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General Bond Graph model for piezoelectric actuators and methodology for experimental identification
Mechatronics 20 (2010) 303–314
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General Bond Graph model for piezoelectric actuators and methodology for experimental identification J.M. Rodriguez-Fortun a,*, J. Orus a, F. Buil a, J.A. Castellanos b a b
Grupo de Investigación Aplicada (GIA-MDPI), Instituto Tecnológico de Aragón, Zaragoza, Spain Departamento de Informática e Ingenierı´a de Sistemas, Centro Politécnico Superior, Universidad de Zaragoza, Spain
a r t i c l e
i n f o
Article history: Received 3 April 2009 Accepted 17 January 2010
Keywords: Piezoelectric actuators Bond Graph Hysteresis Creep effect Rate dependency Identification
a b s t r a c t Piezoelectric actuators are becoming very popular in applications such as nanopositioning, active vibration control or noise reduction, sometimes as part of so called smart structures. The possibility of accurately moving big loads on a micrometric scale and over a wide range of frequencies has resulted in an intense growth of this technology. This interest has prompted the development of a wide variety of mathematical models available for describing their behaviour, whose suitability depends on the specific application involved. This work overcomes this limitation by unifying the different modelization options for stack and stack-based actuators in a general Bond Graph model structure capable of handling the most important physical phenomena observed in these actuators, both linear, such as direct and indirect piezoelectric effects or rate dependence, and nonlinear, such as hysteresis. This model structure represents a basis for specific applications, and can be used for control or simulation purposes thanks to its high generality and adjustment capabilities. The proposed Bond Graph structure graphically shows the power flow between the electrical and mechanical frameworks of the piezoelectric actuator, and uses a modular structure for separately representing the electrical polarization of the material and its macroscopic electrical and mechanical effects. Finally, the model is successfully applied to describe the rate dependent behaviour of the Cedrat Groupe APA-120ML actuator. In this connection, an experimental identification method is described and adapted for implementing hysteresis descriptions based on simple operators or in differential equations in the model structure (O-based and DE-based models). These two typologies cover the majority of the models available, proving the generality of the proposed piezoelectric model for implementing less general or specific phenomenon descriptions into its structure. In consequence, the main contribution of this work is the development of a general framework for modelling piezoelectric actuators, comprising a graphical Bond Graph model and an adjustment procedure, which is flexible enough to embody different representations of the phenomena present in these actuators, and with a modular structure that admits different levels of complexity depending on the phenomena incorporated in the model. Ó 2010 Elsevier Ltd. All rights reserved.
1. Introduction This paper is devoted to modelling the behaviour of piezoelectric actuators operating in one direction of movement, as is the case with stack actuators, both in prestressed and free conditions, and in individual configurations or forming part of more complex elements (smart structures, benders, amplifying structures). The work presented results in a general modelling and identifying framework based on a Bond Graph representation, which clearly describes the power flows and decouply embodies the phenomena inside the actuator. * Corresponding author. Tel.: +34 976 01 1071. E-mail addresses: [email protected] (J.M. Rodriguez-Fortun), [email protected] (J. Orus), [email protected] (F. Buil), [email protected] (J.A. Castellanos). 0957-4158/$ - see front matter Ó 2010 Elsevier Ltd. All rights reserved. doi:10.1016/j.mechatronics.2010.01.004
The introduction of piezoelectric actuators in several technological applications, such as active vibration control, noise reduction, and nanopositioning, has resulted in the development of different models for simulating the behaviour of these materials. The models currently available can be divided into two groups depending on the form of the mathematical description: linear and nonlinear models. The first group mainly comprises the models based on constitutive equations, which consider both direct and indirect piezoelectric effects [22,13,23], and simplified models just considering one of those effects, depending on the final model application [1]. As furtherly explained in Section 2, the models based on constitutive equations can be represented in two different ways depending on its equivalent macroscopic model. Nowadays these descriptions are still used in active vibration control applications [1,2], which do not require high accuracy in position control. A
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further improvement of this sort of model is the inclusion of the creep effect, which sometimes is described in a linear way [3]. The main advantage of linear models is their simplicity, but if a better matching to real behaviour is needed, as is required for nanopositioning applications [3,19], the inclusion of nonlinear descriptions to reproduce hysteresis effect is necessary [25]. The different ways of representing these effects results in the different nonlinear models for piezoelectric actuators. In this connection, two major types of representations can be distinguished for hysteresis according to their mathematical descriptions: the operator based and the differential equation based models (from now on, O-based and DE-based models, respectively). In the first case, the hysteresis curve is obtained by the addition of simple operators, such as the Preisach [4,26] or Ishlinkii [5]; in the second case, an extra state variable is included in the hysteresis definition, which evolves in a different way depending on the direction of movement [6]. These hysteresis models are called phenomenological, in the sense that they are not based on a physical description of the piezoelectric materials. There are other representations that do use a physical basis for representing the hysteresis in the piezoelectric materials and which can also be summarised in the two basic types: the O-based model, such as the domain wall model [7], or the DE-based model, as in the case of the Homogeneized Energy model [8]. Moreover, it has been experimentally proved that the hysteresis phenomenon in piezoelectric materials is rate dependent. Different ways of including the rate dependency in hysteresis models appear in [3], where a damper is added to the model, or in [12], where the weighting factors of a O-based model are made rate dependent. Combined models of hysteresis and creep, which is related with the rate dependence, are described in [3,9–11]. This work classifies the different modelling possibilities, and summarizes them in a general dynamic model structure, which can further be adapted to any simulation or control purpose (some of the models hitherto cited are only intended for describing a static relation between the polarization and the voltage in the piezoelectric materials). In comparison with other nonlinear descriptions of piezoelectric actuators, as [25] which proposes a model not based in the constitutive equations using a DE-based hysteresis model, the proposed model is a general framework incrementally built from the constitutive equations and described in terms of a Bond Graph [16,17] because of its capacity for clearly defining the power flows between the mechanical and the electrical parts. Unlike previous Bond Graph descriptions for piezoelectric actuators [3,14,24], the model proposed considers separately the macroscopic electrical effects (electric charge Q, current i, voltage _ force F), V), and the mechanical effects (displacement x, speed x, while explicitly including the polarization effect in the material _ electric field E). The pro(electric dipole p, its variation with time p, posed model is validated with a real piezoelectric actuator, the APA-120ML produced by the Cedrat Groupe. This is done with a general methodology proposed for the adjustment of the model considering hysteresis and rate dependence, and the model is successfully applied to two different hysteresis descriptions: a DEbased and a O-based model. In Section 2 the linear basis is described. Section 3 discusses the capability for describing nonlinear hysteresis and rate dependency effects observed in piezoelectric actuators. The final model is derived in Section 4, and is applied in Section 5 to the dynamic modelling of the Cedrat APA-120ML amplified actuator.
zoelectric layer as an energy transformer between the mechanical and the electrical parts, taking into account the direct and converse piezoelectric effect, that is to say, the production of an electric field when stress is applied and vice versa.
T ¼ cD S þ hD E ¼ hS þ bs D
ð1Þ
where D T E S cD h bs
electric displacement ðC=m2 Þ stress ðN=m2 Þ electric field ðV=mÞ strain (–) elastic stiffness with constant electric displacement ðN=m2 Þ piezoelectric constant ðV=mÞ impermittivity at constant stress ðVm=CÞ
Integrating and considering constant values for isotropic materials, with the electric field perfectly perpendicular at the surface of the layer in the direction of polarization and behaving as a capacitor, the following assumptions can be taken:
V E ¼ rV l Z Q¼ D@A DA
ð2Þ
A
Therefore, the constitutive equations can be rewritten in terms of the input voltage V, the displacement x, the electric charge Q and the force on the actuator F:
F V
¼
cD Al
h
h
bs Al
!
x Q
where A l x
actuator surface ðm2 Þ layer thickness ðmÞ displacement ðSlÞðmÞ
The elements in the constitutive equations are represented by a more intuitive model in Figs. 1 and 2. In these figures, C i and K i represents the capacitor and the spring used for storing electrical and mechanical energy, respectively, and ei stands for the electromechanical coupling, h. The 2-port C-field in Fig. 3 contains the constitutive equations and serves for storing the energy in the electrical and mechanical parts and interchanging energy between them. The system has two external inputs: an external voltage V e and a force F.
2. Linear models of piezoelectric actuators A piezoelectric layer polarized in a single axis can be described by the constitutive equations assuming isothermal conditions and energy conservation [22,13,23]. These equations describe the pie-
ð3Þ
Fig. 1. Linear description of the piezoactuator according to model 1.
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Now, the expression of the voltage and the force is derived as in [14]:
F V
0 ¼@
K 2 þ a21C
a21C2
a21C2
1 C2
2 2
1 A
x Q
ð7Þ
This model is used in [3]. The parameters in both models are: Ci Ki ai
equivalent capacitor ðFÞ, for i ¼ 1; 2 stiffness ðN=mÞ, for i ¼ 1; 2 electromechanical transformation factor ðm=CÞ or ðV=NÞ, for i ¼ 1; 2
Fig. 2. Linear description of the piezoactuator according to model 2.
Both models just redefines the constants in (3) in terms of macroscopic electrical and mechanical elements. The differences between them arise from the way the coefficients are defined:
Fig. 3. Bond Graph representation of the piezoelectric actuator electromechanical coupling.
The coupling between the electrical and the mechanical part of the piezoelectric material ei described in the off-diagonal terms of the constitutive equations transforms the mechanical energy into electrical energy and vice versa, affecting both the mechanical model of the piezoelectric actuator, characterised by its rigidity K i , and the electrical model, characterised by C i , where i ¼ 1; 2. The two approaches for describing the C-field in terms of the macroscopic stiffness and electric capacitor in Figs. 1 and 2 depends on the assumption for the energy storage in the material: Model 1 (see Fig. 4): the energy of the system U is described as the addition of the mechanical and the electrical part of the actuator:
U¼
1 1 2 Q K 1 ðx a1 Q Þ2 þ 2 2C 1
ð4Þ
The value of the force and the voltage is obtained as @U ; V ¼ @Q and results in [14]: F ¼ @U @x
F V
¼
K1 K 1 a1
K 1 a1 Ka21
!
þ C11
x Q
ð5Þ
This description is used in [14] and in most piezoelectric suppliers technical data. Model 2 (see Fig. 5): the energy of the system is described slightly differently as:
U¼
2 1 1 x Q K 2 x2 þ 2 2C 2 a2
ð6Þ
1. In
model 1 : a1 ¼ Qx , with F ¼ 0. C 1 ¼ QV , with F ¼ 0. K 1 ¼ a1FQ , with x ¼ 0.
2. In
model 2 : a2 ¼ VF , with x ¼ 0. C 2 ¼ QV , with x ¼ 0. K 2 ¼ Fx, with V ¼ 0.
In consequence, using one model or another depends on two factors: the convenience of arranging one test or another and the causality of the model in which the piezoelectric actuator model must be included. In the rest of this work, model 1 is taken as the reference but all conclusions are applicable to model 2. 3. Nonlinear phenomena in piezoelectric actuators The behaviour of the piezoelectric actuators is characterised by the hysteresis and the rate dependency. Even though the rate dependency can be described in terms of linear factors [3], sometimes it is necessary to use nonlinear expressions [10] for better matching the experimental results. For this reason, both effects are included as nonlinear phenomena. 3.1. Hysteresis The hysteresis phenomenon, normally observed in terms of position and voltage or force and voltage, affects both the static and the dynamic operation of actuators, influencing the behaviour of controllers in tracking and positioning applications. As explained in the introduction to this work, there are different models for describing this effect, which is normally placed in the electrical part of the model, assuming that hysteresis is a consequence of reversible and irreversible domain wall motions inside the mate-
Fig. 4. Model 1 of the electromechanical coupling.
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Fig. 5. Model 2 of the electromechanical coupling.
rial [20]. Therefore the effect is represented by a nonlinear voltage and charge relationship, that is to say, a nonlinear capacitor C NL . In this connection, there are two major approaches for defining this new capacitor: Combining the linear capacitor C and a hysteresis model. In this case the hysteresis model just causes a perturbation in the voltage average value represented by the linear capacitor [10]. Another possibility is directly eliminating the capacitor and substituting it with a hysteresis model. This way the model can include the complete phenomena of saturation [7,8], depolarization or inverse polarization [15]. For the purposes of coherency with the nomenclature used in specialised works for ferroelectric ceramics [20], the proposed model uses the following terms for the physical variables when describing the hysteresis phenomenon in the piezoelectric layers: polarization density P and electric field E. The polarization density is defined as the effective dipole per volume unit in the material ðP ¼ Ql=ðlAÞ ¼ Q =AÞ. This transformation assumes that the layer behaves as a capacitor, with a constant electric field as explained in Section 2, without border effects. As indicated in the introduction to this work, two general types of models for the hysteresis effect can be considered only taking into account the shape of the mathematical description: the DEbased and the O-based models. To prove the capability of the proposed model for handling the majority of hysteresis models available, the validation results in following sections are arranged with a DE-based model, the Bouc-Wen [13], and a O-based model, the Ishlinkii [12]. The characteristics of these models are: DE-based models are expressed by means of an extra state zðtÞ _ which evolves differently according to PðtÞ. This way, the hysteresis curve follows a different line upwards and downwards:
EðtÞ ¼ hðzðtÞ; PðtÞÞ _ _ z_ ðtÞ ¼ f ðzðtÞ; PðtÞ; PðtÞÞgð PðtÞÞ
ð8Þ
This structure appears in the Bouc-Wen model, which has the form:
E ¼ kx PðtÞ þ kw zðtÞ _ n1 n _ _ z_ ðtÞ ¼ qðPðtÞ rPðtÞ Þ jzðtÞj zðtÞ þ ðr 1ÞPðtÞjzðtÞj
ð9Þ
where kx ; kw ; q; r; n are adjustable parameters of the model. The O-based model simulates the hysteresis by the addition of simple nonlinear operators:
EðtÞ ¼
n X
wi Hi ðPðtÞÞ
ð10Þ
i¼1
In the case of the Ishlinkii model, the operator Hi is the backlash as shown in Fig. 6:
Hi ðPðtÞÞ ¼ maxðPðtÞ ri ; minðPðtÞ þ r i ; Hi ðPðt 1ÞÞÞÞ where r i ; wi are adjustable parameters.
ð11Þ
Fig. 6. Backlash operator for the Ishlinkii model.
In [12] dead zone operators Sj are applied to the result of the backlash operators for eliminating the symmetry inherent to this operator result:
E¼
m X j¼1
Sj ðxÞ ¼
ws;j Sj
n X
! wi Hi ðPðtÞÞ
i¼1
maxð0; x dj Þ; 8dj > 0 x;
dj ¼ 0
ð12Þ
where dj ; ws;j are adjustable parameters. 3.2. Rate dependency Piezoelectric materials show rate dependency as is demostrated in Fig. 7 and already cited by [11,10]. This phenomenon is related to the creep effect, which is especially important in positioning applications due to the relaxation with time suffered by the material. The relevance of the rate dependency depends on the actuator type and the operation conditions. Although the dynamic model with the resistance/capacitor and the spring/mass already has a certain rate dependency, it is not normally enough for matching the experimental behaviour of the piezoelectric material. For this reason it is necessary to include in the model other elements capable of adding extra parameters to adjust the rate dependency. There are two main ways of including it in the model, both of which can be used at the same time: Using a hysteresis model with rate dependency. There are DEbased models with rate dependency as described in [18], and O-based models as that discussed in [12]. Including a viscous term in the model ðRNL Þ. In the bibliography, this term is normally placed in the mechanical part of the model and can be found as a rate dependant RNL in the force equilibrium as in [10], or as a rate dependant stiffness K modelled by a Voigt–Kelvin model with spring elements ðK i ; i ¼ 0 . . . nÞ and damping elements ðC i ; i ¼ 1 . . . nÞ [3] (Fig. 8).
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Taking into account the considerations described in the previous section for a singular piezoelectric layer, it is possible to build a model for a stack actuator made out of N layers. This model appears in Fig. 10 and contains three different physical domains: electrical macroscopic effect, polarization of the material and mechanical macroscopic effect in order to analyse separately the behaviour of the elements in the actuator. It is important to remark that the polarization density P of the material is substituted in the Bond Graph representation by the electric dipole p ¼ Ql to assure power conservation between the three domains. The following equations are extracted from the Bond Graph model in Fig. 10:
V e ¼ iRe þ V i¼
j¼N X
ij
j¼1
Fig. 7. Effect of the frequency in the position/voltage curves (the position at 7 V is fixed as reference, x ¼ 0)
V ¼ V 1;j þ ij Re;j V 1;j ¼ E1;j ¼ E2;j E3;j lj
ð13Þ
E2;j ¼ C NL;j ðp1;j Þ p_ 1;j ¼ ij l
aj p1;j E3;j lj ¼ F 1;j ¼ K NL;j xj xj1 aj lj F j ¼ M j x€j þ RNL;j ðx_ j x_ j1 Þ þ F 1;j ¼ M j €xj þ F j1
where E1;j E2;j E3;j
Fig. 8. Bond Graph representation of the Kelvin–Voigt model relaxation.
The model proposed can handle both possibilities. The results described in Section 5 have been obtained by using the second option, which is more flexible to adjust, because of the independence between both effects.
4. Final model Piezoelectric stack actuators consist of a pile of N ceramic layers electrically connected in parallel to maximize the displacement of the complete actuator, as shown in Fig. 9.
p1;j Fj FN F0 xj Re Re;j lj aj C NL;j K NL;j RNL;j Mj
electric field in the piezoelectric layer due to the electric part in layer jðV=mÞ effective electric field in the piezoelectric layer jðV=mÞ electric field in the piezoelectric layer due to the mechanical part in layer jðV=mÞ effective electric dipole in the piezoelectric layer jðCmÞ force on the piezoelectric layer jðm=CÞ force externally applied at the piezoelectric F ðm=CÞ force on the basis of the actuator ðm=CÞ displacement of the layer j with respect to an absolute reference placed on the floor ðmÞ electric resistance from amplifier to the piezoelectric actuator electric resistance from power amplifier due to the effective resistance in the layer jðXÞ thickness of the layer jðmÞ electromechanical transformation factor in the layer jðm=CÞ relation between polarization and electric field in layer jðCm2 =VÞ stiffness of the layer jðN=mÞ viscous term between the layer j and j 1ðNs=mÞ mass of the layer jðKgÞ
with j ¼ 1 . . . N. A simplification of the previous model results if all the layers in a stack are assumed to have the same properties. In this situation, if the inertial forces of each layer are disregarded, which is logical given their extremely small size, it can be assumed that each layer behaves in the same way, that is to say:
~xi ¼ ~xi1 ; 8i; with ~xi ¼ xi xi1 pi ¼ pi1 8i With these hypotheses, the Bond Graph takes the shape of Fig. 11 and the equations are reduced to:
V e ¼ iRe þ V
Fig. 9. Connection of the layers comprising the stack actuator.
i ¼ Niw V ¼ E2 E3 l E2 ¼ C NL ðpÞ
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Fig. 10. Complete piezoelectric actuator model with different piezoelectric layers.
Fig. 11. Simplified complete piezoelectric actuator model with equal piezoelectric layers.
p_ ¼ il E3 l ap ¼ F 1 ¼ K NL x a l
ap F ¼ F N ¼ Mw ð€xN þ €xN1 þ . . . þ €x1 Þ þ RNL;1 þ K NL;1 x1 1 l x ap €x _ ~ ¼ M w ð1 þ 2 þ . . . þ NÞ þ RNL;w ðxÞ þ K NL;w N lN N ap _ ¼ M€x þ K NL x þ RNL ðxÞ l
ð14Þ
where iw ; xw ; K NL;w ; Mw ; RNL;w ð~ x_ Þ refers to the value in one single layer and
K NL;w N Nþ1 M¼ Mw 2
This model is flexible enough to represent the complete phenomenology of piezoelectric actuators. It is possible to use it as a reference and a basis for including different representations of creep and hysteresis taking into account the effects of interest (rate dependency, temperature effects,etc.). This model is not intended for representing the polarization inversion caused in a piezoelectric layer when a sufficiently high reverse electric field is applied to it resulting in the typical butterfly-like displacement/voltage figure. To do this, it would be necessary to include a switching factor such as that described in [21]. In the next section the simplified model is applied to a real piezoelectric actuator.
K NL ¼
5. Results with APA-120ML
The rate dependence, which was previously distributed for every layer RNL;j is now concentrated in only one factor RNL , and the same occurs with the capacitor C NL;w representing the hysteresis between P and E, which is concentrated into the element C NL . In a linear case, the concentrated term would be easily calculated as 2 2 C NL ¼ NC NL;w l ¼ NCl , with C the capacitor value in Coulomb/Volt of a single layer given by the supplier.
The model described above has been applied to a Cedrat APA120ML actuator (Fig. 12). Two different hysteresis models are considered: the Bouc-Wen model and the Ishlinkii model as representative of the two major model types. The results show the hysteresis curve in terms of the polarization and electric field in a single layer, taking into account that the behaviour of the other layers is the same according to the approach described in Section 4. This way the results show the real level of polarization in the material. The displacement comparison between the simulated and experimental
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Fig. 12. Cedrat APA-120ML actuator placed on a test bench with an idle current sensor for obtaining the hysteresis voltage–displacement curves.
results refers to the complete actuator and not only a single layer. The APA-120ML is a mechanically amplified actuator, which means that there is a mechanical element for increasing its displacement capabilities. This element is considered in the model as a conservative power transformation and is directly inserted in the transformation between the polarization and the mechanical domain as:
E2 l E2 l ¼ aT ae aT p_1 ae p_1 x_ ¼ ¼ l l F¼
with T being the mechanical amplification. The force F and displacement x is considered after the amplification. The same occurs with the effective stiffness K and mass M, which is the data provided in the technical documentation from the supplier. The parameters of the model are listed in Table 1. 5.1. Adjustment of the rate dependency The two implementations used for modelling the Cedrat APA120ML actuator use rate-independent hysteresis models. However the model is compatible with rate-dependent hysteresis models, the adjustment of these models is complicated because of the coupled effect of both phenomena. To simplify the adjustment process, in this section, a methodology is proposed for decoupling the hysteresis from the rate-dependence phenomenon, so that it is possible to separately adjust both effects. The frequency effect is tuned with RNL , which is adjusted to eliminate the rate dependence between the polarization and electric field observed in the nonlinear capacitor in the frequency range of interest ([5, 50] Hz), which afterwards will be modelled by a rate-independent hysteresis model. The relation between E and P in the nonlinear capacitor E2 ¼ C NL ðPÞ is obtained from measurements of displacement with different voltages in unloaded conditions and from Eq. (14). For F ¼ 0 and RNL ¼ 0, the expressions of P and E are: 2
l d P¼ M 2þK Kae dt Ee ¼
!
Fig. 13. Effect of the frequency in the polarization curves.
From the expressions above and the voltage and displacement measuremens it is possible to obtain the curve between the polarization and the electric field. To avoid numerical problems the sampled xðtÞ data is not directly differentiated, but fitted to a 10 order polynomial which can be differentiated afterwards. Fig. 13 shows that the hysteresis between the polarization and the electric field without the compensating term RNL is influenced to a remarkable degree by the frequency. _ the mathematical description of P and E Selecting RNL ¼ cx, becomes: 2
l d d P¼ M 2þc þK Kae dt dt
!
x Al
! 3 2 Ve Re M d Mae d R d R d ae d x þ þ Ee ¼ þc þ Kae l dt3 Kae l dt l l dt 2 ae l dt l dt ð16Þ The value of c (Table 1) is tuned in order to reduce the rate dependence between the electric field and the polarization, finally obtaining Fig. 14, which in comparison with Fig. 13 shows a much lower dependence on the frequency. The resulting rate-independent curve is the one reproduced by the hysteresis models.
x Al
! 3 2 Ve Re M d Mae d R d x þ þ Kae l dt 3 l l dt2 ae l dt
ð15Þ
Table 1 Parameters of the system. Re
l
a
K
c
M
4:7 X
0:1 mm
0:0474 m=C
10:77 N=lm
3000 Ns=m
160 g
Fig. 14. Elimination of the rate dependence by fixing RNL ¼ c dtd x
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In next two sections, the methodology for fitting the experimental hysteresis curve in Fig. 14 is described and validated for a DE-based model and a O-based model. In both models, the rate dependency of the complete actuator model is validated afterwards by comparing the displacement/voltage curves and displacement/time curve at 5 Hz and 50 Hz. 5.2. DE-based model The Bouc-Wen model in Eq. (9) is used as being representative of the capabilities of the proposed model for implementing DEbased hysteresis modelization. The major problem of these models, and specifically of the Bouc-Wen is their high nonlinearity which makes it difficult to apply optimisation algorithms, having a high sensitivity to the initial seed. To adjust the model a similar procedure to that described in [13] is used, but reducing the number of tests by using a priori knowledge of the actuator behaviour. kx represents the linear correspondence between P and E. This relation is already known from the linear description of the piezoelectric actuator where C NL ¼ C. Therefore, for a single layer
kx ¼
N Cl
Fig. 15. Matching of the piezoactuator layer when using the Bouc-Wen model.
ð17Þ
2
Once kx is known, the sign of zðtÞ corresponds with the sign of the term kw zðtÞ ¼ E kx PðtÞ. The estimation of the remaining parameters is done as: At z ¼ 0, the slope of the curve is:
@ðkw zÞ ¼ kw q ¼ a @P
ð18Þ
Taking the value of the slope at two points with positive zðtÞ, the value of n and kw are obtained:
" # @ðkw zi Þ jkw zi jn ¼a 1 n @P kw with i ¼ 1; 2 and zi > 0; 8i
n¼
log 1 a1
@ðkw z2 Þ @P
log 1 a1
@ðkw z1 Þ @P
logðkw z2 Þ logðkw z1 Þ k w z1 kw ¼ w z1 Þ 1 a1 @ðk@P
ð19Þ
Fig. 16. Voltage/displacement hysteresis curve at 5 Hz with the Bouc-Wen model.
The value of r is estimated with the value of the slope at z3 < 0:
@ðkw z3 Þ ¼ að1 þ 2rjz3 jn jz3 jÞ @P ! n 1 @ðkw z3 Þ jkw z3 jn kw r¼ 1 þ n a @P 2jkw z3 jn kw
ð20Þ
The values estimated above are used as a seed for the final optimisation algorithm, which is intended to reduce the error function of the complete piezoelectric model:
¼
N X
ðxexp ðV i ; wi Þ xðV i ; wi ÞÞ2
ð21Þ
i¼0
with xðV i ; wi Þ and xexp ðV i ; wi Þ being the values of the simulated and experimental displacements for a given voltage V i and frequency wi in the hysteresis loop. Fig. 15 shows the matching of the Bouc-Wen model with the rate-independent hysteresis cycle in Eq. (14). The inherent symmetry of the Bouc-Wen model makes it impossible to match the asymmetry of the experimental results.
Fig. 17. Voltage/displacement hysteresis curve at 50 Hz with the Bouc-Wen model.
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Fig. 18. Displacement evolution at 5 Hz with the Bouc-Wen model.
This symmetry could be eliminated by modifying the standard Bouc-Wen model in a similar way as [3] and reference therein do in a Maxwell model, but it is out of the scope of this work, in which the objective is to prove the compatibility of DE-based models with the proposed modelling framework. The asymmetry is also noticeable in dynamic tests, as can be observed in Figs. 16 and 17. At 50 Hz and 5 Hz the deviation shows the same tendency as in Fig. 15. All the same, the deviation is less evident in a temporal scale as that appearing in Figs. 18 and 19. The simplicity of the model with just five parameters to adjust represents a good compromise in comparison to other strategies which obtain better matching but with a much higher number of parameters, as is the case of the O-based model described in next section. The adjusted parameters are listed in Table 2.
Fig. 19. Displacement evolution at 50 Hz with the Bouc-Wen model.
Table 2 Parameters of the Bouc-Wen model. Parameter
kx
kw
q
r
n
5e12
1.98e5
1.58e7
0.56
1.08
5.3. O-based model The modified Ishlinkii model used in this work is, like the BoucWen, highly nonlinear (backlash, dead zone). This complicates the application of standard optimisation algorithms for estimating the model parameters. Fortunately, the obtention of a first seed for the optimisation in the Ishlinkii model is simplified by the relationship between the slope of the hysteresis curve and the weighting factors, as is mentioned in [12]. Based on this correspondence it is
Fig. 20. Relation between the shape of the hysteresis curve and the value of the adjustable parameters of the Ishlinkii model.
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Fig. 21. Matching of the piezoactuator behaviour when using the Bouc-Wen model. Fig. 23. Voltage/displacement hysteresis curve at 50 Hz with the Ishlinkii model.
Fig. 22. Voltage/displacement hysteresis curve at 5 Hz with the Ishlinkii model. Fig. 24. Displacement evolution at 5 Hz with the Ishlinkii model.
possible to automate the adjustment of the model according to the procedure proposed below (see Fig. 20): The entire input range ðP0 ; PNp Þ is divided into N p points P i and the values of the N p constants ri ; i ¼ 0 . . . N p 1 are fixed at P i =2. The N p constants wi ; i ¼ 1 . . . N p 1 are defined to match the slopes of the curve in the areas I for i ¼ 0 . . . N p =2 and II for i ¼ N p =2 þ 1 . . . N p of the curve. The value of r0 is fixed at 0 and serves to fix the initial slope of the curve. The N p =2 constants dj ; j ¼ 0 . . . N p =2 are defined from ENp =2 þ 1 to ENp . As appears in Fig. 20, dj are relative values with respect to the change of direction value, which means that the model must detect the changes in the electric field sign in order correctly to apply the dead zone constants to the relative electric field. This is necessary to assure that the shape of the hysteresis loop depends on the amplitude of the electric field signal and not on its absolute value. The N p =2 constants ws;j ; j ¼ 0 . . . N p =2 are defined to match the slopes in area III of the curve.
After this first approach, area IV of the curve shows the most significant deviations with respect to the experimental data. To improve this, it is possible to arrange an optimisation of the complete actuator model as with the Bouc-Wen model using the previous adjustment as initial seed for the algorithm. The result of this adjustment is shown in Fig. 21. The frequency has an effect in the curves as appears in Figs. 22 and 23. The slight deviations observed in the hysteresis curve are less patent in the temporal scale (Figs. 24 and 25), as already seen with the DE-based model. The matching with the experimental results is noticeably better than with the DE-based model. However, the O-based model used for the APA-120ML actuator uses fifteen parameters for its adjustment and is formed by a sharp nonlinear backlash and dead zone operators distributed throughout its trajectory, which complicates its mathematical manipulation. The adjusted parameters are listed in Table 3.
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O-based models, such as the Ishlinkii usually have a better matching with the experimental data due to the larger number of parameters to adjust. In contrast, these types of model have harder nonlinearities than DE-based models, such as Bouc Wen, which can complicate their application in some control strategies. However, both typologies have proved their usefulness in control strategies based on the inverse model calculation and feedforward. The adjustment of nonlinear piezoelectric materials is a significant problem due to the difficulty in using standard optimisation algorithms. In this connection, the present work establishes a methodology for the experimental identification of the model parameters. The modelling framework described in this work has succesfully been used in the design of control strategies both for micropositioning and active vibration control. Acknowledgment Thanks to J.R. Sierra for his support and efficient collaboration in the design of the test bench and the obtention of the experimental data used in the present work. Fig. 25. Displacement evolution at 50 Hz with the Ishlinkii model.
References Table 3 Parameters of the Ishlinkii’s model. ri
w1
dj
ws;j
0 2.2e9 5.5e9 1.1e8 1.7e8 2.5e8 1.0e7 1.4e7 1.6e7 1.8e7 2.0e7 2.0e7
6.64e12 1.46e12 0 0.25e12 0 0.75e12 0.80e12 0.61e12 0.43e12 0.42e12 2.74e12 0
0.94e6 1.10e6 1.22e6 1.28e6 1.30e6 1.30e6 0 0 0 0 0 0
0.09 0.15 0.33 1.71 1.33 0 0 0 0 0 0 0
6. Conclusions The present work proposes a general Bond Graph structure for linear and nonlinear modelling of piezoelectric layers with polarization through their thickness, which can handle the most important physical phenomena in their behaviour, such as direct and converse piezoelectric effects, hysteresis, creep and rate dependence. In order to consider the power transformation between the electrics and mechanics and its effect on the piezoelectric material, the model has an intermediate domain where the dipole and electric field in the material are defined. This way, the model merges the macroscopic information, that is to say force, displacement, voltage and current, and obtains an estimation of the material microscopic polarization state, being of great interest for analysing possible material limitations before arranging real tests. Furthermore, the model succeeds in creating a single framework for a complete actuator, considering multiple-layer stacks with or without mechanical amplification. This represents a remarkable advance in comparison with previous approaches mainly devoted to particular cases. Experimentally, the model has proved its flexibility for use with different hysteresis type models by successfully implementing both a Bouc Wen and an Ishlinkii representation for an APA-120ML actuator supplied by the Cedrat Groupe. The use of one type of representation or another mainly depends on the posterior use of the model obtained.
[1] Hanieh AA, Preumont A, Loix N. Piezoelectric Stewart platform for general purpose active damping and precision control. In: 9th European space mechanism and tribology symposium, Liege, Belgium; September 2001. [2] Holterman J, de Vries TJA. Active damping within an advanced microlithography system using piezoelectric smart discs. Mechatronics 2004;14:15–34. [3] Yeh TJ, Hung RF, Lu SW. An integrated physical model that characterizes creep and hysteresis in piezoelectric actuators. Simulat Modell Pract Theory 2008;16:93–110. [4] Pasco Y, Berry A. Consideration of piezoceramic actuator nonlinearity in the active isolation of deterministic vibration. J Sound Vib 2006;289:481–508. [5] Shen JC, Jywe WY, Chiang HK, Shu YL. Precision tracking control of a piezoelectric-actuated system. Prec Eng 2008;32:71–8. [6] Stepanenko Y, Su CY, Intelligent control of piezoelectric actuators. In: Proceedings of the 37th IEEE conference on decision and control, vol. 4; 1998. p. 4234–39. [7] Smith RC, Ounaies Z. A domain wall model for hysteresis in piezoelectric materials. J Intel Mater Syst Struct 2000;11(1):62–79. [8] Smith RC, Hatch AG, Mukherjee B, Liu S. A homogenized energy model for hysteresis in ferroelectric materials: general density formulation. J Intel Mater Syst Struct 2005;16(9):713–32. [9] Kuhnen K, Janocha H. Real-time compensation of hysteresis and creep in piezoelectric actuators. Sensors Actuator, Phys A 2000;79:83–9. [10] Changhai R, Lining S. Hysteresis and creep compensation for piezoelectric actuator in open-loop operation. Sensors Actuator A 2005;122:124–30. [11] Croft D, Shed G, Devasia S. Creep, hysteresis, and vibration compensation for piezoactuators: atomic force microscopy application. ASME J Dyn Syst Measure Control 2001;123:3543. [12] Ang WT, Garmón FA, Khosla PK, Riviere CN. Modeling rate-dependent hysteresis in piezoelectric actuators. In: Proceedings of the 2003 IEEE/RSJ international conference on intelligent robots and systems. Las Vegas, Nevada; October 2003. [13] Gomis-Bellmunt O, Ikhouane F, Castell-Vilanova P, Bergas-Jan J. Modeling and validation of a piezoelectric actuator. Electron Eng 2007;89:629–38. [14] Winters SE, Chung JH, Velinsky SA. Modeling and control of a deformable mirror. J Dyn Syst, Measure, Control 2002;124(2):297–302. [15] Chen W, Lynch CS. A micro-electro-mechanical model for polarization switching of ferroelectric materials. Acta Mater 1998;46(15):5303–11. [16] Kamopp DC, Margolis DL, Rosenberg RC. System dynamics:a unified approach. 2nd ed. John Wiley and sons; 1990. [17] Breedveld PC, Rosenberg RC, Zhou T. Bibliography of bond graph theory and applications. J. Franklin Inst. 1991;328(5/6):1067–87. [18] Oh JH. Semilinear Duhem model for rate-independent and rate-dependent hysteresis. IEEE Trans Autom Control 2005;50(5):631–45. [19] Ang WT, Khosla PK, Riviere CN. Feedforward controller with inverse ratedependent model for piezoelectric actuators in trajectory-tracking applications. IEEE/ASME Trans Mech 2007;12(2):134–42. [20] Bolten D, Bottberg U, Waser R. Reversible and irreversible piezoelectric and ferroelectric response in ferroelectric ceramics and thin films. J Eur Ceram Soc 2004;24:725–32. [21] Chen W, Lynch CS. A micro–electro-mechanical model for polarization switching of ferroelectric materials. Acta Mater 1998;46(15):5303–11. [22] IEEE Ultrasonics, Ferroelectrics, and Frequency Control Society, IEEE Standard on Piezoelectricity 176-1987, IEEE; 1988.
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[23] Damjanovic D. Ferroelectric, dielectric and piezoelectric properties of ferroelectric thin films and ceramics. Rep Prog Phys 1998;61:1267–324. [24] Gawthrop PJ, Bhikkaji B, Moheimani SOR, Physical-model-based control of a piezoelectric tube scanner. In: Proceedings of the 17th IFAC world congress, Seoul, Korea; 2008.
[25] Adriaens HJMTS, de Koning WL, Banning R. Modeling piezoelectric actuators. IEEE/ASME Trans Mech 2000;5(4):331–41. [26] M. Jang, C. Chen and J. Lee, Modeling and control of a piezoelectric actuator driven system with asymmetric hysteresis. In: IEEE international conference on systems and signals; 2005. p. 676–81.
Contents lists available at ScienceDirect
Mechatronics journal homepage: www.elsevier.com/locate/mechatronics
General Bond Graph model for piezoelectric actuators and methodology for experimental identification J.M. Rodriguez-Fortun a,*, J. Orus a, F. Buil a, J.A. Castellanos b a b
Grupo de Investigación Aplicada (GIA-MDPI), Instituto Tecnológico de Aragón, Zaragoza, Spain Departamento de Informática e Ingenierı´a de Sistemas, Centro Politécnico Superior, Universidad de Zaragoza, Spain
a r t i c l e
i n f o
Article history: Received 3 April 2009 Accepted 17 January 2010
Keywords: Piezoelectric actuators Bond Graph Hysteresis Creep effect Rate dependency Identification
a b s t r a c t Piezoelectric actuators are becoming very popular in applications such as nanopositioning, active vibration control or noise reduction, sometimes as part of so called smart structures. The possibility of accurately moving big loads on a micrometric scale and over a wide range of frequencies has resulted in an intense growth of this technology. This interest has prompted the development of a wide variety of mathematical models available for describing their behaviour, whose suitability depends on the specific application involved. This work overcomes this limitation by unifying the different modelization options for stack and stack-based actuators in a general Bond Graph model structure capable of handling the most important physical phenomena observed in these actuators, both linear, such as direct and indirect piezoelectric effects or rate dependence, and nonlinear, such as hysteresis. This model structure represents a basis for specific applications, and can be used for control or simulation purposes thanks to its high generality and adjustment capabilities. The proposed Bond Graph structure graphically shows the power flow between the electrical and mechanical frameworks of the piezoelectric actuator, and uses a modular structure for separately representing the electrical polarization of the material and its macroscopic electrical and mechanical effects. Finally, the model is successfully applied to describe the rate dependent behaviour of the Cedrat Groupe APA-120ML actuator. In this connection, an experimental identification method is described and adapted for implementing hysteresis descriptions based on simple operators or in differential equations in the model structure (O-based and DE-based models). These two typologies cover the majority of the models available, proving the generality of the proposed piezoelectric model for implementing less general or specific phenomenon descriptions into its structure. In consequence, the main contribution of this work is the development of a general framework for modelling piezoelectric actuators, comprising a graphical Bond Graph model and an adjustment procedure, which is flexible enough to embody different representations of the phenomena present in these actuators, and with a modular structure that admits different levels of complexity depending on the phenomena incorporated in the model. Ó 2010 Elsevier Ltd. All rights reserved.
1. Introduction This paper is devoted to modelling the behaviour of piezoelectric actuators operating in one direction of movement, as is the case with stack actuators, both in prestressed and free conditions, and in individual configurations or forming part of more complex elements (smart structures, benders, amplifying structures). The work presented results in a general modelling and identifying framework based on a Bond Graph representation, which clearly describes the power flows and decouply embodies the phenomena inside the actuator. * Corresponding author. Tel.: +34 976 01 1071. E-mail addresses: [email protected] (J.M. Rodriguez-Fortun), [email protected] (J. Orus), [email protected] (F. Buil), [email protected] (J.A. Castellanos). 0957-4158/$ - see front matter Ó 2010 Elsevier Ltd. All rights reserved. doi:10.1016/j.mechatronics.2010.01.004
The introduction of piezoelectric actuators in several technological applications, such as active vibration control, noise reduction, and nanopositioning, has resulted in the development of different models for simulating the behaviour of these materials. The models currently available can be divided into two groups depending on the form of the mathematical description: linear and nonlinear models. The first group mainly comprises the models based on constitutive equations, which consider both direct and indirect piezoelectric effects [22,13,23], and simplified models just considering one of those effects, depending on the final model application [1]. As furtherly explained in Section 2, the models based on constitutive equations can be represented in two different ways depending on its equivalent macroscopic model. Nowadays these descriptions are still used in active vibration control applications [1,2], which do not require high accuracy in position control. A
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further improvement of this sort of model is the inclusion of the creep effect, which sometimes is described in a linear way [3]. The main advantage of linear models is their simplicity, but if a better matching to real behaviour is needed, as is required for nanopositioning applications [3,19], the inclusion of nonlinear descriptions to reproduce hysteresis effect is necessary [25]. The different ways of representing these effects results in the different nonlinear models for piezoelectric actuators. In this connection, two major types of representations can be distinguished for hysteresis according to their mathematical descriptions: the operator based and the differential equation based models (from now on, O-based and DE-based models, respectively). In the first case, the hysteresis curve is obtained by the addition of simple operators, such as the Preisach [4,26] or Ishlinkii [5]; in the second case, an extra state variable is included in the hysteresis definition, which evolves in a different way depending on the direction of movement [6]. These hysteresis models are called phenomenological, in the sense that they are not based on a physical description of the piezoelectric materials. There are other representations that do use a physical basis for representing the hysteresis in the piezoelectric materials and which can also be summarised in the two basic types: the O-based model, such as the domain wall model [7], or the DE-based model, as in the case of the Homogeneized Energy model [8]. Moreover, it has been experimentally proved that the hysteresis phenomenon in piezoelectric materials is rate dependent. Different ways of including the rate dependency in hysteresis models appear in [3], where a damper is added to the model, or in [12], where the weighting factors of a O-based model are made rate dependent. Combined models of hysteresis and creep, which is related with the rate dependence, are described in [3,9–11]. This work classifies the different modelling possibilities, and summarizes them in a general dynamic model structure, which can further be adapted to any simulation or control purpose (some of the models hitherto cited are only intended for describing a static relation between the polarization and the voltage in the piezoelectric materials). In comparison with other nonlinear descriptions of piezoelectric actuators, as [25] which proposes a model not based in the constitutive equations using a DE-based hysteresis model, the proposed model is a general framework incrementally built from the constitutive equations and described in terms of a Bond Graph [16,17] because of its capacity for clearly defining the power flows between the mechanical and the electrical parts. Unlike previous Bond Graph descriptions for piezoelectric actuators [3,14,24], the model proposed considers separately the macroscopic electrical effects (electric charge Q, current i, voltage _ force F), V), and the mechanical effects (displacement x, speed x, while explicitly including the polarization effect in the material _ electric field E). The pro(electric dipole p, its variation with time p, posed model is validated with a real piezoelectric actuator, the APA-120ML produced by the Cedrat Groupe. This is done with a general methodology proposed for the adjustment of the model considering hysteresis and rate dependence, and the model is successfully applied to two different hysteresis descriptions: a DEbased and a O-based model. In Section 2 the linear basis is described. Section 3 discusses the capability for describing nonlinear hysteresis and rate dependency effects observed in piezoelectric actuators. The final model is derived in Section 4, and is applied in Section 5 to the dynamic modelling of the Cedrat APA-120ML amplified actuator.
zoelectric layer as an energy transformer between the mechanical and the electrical parts, taking into account the direct and converse piezoelectric effect, that is to say, the production of an electric field when stress is applied and vice versa.
T ¼ cD S þ hD E ¼ hS þ bs D
ð1Þ
where D T E S cD h bs
electric displacement ðC=m2 Þ stress ðN=m2 Þ electric field ðV=mÞ strain (–) elastic stiffness with constant electric displacement ðN=m2 Þ piezoelectric constant ðV=mÞ impermittivity at constant stress ðVm=CÞ
Integrating and considering constant values for isotropic materials, with the electric field perfectly perpendicular at the surface of the layer in the direction of polarization and behaving as a capacitor, the following assumptions can be taken:
V E ¼ rV l Z Q¼ D@A DA
ð2Þ
A
Therefore, the constitutive equations can be rewritten in terms of the input voltage V, the displacement x, the electric charge Q and the force on the actuator F:
F V
¼
cD Al
h
h
bs Al
!
x Q
where A l x
actuator surface ðm2 Þ layer thickness ðmÞ displacement ðSlÞðmÞ
The elements in the constitutive equations are represented by a more intuitive model in Figs. 1 and 2. In these figures, C i and K i represents the capacitor and the spring used for storing electrical and mechanical energy, respectively, and ei stands for the electromechanical coupling, h. The 2-port C-field in Fig. 3 contains the constitutive equations and serves for storing the energy in the electrical and mechanical parts and interchanging energy between them. The system has two external inputs: an external voltage V e and a force F.
2. Linear models of piezoelectric actuators A piezoelectric layer polarized in a single axis can be described by the constitutive equations assuming isothermal conditions and energy conservation [22,13,23]. These equations describe the pie-
ð3Þ
Fig. 1. Linear description of the piezoactuator according to model 1.
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Now, the expression of the voltage and the force is derived as in [14]:
F V
0 ¼@
K 2 þ a21C
a21C2
a21C2
1 C2
2 2
1 A
x Q
ð7Þ
This model is used in [3]. The parameters in both models are: Ci Ki ai
equivalent capacitor ðFÞ, for i ¼ 1; 2 stiffness ðN=mÞ, for i ¼ 1; 2 electromechanical transformation factor ðm=CÞ or ðV=NÞ, for i ¼ 1; 2
Fig. 2. Linear description of the piezoactuator according to model 2.
Both models just redefines the constants in (3) in terms of macroscopic electrical and mechanical elements. The differences between them arise from the way the coefficients are defined:
Fig. 3. Bond Graph representation of the piezoelectric actuator electromechanical coupling.
The coupling between the electrical and the mechanical part of the piezoelectric material ei described in the off-diagonal terms of the constitutive equations transforms the mechanical energy into electrical energy and vice versa, affecting both the mechanical model of the piezoelectric actuator, characterised by its rigidity K i , and the electrical model, characterised by C i , where i ¼ 1; 2. The two approaches for describing the C-field in terms of the macroscopic stiffness and electric capacitor in Figs. 1 and 2 depends on the assumption for the energy storage in the material: Model 1 (see Fig. 4): the energy of the system U is described as the addition of the mechanical and the electrical part of the actuator:
U¼
1 1 2 Q K 1 ðx a1 Q Þ2 þ 2 2C 1
ð4Þ
The value of the force and the voltage is obtained as @U ; V ¼ @Q and results in [14]: F ¼ @U @x
F V
¼
K1 K 1 a1
K 1 a1 Ka21
!
þ C11
x Q
ð5Þ
This description is used in [14] and in most piezoelectric suppliers technical data. Model 2 (see Fig. 5): the energy of the system is described slightly differently as:
U¼
2 1 1 x Q K 2 x2 þ 2 2C 2 a2
ð6Þ
1. In
model 1 : a1 ¼ Qx , with F ¼ 0. C 1 ¼ QV , with F ¼ 0. K 1 ¼ a1FQ , with x ¼ 0.
2. In
model 2 : a2 ¼ VF , with x ¼ 0. C 2 ¼ QV , with x ¼ 0. K 2 ¼ Fx, with V ¼ 0.
In consequence, using one model or another depends on two factors: the convenience of arranging one test or another and the causality of the model in which the piezoelectric actuator model must be included. In the rest of this work, model 1 is taken as the reference but all conclusions are applicable to model 2. 3. Nonlinear phenomena in piezoelectric actuators The behaviour of the piezoelectric actuators is characterised by the hysteresis and the rate dependency. Even though the rate dependency can be described in terms of linear factors [3], sometimes it is necessary to use nonlinear expressions [10] for better matching the experimental results. For this reason, both effects are included as nonlinear phenomena. 3.1. Hysteresis The hysteresis phenomenon, normally observed in terms of position and voltage or force and voltage, affects both the static and the dynamic operation of actuators, influencing the behaviour of controllers in tracking and positioning applications. As explained in the introduction to this work, there are different models for describing this effect, which is normally placed in the electrical part of the model, assuming that hysteresis is a consequence of reversible and irreversible domain wall motions inside the mate-
Fig. 4. Model 1 of the electromechanical coupling.
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Fig. 5. Model 2 of the electromechanical coupling.
rial [20]. Therefore the effect is represented by a nonlinear voltage and charge relationship, that is to say, a nonlinear capacitor C NL . In this connection, there are two major approaches for defining this new capacitor: Combining the linear capacitor C and a hysteresis model. In this case the hysteresis model just causes a perturbation in the voltage average value represented by the linear capacitor [10]. Another possibility is directly eliminating the capacitor and substituting it with a hysteresis model. This way the model can include the complete phenomena of saturation [7,8], depolarization or inverse polarization [15]. For the purposes of coherency with the nomenclature used in specialised works for ferroelectric ceramics [20], the proposed model uses the following terms for the physical variables when describing the hysteresis phenomenon in the piezoelectric layers: polarization density P and electric field E. The polarization density is defined as the effective dipole per volume unit in the material ðP ¼ Ql=ðlAÞ ¼ Q =AÞ. This transformation assumes that the layer behaves as a capacitor, with a constant electric field as explained in Section 2, without border effects. As indicated in the introduction to this work, two general types of models for the hysteresis effect can be considered only taking into account the shape of the mathematical description: the DEbased and the O-based models. To prove the capability of the proposed model for handling the majority of hysteresis models available, the validation results in following sections are arranged with a DE-based model, the Bouc-Wen [13], and a O-based model, the Ishlinkii [12]. The characteristics of these models are: DE-based models are expressed by means of an extra state zðtÞ _ which evolves differently according to PðtÞ. This way, the hysteresis curve follows a different line upwards and downwards:
EðtÞ ¼ hðzðtÞ; PðtÞÞ _ _ z_ ðtÞ ¼ f ðzðtÞ; PðtÞ; PðtÞÞgð PðtÞÞ
ð8Þ
This structure appears in the Bouc-Wen model, which has the form:
E ¼ kx PðtÞ þ kw zðtÞ _ n1 n _ _ z_ ðtÞ ¼ qðPðtÞ rPðtÞ Þ jzðtÞj zðtÞ þ ðr 1ÞPðtÞjzðtÞj
ð9Þ
where kx ; kw ; q; r; n are adjustable parameters of the model. The O-based model simulates the hysteresis by the addition of simple nonlinear operators:
EðtÞ ¼
n X
wi Hi ðPðtÞÞ
ð10Þ
i¼1
In the case of the Ishlinkii model, the operator Hi is the backlash as shown in Fig. 6:
Hi ðPðtÞÞ ¼ maxðPðtÞ ri ; minðPðtÞ þ r i ; Hi ðPðt 1ÞÞÞÞ where r i ; wi are adjustable parameters.
ð11Þ
Fig. 6. Backlash operator for the Ishlinkii model.
In [12] dead zone operators Sj are applied to the result of the backlash operators for eliminating the symmetry inherent to this operator result:
E¼
m X j¼1
Sj ðxÞ ¼
ws;j Sj
n X
! wi Hi ðPðtÞÞ
i¼1
maxð0; x dj Þ; 8dj > 0 x;
dj ¼ 0
ð12Þ
where dj ; ws;j are adjustable parameters. 3.2. Rate dependency Piezoelectric materials show rate dependency as is demostrated in Fig. 7 and already cited by [11,10]. This phenomenon is related to the creep effect, which is especially important in positioning applications due to the relaxation with time suffered by the material. The relevance of the rate dependency depends on the actuator type and the operation conditions. Although the dynamic model with the resistance/capacitor and the spring/mass already has a certain rate dependency, it is not normally enough for matching the experimental behaviour of the piezoelectric material. For this reason it is necessary to include in the model other elements capable of adding extra parameters to adjust the rate dependency. There are two main ways of including it in the model, both of which can be used at the same time: Using a hysteresis model with rate dependency. There are DEbased models with rate dependency as described in [18], and O-based models as that discussed in [12]. Including a viscous term in the model ðRNL Þ. In the bibliography, this term is normally placed in the mechanical part of the model and can be found as a rate dependant RNL in the force equilibrium as in [10], or as a rate dependant stiffness K modelled by a Voigt–Kelvin model with spring elements ðK i ; i ¼ 0 . . . nÞ and damping elements ðC i ; i ¼ 1 . . . nÞ [3] (Fig. 8).
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Taking into account the considerations described in the previous section for a singular piezoelectric layer, it is possible to build a model for a stack actuator made out of N layers. This model appears in Fig. 10 and contains three different physical domains: electrical macroscopic effect, polarization of the material and mechanical macroscopic effect in order to analyse separately the behaviour of the elements in the actuator. It is important to remark that the polarization density P of the material is substituted in the Bond Graph representation by the electric dipole p ¼ Ql to assure power conservation between the three domains. The following equations are extracted from the Bond Graph model in Fig. 10:
V e ¼ iRe þ V i¼
j¼N X
ij
j¼1
Fig. 7. Effect of the frequency in the position/voltage curves (the position at 7 V is fixed as reference, x ¼ 0)
V ¼ V 1;j þ ij Re;j V 1;j ¼ E1;j ¼ E2;j E3;j lj
ð13Þ
E2;j ¼ C NL;j ðp1;j Þ p_ 1;j ¼ ij l
aj p1;j E3;j lj ¼ F 1;j ¼ K NL;j xj xj1 aj lj F j ¼ M j x€j þ RNL;j ðx_ j x_ j1 Þ þ F 1;j ¼ M j €xj þ F j1
where E1;j E2;j E3;j
Fig. 8. Bond Graph representation of the Kelvin–Voigt model relaxation.
The model proposed can handle both possibilities. The results described in Section 5 have been obtained by using the second option, which is more flexible to adjust, because of the independence between both effects.
4. Final model Piezoelectric stack actuators consist of a pile of N ceramic layers electrically connected in parallel to maximize the displacement of the complete actuator, as shown in Fig. 9.
p1;j Fj FN F0 xj Re Re;j lj aj C NL;j K NL;j RNL;j Mj
electric field in the piezoelectric layer due to the electric part in layer jðV=mÞ effective electric field in the piezoelectric layer jðV=mÞ electric field in the piezoelectric layer due to the mechanical part in layer jðV=mÞ effective electric dipole in the piezoelectric layer jðCmÞ force on the piezoelectric layer jðm=CÞ force externally applied at the piezoelectric F ðm=CÞ force on the basis of the actuator ðm=CÞ displacement of the layer j with respect to an absolute reference placed on the floor ðmÞ electric resistance from amplifier to the piezoelectric actuator electric resistance from power amplifier due to the effective resistance in the layer jðXÞ thickness of the layer jðmÞ electromechanical transformation factor in the layer jðm=CÞ relation between polarization and electric field in layer jðCm2 =VÞ stiffness of the layer jðN=mÞ viscous term between the layer j and j 1ðNs=mÞ mass of the layer jðKgÞ
with j ¼ 1 . . . N. A simplification of the previous model results if all the layers in a stack are assumed to have the same properties. In this situation, if the inertial forces of each layer are disregarded, which is logical given their extremely small size, it can be assumed that each layer behaves in the same way, that is to say:
~xi ¼ ~xi1 ; 8i; with ~xi ¼ xi xi1 pi ¼ pi1 8i With these hypotheses, the Bond Graph takes the shape of Fig. 11 and the equations are reduced to:
V e ¼ iRe þ V
Fig. 9. Connection of the layers comprising the stack actuator.
i ¼ Niw V ¼ E2 E3 l E2 ¼ C NL ðpÞ
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Fig. 10. Complete piezoelectric actuator model with different piezoelectric layers.
Fig. 11. Simplified complete piezoelectric actuator model with equal piezoelectric layers.
p_ ¼ il E3 l ap ¼ F 1 ¼ K NL x a l
ap F ¼ F N ¼ Mw ð€xN þ €xN1 þ . . . þ €x1 Þ þ RNL;1 þ K NL;1 x1 1 l x ap €x _ ~ ¼ M w ð1 þ 2 þ . . . þ NÞ þ RNL;w ðxÞ þ K NL;w N lN N ap _ ¼ M€x þ K NL x þ RNL ðxÞ l
ð14Þ
where iw ; xw ; K NL;w ; Mw ; RNL;w ð~ x_ Þ refers to the value in one single layer and
K NL;w N Nþ1 M¼ Mw 2
This model is flexible enough to represent the complete phenomenology of piezoelectric actuators. It is possible to use it as a reference and a basis for including different representations of creep and hysteresis taking into account the effects of interest (rate dependency, temperature effects,etc.). This model is not intended for representing the polarization inversion caused in a piezoelectric layer when a sufficiently high reverse electric field is applied to it resulting in the typical butterfly-like displacement/voltage figure. To do this, it would be necessary to include a switching factor such as that described in [21]. In the next section the simplified model is applied to a real piezoelectric actuator.
K NL ¼
5. Results with APA-120ML
The rate dependence, which was previously distributed for every layer RNL;j is now concentrated in only one factor RNL , and the same occurs with the capacitor C NL;w representing the hysteresis between P and E, which is concentrated into the element C NL . In a linear case, the concentrated term would be easily calculated as 2 2 C NL ¼ NC NL;w l ¼ NCl , with C the capacitor value in Coulomb/Volt of a single layer given by the supplier.
The model described above has been applied to a Cedrat APA120ML actuator (Fig. 12). Two different hysteresis models are considered: the Bouc-Wen model and the Ishlinkii model as representative of the two major model types. The results show the hysteresis curve in terms of the polarization and electric field in a single layer, taking into account that the behaviour of the other layers is the same according to the approach described in Section 4. This way the results show the real level of polarization in the material. The displacement comparison between the simulated and experimental
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Fig. 12. Cedrat APA-120ML actuator placed on a test bench with an idle current sensor for obtaining the hysteresis voltage–displacement curves.
results refers to the complete actuator and not only a single layer. The APA-120ML is a mechanically amplified actuator, which means that there is a mechanical element for increasing its displacement capabilities. This element is considered in the model as a conservative power transformation and is directly inserted in the transformation between the polarization and the mechanical domain as:
E2 l E2 l ¼ aT ae aT p_1 ae p_1 x_ ¼ ¼ l l F¼
with T being the mechanical amplification. The force F and displacement x is considered after the amplification. The same occurs with the effective stiffness K and mass M, which is the data provided in the technical documentation from the supplier. The parameters of the model are listed in Table 1. 5.1. Adjustment of the rate dependency The two implementations used for modelling the Cedrat APA120ML actuator use rate-independent hysteresis models. However the model is compatible with rate-dependent hysteresis models, the adjustment of these models is complicated because of the coupled effect of both phenomena. To simplify the adjustment process, in this section, a methodology is proposed for decoupling the hysteresis from the rate-dependence phenomenon, so that it is possible to separately adjust both effects. The frequency effect is tuned with RNL , which is adjusted to eliminate the rate dependence between the polarization and electric field observed in the nonlinear capacitor in the frequency range of interest ([5, 50] Hz), which afterwards will be modelled by a rate-independent hysteresis model. The relation between E and P in the nonlinear capacitor E2 ¼ C NL ðPÞ is obtained from measurements of displacement with different voltages in unloaded conditions and from Eq. (14). For F ¼ 0 and RNL ¼ 0, the expressions of P and E are: 2
l d P¼ M 2þK Kae dt Ee ¼
!
Fig. 13. Effect of the frequency in the polarization curves.
From the expressions above and the voltage and displacement measuremens it is possible to obtain the curve between the polarization and the electric field. To avoid numerical problems the sampled xðtÞ data is not directly differentiated, but fitted to a 10 order polynomial which can be differentiated afterwards. Fig. 13 shows that the hysteresis between the polarization and the electric field without the compensating term RNL is influenced to a remarkable degree by the frequency. _ the mathematical description of P and E Selecting RNL ¼ cx, becomes: 2
l d d P¼ M 2þc þK Kae dt dt
!
x Al
! 3 2 Ve Re M d Mae d R d R d ae d x þ þ Ee ¼ þc þ Kae l dt3 Kae l dt l l dt 2 ae l dt l dt ð16Þ The value of c (Table 1) is tuned in order to reduce the rate dependence between the electric field and the polarization, finally obtaining Fig. 14, which in comparison with Fig. 13 shows a much lower dependence on the frequency. The resulting rate-independent curve is the one reproduced by the hysteresis models.
x Al
! 3 2 Ve Re M d Mae d R d x þ þ Kae l dt 3 l l dt2 ae l dt
ð15Þ
Table 1 Parameters of the system. Re
l
a
K
c
M
4:7 X
0:1 mm
0:0474 m=C
10:77 N=lm
3000 Ns=m
160 g
Fig. 14. Elimination of the rate dependence by fixing RNL ¼ c dtd x
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In next two sections, the methodology for fitting the experimental hysteresis curve in Fig. 14 is described and validated for a DE-based model and a O-based model. In both models, the rate dependency of the complete actuator model is validated afterwards by comparing the displacement/voltage curves and displacement/time curve at 5 Hz and 50 Hz. 5.2. DE-based model The Bouc-Wen model in Eq. (9) is used as being representative of the capabilities of the proposed model for implementing DEbased hysteresis modelization. The major problem of these models, and specifically of the Bouc-Wen is their high nonlinearity which makes it difficult to apply optimisation algorithms, having a high sensitivity to the initial seed. To adjust the model a similar procedure to that described in [13] is used, but reducing the number of tests by using a priori knowledge of the actuator behaviour. kx represents the linear correspondence between P and E. This relation is already known from the linear description of the piezoelectric actuator where C NL ¼ C. Therefore, for a single layer
kx ¼
N Cl
Fig. 15. Matching of the piezoactuator layer when using the Bouc-Wen model.
ð17Þ
2
Once kx is known, the sign of zðtÞ corresponds with the sign of the term kw zðtÞ ¼ E kx PðtÞ. The estimation of the remaining parameters is done as: At z ¼ 0, the slope of the curve is:
@ðkw zÞ ¼ kw q ¼ a @P
ð18Þ
Taking the value of the slope at two points with positive zðtÞ, the value of n and kw are obtained:
" # @ðkw zi Þ jkw zi jn ¼a 1 n @P kw with i ¼ 1; 2 and zi > 0; 8i
n¼
log 1 a1
@ðkw z2 Þ @P
log 1 a1
@ðkw z1 Þ @P
logðkw z2 Þ logðkw z1 Þ k w z1 kw ¼ w z1 Þ 1 a1 @ðk@P
ð19Þ
Fig. 16. Voltage/displacement hysteresis curve at 5 Hz with the Bouc-Wen model.
The value of r is estimated with the value of the slope at z3 < 0:
@ðkw z3 Þ ¼ að1 þ 2rjz3 jn jz3 jÞ @P ! n 1 @ðkw z3 Þ jkw z3 jn kw r¼ 1 þ n a @P 2jkw z3 jn kw
ð20Þ
The values estimated above are used as a seed for the final optimisation algorithm, which is intended to reduce the error function of the complete piezoelectric model:
¼
N X
ðxexp ðV i ; wi Þ xðV i ; wi ÞÞ2
ð21Þ
i¼0
with xðV i ; wi Þ and xexp ðV i ; wi Þ being the values of the simulated and experimental displacements for a given voltage V i and frequency wi in the hysteresis loop. Fig. 15 shows the matching of the Bouc-Wen model with the rate-independent hysteresis cycle in Eq. (14). The inherent symmetry of the Bouc-Wen model makes it impossible to match the asymmetry of the experimental results.
Fig. 17. Voltage/displacement hysteresis curve at 50 Hz with the Bouc-Wen model.
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Fig. 18. Displacement evolution at 5 Hz with the Bouc-Wen model.
This symmetry could be eliminated by modifying the standard Bouc-Wen model in a similar way as [3] and reference therein do in a Maxwell model, but it is out of the scope of this work, in which the objective is to prove the compatibility of DE-based models with the proposed modelling framework. The asymmetry is also noticeable in dynamic tests, as can be observed in Figs. 16 and 17. At 50 Hz and 5 Hz the deviation shows the same tendency as in Fig. 15. All the same, the deviation is less evident in a temporal scale as that appearing in Figs. 18 and 19. The simplicity of the model with just five parameters to adjust represents a good compromise in comparison to other strategies which obtain better matching but with a much higher number of parameters, as is the case of the O-based model described in next section. The adjusted parameters are listed in Table 2.
Fig. 19. Displacement evolution at 50 Hz with the Bouc-Wen model.
Table 2 Parameters of the Bouc-Wen model. Parameter
kx
kw
q
r
n
5e12
1.98e5
1.58e7
0.56
1.08
5.3. O-based model The modified Ishlinkii model used in this work is, like the BoucWen, highly nonlinear (backlash, dead zone). This complicates the application of standard optimisation algorithms for estimating the model parameters. Fortunately, the obtention of a first seed for the optimisation in the Ishlinkii model is simplified by the relationship between the slope of the hysteresis curve and the weighting factors, as is mentioned in [12]. Based on this correspondence it is
Fig. 20. Relation between the shape of the hysteresis curve and the value of the adjustable parameters of the Ishlinkii model.
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Fig. 21. Matching of the piezoactuator behaviour when using the Bouc-Wen model. Fig. 23. Voltage/displacement hysteresis curve at 50 Hz with the Ishlinkii model.
Fig. 22. Voltage/displacement hysteresis curve at 5 Hz with the Ishlinkii model. Fig. 24. Displacement evolution at 5 Hz with the Ishlinkii model.
possible to automate the adjustment of the model according to the procedure proposed below (see Fig. 20): The entire input range ðP0 ; PNp Þ is divided into N p points P i and the values of the N p constants ri ; i ¼ 0 . . . N p 1 are fixed at P i =2. The N p constants wi ; i ¼ 1 . . . N p 1 are defined to match the slopes of the curve in the areas I for i ¼ 0 . . . N p =2 and II for i ¼ N p =2 þ 1 . . . N p of the curve. The value of r0 is fixed at 0 and serves to fix the initial slope of the curve. The N p =2 constants dj ; j ¼ 0 . . . N p =2 are defined from ENp =2 þ 1 to ENp . As appears in Fig. 20, dj are relative values with respect to the change of direction value, which means that the model must detect the changes in the electric field sign in order correctly to apply the dead zone constants to the relative electric field. This is necessary to assure that the shape of the hysteresis loop depends on the amplitude of the electric field signal and not on its absolute value. The N p =2 constants ws;j ; j ¼ 0 . . . N p =2 are defined to match the slopes in area III of the curve.
After this first approach, area IV of the curve shows the most significant deviations with respect to the experimental data. To improve this, it is possible to arrange an optimisation of the complete actuator model as with the Bouc-Wen model using the previous adjustment as initial seed for the algorithm. The result of this adjustment is shown in Fig. 21. The frequency has an effect in the curves as appears in Figs. 22 and 23. The slight deviations observed in the hysteresis curve are less patent in the temporal scale (Figs. 24 and 25), as already seen with the DE-based model. The matching with the experimental results is noticeably better than with the DE-based model. However, the O-based model used for the APA-120ML actuator uses fifteen parameters for its adjustment and is formed by a sharp nonlinear backlash and dead zone operators distributed throughout its trajectory, which complicates its mathematical manipulation. The adjusted parameters are listed in Table 3.
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O-based models, such as the Ishlinkii usually have a better matching with the experimental data due to the larger number of parameters to adjust. In contrast, these types of model have harder nonlinearities than DE-based models, such as Bouc Wen, which can complicate their application in some control strategies. However, both typologies have proved their usefulness in control strategies based on the inverse model calculation and feedforward. The adjustment of nonlinear piezoelectric materials is a significant problem due to the difficulty in using standard optimisation algorithms. In this connection, the present work establishes a methodology for the experimental identification of the model parameters. The modelling framework described in this work has succesfully been used in the design of control strategies both for micropositioning and active vibration control. Acknowledgment Thanks to J.R. Sierra for his support and efficient collaboration in the design of the test bench and the obtention of the experimental data used in the present work. Fig. 25. Displacement evolution at 50 Hz with the Ishlinkii model.
References Table 3 Parameters of the Ishlinkii’s model. ri
w1
dj
ws;j
0 2.2e9 5.5e9 1.1e8 1.7e8 2.5e8 1.0e7 1.4e7 1.6e7 1.8e7 2.0e7 2.0e7
6.64e12 1.46e12 0 0.25e12 0 0.75e12 0.80e12 0.61e12 0.43e12 0.42e12 2.74e12 0
0.94e6 1.10e6 1.22e6 1.28e6 1.30e6 1.30e6 0 0 0 0 0 0
0.09 0.15 0.33 1.71 1.33 0 0 0 0 0 0 0
6. Conclusions The present work proposes a general Bond Graph structure for linear and nonlinear modelling of piezoelectric layers with polarization through their thickness, which can handle the most important physical phenomena in their behaviour, such as direct and converse piezoelectric effects, hysteresis, creep and rate dependence. In order to consider the power transformation between the electrics and mechanics and its effect on the piezoelectric material, the model has an intermediate domain where the dipole and electric field in the material are defined. This way, the model merges the macroscopic information, that is to say force, displacement, voltage and current, and obtains an estimation of the material microscopic polarization state, being of great interest for analysing possible material limitations before arranging real tests. Furthermore, the model succeeds in creating a single framework for a complete actuator, considering multiple-layer stacks with or without mechanical amplification. This represents a remarkable advance in comparison with previous approaches mainly devoted to particular cases. Experimentally, the model has proved its flexibility for use with different hysteresis type models by successfully implementing both a Bouc Wen and an Ishlinkii representation for an APA-120ML actuator supplied by the Cedrat Groupe. The use of one type of representation or another mainly depends on the posterior use of the model obtained.
[1] Hanieh AA, Preumont A, Loix N. Piezoelectric Stewart platform for general purpose active damping and precision control. In: 9th European space mechanism and tribology symposium, Liege, Belgium; September 2001. [2] Holterman J, de Vries TJA. Active damping within an advanced microlithography system using piezoelectric smart discs. Mechatronics 2004;14:15–34. [3] Yeh TJ, Hung RF, Lu SW. An integrated physical model that characterizes creep and hysteresis in piezoelectric actuators. Simulat Modell Pract Theory 2008;16:93–110. [4] Pasco Y, Berry A. Consideration of piezoceramic actuator nonlinearity in the active isolation of deterministic vibration. J Sound Vib 2006;289:481–508. [5] Shen JC, Jywe WY, Chiang HK, Shu YL. Precision tracking control of a piezoelectric-actuated system. Prec Eng 2008;32:71–8. [6] Stepanenko Y, Su CY, Intelligent control of piezoelectric actuators. In: Proceedings of the 37th IEEE conference on decision and control, vol. 4; 1998. p. 4234–39. [7] Smith RC, Ounaies Z. A domain wall model for hysteresis in piezoelectric materials. J Intel Mater Syst Struct 2000;11(1):62–79. [8] Smith RC, Hatch AG, Mukherjee B, Liu S. A homogenized energy model for hysteresis in ferroelectric materials: general density formulation. J Intel Mater Syst Struct 2005;16(9):713–32. [9] Kuhnen K, Janocha H. Real-time compensation of hysteresis and creep in piezoelectric actuators. Sensors Actuator, Phys A 2000;79:83–9. [10] Changhai R, Lining S. Hysteresis and creep compensation for piezoelectric actuator in open-loop operation. Sensors Actuator A 2005;122:124–30. [11] Croft D, Shed G, Devasia S. Creep, hysteresis, and vibration compensation for piezoactuators: atomic force microscopy application. ASME J Dyn Syst Measure Control 2001;123:3543. [12] Ang WT, Garmón FA, Khosla PK, Riviere CN. Modeling rate-dependent hysteresis in piezoelectric actuators. In: Proceedings of the 2003 IEEE/RSJ international conference on intelligent robots and systems. Las Vegas, Nevada; October 2003. [13] Gomis-Bellmunt O, Ikhouane F, Castell-Vilanova P, Bergas-Jan J. Modeling and validation of a piezoelectric actuator. Electron Eng 2007;89:629–38. [14] Winters SE, Chung JH, Velinsky SA. Modeling and control of a deformable mirror. J Dyn Syst, Measure, Control 2002;124(2):297–302. [15] Chen W, Lynch CS. A micro-electro-mechanical model for polarization switching of ferroelectric materials. Acta Mater 1998;46(15):5303–11. [16] Kamopp DC, Margolis DL, Rosenberg RC. System dynamics:a unified approach. 2nd ed. John Wiley and sons; 1990. [17] Breedveld PC, Rosenberg RC, Zhou T. Bibliography of bond graph theory and applications. J. Franklin Inst. 1991;328(5/6):1067–87. [18] Oh JH. Semilinear Duhem model for rate-independent and rate-dependent hysteresis. IEEE Trans Autom Control 2005;50(5):631–45. [19] Ang WT, Khosla PK, Riviere CN. Feedforward controller with inverse ratedependent model for piezoelectric actuators in trajectory-tracking applications. IEEE/ASME Trans Mech 2007;12(2):134–42. [20] Bolten D, Bottberg U, Waser R. Reversible and irreversible piezoelectric and ferroelectric response in ferroelectric ceramics and thin films. J Eur Ceram Soc 2004;24:725–32. [21] Chen W, Lynch CS. A micro–electro-mechanical model for polarization switching of ferroelectric materials. Acta Mater 1998;46(15):5303–11. [22] IEEE Ultrasonics, Ferroelectrics, and Frequency Control Society, IEEE Standard on Piezoelectricity 176-1987, IEEE; 1988.
314
J.M. Rodriguez-Fortun et al. / Mechatronics 20 (2010) 303–314
[23] Damjanovic D. Ferroelectric, dielectric and piezoelectric properties of ferroelectric thin films and ceramics. Rep Prog Phys 1998;61:1267–324. [24] Gawthrop PJ, Bhikkaji B, Moheimani SOR, Physical-model-based control of a piezoelectric tube scanner. In: Proceedings of the 17th IFAC world congress, Seoul, Korea; 2008.
[25] Adriaens HJMTS, de Koning WL, Banning R. Modeling piezoelectric actuators. IEEE/ASME Trans Mech 2000;5(4):331–41. [26] M. Jang, C. Chen and J. Lee, Modeling and control of a piezoelectric actuator driven system with asymmetric hysteresis. In: IEEE international conference on systems and signals; 2005. p. 676–81.