Micky Rakotondrabe

Yassine Haddab

Philippe Lutz

Quadrilateral modelling and robust control of a nonlinear piezoelectric cantilever.

Quadrilateral Modelling and Robust Control of a nonLinear Piezoelectric Cantilever

Micky Rakotondrabe, member, IEEE, Yassine Haddab and Philippe Lutz, member, IEEE Abstract-Piezocantilevers are commonly used for the actuation of micromechatronic systems. These systems are generally used to perform micromanipulation tasks which require high positioning accuracy. However, the nonlinearities, i.e. the hysteresis and the creep, of piezoelectric materials and the influence of the environment (vibrations, temperature change, etc.) create difficulties for such a performance to be achieved. Various models have been used to take into account the nonlinearities but they are often complex. In this paper, we study a one degree of freedom piezoelectric cantilever. For that, we propose a simple new model where the hysteresis curve is approximated by a quadrilateral and the creep is considered to be a disturbance. To facilitate the modelling, we first demonstrate that the dynamic hysteresis of the piezocantilever is equivalent to a static hysteresis, i.e. a varying gain, in series with a linear dynamic part. The obtained model is used to synthesize a linear robust controller, making it possible to achieve the performances required in micromanipulation tasks. The experimental results show the relevance of the combination of the developed model and the synthesized robust H∞ controller.

Index Terms-Piezoelectric devices, hysteresis and creep, quadrilateral approximate model, robust control, micromanipulation.

I. INTRODUCTION

P Iezoelectric materials are commonly used for the ac- tuation of microsystems and microrobots. This is due to their good deformation/force ratio, high resolution and low response time. One of their major applications in the microworld is the actuation of microgrippers, as in [START_REF] Heiserman | Piezoelectric polymer micromanipulator[END_REF] [2] [START_REF] Agnus | A smart microrobot on chip: design, identification and modeling[END_REF]. A piezoelectric microgripper (Fig. 1) is generally made up of two piezocantilevers (piezoelectric cantilevers). It is used to perform micromanipulation tasks (manipulation of microobjects). According to the application, the cantilevers may be controlled in position and/or in force. In micromanipulation tasks, the displacement at the tip, i.e. the deflection, of one piezocantilever is often lower than a few tens of microns while the accuracy is submicrometric. Such systems are very sensitive to the environment. For example, environmental vibrations can produce displacements greater than the reference deflections. These environmental vibrations are often caused by surrounding electrical devices and instruments. To perform a micromanipulation task, these disturbances must be taken into account during the design of the controllers.

Due to their small size, piezocantilevers can be damaged during micromanipulation tasks. In this case, they have to be replaced. The controller must then ensure the stability and the performance in spite of the exchange of piezocantilevers, as small geometrical differences always exist between the piezocantilevers. As the size of the piezocantilever in its entirety is relatively small (a few millimeters), such a difference extensively influences the models parameters.

One of the major limitations for piezoelectric actuators to achieve high tracking accuracy is due to their inherent nonlinear behavior. It has been shown that when the deflexion δ of the cantilever becomes large, generally higher than 15% of the maximum field strength [START_REF] Zhong | Optimal control of piezoceramic actuators[END_REF], hysteresis and creep phenomena arise and the performances, notably the accuracy, of the piezoactuators are decreased. To succeed a micromanipulation task and maintain the required performances, the influences of the hysteresis and the creep must be rejected either with open loop or closed-loop techniques.

In the open loop techniques, two linearization methods are used for piezoelectric materials: the charge compensation and the model compensation. As the relation between the applied charge and the deflection is linear, the charge compensation consists in converting the voltage into charge and controlling the piezocantilever through the latter [START_REF] Main | Precision position control of piezoelectric actuators using charge feedback[END_REF]. In the model compensation, the principle is to precisely model the nonlinearities and place the corresponding inverse model in series with the real system. The accuracy of the control depends on the accuracy of the nonlinear model and its inverse. However, open loop techniques are not suitable for disturbed systems and closed-loop techniques should be used.

Concerning closed-loop techniques, feedforward-feedback and feedback methods are used. The feedforward-feedback method consists in placing an inverse hysteresis model in the feedforward loop to cancel the hysteresis, and design a linear feedback controller to improve the performances. In the feedback method, a linear approximate model is used to model the nonlinearities and to design a linear controller.

For both techniques, various nonlinear models and linear approximate models were proposed. For example, a logarithmic function [START_REF] Jung | New open-loop actuating method of piezoelectric actuators for removing hysteresis and creep[END_REF] or the Maxwell model [7] [8] were used to model the creep. To model the hysteresis, the Preisach model is the most accurate [START_REF] Ge | Generalised preisach model for hysteresis nonlinearity of piezoelectric actuators[END_REF] [10] [START_REF] Mrad | A model for voltage to displacement dynamics in piezoceramic actuators subject to dynamic-voltage excitations[END_REF]. However, due to its complexity, the real time implementation of the Preisach model is difficult [START_REF] Smith | A hysteresis model for piezoceramic materials[END_REF]. The same problem also exists in other precise models [START_REF] Gao | Modelling of piezoelectric actuator based nanopositioning system under sinusoidal excitation using multi-polynomial regression[END_REF], such as the Maxwell resistive capacitor model [START_REF] Goldfarb | A lumped parameter electromechanical model for describing the nonlinear behavior of piezoelectric actuators[END_REF] and the Bouc-Wen model (tuning voltage based model) [START_REF] Lining | Tracking control of piezoelectric actuator based on a new mathematical model[END_REF] [START_REF] Low | Modeling of a three-layer piezoelectric bimorph beam with hysteresis[END_REF]. Consequently, linear approximate models were proposed. They are the polynomial model [START_REF] Chonan | Nonlinear hysteresis compensation of piezoelectric ceramic actuators[END_REF] [13] and the variable gain -variable phase model [START_REF] Tsai | Robust tracking control of a piezoactuator using a new approximate hysteresis model[END_REF]. In the polynomial model, polynomial functions are used to approximate the ascending curve and the descending curve of the hysteresis. The control schemes based on this approximation are very easy to implement but as the coefficients of the polynomials are experimentally determined with a sine input at constant frequency and amplitude, the magnitude and rate dependence of the hysteresis were not considered [START_REF] Tsai | Robust tracking control of a piezoactuator using a new approximate hysteresis model[END_REF] [20] [START_REF] Croft | Hysteresis and vibration compensation for piezoactuators[END_REF]. Finally, the variable gain -variable phase model consists in having a linear model with a time-delay. In this approximation, the gain depends on the amplitude of the input while the timedelay depends on its frequency. As the approximate model only considers the hysteresis phenomenon, the creep is not taken into account. A detailed survey of the different control techniques for piezoactuators can be found in [START_REF] Devasia | A survey of control issues in nanopositioning[END_REF].

In this paper we present a simple model easy to identify for a one degree of freedom (dof) piezoelectric cantilever. This model, which does not need to be implemented (as in the inversion compensation method), is used to synthesize low power and low time consuming controllers. The proposed model takes into account the hysteresis and the creep. In this model, the model is approximated by a linear model subjected to uncertainty and with a varying static gain. Before detailing the new approximation, we first demonstrate that the hysteresis only affects the gain of the voltage-deflection relation. Consequently, as the hysteresis is rate-independent, the coefficients of the proposed approximate model are independent of the frequency of the input voltage. However, it will be shown that the static gain is subjected to uncertainty. So, to ensure the performances required in the micromanipulation stated above, a robust H ∞ controller is proposed at the end of the paper.

II. HYSTERESIS AND CREEP IN PIEZOELECTRIC

CANTILEVERS

In this section, we show experimentally that the hysteresis in a piezocantilever only affects the static domain. Such hysteresis is called static hysteresis or rate-independent hysteresis, as its shape is independent of the input signal frequency [START_REF] Mayergoyz | Mathematical models of hysteresis and their applications[END_REF]. The fact that the piezocantilever hysteresis is static is of great interest because the approximate model is independent of the input signal frequency. Before that, we analyze the hysteresis and the creep in a piezocantilever.

For the experiments, a unimorph piezocantilever made up of a PIC151 piezolayer and a copper layer is used. The characteristics of the cantilever are presented in Table I.

The experimental setup (Fig. 2) comprises a computer with Matlab-Simulink software, a DSpace real-time board, a voltage amplifier and a laser sensor with an accuracy of 0.5µm (from KEYENCE).

A. Description of the piezocantilever behaviour

Let Fig. 3 represent a piezocantilever under electrical and mechanical stimulations respectively U and F . The piezocantilever may be a unimorph or a bimorph (made up of two piezolayers) piezoelectric cantilever.

The relation between the deflection δ, the applied force F and the voltage U when the electromechanical part is nonlinear is as follows [START_REF] Pons | Emerging actuator technologies: a micromechatronic approach[END_REF]:

δ(s) = s p • D(s) • F (s) + Γ (U (s), s) (1)
where s p is the elastic constant, D(s) is a dynamic part, s the Laplace variable and D(0) = 1. For the rest, the Laplace Γ (U, s) is an operator that includes the hysteresis and the creep. Since this operator includes the hysteresis, it depends on the past and present values of U . In addition, it may depend on the frequency of U . This is why the notation of Γ has both U and s. Let us study the possible separation of the two nonlinearities inside Γ (U, s).

The creep can be observed when a step voltage is applied to the piezocantilever (Fig. 4-a). The drift tends towards a finite value and its amplitude depends on the step amplitude. If we consider that the drift begins just after the transient part, we can write:

Γ (U, s) = H (U, s) + C r (U, s) • e -T •s (2)
where H (U, s) is the hysteresis operator, C r (U, s) is the creep operator and T is the delay before the creep starts. However, many experiments have shown that the transient part of piezocantilevers is generally below 100ms while its creep response time is more than 3min. The delay can then be neglected from the expression (Eq. 2) (Fig. 4-b). We obtain: In the expression (Eq. 3) and according to Fig. 4-a, the static value and the transient part of the electromechanical transfer is modelled inside the hysteresis term H (U, s). So, the creep C r (U, s) can be considered as a disturbance. Hence, one of the aims of this paper is to approximate the hysteresis term by a simpler one. Before that, let us first demonstrate that H (U, s) can be divided into two more simple terms.

δ = s p • D(s) • F + H (U, s) + C r (U, s) (3)

B. Analysis of the hysteresis H(U, s)

To analyze the hysteresis, we apply a sine voltage input to the piezocantilever and no force is applied to its tip. While the amplitude of the input signal is constant (= 40V ), different frequencies are used. The experimental results show that the shape of the hysteresis depends on the input signal frequency (Fig. 5). Such a hysteresis is called dynamic hysteresis or ratedependent hysteresis [START_REF] Mayergoyz | Mathematical models of hysteresis and their applications[END_REF]. An approximate model of a dynamic hysteresis should have parameters which are also rate-dependent. This will firstly increase the complexity of the model and therefore the complexity of the controller. To avoid that, we propose an easier equivalence of the dynamic hysteresis in this section. We show experimentally that the dynamic hysteresis H(U, s) is equivalent to a static hysteresis H i (U ) followed by the linear dynamical part D(s) as shown in Fig. 6. A static hysteresis has a constant shape whatever the frequency of the input. The experimental steps are as follows.

• Step 1: We first show that the transient part of piezocantilever is independent of the amplitude of the input

1)

Step 1: independence of the transient part.

We apply a square voltage U with a given frequency (f = 0.3Hz) to the piezocantilever. Two amplitudes were used: U = 40V and U = 20V . The corresponding step responses are given in Fig. 7. Besides the creep phenomenon, we notice the similarity of the transient parts for all the responses. From the previous experiments, we conclude that the transient part is independent of the amplitude of the input. In addition, when a repeated step signal (i.e. a square signal) is applied, the transient parts are similar. It can be concluded that a dynamic part D(s) can be separated from a static part which contains the static gain. Let us use the previous notation D(s) for the transient part. To identify D(s), any step response can be used (Fig. 8). It can be approximated by a second order model [START_REF] Haddab | A microgripper using smart piezoelectric actuators[END_REF] [16]:

D(s) = 1 a • s 2 + b • s + 1 (4)
where a = 4.722 × 10 -8 sec 2 is the inertial coefficient and b = 1.304 × 10 -5 sec is the viscous coefficient. In order to confirm that the piezocantilever can be modelled by the dynamic part D(s) in series with a static part, a harmonic analysis is performed. For that, a sine voltage with 40V of amplitude is applyed to the piezocantilever, i.e. to H(U, s). the corresponding magnitude is plotted. After that, the magnitude of the identified D(s) is plotted in the same graph ((Fig. 9)). After scaling the static gain of D(s) in order to allow the comparison, it is shown that the two curves almost coincide with each other. However, in high frequency, i.e. above 10 4 [rad/s], the slopes are different. That is due to the choice of the order of D(s). Using a higher order of D(s), the deviance of the magnitudes is minimized even in high frequency. For example, Fig. 9-star-plot shows the simulation of D(s) with a sixth order model. Finally, it is experimentally shown that the shape of the magnitude of H(U, s) does not depend on the amplitude U . Fig. 10 shows that the magnitude obtained with an amplitude of 40V is similar to the one with 20V except the static gain. The difference between the static gains indicates the presence of a hysteresis inside the static part.

The harmonic analysis demonstrates that H(U, s) is similar to D(s) except for a static gain. As D(s) is amplitudeindependent, the static gain should contain the hysteresis. This is a static hysteresis.

2)

Step 2: comparison of the simulation H i .D(s) and the harmonic experiment results.

In this step, we demonstrate that H (U, s) = H i (U ) • D(s), where H i (U ) is a static hysteresis that represents the gain. For that, we compare the simulation of δ = H i (U )•D(s) with the experimental results of H (U, s) in the (U, δ)-plane. However, while D(s) is given by (Eq. 4), the simulation requires a precise model of the static hysteresis H i (U ). The Bouc-Wen hysteresis model has been chosen due to its accuracy. Applied to piezocantilevers, this model is represented by the following set of equations [START_REF] Low | Modeling of a three-layer piezoelectric bimorph beam with hysteresis[END_REF]:

δ = s p • F + (d p • U -h) ḣ = A bw • U -B bw • U • h -Γ bw • U • |h| (5)
where:

• s p and d p are respectively the elastic constant and the piezoelectric constant, • F is the external force (null for the different experiments and simulations), • h is an internal variable,

• U is the applied voltage, • A bw is a parameter which determines the amplitude of the hysteresis, • B bw et Γ bw are parameters which determine the shape of the hysteresis. The model in the (Eq. 5) can be considered as static at least until 500kHz (checked by simulation) which is very high compared to the cut-off frequency of the whole system. The parameters of the model were identified using the values given in [START_REF] Low | Modeling of a three-layer piezoelectric bimorph beam with hysteresis[END_REF] that were adjusted. We obtain:

                           d p = 1.0773 × 10 -6 [m/V ] s p = 1.931 × 10 -3 [m/N ] A bw = 6.064810 × 10 -7 [m/V ] B bw = 0.00833 V -1 Γ bw = 0.00833 V -1 (6)
The experiments consist in applying a sine voltage to the piezocantilever and tracing the corresponding (U, δ)-plane curve. Since the cantilever includes creep and hysteresis phenomena, the frequency of the sine input signal should be higher than f cc which is the cut-off frequency of the creep. With In order to confirm the previous conclusion, the same experiments and comparison were carried out with a sine voltage of an amplitude of 20V . As shown in Fig. 12, the equivalence While the Bouc-Wen model used for the simulation is a valid approximation of the hysteresis behaviour of the piezocantilever, it contains nonlinear terms and consequently the synthesis of a linear controller is difficult. This is why we propose a new approximate model. The proposed model is linear but with uncertainty. While it is not intended to be implemented, it is used to synthesize a linear controller. In the new version,

f cc ≤ 1 2•π•τc ≈
H (U, s) = H i (U ) • D(s) is still obtained. f=10Hz experimental curve f=100Hz f=200Hz U [V] U [V] U [V] δ [µm] δ [µm] δ [µm] δ [µm] U [V]

III. QUADRILATERAL APPROXIMATION AND IDENTIFICATION OF THE STATIC PART

A. Presentation of the quadrilateral approximation

From the previous result and from (Eq. 3), we have:

δ = s p • D(s) • F + H i (U ) • D(s) + C r (U, s) (7)
In this section, we are going to find a simple model linking the input voltage U and the resulting deflection δ. As a linear dynamic model is required to synthesize a linear controller, the term H i (U )•D(s) is used to derive the model while the creep C r (U, s) and the mechanical terms s p •D(s)•F are considered as disturbances. In fact, the force F is due to the contact with a manipulated object and then behaves like a disturbance. Next, the problem is to seek an approximate model of H i (U ).

For that, we propose a new approximation: the quadrilateral approximation. The principle is detailed below.

A nonlinear curve can be approximated by a piecewise affine function, which is called multilinear approximation. For example, a hysteresis curve can be represented by Fig. 13-a. Assuming that the hysteresis does not reach saturation, it can be sufficiently approximated by four segments (Fig. 13-b). Let (∆ M ) and (∆ m ) represent the two straightlines of the quadrilateral with respectively the maximal and the minimal slopes:

(∆ M ) : δ (t) = α M • U (t) + δ M (.) (∆ m ) : δ (t) = α m • U (t) + δ m (.) (8)
where α M (respectively α m ) represents the maximal (respectively minimal) slope and δ M (.) (respectively δ m (.)) represents the corresponding offset.

Let α O be the middle value of the maximal and the minimal slopes and α E be their radius, such as:

α O = αM +αm 2 α = αM -αm 2 (9)
we propose to replace the static hysteresis H i (U ) by a linear model with a nominal constant slope α O and a new offset δ H :

H i (U ) → α O • U + δ H (10)
Thereby, the hysteresis of the real system has the following characteristics:

H i (U ) = α real • U + δ H α O -α E ≤ α real ≤ α O + α E (11)
While (Eq. 10) is used as a nominal model, the (Eq. 11) indicates that the nominal model has uncertainty relative to the real system. The uncertainty is refered to the nominal static gain α O and its amplitude can be determined with the radius α E . Secondly, δ H can be considered as a disturbance.

B. Identification of the parameters

To identify α O , a (U, δ)-curve is plotted in order to obtain a hysteresis curve. For that, a sine voltage U is applied. On the one hand, the frequency has been chosen to be small (f = 0.1Hz) in order to avoid the effect of the dynamic but should be higher than the creep cut-off frequency. On the other hand, the amplitude is chosen (40V ) to be the maximal range so that the identification is done with the external loop.

When the hysteresis curve derived from the experiments is obtained, a quadrilateral is fitted to it. The sides (straightlines) of the quadrilateral are chosen to connect the vertexes (obtained at U = 40) and the middle points (obtained at U = 0) of the hysteresis curve. At the ventral points, the hysteresis amplitude is maximal. The slope of each side of the quadrilateral is computed. After computing, the lowest slope is chosen to be α m while the highest slope is α M . Fig. 14 presents the hysteresis curve and the two upper sides of the quadrilateral giving the values of α m and α M .

Thereby, the nominal static gain α O and the radius α E are computed using the (Eq. 9). We obtain α O = 502 × 10 -9 [m/V ] and α E = 123 × 10 -9 [m/V ].

C. Performance of the identified model with a varying amplitude

As we can see, the static nominal model is identified from the external loop of the static hysteresis, where the amplitude of the sine voltage is maximal (U = 40V ). Here, we show that when the amplitude varies and internal loops are obtained, the radius α E (and then the uncertainty) lower than the one when the amplitude is maximal. Remember that α E is given by (Eq. 9).

Since the model is calculated from the maximal and minimal slopes of the quadrilateral, the quadrilateral approximation of the internal loops should have slopes that are comprised in [α m , α M ]. So, α E with internal loops are still lower than α E with the external loop. To check that, experiments with varying amplitude sine voltage were performed. The results show that the internal loops are slightly tilted relative to the external loop (Fig. 15). It can be seen that the maximal slopes α internal M of the internal loops are lower than the maximal slope α M of the external loop. However, the minimal slope α m of the external loop is higher than the minimal slopes α internal m of the internal loops. The minimal slopes are never null even if the amplitude of the sine voltage is very small. We have:

   α O < α internal M < α M 0 < α internal m < α m < α O (12)
In spite of that, the values of the minimal slopes are not too different from α m . So, it can be stated that:

α internal m ≈ α m < α O < α internal M < α M (13)
Then, according to the definitions in (Eq. 9) and (Eq. 11) and the result in (Eq. 13) , we can construe that the error of the model on internal loops is lower than the error on the external loop.

IV. COMPLETE MODEL

Using the complete expression (Eq. 7) and the quadrilateral approximation (Eq. 10), we have:

δ = s p • D(s) • F + α O • D(s) • U + δ H + C r (U, s) (14)
The term α O • D(s) • U is a dynamic linear relation between the deflection and the input voltage. It is of great interest to choose it as the nominal model. On the other hand, the force can be considered as a disturbance. Actually, it is due to the contact of the cantilever with a manipulated micro-object. Because the creep is a source of inacuracy in micromanipulation, it can also be considered as a disturbance. Despite the dependence of the creep on the input U , such consideration is possible because the creep C r (U, s) is slow and very small relative to the term

α O • D(s) • U .
So, the nominal model of the piezocantilever is:

δ(s) = α O a • s 2 + b • s + 1 •U (s)+ 1 a • s 2 + b • s + 1 •δ 0 (s) (15)
where:

• the dynamic part D(s) has been replaced by the (Eq. 4),

• the nominal static gain α O completed by an uncertainty represents the real static gain:

α O -α E ≤ α real ≤ α O + α E .
• and the disturbance is defined by:

δ 0 = s p • F + (δ H + C r (U, s)) • D -1 (s) (16)
It can be stated that if the real system is linear, we have:

α E = 0 ⇔ α real = α O δ H = 0 (17)
Fig. 16 represents the scheme of the nominal system. Due to the presence of a disturbance (internal and external), a closed loop control is necessary. The choice of the controller is guided by the characteristics of the model and the specifications according to the micromanipulation tasks:

• the nominal static gain α O is subjected to uncertainty,

• the interchange of piezocantilevers as seen in the first section of this paper is often required, so there may exist other uncertainty types in the model, • as we can see in Fig. 9, the nominal model (second order) neglects fast dynamic. The higher order model fits the real system better than the nominal model, • the effects of the environment (temperature, vibrations, etc.) must be rejected, • finally, the performances required in micromanipulation should be taken into account. To obtain robust stability and robust performances considering the different environmental disturbances and the characteristics of the proposed model, a robust controller best fits our objectives. For that, a H ∞ controller has been synthesized. The performances are compared with those obtained with a P ID controller.

V. H ∞ ROBUST CONTROL OF THE PIEZOCANTILEVER

A H ∞ robust controller is synthesized using the nominal model (Eq. 15). The results are compared with those of a P ID-controller in order to show the robustness. As the controllers are intended to be implemented in an embedded computer, we design discrete controllers. Nevertheless, the best and simplest way to design a discrete H ∞ based optimal controller is via bilinear transformation [START_REF] Chen | Optimal sampled-data control systems[END_REF] [28] [START_REF] Mirkin | On discrete-time H ∞ problem with a strictly proper controller[END_REF]. The main reasons are that continuous analysis is simpler, more standard and has a greater physical sense than the discrete analysis. For these reasons, we use the steps presented in Fig. 17 to synthesize the discrete H ∞ controller. First, we derive a discret model from the (continuous) system in order to take into account the DAC-DCA and the sampling time of the computer. The transformation uses the zero-order-holder (zoh) method. Then, using this discrete model, we derive a pseudo-continuous model. A pseudo-continuous model is the transformation of a discrete model into a continuous model using the Tustin inverse bilinear transformation. It is called pseudo-continuous because it is an image of the initial continuous model using two transformations (the zoh and the Tustin bilinear transformations). In the pseudo-continuous domain, a controller can be synthesized. Finally, the discrete controller is obtained using the Tustin direct bilinear transformation of the synthesized controller. The Tustin bilinear transformation is defined as a bijective application s = f (z), with s the Laplace variable and z the discrete operator, such as:

s = 2 T s • z -1 z + 1 ( 18
)
with T s = 0.2ms the sampling time. The standard H ∞ problem consists in finding an optimal value γ > 0 and a controller K(s) stabilizing the closed-loop scheme of Fig. 19 and guaranteeing the following inequality [START_REF] Balas | ţ-analysis and synthesis toolbox[END_REF]: where F l (., .) is the lower Linear Fractionar Transformation and is defined by F l (P (s), K(s)) = o(s) e(s) . From Fig. 18, we have:

F l (P (s), K(s)) ∞ < γ (19)
o = W 1 • S • δ c -W 1 • S • G • W 2 • i ( 20
)
where S = (1 + K • G) -1 is the sensitivity function.

Using the condition (Ineq. [START_REF] Tsai | Robust tracking control of a piezoactuator using a new approximate hysteresis model[END_REF]) and (Eq. 20), we infer:

W 1 • S ∞ < γ W 1 • S • G • W 2 ∞ < γ ⇔ |S| < γ |W1| |S • G| < γ |W1•W2| (21)
To solve the problem (Eq. 21), we use the Glover-Doyle algorithm which is based on the Riccati equations [31] [32]. The wanted performances are introduced through the weighting functions.

B. Choice of the weighting functions

The choice of the weighting functions is derived from the specifications. The latter have been chosen from general needs in micromanipulation.

1) Choice of W 1 : the transfer function 1 W1 is chosen from the specifications on the tracking performances. These specifications are:

• the maximal response time is lower than 10ms, • the overshoot is null, • and the maximal statical error is lower than 0.1%. For this, we choose:

1 W 1 = 10 -3 • 3 • s + 1 0.003 • s + 1 (22)
so:

W 1 = 10 3 • 0.003 • s + 1 3 • s + 1 (23)
2) Choice of W 2 : here, the specifications relative to the δ 0 disturbance rejection are used. In the (Eq. 21), the transfer between the output and the disturbance is defined as follows:

δ • α O δ 0 = S • G ( 24
)
Replacing δ 0 by the (Eq. 16) and working in the static mode (s = 0), we have:

δ • α O δ 0 = δ • α O s p • F + δ H + C r (U, s) (25)
The previous equation quantifies the influence of the force F , the hysteresis offset δ H and the creep C r (U, s) on the deflection δ in the closed-loop form. Our objective is as follows: when F = 5mN , δ H = 5µm and C r (U, s) = 5µm, the deflection is less than δ = 1µm. These values respectively correspond to:

• the maximal force used to manipulate a micro-object,

• the offset of the quadrilateral segments of the external hysteresis loop obtained with U = 40V , • the creep static value obtained with U = 40V . We have:

S • G (s = 0) = 0.25 × 10 -7 ( 26
)
This is the characteristic in the static mode and should be lower than γ W1•W2 (s = 0). In addition, we specify a cut-off frequency of 1 W1•W2 at 4000 [rad/s] in order to ensure the rejection of a wide range of frequency disturbances. Thus, we choose the following nominal performances:

1 W 1 • W 2 = 0.25 × 10 -7 • 10 -1 • s + 1 1 4000 • s + 1 (27)
We infer:

W 2 = 4 × 10 4 • 1 4000 • s + 1 • (3 • s + 1) (10 -1 • s + 1) • (0.003 • s + 1) (28)

C. Calculation of the controller

The pseudo-continous controller has been calculated with the Matlab software and its Robust Control Toolbox. Afterwards, it has been transformed into a discrete controller using (Eq. 18). We have:

K = 3.9 × 10 5 • z 5 -9.1 × 10 5 • z 4 + 5.3 × 10 5 • z 3 +5.4 × 10 5 • z 2 -9.2 × 10 5 • z + 3.8 × 10 5
z 5 -1.1 • z 4 -1.6 • z 3 + 1.9 • z 2 + 0.6 • z -0.8 (29) and the optimal value of γ is:

γ opt = 1.05 (30)

D. Experimental results

(Eq. 29) has been implemented into the Simulink software and run on the DSpace board such as the sampling time is T s = 0.2ms. First, a step reference, from -25µm to 25µm, is applied. Fig. 20 presents the results. It clearly shows that the wanted performances are obtained, i.e. response time and overshoot.

To evaluate more precisely the performances, harmonic analysis has been performed. The experimental magnitude is then presented in Fig. 21. The results indicate that the bandpass is 300rad/s, it corresponds to a response time nearly equal to 10ms. As there is no resonance, it can be stated that no overshoot and no vibration will appear in the step response.

In order to evaluate the robustness margins of the closed loop system, the Black-Nichols diagram is plotted (Fig. 22). The margin gain and the margin phase can be deduced from the figure. We have: From the margin gain, it can be deduced that the stability of the closed loop system is ensured as long as the gain α real is between 0 and 1.75 × α O = 878.5 × 10 -9 m V . This range includes the identified gains α m and α M that characterizes the quadrilateral (see Fig. 14). That theoretically indicates that the stability is ensured in the applied range of voltage.

VI. CONCLUSION

This paper presents a new method to model a nonlinear piezoelectric cantilever and presents the design of a H ∞ controller to control it.

First, we have demonstrated that the dynamic hysteresis of the piezocantilever is equivalent to a static hysteresis, a varying gain, followed by a linear dynamic system. Then, we have proposed a new approximate model of the static hysteresis. It consists in approximate it by a quadrilateral characterized by the average, the maximal and the minimal slopes and the offset of the different straights. While the average slope is considered as a nominal gain, the offset of the straight have been used to synthesize a H ∞ controller. On the other hand, the creep and the external applied force have been considered as a disturbance and their effect were taken into account during the controller synthesis. Finally, experiments have been carried out and clearly show the robustness of the synthesized H ∞ controller relative to the context of the micromanipulation.

Fig. 1 .

 1 Fig. 1. A piezoelectric microgripper made up of two piezocantilevers. FEMTO-ST Institute, UMR CNRS 6174 -UFC / ENSMM / UTBM Automatic Control and Micro-Mechatronic Systems depart., AS2M 24, rue Alain Savary 25000 Besançon -France mrakoton@femto-st.fr,{yhaddab,plutz}@ens2m.fr Paper type: regular paper.First submission: June 2006. Accepted: Mars 2008

Fig. 3 .

 3 Fig. 3. A piezocantilever under an electrical and a mechanical excitations.

Fig. 4 .

 4 Fig. 4. Creep phenomenon.

Fig. 5 .

 5 Fig. 5. The hysteresis of the piezocantilever is dynamic, i.e. rate-dependent.

Fig. 6 .

 6 Fig.6. The dynamic hysteresis is equivalent to a static hysteresis followed by a linear dynamic part.

Fig. 7 .

 7 Fig. 7. Response of the piezocantilever when a square signal is applied. The different step responses are similar.

Fig. 8 .

 8 Fig. 8. Step response of the real system and of the dynamic model D(s). The static gain of D(s) has been scaled in order to allow the comparison.

Fig. 9 .

 9 Fig. 9. Experimental magnitude of H(U, s) and simulated magnitude of D(s).

Fig. 10 .

 10 Fig. 10. Experimental magnitude of H(U, s) using two different amplitudes.

  1mHz and τ c ≥ 3minutes is the response time of the creep. The plotted (U, δ)-plane curve only corresponds to the hysteresis H (U ). Different values of frequency were used: f = {1Hz, 300Hz, 600Hz, 900Hz} while the amplitude was 40V . The experiments and the simulation are shown in Fig. 11. The comparison results show clearly that the dynamic hysteresis is equivalent to a static hysteresis followed by a dynamic part: H (U, s) = H i (U ) • D(s).

Fig. 11 .

 11 Fig. 11. Comparison of H (U, s) (experimental result) and H i (U ) • D(s) (simulation) with U = 40V .

Fig. 12 .

 12 Fig. 12. Comparison of H (U, s) (experimental result) and H i (U ) • D(s) (simulation) with U = 20V .

Fig. 13 .

 13 Fig. 13. a: multilinear approximation of the hysteresis. b: quadrilateral approximation of the hysteresis.

Fig. 14 .

 14 Fig. 14. Identification of the parameters.

Fig. 15 .

 15 Fig. 15. Hysteresis curve with a varying amplitude and a constant frequency of input voltage.

Fig. 16 .

 16 Fig. 16. Scheme of the nominal system.

Fig. 17

 17 Fig. 17. The steps used to synthesize the discrete controller.

  Fig. 17. The steps used to synthesize the discrete controller.

Fig. 18

 18 Fig.18shows the scheme of the closed-loop system in the pseudo-continuous domain. In this figure, δ c represents the reference deflection. Two weighting functions are introduced: W 1 (s) for the tracking performances and W 2 (s) for the δ 0 disturbance rejection.

Fig. 18 .

 18 Fig. 18. The closed-loop scheme with the weighting transfer functions.

Fig. 19 .

 19 Fig. 19. The standard form.

Fig. 20 .

 20 Fig. 20. Step response of the closed-loop system.

  output magnitude [dB] frequency [rad/s]

Fig. 21 .

 21 Fig. 21. Harmonic experiments of the closed-loop system.

Fig. 22 .

 22 Fig. 22. Black-Nichols diagram.

TABLE I

 I Geometrical parameters of the piezocantilever.

	Dimensions:
	Length	15mm
	Width	2mm
	Total thickness	0.3mm
	PIC151 thickness	0.2mm
	Copper thickness	0.1mm
	Fig. 2. The experimental setup.

  U . The transient part being given by a step response, we use a square signal with different values of amplitude during the experiments. The dynamic part D(s) can then be identified.• Step 2: Following Fig.6-b, we simulate and plot the magnitude of H i .D(s). The aim is to compare the obtained magnitude with that of the piezocantilever which is experimentally performed. For both, we use a sine wave signal U with different values of amplitude. If the two curves coincide, the simplification of the hysteresis such as H(U, s) = H i (U ).D(s) is validated. However, we need a precise static hysteresis model H i for the simulation. For that, we use the Bouc-Wen static hysteresis model because of its accuracy. This model is only used to validate the simplification and will not be implemented or used to synthesize a controller. • Finally, when the relation H(U, s) = H i (U ).D(s) is validated, we propose to approximate the static hysteresis H

i by a new approximate model: the quadrilateral approximation. The aim is to obtain a simple model which is easy to identify. This model is linear contrary to other models (Bouc-Wen, Preisach, etc.) which are nonlinear. However, the proposed model is less accurate than them, so we propose to synthesize a linear robust controller. The uncertainty is easy to derive in the proposed model.

TABLE II

 II Robustness margins.

	Gain margin (dB)	Phase margin ( • )
	]-∞, 35]	87 (at 341rad/s)

ACKNOWLEDGMENT

This project is partially supported by the EUPASSproject (Evolvable Ultra-Precision Assembly SystemS : http://eupass.org/).


Valérie Laforest
email: laforest@emse.fr

Loïc Perrin
email: perrin@emse.fr

Jacques Bourgois
email: bourgois@emse.fr

Study of two sampling procedures for the valorization of metal hydroxide sludge as pollutant trapper

Keywords: valorisation, sampling, polymetallic hydroxide sludge, metal finishing treatment, Cr VI

For the valorisation of metal hydroxide sludge as adsorbent of pollutant, it is necessary to make sure that the adsorption matrix (sludge) has characteristics allowing a reproducibility of the valorisation operation. Taking into account the heterogeneity of this solid waste in terms of nature and metal chemical composition, it is necessary to establish if this heterogeneity will have important consequences on the adsorption yield of these sludge. Thus, the study, is relative to the influence of the composition of metal hydroxide sludge on their adsorption properties. Two sampling procedures are studied: sampling in a continuous way at the drying step or by coring of the final sludge heap. In order to compare the properties of adsorption, each sample is tested according to a definite adsorption protocol. In addition, this study makes it possible to compare the two sampling procedures and to determine their relevance to the sample technique. The first results show that the methods make it possible to obtain close results, both for adsorption and sludge composition. Moreover, successive quartering permits to have a representative sample in terms of metal composition.

Introduction

Industrial aqueous pollution (heavy metals) accounts for 30 to 40% of all industrial pollution. Metal finishing is one of the sectors which contribute mostly to this pollution. Due to its unique production, metal finishing consumes and discharges wastewater containing metallic ions, cyanide, COD, etc.. Therefore, this activity has been bound to the European Integrated Pollution Prevention and Control Directive since 1996, known as the "IPPC Directive" [1]. In France, this directive is applied "as is" for metal finishing workshops through the ministerial emse-00400370, version 1 -30 Jun 2009 order of the 30 th of June 2006. Before 1996, this text was already belonging to the legislation of the officially identified sites under the 2565 codification for the protection of the environment [START_REF]Arrêté du 30 juin 2006 relatif aux installations de traitement de surfaces soumises à autorisation au titre de la rubrique 2565 de la nomenclature des installations classées[END_REF]. This ministerial order defines thresholds in terms of pollutants in the effluent and in terms of flow rate and concentration of specific chemical species for this branch of activity. These workshops must implement curative or preventive Best Available Techniques (BAT) which are another major principle of the IPPC Directive [START_REF] Cikankowitz | Méthodologie d'évaluation des performances environnementales de procédés en vue de les comparer et les valider aux meilleures techniques disponibles[END_REF]. For about 65% of all workshops, initiatives and actions designed to limit environmental impacts are typically curative. Concerning wastewater composed mainly of heavy metals, the detoxication process of wastewater is the precipitation of metal ions in hydroxide. Some species such as chromates may require an oxidation/reduction treatment. After filtration, hydroxides obtained as sludge are disposed of in various sectors. This sequence of processes is described in the BAT reference documents (BREF), on "Best Available Techniques for metal finishing treatment and plastic finishing treatment" [START_REF] Bref | Bureau européen pour l'IPPC. IPPC BREF outline and guide[END_REF]. It is important to note that France produces 180 000 tons of metal sludge per year compared to the data in Europe (950 000 tons per year) (source CETIM). Currently, despite of the existence of processes such as hydrometallurgy for sludge rich in zinc, pyrometallurgy for sludge rich in nickel, thermal reconditioning for sludge with specific characteristics or finally added to cement for sludge riches in aluminium or iron, polymetallic hydroxide sludge (predominance of non-metal) are mostly eliminated in hazardous waste landfills. For these reasons, the objective of this research project is to propose a new method for the valorisation of polymetallic hydroxide sludge that can not be valorised by the methods of treatment mentioned previously because of their characteristics.

Another key problem of the metal finishing industry deals with chromium. Indeed, its' use in metal finishing applications is quite widespread. It gives at a piece different properties such as hardness, aesthetics, and resistance to temperature, corrosion or wear. This versatility justifies that the chromium is an essential element for various applications. The most widespread properties of the chromium metal deposit are their protective and decorative characteristics. It is estimated that 30 000 to 32 000 tons of chromic acid per year are used in metal finishing at the European level. Because of drag-out all along the production line, 20 to 60% are found in industrial aqueous effluents. This ratio depends on the local context of each workshop, namely its size and recycling measures implemented. So, it appears that the decontamination of chromium is very important for the metal finishing industry. This article concerns a part of the project previously presented. In fact, for the valorization of metal hydroxide sludge as pollutant trapper, it is necessary to make sure that the sorption matrix (sludge) has characteristics allowing a reproducibility of the valorization operation. Taking into account the heterogeneity of this solid waste in term of nature and metal chemical composition, it is necessary to establish if this heterogeneity will have important consequences on the sorption yield of this sludge. The study, thus, relates to the influence of the composition of metal hydroxide sludge on their sorption properties. Two sampling procedures are studied: sampling in a continuous way at the drying step or by coring of the final sludge heap. The Cr VI is taken as the pollutant for this study.

Objectives of the study

The purpose of this study is relevant to the poly-metallic hydroxide sludge (pMHS). The objective is to determine what sampling method is most appropriate in order to obtain a chemical composition for sludge as homogeneous as possible. The change in the chemical composition of pMHS could affect the sorption capacity of pollutants. The establishment of a sampling would ensure reproducibility of sorption experiments. It needs to be fast and easy to emse-00400370, version 1 -30 Jun 2009 implement. The experiments of sampling are realized. Its' utility and representativeness depend on the obtained results of composition and adsorption. The main purpose of the waste sampling is similar to the pMHS sampling for the characterization and the determination of their properties. An important feature, which applies to most deposits of waste, is the great diversity of sources, both in time and in space. This will lead to difficulties when it comes to a constant sample. For this reason, different sampling methods were introduced. The Research Center BRGM [5] presents a summary of recommended methods for the final sample according to the state of waste (liquid samples, concentrated paste samples, solid samples). The three main sampling methods [START_REF] Aloueimine | Méthodologie de caractérisation des déchets ménagers à Nouakchott (Mauritanie) -Contribution à la gestion des déchets et outil d'aide à la décision[END_REF] are as follows:

The simple random sampling where the population has the same theoretical chance of being selected in the sample: This sample is perfect if all variations in the population are represented. The stratified random sampling is more complex, and is used for heterogeneous populations. The criterion of stratification must have a close relationship with the studied variable. In other words, people are stratified so that within each party or stratum, fluctuations of the variable (i.e for household waste, standard of living, waste production, household size, ...) are minimal. After this stratification, random sampling (single or systematic) is done in each stratum. The number of samples of each stratum can be determined either by proportionality to the public, either through optimization, ie that the sample is selected so that the variation of the average is the lowest possible for the sample size. The systematic random sampling is sometimes the only solution that can be implemented. Each element of the population is randomly selected. But the drawback of this technique is on the low accuracy of results when the population covered by the sampling, a trend or unknown non-systematic variations.

Concerning pMHS, the sampling method to be used is certainly closer to the stratified random sampling. Indeed, the sludge composition is not constant ("heterogeneous population"). In addition, the sampling method is most representative. Depending on waste nature, different sampling techniques are created:

Clinkers of Incineration plant of household waste [START_REF] Nouvel | Cours de déchets « Echantillonnage et Caractérisation d'un gisement de déchets[END_REF] MODECOM Method for household waste [START_REF] Nouvel | Cours de déchets « Echantillonnage et Caractérisation d'un gisement de déchets[END_REF][START_REF] Morvan | en partenariat avec le Cemagref de Rennes[END_REF] Ground [START_REF] Girard | Sols et Environnement[END_REF][START_REF] Delage | Géomécanique environnementale Sols pollués[END_REF] Sludge: Quartering successive method: the aim is to divide the sample into four parts. Two opposing quarters are excluded, and the two remaining combined. Then this process is repeated until a properly sized sample is obtained. [START_REF]Protocole d'échantillonnage des matières résiduelles fertilisantes (Résidus de fabriques de pâtes et papiers et autres résidus solides[END_REF][START_REF]fixant les prescriptions techniques applicables aux épandages de boues sur les sols agricoles pris en application du décret n° 97-1133 du 8 décembre 1997 relatif à l'épandage des boues issues du traitement des eaux usées[END_REF] In France, the ministerial order of 8 th January, 1998, appendix V «sampling method for a batch of raw sludge [...] and composted» describes the experimental protocol. It consists in ensuring a representative sample, i.e. twenty-five points evenly distributed in different depths and different batch of the sludge to be sampled. The capacity of samples taken is not specified. The samples must be taken outside of the crust and surface areas of water accumulation. The various samples are mixed, homogenized and reduced to a global sample of one kilogram approximately. [START_REF] Trame | Agriculteurs Composteurs de France. Charte de bonnes pratiques de compostage agricole, fiche méthode n°5[END_REF] Finally, in our case study, the sludge collection is done randomly and quartering is used as the sample reduction method.

emse-00400370, version 1 -30 Jun 2009

3 Experimental protocols

Method

After the sludge samples have been taken, they are homogenized and reduced by quartering modeling. Then, sludge is grinded and suspended in a solution containing the pollutant to trap. After that, the mixture is filtrated and the liquid phase is analyzed by ICP. The general protocol used is presented in the figure 1 [START_REF] Lannes | Rapport de projet industriel : Identification et expérimentation d'un protocole d'échantillonnage des Boues d'hydroxydes poly-métalliques[END_REF].

Sample reduction method

The sludge sampling is done directly in the industrial workshop. The sludge reduction is done in the laboratory room using the quartering method. The raw sludge is disposed in a circle which is divided in 4 parts. Two opposite parts are thrown out and the two others are mixed together. This operation is repeated until 50g of sludge are obtained. The initial sludge is noted "sample 0", the sample of the first reduction is named "sample 1" and so on until "sample n". The figure 2

Nota: the notation of each sample is composed by one letter and two numbers. For example the sample written C31 is a sample taking in continuous way for the big bag 3 corresponding at the first reduction and the sample written T26 is a sample taking with the trepan in the big bag 2 corresponding at the sixth reduction.

Sorption protocol

A mass "m" of Cr VI is weighted. A stock solution is then prepared with distilled water in a flask of 500 mL. In order to homogenize, the solution is stirred several times. Then, 25mL are taken and blended with 50 mg of sludge. The solution is stirred during a time "t" and filtrated. The liquid part is analyzed with ICP. Figure 3

Experimental results

Sorption duration

Because of the industrial application targeted by the project, it is necessary to have a laboratory experimental protocol applicable to industrial scale and running. Thus, in order to determine which duration of sorption is optimal, a first range of duration has been tested. Two durations of sorption experiments have been done and are presented here: 2 hours and 13 hours. -The difference of sorption yield seems to be insignificant for an industrial application: 50% for 13 hours of sorption and 55% for 2 hours of sorption (moreover, experimental error is estimated at 4%) Then we have decided to experiment with sorption duration of 2 hours.

Sorption yield

In order to assess if the two sampling methods tested permit to obtain homogeneous sludge, sorption experiments of Cr VI have been conducted. The results presented in this article concern two different big-bags: no. 2 and no. 3. Sorption yields according to [Cr VI ] are presented. Figure 5 presents the results for continuous sampling and the results for the coring method of sampling and both for big-bag 2 Figure presents the results for continuous sampling and the results for the coring method of sampling and both for big-bag 3. Irrespective of the big-bag, one can conclude that the reproducibility of the sorption of Cr VI is higher for the coring method of sampling. Moreover, the reduction steps do not seem useful because results are quite the same for a concentration greater than 100mg/L in Cr VI .

Composition of the big bags

Sorption yield depends on the composition of the sludge and the performance of the precipitation. In order to determine the composition of the sludge, to check the homogeneity of the sample and to determine if a sampling method is more pertinent, analysis of samples have been conducted. Two dissolutions of each sludge sample have been carried out: one in the water and one in hydrochloric acid.

Dissolution in water

The result of sludge dissolution in water is demonstrated in figure 7. Irrespective of the sampling method (continuous way or coring way), one can observe that the chemical species in the solution are the same and samples are more concentrated for the coring way. We can observe that K, Mn, Na, Ni and Zn are present in the sludge.

Dissolution in acid solution

The result of sludge dissolution in HCl is demonstrated in figure 8. Irrespective of the sampling method (continuous way or coring way), one may observe that the chemical species in the solution are the same and their concentrations are quite equal. We can observe that Cr, Cu, Fe, Mn, Na, Ni, Sn and Zn are present in the sludge. The difference of species in the acid solution and aqueous solution is due to the solubility of metal hydroxide for low pH.

Conclusion

The objective of the study presented in this article is the validation of a sampling procedure for the valorization of metal hydroxide sludge as pollutant trapper. The experiments conducted permit to draw the following conclusions:

-Sorption results highlight a better reproducibility of Cr VI sorption for the samples taken with an trepan, -Sample composition is the same and do not allow by themselves to validate the choice of a sampling procedure, -Reduction steps seem to be useless because we have observed that sorption yields are the same whatever the reduction for a concentration in Cr VI greater than 100mg/L Then, if we want to check if the hydroxide sludge have the capacity to trap Cr VI , optimal sampling conditions are:

-A sampling with a trepan -No sampling reduction i.e. adsorption on the raw sample Moreover, it is of interest, on the one hand, a gain on personnel costs for the sampling on an industrial site and, on the other hand, a time gain in preparation before sorption.

Figure

  Figure 1 : general protocol

Figure 2 :

 2 Figure 2 : sampling points and sample reduction method

  presents the stirring and filtration steps. emse-00400370, version 1 -30 Jun 2009

Figure 3 : sorption protocol 3 . 4

 334 Figure 3 : sorption protocol 3.4 Sludge dissolution protocol

Figure 4 :

 4 Figure 4 : sorption yield according to sorption duration One can observe on figure 4 that:-Sorption rate seems to be stable for concentration of Cr VI in the media greater than 100mg/L

Figure 5 : 2 Figure 6 :

 526 Figure 5 : sorption yield according to [Cr VI ] for the big bag 2

  sorption yield according to [Cr(VI)]

Figure 7 :

 7 Figure 7 : big bags' composition in water

Figure 8 :Figure 9 :

 89 Figure 8 : bigbag2 composition in HCl

  presents the step.

		Sampling in a continuous way	Sampling by coring
		Sludge Homogenisation handmade
		to reduce the sample size at 1cm
	emse-00400370, version 1 -30 Jun 2009	Sample reduction by quartering Grinding thinly to a mortar Sludge putting in suspension for a stirring time "t" Preparation of a potassium dichromate solution Filtration Aqueous Phase analyzed by ICP