr><mtr><mtd><mfenced open="|" close="|" separators="|"><mrow><mi>Q</mi><mo stretchy="false">(</mo><mi mathvariant="normal">j</mi><mi>ω</mi><mo stretchy="false">)</mo><msub><mrow><mi>m</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo stretchy="false">(</mo><mi mathvariant="normal">j</mi><mi>ω</mi><mo stretchy="false">)</mo></mrow></mfenced><mo>+</mo><mfenced open="|" close="|" separators="|"><mrow><msub><mrow><mi>L</mi></mrow><mrow><mi>n</mi></mrow></msub><mfenced separators="|"><mrow><mi mathvariant="normal">j</mi><mi>ω</mi></mrow></mfenced></mrow></mfenced><mo>.</mo></mtd></mtr></mtable></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45413074&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45413084&type= 44.02666473 12.78466606 令 <math id="M85"><mfenced open="|" close="|" separators="|"><mrow><msub><mrow><mi>M</mi></mrow><mrow><mi mathvariant="normal">Δ</mi></mrow></msub><mo stretchy="false">(</mo><mi mathvariant="normal">j</mi><mi>ω</mi><mo stretchy="false">)</mo></mrow></mfenced><mo>=</mo><mfenced open="|" close="|" separators="|"><mrow><msup><mrow><msub><mrow><mi>W</mi></mrow><mrow><mi mathvariant="normal">Δ</mi></mrow></msub></mrow><mrow><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msup><mfenced separators="|"><mrow><mi mathvariant="normal">j</mi><mi>ω</mi></mrow></mfenced><mfenced separators="|"><mrow><mn mathvariant="normal">1</mn><mo>+</mo><msub><mrow><mi>L</mi></mrow><mrow><mi>n</mi></mrow></msub><mfenced separators="|"><mrow><mi mathvariant="normal">j</mi><mi>ω</mi></mrow></mfenced></mrow></mfenced></mrow></mfenced><mo>-</mo><mfenced open="|" close="|" separators="|"><mrow><msub><mrow><mi>L</mi></mrow><mrow><mi>n</mi></mrow></msub><mfenced separators="|"><mrow><mi mathvariant="normal">j</mi><mi>ω</mi></mrow></mfenced></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45413078&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45413075&type= 75.18399811 6.26533365 ，当满足： <math display="block" id="M86"><mfenced open="|" close="|" separators="|"><mrow><mi>Q</mi><mo stretchy="false">(</mo><mi mathvariant="normal">j</mi><mi>ω</mi><mo stretchy="false">)</mo><msub><mrow><mi>m</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo stretchy="false">(</mo><mi mathvariant="normal">j</mi><mi>ω</mi><mo stretchy="false">)</mo></mrow></mfenced><mo><</mo><mfenced open="|" close="|" separators="|"><mrow><msub><mrow><mi>M</mi></mrow><mrow><mi mathvariant="normal">Δ</mi></mrow></msub><mo stretchy="false">(</mo><mi mathvariant="normal">j</mi><mi>ω</mi><mo stretchy="false">)</mo></mrow></mfenced><mo>,</mo><mtext> </mtext><mo>∀</mo><mi>ω</mi><mo>∈</mo><msup><mrow><mi mathvariant="bold">R</mi></mrow><mrow><mo>+</mo></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45413093&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45413079&type= 59.43600082 4.74133301 ，此时式（11）式（11）始终成立。考虑权函数 <math id="M87"><msub><mrow><mi>W</mi></mrow><mrow><mi mathvariant="normal">T</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=45413420&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45413418&type= 10.66800022 4.48733330 ： <math display="block" id="M88"><mfenced open="|" close="|" separators="|"><mrow><msub><mrow><mi>W</mi></mrow><mrow><mi mathvariant="normal">T</mi></mrow></msub><mo stretchy="false">(</mo><mi>j</mi><mi>ω</mi><mo stretchy="false">)</mo></mrow></mfenced><mo>></mo><msup><mrow><mfenced open="|" close="|" separators="|"><mrow><msub><mrow><mi>M</mi></mrow><mrow><mi mathvariant="normal">Δ</mi></mrow></msub><mo stretchy="false">(</mo><mi>j</mi><mi>ω</mi><mo stretchy="false">)</mo></mrow></mfenced></mrow><mrow><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msup><mo>,</mo><mtext> </mtext><mo>∀</mo><mi>ω</mi><mo>∈</mo><msup><mrow><mi mathvariant="bold">R</mi></mrow><mrow><mo>+</mo></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45413103&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45413102&type= 53.93266296 5.50333309 . 满足： <math display="block" id="M89"><msub><mrow><mfenced open="‖" close="‖" separators="|"><mrow><msub><mrow><mi>W</mi></mrow><mrow><mi mathvariant="normal">T</mi></mrow></msub><mo stretchy="false">(</mo><mi>s</mi><mo stretchy="false">)</mo><mi>Q</mi><mo stretchy="false">(</mo><mi>s</mi><mo stretchy="false">)</mo><msub><mrow><mi>m</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo stretchy="false">(</mo><mi>s</mi><mo stretchy="false">)</mo></mrow></mfenced></mrow><mrow><mi mathvariant="normal">∞</mi></mrow></msub><mo><</mo><mn mathvariant="normal">1</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45413106&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45413120&type= 40.89400101 5.16466665 ，即可满足闭环系统的稳定条件。至此，对 <math id="M90"><mi>Q</mi><mo stretchy="false">(</mo><mi>s</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45414171&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45414157&type= 7.78933382 3.55599999 的直接优化转化为式（8）式（8）和式（15）式（15）的 H ∞ 函数，以实现周期扰动抑制和稳定的优化性能。本文采用基于Hankel-Toeplitz算子理论的无穷维稳定 H ∞ 控制方法［ 10 10 ， 16 16 ］来求解 <math id="M91"><mi>Q</mi><mo stretchy="false">(</mo><mi>s</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45414171&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45414157&type= 7.78933382 3.55599999 的解析解。 3.1　 Q -滤波器的稳定优化 2 基于推导的优化函数，可将 <math id="M92"><mi>Q</mi><mo stretchy="false">(</mo><mi>s</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45414171&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45414157&type= 7.78933382 3.55599999 的设计问题表示为 H ∞ 优化问题，即： <math display="block" id="M93"><msub><mrow><mi>γ</mi></mrow><mrow><mi mathvariant="normal">o</mi></mrow></msub><mo>:</mo><mo>=</mo><munder><mrow><mi mathvariant="normal">i</mi><mi mathvariant="normal">n</mi><mi mathvariant="normal">f</mi></mrow><mrow><mi>Q</mi><mo stretchy="false">(</mo><mi>s</mi><mo stretchy="false">)</mo><mo>∈</mo><msup><mrow><mi>H</mi></mrow><mrow><mi mathvariant="normal">∞</mi></mrow></msup></mrow></munder><msub><mrow><mfenced open="‖" close="‖" separators="|"><mrow><mfenced open="[" close="]" separators="|"><mrow><mtable><mtr><mtd><msub><mrow><mi>W</mi></mrow><mrow><mi>S</mi></mrow></msub><mo stretchy="false">(</mo><mi>s</mi><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mn mathvariant="normal">1</mn><mo>-</mo><mi>Q</mi><mo stretchy="false">(</mo><mi>s</mi><mo stretchy="false">)</mo><msub><mrow><mi>m</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo stretchy="false">(</mo><mi>s</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mtd></mtr><mtr><mtd><msub><mrow><mi>W</mi></mrow><mrow><mi>T</mi></mrow></msub><mo stretchy="false">(</mo><mi>s</mi><mo stretchy="false">)</mo><mi>Q</mi><mo stretchy="false">(</mo><mi>s</mi><mo stretchy="false">)</mo><msub><mrow><mi>m</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo stretchy="false">(</mo><mi>s</mi><mo stretchy="false">)</mo></mtd></mtr></mtable></mrow></mfenced></mrow></mfenced></mrow><mrow><mi mathvariant="normal">∞</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45413158&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45413156&type= 66.63265991 11.26066685 ，其中 <math id="M94"><msub><mrow><mi>γ</mi></mrow><mrow><mi mathvariant="normal">o</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45413161&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45413174&type= 3.13266683 4.57200003 为最优H ∞ 指标。定义： <math display="block" id="M95"><msub><mrow><mi>C</mi></mrow><mrow><mi mathvariant="normal">o</mi></mrow></msub><mo stretchy="false">(</mo><mi>s</mi><mo stretchy="false">)</mo><mo>=</mo><mfrac><mrow><mi>Q</mi><mo stretchy="false">(</mo><mi>s</mi><mo stretchy="false">)</mo></mrow><mrow><mn mathvariant="normal">1</mn><mo>-</mo><mi>Q</mi><mo stretchy="false">(</mo><mi>s</mi><mo stretchy="false">)</mo><msub><mrow><mi>m</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo stretchy="false">(</mo><mi>s</mi><mo stretchy="false">)</mo></mrow></mfrac></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45413166&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45413180&type= 38.43866730 8.97466660 ，式（16）式（16）转化为延时系统混合灵敏度的优化问题，即： <math display="block" id="M96"><msub><mrow><mi>γ</mi></mrow><mrow><mi mathvariant="normal">o</mi></mrow></msub><mo>:</mo><mo>=</mo><munder><mrow><mi mathvariant="normal">i</mi><mi mathvariant="normal">n</mi><mi mathvariant="normal">f</mi></mrow><mrow><msub><mrow><mi>C</mi></mrow><mrow><mi mathvariant="normal">o</mi></mrow></msub><mo stretchy="false">(</mo><mi>s</mi><mo stretchy="false">)</mo><mtext> </mtext><mi mathvariant="normal">s</mi><mi mathvariant="normal">t</mi><mi mathvariant="normal">a</mi><mi mathvariant="normal">b</mi><mo>.</mo><mtext> </mtext><msub><mrow><mi>m</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo stretchy="false">(</mo><mi>s</mi><mo stretchy="false">)</mo></mrow></munder><msub><mrow><mfenced open="‖" close="‖" separators="|"><mrow><mfenced open="[" close="]" separators="|"><mrow><mtable><mtr><mtd><msub><mrow><mi>W</mi></mrow><mrow><mi mathvariant="normal">S</mi></mrow></msub><mo stretchy="false">(</mo><mi>s</mi><mo stretchy="false">)</mo><msub><mrow><mi>S</mi></mrow><mrow><mi mathvariant="normal">o</mi></mrow></msub><mo stretchy="false">(</mo><mi>s</mi><mo stretchy="false">)</mo></mtd></mtr><mtr><mtd><msub><mrow><mi>W</mi></mrow><mrow><mi mathvariant="normal">T</mi></mrow></msub><mo stretchy="false">(</mo><mi>s</mi><mo stretchy="false">)</mo><msub><mrow><mi>T</mi></mrow><mrow><mi mathvariant="normal">o</mi></mrow></msub><mo stretchy="false">(</mo><mi>s</mi><mo stretchy="false">)</mo></mtd></mtr></mtable></mrow></mfenced></mrow></mfenced></mrow><mrow><mi mathvariant="normal">∞</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45413186&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45413171&type= 54.27133560 11.26066685 ，其中： <math id="M97"><msub><mrow><mi>S</mi></mrow><mrow><mi mathvariant="normal">o</mi></mrow></msub><mo stretchy="false">(</mo><mi>s</mi><mo stretchy="false">)</mo><mo>=</mo><mo stretchy="false">(</mo><mn mathvariant="normal">1</mn><mo>+</mo><msub><mrow><mi>m</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo stretchy="false">(</mo><mi>s</mi><mo stretchy="false">)</mo><msub><mrow><mi>C</mi></mrow><mrow><mi mathvariant="normal">o</mi></mrow></msub><msup><mrow><mo stretchy="false">(</mo><mi>s</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><mrow><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45413192&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45413190&type= 44.02666473 4.65666676 ，定义的灵敏度函数 <math id="M98"><msub><mrow><mi>T</mi></mrow><mrow><mi mathvariant="normal">o</mi></mrow></msub><mo stretchy="false">(</mo><mi>s</mi><mo stretchy="false">)</mo><mo>=</mo><msub><mrow><mi>m</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo stretchy="false">(</mo><mi>s</mi><mo stretchy="false">)</mo><msub><mrow><mi>C</mi></mrow><mrow><mi mathvariant="normal">o</mi></mrow></msub><mo stretchy="false">(</mo><mi>s</mi><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mn mathvariant="normal">1</mn><mo>+</mo><msub><mrow><mi>m</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo stretchy="false">(</mo><mi>s</mi><mo stretchy="false">)</mo><msub><mrow><mi>C</mi></mrow><mrow><mi mathvariant="normal">o</mi></mrow></msub><msup><mrow><mo stretchy="false">(</mo><mi>s</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><mrow><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45413196&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45413195&type= 63.66933060 4.65666676 。回顾基于Hankel-Toeplitz算子理论的无穷维鲁棒 H ∞ 控制方法［ 10 10 ， 16 16 ］，可得最优 H ∞ 控制器为： <math display="block" id="M99"><msub><mrow><mi>C</mi></mrow><mrow><mi mathvariant="normal">o</mi></mrow></msub><mo stretchy="false">(</mo><mi>s</mi><mo stretchy="false">)</mo><mo>=</mo><mfrac><mrow><mi>E</mi><mo stretchy="false">(</mo><mi>s</mi><mo stretchy="false">)</mo><mi>F</mi><mo stretchy="false">(</mo><mi>s</mi><mo stretchy="false">)</mo><mi>L</mi><mo stretchy="false">(</mo><mi>s</mi><mo stretchy="false">)</mo></mrow><mrow><mn mathvariant="normal">1</mn><mo>+</mo><msub><mrow><mi>m</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo stretchy="false">(</mo><mi>s</mi><mo stretchy="false">)</mo><mi>F</mi><mo stretchy="false">(</mo><mi>s</mi><mo stretchy="false">)</mo><mi>L</mi><mo stretchy="false">(</mo><mi>s</mi><mo stretchy="false">)</mo></mrow></mfrac></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45413218&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45413200&type= 46.14333344 8.97466660 . 相应地，求得 <math id="M100"><mi>Q</mi><mo stretchy="false">(</mo><mi>s</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45414171&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45414157&type= 7.78933382 3.55599999 为： <math display="block" id="M101"><mi>Q</mi><mo stretchy="false">(</mo><mi>s</mi><mo stretchy="false">)</mo><mo>=</mo><mfrac><mrow><mi>E</mi><mo stretchy="false">(</mo><mi>s</mi><mo stretchy="false">)</mo><mi>F</mi><mo stretchy="false">(</mo><mi>s</mi><mo stretchy="false">)</mo><mi>L</mi><mo stretchy="false">(</mo><mi>s</mi><mo stretchy="false">)</mo></mrow><mrow><mn mathvariant="normal">1</mn><mo>+</mo><msub><mrow><mi>m</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo stretchy="false">(</mo><mi>s</mi><mo stretchy="false">)</mo><mi>F</mi><mo stretchy="false">(</mo><mi>s</mi><mo stretchy="false">)</mo><mi>L</mi><mo stretchy="false">(</mo><mi>s</mi><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mn mathvariant="normal">1</mn><mo>+</mo><mi>E</mi><mo stretchy="false">(</mo><mi>s</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow></mfrac></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45413229&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45413241&type= 62.90732956 8.97466660 . <math id="M102"><mi>Q</mi><mo stretchy="false">(</mo><mi>s</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45414171&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45414157&type= 7.78933382 3.55599999 应满足相对阶条件（正则条件），以在无需求解被控对象逆模型的情况下实现图4 图4 的周期干扰观测控制，即 <math id="M103"><mi>Q</mi><mo stretchy="false">(</mo><mi>s</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45414171&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45414157&type= 7.78933382 3.55599999 的相对阶应大于/等于 <math id="M104"><msub><mrow><mover accent="true"><mi>P</mi><mo>¯</mo></mover></mrow><mrow><mi>n</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=45413272&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45413256&type= 8.89000034 4.65666676 的相对阶。已知控制器参数 <math id="M105"><mi>E</mi><mo stretchy="false">(</mo><mi>s</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45413425&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45413435&type= 7.70466709 3.55599999 和 <math id="M106"><mi>L</mi><mo stretchy="false">(</mo><mi>s</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45413282&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45413262&type= 7.28133297 3.55599999 的相对阶均为0，参数 <math id="M107"><mi>F</mi><mo stretchy="false">(</mo><mi>s</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45413300&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45413283&type= 7.61999989 3.55599999 的相对阶为 <math id="M108"><msub><mrow><mi>R</mi></mrow><mrow><mi>F</mi></mrow></msub><mo>=</mo><mi mathvariant="normal">m</mi><mi mathvariant="normal">a</mi><mi mathvariant="normal">x</mi><mo stretchy="false">(</mo><msub><mrow><mi>N</mi></mrow><mrow><mi mathvariant="normal">S</mi></mrow></msub><mo>,</mo><msub><mrow><mi>D</mi></mrow><mrow><mi mathvariant="normal">S</mi></mrow></msub><mo stretchy="false">)</mo><mo>+</mo><mi mathvariant="normal">m</mi><mi mathvariant="normal">a</mi><mi mathvariant="normal">x</mi><mo stretchy="false">(</mo><msub><mrow><mi>N</mi></mrow><mrow><mi mathvariant="normal">T</mi></mrow></msub><mo>,</mo><mtext> </mtext><msub><mrow><mi>D</mi></mrow><mrow><mi mathvariant="normal">T</mi></mrow></msub><mo stretchy="false">)</mo><mo>-</mo><mo stretchy="false">(</mo><msub><mrow><mi>D</mi></mrow><mrow><mi mathvariant="normal">S</mi></mrow></msub><mo>+</mo><msub><mrow><mi>D</mi></mrow><mrow><mi mathvariant="normal">T</mi></mrow></msub><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45413305&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45413288&type= 77.38533020 4.48733330 ［ 16 16 ］，其中 <math id="M109"><msub><mrow><mi>N</mi></mrow><mrow><mi mathvariant="normal">S</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45413323&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45413308&type= 4.31799984 3.80999994 ， <math id="M110"><msub><mrow><mi>N</mi></mrow><mrow><mi mathvariant="normal">T</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45413317&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45413314&type= 4.65666676 3.80999994 和 <math id="M111"><msub><mrow><mi>D</mi></mrow><mrow><mi mathvariant="normal">S</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45413322&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45413329&type= 4.31799984 3.72533321 ， <math id="M112"><msub><mrow><mi>D</mi></mrow><mrow><mi mathvariant="normal">T</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45413340&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45413334&type= 4.57200003 3.72533321 分别是 <math id="M113"><msub><mrow><mi>W</mi></mrow><mrow><mi mathvariant="normal">S</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=45413345&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45413343&type= 10.32933331 4.48733330 ， <math id="M114"><msub><mrow><mi>W</mi></mrow><mrow><mi mathvariant="normal">T</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=45413349&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45413357&type= 10.66800022 4.48733330 的分子和分母的阶次，观察式（20）式（20）、式（6）式（6）和式（9）式（9），当选择 <math id="M115"><msub><mrow><mi>W</mi></mrow><mrow><mi mathvariant="normal">T</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=45413420&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45413418&type= 10.66800022 4.48733330 的形式为： <math display="block" id="M116"><msub><mrow><mi>W</mi></mrow><mrow><mi mathvariant="normal">T</mi></mrow></msub><mo stretchy="false">(</mo><mi>s</mi><mo stretchy="false">)</mo><mo>=</mo><msub><mrow><mi>w</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><mo>+</mo><msub><mrow><mi>w</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mi>s</mi><mo>+</mo><msub><mrow><mi>w</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>w</mi></mrow><mrow><mn mathvariant="normal">3</mn></mrow></msub><msup><mrow><mi>s</mi></mrow><mrow><mn mathvariant="normal">3</mn></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45413376&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45413372&type= 53.17066193 4.65666676 ，此时可满足相对阶要求。可确定 <math id="M117"><mi>E</mi><mo stretchy="false">(</mo><mi>s</mi><mo stretchy="false">)</mo><mo>=</mo><mo>-</mo><mfenced separators="|"><mrow><msubsup><mrow><mi>γ</mi></mrow><mrow><mi mathvariant="normal">o</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msubsup><msup><mrow><mi>s</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mo>+</mo><msubsup><mrow><mi>K</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msubsup></mrow></mfenced><msup><mrow><mfenced separators="|"><mrow><msubsup><mrow><mi>γ</mi></mrow><mrow><mi mathvariant="normal">o</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msubsup><msup><mrow><mi>s</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup></mrow></mfenced></mrow><mrow><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45413386&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45413383&type= 46.99000168 5.50333309 ， <math id="M118"><mi>L</mi><mo stretchy="false">(</mo><mi>s</mi><mo stretchy="false">)</mo><mo>=</mo><mo>-</mo><mn mathvariant="normal">1</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45413391&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45413398&type= 17.35666656 3.55599999 ， <math id="M119"><mi>F</mi><mo stretchy="false">(</mo><mi>s</mi><mo stretchy="false">)</mo><mo>=</mo><mi>s</mi><msup><mrow><mfenced separators="|"><mrow><msub><mrow><mi>f</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><mo>+</mo><msub><mrow><mi>f</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mi>s</mi><mo>+</mo><mo>…</mo><mo>+</mo><msub><mrow><mi>f</mi></mrow><mrow><mn mathvariant="normal">4</mn></mrow></msub><msup><mrow><mi>s</mi></mrow><mrow><mn mathvariant="normal">4</mn></mrow></msup></mrow></mfenced></mrow><mrow><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45413394&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45413403&type= 50.88466644 5.50333309 ，其系数由 <math id="M120"><msub><mrow><mi>γ</mi></mrow><mrow><mi mathvariant="normal">o</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45413413&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45413397&type= 3.13266683 4.57200003 ， <math id="M121"><mi>E</mi><mo stretchy="false">(</mo><mi>s</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45413425&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45413435&type= 7.70466709 3.55599999 和 <math id="M122"><msub><mrow><mi>W</mi></mrow><mrow><mi mathvariant="normal">T</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=45413420&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45413418&type= 10.66800022 4.48733330 确定［ 16 16 ］。 3.2　 Q -滤波器的稳定实现 2 延时系统的最优H ∞ 控制器中存在不稳定的零极相消［ 17 17 ］，即 <math id="M123"><mi>E</mi><mo stretchy="false">(</mo><mi>s</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45413425&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45413435&type= 7.70466709 3.55599999 与分母 <math id="M124"><mn mathvariant="normal">1</mn><mo>+</mo><msub><mrow><mi>m</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo stretchy="false">(</mo><mi>s</mi><mo stretchy="false">)</mo><mi>F</mi><mo stretchy="false">(</mo><mi>s</mi><mo stretchy="false">)</mo><mi>L</mi><mo stretchy="false">(</mo><mi>s</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45413458&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45413443&type= 32.00400162 4.48733330 有右半平面 <math id="M125"><msub><mrow><mi mathvariant="bold">C</mi></mrow><mrow><mo>+</mo></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45413461&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45413448&type= 4.48733330 3.72533321 的零点对消。因延时环节 <math id="M126"><msub><mrow><mi>m</mi></mrow><mrow><mn mathvariant="normal">2</mn></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=45413465&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45413454&type= 9.14400005 4.48733330 是无穷维的，无法实现精确的对消。不稳定的零极相消会影响控制器的性能，导致控制系统不稳定。因此，本文重新组合式（20）式（20）中 <math id="M127"><mi>Q</mi><mo stretchy="false">(</mo><mi>s</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45414171&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45414157&type= 7.78933382 3.55599999 的结构，将不稳定的零极相消提取出来得到一个等价的 <math id="M128"><mi>Q</mi><mo stretchy="false">(</mo><mi>s</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45414171&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45414157&type= 7.78933382 3.55599999 结构，并对对消项进行稳定近似，最终获得周期干扰观测控制的稳定实现。提取不稳定的零极相消项得到等价的 <math id="M129"><mi>Q</mi><mo stretchy="false">(</mo><mi>s</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45414171&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45414157&type= 7.78933382 3.55599999 为： <math display="block" id="M130"><mi>Q</mi><mo stretchy="false">(</mo><mi>s</mi><mo stretchy="false">)</mo><mo>=</mo><mfrac><mrow><msub><mrow><mi>R</mi></mrow><mrow><mn mathvariant="normal">1</mn></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>R</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo stretchy="false">(</mo><mi>s</mi><mo stretchy="false">)</mo><msub><mrow><mi>H</mi></mrow><mrow><mi mathvariant="normal">F</mi><mi mathvariant="normal">I</mi><mi mathvariant="normal">R</mi></mrow></msub><mo stretchy="false">(</mo><mi>s</mi><mo stretchy="false">)</mo></mrow></mfrac></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45413498&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45413494&type= 41.14799881 8.97466660 ，其中： <math display="block" id="M131"><msub><mrow><mi>R</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo stretchy="false">(</mo><mi>s</mi><mo stretchy="false">)</mo><mo>=</mo><mfrac><mrow><mn mathvariant="normal">1</mn><mo>+</mo><mfrac><mrow><msubsup><mrow><mi>γ</mi></mrow><mrow><mi mathvariant="normal">o</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msubsup></mrow><mrow><msubsup><mrow><mi>K</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msubsup></mrow></mfrac><msup><mrow><mi>s</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup></mrow><mrow><mfrac><mrow><msubsup><mrow><mi>γ</mi></mrow><mrow><mi mathvariant="normal">o</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msubsup></mrow><mrow><msubsup><mrow><mi>K</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msubsup></mrow></mfrac><mi>s</mi><mi>d</mi><msub><mrow><mi>F</mi></mrow><mrow><mi>l</mi></mrow></msub><mo stretchy="false">(</mo><mi>s</mi><mo stretchy="false">)</mo><mo>-</mo><mfrac><mrow><msub><mrow><mi>γ</mi></mrow><mrow><mi mathvariant="normal">o</mi></mrow></msub></mrow><mrow><msub><mrow><mi>K</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub></mrow></mfrac><mi mathvariant="normal">s</mi><mi mathvariant="normal">i</mi><mi mathvariant="normal">n</mi><mfrac><mrow><mi>T</mi><msub><mrow><mi>K</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub></mrow><mrow><msub><mrow><mi>γ</mi></mrow><mrow><mi mathvariant="normal">o</mi></mrow></msub></mrow></mfrac><mi>s</mi><mo>+</mo><mi mathvariant="normal">c</mi><mi mathvariant="normal">o</mi><mi mathvariant="normal">s</mi><mfrac><mrow><mi>T</mi><msub><mrow><mi>K</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub></mrow><mrow><msub><mrow><mi>γ</mi></mrow><mrow><mi mathvariant="normal">o</mi></mrow></msub></mrow></mfrac></mrow></mfrac></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45413501&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45413499&type= 73.49066925 19.21933365 ，不稳定的对消项为： <math display="block" id="M132"><msub><mrow><mi>H</mi></mrow><mrow><mi mathvariant="normal">F</mi><mi mathvariant="normal">I</mi><mi mathvariant="normal">R</mi></mrow></msub><mo stretchy="false">(</mo><mi>s</mi><mo stretchy="false">)</mo><mo>=</mo><mfrac><mrow><mfrac><mrow><msub><mrow><mi>γ</mi></mrow><mrow><mi mathvariant="normal">o</mi></mrow></msub></mrow><mrow><msub><mrow><mi>K</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub></mrow></mfrac><mi mathvariant="normal">s</mi><mi mathvariant="normal">i</mi><mi mathvariant="normal">n</mi><mfrac><mrow><mi>T</mi><msub><mrow><mi>K</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub></mrow><mrow><msub><mrow><mi>γ</mi></mrow><mrow><mi mathvariant="normal">o</mi></mrow></msub></mrow></mfrac><mi>s</mi><mo>-</mo><mi mathvariant="normal">c</mi><mi mathvariant="normal">o</mi><mi mathvariant="normal">s</mi><mfrac><mrow><mi>T</mi><msub><mrow><mi>K</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub></mrow><mrow><msub><mrow><mi>γ</mi></mrow><mrow><mi mathvariant="normal">o</mi></mrow></msub></mrow></mfrac><mo>+</mo><msub><mrow><mi>m</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo stretchy="false">(</mo><mi>s</mi><mo stretchy="false">)</mo></mrow><mrow><mn mathvariant="normal">1</mn><mo>+</mo><mfrac><mrow><msubsup><mrow><mi>γ</mi></mrow><mrow><mi mathvariant="normal">o</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msubsup></mrow><mrow><msubsup><mrow><mi>K</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msubsup></mrow></mfrac><msup><mrow><mi>s</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup></mrow></mfrac></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45413518&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45413503&type= 68.07199860 19.04999924 ， <math id="M133"><msub><mrow><mi>H</mi></mrow><mrow><mi mathvariant="normal">F</mi><mi mathvariant="normal">I</mi><mi mathvariant="normal">R</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=45414080&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45414090&type= 11.76866722 4.48733330 是一个有限脉冲响应（Finite Impulse Response，FIR）滤波器［ 17 17 ］。直接矩匹配［ 18 18 ］具有给定频率位置拟合精确的优点，本文采用该方法来获得 <math id="M134"><msub><mrow><mi>H</mi></mrow><mrow><mi mathvariant="normal">F</mi><mi mathvariant="normal">I</mi><mi mathvariant="normal">R</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=45413645&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45413636&type= 11.76866722 4.48733330 的稳定近似。考虑矩阵 <math id="M135"><mi mathvariant="bold-italic">A</mi><mo>∈</mo><msup><mrow><mi mathvariant="bold">C</mi></mrow><mrow><mn mathvariant="normal">2</mn><mo>×</mo><mn mathvariant="normal">2</mn></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45413527&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45413524&type= 14.05466747 3.38666677 ， <math id="M136"><mi mathvariant="bold-italic">B</mi><mo>∈</mo><msup><mrow><mi mathvariant="bold">C</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45413538&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45413530&type= 10.24466610 3.38666677 和 <math id="M137"><mi mathvariant="bold-italic">C</mi><mo>∈</mo><msup><mrow><mi mathvariant="bold">C</mi></mrow><mrow><mn mathvariant="normal">1</mn><mo>×</mo><mn mathvariant="normal">2</mn></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45413551&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45413540&type= 13.54666710 3.38666677 ， <math id="M138"><msub><mrow><mi>H</mi></mrow><mrow><mi mathvariant="normal">F</mi><mi mathvariant="normal">I</mi><mi mathvariant="normal">R</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=45414080&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45414090&type= 11.76866722 4.48733330 有如下状态空间表示： <math display="block" id="M139"><msub><mrow><mi>H</mi></mrow><mrow><mi mathvariant="normal">F</mi><mi mathvariant="normal">I</mi><mi mathvariant="normal">R</mi></mrow></msub><mo stretchy="false">(</mo><mi>s</mi><mo stretchy="false">)</mo><mo>=</mo><mover><mrow><mi>C</mi></mrow><mrow><mi>˜</mi></mrow></mover><mo stretchy="false">(</mo><mi>s</mi><mo stretchy="false">)</mo><mfenced open="[" close="]" separators="|"><mrow><mtable><mtr><mtd><mi mathvariant="bold-italic">A</mi></mtd><mtd><mi mathvariant="bold-italic">B</mi></mtd></mtr><mtr><mtd><mi mathvariant="bold-italic">I</mi></mtd><mtd><mn mathvariant="normal">0</mn></mtd></mtr></mtable></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45413563&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45413558&type= 37.33800125 8.55133343 ，其中 <math id="M140"><mover><mrow><mi>C</mi></mrow><mrow><mi>˜</mi></mrow></mover><mo stretchy="false">(</mo><mi>s</mi><mo stretchy="false">)</mo><mo>=</mo><mi>C</mi><mfenced separators="|"><mrow><mi>I</mi><mo>-</mo><msup><mrow><mi mathvariant="normal">e</mi></mrow><mrow><mo>-</mo><mi>T</mi><mfenced separators="|"><mrow><mi>s</mi><mi>I</mi><mo>-</mo><mi>A</mi></mrow></mfenced></mrow></msup></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45413571&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45413567&type= 37.25333405 4.91066647 ，表示式（24）式（24）中的无理部分， <math id="M141"><mfenced open="[" close="]" separators="|"><mrow><mtable><mtr><mtd><mi mathvariant="bold-italic">A</mi></mtd><mtd><mi mathvariant="bold-italic">B</mi></mtd></mtr><mtr><mtd><mi mathvariant="bold-italic">I</mi></mtd><mtd><mn mathvariant="normal">0</mn></mtd></mtr></mtable></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45413582&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45413573&type= 12.10733318 8.55133343 为 <math id="M142"><msup><mrow><mfenced separators="|"><mrow><mi>s</mi><mi mathvariant="bold-italic">I</mi><mo>-</mo><mi mathvariant="bold-italic">A</mi></mrow></mfenced></mrow><mrow><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msup><mi mathvariant="bold-italic">B</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45413595&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45413576&type= 19.64266586 4.74133301 的最小状态空间实现。选择一个非减次矩阵 <math id="M143"><mi mathvariant="bold-italic">S</mi><mo>∈</mo><msup><mrow><mi mathvariant="bold">C</mi></mrow><mrow><mi>N</mi><mo>×</mo><mi>N</mi></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45413600&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45413597&type= 14.39333248 3.38666677 满足特征值 <math id="M144"><mi>σ</mi><mo stretchy="false">(</mo><mi mathvariant="bold-italic">S</mi><mo stretchy="false">)</mo><mo>=</mo><mo stretchy="false">{</mo><msub><mrow><mi>s</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mrow><mi>s</mi></mrow><mrow><mi>N</mi></mrow></msub><mo stretchy="false">}</mo><mo>,</mo><mtext> </mtext><msub><mrow><mi>s</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>∉</mo><mi>σ</mi><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo><mo>,</mo><mtext> </mtext><mi>i</mi><mo>=</mo><mn mathvariant="normal">1</mn><mo>∶</mo><mi>N</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45413604&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45413589&type= 64.43132782 4.57200003 和矩阵 <math id="M145"><mi mathvariant="bold-italic">V</mi><mo>∈</mo><msup><mrow><mi mathvariant="bold">C</mi></mrow><mrow><mn mathvariant="normal">1</mn><mo>×</mo><mi>N</mi></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45413612&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45413609&type= 14.56266594 3.38666677 使得矩阵对 <math id="M146"><mo stretchy="false">(</mo><mi mathvariant="bold-italic">V</mi><mo>,</mo><mi mathvariant="bold-italic">S</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45413625&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45413615&type= 9.73666668 3.55599999 是可观的。存在矩阵 <math id="M147"><msub><mrow><mi mathvariant="bold-italic">C</mi></mrow><mrow><mi mathvariant="normal">Π</mi></mrow></msub><mo stretchy="false">(</mo><mi mathvariant="bold-italic">S</mi><mo stretchy="false">)</mo><mo>∈</mo><msup><mrow><mi mathvariant="bold">C</mi></mrow><mrow><mn mathvariant="normal">1</mn><mo>×</mo><mi>N</mi></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45413620&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45413627&type= 21.59000015 4.65666676 其元素 <math id="M148"><mover><mrow><mi>C</mi></mrow><mrow><mi>˜</mi></mrow></mover><mo stretchy="false">(</mo><msub><mrow><mi>s</mi></mrow><mrow><mi>i</mi></mrow></msub><mo stretchy="false">)</mo><msub><mrow><mi mathvariant="bold-italic">Π</mi></mrow><mrow><mi>i</mi></mrow></msub><mo stretchy="false">(</mo><mi>i</mi><mo>=</mo><mn mathvariant="normal">1</mn><mo>,</mo><mo>…</mo><mo>,</mo><mi>N</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45413640&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45413632&type= 35.64466476 4.91066647 为 <math id="M149"><msub><mrow><mi>H</mi></mrow><mrow><mi mathvariant="normal">F</mi><mi mathvariant="normal">I</mi><mi mathvariant="normal">R</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=45413645&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45413636&type= 11.76866722 4.48733330 在 <math id="M150"><mi>σ</mi><mo stretchy="false">(</mo><mi>S</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45414059&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45414027&type= 7.70466709 3.55599999 的前 <math id="M151"><mi>N</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45413973&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45413989&type= 3.30200005 2.87866688 个0阶矩，其中 <math id="M152"><mi mathvariant="bold-italic">Π</mi><mo>=</mo><mfenced open="[" close="]" separators="|"><mrow><msub><mrow><mi>Π</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>…</mo><msub><mrow><mi>Π</mi></mrow><mrow><mi>N</mi></mrow></msub></mrow></mfenced><mo>∈</mo><msup><mrow><mi mathvariant="bold">C</mi></mrow><mrow><mn mathvariant="normal">2</mn><mo>×</mo><mi>N</mi></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45413673&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45413671&type= 35.05199814 4.06400013 为Sylvester方程的唯一解，即： <math display="block" id="M153"><mi mathvariant="bold-italic">Π</mi><mi mathvariant="bold-italic">S</mi><mo>-</mo><mi mathvariant="bold-italic">A</mi><mi mathvariant="bold-italic">Π</mi><mo>=</mo><mi mathvariant="bold-italic">B</mi><mi mathvariant="bold-italic">V</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45413678&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45413676&type= 27.43199921 2.96333337 . <math id="M154"><msub><mrow><mi>H</mi></mrow><mrow><mi mathvariant="normal">F</mi><mi mathvariant="normal">I</mi><mi mathvariant="normal">R</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=45414080&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45414090&type= 11.76866722 4.48733330 的 <math id="M155"><mi>N</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45413973&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45413989&type= 3.30200005 2.87866688 阶近似为： <math display="block" id="M156"><msub><mrow><mi>H</mi></mrow><mrow><mi>N</mi></mrow></msub><mo stretchy="false">(</mo><mi>s</mi><mo stretchy="false">)</mo><mo>=</mo><mfenced open="[" close="]" separators="|"><mrow><mtable><mtr><mtd><mi mathvariant="bold-italic">S</mi><mo>-</mo><msub><mrow><mi mathvariant="bold-italic">B</mi></mrow><mrow><mi mathvariant="normal">F</mi></mrow></msub><mi mathvariant="bold-italic">V</mi></mtd><mtd><msub><mrow><mi mathvariant="bold-italic">B</mi></mrow><mrow><mi mathvariant="normal">F</mi></mrow></msub></mtd></mtr><mtr><mtd><msub><mrow><mi mathvariant="bold-italic">C</mi></mrow><mrow><mi mathvariant="normal">Π</mi></mrow></msub><mo stretchy="false">(</mo><mi mathvariant="bold-italic">S</mi><mo stretchy="false">)</mo></mtd><mtd><mn mathvariant="normal">0</mn></mtd></mtr></mtable></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45413690&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45413701&type= 39.20066452 9.39799976 ，其中： <math id="M157"><msub><mrow><mi>B</mi></mrow><mrow><mi mathvariant="normal">F</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45414075&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45414072&type= 4.06400013 3.72533321 应满足 <math id="M158"><mi>σ</mi><mo stretchy="false">(</mo><mi mathvariant="bold-italic">S</mi><mo>-</mo><msub><mrow><mi mathvariant="bold-italic">B</mi></mrow><mrow><mi mathvariant="normal">F</mi></mrow></msub><mi mathvariant="bold-italic">V</mi><mo stretchy="false">)</mo><mo>⋂</mo><mi>σ</mi><mo stretchy="false">(</mo><mi mathvariant="bold-italic">S</mi><mo stretchy="false">)</mo><mo>=</mo><mi mathvariant="normal">∅</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45413696&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45413694&type= 39.36999893 4.57200003 和在 <math id="M159"><mi>S</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45413716&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45413713&type= 2.20133328 2.79399991 的特征值处合适的矩匹配条件，以得到稳定的有理近似。相应地，可以得到 <math id="M160"><mi>Q</mi><mo stretchy="false">(</mo><mi>s</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45414171&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45414157&type= 7.78933382 3.55599999 的 <math id="M161"><mi>N</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45413973&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45413989&type= 3.30200005 2.87866688 阶稳定实现为： <math display="block" id="M162"><msub><mrow><mi>Q</mi></mrow><mrow><mi>N</mi></mrow></msub><mo stretchy="false">(</mo><mi>s</mi><mo stretchy="false">)</mo><mo>=</mo><mfrac><mrow><msub><mrow><mi>R</mi></mrow><mrow><mn mathvariant="normal">1</mn></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>R</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo stretchy="false">(</mo><mi>s</mi><mo stretchy="false">)</mo><msub><mrow><mi>H</mi></mrow><mrow><mi>N</mi></mrow></msub><mo stretchy="false">(</mo><mi>s</mi><mo stretchy="false">)</mo></mrow></mfrac></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45413724&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45413722&type= 41.23266602 8.97466660 . 综上所述，对稳定干扰观测控制方法的设计流程如下：（1）通过建模/参数辨识方法获得被控对象的名义模型 <math id="M163"><msub><mrow><mi>P</mi></mrow><mrow><mi>n</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=45413814&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45413813&type= 8.89000034 4.48733330 和相应的建模不确定的界 <math id="M164"><msub><mrow><mi>W</mi></mrow><mrow><mi mathvariant="normal">Δ</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=45413742&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45413740&type= 10.49866676 4.48733330 ；（2）确定周期 <math id="M165"><mi>T</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45413761&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45413759&type= 2.87866688 2.79399991 ；（3）选择形式为式（9）式（9）的适当的性能权函数 <math id="M166"><msub><mrow><mi>W</mi></mrow><mrow><mi mathvariant="normal">S</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=45413750&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45413763&type= 10.32933331 4.48733330 ，根据条件式（14）式（14）确定形式如式（21）式（21）的 <math id="M167"><msub><mrow><mi>W</mi></mrow><mrow><mi mathvariant="normal">T</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=45413768&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45413751&type= 10.66800022 4.48733330 ，求解 H ∞ 优化问题式（18）式（18），确定 <math id="M168"><mi>Q</mi><mo stretchy="false">(</mo><mi>s</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45414171&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45414157&type= 7.78933382 3.55599999 。（4）分解 <math id="M169"><mi>Q</mi><mo stretchy="false">(</mo><mi>s</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45414171&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45414157&type= 7.78933382 3.55599999 ，计算式（27）式（27）中FIR滤波器的 <math id="M170"><mi>N</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45413973&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45413989&type= 3.30200005 2.87866688 阶近似 <math id="M171"><msub><mrow><mi>H</mi></mrow><mrow><mi>N</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=45413794&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45413778&type= 9.90600014 4.48733330 ，得到式（28）式（28）稳定的 <math id="M172"><msub><mrow><mi>Q</mi></mrow><mrow><mi>N</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=45413798&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45413796&type= 9.65200043 4.48733330 。 4 实验 1 实验装置由压电微位移平台、上/下位机、NI PCI 6259数据采集模块、电压放大器、电容位移传感器组件和Simulink/RTW xPC Target实时操控系统组成，如图5 图5 所示。压电微动平台安装在光学隔振基座上以减少不必要的干扰。模拟输入电压由NI数据采集卡的16位D/A转换器产生，随后经高带宽电压放大器放大用于驱动压电平台。平台的输出位置由分辨率为0.5 nm的电容位移传感器捕捉，经传感器模块调理并通过NI数据采集卡的16位A/D转换器传回主机。整套系统基于Simulink/RTW xPC Target实现快速控制原型，采样频率设置为20 kHz。 4.1　模型辨识与控制器设计 2 利用离散傅里叶频率响应法辨识获得压电平台 Y -轴模型的名义参数。对平台 Y -轴施加频率从1~2 000 Hz的正弦波激励信号，同时为减小迟滞非线性对线性动力学参数辨识的影响，选择正弦信号的幅值为50 mV。计算得到的纳米平台实际被控模型 <math id="M173"><mi>P</mi><mo stretchy="false">(</mo><mi>s</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45413823&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45413822&type= 7.53533268 3.55599999 的幅频特性和相频特性曲线如图6 图6 中点线所示。基于式（6）式（6）建立的 <math id="M174"><msub><mrow><mi>P</mi></mrow><mrow><mi>n</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=45413814&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45413813&type= 8.89000034 4.48733330 名义模型结构，通过MATLAB的invfreqs函数，对 <math id="M175"><mi>P</mi><mo stretchy="false">(</mo><mi>s</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45413823&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45413822&type= 7.53533268 3.55599999 的频率响应曲线进行拟合，得到 <math id="M176"><msub><mrow><mi>P</mi></mrow><mrow><mi>n</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=45413814&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45413813&type= 8.89000034 4.48733330 的参数为： <math id="M177"><msub><mrow><mi>K</mi></mrow><mrow><mi mathvariant="normal">p</mi></mrow></msub><mo>=</mo><mn mathvariant="normal">1.514</mn><mo>×</mo><msup><mrow><mn mathvariant="normal">10</mn></mrow><mrow><mn mathvariant="normal">15</mn></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45413830&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45413828&type= 28.87133408 3.97933316 ， <math id="M178"><msub><mrow><mi>a</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo>=</mo><mn mathvariant="normal">13</mn><mtext> </mtext><mn mathvariant="normal">370</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45413849&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45413832&type= 17.94933319 3.80999994 ， <math id="M179"><msub><mrow><mi>a</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>=</mo><mn mathvariant="normal">2.686</mn><mo>×</mo><msup><mrow><mn mathvariant="normal">10</mn></mrow><mrow><mn mathvariant="normal">7</mn></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45413853&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45413851&type= 26.92399979 3.97933316 ， <math id="M180"><msub><mrow><mi>a</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><mo>=</mo><mn mathvariant="normal">3.266</mn><mo>×</mo><msup><mrow><mn mathvariant="normal">10</mn></mrow><mrow><mn mathvariant="normal">11</mn></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45413857&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45413838&type= 27.94000053 3.97933316 和传感器延时 <math id="M181"><msub><mrow><mi>τ</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>=</mo><mn mathvariant="normal">0.001</mn><mtext> </mtext><mn mathvariant="normal">2</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45413862&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45413860&type= 19.64266586 3.80999994 ，拟合结果如图6 图6 中虚线所示。可以看出，拟合模型能够有效地吻合压电微动平台的实验响应。已知 <math id="M182"><mi>Δ</mi><mo stretchy="false">(</mo><mi>s</mi><mo stretchy="false">)</mo><mo>=</mo><msubsup><mrow><mi>P</mi></mrow><mrow><mi>n</mi></mrow><mrow><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msubsup><mo stretchy="false">(</mo><mi>s</mi><mo stretchy="false">)</mo><mfenced separators="|"><mrow><mi>P</mi><mo stretchy="false">(</mo><mi>s</mi><mo stretchy="false">)</mo><mo>-</mo><msub><mrow><mi>P</mi></mrow><mrow><mi>n</mi></mrow></msub><mo stretchy="false">(</mo><mi>s</mi><mo stretchy="false">)</mo></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45413865&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45413843&type= 48.00600052 4.57200003 且不确定的界 <math id="M183"><msub><mrow><mi>W</mi></mrow><mrow><mi mathvariant="normal">Δ</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=45413897&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45413893&type= 10.49866676 4.48733330 满足 <math id="M184"><mfenced open="|" close="|" separators="|"><mrow><msub><mrow><mi>W</mi></mrow><mrow><mi mathvariant="normal">Δ</mi></mrow></msub><mo stretchy="false">(</mo><mi mathvariant="normal">j</mi><mi>ω</mi><mo stretchy="false">)</mo></mrow></mfenced><mo>≥</mo><mfenced open="|" close="|" separators="|"><mrow><mi mathvariant="normal">Δ</mi><mo stretchy="false">(</mo><mi mathvariant="normal">j</mi><mi>ω</mi><mo stretchy="false">)</mo></mrow></mfenced><mo>,</mo><mtext> </mtext><mo>∀</mo><mi>ω</mi><mo>∈</mo><msup><mrow><mi mathvariant="bold">R</mi></mrow><mrow><mo>+</mo></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45413875&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45413873&type= 46.90533447 4.74133301 ，可以确定 <math id="M185"><msub><mrow><mi>W</mi></mrow><mrow><mi mathvariant="normal">Δ</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=45413897&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45413893&type= 10.49866676 4.48733330 为： <math display="block" id="M186"><mtable><mtr><mtd><msub><mrow><mi>W</mi></mrow><mrow><mi mathvariant="normal">Δ</mi></mrow></msub><mo stretchy="false">(</mo><mi>s</mi><mo stretchy="false">)</mo><mo>=</mo><mn mathvariant="normal">4</mn><mo>×</mo><msup><mrow><mn mathvariant="normal">10</mn></mrow><mrow><mo>-</mo><mn mathvariant="normal">12</mn></mrow></msup><msup><mrow><mi>s</mi></mrow><mrow><mn mathvariant="normal">3</mn></mrow></msup><mo>+</mo><mn mathvariant="normal">5.517</mn><mo>×</mo><msup><mrow><mn mathvariant="normal">10</mn></mrow><mrow><mo>-</mo><mn mathvariant="normal">8</mn></mrow></msup><msup><mrow><mi>s</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mo>+</mo></mtd></mtr><mtr><mtd><mn mathvariant="normal">1.774</mn><mo>×</mo><msup><mrow><mn mathvariant="normal">10</mn></mrow><mrow><mo>-</mo><mn mathvariant="normal">4</mn></mrow></msup><mi>s</mi><mo>+</mo><mn mathvariant="normal">0.266</mn><mtext> </mtext><mn mathvariant="normal">1</mn><mo>.</mo></mtd></mtr></mtable></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45413886&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45413883&type= 64.34667206 8.55133343 选择 <math id="M187"><mi>T</mi><mo>=</mo><mn mathvariant="normal">0.02</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45413923&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45413909&type= 14.98600006 2.79399991 。选择形式为式（9）式（9）的性能权函数 <math id="M188"><msub><mrow><mi>W</mi></mrow><mrow><mi mathvariant="normal">S</mi></mrow></msub><mo stretchy="false">(</mo><mi>s</mi><mo stretchy="false">)</mo><mo>=</mo><mn mathvariant="normal">2.5</mn><msup><mrow><mi>s</mi></mrow><mrow><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45413928&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45413926&type= 24.80733490 4.65666676 。根据条件式（14）式（14），确定式（21）式（21）为： <math display="block" id="M189"><mtable><mtr><mtd><msub><mrow><mi>W</mi></mrow><mrow><mi mathvariant="normal">T</mi></mrow></msub><mo stretchy="false">(</mo><mi>s</mi><mo stretchy="false">)</mo><mo>=</mo><mn mathvariant="normal">5.343</mn><mo>×</mo><msup><mrow><mn mathvariant="normal">10</mn></mrow><mrow><mo>-</mo><mn mathvariant="normal">12</mn></mrow></msup><msup><mrow><mi>s</mi></mrow><mrow><mn mathvariant="normal">3</mn></mrow></msup><mo>+</mo><mn mathvariant="normal">8.529</mn><mo>×</mo></mtd></mtr><mtr><mtd><msup><mrow><mn mathvariant="normal">10</mn></mrow><mrow><mo>-</mo><mn mathvariant="normal">8</mn></mrow></msup><msup><mrow><mi>s</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mo>+</mo><mn mathvariant="normal">4.077</mn><mo>×</mo><msup><mrow><mn mathvariant="normal">10</mn></mrow><mrow><mo>-</mo><mn mathvariant="normal">4</mn></mrow></msup><mi>s</mi><mo>+</mo><mn mathvariant="normal">0.891</mn><mtext> </mtext><mn mathvariant="normal">3</mn><mo>.</mo></mtd></mtr></mtable></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45413933&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45413915&type= 57.65800095 8.55133343 求解 H ∞ 优化问题式（18）式（18），获得最优指标 <math id="M190"><msub><mrow><mi>γ</mi></mrow><mrow><mi mathvariant="normal">o</mi></mrow></msub><mo>=</mo><mn mathvariant="normal">0.892</mn><mtext> </mtext><mn mathvariant="normal">716</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45413938&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45413921&type= 23.79133224 4.57200003 。进而得到式（22）式（22）的等价 <math id="M191"><mi>Q</mi><mo stretchy="false">(</mo><mi>s</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45414171&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45414157&type= 7.78933382 3.55599999 和参数 <math id="M192"><msub><mrow><mi>R</mi></mrow><mrow><mn mathvariant="normal">1</mn></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=45413946&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45413943&type= 9.05933285 4.48733330 ， <math id="M193"><msub><mrow><mi>H</mi></mrow><mrow><mi mathvariant="normal">F</mi><mi mathvariant="normal">I</mi><mi mathvariant="normal">R</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=45413950&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45413948&type= 11.76866722 4.48733330 。为了得到稳定的FIR滤波器，首先确定式（25）式（25）中的状态空间矩阵为： <math display="block" id="M194"><mi mathvariant="bold-italic">A</mi><mo>=</mo><mfenced open="[" close="]" separators="|"><mrow><mtable><mtr><mtd><mn mathvariant="normal">0</mn></mtd><mtd><mn mathvariant="normal">1</mn></mtd></mtr><mtr><mtd><mo>-</mo><mn mathvariant="normal">7.842</mn><mtext> </mtext><mn mathvariant="normal">5</mn></mtd><mtd><mn mathvariant="normal">0</mn></mtd></mtr></mtable></mrow></mfenced><mo>,</mo><mtext> </mtext><mi mathvariant="bold-italic">B</mi><mo>=</mo><mfenced open="[" close="]" separators="|"><mrow><mtable><mtr><mtd><mn mathvariant="normal">0</mn></mtd></mtr><mtr><mtd><mn mathvariant="normal">1</mn></mtd></mtr></mtable></mrow></mfenced><mo>,</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45413967&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45413965&type= 49.61466980 8.63599968 <math display="block" id="M195"><mi mathvariant="bold-italic">C</mi><mo>=</mo><mfenced open="[" close="]" separators="|"><mrow><mtable><mtr><mtd><mo>-</mo><mn mathvariant="normal">7.830</mn><mtext> </mtext><mn mathvariant="normal">2</mn></mtd><mtd><mn mathvariant="normal">0.156</mn><mtext> </mtext><mn mathvariant="normal">8</mn></mtd></mtr></mtable></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45413987&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45413969&type= 41.65599823 3.97933316 . 选择 <math id="M196"><mi>N</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45413973&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45413989&type= 3.30200005 2.87866688 ， <math id="M197"><mi>S</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45413995&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45413993&type= 2.20133328 2.79399991 和 <math id="M198"><mi>V</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45414000&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45413998&type= 3.21733332 2.87866688 分别为 <math id="M199"><mi>N</mi><mo>=</mo><mn mathvariant="normal">4</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45413978&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45414002&type= 9.65200043 2.87866688 ， <math id="M200"><mi>S</mi><mo>=</mo><mi mathvariant="normal">d</mi><mi mathvariant="normal">i</mi><mi mathvariant="normal">a</mi><mi mathvariant="normal">g</mi><mo stretchy="false">{</mo><mn mathvariant="normal">0,130,260,390</mn><mo stretchy="false">}</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45413981&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45413980&type= 41.90999985 3.97933316 和 <math id="M201"><mi mathvariant="bold-italic">V</mi><mo>=</mo><mfenced open="[" close="]" separators="|"><mrow><mtable><mtr><mtd><mn mathvariant="normal">1</mn></mtd><mtd><mn mathvariant="normal">1</mn></mtd><mtd><mn mathvariant="normal">1</mn></mtd><mtd><mn mathvariant="normal">1</mn></mtd></mtr></mtable></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45414011&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45414010&type= 29.37933159 3.97933316 ，设计 <math id="M202"><msub><mrow><mi>B</mi></mrow><mrow><mi mathvariant="normal">F</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45414031&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45414018&type= 4.06400013 3.72533321 满足 <math id="M203"><mi mathvariant="normal">d</mi><msub><mrow><mi>H</mi></mrow><mrow><mi>N</mi></mrow></msub><mo>/</mo><mi mathvariant="normal">d</mi><mi>s</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45414037&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45414019&type= 11.59933376 4.14866638 和 <math id="M204"><mi mathvariant="normal">d</mi><msub><mrow><mi>H</mi></mrow><mrow><mi mathvariant="normal">F</mi><mi mathvariant="normal">I</mi><mi mathvariant="normal">R</mi></mrow></msub><mo>/</mo><mi mathvariant="normal">d</mi><mi>s</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45414025&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45414040&type= 13.46199989 4.14866638 在 <math id="M205"><mi>σ</mi><mo stretchy="false">(</mo><mi>S</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45414059&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45414027&type= 7.70466709 3.55599999 处的矩相匹配，即对于所有的 <math id="M206"><msub><mrow><mi>s</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>∈</mo><mo stretchy="false">{</mo><msub><mrow><mi>s</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mrow><mi>s</mi></mrow><mrow><mi>N</mi></mrow></msub><mo stretchy="false">}</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45414065&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45414063&type= 22.43666649 4.48733330 ， <math id="M207"><msubsup><mrow><mi>H</mi></mrow><mrow><mi mathvariant="normal">F</mi><mi mathvariant="normal">I</mi><mi mathvariant="normal">R</mi></mrow><mrow><mi>'</mi></mrow></msubsup><mo stretchy="false">(</mo><msub><mrow><mi>s</mi></mrow><mrow><mi>i</mi></mrow></msub><mo stretchy="false">)</mo><mtext> </mtext><mo>=</mo><mtext> </mtext><msubsup><mrow><mi>H</mi></mrow><mrow><mi>N</mi></mrow><mrow><mi>'</mi></mrow></msubsup><mo stretchy="false">(</mo><msub><mrow><mi>s</mi></mrow><mrow><mi>i</mi></mrow></msub><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45414071&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45414068&type= 28.70199966 4.65666676 ，确定 <math id="M208"><msub><mrow><mi>B</mi></mrow><mrow><mi mathvariant="normal">F</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45414075&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45414072&type= 4.06400013 3.72533321 为： <math display="block" id="M209"><mtable><mtr><mtd><msub><mrow><mi>B</mi></mrow><mrow><mi mathvariant="normal">F</mi></mrow></msub><mo>=</mo><msup><mrow><mn mathvariant="normal">10</mn></mrow><mrow><mn mathvariant="normal">4</mn></mrow></msup><mo>×</mo></mtd></mtr><mtr><mtd><mfenced open="[" close="]" separators="|"><mrow><mo>-</mo><mn mathvariant="normal">0.036</mn><mtext> </mtext><mn mathvariant="normal">6</mn><mo>;</mo><mn mathvariant="normal">0.402</mn><mtext> </mtext><mn mathvariant="normal">2</mn><mo>;</mo><mo>-</mo><mn mathvariant="normal">1.184</mn><mtext> </mtext><mn mathvariant="normal">7</mn><mo>;</mo><mn mathvariant="normal">0.965</mn><mtext> </mtext><mn mathvariant="normal">5</mn></mrow></mfenced><mo>.</mo></mtd></mtr></mtable></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45414057&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45414055&type= 67.30999756 8.89000034 根据式（27）式（27）可以确定 <math id="M210"><msub><mrow><mi>H</mi></mrow><mrow><mi mathvariant="normal">F</mi><mi mathvariant="normal">I</mi><mi mathvariant="normal">R</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=45414080&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45414090&type= 11.76866722 4.48733330 的4阶稳定近似为： <math display="block" id="M211"><mtable><mtr><mtd><msub><mrow><mi>H</mi></mrow><mrow><mn mathvariant="normal">4</mn></mrow></msub><mo stretchy="false">(</mo><mi>s</mi><mo stretchy="false">)</mo><mo>=</mo><mfrac><mrow><mn mathvariant="normal">15</mn><mtext> </mtext><mn mathvariant="normal">529</mn><mo stretchy="false">(</mo><mi>s</mi><mo>+</mo><mn mathvariant="normal">451.5</mn><mo stretchy="false">)</mo></mrow><mrow><mo stretchy="false">(</mo><msup><mrow><mi>s</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mo>+</mo><mn mathvariant="normal">429.8</mn><mi>s</mi><mo>+</mo><mn mathvariant="normal">5.382</mn><mo>×</mo><msup><mrow><mn mathvariant="normal">10</mn></mrow><mrow><mn mathvariant="normal">4</mn></mrow></msup><mo stretchy="false">)</mo></mrow></mfrac><mo>×</mo></mtd></mtr><mtr><mtd><mtext> </mtext><mfrac><mrow><mo stretchy="false">(</mo><msup><mrow><mi>s</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mo>+</mo><mn mathvariant="normal">208.2</mn><mi>s</mi><mo>+</mo><mn mathvariant="normal">1.081</mn><mo>×</mo><msup><mrow><mn mathvariant="normal">10</mn></mrow><mrow><mn mathvariant="normal">5</mn></mrow></msup><mo stretchy="false">)</mo></mrow><mrow><mo stretchy="false">(</mo><msup><mrow><mi>s</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mo>+</mo><mn mathvariant="normal">253.4</mn><mi>s</mi><mo>+</mo><mn mathvariant="normal">8.976</mn><mo>×</mo><msup><mrow><mn mathvariant="normal">10</mn></mrow><mrow><mn mathvariant="normal">4</mn></mrow></msup><mo stretchy="false">)</mo></mrow></mfrac><mo>.</mo><mtext> </mtext><mo stretchy="false">(</mo><mn mathvariant="normal">33</mn><mo stretchy="false">)</mo></mtd></mtr></mtable></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45414093&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45414082&type= 76.45400238 19.72733307 图7 图7 从时域和频域响应的角度比较了 <math id="M212"><msub><mrow><mi>H</mi></mrow><mrow><mi mathvariant="normal">F</mi><mi mathvariant="normal">I</mi><mi mathvariant="normal">R</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=45414096&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45414085&type= 11.76866722 4.48733330 和 <math id="M213"><msub><mrow><mi>H</mi></mrow><mrow><mn mathvariant="normal">4</mn></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=45414106&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45414104&type= 9.39799976 4.48733330 ，验证了近似结果的精准性。通过式（28）式（28）可获得 <math id="M214"><mi>Q</mi><mo stretchy="false">(</mo><mi>s</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45414171&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45414157&type= 7.78933382 3.55599999 的最终稳定应用。 <math id="M215"><mi>Q</mi><mo stretchy="false">(</mo><mi>s</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45414171&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45414157&type= 7.78933382 3.55599999 的相位滞后对抗扰性能的影响可在 <math id="M216"><msub><mrow><mi>m</mi></mrow><mrow><mn mathvariant="normal">1</mn></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=45414148&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45414142&type= 9.14400005 4.48733330 中引入相应的超前项 <math id="M217"><msup><mrow><mi mathvariant="normal">e</mi></mrow><mrow><mo>+</mo><msub><mrow><mi>τ</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><mi>s</mi></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45414130&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45414138&type= 6.01133299 3.04800010 进行消除，即实际应用中， <math id="M218"><msub><mrow><mi>m</mi></mrow><mrow><mn mathvariant="normal">1</mn></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=45414148&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45414142&type= 9.14400005 4.48733330 为 <math id="M219"><msub><mrow><mi>m</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo stretchy="false">(</mo><mi>s</mi><mo stretchy="false">)</mo><mo>=</mo><msup><mrow><mi mathvariant="normal">e</mi></mrow><mrow><mo>-</mo><mo stretchy="false">(</mo><mi>n</mi><mi>T</mi><mo>-</mo><msub><mrow><mi>τ</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>-</mo><msub><mrow><mi>τ</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><mo stretchy="false">)</mo></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45414144&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45414150&type= 29.20999908 4.57200003 ，满足 <math id="M220"><mo stretchy="false">(</mo><msub><mrow><mi>τ</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>+</mo><msub><mrow><mi>τ</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><mo stretchy="false">)</mo><mo>≤</mo><mi>n</mi><mi>T</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45414163&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45414153&type= 22.86000061 4.48733330 ， <math id="M221"><mi>n</mi><mo>∈</mo><msup><mrow><mi mathvariant="bold">N</mi></mrow><mrow><mo>+</mo></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45414167&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45414156&type= 10.24466610 3.38666677 。为验证提出方法的优越性，选择不含干扰观测器的纯反馈控制方法和传统二项式滤波器［ 14 14 ］为 Q -滤波器的周期干扰观测控制方法（Periodic Disturbance Observer， BPDOB）与提出的稳定周期干扰观测控制方法进行了比较，其中，BPDOB中 Q -滤波器的带宽与本文方法的 <math id="M222"><mi>Q</mi><mo stretchy="false">(</mo><mi>s</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45414171&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45414157&type= 7.78933382 3.55599999 相等。 4.2　实验结果 2 首先开展闭环频率响应实验，从频域的角度比较不同方法的扰动抑制性能，得到各方法的闭环系统从 <math id="M223"><msub><mrow><mi>d</mi></mrow><mrow><mi>c</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45414175&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45414173&type= 3.13266683 3.72533321 到 <math id="M224"><mi>y</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45414182&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45414180&type= 2.11666679 3.72533321 （从 <math id="M225"><mi>d</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45414193&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45414184&type= 2.28600001 2.79399991 到 <math id="M226"><mi>u</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45414196&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45414189&type= 1.86266661 2.79399991 ）的频率响应曲线，如图8 图8 所示。可以看出，只采用反馈控制器的系统扰动抑制带宽为75 Hz。由于周期干扰观测器的引入，BPDOB方法和提出的稳定周期干扰观测控制方法比纯反馈方法在50 Hz及其倍频处有更好的抗干扰性能。与BPDOB相比，提出的稳定优化方法在50 Hz基频位置的抗干扰性能提升了7.6 dB。为了验证提出的稳定周期干扰观测控制方法在基频及其倍频位置的扰动估计和抑制性能的优越性，设置参考输入 <math id="M227"><mi>r</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45414219&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45414223&type= 6.51933336 3.55599999 为0，在图4 图4 的输入扰动 <math id="M228"><mi>d</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45414229&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45414220&type= 7.11199999 3.55599999 处（ <math id="M229"><msub><mrow><mi>G</mi></mrow><mrow><mi>u</mi><mi>d</mi></mrow></msub><mo stretchy="false">(</mo><mi>s</mi><mo stretchy="false">)</mo><mo>=</mo><msub><mrow><mi>G</mi></mrow><mrow><mi>d</mi><mi>c</mi><mi>y</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=45414235&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45414232&type= 25.82333374 4.48733330 ）人为添加频率为50，100，200 Hz的正弦扰动信号，采集添加扰动前后0~1.2 s的系统输出和稳态（0.4~1.2 s）扰动估计误差数据，并计算稳态输出的单边频谱，得到的实验结果如图9 图9 ~ 图11 图11 所示。此外，分别计算各方法在稳态时系统输出的均方根（Root Mean Square，RMS）值和扰动估计误差的RMS值，结果如表1 表1 所示。可以明显看出，不含周期干扰观测器的纯反馈方法表现出最差的抗干扰性能，提出的稳定优化方法比BPDOB方法对基频及其倍频处的扰动有更好的抑制。在50 Hz基频位置，稳定周期干扰控制方法相比于纯反馈和BPDOB方法的扰动抑制性能分别有超过99%和50%的提升。 2 Disturbance frequency/Hz 3 RMS output/nm 3 RMS disturbance-estimate error/V Only feedback BPDOB This method Only feedback BPDOB This method 50 1 322.7 4.6 2.2 - 0.001 5 0.000 7 100 834.5 11.3 5.0 - 0.001 9 0.000 9 200 1 392.3 60.7 25.2 - 0.006 3 0.002 6 设置 <math id="M236"><mi>d</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo>=</mo><msub><mrow><mi>d</mi></mrow><mrow><mn mathvariant="normal">4</mn></mrow></msub><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45414341&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45414321&type= 20.06599998 4.48733330 ，开展空载和带100 g 载荷的阶跃响应实验来验证提出的控制方法的稳定性能。如图12 图12 所示，空载和带载时的暂态响应相一致，且计算得到扰动存在时空载和带载的输出误差RMS值均为2.7 nm，验证了稳定优化设计方法优越的不确定补偿性能。最后，在压电微动平台上开展50 Hz和100 Hz的栅格纳米扫描实验，来验证提出方法的轴间耦合补偿性能，给定 Y -轴与扫描轴周期轨迹相匹配的斜坡定位信号为参考信号，扫描轴控制器进行了有效设计，以三角波上升段为扫描段。选择纯反馈和BPDOB方法与提出的稳定优化方法进行比较，得到的实验结果如图13 图13 ~ 图14 图14 所示。只采用反馈控制器时， Y -轴输出出现较大偏差。计算不同控制方法下 Y -轴扫描段的定位误差RMS值，如表2 表2 所示，可以得出，提出的稳定周期干扰控制方法的定位误差明显小于纯反馈和BPDOB方法，显著提升了并联纳米平台轴间周期耦合的补偿性能。 2 Scanning frequency/Hz 3 RMS positioning error during scanning/nm Only feedback BPDOB This method 50 17.6 6.5 3.8 100 25.9 9.6 5.5 5 结论 1 本文提出了一种稳定周期干扰观测控制结构，用于提升带测量滞后的并联压电纳米平台高速栅格扫描运动时的轴间耦合补偿性能。通过构建的性能优化函数和推导的稳定性条件，将最优控制器参数求解问题转化为延时系统的混合灵敏度优化问题，并通过无穷维 H ∞ 控制方法进行求解。对纳米平台系统的实验结果表明，提出的方法相比纯反馈和传统的周期干扰观测控制方法，在基频处抗干扰性能的提升分别超过99%和50%，这为带有时间延迟的纳米运动系统提供了优越且性能稳定的恒速精密扫描解决方案，支持新兴的高速显微成像应用。参考文献 NIKOOIENEJAD N ， MAROUFI M ， MOHEIMANI S O R . 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IEEE ， 2019 ： 2319 - 2324 . doi: 10.23919/ecc.2019.8795745 http://dx.doi.org/10.23919/ecc.2019.8795745 Model reduction by moment matching： case study of a FIR system 10.37188/OPE.20233111.1641 TP273 TH113 A 1004-924X（2023）11-1641-11 张志名,闫鹏.面向纳米栅格扫描的时滞压电微动平台耦合补偿[J].光学精密工程,2023,31(11):1641-1651. 张志名,闫鹏.面向纳米栅格扫描的时滞压电微动平台耦合补偿[J].光学精密工程,2023,31(11):1641-1651. ZHANG Zhiming,YAN Peng.Coupling compensation for nano raster scanning of piezoelectric nano-stages with delays[J].Optics and Precision Engineering,2023,31(11):1641-1651. ZHANG Zhiming,YAN Peng.Coupling compensation for nano raster scanning of piezoelectric nano-stages with delays[J].Optics and Precision Engineering,2023,31(11):1641-1651. 山东省自然科学基金重大基础研究项目（No.ZR2019ZD08） 2023-01-10 2023-02-01 2023-06-10 2023-07-08 38601378 光学精密工程 11 31

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DOI：10.37188/OPE.20233111.1641