﻿ 非线性系统存在过程扰动时故障可分离研究
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 同济大学学报(自然科学版)  2017, Vol. 45 Issue (8): 1183-1190.  DOI: 10.11908/j.issn.0253-374x.2017.08.012 0

### 引用本文

GUO Qiyi, HUANG Shize. Study on the Condition of Separable Fault of Serial Nonlinear System Under the Process Noise[J]. Journal of Tongji University (Natural Science), 2017, 45(8): 1183-1190. DOI: 10.11908/j.issn.0253-374x.2017.08.012.

### 文章历史

1. 同济大学 电子与信息工程学院，上海 201804;
2. 同济大学 道路与交通工程教育部重点实验室，上海 201804

Study on the Condition of Separable Fault of Serial Nonlinear System Under the Process Noise
GUO Qiyi1, HUANG Shize2
1. College of Electronics and Information Engineering, Tongji University, Shanghai 201804, China;
2. Key Laboratory of Road and Traffic Engineering of the Ministry of Education, Tongji University, Shanghai 201804, China
Abstract: Through the detailed derivation, this paper proves that the separability is objective existence and gives the form unity conclusion in the condition of the fault-separation about serial linear system and serial nonlinear system when exist the process noise. This paper also combines the wheels bearing fault of 552732QT electric locomotive, Shao guan 8 electric locomotives, the locomotive SS4G-0446 of the concrete practice of the fault isolation.
Key words: system    state-fault    fault diagnosis    fault-separation
1 问题的提出

 $\left\{ \begin{array}{l} \left[ \begin{array}{l} {x_1}\left( {k + 1} \right)\\ {x_2}\left( {k + 1} \right) \end{array} \right] = \left[ \begin{array}{l} 1,1\\ 1,1 \end{array} \right]\left[ \begin{array}{l} {x_1}\left( k \right)\\ {x_2}\left( k \right) \end{array} \right] + \left[ \begin{array}{l} {b_1}\\ {b_2} \end{array} \right]u\left( k \right)\\ y\left( k \right) = \left[ {1,1} \right]\left[ \begin{array}{l} {x_1}\left( k \right)\\ {x_2}\left( k \right) \end{array} \right] \end{array} \right.$ (1)

 ${\Sigma _1}:\mathit{\boldsymbol{y}}\left( t \right) = p\left( {\mathit{\boldsymbol{x}}\left( t \right),\mathit{\boldsymbol{u}}\left( t \right)} \right)$ (2)

 $\begin{array}{l} {\delta _1}\left( {{\mathit{\boldsymbol{y}}_1}\left( t \right),\mathit{\boldsymbol{u}}\left( t \right)} \right) = \left( {{\rm{or}} \approx } \right)\\ {\delta _1}\left( {{\mathit{\boldsymbol{y}}_2}\left( t \right),\mathit{\boldsymbol{u}}\left( t \right)} \right) \in R\left( {特征集合} \right) \end{array}$ (3)

(1) 如式(2) 所描述的系统，如果对于给定的状态x0是可分辨的，并且通过相应的输出测量可以确认状态x0，则称状态x0是状态可分离的；如果对任意的x1x2Rnx1x2tT，其输出不相同，即：y1(t)≠y2(t)，并且可以通过相应的测量信息y1(t)与y2(t)确认x1x2，则称式(2) 所代表的系统是故障状态可分离的系统，即通过测量信息(y(t), u(t))能够明确系统是哪一个状态引起系统输出异常.

(2) 如式(2) 所示的系统，对初始状态x0Rn ，若存在x0的任意一个开邻域URn，对任意的xU，都存在一个特定的输入控制u(t) ∈Rl，其输出可分辩，并可通过测量y(t)确认x0，则称系统在x0是弱可分离的.

2 存在扰动时的状态故障可分离的基本前提

 \left\{ \begin{align} & {{\mathit{\boldsymbol{x}}}^{'}}\left( t \right)=\mathit{\boldsymbol{Ax}}\left( t \right)+\mathit{\boldsymbol{Bu}}\left( t \right)+\mathit{\boldsymbol{E }}\!\!\omega\!\!\rm{ }\left( t \right) \\ & \mathit{\boldsymbol{y}}\left( t \right)=\mathit{\boldsymbol{Cx}}\left( t \right) \\ \end{align} \right. (4)

 ${{{\mathit{\boldsymbol{\hat{x}}}}}^{'}}\left( t \right)=\mathit{\boldsymbol{Ax}}\left( t \right)+\mathit{\boldsymbol{Bu}}\left( t \right)\mathit{\boldsymbol{+GC}}\left( \mathit{\boldsymbol{x}}\left( t \right)-\mathit{\boldsymbol{\hat{x}}}\left( t \right) \right)$

 $\mathit{\boldsymbol{x}}\left( t \right)={{\rm{e}}^{\left( \mathit{\boldsymbol{A}}-\mathit{\boldsymbol{GC}} \right)\left( t-{{t}_{0}} \right)}}x\left( {{t}_{0}} \right)+\int_{{{t}_{0}}}^{t}{{{\mathit{\boldsymbol{e}}}^{\left( \mathit{\boldsymbol{A}}-\mathit{\boldsymbol{GC}} \right)\tau }}}E\mathit{\boldsymbol{\omega}}\left( \tau \right)\rm{d}\tau$ (5)

 $\mathit{\boldsymbol{y}}\left( t \right)=\mathit{\boldsymbol{\hat{y}}}\left( t \right)=C\int_{{{t}_{0}}}^{t}{{{\rm{e}}^{\left( \mathit{\boldsymbol{A}}-\mathit{\boldsymbol{GC}} \right)\tau }}}E\mathit{\boldsymbol{\omega}}\left( \tau \right)\rm{d}\tau$ (6)

(1) 如果设e1, e2, …, ekRj为一组向量，它的线性组合的全体组成的集合称为e1, e2, …, ek的扩张，记为：

 $\begin{array}{l} {\rm{span}}\left\{ {{\mathit{\boldsymbol{e}}_1},{\mathit{\boldsymbol{e}}_2}, \cdots ,{\mathit{\boldsymbol{e}}_k}} \right\},{\rm{或span}}\left\{ {{\mathit{\boldsymbol{e}}_i},i \in k} \right\}\\ {\rm{或span}}\left\{ {{\mathit{\boldsymbol{e}}_i},i \in k} \right\} = \left\{ {\mathit{\boldsymbol{e}} \in {{\bf{R}}^j},\mathit{\boldsymbol{e}} = \sum\limits_{i = 1}^k {{c_i}{\mathit{\boldsymbol{e}}_i}} } \right\} \end{array}$ (7)

(2) 设矩阵C的列向量为：c1, c2, …, ck，则记矩阵AC张成的子空间为＜A|C＞：＜|A|C＞是以c1, c2, …, ck, c1 A, c2 A, …, ck A, …, c1 An-1, c2 An-1, …, ckAn-1为基扩张成的线性子空间.

(3) 如果令：C:XY是一个线性映射，定义KerCC的核，lmCC的值域，即：

KerC={x:xX, 并且有Cx=0}; lmC={Cx, 对于任意的xX}

(4) 设AXX是线性映射，SX是商空间.如果有：AφS，则称是A-不变的.

 $\mathit{\boldsymbol{C}}\int_{{{t}_{0}}}^{t}{{{\rm{e}}^{\left( \mathit{\boldsymbol{A}}-\mathit{\boldsymbol{GC}} \right)\tau }}E\mathit{\boldsymbol{\omega}}\left( \tau \right)\rm{d}\tau =0}$ (8)

 $\mathit{\boldsymbol{C}}<\mathit{\boldsymbol{A}}-\mathit{\boldsymbol{GC}}\left| \rm{lm}\mathit{\boldsymbol{E}}>=0 \right.$ (9)

① 系统(4) 是状态可观测的，即(A, C)是状态能观测；

② 并且存在矩阵G，使得有C < A-GC|lmE > =0

 $\left( \mathit{\boldsymbol{A}}-\mathit{\boldsymbol{GC}} \right){{\bf{R}}_{0}}\subset {{\bf{R}}_{0}}$

 $\mathit{\boldsymbol{C}}<\mathit{\boldsymbol{A}}-\mathit{\boldsymbol{GC}}\left| \rm{lm}\mathit{\boldsymbol{E}}> \right.\in V\left( \mathit{\boldsymbol{A}},\mathit{\boldsymbol{C}},\rm{Ker}\ C \right)$ (10)

① 系统(4) 是状态可观测的，即(A, C)是状态能观测；

② lmEV*(A, C, Ker C).

 $\text{rank}\left[ {{c}_{1}},{{c}_{2}},\cdots ,{{c}_{k}},{{c}_{1}}A,{{c}_{2}}A,\cdots ,{{c}_{k}}A,\cdots ,{{c}_{1}}{{A}^{n-1}},{{c}_{\text{2}}}{{A}^{n-1}},\cdots ,{{c}_{k}}{{A}^{n-1}} \right]=n,$

 $\left( 1 \right)\mathop {\mathop \cap \limits^n }\limits_{i = 1} \,\rm{Ker}\left( \mathit{\boldsymbol{C}}{{\mathit{\boldsymbol{A}}}^{i-1}} \right)=\varnothing$
 $\left( 2 \right){\rm{lm}}\mathit{\boldsymbol{E}}\subset {{V}^{*}}\left( \mathit{\boldsymbol{A}},\mathit{\boldsymbol{C}},{\rm{Ker}}\;C \right).$
3 存在过程扰动的非线性系统的状态故障可分离条件

 \left\{ \begin{align} & {{\mathit{\boldsymbol{x}}}^{'}}\left( t \right)=\mathit{\boldsymbol{f}}\left( x \right)+\mathit{\boldsymbol{g}}\left( x \right)\mathit{\boldsymbol{u}}\left( t \right)+\mathit{\boldsymbol{e}}\left( x \right)\mathit{\boldsymbol{\omega}}\left( t \right) \\ & \mathit{\boldsymbol{y}}\left( t \right)=\mathit{\boldsymbol{h}}\left( x \right) \\ \end{align} \right. (11)

 $\mathit{\boldsymbol{y}}\left( {t,{x_0},\mathit{\boldsymbol{\omega }}{_1}\left( t \right)} \right) = \mathit{\boldsymbol{y}}\left( {t,{x_0},\mathit{\boldsymbol{\omega }}{_2}\left( t \right)} \right)$ (12)

Mn维微分流形，以C(M)表示M上的所有可微函数的集合，C(x)表示在xM点的所有可微函数组成的集合.则有如下的定义：

(1) 映射L:C(x)→R(实数域)，称为点x的切向量，如果：

(a) L(αf+βh)=αL(f)+βL(h), 任意的α, βR，和任意的f, hC(x)；

(b) L(f, h)=fL(h)+gL(f), 任意的f, hC(x).

TxMM上的点x的全体切向量的集合，在其上以通常的方式定义加法、数乘后构成(上的线性空间，则称之为x点的切空间.

(2) 微分流形M上的分布Δ是一种法则，它将M上的每一点对应于该点切空间TxM的一个子空间Δ(x).若对任意的xM存在x的邻域Ux和其上线性无关的k个向量场X1, X2, …, Xk，使X1(x), X2(x), …, X(x)k构成Δ(x)的基底，则称Δ为可微分布，X1, X2, …, Xk称为分布Δ的局部基底.

(3) 若f, hV(M)(M上的不变子空间全体)，Δ为可微分布，若对任意的xMh(x)∈Δ(x), 记为：hΔ；并且记：[f, Δ]=span{[f, h]:hΔ}，显然[f, Δ]仍然是M上的可微分布.分布Δ称为f的不变分布，若满足：[f, Δ]⊂Δ.

(4) 微分流形M上的分布Δ称为(f, h)不变的，如果存在α(x)，β(x)，使：

 $\left[ \mathit{\boldsymbol{f}}+\mathit{\boldsymbol{h}}\ \mathit{\boldsymbol{ }}\!\!\alpha\!\!\rm{ }\left( x \right),\mathit{\Delta } \right]\subset \mathit{\Delta };\left[ \mathit{\boldsymbol{f}}\ \mathit{\boldsymbol{ }}\!\!\beta\!\!{\rm{ }}\ _{i}^{\rm{T}}\left( x \right),\mathit{\Delta } \right]\subset \mathit{\Delta },i\in r$

(f, h)不变分布是(A, C)-不变子空间的推广，就象f不变分布是A-不变子空间的推广一样.

(5) 如果对任意的向量XΔ，有X=Xhi=0, im，则称分布Δh相容.分布Δh相容等价于Δ∈(span{dhj, jm}).

(1) Δf, e1, …, ei-1, ei+1, …, er的不变分布；

(2) ei∈Δ∈(span{dhj, jm}).其中：dhjhj的微分.

 ${{\mathit{\boldsymbol{\hat{x}}}}^{'}}=\mathit{\boldsymbol{f}}\left( {\mathit{\hat{x}}} \right)+\mathit{\boldsymbol{g}}\left( t \right)\mathit{\boldsymbol{u}}\left( t \right)+\left[ \mathit{\boldsymbol{q}}\left( x \right)\mathit{\boldsymbol{h}}\left( x \right)-\mathit{\boldsymbol{q}}\left( {\mathit{\hat{x}}} \right)\mathit{\boldsymbol{h}}\left( {\mathit{\hat{x}}} \right) \right]$ (13)

 ${{\left( \mathit{\boldsymbol{x}}-\mathit{\boldsymbol{\hat{x}}} \right)}^{'}}=\mathit{\boldsymbol{f}}\left( x \right)-\mathit{\boldsymbol{f}}\left( {\mathit{\hat{x}}} \right)+\mathit{\boldsymbol{q}}\left( x \right)\mathit{\boldsymbol{h}}\left( x \right)-\mathit{\boldsymbol{q}}\left( {\mathit{\hat{x}}} \right)\mathit{\boldsymbol{h}}\left( {\mathit{\hat{x}}} \right)+\mathit{\boldsymbol{g}}\left( x \right)\mathit{\boldsymbol{\omega}}\left( t \right)$ (14)

 $\begin{array}{l} \left( {\mathit{\boldsymbol{x}} - \mathit{\boldsymbol{\hat x}}} \right)' = \left[ {1 - \mathit{\boldsymbol{q}}\left( x \right)} \right]\left[ \begin{array}{l} \mathit{\boldsymbol{f}}\left( x \right)\\ \mathit{\boldsymbol{h}}\left( x \right) \end{array} \right] + \left[ {1, - \mathit{\boldsymbol{q}}\left( {\mathit{\hat x}} \right)} \right]\\ \;\;\;\;\;\;\;\;\;\;\;\;\;\;\left[ \begin{array}{l} \mathit{\boldsymbol{f}}\left( {\mathit{\hat x}} \right)\\ \mathit{\boldsymbol{h}}\left( {\mathit{\hat x}} \right) \end{array} \right] + \mathit{\boldsymbol{g}}\left( x \right)\mathit{\boldsymbol{\omega }}\left( t \right) \end{array}$ (15)

(1) 分布Δf(x), h(x)不变的；

(2) e(x)∈Δ⊂(span{dhi, im})

 $\left[ {\mathit{\boldsymbol{f}},\Delta } \right] \subset \Delta+ {\rm{span}}\left\{ {{h_1},{h_2}, \cdots ,{\mathit{h}_m}} \right\}$
 $\left[ {{\mathit{\boldsymbol{h}}_i},\Delta } \right] \subset \Delta +{\rm{span}}\left\{ {{h_1},{h_2}, \cdots ,{\mathit{h}_m}} \right\}$

 ${L_{{f_1}}}\left( { \cdots \left( {{L_{{f_j}}}\left( {{h_i}} \right)} \right) \cdots } \right),$

 ${\rm{dim}}\left\{ {d\mathit{\boldsymbol{G}}\left( x \right)} \right\} = n$
 $\mathit{\boldsymbol{g}}\left( x \right) \in {\mathit{\Delta }_{\max }};$

4 轮对滚动轴承故障分离实践 4.1 轮对滚动轴承装置故障可分离性分析

 $R\ddot x + Q\dot x + Kx = {F_0}\left( t \right) + {F_a}\left( t \right)$ (16)

R为轴承装置的当量质量；x为接触点径向相对位移；Q为轮对轴承阻尼；K为轮对轴承接触刚度；F0(t)为系统初始激振力，包含列车轮轨接触系统通过转向架给轴承施加的作用力；Fa(t)为局部故障激振力，它包含故障缺陷所产生的激励，受接触刚度和故障函数的影响.由式(16) 可知，轴承系统是一个典型的非线性系统，该系统以车轴转速作为输入，振动信号作为系统的输出.下面对系统发生故障时，系统的振动输出与故障状态特征的关系进行分析.

 ${f_{\rm{k}}}{\rm{ = }}{f_{\rm{n}}}\frac{1}{2}z\left( {1 - \frac{d}{D}\cos \beta } \right)$ (17)

 $K = \frac{1}{2}z\left( {1 - \frac{d}{D}\cos \beta } \right)$ (18)

 ${f_{\rm{k}}}{\rm{ = }}{f_{\rm{n}}}K$ (19)

 $\left\{ \begin{array}{l} {f_{\rm{w}}}{\rm{ = }}{K_{\rm{w}}} \cdot {f_{\rm{n}}}{\rm{ = }}7.195\;2{f_{\rm{n}}},外圈故障\\ {f_{\rm{n}}}{\rm{ = }}{K_{\rm{n}}} \cdot {f_{\rm{n}}}{\rm{ = }}9.804\;8{f_{\rm{n}}},内圈故障\\ {f_{\rm{a}}}{\rm{ = }}{K_{\rm{a}}} \cdot {f_{\rm{n}}}{\rm{ = 2}}{\rm{.929}}\;{\rm{3}}{f_{\rm{n}}},滚子故障 \end{array} \right.$ (20)

4.2 轴承系统故障诊断方法

 图 1 基于第二代小波变换的故障分离示意图 Fig.1 Schematic diagram of fault separation based on second generation wavelet transform

4.3 滚动轴承故障分离案例分析

 图 2 原始信号解调谱 Fig.2 Original signal demodulation spectrum
 图 3 第二代小波包分解信号解调谱 Fig.3 Signal demodulation spectrum of the wavelet transform

 图 4 第二代小波包分解信号解调谱 Fig.4 Signal demodulation spectrum of the second generation of wavelet transform

 图 5 原始信号时域波形 Fig.5 Time domain waveform of original signal
 图 6 小波变换后信号频域波形 Fig.6 Frequency domain waveform of wavelet transform
 图 7 滚动轴承故障解体图 Fig.7 Fault separation diagram of rolling bearings

5 结论

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