基于高斯过程的参数辨识及永磁同步电机模型电流预测控制策略

魏宗恩; 邓永停; 乔延婷; 费强; 李洪文

doi:10.37188/OPE.20233104.0479

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基于高斯过程的参数辨识及永磁同步电机模型电流预测控制策略

微纳技术与精密机械 | 更新时间：2023-02-28

- 基于高斯过程的参数辨识及永磁同步电机模型电流预测控制策略
- Gaussian process-based parameter identification and model current predictive control strategy of PMSM
- 光学精密工程
- 作者机构：
  
  1.中国科学院长春光学精密机械与物理研究所，吉林长春 130033
  2.中国科学院大学，北京 100039
  3.陆军装甲兵学院士官学校，吉林长春 130000
  4.季华实验室，广东佛山 528200
- 作者简介：
  
  [ "魏宗恩（1997-），男，山东日照人，硕士，主要从事机电系统精密控制方面的研究。E-mail：weizongen20@mails. ucas. ac.cn" ]
  
  魏宗恩（1997-），男，山东日照人，硕士，主要从事机电系统精密控制方面的研究。E-mail：weizongen20@mails. ucas. ac.cn
  [ "邓永停（1987-），男，山东潍坊人，博士生导师，研究员，主要从事望远镜精密跟踪控制技术的研究及电子学设计。E-mail：dyt0612@163.com" ]
  
  邓永停（1987-），男，山东潍坊人，博士生导师，研究员，主要从事望远镜精密跟踪控制技术的研究及电子学设计。E-mail：dyt0612@163.com
- 基金信息：
  
  国家自然科学基金资助项目(11973041;12122304);中科院青年创新促进会会员资助项目(2019218)
- DOI：10.37188/OPE.20233104.0479 中图分类号： TM351;TP273
- 纸质出版日期：2023-02-25，
  
  收稿日期：2020-08-30，
  
  修回日期：2020-09-23，
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引用本文
魏宗恩,邓永停,乔延婷等.基于高斯过程的参数辨识及永磁同步电机模型电流预测控制策略[J].光学精密工程,2023,31(04):479-490.

WEI Zongen,DENG Yongting,QIAO Tingting,et al.Gaussian process-based parameter identification and model current predictive control strategy of PMSM[J].Optics and Precision Engineering,2023,31(04):479-490.
魏宗恩,邓永停,乔延婷等.基于高斯过程的参数辨识及永磁同步电机模型电流预测控制策略[J].光学精密工程,2023,31(04):479-490. DOI： 10.37188/OPE.20233104.0479.

WEI Zongen,DENG Yongting,QIAO Tingting,et al.Gaussian process-based parameter identification and model current predictive control strategy of PMSM[J].Optics and Precision Engineering,2023,31(04):479-490. DOI： 10.37188/OPE.20233104.0479.

基于高斯过程的参数辨识及永磁同步电机模型电流预测控制策略 Gaussian process-based parameter identification and model current predictive control strategy of PMSM 1 2 first-author 魏宗恩（1997-），男，山东日照人，硕士，主要从事机电系统精密控制方面的研究。E-mail：weizongen20@mails. ucas. ac.cn http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=40435395&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=40435402&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=40435398&type= 22.01333237 32.00399780 魏宗恩（1997-），男，山东日照人，硕士，主要从事机电系统精密控制方面的研究。E-mail：weizongen20@mails. ucas. ac.cn 1 corresp E-mail： dyt0612@163.com E-mail： dyt0612@163.com 邓永停（1987-），男，山东潍坊人，博士生导师，研究员，主要从事望远镜精密跟踪控制技术的研究及电子学设计。E-mail：dyt0612@163.com http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=40435408&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=40435417&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=40435414&type= 22.01333618 32.00400543 邓永停（1987-），男，山东潍坊人，博士生导师，研究员，主要从事望远镜精密跟踪控制技术的研究及电子学设计。E-mail： dyt0612@163.com dyt0612@163.com 3 4 1 中国科学院长春光学精密机械与物理研究所，吉林长春 130033 中国科学院长春光学精密机械与物理研究所，吉林长春 130033 Changchun Institute of Optics ， Fine Mechanics and Physics ，Chinese Academy of Sciences， Changchun 130033 ， China Changchun Institute of Optics ， Fine Mechanics and Physics ，Chinese Academy of Sciences， Changchun 130033 ， China 中国科学院大学，北京 100039 中国科学院大学，北京 100039 University of Chinese Academy of Sciences ， Beijing 1000339 ， China University of Chinese Academy of Sciences ， Beijing 1000339 ， China 陆军装甲兵学院士官学校，吉林长春 130000 陆军装甲兵学院士官学校，吉林长春 130000 Non-commissioned Officer School of Army Academy of Armed Forces， Changchun 130000， China Non-commissioned Officer School of Army Academy of Armed Forces， Changchun 130000， China 季华实验室，广东佛山 528200 季华实验室，广东佛山 528200 Jihua Laboratory， Foshan 528200 ， China Jihua Laboratory， Foshan 528200 ， China 为了提高永磁同步电机控制系统电流环控制器的性能，降低模型参数失配对控制系统的影响，提出了基于高斯过程参数辨识的永磁同步电机有限集模型预测电流控制策略（FCS-GPMPC）。首先，介绍了永磁同步电机电流预测模型并分析了模型参数失配对系统性能的影响；其次，为简化一般机器学习参数辨识算法中超参数复杂的调试过程，提出了一种基于高斯过程的模型参数的辨识方法；同时，引入预测值的置信区间作为参数预测效果的实时评估参考；最后，将高斯过程参数辨识与基于模型的有限集模型预测电流控制（FCS-MPC）相结合，在得到准确辨识的参数后对系统电流预测模型更新以提高系统鲁棒性和电流环跟踪性能。实验结果显示：在本文训练数据的统计特征下，测试数据均方根误差RMSE为0.002 1， <math id="M1"><msup><mrow><mi>R</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=40435429&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=40435423&type= 3.38666677 2.62466669 达到0.99。在参数波动条件下，与FCS-MPC相比，FCS-GPMPC策略下电流波动度降低了30.5%，电流平均偏移度降低了19.6%，另外对参考电流的阶跃变化，FCS-GPMPC有更好的动态响应。实验结果表明，基于高斯过程的模型预测控制方法可有效抑制模型失配对控制系统的影响，能够提高永磁同步电机控制系统电流控制器性能。 This paper proposes a model predictive control （MPC） method for permanent magnet synchronous motors （PMSMs） based on finite control set Gaussian process MPC （FCS-GPMPC） parameter identification to limit the influence of model mismatches on the control system and to improve the current controller performance of control systems in a PMSM. First， the current PMSM prediction model is introduced and the influence of model parameter mismatches on the system performance is analyzed. Secondly， in order to simplify the complex debugging process of hyperparameters in general machine learning parameter identification algorithms， the GPMPC method is proposed. At the same time， the confidence interval of the predicted value is introduced as a real-time evaluation reference for the parameter prediction effect. Finally， the GP parameter identification method is combined with the FCS-MPC to predict the system current after accurately obtaining the identified parameters. The model is updated to improve system robustness and current loop tracking performance. The experimental results show that under the statistical characteristics of the training data， the root mean square error and of the test data are 0.0021 and 0.99， respectively. Under the condition of parameter fluctuation， compared with FCS-MPC， FCS-GPMPC reduces current fluctuation by 30.5% and the average current offset by 19.6%. In addition， for step changes in the reference current， FCS-GPMPC has a better dynamic response. The proposed GP-MPC can effectively suppress the influence of model mismatch on control systems and can improve the performance of the current controller of PMSM control systems. 永磁同步电机模型预测控制机器学习高斯过程模型失配 Permanent Magnet Synchronous Motor （PMSM） Model Predictive Control （MPC） machine learning Gaussian Process （GP） model mismatch 1 引言 1 永磁同步电机（Permanent Magnet Synchronous Motor， PMSM）因其可靠性高、体积小、控制精度高等优点在光电跟踪设备［ 1 1 - 2 2 1-2 ］、高精度数控机床、航空航天领域得到广泛应用，同时，永磁同步电机也被广泛应用于新能源汽车领域。因此，对高性能的永磁同步电机控制算法［ 3 3 - 4 4 3-4 ］的研究具有重要意义。基于有限控制集的模型预测控制［ 5 5 ］（Finite Control Set Model Predictive Control， FCS-MPC）算法利用代价函数，通过预测不同电压对应的电流来选择逆变器的开关状态。该策略的优点是瞬态响应快、实现简单、无需调制。然而，直接使用系统模型来选择最优控制行为，使得系统在建模错误或参数变化的情况下，出现性能变化［ 6 6 - 8 8 6-8 ］。因此，预测模型误差的影响［ 9 9 - 10 10 9-10 ］是FCS-MPC的重要问题。针对上述问题，传统的方法是建立更为精确的物理模型，或者设计复杂高阶求解器。这增加了系统的复杂度，造成过高的计算负荷。也有许多其他方案被国内外学者提出，其中，无模型预测控制方案［ 11 11 - 12 12 11-12 ］和基于参数辨识的预测模型优化控制方案成果显著。基于参数辨识的预测模型优化控制方案在保证控制性能的前提下进一步提高系统鲁棒性。目前，永磁同步电机参数辨识方法分为离线估计方法［ 13 13 - 14 14 13-14 ］和在线估计方法［ 15 15 - 17 17 15-17 ］。此外，模拟解析方法，特别是有限元分析［ 18 18 ］，也被归类为离线参数估计技术，此技术常被用来作为其他辨识方法实验的实际参考。在线估计方法中，基于机器学习的方法备受关注，其通过数据来进行预测，随着有效数据的积累，机器学习的预测能力可以持续的进化，预测精度不断提高，目前，神经网络（Neural Network， NN）［ 19 19 ］、遗传算法（Genetic Algorithm， GA）［ 20 20 ］和粒子群优化（Particle Swarm Optimization， PSO）［ 21 21 - 22 22 21-22 ］等算法已应用于电机控制的自调试［ 23 23 - 24 24 23-24 ］和状态监测。一般机器学习的算法在系统参数辨识的应用过程中需要进行超参数的调试，如神经网络中的学习率、隐藏层的数量，支持向量机中的正则化常量等。目前，应用于参数辨识机器学习的算法中，大都需要基于经验与不断测试对超参数进行优化，以提高学习的性能和效果，此过程增加了辨识算法的实施难度。另外，机器学习回归算法预测精度的评估需要衡量预测值与实际参数值之间的误差，此“实际的参数值”是基于其他辨识方法得到的，在参数辨识过程中，缺乏实时性的性能指标对预测结果进行评估。高斯过程作为一种“非参数”机器学习技术，其超参数使用极大自然法或蒙特卡洛方法自适应获取，这简化了超参数调试的过程，并增加超参数选取的可解释性。此外，高斯过程是一个具有概率意义的学习机，可对预测输出做出概率解释，其赋予预测结果的置信度指标可用于预测效果的评估。基于上述两点，本文提出基于高斯过程的永磁同步电机参数辨识方法并与有限集电流预测控制相结合，对PMSM进行精确控制。 2 永磁同步电机电流预测模型 1 在理想状态下，PMSM在 <math id="M2"><mi>d</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=40435728&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=40435727&type= 2.28600001 2.79399991 - <math id="M3"><mi>q</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=40436255&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=40436254&type= 1.86266661 3.64066648 轴坐标下的电压方程为： <math display="block" id="M4"><mfenced open="{" close="" separators="|"><mrow><mtable columnalign="left"><mtr><mtd><msub><mrow><mi>v</mi></mrow><mrow><mi>d</mi></mrow></msub><mo>=</mo><msub><mrow><mi>R</mi></mrow><mrow><mi>s</mi></mrow></msub><msub><mrow><mi>i</mi></mrow><mrow><mi>d</mi></mrow></msub><mo>+</mo><mfrac><mrow><mi mathvariant="normal">d</mi><msub><mrow><mi>ψ</mi></mrow><mrow><mi>d</mi></mrow></msub></mrow><mrow><mi mathvariant="normal">d</mi><mi>t</mi></mrow></mfrac><mo>-</mo><msub><mrow><mi>ω</mi></mrow><mrow><mi>r</mi></mrow></msub><msub><mrow><mi>ψ</mi></mrow><mrow><mi>q</mi></mrow></msub></mtd></mtr><mtr><mtd><msub><mrow><mi>v</mi></mrow><mrow><mi>q</mi></mrow></msub><mo>=</mo><msub><mrow><mi>R</mi></mrow><mrow><mi>s</mi></mrow></msub><msub><mrow><mi>i</mi></mrow><mrow><mi>q</mi></mrow></msub><mo>+</mo><mfrac><mrow><mi mathvariant="normal">d</mi><msub><mrow><mi>ψ</mi></mrow><mrow><mi>q</mi></mrow></msub></mrow><mrow><mi mathvariant="normal">d</mi><mi>t</mi></mrow></mfrac><mo>+</mo><msub><mrow><mi>ω</mi></mrow><mrow><mi>r</mi></mrow></msub><msub><mrow><mi>ψ</mi></mrow><mrow><mi>d</mi></mrow></msub></mtd></mtr></mtable></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=40435451&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=40435445&type= 38.77733231 18.45733452 . 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http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=40435495&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=40435491&type= 3.13266683 3.72533321 和 <math id="M11"><msub><mrow><mi>i</mi></mrow><mrow><mi>d</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=40435505&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=40435501&type= 2.45533323 3.72533321 分别为 <math id="M12"><mi>d</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=40435728&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=40435727&type= 2.28600001 2.79399991 轴电压和电流， <math id="M13"><msub><mrow><mi>v</mi></mrow><mrow><mi>q</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=40435519&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=40435513&type= 2.96333337 3.72533321 和 <math id="M14"><msub><mrow><mi>i</mi></mrow><mrow><mi>q</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=40436050&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=40436049&type= 2.28600001 3.72533321 分别为 <math id="M15"><mi>q</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=40436255&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=40436254&type= 1.86266661 3.64066648 轴电压和电流， <math id="M16"><msub><mrow><mi>R</mi></mrow><mrow><mi>s</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=40435542&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=40435537&type= 3.55599999 3.72533321 为定子电阻， <math id="M17"><msub><mrow><mi>L</mi></mrow><mrow><mi>d</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=40435550&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=40435546&type= 3.80999994 3.72533321 和 <math id="M18"><msub><mrow><mi>L</mi></mrow><mrow><mi>q</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=40435560&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=40435556&type= 3.64066648 3.72533321 分别为 <math id="M19"><mi>d</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=40435728&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=40435727&type= 2.28600001 2.79399991 轴和 <math id="M20"><mi>q</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=40436255&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=40436254&type= 1.86266661 3.64066648 轴电感， <math id="M21"><msub><mrow><mi>ψ</mi></mrow><mrow><mi>m</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=40435582&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=40435577&type= 3.80999994 4.48733330 为定子磁链， <math id="M22"><msub><mrow><mi>ω</mi></mrow><mrow><mi>r</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=40435590&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=40435585&type= 3.47133350 3.72533321 为电机的角速度。通过式（1）式（1）和式（2）式（2）可得PMSM连续空间状态方程： <math display="block" id="M23"><mtable><mtr><mtd><mfenced open="[" close="]" separators="|"><mrow><mtable><mtr><mtd><mfrac><mrow><mi mathvariant="normal">d</mi><msub><mrow><mi>i</mi></mrow><mrow><mi>d</mi></mrow></msub></mrow><mrow><mi mathvariant="normal">d</mi><mi>t</mi></mrow></mfrac></mtd></mtr><mtr><mtd><mfrac><mrow><mi mathvariant="normal">d</mi><msub><mrow><mi>i</mi></mrow><mrow><mi>q</mi></mrow></msub></mrow><mrow><mi mathvariant="normal">d</mi><mi>t</mi></mrow></mfrac></mtd></mtr></mtable></mrow></mfenced><mo>=</mo><mfenced open="[" close="]" separators="|"><mrow><mtable><mtr><mtd><mo>-</mo><mfrac><mrow><msub><mrow><mi>R</mi></mrow><mrow><mi>s</mi></mrow></msub></mrow><mrow><msub><mrow><mi>L</mi></mrow><mrow><mi>d</mi></mrow></msub></mrow></mfrac></mtd><mtd><mfrac><mrow><msub><mrow><mi>ω</mi></mrow><mrow><mi>r</mi></mrow></msub><msub><mrow><mi>L</mi></mrow><mrow><mi>q</mi></mrow></msub></mrow><mrow><msub><mrow><mi>L</mi></mrow><mrow><mi>d</mi></mrow></msub></mrow></mfrac></mtd></mtr><mtr><mtd><mo>-</mo><mfrac><mrow><msub><mrow><mi>ω</mi></mrow><mrow><mi>r</mi></mrow></msub><msub><mrow><mi>L</mi></mrow><mrow><mi>d</mi></mrow></msub></mrow><mrow><msub><mrow><mi>L</mi></mrow><mrow><mi>q</mi></mrow></msub></mrow></mfrac></mtd><mtd><mo>-</mo><mfrac><mrow><msub><mrow><mi>R</mi></mrow><mrow><mi>s</mi></mrow></msub></mrow><mrow><msub><mrow><mi>L</mi></mrow><mrow><mi>q</mi></mrow></msub></mrow></mfrac></mtd></mtr></mtable></mrow></mfenced><mfenced open="[" close="]" separators="|"><mrow><mtable><mtr><mtd><msub><mrow><mi>i</mi></mrow><mrow><mi>d</mi></mrow></msub></mtd></mtr><mtr><mtd><msub><mrow><mi>i</mi></mrow><mrow><mi>q</mi></mrow></msub></mtd></mtr></mtable></mrow></mfenced><mo>+</mo></mtd></mtr><mtr><mtd><mfenced open="[" close="]" separators="|"><mrow><mtable><mtr><mtd><mfrac><mrow><mn mathvariant="normal">1</mn></mrow><mrow><msub><mrow><mi>L</mi></mrow><mrow><mi>d</mi></mrow></msub></mrow></mfrac></mtd><mtd><mn mathvariant="normal">0</mn></mtd></mtr><mtr><mtd><mn mathvariant="normal">0</mn></mtd><mtd><mfrac><mrow><mn mathvariant="normal">1</mn></mrow><mrow><msub><mrow><mi>L</mi></mrow><mrow><mi>q</mi></mrow></msub></mrow></mfrac></mtd></mtr></mtable></mrow></mfenced><mfenced open="[" close="]" separators="|"><mrow><mtable><mtr><mtd><msub><mrow><mi>v</mi></mrow><mrow><mi>d</mi></mrow></msub></mtd></mtr><mtr><mtd><msub><mrow><mi>v</mi></mrow><mrow><mi>q</mi></mrow></msub></mtd></mtr></mtable></mrow></mfenced><mo>+</mo><mfenced open="[" close="]" separators="|"><mrow><mtable><mtr><mtd><mn mathvariant="normal">0</mn></mtd></mtr><mtr><mtd><mo>-</mo><mfrac><mrow><msub><mrow><mi>ω</mi></mrow><mrow><mi>r</mi></mrow></msub><msub><mrow><mi>ψ</mi></mrow><mrow><mi>m</mi></mrow></msub></mrow><mrow><msub><mrow><mi>L</mi></mrow><mrow><mi>q</mi></mrow></msub></mrow></mfrac></mtd></mtr></mtable></mrow></mfenced></mtd></mtr></mtable></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=40435596&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=40435592&type= 49.61466980 41.90999985 . 上述连续状态空间方程采用欧拉法进行离散化化处理： <math display="block" id="M24"><mfrac><mrow><mi mathvariant="normal">d</mi><mi>i</mi></mrow><mrow><mi mathvariant="normal">d</mi><mi>t</mi></mrow></mfrac><mo>≈</mo><mfrac><mrow><mi>i</mi><mo stretchy="false">(</mo><mi>k</mi><mo>+</mo><mn mathvariant="normal">1</mn><mo stretchy="false">)</mo><mo>-</mo><mi>i</mi><mo stretchy="false">(</mo><mi>k</mi><mo stretchy="false">)</mo></mrow><mrow><msub><mrow><mi>T</mi></mrow><mrow><mi>s</mi></mrow></msub></mrow></mfrac></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=40435604&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=40435599&type= 33.10466385 8.63599968 ，其中： <math id="M25"><msub><mrow><mi>T</mi></mrow><mrow><mi>s</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=40435615&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=40435610&type= 3.38666677 3.72533321 为采样周期， <math id="M26"><mi>i</mi><mo stretchy="false">(</mo><mi>k</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=40435623&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=40435620&type= 6.51933336 3.55599999 为 <math id="M27"><mi>k</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=40435669&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=40435667&type= 1.69333339 2.79399991 时刻（ <math id="M28"><mi>k</mi><mo>=</mo><mn mathvariant="normal">1,2</mn><mo>,</mo><mn mathvariant="normal">3</mn><mo>,</mo><mo>…</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=40435633&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=40435631&type= 21.50533485 3.21733332 ）电流值， <math id="M29"><mi>i</mi><mo stretchy="false">(</mo><mi>k</mi><mo>+</mo><mn mathvariant="normal">1</mn><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=40435639&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=40435636&type= 12.86933327 3.55599999 为下一采样时刻电流值。因此，PMSM电流预测模型可由式（3）式（3）和式（4）式（4）得出： <math display="block" id="M30"><mfenced open="[" close="]" separators="|"><mrow><mtable><mtr><mtd><msub><mrow><mi>i</mi></mrow><mrow><mi>d</mi></mrow></msub><mo stretchy="false">(</mo><mi>k</mi><mo>+</mo><mn mathvariant="normal">1</mn><mo stretchy="false">)</mo></mtd></mtr><mtr><mtd><msub><mrow><mi>i</mi></mrow><mrow><mi>q</mi></mrow></msub><mo stretchy="false">(</mo><mi>k</mi><mo>+</mo><mn mathvariant="normal">1</mn><mo stretchy="false">)</mo></mtd></mtr></mtable></mrow></mfenced><mo>=</mo><mfenced open="[" close="]" separators="|"><mrow><mtable><mtr><mtd><mn mathvariant="normal">1</mn><mo>-</mo><mfrac><mrow><msub><mrow><mi>T</mi></mrow><mrow><mi>s</mi></mrow></msub><msub><mrow><mi>R</mi></mrow><mrow><mi>s</mi></mrow></msub></mrow><mrow><msub><mrow><mi>L</mi></mrow><mrow><mi>d</mi></mrow></msub></mrow></mfrac></mtd><mtd><mfrac><mrow><msub><mrow><mi>T</mi></mrow><mrow><mi>s</mi></mrow></msub><msub><mrow><mi>ω</mi></mrow><mrow><mi>r</mi></mrow></msub><msub><mrow><mi>L</mi></mrow><mrow><mi>q</mi></mrow></msub></mrow><mrow><msub><mrow><mi>L</mi></mrow><mrow><mi>d</mi></mrow></msub></mrow></mfrac></mtd></mtr><mtr><mtd><mo>-</mo><mfrac><mrow><msub><mrow><mi>T</mi></mrow><mrow><mi>s</mi></mrow></msub><msub><mrow><mi>ω</mi></mrow><mrow><mi>r</mi></mrow></msub><msub><mrow><mi>L</mi></mrow><mrow><mi>d</mi></mrow></msub></mrow><mrow><msub><mrow><mi>L</mi></mrow><mrow><mi>q</mi></mrow></msub></mrow></mfrac></mtd><mtd><mn mathvariant="normal">1</mn><mo>-</mo><mfrac><mrow><msub><mrow><mi>T</mi></mrow><mrow><mi>s</mi></mrow></msub><msub><mrow><mi>R</mi></mrow><mrow><mi>s</mi></mrow></msub></mrow><mrow><msub><mrow><mi>L</mi></mrow><mrow><mi>q</mi></mrow></msub></mrow></mfrac></mtd></mtr></mtable></mrow></mfenced><mfenced open="[" close="]" separators="|"><mrow><mtable><mtr><mtd><msub><mrow><mi>i</mi></mrow><mrow><mi>d</mi></mrow></msub><mo stretchy="false">(</mo><mi>k</mi><mo stretchy="false">)</mo></mtd></mtr><mtr><mtd><msub><mrow><mi>i</mi></mrow><mrow><mi>q</mi></mrow></msub><mo stretchy="false">(</mo><mi>k</mi><mo stretchy="false">)</mo></mtd></mtr></mtable></mrow></mfenced><mo>+</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=40435647&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=40435643&type= 74.84533691 20.15066719 <math display="block" id="M31"><mfenced open="[" close="]" separators="|"><mrow><mtable><mtr><mtd><mfrac><mrow><msub><mrow><mi>T</mi></mrow><mrow><mi>s</mi></mrow></msub></mrow><mrow><msub><mrow><mi>L</mi></mrow><mrow><mi>d</mi></mrow></msub></mrow></mfrac></mtd><mtd><mn mathvariant="normal">0</mn></mtd></mtr><mtr><mtd><mn mathvariant="normal">0</mn></mtd><mtd><mfrac><mrow><msub><mrow><mi>T</mi></mrow><mrow><mi>s</mi></mrow></msub></mrow><mrow><msub><mrow><mi>L</mi></mrow><mrow><mi>q</mi></mrow></msub></mrow></mfrac></mtd></mtr></mtable></mrow></mfenced><mfenced open="[" close="]" separators="|"><mrow><mtable><mtr><mtd><msub><mrow><mi>v</mi></mrow><mrow><mi>d</mi></mrow></msub><mo stretchy="false">(</mo><mi>k</mi><mo stretchy="false">)</mo></mtd></mtr><mtr><mtd><msub><mrow><mi>v</mi></mrow><mrow><mi>q</mi></mrow></msub><mo stretchy="false">(</mo><mi>k</mi><mo stretchy="false">)</mo></mtd></mtr></mtable></mrow></mfenced><mo>+</mo><mfenced open="[" close="]" separators="|"><mrow><mtable><mtr><mtd><mn mathvariant="normal">0</mn></mtd></mtr><mtr><mtd><mo>-</mo><mfrac><mrow><msub><mrow><mi>T</mi></mrow><mrow><mi>s</mi></mrow></msub><msub><mrow><mi>ω</mi></mrow><mrow><mi>r</mi></mrow></msub><msub><mrow><mi>ψ</mi></mrow><mrow><mi>m</mi></mrow></msub></mrow><mrow><msub><mrow><mi>L</mi></mrow><mrow><mi>q</mi></mrow></msub></mrow></mfrac></mtd></mtr></mtable></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=40435653&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=40435650&type= 51.13866425 20.15066719 ，其中： <math id="M32"><msub><mrow><mi>i</mi></mrow><mrow><mi>d</mi></mrow></msub><mo stretchy="false">(</mo><mi>k</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=40435658&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=40435656&type= 8.04333401 4.48733330 ， <math id="M33"><msub><mrow><mi>i</mi></mrow><mrow><mi>q</mi></mrow></msub><mo stretchy="false">(</mo><mi>k</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=40435664&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=40435661&type= 7.87400007 4.48733330 分别为 <math id="M34"><mi>k</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=40435669&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=40435667&type= 1.69333339 2.79399991 时刻的 <math id="M35"><mi>d</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=40435728&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=40435727&type= 2.28600001 2.79399991 轴和 <math id="M36"><mi>q</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=40436255&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=40436254&type= 1.86266661 3.64066648 轴电流， <math id="M37"><msub><mrow><mi>i</mi></mrow><mrow><mi>d</mi></mrow></msub><mo stretchy="false">(</mo><mi>k</mi><mo>+</mo><mn mathvariant="normal">1</mn><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=40435684&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=40435681&type= 14.47799969 4.48733330 ， <math id="M38"><msub><mrow><mi>i</mi></mrow><mrow><mi>q</mi></mrow></msub><mo stretchy="false">(</mo><mi>k</mi><mo>+</mo><mn mathvariant="normal">1</mn><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=40435690&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=40435688&type= 14.30866718 4.48733330 分别为 <math id="M39"><mi>k</mi><mo>+</mo><mn mathvariant="normal">1</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=40435696&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=40435692&type= 8.38199997 2.96333337 时刻的 <math id="M40"><mi>d</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=40435728&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=40435727&type= 2.28600001 2.79399991 轴和 <math id="M41"><mi>q</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=40436255&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=40436254&type= 1.86266661 3.64066648 轴电流。 3 PMSM电流预测模型失配影响 1 将PMSM电流预测模型表达式简化为： <math display="block" id="M42"><mi mathvariant="bold-italic">i</mi><mo stretchy="false">(</mo><mi>k</mi><mo>+</mo><mn mathvariant="normal">1</mn><mo stretchy="false">)</mo><mo>=</mo><msub><mrow><mi>D</mi></mrow><mrow><mi>i</mi></mrow></msub><mi mathvariant="bold-italic">i</mi><mo stretchy="false">(</mo><mi>k</mi><mo stretchy="false">)</mo><mo>+</mo><msub><mrow><mi>D</mi></mrow><mrow><mi>v</mi></mrow></msub><mi mathvariant="bold-italic">v</mi><mo stretchy="false">(</mo><mi>k</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=40435713&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=40435710&type= 44.02666473 4.48733330 ，其中： <math id="M43"><msub><mrow><mi>D</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>=</mo><mo stretchy="false">(</mo><mn mathvariant="normal">1</mn><mo>-</mo><msub><mrow><mi>R</mi></mrow><mrow><mi>s</mi></mrow></msub><msub><mrow><mi>T</mi></mrow><mrow><mi>s</mi></mrow></msub><mo>/</mo><msub><mrow><mi>L</mi></mrow><mrow><mi>s</mi></mrow></msub><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=40435718&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=40435716&type= 30.14133072 4.48733330 ， <math id="M44"><msub><mrow><mi>D</mi></mrow><mrow><mi>v</mi></mrow></msub><mo>=</mo><msub><mrow><mi>T</mi></mrow><mrow><mi>s</mi></mrow></msub><mo>/</mo><msub><mrow><mi>L</mi></mrow><mrow><mi>s</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=40435724&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=40435721&type= 17.52599907 4.14866638 ，表贴式永磁同步电机 <math id="M45"><mi>d</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=40435728&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=40435727&type= 2.28600001 2.79399991 轴和 <math id="M46"><mi>q</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=40436255&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=40436254&type= 1.86266661 3.64066648 轴电感相等，用 <math id="M47"><msub><mrow><mi>L</mi></mrow><mrow><mi>s</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=40435741&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=40435736&type= 3.30200005 3.72533321 表示。空间矢量角度分析，式中预测电流矢量 <math id="M48"><mi mathvariant="bold-italic">i</mi><mo stretchy="false">(</mo><mi>k</mi><mo>+</mo><mn mathvariant="normal">1</mn><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=40435747&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=40435744&type= 12.95400047 3.55599999 由实际电流矢量 <math id="M49"><mi mathvariant="bold-italic">i</mi><mo stretchy="false">(</mo><mi>k</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=40435754&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=40435751&type= 6.51933336 3.55599999 与实际电压矢量 <math id="M50"><mi mathvariant="bold-italic">v</mi><mo stretchy="false">(</mo><mi>k</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=40435759&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=40435757&type= 7.36600018 3.55599999 加权合成。以 <math id="M51"><mi>d</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=40435765&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=40435762&type= 2.28600001 2.79399991 - <math id="M52"><mi>q</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=40435772&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=40435766&type= 1.86266661 3.64066648 轴电感参数为例分析模型失配对系统控制性能的影响。当系统模型未失配时，预测电流可表示为： <math display="block" id="M53"><msup><mrow><mi mathvariant="bold-italic">i</mi></mrow><mrow><mi>r</mi></mrow></msup><mo stretchy="false">(</mo><mi>k</mi><mo>+</mo><mn mathvariant="normal">1</mn><mo stretchy="false">)</mo><mo>=</mo><mfenced separators="|"><mrow><mn mathvariant="normal">1</mn><mo>-</mo><mfrac><mrow><msub><mrow><mi>R</mi></mrow><mrow><mi>s</mi></mrow></msub><msub><mrow><mi>T</mi></mrow><mrow><mi>s</mi></mrow></msub></mrow><mrow><msubsup><mrow><mi>L</mi></mrow><mrow><mi>s</mi></mrow><mrow><mi>r</mi></mrow></msubsup></mrow></mfrac></mrow></mfenced><mi mathvariant="bold-italic">i</mi><mo stretchy="false">(</mo><mi>k</mi><mo stretchy="false">)</mo><mo>+</mo><mfrac><mrow><msub><mrow><mi>T</mi></mrow><mrow><mi>s</mi></mrow></msub></mrow><mrow><msubsup><mrow><mi>L</mi></mrow><mrow><mi>s</mi></mrow><mrow><mi>r</mi></mrow></msubsup></mrow></mfrac><mi mathvariant="bold-italic">v</mi><mo stretchy="false">(</mo><mi>k</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=40435778&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=40435775&type= 59.94400024 9.56733322 ，其中： <math id="M54"><msup><mrow><mi mathvariant="bold-italic">i</mi></mrow><mrow><mi>r</mi></mrow></msup><mo stretchy="false">(</mo><mi>k</mi><mo>+</mo><mn mathvariant="normal">1</mn><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=40435784&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=40435781&type= 14.13933277 3.80999994 为无失配预测电流， <math id="M55"><msubsup><mrow><mi>L</mi></mrow><mrow><mi>s</mi></mrow><mrow><mi>r</mi></mrow></msubsup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=40435792&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=40435789&type= 3.47133350 3.97933316 为电感真实值。当系统模型失配时，预测电流可表示为： <math display="block" id="M56"><msup><mrow><mi mathvariant="bold-italic">i</mi></mrow><mrow><mi>p</mi></mrow></msup><mo stretchy="false">(</mo><mi>k</mi><mo>+</mo><mn mathvariant="normal">1</mn><mo stretchy="false">)</mo><mo>=</mo><mfenced separators="|"><mrow><mn mathvariant="normal">1</mn><mo>-</mo><mfrac><mrow><msub><mrow><mi>R</mi></mrow><mrow><mi>s</mi></mrow></msub><msub><mrow><mi>T</mi></mrow><mrow><mi>s</mi></mrow></msub></mrow><mrow><msubsup><mrow><mi>L</mi></mrow><mrow><mi>s</mi></mrow><mrow><mi>p</mi></mrow></msubsup></mrow></mfrac></mrow></mfenced><mi mathvariant="bold-italic">i</mi><mo stretchy="false">(</mo><mi>k</mi><mo stretchy="false">)</mo><mo>+</mo><mfrac><mrow><msub><mrow><mi>T</mi></mrow><mrow><mi>s</mi></mrow></msub></mrow><mrow><msubsup><mrow><mi>L</mi></mrow><mrow><mi>s</mi></mrow><mrow><mi>p</mi></mrow></msubsup></mrow></mfrac><mi mathvariant="bold-italic">v</mi><mo stretchy="false">(</mo><mi>k</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=40435800&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=40435797&type= 60.28266144 9.56733322 ，其中： <math id="M57"><msup><mrow><mi mathvariant="bold-italic">i</mi></mrow><mrow><mi>p</mi></mrow></msup><mo stretchy="false">(</mo><mi>k</mi><mo>+</mo><mn mathvariant="normal">1</mn><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=40435803&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=40435801&type= 14.30866718 3.80999994 为失配预测电流， <math id="M58"><msubsup><mrow><mi>L</mi></mrow><mrow><mi>s</mi></mrow><mrow><mi>p</mi></mrow></msubsup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=40435809&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=40435806&type= 3.64066648 3.97933316 为电感标定值。由式（7）式（7）和式（8）式（8）可得： <math display="block" id="M59"><msup><mrow><mi mathvariant="bold-italic">i</mi></mrow><mrow><mi>r</mi></mrow></msup><mo stretchy="false">(</mo><mi>k</mi><mo>+</mo><mn mathvariant="normal">1</mn><mo stretchy="false">)</mo><mo>=</mo><mfrac><mrow><msubsup><mrow><mi>L</mi></mrow><mrow><mi>s</mi></mrow><mrow><mi>p</mi></mrow></msubsup></mrow><mrow><msubsup><mrow><mi>L</mi></mrow><mrow><mi>s</mi></mrow><mrow><mi>r</mi></mrow></msubsup></mrow></mfrac><msup><mrow><mi mathvariant="bold-italic">i</mi></mrow><mrow><mi>p</mi></mrow></msup><mo stretchy="false">(</mo><mi>k</mi><mo>+</mo><mn mathvariant="normal">1</mn><mo stretchy="false">)</mo><mo>+</mo><mo stretchy="false">(</mo><mn mathvariant="normal">1</mn><mo>-</mo><mfrac><mrow><msubsup><mrow><mi>L</mi></mrow><mrow><mi>s</mi></mrow><mrow><mi>p</mi></mrow></msubsup></mrow><mrow><msubsup><mrow><mi>L</mi></mrow><mrow><mi>s</mi></mrow><mrow><mi>r</mi></mrow></msubsup></mrow></mfrac><mo stretchy="false">)</mo><mi mathvariant="bold-italic">i</mi><mo stretchy="false">(</mo><mi>k</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=40435815&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=40435811&type= 63.58466721 8.89000034 ，对式（9）式（9）进行 <math id="M60"><mi>z</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=40435901&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=40435896&type= 1.77800000 2.79399991 变换： <math display="block" id="M61"><mfrac><mrow><msup><mrow><mi mathvariant="bold-italic">i</mi></mrow><mrow><mi>p</mi></mrow></msup><mo stretchy="false">(</mo><mi>z</mi><mo stretchy="false">)</mo></mrow><mrow><msup><mrow><mi mathvariant="bold-italic">i</mi></mrow><mrow><mi>r</mi></mrow></msup><mo stretchy="false">(</mo><mi>z</mi><mo stretchy="false">)</mo></mrow></mfrac><mo>=</mo><mfrac><mrow><mi>z</mi><msubsup><mrow><mi>L</mi></mrow><mrow><mi>s</mi></mrow><mrow><mi>r</mi></mrow></msubsup><mo>/</mo><msubsup><mrow><mi>L</mi></mrow><mrow><mi>s</mi></mrow><mrow><mi>p</mi></mrow></msubsup></mrow><mrow><mi>z</mi><mo>+</mo><msubsup><mrow><mi>L</mi></mrow><mrow><mi>s</mi></mrow><mrow><mi>r</mi></mrow></msubsup><mo>/</mo><msubsup><mrow><mi>L</mi></mrow><mrow><mi>s</mi></mrow><mrow><mi>p</mi></mrow></msubsup><mo>-</mo><mn mathvariant="normal">1</mn></mrow></mfrac></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=40435829&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=40435823&type= 36.40666580 8.97466660 . 由此得到此离散系统极点为： <math id="M62"><mi>z</mi><mo>=</mo><mo>-</mo><msubsup><mrow><mi>L</mi></mrow><mrow><mi>s</mi></mrow><mrow><mi>r</mi></mrow></msubsup><mo>/</mo><msubsup><mrow><mi>L</mi></mrow><mrow><mi>s</mi></mrow><mrow><mi>p</mi></mrow></msubsup><mo>+</mo><mn mathvariant="normal">1</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=40435835&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=40435831&type= 25.82333374 4.40266657 ，令 <math id="M63"><mi>m</mi><mo>=</mo><msubsup><mrow><mi>L</mi></mrow><mrow><mi>s</mi></mrow><mrow><mi>r</mi></mrow></msubsup><mo>/</mo><msubsup><mrow><mi>L</mi></mrow><mrow><mi>s</mi></mrow><mrow><mi>p</mi></mrow></msubsup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=40435842&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=40435839&type= 16.59466553 4.40266657 ， <math id="M64"><mi>m</mi><mo>=</mo><mn mathvariant="normal">1</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=40435847&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=40435844&type= 9.56733322 2.79399991 时，系统零极点如图1 图1 （a）所示，系统极点位于原点，冲激响应如图1 图1 （b）所示，电流实现无差拍跟踪。系统模型参数正向失配，并考虑实际情况，在 <math id="M65"><mn mathvariant="normal">0.5</mn><mo><</mo><mi>m</mi><mo><</mo><mn mathvariant="normal">1</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=40435854&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=40435849&type= 19.72733307 2.79399991 的情况下，系统 <math id="M66"><mi>z</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=40435864&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=40435860&type= 1.77800000 2.79399991 域零极点分布图如图1 图1 （c）所示，系统极点位于正实轴且在单位圆内，冲激响应如图1 图1 （d）所示，响应呈指数衰减， <math id="M67"><mi>m</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=40435872&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=40435868&type= 2.87866688 2.79399991 数值与电流响应速度成正相关。系统模型参数负向失配即在 <math id="M68"><mn mathvariant="normal">1</mn><mo><</mo><mi>m</mi><mo><</mo><mn mathvariant="normal">1.5</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=40435892&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=40435886&type= 20.06599998 2.79399991 的情况下，系统 <math id="M69"><mi>z</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=40435901&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=40435896&type= 1.77800000 2.79399991 域零极点分布图如图1 图1 （e）所示，系统极点位于负实轴，冲激响应如图1 图1 （f）所示，电流阶跃响应呈指数衰减， <math id="M70"><mi>m</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=40435910&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=40435907&type= 2.87866688 2.79399991 数值与电流响应速度成负相关。由上述分析可知，系统模型不同失配程度会对系统性能造成不同影响，对于模型预测电流控制，此影响体现在电流控制效果上，图2 图2 展示了在传统FCS-MPC策略下，电感 <math id="M71"><msub><mrow><mi>L</mi></mrow><mrow><mi>s</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=40435916&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=40435913&type= 3.30200005 3.72533321 不同失配程度下 <math id="M72"><mi>q</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=40436255&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=40436254&type= 1.86266661 3.64066648 轴电流表现。如图所示，在电感参数正向失配的情况下， <math id="M73"><mi>q</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=40436255&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=40436254&type= 1.86266661 3.64066648 轴电流参考值相较于实际 <math id="M74"><mi>q</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=40436255&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=40436254&type= 1.86266661 3.64066648 轴电流会产生上偏移现象，而且偏移的程度与电感参数失配程度成正相关。在电感参数负向失配的情况下， <math id="M75"><mi>q</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=40436255&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=40436254&type= 1.86266661 3.64066648 轴电流参考值相较于实际 <math id="M76"><mi>q</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=40436255&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=40436254&type= 1.86266661 3.64066648 轴电流会产生下偏移现象，而且偏移的程度与电感参数失配程度成正相关。通过图2 图2 （a）与图2 图2 （e）、图2 图2 （b）与图2 图2 （f）、图2 图2 （c）与图2 图2 （g）三组对比分析，在失配幅度相同的条件下，电感参数负向失配时会造成更严重的偏移现象。这表明电感变化对FCS-MPC性能的影响是不对称的，在预测模型高估实际电感值的情况下更具危害性。经上述分析可知，电感参数的失配程度与电流实际表现存在某种关系。将电流表现数据作为输入，电感参数的失配程度数据作为输出，通过高斯过程回归模型的训练，可建立起输入变量到输出变量的映射关系。通过此关系，在电流状态已知的情况下可推得模型失配程度，进而求得实际参数值，实现系统模型的在线优化。 4 基于高斯过程的参数优化方案 1 4.1　训练数据的获取与预处理 2 高斯过程作为机器学习的方法，需要量化数据作为支撑。上文描述 <math id="M77"><mi>q</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=40436255&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=40436254&type= 1.86266661 3.64066648 轴电流参考值相较于实际 <math id="M78"><mi>q</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=40436255&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=40436254&type= 1.86266661 3.64066648 轴电流偏移程度，可量化为电流偏移度 <math id="M79"><mi>P</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=40436073&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=40436072&type= 2.62466669 2.79399991 。电流偏移度 <math id="M80"><mi>P</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=40436073&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=40436072&type= 2.62466669 2.79399991 描述 <math id="M81"><mi>q</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=40436255&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=40436254&type= 1.86266661 3.64066648 轴电流 <math id="M82"><msub><mrow><mi>i</mi></mrow><mrow><mi>q</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=40436050&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=40436049&type= 2.28600001 3.72533321 对 <math id="M83"><mi>q</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=40436255&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=40436254&type= 1.86266661 3.64066648 轴参考电流 <math id="M84"><msub><mrow><mi>i</mi></mrow><mrow><mi mathvariant="normal">q</mi><mi mathvariant="normal">r</mi><mi mathvariant="normal">e</mi><mi mathvariant="normal">f</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=40435999&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=40435998&type= 4.40266657 3.72533321 上偏移程度 <math id="M85"><mi>P</mi><mo>_</mo><mi>u</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=40436004&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=40436001&type= 6.18066692 3.30200005 和下偏移程度 <math id="M86"><mi>P</mi><mo>_</mo><mi>d</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=40436007&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=40436006&type= 6.60400009 3.30200005 的比例关系： <math display="block" id="M87"><mfenced open="{" close="" separators="|"><mrow><mtable columnalign="left"><mtr><mtd><mi>P</mi><mo>_</mo><mi>u</mi><mo>=</mo><mfrac><mrow><mn mathvariant="normal">1</mn></mrow><mrow><mn mathvariant="normal">2</mn><mi>n</mi></mrow></mfrac><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>j</mi><mo>=</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi>n</mi></mrow></munderover></mstyle><mrow><mo stretchy="false">[</mo><mroot><mrow><mo stretchy="false">(</mo><msub><mrow><mi>i</mi></mrow><mrow><mi mathvariant="normal">q</mi><mi mathvariant="normal">r</mi><mi mathvariant="normal">e</mi><mi mathvariant="normal">f</mi></mrow></msub><mo>-</mo><msub><mrow><mi>i</mi></mrow><mrow><mi>q</mi></mrow></msub><msubsup><mrow><mo stretchy="false">)</mo></mrow><mrow><mi>j</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msubsup></mrow><mrow/></mroot><mo>+</mo><mo stretchy="false">(</mo><msub><mrow><mi>i</mi></mrow><mrow><mi mathvariant="normal">q</mi><mi mathvariant="normal">r</mi><mi mathvariant="normal">e</mi><mi mathvariant="normal">f</mi></mrow></msub><mo>-</mo><msub><mrow><mi>i</mi></mrow><mrow><mi>q</mi></mrow></msub><msub><mrow><mo stretchy="false">)</mo></mrow><mrow><mi>j</mi></mrow></msub><mo stretchy="false">]</mo></mrow></mtd></mtr><mtr><mtd><mi>P</mi><mo>_</mo><mi>d</mi><mo>=</mo><mfrac><mrow><mn mathvariant="normal">1</mn></mrow><mrow><mi>n</mi></mrow></mfrac><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>j</mi><mo>=</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi>n</mi></mrow></munderover></mstyle><mrow><mo stretchy="false">[</mo><mroot><mrow><mo stretchy="false">(</mo><msub><mrow><mi>i</mi></mrow><mrow><mi mathvariant="normal">q</mi><mi mathvariant="normal">r</mi><mi mathvariant="normal">e</mi><mi mathvariant="normal">f</mi></mrow></msub><mo>-</mo><msub><mrow><mi>i</mi></mrow><mrow><mi>q</mi></mrow></msub><msubsup><mrow><mo stretchy="false">)</mo></mrow><mrow><mi>j</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msubsup></mrow><mrow/></mroot><mo>-</mo><mi>P</mi><mo>_</mo><mi>u</mi><mo stretchy="false">]</mo></mrow></mtd></mtr><mtr><mtd><mi>P</mi><mo>=</mo><mfrac><mrow><mn mathvariant="normal">1</mn></mrow><mrow><mi>λ</mi></mrow></mfrac><mi mathvariant="normal">l</mi><mi mathvariant="normal">n</mi><mo stretchy="false">(</mo><mi>P</mi><mo>_</mo><mi>u</mi><mo>/</mo><mi>P</mi><mo>_</mo><mi>d</mi><mo stretchy="false">)</mo></mtd></mtr></mtable></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=40436014&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=40436010&type= 65.53199768 30.39533424 ，其中： <math id="M88"><mi>n</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=40436063&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=40436061&type= 1.86266661 2.79399991 为单检测周期内数据量， <math id="M89"><mi>λ</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=40436024&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=40436022&type= 1.77800000 2.79399991 为偏移度系数，用作数据标准化处理，初始值设为1。电感值不同失配程度下的 <math id="M90"><mi>q</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=40436255&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=40436254&type= 1.86266661 3.64066648 轴电流偏移度如图3 图3 所示，电流表现通过量化数据分析，电感参数的正、负向失配对电流的影响是不对称的，不同失配程度下电流偏移度 <math id="M91"><mi>P</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=40436073&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=40436072&type= 2.62466669 2.79399991 的数据作为高斯过程回归模型的训练输入之一。 <math id="M92"><mi>q</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=40436255&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=40436254&type= 1.86266661 3.64066648 轴电流表现另一重要量化数据为电流波动度 <math id="M93"><mi>B</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=40436071&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=40436070&type= 2.53999996 2.79399991 ，其描述 <math id="M94"><mi>q</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=40436255&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=40436254&type= 1.86266661 3.64066648 轴电流 <math id="M95"><msub><mrow><mi>i</mi></mrow><mrow><mi>q</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=40436050&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=40436049&type= 2.28600001 3.72533321 以 <math id="M96"><mi>q</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=40436255&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=40436254&type= 1.86266661 3.64066648 轴参考电流 <math id="M97"><msub><mrow><mi>i</mi></mrow><mrow><mi mathvariant="normal">q</mi><mi mathvariant="normal">r</mi><mi mathvariant="normal">e</mi><mi mathvariant="normal">f</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=40436055&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=40436054&type= 4.40266657 3.72533321 为基准的波动程度： <math display="block" id="M98"><mi>B</mi><mo>=</mo><mfrac><mrow><mn mathvariant="normal">1</mn></mrow><mrow><mi>n</mi></mrow></mfrac><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>j</mi><mo>=</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi>n</mi></mrow></munderover></mstyle><mrow><mroot><mrow><mo stretchy="false">(</mo><msub><mrow><mi>i</mi></mrow><mrow><mi mathvariant="normal">q</mi><mi mathvariant="normal">r</mi><mi mathvariant="normal">e</mi><mi mathvariant="normal">f</mi></mrow></msub><mo>-</mo><msub><mrow><mi>i</mi></mrow><mrow><mi>q</mi></mrow></msub><msubsup><mrow><mo stretchy="false">)</mo></mrow><mrow><mi>j</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msubsup></mrow><mrow/></mroot></mrow></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=40436057&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=40436056&type= 34.45933533 9.05933285 . 然而这一指标随电感参数失配程度的变化呈现复杂的变化趋势，以此指标作为高斯过程回归模型的另一训练输入，其变化规律通过高斯过程算法学习。图4 图4 为实时获取部分训练数据集的三维散点图，随着电流偏移度和电流波动度的变化，电感标称值呈螺旋上升的变化趋势。 4.2　高斯过程回归模型 2 区别于传统参数优化方法，高斯过程参数优化作为一种机器学习的方法依赖于有效数据的训练。假设训练数据集有 <math id="M99"><mi>n</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=40436063&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=40436061&type= 1.86266661 2.79399991 个样本，则此数据集可表示为 <math id="M100"><mo stretchy="false">{</mo><mi mathvariant="bold-italic">y</mi><mo>,</mo><mi mathvariant="bold-italic">α</mi><mo>,</mo><mi mathvariant="bold-italic">β</mi><mo>,</mo><mi mathvariant="bold-italic">γ</mi><mo stretchy="false">}</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=40436068&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=40436065&type= 17.69533348 3.97933316 ，其中，电流波动度 <math id="M101"><mi>B</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=40436071&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=40436070&type= 2.53999996 2.79399991 和电流偏移度 <math id="M102"><mi>P</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=40436073&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=40436072&type= 2.62466669 2.79399991 分别用 <math id="M103"><mi>α</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=40436075&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=40436074&type= 1.94733346 2.79399991 ， <math id="M104"><mi>β</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=40436077&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=40436076&type= 2.28600001 3.64066648 来表示， <math id="M105"><mi mathvariant="bold-italic">γ</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=40436079&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=40436078&type= 2.11666679 3.64066648 是参数标称值，以上三项作为训练输入数据， <math id="M106"><mi>y</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=40436081&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=40436080&type= 2.11666679 3.72533321 是参数实际值，作为训练输出数据。测试数据集用 <math id="M107"><mo stretchy="false">{</mo><msup><mrow><mi mathvariant="bold-italic">y</mi></mrow><mrow><mi mathvariant="normal">*</mi></mrow></msup><mo>,</mo><msup><mrow><mi mathvariant="bold-italic">α</mi></mrow><mrow><mi mathvariant="normal">*</mi></mrow></msup><mo>,</mo><msup><mrow><mi mathvariant="bold-italic">β</mi></mrow><mrow><mi mathvariant="normal">*</mi></mrow></msup><mo>,</mo><msup><mrow><mi mathvariant="bold-italic">γ</mi></mrow><mrow><mi mathvariant="normal">*</mi></mrow></msup><mo stretchy="false">}</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=40436083&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=40436082&type= 22.18266678 4.23333359 表示，为表述方便，令多维训练输入数据集 <math id="M108"><mo stretchy="false">{</mo><mi mathvariant="bold-italic">α</mi><mo>,</mo><mi mathvariant="bold-italic">β</mi><mo>,</mo><mi mathvariant="bold-italic">γ</mi><mo stretchy="false">}</mo><mo>=</mo><mi mathvariant="bold-italic">x</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=40436085&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=40436084&type= 20.31999969 3.97933316 ，令多维测试输入数据集 <math id="M109"><mo stretchy="false">{</mo><msup><mrow><mi mathvariant="bold-italic">α</mi></mrow><mrow><mi mathvariant="normal">*</mi></mrow></msup><mo>,</mo><msup><mrow><mi mathvariant="bold-italic">β</mi></mrow><mrow><mi mathvariant="normal">*</mi></mrow></msup><mo>,</mo><msup><mrow><mi mathvariant="bold-italic">γ</mi></mrow><mrow><mi mathvariant="normal">*</mi></mrow></msup><mo stretchy="false">}</mo><mo>=</mo><msup><mrow><mi mathvariant="bold-italic">x</mi></mrow><mrow><mi mathvariant="normal">*</mi></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=40436087&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=40436086&type= 24.80733490 4.23333359 ，测试输出数据表示为 <math id="M110"><msup><mrow><mi mathvariant="bold-italic">y</mi></mrow><mrow><mi mathvariant="normal">*</mi></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=40436089&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=40436088&type= 3.30200005 3.89466691 。与高斯分布描述数据的分布规律类似，高斯过程（GP）描述的是函数的分布。均值函数 <math id="M111"><mi>m</mi><mo stretchy="false">(</mo><mi mathvariant="bold-italic">x</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=40436091&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=40436090&type= 8.89000034 3.55599999 和协方差函数 <math id="M112"><mi>k</mi><mo stretchy="false">(</mo><mi mathvariant="bold-italic">x</mi><mo>,</mo><mi mathvariant="bold-italic">x</mi><mi mathvariant="bold-italic">'</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=40436093&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=40436092&type= 12.44600010 3.72533321 决定了高斯过程全部的统计特征，因此，定义高斯过程为： <math display="block" id="M113"><mi>f</mi><mo stretchy="false">(</mo><mi mathvariant="bold-italic">x</mi><mo stretchy="false">)</mo><mo>∼</mo><mi>G</mi><mi>P</mi><mo stretchy="false">(</mo><mi>m</mi><mo stretchy="false">(</mo><mi mathvariant="bold-italic">x</mi><mo stretchy="false">)</mo><mo>,</mo><mi>k</mi><mo stretchy="false">(</mo><mi mathvariant="bold-italic">x</mi><mo>,</mo><mi mathvariant="bold-italic">x</mi><mi mathvariant="bold-italic">'</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=40436095&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=40436094&type= 45.04266739 4.14866638 . 对于实际问题，还需考虑噪声 <math id="M114"><mi>ε</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=40436097&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=40436096&type= 1.43933344 2.79399991 对观测值的影响，因此，模型描述为： <math display="block" id="M115"><mi mathvariant="bold-italic">y</mi><mo>=</mo><mi>f</mi><mo stretchy="false">(</mo><mi mathvariant="bold-italic">x</mi><mo stretchy="false">)</mo><mo>+</mo><mi mathvariant="bold-italic">ε</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=40436099&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=40436098&type= 20.82799911 3.97933316 ，其中：进一步假设噪声 <math id="M116"><mi mathvariant="bold-italic">ε</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=40436101&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=40436100&type= 1.60866666 2.79399991 符合均值为0，方差为 <math id="M117"><msup><mrow><mi>σ</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=40436103&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=40436102&type= 3.04800010 3.04800010 的高斯分布： <math id="M118"><mi mathvariant="bold-italic">ε</mi><mo>∼</mo><mi>N</mi><mo stretchy="false">(</mo><mn mathvariant="normal">0</mn><mo>,</mo><msubsup><mrow><mi>σ</mi></mrow><mrow><mi>n</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msubsup><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=40436105&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=40436104&type= 19.47333336 4.65666676 ，且其独立于 <math id="M119"><mi>f</mi><mo stretchy="false">(</mo><mi mathvariant="bold-italic">x</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=40436107&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=40436106&type= 8.46666718 3.97933316 。此时，实际函数值 <math id="M120"><mi mathvariant="bold-italic">y</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=40436109&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=40436108&type= 2.20133328 3.64066648 服从高斯过程： <math display="block" id="M121"><mi>f</mi><mo stretchy="false">(</mo><mi mathvariant="bold-italic">x</mi><mo stretchy="false">)</mo><mo>∼</mo><mi>G</mi><mi>P</mi><mo stretchy="false">(</mo><mi>m</mi><mo stretchy="false">(</mo><mi mathvariant="bold-italic">x</mi><mo stretchy="false">)</mo><mo>,</mo><mi>k</mi><mo stretchy="false">(</mo><mi mathvariant="bold-italic">x</mi><mo>,</mo><mi mathvariant="bold-italic">x</mi><mi mathvariant="bold-italic">'</mi><mo stretchy="false">)</mo><mo>+</mo><msubsup><mrow><mi>σ</mi></mrow><mrow><mi>n</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msubsup><msub><mrow><mi>δ</mi></mrow><mrow><mi>i</mi><mi>j</mi></mrow></msub><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=40436111&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=40436110&type= 56.30332947 5.24933338 ，其中： <math id="M122"><msub><mrow><mi>δ</mi></mrow><mrow><mi>i</mi><mi>j</mi></mrow></msub><mo>=</mo><mfenced open="{" close="" separators="|"><mrow><mtable><mtr><mtd><mn mathvariant="normal">0</mn><mo>,</mo><mi>i</mi><mo>≠</mo><mi>j</mi></mtd></mtr><mtr><mtd><mn mathvariant="normal">1</mn><mo>,</mo><mi>i</mi><mo>=</mo><mi>j</mi></mtd></mtr></mtable></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=40436113&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=40436112&type= 20.74333191 10.32933331 。训练样本与一个测试点的联合分布为： <math display="block" id="M123"><mfenced open="[" close="]" separators="|"><mrow><mtable><mtr><mtd><mi mathvariant="bold-italic">y</mi></mtd></mtr><mtr><mtd><mtext> </mtext><msup><mrow><mi>y</mi></mrow><mrow><mi mathvariant="normal">*</mi></mrow></msup></mtd></mtr></mtable></mrow></mfenced><mo>∼</mo><mi>Ν</mi><mfenced separators="|"><mrow><mfenced open="[" close="]" separators="|"><mrow><mtable columnalign="left"><mtr><mtd><msub><mrow><mi>μ</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub></mtd></mtr><mtr><mtd><msub><mrow><mi>μ</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub></mtd></mtr></mtable></mrow></mfenced><mo>,</mo><mfenced open="[" close="" separators="|"><mrow><mtable columnalign="left"><mtr><mtd><mi mathvariant="bold-italic">K</mi></mtd></mtr><mtr><mtd><msub><mrow><mi mathvariant="bold-italic">K</mi></mrow><mrow><mi mathvariant="normal">*</mi></mrow></msub></mtd></mtr></mtable></mrow></mfenced><mfenced open="" close="]" separators="|"><mrow><mtable columnalign="left"><mtr><mtd><msubsup><mrow><mi mathvariant="bold-italic">K</mi></mrow><mrow><mi mathvariant="normal">*</mi></mrow><mrow><mi mathvariant="normal">T</mi></mrow></msubsup></mtd></mtr><mtr><mtd><msub><mrow><mi mathvariant="bold-italic">K</mi></mrow><mrow><mi mathvariant="normal">*</mi><mi mathvariant="normal">*</mi></mrow></msub></mtd></mtr></mtable></mrow></mfenced></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=40436115&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=40436114&type= 39.96266937 10.58333302 ，其中： <math id="M124"><mi mathvariant="bold-italic">K</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=40436117&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=40436116&type= 3.13266683 2.79399991 表示协方差矩阵 <math id="M125"><mi mathvariant="bold-italic">K</mi><mo stretchy="false">(</mo><mi mathvariant="bold-italic">x</mi><mo>,</mo><mi mathvariant="bold-italic">x</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=40436120&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=40436119&type= 13.12333298 3.55599999 ，描述训练数据 <math id="M126"><mi mathvariant="bold-italic">x</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=40436136&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=40436135&type= 2.28600001 2.79399991 内部相关性； <math id="M127"><msubsup><mrow><mi mathvariant="bold-italic">K</mi></mrow><mrow><mi mathvariant="normal">*</mi></mrow><mrow><mi mathvariant="normal">T</mi></mrow></msubsup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=40436128&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=40436126&type= 4.99533367 3.97933316 表示协方差矩阵 <math id="M128"><mi mathvariant="bold-italic">K</mi><mo stretchy="false">(</mo><mi mathvariant="bold-italic">x</mi><mo>,</mo><msup><mrow><mi mathvariant="bold-italic">x</mi></mrow><mrow><mi mathvariant="normal">*</mi></mrow></msup><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=40436133&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=40436130&type= 14.13933277 3.80999994 ，描述训练数据 <math id="M129"><mi mathvariant="bold-italic">x</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=40436136&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=40436135&type= 2.28600001 2.79399991 与测试点 <math id="M130"><msup><mrow><mi mathvariant="bold-italic">x</mi></mrow><mrow><mi mathvariant="normal">*</mi></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=40436166&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=40436164&type= 3.38666677 3.04800010 协差关系； <math id="M131"><msub><mrow><mi mathvariant="bold-italic">K</mi></mrow><mrow><mi mathvariant="normal">*</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=40436147&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=40436143&type= 4.06400013 3.72533321 表示协方差矩阵 <math id="M132"><mi mathvariant="bold-italic">K</mi><mo stretchy="false">(</mo><msup><mrow><mi mathvariant="bold-italic">x</mi></mrow><mrow><mi mathvariant="normal">*</mi></mrow></msup><mo>,</mo><mi mathvariant="bold-italic">x</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=40436152&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=40436149&type= 14.39333248 3.80999994 ； <math id="M133"><msub><mrow><mi mathvariant="bold-italic">K</mi></mrow><mrow><mi mathvariant="normal">*</mi><mi mathvariant="normal">*</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=40436156&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=40436154&type= 4.99533367 3.72533321 表示协方差矩阵 <math id="M134"><mi mathvariant="bold-italic">K</mi><mo stretchy="false">(</mo><msup><mrow><mi mathvariant="bold-italic">x</mi></mrow><mrow><mi mathvariant="normal">*</mi></mrow></msup><mo>,</mo><msup><mrow><mi mathvariant="bold-italic">x</mi></mrow><mrow><mi mathvariant="normal">*</mi></mrow></msup><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=40436162&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=40436159&type= 15.40933418 3.80999994 ，描述测试点 <math id="M135"><msup><mrow><mi mathvariant="bold-italic">x</mi></mrow><mrow><mi mathvariant="normal">*</mi></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=40436166&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=40436164&type= 3.38666677 3.04800010 内部相关性。上述协方差矩阵 <math id="M136"><mi mathvariant="bold-italic">K</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=40436171&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=40436168&type= 3.13266683 2.79399991 中元素 <math id="M137"><msub><mrow><mi>k</mi></mrow><mrow><mi>i</mi><mi>j</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=40436176&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=40436172&type= 2.96333337 3.72533321 是由协方差函数确定的，即 <math id="M138"><msub><mrow><mi>k</mi></mrow><mrow><mi>i</mi><mi>j</mi></mrow></msub><mo>=</mo><mi>k</mi><mo stretchy="false">(</mo><msub><mrow><mi mathvariant="bold-italic">x</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>,</mo><msub><mrow><mi mathvariant="bold-italic">x</mi></mrow><mrow><mi>j</mi></mrow></msub><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=40436181&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=40436178&type= 20.91266632 4.48733330 。根据贝叶斯公式及式（16）式（16）得： <math display="block" id="M139"><msup><mrow><mi>y</mi></mrow><mrow><mi mathvariant="normal">*</mi></mrow></msup><mo stretchy="false">|</mo><mi mathvariant="bold-italic">y</mi><mo>∼</mo><mi>N</mi><mo stretchy="false">(</mo><msub><mrow><mi>μ</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo>+</mo><msub><mrow><mi mathvariant="bold-italic">K</mi></mrow><mrow><mi mathvariant="normal">*</mi></mrow></msub><msup><mrow><mi mathvariant="bold-italic">K</mi></mrow><mrow><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msup><mo stretchy="false">(</mo><mi mathvariant="bold-italic">y</mi><mo>-</mo><msub><mrow><mi>μ</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo stretchy="false">)</mo><mo>,</mo><msub><mrow><mi mathvariant="bold-italic">K</mi></mrow><mrow><mi mathvariant="normal">*</mi><mi mathvariant="normal">*</mi></mrow></msub><mo>-</mo><msub><mrow><mi mathvariant="bold-italic">K</mi></mrow><mrow><mi mathvariant="normal">*</mi></mrow></msub><msup><mrow><mi mathvariant="bold-italic">K</mi></mrow><mrow><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msup><msubsup><mrow><mi mathvariant="bold-italic">K</mi></mrow><mrow><mi mathvariant="normal">*</mi></mrow><mrow><mi mathvariant="normal">T</mi></mrow></msubsup><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=40436185&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=40436184&type= 74.33734131 5.16466665 ，其中， <math id="M140"><mo stretchy="false">[</mo><msub><mrow><mi>μ</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo>+</mo><msub><mrow><mi mathvariant="bold-italic">K</mi></mrow><mrow><mi mathvariant="normal">*</mi></mrow></msub><msup><mrow><mi mathvariant="bold-italic">K</mi></mrow><mrow><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msup><mo stretchy="false">(</mo><mi mathvariant="bold-italic">y</mi><mo>-</mo><msub><mrow><mi>μ</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo stretchy="false">)</mo><mo stretchy="false">]</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=40436190&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=40436187&type= 36.23733139 5.07999992 和 <math id="M141"><mo stretchy="false">[</mo><msub><mrow><mi mathvariant="bold-italic">K</mi></mrow><mrow><mi mathvariant="normal">*</mi><mi mathvariant="normal">*</mi></mrow></msub><mo>-</mo><msub><mrow><mi mathvariant="bold-italic">K</mi></mrow><mrow><mi mathvariant="normal">*</mi></mrow></msub><msup><mrow><mi mathvariant="bold-italic">K</mi></mrow><mrow><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msup><msubsup><mrow><mi mathvariant="bold-italic">K</mi></mrow><mrow><mi mathvariant="normal">*</mi></mrow><mrow><mi mathvariant="normal">T</mi></mrow></msubsup><mo stretchy="false">]</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=40436194&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=40436192&type= 28.95599937 4.65666676 分别为预测均值 <math id="M142"><msup><mrow><mi>y</mi></mrow><mrow><mi mathvariant="normal">*</mi></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=40436303&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=40436301&type= 3.21733332 3.89466691 和预测方差 <math id="M143"><mi mathvariant="normal">v</mi><mi mathvariant="normal">a</mi><mi mathvariant="normal">r</mi><mo stretchy="false">(</mo><msup><mrow><mi>y</mi></mrow><mrow><mi mathvariant="normal">*</mi></mrow></msup><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=40436307&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=40436306&type= 12.02266598 4.23333359 。预测均值即为目标参数预测值，预测方差即为得到此预测值的概率评估。在本文训练数据统计特征下，高斯过程回归模型均值函数选择为零均值函数，核函数选择为Matérn协方差函数： <math display="block" id="M144"><mi>k</mi><mo stretchy="false">(</mo><mi mathvariant="bold-italic">x</mi><mo>,</mo><mi mathvariant="bold-italic">x</mi><mi mathvariant="normal">'</mi><mo stretchy="false">)</mo><mo>=</mo><msubsup><mrow><mi>σ</mi></mrow><mrow><mi>f</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msubsup><mfrac><mrow><msup><mrow><mn mathvariant="normal">2</mn></mrow><mrow><mn mathvariant="normal">1</mn><mo>-</mo><mi>ν</mi></mrow></msup></mrow><mrow><mi>Γ</mi><mo stretchy="false">(</mo><mi>ν</mi><mo stretchy="false">)</mo></mrow></mfrac><mo stretchy="false">(</mo><mroot><mrow><mn mathvariant="normal">2</mn><mi>ν</mi></mrow><mrow/></mroot><mfrac><mrow><mi>d</mi></mrow><mrow><mi>ρ</mi></mrow></mfrac><mo stretchy="false">)</mo><msub><mrow><mi>K</mi></mrow><mrow><mi>ν</mi></mrow></msub><mfenced separators="|"><mrow><mroot><mrow><mn mathvariant="normal">2</mn><mi>ν</mi></mrow><mrow/></mroot><mfrac><mrow><mi>d</mi></mrow><mrow><mi>ρ</mi></mrow></mfrac></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=40436209&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=40436206&type= 62.39933777 9.82133293 ，其中： <math id="M145"><mi>d</mi><mo>=</mo><msup><mrow><mo stretchy="false">(</mo><mi mathvariant="bold-italic">x</mi><mo>-</mo><mi mathvariant="bold-italic">x</mi><mi mathvariant="normal">'</mi><mo stretchy="false">)</mo></mrow><mrow><msup><mrow/><mrow><mi>T</mi></mrow></msup></mrow></msup><mo stretchy="false">(</mo><mi mathvariant="bold-italic">x</mi><mo>-</mo><mi mathvariant="bold-italic">x</mi><mi mathvariant="normal">'</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=40436215&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=40436212&type= 34.62866592 3.72533321 ， <math id="M146"><mfenced open="{" close="}" separators="|"><mrow><msub><mrow><mi>σ</mi></mrow><mrow><mi>f</mi></mrow></msub><mo>,</mo><mi>ρ</mi></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=40436219&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=40436217&type= 9.31333351 4.91066647 为超参数 <math id="M147"><mi mathvariant="bold-italic">θ</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=40436239&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=40436238&type= 2.03200006 2.79399991 。本文用最大似然估计法确定高斯过程核函数中的超参数 <math id="M148"><mi mathvariant="bold-italic">θ</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=40436239&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=40436238&type= 2.03200006 2.79399991 ，似然函数为： <math display="block" id="M149"><mi>p</mi><mo stretchy="false">(</mo><mi mathvariant="bold-italic">y</mi><mo stretchy="false">|</mo><mi mathvariant="bold-italic">x</mi><mo>,</mo><mi mathvariant="bold-italic">θ</mi><mo stretchy="false">)</mo><mo>=</mo><mo>∫</mo><mi>p</mi><mo stretchy="false">(</mo><mi mathvariant="bold-italic">y</mi><mo stretchy="false">|</mo><mi mathvariant="bold-italic">f</mi><mo>,</mo><mi mathvariant="bold-italic">x</mi><mo>,</mo><mi mathvariant="bold-italic">θ</mi><mo stretchy="false">)</mo><mi>p</mi><mo stretchy="false">(</mo><mi mathvariant="bold-italic">f</mi><mo stretchy="false">|</mo><mi mathvariant="bold-italic">x</mi><mo>,</mo><mi mathvariant="bold-italic">θ</mi><mo stretchy="false">)</mo><mi mathvariant="normal">d</mi><mi mathvariant="bold-italic">f</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=40436226&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=40436225&type= 57.91199875 6.60400009 . 取 <math id="M150"><mi mathvariant="bold-italic">f</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=40436229&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=40436227&type= 2.62466669 3.64066648 为高斯过程模型 <math id="M151"><mi mathvariant="bold-italic">f</mi><mo>∼</mo><mi>G</mi><mi>P</mi><mo stretchy="false">(</mo><mn mathvariant="normal">0</mn><mo>,</mo><mi mathvariant="normal">Σ</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=40436232&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=40436230&type= 20.99733353 3.97933316 ，则负对数边际似然为： <math display="block" id="M152"><mtable><mtr><mtd><mi>L</mi><mo>=</mo><mo>-</mo><mi mathvariant="normal">l</mi><mi mathvariant="normal">n</mi><mi>p</mi><mo stretchy="false">(</mo><mi mathvariant="bold-italic">y</mi><mo stretchy="false">|</mo><mi mathvariant="bold-italic">x</mi><mo>,</mo><mi mathvariant="bold-italic">θ</mi><mo stretchy="false">)</mo><mo>=</mo></mtd></mtr><mtr><mtd><mfrac><mrow><mn mathvariant="normal">1</mn></mrow><mrow><mn mathvariant="normal">2</mn></mrow></mfrac><mo stretchy="false">(</mo><msup><mrow><mi mathvariant="bold-italic">y</mi></mrow><mrow><mi mathvariant="normal">T</mi></mrow></msup><msup><mrow><mi mathvariant="normal">Σ</mi></mrow><mrow><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msup><mi mathvariant="bold-italic">y</mi><mo>+</mo><mi mathvariant="normal">l</mi><mi mathvariant="normal">n</mi><mo stretchy="false">|</mo><mi mathvariant="normal">Σ</mi><mo stretchy="false">|</mo><mo>+</mo><mi>n</mi><mi mathvariant="normal">l</mi><mi mathvariant="normal">n</mi><mn mathvariant="normal">2</mn><mi mathvariant="normal">π</mi><mo stretchy="false">)</mo></mtd></mtr></mtable></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=40436236&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=40436235&type= 45.55066681 13.63133430 . 超参数 <math id="M153"><mi mathvariant="bold-italic">θ</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=40436239&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=40436238&type= 2.03200006 2.79399991 的最优解通过 <math id="M154"><mi>L</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=40436241&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=40436240&type= 2.45533323 2.79399991 偏导获得： <math display="block" id="M155"><mfrac><mrow><mo>∂</mo><mi>L</mi></mrow><mrow><mo>∂</mo><mi mathvariant="bold-italic">θ</mi></mrow></mfrac><mo>=</mo><mfrac><mrow><mn mathvariant="normal">1</mn></mrow><mrow><mn mathvariant="normal">2</mn></mrow></mfrac><mfenced separators="|"><mrow><mi mathvariant="normal">t</mi><mi mathvariant="normal">r</mi><mfenced separators="|"><mrow><msup><mrow><mi mathvariant="normal">Σ</mi></mrow><mrow><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msup><mfrac><mrow><mo>∂</mo><mi mathvariant="normal">Σ</mi></mrow><mrow><mo>∂</mo><mi mathvariant="bold-italic">θ</mi></mrow></mfrac></mrow></mfenced><mo>-</mo><msup><mrow><mi mathvariant="bold-italic">y</mi></mrow><mrow><mi>T</mi></mrow></msup><msup><mrow><mi mathvariant="normal">Σ</mi></mrow><mrow><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msup><mfrac><mrow><mo>∂</mo><mi mathvariant="normal">Σ</mi></mrow><mrow><mo>∂</mo><mi mathvariant="bold-italic">θ</mi></mrow></mfrac><msup><mrow><mi mathvariant="normal">Σ</mi></mrow><mrow><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msup><mi mathvariant="bold-italic">y</mi></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=40436243&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=40436242&type= 60.62133789 9.82133293 . 在上述设计的高斯过程算法基础下用训练数据进行在线训练，节省了一般机器学习算法的超参数调试过程，随预测值同时输出的预测方差经预处理后作为预测效果的评估，可实时监测预测状态以及异常数据。高斯过程回归预测模型精度主要受其核函数影响，为了验证本文中核函数相较于其他常用核函数在此数据类型和规模下具有更高的预测精度，随机选择10组测试数据分别在以平方指数协方差函数、有理二次协方差函数以及本文应用的Matérn协方差函数为核函数的高斯过程模型下进行测试。平方指数协方差函数表达式为： <math display="block" id="M156"><mi>k</mi><mo stretchy="false">(</mo><mi mathvariant="bold-italic">x</mi><mo>,</mo><mi mathvariant="bold-italic">x</mi><mi mathvariant="normal">'</mi><mo stretchy="false">)</mo><mo>=</mo><msubsup><mrow><mi>σ</mi></mrow><mrow><mi>f</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msubsup><mi mathvariant="normal">e</mi><mi mathvariant="normal">x</mi><mi mathvariant="normal">p</mi><mfenced separators="|"><mrow><mo>-</mo><mfrac><mrow><mn mathvariant="normal">1</mn></mrow><mrow><mn mathvariant="normal">2</mn><msup><mrow><mi>ι</mi></mrow><mrow><mn 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系统模型参数真实值与预测值对比图如图 5 5 （a）~ 5 5 （c）所示，实验结果显示，以平方指数协方差函数、有理二次协方差函数为核心的高斯过程回归算法均有较高的预测精度，但是在系统参数失配程度较高的情况下，会产生较大的预测误差，本文采用的Matérn核函数，在各种失配状态下均表现高的预测精度。 4.3　基于高斯过程参数辨识的模型预测控制 2 基于高斯过程的永磁同步电机模型预测控制系统框图如图6 图6 所示。系统通过PI控制器进行速度控制，并产生电流参考值。预测电流控制器将对该电流进行跟踪，选择能使代价函数最小化的电压矢量，将其应用于整个采样间隔。在系统运行的过程中进行电流检测，电流检测数据经预处理模块得到电流波动度 B 和电流偏移度 P ，此两项作为高斯过程回归参数优化模块的输入进行系统参数的预测，预测结果将更新预测电流控制器中已失配的参数，使系统恢复未失配状态。系统中高斯过程回归模型是不断更新的，随着训练数据的累积而进化。在不同速度参考值或不同负载条件下，稳定后的 <math id="M158"><mi>q</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=40436255&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=40436254&type= 1.86266661 3.64066648 轴电流表现量化值会有所不同，因此在不同实验条件下的实验数据将会训练新的高斯过程回归模型。高斯过程的训练数据直接由目标算法FCS-MPC获取，因此参数辨识过程对FCS-MPC不会造成类似于其他入侵式辨识算法带来的不利影响，同时，训练得到的回归模型将用来优化FCS-MPC算法，实现了控制与优化的闭环。 5 实验分析 1 实验验证FCS-GPMPC控制策略的有效性，搭建了如图7 图7 所示的实验平台，DSP-TMS320F28335和FPGA-EP3C40F324组成伺服控制器。PMSM参数如表1 表1 所示。参数数值额定功率 <math id="M159"><msub><mrow><mi>P</mi></mrow><mrow><mi>r</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=40436260&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=40436259&type= 3.04800010 3.21733332 5.5 kW 额定转速 <math id="M160"><msub><mrow><mi>S</mi></mrow><mrow><mi>r</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=40436262&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=40436261&type= 2.70933342 3.21733332 250 r/min 额定电流 <math id="M161"><msub><mrow><mi>I</mi></mrow><mrow><mi>r</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=40436264&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=40436263&type= 2.03200006 3.21733332 7 A 磁极对数 <math id="M162"><msub><mrow><mi>P</mi></mrow><mrow><mi>n</mi></mrow></msub></math> 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http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=40436274&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=40436273&type= 1.60866666 3.13266683 轴电感 <math id="M167"><msub><mrow><mi>L</mi></mrow><mrow><mi>s</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=40436276&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=40436275&type= 2.87866688 3.21733332 0.006 45 H 永磁体磁链 <math id="M168"><msub><mrow><mi>ψ</mi></mrow><mrow><mi>m</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=40436278&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=40436277&type= 3.30200005 3.80999994 0.416 81 Wb 首先进行高斯过程回归模型的获取，在给定速度50 <math id="M169"><mi mathvariant="normal">r</mi><mo>/</mo><mi mathvariant="normal">m</mi><mi mathvariant="normal">i</mi><mi mathvariant="normal">n</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=40436282&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=40436281&type= 9.39799976 3.30200005 、零负载条件下，电感参数标称值在以实际参数值为基准 - 25%~25%的范围波动，利用电流波动度等实时数据进行高斯过程回归模型的训练，得到图8 图8 （a）所示拟合曲面，通过电流波动度和电流偏移度即获得电感参数的失配信息。 1;2;3; 为验证回归模型在不同速度条件下的自适应性能，在给定速度100 r/min、零负载条件下进行回归模型自适应获取，得到图8 图8 （b）所示的拟合曲面。为验证高斯过程回归模型在不同负载条件下的自适应性能，在给定速度100 r/min，2 N·m负载条件下进行回归模型自适应获取，得到图8 图8 （c）所示的拟合曲面。实验证明，电机运行条件不同，电流表现和参数失配之间的回归模型会有不同，本文高斯过程方法会在电机不同运行状态下自适应更新回归模型。基于本文方法进行参数辨识的实验，如图9 图9 所示（彩图见期刊电子版），圆点表示该时刻实际参数值，红色曲线表示预测参数值，图中显示，预测参数曲线准确跟踪实际参数值。测试数据集显示，在本文数据类型和规模下，预测误差均控制在2%以内，均方根误差RMSE为0.002 1，已到达千分位级， <math id="M170"><msup><mrow><mi>R</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=40436296&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=40436295&type= 3.89466691 3.04800010 达到0.99，接近于1。高斯过程回归模型在输出预测均值 <math id="M171"><msup><mrow><mi>y</mi></mrow><mrow><mi mathvariant="normal">*</mi></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=40436303&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=40436301&type= 3.21733332 3.89466691 的同时会输出预测方差 <math id="M172"><mi mathvariant="normal">v</mi><mi mathvariant="normal">a</mi><mi mathvariant="normal">r</mi><mo stretchy="false">(</mo><msup><mrow><mi>y</mi></mrow><mrow><mi mathvariant="normal">*</mi></mrow></msup><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=40436307&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=40436306&type= 12.02266598 4.23333359 ，因此置信度为95%的置信区间为： <math id="M173"><mo stretchy="false">[</mo><msup><mrow><mi>y</mi></mrow><mrow><mi mathvariant="normal">*</mi></mrow></msup><mo>-</mo><mn mathvariant="normal">2</mn><mroot><mrow><mi mathvariant="normal">v</mi><mi mathvariant="normal">a</mi><mi mathvariant="normal">r</mi><mo stretchy="false">(</mo><msup><mrow><mi>y</mi></mrow><mrow><mi mathvariant="normal">*</mi></mrow></msup><mo stretchy="false">)</mo></mrow><mrow/></mroot><mo>,</mo><msup><mrow><mi>y</mi></mrow><mrow><mi mathvariant="normal">*</mi></mrow></msup><mo>+</mo><mn mathvariant="normal">2</mn><mroot><mrow><mi mathvariant="normal">v</mi><mi mathvariant="normal">a</mi><mi mathvariant="normal">r</mi><mo stretchy="false">(</mo><msup><mrow><mi>y</mi></mrow><mrow><mi mathvariant="normal">*</mi></mrow></msup><mo stretchy="false">)</mo></mrow><mrow/></mroot><mo stretchy="false">]</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=40436311&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=40436310&type= 57.74266815 5.24933338 。图9 图9 中灰色阴影部分即为预测值的置信区间，可用于预测效果的评估以及异常数据的检测，预测误差与置信区间的波动成正相关。为验证置信区间对异常预测值的检测能力，在系统运行的过程中加入异常值，如图10 图10 所示阴影部分即置信区间迅速扩大，这代表此时刻预测方差出现较大波动，即检测到本组数据中出现异常值或者出现较大预测误差。因此，置信区间的大小可用于预测效果的评估。为进一步验证本文控制策略的稳态性能，在2 N·m负载、100 r/min速度参考值条件下，对比本文控制策略与传统FCS-MPC在参数失配情况下相电流和 q 轴电流状态。图11 图11 （a）、图11 图11 （b）为传统FCS-MPC策略下a相电流波形及其快速傅里叶分析（Fast Fourier Transform， FFT），图11 图11 （c）、图11 图11 （d）为本文控制策略下参数优化后a相电流波形及其快速傅里叶分析。相较于传统FCS-MPC，本文控制策略下的相电流总谐波失真（Total Harmonic Distortion， THD）由4.38%降为3.74%。 1,2;3,4; 考虑电感参数失配程度的极限情况，参数标称值在以实际参数值为基准 - 50%~50%的范围随机波动。传统FCS-MPC算法 q 轴电流响应和基于所提方法参数优化后的 q 轴电流响应分别如图12 图12 （a）、图12 图12 （b）所示。相较于传统FCS-MPC，所提方法 q 轴电流波动度降低30.5%，电流平均偏移度降低19.6%。实验验证FCS-GPMPC控制算法对电流阶跃变化的动态跟踪性能，如图 13 13 ~ 14 14 所示， q 轴参考电流在5 s时由0 A上升到1 A，传统的FCS-MPC算法的响应时间与FCS-GPMPC算法一致，但由于电感参数的不匹配，导致对未来电流的预测不准确，降低了电流跟踪性能。FCS-GPMPC算法及时调整电感参数标称值，因此其对参考电流的阶跃变化有更好的动态响应。 6 结论 1 本文设计了基于高斯过程参数优化的模型预测控制器，高斯过程参数辩识方法简化了其他机器学习算法超参数的调试过程，其波动参数的精确估计提高了模型预测控制器的控制性能。高斯过程回归算法还实时提供了预测结果的置信区间，通过此置信区间，可在线进行系统算法的监测及优化。基于高斯过程参数优化的模型预测控制可实时优化预测模型，对比于传统的FCS-MPC，所提方法对电流的稳态性能和动态性能均有提升。实验结果表明：基于高斯过程回归的参数辩识方法具有较高的精度，在电感参数失配程度不断变化的情况下，实现参数的精确跟踪，预测误差控制在2%，相较于传统FCS-MPC，本文控制策略下的相电流总谐波失真也有所降低，电感失配程度随机变化下 q 轴电流波动度降低30.5%，电流平均偏移度降低19.6%，另外，在参数失配情况下，所提算法对 q 轴参考电流的阶跃变化有更好的动态响应。参考文献邓永停，刘军，李洪文，等 . 基于分段弧形永磁同步电机的4m望远镜控制系统［J］. 光学精密工程， 2020 ， 28 （ 3 ）： 591 - 600 . doi: 10.3788/ope.20202803.0591 http://dx.doi.org/10.3788/ope.20202803.0591 基于分段弧形永磁同步电机的4m望远镜控制系统 DENG Y T ， LIU J ， LI H W ， et al . Control system of 4 meters telescope based on segmented permanent magnet arc synchronous motor ［J］. Opt. Precision Eng. ， 2020 ， 28 （ 3 ）： 591 - 600 . （in Chinese） . doi: 10.3788/ope.20202803.0591 http://dx.doi.org/10.3788/ope.20202803.0591 Control system of 4 meters telescope based on segmented permanent magnet arc synchronous motor 邓永停，李洪文，陈涛 . 2 m级望远镜跟踪架控制系统动态性能分析［J］. 光学精密工程， 2018 ， 26 （ 3 ）： 654 - 661 . doi: 10.3788/ope.20182603.0654 http://dx.doi.org/10.3788/ope.20182603.0654 2 m级望远镜跟踪架控制系统动态性能分析 DENG Y T ， LI H W ， CHEN T . Dynamic analysis of two meters telescope mount control system ［J］. Opt. Precision Eng. ， 2018 ， 26 （ 3 ）： 654 - 661 . （in Chinese） . doi: 10.3788/ope.20182603.0654 http://dx.doi.org/10.3788/ope.20182603.0654 Dynamic analysis of two meters telescope mount control system SARSEMBAYEV B ， SULEIMENOV K ， DO T D . High order disturbance observer based PI-PI control system with tracking anti-windup technique for improvement of transient performance of PMSM ［J］. 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