**2.4 ANFIS basics**

In ANFIS, the adaptive network structure is a multilayer feed-forward network where each node performs a particular function (node function) on incoming signals as well as a set of parameters pertaining to this node. The node functions may vary from node to node, and the choice of each node function depends on the overall input-output function that the adaptive network is required to carry out. Notice that links in an adaptive network indicate the flow direction of signals between nodes; no weights are associated with the links.

Functionally, there are almost no constraints on the node functions of an adaptive network, except piecewise differentiability. Structurally, the only restriction of the network configuration is that it should be of feed-forward type. Due to these minimal restrictions, the adaptive network's applications are immediate and immense in various areas. (Jang, 1993) proposed a class of adaptive networks that are functionally equivalent to fuzzy inference systems. The proposed architecture is referred to as ANFIS, standing for adaptive-networkbased fuzzy inference system.

An adaptive network is a structured network composed by nodes and directional links, which connect nodes, Fig. 4. All or some nodes are adaptive. It means that results depend on nodes' parameters, and the learning rules specify how these parameters must change in order to minimize an error. The adaptive network is constituted by a multilayer feedback network, where each node performs a particular task (node function) on the incoming signals, as well as a set of node parameters.

The ANFIS can be trained by a hybrid learning algorithm (Jang, 1993; Jang & Sun, 1996; Jang & Gulley, 1995). It uses a two-pass learning cycle. In the forward pass, the algorithm uses the least-squares method to identify the consequent parameters on the layer 4. In the backward pass, the errors are propagated backward and the premise parameters are updated by gradient descent.

ANFIS is a tradeoff between neural and fuzzy systems, providing: (*i*) smoothness, due to the Fuzzy interpolation; (*ii*) adaptability, due to the neural net backpropagation; (*iii*) ANFIS however has a strong computational complexity restriction.

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

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

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

In ANFIS, the adaptive network structure is a multilayer feed-forward network where each node performs a particular function (node function) on incoming signals as well as a set of parameters pertaining to this node. The node functions may vary from node to node, and the choice of each node function depends on the overall input-output function that the adaptive network is required to carry out. Notice that links in an adaptive network indicate the flow direction of signals between nodes; no weights are associated

Functionally, there are almost no constraints on the node functions of an adaptive network, except piecewise differentiability. Structurally, the only restriction of the network configuration is that it should be of feed-forward type. Due to these minimal restrictions, the adaptive network's applications are immediate and immense in various areas. (Jang, 1993) proposed a class of adaptive networks that are functionally equivalent to fuzzy inference systems. The proposed architecture is referred to as ANFIS, standing for adaptive-network-

An adaptive network is a structured network composed by nodes and directional links, which connect nodes, Fig. 4. All or some nodes are adaptive. It means that results depend on nodes' parameters, and the learning rules specify how these parameters must change in order to minimize an error. The adaptive network is constituted by a multilayer feedback network, where each node performs a particular task (node function) on the incoming

The ANFIS can be trained by a hybrid learning algorithm (Jang, 1993; Jang & Sun, 1996; Jang & Gulley, 1995). It uses a two-pass learning cycle. In the forward pass, the algorithm uses the least-squares method to identify the consequent parameters on the layer 4. In the backward pass, the errors are propagated backward and the premise parameters are

ANFIS is a tradeoff between neural and fuzzy systems, providing: (*i*) smoothness, due to the Fuzzy interpolation; (*ii*) adaptability, due to the neural net backpropagation; (*iii*) ANFIS

functions used in this scheme must be monotonic functions (lee, 1990).

area, mean of maxima, maximum criterion, etc., (Lee, 1990).

final output is the weighted average of each rule's output.

**2.4 ANFIS basics** 

with the links.

based fuzzy inference system.

updated by gradient descent.

signals, as well as a set of node parameters.

however has a strong computational complexity restriction.

Fig. 4. Set of calculations in ANFIS

where: *xi* is the input into node *i Ai* is the linguistic label *μAi* is the Ai's membership function.

Fuzzy Inference Systems Applied to the Analysis of Vibrations in Electrical Machines 143

 *Fmax* = *Fs*/2 = 1/(2T) (1) where T is the sampling interval. That is, if the frequency *Fmax* of the original signal is divided into two sub-frequency bands, where p / 2 is the highest frequency band, it leads to *Fs*= p and T = 1 / p. To clarify the concept, consider the scheme of underlying filters, which perform the discrete wavelet transformation. Under this concept, for each filtering level the incoming signal is split into low and high frequencies. Since the output from low frequencies is subjected to additional filters, the resolution increases as the spectrum is

The resolution time is reduced because of the decimation that takes place. The abovementioned strategy has been employed to make an inference about the engine's state, using

It is important to emphasize that the main aim of this chapter is the inference system, and to present the structured method for signal processing. The necessary requirements to establish the machine's operating conditions are presented below, and consist in a hybrid method decomposed in two phases. Phase I is the adequacy of the signal, while phase II is the inference or identification procedure, figure 7. Both phases I and II may be represented

The process of the adequacy signal is necessary because the exclusive ANFIS application to

by two functional blocks that perform different treatments to the vibration signals.

minimally invasive faults does not generate a successful inference process

divided again into two sub-bands.

vibration measurements as input

Fig. 6. Time-Frequency resolution of a transformed wavelet

**3. Proposition** 

{*ai, bi, ci*} is the set of parameters. Modifying these parameters, the shape of the bell functions change, so that exhibit different forms of membership functions for the linguistic label *Ai*. ϖ is the level-3 output.

{*pi, qi, ri*}: is the set of parameters, which at this level may be referred to as consequent parameters.

#### **2.5 Vibration analysis by wavelets**

The raw material to make any inference about the machinery's condition is the information captured from vibration signals. The proposition is to utilize the vibration's raw signals to infer about the engine's state. A structured analysis may characterize the nature of the vibration, figure 5.

Wavelets transformation is the disintegration of a signal which becomes represented by means of function approximations and differences, which are divided by levels, figure 6, each of which have different resolutions, being equivalent to filtering the signal through a filter bank. The initial filtering takes the signal and passes it through the first bank, resulting in two signals with different frequency bands (high and low bands).

Fig. 5. Vibration patterns under failure and normal conditions.

The time-frequency resolution of the transformed wavelet satisfies the Nyquist sampling theorem. That is, the maximum frequency component embedded into a signal can be uniquely determined if the signal is sampled at a frequency Fs, which exceeds or equals the double of the signal's maximum frequency *Fmax*. At the limit, if *Fs* = 2*Fmax* then:

$$F\_{\text{max}} = F\_s / 2 = 1 / \text{(2T)}\tag{1}$$

where T is the sampling interval. That is, if the frequency *Fmax* of the original signal is divided into two sub-frequency bands, where p / 2 is the highest frequency band, it leads to *Fs*= p and T = 1 / p. To clarify the concept, consider the scheme of underlying filters, which perform the discrete wavelet transformation. Under this concept, for each filtering level the incoming signal is split into low and high frequencies. Since the output from low frequencies is subjected to additional filters, the resolution increases as the spectrum is divided again into two sub-bands.

The resolution time is reduced because of the decimation that takes place. The abovementioned strategy has been employed to make an inference about the engine's state, using vibration measurements as input

Fig. 6. Time-Frequency resolution of a transformed wavelet

#### **3. Proposition**

142 Fuzzy Inference System – Theory and Applications

{*ai, bi, ci*} is the set of parameters. Modifying these parameters, the shape of the bell functions change, so that exhibit different forms of membership functions for the linguistic label *Ai*.

{*pi, qi, ri*}: is the set of parameters, which at this level may be referred to as consequent

The raw material to make any inference about the machinery's condition is the information captured from vibration signals. The proposition is to utilize the vibration's raw signals to infer about the engine's state. A structured analysis may characterize the nature of the

Wavelets transformation is the disintegration of a signal which becomes represented by means of function approximations and differences, which are divided by levels, figure 6, each of which have different resolutions, being equivalent to filtering the signal through a filter bank. The initial filtering takes the signal and passes it through the first bank, resulting

in two signals with different frequency bands (high and low bands).

Fig. 5. Vibration patterns under failure and normal conditions.

The time-frequency resolution of the transformed wavelet satisfies the Nyquist sampling theorem. That is, the maximum frequency component embedded into a signal can be uniquely determined if the signal is sampled at a frequency Fs, which exceeds or equals the

double of the signal's maximum frequency *Fmax*. At the limit, if *Fs* = 2*Fmax* then:

ϖ is the level-3 output.

vibration, figure 5.

**2.5 Vibration analysis by wavelets** 

parameters.

It is important to emphasize that the main aim of this chapter is the inference system, and to present the structured method for signal processing. The necessary requirements to establish the machine's operating conditions are presented below, and consist in a hybrid method decomposed in two phases. Phase I is the adequacy of the signal, while phase II is the inference or identification procedure, figure 7. Both phases I and II may be represented by two functional blocks that perform different treatments to the vibration signals.

The process of the adequacy signal is necessary because the exclusive ANFIS application to minimally invasive faults does not generate a successful inference process

Fuzzy Inference Systems Applied to the Analysis of Vibrations in Electrical Machines 145

Fig. 8. Induction motor's arrangement

Fig. 9. Vibrations under different conditions

operating conditions:

The proposed hybrid method aims to identify the fault states in rotating machines, distinguishing the smooth operation from failure conditions by measuring vibration signals. Vibration measurements have been monitored in three axes: x, y, and z under the following

#### Fig. 7. Stages of the inference system

In this application, measurements are taken by a 12-bit LIS3L02ASA vibration sensor (accelerometer based on MEMS - Microelectromechanical system), which provides measurement of displacement in three axes. Additionally, to reduce the noise/signal proportion, filtering is added.

Thus, the triaxial accelerometer, is one of the most important parts of the instrumentation system, being located in the engine body, which measures vibrations based on three axes (x, y, z) using a sampling rate of 1500Hz. The ADS7841 is a converter equipped with serial synchronous communication interface with 200KHz conversion rate. After the digitalized data is sent via the RS232 card to capture, the system data acquisition uses a MAX3243 circuit.

The sensor provides vibration measurements in three axes. In this research, it was noticed that the perpendicular axes to the axis of rotation have more useful information to identify a failure occurrence. Thus, in order to optimize the computational load, data from the x-axis were used.
