Hisashi Oishi

Sayaka Mashima

Yutaka Kuwahara

Makoto Takafuji

Kyohei Yoshida

Reiko Oda

Hongdeng Qiu

Hirotaka Ihara
email: ihara@kumamoto-u.ac.jp

Polymer encapsulation and stabilization of molecular gel-based chiroptical information for strong, tunable circularly polarized luminescence film †

We demonstrate a strategic approach for a completely organic, chiroptical polymer film, exhibiting strong (g lum = 0.01-0.05) and broad-band circularly polarized luminescence that can be realized by the stabilization of self-assembly-driven secondary chirality through polymer encapsulation.

Circularly polarized luminescence (CPL) has attracted attention as a next-generation light signal because it carries more information due to its chirality and no angular effect compared to normal and linearly polarized lights. A simple example can be seen in its use as a communication tool by some animals. 1 One technical approach for producing CPL is by converting polarized light from normal to linear using a quarter-wave plate. However, this approach can suffer from serious reductions in light energy with removal of light. The other solution for CPL generation is the use of chiral luminescent compounds that are totally organic. [2][3][4][5][6][7][START_REF] Bosson | [END_REF] Studies using this approach have realized stronger CPL by precise molecular design, skilled organic synthesis, and careful purification. Therefore, gram-scale synthesis has not yet been achieved without few exceptions. [START_REF] Bosson | [END_REF]9 In this work, we describe the fabrication of chiroptical polymer films that exhibit strong CPL and anticipate their possible applications in bio-sensing, chiroptical memory, detector system for security devices, and storage units. 4,10,11 For such purposes, gram-scale synthesis becomes increasingly more important.

We selected a binary system for CPL generation, 6,7 which has a distinct advantage on larger scale. Instead of using specially designed chiral luminescent materials, commercially available, non-chiral luminescent dyes can be used and chirality can be induced by chiral templates. Therefore, the essential problem becomes, rather than the scalability of synthesis and fabrication, the retention of chirality induction in the polymer system. In this study, we demonstrate that strong CPL-exhibiting polymer films can be fabricated by incorporating into the polymer a chirally self-assembled CPL-inducing system consisting of a nonchiral luminescent dye and associated chiral template.

We applied several cyanine dyes including anionic NK-2012 and cationic NK-77 as non-chiral luminescent species for CPL generation. Because these cyanine dyes are very sensitive to the chiral microenvironment, they function as good indicators for chirality evaluation. However, cyanine dyes are often unstable, especially in oxygen atmosphere under sunlight, and therefore, these systems are not suitable for our present purpose. Fig. 1 presents several commercially available fluorescent dyes as new chirality-induction candidates. To induce chirality in these non-chiral dyes, G-ca, having a carboxy head group, was chosen as a self-assembling organogelator (Fig. 1). The advantages of G-ca can be summarized in terms of (1) its good solubility in various organic solvents, which is very important because most of a bulk polymer system is hydrophobic and used in organic media; (2) the enhanced chirality due to the formation of secondary chirality through chiral arrangement;

(3) its ionizability to promote electrostatic interactions with the cationic dyes shown in Fig. 1; and ( 4) its possible gram-scale synthesis via a one-step coupling reaction between L-glutamide 12 and glutalic anhydride.

Mixed systems of G-ca and dyes were prepared by the following procedure: a 0.5 mM solution of G-ca in toluene was prepared by heating at 85 1C; cooling to 25 1C produced a clear solution. If the concentration was higher than a few millimolar in toluene, the solution became a clear gel. This phenomenon has been recognized as molecular gelation. 7,13,14 Fig. 2a shows the TEM image of the 0.5 mM G-ca solution after staining with uranyl acetate. G-ca forms nano-fibrillar aggregates whose smallest diameter is within 50 nm in the photo. This indicates that the gelation behavior is the result of entanglement among the well-developed nano-fibrillar aggregates of G-ca. Next, a methanolic solution of the dye was mixed in a given molar ratio with the G-ca solution. No significant change is observed by TEM, but confocal microscopy provides fibrillar fluorescence images, as shown in Fig. 2b. Since G-ca does not itself fluoresce, the obtained confocal image indicates that the dye is combined with G-ca and the fibrillar aggregation of G-ca is maintained after mixing with the dye.

The chirality of the G-ca/dye mixed systems was investigated by CD spectroscopy. As shown in Fig. 3a, the G-ca/NK-77 system exhibits a new exciton coupling ([y] 573 = 5 Â 10 4 deg cm dmol À1 ) near the absorption band (l max = 570 nm) of NK-77. Since NK-77 is non-chiral, the observed CD can be attributed to induced chirality from the G-ca aggregates. Although this induced CD intensity is not so small, while we have confirmed much larger values (41 Â 10 6 deg cm dmol À1 ) 15 in the case of an anionic cyanine dye, NK-2012, on poly(L-lysine). Therefore, we examined the effect of additives on the G-ca/dye system to promote their interaction.

When an organic amine such as pyridine or triethylamine (TEA) is added to the G-ca/dye system, a distinct increase in CD intensity is observed. Fig. 3b shows the effect of TEA on the CD spectrum. The peak intensity is amplified ca.20 times with the addition of ten equimolar of G-ca. This is probably due to the promotion of electrostatic interactions between the chiral G-ca and non-chiral NK-77 by neutralization of the iodide counter ion of NK-77. Therefore, in this study, TEA was applied in the promotion of induced CD for the other non-chiral dyes.

Polystyrene (PS) was selected as the bulk polymer for film fabrication. The typical fabrication procedure is as follows: a 5 wt% solution of PS was combined with G-ca in hot toluene, and then the methanolic dye solution was added to the mixture. Part of the mixture (400 mL) was drop-cast onto a glass plate (12 mm  25 mm) and dried in air. The thickness of the resultant polymer film was adjusted to be approximately 60 mm. The main composition rate was adjusted to be 2.0 wt% in the G-ca-in-polymer, and to be 1 : 20 in the molar ratio of dye and G-ca. Fig. 2d shows the confocal microscopy images of the PS film containing NK-77 with G-ca. The fibrillar fluorescence image indicates that not only does G-ca self-assemble in the polymer, but also that NK-77 is associated with the G-ca fibrils. Similar fluorescence images (Fig. S1,ESI †) were observed with the other dyes, including acriflavine (AC, l max = 530 nm) and Basic Red-5 (BR-5, l max = 591 nm). Notably, all the PS films are transparent, which means that the diameters of the aggregates are sufficiently small to escape the problem of undesirable light scattering. This is an essential and important requirement for optical applications, and therefore, we conclude that our strategic approach is unquestionably suitable for our research purposes.

Fig. 4 shows the CD spectra of the PS films containing the G-ca/dye systems. Large CD intensities are observed near the absorption bands of the dyes in all films. Since no CDs are observed in the absence of G-ca and the CD intensity is Our CPL generation system has multiple advantages. First, mixed-dye systems can be employed to provide broad-band CPL over a wide range of emission bands. For example, when AC and BR-5 are mixed, the resultant PS film provides two CD signals (Fig. S3, ESI †). Second, broad-band CPL can also be generated by adjusting the chiral microenvironment. For example, when the mixing ratio of AC to G-ca is adjusted, the luminescent peak width increases as shown in Fig. 5: that is, the Stokes shift can be made tunable by promoting the heterogeneity of chiral perturbation. Finally, by encapsulating G-ca in the polymer as well as selecting the proper dye, the CPL phenomenon can be remarkably stabilized. For example, we confirmed that the G-ca/AC polymer system maintained strong CPL over at least a month.

Conclusions

In conclusion, we have established a strong CPL (g lum E 0.05) generation system in a polymer thin film. Our system is advantageous because no chirality is needed in the fluorescent component, meaning that it is free from precise molecular design and complicated synthetic limitations, and therefore possible for industrial-scale application. In addition, the emission bands can be readily tuned by adjusting the molecular orientation of the chiral template system as well as the dye selection. These advantages could expand possible system applications, such as in optical uses (e.g., chiroptical detector for bio-signal sensing), in agriculture, or in solar cells as a reduced-angular wavelength converter.

Fig. 1

 1 Fig. 1 Chemical structures of non-chiral fluorescent compounds and organogelator G-ca used in this study.

Fig. 2

 2 Fig. 2 TEM and confocal micrographs of G-ca aggregates. (a) Initial concentration of G-ca: 0.1 mM in toluene. The dried sample was obtained by drop-casting on Cu mesh, stained with uranyl acetate and dried at room temperature. The confocal microscopy for the solution state (b) and the film (d) were observed with excitation wavelength at 488 nm. (b) G-ca: 0.5 mM and NK-77: 0.01 mM in toluene. (c) and (d) G-ca: 2 wt% in polystyrene. G-ca : NK-77 = 20 : 1.

Fig. 3

 3 Fig. 3 (a) CD spectra of the G-ca with NK-77 in toluene. [G-ca] = 0.5 mM; [NK-77] = 0.01 mM; [TEA] or [pyridine] = 5 mM. (b) Effect of TEA additive in the G-ca/dye system on the CD intensity.

Acknowledgements

This work was supported by the French-Japanese International Associated Laboratory, Chiral Nanostructures for Photonic Applications (LIA-CNPA), and a Japan Society for the Promotion of Science Grant-in-Aid for Scientific Research and the Bilateral Joint Research project.

Conflicts of interest

There are no conflicts to declare.

Notes and references


Anis Kaci
email: anis.kaci@univ-lille.fr

Christophe Giraud-Audine

Frederic Giraud

Michel Amberg

Betty Lemaire-Semail

LQR based MIMO-PID controller for the vector control of an underdamped harmonic oscillator

Keywords: Vector Control (demodulated-modulated control), Piezoelectric transducer, Linear Quadratic Regulator (LQR), Vibration control

Modulated-Demodulated control (or vector control) allows to simultaneously impose amplitude and phase of a resonator. Moreover, the working frequency in the case of discrete-controller is substantially lower than the resonance frequency. However, the design of a such controller can be complex. In this paper, we outline a design directly in the baseband. To do so, the oscillator is modelled as a non-dimensional Multi-Input-Multi-Output system. An optimal control (Linear Quadratic Regulator) framework can then be used to design the controller. Thanks to ad-hoc performances criteria, the weighting matrices are systematically specified according to the desired closed-lop time response. The methodology is validated by an experimental results on a plate actuated using piezoelectric patches.

Introduction

Many medical and industrial processes use resonating electromechanical systems e.g. in nebulizer, echography or ultrasonic welding. In some of these applications [START_REF] Gabai | A rotational traveling wave based levitation device -modelling, design, and control[END_REF][START_REF] Messaoud | Amplitude Control of an Ultrasonic Vibration for a Tactile Stimulator[END_REF], vibration need to be maintained at a given level of amplitude and track the resonance frequency in spite of the non-linearity and exogenous perturbations [START_REF] Davis | Automatic vibration mode selection and excitation; combining modal filtering with autoresonance[END_REF][START_REF] Liu | An Automatic Resonance Tracking Scheme With Maximum Power Transfer for Piezoelectric Transducers[END_REF].

Because these systems often operate at high frequencies, the implementation of a fast and precise closed-loop discrete-controller requires high sampling frequencies and fast computing capabilities. Therefore, some authors rather use open-loop control for which a perfect knowledge of the system is required. In MEMS, Borovic & al. [START_REF] Borovic | Open-loop versus closed-loop control of mems devices: choices and issues[END_REF] conclude that if the device has 2 positions, open-loop control is sufficient, but if the device has to be positioned between 0 and 100 %, then closed-loop control is preferred. For resonant devices, which operate at their resonance frequency, it is important to control the vibration amplitude, because the load can modify the damping or induce a non-linear behaviour [START_REF] Gabai | A rotational traveling wave based levitation device -modelling, design, and control[END_REF]. Since this acoustic load is hardly predictable, closed-loop controls are implemented. Babitsky et al. [START_REF] Babitsky | Autoresonant control of nonlinear mode in ultrasonic transducer for machining applications[END_REF] proposed a self-oscillating circuit to control the vibration at resonance without knowing the dynamic response of the system. In [START_REF] Chen | Neuro-fuzzy speed control of traveling-wave type ultrasonic motor drive using frequency and phase modulation[END_REF], a non-linear controller is designed in order to control the position of a Traveling Wave Ultrasonic Motor. The authors claim superior performances compared to a Proportional-Integral (PI) controller. However, dynamic responses was not studied.

As a consequence of the dramatic cost reduction of some electronic parts, like Digital Signal Processors (DSP) for instance, modulation-demodulation control structure becomes competitive [START_REF] Bazaei | Synthesis of modulated-demodulated control systems[END_REF]. This control consists in demodulating the measure by a sinusoidal signal in order to obtain the in-phase and in-quadrature components, and to modulate controller's outputs by the same signal. This method has shown good results for active damping of structures [START_REF] Karvinen | Modulated-Demodulated Q Control of an Atomic Force Microscope Microcantilever[END_REF]. For Ultrasonic transducer, the method is called Vector Control Method as a reference to the control of Electromagnetic motors, and it has been successfully applied to the control of a Langevin transducer to be robust despite system's non linearities [START_REF] Ghenna | Vector Control Of Piezoelectric Transducers and Ultrasonic Actuators[END_REF]. In this work, good dynamic results are obtained with PI controllers.

The design of the controller is based on a simplified model assuming a weak coupling between the states, which allows to consider two Single Input Single Output -Proportional Integral (SISO-PI). However, this assumption limits the closed-loop dynamics. In order to relax this limitation, this decoupling assumption must be dropped and a Multi Input Multi Output (MIMO) design is then necessary. However, the tuning of the controllers parameters are complicated as the system is MIMO [START_REF] Kadhar | Diversity Controlled Self Adaptive Differential Evolution based design of non-fragile multivariable PI controller[END_REF]. In [START_REF] He | PI/PID controller tuning via LQR approach[END_REF][START_REF] Mukhopadhyay | equivalent of optimal regulator[END_REF][START_REF] Estévez-Sánchez | Design of multiloop PI controllers based on quadratic optimal approach[END_REF][START_REF] Das | LQR based improved discrete PID controller design via optimum selection of weighting matrices using fractional order integral performance index[END_REF][START_REF] Das | Multi-objective LQR with optimum weight selection to design FOPID controllers for delayed fractional order processes[END_REF], the authors propose to use the Linear Quadratic Regulator (LQR) methodology to derive the SISO or MIMO-PID controllers. However, the choice of the Q and R matrices is still left to the designers judgement and experience, or using optimization procedure [START_REF] Das | LQR based improved discrete PID controller design via optimum selection of weighting matrices using fractional order integral performance index[END_REF].

In this paper, the model of the oscillator in the baseband is reformulated to take into account the higher dynamics. A non-dimensional model is then derived for the sake of generality. Performances criteria describing the desired closed loop dynamic and the transient vibration envelop are then proposed. Based on these criteria, the weighting matrices coefficients can be directly expressed as a function of the desired closed-loop time response and thus the PID controller coefficients can be deduced.

The rest of the paper is organized as follows. In section 2, the normalized dynamic model of the resonant transducer in a demodulated form is presented.

In section 3, the method to implement MIMO-PID controller based on the LQR design is recalled. The performances indexes are introduced, and their relation with the weighting matrices is studied, yielding a systematic procedure to design the controller. In the following part,the experimental set-up and protocol are explained. Experimental tests confirming the methodology are then presented.

Finally, the paper ends with the conclusion in section 5.

Dynamic modelling

State space representation in the demodulated space

We consider a damped linear harmonic oscillator, which dynamic is given in a normalized form as follows:

η(t) + 2ξω n η(t) + ω 2 n η(t) = ω 2 n f (t) (1)
Where η(t) is the deflection of the resonator, ξ the damping factor, ω n the resonance frequency, and f (t) the external normalized force applied to the system which produces its oscillation. Actually, due to normalization f (t) is rather the quasi-static displacement under the real load and it is expressed in meter (m).

This model is actually general, and can represent for instance the dynamic of the behaviour of a given mode once projected in the modal basis either for a lumped parameters or a continuous system (see [17, p. 25] or [18, p.440] for instance). When the displacement η and the external force f are purely harmonic, one we can use their complex form as in the following equations:

η(t) = (H d (t) + jH q (t))e jωt (2)
f (t) = (F d (t) + jF q (t))e jωt (3)
Where H d and H q are respectively the real and imaginary part of the deflection, F d and F q are the real and imaginary part of the external force and ω is the excitation frequency which is also the pulsation of the resonator in this paper.

By deriving two times the Eq.2 η = [( Ḣd + j Ḣq ) + jω(H d + jH q )]e jωt (4)

η = [( Ḧd + j Ḧq ) + 2jω( Ḣd + j Ḣq ) -ω 2 (H d + jH q )]e jωt (5)
By replacing the Eq. (2) (3) (4) and ( 5) into (1) a system of two differential equation are obtained. The resulting equations can be expressed as a state space

ẋ = Ax + Bu         Ḣd Ḣq Ḧd Ḧq         =         0 0 1 0 0 0 0 1 ω 2 -ω 2 n 2ξωω n -2ξω n 2ω -2ξωω n ω 2 -ω 2 n -2ω -2ξω n                 H d H q Ḣd Ḣq         +         0 0 0 0 ω 2 n 0 0 ω 2 n           F d F q   y = Cx =   1 0 0 0 0 1 0 0           H d H q Ḣd Ḣq         (6)
Where the output is

y = [H d , H q ] T .
Hence, by applying the transformation of the Eq. ( 2) and Eq. ( 3), we obtain an equivalent model of the harmonic oscillator into the demodulated state space, which dynamic depends on the difference between the resonance frequency ω n and the operating frequency ω. The equivalent scheme of this representation is shown in the Fig. 1. The Fig. 2 shows an example of demodulated signals.

Normalized Model

For sake of generality, the state space representation of the Eq. ( 6) is written in a normalized form that depends only on the damping ξ and the relative frequency Ω = ω ωn . For that purpose, we define a new time base θ = ω n t, and the state variables are revised into:

H nd = H d H nd = dH nd dθ = Ḣd ω n H nd = d 2 H nd dθ 2 = Ḧd ω 2 n ( 7
)
H nq = H q H nq = dH nq dθ = Ḣq ω n H nq = d 2 H nq dθ 2 = Ḧq ω 2 n ( 8
)
Introducing these new variables into the Eq. ( 6), yields:

ẋn = A n x n + B n u n         H nd H nq H nd H nq         =         0 0 1 0 0 0 0 1 Ω 2 -1 2ξΩ -2ξ 2Ω -2ξΩ Ω 2 -1 -2Ω -2ξ                 H nd H nq H nd H nq         +         0 0 0 0 1 0 0 1           F d F q   (9)
To analyze the dynamic of the system in its demodulated space, we first calculate the characteristic polynomial of the plant

P (λ n ) = det(λ n I -A n )
which gives:

P (λ n ) = λ 4 n +4ξλ 3 n +(4ξ 2 +2Ω 2 +2)λ 2 n +4ξ(1-Ω 2 )λ n +Ω 2 (Ω 2 +4ξ 2 -2)+1 (10)
The eigen values of this plant do not have general form. However, for the specific case of a low damped oscillator (ξ < 1), they can be approximated by:

λ n(1,2) = -ξ ± j(Ω + 1 -ξ 2 ) λ n(3,4) = -ξ ± j(Ω -1 -ξ 2 ) (11)
Hence, compared with the system in its natural state space, the poles in the demodulated form resembles the poles of a second order system to which are added the demodulation term ±jω (or ±jΩ for normalised model) [START_REF] Karvinen | Modulated-Demodulated Q Control of an Atomic Force Microscope Microcantilever[END_REF]. By operating the resonator close to its resonant frequency (Ω = 1), 2 poles of the system get closer to the real axis, as depicted in Fig. 3. This normalized representation of the system in the demodulated state space allows us to consider the general design of the controllers, as presented in the next section of this paper.

General design of the controllers

State feedback to MIMO-PID

In this work, we present a method to control the resonator's oscillation using MIMO-PID state feedback controller in the demodulated state space, and the parameters of this controller are calculated using an LQR approach.

Let's consider the augmented state space representation which yields:

ẋn = Ân xn + Bn u n               H nd H nq H nd H nq H nd H nq               =               0 0 0 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0      A n                                  H nd H nq H nd H nq H nd H nd               +            0 0 0 0      B n                   F d F q   (12)
Therefore, the MIMO-PID controller consists then to write the input u n = [F d , F q ] t as:

u n = K n xn = K n (x nref -xn ) = K ni K np K nd xn =   k ni1 k ni2 k ni3 k ni4     Hnd Hnq   +   k np1 k np2 k np3 k np4     Hnd Hnq   +   k nd1 k nd2 k nd3 k nd4     H nd H nq   (13)
Where xn = [ Hnd , Hnq , Hnd , Hnq , H nd , H nq ] and where Hnd = H nd -H dref and Hnq = H nq -H qref . The design of this controller needs then 12 parameters to be chosen to give an optimized behaviour of the system. In the following section, we apply the LQR method to optimally determine the value of these parameters.

PID Controller tuning via LQR approach

To get an optimal K n feedback matrix, one of the most popular methods is to use an LQR approach, as it allows to reduce the transient of tracking error (response time/overshoot) while minimizing the control effort [START_REF] Preumont | Vibration Control of Active Structures: An Introduction, 4th Edition, Solid Mechanics and Its Applications[END_REF]. The LQR approach consists in finding a state feedback (13) minimizing the following criteria J:

J = ∞ 0 (x T n Qx n + u T n Ru n )dt ( 14
)
Where Q is semi positive definite (Q ≥ 0) and R positive definite (R > 0).

These matrix can be tuned to get the desired performance while respecting the actuators maximum effort. It can be shown that a solution to this optimization problem, is to give K n the following expression [START_REF] Preumont | Vibration Control of Active Structures: An Introduction, 4th Edition, Solid Mechanics and Its Applications[END_REF]:

K n = R -1 BT n P ( 15
)
where P is the solution of the algebraic Riccati equation (P ≥ 0)

P Ân + ÂT n P + Q -P Bn R -1 BT n P = 0 (16)
Tuning the closed loop system is then achieved through the matrices R and Q. However, in this system, since H nd and H nq are two component of the same signal, there is no reason to favour the imaginary part over the real part (and vice-versa), and the matrices R and Q will be chosen as follows:

Q = diag[q i , q i , q p , q p , 0, 0] R = diag[1, 1] (17)
Where q i and q p are the weighting gains associated respectively with the integrals ( Hnd dθ, Hnq dθ) and proportionals ( Hnd , Hnq ); the weighting gains associated with the derivatives are left equal to zero in order to reduce the amplification of measurement noise. It is a known effect due to the amplification of the signals at high frequencies, where the noise level is higher [19, p.615]. In vibration amplitude control, the objective is to achieve an exponentially convergent envelop with a given rising time, from 10% to 90% of the steady state value. To quantify the effect of the weighting gains, we define the following speed and shape criteria:

• lg(α) = lg( tr trc ) is the decimal logarithm of the ratio between the rising time in open-loop (t r ) and in closed-loop (t rc ), this indicator is defined as the acceleration factor. Since the study assumes that the operating frequency is at resonance Ω = 1, the dominant poles (λ n [START_REF] Davis | Automatic vibration mode selection and excitation; combining modal filtering with autoresonance[END_REF][START_REF] Liu | An Automatic Resonance Tracking Scheme With Maximum Power Transfer for Piezoelectric Transducers[END_REF] ) are then purely real, according to Eq.11. Hence, t r is approximately t r 2.2/(ξω n ).

We can also express α as lg(α) = lg( ξc ξ ), where ξ c is the closed-loop damping.

• lg(γ) = lg( trc 2.2τc ), where τ c is defined as the settling time from 0% to 63.8% of the steady state value. This value was chosen because it corresponds to the time constant of a first order system.

An acceptable closed loop dynamic is reached if the desired rise time in closed loop is obtained. Moreover the time response should be exponentially convergent which is achieved if the closed loop rise time (10% -90%) equals 2.2 times the closed loop time constant (0% -63.8%). Thus, the following condition lg(γ) = 0 should be respected. In practice, the response is considered to be exponentially convergent for |lg(γ)| < 0.02.

To illustrate the effect of q i and q p , the fig. 4(a-c) shows the criteria lg(α) and lg(γ) as a function of the weighting gains q i and q p , for ξ = 10 -4 and Ω = 1. The fig. 4 (a) shows that in order to have a given α there is several couple q i and q p that lead to the same result. However, considering time response depicted in the Fig. 4 (c), one can observe that the exponential convergence is not guaranteed in the region A and C of the Fig. 4 (b). Hence, considering that this exponential convergence is obtained if |lg(γ)| < 0.02, we remove from 4-a the zones A and C depicted on the 4-b where this condition is not fulfilled, giving rise to fig. 4-d.

This can be ensured by constraining q i and q p along a line as depicted in black on the same figure. On this line, we write lg(q i ) = a × lg(q p ) + b with a = 1 and b = -8.

As shown in the Fig. 4-(d:i), the same procedure is performed for ξ ∈ 10 -7 , 10 -2 and for each plot the values of a and of the coordinates [lg(q p0 ), lg(q i0 )] for which lg(α) = 0 (i.e t r = t rc ) are taken; the results are given in the table 1.

From the Tab 1, several approximations can be made. First, a is supposed Vibration amplitude ( m) [lg(q p0 ), lg(q i0 )] where lg(α) = 0 to be invariant and equal to 1, leading to:

-5 -4 -3 -2 lg(q p ) -9 -8.5 -8 -7.5 -7 -6.5 -6 -5.5 lg(q i ) -7 -6 -5 -4 lg
A B C (c) (b) (a) (d) (e) (f) (g) (h) (i)
lg q i q i0 = lg q p q p0 (18)
Second, lg(q p0 ) and lg(q i0 ) can be approximated by a linear interpolation and we write:

lg(q p0 ) = 2 × lg(ξ) + 0.6021 ≈ 2 × lg(ξ) + lg(4) = lg(4ξ 2 ) lg(q i0 ) = 4 × lg(ξ) + 0.6021 ≈ 4 × lg(ξ) + lg(4) = lg(4ξ 4 ) (19)
At this point, the methodology gives the weighting parameters such that a system with damping ξ will have the same dynamic in closed-loop as in openloop. So, the next step is to establish a relation between lg(α), lg(q p ), lg(q i ), while imposing the Eq. [START_REF] Meirovitch | Fundamentals of Vibrations[END_REF]. By this way, the weighting parameters can be calculated for any system damping ξ ∈ [1 × 10 -7 , 1 × 10 -2 ] and for any desired dynamic α. The relation between δ = lg(q p /q p0 ) and lg(α) for different values of ξ is shown in the Tab 2 . From the values in this table, it can be observed that for a given δ, lg(α) is almost invariant (less than 0.1% for ξ ∈ [1 × 10 -7 , 1 × 10 -2 ]), while, for a given ξ, δ varies almost linearly. Therefore, a linear interpolation can be applied as follows lg(α) = a α × lg(q p /q p0 ) + b α .

Tab 3 gathers the coefficient of the linear regression within a considered range for ξ ∈ [1 × 10 -7 , 1 × 10 -2 ] and δ ∈ [-2, 2]. It can be observed that 1) a α is constant, 2) b α is small compared to the practical range of lg(α), 3) the norm of the residuals ||e|| decreases with ξ and are always smaller than 2.04 × 10 -2 .

Since b α is small enough, the relation between lg(α) and lg(q p /q p0 ) is further δ ξ 10 -7 10 -6 10 -5 10 -4 10 -3 10 -2 -2.0 -1.0000 -1.0000 -1.0000 -1.0000 -1.0001 -0.9998 -1.5 -0.7500 -0.7500 -0.7500 -0.7500 -0.7501 -0.7498 -1.0 -0.5000 -0.5000 -0.5000 -0.5000 -0.5001 -0.4997 -0.

lg(α) = a α × lg q p q p0 = 1 2 × lg q p q p0 (20)
Considering a range of αξ ≤ 10 -1 , the Eq. ( 20) gives α with an accuracy of at least 99.5%. By replacing the Eq.( 19) and the Eq.( 18) into the Eq.( 20), we can write an empirical formula of the weighting gains in function of the desired

dynamic lg(q p ) = 2 × lg(α) + lg(4ξ 2 ) lg(q i ) = 2 × lg(α) + lg(4ξ 4 ) (21)
After simplifying the equations we get

q p = (2αξ) 2 = (2ξ c ) 2 q i = (2αξ 2 ) 2 = (2ξ c ξ) 2 (22)
At this point, the weighting coefficients q p and q i can be expressed directly from the system parameter ξ and the design parameter α which sets the closed-loop dynamic. It is now possible to use them in order to define systematically the Q and R matrices of Eq. ( 17) and proceed to solve the Riccati equation for the normalized system yielding the state feedback matrix K n of the Eq.( 13).

The normalized controller is calculated by

  F d F q   =   k ni1 k ni2 k ni3 k ni4     Hnd Hnq   +   k np1 k np2 k np3 k np4     Hnd Hnq   +   k nd1 k nd2 k nd3 k nd4     H nd H nq   (23)
The last step is to define the actual gains.

  F d F q   = ω n   k ni1 k ni2 k ni3 k ni4     Hd Hq   +   k np1 k np2 k np3 k np4     Hd Hq   + 1 ω n   k nd1 k nd2 k nd3 k nd4     Ḣd Ḣq   (24)
The resulting closed loop control scheme is shown in the Fig. 5 4. Experimental set-up and protocol

Experimental set-up

In this experimental validation, the strategy is used to control a resonant plate actuated by piezoelectric ceramics (see Fig. 6-a) used in a haptic tactile (Texas instrument). To achieve the voltages required to operate the ceramics (up to 200 V peak), the output voltage of the amplifier is further amplified by a resonant circuit realized by introducing an inductance in series that interact with the blocked capacitance of the piezoelectric ceramics. The inductance is designed to ensure that the electric resonant frequency matches approximately the mechanical resonance. The actual tactile feedback device consists in a 18 × 119 × 2mm 3 aluminum plate to exploit the first plate bending mode at 24 860 Hz. The identified mode shape, identified using a laser vibrometer (Polytec OFV 505), is depicted in Fig. 6-b). Below the plate, four piezoelectric ceramics (16 × 4 × 0.5mm 3 ) are glued, three of them are used to actuate the plate. The remaining is connected to one of the STM32 ADC inputs and is used as a vibration sensor.

Control scheme

The control implemented (shown in the Fig. 7) in this experiment is similar to the one described in [START_REF] Ghenna | Vector Control Of Piezoelectric Transducers and Ultrasonic Actuators[END_REF] which involves two asynchronous process. First, Direct Digital Synthesis (DDS), with frequency 1 MHz (T s =1 µs), generates the voltage reference that is then amplified to supply the actuators. Since for

= (V d + jV q )e jωt (25)
The voltage is generated using the real part of eq.25

v (t k ) = V d cos (ωt k ) -V q sin (ωt k ) (26)
Where t k = kT s and k is the sample number. Similarly to [START_REF] Ghenna | Vector Control Of Piezoelectric Transducers and Ultrasonic Actuators[END_REF] the discrete value of H d and H q are calculated as follows

H d (t k ) = Ns-1 n=0 η(t k-n ) cos (ωt k-n ) (27) H q (t k ) = Ns-1 n=0 η(t k-n ) sin (ωt k-n ) (28)
where N s is the number of samples over a period of the oscillation T = 2π ω . A slower process is implemented at a rate of 10 kHz. In closed-loop, it calculates and updates the voltages components V d , V q . In open loop, it is used for dynamic identification and tests, as discussed in the following section.

Parameters identification

In some circumstances, the parameters can vary due to environmental variations such as temperature or modified boundary conditions during tests. In Once this acquisition performed, the damping factor ξ is calculated using the extrema of the real part H d . Let ω 1 and ω 2 be the angular frequency where H d is maximum and minimum (cf Fig. 8) respectively, then, one can verify that:

ξ = 1 2 1 - ω 2 1 ω 2 2 1 + ω 2 1 ω 2 2 (29)
Since the argument of H = H d + jH q are close to -π 4 and -3π 4 respectively for ω 1 and ω 2 if the damping ξ is small, these points are simple to identify in practice. With the same assumption on the damping, one can consider that the resonance frequency ω n corresponds to the maximum gain frequency. Thus, the static gain g is deduced from:

g = -2 × ξ × H(jω n ) V (jω n ) (30)

Controller implementation

The method discussed so far yields a PID, even if the weighting associated with the derivative action is kept nul in Eq.( 17) . The derivative action can be a problem as it tends to amplify measurement noise. The case of continuous and discrete implementation of the control are examined in Fig. no instability is possible. This contrasts with discrete case, where the high frequency poles remain mostly unchanged with regards to the choice of α for the PI and PID controllers. Hence, in this work, the derivative action is omitted in the following tests.

Results

In order to validate the control approach presented in this paper, a comparison between different desired dynamics (α=[0.25, 0.5, 1.0, 1.5, 2.0, 2.5, 3.0, 3.5, 4.0]) and experimental obtained dynamics is presented. The value of α are used to calculate the MIMO-PID gains by using the approach developed in the previous section. Since the implemented controller is discretized, we consider that a MIMO-PI is sufficient in order to have the same performances. Indeed, the Fig. 9 where the time response is slower, equal or faster than the natural time response of the plate. On the same figure, the theoretical responses are superimposed, showing a good agreement between prediction and measurement. Oscillations can also be observed, which are more important for higher α. They are caused by the spillover effect due to neighbour modes. Since the vibrations signal are demodulated, the oscillation frequency corresponds to the difference between the controlled mode and the neighbour mode. A large choice of α induces a larger closed-loop bandwidth and thus increases the spillover effect, which is not considered in the model [START_REF] Babitsky | Autoresonant control of nonlinear mode in ultrasonic transducer for machining applications[END_REF].

To further assess the theory discussed above, the Fig. 11 depicts the logarithm of the time response to a reference for various α. The tendency is linear as expected. Based this figure, the slope α exp is identified and compared to the theoretical values α on table 4. However, it can be noted that for low values of α, the discrepancy between theory and measurement can be large. Actually, this is due to the fact the gain implemented to slow down the transient are small. As a consequence, the voltage applied is small, and results in a coarse output voltage once converted by the digital to analog conversion. Hence, the seemingly constant output at start-up and steady-state that can be observed for α = 0.25, and results in a rough linear approximation. For high values of α, however, the good agreement proves that the simplifying hypothesis proposed in the theory are verified and it demonstrates the validity of the method.

Conclusion

A method to design systematically the controller for a resonator in a rotating frame was discussed. First, the model was presented in a state space form, and the design of the controller is based on a LQR optimal controller. The novelty of the proposed design is to relate directly the weight matrices Q and R to relevant performances indexes that guarantee a given time response and an envelop of the vibration during the transient. This reveals a simple relationship linking the α index, that governs the time response, to the weighting factors implemented in the Q and R matrices. The experimental results demonstrate a good agreement between the theory and the measurements despite some technological limitation such as the digital/analog conversion. A second issue is the spillover effect that limit the practical choice for α.

Figure 1 :

 1 Figure 1: The equivalent model of a damped harmonic oscillator in a demodulated form

Figure 2 :

 2 Figure 2: Example of a Demodulated signals : (left) example of two vibration signals with the same envelope and with different phases, and their (right) real and imaginary parts when demodulated.

Figure 3 :

 3 Figure 3: Eigenvalues of a second order system, in a normalized time base, before and after demodulation (respectively circles and crosses).

Figure 4 :

 4 Figure 4: Closed loop performances as a function of the weighting parameters (qp and q i ) : (a) relation between the acceleration factor lg(α) and the weighting parameters for ξ = 10 -4 , (b) relation between the indicator lg(γ) and the weighting parameters for ξ = 10 -4 , (c) examples of closed-loop time responses when the weighting parameters are chosen in zone A,B and C and for ξ = 10 -4 , (d-i) relation between the acceleration factor lg(α) and the weighting parameters by eliminating all the zone where |lg(γ)| > 0.02 for different value of ξ

  For a specific plant with the resonance frequency ω n , they are deduced from the obtained generalized controller by applying the variable change Hnd = Hd = H dref -H d , Hnq = Hq = H qref -H q , Hnd dθ = ω n Hd dt , Hnq dθ = ω n Hq dt, H nd = d Hd dθ = Ḣd ωn and H nq = d Hq dθ = Ḣq ωn . Finally, the implemented controller is written

Figure 5 :

 5 Figure 5: Closed loop control scheme

Figure 6 :

 6 Figure 6: (a) The vibrating plate and it's control unit, (b) identified mode shape φ k (x, y) of the 1x1 bending mode

Figure 7 :

 7 Figure 7: Feedback control scheme

Figure 8 :

 8 Figure 8: Frequency response of the plate with different input voltage and frequency response of the nominal model

Figure 9 :

 9 figure, the closed loop poles are normalized by Ω in order to compare oscillators with different dynamics. Concerning the discrete case, the sampling time used in the computation of the poles is chosen large enough to respect the Nyquist frequency with regards to the dynamics αξΩ.Considering the continuous case, if the derivative actions resulting from the LQR procedure are omitted, the high frequency poles at 2Ω are shifted toward the right as the value of α is increased rapidly resulting in an unstable pole pair. By contrast, including the derivative action induce the opposite behaviour, hence

Figure 10 :

 10 Figure 10: Step response for α = (0.5, 1.0, 2.0)

Figure 11 :

 11 Figure 11: Logarithm of the time response to a reference (log(1 -H d H dref)) for various α, the slope of the fitted lines corresponds to the closed-loop time constant 1/(ξcωn)

Table 1 :

 1 Polynomial's parameters (a,b) of the relation lg(q i ) = a × lg(qp) + b and coordinates

		-7	-6	-5	-4	-3	-2
	a	1.0000	1.0000	1.0000	0.9999	1.0000	0.9913
	b	-13.9950 -12.0000 -10.0000 -7.9894	-6.0000 -4.0292
	lg(q p0 ) -13.3939 -11.4177 -9.3933	-7.3933	-5.4177 -3.3708
	lg(q i0 )	-27.3889 -23.7177 -19.3933 -15.3820 -11.4177 -7.3708

Table 2 :

 2 Relation between δ = lg(qp/q p0 ) and lg(α) for different values of ξ ∈ [10 -7 , 10 -2 ]

	lg(ξ)	-7	-6	-5	-4	-3	-2
	a α	0.5000	0.5000	0.5000	0.5000	0.5000	0.5000
	b α	1.34e -4 -1.23e -4 4.24e -5 4.36e -5 4.30e -5 4.30e -5
	||e||	2.04e -2	2.8e -3	2.07e -4 2.56e -5 4.33e -6 1.16e -6

Table 3 :

 3

Polynomial's parameters (aα, bα) of the relation lg(α) = aα × lg(qp/q p0 ) + bα for different values of ξ ∈ [10 -7 , 10 -2 ], and the norm of the residuals ||e|| simplified as:

Table 4 :

 4 Experimental acceleration ratios αexp and the error (in percent)

Acknowledgement

The authors would like to thank the CNRS/IRCICA-USR 3380 research center, which hosted the research.