VECTOR HYSTERESIS MODEL BASED ON NEURAL NETWORK

Transcription

1 Kuczmann, Ivány, VECTOR HYSTERESIS MODEL BASED ON NEURAL NETWORK 1 VECTOR HYSTERESIS MODEL BASED ON NEURAL NETWORK Mlós KUCZMANN, Amála IVÁNYI Budapest Unversty of Technology and Economcs Department of Electromagnetc Theory H-151 Budapest, Hungary (+36 1) , (+36 1) , uczman@evtsz1.evt.bme.hu Abstract- On the bass of the Kolmogorov-Arnold theory the feedforward type artfcal neural networs are able to approxmate any nd of nonlnear, contnuous functons represented by ts dscrete set of measurements. A neural networ (NN) based scalar hysteress model has been constructed prelmnarly on the functon approxmaton ablty of NNs. An f-then type nowledge-base represents the propertes of the hysteress characterstcs. Vectoral generalzaton and to descrbe sotropc and ansotropc magnetc materals n two and three dmensons wth an orgnal dentfcaton method has been ntroduced n ths paper. Index Terms-- hysteress characterstcs, Everett surface, vector hysteress, neural networs. 1. Introducton Smulaton of hysteress characterstcs of magnetc materals needs to be mplemented nto electromagnetc feld smulaton software tools to predct the behavor of dfferent types of magnetc equpments. Hysteress characterstcs of magnetc materals can be descrbed by a nonlnear, multvalued relaton between the magnetc feld ntensty and the magnetzaton H t M t. Many assumptons and hypotheses have been developed snce the last perod of magnetc research, as the classcal Presach model (Ivány, 1997)(Füz, 000)(Mayergoyz, 1991), the Jles-Atherton model (Ivány, 1997), the Stoner-Wohlfarth

2 Kuczmann, Ivány, VECTOR HYSTERESIS MODEL BASED ON NEURAL NETWORK model (Ivány, 1997) and some new approaches based on NNs (Serpco, 1998)(Adly, 1998)(Adly, 000)(Kuczmann, 001)(Kuczmann, 00a)( Kuczmann, 00b)( Kuczmann, 00c). A mathematcal hysteress operator wth contnuous output bult on the functon approxmaton ablty of feedforward type NNs and ts vectoral generalzaton for sotropc and ansotropc magnetc materals n two and three dmensons have been nvestgated. The applcablty of the developed method s llustrated n the fgures.. Neural Networs for Functon Approxmaton NNs are parallel nformatonprocessng systems, mplemented by hardware or software (Chrstodoulou, 001). Usng the technque of NNs s a qute new and very attractve calculaton method, moreover a powerful mathematcal tool. One of the wde pallets of applcatons of NNs s the functon approxmaton, based on the theorem of Kolmogorov- Arnold, can be represented by a feedforward type NN wth at least two hdden layers wth nonlnear actvaton functons. Feedforward type NNs consst of elementary processng elements (neurons) collected nto layers. The output of an ndvdual neuron y s the output of a nonlnear, dfferentable actvaton functon of the nput values x, the neuron, moreover s w T x y ψs, where s s the lnear combnaton b, where w and b are the weght coeffcents and the bas of ψ s typcally the bpolar sgmod actvaton functon. Neurons n the th j layer are connected nto the neurons n the th j 1 layer by weghts W. Selectng the transfer functon of the ndvdual neurons, the only one degree of freedom left s the settng of weghts W, determned by a convergent, teratve algorthm, called tranng process to mnmze a sutably defned error (as mean square error, MSE, sum squared error, SSE, etc.) between the desred output value measured on a real plant and the answer of the networ. Tranng patterns can be collected as nput-output pars n the general form N x,t, 1,..., N, where f t and N s the number of measured ponts. x 3. The Scalar Hysteress Model The developed NN model of scalar hysteress characterstcs conssts of a system of two feedforward type NNs wth bpolar sgmod transfer functons and a heurstc f-then type nowledge-base about the hysteress phenomena (Kuczmann, 001)(Kuczmann, 00a)( Kuczmann, 00b)( Kuczmann, 00c). Let us suppose that the vrgn curve and a set of the frst order reversal branches are avalable. It s enough to use only the descendng (or ascendng) branches, because of the symmetry of hysteress characterstcs. Hysteress curves

3 Kuczmann, Ivány, VECTOR HYSTERESIS MODEL BASED ON NEURAL NETWORK 3 are normalzed wth the nducton value (or magnetzaton value) n saturaton state B s (or M s ) and the approprate magnetc feld ntensty H m. Hysteress characterstcs s a multvalued functon, t results n dffcultes when usng feedforward type NNs. If a new dmenson s ntroduced to the measured and normalzed transton curves, multvalued functon can be represented by a sngle-valued surface. The downgrade part of hysteress characterstcs can be descrbed at a turnng pont H tp wth a real parameter, determned as 1 Htp. The effect of ths preprocessng technque can be seen n Fg.1 for measured descendng curves. After preprocessng, functon approxmaton can be wored out by feedforward type NNs traned by the Levenberg-Marquardt bacpropagaton method (Chrstodoulou, 001). The anhysteretc curve wth 41 tranng ponts can be approxmated by a NN wth 8 neurons n one hdden layer, and the preprocessed frst order reversal branches (about 500 data pars) are approached by a NN wth 7, 11 and 6 processng elements n 3 hdden layers. Tranng of NNs taes about twenty mnutes ( MSE ) on a Celeron 566MHz computer (19Mbyte RAM), usng the Neural Networ Toolbox of MATLAB. TAKE IN FIG.1. Applyng NNs, relatonshp between magnetzaton M and magnetc feld ntensty H can be performed n analytcal formula, M H H. Memory mechansm of magnetc materals s realzed by an addtonal algorthm based on heurstcs. It s the nowledge-base for the propertes of hysteress phenomena. Magnetzaton at a smulaton step responsed by the NN s constructed on the actual value of the magnetc feld ntensty, the approprate value of parameter and the set of turnng ponts. Turnng ponts n the ascendng and descendng branches are stored n the memory as H tp M tp T,,. It s a mathematcal representaton of the prehystory of a magnetc materal and denoted by MATRIX. Turnng ponts can be detected by the evaluaton of a sequence of H, 1 H tp H 1, generated by a tapped delay lne (TDL). After detectng a turnng pont H and storng t n the memory, the am s to select an approprate transton curve for the detected turnng pont calculated by the regula fals method. Condtons are collected n the selecton rules, to choose the sutable NN. After detectng a turnng pont, generally denoted by desc MATRIX ( H H, the algorthm for mnor loops can be summarzed as follows. If START tp MATRIX asc ) has more columns and magnetc feld ntensty s ncreasng (decreasng) at a smulaton step, the actual mnor loop must be closed at the last stored value

4 Kuczmann, Ivány, VECTOR HYSTERESIS MODEL BASED ON NEURAL NETWORK 4 of desc GOAL H tp asc H ( H ) whch can be found n the last column of GOAL H tp MATRIX desc ( MATRIX asc normalzed magnetzaton desc GOAL M tp ). Denote ths column of the correspondng MATRIX wth C. The value of asc M ( M ) n the GOAL M tp M responsed by the neural model at H GOAL must be equal to th C column of the accordng MATRIX. It s the condton for closng the mnor loops, can be satsfed by the correcton functon NN H, M, where H, M M M and decreasng lnearly. After START START GOAL closng a mnor loop, the approprate columns of MATRIX must be erased. The bloc representaton of the model can be seen n Fg.. TAKE IN FIG.. The expermented NN model of hysteress can be used as a mathematcal, contnuous scalar model to smulate the behavor of magnetc materals. Hysteress characterstcs predcted by the developed model have been compared wth measurements as plotted n Fg.3. Accommodaton property also can be smulated, when H H M 1 s appled as nput of the model, where s the movng parameter. The dfferental GOAL susceptblty, dff dm dh can be performed n analytcal, closed form by the chan rule when applyng NNs, dm dh H dh dh. TAKE IN FIG The Isotropc Vector Model The vector NN model s constructed as a superposton of scalar NN models n gven drectons e (Mayergoyz, 1991)(Ragusa, 000). The magnetzaton vector can be expressed n two dmensons as t e H H M d, (1) where M H s the magnetzaton n the drecton H e, H cos H H and H s the drecton of magnetc feld ntensty vector H. The functons H H depend on the polar angle f the magnetc materal presents ansotropy, otherwse t s -ndependent. In computer realzaton t s useful to dscretze the nterval, n 1, where 1,..., n (Fg.4) and n s the number of drectons. In three dmensons a smlar expresson can be obtaned, as

5 Kuczmann, Ivány, VECTOR HYSTERESIS MODEL BASED ON NEURAL NETWORK 5 t e, H H, M d d, () where T H a a a H H H and the drectons are gven as, 1 3, x y z a a, a 1, generated by the cosahedron (Fg.5). The angles and are 1e1 ae a3e3 measured from the x-axs and the x-y plane. TAKE IN FIG.4 TAKE IN FIG.5. After measurng the Everett surface n the x drecton, the followng condton can be obtaned between the measured scalar Everett surface F, and the unnown vector Everett functon, E : n, cos cos, cos F. (3) 1 E In sotropc case the vector Everett surface s unque for all drectons. Expresson (3) can be solved numercally and can be rewrtten n the form F n1, l cos E cos, cos 1 1 l n 1 cos E cos, cos, l (4) where N N, N Everett table s 1 N 1 l N,, l 1,..., N 1 and the sze of l N. Expresson (4) contans n 1 nown and n unnown ponts n the Everett surface as llustrated n Fg.6. The frst sum contans nown values of the Everett functon, whle the second one contans the unnown value of the vector Everett surface. TAKE IN FIG.6. the F If 0, 1,..., N 1 l l and assumng lnear nterpolaton n the surface, n1,0 cos E 1,0 E,0 E 1,0 cos j j j 1 j j j 1 1 E E,,01 b c a c E,0a c b c (5) formulaton can be got, where c cos, 1 1 c cos 1, and a, b 1 and n j. From (5), value of,0 n E can be

6 Kuczmann, Ivány, VECTOR HYSTERESIS MODEL BASED ON NEURAL NETWORK 6 expressed. If l 0, a smlar mathematcal formulaton can be obtaned. Frstly, let us assume that the pont cos, cos 1 l s bounded by nown ponts A x 1 1, y1, z1, B x, y z and x, y z, C n the vector Everett surface. The value of the Everett surface 3 3, 3 n ths coordnate can be expressed usng lnear nterpolaton n the gven trangle A, B, C. Unnown values can be expressed after some smple mathematcal formulatons. Because of symmetry of the hysteress characterstcs, t s enough to calculate the half of the Everett surface. The frst order reversal curves of vector NN model can be calculated from the dentfed vector Everett surface. The reversal curves can be approxmated by the scalar NN model. Let us assume that, measured hysteress curve and the correspondng Everett surface s gven n the x drecton. Smulaton results for reversal curves obtaned from the dentfed Everett surface n two dmensons (0 drectons) and three dmensons (4 drectons) can be seen n Fg.7 and n Fg.8. Expermental results are denoted by ponts, hysteress characterstcs gven by the NN model s denoted by dashed lne. TAKE IN FIG.7 TAKE IN FIG The Ansotropc Vector Model In ansotropc case the scalar Everett surface s dependng on the drecton (Ragusa, 000). It s dffcult to tae nto account the angular dependence of the Everett surface n the dentfcaton tas, therefore the Fourer expanson has been appled, so the same dentfcaton procedure can be used as n sotropc case, because the Fourer seres are not dependng on the angle. In two dmensons, the Everett surface F,, -perodc wth respect to and an even functon, F F F s f 0 and represent the easy axs and the hard axs respectvely. Let us assume that the Everett functons are avalable n n gven drectons n the nterval 0, from measurements. The angular dependence of the Everett surface can be handled by the Fourer expanson as F C cos, (6)

7 Kuczmann, Ivány, VECTOR HYSTERESIS MODEL BASED ON NEURAL NETWORK 7 where functon, C s the th harmonc component, whch s ndependent on the C angle. Usng the trapezodal formula for ntegraton, the Fourer components can be calculated as and C C n 1 F0 F F 0 1 where n 1 1 F 0 F 1 F cos n for the Fourer harmoncs, C n 1 1 (7) (8). The same dentfcaton process can be appled as n sotropc case, cos cos E cos, cos, (9) where 1 n and n 1 n s the number of drectons. In ths study two angular harmoncs ( 0, 1) have been used. In three dmensons a smlar process can be appled. Assumng the same condtons as n D model, the Everett surface, F,,, Fourer expanson accordng to and n the form F C cosm cosn n m mn F also can be represented by a,, (10) where functon, and C s the harmonc component, can be calculated as mn C mn 4 C 00 F, d d (11) C, cos cos mn F m n d d, (1) 0 0 where 0,, 0,, M 1 and N 1, M N measurements must be used. The dentfcaton process must be appled for the followng expresson: C mn n 1, cos cosm cosn E, (13) mn

8 Kuczmann, Ivány, VECTOR HYSTERESIS MODEL BASED ON NEURAL NETWORK 8 where E cos, cos mn E mn, n s the number of drectons, generated by the cosahedron and s the angle between the th drecton gven by the cosahedron and the x- axs. In ths study two angular harmoncs accordng to the angles and ( m, n 0, 1) have been used. The ntroduced methods have been appled to smulate ansotropc magnetc materals. In two dmensons ten dfferent scalar hysteress characterstcs were avalable, generated by the ellptcal nterpolaton functon F F x cos F sn, (14) y where F F,, and F,,,,, x F x F are the Everett surfaces n y F y the rollng and transverse drectons and 0,. In three dmensons 5 5 scalar hysteress characterstcs have been assumed. In practce, these data sets can be replaced by measurements. Smulaton results for reversal curves obtaned from the dentfed Everett surface n two dmensons (18 drectons) and three dmensons (9 drectons) can be seen n Fg.9. a, b and Fg.10. a, b, and c (trval condtons n the drectons of x, y and z axes). TAKE IN FIG.9 TAKE IN FIG Some Propertes of the Vector Model Applyng rotatonal magnetc feld ntensty wth gven ampltude and wth lnearly ncreasng ampltude, output of the D vector NN sotropc and ansotropc model has been plotted n Fg.11 and n Fg.1. The specmen s magnetzed to a gven value n the rollng drecton, and then the magnetc feld ntensty s rotated eepng ts magntude constant. In Fg.1, the vector of magnetzaton gradually approaches the regme of unform rotaton. TAKE IN FIG.11 TAKE IN FIG.1 TAKE IN FIG.13 The poston of the vector of magnetzaton for a lnear exctaton ( H 60 ansotropc case can be seen n Fg.13.a and ts projectons n the easy and n the hard axs n Fg.13.b and c respectvely. ) n

9 Kuczmann, Ivány, VECTOR HYSTERESIS MODEL BASED ON NEURAL NETWORK 9 7. Conclusons A NN model for magnetc hysteress based on the functon approxmaton ablty of NNs has been expermented. The anhysteretc magnetzaton curve and a set of the frst order reversal branches must be measured on a magnetc materal. Introducng an addtonal parameter solves a fundamental problem of smulatng hysteress characterstcs, that s the multvalued property. The magnetzaton becomes a sngle valued functon of two varables and an f-then type nowledge-base can be used for smulatng dfferent phenomena of magnetc materals. Ths method has been generalzed n two and three dmensons wth an orgnal dentfcaton process. Am of further research s to develop the measurements of hysteress characterstcs n two dmensons and comparng the smulaton technque wth these results. References Adly, A. A. Abd-El-Hazf, S. K. (1998) Usng Neural Networs n the Identfcaton of Presach-Type Hysteress Models, IEEE Trans. on Magn., vol.34, pp Adly, A. A. Abd-El-Hafz, S. K. Mayergoyz, I. D. (000) Identfcaton of Vector Presach Models from Arbtrary Measured Data Usng Neural Networs, Journal of Appled Physcs, vol.87, No.9, pp Chrstodoulou, C. Georgopoulos, M. (001) Applcatons of Neural Networs n Electromagnetcs, Artech House, Norwood. Füz, J. (000) Ansotropc Vector Presach Partcle, Journal of Magnetsm and Magnetc Materals, vol , 000, pp Ivany, A. (1997), Hysteress Models n Electromagnetc Computaton, Aadéma Kadó, Budapest. Kuczmann, M. Ivány, A. (001) Vector Neural Networ Hysteress Model, Physca B, vol.306, pp Kuczmann, M. Ivány, A. (00a) A New Neural-Networ-Based Scalar Hysteress Model, IEEE Trans. on Magn., vol.38, no., pp Kuczmann, M. Ivány, A. (00b) Neural Networ Model of Magnetc Hysteress, Compel, vol.1, no.3, pp Kuczmann, M. Ivány, A. (00c) Neural Networ Based Vector Hysteress Operator, Proceedngs of the Tenth Bennal IEEE CEFC, Peruga, Italy, June 16-19, p.59. Mayergoyz, I. D. (1991) Mathematcal Models of Hysteress, Sprnger. Ragusa, C. Repetto, M. (000) Accurate Analyss of Magnetc Devces wth Ansotropc Vector Hysteress, Physca B 75, pp Serpco, C. Vsone, C. (1998) Magnetc Hysteress Modelng va Feed-Forward Neural Networs, IEEE Trans. on Magn., vol.34, pp Acnowledgement: The research wor s sponsored by the Hungaran Scentfc Research Fund, OTKA 00, Pr. No. T /00 and measurement system s sponsored by OMFB-0047/001.

10 Kuczmann, Ivány, VECTOR HYSTERESIS MODEL BASED ON NEURAL NETWORK 10 Fg. 1. Measured and preprocessed frst order reversal curves

11 Kuczmann, Ivány, VECTOR HYSTERESIS MODEL BASED ON NEURAL NETWORK 11 Fg.. Bloc representaton of the scalar model

12 Kuczmann, Ivány, VECTOR HYSTERESIS MODEL BASED ON NEURAL NETWORK 1 Fg.3. Comparsons wth measurements

13 Kuczmann, Ivány, VECTOR HYSTERESIS MODEL BASED ON NEURAL NETWORK 13 Fg.4. Defnton of drectons n two dmensons

14 Kuczmann, Ivány, VECTOR HYSTERESIS MODEL BASED ON NEURAL NETWORK 14 Fg.5. Icosahedron to generate the drectons of the 3D model

15 Kuczmann, Ivány, VECTOR HYSTERESIS MODEL BASED ON NEURAL NETWORK 15 Fg.6. Illustraton for the calculaton of the Everett surface

16 Kuczmann, Ivány, VECTOR HYSTERESIS MODEL BASED ON NEURAL NETWORK 16 Fg.7. Identfcaton result n two dmensons Fg.8. Identfcaton result n three dmensons

17 Kuczmann, Ivány, VECTOR HYSTERESIS MODEL BASED ON NEURAL NETWORK 17 (a) (b) Fg. 9. Identfcaton result n two dmensons

18 Kuczmann, Ivány, VECTOR HYSTERESIS MODEL BASED ON NEURAL NETWORK 18 (a) (b) (c) Fg. 10. Identfcaton result n three dmensons

19 Kuczmann, Ivány, VECTOR HYSTERESIS MODEL BASED ON NEURAL NETWORK 19 Fg.11. Smulated H and M loc for sotropc case

20 Kuczmann, Ivány, VECTOR HYSTERESIS MODEL BASED ON NEURAL NETWORK 0 Fg.1. Smulated M loc for ansotropc case

21 Kuczmann, Ivány, VECTOR HYSTERESIS MODEL BASED ON NEURAL NETWORK 1 (a) (b) (c) Fg.13. Ansotropc magnetc materal n lnear exctaton