Guillermo Ahumada

Jocelyn Oyarce

Thierry Roisnel

Samia Kahlal

María Angélica

Del Valle

David Carrillo

Jean-Yves Saillard

Jean-René Hamon
email: jean-rene.hamon@univ-rennes1.fr

Carolina Manzur
email: cecilia.manzur@pucv.cl

Prof Jean-Yves

Synthesis, structures, electrochemical and quantum chemical investigations of Ni(II) and Cu(II) complexes with a tetradentate Schiff base derived from 1-(2-thienyl)-1,3-butanedione †, ‡

Double condensation of the newly prepared 1-(2-thienyl)-1,3-butanedione with ethylenediamine lead to the formation of the novel symmetrical Schiff base proligand bearing two potentially electropolymerizable 2-thienyl groups. Both organic species were obtained in 55 and 80% yields, respectively. They exist as their respective keto-enol and enaminone tautomeric forms that were computed to be more stable by 8.6 and 30.3 kcal/mol than their βdiketone and keto-imine isomers. The corresponding Ni(II) and Cu(II) complexes featuring a N2O2-tetradentate Schiff base ligand were readily synthesized upon reaction of the diprotic Schiff base precursor with the appropriate hydrated metal acetates, and isolated as neutral, air and thermally stable solids in excellent yields (84 and 90%). The four compounds were characterized using various analytical and spectroscopic methods, and by X-ray diffraction study for the two coordination complexes. Both Ni(II) and Cu(II) metal ions are fourcoordinated and adopt a perfect square planar environment (τ4 value of 0.036 and 0.025) with two nitrogen and two oxygen atoms as donors. Both complexes were analyzed by cyclic voltammetry experiments showing a decrease of the current response per cycle, indicating the formation of oligomeric units. This was verified by their doping/undoping responses.

Optimized geometries of the four compounds as well as the electronic structures of the two Ni(II) and Cu(II) complexes and their respective cations have been analysed through DFT calculations, allowing providing a consistent view of their structure and properties. TDDFT calculations have been used to interpret the major features of the UV-vis. spectra.

Introduction

Almost 130 years ago, Combes reported the first and simplest symmetrical Schiff base derivative resulting from condensation of acetylacetone with ethylenediamine in the 2:1 molar ratio, as well as its corresponding copper(II) complex obtained upon reaction with cupric acetate. 1 Since then, the preparation and study of the class of acyclic N2O2 tetradentate based Schiff base ligands and their complexes greatly participated to the booming of a very fertile and dynamic research field in coordination chemistry. [START_REF] Hernandez-Molina | Acyclic and Macrocyclic Schiff Base Ligands, Comprehensive Coordination Chemistry II[END_REF][START_REF] Vigato | [END_REF][4] The versatility of Schiff base ligands arises from their structural and electronic variations, achieved by varying the stereo-electronic nature of the substituents incorporated on their skeletons via the β-diketone starting materials. 5 Those easily synthetized and easily tunable dicarbonyl building blocks, presenting a keto-enol tautomerism, [6][7][8] are readily tailored to bring about donor, acceptor, polymerizable, or electroactive groups at the three carbon atoms constituting the propane-1,3-dione skeleton. [7][8][9][10][11][12][13][14] β-diketones containing a thiophene group are of particular interest because they can be used as starting materials for the construction of Schiff base derivatives and metal complexes tagged with electroactive thiophene moieties, thus allowing electrochemical polymerization to conveniently enable the direct preparation of an insoluble polymeric coating on the surface of the electrode. [15][16][17] In this context, we have recently disclosed the generation and characterization of a polythienyl-containing β-diketonate copper(II) compound and its deposition on an electrode, upon electropolymerization of the [Cu(II)-bis-{1-(2-thienyl)-3-(3-thienyl)propane-1,3-dionato}] complex using potentiodynamic method. 18 By contrast, no deposits of polymeric species on the electrode surface could be observed upon changing the thiophen-3-yl group for the more electron-withdrawing 4-fluorophenyl unit. 19 A similar observation was made with a series of cobalt(II), nickel(II) and copper(II) complexes of a fluorinated Schiff base ligand formally resulting from the condensation of 2-thenoyltrifluoroacetone (TTA) with 1,2diaminoethane; cyclic voltammograms of these complexes displayed indeed oxidation waves, but none of them formed polymer films upon repeated cycling. 20 Those experiments clearly demonstrate that the electronics play an important role in the final outcome of the electropolymerization process. Therefore, pursuing along this line, we thought that such electropolymerization could be applied to first raw transition metal complexes with the symmetrically-substituted electron-rich Schiff bases 2 obtained by double condensation of ethylenediamine with the known 2-thenoylacetone 1, that mainly exists as its keto-enol tautomer (Z)-4-Hydroxy-4-(2-thienyl)-3-buten-2-one 10,11 (see formulas in Scheme 1).

Results and Discussion

Synthesis and characterization

The asymmetric β-diketone 1 was prepared in a two-step procedure (see Scheme 1) and isolated in 55% yield as a white solid that decomposes above 50 °C. 2-acetylthiophene was first treated with potassium tert-butoxide and ethyl acetate at ambient temperature, 21 followed by a templated purification procedure of the formed β-diketone using copper(II) acetate (see Experimental). In a recent work, Prabhu et al. 11 used NaH (60% suspension in paraffin oil) as the base initiator in THF at 0 °C, and obtained 1 as a yellow liquid (no yield communicated).

Compound 1 has also been synthetized by the LDA-mediated reaction of acetone with 2thiophenecarbonyl chloride in THF at -78 °C and isolated in 34% yield as a yellowish oil. 10 Although we prepared compound 1 as a white solid whose high purity was established by combustion analysis, its IR, 1 H and 13 C NMR spectral data (see Experimental) fully agree with the reported keto-enol structure (Z)-4-Hydroxy-4-(2-thienyl)-3-buten-2-one elucidated by 2D NMR method. 10 Only this isomer is found in solution and in the solid-state. Note that in the case of the NaH-mediated reaction, the isolated product is a 85/15 keto-enol/diketone tautomeric mixture. The symmetrically-substituted Schiff base 2 was readily synthesized by condensation of 1,2-diaminoethane with 2 equivalents of 1 in dry ethanol for 18 h at ambient temperature (Scheme 1). Compound 2 was isolated in 80% yield as an air and moisture sensitive yellow powder. Like 1, it is better to store 2 neat under a nitrogen atmosphere at low temperature (-30 °C). All attempts to increase the yield or to reduce the reaction time by either heating or using catalyst such as acetic acid lead to decomposition of the final product. Schiff base 2 is highly soluble in common organic solvents (CH2Cl2, THF, Et2O, n-hexane).

Complexes 3 and 4 were facilely obtained by reacting the Schiff base proligand 2 with either hydrated nickel or copper acetate salt in methanol at ambient temperature for 4 h (Scheme 2). The two compounds were isolated as air and thermally stable, and moisture insensitive on storage under ordinary conditions, light brown 3 and green 4 solids in excellent yields of 84 and 90%, respectively. They show good solubility in polar organic solvents such as dichloromethane, tetrahydrofuran and methanol, but are insoluble in diethyl ether and hydrocarbon solvents. A c c e p t e d m a n u s c r i p t found values for carbon, hydrogen, nitrogen, and sulfur are in good agreement with each other, which established that the four species are amply pure in bulk. The proposed structures are based on spectroscopic characteristics (FT-IR, UV-vis., 1 H and 13 C NMR of diamagnetic 1-3), and single crystal X-ray diffraction studies of both complexes 3 and 4. Moreover, the structure of the paramagnetic Cu(II) derivative 4, was confirmed by HRMS ESI + , where the sodium aggregate appeared at m/z = 444.00 (see Experimental).

The solid-state FT-IR spectrum of the β-diketone 1 exhibited a strong band at 3460 cm - 1 assigned to the ν(O-H) stretching mode, and a series of three strong to medium intensity band at 1615, 1519 and 1413 cm -1 attributed to the C … O and C … C stretching vibrations. The lack of ν(C=O) absorption observed at 1721 cm -1 for a neat liquid sample of 1, 11 clearly suggests that the presently prepared 1 does only exist as its keto-enol tautomeric form in the solid-state. In the spectrum of the free Schiff base 2, the weak band observed at 3444 cm -1 is due to the ν(N-H) vibration, thus indicating that the enamine tautomer is the predominant form in the solidstate. 22 Three strong characteristic absorptions due to the [O … C … C … C … N] enaminone core are also observed at 1595, 1558 and 1529 cm -1 . In the spectra of complexes 3 and 4, those bands decrease in energy by 14/7, 28/31 and 21/23 cm -1 , respectively, in agreement with coordination of the dianionic Schiff base ligand 2 2-to the Ni(II) and Cu(II) ions through the oxygen and nitrogen atoms, thus forming a six-membered heterometallacycle. Furthermore, this is confirmed by the disappearance of the (N-H) absorption band seen in precursor 2. Finally, each compound showed a medium intensity band in the range 705-750 cm -1 assigned to the outof-plane bending vibration mode of the C-H groups of the 2-thiophenyl unit. [START_REF] Molina | Thiophenes and their Benzo Derivatives: Structure, Comprehensive Heterocyclic Chemistry III[END_REF] The 1 H NMR spectrum of compound 1 (Fig. S1, ESI † ) showed two sharp and one broad signals at H 2.11, 6.29 and 15.80 ppm with integral ratio 3:1:1 attributed to the methyl, methine and bridging enol protons, respectively. That of the free Schiff base 2 (Fig. S1,ESI † ) presented a similar peak and integration pattern with the protons of the methyl, methine and intramolecularly hydrogen bonded amine groups showing up upfield shifted at H 2.03, 5.56 and 11.17 ppm, respectively. Additionally, the single peak of the four magnetically equivalent protons of the ethylene linker was seen at H 3.51 ppm. The downfield chemical shifts observed in 1 and 2 clearly suggest that both their respective enol and enaminone forms that are stabilized by intramolecular hydrogen bonding predominate in solution. Furthermore, the molecular structure of 2 was unambiguously established by 2D-HMBC 1 H-13 C NMR experiments. The A c c e p t e d m a n u s c r i p t 2D-spectrum (Fig. S2,ESI † ) showed that the enamino carbon signal at C 164.58 ppm can be correlated with the methyl, methylene and methine proton resonances, thus firmly demonstrating that the acetyl carbonyl group of 1 reacted with the amino group of 1,2diaminoethane to form the C-N bond of the enamine fragment. By contrast, the carbonyl carbon signal at C 181.28 ppm was found to only correlate with that of the methine proton. This proposed structure of 2 was later confirmed as its doubly deprotonated ligand form in the crystal structures of complexes 3 and 4 (see below). This Schiff condensation, taking place at the acetyl carbonyl carbon of 1, sharply contrasts with the opposite situation observed with the related 2thenoyltrifluoroacetone, where the condensation occurred at the thenoyl carbonyl carbon. [START_REF] Asiri | [END_REF] The crystal structure of the enol form of the fluorinated diketone starting material showed that the hydroxyl group is adjacent to the CF3 unit. 25 In the 1 H NMR spectrum of complex 3 (Fig. S1,ESI † ), the signal of the amine proton vanished while those of methyl, methylene and methine groups are slightly upfield shifted, appearing at H 1.96, 3.08 and 5.55 ppm, respectively. The proton-decoupled 13 C NMR spectral data (see Experimental for complete assignments) confirms both the proposed keto-enol and keto-enamine structures of 1 and 2, respectively, with resonances attributed to the methine carbon nucleus observed at C 96.87 and 92.61 ppm, as well as the symmetrical nature of the free Schiff base 2 and its Ni(II) complex 3 with the nine lines observed in their respective spectra (Figure S3, esi † ).

Electronic absorption spectra

The UV-vis. spectra of the 2-thenoylacetone 1 (Fig. S4, ESI † ), of the Schiff base proligand 2 and its related metal(II) complexes 3 and 4 (Fig. 1) were recorded in dichloromethane solutions at 20 °C. Absorption spectral data are gathered in Table 1. In the spectrum of proligand 2, the observed maximum absorptions at 282 nm, and 362 nm are due to →* transitions, according to the TD-DFT simulated results (see below). The peak at 257 nm has a substantial →(CH2CH2)* character, due to its non-planar structure (see below and Fig. S5,ESI † ). TDDFT indicates also that these transitions do not exhibit any significant n→* character. Although the agreement between the experimental and simulated spectra of 3 and 4 is not as good as for 1 and 2, →* transitions were also found in the spectra of the complexes (LLCT in Table 1). The low energy band of 3 at 564 nm was not reproduced by TD-DFT and A c c e p t e d m a n u s c r i p t can be tentatively assigned to a forbidden d-d transition. That at 412 nm can be indexed as of MLCT character. In the case of 4, the two low-energy bands are found to be of LMCT nature. a Spectra simulated assuming Gaussian half-height bandwidths of 3000 cm -1

X-ray Crystallographic Study

Crystals of complexes 3 and 4 suitable for X-ray diffraction analysis were obtained by slow evaporation of their respective saturated ethanolic solution. A perspective view of 3 is presented in Fig. 2. The molecular structure of 4, whose structure determination resulted in a poor resolution (Rint = 0.1449 and R1 = 0.1103), is displayed in Fig. S6 (ESI † ), showing the atom connectivity. Selected bond distances and angles of the first metal(II) coordination sphere of 3 and 4 are given in Table 2, while those of the Schiff base ligands of the two complexes are provided in Table S1 (ESI † ). Both complexes crystallize in the monoclinic crystal system in the centrosymmetric space group P21/n with a single molecule in the asymmetric unit. As expected, the X-ray diffraction study of 3 and 4 confirm the proposed structure of proligand 2 (see 2D NMR above), and that both complexes consist of a metal(II) ion tetracoordinated by the N2O2 donor set of the same dianionic Schiff base ligand symmetrically substituted with methyl and 2-thienyl groups, forming monomeric units. Both complexes are isostructural species that differ only by the nature of the M(II)-centered metal ion, M = Ni for 3 and Cu for 4. The 2-thienyl rings are strongly disordered thus precluding any further discussion of their metrical parameters. We previously observed such a disorder, albeit much weaker, in structures of thiophen-containing compounds solved at low temperature. [18][19][20] All close intermolecular contacts correspond to van der Waals interactions. 2) of 1.850, 1.856, 1.911 and 1.930 Å, respectively. The extent of deformation in a tetracoordinated complex can be quantified by the four-coordinate geometry τ4 index. 26 This parameter that measures the extent of distortion between a perfect square planar geometry

Electrochemical studies

The electrochemical oxidation of complexes 3 and 4 was investigated by cyclic voltammetry (CV) using acetonitrile as solvent, at ambient temperature. All redox potentials are quoted versus saturated calomel electrode (SCE). CV measurements were carried out in the potential range previously established as optimal for each compound: -0.2 to 1.8 V for complex 3 and 0.1 to 1.8 V for complex 4, respectively. Both complexes present a similar potentiodynamic behavior where two oxidation processes were found, the first one, E1, around 1.0 V and a second one, E2, around 1.5 V (figs. 3 and4). The first peak is attributed to the oxidation of the metal center, whose peak of the respective reduction, when reversing the potential sweep, is observed in the first case (compound 3) at around 1.0 V, and in the second case (compound 4) near 0.8 V. This shows that the first redox process of 3 is electrochemically A c c e p t e d m a n u s c r i p t reversible (ΔEp < 60 mV), so that its potentials do not vary depending on the number of successive cycles, however, in the case of compound 4, it is a quasi-reversible process (ΔEp > 60 mV), which is manifested not only by the difference between its oxidation and reduction peak potentials, but also by the reduction potential, which moves towards less positive values depending on the number of cycles. At the same time, in both cases the currents of these peaks are decreasing, and practically disappearing in cycle 20.

The second oxidation peak is attributed to the oxidation of the thienyl moiety. In both cases, in the response of the modified electrodes a reduction process is observed around 0.0 V and it could be attributed to the p-doping/undoping process characteristic of modified electrodes with deposits of oligomers or conducting polymers. It is important to note that, also in this oxidation process, responsible for the formation of oligomers of chains that are lengthened by successive scans, the current is decreasing. This is attributable to the low conductivity of the deposit formed and/or to the fact that the oligomeric structure prevents the growth of the chains enough to form the polymer, reaching only oligomer levels. After 20 cycles, the working electrodes were transferred to a solution containing only the supporting electrolyte in order to analyze its responses. As it is shown in the right part of Figs. 3 and4, a p-doping/undoping process is observed, with current responses resembling that of analogous deposits prepared under similar conditions from complexes previously synthesized by us. 18,29,30 In any case, it should be noted that both deposits have a fully reversible charge response, as is the case with the other thiophene derivatives studied, they have been proposed to be tested as battery cathodes. 30,31 These results, which do not seem very auspicious in that sense, will allow to propose structural modifications in search of the preparation of metal containing polymers.

Theoretical investigations

DFT calculations were performed on compounds 1-4, in order to shed some light on their electronic structure and understand their structure and properties. Details of the calculations are given in the Experimental section. Their fully optimized geometries are provided as xmol files in the ESI † . Consistently with the experimental observations (see above), the keto-enol and enaminone forms of 1 and 2 were found more stable than their alternative tautomers by 8.6 and 30.3 kcal•mol -1 , respectively. It is noteworthy that the free ligand 2 (Scheme 1) does not adopt a planar structure, but affords a rotation around the CH2-CH2 bridge in such a way the two pseudo 6-membered heterocycles are far from each other (see Fig. S5,

3.

With one more electron, the dx2-y2-based orbital of 4 is singly occupied, i.e. splits into one occupied and one unoccupied spinorbital. It should be noted that the 3d-type levels of 4 are lying far below the HOMO, a situation common for such Cu(II) square-planar species. 19,20 Calculations on the oxidized forms of 3 and 4 were also performed to better understand their electrochemical behavior. The computed spin densities of 3 +/2+ and 4 2+ are shown in Fig. 6. 4 + was found to be a closed shell isoelectronic to 3. The shape of the 3 + spin density indicates that the first oxidation occurs on the HOMO of 3 which has an important metal character. The ground state of 3 2+ was found to be a triplet, with the second oxidation involving the HOMO-2. However, substantial electron reorganization occurs upon oxidation, which relocates spin density on the ligands, with a significant participation of the thienyl moieties, in full agreement with the electrochemical data (see above). Similarly, the first oxidation of 4 occurs at the HOMO. The ground state of 4 2+ was found to be a doublet and resulting also from an important reorganization of the spin density on the thienyl groups.

A c c e p t e d m a n u s c r i p t TD-DFT calculations were performed on compounds 1-4 (see Table 1 and Fig. S8, ESI † ) in order to index their optical transitions of low energy. As said above, the computed low- This difference comes from the fact that the d-type levels of 4 lie at much lower energy than those of 3 (see Figure 5).

Conclusions

In

Experimental Materials and physical measurements

Reactions were performed under dry dinitrogen or argon atmosphere using standard Schlenk techniques. Solvents were dried and distilled according to standard procedures. [START_REF] Armarego | Purification of Laboratory Chemicals[END_REF] All starting materials were purchased from commercial sources and used as received.

Synthesis of 1-(2-thienyl)-1,3-butanedione (1)

A Schlenk tube was charged with a magnetic stirring bar, 1.00 g (7.93 mmol) of 2-acetyl thiophene and 30 mL of dry THF. The solution was stirred for 5 min. before adding 0.89 g (7.93 mmol) of t-BuOK, turning the solution from translucent to purple color.

Synthesis of the Ni(II) Schiff base complex 3

A Schlenk tube was charged with a magnetic stirring bar, 1.0 g (2.30 mmol) of 2, 0.57 g (2.30 mmol) of nickel(II) acetate tetrahydrate and 15 mL of methanol. The reaction mixture was stirred for 4 h at r.t. Then, the resulting solution was evaporated under reduced pressure and the solid residue was washed with 100 mL of water and filtered off using a glass frit. The collected material was dissolved in CH2Cl2 and dried over MgSO4. The suspension was filtered and evaporated under reduced pressure. The product was dried under vacuum, affording 0.81 g

A

  c c e p t e d m a n u s c r i p t Herein, we report on a new template synthesis of 1, the preparation of the Schiff base proligand 2 and its reactions with the appropriate metal(II) acetate salt in order to synthesize the N2O2-tetradentate Schiff base complexes of Ni(II) 3 and Cu(II) 4 (see formulas in Schemes 2). All the compounds were fully characterized by analytical and spectroscopic techniques. Both complexes 3 and 4 were authenticated by single crystal X-ray diffraction analysis. In addition, the geometries and electronic properties of 1-4 were investigated by density functional (DFT) and time-dependent DFT (TD-DFT) calculation. Electrochemical study of complexes 3-4 was carried out by cyclic voltammetry, showing the formation of low conductivity oligomeric units that possess a fully reversible charge response.

11

 11

A c c e p t e d m a n u s c r i p t Scheme 1 .

 1 Scheme 1. Synthesis of the β-diketone 1 (keto-enol tautomer) and its derivatized Schiff base 2.

Scheme 2 .

 2 Scheme 2. Synthesis of the symmetrically-substituted Schiff base complexes 3 and 4.

Fig. 1

 1 Fig. 1 UV spectra (top) and Vis spectra (bottom) of proligand 2 (solid line) and complexes 3 (dashed line) and 4 (dotted line), measured in CH2Cl2 solutions at 20 °C.

Fig. 2

 2 Fig.2Molecular structure of the Ni(II) complex 3 with a selected atom numbering scheme.Thermal ellipsoids are drawn at the 50% probability level.

(

  τ4 = 0) and a perfect tetrahedral geometry (τ4 = 1), is obtained by the formula: τ4 = [360° -(α + β)]/141°, α and β being the two largest bond angles of the metal coordination sphere. The Ni(II) complex 3 has a τ4 value of 0.036 over 0.025 for the Cu(II) congener 4. The central Ni(II) and Cu(II) atoms barely deviate from the basal plane by 0.019(1) Å and 0.024(3) Å, respectively. The angular summations of 359.97° (for 3) and 359.90° (for 4) are equal to the idealized value of 360° within the experimental error. Although the molecules are packed to form centrosymmetric pairs in which the metal centers are superimposed on top of each other and the nitrogens of the coordination pocket of one molecule are aligned with the oxygens of the second molecule and vice versa (Fig. S7, ESI † ), there are no significant interactions in axial positions. The closest contacts are 3.4871(15) Å for Ni(1)•••Ni(1) #1 and 3.4490(12) Å for Cu(1)•••Cu(1) #1 (#1: 1-x,1-y,1-z). Interestingly, such M•••M separations are of the same order as those we previously observed for nickel and copper complexes with unsymmetricallysubstituted N2O2-tetradentate metallocene-containing Schiff base ligands. 27,28

A

  c c e p t e d m a n u s c r i p t On the other hand, the atoms that constitute the two six-membered heterometallacycles [M(1)-O(1)-C(6)C(7)-C(8)-N(1)]/[M(1)-O(2)-C(16)C(17)-C(18)-N(2)] (Fig. 2) formed upon the coordination of the doubly deprotonated Schiff base 2 to the metal(II) ion, are nearly coplanar in both 3 and 4, with rms deviations of 0.040/0.008 Å and 0.031/0.010 Å, respectively. This flatness, along with the O-C, C-C and C-N bond falling between single and double bond lengths and bond angles of sp 2 hybridized atoms (Table S1, ESI † ), suggests a significant electron delocalization within the chelating rings. These two six-membered metallacycles are also almost coplanar with their respective attached 2-thienyl rings, making dihedral angles of 2.66(9)/11.90(3)° for 3 and 5.20(9)/11.85(5)° for 4. Moreover, the five-membered metallocycle [M(1)-N(1)-C(10)-C(20)-N(2)] of both complexes 3 and 4 assumes an envelope conformation with C(10) as the flap atom (Fig. 2), deviating from the mean [N(1)-M(1)-N(2)-C(20)] plane by 0.401(7) and 0.3785(15) Å, respectively.

A c c e p t e d m a n u s c r i p t Table 2 .

 2 Selected X-ray and corresponding DFT-optimized (in square brackets) bond distances (Å) and angles (°) for the first metal(II) coordination sphere of compound 3 and 4. a A M = Ni, 3; Cu, 4.

Fig. 3

 3 Fig. 3 Voltamperometric profiles of compound 3 (left) registered during electro-oxidation of the complex by 20 successive potentiodynamic scans under the working conditions detailed in Experimental; deposit response at the right (5 successive voltamperometric scans).

Fig. 4

 4 Fig. 4 Voltamperometric profile of compound 4 (left) registered during electro-oxidation of the complex by 20 successive potentiodynamic scans under the working conditions detailed in Experimental; deposit response at the right (5 successive voltamperometric scans).

A c c e p t e d m a n u

  s c r i p t ESI † ). The optimized structures of 3 and 4 are in a very good agreement with the experimental ones (see selected metrical data in Tables2 and S1, ESI † ). 3 and 4 are 16-and 17-electron square-planar transition-metal complexes, respectively. It follows that the ground-state of 3 is a singlet, whereas that of 4 is a doublet. The MO diagrams of 3 and 4 are shown in Fig. 5. Due to the spin-unrestricted nature of the calculations, each Kohn-Sham MO level in the diagram of 4 is split into two spinorbitals: one spin-up (left) and one spin-down (right). Considering first the closed-shell situation of 3, its 3d atomic level generates in the complex a set of four nonbonding or weakly antibonding metal-based MOs which are occupied (Ni(II) = d 8 ) and a antibonding dx2-y2 orbital which is vacant. The latter is the LUMO, whereas three among the former constitute the three highest occupied orbitals, the fourth one being the HOMO-5. The substantial computed HOMO-LUMO gap is consistent with the stability and diamagnetism of

Fig. 5

 5 Fig. 5 Kohn-Sham orbital diagrams of 3 and 4. The localizations (in %) are given in the following order: metal/ligand. HOSO: highest occupied spin-orbital; LUSO: lowest unoccupied spin-orbital.

Fig. 6

 6 Fig. 6 Computed spin densities of the ground states of 3 +/2+ and 4 2+ .

A c c e p t e d m a n u s c r i p t energy transitions of 3 and 4

 4 have different character, respectively of MLCT and LMCT nature.

  this contribution, two new Ni(II) and Cu(II) complexes featuring a symmetricallysubstituted N2O2-tetradentate Schiff base ligand derived from the double condensation of 1-(2thienyl)-1,3-butanedione with ethylenediamine, were prepared. The two organic species and the two coordination compounds were fully characterized by a variety of analytical and spectroscopic methods, as well as by X-ray crystallography for the nickel and copper complexes. 2D NMR experiments established that the Schiff condensation took place at the acetyl carbon of the diketone while geometry optimization depicted a non co-planarity of the two formed six-membered enaminone rings. Both the Ni(II) and Cu(II) metal ions adopt a perfect square planar geometry with partial delocalization of bonding electron density throughout the six membered heterometallacycle frameworks. Geometry, bonding, electronic structure and electrochemical properties have been rationalized by DFT calculations. Indexation of the optical transitions were supported by TD-DFT calculations. Cyclic voltammetry exhibits a similar potentiodynamic behavior for the Ni (II) and Cu (II) complexeswith two oxidation processes occurring in each case. The first reversible event is associated to M(II)/M(III) redox couple while the second oxidation process is responsible for the formation of a low conductivity deposit of oligomeric units. Extension of the conjugation between the electroactive thienyl group and the metal core could facilitate the electrochemically-induced polymerization by the stabilization of the radical cation formed in the process, and also increase the conductivity of the final deposit. Work is currently being devoted toward this goal.

( 1 .

 1 93 mmol, 84% yield) of a light brown powder. Suitable single crystals for structure determination by X-ray diffraction were obtained by slow evaporation of an ethanolic solution of 3. m.p. 288-289 °C. Found: C, 51.74; H, 4.37; N, 6.58; S, 15.01. Calcd for C18H18N2NiO2S2, requires: C, 51.82; H, 4.35; N, 6.72; S, 15.37%. FT-IR (KBr, cm -1 ): 3100 (w), 3070 (w) (C-H arom); 2990(w), 2941 (w), 2845 (w) (C-H aliph); 1581(s), 1530(s) and 1508(vs) (C … O), A c c e p t e d m a n u s c r i p t A c c e p t e d m a n u s c r i p t

Table 1 .

 1 Experimental and computed electronic absorption data of compounds 1-4.

			m a n u s c r i p t
	λmax (exp.) / A c c e p t e d λmax (simul.) / nm a	Major transition involved
	Compd.	(CH2Cl2) nm (log )	(from TD-DFT) (rel. absorbance)	(from TD-DFT)
	1	265 (4.22) 320 (4.61)	294 (8491) 337 (23433)	→* →*
	2	257 (4.19) 282 (4.04)	258 (8878) 302 (27475)	→* →*

  the purity of the organic compounds was carried out by TLC performed on silica coated Merck kieselgel 60F254 0.25 mm plates and visualized by UV irradiation at 254 nm. Solid-state FT-IR spectra were recorded on a Perkin-Elmer Model 1600 FT-IR spectrophotometer with KBr disks in the 4000 to 450 cm -1 range. Uv/Vis spectra were recorded using a Thermo Scientific, Helios Omega spectrophotometer.1 H and 13 C NMR spectra were acquired at 298 K with a Bruker 300 FT-NMR spectrometer. Chemical shifts () are reported in parts per million (ppm, ), and referenced to the residual deuterated solvent peaks. Coupling constants (J) are reported in Hertz (Hz), and integrations are reported as the number of protons.1 H and 13 C NMR chemical shift assignments are supported by data obtained from 1 H-1 H COSY and 1 H-13 C HSQC NMR experiments. High Resolution Electrospray Mass Spectra (HR-ESI-MS) were collected on a Bruker MAXI 4G spectrometer at the Centre Régional de Mesures Physiques de l'Ouest (CRMPO) at the Université de Rennes 1, France. Elemental analyses were conducted on a Thermo-Finnigan Flash EA 1112 CHNS/O analyzer by the Microanalytical Service of the CRMPO. Melting points were measured in evacuated capillaries on a Kofler Bristoline melting point apparatus and were not corrected.

	m a n u s c r i p t
	A c c e p t e d
	Checking of

  aliph); 1615 (vs), 1519 (m), 1413(m) (C … O) and (C … C); 722 (m) (C-H SC4H3).1 H NMR (300 MHz, (CD3)2CO): H 2.11 (s, 3 H, CH3), 6.29 (s, 1 H, CH=C), 7.21 (dd, JH,H = 4.9 and 3.8 Hz, 1 H, C4H3S), 7.86 (dd, JH,H = 4.4 and 1.4 Hz, 2 H, C4H3S), 15.80 (s, 1 H, O-H). 13 C{ 1 H} NMR (75.48 MHz, (CD3)2CO): C 23.44 (CH3), 96.87 (CH=C), 129.34 (C4H3S), 131.67 (C4H3S), 134.07 (C4H3S), 142.64 (Cipso, C4H3S), 183.35 (CH=C), 187.92 (C=O). A Schlenk tube was loaded with a magnetic stirring bar, 1.00 g (5.94 mmol) of 1 and 30 mL of dry ethanol. The solution was stirred for 5 min, and 0.20 mL (2.97 mmol) of 1,2diaminoethane was added dropwise by syringe. The reaction mixture was stirred at r.t. for 18 h. The volatiles were evaporated and the solid residue dried under vacuum, providing 0.86 g (2.38 mmol, 80% yield) of a yellow powder. m.p. 73-74 °C (dec). found: C, 59.68; H, 5.66; N, 7.63; S, 17.50. Calcd for C18H20N2O2S2, requires: C, 59.97; H, 5.59; N, 7.77; S, 17.79%. FT-IR (KBr, cm -1 ): 3444 (w) (N-H); 3082 (vw), 3066(vw) (C-H arom); 2980(vw), 2940 (vw), H NMR (300 MHz, CDCl3): H 2.03 (s, 6 H, CH3), 3.51 (s, 4 H, CH2), 5.56 (s, 2 H, CH=C), 7.04 (dd, JH,H = 4.9 and 3.7 Hz , 2 H, SC4H3), 7.43 (dd, JH,H = 4.9 and 1.1 Hz, 2 H, SC4H3), 7.51 (dd, JH,H = 3.7 and 1.1 Hz, 2 H, SC4H3), 11.17 (s, 2 H, N-H). 13 C{ 1 H} NMR (75.48 MHz, CDCl3): C 19.18 (CH3), 43.81 (CH2), 92.61 (-CH=C), 127.43 (C4H3S), 127.75 (C4H3S), 129.98 (C4H3S), 146.85 (Cipso, C4H3S), 164.58 (CH=C), 181.28 (C=O).

	Synthesis of the Schiff base proligand 2
	m a n u s c r i p t
	The reaction mixture was stirred for 45 min. Then, 1.94 mL (19.8 mmol) of ethyl acetate was added dropwise by syringe. After 30 min. of stirring, the volatiles were removed under vacuum, the solid residue was washed with 50 mL of an 1% aqueous HCl solution and extracted with diethyl ether (3 x 100 mL). The extracts were combined and evaporated to dryness. The solid material was absorbed on a column packed with silica gel (grade 60). Elution was carried out with hexane/ethyl acetate (24:1 v/v), and the collected colorless solution taken to dryness, affording 0.80 g of a white product. The white product was transferred into a new Schlenk tube along with 475 mg (2.38 mmol) of copper (II) acetate monohydrate and dissolved in 10 mL of methanol. The solution was stirred at 60 °C for 3 h. After cooling, 50 mL of an aqueous 1% A c c e p t e d
	HCl solution was added and the reaction mixture extracted with dichloromethane (3 x 100 mL).
	The extracts were combined, dried over anhydrous MgSO4, filtered off and finally dried under

vacuum, affording 0.73 g (4.34 mmol, 55% yield) of pure 1. m.p. 51-52 °C (dec). Found: C, 57.07; H, 4.77; S, 18.93. Calcd for C8H8O2S, requires: C, 57.12; H, 4.79; S, 19.06%. FT-IR (KBr, cm -1 ): 3460(vs) (O-H); 3110(w), 3089(w) (C-H arom); 2969(vw), 2914(vw), (C-H 2876 (vw) (C-H aliph); 1595(vs), 1558(s), 1529(s) (C … O), (C … N) and/or (C … C); 705 (m) (C-H C4H3S). 1

(39295) →*

Acknowledgments

We thank F. Lambert (CRMPO, Rennes) for helpful assistance with MS measurements. Financial support from the Fondo Nacional de Desarrollo Científico y Tecnológico [FONDECYT (Chile), grant no. 1140903 (C.M. and D.C.)], FONDEQUIP [EQM130154], the Vicerrectoría de Investigación y Estudios Avanzados, Pontificia Universidad Católica de Valparaíso, Chile (C.M. and D.C.), the CNRS and the Université de Rennes 1 is gratefully acknowledged. The GENCI-CINES and GENCI-IDRISS French national centers are acknowledged for computational resources (Grant a0010807367). This research has been A c c e p t e d m a n u s c r i p t performed as part of the Chilean-French International Associated Laboratory "Multifunctional Molecules and Materials" (LIA M3 -CNRS N° 1207). G.A. and J.O. thank the CONICYT (Chile) for support of graduate fellowships N°21120098 and N°21171363, respectively.

Synthesis of the Cu(II) Schiff base complex 4

Complex 4 was synthesized following an identical procedure to that described above for 3, using in this case 0.46 g (2.30 mmol) of copper(II) acetate monohydrate. Yield: 0.87 g

X-ray crystal structure determination

A crystal of appropriate size and shape of each of the compounds 3 and 4 was mounted on top of a glass fiber in a random orientation. X-ray intensity data for both compounds were collected at 295(2) K on a D8 VENTURE Bruker AXS diffractometer equipped with a multilayer monochromated Mo-K radiation ( = 0.71073 Å) and a CMOS Photon100 detector.

The two structures were solved by dual-space algorithm using the SHELXT program, [START_REF] Sheldrick | [END_REF] and then refined with full-matrix least square methods based on F2 (SHELXL-2014). 34 The disorder observed for the carbon and sulfur atoms of the thiophene fragment in the two compounds was modelled using two positions per carbon and sulfur atom with refined occupancy factors of 0.65/0.35 and 0.75/0.25 in 3, and 0.71/0.29 and 0.80/0.20 in 4. For the two compounds, nonhydrogen atoms were refined with anisotropic displacement parameters. H atoms were included in their calculated positions, assigned fixed isotropic thermal parameters and constrained to ride on their parent atoms. A summary of the details about crystal data, collection parameters and refinement are documented in Table 3, and additional crystallographic details are in the CIF files. ORTEP views were drawn using Olex2 software. 35 A c c e p t e d m a n u s c r i p t

Electrochemical synthesis and characterization

Cyclic voltammetry (CV) measurements (electropolymerization and in situ characterization) were performed using a platinum disc working electrode (7•10 -2 cm 2 geometric area), Ag|AgCl in tetramethylammonium chloride solution, to match the potential of a saturated calomel electrode (SCE) as reference, and a platinum wire of the large geometric area as auxiliary electrode, perturbing potentiodynamically (CV), by successive cycles at a potential scan rate of 100 mV•s -1 . The concentration of the complex under investigation was 10 -3 mol•L -1 in acetonitrile solution containing 0.1 mol•L -1 supporting electrolyte. Ferrocene (Fc) was added as an internal reference at the end of each experiment. The Fc|Fc + couple was located at E½ = 0.382 V vs. SCE in acetonitrile and at E½ = 0.507 V vs. Cp*2Fe +/0 (Cp* =

A c c e p t e d m a n u s c r i p t

C5Me5) which is independent of the solvent and supporting electrolyte, 36 where E½ was calculated from the average of the oxidation and reduction peak potentials. Anhydrous acetonitrile (CH3CN, 99.8%, Aldrich) stored over molecular sieves and tetrabutylammonium tetrafluoroborate (TBATFB, 99.0 % Aldrich) previously dried at 120 °C, were used as solvent and supporting electrolyte, respectively. All electrochemical measurements were accomplished on a CHI660E potentiostat at ambient temperature (20 °C) under high purity argon atmosphere, in a three-compartment, three-electrode anchor-type electrochemical cell. The response of the after modification with the deposit produced by electro-oxidation of the metal complexes were studied in 10 -1 mol•L -1 TBATFB in CH3CN.

Computational details

DFT calculations were carried out for compounds 1-4 using the ADF2016 package, 37,38 employing the PW91 functional 39 and the TZ2P basis set. 40 Spin-unrestricted calculations were

performed on the open-shell systems. The geometry optimizations in vacuum were carried out assuming the X-ray structures as input data, without any symmetry constraint. The optimized geometries were characterized as true minima on the potential energy surface using vibrational frequency calculations (no imaginary values). The UV-vis. transitions were calculated by means of TD-DFT calculations on the optimized geometries, at the same PW91/TZ2P level of theory.

Conflicts of Interest

The authors declare no conflict of interest.


Olivier Huber

Vincent Acary
email: vincent.acary@inria.fr

Bernard Brogliato
email: bernard.brogliato@inria.fr

Lyapunov stability analysis of the implicit discrete-time twisting control algorithm

Keywords: sliding mode control, sampled-data system, discrete-time Lyapunov stability, implicit discretization, finitetime stability, robust control

An implicit discrete-time version of the twisting sliding-mode control algorithm is considered. The framework of variational inequalities is used to define the control input values. This provides the foundation for both the analysis of the controller and the numerical computations. The controller is shown to be well-defined and the discrete-time closed-loop system's fixed point is finite-time globally stable in the sense of Lyapunov. The analysis is led in the unperturbed case, and numerical simulations demonstrate the efficiency of the proposed controller when a disturbance acts on the system.

I. INTRODUCTION

Sliding-mode control (SMC) is widely appreciated for its robustness and ease of implementation. One of its drawbacks is the so-called chattering phenomenon, which consists of unwanted, high-frequency oscillations in the output (the sliding variable) and in the input (whose shape is a high-frequency bang-bang-like signal). Especially it has been recognized that the time-discretization of such set-valued controllers can have crucial consequences on the chattering [START_REF] Yan | Euler's discretization effect on a twisting algorithm based sliding mode control[END_REF], that is in this case a numerical chattering. Roughly speaking, explicit discretizations of the set-valued controller yield numerical chattering, even in the absence of perturbations. The regularization of the set-valued controller (usually a signum multifunction) at zero, is often considered as a universal remedy to chattering. However, it has severe drawbacks: it introduces additional control parameters (the regularization slopes), whose tuning is not so clear even in the case of a single attracting surface [START_REF] Huber | Experimental results on implicit and explicit time-discretization of equivalent-control-based sliding-mode control[END_REF], and in the case of co-dimension ≥ 2 sliding surface, regularizations become even harder to analyze [START_REF] Guglielmi | Classification of hidden dynamics in discontinuous dynamical systems[END_REF], [START_REF] Dieci | Regularizing piecewise smooth differential systems: co-dimension 2 discontinuity surface[END_REF]. Moreover, the precision is decreased and one has to choose a regularization with a steep slope, and a very small sampling period. In discrete-time, the choice of the sampling period as a function of this slope is not clear, even in the simplest cases [START_REF] Huber | Experimental results on implicit and explicit time-discretization of equivalent-control-based sliding-mode control[END_REF]. Recently, an alternative to the explicit discretization method has been introduced, which is based on an implicit discretization of the set-valued controller [START_REF] Acary | Chattering-free digital slidingmode control with state observer and disturbance rejection[END_REF], and takes the form in the simplest cases of a simple projection over the interval [-1, 1]. It has been successfully implemented and tested for first order SMC in [START_REF] Huber | Implicit discretetime twisting controller without numerical chattering: Analysis and experimental results[END_REF], [START_REF] Huber | Experimental results on implicit and explicit time-discretization of equivalent-control-based sliding-mode control[END_REF], [START_REF] Huber | Lyapunov stability and performance analysis of the implicit discrete sliding mode control[END_REF], where it is demonstrated that the chattering at both the input and output signals, is drastically decreased compared with the explicit method. In particular it has been proved [START_REF] Efimov | Realization and discretization of asymptotically stable homogeneous systems[END_REF] that the explicit Euler method may yield instability, while the implicit one enjoys global stability. The twisting and super-twisting algorithms were tackled in [START_REF] Acary | Chattering-free digital slidingmode control with state observer and disturbance rejection[END_REF], [START_REF] Huber | Implicit discretetime twisting controller without numerical chattering: Analysis and experimental results[END_REF], with preliminary analytical and experimental results. The main properties of the implicit approach (which are impossible to obtain with the explicit one, or with a saturation) are: (i) a rigorous definition of the sliding-surface in discrete-time, (ii) global Lyapunov stability, (iii) finite-time convergence to the sliding-surface, (iv) convergence of the input to its continuous-time counterpart, (v) insensitivity of the control input with respect to the gains in the sliding-mode phase (in accordance with the continuous-time analysis using for instance Filippov's framework), (vi) robustness with respect to a large class of disturbances, and (vii) the possibility to choose large sampling periods without deteriorating too much the performance. Properties (v) (vi) (vii) have been validated experimentally in [START_REF] Huber | Implicit discretetime twisting controller without numerical chattering: Analysis and experimental results[END_REF], [START_REF] Huber | Experimental results on implicit and explicit time-discretization of equivalent-control-based sliding-mode control[END_REF], [START_REF] Huber | Lyapunov stability and performance analysis of the implicit discrete sliding mode control[END_REF]. In parallel with discrete-time SMC analysis, a great deal of work has been dedicated in the past twenty years, to the analysis of the higher-order SMC (HOSMC), a class of SMC introduced by A. Levant [START_REF] Levant | Sliding order and sliding accuracy in sliding mode control[END_REF], [START_REF] Emelyanov | Higher-order sliding modes in binary control systems[END_REF], [START_REF] Levantovsky | Second order sliding algorithms: their realization[END_REF], [START_REF] Levant | Higher-order sliding modes, differentiation and outputfeedback control[END_REF]. In particular, finding Lyapunov functions for the order-two twisting and super-twisting algorithms, has been a challenge [START_REF] Orlov | Finite time stability and robust control synthesis of uncertain switched systems[END_REF], [START_REF] Moreno | Strict Lyapunov functions for the supertwisting algorithm[END_REF]. HOSMC is known to enable reduction of chattering, at the price however of tolerating only a smaller class of disturbances than the classical, first order SMC. But it is noteworthy that when discretized with an explicit method, HOSMC may also suffer from severe chattering effects, see [START_REF] Yan | Euler's discretization effect on a twisting algorithm based sliding mode control[END_REF] and experiments on the twisting algorithm in [START_REF] Huber | Implicit discretetime twisting controller without numerical chattering: Analysis and experimental results[END_REF]. The goal of this article is to analyze in detail the implicit discretization of the twisting algorithm (in the disturbance-free case) and to show that, under a modification of the basic algorithm used in [START_REF] Huber | Implicit discretetime twisting controller without numerical chattering: Analysis and experimental results[END_REF]Equation (5)], Lyapunov global stability and finitetime convergence results can be obtained, hence extending the results in [START_REF] Orlov | Finite time stability and robust control synthesis of uncertain switched systems[END_REF] to the discrete-time case. As demonstrated experimentally in [START_REF] Huber | Implicit discretetime twisting controller without numerical chattering: Analysis and experimental results[END_REF], the unmodified version of the implicit twisting scheme (called in this article the regular implicit control), results in a much better behavior (with important chattering reduction) than its explicitly discretized version. However, as shown in [START_REF] Huber | Analysis and implementation of discrete-time sliding mode control[END_REF], the regular implicit control yields always, in theory, oscillations around the origin. The objective of this article is to prove that stronger stability properties can be obtained when the regular implicit discrete-time scheme is suitably modified, where the goal may also be to improve the precision of the closed-loop system.

This article is organized as follows: the discrete-time controller is introduced in Section II. The well-posedness (ex-istence and uniqueness of the controller) is shown in Section III, and the stability analysis of the closed-loop system is established in Section IV. The computation of the control law and simulations are presented in Section V, on a system with perturbations. Conclusions are given in Section VI. Some proofs are in the Appendix.

Notation and definitions: The tools used in this article are classical tools from Convex Analysis (see [START_REF] Rockafellar | Variational Analysis, ser. Grundlehren der mathematischen Wissenschaften[END_REF], [START_REF] Hiriart-Urruty | Fundamentals of Convex Analysis[END_REF]), and from Complementarity Theory ( [START_REF] Cottle | The Linear Complementarity Problem, ser[END_REF], [START_REF] Facchinei | Finite-Dimensional Variational Inequalities and Complementarity Problems[END_REF]). Let K ⊆ R p be a closed non-empty convex set. Its normal cone N K is defined as follows: for any

x ∈ K, N K (x) = {d ∈ R p : d, y -x ≤ 0, ∀y ∈ K}. The indicator function of K ⊆ R p is defined as: δ K (x) = 0 if x ∈ K, δ K (x) = +∞ if x ∈ K. The support function σ K is defined as: σ K (y) = sup x∈K y, x . It is the conjugate of the indicator function, that is σ K (y) = δ * K (y) = sup x∈R p [ x, y -δ K (x)]. Let f : R n → R ∪ {±∞} be convex, proper (that is f (x)
< +∞ for at least one x, and f (x) > -∞ for all x), and lower semi-continuous (lsc) function (that is, its epigraph epi f :

= {(x, α) ∈ R p × R : f (x) ≤ α} is a closed set). Its subdifferential ∂f is defined as: ∂f (x) := {g : f (z) ≥ f (x) + g, z -x , ∀z}.
When K is closed convex non-empty, then δ K is proper, convex and lsc, and one has ∂δ K (x) = N K (x) for all x ∈ K. Using the definition, it is easy to see that N K (-x) = -N -K (x).

∈ ∂δ * K (x) = ∂σ K (x) ⇔ x ∈ N K (y) for all x, y ∈ R p . The set-valued signum function is Sgn(x) = -1 if x < 0, Sgn(x) = 1 if x > 0, Sgn(0) = [-1, 1]. Let g(x) = |x|, x ∈ R, one has Sgn(x) = ∂g(x) for all x ∈ R. Then g(x) = δ * [-1,1] (x), so that y ∈ Sgn(x) ⇔ x ∈ ∂δ [-1,1] (y) = N [-1,1] (y) = [0, +∞) if y = 1, (-∞, 0] if y = -1, 0 if y ∈ [-1, 1].
Let the unit ball for the maximum norm be defined as

B ∞ = {x ∈ R n : ||x|| ∞ := max 1≤i≤n |x i | ≤ 1}. Fact 2. Let x ∈ R n . We have δ * B∞ = |x 1 |+|x 2 |+. . .+|x n | = ||x|| 2 ,
and ∂δ * B∞ = (Sgn(x 1 ), Sgn(x 2 ), . . . , Sgn(x n )) T . Now let us give an explicit expression for the normal cone in the polyhedral case. Fact 3. [17, p. 67] Let K be a closed convex polyhedron defined as:

K = {x ∈ R n | Hx ≤ b} , with H ∈ R m×n , b ∈ R m .
The normal cone at a point x ∈ K is generated by the outward normals of the actives constraints: N K (x) = H T α• r, r ≥ 0 , with α ∈ {1, . . . , m} the set of active constraints, that is for all i ∈ α, we have

H i• x = b i .

II. DISCRETE-TIME TWISTING CONTROLLER

Let us quickly recall the basics of the twisting algorithm. We deal in this section with the unperturbed case. The case with a perturbation will be tackled in Section V. In the following, is its shown that a "naive" implicit discretization of the twisting algorithm, called the regular implicit control does not allow one to get the asymptotic stability. Thus, a modified algorithm is proposed. In the general setting, the sliding variable is supposed to be twice differentiable and has dynamics given by σ = a(x, t) + g s (x, t)u -u ∈ a Sgn(σ) + b Sgn( σ), with a > b > 0, and x is the state of the original plant. To simplify the analysis let us consider the case of a double integrator without perturbation, that is

σ ∈ -a Sgn(σ) -b Sgn( σ).
Recasting this as a first order system, we get

Σ = AΣ + Bλ with A = 0 1 0 0 , B = 0 0 a b and Σ := σ σ , -λ ∈ Sgn σ σ = Sgn Σ.
The condition a > b > 0 ensures the finite-time global Lyapunov stability of the closed-loop fixed point, see [START_REF] Orlov | Finite time stability and robust control synthesis of uncertain switched systems[END_REF].

Let us discretize the dynamics using the ZOH method. The discontinuous control input is implemented using the following discretization:

Σ k+1 = A * Σ k + B * λ k+1 with λ k+1 = λ 1,k+1 λ 2,k+1 (1)
and

A * = 1 h 0 1 , B * = h h 2 1 a b . (2)
The piecewise constant control input value, to be calculated at time t k and applied on [t k , t k+1 ) with h := t k+1 -t k > 0 constant, is given by u k = aλ 1,k+1 + bλ 2,k+1 . The discretetime dynamics is thus given by

(a) σ k+1 = σ k + h σk + h 2 2 u k (b) σk+1 = σk + hu k . (3)
The precise definition of the relationship between λ k+1 and Σ k+1 is formalized later. First let us state a result which will be used later in Section IV for the stability analysis, and which concerns the set of points reaching the origin in one step.

Lemma 1. Suppose that the piecewise constant control law is such that u k can be freely chosen in [-u M , u M ] whenever Σ k+1 = 0. Then the origin of the closed-loop system (3) is only reachable from the line segment S 0 defined as

S 0 := (σ k , σk ) ∈ R 2 : σ k + h 2 σk = 0, | σk | ≤ hu M .
The proof is in Appendix A. Therefore, a necessary condition for the global asymptotic stability of the closedloop system is that S 0 is an attracting surface. If we were to perform a straightforward implicit discretization of the inclusion -λ ∈ Sgn Σ as done in [START_REF] Huber | Implicit discretetime twisting controller without numerical chattering: Analysis and experimental results[END_REF], we would get

-λ k+1 ∈ Sgn(Σ k+1 ) ⇐⇒ -Σ k+1 ∈ N H (λ k+1 ), (4)
with H := [-1, 1] 2 . Note that inclusion (4) together with (1) defines the regular implicit control. It has been shown in [START_REF] Huber | Analysis and implementation of discrete-time sliding mode control[END_REF] that S 0 is not an attracting surface with a controller defined from the implicit discretization in (4), and that the set of points that can reach the origin is a set of measure zero. Whence, let us state a more general type of control law than (4), by setting

-Σ k+1 ∈ N K (λ k+1 ), (5)
with K a polyhedral compact set. Then, we obtain from this inclusion and (1) the generalized equation (GE):

0 ∈ A * Σ k + B * λ k+1 + N K (λ k+1 ). (6)
This type of inclusion is also known as an Affine Variational Inequality (AVI). Let us denote by SOL(M, q, K) the set of solutions to the AVI 0 ∈ M x + q + N K (x). Therefore, we shall write λ k+1 ∈ SOL(B * , A * Σ k , K) whenever λ k+1 is a solution to [START_REF] Huber | Implicit discretetime twisting controller without numerical chattering: Analysis and experimental results[END_REF]. The sliding variables and control input are then given as follows:

Σ k+1 = AΣ k + B * λ k+1 (7) u k = aλ 1,k+1 + bλ 2,k+1 , whenever λ k+1 ∈ SOL(B * , A * Σ k , K).
Let us now study the existence and uniqueness of a solution to the AVI in [START_REF] Huber | Implicit discretetime twisting controller without numerical chattering: Analysis and experimental results[END_REF].

III. THE EXTENDED IMPLICIT TWISTING CONTROLLER A. Definition of the control law

As eluded to above, the basic idea in ( 5) is to consider the variable λ k+1 to be defined as

-Σ k+1 ∈ N K (λ k+1 ) ⇐⇒ -λ k+1 ∈ ∂δ * -K (Σ k+1 ), (8)
with K a bounded polytopic convex set. This can be seen as a generalization of the inclusion -λ k+1 ∈ Sgn(Σ k+1 ), which can be rewritten equivalently as 4) and Fact 1). With the dynamics [START_REF] Huber | Lyapunov stability and performance analysis of the implicit discrete sliding mode control[END_REF], the relation ( 8) is transformed into an AVI of the type ( 6). Remark 1. The controller calculated from ( 6) is nonanticipative, since the solution of (6) depends only on Σ k and h.

-λ k+1 ∈ ∂δ * [-1,1] 2 (Σ k+1 ) given that Sgn(•) = ∂δ * [-1,1] 2 (•) (see (
Lemma 1 provides us with an interesting insight on how to define the control input, such that the origin can be reached from a set of initial conditions with measure greater than zero in R 2 . Using this approach based on AVI is interesting since we want to be able to design a control law that steers a set with positive measure to the origin. To achieve this, we choose a set K, in which λ k+1 takes its values, that is not the box [-1, 1] 2 . We impose that a half-line containing a part of -S 0 belongs to the normal cone to K. Let us define K as a convex polytope:

K := {x ∈ R 2 | Ex ≤ b} with E ∈ R 4×2 and b ∈ R 4 . (9)
From Fact 3, we know that the normal cone is generated by the rows of E. Hence, a simple way to have a half-line containing a part of -S 0 as at least one line of the normal cones, is to include it as a constraint, that is at least one row of E has to be proportional to (h/2, -1). To be more concrete, the square

[-1, 1] 2 admits the representation x ∈ R 2 | Hx ≤ b with H =     1 0 0 1 -1 0 0 -1     and b =     1 1 1 1     . We propose to use instead the matrix E(h) =     1 0 -h/2 1 -1 0 h/2 -1    
in [START_REF] Levant | Sliding order and sliding accuracy in sliding mode control[END_REF], where E(0) = H. The choice of the vector b depends on the constraints we want to impose on the control inputs. Let us discuss three possible choices:

b 1 = b, b 2 (h) =     1 1 -h/2 1 1 -h/2     , b 3 (h) =     1 1 + h/2 1 1 + h/2     . (10) which satisfy b 1 = b 2 (0) = b 3 (0) = b. With b 1 , we obtain a parallelogram, which is not contained in the square [-1, 1] 2 .
If the original control constraints were important to respect, then by using the vector b 2 , the resulting set is a parallelogram contained in the unit square. Finally, another choice could be b 3 , which gives a set containing the original square. Note that all those sets are symmetric with respect to the origin. Solving ( 6) with K defined by ( 9), gives us the value of the extended implicit twisting control input.

B. Existence and uniqueness of the control input

Let us now investigate the well-posedness of the extended implicit twisting controller. In the following, the sliding variables have the dynamics (3) and the control variables are defined by the inclusion (8), with the set K defined as in [START_REF] Levant | Sliding order and sliding accuracy in sliding mode control[END_REF], and b 1 choosen as [START_REF] Emelyanov | Higher-order sliding modes in binary control systems[END_REF]. The transformation from

B ∞ = [-1, 1]
2 to K is given by

L = 1 0 h 2 1 . (11)
Recall that the control input is defined as u k = aλ 1,k+1 + bλ 2,k+1 , with λ k+1 a solution to [START_REF] Huber | Implicit discretetime twisting controller without numerical chattering: Analysis and experimental results[END_REF]. The well-posedness of the controller is now investigated.

Lemma 2. The system composed of the double integrator system and the implicit twisting controller as defined in (8) enjoys the existence and uniqueness of Σ k+1 and u k . Moreover, if Σ k+1 = 0, then λ k+1 is also unique.

The proof is in Appendix A. Remark 2. In this section and the foregoing one, it has been proved that the controller exists and is unique and non anticipative. A way to calculate it on a real implementation, is indicated in Section V-A.

IV. STABILITY ANALYSIS

Let us turn our attention to the stability analysis of the discrete-time closed-loop system (3) (8) (9) with b 1 in [START_REF] Emelyanov | Higher-order sliding modes in binary control systems[END_REF]. It is clear that the origin is a fixed point, and we are now going to prove its global stability (which in passing proves that it is the unique fixed point).

Theorem 4. Let 0 < h < 2 and a > 1 + h 2 b > 0.
The origin is the unique equilibrium of the discrete-time system (3) with the controller given by (7) [START_REF] Efimov | Realization and discretization of asymptotically stable homogeneous systems[END_REF] and is globally Lyapunov finite-time stable.

Proof. The remaining part of this section is dedicated to the proof of this theorem. Before searching for a candidate Lyapunov function, let us provide some relations between the variables used in the twisting controller (3) and [START_REF] Efimov | Realization and discretization of asymptotically stable homogeneous systems[END_REF]. First note that we can relate the support functions of the set K and of [-1, 1]

2 as follows:

δ * -K (x) = sup y∈-K y, x = sup z∈-[-1,1] 2 Lz, x = sup z∈[-1,1] 2 z, L T x = δ * [-1,1] 2 (L T x).
Using the chain rule for proper lower semi-continuous convex functions [START_REF] Rockafellar | Convex Analysis[END_REF]Theorem 23.9], we get: for all Σ ∈ R 2 ,

∂δ * -K (Σ) = L ∂δ * [-1,1] 2 (L T Σ
). Thus, using [START_REF] Levantovsky | Second order sliding algorithms: their realization[END_REF], the relation ( 8) can be rewritten at step k as:

-λ 1,k ∈ Sgn(σ k + h 2 σk ) (12)
-λ 2,k ∈ Sgn( σk ) + {-h 2 λ 1,k }. ( 13
)
This gives rise to the following bounds:

|λ 1,k | ≤ 1 |λ 2,k | ≤ 1 + h 2 ( 14
)
|u k | ≤ a + b(1 + h 2 )
. Now, let us provide additional relations between the controller and the sliding variables. The inclusion ( 12) can be inverted as:

σ k + h 2 σk ∈ N [-1,1] (-λ 1,k ), ⇐⇒ ∀λ 1 ∈ [-1, 1], (λ 1 + λ 1,k ) σ k + h 2 σk ≤ 0. (15)
Note that from (12), we have

σ k + h 2 σk = -λ 1,k σ k + h 2 σk . (16)
Also, whenever h < 2, the relation [START_REF] Orlov | Finite time stability and robust control synthesis of uncertain switched systems[END_REF] implies that

-λ 2,k σk > 0 if σk = 0 and -λ 2,k σk = 0 if σk = 0, (17)
since for σk = 0, the first term in the right-hand side of (13) always dominates the second one. Now that we have those relations ready for use, let us propose the Lyapunov function candidate:

V k := V (σ k , σk ) = a σ k + h 2 σk + 1 2 σ2 k - h 2 bλ 2,k σk . ( 18
)
Using ( 16) yields

V k = -aλ 1,k (σ k + h 2 σk ) + 1 2 σ2 k -h 2 bλ 2,k σk = -aλ 1,k 1 2 ( σk -ahλ 1,k -hbλ 2,k ) Σ k . ( 19
)
Starting from [START_REF] Cottle | The Linear Complementarity Problem, ser[END_REF] and using [START_REF] Hiriart-Urruty | Fundamentals of Convex Analysis[END_REF], it is easy to assess that if h < 2, then V k (•) is positive everywhere except at the origin where it vanishes, and that it is also radially unbounded. The remaining part is to prove that this function decreases between two iterates, that is

∆V k := V k+1 -V k < 0 whenever V k = 0.
Recall that the dynamics of the system is in [START_REF] Guglielmi | Classification of hidden dynamics in discontinuous dynamical systems[END_REF]. First note that inserting (3)(b) in [START_REF] Facchinei | Finite-Dimensional Variational Inequalities and Complementarity Problems[END_REF], we can write V k+1 = -aλ 1,k+1 σ k+1 + 1 2 σk σk+1 . Now we investigate the evolution of V k :

∆V k = -aλ 1,k+1 σ k + h σk + h 2 2 (aλ 1,k+1 + bλ 2,k+1 ) +aλ 1,k σ k + 1 2 ( σk σk+1 -σk ( σk -ahλ 1,k -hbλ 2,k )),
where we used [START_REF] Facchinei | Finite-Dimensional Variational Inequalities and Complementarity Problems[END_REF] to get the last term. Using (3)(b), we substitute the terms tagged with to get

∆V k = a(λ 1,k -λ 1,k+1 )σ k -ah 2 λ 1,k+1 ( σk+1 + σk ) + h 2 (aλ 1,k+1 + bλ 2,k+1 + aλ 1,k + bλ 2,k ) σk = a(λ 1,k -λ 1,k+1 )(σ k + h 2 σk ) + h 2 (-aλ 1,k+1 σk+1 +(aλ 1,k+1 + bλ 2,k+1 + bλ 2,k ) σk ).
Let us replace σk+1 with its expression in (3)(b), to obtain:

∆V k = a(λ 1,k -λ 1,k+1 )(σ k + h 2 σk ) + h 2 (-ahλ 1,k+1 (aλ 1,k+1 + bλ 2,k+1 ) + (bλ 2,k+1 + bλ 2,k ) σk ).
Using again relation (3)(b) to replace the term λ 2,k+1 σk , we get

∆V k = a(λ 1,k -λ 1,k+1 )(σ k + h 2 σk )+ h 2 -ahλ 1,k+1 (aλ 1,k+1 + bλ 2,k+1 ) + bλ 2,k+1 σk+1 -hbλ 2,k+1 (aλ 1,k+1 + bλ 2,k+1 )) + bλ 2,k σk .
A final rearrangement in the second term yields

∆V k = a(λ 1,k -λ 1,k+1 )(σ k + h 2 σk ) -h 2 2 (aλ 1,k+1 + bλ 2,k+1 ) 2 + bh 2 (λ 2,k+1 σk+1 + λ 2,k σk ). (20)
Let us analyze the last equality term by term: using [START_REF] Huber | Analysis and implementation of discrete-time sliding mode control[END_REF] with the choice λ 1 = -λ k+1 , the first term is nonpositive. The second term is clearly nonpositive and the third one too, using the relation [START_REF] Hiriart-Urruty | Fundamentals of Convex Analysis[END_REF]. Thus, ∆V k ≤ 0. Let us show now that the variation ∆V k is negative as long as the origin is not reached. The second term in [START_REF] Rockafellar | Convex Analysis[END_REF]

σ k+1 + h 2 σk+1 = 0. (22)
Using ( 21) in the dynamics (3)(b), we get that in such a case σk+1 = σk . Going back to the analysis of (20), using [START_REF] Hiriart-Urruty | Fundamentals of Convex Analysis[END_REF], the last term in the right-hand side is zero if and only if σk+1 = σk = 0, which combined with [START_REF] Kim | A structure-preserving pivotal method for affine variational inequalities[END_REF] implies that σ k+1 = 0, and by ( 12) that σ k = 0. Whence, ∆V k can be zero only when the system has already reached the origin. Otherwise, ∆V k < 0.

Let us now prove the finite-time stability. Remember that the three terms in the right-hand side of ( 20) are all nonpositive. Hence, we can find an upper bound of ∆V k by considering only one of those terms. Let us take a closer look at the second one:

-

h 2 2 (aλ 1,k+1 + bλ 2,k+1 ) 2 = -h 2 2 u 2 k .
We have shown that 22)). Suppose that σ k + h 2 σk = 0. Then 0 = u k ∈ ±(a-h 2 b)+b Sgn( σk+1 ). The second gain condition in the theorem statement gives us a -b h 2 -b > 0. Therefore, we get that u

u k = 0 ⇔ |λ 1,k+1 | < 1 ⇔ σ k+1 + h 2 σk+1 = 0. Thus u k = 0 ⇒ |λ 1,k+1 | = 1 and σ k+1 + h 2 σk+1 = 0 (using (
2 k ≥ (a -(1 + h 2 )b) 2 , so that ∆V k ≤ - h 2 2 u 2 k ≤ - h 2 2 a -(1 + h 2 )b 2 < 0. (23)
Inequality [START_REF] Acary | Siconos. a software for modeling and simulation of nonsmooth dynamical systems[END_REF] holds everywhere outside the line σ k + h 2 σk = 0. From Lemma 1, we know that if the state of the system belongs to S 0 , the origin is reached at the next time instant. If we prove that this segment is reachable in finite-time from any initial conditions, then the origin is globally finite-time reachable. Hence, given that ( 23) holds everywhere except on the line [START_REF] Kim | A structure-preserving pivotal method for affine variational inequalities[END_REF], we just need to bound ∆V k away from 0 if Σ k+1 belongs to the line ( 22) minus S 0 . Thus, suppose that Σ k+1 belongs to the line [START_REF] Kim | A structure-preserving pivotal method for affine variational inequalities[END_REF], and that | σk+1 | > h(a + b). In the third term in the right-hand side of (20), and using (12) (13) we get:

λ 2,k+1 σk+1 = -| σk+1 | -h 2 λ 1,k+1 σk+1 ≤ -(1 -h 2 )| σk+1 | ≤ -h(1 -h 2 )(a + b).
Hence, disregarding all other terms (which are all non positive) but λ 2,k+1 σk+1 in the right-hand side of (20), we obtain the upperbound:

∆V k ≤ -bh 2 2 (1 -h 2 )(a + b), (24)
the right-hand side of (24) being negative for all 0 < h < 2. For all k ≥ 0, ∆V k is smaller than the maximum of the righthand side of ( 23) and ( 24). Both quantities being negative constants, the finite-time convergence to S 0 holds, hence the finite-time convergence to the origin.

Remark 3. The discrete-time Lyapunov function V k (•) is close to its continuous-time counterpart used in [START_REF] Orlov | Finite time stability and robust control synthesis of uncertain switched systems[END_REF], which is given by a|σ| + σ2 /2.

Corollary 1. Let 0 < h < 2 and a > 1 + h 2 b > 0. The origin is the unique equilibrium of the closed-loop system, consisting of a double-integrator with the piecewise constant controller obtained from (7)-( 8), and it is globally Lyapunov finite-time stable.

Proof. The ZOH discretization being exact, we know by the previous Theorem that there exists k 0 ∈ N such that Σ k0 = 0. Then for all k > k 0 , Σ k = 0, with the control input u k = 0, as we can easily infer from (3). On each sampling interval [t k , t k+1 ], k > k 0 , the continuous-time system has the dynamics

Σ = 0 1 0 0 Σ + 0 0 a b 0 = 0.
This concludes the proof.

Remark 4. From (3) it follows that once the origin has been attained, then the (unperturbed) discrete-time system stays at the origin for all future times. Therefore, there is no numerical chattering with the implicit controller, contrarily to its explicit counterpart [START_REF] Yan | Euler's discretization effect on a twisting algorithm based sliding mode control[END_REF]. Let us illustrate this point numerically on the double integrator: on Fig. 1 the phase plot of the explicitly discretized control law,

u k = -a sgn(σ k ) -b sgn( σk ),
and the enhanced implicit twisting given by ( 7)-( 8). This is confirmed experimentally in [START_REF] Huber | Implicit discretetime twisting controller without numerical chattering: Analysis and experimental results[END_REF] with the regular implicit twisting controller calculated from (4). The enhanced implicit twisting controller is expected to provide even better results. The regular implicit twisting algorithm has also been analyzed in [START_REF] Acary | Chattering-free digital slidingmode control with state observer and disturbance rejection[END_REF]Propositions 7 and 8], including a perturbation. It is shown therein that during the discrete-time sliding mode, the disturbance is attenuated with a factor h 2 on σ k and with a factor h on σk .

V. ENHANCED IMPLICIT TWISTING WITH PERTURBATIONS A. Computation of the control input

Let us now tackle the case with perturbations, with the same setup as before: we consider the ZOH discretization of a double integrator. Let us denote by Σ k and Σ k+1 the sliding variables for the real plant, i.e., with perturbations, whereas the sliding variables Σ k and Σ k+1 are the variables of the nominal system, i.e., without perturbation. The latter obey the same system (1), whereas Σ k+1 is given by:

Σ k+1 = A * Σ k + h 2 /2 h u k + d with u k = aλ 1,k+1 + bλ 2,k+1 and -λ k+1 ∈ ∂δ * -K (Σ k+1
), where R 2 d = O(h) captures the contribution from the perturbations. At each timestep, the value Σ k of the nominal system is set to the value Σ k , measured (or estimated) on the real system. The inclusion -λ k+1 ∈ ∂δ * -K (Σ k+1 ) combined with the one step relation

Σ k+1 = A * Σ k + B * λ k+1 gives the AVI 0 ∈ A * Σ k + B * λ k+1 + N K (λ k+1 ). (25)
Again, a solution to this AVI is computed with data known at time t k . Therefore, the control law is causal. Note that from these equations, it is easy to see that the discrepancy on one time step between Σ k+1 and Σ k+1 , is d.

Let us now detail how a solution to the AVI (25) can be computed. Since we are dealing with an AVI of dimension 2, it is possible to apply a simple algorithm. Note that with our choice of K, each λ i can take values in 3 sets:

λ 1 in {1}, {-1} and [-1, 1], and λ 2 in {1 + h 2 λ 1 }, {-1 + h 2 λ 1 } and [-1 + h 2 λ 1 , 1 + h 2 λ 1 ]
. Each corresponds to the arguments of the Sgn multifunction to be either nonnegative, nonpositive and 0. This gives 9 possible cases to consider. Therefore, it is tractable to find a solution by enumerating the candidates. Thanks to the result of existence and uniqueness of the control input in Lemma 2, this approach is well defined. Let us first describe the enumeration procedure, before giving some heuristic to find a solution in a faster way.

First, whenever λ is one of the extreme point of K, then checking that it is a solution consists of computing Σ k+1 based on that value of λ, and check that the set (Sgn(σ k+1 ), Sgn( σk+1 ) + h 2 Sgn(σ k+1 )) T contains that value. When the candidate λ belongs to the boundary of set K, the procedure consists of computing u k as a solution to either σ k+1 = 0 or σk+1 = 0, and checking that there exists a λ that can give that value u k . Whenever we consider a candidate λ with λ 1 = 0 (resp. λ 2 ), we compute the value of u k solution to σ k+1 = 0 (resp. σ = 0). Then, we compute the value of λ 1 (resp. λ 2 ) giving that control input value u k :

-

λ = (1, λ 2 ), then λ 2 ∈ [-1 + h 2 , 1 + h 2 ]. -λ = (-1, λ 2 ), then λ 2 ∈ [-1 -h 2 , 1 -h 2 ]. For the case λ = (λ 1 , λ 2 ) such that λ 2 -h
2 λ 1 = ±1, then we combine this with the equation aλ 1 + bλ 2 = u k . It is easy to check that this 2 × 2 system always has a unique solution, our candidate λ. To check that it is in the constraint set K, it is sufficient that

λ 1 ∈ [-1, 1].
Finally, the last case is when Σ k+1 = 0 for which λ ∈ K. For this last case, the method consists in computing u k as the solution of σk+1 . If |u k | < a+b( 1+ h 2 ), then the value of σ k+1 is checked to be 0. If the latter condition holds, then the control input value is u k . Indeed, one can check that whenever

|u k | ≤ a + b(1 + h 2 ), the vector λ such that λ 1 = u k /(a + b(1 + h 2 ) and λ 2 = (1 + h
2 )λ 1 belongs to K, and provides the required control input value.

A simple and efficient heuristic consists in trying first the value of λ = (sgn(σ k ), sgn(σ k+1 ) + h 2 sgn(σ k+1 )) T , since the value of the control is likely not to change, especially away from the origin. Note that we used the single-valued signum function. If this candidate fails, then the remaining 8 combinations are tried.

B. Numerical simulations

Let us illustrate the above developments on a disturbed double integrator. The control input value is supposed to be piecewise constant: u(t) = u k for t ∈ [t k , t k+1 ), k ≥ 0. Three control laws are compared, with G and β such that a = G and b = Gβ:

explicit control: u k = -G(sgn(σ k ) + β sgn( σk )) -regular implicit control: u k = aλ 1,k+1 + bλ 2,k+1 , and

λ k+1 ∈ SOL(A * Σ k , B * , [-1, 1] 2 ).
enhanced implicit control: u k = aλ 1,k+1 + bλ 2,k+1 , and

λ k+1 ∈ SOL(A * Σ k , B * , K).
The plant dynamics with a perturbation, σ(t) = u(t) + α sin(ωt), is integrated using the Lsodar method from ODE-PACK, for a high fidelity integration process mimicking the continuous-time plant. In the following, ω = 50 and α = 1. Since B * is not symmetric positive definite, the AVI cannot be solved with a projection as for first-order SMC [START_REF] Huber | Lyapunov stability and performance analysis of the implicit discrete sliding mode control[END_REF], [START_REF] Huber | Experimental results on implicit and explicit time-discretization of equivalent-control-based sliding-mode control[END_REF]. However, the algorithm from [START_REF] Cao | A pivotal method for affine variational inequalities[END_REF] can always find a solution.

Thanks to the linear nature of the problem, the maximum number of iterations is nine [START_REF] Kim | A structure-preserving pivotal method for affine variational inequalities[END_REF]. Furthermore, each iteration requires solving a linear system with only two variables. Hot-start strategy is also very easy to set up, since in the reaching phase the values of λ do not frequently change.

The implementation of the proposed implicit control is available in the open-source software SICONOS and its module SICONOS/CONTROL [START_REF] Acary | Siconos. a software for modeling and simulation of nonsmooth dynamical systems[END_REF]. Otherwise, it is possible to perform the simulation with the algorithm described in Section V-A.

Let us first show the control input values in Fig. 2, for the three control laws. The explicit control law behaves as a bang-bang signal, while the regular implicit twisting has a control magnitude about twice as big as the enhanced implicit control law. Let us now analyze Figures 3 and4. The enhanced implicit control law has the best performance when taking into account the control input effort, the chattering and the precision (on both σ and σ). The control effort is defined as {u k } 1 , the chattering is the total variation (TV) of that sequence: T V (u) := k |u k+1 -u k |. The chattering of the sliding variables is also defined as the total variation of the sequences {σ k } and { σk }. The following trends can be observed: the enhanced implicit control law always outperforms the other two. It provides the best precision, in the same range as the regular implicit control law (magenta crosses), but each time with less chattering or less control effort. As expected, the closed-loop system with the explicitly discretized control law delivers the worst performances.

The continuous time twisting algorithm enjoys the invariance of the control input with respect to the gain in the sliding phase, given that the magnitude of the control input is large enough to overcome the perturbation. This is illustrated in Fig. 4, which is obtained as follows: for each gain value, simulations for 49 initial conditions in a ball of radius 2e -2 around (1, 2) are performed. The worst case performance is then used to create this picture. This methodology is needed since the controller obtained by a regular implicit discretization is very sensitive to the initial value, as documented in [15, Section 2.3.4]. From Fig. 4 it is clear that the enhanced implicit controller is not sensitive with respect to an increase of the control gain, hence confirming a property of the implicit firstorder discrete-time SMC [START_REF] Huber | Experimental results on implicit and explicit time-discretization of equivalent-control-based sliding-mode control[END_REF], [START_REF] Huber | Lyapunov stability and performance analysis of the implicit discrete sliding mode control[END_REF]. On the other hand, both the explicitly discretized controller and the regular implicit controller are sensitive to an increase of the gain, linearly for the errors and the control effort.

Finally, let us investigate the influence of the sampling period on the closed-loop performance, depicted in Fig. 5. The average error on σ is quadratic for all control laws. Moreover the results about the control chattering show that the enhanced implicit control law achieves a much better performance than the other two controllers: its chattering level remains constant, whereas it is inversely proportional to the sampling period in the other two cases. This is also coherent with results obtained in [START_REF] Huber | Lyapunov stability and performance analysis of the implicit discrete sliding mode control[END_REF], [START_REF] Huber | Experimental results on implicit and explicit time-discretization of equivalent-control-based sliding-mode control[END_REF] for the first order SMC.

VI. CONCLUSION

In this article, an enhanced implicit discrete-time twisting sliding-mode controller is studied. Fundamental properties are investigated: existence and uniqueness of the controller at each step, and the global finite-time Lyapunov stability of the closed-loop system. Affine variational inequalities are at the core of the analysis and computational algorithms. Numerical comparisons illustrate the superior performances of Proof of Lemma 2. The existence of a solution to the AVI ( 6) is ensures by the continuity of the functional L defined as λ → A * Σ k + B * λ, and the compactness of the convex set K. The outline of the proof (that can be found in [19, Corollary 2.2.5, p. 148]), is as follows: consider the mapping Λ → Π K (Λ -L(Λ)), with Π K the projector onto K. Since K is convex, it is single valued from K → K. By Brouwer's fixed-point theorem, there exists a fixed point Λ 0 . From [START_REF] Rockafellar | Variational Analysis, ser. Grundlehren der mathematischen Wissenschaften[END_REF]Prop. 6.17], Π -1 K = (I +N K ), in the sense of set-valued mappings. Hence, Π -1 K (Λ 0 ) = Λ 0 + N K (Λ 0 ) = {Λ 0 -L(Λ 0 )}. Rearranging terms gives that -L(Λ 0 ) ∈ N K (Λ 0 ), whence Λ 0 is a solution to the AVI.

For the uniqueness property, note that since B * , from (2), is a rank-one matrix, L T B * L is also a rank-one matrix given by the outer product L T B * L = h(h, 1) T (a + h 2 b, b). This matrix is singular with positive diagonal elements. By construction its columns are linearly dependent. Hence, we have by [START_REF] Huber | Analysis and implementation of discrete-time sliding mode control[END_REF]Proposition 2.3.4 and Remark 2.3.5], the uniqueness of Σ k+1 . As shown in the proof of that proposition, this implies that the difference between any two solutions lies the kernel of the linear mapping, here B * . Note that ker B * = span (b, -a) T and that u k is defined as (a, b)λ. It is easy to see that any element of ker B * is orthogonal to (a, b) T , which ensures the uniqueness of u k . For the last part of the statement, the uniqueness of Σ k+1 implies that if λ 1 k+1 and λ 2 k+1 are two distinct solutions, then their opposites are both in the set ∂δ * -K (Σ k+1 ) (see [START_REF] Efimov | Realization and discretization of asymptotically stable homogeneous systems[END_REF]). This implies that λ 1 k+1 and λ 2 k+1 are such that -λ 1 k+1 , Σ k+1 = -λ 2 k+1 , Σ k+1 = δ * -K (Σ k+1 ). Whence, ∆λ k+1 := λ 1 k+1 -λ 2 k+1 is orthogonal to Σ k+1 . But we know that ∆λ ∈ ker M = span ( -b a ), which means that Σ k+1 = s( a b ), for some s ∈ R. We are interested in the case Σ k+1 = 0 which means that s = 0. Given that a > b > 0, we get that:

∂δ * -K (Σ k+1 ) = arg sup y∈-K y, Σ k+1 =      1 1+ h 2 if s > 0 -1 -1- h 2
if s < 0.

The two sets in the right-hand side are both singletons: the solution λ k+1 of the AVI is therefore unique whenever Σ k+1 = 0.

Fact 1 .

 1 The subdifferentials of the indicator and the support function are inverses, that is y

Fig. 1 :

 1 Fig.1: The blue trajectory is with the enhanced implicit control law, the red one is obtained with an explicit discretization. The timestep is h = 10ms.

Fig. 2 :Fig. 3 :

 23 Fig. 2: Control inputs for the three control laws

Fig. 4 :Fig. 5 :

 45 Fig. 4: Evolution of the performance of the controllers with respect to the gain.

  is zero if and only if

	aλ 1,k+1 + bλ 2,k+1 = 0.	(21)

Using

[START_REF] Moreno | Strict Lyapunov functions for the supertwisting algorithm[END_REF]

and the second gain condition in Theorem 4, it follows that (21) has a solution if and only if |λ 1,k+1 | < 1. Thus, from (12) at step k + 1, (21) holds if and only if

The authors acknowledge the support of the ANR grant CHASLIM (ANR-11-BS03-0007).