**2. Preliminaries**

In this research, historical developments around the vibration analysis have been reviewed, while the use of emerging technologies are proposed to identify failures in rotating electrical machines. Through a wavelet decomposition, it is possible to extract information that enables the detection of signal changes under significant vibrations, affecting the equipments' useful life. The vibration signals have been utilized to detect failures in rotating electrical machines. However, the use of Fourier-based techniques is not practical, because such techniques need stable and long-term records.

No given rules exist to allow characterization of the type of machine, size, or even some specific operating characteristics through vibration patterns. It is relevant to establish strategies able to identify a failure, and even to differentiate among the types of failures. Thus, the neural networks may be quite useful. Through learning elements, neural networks are able to infer the actual conditions of the system under analysis. In this application, the Adaptive Network Based Fuzzy Inference System (ANFIS) has been selected for such purposes.

ANFIS is an Artificial Neuro-Fuzzy Inference System, which is functionally equivalent to fuzzy inference systems. It represents a Sugeno-Tsukamoto fuzzy model, that uses a hybrid learning algorithm (Omar, 2010; Jang, 1993; Jang & Sun, 1996; Bonissone & Badami & Chiang & Knedkar & Schutter, 1996; Jang & Gulley, 1995; Michie & Spregelhart & Taylor, 1994).

#### **2.1 Fuzzy inference systems**

It is necessary to study other alternatives because the system models based on conventional mathematical tools, like differential equations, is not well suited for dealing with ill-defined and uncertain systems (Proakis, 2001). Through the use of vibration signals, it is possible to implement tools able to differentiate characteristics to establish the electrical machine's conditions. A fuzzy inference system employing fuzzy *if-then* rules can model the qualitative aspects of human knowledge and reasoning processes without employing precise quantitative analyses. The fuzzy modeling or fuzzy identification, was first explored systematically by Takagi and Sugeno (Takagi & Sugeno, 1985). There are some basic aspects of this approach that require some comments. In particular:


### Fig. 1. Basic inference system

### **2.2 Fuzzy i***f-then* **rules**

136 Fuzzy Inference System – Theory and Applications

3. The precise analysis of a problem at a given frequency depends on the presence of one or more related frequencies. In the current methods, an important difficulty is the need to monitor through sophisticated sensors. Additionally, failures detection depends on

Different detection techniques for machines' state monitoring have been studied. Some techniques are based on analyzing electrical signals, some others are based on vibration measurements, and some combine them. In this paper, vibration measurements are used for

Vibrations must be properly evaluated, especially those associated to rotating machinery. Capturing vibration patterns, using identification techniques and signal processing, distinctive signatures for failures detection can be set. This could help to anticipate the occurrence of equipment damage, and therefore, corrective actions can be taken to avoid the high cost of a partial or total machinery replacement, as well as economic expenses caused

In this research, historical developments around the vibration analysis have been reviewed, while the use of emerging technologies are proposed to identify failures in rotating electrical machines. Through a wavelet decomposition, it is possible to extract information that enables the detection of signal changes under significant vibrations, affecting the equipments' useful life. The vibration signals have been utilized to detect failures in rotating electrical machines. However, the use of Fourier-based techniques is not practical, because

No given rules exist to allow characterization of the type of machine, size, or even some specific operating characteristics through vibration patterns. It is relevant to establish strategies able to identify a failure, and even to differentiate among the types of failures. Thus, the neural networks may be quite useful. Through learning elements, neural networks are able to infer the actual conditions of the system under analysis. In this application, the Adaptive Network Based Fuzzy Inference System (ANFIS) has been

ANFIS is an Artificial Neuro-Fuzzy Inference System, which is functionally equivalent to fuzzy inference systems. It represents a Sugeno-Tsukamoto fuzzy model, that uses a hybrid learning algorithm (Omar, 2010; Jang, 1993; Jang & Sun, 1996; Bonissone & Badami & Chiang & Knedkar & Schutter, 1996; Jang & Gulley, 1995; Michie & Spregelhart &

It is necessary to study other alternatives because the system models based on conventional mathematical tools, like differential equations, is not well suited for dealing with ill-defined and uncertain systems (Proakis, 2001). Through the use of vibration signals, it is possible to implement tools able to differentiate characteristics to establish the electrical machine's conditions. A fuzzy inference system employing fuzzy *if-then* rules can model the qualitative aspects of human knowledge and reasoning processes without employing precise

the load's inertia.

monitoring purposes.

by their unavailability.

selected for such purposes.

**2.1 Fuzzy inference systems** 

Taylor, 1994).

such techniques need stable and long-term records.

**2. Preliminaries** 

Fuzzy *if-then* rules or fuzzy conditional statements are expressions of the form *IF* A *THEN* B, where A and B are labels of fuzzy sets (Zadeh, 1965) characterized by appropriate membership functions. Due to their concise form, fuzzy *if-then* rules are often employed to capture the imprecise modes of reasoning that play an essential role in the human ability to make decisions in an environment of uncertainty and imprecision. An example that describes a simple fact is:

#### *If vibration is high, it is possible the bars' failure*

where *vibration* and *failure* are linguistic variables (Jang, 1994); *high* (*small*) are linguistic values or labels that are characterized by membership functions.

A different form of fuzzy *if-then* rules, proposed by (Omar, 2010; Takagi & Sugeno, 1985, as cited in Jang, 1993), have fuzzy sets involved only in the premise part. Both types of fuzzy *if-then* rules have been used extensively in both modeling and control. Through the use of linguistic labels and membership functions, a fuzzy *if-then* rule can easily capture the spirit of a "*rule of thumb*" used by humans. From another point of view, due to the qualifiers on the premise parts, each fuzzy *if-then* rule can be viewed as a local description of the system under consideration. Fuzzy *if-then* rules form a core part of the fuzzy inference system described in the following.

Fuzzy Inference Systems Applied to the Analysis of Vibrations in Electrical Machines 139

Fig. 3. Flowchart of the followed inference strategy

#### **2.3 Fuzzy inference system structure for vibration analysis**

Fuzzy inference systems are also known as fuzzy-rule-based systems, fuzzy models, fuzzy associative memories (FAM), or fuzzy controllers when used as controllers. Basically, a fuzzy inference system is composed by five functional blocks (Jang, 1993), Fig. 2.

Fig. 2. Fuzzy inference system structure


Frequently, the rule base and the database (e.g. vibrations data in different conditions) are jointly referred to as the *knowledge base*.

The steps of fuzzy logic (inference operations upon fuzzy *if-then* rules) performed by fuzzy inference systems for machine's diagnoses are shown in Figure 3.

Several types of fuzzy logic have been proposed in the open research. Depending on the types of fuzzy reasoning and fuzzy *if-then* rules employed, most fuzzy inference strategies may be classified as follows (Jang, 1993).

Fuzzy inference systems are also known as fuzzy-rule-based systems, fuzzy models, fuzzy associative memories (FAM), or fuzzy controllers when used as controllers. Basically, a

fuzzy inference system is composed by five functional blocks (Jang, 1993), Fig. 2.

**2.3 Fuzzy inference system structure for vibration analysis** 

Fig. 2. Fuzzy inference system structure

fuzzy rules.

crisp output.

with linguistic values.

jointly referred to as the *knowledge base*.

may be classified as follows (Jang, 1993).

i. A rule base containing a number of fuzzy *if-then* rules.

inference systems for machine's diagnoses are shown in Figure 3.

ii. A database which defines the membership functions of the fuzzy sets used in the

v. A defuzzification interface which transforms the fuzzy results of the inference into a

Frequently, the rule base and the database (e.g. vibrations data in different conditions) are

The steps of fuzzy logic (inference operations upon fuzzy *if-then* rules) performed by fuzzy

Several types of fuzzy logic have been proposed in the open research. Depending on the types of fuzzy reasoning and fuzzy *if-then* rules employed, most fuzzy inference strategies

 iii. A decision-making unit which performs the inference operations on the rules. iv. A fuzzification interface which transforms the crisp inputs into degrees of match

Fig. 3. Flowchart of the followed inference strategy

Fuzzy Inference Systems Applied to the Analysis of Vibrations in Electrical Machines 141

Fig. 4. Set of calculations in ANFIS

*μAi* is the Ai's membership function.

*xi* is the input into node *i Ai* is the linguistic label

where:

Type 1: The overall output is the weighted average of each rule's crisp output induced by the rule's firing strength and output membership functions. The output membership functions used in this scheme must be monotonic functions (lee, 1990).

Type 2: The overall fuzzy output is derived by applying maximization operation to the qualified fuzzy outputs, each of which is equal to the minimum of firing strength and the output membership function of each rule. Various schemes have been proposed to choose the final crisp output based on the overall fuzzy output; some of them are the centroid of area, mean of maxima, maximum criterion, etc., (Lee, 1990).

Type 3: In (Lee, 1990, as cited in Takagi & Sugeno, 1985) fuzzy *if-then* rules are used. The output of each rule is a linear combination of input variables plus a constant term, and the final output is the weighted average of each rule's output.
