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Identification of time-varying systems using multi-scale radial basis function
LIU Qing, LI Yang
School of Automation Science and Electrical Engineering, Beijing University of Aeronautics and Astronautics, Beijing 100191, China
Abstract: A time-varying autoregressive model with time-varying coefficients was investigated to identify linear system parameters from nonstationary time series. The basis function of multi-scale radial basis function (MRBF) was employed, and the identification of nonstationary modeling problem was then simplified to a linear time-invariant modeling problem. Particle swarm optimization (PSO) algorithm was applied to search the optimal RBF scales for the estimation of time-varying system parameters. The basis functions of RBF can better estimate time-varying parameters with a variety of dynamic process because optimal different RBF scales with good local properties can be effectively adjusted by the PSO algorithm. One simulation example of second-order time-varying autoregressive model with time-varying parameters involved different waveform was presented to show the effectiveness of the proposed method. Compared with classical approaches of time-varying parametric estimations such as recursive least square algorithms and the expansion approach of Legendre polynomial basis function, the identification results of time-varying parameters can be more accurately estimated which validates the effectiveness of the proposed time-varying modeling method.
Key words: time-varying autoregressive model     recursive least squares algorithm     Legendre basis function     multi-scale radial basis function     particle swarm optimization     parameter identification

1 时变自回归参数建模法 1.1 时变自回归模型

p阶时变自回归参数模型输出为

1.2 时变系数辨识

2 径向基函数 2.1 高斯径向基函数

2.2 RBF维数选择

2.3 RBF中心的选择

RBF的中心决定了RBF在整个时变系数估计的位置.为了使RBF分布到整个时变系数中,以确保RBF对时变系数的所有局部进行准确估计,本文将RBF的中心均匀分布到时变系数中,设第k个RBF公式及RBF的中心分别表示为

2.4 RBF尺度的选择

RBF的候选尺度为

RBF-PSO算法主要步骤包含:

1) 初始化:包括初始化粒子、粒子个数、适应度值、局部最优粒子、全局最优粒子、粒子最大值和最小值、速度最大值、迭代次数等.取随机整数为粒子赋初始值,多种尺度的选取可根据参数变化来确定.初始适应度值设为0,粒子数和迭代次数可根据参数变化特征来调整.

2) 计算每个粒子的适应度值.即将粒子代入式(13),得到RBF的候选尺度,再代入RBF公式(11),得到多尺度RBF,将MRBF展开式方法对时变模型进行参数估计,得到测量数据y的模型预测值ŷ,最后根据式(10)计算相关指数,得到适应度值.

3) 更新局部最优粒子和全局最优粒子:找到适应度最大的粒子,该粒子作为局部最优粒子,若其适应度值比全局最优粒子大则赋值给全局最优粒子.

4) 更新速度和粒子:根据局部最优粒子和全局最优粒子更新速度和粒子.速度更新公式和粒子更新公式[16]分别为

5) 返回步骤2),重复执行步骤2)~步骤4)直到达到最大迭代次数.最后得到的全局最优粒子即为最终选择的粒子,代入式(13)得到径向基函数的最优尺度.

3 仿真实验 3.1 时变自回归模型

3.2 实验结果分析及性能评估

 图 1 相关指数随RBF维数变化 Fig. 1 Correlation index changing with RBF dimension

 图 2 MRBF展开式方法中选取的最优RBF Fig. 2 Selected optimal RBF in MRBF expansion method

 图 3 基于不同方法的时变系数辨识结果 Fig. 3 Identification results of time-varying parameters using different methods

 参数辨识方法 MAE NRMSE a1 a2 a1 a2 递推最小二乘 0.0689 0.0635 0.4929 0.4580 勒让德展开法 0.0488 0.0541 0.3676 0.3844 本文方法 0.0419 0.0410 0.3043 0.2969
4 结 论

1) 针对时变系数包含多种波形的时变系统,本文提出了一种新型的基于MRBF-PSO算法的时变系统辨识方法.该算法兼顾了多尺度径向基函数的局部分析能力及粒子群优化算法的最优估计等优点,可以准确、迅速地识别时变系统的参数,有效提高时变系统的动态跟踪能力.

2) 仿真实验结果表明,与常用的时变系数辨识算法如递推最小二乘算法、勒让德基函数展开法相比,对于在强噪声干扰下的时变系统,该新型的时变系数估计方法仍能取得良好的参数辨识效果,为时变系统建模方法的时变系数估计提供了一种新的思路.

3) 时变系统的参数辨识问题是学术界研究的难点,该算法可以有效识别包含多种波形的时变系统参数,具有普遍适用性,因而可以用于大多数时变系统的动力学问题分析.为了进一步验证提出方法的性能及工程实用性,除了仿真实验验证该算法具有良好的识别精度之外,在接下来的研究工作中,继续研究该算法对实际工程系统如高速飞行器、高速列车以及大型的柔性航天结构等动力学问题的应用前景及工程实用价值.

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#### 文章信息

LIU Qing, LI Yang

Identification of time-varying systems using multi-scale radial basis function

Journal of Beijing University of Aeronautics and Astronsutics, 2015, 41(9): 1722-1728.
http://dx.doi.org/10.13700/j.bh.1001-5965.2014.0693