**Using Wavelets for Feature Extraction and Self Organizing Maps for Fault Diagnosis of Nonlinear Dynamic Systems**

Héctor Benítez-Pérez, Jorge L. Ortega-Arjona and Alma Benítez-Pérez

Additional information is available at the end of the chapter

http://dx.doi.org/10.5772/50235

## **1. Introduction**

16 Applications of Self-Organizing Maps

42 Applications of Self-Organizing Maps

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[3] A. Ultsch. U\*-matrix: a tool to visualize clusters in high dimensional data. Technical Report 36,

[4] J. Vesanto. SOM-based data visualization methods. *Intelligent Data Analysis*, 3:111–126, 1999.

[5] S. Kaski, J. Nikkila, and T. Kohonen. Methods for interpreting a self-organized map in data analysis. In *Proceedings of European Symposium on Artificial Neural Networks*, Bruges, Belgium,

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[7] Hujun Yin. ViSOM-a novel method for multivariate data projection and structure visualization.

[8] Mu-Chun Su and Hsiao-Te Chang. A new model of self-organizing neural networks and its application in data projection. *IEEE Transactions on Neural Networks*, 123(1):153–158, 2001.

[9] Lu Xu, Yang Xu, and Tommy W.S. Chow. PolSOM-a new method for multidimentional data

[10] R. Kamimura. Self-enhancement learning: target-creating learning and its application to

[11] R. Kamimura. Constrained information maximization by free energy minimization. *International*

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References

1998.

Fault diagnosis has been established in two main approaches: model-based fault diagnosis and model-free fault diagnosis. Present paper focuses on the later, mainly as an extension of the approach proposed in [17]. The challenge here is to classify faults at early stages, with an accurate response. However, as the term model-free implies, a model for the plant is not available neither for fault-free nor for fault-present scenarios. The objective, thus, is to classi‐ fy faults based on system's response and the related signal analysis, in terms of dilation and shift decomposition, as obtained by a wavelets approach. So, self-organizing maps (SOM) are proposed as a powerful nonlinear neural network to achieve such a fault classification.

Several strategies have been proposed for feature extraction using wavelets. For instance, [1] presents a wavelet packet feature extraction, based on the analysis and measure of a "distance" between the energy distribution of some signal classes and the proper classifi‐ cation by the use of fuzzy sets. Alternatively, [2] proposes the use of wavelets as a strat‐ egy of parametric system identification, giving prime emphasis to wavelet properties and parameter relations. The idea of using wavelets for fault classification is a powerful proce‐ dure for feature extraction of several scenarios, even in the case of frequency and power shifts. [3] and [4] have explored this approach for process system, in which practical re‐ sults are satisfactory, regardless of the classification. Moreover, several other strategies us‐ ing wavelets have been proposed for abnormal signal detection, like that presented in [5], in which a parasitic wavelet transform is proposed. Further research in the same direction is followed in [6], in which a cubic spline methodology is proposed for the boundary

© 2012 Benítez-Pérez et al.; licensee InTech. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. © 2012 Benítez-Pérez et al.; licensee InTech. This is a paper distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

problem, although the results of this approach tend to be just local, linear models. An al‐ ternative strategy for auto-correlation and signal discovery is proposed in [7], following some multi-class wavelet support vector machines. In all these methodologies, wavelets are used as a technique for feature extraction; however, none of the above have presented any enhancement for pattern classification. Here, an enhancement for pattern classifica‐ tion is proposed, in order to isolate different scenarios.

selected patterns. The importance of the methodology is the capacity for fault isolation at unknown scenarios, with enough time to pursue modification in terms of system safety. In this sense, time response is determined in terms of the sampled window and the classifica‐ tion of current analyzed data. It is necessary to establish a feasible relationship among the sampling window, the time taken to process information, and the accuracy to classify a par‐ ticular scenario. This can be achieved by following a frequency analysis of some selected scenarios, in order to find such a relationship. However, this is a non-homogeneous strategy for any scenario. Further work needs to be done in terms of data analysis and sampling ca‐

Using Wavelets for Feature Extraction and Self Organizing Maps for Fault Diagnosis of Nonlinear Dynamic Systems

http://dx.doi.org/10.5772/50235

45

Based on this extensive review, the current approach divides the process of fault isolation using SOM techniques into a two stages process. The first stage is a basic construction of the map, as pattern clusters, using SOM. The second stage is a labeling process that identifies scenarios of the system. Following this, the current approach proposes the classification of time-varying faults within a bounded time window, using the wavelet decomposition inher‐ ent response. The objective, hence, is to establish an approach for fault localization, based on feature extraction and clustering, by considering diverse fault-present and fault-free scenar‐ ios. The novelty of this approach is on signal classification for time varying scenarios, under unknown consideration. The proposed system is limited to certain signals conditions, such as coupled noise and frequency response. In this case, frequency dispersion should be bounded, regardless of time variance. The main advantage of this approach is related to fault isolation through signal decomposition, and classification in a bounded time response.

Wavelet transform (WT) is an alternative method for processing transient, non-stationary signals simultaneously in time-scale domains [11]. Wavelets are used to decompose a signal into different *scale factors*. Wavelet approach provides a more natural description of the sig‐ nal, in terms of a composition of a set of "typical signals", or wavelets. In fact, the WT is the correlation between a signal and a set of basic wavelets, proposed from a basic "mother wavelet" *h(t)*, chosen in order to analyze a specific transient signal of finite energy. Thus, a complete orthogonal set of "daughter wavelets" *h a,b (t)* is generated from *h(t)* by two opera‐

Formally, the dilation and shift, as wavelet coefficients of the signal, are defined by:

−*∞*

*s*(*t*)*ha*,*<sup>b</sup>*

\* (*t*)*dt* (1)

*∞*

*WS* (*a*, *b*)= *∫*

pabilities to recognize scenarios, either known or unknown.

**2. Background**

**2.1. The Wavelet Transform**

tions: a dilation *a* and shift *b*.

On the other hand, the use of neural networks for feature extraction only presents the disad‐ vantage of inherent data uncertainties and large quantity of necessary data. Different pro‐ posals have previously explored similar strategies. For example, [8] proposes feature extraction using local parametric models, giving valuable results; however, there is a draw‐ back of a bounded system response. This strategy for fault diagnosis integrates an ART2A network and a Kohonen neural network. The objective is to combine both strategies in order to generate two subsystems capable of overcoming glitches and redundant data representa‐ tions [8]. Both subsystems, based on the ART2A topology and the Kohonen neural network, are used to perform a learning strategy. This strategy allows on-line fault diagnosis, with the inherent uncertainty of SOM variation due to the plasticity-stability dilemma. A fundamen‐ tal work has been introduced in [9], in which an extended review is provided regarding top‐ ics related to sensors patterns and stability-plasticity trade-offs, inherent to an ART2A network. Interesting comments have been included about how a time window data can be monitored, in order to identify abnormal situations, as well as how data should be treated in terms of normalization, time scaling, and filtering, and their comparison prior to declare a winner selection. Further developments are addressed in [10], focusing on the use of a paral‐ lel ART2A network approach, based on a wavelet decomposition in which clustering is de‐ fined in the wavelet domain, although it is not proposed for a dynamical system.

Feature extraction for dynamical systems based on wavelets presents the advantage of scale decomposition, allowing several possibilities of fault detection depending on the scale of the fault. Similarly, fault detection can be easily engaged if a source of information is decom‐ posed into several fruitful components. These components are taken as parameter vectors, where several signal conditions are highlighted depending on the resolution. Further, these components need to be combined in a fair strategy, in order to classify similar behaviors. To do so, the use of SOM is proposed, in which each vector is processed as a consecutive input. The result of this classification gives a number of selected patterns, depending on the learn‐ ing rate, and regarding a time window. Nevertheless, using this technique, the plasticity-sta‐ bility dilemma is still not solved.

As stated before, feature extraction presents an inherently extrapolated method to determine several characteristics (like geometry differences) related to the monitored signal, based on a scale factor. The most typical characteristics are those related to frequency and phase modi‐ fication, which are multiplicative faults in terms of fault detection. Amplitude change is de‐ tected on the general modification from wavelets scales, and the consequent change on the selected patterns. The importance of the methodology is the capacity for fault isolation at unknown scenarios, with enough time to pursue modification in terms of system safety. In this sense, time response is determined in terms of the sampled window and the classifica‐ tion of current analyzed data. It is necessary to establish a feasible relationship among the sampling window, the time taken to process information, and the accuracy to classify a par‐ ticular scenario. This can be achieved by following a frequency analysis of some selected scenarios, in order to find such a relationship. However, this is a non-homogeneous strategy for any scenario. Further work needs to be done in terms of data analysis and sampling ca‐ pabilities to recognize scenarios, either known or unknown.

Based on this extensive review, the current approach divides the process of fault isolation using SOM techniques into a two stages process. The first stage is a basic construction of the map, as pattern clusters, using SOM. The second stage is a labeling process that identifies scenarios of the system. Following this, the current approach proposes the classification of time-varying faults within a bounded time window, using the wavelet decomposition inher‐ ent response. The objective, hence, is to establish an approach for fault localization, based on feature extraction and clustering, by considering diverse fault-present and fault-free scenar‐ ios. The novelty of this approach is on signal classification for time varying scenarios, under unknown consideration. The proposed system is limited to certain signals conditions, such as coupled noise and frequency response. In this case, frequency dispersion should be bounded, regardless of time variance. The main advantage of this approach is related to fault isolation through signal decomposition, and classification in a bounded time response.

## **2. Background**

problem, although the results of this approach tend to be just local, linear models. An al‐ ternative strategy for auto-correlation and signal discovery is proposed in [7], following some multi-class wavelet support vector machines. In all these methodologies, wavelets are used as a technique for feature extraction; however, none of the above have presented any enhancement for pattern classification. Here, an enhancement for pattern classifica‐

On the other hand, the use of neural networks for feature extraction only presents the disad‐ vantage of inherent data uncertainties and large quantity of necessary data. Different pro‐ posals have previously explored similar strategies. For example, [8] proposes feature extraction using local parametric models, giving valuable results; however, there is a draw‐ back of a bounded system response. This strategy for fault diagnosis integrates an ART2A network and a Kohonen neural network. The objective is to combine both strategies in order to generate two subsystems capable of overcoming glitches and redundant data representa‐ tions [8]. Both subsystems, based on the ART2A topology and the Kohonen neural network, are used to perform a learning strategy. This strategy allows on-line fault diagnosis, with the inherent uncertainty of SOM variation due to the plasticity-stability dilemma. A fundamen‐ tal work has been introduced in [9], in which an extended review is provided regarding top‐ ics related to sensors patterns and stability-plasticity trade-offs, inherent to an ART2A network. Interesting comments have been included about how a time window data can be monitored, in order to identify abnormal situations, as well as how data should be treated in terms of normalization, time scaling, and filtering, and their comparison prior to declare a winner selection. Further developments are addressed in [10], focusing on the use of a paral‐ lel ART2A network approach, based on a wavelet decomposition in which clustering is de‐

fined in the wavelet domain, although it is not proposed for a dynamical system.

bility dilemma is still not solved.

Feature extraction for dynamical systems based on wavelets presents the advantage of scale decomposition, allowing several possibilities of fault detection depending on the scale of the fault. Similarly, fault detection can be easily engaged if a source of information is decom‐ posed into several fruitful components. These components are taken as parameter vectors, where several signal conditions are highlighted depending on the resolution. Further, these components need to be combined in a fair strategy, in order to classify similar behaviors. To do so, the use of SOM is proposed, in which each vector is processed as a consecutive input. The result of this classification gives a number of selected patterns, depending on the learn‐ ing rate, and regarding a time window. Nevertheless, using this technique, the plasticity-sta‐

As stated before, feature extraction presents an inherently extrapolated method to determine several characteristics (like geometry differences) related to the monitored signal, based on a scale factor. The most typical characteristics are those related to frequency and phase modi‐ fication, which are multiplicative faults in terms of fault detection. Amplitude change is de‐ tected on the general modification from wavelets scales, and the consequent change on the

tion is proposed, in order to isolate different scenarios.

44 Developments and Applications of Self-Organizing Maps Applications of Self-Organizing Maps

#### **2.1. The Wavelet Transform**

Wavelet transform (WT) is an alternative method for processing transient, non-stationary signals simultaneously in time-scale domains [11]. Wavelets are used to decompose a signal into different *scale factors*. Wavelet approach provides a more natural description of the sig‐ nal, in terms of a composition of a set of "typical signals", or wavelets. In fact, the WT is the correlation between a signal and a set of basic wavelets, proposed from a basic "mother wavelet" *h(t)*, chosen in order to analyze a specific transient signal of finite energy. Thus, a complete orthogonal set of "daughter wavelets" *h a,b (t)* is generated from *h(t)* by two opera‐ tions: a dilation *a* and shift *b*.

Formally, the dilation and shift, as wavelet coefficients of the signal, are defined by:

$$\mathcal{W}\_S(a,b) = \bigcup\_{\sim}^{\sim} s(t)h\_{a,b}^{\prime}(t)dt\tag{1}$$

where *s(t)* is the current signal, and the function *h a,b (t)* defined as:

$$h\_{a,b}(t) = a^{-1/2} h\left(\frac{t-b}{a}\right) \tag{2}$$

raw resolution 2<sup>−</sup> *<sup>j</sup>*−<sup>1</sup>

of signal *f* over the space V*<sup>j</sup>*

On the other hand, if the resolution2<sup>−</sup> *<sup>j</sup>*

*f* ∈V0, it may be decomposed as:

The approximation of *f* in the resolution 2<sup>−</sup> *<sup>j</sup>*

*scale function, where the Fourier Transformation is:*

*φ*

^(*<sup>ω</sup>*)= *<sup>θ</sup>*

( ∑ *k*=−*∞* +*∞* |*θ*

with

V*j*

*scale*.

merically stable.

of the signal converges to the original signal:

. If the functions are dilated in V*<sup>j</sup>* by 2, then the details are amplified by a

*f* =0. (4)

http://dx.doi.org/10.5772/50235

47

*f* =0. (5)

*a n θ*(*t* −*n*) (6)


is defined as the orthogonal projection *P*<sup>V</sup> *<sup>j</sup>*

1/2 (8)

*f* over

. Hence,

tends to+*∞*property (v) forces that the approximation

factor of 2. So (iii) defines an approximation of a raw resolution when the resolution 2<sup>−</sup> *<sup>j</sup>* tends to be cero. (iv) implies that all the details have been lost, meaning that the projection

Using Wavelets for Feature Extraction and Self Organizing Maps for Fault Diagnosis of Nonlinear Dynamic Systems

when *j* → + *∞*is zero:

lim *j*→+*∞*

lim *j*→−*∞*

*f* (*t*)= ∑ *n*=−*∞* +*∞*

*A f* <sup>2</sup> ≤ ∑

*n*=−*∞* +*∞*

This last expression guarantees that the expansion of the signal over {*θ*(*t* −*n*):*n* ∈*Z*} is nu‐

the following theorem allows for an orthogonal Riesz base{*θ*(*t* −*n*):*n* ∈*Z*}, and it builds an orthogonal basis for each space V*<sup>j</sup>* where dilation and transferring over a function *ϕ*, named

*Theorem* (*Goswami Jaideva, 1999*)*. Being* {V *<sup>j</sup>* : *j* ∈*Z*} *an approximation of multi-resolution and ϕ a*

^(*<sup>ω</sup>*)

^(*<sup>ω</sup>* <sup>+</sup> <sup>2</sup>*kπ*)| <sup>2</sup>

)

. To compute this projection, an orthogonal base should be find over the space V*<sup>j</sup>*

*P*<sup>V</sup> *<sup>j</sup>*

*f* −*P*<sup>V</sup> *<sup>j</sup>*

Finally, the existence of a Riesz base {*θ*(*t* −*n*):*n* ∈*Z*} of V*<sup>0</sup>* provides a discretization theorem. Function *θ* can be interpreted as a cell with unitary resolution. To compute several resolu‐ tions of signal, it is necessary then to compute the orthogonal components over different spaces {V *<sup>j</sup>* : *<sup>j</sup>* <sup>∈</sup>*Z*} of L<sup>2</sup><sup>⊄</sup> According to the definition of a Riesz base, there are *A,B > 0* that if

The information used in this approach is based on both *a* and *b*, as expansion coefficients. Commonly, the mother wavelet is considered as a Daubechies signal. The wavelets automat‐ ically adapt to the different components of a signal, using a small window (large scale) to search for brief high-frequency components, and large window (low scale) to look for long lived, low-frequency components. The shape of low- and high-frequency components is de‐ termined by the mother wavelet. A further and deeper revision of the wavelet technique may be found in [12].

Wavelets can be represented using a function *ψ*, and the familyℑof expanded and translated wavelets are expressed as :

$$\mathfrak{D} = \left| \psi\_{\,j,n}(t) = \frac{1}{\sqrt{2^{\frac{1}{j}}}} \psi\left(\frac{t - 2^{\frac{1}{j}}n}{2^{\frac{1}{j}}}\right); j, \; n \in \mathbb{Z} \right| \tag{3}$$

which performs an orthogonal baseL2⊄. Orthonormal wavelets are obtained by expanding this by a factor 2 *<sup>j</sup>* , allowing variations of the signal in a 2<sup>−</sup> *<sup>j</sup>* resolution. The construction of these bases permits the study of multi-resolution of a signal. Formally, the approximation of a function with a resolution 2<sup>−</sup> *<sup>j</sup>* is defined as an orthogonal projection over a spaceV *<sup>j</sup>* <sup>⊂</sup>L2<sup>⊄</sup> The space Vj groups every possible approximation with resolution 2<sup>−</sup> *<sup>j</sup>* . Remember that an orthogonal projection from function *f* is a function *f <sup>j</sup>* ∈V *<sup>j</sup>* which is minimized *f* − *f <sup>j</sup>* .

*Definition.* (*Multi-Resolution*) *A family of closed subspaces* {V *<sup>j</sup>* : *<sup>j</sup>* <sup>∈</sup>*Z*} *of L<sup>2</sup> <sup>⊄</sup> is a multi-resolution approximation, if it satisfies the following properties:*

$$\begin{array}{cc} \textbf{i.} & \text{For j. } k \in \mathbb{Z}; f(t) \in \textbf{V}\_{\not\equiv} \bullet f\left(t - \mathbf{2}^{\not\equiv} k\right) \in \textbf{V}\_{\not\equiv} \end{array}$$

$$\begin{array}{cc} \text{iii} & \text{For } j \in \mathbb{Z}, \mathbf{V}\_{j^{\*1}} \subset \mathbf{V}\_{j} \end{array}$$

$$\mathbf{iii.}$$

$$\begin{array}{ll} \text{iii.} & \text{For } j \in \mathbb{Z}, f(t) \in \mathbb{V}\_{\neq} \bullet \; f\left(\frac{t}{2}\right) \in \mathbb{V}\_{\neq 1}. \end{array}$$


$$\mathbf{v}\mathbf{i}.\qquad\text{there is }\theta\text{ such that}\\
\{\theta(t-n): n\in\mathbb{Z}\}\text{is a Riesz base of }\mathbf{V}\_0\text{ .}$$

Property (i) states that the subspace V*<sup>j</sup>* is invariant in any translation proportional to scale 2 *<sup>j</sup>* Property (ii) is causal, since the resolution 2<sup>−</sup> *<sup>j</sup>* owns the necessary information to calculate a raw resolution 2<sup>−</sup> *<sup>j</sup>*−<sup>1</sup> . If the functions are dilated in V*<sup>j</sup>* by 2, then the details are amplified by a factor of 2. So (iii) defines an approximation of a raw resolution when the resolution 2<sup>−</sup> *<sup>j</sup>* tends to be cero. (iv) implies that all the details have been lost, meaning that the projection of signal *f* over the space V*<sup>j</sup>* when *j* → + *∞*is zero:

$$\lim\_{j \to \infty} \|\|P\_{\mathcal{V}\_j} f\|\| = 0. \tag{4}$$

On the other hand, if the resolution2<sup>−</sup> *<sup>j</sup>* tends to+*∞*property (v) forces that the approximation of the signal converges to the original signal:

$$\lim\_{j \to \! - \! \! - \! \! - \! \! \! -} \|f - P\_{\text{V}\_j} f\| = 0. \tag{5}$$

Finally, the existence of a Riesz base {*θ*(*t* −*n*):*n* ∈*Z*} of V*<sup>0</sup>* provides a discretization theorem. Function *θ* can be interpreted as a cell with unitary resolution. To compute several resolu‐ tions of signal, it is necessary then to compute the orthogonal components over different spaces {V *<sup>j</sup>* : *<sup>j</sup>* <sup>∈</sup>*Z*} of L<sup>2</sup><sup>⊄</sup> According to the definition of a Riesz base, there are *A,B > 0* that if *f* ∈V0, it may be decomposed as:

$$f\left(t\right) = \sum\_{n=-\infty}^{+\infty} a \mathbf{f}[n] \Theta(t-n) \tag{6}$$

with

where *s(t)* is the current signal, and the function *h a,b (t)* defined as:

46 Developments and Applications of Self-Organizing Maps Applications of Self-Organizing Maps

<sup>ℑ</sup>={*ψj*,*n*(*t*)= <sup>1</sup>

orthogonal projection from function *f* is a function *f <sup>j</sup>* ∈V *<sup>j</sup>*

*f* (*t* −2 *<sup>j</sup>*

<sup>2</sup> ) <sup>∈</sup><sup>V</sup> *<sup>j</sup>*+1

*<sup>f</sup>*( *<sup>t</sup>*

**vi.** *there isθsuch that*{*θ*(*t* −*n*):*n* ∈*Z*}*is a Riesz base of* V*<sup>0</sup> .*

*approximation, if it satisfies the following properties:*

V *<sup>j</sup>* ={0}

<sup>V</sup> *<sup>j</sup>* =L2<sup>⊄</sup>

Property (ii) is causal, since the resolution 2<sup>−</sup> *<sup>j</sup>*

**i.** *Forj*, *<sup>k</sup>* <sup>∈</sup>*Z, <sup>f</sup>* (*t*)∈<sup>V</sup> *<sup>j</sup>*

**ii.** *Forj* ∈*Z,*V *<sup>j</sup>*+1⊂V *<sup>j</sup>*

**iii.** *Forj* <sup>∈</sup>*Z, <sup>f</sup>* (*t*)∈<sup>V</sup> *<sup>j</sup>*

V *<sup>j</sup>* = ∩ *j*=−*∞* +*∞*

V *<sup>j</sup>* = ∪ *j*=−*∞* +*∞*

*j*→+*∞*

*j*→−*∞*

2 *j*

, allowing variations of the signal in a 2<sup>−</sup> *<sup>j</sup>*

groups every possible approximation with resolution 2<sup>−</sup> *<sup>j</sup>*

*k*)∈V *<sup>j</sup>*

may be found in [12].

this by a factor 2 *<sup>j</sup>*

The space Vj

**iv.** lim

**v.** lim

wavelets are expressed as :

*ha*,*b*(*t*)=*<sup>a</sup>* <sup>−</sup>1/2

*<sup>h</sup>* ( *<sup>t</sup>* <sup>−</sup>*<sup>b</sup>*

The information used in this approach is based on both *a* and *b*, as expansion coefficients. Commonly, the mother wavelet is considered as a Daubechies signal. The wavelets automat‐ ically adapt to the different components of a signal, using a small window (large scale) to search for brief high-frequency components, and large window (low scale) to look for long lived, low-frequency components. The shape of low- and high-frequency components is de‐ termined by the mother wavelet. A further and deeper revision of the wavelet technique

Wavelets can be represented using a function *ψ*, and the familyℑof expanded and translated

*<sup>ψ</sup>*( *<sup>t</sup>* <sup>−</sup><sup>2</sup> *<sup>j</sup> n*

which performs an orthogonal baseL2⊄. Orthonormal wavelets are obtained by expanding

these bases permits the study of multi-resolution of a signal. Formally, the approximation of a function with a resolution 2<sup>−</sup> *<sup>j</sup>* is defined as an orthogonal projection over a spaceV *<sup>j</sup>* <sup>⊂</sup>L2<sup>⊄</sup>

*Definition.* (*Multi-Resolution*) *A family of closed subspaces* {V *<sup>j</sup>* : *<sup>j</sup>* <sup>∈</sup>*Z*} *of L<sup>2</sup> <sup>⊄</sup> is a multi-resolution*

Property (i) states that the subspace V*<sup>j</sup>* is invariant in any translation proportional to scale 2 *<sup>j</sup>*

*<sup>a</sup>* ) (2)

<sup>2</sup> *<sup>j</sup>* ): *<sup>j</sup>*, *<sup>n</sup>* <sup>∈</sup>*Z*} (3)

resolution. The construction of

which is minimized *f* − *f <sup>j</sup>* .

owns the necessary information to calculate a

. Remember that an

$$\|A\| \|f\|\|^2 \le \sum\_{n=-\infty}^{+\infty} \|a\mathbb{I}n\mathbb{I}\|^{\frac{1}{2}} \le B\|\|f\|\|^2\tag{7}$$

This last expression guarantees that the expansion of the signal over {*θ*(*t* −*n*):*n* ∈*Z*} is nu‐ merically stable.

The approximation of *f* in the resolution 2<sup>−</sup> *<sup>j</sup>* is defined as the orthogonal projection *P*<sup>V</sup> *<sup>j</sup> f* over V*j* . To compute this projection, an orthogonal base should be find over the space V*<sup>j</sup>* . Hence, the following theorem allows for an orthogonal Riesz base{*θ*(*t* −*n*):*n* ∈*Z*}, and it builds an orthogonal basis for each space V*<sup>j</sup>* where dilation and transferring over a function *ϕ*, named *scale*.

*Theorem* (*Goswami Jaideva, 1999*)*. Being* {V *<sup>j</sup>* : *j* ∈*Z*} *an approximation of multi-resolution and ϕ a scale function, where the Fourier Transformation is:*

$$\widehat{\phi}(\omega) = \frac{\widehat{\Theta}(\omega)}{\left(\sum\_{k=-\infty}^{+\infty} |\widehat{\Theta}(\omega + 2k\pi)|^2\right)^{1/2}} \tag{8}$$

*Defining*

$$
\varphi \rho\_{j,n}(t) = \frac{2}{\sqrt{2^{j}}} \varphi \left( \frac{t - n}{2^{j}} \right) \tag{9}
$$

where the inner products are:

lution product:

with *<sup>φ</sup>*¯ *<sup>j</sup>*

(*t*)= 2<sup>−</sup> *<sup>j</sup>*

energy in the interval −2<sup>−</sup> *<sup>j</sup>*

giving a discrete approximation of scale 2*<sup>j</sup>*

*φ*(2<sup>−</sup> *<sup>j</sup>*

filter of function *f,* from sampling 2*<sup>j</sup>*

**2.2. Self Organizing Maps (SOM)**

*a n* =*∫* −*∞* +*∞* *<sup>f</sup>* (*t*) <sup>1</sup> 2 *j*

−*π*, *π* , and as a consequence, the Fourier Transform 2 *<sup>j</sup>*

*π*, 2<sup>−</sup> *<sup>j</sup>*

presents a homogenous response suitable for noise cancellation.

of bi-dimensional Gaussian functions, as described by Equation 18.

2) =exp( −0.5 \*

*h* (*i* 1, *i*

modified. Each neuron also has a weight vector (*wi*

lows for modifications of neighbor neurons of a SOM.

.

*<sup>φ</sup>*( *<sup>t</sup>* <sup>−</sup><sup>2</sup> *<sup>j</sup> n* <sup>2</sup> *<sup>j</sup>* )*dt* <sup>=</sup> *<sup>f</sup>* \* *<sup>φ</sup>*¯ *<sup>j</sup>*

Using Wavelets for Feature Extraction and Self Organizing Maps for Fault Diagnosis of Nonlinear Dynamic Systems

*t*).The energy of the Fourier Transform *φ*

The purpose of Kohonen's SOM is to capture the topology and probability distribution of some

Different types of grid may be used to represent data, although the one shown in Figure 2

The neighborhood function regarding a rectangular grid, such as this one, is based on a set

where *i <sup>1</sup>* and *i <sup>2</sup>* represent the indices of each neuron, and *σ*is the standard deviation of each Gaussian distribution, which determines how the neighbor neurons of a winner neuron are

ron is modified by an input updating. Thus, *h(i <sup>1</sup> ,i <sup>2</sup> )* is the Gaussian representation that al‐

1)<sup>2</sup> + (*i* 2 *win* −*i* 2)2

*j*

(*i* 1 *win* −*i*

input data (Figure 1) [13][14]. First, a topology of SOM is defined as a rectangular grid [15].

*aj n* = *f* , *φj*,*<sup>n</sup>* (16)

(2 *<sup>j</sup>*

*φ* ^ \*(2 *<sup>j</sup>*

*π* . Then, the discrete approximation is *aj n* , as a low pass

. This last equation can be expressed as the convo‐

*<sup>ω</sup>*) of *φ*¯ *<sup>j</sup>*

*<sup>σ</sup>* <sup>2</sup> ) (18)

), which represents how the actual neu‐

*n*) (17)

http://dx.doi.org/10.5772/50235

49

^ is typically concentrated in

(*t*) concentrates its

*Then the family*{*θ <sup>j</sup>*,*<sup>n</sup>* :*n* ∈*Z*}*is an orthonormal base of*V *<sup>j</sup>* ∀ *j* ∈*Z*

#### *Demonstration*

The objective is to build an orthonormal base, and therefore, a function *φ* ∈V0. Now, this function is expanded in terms of the Reisz base {*θ*(*t* −*n*):*n* ∈*Z*}:

*φ*(*t*)= ∑ *k*=−*∞* +*∞ a n θ*(*t* −*n*) (10)

Computing the Fourier transform for this last function, it is obtained that:

$$
\stackrel{\frown}{\varphi}(\omega) = \stackrel{\frown}{a}(\omega)\stackrel{\frown}{\Theta}(\omega) \tag{11}
$$

where*a* ^is a Fourier series with period 2π and finite energy; *<sup>a</sup>* ^ is expressed in terms of the or‐ thogonal condition of {*φ*(*<sup>t</sup>* <sup>−</sup>*n*):*<sup>n</sup>* <sup>∈</sup>*Z*} into the Fourier dominion, being *φ*¯(*t*)=*<sup>φ</sup>* \* (−*t*) It is nec‐ essary that for any *n*, *p* ∈*Z*:

$$\{\not{\varphi}(t-n),\ \varphi(t-p)\} = \left[\not{\varphi}(t-n)\varphi^\*(t-p)dt = \varphi\*\overline{\varphi}(p-n) = \delta\!\!/ n\!\right] \tag{12}$$

If the Fourier transform is obtained from this, it determines the following equation:

$$\sum\_{k=-\infty}^{+\infty} |\stackrel{\frown}{\phi}(\omega + 2k\pi)| \,^2 = 1 \tag{13}$$

On the other hand, the Fourier transform of *φ*(*t*)∗*<sup>φ</sup>*¯(*t*) is |*<sup>φ</sup>* ^(*ω*)| <sup>2</sup> , and therefore:

$$\widehat{\hat{a}}(\omega) = \frac{1}{\left(\sum\_{\nu=\omega}^{\ast \ast} |\,\hat{\partial}(\omega + 2k\pi) \mid^2\right)^{\frac{1}{2}}} \tag{14}$$

To find an approximation of *f* over the space V*<sup>j</sup>* , it is necessary to expand the function in terms of the orthogonal base from the scale function:

$$P\_{V\_j} f = \sum\_{n=-\infty}^{\star \alpha} \{ f \llcorner \phi \llcorner \Box \phi \llcorner \Box \phi \llcorner \Box \phi \llcorner \Box \phi \llcorner \Box \phi \llcorner \Box \phi \llcorner \Box \phi \llcorner \Box \phi \llcorner \Box \phi \llcorner \Box \phi \llcorner \Box \phi \llcorner \Box \phi \llcorner \Box \phi \llcorner \Box \phi \llcorner \Box \phi \llcorner \Box \phi \llcorner \Box \phi \llcorner \Box \phi \llcorner \Box \phi \llcorner \Box \phi$$

Using Wavelets for Feature Extraction and Self Organizing Maps for Fault Diagnosis of Nonlinear Dynamic Systems http://dx.doi.org/10.5772/50235 49

where the inner products are:

*Defining*

*Demonstration*

where*a*

essary that for any *n*, *p* ∈*Z*:

*φ*(*t* −*n*), *φ*(*t* − *p*) =*∫*

*<sup>φ</sup>j*,*n*(*t*)= <sup>2</sup>

*Then the family*{*θ <sup>j</sup>*,*<sup>n</sup>* :*n* ∈*Z*}*is an orthonormal base of*V *<sup>j</sup>*

48 Developments and Applications of Self-Organizing Maps Applications of Self-Organizing Maps

function is expanded in terms of the Reisz base {*θ*(*t* −*n*):*n* ∈*Z*}:

*φ*(*t*)= ∑ *k*=−*∞* +*∞*

Computing the Fourier transform for this last function, it is obtained that:

*φ* ^(*ω*)= *<sup>a</sup>*

thogonal condition of {*φ*(*<sup>t</sup>* <sup>−</sup>*n*):*<sup>n</sup>* <sup>∈</sup>*Z*} into the Fourier dominion, being *φ*¯(*t*)=*<sup>φ</sup>* \*

*φ*(*t* −*n*)*φ* \*

If the Fourier transform is obtained from this, it determines the following equation:

^is a Fourier series with period 2π and finite energy; *<sup>a</sup>*

−*∞* +*∞*

∑ *k*=−*∞* +*∞* |*φ*

^(*ω*)= <sup>1</sup> ( ∑ *k*=−*∞* +*∞* |*θ*

^(*<sup>ω</sup>* <sup>+</sup> <sup>2</sup>*kπ*)| <sup>2</sup>

) 1

On the other hand, the Fourier transform of *φ*(*t*)∗*<sup>φ</sup>*¯(*t*) is |*<sup>φ</sup>*

*a*

To find an approximation of *f* over the space V*<sup>j</sup>*

terms of the orthogonal base from the scale function:

*PVj*

*f* = ∑ *n*=−*∞* +*∞*

2 *j*

*<sup>φ</sup>*( *<sup>t</sup>* <sup>−</sup>*<sup>n</sup>*

The objective is to build an orthonormal base, and therefore, a function *φ* ∈V0. Now, this

^(*ω*)*<sup>θ</sup>*

∀ *j* ∈*Z*

<sup>2</sup> *<sup>j</sup>* ) (9)

*a n θ*(*t* −*n*) (10)

^(*ω*) (11)

(*<sup>t</sup>* <sup>−</sup> *<sup>p</sup>*)*dt* <sup>=</sup>*<sup>φ</sup>* <sup>∗</sup>*<sup>φ</sup>*¯(*<sup>p</sup>* <sup>−</sup>*n*)=*<sup>δ</sup> <sup>n</sup>* (12)

, and therefore:

<sup>2</sup> (14)

, it is necessary to expand the function in

*f* , *φj*,*<sup>n</sup> φj*,*<sup>n</sup>* (15)

^(*<sup>ω</sup>* <sup>+</sup> <sup>2</sup>*kπ*)| <sup>2</sup> =1 (13)

^(*ω*)| <sup>2</sup>

^ is expressed in terms of the or‐

(−*t*) It is nec‐

$$a\_{\!\!\!\!\!\!\!\!\!\/} \!\!\!\!\!\!\!\!\/) = \{f \;\!\!\/\/\,\!\/\,\!\/}\_{\!\!\/\,\!\/, \,\!\/)} \tag{16}$$

giving a discrete approximation of scale 2*<sup>j</sup>* . This last equation can be expressed as the convo‐ lution product:

$$a\{n\mathbf{1}\} = \left| \int\_{-\infty}^{+\infty} f\left(t\right) \frac{1}{\sqrt{2^{\frac{1}{j}}}} \varphi\left(\frac{t - 2^{\frac{j}{2}} n}{2^{\frac{1}{j}}}\right) dt = f \ast \overline{\varphi}\_{\frac{j}{2}}\{2^{\frac{1}{j}} n\} \tag{17}$$

with *<sup>φ</sup>*¯ *<sup>j</sup>* (*t*)= 2<sup>−</sup> *<sup>j</sup> φ*(2<sup>−</sup> *<sup>j</sup> t*).The energy of the Fourier Transform *φ* ^ is typically concentrated in −*π*, *π* , and as a consequence, the Fourier Transform 2 *<sup>j</sup> φ* ^ \*(2 *<sup>j</sup> <sup>ω</sup>*) of *φ*¯ *<sup>j</sup>* (*t*) concentrates its energy in the interval −2<sup>−</sup> *<sup>j</sup> π*, 2<sup>−</sup> *<sup>j</sup> π* . Then, the discrete approximation is *aj n* , as a low pass filter of function *f,* from sampling 2*<sup>j</sup>* .

#### **2.2. Self Organizing Maps (SOM)**

The purpose of Kohonen's SOM is to capture the topology and probability distribution of some input data (Figure 1) [13][14]. First, a topology of SOM is defined as a rectangular grid [15].

Different types of grid may be used to represent data, although the one shown in Figure 2 presents a homogenous response suitable for noise cancellation.

The neighborhood function regarding a rectangular grid, such as this one, is based on a set of bi-dimensional Gaussian functions, as described by Equation 18.

$$\ln\left(i\_{1'}\ i\_2\right) = \exp\left(-0.5\*\frac{(i\_1^{win}-i\_1)^2+(i\_2^{win}-i\_2)^2}{\sigma^2}\right) \tag{18}$$

where *i <sup>1</sup>* and *i <sup>2</sup>* represent the indices of each neuron, and *σ*is the standard deviation of each Gaussian distribution, which determines how the neighbor neurons of a winner neuron are modified. Each neuron also has a weight vector (*wi j* ), which represents how the actual neu‐ ron is modified by an input updating. Thus, *h(i <sup>1</sup> ,i <sup>2</sup> )* is the Gaussian representation that al‐ lows for modifications of neighbor neurons of a SOM.

The updating process of the weight matrix is performed as shown by Equation 19.

*old* + *η* \* *h* (*i*

1, *i*

Using Wavelets for Feature Extraction and Self Organizing Maps for Fault Diagnosis of Nonlinear Dynamic Systems

where*η* represents a constant value equals to 0.7. This parameter can be tuned as a learning

The proposed organization for the actual model is shown in Figure 3. The model is divided into two stages: first, it is performed offline, where the SOM is trained; second, the fault di‐

Feedback of 100 Epochs

The SOM is trained offline, following the output signal, which is decomposed into sever‐ al levels, as proposed by wavelets feature extraction scales. The observed dynamical sys‐ tem decomposition should perform this regarding its input, by considering a diversity of frequencies. The decomposition is proposed for several wavelets scales as feature extrac‐ tion, where just one is chosen based on repeatability in similar scenarios, either fault-

Several parameters need to be tuned for the SOM and wavelet feature extraction, such as:

Plant

Feature Extraction

Wavelets Obs

Persistant Input Output

2)\*(*I* −*wj*

*old* ) (19)

http://dx.doi.org/10.5772/50235

51

SOM (Learning Stage)

*wj new* <sup>=</sup>*wj*

parameter. Here, *I* represents the current input vector.

**3. Main approach**

agnosis procedure is performed online.

**Figure 3.** Organization for the model.

**•** Length of the sampling window *k*.

**•** Number of wavelet decomposition levels.

present or fault-free.

**•** Learning value η.

**•** Vigilance threshold β.

**Figure 1.** Topology Network of a SOM.

Equation 20 is the basis of the SOM. In this equation, the weight matrix is updated based on the bi-dimensional indexing, namely *h(i <sup>1</sup> ,i <sup>2</sup> ).* This equation is used during training (offline) stage of the SOM. Moreover, this bi-dimensional function allows the weight ma‐ trix to be updated in a global way, rather than just to update the weight vector associated to a winner neuron.

For updating the SOM, an inner product is performed between the weight matrix *W* and the input vector (*I*), in order to define a winner neuron. Having calculated this product, the maximum value is determined by the comparison between each scalar from resultant vector. This value is declared as the winner, just as in the technique known as "the winner take all". The related bi-dimensional index (Figure 2) is calculated in order to determine how the weight matrix is modified.

The updating process of the weight matrix is performed as shown by Equation 19.

$$\left(w\_{j}^{new} = w\_{j}^{old} + \eta \ast^{\*} h \left(i\_{1}, i\_{2}\right) \ast \left(I - w\_{j}^{old}\right)\right) \tag{19}$$

where*η* represents a constant value equals to 0.7. This parameter can be tuned as a learning parameter. Here, *I* represents the current input vector.

## **3. Main approach**

**Figure 1.** Topology Network of a SOM.

50 Developments and Applications of Self-Organizing Maps Applications of Self-Organizing Maps

**Figure 2.** Index Grid for noise cancellation.

weight matrix is modified.

to a winner neuron.

Equation 20 is the basis of the SOM. In this equation, the weight matrix is updated based on the bi-dimensional indexing, namely *h(i <sup>1</sup> ,i <sup>2</sup> ).* This equation is used during training (offline) stage of the SOM. Moreover, this bi-dimensional function allows the weight ma‐ trix to be updated in a global way, rather than just to update the weight vector associated

For updating the SOM, an inner product is performed between the weight matrix *W* and the input vector (*I*), in order to define a winner neuron. Having calculated this product, the maximum value is determined by the comparison between each scalar from resultant vector. This value is declared as the winner, just as in the technique known as "the winner take all". The related bi-dimensional index (Figure 2) is calculated in order to determine how the

The proposed organization for the actual model is shown in Figure 3. The model is divided into two stages: first, it is performed offline, where the SOM is trained; second, the fault di‐ agnosis procedure is performed online.

**Figure 3.** Organization for the model.

The SOM is trained offline, following the output signal, which is decomposed into sever‐ al levels, as proposed by wavelets feature extraction scales. The observed dynamical sys‐ tem decomposition should perform this regarding its input, by considering a diversity of frequencies. The decomposition is proposed for several wavelets scales as feature extrac‐ tion, where just one is chosen based on repeatability in similar scenarios, either faultpresent or fault-free.

Several parameters need to be tuned for the SOM and wavelet feature extraction, such as:


During the offline stage, the SOM is trained based on known fault-present and fault-free sce‐ narios with a local decomposition wavelet strategy, using a mother wavelet known as Dau‐ bechies 4. This means that four decomposition levels (scales) are produced. The Daubechies 4 is a wavelet particularly chosen here due to the case study response, as it is discussed later.

For fault-free scenarios, four decomposed levels have a particular powerful response, being different of the fault-present scenarios, as shown in the next section. In this case, similar pat‐ terns are stated to occur when a bounded fault scenario is presented. In both, fault-free and fault-present cases, and for four levels, Daubechies 4 presents a trustable response.

Time

http://dx.doi.org/10.5772/50235

53

Sampling Time Window

Sampling Time Window

**Figure 6.** Results produced during the online stage.

Sampling Time Window

Sampling Time Window

Using Wavelets for Feature Extraction and Self Organizing Maps for Fault Diagnosis of Nonlinear Dynamic Systems

Data Taken by Sampling Window

Normalized Local Data

SOM Network Classification

During the online stage, the SOM does not perform any learning procedure. The compari‐ son between already classified patterns against current scales vectors, produced by wave‐ lets, is performed by an inner product, where the minimal value obtained amongst all patterns is defined as the winner, and compared with β. If the winner value is smaller than β, this is a correct winner; otherwise, the SOM is not capable of performing this comparison. The value β is called *error*, for the purposes of model evaluation. Similarly to the offline stage, the online stage takes four major steps, as shown in Figure 5, in order to produce a result. The main difference, thus, is that here SOM does evaluate the resultant

**Figure 7.** Proposed main procedure for the model-free fault diagnosis.

Wavelet Feature Extraction

Selected Patterns and Produced Results

*w w w w*

Current data has a pre-treatment as input/output responses from the plant. Input and out‐ put are locally normalized before they are processed by wavelets, to extract certain features. To take the advantage of this situation, it is necessary that the learning law of the SOM is suitable to perform an accurate response with respect to the case study.

**Figure 4.** Three steps of the offline stage.

Three steps are defined during the offline stage, as shown in Figure 4, in terms of data proc‐ essing: first, a sampling window of input and output data is taken and normalized; next, this information is processed by the wavelet module, to perform feature extraction; finally, the local matrix is classified by the SOM in learning mode. In this case, it is necessary to pro‐ vide with enough information and a diversity of scenarios (fault-present and fault-free) to the SOM, in order to ensure a suitable fault identification.

Using Wavelets for Feature Extraction and Self Organizing Maps for Fault Diagnosis of Nonlinear Dynamic Systems http://dx.doi.org/10.5772/50235 53

**Figure 6.** Results produced during the online stage.

During the offline stage, the SOM is trained based on known fault-present and fault-free sce‐ narios with a local decomposition wavelet strategy, using a mother wavelet known as Dau‐ bechies 4. This means that four decomposition levels (scales) are produced. The Daubechies 4 is a wavelet particularly chosen here due to the case study response, as it is discussed later.

For fault-free scenarios, four decomposed levels have a particular powerful response, being different of the fault-present scenarios, as shown in the next section. In this case, similar pat‐ terns are stated to occur when a bounded fault scenario is presented. In both, fault-free and

Current data has a pre-treatment as input/output responses from the plant. Input and out‐ put are locally normalized before they are processed by wavelets, to extract certain features. To take the advantage of this situation, it is necessary that the learning law of the SOM is

Extraction SOM Learning Stage

SOM

Three steps are defined during the offline stage, as shown in Figure 4, in terms of data proc‐ essing: first, a sampling window of input and output data is taken and normalized; next, this information is processed by the wavelet module, to perform feature extraction; finally, the local matrix is classified by the SOM in learning mode. In this case, it is necessary to pro‐ vide with enough information and a diversity of scenarios (fault-present and fault-free) to

Classification Decision Making

Time

Time

fault-present cases, and for four levels, Daubechies 4 presents a trustable response.

suitable to perform an accurate response with respect to the case study.

Sampling Window Wavelet Feature

Sampling Window Wavelet Feature

the SOM, in order to ensure a suitable fault identification.

Extraction

**Figure 4.** Three steps of the offline stage.

52 Developments and Applications of Self-Organizing Maps Applications of Self-Organizing Maps

**Figure 5.** Three steps of the online stage.

**Figure 7.** Proposed main procedure for the model-free fault diagnosis.

During the online stage, the SOM does not perform any learning procedure. The compari‐ son between already classified patterns against current scales vectors, produced by wave‐ lets, is performed by an inner product, where the minimal value obtained amongst all patterns is defined as the winner, and compared with β. If the winner value is smaller than β, this is a correct winner; otherwise, the SOM is not capable of performing this comparison. The value β is called *error*, for the purposes of model evaluation. Similarly to the offline stage, the online stage takes four major steps, as shown in Figure 5, in order to produce a result. The main difference, thus, is that here SOM does evaluate the resultant local model, generated by wavelet feature extraction. The decision making module inher‐ ent in SOM determines if this classification is valid or not. Notice that for the current pur‐ poses, time consumption is neglected.

*Fault 1:* Backlash of 0.01, followed by a dead zone between {-0.1 0.1}

*Fault 2:* Backlash of 0.91, followed by a dead zone between {-0.3 0.4}

where η =0.1952 for this case study

MATLAB Function Fault Injection

double double

**Figure 8.** Schematic diagram of the system for the case study.

MATLAB Function Time Delay variable input

double

model of the current response.

**5. Case study results**

double double double

double double

Step

double double

0 Constant

Sine Wave

First Input1

double

cific ranges:

0.01*β*0.99 0.01*η*0.99 2*a*, *b*6 *k* =10

For this case study, the parameters to be tuned (such as β, η, a, b and k) are bounded to spe‐

Using Wavelets for Feature Extraction and Self Organizing Maps for Fault Diagnosis of Nonlinear Dynamic Systems

tmp3 temporal value 3

double

These ranges are arbitrarily fixed, since there is no further information regarding these val‐ ues. In order to obtain certain selection according to fault presence and dynamic system re‐ sponse, a testing (response) from the tuples integrated by the combination of these values is performed, as shown in the next section. Notice that the pattern selected by the SOM is a representation of the most suitable approximation of current feature extraction, that is, a

The present results are referred in terms of pattern construction and feature extraction, con‐ sidering several known fault-present and fault-free scenarios. Regarding this, Figure 9 shows the output response of the benchmark during a fault-free scenario. The response of the system is inherently stable, due to the inner local control. In this case, the sampling time window has a duration of 10 seconds, representing a period of system response. This re‐

Demux

double

double

0 Clock

y(n)=Cx(n)+Du(n) x(n+1)=Ax(n)+Bu(n) Dynamics

double

In1 In2 Out1 UIO

PID Residual Output PID Controller out

> Backlash1 Backlash

double

double double

t Time Terminator

double

double

http://dx.doi.org/10.5772/50235

Multiport Switch

in1 Fault Injection Value

Dead Zone1 Dead Zone

double double double double

double

MATLAB Function WAVELET ¬ SOM EVALUATION

out\_obs UIO OUT

in Input Value 55

residual

Output Value

The results of the online stage are produced every *w* sampling time windows, which is the time taken by the application in order to produce a result (Figure 6). Such a time win‐ dow *w* becomes crucial in terms of the frequency of the plant. Further, the sampling win‐ dow is a multiple of inherent period of the system. In fact, the plant response during fault-present and fault-free scenarios need to be bounded to this parameter, in order to guarantee a reliable response.

The main procedure followed here is shown in Figure 7, in which the normalized, local preprocessing stage is executed per local sample data; after that, the wavelet feature extraction is performed as data decomposition. The results related to different levels are processed by the SOM per level, where just one would be the winner, and learned by the SOM.

## **4. The case study**

The case study is integrated by a system, as it is shown in Figure 8. It consists of a multipleinput, single-output (MISO) system, with a PID controller, and a switching fault injection procedure. As it may be noticed, the dynamics of this plant tends to be quite slow in com‐ parison with the occurrence of faults, according to the dynamics of the plant and the fault scenarios. This characteristic is crucial for the construction of the local model and the feature extraction, in order to produce a fruitful fault diagnosis procedure. Case study is linear and modeled through model-based techniques; however, when a fault is present, its dynamics becomes nonlinear. This nonlinear behavior tends to be extremely difficult to be diagnosed by classical strategies, like for example, unknown input observers [8]. This scenario has been presented for local system identification techniques and for a global classification strategy, in order to have an accurate fault diagnosis strategy [16]. However, the strategy is depend‐ ent on the persistent excitation of certain frequency responses. Alternatively, the proposed strategy here overcomes such a frequency dependency, since the only obvious dependency is the sampling period. This strategy is based on the sensibility of the feature extraction strategy, which uses Daubechies 4 wavelets (db4).

The dynamics of the case study are expressed as vectors of the state space representation:

k1= [0.1; -20.000]; e=[0; 0.001];

b=[1.8000 -2.1000; 0.9000 0.8600];

A=[1.1000 0; 0 2.1000];

c=[0 1];

where A, b, c, and e belong to the case study model, and k1 to the PID controller. The types of faults are:

*Fault 1:* Backlash of 0.01, followed by a dead zone between {-0.1 0.1}

*Fault 2:* Backlash of 0.91, followed by a dead zone between {-0.3 0.4}

For this case study, the parameters to be tuned (such as β, η, a, b and k) are bounded to spe‐ cific ranges:

0.01*β*0.99 0.01*η*0.99 2*a*, *b*6 *k* =10 where η =0.1952 for this case study

local model, generated by wavelet feature extraction. The decision making module inher‐ ent in SOM determines if this classification is valid or not. Notice that for the current pur‐

The results of the online stage are produced every *w* sampling time windows, which is the time taken by the application in order to produce a result (Figure 6). Such a time win‐ dow *w* becomes crucial in terms of the frequency of the plant. Further, the sampling win‐ dow is a multiple of inherent period of the system. In fact, the plant response during fault-present and fault-free scenarios need to be bounded to this parameter, in order to

The main procedure followed here is shown in Figure 7, in which the normalized, local preprocessing stage is executed per local sample data; after that, the wavelet feature extraction is performed as data decomposition. The results related to different levels are processed by

The case study is integrated by a system, as it is shown in Figure 8. It consists of a multipleinput, single-output (MISO) system, with a PID controller, and a switching fault injection procedure. As it may be noticed, the dynamics of this plant tends to be quite slow in com‐ parison with the occurrence of faults, according to the dynamics of the plant and the fault scenarios. This characteristic is crucial for the construction of the local model and the feature extraction, in order to produce a fruitful fault diagnosis procedure. Case study is linear and modeled through model-based techniques; however, when a fault is present, its dynamics becomes nonlinear. This nonlinear behavior tends to be extremely difficult to be diagnosed by classical strategies, like for example, unknown input observers [8]. This scenario has been presented for local system identification techniques and for a global classification strategy, in order to have an accurate fault diagnosis strategy [16]. However, the strategy is depend‐ ent on the persistent excitation of certain frequency responses. Alternatively, the proposed strategy here overcomes such a frequency dependency, since the only obvious dependency is the sampling period. This strategy is based on the sensibility of the feature extraction

The dynamics of the case study are expressed as vectors of the state space representation:

where A, b, c, and e belong to the case study model, and k1 to the PID controller. The types

the SOM per level, where just one would be the winner, and learned by the SOM.

poses, time consumption is neglected.

54 Developments and Applications of Self-Organizing Maps Applications of Self-Organizing Maps

guarantee a reliable response.

**4. The case study**

k1= [0.1; -20.000];

A=[1.1000 0; 0 2.1000];

b=[1.8000 -2.1000; 0.9000 0.8600];

e=[0; 0.001];

c=[0 1];

of faults are:

strategy, which uses Daubechies 4 wavelets (db4).

**Figure 8.** Schematic diagram of the system for the case study.

These ranges are arbitrarily fixed, since there is no further information regarding these val‐ ues. In order to obtain certain selection according to fault presence and dynamic system re‐ sponse, a testing (response) from the tuples integrated by the combination of these values is performed, as shown in the next section. Notice that the pattern selected by the SOM is a representation of the most suitable approximation of current feature extraction, that is, a model of the current response.

## **5. Case study results**

The present results are referred in terms of pattern construction and feature extraction, con‐ sidering several known fault-present and fault-free scenarios. Regarding this, Figure 9 shows the output response of the benchmark during a fault-free scenario. The response of the system is inherently stable, due to the inner local control. In this case, the sampling time window has a duration of 10 seconds, representing a period of system response. This re‐ sponse produces the level decomposition presented in Figure 10, in which four wavelet lev‐ els are used to decompose the data. Since response is fairly stable, the main differences amongst levels are not significant neither regarding to power, nor regarding to frequency selection. Based on this approximation, the selected patterns should be similar, equidistant, and close related; otherwise, an unknown response has appeared.

**Figure 11.** SOM for the fault-free scenario.

**Figure 12.** Selected patterns per level for the fault-free scenario. The winner patterns are at the first level.

The fault-free scenario shows the response of a damped system, in which four decomposed levels tends to be similar in terms of power values. For this scenario, the patterns are classi‐ fied by the SOM as shown in Figure 11. The total number of patterns is 125, from which 55 patterns are "zoom" (enlarged) patterns with a similar power response, given the wavelets feature extraction. Remember that Daubechies 4 has been selected for feature extraction, since this wavelet is stable enough in terms of feature extraction repetition for same scenar‐ ios. In Figure 11, the amplitude of 55 patterns reflects a similar response amongst them. Moreover, there is a region around patterns 30 to 45, whose amplitude is close to zero. No‐

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**Figure 9.** Fault-free response for the current example.

**Figure 10.** Wavelets feature extraction for fault-free scenario.

Using Wavelets for Feature Extraction and Self Organizing Maps for Fault Diagnosis of Nonlinear Dynamic Systems http://dx.doi.org/10.5772/50235 57

**Figure 11.** SOM for the fault-free scenario.

sponse produces the level decomposition presented in Figure 10, in which four wavelet lev‐ els are used to decompose the data. Since response is fairly stable, the main differences amongst levels are not significant neither regarding to power, nor regarding to frequency selection. Based on this approximation, the selected patterns should be similar, equidistant,

and close related; otherwise, an unknown response has appeared.

56 Developments and Applications of Self-Organizing Maps Applications of Self-Organizing Maps

**Figure 9.** Fault-free response for the current example.

**Figure 10.** Wavelets feature extraction for fault-free scenario.

**Figure 12.** Selected patterns per level for the fault-free scenario. The winner patterns are at the first level.

The fault-free scenario shows the response of a damped system, in which four decomposed levels tends to be similar in terms of power values. For this scenario, the patterns are classi‐ fied by the SOM as shown in Figure 11. The total number of patterns is 125, from which 55 patterns are "zoom" (enlarged) patterns with a similar power response, given the wavelets feature extraction. Remember that Daubechies 4 has been selected for feature extraction, since this wavelet is stable enough in terms of feature extraction repetition for same scenar‐ ios. In Figure 11, the amplitude of 55 patterns reflects a similar response amongst them. Moreover, there is a region around patterns 30 to 45, whose amplitude is close to zero. No‐ tice that the pattern response tends to be similar through time. In Figure 12, the related pat‐ terns per level are classified by the SOM for this fault-free scenario. In this case, level 1 presents most of the variations with respect to the observer scenario. This is reflected by the number of created patterns regarding the rest of the levels: level 1 proposes 125 patterns, while the rest of the levels propose mostly 2.

which clearly exposes the presence of the fault. In Figure 16, the first level presents a very noticeable difference between the two responses of the whole scale presentation. More‐ over, the response in terms of patterns classified by the SOM tends to increase, but it is

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still stable (Figure 17).

**Figure 14.** System response during fault present scenario.

**Figure 15.** Detail of the response for the first fault present scenario.

Figure 13 presents the response of the system and the selected patterns that vary from 0.1 to 0.8, in magnitude terms, presenting a close relation between the winner patterns. It is inter‐ esting to observe that the most selected pattern has a magnitude of 0.6.

**Figure 13.** Fault-free scenario and selected patterns for the case study.

For the fault-present scenario, two types of faults are injected to the system (see Section 4). First, let us consider Fault 1, which is an increment of sudden amplitude, as shown in Figure 14. In this case, the fault scenario is presented from 400 to 800 seconds, and it is related to a 10% increment of the amplitude. A detail of the response for this scenario is shown in Figure 15. Observe that the system response presents just an oscillation, and not a clear stage of fault condition. However, the expected response is not desirable according to system dynamics.

Figure 16 shows the features extraction results for this scenario. In this case, power values are modified around 500 seconds, in comparison with the fault-free response (Figure 10). This difference is a clear modification of phase around the 500 seconds. This slight phase modification is the result of the increment of amplitude for this fault scenario. Regarding the patterns that are classified by the SOM for this fault scenario, the number of patterns simply does not augment, but the feature extraction results are different. This is shown in the amplitude of the patterns and the selected patterns, especially in pattern number 15, which clearly exposes the presence of the fault. In Figure 16, the first level presents a very noticeable difference between the two responses of the whole scale presentation. More‐ over, the response in terms of patterns classified by the SOM tends to increase, but it is still stable (Figure 17).

**Figure 14.** System response during fault present scenario.

tice that the pattern response tends to be similar through time. In Figure 12, the related pat‐ terns per level are classified by the SOM for this fault-free scenario. In this case, level 1 presents most of the variations with respect to the observer scenario. This is reflected by the number of created patterns regarding the rest of the levels: level 1 proposes 125 patterns,

Figure 13 presents the response of the system and the selected patterns that vary from 0.1 to 0.8, in magnitude terms, presenting a close relation between the winner patterns. It is inter‐

For the fault-present scenario, two types of faults are injected to the system (see Section 4). First, let us consider Fault 1, which is an increment of sudden amplitude, as shown in Figure 14. In this case, the fault scenario is presented from 400 to 800 seconds, and it is related to a 10% increment of the amplitude. A detail of the response for this scenario is shown in Figure 15. Observe that the system response presents just an oscillation, and not a clear stage of fault condition. However, the expected response is not desirable according

Figure 16 shows the features extraction results for this scenario. In this case, power values are modified around 500 seconds, in comparison with the fault-free response (Figure 10). This difference is a clear modification of phase around the 500 seconds. This slight phase modification is the result of the increment of amplitude for this fault scenario. Regarding the patterns that are classified by the SOM for this fault scenario, the number of patterns simply does not augment, but the feature extraction results are different. This is shown in the amplitude of the patterns and the selected patterns, especially in pattern number 15,

esting to observe that the most selected pattern has a magnitude of 0.6.

while the rest of the levels propose mostly 2.

58 Developments and Applications of Self-Organizing Maps Applications of Self-Organizing Maps

**Figure 13.** Fault-free scenario and selected patterns for the case study.

to system dynamics.

**Figure 15.** Detail of the response for the first fault present scenario.

of the fault in fast response. There is a small magnitude variation around patterns 2 to 3, at the beginning of the fault scenario, which is resolved at the fifth sampling window. The val‐ ues of the selected patterns in terms of magnitude are quite defined around 3, which is clear‐

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ly different in comparison with the other two previous scenarios.

**Figure 18.** Detail of the response for the second fault present scenario.

**Figure 19.** General response for the second fault present scenario and the selected pattern.

**Figure 16.** Wavelets feature extraction for the fault present scenario.

**Figure 17.** SOM for the fault present scenario.

For the case second fault scenario, Fault 2, Figure 18 shows a detail of the output response of the system, in which an oscillation tends to be larger than that present in the first fault sce‐ nario. Although, this behavior is quite erratic, the case study is still measurable, as shown in Figure 19. In this figure, the patterns are selected and quite defined in terms of the presence of the fault in fast response. There is a small magnitude variation around patterns 2 to 3, at the beginning of the fault scenario, which is resolved at the fifth sampling window. The val‐ ues of the selected patterns in terms of magnitude are quite defined around 3, which is clear‐ ly different in comparison with the other two previous scenarios.

**Figure 18.** Detail of the response for the second fault present scenario.

**Figure 16.** Wavelets feature extraction for the fault present scenario.

60 Developments and Applications of Self-Organizing Maps Applications of Self-Organizing Maps

**Figure 17.** SOM for the fault present scenario.

For the case second fault scenario, Fault 2, Figure 18 shows a detail of the output response of the system, in which an oscillation tends to be larger than that present in the first fault sce‐ nario. Although, this behavior is quite erratic, the case study is still measurable, as shown in Figure 19. In this figure, the patterns are selected and quite defined in terms of the presence

**Figure 19.** General response for the second fault present scenario and the selected pattern.

Figure 20 shows the selected patterns per level, which present an increment in the number of patterns. Now, the first level has 200 patterns, while the rest of three levels have around 80 patterns. Moreover, the amplitude of learned and winner patterns are around 0.15 for the first level, and close to zero for the second, third, and fourth levels, in the majority of learned patterns (red patterns).

Figure 21 shows a combination of the three scenarios: the fault-free scenario from time 0 to 400 seconds, the first fault scenario from 400 to 700 seconds, and the second fault scenario from 1100 to 1500 seconds, and again from 2000 to 2500 seconds. From this combination, several patterns have been selected, as shown by the upper diagram of Figure 21. These se‐ lected patterns, depicted with their amplitude, show a clear definition of the three fault sce‐ narios: the fault-free scenario is related to the patterns measured at 0.7, the first fault-present scenario is related with the two patterns measured at 0.4 and 0.9, respectively, and the sec‐ ond fault-present scenario is related with the three patterns whose values are 1.4, 1.6 and

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2.0. The richness is quite obvious in this scenario, due to the oscillation present in it.

Finally, Figure 22 shows the levels of the four wavelets, depicted as learned patterns, in which the first level has 190 winner patterns, while the other three levels have 60 winner patterns. Observe that the richness of the combined winner patterns determines a clear clas‐ sification of the patterns, as shown in Figure 21. Moreover, the current amplitude is quite

The present work shows a strategy for fault diagnosis based on local feature extraction and global system classification, in which different parameters play an important role in this condition monitoring proposal. The number of samples with respect to sampling window, the learning and vigilance parameters, and the number of extracted features has been tuned by an extensive review of every possible scenario. This approach enhances the capabilities of the simple use of a neural network for pattern classification. This strategy has shown an al‐ ternative for classification of abnormal situations, with no information from current case

**Figure 22.** Selected patterns per wavelet levels for the three scenarios.

**6. Concluding remarks**

similar and rich for the first level, as presented in the three scenarios.

**Figure 20.** Selected patterns per level for the second fault present scenario.

**Figure 21.** Current output response for the fault-present and fault-free scenarios.

Figure 21 shows a combination of the three scenarios: the fault-free scenario from time 0 to 400 seconds, the first fault scenario from 400 to 700 seconds, and the second fault scenario from 1100 to 1500 seconds, and again from 2000 to 2500 seconds. From this combination, several patterns have been selected, as shown by the upper diagram of Figure 21. These se‐ lected patterns, depicted with their amplitude, show a clear definition of the three fault sce‐ narios: the fault-free scenario is related to the patterns measured at 0.7, the first fault-present scenario is related with the two patterns measured at 0.4 and 0.9, respectively, and the sec‐ ond fault-present scenario is related with the three patterns whose values are 1.4, 1.6 and 2.0. The richness is quite obvious in this scenario, due to the oscillation present in it.

**Figure 22.** Selected patterns per wavelet levels for the three scenarios.

Finally, Figure 22 shows the levels of the four wavelets, depicted as learned patterns, in which the first level has 190 winner patterns, while the other three levels have 60 winner patterns. Observe that the richness of the combined winner patterns determines a clear clas‐ sification of the patterns, as shown in Figure 21. Moreover, the current amplitude is quite similar and rich for the first level, as presented in the three scenarios.

## **6. Concluding remarks**

Figure 20 shows the selected patterns per level, which present an increment in the number of patterns. Now, the first level has 200 patterns, while the rest of three levels have around 80 patterns. Moreover, the amplitude of learned and winner patterns are around 0.15 for the first level, and close to zero for the second, third, and fourth levels, in the majority of

learned patterns (red patterns).

62 Developments and Applications of Self-Organizing Maps Applications of Self-Organizing Maps

**Figure 20.** Selected patterns per level for the second fault present scenario.

**Figure 21.** Current output response for the fault-present and fault-free scenarios.

The present work shows a strategy for fault diagnosis based on local feature extraction and global system classification, in which different parameters play an important role in this condition monitoring proposal. The number of samples with respect to sampling window, the learning and vigilance parameters, and the number of extracted features has been tuned by an extensive review of every possible scenario. This approach enhances the capabilities of the simple use of a neural network for pattern classification. This strategy has shown an al‐ ternative for classification of abnormal situations, with no information from current case study as state space response. Furthermore, pattern databases have been constructed, based on several selected scenarios, which have been obtained offline per scenario. This initial in‐ formation is basic in order to obtain an accurate model of the system under study. This strat‐ egy shows how model-free techniques can be implemented for fault diagnosis, and the need of a large amount of data, as well as the extensive review of multiple parameters. The main contribution here is the use of several pre-processing data stages, in order to conform suita‐ ble and accurate information to be processed by the neural network. In addition, a database of several scenarios needs to be used for a trustable fault diagnosis strategy. In fact, it is nec‐ essary to define a heuristic index that allows the differentiation between scenarios. Here, the use of one specific time frequency distribution approach over the rest of the current algo‐ rithms is pursued. Moreover, this strategy could address a proper dynamic non-linear sys‐ tem for online classification of unknown scenarios.

[2] Billings, S., & Wei, H. (2005). A New Class of Wavelet Networks for Nonlinear Sys‐

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## **Acknowledgements**

The authors would like to thank the financial support of DISCA-IIMAS-UNAM, ICYTDF PICCO10-53, and UNAM-PAPIIT (IN103310) Mexico, in connection with this work. Further‐ more, authors would like to thank the valuable help of Mr. Adrian Duran during the con‐ struction of this chapter.

## **Author details**

Héctor Benítez-Pérez1\*, Jorge L. Ortega-Arjona2 and Alma Benítez-Pérez3

\*Address all correspondence to: hector@uxdea4.iimas.unam.mx

1 Departamento de Ingeniería de Sistemas Computacionales y Automatización, Instituto de Investigaciones en Matemáticas Aplicadas y en Sistemas, Universidad Nacional Autónoma de México, México

2 Departamento de Matemáticas, Facultad de Ciencias, Universidad Nacional Autónoma de México, México

3 CECYT 11 "WILFRIDO MASSIEU", Instituto Politécnico Nacional, México

## **References**

[1] Li, D., Pedrycz, W., & Pizzu, N. (2005). Fuzzy Wavelet Packet based Feature Extrac‐ tion Method and Its Application to Biomedical Signal Classification. *IEEE Transaction on Biomedical Engineering*, 52(6), 1132-1139.

[2] Billings, S., & Wei, H. (2005). A New Class of Wavelet Networks for Nonlinear Sys‐ tem Identification. *IEEE Transaction on Neural Networks*, 14(4), 862-874.

study as state space response. Furthermore, pattern databases have been constructed, based on several selected scenarios, which have been obtained offline per scenario. This initial in‐ formation is basic in order to obtain an accurate model of the system under study. This strat‐ egy shows how model-free techniques can be implemented for fault diagnosis, and the need of a large amount of data, as well as the extensive review of multiple parameters. The main contribution here is the use of several pre-processing data stages, in order to conform suita‐ ble and accurate information to be processed by the neural network. In addition, a database of several scenarios needs to be used for a trustable fault diagnosis strategy. In fact, it is nec‐ essary to define a heuristic index that allows the differentiation between scenarios. Here, the use of one specific time frequency distribution approach over the rest of the current algo‐ rithms is pursued. Moreover, this strategy could address a proper dynamic non-linear sys‐

The authors would like to thank the financial support of DISCA-IIMAS-UNAM, ICYTDF PICCO10-53, and UNAM-PAPIIT (IN103310) Mexico, in connection with this work. Further‐ more, authors would like to thank the valuable help of Mr. Adrian Duran during the con‐

1 Departamento de Ingeniería de Sistemas Computacionales y Automatización, Instituto de Investigaciones en Matemáticas Aplicadas y en Sistemas, Universidad Nacional Autónoma

2 Departamento de Matemáticas, Facultad de Ciencias, Universidad Nacional Autónoma de

[1] Li, D., Pedrycz, W., & Pizzu, N. (2005). Fuzzy Wavelet Packet based Feature Extrac‐ tion Method and Its Application to Biomedical Signal Classification. *IEEE Transaction*

3 CECYT 11 "WILFRIDO MASSIEU", Instituto Politécnico Nacional, México

and Alma Benítez-Pérez3

tem for online classification of unknown scenarios.

64 Developments and Applications of Self-Organizing Maps Applications of Self-Organizing Maps

Héctor Benítez-Pérez1\*, Jorge L. Ortega-Arjona2

\*Address all correspondence to: hector@uxdea4.iimas.unam.mx

*on Biomedical Engineering*, 52(6), 1132-1139.

**Acknowledgements**

struction of this chapter.

**Author details**

de México, México

México, México

**References**


[16] Blanke, M., Kinnaert, M., Lunze, J., & Staroswicki, . (2003). *Diagnosis and Fault Toler‐ ant Control*, Springer-Verlag.

**Chapter 4**

**Ex-Post Clustering of Brazilian Beef Cattle Farms Using**

The beef cattle production system is the set of technologies and management practices, ani‐ mal type, purpose of breeding, breed group and the eco-region where the activity is devel‐ oped. The central structure in the beef cattle production chain is the biological system of beef production, including the stages of creation (cow-calf production, stocker production, feedlot beef production) and their combinations. The cow-calf phase is the less profitable ac‐ tivity and the one that has the higher risk. However, it supports the entire structure of the

Although it is undeniable that a systemic view in agriculture is important, it is not yet estab‐ lished in the Brazilian agricultural research. In this study, by using a non parametric techni‐ que known as Data Envelopment Analysis (DEA) and Self-Organizing Maps (SOMS), we intend to cluster cattle breeders of some Brazilian municipalities. The objective is to group the farmers according to their efficiency profiles regarding the decisions related to the com‐ position of their production systems, which are focused on the cow-calf phase. The farmers'

Efficiency is a relative concept: we compare the amount produced by a productive unit or firm given the available resources, to the maximum quantities that would have been pro‐ duced with the same amount of resources. There are different approaches to measure effi‐ ciency. The so-called parametric methods assume a pre-defined functional relationship between resources (inputs) and products (outputs). These methods usually use averages to

> © 2012 Soares de Mello et al.; licensee InTech. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

© 2012 Soares de Mello et al.; licensee InTech. This is a paper distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use,

distribution, and reproduction in any medium, provided the original work is properly cited.

decisions have direct impacts on the expenditures and on the income reached.

**Soms and Cross-Evaluation Dea Models**

João Carlos Correia Baptista Soares de Mello, Eliane Gonçalves Gomes, Lidia Angulo Meza,

Additional information is available at the end of the chapter

http://dx.doi.org/10.5772/51324

**1. Introduction**

production system.

Luiz Biondi Neto, Urbano Gomes Pinto de Abreu, Thiago Bernardino de Carvalho and Sergio de Zen

[17] Benítez-Pérez, Héctor, & Benitez-Perez, Alma. (2010). The use of WAVELET strategy and Self Organizing Maps for Feature Extraction and Classification for Fault Diagno‐ sis. *International Journal of Innovative Computing, Information and Control, IJICIC*, 6(11), 4923-4936.
