Fault Reconstruction and Identification

Abstract

This chapter introduces fault reconstruction and identification methods, particularly for those processes whose data follow non-Gaussian distributions. For non-Gaussian fault detection, recent work has demonstrated the effectiveness of Independent Component Analysis (ICA) to extract non-Gaussian source signal and Support Vector Data Description (SVDD) to determine control limits for associated monitoring statistics. Here, a fault reconstruction method is introduced based on the SVDD model, which is similar to the recently developed reconstruction methods upon the PCA model. For fault identification, the traditional PCA-based similarity factor is extended to the non-Gaussian case, which is developed upon the ICA model, namely ICA similarity factor. Furthermore, with the introduction of the FA model, an additional noise similarity factor has also been defined. By combining these different similarity factors, a mixture similarity factor is developed for fault identification under different situations.

Appendix: Iterative Reconstruction Procedure in the Feature Space

Section 4.2 showed that z was reconstructed without the consideration of noises and outliers. To develop a robust iterative reconstruction scheme, Eq. (4.9) can be rewritten as follows

$$ \mathbf{f}(t+1)={{({{\mathbf{\Theta }}_{j}}^{T}{{\mathbf{W}}^{T}}\mathbf{W}{{\mathbf{\Theta }}_{j}})}^{-1}}\frac{{{\mathbf{\Theta }}_{j}}^{T}{{\mathbf{W}}^{T}}\sum\limits_{i=1}^{K}{{{\alpha }_{i}}K(\mathbf{s}{{\mathbf{v}}_{i}},\mathbf{\tau }(t))(\hat{\mathbf{s}}-\mathbf{s}{{\mathbf{v}}_{i}})}}{\sum\limits_{i=1}^{K}{{{\alpha }_{i}}K(\mathbf{s}{{\mathbf{v}}_{i}},\mathbf{\tau }(t))}} $$

(4.35)

where

$$ \mathbf{\uptau }(t)=\left\{ \begin{array}{*{35}{l}} {\hat{\mathbf{s}}} & t=0\\ \hat{\mathbf{s}}-\mathbf{W}{{\mathbf{\Theta }}_{j}}\mathbf{f}(t-1) & t>0\end{array} \right. $$

(4.36)

However, according to Takahashi and Kurita (2002), above updating procedure may fail during the iteration unless τ(0) locates in the near neighborhood of ŝ*. To prevent this issue, we introduce a robust form of the update in Eq. (4.29), which is given by

$$ \mathbf{\tau }(t)=\left\{ \begin{array}{*{35}{l}} {\hat{\mathbf{s}}} & t=0\\ \mathbf{B}(t)\hat{\mathbf{s}}+[\mathbf{I}-\mathbf{B}(t)](\hat{\mathbf{s}}-\mathbf{W}{{\mathbf{\Theta }}_{j}}\mathbf{f}(t-1)) & t>0\end{array} \right. $$

(4.37)

where I is a d ´ d identity matrix and the d ´ d matrix B(t) is defined as diag{b ₁(t), …, b _d(t) }. Each element B(t) denotes the “certainty” of ŝ and can be estimated by the difference between sv and its corresponding reconstruction ŝ(t − 1) = s − W Θ _j f(t − 1)

$$ {{b}_{i}}(t)=\exp \left({-}\frac{1}{2}\frac{-{{[{{{\hat{\mathbf{s}}}}_{i}}-{{{\hat{\mathbf{s}}}}_{i}}(t-1)]}^{2}}}{\sigma _{i}^{2}} \right) $$

(4.38)

where i = 1, 2, … d and the parameter σ _i is the standard deviation of the differences, given by (Takahashi and Kurita 2002)

$$ {{\sigma }_{i}}=1.4826\left( 1+\frac{5}{K-1} \right)\text{me}{{\text{d}}_{j}}\sqrt{\varepsilon _{ji}^{2}} $$

(4.39)

where ε ² _ji is the squared error between the ith component of the jth support vector and the reconstructed value, and $\text{me}{{\text{d}}_{j}}\sqrt{\varepsilon _{\textit{ji}}^{2}}$ is the median of $\sqrt{\varepsilon _{\textit{ji}}^{2}}.$ The next iteration implies that the original value s is used for the next step if it has a high “certainty.” Otherwise, the reconstructed value ŝ(t − 1) is used as its estimation. The iteration has converged if ||τ(t) − τ(t − 1)|| < e or if ||f(t) − f(t − 1)|| < e with e being the convergence criterion, e.g., 10⁻⁵.