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Implementation of magnetostriction Preisach-type models using orthogonally coupled hysteresis operators
ARTICLE IN PRESS
Physica B 403 (2008) 425–427 www.elsevier.com/locate/physb
Implementation of magnetostriction Preisach-type models using orthogonally coupled hysteresis operators A.A. Adlya,, S.K. Abd-El-Hafizb a
Electrical Power and Machines Department, Faculty of Engineering, Cairo University, Giza 12211, Egypt b Department of Engineering Mathematics, Faculty of Engineering, Cairo University, Giza 12211, Egypt
Abstract Magnetic materials exhibiting gigantic magnetostriction are currently being used in various actuator devices and vibration damping applications. Recently, a new family of efficient Preisach-type vector hysteresis models having coupled elementary operators has been introduced. The purpose of this paper is to extend the applicability of those recently introduced models to magnetostriction simulation. Details of the model, its identification, and experimental testing are presented in the paper. r 2007 Elsevier B.V. All rights reserved. Keywords: Magnetostriction simulation; Vector Preisach model; Terfenol
1. Introduction Magnetic materials exhibiting gigantic magnetostriction such as Terfenol are currently being widely used in various actuator devices and vibration damping applications. As a result, considerable efforts have been directed towards the development of models that can accurately simulate such a phenomenon (refer, for instance, to Refs. [1–3]). Recently, it has been demonstrated that elementary hysteresis operators may be implemented using a two-node discrete Hopfield neural network (DHNN) having step activation functions and positive feedback weights [4]. Based upon this DHNN realization of hysteresis operators, a computationally efficient vector Preisach-type model has been proposed in Ref. [5]. The purpose of this paper is to extend the applicability of recently introduced efficient Preisach-type models to magnetostriction simulation. Implementation efficiency of the proposed model stems from its assembly from only two scalar models having orthogonally coupled elementary operators. Details of the suggested model, its identification
Corresponding author. Tel.: +2 02 25163364; fax: +2 02 35723486.
E-mail addresses: [email protected] (A.A. Adly), salwahafi[email protected] (S.K. Abd-El-Hafiz). 0921-4526/$ - see front matter r 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.physb.2007.08.066
mechanism, and experimental testing are given in the following sections of the paper. 2. Model implementation It has been shown in Ref. [4] that a single elementary hysteresis operator may be realized via a two-node DHNN. For this DHNN, the external input is denoted by Ix and the outputs, UAx and UBx, are binary variables A{1, 1}. Node output values may change as a result of the external input Ix until the network state converges to a minimum energy [6]. Loop width may be controlled by the feedback weight, kJ. Moreover, the loop center can be shifted with respect to the x-axis by introducing an offset Qx to its external input Ix. In other words, the elementary hysteresis operator switching up and down values, regularly denoted by a and b [7], would generally be equivalent to (Qx+kJ) and (QxkJ), respectively. Consider the four-node DHNN network reported in Ref. [5]. According to Refs. [1,5] this network is capable of realizing a couple of hysteresis operators whose inputs (i.e., Ix and Iy) correspond to the field H and stress s. More specifically, combination of outputs of nodes Ax and Bx can represent magnetization M or strain l. For the considered four-node DHNN network the symbol k? denotes the feedback between nodes corresponding to
ARTICLE IN PRESS A.A. Adly, S.K. Abd-El-Hafiz / Physica B 403 (2008) 425–427
H- and s-driven operators. Once more, state of this DHNN network converges to a minimized energy [6]. A vector Preisach-type model may be, thus, constructed from a couple of scalar models whose elementary hysteresis operators are realized using the suggested DHNN. As previously mentioned, variation of the widths and centers of the ensemble of rectangular hysteresis operators may be achieved by imposing the appropriate feedback weights and offsets, respectively. Implementation efficiency (memory wise and computation wise) of the proposed magnetostriction model is hence achieved by dealing with a couple of scalar Preisach-type models instead of a collection of models oriented along all possible directions as suggested in Ref. [1]. In accordance with the basic configuration of a vector Preisach-type model [7] the suggested model may be implemented using a modular combination of DHNN blocks via the linear neural network structure (DHNNLNN) reported in Ref. [5]. For that DHNN-LNN offset values Qi and feedback factors kJi corresponding to the different DHNN blocks should cover the complete ab plane region under consideration. More precisely, for the DHNN #i representing rectangular operators whose switching up and down thresholds are ai and bi, respectively, we have (1)
The modular DHNN under consideration is expected to evolve—as a result of any applied input—by changing output values (states) of the operator blocks. The network eventually converges to a minimum of the quadratic energy function given by h i 9 8 ai þbi > > þ H þ U Þ ðU > > Axi Bxi 2 > > > > > > h i > > > > ai þbi > > þ U Þ þ s ðU > > Ayi Byi > > 2 > > > > h i > > > > a b > > i i = < N U þ U X Axi Bxi 2 . (2) E¼ h i > > ai bi > i¼1 > þ U U > > Ayi Byi 2 > > > > > > > > > > k? > > > þ ðU U ÞðU þ U Þ > > Axi Bxi Ayi Byi > 2 > > > > > > > ; : þ k2? U Ayi U Byi ðU Axi þ U Bxi Þ > The output, on the other hand, may be expressed as N X U Axi þ U Bxi Output ¼ mi . (3) 2 i¼1 Within this implementation, the model identification is reduced to the determination of the various mi and k? values that would result in a best fit to experimental data corresponding to any material under consideration. 3. Model identification and experimental testing Numerical implementation of the proposed model has been carried out to separately simulate M(H,s) and l(H,s)
Meas (0)
Comp (0)
Meas (1)
Comp (1)
1.00 M (Normalized)
Qi ¼ ðai þ bi Þ=2 and kjji ¼ ðai bi Þ=2.
variations. Throughout the simulations, 930 uniformly dispersed elementary operators were utilized. Because of its importance in industrial applications, Terfenol was chosen as the test sample material. First, model unknowns had to be identified using measured M–H and l–H curves for different stress values. Being realized by the pre-described DHNN-LNN configurations, identification process of the two vector Preisach-type models is actually executed using an automated training algorithm. Thus, potential complications embedded in the analytical identification of such a model are avoided. Moreover, identification may be carried out using any available sets of M(H,s) and l(H,s) data. For each DHNN-LNN the identification process is carried out by first assuming some k? value and finding out appropriate values for the unknowns mi for i ¼ 1,2,y,N. To determine these appropriate mi values, training using the available scalar data provided to the network and the leastmean-square (LMS) algorithm may be applied to the linear neuron whose output represents M or l (refer to Ref. [6]). The outcome of the identification process is shown in Figs. 1 and 2 for an assumed k? value of 5. In those figures stress and field values were normalized to 1.38 KN/cm2 and 3000 Oe, respectively. In order to check the model capability of predicting M–H and l–H curves for intermediate stress values
0.75 0.50 0.25 0.00 -0.25
0.00
0.25
0.50
0.75
1.00
H (Normalized)
Fig. 1. Comparison between supplied and computed M–H curves for different stress values at the end of the training (i.e., identification) process.
Meas (0)
Comp (0)
Meas (1)
Comp (1)
1.0 Strain (Normalized)
426
0.5 0.0 -0.5 -1.0
-1.0
-0.5
0.0
0.5
1.0
H (Normalized)
Fig. 2. Comparison between supplied and computed l–H curves for different stress values at the end of the training (i.e., identification) process.
ARTICLE IN PRESS A.A. Adly, S.K. Abd-El-Hafiz / Physica B 403 (2008) 425–427 Meas (0.5)
Comp (0.5)
427
4. Conclusion
M (Normalized)
1.00 0.75 0.50 0.25 0.00 -0.25
0.00
0.25
0.50
0.75
1.00
H (Normalized)
Fig. 3. Comparison between measured and computed M–H limiting curves for a normalized stress value of 0.5.
Meas (0.5)
Comp (0.5)
Strain (Normalized)
1.0 0.5 0.0 -0.5 -1.0 -1.0
-0.5
0.0 H (Normalized)
0.5
1.0
It can be concluded that the proposed implementation of the vector Preisach-type model can lead to good simulation results of M–H and l–H curves under different stress values. Moreover, this approach leads to a significant reduction in computational resources required while facilitating the model identification procedure. This is because, according to this efficient implementation, the vector Preisach-type model can be constructed using only two scalar models whose elementary hysteresis operators are orthogonally interrelated. Quantitatively, the proposed model consumes almost 2/N of the computational resources and time required by a conventional vector Preisach-type model constructed by superimposing N scalar models having the same number of elementary hysteresis operators and whose orientations are equally distributed along all possible 2D directions. Moreover, the identification problem for such an implementation may be carried out in an automated manner using well-established neural network algorithms and for any available set of experimental data. It is believed that this type of implementation may have wide applications in other coupled modeling areas. It is also believed that further enhancement of this implementation is worth investigating in the future.
Fig. 4. Comparison between measured and computed l–H limiting curves for a normalized stress value of 0.5.
References corresponding computations were performed and compared to corresponding measurements for a normalized stress value of 0.5. Such a comparison is shown in Figs. 3 and 4 and demonstrates good qualitative match between measured and computed curves. It should be pointed out that results of Fig. 4 may be significantly improved by increasing the number of elementary hysteresis operators comprising each scalar model.
[1] A.A. Adly, I.D. Mayergoyz, IEEE Trans. Magn. 32 (1996) 4773. [2] D. Davino, C. Natale, S. Pirozzi, C. Visone, J. Magn. Magn. Mater. 272–276 (2004) 781. [3] A.A. Adly, D. Davino, C. Visone, Physica B 372 (2006) 207. [4] A.A. Adly, S.K. Abd-El-Hafiz, J. Magn. Magn. Mater. 263 (2003) 301. [5] A.A. Adly, S.K. Abd-El-Hafiz, IEEE Trans. Magn. 42 (2006) 1518. [6] S. Haykin, Neural Networks: A Comprehensive Foundation, PrenticeHall, New Jersey, 1999. [7] I.D. Mayergoyz, Mathematical Models of Hysteresis, Springer, New York, 1991.
Physica B 403 (2008) 425–427 www.elsevier.com/locate/physb
Implementation of magnetostriction Preisach-type models using orthogonally coupled hysteresis operators A.A. Adlya,, S.K. Abd-El-Hafizb a
Electrical Power and Machines Department, Faculty of Engineering, Cairo University, Giza 12211, Egypt b Department of Engineering Mathematics, Faculty of Engineering, Cairo University, Giza 12211, Egypt
Abstract Magnetic materials exhibiting gigantic magnetostriction are currently being used in various actuator devices and vibration damping applications. Recently, a new family of efficient Preisach-type vector hysteresis models having coupled elementary operators has been introduced. The purpose of this paper is to extend the applicability of those recently introduced models to magnetostriction simulation. Details of the model, its identification, and experimental testing are presented in the paper. r 2007 Elsevier B.V. All rights reserved. Keywords: Magnetostriction simulation; Vector Preisach model; Terfenol
1. Introduction Magnetic materials exhibiting gigantic magnetostriction such as Terfenol are currently being widely used in various actuator devices and vibration damping applications. As a result, considerable efforts have been directed towards the development of models that can accurately simulate such a phenomenon (refer, for instance, to Refs. [1–3]). Recently, it has been demonstrated that elementary hysteresis operators may be implemented using a two-node discrete Hopfield neural network (DHNN) having step activation functions and positive feedback weights [4]. Based upon this DHNN realization of hysteresis operators, a computationally efficient vector Preisach-type model has been proposed in Ref. [5]. The purpose of this paper is to extend the applicability of recently introduced efficient Preisach-type models to magnetostriction simulation. Implementation efficiency of the proposed model stems from its assembly from only two scalar models having orthogonally coupled elementary operators. Details of the suggested model, its identification
Corresponding author. Tel.: +2 02 25163364; fax: +2 02 35723486.
E-mail addresses: [email protected] (A.A. Adly), salwahafi[email protected] (S.K. Abd-El-Hafiz). 0921-4526/$ - see front matter r 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.physb.2007.08.066
mechanism, and experimental testing are given in the following sections of the paper. 2. Model implementation It has been shown in Ref. [4] that a single elementary hysteresis operator may be realized via a two-node DHNN. For this DHNN, the external input is denoted by Ix and the outputs, UAx and UBx, are binary variables A{1, 1}. Node output values may change as a result of the external input Ix until the network state converges to a minimum energy [6]. Loop width may be controlled by the feedback weight, kJ. Moreover, the loop center can be shifted with respect to the x-axis by introducing an offset Qx to its external input Ix. In other words, the elementary hysteresis operator switching up and down values, regularly denoted by a and b [7], would generally be equivalent to (Qx+kJ) and (QxkJ), respectively. Consider the four-node DHNN network reported in Ref. [5]. According to Refs. [1,5] this network is capable of realizing a couple of hysteresis operators whose inputs (i.e., Ix and Iy) correspond to the field H and stress s. More specifically, combination of outputs of nodes Ax and Bx can represent magnetization M or strain l. For the considered four-node DHNN network the symbol k? denotes the feedback between nodes corresponding to
ARTICLE IN PRESS A.A. Adly, S.K. Abd-El-Hafiz / Physica B 403 (2008) 425–427
H- and s-driven operators. Once more, state of this DHNN network converges to a minimized energy [6]. A vector Preisach-type model may be, thus, constructed from a couple of scalar models whose elementary hysteresis operators are realized using the suggested DHNN. As previously mentioned, variation of the widths and centers of the ensemble of rectangular hysteresis operators may be achieved by imposing the appropriate feedback weights and offsets, respectively. Implementation efficiency (memory wise and computation wise) of the proposed magnetostriction model is hence achieved by dealing with a couple of scalar Preisach-type models instead of a collection of models oriented along all possible directions as suggested in Ref. [1]. In accordance with the basic configuration of a vector Preisach-type model [7] the suggested model may be implemented using a modular combination of DHNN blocks via the linear neural network structure (DHNNLNN) reported in Ref. [5]. For that DHNN-LNN offset values Qi and feedback factors kJi corresponding to the different DHNN blocks should cover the complete ab plane region under consideration. More precisely, for the DHNN #i representing rectangular operators whose switching up and down thresholds are ai and bi, respectively, we have (1)
The modular DHNN under consideration is expected to evolve—as a result of any applied input—by changing output values (states) of the operator blocks. The network eventually converges to a minimum of the quadratic energy function given by h i 9 8 ai þbi > > þ H þ U Þ ðU > > Axi Bxi 2 > > > > > > h i > > > > ai þbi > > þ U Þ þ s ðU > > Ayi Byi > > 2 > > > > h i > > > > a b > > i i = < N U þ U X Axi Bxi 2 . (2) E¼ h i > > ai bi > i¼1 > þ U U > > Ayi Byi 2 > > > > > > > > > > k? > > > þ ðU U ÞðU þ U Þ > > Axi Bxi Ayi Byi > 2 > > > > > > > ; : þ k2? U Ayi U Byi ðU Axi þ U Bxi Þ > The output, on the other hand, may be expressed as N X U Axi þ U Bxi Output ¼ mi . (3) 2 i¼1 Within this implementation, the model identification is reduced to the determination of the various mi and k? values that would result in a best fit to experimental data corresponding to any material under consideration. 3. Model identification and experimental testing Numerical implementation of the proposed model has been carried out to separately simulate M(H,s) and l(H,s)
Meas (0)
Comp (0)
Meas (1)
Comp (1)
1.00 M (Normalized)
Qi ¼ ðai þ bi Þ=2 and kjji ¼ ðai bi Þ=2.
variations. Throughout the simulations, 930 uniformly dispersed elementary operators were utilized. Because of its importance in industrial applications, Terfenol was chosen as the test sample material. First, model unknowns had to be identified using measured M–H and l–H curves for different stress values. Being realized by the pre-described DHNN-LNN configurations, identification process of the two vector Preisach-type models is actually executed using an automated training algorithm. Thus, potential complications embedded in the analytical identification of such a model are avoided. Moreover, identification may be carried out using any available sets of M(H,s) and l(H,s) data. For each DHNN-LNN the identification process is carried out by first assuming some k? value and finding out appropriate values for the unknowns mi for i ¼ 1,2,y,N. To determine these appropriate mi values, training using the available scalar data provided to the network and the leastmean-square (LMS) algorithm may be applied to the linear neuron whose output represents M or l (refer to Ref. [6]). The outcome of the identification process is shown in Figs. 1 and 2 for an assumed k? value of 5. In those figures stress and field values were normalized to 1.38 KN/cm2 and 3000 Oe, respectively. In order to check the model capability of predicting M–H and l–H curves for intermediate stress values
0.75 0.50 0.25 0.00 -0.25
0.00
0.25
0.50
0.75
1.00
H (Normalized)
Fig. 1. Comparison between supplied and computed M–H curves for different stress values at the end of the training (i.e., identification) process.
Meas (0)
Comp (0)
Meas (1)
Comp (1)
1.0 Strain (Normalized)
426
0.5 0.0 -0.5 -1.0
-1.0
-0.5
0.0
0.5
1.0
H (Normalized)
Fig. 2. Comparison between supplied and computed l–H curves for different stress values at the end of the training (i.e., identification) process.
ARTICLE IN PRESS A.A. Adly, S.K. Abd-El-Hafiz / Physica B 403 (2008) 425–427 Meas (0.5)
Comp (0.5)
427
4. Conclusion
M (Normalized)
1.00 0.75 0.50 0.25 0.00 -0.25
0.00
0.25
0.50
0.75
1.00
H (Normalized)
Fig. 3. Comparison between measured and computed M–H limiting curves for a normalized stress value of 0.5.
Meas (0.5)
Comp (0.5)
Strain (Normalized)
1.0 0.5 0.0 -0.5 -1.0 -1.0
-0.5
0.0 H (Normalized)
0.5
1.0
It can be concluded that the proposed implementation of the vector Preisach-type model can lead to good simulation results of M–H and l–H curves under different stress values. Moreover, this approach leads to a significant reduction in computational resources required while facilitating the model identification procedure. This is because, according to this efficient implementation, the vector Preisach-type model can be constructed using only two scalar models whose elementary hysteresis operators are orthogonally interrelated. Quantitatively, the proposed model consumes almost 2/N of the computational resources and time required by a conventional vector Preisach-type model constructed by superimposing N scalar models having the same number of elementary hysteresis operators and whose orientations are equally distributed along all possible 2D directions. Moreover, the identification problem for such an implementation may be carried out in an automated manner using well-established neural network algorithms and for any available set of experimental data. It is believed that this type of implementation may have wide applications in other coupled modeling areas. It is also believed that further enhancement of this implementation is worth investigating in the future.
Fig. 4. Comparison between measured and computed l–H limiting curves for a normalized stress value of 0.5.
References corresponding computations were performed and compared to corresponding measurements for a normalized stress value of 0.5. Such a comparison is shown in Figs. 3 and 4 and demonstrates good qualitative match between measured and computed curves. It should be pointed out that results of Fig. 4 may be significantly improved by increasing the number of elementary hysteresis operators comprising each scalar model.
[1] A.A. Adly, I.D. Mayergoyz, IEEE Trans. Magn. 32 (1996) 4773. [2] D. Davino, C. Natale, S. Pirozzi, C. Visone, J. Magn. Magn. Mater. 272–276 (2004) 781. [3] A.A. Adly, D. Davino, C. Visone, Physica B 372 (2006) 207. [4] A.A. Adly, S.K. Abd-El-Hafiz, J. Magn. Magn. Mater. 263 (2003) 301. [5] A.A. Adly, S.K. Abd-El-Hafiz, IEEE Trans. Magn. 42 (2006) 1518. [6] S. Haykin, Neural Networks: A Comprehensive Foundation, PrenticeHall, New Jersey, 1999. [7] I.D. Mayergoyz, Mathematical Models of Hysteresis, Springer, New York, 1991.