Harkaitz Urreta
email: hurreta@ideko.es

Gorka Aguirre

Pavel Kuzhir

Luis Norberto Lopez De Lacalle

Seals based on magnetic fluids for high precision spindles of machine tools

Keywords: Magnetic fluids, ferrofluids, magnetorheological fluids, magnetic fluid sealing, machine tool spindle

The research work reported in this paper is focused on the use of magnetic fluids as active elements in seals for improving sealing capacity and minimizing friction torque, with application to the spindles of high precision machine tools. The prototype design was optimized following numerical computation of the magnetic field in the rings of the seal. Two magnetic fluids were analysed for their use in the seals: a ferrofluid and magnetorheological fluid. The sealing capacity of the MRF based seals was higher than 45kPa per ring, but the friction of the seal in the bearing was 8 N•m, too large for the use in precision spindles of machine tools. The ferrofluid seal achieved sealing capacity around 9kPa per ring, good enough to be used in the spindles of machine tools, with a friction 0.25 N•m. The feasibility of using ferrofluids for developing high performance seals for high precision spindles and the validity of the simulation models has been demonstrated experimentally.

INTRODUCTION

The introduction is divided in three sections, 1) a brief introduction to classic seals and labyrinths for high precision spindles in machine tools, 2) behavior of magnetic fluids and their technological applications, and 3) seals for spindles and bearing houses based on magnetic fluids.

Seals and labyrinths for high precision machine tools

High precision spindles of machine tools, as used in grinding machines, usually implement hydrostatic bearings in their main shafts (wheelhead, workhead or tailstock) in order to ensure the highest quality and accuracy of the grinded parts, since hydrostatic lubrication provides best performance in terms of positioning resolution, damping capacity and motion smoothness [START_REF] Slocum | Precision Machine Design[END_REF]. Hydrostatic lubrication requires handling pressurized oil within the machine, fed through a manifold into the bearing and back into the tank, and the sealing of the whole system is a critical issue in the reliability of the bearings [START_REF] Bassani | Hydrostatic Lubrication[END_REF].

The sealing of the bearing housing must be ensured whether the shaft is at rest or rotating at nominal velocity, avoiding the contamination of the lubricant oil with the cutting fluid, chips from machining, abrasive micro grains from grinding, and any other potential contaminant within the machine tool. A wide review of available sealing technologies was given by Müller and Nau in their book about the principles and applications of fluid sealings [START_REF] Müller | Fluid Sealing Technology: Principles and Applications[END_REF].

One of the most active fields in the study of fluid seals is turbomachinery and rotordynamics, focused on energy applications [START_REF] Vance | Rotordynamics of Turbomachinery[END_REF]. CFD studies were carried out to analyse the force perturbation in the system, key issue to ensure fluid stability [START_REF] Hirano | Application of Computational Fluid Dynamics Analysis for Rotating Machinery-Part II: Labyrinth Seal Analysis[END_REF], and those forces were also studied from the tribology point of view, where the forces are reduced with pattern based surface microtexturing [START_REF] Yu | Frictional characteristics of mechanical seals with a laser-textured seal face[END_REF].

In the particular case of hydrostatic journal bearings lubricated with low viscosity fluids, as in the lubrication of very high speed bearings sealing (i.e. CBN grinding applications), it is very important to avoid leakage in the spindle. An extreme case due to very low viscosity is the water based hydrostatic lubrication, as presented in [START_REF] Wang | Characteristics of hydrostatic bearing/seal parts for water hydraulic pumps and motors. Part 1: Experiment and theory[END_REF].

Therefore, high sealing capacity and low friction force are the main performance criteria for bearing seals. A solution based on magnetic fluids is proposed in this work.

Magnetic fluids, behaviour and applications

Magnetic fluids are materials whose physical properties can be tuned by external magnetic fields. Two main groups can be distinguished: ferrofluids and magnetorheological fluids, MRF hereinafter. All magnetic fluids are formed by: carrier fluid, additives to improve fluid stability, and magnetic particles. The size of the magnetic particles, nanometric or micrometric, categorize them as ferrofluids or MRF respectively. Ferrofluids are compose by nanometric (5-12nm diameter) particles, and due to their size and energy balance (gravity < particle's thermal energy) the colloidal suspension is stable in time. These particles are usually formed by iron or cobalt oxides, covered with surfactants (like oleic acid) to avoid aggregation and final settling of the particles. The most relevant reference in ferrofluids behaviour is a book by Rosenweig (Rosensweig 1985), while a wide overview of the synthesis, characterization and application of ferrofluids is presented in the compilation book by Odenbach [START_REF] Odenbach | Magnetoviscous Effects in Ferrofluids[END_REF].

The MRF are similar to ferrofluids but the magnetic particles are in the range of micrometres, within 1 to 10µm, mainly made of carbonyl iron powder (CIP). They are strongly unstable, settling of the particles is inevitable, but to avoid agglomeration and to have a quick restoration of homogeneous MRF, the fluids include several additives, usually kept confidential by commercial manufacturers.

Engineering applications of magnetic fluids range from active dampers, bearings lubrication, hydraulic actuators to sealing of spindles [START_REF] Carlson | COMMERCIAL MAGNETO-RHEOLOGICAL FLUID DEVICES[END_REF]. In this research paper the application to spindle seals will be analysed.

Face sealing based on magnetic fluids for machine tool spindles

Magnetic fluids based seals were presented in the early 1980's where the basic design parameters were established [START_REF] Walowit | Analysis of Magnetic-Fluid Seals[END_REF], such as the gap, size and the distance between the rings. Simulations of those seals with non-linear models to consider the real magnetization curve were discussed [START_REF] Park | Numerical algorithm for analyzing the magnetic fluid seals[END_REF]. Looking for a better response in the ferrofluids based seals, the centrifugal force was analysed in the case of high speed systems [START_REF] Zou | Numerical analysis on the action of centrifuge force in magnetic fluid rotating shaft seals[END_REF], computing Reynolds equations together with the magnetic pressure to consider high speed effects in the fluid.

A theoretical study of the sealing capacity of large seals was presented , including centrifugal, gravitational and magnetic effect [START_REF] Meng | An analysis on the magnetic fluid seal capacity[END_REF]. The main parameters to be considered in the design were identified: Width of the rings, distance between them and the effect of the gap and magnetic fluid volume. Other papers went further in the theoretical study and analysis of ferrofluids based seals with free surface consideration ( [14]- [START_REF] Ravaud | Magnetic pressure and shape of ferrofluid seals in cylindrical structures[END_REF]) discussing the self-sealing capacity of ferrofluids for applications such as bearings.

The use of MRFs for sealing applications was considered by Vardarajan [START_REF] Iyengar | Wear Testing of Seals in Magneto-Rheological Fluids©[END_REF], where the pressurization and the wear generated in the rings by the magnetic fluids with strong yield stress was studied.

Finally a deep theoretical and experimental study of ferrofluids based seal was presented in [START_REF] Horak | Experimental and numerical determination of the static critical pressure in ferrofluid seals[END_REF], discussing the maximum pressure achieved by the rings and the recovery pressure after the fluid seal was broken, once the maximum pressure was overcome.

This research work was focused in the use of magnetic fluid seals for the application of high precision spindles in machine tools. The work was based on the models presented in the literature, two commercial magnetic fluids that were characterized experimentally in the research, and a final construction of a prototype that was used to validate the seal in close to industrial conditions. A section view of a magnetic fluid based seal is described below following the numeration of Figure 1: 1 Semi-shaft: Rotating shaft. Sealing rings (part 2) is assembled in the semi-shaft. In Section 2, the experimental characterization of the magnetic fluids used in the research is performed; Section 3 deals with the modelling of the seals; Section 4 presents the test bench and the prototype for experimental validation; Section 5 summarizes the results and finally in Section 6 the main conclusions are outlined.

CHARACTERIZATION

OF FERROFLUIDS AND MAGNETORHEOLOGICAL FLUIDS

Most magnetic fluid based seals found in the literature use ferrofluids, but there are also some solutions based on MRF. In order to analyse both alternatives, the following commercial magnetic fluids were used in this research work: a ferrofluid from FERROTEC® APG s10n and a magnetorheological fluid from LORD® Corp. model MRF 122-2ED. Data given by manufacturers are usually approximate values, both for magnetic and rheological behaviour, and therefore a detailed characterization of these fluids was performed, looking at the magnetic response and the rheological properties under the magnetic field and the temperature. The characterization of those fluids was essential to have an accurate computation of fluids magnetization and sealing capacity of the rings, as it is presented in section 3.

The magnetic characterization was performed with a Magnet-Physik Permagraph L magnetometer and the electromagnet EP-3, from which the magnetic field strength, flux density (Figure 2), and magnetic fluid permeability were obtained (Figure 3). The magnetorheological characterization was carried out with a Thermo HAAKE RheoStress RS150, with a magnetic module and a thermally controlled plate.

The values for ferrofluids characterization are shown in Figure 5, where magnetic tests were done at room temperature and ferrofluid temperature characterization without magnetic field:

DESIGN AND MODELLING OF SEALING WITH MAGNETIC FLUIDS

A model of the magnetic fluid based seal was developed to support the prototype design process (see Section 4), and they were later validated with the experimental evaluation (see Section 5). The model considers the magnetic field in the seal and the pressure in the fluid.

Magnetic simulations were performed to determine the magnetic field strength and flux density with the FEMM finite element software. Then, fluid pressure was simulated based on the model presented by Park, Kim, Hahn and Lee, [START_REF] Park | Numerical algorithm for analyzing the magnetic fluid seals[END_REF], solved by integration in in-house software. Those results are based on the experimental values of magnetic fluids characterization, section 2.

Design of seals with magnetic fluids

The ring of magnetic fluids based seal, as presented in Figure 1 (part 2), is manufactured by different groves and edges. In this section the shape of these ring edges was discussed and designed. Four different designs were studied for the ring, which are presented in Fig. 6: a) plain seal, b) square rings, c) saw shape and d) triangular shape. In all those cases the gap, overall geometry and magnet were the same, so only the effect of rings shape was analysed. The magnetic flux density was simulated in the fluid for these four different shapes,

where the field is generated by a permanent magnet. The simulations were carried out considering axis-symmetry assumption (cylindrical structure like shown in the section view of Figure 1), and magnetic model of the seal is depicted in Figure 7: 1. Considering the higher magnetic flux density achieved (from 0.359T to 0.510T), square rings were selected for the development of a magnetic fluid seal prototype.

Sealing capacity of magnetic fluid

The final step in the modelling of magnetic fluids seals was to determine the sealing capacity (pressure difference withstood by the rings) that they could achieve. In the case of MRF based seals, the sealing capacity is defined by the yield stress (function of magnetic field), which was experimentally described in Section 2 by Equation (1) and experimental results summarized in Figure 4.

For ferrofluids, the pressure achievable by each ring of the seal was determined following Equation ( 4), [START_REF] Horak | Experimental and numerical determination of the static critical pressure in ferrofluid seals[END_REF].

2 1 0 max H m H p MdH        (3)   0 2 1 max ( ) mS p M H H   (4)
Where (p m ) is the pressure in the fluid due to the magnetic field, (µ 0 ) is the permeability in the vacuum, (M s ) the saturation magnetization of the fluid and (H) the magnetic field intensity, H 1 and H 2 are the values of the magnetic field intensity on both free ferrofluid surfaces within one sealing stage.

In Equation (3), M is function of H, as shown in Figure 8. If the variation of magnetization with field is neglected at high fields within the bearing seal, and the magnetization is put equal to saturation magnetization (M s ), than one can use Equation ( 4). The maximum pressure corresponds to the maximal difference of fields H 1 and H 2 and is calculated imposing a certain volume of the ferrofluid ring and displacing this volume along the shaft direction by small steps.

The model was fed by the magnetic characterization of the ferrofluid (section 2) and the magnetic field simulation carried out in section 3.1. Finally, the sealing capacity is determined calculating the integral in Eq. ( 3) as presented graphically in Figure 8, and finding its maximum for a given volume of the ferrofluid contained in one sealing stage: 3)).

The nature and therefore the response of MRF are different from ferrofluids so the calculations were based on the yield stress in the rings and the average pressure achievable by the seal.

Once the yield stress was obtained (from magnetic simulations and fluids experimental analysis), the pressure in the ring was achieved with the computation of Equation ( 5):

  00 2 ( ) ( ) m i i e e F R H dz R H dz      (5)
Where R i and R e are the shaft radius and the internal radius of the magnetic poles; H i and H ethe magnetic field intensity on the shaft and magnetic pole surfaces. If the field H does not change significantly across the gap of the seal (H i ≈ H e ≈ H), Equation ( 6) is applied:

0 2 ( ) ( ) m i e F R R H dz    (6)
Finally, the fluid magnetic force (F m ) was divided by the annular area (A r ) of the ring and gapto get the sealing pressure (p m ), or sealing capacity, as presented in Equation [START_REF] Wang | Characteristics of hydrostatic bearing/seal parts for water hydraulic pumps and motors. Part 1: Experiment and theory[END_REF].

𝑝 𝑚 = 𝐹 𝑚 𝐴 𝑟 ⁄ (7)

DESING OF SEAL PROTOTYPE AND TEST BENCH

The experimental analysis was carried on a test bench (see Figure 9) where the prototype of magnetic fluid seal was analysed under variable operating conditions (rotational speed and pressurized air). The magnetic fluid based seal prototype was located in the semi-shaft, which can rotate up to 3000 rpm with a run-out below 5µm. The pressurization of the rings, tested with an air inlet in the body, was evaluated in function of the rotational speed.

Based on the results from magnetic calculations, a prototype of magnetic fluid seal was designed. This prototype is analysed in the test bench presented in Figure 9, assembled in the area marked with a circle. In Figure 10 a section view of the prototype with main dimensions is presented. The magnetic field was created with a strontium ferrite permanent magnet located in the sealing body and having an axial polarization (120mm external diameter, 60mm internal diameter and 12mm thickness). The average magnetic field strength of the magnet (given by manufacturer) was around 260kA/m, when flux density achieved is around 0.4T. This magnetic field was guided with two annular poles towards the rings of the sealing, 8 rings in this case as shown in Figure 10. The magnetic field in the rings and a photo of a disassembled seal are shown in Figure 11.

RESULTS AND DISCUSSION

The experimental tests to analyse the behaviour of magnetic fluids based seals are presented in this section. The section is structured in three parts: Analysis of the friction torque, sealing capacity of MRF and sealing capacity of ferrofluids.

Friction torque analyse

Friction torque was evaluated for two different magnetic fluids, FERROTEC APG s10n and LORD MRF122. In addition, two other magnetic fluids were tested: a dilution of 30% and 50% of MRF 122. Finally, a lubricant oil fluid (viscosity ISO-VG 46) and the torque without any fluid were also tested to determine friction of the kinematic chain. Results are shown in Figure 12 . The friction torque with MRF fluids, even under diluted concentration, is very high, between 2 and 8 N•m at 100rpm, too large to be used as sealing for spindles of machine tools. The torque with ferrofluid in the seal was slightly higher than base friction, around 0.25N•m, at the same rotation speed of the shaft eligible to be used in machine tool applications.

Figure 12 Friction torque in the sealing with different magnetic fluids at 100rpm.

MRF based sealing results

Tests were carried out to determine the sealing capacity of the system, evaluated as the pressure difference that it can sustain. The experimental procedure was the same for both magnetic fluids, MRF and ferrofluids (section 5.3), and it was the following the scheme shown in the Figure 13: the sealing body was filled gradually with compressed air, controlling the inlet air with a flow valve. The flow rate and pressure were measured with a flowmeter and a pressure gauge, respectively. This set-up was used for the experimental validation of the magnetic seals, and the characterization of sealing capacity of the tested magnetic fluids. Once the maximum pressure was reached two behaviours were shown: In the case of MRF the pressure stayed constant, and with the ferrofluids the magnetic seals broke (falling down the pressure) and the flowrate increased instantaneously. After a recovery time the seal was formed again and the air pressure increased again till the previous limit (maximum pressure) periodically.

The prototype was composed of eight sealing rings, therefore the sealing capacity of each ring, evaluated with the integration of equation ( 4) for ferrofluids and Equation

Erreur ! Source du renvoi introuvable. and Erreur ! Source du renvoi introuvable.

for MRF, it was multiplied by eight. The results from model and the experimental tests are summarized and compared in Table 2: In the case of MRF the pressure for each ring goes from 41.2kPa to 52.5kPa in function of the magnetic field in the fluid, which as shown in Figure 6, is not constant over the ring. For the case of 30% dilution, assuming a proportional reduction of yield stress, and therefore the sealing capacity, the theoretical values go from 12.36kPa to 15.75kPa. Summarizing, the theoretical sealing capacity of MRF for all system with eight rings was: 336.8Pa (3.3bar) to 420kPa (4.2bar), and for 30% diluted MRF: 98.9kPa (0.99bar) and 126kPa (1.26bar).

Figure 14 Seal capacity of the system with MRF.

As shown in the Figure 14, the pressure achieved within the tests was around 3.2bar slightly lower than the expected values, which go from 3.3 to 4.1bar following the simulations and calculations described above. Due to the high friction torque of MRF, a diluted fluid at 30% was analysed, where the results are shown in Figure 15. In this case the experimental value matches with expected pressure, from 0.99 to 0.126bar. Those results were obtained at rest because the pressure in the rings was the same for analysed rotational speed, up to 100rpm. Rupture and recovery pressure were not detected as in ferrofluid based seals, (see section 5.3).

MRF based seals have high sealing capacity, but the friction torque presented in the system is not compatible with machine tool spindle application.

Ferrofluid based seals results

In Figure 16 the pressure value estimated in each one of the ring filled with ferrofluids is shown, where as calculated the average pressure value was around 9kPa, which means a total pressure capacity (by eight), around 72kPa (0.72bar) in the prototype. The summary of results is presented in Table 3:

Recovery pressure, as shown in Figure 17, was the pressure where the magnetic seals (after rupture at maximum pressure) got again pressurization, recovering sealing capacity. These values are a very good pressurization capacity to be used in machine tool's spindles, which together with the low friction torque in the shaft it could be considered as a good candidate for industrial application. In Figure 17, Figure 18 and Figure 19 the evolution of air pressure and flow rate for 298 0 rpm, 50rpm and 100rpm respectively are shown. 299 As shown in Figure 17, the maximum pressure of the sealing matched quite well with the estimated value, around 72000Pa. When the shaft was rotating (see Figure 18), the difference between maximum pressure and recovery pressure was reduced, and at 100r/min (see Figure 19), both pressures were almost equal, converging at a value around 0.6bar. Above this rotational speed it was not noted variations in the results.

CONCLUSIONS

The use of magnetic fluids as active elements in seals was analysed in this research work. A close to industrial application prototype has been developed, and the validity of the simulation models and the final performance of the systems were demonstrated experimentally.

 Two magnetic fluids (a ferrofluid and a MRF) were experimentally analysed, obtaining deeper information about their behaviour than that offered by the manufacturers. Such a detailed experimental characterization is highly recommended for any studies in the field of magnetic fluids.

 MRF and ferrofluid based seal models were developed, simulating the magnetic field and the sealing capacity measured in pressure per ring.

 The seal based on MRF showed very good sealing performance (over 3 bar) but with an excessive friction torque (8Nm). Even diluted at 30%, sealing was still good but the torque was too high for machine tool applications.

 The seal based on ferrofluids demonstrated a good enough sealing capacity (0.9bar) and with a friction torque slightly higher than without the seal.

 The seal based on ferrofluids showed thus true potential for industrialization in machine tool spindles.

2

  Sealing rings: Ferromagnetic part filled with magnetic fluid. Sealing rings are machined in this part, and magnetic fluid seals are formed in the edges. Different ring edge shapes are studied in the research. 3 Magnetic poles: Steel made rings to conduct magnetic field from the magnets (part 4) to the seal rings (part 2).

4

  Magnets: Strontium ferrite made permanent magnets 5 Seal body: Main structure of the seal made in aluminium. Air pressure is achieved into the body, air inlet (orifice 7). 6 Assembly cover: Cover ring to assemble the sealing prototype. 7 Air inlet: Compressed air inlet to the main body. Interior volume of seal body (5) is pressurized with compressed air, introduced by air inlet (7), with sealing capacity given by sealing rings (2).

Figure 1

 1 Figure 1 Magnetic fluids sealing, section view with parts description.

Figure 2 Figure 3 1 )Figure 4

 2314 Figure 2 Magnetic characterization of the magnetic fluids: flux density and magnetic field strength.

Ferrofluids

  rheology was characterized following tuneable Newtonian fluid (2), where fluid dynamic viscosity (η) it is in function of temperature (T) and applied magnetic field (H). η = 𝜂 𝑚 (𝐻, 𝑇)

Figure 5

 5 Figure 5 Rheological characterization of ferrofluid: viscosity and temperature.

Figure 6

 6 Figure 6 Magnetic field distribution for different rings shape studied in the paper : a) plain seal, b) square rings, c) saw shape and d) triangular rings.

Figure 7

 7 Figure 7 Magnetic fluid seal magnetic model with parts description and detail of mesh.

Figure 8

 8 Figure 8 Fluid magnetization and magnetic field strength, pressure computation (Equation (3)).

Figure 9

 9 Figure 9 Test bench with the prototype of magnetic fluid based seal.

Figure 10

 10 Figure 10 Magnetic fluids sealing, main dimensions

Figure 11

 11 Figure 11 Magnetic fluids sealing, working principle (a) and manufactured set-up (b)

Figure 13 Friction

 13 Figure 13 Friction torque in the sealing with different magnetic fluids.

Figure 15

 15 Figure 15 Seal capacity of the rings with MRF at 30% dilution.

Figure 16

 16 Figure 16 Sealing capacity of each ring with ferrofluid FERROTEC APG s10n.

Figure 17

 17 Figure 17 Sealing capacity at rest with FERROTEC APG s10n

Figure 18

 18 Figure 18 Sealing capacity at rest with FERROTEC APG s10n

Table 1

 1 Magnetic flux density (B-Tesla) values in the fluid.

			Control points in the model
	Shape	1	2	3	4	5	B min	B max	B ave
	Plain	0.4167	0.4719	0.4661	0.4620	0.3443	0.3433	0.4760	0.4167
	Square 0.4481	0.3770	0.4641	0.5026	0.3832	0.3591	0.5105	0.4398
	Saw	0.4885	0.4206	0.4603	0.4127	0.3534	0.3534	0.4939	0.4236
	Triang. 0.4620	0.4508	0.4459	0.4424	0.3486	0.3486	0.4729	0.4107

Table 2

 2 MRF based seal, sealing capacity (Pa).

			MRF 122 (dilution 30%)	MRF 122
	Sealing capacity (kPa)
			per ring	Seal	per ring	Seal
	Calculated sealing	Min	12.36	98.88	41.2	336.8
	capacity	Max	15.75	126.0	52.5	420.0
	Experimental value in the	----	130.0	----	320.0
	seal (shaft at rest)

Table 3

 3 Ferrofluid based seal, sealing capacity (Pa).

				Ferrofluid
	Sealing capacity (kPa)
			per ring	Seal
	Calculated sealing capacity		9	72
	Experimental (at rest)	Max	----	70
		Recovery	----	48
	Experimental (50r/min)	Max	----	68
		Recovery	----	58
	Experimental (100r/min)	Max	----	60
		Recovery	----	58


Hinda Ezzaier

Jessica Alves Marins

Sébastien Schaub

Ben Haj Amara

Pavel Kuzhir
email: kuzhir@unice.fr

Jéssica Alves Marins

Ben Haj

Capture of magnetic nanoparticles on ordered magnetizable arrays: A parametric study

teaching and research institutions in France or abroad, or from public or private research centers.

Introduction

Small scale magnetic separation of magnetic particles has found numerous biomedical applications, such as cell separation [START_REF] Zborowski | Magnetic Cell Separation[END_REF], protein purification [START_REF] Okoli | [END_REF], control drug delivery and release [3] and immunoassays [4]. In most of these applications, micron sized beads bearing appropriate chemical groups on their surface are used, while the use of nanoparticles is limited because of strong Brownian motion hindering attractive interactions with magnetized collectors. However, phase separation or aggregation induced by an applied magnetic field have been found to significantly enhance these interactions leading to stronger magnetophoretic mobility of particles [5] and better separation efficiency [6]. The reason is that the magnetic force experienced by micron-sized aggregates is much larger than that of individual nanoparticles. This will likely allow replacement of microbeads by nanoparticles in aforementioned applications for which high specific surface and good colloidal stability are unconditional advantages of nanoparticles with respect to microbeads.

In our previous work, we have demonstrated an efficient capture of magnetic nanoparticles of a mean size about 80 nm on an ordered array of nickel micropillars incorporated into a microfluidic channel [7]. The micropillars were magnetized by an external uniform magnetic field of an intensity 6-10 kA/m. However, the capture efficiency  (related to the ratio of the inlet-to-outlet particle concentration) appeared to be a strongly decreasing function of the flow speed and of the Mason number:

1.78 Ma   that was explained by flow-induced rupture of the particle aggregates.

To optimize the capture efficiency keeping low values of the applied magnetic field, we have to establish and understand the effects of numerous parameters (other than Ma) on the nanoparticle capture, such as dipolar coupling parameter, nanoparticle concentration, magnetic field orientation and geometry of micropillar arrays. This work is dedicated to an extensive parametric study of the magnetic separation process on ordered magnetizable arrays. To this purpose, we visualize the process of accumulation of magnetic nanoparticles around the micropillars when the nanoparticle suspension flows across the micropillar array in the presence of an externally applied magnetic field. Growth of the nanoparticle deposits with time allows analyzing the capture efficiency as function of numerous dimensionless parameters. In the next Section 2, we briefly describe the experimental setup and protocol. The results of the visualization experiments and parametric studies are presented in Section 3. The conclusions and perspectives are outlined in Section 4.

Experimental setup

A sketch of the microscale channel equipped with an array of nickel micropillars is presented in Fig. 1. Briefly, micropillar arrays were created on a glass substrate by electroplating and soft photolithography, as described in [7]. Square and hexagonal arrays were used with the micropillar expected diameter, expected height and the distance between the pillar axes equal to d m =50 µm, h=50 µm and =150 µm, respectively. The flow channel was created by squeezing a polyvinyl seal between a Plexiglas substrate and the glass cover bearing micropillars. The channel height was approximately adjusted to H≈2h=100 µm by four screws pressing the glass cover to the Plexiglas substrate through an aluminium lid not shown in Fig. 1. This design has an advantage of easy disassembling required for frequent cleaning of the cell, as opposed to permanently sealed microfluidic channel used in [7].

Water dispersions of iron oxide nanoparticles (ferrofluids) were synthesized by co-precipitation of iron salts in alkali medium and stabilized by a double layer of oleate ions, as described in detail in [8]. Using this method, instead of isolated nanoparticles (8-12 nm), permanent isotropic nanoclusters composed of numerous nanoparticles were obtained. These nanoclusters possessed an induced magnetic moment and had zero remnant magnetization. Since nanoclusters (40-70 nm) are the smallest structural units involved into magnetic separation process, they will be hereinafter called nanoparticles in order to avoid confusion with fieldinduced aggregates. The two synthesized ferrofluids, labelled as F1 and F2, contained nanoparticles of a mean diameter of d n =68 nm and 42nm, respectively, as measured by dynamic light scattering and confirmed by transmission electron microscopy. The primary ferrofluids F1 and F2 were diluted at desired particle concentrations by distilled water.

Fig. 1. Microscale channel with a micropillar array

A dilute suspension of magnetic nanoparticles was pushed through the channel with a desired flow rate Q using a syringe pump Harvard PHD Ultra (United States). The flow velocity u was estimated as u=Q/S, where S is the channel cross-section. The channel was settled on the stage of an inverted microscope Diaphot (Nikon, Japan). Images were collected in transmission light with a four-fold objective and a CMOS camera PL-B742U (PixelLINK, Canada). An external uniform magnetic field was generated with a pair of Helmholtz coils equipped with iron yokes and placed around the microscope. The angle between the external field H 0 and the main flow was changed in the range =0-90° by turning the flow cell on the microscope stage.

The following experimental protocol was adopted. First, the suspension was pushed through the channel at a desired velocity u in the absence of magnetic field for 5 minutes. Then, a magnetic field of a desired intensity H 0 was applied. The nickel micropillars got magnetized and started to attract magnetic nanoparticles and separate them from the suspending liquid (water). Snapshots of nanoparticle accumulation around micropillars (such as shown in Fig. 2) were taken every minute for 60 minutes from the moment of field application. For quantification of the acquired time-lapses, we developed a macro under ImageJ to automatically segment the pillars and their neighbourhood. The macro measured the surface of nanoparticle deposits per pillar versus time. The relative deposit area was defined as the ratio of the whole area occupied by nanoparticle deposits to the whole area of the micropillar array. The image processing allowed obtaining experimental dependencies of the relative deposit area, s, versus the elapsed time t. In each case, the experimental s(t) curves were satisfactorily fitted to the following semi-empirical filtration law [7]:

0 ( ) 1 exp m m ut s t s sL              (1)
where  0 is the nanoparticle volume fraction at the channel inlet; ≈0.70 is the internal volume fraction of the deposit; L=1.3-1.5 mm -array's dimension in the direction of flow; =ln( 0 / out ) is the capture efficiency in the beginning of separation (t=0), this magnitude is related to the inlet ( 0 ) and outlet ( out ) nanoparticle volume fractions; and s m is the maximum relative deposit area at long times (t), this magnitude characterizes the maximum amount of nanoparticles that the filter is able to capture and is hereinafter called the retention capacity. Typical s(t) curves have been presented in our previous work [7]. The magnitudes  and s m are two fitting parameters depending on microscopic details of the particle capture. These magnitudes are analyzed in Section 3.2 as a function of different dimensionless parameters governing separation.

Results and discussion

Visualisation of particle deposits

Some snapshots showing nanoparticle deposits around the micropillars are shown in Fig. 2. The upper row of the snapshots shows the top view of the micropillar arrays without captured nanoparticles, the middle and the bottom rows show nanoparticle deposits for the sample F1 at the flow speeds u=7.1410 -4 m/s and 2.1410 -3 m/s, respectively, at an applied magnetic field H 0 =13.5 kA/m, inlet particle volume fraction  0 =0.3% and elapsed time t=60 min. Different columns correspond to different angles  between the flow (whose direction is indicated by white arrows) and the magnetic field (horizontal for each snapshot). The three left columns show the capture on the square array, while the right three columns -on the hexagonal array.

In most of the snapshots, the nanoparticle deposits exhibit wavy patterns interconnecting the neighbouring micropillars and having a wavelength corresponding to the distance between them. In the beginning of the capture process (not shown in Fig. 2), the nanoparticles are condensed in the vicinity of the micropillars and form concentrated elliptic deposits, which grow in the direction of the applied magnetic field until they touch each other. Percolation of primary elliptic deposits should explain wavy shape of the final deposits governed When the flow makes some angle 0°<<90° with respect to the magnetic field H 0 , the nanoparticle deposits are no longer aligned with the applied magnetic field but show stair-like patterns making some angle with H 0 different from . Again, these patterns appear as a result of percolation of neighbouring deposits which find the optimal pathways for percolation depending on hydrodynamic drag and magnetic interactions between primary deposits. So, the descending stair-like pattern for the square array at =60° is formed by linking primary deposits in the direction of flow [2 nd column from the left in Fig. 2], while ascending pattern for the hexagonal array at =45° is formed by linking the deposits in the direction of the applied field [2 nd column from the right in Fig. 2]. In the case of the flow perpendicular to the magnetic field (=90°), the patterns are quite similar to those observed for the flow parallel to the field (=0°) except some asymmetry with respect to the horizontal axis (magnetic field direction) [3 rd and the last columns of Fig. 2].

Comparing the middle and the bottom rows of Fig. 2, it is seen that the size of the deposits decreases with the flow speed because of the larger hydrodynamic stresses acting on the deposit surface and washing away nanoparticles. Comparing the left three columns with the right three right three columns, it is quite obvious that the hexagonal array generally captures more nanoparticles than the square array. This can be explained by simple geometrical considerations. In fact, at the same distance between micropillars and at the same size of the primary elliptical deposits, the distance between the surfaces of primary deposits is closer for the hexagonal array allowing easier percolation and junction at several points, as in the case of honeycomb structure at =0°. In summary, the array geometry and mutual orientation of the flow and of the magnetic field are crucial parameters defining the structure of the nanoparticle deposits and the retention capacity s m [Section 2]. The parameters  and s m quantifying magnetic separation will be inspected in details in the next Section 3.2.

Capture efficiency and retention capacity

Experiments show that the micropillars capture a small amount of nanoparticles in the absence of field, while the capture is significant in the presence of moderate magnetic fields 4.8≤H 0 ≤13.5 kA/m. This signifies that colloidal interactions are negligible as compared to magnetic ones. In this case, the capture of magnetic nanoparticles is mostly governed by hydrodynamic (F h ) and magnetic (F m ) forces and can be affected by Brownian motion. Comparison of orders of magnitudes of these forces (or energies U) allows definition of the three following dimensionless parameters: (a) Mason number dependencies of  and s m are shown in Fig. 3 for two different particle diameters d n (samples F1 with d n =68 nm and F2 with d n =42 nm) and three different values of the applied magnetic field H 0 while the particle concentration ( 0 =0.3%), flow orientation (=0°) and arrays' geometry (square) are kept constant. Variations of d n and H 0 result in a variation of the dipolar coupling parameter in the range 0.13.7. Three experimental  versus Ma curves, corresponding to ≥0.5, are gathered along a straight line in log-log scale and fitted by a scaling law Ma -2 (dashed line on Fig. 3a), while the fourth curve with ≈0.1 shows a weaker Mason number dependency. Experiments show that nanoparticle suspensions undergo phase separation with appearance of bulk aggregates at ≥0.5 and  0 =0.3% 1 .

Intense shear flows around the micropillars can disrupt nanoparticles from aggregates such that the aggregate size is expected to be a decreasing function of the flow speed u and of the Mason number. The combination of this effect with increasing convective flux washing away the aggregates from the micropillars could explain a strong decrease of the capture efficiency with Mason number (Ma -2 ) for ≥0.5. The theoretical model assuming chain-like clusters instead of observed bulk needle-like aggregates predicts a quite similar behaviour (Ma -1.7 ) [7]. Furthermore, collapse of the experimental Mason number dependencies for ≥0.5

signifies that the capture is governed by the interplay between magnetic and hydrodynamic forces, while the Brownian motion of bulk aggregates is expected to be very weak. Nanoparticle suspensions do not exhibit phase separation at =0.1 and  0 =0.3%. In this case, the capture efficiency of individual nanoparticles is only defined by a compromise between convective, magnetophoretic and diffusive fluxes. A simple model neglecting diffusive flux predicts the following Mason number dependency in the limit Ma>>1 [7]:

1 2 16 (1 ) m cL Ma cd    ( 2
)
where c is the volume fraction of the micropillars with respect to the total array's volume; c≈0.09 and 0.10 for the square and hexagonal arrays, respectively. The theoretical dependency [Eq. ( 2), solid line in Fig. 3a] agrees relatively well with experimental points for ≈0.1

1 Field-induced phase separation is possible at such low values of  because both samples F1 and F2 have relatively broad particle size distribution; the largest particles with the size above 100 nm are assumed to form the dense phase (aggregates) while the smallest ones remain in a dilute phase, as suggested in numerous works -see for example [9].

[open triangles] without adjustable parameters. Thus, the single particle model supports a smoother decrease of the capture efficiency with growing Mason numbers (Ma -1 ) at ≈0.1, as opposed to the case of phase separating colloid (Ma -2 ) at ≥0.5. The retention capacity fits to an approximate scaling law s m Ma -0.6±0.1 but the data are much more dispersed around the fitting curve [dashed line on Fig. 3b]. The maximum size of deposits is expected to decrease with increasing diffusive nanoparticle flux responsible for nanoparticle "evaporation" from the deposit surface. This could explain generally lower retention capacities for lower dipolar coupling parameters , as observed in Fig. 3b.

The effect of the particle volume fraction  0 is inspected in Fig. 4 where Mason number dependencies of  and s m are shown for different values of  0 and for two samples F1 and F2 at constant magnetic field intensity (H 0 =13.5 kA/m), the flow orientation (=0°) and the array's geometry (square). Experiments show that the suspensions undergo phase separation for all the curves presented in Fig. 4. The Mason number dependencies of the capture efficiency are spread around a master curve with a scaling law Ma -2 [dashed line in Fig. 4a] inherent for phase separating colloids, while there is no clear effect of the particle volume fraction. This is in agreement with the scenario of destructible aggregates [7] where their size and interaction with micropillars depend entirely on Mason number but not on particle concentration, at least in the beginning of separation when the dilute limit applies in the whole volume of the filter. The Mason number dependencies of the retention capacity follow previously found scaling law s m Ma -0.6±0.1 but no longer collapse on a single curve for different particle concentrations and show an increase of s m with increasing  0 [Fig. 4b]. This is probably because the dilute limit approach is no longer valid for the concentrated deposits and their size is expected to grow with the inlet particle concentration because of enhanced magnetic interactions between nanoparticles that result in larger deposit sizes, as has been previously shown for separation on single collectors [8].

The effect of the array's geometry is shown in Fig. 5 where  and s m versus Ma dependencies are plotted for square and hexagonal arrays and for two different particle sizes (samples F1 and F2) at constant nano-Fig. 5. The same as in Fig. 3 but for different array geometries (square and hexagonal) and for two different samples (F1 and F2) at fixed values of H 0 =13.5 kA/m, =0°, and  0 =0.3%. The dashed line in (a) corresponds to a fit of all the data by a single curve Ma -2 , while two dashed curves in (b) correspond to the fit of the data for the square and hexagonal arrays by two distinct curves, s m Ma -0.6±0.1 and s m Ma -0.34±0.11 , respectively. particle volume fraction ( 0 =0.3%), field intensity (H 0 =13.5 kA/m) and the flow orientation (=0°). Fieldinduced phase separation is observed for all the experimental data on Fig. 5. Again, the data on the capture efficiency collapse on a single master curve Ma -2 [dashed line on Fig. 5a], while the data on the retention capacity are regrouped along two different curves corresponding to both considered geometries: s m Ma -0.60±0.10 for square arrays and s m Ma -0.34±0.11 for hexagonal arrays [dashed curves on Fig. 5b]. The array's geometry does not influence the initial capture efficiency  likely because, in the beginning of separation, the nanoparticle trajectories in the vicinity of a given micropillar are expected to be weakly influenced by the geometrical disposition of the neighbouring pillars provided that they occupy only about 10% of the array volume (c≈0.1). On the other hand, the hexagonal array has a larger retention capacity than the square array (in agreement with observations on Fig. 2) likely because hexagonal arrays favour more easily percolation of the primary elliptical deposits, as has been supposed in Section 3.1.

Conclusions

We presented a detailed experimental study of the effect of different dimensionless parameters on the magnetic separation of medium-sized nanoparticles (40-70 nm) on ordered arrays of magnetizable micropillars. Visualization of suspension flows across micropillar arrays allowed us to follow formation and growth of nanoparticles deposits and extract two major parameters describing this processthe capture efficiency  in the beginning of separation and the retention capacity s m of the array. It has been shown that the capture efficiency depends mostly on Mason number and on the mutual orientation between the flow and the magnetic field, being nearly independent of the dipolar coupling parameter (in the range 0.53.7), of the nanoparticle volume fraction (at 0.075 0 0.3%) and on the array's geometry (square or hexagonal). The Mason number behaviour changing from Ma -2 to Ma -1 has been explained by transition from destructible nanoparticle aggregates (appearing as a result of field-induced phase separation) at ≥0.5 to isolated nanoparticles at <0.5. A non monotonic dependence on the angle  with a minimum of  at intermediate angles is less clear and has been tentatively explained by an effect of  on mechanical stability of aggregates near the micropillars. The retention capacity has also been found to be a decreasing function of Mason number (typically s m Ma -0.6 ) however it also depends on all other parameters (increases with  and  0 and is larger for the hexagonal array than for the square array). Tuning the array's geometry and mutual orientation of the flow and of the magnetic field allows generation of particular patterns of nanoparticle deposits minimally affected by the flow and providing a maximized retention capacity. This could be useful for development of microscale magnetic separators for biomedical applications.

For the deeper understanding of the above reported effects, numerical modelling of the separation process incorporating trajectory analysis and field-induced aggregation has to be conducted in future.

Fig. 2 .

 2 Fig. 2. Snapshots of nanoparticles deposits around micropillars. The upper row shows naked micropillars without captured particles, the second and the bottom rows show nanoparticle deposits for the sample F1 at the flow speed u=7.14x10 -4 m/s and 2.14x10 -3 m/s, respectively and for H 0 =13.5 kA/m,  0 =0.3% and t=60 min. The applied magnetic field is horizontal for each snapshot. by the stress balance on their surface. The flow intensity and orientation influence the shape of the deposits which evolve to a straight stripe pattern with increasing speed in the case of the square array with =0° [bottom left snapshot in Fig. 2] or to a honeycomb pattern in the case of the hexagonal array at =0° [4 th column from the left of Fig.2]. The last pattern with the highest deposit area is possible thanks to a gap of a thickness H-h≈50 µm between the end surface of micropillars and the bottom Plexiglas surface of the channel [Fig.1] allowing the liquid to pass through the channel.When the flow makes some angle 0°<<90° with respect to the magnetic field H 0 , the nanoparticle deposits are no longer aligned with the applied magnetic field but show stair-like patterns making some angle

Fig. 3 .

 3 Fig. 3. Mason number dependency of the capture efficiency (a) and retention capacity (b) at different dipolar coupling parameter  and fixed values of  0 =0.3%, =0° and for the square array. Solid line in (a) stands for the prediction of the single particle model [Eq. (2)]. Dashed lines in (a) and (b) correspond to a power-law fit of the experimental data.

Fig. 4 .

 4 Fig. 4. The same as in Fig.3 but at different particle volume fractions  0 and for two different samples (F1 and F2) at fixed values of H 0 =13.5 kA/m, =0° and for the square array. The dashed line in (a) corresponds to a fit of all the data by a single curve Ma -2 , while two dashed curves in (b) correspond to the fit of the data at  0 =0.3% and  0 =0.075% by two distinct curves having the same scaling behaviour s m Ma -0.6±0.1 .

Fig. 6 .

 6 Fig. 6. Dependences of the capture efficiency and retention capacity on the angle  between the flow and the magnetic field for different values of Ma, at fixed values of H 0 =13.5 kA/m,  0 =0.3%, and for the sample F1 and the square array. The effect of the flow orientation with respect to the applied magnetic field is inspected on Fig. 6 where  and s m are plotted as function of the angle  for different values of Mason number. The capture efficiency exhibits a minimum at some intermediate angles [Fig. 6a]. It can be shown that the change of the field orientation significantly changes the local distribution of Mason numbers in the vicinity of micropillars that affects mechanical stability of the nanoparticle aggregates and modifies their size and therefore their capture efficiency. This scenario has to be studied in detail in to confirm if it can explain a non-monotonic  versus  function. The effect of the flow orientation on the retention capacity is even less clear because there is a minimum of s m ()-function for Ma=895 but there is no pronounced effect at Ma=229 and 449 [Fig. 6b].

  Pe, varying all other parameters, we realize that different data better collapse when plotted against Mason number. Thus, in what follows, the magnitudes  and s m will be mostly analysed as function of Mason number.

												Mason
	number	/ Ma F F h m 	0  6	ud	m	/(	22 n H d 0 0   n	)	; (b)
	dipolar			coupling			parameter
		/( U k T m B  )	2   00 n H d 2 3 n	/(12 k T B	)	; and (c) Péclet
	number	/ Pe ud D m n 		0  3	/( ud d k T n m B	) 6  	Ma	/		n	,

where  0 ≈10 -3 Pas is the suspending liquid (water) viscosity;  0 =410 -7 H/m is the vacuum magnetic permeability;  n ≈0.9 is the magnetic contrast factor of nanoparticles; k B T≈410 21 J is the thermal agitation energy; and D n is the Brownian diffusion coefficient of nanoparticles.

Inspecting dependencies of the capture efficiency  and retention capacity s m on one of the parameters Ma,  and

The authors acknowledge C. Claudet and F. Lippi for the channel design and fabrication. This work has been supported by French RENATECH network and its FEMTO-ST technological facility. P.K. acknowledges UCA Jedi through its "action du site" and "AAP Academy 2" projects ImmunoMag and MagFilter; H.E. acknowledges FSB through its PhD collaboration fellowship and J.A.M is grateful to CNPq though its postdoctoral fellowship (No. 203100/2014-0).


Boukhalfa Abdelouaheb

Khaber Farid

Essounbouli Najib

Synergetic Adaptive Fuzzy Control for a Class of Nonlinear Discrete-time Systems

Introduction

In the past decades, considerable e↵orts have been devoted to the enhancement of nonlinear systems and their stability and nonlinear phenomena. Although the stability of these systems can be improved by Lyapunov analysis. Because linear superposition is no longer available, explicit formula are di cult to come by, numerical approximations are not always su ciently accurate [START_REF] Ho | Adaptive fuzzy sliding mode control with chattering elimination for nonlinear SISO systems[END_REF]. The advent of powerful computers has fomented a veritable revolution in our understanding of nonlinear systems. Indeed, many of the most important modern analytical techniques drew their inspiration from early computer-aided investigations of nonlinear systems [START_REF] Olver | Indirect Adaptive Fuzzy Control for a Class of Nonlinear Discrete-Time System[END_REF]. Most systems are nonlinear with characteristics that change with time, since that, in a dynamic operating mode, we cannot guaranteed the strong performance of the linearized model-based controllers of nonlinear systems. Therefore, the nonlinearities of nonlinear systems are required for an intelligent controller. To improve the behavior of nonlinear controllers, many techniques have been proposed for their design, such as fuzzy logic [START_REF] Erginer | Design and Implementation of a Hybrid Fuzzy Logic Controller for a Quadrotor VTOL Vehicle[END_REF], adaptive antiswing control strategy for crane systems [START_REF] Sun | Nonlinear Antiswing Control for Crane Systems With Double-Pendulum Swing E↵ects and Uncertain Parameters: Design and Experiments[END_REF], artificial neural networks [START_REF] Yu | RBF Neural Networks-Based Robust Adaptive Tracking Control for Switched Uncertain Nonlinear Systems[END_REF][START_REF] Dexu | Adaptive Robust Control based on RBF Neural Networks for Duct Cleaning Robot[END_REF], adaptive switched non-strict-feedback nonlinear systems [START_REF] Li | Adaptive output-feedback control design with prescribed performance for switched nonlinear systems[END_REF], metaheuristic algorithms [START_REF] Roeva | PID Controller Tuning based on Metaheuristic Algorithms for Bioprocess Control[END_REF], nonlinear designs using variable structure control (sliding mode, synergetic control), design of an adaptive fuzzy control for stochastic nonlinear systems with unmeasured states [START_REF] Li | Adaptive Fuzzy Control Design for Stochastic Nonlinear Switched Systems With Arbitrary Switchings and Unmodeled Dynamics[END_REF][START_REF] Wu | Observer-Based Adaptive Fault-Tolerant Tracking Control of Nonlinear Nonstrict-Feedback Systems[END_REF], controlling underactuated systems [START_REF] Sun | Amplitude-Saturated Nonlinear Output Feedback Antiswing Control for Underactuated Cranes With Double-Pendulum Cargo Dynamics[END_REF], stabilizing control strategy for underactuated ship-mounted crane systems [START_REF] Sun | Nonlinear Stabilizing Control for Ship-Mounted Cranes With Ship Roll and Heave Movements: Design, Analysis, and Experiments[END_REF], and many other nonlinear control techniques [START_REF] Ruiyun | Adaptive control of discrete-time state-space T-S fuzzy systems with general relative degree[END_REF][START_REF] Wu | Adaptive Fuzzy Control for Nonlinear Networked Control Systems[END_REF]. The advantage of fuzzy logic control methods is that we can obtain and adjust online the parameters of the system. In order to make the design more adaptable, the combination between fuzzy logic and other control methods such as synergetic control is made [START_REF] Bouchama | Reaching phase free adaptive fuzzy synergetic power system stabilizer[END_REF]. To reduce the algorithm complexity, simplified models of nonlinear systems are used. Furthermore, reduced computational burdens for realistic models are required to design a robust controller under di↵erent operating conditions. An adaptive nonlinear fuzzy control technique has obtained a great attention based on the universal approximation theorem, however, most results are limited to continuous nonlinear systems [START_REF] Olver | Indirect Adaptive Fuzzy Control for a Class of Nonlinear Discrete-Time System[END_REF], which cannot be extended to discrete-time nonlinear systems directly. The implementation of a fuzzy control algorithms in digital calculators involve the loss of some advantages of continuous time controllers. However, it should be mentioned that the used controller is a discrete-time systems [START_REF] Da-Wei | Non-fragile H 1 fuzzy filtering for discrete-time nonlinear systems[END_REF]. In [START_REF] Ruiyun | Stable indirect adaptive control based on discrete-time T-S fuzzy model[END_REF], an indirect fuzzy adaptive controller algorithm for uncertain nonlinear systems was developed where Takagi-Sugeno (T-S) fuzzy input output model was used to approximate the nonlinear system dynamics, an adaptive feedback controller model was designed [START_REF] Haitao | Fuzzy Sliding Mode Control of Robotic Manipulators With Kinematic and Dynamic Uncertainties[END_REF], [START_REF] Larguech | Fuzzy Sliding Mode Control for Turbocharged Diesel Engine[END_REF] using the (T-S) fuzzy models. Authors in [START_REF] Ruiyun | Adaptive control schemes for discrete-time T-S fuzzy systems with unknown parameters and actuator failures[END_REF] had developed a solution framework and control design for adaptive control discrete-time input-output multiple-delay T-S fuzzy systems. We have proposed a new adaptive fuzzy synergetic control scheme for uncertain nonlinear discrete-time systems. So we have organized our work as follows. Section 2 presents the basics in the design of a synergetic control, in Section 3, the nonlinear systems described by discrete-time equations for which the design of a synergetic controller is performed. Fuzzy IF-THEN rules are presented in section 4 which show how an adaptive synergetic controller can estimates the nonlinear dynamics of the nonlinear systems based on the intelligent approach of the fuzzy logic. In section 5, through the Lyapunov stability analysis, we demonstrate the stability of the nonlinear discrete-time system and derive the adaptation law. The e↵ectiveness of the proposed adaptive synergetic fuzzy controller (ASFC) is illustrated by the design for a real world example in section 6. In Section 7, conclusion and some perspectives are given.

Basics in Synergetic Control Design

The synergetic control is a novel nonlinear control technique that take into account the nonlinearities of system in the control design. A systematic design procedure which yields control laws suitable for digital implementation are o↵ered [START_REF] Kolesnikov | Modern Applied Control Theory:Synergetic Approach in Control Theory[END_REF][START_REF] Kolesnikov | Synergetic synthesis of dc-dc boost converter controllers: Theory and experimental analysis[END_REF]. Moreover, synergetic control not only gives constant switching frequency operation, but also provides asymptotic stability with respect to the required operating conditions, and robustness in parameters variation of the system [START_REF] Ho | Adaptive fuzzy sliding mode control with chattering elimination for nonlinear SISO systems[END_REF][START_REF] Bouchama | Reaching phase free adaptive fuzzy synergetic power system stabilizer[END_REF]. The designer can select the characteristics of the macro-variable according to performances and control specifications (overshoot, control signal limits, etc...) which make the design more robust. Thus the control law, will not cause chattering phenomena as in the sliding mode control approach. The parameters were optimized and the implementation is easy to realize due to use of measurable variables in the control law. Let's consider an nth order dynamic nonlinear systems described as :

x(k + 1) = f (x(k), u(k), k) ( 1 )
Where x(k) represents the system vector, u(k) the control input vector and f is an nonlinear function. The synthesis of a synergetic controller starts with the selection of a function of the system state variables, which is called the macro-variable and depend on the state variables.

= (x(k), k) ( 2 )
The control objective is to force the system state to operate on the manifold = 0. The designer can select the characteristics of the macro-variable according to performances and control specifications (overshoot, control signal limits, etc...). In continuous-time synergetic approach, to control theory (SACT) procedure, T ˙ + = 0 defines the speed and trajectory of convergence to the invariant manifold [START_REF] Qian | Analysis of the synergetic control based on variable structure and application of power electronics[END_REF]. Considering sampling period T s , the discrete counterpart is derived as follows :

T h (k + 1) (k) T s i + (k) = 0 ( 3
)
Where T is a design parameter that specifies the convergence speed to the manifold. Equation (3) can be rewritten as :

T T s ⇣ T T s T ⌘ (k + 1) + (k) = 0 (4)

Design of Synergetic Controller

Consider a discrete-time nonlinear systems which have the state space representation :

8 > > > > > > > < > > > > > > > : x 1 (k + 1) = x 2 (k) x 2 (k + 1) = x 3 (k) . . . x n 1 (k + 1) = x n (k) x n (k + 1) = f (x(k)) + u(k) + d(k) y(k) = x 1 (k) (5)
where

f (x(k)) is a nonlinear function, x(k) = [x 1 (k),
x 2 (k), ..., x n (k)] T 2 R n is the state vector of the systems which is assumed to be available for measurement, u(k) 2 R and y(k) 2 R are respectively, the input and the output of the system, and d(k) is the external disturbance which is assumed to be bounded. Define the tracking errors as :

e 1 (k) = x 1 (k) y d (k) e 2 (k) = x 2 (k) y d (k + 1) . . . e n (k) = x n (k) y d (k + n 1)
where y d (k) denote the reference trajectory. The tracking error equation is given as :

e(k + 1) = Ae(k) + B h f (x(k)) + u(k) y d (k + n) + d(k) i (6)
The objective of the control is to design a controller for the state x 1 (k) to track a desired reference signal y d (k) in the presence of external disturbance and uncertainties d(k). The control u(k) is calculated according to (4) and ( 6), which gives a control signal that ensures a specified properties. Let's define the macro-variable as :

(k) =  1 e 1 (k) + e 2 (k) = n 1 X i=1  i e i (k) + e n (k) ( 8 )
(k + 1) =  1 e 1 (k + 1) + e 2 (k + 1) (9)
Where  1 is the SACT controller parameter.

Where :

e 1 (k + 1) = x 1 (k + 1) y d (k + 1) (10)
e 2 (k + 1) = x 2 (k + 1) y d (k) ( 1 1 )
(k + 1) =  1 x 1 (k + 1)  1 y d (k + 1) +x 2 (k + 1) y d (k) (12)
(k + 1) =  1 x 2 (k)  1 y d (k + 1) + f (x(k)) + u(k) + d(k) y d (k) (13)
The incremental change in (k) can be expressed by :

(k + 1) = (k + 1) (k) = n 1 X i=1  i e i (k + 1) + e n (k + 1) n 1 X i=1  i e i (k) e n (k) (k + 1) = n 1 X i=1  i e i (k + 1) + x n (k + 1)
y d (k + n) n 1 X i=1  i e i (k) e n (k) (k + 1) = n 1 X i=1  i e i (k + 1) + f (x(k)) y d (k + n) + u eq (k) n 1 X i=1  i e i (k) e n (k) (14)
Let's take

↵ = T T s ⇣ T T s T ⌘ (15)
Combining equations ( 12) and (4) yields :

↵ h  1 x 2 (k)  1 y d (k + 1) + f (x(k)) + u(k) + d(k) y d (k) i + (k) = 0 (16)
The synergetic control law is then deduced and given as [START_REF] Ruiyun | Stable indirect adaptive control based on discrete-time T-S fuzzy model[END_REF] :

u(k) = f (x(k))  1 x 2 (k) +  1 y d (k) + y d (k) d(k) 1 ↵ (k) (17)
If f (x(k)) is known, we can easily construct the synergetic control law [START_REF] Ruiyun | Stable indirect adaptive control based on discrete-time T-S fuzzy model[END_REF]. However, this controller contains a constraint : knowledge of f (x(k)) which is not always possible making the implementation impossible. Therefore, to overcome this obvious problem, an adaptive synergetic fuzzy controller using fuzzy logic system is proposed in the next section.

Adaptive Synergetic Fuzzy Controller

Since f (x(k)) is unknown, we cannot implement the ideal controller [START_REF] Ruiyun | Stable indirect adaptive control based on discrete-time T-S fuzzy model[END_REF], we suppose that a fuzzy system can approximate f (x(k)). A fuzzy system is a collection of IF-THEN rules in the form :

R (l) : IF x 1 is F l 1 and ... and x n is F l n THEN y is G l (18)
Where x = (x 1 , ..., x n ) T is the input of the fuzzy systems, and y is it's output, F l i and G l are fuzzy sets, for l = 1, ..., m. Using singleton fuzzification method, product inference, and center-average defuzzification, y(x) is given by [START_REF] Wang | Stable adaptive fuzzy control of nonlinear systems[END_REF][START_REF] Kosko | Fuzzy Systems as Universal Approximators[END_REF][START_REF] Kadri | Comparison of fuzzy identification schemes for robust control performance of an adaptive fuzzy controller[END_REF][START_REF] Zaki | Embedded indirect adaptive fuzzy controller based on T-S fuzzy inverse model[END_REF] :

y(x) = m X j=1 y j ⇣ n Y i=1 µ F j i (x i ) ⌘ m X j=1 n Y i=1 µ F j i (x i ) (19)
Where µ F j i (x i ) is the membership function of the linguistic variable x i and y j is the point at which the membership function of G l achieves its maximum value. Introducing the concept of fuzzy basis function vector ⇠(x), y(x) is given by [START_REF] Larguech | Fuzzy Sliding Mode Control for Turbocharged Diesel Engine[END_REF] and can be rewritten as :

y(x) = ✓ T ⇠(x) = ⇠(x) T ✓ (20)
Where

✓ = [y 1 , ..., y m ] T , ⇠(x) = [⇠ 1 (x), ..., ⇠ m (x)]
T are the fuzzy basis functions given by :

⇠ j (x) = n Y i=1 µ F j i (x i ) m X j=1 n Y i=1 µ F j i (x i ) (21)
Using the universal approximation theorem, we can use f ( [START_REF] Ruiyun | Adaptive control schemes for discrete-time T-S fuzzy systems with unknown parameters and actuator failures[END_REF] in order to approximate the function f (x(k)). Therefore, a new control law is obtained :

x(k)|✓ f ) = ✓ T f ⇠ f (x(k)) in the form of
u c (k) = f (x(k)) x 2 (k) + y d (k) + y d (k) d(k) 1 ↵ (k) (22)

Stability Analysis

Theorem : Taking in account the control problem of the nonlinear system (5), with the robust adaptive fuzzy synergetic controller given by ( 22), f is used and the parameter vector ✓ f (k) can be tuned by the adaptive law given in ( 23), the signals in the closed-loop are ultimately bounded and the tracking error converges to neighborhood of origin [START_REF] Wang | Adaptive fuzzy systems and control :design and stability analysis[END_REF][START_REF] Sheng | Robust adaptive fuzzy control of compressor surge using backstepping[END_REF].

✓ f (k) = ⇠(k) (k) ( 2 3 )
Where is a real constant determining the adaptation rate. Let us defining the optimal parameter vector :

✓ ⇤ f = arg min ✓ f 2⌦ ✓ f ( sup x2⌦x f (x(k)) f (x(k)|✓ f ) ) (24)
Where ⌦ ✓ f and ⌦ x are constraint sets for ✓ f and x respectively. The minimum approximation error is as follows :

"(k) = f (x(k)) f (x(k)|✓ ⇤ f ) + d(k) ( 2 5 )
The fuzzy control u(k), is chosen in the closed-loop system as :

u(k) = u c (k) + u r (k) + u (k) = u c (k) + u r (k) ⌧ (k) ( 26
)
Where the term u r (k) is a robust controller which is used to attenuate the external disturbance.

u r (k) = 1 2 h (k) + ⇣ (k) 2 4A 0 (k) ⌘ 1/2 i (k)
With A 0 (k) and (k) to be defined later.

Then substituting ( 22)-( 25) into [START_REF] Sun | Nonlinear Stabilizing Control for Ship-Mounted Cranes With Ship Roll and Heave Movements: Design, Analysis, and Experiments[END_REF], after simple manipulation, we obtain :

(k + 1) = u (k) + u r (k) + ✓T f (k)⇠(k) + "(k) ( 2 7 )
Let ✓f (k) = ✓ ⇤ f ✓ f (k). Now consider the candidate Lyapunov function :

V (k) = 1 2 ⇣ 2 (k) + 1 ✓T f (k 1) ✓f (k 1) ⌘ (28)
V (k + 1) can be calculated as :

V (k + 1) = V (k + 1) V (k) = 1 2 2 (k + 1) 1 2 2 (k) + 1 2 ✓T f (k) ✓f (k) 1 2 ✓T f (k 1) ✓f (k 1) (29)
Let

✓ tf = 1 2 ✓T f (k) ✓f (k) 1 2 ✓T f (k 1) ✓f (k 1) (30)
By using [START_REF] Xiao | A novel single fuzzy approximation based adaptive control for a class of uncertain strict-feedback discrete-time nonlinear systems[END_REF], we can rewrite (29) as :

V (k + 1) = 1 2 2 (k + 1) 1 2 2 (k) + ✓ tf = 1 2 ( (k + 1) + (k)) 2 1 2 2 (k) + ✓ tf = 1 2 2 (k + 1) + (k) (k + 1) + ✓ tf = 1 2 2 (k + 1) + (k) h u (k) + u r (k) + ✓ T f (k)⇠(k) i + ✓ tf V (k + 1) = 1 2 2 (k + 1) + (k)u (k) + (k)u r (k) + (k)✓ T f (k)⇠(k) + ✓ tf (31)
From [START_REF] Xiao | A novel single fuzzy approximation based adaptive control for a class of uncertain strict-feedback discrete-time nonlinear systems[END_REF], ✓ tf can be expressed as :

✓ tf = 1 2 ⇣ ✓T f (k) ✓f (k) h ✓f (k) ✓f (k) i T ⇥ h ✓f (k) ✓f (k) i⌘ ✓ tf = 1 ✓ T f (k) ✓(k) 1 2 ✓T f (k) ✓f (k) (32)
Substituting (32) into (31), yields :

V (k + 1) = 1 2 2 (k + 1) + (k)u (k) + (k)u r (k) + "(k) (k) + ✓T f (k) (k) + ✓ tf = 1 2 2 (k + 1) + (k)u (k) + (k)u r (k) + "(k) (k) + ✓ T f (k) h ⇠(k) (k) 1 ✓f (k) i 1 2 ✓T f (k) ✓f (k) (33)
Using the adaptive law ( 23), we get :

V (k + 1) = 1 2 2 (k + 1) + (k).u (k) + (k).u r (k) + "(k) (k) 1 2 ✓T f (k) ✓f (k) (34)
From ( 27), we have :

| (k + 1)|  |u (k)| + |u r (k)| + | ✓T f (k)⇠(k)| + |"(k)|  s + s f k⇠(k)k + |u r (k)| + s "  |A 0 (k)| + |u r (k)| (35) With A 0 (k) = s + s f k⇠(k)k + s "
Taking the square from both sides of (35), we get :

| (k + 1)| 2  |u r (k)| 2 + 2|A 0 (k)||u r (k)| + 2| (k)||u r (k)| + |A 0 (k)| 2 2 (k)|u r (k)|  h (k) + ⇣ (k) 2 4A 0 (k) ⌘ 1/2 i | (k)| (36) Where (k) = 2(A 0 (k) | (k)|).
Therefore, (34) becomes as follows :

V (k + 1) = "(k) (k) + (k)u (k) 1 2 ✓T f (k) ✓f (k) (37)
Because (k)u (k) < 0, ✓T f (k) ✓f (k) > 0, and based on the universal approximation theorem, the term "(k) (k) is very small. So we have :

V (k + 1)  0

Results and discussions

A simulation is carried out for a real nonlinear discrete-time system model [START_REF] Xiao | A novel single fuzzy approximation based adaptive control for a class of uncertain strict-feedback discrete-time nonlinear systems[END_REF], given in the canonical form of (5).

<

:

x 1 (k + 1) = x 2 (k) x 2 (k + 1) = f (x(k)) + (K/T )u(k) + d(k) y(k) = x 1 (k) (38) Where f (x(k)) = [a 1 x 2 + a 2 x 3
2 (k)]/T represents the system dynamics. y d (k) = sin(k⇡/20) is the trajectory of reference.

d(k) = ( 0, if k  500 0.1 tanh(0.5k), if k > 500 (39)
represents the external disturbance, the initial conditions are chosen as : [x 1 (0), x 2 (0)] T = [0, 0], a 1 = 0.5, a 2 = 30, K = 0.5, T = 64. The membership functions for the system states x i , i = 1, 2 are selected as :

µ(x i ) = e ( 0.5(x i +6 2(j+1)) 2 ) , j = 1, ..., 5, = 10 3 , the sampling period is T s = 0.02s. The simulation is carried out where the time evolutions of the variable x 1 (k) and the trajectory reference y d (k) is shown on figure 1. From this figure, a good tracking performance is obtained, where the total sampling number is 3000. Figure 2 indicates that the proposed adaptive Synergetic Fuzzy Controller u(k) is bounded. The Trajectory of the macro-variable (k) is illustrated in figure 3 which converges to zero as k increases. Figure 4 indicates the time evolutions of the dynamics of f (k) and it's estimate f (k), it is shown that ✓ T f ⇠ f (x(k)) approaches f (k) as k increases, and the estimation error converges to neighborhood of zero. Figure 5 shows the norm of adaptive parameter vector ✓ f (k) which is bounded. Figure 6 illustrate that if we change the dynamic of the system by amont of 10% of its actual value, the tracking performance is upheld. Comparing our results with those of [START_REF] Xiao | A novel single fuzzy approximation based adaptive control for a class of uncertain strict-feedback discrete-time nonlinear systems[END_REF], we can see that better tracking performance is obtained in this paper. The form of the curves in control input of our study is smoother than in [START_REF] Xiao | A novel single fuzzy approximation based adaptive control for a class of uncertain strict-feedback discrete-time nonlinear systems[END_REF]. Thus, in conclusion, all signals in the closed-loop system are bounded, illustrating the e↵ectiveness of the proposed technique.

Conclusion

The work considered in this paper consists of a robust synergetic based adaptive fuzzy controller designed for a class of nonlinear discrete-time systems. Stability analysis is based on Lyapunov theory, we have shown that all the signals in the closed-loop system are bounded and the tracking error is very small. As future work, we will engage to realize a state observer in the case, where the system states are not all available, and we will use the same algorithm for chaotic nonlinear systems such as Henon chaos system. The output trajectories of x 1 (k) and y d (k)

x 1 (k) y d (k) The nonlinear function f(k) and its estimate f(k) The output trajectories of x 1 (k) and y d (k)

f(k) f(k)
x 1 (k) y d (k)

Figure 1 :Figure 2 :

 12 Figure 1: Evolution of output trajectories of x 1 (k) and y d (k).

Figure 3 :

 3 Figure 3: Trajectory of the Macro-variable (k).

Figure 4 :

 4 Figure 4: The nonlinear function f (k) and it's estimation f (k).

Figure 5 :

 5 Figure 5: Norm of ✓ f (k).

Figure 6 :

 6 Figure 6: parametric uncertainties f (k) + f (k).

Acknowledgements

The authors would like to thank the Editor and the anonymous reviewers for their most valuable comments and suggestions. Without these comments, the paper would not be improved to its present quality.