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非对称偏斜噪声条件下一种鲁棒概率系统辨识算法研究

刘鑫 陈强 王兰豪 代伟

刘鑫, 陈强, 王兰豪, 代伟. 非对称偏斜噪声条件下一种鲁棒概率系统辨识算法研究. 自动化学报, xxxx, xx(x): x−xx doi: 10.16383/j.aas.c211127
引用本文: 刘鑫, 陈强, 王兰豪, 代伟. 非对称偏斜噪声条件下一种鲁棒概率系统辨识算法研究. 自动化学报, xxxx, xx(x): x−xx doi: 10.16383/j.aas.c211127
Liu Xin, Chen Qiang, Wang Lan-Hao, Dai Wei. Research on robust probabilistic system identification method with asymmetric and skewed measurement noise. Acta Automatica Sinica, xxxx, xx(x): x−xx doi: 10.16383/j.aas.c211127
Citation: Liu Xin, Chen Qiang, Wang Lan-Hao, Dai Wei. Research on robust probabilistic system identification method with asymmetric and skewed measurement noise. Acta Automatica Sinica, xxxx, xx(x): x−xx doi: 10.16383/j.aas.c211127

非对称偏斜噪声条件下一种鲁棒概率系统辨识算法研究

doi: 10.16383/j.aas.c211127
基金项目: 国家自然科学基金(62103134, 62373361, 52304309), 国家重点研发计划(2022YFB3304700), 中国博士后科学基金(2023M743776)资助
详细信息
    作者简介:

    刘鑫:中国矿业大学人工智能研究院副教授. 2019年获得哈尔滨工业大学控制科学与工程专业博士学位. 主要研究方向为系统辨识,数据驱动的工程建模,软测量方法.E-mail: 15B904027@hit.edu.cn

    陈强:中国矿业大学信息与控制工程学院硕士研究生. 主要研究方向为系统辨识. E-mail: qiangchen@cumt.edu.cn

    王兰豪:中国矿业大学炼焦煤资源绿色开发全国重点实验室副教授. 主要研究方向为复杂工业过程的工艺参数检测,优化决策与智能控制. 本文通信作者. E-mail: wanglanhao888@163.com

    代伟:中国矿业大学信息与控制工程学院、人工智能研究院教授. 主要研究方向为复杂工业过程建模,运行优化与控制. E-mail: weidai@cumt.edu.cn

Research on Robust Probabilistic System Identification Method With Asymmetric and Skewed Measurement Noise

Funds: Supported by National Natural Science Foundation of China (62103134, 62373361, 52304309), National Key Research and Development Program of China (2022YFB3304700), and China Postdoctoral Science Foundation (2023M743776)
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    Author Bio:

    LIU Xin Associate professor at the Artificial Intelligence Research Institute, China University of Mining and Technology. He received his Ph.D. degree in control science and engineering from Harbin Institute of Technology in 2019. His research interest covers system identification, data-driven process modeling, and soft sensor development

    CHEN Qiang Master student at the School of Information and Control Engineering, China University of Mining and Technology. Her main research interest is system identification

    WANG Lan-Hao Associate professor at National Key Laboratory for Green Development of Coking Coal Resources, China University of Mining and Technology. His research interest covers process parameter detection, optimal decision making, and intelligent control of complex industrial process. Corresponding author of this paper

    DAI Wei Professor at the School of Information and Control Engineering and Artificial Intelligence Research Institute, China University of Mining and Technology. His research interest covers modeling, operational optimization and control for complex industrial process

  • 摘要: 在现有的系统辨识算法中, 常用的高斯、学生氏t、拉普拉斯等噪声分布均呈现出对称的统计特性, 难以描述非对称性、有偏的输出噪声, 使得在非对称偏斜噪声条件下算法的性能下降. 基于此, 研究了一类广义双曲倾斜学生氏t(Generalized hyperbolic skew student's t, GHSkewt)分布, 并在非对称偏斜噪声条件下提出了一种线性系统鲁棒辨识算法. 首先, 对GHSkewt分布的重尾特性和偏斜特性进行了详细阐述, 在数学上证明了标准学生氏t分布可以看作是GHSkewt分布的一个特例; 其次, 引入隐含变量将GHSkewt分布进行数学分解以方便算法的推导和实现; 最后, 在期望最大化(Expectation-maximization, EM)算法下重构具有隐含变量系统的代价函数, 通过迭代优化的方式不断从被污染数据集中学习过程的动态特性和噪声分布, 实现噪声参数和模型参数的联合估计. 利用数值例子和质量弹簧阻尼机械系统验证了算法的鲁棒性、有效性和稳定性.
  • 图  1  不同自由度参数${\upsilon}$和偏斜参数${\beta}$下的GHSkewt分布和其它对称分布

    Fig.  1  GHSkewt distribution with different degrees of freedom parameters ${\upsilon}$ and skewed parameters ${\beta}$, and other symmetric distributions

    图  2  真实输出和含15%异常值比例输出

    Fig.  2  The real output and output with 15% outlier points

    图  3  偏斜噪声${{\boldsymbol{G}}_0}$的概率密度

    Fig.  3  The probability density of skewed noise ${{\boldsymbol{G}}_0}$

    图  4  偏斜噪声${{\boldsymbol{G}}_0}$下参数估计轨迹

    Fig.  4  Trajectories of the parameter estimates with skewed noise ${{\boldsymbol{G}}_0}$

    图  5  偏斜噪声${{\boldsymbol{G}}_0}$下参数估计的RPEE、RMSE和${\rm{R}}^2$值

    Fig.  5  The RPEE、RMSE and ${\rm{R}}^2$ values of the parameter estimates with skewed noise ${{\boldsymbol{G}}_0}$

    图  6  本文算法的代价函数轨迹

    Fig.  6  Trajectory of the cost function for the developed approach

    图  7  不同偏斜噪声下蒙特卡洛仿真的RPEE曲面

    Fig.  7  The RPEE surface for the monte carlos with different levels of skewed noise

    图  8  参数初始值不同时蒙特卡洛仿真的参数估计轨迹

    Fig.  8  Trajectories of the parameter estimates for the monte carlos with different parameter initial values

    图  9  质量弹簧阻尼系统

    Fig.  9  The mass-spring-damper system

    图  10  质量弹簧阻尼系统的输入输出数据

    Fig.  10  The input and output of the mass-spring-damper system

    图  11  质量弹簧阻尼系统的真实输出和含偏斜噪声输出

    Fig.  11  The real output and output with skewed noise of the mass-spring-damper system

    图  12  自我验证输出估计对比

    Fig.  12  Output estimation comparison of self-verification

    图  13  交叉验证输出估计对比

    Fig.  13  Output estimation comparison of cross-verification

    图  14  交叉验证输出估计误差

    Fig.  14  Output estimation errors of cross-verification

    表  1  异常值比例为15%时蒙特卡洛仿真参数估计结果的均值和标准差

    Table  1  The mean and standard deviation of the monte carlo parameter estimation results with 15% of outlier points

    算法 参数$ {a_1} $ 参数$ {a_2} $ 参数$ {b_1} $ 参数$ {b_2} $
    均值 标准差 均值 标准差 均值 标准差 均值 标准差
    Laplace-Iden −0.575 1 $ {3.6 \times 10^{ - 4}} $ 0.773 9 $ {1.9 \times 10^{-4}} $ 0.508 3 $ {2.8 \times 10^{-4}} $ 0.408 5 $ {2.7 \times 10^{-4}} $
    St-Iden −0.587 8 $ {3.2 \times 10^{-8}} $ 0.789 9 $ {1.1 \times 10^{-8}} $ 0.504 3 $ {6.5 \times 10^{-8}} $ 0.402 8 $ {2.0 \times 10^{-8}} $
    GHSkewt-Iden −0.587 6 $ {6.5 \times 10^{-7}} $ 0.789 9 $ {2.5 \times 10^{-7}} $ 0.504 2 $ {1.1 \times 10^{-7}} $ 0.403 0 $ {4.0 \times 10^{-7}} $
    真实值 −0.600 0 0.800 0 0.500 0 0.400 0
    下载: 导出CSV

    表  2  不同异常值比例下蒙特卡洛仿真的平均RPEE、RMSE和$ {\rm{R}}^2 $值

    Table  2  The averaged RPEE、RMSE and $ {\rm{R}}^2 $ values for the monte carlos with different ratios of outlier points

    异常值比例 Laplace-Iden St-Iden GHSkewt-Iden
    RPEE (%) RMSE $ {\rm{R}}^2 $ RPEE (%) RMSE $ {\rm{R}}^2 $ RPEE (%) RMSE $ {\rm{R}}^2 $
    5 (%) 0.71 0.014 4 0.999 5 0.61 0.010 4 0.999 8 0.39 0.007 2 0.999 9
    10 (%) 2.44 0.047 5 0.994 9 1.20 0.018 6 0.999 2 1.13 0.019 2 0.999 2
    15 (%) 3.20 0.056 4 0.992 8 1.40 0.026 7 0.998 4 1.42 0.027 1 0.998 3
    20 (%) 5.33 0.088 0 0.982 6 2.70 0.049 9 0.994 4 2.74 0.050 3 0.994 3
    下载: 导出CSV

    表  3  不同偏斜噪声下蒙特卡洛仿真的平均RPEE、RMSE和$ {\rm{R}}^2 $值

    Table  3  The averaged RPEE、RMSE and $ {\rm{R}}^2 $ values for the monte carlos with different levels of skewed noise

    偏斜噪声 Laplace-Iden St-Iden GHSkewt-Iden
    RPEE (%) RMSE $ {\rm{R}}^2 $ RPEE (%) RMSE $ {\rm{R}}^2 $ RPEE (%) RMSE $ {\rm{R}}^2 $
    $ {\boldsymbol{G}}_1 $ 10.93 0.261 5 0.841 6 7.19 0.188 9 0.917 5 1.79 0.031 5 0.997 8
    $ {\boldsymbol{G}}_2 $ 16.02 0.338 1 0.735 6 12.39 0.282 5 0.815 5 3.99 0.069 9 0.988 4
    $ {\boldsymbol{G}}_3 $ 11.78 0.260 7 0.842 6 9.40 0.218 4 0.888 6 4.41 0.077 9 0.985 5
    下载: 导出CSV

    表  4  参数初始值不同时蒙特卡洛仿真的均值和标准差

    Table  4  The mean and st andard deviation of the monte carlos with different parameter initial values

    模型参数 真实值 均值 标准差
    $ {{a_1}} $ −0.600 0 −0.586 0 0.004 5
    $ {{a_2}} $ 0.800 0 0.787 2 0.003 3
    $ {{b_1}} $ 0.500 0 0.502 8 0.006 8
    $ {{b_2}} $ 0.400 0 0.403 7 0.008 0
    下载: 导出CSV

    表  5  自我验证的RMSE和$ {\rm{R}}^2 $值

    Table  5  The RMSE and $ {\rm{R}}^2 $ values for self-verification

    质量块位置 Laplace-Iden St-Iden GHSkewt-Iden
    RMSE $ {\rm{R}}^2 $ RMSE $ {\rm{R}}^2 $ RMSE $ {\rm{R}}^2 $
    $ {\boldsymbol{x}} \ ({\rm{m}})$ 0.079 5 0.937 3 0.060 7 0.963 5 0.013 7 0.998 1
    下载: 导出CSV

    表  6  交叉验证的RMSE和$ {\rm{R}}^2 $值

    Table  6  The RMSE and $ {\rm{R}}^2 $ values for cross-verification

    质量块位置 Laplace-Iden St-Iden GHSkewt-Iden
    RMSE $ {\rm{R}}^2 $ RMSE $ {\rm{R}}^2 $ RMSE $ {\rm{R}}^2 $
    $ {\boldsymbol{x}} \ ({\rm{m}})$ 0.089 4 0.934 5 0.067 4 0.962 7 0.014 8 0.998 2
    下载: 导出CSV
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  • 收稿日期:  2023-10-09
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