Yachao Zhu

Khalil Rajouâ

Steven Le Vot

Olivier Fontaine

Patrice Simon

Frédéric Favier
email: frederic.favier@umontpellier.fr

Modifications of MXene Layers for Supercapacitors

Keywords: MXene, foam, layer modification, pseudocapacitive behavior

The re-stacking of Ti3C2Tx-MXene layers has been prevented by using two different approaches: a facile hard templating method and a pore-forming approach. The expanded MXene obtained by using MgO nanoparticles as hard templates displayed an open morphology based on crumpled layers. The corresponding electrode material delivered 180 F g -1 of capacitance at 1 A g -1 and maintained 99 % of its initial capacitance at 5 A g -1 over five thousand charge-discharge cycles. On the other hand, the MXene foam prepared after heating a MXene-urea composite at 550°C, showed numerous macropores on the surface layer and a complex open 3D inner-architecture. Thanks to this foamy porous structure, the binder-free electrode based on the resulting MXene foam displayed a great capacitance of 203 F g -1 at 5 A g -1 current density, 99 % of which was retained after five thousand cycles. In comparison, the pristine MXene -based electrode delivered 82 F g -1 , only, in the same operating conditions. An asymmetric device built on a negative MXene foam electrode and a positive MnO2 electrode exhibited an attractive energy density of 16.5 Wh kg -1 (or 10 Wh L -1 ) and 160 W kg -1 (or 8.5 kW L -1 ) power density. Altogether, the enhanced performances of these nano-engineered 2D materials are a clear demonstration of the efficiency of the chosen synthetic approaches to work out the re-stacking issue of MXene layers.

Introduction

In the wake of graphene 1 , other two dimensional materials 2 , such as, δ-MnO2 3, 4, 5 , MoS2 6,7 and MXene 8, 9, , have received a surge of interest from the material science community, as they offer, together with their unique "planar" physical peculiarities, an infinite number of surface chemistry opportunities, especially when compared with their carbonaceous flagship. 11 These opportunities have been explored in many application fields including electrochemical energy storage. [12][13][14] In the corresponding devices, both basal planes and defected edges are usually contributing to charge storage. 15 As a result, one can take profit of the strong in-plane covalent bonding and weak out-of-plane Van der Waals interactions between layers 16 , to develop chemical strategies to provide or tune further active sites. These proceed by many chemical and physical methods including exfoliation 17 , intercalation 18 and hybridization 19 . The main idea is to prevent the natural tendency of the individual 2D layers to restack to minimize the surface energy just as in the pristine material(s). By chemical engineering at the nanoscale, associating layers of 2D materials of different chemical compositions and physical properties offers wonderful opportunities to take advantage of resulting synergistic effects 20 . Potential combinations are unnumbered, imagination is probably the limit.

As such, in the resulting composite, expanded, hybrid… materials, the opened 2D space between the layers can, for example, provide a "path" strongly suited for ion adsorption and transport. These characteristics have made 2D materials as very attractive electrode materials in batteries 21 and supercapacitors 22 . Two different kinds of supercapacitors (SC), can be distinguished, on the basis of the corresponding charge storage mechanism: electrostatic interactions in electrical double layer capacitors (EDLCs), and fast redox reactions near the surface material in pseudocapacitors. 23 With 2D materials, this classification remains as graphene is probably mostly EDLC type while exfoliated MnO2 birnessite is pseudocapacitive. [24][25][26][27][28][29] Composites obtained by stacking of EDLC and pseudocapacitive, EDLC and battery type, or battery type and pseudocapative 2D materials have been recently reported. [30][31][START_REF] Wang | Yanfang Song[END_REF][33] However, the anisotropy of material properties could also be a critical drawback for 2D materials, especially in energy storage electrode materials. Although, both in-plane ionic and electronic conductivities can be remarkable, through-plane conductivities can be fairly limited. Depending on the layer orientation toward the current collector surface, perpendicular or parallel, electrode overall performance can be down-graded.

MXene 34 , as a remarkable 2D material, containing a conductive carbide core along with transition metal oxide-like surfaces and intercalated water molecules, 35 triggered much attention and attracted worldwide researches in the field of energy storage and more specifically for supercapacitors. [36][37][38][39][40] MXene is prepared from the corresponding MAX phase by a chemical etching method, usually using a fluorine-containing solution. MAX phases are layered ternary carbides and nitrides with the formula Mn+1AXn, where M is an early transition metal (such as Ti, V and Nb), A is an element from A-group (usually Al or Si) and X is carbon and/or nitrogen. 8,41,42 Ti3C2Tx-MXene could be obtained by etching Ti3AlC2 in a mixture of LiF and HCl. Tx stands for the termination moieties at the layer surface. Their chemical nature depends on the etching process. They have a strong impact on the electrolyte/electrode interface especially through the hydrophilic/hydrophobic surface balance. Ti3C2Tx-MXene also shows up to ≈ 6700 S.cm -1 of metallic conductivity which is highly favorable to fast electron transfer. [43][44][45] However, as for other 2D materials, the MXene flakes tend to restack during the preparation process, resulting in a drastic loss of the developed electroactive surface area, and hindering the electrolyte ion access into the electrode bulk 15 . To prevent this re-stacking issue and simultaneously enhance the through-plane ionic conductivities, alternative methods based, for instance, on the modification of the layer morphology and texture have to be considered.

In this study, exfoliated Ti3C2Tx-MXene was first prepared by a chemical etching method from Ti3AlC2 corresponding MAX phase. To prevent the re-stacking of the resulting individual layers, several routes were explored. First, nano-sized MgO particles were used as solid spacer. After removal of the particles adsorbed at the layer surface, electrolytic ions were able to be efficiently transported in between the layers of the resulting expanded MXene during the charge-discharge process. Therefore, prepared expanded MXene (EM) electrodes showed enhanced electrochemical performances when compared to regular MXene. Alternatively, urea was used as molecular spacer or template. After a thermal treatment under argon atmosphere, a Ti3C2Tx-MXene foam (MF) was obtained. The resulting MF electrode showed an attractive and seriously improved capacitance performance, especially at high rate. When associated to a MnO2 positive-electrode in an MF//MnO2 asymmetric device, an attractive energy density of 16.5 Wh kg -1 (or 10 Wh L -1 ) was obtained at 160 W kg -1 (or 8.5 kW L -1 ) power density.

Experimental details

Preparation of MXene-Ti3C2Tx suspension and pure MXene: 1 g LiF (Sigma, 99.98%) and 20 ml 9 M HCl (Sigma, 37%) solution were mixed in a plastic beaker and stirred for a few of minutes. 1 g Ti 3AlC2 powders was then slowly added to the solution. The reaction temperature was kept at 35 °C for 24 h under constant stirring. The resulting Ti3C2Tx flakes were washed with water and separated by centrifugation until the pH value was ~ 6. The flakes were dispersed in 250 ml H2O and treated by sonication for 1 h. Finally, the resulting Ti3C2Tx was recovered by a 1 h centrifuge step at 3500 rpm. The average size of resulting MXene flakes is centered at 0.2 and 2 µm (Fig S1). The pure MXene film was prepared by a simple vacuum filtration procedure on a PTFE membrane by using 3500 rpm rotating speed.

Preparation of expanded MXene by hard templating method: 23.26 ml Ti3C2Tx MXene suspension (50 mg, 2.15 mg/ml) were dispersed in 26.74 ml, 200 mg MgO (Sigma, nanopowder ≤ 50 nm particle diameter) solution, and the solution was stirred for 24 h. During the filtration process, 40 ml, 3 M acetic acid (Sigma, 99.8%) were slowly added to remove the MgO template. After washing three times with pure water, the film was dried at 50 °C under vacuum.

Preparation of the MXene foam and overlapped MXene foam: 23.26 ml Ti3C2Tx MXene suspension (50 mg, 2.15 mg.ml -1 ) was mixed with 23.26 ml urea (Sigma) solution (200 mg with or without 0.1 M HCl).The resulting solution was stirred for 2 h. A thick disk was obtained by vacuum filtration. After drying it at 50 °C under vacuum, the resulting film was placed in a porcelain crucible and treated at 550 °C for 2 hours under argon atmosphere.

Preparation of MnO2: 0.1272 g MnCl2 (Merck Schuchardt, anhydrous) was dispersed in 20 mL H2O under continuous stirring. The same was done with 0.24 g KMnO4 (Sigma, 99.0%). The KMnO4 solution was slowly added to MnCl2 solution. The resulting mixture was heated at 100 o C and kept for 6 h at this temperature under constant stirring. After filtration and rinsing with water, the powder was obtained by drying at 80 o C overnight. From the XRD pattern, the prepared powder was assigned to -MnO2 (Figure S2). Corresponding SEM and TEM images are depicted in Figures S3 andS4, respectively.

Material characterization:

The crystal phase and structure of the prepared material films were examined by X-ray diffraction (XRD) using a Phillips X'Pert diffractometer with Cu Kα radiation (λ=1.5405Å).

Morphologies were imaged by using a JEOL JSM-6300F scanning electron microscope (SEM). The element composition was analyzed by Energy-dispersive X-ray spectroscopy (EDX) in SEM. The thermal decomposition of MF was followed by thermo-gravimetry coupled mass-spectrometry (TG-MS) under Ar from room temperature to 800 °C (rate 5 °C/min).

Electrochemical characterization: Electrochemical measurements were performed in three electrode systems, symmetric device and asymmetric at ambient temperature by using a VMP3 multi-channel Bio-Logic electrochemical workstation. In three-electrode system, platinum foil and Hg/HgO/OH -electrode were used as counter and reference electrode respectively. The working electrode was assembled by two clean nickel foam with sandwiched binder-free MXene film in between two stainless steel under 10 Mpa for 30 seconds. Finally, the system was tested in 1 M KOH electrolyte. In the symmetric device, two identical working electrodes were prepared as follows. The electrode was prepared by mixing the active material, carbon black and Polytetrafluoroethylene (PTFE) with a mass ratio of 60/30/10. The mixture was dispersed in a proper volume of ethanol under constant stirring. Then, the mixture was evaporated at 60 °C to get a slurry. With the adapted behavior, the slurry was rolled and, when dry, pressed in between two stainless steel under 10 Mpa for a few tens of second. Finally, the cut suitable film was assembled into the Swagelok device. For asymmetric device, MnO2 was used as a positive electrode. For individual electrodes, capacitances are relative to the mass of active material while for devices, they are relative to the cu;ulative mass of active materials in both electrodes. The specific synthetic routes are depicted in Figure 1. A suspension of exfoliated MXene flakes was first prepared as described in the experimental section (see Experimental details below). In the present study, it was used as starting material for the preparation of other MXene-based materials. The drastic changes in the corresponding XRD patterns shown in Figure 2 are a crystallographic proof of the conversion of Ti3AlC2 MAX phase (Fig. 2a) to MXene (Fig. 2b). As usually observed for 2D materials, only the peak series characteristic of the layered structure of MXene remains after conversion while the other peaks, especially at about 39° 2, disappear. An inter-layer distance of 11.15 Å was calculated for the prepared MXene from the 2 position of the (002) diffraction peak at 7.87° 2. From Ti3AlC2 MAX phase (Fig. 3a) to MXene (Fig. 3b), the more open and disordered 2D morphology is also evidenced by looking at the corresponding SEM micrographies. To prevent the re-stacking of the MXene layers and eventually to get an even more open 2D structure, magnesium oxide was considered as a solid nano-spacer (Fig. 1, middle). Such a hard template has already been used for the preparation of carbons with hierarchical porosities including activated or doped carbons, graphene... 46,47,48 It appears as suited for this purpose as it is easy to prepare and/or commercially available as nanoparticles that efficiently absorb material precursors, either in molecular, polymeric or solid states, through Van der Waals interactions and hydrogen bonds. Moreover, as such a template, it can be easily removed using a mild acidic post-treatment. MgO particles were first homogeneously dispersed in the suspension of Ti3C2Tx-MXene flakes by ultrasonication and vigorous stirring. Once the composite suspension filtrated, MgO nanoparticles are randomly and uniformly distributed in between the MXene layers. Commercial MgO particles were actually chosen for their diameter below 50 nm to fit the MXene flake size distribution (at 0.2 and 2 µm, Fig S1). This flake/particle size ratio promoted a consistent 2D sandwich composite structure build on flakes large enough to cover "many" particles instead of a simple/single particle wrapping that would restrain the development of the targeted expanded framework. Afterwards, MgO particles were slowly "digested" by gently pouring an acetic acid solution, and progressively washed away. The resulting film of expanded MXene was recovered after washing and drying. When comparing the XRD pattern of the expanded MXene as shown in Figure 2 with that of the exfoliated MXene used as precursor, the structural impact of the MgO hard templating is obvious: While the MXene layered structure remains, despite a severe amorphization of the material related to the loss of the long range order, a strong shift of the (002) peak towards lower angles is observed (Fig. 2c). As such, the interlayer distance increases from 11.15 to 13.60 Å along the c-axis thanks to hard templating with MgO nanoparticles. interconnected channels and conductive walls, is anticipated to promote suitable ion diffusion and electronic percolation through the whole material volume.

On the other hand, urea has been widely used as a reactant to synthesize N-doped materials through a pyrolysis process, including nanostructured metal oxides 49 , carbides 50 , carbons 51,52 , etc. Although, such a doping is a conventional strategy in semiconductor processes, it has been more recently introduced in the synthesis of energy conversion and storage electrode materials, for oxygen reduction 53 , supercapacitors 54 or LiS batteries 55 , leading to enhanced electrochemical performances and stability. After mixing the Ti3C2Tx-MXene with urea, a thick disk was obtained by vacuum filtration. The re-staking of MXene layers is prevented by urea molecules trapped in between. In such a slightly acidic medium, cationic protonated urea strongly absorbs at the negatively charged MXene flakes and the layers are efficiently kept apart when going to solid state. The resulting MXene@urea composite material was pyrolyzed at 550 °C under argon atmosphere. All synthetic details are given in the experimental part above. SEM micrographies of the pyrolized Mxene@urea composites are depicted in Figure 3e, 3f and Figure 4b to 4d. Although the side view (Fig. 3e and3f) is characteristic of the opening of the structure of the material induced by the synthetic process, the layered morphology of the pristine MXene (Fig. 3b) is hardly identified in the resulting disordered honey-comb structure. Moreover, discrepancies in the top views in Figure 4 of MXene and the resulting material are spectacular. While MXene shows a flat and "plain" surface build on Ti3C2Tx layers (Fig. 4a), the pyrolysis of MXene/urea composite obviously generate numerous macro-pores through the layers, i.e. perpendicular to the surface (Fig. 4b-4d). The resulting 3D structure based on a disordered self-assembling of exfoliated and porous layers displays a complex foam architecture. When comparing with the pristine MXene, the (002) peaks of MXene Foam (MF) slightly shift towards greater 2 angles, suggesting a decrease of the inter-layer distance induced by the synthetic process (Fig. 2d). Such a shrinking can be assigned to the de-intercalation of H2O molecules during the thermal treatment, either free or solvating urea molecules, originally trapped in between the MXene layers. Moreover, during the process, MXene flakes are self-assembling as a foam structure by a pore-forming process that is confirmed by the appearance of a (110) peak at about 60.85° 2. This peak is characteristic of the As already mentioned, many authors have reported on the use of urea to N-dope various materials including carbons and carbides. To the best of our knowledge, any of these have mentioned such a drastic impact on the resulting material morphology, especially on pore generation. In the present case, the reactivity of urea towards Ti3C2Tx layers at 550 °C can be assigned to the pore forming process and generation of the complex 3D foam structure. Surprisingly, it was not possible to demonstrate the presence of nitrogen as dopant in the resulting material by SEM-EDX (Fig. S5 and S6 and Table S1). The TG-MS analysis performed on an MXene/urea composite up to 800 °C, confirms the thermal decomposition of urea as ammonia, carbon monoxide, carbon dioxide and nitrous oxides. The mechanism of formation of MXene foams involves these gases as generated CO2 etches the carbon from Ti3C2.2Tx-MXene (as calculated from EDX measurements in Table S1) to drill holes in the layers, resulting in a Ti3C1.7Tx-MXene foam. Generated gases, while confined during the thermal treatment, are suspected to induce mechanical deformations of the layers before to escape from the complex macroporous structure. Electrochemical performance of prepared electrode materials was investigated in three electrode configuration in 1M KOH electrolyte. In Figure 5a, the roughly rectangular shape of CVs at 5 mV s -1 demonstrates the pseudocapacitive behavior of prepared MXene-based materials. A similar behavior was also observed in 1M Na2SO4 neutral electrolyte while in acetonitrile organic electrolyte, the electrochemical activity was limited (Figure S7). As demonstrated by the CVs of the MXene foam-based electrode in Fig. 5b, this behavior remains at higher scan rate, up to 100 mV s -1 at least, as only slight distortions are observed. In the series, a quick comparison of the corresponding integral area at 5 mV s -1 scan rate suggests that the MXene foam -based electrode presents the greatest capacitance. This was confirmed by the galvanostatic charge-discharge measurements at 5 A g -1 (Fig. S8a) used for specific capacitance calculations. The specific capacitance of pristine MXene is the lowest at 80 F g -1 . For expanded MXene, capacitances is 112 F g -1 . Finally, the greatest specific capacitance at 5 A g -1 was measured at 203 F g -1 (811 mF cm -2 ) for the MXene foam -based electrode. This enhancement can certainly be assigned to the progressive opening of the material structures in the series leading to a larger electrochemical interface and greater number of sites available for electrolytic ions involved in charge compensation. The improvement in the ion transport capabilities of the materials is also confirmed by the specific capacitance retention at higher scan rate (Fig. 5b). In addition, the achieved capacitance of MXene foam in this paper exceeds other MXene-based supercapacitors. For instance, Zhao et al. 57 prepared a composite, in which RGO plays a role of conductive "bridge" to link with Ti3C2Tx blocks, thus showing a specific capacitance of 154 F g -1 at 2 A g -1 . Zhu and co-workers 58 decorated the TiO2 on the Ti3C2 layers by using a simple in situ hydrolysis and heat-treatment process. The synthesized composite revealed a high specific capacitance of 143 F g -1 at 5 mV s -1 . Yang et al. 59 deposited the binder-free Ti3C2

MXene/carbon nanotubes films onto graphite paper by the electrophoretic deposition method. The asprepared electrode displayed 134 F g -1 at 1 A g -1 . Shi et al. 60 synthesized the MXene foam by using a thermal treatment method to restrain the restacking issue. The obtained MXene foam exhibited a high capacitance of 123 F g -1 at 5 mV s -1 . Starting with pristine MXene electrode at 48 F g -1 , specific capacitances at 20 A g -1 ranked about the same as the Expanded MXene electrode shows 70 F g -1 and

MXene Foam a remarkable 125 F g -1 specific capacitances (Fig. 5c and Fig. S8b). Moreover, the observed asymmetry of the galvanostatic curves is characteristic of limited Coulombic efficiencies. This behavior is actually shared by many MXene-based electrode materials. 61,62,63,64 Although a poor 65 % Coulombic efficiency was measured for the MXene foam electrode at low current density (5 Ag -1 ), it reached up to 90% at 20 A.g -1 . For pure MXene and Expanded MXene -based electrodes, fair Coulombic efficiencies of 99 and 95 % were measured, respectively. These values actually fit within the values reported in the literature with, in average, 81 % Coulombic efficiency when titanium carbide MXene is the active material in such pseudocapacitive electrodes (Table S2). Moreover, Coulombic efficiencies as low as 60 % were reported for an electrode based on a Ti2CTx/carbon nanotube film [START_REF] Li | The facile synthesis of layered Ti2C MXene/carbon nanotube composite paper with enhanced electrochemical properties[END_REF] . In the present study, limitations in Coulombic efficiencies are assigned to parasitic redox reactions, more specifically those involving the electrolyte, such as water reduction, and/or the MXene surface groups. The Faradaic origin of this (slow) phenomenon is supported by the improvement of the measured Coulombic efficiency as the scan rate (or current density) is increased (Fig. S9a). The detrimental effects of such parasitic Faradaic reactions are obviously related to the developed active surface area of the corresponding electrode material. As such they are progressively more marked for the electrode based on pure MXene, then expanded MXene, and finally MXene foam. Calculated Coulombic efficiencies were normalized by the active surface area of each electrode and plotted as a function of the applied current density (Fig. S9b). For all the electrodes in the series, the behavior of the normalized Coulombic efficiency is the same when increasing the applied current density, confirming the dependence of the Coulombic efficiency towards the electroactive surface area and the unique Faradaic origin of the parasitic reaction impeding the electrode Coulombic efficiencies. By the end, the improvement of the Coulombic efficiency with charge/discharge rate makes these materials as remarkably suited for supercapacitor application, especially when using an MXene foam -based electrode for which the improvement is the greatest. A more detailed analysis was done on the kinetics of the charge storage in the various prepared MXene-based electrode materials. First, the Log-Log current versus scan rate curves were plotted from CV data at various scan rates (Fig. S10). b slopes were calculated at 1.06 (pure MXene), 0.99 (Expanded MXene) and 1.02 (MXene foam) and correspond to i=a v b relationship [START_REF] Lai | [END_REF]67,68 . Being close to 1, they suggest the charge storage mechanism in the corresponding materials to mostly proceed through fast surface-controlled processes, hardly limited by diffusion of electrolytic ions in the electrode bulk. Second, the stronger contribution of the surface material was especially confirmed for the MXene foam electrode by separating capacitive (surface) versus diffusion-limited currents in the corresponding CVs operated at various scan rates (Fig. S11). The corresponding capacitive ratio are reported as an histogram in Fig. S12. It should be noticed that it is 71% at 5 mV s -1 , implying a significant capacitive/surface contribution to the overall measured capacitance, even at such a low scan rate. Furthermore, the capacitive contribution increases while raising the scan rate, suggesting that, at high rates, MXene foam stores charges mostly through surface reactions in the EDL.

Nyquist plots shown in Fig. S13 The electrochemical performances of the prepared electrode materials can be rationalized by using Gogotsi statement on the various strategies to prevent nanosheet re-stacking to fully use the developed surface area and enhance the electrolytic ion diffusion and transport. 15 As shown in Figure 6a, the flat layers of the pristine MXene are the less favorable material design in the series as nothing, in such a case, can prevent the natural trend of the layers to self-assemble in a re-stacked structure. The diffusion of the electrolytic ions is hindered by the limited inter-layer space and most of the electrochemically active surface corresponds to the external surface of the material. Ion diffusion perpendicular to the layer surface is only possible along the edges of the MXene particle. As such, corresponding electrochemical performances, especially specific capacitance and rate capability, are fair but also leave many room for improvement. In contrast, the morphology of the expanded MXene is inherited by the shape and the size of the MgO particles used as hard template. The open structure is promoted by the assembling of a composite made of exfoliated MXene layers wrapping MgO particles. After hard template removal in mild acidic conditions, its footprint remains, providing the assembled layers with a crumpled morphology and an enlarged interlayer space. The porosity generated in between the curved layers is obviously more favorable for in-plane fast ion transport and access to more active sites. As a consequence, both electrochemically active surface area and ion diffusion are greatly improved. So are the corresponding electrochemical performances. Unfortunately, diffusion through the plane is still limited to particle edges.

This limitation can be overpassed by introducing porosity on the layers. The thermal decomposition of urea between or at MXene layers generate holes and promote a structure built on the assembly of curved porous layers. The resulting electrochemical performances of the prepared foam take advantage of this complexe 3D structure showing simultaneously large interlayer spaces and many connexions through the layers facilitating access of the electrolytic ions to the entire volume of the electrode. The electrochemical performance of a symmetric device based on MXene foam -based electrodes (MF//MF) are shown in Figure 7 and Fig. S15. The CV curves in KOH 1M at various scan rates are displayed in Fig. 7(a). A decent device rate capability is anticipated since, up to 300 mV s -1 , distortions of the rectangular CV shape measured at low scan rate are very limited. Specific capacitance values were calculated from galvanostatic charging-discharging measurements at various current densities. Up to 20 A g -1 the signal kept the expected triangular shape with limited Ohmic drop (Fig. S15a). Device energy and power densities were obtained from these measurements and reported in the Ragone plot shown in Fig. S15b. With electrode loadings at about 3.5 mg cm -2 , the energy density of symmetric MF//MF device is about 3.6 Wh kg -1 at 350 W kg -1 of power density. When relative to the cumulative volume of MF in both electrodes, it translates as 2.5 Wh L -1 at 2.5 kW L -1 . Although these performances are limited, the aim of these measurements, either in 3-and 2-electrode setups, is to highlight the specific behavior of MF electrode, notwithstanding the potential limitations by any other "counter" electrode material. The cycle stability of device was measured at 5 A.g -1 over 8000 charge-discharge cycles. The calculated 87 % capacitance retention confirm the fair stability of the electrode materials when operated in the given conditions (Fig. 7b). However, despite the promising electrochemical behavior of the electrode material, in such MF//MF symmetric device, the operating voltage is limited to 0.7 V with a severe impact on the available energy density, especially. To address this issue, an asymmetric device was considered. It was built on a MnO2 positive electrode (Fig. S16) and a MF negative electrode. Electrochemical results in KOH 1M are summarized in Figure 8. In comparison with the symmetric MF//MF device, MF//MnO2 shows a great enhancement as a 1.6 V cell voltage was obtained. Both CV and galvanostatic series of curves are characteristic of the pseudocapacitive behavior of the electrode components but the signal distortions are characteristic of lower rate capabilities (Fig. 8a and8b). This was confirmed when calculating the device power density at 160 W Kg -1 or 8.5 kW L -1 . In contrast the effect of the larger cell voltage is expressed by an enhanced energy density at 16.5 Wh kg -1 , 2 to 3 times greater than that of a C//C symmetric device based on activated carbon electrodes in aqueous electrolyte 69 . When considering the volume of active materials in both electrode, the volumetric energy density reached 10 Wh L -1 (4 times greater than that of a C//C device). During the first 5000 charge-discharge cycles, the device capacitance progressively increased to reach more than 110% of the initial value at 25 F g -1 (Fig. 8d). This increase can be assigned to a slow and progressive impregnation of both electrodes upon cycling.

Conclusion

To address the re-stacking issue of exfoliated MXene layers, we successfully prepared and engineered the expanded MXene and MXene foam materials by a hard template approach and a pore-forming method, respectively. As electrode materials, the binder-free MXene-based materials showed promising electrochemical performances in KOH 1M, either in terms of specific capacitance, rate capability and long term cycling. When moving from MXene to expanded MXene and MXene foam, the observed great improvement was assigned to the increase of the developed electroactive area and ion transport capability promoted by the morphology opening with larger interlayer space and pores through the MXene layers.

MF was tested as electrode material in devices. Both symmetric MF//MF and asymmetric MF//MnO 2 devices displayed attractive rate capability and excellent cycle stability. Moreover, MF//MnO2 device exhibited a fair energy density of 16.5 Wh kg -1 and 10 Wh L -1 .

Figure 1 .

 1 Figure 1. Synthesis scheme of the preparation of various forms of MXene, as re-stacked material after exfoliation (top), expanded MXene after re-stacking in presence of MgO nanoparticles (middle), and MXene foam from thermal treatment in presence of urea (bottom).

Figure 2 .

 2 Figure 2. XRD patterns of Ti3AlC2 MAX phase (a), MXene (b), Expanded MXene (c), and Mxene Foam (d).

Figure 3 .

 3 Figure 3. Side views from SEM images of Ti3AlC2 MAX phase (a), Ti3C2Tx-MXene (b), Expanded Ti3C2Tx-MXene (c, d) and Ti3C2Tx-Mxene foam (e, f).

Figure 4 .

 4 Figure 4. The SEM images of pristine MXene (a) and MXene Foam (b-d) surface view.

Figure 5 .

 5 Figure 5. Electrochemical characteristics of binder-free electrodes in three configurations using 1 M KOH as electrolyte. (a) Comparison of CV curves of the prepared materials at 5 mV s -1 scan rate. (b) CV curves of MXene foam electrode at various scan rates. (c) Comparison of the specific capacitances of the prepared material as a function of the applied current densities. (d) Comparison of the capacity retention of pristine MXene, expanded MXene and MXene foam at 5 A g -1 over 5000 charge-discharge cycles. Dashed curve for MXene Foam corresponds to data withdrawn because of equipment failure.

  confirm the similar behaviour of the prepared electrodes at high frequency, especially in terms of charge transfer resistance, but the straight and sharp increase of the imaginary part of the impedance (-Z'') of the MF -based electrode in the low frequency range confirms this material attractive capacitive performance. On the other hand, the cycle stability was tested for 5000 cycles by galvanostatic charge-discharge measurements at 5 A g -1 current density. The results are shown in Fig. 5d for MXene-based electrodes. Dashed curve, especially between cycles #2000 and #3000, corresponds to data withdrawn because of equipment failure. Finally, the pristine MXene keeps 94 % of its initial specific capacitance, while expanded MXene and MXene foam show excellent 99 % capacitance retention.

Figure 6 .

 6 Figure 6. Schematic diagram of ion transport in different morphologies' electrode materials in supercapacitor.

Figure 7 .

 7 Figure 7. The electrochemical characteristic of symmetric MF//MF device. (a) CV curves at various scan rates. (b) Cycling performance over 8k cycles of charge discharge at 5 A g -1 .

Figure 8 .

 8 Figure 8. The electrochemical performance of asymmetric MF//MnO2 device. (a) CV curves at various scan rates. (b) Galvanostatic charge-discharge test at various current densities. (c) Charge-discharge cycling performance at 5 A g -1 over 5k cycles.

Acknowledgements

Y. C. ZHU (NO. 201606240097) is supported by China Scholarship Council (CSC). We thank C. Bodin and P. Lannelongue for help with the electrode preparation.


Stéphane Gorsse

Yung-Ta Chen

Wei-Che Hsu

Hideyuki Murakami

An-Chou Yeh

Modeling the precipitation processes and the formation of hierarchical microstructures in a single crystal high entropy superalloys

Although superior high temperature tensile yield strength of high entropy superalloys (HESAs) arises from their hierarchical microstructure, the precipitation processes driving its formation remains unclear. In the present study, we analyze the kinetics of '-and-precipitations by treating the concurrent nucleation, growth and coarsening using common computational thermodynamic and kinetic tools to simulate the microstructure genesis and evolution in HESA during thermal treatments.

The ability of the simulations to reproduce the experimentally observed microstructure parameters is evaluated. Temperature-time-transformation (TTT) diagrams are calculated to serve as guidelines for further optimization of the hierarchical microstructure of HESAs.

Ni-based superalloys have been the best answer so far to the demand for high-temperature structural materials in modern gas turbine engines used in the aerospace and energy industries. This demand-driven material requirement mainly raised from the economic advantages of increased efficiency at higher operative temperature. Nowadays, ecologic imperative on cutting down in CO 2 emissions of the aircraft sector and gas-fired power plants represents a further strong incentive for the research and development of new alloys with even better high-temperature capability than Nibased superalloys (i.e. enhanced resistances to high-temperature creep and high-temperature oxidation).

Excellent high-temperature mechanical properties of Ni-based superalloys result from their -′ microstructure, i.e. Ni 3 Al-based precipitates (′) with an L1 2 -ordered structure coherently embedded in a disordered  matrix phase with a face-centered cubic (FCC-A1) structure [START_REF] Reed | The Superalloys Fundamentals and Applications[END_REF]. Because of its great success in Ni-superalloys, the -′ (A1+L1 2 ) microstructure is regarded as a template for the development of two new classes of high-temperature alloys whose underlying concepts have been discussed in recent papers, i.e. high entropy superalloys (HESAs) [START_REF] Tsao | [END_REF][3][4] and refractory superalloys (RSAs) [5,6]. In brief, by shifting from the center of hyper-dimensional composition space, HESAs fill the compositional gap between equimolar high entropy alloys (HEAs) and Ni-based superalloys, and RSAs are BCC(A2)-based refractory complex concentrated alloys (RCCAs) with B2-ordered phase.

While the transformation pathways responsible for forming the A2+B2 microstructure are unknown and that available thermodynamic databases do not include or fail to correctly describe the B2 phase in refractory systems, the development of HESAs could benefit from well-established Ni-based thermodynamic and kinetics CALPHAD databases.

The general intention of this article is to assess the agreement between observations of the -′ microstructure in HESA with CALPHAD calculations of the precipitation process using common computational thermodynamic and available databases. For that, we will rely on the detailed information on the precipitation kinetics in one HESA recently published by Chen et al. [7]. Our interest in this HESA is spotted by the formation of a hierarchical microstructure comprising secondary  spherical particles formed inside primary ′ cuboid precipitates and its elevated temperature tensile yield strength which overcomes that of advanced superalloys such as the CMSX-4 benchmark. Hierarchical microstructures are not specific to HESA, they were also observed in several aged Ni-based superalloys such as IN-738 [8], Mar-M200, IN-100 [9], and CMSX-4 [10], as well as in model ' precipitation hardened ternary alloy such as Ni-Al-Ti [11,12], Ni-Si-Fe [13,14], Ni-Al-Fe [13], and Ni-Al-Si [15]. Their formation was first explained by H. F. Merrick [8] and Oblak [9] as the precipitation of  within ' particles during aging. In the following, we present the results of the investigation on the precipitation kinetics and the predicted temperature-time-transformation (TTT) diagram for the phase separation of the  matrix and of ' coherent precipitates causing the hierarchical /' two-phase microstructure in HESA.

In this work, the nucleation, growth and coarsening of precipitates during cooling and thermal treatments in HESA was treated using the Thermo-Calc Precipitation Module (TC-PRISMA) integrated into a programming workflow with TC-Python, the thermodynamic database TCNI9 and the kinetic database MOBNI5 from Thermo-Calc [16]. TCNI9 includes the assessed thermodynamic descriptions for all the ternary systems with Ni comprised in the present Ni-rich HESA. TC-PRISMA model is built on Langer-Schwartz theory [17] and the Kampmann-Wagner numerical approach [18]. The effects of interfaces on equilibrium solubilities of small particles are accounted for using the Gibbs-Thomson equation. The simulations of diffusion-controlled precipitation kinetics and temperature-timetransformation (TTT) diagrams for  and ' in HESA were performed through the TC-Python Software Development Kit (SDK) allowing to couple TC-PRISMA with other numerical packages for data visualization.

The nominal composition of the HESA is listed in Table 1 [7]. Single crystal HESA samples were fabricated along [001] orientation by Bridgman method from Superalloys and High Temperature Materials Group in National Institute for Materials Science (NIMS), Japan. The as-cast samples were homogenized and solutioned at 1500 K for 20 h followed by air cooling to room temperature. After solution treatment, aging window (temperature and time space) for the formation of  particles inside the primary ' precipitates were examined at 723, 923, 1023, 1123 or 1223 K for 20 hours, or at 1023 K for 10, 20, 50, 100 or 200 hours followed by water quench. The presence of nano-sized  particles were characterized by Scanning Electron Microscope (SEM: JEOL 7200F) with the samples prepared by grinding and electro-etching in 20% H 3 PO 4 + 80% H 2 O solution at 2.5 Volts. Based on the SEM images, the average radius and number densities of  particles were determined by Nano-Measurer software. Additional structural information was revealed by Transmission Electron Microscope (TEM: JEOL JEM-F200, 200 kV) with the thin foils prepared by twinjet polishing in the solution containing 10% HClO 4 + 90% C 2 H 5 OH under 30 Volts / -30 o C. -b show the SEM image, the dark field TEM image and the corresponding electron diffraction pattern of single crystal HESA with the composition shown in Table 1, after homogenization and solutioning at 1500 K for 20 h followed by air cooling to room temperature according to the temperature profile shown in Figure 2a. The microstructure consists of ' precipitates, with a volume fraction of 0.68, embedded in the  matrix. Superlattice diffraction pattern observed on the [001] zone axis indicates that these precipitates possess a L1 2 structure. The ' precipitates exhibit two different morphologies, i.e. large primary cuboids and small secondary spherical particles. The size of cuboid precipitates was taken as the radius of a circle with equivalent area. The estimate from image analysis gives 72 nm of equivalent radius for the primary ' and 0.5 nm for the secondary '. The chemical composition of the primary and secondary ' precipitates formed after air cooling was investigated by atom probe tomography and listed in Table 1.

Figures 1a

Table 2 summarizes the subsequent isothermal aging treatments performed on the air cooled HESA.

Aging at 1023 K for 20 h followed by water quenching causes the precipitation of nano-sized  particles with spherical shape inside the ' cuboidal primary precipitates, resulting in the formation of the hierarchical microstructure (Figs. 1c-g). No  particle is visible inside ' for aging performed at 723, 923, 1123 or 1223 K. The precipitation of  inside ' is already visible after 10 h at 1023 K, while the  particles increase in size and decrease in number density during prolonged aging, to finally disappear after 200 h. To compare observations and simulations, we converted the apparent particle radius, , and number of particles per unit area, , from 2D sections into 3D spheres of radius [19], and number density per unit volume, [20] (Table 3).

Transformation upon continuous cooling -The regularity of the microstructure shown in Figure 1a illustrates the morphological adaptation of the ' precipitates inside the  matrix resulting from the interplay between coherency stresses (stress fields arising from coherency strains due to the lattice misfit between  and ') and interfacial energy. To capture the genesis and evolution of this Figure 2b shows the calculated particle size distribution and the cubic factor [24] of the '

precipitates formed during continuous cooling from the solvus temperature according to the temperature profile in Figure 2a. Upon cooling, while the primary ' precipitates are growing, they undergo successive changes in shape from a spherical shape, when interfacial energy is predominant, to a cuboid morphology, when coherency energy dominates. The calculated radius of the primary and secondary precipitates is in good agreement with the measurements (Fig. 2b), and the simulation captures well the bimodal distribution of the precipitate's sizes according to two different morphologies.

Further insights into the precipitation process are provided by the calculated TTT diagram in Figure 3.

In addition to the temperature-time combinations calculated for a volume fraction of 1% of ' (' start curve), the predicted temporal evolution of the radius, volume fraction, and morphology of '

precipitates are represented in the temperature-time space for both isothermal (aging) and nonisothermal (continuous cooling from the solvus temperature) treatments. According to calculations, the precipitation kinetic of ' occurs very rapidly in HESA, at a slightly shorter timescale compared to most of the Ni-based superalloys. For example, the air cooling process of the CM247LC Ni-based superalloy results in ' radius around 50 nm [START_REF] Chiou | Mater. Sci. Forum[END_REF] while it is 72 nm in HESA after air cooling (note that in [7] the reported value of 143 nm represents the diameter of primary ' while in the present paper we refer to the radius). The radius of ' in the FGH98 Ni-based superalloys is about 40 nm after cooling from the solvus temperature at a rate of 10.8 K/s [START_REF] Huang | [END_REF], while in HESA the radius of ' is twice larger (72 nm) for a cooling twice faster (about 20 K/s between the solvus temperature and 1000 K).

The higher precipitation kinetics in HESA can be attributed to faster chemical diffusion due to less refractory contents than in conventional Ni-based superalloys. As shown in Figure 3, even quenching 100 times faster than air cooling cannot avoid crossing the ' start curve; it takes only few seconds for ' to precipitate and grow in HESA. This is also observed for conventional Ni-based superalloys such as CM247LC, for which fast quenching process, such as forced argon cooling, cannot avoid nucleation of ' precipitates and result in a refinement of the precipitation structure (higher number density and smaller size of ' particles) [START_REF] Chiou | Mater. Sci. Forum[END_REF].

Hierarchical precipitation during aging -We find that ' precipitates formed during air cooling are supersaturated in Co, Cr and Fe (Table 1), therefore driving the formation of  particles inside ' during aging at 1023 K (Table 2, Figures 1c-g). Figure 4 shows the calculated TTT diagram for the precipitation of  from ' with composition measured by APT (Table 1), which was reported in previous work [7]. The  start and the 50% completion curves indicate that the precipitation of  inside ' occurs at lower temperatures and with lower rates than for ' from . The isothermal aging treatments applied to HESA after cooling (Table 2) are represented at the corresponding temperature-time coordinates. When the coordinate lies inside the  start curve, the experimental point is denoted by a circle whose radius is proportional to the calculated average radius of the  precipitates and the color indicates their number density, . In contrast, when the aging falls outside the  start curve (i.e. in the supersaturated ' region of the temperature-time space), it is shown as black diamond.

The assessment of the validity of the predictions presented in Figure 4 is done by comparing three features from the observations: (1) the occurrence or not of the phase separation (i.e. formation of the hierarchical microstructure), (2) the average size, and (3) the number density of the  particles inside ' precipitates as a function of the temperature and time of the thermal treatments. Firstly, the calculated TTT diagram (Figure 4) reveals that the 723K/20h and 1223K/20h temperature-time coordinates remain in the metastable supersaturated ' region which explains that phase separation is not experimentally observed. Even though ' could eventually serves as a matrix for  precipitates for both these temperatures, at 1223 K there is not enough driving force while at 723 K there is a lack of atomic mobility for the precipitation to occur. By contrast, thermal treatments performed at 1023 K are located inside the  start curve where the kinetics are at the highest, which is consistent with the observations. At 923K/20h and 1123K/20h, the results of the calculations give new insights about the absence of hierarchical structure for these temperature-time coordinates: at 923K/20h  precipitates are two small (less than 1 nm) for SEM resolution, and at 1123K/20h  precipitates are too few (about 1  particle every 100  precipitates) to be caught during observation. Secondly, there is an overall good agreement between measured and calculated values for the radius of the  particles formed inside ' precipitates after aging at 1023 K. The deviation between observed and calculated values of the radius is less than 20%. Thirdly, measured and predicted values of the number density of the  particles are of the same order of magnitude and follow the same decreasing trend with the duration of the thermal treatment, from about 10 23 after 10 h for both, to 10 20 m -3 and 10 22 m -3 after 200 h, respectively, for the experimental observation and the prediction. This assessment suggests that the present model based on TC-PRISMA, and operating the thermodynamic database TCNI9 and the kinetic database MOBNI5 from Thermo-Calc Software, captures reasonably well the processing pathway inducing the formation and evolution of the hierarchical microstructure, including the kinetics of ' precipitation from  matrix and  precipitation from ' precipitates acting as a matrix. To verify the predictive accuracy of the model, we have simulated unseen data used as a testing set. Figure 5 shows the model's performance for the radius and the volume fraction of '

precipitates formed during aging treatments at 800, 950, and 1050°C for 500 h. The main discrepancy related to the calculated ' volume fraction, which is underestimated by 20% at the highest temperature, suggests that the equilibrium tie-lines are not perfectly estimated from the TCNI9 thermodynamic database. Despite this, the overall fair agreement between experiments and predictions encourages confidence in the model. These results make it possible to consider further guidance of the design of future advanced high-temperature HESAs using computational tools. For example, according to Chen et al. [7] the maximum strengthening contribution arising from the hierarchical microstructure is obtained with  particles of 3.6 nm in radius associated to the equilibrium volume fraction of about 8%. According to the predictions, ' precipitates should undergo phase separation between 950-1000 K for 300 h to achieve these optimal values for peak strength.

This result suggests that tuning HESA composition to favor a higher proportion of  particles at the optimal size should be preferably explored as a better alternative to very long thermal aging for further improvement of the high-temperature yield strength.

Common computational thermodynamic and kinetic tools have been used to successfully rationalize the genesis and evolution of the hierarchical -' microstructure in a high entropy superalloy. The nucleation, growth, and coarsening of (i) ' precipitates from supersaturated  during continuous cooling, and (ii) the formation of  particles inside supersaturated ' cuboid precipitates acting as a supersaturated matrix during subsequent isothermal aging, were modeled separately, and the corresponding temperature-time-transformation (TTT) were simulated.

Several specific outcomes can be highlighted:

 The precipitation kinetics, the size, the morphology, the volume fraction and the number density of the  and ' precipitates formed under isothermal and non-isothermal treatments in HESA can be predicted with a good agreement with experimental observations, which validates the use of thermodynamic and kinetic databases initially developed for Ni-based alloys.

 The simulations for the ' precipitation in HESA show a shorter timescale in comparison to conventional Ni-based superalloys.

 The simulations for the  precipitation from ' show isothermal aging performed at 1023 K promotes the formation of the hierarchical microstructure with features close to the peak strength. Gaining better high temperature tensile yield strength in this HESA might be achieved by lowering the aging temperature but with an increase of the aging time.

 While the development of HESAs, and more generally of HEAs, still relies mainly on the study of phase equilibria, the present work suggests that available computational thermodynamic and kinetic tools can be considered for detailed analysis and predictions of the precipitation processes. This is important to better control and optimize the microstructures and the resulting mechanical properties throughout appropriated thermal treatments and composition tuning. for spherical shape to for cubic shape [24]. Red dashed lines show the uncertainty interval for the measured radius of the primary and secondary ' precipitate. Circles and rounded squares lying on dashed lines are calculated for non-isothermal treatments, i.e.

Tables

AC stands for air cooling and Q for quenching at a rate 100 times AC.

  microstructure it is necessary to consider the elastic anisotropy of the  matrix, being known that the <100> directions are less rigid than the <110> and <111> directions. Values of elastic constants C 11 = 250 GPa, C 12 = 160 GPa and C 44 = 130 GPa were taken from Wen et al.[21] who investigated one Nibased superalloy with a composition close to that of the present HESA composition. Interfacial energy between ' and  was tuned by matching precipitate sizes measured by image analysis. The obtained value of 0.08 J/m 2 is similar with typical values found in Ni-based superalloys such as TMW-4M3 and NIMONIC exhibiting high volume fraction of' phase (above 50%)[22]. On the other hand, lower values of interfacial energy, about 0.02 J/m 2 , is generally observed in Ni-based superalloys with a low volume fraction of ' precipitates such as in the Alloy 617 (about 6.4%)[23]. This observation suggests a positive correlation between the '/ interfacial energy and the volume fraction of' precipitation in Ni-based superalloys.

FiguresFigure 1 .

 1 Figures

Figure 2 .

 2 Figure 1. SEM and TEM observations of HESA after heat treatments. (a) SEM image of the

Figure 3 .

 3 Figure 3. TTT diagram for the precipitation of ' from the  matrix. The supersaturated  matrix

Figure 4 .

 4 Figure 4. TTT diagram for the precipitation of  from ' primary precipitates. The supersaturated '

Figure 5 .

 5 Figure 5. Observed vs predicted radius of ' precipitates from  after thermal treatments at 800, 950,

Table 1 .

 1 Chemical compositions of HESA and different phases analyzed by APT (at.%)[7]

		Al	Ti	Nb	Ni	Co	Cr	Fe	Mo	W
	Nominal	10.2	5.8	1.2	48.3	16.9	7.4	8.9	0.9	0.4
	After air cooling from the solvus temperature
	Primary '	13.1	8.3	1.3	56.6	13.2	2.3	4.5	0.5	0.2
	Secondary ' 14.5	8.6	0.7	60.6	12.1	0.8	1.7	0.8	0.2
	Calculated equilibrium composition of ' at 1023 K
	Eq. '	14.1	8.5	1.5	60.2	12.4	0.7	2.4	0.2	<0.01

Table 2 .

 2 Observed precipitation of  inside ' after isothermal aging of air cooled HESA. N stands for no precipitation, Y stands for precipitation.

		Time
	Temperature	10 h	20 h	50 h	100 h 200 h
	1223 K		N
	1123 K		N
	1023 K	Y	Y	Y	Y	Y
	923 K		N
	723 K		N

Table 3 .

 3 Quantities measured by SEM for  particles inside ' cuboid precipitates and their conversion into 3D equivalents. and are the apparent and 3D radius,

	and	are the


Andrea Bisoffi

Ruud Beerens

Luca Zaccarian

Maurice Heemels

Henk Nijmeijer

N Van De Wouw

W P M H Heemels

Hybrid model

Keywords: hybrid systems, nonlinear systems, Coulomb friction, Lyapunov methods, global asymptotic stability, PID control

come   L'archive ouverte pluridisciplinaire

INTRODUCTION

In this paper, we present a hybrid model formulation for a proportional-integral-derivative (PID) controlled singlemass motion system subject to Coulomb friction. Friction is a performance-limiting factor in many high-precision positioning systems in terms of the achievable setpoint accuracy and settling times. Many different control strategies have been developed for frictional systems (e.g., model-based compensation, see [START_REF] Armstrong-Hélouvry | A survey of models, analysis tools and compensation methods for the control of machines with friction[END_REF], impulsive control, see van de [START_REF] Van De Wouw | Robust impulsive control of motion systems with uncertain friction[END_REF], or sliding mode control, see [START_REF] Bartolini | A survey of applications of 2nd-order sliding mode control to mechanical systems[END_REF]). PID control, however, is still used in the vast majority of industrial motion systems, since the integrator action is able to compensate for unknown static friction. However, it has performance limitations (e.g., long settling times, see [START_REF] Beerens | Hybrid PID control for transient performance improvement of motion systems with friction[END_REF]). The popularity of the PID controller for mechatronics applications in industry, however, motivates the development of hybrid add-ons, such as reset control strategies, to complement the classical PID controller and improve its baseline performance.

Coulomb friction is often modeled by a set-valued force law, so that the resulting dynamical model of a PID-This research is part of the research programme High Tech Systems and Materials (HTSM), which is supported by NWO domain Applied and Engineering Sciences and partly funded by the Dutch Ministry of Economic Affairs and is also supported by ANR via project HANDY, number ANR-18-CE40-0010.

controlled mass subject to friction yields a differential inclusion, see, e.g., (Acary and Brogliato, 2008, Sec. 1.3). When appropriately tuned, the PID control guarantees global asymptotic stability (GAS) of a constant position reference, see [START_REF] Bisoffi | Global asymptotic stability of a PID control system with Coulomb friction[END_REF].

On the other hand, the PID-controlled frictional system evolves, in some sense, discrete-wise by toggling between the logical states of stick and slip. The stick phase is characterized by zero velocity and the control force not exceeding the static friction. The integrator action then builds up the control force by integrating the position error and eventually compensating for the (unknown) static friction, so that a slip phase (i.e., a phase with nonzero velocity) occurs. These logical states can be modeled as suitable subsystems of the differential inclusion, see ( Åström and Canudas-de-Wit, 2008, p. 106), which motivates the development of an equivalent hybrid representation of the differential inclusion model. The main motivation for such a hybrid model representation is the fact that it favors a simplified stability analysis, compared to the one in [START_REF] Bisoffi | Global asymptotic stability of a PID control system with Coulomb friction[END_REF], as we will show in Section 5. In particular, the hybrid model allows employing a smooth Lyapunov function in order to prove GAS of a constant position reference, instead of the discontinuous one used in [START_REF] Bisoffi | Global asymptotic stability of a PID control system with Coulomb friction[END_REF]. Moreover, the hybrid model facilitates a simplified stability analysis of the frictional system complemented with reset control strategies, compared to the analysis framework in [START_REF] Bisoffi | Global asymptotic stability of a PID control system with Coulomb friction[END_REF]. Reset strategies added to a classical PID controller for frictional systems have been shown to be promising in [START_REF] Beerens | Hybrid PID control for transient performance improvement of motion systems with friction[END_REF]. This paper has the following contributions. First, we present a hybrid automaton model for a PID-controlled mass subject to Coulomb friction. We show that the solution set of the differential inclusion model is contained in the solution set of the hybrid model, for initial conditions in arbitrarily large compact sets. In particular, we prove that the solutions to the differential inclusion enjoy inherently a semiglobal dwell-time between the stick and slip phases. Second, we prove GAS of the setpoint for the hybrid model through a Lyapunov analysis, and provide, as a corollary, an alternative, simplified proof of GAS for the differential inclusion model, by using the above property of solution sets. Third, we show that the proposed hybrid model is useful for PID-based reset control design and analysis, by presenting a novel reset controller and a simulation example.

After presenting the physical model in Section 2, we show in Section 3 that the solutions to this model enjoy a certain dwell-time property and introduce in Section 4 the hybrid automaton model describing (semiglobally) the same behavior. A stability analysis is in Section 5, and Section 6 illustrates the advantages of the hybrid automaton model for reset design.

Notation. B is the closed unit ball, of appropriate dimensions, in the Euclidean norm. For w 1 ∈ R n1 and w 2 ∈ R n2 , (w 1 , w 2 ) := w 1 w 2 . The function sign : R → [-1, 1] is defined as sign(0) := 0, sign(x) := x/|x| for x = 0. The set-valued mapping Sign : R ⇒ [-1, 1] is defined as Sign(0) := [-1, 1], Sign(x) := {sign(x)} for x = 0. For c > 0, the function dz c : R → R is defined as dz c (x) := 0 for

x ∈ [-c, c], dz c (x) := x -c sign(x) for x / ∈ [-c, c].
In particular, dz : R → R is defined as dz(x) := dz 1 (x) for all x. For a hybrid solution ψ (Goebel et al., 2012, Def. 2.6) with hybrid time domain dom ψ (Goebel et al., 2012, Def. 2.3), the function j(•) is defined as j(t) := min (t,j)∈dom ψ j. For a generic hybrid system H = (F, C, G, D) in [START_REF] Goebel | Hybrid dynamical systems[END_REF], S H (x 0 ) denotes the set of all maximal solutions to H from the initial condition x 0 . ∨, ∧ denote the logical operators or, and. For x ∈ R n and a closed set A ⊂ R n , |x| A := inf y∈A |x -y|.

PHYSICAL MODEL AND MAIN RESULT

A PID controller drives a 1-degree-of-freedom mass m on a horizontal plane with position z 1 and velocity z 2 , and subject to a friction force taking values from a set-valued force law z 2 ⇒ Ψ(z 2 ) := -Fs Sign(z 2 ) -ᾱz 2 , where Fs is the static Coulomb friction and ᾱ ≥ 0 is the viscous friction coefficient. With the goal of attaining the constant setpoint (z 1 , z 2 ) = (r, 0), the described model reads ż1

= z 2 , (1a) ż2 ∈ 1 m Ψ(z 2 ) -kp (z 1 -r) -ki z 3 -kd z 2 , ( 1b
) ż3 = z 1 -r, (1c)
where z 3 is the integral state of the PID controller and kp , ki , kd represent its proportional, integral and derivative gains, respectively. As in [START_REF] Beerens | Hybrid PID control for transient performance improvement of motion systems with friction[END_REF], we introduce mass-normalized parameters to favor clarity

k p := kp m , k d := kd + ᾱ m , k i := ki m , F s := Fs m ,
(2) and we adopt the convenient change of coordinates

x := (σ, φ, v) = -ki(z1-r) -kp(z1-r)-kiz3 z2 , (3)
where σ is a generalized position error, φ corresponds to the proportional and integral actions of the PID controller, and v is the velocity. Further details on the model derivation and the reasons for the change of coordinates can be found in [START_REF] Beerens | Hybrid PID control for transient performance improvement of motion systems with friction[END_REF]. After the reparametrization in (2) and the change of coordinates in (3), the model in ( 1) is equivalent to Ĥ :

   σ φ v   ∈   -k i v σ -k p v φ -k d v -F s Sign(v)   = 0 0 -k i 1 0 -k p 0 1 -k d   σ φ v  - 0 0 F s Sign(v) =: Ax -b Sign(v), x ∈ Ĉ := R 3 . ( 4
)
Although Ĥ has an empty jump set in the terminology of [START_REF] Goebel | Hybrid dynamical systems[END_REF], we call it a hybrid system since we will consider its (truly) hybrid counterpart below (see ( 21)). We consider F s > 0 and assume that

k i > 0, k p > 0, k p k d > k i , (5)
which is equivalent, by the Routh-Hurwitz criterion, to the origin being globally asymptotically stable for (4) in the frictionless case, i.e., F s = 0. The assumption in (5) corresponds then to tuning a stabilizing PID controller and is hence not restrictive. Consider for Ĥ in (4) the attractor  := {x = (σ, φ, v) : σ = 0, | φ| ≤ F s , v = 0}, (6) corresponding to the set of all possible equilibria. We state then the next result. Theorem 1. Under the controller gain selection in (5),  in (6) is globally asymptotically stable for Ĥ in (4).

System Ĥ in (4) coincides with the model in [START_REF] Bisoffi | Global asymptotic stability of a PID control system with Coulomb friction[END_REF] where Theorem 1 was proven via a discontinuous Lyapunov function. The motivation of this paper is to provide an alternative hybrid proof of Theorem 1 (see Section 5.2), based on the hybrid model H δ presented below (see ( 21)). The ingredients of this proof are: (a) the solutions to Ĥ have a semiglobal dwell time (see Lemma 1) and are bounded (see Lemma 4), (b) the solutions to Ĥ are semiglobally contained in the solutions to H δ (see Lemma 2), (c) the stability analysis of the hybrid model H δ (see Section 5.1) is significantly simplified by the smooth, instead of discontinuous, Lyapunov function. Moreover, Section 6 sketches the benefit of H δ for designing reset laws to improve the PID-control performance.

SEMIGLOBAL DWELL TIME OF SOLUTIONS

Given the sets (intuitively associated with incipient slip or stick phases) 9) we summarize the results from [START_REF] Bisoffi | Global asymptotic stability of a PID control system with Coulomb friction[END_REF], Lemma 1 and Claim 1) that we exploit in this paper. Fact 1. (i) From each initial condition x0 ∈ R 3 , there exists a unique and complete solution x to Ĥ. (ii) For each initial condition x0 = (σ 0 , φ0 , v0 ) ∈ Ŝ1 or with v0 > 0 (x 0 = (σ 0 , φ0 , v0 ) ∈ Ŝ-1 or with v0 < 0, respectively), there exists T > 0 such that the unique solution x to Ĥ coincides over [0, T ] with the unique solution xl := (σ l , φl , vl )

Ŝ1 := {x : v = 0 ∧ φ > F s ∨ ( φ = F s ∧ σ > 0) } (7) Ŝ-1 := {x : v = 0 ∧ φ < -F s ∨ ( φ = -F s ∧ σ < 0) } (8) Ŝ0 := {x : v = 0 ∧ σ > 0 ∧ φ ∈ [-F s , F s )} ∪ {x : v = 0 ∧ σ < 0 ∧ φ ∈ (-F s , F s ]}, (
to ẋl = Ax l -b ( ẋl = Ax l + b, respectively), (10)
with A and b in (4). (iii) For each initial condition x0 ∈ Ŝ0 or x0 = (0, φ0 , 0) with | φ0 | ≤ F s , there exists T > 0 such that the unique solution x to Ĥ coincides over [0, T ] with the unique solution x := (σ, φ, v) to

ẋ := ( σ, φ, v) = (0, σ, 0). ( 11
)
The interval [0, T ] in Fact 1(ii) might not be uniform with respect to the initial conditions, and we need this uniformity for building a hybrid model of ( 4) with desirable stability properties. Indeed, we have the property of solutions to Ĥ established in the next lemma, which entails a uniform dwell-time in a semiglobal fashion. Note that the presence of a guaranteed uniform dwell time allows ruling out Zeno phenomena (see [START_REF] Goebel | Hybrid dynamical systems[END_REF] and references therein) in the following. Lemma 1. Under selection (5), for each M > 0, there exists δ > 0 such that for each initial condition x0 ∈ Ŝ1 ∩ M B (x 0 ∈ Ŝ-1 ∩ M B, respectively), the unique complete solution x = (σ, φ, v) to Ĥ from the initial condition x0 satisfies v(t) ≥ 0 (v(t) ≤ 0, respectively) for all t ∈ [0, δ].

Proof. We consider only the case x0 ∈ Ŝ1 , because the case x0 ∈ Ŝ-1 is proven with similar arguments. By Fact 1(ii), the solution x satisfies ẋ = Ax -b on some [0, T ]. The last equation can be made homogeneous as ẋ = Ax with x = (σ, φ, ṽ) := (σ, φ -F s , v), ( 12) so that x ∈ Ŝ1 if and only if x ∈ S1 , where S1 :

= {x : ṽ = 0 ∧ φ > 0 ∨ ( φ = 0 ∧ σ > 0) }. (13)
Consider an initial condition x0 ∈ S1 and note that 0 / ∈ S1 , hence |x 0 | = 0. Because the dynamics in ( 12) is homogeneous, consider the unitary-norm initial condition ñ0 := x0 /|x 0 | without loss of generality. We parameterize ñ0 as ñ0 = (cos( θ0 ), sin( θ0 ), 0) with θ0 ∈ [0, π) (π is excluded because φ0 = 0 and σ0 < 0 do not belong to S1 ). Then, the solution to (12) with initial condition ñ0 is given by x(t) = (σ(t), φ(t), ṽ(t)) = exp(At)ñ 0 , and by expanding the matrix exponential into its powers, straightforward computations yield 14) where O(t 3 ) denotes the terms of order t 3 or higher in the Taylor expansion. Based on (14), we show below for each θ0 ∈ [0, π) the existence of T v > 0 such that ṽ(t) > 0 for all t ∈ (0, T v ), ṽ(T v ) = 0.

ṽ(t) = cos( θ0 )( t 2 2 +O(t 3 ))+sin( θ0 )(t-k d t 2 2 +O(t 3 )), (
(15)

Then, we prove the existence of a uniform δ. For θ0 ∈ [0, π/2], we are considering a compact set of values of θ0 where both cos( θ0 ) and sin( θ0 ) are nonnegative. Since in ( 14) the coefficients of their dominant powers (t 2 /2 and t, respectively) are a positive constant, δ 1 can be found independently of θ 0 so that both Taylor expansions are (strictly) positive on (0, δ 1 ). As a consequence, T v ≥ δ 1 and the claim of the lemma is proven for θ0 ∈ [0, π/2]. For θ0 ∈ (π/2, π), the existence of T v > 0 in (15) follows from the positive coefficient sin( θ0 ) of the dominant power t in ( 14), but uniformity of δ might not hold. Note that θ0 ∈ (π/2, π) corresponds to σ0 < 0, which we then assume in the rest of the proof without loss of generality. This implies by ( 12), ( 15) and 16) Since A is Hurwitz by ( 5) and x satisfies (12) until t = T v by Fact 1(ii), there exist positive c and λ such that

k i > 0 in (5) that σ(T v ) = σ0 + Tv 0 -k i ṽ(τ ) dτ ≤ σ0 < 0. (
|x(t)| = | exp(At)x 0 | ≤ c exp(-λt)|x 0 |. Define δ 2 := F s /(|A|cM ) > 0 (17) where |A| > 0 is the (induced) 2-norm of matrix A. If T v ≥ δ 2 , the claim of the lemma is proven. If T v < δ 2 , we
show that the velocity remains zero for a uniform interval. Note first that v(T v ) ≤ 0 otherwise ṽ would not cross zero as in (15). Hence 0

≥ v(T v ) = φ(T v ) -k d ṽ(T v ) = φ(T v ), so φ(T v ) ≤ 0. Also, φ(T v ) > -F s as we conclude in (18). Indeed, for t ∈ [0, T v ] (x 0 ∈ Ŝ1 ∩ M B implies x0 ∈ M B) | φ(t)| = | d dt [ 0 1 0 ] exp(At)x 0 | = | [ 0 1 0 ] A exp(At)x 0 | ≤ |A|c exp(-λt)|x 0 | ≤ |A|cM = F s /δ 2 by (17). Hence, for t ∈ [0, T v ], φ(t) = φ0 + t 0 φ(τ )dτ ≥ φ0 - t 0 | φ(τ )|dτ ≥ φ0 -t Fs δ2 . In particular then, φ(T v ) ≥ φ0 -T v Fs δ2 ≥ -Tv δ2 F s > -F s ( 18
)
because φ0 ≥ 0 and T v < δ 2 . Then, ṽ(T v ) = 0 by (15), σ(T v ) < 0 by ( 16) and

-F s < φ(T v ) ≤ 0, i.e., v(T v ) = 0, σ(T v ) < 0, 0 < φ(T v ) ≤ F s . ( 19
)
We may now continue the solution to Ĥ from the "initial" condition in (19). Fact 1(iii) shows that the solution x to Ĥ coincides with the solution x to (11) as long as φ ≥ -F s . Since, from (11), φ(t

) = φ(T v ) + σ(T v )(t -T v ) with t ≥ T v , φ(T v ) = φ(T v ) ∈ (0, F s ] and |σ(T v )| = |σ(T v )| = |σ(T v )| ≤ |x(T v )| ≤ c exp(-λT v )|x 0 | ≤ cM , we have φ(t) ∈ [-F s , F s ] for all t ∈ [T v , T v + δ 3 ] with δ 3 := Fs cM > 0.
In turn, the solution component v to Ĥ remains zero over the same interval, whose uniform length is δ 3 . Therefore, δ := min{δ 1 , δ 2 , δ 3 , 2F s /M } > 0 (20) proves the lemma, where 2F s /M has been introduced without loss of generality for the sake of Lemma 2.

PROPOSED HYBRID MODEL

With the quantity δ from Lemma 1 in mind, we introduce a hybrid model enabling a hybrid-based proof of Theorem 1. Consider δ > 0, and for the state

x := (σ, φ, v, τ, q) ∈ R 3 × [0, 2δ] × {-1, 0, 1}, (21a)
define the hybrid system H δ as where the flow and jump maps are given by

H δ :    ẋ = f (x), x ∈ C := C slip ∪ C stick x + ∈ G(x), x ∈ D := i∈{1,-1,0} D i (21b) (21c) C slip q = 1 C slip q = -1 C stick q = 0 x ∈ D -1 x ∈ D 1 x ∈ D 0 x ∈ D 0 σ φ v C slip D 0 σ φ v q = 0 C slip q = -1 q = 1 D 0 D 1 D -1 σ φ v C stick v = 0 -Fs Fs Fs -Fs
f (x) :=      -k i v σ -k p v -k d v + |q|φ -qF s 1 -dz(τ /δ) 0      , G(x) := i : x∈Di {g i (x)}, ( 21d
)
the different jump maps are given by

g 1 (x) := σ φ v 0 1 , g -1 (x) := σ φ v 0 -1 , g 0 (x) := σ φ v τ 0 (21e)
and the flow and jump sets are given by

C slip := {x : |q| = 1, qv ≥ 0} (21f) C stick := {x : q = 0, v = 0, |φ| ≤ F s } (21g) D 1 := {x : q = 0, v = 0, φ ≥ F s , τ ∈ [δ, 2δ]} (21h) D -1 := {x : q = 0, v = 0, φ ≤ -F s , τ ∈ [δ, 2δ]} (21i) D 0 := {x : |q| = 1, v = 0, qφ ≤ F s }.
(21j) The flow and jump maps for τ ensure the invariance of the set [0, 2δ] for τ , as per (21a). Since D i ∩ D k = ∅ for i, k ∈ {-1, 0, 1} and i = k, G is actually always a single-valued mapping. A pictorial representation of ( 21) is in Figure 1, which gives a clear hybrid automaton interpretation of (21).

As an important observation, the first three components of the flow map in (21d) coincide in C slip and C stick with the affine right-hand sides in ( 10) and ( 11), to which the differential inclusion in (4) has been shown to "reduce" in Fact 1. Then, it is intuitive that a solution H δ captures the solution to Ĥ when the condition τ ∈ [δ, 2δ] is absent from (21h)-( 21i). In such a case, however, (21) would also have nonconverging Zeno solutions (e.g., with v = 0, φ = F s , σ = 0) that would nullify the desired attractivity property of Section 5.1. The timer in H δ prevents these Zeno solutions, and translates the inherent dwell-time property of solutions to (4) in Lemma 1. Indeed, after solutions to H δ exit a stick phase and enter a slip phase jumping from D 1 or D -1 , the timer is reset to zero via g 1 or g -1 and enforces that a time δ passes before solutions exit a stick phase again (due to the condition τ ∈ [δ, 2δ]), which corresponds to the property of solutions to (4) in Lemma 1. For the sake of Lemma 2, note that due to Fact 1(i), for each x0 ∈ R 3 , the set of maximal solutions S Ĥ(x 0 ) contains one single element x. Lemma 2. Under selection (5), for each M > 0, there exists δ > 0 such that for each initial condition x0 ∈ R 3 ∩ M B, there exist τ 0 and q 0 such that, for some solution x = (σ, φ, v, τ, q) ∈ S H δ ((x 0 , τ 0 , q 0 )), for all t ≥ 0 σ(t, j(t)) = σ(t), φ(t, j(t)) = φ(t), v(t, j(t)) = v(t), ( 22) where x = (σ, φ, v) ∈ S Ĥ(x 0 ). The intuition behind Lemma 2 is that there exists one solution to (21) that can jump so as to reproduce the solution to (4), although there might be other solutions to H δ that are not complete. The proof of Lemma 2 is in Appendix A.

STABILITY ANALYSIS

In this section, we first provide a stability analysis for H δ and then, as a result of the relationship between the solution sets of Ĥ and H δ shown in Section 4, we obtain a hybrid proof of Theorem 1.

Stability analysis of A for H δ

We study the global asymptotic stability of the following attractor for H δ in ( 21)

A := {(0, φ, 0, τ, q) : τ ∈ [0, 2δ], q ∈ {-1, 0, 1}, φ ∈ F s Sign(q)},
(23) which is compact. Note that  in ( 6) is essentially the "projection" of A for Ĥ in the directions σ, φ, v. Consider the smooth Lyapunov function

V (x) := σ v k d k i -1 -1 k p σ v +|q|(φ-qF s ) 2 +(1-|q|)(dz Fs (φ)) 2 .
(24) From ( 5) and x as in (21a), A corresponds to the zero-level set of V . We can prove the following relations. Lemma 3. Consider the hybrid system H δ in (21) with δ > 0. Then, V in (24) satisfies:

∇V (x),f (x) = -2(k p k d -k i )v 2 ∀x ∈ C slip ∪ C stick (25a) V (g 1 (x)) -V (x) = 0 ∀x ∈ D 1 (25b) V (g -1 (x)) -V (x) = 0 ∀x ∈ D -1 (25c) V (g 0 (x)) -V (x) = (dz Fs (φ)) 2 -(φ -qF s ) 2 ≤ 0 ∀x ∈ D 0 . ( 25d
)
Proof. As for (25a), some computations yield 24) and ( 21d). (25a) follows readily from ( 26) by considering separately the cases of |q| = 1 (C slip ) and q = 0 (C stick ). Next,consider (25b). D 1 in (21h) and 25c) follows analogously. (25d) follows from (21j) and g 0 in (21e) updating q from |q| = 1 to 0.

∇V (x), f (x) = 2 [ σ v ] k d k i -1 -1 kp -kiv -k d v+|q|φ-qFs + 2 |q|(φ -qF s ) + (1 -|q|) dz Fs (φ) (σ -k p v) (26) from (
g 1 in (21e) yield V (g 1 (x)) -V (x) = (φ -F s ) 2 -(dz Fs (φ)) 2 , which is zero for φ ≥ F s . (
The next proposition establishes GAS of A for H δ . Proposition 1. Under selection (5), consider H δ in (21) with δ > 0. A in (23) is globally asymptotically stable for H δ .

Proof. The proof exploits (Seuret et al., 2019, Thm. 1). The set A in ( 23) is compact and H δ in (21) satisfies the hybrid basic conditions (Goebel et al., 2012, Assumption 6.5). G(D ∩ A) ⊂ A follows from the fact that σ, φ, v remain constant across jumps, and the Lyapunov nonincrease conditions hold due to Lemma 3.

Hence, we only need to prove that no complete solution x bad to H δ keeps V constant and nonzero. To this end, first note that the dwell time property enforced by τ implies that complete solutions exhibit an infinite amount of flow (cf. Fig. 1,top). Due to the decrease property in (25a), and since v remains constant across jumps, the only possibility for x bad to exist is that it flows outside A (where V is nonzero) with velocity v identically equal to zero (otherwise V would decrease from (25a) in any arbitrarily small interval of variation of v). Such flowing (with v ≡ 0) is impossible in C slip \A due to the linear representation in ( 10) and ( 12), and the pair ([ 0 0 1 ] , A) being observable. Hence, x bad could flow indefinitely only in C stick . However, this is also impossible because the components (σ, φ, v) of any solution flowing in C stick \A obey dynamics (11) and, since the σ component of x bad must be nonzero (otherwise x bad would be in A), the φ component of x bad would grow unbounded, thereby contradicting the fact that x bad keeps flowing in C stick \A. hence, such solution x bad cannot exist and the proof is completed.

Hybrid proof of Theorem 1

The proof of Proposition 1 is significantly simplified because the automaton model H δ allows using the smooth Lyapunov function V in (24) (rather than the discontinuous Lyapunov function used in [START_REF] Bisoffi | Global asymptotic stability of a PID control system with Coulomb friction[END_REF]). The proof of Theorem 1 is completed here through Lemma 2, Proposition 1, and the next straightforward property of Ĥ. Lemma 4. Under selection (5), solutions to Ĥ in (4) are bounded.

Proof. By (5) (i.e., A is Hurwitz), the system ẋl = Ax lbν is exponentially stable, hence bounded-input-boundedoutput stability of this linear system implies that solutions to Ĥ are bounded (consider (4) with input |ν| ≤ 1). Proof of Theorem 1. Preliminarily, define Â5 := {(σ, φ, v, τ, q) : σ = 0, | φ| ≤ F s , v = 0, τ ∈ [0, 2δ], q ∈ {-1, 0, 1}}, i.e., Â in (6) written as a subset of R 5 (instead of R 3 ). Clearly Â5 ⊃ A with A in (23). Then,

|x| A ≥ inf y∈ Â5 |x -y| = inf y∈ Â5 |(x, τ, q) -y| = |x| Â. (27)
For the proof, we need to show stability and global attractivity of Â, where the latter entails by (Goebel et al., 2012, Def. 7.1) that for each solution x with x(0) ∈ R 3 , x is bounded and satisfies lim t→∞ |x(t)|  = 0 (because solutions are complete from Fact 1(i)). Boundedness of solutions is guaranteed by Lemma 4. Indeed, for each M > 0, there exists M ≥ M , such that x(0) ∈ M B implies, by Lemma 4, that the range rge x ⊂ M B. Then, Lemma 2 guarantees that there exists δ > 0 so that the solution x (with rge x ⊂ M B) coincides with the first three components of one solution x to H δ , as in ( 22). Global asymptotic stability of A for H δ (Proposition 1) and completeness of such x (due to ( 22)) ensure lim t→∞ |x(t, j(t))| A = 0 and hence, by ( 27), lim t→∞ |x(t)|  = 0. Since both H δ and Ĥ satisfy the hybrid basic conditions (Goebel et al., 2012, Ass. 6.5), global asymptotic stability of A for H δ in Proposition 1 actually implies uniform global asymptotic stability (Goebel et al., 2012, Thm. 7.12), and uniform global attractivity. Hence, also  is uniformly globally attractive. By (4) and Fact 1(iii),  is strongly forward invariant because for every solution x ∈ S Ĥ( Â), rge x ⊂ Â. Then  is stable by (Goebel et al., 2012, Prop. 7.5). 2018)), we design from (21) a reset integrator to reduce the overshoot of its linear counterpart. In particular, we change sign to the controller state φ when the generalized position error σ crosses zero. Such a design ensures that the PI control force always points in the direction of the setpoint. Secondly, a scaling of φ by a factor α ∈ [0, 1) is required when the mass hits zero velocity, after overshooting the reference, to leave intact the global asymptotic stability properties.

AUTOMATON MODEL

As a baseline, we consider system H δ in (21), and we augment it with suitable jump sets and maps, corresponding to the described controller resets. Let us introduce a boolean state b ∈ {-1, 1}, characterizing whether the mass moves towards the reference (b = 1), or away from it (b = -1), when initialized properly. Consider the augmented state x r := (x, b), and the resulting hybrid system including the controller resets:

H r :      ẋr = f r (x r ), x r ∈ C r x + r ∈ p : xr∈Dp,r {g p,r (x r )}, x r ∈ p∈{1,-1,0,φ,α} D p,r (28a) (28b)
with flow map f r (x r ) := (f (x), 0), and jump maps g 1,r (x r ) := (g 1 (x), b), g -1,r (x r ) := (g -1 (x), b), g 0,r (x r ) := (g 0 (x), b), g φ,r (x r ) := (σ, -φ, v, τ, q, -1), and g α,r (x r ) := (σ, αφ, v, τ, q, 1). The flow and jump sets are given by

C r := {(x, b) : x ∈ C, b ∈ {-1, 1}}, D 1,r := {(x, b) : x ∈ D 1 , b = 1}, D -1,r := {(x, b) : x ∈ D -1 , b = 1}, D 0,r := {(x, b) : x ∈ D 0 , b = 1}, D φ,r := {(x, b) : |q| = 1, qv ≥ 0, b = 1, σ = 0}, D α,r := {(x, b) : x ∈ D 0 , b = -1},
where D φ,r and D α,r are the jump sets implementing the resets. We leave the robust detection of σ = 0 and v = 0 in D φ,r and D α,r as future work (with an approach similar to (Beerens et al., 2018, Remark 1) and drawing from the ideas in (Goebel et al., 2012, Example 4.18 and Chap. 4)).

Computer simulations of H δ (i.e., with a classical PID controller) and of H r in (28) (i.e., with a reset enhancement) are performed with δ sufficiently small. For a unitary mass, we consider F s = 0.981 m/s 2 , and select k p = 18 s -2 , k i = 35 s -3 , k d = 2.5 s -1 (satisfying (5)), α = 0.95. The generalized position error σ, and the PI control force φ (illustrating the controller resets) are depicted in Fig. 2, where the red dash-dotted line corresponds to the solution to H δ . Such solution enjoys the stability properties of Theorem 1, but its settling time and overshoot are significantly reduced by using the reset controller, while maintaining global asymptotic stability. Indeed, the developments in this paper pave the way for an analysis that shows 1) global asymptotic stability for H r , and 2) that global asymptotic stability for H r implies global asymptotic stability for the corresponding inclusion model, augmented with the proposed reset controller. In particular, the smooth Lyapunov function in (24) can be extended to incorporate also b.

CONCLUSIONS

For PID-controlled motion systems subject to Coulomb friction, we introduced a hybrid model exploiting some intrinsic semiglobal dwell-time between stick and slip enjoyed by the original solutions. This new model allows to simplify the proof of GAS, which can now rely on a smooth Lyapunov function, rather than the discontinuous one used in previous work. The model and the smooth function are important preliminary steps towards proving the effectiveness of certain hybrid resetting rules for the integrator state, providing improved closed-loop performance in mechatronics systems. While simulations illustrate this fact here, a complete proof of GAS for the PID scheme with resets is left as future work.

Appendix A. PROOF OF LEMMA 2

If a generic solution x = (σ, φ, v, τ, q) ∈ S H δ flows, its components (σ, φ, v) satisfy also the differential inclusion in (4). Indeed, flow according to (21d) is possible only on a subset of C slip ∪ C stick given by S a ∪ S b ∪ S c defined as

S a := {x : (q = 1 ∧ v > 0) ∨ (q = 1 ∧ v = 0 ∧ φ > F s ) ∨ (q = 1 ∧ v = 0 ∧ φ = F s ∧ σ ≥ 0)} S b := {x : (q = 0 ∧ v = 0 ∧ φ ∈ [-F s , F s ) ∧ σ > 0) ∨ (q = 0 ∧ v = 0 ∧ φ ∈ [-F s , F s ] ∧ σ = 0) ∨ (q = 0 ∧ v = 0 ∧ φ ∈ (-F s , F s ] ∧ σ < 0)} S c := {x : (q = -1∧v < 0)∨(q = -1∧v = 0 ∧φ < -F s )∨(q = -1∧v = 0∧φ = -F s ∧σ ≤ 0)}. (A.1) Define f from f (x) =: ( f (x), 1 -dz(τ /δ), 0). On S a , S b , S c , f in (21d) is such that, respectively, f (x) = A σ φ v -b, f (x) = 0 σ 0 , f (x) = A σ φ v + b (with A and
b in (4)). By Fact 1, the solution to (4) coincides with the unique solution of the affine systems given by ( 10) and ( 11), which coincide with the vector fields f obtained from (21d). Since solutions to H δ are locally absolutely continuous by (Goebel et al., 2012, Defs. 2.6 and 2.4), this is sufficient to conclude that the components (σ, φ, v) of x ∈ S H δ satisfy the differential inclusion in (4) when flowing. Moreover, since the jump map in (21e) is such that (σ + , φ + , v + ) = (σ, φ, v), (22) holds for x ∈ S H δ for all t ∈ [0, sup t dom x].

It now remains to prove that there exists a complete solution to H δ . To this end, the following are made based on the above discussion on the sets in (A.1). Whenever the solution x = (σ, φ, v, τ, q) cannot flow with the given q (i.e., x(t, j) ∈ (C stick ∪ C slip )\(S a ∪ S b ∪ S c )), the designed jump sets in (21h)-(21j) without the condition τ ∈ [δ, 2δ] would ensure that x can jump and, after at most two jumps, q takes a value such that the flow map f corresponds to the affine system f according to which (4) evolves; in particular, the complete solution to (4) augmented with suitable components τ and q would be a solution to (21), and (22) would hold. Without the condition τ ∈ [δ, 2δ], however, (21) would also have solutions with Zeno behavior nullifying the desired stability property in Proposition 1. Then, it remains to prove that for each x0 ∈ M B such that x 0 = (x 0 , τ 0 , q 0 ) ∈ D 1 ∪D -1 , a solution x ∈ S H δ (x 0 ) evolves without jumping from D 1 ∪ D -1 for at least t ∈ (0, δ), so that after that time, jumps from D 1 ∪ D -1 are enabled again thanks to the flow map for τ . We address the case x 0 ∈ D 1 because x 0 ∈ D -1 follows from similar arguments. Define the set L 1 := {x = (σ, F s , 0, τ, 0) : σ ≤ 0}. Consider first x 0 = (x 0 , τ 0 , 0) ∈ D 1 ∩ L 1 , x0 ∈ M B. A solution x ∈ S H δ (x 0 ) can flow in C stick for at least [0, 2F s /M ], and actually x(t, 0) ∈ C stick \(D 1 ∪ D -1 ) for t ∈ (0, 2F s /M ) with 2F s /M ≥ δ by (20) in the proof of Lemma 1, and this shows that x can evolve without jumping from D 1 ∪ D -1 for at least t ∈ (0, δ). Consider second x 0 = (x 0 , τ 0 , 0) ∈ D 1 \L 1 , x0 ∈ M B. By selecting τ 0 = δ and q 0 = 0, we have τ (0, 1) = 0 and q(0, 1) = 1. By using T v defined in (15) in the proof of Lemma 1, we have q(t, 1) = 1 for t ∈ [0, T v ] and q(T v , 2) = 0 corresponding to a jump from D 0 . Then, x(t, 2) ∈ C stick for t ∈ [T v , T s ] with some time T s . Lemma 1 guarantees that T v ≥ δ or T s ≥ δ, which shows that x can evolve without jumping from D 1 ∪ D -1 for at least t ∈ (0, δ).

Fig. 1 .

 1 Fig. 1. Top: hybrid automaton underlying (21). Bottom: "projections" to the (σ, φ, v) space of the flow and jump sets in (21f)-(21j).

Fig. 2 .

 2 Fig. 2. Simulated response of σ (top) and φ (bottom), for H r (blue, solid) and H δ (red, dash-dotted).

  H δ FOR RESET DESIGN Motivated by improving performance of mechatronics systems through PID-based reset controllers (see, e.g.,Beerens et al. (