Tatiana Starciuc

Yannick Guinet

Alain Hedoux
email: alain.hedoux@univ-lille.fr

Evgenyi Shalaev
email: evgenyi.shalaev@allergan.com

Water Content Thresholds in Glycerol/Water 1 System: Low-and High-Wavenumber Raman 2 Spectroscopy Study 3

Water structure is the key in such diverse processes as protein folding, catalysis of chemical reactions, and stability of pharmaceuticals and biopharmaceuticals. In the current investigation, glycerol-water mixtures are studied by low-and high-wavenumber Raman spectroscopy. Three water content regions, which are separated by two water content thresholds (WT), WT1 at 6.1±0.7 wt% water and WT2 at 18.6±4.4 wt% water, are identified, based on the trends in the hydrogen bond strength and the cage effect. It is hypothesized that WT1 corresponds to a transition from predominantly unclustered water molecules to small water clusters, and is associated with facilitation of proton transfer as water content exceeds WT1. The WT2 is proposed to mark formation of large (>100 molecules) water clusters, which can form stable nucleus of hexagonal ice at lower temperatures. These findings, if confirmed for other polyhydroxycompounds, could facilitate development of novel pharmaceuticals and vaccines, and also contribute to fundamental understanding of mechanisms of cryo-and lyo-protection of proteins, viruses, and cells.

Introduction.

Water, "a matrix of life" [START_REF] Szent-Gyorgyi | Biology and Pathology of Water[END_REF][START_REF] Franks | Water: a Matrix of Life[END_REF], is an essential component of all living organisms, as well as countless industrial processes and products. Studies of water structure and its distribution on molecular scale are critical for various fields, e.g., catalysis of chemical reaction [START_REF] Ribe | Water-Accelerated Organic Transformations[END_REF], interaction of drugs with targets [START_REF] Nittinger | Water Molecules in Protein-Ligand Interfaces. Evaluation of Software Tools and SAR Comparison[END_REF], and stability of pharmaceuticals and biopharmaceuticals [START_REF] Shalaev | How Does Residual Water Affect the Solid-State Degradation of Drugs in the Amorphous State?[END_REF], to name a few. Water relationships in polyhydroxycompounds (such as sugars and polyalcohols) are of special interest for biological, pharmaceutical and food systems, with multiple conferences and publications devoted to this subject (e.g., [START_REF]International Symposium on the Properties of Water[END_REF]).

Between different materials studied, water/glycerol mixtures represent a popular system for investigation of water distribution and structure. Glycerol is known to protect various biological objects against freezing [START_REF] Salt | Natural Occurrence of Glycerol in Insects and its Relation to their Ability to Survive Freezing[END_REF] and has been used for cryoprotection of cells and organs [START_REF] Sztein | History of Cryobiology, with Special Emphasis in Evolution of Mouse Sperm Cryopreservation[END_REF] and as an additive in stabilization of freezedried proteins [START_REF] Cicerone | Stabilization of Proteins in Solid Form[END_REF]. Studies of glycerol-water mixtures are also relevant to fundamental science, e.g., to polyamorphism of water [START_REF] Bachler | Glass Polymorphism in Glycerol-Water Mixtures: II. Experimental studies[END_REF]. Finally, this is a practically convenient model aqueous system, as glycerol-water mixtures can be easily prepared over the entire concentration range, and glycerol remains amorphous at essentially all practical experimental conditions. It has been demonstrated that distribution and structure of water in glycerol-water mixtures depend on the composition [START_REF] Hayashi | Slow and Fast Dynamics in Glycerol-Water Mixtures[END_REF][START_REF] Sudo | Broadband Dielectric Study of a-b Separation for Supercooled Glycerol-Water Mixtures[END_REF][START_REF] Dashnau | Hydrogen Bonding and the Cryoprotective Properties of Glycerol/Water Mixtures[END_REF][START_REF] To | Excess Chemical Potentials, Excess Partial Molar Enthalpies, Entropies, Volumes, and Isobaric Thermal Expansivities of Aqueous Glycerol at 25°C[END_REF][START_REF] Nakagawa | Molecular Basis of Water Activity in Glycerol-Water Mixtures[END_REF], with transitions between different regions observed at various water contents values ranging from approx. 14 to 53 wt%, as briefly summarized in the Discussion section below. In addition, a change in water clustering patterns between glycerol with 4.7 wt% water vs ≥16.4 wt% has been reported [START_REF] Towey | Molecular Insight into the Hydrogen Bonding and Microsegregation of a Cryoprotectant Molecule[END_REF][START_REF] Grasso | Exploring the Behaviour of Water in Glycerol Solutions by Using Delayed Luminescence[END_REF], with predominantly nonclustered water molecules detected at the lower water content, whereas clusters were found to dominate at ≥16.4 wt% water.

In the current study, glycerol is used as a model polyhydroxycompound to investigate the hydrogen bonding patterns as a function of water content, with a special attention to lower water content region. Glycerol-water mixtures with 0 to 40 wt% water are studied by low-and highwavenumber Raman spectroscopy, to investigate inter-and intra-molecular vibrations.

Intermolecular O-H…H stretching vibrations are analyzed in the low-wavenumber (LW) region, while intramolecular O-H stretching vibrations are analyzed in the high wavenumber (HW) range. Raman spectroscopy has been used to investigate water clusters and hydrogen bonding in various systems, including, e.g., pure water [START_REF] Walrafen | Linearity between Structural Correlation Length and Correlated-Proton Raman Intensity from Amorphous Ice and Supercooled Water up to Dense Supercritical Steam[END_REF], water mixtures with hydrophobic [START_REF] Sassi | Water/Alcohol Mixtures: A Spectroscopic Study of the Water-Saturated 1-Octanol Solution[END_REF][START_REF] Nickolov | Hydrogen Bonding in Concentrated Aqueous Solutions of 1,2-Dimethoxyethane: Formation of Water Clusters[END_REF] and hydrophilic [START_REF] Nickolov | Hydrogen Bonding in Concentrated Aqueous Solutions of 1,2-Dimethoxyethane: Formation of Water Clusters[END_REF] co-solvents, aqueous electrolyte solutions [START_REF] Frosch | From Water Clustering to Osmotic Coefficients[END_REF], and confined systems [START_REF] Guégan | Confinement Effects on Water Structure in Membrane Lyotropic Phases[END_REF].

Materials and Methods.

Raman Spectroscopy in the high-wavenumber region. All investigations were carried out using Renishaw InVia Raman spectrometer (Renishaw plc, Wottonunder-Edge, Gloucestershire, UK). The 514.5 nm line of an argon laser Modu-Laser (Centerville, UT), was focused via a ×50 long working distance Leica objective (Leica microsystems, SAS, Nanterre, France). The spectra were collected in backscattering geometry with a resolution of 2 cm -1 in the 2000 -4000 cm -1 frequency range.

Glycerol-water mixtures were prepared by equilibrating glycerol (Fluka/Merk with ≥ 99.5% purity determined by gas chromatography) at relative humidity (RH) values from 0.8 to 70 %, using GenRH-ambient generator (Surface Measurement Systems Ltd., London UK).

Raman spectra were collected every 1 % RH between 0.8 and 10 % RH, 2 % between 10 and 20 % RH and 5 % between 20 and 70 % RH, after equilibration of about 15 min at each corresponding RH. The precision of the GenRH generator is ± 0.8 %. The experimental spectra were fitted in 2000 -4000 cm -1 spectral region, to 5 Gaussian peaks, after baseline correction and area normalization in 2200 -3000 cm -1 spectral range.

In the analysis of the spectra intensities, it was noted that O-H stretching bands of water can overlap with that for polyhydroxycompounds in the region of 2800-3800 1/cm. 23 Therefore, the relative intensity for both HW-band I and HW-band II might not be directly quantitatively proportions to the relative fraction of the corresponding OH groups of water molecules.

Nevertheless, an increase in an intensity of a Raman signal would probably correspond to an increase in corresponding population of water, at least in a semi-quantitative manner.

Low wavenumber Raman spectroscopy. All investigations, in the 10 -600 cm -1 spectral range, were performed in backscattering geometry to obtain non polarized light-scattering spectrum, under a scattering angle θ = 180 °, using a solid diode laser at 514.5 nm. The very high-dispersive XY Dilor spectrometer is composed of three gratings. The choice of experimental conditions, i.e. 514.5 nm laser line, the width of entrance and exit slits opened at 200 µm, allows a rejection of the exciting line down to 5 cm -1 , and leads to a resolution slightly lower than 2 cm -1 in the low-frequency region.

Glycerol-water mixtures with 0 to 40 wt% water were obtained by mixing water (ultrapure water prepared in the lab using Purelab Flex system purchased from ELGA LabWater, High Wycombe, UK) with glycerol, and loaded in spherical Pyrex cell and hermetically sealed.

The mixtures were agitated for at least 15 minutes before the measurements. All spectra were recorded at room temperature.

The Raman intensity, in low frequency region, is related to the vibrational density of states (VDOS, ) according to [START_REF] Galeener | Theory for the First-order Vibrational Spectra of Disordered Solids[END_REF][START_REF] Shuker | Raman-scattering Selection-rule Breaking and the Density of States in Amorphous Materials[END_REF]:

(

where and correspond respectively to the light-vibration coefficient and the vibrational density of states (VDOS), with . The low-wavenumber spectrum of glycerol -water mixtures is typical of a highly disordered molecular system, with overlapping of the relaxational and vibrational contributions within the 10-70 cm -1 . In order to obtain only the relaxational contribution, is then transformed in Raman susceptibility χ"(ω) according to [START_REF] Galeener | Theory for the First-order Vibrational Spectra of Disordered Solids[END_REF][START_REF] Shuker | Raman-scattering Selection-rule Breaking and the Density of States in Amorphous Materials[END_REF]:

(

The susceptibility spectra can be fitted using a lognormal distribution for the lowwavenumber band (LW-band I) and a Gaussian function for the higher wavenumber band (LWband II). The relative intensities of LW-band I and LW-band II should be considered as semiquantitative values, because of the normalization procedure for LW Raman spectra. The normalization was necessarily performed by Raman intensities for different water-glycerol concentrations, and therefore, while a qualitative trend in the intensity values with water content is expected to be reliable, the intensity values should not be used to quantify fractions of corresponding structures.

Results.

Representative HW-Raman spectra of glycerol samples equilibrated at different relative humidity (RH) values, as well as bulk water, are shown in Figure 1a. The spectrum of pure water (blue solid line) can be interpreted as being composed of three distinctive bands [START_REF] Lerbret | Influence of Homologous Disaccharides on the Hydrogen-Bond Network of Water: Complementary Raman Scattering Experiments and Molecular Dynamics Simulations[END_REF]. The positions of these bands are localized by dashed lines in Figure 1a, i.e. HW-band I at 3200 -3300 cm -1 , HW-band II at 3300 -3400 cm -1 and HW-band III at 3600 -3700 cm -1 . HW-bands I and II were interpreted from a percolation model [START_REF] D'arrigo | Raman Scattering and Structure of Normal and Supercooled Water[END_REF] as corresponding to tetrabonded water molecules (HW-band I) and to molecules involved in distorted hydrogen-bonding (H-bonding) (HW-band II). The HW-band III component is assigned to the stretching vibrations of non-Hbounded O-H groups, i.e. water molecules with at least one "dangling" (not H-bound) proton [START_REF] Walrafen | Linearity between Structural Correlation Length and Correlated-Proton Raman Intensity from Amorphous Ice and Supercooled Water up to Dense Supercritical Steam[END_REF][START_REF] Nickolov | Hydrogen Bonding in Concentrated Aqueous Solutions of 1,2-Dimethoxyethane: Formation of Water Clusters[END_REF][START_REF] Monosmith | Temperature Dependence of the Raman OH-Stretching Overtone from Liquid Water[END_REF]. In glycerol samples equilibrated at RH > 20, spectra have 2 distinctive peaks, corresponding to HW-band I and HW-band II of pure water (Figure 1a). In the samples equilibrated at lower RH, the 2 peaks are less pronounced; in addition, spectra recorded between 0 -20 %RH are similar to each other and to the spectrum of anhydrous glycerol. Based on the visual appearance of the spectra, there is a distinct change at RH of 20% (corresponds to approx.

6.7 wt% water content, based on the water sorption isotherm [START_REF] Ninni | Water Activity in Polyol Systems[END_REF]).

Deconvolution of the HW-Raman spectra of glycerol and water-glycerol mixtures (Figure S1, Supporting information) consistently produces 2 peaks, HW-band I and HW-band II, whereas the HW-band III is not detectable. This is an interesting observation per se, as the HWband III has been observed in pure water and other aqueous systems, e.g., water/1,2dimethoxyethane solutions [START_REF] Nickolov | Hydrogen Bonding in Concentrated Aqueous Solutions of 1,2-Dimethoxyethane: Formation of Water Clusters[END_REF]. The lack of this band in water-glycerol mixtures indicates that both protons in the majority of water molecules are involved in H-bounds. The band position LW Raman susceptibility spectra of pure water, anhydrous glycerol, as glycerol/water mixtures are presented in Figure 2a. LW-Band I is assigned to the caging effect in liquids [START_REF] Idrissi | Study of Aqueous Acetone Solution at Various Concentrations: Low-Frequency Raman and Molecular Dynamics Simulations[END_REF],

and a shift in the position of this peak reflects change in its relative strengths. A higher wavenumber for the LW-band I would be indicative of a stronger cage effect, i.e., the stiffening of the cage formed by the neighboring molecule [START_REF] Idrissi | Study of Aqueous Acetone Solution at Various Concentrations: Low-Frequency Raman and Molecular Dynamics Simulations[END_REF]. For glycerol, LW-Band I is detected at lower frequency comparing with water spectrum, reflecting a less rigid cage formed by the close molecular neighboring in glycerol. In pure water, LW-Band II is commonly attributed to intermolecular O-H…O stretching vibrations in the H-bond network of the tetrahedrally coordinated water clusters [START_REF] Agmon | Tetrahedral Displacement: The Molecular Mechanism behind the Debye Relaxation in Water[END_REF], and reflects restricted translation of 2 water molecules against each other along O-H…O parallel direction [START_REF] Magazu | Vibrational Dynamics of Water Molecules Confined within Trehalose H-bond Imposed Networks: A Raman Response[END_REF]. In the glycerol-water mixtures, LW-Band II corresponds to intermolecular O-H…O stretching vibrations involving both water and glycerol, i.e., (water)O-H…O(water), (water)O-H…O(glycerol), and (glycerol)O-H…O(glycerol). The peak at 250 cm -1 is attributed to intramolecular vibrations in glycerol.

Raman susceptibility spectra of glycerol-water mixtures (Figure 2a) are similar at water contents of 0 to 15 wt%, whereas changes are observed in the LW-Band II spectral region as water content increased to 20 wt% water and above, with the LW-Band II clearly observed. The LW-Band I is generally well described by a lognormal shape [START_REF] Hédoux | Recent Developments in the Raman and Infrared Investigations of Amorphous Pharmaceuticals and Protein Formulations: A review[END_REF], while a Gaussian is used for fitting the contribution of LW-band II. An example of the fitting is provided in Figure S1b (Supporting Information). The plots of the position of LW-band I and LW-band II as function of water content are presented in Figure 2b, and the intensity trend for LW-band II is shown in Figure S3 (Supporting Information). The intensity of LW-band II in pure water has been related to a fraction of water molecules participating in the tetrahedral and distorted tetrahedral H-bond network structure [START_REF] Krishnamurthy | Low-frequency Raman Spectrum of Supercooled Water[END_REF], although interpretation of the intensity in waterglycerol mixtures is not straightforward, as discussed briefly in the Supporting Information. The trends in the LW-Raman spectra wavenumbers show 3 ranges of water contents, similar to the HW-Raman results. The wavenumbers for both LW-band I and LW-band II at lower water contents (below 6 wt%) are approximately constant, whereas the wavenumbers increase with water content in the mid-water content range (8 to 15 wt%). A shift toward the high frequencies for LW-band II indicates a strengthening of the O-H…O H-Bond (and decrease in the distance between the nearest-neighbor H-bonded oxygen atoms [START_REF] Bansil | Spectroscopy of Supercooled Water[END_REF]) as water content increasing. The increase in the LW-band I wavenumber in the mid-water content region (approx. 7 to 15 wt%) indicates a stronger cage effect. At higher water contents, there is a change in water-dependence for both bands, with downward and minor upward trends noticed for LW-band I and LW-band II, respectively.

Discussion.

Overall, both HW-and LW-Raman spectra (Figures 1,2) allow to distinguish 3 ranges in the water content. In the low-water content region, both H-bond strengths and the cage effect are essentially independent of water content. The medium water content range is characterized by a gradual increase in the strength of hydrogen bonds with water content; the cage effect is also getting stronger. A transition from the medium-to high-water content region is characterized by a weaker water-dependence of the H-bond strength, and a reversed trend for the cage effect.

Water content threshold-1 (WT1), which marks transition from the low-water to medium-water region, takes place at water content of 6.1±0.7 wt%, with the values of WT1 similar between all 4 wavenumber datasets (Table S1, Supporting Information). For WT2, which divides mediumand high-water content ranges, values obtained from LW-Raman (15 wt% water) are lower than that from HW-Raman spectra (20 and 24 wt%). The difference could be related, at least in part, to the uncertainty in the HW-based threshold value (in particular in the HW-band II) as the transition is spread over a range of water contents (Figure 1), while the LW data show a welldefined transition (Figure 2). The WT1 and WT2 values from the present Raman study are marked in Figure 3, along with the information on other properties of this system from the literature. Note that the definitions of water concentration regions and specific values for water thresholds are based on the trends in H-bond strength from this investigation, whereas published studies are invoked to discuss relevant changes in the water clustering patterns, as well as features of proton transfer and freezing behavior.

The 3 composition regions in water-glycerol system can be described as follows. At low water contents (water content < WT1), water molecules are dispersed in glycerol matrix and predominantly non-clustered, although small water clusters could also be present. E.g., in glycerol with 4.7 wt% water, 80% of water molecules are present as monomers, with 15% forming dimers, and the remaining molecules belonging to larger (up to 6 molecules) water clusters [START_REF] Towey | Molecular Insight into the Hydrogen Bonding and Microsegregation of a Cryoprotectant Molecule[END_REF]. In this water content range, increase in water content does not result in any major structural changes, with additional water molecules forming predominantly water-glycerol Hbonds, strength of which is independent of the water content. The relative strength of H-bonds can be estimated from the analysis of intermolecular vibrations (in the low-wavenumber region)

or from the analysis of intramolecular vibrations (HW-region). interaction (ww interaction) at 13.8 wt% water [START_REF] Hayashi | Slow and Fast Dynamics in Glycerol-Water Mixtures[END_REF]; threshold for water-water cooperative domains (wwCD) at 22.3 wt% [START_REF] Hayashi | Slow and Fast Dynamics in Glycerol-Water Mixtures[END_REF] 1 and >19.3 wt% [START_REF] Sudo | Broadband Dielectric Study of a-b Separation for Supercooled Glycerol-Water Mixtures[END_REF]; water-water H-bonding percolation threshold at approx. 35 wt% water (ww percolation) [START_REF] Moberg | The End of Ice I[END_REF]; transition from mixing Zone III (glycerol clusters) to Zone II (both glycerol and water clusters) at 26.6 wt% [START_REF] To | Excess Chemical Potentials, Excess Partial Molar Enthalpies, Entropies, Volumes, and Isobaric Thermal Expansivities of Aqueous Glycerol at 25°C[END_REF]; freezing threshold at 20.9% [START_REF] Chen | Kinetics of Proton Migration in Liquid Water[END_REF][START_REF] Lai | Chemical Stability of Peptides in Polymers. 2. Discriminating between Solvent and Plasticizing Effects of Water on Peptide Deamidation in Polyvinylpyrrolidone[END_REF] and >35 wt% water [START_REF] Nakagawa | Molecular Basis of Water Activity in Glycerol-Water Mixtures[END_REF]. Not shown: appearance of bulk-like water above approx. 53 wt% water [START_REF] Dashnau | Hydrogen Bonding and the Cryoprotective Properties of Glycerol/Water Mixtures[END_REF], and transition from mixing Zone II to Zone I at 52.6

wt% water [START_REF] To | Excess Chemical Potentials, Excess Partial Molar Enthalpies, Entropies, Volumes, and Isobaric Thermal Expansivities of Aqueous Glycerol at 25°C[END_REF].

As water content increases above the WT1 threshold, there is an increase in water clusters size and their connectivity, with corresponding strengthening of the hydrogen bonds in the H-bond network of water. The clusters are still relatively small, with the number of water molecules in clusters in single and double-digit. In one example (16.4 wt% water), approx. 50% water molecules were observed to form clusters, with the majority of water molecules belonging to clusters with less than 50 water molecules [START_REF] Towey | Molecular Insight into the Hydrogen Bonding and Microsegregation of a Cryoprotectant Molecule[END_REF]. H-bond strength increases with the increase in water content, which could be related to formation of 3-dimensional network of hydrogen bonds.

Note that small water clusters (3-5 water molecules) tend to form stable cyclic structures, while 3-dimensional structures are formed with more than 6 water molecules [START_REF] Wang | Energetics Competition in Centrally Four Coordinated Water Clusters and Raman Spectroscopic Signature for Hydrogen Bonding[END_REF][START_REF] Saykally | Pinning Down the Water Hexamer[END_REF]. The increase in the water clusters size and connectivity could have a major impact on such fundamental process as proton transfer, which occurs via proton hopping along a chain of water molecules connected by hydrogen bonds [START_REF] Cukierman | Grotthuss! and other Unfinished Stories[END_REF]. Small water clusters (3-4 water molecules) were shown to facilitate intra-cluster proton transfer in phenol-water clusters in the gas phase [START_REF] Katada | Infrared Spectroscopy of Protonated Phenol-Water Clusters[END_REF]. Similarly, cluster size of at least 3 water molecules was determined to be essential for proton transfer from water to ketyl radical anion [START_REF] Biswas | Proton transfer from water to ketyl radical anion: Assessment of critical size of hydrated cluster and free energy barrier in solution from first principles simulations[END_REF], and at least 2 water molecules for proton transfer from water cluster to anilide anion [START_REF] Roohi | Proton transfer from H2O to p-substituted anilide anion: can the size of water cluster influence the N-⋅⋅⋅H-OH→N-H⋅⋅⋅OH-switching[END_REF]. Larger water clusters of approx. 20 molecules may be needed to facilitate proton hopping event in the protonated water-dimer complex (Zundel cation) [START_REF] Chen | Kinetics of Proton Migration in Liquid Water[END_REF]. In glycerol, such clusters could form when water content exceeds WT1, thus marking a threshold for an onset of a more efficient proton transfer, and therefore acceleration of proton-transfer-related processes. Correspondingly, for a proton-transfer-dependent reaction in glycerol/water mixtures, rate constant would be expected to be approximately constant at lower water contents (below the WT1), and accelerates as water content exceeds WT1. An example of such "hockey-stick" pattern has been indeed observed for deamidation reaction of a peptide in water-glycerol mixtures [START_REF] Lai | Chemical Stability of Peptides in Polymers. 2. Discriminating between Solvent and Plasticizing Effects of Water on Peptide Deamidation in Polyvinylpyrrolidone[END_REF]. Deamidation of proteins and peptides is of a fundamental importance in both invivo processes and in pharmaceutical systems, and it involves water-catalyzed proton transfer as the first reaction step. Deamidation rate of the hexapeptide was approximately the same at water contents of 2 and 6 wt%, while increasing at 18 wt% [START_REF] Lai | Chemical Stability of Peptides in Polymers. 2. Discriminating between Solvent and Plasticizing Effects of Water on Peptide Deamidation in Polyvinylpyrrolidone[END_REF] (see Supporting Information -Figure S4, and [START_REF] Ohtake | Effect of Water on the Chemical Stability of Amorphous Pharmaceuticals: 2. Deamidation of peptides and proteins[END_REF]). The water-independent deamidation rate constant at 2-6% water (i.e., below the WT1) is consistent with water-independent strength of hydrogen bond in the low-water content region of this study, whereas the increase in the deamidation rate above WT1 coincides with strengthening of hydrogen bonds and increase in the size and connectivity of water clusters, presumably facilitating proton transfer.

A further increase in water content is associated with additional changes in physical properties of glycerol-water mixtures, including freezing behavior. Briefly, cooling glycerol solutions with water content lower than a certain value ("freezing threshold") do not result in ice formation even after prolonged annealing at temperatures below equilibrium melting points, whereas they would exhibit freezing at water contents above the freezing threshold. Values for the freezing threshold (which is also closely related to the composition of the maximally freezeconcentrated solution) in water-glycerol system vary depending on experimental conditions, ranging from 20.9% water [START_REF] Lusena | Ice Propagation in Glycerol Solutions at Temperatures Below -40[END_REF][START_REF] Luyet | Study by Differential Thermal Analysis of the Temperatures of Instability of Rapidly Cooled Solutions of Glycerol, Ethylene Glycol, Sucrose and Glycose[END_REF] to >35 wt% water [START_REF] Nakagawa | Molecular Basis of Water Activity in Glycerol-Water Mixtures[END_REF]. It has been suggested [START_REF] Shalaev | Freezing of Aqueous Solutions and Chemical Stability of Amorphous Pharmaceuticals: Water Clusters Hypothesis[END_REF] that the freezing threshold in aqueous polyhydroxycompounds is related to the size of water clusters, with the freezing commencing when water cluster size exceeds a critical value, probably at least 90 to 150 molecules in order to form stable nucleus of ice I [START_REF] Moberg | The End of Ice I[END_REF]. Based on a wide-angle neutron scattering study [START_REF] Towey | Molecular Insight into the Hydrogen Bonding and Microsegregation of a Cryoprotectant Molecule[END_REF], increase in the water content from approx. 16 to 37 wt% results in transformation from smaller water clusters at 16.4 wt% water (single-and double-digit number of water molecules) to large water clusters of hundreds and thousands of molecules at 37 wt% water. Therefore, the water clusters at 16.4 wt% water are smaller than stable nucleus of ice, whereas the water clusters in glycerol at the higher water content are sufficiently large to form the stable nucleus. We propose that the Raman-defined WT2, which is estimated to be 18.6±4.4

wt%, corresponds to the formation of large water clusters with the size sufficient to form stable ice crystal nucleus (>100 molecules) when cooled below the freezing point, and therefore to the "freezing threshold".

Figure 3 provides also a summary on water content thresholds which separate areas with different properties in glycerol-water system, combining this study with the literature reports.

The WT2 from this study coincides with water-water cooperative domains from dielectric spectroscopy [START_REF] Hayashi | Slow and Fast Dynamics in Glycerol-Water Mixtures[END_REF][START_REF] Sudo | Broadband Dielectric Study of a-b Separation for Supercooled Glycerol-Water Mixtures[END_REF], as well as with the thermodynamically-defined transition from mixing zone III to mixing zone II [START_REF] Dashnau | Hydrogen Bonding and the Cryoprotective Properties of Glycerol/Water Mixtures[END_REF]. The closest literature analogue for the WT1 is probably the onset of water-water interactions, which was determined by the broadband dielectric spectroscopy to be approx. 14 wt% water [START_REF] Hayashi | Slow and Fast Dynamics in Glycerol-Water Mixtures[END_REF]; a lower value of approx. 6 wt% is obtained in this study. Note that one wouldn't expect an exact correspondence in the water content thresholds obtained from different methods. One potential reason could be related to differences in timescale between different experimental techniques applied. E.g., the timescale for hydrogen bonding as measured by Raman spectroscopy is 10 -13 sec [START_REF] Shalaev | Freezing of Aqueous Solutions and Chemical Stability of Amorphous Pharmaceuticals: Water Clusters Hypothesis[END_REF], whereas dielectric spectroscopy covers timescale range of 10 2 to 10 -8 sec [START_REF] Hayashi | Slow and Fast Dynamics in Glycerol-Water Mixtures[END_REF], and the mixing zones [START_REF] To | Excess Chemical Potentials, Excess Partial Molar Enthalpies, Entropies, Volumes, and Isobaric Thermal Expansivities of Aqueous Glycerol at 25°C[END_REF], which are determined from the thermodynamics analysis, reflect even longer (the infinite) timescale.

Conclusion.

To summarize, Raman spectroscopy investigation of hydrogen bonding strength and cage effect in glycerol-water mixtures reveals 3 composition regions, which are divided by 2 thresholds, WT1 (approx. 6 wt% water) and WT2 (15 to 24 wt% water, depending on the spectral region). It is proposed that WT1 corresponds to a transition from predominantly unclustered water molecules to water clusters with single-and double-digit number of molecules, and is also associated with facilitation of proton transfer, resulting in "hockey-stick" degradation patterns reported for various amorphous freeze-dried pharmaceuticals [START_REF] Ohtake | Effect of Water on the Chemical Stability of Amorphous Pharmaceuticals: 2. Deamidation of peptides and proteins[END_REF]. The WT2 marks formation of large water clusters of a >100 water molecules, which can form stable nuclei of hexagonal ice at lower temperatures and therefore associated with a change in the freezing behavior of glycerol-water mixtures. These findings, if further confirmed for other polyhydroxycompounds (sugar alcohols and sugars), could lead to improvements in understanding mechanisms of action of cryo-and lyo-protectants, and benefit practical fields of cryobiology, pharmaceutical products, and vaccines. In particular, future studies would focus on confirmation of the WT1 value by a range of methods (including terahertz and FTIR spectroscopy) in various systems, as it could be directly transferred to improved predictions of stability of amorphous pharmaceuticals and biopharmaceuticals.

Acknowledgements.

This study was initiated in collaboration with Johanna Kölbel and Axel Zeitler (Cambridge University) and Ke Wu (Allergan PLC), as presented elsewhere [START_REF] Kölbel | Is there a Water Content Threshold for Cluster Formation? Presented at: Freeze-drying of Pharmaceuticas and Biologicals: Short Course and Conference[END_REF].

  Figure S2 (Supporting Information), respectively, as a function of water content. As water

Figure 1 :

 1 Figure 1 : (a): Representative high-wavenumber Raman spectra of water and water-glycerol

Figure 2 :

 2 Figure 2: (a): Representative Raman spectra of glycerol, water, and water-glycerol mixtures in

  For O -H … O H-bonds the strength is probed by O … H intermolecular vibrations between 2 neighboring molecules, and the strength is proportional to the energy transfer in the scattering process, i.e. proportional to the OH vibration frequency. In the HW range, the frequency of the internal O-H vibration decreases when the oxygen atom of the neighboring associated molecule via H-bonds is closer, i.e. when the H-bond is strengthened.

Figure 3 .

 3 Figure 3. Composition ranges in glycerol/water system from this study (vertical color lines, see

KEYWORDS water structure; glycerol; Raman spectroscopy; hydrogen bonding; water clusters; proton transfer.


Karima Tebani
email: tebanikarima@yahoo.fr

Said Amari

Min-Plus realizable control design for partially observable timed event graphs under marking constraints

Keywords: Discrete Event Systems, Timed event graphs, Min-Plus algebra, Marking constraints, Realizable control laws

This paper deals with a control problem of discrete event systems subject to capacity constraints. Combined use of timed event graphs and Min-Plus algebra is a well-known approach and efficient for handling timed behavior and mathematical modelling of discrete event systems. However, in current literature, many of control approaches assume that system states are fully observable, which is not the case in our study. Hence, we propose in this paper a feedback control method to guarantee the respect of marking constraints imposed for some paths of partially observable timed event graphs. We demonstrate that if each loop of the considered TEG contains at last one observable transition, we can derive a realisable control law satisfying a set of constraints.

Introduction

In many industrial contexts, we find different critical specifications that must be met to ensure the proper functioning of the systems. These functional constraints can take several forms and they are often encountered in various real applications like flexible manufacturing workshops including thermal or chemical treatments, real-time systems, urban or rail transport and networked automation systems. In this work we treat a class of deterministic controlled discrete event systems subject to strict constraints. In the literature, some authors focused in their research on the respect of durations in some sensitive areas [START_REF] Amari | Max-Plus Control Design for Temporal Constraints Meeting in Timed Event Graphs[END_REF], [START_REF] Atto | Control of Discrete Event Systems with Respect to Strict Duration: Supervision of an Industrial Manufacturing Plant[END_REF], [START_REF] Amari | Output feedback control of discrete processes under time constraint: application to cluster tools[END_REF], [START_REF] Tebani | Closed-loop control of constrained discrete event systems: application to a networked automation system[END_REF] where the objective is to satisfy temporal constraints imposed on the behavior of systems. Others considered specifications related to capacities which are described by marking constraints or forbidden states in Petri nets [START_REF] Giua | Generalized Mutual Exclusion Constraints on Petri Nets with Uncontrollable Transitions[END_REF], [START_REF] Atli | Supervisory control of timedplace marked graph based on Marking Exclusion Constraint[END_REF], [START_REF] Zhao | Suboptimal liveness-enforcing supervisor design for a class of generalized Petri nets using partial siphon enumeration and mathematical programming[END_REF], [START_REF] Yamalidou | Feedback control Petri net based on place invariant[END_REF]. In our study, we are specifically interested to the problem of control under constraints for discrete event systems represented by timed event graphs. It's well known that dioids are an algebraic structure that can be used to formally generate linear models for a class of timed discreteevent systems [START_REF] Baccelli | Synchronization and Linearity: An algebra for Discrete Event Systems[END_REF]. Combined use of TEG and Max-Plus (or Min-Plus) is a well-known approach in the literature initiated back in the 1960's [START_REF] Cuninghame-Green | Process synchronisation in a steelworks-a problem of feasibility[END_REF] and it is still a very active field of research. This framework has been successfully applied to solve diverse control problems, such as dynamic scheduling [START_REF] Bonhomme | Scheduling and control of real-time systems based on a token player approach[END_REF], [START_REF] De Schutter | Model predictive control for max-plus-linear discrete event systems[END_REF], [START_REF] Kim | Schedulability analysis of timeconstrained cluster tools with bounded time variation by an extended petri net[END_REF], synchronization of switching models [START_REF] Lopes | Modeling and control of legged locomotion via switching max-plus models[END_REF], [START_REF] Lopes | On the synchronization of cyclic discrete-event systems[END_REF], trajectory tracking [START_REF] Cottenceau | Synthesis of Greatest linear feedback for timed event graphs in dioid[END_REF], disturbance decoupling problem [START_REF] Shang | An integrated control strategy in disturbance decoupling of maxplus linear systems with applications to a high throughput screening system in drug discovery[END_REF], or just-in-time control [START_REF] Houssin | Just in time control of constrained (max, +)-linear systems: Discrete Event Dynamic Systems[END_REF], [START_REF] Lhommeau | A non-linear set-membership approach for the control of discrete event systems[END_REF]. Both open loop [START_REF] Shang | An integrated control strategy in disturbance decoupling of maxplus linear systems with applications to a high throughput screening system in drug discovery[END_REF] and feedback [START_REF] Lhommeau | Interval analysis and dioid: application to robust controller design for timed event graphs[END_REF], [START_REF] Andrade | On the control of max-plus linear system subject to state restriction[END_REF], [START_REF] Hardouin | super-eingenvector approach to control constrained max-plus linear systems[END_REF] approaches have been considered to solve those problems. Significant efforts have also been made to apply model predictive control techniques to Max-Plus linear systems [START_REF] De Schutter | Model predictive control for max-plus-linear discrete event systems[END_REF], [START_REF] Xu | Optimistic optimization for model predictive control of max-plus linear systems[END_REF]. Other methods are also developed to design a controller to guarantee that a Max-Plus system evolves without violating time restrictions imposed on the state [START_REF] Andrade | On the control of max-plus linear system subject to state restriction[END_REF], or even characterized by a semimodule [START_REF] Hardouin | super-eingenvector approach to control constrained max-plus linear systems[END_REF]. The feedback control problem of Max-Plus linear systems under temporal constraints has been addressed using a temporal approach based on daters like in [START_REF] Amari | Max-Plus Control Design for Temporal Constraints Meeting in Timed Event Graphs[END_REF], [START_REF] Andrade | Some results on the feedback control of max-plus linear systems under state constrains[END_REF]. The objective of calculated control laws is to ensure maximum time limits of the sojourn for tokens in some places of a TEG. In [START_REF] Tebani | Closed-loop control of constrained discrete event systems: application to a networked automation system[END_REF], authors proposed a closed-loop control to satisfy a critical response time for networked automation systems based on timed event graphs and Max-Plus algebra. Others approaches deal with the constraint control of max-plus linear systems by using its transformed version based on power series [START_REF] Cottenceau | Modeling and control of weight-balanced timed event graphs in dioids[END_REF], [START_REF] Houssin | Control of (max,+)-linear systems minimizing delays, Discrete Event Dynamic Systems[END_REF] which is similar to Z transform for conventional linear system theory. An alternative method was proposed in [START_REF] Katz | Max-plus (A, B)-invariant spaces and control of timed discrete-event systems. Automatic Control[END_REF], where the constraints satisfaction is expressed as the (A, B)invariance of a semimodule. The control problem is solved by computing the maximal of such semimodules, included in a domain defined by constraints. In this paper, we address the feedback control problem for TEGs under marking constraints by means of Min-Plus algebra formalisms. Historically, the question is originally treated by Ramadge and Wonham [START_REF] Ramadge | The control of discrete event systems, Dynamics of discrete event systems[END_REF], [START_REF] Ramadge | Modular feedback logic for discrete event systems[END_REF], using finite automata and formal languages. The synthesized controller (supervisor) has the role to inhibit controlled events in order to satisfy the specifications. Other approaches are based on untimed Petri nets tools [START_REF] Murata | Petri nets: properties, analysis and application[END_REF], where the constraints are formulated as a problem of linear marking constraints. The role of the controller is to limit the number of tokens in the some places, which can represent the bounded stock in the industrial context: the buffer memory in communication systems or stations in transportation networks. In [START_REF] Giua | Generalized Mutual Exclusion Constraints on Petri Nets with Uncontrollable Transitions[END_REF], [START_REF] Yamalidou | Modeling and optimal control of discrete-event chemical processes using petri nets[END_REF], [START_REF] Yamalidou | Feedback control Petri net based on place invariant[END_REF], authors express various specifications to be respected by linear inequalities on markings. Monitor places are synthesized using the P-invariants property of Petri nets. Another approach to solve the forbidden state problem in discrete event systems modeled by Petri nets with unobservable transitions is proposed in [START_REF] Dideban | Controller synthesis with highly simplied linear constraints[END_REF]. This method is based on a linear constraint transformation, authors deal with the problem of forbidden states in safe Petri nets and they establish a maximally permissive controller without the time. Time is not taken into account in the system model and control synthesis in existing contribution on capacity constraints. Practically for the dynamic study of systems, time parameter is very important and must be integrated into modelling and control laws. Some studies based on timed Petri nets models dealing with the control under marking constraints appeared during these last years. We can cite for example the approach [START_REF] Tebani | State Feedback Control for a Class of Timed Petri Nets Subject to Marking Constraints[END_REF] based on the TEG and the Min-Plus algebra proposed for the control of TEGs under linear marking constraints. Authors considered the case where all transitions of the TEG are not controllable but all are observable. The problem posed in the case of TEGs with some non observable transitions is that the calculated controller is not always realizable. For this, it is necessary to assure that established control laws are given in function of observable transitions. The present paper deals with the design of feedback control laws for timed discrete event systems subject to capacity constraints using Min-Plus models. This study directly extends and improves the methods recently developed in [START_REF] Tebani | State Feedback Control for a Class of Timed Petri Nets Subject to Marking Constraints[END_REF], which have a strong limitation that all transitions of the TEG are observable and measurable. In this work, we propose a new control method of partially observable TEGs subject to linear marking constraints. The objective is to provide realizable control laws to guarantee different specifications required on markings of TEGs. The calculated laws are expressed only in terms of observable transition states. The remainder of this paper is organized as follows. Section 2 recalls some basic notions of TEGs and their linear Min-Plus algebra representations well as so some definitions and hypothesis that are needed for the comprehension of the sequel of document. Problem of limited capacities and linear marking constraints is presented in section 3 where the specifications are expressed in terms of linear inequalities in Min-Plus algebra. Section 4 is dedicated to the description of the proposed approach. Finally, section 5 concludes this paper with some perspectives for future work.

PRELIMINARIES

Min-Plus algebra

A monoid is a set, say D, endowed with an internal law, noted ⊕ which is associative, commutative and has a neutral element denoted ⊕ (i.e., ∀a ∈ D, a ⊕ ε = ε ⊕ a = ε), ⊗ is associative, admits a unit element denoted e = 0, and distributes over ⊕; and ε is absorbing for the multiplicative law (∀a ∈ D, a ⊗ ε = ε ⊗ a = ε). The semiring is commutative when ⊗ is commutative. A dioid (D, ⊕, ⊗) is a semiring with an idempotent internal law (i.e., ∀a ∈ D, a ⊕ a = a). The dioid is said to be commutative if the second law ⊗ is commutative. The Min-Plus algebra is defined as (R ∪ -∞, + ∞, min, +). This semiring, denoted Rmin , is a commutative dioid, the law ⊕ is the operator min with neutral element ε = +∞, and the second law ⊗ is the usual addition, with neutral element e = 0. ⊗ is abbreviated by • (dot) for convenience. (+, ×) refers to the usual addition and multiplication. As in classical algebra, the binary operations can be extended to the matrix case in dioid. For a matrix A ∈ R(m×n) , the element of the i th row and j th column of the matrix A is denoted by A ij . For any three matrix defined by A, B ∈ R(m×n) min and C ∈ R(n×p) min in Min-Plus algebra Rmin , we define: Addition A ⊕ B:

[A ⊕ B] ij = A ij ⊕ B ij = min(A ij , B ij ) Multiplication A ⊗ C : [A ⊗ C] ij = n k=1 A ik ⊗ C kj = min k=1,..,n (A ik + C kj ).
Theorem 1 [START_REF] Baccelli | Synchronization and Linearity: An algebra for Discrete Event Systems[END_REF] The implicit equation x = A • x ⊕ B defined over a complete dioid D admits x = A * • B as a least solution with A * is the Kleene star of the matrix A, such as: ∀A ∈ D, A * = i>0 A i where A i = A i-1 ⊗ A and A i = I I is the unit matrix, with entries equal to e on the diagonal, and ε else.

The reader is invited to consult [START_REF] Baccelli | Synchronization and Linearity: An algebra for Discrete Event Systems[END_REF] and [START_REF] Baccelli | Recursive equations and basic properties of timed Petri nets[END_REF] for a complete presentation of the following theoretical recalls.

2.2 Timed Event Graphs (TEGs) and linear Min-Plus models

Timed Event Graphs (TEGs)

An event graph is a Petri net where each place has exactly one upstream and one downstream transition. A timed event graph (TEG) is an event graph with delays associated to places, the holding times, and/or transitions, the firing times. An event is the firing of a transition. In the present work, we use the TEG, where the delays are associated only with the places and we adopted the following notations: -P will denote the set of places of the considered graph and T is a set of transitions.

-The internal transitions (transition having at least one upstream place) are denoted by T i (with T i = N ), and the source transitions or the input transitions (transition having no upstream place) are represented by T uc (with

T uc = µ).
-t j denotes the j th transition of the graph, -T ω denote the set of the observable transitions and T β denote the set of the unobservable transitions, where T ω + T β = N , -p ij denotes the place linking t j to t i when it exists, -An elementary path between two internal transitions t j and t i of a graph is an oriented alternating sequence of transitions and places successively connected by an arc, of the form (t j , p 1j , t 1 , p 21 , t 2 , p 32 , ..., t i ). This path is denoted by ρ. The cumulated token in this path is given by m ρ and their timing is denoted by τ ρ where this timing is given by the sum of the places of the path timing.

-Place of a TEG is a particularly case of the path with a longer is 1.

-An elementary path between the input transition t u and the internal transitions t j is an oriented alternating sequence of transitions and places successively connected by an arc. This path is denoted by α. The cumulated tokens in this path is given by m α and its timing is denoted by τ α .

-For a number m ij of tokens, several m ij -token paths connecting t j to t i can exist in general.

t j mij ,τij
-----→ t i denotes the m ij -token path with the maximal delay, which is denoted by τ ij , -An empty path contains no token, -A loop in Petri net is a path whose start node coincides with the arrival node.

-We note that the transitions without downstream places are outputs and those without upstream places are inputs. Others are simply called internal transitions. Inputs are also referred to as controls.

Linear Min-Plus models

A TEG being a directed graph, its dynamic behavior can be described by the linear equations thanks to counter functions x i (t) associated for each transition t i of the graph. The counter functions u(t) attributed to the input transition t u , with the function x i (t) represents the firing number of the transition t i at time T. Generally, the behavior of the TEG is represented by a following equation:

x(t) = τ max τ =0 (A τ • x(t -τ ) ⊕ B τ • u(t -τ )) (1)
where A τ ∈ RN×N min is the matrix whose term A τ,ij it is equal to m ij which is the number of the initial marking of the place p ij if it exists and ε else. B τ ∈ RN×µ min corresponds to the initial markings of the exit places of the source transitions. It is known that, for a live TEG, by decomposing each place into several places which are timed with values less than or equal to 1, the state vector can be extended and rewriting under the following explicit form:

x(t) = A • x(t -1) ⊕ B • u(t) (2)
where

A = A * 0 • A 1 and B = A * 0 • B 0 .
The detailed procedure for the Min-Plus equations representation of timed event graphs is illustrated in the application example in section 4.3 of the page 13. Finally, for any integer τ such that τ ≥ 1, by doing τ substitutions in (2), one obtains:

x(t) = A τ • x(t -τ ) ⊕ τ -1 k=0 A k • B • u(t -k) (3)
Property 1 : In the TEG with the assumptions that each loop of this graph contains at last one observable transition, for any unobservable transition t i ∈ T β , there exists a path δ connecting this transition to an observable transition t j ∈ T ω . The counter associated to the unobservable transition t i can be expressed in function of the observable transition t j by the following expression:

x i (t) ≤ m δ • x j (t -τ δ ), (4)
where m δ represents the number of tokens in the path δ, and τ δ corresponds to the timing of this path.

Definition 1 (Observability). A transition t i is observable if it is either a control or it is linked directly to an output transition. A TEG is fully observable if all his internal transitions t i are observable.

Definition 2 (Partially observable TEG)

A graph G is said to be partially observable if it has at least one unobservable transition.

Definition 3 (Structural controllability)

An event graph is structurally controllable if every internal transition can be reached by a path from at least one input transition.

Definition 4 (Valid control)

The control law u(t) is said to be a validated controller if it grantees the constraints imposed to the system without taking into account any other system specification.

Definition 5 (Realizable control)

A control law is said to be realizable if it is a valid control expressed by a causal and observable expression.

3 Problem of limited capacities and linear marking constraints

Limited capacities problem

To simplify the explanation and show the motivations of the work, we present a simple information transmission example with two tasks A and B in series (see Fig. 1). We consider that the execution time of the task A is fixed to 1 milliseconds (ms), the execution time of the task B is fixed to 3 ms, and the transmission delay of i 1 , is 2 ms. The considered system can be modeled by the TEG shown in the Fig. 2. Let us presume that the transitions (t u , t 1 , t 3 ) are the only observable transitions of this graph and we consider that the transmission line cannot transfer more than one information at a time which implies that the number of tokens in the place p 32 cannot exceed 1 token.

The cycle time of the task B is greater than the cycle time of both the task A and the delay of transmission. This leads to exceeding a maximum number of the information transmitted in the transmission line. For the case where the internal transitions of the graph are controllable, the solution of this problem is intuitive; it is enough to add a place p 23 marked with 1 tokens for example, to ensure that the number of the tokens in the place p 32 does not exceed 1 tokens. The resolution of the problem in the case where the input transition of the place under constraint is uncontrollable is not obvious. Therefore, it is necessary, to have other techniques to guarantee these marking constraints. The objective of this paper is to synthesise control laws to satisfy the marking constraints imposed on the some path of the partial observable timed event graph with uncontrollable transitions.

Linear marking constraints

We consider a discrete event system modeled by a timed event graph. We assume that the path ρ of this graph is subject to a marking constraint, where the path ρ cannot contain more than b tokens throughout the system operation. We denote by t j and t i respectively the input and the output transition of the constrained path, and by M ρ (t) the number of the tokens in this path at time t. The marking constraint can be described by the following inequality:

M ρ (t) ≤ b (5)
In order to express this constraint by a linear inequality in the Min-Plus algebra, we associate with each transition of the graph a counter function and we denote by x i (t) and x j (t) respectively the counter associated to the transitions t i and t j . By definition, the counter x j (t) represents the number of firing of the transition t j at time t. Then, in Min-Plus algebra M ρ (t) can be represented by ((x j (t) -x i (t)) + M 0ρ ), such that M 0ρ is the initial marking of the path ρ.

The inequality (5) can be written as:

x j (t) -x i (t) + M 0ρ ≤ b (6)
This marking constraint can be expressed by a linear inequality in Min-Plus algebra as:

x j (t) ≤ x i (t) + (b -M 0ρ ) (7)
4 Control synthesis

Realizable feedback controls

In this section, we propose a formal approach for the control of the TEG subject to one or several marking constraints. As expressed previously, the proposed approach is based on the use of the TEG and Min-Plus algebra for synthesizing a realizable control law on the form u(t) = F • x(t -1). The role of the calculated controller is to ensure the satisfaction of the marking constraints imposed on the considered partially observable TEG. We consider that the firing of the transitions of the TEG is performed as soon as possible. The transitions of this considered TEG are not all controllable i.e only the input transitions are controllable. First, we consider the case where the system has a single input and then we extend the results to the case where the system has several inputs. In this work we consider that the marking constraints concerns the places situated on the same path and we consider the following Hypothesis:

Hypothesis We assume in the following that each loop of the considered timed event graph contains at last one observable transition.

Single control a.Only one marking constraint

In this first case, we consider a timed event graph with the state equation of this system is given by the equation ( 2). The path ρ connecting the transition t j to the transition t i of this graph is subject to a marking constraint. This constraint is given by the inequality [START_REF] Bonhomme | Scheduling and control of real-time systems based on a token player approach[END_REF]. We assume that there exists at least a path α from the control transition t u to the transition t j , where t j is the upstream transition of the constrained path ρ, and we denote by m α and τ α respectively the marking and the timing of this path. From the definition of t u , we have:

x j (t) ≤ (A τα • B) j • u(t -τ α ) (8)
Taking τ = φ in the equation ( 3), x i (t) can be expressed by:

x i (t) = [ N r=1 A φ ir • x r (t -φ)] ⊕ [⊕ φ-1 k=0 (A k • B) i • u(t -k)], (9)
where φ ≥ 1, A ∈ RN×N min , N represents the number of the internal transition of the TEG and A ir indicates the i th row of the matrix A φ . For every integer φ ≥ 1, we take into account the Property 1 and we have the following result.

Theorem 2 Taking φ = τ α + 1, the equation:

u(t) = F 1 • x r (t -τ δz -1) ⊕ F 2 • x r (t -1) (10)
with,

F 1 = ⊕ β z=1 ⊕ r∈ω [(b + m δz -M 0ρ ) -(A τα • B)] j • A (τα+1) ir and F 2 = ⊕ r∈ω [b -M 0ρ -(A τα • B) j ] • A (τα+1) ir
defines realizable controls which guarantee that the constraint ( 7) is satisfied if the following condition is verified:

(A τα • B) j ≤ (b -M 0ρ )(A k • B) i f or k = 0 to τ α (11)
Proof Let G be a timed event graph and ρ a path under marking constraint with M ρ (t) ≤ b

As we saw previously this constraint can be expressed by the following inequality:

x j (t) ≤ x i (t) + ((b -M 0ρ ) (12)
We consider control laws defined by the following inequality (13):

(A τα • B) j • u(t -τ α ) ≤ x i (t) + (b -M 0ρ ) (13)
It is clear that the combination of inequalities ( 13) and ( 8) implies the satisfaction of the marking constraint [START_REF] Dideban | Controller synthesis with highly simplied linear constraints[END_REF]. In other words, if we choose control laws that verify the inequality ( 13) and as the expression ( 8) that describes the path between two transitions t u and t j is always true, then by the transitivity relation, we conclude that the marking constraint ( 12) is guaranteed.

The next step of this demonstration consists to replace the counter function x i (t) in the inequality ( 13) by its expression given by the equation [START_REF] Cottenceau | Modeling and control of weight-balanced timed event graphs in dioids[END_REF]. After this change, the inequation ( 13) is divided into both conditions ( 14) and ( 15) which are as follows:

(B τα • B) j • u(t -τ α ) ≤ (b -M 0ρ ) + [⊕ N r=1 A φ ir • x r (t -φ)] (14)
(A τα • B) j • u(t -τ α ) ≤ (b -M 0ρ ) + [⊕ φ-1 k=0 (A k • B) i • u(t -k)] (15)
We can easily see that in order to check the satisfaction of inequality [START_REF] Giua | Generalized Mutual Exclusion Constraints on Petri Nets with Uncontrollable Transitions[END_REF], it is sufficient to ensure that the two inequalities ( 14) and ( 15) if they are simultaneously true.

The purpose is to synthesize a realizable control law in the form u(t) = F • x(t -1). In order to obtain the controller in this form, we take φ = τ α + 1, and the fact that u(t) ≥ u(t -1), the inequality ( 15) can be given as:

(A τα • B) j ≤ (b -M 0ρ ) + (A k • B) i
This inequality represents the condition of the Theorem 2. The control inequality described in the Theorem 2 derived from the inequality (15), represents the control laws which satisfy the marking constraint [START_REF] Dideban | Controller synthesis with highly simplied linear constraints[END_REF], if the condition is ( 14) is satisfied.

The formula:

u(t) = ⊕ N r=1 (b -M 0ρ -(A τα • B) j ) • A (τα+1) ir • x r (t -1)
describes a valid controller for a partially observable TEG, but it is not a realizable controller, since the formula of this controller is given in function of the internal transitions of the TEG which contains the unobservable transitions. Since each loop of the considered TEG contains at last one observable transition, this controller can be transformed into a realizable controller by applying the result of the proprty1. For this, we divide the controller formula into two parts, one part according to the observable transitions and the other part according to the unobservable transitions as described by the inequality [START_REF] Kim | Schedulability analysis of timeconstrained cluster tools with bounded time variation by an extended petri net[END_REF]:

u(t) = [⊕ r∈β (b -M 0ρ -(A τα • B) j ) • A (τα+1) ir • x r (t -1)]⊕ (16) [⊕ r∈ω (b -M 0ρ )(A τα • B) j • A (τα+1) ir ] • x r (t -1).
We take into account the results described in the Property 1, where each counter function associated to the unobservable transition can be expressed in function of the counter function associated with observable transition. The equation that describes the calculated control law in function of the observable transitions is as follows:

u(t) = ⊕ β z=1 [⊕ r∈ω [b + m δz -M 0ρ ] -(A τα • B) j • A (τα+1) ir • x r (t -τ δz -1)]⊕ [⊕ r∈ω [(b -M 0ρ -(A τα • B) j ) • A (τα+1) ir ] • x r (t -1)]
b.Several constraints We assume now a timed event graph with one input transition and subject to S marking constraints. These constraints are imposed on S paths of this graph denoted ρ s for s = 1 to S. Let M 0ρs and τ ρs respectively represent the initial marking and the timing of the path ρ s . We denote by t d and t d respectively the input and the output transition of the path ρ s . x d and x d indicate respectively the corresponding counters. We take into account the fact that the marking constraints concern only the places lacated on the same path and we assume that for each s, with s = 1 to S, there is at least one path denoted α s . This path connects the input transition t u to the transition t d , which represents the input transition of the constrained path ρ s . We denote by τ αs the timing of the path α s . The different marking constraints imposed in this graph are expressed by the following inequality:

x d (t) ≤ (b s -M 0ρs ) • x d (t) f or s = 1 to S (17)
Theorem 3 The following control law:

u(t) = ⊕ s≥1 u s (t), (18)
with

u s (t) = F 1s • x r (t -τ δz -1) ⊕ F 2s • x r (t -1) (19)
where

F 1s = β z=1 [⊕ r∈ω [(b s + m δz -M 0ρs ) -(A τα s • B) d • A (τα s +1) ir ]] and F 2s = [⊕ r∈ω [b s -M 0ρs -(A τα s • B) d ) • A (τα s +1) d r
]. ensures the satisfaction of marking constraints [START_REF] Kim | Schedulability analysis of timeconstrained cluster tools with bounded time variation by an extended petri net[END_REF] if the condition below:

(A τα s • B) d ≤ (b s -M 0ρs )(A k • B) d (20)
is verified for each s = 1 to S.

Proof According to the results obtained in the section 4.1, the controller u(t) described in the Theorem 2 guarantees the satisfaction of the marking constraint [START_REF] Bonhomme | Scheduling and control of real-time systems based on a token player approach[END_REF] if the condition (11) is verified. Then, for S marking constraints the control law u s (t) prevents the violation of the S th constraint if the condition (20) holds. The control laws which guarantee de satisfaction of the S marking constraints [START_REF] Kim | Schedulability analysis of timeconstrained cluster tools with bounded time variation by an extended petri net[END_REF] are given by the following equation: u(t) = ⊕ s≥1 u s (t), for s = 1 to S.

Several controls

We now generalize the previous approach to solve the problem of marking constraints in TEGs with several input transitions. The behavior of the considered TEG is described by the state equation ( 2), with u(t) ∈ Rµ min and B ∈ RN×µ min , µ represents the number of the input transition, with µ ≥ 1. We associate with each source transition a counter noted u l (t), for l = 1 to µ.We assume that this graph is subject to marking constraints related for the same path. First, we consider the case where this graph is subject to one constraint, and then we extend the results to the case where several constraints are imposed to this graph. a.Single constraint We suppose in this sub-section a TEG with µ input transitions, and subject to one marking constraint. This constraint is imposed on a single path of this graph denoted ρ. We denote by t j and t i respectively the input and the output transition of the path ρ. M 0ρ and τ ρ represent respectively initial marking and the timing of the path ρ. This constraint can be described by the following inequality:

M ρ (t) ≤ b (21)
The aim of the approach is to compute the components of the vector u(t) which will satisfy the marking constraint [START_REF] Lopes | Modeling and control of legged locomotion via switching max-plus models[END_REF]. We always assume that the input transition t j of the constrained path ρ is connected to at least one input transition t uc of the considered TEG. This path is noted α c with c = 1 to µ. The timing of this path is denoted by τ αc .

Theorem 4 Let G be a partially observable TEG with µ input transitions, whose dynamics is modeled by the equation ( 2). This graph is subject to a marking constraint represented in [START_REF] Lopes | Modeling and control of legged locomotion via switching max-plus models[END_REF].

The marking constraint is satisfied if there exists an index c such as

u c (t) = F 1c • x r (t -τ δz -1) ⊕ F 2c • x r (t -1). ( 22
)
where

F 1c = ⊕ β z=1 [⊕ r∈ω [(b + m δz -M 0ρ ) -(A τα • B) jc • A τα+1 ir ) F 2c = ⊕ r∈ω [(b -M 0ρ -(A τα • B) jc )] • A (τα+1) ir
With u l (t) = ε for l = c and if the condition given by

(A τα s • B) dc ≤ (b -M 0ρ0 )(A k • B) d l (23)
for l = 1 to µ is satisfied.

Proof The procedure to calculate the control laws u c (t) for a given c ∈ {1, ..., µ}, is similar that the controller synthesis u(t) proposed in the Theorem 2. In this case, the formula [START_REF] Lopes | On the synchronization of cyclic discrete-event systems[END_REF] of the control law u c (t) is given if the condition ( 21) is satisfied for an index c and then the other controls can be forced to ε.

b.Multiple constraints In this section, we extend the work to deal with the general case, where a partially observable TEG has µ input transitions, and is subject to s marking constraints with s = 1 to S. In this case as well we assume that each path ρ s is connected to at least one input transition by a path α s where ρ s represents the constrained path. The marking constraints are given by the following inequality:

M ρs (t) ≤ b s , s = 1 to S. (24)
The following Theorem gives the results for this study.

Theorem 5 Let G be a partially observable TEG with µ input transitions and is subject to S constraints of the form [START_REF] Andrade | On the control of max-plus linear system subject to state restriction[END_REF]. The following control law:

u(t) = S s=1 u cs (t) (25)
requires compliance with the constraints [START_REF] Andrade | On the control of max-plus linear system subject to state restriction[END_REF] if the condition holds true for each s = 1 to S Proof The control law [START_REF] Andrade | Some results on the feedback control of max-plus linear systems under state constrains[END_REF] provided from the Theorem 5, ensures the respect of one constraint. For the case where we have several constraints, the controller represented by the equation ( 22) ensures the satisfaction of all considered constraints (25) if the condition ( 23) is verified for each constraint.

Example

We consider the example of the timed event graph described in the section 3. We suppose that the place p 32 subject to a marking constraint, as it was represented in the section 3, where the number of tokens in the place p 32 cannot exceed 1 token. This constraint can be reformulated in the form of the following inequality:

x 2 (t) ≤ x 3 (t) + 1 (26)
We associate for each transition of the TEG, represented by the Fig. 2, a counter function (θ i (t)) for i = 1 to i = 4. The behavior of this graph can be described by following expressions:

       θ 1 (t) = min(0 + u(t)), 1 + θ 2 (t), θ 2 (t) = 0 + θ 1 (t -1), θ 3 (t) = min(0 + θ 2 (t -2)), 1 + θ 4 (t), θ 4 (t) = 0 + θ 3 (t -3),
Using Min-Plus algebra operator (⊕) for the min operator and (•) for the classical sum +, these equations can be written as:

       θ 1 (t) = u(t) ⊕ 1 • θ 2 (t), θ 2 (t) = θ 1 (t -1), θ 3 (t) = θ 2 (t -2) ⊕ 1 • θ 4 (t), θ 4 (t) = θ 3 (t -3),
We finally get to the following matrix notation:

θ(t) = A 0 • θ(t) ⊕ A 1 • θ(t -1) ⊕ A 2 (t -2) ⊕ A 3 (t -3) ⊕ B 0 • u(t) (27)
with:

θ(t) =     θ 1 (t) θ 2 (t) θ 3 (t) θ 4 (t)     , A 0 =     ε 1 ε ε ε ε ε ε ε ε ε 1 ε ε ε ε.     , A 1 =     ε ε ε ε e ε ε ε ε ε ε ε ε ε ε ε.     , A 3 =     ε ε ε ε ε ε ε ε ε e ε ε ε ε ε ε     , A 4 =     ε ε ε ε ε ε ε ε ε ε ε ε ε ε e ε     , B 0 =     e ε ε ε    
Using the Theorem 1, the implicit equation ( 27) can be replaced by the following explicit solution,

θ(t) = A * 0 • A 1 • θ(t -1) ⊕ A * 0 • A 2 • θ(t -2) ⊕ A * 0 • A 3 • θ(t -3) ⊕ A * 0 • B 0 • u(t)
where A * 0 is the Kleene star of A 0 .

Our objective is to calculate a realizable controller which satisfies this marking constraint. For this we apply Theorem 2. In order to obtain the state equation of this system, we decompose each place of the TEG shown in Fig. 2 with time greater than 1 time unit to several places timed to 1 time unit. The initial graph of the information transmission network (see the Fig. 2) has been extended to the new timed event graph represented by the Fig. 3). The place p 32 and the place p 43 timed respectively with 2 and 3 time units has been split respectively to 2 and 3 places timed to 1 time unit, and the intermediate transition t 5 , t 6 and t 7 has been added.

Fig. 3 The extended TEG We apply for this example the Min-Plus modeling approach of TEG described in section 2.2.2. The resulting state representation of the extended graph is given below:

x(t) =           1 ε ε ε ε ε ε 1 ε ε ε ε ε ε ε ε ε ε ε 1 ε ε ε ε ε ε 1 ε e ε ε ε ε ε ε ε e ε ε ε ε ε ε ε ε ε e ε           • x(t -1) ⊕           e ε ε ε ε ε ε           • u(t). ( 28
)
We check the compliance of required conditions by the Theorem 2 for the existence of the control laws which guarantee the constraint formalized by the inequality 26. There exists a path α from transition t u to transition t 2 , with timing τ α = 1. Then φ = τ α + 1 = 2. We have also, (A k • B) i = [e, ε], for k = 0 to 1, where (A τα • B) j = 1, where the condition of the Theorem 2 is satisfied. We have also each loop of the considered TEG containing at last one observable transition. According to the Theorem 2, there exists a realizable control law which satisfies the marking constraint. For this example, we have A 2 3r = [ε e ε ε ε 1 ε]. When take into account the equation ( 10), the control law is written as follows:

u(t) = (1 -1) • [e • x 1 (t -1 -1) ⊕ 1 • x 3 (t -2 -1) = 1 • x 1 (t -2) ⊕ 1 • x 3 (t -3)
Using the propriety of the Min-Plus algebra, this controller can be simplified to:

u(t) = 1 • x 1 (t -2)
This feedback control is represented by a marked and a timed monitor place in Fig. 4 (place represented by a double circle) Fig. 4 The controlled TEG of the example

Conclusion

This paper presents a new approach to design a controller which satisfies marking constraints for partially observable timed event graphs (TEGs). In these studies, the behavior of TEGs is described by linear Min-Plus equations and the different marking marking constraints are translated by inequalities in Min-Plus algebra. We have established a sucient condition for the existence of control laws which guarantee the respect of these tokens capacity constraints in some paths in TEGs. To develop this control strategy, we have taken into account the hypothesis where for each loop of the TEG considered, there exists always a path connecting the unobservable transition to the observable transition. For this, we have calculated realizable feedback controls expressed in function of observable counters with the aim of ensuring that marking constraints are met. We first considered the case where the problem is to limit the number of tokens in one place of TEG. The approach is then extended to the case where the control law imposes a upper bound of tokens in a sequence of places. The results in this paper include the single control and the multivariable control. For future work, it is important to relax the hypothesis of the cyclicity for TEG and it would be interesting to calculate the optimal control under a given criterion. We believe that this contribution could be valuable in various applications as well notably in flexible manufacturing workshops, communication processes and transportation networks.

Fig. 1 Fig. 2

 12 Fig. 1 Example of an information transmission network