H Delmas
email: henri.delmas@ensiacet.fr

C Creanga

C Julcour-Lebigue

A.-M Wilhelm

AD-OX: A sequential oxidative process for water treatment-adsorption and batch CWAO regeneration of activated carbon

Keywords: Adsorption, Catalytic oxidation, Activated carbon, Water treatment, Multiphase reactions, Packed bed, Mathematical modelling

A sequential process for water treatment involving usual adsorption on activated carbon (AC) followed by wet air catalytic oxidation of the adsorbed pollutants has been carried out in a fixed bed reactor with a mixture of two model pollutants. The first step achieves water purification while the second one reduces the organic pollution but also, more importantly, performs some AC in situ regeneration.

The experimental work has been done with AC yet extensively used and stabilized by long range continuous oxidation. The two steps have been analysed successively showing very important drop of adsorption capacity with respect to fresh AC but efficient oxidative partial regeneration. As expected with used AC no more evolution occurs in between two consecutive runs. The first step of competitive adsorption has been simulated by a model leading to higher diffusivities than estimations based on correlations. The main features of the complex second step, involving simultaneous non isothermal desorption and three phase catalytic reaction, are qualitatively explained.

Introduction

Wastewater treatment is a growing issue for sustainable development. Specially, cheap and reliable processes should be developed to match the future specifications on both the toxic organic compounds and the global Chemical Oxygen Demand (COD). The most usual processes to carry out industrial water treatment are biodegradation and/or physicochemical treatment. To achieve further separation of the less biodegradable organic compounds, adsorption on activated carbon is usually carried out as a final treatment. Nevertheless this powerful technique, with regards to organic pollutant separation, highly suffers from economy and sustainability aspects, as the saturated activated carbon is either hardly regenerated -by non environmental friendly ex situ high temperature processes -or even, most often for industrial wastewater, it becomes itself a solid waste to be incinerated. Therefore new methods have been investigated by researchers for the treatment and recovery of spent activated carbons (ACs). They are mostly based on chemical regeneration, performed either by desorption of fixed pollutants using specific solvents or by decomposition of the adsorbates with oxidizing/reducing agents. Extraction with solvents requires further separation such as distillation to remove the extracted contaminants and recycle the solvents. Advanced oxidation processes (AOPs) using hydrogen peroxide or Fenton's reagent are quite promising technologies for the recovery of used carbon owing to their potency to degrade a wide range of organic pollutants by the generation of very reactive and non selective free hydroxyl radicals [START_REF] Toledo | Application of Fenton's reagent to regenerate activated carbon saturated with organochloro compounds[END_REF][START_REF] Huling | Fenton-driven chemical regeneration of MTBE-spent GAC[END_REF][START_REF] Okawa | Regeneration of granular activated carbon with adsorbed trichloroethylene using wet peroxide oxidation[END_REF]. However they require extra reagents which greatly increase their operating cost. Ozone utilization leads to interactions between ozone and AC which may affect its structural and chemical properties resulting in a decrease of its adsorption capacity [START_REF] Alvarez | Comparison between thermal and ozone regenerations of spent activated carbon exhausted with phenol[END_REF]. Impregnation of the ACs with metal oxides has also been proposed to promote decomposition of adsorbed contaminants by catalytic air oxidation [START_REF] Matatov-Meytal | Abatement of Pollutants by Adsorption and Oxidative Catalytic Regeneration[END_REF][START_REF] Sheintuch | Comparison of catalytic processes with other regeneration methods of activated carbon[END_REF]. Recently commercial AC has been successfully applied as a direct catalyst in Catalytic Wet Air Oxidation (CWAO) for the destruction of phenolic compounds [START_REF] Fortuny | Wet air oxidation of phenol using active carbon as catalyst[END_REF][START_REF] Eftaxias | Catalytic wet air oxidation of phenol over active carbon catalyst. Global kinetic modelling using simulated annealing[END_REF][START_REF] Suwanprasop | Scaleup and modeling of fixed bed reactors for the catalytic phenol oxidation over adsorptive active carbon[END_REF][START_REF] Suarez-Ojeda | Catalytic wet air oxidation of substituted phenols using active carbon catalyst[END_REF][START_REF] Santos | Catalytic wet air oxidation of phenol on active carbon: stability, phenol conversion and mineralization[END_REF]. AC can even perform better than supported catalysts based on the transition metals [START_REF] Fortuny | Wet air oxidation of phenol using active carbon as catalyst[END_REF], probably due to its high adsorption capacity combined to oxygen-containing surface groups. In this work a sequential process [START_REF] Delmas | Sequential process of adsorption and catalytic oxidation on activated carbon for (pre)treatment of water polluted by non-biodegradable organic products[END_REF][START_REF] Polaert | Phenol wastewater treatment by a two-step adsorption-oxidation process on activated carbon[END_REF] for (post)treatment of water polluted by non biodegradable organic compounds has been investigated. It is based on hybridizing classical adsorption on a fixed bed of activated carbon followed by batch wet catalytic oxidation at higher temperature and pressure on the same bed of AC, which is then regenerated in situ. The basic idea is to take advantage of both operations: 1-efficient and cheap water purification at room temperature by adsorption, 2-effective pollutant degradation by batch air oxidation achieving simultaneously some AC regeneration for the next adsorption. Compared to usual continuous oxidation [START_REF] Stüber | Carbon materials and catalytic wet air oxidation of organic pollutants in wastewater[END_REF] only a small fraction of the wastewater to be treated has to be heated in this batch CWAO and very high conversion is no longer required as water quality is provided by the adsorption step. These advantages are more valuable at low pollutant concentration, i.e. after conventional biological treatment. When comparing now with usual adsorption with spent AC significant AC savings are expected. This new process should be especially beneficial in the highest concentration range of usual adsorption and/or in the lower concentration range of continuous CWAO process.

In most academic works dealing with oxidative water treatments, model pollutants have been used for analytical and scientific reasons in order to better quantify the process efficiency and its possible by-products. Among the model pollutants phenol is undoubtedly the most popular one [START_REF] Pintar | Catalytic liquid-phase oxidation of phenol aqueous solutions. A kinetic investigation[END_REF][START_REF] Stüber | Catalytic wet air oxidation of phenol using active carbon: performance of discontinuous and continuous reactors[END_REF]; nevertheless a few papers have investigated the catalytic oxidation of substituted phenols: 2-and 4-chlorophenol and 4-nitrophenol on supported metal oxides [START_REF] Pintar | Catalytic oxidation of aqueous p-chlorophenol and p-nitrophenol solutions[END_REF][START_REF] Posada | Catalytic wet air oxidation of aqueous solutions of substituted phenols[END_REF], 2aminophenol, salicylic acid, 5-sulfo salicylic acid, nitrophenols, cresols and chlorophenols on AC [START_REF] Suarez-Ojeda | Catalytic wet air oxidation of substituted phenols using active carbon catalyst[END_REF][START_REF] Tukac | Catalytic wet oxidation of substituted phenols in the trickle bed reactor[END_REF][START_REF] Santos | Wet oxidation of phenol, cresols and nitrophenols catalyzed by activated carbon in acid and basic media[END_REF], 2-chlorophenol, 4-coumaric, 4-hydroxyphenylacetic and 4-hydroxybenzoic acids over Pt and Ru supported catalysts [START_REF] Kojima | Catalytic wet oxidation of o-chlorophenol at mild temperatures under alkaline conditions[END_REF][START_REF] Li | Catalytic wet air oxidation of aqueous solution of 2chlorophenol over Ru/zirconia catalysts[END_REF][START_REF] Pham Minh | Degradation of olive oil mill effluents by catalytic wet air oxidation: 1-Reactivity of p-coumaric acid over Pt and Ru supported catalysts[END_REF][START_REF] Pham Minh | Degradation of olive oil mill effluents by catalytic wet air oxidation: 2-Oxidation of p-hydroxyphenylacetic and phydroxybenzoic acids over Pt and Ru supported catalysts[END_REF]. 4-hydroxybenzoic acid (4HBA) is of special interest as typically found in oxidative treatments of wastes of olive oil industry [START_REF] Beltran-Heredia | Comparison of the degradation of p-hydroxybenzoic acid in aqueous solution by several oxidation processes[END_REF]. It is considered especially toxic and refractory to usual wastewater biological treatment [START_REF] Gonzalez | Studies on antibacterial activity of waste waters from olive oil mills (alpechin): inhibitory activity of phenolic and fatty acids[END_REF]. It is also an unexpected intermediate product of phenol oxidation on AC [START_REF] Eftaxias | Catalytic wet air oxidation of phenol over active carbon catalyst. Global kinetic modelling using simulated annealing[END_REF][START_REF] Suwanprasop | Scaleup and modeling of fixed bed reactors for the catalytic phenol oxidation over adsorptive active carbon[END_REF]. Most often wastewaters are mixtures of several pollutants. To investigate the feasibility of the AD-OX process with complex mixtures, a model mixture of phenol and 4HBA has been treated in this work, following previous studies on batch and continuous CWAO of phenol [START_REF] Suwanprasop | Scaleup and modeling of fixed bed reactors for the catalytic phenol oxidation over adsorptive active carbon[END_REF] and 4-hydroxybenzoic acid [START_REF] Creanga Manole | Catalytic oxidation of 4-hydroxybenzoic acid on activated carbon in batch autoclave and fixed bed reactor[END_REF][START_REF] Creanga Manole | Catalytic wet air oxidation of aqueous organic mixtures[END_REF] on activated carbon.

Experimental

The experimental set-up (Figure 1) has been derived from a continuous CWAO equipment, yet detailed in Suwanprasop et al. [START_REF] Suwanprasop | Scaleup and modeling of fixed bed reactors for the catalytic phenol oxidation over adsorptive active carbon[END_REF]. The main change concerns the liquid flow which may successively enter and leave the reactor, during the adsorption step, or be recycled through a batch loop including a pressurized tank (3), during the heating, reaction and cooling steps.

The fixed bed reactor (1) consists of a jacketed column of 1.2 m high and 0.025 m internal diameter. It is packed with 0.325 kg of activated carbon particles (Merck 2514, [1.25-1.60]⋅10 -3 m sieved fraction). Note that the experiments have not been carried with fresh activated carbon. In order to avoid an unknown period of AC stabilization, this work was indeed achieved with aged AC, yet extensively used in the same fixed bed (similar temperature and pressure conditions, same model pollutants) for many continuous oxidation runs, thus undergoing continuous simultaneous adsorption -oxidation for more than 300hrs. It has been verified that this used AC has reached a quasi steady state by repeating experiments in reference conditions [START_REF] Creanga Manole | Catalytic wet air oxidation of aqueous organic mixtures[END_REF]. It has also been estimated that AC consumption by direct oxidation is very limited as discussed below. The weight of the aged AC used in this work was 20% higher than the fresh AC introduced in the reactor before preliminary continuous oxidation runs. On another hand the weight loss during TGA experiments was approximately 17% between 100°C and 700°C on aged AC (20% heavier) and 1.5% on fresh AC. So the TGA weight loss difference (about 19%) is approximately equal to the weight increase from fresh to aged AC, suggesting that very few carbon has been consumed during all the oxidation runs (continuous and batch at the end).

Nine temperature probes (T) and eight sample valves [START_REF] Suarez-Ojeda | Catalytic wet air oxidation of substituted phenols using active carbon catalyst[END_REF] are located at different reactor heights to get axial profiles of temperature and liquid concentrations respectively. Additionally, liquid sampling is done after the gas-liquid separator. The reactor can operate with either down flow or up flow of gas-liquid according to the position of five three way valves (V1 to V5) located along the gas and liquid circuits. A flexible grid is put at the top of the reactor to prevent fluidization of particles during the up flow operation. Here only up flow has been used. During the adsorption step the solution of equimolar mixture of phenol and 4-hydroxybenzoic acid (25.8 mol.m -3 each) is fed to the column at atmospheric pressure by a dosing pump (8) from a weighted tank [START_REF] Sheintuch | Comparison of catalytic processes with other regeneration methods of activated carbon[END_REF]. During the batch reaction step the same pump is used to recycle the reaction mixture at higher temperature and pressure through the pressurized tank (3) and gas supply is achieved by two mass flow-controllers allowing different gas mixtures. The wall temperature is set at 413 K and the total operating pressure at 6 bar. Gas flow rate is 2.78⋅10 -5 m 3 .s -1 (NTP), corresponding to an inlet gas velocity of about 10 -2 m.s -1 . By using a 85/15 composition of the O 2 /N 2 inlet mixture, oxygen partial pressure is adjusted at 2 bar (equilibrium pressure at reactor outlet). Liquid space time varies from 0.1 to 0.7 h (liquid velocities of 0.25⋅10 -3 to 1.2⋅10 -3 m.s -1 ). These conditions correspond to the bubble flow regime or transition to pulsed flow regime in the selected up flow mode. During the two steps, liquid samples are regularly withdrawn at the bed outlet. They are analyzed by HPLC using a C18 reverse phase column (ProntoSIL C18 AQ) and a dual wavelength UV detector (UV2000 Thermo Finnigan) to obtain concentrations of phenol and 4HBA. The separation of those compounds from the oxidation intermediates is achieved with a mobile phase of variable composition programmed at 1 mL/min, using acidified deionised water and acetonitrile as eluants. Three successive cycles -adsorption steps at room temperature, followed by heating and pressurization, then batch oxidation with liquid recycle, and final cooling prior to next adsorption -have been performed for the phenol-4HBA mixture at different liquid flow rates.

Results and discussion

Adsorption.

In this work, as shown on breakthrough curves (Figure 2), the adsorption step has been achieved up to complete AC saturation, contrary to a true AD-OX operation where the polluted water fed should be stopped when the pollutant concentration reaches the accepted threshold. Indeed the complete saturation provides a better knowledge of the adsorption performance of this used activated carbon and allows the comparison with simulations of the adsorption step, while in the actual AD-OX process this step should stop as soon as the pollutant concentration reaches the accepted limit. First it should be noticed that the two consecutive experiments give very reproducible breakthrough curves, which confirms the stabilization of the activated carbon both as adsorbent and as oxidation catalyst. It should be remembered that this stable behavior may only be observed on used catalyst while fresh activated carbon undergoes a severe drop of adsorption capacity after the first oxidation [START_REF] Creanga Manole | Catalytic oxidation of 4-hydroxybenzoic acid on activated carbon in batch autoclave and fixed bed reactor[END_REF]. Concerning adsorption selectivity, phenol adsorbs faster but is progressively removed by 4HBA in agreement with competitive adsorption data showing preferential 4HBA adsorption [START_REF] Manole | AD-OX Process for remediation of non biodegradable organic pollutants (by adsorption then catalytic oxydation)[END_REF]. This selectivity can be explained by the much lower solubility of 4HBA in water (about 20 times less than phenol). The dynamics of this competitive adsorption has been simulated by a model including adsorption equilibria (isotherms), pore diffusion, liquid-solid mass transfer and liquid axial dispersion through series of CSTR with the following assumptions:

-Surface diffusion in pores is not taken into account (only usual pore diffusion).

-External liquid-solid mass transfer coefficient is calculated by Ranz and Levenspiel correlation [START_REF] Kunii | Fluidization Engineering[END_REF].

-The number of CSTR in series is calculated with the Bodenstein Number (or particle Peclet Number) of 0.5 as usually accepted. Here, it gives 241 tanks in series.

-Adsorption is very fast: at any depth in pores local adsorption equilibrium is assumed. This equilibrium is represented by the two simplified Langmuir adsorption isotherms (eq. 1 and 2 where A stands for 4HBA and B for phenol):

B B L A A L A A L A C K C K C K q q ⋅ + ⋅ + ⋅ ⋅ = , , , max 1 (1) B B L A A L B B L B C K C K C K q q ⋅ + ⋅ + ⋅ ⋅ = , , , max 1 (2)
which have been proved convenient for this competitive adsorption on the fresh Merck activated carbon. In this model the initial conditions assume no remaining phenol and 4HBA on the activated carbon though the previous batch oxidation cannot be completely achieved (it is stopped when the concentration of pollutant in the liquid phase reaches 10% or less of initial feed concentration, cf. Figure 5).

The local mass balances inside a spherical AC particle of the i th tank gives

        ∂ ∂ ⋅ ∂ ∂ ⋅ = ∂ ∂ ⋅ + ∂ ∂ ⋅ r C r r r D t q t C i A A e i A p i A p , 2 2 , , , ' ' ' ρ ε (3)         ∂ ∂ ⋅ ∂ ∂ ⋅ = ∂ ∂ ⋅ + ∂ ∂ ⋅ r C r r r D t q t C i B B e i B p i B p , 2 2 , , , ' ' ' ρ ε (4)
ε' p et ρ' p are the porosity and apparent density of aged AC (ε' p = 0.31; ρ' p = 1251 kg.m -3 ) which are found significantly different from that of fresh AC (ε p = 0.59; ρ p = 1032 kg.m -3 ).

Those properties have been calculated from pore volume and structural density measured with nitrogen sorption analyzer and helium pycnometer respectively.

The effective pore diffusivity of compound j (D' e , j ) is obtained from its molecular diffusivity in water (D m,j ) according to:

j m p p j e D D , , ' ' τ ε = (5)
with τ p the pore tortuosity of the catalyst particles.

Boundary conditions

∀ t, r = 0, 0 r C i A, = ∂ ∂ (spherical symmetry) (6) 0 r C i B, = ∂ ∂ (7) t, r = d p /2, ( ) 2 / , , , , , ' p d r i A A e i SA i LA A LS r C D C C k =         ∂ ∂ ⋅ = - ⋅ (8) ( ) 2 / , , , , , ' p d r i B B e i SB i LB B LS r C D C C k =         ∂ ∂ ⋅ = - ⋅ (9)
The mass balance on the liquid phase (out of particles) of the i th tank gives:

( ) ( ) ( ) 1 , , , , , , 1 6 -
- ⋅ - ⋅ - ⋅ - ⋅ ⋅ = ⋅ ⋅ i LA i LA L cell R i LA i SA B p A LS i LA cell R B C C Q N V C C d k dt dC N V ε ε (10) ( ) ( ) ( ) 1 , , , , , , 1 6 -
- ⋅ - ⋅ - ⋅ - ⋅ ⋅ = ⋅ ⋅ i LB i LB L cell R i LB i SB B p B LS i LB cell R B C C Q N V C C d k dt dC N V ε ε (11)
A four point orthogonal collocation method has been used to approximate spatial derivatives (inside the pellet) and the resulting system of equations has been solved using the Gear method. A Gauss Newton technique for optimizing the unknown (isotherm) parameters by matching the exit concentration profiles of the two adsorbed species is then carried out.

Concerning the result analysis the main point is that the model cannot fit the experimental breakthrough curves when using effective pore diffusivities based on Wilke-Chang correlation for molecular diffusivity of 4HBA and phenol in water, and usual values of tortuosity factor (τ p = 2-3). It should be noticed that the same problem occurred when trying to simulate the dynamics of batch adsorption experiments of fresh AC in stirred vessel [START_REF] Manole | AD-OX Process for remediation of non biodegradable organic pollutants (by adsorption then catalytic oxydation)[END_REF]. In both cases effective diffusivities should be significantly increased. Here diffusivities are thus also optimized, keeping the 4HBA/phenol diffusivity ratio as obtained from Wilke-Chang correlation. All optimized data are presented in Table 1.

Optimization tests have proved the three adsorption parameters to be less sensitive than diffusivity, here varied through the pore tortuosity factor τ p while keeping Wilke-Chang correlation. Figure 3 shows the best fit of the model with the breakthrough curves, with faster initial adsorption of phenol due to higher diffusivity, followed by its removal due to higher 4HBA affinity to AC. The best fit being obtained with τ p = 0.64 means that optimized diffusivities are at least three times higher than expected ones. This significant disagreement may be attributed not only to this questionable correlation but even more to the particle size distribution and measurement and to the spherical particle hypothesis, as well as to the possible effect of surface diffusion.

An important and negative result should be pointed out: this used AC has lost a large part of its potential adsorption capacity as it has been divided by about 6 compared to the same fresh AC. Indeed according to co-adsorption results using different initial proportions of the two pollutants and modelling of the corresponding data by a generalized Langmuir isotherm, the theoretical adsorption capacity of fresh AC in equilibrium with feed concentrations should be q e,phenol = 0.17 mol.kg -1 AC and q e,4HBA = 2.09 mol.kg -1 AC [START_REF] Manole | AD-OX Process for remediation of non biodegradable organic pollutants (by adsorption then catalytic oxydation)[END_REF]. The lower adsorption capacity of aged AC (cf. Table 1) is due to accumulation of heavy polyphenols irreversibly adsorbed on AC and hardly oxidized later, resulting in a significant loss of BET surface area, from 980 m 2 /g to 250 m 2 /g. Similar drop of adsorption capacity has been already observed on various ACs using various aromatic pollutants even after much less oxidation time. Successive adsorption and oxidation runs in autoclave reactor, starting with fresh AC, suggest that the main adsorption drop occurs after the first oxidation followed by quasi steady behaviour both for oxidation and adsorption.

Finally the optimized values not only match reasonably well with the breakthrough curves but also with adsorption equilibrium data of 4HBA measured on this aged AC after it has been washed with bidistilled water during several days -so that to eliminate all reversibly-adsorbed components -and dried (Figure 4).

Oxidation

Starting from complete AC saturation, batch oxidations were performed, at p O2 = 2 bar and T = 413 K, and at two different liquid flow rates (5.56⋅10 -7 and 1.39⋅10 -7 m 3 .s -1 ). Batch oxidation curves are shown in Figure 5. Due to significant initial desorption at higher temperature (413 K instead of 298 K, approximately as on Fig. 4, where this temperature effect is shown for fresh AC) the concentration profiles start at higher concentrations than 25.8 mol.m -3 , the equimolar concentration of pollutants. This batch CWAO being carried out during three days with stops by night, some peaks are found when starting heating again (e.g at 550 min) due to subsequent desorption. A clear understanding of the two concentration variations with time, based on a complete modeling of all phenomena is not yet possible contrary to continuous CWAO.

First, it should be recalled that liquid concentrations vary due to two main reasons: oxidation and desorption. Due to the relative high solid loading of this kind of batch recycle reactor, the two compounds are mostly adsorbed on AC and only a small fraction appears in the aqueous phase. Adsorption isotherms for the mixture on this particular used AC at high temperature are not available. Nevertheless they have been performed in stirred autoclave on the same Merck AC but being stabilized after several oxidation runs in the autoclave using thus a different liquid/solid ratio than in the fixed bed [START_REF] Creanga Manole | Catalytic wet air oxidation of aqueous organic mixtures[END_REF].

Second, intrinsic kinetics have been obtained again with slightly differently aged ACs for both 4HBA and 4HBA-phenol mixture [START_REF] Creanga Manole | Catalytic wet air oxidation of aqueous organic mixtures[END_REF]. This paper enlightens a commonly ignored effect of adsorption on catalytic reaction kinetics determination from batch measurements: the mass balance should take into account the variation of the adsorbed reactant content on the catalyst and not only the concentration variations in the liquid phase. It was shown that intrinsic kinetics derived when ignoring this variable adsorbed content may lead to large errors concerning the rate constants. Here due to the very high solid loading this adsorption effect is predominant. So, in this AD-OX oxidation step, the much faster phenol concentration decrease during oxidation is due to lower adsorbed amount and not to intrinsic kinetics, which proved phenol and 4HBA to be very similarly oxidized [START_REF] Creanga Manole | Catalytic wet air oxidation of aqueous organic mixtures[END_REF]. Faster oxidation at low liquid flow rate may appear surprising. Here again a complete explanation through a model would be useful. Nevertheless it should be pointed out that low liquid recycle flow corresponds to higher temperature profile in the fixed bed reactor, as presented on Figure 6, showing axial temperature profiles when steady state has been reached, while Figure 7 shows the temperature transients after starting heating. This steeper temperature profile at low liquid flow rate is likely the most relevant explanation of the better performance of the oxidation step in this condition: a larger part of the bed is catalytically active.

Conclusions

A sequential oxidation-adsorption process using a commercial activated carbon without pretreatment and operating in mild conditions has been applied for the remediation of a pollutant mixture. Due to complex simultaneous non isothermal oxidation and desorption phenomena on aged AC the oxidative regeneration step cannot be simulated, contrary to continuous oxidation [START_REF] Suwanprasop | Scaleup and modeling of fixed bed reactors for the catalytic phenol oxidation over adsorptive active carbon[END_REF][START_REF] Manole | AD-OX Process for remediation of non biodegradable organic pollutants (by adsorption then catalytic oxydation)[END_REF] and to isothermal adsorption. Nevertheless the main features may be qualitatively understood.

Compared to fresh AC the adsorption capacity with this used AC has been dramatically decreased to about 15%. Nevertheless this limited adsorption capacity may be fully recovered by batch CWAO, AC playing then the role of catalyst of its own regeneration. In the mini pilot plant previously built for continuous CWAO the adsorption time is extremely low (1/2 h) compared to the regeneration time (25 h). A convenient plant to investigate this process on actual industrial wastewater should then have a much higher AC loading to increase the adsorption time while the batch reaction time is constant. For example to have similar adsorption and heating-oxidation-cooling times the AC weight should be fifty to hundredfold. A two parallel column plant with alternative adsorption -regeneration steps should then be operated to carry out continuous water treatment.

Notation

C j concentration of compound j (mol.m -3 ) C Lj concentration of compound j in the bulk liquid phase (mol.m -3 ) C Sj liquid-phase concentration of compound j on the catalyst surface (mol.m -3 ) D' e,j effective diffusion coefficient of compound j in aged AC pores (m 2 .s -1 ) D m,j molecular diffusivity of compound j in water (m 2 .s -1 ) d p catalyst particle diameter (m) K L,j isotherm constant of compound j (m 3 .mol -1 ) k LS,j j-compound liquid -solid mass transfer coefficient (m.s -1 ) N cell number of CSTR in series p O2 oxygen partial pressure (bar) q e,j adsorbed amount of compound j on AC in equilibrium with feed concentration (mol.kg -1 AC ) q j adsorbed amount of compound j on AC (mol.kg -1 AC ) Q L liquid flow rate (m 3 .s -1 ) q max maximum adsorption capacity of AC (mol.kg -

FIGURE CAPTIONS Figure 1 .

 CAPTIONS1 FIGURE CAPTIONS Figure 1. AD-OX experimental set up. 1 jacketed packed bed column, 2 gas-liquid separator, 3 pressurized liquid storage tank, 4 condenser, 5 gas massflow controllers, 6 feed tank, 7 balance, 8 dosing pump, 9 sampling device, 10 liquid sample valves, 11 pneumatic valve, 12 expansion vase, 13 gear pump, 14 heater, 15 cooling exchanger, 16* cooling exchanger, 17* anti-pulsatory vessel, 18* flowmeter, V1-V5 three-way valves for up-or down-flow mode.

Figure 2 .

 2 Figure 2.Successive breakthrough curves with equimolar phenol-4HBA mixture (Q L =8.33⋅10 -7 m 3 .s -1 , T=298K). 1 st breakthrough: 4HBA ( ) and phenol ( ) outlet concentrations; 2 nd breakthrough: 4HBA ( ) and phenol ( ) outlet concentrations.

Figure 3 .

 3 Figure 3. Comparison of experimental and simulated breakthrough curves. Dots: experimental 4HBA ( ) and phenol ( ) outlet concentrations; lines: simulated breakthrough curves for 4HBA (dashed) and phenol (continuous).

Figure 4 .

 4 Figure 4. Adsorption isotherms of 4HBA on fresh AC at 298 K ( ) and 423 K ( ) and on aged AC at 298 K ( ) [continuous lines: Langmuir models for fresh AC, dashed line: fitted generalized Langmuir model (C e,phenol = 0) for aged AC].

Figure 5 .

 5 Figure 5. Evolution of 4HBA ( ) and phenol ( ) concentration during batch CWAO at two liquid recycle flow rates: Q L = 5.56⋅10 -7 m 3 .s -1 (gray filled symbols) and Q L = 1.39⋅10 -7 m 3 .s -1 (black filled symbols) [t=0 corresponds to the introduction of O 2 /N 2 mixture].

Figure 6 .

 6 Figure 6. Steady state axial temperature profiles at two liquid recycle flow rates: Q L = 5.56⋅10 -7 m 3 .s -1 ( ) and Q L = 1.39⋅10 -7 m 3 .s -1 ( ).

Figure 7 .

 7 Figure 7. Time evolution of temperature at different axial positions after starting heating: Z=0 ( ), Z=0.12 m ( ), Z=0.36 m ( ), Z=0.6 m ( ), Z=0.84 m ( ), Z=1.08 m ( ), Z=1.2 m ( ) [Q L = 1.39⋅10 -7 m 3 .s -1 ].

TABLE CAPTIONS Table 1 .

 CAPTIONS1 Optimized parameters of generalized Langmuir adsorption of 4HBA -phenol and corresponding adsorption capacities at equilibrium with the equimolar solution.

Table 1

 1

	q max		K L,phenol	K L,4HBA		q e,phenol	q e,4HBA
	(mol.kg -1	AC )	(m 3 .mol -1 )	(m 3 .mol -1 )	τ p	(mol.kg -1	AC )	(mol.kg -1	AC )
	0.65		0.013	0.039	0.64	0.092		0.28

Acknowledgements

EC through the 6 th FP "REMOVALS" and ANR-Precodd "PHARE" are acknowledged.


Alexandre Seuret
email: aseuret@laas.fr

João M Gomes
email: jmgomes@ece.ufrgs.br

Da Silva

Taking into account period variations and actuators saturation in sampled-data systems

Keywords: Linear systems, saturation, sampled-data systems, aperiodic sampling

This paper deals with the problem of stability and stabilization of sampled-data systems under asynchronous samplings and actuators saturation. The method is based, on the first hand, on the use of a novel class of Lyapunov functionals whose derivative is negative along the trajectories of the continuoustime model of the sampled data system. It is shown that this fact guarantees that a quadratic Lyapunov function is strictly decreasing for the discrete-time asynchronous system. On the other side, the control saturation is taken into account from the use of a modified sector condition. These ingredients lead to the formulation of improved LMI conditions that can be cast in optimization problems aiming at enlarging estimates of the region of attraction of the closed-loop system or maximizing the bounds on the sampling period jitter for which stability and stabilization are ensured.

Introduction

In the past decade, a large attention has been devoted to sampled-data systems [START_REF] Hespanha | A survey of recent results in networked control systems[END_REF]. This area addresses the problem of stability and stabilization of systems evolving in continuous-time whereas the controller delivers control inputs in discrete-time. Although the theory for linear sampled-data control systems is rather well established for the case of a constant sampling period [START_REF] Åstrom | Computer Controlled Systems[END_REF], the problem of considering asynchronous sampling is an open domain of research. In particular, the interest to study sample-data systems has increased with the spreading of the so-called networked control systems [START_REF] Zampieri | Trends in networked control systems[END_REF]. In such applications several distributed plants are controlled over a communication network and the controllers can be implemented in a decentralized way (i.e. in local processors). In this case, communication delays, package losses and heavy temporary load of computation in a processor can lead to significant variations on the sampling time. These variations can dramatically affect the stability properties of the control system. It is then clear the motivation for developing robust stability and stabilization conditions which take into account the variations on the sampling period.

The sampling in a control loop can be modeled by a time-varying delay on the plant control input while considering a continuous-time dynamics. In this case, the delay variation rate between two sampling instants is equal to one. Thus, a fundamental problem regards the determination of the bounds on the delay variation (corresponding to the sampling period jitter), for which stability of the closed-loop system can be kept. Considering this perspective, we can cite for example the approach proposed in [START_REF] Fridman | Robust sampled-data stabilization of linear systems: An input delay approach[END_REF], and the improvements made in [START_REF] Naghshtabrizi | Exponential stability of impulsive systems with application to uncertain sampled-data systems[END_REF] and [START_REF] Fridman | A refined input delay approach to sampled-data control[END_REF].

On the other side, many techniques have been proposed in the literature to deal with the stabilization of time-delay systems subject to actuator saturation. In this context, we can cite, for instance: [START_REF] Niculescu | Robust stabilization for uncertain time-delay systems containing saturating actuators[END_REF] and [START_REF] Oucheriah | Global stabilization of a class of linear continuous time delay systems with saturating controls[END_REF], where globally stabilizing control laws are proposed; and [START_REF] Tarbouriech | Synthesis of controllers for continuous-time delay systems with saturating controls via LMIs[END_REF], [START_REF] Cao | Stability analysis of linear time-delay systems subject to input saturation[END_REF] and [START_REF] Fridman | Robust sampled-data stabilization of linear systems: An input delay approach[END_REF], where the regional stabilization problem is considered. These works are mainly concerned by state delayed systems. The stabilization conditions consider delay independent approaches (which allow to address time-varying delays, but in a possible conservative way) and delay dependent conditions considering fixed delays. Recently, in [START_REF] Gomes Da | Stabilisation of neutral systems with saturating control inputs[END_REF] and [START_REF] Zhang | Delay-range-dependent control synthesis for time-delay systems with actuator saturation[END_REF], results considering systems presenting time-varying delays on the states have been proposed. On the other hand, we can note a lack of results considering input delays, and, in particular, time-varying ones. In [START_REF] Gomes Da | Anti-windup design with guaranteed regions of stability: an LMI-based approach[END_REF] and [START_REF] Tarbouriech | Delay-dependent anti-windup loops for enlarging the stability region of time-delay systems with saturating inputs[END_REF], anti-windup techniques are proposed for systems with fixed input delays.

In this paper we are interested in the problem of stabilizing a sampled-data system taking into account the possible variations on the sample period (due to packet losses in the network) and also the fact that the signals provided by the actuators are bounded (i.e saturating plant inputs). The method is based on the use of a particular functional that, differently from the Lyaunov-Krasovskii based approaches adopted for instance in [START_REF] Fridman | Robust sampled-data stabilization of linear systems: An input delay approach[END_REF], [START_REF] Naghshtabrizi | Exponential stability of impulsive systems with application to uncertain sampled-data systems[END_REF] and [START_REF] Fridman | A refined input delay approach to sampled-data control[END_REF], does not need to be positive definite. It is shown that if the time-derivative of this functional along the trajectories of the continuous-time model is strictly negative, then a quadratic Lyapunov function is strictly decreasing for the discrete-time asynchronous system. On the other hand, the control saturation is taken into account from the use of a generalized sector condition. These ingredients lead to the formulation of LMI conditions that can be cast in optimization problems aiming at enlarging estimates of the region of attraction of the closed-loop system or maximizing the bounds on sampling period jitter for which the stability is kept.

The paper is organized as follows. The next section describes the problem formulation. Section 3 presents some preliminary lemmas on a LMI manipulation, on the generalized sector condition. Section 5 provides a generic result to ensure the stability of sample-data systems with constrained control signals. Based on this result, LMI-based stabilization conditions allowing to compute a state feedback gain in order to guarantee the regional (local) or the global stability of the closed-loop sample data system in the presence of actuator saturation is provided in section 5. In section 6, some optimization problems are proposed. Some numerical examples illustrating the potentialities of the method are provided in section 7.

Notation. Throughout the article, the sets N, R + , R n , R n×n and S n denote respectively the set of positive integers, positive scalars, n-dimensional vectors, n × n matrices and symmetric matrices of R n×n . For a given positive scalar, T 2 , define K as the set of continuous functions from an interval [0, T ] to R n , where T is a positive scalar less than T 2 . The notations | • | and • stand for the absolute value of a scalar and for the Euclidean norm of a vector, respectively. The superscript ' T ' stands for matrix transposition. The notation P > 0 for P ∈ S n means that P is positive definite. For any positive integer j ≤ n any vector x ∈ R n and any matrix A ∈ R n×n , the notations A j , x j and He{A} > 0 refer to the j th line of matrix A, the j th component of vector x and A + A T > 0, respectively. The symbols I and 0 represent the identity and the zero matrices of appropriate dimension. Co{•} denotes a convex hull.

Problem formulation

Let {t k } k∈N be an increasing sequence of positive scalars such that k∈N [t k , t k+1 [= [0, + ∞[. Assume that there exist two positive scalars T 1 ≤ T 2 such that the difference between two successive sampling instants

T k = t k+1 -t k satisfies ∀k ∈ N, 0 ≤ T 1 ≤ T k ≤ T 2 . ( 1
)
Consider the linear system with a sampled-data input

∀t ∈ [t k , t k+1 [, ẋ(t) = Ax(t) + Bu(t k ), (2)
where x ∈ R n and u ∈ R m represent the state variable and the input vector. The matrices A and B are assumed to be constant, known and of appropriate dimension. We suppose that the input vector u is subject to amplitude limitations defined by:

|u i | ≤ u 0i , u 0i > 0, i = 1, ..., m. ( 3
)
Consider a linear state feedback control law u(t) = Kx(t), where K ∈ R m×n . Due to the control bounds defined in (3), the effective control signal to be applied to the system is given by

u(t) = sat(Kx(t k )), t k ≤ t < t k+1 , ( 4
)
with u i (t) = sat(K i x(t k )) = sign(K i x(t k ))) min{u 0i , |K i x(t k )|}, i = 1, . . . , m. Hence, the closed-loop system reads ẋ(t) = Ax(t) + Bsat(Kx(t k )). (5)
As noticed in [START_REF] Yamamoto | A function space approach to sampled data control systems and tracking problems[END_REF], integrating the previous differential equation over a sampling period, the dynamics of the system satisfy ∀τ ∈ [0,

T k ], x(t k + τ ) = Γ(τ )x(t k ) + Υ(τ )sat(Kx(t k )), Γ(τ ) = e Aτ , Υ(τ ) = τ 0 e A(τ -θ) dθB. (6)
In particular, if the sampling period is constant, i.e. T k = T ∀k ∈ N, the dynamics become

x(t k+1 ) = Γ(T )x(t k ) + Υ(T )sat(Kx(t k )),
and stability of the closed-loop system can be analyzed by the techniques proposed, for instance in [START_REF] Gomes Da | Local stabilization of discrete-time linear systems with saturating controls: an lmi-based approach[END_REF]. Note that in the absence of the control saturation, it suffices that matrix (Γ(T ) + Υ(T )K) has all eigenvalues inside the unit circle to ensure the asymptotic stability of the closed-loop system. On the other hand, if the sampling period is time-varying, this does not hold anymore. Relevant stability analysis based on an uncertain representation of (Γ(T k ) + Υ(T k )K) (without saturations) have been already investigated for instance in [START_REF] Hetel | Robust stability and control of switched linear systems[END_REF][START_REF] Oishi | Stability and stabilization of aperiodic sampled-data control systems: An approach using robust linear matrix inequalities[END_REF][START_REF] Suh | Stability and stabilization of nonuniform sampling systems[END_REF]. However, extensions to systems with saturated control input lead to additional difficulties due to the definition of Γ(T k ) and Υ(T k ).

Concerning the analysis based directly on the continuous-time model [START_REF] Cao | Stability analysis of linear time-delay systems subject to input saturation[END_REF], sufficient stability conditions were firstly designed using the Lyapunov-Kravovskii theorem for time-delay systems in [START_REF] Fridman | Robust sampled-data stabilization of linear systems: An input delay approach[END_REF] and refined in [START_REF] Naghshtabrizi | Exponential stability of impulsive systems with application to uncertain sampled-data systems[END_REF][START_REF] Mirkin | Some remarks on the use of time-varying delay to model sample-and-hold circuits[END_REF][START_REF] Fridman | A refined input delay approach to sampled-data control[END_REF][START_REF] Seuret | Stability analysis for sampled-data systems with a time-varying period[END_REF]. This approach consists in modeling the sampling effect as an input varying delay and to apply results issued from the time-delay systems theory to derive stability criteria expressed with respect to the continuous-time model. However, this approach is still conservative in comparison to discrete-time approaches.

Although the system dynamics are considered to be linear, due to the control saturation, the closed-loop system is nonlinear. Hence, the determination of a global stabilizing controller is possible only when some stability assumptions are verified by the open-loop system (u(t) = 0)(see [START_REF] Lin | Semi-global exponential stabilization of linear systems subject to input saturation via linear feedback[END_REF][START_REF] Sussmann | A general result on the stabilization of linear systems using bounded controls[END_REF]). When this hypothesis is not verified, it is only possible to achieve semi-global or local/regional stabilization. In this case, given a stabilizing matrix K, we associate a region of attraction to the equilibrium point x e (t) ≡ 0 of the system [START_REF] Fridman | Robust sampled-data stabilization of linear systems: An input delay approach[END_REF]. The region of attraction corresponds to all initial conditions x 0 ∈ R such that the corresponding trajectories of the system (5) converge asymptotically to the origin [START_REF] Khalil | Nonlinear Systems[END_REF]. Since the determination of the exact region of attraction is practically impossible, a problem of interest is to ensure asymptotic stability for a set of admissible initial conditions x 0 . Hence, from the considerations above, in this paper, we are interested in studying the stabilization problems stated as follows.

P1. Given T 1 , T 2 , find K and a set of admissible initial conditions, as large as possible, for which asymptotic stability of the closed-loop system (5) is ensured.

P2. Maximize the bound on the delay T 2 , for which asymptotic stability of the closed-loop system (5) can be ensured for some set of admissible initial conditions.

Regarding a networked control system, problem P1 can be seen as a controller design problem, which is defined from the network schedule constraints. On the other hand, problem P2 can be seen as a network design problem. In this case, the network designer will consider the maximum allowable jitter (for which stability of the system can be guaranteed) as a constraint in the network scheduling. Of course, when it is possible, the objective will be global stabilization of the closed-loop system. Otherwise, a set of admissible initial conditions, included in the region of attraction of the closed-loop system, will be defined. This set can be seen as an estimate of the actual region of attraction and defines a region of "safe initialization" for the system.

In order to develop conditions to solve problems P1 and P2, in the sequel a particular notation is adopted. For all integers k ∈ N, a function χ k ∈ K can be defined such that equation ( 5) can be equivalently represented by:

∀τ ∈ [0, T k ], χ k (τ ) = x(t k + τ ), χk (τ ) = d dτ χ k (τ ) = Aχ k (τ ) + Bsat(Kχ k (0)). ( 7
)

Preliminaries

The following lemmas will be used in the sequel.

Lemma 1 Let P be a positive definite matrix and define P = M T P M , with M being a nonsingular matrix.

If P 0 I I M + M T -P > 0, ( 8
)
then P < P 0 .

Proof. If P > 0 it follows that (P -1 -M ) T P (P -1 -M ) ≥ 0, which implies that P -1 ≥ (M T + M -P ), or equivalently, P ≤ (M T + M -P ) -1 . Hence, from Schur's complement, if ( 8) is verified it follows that P 0 > (M + M + P ) -1 , which implies that P < P 0 .

2

Using the notation χ k introduced in ( 7), the following dead-zone function is defined

ψ(Kχ k (0)) = ψ k = Kχ k (0) -sat(Kχ k (0)). ( 9
)
Note that, ψ(Kχ k (0)) corresponds to a decentralized dead-zone nonlinearity. Considering the function ψ(Kχ k (0)), the closed-loop system can be re-written as

χk (τ ) = Aχ k (τ ) + BKχ k (0) -Bψ(Kχ k (0)).
Consider now a matrix G ∈ R m×n and define the polyhedral set

S = {x ∈ R n ; |(K i -G i )x| ≤ u 0i , i = 1, ..., m}.
The following Lemma from [START_REF] Gomes Da | Anti-windup design with guaranteed regions of stability: an LMI-based approach[END_REF], concerning the nonlinearity ψ(Kχ k (0)) is recalled.

Lemma 2 Consider the function ψ(Kχ k (0)) defined in [START_REF] Gomes Da | Anti-windup design with guaranteed regions of stability: an LMI-based approach[END_REF]. If χ k (0) ∈ S then the relation

ψ T (Kχ k (0))U [ψ(Kχ k (0)) -Gχ k (0)] ≤ 0, ( 10
)
is verified for any matrix U ∈ R m×m diagonal and positive definite.

The result in Lemma 2 can be seen as a generalized sector condition. As will be seen in the sequel, differently from the classical sector condition (used for instance in [START_REF] Tarbouriech | Delay-dependent anti-windup loops for enlarging the stability region of time-delay systems with saturating inputs[END_REF]), this condition will allow to obtain stability conditions directly in an LMI form. For notation simplicity, in the sequel we denote ψ(Kχ k (0)) as ψ k .

Asymptotic stability of saturated and sampled-data systems

This section is motivated by the difference between the discrete and continuous-time Lyapunov theorems.

As the problem of sampled-data systems is at the boundary of the discrete and the continuous-time theories, it is important to focus on the connections between them. More specifically, the main idea of this section consists in developing novel stability conditions for sampled-data systems with control saturation, modeled in continuous-time, using the discrete-time Lyapunov theorem.

Theorem 1 Consider given matrices K and G in R m×n and any positive definite diagonal matrix U in R m×m . Let T 1 and T 2 , T 1 < T 2 , be two positive scalars and V : R n → R + be a function for which there exist real numbers 0 < µ 1 < µ 2 and p > 0 such that

∀x ∈ R n , µ 1 |x| p ≤ V (x) ≤ µ 2 |x| p . ( 11
)
and such that, for all i = 1, . . . , m and

x ∈ R n x T (K i -G i ) T (K i -G i )x ≤ u 2 0i V (x). ( 12
)
Then, the two following statements are equivalent.

(i) For all k ∈ N, T k ∈ [T 1 , T 2 ]
, the increment of the Lyapunov function satisfies

∆V (k) -2T k ψ T k U [ψ k -Gχ k (0)] < 0;
where

∆V (k) = V (χ k (T k )) -V (χ k (0)) (ii) There exists a continuous functional V 0 : [0, T 2 ] × K → R which satisfies for all z ∈ K ∀T k ∈ [T 1 , T 2 ] V 0 (T k , z) = V 0 (0, z). ( 13
)
and such that, for all k ∈ N,

T k ∈ [T 1 , T 2 ] and τ ∈ [0 T k [ and Ẇ(τ, χ k ) = d dτ [V (χ k (τ )) + V 0 (τ, χ k ) -2τ ψ T k U [ψ k -Gχ k (0)]] < 0, (14)
Moreover, if one of these two statements is satisfied, then, for all initial conditions x(0) = χ 0 (0) in the set E defined by

E = {x ∈ R n ; V (x) ≤ 1} , ( 15
)
the solutions of system [START_REF] Cao | Stability analysis of linear time-delay systems subject to input saturation[END_REF] with the saturated and sampled control law ( 4) converge asymptotically to the origin.

Proof. Consider a positive integer k and τ ∈ [0, T k [. Assume (ii) is satisfied. Integrating Ẇ over the interval [0 T k [ and assuming that ( 13) and ( 14) hold, this directly implies ∆V (k

) < 2T k ψ T k U [ψ k -Gχ k (0)
] and then (i) holds.

Assume now that (i) is satisfied. Inspired by Lemma 2 in [START_REF] Peet | Positive forms and stability of linear time-delay systems[END_REF], consider the functional

V 0 (τ, χ k ) = -V (χ k (τ )) + τ /T k ∆V (k). Indeed, V 0 is a functional since it is expressed with respect to ∆V (k) which depends on the function χ k (0), χ k (T k ) and χ k (τ ) for all τ ∈ [0, T k ].
By simple computations, it is easy to obtain that this functional satisfies [START_REF] Khalil | Nonlinear Systems[END_REF] and that

Ẇ(τ, χ k ) = 1/T k ∆V (k) -2T k ψ T k U [ψ k -Gχ k (0)] .
Then Ẇ is negative if (i) holds. This proves the equivalence between (i) and (ii). Since x(0) belongs to E, inequality [START_REF] Hindi | Analysis of linear systems with saturation using convex optimization[END_REF] directly implies that |(K i -G i )χ 0 (0)| ≤ u i0 . Consequently χ 0 (0) also belongs to S and, according to Lemma 2, the quantity ψ T 0 U [ψ 0 -Gχ 0 (0)] is negative. Hence the increment of the Lyapunov function ∆V (0) is negative and χ 0 (T 0 ) = χ 1 (0) also belongs to E. Repeating the reasoning for k = 1, 2 . . ., we conclude that χ k (0) ∈ E and ψ T k U [ψ k -Gχ k (0)] ≤ 0, for all k ∈ N. From the discrete-time Lyapunov theorem, the equilibrium of the discrete-time system is asymptotically stable and χ k (0) tends to zero as k tends to infinity.

The end of the proof consists in ensuring that the solutions of the continuous-time system are not diverging within a sampling period. From ( 6) and [START_REF] Hetel | Robust stability and control of switched linear systems[END_REF], the inequality

V (χ k (τ )) ≤ µ 2 |Γ(τ )χ k (0) + Υ(τ )sat(Kχ k (0))| p , ≤ µ 2 (|Γ(τ )| p + |Υ(τ )K| p )|χ k (0)|,
µ m such that, |Γ(τ )| p + |Υ(τ )K| p ≤ µ m . This ensures V (χ k (τ )) ≤ µ 2 µ m |χ k (0)| p , for all τ in [0, T k ].
This proves that the continuous Lyapunov function uniformly and asymptotically tends to zero.

2

In the literature several articles have introduced functionals satisfying the requirements of Theorem 1 (see for instance [START_REF] Naghshtabrizi | Exponential stability of impulsive systems with application to uncertain sampled-data systems[END_REF][START_REF] Fridman | A refined input delay approach to sampled-data control[END_REF]). However, those results use the Lyapunov-Krasovskii theorem which requires necessarily the positive definiteness of the functional V 0 . Theorem 1 relaxes the constraint on the positivity of the functional. The only requirement to ensure stability is on the discrete-time Lyapunov function V , and [START_REF] Khalil | Nonlinear Systems[END_REF].

A graphical illustration of Theorem 1 is shown in Figure 1. The main idea consists in showing the equivalence between the conditions on the decreasing increment ∆V the existence of a continuous functional W which coincides with the Lyapunov function V at the sampling instants and which is strictly decreasing within all sampling intervals. The main contribution of Theorem 1 is that the introduction of the functional W allows the Lyapunov function V to be locally increasing.

(k) = V (χ k (T k )) -V (χ k (0)) < 0 and
Under the conditions of Theorem 1, it follows that the set E is included in the region of attraction of the closed-loop system [START_REF] Fridman | Robust sampled-data stabilization of linear systems: An input delay approach[END_REF]. Note this set is not necessarily positively invariant for the continuous-time system, but it is indeed for the discrete-time system, i.e. if χ 0 (0) ∈ E it follows that χ k (0) ∈ E, for all k > 0. Nonetheless, the convergence of the continuous-time system trajectories to the origin is guaranteed for any initial condition belonging to E, i.e. the regional asymptotic stability is ensured. On the other hand, when the open-loop system is asymptotically stable, we may attempt to compute a control law that globally stabilizes the origin of the closed-loop system. The following corollary gives support to this case.

Corollary 1 Consider a given matrix K in R m×n and any positive definite diagonal matrix U in R m×m . Let T 1 and T 2 , T 1 < T 2 , be two positive scalars and V : R n → R + be a function for which there exist real numbers 0 < µ 1 < µ 2 and p > 0 such that condition [START_REF] Hetel | Robust stability and control of switched linear systems[END_REF] holds. Then the two following statements are equivalent.

(i) For all k ∈ N, T k ∈ [T 1 , T 2 ]
, the increment of the Lyapunov function satisfies

∆V (k) -2T k ψ T k U [ψ k -Kχ k (0)] < 0;
where

∆V (k) = V (χ k (T k )) -V (χ k (0)) (ii) There exists a continuous functional V 0 : [0, T 2 ] × K → R which satisfies for all z ∈ K ∀T k ∈ [T 1 , T 2 ] V 0 (T k , z) = V 0 (0, z). ( 16
)
and such that, for all k ∈ N,

T k ∈ [T 1 , T 2 ] and τ ∈ [0 T k ] and Ẇ(τ, χ k ) = d dτ [V (χ k (τ )) + V 0 (τ, χ k ) -2τ ψ T k U [ψ k -Kχ k (0)]] < 0. ( 17
)
Moreover, if one of these two statements is satisfied, then the origin of system [START_REF] Cao | Stability analysis of linear time-delay systems subject to input saturation[END_REF] with the saturated and sampled control law ( 4) is globally asymptotically stable.

Proof. The proof follows the same lines of the one of Theorem 1. The difference is that now, from the definition of the dead-zone function ψ k , the relation ψ T k U [ψ k -Kχ k (0)]] ≤ 0 holds for any χ k (0). Then, it is ensured that ∆V (k) < 0, for any χ k (0) ∈ R n , and global stability follows.

2

Based on an appropriate choice of the functional V 0 and the application of Theorem 1 and Corollary 1, we derive LMI-based conditions for the stabilization of sampled data systems (1) under the saturated control law defined in (4), both in regional and global contexts.

Remark 1

The technique to handle saturation applied here follows the concepts used in the problem of absolute stability [START_REF] Khalil | Nonlinear Systems[END_REF]. Basically, the idea consists in relaxing the conditions for the satisfaction of the decreasing of the Lyapunov function just for nonlinearities satisfying a sector bound condition. It is however important to emphasize that, as pointed in [START_REF] Gomes Da | Anti-windup design with guaranteed regions of stability: an LMI-based approach[END_REF], considering deadzone nonlinearities, the relation [START_REF] Hespanha | A survey of recent results in networked control systems[END_REF] can be viewed as a generalized sector condition which encompasses the classical one used, for instance in [START_REF] Hindi | Analysis of linear systems with saturation using convex optimization[END_REF] and [START_REF] Kiyama | On the use of multi-loop circle for saturating control synthesis[END_REF]. This fact leads to a clear reduction of the conservatism in stability and stabilization conditions for systems with saturation nonlinearities. Moreover, as it will be clear in the sequel, the synthesis conditions can be obtained in an LMI form, while the classical approach leads to BMI conditions (see [START_REF] Tarbouriech | Stability and Stabilization of Linear Systems with Saturating Actuators[END_REF], Section 1.7.2, for a comprehensive discussion about this matter)

Stabilization of sampled-data systems under input saturation

In this section, asymptotic stabilization conditions (both in regional as well global contexts) of the sampleddata system (2) with the saturating control law given by ( 4) are derived from the results of Theorem 1. These conditions allow the computation of a gain K that ensures the asymptotic stability of the closed-loop system.

Theorem 2 For given positive scalars T 1 < T 2 , assume that there exist positive definite matrices P , R ∈ S n , a matrix S1 ∈ R n×n , a matrix X ∈ S n+m , a positive definite diagonal matrix Ũ ∈ S m , matrices Ỹ , S2 ∈ R n×n and Ñ ∈ R (3n+m)×n , two matrices K and G ∈ R m×n and a positive scalar that satisfy, for i = 1, 2

Ψ 1 (T i ) = Π1 + T i Π2 + T i Π3 < 0, Ψ 2 (T i ) = Π1 -T i Π3 T i Ñ * -T i R < 0, (18)
Ψ 3j (u 0j ) = P ( K -G) T j * u 2 0j ≥ 0, ∀j = 1, . . . , m, ( 19
)
with Π1 = He M T 1 P M 3 -M T 12 S2 M 2 -Ñ M 12 -M T 12 S1 M 12 +He ( M T 1 + M T 3 )(A Ỹ M 1 + B KM 2 -Ỹ M 3 -B Ũ M 4 ) -2M T 4 Ũ M 4 + He{M T 4 GM 2 }, Π2 = M T 3 RM 3 + He M T 3 ( S1 M 12 + S2 M 2 ) Π3 = M T 24 XM 24 , ( 20
)
where 1 M 1 = I 0 0 0 , M 2 = 0 I 0 0 , M 3 = 0 0 I 0 , M 4 = 0 0 0 I , M 12 = M 1 -M 2 , M 24 = [M T 2 M T 4 ] T .
Then, for all initial conditions x 0 = x(0) belonging to the ellipsoidal set

E P = x ∈ R n ; x T P x ≤ 1 ,
where P = Ỹ -T P Ỹ -1 , the corresponding trajectories of the system (2) under the saturated control law defined in ( 4) with K = K Ỹ -1 converge asymptotically to the origin for any asynchronous sampling satisfying [START_REF] Alamo | Estimation of the domain of attraction for saturated discrete-time systems[END_REF].

Proof. Introduce a quadratic Lyapunov function candidate defined, for any x in R n , by V (x) = x T P x, where P is a symmetric positive definite matrix from S n . Thus , the function V satisfies [START_REF] Hetel | Robust stability and control of switched linear systems[END_REF] since it has a quadratic form.

Considering now the result of Theorem 1, the idea is to prove that

∆V k = V (χ k (T k )) -V (χ k (0)) - 2T k ψ T k U [ψ k -Gχ k (0)] < 0, for all k ∈ N.
With this aim an appropriate functional V 0 satisfying ( 13) and ( 14) must be chosen. A candidate of such type of functionals is defined for all τ ∈ [0 T k ], as follows:

V 0 (τ, χ k ) = (T k -τ )(χ k (τ ) -χ k (0)) T [S 1 (χ k (τ ) -χ k (0)) + 2S 2 χ k (0)] +(T k -τ )τ χ k (0) ψ k T X χ k (0) ψ k + (T k -τ ) τ 0 χT k (θ)R χk (θ)dθ, with S 1 ∈ R n×n , S 2 ∈ R n×n , X ∈ S n+m , R ∈ S n , R > 0.
The first step of the proof focuses on the fact that the functional V 0 satisfies the condition [START_REF] Khalil | Nonlinear Systems[END_REF]. Since (T k -τ ) = 0 when τ = T k , χ k (τ ) -χ k (0) = 0 when τ = 0 and the integral term are equal to zero when τ = 0, it follows that the functional V 0 (τ, χ k ) is equal to zero at τ = 0 and τ = T k . Thus, the functional V 0 satisfies condition [START_REF] Khalil | Nonlinear Systems[END_REF] and, moreover, it is continuous at all sampling instants and differentiable over [0 T k [. The end of the proof consists in showing that inequality [START_REF] Kiyama | On the use of multi-loop circle for saturating control synthesis[END_REF] hold. The following equality is then obtained

Ẇ(τ, χ k ) = 2χ T k (τ )P χk (τ ) -2ψ T k U [ψ k -Gχ k (0)] +(T k -τ ) χT k (τ )[R χk (τ ) + 2S 1 (χ k (τ ) -χ k (0)) + 2S 2 χ k (0)] -(χ k (τ ) -χ k (0)) T [S 1 (χ k (τ ) -χ k (0)) + 2S 2 χ k (0)] +(T k -2τ ) χ k (0) ψ k T X χ k (0) ψ k - τ 0 χT k (θ)R χk (θ)dθ. ( 21
)
Consider the extended vector

ξ k (τ ) = [χ T k (τ ) χ T k (0) χT k (τ ) ψ T k ]
T , for all τ ∈ [0, T k [, and a matrix N ∈ R (3n+m)×n . Since R is assumed to be positive definite and thus non singular, the product ( χT

k (θ) - R -1 N T ξ k (τ )) T R( χk (θ) -R -1 N T ξ k (τ )) is positive for all τ ∈ [0, T k [ and all θ ∈ [0, τ [. Integrating its development over [0, τ ], the following inequality is obtained τ 0 χT k (θ)R χk (θ)dθ -2ξ T k (τ )N (χ k (τ ) -χ k (0)) + τ ξ T k (τ )N R -1 N T ξ k (τ ) ≥ 0. ( 22
)
On the other hand, there exists a coupling relation between the components of the augmented vector ξ k , which is given by

2(χ T k (τ )Y T 1 + χT k (τ )Y T 2 )(Aχ k (τ ) -χk (τ ) + BKχ k (0) -Bψ k ) = 0,
for any square matrices Y 1 and Y 2 ∈ R n×n . The following null term can be added to inequality [START_REF] Oucheriah | Global stabilization of a class of linear continuous time delay systems with saturating controls[END_REF]. This can be interpreted as the use of the descriptor approach introduced in [START_REF] Fridman | A descriptor system approach to H ∞ control of linear time-delay systems[END_REF]. Combining this inequality to equations ( 21) and ( 22), one obtains:

Ẇ(τ, χ k ) ≤ ξ T k (τ )[Π 1 + (T k -τ )Π 2 + τ N R -1 N T + (T k -2τ )Π 3 ]ξ k (τ ). ( 23
)
where

Π 1 = He{M T 1 P M 3 -M T 12 S 2 M 2 -N M 12 } -M T 12 S 1 M 12 +He{(M T 1 Y T 1 + M T 3 Y T 2 )M 0 + M T 4 U GM 2 } -2M T 4 U M 4 Π 2 = M T 3 RM 3 + He{M T 3 (S 1 M 12 + S 2 M 2 )} Π 3 = M T 24 XM 24 , ( 24
)
where M 0 = A BK -I -B , Since the matrices K and G are decision variables, the previous inequality is not an LMI. Indeed the matrix Π 1 contains products of matrices variables (Y 1 with K and Y 2 with K). There exists several transformations to derive an LMI condition that allows to compute a gain K, while guaranteeing that Ẇ(τ, χ k ) < 0. A review of existing methods is given in [START_REF] Seuret | Commande et observation des systèmes retards variables: Théorie et applications[END_REF]. In this paper, a method inspired from [START_REF] Fridman | Descriptor discretized Lyapunov functional method: Analysis and design[END_REF] is applied. Consider the contributions corresponding to terms of the form " χT k (τ )(.) χk (τ ) " in Ẇ. This is given by

M 3 Π 1 + (T k -τ )Π 2 + τ N R -1 N T + (T k -2τ )Π 3 M T 3 . This leads to -Y 2 -Y T 2 + (T k -τ )R + τ N 3 R -1 N T 3 , where N 3 ∈ R n×n is a component of N .
Since we are looking for a negative contribution of this term, a necessary conditions is that Y 2 is non singular. Additionally, U is assumed to be a diagonal positive definite matrix. Thus it is possible to define the matrices Ỹ = Y -1 2 , Ũ = U -1 and Ξ = diag{ Ỹ , Ỹ , Ỹ , Ũ }. Consider now the vector ξ = Ξ -1 ξ. Rewriting (23), using the new variable ξ leads to

Ẇ(τ, χ k ) ≤ ξT [Ξ T Π 1 Ξ + (T k -τ )Ξ T Π 2 Ξ + τ Ξ T N R -1 N T Ξ + (T k -2τ )Ξ T Π 3 Ξ] ξ.
From the definition of the matrices M i for i = 1, . . . , 4, one has

M 1 Ξ = Ỹ M 1 , M 2 Ξ = Ỹ M 2 , M 3 Ξ = Ỹ M 3 , M 4 Ξ = Ũ M 4 , M 12 Ξ = Ỹ M 12 , M 24 Ξ = Ỹ 0 0 Ũ M 24 , M 0 Ξ = A Ỹ M 1 + BK Ỹ M 2 -Ỹ M 3 -B Ũ M 4 .
Setting now Y 1 = Y 2 and considering the following change of variables P = Ỹ

T P Ỹ , S1 = Ỹ T S 1 Ỹ , S2 = Ỹ T S 2 Ỹ , X = Ỹ 0 0 Ũ T X Ỹ 0 0 Ũ , R = Ỹ T R Ỹ , Ñ = Ξ T N Ỹ , K = K Ỹ and G = G Ỹ , the following inequality is obtained: Ẇ(τ, χ k ) ≤ ξT [ Π1 + (T k -τ ) Π2 + τ Ñ R-1 Ñ T + (T k -2τ ) Π3 ] ξ. ( 25
)
where Π1 , Π2 and Π3 are defined in [START_REF] Oishi | Stability and stabilization of aperiodic sampled-data control systems: An approach using robust linear matrix inequalities[END_REF]. To prove that Ẇ is negative definite for all τ , note that the right hand side of equation ( 25) is affine with respect to the variable τ in [0, T k ]. Then, by convexity, it suffices to ensure that the right hand side of ( 25) is negative for τ = 0 and τ = T k (see [START_REF] Naghshtabrizi | Exponential stability of impulsive systems with application to uncertain sampled-data systems[END_REF] for more detail). This fact leads to the inequalities

Π1 + T k ( Π2 + Π3 ) < 0, and Π1 -T k Π3 + T k Ñ R-1 Ñ T < 0.
Applying the same argument on T k in the interval [T 1 , T 2 ] and by using the Schur's complement, conditions Ψ 1 (T i ) < 0 and Ψ 2 (T i ) < 0 given in [START_REF] Naghshtabrizi | Exponential stability of impulsive systems with application to uncertain sampled-data systems[END_REF] are obtained for i = 1, 2.

Right and left-multiplying ( 19) by diag{Y 2 , I} and next applying Schur's complement, we conclude that ( 19) is equivalent to [START_REF] Hindi | Analysis of linear systems with saturation using convex optimization[END_REF]. Hence, it follows that E P ⊂ S and the condition [START_REF] Hespanha | A survey of recent results in networked control systems[END_REF]

is verified ∀χ k (0) ∈ E P .
Hence, by virtue of Theorem 1 the satisfaction of conditions ( 18)-( 19) ensures the asymptotic convergence of the trajectories to the origin, provided that x(0) ∈ E P .

2

Concerning the global stabilization, the following result based on the Corollary 1 can be stated.

Corollary 2 For given positive scalars T 1 < T 2 , assume that there exist positive definite matrices P , R ∈ S n , a matrix S1 ∈ R n×n , a matrix X ∈ S n+m , a positive definite diagonal matrix Ũ ∈ S m , matrices Ỹ , S2 ∈ R n×n and Ñ ∈ R 2n×n and a positive scalar that satisfy, for i = 1, 2

Ψ 1 (T i ) < 0 and Ψ 2 (T i ) < 0 ( 26
)
with Π1 , Π2 and Π3 as in Theorem 2 but with G = K.

Then, the origin of system (2) under the saturated control law defined in (4) with K = K Ỹ -1 is globally asymptotically stable for any asynchronous sampling satisfying [START_REF] Alamo | Estimation of the domain of attraction for saturated discrete-time systems[END_REF].

Proof. It suffices to replace G by K in the proof of Theorem 2.

2

Remark 2 The strategy to handle the saturation effects in the derived stabilization conditions is based on the use of a generalized sector condition. Nonetheless, it should be pointed out that the results presented can be straightforwardly re-derived considering a polytopic modeling for the saturation term (see [START_REF] Tarbouriech | Stability and Stabilization of Linear Systems with Saturating Actuators[END_REF], Section 1.7). For instance, if we consider the model used in [START_REF] Alamo | Estimation of the domain of attraction for saturated discrete-time systems[END_REF] and [START_REF] Zhou | An improved treatment of saturation nonlinearity with its application to control of systems subject to nested saturation[END_REF], with B j , denoting the j th column of matrix B, and the sets S r and S c r , r = 1, . . . , 2 m of indices j = 1, . . . , m, with S 1 = ∅, defined as in [START_REF] Alamo | Estimation of the domain of attraction for saturated discrete-time systems[END_REF], it follows that [START_REF] Tarbouriech | Stability and Stabilization of Linear Systems with Saturating Actuators[END_REF]:

Bsat(Kx) ∈ Co{ j∈S c r B j K j x + j∈S r B j H r,j x, r = 1, . . . , 2 m } ( 27
)
provided that the state belongs to the set S = x ∈ ∩ 2 m r=2 S(H r , u 0 ), with

S(H r , u 0 ) = {x ∈ n ; |H r,j x| ≤ u 0j , ∀j ∈ S r }
Hence, following the same steps used in the proof of Theorem 2, new stabilization conditions can be basically obtained by: 1. eliminating the term ψ T k U [ψ k -Kχ k (0)] < 0 from ( 14), this will imply the elimination of the terms in variables Ũ and G appearing in Ψ 1 (T i ) and Ψ 2 (T i ).

2. replacing the term B KM 2 in Π1 by ( j∈S c r B j Kj + j∈S r B j Hr,j )M 2 , with Hr,j = H r,j Ỹ -in this case, the conditions Ψ 1 (T i ) < 0 and Ψ 2 (T i ) < 0 would be replaced, respectively, by Ψ 1,r (T i ) < 0 and Ψ 2,r (T i ) < 0, for r = 1, . . . , 2 m -this corresponds to ensure Ẇ < 0 on the vertices of the convex hull [START_REF] Åstrom | Computer Controlled Systems[END_REF].

3. replacing the inclusion condition E P ⊂ S by E P ⊂ S -this former inclusion can be verified by a set of inequalities similar to [START_REF] Niculescu | Robust stabilization for uncertain time-delay systems containing saturating actuators[END_REF], just by replacing ( Kj -Gj ) by Hr,j .

In some cases, a small improvement (when the objective is the maximization of the set E P ) can be obtained with this kind of modeling at the expense of a higher numerical complexity (see a nice analysis in this sense in [START_REF] Zhou | An improved treatment of saturation nonlinearity with its application to control of systems subject to nested saturation[END_REF] and also in [START_REF] Tarbouriech | Stability and Stabilization of Linear Systems with Saturating Actuators[END_REF], Section 3.2.3). Basically, this comes from the fact that LMIs Ψ 1 (T i ) < 0 and Ψ 2 (T i ) < 0 would be replaced, each one, by 2 m LMIs.

On the other hand, it should be mentioned that the sector approach allows to cope with the global stabilization and the results for the anti-windup synthesis (which can be seen as a future extension of the present work) would be convex, which is not the case with the polytopic representations of the saturation function (see Remark 6.11 in [23]).

The facts above justify our choice by the use of a generalized sector condition approach, instead of the polytopic one, to model the control saturation effects.

Remark 3

The LMI conditions which are provided in Theorem 2 and Corollary 2 are linear with respect to the matrices A and B which characterizes the dynamics of the system. Then direct extensions of these two results are provided in the case of systems with polytopic uncertainties defined by some positive scalars λ i 's such that M j=1 λ j = 1 and

[A B] = M l=1 λ l [A l B l ]. ( 28
)
Note that the λ l 's could either be constant parameters or time-varying but in both cases they are considered unknown.

Optimization problems

From the theoretical result given by Theorem 2 and Corollary 2, this section provides the formulation of some optimization problems to address the stabilization problems P1 and P2 defined in Section 2.

Maximization of the estimate of the region of attraction

For a given u 0 and an asynchronous sampling satisfying (1) with given T 1 and T 2 , the objective is to determine K that leads to the largest set of initial conditions for which the conditions of Theorem 2 are satisfied. In other words, we should find K such that E P is maximized considering some size criterion. For instance, we can maximize the minimal axis of E P , which corresponds to minimize the maximal eigenvalue of P . This can be accomplished from the following optimization problem min δ subject to [START_REF] Naghshtabrizi | Exponential stability of impulsive systems with application to uncertain sampled-data systems[END_REF], [START_REF] Niculescu | Robust stabilization for uncertain time-delay systems containing saturating actuators[END_REF]

δI I I Ỹ + Ỹ T -P > 0. ( 29
)
Note that, from Lemma 1, the last inequality above ensures that P < δI, which ensures that λ max (P ) < δ.

Maximization of the sampling period

Define a region of admissible initial states

E 0 = x ∈ R n , x T P 0 x ≤ 1 .
Hence, given T 1 the idea is to find the maximal T 2 for which it is possible to compute K such that asymptotic stability is ensured for all initial conditions belonging to E 0 . An upper-bound on the maximal T 2 can therefore be obtained from the following optimization problem max T 2 subject to (18), ( 19)

P 0 I I Ỹ + Ỹ T -P > 0. (30)
Note that, from Lemma 1, the last inequality above ensures that P < P 0 , which ensures that E 0 ⊂ E P .

On the other hand, when the open-loop is asymptotically stable, a global stabilizing gain K leading to a maximized bound the sampling period jitter T 2 can be obtained by solving the following optimization problem: max T 2 subject to [START_REF] Seuret | A novel stability analysis of linear systems under asynchronous samplings[END_REF] (31)

Optimization of the actuator size

Given bounds T and T 2 for the sampling period, the idea in this case is to design the control law in order to achieve a guaranteed stability for a given set E 0 of admissible initial conditions with less costly actuators.

In general the cost of the j th actuator is directly related with its size, which can be measured by the bound u 0j . Hence, setting ūj = u 2 0j , j = 1, . . . , m, this problem can be addressed from the solution of the following optimization problem. min m j=1 c j ūj subject to [START_REF] Naghshtabrizi | Exponential stability of impulsive systems with application to uncertain sampled-data systems[END_REF], ( 19)

P 0 I I Ỹ + Ỹ T -P > 0. ( 32
)
with the coefficients c j being the weights of the size/cost of each actuator in the composition of the cost function.

Remark 4 For a fixed , the constraints in ( 29), [START_REF] Tarbouriech | Synthesis of controllers for continuous-time delay systems with saturating controls via LMIs[END_REF], [START_REF] Tarbouriech | Delay-dependent anti-windup loops for enlarging the stability region of time-delay systems with saturating inputs[END_REF] and [START_REF] Yamamoto | A function space approach to sampled data control systems and tracking problems[END_REF] are LMIs. Then the optimal solution of these problems can be easily approached by solving LMI-based problems on a grid in .

Remark 5 The result of Theorem 2 concerns the synthesis of a stabilizing gain K. Nonetheless, the conditions [START_REF] Naghshtabrizi | Exponential stability of impulsive systems with application to uncertain sampled-data systems[END_REF], and ( 19) can also be used for analysis purposes, when the gain K is given. It suffices to replace K by K Ỹ . In this case, problems ( 29) and ( 30) can be solved, to determine an estimate of the region of attraction of the closed-loop system or to determine a bound on the admissible sampling period jitter, respectively. Note that the conditions are still LMIs for a fixed .

Illustrative Examples

Example 1

We consider (2) with the following matrices, taken from [START_REF] Michiels | Using delays and time-varying gains to improve the static output feedback stabilizability of linear systems : a comparison[END_REF] : and where u 0 = 5. The control gain K is chosen as [1 0]. This system is unstable with a continuous-time state feedback control u(t) = Kx(t). However, it was proven in [START_REF] Seuret | A novel stability analysis of linear systems under asynchronous samplings[END_REF] that the closed-loop system using a sampled version of the same control law becomes stable if the sampling period becomes sufficiently large. We will show now that the same behavior appears even if the system is subject to input saturation.

A = 0 1 -2 0.1 , B = 0 1 , 0 0.5 1
In order to asses the stability of the closed-loop system with the given gain K, we can consider the optimization problem [START_REF] Sussmann | A general result on the stabilization of linear systems using bounded controls[END_REF] with the variable K replaced by K Ỹ (see Remark 5). In this case, Figure 2 shows the influence of the maximum allowable sampling period on the size (measured in terms of δ) of the set of admissible initial states for which the stability can be ensured (i.e. the estimate of the region of attraction). As mentioned above, the set of admissible initial conditions is empty for small sampling periods. This comes from the fact that this particular system is not stable for small sampling periods. Moreover, it should be noticed that the case of constant sampling periods (i.e. T 1 = T 2 ) leads to larger sets when compared to the case of asynchronous sampling.

Stabilization example

We consider (2) with the following matrices (taken from [START_REF] Cao | Stability analysis of linear time-delay systems subject to input saturation[END_REF], where h = 0): Applying Theorem 2 with = 1.3, it follows that the asymptotic stability of the system is ensured with the controller gain K = [-1.7491 0.5417] for any asynchronous sampling period satisfying (1) with T 1 = 0 and T 2 ≤ 0.75. The obtained set of admissible initial conditions in this case is given by E 2 = {x ∈ R 2 ; x T 0.4450 0.2307 0.2307 21.0091

x ≤ 1}. The two obtained sets of admissible initial states (estimates of the region of attraction) are depicted in Figure 3. It can be noticed that the set obtained using Theorem 2 is significantly larger than the one obtained from the result in [START_REF] Fridman | Robust sampled-data stabilization of linear systems: An input delay approach[END_REF]. This proves the efficiency of the method. It should be pointed out that, although a different modeling (i.e. a convex hull approach) for the saturation is considered in [START_REF] Fridman | Robust sampled-data stabilization of linear systems: An input delay approach[END_REF], the reduction of conservatism is mainly due to the approach of dealing with stability, i.e. without using the Lyapunov-Krasovskii theorem. Indeed, similar improvements would be achieved if we consider the polytopic (convex hull) modeling for the saturated term adopted in [START_REF] Fridman | Robust sampled-data stabilization of linear systems: An input delay approach[END_REF], at the expense of a larger computational burden (see Remark 2).

Furthermore, solving problem [START_REF] Tarbouriech | Synthesis of controllers for continuous-time delay systems with saturating controls via LMIs[END_REF] it is possible to design a stabilizing state control feedback for all asynchronous samplings satisfying with T 1 = 0 and T 2 = 1.99 (obtained with = 0.1).

Conclusion

This article proposes a novel constructive stabilization criterion for sampled and saturated controlled systems based on the discrete-time Lyapunov theorem. Another important feature of the proposed approach regards the fact of taking explicitly into account control limitations (saturation) in the stabilization problem. Moreover, as illustrated in the example the method provides less conservative results than existing approaches reported in the literature.

  is ensured, for all integer k, and τ ∈ [0, T k ]. Noting that the functions Γ(.) : [0, T k ] → R n×n and Υ(.) : [0, T k ] → R n×m are continuous and consequently bounded over [0, T k ], there exists a positive scalar

Figure 1 :

 1 Figure 1: Illustration of the proof of Theorem 1

2 Figure 2 :

 22 Figure 2: Evolution of 1/ √ δ with respect to the maximum available sampling period T 2 for different cases of T 1 .

  and where u 0 = 5. From[START_REF] Fridman | Robust sampled-data stabilization of linear systems: An input delay approach[END_REF], the gain K = [-1.696 0.523] ensures local stability of the closed loop for constant sampling periods (T = T k = T 1 = T 2 , ∀k), such that T ≤ 0.75. The obtained set of admissible initial conditions is defined byE 1 = {x ∈ R 2 ; x T 5.6933.10 4 -1.8807.10 4 -1.8807.10 4 0.8653.10 4 x ≤ 1}.

Figure 3 :

 3 Figure 3: Set of admissible initial conditions using Theorem 3.3 from [5] (continuous-line) and using Theorem 2 (dashed-line).

The matrices M i are not of the same dimension. The notation 0 and I correspond to the zero and identity matrices of appropriate dimension.

This author was partially supported by CAPES (BEX 3594-10-6) and CNPq (grant 306210/2009-6/PQ), Brazil.