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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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 . 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