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 自动化学报  2018, Vol. 44 Issue (9): 1569-1589 PDF

1. 北京工业大学信息学部 北京 100124;
2. 计算智能与智能系统北京市重点实验室 北京 100124;
3. 流程工业综合自动化国家重点实验室 沈阳 110004

Modeling Multiple Components Mechanical Signals by Means of Virtual Sample Generation Technique
TANG Jian1,2,3, QIAO Jun-Fei1,2, CHAI Tian-You3, LIU Zhuo3, WU Zhi-Wei3
1. Faculty of Information Technology, Beijing University of Technology, Beijing 100124;
2. Beijing Key Laboratory of Computational Intelligence and Intelligent System, Beijing 100124;
3. State Key Laboratory of Synthetical Automation for Process Industries, Northeastern University, Shenyang 110004
Manuscript received : April 16, 2017, accepted: June 22, 2017.
Foundation Item: Supported by National Natural Science Foundation of China (61573364, 61703089), State Key Laboratory of Synthetical Automation for Process Industries (PAL-N201504), and State Key Laboratory of Process Automation in Mining & Metallurgy Beijing Key Laboratory of Process Automation in Mining & Metallurgy (BGRIMM-KZSKL-2017-07)
Corresponding author. TANG Jian  Professor at Beijing University of Technology. His research interest covers intelligent control and modeling for complex industrial processes, and data driven-based soft sensor. Corresponding author of this paper.
Abstract: Mechanical vibration & acoustic signals with characteristics of multiple components, nonstationarity and nonlinearity are always used to construct the data-driven soft sensor model of industrial processes. It is one of the main approaches to measure the difficulty-to-measure process parameters inside those high energy consumption mechanical devices. Duo to the complexity of the production mechanism of these mechanical signals, most of these soft sensor models are difficult to be explained. Moreover, the characteristics of the industrial process' continuous running and the mechanical equipment' operation modes lead to the difficulty of high economic cost and long period waiting to obtain sufficient training samples. To solve these problems, a new multi-component mechanical signal modeling method based on virtual sample generation (VSG) technology is proposed. Firstly, the mechanical signals are processed into a set of sub-signals with different time scales by using adaptive multi-component signal decomposition technique; then these sub-signals are transferred to high dimensional multi-scale spectral data. Secondly, an improved selective ensemble kernel partial least squares (SENKPLS) algorithm that suits to model small sample high dimensional data is used to construct a feasibility-based programming (FBP) model with the true training samples; then prior knowledge, FBP models and information entropy are integrated to produce virtual training samples. Thirdly, mutual information (MI) method is used to select the spectral features of the new mixing training samples based on the true and virtual ones. Finally, a soft sensor model is built by using these reduced mixing spectral data. Near-infra spectra data and mechanical vibration and acoustic singals of a laboratory-scale ball mill in grinding process validate the reasonability and effectiveness of the proposed VSG techniques and multi-component mechanical signals-based modeling approach.
Key words: Multi-component mechanical signal     high dimentional spectra data     difficulty-to-measure process parameters     data-driven modeling     virtual sample generation (VSG)

0.1 多组分机械信号建模

0.2 建模样本非完备

0.3 本文研究动机

1 相关知识

1.1 面向多组分机械信号的建模 1.1.1 多组分机械信号自适应分解

 $$${\pmb{x}}_{}^{\rm{t}} = \sum\limits_{{j_{}} = 1}^{{J_{{\rm{EMD}}}}} {{\pmb{x}}_{{\rm{EM}}{{\rm{D}}_j}}^{\rm{t}} + r_{{J_{{\rm{EMD}}}}}^{\rm{t}}}$$$ (1)

EMD在处理机械振动和振声等多组分信号较传统FFT和小波变换具有明显优势, 但也存在虚假人工成分导致的模态混叠、分解端点效应、子信号非严格正交、有效子信号数量有限等问题.基于白噪声统计属性的EEMD可以有效克服EMD的模态混叠问题, 其基本思路是加入影响整个时频空间的白噪声$A_{\rm noise }$, 重复进行整个EMD分解过程$M$次后进行平均. EEMD和EMD之间的关系表示为

 $$$\left\{ {\begin{array}{*{20}{c}} {{\pmb{x}}_{{\rm{EEM}}{{\rm{D}}_j}}^{\rm{t}} = \dfrac{1}{M}\sum\limits_{m = 1}^M {({\pmb{x}}_{{\rm{EM}}{{\rm{D}}_j}}^{\rm{t}})_m^{}} }\\ {{\pmb r}_{{\rm{EEMD}}}^{\rm{t}} = \dfrac{1}{M}\sum\limits_{m = 1}^M {({\pmb r}_{{\rm{EM}}{{\rm{D}}_J}}^{\rm{t}})_m^{}} } \end{array}} \right.$$$ (2)

1.1.2 高维谱数据的维数约简

1.1.3 基于改进SENKPLS的高维谱数据建模

PLS/KPLS算法除用于提取与输入输出均相关的潜在特征外, 也可直接构建潜结构模型, 类似于人脑逐层进行特征抽取进行认知的机制.面对多源机械振动/振声信号特征子集, 基于"操纵输入特征"的集成构造策略, 文献[24]从选择性信息融合的视角构建SENKPLS模型.

 $$$K_{}^j = \Phi ^{\rm{T}}{({({\pmb{z}}_{}^j)_l})}\Phi ({({\pmb{z}}_{}^j)_m}), \begin{array}{*{20}{c}} {}&{l, m} \end{array} = 1, 2, \cdots, k$$$ (3)

 $$$\tilde K_{}^j = \left(I - \frac{1}{k}{{\pmb l}_k}{\pmb l}_k^{\rm{T}}\right)K_{}^j\left(I - \frac{1}{k}{{\pmb l}_k}{\pmb l}_k^{\rm{T}}\right)$$$ (4)

 ${\hat{\pmb y}}_{{\rm{valid}}}^j = f^j(\{ {({\pmb{z}}_{{\rm{valid}}})_l}\} _{l = 1}^{{k^{{\rm{valid}}}}})=\\ \ \ \ \ \ \ \ \tilde K_{{\rm{valid}}}^jU_{}^j{\left( {{{(T_{}^j)}^{\rm{T}}}\tilde K_{}^jU_{}^j} \right)^{ - 1}} {(T_{}^j)^{\rm{T}}}{\pmb{y}}_{}^j$ (5)
 $\tilde K_{{\rm{valid}}}^j =\left (K_{{\rm{valid}}}^jI - \frac{1}{k}{{\pmb l}_{{k^{{\rm{valid}}}}}}{\pmb l}_k^{\rm{T}}K_{}^j\right)\left(I - \frac{1}{k}{{\pmb l}_k}{\pmb l}_k^{\rm{T}}\right)$ (6)
 $K_{{\rm{valid}}}^j = K({({{\pmb{z}}_{{\rm{valid}}}})_l}, {({\pmb{z}}_{}^j)_m})$ (7)

 $$${\pmb e}_{{\rm{valid}}}^j = {\hat{\pmb y}}_{{\rm{valid}}}^j - {{\pmb y}_{^{{\rm{valid}}}}}$$$ (8)

$j{\rm{th}}$个和第$s{\rm{th}}$个候选子模型的相关系数为

 $$$c_{js}^{{\rm{valid}}} = \frac{{\sum\limits_{l = 1}^{{k^{{\rm{valid}}}}} {{\pmb e}_{{\rm{valid}}}^j(j, {k^{{\rm{valid}}}}) \cdot {\pmb e}_{{\rm{valid}}}^s(s, {k^{{\rm{valid}}}})} }}{{{k^{{\rm{valid}}}}}}$$$ (9)

 $$${C^{{\rm{valid}}}} = {\left[{\begin{array}{*{20}{c}} {c_{11}^{{\rm{valid}}}}&{c_{12}^{{\rm{valid}}}}& \cdots &{c_{1J}^{{\rm{valid}}}}\\ {c_{21}^{{\rm{valid}}}}&{c_{22}^{{\rm{valid}}}}& \cdots &{c_{2J}^{{\rm{valid}}}}\\ \vdots&\vdots &{c_{js}^{{\rm{valid}}}}& \vdots \\ {c_{J1}^{{\rm{valid}}}}&{c_{J2}^{{\rm{valid}}}}& \cdots &{c_{JJ}^{{\rm{valid}}}} \end{array}} \right]_{J \times J}}$$$ (10)

 $$$\xi _j^{} = \begin{cases} 1, &w_j^* \ge \lambda \\ 0, &{w_j^* < \lambda } \end{cases}$$$ (11)

$\xi _j^{} = 1$的候选子模型选择为集成子模型, 并将其数量记为${J^*}$, 即SEN模型的集成尺寸.将第${j^*}{\rm{th}}$集成子模型记为$f_{}^{{j^*}}( \cdot )$, 其输出为

 $$${\hat{\pmb y}}_{{\rm{valid}}}^{{j^*}} = f_{}^{{j^*}}(\{ {({\pmb{z}}_{{\rm{valid}}}^{})_l}\} _{l = 1}^{{k^{{\rm{valid}}}}})$$$ (12)

 $$$\begin{array}{l}\min\quad \sigma _{}^2 = \sum\limits_{{j^*} = 1}^{{J^*}} {(W_{{j^*}}^{{\rm{AWF}}})_{}^2\sigma _{{j^*}}^2} \\ ~~~~\qquad{{\rm{s.\, t.}}}\quad{\sum\limits_{{j^*} = 1}^{{J^*}} {W_{{j^*}}^{{\rm{AWF}}}} = 1}, ~~ 0 \le W_{{j^*}}^{{\rm{AWF}}} \le 1 \end{array}$$$ (13)

 $$$W_{{j^*}}^{{\rm{AWF}}} = \frac{1}{{{{(\sigma _{{j^*}}^{})}^2}\sum\limits_{{j^*} = 1}^{{J^*}} {\frac{1}{{{{(\sigma _{{j^*}}^{})}^2}}}} }}$$$ (14)

 $$$\hat y_{{\rm{test}}}^{} = \sum\limits_{{j^*} = 1}^{{J^*}} {W_{{j^*}}^{{\rm{AWF}}}} \hat y_{{\rm{test}}}^{{j^*}}$$$ (15)

1.2 面向建模样本非完备的VSG技术 1.2.1 小样本数据集概述

 $$$\alpha = \frac{{{n_{{\rm{sample}}}}}}{{{p_{{\rm{feature}}}}}}$$$ (16)

1.2.3 VSG的关注问题

1) 确定虚拟样本产生策略.方法包括:利用先验知识、扰动原始样本、对输入数据添加噪声等[67];

2) 确定虚拟样本输入.方法包括:先验知识[31]、在真实样本输入点的超域内随机选取[40]、函数化虚拟群体[68]、基于真实样本输入间隔的信息分散技术[41]、间隔核密度估计[37]、Mega趋势分散(MTD)函数[34]、基于模糊数据集的成员函数[69]、组虚拟样本产生[38]、基于模糊理论的产生趋势分散(GTD)[70]、基于高斯分布[67]、基于GA和BPNN[42];

3) 确定虚拟样本输出.方法包括:平均神经网络的输出[40]、基于实际样本输出间隔的信息分散技术[41]和基于GA和BPNN的方法[42]等;

4) 确定虚拟样本数量.目前, 最优化虚拟样本数量的确定主要基于实验数据确定.如何在理论上指导或确定优化的虚拟样本数量还是个开放性难题.

2 基于VSG的多组分机械信号建模策略

 图 1 基于VSG的多组分机械信号建模策略 Figure 1 Multi-component mechanical signal modeling strategy based on VSG

1) 多尺度谱数据获取模块:其输入为真实的时域机械振动/振声信号, 输出为真实的频域多尺度训练样本; 主要功能是将包含若干数据点的多组分时域信号经自适应分解和时频域转换得到多源多尺度高维谱数据.

2) 虚拟样本产生模块:是本文所提方法的核心模块, 其输入为真实的训练样本和先验知识, 输出为混合训练样本; 主要功能包括:面向IMF的VSG, 基于信息墒加权IMF虚拟样本输出, 以及虚拟样本合成.

3) 谱特征自适应选择:其输入为混合训练样本, 输出为经特征选择的约简混合训练样本; 主要功能是自适应地选择有价值的多源多尺度子信号及其谱特征.

4) 软测量模型构建:其输入为约简混合样本, 输出为难以检测过程参数的预测值; 主要功能是构建基于"操纵训练样本"策略的适合于高维谱数据的SENKPLS模型.

3 基于VSG的多组分机械信号建模实现 3.1 多尺度谱数据获取模块

 $$${\pmb{x}}_{\rm{V}}^{\rm{t}}\xrightarrow{{f^{\rm{DCOM(\cdot)}}}}\sum\limits_{{j_{\rm{V}}} = 1}^{J_{\rm{V}}^{{\rm{all}}}} {({\bar{\pmb c}}_{\rm{V}}^{\rm{t}})_{{{{j}}_{\rm{V}}}}^{} + {{({{\bar r}_{\rm{V}}})}_{J_{\rm{V}}^{{\rm{all}}}}}}$$$ (20)
 $$${\pmb{x}}_{\rm{A}}^{\rm{t}}\xrightarrow{{f^{\rm{DCOM(\cdot)}}}}\sum\limits_{{j_{\rm{A}}} = 1}^{J_{\rm{A}}^{{\rm{all}}}} {({\bar{\pmb c}}_{\rm{A}}^{\rm{t}})_{{{{j}}_{_{\rm{A}}}}}^{} + {{({{\bar r}_{\rm{A}}})}_{J_{\rm{A}}^{{\rm{all}}}}}}$$$ (21)

 \begin{align} {\bar{\pmb c}}_{_{{\rm{IMF}}}}^{\rm{t}} = \, &[({\bar{\pmb c}}_{\rm{V}}^{\rm{t}})_{\rm{1}}^{}, \cdots, ({\bar{\pmb c}}_{\rm{V}}^{\rm{t}})_{{j_{\rm{V}}}}^{}, \cdots, ({\bar{\pmb c}}_{\rm{V}}^{\rm{t}})_{{J_{\rm{V}}}}^{}, ({\bar{\pmb c}}_{\rm{A}}^{\rm{t}})_{\rm{1}}^{}, \cdots, \nonumber\\ & ({\bar{\pmb c}}_{\rm{A}}^{\rm{t}})_{{j_{\rm{A}}}}^{}, \cdots, ({\bar{\pmb c}}_{\rm{A}}^{\rm{t}})_{{J_{\rm{A}}}}^{}] = \nonumber\\ &[{\bar{\pmb c}}_1^{\rm{t}}, \cdots, {\bar{\pmb c}}_{{j_{{\rm{IMF}}}}}^{\rm{t}}, \cdots, {\bar{\pmb c}}_{{J_{{\rm{IMF}}}}}^{\rm{t}}] \end{align} (22)

 $$${\bar{\pmb c}}_{{j_{{\rm{IMF}}}}}^{\rm{t}}\xrightarrow{{f^{\rm{Trans(\cdot)}}}}{\pmb{z}}_{{j_{{\rm{IMF}}}}}^{{\rm{True}}}$$$ (23)

 $$${\pmb{z}}_{}^{{\rm{True}}} = [{\pmb{z}}_{{1_{{\rm{IMF}}}}}^{{\rm{True}}}, \cdots, {\pmb{z}}_{{j_{{\rm{IMF}}}}}^{{\rm{True}}}, \cdots, {\pmb{z}}_{{J_{{\rm{IMF}}}}}^{{\rm{True}}}]$$$ (24)

3.2 虚拟样本产生(VSG)模块 3.2.1 VSG模块的结构与功能

 图 2 虚拟样本产生(VSG)模块的结构 Figure 2 Structure of the virtual sample generation (VSG) module

3.2.2 VSG模块的算法实现

1) 面向IMF的VSG子模块

 \begin{align} ({\pmb{z}}_{{j_{{\rm{IMF}}}}}^{{\rm{VSG}}})_{{{l'}_{{\rm{VSG}}}}}^{} = \, &({\pmb{z}}_{{j_{{\rm{IMF}}}}}^{{\rm{VSG}}})_{{\rm{low}}}^{} +\nonumber\\ & \frac{{\left( {({\pmb{z}}_{{j_{{\rm{IMF}}}}}^{{\rm{VSG}}})_{{\rm{high}}}^{} - ({\pmb{z}}_{{j_{{\rm{IMF}}}}}^{{\rm{VSG}}})_{{\rm{low}}}^{}} \right){{l'}_{{\rm{VSG}}}}}}{{{N_{{\rm{VSG}}}}}} \end{align} (25)

 $$$k'_{{j_{{\rm{IMF}}}}}= {k_{{\rm{VSG}}}}({N_{{\rm{VSG}}}} - 1)$$$ (26)

 $$$(y_{{j_{{\rm{IMF}}}}}^{{\rm{VSG}}})_{{{l'}_{{\rm{VSG}}}}}^{} = f_{{j_{{\rm{IMF}}}}}^{{\rm{FBP}}}\left( {({\pmb{z}}_{{j_{{\rm{IMF}}}}}^{{\rm{VSG}}})_{{{l'}_{{\rm{VSG}}}}}^{}} \right)$$$ (27)

 $$$y_{{\rm{low}}}^{{\rm{VSG}}} \le (y_{{j_{{\rm{IMF}}}}}^{{\rm{VSG}}})_{{{l'}_{{\rm{VSG}}}}}^{} \le y_{{\rm{high}}}^{{\rm{VSG}}}$$$ (28)

 $$$(S_{{j_{{\rm{IMF}}}}}^{{\rm{VSG}}})_{{{l'}_{{\rm{VSG}}}}}^{} = \{ ({\pmb{z}}_{{j_{{\rm{IMF}}}}}^{{\rm{VSG}}})_{{{l'}_{{\rm{VSG}}}}}^{}, (y_{{j_{{\rm{IMF}}}}}^{{\rm{VSG}}})_{{{l'}_{{\rm{VSG}}}}}^{}\}$$$ (29)

 $$$\left. {\begin{array}{*{20}{l}} {S_{{\rm{True}}}^{{j_{{\rm{IMF}}}}}}\\ {Know}\\ {f_{{j_{{\rm{IMF}}}}}^{{\rm{FBP}}}( \cdot )} \end{array}} \right\}\xrightarrow{{f_{j_{\rm IMF}}^{\rm VSG}(\cdot)}} S_{{j_{{\rm{IMF}}}}}^{{\rm{VSG}}} = \left\{ {{\pmb{z}}_{{j_{{\rm{IMF}}}}}^{{\rm{VSG}}}, {\pmb y}_{{j_{{\rm{IMF}}}}}^{{\rm{VSG}}}} \right\}$$$ (31)

 \begin{align} S_{{\rm{All}}}^{{\rm{VSG}}} =\,&\{ Z_{{\rm{All}}}^{{\rm{VSG}}}, y_{{\rm{All}}}^{{\rm{VSG}}}\} = \{ S_{{j_{{\rm{IMF}}}}}^{{\rm{VSG}}}\} _{{j_{{\rm{IMF}}}} = 1}^{{J_{{\rm{IMF}}}}}= \nonumber\\ &\{ \{ {({\pmb{z}}_{{j_{{\rm{IMF}}}}}^{{\rm{VSG}}})_{{{l'}_{{\rm{VSG}}}}}}, {(y_{{j_{{\rm{IMF}}}}}^{{\rm{VSG}}})_{{{l'}_{{\rm{VSG}}}}}}\} _{{{l'}_{{\rm{VSG}}}}}^{k'}\} _{{j_{{\rm{IMF}}}} = 1}^{{J_{{\rm{IMF}}}}} \end{align} (32)

2) 基于信息墒加权的虚拟样本输出子模块

 $$${(y_{}^{{\rm{VSG}}})_{{{l'}_{{\rm{VSG}}}}}} = f_{{\rm{IMF}}}^{{\rm{Weight}}}( \cdot ) = \sum\limits_{{j_{{\rm{IMF}}}} = 1}^{{J_{{\rm{IMF}}}}} {w_{{j_{{\rm{IMF}}}}}^{{\rm{VSG}}}(y_{{j_{{\rm{IMF}}}}}^{{\rm{VSG}}})_{{{l'}_{{\rm{VSG}}}}}^{}}$$$ (33)

 $$$w_{{j_{{\rm{IMF}}}}}^{{\rm{VSG}}} = \frac{1}{{{J_{{\rm{IMF}}}} - 1}}\left( {1 - \frac{{1 - E_{{j_{{\rm{IMF}}}}}^{}}}{{\sum\limits_{{j_{{\rm{IMF}}}} = 1}^{{J_{{\rm{IMF}}}}} {(1 - E_{{j_{{\rm{IMF}}}}}^{})} }}} \right)$$$ (34)

 $$$E_{{j_{{\rm{IMF}}}}}^{} = \frac{1}{{\ln k}}\sum\limits_{l = 1}^k {\frac{{{{(e_{{j_{{\rm{IMF}}}}}^{})}_l}}}{{\left(\sum\limits_{l = 1}^k {{{(e_{{j_{{\rm{IMF}}}}}^{})}_l}} \right)}}\ln \frac{{{{(e_{{j_{{\rm{IMF}}}}}^{})}_l}}}{{\sum\limits_{l = 1}^k {{{(e_{{j_{{\rm{IMF}}}}}^{})}_l}} }}}$$$ (35)
 $$${(e_{{j_{{\rm{IMF}}}}}^{})_l} = \left\{ {\begin{array}{*{20}{l}} {\frac{{{{(\hat y_{{j_{{\rm{IMF}}}}}^{{\rm{True}}})}_l} - y_l^{{\rm{True}}}}}{{y_l^{{\rm{True}}}}}, \begin{array}{*{20}{l}} {}&{}&{0 \le \left| {\left| {\frac{{{{(\hat y_{{j_{{\rm{IMF}}}}}^{{\rm{True}}})}_l} - y_l^{{\rm{True}}}}}{{y_l^{{\rm{True}}}}}} \right|} \right|} \end{array} < 1}\\ {1\begin{array}{*{20}{l}} {\begin{array}{*{20}{l}} {}&{} \end{array}}&{\begin{array}{*{20}{l}} {}&{}&{\begin{array}{*{20}{l}} {}&{} \end{array}\left| {\left| {\frac{{{{(\hat y_{{j_{{\rm{IMF}}}}}^{{\rm{True}}})}_l} - y_l^{{\rm{True}}}}}{{y_l^{{\rm{True}}}}}} \right|} \right|} \end{array} \ge 1} \end{array}} \end{array}} \right.$$$ (36)

 $$$f_{}^{{\rm{VSG}}}( \cdot ) = \left\{ {\begin{array}{*{20}{l}} {\{ \{ {{({\pmb{z}}_{{j_{{\rm{IMF}}}}}^{{\rm{True}}})}_l}\} _{l = 1}^k\} _{{j_{{\rm{IMF}}}} = 1}^{{J_{{\rm{IMF}}}}}}\xrightarrow{{k_{\rm now}}}\{ \{ {{({\pmb{z}}_ {{j_{{\rm{IMF}}}}}^{{\rm{VSG}}})}_{{{l'}_{{\rm{VSG}}}}}}\} _{{{l'}_ {{\rm{VSG}}}}}^{k'}\} _{{j_{{\rm{IMF}}}} = 1}^{{J_{{\rm{IMF}}}}}\\ {\left. {\begin{array}{*{20}{l}}\!\!\! \{ \{ {{({\pmb{z}}_{{j_{{\rm{IMF}}}}}^{{\rm{VSG}}})}_{{{l'}_{{\rm{VSG}}}}}}\} _{{{l'}_{{\rm{VSG}}}}}^{k'}\} _{{j_{{\rm{IMF}}}} = 1}^{{J_{{\rm{IMF}}}}}\\ \!\!\!\{ {{(y_{}^{{\rm{True}}})}_l}\} _{l = 1}^k \end{array}} \right\}}{\xrightarrow{{f_{j_{\rm IMF}}^{\rm FBP}(\cdot)}}} \{ \{ {{(y_{{j_{{\rm{IMF}}}}}^{{\rm{VSG}}})}_{{{l'}_ {{\rm{VSG}}}}}}\} _{{{l'}_{{\rm{VSG}}}}}^{k'}\} _{{j_{{\rm{IMF}}}} = 1}^{{J_{{\rm{IMF}}}}}\\ {\left. {\begin{array}{*{20}{l}} \!\!\!{\{ \{ {{(y_{{j_{{\rm{IMF}}}}}^{{\rm{VSG}}})}_{{{l'}_{{\rm{VSG}}}}}}\} _{{{l'}_{{\rm{VSG}}}}}^{k'}\} _{{j_{{\rm{IMF}}}} = 1}^{{J_{{\rm{IMF}}}}}}\\ \!\!\!{\{ {{(y_{}^{{\rm{True}}}, \hat y_{{j_{{\rm{IMF}}}}}^{{\rm{True}}})}_l}\} _{l = 1}^k} \end{array}} \right\}}{\xrightarrow{{f_{\rm IMF}^{\rm Weight}(\cdot)}}} \{ {{(y_{}^{{\rm{VSG}}})}_{{{l'}_{{\rm{VSG}}}}}}\} _{{{l'}_{{\rm{VSG}}}} = 1}^{k'} \end{array}} \right.$$$ (37)

3) 虚拟样本合成子模块

 $$$S_{}^{{\rm{VSG}}} = \{ {(\{ {\pmb{z}}_{{j_{{\rm{IMF}}}}}^{{\rm{VSG}}}\} _{{j_{{\rm{IMF}}}} = 1}^{{J_{{\rm{IMF}}}}})_{{{l'}_{{\rm{VSG}}}}}}, {(y_{}^{{\rm{VSG}}})_{{{l'}_{{\rm{VSG}}}}}}\} _{{{l'}_{{\rm{VSG}}}} = 1}^{k'}$$$ (38)

 \begin{align} S_{}^{{\rm{Mix}}} =\,&f_{}^{{\rm{Mix}}}( \cdot ) = \{ S_{}^{{\rm{True}}}; S_{}^{{\rm{VSG}}}\}= \nonumber\\ & \{ \{ {(\{ {\pmb{z}}_{{j_{{\rm{IMF}}}}}^{{\rm{True}}}\} _{{j_{{\rm{IMF}}}} = 1}^{{J_{{\rm{IMF}}}}})_l}, {(y_{}^{{\rm{True}}})_l}\} _{l = 1}^k;\nonumber\\ & \{ {(\{ {\pmb{z}}_{{j_{{\rm{IMF}}}}}^{{\rm{VSG}}}\} _{{j_{{\rm{IMF}}}} = 1}^{{J_{{\rm{IMF}}}}})_{{{l'}_{{\rm{VSG}}}}}}, {(y_{}^{{\rm{VSG}}})_{{{l'}_{{\rm{VSG}}}}}}\} _{{{l'}_{{\rm{VSG}}}} = 1}^{k'}\} =\nonumber\\ & \{ \{ \{ {(\{ {\pmb{z}}_{{j_{{\rm{IMF}}}}}^{{\rm{True}}}\} _{{j_{{\rm{IMF}}}} = 1}^{{J_{{\rm{IMF}}}}})_l}\} _{l = 1}^k;\nonumber\\ &\{ {(\{ {\pmb{z}}_{{j_{{\rm{IMF}}}} }^{{\rm{VSG}}}\} _{{j_{{\rm{IMF}}}} = 1}^{{J_{{\rm{IMF}}}}})_{{{l'}_{ {\rm{VSG}}}}}}\} _{{{l'}_{{\rm{VSG}}}} = 1}^{k'}\}, \nonumber\\ &\{ \{ {(y_{}^{{\rm{True}}})_l}\} _{l = 1}^k;\{ {(y_{}^ {{\rm{VSG}}})_{{{l'}_{{\rm{VSG}}}}}}\} _{{{l'}_{{\rm{VSG}}}} = 1}^{k'}\} \} =\nonumber\\ & \{ \{ {\pmb{z}}_{{l^{mix}}}^{{\rm{mix}}}\} _{{l^{mix}} = 1}^{k + k'}, \{ y_{l^{mix}}^{{\rm{mix}}}\} _{{l^{mix}} = 1}^{k + k'}\} = \nonumber\\ & \{ {Z^{{\rm{mix}}}}, {\pmb y}^{{\rm{mix}}}\} \end{align} (39)

3.2.3 VSG模块的准确性分析与适应性讨论

VSG模块是保证本文所提多组分机械信号建模方法的准确性并具有良好预测性能的关键环节.此处针对产生虚拟样本过程中的关键环节进行准确性分析和适用性讨论.

1) 面向IMF构建的FBP模型:此处采用的建模方法是适合于小样本高维数据的SENKPLS算法. KPLS算法是PLS的核版本, 能够有效地处理输入变量间的共线性问题, 其用于构建内层模型的潜在变量远远小于原始输入特征数量, 进而能够保证基于IMF的高维谱数据构建的FBP模型的泛化性能.另外, 基于"操纵训练样本"集成构造策略的SEN算法选择有价值训练样本构造软测量模型, 其与KPLS算法的结合, 进一步增强了FBP模型的泛化性能.

2) 面向IMF的虚拟样本输入:基于多源多尺度IMF的真实训练样本子集是基于机械振动/振声信号经自适应分解和时频变换获得; 虽然这些高维谱变量难以得到合理的具体解释, 但原始的机械振动/振声样本均有明确的物理含义; 在产生虚拟样本输入值时, 两个真实训练样本间所划分间隔的大小也是依据先验知识确定的; 并且式(25)保证了其合理的取值范围.这些约束保证了虚拟样本输入值的准确性.

3)面向IMF的虚拟样本输出产生的准确性:这些不同的IMF基于各自的FBP模型所产生的虚拟输出值的范围由式(28)及其相应的策略予以保证, 由图 2所给出的"面向IMF的VSG"中的算法流程给出了更为清晰的描述, 进而保证了虚拟样本输出值的准确性.

4) 基于信息墒加权的虚拟样本输出:这些多源多尺度IMF均是由原始的机械振动/振声信号分解得到的, 显然这些IMF的虚拟样本输入应该对应统一且唯一的虚拟样本输出值, 需要将不同IMF的虚拟样本输出值进行加权, 式(33)~(36)采用信息墒加权, 由FBP模型的预测误差得到不同IMF虚拟样本输出值的权系数, 从而保证了最终的虚拟样本输出值的准确性.

5) VSG模块的适用性讨论:在VSG过程中, FBP模型的准确性比较重要, 这就要求真实的训练样本虽然稀少, 但尽量保证较宽的工况覆盖范围; VSG模块的所描述的算法适用于已经将原始多组分信号分解为多个不同的子信号, 并且这些子信号的输入变量间具有较强的共线性的情况; 同时, "面向IMF的VSG子模块"中所描述的算法适用于具有高维共线性特性的真实训练数据产生虚拟样本.

3.3 谱特征自适应选择模块

 \begin{align} Z_{}^{{\rm{mix}}} = \, &[Z_{{1_{{\rm{IMF}}}}}^{{\rm{mix}}}, \cdots, Z_{{j_{{\rm{IMF}}}}}^{{\rm{mix}}}, \cdots, Z_{{J_{{\rm{IMF}}}}}^{{\rm{mix}}}]= \nonumber\\ & [({\pmb{z}}_1^{{\rm{mix}}}), \cdots, ({\pmb{z}}_{}^{{\rm{mix}}})_p^{}, \cdots, ({\pmb{z}}_P^{{\rm{mix}}})] \end{align} (40)

 \begin{align} &Muin\left( {{{\pmb y}^{{\rm{mix}}}};{{({\pmb{z}}_{}^{{\rm{mix}}})}_p}} \right) = \int\int p({{({\pmb{z}}_{}^{{\rm{mix}}})}_p})\times\nonumber\\ &\qquad \log \frac{{p({{\pmb y}^{{\rm{mix}}}}, {{({\pmb{z}}_{}^{{\rm{mix}}})}_p})}} {{p({{({\pmb{z}}_{}^{{\rm{mix}}})}_p})p({{\pmb y}^{{\rm{mix}}}})}}{\rm{d}} ({({\pmb{z}}_{}^{{\rm{mix}}})_p}){\rm{d}}{{\pmb y}^{{\rm{mix}}}}=\nonumber\\ &\qquad H{\rm{(}}{{\pmb y}^{{\rm{mix}}}}{\rm{)}} - H{\rm{(}}{{\pmb y}^{{\rm{mix}}}} {\bf{|}}{({\pmb{z}}_{}^{{\rm{mix}}})_p}{\rm{)}} \end{align} (41)

 图 3 NIR真实训练样本输入和输出 Figure 3 Inputs and outputs of NIR training samples

4.1.2 谱数据的VSG结果

 图 4 KLVs数量与RMSE的关系 Figure 4 Relationships between KLVs$'$ number and RMSE
 图 5 核半径与RMSE的关系 Figure 5 Relationships between kernel radiu and RMSE

 图 6 NIR虚拟训练样本的输入和输出 Figure 6 Inputs and outputs of NIR virtual training samples

4.1.3 基于VSG的模型性能比较结果

 图 7 基于不同数量的虚拟样本构建模型的训练误差 Figure 7 Training errors of the constructed model based on virtual samples with different numbers

4.2 基于球磨机机械信号的建模方法验证 4.2.1 磨矿过程及磨机实验描述

1) 磨矿工艺与磨机负荷参数磨矿过程是整个选矿流程中最为重要的作业环节.国内选矿行业广泛应用两段式闭式磨矿回路(Grinding circuit, GC)工艺.其中一段GA (GC Ⅰ)的工艺流程如图 8所示.

 图 8 某选矿厂一段磨矿回路(GC Ⅰ)工艺流程 Figure 8 Flow chart of the grinding circuit Ⅰ (GC Ⅰ) of some mineral grinding process

2) 实验球磨机筒体振动/振声数据采集与描述

4.2.2 多尺度IMF获取结果

 $RMSRE{P_{0.632}} = 0.632RMSRE{P_{{\rm{BCV}}}} + \nonumber\\ \ \ \ \ \ (1 - 0.632)RMSRE{P_{{\rm{app}}}}$ (46)
 $RMSRE{P_{{\rm{BCV}}}} = \nonumber\\ \sqrt {\frac{1}{{{k^{{\rm{mix}}}}}}\sum\limits_{{l^{{\rm{mix}}}} = 1}^{{k^{{\rm{mix}}}}} {\frac{1}{{{R_{ - {l^{{\rm{mix}}}}}}}}} \sum\limits_{r: {l^{{\rm{mix}}}} \notin k_r^*} {{{\left( {\frac{{f_r^*({\pmb{z}}_{{l^{{\rm{mix}}}}}^{{\rm{mix}}}) - y_{{l^{{\rm{mix}}}}}^{{\rm{mix}}}}}{{y_{{l^{{\rm{mix}}}}}^{{\rm{mix}}}}}} \right)}^2}} }$ (47)
 $RMSRE{P_{{\rm{app}}}} = \nonumber\\ \sqrt {\frac{1}{{{k^{{\rm{mix}}}}}}\sum\limits_{{l^{{\rm{mix}}}} = 1}^{{k^{{\rm{mix}}}}} {{{\left( {\frac{{f_{{k^{{\rm{mix}}}}}^{}({\pmb{z}}_{{l^{{\rm{mix}}}}}^{{\rm{mix}}}) - y_{{l^{{\rm{mix}}}}}^{{\rm{mix}}}}}{{y_{{l^{{\rm{mix}}}}}^{{\rm{mix}}}}}} \right)}^2}} }$ (48)

1) 本文所提方法的平均预测性能随虚拟样本数量的增加而增加.本文方法在虚拟样本的数量为81时, 其平均预测误差为0.1290, 与文献[24]的最佳的平均预测误差0.1265接近.另外, 文献[24]并未对多组分信号进行自适应分解, 在提高软测量模型的可解释性和深入理解磨机研磨机理等方面弱于本文所提方法.本文方法在平均预测性能上也强于基于单尺度单模型的PLS/KPLS方法和基于多尺度频谱选择性信息融合的SENKPLS方法.

2) 在所有的软测量模型中, 本文所提方法具有最佳的预测稳定性, 这在工业实际中具有较高的应用价值.文献[24]虽然具有预测误差性能的最小值, 但该方法同时也具有预测误差的最大方差(0.0677), 是本文所提建模方法的至少2倍.显然, 文献[24]的预测性能的稳定性较差.本文所提方法的测试误差的方差与的关系如图 13所示.

 图 13 基于不同${N_{{\rm{VSG}}}}$值构建的软测量模型测试误差的方差 Figure 13 Variance of the testing errors based on soft sensor models using different ${N_{{\rm{VSG}}}}$ values

3) 本文所提方法的软测量模型预测误差的均值和最大值随着虚拟样本数量的增加而降低, 如当采用81个虚拟样本时, 预测误差的均值和最大值与无虚拟样本时进行比较, 分别从0.1708和0.2829降低到了0.1290和0.1749.本文所提方法的测试误差与的关系如图 14所示.

 图 14 基于不同${N_{{\rm{VSG}}}}$值构建的软测量模型的测试误差 Figure 14 Testing errors based on soft sensor models using different ${N_{{\rm{VSG}}}}$ values

5 结论