自动化学报  2017, Vol. 43 Issue (11): 1920-1930   PDF    
基于SWT与等价空间的LDTV系统故障检测
薛婷1, 钟麦英2     
1. 北京航空航天大学 仪器科学与光电工程学院 北京 100190;
2. 山东科技大学 电气与自动化工程学院 青岛 266590
摘要: 为提高基于等价空间的线性离散时变(Linear discrete time-varying,LDTV)系统故障检测的检测性能,本文提出一种基于平稳小波变换(Stationary wavelet transform,SWT)与等价空间的LDTV系统故障检测方法.通过引入SWT对基于低阶等价关系构造的残差进行多尺度滤波,将残差产生器设计转化为不同尺度下的多目标最优化问题,保证了各尺度下残差对干扰鲁棒性和对故障灵敏性指标的最小化,同时利用SWT快速算法获得一组多尺度残差信号.进一步,对产生的多尺度残差信号进行多分辨率分析,从而实现较宽频率范围内故障信号的检测,有效降低了故障漏报率.最后,通过仿真实验验证了本文方法的有效性.
关键词: 故障检测     等价空间     平稳小波变换     LDTV系统    
SWT and Parity Space Based Fault Detection for Linear Discrete Time-varying Systems
XUE Ting1, ZHONG Mai-Ying2     
1. School of Instrumentation Science and Optoelectronics Engineering, Beihang University, Beijing 100190;
2. College of Electrical Engineering and Automation, Shandong University of Science and Technology, Qingdao 266590
Manuscript received : June 21, 2016, accepted: September 19, 2016.
Foundation Item: Supported by National Natural Science Foundation of China (61333005, 61733009, 61421063)
Author brief: XUE Ting She received her master degree from Beihang University in 2016. Her main research interest is control systems fault detection
Corresponding author. ZHONG Mai-Ying Professor at Shandong University of Science and Technology. She received her Ph. D. degree in control theory and control engineering from Northestern University in 1999. Her research interest covers model based fault diagnosis, fault tolerant systems and their application. Corresponding author of this paper
Recommended by Associate Editor JIANG Bin
Abstract: This paper deals with fault detection (FD) for linear discrete time-varying (LDTV) systems by combining stationary wavelet transform (SWT) with parity space based method, so as to improve the FD performance. By employing SWT to filter the residual generated with low order parity relation, the design of residual generator can be formulated as multi-objective optimization problem at each scale of SWT, so that minimum ratio criterion of robustness to unknown input and sensitivity to faults are assured, and a bank of multi-scale residuals are obtained by applying the SWT recursive algorithm. Moreover, by analysing the generated residual signals at multiscale, the faults within a broader frequency band can be detected and a lower miss detection rate can be achieved. Finally, a numerical example is given to verify the effectiveness of the proposed approach.
Key words: Fault detection     parity space     stationary wavelet transform (SWT)     linear discrete time-varying (LDTV) systems    

控制系统在实际运行过程中不可避免会发生故障, 故障检测(Fault detection, FD)技术是提高控制系统可靠性、降低事故风险的重要手段.在实际工业过程中, 受控系统通常具有时变特性, 如化工过程、飞行控制系统、卫星导航系统等, 且由于运行环境的影响和系统复杂度的不断增加, 控制系统故障检测的难度也日益增大.因此, 针对线性离散时变(Linear discrete time-varying, LDTV)系统故障检测的研究具有重要理论和实际意义[1-3].现有的FD方法主要包括基于解析模型的方法[4-9]、基于信号处理的方法[10-11]和基于知识的方法[12].根据残差产生方式的不同, 基于解析模型的方法又可分为状态估计法、参数估计法和等价空间方法.目前, 基于状态估计的LDTV系统故障检测滤波器(Fault detection filter, FDF)方法已有较多研究[7-9].如文献[7]基于$H_{\infty}/H_{\infty}$$H_\_/H_{\infty}$指标最大化设计FDF, 改善了LDTV系统多量测量丢失或量化不精确情况下的FD性能; 文献[8]将基于$H_{\infty}$的FDF设计转化为正定二次型最小化问题进行求解, 并给出了相应递归算法; 文献[9]利用最小最大化方法使零误报率情况下非可检测故障集合最小, 给出了一种LDTV系统FD集成设计方法等.与状态估计方法相比, 基于等价空间的FD可以直接构造残差产生器, 具有残差与系统初始状态解耦的优点, 是目前应用较为广泛的重要FD方法之一[13-21].然而, 已有的基于等价空间的故障检测大多针对线性时不变(Linear time invariant, LTI)系统开展研究[13, 16-21], 并且在等价关系阶数较低时, FD性能较差, 而等价关系阶数的提高则使得在线计算量较大.特别是, 高阶等价向量在频域等效为窄频带滤波器, 导致故障漏报率较高[19-20].

另一方面, 小波变换是具有多分辨率分析(Multi-resolution analysis, MRA)特性的一种时频分析工具, 可采用多种快速算法实现, 在故障诊断领域得到了广泛应用[22-24].针对传统基于等价空间方法存在的缺陷, 文献[20]提出将平稳小波变换(Stationary wavelet transform, SWT)与等价空间方法相结合, 针对LTI系统设计残差产生器, 通过对基于低阶等价关系产生的残差进行多尺度SWT, 产生一组处于不同频带的多个残差信号, 以较小的在线计算量获得了满意的性能指标和低的漏报率.文献[21]提出一种改进的SWT与等价空间FD系统设计方法, 并基于无人机飞行控制系统仿真平台进行了故障检测实验研究.

与LTI系统不同, 基于等价空间的LDTV系统故障检测需要在每个时刻点在线更新系统相关矩阵及等价向量, 在线计算量将随着等价关系阶数的增大而急剧增加.现有的基于等价空间的LDTV系统故障诊断研究成果相对较少[25-26], 文献[25]提出了一种迭代算法更新系统参数矩阵, 以降低在线计算量; 文献[26]利用Krein空间投影技术建立了等价空间与观测器方法之间的关系, 提出了一种快速故障检测的递归实现算法.事实上, 基于SWT的优良特性, 文献[20]给出了改善等价空间方法故障检测性能的较好思路, 但其研究对象为LTI系统, 并侧重于减小在线计算量, 故障检测系统设计及实现相对简单, 如何进一步将SWT与等价空间相结合的方法推广应用于LDTV系统, 提高可检测故障信号的频率范围, 并实现故障的快速检测仍然是挑战性难题.

本文将在文献[20-21]的基础上开展基于SWT与等价空间的LDTV系统故障检测研究.首先根据低阶等价关系构造残差, 并引入SWT对残差进行多尺度滤波, 将残差产生器设计转化为不同尺度下的多目标最优化问题, 进而获得一组多尺度残差信号.进一步, 通过选取适当的残差评价函数及阈值, 对各尺度残差进行多分辨率分析, 从而实现较宽频率范围内故障信号的检测, 有效降低漏报率.

1 问题描述

考虑如下所示LDTV系统:

$ \begin{equation} \left\{ {\begin{array}{l} {\pmb x}(k+1)=A(k){\pmb x}(k)+B(k){\pmb u}(k)+ \\ \quad \quad \quad \quad B_d(k){\pmb d}(k)+B_f(k){\pmb f}(k) \\ {\pmb y}(k)=C(k){\pmb x}(k)+D_d(k){\pmb d}(k)+D_f(k){\pmb f}(k) \\ \end{array}} \right. \end{equation} $ (1)

式中, ${\pmb x}(k)\in {\bf R}^n$为系统状态变量, ${\pmb u}(k)\in {\bf R}^m$为输入, ${\pmb y}(k)\in {\bf R}^p$为输出, ${\pmb d}(k)\in {\bf R}^q$为未知干扰, ${\pmb f}(k)\in {\bf R}^l$为待检测故障; ${\pmb d}(k)$${\pmb f}(k)$均为$L_2$范数有界信号; $A(k)$$B(k)$$C(k)$$B_d(k)$$B_f(k)$$D_d(k)$$D_f(k)$为已知相应维数的时变系统矩阵.

在给定的时间窗口$[k-s, k] (s>0)$内, 由式(1)可得如下等价关系:

$ \begin{equation} \begin{array}{l} {\pmb y}_s(k)-H_{us}(k){\pmb u}_s(k)= H_{os}(k){\pmb x}(k-s)+ \\ \quad \quad \quad H_{ds}(k){\pmb d}_s (k)+H_{fs}(k){\pmb f}_s (k) \\ \end{array} \end{equation} $ (2)

其中

$ \begin{array}{l} {\pmb y}_s (k)=\left[{{\pmb y}^{\rm T }(k-s), {\pmb y}^{\rm T }(k-s+1), \cdots, {\pmb y}^{\rm T }(k)} \right]^{\rm T }\\ {\pmb u}_s (k)=\left[{{\pmb u}^{\rm T }(k-s), {\pmb u}^{\rm T }(k-s+1), \cdots, {\pmb u}^{\rm T }(k)} \right]^{\rm T }\\ {\pmb d}_s (k)=\left[{{\pmb d}^{\rm T }(k-s), {\pmb d}^{\rm T }(k-s+1), \cdots, {\pmb d}^{\rm T }(k)} \right]^{\rm T }\\ {\pmb f}_s (k)=\left[{{\pmb f}^{\rm T }(k-s), {\pmb f}^{\rm T }(k-s+1), \cdots, {\pmb f}^{\rm T }(k)} \right]^{\rm T }\\ \end{array} $
$ { H_{os}(k)=\left[{{\begin{array}{c} {C(k-s)} \\ {C(k-s+1)A(k-s)} \\ \vdots \\ {C(k-1)A(k-2)\cdots A(k-s+1)A(k-s)} \\ {C(k)A(k-1)\cdots A(k-s+1)A(k-s)} \\ \end{array} }} \right]} $
$ {H_{ds}(k)= }\\ {\left[{{\begin{array}{*{20}c} {(1, 1)} & 0 &0 &{\cdots} &0 \ \\ {(2, 1)} &{(2, 2)} &0 &{\cdots} &0 \ \\ {(3, 1)} &{(3, 2)} &{(3, 3)} &\ddots &\vdots \ \\ \vdots & \vdots & \vdots &\ddots &0 \ \\ {(s+1, 1)} \!\!&\! \!\!{\cdots} \!&\! \!{(s+1, s-1)} \!&\! \!{(s+1, s)} \!&\!\!{(s+1, s+1)} \\ \end{array} }} \right]} $
$ \begin{array}{l} (i, i)=D_d(k-s+i-1), i=1, 2\cdots, s+1 \\ (2, 1)=C\left( {k-s+1} \right)B_d (k-s) \\ (3, 1)=C(k-s+2)A(k-s+1)B_d (k-s) \\ (3, 2)=C(k-s+2)B_d (k-s+1) \\ (s+1, 1)=C(k)A(k-1)\cdots A(k-s+1)\times\\ \qquad B_d (k-s) \\ (s+1, s-1)=C(k)A(k-1)B_d (k-2) \\ (s+1, s)=C(k)B_d (k-1) \\ \end{array} $

$H_{ds}(k)$中的$\left\{{D_d, B_{ d}}\right\}$分别替换为$\left\{{0, B}\right\}$$\left\{{D_f, B_f}\right\}$即可得到$H_{us}(k)$$H_{fs}(k)$.

基于等价空间的FD包括残差产生与残差评价, 首先考虑如下基于等价关系的残差产生器:

$ \begin{equation} \begin{split} r_s(k)=\,&{\pmb w}_s(k)\left[{{\pmb y}_s(k)-H_{us}(k){\pmb u}_s(k)}\right]= \\ &{\pmb w}_s(k)\left[{H_{ds}(k){\pmb d}_s(k)+H_{fs}(k){\pmb f}_s(k)}\right] \\ \end{split} \end{equation} $ (3)

式中, $r_s(k)$为残差信号, $s$为等价关系阶数, ${\pmb w}_s(k)\in {\bf R}^{p(s+1)}$为等价向量, 满足等价空间$P_s (k)$的约束条件, 即$P_s(k)=\left\{{{\pmb w}_s(k)\left|{{\pmb w}_s (k)H_{os}(k)=0}\right.}\right\}$.将等价空间$P_s(k)$的基向量记为$N_b(k)$, 则${\pmb w}_s(k)$可表示为${\pmb w}_s(k)={\pmb\upsilon} _s(k)N_b(k)$, 令

$ \mbox{$\bar{H}_{ds}(k)=N_b(k)H_{ds}(k), \bar{H}_{fs}(k)=N_b(k)H_{fs}(k)$} $

进一步可得:

$ \begin{equation} r_s (k)={\pmb\upsilon}_s (k)\left[{\bar{H}_{ds}(k){\pmb d}_s(k)+\bar{H}_{fs}(k){\pmb f}_s(k)}\right] \end{equation} $ (4)

将文献[5]的残差产生器设计方法推广应用于LDTV系统, 并将其归结为如下最小化问题:

$ \begin{equation} J_s(k)=\mathop{\min}\limits_{{\pmb\upsilon} _s (k)}\frac{{\pmb\upsilon} _s(k)\bar{H}_{ds}(k)\bar {H}_{ds}^{\rm T }(k){\pmb\upsilon}_s^{\rm T } (k)}{{\pmb\upsilon}_s(k)\bar{H}_{fs} (k)\bar{H}_{fs}^{\rm T }(k){\pmb\upsilon}_s^{\rm T }(k)} \end{equation} $ (5)

从而可采用奇异值分解(Singular value division, SVD)获得满足式(5)的${\pmb\upsilon}_s(k)$最优解.计算最优等价向量${\pmb w}_s(k)$, 将其代入式(3)即可得到残差$r_s(k)$.

残差评价是FD的另一重要任务.选取适当的残差评价函数$J_{r_s}(k)$及阈值$J_{th}$, 根据如下逻辑关系判断故障的发生:

$ \begin{equation} \left\{ {\begin{array}{lc} J_{r_s } (k)\le J_{th, s}, \hfill&\mbox{正常} \hfill \\ J_{r_s } (k)> J_{th, s}, \hfill&\mbox{故障} \hfill \\ \end{array}} \right. \end{equation} $ (6)

值得提出的是, 与文献[20]中所示LTI情况类似, 等价关系阶数较低时通常难以获得较满意的FD性能, 而等价关系阶数较高时的残差在线计算量大, 且漏报率较高.

另一方面, 与LTI系统不同, 基于等价空间LDTV系统FD在各个时刻均需要在线更新矩阵$H_{\xi s}(k)~(\xi =o, d, f, u)$及等价向量${\pmb w}_s(k)$, 且在线计算量也随着等价关系阶数的增大而急剧增加.因此, 针对LDTV系统的等价空间FD而言, 仍然面临在线计算量大、难以工程实现的问题[26].

本文将在文献[20-21]的基础上, 应用SWT的MRA特性和快速实现算法, 针对LDTV系统提出一种基于SWT与等价空间的LDTV系统FD设计方法, 为实现快速检测宽频率范围故障信号提供解决手段.

注1.  在每一$k$时刻, 通过式(5)所示性能指标最小化获得的最优等价向量${\pmb w}_s(k)$在频域等效为一个后置滤波器, 且其频带宽度随着等价关系阶数$s$的增大而变窄[19], 使得残差$r_s(k)$对于频率偏离该窄频带中心频率较大的故障信号的灵敏度降低, 以至于在该故障信号较小或干扰较大时, 残差评价函数$J_{r_s}(k)$难以实现较宽频率范围内故障信号的高性能检测, 导致较高的故障漏报率.

2 基于SWT与等价空间的FD系统设计

基于SWT与等价空间的FD系统设计主要包括残差产生和残差评价两个阶段.首先, 引入SWT对基于等价空间的残差进行多尺度滤波, 将残差产生器设计转化为不同尺度下的多目标优化问题, 产生对不同频率故障信息灵敏的多个残差信号.其次, 针对不同尺度下的残差, 提出基于多分辨率分析的残差评价策略.

2.1 多尺度残差信号设计

残差产生器设计是基于SWT与等价空间实现LDTV系统故障检测的首要任务. SWT是小波变换的一种快速实现算法, 具有时移不变性, 且能够提供一组时域和频域带宽均较窄的线性滤波器[20, 27].

对于残差信号${r_s(k)}$, 其$j$尺度SWT的逼近系数$\{WT_{r_s (k)}^a (j, k)\}$及细节系数$\{WT_{r_s (k)}^d (j, k)\}$可看作序列信号$\left\{ {r_s (k)} \right\}$与线性滤波器的卷积输出[27], 即:

$ \begin{equation} \left\{ {\begin{array}{l} WT_{r_s (k)}^a (j, k)=r_s (k)\ast {\pmb g}_{l.j} \\ WT_{r_s (k)}^d (j, k)=r_s (k)\ast {\pmb g}_{b.j} \\ \end{array}} \right. \end{equation} $ (7)

式中, “$\ast $”表示卷积运算, ${\pmb g}_{l, j} ={\pmb l}_1 \ast {\pmb l}_2 * \cdots \ast {\pmb l}_{j-1} \ast {\pmb l}_j $${\pmb g}_{b, j} ={\pmb l}_1 \ast {\pmb l}_2 \cdots \ast {\pmb l}_{j-1} \ast {\pmb h}_j $分别为$j$尺度窄频带离散低通滤波器和带通滤波器, 其中${\pmb l}_1$${\pmb h}_1$是由小波基函数决定的初始低通和高通滤波器, 且有:

$ \begin{equation} \left\{ {\begin{array}{l} {\pmb h}_j (i)={\pmb h}_1 (2^{j-1}i) \\ {\pmb l}_j (i)={\pmb l}_1 (2^{j-1}i) \\ \end{array}} \right. \end{equation} $ (8)

根据SWT的时移不变性, $\{WT_{r_s (k)}^a(j, k)\}$$\{WT_{r_s (k)}^d (j, k)\}$可通过如下所示递归算法得到[28]:

$ \begin{equation} \label{eq9} \left\{ {\begin{array}{l} WT_{r_s (k)}^a (j, k)=\sum\limits_i {{\pmb l}_1 (i)WT_{r_s (k)}^a (j-1, k\mbox{+}2^ji)} \\ WT_{r_s (k)}^d (j, k)=WT_{r_s (k)}^a (j-1, k)-\\\quad \quad\quad WT_{r_s (k)}^a (j, k) \\ \end{array}} \right. \end{equation} $ (9)

通过对$r_s (k)$进行$j_m$尺度SWT滤波, 可将其分解为不同尺度下处于不同频带的一组残差信号, 从而得到如下所示多尺度残差产生器:

$ \begin{equation} \left\{ {\begin{array}{l} r_s (k)={\pmb w}_s (k)\left[{{\pmb y}_s (k)-H_{us} (k){\pmb u}_s (k)} \right] \\ r_{s, j} (k)=WT_{r_s (k)}^d \left( {j, k} \right), \quad j=1, 2, \cdots, j_m \\ r_{s, j_m +1} (k)=WT_{r_s (k)}^a \left( {j_m, k} \right) \\ \end{array}} \right. \end{equation} $ (10)

式中, $r_{s, j}(k), j=1, 2, \cdots, j_m $$r_{s, j_m+1}$分别为$r_s (k)$的高频细节分量及低频逼近分量, 近似频率范围分别为$B_j^d =\left[2^{-j}F_0, 2^{-j+1}F_0 \right], j=1, 2, \cdots, j_m, B_{j_m }^a =\left[0, 2^{-j_m }F_0 \right]$, $F_0 $为系统采样频率.当等价关系阶数$s$、小波基函数及最高分解尺度$j_m $选定后, 多尺度残差产生器设计即可归结为等价向量${\pmb w}_s (k)$的求解问题.

注意到, 如果能将式(10)所示残差$r_{s, j} (k), j=1, 2, \cdots, j_m+1$表示为式(4)所示形式, 则可将等价向量${\pmb w}_s(k)$的设计转化为式(5)形式的性能指标最优化问题进行求解, 从而确保各尺度下残差对干扰的鲁棒性和对故障的灵敏性.为此, 根据式(4)和(7), 式(10)所示残差可写为如下形式:

$ \begin{equation} \left\{ {\begin{array}{l} r_s (k)={\pmb\upsilon} _s (k)\left[{\bar {H}_{ds} (k){\pmb d}_s (k)+\bar {H}_{fs} (k){\pmb f}_s (k)} \right] \\ r_{s, j} (k)=r_s (k)\ast {\pmb g}_{b, j}, \quad j=1, 2, \cdots, j_m \\ r_{s, j_m +1} (k)=r_s (k)\ast {\pmb g}_{l, j_m } \\ \end{array}} \right. \end{equation} $ (11)

根据卷积运算的性质进一步可得到:

$ \begin{equation} \left\{ {\begin{array}{l} r_{s, j} (k)=\sum\limits_{i=0}^{\alpha _j } {{\pmb g}_{b, j} (i)r_s (k-i)}, {\begin{array}{*{20}c} {~~ j=1, 2, \cdots, j_m } \hfill&\hfill \\ \end{array} } \\ r_{s, j_m +1} (k)=\sum\limits_{i=0}^{\alpha _{j_m }} {{\pmb g}_{l, j_m +1}(i)r_s(k-i)} \\ \end{array}} \right. \end{equation} $ (12)

式中, $\alpha _j $为线性滤波器的时域长度, 当$i\ge\alpha_j$时, ${\pmb g}_{b, j} (i)\to 0$; 当$i\ge \alpha _{j_m } $时, ${\pmb g}_{l, j_m } (i)\to 0$.

将式(11)所示$r_s(k)$代入式(12), 则有:

$ \begin{equation} \left\{ {\begin{array}{l} r_{s, j} (k)=\sum\limits_{i=0}^{\alpha _j } {{\pmb g}_{b, j} (i){\pmb\upsilon} _s (k-i)\left[{\bar {H}_{ds}(k-i)}\times \right.} \\ \qquad\left. {{\pmb d}_s (k-i)+\bar {H}_{fs} (k-i){\pmb f}_s (k-i)} \right], \\ \qquad~~\quad\qquad\qquad j=1, 2, \cdots, j_m \\ r_{s, j_m +1} (k)=\sum\limits_{i=0}^{\alpha _{j_m } } {{\pmb g}_{l, j_m }(i){\pmb\upsilon} _s (k-i)\left[{\bar {H}_{ds}(k-i)\times } \right.} \\ \qquad \left. {{\pmb d}_s (k-i)+\bar {H}_{fs} (k-i){\pmb f}_s (k-i)} \right] \\ \end{array}} \right. \end{equation} $ (13)

定义零矩阵${\bf0}_d \in {\bf R}^{q(s+1)\times q}$, ${\bf0}_f \in {\bf R}^{l(s+1)\times l}$, 单位矩阵${\pmb I}_d \in {\bf R}^{q(s+1)\times q(s+1)}$, ${\pmb I}_f \in {\bf R}^{l(s+1)\times l(s+1)}$, 以及

$ \begin{eqnarray*} &M_{b, j}^{ f} (i)=\left[{\bf0}_f \cdots {\bf0}_f, {\pmb g}_{b, j} (i){\pmb I}_{ f}, {\bf0}_f \cdots {\bf0}_f \right], \\& \quad\qquad\qquad\qquad\qquad j=1, 2, \cdots, j_m\\ &M_{b, j}^d (i)=\left[{{\bf0}_d \cdots {\bf0}_d, {\pmb g}_{b, j} (i){\pmb I}_{ d}, {\bf0}_d\cdots {\bf0}_d } \right], \\& \quad\qquad\qquad\qquad\qquad j=1, 2, \cdots, j_m\\ &M_{l, j_m }^{ f} (i)=\left[{{\bf0}_f \cdots {\bf0}_f, {\pmb g}_{l, j_m } (i){\pmb I}_f, {\bf0}_f \cdots {\bf0}_f } \right]\\ & M_{l, j_m }^d (i)=\left[{{\bf0}_d \cdots {\bf0}_d, {\pmb g}_{l, j_m } (i){\pmb I}_d, {\bf0}_d \cdots {\bf0}_d } \right] \end{eqnarray*} $

以式(13)中第一个等式右侧第一项为例进行化简.当$0\le i\le\alpha _j $时, 有:

$ \begin{align} {\pmb g}_{b, j}(i)&{\pmb\upsilon} _s (k-i)\bar {H}_{ds} (k-i){\pmb d}_s (k-i) =\nonumber\\& {\pmb\upsilon} _ s(k-i)\bar{H}_{ds}(k-i)\times\nonumber\\& \left[{\underbrace {{\bf0}_d \cdots {\bf0}_d }_{(\alpha _j-i)}, {\pmb g}_{b, j} (i){\pmb I}_d, \underbrace {{\bf0}_{ d} \cdots {\bf0}_d }_i} \right] \times \nonumber\\& \left[{\pmb d}^{\rm T }(k-s-\alpha _j )\cdots{{\pmb d}^{\rm T }(k-s-i-1)}, \right. \nonumber\\&\left. {\pmb d}_s^{\rm T }(k-i), {{\pmb d}^{\rm T }(k-i+1)} \cdots{{\pmb d}^{\rm T }(k)} \right]^{\rm T } =\nonumber\\& {\pmb\upsilon} _s(k-i)\bar {H}_{ds} (k-i)M_{b, j}^d (i){\pmb d}_{s+\alpha _j}(k) \end{align} $ (14)

从而

$ \begin{align} \sum\limits_{i=0}^{\alpha _j }& {\pmb g}_{b, j} (i){\pmb\upsilon} _s (k-i)\bar {H}_{ds} (k-i){\pmb d}_s (k-i) =\nonumber\\& \left\{ {\sum\limits_{i=0}^{\alpha _j } {{\pmb\upsilon} _s (k-i)\bar {H}_{ds} (k-i)M_{b, j}^d (i)} } \right\} {\pmb d}_{s+\alpha _j } (k)= \nonumber\\& \bigg\{ {{\pmb\upsilon} _s (k)\bar {H}_{ds} (k)M_{b, j}^d(0)+ }\nonumber\\& {\sum\limits_{i=1}^{\alpha _j } {{\pmb\upsilon} _s (k-i)\bar {H}_{ds} (k-i)M_{b, j}^d (i)} } \bigg\}{\pmb d}_{s+\alpha _j } (k) =\nonumber\\& \left[{{\pmb\upsilon} _s (k), 1} \right]\times \nonumber\\& \left[ {{\begin{array}{*{20}c} \quad\quad\quad {\bar {H}_{ds} (k)M_{b, j}^d (0)} \hfill \\ {\sum\limits_{i=1}^{\alpha _j } {{\pmb\upsilon} _s (k-i)\bar {H}_{ds} (k-i)M_{b, j}^d (i)} } \hfill \\ \end{array} }}\right]\times \nonumber\\& {\pmb d}_{s+\alpha _j} (k) ={{\pmb\upsilon} }'_s (k)\bar {H}_{ds}^{b, j} (k){\pmb d}_{s+\alpha _j } (k) \end{align} $ (15)

其中

$ {{\pmb\upsilon} }'_s (k)=\left[{{\begin{array}{*{20}c} {{\pmb\upsilon} _s (k), } \hfill&1 \hfill \\ \end{array}}} \right]\\ \bar {H}_{ds}^{b, j}(k)=\left[{{\begin{array}{*{20}c} \quad\quad\quad {\bar {H}_{ds} (k)M_{b, j}^d (0)} \hfill \\ {\sum\limits_{i=1}^{\alpha _j } {{\pmb\upsilon} _s (k-i)\bar {H}_{ds}(k-i)M_{b, j}^d (i)} } \hfill \\ \end{array} }} \right] $

同理, 采用相同的方式对式(13)中的其余项进行化简, 则式(13)可进一步表示为

$ \begin{equation} \left\{ {\begin{array}{l} \begin{split} r_{s, j}(k)= \, &{{\pmb\upsilon} }'_s(k)\left\{ {\bar {H}_{ds}^{b, j}(k){\pmb d}_{s+\alpha_j }(k)+} \right.\\ &\left.{\bar {H}_{fs}^{b, j}(k){\pmb f}_{s+\alpha _j }(k)}\right\}, \quad j=1, 2, \cdots, j_m \end{split} \\ \begin{split} r_{s, j_{m+1}}(k)= \, &{{\pmb\upsilon} }'_s (k)\left\{ {\bar {H}_{ds}^{l, j_m}(k){\pmb d}_{s+\alpha _{j_m }}(k)+} \right.\\ &\left.{\bar {H}_{fs}^{l, j_m} (k){\pmb f}_{s+\alpha_{j_m } }(k)} \right\} \\ \end{split} \end{array}} \right. \end{equation} $ (16)

式中, $\bar{H}_{ds}^{l, j_m}(k)$$\bar {H}_{fs}^{b, j}(k)$$\bar{H}_{fs}^{l, j_m}(k)$分别由$\bar {H}_{ds}^{b, j}(k)$中的$\left\{{\bar{H}_{ds} (k), M_{b, j}^d (i)} \right\}$替换为$\left\{ {\bar{H}_{ds} (k), M_{l, j_m }^d (i)} \right\}$$\left\{{\bar{H}_{fs}(k), M_{b, j}^{ f} (i)}\right\}$$\left\{ {\bar {H}_{fs}(k), M_{l, j_m}^{ f} (i)}\right\}$得到.

至此, 式(16)所示残差与(4)已具有相同的形式, 则可采用如下所示性能指标$J_{s, j}(k), j=1, 2, \cdots, j_m+1$, 将残差产生器设计转化不同尺度下的多目标最优化问题:

$ \begin{equation} \left\{ {\begin{array}{l} %\begin{split} J_{s, j} (k)=\mathop {\min }\limits_{{{\pmb\upsilon} }'_s (k)} \frac{{{\pmb\upsilon} }'_s (k)\bar {H}_{ds}^{b, j} (k)\left( {\bar {H}_{ds}^{b, j} (k)}\right)^{\rm T }\left( {{{\pmb\upsilon} }'_s (k)} \right)^{\rm T }}{{{\pmb\upsilon} }'_s(k)\bar {H}_{fs}^{b, j}(k) \left( {\bar {H}_{fs}^{b, j}(k)}\right)^{\rm T } \left({{{\pmb\upsilon} }'_s(k)} \right)^{\rm T }}, \\ \qquad \qquad \qquad \qquad \qquad j=1, 2, \cdots, j_m \\ %\end{split}\\ % \begin{split} J_{s, j_m +1} (k)=\mathop {\min }\limits_{{{\pmb\upsilon} }'_s (k)} \frac{{{\pmb\upsilon} }'_s (k)\bar {H}_{ds}^{l, j_m } (k)\left( {\bar {H}_{ds}^{l, j_m } (k)} \right)^{\rm T }\left( {{{\pmb\upsilon} }'_s (k)} \right)^{\rm T }}{{{\pmb\upsilon} }'_s (k)\bar {H}_{fs}^{l, j_m } (k)\left( {\bar {H}_{fs}^{l, j_m } (k)} \right)^{\rm T }\left( {{{\pmb\upsilon} }'_s (k)} \right)^{\rm T }} \\ %\end{split} \end{array}} \right. \end{equation} $ (17)

采用SVD方法求解式(17), 得到的$j_m +1$${{\pmb\upsilon} }'_s(k)$可分别使对应尺度的性能指标最小, 且通常情况下互不相等.为保证各尺度下所得残差对应性能指标均最优, 则应保留全部${{\pmb\upsilon}}'_s(k)$.为便于表述, 将不同$j$对应的${{\pmb\upsilon} }'_s(k)$记为${{\pmb\upsilon}}'_{s, j}(k), j=1, 2, \cdots, j_m +1$, 并化简为${{\pmb\upsilon} }'_{s, j}(k)=[{{\pmb\upsilon} _{s, j}(k), } 1]$的形式.对于不同的$j$, 分别将${\pmb w}_s (k)={\pmb\upsilon} _{s, j}(k)N_b(k)$代入式(10)求解$r_s(k)$, 则可根据式(9)所示递归算法计算残差$r_{s, j}(k), j=1, 2, \cdots, j_m +1$.

现将残差产生器设计步骤总结如下:

步骤1.  选取适当等价关系阶数$s$、小波基函数及最高分解尺度$j_m$, 确定初始滤波器${\pmb l}_1 $${\pmb h}_1 $.

步骤2.  计算${\pmb g}_{b, j}$, $j=1, 2, \cdots, j_m$, ${\pmb g}_{l, j_m }$及矩阵$M_{b, j}^d$$M_{b, j}^f$, $j=1, 2, \cdots, j_m$, $M_{l, j_m }^d$$M_{l, j_m }^f$.

步骤3.  令$k=k_0~(k_0 \ge s)$, 计算$H_{\xi s} (k)~(\xi =$ $o, { d}, { f}, { u})$$\bar {H}_{ds} (k)$$\bar {H}_{fs}(k)$.求解式(5)得到${\pmb\upsilon} _s (k)$, 并令${\pmb\upsilon} _{s, j} (k)={\pmb\upsilon} _s (k), j=1, 2, \cdots, j_m +1$.计算等价向量${\pmb w}_s (k)$及残差$r_s(k)$.令$WT_{r_s(k)}^a(0, k)=r_s(k)$, 根据式(9)所示递归算法计算$r_{s, j}(k)$, $j=1, 2, \cdots, j_m+1$.

步骤4.  令$k=k+1$, 采用文献[25]提出的递归算法更新$H_{\xi s} (k)~(\xi =o, d, f, u)$, 求解$\bar {H}_{ds} (k)$$\bar {H}_{fs} (k)$$\bar {H}_{ds}^{b, j} (k)$, $\bar {H}_{fs}^{b, j} (k)$, $j=1, 2, \cdots, j_m $$\bar {H}_{ds}^{l, j_m } (k)$$\bar {H}_{fs}^{l, j_m } (k)$.

步骤5.  求解式(17)所示多目标最优化问题, 得到${\pmb\upsilon} _{s, j} (k), j=1, 2, \cdots, j_m +1$, 令$j=1$.

步骤6.  计算${\pmb w}_s (k)={\pmb\upsilon} _{s, j}(k)N_b(k)$$r_s(k)={\pmb w}_s(k)\left[{{\pmb y}_s(k)-H_{us}(k){\pmb u}_s(k)}\right]$, 并令初始系数$WT_{r_s (k)}^a(0, k)=r_s(k)$, 进而根据递归算法式(9)计算$WT_{r_s (k)}^d (j, k)$.

步骤7.  若$j<j_m $, 令$j=j+1$, 执行步骤6.从而可得到残差$r_{s, j} (k)=WT_{r_s (k)}^d (j, k)$, $j=1, 2, \cdots, j_m $.

步骤8.  计算${\pmb w}_s (k)={\pmb\upsilon} _{s, j_m +1} (k)N_b (k)$$r_s (k)={\pmb w}_s (k)[{\pmb y}_s (k)-H_{us} (k){\pmb u}_s (k)]$, 令$WT_{r_s(k)}^a (0, k)= \quad r_s (k)$, 采用递归算法(9)可得到$r_{s, j_m +1}(k)=WT_{r_s (k)}^d (j_m, k)$.

步骤9.  重复步骤4至8, 得到不同$k$时刻各尺度下的残差信号$r_{s, j} (k), j=1, 2, \cdots, j_m +1$.

需要指出的是, 通过引入SWT将多尺度残差产生器设计转化为各尺度下的性能指标最小化问题进行求解, 确保了各尺度残差对相应频率范围内故障信号较好的灵敏性和对干扰的鲁棒性.从而可实现较宽频率范围内故障信号的检测, 有效降低故障漏报率.另外, 引入SWT将导致较大的附加计算量, 不利于FD系统的在线实现.因此, 为尽可能地减小在线计算量, 提高故障检测的快速性, 应尽量选取紧支撑特性较强的小波基函数, 且最高分解尺度$j_m$不宜过大.

注2.  在选取小波基函数及最高分解尺度$j_m$时, 目前并无一套统一的固定规则, 通常需结合小波基函数的数学特性及实际需求进行选取.这里考虑在获得较好FD性能的前提下尽可能地减小在线计算量.由式(9)和(17)可知, 当等价关系阶数$s$确定后, 引入SWT所带来的附加在线计算量主要由小波分解初始滤波器${\pmb l}_1$${\pmb h}_1$的时域宽度以及最高分解尺度$j_m$决定.紧支撑特性是小波基函数时域支集长度的度量, 紧支撑特性越强, 则小波基所对应初始滤波器的时域宽度越小, 其带来的附加在线计算量也越小, 故而应选取具有较强紧支撑特性的小波作为SWT的基函数.另外, 增大最高分解尺度$j_m$可实现更高分辨率下的残差分析及评价, 并改善多尺度残差所对应最优性能指标, 但在线计算量也将随之增大, 且低通滤波器${\pmb g}_{l, j_m}$的带宽变窄, 引起较大的检测延时, 不利于实时故障检测.因此, SWT的最高分解尺度$j_m$不宜过大.

2.2 多分辨率残差分析

根据上述步骤产生残差信号后, 需选取适当的残差评价函数及阈值进行残差评价.本文采用有限时间窗口内各尺度残差信号的二范数作为残差评价函数, 即

$ \begin{equation} J_{r_{s, j} } (k)=\left\| {r_{s, j} (k)} \right\|_{2, N}, \quad j=1, 2, \cdots, j_m+1 \end{equation} $ (18)

式中, $\left\| {r_{s, j} (k)} \right\|_{2, N} =\sqrt{\sum_{k-N}^k{r_{s, j}^{\rm T}(i)r_{s, j}(i)}}$, $N$为移动时间窗口长度.

相应地, 阈值应取为无故障情况下残差评价函数的上确界.考虑到在系统无故障(即${\pmb f}=0$)时, $r_s(k)={\pmb\upsilon} _s (k)\bar{H}_{ds}(k){\pmb d}_s(k)$, 采用SWT对$r_s (k)$进行$j_m$尺度滤波, $r_s(k)$被由粗及精地分解为不同分辨率(尺度)下、处于不同频带的一组残差信号$r_{s, j}(k), j=1, 2, \cdots, j_m+1$.显然, 不同分辨率下的残差在有限时间步长内的范数上界通常是不同的.因此, 这里针对获得的$j_m+1$个不同分辨率下的残差评价函数分别设定阈值, 以降低故障漏报率, 即

$ \begin{equation} J_{th, s, j} =\mathop {\sup }\limits_{{\pmb f}=0} \left\| {r_{s, j} (k)} \right\|_{2, N}, \quad j=1, 2, \cdots, j_m +1 \end{equation} $ (19)

在系统运行过程中, 当存在$j(1\le j\le j_m+1)$使得残差评价函数$J_{r_{s, j}}(k)$高于对应阈值$J_{th, s, j}$时, 即认为LDTV系统发生故障, 判定逻辑如下所示:

$ \begin{equation} \left\{ {\begin{array}{ll} J_{r_{s, j} }(k)\le {J_{th, s, j} }, \forall j=1, 2, \cdots, j_m+1,&\mbox{正常} \\ J_{r_{s, j} }(k)> {J_{th, s, j} }, \forall j=1, 2, \cdots, j_m+1,&\mbox{故障} \\ \end{array}} \right. \end{equation} $ (20)
3 仿真实验

为验证本文所提方法的有效性, 考虑LDTV系统(1), 系统矩阵如下所示:

$ A(k)=\left[{{\begin{array}{*{20}c} {0} &{1-0.2\cos (k)} &{0} \\ {0} &{0.45-{\rm e}^{-\frac{k}{60}}} &{0.6} \\ {-1} &{0} & {0} \\ \end{array} }} \right] $
$ B(k)=\left[{\begin{array}{l} 1 \\ 1 \\ 0 \\ \end{array}} \right] $
$ C(k)=\left[{{\begin{array}{*{20}c} 1 & 0.9-\dfrac{1}{k+1} &0 \\ \end{array} }} \right] $
$ B_d (k)=\left[{{\begin{array}{*{20}c} {0.8} &0 \\ {0.1} &0 \\ 0&1 \\ \end{array} }} \right], \quad B_f (k)=\left[{{\begin{array}{*{20}c} 1 \hfill \\ 0 \hfill \\ 1 \hfill \\ \end{array} }} \right] $
$ D_d (k)=\left[{{\begin{array}{*{20}c} 0 \hfill&{0.5} \hfill \\ \end{array} }} \right], \quad D_f (k)=0.6 $

${\pmb u}(k)=1$, ${\pmb d}(k)=\left[{{\begin{array}{*{20}c} {{ d}_1 (k)},&{{ d}_2(k)}\end{array} }}\right]$, 其中${ d}_1(k)$为方差0.5$^{2}$的零均值高斯白噪声, ${ d}_2 (k)=0.5\cos (k)$.在Matlab仿真环境下, 分别考虑上述LDTV系统发生如下所示两种故障的情况:

1) 阶跃故障:

$ \begin{equation} {\pmb f}(k)=\left\{ \begin{array}{l} 0.2, \quad ~~~k\in \left[{200, 500} \right]\\ -0.2, \quad k\in \left[{700, 1\, 000} \right]\\ 0, \quad \quad~~ \mbox{其他}\\ \end{array} \right. \end{equation} $ (21)

2) 正弦故障:

$ \begin{equation} {\pmb f}(k)=\left\{ \begin{array}{l} 0.4\sin (0.1\pi k), \quad k\in \left[{200, 500} \right] \\ 0.4\sin (0.2\pi k), \quad k\in \left[{700, 1\, 000} \right] \\ 0, \qquad \qquad~ \qquad \mbox{其他} \\ \end{array} \right. \end{equation} $ (22)

选取$s=4$, SWT的基函数为db1小波, 最高分解尺度$j_m =5$, 移动步长$N=10$, 采用本文方法对上述故障进行检测, 并与$s=4$$s=15$时的等价空间方法进行比较, 其残差评价函数记为$J_{r_s } (k)=\left\| {r_s (k)} \right\|_{2, N} $, 阈值记为$J_{th, s} = \sup\limits_{{\pmb f}=0} \left\| {r_s (k)} \right\|_{2, N} $.仿真结果如图 1图 2所示.

图 1 阶跃故障检测结果 Figure 1 The FD results of step fualt
图 2 正弦故障检测结果 Figure 2 The FD results of sine fault

图 1可见, 采用本文方法获得的残差评价函数$J_{r_{4, 3} } (k)$$J_{r_{4, 4} } (k)$$J_{r_{4, 5} } (k)$均可较好地检测到故障.基于等价空间的方法中, 残差评价函数$J_{r_{15} }(k)$可较好地检测到故障, 而$s=4$时, 因相应性能指标较差, $J_{r_4 } (k)$的检测效果相对较差.

图 2(a)可以看出, 当故障信号的角频率为$\omega _1=0.1\pi$时, 残差评价函数$J_{r_{4, 2}}(k)$$J_{r_{4, 3}} (k)$$J_{r_{4, 4}} (k)$$J_{r_{4, 5} } (k)$$J_{r_{4, 6} }(k)$均可较好地检测到故障, 当故障信号角频率为$\omega _2 =0.2\pi $时, $J_{r_{4, 2} } (k)$$J_{r_{4, 3} } (k)$$J_{r_{4, 6} } (k)$仍可保证较好的检测效果.如图 2(b)所示, 基于等价空间的方法中, $J_{r_4 } (k)$可检测到角频率为$\omega _1$$\omega _2 $的故障, 而$J_{r_{15}}(k)$则仅能较好地检测角频率为$\omega_1$的故障.由此可见, 与基于等价空间的方法相比, 本文方法可高效检测较宽频率范围内的故障信号, 有效降低了故障漏报率.

为进一步验证未知输入信号${\pmb d}(k)$增大情况下本文方法的有效性, 分别考虑当高斯白噪声$d_1(k)$的方差增大为$0.7^2$${ d}_2(k)=0.5\cos (k)$${ d}_1(k)$方差为$0.5^2$${ d}_2 (k)=1.0\cos(k)$的情况, 对式(22)所示正弦故障信号进行检测, 检测结果如图 3图 4所示.

图 3$d_1(k)$方差为0.7$^2$时的正弦故障检测结果 Figure 3 The FD results of sine fault with the variance of $d_1(k)$ rising to $0.7^2$
图 4$d_2(k)=1.0\cos(k)$时的正弦故障检测结果 Figure 4 The FD results of sine fault with the $d_2(k)=1.0\cos(k)$

图 3图 4可见, 当干扰信号的幅度增大时, 采用本文方法获得的残差评价函数$J_{r_{4, 3}} (k)$$J_{r_{4, 4} }(k)$$J_{r_{4, 6}}(k)$仍可较好地检测到故障.相比之下, 传统等价空间方法得到的$J_{r_{15} }(k)$仍旧只能检测到角频率为$\omega_1 $的故障, 而$J_{r_4 }(k)$已难以检测到故障.由此可见, 本文提出的方法对干扰具有较强的鲁棒性, 在强噪声环境下仍可较好地检测较宽频带内的故障信号.

4 结论

本文将SWT与等价空间方法相结合对LDTV系统FD问题进行了研究.通过引入SWT对基于低阶等价关系构造的残差进行多尺度滤波, 将残差产生器设计转化为不同尺度下的多目标最优化问题, 并利用SWT快速算法获得一组多尺度残差信号.在残差评价阶段, 对多尺度残差进行多分辨率分析与评价, 从而实现LDTV系统故障检测.仿真结果表明, 与传统基于等价空间的方法相比, 该方法可实现较宽频率范围内故障信号的检测, 有效降低了故障漏报率, 并且对干扰具有较强的鲁棒性.如何进一步简化该方法的在线实现, 减小在线计算量仍有待下一步研究.

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