﻿ 非线性多项式模型结构与参数一体化辨识<sup>*</sup>
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Nonlinear polynomial model's structure and parameter integration identification
JIA Weizhou, PENG Jingbo, XIE Shousheng, LIU Yunlong, LI Tenghui, HE Dawei
Aeronautics and Astronautics Engineering Institute, Air Force Engineering University, Xi'an 710038, China
Received: 2017-07-03; Accepted: 2017-10-20; Published online: 2017-11-07 10:17
Foundation item: National Natural Science Foundation of China (51476187, 51506221)
Corresponding author. PENG Jingbo, E-mail:pjb1209@126.com
Abstract: An integration algorithm of nonlinear polynomial model structure identification and parameter identification was proposed for the linear parametric polynomial assembled model, which had wider significance in the field of nonlinear systems. The algorithm combined optimal-selecting process based on contribution items with poor-eliminating process based on redundant items in structure identification. In the optimal-selecting process, the recursive modified Gram-Schmidt (RMGS) algorithm based on output vector residual was used to select the better terms in the vector space, and some redundant non-model terms were allowed to be selected, according to the maximizing drop of the output vector projection residual. In the poor-eliminating process, the algorithm adopted the model structure poor-eliminating strategy based on modified orthogonal sequence to deal with the contribution of the orthogonal vector equally. The structure items with small contribution to the actual output were deleted from the optimal set. The structure and parameters were determined by the system completeness index. Two examples of typical nonlinear polynomial model identification simulation demonstrate the effectiveness of the algorithm.
Key words: nonlinear system identification     polynomial model     integration identification     recursive modified Gram-Schmidt (RMGS) algorithm     modified orthogonal sequence

1 线参数多项式组合模型描述

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ϕi(x)∈u(k-j)，j=1, 2, …, m，则模型为脉冲响应模型，u为与输入有关的结构项，k为采样时刻；若ϕi(x)∈{u(k-i), y(k-j)}，i=0, 1, …, m1j=1, 2, …, m2，则模型为AMAX模型，y为与输出有关的结构项；若ϕi(x)∈{y(k-j)}，j=1, 2, …, m2，则模型为自回归(AR)模型；若ϕi(x)∈{u(j)，u(j)u(k)，u(j)u(k)u(l), …}，j, k, l=1, 2, …, n，则模型为Volterra级数模型；若ϕi(x)∈{u(j), y(j), u(j)u(k), u(j)y(k), y(j)y(k), …}，j, k=1, 2, …, n，则模型为非线性NARMAX模型；若ϕi(x)∈{K(x, x1), K(x, x2), …, K(x, xl), b}，其中x1, x2, …, xll个训练样本，K(·)为样本输入的非线性映射，则模型为基于核函数映射的非线性模型，b为常数；若ϕi(x)∈{(x, cos x, ex, ln x, …)，[+, /, ∑·, …]}，其中(·)为对输入向量非线性描述，[·]为非线性运算符，则模型为非线性多项式组合模型。

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2 基于向量空间投影理论的正交化分析

1) Y(Vr)=Y(Vr-1)+δYVr-1δϕiVr-1

2) ||Y(Vr)||2=||Y(Vr-1)||2+||δYVr-1δϕiVr-1||2

Vr=Vr-1+span(δϕiVr-1), 得Y(Vr)=Y(Vr-1+span(δϕiVr-1))=Y(Vr-1)+YδϕiVr-1；又由Y(Vr-1)⊥span(δϕiVr-1)，span(δϕiVr-1)⊥Vr-1Vr-1δϕiVr-1Y(Vr-1)⊥δϕiVr-1；即YδϕiVr-1=YδϕiVr-1-Y(Vr-1δϕiVr-1=δYVr-1δϕiVr-1。故Y(Vr)=Y(Vr-1)+δYVr-1δϕiVr-1

Vr-1δϕiVr-1，得||Y(Vr)||2=||Y(Vr-1)||2+||YδϕiVr-1||2=||Y(Vr-1)||2+||δYVr-1δϕiVr-1||2。  证毕

δYVr=Y-Y(Vr)=Y-(Y(Vr-1)+Y·δϕiVr-1)=δYVr-1-YδϕiVr-1；又Y(Vr-1)⊥δϕiVr-1δYVr=δYVr-1-YδϕiVr-1+Y(Vr-1)δϕiVr-1=δYVr-1-δYVr-1δϕiVr-1

 图 1 大向量正交误差传播 Fig. 1 Orthogonal error propagation of large vectors

MGS算法[12]由于每一次迭代选择的向量是ϕik而非原始向量。而ϕik的获得是基于原始向量已正交化后的向量，可理解为“小向量面向小向量正交化”的过程，故误差较小，如图 2所示。

 图 2 小向量正交误差传播 Fig. 2 Orthogonal error propagation of small vectors

3 多项式模型的结构和参数一体化辨识 3.1 模型结构优选分析与停止条件

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1) 监视残差比，设置阈值，当残差比小于阈值时停止。

2) 监视残差化输出，同样通过设置阈值项进行判别。

3.2 基于RMGS算法的模型结构优选策略

RMGS算法的迭代流程如下：

Rnewθnew=Gnew回代，回代结果对应序号{λ(1), λ(2), …, λ(m-1)}的原始结构项系数。

3.3 基于改进正交化次序模型结构劣汰策略

Qnewi的求解可重新执行一次正交化过程，但会占用大量计算资源。考虑到上述迭代过程中已进行正交化，因此通过适当的变形即可求出Qnewi。下面给出推理过程：

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4 仿真算例

4.1 算例1

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 图 3 残差减小比率(算例1) Fig. 3 Residual decrease ratio (Example 1)

 图 4 贡献因子变化(算例1) Fig. 4 Change of contribution factor (Example 1)

 模型 ϕ2 ϕ1 ϕ24 ϕ3 ϕ40 ϕ11 ϕ12 ϕ13 ϕ26 ϕ19 ϕ5 实际模型 0.8 1.0 -1.8 0.5 4.6 1.0 1.0 1.0 1.0 0.8 -0.1 MTGS算法辨识模型 0.844 2 0.997 1 -1.706 9 0.496 7 4.633 4 0.982 5 1.015 3 0.998 5 1.000 0 0.798 7 -0.101 5 CGS算法辨识模型 0.879 3 1.055 6 -1.706 9 0.526 5 4.181 2 0.894 6 0.993 0 0.988 3 1.020 5 0.786 9 -0.054 9 MGS算法辨识模型 0.874 2 0.993 8 -1.775 9 0.496 2 4.636 4 1.007 7 1.003 3 0.984 9 0.994 4 0.810 2 -0.106 3

 图 5 辨识模型与实际模型仿真对比(算例1) Fig. 5 Comparison of identifiation model and actual modelsimulation(Example 1)
4.2 算例2

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 图 6 残差减小比率(算例2) Fig. 6 Residual decrease ratio (Example 2)

 图 7 贡献因子变化(算例2) Fig. 7 Change of contribution factor (Example 2)

 模型 ϕ5 ϕ11 ϕ4 ϕ1 ϕ10 ϕ2 实际模型 0.555 0.2 -0.4 0.049 -0.1 0.022 MTGS算法辨识模型 0.567 1 0.199 8 -0.397 8 0.048 9 -0.100 0 0.022 0 CGS算法辨识模型 0.562 950 0.199 490 -0.369 880 0.049 000 -0.100 180 0.020 574 MGS算法辨识模型 0.543 270 0.200 060 -0.429 500 0.048 992 -0.099 997 0.023 458

 图 8 残差减小比率(算例2第2次仿真数据) Fig. 8 Residual decrease ratio (data in the second simulation for Example 2)

 图 9 贡献因子变化(算例2第2次仿真数据) Fig. 9 Change of contribution factor (data in the second simulation for Example 2)

 模型 ϕ4 ϕ11 ϕ5 ϕ1 ϕ10 ϕ2 实际模型 -0.4 0.2 0.555 0.049 -0.1 0.022 0 MTGS算法辨识模型 -0.391 7 0.208 8 0.554 8 0.048 9 -0.100 0 0.022 0

 图 10 辨识模型与实际模型仿真对比(算例2) Fig. 10 Comparison of identification model and actual model simulation (Example 2)
5 结论

1) 采用RMGS算法可以从一个满覆盖的向量空间集合Vm中寻找出一个次优且满足精度需求的子空间集合Vr

2) 基于改进正交化次序的模型结构劣汰策略，在包含核心结构项和少量冗余项的优选集中筛选出对实际输出贡献相对较小的结构项，以系统完备性指标为约束，确认了系统模型的结构与参数，稳妥地保护了随正交化次数增加可能会漏选的非显著模型项。

3) 对2个典型的非线性多项式模型进行仿真验证。无论从辨识模型的模型系数还是基于辨识模型的再次仿真对比结果，都说明MTGS算法比CGS算法和MGS算法更具有效性且计算时间短。

4) 需要说明的是，合理地定量选择阈值仍需进一步研究。若设置过大，则会将某些微弱贡献项漏选；若设置过小，则又可能无端增加迭代次数，增加计算时间。

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文章信息

JIA Weizhou, PENG Jingbo, XIE Shousheng, LIU Yunlong, LI Tenghui, HE Dawei

Nonlinear polynomial model's structure and parameter integration identification

Journal of Beijing University of Aeronautics and Astronsutics, 2018, 44(6): 1303-1311
http://dx.doi.org/10.13700/j.bh.1001-5965.2017.0442