Ye Zhang

Qimeng Wang

Ziyan Wang

Difei Zhang

Jieyu Gu

Dr Kewei Ye

Dr Dandan Su

Yan Zhang
email: zhangyanyz@jiangnan.edu.cn

Jinghua Chen
email: chenjinghua@jiangnan.edu.cn

Mihail Barboiu
email: mihail-dumitru.barboiu@umontpellier.fr

Y Zhang

Prof J Chen

Strong, Self-Healing Gelatin Hydrogels Cross-Linked by Double Dynamic Covalent Chemistry

Keywords: dynamic covalent chemistry, Diels-Alder (DA) reaction, gelatin, hydrogels, imine

Hydrogels constructed from natural sources received increased attention and applied in biomedical fields. They are protein or polysaccharide cross-linked scaffolds that display water retention and are able to recognize host cargos. Their excellent biocompatibility does not always combine with high mechanical strength (up to 136 kPa) and thermostability, making them less useful in biomedical applications. This paper reports biocompatible gelatin hydrogels, double cross-linked via imine and Diels-Alder (DA) dynamic covalent frameworks. They showed integrated advantages of adjustable and durable mechanical strength, good thermal stability, biocompatibility for promoting cell growth and reasonable degradable rate. These hydrogels possess remarkable self-healing property, acid/alkali resistance at 65 °C and good integrity in organic solvents at 130 °C, holding great potentials for biomedical applications in the areas such as cartilage regeneration, articular reconstruction or soft robotics.

Hydrogels are fibrillar functional soft materials with high surface area, great responsiveness and valuable storage and delivery abilities for biological targets. Thanks to their large content of water inside their three-dimensional (3D) polymeric networks and their good integrity under physiological conditions, hydrogels have been widely applied in biomedical fields, including wound dressing, [1] biosensors, [2] controlled drug release, [3] etc. Recent developments in areas of soft robotics, [4] tissue engineering and regenerative medicine, [5] show that biocompatibility is one of the key features for hydrogel application. In this respect, low toxic natural polymers, such as collagen, chitosan, silk and cellulose, [6] have become better options to synthetic polymers.

Gelatin is a natural protein, derived from collagen hydrolysis, which shows good biocompatibility, proper biodegradability, nonantigenicity and comparatively low costs. [7] As a typical thermoreversible gel, pure gelatin transforms into gel state upon cooling (t < 30 °C) and becomes a type of popular raw material for soft robotic applications. For example, entirely degradable gelatin bio-gels have been applied for electronic skin-E-skin and gelatin methacrylate hydrogels present strain-sensitive conductivity. [8] However, most of these protein-based hydrogels still lack desirable mechanical tunability and stability, which may bring obstacles for practical applications. In order to address this issue, various chemical modifications or cross-linking methods have been used to improve the gelatin hydrogels. [9] Compared to strong covalent connections, reversible covalent bonds can not only strengthen the cross-linking of the polymeric networks, but also provide the dynamic rearrangement of their backbones, resulting in self-healing, stimuli-responsive and bio-interactive features of the hydrogels. [10] Typical dynamic covalent chemistry includes imine, acyl-hydrazone, boronic ester, disulfide bonds and Diels-Alder (DA) reactions. [11] Imine reaction is one of the most commonly used for its mild conditions, fast equilibration rate and good biocompatibility. [12] Biomaterials cross-linked by imine bonds have been frequently reported with properties of self-healing, pH responsiveness and found wide applications in anti-bacterial, wound healing, adhesive dressing, 3D cell-printing studies. [13] However, imine-based materials may not survive to acidic conditions from either metabolic or extreme skin surface environments. [14] Diels-Alder (DA) reaction is a cycloaddition reaction with "click" characteristics, which shows reversibility at high temperature of around 110 °C. [15] For these reasons, in the current work, we combine imine and DA reversible covalent bonds to design and synthesize double cross-linked gelatin hydrogels with integrated multiple functions: tough yet biocompatible, thermostable yet degradable, durable yet self-healing, acid/alkali resistance and organic solvents tolerance. The presence of double imine/DA connections further increase the cross-linking of such double network DN, hydrogels, meanwhile, introduced good thermostability, adjustable mechanical strength (up to 136 kPa), acid/alkali resistance at 65 °C and good integrity in DMF and DMSO solvents at 130 °C. The originality of this work is not based on reversible crosslinking points involved in hydrogel structures, or on the use of gelatin as natural sources of material, but rather on the very interesting chemical and mechanical properties of the resulting materials. Moreover, the biocompatibility of the DN hydrogels was confirmed by MTT assays and confocal laser scanning microscopy, indicating great potentials for applications in various biomedical related or extreme E-skin conditions.

The natural gelatin possesses chemical reactivity via different side groups. In the current work, we decided to increase the number of amino groups for further imine crosslinking on gelatin backbone via the amidation reaction with diethylenetriamine. The modified amino-gelatine, AGel lost the gel state and appears as fluid at r.t. and 15 wt % (Figure S1). Two bands for amide group at 1635 and 1540 cm -1 were observed in FTIR spectra of gelatin and AGel, while the peak intensities of AGel were obviously not useful to prove the successful attachment of diethylenetriamine (Figure S2a). Thereafter, stoichiometric point method was adopted to measure the degree of the amino substitution. As a result, the amino content of the AGels with glue strength ~100 g Bloom and ~240 g Bloom were named as DN1 and DN2, respectively and were determined to be 1.027 mmol/g and 0.958 mmol/g, respectively, with no significant differences observed between these two types. The Bloom number indicates the firmness or softness of a specific type of gelatin and is mainly related to its viscosity.

To construct the DN hydrogels through both imine and DA reactions were used: a) a 25 mM ethanolic solution of 1 eq. of 1,3,5-benzenetricarboxaldehyde and 0.6 eq. of 4-arm-amino-terminated-poly ethyleneglycol-PEG2000 were first mixed, so after the equilibration 0.6 eq. of aldehyde remains unreacted; then the mixture was added to amino gelatin probes resulting in the construction of the first dynamic imine framework. B) the rest of the amino groups in functionalized gelatin was further reacted with increasing amounts of 2-furaldehyde and subsequently, stoichiometric amounts of MAL-PEG-MAL (2 : 1, mol : mol) were added to construct the second dynamic framework within hydrogels DN1-1 to DN1-3 and DH2-1 to DN2-3 with increasing DA connections (Figure 1a,b, Table S1). The imine formation between 4-arm PEG and 2-furaldehyde it is less favorable as subsequent cross linking between the two networks via transamination is substantially reduced in the presence of excess of amino groups on gelatin backbones.

FTIR spectroscopy was also used to emphasize the structural changes of the prepared DN hydrogels, after the double cross-linking steps, (Figure S2b). In DN1-3, the vibration band of C=C at 1463 and 1466 cm -1 corresponding to the furan and maleimide groups, decrease reminiscent with the DA reaction between 2-furaldehyde and MAL-PEG-MAL. Moreover, the vibration band of aldehyde at 1692 cm -1 was almost disappearing after cross-linking with AGel, indicating the high conversions toward the imine-bond formation.

Then, compressive tests were performed to evaluate the mechanical strength and compressibility of the DN hydrogels (Figure 2). With the increasing amounts of DA constituents (2furaldehyde and MAL-PEG-MAL) compressive modulus of the hydrogels increased from 52.91 kPa to 136 kPa.

Among these, DN2-3 showed the largest compressive strength of 136 kPa and a breaking strain of 65%, indicating that the dynamic double cross-linked networks were soft and easy to extend. Thereafter, a strain of 5-10% was used to determine the compressive modulus. For the AGel with both glue strengths, the same trend was observed: higher contents of DAs led to higher compressive moduli from 12±4.5 to 53±9 kPa. Meanwhile, the obtained compressive moduli were higher for DN1 series with ~100 g Bloom than DN2 series. Subsequently, the DN1-3 was selected to characterize the shape recovery and anti-fatigue properties with loading and unloading processes. Moreover, the DN1-3 was subjected to compression loading-unloading tests under 40% strain. After 20 or 50 consecutive tests, the compressive strength of the hydrogels restored to 91% and 79.9% of its original strength, respectively, without any rupture, demonstrating good fatigue resistance and self-recovery ability (Figure 2c-d). As shown in Figure 2e, after removing the external force, the shape of DN1-3 is restored immediately. Then, DN1-3 was selected to carry out the macroscopic selfhealing experiments. The hydrogels were made into round-or cylindrical-shapes and cut into different pieces. After contacting the pieces for 30 min at 37 °C, DN1-3 not only bore its own weight (Figure 3a), but also kept as a whole piece when being bent for nearly 90°, showing good toughness and recombining ability (Figure 3b). Moreover, the hydrogel did not break off from the cutting point when being stretched, indicating its ability to withstand considerable tension after healing (Figure 3c). The thermal stability of the DN1 hydrogels were subsequently examined. From the thermogravimetric analysis (TGA), there were two significant weight losses (Figure S3a): the first one from 20 °C to 150 °C indicate the loss of water, while the second one extending from 200 °C to 450 °C is related to the decomposition of the cross-linking molecules and the aminogelatin. In addition, the maximum degradation temperature (Tmax) was examined for the DN1 series, the gelatin and AGel. For AGel, the Tmax increased from 305.5 °C to 321.4 °C (Figure S3b), indicating the increase of thermal stability. Owing to the double cross-linking effect of imine and DA bonds, the Tmax of the hydrogel DN1-3 increased to 402.5°C. Differential scanning calorimetry (DSC) analysis show that two endothermic peaks at ~ 150 °C and 279 °C were detected for all samples; they relate to the volatilization of water and the content of triple helix in gelatin, respectively (Figure S3c). The last but not least, the second endothermic peak shifted from 278.7 °C to 285.5 °C for AGel, and further increased from 290.3 °C to 292.9 °C with the higher content of 2-furaldehyde and MAL-PEG-MAL, indicating that the amino-modification and the double cross-linking have a great impact on the helical structure of gelatin.

It is a great challenge for natural-sourced biomaterials to adapt to extreme environments either from inside the body or on the skin surfaces. Therefore, the hydrogels of DN1 series containing various amounts of 2-furaldehyde and MAL-PEG-MAL were immersed in HCl (pH=1), PBS (pH=7.4) and NaOH (pH=11) solutions at 37 °C, and their swelling properties were evaluated. Decreasing swelling ratios (DN1-1 > DN1-2 > DN1-3) were observed with the increased amounts of 2-furaldehyde and MAL-PEG-MAL and higher cross-linking degrees (Figure 3d). The impacts of tested medium had a greater influence on the swelling behaviors of the hydrogels than the cross-linking degree of the structures (Figure 3e). For each individual hydrogel, its equilibrated swelling ratios in acidic medium was much higher than that of PBS and alkaline medium, which can be explained by the hydrolysis of imine bonds at low pH, resulting in collapse of the imine network. On the other hand, the equilibrated swelling ratios for all the DN1 hydrogels were recorded as around 30% at the beginning, suggesting the immediate loss of water in PBS, and the values maintained lower than 100% after reaching equilibrium. Considering this feature, the hydrogel DN1-3 was immersed in DMF and DMSO respectively at 37 °C in order to evaluate its solvent resistance. The results showed that the hydrogel displayed good antiswelling property in PBS and DMF, with equilibrated swelling ratio even <100%. Comparatively higher swelling ratios in HCl solution (pH=1) and DMSO were recorded, but still showing a high structural integrity owing to the stable DA network. Figure 3f showed the photos taken during the swelling process of the DN1-3 at 37 °C, which were in consistent with the recorded swelling curves (Figure S4).

Since DA reaction is thermally sensitive, it is necessary to reveal the influence of temperature on the stability of the DN hydrogels. In order to detect their responses to external stimuli, the DN1-3 were severally immersed in hydrochloric acid solution (pH=1), PBS (pH=7.4) and sodium hydroxide solution (pH=11) and kept at 65 °C for 6 h, and in DMF and DMSO at 130 °C for 6 h because of their higher boiling points. When the temperature increased, the volume of the hydrogels decreases significantly, especially in HCl solution and DMSO (Figure 3g), indicating partial breakage of imine bonds in acidic environment and partial rupture of DA bonds at high temperature of 130 °C, respectively. However, the hydrogels still remained one whole piece during the processes, demonstrating good thermostability and integrity of the materials, which were still benefited from the double cross-linked structures. The cross-sectional morphology of freeze-dried hydrogels DN1, visualized by Scanning Electronic Microscopy-SEM, revealed dense structures, which can be related to the good mechanical and swelling properties of the hydrogels. The small holes in the cross section were probably caused by bubbles in the solution during the preparation process. To explain the impacts of DA constituents on the in vitro degradation rates of the hydrogels, the mass loss profiles of the DN1 in PBS (pH=7.4) at 37 °C shown the slow degradation rates within the first 7 days (Figure 3h). On the 15 day, the DN1-1 was completely degraded, while the DN1-2 and DN1-3 show remaining 69.4% and 83.9% mass residue, respectively. On 26 days, the hydrogels DN1-2 and DN1-3 still contained mass residues of 45.8% and 69.7%, respectively, and both maintained good shape integrity (Figure 3i). These results indicated that the increased contents of DA constituents and higher cross-linking slowed down the degradation process.

The MTT assay was used to evaluate the cytotoxicity of the DN hydrogels. NIH 3T3 cells were treated with gradient-diluted hydrogel leachates (100%, 50%, 25% and 12.5% of C0). After 24 h of incubation, all the samples provided more than 100% cell viability (Figure 4a). After 72 h of incubation, the same trend was observed, and the cell survival rates were even higher than those at 24 h (Figure 4b), indicating continuous cell growth within the DN hydrogels that demonstrate excellent biocompatibility. Further, the DN hydrogels were put in direct contact with the NIH 3T3 cells and incubated for 72 h to investigate their cytotoxicity, while all the hydrogels maintained good integrity during the whole testing period. Thereafter, the cells were stained with Calcein-AM (green-fluorescence) and PI (red-fluorescence) for the observation of live and dead cells, respectively. As shown by using confocal laser scanning microscopy, all the tested hydrogels revealed similar fluorescence for the live cells with the spindle-shaped morphology similar to the reference sample in pure medium. Moreover, almost no dead cells (red spots) were observed, confirming the negligible cytotoxicity of the DN hydrogels (Figure 4c-f).

In conclusion, a chemically modified amino-enriched gelatin was used to prepare acid-, alkali-and solvent-resistant hydrogels via dynamic imine and Diels-Alder (DA) reversible reactions. The use of double dynamic covalent frameworks to cross link the protein backbones led to strong self-healing property of the hydrogels. The insertion of a secondary reversible DA bonds provided stability to the network structure and up to 3-fold improved the mechanical strength, leading to good fatigueresistance and self-recovery capacities. After 50 consecutive compression loading and unloading tests, the compressive strength of the DN hydrogel was still restored to 79.9% of its original strength without rupture. Meanwhile, the DN hydrogel also exhibited good stability in response to pH and temperature changes, still maintained shape integrity in acidic/alkali solutions at 65 °C or in organic solvents at 130 °C. Importantly, they showed good biocompatibility, as confirmed by MTT assay and confocal laser scanning microscopy. With all these synergetic properties integrated in one type of materials, the present DN hydrogels hold great potentials in various fields of biomedical or soft robotic applications.

Figure 1 .

 1 Figure 1. a) Synthesis of amino functionalized-gelatin, A-Gel and of the hydrogels DN1-3 via synergistic imine/DA reversible reactions; b) schematic illustration of the double cross-linked dynamic network and the picture of the resulted hydrogel.

Figure 2 .

 2 Figure 2. a) The compression strain-stress curve of the DN hydrogels; b) the compressive modulus; c) Loading-unloading continuous compression tests and d) the stress recovery property of hydrogel DN1-3 at 40% strain for 50 cycles; e) photograph of the loading-unloading processes for compression tests.

Figure 3 .

 3 Figure 3. The self-healing behaviors of the hydrogel DN1-3: a) with manual hanging; b) with manual bending and c) with manual stretching of the healed hydrogel for 30 min. Some samples were stained with Rhodamine B for better visualization effect. Swelling ratio at equilibrium of hydrogels DN1 d) in HCl (pH=1), PBS and NaOH (pH=11) solutions at 37 °C; e) in HCl (pH=1), PBS, NaOH (pH=11), DMF and DMSO solutions; f) photographs showing the state of DN1-3 in different solutions at 37 °C for 6 h, 1 d and 4 d, respectively; g) photographs of swelling in different solutions at high temperature for 6 h; h) degradation behavior of DN1 hydrogel in PBS (pH=7.4) at 37 °C; i) photographs of the initial freeze-dried hydrogels and hydrogels after degradation for 26 d.

Figure 4 .

 4 Figure 4. Cell viabilities of NIH 3T3 cells incubated with the the double cross-linked hydrogels, measured by the MTT assay for a) 24 h and b) 72 h. Confocal laser scanning microscopy (CLSM) images of NIH 3T3 cells cultured on the double cross-linked hydrogels: c) control, d) DN1-1, e) DN1-2 and f) DN1-3.

Acknowledgements

This work was supported by National Natural Science Foundation of China (31700708, 22008090), Natural Science Foundation of Jiangsu Province (BK20180625), National Firstclass Discipline Program of Light Industry Technology and Engineering (LITE2018-20) and M-ERA NET 2019, Agence Nationale de la Recherche ANR-20-MERA-0001-01, SMARTMATTER.

Entry for the Table of Contents

Double dynamic covalent gelatin hydrogels with integrated functions: tough mechanical strength (up to 136 kPa) yet biocompatible, thermostable yet degradable, durable yet self-healing, plus properties of acid/alkali resistance at 65 °C and organic solvent tolerance at 130 °C. These hydrogels are biocompatible revealing similar fluorescence for the live cells with the spindle-shaped morphology similar to the reference sample in pure medium.


Anna Shvarts
email: a.y.shvarts@uu.nlm.doorman@uu.nlr.a.alberto@uu.nl

Michiel Doorman

Rosa Alberto

Concrete-abstract-new-concrete: Freudenthal and Davydov in advancing embodied design framework

Freudenthal and Davydov-two giants of mathematics education research-had a similar sophisticated vision on the position of mathematical models and symbols in abstract-concrete dialectics. In our vertical analysis of those approaches, we integrate them in a spiral vision of mathematics learning. Starting from concrete enactment, students abstract previously latent aspects of their reality. Further, students fixate abstract ideas in mathematical artifacts, which in turn enable new concrete experiences. We show how local integration of two theoretical approaches can support an empirical analysis of embodied design for proportions and act as backing a design heuristic for embodied technologically supported activities.

Introduction

The relation between abstraction and concrete experience inevitably lies in the core of mathematics education studies. Should concrete experience become a starting point for teaching abstraction? Can abstract ideas be derived from empirical observation or enactment with concrete objects, or shall abstract ideas be exposed to students directly, through 'symbolically structured environments' [START_REF] Coles | Re-thinking 'concrete to abstract' in mathematics education: Towards the use of symbolically structured environments[END_REF]? Traditional views on abstract and concrete-established within inductive empirical science-are questioned in contemporary investigations of mathematics learning, particularly by those who take an embodied stand.

The variety of visions on the problem of abstraction in mathematics learning creates a need to base theoretical work on a solid ground. In this paper, we coordinate the approaches of two giants in the field of mathematics education research, Davydov and Freudenthal, with respect to the role of mathematical artifacts and students' concrete experiences in teaching abstract ideas. Revealed similarities in the views of such major yet independent figures makes, in our opinion, the emerging theoretical proposal particularly strong and related design heuristics well backed up. The compatibility of views is based on Davydov's approach belonging to the Marxist tradition of cultural-historical activity theory, which can be aligned with Freudenthal's approach to mathematics as a human activity of mathematizing. Neither Davydov, nor Freudenthal used the term artifacts, however symbols, models, and visuals lied in the core of their ideas. We refer to these and any other instances of material culture developed within mathematical activity as mathematical artifacts. We interpret cultural artifacts as a broader category of entities developed within cultural practices.

The paper consists of two main parts. Firstly, we integrate Davydov's and Freudenthal's thinking into a joint view on concrete and abstract. For this local integration of theories (Bikner-Ahsbahs & Prediger, 2014) we conduct a vertical analysis (Shvarts & Bakker, 2021) as we go beyond comparing current states of the theories and dive into the history and philosophical roots of their development. Secondly, we apply the results of this theoretical analysis to guide an analysis of students' interaction with (cultural) artifacts within embodied design [START_REF] Abrahamson | Learning is moving in new ways: The ecological dynamics of mathematics education[END_REF] and further suggest some design heuristics for technologically supported embodied activities. Two research questions guide our study: (1) What are Freudenthal's and Davydov's positions towards the role of mathematical artifacts in facilitating concrete experiences and abstract ideas? (2) How to introduce mathematical artifacts in embodied designs informed by the integrated Freudenthal-Davydov approach to mathematical abstraction?

Freudenthal and Davydov: concrete, abstraction, and mathematical artifacts

Freudenthal dedicated quite some publications to Davydov's approach [START_REF] Freudenthal | Soviet research on teaching algebra at the lower grades of the elementary school[END_REF][START_REF] Freudenthal | Bastiaan's experiments on archimedes' principle[END_REF][START_REF] Freudenthal | Lessen van Sovjet rekenonderwijskunde [Lessons of Soviet mathematics educational science[END_REF]. Moreover, the Davydov's curriculum has been implemented in some Dutch schools, and compared to Wiskobas-a curriculum inspired by Freudenthal [START_REF] Nelissen | Kinderen leren wiskunde: een studie over constructie en reflectie in het basisonderwijs[END_REF]. In our view, Freudenthals' interest in Davydov is rooted in a deep agreement on the research and teaching methods. Within both research programs, intensive investigatory implementations were conducted in schools and referred to as formative experiments in Russia and developmental research in the Netherlands, presenting historical variants of what we know now as design research [START_REF] Bakker | Design research in education. A practical guide for early career researchers[END_REF]. As we explain below, the similarities in the teaching methods convey insights on abstraction, the role of cultural artifacts, and the progressive development of concrete experiences.

(1) An abstraction is not based on recollection of empirical impressions

The fundamental innovation of the Davydov approach lies in an intensive critique of empirical thinking and pedagogy that treats abstraction as deriving from empirical examples [START_REF] Davydov | Types of generalization in instruction: Logical and psychological problems in the structuring of school curricula[END_REF]. As van Oers (2019) explains, Ernest Cassirer was apparently the first to criticize this type of abstraction because of the impossibility to limit the set of empirical observations from which to derive abstract qualities. Per [START_REF] Davydov | Types of generalization in instruction: Logical and psychological problems in the structuring of school curricula[END_REF], new classes of objects are created within human practical activity, and theoretical thinking later describes those classes not through empirical observation but through transformative actions that reveal otherwise hidden properties. In the course of learning, students are to "develop special object-related actions by which they can disclose in the instructional material and reproduce in models the essential connection in an entity, then study its properties" (p. 174). Analyzing Davydov's approach, Freudenthal highly appreciated this perspective on abstraction and stressed that "abstraction and generality are-in many casesnot reached by abstracting and generalizing from a large number of concrete and special cases" (Freudenthal, 1974, p. 412). Later, Freudenthal tried out Davydov's approach of deriving arithmetic operations from practical actions with magnitudes-such as length and volume-with his grandson and found this approach to be effective [START_REF] Freudenthal | Bastiaan's experiments on archimedes' principle[END_REF](Freudenthal, , 2002b, p. 102), p. 102).

(2) Children need to reinvent mathematical models and symbols Another point of the clear coordination between the approaches of Davydov and Freudenthal lies in addressing the role of mathematical models and symbols. Per Freudenthal, the mathematical activity consists of progressive schematization and algorithmization of solving problems that are meaningful for students. Those schematizations and algorithmizations are later preserved in the form of mathematical models and formalized rules (Freudenthal, 2002a). The rules preserve the history of problem-solving for those who came up with them in their own problem-solving. So, the only way for the learners to meaningfully extend their understanding of reality through mathematics, lies in reinventing mathematical rules and symbols. Otherwise, "having been imposed, they [rules and symbols], never had a real chance to develop into common sense of a higher order" (p. 8).

Davydov similarly assigned a primary role in scientific thinking to models, symbols, and signs. "Symbols and signs, as well as mixed forms of them, serve as the material means of idealizing and constructing scientific objectness" (Davydov, 1990, p. 121). Constructing these material means (artifacts) is exactly a process of abstraction, which fixates (reifies) the essential (for a practical activity) aspects of the object under investigation: "The construction of this new object [idealized model] functions as a certain mode of activity-as abstraction" (p. 117). In learning, children pass through a quasi-investigation, in which they uncover an essential (theoretical, mathematical) aspect of an object and reproduce it "in particular object-related, graphic, or symbolic models" (p. 174).

(

3) Progressive development of concrete experiences

The origins of Freudenthal's ideas lie in the observation that mathematics education tends to inverse the development of mathematical ideas by presenting students the final products. He referred to his approach as phenomenological, and-although he insisted on the divergence with Husserl, Hegel, and Heidegger (Freudenthal, 2002b, p. 28)-he was apparently essentially influenced by phenomenological thinking. This influence can be traced in his ideas of developing common sense: in "the course of life, common sense generates common habits, in particular, where arithmetic is concerned, algorithms and patterns of actions and thoughts, initially supported by paradigms, which in the long run are superseded by abstractions" (Freudenthal, 2002a, p. 7). So, mathematical abstraction, such as arithmetic, derives from the common sense experience of acting and thinking. Further, those mathematical abstractions support later common sense experiences: "These products of common sense acquire in turn the behavioural status of common sense, while their common sense ancestry may have even been forgotten" (p. 7). Per Freudenthal, good mathematical education develops students' ability to see reality mathematically; mathematical symbolism is a lens for this newly developed common sense.

We find Freudenthal's idea of developing common sense to be close to the dialectical materialist method of ascending from abstract to concrete that Davydov exploited. This method does not mean presenting abstraction from the beginning. As Ilyenkov explains, "the ascent from the abstract to the concrete without its opposite, without the ascent from the concrete to the abstract would become a purely scholastic linking up of ready-made meager abstractions borrowed uncritically" (Ilyenkov, 2008, p. 137-138). So, abstraction starts from concrete experience, as well as progresses towards concrete experience: "the ascent from the concrete to the abstract and the ascent from the abstract to the concrete, are two mutually assuming forms of theoretical assimilation of the world, of abstract thinking" (p. 137). However, those two directions are not forward and backward. Abstraction reveals the latent aspects of initially experienced concrete reality, and those aspects further become salient in the theoretically grounded concrete perception of the objects. So, ascending from the abstract to concrete does not mean a detachment from the initial concrete experiences but a transformation of perception towards seeing concrete objects in a new way-through the lens of abstraction, facilitated by the artifacts as if superimposed on the perceptual field.

Concluding the theoretical analysis

Answering our first research question, we interpret Freudenthal's and Davydov's positions towards concrete experiences and abstract notions preserved by mathematical artifacts as converging in the following vision of the learning process. Students derive an abstract understanding from the concrete experiences within a specially organized practical problem-solving activity. This practical transformative activity elicits latent aspects of the world, which students fixate in cultural artifacts, such as mathematical models and symbols (movement from concrete to abstract). Having constituted an abstraction supported by the artifacts, students can put these artifacts into action and distinguish new initially latent aspects (movement from abstract to new concrete). In this iteration, students develop their common sense (in Freudenthal's words) or ascend from abstract to concrete (in Davydov's words) in establishing a theoretical vision of an object. Thus, we see students' development as a spiral: from concrete approaching the world in practical activities to abstracting latent aspects and fixating them in mathematical artifacts, and further towards establishing new-concrete perception mediated by those artifacts. From this approach, cultural artifacts emerge as reifications of the actions, which have elicited abstract features. Students need to actively constitute those artifacts to preserve the history of initial concrete enactment.

Concrete-abstract-new-concrete in implementing embodied design

Embodied action-based design is one of the quickly developing paradigms related to radicalembodied-enactivist-phenomenological reconsiderations within cognitive science [START_REF] Abrahamson | Learning is moving in new ways: The ecological dynamics of mathematics education[END_REF]. The learning sequence in this design genre consists of a few major steps [START_REF] Alberto | Action-based embodied design for mathematics learning: A decade of variations on a theme[END_REF][START_REF] Abrahamson | The future of embodied design for mathematics teaching and learning[END_REF]. At first, students are invited to solve a motor problem, i.e. discover a new coordination between their hands based on continuous feedback, and uncover the rule of positive feedback to their actions. Later, artifacts are introduced into the problem space, and students are guided towards the quantification of their experiences. Within this paradigm, the researchers have intensively questioned the position of embodied activities and cultural artifacts within abstract-concrete dialectics. In particular, they consider a sensorimotor scheme as "the epistemological core of mathematical learning and knowing" [START_REF] Rosen | Tradeoffs of situatedness: iconicity constrains the development of content-oriented sensorimotor schemes[END_REF](Rosen et al., , p. 1509)), which can be further developed in both directions: towards abstract notions through semiotic signification by cultural artifacts, and towards concrete situations through providing context. Our empirical analysis of embodied activities through the lens of a joint view of Freudenthal and Davydov hints towards another role of the artifacts in abstract-concrete dialectics and further advances the design framework.

Stage 1. Action-based abstraction: Seeing new structures in concrete embodied experience

When solving a motor problem, students discover new abstract qualities,-at first at the embodied level and later in conversations with tutors-such as a proportional relation between the length of two bars, or a coordination of a unit circle circumference and an x-coordinate of a sine graph. Solving a motor problem is coherent with Freudenthal's and Davydov's ideas about abstraction as emerging from a goal-oriented practical actions. Technological environments restrict students' degrees of freedom, thus facilitating quasi-investigation, as Davydov would insist. Although restricted, the students appear to come up with a multiplicity of personal strategies and perceptual orientations, thus meeting Freudenthal's idea of reinventing rather than exposing culture. So, initial embodied enactment with concrete, tangible objects enables the discovery of abstract mathematical relations.

Stage 2. New-concrete: Looking through the lens of emerging cultural artifacts

While the phase of solving a motor problem has been extensively studied, researchers paid relatively less attention to the stage when artifacts are introduced. Therefore, we bring forth a small classroom episode from an experimental tryout of an embodied action-based design for proportions in an ordinary third grade (8-9 years old) classroom in the Netherlands (see a description of the data collection in Alberto, van Helden, Bakker, submitted). Four student pairs were video recorded and the following episode is selected to provide the best insights on the use of mathematical artifacts in establishing new understanding. Two girls (Iris and Frida) collaborated in the tablet-based activity: they manipulated two bars on a screen, which turned green when their lengths were in a particular fixed ratio. The girls were required to keep the bars green while moving and later "to guess a code" that determines the bars' green color, thus describing the proportional relation between the green bars' lengths. In the analysis, we contrast the use of two cultural artifacts: a dice, which was spontaneously appropriated by the students, and a grid, which was imposed by the educators. By the moment of the episode, the girls have already solved the task with the bars being in ratio 1:2. In order to see that the length of one bar is doubled in the length of another bar, the girls spontaneously used a dice that was occasionally lying on a table: They positioned the dice in the middle of the big bar, marking that small bar fits in it two times (Fig. 1a). In the next task, the bars turned green at a ratio 1:4, but the girls did not know this yet. Adjusting one hand upwards somewhat slower than the other one, Iris found several green positions. She exhibited a general abstract strategy of maintaining two lengths in the same proportional relation, which needs to be concretized in quantifying the exact relation. Iris again took a dice and marked the length of the shorted bar on the longer bar (Fig. 1b), thus marking a unit of measurement that would help assess how many times the small bars would fit into the big one-a concretization of the abstract relation of "fitting into the other one." Then both girls tried to measure with their fingers how many times the length of the short bar would fit into the long bar (Fig. 1 c,d). The relation between two bars is seen through the lens of a cultural artifact, a dice, which here means marking down a measurement unit. The dice served as a reification of previous sensory-motor coordination; it allowed for distinguishing a new-concrete, new previously invisible aspects of the bars, namely a measurement unit that helps assess how many times one bar fits in another bar. A common sense action, in Freudenthal's terms, developed for Iris towards seeing the big bar as containing some number of small bars. Iris ascended from abstract coordination of two lengths to concrete quantification of their relation mediated by a dice (in a very physical sense). Unfortunately, a dice was barely an appropriate artifact for marking the small bar length: its own size distorted the measurement. As a result, the girls efficiently exploit an abstract idea of proportional relation as "fitting some number times in" but miscalculated the relation as 1:3 (combining two possible mistakes, see Fig 1 с,d).

The activity progressed towards the next stage of introducing cultural artifacts [START_REF] Abrahamson | The future of embodied design for mathematics teaching and learning[END_REF], in which the girls were asked to confirm their code using an imposed grid (Fig. 1 e,f,g).

With the help of a teacher, two bars were positioned at lines 8 and 2 and the bars were green. The teacher asked: "What does it mean?" expecting that 2:8 relation was obvious enough to dissolve students' 1:3 hypothesis. "Three times smaller" was the answer. The teacher invited the students to check: "Does it fit in three times? Look?" Frida did not use the grid, but took the dice (!), and marked the length of a small bar on the large bar by a horizontal gesture (Fig. 1e). Despite ignoring the imposed grid, by this horizontal gesture, Frida re-invented the functionality of grid's horizontal lines, as both artifacts serve the same function of marking equal units of measurement. The dice was big, and an approximate measurement led to the answer 1:3 again. Supporting the use of the imposed mathematical artifact, the teacher guided the students' perception towards the grid [START_REF] Abrahamson | Learning is moving in new ways: The ecological dynamics of mathematics education[END_REF]: She gestured the horizontal alignment of the large bar and number 8 and then pointed at number 2, Frida read the numbers (Fig. 1f). However, their relation did not guide further enactment. Following the sequence of the teacher's gestures from top to bottom, Iris made a new measuring attempt counting from the top without clear measurement unit (Fig. 1g). She came up with an answer 2,5. The initial abstraction of "fitting in" was lost, and the students could not concretize (quantify) abstract proportional relation using the grid.

The dice was a natural continuation of the students' thinking and bodily enactment (see Shvarts et al., 2021 for conceptualization of this situation as a body-artifact functional system), and it allowed the girls' common sense development. By exploiting the dice, the girls could mark a measurement unit and concretize an abstract embodied idea of proportional relations in the given situation. Contrary, an imposed grid was not reinvented and stayed alien to the emerging abstraction. The teacher could see the bars' proportional relation naturally through the grid, while the girls could use the grid in this way. Their phenomenological realm did not the grid, contrary, a dice that became a mediator for a new concrete, i.e. for distinguishing new mathematical aspects of reality.

Towards a new design heuristic

As the theoretical and empirical analyses reveal, a cultural artifact might become a reification of practical actions, which helped to distinguish an abstract relation-a proportional relation between the bars, tangible as "small bar fitting into the big one." Importantly, an imposed mathematical artifact (a grid, see [START_REF] Abrahamson | The future of embodied design for mathematics teaching and learning[END_REF] did not fulfill this function, even with the teacher's guidance. Another artifact (a dice) spontaneously was appropriated by the students to reify an action of distinguishing a unit of measurement and served as an instrument in concretizing the ratio. Yet, this other artifact was barely appropriate for fulfilling this function. A design solution to this dilemma might be in creating an environment where students could spontaneously find suitable materials for creating the target artifacts. Such material might include thin sticks to mark the horizontal position of a small bar, paper stripes for creating a measuring unit and overlaying it on the big bar, or even a ruler, which was spontaneously and efficiently appropriated by some other pairs in the study. This way, a classroom can be enriched by appropriate means for progressive mathematizing/modeling of the situation, which could support establishing the perception and use of new mathematical aspects of concrete situations, thus distinguishing new concrete.

Concluding remarks

Freudenthal and Davydov are unique figures by the scale of their influence on the mathematics education and educational psychology communities. However, the program of each of them does not flourish nowadays in the countries where they were working despite intensive and successful experiment-based elaborations. Our analysis brings forth the complexity of their understanding of mathematical abstraction and concrete experience. Those approaches aim to develop in students a new ability to see an object concretely within its mathematical interrelations, i.e., developing a new common sense. From a concrete action-based experience, students come to distinguish abstract relations that are later reified in cultural artifacts. Further, the artifacts come to illuminate their newconcrete experiences. We hope that contemporary technologies can support students and teachers in fulfilling the aim of learning to see the world mathematically. This type of mathematics learning is in particular valuable for the 21st century with routine calculations being outsourced to the machines and increasing importance of skills such as mathematical modeling and recognizing mathematical patterns in everyday situations [START_REF] Gravemeijer | Reconsidering mathematics education for the future[END_REF].

Looking back at the interaction of theoretical ideas and design heuristics, we notice that we used two prominent theoretical approaches as a way of backing a design idea that has been already emerging in our design work and empirical data. We uncovered the essential coherence of two approaches in seeing cultural artifacts as instruments that transform students' concrete experiences. The fact that those approaches are widely recognized as highly valuable strengthens the design heuristic of re-inventing mathematical artifacts and its potential for curriculum design.

Fig. 1 .

 1 Fig. 1. (a, b, c, d): Seeing proportional relation through a dice. (e, f, g,): Missing proportional relation when applying a grid


M N Chernodub

Instantons in rotating finite-temperature Yang-Mills gas

Instantons in rotating finite-temperature Yang-Mills gas M. N.

The recent observation of highly-vortical quark-gluon plasma [1] has attracted attention to the properties of relativistically rotating quark-gluon matter. The vorticity of the quark-gluon plasma can be probed experimentally via spin polarization of the hadronized matter [2,3]. Theoretical analysis indicates that rotation can also affect the thermodynamic properties and modify the phase diagram of hot QCD [4][5][6][7][8][9][10][11][12][13][14][15][16]. Although most of the theoretical studies were carried out in the much simplifying approximation of rigid rotation, a global picture of the phase diagram of rigidly-rotating quark-gluon plasma is still missing.

In the absence of rotation, the finite-temperature QCD transition appears to unify the chiral and deconfining crossovers that occur close to each other [17]. However, the theoretical results on chiral and deconfining transitions in rotating plasma contain certain controversies.

All existing theoretical studies agree that vorticity reduces the critical temperature of the chiral phase transition [4][5][6][7][8][9][10][11]. The physical mechanism of this phenomenon takes its roots in the Barnett effect [18] discovered in 1915: the rotation aligns the spins of quarks and antiquarks along the rotation axis and suppresses the pairing of fermions in the scalar channel, thus inhibiting the quark condensate [5]. Thus, the faster rotation, the lower the chiral critical temperature.

However, the first-principle numerical simulations of purely gluonic matter show a different picture implying that the critical temperature of the deconfining transition should increase with the angular velocity [12,14]: the faster rotation, the higher the deconfining critical temperature. At the same time, other independent approaches to the deconfinement problem in rotating quarkgluon plasma, a holographic technique of Ref. [10] and the hadron-resonance gas used in Ref. [15], indicate, in a contradictory manner as well, that the deconfining critical temperature should decrease with the angular velocity. Moreover, the Tolman-Ehrenfest argument, re-derived in a confining model in Ref. [13], suggests that rotation leads to yet another outcome: a qualitative change of the QCD phase diagram featuring a new mixed confiningdeconfining phase with both phases present at the same time in a broad domain of the parameters of the model. Thus, the connection between the chiral and confining properties and intrinsic coherent explanation of the confining mechanism are missing.

Let us now look briefly at the problems of chiral symmetry breaking and confinement from the point of view of the topology of the Yang-Mills gauge group1 . The natural topological objects of zero-temperature Euclidean Yang-Mills theory are instantons representing self-dual solutions to the classical Yang-Mills equations of motion [22]. The Euclidean instanton configuration can be associated with a process in a Minkowski spacetime.

The instanton is a solution to the classical equations of motion,

∂ µ F µν + [A µ , F µν ] = 0 for the Yang-Mills field A µ = gt a A a µ
, where t a are the generators of the SU (N c ) gauge group with a = 1, . . . N 2 c -1. The selfdual instanton solutions are determined by the relation Fµν ≡ µναβ Fαβ = ±F µν imposed on the field strength tensor

F µν = ∂ µ A ν -∂ ν A µ +[A µ , A ν ]. A convenient form of a general instanton solution is A a µ = -η a µν ∂ ν ln φ , (1)
where ηa µν is the 't Hooft symbol [23] and φ is the scalar potential which satisfies the equation φ = 0.

The topology of the Euclidean gluon configuration is characterized by the integer-valued Pontryagin index (topological charge),

Q = 1 16π 2 d 4 x Tr Fµν F µν = ± 1 8π 2 S , (2)
where the Euclidean action is:

S = 1 2g 2 d 4 x d 4 x s(x), s(x) = - log φ. (3)
An N -instanton configuration is given by the potential:

φ = 1 + N k=0 λ 2 k (x -x k ) 2 , (4)
where k th instanton is characterized by its size λ k and position x k .

The role of instantons in QCD is impossible to overstate [24]. Instantons are believed to be responsible for the chiral symmetry breaking since they host exact fermionic zero modes. An instanton gas or liquid lifts the degeneracy of zero modes producing a finite density of near-zero-mode states that, in turn, lead straightforwardly to the breaking of the chiral symmetry [25]. As temperature increases, the instantons form instantonic molecules (see, for example, Ref. [26]), which possess a gapped fermionic spectrum. Thus, the instanton picture of the QCD vacuum correctly implies the chiral symmetry breaking at low temperatures and reproduce the chiral symmetry restoration at high temperature.

At finite temperature T , the Euclidean time direction τ ≡ -it is compactified into a circle S 1 of the length β = 1/T with periodic boundary conditions imposed on the gluon fields,

A µ (x, τ ) = A µ (x, τ + β) . (5)
At finite temperature, the instanton solutions become periodic instantons, so-called Harrington-Shepard (HS) calorons [27,28]. The periodicity of the gluon field (5) translates into the periodicity of the scalar potential (1):

φ(x, τ + β) = φ(x, τ ) . (6)
The solution (4) for a single periodic HS instanton (6) corresponds to infinite amount of copies of the same radius λ k = λ with the positions x k = (x 0 , τ 0 + βk). One gets for the HS scalar potential:

φ HS β = 1 + k∈Z λ 2 (x -x 0 ) 2 + (τ -τ 0 -kβ) 2 , (7)
or, explicitly:

φ HS β = 1 + πλ 2 β H(x -x 0 , τ -τ 0 , β) , (8)
where we introduced the Harrington-Shepard function:

H(x, τ, β) = 1 |x| sinh(2π|x|/β) cosh(2π|x|/β) -cos(2πτ /β) . (9)
In the low-temperature limit, β → ∞, the HS solution (8) reduces to a zero-temperature single-instanton configuration with the topological charge Q = ±1:

lim β→∞ φ HS β = φ inst ≡ 1 + λ 2 (x -x 0 ) 2 + (τ -τ 0 ) 0 , (10)
In the opposite limit of high temperatures,

lim β→∞ φ HS β = φ mon BPS ≡ 1 + πλ 2 β 1 |x -x 0 | , (11)
one finds the asymptotics of a dyon [29]: the caloron solution loses its dependence on the time coordinate and becomes a static, three-dimensional dyon-like configuration in the Bogomolny-Prasad-Sommerfield (BPS) limit [30,31]. Despite its explicit self-duality, the BPS solution and its asymptotic are often referred to as a BPS monopole. One can also obtain the BPS expression (11) directly from Eq. ( 8) replacing over the individual instantons by the integral, n → dn, and performing the integration explicitly.

Monopole degrees of freedom provide the basis for one of the most popular mechanisms of color confinement which suggest that the confining force between a quark and an anti-quark appears due to a condensate of monopole-like gluonic configurations. At high temperature, the condensate evaporates and the color confinement disappears in agreement with the existing phenomenology [19,20]. This scenario, however, cannot work with the static BPS monopoles inherent to the high-temperature limit since any ensemble of the static monopoles contribute trivially to the confining order parameter, the Polyakov loop P x = exp{ β 0 dτ A 4 (x, τ )} (and at high temperature, the confining property is lost anyway). The simplest instanton-gas picture of gluon vacuum cannot explain the color confinement as well [32].

At finite temperature, the BPS monopoles reveal themselves explicitly [33] as constituents of the so-called Kraan-van Baal-Lee-Lu (KvBLL) calorons [34][35][36][37]. The KvBLL caloron is a generalization of the periodic HS instanton [27,28] extended to configurations that possess a nontrivial holonomy at spatial infinity, P ∞ ≡ lim x→∞ P x = 1l. A gas of the KvBLL calorons has been shown to produce the confinement property [38] similarly to confining models based on dyon configurations [39,40] Given the promising applications of instantons, calorons, and their holonomically twisted counterparts to chiral symmetry breaking and confinement of QCD, in this article, we construct the finite-temperature instanton (caloron) solutions in the rotating Yang-Mills vacuum regarding their possible relevance to vortical quarkgluon plasma. We work with the case of trivial holonomy, P ∞ ≡ 1l, and consider Yang-Mills theory with SU (2) gauge group. The generalization to SU (N c ) is straightforward.

Consider a system that rotates with the angular velocity Ω about the axis z. In cylindrical coordinates of the inertial laboratory frame in Minkowski spacetime, (ρ, ϕ, z, t), a uniform rotation of an object is encoded in the uniform growth (modulo 2π) of the polar coordinate, φ = [ϕ -Ωt] 2π where the tilded coordinate refers to the system co-rotating with the object. The causality principle restricts the maximal extent of the object in the radial coordinate, |Ωρ| < 1 to guarantee the positivity of the purely time component of the metric g tt (ρ) = 1-Ω 2 ρ 2 (we remind that c = 1 in our units).

After a Wick rotation to the Euclidean spacetime, the time coordinate becomes imaginary. The rotation becomes imaginary as well: the angular frequency Ω becomes a purely imaginary parameter, Ω I = -iΩ [12-14, 16, 41]. The imaginary nature of rotation makes it possible to avoid the sign problem and opens a way for first-principle lattice simulations of rotating systems [12,14,41]. Under the Wick transformation, the real rotation in the real-time maps to the imaginary rotation with the imaginary angular frequency Ω I in the imaginary time τ such that Ωt → Ω I τ . In terms of the polar coordinate, the rotation in Euclidean space becomes φ = [ϕ -Ω I τ ] 2π , implying that the periodic constraint for the gluon field (5) should be replaced, in a rotating medium, by the rotationally twisted periodicity:

A µ (ρ, ϕ, z, τ ) = A µ (ρ, ϕ -Ω I β, z, τ + β) . (12)
The uniform imaginary rotation with arbitrarily large Ω I does not violate causality because there is no notion of causality in the Euclidean spacetime. In other words, the light cone does not exist in the Euclidean space (see also a discussion in Ref. [16]). Moreover, at finite temperature, the theory is periodic in Ω I since the spin-statistic relations for bosonic and fermionic theories imply the equivalence of the thermal state with respect to the following transformations [13]:

Ω I → Ω I + 2πβ -1 k, (bosons), (13)
Ω I → Ω I + 4πβ -1 k, (fermions), (14)
where k ∈ Z. The first relation for bosonic fields is seen from Eq. ( 12). For fermionic fields possessing anti-periodic boundary conditions, the system returns to its state after a double rotation. Interestingly, at Ω I = 2πβ -1 , the fermions become ghost particles as they are converted to spinors obeying bosonic statistics [13].

How the imaginary rotation affects the periodic instanton (caloron)? The scalar potential φ is now determined by the rotationally twisted condition (12) which replaces the finite-temperature periodic constraint (6):

φ(ρ, ϕ, z, τ ) = φ(ρ, ϕ -Ω I β, z, τ + β) . (15)
In order to satisfy Eq. ( 15) we supplement the sum (7) with the rotational twists. A single caloron under the imaginary rotation is thus described by the potential:

φ β,Ω I = 1 + λ 2 k∈Z ρ 2 + ρ 2 0 -2ρρ 0 cos(ϕ -ϕ 0 + Ω I βk) +(z -z 0 ) 2 + (τ -τ 0 -βk) 2 -1 , ( 16
)
where we use the cylindrical coordinate system and consider the system rotating about the axis z centered at the origin ρ = 0. The original 4-position of the "seed" instanton is x µ 0 = (ρ 0 , ϕ 0 , z 0 , τ 0 ). Since we will be considering mainly the solutions seeded by a single instanton, we put, for simplicity, the seed instanton at the non-negative part of the x axis, ϕ 0 = 0. A generalization to the manyinstanton case is straightforward and follows identically the general prescription of Eq. ( 4).

In the low-temperature limit, β → ∞, the rotated instanton ( 16) reduces to the one-instanton solution (10) regardless of the value of Ω I .

If the instanton is centered at the rotation axis, ρ 0 = 0, then this spherically symmetric system is insensitive to the rotation, and the solution ( 16) reduces to the caloron solution given by the finite-temperature periodic instanton (8). Below we consider several nontrivial cases with ρ 0 = 0.

The explicit form of the potential ( 16) can also be given for a set of imaginary angular frequencies Ω I = 2πβ -1 κ where κ is a rational number. Obviously, at Ω I = 2π/β (κ = 1), the system is insensitive to rotation due to the bosonic symmetry (13). The first non-trivial case is given by Ω I = π/β (κ = 1/2), which gives us cos(ϕ + Ω I βk) = (-1) k cos ϕ in the denominator of the sum (16). The sum in Eq. ( 16) thus decouples into two sums over even and odd k with the result:

φ β,Ω I Ω I = π β = 1 + πλ 2 β H(x -x 0 , τ -τ 0 , 2β) +H(x + x 0 , τ -τ 0 + β, 2β) , ( 17
)
where H is the HS function (9). The sense of Eq. ( 17) is clear: the π-rotating (flipping) instanton chain consists of two sub-chains that can be summed over independently. This example provides us with the idea of how to build the rotating instanton solution for any rational (in units of 2π/β ≡ 2πT ) angular frequency:

Ω I = 2π β m n , m = 0, . . . n -1, n = 1, 2, . . . , (18)
where the ratio m/n represents an irreducible fraction.

The general solution for the angular frequency ( 18) is:

φ β,Ω I Ω I = 2π β m n = 1 + πλ 2 β ( 19
) × n-1 l=0 H x -x(lm) n , τ -τ 0 + lβ, nβ ,
where we introduced the set of images x(k) n of the position of the seed instanton, x 0 = (x 0 ≡ ρ 0 , 0, z 0 ), rotated about the z axis by the angle 2πk/n:

x(k) n = x 0 cos 2πk n , x 0 sin 2πk n , z 0 T . ( 20
)
In the simplest nontrivial case, x The geometric localization of the topological charge (2) and action (3) densities in the classical self-dual configurations closely follows the behavior of the scalar function φ. In order to get an insight into the effect of the rotation on the localization properties of the solutions, we show in Fig. 1 the scalar potential φ in the spatial volume for various rational frequencies Ω I .

The rotation produces several fascinating effects: 1. First of all, the imaginary rotation with the rational frequency generates exactly n identical lumps over the spatial ring, where n is the denominator of the irreducible fraction m/n of the frequency (18). Since the lumps represent identical structures, their charge is fractionalized. Each lump carries the topological charge of 1/n of the original solution.

Ω I = 2π β ⋅ 1 2 Ω I = 2π β ⋅ 1 3 Ω I = 2π β ⋅ 1 4 Ω I = 2π β ⋅ 1 5 Ω I = 0 x y z
2. The number of the lumps does not depend on the numerator m that determines the rational frequency (18). This property implies that the rotation with the frequencies, for example, Ω I = 1 5 •2πT and Ω I = 2 5 • 2πT lead to the identical solutions. 3. As one can see from the analytical formula (19), the rotation with the rational m/n frequency effectively cools down the caloron configuration by reducing its temperature from T down to T /n.

Thus, the imaginary rotation with the rational (in units of 2πT ) angular frequency leads to the fractionalization and delocalization of the topological charge of the finite-temperature caloron and the temperature drop in the solution. The latter property might imply that the imaginary rotation effectively decreases the temperature in the topological sector of the gluonic plasma.

We suggest that a rotating caloron at an irrational frequency (for example, at Ω I = √ 2πT or Ω I = 0.1T ) reduces, regardless of the temperature T , to a zerotemperature configuration with an infinite number of instantons placed along a circular ring of a finite radius. The radius is equal to the distance from the seed instanton's center to the rotation axis. This "instanton-ring" solution, known to produce a monopole along the ring [42], can provide us with another intriguing link between rotation and the problem of color confinement.

The strong sensitivity of the basic properties of the rotating calorons, such that the effective temperature and the localization of topological charge, calls for a careful study of the analytical continuation of observables from the imaginary to real angular frequencies, Ω I → Ω, at least in the topological sector of Yang-Mills theory.

For example, the angular frequencies Ω I = 2πT /10 and Ω I = 2πT • 99/1000, that differ only by 1%, produce substantially different caloron configurations that feature, respectively, 10 and 1000 constituents each of which carries 1/10 and 1/1000 fraction of the unit topological charge.

Finally, let us consider the high-temperature limit of the rotating instanton solution. In the non-rotating case, the HS periodic instanton (8) reaches its BPS monopole limit (11) when the inverse temperature β becomes smaller than the typical length scale λ of the solution, β λ. For solution (8), this scale corresponds to a smaller length out of the two length parameters, λ ∼ |x -x 0 | ∼ |τ -τ 0 |, which in dynamical configurations, are dictated by the scale factor λ in Eq. (7).

The rotating instanton (19) is characterized by three parameters: its size λ, the inverse temperature scale β, and the imaginary angular frequency Ω I . While the hierarchy β λ is rigidly fixed by the requirement of the high-T limit, the properties of the high-T solution may vary significantly depending on the value of the frequency Ω I .

In the case of slow rotation, the frequency is small compared to the length scales of the solution, Ω I λ 1. Assuming rationality of the angular velocity ( 18), the high-T limit is achieved in the limit mβ λ, where m is defined as the denominator in the irreducible fraction in Eq. ( 18). The scalar potential of the slowly rotating high-temperature instanton is then obtained by replacing the HS functions in Eq. ( 19) with their BPS limits (11):

φ highT slow Ω I = 2π β m n = 1 + πλ 2 β 1 |x - x(m) n | , (21)
where the monopole positions x(k) n is given in Eq. (20). Thus, the high-temperature limit of the slowly rotating solution ( 19) is a simple sum of the static BPS monopoles. On the contrary, the limit of the rapidly rotating instanton, Ω I λ 1, is a nontrivial example of an analytically solvable high-temperature (β λ) solution pro-vided 2 Ω I β 2π. In this limit, the term Ω I βk under the cosine function in the sum over (16) represents a "fast" variable that oscillates more rapidly compared to a "slow" evolution of the rest of the expression. Performing first the integration averaging over a full period of the fast variable and then taking the integral over the slow variable, we get for the scalar potential of the rapidly rotating caloron the following analytical expression:

φ • (ρ, z) = 1 + 2λ 2 K - 4ρρ0 (ρ-ρ0) 2 +(z-z0) 2 β (ρ -ρ 0 ) 2 + (z -z 0 ) 2 , ( 22
)
where K(x) is complete elliptic integral of the first kind 3 . We call the high-temperature rapidly-rotating caloron solution ( 22) "circulon".

x y p l a n e Remarkable, the circulon solution (22) does not depend on the angular frequency Ω I implying that Eq. ( 22) is an ultimate configuration in the limit of a high-frequency rotation. the high-T high-Ω I solution (22) possesses the axial symmetry implying that the original caloron gets smeared uniformly by the rotation. The core of the circulon forms a ring at (ρ, z) = (ρ 0 , z 0 ), Fig. 2.

At large distances r δ ≡ (ρ -ρ 0 ) 2 + (z -z 0 ) 2 from the ring (ρ, z) = (ρ 0 , z 0 ), the circulon potential ( 22) acquires the asymptotic form of the BPS monopole (11):

φ • = 1 + πλ 2 β 1 r δ + . . . , r δ ρ 0 , (23)
where the ellipsis denotes the subleading O(r -3 δ ) terms. The circulon potential (22) has a logarithmic divergence close to its ring-shaped core:

φ • = 1 - λ 2 βρ 0 log r δ 8ρ 0 + . . . , r δ ρ 0 , (24)
2 These three hierarchical relations are consistent with each other. For example, one can take β = 0.1 fm, λ = 1 fm, Ω I = 10 fm -1 . 3 In other words, K(x) = where the ellipsis corresponds to O(r 2 δ ) corrections. Therefore, the rapid rotation softens the point-like divergent pole of the static BPS solution (11) down to the logarithmic singularity (24).

A generalization of our construction to many-circulon solutions is straightforward. An example of a serious twocirculon configuration is shown in Fig. 3. In summarizing, we constructed the finite-temperature instanton (caloron) solution in Euclidean Yang-Mills theory subjected to imaginary rotation. The solution with a rational-valued angular frequency Ω I can be represented as an ensemble of the Harrington-Shepard calorons arranged equidistantly along a ring (19). We demonstrated that the imaginary rotation cools down the caloron solution, reducing its temperature and leading, at the same time, to the delocalization and subsequent fractionalization of its topological charge. In the high-temperature limit, a slowly rotating caloron reduces to an ensemble of static BPS monopoles. A fast imaginary rotation (which still belongs to a physically reasonable domain with Ω I 2πT ) produces the "circulon" solution with a toroidal core. The circulon potential can explicitly be expressed via the elliptic integral (22).

2 ≡

 2 -x 0 enters the n = 2 solution (17) as the mirror image of the position x 0 of the seed instanton.

Figure 1 .

 1 Figure 1. Equipotential lines of the scalar potential (19) of the rotating caloron solution at fixed τ = β/2 and various imaginary angular frequencies ΩI .

Figure 2 .

 2 Figure 2. Equipotential lines for the topological charge (action density) of (left) the HS caloron solution (8), (9) and (right) the circulon solution(22). Fast imaginary rotation smears the caloron over the ring and produces the circulon.

  2 sin 2 θ .

Figure 3 .

 3 Figure 3. Equipotential lines of the topological charge (action) density of a two-circulon solution with the cores at (ρ0, z0) = (λ, 0), (3λ, 6λ) and inverse temperature β = 0.1λ.

Here, we do not consider alternative approaches based on partial gauge fixing of non-Abelian symmetry and subsequent use of topology of the residual Abelian symmetry group to uncover the topological content of the gluonic configurations[19][20][21].


Rodolfo Orjuela

Benoît Marx

José Ragot

Didier Maquin
email: didier.maquin@ensem.inpl-nancy.fr

{rodolfo Orjuela

Benoit Marx

Jose Ragot

Observer design for nonlinear systems described by multiple models

Keywords: state estimation, nonlinear systems, sensor faults, multiple models, decoupled multiple model, heterogeneous local model networks

This paper addresses both analysis and observer design for nonlinear systems modelled by decoupled multiple models. With respect to classic used multiple models, the decoupled multiple model is characterized by heterogeneous submodels in the sense that their state spaces may be of various dimensions. Thanks to this fact, flexibility and generality is introduced in the modelling stage. The main contribution of the paper is the development of new sufficient conditions on LMI form for ensuring the exponential convergence towards zero of the estimation error in the continuous and in the discrete-time. The new proposed conditions enable to obtain a better decay rate with respect to the existing conditions. The validity of the proposed methodology and its application to sensor faults detection and isolation is illustrated by an academic example.

INTRODUCTION

Context. Nowadays, fault diagnosis systems is becoming unavoidable in order to ensure the safety of systems and operators. In order to setting up a fault diagnosis system strategy, sensitive signals to faults must be available. These signals can be obtained, for example, with the help of state estimation techniques [START_REF] Patton | Issues of fault diagnosis for dynamic systems[END_REF], [START_REF] Isermann | Fault-Diagnosis Systems. An Introduction from Fault Detection to Fault Tolerance[END_REF]. Indeed, the state estimation of a system provided by an observer, is an intermediary stage to estimate the characteristic variables of the system such as its outputs. These estimations can be considered as an analytical source of redundancy of the system. With the help of this source of redundancy, fault diagnosis tasks (detection, isolation and identification) can be effectively accomplished by analysing the signals sensitive to faults (residual signals) generated by the discrepancies between the measured outputs of the system and their estimated. Indeed, if the residual signal deviates from zero then an abnormal behaviour of the system can be detected. Consequently, the observer design plays an important role in the described fault diagnosis strategy.

Note however that the state estimation problem (i.e. observer design) can be particularly arduous when the dynamic behaviour of the system must be represented by nonlinear models. A nonlinear representation of the system is generally needed when the whole operating space of the system must be considered (global modelling) instead of a reduced operating space for example around an operating point (local modelling). This situation frequently arise in a fault diagnosis strategy because all admissible dynamic behaviours of the system must be taken into consideration. In order to capture highly nonlinear dynamic behaviours, in a wide operational range of the system, nonlinear representations based on local model networks also known as multiple model approach can be used as an appropriate global modelling tool [START_REF] Murray-Smith | Multiple model approaches to modelling and control[END_REF]. On the other hand, by means of this representation the specific analysis of the non-linearity of the system can be avoided and the observer design can be in this way simplified.

Tools and proposed method. In the multiple model approach the use of an only one complex model is avoided by introducing a set of submodels correctly interconnected by means of an interpolation mechanism. The multiple models are able to accurately capture the nonlinear behaviour of a large class of systems in the whole operating space. In this modelling strategy, the operating space of the system under investigation is decomposed in a finite number of operating zones. The dynamic behaviour of the system, inside each operating zone, is then modelled by a local model (a submodel), often a linear model of reduced dimension. Finally, according to the current operating point of the system, the contribution of each submodel is more or less taken into consideration thanks to weighting functions.

In this modelling approach, two main structures of multiple models can be considered [START_REF] Filev | Fuzzy modeling of complex systems[END_REF]. In the first one, the submodels have the same state-space and consequently the multiple model is composed of homogeneous submodels. The well known Takagi-Sugeno model proposed, in a fuzzy modelling framework, by [START_REF] Takagi | Fuzzy identification of systems and its application to modelling and control[END_REF] and in a multiple model modelling framework by [START_REF] Johansen | Constructing NARMAX models using ARMAX model[END_REF] is a good example of this kind of multiple model. In the second structure, the multiple model uses heterogeneous submodels, in the sense that the submodels do not share the same state-space. This second structure, so-called decoupled multiple model, will be employed in the present work.

Related work.

The main interest for employing heterogeneous submodels lies in the fact that the order (i.e. the state space dimension) of each submodel can be different. Consequently, the degree of generality of the decoupled multiple model is undoubtedly increased because a non uniform decomposition of the operating space can be made, i.e. the complexity of the system in each operating zone may be different. Nonlinear systems identification using this multiple model is presented in [START_REF] Venkat | Identification of complex nonlinear processes based on fuzzy decomposition of the steady state space[END_REF][START_REF] Vinsonneau | Extended global total least square approach to multiple-model identification[END_REF][START_REF] Orjuela | Nonlinear system identification using uncoupled state multiple-model approach[END_REF] and the design of control laws in [START_REF] Gawthrop | Continuous-time local state local model networks[END_REF][START_REF] Gatzke | Multiple model approach for CSTR control[END_REF][START_REF] Gregorcic | Control of highly nonlinear processes using self-tuning control and multiple/local model approaches[END_REF][START_REF] García-Nieto | Nonlinear predictive control based on local model networks for air management in diesel engines[END_REF]. Note however that the state estimation problem of nonlinear systems using this multiple model has been poorly considered in the literature. To the best of the authors' knowledge, only in the few recent works [START_REF] Uppal | A hybrid neurofuzzy and de-coupling approach aplied to the DAMADICS benchmark problem[END_REF][START_REF] Uppal | A neuro-fuzzy multiple-model observer approach to robust fault diagnosis based on the DAMADICS benchmark problem[END_REF] this structure of multiple is employed in a fault detection and isolation scheme for nonlinear dynamic systems (in these works, the proposed structure is namely Neuro-Fuzzy Decoupling Fault Detection Scheme NFDFDS). However, only a note of the convergence estimation error is proposed in these works and no analytic proof of the convergence of the estimation error is given.

Contribution. This paper refers to the development of new conditions for ensuring exponential convergence towards zero of the estimation error as a further development of previous works [START_REF] Orjuela | State estimation of nonlinear discrete-time systems based on the decoupled multiple model approach[END_REF][START_REF] Orjuela | State estimation for nonlinear systems using a decoupled multiple model[END_REF]. With respect to these works, the proposed conditions makes it possible to improve the dynamic performances of the observer. In particular, the convergence velocity of the estimation error can be improved because a better decay rate can be obtained. The analytic proof of the estimation error convergence is proposed using the Lyapunov theory and the existence condition of the observer, in both continuous and discrete-time, are formulated using linear matrix inequalities (LMIs) [START_REF] Boyd | Linear Matrix Inequalities in System and Control Theory[END_REF].

Paper organisation. The decoupled multiple model is presented in section 2. In section 3, sufficient conditions for observer design are remember and new conditions for ensuring the exponential convergence of the estimation error are proposed. Finally, in section 4, the validity of the proposed methodology and its application to sensor faults detection and isolation is illustrated by an academic example.

Notations. P > 0 (P < 0) denotes a positive (negative) definite matrix P; X T denotes the transpose of matrix X and I is the identity matrix of appropriate dimension and diag{} is a block diagonal matrix of appropriate dimension.

ON THE DECOUPLED MULTIPLE MODEL

Modelling complex system by a rigorous mathematical model can be very arduous and even if this model exist its systematic exploitation can be delicate because this model is generally nonlinear. Multiple models offer an excellent way to avoid partially these difficulties. Definition 1. (Operator δ ). The operator δ is the time derivative for continuous-time models, i.e. δ (x(t)) = ẋ(t), and the shift operator for discrete-time models, i.e. δ (x(t)) = x(t + 1).

The structure of the decoupled multiple model used in the sequel is given by [START_REF] Orjuela | State estimation for nonlinear systems using a decoupled multiple model[END_REF]:

δ (x i (t)) = A i x i (t) + B i u(t) , ( 1a
)
y i (t) = C i x i (t) , (1b)
y(t) = L ∑ i=1 µ i (ξ (t))y i (t) , (1c)
where x i ∈ R n i and y i ∈ R p are respectively the state vector and the output of the i th submodel; y ∈ R p is the output of the multiple model. The matrices A i ∈ R n i ×n i , B i ∈ R n i ×m and C i ∈ R p×n i are known and constant. Note that the dimensions n i of each state vector x i can be different. Indeed, the submodel contribution is taken into account via a weighted sum of the submodel outputs and consequently the submodels do not share the same state space.

The weighting functions µ i (ξ (t)) are associated to each operating zone. They satisfy the following convex sum properties:

L ∑ i=1 µ i (ξ (t)) = 1 and 0 ≤ µ i (ξ (t)) ≤ 1 , ∀i = 1...L, ∀t . (2)
The weighting functions quantify the relative contribution of each submodel to the global model according to the current operating point of the system. The current operating point is taken into account by means of the so-called decision variable ξ (t) which is a characteristic variable of the system (e.g. inputs and/or measured variables). This variable is considered as an accessible signal free of faults. For simplicity, in the sequel, we shall simply write µ i (ξ (t)) = µ i (t).

Comments on the multiple model representation. The main feature of this multiple model is that the state space dimension of submodels can be different and thanks to this fact, complex systems with variable structure in the operating range can be well modelled. Indeed, the complexity of each submodel can be well adapted to the complexity of the nonlinear system in each operating zone. Thanks to this fact, the number of parameters needed for modelling the system can be reduced and the well known problem of curse of dimensionality of the model can be reduced in a good way. From a practical point of view, matrices A i , B i and C i describe the local behaviour of the system inside of the ith operating zone. Intuitively, the number of submodels is given by the granularity of the operating space decomposition. Multiple model parameters can be obtained by using the direct linearisation of an a priori nonlinear model around some operating points, or alternatively by using an identification procedure as proposed in [START_REF] Venkat | Identification of complex nonlinear processes based on fuzzy decomposition of the steady state space[END_REF][START_REF] Vinsonneau | Extended global total least square approach to multiple-model identification[END_REF][START_REF] Orjuela | Nonlinear system identification using uncoupled state multiple-model approach[END_REF].

OBSERVER DESIGN

In this section the observer design based on decoupled multiple model is investigated. New convergence conditions of the estimation error are established based on our previous works [START_REF] Orjuela | State estimation of nonlinear discrete-time systems based on the decoupled multiple model approach[END_REF][START_REF] Orjuela | State estimation for nonlinear systems using a decoupled multiple model[END_REF].

Preliminaries

For the simplicity of manipulations, the decoupled multiple model ( 1) is rewritten in the following compact form:

δ (x(t)) = Ãx(t) + Bu(t) , (3a)
y(t) = C(t)x(t) , (3b)
where:

x(t) = x T 1 (t) • • • x T i (t) • • • x T L (t) T ∈ R n , n = L ∑ i=1 n i , (4a)
à = diag {A 1 • • • A i • • • A L } , (4b)
B = B 1 T • • • B i T • • • B L T T , (4c) C(t) = [µ 1 (t)C 1 • • • µ i (t)C i • • • µ L (t)C L ] . ( 4d
)
Remark 1. The matrix C(t) can be rewritten as a weighted sum of matrices as follows:

C(t) = L ∑ i=1 µ i (t) Ci , (5)
where Ci is a constant block matrix given by:

Ci = [0 • • • C i • • • 0] (6)
such that the term C i is found on the i th block column of Ci .

Observer structure

The proposed observer based on the decoupled multiple model has the following form:

δ ( x(t)) = Ã x(t) + Bu(t) -K(y(t) -ŷ(t)) , (7a)
ŷ(t) = C(t) x(t) , (7b)
where x(t) is the state estimation and ŷ(t) the output estimation and K ∈ R n×p is the observer gain to be determined such as the exponential convergence of x(t) towards x(t) is guaranteed. Note that the exponential convergence is investigated in order to introduce dynamic performances of the observer.

Remark 2. The outputs y i (t) of the submodels are "artificial modelling signals" only used in order to provide an approximation of the output of the real system. Therefore the outputs y i (t) cannot be employed as accessible signals for driving an observer.

Convergence condition of the estimation error

Let us define the state estimation error:

e(t) = x(t) -x(t) . (8)
The estimation error dynamics is obtained by applying the operator δ to (8), and by using the equations ( 3) and (7):

δ (e(t)) = A obs (t)e(t) ,

where A obs (t) is given by:

A obs (t) = L ∑ i=1 µ i (t)φ i , (10)
φ i = Ã + K Ci . ( 11
)
Remark 3. The observer design based on each submodel cannot guarantee in a general way the convergence towards zero of the estimation error. Indeed, the matrix A obs (t) is a timevarying matrix and consequently the combination of the submodel contributions must be taken into consideration in the observer design stage. The observer gain K must be selected in order to ensure the convergence of the estimation error for an arbitrary combination of the submodel contributions and for any initial conditions. For this purpose, the Lyapunov method is employed.

Theorem 1. (Continuous-time case). Consider the decoupled multiple model (3) and the observer (7). The exponential convergence of the estimation error ( 8) is guaranteed if there exists a symmetric and positive definite matrix P, a matrix G and a positive scalar α such that:

( Ã + αI) T P + P( Ã + αI) + (G Ci ) T + G Ci < 0, i = 1...L (12)
where α is the decay rate. The observer gain is given by K = P -1 G.

Proof. See [START_REF] Orjuela | State estimation for nonlinear systems using a decoupled multiple model[END_REF]. ✷ Theorem 2. (Discrete-time case). Consider the decoupled multiple model (3) and the observer (7). The exponential convergence of the estimation error ( 8) is guaranteed if there exists a symmetric and positive definite matrix P, a matrix G and a positive scalar α such that:

(1 -2α)P ÃT P + CT i G T P Ã + G Ci P > 0, i = 1...L ( 13
)
where α is the decay rate. The observer gain is given by K = P -1 G.

Proof. See [START_REF] Orjuela | State estimation of nonlinear discrete-time systems based on the decoupled multiple model approach[END_REF]. ✷

Comments and discussion on the observability issues. The augmented matrices à and Ci have been introduced in order to establish the observer existence conditions of the observer.

The exponential convergence of the estimation error is carried out by considering a decay rate α different to zero. Note however that the choice of this decay rate can be limited by the particular form of matrices à and Ci used in theorems 1 and 2. Indeed, all columns of matrix Ci are equal to zero except the i th block column. Consequently, eigenvalues assignment of matrix A obs (t) may be limited because de pair ( Ã, Ci ) is not observable.

In order to partially avoid this problem new convergence conditions are established by introducing new augmented matrices.

New convergence conditions

In this section the main contribution of the paper are presented. New convergence conditions of the estimation error are proposed by considering the matrix A obs (t), in (9), as constant matrix with some "artificial" norm-bounded uncertainties due to the weighting functions. For this purpose, the mean value of the matrices Ci is introduced:

C0 = 1 L L ∑ i=1 Ci = 1 L [C 1 C 2 • • • C L ] . (14)
By taking into consideration

Ci = Ci + C0 -C0 , (15)
= Ci + C0 , (16)
the matrix A obs (t), defined by (10), becomes

A obs (t) = Ã + L ∑ i=1 µ i (t) K Ci , (17)
= Ã + K C0 + L ∑ i=1 µ i (t) K Ci ( 18
)
where Ci = Ci -C0 . Now, the matrix A obs (t) can then considered as constant matrix with norm-bounded uncertainties due to the weighting functions. Lemma 1. For any constant real matrices X and Y with appropriate dimensions, a matrix function F(t) bounded-norm, i.e. F T (t)F(t) ≤ I, then the following property holds for any positive matrix Q XF(t)Y +Y T F T (t)X T ≤ XQ -1 X T +Y T QY . Lemma 2. (Schur complement). The following two inequalities are equivalent:

(1)

Q S S T R > 0 where Q = Q T and R = R T (2) R > 0, Q -SR -1 S T > 0.
The convergence towards zero of the estimation error can be investigated as previously by considering the following candidate quadratic Lyapunov function:

V (t) = e T (t)Pe(t), P > 0 P = P T . ( 19
)
Continuous-time case Theorem 3. (Continuous-time case). Consider the decoupled multiple model ( 3) and the observer ( 7). The exponential convergence towards zero of the estimation error ( 8) is guaranteed if there exists symmetric and positive definite matrices P and Q, a matrix G and a positive scalar α such that:

  ∆ Ḡ CT Q ḠT -Q 0 Q C 0 -Q   < 0 ( 20
)
where

∆ = P( Ã + αI) + ( Ã + αI) T P + G C0 + (G C0 ) T , C0 = 1 L L ∑ i=1 Ci , Ḡ = [G • • • G • • • G] , C = CT 1 • • • CT i • • • CT L T , Ci = Ci -C0 .
where α is the decay rate and the observer gain is given by K = P -1 G.

Proof. The exponential convergence towards zero of the estimation error is guaranteed if the following inequality is ensured [START_REF] Boyd | Linear Matrix Inequalities in System and Control Theory[END_REF]]

∃ α > 0 : V (t) < -2αV (t) , ( 21
)
where α is the so called decay rate.

By using the Lyapunov function ( 19) and the definition of δ (e(t)), given by ( 9), then the inequality ( 21) is ensured if the following inequality holds:

PA obs (t) + A T obs (t)P + 2αP < 0 . ( 22
)
Considering the new definition (18) of A obs (t) then the inequality ( 22) becomes

P Ã + G C0 + ÃT P + (G C0 ) T + 2αP + L ∑ i=1 µ i (t)G Ci + L ∑ i=1 µ i (t)(G Ci ) T < 0 , ( 23
)
where G = P K. The previous inequality can be rewritten as follows:

P Ã + ÃT P + G C0 + (G C0 ) T + 2αP + ḠF(t) C + ( ḠF(t) C) T < 0 , (24)
where

Ḡ = [G • • • G • • • G] , (25a)
F(t) = diag {µ 1 (t)I • • • µ i (t)I • • • µ L (t)I} , (25b) C = CT 1 • • • CT i • • • CT L T . ( 25c
)
Let us notice that the matrix F(t) in ( 24) is norm-bounded (i.e. F T (t)F(t) ≤ I) because the weighting functions satisfy 0 ≤ µ i (t) ≤ 1.

With the help of lemma 1 then the inequality ( 24) is guaranteed if the following inequality holds:

P( Ã + αI) + ( Ã + αI) T P + G C0 + (G C0 ) T + ḠQ -1 ḠT + CT Q C < 0 . ( 26
)
Note that the previous inequality is not in a LMI form. However, using two times the Schur complement (see lemma 2) the inequality ( 26) becomes an LMI and this completes the proof of theorem 3. ✷

Discrete-time case

Theorem 4. (Discrete-time case). Consider the decoupled multiple model ( 3) and the observer ( 7). The exponential convergence towards zero of the estimation error ( 8) is guaranteed if there exists symmetric and positive definite matrices P and Q, a matrix G and a positive scalar α such that:

  (1 -2α)P -CT Q C (P Ã + G C0 ) T 0 P Ã + G C0 P Ḡ 0 ḠT Q   > 0 (27)
where

C0 = 1 L L ∑ i=1 Ci , Ḡ = [G • • • G • • • G] , C = CT 1 • • • CT i • • • CT L T , Ci = Ci -C0 .
where α is the decay rate and the observer gain is given by K = P -1 G.

Proof. The exponential convergence towards zero of the estimation error in the discrete-time case is investigated in a similar way. Consider the same candidate quadratic Lyapunov function as (19). The exponential convergence is guaranteed if the following inequality is ensured

∃ α > 0 : ∆V (t) < -2αV (t) , (28)
where ∆V (t) = V (t + 1) -V (t).

By using the Lyapunov function ( 19) and the estimation error dynamics given by ( 9), this inequality becomes

A T obs (t)PA obs (t) -(1 -2α)P < 0 . (29)
By applying the Schur complement the obtained inequality ( 29) becomes

(1 -2α)P A T obs (t)P PA obs (t) P > 0 .

Using the expression of A obs (t) given by ( 18), this last expression can be rewritten as follows:

(

1 -2α)P ∆ T ∆ P + XF(t)Y +Y T F(t)X T > 0 (31)
where

∆ = P Ã + G C0 , G = P K , X = 0 ḠT T , Y = C 0 . ( 32
)
and where matrices Ḡ, F(t) and C are defined in (25). The proof of theorem 4 is completed by using the lemma 1 and the Schur complement. ✷

Comments and discussion on the observability issues. Notice that in the previous theorems the matrix Ci is not directly used for decay rate selection. Indeed, the supplementary matrix C0 introduce some degrees of freedom for decay rate choice because the rank of C0 is greater than the rank of matrices Ci and the provides a better observability of the system. Indeed, the pair ( Ã, C0 ) in theorems 3 and 4 are observable.

SIMULATION EXAMPLE

Consider the continuous-time decoupled multiple model with L = 2 different dimension submodels given by:

A 1 = -2.0 0.5 0.6 -0.3 -0.9 -0.5 -1.0 0.6 -0.8 , A 2 = -0.8 -0.4 0.1 -1.0 , B 1 = [1.0 0.8 0.5] T , B 2 = [-0.5 0.8] ,
C 1 = 0.9 -0.8 -0.5 -0.4 0.6 0.7 , C 2 = -0.8 0.6 0.4 -0.7 .

Here, the decision variable ξ (t) is the input signal u(t) ∈ [0, 1]. The weighting functions are obtained from normalised Gaussian functions:

µ i (ξ (t)) = η i (ξ (t))/ L ∑ j=1 η j (ξ (t)) , (33)
η i (ξ (t)) = exp -(ξ (t) -c i ) 2 /σ 2 , (34)
with the σ = 0.5, c 1 = 0.25 and c 2 = 0.75. State estimation errors provided by these two observers are depicted in figures 1 and 2. Let us notice that the initial conditions of the multiple model are not null and the initial conditions of the observers are null. It can be seen from these figures that the observer design using theorem 3 provides a better decay rate and good dynamics performances of the estimation errors.

However a great decay rate is obtained by a high gain and this fact is at the origin of undesirable phenomenons, for example an amplification of the noise. The choice of the observer gain is subject to a trade-off between the noise amplification and the decay rate.

Application to sensor faults diagnosis

The state estimation provided by the proposed observer can be used in a classic way for fault diagnosis of nonlinear systems represented by means of a decoupled multiple model, as proposed in [START_REF] Orjuela | State estimation for nonlinear systems using a decoupled multiple model[END_REF]. For example, sensor fault detection and isolation task can be well achieved by structuring judiciously the residual signals with the aid of an observer schemes (e.g. Dedicated and/or Generalized Observer Schemes) and by using an appropriated logical decision rule [START_REF] Patton | Issues of fault diagnosis for dynamic systems[END_REF], [START_REF] Isermann | Fault-Diagnosis Systems. An Introduction from Fault Detection to Fault Tolerance[END_REF].

Here, two dedicated observers are designed by using the new conditions proposed by the theorem 3. The first dedicated observer is driven only by the first output of the system and the second one by the second output. Each dedicated observer provides an estimation of the system outputs, noted by ŷobs i (t) where i is the number of the estimated output and obs is the considered observer. Hence, signals sensitive to sensor faults acting on the system, residuals r i,obs (t), can be easily generated by considering the discrepancies between the measured outputs of the system y i (t) and their estimated ŷobs i (t). Sensor faults detection and isolation are accomplished by considering the simultaneous configuration of the obtained residual signals r i,obs (t) (see the fault signatures of different faults).

r 1,1 r 2,1 r 1,2 r 2,2 δ 1 ? ? 1 0 δ 2 0 1 ? ?
Table 1. Fault signatures of different faults

The fault signatures of different faults δ i is built as follows.

Considering that y 1 (t) is free of fault, if a sensor fault δ 2 acts on the second output of the system then the sensitive signals obtained with the help of the first observer are, in theory, r 1,1 (t) = 0 and r 2,1 (t) = 1. Note that the configuration of the residual signals generated by the second observer r i,2 (t) cannot be taken into consideration because this observer is driven by the corrupted output y 2 (t). Consequently, configuration of r i,2 (t) is represented by " ? " in the fault signatures table 1. When a sensor fault δ 1 acts on the first output then the residuals obtained with the help of the second observer are r 1,2 (t) = 1 and r 2,2 (t) = 0. In the same way, the residual signals r i,1 (t) are not taken into consideration (configuration of r i,1 (t) is represented by " ? "). A logical decision rule can be then established on the basis of the proposed configuration of the residuals r i,obs (t) summarized in table 1 (see [START_REF] Orjuela | State estimation for nonlinear systems using a decoupled multiple model[END_REF] for details). In this simple example, two constant sensor faults δ 1 and δ 2 act- ing respectively on output 1 and output 2 are considered. In the simulation sensor noise are also added on the system outputs. The time evolution of the residual signals are plotted on figure 3. Considering the evolution of the residual signals r i,2 (t). In the time-interval t = 2s and t = 3s, these residual signals are r 1,2 (t) = 1 and r 2,2 (t) = 0. Consequently, a sensor fault acting on y 1 (t) can be well detected in this time-interval. On the other hand, the residual signals r i,1 (t), in the time-interval t = 4s and t = 5s, are r 1,1 (t) = 0 and r 2,1 (t) = 1. Consequently, a sensor fault acting on y 2 (t) can be well detected in this time-interval.

CONCLUSIONS

This paper has proposed a new way for designing a proportional observer based on a decoupled multiple model. Systematic procedure, based on the LMI framework, has been established in order to design an observer which ensures the exponential convergence of the estimation error. The new proposed conditions allow to obtain a better decay rate with respect to the existing conditions. The effectiveness of the proposed approach is illustrated via a simulation example. The application of the proposed observer in a sensor fault detection and isolation is also illustrated.

The feasibility of the new proposed conditions with respect to the previous ones must be investigated in the future in order to stablish the conservatism of the proposed conditions with respect to the previous ones. The use of the proposed approach in a FDI strategy must be investigated in detail in order to ensure, for example, an attenuation level with respect to disturbances and a maximal amplification level with respect to faults.

Figure 2 .

 2 Figure 1. State estimation errors of submodel 1

Figure 3 .

 3 Figure 3. Evolution of the residual signals


Laurence Mathieu

A Keraval

J.-C Block

Jean-François Mougel

Denis Mathieu

Pierre-Olivier Jost

Julien Jost

Stéfanie Machicado

Assessment of low-power ultrasound device on biofilm formation and carbonate deposition in heat exchanger: a lab study

des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

CleanUS -EU / FEDER-FSE Lorraine et Massif des Vosges, et Région Grand Est

Bio-fouling of any material in contact with water results in a range of adverse issues for industrial engineered water systems such as decrease in water transport efficiency, heat exchange limitation, corrosion, increase maintenance costs, and environmental impact (indeed traditional cleaning strategies used sometimes unfriendly chemicals). Alternative procedures (less drastic, eco-aware, low water footprint, material integrity respect) are required especially for reducing environmental impacts. Here we assessed the ultrasound efficacy on heat exchangers, which experience problems with biofouling formation (mainly carbonate deposition, biofilm formation, and 'sludge' accumulation).

Materials

Background and objectives

In the heat exchanger submitted to ultrasounds (cooling water side): -Deposition of calcium carbonates on the plates was decreased by a factor 20 to 300 (according to the power of the ultrasound transducer) comparatively to the control without ultrasounds.

-Biomass on the plates (number of bacterial cells) was decreased by a factor 7 (whatever the power of the ultrasound transducer) comparatively to the control without ultrasounds.

  This 2016-2017 project funded by the EU (FEDER-FSE Lorraine and Massif des Vosges) and the Région Grand Est was carried out by 1 Laboratoire de Chimie Physique et Microbiologie pour l'Environnement and EPHE, LCPME, UL-CNRS, Villers-les-Nancy,F; 2 SO.GE.EAU, Gérardmer, F; 3 Ascometal, Hagondange, F; 4 Sofchem, Rueil-Malmaison, F; 5 HYDREOS, Tomblaine, F.


Camille Sleth

François Bauzou

Claudia De Cristo

Fanny Alkar

Pauline Joly-Monrigal

Clément Jeandel

Jérôme Cottalorda

Djamel Louahem M'sabah

Marion Delpont

Is there a