P Y Bryk

S Bellizzi

R Côte
email: cote@lma.cnrs-mrs.fr

Experimental study of a hybrid electro-acoustic nonlinear membrane absorber

Keywords: Noise Reduction, Hybrid Absorber, Nonlinear Energy Sink, Electroacoustic Absorber, High Sound Level, Low Frequency

A hybrid electro-acoustic nonlinear membrane absorber working as a nonlinear energy sink (here after named EA-NES) is described. The device is composed of a thin circular visco-elastic membrane working as an essentially cubic oscillator. One face of the membrane is coupled to the acoustic field to be reduced and the other face is enclosed. The enclosure includes a loudspeaker for the control of the acoustic pressure felt by the rear face of the membrane through proportional feedback control. An experimental set-up has been developed where the EA-NES is weakly coupled to a linear acoustic system. The linear acoustic

system is an open-ended tube, coupled on one side to the EA-NES by a box, and on the other side to a source loudspeaker by another box. Only sinusoidal forcing is considered. It is shown that the EA-NES is able to perform resonance capture with the acoustic field, resulting in noise reduction by targeted energy transfer, and to operate in a large frequency band, tuning itself passively to any linear system. We demonstrate the ability of the feedback gain defining the active loop to modify the resonance frequency of the EA-NES, which is a key factor to tune the triggering threshold of energy pumping. The novelty of this work is to use active control combined to passive nonlinear transfer energy to improve it. In this paper, only experimental results are analyzed.

Introduction

The reduction of noise and vibration at low frequencies is still nowadays a main issue in many fields of engineering. In order to overpass this issue, a new concept of absorbers including nonlinear behavior has been developed in the past decade. This type of absorbers is based on the principle of the "Targeted Energy Transfer" (TET) also named "energy pumping" [START_REF] Vakakis | Nonlinear targeted energy transfer in mechanical and structural systems[END_REF]. TET is an irreversible transfer of the vibrational energy from an input subsystem to a nonlinear attachment (the absorber) called Nonlinear Energy Sink (NES). TET permits to reduce undesirable large vibration amplitudes of structures or acoustic modes. Nonlinear energy transfer results from nonlinear mode bifurcations, or through spatial energy localization by formation of nonlinear normal modes.

The phenomena can be described as a 1:1 resonance capture [START_REF] Vakakis | Energy pumping in nonlinear mechanical oscillators: Part II -Resonance capture[END_REF] and, considering harmonic forcing, as response regimes characterized in terms of periodic and Strongly Modulated Responses (SMR) [START_REF] Starosvetsky | Dynamics of a strongly nonlinear vibration absorber coupled to a harmonically excited two-degree-of-freedom system[END_REF].

The basic NES generally consists of a lightmass, an essentially nonlinear spring and a viscous linear damper. In the field of structural vibration, a wide variety of NES designs with different types of stiffness (cubic , non-polynomial, non-smooth nonlinearities...) has been proposed [START_REF] Gourdon | Nonlinear energy pumping under transient forcing with strongly nonlinear coupling: Theoretical and experimental results[END_REF][START_REF] Sigalov | Resonance captures and targeted energy transfers in an inertially-coupled rotational nonlinear energy sink[END_REF][START_REF] Gourc | Quenching chatter instability in turning process with a vibro-impact nonlinear energy sink[END_REF][START_REF] Mattei | Nonlinear targeted energy transfer of two coupled cantilever beams coupled to a bistable light attachment[END_REF]. In the acoustic field, to the best of our knowledge only one type of vibro-acoustic NES design has been tested, see the series of papers [START_REF] Cochelin | Experimental evidence of energy pumping in acoustics[END_REF][START_REF] Bellet | Experimental study of targeted energy transfer from an acoustic system to a nonlinear membrane absorber[END_REF][START_REF] Mariani | Toward an adjustable nonlinear low frequency acoustic absorber[END_REF][START_REF] Cote | Experimental evidence of simultaneous multi-resonance noise reduction using an absorber with essential nonlinearity under two excitation frequencies[END_REF][START_REF] Shao | Theoretical and numerical study of targeted energy transfer inside an acoustic cavity by a non-linear membrane absorber[END_REF]. It was demonstrated that a passive control of sound at low frequency can be achieved using a vibroacoustic coupling between the acoustic field (the primary system) and a geometrically nonlinear thin clamped structure (the NES). In [START_REF] Cochelin | Experimental evidence of energy pumping in acoustics[END_REF][START_REF] Bellet | Experimental study of targeted energy transfer from an acoustic system to a nonlinear membrane absorber[END_REF][START_REF] Shao | Theoretical and numerical study of targeted energy transfer inside an acoustic cavity by a non-linear membrane absorber[END_REF], the thin baffled structure consists of a simple thin circular latex (visco-elastic) membrane whereas in [START_REF] Mariani | Toward an adjustable nonlinear low frequency acoustic absorber[END_REF] a loudspeaker used as a suspended piston is considered. In both cases, the thin baffled structure has to be part of the frontier of the closed acoustic domain (to be controlled). Hence only one face (named the front face) is exposed to the acoustic field whereas the other face (the rear face) radiates outside. Hence this type of devices has to be modified to be used in cavity noise reduction.

A simple way to do this is to enclose the rear face of the thin clamped structure limiting the sound radiation. This principle has been used to design electroacoustic absorbers based on the use of an enclosed loudspeaker including an electric load that shunts the loudspeaker electrical terminals [START_REF] Boulandet | Optimization of electroacoustic absorbers by means of designed experiments[END_REF]. An electroacoustic absorber can either be passive or active in terms of their noise suppression characteristics including as in [START_REF] Lissek | Electroacoustic absorbers : Bridging the gap between shunt loudspeaker and active sound absorption[END_REF][START_REF] Boulandet | Toward broad band electroacoustic resonators through optimized feedback control strategies[END_REF] pressure and/or velocity feedback techniques. Loudspeakers have also been used to design devices to control normal surface impedance [START_REF] Lacour | Preliminary experiments on noise reduction in cavities using active impedance changes[END_REF]. Two approaches have been developed. The first is referred to as direct control: the acoustic pressure is measured close to the diaphragm of the loudspeaker and used to produce the desired impedance. In the second approach, passive and active means are combined: the rear face of a porous layer is actively controlled so as to make the front face normal impedance take a prescribed value.

In this paper, a hybrid passive/active nonlinear absorber is developed. The absorber is composed of a clamped thin circular visco-elastic membrane with its rear face enclosed. The acoustic field inside the hood (i.e the acoustic load of the rear face) is controlled using a loudspeaker with proportional feedback control. Three objective are assigned. Firstly, the device has to be designed such that it can be used inside a cavity. Secondly, noise reduction must mainly result from TET due to the nonlinear behavior of the membrane, thus defining a new concept of NES. Thirdly, the control loudspeaker has to be used as a linear electroacoustic absorber inside the hood. The control loudspeaker does not act directly on the acoustic field to be reduced. It only modifies the relative acoustic load exciting the membrane. This absorber is here after named hybrid electroacoustic NES (EA-NES).

The paper is organized as follows. In Section 2, the principle of functioning of the EA-NES under study is described considering each sub-structure separately.

In Section 3, we first describe the experimental set-up. It is composed of an acoustic field (in a pipe, excited by a loudspeaker) coupled to the EA-NES. Then we check the stability analysis of the feedback loop and perform a frequency analysis under broadband excitation. In Section 4, we analyze in detail the responses under sinusoidal excitations and we bring some confirmations on the efficiency of the EA-NES.

The Hybrid Electro-Acoustic NES

General presentation

The EA-NES is shown in Fig. 1 The EA-NES is based on the conjugate functioning of three elements :

• The clamped membrane that interacts with the acoustic field in order to provide noise attenuation in its non-linear range;

• The hood by which the EA-NES can work inside a surrounding acoustic field unlike previous developed NES (see for example [START_REF] Bellet | Experimental study of targeted energy transfer from an acoustic system to a nonlinear membrane absorber[END_REF]);

• The feedback loop that reduces the pressure in the hood allows to use a small hood volume and also to tune the stiffness and damping linear behaviour of the EA-NES.

2.2. About each subsystem of the EA-NES

The clamped membrane

The clamped membrane with its supporting device is shown in Fig. 1(right).

It was already used in [START_REF] Bellet | Experimental study of targeted energy transfer from an acoustic system to a nonlinear membrane absorber[END_REF]. The device allows to change the diameter of the membrane (from 40 to 80 mm). It includes a sliding system used to apply a constant and permanent in-plan pre-stress to the membrane. Once the prestress is set, the membrane is clamped to the supporting device. Applying an in-plane pre-stress modifies the modal component of the associated underlying linear system.

Coupled to an undesirable acoustic field, the clamped membrane device can be used as an acoustic NES absorber [START_REF] Bellet | Experimental study of targeted energy transfer from an acoustic system to a nonlinear membrane absorber[END_REF] to reduce the acoustic pressure.

Direct [START_REF] Shao | Theoretical and numerical study of targeted energy transfer inside an acoustic cavity by a non-linear membrane absorber[END_REF] or indirect [START_REF] Bellet | Experimental study of targeted energy transfer from an acoustic system to a nonlinear membrane absorber[END_REF] couplings are possible. If the clamped membrane device is properly designed, TET is obtained thanks to resonance capture phenomena.

Resonance capture occurs between two nonlinear modes resulting of a weak coupling between the nonlinear subsystem (the NES) and the linear subsystem (the acoustic field). At low excitation level, the two nonlinear modes coincide respectively with the linear mode of the NES and the target mode of the acoustic field. Hence, when the in-plane pre-stress of the clamped membrane sets the resonance frequency of the NES lower than the resonance frequency of the target mode of the acoustic field, TET is possible above a threshold excitation level.

Furthermore, Bellet [START_REF] Bellet | Vers une nouvelle technique de controle passif du bruit : Absorbeur dynamique non lineaire et pompage energetique[END_REF] has also shown that the gap between the two resonance frequencies has an impact on the threshold of the TET: the closer are the two resonance frequencies, the smaller is the threshold of the TET.

As a subsystem of the EA-NES, the clamped membrane provides the coupling between the EA-NES and the acoustic field to reduce. It is also responsible for the resonance capture phenomena (as a nonlinear component). The other subsystems (the hood and the feedback loop) must preserve the essential prop-erties of a NES, namely a weakly damped system with a hardening nonlinear mode, and with a frequency at low vibratory amplitude smaller than the resonance frequency of the target mode of the acoustic field.

The hood

The hood is a wooden cubic box with the clamped membrane fixed on one face and an enclosed control loudspeaker on the opposite face (see Fig. 1(left)).

The hood has been added to meet two key objectives. Firstly, with hood, the EA-NES can be used to reduce the noise in a cavity where only the front face of the clamped membrane is load by the undesirable acoustic field. Secondly, the hood creates a difference of pressure between the two faces of the membrane, which permits to place it anywhere inside the acoustic field.

When the acoustic wavelength is large in comparison with the largest dimension of the hood, the acoustic field can be considered as homogeneous in the hood and the acoustic pressure p e (t) can be related to the relative variation of the volume as

p e (t) = -ρ a c 2 0 ∆V e (t) V e (1)
where V e denotes the volume of the box at rest, ∆V e (t) the variation of the volume, ρ a the volumetric mass of the air and c 0 the celerity of sound in the air.

Assuming the motion of the membrane primarily defined on the first transversal mode, Eq. (1) reduces to

p e (t) = ρ a c 2 0 V e (S e m x m (t) -S e ls x ls (t)) (2)
where x m denotes the transversal membrane motion with S e m the effective area of the membrane and x ls denotes the transversal motion of the diaphragm of the loudspeaker with S e ls the effective area. Eq. ( 2) means that the air inside the hood is equivalent to an acoustic compliance that resists to the motion of the membrane and hardens it. This compliance being in return inversely related to the volume V e , it results that the smaller is the volume of hood, the higher is the resonance frequency of the underlying linear EA-NES. Finally Eq. ( 2) also shows that the acoustic pressure p e (t) can be reduced to zero if x ls and x m are in-phase. The feedback loop is composed of an enclosed loudspeaker inside the hood, a microphone placed at the geometrical center of the hood and a unit control (see The reference of the loop (u t (t) = 0) means a pressure reduction p e (t) until zero, which is only possible with an infinite value of K as illustrated by the following total loop transfer in Laplace variables

P e (s) = (1 -H p (s)KH F (s)H s (s)) -1 P p (s) (3)
between the acoustic pressure p p (t) and the acoustic pressure p e (t). However the choice of the gain K is limited by instability phenomena which exist usually for positive gain K at high frequencies, resulting from the acoustic modes of the cavity and for negative gain K at low frequency (< 100 Hz), resulting from the control loudspeaker and membrane dynamics. The influence of the acoustic modes of the cavity is reduced by placing the microphone at the geometrical center of the hood. This location is near the node of pressure of some of the first modes.

Stability margin analysis results from the properties of the Open Loop Transfer Function (OLTF) between (t) and u M (t) defined by

H F (s)H S (s)H p (s)K (see Fig. ( 2
)
). The variation range of the gain values can be increased by selecting adequately the filter H F (s). It is that we do in the next section.

Choice and sizing of components of the EA-NES

Our objective is to design a EA-NES able to interact with primary acoustic fields in the frequency range [40, 100] Hz.

The clamped latex membrane was designed following the recommendations discussed in [START_REF] Bellet | Experimental study of targeted energy transfer from an acoustic system to a nonlinear membrane absorber[END_REF]. The membrane has a radius of 0.05 m and a thickness of 0.0002 m. These dimensions guarantee a proper functioning of the clamped latex membrane as a NES when the membrane is coupled to a resonant tube [START_REF] Bellet | Experimental study of targeted energy transfer from an acoustic system to a nonlinear membrane absorber[END_REF][START_REF] Cote | Experimental evidence of simultaneous multi-resonance noise reduction using an absorber with essential nonlinearity under two excitation frequencies[END_REF] without hood (alone). The sliding system is adjusted such that the pre-stress of the membrane gives the coupled setup (EA-NES and primary system) a resonance associated to the EA-NES at a frequency around 70 Hz (see Fig. 5).

The

Experimental set-up and stability analysis

Experimental set-up

The experimental set-up shown in Fig. 3 consists in a vibroacoustic system (also named primary system) coupled to the EA-NES. The same set-up was used in [START_REF] Bellet | Experimental study of targeted energy transfer from an acoustic system to a nonlinear membrane absorber[END_REF]. The primary system is made of an open interchangeable U-shaped tube which length L can be adjusted, coupled at each end to coupling boxes. The diameter of the tube is 0.095 m. One coupling box (box 1) contains an acoustic source, and the EA-NES (as described in the previous section) is mounted on one face of the other coupling box (box 2). The volume of the coupling box During the measurement, a target voltage signal e(t) from a generator (not shown in Fig. 3) and a power amplifier (TIRA, BAA120) provide an input current or input voltage signal to the source loudspeaker (depending the selected driving mode: current-or voltage-feedback control mode). The responses of the system are recorded simultaneously (using a multi-channel analyzer/recorder OROS, OR38): the acoustic pressures p tube (t) (at the mid-length of the tube), The lowest resonance around 20 Hz results from the whole coupled system (the tube acts as an acoustic mass coupled to the coupling boxes acting like springs). The next two resonances, around 72 Hz and 87 Hz, as seen in Fig. 5, are respectively assigned to the EA-NES and to the first mode of the tube alone.

1 is V 1 = 27 × 10 -3 m
Beyond 100 Hz, the resonance frequencies coming from the highest modes of the tube and the coupling boxes appear. From now on we will focus on a frequency span below 100 Hz.

Stability of the feedback loop

As a preliminary step, the OLTF, H F (s)H S (s)H p (s)K at s =j2πf , is measured with the EA-NES coupled to the tube of length L 1 with a unity gain (K = 1) (see Fig. 3 Selecting the gain K in the range [0, 200] results in an OLTF with gain margin greater than or equal to 6 dB, the 0 dB gain margin corresponding 260 to K = 400. Gain margins and phase margins with the associated critical frequencies are reported Table 1 for several values of K. Similarly selecting the gain K in the range [-64, 0] results in an OLTF with gain margin from to 0 dB to 36 dB. . Note that, one can also modify the use of the feedback loop in order to amplify p e (t) instead of reducing it (hardening the clamped membrane behaviour).

K f Gm (Hz) G m (dB) f Pm (Hz) P m ( • ) L 1 L 2 L 1 L 2 L 1 L 2 L 1 L 2 -1 75
To achieve this goal, the loudspeaker is fed with a negative value for the gain K. This case is equivalent to study the feedback loop with OLTF shifted of 180 degrees, which gives different phase and gain margins reported in Table 1 for K = -1. One can notice that the gain margin for K = 1 is higher than the gain margin with K = -1. It means the feedback loop can soften more the membrane than harden it.

The gain K as a tuning parameter of the EA-NES

The objective of this section is to verify that the gain K in the feedback loop can be used to tune the modal component associated with the EA-NES without disturbing the modal component associated with the primary system.

The influence of the gain K on the behavior of the EA-NES is analyzed after measuring the FRF p 2 /e. Here also the source loudspeaker is driven in voltagefeedback control mode. It is excited with a low-level band-limited white-noise. In terms of modal parameters, these results confirm that the gain K affects simultaneously the resonance frequency and the associated damping ratio of the modal component assigned to the EA-NES. Increasing the gain K reduces the resonance frequency and simultaneously increases the damping ratio. Finally, as expected, negative values of the gain K increase the resonance frequency. NES is able to work as a linear electroacoustic absorber inside the hood with any linear primary system having its resonance frequency in a large frequency range.

Study at high excitation level: efficiency of the EA-NES from TET

Now let us look at the behavior of the coupled system under a sinusoidal forcing when the frequency and the amplitude of the sinusoidal forcing vary.

Here the source loudspeaker is driven using a current-feedback control mode reducing the dissipation introduced in the system by the source loudspeaker [START_REF] Bortoni | Effects of acoustic damping on current-driven loudspeakers[END_REF].

During the measurement, a target signal e(t) from a generator (TTI TGA1244) (not shown in Fig. 3) and a power amplifier (TIRA, BAA120) provide an input current signal to the source loudspeaker. The target signal e(t) is of the form

e(t) = E cos(2πf e t + φ e ) ( 4
)
where f e denotes the excitation frequency and E is the associated excitation amplitude. The phase φ e is introduced arbitrarily by the signal generator.

A measurement run consists in making a series of experiments, where the value of the scanning frequency f e is updated for each experiment, while the other parameter E remains unchanged. The duration of an experiment must be limited for practical reasons, but must be long enough to capture the physics of the response. We have chosen a duration of 13 s for each experiment. There In order to characterize the pumping phenomenon, we first describe in details the results obtained with the EA-NES with K = 200 coupled to the tube of L 1 length, followed by an analysis of the influence of K. Then, the experiment involving the tube of L 2 length is considered.

EA-NES with the tube of L 1 length and K = 200

The behavior of the system with the tube of L 1 length and K = 200 under sinusoidal excitation, scanned around the first mode of the tube is presented. Concerning the command level of the source loudspeaker, the power amplifier driving the loudspeaker in current-feedback mode provides a nearly constant source loudspeaker current i s (t) as shown in Fig. 9 where the Root Mean Square (RMS) values of i s (t) are plotted versus frequency for some excitation amplitudes. Hence the source loudspeaker is not modified by the system (tube+EA-NES) and it plays its full role as a controlled source.

Concerning the system response, the RMS value in the steady state regime is extracted from each measurement of the acoustic pressure p tube (t) and it is plotted in Fig. 10(a peak around f = 83.5 Hz, smaller than the resonance frequency observed at low excitation level. These behaviors were classically reported considering NES analysis [START_REF] Bellet | Experimental study of targeted energy transfer from an acoustic system to a nonlinear membrane absorber[END_REF] and they can be attributed to the pumping phenomenon and TET from the primary system to the NES. TET signatures are of two types. First, TET occurs only when the primary linear system reaches a certain vibration energy threshold and secondly TET is associated to quasi-periodic response regime with a slow evolution of the amplitudes (also named Strongly Modulated Regimes (SMR) [START_REF] Starosvetsky | Dynamics of a strongly nonlinear vibration absorber coupled to a harmonically excited two-degree-of-freedom system[END_REF]). This point can be confirmed by an analysis of energy conversion.

To analyze the energy conversion occurring from the fundamental frequency f e , to the harmonic frequencies (kf e for the integer k > 1) and to the non harmonic frequencies (f = kf e ), the Fundamental Conversion Ratio (FCR), the Harmonic Conversion Ratio (HCR) and the Non Harmonic Conversion Ratio (NHCR) are used. The definition are these indicators are recalled in [START_REF] Cote | Experimental evidence of simultaneous multi-resonance noise reduction using an absorber with essential nonlinearity under two excitation frequencies[END_REF]. For each signal, FCR, HCR and NHCR are obtained from Fourier analysis estimated from the steady state response. Basically, the FCR is the proportion of signal energy at the source frequency. For a linear system this indicator should be 100%. The HCR is the proportion of signal at the integer harmonics of the source frequency. It can give information about nonlinear effects like saturation.

The NHCR is the proportion of signal frequencies out of the integer harmonics of the source frequency. It can give information about nonlinear effects like a loss of periodicity. 10(d)). This transfer of energy increases with the excitation amplitude but the associated RMS level of the acoustic pressure in the tube remains limited (see Fig. 10(a)). In this domain, the responses of the system are no more periodic and are replaced by so-called SMR. This domain is the domain where the EA-NES works well.

Another amplitude-frequency domain which is detached from the previous domain is also visible in Fig. 10(c) at low frequencies and high amplitudes showing that, in this case, energy can be transferred from the fundamental frequency to the high harmonic frequencies. In this domain, the response of the system is periodic but unfortunately the associated RMS level increases, which leads to undesired periodic regimes. This domain is the domain where is EA-NES not efficient for noise reduction and it is characterized by the appearance of undesired periodic regimes. The threshold for the appearance of undesired periodic regimes, here denoted EA-NES leads to a two-to-three decrease (6 to 10 dB) in the acoustic pressure.

The influence of the gain K is similar to the influence of the linear stiffness of a classical NES [START_REF] Bellet | Experimental study of targeted energy transfer from an acoustic system to a nonlinear membrane absorber[END_REF].

4.3. EA-NES with the tube of L 2 length: influence of K

The same analysis was also conducted with a tube of length L 2 . A very important property of this EA-NES can be first highlighted: its ability to adapt and tune itself to the resonance frequency of different linear systems. To demonstrate experimentally this property we report in Fig. 16 the ridge lines of the acoustic pressure p tube measured on the system with the tube of L 1 length and on the system with the tube of L 2 length using the the EA-NES with the same gain value K = 50. In both cases, we observe that EA-NES works well and the associated efficiency span ([E s , E S ])are [0.5, 1.1] and [1, 2.2] respectively. However it appears that the EA-NES does not perform noise reduction with the same efficiency for the two tubes. Indeed, with the tube of L 1 length the acoustic pressure in the efficiency span is near 750 Pa whereas is about 1500 Pa for the tube of length L 2 . This difference is due to the fact that the tube of L 2 length has a higher resonance frequency than the tube of L 1 length. In consequence, more energy has to be provided to the tube of length L 2 so that resonance capture occurs [START_REF] Bellet | Vers une nouvelle technique de controle passif du bruit : Absorbeur dynamique non lineaire et pompage energetique[END_REF]. A complete excursion as in case of tube of L 1 length was not be possible due to a limitation of the acoustic source performance used in this setup.

Finally, the maximum power consumed by the control loudspeaker are displayed in Fig. 18 for several values of the gain K. For all cases, the power consumed by the loudspeaker increases with the excitation amplitude and slowly varies with the gain K for positive values. The maximum power consumed by the loudspeaker is inferior to 1.5 W rms . Thus the EA-NES requires a limited amount energy to work, which is a good asset for industrial applications.

Conclusion

A new acoustic NES with a controlled acoustic load has been presented.

The control of the linear stiffness of the membrane by the help of a feedback loop has been validated experimentally. Furthermore this NES has been tested coupled to two different tubes and has performed acoustic pumping with various efficiencies depending on the gain K. It appears that there is no optimum value of the gain K because it depends on the excitation level. Indeed one can obtain either a low threshold but a short span when f N ES is close to f T , or a high threshold with a large span when f N ES is distant from f T . However, unlike the previous passive NES, the frequency of the EA-NES can be easily modified in real time. Future work is ongoing, focusing on using a gain K variable.

It would allow to optimize K according to the excitation in order to get the strongest sound attenuation. We work also on the integration of the EA-NES in a cavity, with a narrow band noise excitation, in the framework of progress towards applications.

  . It is composed of a clamped circular latex membrane with one face (the front face) exposed to the acoustic field to be reduced and the other one (the rear face) enclosed. The hood includes a feedback loop composed of a microphone, a loudspeaker and a unit control. The feedback loop controls the acoustic pressure inside the hood and seen by the rear face of the membrane.

Figure 1 :

 1 Figure 1: EA-NES: (a) Schematic representation and (b) Front face.

Fig. 1 (

 1 Fig.1(left)). The feedback loop is based on a proportional controller following the block-diagram displayed in Fig.2where H p (the plant transfer) denotes the transfer function between the tension u c (t) applied to the control loudspeaker (voltage-feedback control) and the acoustic pressure p e (t) and H s the transfer function characterizing the microphone. A part of the acoustic pressure p e (t) is due to p p (t) (perturbation term) resulting from the undesirable acoustic field acting on the front face of the clamped membrane. The unit control includes an analogue band-pass filter H F and a controller sets a scalar (real) gain K.

  dimensions of the enclosure are 0.38 m×0.22 m×0.22 m, which gives V e = 0.018 m 3 . The membrane and the control loudspeaker are fixed respectively on the opposite faces with size 0.22 m × 0.22 m. Note that the pressure p e (t) inside the volume V e is equivalent to a compliance as long as the acoustic wavelength is large in comparison with the largest dimension of the enclosure. With λ > 10 × 0.38 one obtains a maximum frequency of f max = 89.5 Hz. Efficient control loudspeakers need the following properties: high resonance frequency of the driver part (such that the resonance frequency of the enclosed loudspeaker is above the working frequency band of the EA-NES), large force factor, linear behavior for large excursion of the diaphragm and effective area of the diaphragm as large as possible compared to the effective area of the latex membrane. Note that the area of the diaphragm is also limited by the size of the boxes. The BEYMA 8P300Fe/N (8 Inch) loudspeaker family was selected corresponding to the following Thiele/Small parameter values: m LS = 19.4 10 -3 kg, c LS = 1.7 Nsm -1 and k LS = 2834.9 Nm -1 , S e LS = 0.022 m 2 , R e = 6.6 Ω and Bl = 9.21 NA -1 . The rear enclosure of the loudspeaker is chosen as V LS = 0.0248 m 3 . The effective diaphragm area is six time larger than the effective latex membrane, so for the same volume variation the displacement of the diaphragm is reduced to the same extent with respect to the displacement of the membrane (see Eq. (2)), which should happen in case of a perfect control. The resonance frequency of the driver part alone of the control loudspeaker is 60.8 Hz. It increases to 95 Hz when V e and V LS are taken into account. Finally the other elements of the control unit are a G.R.A.S 40BH microphone chosen for its high dynamics (until 181 dB SPL), an analogic band-pass filter KEMO Benchmaster VBF8, and a TIRA BAA 120 amplifier used to set the gain K.

3 Figure 3 :

 33 Figure 3: Experimental set-up under study: (a) Picture and (b) Schema (the blue lines define the configuration used to measure the OLTF and the same notations as in Fig. 2 were used).

p 2

 2 (t) (inside the box 2) and p e (t) (inside the hood of the EA-NES) with three microphones (GRAS, 40BH), (see Fig.3, right). Also recorded (not shown on the schema) are the source loudspeaker current i s (t) and voltage e s (t) responses and the control loudspeaker current i c (t) and voltage responses e c (t). The sampling frequency is f s = 8192 Hz.

Figure 4

 4 Figure 4 shows the Frequency Response Function (FRF) denoted p 2 /e between the voltage applied to the source loudspeaker e(t) (in voltage-feedback

Figure 4 :

 4 Figure 4: FRF p 2 /e measured with the EA-NES with K = 0 coupled to the tube of L 1 length: (a) Modulus and (b) Phase.

Figure 5

 5 Figure 5 shows the FRF p tube /e and p 2 /e measured with the tube of L 1 length and with the tube of L 2 length and imposing in both cases the control gain K = 0. For the two tubes, two resonance peaks appear in the frequency band [40, 120] Hz. In both cases (L 1 and L 2 lengths), the first resonance peak is localized around the frequency 70 Hz and can be attributed to the EA-NES.The second resonance peak (around the frequency 87 Hz for the tube of L 1 length and around the frequency 99 Hz for the tube of L 2 length) is primarily attributed to the tube (in accordance with L 1 < L 2 ). In both case, they exhibit high response levels.

Figure 5 :

 5 Figure 5: FRFs (a,c) p tube /e and (b,e) p 2 /e measured with the EA-NES with K = 0 coupled to the tube of L 1 (blue curves) and L 2 (red dashed curves) length: (a,b) Modulus and (c,d) Phase.

2 -Figure 6 :

 26 Figure 6: Open loop transfer function in the Nysquist domain measured with the EA-NES with unity gain (K = 0) coupled to the tube of L 1 (red dashed curves) and L 2 (blue curves) length.

Figure 7

 7 Figure7shows the modulus of the FRF p 2 /e with the tube of L 1 length for several values of the gain K. Also reported is the FRF obtained by replacing the EA-NES with the clamped membrane NES alone (no hood). First of all, we

Figure 7 :

 7 Figure 7: FRF p 2 /e measured with the EA-NES for several values of the gain K and with a clamped membrane NES (red continuous line) coupled to the tube of L 1 length.

Figure 8 :

 8 Figure 8: FRF p 2 /e measured with the EA-NES for several values of the gain K coupled to the tube of L 2 length.

  are two steps in an experiment. The first step lasts 3 s with no source signal. It permits us to get null initial conditions, whatever happened before. The second step lasts 10 s with the source on, but we record only the last 7 s, the first 3 s include generally the transient effects of the excitation. A measurement test consists in making a series of runs where the value of the amplitude E is updated for each run. Six tests were performed with the tube of L 1 length with the following six values of the gain K: -40, 0, 50, 100, 200 and 400. Five tests were performed with the tube of L 2 length with the following five values of the gain K: -40, 0, 50, 65 and 100. The frequency step δf = 0.25 Hz was used to define runs in the frequency band [80, 93] Hz for the tube of L 1 length and [90, 105] Hz for the tube of L 2 length. The step in the amplitude band [0.01414, E max ] is equal to 0.07 with E max = 2.9 for the tube of L 1 length and E max = 4 for the tube of L 2 length.

Figure 9 :

 9 Figure 9: RMS values of the source loudspeaker current is(t) measured with the EA-NES with K = 200 coupled to the tube of L 1 length for several values of the excitation amplitude E versus excitation frequency.

Figure 10 :

 10 Figure 10: System with the EA-NES with K = 200 coupled to the tube of L 1 length: (a) RMS values, (b) FCR, (c) HCR and (d) NHCR of the steady state regime of p tube as a surface level according to frequency and excitation amplitude.

  Fig. 10(b)). The corresponding response regimes are periodic resulting from the linear behavior of the EA-NES. By increasing the excitation amplitude, an amplitude-frequency domain appears, characterized by steady state responses, a fraction of energy of which is transferred from the fundamental frequency to the non harmonic frequency domain (see Fig. 10(d)). This transfer of energy

Figure 11 :

 11 Figure 11: System with the EA-NES with K = 200 coupled to the tube of L 1 length: Steady state responses of (a) p tube and (b) pe versus time and (c) parametric plot (p tube , pe) obtained with the excitation frequency fe = 86.75 and excitation amplitude E = 0.29.

Figure 12 :

 12 Figure 12: Idem as Fig. 11 with the excitation frequency fe = 86.75 and excitation amplitude E = 1.5.

Figure 13 :Figure 14 :

 1314 Figure 13: Idem as Fig. 11 with the excitation frequency fe = 83.5 and excitation amplitude E = 2.24.

Figure 16 :

 16 Figure 16: System with the EA-NES with the gain K = 50 coupled to the tube of (a) L 1 and (b) L 2 length: Ridge line (blue line) of the acoustic pressure p tube and corresponding resonance frequencies (red with bullets line, y-axis on right side) versus the excitation amplitude.

Figure 17 :

 17 Figure 17: System with the EA-NES with several values of gain K coupled to the tube of L 2 length: Ridge line of the RMS values of p tube according to the level of excitation.

Figure 17 )

 17 Figure 17) shows the ridge lines of the acoustic pressure p tube obtained with five values of gain K (= -40, 0, 50, 65 and 100). The results are similar to that observed with the tube of length L 1 for the configuration with the EA-NES with K = -40, 0, and 50 where the thresholds E s and E S are visible. For the configuration with the EA-NES with K > 50, only the threshold E s are reached.

Figure 18 :

 18 Figure 18: System with the EA-NES with several values of gain K coupled to the tube of L 2 length: Maximum value of power consumed by the control loudspeaker depending on the level of excitation.

Table 1 :

 1 Gain Gm and phase Pm margins and associated frequencies f Gm and f Pm of the feedback loop measured with the EA-NES for several values of the gain K coupled to the tubes of L 1 and L 2 length.

			76.25 36.1 36.9	-	-	360 360
	1	441	436	52. 52.7	-	-	360 360
	50 441	436	18. 18.7	-	-	360 360
	100 441	436	12. 12.7 131.9 133 93 102
	200 441	436	6.	6.7	207	203 45	40

E S , is defined as the excitation amplitude where the variation of the acoustic pressure again increases corresponding to an abrupt change in the resonance frequency resulting in a smaller value. As shown in Fig. 7,E

Acknowledgment

The first author acknowledges DGA-France for the financial support.


Martial Sauceau Nicolas Le Moigne
email: nicolas.le-moigne@mines-ales.fr

Mohamed Benyakhlef

Rabeb Jemai

Jean-Charles Benezet

José-Marie Lopez-Cuesta

Élisabeth Rodier

Jacques Fages

Jean-Charles Bénézet

Elisabeth Rodier

Processing and characterization of PHBV/clay nano-biocomposite foams by supercritical CO2 assisted extrusion

Keywords: polyhydroxyalkanoates, nanocomposite, foam, supercritical fluid

Introduction

Bio-based polymers like polyhydroxyalkanoates (PHAs) are marketed as eco-friendly alternatives to the currently widespread non-degradable oil-based thermoplastics, due to their natural and renewable origin, their biodegradability and biocompatibility. Poly 3hydroxybutyrate (PHB) properties are similar to various synthetic thermoplastics like polypropylene and hence it can be used alternatively in several applications, especially for agriculture, packaging but also biomedicine where biodegradability and biocompatibility are of great interest. However, some drawbacks have prevented its introduction to the market as an effective alternative to the oil-based thermoplastics. PHB is indeed brittle and presents a slow crystallization rate and a poor thermal stability which makes it difficult to process [START_REF] Bordes | Nano-biocomposites: biodegradable polyester/nanoclay systems[END_REF][START_REF] Cabedo | Studying the degradation of polyhydroxybutyratecovalerate during processing with clay-based nanofillers[END_REF]. In order to improve the PHB properties, several kinds of PHAs copolymers have been described in the literature such as the Poly(3-hydroxybutyrate-co-3-hydroxyvalerate) (PHBV) with various hydroxyvalerate (HV) contents and molecular weights, which present better mechanical properties, lower melting temperature and an extended processing window [START_REF] Bordes | Effect of clay organomodifiers on degradation of polyhydroxyalkanoates[END_REF]. Improved properties can be also obtained by the addition of nanoparticles of layered silicates such as clays. Indeed, clay minerals present high aspect ratio and specific surface, and can be dispersed in small amounts in polymer matrices to prepare nanocomposites with improved thermal stability, mechanical properties or barrier properties [START_REF] Ray | Polymer/layered silicate nanocomposites: a review from preparation to processing[END_REF]. One of the key parameters is the clay dispersion that can be controlled by the elaboration route either the solvent intercalation, the in-situ intercalation or the melt intercalation; the latter being preferred for a sustainable development since it limits the use of organic solvents [START_REF] Bordes | Nano-biocomposites: biodegradable polyester/nanoclay systems[END_REF]. The organomodifiers inserted in clays interlayer spaces to improve polymer / clay affinity and the polymer chains intercalation have a strong influence on the dispersion but it has been also shown to catalyse the PHBV degradation during processing [START_REF] Bordes | Effect of clay organomodifiers on degradation of polyhydroxyalkanoates[END_REF][START_REF] Hablot | Thermal and thermomechanical degradation of PHB-based multiphase systems[END_REF]. The use of supercritical fluids have recently appeared as an innovative way to improve clay dispersion leading new "clean and environment friendly" processes. Supercritical carbon dioxyde (sc-CO 2 ) has favorable interaction with polymers and it has the ability to dissolve in large quantities and to act as a plasticizer, which modify drastically polymer properties (viscosity, interfacial tension, …). In addition, the dissolved sc-CO 2 can acts as a foaming agent during processing. It is therefore possible to control pore generation and growth by controlling the operating conditions [START_REF] Nikitine | Controlling the structure of a porous polymer by coupling supercritical CO 2 and single screw extrusion process[END_REF][START_REF] Sauceau | New challenges in polymer foaming: a review of extrusion processes assisted by supercritical carbon dioxide[END_REF], and to generate low density porous structures of interest for the lightening of packaging or the storage of active ingredients, e.g. for drug release applications. All these features make sc-CO 2 also able to modify the nanoparticles dispersion inside polymer matrices, which in turn has an effect on the foam structure. Improved dispersion of clays and modified porous structures of synthetic polymers have been obtained with sc-CO 2 , mainly in batch processes via the in-situ intercalation method. Only few studies have reported on the preparation of nanocomposites systems by sc-CO2 assisted continuous processes [START_REF] Sauceau | New challenges in polymer foaming: a review of extrusion processes assisted by supercritical carbon dioxide[END_REF], more easily adaptable for an industrial scale-up. Zhao et al. [START_REF] Zhao | Processing and characterization of solid and microcellular poly(lactic acid)/polyhydroxybutyrate-valerate (PLA/PHBV) blends and PLA/PHBV/ Clay nanocomposites[END_REF][START_REF] Zhao | Morphology and Properties of Injection Molded Solid and Microcellular Polylactic Acid/Polyhydroxybutyrate-Valerate (PLA/ PHBV) Blends[END_REF] have recently investigated the possibility to use a supercritical N 2 assisted injection molding process to develop microcellular PLA/PHBV clay nano-biocomposites. The results showed a decrease of the average cell size and an increased cell density with the addition of clays in PLA/PHBV blends. Rheological behaviour of the PLA/PHBV/clays nanocomposites suggested good dispersion of the clays within the matrix. In this study, we developed a continuous sc-CO 2 assisted extrusion process to prepare PHBV/clays nano-biocomposite foams by two methods: a one-step method based on the direct foaming of physical PHBV / clays mixtures, and a two-step method based on the foaming of PHBV / clays mixtures prepared beforehand by twin-screw extrusion. The structures obtained are characterized in terms of clay dispersion, matrix crystallization, porosity and pore size distribution and density, and discussed as regard to the processing conditions such as temperature, shearing/pressure, CO 2 mass fraction.

Materials and methods

Materials

PHBV with a HV content of 13 wt%, nucleated with boron nitride and plasticized with 10 % of a copolyester was purchased from Biomer (Germany). The weight-average molecular weight is 600 kDa. The clay used is an organo-modified montmorillonite (MMT), Cloisite C30B (C30B), produced by Southern Clay Products, Inc. (USA). To limit hydrolysis of PHBV upon processing, C30B and PHBV were dried at 80°C before use.

Preparation of PHBV / C30B extruded mixtures and physical mixtures

PHBV based nanocomposites containing 2.5% w/w C30B and PHBV based masterbatches containing 10%, 20% w/w C30B were prepared by melt intercalation using a co-rotating twin-screw extruder BC21 (Clextral, France) having a L/D (length to diameter ratio) of 48. A parabolic temperature profile not exceeding 165°C was used to limit thermal degradation of PHBV. The mixing and the dispersion of the C30B clays within the PHBV matrix were ensured by two kneading sections. Extrudates were water-cooled at the exit of the die, and dried overnight at 50°C under vacuum. About 2.5 kg of granules were collected for each batch. All the batches were moulded by injection with a Krauss Maffei KM-50-180-CX into test specimens. The barrel to die temperature profile was 40 to 165°C. PHBV based masterbatches containing 10%, 20% w/w C30B were diluted to 2.5 % w/w C30B in the injection molding machine to analyze the effect of the dilution on the nanocomposite structures and properties. As will be shown in the following, this dilution procedure was also used to produce PHBV / 2.5% nanocomposite foams by sc-CO 2 assisted extrusion. In addition to extruded mixtures, physical mixtures of PHBV pellets coated with 2.5% w/w of C30B were prepared with a simple manual batch mixing by placing a mixture of both components in a stainless steel rotative drum for 10 minutes (Faraday cage linked to Keithley 6514 electrometer). These physical mixtures of PHBV pellets / 2.5% C30B were then also foamed by sc-CO 2 assisted extrusion.

Foaming by sc-CO 2 assisted extrusion

Figure 1 shows the experimental set up, which has previously been described elsewhere [START_REF] Nikitine | Controlling the structure of a porous polymer by coupling supercritical CO 2 and single screw extrusion process[END_REF][START_REF] Kamar | Biopolymer foam production using a (sc-CO 2 ) assisted extrusion process[END_REF]. The single-screw extruder (Rheoscam, SCAMEX) has a screw diameter of 30 mm and a length to diameter ratio (L/D) of 37. It is equipped with four static mixer elements (SMB-H 17/4, Sulzer, Switzerland). Sensors allow measuring the temperature and the pressure of the polymer during the extrusion process. CO 2 (N45, Air liquide) is pumped from a cylinder by a syringe pump (260D, ISCO, USA) and then introduced at constant volumetric flow rate. The pressure, the temperature and the volumetric sc-CO 2 flow rate are measured within the syringe pump. Sc-CO 2 density, obtained with the equation of state established by Span and Wagner [START_REF] Span | A new equation of state for carbon dioxide covering the fluid region from the triplepoint temperature to 1100 K at pressures up to 800 MPa[END_REF], is used to calculate mass flow rate and thus the sc-CO 2 mass fraction w CO2 . Once steady state conditions are reached with the chosen operating conditions, extrudates are collected and water-cooled at ambient temperature. Several samples were collected during each experiment in order to check the homogeneity of the extrudates. Experimental conditions chosen after preliminary trials (not described here) are summarized in Table 1. The lowest possible screw speed was selected to increase the residence time of the mixtures, and thus the mixing time. For all experiments, T a , T b and T c were fixed at 160°C to ensure the melting of the PHBV matrix while limiting thermal degradation. Moreover, the polymer temperature was reduced at the exit of the die in order to favour foaming. T d and T e were fixed at 140°C and T f not higher than 140°C.

Series

Structural characterizations

The structures of nanocomposites and masterbatches were observed with a Quanta 200 FEG (FEI Company) electron microscope in transmission mode (STEM). Ultra-thin specimens of 70 nm thicknesses were cut from the middle section of moulded specimens and deposited on Cu grids. The wide angle X-Ray diffraction (WAXD) were performed using an AXS D8 Advance diffractometer (Bruker, Germany) equipped with a Cu cathode (λ = 1.5405 Å). The interlayer distance d 001 of the C30B clays were determined from the (001) diffraction peak using Bragg's law. Porosity ε is defined as the ratio of void volume to the total volume of the sample and can be calculated by the Equation ( 1):

ε = 1 -ρ app / ρ p (1)
Where ρ app is the apparent density calculated from the weight of the samples and their volumes evaluated by measuring their diameter and length with a vernier (Facom, France). ρ p is the solid polymer density, determined by helium pycnometry (Micromeretics, AccuPYC 1330), which is about 1.216. The fracture surfaces of foamed extrudates were sputter coated with gold and observed using an Environmental Scanning Electronic Microscope XL30 ESEM FEG (Philips, Netherlands). On the basis of the images obtained, a mean diameter D cell is determined and a cell density N cell per volume unit of unfoamed sample were calculated according to Equation (2):

N cell = (N 3/2 / A) × (ρ p / ρ app ) (2)
N cell is the number of cells on an SEM image, A the area of the image, ρ app the apparent density of the foam and ρ p the solid polymer density.

Results and discussion

Structure and rheology of extruded PHBV / C30B nanocomposites and masterbatches

The structures of the PHBV / C30B nanocomposites and masterbatches were investigated by two complementary methods, WAXD and STEM. The WAXD measurements allowed to determine the interlayer distance of the C30B clays within the PHBV matrix. The STEM gave a direct visualization of the clay dispersion within the matrix. As shown on Figure 2a, WAXD patterns showed a diffraction peak at 2θ = 4.6° for C30B clays, which corresponds to an interlayer distance of 18 Å. Two diffraction peaks were detected on the patterns of the PHBV / 2.5, 10 and 20 wt% C30B mixtures. For the PHBV / 10% and 20% C30B masterbatches, the peak at d 001 distance of 17 -18 Å corresponding to the initial interlayer distance of C30B, suggests that a part of the clays is still aggregated. The second peak observed at d 001 distance of 38 Å, 36.1 Å, 34 Å for PHBV / C30B containing 2.5, 10 and 20 wt% C30B, respectively, indicates an important intercalation of the polymer chains within the interlayer space of the clays. Similar results were found by Choi et al. [START_REF] Choi | Preparation and characterization of poly(hydroxybutyrate-co-hydroxyvalerate)-organoclay nanocomposites[END_REF] and Bordes et al. [START_REF] Bordes | Structure and properties of PHA/clay nano-biocomposites prepared by melt intercalation[END_REF] for PHBV / 2 -3% C30B prepared by melt intercalation. The intensity of the peak at 38 Å for the PHBV / 2.5% C30B is particularly weak suggesting a possible exfoliation of the clays. Moreover, masterbatches have been diluted in a single-screw injection press to analyse the effect on the clay dispersion. The WAXD patterns are very similar for the PHBV / 2.5% C30B nanocomposite and the PHBV / 2.5% C30B nanocomposites diluted from the masterbatches. These results were confirmed by the STEM pictures, on which mostly intercalated and possibly exfoliated layered structures are observed for both PHBV / 2.5% C30B (Figure 2b) and PHBV / 2.5% C30B diluted from the 20% C30B masterbatch (Figure 2c). The rheological behaviour of extruded PHBV and PHBV / C30B nanocomposites was compared to unprocessed PHBV. The zero-shear rate viscosity of the extruded PHBV was strongly decreased after the extrusion process. This can be directly related to a decrease of the average molecular weight of the PHBV chains due to thermal degradation and shearing upon extrusion. When adding 2.5% C30B, a strong shear thinning behaviour was observed at high pulsations, highlighting the catalytic degradation effect of the clays on the PHBV matrix. The rheological behaviour at high pulsations is indeed known to be mainly determined by the macromolecular structure of the polymer matrix with little influence of the clays. When diluting the 20% C30B masterbatch to 2.5% C30B, it was interesting to observe that at high pulsations, the viscosity of the unprocessed PHBV was recovered due to the input of unprocessed and hence non-degraded PHBV in the mixture. This preliminary study on the nanostructure and the rheological behaviour of the PHBV / 2.5% C30B nanocomposites and those produced from the masterbatches allowed to demonstrate that the dilution of the masterbatches to lower clay contents in a single-screw apparatus is a good approach to prepare PHBV / C30B nanocomposites with good dispersion and limited degradation. The good dispersion capacity of the clays is attributed to the physico-chemical interactions between PHBV and C30B that originates from strong hydrogen bonding between the ester carbonyl groups of PHBV and the hydroxyl groups in the interlayer space of C30B [START_REF] Choi | Preparation and characterization of poly(hydroxybutyrate-co-hydroxyvalerate)-organoclay nanocomposites[END_REF]. In the following, this dilution procedure has thus been used to prepare PHBV / 2.5% C30B nanocomposites foams, i.e. PHBV / 20% C30B was diluted to 2.5% C30B with unprocessed PHBV during the sc-CO2 assisted single-screw extrusion. The obtained foams were compared to PHBV / 2.5% C30B foams based on physical mixtures.

Processing and characterization of the PHBV / clays nano-biocomposite foams 3.2.1. Effect of the sc-CO 2 mass fraction on the PHBV and PHBV / 2.5% C30B foams

The effect of the sc-CO 2 mass fraction on the porosity ε is represented on Figure 3a. In all cases, porosity decreases with increasing sc-CO 2 mass fraction, which is rather astonishing since more sc-CO 2 is theoretically available for nucleation. This could be explained by a faster cooling of the extrudates, as sc-CO 2 pressure drop is endothermic. The fast cooling thus increases the stiffness of the polymer which limits the growth of the cells and hence the expansion and the porosity. A part of the sc-CO 2 could also be in excess [START_REF] Han | Continuous microcellular polystyrene foam extrusion with supercritical CO 2[END_REF] due to the formation of PHBV crystals upon cooling that limits the diffusion. This excess of sc-CO 2 diffuses to the walls of the extruder and does not participate to cell nucleation. This also decreases the processing pressures and temperatures, which in turn limits the expansion. SEM pictures of the extrudates of neat PHBV and physical and extruded PHBV / 2.5% C30B mixtures are represented on Figure 4. When sc-CO 2 mass fraction increases, pores become smaller, more numerous and regular. These observations are illustrated by the evolution of the cell density and the mean cell diameter on Figure 3b andc. It confirms that the presence of more sc-CO 2 increases nucleation (higher cell density) but accelerates the cooling and thus limits the growth and the coalescence of the pores (lower cell diameter). The sc-CO 2 mass fraction must thus be optimized to promote nucleation while keeping enough growth and expansion. Generally, the PHBV and PHBV / 2.5% C30B foam structure dependency to sc-CO 2 mass fraction can be summarized as follows:

(i) Low sc-CO 2 mass fractions induce higher processing pressures and temperatures that promote higher pressure drop, extensive growth and higher porosity, but also less nucleation and poor homogeneity due to the coalescence of the pores. (ii) High sc-CO 2 mass fractions promote nucleation and homogeneity, but also induce lower processing pressures and temperatures, that limit the growth and porosity due to the increased stiffness of the frozen polymer.

Clays / Foaming interrelationships and effect on the nanocomposite foams structure

As shown on Figure 4, the clay particles are inhomogeneously dispersed in the extrudates based on the physical PHBV / 2.5% C30B mixtures (series 3). Several aggregates of 10 µm to 350 µm are observed whatever the sc-CO 2 mass fraction. The effect of the sc-CO 2 on the properties of the polymer matrix and the presence of the static mixer were thus not sufficient to induce the intercalation of the polymer chains within the interlayer space of the C30B clays and their dispersion during single-screw extrusion. The static mixer indeed enhances the distributive mixing of the sc-CO 2 and the clays but only have little dispersive efficiency. Concerning the extruded PHBV / 2.5% C30B mixtures (series 4), a very good dispersion of the C30B is observed with no visible aggregates (Figure 4). This supports that the prior preparation of a PHBV / 20% C30B masterbatch and its further dilution to 2.5% C30B during the sc-CO 2 assisted extrusion process is a necessary step to obtain a good dispersion. WAXD patterns (not shown) revealed that the position of the intercalation peak at 36.2° for series 4 remains unchanged by the foaming process whatever the sc-CO 2 mass fraction, meaning that no significant enhanced intercalation occurs with the plasticization of the matrix induced by the sc-CO 2 . A possible improvement of the dispersion capacity of the clays could be obtained by improving their 'CO 2 -philicity' with a surfactant bearing together hydroxyl groups to conserve a good affinity with PHBV and a 'CO 2 -philic' carbonyl group. Good dispersion of the clays has been shown to favour homogeneous nucleation, limit the coalescence, and hence give porous structures with higher cell density [START_REF] Ngo | Processing of nanocomposite foams in supercritical carbon dioxide. Part I: Effect of surfactant[END_REF][START_REF] Zeng | Polymer-Clay Nanocomposite Foams Prepared Using Carbon Dioxide[END_REF]. As shown on Figures 3 and4 at low sc-CO 2 mass fraction (w CO2 < 1.5%), extruded PHBV / 2.5% C30B mixtures based foams indeed appear less heterogeneous with rather smaller cell diameter as compared to neat PHBV foams, while showing equivalent porosity. At high sc-CO 2 mass fraction (w CO2 .> 3.5%), the presence of the clays decreases significantly the porosity. The diffusion of the sc-CO 2 within the PHBV matrix may be slightly hampered by the C30B clays, which are known to be barrier to gas and fluid in polymeric materials [START_REF] Ray | Biodegradable polymers and their layered silicate nanocomposites: In greening the 21 st century materials world[END_REF]. Consequently, the excess of sc-CO 2 diffuses to the walls of the extruder and accelerates the cooling of the extrudate, limiting the growth of the pores and the expansion of the foams.

Conclusions

A continuous sc-CO 2 assisted extrusion process has been developed to prepare PHBV/clays nano-biocomposite foams. The prior preparation of a PHBV / 20% C30B masterbatch and its further dilution during the sc-CO 2 assisted single-screw extrusion process are necessary steps to obtain good clay dispersion and limited PHBV degradation. By controlling the sc-CO 2 mass fraction in a narrow window, good clay dispersion appears to favour homogeneous nucleation while limiting the coalescence, and hence allows to obtain PHBV/clays nanobiocomposite foams with better homogeneity and porosity higher than 50 %.

Figure 1 .

 1 Figure 1. Experimental device used for the foaming by sc-CO 2 assisted single-screw extrusion

Figure 2 .

 2 Figure 2. WAXD patterns (a) and STEM pictures of PHBV / 2.5% C30B (b), and PHBV / 2.5% C30B diluted from the 20% C30B masterbatch (c).

Figure 3 .

 3 Figure 3. Evolution of (a) the porosity ε, (b) the cell density N cell and (c) the mean diameter D cell as a function of the CO 2 mass fraction for foams series 2, 3 and 4.

Figure 4 .

 4 Figure 4. SEM pictures for foams series 2, 3 and 4 at different sc-CO 2 mass fraction.

Table 1 .

 1 Experimental conditions used for the foaming by sc-CO2 assisted single-screw extrusion

		Material	Screw speed (rpm)	T f (°C)	w CO2 (mass %)	Static mixer	Die length (mm)	Die diameter (mm)
	2	Neat PHBV	30	140	0 to 4	no	20	1
	3	PHBV / 2.5% C30B (physical mixture)	40	140	0 to 3	yes	5	0.5
	4	PHBV / 2.5% C30B (extruded mixture)	55	140	0 to 4	yes	20	1


On hyper-exponential output-feedback stabilization of a double integrator by using artificial delay

The problem of output-feedback stabilization of a double integrator is revisited with the objective of achieving the rates of convergence faster than exponential. It is assumed that only position is available for measurements, and the designed feedback is based on the output and its delayed values without an estimation of velocity. It is shown that by selecting the closedloop system to be homogeneous with negative or positive degree it is possible to accelerate the rate of convergence in the system at the price of a small steady-state error. Efficiency of the proposed control is demonstrated in simulations.

I. INTRODUCTION

The design of regulators for dynamical systems is a fundamental and complex problem studied in the control theory. An important feature of different existing methods for control synthesis is the achievable quality of transients and robustness against exogenous perturbations and noises. Very frequently the design methods are oriented on various canonical models, and the linear ones are the most popular. Then the double integrator is a conventional benchmark system, since the tools designed for it can be easily extended to other more generic models. If non-asymptotic rates of convergence (i.e. finite-time or fixed time [START_REF] Bernuau | On homogeneity and its application in sliding mode[END_REF]) are needed in the closed-loop system, then usually homogeneous systems come to the attention as canonical dynamics, which include linear models as a subclass.

The theory of homogeneous dynamical systems is welldeveloped for continuous time-invariant differential equations [START_REF] Bacciotti | Liapunov Functions and Stability in Control Theory[END_REF], [START_REF] Bhat | Geometric homogeneity with applications to finite-time stability[END_REF], [START_REF] Kawski | Progress in systems and control theory: New trends in systems theory[END_REF], [START_REF] Zubov | On systems of ordinary differential equations with generalized homogenous right-hand sides[END_REF] or time-delay systems [START_REF] Efimov | Development of homogeneity concept for time-delay systems[END_REF], [START_REF] Efimov | Weighted homogeneity for time-delay systems: Finite-time and independent of delay stability[END_REF] (applications of the conventional homogeneity theory to analysis of timedelay systems considering delay as a kind of perturbation have been considered in [START_REF] Aleksandrov | On the asymptotic stability of solutions of nonlinear systems with delay[END_REF], [START_REF] Asl | Analytical solution of a system of homogeneous delay differential equations via the Lambert function[END_REF], [START_REF] Bokharaie | D-stability and delayindependent stability of homogeneous cooperative systems[END_REF], [START_REF] Diblik | Asymptotic equilibrium for homogeneous delay linear differential equations with l-perturbation term[END_REF]). The main feature of a homogeneous system (described by ordinary differential equation) is that its local behavior of trajectories is the same as global (local attractiveness implies global asymptotic stability, for example [START_REF] Bernuau | On homogeneity and its application in sliding mode[END_REF]), while for time-delay homogeneous D. Efimov systems the independent on delay (IOD) stability follows [START_REF] Efimov | Weighted homogeneity for time-delay systems: Finite-time and independent of delay stability[END_REF], with certain robustness to exogenous inputs in both cases. The rate of convergence for homogeneous ordinary differential equations is related with degree of homogeneity [START_REF] Bernuau | On homogeneity and its application in sliding mode[END_REF], but for time-delay systems the links are not so straightforward [START_REF] Efimov | Comments on finite-time stability of time-delay systems[END_REF]. In addition, the homogeneous stable/unstable systems admit homogeneous Lyapunov functions [START_REF] Zubov | On systems of ordinary differential equations with generalized homogenous right-hand sides[END_REF], [START_REF] Rosier | Homogeneous Lyapunov function for homogeneous continuous vector field[END_REF], [START_REF] Efimov | Oscillations conditions in homogenous systems[END_REF].

Analysis of delay influence on the system stability is vital in many cases [START_REF] Gu | Stability of Time-Delay Systems[END_REF], [START_REF] Fridman | Introduction to Time-Delay Systems: Analysis and Control[END_REF]. Despite of variety of applications, most of them deal with the linear time-delay models, which is originated by complexity of stability analysis for time-delay systems [START_REF] Fridman | Introduction to Time-Delay Systems: Analysis and Control[END_REF]. However, in some cases introduction of a delay may lead to an improvement of the system performance [START_REF] Fridman | Delay-induced stability of vector secondorder systems via simple Lyapunov functionals[END_REF], [START_REF] Fridman | Stabilization by using artificial delays: An LMI approach[END_REF].

The goal of this work is to develop the results obtained in [START_REF] Fridman | Delay-induced stability of vector secondorder systems via simple Lyapunov functionals[END_REF], [START_REF] Fridman | Stabilization by using artificial delays: An LMI approach[END_REF] for linear systems to a nonlinear homogeneous case restricting for brevity the attention to the case of the double integrator model. A design method is proposed, which uses position and its delayed values for practical output stabilization with hyper-exponential convergence rates.

The outline of this work is as follows. The preliminary definitions and homogeneity concept for time-delay systems are given in Section II. The problem statement and the control design and stability analysis are presented in sections III and IV, respectively. An example is considered in Section V.

II. PRELIMINARIES

Consider an autonomous functional differential equation of retarded type with inputs [START_REF] Kolmanovsky | Stability of functional differential equations[END_REF]:

ẋ(t) = f (x t , d(t)), t ≥ 0 (1)
where

x(t) ∈ R n and x t ∈ C [-τ,
; f : C [-τ,0] × R m → R n is
a continuous function ensuring forward uniqueness and existence of the system solutions, f (0, 0) = 0. We assume that for the initial functional condition

x 0 ∈ C [-τ,0] and d ∈ L m ∞ the system
(1) admits a unique solution x(t, x 0 , d), which is defined on some time interval [-τ, T ) for T > 0.

The upper right-hand Dini derivative of a locally Lipschitz continuous functional V : C [-τ,0] → R + along the system (1) solutions is defined as follows for any φ ∈ C [-τ,0] and d ∈ R m :

D + V (φ, d) = lim h→0 + sup 1 h [V (φ h ) -V (φ)],
where φ h ∈ C [-τ,0] for 0 < h < τ is given by

φ h = φ(θ + h), θ ∈ [-τ, -h) φ(0) + f (φ, d)(θ + h), θ ∈ [-h, 0]. A continuous function σ : R + → R + belongs to class K if it is strictly increasing and σ(0) = 0; it belongs to class K ∞ if it is also radially unbounded. A continuous function β : R + × R + → R + belongs to class KL if β(•, r) ∈ K and β(r, •) is a strictly decreasing to zero for any fixed r ∈ R + .
The symbol 1, m is used to denote a sequence of integers 1, ..., m. For a symmetric matrix P ∈ R n×n , the minimum and maximum eigenvalues are denoted as λ min (P ) and λ max (P ), respectively.

A. ISS of time delay systems

The input-to-state stability (ISS) property is an extension of conventional stability paradigm to the systems with external inputs [START_REF] Pepe | A Lyapunov-Krasovskii methodology for ISS and iISS of time-delay systems[END_REF], [START_REF] Teel | Connections between Razumikhin-type theorems and the ISS nonlinear small gain theorem[END_REF]. Definition 1. [START_REF] Pepe | A Lyapunov-Krasovskii methodology for ISS and iISS of time-delay systems[END_REF] The system (1) is called ISS, if for all x 0 ∈ C [-τ,0] and d ∈ L m ∞ the solutions are defined for all t ≥ 0 and there exist β ∈ KL and γ ∈ K such that

|x(t, x 0 , d)| ≤ β( x 0 , t) + γ(||d|| ∞ ) ∀t ≥ 0. Definition 2. [20] A locally Lipschitz continuous functional V : C [-τ,0] → R + is called ISS Lyapunov-Krasovskii functional for the system (1) if there exist α 1 , α 2 ∈ K ∞ and α, χ ∈ K such that for all φ ∈ C [-τ,0] and d ∈ R m : α 1 (|φ(0)|) ≤ V (φ) ≤ α 2 ( φ ), V (φ) ≥ χ(|d|) =⇒ D + V (φ, d) ≤ -α(V (φ)).
Theorem 1. [START_REF] Pepe | A Lyapunov-Krasovskii methodology for ISS and iISS of time-delay systems[END_REF] If there exists an ISS Lyapunov-Krasovskii functional for the system (1), then it is ISS with γ = α -1 1 • χ.

B. Homogeneity

For any r i > 0, i = 1, n and λ > 0, define the dilation matrix Λ r (λ) = diag{λ ri } n i=1 and the vector of weights r = [r 1 , ..., r n ] T . Definition 3. [START_REF] Efimov | Homogeneity for time-delay systems[END_REF] The function g :

C [-τ,0] → R is called r-homogeneous (r i > 0, i = 1, n), if for any φ ∈ C [-τ,0] the relation g(Λ r (λ)φ) = λ ν g(φ)
holds for some ν ∈ R and all λ > 0.

The vector field f :

C [-τ,0] → R n is called r- homogeneous (r i > 0, i = 1, n), if for any φ ∈ C [-τ,0] the relation f (Λ r (λ)φ) = λ ν Λ r (λ)f (φ)
holds for some ν ≥ -min 1≤i≤n r i and all λ > 0.

In both cases, the constant ν is called the degree of homogeneity.

The introduced notion of homogeneity in C [-τ,0] is reduced to the standard one in R n [START_REF] Zubov | On systems of ordinary differential equations with generalized homogenous right-hand sides[END_REF] under a vector argument substitution.

For any x ∈ R n the homogeneous norm can be defined as follows

|x| r = n i=1 |x i | /ri 1/ , ≥ max 1≤i≤n r i .
For all x ∈ R n , its Euclidean norm |x| is related with the homogeneous one:

σ r (|x| r ) ≤ |x| ≤ σ r (|x| r ) for some σ r , σ r ∈ K ∞ .
The homogeneous norm in the Banach space has the same homogeneity property that is ||Λ r (λ)φ|| r = λ||φ|| r for all

φ ∈ C [a,b] . In C [-τ,0]
, for a radius ρ > 0, denote the corresponding sphere S τ ρ = {φ ∈ C [-τ,0] : ||φ|| r = ρ} and the closed ball B τ ρ = {φ ∈ C [-τ,0] : ||φ|| r ≤ ρ}. An advantage of homogeneous systems described by nonlinear ordinary differential equations is that any of its solution can be obtained from another solution under the dilation re-scaling and a suitable time parameterization. A similar property holds for functional homogeneous systems. Proposition 1. [START_REF] Efimov | Weighted homogeneity for time-delay systems: Finite-time and independent of delay stability[END_REF] Let x(t, x 0 ) be a solution of the rhomogeneous system

dx(t)/dt = f (x t ), t ≥ 0, x t ∈ C [-τ,0] (3)
with the degree ν for an initial condition x 0 ∈ C [-τ,0] , τ ∈ (0, +∞). For any λ > 0 the functional differential equation

dy(t)/dt = f (y t ), t ≥ 0, y t ∈ C [-λ -ν τ,0] (4)
has a solution y(t, y 0 ) = Λ r (λ)x(λ ν t, x 0 ) with the initial condition

y 0 ∈ C [-λ -ν τ,0] , y 0 (s) = Λ r (λ)x 0 (λ ν s) for s ∈ [-λ -ν τ, 0].
In [START_REF] Efimov | Development of homogeneity concept for time-delay systems[END_REF], using that result it has been shown that for (3) with ν = 0 the local asymptotic stability implies global one (for the ordinary differential equations even more stronger conclusion can be obtained: local attractiveness implies global asymptotic stability [START_REF] Bernuau | On homogeneity and its application in sliding mode[END_REF]). For time-delay systems with ν = 0 that result has the following correspondences: Lemma 1. [START_REF] Efimov | Weighted homogeneity for time-delay systems: Finite-time and independent of delay stability[END_REF] Let the system (3) be r-homogeneous with degree ν = 0 and globally asymptotically stable for some delay 0 < τ 0 < +∞, then it is globally asymptotically stable for any delay 0 < τ < +∞ (i.e. IOD).

Corollary 1.

[7] Let the system (3) be r-homogeneous with degree ν and asymptotically stable with the region of attraction B τ ρ for some 0 < ρ < +∞ for any value of delay 0 ≤ τ < +∞, then it is globally asymptotically stable IOD.

Corollary 2.

[7] Let the system (3) be r-homogeneous with degree ν < 0 and asymptotically stable with the region of attraction B τ ρ for some 0 < ρ < +∞ for any value of delay 0 ≤ τ ≤ τ 0 with 0 < τ 0 < +∞, then it is globally asymptotically stable IOD.

Corollary 3.

[7] Let the system (3) be r-homogeneous with degree ν > 0 and the set B τ ρ for some 0 < ρ < +∞ be uniformly globally asymptotically stable for any value of delay 0 ≤ τ ≤ τ 0 , 0 < τ 0 < +∞ 1 , then (3) is globally asymptotically stable (at the origin) IOD.

III. PROBLEM STATEMENT

Consider the double integrator system:

ẋ1 (t) = x 2 (t), ẋ2 (t) = u(t), (5)
y(t) = x 1 (t),
where x 1 (t) ∈ R and x 2 (t) ∈ R are the position and velocity, respectively, u(t) ∈ R is the control input and y(t) ∈ R is the output available for measurements. The goal is to design a static output-feedback control practically stabilizing the system with a hyper-exponential convergence rate, i.e. with a convergence faster than any exponential.

IV. MAIN RESULTS

The solution considered in this paper is the delayed nonlinear controller

u(t) = -(k 1 + k 2 ) y(t) α + k 2 y(t -h) α , (6)
where y α = |y| α sign(y), k 1 > 0 and k 2 > 0 are tuning gains, α > 0, α = 1 is a tuning power and h > 0 is the delay (if α = 1 then the control ( 6) is linear and it has been studied in [START_REF] Fridman | Delay-induced stability of vector secondorder systems via simple Lyapunov functionals[END_REF], [START_REF] Fridman | Stabilization by using artificial delays: An LMI approach[END_REF]). The restrictions on selection of 1 In this case for any 0 ≤ τ ≤ τ 0 , any ε > 0 and κ ≥ 0 there is 0 ≤ T ε κ,τ < +∞ such that |x(t, x 0 )|r ≤ ρ + ε for all t ≥ T ε κ,τ for any x 0 ∈ B τ κ , and |x(t, x 0 )|r ≤ στ (||x 0 ||r) for all t ≥ 0 for some function στ ∈ K∞ for all

x 0 ∈ C [-τ,0] .
these parameters and the conditions to check are given in the following theorem. Theorem 2. For any k 1 > 0, k 2 > 0, h 0 > 0, if the system of linear matrix inequalities

Q ≤ 0, P > 0, q > 0, (7)
Q =    Q 11 k 2 Zb Zb k 2 b Z k 2 2 h 2 -4 e -h h 2 q qh 2 k 2 b Z qh 2 k 2 qh 2 -γ    , Q 11 = A P + P A + qh 2 A bb A + P, Z = P + qh 2 A , A = 0 1 -k 1 -k 2 h , b = 0 1
is feasible for some > 0, γ > 0 and any 0 < h ≤ h 0 , then for any 0 < η < +∞ there exists ∈ (0, 1) sufficiently small such that the system (5), ( 6) is a) globally asymptotically stable with respect to the set B 2h η for any α ∈ (1 -, 1); b) locally asymptotically stable at the origin from B 2h η for any α ∈ (1, 1 + ).

All proofs are omitted due to space limitations. Note that for any α ≥ 0 the closed-loop system (5), ( 6) is rhomogeneous for r 1 = 1 and r 2 = α+1 2 with the degree ν = α-1 2 , then the result of Proposition 1 can be used for substantiation.

The requirement that the matrix inequalities [START_REF] Efimov | Weighted homogeneity for time-delay systems: Finite-time and independent of delay stability[END_REF] have to be verified for any 0 < h ≤ h 0 may be restrictive for given gains k 1 and k 2 , then another local result can be obtained by relaxing this constraint. Corollary 4. For any k 1 > 0, k 2 > 0 and 0 < h 1 < h 0 , let the system of linear matrix inequalities (7) be verified for some > 0 and all h 1 ≤ h ≤ h 0 . Then for any 0 < ρ 1 < +∞ there exist ∈ (0, 1) sufficiently small and ρ 2 > ρ 1 such that the system (5), ( 6) is asymptotically stable with respect to the set B 2h ρ1 with the region of attraction B 2h ρ2 for any α ∈ (1 -, 1 + ).

Remark 1. The result of Theorem 2 complements corollaries 2 and 3.

Note that in all cases, for ν = 0, the global stability at the origin cannot be obtained in ( 5), (6) (due to homogeneity of the system, following the result of Lemma 1 the globality implies IOD result), while in the linear case with ν = 0 such a result is possible to derive for any 0 < h ≤ h 0 . Then it is necessary to justify a need in the control with ν = 0 comparing to the linear feedback with the same gains. An answer to this question is presented in the following result, and to this end denote for the system (5), (6):

T (α, ρ 1 , ρ 2 , h) = arg sup t≥T -h sup x0∈S 2h ρ 2 |x(t, x 0 )| r ≤ ρ 1
as the time of convergence of all trajectories initiated on the sphere S 2h ρ2 to the set B 2h ρ1 provided that the delay h and the power α applied in the feedback. Proposition 2. For given k 1 > 0, k 2 > 0, h 0 > 0, let the system of linear matrix inequalities [START_REF] Efimov | Weighted homogeneity for time-delay systems: Finite-time and independent of delay stability[END_REF] be verified for some > 0 and any 0 < h ≤ h 0 . Then there exist ∈ (0, 1) sufficiently small and 0 < ρ 1 < ρ 2 < +∞ such that in the system (5), ( 6) In other words, the result above claims that for any fixed feedback gains k 1 and k 2 , if the conditions of Theorem 2 are satisfied, then the nonlinear closed-loop system ( 5), ( 6) with ν = 0 (α = 1) is always converging faster than its linear analog with ν = 0 (α = 1) between properly selected levels ρ 1 and ρ 2 (which values depend on smaller or high than 1 is α) for a delay h .

T (α, ρ 1 , ρ 2 , h ) < T (1, ρ 1 , ρ 2 , h ) (8
The result of Proposition 2 provides a motivation for using nonlinear control in this setting: playing with degree of homogeneity of the closed-loop system it is possible to accelerate the obtained linear feedback by fixing the gains and delay values, but introducing an additional power tuning parameter. Note that another, conventional solution, which consists in gains k 1 and k 2 increasing for acceleration, may be infeasible for the given delay value h 0 . Let us consider some results of application of the proposed control and an illustration of the obtained acceleration. the matrix inequalities (7) are satisfied for h 1 < h ≤ h 0 with h 1 = 5 × 10 -4 , and the results of verification are presented in Fig. 1. Thus, all conditions of Corollary 4 are verified. The errors of regulation obtained in simulation of the system (5), [START_REF] Efimov | Development of homogeneity concept for time-delay systems[END_REF] with delay h 0 for different initial conditions with α = 0.8 and α = 1.2, in comparison with the linear controller with α = 1, are shown in figures 2 and 3, respectively (the solid lines represent the trajectories of the system with α = 1 and the dashed ones correspond to α = 1, since the plots are given in a logarithmic scale, then the latter trajectories are close to straight lines). As we can conclude, in the nonlinear case the convergence is much faster than in the linear one close to the origin for α ∈ (0, 1) and far outside for α > 1, which confirms the statement of Proposition 2. Note that the value of η (the radius of the set to which the trajectories converge for α < 1 or from which they converge to the origin for α > 1) is not restrictive.

VI. CONCLUSIONS

The paper addresses the problem of output stabilization of the double integrator using a nonlinear delayed feedback by obtaining hyper-exponential (faster than any exponential) rates of convergence. The control does not need an estimation of velocity, and the applicability of the approach can be checked by resolving linear matrix inequalities. The efficiency of the proposed approach is demonstrated in simulations and a comparison with a linear controller is carried out.

  The homogeneous norm is a rhomogeneous function of degree one: |Λ r (λ)x| r = λ|x| r for all x ∈ R n . Similarly, for any φ ∈ C [a,b] , -∞ ≤ a < b ≤ +∞ the homogeneous norm can be defined as follows ||φ|| r = n i=1 ||φ i || /ri 1/ , ≥ max 1≤i≤n r i , and there exist two functions ρ r , ρ r ∈ K ∞ such that for all φ ∈ C [a,b] ρ r (||φ|| r ) ≤ ||φ|| ≤ ρ r (||φ|| r ).

)

  for some 0 < h ≤ h 0 and any α ∈ (1-, 1) or α ∈ (1, 1+ ), provided that sup 0<h≤h0 T (α, 0.5, 1, h) ≤ T α for some T α ∈ R + and all α ∈ (1 -, 1 + ).

  3.73 × 10 -2 , q = 1.5 × 10 -11

Figure 1 .Figure 2 .

 12 Figure 1. The results of verification of (7) for different h

Figure 3 .

 3 Figure 3. Trajectories of stabilized double integrator with α = 1.2

  , W. Perruquetti and J.-P. Richard are at Inria, Non-A team, Parc Scientifique de la Haute Borne, 40 av. Halley, 59650 Villeneuve d'Ascq, France and CRIStAL (UMR-CNRS 9189), Ecole Centrale de Lille, BP 48, Cité Scientifique, 59651 Villeneuve-d'Ascq, France.

E. Fridman is with School of Electrical Engineering, Tel-Aviv University, Tel-Aviv 69978, Israel. D. Efimov is with Department of Control Systems and Informatics, Saint Petersburg State University of Information Technologies Mechanics and Optics (ITMO), 49 Kronverkskiy av., 197101 Saint Petersburg, Russia. This work was partially supported by the Government of Russian Federation (Grant 074-U01), the Ministry of Education and Science of Russian Federation (Project 14.Z50.31.0031) and by Israel Science Foundation (grant No 1128/14).