Elena-Laura Ursu

Gabriela Gavril

Simona Morariu

Mariana Pinteala

Mihail Barboiu

Alexandru Rotaru

Single-Walled Carbon Nanotubes -G-quadruplex Hydrogel Nanocomposite Matrixes for Cell Support Applications

INTRODUCTION

During the last decade supramolecular hydrogels, including self-assembled supramolecular hydrogels have received tremendous attention in the view of new materials design, preparation and applications [START_REF] Hirst | High-tech applications of self-assembling supramolecular nanostructured gel-phase materials: from regenerative medicine to electronic devices[END_REF][START_REF] Buerkle | Supramolecular gels formed from multi-component low molecular weight species[END_REF][START_REF] Sangeetha | Supramolecular gels: Functions and uses[END_REF][START_REF] Kiyonaka | Semi-wet peptide/protein array using supramolecular hydrogel[END_REF][START_REF] Lloyd | Anion-tuning of supramolecular gel properties[END_REF][START_REF] Dankers | Hierarchical formation of supramolecular transient networks in water: a modular injectable delivery system[END_REF][START_REF] Appel | Sustained release of proteins from high water content supramolecular polymer hydrogels[END_REF]. These three-dimensional materials are typically constructed involving relatively weak crosslinking forces such as hydrogen bonding, metal coordination, host-guest interaction, π -π interaction, etc. [START_REF] Du | Hierarchical hydrogen bonds directed multi-functional carbon nanotube-based supramolecular hydrogels[END_REF][START_REF] Du | Synthesis of conducting polymer hydrogels with 2D building blocks and their potential-dependent gel-sol transitions[END_REF][START_REF] Wang | A Strong Bio-Inspired Layered PNIPAM-Clay Nanocomposite Hydrogel[END_REF][START_REF] Ma | Aromatic-Aromatic Interactions Induce the Self-Assembly of Pentapeptidic Derivatives in Water To Form Nanofibers and Supramolecular Hydrogels[END_REF][START_REF] Harada | Cyclodextrin-based supramolecular polymers[END_REF][START_REF] Meazza | Halogen-bonding-triggered supramolecular gel formation[END_REF][START_REF] O'leary | Multi-hierarchical self-assembly of a collagen mimetic peptide from triple helix to nanofibre and hydrogel[END_REF] An interesting direction in the design of self-assembled supramolecular hydrogels plays guanosine-based hydrogels which involve the formation of guanosine quartets as the gelation trigger [START_REF] Bang | Untersuchungen über die Guanylsäre[END_REF][START_REF] Gellert | Helix formation by guanylic acid[END_REF][START_REF] Davis | G-quartets 40 years later: from 5'-GMP to molecular biology and supramolecular chemistry[END_REF][START_REF] García-Arriaga | Isostructural Self-Assembly of 2′-Deoxyguanosine Derivatives in Aqueous and Organic Media[END_REF][START_REF] Ciesielski | Dynamers at the solid-liquid interface: controlling the reversible assembly/reassembly process between two highly ordered supramolecular guanine motifs[END_REF]. We and others have recently considerably improved the approaches for the formation of transparent and strong guanosine-based hydrogels with high water retention values [START_REF] Peters | A G4•K+ Hydrogel Stabilized by an Anion[END_REF][START_REF] Peters | G4-Quartet•M+ Borate Hydrogels[END_REF][START_REF] Rotaru | G-Quartet hydrogels for effective cell growth applications[END_REF]. More precisely, the formation of boric or boronic esters has been utilized in the preparation of guanosine dimers which afterwards formed stable hydrogels in the presence of guanosine quartet stabilizing metals. Subsequently, some of the prepared hydrogels have successfully been tested as cell growing supports [START_REF] Rotaru | G-Quartet hydrogels for effective cell growth applications[END_REF]. The cell supporting properties of guanosine diboronic hydrogels (GDH) were strictly dependent on the water retention ability of the investigated hydrogels. It was found that GDH containing Mg 2+ and Ba 2+ serve as better cell supports in comparison to K + GDH since Ba 2+ and Mg 2+ GDH water retention ability was three and, correspondingly fifteen times better than K + GDH. Unfortunately, high water retention ability of Ba 2+ and Mg 2+ GDH has led to the hydrogel structure instability which resulted in difficulties of hydrogels handling during the cell experiments. Additionally, the Mg 2+ crosslinking property has led to the strong decrease in self-assembly properties of GDH. To overcome these disadvantages, we investigated alternative crosslinking agents to both improve the GDH properties and increase the water retention. Nanofillers including clay, nanotubes or graphene oxides have already been reported to form corresponding hydrogel composites enhancing hydrogels mechanical properties by intercalation into the hydrogel polymeric or supramolecular structure [START_REF] Haraguchi | Nanocomposite Hydrogels: A Unique Organic-Inorganic Network Structure with Extraordinary Mechanical, Optical, and Swelling/De-swelling Properties[END_REF][START_REF] Bhattacharya | Soft-Nanocomposites of Nanoparticles and Nanocarbons with Supramolecular and Polymer Gels and Their Applications[END_REF][START_REF] Liu | Cross-linkable graphene oxide embedded nanocomposite hydrogel with enhanced mechanics and cytocompatibility for tissue engineering[END_REF]. Among the mentioned materials, carbon nanotubes (CNTs) have proved their strong efficiency in reinforcing hydrogel structures, leading to hydrogel composites with superior properties [START_REF] Bhattacharya | Soft-Nanocomposites of Nanoparticles and Nanocarbons with Supramolecular and Polymer Gels and Their Applications[END_REF][START_REF] Kovtyukhova | Individual Single-Walled Nanotubes and Hydrogels Made by Oxidative Exfoliation of Carbon Nanotube Ropes[END_REF][START_REF]Structure of Semidilute Single-Wall Carbon Nanotube Suspensions and Gels -Nano Letters[END_REF][START_REF] Tan | Supramolecular Hydrogel of Bile Salts Triggered by Single-Walled Carbon Nanotubes[END_REF][START_REF] Adewunmi | Carbon Nanotubes (CNTs) Nanocomposite Hydrogels Developed for Various Applications: A Critical Review[END_REF]. Kouser et al have recently reported the facile synthesis of biocompatible nanocomposite hydrogels using microporous multi wall carbon nanotubes, dispersed chitosan, acrylonitrile N, N´-methylenebisacrylamide and linseed polyol to yield materials with significantly enhanced strength and high swelling capacity [START_REF] Kouser | Biocompatible and mechanically robust nanocomposite hydrogels for potential applications in tissue engineering[END_REF]. The non-toxic and biocompatible nature of the obtained materials makes them excellent candidates in the field of tissue engineering. In another study, Bellingeri et al [START_REF] Bellingeri | Nanocomposites based on pH-sensitive hydrogels and chitosan decorated carbon nanotubes with antibacterial properties[END_REF] have recently investigated the properties of a novel nanocomposite made of chitosan decorated carbon nanotubes and acrylamide-co-acrylic acid hydrogels as promising substrates for biomedical applications. The new materials have shown strong pH-response, antimicrobial activity and were found to be cytocompatible. These and other examples suggested the possibility of CNT application in improving the overall properties of GDH.

In the current work, we have investigated the protocol which permits not only dispersion of singlewalled carbon nanotubes (SWNTs) by GDH in liquid state, but also their successful introduction into the GDH matrix (Figure 1). The SWNTs strong impact on the overall GDH properties, including water retention ability and cytotoxicity compared to the original GDH hydrogel were investigated and described. Moreover, the introduction of SWNTs, which is useful for the formation and stabilization of 3D supramolecular hybrid hydrogels, led to the possibility of using GDH/SWNT nanocomposites as an improved cell growing support.

MATHERIALS AND METHODS

Materials

All commercially available reagents used in the preparation of the GDH hydrogels were purchased in their highest purity grade and used without further purification. The SWNT® SG 76 single-wall carbon nanotubes obtained by catalytic chemical vapor deposition method (CoMoCAT™) were purchased from Sigma Aldrich (Munich, Germany). NHDF (normal human dermal fibroblasts) cells were acquired from PromoCell. MTS assay (CellTiter 96 AQueous One Solution Cell Proliferation Assay) was acquired from Promega.

Synthesis of GDH hydrogel-SWNT composites

GDH hydrogel-SWNT composites hydrogels were prepared utilizing an adapted synthetic protocol reported previously [START_REF] Rotaru | G-Quartet hydrogels for effective cell growth applications[END_REF]. Thus, in a vial, an amount of 0.2 g (1 eq) of guanosine was mixed with 0.0584 g (0.5 eq) of 1,4-benzene diboronic acid. Distilled water (7.6 mL) and 400 µL containing 0.0296 g LiOH, (1 eq, stock solution: 0.148 g in 2 mL distilled water) are added and the mixture was sonicated for several minutes until all the components were dispersed. The suspension was then heated on an oil bath preheated to 120°C until the solution becomes transparent. To avoid pressure in the vial upon heating, a syringe needle in the cap of the vial was used. After cooling to room temperature, 4 mg of SWNT were added and the solution was further sonicated to homogenously disperse the carbon nanotubes, yielding a stable black suspension. This solution was preserved and used as SWNT stock solution (SWNT-Li stock) in further experiments.

Separately, in a glass vial, 0.05g, 0.2 mmol (1 eq) of guanosine was mixed with 0.0146 g, 0.1 mmol (0.5 eq) of benzene-1,4-diboronic acid. Distilled water (1.9 mL) was added, and the mixture was sonicated for a few minutes until all the components were dispersed. The suspension was then heated in an oil bath preheated to 120 o C until the solution becomes transparent. To avoid pressure in the vial upon heating a syringe needle in the plastic cap of the vial was used. Next, 100 µL, 0.01 g, KOH (1 eq, stock solution of KOH: 0.66 g in 6.6 mL distilled water) was added and the mixture heated and stirred for a few more minutes, followed by the addition of the SWNT-Li stock solution (200 μL, 600 μL, 1000 μL, 1400 μL, 2000 μL, containing correspondingly 0.1 mg, 0.3 mg, 0.5 mg, 0.7 mg or 1 mg SWNT). The obtained composites were cooled down to room temperature and sonicated for 30 min to give light brown to dark black hydrogel nanocomposites.

Characterization

Water retention experiments. For each of the obtained GDH-SWNT sample, a water retention experiment was performed. Each sample was brought to the maximum water retention by the consecutive addition of water to the initial sample volume (2 mL), followed by vigorous mixing, incubation at room temperature (2 hours) and check for self-sustainability by inverting the sample vial.

Rheological measurements. The rheological properties of GDH-SWNT hydrogels were determined at 25°C on a MCR302 Anton-Paar rheometer equipped with a Peltier device. All measurements were performed using plane-plane geometry (diameter of 25 mm) and an anti-evaporation device. Firstly, the amplitude sweep tests were carried out at a constant frequency () of 10 rad/s in the shear stress () range of 0.001 Pa -50 Pa, in order to determine the linear viscoelastic regime (LVR) for each sample. Then, the following tests were carried out: a) frequency sweep measurements at a constant  value from LVR in the frequency range of 0.05 rad/s -100 rad/s to determine the values of the storage (G') and loss (G") moduli; b) stress relaxation measurements in order to establish the gel strength; c) oscillatory step tests at 5 rad/s carried out alternatively for 200 s at low strain (5%) and 100 s at very high strain (1000%) and the structure recovery was established in each case.

Raman spectroscopy. Raman spectra for SWNT hybrid hydrogels were recorded using an inVia Raman confocal microscope (Renishaw, UK) equipped with a He-Ne laser at 633 nm and a RenCam CCD detector coupled to a Leica DM 2500 M microscope. All measurements were performed in backscattering geometry using a 50× objective, at room temperature and atmospheric pressure.

Scanning Electron Microscopy (SEM).

Morphological studies of SWNT hybrid hydrogels were performed using a Quanta 200-FEI scanning electron microscope working in low vacuum mode, at 20 kV with LFD detector. Sample of hydrogels were freeze-dried prior to measurements.

Powder X-ray diffraction (PXRD). X-ray diffraction characterization of the lyophilized samples was carried out using a D8 Advance Bruker AXS device. The X-rays were generated using a Cu Kα source with an emission current of 36 mA and a voltage of 30 kV.

MTS assay.

The cytotoxicity of the GDH-SWNT hydrogels was tested in normal human dermal fibroblasts (NHDF) cells using CellTiter 96®AQueousOne Solution Cell Proliferation Assay (MTS, Promega). Cells were expanded and maintained in alpha-MEM medium (Lonza) supplemented with 10% fetal bovine serum (FBS, Gibco) and 1% Penicillin-Streptomycin-Amphotericin B mixture (Lonza) at 37°C and 5% CO2 under humidified atmosphere. For the assay, NHDF cells (PromoCell) were plated in 96-well format (5 x 10 3 cells/well in 100 µL cell culture medium) for 24 h to ensure cell adhesion. The next day the media was aspirated and replaced with the GDH solutions. For this, aliquots of 20 µL of each hydrogel (with total water volume of 5 mL) were mixed with 80 μL cell culture medium in order to obtain the desired concentrations of hydrogels. Control wells received 80 µL cell culture medium mixed with 20 µL water to ensure the control cells and the treated ones are maintained in the same conditions. Also, wells with only cell culture medium and water and wells with hydrogel solution, but/and no cells were kept in the same conditions to serve as blank values, and their additional absorbance was subtracted from the results. After 44 hours of incubation, a volume of 20 µL of CellTiter 96®AQueousOne Solution reagent was pipetted to each well and the plates were returned to the incubator. After another 4 hours the absorbance at 490 nm was recorded with a plate reader (EnSight, PerkinElmer). The cell viability was calculated and expressed as an average percentage relative to the control culture (considered 100%).

The cytotoxicity of GDH-SWNT at maximum water retention capacity (GDH (5mL), GDH-SWNTs: 0.1 mg SWNT (6 mL), 0.3 mg SWNT (7mL), 0.5 mg SWNT (9 mL), 0.7 mg SWNT (11 mL) and 1 mg SWNT (12 mL)) and the cytotoxicity of the SWNT-Li stock was measured using the previously described protocol with the specification that in this case, aliquots of 10 µL of each hydrogel were mixed with 90 μL cell culture medium in order to obtain the desired concentrations of hydrogels. The SWNT-Li stock was diluted with cell culture medium in order to match the concentrations of the hydrogels. GDH-SWNT cell culture studies. Aliquots of 100 μL of each hydrogel were dispensed in 96 well plates, treated with 3X TAE Mg 2+ buffer at 7.4 pH in order to adjust the pH of the samples, followed by treatment with MEM cell culture medium to prepare the hydrogels for cell seeding. NHDF (Normal Human Dermal Fibroblasts) cells were seeded in concentration of 8 x 10 3 cells/well, using the droplet seeding technique. At this point the cells were incubated at 37°C in a humidified incubator, in order to promote cell adhesion to the surface of the hydrogel. After 4 hours, 100 µL of MEM cell culture medium was added and the samples were further incubated. At 24 hours of incubation the cells were stained using a live/dead assay. A mixture of calcein AM (5μM) and propidium iodide (500 nM) was used to selectively stain live and dead cells. After 30 minutes of incubation, in the humidified incubator, at 37°C in 5% CO 2 cells were then washed with PBS and imaged in fresh cell culture medium using an inverted Leica DMI 3000 B microscope equipped with GFP and N2.1 filters.

Additionally, a second staining protocol was used, employing the indolizinyl-pyridinium salt/βcyclodextrin inclusion complex as a staining agent developed in our group [START_REF] Pricope | Novel cyclodextrin-based pH-sensitive supramolecular host-guest assembly for staining acidic cellular organelles[END_REF]. The hydrogel samples were prepared and treated as previously described. Aliquots of 400 µL of each hydrogel were dispensed in 24 well plates. NHDF (Normal Human Dermal Fibroblasts) cells were seeded in a concentration of 5 x 10 4 cells/well, using the droplet seeding technique. After 4 hours of incubation, 400 µL of MEM cell culture medium was added and the samples were further incubated. The indolizinyl-pyridinium salt/βcyclodextrin inclusion complex (20 µL, 3.4 mM) was added to the cell suspension at seeding time. The stained cells were subsequently imaged at 4 and 24 hours of incubation.

RESULTS AND DISCUSSIONS

In order to disperse and introduce the SWNTs inside the hydrogel structure we have developed, based on our previous expertize [START_REF] Rotaru | G-Quartet hydrogels for effective cell growth applications[END_REF], the following strategy: first, the guanosine-boronic acid dimer solution was synthesized using standard conditions, but instead of potassium hydroxide which typically leads to the reaction and subsequent spontaneous gelation, we used lithium hydroxide which allows for the formation of the dimer but the lithium cation is not able to stabilize the guanosine quartet. Next, 4 mg of SWNTs were added and the solution (8 mL) was sonicated for 60 min to yield stable black suspension. Thus obtained SWNT-Li stock suspension was next used to introduce controlled amount of SWNTs inside the preformed GDH.

Typically, GDH were obtained using previously reported conditions [START_REF] Rotaru | G-Quartet hydrogels for effective cell growth applications[END_REF] and the needed amount of the SWNTs was achieved by adding corresponding amounts of the SWNT-Li stock solution (equivalent of 0.1 mg, 0.3 mg, 0.5 mg, 0.7 mg and 1 mg). The obtained light-brown to black suspensions were first tested for water retention ability (Figure 2). For the investigated GDH and GDH-SWNT sample the following maximum volumes were obtained: GDH (5mL), GDH-SWNT: 0.1 mg GDH-SWNT (6 mL), 0.3 mg GDH-SWNT (7mL), 0.5 mg GDH-SWNT (9 mL), 0.7 mg GDH-SWNT (11 mL) and 1 mg GDH-SWNT (12 mL). Analyzing the obtained results we could clearly observe the increase in water retention ability (self-sustaining property) with the increase in the SWNT amount in GDH-SWNT composite hydrogels. The starting GDH was selfsupportive at up to 5 mL of overall hydrogel volume, while the SWNT containing hydrogels were still self-sustained at volumes of up to 12 mL depending on the amount of SWNT. The considerable increase in water content could be explained by the reinforcing of the GDH matrix by SWNTs, leading to the combined supramolecular skeleton with pore size increasing with the absorbed water amount.

Insights on GDH and GDH-SWNT hydrogel morphology at their maximum water content were obtained by scanning electronic microscopy (SEM) on freeze-dried samples (Figure S1). The morphology of the hydrogels observed in SEM changed dramatically after the addition of first 0.1 mg of SWNTs to the GDH (Figure S1 a,b). The formation of larger distances between the agglomerations suggests the formation of larger pores inside the GDH. Interestingly, subsequent changes in the GDH-SWNT morphology with the addition of larger quantities of SWNTs were not so evident by SEM (Figure S1, c-f), the images revealing non-uniform morphologies all over the investigated surfaces. Overall, the obtained images showed a typical microporous morphology with interconnected, micrometric pores. In the case of starting GDH the pores were denser, with smaller sizes (about 1 µm). Pore sizes proportionately increased with increasing concentration of SWNT as resultant images revealed an average pore size of 15 μm (for 1 mg of SWNT added in hydrogel).

To further confirm the implication of SWNTs into the GDH matrix and their influence on the internal π-π stacking interactions between the successive G-quartet stacks within the supramolecular hydrogel structure, we have performed the PXRD studies (Figure 4). The PXRD diffraction pattern of the dry GDH showed characteristic peaks at 2θ ≈ 4, 18.9 and 26.8° representing G-quartet stacks [START_REF] Peters | G4-Quartet•M+ Borate Hydrogels[END_REF]. In case of GDH with the maximum amount of SWNTs (1 mg), the PXRD diffraction pattern presented similar three peaks at 2θ ≈ 4, 18.9, 26.8 o , but with the slight changes in the intensity and the shifting to the lower angles of the first band, corresponding to a higher d-spacing.

Additionally, an increase in the band intensity at the 2θ ≈ 26.8 o was observed, owned to the overlapping of the G-quartet stacks peak and high intensity characteristic SWNTs sharp peak at 2θ = ~25° [START_REF] Zhou | Defects in carbon nanostructures[END_REF]. A much lower intensity of this combined peak in comparison to the reported high intensities of the individual SWNTs peak could be explained by its masking by the GDH supramolecular matrix, a phenomenon also observed in case of hydrogels containing SWNTs [START_REF]Materials | Free Full-Text | Solid-State Synthesis of Polyaniline/Single-Walled Carbon Nanotubes: A Comparative Study with Polyaniline/Multi-Walled Carbon Nanotubes[END_REF] and multi-walled carbon nanotubes [START_REF] Kouser | Biocompatible and mechanically robust nanocomposite hydrogels for potential applications in tissue engineering[END_REF].

To investigate the uniform incorporation of dispersed SWNTs in hydrogel structure, Raman measurements of the GDH as a reference and GDH-SWNTs with different amounts SWNTs were performed (Figure 5). In order to confirm the gel-like behavior on the macroscopic scale of the obtained GDH-SWNTs samples and to establish their mechanical properties, rheological behavior of the composites was monitored (Figure 6). All samples have shown gel-like properties with values of the storage modulus above those of loss modulus (G' > G") (Figure 6 a). The viscoelastic moduli increased with about 3 orders of magnitude after addition of 1 mg SWNT against the original GDH. The limiting value of the shear stress,  L , from which hydrogels structure started to change, moved to higher values as the SWNTs content increased as a result of strong interactions developed between the components.

As shown in Figure 6 b, the frequency sweep measurements evidenced the increase of G' modulus by addition of SWNTs. Thereby, G′, determined at 1 rad/s, increases from 7.55 Pa for the sample free SWNTs to 843 Pa for that containing the maximum amount (1 mg) of SWNTs. This is consistent with the water retention data when the material holds its weight at higher amounts of water depending on SWNTs amount.

Considering the Chambon-Winter theories [START_REF] Winter | Analysis of Linear Viscoelasticity of a Crosslinking Polymer at the Gel Point[END_REF][START_REF] Chambon | Linear Viscoelasticity at the Gel Point of a Crosslinking PDMS with Imbalanced Stoichiometry[END_REF][START_REF] Winter | Rheology of Polymers Near Liquid-Solid Transitions[END_REF], for a critical gel characterized by G'()   n and G"()   n (0 < n < 1), the stress relaxation modulus G(t) shows the following power-law:

G(t)=S•t -n
where n and S represent the relaxation exponent and gel strength parameter, respectively. The critical gel condition, where G' and G" were approximately parallel, was fulfilled only by the GDH-SWNTs The recovery capacity of the mechanical properties was investigated at 5 rad/s, in three consecutive steps, in which the strain () is changed alternately from 5% (low strain) -1000% (above the deformation limit) -5%. Initially, at low strain, G' values were higher than G". By applying a high strain above a deformation limit, the network structure is broken, G" becomes higher than G' and the samples gains the liquid-like properties (Figure 6 d). When the strain returns to small value, the gel structure is recovered quickly within seconds and G' exceeds again G". In evolution of the elastic recovery degree with SWNTs content (Figure 6 e) three domains can be delimited: (i) up to 0.7 mg SWNTs, the elastic recovery degree remains around 84%, (ii) between 0.7 mg and 0.9 mg SWNTs the elastic recovery increases abruptly and

(iii) above 0.9 mg SWNTs the regeneration of the initial structure is completely (100%). The obtained rheological data demonstrate that the GDH-SWNT hydrogels with SWNTs content greater than 0.9 mg show the superior viscoelastic properties. The insertion of the SWNTs inside the GDH matrix plays a crucial role in defining the structure and mechanical properties of these GDH samples.

After optimizing the synthesis protocol for the GDH-SWNT and the positive results obtained from characterization studies, the next step in our research was to test the hydrogels for cell supporting applications. As the original GDH showed promising results [START_REF] Rotaru | G-Quartet hydrogels for effective cell growth applications[END_REF] in this field, we expected that the addition of SWNTs would offer an advantage in cell culture studies as the water retention capacity of the hydrogel and the overall hydrogel properties were significantly improved. First, the cytotoxicity study was performed for GDH and all GDH-SWNTs samples only at GDH maximum water retention volume (5 mL). As a result, we were able to assess the cytotoxic effect of the SWNTs in the context of the hydrogels (Figure 7). As shown in Figure 7, the increasing of SWNT in GDH from 0.1 to 1 mg led to a 20% reduction in cell viability. The decreasing in cell viability due to the presence of SWNTs was further confirmed by the evaluation of the GDH-Li and SWNT-Li solutions with corresponding amounts of SWNTs (Figure 8, grey). This experiment has clearly confirmed the cytotoxic effect of the increasing concentrations of SWNTs in comparison to the reference toxicity of the GDH-Li (100% viability). These results are in accordance with the results also reported by other researchers [START_REF] Cui | Effect of single wall carbon nanotubes on human HEK293 cells[END_REF][START_REF] Manna | Single-walled carbon nanotube induces oxidative stress and activates nuclear transcription factor-kappaB in human keratinocytes[END_REF].

Next, we have investigated the cytotoxicity of GDH-SWNT at maximum water retention capacity (GDH (5mL), GDH-SWNTs: 0.1 mg SWNT (6 mL), 0.3 mg SWNT (7mL), 0.5 mg SWNT (9 mL), 0.7 mg SWNT (11 mL) and 1 mg SWNT (12 mL)). The results (Figure 8, black) have revealed the fact that the increased water retention ability of the GDH given by the addition of SWNT allowed the counteraction of the SWNTs cytotoxic effect. When the water volume was increased with the increasing of SWNT concentration we can observe that the GDH-SWNTs resulted in the same cell viability percentage (70%), and most importantly the cell viability was 10% higher when compared to the starting GDH at maximum water retention capacity (Figure 8). We have continued with the investigation on the cell supporting properties of the investigated GDH-SWNTs. For these studies, 6 hydrogels with increasing concentrations of SWNT with the maximum water capacity were prepared according to the established synthesis protocol. NHDF cells were cultivated on the surface of these hydrogels using the droplet seeding technique. This implies that the cells were seeded in a very small amount of culture medium (usually up to a few µL) and after it is ensured a period of up to four hours in the incubator, without adding the rest of the cell culture medium, in order to promote cell adhesion to the surface of the hydrogel. Then the rest of the cell culture medium is added and the cells are further incubated in proper conditions. Subsequently, two separate staining protocols were used and compared in these experiments. First, the live/dead assay was used as a qualitative, complementary method to the quantitative MTS assay (Figure 9). By applying this, we were able to selectively stain and visualize living cells (using calcein AM, green cells) and dead cells (using propidium iodide, red cells) in all the investigated GDH and GDH-SWNTs samples incubated with cells. The obtained pictures were then overlaid by merging the green and red fluorescence channels (using ImageJ) resulting in pictures that represent both the dead and the viable cells (Figure 9 a-f). This fluorescent staining also allowed us to better visualize the adhered cells, that otherwise would be difficult to image in bright field, since they are transparent [START_REF] Thorn | A quick guide to light microscopy in cell biology[END_REF]. The elongated morphology of the viable cells (green) denotes the attachment on the hydrogels surface. The pictures analysis has confirmed the results obtained in the cytotoxicity studies,

showing an overwhelmingly larger number of viable cells (green) in comparison to the dead cells (red), in all the investigated samples after 24 hour of incubation. As the fluorescence of the commercial calcein AM used to stain live cells is usually retained poorly in time [START_REF] Miles | Cell-based assays using calcein acetoxymethyl ester show variation in fluorescence with treatment conditions[END_REF] and could have a genotoxic effect [START_REF] Ge | Standard fluorescent imaging of live cells is highly genotoxic[END_REF] due to the exposure of samples to light during imaging, a second imaging protocol was chosen: the use of indolizinyl-pyridinium salt/β-cyclodextrin inclusion complex (IPSCD) [START_REF] Pricope | Novel cyclodextrin-based pH-sensitive supramolecular host-guest assembly for staining acidic cellular organelles[END_REF] as cell staining agent developed in our group. Although this compound was developed for selective labeling of acidic organelles in cells, the absence of cytotoxicity, cellular permeability and increased long-lived intracellular fluorescence makes it a perfect candidate for imaging cells for over 24 hours without having to repeat the staining protocol. By applying this protocol, we were able to image the seeded cells at 4 and 24 hours after incubation and monitor their progress in time (Figures 10 andS2). The analysis of the images obtained after 4 and 24 hours of incubation and stained with IPSCD has confirmed the results obtained by live/dead assay staining that the seeded NHDF cells were able to succesfully adhere to the surface of the hydrogels, presenting a typical fusiform appearance at all the investigated GDH-SWNTs samples.

Conclusions

We have successfully developed supramolecular GDH-SWNTs composites which exhibited much better mechanical and water retention properties compared with the starting GDH. The successful incorporation of SWNTs inside the GDH matrix was investigated by SEM, powder XRD and Raman techniques, while the hydrogels mechanical properties were in detail investigated and compared by rheological studies. The favorable cytotoxicity results for all the investigated nanocomposites and the characterization results showing an increase in the pore size after the addition of SWNTs to the GDH, made them ideal materials for cell support applications. NHDF cells have been effectively seeded and then imaged by means of two separate techniques (commercial live-dead essay and the indolizinyl-pyridinium salt/β-cyclodextrin inclusion complex essay developed in our group), showing successful attachment and cell viability in time confirmed by both utilized assays. The promising obtained results regarding cytotoxicity and cell supporting properties, together with the peculiar properties of supramolecular hydrogels to be responsive to various external stimuli, make the investigated SWNT composite hydrogels interesting materials for biomedical applications in the field of tissue engineering.

Figure 1 .

 1 Figure 1. Schematic representation of SWNTs dispersion by GDH matrix leading to new composite hydrogels with increased water retention ability.

Figure 2 .

 2 Figure 2. Water retention ability of GDH and GDH-SWNT hydrogels. The following volumes for the sample were determined: GDH (5mL), GDH-SWNT: 0.1 mg GDH-SWNT (6 mL), 0.3 mg GDH-SWNT (7mL), 0.5 mg GDH-SWNT (9 mL), 0.7 mg GDH-SWNT (11 mL) and 1 mg GDH-SWNT (12 mL).

Figure 4 .

 4 Figure 4. X-ray diffraction patterns of freeze-dried GDH (red) and GDH-SWNT composite (black) with 1 mg content of SWNTs.

Figure 5 .

 5 Figure 5. Raman spectra of GDH and GDH-SWNTs containing 0.1, 0.3, 0.5, 0.7 and 1 mg SWNTs.

1 )

 1 containing more than 0.3 mg SWNTs. S values determined by stress relaxation measurements showed an abrupt rise above 0.7 mg SWNTs (Figure6 c).

Figure 6 .

 6 Figure 6. Rheology data: (a) variation of G' as a function of shear stress at 10 rad/s; (b) dependence of G' and G" on the oscillatory frequency in linear viscoelastic regime; (c) effect of SWNTs content on the gel strength; (d) G' and G" values in continuous step strain (5% -1000% -5%) measurements at 5 rad/s and (e) effect of SWNTs content on the elastic recovery.

Figure 7 .

 7 Figure 7. SWNT concentration dependence of cytotoxicity for GDH and GDH-SWNT (0.1, 0.3, 0.5, 0.7 and 1 mg SWNT) samples at a fixed volume (5 mL).

Figure 8 .

 8 Figure 8. Cytotoxicity results for the investigated SWNT-Li suspension (grey) and GDH-SWNTs (black) with different SWNT content.

Figure 9 .

 9 Figure 9. Fluorescence mycroscopy images of NHDF cells incubated for 24 hours and stained using live/dead assay on (a) GDH, (b-f) GDH-SWNTs at different SWNTs amounts at maximum water retention capacity.

Figure 10 .

 10 Figure 10. Fluorescence mycroscopy images of NHDF cells seeded on GDH (a) and GDH-SWNTs (b-f) after 4 hours of incubation, stained with IPSCD.

Acknowledgements

This publication is part of a project that has received funding from the European Union's Horizon 2020 research and innovation programme under grant agreement N o 667387 WIDESPREAD 2-2014 SupraChem Lab. This work was also supported by a grant of the Ministry of Research and Innovation, CNCS -UEFISCDI, project number PN-III-P4-ID-PCCF-2016-0050, within PNCDI III.


Didier Maquin
email: dmaquin@ensem.u-nancy.fr

Marie Luong
email: mluong@ensem.u-nancy.fr

José Ragot
email: jragot@ensem.u-nancy.fr

OBSERVABILITY ANALYSIS AND SENSOR PLACEMENT

Keywords: Sensor placement, observability, redundancy, reliability, control system design, sensor failures, measurement system

The quality of the state estimation of a system is strongly conditioned by the number of measurements obtained from this system. Here, this dependence is studied by specifying especially the influence of the number and the position of sensors as well as of their precision. When some variables are necessary for the control, the possibility of failure of some sensors is analysed from the calculation of the whole system reliability. The reciprocal problem is also presented: what is the measurement system allowing to respect some constraints on the observability, the redundancy and the reliability.

INTRODUCTION

Nowadays, the conception of a measurement system is of fundamental importance. Indeed, the position of sensors and their number condition the possibility to observe a process, to estimate its state and consequently to determine the type of control to apply. Studying the observability of a system consists in answering the question: can we determine the value of variables of a system from a given set of measurements ? If the process is non observable, observable and non observable parts may however be determined. More generally, the posed problem is that one of diagnosis which consists in estimating the state of a system, deciding if it is normal or abnormal, judging the gravity of the abnormal state and with this appreciation deciding an action to be undertake in order to preserve the system functionalities. As we can quite imagine, this estimation is possible only under some conditions and especially requires to have a minimum amount of information that is to say a minimum amount of sensors; these sensors have to be judiciously placed in order to give an "image" of the totality of the process. Besides, the precision of collected information, closely linked to the precision of sensors and to their good functioning, conditions the precision of the diagnosis very largely. Finally, it is necessary to consider the realistic case in which initially provided sensors can fail. Consequently to these failures, the diagnosis of the state of the process can be performed only if variables measured by these faulty sensors can be estimated. Considering these remarks, one can assert that the choice of the instrumentation (number, position and characteristics of sensors) is an essential factor of the outcome of diagnosis. In the following, our purpose is limited to the study of system described by linear equations (for example, they can correspond to equations of matter or energy conservation), having concern for the simplicity, the generalisation to more complex models does not create major problems. Many works on observability analysis of steady-state systems have already been published. Firsts of them have focused on the linear system study (Vaclaveck et al., 1969); afterwards, [START_REF] Stanley | Observability and redundancy classification in process networks[END_REF] and [START_REF] References Crowe | Reconciliation of process flow rates by matrix projection, part 1: the linear case[END_REF] have largely contributed to develop this analysis. Algorithms of observability that have been developed are generally based on graph theory [START_REF] Mah | Reconciliation and rectification of process flow and inventory data[END_REF] or on a classification of variables from a projection matrix [START_REF] Crowe | Observability and redundancy of process data for steady-state reconciliation[END_REF], [START_REF] Ragot | Observability and redundancy decomposition -application to diagnosis[END_REF]. For systems described by linear-bilinear equations the first studies of observability are due to Vaclaveck and Loucka (1976), [START_REF] Romagnoli | On the rectification of measurement errors for complex chemical plants. Steady-state analysis[END_REF]. [START_REF] Kretsovalis | Observability and redundancy classification in generalized process networks[END_REF] have then extended this study by using graph theory. Concurrently, [START_REF] Darouach | Observabilité et validation des données des systèmes de grande dimension, application à l'équilibrage des bilans de mesures[END_REF] has formalised the study of the observability of bilinear systems in the matrix form; the method employed is summarised to a sequential study of the observability of linear systems [START_REF] Maquin | Observability and data validation of bilinear systems[END_REF]. More recently, [START_REF] Ragot | Validation de données et diagnostic[END_REF] proposed an algorithm based on the simultaneous analysis of linear and bilinear equations.

After the analysis phase, making in evidence not only redundancy but also the possible weaknesses of the measurement system, it is advisable to propose modifications of the existent sensor placements or even sensor addition. These modifications are studied considering precise objectives: to render observable a particular variable, to increase the degree of redundancy of another one, to enhance the precision of the estimation of a particularly useful variable for the control of the process, to tolerate sensor failures. Precisely, our purpose is to analyze the different criteria allowing to define the choice of the measurement system.

SYSTEM STATE ESTIMATION

The state estimation of a system is, in general, proceeded by data reconciliation with respect to the exact and known supposed model of the system. Indeed, measurements provided from inputs and outputs of the system do not verify its constraint equations. It is then advisable to correct these measurements, therefore to estimate probable values of the corresponding variables, so as to meet these equations. In what follows, we note: X * the actual value vector (∈ R m ) X the measurement vector (∈ R m ) X ^ the estimation vector (∈ R m ) A the constraint matrix (∈ R n.m ) V the variance-covariance matrix of measurement errors Assuming that the measurement errors are zero-mean and considering the case where the variance-covariance matrix V is known, the maximum likelihood estimation X ^ is solution of the optimization problem:

min Φ = 1 2 || X -X ^ || 2 V -1 (1a) subject to A X ^ = 0 (1b)
This very classic problem can be solved by different techniques. All, they give the same result with numerical errors caused by truncatures appearing in the different calculations:

X ^ = P X (2a) P = I -V A T (A V A T ) -1 A (2b)
The variance-covariance matrix of X ^ is given by:

V ^ = P V (3)
This expression provides the precision of estimations and therefore defines their confidence interval; as direct consequence, estimated confidence intervals of corrective terms of measurements can also be used to detect possible anomalies [START_REF] Ragot | Validation de données et diagnostic[END_REF].

Most of systems are partially observed only and the number of available measurements is in general much inferior than that of variables whose state is to be estimated. The estimation problem has therefore to be reexamined considering this constraint. The measurement vector is then linked to the actual value vector and the measurement error vector by:

X = H X * + ε (4)
The estimation X ^ is solution of the optimization problem described by equation (1). The conditions of existence for this solution X ^ are closely related to the rank of the observability matrix (H T A T ). If this rank is equal to the dimension of X [START_REF] Darouach | Observabilité et validation des données des systèmes de grande dimension, application à l'équilibrage des bilans de mesures[END_REF], the system state is totally observable and its estimation can be written as:

X ^ = (R -R A T (A R A T ) -1 A R) H T V -1 X (5a) R = (H T V -1 H + A T A) -1 (5b)
This expression shows that the estimation depends on the measurement system (number and placement of sensors through the matrix H), the measurements themselves X and their precisions V; it is necessary to specify this dependence so as to improve the quality of these estimations (see section 4).

SYSTEM OBSERVABILITY

In practice, it is interesting to test the observability condition directly from the network structure. Different solutions have been proposed among which that using the analysis of cycle matrix associated to the system graph which is simple and elegant to implement [START_REF] Gomolka | Contribution à l'analyse de l'observabilité des systèmes linéaires décrits par des équations de bilan[END_REF]; here, we remind the principle of this analysis.

Cycle matrix of a system

Let us consider the graph (Fig. 1) of a simple process, where the node '0' represents the environment one (for a system of matter transportation for instance, the environment node connects entry and exit flows of this system). One wishes to enumerate the observable variables of this system, more precisely to classify variables in deducible ones and redundant ones.

1 2 3 4 5 6 7 1 . 1 1 1 1 . . CF = 2 1 1 . . . 1 . 3 . 1 1 . . . 1

Fig. 2. Fundamental cycle matrix

A mathematical operation performed on the fundamental cycle matrix CF allows all the cycles contained in a graph to be enumerate. By definition, a fundamental cycle matrix constitutes a basis of cycles. By making a logical xor between two rows of CF (which represent fundamental cycles), one can list all the cycles contained in a graph. For our example, we deduce the following cycle matrix C:

1 2 3 4 5 6 7 1 . 1 1 1 1 . . 2 1 1 . . . 1 . 3 . 1 1 . . . 1 C = 4 1 . 1 1 1 1 . 5 . . . 1 1 . 1 6 1 . 1 . . 1 1
Fig. 3. Matrix of all the cycles of the graph

Observability analysis

The previous cycles are constituted of arcs corresponding to measured or unmeasured variables. Depending on the number of measured variables per cycle, one can define the observability of all the system variables. The cyclebased observability study is realised through four elementary rules [START_REF] Turbatte | Conception de l'architecture d'un système d'instrumentation -Application à un procédé pétrochimique[END_REF] It is important to note that changing the measured variable list (sensor addition, withdrawal or moving) is simple to take into account. Indeed, the preceding analysis is based on the enumeration of cycles which is carried out independently of measurements. Now, the conception of measurement system profit by this remark.

Synthesis of a measurement system according to the observability concept

The preceding analysis has allowed to enumerate deducible variables, redundant ones and those that cannot be estimated. When the "observability degree" of all the variables is judged unsatisfied, it is advisable to measure supplementary variables by placing a certain number of sensors judiciously. This choice can be made by analyzing the cycle matrix of the system. In a very general manner this latter can be put under the form: -Measured and non estimable variables are part of the bloc 1 (matrix Cme _ ). There is only a '1' by row in this submatrix (rule II).

-Estimable and measured variables are part of the submatrix 10 (matrix Cme). In this submatrix, there is at least two measured variables by row (rule I).

-Unmeasured and non deducible variables are ordered in the submatrix 7 (matrix Cm _ e _ ). This submatrix intervenes in a "row" where no variable is measured; all the other submatrices are null (rule IV).

-Unmeasured and deducible variables intervene in submatrices 4 and 12 (matrices C'm _ e and Cm _ e). These submatrices are part of a "row" containing measured variable submatrices -submatrices 9 and 10 -(rule II).

Consider the situation where some variables a priori are measured (list L m ); different criteria can be envisaged to complete this list so as to increase the observability degree of the system. For example, one can wish that some variables become deducible, or that others become redundant; moreover, one can impose that some of these variables could not be measured.

As an example, consider the case where L m = {1} and where the variable 7 cannot be measured; one wishes to render observable all the variables of the system with respect to these constraints. In the cycle matrix of the fig. 3, one observes that the definition of L m allows to enumerate 3 cycles (2, 4 and 6) containing the measured variable 1. Therefore, it is necessary that the remaining cycles (1, 3 and 5) contain each a measured variable at least in order to insure the observability of all the variables of the system. To obtain that and to minimise the number of supplementary sensors, a couple can be chosen among next ones: {2, 4}, {2, 5}, {2, 7}, {3, 4}, {3, 5}, {3, 7}, {4, 5}, {5, 7} Among these eight solutions, the choice will be able to be made according to a complementary criterion, for example of minimal cost, maximal reliability or desired estimation precision.

ESTIMATION ANALYSIS

The estimation procedure provides the state of the system through the measurements and their respective precisions. Though, it is very useful to complete this study by characterising the estimations by their sensitivity with respect to the parameters describing the process. As process equations a priori are supposed exact, the sensitivity study is here reduced to the influence of a modification of the measurements and their number.

Sensitivity to a small variation of the measurement precision

A variation ΔV of the measurement variance induces a corresponding estimation variation Δ X ^ whose expression is deduced from (2):

X ^ + ΔX ^ = (I -(V+ΔV) A T (A (V+Δ V) A T ) -1 A) X (6)
After making some tedious calculations and taking into account small variations such that ΔV << V, the relationship ( 6) is then reduced to:

ΔX ^ = P Δ V V -1 (X ^ -X) (7)
By noting p .i the i th column of the projection matrix P, v ii the i th diagonal term of V and x i the i th component of X, the vector ΔX ^ can be written as:

ΔX ^ = ∑ i=1 v p .i Δv ii v ii (x ^i -x i ) (8)
Thus, the vector of estimation sensitivity with respect to the variance of the i th measurement can be written:

S = ∂X ∂v ii = p .i 1 v ii (x ^i -x i ) (9)
In this calculation, one can avoid utilising the terms p .i of the projection matrix by eliminating the matrix P by using (3). We obtain:

ΔX ^ = V ^ V -1 ΔV V -1 (X ^ -X) (10)
By performing the different matrix products that appear in (11), coefficients of sensitivity can be expressed under the form:

s ij = ∂x ^i ∂v jj = v ^ ji x ^i -x i v 2 ii (11)
Using this relationship, the ratio of estimation sensitivities of two variables j and k when the variance of the variable i varies can be deduced:

s ij s ik = v ^ji v ^ki (12)
Thus, when the measurement variance changes, all the estimations are modified; the estimation that undergoes the greatest variation is that whose variance is the most important. There, it concerns an intuitive result whose equation ( 12) gives the theoretical justification.

Sensitivity with respect to any variation of the measurement precision

To simplify this study, only one variance is modified of the quantity δ at a time (for example the i th ). Thus, we note:

ΔV = δ E E T (13)
where all the components of the vector E are null except for the i th which is equal to one. After modifying the measurement variance, estimations are always expressed by the relationship (6). Using the lemma of matrix inversion and with K = (A V A T ) -1 we obtain:

ΔX ^ = δ 1 + δ E T A T K A E P E E T V -1 (X ^ -X) (14)
This relationship therefore quantifies the influence of a variation of the measurement variance on the state estimation of the system.

ANALYSIS OF THE ESTIMATION PRECISION

We propose to examine the influence of any amplitude variations of the measurement precision on the estimation variance. An interesting limit case occurs where this variation is of infinite amplitude; that case practically corresponds to suppressing the corresponding measurement and therefore to testing the influence of a measurement on the precision of estimations.

As previously, we modify one variance of the quantity δ at a time (the i th for example). Thus, we note:

ΔV = δ E E T (15)
where all the components of the vector E are null except for the i th which is equal to one. After modifying the measurement variance, estimation variance is still the following:

V ^ + ΔV ^ = (V + ΔV) (I -A T (A(V + Δ V)A T ) -1 A(V + Δ V)) (16)
After usual simplification, we obtain:

ΔV ^ = δ 1 + δ E T A T K A E P E E T P T (17)
This formula is not an approximation; it exactly expresses the influence of a modification δ of the measurement variance on the variance of estimations. In the particular case where this modification has a weak amplitude, we have:

ΔV ^ = δ P E E T P T (18)
Finally, a more elegant form can be obtained by using the variance of estimations:

V ^ = V -V A T (AVA T ) -1 A V (19)
From ( 19), the following expression is extracted:

A T K A = V -1 -V -1 V ^ V -1 (20)
Thus ( 18) implies:

ΔV ^ = δ 1 + δ E T (V -1 -V -1 V ^ V -1 ) E V ^ V -1 E E T V -1 V ^ (21)
So, taking into account the expression of the vector E, the general term Δv ^ pq can be written as:

Δv ^ pq = δ v 2 ii + δ (v ii -v ^ii ) v ^ pi v ^ qi (22)
Thus, the diagonal coefficients of this matrix have the following expression:

Δv ^ kk = δ v 2 ii + δ (v ii -v ^ii ) v ^2 kk (23)
and they represent the variation of the estimation variances according to a modification of amplitude δ of a measurement variance.

Application to the suppression of sensors

The particular case where the amplitude variation δ is infinite corresponds to an interesting situation. Since δ is directly related to the measurement variance, it is equivalent to consider that the precision of this measurement is equal to zero; the weight of this measure in the criterion of estimation therefore becomes null. Thus, with a roundabout manner which is simple to implement though, we can study the influence of the suppression of a measure on the variance of estimated variables.

Let us explain therefore the limit of ( 22), when δ tends to infinity:

Δv ^ pq = 1 (v ii -v ^ii ) v ^ pi v ^ qi ( 24
)
From this expression, one may deduce the terms related to the modification of the estimation variances:

Δv ^ kk = 1 (v ii -v ^ii ) v ^2 kk (25)
Especially, for the i th variable (whose measurement has a infinite variance), the modification of the variance of estimations is the following:

(33a) V k+1 22 = V ^ 22 + V k 11 N T H(V k 11 ,V k 22 ) N V k 22 (33b)
Practically, the values of V ^ 11 and V ^ 22 are given and the stable solution of (33) is searched. The equations (32b) and (32c) then give the estimation variances that have not been able to be imposed.

RELIABILITY OF A MEASUREMENT SYSTEM

For a real process, the calculation of estimations as well as that of their sensitivity that has been presented supposes that the measurements are available that is to say that sensors are able to deliver them. In some cases, the failure of a sensor does not involves necessarily the impossibility to calculate the estimation; this notion was specified in the section 2 by defining the condition to meet this and giving the expression of the estimation according to available measurements. If this condition of observability is not satisfied, the complete state estimation of the system is not possible. These situations have to be specified and can be characterised by an indication of reliability of the measurement system. A second motivation of this calculation of reliability stem from the necessity to have a certain number of measured variables allowing a security for the controlling of the process. Necessary variables for this control are in general directly measured, but it is then necessary to insure that a variable remains deducible if the associated sensor becomes failing. The situation complicates with the number of failures one can accept while preserving the observability of this necessary variable list for the control of the system. Let us present now the mean to evaluate the reliability of a measurement system.

By definition, the reliability of a sensor is the probability r(t) that at the instant t, there is no failure knowing that no failure had occurred at the instant t = 0. This probability may be expressed analytically by using, for example, the law of Poisson:

r(t) = exp(-λt) ( 34
)
where λ is a constant representing the rate of sensor failure. As an extension, one may define the reliability of a system constituted of n sensors of respective reliabilities r i (t): R(t) = f(r 1 (t), ..., r n (t)) (35)

In the following, without loss of generality, the sensor failure rates are chosen equal. In practice, the evaluation of the system reliability uses the Mean Time To Failure function defined by the expression:

MTTF = ⌡ ⌠ 0 ∞ R(t) dt (36)
Let us consider the reliability of a system (fig. 1) supplied with five sensors on arcs 1, 3, 5, 6 and 7 and whose variables are linked by the two redundancy equations: q 1 -q 6 = 0 (37a) q 3 -q 5 -q 7 = 0 (37b)

Let us assume that measured variables are all necessary for the elaboration of the control of the system or to its supervision. That means that a failure of one or several sensors is tolerated provided that the corresponding information could be deduced from information of remaining sensors.

If the redundancy expressed in the equations (37) is not considered the probability that all sensors function, therefore that they do not fail, is: R 0 (t) = (r(t)) 5 = exp(-5λt)

Therefore, the MTTF of the measurement system is:

MTTF 0 = 0.20/λ (39)
If redundancy equations are taken into account, then some failures can be tolerated. Indeed if one of sensors intervening in the redundancy equations (37) fails, then the information that it delivers can be replaced by an estimation obtained by simple deduction. Generally, sensors whose failures are tolerated are those appearing in the redundancy equations of a system. Equations (37) show that, if a sensor fails, the value of the corresponding variable can be easily deduced from the remaining measurements. The probability that one sensor fails and that the four others well function is: R 1 (t) = (1 -r(t)) r(t) 4 (40)

There are five possibilities of one sensor failure, the corresponding measurement being deducible.

Notice also that some failures affecting two sensors simultaneously are tolerable. Indeed, if sensors 1 and 7 fail for example, the observability of variables 1, 3, 5, 6 and 7 remains. For the same reason, other couples of failing sensors are acceptable: 1 and 5, 1 and 3, 6 and 7, 5 and 6 and 6 and 3. Therefore, there are six possibilities involving two sensor failures, the corresponding measurements being deducible. The probability that two sensors are simulta-neously defective and that the three others function is: R 2 (t) = (1 -r(t)) 2 r(t) 3 (41)

The case of more than two simultaneous sensor failures cannot be envisaged because some variables become unobservable. Summarising, the probability that variables 1, 3, 5, 6 and 7 are observable (by direct measurement or by deduction) is therefore: R(t) = R 0 (t) + 5 R 1 (t) + 6 R 2 (t) (42)

Applying the definition (36), the global MTTF for the measurement system yields: MTTF = 0.65/λ (43)

The comparison of the results (39) and ( 43) shows that a significant increasing of the measurement system reliability is achieved by taking into account the redundancy of information stemming from equations of the physical system. This simple example also points out the relationships between reliability, redundancy and observability notions. More generally, the influence of the modifications of the positions and the number of sensors on the MTTF can be studied. When several sensor configurations lead to the same degree of observability of the system, then one can choose the configuration offering the maximal MTTF. Alternately, we can define the measurement system guaranteeing a certain degree of reliability and maximizing the observability.

CONCLUSION

For the class of linear systems, we have analysed the influence of the measurement system on the state estimation of a system. This latter is especially sensitive to the number of measurements, to their distribution and to their precision. In a phase of conception, we have shown that it was possible to impose some constraints on the estimation quality and to respect them by a judicious choice of measurement system. This step can be performed on an already supplied sensor system but also on a non instrumented system. This technique could then be integrated as early as the conception phase of the process itself, the instrumentation being then part of the system. It would be suitable to extend the scope of the analysis performed to the case of process described by more complex models. Especially, the case of dynamic linear processes could be analysed by an analogous approach.

Fig. 1 .

 1 Fig. 1. Nodes and arcs of a graph The fundamental cycle matrix CF of the fig. 2 is obtained from the fig. 1. In this matrix CF, each column corresponds to an arc and each row to a fundamental cycle.

Fig. 4 .

 4 Fig. 4. General form of the cycle matrixThe above-mentioned decomposition yields the next classification of variables:

that therefore corresponds to the new variance:

The whole established relationships are useful to analyse the results of the data reconciliation. These relationships provide this analysis a way to appreciate the contribution of the measurements and their respective variances to the construction of their estimations and variances.

Synthesis of the measurement system from the estimation precision

The preceding study has shown the influence of the precision of measurements on that of their estimations.

Conversely, one can seek to determine the measurement variance (therefore specify the corresponding sensor precision) to obtain a given variance of the estimations.

Remind the expression of the estimation variance:

Knowing V, can we then determine V ^ ? To solve this problem, notice first of all that V ^ cannot be arbitrarily chosen. This matrix indeed might verify the constraint:

To exploit easily this constraint, let us use an equivalent form of A obtained by extraction of its regular part and let us partition correspondingly the matrix V (chosen diagonal) and V ^ :

By expressing (28) from ( 30) and (31), we obtain:

with:

Then, the equations (32a) and (32d) that constitute a non linear system with respect to V 11 and V 22 can be solved by using, for example, an algorithm with direct iterations:


S Mariano

Senior Member, IEEE F Blanchini

Member, IEEE S Formentin

Fellow, IEEE L Zaccarian

Asymmetric state feedback for linear plants with asymmetric input saturation

Keywords: Constrained control, Lyapunov methods, Stability of nonlinear systems

We consider a linear plant with decentralized input saturation, whose limits are not necessarily symmetric. We propose a nonlinear static state feedback stabilizer that is asymmetric, in such a way that the non-symmetric nature of the saturation is fully exploited in the control design, for larger regions of attraction. We show by example that the proposed technique provides significantly larger regions of attractions as compared to the symmetric solution results, in a case where the positive and negative saturation limits are different.

I. INTRODUCTION AND BACKGROUND

Most existing works dealing with the design of a state feedback (locally or globally) stabilizing the origin of a linear system with input saturation address the simplified problem of symmetric saturation (see, e.g., the extensive surveys in the books [START_REF] Hu | Control systems with actuator saturation: analysis and design[END_REF], [START_REF] Zaccarian | Modern anti-windup synthesis: control augmentation for actuator saturation[END_REF], [START_REF] Tarbouriech | Stability and stabilization of linear systems with saturating actuators[END_REF]). Nevertheless, in many practical situations (see, e.g., the applications in [START_REF] Todeschini | Nonlinear pressure control for BBW systems via dead zone and anti-windup compensation[END_REF], [START_REF] Zabi | Time-optimal control for the induction phase of anesthesia[END_REF]), the two limits are substantially different and the typical solution adopted in the literature is to disregard achievable control performance by conservatively focusing on the smallest limit.

While the majority of the existing results deals with the symmetric saturated case, a few papers have been published addressing the nonsymmetric case more directly. In particular, several continuous-time and discrete-time techniques have been proposed in [START_REF] Benzaouia | Constrained stabilization: an enlargement technique of positively invariant sets[END_REF] and references therein, based on the development of non-symmetric Lyapunov functions in the presence of linear (symmetric) stabilizers with nonsymmetric saturation (these, in turn, being developments of the pioneering ideas reported in [START_REF] Benzaouia | Existence of non-symmetrical Lyapunov functions for linear systems[END_REF]). Similar approaches have been followed in [START_REF] Hu | Null controllability and stabilization of linear systems subject to asymmetric actuator saturation[END_REF], [START_REF] Hu | Null controllability and stabilization of linear systems subject to asymmetric actuator saturation[END_REF], even though an informative study is carried out therein illustrating how stretched the nullcontrollability set is, whenever the input saturation limits are unbalanced (this being a main motivation for the approach adopted here). Another line of research relies on shifting the coordinate system in such a way that the saturation becomes symmetric, but the drawback of this is that the point being stabilized is not anymore the origin (see the Research supported in part by the Agence Nationale de la Recherche (ANR) via grant Hybrid And Networked Dynamical sYstems (HANDY), number ANR-18-CE40-0010.

Simone Mariano is with Université de Lorraine, CNRS, CRAN, F-54000 Nancy, France. Franco Blanchini is with Dipartimento di Matematica e Informatica, University of Udine, Via Delle Scienze 208, Udine, Italy. Simone Formentin is with Dipartimento di Elettronica, Informazione e Bioingegneria, Politecnico di Milano, Piazza Leonardo da Vinci 32, 20133 Milano, Italy. Luca Zaccarian is with CNRS, LAAS, 7 avenue du colonel Roche, F-31400 Toulouse, France, Univ. de Toulouse, LAAS, F-31400 Toulouse, France, and Dip. di Ingegneria Industriale, University of Trento, Italy.

recent book [3, ch. 8] and references therein). Perhaps the most sophisticated solution to the asymmetric bounded stabilization problem is given in [START_REF] Yuan | Switching control of linear systems subject to asymmetric actuator saturation[END_REF], where the authors propose a switching dynamical control scheme, designed by solving suitable linear matrix inequalities and capable of exploiting the available range of the control action on both sides of the saturation levels. A parallel approach has been proposed in [START_REF] Li | An asymmetric Lyapunov function for linear systems with asymmetric actuator saturation[END_REF] and [START_REF] Groff | Regional stability of discrete-time linear systems subject to asymmetric input saturation[END_REF], addressing respectively continuous-time and discrete-time systems, where suitable LMIs addressing each set in the partition ensure desirable properties of a piecewise quadratic Lyapunov function with symmetric stabilizing saturated linear feedbacks.

Different from the above mentioned approaches, inspired by the stretched shapes of the null controllability regions reported in [START_REF] Hu | Null controllability and stabilization of linear systems subject to asymmetric actuator saturation[END_REF], [START_REF] Hu | Null controllability and stabilization of linear systems subject to asymmetric actuator saturation[END_REF], the control design technique proposed in this paper corresponds to a nonlinear asymmetric stretch of a symmetric control law, which is capable of providing enlarged certified stability regions (subsets of the domain of attraction), as compared to the symmetric counterpart. The core intuition behind the proposed controller is to address the control design in shifted state coordinates under which the saturation limits would be symmetric, and then once the dynamics is analysed in that symmetric shifted setting, use the non-symmetric solution of [START_REF] Blanchini | Any domain of attraction for a linear constrained system is a tracking domain of attraction[END_REF] for stabilizing the origin (the origin, in those coordinates, corresponds to a shifted, off-center equilibrium).

II. SYMMETRIC STABILIZER

Consider the linear saturated continuous-time plant:

ẋ = Ax + B sat(u), (1)
where x ∈ R n is the plant state, u ∈ R m is the unconstrained plant input, A ∈ R n×n , B ∈ R n×m and sat denotes the standard vector saturation having components sat i (u i ) = max(min(u + i , u i ), -u - i ), being u + i > 0 and u - i > 0 the upper and lower bounds, respectively, on input channel i. Also denote by u + and u -, respectively, the vectors obtained by stacking together the elements u + i and u - i , respectively, for all i = 1, . . . , m. To suitably stabilize the origin of (1), a standard assumption is that the origin is in the interior of the subset of R m where the input is not saturated (namely the set where u = sat(u)). Such an assumption, formalized below, is generally necessary to achieve local exponential stabilization of (1) when A is not Hurwitz (unless redundant input directions are available).

Assumption 1: All the elements of vectors u + and u -are positive, that is, u + > 0 and u -> 0, where the inequalities should be understood componentwise.

Asymmetric saturation limits may emerge due to shifted coordinate systems with constant setpoints, or simply due to specific actuators with asymmetric stroke. For the proposed paradigm to make sense, we assume the following.

Assumption 2: Matrix A is non-singular. Under the assumption that x be accessible for measurement, a nonlinear state-feedback controller of the form

u = Kx + L dz(u), (2)
can be designed based on the plant dynamics, in order to induce suitable closed-loop properties when interconnected to the plant [START_REF] Benzaouia | Constrained stabilization: an enlargement technique of positively invariant sets[END_REF]. In [START_REF] Benzaouia | Existence of non-symmetrical Lyapunov functions for linear systems[END_REF], dz denotes the deadzone function, defined as dz(u) := u -sat(u). The resulting nonlinear control system obtained by lumping together (1) and ( 2) can be compactly written as

ẋ = (A + BK)x -(B -BL) dz(u) u = Kx + L dz(u). (3)
We call (2) symmetric stabilizer because in a neighbourhood of the origin in dynamics (3) is symmetric. In the case where the state x is not accessible the state feedback solution (2) can be replaced by a dynamic output feedback solution with anti-windup, such as in [START_REF] Dai | Output feedback design for saturated linear plants using deadzone loops[END_REF]. Extending our state-feedback approach to that case is an interesting future direction. The control parameters K ∈ R m×n and L ∈ R m×m can be designed according to several different criteria. In this paper, we address the problem of simultaneous stabilization and maximization of a quadratic estimate of the Basin of Attraction of the origin for (3). To this end, a slight extension (mentioned in the proof below) of the classical results proven in [START_REF] Gomes Da | Anti-windup design with guaranteed regions of stability: an LMI-based approach[END_REF], [START_REF] Hu | Control systems with actuator saturation: analysis and design[END_REF], [START_REF] Hu | Anti-windup synthesis for linear control systems with input saturation: achieving regional, nonlinear performance[END_REF] and [17, §3.2.1] allow to state the following theorem.

Theorem 1 (Symmetric stabilizer): Given plant (1), define the lowest limits u := min(u + , u -), where the minimum should be computed componentwise. Then, given any solution to the following convex optimization:

max Q∈R n×n ,W,Y ∈R m×n , U,X∈R m×m log det(Q) subject to: (4a) Q = Q T > 0, U > 0 diagonal (4b) He AQ + BW -BU + BX W + Y X -U < 0 (4c) u 2 k Y [k] Y T [k] Q ≥ 0, k = 1, . . . , m (4d)
where He(Z) = Z +Z T , Y [k] denotes the k-th row of matrix Y and u k denotes the k-the entry of vector u, by selecting

K = W Q -1 , L = XU -1 , P = Q -1 , (5)
the nonlinear algebraic loop in (2) is well posed (in the sense that its solution is unique and Lipschitz) and the origin is locally exponentially stable, with basin of attraction containing the set:

E s (P ) := {x : x T P x ≤ 1}. (6
) Proof: With symmetric saturation limits, that is ȗ = u + = u -, the proof follows the generalized (local) sector condition approach of [START_REF] Gomes Da | Anti-windup design with guaranteed regions of stability: an LMI-based approach[END_REF], [START_REF] Hu | Control systems with actuator saturation: analysis and design[END_REF], [START_REF] Hu | Anti-windup synthesis for linear control systems with input saturation: achieving regional, nonlinear performance[END_REF]. However, the same exact proofs also apply to the case u ≤ u + and u ≤ u -, which holds from the definition of u.

The above result represents a computationally efficient way to design a stabilizing controller which also maximizes the basin of attraction for the closed loop (3), and ( 4) is feasible (see, e.g., [START_REF] Hu | Anti-windup synthesis for linear control systems with input saturation: achieving regional, nonlinear performance[END_REF]) for any stabilizable pair (A, B) under Assumption 1. However, the guaranteed estimate can be quite conservative when the upper and lower limits of the inputs are different, namely when u + k = u - k for some k.

III. ASYMMETRIC STABILIZER

Let us define the average saturation range ū and the average saturation center u • as:

ū :=(u + + u -)/2, (7a) u • :=(u + -u -)/2. ( 7b
)
Due to Assumption 2, we may introduce the mid-range point

x • ∈ R n satisfying x • := -A -1 Bu • , Ax • + Bu • = 0. (8)
In particular, x • characterizes an equilibrium point induced by input u • . For (A, B) to be stabilizable it is necessary that rank[A B] = n (from the PBH test applied to the zero eigenvalue). When u • = 0 (as customary in the asymmetric case

u + = u -), x • satisfying the second equation in (8) exists only if rank[A] = rank[A B]
= n, which reveals that Assumption 2 is necessary (and sufficient) under a stabilizability condition.

Based on the quantities introduced above we propose an alternative nonlinear (and asymmetric) stabilizer based on a scalar λ ∈ [0, 1], whose expression generalizes the one of Theorem 1 (which becomes a special case when λ = 0) , having the following form

u = κ(x) := (9a) Kx + γ(x)(λδ • + (L -I) dz(ν(x) + λδ • )), if x = 0, Kx, if x = 0,
where δ • := u • -Kx • , the function γ is defined in (9b) at the top of the next page, and the function ν is the solution of the following nonlinear algebraic loop:

ν = Kx γ(x) + L dz(ν + λδ • ). ( 9c
)
Remark 1: The algebraic loop appearing in (9c) can be eliminated by imposing X = 0 in the matrix inequalities (4), which results in L = 0. In that case we get u = Kx in (3) and ν = Kx γ(x) , which in turns implies that function κ in (9a) reduces to

κ(x) = Kx + γ(x)(λδ • -dz( Kx γ(x) + λδ • )), if x = 0, Kx, if x = 0, (10)
corresponding to an explicit nonlinear state feedback that is easier to implement in real-time applications. For the case where X = 0 (and then L = 0), the algebraic loop (9c) (equivalently, the one in ( 2)) can be addressed using the

γ(x) :=        min    1, (1 -λ|x • | P ) -1   - x T • P x |x • | P + (1 -λ|x • | P ) |x| 2 P λ|x • | P + x T • P x |x • | P 2      , if λx • = 0, 1, if λx • = 0, (9b)
methods in [START_REF] Zaccarian | Modern anti-windup synthesis: control augmentation for actuator saturation[END_REF]Sec. 2.3.7] or the equivalent QP optimization discussed in [START_REF] Syaichu-Rohman | A multivariable nonlinear algebraic loop as a QP applications to MPC[END_REF]. We emphasize that the algebraic loop is especially useful for the multi-input case m > 1, see, e.g., [START_REF] Mulder | Multivariable anti-windup controller synthesis using linear matrix inequalities[END_REF]. For the case m = 1 any nonzero selection of scalars (X, U ) = (0, 0) in ( 4) can be exchanged for the alternative choice ( X, Ū ) = (0, U -X), thus removing algebraic loop and getting the same performance. This is clearly not possible when m > 1 because U must be diagonal.

• The following lemma establishes a few useful properties of the proposed asymmetric stabilizer [START_REF] Hu | Control systems with actuator saturation: analysis and design[END_REF].

Lemma 1: If λ 2 x T • P x • < 1 and K and L arise from the construction in Theorem 1, then the following holds:

1) function γ in (9b) is locally bounded, continuous and locally homogeneous of degree one, in particular, for each x ∈ R n and each scalar s ≥ 1, γ(sx) = min{1, sγ(x)}; 2) for each x = 0, the algebraic loop in (9c) is well posed (it has a unique and Lipschitz solution x → ν(x)); 3) function κ in (9a) is continuous.

Proof: Item 1. Under the stated assumption, λ|x

• | P := λ x T • P x • < 1
, therefore all the terms in (9b) are bounded for each value of x. Local homogeneity and the consequent continuity is easily verified by inspection.

Item 2. If x = 0, the algebraic loop in (9c) becomes:

ν + λδ • = Kx γ(x) + λδ • + L dz(ν + λδ • ), (11)
which coincides with the algebraic loop in (3) (with u = ν + λδ • and Kx replaced by the external input Kx γ(x) + λδ • ). The well-posedness of this nonlinear algebraic loop is established in Theorem 1.

Item 3. The continuity property of κ away from the origin immediately follows from the fact that it is a composition of Lipschitz functions. At the origin, first note that when λx • = 0 the stabilizer coincides with the symmetric one of Theorem 1 and continuity follows from well-posedness. When λx • = 0, close to the origin we have γ(x) = |x|γ x |x| , meaning that the explicit solution ν(x) is locally constant around rays close to the origin. Therefore ν is uniformly upper bounded around the origin and the multiplicative γ(x) in (9a), together with the fact that lim x→0 γ(x) = 0, guarantees continuity of κ at the origin.

Starting from the above statements, we derive the next theorem, the main contribution of this paper.

Theorem 2 (Asymmetric stabilizer): Consider plant (1) and the average saturation center u • in (7b). Take any solution to the convex optimization (4a)-(4c) augmented with Then the asymmetric nonlinear control law [START_REF] Hu | Control systems with actuator saturation: analysis and design[END_REF] with selection λ = √ µ and ( 5) is globally well defined and locally exponentially stabilizes the origin with basin of attraction containing the neighbourhood of the origin:

µ µx T • µx • Q > 0, u 2 k + µ|u •,k |(2u k + |u •,k |) Y [k] Y T [k] Q ≥ 0, (12)
E a (P, λx • ) := {x : (x -λx • ) T P (x -λx • ) ≤ 1}. ( 13
)
Proof: Using a congruence transformation in the left inequality and µ ≤ λ 2 ≤ λ in the right inequality, constraints [START_REF] Massimetti | Linear discretetime global and regional anti-windup: an LMI approach[END_REF] imply

1 λx T • λx • Q > 0, (u k + λ|u •,k |) 2 Y [k] Y T [k] Q ≥ 0, (14)
k = 1, . . . , m. The left inequality of ( 14) can be rearranged via a Schur complement to show that λ 2 x T • P x • < 1, so that Lemma 1 applies and the feedback selection satisfies suitable regularity conditions.

Consider now the change of coordinates

x = x -λx • , (15a) ũ
= u -λu • . (15b)
The plant dynamics (1) in the reference framework defined in [START_REF] Mulder | Multivariable anti-windup controller synthesis using linear matrix inequalities[END_REF] can be rewritten as

ẋ = Ax + B sat(u) -(Aλx • + Bλu • ) (16) = Ax + B sat(u) -Bλu • = Ax + Bσ(ũ),
where σ(ũ) := sat(ũ + λu • ) -λu • is a vector saturation function having lower and upper limits u - σ := u -+ λu • and u + σ := u + -λu • , respectively, whose components satisfy

u k + λ|u •,k | ≤ min{u - σ,k , u + σ,k } (in particular, from (7a)
, with λ = 1 we recover the symmetric case u - σ = u + σ = ū). Since any solution to (4a)-(4c), ( 12) satisfies ( 4) with (4d) replaced by the right inequalities in [START_REF] Massimetti | Linear discretetime global and regional anti-windup: an LMI approach[END_REF], then Theorem 1 guarantees that K and L selected as in [START_REF] Gomes Da | Anti-windup design with guaranteed regions of stability: an LMI-based approach[END_REF] are such that the following algebraic loop is well posed:

v = Kz + L dz σ (v), (17a)
where dz σ (v) := v -σ(v). Moreover, introducing the feedback function

φ(z) := σ(v) = v -dz σ (v) (17b) = Kz + (L -I) dz σ (v), (17c)
the feedback stabilizer ũ = φ(x) stabilizes the origin x = 0 of ( 16) with basin of attraction including the set

Ẽs (P ) := {x : xT P x ≤ 1} = E a (P, λx • ). (18)
In the rest of the proof, we use the following result, which is a generalization of [4, Thm 3.3] with the explicit expression for ellipsoidal domains given in [4, §6.1].

Lemma 2: [START_REF] Blanchini | Any domain of attraction for a linear constrained system is a tracking domain of attraction[END_REF] Consider plant ( 16) and any point x * ∈ Ẽs (P ) satisfying Ax * + Bu * = 0 for some suitable vector u * . Then the input selection

ũ = u * + Ψ(x, x * ) φ x * + x -x * Ψ(x, x * ) -u * (19a) Ψ(x, x * ) := x * T P ξ |x * | P + x * T P ξ |x * | P 2 + w ξ T P ξ |x * | P w , (19b)
with ξ = x -x * and w = 1 -|x * | P , ensures that Ẽs (P ) is forward invariant and all solutions starting in Ẽs (P ) uniformly exponentially converge to the equilibrium x = x * . Applying Lemma 2 with the selection x * = -λx • (and consequently, from [START_REF] Groff | Regional stability of discrete-time linear systems subject to asymmetric input saturation[END_REF], u * = -λu • ), we obtain that the control law [START_REF] Yuan | Switching control of linear systems subject to asymmetric actuator saturation[END_REF] induces forward invariance of Ẽs (P ) and uniform exponential convergence to x * = -λx • . In particular, since dynamics ( 16) is linear and x = x * is an equilibrium for the closed loop with [START_REF] Yuan | Switching control of linear systems subject to asymmetric actuator saturation[END_REF], then the set (the singleton) {x = x * } is forward invariant and applying, for example,1 [7, Prop. 7.5], we obtain local asymptotic stability of x * for dynamics ( 16), [START_REF] Yuan | Switching control of linear systems subject to asymmetric actuator saturation[END_REF] with domain of attraction including Ẽs (P ).

The proof is completed by first noting that, in the original coordinates x, the compact set {x = x * } coincides with the origin (this trivially follows from (15a)) and set Ẽs (P ) coincides with set E a (P, λx • ) (as already remarked in ( 18)). Then, we observe that selection [START_REF] Yuan | Switching control of linear systems subject to asymmetric actuator saturation[END_REF] coincides with (9a), (9b), as shown next. In particular, the following relations hold for the quantities in (19b): ξ = x -λx • + λx • = x and w = 1 -λ|x • | P . Then comparing (9b) with (19b) we immediately get Ψ(x, x * ) = γ(x).

Moreover, inspired by (9c), we may compute:

dz σ (v) := v -σ(v) = v -(sat(v + λu • ) -λu • ) = dz(v + λu • ) = dz(ν + λδ • ), (20)
where we selected ν = v + Kλx • . Equation ( 20) may be used to show that (9c) coincides with (17a) when evaluating [START_REF] Tarbouriech | Stability and stabilization of linear systems with saturating actuators[END_REF] with z = x * + x-x * Ψ(x,x * ) = x γ(x) -λx • . Indeed, one gets from (17a) and [START_REF] Zabi | Time-optimal control for the induction phase of anesthesia[END_REF],

ν = v + Kλx • = K x γ(x) -λx • + L dz(ν + λδ • ) + Kλx • .
Finally, we may rearrange (19a) to obtain (9a), by using again (20) and also using (15b) as follows:

u = ũ + λu • = γ(x) φ x * + x -x * γ(x) -u * = γ(x) φ -λx • + x γ(x) + λu • = γ(x) -Kλx • + Kx γ(x) + (L -I) dz(ν + λδ • ) + λu • ,
which coincides with (9a), thus completing the proof. Remark 2: Replacing Assumption 2 with A -I being invertible, discrete-time extensions of our results can be derived. to this end, the LMI-based design (17) should be revised using [START_REF] Massimetti | Linear discretetime global and regional anti-windup: an LMI approach[END_REF], while Lemma 2 is valid in both continuous and discrete time [START_REF] Blanchini | Any domain of attraction for a linear constrained system is a tracking domain of attraction[END_REF].

• We emphasize that with λ = 0, one gets γ(x) ≡ 1 from (9b). Then feedback (1), ( 9) coincides with the symmetric solution (1), [START_REF] Benzaouia | Existence of non-symmetrical Lyapunov functions for linear systems[END_REF]. Indeed, the selection of ν in (9c) coincides with the selection of u in (2) (called u s hereafter, to distinguish it from u in (9a)). As a consequence using ν = u s , we get from (9a)

u = Kx + L dz(u s ) -dz(u s ) = sat(u s ),
where we used sat(u s ) = u s -dz(u s ).

Moreover, λ = 0 implies E a (P, λx • ) = E a (P, 0) (see again ( 8)), which means that, by comparing ( 6) with [START_REF] Li | An asymmetric Lyapunov function for linear systems with asymmetric actuator saturation[END_REF], in this special case E a (P, λx • ) = E s (P ). Then, the statement of Theorem 2 coincides with that of Theorem 1.

In the general case when u + k = u - k for all k, we can prove that Theorem 2 provides an estimate E a (P, λx • ) of the domain of attraction whose size is always larger than the size of estimate E s (P ) given by Theorem 1, as stated next.

Proposition 1:

If u + k = u - k for all k = 1, . . . , m, then the optimal values J * 1 := log det(Q * 1 ) and J * 2 := log det(Q * 2 )
of the convex optimizations in Theorems 1 and 2, respectively, satisfy J * 1 < J * 2 . Moreover, using η := min

k ū2 k u 2 k > 1, if an optimal solution (Q, Y, W, U, X) = (Q * 1 , Y * 1 , W * 1 , U * 1 , X * 1 ) from Theorem 1 satisfies x T • (Q * 1 ) -1 x • < η, then (Q * 2 , Y * 2 , W * 2 , U * 2 , X * 2 , µ * ) = (ηQ * 1 , ηY * 1 , ηW * 1 , ηU * 1 , ηX * 1 , 1
) is a feasible solution to the constraints of Theorem 2.

Remark 3: In addition to stating that Theorem 2 always provides better estimates than Theorem 1, Proposition 1 states that, whenever x T

• (Q * 1 ) -1 x • < η (namely x • falls into the corresponding ellipsoidal estimate scaled by η), then Theorem 2 provides an ellipsoidal estimate that is at least η times larger than the one of Theorem 1. That estimate, moreover, is obtained with the same gains K and L as those of Theorem 1, because η cancels out in the two left equations of [START_REF] Gomes Da | Anti-windup design with guaranteed regions of stability: an LMI-based approach[END_REF]. We also emphasize that if u is scalar, then the feasible solution 12) of Theorem 2, there exists a small enough µ 0 > 0 satisfying the left inequality of ( 12) with Q = (1 + µ 0 )Q * 1 , and introducing an increase on each one of the (1,1) terms of the right inequalities of ( 12) by at least a factor

(Q * 2 , Y * 2 , W * 2 , U * 2 , X * 2 , µ * ) characterized in Proposition 1 is also optimal. • Proof: Consider an optimal solution (Q * 1 , Y * 1 , W * 1 , U * 1 , X * 1 ) from Theorem 1. If u + k = u - k for all k, then from (7b), also |u •,k | > 0 for all k. As a consequence, selecting (Q, Y, W, U, X) = (Q * 1 , Y * 1 , W * 1 , U * 1 , X * 1 ) in the conditions (4a)-(4c), (
η 0 := min 1 + µ 0 , min k u 2 k +µ|u •,k |(2u k +|u •,k |) u 2 k > 1.
Since the remaining inequalities in (4b), (4c) are homogeneous in the decision variables, a feasible solution to (4a)-(4c),

(12) of Theorem 2 is then (Q 2 , Y 2 , W 2 , U 2 , X 2 , µ) = (η 0 Q * 1 , η 0 Y * 1 , η 0 W * 1 , η 0 U * 1 , η 0 X * 1 , µ 0 ), for which it is evident that log det(Q 2 ) > log det(Q * 1
). The proof of the second statement of the proposition is similar because, with µ = 1, entry (1,1) of the right inequality in [START_REF] Hu | Anti-windup synthesis for linear control systems with input saturation: achieving regional, nonlinear performance[END_REF]

becomes u 2 k + 2|u •,k |u k + |u •,k | 2 = (u k + |u •,k |) 2 = u 2 k . IV. SIMULATION EXAMPLE
The asymmetric stabilizer proposed in Theorem 2 is here compared with the standard symmetric solution of Theorem 1 on the benchmark example in Section 4.1 of [START_REF] Yuan | Switching control of linear systems subject to asymmetric actuator saturation[END_REF]. The plant dynamics is given by (1) with

A = 0.6 -0.8 0.8 0.6 , B = 1 3 (21)
and the saturation limits are u + = 0.5 and u -= 1.5. Unlike [START_REF] Yuan | Switching control of linear systems subject to asymmetric actuator saturation[END_REF], the state is supposed to be fully available, therefore we do not compare our results to those of [START_REF] Yuan | Switching control of linear systems subject to asymmetric actuator saturation[END_REF], which use an output feedback scheme. It is however clear that exponential instability of ( 21) imposes constraints on the null controllability region and due to this reason the symmetric stability sets are comparable to those in [19, Fig. 1, §.4.1]. Solving the convex optimization problem (4), the symmetric stabilizer of Theorem 1 leads to the certified domain of attraction E s (P 1 ) illustrated by the black ellipsoid in Figure 1. Following the indications of Remark 1 we may design K with L = 0. For this single input example, we obtain the same estimate of the basin of attraction as emphasized at the end of Remark 1. The certified domain of attraction E a (P 2 , x • ) associated with the asymmetric stabilizer of Theorem 2 is instead shown in Figure 1 by the solid red ellipsoid. As suggested in Remark 3, since x • is small enough, the same result is obtained by either optimizing (4a)-(4c), [START_REF] Hu | Anti-windup synthesis for linear control systems with input saturation: achieving regional, nonlinear performance[END_REF] (which gives µ * = 1) or scaling the solution of Theorem 1. It can be easily noticed that, as expected, its size is considerably increased in comparison to the one of Theorem 1.

Remark 4: If L = 0, the algebraic loop (9c) may yield implementation problems. To address this in our simulations, the following lemma has been derived and used.

• Lemma 3: With a scalar input u ∈ R, the solution of the algebraic loop (9c) is given by the explicit equation

ν = Kx γ(x) + L(I -L) -1 dz δ • + Kx γ(x) . ( 22
)
Proof: The proof can be straightforwardly derived by inspection, by comparing (9c) and ( 22) for values of the input above and below the upper and lower thresholds.

Two solutions starting outside the black ellipsoid and inside the red one are reported at the left of Figure 1, showing that, despite being outside the black guaranteed domain of attraction, the blue initial condition is inside both basins of attraction. At the right of Figure 1 we see that all the proposed constructions induce an initially saturated input. With reference to the designs having L = 0, Figure 2 shows green shaded areas where the input is saturated for Theorem 1 (left) and Theorem 2 (right). Figure 2 also shows accurate Comparing the red and black dashed ovals of Figure 2, we notice similar basins of attraction induced by the two stabilizers. However, from Figure 2 the certified domain of attraction (C.DoA) associated with the asymmetric stabilizer is a significantly less conservative estimate of the actual basin of attraction (red dashed and solid lines), as compared to the symmetric case (black dashed and solid lines). This property may be fundamental in critical applications, where operating within a guaranteed domain of attraction is mandatory.

V. CONCLUDING REMARKS

A nonlinear asymmetric stabilizer has been proposed to guarantee regional stability of a linear system subject to asymmetric input saturation. The proposed paradigm exploits the theory developed in the context of symmetric saturation, by nonlinearly combining a symmetric stabilizer in shifted coordinates with a local stabilizer. The presented numerical example shows that the arising nonlinear controller provides a significant increase of the maximal certified domain of attraction, as compared to the classical symmetric approach. Future work includes addressing other performance metrics such as (regional) L 2 gains or anti-windup features using asymmetric stabilizers, in addition to addressing the dynamic output feedback design problem.

APPENDIX: PROOF OF LEMMA 2

We re-state the result in Lemma 2 by converting the notation to that one used in [START_REF] Blanchini | Any domain of attraction for a linear constrained system is a tracking domain of attraction[END_REF], corresponding to the following substitutions: x * → x, u * → ū, x → x, ξ → y.

Lemma 4: [START_REF] Blanchini | Any domain of attraction for a linear constrained system is a tracking domain of attraction[END_REF] Consider plant ẋ = Ax + Bσ and any point x = 0 satisfying Ax + B ū = 0 and |x| P < 1 for some suitable vector ū and some positive definite matrix P . Assume furthermore that a continuous feedback σ = φ(x) is such that function x T P x strictly decreases along solutions in the donut-shaped set {x : x T P x ∈ [|x| P , 1]}.

Then, defining y := x -x and w := 1 -|x| P , the input selection ensures that E(P, 1) := {x : x T P x ≤ 1} is forward invariant and all solutions starting in this set uniformly exponentially converge to the equilibrium x = x. Sketch of the proof. The proof is a slight adaptation of the proof in [START_REF] Blanchini | Any domain of attraction for a linear constrained system is a tracking domain of attraction[END_REF]. Similar to the construction in [4, Section 3], the function (x, x) → Ψ(x, x) is a Minkovski functional whose value is zero for x = x, and it is one on the ellipse ∂E(P, |x| P ) := {x : x T P x = |x| P } (which is a reduced version of the boundary ∂E(P, 1)). In particular, for each point x ∈ E(P, |x| P ), we determine the value of Ψ(x, x) by imposing the following scaling condition (where the arguments of Ψ have been omitted for simplicity):

σ = ū + Ψ(x, x) φ x + x - x Ψ(x, x) -ū (
x + x - x Ψ T P x + x - x Ψ = |x| P . (24)
Multiplying both sides by Ψ 2 and solving the equation above for Ψ, we get two solutions for each x = x. One of them is negative and is disregarded. The second one corresponds to Ψ = -x T P y -(x T P y) 2 -(x T P x -|x| P )y T P y xT P x -|x| P .

Dividing both terms by |x| P = 0 and rearranging, we get the expression in the statement of the lemma. Following the proof technique in [START_REF] Blanchini | Any domain of attraction for a linear constrained system is a tracking domain of attraction[END_REF] we obtain that Ψ is a good Lyapunov function within the set E(P, |x| P ) := {x : x T P x ≤ |x| P }. In particular, by definition of Ψ in (24), for each point x in E(P, |x| P )\{x} we have that x := x+ x-x Ψ is on the boundary of E(P, |x| P ), and then the stabilizer in (23) is the convex combination of ū and φ(x). Due to this fact and because 1) the gradient of Ψ(x, x) is constant along any ray x + λ(x -x) intersected with E(P, |x| P ) (see [4, Lemma 4.2, item (i)]), and 2) the gradient is aligned with that of x T P x on the ellipsoid {x : x T P x = |x| P }, the assumption of strict decrease of x T P x on {x : x T P x = |x| P } is enough to conclude strict decrease of Ψ(x, x) in E(P, |x| P ).

In the donut {x : x T P x ∈ [|x| P , 1]}, the second element in the max reported in the definition of Ψ in ( 23) is larger than one (by construction), and then Ψ(x, x) = 1 implying (again from (23)) that u = φ(x), which is assumed to induce decrease of x T P x in the considered donut.

k = 1 ,

 1 . . . , m, and the extra decision variable µ ∈ [0, 1].

Fig. 1 .

 1 Fig.1. Left: certified domains of attraction from Theorem 1 (inner black ellipsoid) and Theorem 2 (outer red ellipsoid, centered at x•). Right: time histories of the input sat(u) for the four designs of Theorems 1 and 2, with and without the algebraic loop.

Fig. 2 .

 2 Fig. 2. Actual basins of attraction (BoA) associated with the symmetric design of Theorem 1 (black dashed oval) and the asymmetric design of Theorem 2 (red dashed oval). The quadratic estimates are shown with solid lines. Diverging and converging solutions are also reported in both plots. The green shaded area characterize subsets of the BoA where the symmetric feedback (left) and the asymmetric feedback (right) are saturated.

Note that the cited result[START_REF] Goebel | Hybrid Dynamical Systems: modeling, stability, and robustness[END_REF] Prop. 7.5] is much more general than what we need here. Indeed, the "strong" forward invariance assumed there refers the possibility that solutions be non-unique (whereas here solutions are unique and we simply talk about "forward invariance"). Moreover, the result in [7, Prop. 7.5] is for general hybrid systems, whereas we use it here in the special case of purely continuous-time systems.