Didace Habineza

Micky Rakotondrabe

Yann Le Gorrec

Characterization, Modeling and H ∞ Control of n-DOF Piezoelectric Actuators: Application to a 3-DOF Precise Positioner

Keywords: Multi-axis (multi-degrees of freedom) piezoelectric actuators, hysteresis, creep, badly-damped oscillations, cross-couplings, characterization, modeling, H ∞ control

    Characterization modeling and H∞ control of n-DOF Piezoelectric actuators : Application to a 3-DOF precise positioner.

INTRODUCTION

Piezoelectric based actuators are very recognized as basis of precise positioning systems in nano/micro scale tasks. Comparatively to magnetostrictive, electrostatic and thermal actuators, piezoelectric actuators provide a higher positioning resolution, a large bandwidth and an ease of manufacturing and integration in positioning systems such as scanning and atomic force microscopy [START_REF] Binnig | Atomic Force Microscope[END_REF][START_REF] Binnig | Single-tube three-dimensional scanner for scanning tunneling microscopy[END_REF][START_REF] Santosh | A survey of Control Issues in Nanopositioning[END_REF] or in microrobotics [START_REF] Agnus | Robotic Microassembly and micromanipulation at FEMTO-ST[END_REF]. In addition, beyond Authors are with FEMTO-ST Institute, AS2M department, Université Bourgogne Franche-Comté/ Université de Franche-Comté/CNRS/ENSMM, 24 rue Savary, 25000 Besanc ¸on, France.

This work has been supported by the national ANR-JCJC C-MUMS-project (National young investigator project ANR-12-JS03007.01: Control of Multivariable Piezoelectric Microsystems with Minimization of Sensors). This work has also been supported by the Labex ACTION project (contract "ANR-11-LABX-01-01"). their actuation ability, piezoelectric actuators offer the possibility to be used also as sensors, by exploiting simultaneously piezoelectric direct and converse effects. Such possibility is called self-sensing [START_REF] Dosch | A Self-Sensing Piezoelectric Actuator for Collocated Control[END_REF][START_REF] Moheimani | A new piezoelectric tube scanner for simultaneous sensing and actuation[END_REF][START_REF] Rakotondrabe | Dynamic displacement self-sensing and robust control of cantilevered piezoelectric actuators dedicated to microassembly tasks[END_REF][START_REF] Rakotondrabe | Simultaneous displacement and force self-sensing in piezoelectric actuators and applications to robust control of the displacement[END_REF].

The most used piezoelectric actuators are piezocantilevers which are cantilevers with rectangular section and which are mainly used for microassembly and micromanipulation tasks [START_REF] Agnus | Robotic Microassembly and micromanipulation at FEMTO-ST[END_REF][START_REF] Rakotondrabe | Robust Feedforward-Feedback Control of a Nonlinear and Oscillating 2-DOF Piezocantilever[END_REF], piezostacks which are used for positioning and switching tasks [START_REF] Goldfarb | Modeling Piezoelectric Stack Actuators for Control of Micromanipulation[END_REF][START_REF] Mitsuhashi | Relay Using Multilayer Piezoelectric Actuator[END_REF], and piezo tube scanners (PTS) which are mainly used in scanning probe and atomic force microscopy [START_REF] Binnig | Single-tube three-dimensional scanner for scanning tunneling microscopy[END_REF].

Despite the advantages stated above, piezoelectric based actuators exhibit nonlinearities (hysteresis and creep), badly damped oscillations and high sensitivity to the thermal variation [START_REF] Santosh | A survey of Control Issues in Nanopositioning[END_REF][START_REF] Vandervoort | Full temperature calibration from 4 to 300 K of the voltage response of piezoelectric tube scanner PZT-5A for use in scanning tunneling microscopes[END_REF][START_REF] Rakotondrabe | Smart materials-based actuators at the micro/nano-scale: characterization, control and applications[END_REF]. In addition, when dealing with multi-axis actuators, there are crosscouplings between the different axes that cause additional positioning inaccuracy [START_REF] Chung | Adaptive Sliding Mode Control of Piezoelectric Tube Actuator With Hysteresis, Creep and Coupling Effect[END_REF][START_REF] Habineza | Bouc-Wen Modeling and Feedforward Control of multivariable Hysteresis in Piezoelectric Systems: Application to a 3-DoF Piezotube scanner[END_REF][START_REF] Rakotondrabe | Modeling and Compensation of Multivariable Creep in multi-DOF Piezoelectric Actuators[END_REF].

To tackle these disadvantages, many research on their control have been conducted which include openloop and closed-loop techniques. Open-loop control of piezoelectric actuators is essentially based on model inversion, where the considered behavior (hysteresis, creep or oscillations) is first modelled and then the inverse model or its approximation is used as compensator, see for instance [START_REF] Santosh | A survey of Control Issues in Nanopositioning[END_REF][START_REF] Rakotondrabe | Smart materials-based actuators at the micro/nano-scale: characterization, control and applications[END_REF][START_REF] Habineza | Bouc-Wen Modeling and Feedforward Control of multivariable Hysteresis in Piezoelectric Systems: Application to a 3-DoF Piezotube scanner[END_REF][START_REF] Rakotondrabe | Complete open loop control of hysteretic, creeped and oscillating piezoelectric cantilever[END_REF][START_REF] Rakotondrabe | Modeling and Compensation of Multivariable Creep in multi-DOF Piezoelectric Actuators[END_REF][START_REF] Croft | Creep, hysteresis, and vibration compensation for piezoactuators: atomic force microscopy application[END_REF][START_REF] Jung | New openloop actuating method of piezoelectric actuators for removing hysteresis and creep[END_REF][START_REF] Leang | Feedforward control of piezoactuators in atomic force microscope systems[END_REF][START_REF] Rakotondrabe | Classical Prandtl-Ishlinskii modeling and inverse multiplicative structure to compensate hysteresis in piezoactuators[END_REF][START_REF] Rakotondrabe | Bouc-Wen modeling and inverse multiplicative structure to compensate hysteresis nonlinearity in piezoelectric actuators[END_REF][START_REF] Janaideh | An analytical generalized Prandtl-Ishlinskii model inversion for hysteresis compensation in micropositioning control[END_REF] and references herein. This kind of control suffers from the lack of robustness face to model uncertainties and to external disturbances. Therefore, many works proposing feedback control, or open-loop combined with feedback techniques, have been raised, see for instance [START_REF] Santosh | A survey of Control Issues in Nanopositioning[END_REF][START_REF] Tan | Micropositioning of linear-piezoelectric motors based on a learning nonlinear PID controller[END_REF][START_REF] Kouno | A fast response piezoelectric actuator for servo correction of systematic errors in precision machining[END_REF][START_REF] Tamer | Feedback control of piezoelectric tube scanners[END_REF][START_REF] Lei | Discrete composite control of piezoelectric actuators for high-speed and precision scanning[END_REF][START_REF] Salapaka | Design, Identification and Control of a Fast Nanopositioning Device[END_REF][START_REF] Chuang | Robust H Infinity Control of Hysteresis in a Piezoelectric Stack Actuator[END_REF][START_REF] Rakotondrabe | Quadrilateral modelling and robust control of a nonlinear piezoelectric cantilever[END_REF][START_REF] Rakotondrabe | Plurilinear modeling and discrete -synthesis control of a hysteretic and creeped unimorph piezoelectric cantilever[END_REF][START_REF] Leang | Feedbacklinearized inverse feedforward for creep, hysteresis, and oscillation compensation in AFM piezoactuators[END_REF][START_REF] Khadraoui | Design of RST-structured controller for parametric uncertain system using interval analysis: application to piezocantilever[END_REF] and references herein. However, these works are only in concern with monovariable modeling and control.

Feedback control of multi-axis piezoelectric actuators has been investigated in [START_REF] Rakotondrabe | Smart materials-based actuators at the micro/nano-scale: characterization, control and applications[END_REF][START_REF] Kuiper | Model-based feedback controller design for dual actuated atomic force microscopy[END_REF][START_REF] Kenton | Design and Control of a Three-Axis Serial Kinematic High-Bandwidth Nanopositioner[END_REF][START_REF] Al-Thani Mahmood | Two Sensor Based H-Infinity Control of a Piezoelectric Tube Scanner[END_REF][START_REF] Daniele | Piezoelectric scanners for atomic force microscopes : design of lateral sensors, identification and control[END_REF][START_REF] Lee | Robust broadband nanopositioning: fundamental trade-offs, analysis, and design in a two-degree-of-freedom control framework[END_REF]. In these works, various feedback control techniques, including H ∞ , have been applied but the cross-couplings between the different axes of the actuator were not explicitely accounted for and were not identified for further consideration during the controller design. In [START_REF] Yong | Reducing cross-coupling in a compliant XY nanopositioner for fast and accurate raster scanning[END_REF] [40] [START_REF] Aphale | A robust loop-shaping approach to fast and accurate nanopositioning[END_REF], the cross-couplings were identified but the controllers were finally synthesized for frequency ranges outside the validity of these cross-couplings. Finally, an explicit and simultaneous consideration of the hysteresis, the creep, the dynamics and the crosscouplings in a generalized n-DOF (degrees of freedom),

has not yet been treated.

The novelty of this paper is threefold: 1) simultaneous and explicit modeling of the hysteresis, of the creep, of the dynamics and of the cross-couplings in multi-DOF piezoelectric actuators, 2) systematic way to synthesize a H ∞ controller for the yielded model, 3) and valuability of the proposed modeling and control techniques for any number of axes (n-DOF). First, we propose a systematic approach to yield n uncertain and perturbed linear monovariable systems from the initial n-DOF multivariable system. In the yielded model, the hysteresis is approximated by a linear gain followed by a direct multiplicative structured uncertainty and the creep and the cross-couplings are rassembled in a well defined and bounded fictive external disturbance. Afterwards, we propose n decoupled H ∞ controllers that explicitely account for the uncertainties, the disturbances and the specified performances. Thanks to the controllers, it is therefore possible to ensure the specifications on reference tracking in presence of the hysteresis, the creep, the badly damped vibrations and the cross-couplings. Experimental tests on a 3-DOF piezotube actuator have been carried out and demonstrated the efficiency of the proposed modeling and control approach.

The paper is organized as follows: Section II reminds the modeling for monovariable piezoelectric actuators, and proposes the extension of this latter into a multivariable modeling valuable for n-DOF piezoelectric actuators. Section III is devoted to the H ∞ control of multivariable piezoelectric actuators by using the model proposed in section II. In Section IV, the proposed modeling and H ∞ control are applied to a 3-DOF positioner. Section V evaluates the performances of the proposed H ∞ controller through various experimental tests. Finally, section VI concludes the paper.

Modeling of n-DOF piezoelectric actuators 2.1 Remind of the monovariable modeling

According to [START_REF] Pons | Emerging actuator technologies: a micromechatronic approach[END_REF], the deflection y of a cantilevered piezoelectric actuator with a nonlinear electromechanical part and subjected to an external force F and a voltage U (Fig. 1), can be expressed as: where s p , D f and F represent the cantilever elastic constant, the mechanical dynamic part and the applied force, respectively. s represents the Laplace variable and Γ an operator which includes dynamic (or ratedependent) hysteresis and creep nonlinearities affecting the cantilever. In Eq. 1, the operator Γ is a function of U and s since hysteresis depends upon the frequency or the rate and the history of the input U. In addition, the creep effect can be modelled by using dynamic transfers, which are functions of the Laplace variable s [START_REF] Rakotondrabe | Complete open loop control of hysteretic, creeped and oscillating piezoelectric cantilever[END_REF][START_REF] Croft | Creep, hysteresis, and vibration compensation for piezoactuators: atomic force microscopy application[END_REF].

y = s p D f (s) F + Γ(U, s), (1)
The force F is due to the contact between the actuator and the manipulated objects or other mechanical disturbances in the working environment. For the remainder of this paper, we consider F to be negligible and keep the case where the actuator is driven only by the voltage U: y = Γ(U, s).

(

) 2
Let us consider the modeling strategy developed in [START_REF] Rakotondrabe | Quadrilateral modelling and robust control of a nonlinear piezoelectric cantilever[END_REF] where the nonlinear operator Γ(U, s) is splitted into a dynamic or rate-dependent hysteresis operator H(U, s) and a creep model C r (s), that is:

y = Γ(U, s) = H(U, s) dynamic hysteresis +C r (s) U creep . (3)
As classically admitted [START_REF] Santosh | A survey of Control Issues in Nanopositioning[END_REF], the dynamic hysteresis H(U, s) can be decomposed into a nonlinear static part H s (U) followed by a linear dynamic part D(s). The hysteresis operator H(U, s) can therefore be decomposed as in Eq. 4, where H s (U) represents the static (or rate-independent) hysteresis and D(s) the normalized dynamic part (D(s = 0) = 1).

H(U, s) = H s (U) D(s) (4)
The rate-independent hysteresis part H s (U) can be modelled by using classical nonlinear models such as Preisach [START_REF] Preisach | Ueber die magnetische Nachwirkung[END_REF], Prandtl-Ishlinskii [START_REF] Prandtl | Ein Gedankenmodell zur kinetischen Theorie der festen Korper[END_REF] [45], Bouc-Wen [START_REF] Bouc | Forced vibration of mechanical systems with hysteresis[END_REF][47], etc. However, for sake of simplicity, [START_REF] Rakotondrabe | Quadrilateral modelling and robust control of a nonlinear piezoelectric cantilever[END_REF] proposes quadrilateral modeling, which allows to determine a linear model for hysteresis, followed by a well modelled uncertainty. The proposed model has advantage of being simple and easy to identify, comparably to the aforementioned nonlinear models. This is particularly useful for multi-DOF systems where complex models lead to a difficult and long identification procedure.

Quadrilateral approximation consists in approximating a nonlinear curve by a piecewise affine function. Hence, a non saturated hysteresis curve is sufficiently modelled by a four sided polygon as in Fig. 2. This is generally the case for piezoelectric based actuators which exhibit non-saturated hysteresis within the input voltage (U) operating range, differently to magnetostrictive or memory alloy actuators [START_REF] Janaideh | A generalized asymmetric play hysteresis operator for modeling hysteresis nonlinearities of smart actuators[END_REF].

From this quadrilateral model, let us consider K max and K min as the maximal and minimal slopes among the slopes of the four segments of the quadrilateral ABCD (slopes of straight lines Y M and Y m , respectively). We can now define three parameters: a static gain (α) which is the average (middle value) between K max and K min , the uncertainty denoted δ on the static gain and a radius r as in Eq. 5.

     α = K max +K min 2 r = K max -K min 2 -r ≤ δ ≤ r (5)
Hence, an uncertain affine model can be constructed as approximation of the rate-independent hysteresis:

H s (U) = (α + δ ) U + y 0 , (6)
where y 0 is the (hysteresis) offset (see Fig. 2). By replacing H s (U) in Eq. 4 with Eq. 6, we obtain the approximated dynamic model of the hysteresis:

H(U, s) = (α + δ ) D(s) U + y H , (7)
with y H = D(s) y 0 which we will consider as a fictive disturbance further.

The identification of K max and K min permitting to yield α and δ is performed on a hysteresis curve captured at a frequency judiciously chosen, in order to isolate the rate-independent hysteresis from the phase lag and creep effects. In fact, at high frequency, the pure static hysteresis is affected by a phase shift between the input voltage and the actuator deflection. On the other hand, at a very low frequency, additional increase in deflection, due to the creep effect, will affect the rateindependent hysteresis [START_REF] Rakotondrabe | Complete open loop control of hysteretic, creeped and oscillating piezoelectric cantilever[END_REF].

The dynamic part D(s) is identified from the step response of the actuator. In this case, systems identification techniques are used to derive the corresponding transfer function. Then, this derived transfer function is factorized in order to be normalized, i.e. in order to have D(s = 0) = 1.

Let us now combine the hysteresis approximation in Eq. 7 with the creep model in Eq. 3. We have:

y = (α + δ ) D(s) U + y H +C r (s) U, (8)
which can also be factorized as:

y = 1 + δ α α D(s) U + y H +C r (s) U. (9)
This type of uncertainty can be expressed by using a direct multiplicative structure represented by a normalized uncertainty ∆ weighted by a function W as:

y = (1 + ∆(s) W (s)) α D(s) U + y H +C r (s) U disturbance d , (10) with ||∆|| ∞ ≤ 1 W = r α = K max -K min 2 α . (11)
The implementation of the model in Eq. 10 is represented by Fig. 3.

U + y G = α D(s) + ∆ W + + d

Extension to multivariable modeling

In this section, we propose to extend the monovariable modeling described above into a multivariable modeling, by considering a square multi-DOF piezoelectric system, with n inputs and n outputs as depicted in Fig. 4.

Let the monovariable representation in Eq. 10 be expressed as: From the first equation of Eq. 12, we propose the structure of Eq. 13 as a multivariable model, relating the input vector

y = k D(s) U + y H +C r (s) U k = (1 + ∆(s) W (s)) α. (12)
U = (U 1 U 2 • • •U j • • •U n ) T and the output vector y = (y 1 y 2 • • • y j • • • y n ) T .      y 1 y 2 . . . y n      y =      k 11 k 12 • • • k 1n k 21 k 22 • • • k 2n . . . . . . . . . k n1 k n2 • • • k nn      K •      D 11 (s) D 12 (s) • • • D 1n (s) D 21 (s) D 22 (s) • • • D 2n (s) . . . . . . . . . D n1 (s) D n2 (s) • • • D nn (s)      D(s)      U 1 U 2 . . . U n      U +      y H 1 y H 2 . . . y H n      y H +      C r 11 (s) C r 12 (s) • • • C r 1n (s) C r 21 (s) C r 22 (s) • • • C r 2n (s) . . . . . . . . . C r n1 (s) C r n2 (s) • • • C r nn (s)      C r (s)      U 1 U 2 . . . U n      U ( 13
)
where the operation K •D(s) represents the Hadamard product between the two matrices:

(K • D(s)) i, j = k i j D i j (s).
By developing Eq. 13, we can derive equation relating each single output y i to the input U j and the offset y H i as follows:

y i = n ∑ j=1 k i j D i j (s)U j hysteresis + y H i direct hysteresis offset + n ∑ j=1 C r i j (s)U j creep . (14)
The subscripts i and j denote the considered output inside the vector y and the considered input inside the vector U, respectively. When i = j we have the direct transfers and when i = j, we have the cross-couplings.

By separating the direct and the cross-coupling in the hysteresis and creep terms, Eq. 14 becomes:

y i = k ii D ii (s) U i direct hysteresis + n ∑ j=1 j =i k i j D i j (s)U j couplings hysteresis + y H i direct hysteresis offset + C r ii (s) U i direct creep + n ∑ j=1 j =i C r i j (s)U j couplings creep . (15)
Remind that the modeling strategy in this paper consists in considering the hysteresis offset, the creep and the cross-couplings as external fictive disturbances. Hence, Eq. 15 becomes:

y i = k ii D ii (s) U i + y H i + d Cr ii + n ∑ j=1 j =i d H i j + n ∑ j=1 j =i d Cr i j disturbance d i , (16)
with y H i the direct hysteresis offset, and d Cr ii , d H i j , and d Cr i j , standing for the worst cases of direct creep, coupling hysteresis and coupling creep, respectively. By worst case, we mean the maximal direct creep, the maximal couplings hysteresis and the maximal couplings creep obtained over the range of utilization, i.e. obtained with the maximal input voltage. Doing this will allow us to further calculate a controller which is robust even for the worst case.

Furthermore, accounting for the cross-coupling effect as a disturbance permits to study each axis i (1 ≤ i ≤ n) individually. In the sequel, the subscript ii will be simply written as i.

Let us replace k i in Eq. 16 with the second equation of Eq. 12. The final model for axis i whose the blockdiagram is shown is Fig. 5, is:

y i = (1 + ∆ i (s) W i (s)) α i D i (s) U i + d i d i = |y H i | + |d Cr i | + ∑ n j=1 j =i |d H i j | + ∑ n j=1 j =i |d Cr i j |. (17)
Finally, from Eq. 17, the decoupled, linear and dynamic model, for a system with n inputs and n outputs, is derived (Eq. 18):

     y 1 y 2 . . . y n      =      diag      (1 + ∆ 1 (s)W 1 (s))α 1 D 1 (s) (1 + ∆ 2 (s)W 2 (s))α 2 D 2 (s) . . . (1 + ∆ n (s)W n (s))α n D n (s)                U 1 U 2 . . . U n      +      d 1 d 2 . . . d n      (18)
U i + y i G i = α i D i (s) + ∆ i W i + + d i

H ∞ control of n-DOF piezoelectric actuators

This section is devoted to a systematic way of controller synthesize for the n-DOF piezoelectric actuator presented above and which was approximated by n uncertain linear and perturbed monovariable system as described in Eq. 18. For that, we need first to characterize the hysteresis, the creep, the dynamics and the cross-couplings. This characterization is important to identify the model parameters and to characterize the uncertainties and the disturbance. Finally, after giving some specifications for the desired closed-loop performances, the H ∞ synthesis problem is presented.

Characterization of n-DOF piezoelectric actuators

The steps followed to characterize the hysteresis, the creep, the dynamics and the cross-couplings for the n-DOF piezoelectric actuator are presented in this subsection.

Hysteresis characterization

In order to obtain experimental hysteresis curves for the direct transfers and for the cross-couplings, we apply a repetitive (sinusoidal or triangular) input voltage U j (starting by j = 1) to the n-DOF system and leave all the remaining inputs (i.e. all U k with k = j and 1 ≤ k ≤ n) null. A sinusoidal shaped input will yield a hysteresis corresponding to the frequency of the excitation signal whilst a triangular input will yield a hysteresis corresponding to different frequencies (the excitation frequency and the harmonics). In general, triangular input is employed when the final utilization of the actuator will employ such signal shape, like in atomic force microscopy application [START_REF] Santosh | A survey of Control Issues in Nanopositioning[END_REF]. Then, we capture the actuator deflections y i due to this applied voltage U j . The deflection y i(i= j) = f(U j ) represents the direct hysteresis and all the remaining y i(i = j) = f(U j ) are the cross-couplings. To obtain the hysteresis map for the whole system, we repeat the same procedure for all U j (2 ≤ j ≤ n).

Creep characterization

The creep nonlinearity is characterized by following the same steps than those of the hysteresis but, instead of utilizing a sine signal, step inputs are employed. The creep is observed as the slow drift on the step responses.

For instance, in a piezoelectric cantilevered actuator with rectangular section, this drift can be observed over several tens of minutes [START_REF] Rakotondrabe | Modeling and Compensation of Multivariable Creep in multi-DOF Piezoelectric Actuators[END_REF]. The observed creep at the output y i due to a step input U j represents the direct creep when i = j, and represents the cross-couplings when i = j.

Dynamics characterization

Similarly to the creep, the dynamics are observed by applying step inputs. Then they are taken as the transient part in a very short period before the creep starts.

Specifications and weighting functions

From the system characterization above, let us specify the performances we desire for the closed-loop, in terms of tracking performances, command moderation and rejection of the disturbances. For each monovariable system schematized in Fig. 5 and controlled by a controller C i (s), we systematically introduce three weighting functions called W i e , W i U and W i d related to the signals e i (error), U i (input signal) and d i (disturbance), respectively. We therefore obtain the augmented closed-loop in Fig. 6. In this, y ire f is the reference input, ẽi and Ũi are the new outputs (also called weighted outputs) and di is a new exogenous input. Applying the stability condition when having a direct multiplicative structure uncertainty (see for instance [START_REF] Zhou | Essentials of robust control[END_REF]), the uncertainty in Fig. 5 can be moved at the output signal y i in order to have only the weighting W i in the augmented closed-loop. We thus otbain the new augmented closed-loop schematized in Fig. 7. This latter scheme is used in next sections to define the standard H ∞ problem and to compute the controller.

The desired specifications and corresponding weightings functions are defined as follows:

y ire f + + y i - C i G i + e i W i e ẽi U i W i U Ũi ∆ i W i + + W i d di
Figure 6: The closed-loop system augmented by the proposed weightings. ∆ i represents the normalized model uncertainty (||∆ i || ∞ ≤ 1).

y ire f + y i - C i G i + e i W i e ẽi U i W i U Ũi + W i d di W i ĩ Figure 7:
The representation from which the standard H ∞ problem is defined (equivalent representation of the scheme of Fig. 6 in terms of stability condition).

• Tracking performances: in order to achieve good tracking performances for the closed-loop, we impose a maximal settling time t r i , a statical error less than ε i and finally, a zero overshoot (ν i ), for each axis i of the actuator. A number of schemes to derive the gabarit 1 W i e (the inverse of the weighting W i e ) has been proposed to account for these specifications [START_REF] Rakotondrabe | Smart materials-based actuators at the micro/nano-scale: characterization, control and applications[END_REF] [50] [START_REF] Berard | La commande multivariable-Application au pilotage d'un avion[END_REF]. In this paper, we use the following gabarit:

1 W i e = θ i s + 3ε i /t r i s + 3/t r i , (19)
whith

θ i = 1 + ν i . For zero overshoot, θ i = 1.
• Moderation of command signal U j : the objective of the command moderation is to limit the input control U j and thus, avoid over voltages which may saturate or destroy the actuator. Let us impose a maximal range of voltage (U max j ) for the range of reference input y max ire f . Thus, the gabarit for the command moderation can be calculated as follows:

1

W i U = U max i y max ire f . ( 20
)
• Rejection of disturbances d i : the aim of this specification is to impose the performances on how to reject the disturbances d i . For that, we set the settling time for disturbance rejection to be less than t rd i for all the axis. In addition, we set the effect of the maximum disturbance d i to the corresponding output y i to be less than ρ = |y ire f -y i | [µm]. The statical error under the effect of disturbance (ε d i ) is then given by:

ε d i = ρ d wc i , (21)
where d wc i represents the worst case of the disturbance d i , calculated from the second equation of Eq. 17. From all that, we yield the gabarit related to the disturbance rejection:

1 W i e W i d = θ i s + 3ε d i /t rd i s + 3/t rd i (22)
• The uncertainty weightings: the uncertainty weightings (W i ) are directly calculated from the second equation of Eq. 11.

Standard H ∞ problem and computation of the controller

Having given the specifications and transformed them into gabarits (and therefore into weightings), we will use them in this subsection to synthesize the controllers C i (s) by employing the standard H ∞ technique. The objective is to come into a matricial controller C(s) of the structure represented by Eq. 23, able to satisfy the specifications imposed in the previous section.

C(s) = diag(C i (s)) (23)
The coupling terms in the controller C(s) are null due to the fact that the cross-couplings are already accounted in the disturbances d i for each axis model.

From the augmented closed-loop in Fig. 7, we yield the standard scheme for each axis as depicted in Fig. 8.

Thus, the standard H ∞ problem consists in finding a controller C i (s) stabilizing the closed loop of Fig. 8 and satisfying the following constraint:

||F l (P i ,C i )|| ∞ < γ i , (24)
where γ i > 0 is the performances gain and F l (P i ,C i ) the lower linear fractional transformation transfer, which

W e i e i y iref d i W d i U i i W U i U i e i + + + - ~P i C i W i
Gi links the exogenous inputs y ire f di T with the weighted outputs ẽi Ũi ĩ T . It is defined by:

  ẽi Ũi ẽi   = [F l (P i ,C i )] y ire f di (25)
i.e.

F l (P i ,C i ) =   W i e S i -W i e S i W i d W i U C i S i -W i U C i S i W i d W i G i C i S i W i S i W i d   , (26)
where S i represents the sensitivity function, given by:

S i = (1 + G i C i ) -1 (27)
By combining Eq. 24 with Eq. 26, the problem becomes of finding the controller C i (s) such that the following condition is satisfied:

  ||W i e S i || ∞ < γ i || -W i e S i W i d || ∞ < γ i ||W i U C i S i || ∞ < γ i || -W i U C i S i W i d || ∞ < γ i ||W i G i C i S i || ∞ < γ i ||W i S i W i d || ∞ < γ i   , (28)
which is also satisfied if the following inequalities are respected:

    |S i | < | 1 W i e |γ i |S i | < | 1 W i e W i d |γ i |C i S i | < | 1 W i U |γ i |C i S i | < | 1 W i U W i d |γ i |G i C i S i | < | 1 W i |γ i S i | < | 1 |W i W i d |γ i     . (29)
Each controller C i (s) of each problem in Eq. 24, i = 1 → n, can be afterwards calculated, in this case, by using the Doyle-Glover algorithm [START_REF] Glover | State-space formulae for all stabilizing controllers that satisfy an H ∞norm bound and relations to risk sensivity[END_REF] [START_REF] Doyle | State-space solutions to standard H 2 and H ∞ control problems[END_REF]. We use a dSPACE board and a computer (with Matlab Simulink) to generate the control signals U x , U y and U z and to further implement the controller. A high voltage amplifier (±200V ) amplifies these driving signals before supplying the piezotube. The displacements x, y and z are measured with three inductive displacement sensors from IBS company (ECL202). These sensors have a bandwidth of 15kHz, a resolution of 40nm and a measurement range of 500µm.

Hysteresis characterization

Hysteresis has been characterized by using the procedure described in Section 3.1.

We apply first a sine input signal U x of 200V and leave all other input signals (U y and U z ) equal to zero. The frequency of the sine input signal was chosen to be f = 0.1Hz. As stated in Section 2.1, the frequency at which the rate-independent hysteresis is characterized has to be judiciously chosen, in order to isolate the rateindependent hysteresis from the phase lag and creep effects. From the characteristics of the actuator used, the frequency f = 0, 1Hz was found to be a good compromise. The application of the signal U x causes the bending of the tube along X axis. However, we also observe bendings along Y and Z axis due to the cross-couplings presence. These cross-couplings can be caused by the minor misalignment of the electrodes of the actuator which results in a diagonal bending instead of purely axial bendings. Fig. 11-(a),(d) and (g), depict the displacement x vs. U x , y vs. U x and z vs. U x , respectively. Afterwards, we repeat the previous step by using a sine input U y and by letting U x and U z be equal to zero. Fig. 11-(b),(i) and (h), depict x vs. U y , y vs. U y and z vs. U y , respectively. Finally, we use a sine input U z and let U x and U y be equal to zero. Fig. 11-(c),(f) and (i) depict x vs. U z , y vs. U z and z vs. U z , respectively.

The curves in Fig. 11-(a),(e) and (i) represent the direct hysteresis. The remaining curves represent the couplings hysteresis.

Z[µm] h H (a) (b) (c) (d) (e) (f) (g) (h) (i)

Creep characterization

The creep nonlinearity was characterized by applying a step inputs of amplitude equal to 200V . In this paper, the creep is observed over 600s. Fig. 12 depict the results where we observe direct creeps (Fig. 12-(a),(e) and (i)) and the couplings creeps (Fig. 12-(b),(c),(d),(f),(g) and (h)). As we can see, the creep can have positive or negative evolution.

Dynamics characterization

Similarly to the creep, the dynamics are observed by applying step inputs of amplitude equal to 200V and observing the step response during a short period of about 20ms, corresponding to the transient part.

System modeling

We follow the modeling procedure in Section 2.2, in order to propose a linear, dynamic and decoupled For that, the initial nonlinear, oscillating and coupled 3-DOF system pictured in Fig. 14-(a) is rewritten to be equivalent to the system in Fig. 14-(b), which contains 3 monovariable systems each one with uncertainty and external fictive disturbance. Derived from Eq. 18, the model corresponding to the 3-DOF piezotube is: [START_REF] Rakotondrabe | Quadrilateral modelling and robust control of a nonlinear piezoelectric cantilever[END_REF] where W i , α i , D i (s) and d i (i ∈ {x, y, z}) are the parameters to be identified.

Z [µm] (a) (b) (c) (d) (e) (f) (g) (h) (i) U x x U x y U x z U y x U y z U y y U z x U z y U z z A m Af
U x x U x y U x z U y x U y z U y y U z x U z y U z z (a) (b) (c) (d) (e) (f) (g) (h) (i) A m A f
  x y z   =   diag   (1 + ∆ x (s)W x (s))α x D x (s) (1 + ∆ y (s)W y (s))α y D y (s) (1 + ∆ z (s)W z (s))α z D z (s)       U x U y U z   +   d x d y d z   ,

Identification of the model parameters

Identification of α i

Fig. 15 represents the quadrilateral modeling applied to each individual direct hysteresis (Fig. 11-(a),(e) and (i)), and the numerical values of slopes K max and K min for each axis.

By using the first equation of Eq. 5, the numerical values of α i are obtained:

     α x = 0.1400 α y = 0.1502 α z = 0.0150. (31)

Identification of dynamics D i (s)

According to the model in Eq. 30, the dynamics to be identified are D x (s), D y (s) and D z (s). By applying the ARMAX method to the individual experimental step responses of Fig. 13-(a),(e) and (i), we obtain transfer functions G 0 x (s), G 0 y (s) and G 0 z (s) (see Eq. 32), which are afterwards normalized in order to extract normalized dynamics D x (s), D y (s) and D z (s) (see Eq. 33).

                                     G 0 x (s) = 364.69(s+21.49)(s 2 +6371s+9.044×10 7 ) (s+5784)(s+20.26)(s 2 +728.4s+4.043×10 7 ) = 0.1496 2438s 3 +1.558×10 7 s 2 +2.208×10 11 s+4.738×10 12 s 4 +6533s 3 +4.478×10 7 s 2 +2.348×10 11 s+4.738×10 12 = 0.1496 D x (s) G 0 y (s) = 426.29(s+12.81)(s 2 -3406s+2.03×10 8 ) (s+8832)(s+11.91)(s 2 +374.1s+6.893×10 7 )
= 0.1529 2787s 3 -9.459×10 6 s 2 +5.658×10 11 s+7.251×10 12

s 4 +9218s 3 +7.234×10 7 s 2 +6.096×10 11 s+7.251×10 12 = 0.1529 D y (s) G 0 z (s) = 3.2654×10 12 (s+3.367×10 4 )(s 2 +3.308×10 4 s+6.48×10 9 ) = 0.0150 2.182×10 14 s 3 +6.675×10 4 s 2 +7.594×10 9 s+2.182×10 14 = 0.0150 D z (s) (32)
     D x (s) = 2438s 3 +1.558×10 7 s 2 +2.208×10 11 s+4.738×10 12 s 4 +6533s 3 +4.478×10 7 s 2 +2.348×10 11 s+4.738×10 12 D y (s) = 2787s 3 -9.459×10 6 s 2 +5.658×10 11 s+7.251×10 12 s 4 +9218s 3 +7.234×10 7 s 2 +6.096×10 11 s+7.251×10 12 D z (s) = 2.182×10 14 s 3 +6.675×10 4 s 2 +7.594×10 9 s+2.182×10 14 (33) Notice that D x (s = 0) = D y (s = 0) = D z (s = 0) = 1.

Identification of disturbances d i

In this section, we identify numerically the maximum amount of external disturbances (worst case) affecting each axis.

Considering a 3-DOF system of inputs U x ,U y ,U z and outputs x, y, z, the disturbances expressed by the second equation of Eq. 17 are calculated as follows: (34) From the characterization of the hysteresis and of the creep in figures Fig. 11 and Fig. 12, the numerical values of disturbances d x , d y and d z are computed as:

     d x = |y H x | +
                   d x = |5| + |5.2| + |1.5| + | -0.4| + | -0.2| + | -0.3| = 12.6µm d y = |5| + |4| + | -0.5| + |0.6| + | -0.1| + | -1.2| = 11.4µm d z = |0.6| + |0.5| + |1.5| + |0.8 + |0.3| + |0.15|| = 3.8µm. (35)

Identification of the uncertainty weightings W i

The Uncertainty weightings W i (i ∈ {x, y, z}) are calculated from the second equation of Eq.11, with K max i and K min i identified from the quadrilateral models in Fig. 15. We have:

     W x = 0.1786 W y = 0.1814 W z = 0.2000 (36)
4.5 H ∞ control of the 3-DOF piezoelectric positioner

Specifications

• Tracking performances: in order to improve the tracking performances of the actuator, we impose a settling time less than 10ms for X and Y axis, and less than 0.5ms for Z axis; a statical error less than 10%; and finally a zero overshoot for all the three axes.

• Command moderation performances: the driving voltages of the actuator should be limited to avoid its saturation. For that, we impose a maximal input voltage amplitude (U x , U y and U z ) of ±200V for the maximal range of displacement. From the hysteresis identification in Fig. 11, this range is ±30µm for both X and Y axis, and ±3µm for Z axis.

• Rejection of disturbances d x , d y and d z : we set the settling time of disturbance rejection to be less than 5ms for all the three axis. In addition, we set the effect of the maximal disturbance d wc i for each axis, to be less than 1µm.

Based on the schemes described in Section 3.2 and the gabarits in Eq. 29, these specifications have been transcribed into the following gabarits:

       1 W x e = s+60 s+600 ; 1 W x U = 200 30 ; 1 W x e W x d = s+47.62 s+600 1 W y e = s+30 s+300 ; 1 W y U = 200 30 ; 1 W y e W y d = s+5.263 s+60 1 W z e = s+60 s+6000 ; 1 W z U = 200 30 ; 1 W z e W z d = s+1558 s+6000 (37)

Calculation of the controller

The objective is to calculate the controller C(s) such that:

C =   C x (s) 0 0 0 C y (s) 0 0 0 C z (s)   . (38)
Hence, the H ∞ problem of Eq. 29 has been solved for each axis i (with i = {x, y, z}) and the three controllers C x (s), C y (s) and C z (s) have been obtained:

                             C x (
s) = 39897(s+5784)(s+600)(s+54)(s+20)(s 2 +728.4s+4.0×10 7 ) (s+1.7×10 4 )(s+600)(s+60)(s 2 +35s+692.9)(s 2 +5933s+6.3×10 7 ) γ x-opt = 1.632359 C y (s) = 11358(s+8832)(s+300)(s+11.9)(s+8.4)(s 2 +374.1s+6.8×10 7 ) (s+2.1×10 4 )(s+346.3)(s+30)(s+13)(s+1.0)(s 2 +5393s+1.5×10 8 ) γ y-opt = 1.483102 C z (s) = 2.8×105 (s+2×10 5 )(s+3.3×10 4 )(s+6×10 3 )(s+1257) (s+2×10 5 )(s+3.9×10 4 )(s+6×10 3 )(s+1627) × (s 2 +3×10 4 s+6.4×10 9 ) (s+60)(s 2 +3×10 4 s+6.6×10 9 ) γ z-opt = 2.424572 [START_REF] Yong | Reducing cross-coupling in a compliant XY nanopositioner for fast and accurate raster scanning[END_REF] where γ i-opt (i ∈ {x, y, z}) are the optimal values for the performances gains.

T i = 1 -S i = (1 + G i C i ) -1 G i C i

Hysteresis, creep and dynamics for the closedloop system

After the simulation step, the calculated controller has been implemented (Fig. 17) by using Simulink-MATLAB software , and hysteresis, creep and dynamics for the closed-loop system have been characterized. To characterize the hysteresis for the closed-loop system, input sine reference signals x re f and then y re f with an amplitude of 25µm and a frequency f = 0.1Hz have been applied successively. For z re f , we used 3µm and f = 0.1Hz. The resulting outputs x, y and z are reported in Fig. 18. We notice the ability of the calculated controller to suppress hysteresis of direct transfers and the cross-couplings amplitudes.

-2 -1 0 1 2 -2 -1 0 1 2 Zd[µm] Z[µm] (a) (b) (c) (d) (e) (f) (g) (h) (i)
Figure 18: Verification of the hysteresis for the closedloop.

To observe the creep and dynamics for the closedloop system, step inputs of 25µm for x re f and y re f , and of 3µm for z re f have been applied successively. While the creep are observed during 600s, the dynamics are observed during only 0.2ms .

The verification of the creep and dynamics for the closed-loop system is reported in Fig. 19 and Fig. 20, respectively.

In Fig. 19, we notice the suppression of the creep and the stability of the closed-loop system during the observed period. The eight disturbances d k (k = 1 • • • 8) represent impulses applied intentionally (manually) to see the impact of disturbances on the controlled system. The manual impulses are applied to the mechanical support of the actuator. As we can see, disturbances are quickly rejected. Finally, dynamics in Fig. 20 show the convenient experimental settling times, the statical errors and the overshoot thanks to the calculated controller.

(g) (h) (i) (c) (d) (e) (f)

Comparison between the open-loop and closed-loop systems: numerical values of hysteresis, creep, oscillations and cross-couplings

To evaluate the impact of the calculated controller, various evaluation techniques, including graphical comparison of the system responses before and after the controller implementation, can be used.

In this subsection, we evaluate the effect of the calculated controller by comparing the numerical values of hysteresis, creep, oscillations and cross-couplings, before (open-loop) and after the implementation of the controller (closed-loop).

In this paper, to quantify numerically the hysteresis, we use the ratio (h/H) × 100%, with h and H (see Fig. 11-(a)), the maximal residual deflection and the maximal (peak-to-peak) deflection, respectively.

To quantify the creep and oscillations, we use the expression [(A m -A f )/A f ] × 100%. For the creep, 12 and Fig. 13). For the closed-loop system, they are calculated based on results in Fig. 18, Fig. 19 and Fig. 20.

Table 1: Numerical values of hysteresis, creep and oscillations, for direct transfers.

Open-loop

Closed-loop Hyst. Creep Oscil.

Hyst. Creep Oscil. X 16.2% 17.0% 42.1% 0.01% 0.31% 1.62% Y 17.1% 7.00% 50.6% 0.01% 0.04% 0.06% Z 25.8% 12.9% 6.40% 0.00% 0.01% 3.31%

As we can see, the comparison in Tab. 1 shows that hysteresis, which was more than 16.2% in direct transfers for the open-loop system, is removed ( 0.0%). The amount of the creep decreases from more than 7.0% to less than 0.4%, and the oscillations from 42.19% to less than 3.5%.

To evaluate the cross-couplings, we quantify their amplitudes with respect to the corresponding direct transfers. The cross-coupling amplitude for axis i (i ∈ {x, y, z}) is defined by the ratio (H i U j /H i U i ) × 100%, with H i U j the maximal deflection for the cross-coupling U j → i ( j = i), and H i U i the maximal deflection of the corresponding direct transfer U i → i (the maximal deflection that we would obtain by applying U i ).

The cross-couplings amplitudes (Tab. 2) have been calculated based on Fig. 11 (for the open-loop system) and Fig. 18 (for the closed-loop system). We notice that the calculated controller lowered considerably the cross-couplings amplitudes (from 6% to less than 0.1% for X axis, from 1.9% to 0.0% for Y axis, and from 38.7% to 4.0% for Z axis).

Helix trajectories tracking

To test the ability of the closed-loop to track complex trajectories, we have forced the tip of the piezoelectric tube to track an helix shaped trajectory, by applying simultaneously a sine wave x re f of amplitude 15µm, a cosine wave y re f of amplitude 15µm, both at at f = 0.1Hz, and a staircase signal z re f increasing from 0µm to 3µm. Fig. 21 shows that the helix trajectory tracking is successfully achieved, with the tracking error less than 0.6µm for X and Y axis, and less than 0.1µm for Z axis.

Conclusion

The characterization, the modeling and the H ∞ control of n-DOF piezoelectric actuators has been treated in this paper, with an application to a 3-DOF nanopositioner. For that, instead of using a nonlinear and coupled multivariable system, we proposed to decouple the system and derive n monovariable linear systems where the hysteresis, the creep and the cross-couplings are accounted in a direct multiplicative structured uncertainty and external fictive disturbance. The proposed technique allowed to calculate H ∞ controller for each axis independently, with an explicit consideration of the cross-coupling effect. Additionally to the crosscouplings, to the creep and to the hysteresis nonlinearity, piezoelectric actuators exhibit badly damped oscillations which are considered as dynamics during the proposed modeling. The experimental tests carried out on a 3-DOF piezoelectric tube scanner have shown that the calculated controllers rejected the effect of the external disturbance successfully, and therefore the effect of the hysteresis, the creep and the crosscouplings. Furthermore, the results on an helix trajectory tracking confirmed the efficiency of the proposed control technique for spacial precise positioning and tracking. Nevertheless, the current research focused on modeling and control of piezoelectric systems with the same number of inputs and outputs. In future works, the proposed technique will be extended to under and over actuated systems modeling and control. Moreover, our future works will include the combination of the proposed technique with feedforward control schemes, in order to increase the overall bandwidth of the closed-loop system and to achieve the high tracking performances at very high frequencies.

FFigure 1 :

 1 Figure 1: Deflection y of a piezoelectric cantilever driven by a force F and a voltage U .

Figure 2 :

 2 Figure 2: Quadrilateral modeling of hysteresis.

Figure 3 :

 3 Figure 3: Implementation of the linear and dynamic model for the deflection y of a piezoelectric actuator driven by a voltage U. ∆ represents the normalized model uncertainty (||∆|| ∞ ≤ 1) and d the disturbance to the system, which includes the creep and hysteresis offset.

Figure 5 :

 5 Figure 5: Implementation of the proposed model for axis i (1 ≤ i ≤ n): cross-couplings, creep and direct hysteresis offsets are included in the disturbance d i . ∆ i represents the normalized model uncertainty (||∆ i || ∞ ≤ 1).

Figure 8 :

 8 Figure 8: The standard scheme block diagram for each axis i.

4 9 .

 9 Application to a 3-DOF nanopositioner 4.1 Presentation of the experimental setup The 3-DOF positionner used is a piezotube scanner (PT230.94 from PIceramic company) which can provide deflections along X, Y and Z axis as shown in Fig.In fact, when a positive voltage +U is applied to +x electrode and a negative voltage -U to -x electrode, the first sector expands while the second one contracts. This expansion and contraction result in a deflection of the overall piezotube along the X direction (Fig.9-(c)). The same tube deflection is obtained along the Y direction if the voltages are applied to +y and -y electrodes (Fig.9-(d)). The tube extension along Z axis is obtained by applying the same voltage (+U or -U) simultaneously on the four electrodes +x, -x, +y and -y (Fig.9-(e)).

Figure 9 :

 9 Figure 9: Piezoelectric tube structure and operation: (a) Top view, showing four external electrodes and an internal ground electrode, separated by piezoelectric material; (b) A perspective view of the piezoelectric tube; (c) deflection along X axis; (d) deflection along Y axis, and (e) elongation along Z axis.

Fig. 10

 10 presents the experimental setup.

4. 2

 2 Characterization the 3-DOF piezoelectric positioner 4.2.

Figure 11 :

 11 Figure 11: Hysteresis characterization with input sinusoidal signals of amplitude 200V and of frequency 0.1Hz. Notations h and H in (a) are used later for computation of numerical values of hysteresis.

Fig. 13 -

 13 (a),(e) and (i) show the direct step responses while Fig. 13-(b),(c),(d),(f),(g) and (h), show the cross-couplings step responses. These figures demonstrate the lightly damped oscillations in the different responses.

Figure 12 :

 12 Figure 12: Creep characterization: step responses obtained with input steps of amplitude 200V observed over a period of 600s. Notations A f and A m in (a) are used later for computation of numerical values of creep.

Figure 13 :

 13 Figure 13: Dynamics characterization: transient responses obtained with step inputs of amplitude 200V and observed during 20ms. Notations A f and A m in (a) are used later to numerically quantify the oscillations.

Figure 14 :

 14 Figure 14: (a): the nonlinear, oscillating and coupled 3-DOF system. (b): approximation into 3 monovariable systems with external fictive disturbance.

Figure 15 :

 15 Figure 15: Experimental hysteresis curves (blue-solid line), quadrilateral modeling (red-dashed line) and numerical values of slopes K max and K min , for each axis: (a), (b) and (c) represent hysteresis x vs U x , y vs U y and z vs U z , respectively.

Figure 16 :

 16 Figure 16: Open-and closed-loop frequency responses. (a), (b) and (c) for X, Y and Z axis, respectively.

Figure 17 :

 17 Figure 17: The closed-loop control of the 3-DOF positioner.

Figure 19 :

 19 Figure 19: Verification of the creep for the closed-loop.

Figure 20 :

 20 Figure 20: Verification of the transient parts for the closed-loop.

Figure 21 :

 21 Figure 21: Helix trajectory tracking.

  |d Cr xx | + |d H xy | + |d H xz | + |d Cr xy | + |d Cr xz | d y = |y H y | + |d Cr yy | + |d H yx | + |d H yz | + |d Cr yx | + |d Cr yz | d z = |y H z | + |d Cr zz | + |d H zx | + |d H zy | + |d Cr zx | + |d Cr zy |.

Table 2 :

 2 Cross-couplings amplitudes, for open-loop and closed-loop systems.

		Open-loop	Closed-loop
	X axis Y axis Z axis	H x Uy H x Ux H x Uz H x Ux H y Ux H y Uy H y Uz H y Uy H z Ux H z Uz H z Uy H z Uz	= 3.4 56.5 = 6.0% = 0.7 56.5 = 1.2% = 1.1 60.2 = 1.8% = 1.2 60.2 = 1.9% = 2.4 6.2 = 38.7% = 1.2 6.2 = 19.3%	H x y re f H x x re f H x z re f H x x re f H y x re f H y y re f H y z re f H y y re f H z x re f H z z re f H z y re f H z z re f	= 0.04 24.6 = 0.1% = 0.02 24.6 = 0.0% = 0.02 24.9 = 0.0% = 0.01 24.9 = 0.0% = 0.1 2.5 = 4.0% = 0.1 2.5 = 4.0%

Controller implementation and experimental tests5.1 Simulation of the closed-loop frequency responseBefore implementing the controller, the bandwidth of the closed-loop has been evaluated. From the sensitivity function S i (Eq. 27), let T i (Eq. 40) be the closed-loop transfer function for axis i (i = {x, y, z}).


Vincent Chalvet

Didace Habineza
email: didace.habineza@femto-st.fr

Micky Rakotondrabe

Cédric Clévy

Presentation and characterization of novel thick-film PZT microactuators

Keywords: piezoelectric, cantilever, thinning, hysteresis, creep, dynamics, characterization

We propose in this paper the characterization of a new generation of piezoelectric cantilevers called thick-films piezoelectric actuators. Based on the bonding and thinning process of a bulk PZT layer onto a silicon layer, these cantilevers can provide better static and dynamic performances compared to traditional piezocantilevers, additionally to the small dimensions.

Introduction

PZT (lead, zirconate, titanate) material is widely used in the microworld for the design of highly performant actuators and sensors. Thanks to their high displacement resolution and high bandwidth, pizoelectric actuators are prized for several applications such as in microrobotic domain (cantilevers for microtweezers [START_REF] Rakotondrabe | Development and force/position control of a new hybrid thermopiezoelectric microgripper dedicated to micromanipulation tasks[END_REF][START_REF] Agnus | Robotic Microassembly and micromanipulation at FEMTO-ST[END_REF], AFM piezo-scanner [START_REF] Zhang | Imagebased hysteresis modeling and compensation for an afm piezo-scanner[END_REF], etc) or biomedical applications [START_REF] Lescano | Kinematic analysis of a meso-scale parallel robot for laser phonomicrosurgery[END_REF]. Despite their high resolution, piezoelectric actuators present several well known nonlinearities, such as hysteresis or creep. Many studies are currently trying to overcome these nonlinearities by using compensation techniques [START_REF] Rakotondrabe | Classical Prandtl-Ishlinskii modeling and inverse multiplicative structure to compensate hysteresis in piezoactuators[END_REF][START_REF] Devasia | A survey of control issues in nanopositioning[END_REF][START_REF] Croft | Creep, hysteresis and vibration compensation for piezoactuators: atomic force microscopy application[END_REF][START_REF] Rakotondrabe | Bouc-Wen modeling and inverse multiplicative structure to compensate hysteresis nonlinearity in piezoelectric actuators[END_REF][START_REF] Habineza | Bouc-Wen Modeling and Feedforward Control of multivariable Hysteresis in Piezoelectric Systems: Application to a 3-DoF Piezotube scanner[END_REF][START_REF] Bashash | A Polynomial-Based Linear Mapping Strategy for Feedforward Compensation of Hysteresis in Piezoelectric Actuators[END_REF][START_REF] Eddy | Actuator bias prediction using lookup-table hysteresis modeling[END_REF][START_REF] Dubra | Preisach classical and nonlinear modeling of hysteresis in piezoceramic deformable mirrors[END_REF][START_REF] Venkataraman Iyer | Approximate Inversion of the Preisach Hysteresis Operator With Application to Control of Smart Actuators[END_REF][START_REF] Hughes | Preisach modeling and compensation for smart material hysteresis[END_REF], or through feedback control strategies (see [START_REF] Devasia | A survey of control issues in nanopositioning[END_REF][START_REF] Rakotondrabe | Smart materials-based actuators at the micro/nano-scale: characterization, control and applications[END_REF] and references herein).

One of the mainly used piezoactuators is the unimorph cantilever, it is made of one piezoelectric active layer bonded onto a passive elastic layer, resulting in an out of plane displacement when an electric field is applied between the two sides of the piezoelectric layer. Different technologies are used to create PZT piezoactuators. The first one uses bulk PZT layer resulting in thick cantilevers (> 200 μm) allowing the generation of large bandwidth but with small displacements. In counterpart, another fabrication process consists in the deposition of thin layer of piezoelectric materials (in the range of some micrometers), resulting in thin film cantilevers generating high displacements but with small bandwidth. In [START_REF] Bienaim E | Static / dynamic trade-off performance of PZT thickfilm micro-actuators[END_REF], we proposed a new fabrication approach resulting in the generation of thick-film piezoactuators, with novel performances both in the static and dynamic domains. In fact, the actuators developed with this technology exhibit performances that are taken from the advantages of the classical bulk PZT (large bandwidth) and from those of the thin films (large displacements).

Linear assumption and modeling of the actuators have been given for that. In this paper, we deal with the characterization of this novel generation of piezoelectric actuators, for both static and dynamics aspects in order to demonstrate that at the optimal operating range, they also exhibit hysteresis nonlinearities similarly to classical piezoelectric cantilever actuators.

Proposed piezoelectric actuator

The fabrication process of the piezoelectric actuator, detailed in [START_REF] Bienaim E | Static / dynamic trade-off performance of PZT thickfilm micro-actuators[END_REF], is based on the gold bonding (at room temperature) of a PZT bulk layer (200 μm thickness) onto a SOI (Silicon On Insulator) wafer. The SOI composed of a 5 μm device silicon layer, a 1 μm buried oxide layer (insulator), and a 500 μm handle silicon layer. Mechanical thinning and polishing of the PZT layer is then performed. A thin gold film is then deposited on the top surface of the PZT layer, which serves as a top electrode for the cantilever actuation. The gold layer used for the bonding is also used as the bottom electrode. Cantilevers are then sliced into beams with a saw, and the handle layer of the SOI wafer is etched through DRIE (Deep Reactive Ion Etching) process.

The realized unimorph cantilevers had a PZT layer of h p = 25 μm thick and a silicon elastic layer of h s = 5 μm. The length of the cantilevers is L = 4 mm, and the width w varies from 150 to 750 μm (Figure 1).

Characterization of the piezoelectric actuator

In the sequel, we use a piezoelectric cantilever actuator of 150μm of width. The characterization setup is composed of: • the piezoelectric actuator, • an optical sensor (LC2420 from Keyence companyn) capable of measuring the deflection of the cantilever (displacement) with a resolution of tens of nanometer and bandwidth in excess of 5kHz,

• and a computer and a dSPACE board used to generate the input voltage and to acquire the measurement. The sampling time is tuned to be 50μs.

Hysteresis characterization

In [START_REF] Bienaim E | Static / dynamic trade-off performance of PZT thickfilm micro-actuators[END_REF], the modeling used to estimate the performances of the actuator was linear. Here, we provide deeper characterization of the actuators which demonstrate that they are nonlinear when increasing the operating range. First, a sine input voltage of amplitude 10V is applied to the cantilever actuator used for the experiment. The frequency is chosen to be low relative to the resonant frequency in order to avoid the phase-lag effect and thus to avoid a distorted curve. However, the frequency should not be too low in order to avoid the creep effect which is another nonlinearity found at very low frequency or very low rate [START_REF] Rakotondrabe | Smart materials-based actuators at the micro/nano-scale: characterization, control and applications[END_REF]. In this case, we choose 0.1Hz which is sufficiently low face to the first resonant frequency of the cantilever and sufficiently high relative to the creep frequency domain. Figure 2 depicts the output displacement obtained from the actuator versus the input voltage. As depicted, the actuator strongly exhibit hysteresis whose amplitude exceeds h/H≈ 14%. We can also see that the gain obtained with the novel actuator is in excess of 3.7 μm/V which is much larger than that of traditional piezoelectric cantilever actuators based on PZT. The gain of these latters are generally less than 1.5μm/V [START_REF] Rakotondrabe | Smart materials-based actuators at the micro/nano-scale: characterization, control and applications[END_REF].

To go further on the characterization of the gain of the novel actuator, we apply higher amplitude than 10V for the input voltage. Figure 3 depicts the gain which corresponds to the ratio between the amplitude of the output displacement and the amplitude of the input voltage. In the same figure, we also plot the linear model that is used to estimate the displacement in [START_REF] Bienaim E | Static / dynamic trade-off performance of PZT thickfilm micro-actuators[END_REF]. As we can observe, the gain increases substantially with the input voltage amplitude. Furthermore, the static performance of the actuator is underestimated with the linear model. These experimental characterization results highlight the static performances of the novel actuator.

Creep characterization

Classical piezoelectric cantilever actuators are also known to exhibit creep nonlinearity [START_REF] Rakotondrabe | Smart materials-based actuators at the micro/nano-scale: characterization, control and applications[END_REF]. The creep is a nonlinear phenomenon observed at very low frequency or very low rate input voltage. To characterize or to model a creep, the most employed procedure consists in applying a step input voltage and observing the output response for a long period duration. Here, a step input voltage of 10V amplitude is applied. Figure 4 depicts the results which show a creep of the actuator up to c/C ≈ 70% when observed during 10 minutes. As a conclusion, similarly to the hysteresis, it is shown that the novel actuators exhibit nonlinear phenomenon and the linear model as utilized in [START_REF] Bienaim E | Static / dynamic trade-off performance of PZT thickfilm micro-actuators[END_REF] is not sufficient to estimate their performances.

Dynamic characterization

Here, we characterize the dynamic property of the actuator. For that, additionally to the experimental materials listed above, we use a SR785 spectrum analyzer (from Stanford Research Systems company). A swept sine input voltage of frequency ranging between 0.1Hz and 2kHz is generated thanks to the spectrum analyzer and the bode diagram. Figure 5 depicts the resulting frequency response of the actuator. It shows that the first resonant frequency of the actuator is around 1100Hz. This is almost twice more than classical piezoelectric cantilever actuators, see for instance in the actuators in [START_REF] Rakotondrabe | Smart materials-based actuators at the micro/nano-scale: characterization, control and applications[END_REF] which exhibit a first resonant frequency less than 600Hz. Another way to characterize the dynamic behavior of the actuator is to excite it with a brusque input voltage. A step input voltage of 10V amplitude is therefore applied. The step response is depicted in Figure 6 which shows a badly oscillating behavior of the actuator. This behavior is also observed in classical piezoelectric cantilever actuator [START_REF] Rakotondrabe | Smart materials-based actuators at the micro/nano-scale: characterization, control and applications[END_REF] and is principally due to two reasons: i) the cantilever structure, ii) and the high stiffness of the PZT piezoelectric material.

Conclusion

This paper dealt with the static and dynamic characterization of a novel piezoelectric cantilever actuator. Along the paper, it is shown that the actuator exhibit much larger gain (in excess of 3.7μm/V) and higher dynamics (first resonant frequency 1100Hz) than in classical piezoelectric PZT cantilever actuators. During the characterization, it is also shown that the classical properties, i.e. hysteresis and creep nonlinearities and badly damped oscillations, of piezoelectric actuators are found in this novel generation of actuators.

  (a) Design of the cantilever. (b) Photo of a set of cantilevers.

Figure 1 :

 1 Figure 1: Thick-film PZT unimorph cantilever.

Figure 2 :

 2 Figure 2: Experimental characterization of the hysteresis nonlinearity.

Figure 3 :

 3 Figure 3: Experimental characterization of the gain of the actuator.

Figure 4 :

 4 Figure 4: Experimental characterization of the creep nonlinearity.

Figure 5 :

 5 Figure 5: Frequency response of the actuator.

Figure 6 :

 6 Figure 6: Step response of the actuator.

Aknowledgments

This work is supported by the national ANR-JCJC C-MUMS-project (National young investigator project ANR-12-JS03007.01: Control of Multivariable Piezoelectric Microsystems with Minimization of Sensors). This work has been supported by the Labex ACTION project (contract "ANR-11-LABX-01-01").