ARTICLE IN PRESS

Neurocomputing 70 (2007) 749–761 www.elsevier.com/locate/neucom

Hybrid genetic algorithms for parameter identification of a hysteresis model of magnetostrictive actuators Jiaju Zhenga,, Shuying Caob, Hongli Wanga, Wenmei Huangb a

School of Mechanical Engineering, Tianjin University, Tianjin 300072, China School of Electrical Engineering and Automation, Hebei University of Technology, Tianjin 300130, China

b

Available online 14 October 2006

Abstract In this paper, we present an improved hysteresis model for magnetostrictive actuators. To obtain optimal parameters of the model, we study two distinct hybrid strategies: namely, employing a gradient algorithm as a local search operation of a genetic algorithm (GA), and taking the best individual of a GA as the initial value of a gradient algorithm. Here, two different gradient algorithms, a well-known Levenberg–Marquardt algorithm (LMA) and a novel Trust-Region algorithm (TRA), are investigated. Finally, the proposed four hybrid genetic algorithms (HGAs) are applied to identify parameters of the improved model. The simulation and experimental results show the performances of the HGAs and the improved hysteresis model. r 2006 Elsevier B.V. All rights reserved. Keywords: Hybrid genetic algorithms; Hysteresis; Magnetostrictive actuator; Parameter identification

1. Introduction Magnetostriction is the phenomenon of strong coupling between magnetic properties and mechanical properties of magnetostrictive materials: strains are generated in response to an applied magnetic field, whereas mechanical stresses in the materials produce measurable changes in magnetization. This phenomenon can be used for actuation and sensing. Magnetostrictive actuators made of giant magnetostrictive materials (e.g. Terfenol-D), with large strain, high force, fast speed response and nanometer solution, have a wide range of potential applications in the micro-positioning, robotics, ultrasonics and vibration control, etc. However, like piezoelectric and shape memory alloy actuators, magnetostrictive actuators display dominant hysteresis. To illustrate, we consider the prototypical magnetostrictive actuator depicted in Fig. 1 and detailed in [5,8,9]. By varying the current in the solenoid, the variable magnetic field in the Terfenol-D rod is generated, and then Corresponding author.

E-mail addresses: [email protected] (J. Zheng), [email protected] (S. Cao). 0925-2312/$ - see front matter r 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.neucom.2006.10.010

the strain and force are provided by the Terfenol-D rod in response to the magnetic field. To attain bidirectional strain and force, a DC bias magnetic field is provided either by a surrounding permanent magnet or through a prescribed DC current to the solenoid. The prestress bolt and the spring washer maintain the rod in a state of compression. It is noted that the magnetostrictive actuators adopted in [7,24,31,32] are similar to the actuator depicted in Fig. 1. As detailed in [5,7–9,24,31,32], the actuators exhibit dominant hysteresis and nonlinearitys in the relationship between the input magnetic field and the generated strain. Moreover, the hysteresis behavior of the actuators at low frequencies is frequency independent: roughly speaking, the shape of the hysteresis loop does not depend on the frequency of the input magnetic field. This is no longer the case when the operating frequency gets high, due to the dynamic effects such as eddy currents loss, anomalous loss and structural dynamic mechanical behavior of the actuators. This fact is studied in [9], where the frequencydependent dynamic hysteresis of an actuator is investigated by measuring the hysteresis loops in a wide frequency range (10–2000 Hz). The experimental results in [31,32] also show the frequency-dependent hysteresis characteristics of magnetostrictive actuators. Such hysteresis characteristics must

ARTICLE IN PRESS J. Zheng et al. / Neurocomputing 70 (2007) 749–761

750

Solenoid

Terfenol-D Rod

Spring Washer Push Rod End Mass

Motion Output Prestress Bolt

Permanent Magnet

Fig. 1. Schematic diagram of Terfenol-D magnetostrictive actuator.

severely limit the actuator system’s performances such as undesirable position error, oscillation or instability. To analyze, control and use the actuators, it is necessary to establish an accurate hysteresis model for the actuators. Hysteresis models for magnetostrictive actuators can be roughly classified into Preisach models [1,24,26,31] and energy-based models [5,8,32]. Preisach models approach hysteresis through expansions based on a variety of multivalued kernels and have the advantage of generality. However, Preisach models are developed based on the firstorder reversal curves that are measured under the fixed operating conditions (e.g. drive frequency, mechanical prestress and external load), and require a large number of nonphysical parameters to be identified, thus the models are not easily adapted to the changing operating conditions. It is obvious that the accuracy of Preisach model deteriorates due to the changing frequency [31]. The energy-based models are established based on the static Jiles–Atherton (JA) domain wall hysteresis model for ferromagnetic materials [12–14]. The JA model is loworder and has a small number of physical parameters, and can generally satisfy some modeling requirements including accuracy, speed and good convergence [25]. Moreover, the JA model is sufficiently general to encompass ferroelectric and piezoelectric materials [28,30] and ferroelastic compounds [20] so it provides a unified framework for quantifying hysteresis and constitutive nonlinearities in a broad range of ferric compounds [29]. However, at higher frequencies, the energy-based models [5,8,32] and Preisach models [1,24,26,31] must badly degenerate due to the unmodeled frequency-dependent dynamic effects such as eddy current and anomalous losses. In this paper, we adopt JA model to establish an energy-based hysteresis model of magnetostrictive actuators, and identify the parameters of the model for generalization to dynamic hysteresis loops. A lot of researches have shown that the parameters of the JA model are very difficult to be determined because they are strongly interrelated nonlinear parameters. Jiles et al. [14] derive several equations to obtain parameters of the JA model. Because the equations are implicit, convergence problems may occur. Moreover, a better understanding of physical effects of the parameters is necessary to establish the ‘‘fixed reference points’’ [14] and achieve the complete identification process. Calkins et al. [5] adopt the JA model

and the magnetostriction model to establish a quasi-static energy-based hysteresis model for magnetostrictive actuators, and optimize parameters of the model in two steps: a sequential quadratic programming followed by a leastsquares fitting. Some discrepancies between the calculated and experimental data can be caused due to a disjointed two-step identification procedure. Recently, researchers [2,16,33] have applied random optimization methods, such as simulated annealing method and genetic algorithm (GA), to solve the problem of parameter identification of the energy-based hysteresis models. Wilson et al. [33] use a binary coded GA and a simulated annealing method to determine the parameters of the JA model, and the statistical analysis shows that the result obtained by the GA is better on average than the one obtained by the simulated annealing method. Leite et al. [16] adopt a real coded GA to improve the precision of parameter identification of the JA model. Almeida et al. [2] show that the GA can automatically obtain the parameters of the quasi-static hysteresis model [5] with a single integrated procedure. However, GA’s disadvantages, such as premature convergence and calculation inefficiency, have also been shown in [2,33]. For instance, the simulation results in [2] show that the error in some parameters is as high as 18% after the GA long time calculation. To overcome the disadvantages of the pure GA, researchers [17,18,27] have begun to consider different combinations of the GA and other optimization algorithms such as the Newton–Raphson, tabu search, and neural network. In this paper, we establish an improved hysteresis model for magnetostrictive actuators by extending the quasi-static hysteresis model [5], and propose four hybrid genetic algorithms (HGAs) to identify parameters of the improved model. More specifically, the improved hysteresis model is established based on the quasi-static hysteresis model [5] and the structural dynamic mechanical model of magnetostrictive actuators. Four HGAs adopt two distinct hybrid strategies: namely, employing a gradient algorithm as a local search operation of a GA, and taking the best individual of a GA as the initial value of a gradient algorithm. In this study, two different gradient algorithms, a well-known Levenberg–Marquardt algorithm (LMA) and a novel Trust-Region algorithm (TRA), are investigated. Finally, four HGAs are used to update parameters of the improved model at different frequencies of the input magnetic field. In this way, we can avoid complex experimental procedure described in [5,14] for static loops, and we can extend the improved model to dynamic hysteresis model by assuming that each frequency-dependent loop can be regarded as a static loop with different parameters. Also, we can avoid the computational costs to be incurred by using the finite element method to evaluate the eddy current and anomalous losses. This paper is organized as follows. Section 2 presents an improved hysteresis model of magnetostrictive actuators. In Section 3, four HGAs are studied and developed. Section 4 gives some simulation results to demonstrate the

ARTICLE IN PRESS J. Zheng et al. / Neurocomputing 70 (2007) 749–761

performances of four HGAs. Section 5 gives some experimental results to demonstrate the effectiveness of the HGAs and the improved hysteresis model. Finally, the conclusions are given in Section 6. 2. Hysteresis model of magnetostrictive actuators In this section, we develop an improved model for magnetostrictive actuators by extending the quasi-static hysteresis model [5]. The model is constructed in two steps. In the first step, the quasi-static hysteresis model is used to quantify the relationship among the input current, the generated magnetization and magnetostriction. In the second step, a structural dynamic mechanical model is used to quantify displacement and strain of actuators in response to the magnetostriction. 2.1. Magnetization and magnetostriction models The magnetization model in [5] is based upon the static JA hysteresis model for ferromagnetic materials [12–14]. The JA model is based on the quantification of energy losses due to domain wall intersections with inclusions or pinning sites in the material (the transition regions between domains are termed domain walls). For a material that is free from inclusions, the domain wall movement is reversible which leads to anhysteretic magnetization Man. However, materials such as Terfenol-D contain the second phase, which impedes domain wall movement. At low field levels, domain wall movement about pinning sites is reversible and yields a reversible magnetization Mrev. At higher drive levels, domain walls intersect remote pinning sites which provides an irreversible component Mirr. To characterize the total magnetization M in the Terfenol-D rod, we consider first the effective field He in the Terfenol-D rod. The effective field is given by ~ H e ¼ H þ aM,

(1)

where H ¼ HAC+Hb ¼ nI is the magnetic field generated by a solenoid having n turns per unit length with an input current I (Here HAC is AC magnetic field, Hb is DC bias magnetic field). The parameter a~ quantifies magnetic and stress interactions. Through thermodynamic considerations, the anhysteretic magnetization is then defined in terms of the Langevin function     He a M an ¼ M s coth , (2)  He a where Ms denotes the saturation magnetization of the Terfenol-D rod and a is a parameter which characterizes the shape of the anhysteretic curve. Energy balancing (see [13]) is then used to quantify the irreversible, reversible and total magnetizations through the expressions dM irr M an  M irr ¼ dM , ~ an  M irr Þ dH d1 k  aðM

(3)

751

M rev ¼ cðM an  M irr Þ,

(4)

M ¼ M irr þ M rev ,

(5)

where d1 takes the value 1 or 1 based on the sign of dH/ dt. k and c are, respectively, average energy of pinning sites and reversibility coefficient of the Terfenol-D rod. To suppress unphysical solutions (dM irr =dHo0), dM is defined as 8 _ > < 0 Ho0 and ðM an  MÞ40; _ dM ¼ 0 H40 and ðM an  MÞo0; > : 1 otherwise: The magnetostriction model in [5] is based upon quadratic model [12]. Because the magnetostriction l indicates the relative change in length of the Terfenol-D rod, it provides a measure of strains generated by the Terfenol-D rod. As detailed in [12], consideration of the potential energy for the system yields a quadratic model l¼

3 ls M2, 2 M 2s

(6)

where ls is the saturation magnetostriction of the Terfenol-D rod. 2.2. Structural dynamic mechanical model From the magnetostrictive actuator in Fig. 1, the mechanical load of Terfenol-D rod consists of the push rod, spring washer and end mass. Ar, lr, r, CD, E are, respectively, cross-sectional area, length, density, internal damping coefficient and nominal Young’s modulus of the Terfenol-D rod. Assume that the left end of the rod is fixed while the right end is constrained by the mechanical load, whose mass is Ml, stiffness is Kl and damping coefficient is Cl. Also, assume that stress s and strain e are spatially uniform, thus the rod output displacement is y ¼ elr and output force is F ¼ sAr ¼ ðM l y€ þ C l y_ þ K l yÞ.

(7)

The linear strain equation for the Terfenol-D rod is given as [6]  ¼ s=E þ dH,

(8)

where d is the magnetoelastic coupling coefficient. It is noted in (8) that the generated strains are dependent upon both the elastic properties of the material (modeled by s/E) and the magnetic inputs (modeled by dH). Considering hysteretic nonlinearity of the magnetic field, the term dH in (8) is replaced by the magnetostriction, thus we get the nonlinear strain equation  ¼ s=E þ l.

(9)

When the mass and damping of the Terfenol-D rod are considered, (9) becomes  ¼ s=E þ l  ðC D =EÞ_ 

rl 2r =E €. 3

(10)

ARTICLE IN PRESS J. Zheng et al. / Neurocomputing 70 (2007) 749–761

752

The combination of (7), (10) and y ¼ elr yields the strain dynamic equation M z € þ C z _ þ K z  ¼ Ar El=l r ,

(11)

where Mz ¼ rlrAr/3+Ml, Cz ¼ CDAr/lr+Cl, Kz ¼ ArE/lr+Kl. Mz, Cz, Kz are, respectively, the equivalent mass, damping and stiffness of the magnetostrictive actuator. Laplace transform for (11) produces displacement transfer function y ¼ l r ¼

1 ðAr ElÞ. M z s2 þ C z s þ K z

(12)

From (12), the frequency characteristic of strain versus magnetostriction is given as GðjwÞ ¼

 Ar E=l r ¼ AðwÞejjðwÞ ¼ , 2 l M z ðjwÞ þ C z ðjwÞ þ K z

(13)

where AðwÞ ¼ j=lj and jðwÞ ¼ ffð=lÞ are, respectively, the magnitude and phase frequency characteristics of strain versus magnetostriction. Eqs. (1)–(6) and (12) can be combined to yield displacement and strain of the magnetostrictive actuator in response to the input current I. 3. HGAs for parameter identification of hysteresis model In this section, we first present the parameter identification principle of the improved hysteresis model. Then, we describe three optimization algorithms: the LMA, TRA and GA. Further, we propose four HGAs by using the LMA, TRA and GA. 3.1. Parameter identification principle As detailed in [5,7–9,32], the Terfenol-D material properties and actuator performances are strongly coupled, and the Terfenol-D material properties vary greatly with the operating conditions of magnetostrictive actuators. Thus, for an actual magnetostrictive actuator under a given operating condition, we must estimate the unknown parameters of the improved model (1)–(6) and (12). In the model, the parameters to be estimated are   y ¼ M s ls a k a~ c E K z M z C z . (14) The values of y are estimated through minimization of the objective function EðyÞ ¼

Q Q X X ~ yÞÞ2 ¼ ðyðkÞ  yðk; e2 ðk; yÞ, k¼1

s:t:d r pyr pbr ;

eðyÞ ¼ ½ eð1; yÞ

ð15Þ

where k is the kth sample time, Q is the number of all ~ yÞ samples, y(k) is the measured (or true) displacement, yðk; ~ yÞ is the is the calculated displacement, eðk; yÞ ¼ yðkÞ  yðk; displacement error, yr is the rth element of the parameter vector y, dr and br are the lower and upper bounds on yr.

eð2; yÞ   

eðQ; yÞT .

(16)

The function (15) is minimized using the following HGAs. 3.2. Gradient algorithms Gradient algorithms have shown their effectiveness in many nonlinear optimization problems. To minimize the objective function E(y), many gradient algorithms adopt the quadratic function q(d). The quadratic function q(d) is defined by the first two terms of the Taylor approximation to the objective function E(y) at y Eðy þ dÞ  qðdÞ ¼ EðyÞ þ gT d þ 12dT Bd,

(17)

where g is the gradient of E(y), B is an approximation of the real Hessian matrix of E(y). For the objective function E(y) defined in (15), g and B can be written as g ¼ J T ðyÞeðyÞ,

(18)

B ¼ J T ðyÞJðyÞ þ SðyÞ, (19) PQ where SðyÞ ¼ k¼1 eðk; yÞBðk; yÞ, e(y) is the displacement error vector defined in (16), J(y) is the Jacobian matrix of e(y), i.e. 3 2 qeð1; yÞ qeð1; yÞ qeð1; yÞ  6 qy1 qy2 qy10 7 7 6 6 qeð2; yÞ qeð2; yÞ qeð2; yÞ 7 7 6  6 qy qy2 qy10 7 1 7. (20) JðyÞ ¼ 6 7 6 .. .. .. 7 6 7 6 .  . . 7 6 4 qeðQ; yÞ qeðQ; yÞ qeðQ; yÞ 5  qy1 qy2 qy10 3.2.1. Levenberg–Marquardt algorithm It is known that the Newton-type gradient algorithms (such as the steepest descend, Quasi-Newton, Gauss–Newton and Levenberg–Marquardt, etc.) usually have strong local search ability. Especially, the LMA not only has Newton method’s fast convergence, but also has numeric stability even when the Hessian matrix of the objective function is indefinite, singular, or ill-conditioned. Therefore, the LMA has been effectively applied to many nonlinear optimization problems. Newton algorithm generates a step d by minimizing the quadratic function q(d) defined in (17)   min qðdÞ ¼ min EðyÞ þ gT d þ 12dT Bd , (21) d

k¼1

r ¼ 1; 2; . . . ; 10,

Let e(y)be the displacement error vector, i.e.

d

rqðdÞ ¼ g þ Bd ¼ 0.

(22)

From (22), the step d can be written as d ¼ B1 g.

(23)

Eq. (23) shows that we must construct B matrix so that it is positive definite. If we do not have B positive definite,

ARTICLE IN PRESS J. Zheng et al. / Neurocomputing 70 (2007) 749–761

one possibility is to use Bnew ¼ B+hIe (h is a positive scalar parameter, Ie is an identity matrix), such that Bnew is positive definite. Thus, we retrieve the LMA. The LMA computes a step d by the formula d ¼ ðB þ hI e Þ1 g.

(24)

d ¼ ½JðyÞT JðyÞ þ hI e 1 JðyÞeðyÞ.

(25)

Given a current iterate yi, the LMA finds the next iterate yi+1 yiþ1 ¼ yi þ d,

approximation of the objective function E(y). In the standard TRA [23], the function q(d) is defined as (17), the trust region is spherical in shape, and the step d is computed by solving the TRA subproblem   min qðdÞ ¼ min EðyÞ þ gT d þ 12dT Bd ; s:t:jjdjj2 pDi , (28) d

Substituting (18), (19) into (24) and assuming SðyÞ ¼ 0 give [19,22]

(26)

where i is the iteration number. As shown in (25), when h tends to infinity, the LMA tends toward the steepest descent method. When h is zero, the LMA becomes the Newton method. For intermediate value of h, the LMA will be a combination of the steepest descent method and the Newton method. Here, the LMA uses a technique which adapts the value of h during the optimization. If the iteration is successful (E(yi+d)oE(yi)), h is decreased to exploit more curvature information contained inside B. If the iteration is unsuccessful (E(yi+d)4E(yi)), h is increased in order to follow closely the gradient. The parameter h is modified by ( hi =10; Eðyi þ dÞoEðyi Þ; hiþ1 ¼ (27) 10hi ; Eðyi þ dÞ4Eðyi Þ; where i is the iteration number. In the paper, h1 ¼ 0.01 as an initial value. In short, when a negative curvature is encountered (B negative definite), the LMA can compute d based on an arbitrary perturbation of B (d is the solution of Eq. (24): d ¼ ðB þ hI e Þ1 g). The parameter identification procedure of the LMA is described in Procedure LMA. Procedure LMA. The parameter identification procedure of the LMA. S1: The iteration number T3, initial parameter h1 and initial parameters y1 are quantified. Let i ¼ 1. ~ yi Þ, S2: The parameters yi are used to calculate their yðk; E(yi) by (1)–(6),(12) and (15). S3: Compute J(y) by (20). S4: Compute d by (25). S5: The parameters yi+1 ¼ yi+d are used to calculate ~ yiþ1 Þ, E(yi+1) by (1)–(6), (12) and (15). If their yðk; E(yi+d)oE(yi), then hiþ1 ¼ hi =10 and yiþ1 ¼ yi þ d; otherwise hiþ1 ¼ 10hi and yi+1 ¼ yi. S6: Termination of the LMA. If i4T3, terminate the LMA and record yi+1; otherwise, i ¼ i+1, go to S3. 3.2.2. Trust-Region algorithm The TRA is a novel gradient algorithm [3,4,23], and it generates a step d at each iteration by minimizing a simple function q(d) while the step d is limited to lie within a trust region in which the function q(d) is expected to be a good

753

d

where i is the iteration number, Di is the trust-region radius and ||.||2 is the l2-norm. The key questions in defining the TRA to minimize E(y) are how to solve d of the TRA subproblem and how to modify Di. As detailed in [3,4] a way to solve d of the TRA subproblem is to solve a two-dimensional subspace S ¼ /s1, s2S, where s1 is the direction of the gradient g, and s2 is either an approximate Newton direction or a direction of negative curvature. The S is solved by a Preconditioned Conjugate Gradient (PCG) method. Once S has been computed, the work to solve d is trivial. After d is solved, the objective function E(yi+d), E(yi) and the ‘‘degree of agreement’’ ri between E(y) and q(d) are computed and judged. If E(yi+d)4E(yi) or ri is too small, the next iterate yi+1 remains unchanged yi+1 ¼ yi and Di is shrunk since the model is judged to be a poor representation of the function E(y) inside the region defined by Di. Otherwise, the step d is judged to be a step of good quality (i.e. sufficiently reduces E(y)), then the next iterate yi+1 is updated by yi+1 ¼ yi+d and the trust-region radius Di may be increased to try a more aggressive step on the next iteration. Taylor’s theorem ensures that as we shrink Di, the function q(d) will more accurately approximate the function E(y). Thus for a sufficiently small Di, an acceptable step d should be found. Here, Di is given by 8 > < Di =2; ri o0:1; 0:1pri p0:75; Diþ1 ¼ Di ; (29) > : 2D ; r 40:75; i i where the ‘‘degree of agreement’’ ri between E(y) and q(d) are computed by ri ¼ DE=Dq,

(30)

where DE is the practical descend value of E(y), i.e. DE ¼ Eðyi Þ  Eðyi þ dÞ

(31)

and Dq is the estimated descend value of E(y), i.e. Dq ¼ qð0Þ  qðdÞ.

(32)

The parameter identification procedure of the TRA is described in Procedure TRA. Procedure TRA. the parameter identification procedure of the TRA. S1: The iteration number T3, initial trust-region radius D1 and initial parameters y1 are quantified. Let i ¼ 1. ~ yi Þ, S2: The parameters yi are used to calculate their yðk; E(yi) by (1)–(6), (12) and (15). S3: Compute d in (28) by the PCG method.

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J. Zheng et al. / Neurocomputing 70 (2007) 749–761

S4: The parameters yi+1 ¼ yi+d are used to calculate ~ yiþ1 Þ, E(yi+1) by (1)–(6), (12) and (15). If their yðk; E(yi+d)4E(yi) or rio0.1, then yi+1 ¼ yi; otherwise yi+1 ¼ yi+d. S5: Compute DE, Dq, ri and Di+1 by (29)–(32). S6: Termination of the TRA. If i4T3, terminate the TRA and record yi+1; otherwise, i ¼ i+1, go to S3. 3.3. Genetic algorithm By evolving a population of individuals and adopting simple genetic operations (such as selection, crossover and mutation), GA is endowed with the global search capability of rapidly locating suboptimal solutions for difficult problems [10,11,15,21]. As a directed random search algorithm, GA prevails over the gradient algorithms, especially in complex, multimodal and noisy situations since it has no requirement on the derivatives of the objective functions. Moreover, GA is known to be more efficient than other random search algorithms for their capability of accumulating the useful information during the search process. Therefore, GA has been widely applied in many optimization problems [10,11,15,21]. In this paper, a hybrid coded GA is adopted. Its details will be given as follows. 3.3.1. Encoding scheme and individual It is well-known that the encoding scheme plays the critical part in the effectiveness of the GA because the individuals and genetic operations in the GA are dependent on the encoding scheme. In this paper, we adopt the hybrid encoding scheme. At the beginning of the GA, a binary coded GA (BCGA), with big population size q1 and short coded length, is used to produce adequate diversity population, thus ensures that the GA can fast find a better region of the solutions. After about 5 iterations of the BCGA, a floating-point coded GA (FCGA), with small population size q2, is used to search for the better solutions. The hybrid coded GA is paid attention to both the advantages of the BCGA and FCGA. It not only has higher convergence speed, but also has higher search precision. When the binary encoding scheme is used, each variable yr (r ¼ 1,2, y, 10) is encoded as a p-bit binary string vr, and these strings are concatenated to form an individual v ¼ [v1, v2, y, v10]. Thus, the length of the individual is 10p. When the individual is decoded, the 10p-size individual is divided into 10 p-bit binary strings vr, and each binary string vr is converted to variable yr by vr ðbr  d r Þ; r ¼ 1; 2; . . . ; 10. (33) yr ¼ d r þ p 2 1 where dr and br are the lower and upper bounds on yr. When a floating-point encoding scheme is used, we define the individual as x ¼ ½x1 ; x2 ; . . . ; x10  ¼ ½y1 ; y2 ; . . . ; y10 .

(34)

Eq. (34) shows that xr ¼ yr 2 ½d r ; br  (r ¼ 1,2,y,10), yr is the rth element of the parameter vector y defined in (14). 3.3.2. Fitness function The performance of each individual is evaluated according to its fitness. After generations of evolution, it is expected that the GA converges, and a best individual with largest fitness (or smallest error) representing the optimal solution to the problem is obtained. The fitness function is defined as pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi F ¼ 1=ðc1 þ EðyÞÞ, (35) where c1 is a positive constant to avoid singularity when E(y) tends to zero. 3.3.3. Selection The individuals selected into the new population are based on the elitist roulette wheel selection method. The elitist strategy is to select two individuals with the highest fitness and pass them onto the next generation without crossover and mutation. The roulette wheel selection can be done by assigning a probability psj of the jth individual Fj psj ¼ Pq

j¼1 F j

;

j ¼ 1; 2; . . . ; q,

(36)

where Fj is the fitness function of the jth individual, q is the population size. The cumulative probability F^ j for the jth individual is defined as F^ j ¼

q X

F j;

j ¼ 1; 2; . . . ; q.

(37)

j¼1

The selection process starts by randomly generating a nonzero floating-point number, wA[0,1]. Then, the jth individual is selected if F^ j1 owpF^ j . It can be observed from this selection process that an individual having a larger Fj will have a higher chance to be selected. Consequently, the best individuals will get more offspring, the average will stay and the worst will die off. 3.3.4. Crossover and mutation The crossover and the mutation operations are to generate some new individuals (offspring) from their parents after the selection process. In the BCGA, the well-known single-point crossover and simple mutation operations are used. In the FCGA, the arithmetic crossover and non-uniform mutation operations are used. In the arithmetic crossover operation, if two parents x ¼ [x1, x2, y, x10] and z ¼ [z1, z2, y, z10] are selected for crossover, then the offspring x0 ¼ ðx01 ; x02 ; . . . ; x010 Þ and z0 ¼ ðz01 ; z02 ; . . . ; z010 Þ are calculated by x0r ¼ xr Z þ zr ð1  ZÞ zr ¼ xr ð1  ZÞ þ zr Z

;

r ¼ 1; 2; . . . ; 10,

(38)

where Z 2 ½0; 1 is a random number. In the non-uniform mutation operation, if the parent x ¼ [x1, x2, y, x10] is selected for mutation, then the

ARTICLE IN PRESS J. Zheng et al. / Neurocomputing 70 (2007) 749–761

offspring x0 ¼ ðx01 ; x02 ; . . . ; x010 Þ is calculated by ( xr þ Dðt; br  xr Þ if t ¼ 0 0 xr ¼ ; r ¼ 1; 2; . . . ; 10, xr  Dðt; xr  d r Þ if t ¼ 1 (39) b

Dðt; yÞ ¼ yð1  mð1ði=TÞÞ Þ,

(40)

where t is a binary random number, m 2 ½0; 1 is a random number, i is the current iteration number, T is the maximum iteration number , dr and br are the lower and upper bounds on xr, and b is a positive constant. In this study, b ¼ 5 is selected. This mutation operation givens a value x0r 2 ½d r ; br  such that the probability of returning a value close to xr increases as the algorithm advances. 3.4. Hybrid genetic algorithms Gradient algorithms have shown their effectiveness in many nonlinear optimization problems. However, problems such as local minimum and sensitivity to initial conditions still remain due to the nature of the gradient search. Evolutionary algorithms including GA are global search algorithms but suffer from fine-tuning inefficiency due to the nature of the random search. By analyzing the different strengths of GA and gradient algorithms, we study two distinct hybrid strategies: the first hybrid strategy is to employ a gradient algorithm as a local search operation of a GA, and the second hybrid strategy is to take the best solution of a GA as the initial value of a gradient algorithm. In this study, two different gradient algorithms, a well-known LMA and a novel TRA, are investigated. Finally, the proposed four HGAs, namely GATR1, GALM1, GATR2 and GALM2, are applied to identify parameters of

the improved hysteresis model. Here, GATR1/GALM1 employs the TRA/LMA as a local search operation of the FCGA, GATR2/GALM2 takes the best individual of the FCGA as the initial value of the TRA/LMA. The flow chart of the GATR1/GALM1 is shown in Fig. 2, the parameter identification procedures of the GATR1/GALM1 and GATR2/GALM2 are described as follows. Procedure GATR1/GALM1. the parameter identification procedure of the GATR1/GALM1. Step 1: Initialization. Firstly, the BCGA iteration number T1, the FCGA iteration number T2, crossover probability pc, mutation probability pm, BCGA population size q1, FCGA population size q2 and binary bit number p are quantified. Secondly, an initial population of q1 individuals is created randomly under binary encoding scheme. Let i ¼ 1. Step 2: Evaluation of all individuals. The q1 individuals ~ yi Þ, E(yi) and F by are used to calculate their yðk; (1)–(6),(12), (15) and (35), and the best individual yb is obtained. Step 3: Selection, crossover and mutation operations. If i4T1, the selection, crossover, and mutation operations of FCGA are to generate q2 new individuals (offspring) from their q2 parent individuals; otherwise, the selection, crossover, and mutation operations of BCGA are to generate q1 new individuals (offspring) from their q1 parent individuals. Step 4: Evaluation of new individuals using Step 2. Step 5: If i4T1, go to Step 6; otherwise, go to Step 7. Step 6: TRA/LMA Local search operation. In the GATR1, the process of the TRA local search is: (1) Take yb as an initial value of the TRA; (2) Perform orderly S2, S3, S4, S5, S6 in Procedure TRA; (3) Finally, yi+1 is taken as the best individual yb of the next generation.

Start i=1, Initialization Evaluation of all individuals, record b N

Y

i>T1?

BCGA produces q1 individuals

FCGA produces q2 individuals

Evaluation of new individuals, record b

Evaluation of new individuals, record b TRA/LMA Local search operation, record b i = i+ 1

N

755

i>(T1+T2)? Y End

Fig. 2. Flow chart of the GATR1/GALM1.

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In the GALM1, the process of the LMA local search is: (1) Take yb as an initial value of the LMA; (2) Perform orderly S2, S3, S4, S5, S6 in Procedure LMA; (3) Finally, yi+1 is taken as the best individual yb of the next generation. Step 7: Termination of the GATR1/GALM1. If i4(T1+T2), terminate the GATR1/GALM1 and record yb; otherwise, i ¼ i+1, go to Step 3.

Procedure GATR1/GALM1. the parameter identification procedure of the GATR2/GALM2. Step 1: The BCGA runs T1 iterations to search for a better region of the solutions. Step 2: The FCGA runs T2 iterations to search for the better solutions. Step 3: The TRA/LMA takes the best individual of the FCGA as the initial value, and runs T3 iterations to obtain optimal solution. 4. Simulation results and analysis In the simulation study, the parameters to be estimated are listed in Table 1, and the input current is I ¼ (0.77sin(200pt)+1.53) A. The sample values of current I(k) and displacement y(k) are obtained by the numerical

computation of (1)–(6) and (12). The parameters of four HGAs are: q1 ¼ 120, q2 ¼ 30, pc ¼ 0.8, pm ¼ 0.05, p ¼ 16, T1 ¼ 5, T2 ¼ 295, T3 ¼ 350. Hence, the iteration number of GATR1/GALM1 is T ¼ 300 and the iteration number of GATR2/GALM2 is T ¼ 650. Obviously, the calculating quantity of GATR2/GALM2 is bigger than that of GATR1/GALM1. The results obtained from running four HGAs are listed in Table 1, the iterative processes of the maximum fitness are shown Fig. 3, and the iterative processes of parameters are shown in Fig. 4. Here, Remax is the maximum relative error of ten parameters, emax is the maximum displacement ~ yÞ, and Fmax is the maximum error between y(k) and yðk; fitness. From the analysis of Table 1, Figs. 3 and 4, some conclusions can be drawn: (1) The values of Fmax, Remax and emax show that the identification performance of GATR1 is best, the performance of GATR2 takes second place, and the performance of GALM1/GALM2 is worst. (2) The GATR1 can obtain very accurate parameters with a rather fast speed while the GALM1 cannot obtain the accurate parameters. The reason lies in the difference of the search mechanism of the LMA and TRA (see 3.2.1.and 3.2.2.). The TRA can ensure that the model function q(d) more accurately approximate

Table 1 Parameter identification results of the improved hysteresis model using four HGAs

Ms (A/m) ls a (A/m) k (A/m) a~ c E (N/m2) Kz (N/m) Mz (kg) Cz (Ns/m) Remax emax (mm) Fmax

Parameter range

Actual value

7.4e+5–8.0e+5 1.0e3–1.2e-3 3506–10518 1642–4925 0.1–0 0–0.3 2.5e+10–3.5e+10 3.0e+7–4.0e+7 0.46–0.62 3650–4950

7.65e+5 1.005e3 7012 3283 0.01 0.18 3.0e+10 3.36e+7 0.55 4300

(a)

Maximum fitness

Maximum fitness

1500

1000

500

0

0

Identified value

100 200 300 Iteration number

GALM1

GATR2

GALM2

7.47e+5 1.022e3 7029 3284.1 0.0102 0.18013 2.95e+10 3.358e+7 0.5467 4290.3 2.3% 0.0018 1416.8

7.55e+5 0.955e3 7166.1 3060 0.01004 0.18408 3.28e+10 3.563e+7 1.149 5224.2 108.9% 1.2073 12.876

7.70e+5 1.154e3 6617.4 3240 0.01138 0.17596 2.59e+10 3.414e+7 0.57997 4589.1 14.84% 0.0404 45.581

7.81e+5 1.099e3 6592.6 3017.6 0.01095 0.189 2.72e+10 3.566e+7 1.2035 5702.5 118.81% 1.2149 24.61

50

25

20

20

15 10 5 0

(b)

GATR1

Maximum fitness

Parameter

0

200 400 600 Iteration number

15 10 5 0

(c)

0

200 400 600 Iteration number

Fig. 3. Iterative processes of the maximum fitness using four HGAs: (a) GATR1; (b) GATR2; (c) GALM1 (solid line); GALM2 (dot-and-dash line).

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Fig. 4. Iterative processes of parameters of the improved hysteresis model using four HGAs.

the objective function E(y) in the trust-region radius Di. Thus, the TRA is prone to find an acceptable step d and a better individual yi+1, and finally ensures that the GATR1 has the progressional force all along. However, as an improved Newton method, the LMA is often difficult to find the step d due to the strong nonlinearity of the objective function and the strong interrelation among nonlinear parameters. Even if the LMA can find the step d, the LMA also possibly can not find a better individual because in the LMA, the quadratic function q(d) sometimes does not fit the objective function E(y). Hence, the LMA can not efficiently increase the fitness function of GALM1 and improve the best individual of GALM1. For example in GALM1, Fmax slowly increases to 12.876 after the GALM1 run 200 iterations and after 200 iterations,

Fmax is almost changeless. Moreover, it is noted that the parameters Mz and Cz have inclination to depart from the actual values when Fmax increases. (3) Similarly, the difference of the search mechanism of the TRA and LMA also leads to the fact that the GATR2 have better performances than GALM2. (4) The GATR1 vastly outperforms the GATR2 in the precision and speed of parameter identification. The reason lies in two distinct hybrid strategies of GATR1 and GATR2. In the GATR1 using TRA as a local search operation of FCGA, the excellent local search ability of the TRA can constantly improve the best individual so that the GATR1 has the progressional force all along and thus can obtain very accurate parameter value with very fast speed. The GATR2 adopts the ‘‘BCGA-FCGA-TRA’’ in serial manner

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so that every algorithm independently searches and disadvantages of every algorithm cannot be overcome. For example in the GATR2, Fmax slowly increases to 5.94 after the FCGA run 100 iterations and is nearly changeless after 100 iterations due to the weak local search ability of the FCGA; Fmax slowly increases to 45.51 after the TRA run 100 iterations because the TRA can not efficiently solve the strong interrelation among nonlinear parameters.

5. Experimental validation and analysis Here, the magnetostrictive actuator developed by Hebei University of Technology is called as Actuator1 shown in Fig. 5(a), and the magnetostrictive actuator developed by Iowa State University is called as Actuator2 shown in Fig. 1. In order to validate performance of the improved hysteresis model and GATR1, comparisons are made with the measured hysteresis curves of Actuator1 and Actuator2. 5.1. Actuator1 experimental validation The high precision experimental setup of Actuator1 is shown in Fig. 5(b). It consists of a TMS320C31-based digital signal processor (DSP) board, a exact digital control constant current source, a 12-bit A/D, and a CAPANCDT S600-0.2 capacitance displacement sensor, whose precision is 8 nm at low frequencies (below 30 Hz).

With the experimental setup, we measured a quasi-static displacement hysteresis curve under DC magnetic bias Hb ¼ 0 A/m. Note that the input current of the Actuator1 were produced at 2 Hz frequency, and the output displacement data were also collected at 2 Hz frequency. The known parameters of Actuator1 are Ar ¼ 7.85  105 m2, lr ¼ 85 mm and n ¼ 48 471/m. The measured curve of Actuator1 is plotted in Fig. 6. With the measured curve of Actuator1, GATR1 runs to identify the parameters of the improved hysteresis model. The repeated identification results are summarized in Table 2, where emax is the maximum displacement error between y(k) ~ yÞ. From Table 2, the repeated identification values and yðk; of the parameters are almost same, and emax is about 2.7 mm.

Table 2 Repeated identification results of parameters of the improved hysteresis model using GATR1 Parameter

No. 1

No. 2

No. 3

Ms (A/m) ls a (A/m) k (A/m) a~ c E (N/m2) Kz (N/m) Mz (kg) Cz (Ns/m) emax (mm)

7.60e+5 1.10e3 1925 7541.5 0.0155 0.10026 2.54e+10 4.80e+7 — — 2.7351

7.60e+5 1.10e3 1917.4 7552.9 -0.0157 0.10043 2.60e+10 4.94e+7 — — 2.7671

7.62e+5 1.10e3 1934.6 7549.9 0.0158 0.10014 2.60e+10 4.97e+7 — — 2.7519

Prestress Bolt Terfenol-D Rod Cap Spring Washer

TMS320C31 DSP

Push Rod End Mass

A/D

I y

Motion Output Water Cooling Tube (a)

Constant Current Source

Capacitance Sensor

Actuator1

Solenoid Magnetic Yoke (b)

Fig. 5. (a) Schematic diagram of Actuator1. (b) Experimental setup of Actuator1.

Fig. 6. (a) Measured and calculated hysteresis curves for Actuator1 under Hb ¼ 0 A/m and 2 Hz frequency. (b) Model calculation errors.

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It is noted that the values of Mz and Cz cannot be estimated because the quasi-static (2 Hz) displacement data are barely sensitive to variations in Mz and Cz according to (12). When emax is 2.7351 mm, the comparison of the measured and calculated curve is shown in Fig. 6. It is found that the calculated result is in a better agreement with the experimental result. These results verify the practical performance of the improved hysteresis model and GATR1. 5.2. Actuator2 experimental validation With a Techron 7780 amplifier, a Tektronix 2642A spectrum analyzer and a linear variable differential transformer(Lucas LVM-10 LVDT), [9] measured strain hysteresis curves of Actuator2 under DC magnetic bias Hb ¼ 16 kA/m and AC magnetic field HAC ¼ 8 sin (2pft) kA/m with different frequencies from 10 to 2000 Hz (see [9] for details). The known parameters for Actuator2 are Ar ¼ 1.2668  104 m2 and lr ¼ 115 mm. The measured curves of Actuator2 at five different frequencies are shown in Fig. 7. With these measured curves, the parameters estimated by GATR1 are listed in Table 3. In Table 3, at quasi-static frequency f ¼ 10 Hz, Mz and Cz cannot be estimated because the quasi-static strain data are barely sensitive to variations in Mz and Cz according to (12). Furthermore, it is noted that the parameters Ms, E, Mz, Cz and Kz almost do not vary with frequency f, but the five parameters of JA model, ls, a, k, a~ and c vary greatly with frequency f due to the frequency-dependent property of the eddy current and anomalous losses. The magnetostriction and strain calculated by (1)–(6) and (12) with the parameters in Table 3 are also shown in

759

Fig. 7. By (13) with the parameters E ¼ 2.22e+10 N/m2, Mz ¼ 0.54 kg, Cz ¼ 2010 Ns/m and Kz ¼ 2.64e+7 N/m, the calculated frequency characteristics of strain versus magnetostriction are shown in Fig. 8. From the analysis of Figs. 7 and 8, some conclusions can be drawn: (1) The magnetostriction loop exhibits sigmoid shape at 10 Hz, then gradually becomes bigger from 800 to 1250 Hz, and subsequently becomes smaller at 2000 Hz. The shape changes of the magnetostriction loops are caused by the changes of the parameters ls, a, k, a~ and c in the improved hysteresis model. The changing processes of the magnetostriction loops show the frequency-dependent characteristic of the eddy current and anomalous losses of the magnetostrictive actuator. Table 3 Parameter identification results of the improved hysteresis model using GATR1 Parameter Frequency f

Ms(A/m) ls a(A/m) k (A/m) a~ c E (N/m2) Kz (N/m) Mz (kg) Cz (Ns/m)

10 Hz

800 Hz

1000 Hz

1250 Hz

2000 Hz

7.65e+5 1.005e3 7012 3283 0.010 0.180 2.220e+10 2.6454e+7 — —

7.61e+5 1.067e3 7156.8 7495.7 0.0001 0.172 2.221e+10 2.6455e+7 0.5443 2011.1

7.66e+5 1.500e3 8975.6 4799.6 0.013 0.038 2.220e+10 2.6454e+7 0.5439 2009.9

7.67e+5 1.892e-3 3458.2 10217 0.0001 0.0003 2.219e+10 2.644e+7 0.5445 2010

7.68e+5 2.002e3 6771.2 15740 0.058 0.1989 2.222e+10 2.647e+7 0.5441 2010

Fig. 7. Comparison of the measured and calculated hysteresis curves for Actuator2 under Hb ¼ 16 kA/m and HAC ¼ 8sin(2pft) kA/m with different frequencies.

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Fig. 8. Frequency characteristics of strain versus magnetostriction for Actuator2. (a) Magnitude frequency characteristic. (b) Phase frequency characteristic.

(2) The calculated magnetostriction and strain loops show slight difference at 10 Hz, but show obvious difference from 800 to 2000 Hz. The difference is caused by the dynamic mechanical behavior of the magnetostrictive actuator, and is dependent on the frequency characteristics shown in Fig. 8. It is obvious from Fig. 8 that at f ¼ 10 Hz, the small magnitude difference D ¼ ||e||l|| ¼ |A(w)|l||l||E|0.93|l||l|| ¼ 0.07|l| and the small phase f(w)E0.160 must lead to slight difference between the magnetostriction and strain loops. Similarly, when 800 Hzpfp1250 Hz, the big magnitude difference D (DE0.57|l| at f ¼ 800 Hz, DE0.77|l| at f ¼ 1000 Hz, DE0.31|l| at f ¼ 1250 Hz, DE0.62|l| at f ¼ 2000 Hz) and the big phase f(w) (f(w)E42.760 at f ¼ 800 Hz, f(w)E62.590 at f ¼ 1000 Hz, f(w)E114.380 at f ¼ 1250 Hz, f(w)E157.160 at f ¼ 2000 Hz ) must lead to obvious difference between the magnetostriction and strain loops. Especially at 1250 and 2000 Hz, the phase f(w) is up to 114.380 and 157.160, and thus the direction of the average slope of the strain loops switches from positive to negative. These results show that the structural dynamic mechanical behavior of the magnetostrictive actuator plays an important role at higher frequencies. (3) It is obvious that the calculated strain loops are in a good agreement with the measured strain loops from 10 to 2000 Hz. (4) The above results show that the GATR1 can automatically update the parameters of the improved hysteresis model, and the improved model with five frequency-dependent parameters has clear physical meaning and has good accuracy, thus will help the analysis of the magnetostrictive actuator in a wide frequency range. 6. Conclusions This paper establishes an improved hysteresis model for magnetostrictive actuators based on the JA model, the magnetostriction model and a linear structural dynamic mechanical model, and proposes four HGAs, namely GATR1, GALM1, GATR2 and GALM2, to estimate the parameters of the model. The simulation results show that GATR1 vastly excels GALM1, GATR2 and GALM2 in

the precision and speed of parameter identification. The reason why GALM1, GATR2 and GALM2 are not as efficient as GATR1 is thoroughly analyzed. The experimental results show that the GATR1 can automatically update the parameters of the improved hysteresis model at different frequencies, thus can extend the model to a dynamic hysteresis model. The extended dynamic hysteresis model has clear physical meaning, and can accurately describe the relationship between the input magnetic field and the output strain for magnetostrictive actuators in a wide frequency range, thus has very strong practicability. Acknowledgements The authors acknowledge the financial support from the National Natural Science Foundation of China (Project no.50371025). References [1] A. Adly, I. Mayergoyz, A. Bergqvist, Preisach modeling of magnetostrictive hysteresis, J. Appl. Phys. 69 (8) (1991) 5777–5779. [2] L.A.L. Almeida, G.S. Deep, A.M.N. Lima, H. Neff, Modeling a magnetostrictive transducer using genetic algorithm, J. Magn. Magn. Mater. 266–230 (2001) 1262–1264. [3] M.A. Branch, T.F. Coleman, Y. Li, A subspace, interior, and conjugate gradient method for large-scale bound-constrained minimization problems, SIAM J. Sci. Comput. 21 (1) (1999) 1–23. [4] R.H. Byrd, R.B. Schnabel, G.A. Shultz, Approximate solution of the trust region problem by minimization over two-dimensional subspaces, Math. Program. 40 (1988) 247–263. [5] F.T. Calkins, R.C. Smith, A.B. Flatau, Energy-based hysteresis model for magnetostrictive transducers, IEEE Trans. Magn. 36 (2) (2000) 429–439. [6] A.E. Clark, Magnetostrictive rare earth-Fe2 compounds, in: E.P. Wohlfarth (Ed.), Ferromagnetic Materials, vol. 1, North-Holland Publishing Company, Amsterdam, 1980, pp. 531–589 (Chapter 7). [7] M.J. Dapino, A.B. Flatau, F.T. Calkins, Statistical analysis of Terfenol-D material properties, in Proceedings of the SPIE Smart Structures and Materials, vol. 3041, 1997, pp. 256–267. [8] M.J. Dapino, R.C. Smith, A.B. Flatau, Structural magnetic strain model for magnetostrictive transducers, IEEE Trans. Magn. 36 (3) (2000) 545–556. [9] L.E. Faidley, B.J. Lund, A.B. Flatau, F.T. Calkins, Tefernol-D elasto-magnetic properties under varied operating conditions using hysteresis loop analysis, in Proceedings of the SPIE Smart Structures and Integrated Systems, paper #92, 3329 (3), 1998.

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[31] X.B. Tan, J.S. Baras, Modeling and control of hysteresis in magnetostrictive actuators, Automatica 40 (2004) 1469–1480. [32] R. Venkataraman, Modeling and adaptive control of magnetostrictive actuators, Ph.D. Thesis, University of Maryland, College Park, 1999. [33] P.R. Wilson, J.N. Ross, A.D. Brown, Optimizing the Jiles–Atherton model of hysteresis by a genetic algorithm, IEEE Trans. Magn. 37 (2) (2001) 989–993. Jiaju Zheng received the B.E. degree in electronic engineering from XiDian University, China, and the M.Sc. degree in materials from Hebei University of Technology, China, in 1988 and 2005, respectively. He is currently a Ph.D. student at Tianjin University. His current research interests include neural networks, fuzzy systems, evolutionary algorithms, intelligent control, embedded system, and their applications in the modeling, identification and control of nonlinear system. Shuying Cao received the B.E. degree and the M.E. degree in automatic theory and application from Huabei Institute of Technology, China, in 1993 and 1996, respectively, and the Ph.D. degree in electric theory and new technology from Hebei University of Technology, China, in 2004. She is currently an associate professor at Hebei University of Technology. Her main research interests include evolutionary computation, neural networks, hybrid computational intelligence and their applications in the modeling and control of nonlinear system. Hongli Wang received the B.E. degree in engineering mechanics from Tsinghua University, China, in 1967, and the M.E. degree in mechanics from Tianjin University in 1981. She is currently a professor and Ph.D. advisor at Tianjin University. Her main research interests include evolutionary computation, neural networks, intelligent control, and their applications in the modeling and control of nonlinear system. She has published numerous papers and won some important scientific awards in the area of nonlinear vibration control.

Wenmei Huang received the B.E. degree and the M.E. degree in automatic theory and application from Hebei University of Technology, China, in 1994 and 2001, respectively, and the Ph.D. degree in electric theory and new technology from Hebei University of Technology, China, in 2005. She is currently a teacher at Hebei University of Technology. Her main research interests include fuzzy systems, magnetic device design, and intelligent control.