Journal of Magnetism and Magnetic Materials 266}230 (2001) 1262}1264
Modeling a magnetostrictive transducer using genetic algorithm L.A.L. Almeida , G.S. Deep *, A.M.N. Lima, H. Ne! Departamento de Engenharia Ele& trica, da Universidade Federal da Bahia, Brazil Departamento de Engenharia Ele& trica, Universidade Federal da Parai! ba, Caixa Postal 10.004, 58109-970 Campina Grande, PB, Brazil
Abstract This work reports on the applicability of the genetic algorithm (GA) to the problem of parameter determination of magnetostrictive transducers. A combination of the Jiles}Atherton hysteresis model with a quadratic moment rotation model is simulated using known parameters of a sensor. The simulated sensor data are then used as input data for the GA parameter calculation method. Taking the previously known parameters, the accuracy of the GA parameter calculation method can be evaluated. 2001 Elsevier Science B.V. All rights reserved. Keywords: Magnetostriction; Magnetic hysteresis; Magnetization models
Modeling of magnetostrictive transducers requires a precise characterization of the relation between input current and resulted strain. Commonly, it is based upon a combination of the Jiles}Atherton hysteresis model with a quadratic moment rotation model [1]. To determine parameters of the ferromagnetic hysteresis, Jiles derived several equations presented as an inversion algorithm [2] that uses previously selected slopes and intercepts obtained from experimental data. Another approach from Calkins [1] applied nonlinear optimization to "nd the parameters of the Jiles}Atherton hysteresis model in magnetostrictive transducers, and the parameters were estimated, applying a constrained optimization based upon sequential quadratic programming [4]. In the inversion algorithm proposed by Jiles, some of the equations needed for determining the parameters are implicit and convergence problems may occur. Also, there are some di$culties associated with the presence of noise in the experimental data. The absence of an explicit regression procedure for curve "tting makes the solution
* Corresponding author. Tel.: #55-83-310-1146; fax: #5583-310-1015. E-mail address:
[email protected] (G.S. Deep).
very sensitive for measurement noise and to small changes in the recorded magnetic data. Additionally, a better understanding of physical e!ects of the parameters is necessary to establish the `"xed reference pointsa [2] and achieve the complete identi"cation process. Subsequently to the inversion algorithm proposed by Jiles, the magnetostriction parameter needs to be determined. The optimization procedure employed by Calkins was implemented in two steps: a sequential quadratic programming followed by a least-squares "tting [1]. Some discrepancy between simulation and experimental data were observed due to a disjoint two-step identi"cation procedure. These e!ects are common in strongly nonlinear problems, causing either unsuitable or impractical solutions when using classical nonlinear optimization methods [5]. Evolutionary computation techniques can cope with situations, where the objective function and constraints are not analytically tractable, or are not given in closed form [6]. Genetic algorithm (GA), introduced by Holland and well described by Goldberg [6] is by far the most employed method of evolutionary computation to deal with parameter optimization problems. In contrast to the methods described above, GA is able to solve the problem of parameter identi"cation of the combined magnetostrictive transducer model referred to above with a single integrated procedure. As a globally convergent
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L.A.L. Almeida et al. / Journal of Magnetism and Magnetic Materials 226}230 (2001) 1262}1264
method, GA can jointly estimate the parameters with minimal user intervention during the identi"cation process. The GA robustness eliminates the possibility of the algorithm to be trapped in a local minimum. Also, unfeasible solutions can be destroyed during evolution process. In contrast to this, the nonlinear parameter optimization procedures usually cannot continue optimization. GA is a domain-independent search and parameter optimization algorithm, based on the concept of natural selection and inheritance of parental genetic information [6]. It searches for an optimal solution K , by manipula% ting a population of N di!erent solutions candidates "[ 2 ]. Each solution candidate , G individually corresponding to a sample point from the search space. In a similar way to chromosome structure, individuals or solutions are represented as "xed-length binary strings over the alphabet 0,1, with bits representing natural genes. The standard implementation of a genetic algorithm utilizes three basic operators: reproduction, crossover and mutation. The algorithm starts with an initial population that is generated randomly but restricted to D -RN where p"dim( ) and D repG resents the feasible search subspace. In the case of realvalued parameters, the binary string representing solution candidate is divided into segments being a binary G representation of each parameter. Next step is to compute "tness function J( ) for each individual. The reproG duction operator assigns to each individual a reproduction probability that is proportional to G J( ). The candidates for a new population are selected G based on this probability, allowing that the "t individuals have higher number of o!spring in the succeeding generation. This cycle continues, until GA reaches a maximum number of generations or a minimum J( ) is G found. The search properties may be controlled by changing the probabilities of crossover and mutation. In the Jiles}Atherton model [3] the e!ective "eld H (t) of a device excited by a current source nI(t) and with domain interactions is de"ned as H (t)"H(t)#M(t) with H(t)"nI(t). The total magnetization M(t) can be decomposed into two di!erent components. The component of anhysteretic magnetization can be expressed by
M (t)"M coth
H (t) a ! , a H (t)
where a is a shape parameter and M is the saturation magnetization. The energy lost to pinning is expressed as a function of the irreversible change in magnetization leading to the equation for the di!erential irreversible susceptibility
dM dI M (t)!M (t) "n , dt dt k!(M (t)!M (t))
(1)
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Fig. 1. Comparison between the `measureda strain and the calculated strain.
where k is the average energy required to break pinning sites depending on "eld direction and "#1 if dH/dt'0 or "!1 if dH/dt(0. The component of reversible magnetization M (t)"c(M (t)!M (t)), where c is the reversibility coe$cient, can then be used to de"ne the total magnetization as M(t)"M (t)#M (t) (2) and, consequently, the magnetostriction "¸/¸ can be de"ned as 3 (t)" M(t) (3) 2M with representing the saturation magnetostriction [3]. The parameter optimization problem consists in determining a vector of six parameters " [M c k a ]2 that minimizes the "tness function given by , J()" ((t )!K (t ,)), (4) G G G where (t ) and K (t ,) represent the measured and estiG G mated magnetostriction, respectively. The optimal parameter vector is obtained solving K , " arg min J(). % FZDF ER In the simulation study, the `measureda values of the magnetostriction have been calculated using the previous equations. These values were supplied to the GA which provides K , that is constrained to the physically pos% sible region D . In this case, the binary string representF ing solution candidate is divided into six segments of G equal length, each segment being a binary representation of each component of . In the simulation study the `truea values of the strain data as well the model parameters, that are given in
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L.A.L. Almeida et al. / Journal of Magnetism and Magnetic Materials 226}230 (2001) 1262}1264
Table 1 Performance of the genetic algorithm.
a b c d
IP
NG
M
c
k
a
J()
} 10 20 20
} 15 20 50
1.00 1.28 0.92 0.82
0.092 0.108 0.092 0.104
0.0015 0.0018 0.0016 0.0015
0.130 0.133 0.129 0.131
0.095 0.099 0.101 0.095
0.830 0.844 0.851 0.825
} 0.048 0.029 0.021
Table 1, are known and the accuracy of parameter optimization procedure can be easily assessed. The `measureda data used for testing the GA were obtained from a simulated curve (t);H(t). Wherever required, the numerical integrations were implemented with the fourthorder Runge}Kutta method. Table 1 shows the results obtained with the GA. Row a of Table 1 provides the `truea values for comparison purposes. The columns tagged with IP and NG stand, respectively, for the size of the initial population and the number of generations used to obtain the parameters given in the columns M to . Also, the resulting cost function, is provided in the column J(). Note that the accuracy of parameters increases when the size of the initial population and the number of generations increase. However, as illustrated in row d of Table 1 even with a relatively small cost function the error in some parameters can be as high as 18%. Fig. 1 shows the comparison between the `measureda strain and the calculated strain using the parameters obtained with the GA (row d of Table 1). The inset of Fig. 1 shows that the maximum di!erence between these two curves is at most 0.5%.
The results obtained in this simulation study have demonstrated that it is feasible to use GA to solve the parameter optimization of the combined Jiles}Atherton model. Furthermore, the use of GA provides an integrated procedure that requires minimal user intervention to provide the solution.
References [1] F.T. Calkins, R.C. Smith, A.B. Flatau, An energy-based hysteresis model for magnetostrictive transducers, NASAICASE Report No. 97-60, November 1997. [2] D.C. Jiles, J.B. Thoelke, M.K. Devine, IEEE Trans. Magn. 28 (1992) 27. [3] D.C. Jiles, Introduction to Magnetism and Magnetic Materials, Chapman & Hall, New York, 1991. [4] D.G. Luenberger, Linear, and Nonlinear Programming, Addison-Wesley, Reading, MA, 1989. [5] J.-H. Kim, H. Myung, IEEE Trans. Evol. Comput 1 (2) (1997) 3. [6] D.E. Goldberg, Genetic Algorithm in Search, Optimization and Machine Learning, Addison-Wesley, Reading MA, 1989.
