Partial approximate least absolute deviation for nonlinear system identification based on radial basis function

doi:10.11949/j.issn.0438-1157.20180857

Abstract

Abstract:

For nonlinear system with nonlinear correlation of input signals, a partial approximate least absolute deviation based on radial basis function is proposed. In this paper, the observation data matrix is first extended by columns, and the output of the RBF network is used as an extension of the observed data matrix. Then, the extended observation data matrix and the output matrix are linearly regressed by using the partial approximate least absolute deviation. An approximate least absolute deviation objective function is established by introducing a deterministic function to replace the absolute value under certain situations. The proposed method can overcome the disadvantage of large square residual of least square criterion when the identification data is disturbed by the impulse noise which obeys symmetrical alpha stable (SαS) distribution. By adopting principal component analysis to eliminate the nonlinear correlation among the elements of data vector of nonlinear systems, the unique solution of model parameters can be easily acquired by the proposed method. The simulation experiments show that the proposed algorithm can directly recognize the nonlinear system with nonlinear correlation of input signals and suppress the influence of impulse noise.

Key words: nonlinear system, radial basis function, principal component analysis, partial approximate least absolute deviation, impulse noise

摘要：

针对输入信号非线性相关的非线性系统，提出了基于径向基函数的近似偏最小一乘准则辨识算法。首先对观测数据矩阵进行列扩展，以径向基函数（radial basis function，RBF）网络的输出作为观测数据矩阵的扩展项，然后利用近似偏最小一乘算法对扩展的观测矩阵和输出矩阵进行线性回归。近似偏最小一乘算法用确定性可导函数近似代替残差绝对值，可以抑制对称α稳定（symmetrical alpha stable，SαS）分布的尖峰噪声。同时，通过主成分分析去除非线性系统数据向量矩阵之间的非线性相关，得出模型参数的唯一解。仿真实验表明，本文算法可以对输入信号存在非线性相关的非线性系统进行直接辨识，抑制了尖峰噪声对辨识结果的影响，具有优良的稳健性。

关键词: 非线性系统, 径向基函数, 主成分分析, 近似偏最小一乘, 尖峰噪声

CLC Number:

TP 273

Baochang XU, Hua ZHANG, Jinshan WANG. Partial approximate least absolute deviation for nonlinear system identification based on radial basis function[J]. CIESC Journal, 2019, 70(2): 653-660.

徐宝昌, 张华, 王金山. 基于径向基函数的非线性系统近似偏最小一乘准则辨识算法[J]. 化工学报, 2019, 70(2): 653-660.

Figures/Tables 10

Fig.1 RBF structure

Fig.2 Flow diagram of RBF-PALAD algorithm

Fig.3 Relative error curve when white noise exists only

Table 1 Identified results when white noise exists only

Algorithm	a₁₁	a₂₂	b₁₁	b₂₂	c₁₂	c₂₂	$δ$ /%
RBF-PALAD	0.9625	0.9029	0.0613	-0.1387	0.3342	2.5840	1.65
RBF-PLS	0.9610	0.9036	0.0638	-0.1740	0.2464	2.0926	0.59
RBF-ALAD	NaN	NaN	NaN	NaN	NaN	NaN	NaN
true value	0.9608	0.9048	0.0699	-0.1748	0.2735	2.0381	0

Table 1 Identified results when white noise exists only

Algorithm	a₁₁	a₂₂	b₁₁	b₂₂	c₁₂	c₂₂	$δ$ /%
RBF-PALAD	0.9625	0.9029	0.0613	-0.1387	0.3342	2.5840	1.65
RBF-PLS	0.9610	0.9036	0.0638	-0.1740	0.2464	2.0926	0.59
RBF-ALAD	NaN	NaN	NaN	NaN	NaN	NaN	NaN
true value	0.9608	0.9048	0.0699	-0.1748	0.2735	2.0381	0

Fig.4 Relative error curve when α=1.5

Fig.5 Relative error curve when α=1.2

Fig.6 Relative error curve when α=0.9

Table 2 Identified results when α=1.5

Algorithm	a₁₁	a₂₂	b₁₁	b₂₂	c₁₂	c₂₂	δ/%
RBF-PALAD	0.9589	0.9088	0.0477	-0.1188	0.3742	3.1502	5.37
RBF-PLS	0.9541	0.9173	0.0076	-0.2887	2.0352	1.2742	6.39
true value	0.9608	0.9048	0.0699	-0.1748	0.2735	2.0381	0

Table 3 Identified results when α=1.2

Algorithm	a₁₁	a₂₂	b₁₁	b₂₂	c₁₂	c₂₂	δ/%
RBF-PALAD	0.9605	0.9058	0.0482	-0.1950	0.3903	1.5724	2.52
RBF-PLS	0.9543	0.9132	-0.0345	-0.4892	-0.10835	0.6921	17.47
true value	0.9608	0.9048	0.0699	-0.1748	0.2735	2.0381	0

Table 4 Identified results when α=0.9

Algorithm	a₁₁	a₂₂	b₁₁	b₂₂	c₁₂	c₂₂	δ/%
RBF-PALAD	0.9195	0.0596	1.0655	-0.1284	1.7075	3.0427	7.55
RBF-PLS	0.9571	0.9023	-0.2999	-1.3921	-0.3546	0.1617	70.00
true value	0.9608	0.9048	0.0699	-0.1748	0.2735	2.0381	0

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