Identification of Wiener Systems Based on LVW-LM-QN Algorithm

doi:10.12068/j.issn.1005-3026.2025.20240061

Abstract

Abstract:

Wiener systems， consisting of a linear dynamic subsystem and a static nonlinear subsystem in series， find extensive application in process industries such as petroleum and chemical engineering. Obtaining the model of Wiener systems holds significant importance. A nonlinear system identification method based on the linear variable weight–Levenberg Marquardt–quasi Newton （LVW-LM-QN） algorithm for Wiener systems was proposed. The Wiener system was divided into two subsystems for separate processing. For the linear dynamic part， the subspace identification method with the canonical variate analysis （CVA） algorithm was used for parameter estimation， whereas for the subsequent nonlinear static part， the LVW-LM-QN algorithm was employed for identification. Finally， the method was evaluated through numerical examples and an application case of liquid level control in a two-tank system， and the effectiveness and accuracy of the proposed method were verified by the simulation results.

Key words: Wiener system, subspace method, LVW-LM-QN algorithm, neural network, CVA algorithm

CLC Number:

N 945.14

Miao YU, Wan-li WANG, Yong-tao WEI. Identification of Wiener Systems Based on LVW-LM-QN Algorithm[J]. Journal of Northeastern University(Natural Science), 2025, 46(10): 27-35.

Figures/Tables 11

Fig.1 Block diagram of Wiener system

Fig.2 Wiener model of nonlinear subsystem as a neural network

Table 1 LVW-LM-QN identification algorithm

常量初始化：迭代最大值k_max，神经元权重 w₁， w₂，偏置参数b₁，b₂，算法停止标准e₁，e₂，学习率最值α_max，α_min， $μ$ 值,矩阵 B

found= $g (k) ∞ ≤ e 1$

While(not found)and(k $≤$ k_max)

k=k+1;

通过式(42)计算学习率

if method=LM

通过式(38)计算h_1m

初始化信赖域半径Δ

if $h 1 m ≤ e 2 w + e 2$

found=true

else

通过式(43)更新w

通过式(45)计算增益率 $ρ$

if $ρ$ $>$ 0

better=true

if连续3次迭代都满足式(47)，

则method切换到QN

end if

更新 $μ$ 值

end if

else if method=QN

通过式(40)计算

if $h q n ≤ e 2 w + e 2$

found=true

Table 1 LVW-LM-QN identification algorithm

常量初始化：迭代最大值k_max，神经元权重 w₁， w₂，偏置参数b₁，b₂，算法停止标准e₁，e₂，学习率最值α_max，α_min， $μ$ 值,矩阵 B

found= $g (k) ∞ ≤ e 1$

While(not found)and(k $≤$ k_max)

k=k+1;

通过式(42)计算学习率

if method=LM

通过式(38)计算h_1m

初始化信赖域半径Δ

if $h 1 m ≤ e 2 w + e 2$

found=true

else

通过式(43)更新w

通过式(45)计算增益率 $ρ$

if $ρ$ $>$ 0

better=true

if连续3次迭代都满足式(47)，

则method切换到QN

end if

更新 $μ$ 值

end if

else if method=QN

通过式(40)计算

if $h q n ≤ e 2 w + e 2$

found=true

Fig.3 Block diagram of proposed identification method

Fig.4 Input and output data of Wiener system

Fig.5 Output signals of Wiener system and different methods

Table 2 RMSE and MAPE of different identification

方法	RMSE	MAPE
Subspace-based	10.574 6	7.377 9
LM	1.627 3	1.281 1
LVW-LM-QN	0.497 4	0.317 1

Fig.6 Input and output data of two-tank system

Fig.7 Output signals of two-tank system and different methods

Table 3 RMSE and MAPE of different identification

方法	RMSE	MAPE
Subspace-based	0.116 6	0.627 7
LM	0.123 2	0.854 3
LVW-LM-QN	0.042 5	0.242 7

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