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As a result from the demanding of process safety, reliability and environmental constraints, a called of fault detection and diagnosis system become more and more important. In this article some basic aspects of TSK (Takigi Sugeno Kang) neuro-fuzzy techniques for the prognosis and diagnosis of manufacturing systems are presented. In particular, a neuro-fuzzy model that can be used for the identification and the simulation of faults prognosis models is described. The presented model is motivated by a cooperative neuro-fuzzy approach based on a vectorized recurrent neural network architecture. The neuro-fuzzy architecture maps the residuals into two classes: a one of fixed direction residuals and another one of faults belonging to rotary kiln.

Failure may cause large amount of loss. Therefore, fault diagnosis and prognosis system is very important for safe operation and preventing rescue. Recent progress in the field of diagnostics of manufacturing systems (MS) drives is a result of broadly conceived basic research carried out over many years.

According to [

Fuzzy neural network (FNN) approach has become a powerful tool for solving real-world problems in the area of forecasting, identification, control, image recognition and others that are associated with high level of uncertainty [

The Neuro-fuzzy model combines, in a single framework, both numerical and symbolic knowledge about the process. Automatic linguistic rule extraction is a useful aspect of NF especially when little or no prior knowledge about the process is available [1,3]. For example, a NF model of a non-linear dynamical system can be identified from the empirical data.

This model can give us some insight about the on linearity and dynamical properties of the system.

The most common NF systems are based on two types of fuzzy models TSK [4,5] combined with NN learning algorithms. TSK models use local linear models in the consequents, which are easier to interpret and can be used for control and fault diagnosis [

Neuro-fuzzy networks by intrinsic nature can handle limited number of inputs. When the system to be identified is complex and has large number of inputs, the fuzzy rule base becomes large.

NF models usually identified from empirical data are not very transparent. Transparency accounts a more meaningful description of the process i.e. less rules with appropriate membership functions. In ANFIS [

Hierarchical NF networks can be used to overcome the dimensionality problem by decomposing the system into a series of MISO and/or SISO systems called hierarchical systems [

The criteria on which to build a NF model are based on the requirements for faults diagnosis and the system characteristics. The function of the NF model in the FDI scheme is also important i.e. Preprocessing data, Identification (Residual generation) or classification (Decision Making/Fault Isolation).

For example a NF model with high approximation capability and disturbance rejection is needed for identification so that the residuals are more accurate.

Whereas in the classification stage, a NF network with more transparency is required.

The following characteristics of NF models are important:

Approximation/Generalisation capabilities;

Transparency: Reasoning/use of prior knowledge/rules;

Training Speed/Processing speed;

Complexity;

Transformability: To be able to convert in other forms of NF models in order to provide different levels of transparency and approximation power.

Adaptive learning.

Two most important characteristics are the generalising and reasoning capabilities. Depending on the application requirement, usually a compromise is made between the above two.

In order to implement this type of Neuro-Fuzzy Systems for Fault Diagnosis and Prognosis and exploited to diagnose of dedicated production system we have to propose data-processing software NEFDIAG (NeuroFuzzy Diagnosis).

The Takagi-Sugeno type fuzzy rules are discussed in detail in Subsection A. In Subsection B, the network structure of FENN is presented.

Recently, more and more attention has paid to the Takagi-Sugeno type rules [

Rule r:

where is the inner is the inner state vector of the nonlinear system;

is the input vector to the system, and N, M are the dimensions;

are linguistic terms (fuzzy sets) defining the conditions for x_{i} and u_{j} respectively, according to Rule r;

is a matrix of and

of

When considered in discrete time, such as modeling using a digital computer, we often use the discrete statespace equations instead of the continuous version. Concretely, the fuzzy rules become:

Rule r:

where is the discrete sample of state vector at discrete time t. In following discussion we shall use the latter form of rules.

In both forms, the output of the system is always defined as:

where C= (c_{ij})_{Px Xis} a matrix of P × N, and P is the dimension of output vector Y.

The fuzzy inference procedure is specified as below. First, we use multiplication as operation AND to get the firing strength of Rule r:

where^{.}and are the membership functions of

respectively? After normalization of the firing strengths, we get (assuming R is the total number of rules)

where S is the summation of firing strengths of all the rules, and h_{r} is the normalized firing strength of Rule r. When the defuzzification is employed, we have

where

Using equation (4), the system state transient equation, we can calculate the next state of system by current state and input.

The main idea of this model is to combine simple feed forward fussy systems to arbitrary hierarchical models.

The structure of recurrent Neuro-fuzzy systems is presented in figure 1.

In this network, input nodes which accept the environment inputs and context nodes which copy the value of the state-space vector from layer 3 are all at layer 1 (the Input Layer). They represent the linguistic variables known as u_{j} and x_{i} in the fuzzy rules. Nodes at layer 2 act as the membership functions, translating the linguistic variables from layer 1 into their membership degrees. Since there may exist several terms for one linguistic variable, one node in layer 1 may have links to several nodes in layer 2, which is accordingly named as the term nodes. The number of nodes in the Rule Layer (layer 3) and the one of the fuzzy rules are the same—each node represents one fuzzy rule and calculates the firing strength of the rule using membership degrees from layer 2. The connections between layer 2 and layer 3 correspond with the antecedent of each fuzzy rule. Layer 4, as the Normalization Layer, simply does the normalization of the firing strengths. Then with the normalized firing strengths h_{r}, rules are combined at layer 5, the Parameter Layer, where A and B become available. In the Linear System Layer, the 6th layer, current state vector X(t) and input vector U(t) are used to get the next state X(t + 1), which is also fed back to the context nodes for fuzzy inference at time (t + 1). The last layer is the Output Layer, multiplying X(t + 1) with C to get Y(t + 1) and outputting it.

Next we shall describe the feed forward procedure of TNFS by giving the detailed node functions of each layer, taking one node per layer as example. We shall use notations like to denote the i^{th} input to the node in layer k, and o[k] the output of the node in layer k. Another issue to mention here is the initial values of the context

nodes. Since TNFS is a recurrent network, the initial values are essential to the temporal output of the network. Usually they are preset to 0, as zero-state, but non-zero initial state is also needed for some particular case.

Layer 1. There is only one input to each node at layer 2. The Gaussian function is adopted here as the membership function:

where c^{r} and s^{r} give the center (mean) and width (variation) of the corresponding u[

Layer 2. This layer has several nodes, one for figuring matrix A and the other for B. Though we can use many nodes to represent the components of A and B separately, it is more convenient to use matrices. So with a little specialty, its weights of links from layer 4 are matrices A^{r} (to node for A) and B^{r} (to node for B). It is also fully connected with the previous layer. The functions of nodes for A and B are respectively.

Layer 3. The Linear System Layer has only one node, which has all the outputs of layer 1 and layer 2 connected to it as inputs. Using matrix form of inputs and output, we have

The first step in building a prognostics system, as published in the ISO standard, is the identification of the set of failure modes (FM), their influence factors on each other and the detection measures (descriptors) that allow to track the evolution of the degradation. The international standard IEC 60812 [

The network structure is build in three steps:

Step 1. The determination of fuzzy subsets for every input variable. The initial values of the centres and variances characterising the membership functions of the first layer down, can be arbitrarily established (equidistant on the domain of definition of the linguistic variable) or applying a clustering algorithm of the type Fuzzy C-Means.

Step 2. Obtain the minimal dimension of the rule base.

The extraction of most significant rule that determines the number of the nodes in the second layer.

Step 3. Optimization of the parameters of rules determined at Step 2. The objective is to alternate the parameter values (c,w) of the network in order to improve the rule base minimizing the quadratic criteria of performance,

To test the quality of the model, several actions were generated, and fixed goals were defined. The goals were defined in a way that the results were understood without ambiguity by human knowledge, the figure 2 illustrate the fuzzy base rules with 3 parametres to classify the defaults modes In order to illustrate the learning effect of the proposed immune based FNN (IM-FNN), we use One of the most important types of systems present in the process industry is workshop of SCIMAT clinker. A fault in a workshop of SCIMAT clinker may lead to a halt in production for long periods of time. Apart from these economic considerations faults may also have security implications. A fault in an actuator may endanger human lives, as in the case of a fault in an elevator’s emergency brakes or in the stems position control system of a nuclear power plant [9,12]. The design and performance testing of fault diagnosis systems for industrial process often requires a simulation model since the actual system is not available to generate normal and faulty operational.

In figure 3 the detection of fault mode in the rotary kiln is observed with the classification after training the neuro-fuzzy system.

data needed for design and testing, due to the economic and security reasons that they would imply. Accord-

ing to this figure 4 we can say the prediction of a faults is a complex problem and need the correction of inverse problem.

The successful of implementing neuron-fuzzy is heavily depends on prior knowledge of the system and the training data. In the intrinsic nature, the neuro-fuzzy only can handle a limited number of inputs and can usually be identified in a not very transparent way from the empirical data [