函数模型误差弹性自适应滤波修正

刘慧敏; 王振杰; 单瑞; 陆凯

doi:10.13203/j.whugis20200469

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函数模型误差弹性自适应滤波修正

BDS/GNSS精密数据处理 | 更新时间：2023-06-29

- 函数模型误差弹性自适应滤波修正
- Resilient Adaptive Filtering Based on Model Modification and Underwater Application
- 武汉大学学报（信息科学版）
- 作者机构：
  
  1.中国地质调查局青岛海洋地质研究所，山东青岛，266071
  2.青岛海洋科学与技术试点国家实验室海洋矿产资源评价与探测技术功能实验室，山东青岛，266071
  3.中国石油大学（华东）海洋空间与信息学院，山东青岛，266580
- 作者简介：
  
  刘慧敏，博士，助理研究员，主要从事海洋大地测量数据处理理论研究。pcliuhn@foxmail.com
  
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  单瑞，博士，助理研究员。shanrui416@163.com
  
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- 基金信息：
  
  中国地质调查局项目(DD20191003);国家自然科学基金(41406115);青岛市市南区科技发展资金项目(2016-3-015-ZH)
- DOI：10.13203/j.whugis20200469
  中图分类号： P229
- 纸质出版日期：2023-01-05，
  
  收稿日期：2022-03-05，
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引用本文
刘慧敏,王振杰,单瑞，等.函数模型误差弹性自适应滤波修正[J].武汉大学学报（信息科学版）,2023,48(1):146-154.

LIU Huimin,WANG Zhenjie,Shan Rui,et al.Resilient Adaptive Filtering Based on Model Modification and Underwater Application[J].Geomatics and Information Science of Wuhan University,2023,48(1):146-154.
刘慧敏,王振杰,单瑞，等.函数模型误差弹性自适应滤波修正[J].武汉大学学报（信息科学版）,2023,48(1):146-154. DOI：10.13203/j.whugis20200469

LIU Huimin,WANG Zhenjie,Shan Rui,et al.Resilient Adaptive Filtering Based on Model Modification and Underwater Application[J].Geomatics and Information Science of Wuhan University,2023,48(1):146-154. DOI：10.13203/j.whugis20200469

函数模型误差弹性自适应滤波修正 Resilient Adaptive Filtering Based on Model Modification and Underwater Application 1 2 first-author 刘慧敏，博士，助理研究员，主要从事海洋大地测量数据处理理论研究。pcliuhn@foxmail.com 刘慧敏，博士，助理研究员，主要从事海洋大地测量数据处理理论研究。pcliuhn@foxmail.com pcliuhn@foxmail.com 3 1 2 corresp 单瑞，博士，助理研究员。shanrui416@163.com 单瑞，博士，助理研究员。shanrui416@163.com shanrui416@163.com 1 2 中国地质调查局青岛海洋地质研究所，山东青岛，266071 中国地质调查局青岛海洋地质研究所，山东青岛，266071 China Geological Survey Qingdao Institute of Marine Geology, Qingdao 266071, China China Geological Survey Qingdao Institute of Marine Geology, Qingdao 266071, China 青岛海洋科学与技术试点国家实验室海洋矿产资源评价与探测技术功能实验室，山东青岛，266071 青岛海洋科学与技术试点国家实验室海洋矿产资源评价与探测技术功能实验室，山东青岛，266071 Laboratory for Marine Mineral Resources, Qingdao National Laboratory for Marine Science and Technology, Qingdao 266071, China Laboratory for Marine Mineral Resources, Qingdao National Laboratory for Marine Science and Technology, Qingdao 266071, China 中国石油大学（华东）海洋空间与信息学院，山东青岛，266580 中国石油大学（华东）海洋空间与信息学院，山东青岛，266580 China University of Petroleum (East China), Qingdao 266580, China China University of Petroleum (East China), Qingdao 266580, China 多源导航信息融合过程中，观测模型和动力学模型随时间和空间变化复杂，高精度的动态载体导航与定位需要观测模型和动力学模型具有准实时或实时修正的能力。针对包含观测模型误差以及动力学模型误差的滤波系统，提出了一种基于信息滤波的弹性自适应滤波算法。所提算法以不含模型误差的标准信息滤波器为主滤波器，分别构造了观测函数模型及动力学函数模型误差补偿滤波器，对两类模型误差进行补偿。所提方法强调模型补偿项的弹性自适应估计和状态参数的弹性组合，提高了时变模型误差估计的稳定性。半物理仿真实验结果表明，基于函数模型补偿的弹性自适应滤波算法可以有效地估计观测模型和载体动力学模型误差项，水下拖体的三维位置偏差在0.2 m以内，两类模型误差的影响基本消除，明显提高了载体动态参数的估计精度。 Objectives 2 Multi-sensor information integration provides important support for positioning, navigation and timing services in extreme environments. High-precision dynamic vehicle navigation and localization requires observation models and dynamic models with the capability of quasi-real-time or real-time correction. In the process of multi-source navigation information fusion, the observed model and noise of navigation sensors vary in time and space, so it is of great theoretical importance and practical value to construct accurate functional and stochastic models. Methods 2 A resilient adaptive filtering algorithm based on information filtering is proposed for filtering systems containing both observational and dynamic model errors. In the proposed algorithm, the standard information filter without model errors is used as the primary filter, and error compensation filters for the observed and dynamic function models are constructed to compensate the errors of the two types of models, respectively. Results 2 The proposed approach emphasizes the elastic adaptive estimation of the model compensation term and the elastic combination of the state parameters, which improves the stability of the error estimates for time-varying models. Semi physical simulation experimental results show that the elasticity of the adaptive filter algorithm based on function model compensation can effectively estimate the observation model error term and vehicle dynamics model, with the error of three-dimensional position of the underwater towed body deviation less than 0.2 m. Conclusions 2 It can fundamentally eliminate the influence of two kinds of model errors and clearly improve the estimation accuracy of vehicle dynamic parameters. 观测模型误差动力学模型误差信息滤波弹性自适应滤波模型补偿 observation model bias dynamic model bias information filtering resilient adaptive filtering model compensation 多传感器的信息集成和数据融合为极端环境下的定位、导航和授时（positioning，navigation and timing，PNT）服务提供了一种重要思路。在多源导航信息融合过程中，导航传感器的观测模型和噪声随时间和空间变化复杂，构建准确的函数模型和随机模型具有重要的理论意义和实用价值。为了提升信息融合后的精度和稳定性，许多学者从数据处理策略或模型精化角度不断尝试改进。而弹性PNT以函数模型弹性调整和随机模型弹性优化为手段，融合生成适应多种复杂环境的PNT信息，因而具备高可用性、高连续性和高可靠性，成为近年来的研究热点［ 1 1 - 2 2 1-2 ］。 Kalman滤波算法是一种基于理想的高斯噪声假设的最优估计方法，非高斯系统噪声和观测噪声会造成滤波的不稳定，严重时会造成滤波的发散［ 3 3 - 4 4 3-4 ］。处于不同时空下的动态载体，其导航传感器的工作模式和观测精度随时空变化而变化，其观测模型和载体动力学模型都应具有准实时或实时修正的能力。因此，在实际滤波处理中，在对不同导航传感器误差性质的认识基础上，应尽可能利用多源导航传感器的冗余观测信息，附加函数模型误差的拟合模型补偿或者通过方差膨胀技术进行随机模型补偿。在函数模型补偿方面，经过数十年不断探索，大量文献直接将与时间相关的模型误差通过一阶或二阶自相关模型进行建模，并在Kalman滤波中使用增广参数向量与状态参数一同估计［ 5 5 - 8 8 5-8 ］。这种增广形式操作简便，但容易简化模型误差的随机属性，忽略模型误差较强的时变特性。此外，也有学者通过移动窗口拟合系统误差，使用历史新息向量估计函数模型的修正量，并对状态参数估值进行修正并补偿函数模型误差［ 9 9 ］。但利用移动窗口补偿模型误差同样容易造成模型误差的过度平均，忽略模型误差较强的时变特性。此外，在随机模型补偿方面，有学者通过构造4种动力学模型误差学习统计量，对动力学模型误差进行弹性调整［ 10 10 - 11 11 10-11 ］。但当观测信息太少或不可靠时，求得的自适应因子易出现偏差，容易造成部分滤波解异常。本文基于信息滤波理论和抗差自适应滤波理论，提出一种基于函数模型的弹性自适应滤波构建方法。所提滤波器由不含模型误差的状态参数滤波、观测模型误差和动力学模型误差修正滤波组成。其弹性组合强调模型补偿项的弹性自适应估计以及与状态参数的独立性和交互性，强调特殊工作环境下的自适应模型选择。结合某海域水下拖曳系统的水下导航数据进行半物理仿真，验证了所提方法的有效性。 1　函数模型弹性自适应滤波原理 1 含动力学和观测模型误差的系统为： <math display="block" id="M1"><mfenced open="{" close="" separators="|"><mrow><mtable columnalign="left"><mtr><mtd><msub><mrow><mi mathvariant="bold-italic">x</mi></mrow><mrow><mi>k</mi></mrow></msub><mo>=</mo><msub><mrow><mi mathvariant="bold-italic">Φ</mi></mrow><mrow><mi>k</mi></mrow></msub><msub><mrow><mi mathvariant="bold-italic">x</mi></mrow><mrow><mi>k</mi><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msub><mo>+</mo><msub><mrow><mi 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columnalign="left"><mtr><mtd><msub><mrow><mi mathvariant="bold-italic">Φ</mi></mrow><mrow><mi>k</mi></mrow></msub></mtd><mtd><msub><mrow><mi mathvariant="bold-italic">Γ</mi></mrow><mrow><mi>k</mi></mrow></msub></mtd><mtd/></mtr><mtr><mtd/><mtd><msub><mrow><mi mathvariant="bold-italic">I</mi></mrow><mrow><mi>p</mi><mo>×</mo><mi>p</mi></mrow></msub></mtd><mtd/></mtr><mtr><mtd/><mtd/><mtd><msub><mrow><mi mathvariant="bold-italic">I</mi></mrow><mrow><mi>q</mi><mo>×</mo><mi>q</mi></mrow></msub></mtd></mtr></mtable></mrow></mfenced><mfenced open="[" close="]" separators="|"><mrow><mtable columnalign="left"><mtr><mtd><msub><mrow><mi mathvariant="bold-italic">x</mi></mrow><mrow><mi>k</mi><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msub></mtd></mtr><mtr><mtd><msub><mrow><mi mathvariant="bold-italic">θ</mi></mrow><mrow><mi>k</mi><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msub></mtd></mtr><mtr><mtd><msub><mrow><mi mathvariant="bold-italic">s</mi></mrow><mrow><mi>k</mi><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msub></mtd></mtr></mtable></mrow></mfenced><mo>+</mo><mfenced open="[" close="]" separators="|"><mrow><mtable columnalign="left"><mtr><mtd><msub><mrow><mi mathvariant="bold-italic">w</mi></mrow><mrow><mi>k</mi></mrow></msub></mtd></mtr><mtr><mtd><msubsup><mrow><mi mathvariant="bold-italic">ε</mi></mrow><mrow><mi>k</mi></mrow><mrow><mi>θ</mi></mrow></msubsup></mtd></mtr><mtr><mtd><msubsup><mrow><mi mathvariant="bold-italic">ε</mi></mrow><mrow><mi>k</mi></mrow><mrow><mi>s</mi></mrow></msubsup></mtd></mtr></mtable></mrow></mfenced></mtd></mtr><mtr><mtd><msub><mrow><mi mathvariant="bold-italic">y</mi></mrow><mrow><mi>k</mi></mrow></msub><mo>=</mo><mo stretchy="false">[</mo><mtable columnalign="left"><mtr><mtd><msub><mrow><mi mathvariant="bold-italic">C</mi></mrow><mrow><mi>k</mi></mrow></msub></mtd><mtd><mn mathvariant="normal">0</mn></mtd><mtd><msub><mrow><mi mathvariant="bold-italic">B</mi></mrow><mrow><mi>k</mi></mrow></msub></mtd></mtr></mtable><mo stretchy="false">]</mo><mfenced open="[" close="]" separators="|"><mrow><mtable columnalign="left"><mtr><mtd><msub><mrow><mi mathvariant="bold-italic">x</mi></mrow><mrow><mi>k</mi></mrow></msub></mtd></mtr><mtr><mtd><msub><mrow><mi mathvariant="bold-italic">θ</mi></mrow><mrow><mi>k</mi></mrow></msub></mtd></mtr><mtr><mtd><msub><mrow><mi mathvariant="bold-italic">s</mi></mrow><mrow><mi>k</mi></mrow></msub></mtd></mtr></mtable></mrow></mfenced><mo>+</mo><msub><mrow><mi mathvariant="bold-italic">v</mi></mrow><mrow><mi>k</mi></mrow></msub></mtd></mtr></mtable></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45116585&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45116582&type= 62.14533615 29.97200012 式中， <math id="M17"><mi mathvariant="bold-italic">I</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45116589&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45116587&type= 1.86266661 2.79399991 表示单位阵； <math id="M18"><msubsup><mrow><mi mathvariant="bold-italic">ε</mi></mrow><mrow><mi>k</mi></mrow><mrow><mi>θ</mi></mrow></msubsup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45116608&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45116600&type= 2.70933342 3.97933316 和 <math id="M19"><msubsup><mrow><mi mathvariant="bold-italic">ε</mi></mrow><mrow><mi>k</mi></mrow><mrow><mi>s</mi></mrow></msubsup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45116611&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45116606&type= 2.62466669 3.97933316 分别为动力学模型和观测模型随机误差。增广后的式（2）式（2）可用经典Kalman滤波直接进行估计。当模型误差具有较强时变性时，增广Kalman滤波简化了模型误差的函数模型，往往滤波效果不佳。当估值 <math id="M20"><msubsup><mrow><mover accent="true"><mi mathvariant="bold-italic">x</mi><mo>^</mo></mover></mrow><mrow><mi>k</mi><mo>-</mo><mn mathvariant="normal">1</mn></mrow><mrow/></msubsup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45116628&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45116614&type= 6.85799980 4.06400013 、 <math id="M21"><msubsup><mrow><mi mathvariant="bold-italic">P</mi></mrow><mrow><mi>k</mi><mo>-</mo><mn mathvariant="normal">1</mn></mrow><mrow/></msubsup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45116632&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45116619&type= 7.11199999 3.97933316 、 <math id="M22"><msub><mrow><mover accent="true"><mi mathvariant="bold-italic">θ</mi><mo>^</mo></mover></mrow><mrow><mi>k</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45116641&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45116639&type= 2.96333337 4.65666676 和 <math id="M23"><msub><mrow><mover accent="true"><mi mathvariant="bold-italic">s</mi><mo>^</mo></mover></mrow><mrow><mi>k</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45116647&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45116643&type= 2.45533323 3.80999994 已知时，式（2）式（2）的广义最小二乘解为： <math display="block" id="M24"><msubsup><mrow><mover accent="true"><mi mathvariant="bold-italic">x</mi><mo>^</mo></mover></mrow><mrow><mi>k</mi></mrow><mrow/></msubsup><mo>=</mo><msubsup><mrow><mi mathvariant="bold-italic">N</mi></mrow><mrow><mi>k</mi></mrow><mrow><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msubsup><mo stretchy="false">(</mo><msubsup><mrow><mi mathvariant="bold-italic">C</mi></mrow><mrow><mi>k</mi></mrow><mrow><mi mathvariant="normal">T</mi></mrow></msubsup><msubsup><mrow><mi mathvariant="bold-italic">R</mi></mrow><mrow><mi>k</mi></mrow><mrow><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msubsup><msub><mrow><mi mathvariant="bold-italic">y</mi></mrow><mrow><mi>k</mi></mrow></msub><mo>-</mo><msubsup><mrow><mi mathvariant="bold-italic">C</mi></mrow><mrow><mi>k</mi></mrow><mrow><mi mathvariant="normal">T</mi></mrow></msubsup><msubsup><mrow><mi mathvariant="bold-italic">R</mi></mrow><mrow><mi>k</mi></mrow><mrow><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msubsup><msub><mrow><mi mathvariant="bold-italic">B</mi></mrow><mrow><mi>k</mi></mrow></msub><msub><mrow><mover accent="true"><mi mathvariant="bold-italic">s</mi><mo>^</mo></mover></mrow><mrow><mi>k</mi></mrow></msub><mo>+</mo><msubsup><mrow><mi mathvariant="bold-italic">P</mi></mrow><mrow><mi>k</mi><mo stretchy="false">|</mo><mi>k</mi><mo>-</mo><mn mathvariant="normal">1</mn></mrow><mrow><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msubsup><msubsup><mrow><mover accent="true"><mi mathvariant="bold-italic">x</mi><mo>^</mo></mover></mrow><mrow><mi>k</mi><mo stretchy="false">|</mo><mi>k</mi><mo>-</mo><mn mathvariant="normal">1</mn></mrow><mrow/></msubsup><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45116651&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45116656&type= 76.19999695 5.07999992 <math display="block" id="M25"><mfenced open="{" close="" separators="|"><mrow><mtable columnalign="left"><mtr><mtd><msubsup><mrow><mover accent="true"><mi mathvariant="bold-italic">x</mi><mo>^</mo></mover></mrow><mrow><mi>k</mi><mo stretchy="false">|</mo><mi>k</mi><mo>-</mo><mn mathvariant="normal">1</mn></mrow><mrow/></msubsup><mo>=</mo><msub><mrow><mi mathvariant="bold-italic">Φ</mi></mrow><mrow><mi>k</mi></mrow></msub><msubsup><mrow><mover accent="true"><mi mathvariant="bold-italic">x</mi><mo>^</mo></mover></mrow><mrow><mi>k</mi><mo>-</mo><mn mathvariant="normal">1</mn></mrow><mrow/></msubsup><mo>+</mo><msub><mrow><mi mathvariant="bold-italic">Γ</mi></mrow><mrow><mi>k</mi></mrow></msub><msub><mrow><mover accent="true"><mi mathvariant="bold-italic">θ</mi><mo>^</mo></mover></mrow><mrow><mi>k</mi></mrow></msub></mtd></mtr><mtr><mtd><msubsup><mrow><mi mathvariant="bold-italic">P</mi></mrow><mrow><mi>k</mi><mo stretchy="false">|</mo><mi>k</mi><mo>-</mo><mn mathvariant="normal">1</mn></mrow><mrow/></msubsup><mo>=</mo><msub><mrow><mi mathvariant="bold-italic">Φ</mi></mrow><mrow><mi>k</mi></mrow></msub><msubsup><mrow><mi mathvariant="bold-italic">P</mi></mrow><mrow><mi>k</mi><mo>-</mo><mn mathvariant="normal">1</mn></mrow><mrow/></msubsup><msubsup><mrow><mi mathvariant="bold-italic">Φ</mi></mrow><mrow><mi>k</mi></mrow><mrow><mi mathvariant="normal">T</mi></mrow></msubsup><mo>+</mo><mi mathvariant="bold-italic">Q</mi><mmultiscripts><mrow/><mprescripts/><mrow><mi>k</mi></mrow><mrow/></mmultiscripts></mtd></mtr><mtr><mtd><msub><mrow><mi mathvariant="bold-italic">N</mi></mrow><mrow><mi>k</mi></mrow></msub><mo>=</mo><mo stretchy="false">(</mo><msubsup><mrow><mi mathvariant="bold-italic">C</mi></mrow><mrow><mi>k</mi></mrow><mrow><mi mathvariant="normal">T</mi></mrow></msubsup><msubsup><mrow><mi mathvariant="bold-italic">R</mi></mrow><mrow><mi>k</mi></mrow><mrow><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msubsup><msub><mrow><mi mathvariant="bold-italic">C</mi></mrow><mrow><mi>k</mi></mrow></msub><mo>+</mo><msubsup><mrow><mi mathvariant="bold-italic">P</mi></mrow><mrow><mi>k</mi><mo stretchy="false">|</mo><mi>k</mi><mo>-</mo><mn mathvariant="normal">1</mn></mrow><mrow><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msubsup><mo stretchy="false">)</mo></mtd></mtr></mtable></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45116664&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45116668&type= 41.40200043 16.67933464 式中， <math id="M26"><msubsup><mrow><mover accent="true"><mi mathvariant="bold-italic">x</mi><mo>^</mo></mover></mrow><mrow><mi>k</mi><mo stretchy="false">|</mo><mi>k</mi><mo>-</mo><mn mathvariant="normal">1</mn></mrow><mrow/></msubsup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45116684&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45116666&type= 8.04333401 4.06400013 为 <math id="M27"><msubsup><mrow><mover accent="true"><mi mathvariant="bold-italic">x</mi><mo>^</mo></mover></mrow><mrow><mi>k</mi><mo>-</mo><mn mathvariant="normal">1</mn></mrow><mrow/></msubsup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45116677&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45116688&type= 6.85799980 4.06400013 和 <math id="M28"><msub><mrow><mover accent="true"><mi mathvariant="bold-italic">θ</mi><mo>^</mo></mover></mrow><mrow><mi>k</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45116726&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45116710&type= 2.96333337 4.65666676 经动力学模型预测的线性组合； <math id="M29"><msubsup><mrow><mi mathvariant="bold-italic">P</mi></mrow><mrow><mi>k</mi><mo stretchy="false">|</mo><mi>k</mi><mo>-</mo><mn mathvariant="normal">1</mn></mrow><mrow/></msubsup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45116713&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45116694&type= 8.38199997 3.97933316 为预测状态协方差阵。 <math id="M30"><msubsup><mrow><mover accent="true"><mi mathvariant="bold-italic">x</mi><mo>^</mo></mover></mrow><mrow><mi>k</mi></mrow><mrow/></msubsup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45116718&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45116716&type= 3.38666677 4.06400013 由 <math id="M31"><msubsup><mrow><mover accent="true"><mi mathvariant="bold-italic">x</mi><mo>^</mo></mover></mrow><mrow><mi>k</mi><mo>-</mo><mn mathvariant="normal">1</mn></mrow><mrow/></msubsup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45116708&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45116707&type= 6.85799980 4.06400013 、 <math id="M32"><msub><mrow><mover accent="true"><mi mathvariant="bold-italic">θ</mi><mo>^</mo></mover></mrow><mrow><mi>k</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45116726&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45116710&type= 2.96333337 4.65666676 和 <math id="M33"><msub><mrow><mover accent="true"><mi mathvariant="bold-italic">s</mi><mo>^</mo></mover></mrow><mrow><mi>k</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45116746&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45116727&type= 2.45533323 3.80999994 共同确定，且有： <math display="block" id="M34"><msubsup><mrow><mi mathvariant="bold-italic">P</mi></mrow><mrow><mi>k</mi></mrow><mrow/></msubsup><mo>=</mo><msubsup><mrow><mi mathvariant="bold-italic">N</mi></mrow><mrow><mi>k</mi></mrow><mrow><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msubsup><mo>=</mo><mo stretchy="false">(</mo><msubsup><mrow><mi mathvariant="bold-italic">C</mi></mrow><mrow><mi>k</mi></mrow><mrow><mi mathvariant="normal">T</mi></mrow></msubsup><msubsup><mrow><mi mathvariant="bold-italic">R</mi></mrow><mrow><mi>k</mi></mrow><mrow><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msubsup><msub><mrow><mi mathvariant="bold-italic">C</mi></mrow><mrow><mi>k</mi></mrow></msub><mo>+</mo><msubsup><mrow><mi mathvariant="bold-italic">P</mi></mrow><mrow><mi>k</mi><mo stretchy="false">|</mo><mi>k</mi><mo>-</mo><mn mathvariant="normal">1</mn></mrow><mrow><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msubsup><msup><mrow><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=45116733&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45116749&type= 52.83200073 4.65666676 考虑将 <math id="M35"><msubsup><mrow><mover accent="true"><mi mathvariant="bold-italic">x</mi><mo>^</mo></mover></mrow><mrow><mi>k</mi></mrow><mrow/></msubsup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45116736&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45116755&type= 3.38666677 4.06400013 分解为 <math id="M36"><msubsup><mrow><mover accent="true"><mi mathvariant="bold-italic">x</mi><mo>^</mo></mover></mrow><mrow><mi>k</mi></mrow><mrow/></msubsup><mo>=</mo><msubsup><mrow><mover accent="true"><mi mathvariant="bold-italic">x</mi><mo>^</mo></mover></mrow><mrow><mi>k</mi></mrow><mrow><mi>a</mi></mrow></msubsup><mo>+</mo><msubsup><mrow><mover accent="true"><mi mathvariant="bold-italic">x</mi><mo>^</mo></mover></mrow><mrow><mi>k</mi></mrow><mrow><mi>θ</mi></mrow></msubsup><mo>-</mo><msubsup><mrow><mover accent="true"><mi mathvariant="bold-italic">x</mi><mo>^</mo></mover></mrow><mrow><mi>k</mi></mrow><mrow><mi>s</mi></mrow></msubsup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45116740&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45116738&type= 27.94000053 4.14866638 ，且3个参数分别满足： <math display="block" id="M37"><msubsup><mrow><mover accent="true"><mi mathvariant="bold-italic">x</mi><mo>^</mo></mover></mrow><mrow><mi>k</mi></mrow><mrow><mi>a</mi></mrow></msubsup><mo>=</mo><msubsup><mrow><mi mathvariant="bold-italic">N</mi></mrow><mrow><mi>k</mi></mrow><mrow><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msubsup><mo stretchy="false">(</mo><msubsup><mrow><mi mathvariant="bold-italic">C</mi></mrow><mrow><mi>k</mi></mrow><mrow><mi mathvariant="normal">T</mi></mrow></msubsup><msubsup><mrow><mi mathvariant="bold-italic">R</mi></mrow><mrow><mi>k</mi></mrow><mrow><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msubsup><msub><mrow><mi mathvariant="bold-italic">y</mi></mrow><mrow><mi>k</mi></mrow></msub><mo>+</mo><msubsup><mrow><mi mathvariant="bold-italic">P</mi></mrow><mrow><mi>k</mi><mo stretchy="false">|</mo><mi>k</mi><mo>-</mo><mn mathvariant="normal">1</mn></mrow><mrow><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msubsup><msub><mrow><mi mathvariant="bold-italic">Φ</mi></mrow><mrow><mi>k</mi></mrow></msub><msubsup><mrow><mover accent="true"><mi mathvariant="bold-italic">x</mi><mo>^</mo></mover></mrow><mrow><mi>k</mi><mo>-</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi>a</mi></mrow></msubsup><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45116773&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45116742&type= 57.40399933 5.07999992 <math display="block" id="M38"><msubsup><mrow><mover accent="true"><mi mathvariant="bold-italic">x</mi><mo>^</mo></mover></mrow><mrow><mi>k</mi></mrow><mrow><mi>θ</mi></mrow></msubsup><mo>=</mo><msubsup><mrow><mi mathvariant="bold-italic">N</mi></mrow><mrow><mi>k</mi></mrow><mrow><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msubsup><mo stretchy="false">(</mo><msubsup><mrow><mi mathvariant="bold-italic">P</mi></mrow><mrow><mi>k</mi><mo stretchy="false">|</mo><mi>k</mi><mo>-</mo><mn mathvariant="normal">1</mn></mrow><mrow><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msubsup><msub><mrow><mi mathvariant="bold-italic">Γ</mi></mrow><mrow><mi>k</mi></mrow></msub><msub><mrow><mover accent="true"><mi mathvariant="bold-italic">θ</mi><mo>^</mo></mover></mrow><mrow><mi>k</mi></mrow></msub><mo>+</mo><msubsup><mrow><mi mathvariant="bold-italic">P</mi></mrow><mrow><mi>k</mi><mo stretchy="false">|</mo><mi>k</mi><mo>-</mo><mn mathvariant="normal">1</mn></mrow><mrow><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msubsup><msub><mrow><mi mathvariant="bold-italic">Φ</mi></mrow><mrow><mi>k</mi></mrow></msub><msubsup><mrow><mover accent="true"><mi mathvariant="bold-italic">x</mi><mo>^</mo></mover></mrow><mrow><mi>k</mi><mo>-</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi>θ</mi></mrow></msubsup><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45116779&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45116775&type= 58.92800140 5.33400011 <math display="block" id="M39"><msubsup><mrow><mover accent="true"><mi mathvariant="bold-italic">x</mi><mo>^</mo></mover></mrow><mrow><mi>k</mi></mrow><mrow><mi>s</mi></mrow></msubsup><mo>=</mo><msubsup><mrow><mi mathvariant="bold-italic">N</mi></mrow><mrow><mi>k</mi></mrow><mrow><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msubsup><mo stretchy="false">(</mo><msubsup><mrow><mi mathvariant="bold-italic">C</mi></mrow><mrow><mi>k</mi></mrow><mrow><mi mathvariant="normal">T</mi></mrow></msubsup><msubsup><mrow><mi mathvariant="bold-italic">R</mi></mrow><mrow><mi>k</mi></mrow><mrow><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msubsup><msub><mrow><mi mathvariant="bold-italic">B</mi></mrow><mrow><mi>k</mi></mrow></msub><msub><mrow><mover accent="true"><mi mathvariant="bold-italic">s</mi><mo>^</mo></mover></mrow><mrow><mi>k</mi></mrow></msub><mo>+</mo><msubsup><mrow><mi mathvariant="bold-italic">P</mi></mrow><mrow><mi>k</mi><mo stretchy="false">|</mo><mi>k</mi><mo>-</mo><mn mathvariant="normal">1</mn></mrow><mrow><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msubsup><msub><mrow><mi mathvariant="bold-italic">Φ</mi></mrow><mrow><mi>k</mi></mrow></msub><msubsup><mrow><mover accent="true"><mi mathvariant="bold-italic">x</mi><mo>^</mo></mover></mrow><mrow><mi>k</mi><mo>-</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi>s</mi></mrow></msubsup><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45116767&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45116781&type= 60.36733627 4.65666676 式中， <math id="M40"><msubsup><mrow><mover accent="true"><mi mathvariant="bold-italic">x</mi><mo>^</mo></mover></mrow><mrow><mi>k</mi></mrow><mrow><mi>a</mi></mrow></msubsup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45116770&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45116768&type= 3.64066648 4.06400013 为式（1）式（1）不考虑动力学模型误差 <math id="M41"><msub><mrow><mi mathvariant="bold-italic">Γ</mi></mrow><mrow><mi>k</mi></mrow></msub><msub><mrow><mi mathvariant="bold-italic">θ</mi></mrow><mrow><mi>k</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45116803&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45116772&type= 6.94266701 3.72533321 和观测模型误差 <math id="M42"><msub><mrow><mi mathvariant="bold-italic">B</mi></mrow><mrow><mi>k</mi></mrow></msub><msub><mrow><mi mathvariant="bold-italic">s</mi></mrow><mrow><mi>k</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45116791&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45116789&type= 6.60400009 3.72533321 的状态参数估计，可用信息滤波直接递归计算。若用滤波解算的 <math id="M43"><msub><mrow><mover accent="true"><mi mathvariant="bold-italic">θ</mi><mo>^</mo></mover></mrow><mrow><mi>k</mi><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45116807&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45116792&type= 6.51933336 4.65666676 代替 <math id="M44"><msub><mrow><mi mathvariant="bold-italic">θ</mi></mrow><mrow><mi>k</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45116798&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45116796&type= 2.96333337 3.72533321 对 <math id="M45"><msubsup><mrow><mover accent="true"><mi mathvariant="bold-italic">x</mi><mo>^</mo></mover></mrow><mrow><mi>k</mi></mrow><mrow><mi>θ</mi></mrow></msubsup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45116811&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45116801&type= 3.55599999 4.06400013 估计，并用 <math id="M46"><msub><mrow><mover accent="true"><mi mathvariant="bold-italic">s</mi><mo>^</mo></mover></mrow><mrow><mi>k</mi><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45116818&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45116813&type= 6.01133299 3.80999994 代替 <math id="M47"><msub><mrow><mi mathvariant="bold-italic">s</mi></mrow><mrow><mi>k</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45116820&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45116815&type= 2.45533323 3.72533321 对 <math id="M48"><msubsup><mrow><mover accent="true"><mi mathvariant="bold-italic">x</mi><mo>^</mo></mover></mrow><mrow><mi>k</mi></mrow><mrow><mi>s</mi></mrow></msubsup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45116817&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45116822&type= 3.30200005 4.06400013 进行估计，则式（7）式（7）、式（8）式（8）可写为： <math display="block" id="M49"><mfenced open="{" close="" separators="|"><mrow><mtable columnalign="left"><mtr><mtd><msubsup><mrow><mover accent="true"><mi mathvariant="bold-italic">x</mi><mo>^</mo></mover></mrow><mrow><mi>k</mi></mrow><mrow><mi>θ</mi></mrow></msubsup><mo>=</mo><msubsup><mrow><mi mathvariant="bold-italic">N</mi></mrow><mrow><mi>k</mi></mrow><mrow><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msubsup><mo stretchy="false">(</mo><msubsup><mrow><mi mathvariant="bold-italic">P</mi></mrow><mrow><mi>k</mi><mo stretchy="false">|</mo><mi>k</mi><mo>-</mo><mn mathvariant="normal">1</mn></mrow><mrow><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msubsup><msub><mrow><mi mathvariant="bold-italic">Γ</mi></mrow><mrow><mi>k</mi></mrow></msub><msub><mrow><mover accent="true"><mi mathvariant="bold-italic">θ</mi><mo>^</mo></mover></mrow><mrow><mi>k</mi><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msub><mo>+</mo><msubsup><mrow><mi mathvariant="bold-italic">P</mi></mrow><mrow><mi>k</mi><mo stretchy="false">|</mo><mi>k</mi><mo>-</mo><mn mathvariant="normal">1</mn></mrow><mrow><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msubsup><msub><mrow><mi mathvariant="bold-italic">Φ</mi></mrow><mrow><mi>k</mi></mrow></msub><msubsup><mrow><mover accent="true"><mi mathvariant="bold-italic">x</mi><mo>^</mo></mover></mrow><mrow><mi>k</mi><mo>-</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi>θ</mi></mrow></msubsup><mo stretchy="false">)</mo><mo>+</mo><msubsup><mrow><mi>Δ</mi></mrow><mrow><mi>k</mi></mrow><mrow><mi>θ</mi></mrow></msubsup></mtd></mtr><mtr><mtd><msubsup><mrow><mover accent="true"><mi mathvariant="bold-italic">x</mi><mo>^</mo></mover></mrow><mrow><mi>k</mi></mrow><mrow><mi>s</mi></mrow></msubsup><mo>=</mo><msubsup><mrow><mi mathvariant="bold-italic">N</mi></mrow><mrow><mi>k</mi></mrow><mrow><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msubsup><mo stretchy="false">(</mo><msubsup><mrow><mi mathvariant="bold-italic">C</mi></mrow><mrow><mi>k</mi></mrow><mrow><mi mathvariant="normal">T</mi></mrow></msubsup><msubsup><mrow><mi mathvariant="bold-italic">R</mi></mrow><mrow><mi>k</mi></mrow><mrow><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msubsup><msub><mrow><mi mathvariant="bold-italic">B</mi></mrow><mrow><mi>k</mi></mrow></msub><msub><mrow><mover accent="true"><mi mathvariant="bold-italic">s</mi><mo>^</mo></mover></mrow><mrow><mi>k</mi><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msub><mo>+</mo><msubsup><mrow><mi mathvariant="bold-italic">P</mi></mrow><mrow><mi>k</mi><mo stretchy="false">|</mo><mi>k</mi><mo>-</mo><mn mathvariant="normal">1</mn></mrow><mrow><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msubsup><msub><mrow><mi mathvariant="bold-italic">Φ</mi></mrow><mrow><mi>k</mi></mrow></msub><msubsup><mrow><mover accent="true"><mi mathvariant="bold-italic">x</mi><mo>^</mo></mover></mrow><mrow><mi>k</mi><mo>-</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi>s</mi></mrow></msubsup><mo stretchy="false">)</mo><mo>+</mo><msubsup><mrow><mi>Δ</mi></mrow><mrow><mi>k</mi></mrow><mrow><mi>s</mi></mrow></msubsup></mtd></mtr></mtable></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45116834&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45116833&type= 74.08333588 11.00666618 式中， <math id="M50"><msubsup><mrow><mi>Δ</mi></mrow><mrow><mi>k</mi></mrow><mrow><mi>θ</mi></mrow></msubsup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45116838&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45116836&type= 3.72533321 3.97933316 为估值 <math id="M51"><msub><mrow><mover accent="true"><mi mathvariant="bold-italic">θ</mi><mo>^</mo></mover></mrow><mrow><mi>k</mi><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45116828&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45116826&type= 6.51933336 4.65666676 代替 <math id="M52"><msub><mrow><mover accent="true"><mi mathvariant="bold-italic">θ</mi><mo>^</mo></mover></mrow><mrow><mi>k</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45116941&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45116939&type= 2.96333337 4.65666676 的补偿项，反映动力学模型误差随时间变化对滤波的影响； <math id="M53"><msubsup><mrow><mi>Δ</mi></mrow><mrow><mi>k</mi></mrow><mrow><mi>s</mi></mrow></msubsup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45116842&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45116840&type= 3.64066648 3.97933316 为估值 <math id="M54"><msub><mrow><mover accent="true"><mi mathvariant="bold-italic">s</mi><mo>^</mo></mover></mrow><mrow><mi>k</mi><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45116848&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45116843&type= 6.01133299 3.80999994 代替 <math id="M55"><msub><mrow><mover accent="true"><mi mathvariant="bold-italic">s</mi><mo>^</mo></mover></mrow><mrow><mi>k</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45116945&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45116943&type= 2.45533323 3.80999994 的补偿项，反映观测模型误差随时间变化对滤波的影响。设任意时刻都存在系数阵 <math id="M56"><msub><mrow><mi mathvariant="bold-italic">Τ</mi></mrow><mrow><mi>k</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45116890&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45116888&type= 3.64066648 3.72533321 ，使线性关系 <math id="M57"><msubsup><mrow><mover accent="true"><mi mathvariant="bold-italic">x</mi><mo>^</mo></mover></mrow><mrow><mi>k</mi></mrow><mrow><mi>θ</mi></mrow></msubsup><mo>=</mo><msub><mrow><mi mathvariant="bold-italic">Τ</mi></mrow><mrow><mi>k</mi></mrow></msub><msub><mrow><mover accent="true"><mi mathvariant="bold-italic">θ</mi><mo>^</mo></mover></mrow><mrow><mi>k</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45116859&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45116857&type= 15.15533447 4.91066647 成立［ 12 12 - 14 14 12-14 ］；设在任意时刻都存在系数阵 <math id="M58"><msub><mrow><mi mathvariant="bold-italic">F</mi></mrow><mrow><mi>k</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45116909&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45116892&type= 3.64066648 3.72533321 ，使线性关系 <math id="M59"><msubsup><mrow><mover accent="true"><mi mathvariant="bold-italic">x</mi><mo>^</mo></mover></mrow><mrow><mi>k</mi></mrow><mrow><mi>s</mi></mrow></msubsup><mo>=</mo><msub><mrow><mi mathvariant="bold-italic">F</mi></mrow><mrow><mi>k</mi></mrow></msub><msub><mrow><mover accent="true"><mi mathvariant="bold-italic">s</mi><mo>^</mo></mover></mrow><mrow><mi>k</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45116845&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45116864&type= 14.56266594 4.06400013 成立。则式（9）式（9）可写为： <math display="block" id="M60"><mfenced open="{" close="" separators="|"><mrow><mtable columnalign="left"><mtr><mtd><msub><mrow><mi mathvariant="bold-italic">Τ</mi></mrow><mrow><mi>k</mi></mrow></msub><msub><mrow><mover accent="true"><mi mathvariant="bold-italic">θ</mi><mo>^</mo></mover></mrow><mrow><mi>k</mi></mrow></msub><mo>=</mo><msubsup><mrow><mi mathvariant="bold-italic">N</mi></mrow><mrow><mi>k</mi></mrow><mrow><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msubsup><mo stretchy="false">(</mo><msubsup><mrow><mi mathvariant="bold-italic">P</mi></mrow><mrow><mi>k</mi><mo stretchy="false">|</mo><mi>k</mi><mo>-</mo><mn mathvariant="normal">1</mn></mrow><mrow><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msubsup><msub><mrow><mi mathvariant="bold-italic">Φ</mi></mrow><mrow><mi>k</mi></mrow></msub><msub><mrow><mi mathvariant="bold-italic">Τ</mi></mrow><mrow><mi>k</mi><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msub><msub><mrow><mover accent="true"><mi mathvariant="bold-italic">θ</mi><mo>^</mo></mover></mrow><mrow><mi>k</mi><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msub><mo>+</mo></mtd></mtr><mtr><mtd><mtext> </mtext><msubsup><mrow><mi mathvariant="bold-italic">P</mi></mrow><mrow><mi>k</mi><mo stretchy="false">|</mo><mi>k</mi><mo>-</mo><mn mathvariant="normal">1</mn></mrow><mrow><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msubsup><msub><mrow><mi mathvariant="bold-italic">Γ</mi></mrow><mrow><mi>k</mi></mrow></msub><msub><mrow><mover accent="true"><mi mathvariant="bold-italic">θ</mi><mo>^</mo></mover></mrow><mrow><mi>k</mi><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msub><mo stretchy="false">)</mo><mo>+</mo><msubsup><mrow><mi>Δ</mi></mrow><mrow><mi>k</mi></mrow><mrow><mi>θ</mi></mrow></msubsup></mtd></mtr><mtr><mtd><msub><mrow><mi mathvariant="bold-italic">F</mi></mrow><mrow><mi>k</mi></mrow></msub><msub><mrow><mover accent="true"><mi mathvariant="bold-italic">s</mi><mo>^</mo></mover></mrow><mrow><mi>k</mi></mrow></msub><mo>=</mo><msubsup><mrow><mi mathvariant="bold-italic">N</mi></mrow><mrow><mi>k</mi></mrow><mrow><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msubsup><mo stretchy="false">(</mo><msubsup><mrow><mi mathvariant="bold-italic">C</mi></mrow><mrow><mi>k</mi></mrow><mrow><mi mathvariant="normal">T</mi></mrow></msubsup><msubsup><mrow><mi mathvariant="bold-italic">R</mi></mrow><mrow><mi>k</mi></mrow><mrow><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msubsup><msub><mrow><mi mathvariant="bold-italic">B</mi></mrow><mrow><mi>k</mi></mrow></msub><msub><mrow><mover accent="true"><mi mathvariant="bold-italic">s</mi><mo>^</mo></mover></mrow><mrow><mi>k</mi><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msub><mo>+</mo></mtd></mtr><mtr><mtd><mtext> </mtext><msubsup><mrow><mi mathvariant="bold-italic">P</mi></mrow><mrow><mi>k</mi><mo stretchy="false">|</mo><mi>k</mi><mo>-</mo><mn mathvariant="normal">1</mn></mrow><mrow><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msubsup><msub><mrow><mi mathvariant="bold-italic">Φ</mi></mrow><mrow><mi>k</mi></mrow></msub><msub><mrow><mi mathvariant="bold-italic">F</mi></mrow><mrow><mi>k</mi><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msub><msub><mrow><mover accent="true"><mi mathvariant="bold-italic">s</mi><mo>^</mo></mover></mrow><mrow><mi>k</mi><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msub><mo stretchy="false">)</mo><mo>+</mo><msubsup><mrow><mi>Δ</mi></mrow><mrow><mi>k</mi></mrow><mrow><mi>s</mi></mrow></msubsup></mtd></mtr></mtable></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45116879&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45116847&type= 52.66266251 22.77533340 式（10）式（10）分别建立了动力学模型和观测模型误差估计与对应状态估计的线性递归表达。设动力学模型误差的估值 <math id="M61"><msub><mrow><mover accent="true"><mi mathvariant="bold-italic">θ</mi><mo>^</mo></mover></mrow><mrow><mi>k</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45116880&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45116867&type= 2.96333337 4.65666676 满足 <math id="M62"><msub><mrow><mover accent="true"><mi mathvariant="bold-italic">θ</mi><mo>^</mo></mover></mrow><mrow><mi>k</mi></mrow></msub><mo>=</mo><msub><mrow><mover accent="true"><mi mathvariant="bold-italic">θ</mi><mo>^</mo></mover></mrow><mrow><mi>k</mi><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msub><mo>+</mo><mi mathvariant="normal">δ</mi><msub><mrow><mover accent="true"><mi mathvariant="bold-italic">θ</mi><mo>^</mo></mover></mrow><mrow><mi>k</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45116871&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45116881&type= 23.70666695 4.74133301 ，观测模型误差的估值 <math id="M63"><msub><mrow><mover accent="true"><mi mathvariant="bold-italic">s</mi><mo>^</mo></mover></mrow><mrow><mi>k</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45116882&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45116873&type= 2.45533323 3.80999994 满足 <math id="M64"><msub><mrow><mover accent="true"><mi mathvariant="bold-italic">s</mi><mo>^</mo></mover></mrow><mrow><mi>k</mi></mrow></msub><mo>=</mo><msub><mrow><mover accent="true"><mi mathvariant="bold-italic">s</mi><mo>^</mo></mover></mrow><mrow><mi>k</mi><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msub><mo>+</mo><mi mathvariant="normal">δ</mi><msub><mrow><mover accent="true"><mi mathvariant="bold-italic">s</mi><mo>^</mo></mover></mrow><mrow><mi>k</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45116876&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45116874&type= 22.18266678 3.89466691 ， <math id="M65"><mi mathvariant="normal">δ</mi><msub><mrow><mover accent="true"><mi mathvariant="bold-italic">θ</mi><mo>^</mo></mover></mrow><mrow><mi>k</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45116894&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45116893&type= 4.57200003 4.65666676 和 <math id="M66"><mi mathvariant="normal">δ</mi><msub><mrow><mover accent="true"><mi mathvariant="bold-italic">s</mi><mo>^</mo></mover></mrow><mrow><mi>k</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45116898&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45116896&type= 4.06400013 3.80999994 分别为动力学模型和观测模型误差时变项，则式（10）式（10）可改写为： <math display="block" id="M67"><mfenced open="{" close="" separators="|"><mrow><mtable columnalign="left"><mtr><mtd><msub><mrow><mi mathvariant="bold-italic">Τ</mi></mrow><mrow><mi>k</mi></mrow></msub><mo stretchy="false">(</mo><msub><mrow><mover accent="true"><mi mathvariant="bold-italic">θ</mi><mo>^</mo></mover></mrow><mrow><mi>k</mi><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msub><mo>+</mo><mi mathvariant="normal">δ</mi><msub><mrow><mover accent="true"><mi mathvariant="bold-italic">θ</mi><mo>^</mo></mover></mrow><mrow><mi>k</mi></mrow></msub><mo stretchy="false">)</mo><mo>=</mo></mtd></mtr><mtr><mtd><mtext> </mtext><msubsup><mrow><mi mathvariant="bold-italic">N</mi></mrow><mrow><mi>k</mi></mrow><mrow><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msubsup><mo stretchy="false">(</mo><msubsup><mrow><mi mathvariant="bold-italic">P</mi></mrow><mrow><mi>k</mi><mo stretchy="false">|</mo><mi>k</mi><mo>-</mo><mn mathvariant="normal">1</mn></mrow><mrow><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msubsup><msub><mrow><mi mathvariant="bold-italic">Φ</mi></mrow><mrow><mi>k</mi></mrow></msub><msub><mrow><mi mathvariant="bold-italic">Τ</mi></mrow><mrow><mi>k</mi><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msub><msub><mrow><mover accent="true"><mi mathvariant="bold-italic">θ</mi><mo>^</mo></mover></mrow><mrow><mi>k</mi><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msub><mo>+</mo><msubsup><mrow><mi mathvariant="bold-italic">P</mi></mrow><mrow><mi>k</mi><mo stretchy="false">|</mo><mi>k</mi><mo>-</mo><mn mathvariant="normal">1</mn></mrow><mrow><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msubsup><msub><mrow><mi mathvariant="bold-italic">Γ</mi></mrow><mrow><mi>k</mi></mrow></msub><msub><mrow><mover accent="true"><mi mathvariant="bold-italic">θ</mi><mo>^</mo></mover></mrow><mrow><mi>k</mi><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msub><mo stretchy="false">)</mo><mo>+</mo><msubsup><mrow><mi>Δ</mi></mrow><mrow><mi>k</mi></mrow><mrow><mi>θ</mi></mrow></msubsup></mtd></mtr><mtr><mtd><msub><mrow><mi mathvariant="bold-italic">F</mi></mrow><mrow><mi>k</mi></mrow></msub><mo stretchy="false">(</mo><msub><mrow><mover accent="true"><mi mathvariant="bold-italic">s</mi><mo>^</mo></mover></mrow><mrow><mi>k</mi><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msub><mo>+</mo><mi mathvariant="normal">δ</mi><msub><mrow><mover accent="true"><mi mathvariant="bold-italic">s</mi><mo>^</mo></mover></mrow><mrow><mi>k</mi></mrow></msub><mo stretchy="false">)</mo><mo>=</mo></mtd></mtr><mtr><mtd><mtext> </mtext><msubsup><mrow><mi mathvariant="bold-italic">N</mi></mrow><mrow><mi>k</mi></mrow><mrow><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msubsup><mo stretchy="false">(</mo><msubsup><mrow><mi mathvariant="bold-italic">C</mi></mrow><mrow><mi>k</mi></mrow><mrow><mi mathvariant="normal">T</mi></mrow></msubsup><msubsup><mrow><mi mathvariant="bold-italic">R</mi></mrow><mrow><mi>k</mi></mrow><mrow><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msubsup><msub><mrow><mi mathvariant="bold-italic">B</mi></mrow><mrow><mi>k</mi></mrow></msub><msub><mrow><mover accent="true"><mi mathvariant="bold-italic">s</mi><mo>^</mo></mover></mrow><mrow><mi>k</mi><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msub><mo>+</mo><msubsup><mrow><mi mathvariant="bold-italic">P</mi></mrow><mrow><mi>k</mi><mo stretchy="false">|</mo><mi>k</mi><mo>-</mo><mn mathvariant="normal">1</mn></mrow><mrow><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msubsup><msub><mrow><mi mathvariant="bold-italic">Φ</mi></mrow><mrow><mi>k</mi></mrow></msub><msub><mrow><mi mathvariant="bold-italic">F</mi></mrow><mrow><mi>k</mi><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msub><msub><mrow><mover accent="true"><mi mathvariant="bold-italic">s</mi><mo>^</mo></mover></mrow><mrow><mi>k</mi><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msub><mo stretchy="false">)</mo><mo>+</mo><msubsup><mrow><mi>Δ</mi></mrow><mrow><mi>k</mi></mrow><mrow><mi>s</mi></mrow></msubsup></mtd></mtr></mtable></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45116902&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45116900&type= 73.91400146 22.35199928 若令 <math id="M68"><msubsup><mrow><mi>Δ</mi></mrow><mrow><mi>k</mi></mrow><mrow><mi>θ</mi></mrow></msubsup><mo>=</mo><msub><mrow><mi mathvariant="bold-italic">Τ</mi></mrow><mrow><mi>k</mi></mrow></msub><mi mathvariant="normal">δ</mi><msub><mrow><mover accent="true"><mi mathvariant="bold-italic">θ</mi><mo>^</mo></mover></mrow><mrow><mi>k</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45116904&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45116884&type= 17.01799965 4.91066647 ，且 <math id="M69"><msubsup><mrow><mi>Δ</mi></mrow><mrow><mi>k</mi></mrow><mrow><mi>s</mi></mrow></msubsup><mo>=</mo><msub><mrow><mi mathvariant="bold-italic">F</mi></mrow><mrow><mi>k</mi></mrow></msub><mi mathvariant="normal">δ</mi><msub><mrow><mover accent="true"><mi mathvariant="bold-italic">s</mi><mo>^</mo></mover></mrow><mrow><mi>k</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45116906&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45116905&type= 16.42533302 4.06400013 ，则得到 <math id="M70"><msub><mrow><mi mathvariant="bold-italic">Τ</mi></mrow><mrow><mi>k</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45116890&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45116888&type= 3.64066648 3.72533321 和 <math id="M71"><msub><mrow><mi mathvariant="bold-italic">F</mi></mrow><mrow><mi>k</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45116909&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45116892&type= 3.64066648 3.72533321 的递归表达形式，即： <math display="block" id="M72"><mfenced open="{" close="" separators="|"><mrow><mtable columnalign="left"><mtr><mtd><msub><mrow><mi mathvariant="bold-italic">Τ</mi></mrow><mrow><mi>k</mi></mrow></msub><mo>=</mo><msubsup><mrow><mi mathvariant="bold-italic">N</mi></mrow><mrow><mi>k</mi></mrow><mrow><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msubsup><mo stretchy="false">(</mo><msubsup><mrow><mi mathvariant="bold-italic">P</mi></mrow><mrow><mi>k</mi><mo stretchy="false">|</mo><mi>k</mi><mo>-</mo><mn mathvariant="normal">1</mn></mrow><mrow><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msubsup><msub><mrow><mi mathvariant="bold-italic">Φ</mi></mrow><mrow><mi>k</mi></mrow></msub><msub><mrow><mi mathvariant="bold-italic">Τ</mi></mrow><mrow><mi>k</mi><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msub><mo>+</mo><msubsup><mrow><mi mathvariant="bold-italic">P</mi></mrow><mrow><mi>k</mi><mo stretchy="false">|</mo><mi>k</mi><mo>-</mo><mn mathvariant="normal">1</mn></mrow><mrow><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msubsup><msub><mrow><mi mathvariant="bold-italic">Γ</mi></mrow><mrow><mi>k</mi></mrow></msub><mo stretchy="false">)</mo></mtd></mtr><mtr><mtd><msub><mrow><mi mathvariant="bold-italic">F</mi></mrow><mrow><mi>k</mi></mrow></msub><mo>=</mo><msubsup><mrow><mi mathvariant="bold-italic">N</mi></mrow><mrow><mi>k</mi></mrow><mrow><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msubsup><mo stretchy="false">(</mo><msubsup><mrow><mi mathvariant="bold-italic">C</mi></mrow><mrow><mi>k</mi></mrow><mrow><mi mathvariant="normal">T</mi></mrow></msubsup><msubsup><mrow><mi mathvariant="bold-italic">R</mi></mrow><mrow><mi>k</mi></mrow><mrow><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msubsup><msub><mrow><mi mathvariant="bold-italic">B</mi></mrow><mrow><mi>k</mi></mrow></msub><mo>+</mo><msubsup><mrow><mi mathvariant="bold-italic">P</mi></mrow><mrow><mi>k</mi><mo stretchy="false">|</mo><mi>k</mi><mo>-</mo><mn mathvariant="normal">1</mn></mrow><mrow><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msubsup><msub><mrow><mi mathvariant="bold-italic">Φ</mi></mrow><mrow><mi>k</mi></mrow></msub><msub><mrow><mi mathvariant="bold-italic">F</mi></mrow><mrow><mi>k</mi><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msub><mo stretchy="false">)</mo></mtd></mtr></mtable></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45116912&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45116910&type= 60.28266144 10.41399956 式（9）式（9）可重新表示为： <math display="block" id="M73"><mfenced open="{" close="" separators="|"><mrow><mtable columnalign="left"><mtr><mtd><msubsup><mrow><mover accent="true"><mi mathvariant="bold-italic">x</mi><mo>^</mo></mover></mrow><mrow><mi>k</mi></mrow><mrow><mi>θ</mi></mrow></msubsup><mo>=</mo><msubsup><mrow><mi mathvariant="bold-italic">N</mi></mrow><mrow><mi>k</mi></mrow><mrow><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msubsup><mo stretchy="false">(</mo><msubsup><mrow><mi mathvariant="bold-italic">P</mi></mrow><mrow><mi>k</mi><mo stretchy="false">|</mo><mi>k</mi><mo>-</mo><mn mathvariant="normal">1</mn></mrow><mrow><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msubsup><msub><mrow><mi mathvariant="bold-italic">Γ</mi></mrow><mrow><mi>k</mi></mrow></msub><msub><mrow><mover accent="true"><mi mathvariant="bold-italic">θ</mi><mo>^</mo></mover></mrow><mrow><mi>k</mi><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msub><mo>+</mo><msubsup><mrow><mi mathvariant="bold-italic">P</mi></mrow><mrow><mi>k</mi><mo stretchy="false">|</mo><mi>k</mi><mo>-</mo><mn mathvariant="normal">1</mn></mrow><mrow><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msubsup><msub><mrow><mi mathvariant="bold-italic">Φ</mi></mrow><mrow><mi>k</mi></mrow></msub><msubsup><mrow><mover accent="true"><mi mathvariant="bold-italic">x</mi><mo>^</mo></mover></mrow><mrow><mi>k</mi><mo>-</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi>θ</mi></mrow></msubsup><mo stretchy="false">)</mo><mo>+</mo></mtd></mtr><mtr><mtd><msub><mrow><mi mathvariant="bold-italic">Τ</mi></mrow><mrow><mi>k</mi></mrow></msub><mo stretchy="false">(</mo><msub><mrow><mover accent="true"><mi mathvariant="bold-italic">θ</mi><mo>^</mo></mover></mrow><mrow><mi>k</mi></mrow></msub><mo>-</mo><msub><mrow><mover accent="true"><mi mathvariant="bold-italic">θ</mi><mo>^</mo></mover></mrow><mrow><mi>k</mi><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msub><mo stretchy="false">)</mo></mtd></mtr><mtr><mtd><msubsup><mrow><mover accent="true"><mi mathvariant="bold-italic">x</mi><mo>^</mo></mover></mrow><mrow><mi>k</mi></mrow><mrow><mi>s</mi></mrow></msubsup><mo>=</mo><msubsup><mrow><mi mathvariant="bold-italic">N</mi></mrow><mrow><mi>k</mi></mrow><mrow><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msubsup><mo stretchy="false">(</mo><msubsup><mrow><mi mathvariant="bold-italic">C</mi></mrow><mrow><mi>k</mi></mrow><mrow><mi mathvariant="normal">T</mi></mrow></msubsup><msubsup><mrow><mi mathvariant="bold-italic">R</mi></mrow><mrow><mi>k</mi></mrow><mrow><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msubsup><msub><mrow><mi mathvariant="bold-italic">B</mi></mrow><mrow><mi>k</mi></mrow></msub><msub><mrow><mover accent="true"><mi mathvariant="bold-italic">s</mi><mo>^</mo></mover></mrow><mrow><mi>k</mi><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msub><mo>+</mo><msubsup><mrow><mi mathvariant="bold-italic">P</mi></mrow><mrow><mi>k</mi><mo stretchy="false">|</mo><mi>k</mi><mo>-</mo><mn mathvariant="normal">1</mn></mrow><mrow><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msubsup><msub><mrow><mi mathvariant="bold-italic">Φ</mi></mrow><mrow><mi>k</mi></mrow></msub><msubsup><mrow><mover accent="true"><mi mathvariant="bold-italic">x</mi><mo>^</mo></mover></mrow><mrow><mi>k</mi><mo>-</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi mathvariant="bold-italic">s</mi></mrow></msubsup><mo stretchy="false">)</mo><mo>+</mo></mtd></mtr><mtr><mtd><msub><mrow><mi mathvariant="bold-italic">F</mi></mrow><mrow><mi>k</mi></mrow></msub><mo stretchy="false">(</mo><msub><mrow><mover accent="true"><mi mathvariant="bold-italic">s</mi><mo>^</mo></mover></mrow><mrow><mi>k</mi></mrow></msub><mo>-</mo><msub><mrow><mover accent="true"><mi mathvariant="bold-italic">s</mi><mo>^</mo></mover></mrow><mrow><mi>k</mi><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msub><mo stretchy="false">)</mo></mtd></mtr></mtable></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45116907&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45116914&type= 69.76532745 22.35199928 将（12）代入式（13）式（13）并化简可得： <math display="block" id="M74"><mfenced open="{" close="" separators="|"><mrow><mtable columnalign="left"><mtr><mtd><msubsup><mrow><mover accent="true"><mi mathvariant="bold-italic">x</mi><mo>^</mo></mover></mrow><mrow><mi>k</mi></mrow><mrow><mi>θ</mi></mrow></msubsup><mo>=</mo><msubsup><mrow><mi mathvariant="bold-italic">N</mi></mrow><mrow><mi>k</mi></mrow><mrow><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msubsup><mo stretchy="false">(</mo><msubsup><mrow><mi mathvariant="bold-italic">P</mi></mrow><mrow><mi>k</mi><mo stretchy="false">|</mo><mi>k</mi><mo>-</mo><mn mathvariant="normal">1</mn></mrow><mrow><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msubsup><msub><mrow><mi mathvariant="bold-italic">Γ</mi></mrow><mrow><mi>k</mi></mrow></msub><msub><mrow><mover accent="true"><mi mathvariant="bold-italic">θ</mi><mo>^</mo></mover></mrow><mrow><mi>k</mi></mrow></msub><mo>+</mo></mtd></mtr><mtr><mtd><mtext> </mtext><msubsup><mrow><mi mathvariant="bold-italic">P</mi></mrow><mrow><mi>k</mi><mo stretchy="false">|</mo><mi>k</mi><mo>-</mo><mn mathvariant="normal">1</mn></mrow><mrow><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msubsup><msub><mrow><mi mathvariant="bold-italic">Φ</mi></mrow><mrow><mi>k</mi></mrow></msub><mo stretchy="false">(</mo><msubsup><mrow><mover accent="true"><mi mathvariant="bold-italic">x</mi><mo>^</mo></mover></mrow><mrow><mi>k</mi><mo>-</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi>θ</mi></mrow></msubsup><mo>+</mo><msub><mrow><mi mathvariant="bold-italic">Τ</mi></mrow><mrow><mi>k</mi><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msub><mo stretchy="false">(</mo><msub><mrow><mover accent="true"><mi mathvariant="bold-italic">θ</mi><mo>^</mo></mover></mrow><mrow><mi>k</mi></mrow></msub><mo>-</mo><msub><mrow><mover accent="true"><mi mathvariant="bold-italic">θ</mi><mo>^</mo></mover></mrow><mrow><mi>k</mi><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msub><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mtd></mtr><mtr><mtd><msubsup><mrow><mover accent="true"><mi mathvariant="bold-italic">x</mi><mo>^</mo></mover></mrow><mrow><mi>k</mi></mrow><mrow><mi>s</mi></mrow></msubsup><mo>=</mo><msubsup><mrow><mi mathvariant="bold-italic">N</mi></mrow><mrow><mi>k</mi></mrow><mrow><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msubsup><mo stretchy="false">(</mo><msubsup><mrow><mi mathvariant="bold-italic">C</mi></mrow><mrow><mi>k</mi></mrow><mrow><mi mathvariant="normal">T</mi></mrow></msubsup><msubsup><mrow><mi mathvariant="bold-italic">R</mi></mrow><mrow><mi>k</mi></mrow><mrow><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msubsup><msub><mrow><mi mathvariant="bold-italic">B</mi></mrow><mrow><mi>k</mi></mrow></msub><msub><mrow><mover accent="true"><mi mathvariant="bold-italic">s</mi><mo>^</mo></mover></mrow><mrow><mi>k</mi></mrow></msub><mo>+</mo></mtd></mtr><mtr><mtd><mtext> </mtext><msubsup><mrow><mi mathvariant="bold-italic">P</mi></mrow><mrow><mi>k</mi><mo stretchy="false">|</mo><mi>k</mi><mo>-</mo><mn mathvariant="normal">1</mn></mrow><mrow><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msubsup><msub><mrow><mi mathvariant="bold-italic">Φ</mi></mrow><mrow><mi>k</mi></mrow></msub><mo stretchy="false">(</mo><msubsup><mrow><mover accent="true"><mi mathvariant="bold-italic">x</mi><mo>^</mo></mover></mrow><mrow><mi>k</mi><mo>-</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi>s</mi></mrow></msubsup><mo>+</mo><msub><mrow><mi mathvariant="bold-italic">F</mi></mrow><mrow><mi>k</mi><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msub><mo stretchy="false">(</mo><msub><mrow><mover accent="true"><mi mathvariant="bold-italic">s</mi><mo>^</mo></mover></mrow><mrow><mi>k</mi></mrow></msub><mo>-</mo><msub><mrow><mover accent="true"><mi mathvariant="bold-italic">s</mi><mo>^</mo></mover></mrow><mrow><mi>k</mi><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msub><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mtd></mtr></mtable></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45116924&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45116923&type= 62.39933777 22.77533340 同式（9）式（9）和式（10）式（10）相比，式（14）式（14）不仅考虑了 <math id="M75"><mi mathvariant="bold-italic">θ</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45117029&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45117018&type= 2.03200006 2.79399991 和 <math id="M76"><mi mathvariant="bold-italic">s</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45117033&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45117031&type= 1.43933344 2.79399991 在 k 时刻的估值 <math id="M77"><msub><mrow><mover accent="true"><mi mathvariant="bold-italic">θ</mi><mo>^</mo></mover></mrow><mrow><mi>k</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45116918&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45116917&type= 2.96333337 4.65666676 和 <math id="M78"><msub><mrow><mover accent="true"><mi mathvariant="bold-italic">s</mi><mo>^</mo></mover></mrow><mrow><mi>k</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45116921&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45116931&type= 2.45533323 3.80999994 ，其右边第二项还基于当前 <math id="M79"><msub><mrow><mover accent="true"><mi mathvariant="bold-italic">θ</mi><mo>^</mo></mover></mrow><mrow><mi>k</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45116941&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45116939&type= 2.96333337 4.65666676 和 <math id="M80"><msub><mrow><mover accent="true"><mi mathvariant="bold-italic">s</mi><mo>^</mo></mover></mrow><mrow><mi>k</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45116945&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45116943&type= 2.45533323 3.80999994 的估值对 <math id="M81"><msubsup><mrow><mover accent="true"><mi mathvariant="bold-italic">x</mi><mo>^</mo></mover></mrow><mrow><mi>k</mi><mo>-</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi>θ</mi></mrow></msubsup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45116935&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45116933&type= 6.85799980 4.06400013 以及 <math id="M82"><msubsup><mrow><mover accent="true"><mi mathvariant="bold-italic">x</mi><mo>^</mo></mover></mrow><mrow><mi>k</mi><mo>-</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi>s</mi></mrow></msubsup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45116948&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45116946&type= 6.85799980 4.06400013 进行了修正，提高了对模型误差的估计精度。结合式（6）~ 式（8）式（8）和式（14）式（14），得到顾及动力学模型误差和观测模型误差的弹性自适应滤波系统： <math display="block" id="M83"><msubsup><mrow><mover accent="true"><mi mathvariant="bold-italic">x</mi><mo>^</mo></mover></mrow><mrow><mi>k</mi></mrow><mrow/></msubsup><mo>=</mo><msubsup><mrow><mover accent="true"><mi mathvariant="bold-italic">x</mi><mo>^</mo></mover></mrow><mrow><mi>k</mi></mrow><mrow><mi>a</mi></mrow></msubsup><mo>+</mo><msup><mrow><mi>β</mi></mrow><mrow><mi>θ</mi></mrow></msup><msubsup><mrow><mover accent="true"><mi mathvariant="bold-italic">x</mi><mo>^</mo></mover></mrow><mrow><mi>k</mi></mrow><mrow><mi>θ</mi></mrow></msubsup><mo>-</mo><msup><mrow><mi>β</mi></mrow><mrow><mi>s</mi></mrow></msup><msubsup><mrow><mover accent="true"><mi mathvariant="bold-italic">x</mi><mo>^</mo></mover></mrow><mrow><mi>k</mi></mrow><mrow><mi>s</mi></mrow></msubsup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45116955&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45116953&type= 34.88266373 4.91066647 <math display="block" id="M84"><mfenced open="{" close="" separators="|"><mrow><mtable columnalign="left"><mtr><mtd><msubsup><mrow><mover accent="true"><mi mathvariant="bold-italic">x</mi><mo>^</mo></mover></mrow><mrow><mi>k</mi></mrow><mrow><mi>a</mi></mrow></msubsup><mo>=</mo><msubsup><mrow><mi mathvariant="bold-italic">N</mi></mrow><mrow><mi>k</mi></mrow><mrow><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msubsup><mo stretchy="false">(</mo><msubsup><mrow><mi mathvariant="bold-italic">C</mi></mrow><mrow><mi>k</mi></mrow><mrow><mi mathvariant="normal">T</mi></mrow></msubsup><msubsup><mrow><mi mathvariant="bold-italic">R</mi></mrow><mrow><mi>k</mi></mrow><mrow><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msubsup><msub><mrow><mi mathvariant="bold-italic">y</mi></mrow><mrow><mi>k</mi></mrow></msub><mo>+</mo><msubsup><mrow><mi mathvariant="bold-italic">P</mi></mrow><mrow><mi>k</mi><mo stretchy="false">|</mo><mi>k</mi><mo>-</mo><mn mathvariant="normal">1</mn></mrow><mrow><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msubsup><msub><mrow><mi mathvariant="bold-italic">Φ</mi></mrow><mrow><mi>k</mi></mrow></msub><msubsup><mrow><mover accent="true"><mi mathvariant="bold-italic">x</mi><mo>^</mo></mover></mrow><mrow><mi>k</mi><mo>-</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi>a</mi></mrow></msubsup><mo stretchy="false">)</mo></mtd></mtr><mtr><mtd><msubsup><mrow><mover accent="true"><mi mathvariant="bold-italic">x</mi><mo>^</mo></mover></mrow><mrow><mi>k</mi></mrow><mrow><mi>θ</mi></mrow></msubsup><mo>=</mo><msubsup><mrow><mi mathvariant="bold-italic">N</mi></mrow><mrow><mi>k</mi></mrow><mrow><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msubsup><mo stretchy="false">(</mo><msubsup><mrow><mi mathvariant="bold-italic">P</mi></mrow><mrow><mi>k</mi><mo stretchy="false">|</mo><mi>k</mi><mo>-</mo><mn mathvariant="normal">1</mn></mrow><mrow><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msubsup><msub><mrow><mi mathvariant="bold-italic">Γ</mi></mrow><mrow><mi>k</mi></mrow></msub><msub><mrow><mover accent="true"><mi mathvariant="bold-italic">θ</mi><mo>^</mo></mover></mrow><mrow><mi>k</mi><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msub><mo>+</mo><msubsup><mrow><mi mathvariant="bold-italic">P</mi></mrow><mrow><mi>k</mi><mo stretchy="false">|</mo><mi>k</mi><mo>-</mo><mn mathvariant="normal">1</mn></mrow><mrow><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msubsup><msub><mrow><mi mathvariant="bold-italic">Φ</mi></mrow><mrow><mi>k</mi></mrow></msub><msubsup><mrow><mover accent="true"><mi mathvariant="bold-italic">x</mi><mo>^</mo></mover></mrow><mrow><mi>k</mi><mo>-</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi>θ</mi></mrow></msubsup><mo stretchy="false">)</mo><mo>+</mo></mtd></mtr><mtr><mtd><mtext> </mtext><msub><mrow><mi mathvariant="bold-italic">Τ</mi></mrow><mrow><mi>k</mi></mrow></msub><mo stretchy="false">(</mo><msub><mrow><mover accent="true"><mi mathvariant="bold-italic">θ</mi><mo>^</mo></mover></mrow><mrow><mi>k</mi></mrow></msub><mo>-</mo><msub><mrow><mover accent="true"><mi mathvariant="bold-italic">θ</mi><mo>^</mo></mover></mrow><mrow><mi>k</mi><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msub><mo stretchy="false">)</mo></mtd></mtr><mtr><mtd><msubsup><mrow><mover accent="true"><mi mathvariant="bold-italic">x</mi><mo>^</mo></mover></mrow><mrow><mi>k</mi></mrow><mrow><mi>s</mi></mrow></msubsup><mo>=</mo><msubsup><mrow><mi mathvariant="bold-italic">N</mi></mrow><mrow><mi>k</mi></mrow><mrow><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msubsup><mo stretchy="false">(</mo><msubsup><mrow><mi mathvariant="bold-italic">C</mi></mrow><mrow><mi>k</mi></mrow><mrow><mi mathvariant="normal">T</mi></mrow></msubsup><msubsup><mrow><mi mathvariant="bold-italic">R</mi></mrow><mrow><mi>k</mi></mrow><mrow><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msubsup><msub><mrow><mi mathvariant="bold-italic">B</mi></mrow><mrow><mi>k</mi></mrow></msub><msub><mrow><mover accent="true"><mi mathvariant="bold-italic">s</mi><mo>^</mo></mover></mrow><mrow><mi>k</mi><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msub><mo>+</mo><msubsup><mrow><mi mathvariant="bold-italic">P</mi></mrow><mrow><mi>k</mi><mo stretchy="false">|</mo><mi>k</mi><mo>-</mo><mn mathvariant="normal">1</mn></mrow><mrow><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msubsup><msub><mrow><mi mathvariant="bold-italic">Φ</mi></mrow><mrow><mi>k</mi></mrow></msub><msubsup><mrow><mover accent="true"><mi mathvariant="bold-italic">x</mi><mo>^</mo></mover></mrow><mrow><mi>k</mi><mo>-</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi>s</mi></mrow></msubsup><mo stretchy="false">)</mo><mo>+</mo></mtd></mtr><mtr><mtd><mtext> </mtext><msub><mrow><mi mathvariant="bold-italic">F</mi></mrow><mrow><mi>k</mi></mrow></msub><mo stretchy="false">(</mo><msub><mrow><mover accent="true"><mi mathvariant="bold-italic">s</mi><mo>^</mo></mover></mrow><mrow><mi>k</mi></mrow></msub><mo>-</mo><msub><mrow><mover accent="true"><mi mathvariant="bold-italic">s</mi><mo>^</mo></mover></mrow><mrow><mi>k</mi><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msub><mo stretchy="false">)</mo></mtd></mtr></mtable></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45116949&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45116958&type= 69.42666626 28.10933495 式（16）式（16）显然需要构造模型误差估值 <math id="M85"><msub><mrow><mover accent="true"><mi mathvariant="bold-italic">θ</mi><mo>^</mo></mover></mrow><mrow><mi>k</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45116961&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45116960&type= 2.96333337 4.65666676 和 <math id="M86"><msub><mrow><mover accent="true"><mi mathvariant="bold-italic">s</mi><mo>^</mo></mover></mrow><mrow><mi>k</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45116968&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45116952&type= 2.45533323 3.80999994 的递归解，而在实际的滤波场景中，应尽可能避免 <math id="M87"><mi mathvariant="bold-italic">θ</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45116966&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45116969&type= 2.03200006 2.79399991 和 <math id="M88"><mi mathvariant="bold-italic">s</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45116972&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45116967&type= 1.43933344 2.79399991 的同时出现，以保证滤波系统的可观测性。 <math id="M89"><msup><mrow><mi>β</mi></mrow><mrow><mi mathvariant="bold-italic">θ</mi></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45116976&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45116974&type= 3.47133350 3.89466691 、 <math id="M90"><msup><mrow><mi>β</mi></mrow><mrow><mi mathvariant="bold-italic">s</mi></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45116987&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45116985&type= 3.21733332 3.89466691 分别表示动力学模型以及观测模型修正值的比例系数，且 <math id="M91"><msup><mrow><mi>β</mi></mrow><mrow><mi>θ</mi></mrow></msup><mo>∈</mo><mo stretchy="false">[</mo><mn mathvariant="normal">0,1</mn><mo stretchy="false">]</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45116981&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45116978&type= 16.42533302 4.23333359 ， <math id="M92"><msup><mrow><mi>β</mi></mrow><mrow><mi>s</mi></mrow></msup><mo>∈</mo><mo stretchy="false">[</mo><mn mathvariant="normal">0,1</mn><mo stretchy="false">]</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45117000&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45116998&type= 16.08666801 4.23333359 。在滤波最初期，由于模型误差估计的初始值与实际不符，比例系数可适当减小；随着滤波时间的增加，比例系数可迅速调至1，保证滤波的稳定性。 2　函数模型误差的自适应估计 1 考虑滤波系统满足可观测性的条件，且模型误差 <math id="M93"><mi mathvariant="bold-italic">θ</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45117029&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45117018&type= 2.03200006 2.79399991 和 <math id="M94"><mi mathvariant="bold-italic">s</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45117033&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45117031&type= 1.43933344 2.79399991 不同时出现时的自适应滤波构建。两种常见的模型误差变化情景为：（1）模型误差随时间变化，但一定时间内稳定或呈稳定趋势变化；（2）模型误差出现突然的跳变，跳变后保持一段时间内稳定或稳定变化。针对以上两种情况，分别设计对应的自适应滤波算法。定义 <math id="M95"><msub><mrow><mi mathvariant="bold-italic">V</mi></mrow><mrow><mi>k</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45117199&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45117197&type= 4.06400013 3.80999994 为式（1）式（1）中不考虑模型误差项的标准Kalman滤波新息向量： <math display="block" id="M96"><msub><mrow><mi mathvariant="bold-italic">V</mi></mrow><mrow><mi>k</mi></mrow></msub><mo>=</mo><msub><mrow><mi mathvariant="bold-italic">y</mi></mrow><mrow><mi>k</mi></mrow></msub><mo>-</mo><msub><mrow><mi mathvariant="bold-italic">C</mi></mrow><mrow><mi>k</mi></mrow></msub><msub><mrow><mi mathvariant="bold-italic">Φ</mi></mrow><mrow><mi>k</mi></mrow></msub><msubsup><mrow><mover accent="true"><mi mathvariant="bold-italic">x</mi><mo>^</mo></mover></mrow><mrow><mi>k</mi><mo>-</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi>a</mi></mrow></msubsup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45116996&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45117015&type= 31.91933441 4.91066647 当不存在动力学模型以及观测模型误差项 <math id="M97"><mi mathvariant="bold-italic">θ</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45117029&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45117018&type= 2.03200006 2.79399991 和 <math id="M98"><mi mathvariant="bold-italic">s</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45117033&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45117031&type= 1.43933344 2.79399991 时， <math id="M99"><msub><mrow><mi mathvariant="bold-italic">V</mi></mrow><mrow><mi>k</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45117199&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45117197&type= 4.06400013 3.80999994 应满足 <math id="M100"><msub><mrow><mi mathvariant="bold-italic">V</mi></mrow><mrow><mi>k</mi></mrow></msub><mo>∈</mo><mi>N</mi><mo stretchy="false">(</mo><mn mathvariant="normal">0</mn><mo>,</mo><msub><mrow><mi mathvariant="bold-italic">Σ</mi></mrow><mrow><mi>k</mi></mrow></msub><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45117041&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45117039&type= 21.92866707 4.48733330 ， <math id="M101"><msub><mrow><mi mathvariant="bold-italic">Σ</mi></mrow><mrow><mi>k</mi></mrow></msub><mo>=</mo><msubsup><mrow><mi mathvariant="bold-italic">R</mi></mrow><mrow><mi>k</mi></mrow><mrow/></msubsup><mo>+</mo><msub><mrow><mi mathvariant="bold-italic">C</mi></mrow><mrow><mi>k</mi></mrow></msub><msubsup><mrow><mi mathvariant="bold-italic">P</mi></mrow><mrow><mi>k</mi><mo stretchy="false">|</mo><mi>k</mi><mo>-</mo><mn mathvariant="normal">1</mn></mrow><mrow/></msubsup><msubsup><mrow><mi mathvariant="bold-italic">C</mi></mrow><mrow><mi>k</mi></mrow><mrow><mi mathvariant="normal">T</mi></mrow></msubsup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45117058&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45117025&type= 34.12067032 4.06400013 。 2.1　动力学模型误差的自适应估计 2 假设仅存在动力学模型误差 <math id="M102"><msub><mrow><mi mathvariant="bold-italic">θ</mi></mrow><mrow><mi>k</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45117048&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45117045&type= 2.96333337 3.72533321 时，结合 <math id="M103"><msubsup><mrow><mover accent="true"><mi mathvariant="bold-italic">x</mi><mo>^</mo></mover></mrow><mrow><mi>k</mi><mo>-</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi>θ</mi></mrow></msubsup><mo>=</mo><msub><mrow><mi mathvariant="bold-italic">Τ</mi></mrow><mrow><mi>k</mi><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msub><msub><mrow><mover accent="true"><mi mathvariant="bold-italic">θ</mi><mo>^</mo></mover></mrow><mrow><mi>k</mi><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45117060&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45117049&type= 25.56933212 4.91066647 ， <math id="M104"><msub><mrow><mi mathvariant="bold-italic">V</mi></mrow><mrow><mi>k</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45117199&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45117197&type= 4.06400013 3.80999994 同样满足： <math display="block" id="M105"><mtable><mtr><mtd><msub><mrow><mi mathvariant="bold-italic">V</mi></mrow><mrow><mi>k</mi></mrow></msub><mo>=</mo><msub><mrow><mi mathvariant="bold-italic">C</mi></mrow><mrow><mi>k</mi></mrow></msub><msub><mrow><mi mathvariant="bold-italic">Φ</mi></mrow><mrow><mi>k</mi></mrow></msub><msub><mrow><mi mathvariant="bold-italic">Τ</mi></mrow><mrow><mi>k</mi><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msub><msub><mrow><mover accent="true"><mi mathvariant="bold-italic">θ</mi><mo>^</mo></mover></mrow><mrow><mi>k</mi><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msub><mo>+</mo><msub><mrow><mi mathvariant="bold-italic">C</mi></mrow><mrow><mi>k</mi></mrow></msub><msub><mrow><mi mathvariant="bold-italic">Γ</mi></mrow><mrow><mi>k</mi></mrow></msub><msub><mrow><mover accent="true"><mi mathvariant="bold-italic">θ</mi><mo>^</mo></mover></mrow><mrow><mi>k</mi><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msub><mo>=</mo></mtd></mtr><mtr><mtd><mo stretchy="false">(</mo><msub><mrow><mi mathvariant="bold-italic">C</mi></mrow><mrow><mi>k</mi></mrow></msub><msub><mrow><mi mathvariant="bold-italic">Φ</mi></mrow><mrow><mi>k</mi></mrow></msub><msub><mrow><mi mathvariant="bold-italic">Τ</mi></mrow><mrow><mi>k</mi><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msub><mo>+</mo><msub><mrow><mi mathvariant="bold-italic">C</mi></mrow><mrow><mi>k</mi></mrow></msub><msub><mrow><mi mathvariant="bold-italic">Γ</mi></mrow><mrow><mi>k</mi></mrow></msub><mo stretchy="false">)</mo><msub><mrow><mover accent="true"><mi mathvariant="bold-italic">θ</mi><mo>^</mo></mover></mrow><mrow><mi>k</mi><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msub></mtd></mtr></mtable></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45117074&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45117057&type= 54.10200119 10.75266743 式（17）式（17）与式（18）式（18）左右两边分别相减，得到： <math display="block" id="M106"><mtable><mtr><mtd><msub><mrow><mi mathvariant="bold-italic">y</mi></mrow><mrow><mi>k</mi></mrow></msub><mo>-</mo><msub><mrow><mi mathvariant="bold-italic">C</mi></mrow><mrow><mi>k</mi></mrow></msub><mo stretchy="false">(</mo><msub><mrow><mi mathvariant="bold-italic">Φ</mi></mrow><mrow><mi>k</mi></mrow></msub><msubsup><mrow><mover accent="true"><mi mathvariant="bold-italic">x</mi><mo>^</mo></mover></mrow><mrow><mi>k</mi><mo>-</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi>a</mi></mrow></msubsup><mo stretchy="false">)</mo><mo>-</mo><mo stretchy="false">(</mo><msub><mrow><mi mathvariant="bold-italic">C</mi></mrow><mrow><mi>k</mi></mrow></msub><msub><mrow><mi mathvariant="bold-italic">Φ</mi></mrow><mrow><mi>k</mi></mrow></msub><msub><mrow><mi mathvariant="bold-italic">Τ</mi></mrow><mrow><mi>k</mi><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msub><mo>+</mo><msub><mrow><mi mathvariant="bold-italic">C</mi></mrow><mrow><mi>k</mi></mrow></msub><msub><mrow><mi mathvariant="bold-italic">Γ</mi></mrow><mrow><mi>k</mi></mrow></msub><mo stretchy="false">)</mo><msub><mrow><mover accent="true"><mi mathvariant="bold-italic">θ</mi><mo>^</mo></mover></mrow><mrow><mi>k</mi><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msub><mo>=</mo></mtd></mtr><mtr><mtd><msub><mrow><mi mathvariant="bold-italic">y</mi></mrow><mrow><mi>k</mi></mrow></msub><mo>-</mo><msub><mrow><mi mathvariant="bold-italic">C</mi></mrow><mrow><mi>k</mi></mrow></msub><mo stretchy="false">(</mo><msub><mrow><mi mathvariant="bold-italic">Φ</mi></mrow><mrow><mi>k</mi></mrow></msub><msubsup><mrow><mover accent="true"><mi mathvariant="bold-italic">x</mi><mo>^</mo></mover></mrow><mrow><mi>k</mi><mo>-</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi>a</mi></mrow></msubsup><mo>+</mo><msub><mrow><mi mathvariant="bold-italic">Φ</mi></mrow><mrow><mi>k</mi></mrow></msub><msubsup><mrow><mover accent="true"><mi mathvariant="bold-italic">x</mi><mo>^</mo></mover></mrow><mrow><mi>k</mi><mo>-</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi>θ</mi></mrow></msubsup><mo>+</mo><msub><mrow><mi mathvariant="bold-italic">Γ</mi></mrow><mrow><mi>k</mi></mrow></msub><msub><mrow><mover accent="true"><mi mathvariant="bold-italic">θ</mi><mo>^</mo></mover></mrow><mrow><mi>k</mi><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msub><mo stretchy="false">)</mo><mo>=</mo></mtd></mtr><mtr><mtd><msub><mrow><mi mathvariant="bold-italic">y</mi></mrow><mrow><mi>k</mi></mrow></msub><mo>-</mo><msub><mrow><mi mathvariant="bold-italic">C</mi></mrow><mrow><mi>k</mi></mrow></msub><msubsup><mrow><mover accent="true"><mi mathvariant="bold-italic">x</mi><mo>^</mo></mover></mrow><mrow><mi>k</mi></mrow><mrow/></msubsup><mo>=</mo><mn mathvariant="normal">0</mn></mtd></mtr></mtable></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45117078&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45117076&type= 72.64399719 17.69533348 则 <math id="M107"><msub><mrow><mover accent="true"><mi mathvariant="bold-italic">θ</mi><mo>^</mo></mover></mrow><mrow><mi>k</mi><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45117069&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45117068&type= 6.51933336 4.65666676 和 <math id="M108"><msub><mrow><mi mathvariant="bold-italic">V</mi></mrow><mrow><mi>k</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45117199&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45117197&type= 4.06400013 3.80999994 可表示为： <math display="block" id="M109"><mtable columnalign="left"><mtr><mtd><msub><mrow><mi mathvariant="bold-italic">V</mi></mrow><mrow><mi>k</mi></mrow></msub><mo>=</mo><msub><mrow><mi mathvariant="bold-italic">y</mi></mrow><mrow><mi>k</mi></mrow></msub><mo>-</mo><msub><mrow><mi mathvariant="bold-italic">C</mi></mrow><mrow><mi>k</mi></mrow></msub><mo stretchy="false">(</mo><msub><mrow><mi mathvariant="bold-italic">Φ</mi></mrow><mrow><mi>k</mi></mrow></msub><msubsup><mrow><mover accent="true"><mi mathvariant="bold-italic">x</mi><mo>^</mo></mover></mrow><mrow><mi>k</mi><mo>-</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi>a</mi></mrow></msubsup><mo stretchy="false">)</mo><mo>=</mo></mtd></mtr><mtr><mtd><mo stretchy="false">(</mo><msub><mrow><mi mathvariant="bold-italic">C</mi></mrow><mrow><mi>k</mi></mrow></msub><msub><mrow><mi mathvariant="bold-italic">Φ</mi></mrow><mrow><mi>k</mi></mrow></msub><msub><mrow><mi mathvariant="bold-italic">Τ</mi></mrow><mrow><mi>k</mi><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msub><mo>+</mo><msub><mrow><mi mathvariant="bold-italic">C</mi></mrow><mrow><mi>k</mi></mrow></msub><msub><mrow><mi mathvariant="bold-italic">Γ</mi></mrow><mrow><mi>k</mi></mrow></msub><mo stretchy="false">)</mo><msub><mrow><mover accent="true"><mi mathvariant="bold-italic">θ</mi><mo>^</mo></mover></mrow><mrow><mi>k</mi><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msub><mo>≜</mo><msub><mrow><mi mathvariant="bold-italic">D</mi></mrow><mrow><mi>k</mi></mrow></msub><msub><mrow><mover accent="true"><mi mathvariant="bold-italic">θ</mi><mo>^</mo></mover></mrow><mrow><mi>k</mi><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msub></mtd></mtr></mtable></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45117072&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45117070&type= 53.17066193 11.09133339 结合式（20）式（20），用序贯最小二乘（recursive least squares，RLS）递归计算，得： <math display="block" id="M110"><mfenced open="{" close="" separators="|"><mrow><mtable columnalign="left"><mtr><mtd><msubsup><mrow><mi mathvariant="bold-italic">Ψ</mi></mrow><mrow><mi>k</mi></mrow><mrow><mi>θ</mi></mrow></msubsup><mo>=</mo><mo stretchy="false">(</mo><msub><mrow><mi mathvariant="bold-italic">Σ</mi></mrow><mrow><mi>k</mi></mrow></msub><mo>+</mo><msub><mrow><mi mathvariant="bold-italic">D</mi></mrow><mrow><mi>k</mi></mrow></msub><msubsup><mrow><mi mathvariant="bold-italic">Λ</mi></mrow><mrow><mi>k</mi><mo>-</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi>θ</mi></mrow></msubsup><msubsup><mrow><mi mathvariant="bold-italic">D</mi></mrow><mrow><mi>k</mi></mrow><mrow><mi mathvariant="normal">T</mi></mrow></msubsup><msup><mrow><mo stretchy="false">)</mo></mrow><mrow><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msup></mtd></mtr><mtr><mtd><msubsup><mrow><mi mathvariant="bold-italic">G</mi></mrow><mrow><mi>k</mi></mrow><mrow><mi>θ</mi></mrow></msubsup><mo>=</mo><msubsup><mrow><mi mathvariant="bold-italic">Λ</mi></mrow><mrow><mi>k</mi><mo>-</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi>θ</mi></mrow></msubsup><msubsup><mrow><mi mathvariant="bold-italic">D</mi></mrow><mrow><mi>k</mi></mrow><mrow><mi mathvariant="normal">T</mi></mrow></msubsup><msubsup><mrow><mi mathvariant="bold-italic">Ψ</mi></mrow><mrow><mi>k</mi></mrow><mrow><mi>θ</mi></mrow></msubsup></mtd></mtr><mtr><mtd><msubsup><mrow><mi mathvariant="bold-italic">Λ</mi></mrow><mrow><mi>k</mi></mrow><mrow><mi>θ</mi></mrow></msubsup><mo>=</mo><msubsup><mrow><mi mathvariant="bold-italic">Λ</mi></mrow><mrow><mi>k</mi><mo>-</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi>θ</mi></mrow></msubsup><mo>-</mo><msubsup><mrow><mi mathvariant="bold-italic">Λ</mi></mrow><mrow><mi>k</mi><mo>-</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi>θ</mi></mrow></msubsup><msubsup><mrow><mi mathvariant="bold-italic">D</mi></mrow><mrow><mi>k</mi></mrow><mrow><mi mathvariant="normal">T</mi></mrow></msubsup><msubsup><mrow><mi mathvariant="bold-italic">Ψ</mi></mrow><mrow><mi>k</mi></mrow><mrow><mi>θ</mi></mrow></msubsup><msub><mrow><mi mathvariant="bold-italic">D</mi></mrow><mrow><mi>k</mi></mrow></msub><msubsup><mrow><mi mathvariant="bold-italic">Λ</mi></mrow><mrow><mi>k</mi><mo>-</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi>θ</mi></mrow></msubsup></mtd></mtr><mtr><mtd><msubsup><mrow><mover accent="true"><mi mathvariant="bold-italic">θ</mi><mo>^</mo></mover></mrow><mrow><mi>k</mi></mrow><mrow><mi mathvariant="normal">R</mi><mi mathvariant="normal">L</mi><mi mathvariant="normal">S</mi></mrow></msubsup><mo>=</mo><msubsup><mrow><mover accent="true"><mi mathvariant="bold-italic">θ</mi><mo>^</mo></mover></mrow><mrow><mi>k</mi><mo>-</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi mathvariant="normal">R</mi><mi mathvariant="normal">L</mi><mi mathvariant="normal">S</mi></mrow></msubsup><mo>+</mo><msubsup><mrow><mi mathvariant="bold-italic">G</mi></mrow><mrow><mi>k</mi></mrow><mrow><mi>θ</mi></mrow></msubsup><mo stretchy="false">(</mo><msub><mrow><mi mathvariant="bold-italic">V</mi></mrow><mrow><mi>k</mi></mrow></msub><mo>-</mo><msub><mrow><mi mathvariant="bold-italic">D</mi></mrow><mrow><mi>k</mi></mrow></msub><msubsup><mrow><mover accent="true"><mi mathvariant="bold-italic">θ</mi><mo>^</mo></mover></mrow><mrow><mi>k</mi><mo>-</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi mathvariant="normal">R</mi><mi mathvariant="normal">L</mi><mi mathvariant="normal">S</mi></mrow></msubsup><mo stretchy="false">)</mo></mtd></mtr></mtable></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45117106&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45117104&type= 52.15466690 20.91266632 式（21）式（21）中， <math id="M111"><msubsup><mrow><mover accent="true"><mi mathvariant="bold-italic">θ</mi><mo>^</mo></mover></mrow><mrow><mi>k</mi></mrow><mrow><mi mathvariant="normal">R</mi><mi mathvariant="normal">L</mi><mi mathvariant="normal">S</mi></mrow></msubsup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45117109&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45117094&type= 6.18066692 4.91066647 表示动力学模型误差的序贯最小二乘估计，并考虑了动力学模型误差的稳定性。若考虑系统噪声随时间的趋势性变化，可在式（21）式（21）中引入遗忘因子 <math id="M112"><msup><mrow><mi>λ</mi></mrow><mrow><mi>θ</mi></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45117115&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45117097&type= 2.70933342 3.04800010 ， <math id="M113"><msup><mrow><mi>λ</mi></mrow><mrow><mi>θ</mi></mrow></msup><mo>∈</mo><mo stretchy="false">[</mo><mn mathvariant="normal">0,1</mn><mo stretchy="false">]</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45117099&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45117098&type= 15.40933418 3.80999994 ，增大当前残差信息对系统噪声估计的影响。此时的自适应RLS算法表示为： <math display="block" id="M114"><mfenced open="{" close="" separators="|"><mrow><mtable columnalign="left"><mtr><mtd><msubsup><mrow><mover accent="true"><mi mathvariant="bold-italic">Ψ</mi><mo>¯</mo></mover></mrow><mrow><mi>k</mi></mrow><mrow><mi>θ</mi></mrow></msubsup><mo>=</mo><mo stretchy="false">(</mo><msup><mrow><mi>λ</mi></mrow><mrow><mi>θ</mi></mrow></msup><msub><mrow><mi mathvariant="bold-italic">Σ</mi></mrow><mrow><mi>k</mi></mrow></msub><mo>+</mo><msub><mrow><mi mathvariant="bold-italic">D</mi></mrow><mrow><mi>k</mi></mrow></msub><msubsup><mrow><mover accent="true"><mi mathvariant="bold-italic">Λ</mi><mo>¯</mo></mover></mrow><mrow><mi>k</mi><mo>-</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi>θ</mi></mrow></msubsup><msubsup><mrow><mi mathvariant="bold-italic">D</mi></mrow><mrow><mi>k</mi></mrow><mrow><mi mathvariant="normal">T</mi></mrow></msubsup><msup><mrow><mo stretchy="false">)</mo></mrow><mrow><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msup></mtd></mtr><mtr><mtd><msubsup><mrow><mover accent="true"><mi mathvariant="bold-italic">Λ</mi><mo>¯</mo></mover></mrow><mrow><mi>k</mi></mrow><mrow><mi>θ</mi></mrow></msubsup><mo>=</mo><mo stretchy="false">(</mo><msup><mrow><mi>λ</mi></mrow><mrow><mi>θ</mi></mrow></msup><msup><mrow><mo stretchy="false">)</mo></mrow><mrow><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msup><msubsup><mrow><mover accent="true"><mi mathvariant="bold-italic">Λ</mi><mo>¯</mo></mover></mrow><mrow><mi>k</mi><mo>-</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi>θ</mi></mrow></msubsup><mo>-</mo><mo stretchy="false">(</mo><msup><mrow><mi>λ</mi></mrow><mrow><mi>θ</mi></mrow></msup><msup><mrow><mo stretchy="false">)</mo></mrow><mrow><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msup><msubsup><mrow><mover accent="true"><mi mathvariant="bold-italic">Λ</mi><mo>¯</mo></mover></mrow><mrow><mi>k</mi><mo>-</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi>θ</mi></mrow></msubsup><msubsup><mrow><mi mathvariant="bold-italic">D</mi></mrow><mrow><mi>k</mi></mrow><mrow><mi mathvariant="normal">T</mi></mrow></msubsup><msubsup><mrow><mover accent="true"><mi mathvariant="bold-italic">Ψ</mi><mo>¯</mo></mover></mrow><mrow><mi>k</mi></mrow><mrow><mi>θ</mi></mrow></msubsup><msubsup><mrow><mi mathvariant="bold-italic">D</mi></mrow><mrow><mi>k</mi></mrow><mrow/></msubsup><msubsup><mrow><mover accent="true"><mi mathvariant="bold-italic">Λ</mi><mo>¯</mo></mover></mrow><mrow><mi>k</mi><mo>-</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi>θ</mi></mrow></msubsup></mtd></mtr></mtable></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45117122&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45117121&type= 70.18866730 9.99066734 考虑部分模型误差的突变特性，利用式（17）式（17）中的新息向量 <math id="M115"><msub><mrow><mover><mrow><mi mathvariant="bold-italic">v</mi></mrow><mrow><mi>˜</mi></mrow></mover></mrow><mrow><mi>k</mi></mrow></msub><mo>=</mo><msub><mrow><mi mathvariant="bold-italic">V</mi></mrow><mrow><mi>k</mi></mrow></msub><mo>-</mo><msub><mrow><mi mathvariant="bold-italic">D</mi></mrow><mrow><mi>k</mi></mrow></msub><msubsup><mrow><mover accent="true"><mi mathvariant="bold-italic">θ</mi><mo>^</mo></mover></mrow><mrow><mi>k</mi><mo>-</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi mathvariant="normal">R</mi><mi mathvariant="normal">L</mi><mi mathvariant="normal">S</mi></mrow></msubsup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45117126&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45117123&type= 27.26266861 4.99533367 和二段自适应因子［ 15 15 - 16 16 15-16 ］，自适应RLS2算法为： <math display="block" id="M116"><mfenced open="{" close="" separators="|"><mrow><mtable columnalign="left"><mtr><mtd><msubsup><mrow><mover><mrow><mi mathvariant="bold-italic">Ψ</mi></mrow><mrow><mi>˜</mi></mrow></mover></mrow><mrow><mi>k</mi></mrow><mrow><mi>θ</mi></mrow></msubsup><mo>=</mo><mo stretchy="false">(</mo><msub><mrow><mi>α</mi></mrow><mrow><mi>k</mi></mrow></msub><msub><mrow><mi mathvariant="bold-italic">Σ</mi></mrow><mrow><mi>k</mi></mrow></msub><mo>+</mo><msub><mrow><mi mathvariant="bold-italic">D</mi></mrow><mrow><mi>k</mi></mrow></msub><msubsup><mrow><mover><mrow><mi mathvariant="bold-italic">Λ</mi></mrow><mrow><mi>˜</mi></mrow></mover></mrow><mrow><mi>k</mi><mo>-</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi>θ</mi></mrow></msubsup><msubsup><mrow><mi mathvariant="bold-italic">D</mi></mrow><mrow><mi>k</mi></mrow><mrow><mi mathvariant="normal">T</mi></mrow></msubsup><msup><mrow><mo stretchy="false">)</mo></mrow><mrow><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msup></mtd></mtr><mtr><mtd><msubsup><mrow><mover><mrow><mi mathvariant="bold-italic">Λ</mi></mrow><mrow><mi>˜</mi></mrow></mover></mrow><mrow><mi>k</mi></mrow><mrow><mi>θ</mi></mrow></msubsup><mo>=</mo><msubsup><mrow><mi>α</mi></mrow><mrow><mi>k</mi></mrow><mrow><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msubsup><msubsup><mrow><mover><mrow><mi mathvariant="bold-italic">Λ</mi></mrow><mrow><mi>˜</mi></mrow></mover></mrow><mrow><mi>k</mi><mo>-</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi>θ</mi></mrow></msubsup><mo>-</mo><msubsup><mrow><mi>α</mi></mrow><mrow><mi>k</mi></mrow><mrow><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msubsup><msubsup><mrow><mover><mrow><mi mathvariant="bold-italic">Λ</mi></mrow><mrow><mi>˜</mi></mrow></mover></mrow><mrow><mi>k</mi><mo>-</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi>θ</mi></mrow></msubsup><msubsup><mrow><mi mathvariant="bold-italic">D</mi></mrow><mrow><mi>k</mi></mrow><mrow><mi mathvariant="normal">T</mi></mrow></msubsup><msubsup><mrow><mover><mrow><mi mathvariant="bold-italic">Ψ</mi></mrow><mrow><mi>˜</mi></mrow></mover></mrow><mrow><mi>k</mi></mrow><mrow><mi>θ</mi></mrow></msubsup><msubsup><mrow><mi mathvariant="bold-italic">D</mi></mrow><mrow><mi>k</mi></mrow><mrow/></msubsup><msubsup><mrow><mover><mrow><mi mathvariant="bold-italic">Λ</mi></mrow><mrow><mi>˜</mi></mrow></mover></mrow><mrow><mi>k</mi><mo>-</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi>θ</mi></mrow></msubsup></mtd></mtr></mtable></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45117130&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45117133&type= 63.33066940 10.49866676 其中， <math id="M117"><msub><mrow><mi>α</mi></mrow><mrow><mi>k</mi></mrow></msub><mo>≈</mo><mfenced open="{" close="" separators="|"><mrow><mtable columnalign="left"><mtr><mtd><mn mathvariant="normal">1</mn><mo>,</mo><mi mathvariant="normal">Δ</mi><msub><mrow><mover><mrow><mi>v</mi></mrow><mrow><mi>˜</mi></mrow></mover></mrow><mrow><mi>k</mi></mrow></msub><mo>≤</mo><mi>c</mi></mtd></mtr><mtr><mtd><mfrac><mrow><mi>c</mi></mrow><mrow><mi mathvariant="normal">Δ</mi><msub><mrow><mover><mrow><mi>v</mi></mrow><mrow><mi>˜</mi></mrow></mover></mrow><mrow><mi>k</mi></mrow></msub></mrow></mfrac><mo>,</mo><mi mathvariant="normal">Δ</mi><msub><mrow><mover><mrow><mi>v</mi></mrow><mrow><mi>˜</mi></mrow></mover></mrow><mrow><mi>k</mi></mrow></msub><mo>≥</mo><mi>c</mi></mtd></mtr></mtable></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45117139&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45117131&type= 29.71800041 14.39333248 ； <math id="M118"><mi mathvariant="normal">Δ</mi><msub><mrow><mover><mrow><mi>v</mi></mrow><mrow><mi>˜</mi></mrow></mover></mrow><mrow><mi>k</mi></mrow></msub><mo>=</mo><mfrac><mrow><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>i</mi><mo>=</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi>s</mi></mrow></munderover></mstyle><mrow><msubsup><mrow><mover><mrow><mi>v</mi></mrow><mrow><mi>˜</mi></mrow></mover></mrow><mrow><mi>k</mi><mo>,</mo><mi>i</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msubsup></mrow></mrow><mrow><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>i</mi><mo>=</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi>s</mi></mrow></munderover></mstyle><mrow><msubsup><mrow><mi>σ</mi></mrow><mrow><mi>k</mi><mo>,</mo><mi>i</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msubsup></mrow></mrow></mfrac></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45117146&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45117148&type= 19.98133469 18.54199982 ； s 为观测值个数； <math id="M119"><msubsup><mrow><mi>σ</mi></mrow><mrow><mi>k</mi><mo>,</mo><mi>i</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msubsup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45117152&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45117164&type= 4.31799984 3.97933316 为 <math id="M120"><msub><mrow><mi mathvariant="bold-italic">Σ</mi></mrow><mrow><mi>k</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45117154&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45117168&type= 3.89466691 3.72533321 的对角线元素； <math id="M121"><mi>c</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45117158&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45117173&type= 1.52400005 2.79399991 为经验常数，通常按经验设为 <math id="M122"><mi>c</mi><mo>∈</mo><mo stretchy="false">[</mo><mn mathvariant="normal">1.5,2</mn><mo stretchy="false">]</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45117176&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45117160&type= 17.10266685 3.55599999 。式（23）式（23）的意义在于减弱之前模型误差估计对当前模型误差估计影响，提高突变后的模型误差的收敛速度。 2.2　观测模型误差的自适应估计 2 假设仅存在观测模型误差项 <math id="M123"><msub><mrow><mi mathvariant="bold-italic">s</mi></mrow><mrow><mi>k</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45117193&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45117178&type= 2.45533323 3.72533321 时，且结合式 <math id="M124"><msubsup><mrow><mover accent="true"><mi mathvariant="bold-italic">x</mi><mo>^</mo></mover></mrow><mrow><mi>k</mi><mo>-</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi>s</mi></mrow></msubsup><mo>=</mo><msub><mrow><mi mathvariant="bold-italic">F</mi></mrow><mrow><mi>k</mi><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msub><msub><mrow><mover accent="true"><mi mathvariant="bold-italic">s</mi><mo>^</mo></mover></mrow><mrow><mi>k</mi><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45117185&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45117183&type= 25.23066711 4.06400013 ，新息向量 <math id="M125"><msub><mrow><mi mathvariant="bold-italic">V</mi></mrow><mrow><mi>k</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45117199&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45117197&type= 4.06400013 3.80999994 同样满足： <math display="block" id="M126"><msub><mrow><mi mathvariant="bold-italic">V</mi></mrow><mrow><mi>k</mi></mrow></msub><mo>=</mo><msub><mrow><mi mathvariant="bold-italic">B</mi></mrow><mrow><mi>k</mi></mrow></msub><msub><mrow><mover accent="true"><mi mathvariant="bold-italic">s</mi><mo>^</mo></mover></mrow><mrow><mi>k</mi></mrow></msub><mo>-</mo><msub><mrow><mi mathvariant="bold-italic">C</mi></mrow><mrow><mi>k</mi></mrow></msub><msub><mrow><mi mathvariant="bold-italic">Φ</mi></mrow><mrow><mi>k</mi></mrow></msub><msubsup><mrow><mover accent="true"><mi mathvariant="bold-italic">x</mi><mo>^</mo></mover></mrow><mrow><mi>k</mi><mo>-</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi>s</mi></mrow></msubsup><mo>=</mo><msub><mrow><mi mathvariant="bold-italic">B</mi></mrow><mrow><mi>k</mi></mrow></msub><msub><mrow><mover accent="true"><mi mathvariant="bold-italic">s</mi><mo>^</mo></mover></mrow><mrow><mi>k</mi></mrow></msub><mo>-</mo><msub><mrow><mi mathvariant="bold-italic">C</mi></mrow><mrow><mi>k</mi></mrow></msub><msub><mrow><mi mathvariant="bold-italic">Φ</mi></mrow><mrow><mi>k</mi></mrow></msub><msub><mrow><mi mathvariant="bold-italic">F</mi></mrow><mrow><mi>k</mi><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msub><msub><mrow><mover accent="true"><mi mathvariant="bold-italic">s</mi><mo>^</mo></mover></mrow><mrow><mi>k</mi><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45117203&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45117201&type= 73.23667145 4.14866638 式（17）式（17）与式（24）式（24）左右两边相减得： <math display="block" id="M127"><msub><mrow><mi mathvariant="bold-italic">y</mi></mrow><mrow><mi>k</mi></mrow></msub><mo>-</mo><msub><mrow><mi mathvariant="bold-italic">C</mi></mrow><mrow><mi>k</mi></mrow></msub><msub><mrow><mi mathvariant="bold-italic">Φ</mi></mrow><mrow><mi>k</mi></mrow></msub><mo stretchy="false">(</mo><msubsup><mrow><mover accent="true"><mi mathvariant="bold-italic">x</mi><mo>^</mo></mover></mrow><mrow><mi>k</mi><mo>-</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi>a</mi></mrow></msubsup><mo>-</mo><msubsup><mrow><mover accent="true"><mi mathvariant="bold-italic">x</mi><mo>^</mo></mover></mrow><mrow><mi>k</mi><mo>-</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi>s</mi></mrow></msubsup><mo stretchy="false">)</mo><mo>-</mo><msub><mrow><mi mathvariant="bold-italic">B</mi></mrow><mrow><mi>k</mi></mrow></msub><msub><mrow><mover accent="true"><mi mathvariant="bold-italic">s</mi><mo>^</mo></mover></mrow><mrow><mi>k</mi></mrow></msub><mo>=</mo><mn mathvariant="normal">0</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45117206&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45117205&type= 55.79533768 5.07999992 当观测模型误差有一定连续性时，同样可构造类似于序贯最小二乘的递归计算形式。参考式（21）式（21），得到： <math display="block" id="M128"><mfenced open="{" close="" separators="|"><mrow><mtable columnalign="left"><mtr><mtd><msubsup><mrow><mi mathvariant="bold-italic">Ψ</mi></mrow><mrow><mi>k</mi></mrow><mrow><mi>s</mi></mrow></msubsup><mo>=</mo><mo stretchy="false">(</mo><msub><mrow><mi mathvariant="bold-italic">Σ</mi></mrow><mrow><mi>k</mi></mrow></msub><mo>+</mo><msub><mrow><mi mathvariant="bold-italic">B</mi></mrow><mrow><mi>k</mi></mrow></msub><msubsup><mrow><mi mathvariant="bold-italic">Λ</mi></mrow><mrow><mi>k</mi><mo>-</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi>s</mi></mrow></msubsup><msubsup><mrow><mi mathvariant="bold-italic">B</mi></mrow><mrow><mi>k</mi></mrow><mrow><mi mathvariant="normal">T</mi></mrow></msubsup><msup><mrow><mo stretchy="false">)</mo></mrow><mrow><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msup></mtd></mtr><mtr><mtd><msubsup><mrow><mi mathvariant="bold-italic">G</mi></mrow><mrow><mi>k</mi></mrow><mrow><mi>s</mi></mrow></msubsup><mo>=</mo><msubsup><mrow><mi mathvariant="bold-italic">Λ</mi></mrow><mrow><mi>k</mi><mo>-</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi>s</mi></mrow></msubsup><msubsup><mrow><mi mathvariant="bold-italic">B</mi></mrow><mrow><mi>k</mi></mrow><mrow><mi mathvariant="normal">T</mi></mrow></msubsup><msubsup><mrow><mi mathvariant="bold-italic">Ψ</mi></mrow><mrow><mi>k</mi></mrow><mrow><mi>s</mi></mrow></msubsup></mtd></mtr><mtr><mtd><msubsup><mrow><mi mathvariant="bold-italic">Λ</mi></mrow><mrow><mi>k</mi></mrow><mrow><mi>s</mi></mrow></msubsup><mo>=</mo><msubsup><mrow><mi mathvariant="bold-italic">Λ</mi></mrow><mrow><mi>k</mi><mo>-</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi>s</mi></mrow></msubsup><mo>-</mo><msubsup><mrow><mi mathvariant="bold-italic">Λ</mi></mrow><mrow><mi>k</mi><mo>-</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi>s</mi></mrow></msubsup><msubsup><mrow><mi mathvariant="bold-italic">B</mi></mrow><mrow><mi>k</mi></mrow><mrow><mi mathvariant="normal">T</mi></mrow></msubsup><msubsup><mrow><mi mathvariant="bold-italic">Ψ</mi></mrow><mrow><mi>k</mi></mrow><mrow><mi>s</mi></mrow></msubsup><msub><mrow><mi mathvariant="bold-italic">B</mi></mrow><mrow><mi>k</mi></mrow></msub><msubsup><mrow><mi mathvariant="bold-italic">Λ</mi></mrow><mrow><mi>k</mi><mo>-</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi>s</mi></mrow></msubsup></mtd></mtr><mtr><mtd><msubsup><mrow><mover accent="true"><mi mathvariant="bold-italic">s</mi><mo>^</mo></mover></mrow><mrow><mi>k</mi></mrow><mrow><mi mathvariant="normal">R</mi><mi mathvariant="normal">L</mi><mi mathvariant="normal">S</mi></mrow></msubsup><mo>=</mo><msubsup><mrow><mover accent="true"><mi mathvariant="bold-italic">s</mi><mo>^</mo></mover></mrow><mrow><mi>k</mi><mo>-</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi mathvariant="normal">R</mi><mi mathvariant="normal">L</mi><mi mathvariant="normal">S</mi></mrow></msubsup><mo>+</mo><msubsup><mrow><mi mathvariant="bold-italic">G</mi></mrow><mrow><mi>k</mi></mrow><mrow><mi>s</mi></mrow></msubsup><mo stretchy="false">(</mo><msub><mrow><mi mathvariant="bold-italic">V</mi></mrow><mrow><mi>k</mi></mrow></msub><mo>+</mo><msub><mrow><mi mathvariant="bold-italic">C</mi></mrow><mrow><mi>k</mi></mrow></msub><msub><mrow><mi mathvariant="bold-italic">Φ</mi></mrow><mrow><mi>k</mi></mrow></msub><msub><mrow><mi mathvariant="bold-italic">F</mi></mrow><mrow><mi>k</mi><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msub><msubsup><mrow><mover accent="true"><mi mathvariant="bold-italic">s</mi><mo>^</mo></mover></mrow><mrow><mi>k</mi><mo>-</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi mathvariant="normal">R</mi><mi mathvariant="normal">L</mi><mi mathvariant="normal">S</mi></mrow></msubsup><mo>-</mo><msub><mrow><mi mathvariant="bold-italic">B</mi></mrow><mrow><mi>k</mi></mrow></msub><msubsup><mrow><mover accent="true"><mi mathvariant="bold-italic">s</mi><mo>^</mo></mover></mrow><mrow><mi>k</mi><mo>-</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi mathvariant="normal">R</mi><mi mathvariant="normal">L</mi><mi mathvariant="normal">S</mi></mrow></msubsup><mo stretchy="false">)</mo></mtd></mtr></mtable></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45117189&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45117208&type= 75.86132813 20.23533440 式中， <math id="M129"><msubsup><mrow><mover accent="true"><mi mathvariant="bold-italic">s</mi><mo>^</mo></mover></mrow><mrow><mi>k</mi></mrow><mrow><mi mathvariant="normal">R</mi><mi mathvariant="normal">L</mi><mi mathvariant="normal">S</mi></mrow></msubsup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45117190&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45117211&type= 5.41866684 4.06400013 表示系统误差的序贯最小二乘估计。式（26）式（26）考虑了观测模型误差的定常特性，利用RLS的序贯解对模型误差进行估计。若考虑观测模型误差随时间的趋势性变化，可在式（26）式（26）中引入遗忘因子 <math id="M130"><mi>λ</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45117527&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45117526&type= 1.77800000 2.79399991 ， <math id="M131"><mi>λ</mi><mo>∈</mo><mo stretchy="false">[</mo><mn mathvariant="normal">0,1</mn><mo stretchy="false">]</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45117227&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45117218&type= 14.47799969 3.55599999 ，增大当前残差信息对模型误差估计的影响。此时的自适应RLS3算法表示为： <math display="block" id="M132"><mfenced open="{" close="" separators="|"><mrow><mtable columnalign="left"><mtr><mtd><msubsup><mrow><mover accent="true"><mi mathvariant="bold-italic">Ψ</mi><mo>¯</mo></mover></mrow><mrow><mi>k</mi></mrow><mrow><mi>s</mi></mrow></msubsup><mo>=</mo><mo stretchy="false">(</mo><msup><mrow><mi>λ</mi></mrow><mrow><mi>s</mi></mrow></msup><msub><mrow><mi mathvariant="bold-italic">Σ</mi></mrow><mrow><mi>k</mi></mrow></msub><mo>+</mo><msup><mrow><mi>λ</mi></mrow><mrow><mi>s</mi></mrow></msup><msub><mrow><mi mathvariant="bold-italic">C</mi></mrow><mrow><mi>k</mi></mrow></msub><msub><mrow><mi mathvariant="bold-italic">Φ</mi></mrow><mrow><mi>k</mi></mrow></msub><msub><mrow><mi mathvariant="bold-italic">F</mi></mrow><mrow><mi>k</mi><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msub><msubsup><mrow><mi mathvariant="bold-italic">Λ</mi></mrow><mrow><mi>k</mi><mo>-</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi>s</mi></mrow></msubsup><msubsup><mrow><mi mathvariant="bold-italic">F</mi></mrow><mrow><mi>k</mi><mo>-</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi mathvariant="normal">T</mi></mrow></msubsup><msubsup><mrow><mi mathvariant="bold-italic">Φ</mi></mrow><mrow><mi>k</mi></mrow><mrow><mi mathvariant="normal">T</mi></mrow></msubsup><msubsup><mrow><mi mathvariant="bold-italic">C</mi></mrow><mrow><mi>k</mi></mrow><mrow><mi mathvariant="normal">T</mi></mrow></msubsup><mo>+</mo></mtd></mtr><mtr><mtd><mtext> </mtext><msub><mrow><mi mathvariant="bold-italic">B</mi></mrow><mrow><mi>k</mi></mrow></msub><msubsup><mrow><mover accent="true"><mi mathvariant="bold-italic">Λ</mi><mo>¯</mo></mover></mrow><mrow><mi>k</mi><mo>-</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi>s</mi></mrow></msubsup><msubsup><mrow><mi mathvariant="bold-italic">B</mi></mrow><mrow><mi>k</mi></mrow><mrow><mi mathvariant="normal">T</mi></mrow></msubsup><msup><mrow><mo stretchy="false">)</mo></mrow><mrow><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msup></mtd></mtr><mtr><mtd><msubsup><mrow><mover accent="true"><mi mathvariant="bold-italic">Λ</mi><mo>¯</mo></mover></mrow><mrow><mi>k</mi></mrow><mrow><mi>s</mi></mrow></msubsup><mo>=</mo><mo stretchy="false">(</mo><msup><mrow><mi>λ</mi></mrow><mrow><mi>s</mi></mrow></msup><msup><mrow><mo stretchy="false">)</mo></mrow><mrow><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msup><msubsup><mrow><mover accent="true"><mi mathvariant="bold-italic">Λ</mi><mo>¯</mo></mover></mrow><mrow><mi>k</mi><mo>-</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi>s</mi></mrow></msubsup><mo>-</mo><mo stretchy="false">(</mo><msup><mrow><mi>λ</mi></mrow><mrow><mi>s</mi></mrow></msup><msup><mrow><mo stretchy="false">)</mo></mrow><mrow><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msup><msubsup><mrow><mover accent="true"><mi mathvariant="bold-italic">Λ</mi><mo>¯</mo></mover></mrow><mrow><mi>k</mi><mo>-</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi>s</mi></mrow></msubsup><msubsup><mrow><mi mathvariant="bold-italic">B</mi></mrow><mrow><mi>k</mi></mrow><mrow><mi mathvariant="normal">T</mi></mrow></msubsup><msub><mrow><mi mathvariant="bold-italic">Ψ</mi></mrow><mrow><mi>k</mi></mrow></msub><msub><mrow><mi mathvariant="bold-italic">B</mi></mrow><mrow><mi>k</mi></mrow></msub><msubsup><mrow><mover accent="true"><mi mathvariant="bold-italic">Λ</mi><mo>¯</mo></mover></mrow><mrow><mi>k</mi><mo>-</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi>s</mi></mrow></msubsup></mtd></mtr></mtable></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45117230&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45117239&type= 70.69667053 15.57866764 式（27）式（27）的意义在于减弱之前模型误差估计对当前模型误差的估计影响，提高突变后的模型误差的收敛速度。 3　函数模型弹性自适应滤波构建 1 自适应滤波涉及到对不同的模型误差进行判别，其判别方法随模型误差的性质以及观测条件变化而做相应的设计［ 17 17 - 18 18 17-18 ］。将滤波新息和观测残差作为检验统计量外推得到估计值，至今仍是模型误差检测研究的重点方向。本文省略了对模型误差的判别过程，仅给出函数模型弹性自适应滤波构建方法。具体的滤波算法如下： 1）滤波初始化 <math display="block" id="M133"><msubsup><mrow><mover accent="true"><mi mathvariant="bold-italic">x</mi><mo>^</mo></mover></mrow><mrow><mn mathvariant="normal">0</mn></mrow><mrow><mi>a</mi></mrow></msubsup><mo>∼</mo><mi>N</mi><mo stretchy="false">(</mo><msub><mrow><mi mathvariant="bold-italic">x</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><mo>,</mo><msub><mrow><mi mathvariant="bold-italic">P</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45117234&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45117241&type= 23.11400032 4.65666676 ， <math display="block" id="M134"><msubsup><mrow><mover accent="true"><mi mathvariant="bold-italic">x</mi><mo>^</mo></mover></mrow><mrow><mi>k</mi><mo>-</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi>θ</mi></mrow></msubsup><mo>=</mo><mn mathvariant="normal">0</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45117253&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45117236&type= 13.54666710 4.06400013 ， <math display="block" id="M135"><msubsup><mrow><mover accent="true"><mi mathvariant="bold-italic">x</mi><mo>^</mo></mover></mrow><mrow><mi>k</mi><mo>-</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi>s</mi></mrow></msubsup><mo>=</mo><mn mathvariant="normal">0</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45117257&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45117255&type= 13.54666710 4.06400013 ， <math display="block" id="M136"><msub><mrow><mi mathvariant="bold-italic">Τ</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><mo>=</mo><mn mathvariant="normal">0</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45117260&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45117247&type= 10.49866676 3.72533321 ， <math display="block" id="M137"><msub><mrow><mi mathvariant="bold-italic">F</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><mo>=</mo><mn mathvariant="normal">0</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45117264&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45117262&type= 10.49866676 3.72533321 ， <math display="block" id="M138"><msubsup><mrow><mi mathvariant="bold-italic">Λ</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow><mrow><mi>θ</mi></mrow></msubsup><mo>=</mo><msup><mrow><mi>ω</mi></mrow><mrow><mi>θ</mi></mrow></msup><msub><mrow><mi mathvariant="bold-italic">I</mi></mrow><mrow><mi>p</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45117266&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45117252&type= 15.49400043 3.97933316 ， <math display="block" id="M139"><msubsup><mrow><mi mathvariant="bold-italic">Λ</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow><mrow><mi>s</mi></mrow></msubsup><mo>=</mo><msup><mrow><mi>ω</mi></mrow><mrow><mi>s</mi></mrow></msup><msub><mrow><mi mathvariant="bold-italic">I</mi></mrow><mrow><mi>q</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45117283&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45117271&type= 15.23999977 3.97933316 ， <math display="block" id="M140"><msubsup><mrow><mover accent="true"><mi mathvariant="bold-italic">θ</mi><mo>^</mo></mover></mrow><mrow><mn mathvariant="normal">0</mn></mrow><mrow/></msubsup><mo>=</mo><mn mathvariant="normal">0</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45117287&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45117273&type= 9.82133293 4.91066647 ， <math display="block" id="M141"><msubsup><mrow><mover accent="true"><mi mathvariant="bold-italic">s</mi><mo>^</mo></mover></mrow><mrow><mn mathvariant="normal">0</mn></mrow><mrow/></msubsup><mo>=</mo><mn mathvariant="normal">0</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45117276&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45117289&type= 9.31333351 4.06400013 2）状态预报 <math display="block" id="M142"><mfenced open="{" close="" separators="|"><mrow><mtable columnalign="left"><mtr><mtd><msubsup><mrow><mover accent="true"><mi mathvariant="bold-italic">x</mi><mo>^</mo></mover></mrow><mrow><mi>k</mi><mfenced open="|" close="" separators="|"><mrow><mi>k</mi><mo>-</mo><mn mathvariant="normal">1</mn></mrow></mfenced></mrow><mrow><mi>a</mi></mrow></msubsup><mo>=</mo><msub><mrow><mi mathvariant="bold-italic">Φ</mi></mrow><mrow><mi>k</mi></mrow></msub><msubsup><mrow><mover accent="true"><mi mathvariant="bold-italic">x</mi><mo>^</mo></mover></mrow><mrow><mi>k</mi><mo>-</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi>a</mi></mrow></msubsup></mtd></mtr><mtr><mtd><msubsup><mrow><mi mathvariant="bold-italic">P</mi></mrow><mrow><mi>k</mi><mo stretchy="false">|</mo><mi>k</mi><mo>-</mo><mn mathvariant="normal">1</mn></mrow><mrow/></msubsup><mo>=</mo><msub><mrow><mi mathvariant="bold-italic">Φ</mi></mrow><mrow><mi>k</mi></mrow></msub><msubsup><mrow><mi mathvariant="bold-italic">P</mi></mrow><mrow><mi>k</mi><mo>-</mo><mn mathvariant="normal">1</mn></mrow><mrow/></msubsup><msubsup><mrow><mi mathvariant="bold-italic">Φ</mi></mrow><mrow><mi>k</mi></mrow><mrow><mi mathvariant="normal">T</mi></mrow></msubsup><mo>+</mo><mi mathvariant="bold-italic">Q</mi><mmultiscripts><mrow/><mprescripts/><mrow><mi>k</mi></mrow><mrow/></mmultiscripts></mtd></mtr><mtr><mtd><msub><mrow><mi mathvariant="bold-italic">N</mi></mrow><mrow><mi>k</mi></mrow></msub><mo>=</mo><mo stretchy="false">(</mo><msubsup><mrow><mi mathvariant="bold-italic">C</mi></mrow><mrow><mi>k</mi></mrow><mrow><mi mathvariant="normal">T</mi></mrow></msubsup><msubsup><mrow><mi mathvariant="bold-italic">R</mi></mrow><mrow><mi>k</mi></mrow><mrow><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msubsup><msub><mrow><mi mathvariant="bold-italic">C</mi></mrow><mrow><mi>k</mi></mrow></msub><mo>+</mo><msubsup><mrow><mi mathvariant="bold-italic">P</mi></mrow><mrow><mi>k</mi><mo stretchy="false">|</mo><mi>k</mi><mo>-</mo><mn mathvariant="normal">1</mn></mrow><mrow><msup><mrow/><mrow><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msup></mrow></msubsup><mo stretchy="false">)</mo></mtd></mtr></mtable></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45117279&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45117293&type= 41.40200043 16.67933464 3）状态更新 <math display="block" id="M143"><mfenced open="{" close="" separators="|"><mrow><mtable columnalign="left"><mtr><mtd><msubsup><mrow><mover accent="true"><mi mathvariant="bold-italic">x</mi><mo>^</mo></mover></mrow><mrow><mi>k</mi></mrow><mrow><mi>a</mi></mrow></msubsup><mo>=</mo><msubsup><mrow><mi mathvariant="bold-italic">N</mi></mrow><mrow><mi>k</mi></mrow><mrow><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msubsup><mo stretchy="false">(</mo><msubsup><mrow><mi mathvariant="bold-italic">C</mi></mrow><mrow><mi>k</mi></mrow><mrow><mi mathvariant="normal">T</mi></mrow></msubsup><msubsup><mrow><mi mathvariant="bold-italic">R</mi></mrow><mrow><mi>k</mi></mrow><mrow><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msubsup><msub><mrow><mi mathvariant="bold-italic">y</mi></mrow><mrow><mi>k</mi></mrow></msub><mo>+</mo><msubsup><mrow><mi mathvariant="bold-italic">P</mi></mrow><mrow><mi>k</mi><mo stretchy="false">|</mo><mi>k</mi><mo>-</mo><mn mathvariant="normal">1</mn></mrow><mrow><msup><mrow/><mrow><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msup></mrow></msubsup><msub><mrow><mi mathvariant="bold-italic">Φ</mi></mrow><mrow><mi>k</mi></mrow></msub><msubsup><mrow><mover accent="true"><mi mathvariant="bold-italic">x</mi><mo>^</mo></mover></mrow><mrow><mi>k</mi><mo>-</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi>a</mi></mrow></msubsup><mo stretchy="false">)</mo></mtd></mtr><mtr><mtd><msubsup><mrow><mi mathvariant="bold-italic">P</mi></mrow><mrow><mi>k</mi></mrow><mrow/></msubsup><mo>=</mo><msubsup><mrow><mi mathvariant="bold-italic">N</mi></mrow><mrow><mi>k</mi></mrow><mrow><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msubsup></mtd></mtr></mtable></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45117302&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45117299&type= 59.35132980 10.41399956 4）模型修正量估计 <math display="block" id="M144"><mfenced open="{" close="" separators="|"><mrow><mtable columnalign="left"><mtr><mtd><msub><mrow><mi mathvariant="bold-italic">V</mi></mrow><mrow><mi>k</mi></mrow></msub><mo>=</mo><msub><mrow><mi mathvariant="bold-italic">y</mi></mrow><mrow><mi>k</mi></mrow></msub><mo>-</mo><msub><mrow><mi mathvariant="bold-italic">C</mi></mrow><mrow><mi>k</mi></mrow></msub><mo stretchy="false">(</mo><msub><mrow><mi mathvariant="bold-italic">Φ</mi></mrow><mrow><mi>k</mi></mrow></msub><msubsup><mrow><mover accent="true"><mi mathvariant="bold-italic">x</mi><mo>^</mo></mover></mrow><mrow><mi>k</mi><mo>-</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi>a</mi></mrow></msubsup><mo stretchy="false">)</mo></mtd></mtr><mtr><mtd><msub><mrow><mi mathvariant="bold-italic">Σ</mi></mrow><mrow><mi>k</mi></mrow></msub><mo>=</mo><msubsup><mrow><mi mathvariant="bold-italic">R</mi></mrow><mrow><mi>k</mi></mrow><mrow/></msubsup><mo>+</mo><msub><mrow><mi mathvariant="bold-italic">C</mi></mrow><mrow><mi>k</mi></mrow></msub><msubsup><mrow><mi mathvariant="bold-italic">P</mi></mrow><mrow><mi>k</mi><mo stretchy="false">|</mo><mi>k</mi><mo>-</mo><mn mathvariant="normal">1</mn></mrow><mrow><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msubsup><msubsup><mrow><mi mathvariant="bold-italic">C</mi></mrow><mrow><mi>k</mi></mrow><mrow><mi mathvariant="normal">T</mi></mrow></msubsup></mtd></mtr></mtable></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45117305&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45117303&type= 36.83000183 10.32933331 （1）不存在模型误差时， <math id="M145"><msup><mrow><mi>β</mi></mrow><mrow><mi>θ</mi></mrow></msup><mo>=</mo><mn mathvariant="normal">0</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45117314&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45117313&type= 10.07533360 3.89466691 ， <math id="M146"><msup><mrow><mi>β</mi></mrow><mrow><mi>s</mi></mrow></msup><mo>=</mo><mn mathvariant="normal">0</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45117319&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45117316&type= 9.82133293 3.89466691 ，则有： <math display="block" id="M147"><msubsup><mrow><mover accent="true"><mi mathvariant="bold-italic">θ</mi><mo>^</mo></mover></mrow><mrow><mi>k</mi></mrow><mrow/></msubsup><mo>=</mo><mn mathvariant="normal">0</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45117325&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45117321&type= 9.73666668 4.91066647 ， <math display="block" id="M148"><msubsup><mrow><mover accent="true"><mi mathvariant="bold-italic">s</mi><mo>^</mo></mover></mrow><mrow><mi>k</mi></mrow><mrow/></msubsup><mo>=</mo><mn mathvariant="normal">0</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45117332&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45117328&type= 9.14400005 4.06400013 ， <math display="block" id="M149"><msubsup><mrow><mover accent="true"><mi mathvariant="bold-italic">x</mi><mo>^</mo></mover></mrow><mrow><mi>k</mi></mrow><mrow><mi>θ</mi></mrow></msubsup><mo>=</mo><mn mathvariant="normal">0</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45117346&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45117312&type= 10.32933331 4.06400013 ， <math display="block" id="M150"><msubsup><mrow><mover accent="true"><mi mathvariant="bold-italic">x</mi><mo>^</mo></mover></mrow><mrow><mi>k</mi></mrow><mrow><mi>s</mi></mrow></msubsup><mo>=</mo><mn mathvariant="normal">0</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45117351&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45117349&type= 9.99066734 4.06400013 （2）仅存在动力学模型误差时， <math id="M151"><msup><mrow><mi>β</mi></mrow><mrow><mi>θ</mi></mrow></msup><mo>≠</mo><mn mathvariant="normal">0</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45117355&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45117336&type= 10.07533360 3.97933316 ， <math id="M152"><msup><mrow><mi>β</mi></mrow><mrow><mi>s</mi></mrow></msup><mo>=</mo><mn mathvariant="normal">0</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45117360&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45117339&type= 9.82133293 3.89466691 ，则有： <math display="block" id="M153"><mfenced open="{" close="" separators="|"><mrow><mtable columnalign="left"><mtr><mtd><msub><mrow><mi mathvariant="bold-italic">Τ</mi></mrow><mrow><mi>k</mi></mrow></msub><mo>=</mo><msubsup><mrow><mi mathvariant="bold-italic">N</mi></mrow><mrow><mi>k</mi></mrow><mrow><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msubsup><mo stretchy="false">(</mo><msubsup><mrow><mi mathvariant="bold-italic">P</mi></mrow><mrow><mi>k</mi><mo stretchy="false">|</mo><mi>k</mi><mo>-</mo><mn mathvariant="normal">1</mn></mrow><mrow><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msubsup><msub><mrow><mi mathvariant="bold-italic">Φ</mi></mrow><mrow><mi>k</mi></mrow></msub><msub><mrow><mi mathvariant="bold-italic">Τ</mi></mrow><mrow><mi>k</mi><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msub><mo>+</mo><msubsup><mrow><mi mathvariant="bold-italic">P</mi></mrow><mrow><mi>k</mi><mo stretchy="false">|</mo><mi>k</mi><mo>-</mo><mn mathvariant="normal">1</mn></mrow><mrow><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msubsup><msub><mrow><mi mathvariant="bold-italic">Γ</mi></mrow><mrow><mi>k</mi></mrow></msub><mo stretchy="false">)</mo></mtd></mtr><mtr><mtd><msub><mrow><mi mathvariant="bold-italic">D</mi></mrow><mrow><mi>k</mi></mrow></msub><mo>=</mo><mo stretchy="false">(</mo><msub><mrow><mi mathvariant="bold-italic">C</mi></mrow><mrow><mi>k</mi></mrow></msub><msub><mrow><mi mathvariant="bold-italic">Φ</mi></mrow><mrow><mi>k</mi></mrow></msub><msub><mrow><mi mathvariant="bold-italic">Τ</mi></mrow><mrow><mi>k</mi><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msub><mo>+</mo><msub><mrow><mi mathvariant="bold-italic">C</mi></mrow><mrow><mi>k</mi></mrow></msub><msub><mrow><mi mathvariant="bold-italic">Γ</mi></mrow><mrow><mi>k</mi></mrow></msub><mo stretchy="false">)</mo></mtd></mtr><mtr><mtd><msubsup><mrow><mi mathvariant="bold-italic">Ψ</mi></mrow><mrow><mi>k</mi></mrow><mrow><mi>θ</mi></mrow></msubsup><mo>=</mo><mo stretchy="false">(</mo><msubsup><mrow><mi>λ</mi></mrow><mrow><mi>k</mi></mrow><mrow><mi>θ</mi></mrow></msubsup><msub><mrow><mi mathvariant="bold-italic">Σ</mi></mrow><mrow><mi>k</mi></mrow></msub><mo>+</mo><msub><mrow><mi mathvariant="bold-italic">D</mi></mrow><mrow><mi>k</mi></mrow></msub><msubsup><mrow><mi mathvariant="bold-italic">Λ</mi></mrow><mrow><mi>k</mi><mo>-</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi>θ</mi></mrow></msubsup><msubsup><mrow><mi mathvariant="bold-italic">D</mi></mrow><mrow><mi>k</mi></mrow><mrow><mi mathvariant="normal">T</mi></mrow></msubsup><msup><mrow><mo stretchy="false">)</mo></mrow><mrow><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msup></mtd></mtr><mtr><mtd><msubsup><mrow><mi mathvariant="bold-italic">G</mi></mrow><mrow><mi>k</mi></mrow><mrow><mi>θ</mi></mrow></msubsup><mo>=</mo><msubsup><mrow><mi mathvariant="bold-italic">Λ</mi></mrow><mrow><mi>k</mi><mo>-</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi>θ</mi></mrow></msubsup><msubsup><mrow><mi mathvariant="bold-italic">D</mi></mrow><mrow><mi>k</mi></mrow><mrow><mi mathvariant="normal">T</mi></mrow></msubsup><msubsup><mrow><mi mathvariant="bold-italic">Ψ</mi></mrow><mrow><mi>k</mi></mrow><mrow><mi>θ</mi></mrow></msubsup></mtd></mtr><mtr><mtd><msubsup><mrow><mi mathvariant="bold-italic">Λ</mi></mrow><mrow><mi>k</mi></mrow><mrow><mi>θ</mi></mrow></msubsup><mo>=</mo><mo stretchy="false">(</mo><msubsup><mrow><mi>λ</mi></mrow><mrow><mi>k</mi></mrow><mrow><mi>θ</mi></mrow></msubsup><msup><mrow><mo stretchy="false">)</mo></mrow><mrow><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msup><msubsup><mrow><mi mathvariant="bold-italic">Λ</mi></mrow><mrow><mi>k</mi><mo>-</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi>θ</mi></mrow></msubsup><mo>-</mo><mo stretchy="false">(</mo><msubsup><mrow><mi>λ</mi></mrow><mrow><mi>k</mi></mrow><mrow><mi>θ</mi></mrow></msubsup><msup><mrow><mo stretchy="false">)</mo></mrow><mrow><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msup><msubsup><mrow><mi mathvariant="bold-italic">Λ</mi></mrow><mrow><mi>k</mi><mo>-</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi>θ</mi></mrow></msubsup><msubsup><mrow><mi mathvariant="bold-italic">D</mi></mrow><mrow><mi>k</mi></mrow><mrow><mi mathvariant="normal">T</mi></mrow></msubsup><msubsup><mrow><mi mathvariant="bold-italic">Ψ</mi></mrow><mrow><mi>k</mi></mrow><mrow><mi>θ</mi></mrow></msubsup><msub><mrow><mi mathvariant="bold-italic">D</mi></mrow><mrow><mi>k</mi></mrow></msub><msubsup><mrow><mi mathvariant="bold-italic">Λ</mi></mrow><mrow><mi>k</mi><mo>-</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi>θ</mi></mrow></msubsup></mtd></mtr><mtr><mtd><msubsup><mrow><mover accent="true"><mi mathvariant="bold-italic">θ</mi><mo>^</mo></mover></mrow><mrow><mi>k</mi></mrow><mrow/></msubsup><mo>=</mo><msubsup><mrow><mover accent="true"><mi mathvariant="bold-italic">θ</mi><mo>^</mo></mover></mrow><mrow><mi>k</mi><mo>-</mo><mn mathvariant="normal">1</mn></mrow><mrow/></msubsup><mo>+</mo><msubsup><mrow><mi mathvariant="bold-italic">G</mi></mrow><mrow><mi>k</mi></mrow><mrow><mi>θ</mi></mrow></msubsup><mo stretchy="false">(</mo><msub><mrow><mi mathvariant="bold-italic">V</mi></mrow><mrow><mi>k</mi></mrow></msub><mo>-</mo><msub><mrow><mi mathvariant="bold-italic">D</mi></mrow><mrow><mi>k</mi></mrow></msub><msubsup><mrow><mover accent="true"><mi mathvariant="bold-italic">θ</mi><mo>^</mo></mover></mrow><mrow><mi>k</mi><mo>-</mo><mn mathvariant="normal">1</mn></mrow><mrow/></msubsup><mo stretchy="false">)</mo></mtd></mtr></mtable></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45117374&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45117363&type= 70.61199951 32.00400162 （3）仅存在观测模型误差时， <math id="M154"><msup><mrow><mi>β</mi></mrow><mrow><mi>θ</mi></mrow></msup><mo>=</mo><mn mathvariant="normal">0</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45117379&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45117364&type= 10.07533360 3.89466691 ， <math id="M155"><msup><mrow><mi>β</mi></mrow><mrow><mi>s</mi></mrow></msup><mo>≠</mo><mn mathvariant="normal">0</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45117366&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45117384&type= 9.82133293 3.97933316 ，则有： <math display="block" id="M156"><mfenced open="{" close="" separators="|"><mrow><mtable columnalign="left"><mtr><mtd><msub><mrow><mi mathvariant="bold-italic">F</mi></mrow><mrow><mi>k</mi></mrow></msub><mo>=</mo><msubsup><mrow><mi mathvariant="bold-italic">N</mi></mrow><mrow><mi>k</mi></mrow><mrow><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msubsup><mo stretchy="false">(</mo><msubsup><mrow><mi mathvariant="bold-italic">C</mi></mrow><mrow><mi>k</mi></mrow><mrow><mi mathvariant="normal">T</mi></mrow></msubsup><msubsup><mrow><mi mathvariant="bold-italic">R</mi></mrow><mrow><mi>k</mi></mrow><mrow><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msubsup><msub><mrow><mi mathvariant="bold-italic">B</mi></mrow><mrow><mi>k</mi></mrow></msub><mo>+</mo><msubsup><mrow><mi mathvariant="bold-italic">P</mi></mrow><mrow><mi>k</mi><mo stretchy="false">|</mo><mi>k</mi><mo>-</mo><mn mathvariant="normal">1</mn></mrow><mrow><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msubsup><msub><mrow><mi mathvariant="bold-italic">Φ</mi></mrow><mrow><mi>k</mi></mrow></msub><msub><mrow><mi mathvariant="bold-italic">F</mi></mrow><mrow><mi>k</mi><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msub><mo stretchy="false">)</mo></mtd></mtr><mtr><mtd><msubsup><mrow><mi mathvariant="bold-italic">Ψ</mi></mrow><mrow><mi>k</mi></mrow><mrow><mi>s</mi></mrow></msubsup><mo>=</mo><mo stretchy="false">(</mo><msubsup><mrow><mi>λ</mi></mrow><mrow><mi>k</mi></mrow><mrow><mi>s</mi></mrow></msubsup><msub><mrow><mi mathvariant="bold-italic">Σ</mi></mrow><mrow><mi>k</mi></mrow></msub><mo>+</mo><msubsup><mrow><mi>λ</mi></mrow><mrow><mi>k</mi></mrow><mrow><mi>s</mi></mrow></msubsup><msub><mrow><mi mathvariant="bold-italic">C</mi></mrow><mrow><mi>k</mi></mrow></msub><msub><mrow><mi mathvariant="bold-italic">Φ</mi></mrow><mrow><mi>k</mi></mrow></msub><msub><mrow><mi mathvariant="bold-italic">F</mi></mrow><mrow><mi>k</mi><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msub><msubsup><mrow><mi mathvariant="bold-italic">Λ</mi></mrow><mrow><mi>k</mi><mo>-</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi mathvariant="normal">s</mi></mrow></msubsup><msubsup><mrow><mi mathvariant="bold-italic">F</mi></mrow><mrow><mi>k</mi><mo>-</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi mathvariant="normal">T</mi></mrow></msubsup><msubsup><mrow><mi mathvariant="bold-italic">Φ</mi></mrow><mrow><mi>k</mi></mrow><mrow><mi mathvariant="normal">T</mi></mrow></msubsup><msubsup><mrow><mi mathvariant="bold-italic">C</mi></mrow><mrow><mi>k</mi></mrow><mrow><mi mathvariant="normal">T</mi></mrow></msubsup><mo>+</mo></mtd></mtr><mtr><mtd><mtext> </mtext><msub><mrow><mi mathvariant="bold-italic">B</mi></mrow><mrow><mi>k</mi></mrow></msub><msubsup><mrow><mi mathvariant="bold-italic">Λ</mi></mrow><mrow><mi>k</mi><mo>-</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi>s</mi></mrow></msubsup><msubsup><mrow><mi mathvariant="bold-italic">B</mi></mrow><mrow><mi>k</mi></mrow><mrow><mi mathvariant="normal">T</mi></mrow></msubsup><msup><mrow><mo stretchy="false">)</mo></mrow><mrow><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msup></mtd></mtr><mtr><mtd><msubsup><mrow><mi mathvariant="bold-italic">G</mi></mrow><mrow><mi>k</mi></mrow><mrow><mi>s</mi></mrow></msubsup><mo>=</mo><msubsup><mrow><mi mathvariant="bold-italic">Λ</mi></mrow><mrow><mi>k</mi><mo>-</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi>s</mi></mrow></msubsup><msubsup><mrow><mi mathvariant="bold-italic">B</mi></mrow><mrow><mi>k</mi></mrow><mrow><mi mathvariant="normal">T</mi></mrow></msubsup><msubsup><mrow><mi mathvariant="bold-italic">Ψ</mi></mrow><mrow><mi>k</mi></mrow><mrow><mi>s</mi></mrow></msubsup></mtd></mtr><mtr><mtd><msubsup><mrow><mi mathvariant="bold-italic">Λ</mi></mrow><mrow><mi>k</mi></mrow><mrow><mi>s</mi></mrow></msubsup><mo>=</mo><mo stretchy="false">(</mo><msubsup><mrow><mi>λ</mi></mrow><mrow><mi>k</mi></mrow><mrow><mi>s</mi></mrow></msubsup><msup><mrow><mo stretchy="false">)</mo></mrow><mrow><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msup><msubsup><mrow><mi mathvariant="bold-italic">Λ</mi></mrow><mrow><mi>k</mi><mo>-</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi>s</mi></mrow></msubsup><mo>-</mo><mo stretchy="false">(</mo><msubsup><mrow><mi>λ</mi></mrow><mrow><mi>k</mi></mrow><mrow><mi>s</mi></mrow></msubsup><msup><mrow><mo stretchy="false">)</mo></mrow><mrow><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msup><msubsup><mrow><mi mathvariant="bold-italic">Λ</mi></mrow><mrow><mi>k</mi><mo>-</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi>s</mi></mrow></msubsup><msubsup><mrow><mi mathvariant="bold-italic">B</mi></mrow><mrow><mi>k</mi></mrow><mrow><mi mathvariant="normal">T</mi></mrow></msubsup><msubsup><mrow><mi mathvariant="bold-italic">Ψ</mi></mrow><mrow><mi>k</mi></mrow><mrow><mi>s</mi></mrow></msubsup><msub><mrow><mi mathvariant="bold-italic">B</mi></mrow><mrow><mi>k</mi></mrow></msub><msubsup><mrow><mi mathvariant="bold-italic">Λ</mi></mrow><mrow><mi>k</mi><mo>-</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi>s</mi></mrow></msubsup></mtd></mtr><mtr><mtd><msubsup><mrow><mover accent="true"><mi mathvariant="bold-italic">s</mi><mo>^</mo></mover></mrow><mrow><mi>k</mi></mrow><mrow/></msubsup><mo>=</mo><msubsup><mrow><mover accent="true"><mi mathvariant="bold-italic">s</mi><mo>^</mo></mover></mrow><mrow><mi>k</mi><mo>-</mo><mn mathvariant="normal">1</mn></mrow><mrow/></msubsup><mo>+</mo><msubsup><mrow><mi mathvariant="bold-italic">G</mi></mrow><mrow><mi>k</mi></mrow><mrow><mi>s</mi></mrow></msubsup><mo stretchy="false">(</mo><msub><mrow><mi mathvariant="bold-italic">V</mi></mrow><mrow><mi>k</mi></mrow></msub><mo>+</mo><msub><mrow><mi mathvariant="bold-italic">C</mi></mrow><mrow><mi>k</mi></mrow></msub><msub><mrow><mi mathvariant="bold-italic">Φ</mi></mrow><mrow><mi>k</mi></mrow></msub><msub><mrow><mi mathvariant="bold-italic">F</mi></mrow><mrow><mi>k</mi><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msub><msubsup><mrow><mover accent="true"><mi mathvariant="bold-italic">s</mi><mo>^</mo></mover></mrow><mrow><mi>k</mi><mo>-</mo><mn mathvariant="normal">1</mn></mrow><mrow/></msubsup><mo>-</mo><msub><mrow><mi mathvariant="bold-italic">B</mi></mrow><mrow><mi>k</mi></mrow></msub><msubsup><mrow><mover accent="true"><mi mathvariant="bold-italic">s</mi><mo>^</mo></mover></mrow><mrow><mi>k</mi><mo>-</mo><mn mathvariant="normal">1</mn></mrow><mrow/></msubsup><mo stretchy="false">)</mo></mtd></mtr><mtr><mtd><msubsup><mrow><mover accent="true"><mi mathvariant="bold-italic">x</mi><mo>^</mo></mover></mrow><mrow><mi>k</mi></mrow><mrow><mi>s</mi></mrow></msubsup><mo>=</mo><msubsup><mrow><mi mathvariant="bold-italic">N</mi></mrow><mrow><mi>k</mi></mrow><mrow><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msubsup><mo stretchy="false">(</mo><msubsup><mrow><mi mathvariant="bold-italic">C</mi></mrow><mrow><mi>k</mi></mrow><mrow><mi mathvariant="normal">T</mi></mrow></msubsup><msubsup><mrow><mi mathvariant="bold-italic">R</mi></mrow><mrow><mi>k</mi></mrow><mrow><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msubsup><msub><mrow><mi mathvariant="bold-italic">B</mi></mrow><mrow><mi>k</mi></mrow></msub><msub><mrow><mover accent="true"><mi mathvariant="bold-italic">s</mi><mo>^</mo></mover></mrow><mrow><mi>k</mi><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msub><mo>+</mo><msubsup><mrow><mi mathvariant="bold-italic">P</mi></mrow><mrow><mi>k</mi><mo stretchy="false">|</mo><mi>k</mi><mo>-</mo><mn mathvariant="normal">1</mn></mrow><mrow><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msubsup><msub><mrow><mi mathvariant="bold-italic">Φ</mi></mrow><mrow><mi>k</mi></mrow></msub><msubsup><mrow><mover accent="true"><mi mathvariant="bold-italic">x</mi><mo>^</mo></mover></mrow><mrow><mi>k</mi><mo>-</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi>s</mi></mrow></msubsup><mo stretchy="false">)</mo><mo>+</mo></mtd></mtr><mtr><mtd><mtext> </mtext><msub><mrow><mi mathvariant="bold-italic">F</mi></mrow><mrow><mi>k</mi></mrow></msub><mo stretchy="false">(</mo><msub><mrow><mover accent="true"><mi mathvariant="bold-italic">s</mi><mo>^</mo></mover></mrow><mrow><mi>k</mi></mrow></msub><mo>-</mo><msub><mrow><mover accent="true"><mi mathvariant="bold-italic">s</mi><mo>^</mo></mover></mrow><mrow><mi>k</mi><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msub><mo stretchy="false">)</mo></mtd></mtr></mtable></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45117391&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45117389&type= 72.81333160 42.41799927 式中， <math id="M157"><msubsup><mrow><mi>λ</mi></mrow><mrow><mi>k</mi></mrow><mrow><mi>θ</mi></mrow></msubsup><mi mathvariant="normal">、</mi><msubsup><mrow><mi>λ</mi></mrow><mrow><mi>k</mi></mrow><mrow><mi>s</mi></mrow></msubsup><mo>∈</mo><mo stretchy="false">[</mo><mn mathvariant="normal">0,1</mn><mo stretchy="false">]</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45117372&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45117371&type= 20.40466690 4.65666676 。对于渐变模型误差，其值可设为接近1的值，对于突变模型误差，选择方法参考式（23）式（23）。 5）修正状态参数 <math display="block" id="M158"><msubsup><mrow><mover accent="true"><mi mathvariant="bold-italic">x</mi><mo>^</mo></mover></mrow><mrow><mi>k</mi></mrow><mrow/></msubsup><mo>=</mo><msubsup><mrow><mover accent="true"><mi mathvariant="bold-italic">x</mi><mo>^</mo></mover></mrow><mrow><mi>k</mi></mrow><mrow><mi>a</mi></mrow></msubsup><mo>+</mo><msup><mrow><mi>β</mi></mrow><mrow><mi>θ</mi></mrow></msup><msubsup><mrow><mover accent="true"><mi mathvariant="bold-italic">x</mi><mo>^</mo></mover></mrow><mrow><mi>k</mi></mrow><mrow><mi>θ</mi></mrow></msubsup><mo>-</mo><msup><mrow><mi>β</mi></mrow><mrow><mi>s</mi></mrow></msup><msubsup><mrow><mover accent="true"><mi mathvariant="bold-italic">x</mi><mo>^</mo></mover></mrow><mrow><mi>k</mi></mrow><mrow><mi>s</mi></mrow></msubsup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45117407&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45117405&type= 34.88266373 4.91066647 式（28）~ 式（35）式（35）中， <math id="M159"><msubsup><mrow><mover accent="true"><mi mathvariant="bold-italic">x</mi><mo>^</mo></mover></mrow><mrow><mi>k</mi></mrow><mrow><mi>a</mi></mrow></msubsup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45117421&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45117402&type= 3.64066648 4.06400013 作为主要滤波器，保证滤波系统的基础适应性，当判定不含模型误差时， <math id="M160"><mi>β</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45117410&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45117408&type= 2.28600001 3.64066648 可设为0。图1 图1 为函数模型弹性自适应滤波参数变化示意图。从图1 图1 可以看出，当不存在两类模型误差时，信息滤波器直接参与状态参数的解算，此时估计结果为无偏估计；当存在观测模型误差时，进行观测模型误差滤波修正，最终滤波参数输出应减去观测模型修正项；当存在动力学模型误差时，启动动力学模型误差滤波进行修正，最终滤波参数输出应加上动力学模型修正项。本文方法较传统的增广滤波法可避免反复增维或降维运算，更能实时或准实时地自适应估计模型误差修正值，提高滤波精度。 4　实验验证与分析 1 为验证和分析本文所提函数模型弹性自适应滤波算法，结合某海域1 000 m水深实测的水下拖曳系统释放和回收的水下导航数据进行半物理仿真。水下拖体入水后由调查船通过缆绳进行拖曳作业，调查船和水下拖体的运动轨迹如图2 图2 所示，包括北（N）、东（E）、天（U）3个方向上的位置。仿真时间为6 000 s（约1.5 h）。船上搭载了Subsea7公司的Veripos星站差分系统，其定位精度约为0.3 m。水下声学定位数据由法国IxBlue公司的GAPS型超短基线系统（ultra short baseline system，USBL）提供，其测向精度约为0.15°，经声速改正后测距精度约为0.2 m。多普勒计程仪（Doppler velocity log，DVL）在距离海底200 m左右时进行底锁定，并测量相对于海底的速度，测速精度约为0.02 m/s。PHINS惯导数据具体型号为Phins-subsea III型，其中陀螺型号为fog-120，加速度型号为qa-700。在无辅助传感器条件下，60 s和120 s导航精度分别为0.06 m和0.3 m，横摇角和纵摇角均方根（root mean square，RMS）为0.01 （°）/s。由于技术封锁和其他条件限制，USBL的原始相位数据以及DVL的水跟踪数据暂时无法获得，本文结合其测量原理进行半物理仿真。仿真策略如下： 1）结合PHINS惯导数据以及USBL、DVL和深度计数据，采用IXBLUE公司商用后处理软件DELPH的前后双向滤波后处理导航结果作为动态参考。 2）DVL/USBL半物理仿真：（1）USBL测向数据利用平面波近似（plane wave，PW）原理进行仿真，并添加10 -4 量级的相位时间延迟噪声，USBL声学发射和接收单元相对位置如图3 图3 所示。USBL测距数据在底锁定时添加因环境剧烈变化造成10 m的固定偏差，其测向误差和PW定位误差如图4 图4 所示。（2）DVL水跟踪数据仿真通过参考速度结合仿真洋流数据经姿态转换成载体坐标系下的速度。DVL洋流速度为0.1~0.4 m/s，其方向及变化如图5 图5 所示，其中绿色表示底锁定成功，当洋流速度低于0.1 m/s时，不考虑其影响（洋流发生时刻为200~3 000 s）。半物理仿真实验采用USBL和DVL组合导航，并直接将DVL测量的速度作为状态模型的输入，观测方程则采用USBL的PW近似公式。当DVL测速和USBL定位信息有界时，滤波系统也为有界的。USBL通过测距和测向计算声学基阵和声信标的相对位置，其通过测定声波到达不同声基阵接收单元的时间差（或相位差），可计算探头与目标之间的相对方位。本文滤波估计的主要参数包括水下拖体的位置、声学测距偏差和洋流速度。为验证算法的有效性，采用4种方案进行对比分析，方案1为标准Kalman滤波算法，方案2为参数扩增法Kalman滤波算法，方案3为影响函数修正法，方案4为本文方法。方案2和方案3的具体细节参见文献［ 5 5 ］。自适应因子选择首先设 <math id="M161"><msubsup><mrow><mi>λ</mi></mrow><mrow><mi>k</mi></mrow><mrow><mi>θ</mi></mrow></msubsup><mo>=</mo><msubsup><mrow><mi>λ</mi></mrow><mrow><mi>k</mi></mrow><mrow><mi>s</mi></mrow></msubsup><mo>=</mo><mi>λ</mi><mo>=</mo><mn mathvariant="normal">0.99</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45117488&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45117468&type= 28.53266525 3.97933316 ，滤波后的模型未知偏差参数如图5 图5 所示。对比方案3和真值，由于本算例中的观测噪声量级大于洋流对定位偏差的影响，因此采用参数扩增法并不能有效地估计洋流，但能一定程度改善滤波的精度。对于声学测距偏差，由于其量级远高于其随机噪声，采用参数扩增法能一定程度估计出其偏差值。本文方法相比真值可以迅速利用模型误差和标准Kalman滤波新息向量直接递推函数关系，利用RLS法递归求解准确的模型偏差值。模型未知偏差解算结果对比如图6 图6 所示。对比本文方法滤波后的模型偏差修正值与模型偏差值，其中模型偏差值的毛刺主要是受随机噪声分量的影响。滤波后的 <math id="M162"><msubsup><mrow><mover accent="true"><mi mathvariant="bold-italic">x</mi><mo>^</mo></mover></mrow><mrow><mi>k</mi></mrow><mrow><mi>θ</mi></mrow></msubsup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45117472&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45117470&type= 3.55599999 4.06400013 和 <math id="M163"><msubsup><mrow><mover accent="true"><mi mathvariant="bold-italic">x</mi><mo>^</mo></mover></mrow><mrow><mi>k</mi></mrow><mrow><mi>s</mi></mrow></msubsup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45117489&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45117474&type= 3.30200005 4.06400013 分别对应洋流和测距偏差对状态参数的修正值。在200~3 000 s时，启动动力学模型误差滤波进行修正，最终滤波参数输出应加上动力学模型修正项，在3 000~6 200 s时，进行观测模型误差滤波修正，最终滤波参数输出应减去观测模型修正项。对比本文方法，模型偏差修正值解算可以直接求解N、E、U方向的模型偏差修正值，解算结果与模型偏差真值相符合，如图7 图7 所示。4种方案的位置偏差结果对比如图8 图8 所示，4种方案解算的偏差序列RMS见表1 表1 。方案 N方向RMS/m E方向RMS/m U方向RMS/m 1 1.370 1.437 6.865 2 0.575 0.604 0.790 3 0.809 0.375 0.284 4 0.155 0.189 0.120 由图7 图7 、图8 图8 和表1 表1 可知，方案1的位置计算误差的变化基本对应了图6 图6 中的模型偏差值，即两类模型误差对滤波造成的影响，显然滤波精度极大地受模型偏差影响，特别在U方向上，受观测结构的影响极大。表2 表2 统计了方案2和方案4对模型偏差解算的RMS。方案2采用参数扩增法，受量测噪声量级的影响，其滤波误差有一定程度波动，但对比方案2，方案4在一定程度上改善了滤波精度。从表1 表1 可以看出，方案3基于前一时刻残差序列，对于U方向的滤波精度改善明显，但由于未考虑模型误差的时间连续性，当随机噪声量级可以淹没模型偏差时，滤波的性能受到明显影响。方案 N方向RMS/m <math id="M164"><mi mathvariant="normal">E</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45117517&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45117516&type= 1.94733346 2.11666679 方向RMS/m <math id="M165"><mi mathvariant="normal">U</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45117519&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45117503&type= 2.28600001 2.20133328 方向RMS/m 测距/m 2 1.324 0.745 3.850 3.848 4 0.151 0.189 0.127 0.128 为验证新算法对于突变的模型偏差的估计性能，特别是不同自适应因子对滤波结果的影响。将洋流速度方向在1 500 s时调换，对比不同自适应因子的滤波结果。模型未知偏差突变后解算对比结果如图9 图9 所示。在1 500 s时，洋流的速度发生剧烈的变化；当 <math id="M166"><mi>λ</mi><mo>=</mo><mn mathvariant="normal">0.99</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45117506&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45117505&type= 13.29266548 2.79399991 时，滤波的收敛速度较慢，但收敛后的精度较高；当 <math id="M167"><mi>λ</mi><mo>=</mo><mn mathvariant="normal">0.96</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45117523&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45117507&type= 13.29266548 2.79399991 时，滤波的收敛速度较快，但收敛后的精度较低。结合不同的 <math id="M168"><mi>λ</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45117527&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45117526&type= 1.77800000 2.79399991 可以有效地权衡渐变或突变模型偏差对滤波的影响。当模型偏差发生突变时，可降低 <math id="M169"><mi>λ</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45117527&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45117526&type= 1.77800000 2.79399991 以增加当前观测对模型偏差的估计精度。 5　结　语 1 本文分析了模型偏差项和状态参数修正项之间的函数传递关系，提出了一种基于模型修正的弹性自适应滤波算法。该算法可以顾及两类模型偏差的时变性和突变性，经测试能有效地应用在水下USBL/DVL组合导航案例中，精确地估计洋流和测距偏差值。综合来看，本文提出的基于模型修正的弹性自适应滤波算法可以修正洋流和测距偏差对滤波造成的影响，半物理仿真实验结果表明，所提方法的三维偏差在0.2 m以内，两类模型误差的影响基本消除。所提方法适用于含时变模型偏差的导航滤波方案设计，为水下组合导航研究提供了一定的参考，在未来仍有极大的研究潜力。参考文献 Bian Hongwei ， Xu Jiangning ， He Hongyang ， et al . The Concept of Resilience of National Comprehensive PNT System ［J］. Geomatics and Information Science of Wuhan University ， 2021 ， 46 （ 9 ）： 1265 - 1272 . The Concept of Resilience of National Comprehensive PNT System 卞鸿巍，许江宁，何泓洋，等 . 国家综合PNT体系弹性概念［J］. 武汉大学学报（信息科学版）， 2021 ， 46 （ 9 ）： 1265 - 1272 . 国家综合PNT体系弹性概念 Jing Guifei . Analysis of Connotation and Framework of Network Management System of Comprehensive PNT ［J］. Journal of Geomatics ， 2020 ， 45 （ 6 ）： 1 - 6 . Analysis of Connotation and Framework of Network Management System of Comprehensive PNT 景贵飞 . 综合时空体系网络管理系统内涵及框架分析［J］. 测绘地理信息， 2020 ， 45 （ 6 ）： 1 - 6 . 综合时空体系网络管理系统内涵及框架分析 Ou Jikun ， Chai Yanju ， Yuan Yunbin . Adaptive Filtering for Kinematic Positioning by Selection of the Parameter Weights ［C］// Progress in Geodesy and Geodynamics . Hubei Association of Science and Technology ， 2014 ： 839 - 846 . Adaptive Filtering for Kinematic Positioning by Selection of the Parameter Weights 欧吉坤，柴艳菊，袁运斌 . 自适应选权滤波［C］ //《大地测量与地球动力学进展》论文集. 湖北省科学技术协会， 2004 ： 839 - 846 . 自适应选权滤波 Cui Xianqiang ， Yang Yuanxi ， Gao Weiguang . Comparison of Adaptive Filter Agrithmetics in Controlling Influence of Colored Noises ［J］. Geomatics and Information Science of Wuhan University ， 2006 ， 31 （ 8 ）： 731 - 735 . Comparison of Adaptive Filter Agrithmetics in Controlling Influence of Colored Noises 崔先强，杨元喜，高为广 . 多种有色噪声自适应滤波算法的比较［J］. 武汉大学学报（信息科学版）， 2006 ， 31 （ 8 ）： 731 - 735 . 多种有色噪声自适应滤波算法的比较 Zhao Changsheng ， Tao Benzao . Robust Kalman Filtering of Linear System with Colored Noises ［J］. Geomatics and Information Science of Wuhan University ， 2007 ， 32 （ 10 ）： 880 - 882 . Robust Kalman Filtering of Linear System with Colored Noises 赵长胜，陶本藻 . 有色噪声作用下的抗差卡尔曼滤波［J］. 武汉大学学报（信息科学版）， 2007 ， 32 （ 10 ）： 880 - 882 . 有色噪声作用下的抗差卡尔曼滤波 Alma M ， Darouach M . Adaptive Observers Design for a Class of Linear Descriptor Systems ［J］. Automatica ， 2014 ， 50 （ 2 ）： 578 - 583 . Adaptive Observers Design for a Class of Linear Descriptor Systems Liu Y S . Robust Adaptive Observer for Nonlinear Systems with Unmodeled Dynamics ［J］. Automatica ， 2009 ， 45 （ 8 ）： 1891 - 1895 . Robust Adaptive Observer for Nonlinear Systems with Unmodeled Dynamics Xiao Yuanbi ， Peng Rencan ， Bao Jingyang ， et al . Sounding Velocity Intergrated Error Correction Method of Multi-Beam Data Based on Kalman Filtering ［J］. Geomatics and Information Science of Wuhan University ， 2020 ， 45 （ 9 ）： 1461 - 1468 . Sounding Velocity Intergrated Error Correction Method of Multi-Beam Data Based on Kalman Filtering 肖元弼，彭认灿，暴景阳，等 . 利用卡尔曼滤波改正多波束数据声速整体误差［J］. 武汉大学学报（信息科学版）， 2020 ， 45 （ 9 ）： 1461 - 1468 . 利用卡尔曼滤波改正多波束数据声速整体误差 Yang Yuanxi ， Gao Weiguang . Comparison of Two Fading Filters and Adaptively Robust Filter ［J］. Geo⁃Spatial Information Science ， 2007 ， 10 （ 3 ）： 200 - 203 . Comparison of Two Fading Filters and Adaptively Robust Filter Yang Yuanxi ， Ren Xia ， Xu Yan . Main Progress of Adaptively Robust Filter with Applications in Navigation ［J］. Journal of Navigation and Positioning ， 2013 ， 1 （ 1 ）： 9 - 15 . Main Progress of Adaptively Robust Filter with Applications in Navigation 杨元喜，任夏，许艳 . 自适应抗差滤波理论及应用的主要进展［J］. 导航定位学报， 2013 ， 1 （ 1 ）： 9 - 15 . 自适应抗差滤波理论及应用的主要进展 Sui Lifen ， Liu Yanyu ， Wang Wei . Adaptive Sequential Adjustment and Its Application ［J］. Geomatics and Information Science of Wuhan University ， 2007 ， 32 （ 1 ）： 51 - 54 . Adaptive Sequential Adjustment and Its Application 隋立芬，刘雁雨，王威 . 自适应序贯平差及其应用［J］. 武汉大学学报（信息科学版）， 2007 ， 32 （ 1 ）： 51 - 54 . 自适应序贯平差及其应用 Zhang Q H . Adaptive Observer for Multiple-Input-Multiple-Output （MIMO） Linear Time-Varying Systems ［J］. IEEE Transactions on Automatic Control ， 2002 ， 47 （ 3 ）： 525 - 529 . Adaptive Observer for Multiple-Input-Multiple-Output （MIMO） Linear Time-Varying Systems Zhang Q H . Adaptive Kalman Filter for Actuator Fault Diagnosis ［J］. Automatica ， 2018 ， 93 ： 333 - 342 . Adaptive Kalman Filter for Actuator Fault Diagnosis Wang Zhenjie ， Liu Huimin ， Shan Rui ， et al . Hierarchical Adaptive Information Filtering Algorithm for System Noise and Observation Noise ［J］. Geomatics and Information Science of Wuhan University ， 2021 ， 46 （ 1 ）： 88 - 95 . Hierarchical Adaptive Information Filtering Algorithm for System Noise and Observation Noise 王振杰，刘慧敏，单瑞，等 . 顾及系统噪声和观测噪声的分级自适应信息滤波算法［J］. 武汉大学学报（信息科学版）， 2021 ， 46 （ 1 ）： 88 - 95 . 顾及系统噪声和观测噪声的分级自适应信息滤波算法 Yang Yuanxi ， Xu Tianhe . An Adaptive Kalman Filter Combining Variance Component Estimation with Covariance Matrix Estimation Based on Moving Window ［J］. Geomatics and Information Science of Wuhan University ， 2003 ， 28 （ 6 ）： 714 - 718 . An Adaptive Kalman Filter Combining Variance Component Estimation with Covariance Matrix Estimation Based on Moving Window 杨元喜，徐天河 . 基于移动开窗法协方差估计和方差分量估计的自适应滤波［J］. 武汉大学学报（信息科学版）， 2003 ， 28 （ 6 ）： 714 - 718 . 基于移动开窗法协方差估计和方差分量估计的自适应滤波 Yang Y X ， Gao W G . An Optimal Adaptive Kalman Filter ［J］. Journal of Geodesy ， 2006 ， 80 （ 4 ）： 177 - 183 . An Optimal Adaptive Kalman Filter Ouyang Wenyi ， Jiang Weiping ， Zhou Xiaohui ， et al . Analysis and Comparison Filtering Methods for Common Mode Error Elimination of GNSS Coordinate Series ［J］. Journal of Geomatics ， 2021 ， 46 （ S1 ）： 105 - 108 . ［1］ Analysis and Comparison Filtering Methods for Common Mode Error Elimination of GNSS Coordinate Series 欧阳文一，姜卫平，周晓慧，等 . 用于GNSS坐标序列共模误差剔除的滤波方法分析对比［J］. 测绘地理信息， 2021 ， 46 （ S1 ）： 105 - 108 . 用于GNSS坐标序列共模误差剔除的滤波方法分析对比 Tsai J S H ， Lin M H ， Zheng C H ， et al . Actuator Fault Detection and Performance Recovery with Kalman Filter-Based Adaptive Observer ［J］. International Journal of General Systems ， 2007 ， 36 （ 4 ）： 375 - 398 . Actuator Fault Detection and Performance Recovery with Kalman Filter-Based Adaptive Observer 10.13203/j.whugis20200469 P229 A 1671-8860（2023）01-0146-09 刘慧敏,王振杰,单瑞，等.函数模型误差弹性自适应滤波修正[J].武汉大学学报（信息科学版）,2023,48(1):146-154.DOI: 10.13203/j.whugis20200469 http://dx.doi.org/10.13203/j.whugis20200469 刘慧敏,王振杰,单瑞，等.函数模型误差弹性自适应滤波修正[J].武汉大学学报（信息科学版）,2023,48(1):146-154.DOI:10.13203/j.whugis20200469 LIU Huimin,WANG Zhenjie,Shan Rui,et al.Resilient Adaptive Filtering Based on Model Modification and Underwater Application[J].Geomatics and Information Science of Wuhan University,2023,48(1):146-154.DOI: 10.13203/j.whugis20200469 http://dx.doi.org/10.13203/j.whugis20200469 LIU Huimin,WANG Zhenjie,Shan Rui,et al.Resilient Adaptive Filtering Based on Model Modification and Underwater Application[J].Geomatics and Information Science of Wuhan University,2023,48(1):146-154.DOI:10.13203/j.whugis20200469 中国地质调查局项目（DD20191003）；国家自然科学基金（41406115）；青岛市市南区科技发展资金项目（2016-3-015-ZH）。 2022-03-05 2023-01-05 2023-06-29 33226205 武汉大学学报（信息科学版） 1 48

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函数模型误差弹性自适应滤波修正

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DOI：10.13203/j.whugis20200469