© 2007 Science Publications Qualitative Ferromagnetic Hysteresis Modeling

In determining the electromagnetic properties of magnetic materials, hysteresis modeling is of high importance. Many models are available to investigate those characteristics but they tend to be complex and difficult to implement. A new qualitative hysteresis model for ferromagnetic core presented, based on the function approximation capabilities of adaptive neuro-fuzzy inference system (ANFIS). The proposed ANFIS model combined the neural network adaptive capabilities and the fuzzy logic qualitative approach can restored the hysteresis curve with a little RMS error. The model accuracy was good and can be easily adapted to the requirements of the application by extending or reducing the network training set and thus the required amount of measurement data.


INTRODUCTION
Analysis of electrical machines requires a computationally efficient hysteresis model describing the nonlinear relation between the magnetic induction and the magnetic field strength in the ferromagnetic core of the machine. However, there exist many approaches to develop a mathematical model to describe the hysteretic relationship between the magnetization M and the magnetic field H. the first approach was the hysteresis model of Preisach invented in the 1935 [1] and the second is the Jiles-Atherton (JA) model [2] . Artificial intelligence has also been applied to the modeling of magnetic hysteresis and parameters identification of these models such as neural network and genetic algorithm [3,4,5,6,7,8,9,10,11,12,13] . Like neural networks, fuzzy logic can be conveniently used to approximate any arbitrary functions [14,15,16] . Neural networks can learn from data, but knowledge learned can be difficult to understand. Models based on fuzzy logic are easy to understand, but they do not have learning algorithms; learning has to be adapted from other technologies. A Neuro-Fuzzy model can be defined as a model built using a combination of fuzzy logic and neural networks. Recently, there has been a remarkable advance in the development of Neuro-Fuzzy models, as it is described in [17,18,19] . One of the most popular and well documented Neuro-Fuzzy systems is ANFIS, which has a good software support [20] . Jang [21,22,23] present the ANFIS architecture and application examples in modeling a nonlinear function, dynamic system identification and a chaotic time series prediction. Given its potential in building fuzzy models with good prediction capabilities, the ANFIS architecture was chosen for modeling magnetic hysteresis in this work. In the following sections information is given about adaptive neuro-Fuzzy modeling, the JA model for magnetic material testing system, the selection of ways to modeling the hysteresis phenomena with neuro-Fuzzy modeling, results and conclusions.

JILES-ATHERTON HYSTERESIS MODEL
Formulation: The Jiles-Atherton model is a physically based model that includes the different mechanisms that take place at magnetization of a ferromagnetic material.
The magnetization M is represented as the sum of the irreversible magnetization M irr due to domain wall displacement and the reversible magnetization M rev due to domain wall bending [2] . The rate of change of the irreversible part of the magnetization is given by.
The anhysteretic magnetization M an in (1) follows the Langevin function [3] , which is a nonlinear function of the effective field: The rate of change of the reversible component is proportional to the rate of the difference between the hysteretic component and the total magnetization [4] . Consequently, the differential of the reversible magnetization is: Combining the irreversible and reversible components of magnetization, the differential equation for the rate of change of the total magnetization is given by: Before using the J-A model, five parameters must be determined: α: a mean field parameter defining the magnetic coupling between domains in the material, and is required to calculate the effective magnetic field, H e (2) composed by the applied external field and the internal magnetization. M s : magnetic saturation a : langevin parameter These two parameters defined a Langevin function needed in the equation describing anhysteretic curve. k : parameter defining the pinning site density of domain walls. It is assumed to be the major contribution to hysteresis. c : parameter defining the amount of reversible magnetization due to wall bowing and reversal rotation, included in the magnetization process. δ is a directional parameter and takes +1 for increasing field (dH/dt>0) and -1 for decreasing field (dH/dt<0). A. Parameter Identification B.1 Anhysteretic Susceptibility: The anhysteretic susceptibility at the origin, can be used to define a relationship between M s , a and α The hysteresis loss parameter k can be determined from the coercivity H c and the differential susceptibility at the coercive point χ an (H c ).
The coupling parameter α can be determined independently if a is known by using the remanence magnetization M r and the differential susceptibility at remanence. An adaptive Neuro-Fuzzy inference system is a cross between an artificial neural network and a fuzzy inference system. An artificial neural network is designed to mimic the characteristics of the human brain and consists of a collection of artificial neurons. An adaptive network is a multi-layer feed-forward network in which each node (neuron) performs a particular function on incoming signals. The form of the node functions may vary from node to node. In an adaptive network, there are two types of nodes: adaptive and fixed. The function and the grouping of the neurons are dependent on the overall function of the network. . Based on the ability of an ANFIS to learn from training data, it is possible to create an ANFIS structure from an extremely limited mathematical representation of the system.

Architecture of ANFIS:
The ANFIS is a fuzzy Sugeno model put in the framework of adaptive systems to facilitate learning and adaptation [18] . Such framework makes the ANFIS modeling more systematic and less reliant on expert knowledge. To present the ANFIS architecture, we suppose that there are two input linguistic variables (x, y) and each variable has two fuzzy sets (A 1 , A 2 ) and ( B 1 ,B 2 ) as is indicated in fig.1, in which a circle indicates a fixed node, whereas a square indicates an adaptive node. Then a Takagi-Sugeno-type fuzzy if-then rule could be set up as: Fig. 1: ANFIS architecture f i are the outputs within the fuzzy region specified by the fuzzy rule. p i , q i and r i are the design parameters that are determined during the training process. Some layers of ANFIS have the same number of nodes, and nodes in the same layer have similar functions.
Output of nodes in layer-l is denoted as 1 i O , where l is the layer umber and i is neuron number of the next layer. The function of each layer is described as follows: Layer 1: In this layer, all the nodes are adaptive nodes. The outputs of layer 1 are the fuzzy membership grade of the inputs, which are given by: where a i , b i and c i are the parameters of the membership function, governing the bell shaped functions accordingly.
• Layer 2: Each node computes the firing strengths of the associated rules. The output of nodes in this layer can be presented as: Layer 3: In this third layer, the nodes are also fixed nodes. They play a normalization role to the firing strengths from the previous layer. The outputs of this layer can be represented as: ω ω (15) which are the so-called normalized firing levels.
• Layer 4: The output of each adaptive node in this layer is simply the product of the normalized firing level and a first order polynomial (for a first order Sugeno model). Thus, the outputs of this layer are given by:  (17) From the architecture of ANFIS, we can observe that there are two adaptive layers the first and the fourth. In the first layer, there are three modifiable parameters {a i , b i , c i }, which are related to the input membership functions. These parameters are the so-called premise parameters. In the fourth layer, there are also three modifiable parameters {p i ,q i ,r i }, pertaining to the first order polynomial. These parameters are so-called consequent parameters [21][22] .
Learning algorithm of ANFIS: The task of training algorithm for this architecture is tuning all the modifiable parameters to make the ANFIS output match the training data. Note here that ai, bi and ci describe the sigma, slope and the center of the bell MF's, respectively. If these parameters are fixed, the output of the network becomes: This is a linear combination of the modifiable parameters. For this observation, we can divide the parameter set S into two sets: S=set of total parameters, S 1 =set of premise (nonlinear) parameters, S 2 =set of consequent (linear) parameters ⊕ : Direct sum For the forward path (see Fig 1), we can apply least square method to identify the consequent parameters. Now for a given set of values of S 1 , we can plug training data and obtain a matrix equation: y A = Θ (22) where Θ contains the unknown parameters in S 2 . This is a linear square problem, and the solution for Θ, which is minimizes y A = Θ , is the least square estimator: we can use also recursive least square estimator in case of on-line training. For the backward path (see Fig. 1), the error signals propagate backward. The premise parameters are updated by descent method, through minimising the overall quadratic cost function (24) in a recursive manner with respect Θ (S2) . The update of the parameters in the i th node in layer L th layer can be written as: where η is the learning rate and the gradient vector being the node's output and i L, ε is the backpropagated error signal. Fig.2   Where µ 0 =4.π.10 -7 (H/m) is the permeability of free space and µ r is the relative permeability. The B(H) curve result of simulation of the Jiles-Atherton model will be used as 'experimental data' to be approximate by proposed Neuro-Fuzzy model.

Proposed model:
In this section, the learning ability of ANFIS is verified by approximating a hysteresis of magnetic material. The data set used as input/output pairs for Anfis was generated by Jiles Atherton model for ferrite core described in [24] with sinusoidal magnetic field as an input H(t) and magnetic field B(t) as output Fig. (3.a & b)   The number of MFs assigned to each input of the ANFIS was set to two bell type, so the number of rules is 04. The training was run for 100 iterations, the network performance were evaluated on the checking set after every iteration, by calculating the root-meansquare errors (RMSE): Where k is the pattern number, k=1,…K. The RMSE was also evaluated on training data set in every iteration. The optimal number of iteration was obtained when checking RMSE has reached its minimum value 0.0069 after 11 epochs. See fig.5.  Table.II summarize all characteristics of the network used.  The ANFIS shown in Fig.1 was implemented by using MATLAB software package ( MATLAB version 6.5 with fuzzy logic toolbox), it uses 346 training data in 100 training periods and the step size for parameter adaptation had an initial value of 0.1. The steps of parameter adaptation of the ANFIS are shown in Fig.7.