Marine Moussier

Valérie Guénard-Lampron

Kevin Lachin

Gabrielle Moulin

Sylvie L Turgeon

Camille Michon

Delphine Huc-Mathis

Véronique Bosc

What do stirred yogurt microgels look like? Comparison of laser diffraction, 2D dynamic image analysis and 3D reconstruction

Keywords: Size, Shape, Stirred yogurt microgels, Laser diffraction, 2D dynamic image analysis, 3D reconstruction

Highlights

Microgel size distributions were similar using LD, 2D and 3D techniques. Microgels were non-spherical, rough and heterogeneous with the 2D and 3D analyses. Estimations of the fractal dimension were more reliable using LD and 3D than 2D. LD was relevant, fast and versatile in accessing size and fractal dimension. 2D was faster and 3D more accurate in accessing both shape and fractal dimension.

Introduction

From a structural point of view, stirred yogurts are concentrated dispersions of microgels (soft particles) whose diameters range from 10 to 100 µm [START_REF] Sodini | The Relative Effect of Milk Base, Starter, and Process on Yogurt Texture: A Review[END_REF][START_REF] Van Marle | Structure and rheological properties of yogurt gels and stirred yogurt[END_REF]. In presence of fat, each microgel can itself be considered as an emulsion-filled gel, with fat dispersed as droplets that interact with the protein network via the interface located on the surface of fat globules and mainly composed of milk proteins. The term "microgels" is used for the entities obtained after the set yogurt is stirred. Structurally speaking, these microgels are "aggregates" of primary particles of fat droplets and proteins (mainly whey protein / casein micelle complexes). It is established that stirring causes profound changes in the textural and sensory properties of the yogurts by breaking the continuous gel (i.e. set yogurt) into microgels (soft particles) [START_REF] Cayot | Creaminess in relation to consistency and particle size in stirred fat-free yogurt[END_REF][START_REF] Lee | Impact of Gelation Conditions and Structural Breakdown on the Physical and Sensory Properties of Stirred Yogurts[END_REF]. [START_REF] Shewan | Review of techniques to manufacture micro-hydrogel particles for the food industry and their applications[END_REF] have also demonstrated that the properties of soft particle concentrated dispersions are directly impacted by the properties of the dispersed particles (microgels in the case of stirred yogurts): their hardness, size distribution or shape. Having access to reliable data on particle size distribution (PSD), the distributions of shape factors and average mass fractal dimension of the microgels is thus of interest to understand the textural properties of stirred yogurts.

Laser diffraction (LD) particle size analysis (or static light scattering) is commonly used to access the size distribution of stirred yogurt microgels [START_REF] Chung | Development of Reduced-calorie foods : Microparticulated whey proteins as fat mimetics in semi-solid food emulsions[END_REF][START_REF] Hahn | Post-processing of concentrated fermented milk: Influence of temperature and holding time on the formation of particle clusters[END_REF][START_REF] Huc | Design of a multi-scale texture study of yoghurts using rheology, and tribology mimicking the eating process and microstructure characterisation[END_REF][START_REF] Nöbel | Microgel particle formation in yogurt as influenced by sonication during fermentation[END_REF]. This technique measures particles ranging from 0.02 to 2,000 µm in diameter. To do so, a laser beam of known wavelength (λ = 633 nm) irradiates the suspension to be analyzed, and detectors located at specific angles collect the intensity of the light scattered by the particles. Assuming spherical particles with homogenous composition, the software then uses the Mie theory to deduce a theoretical PSD from the light scattering results obtained with LD. To successfully use the Mie theory, knowledge of the refractive and absorbance indexes of the dispersed medium is required (Malvern Instruments Ltd., 2007). In the specific case of the stirred yogurt microgels, these optical indexes are difficult to access. The question of the consistency of the LD size measurement of complex systems like stirred yogurt microgels thus naturally arises. As this technique does not enable access to shape, other techniques of image analysis can be used, compared and possibly combined with LD to obtain the most accurate results possible.

2D dynamic image analysis is a recently developed technique that enables precise access to the PSD and to the shape of different types of particles [START_REF] Carugo | Benefits of polidocanol endovenous microfoam (Varithena®) compared with physician-compounded foams[END_REF][START_REF] Mallipeddi | Use of fine particle ethylcellulose as the diluent in the production of pellets by extrusion-spheronization[END_REF][START_REF] Perez | Dynamic Image Analysis to Evaluate Subvisible Particles during Continuous Drug Infusion in a Neonatal Intensive Care Unit[END_REF]. This technique can be compared to a modern microscope using a pulsed light source and a high speed mega-pixel camera [START_REF] Köhler | Investigations on non-Spherical Reference Material Using Laser Diffraction and Dynamic Image Analysis[END_REF][START_REF] List | Dynamic Image Analysis extended to Fine and Coarse Particles[END_REF]. Unlike laser diffraction analysis, image analysis directly records the properties of the image of each particle to determine diameter and shape factors. 2D image analysis thus appears to be an appropriate tool to access data concerning the morphology of stirred yogurt microgels. Many shape factors are described in the literature for non-spherical microgels (convexity, roundness, circularity, sphericity or roughness) and the definitions of these factors depends on the equipment and analytical technique used [START_REF] Hentschel | Selection of descriptors for particle shape characterization[END_REF][START_REF] Podczek | A shape factor to assess the shape of particles using image analysis[END_REF][START_REF] Yan | Inferring 3D particle size and shape characteristics from projected 2D images: Lessons learned from ellipsoids[END_REF]. However, the roughness index was the most often used, because it is relevant regarding surface heterogeneity.

A variety of microscopic techniques are used to assess the microstructure of stirred yogurts [START_REF] Mortazavian | Application of advanced instrumental methods for yogurt analysis[END_REF]. These include transmission electron microscopy (TEM), scanning electron microscopy (SEM) and confocal laser scanning microscopy (CLSM), sometimes associated with image analysis [START_REF] Torres | Using fractal image analysis to characterize microstructure of low-fat stirred yoghurt manufactured with microparticulated whey protein[END_REF]. Both TEM and SEM are high resolution techniques, but sample preparation can be complex and quite expensive. Moreover, structure artefacts due to the sample preparation are very often suspected. CLSM is a low-invasive alternative requiring the staining of the compounds to be observed. In particular, this technique makes it possible to obtain a series of two-dimensional images (x, y) by z-stacking. Using the appropriate software, these images can be compiled and computed into a 3D representation. This technique was recently applied to food systems such as continuous model gels (whey protein isolate/polysaccharide) (van den [START_REF] Van Den Berg | Quantification of a 3D structural evolution of food composites under large deformations using microrheology[END_REF] or soft apple cells [START_REF] Leverrier | Assessment of deformability of soft plant cells by 3D imaging[END_REF]. However, to the best of our knowledge, CLSM associated with 3D reconstruction has not yet been used to assess the size and shape of the microgels of stirred yogurts.

Laser diffraction, 2D dynamic image analysis and 3D reconstruction can also be used to extract information on the structure of the samples by means of the mass fractal dimension (Df). This structural parameter is closely linked to the concept of fractal geometry (object having a structure independent of the scale of observation) and thus compactness. Fractal geometries were first mathematically introduced by [START_REF] Mandelbrot | Les objets fractals, forme, hasard et dimension[END_REF] in the mid-1970s and later used in the field of colloid and aggregates, thus opening a new way of characterizing the structure of aggregates in terms of occupancy rate and compaction of the structure in the volume of the aggregates [START_REF] Andoyo | Spatial arrangement of casein micelles and whey protein aggregate in acid gels : Insight on mechanisms[END_REF][START_REF] Mellema | Effects of structural rearrangements on the rheology of rennet-induced casein particle gels[END_REF] or roughness and sphericity of the aggregates [START_REF] Raper | Measurement of aggregate fractal dimensions using static light scattering[END_REF][START_REF] Torres | Using fractal image analysis to characterize microstructure of low-fat stirred yoghurt manufactured with microparticulated whey protein[END_REF]. It has been accepted for many years [START_REF] Forrest | Long-range correlations in smoke-particle aggregates[END_REF]) that aggregates can be described as fractal-like structures, meaning their mass scales with a characteristic radius through the use a specific dimension named the mass fractal dimension. Unlike the topological dimension, which is stricly an integer (between 1 and 3), the fractal dimension is usually a non-integer number. The use of accurate fractal dimensions thus makes it possible to replace the conventional sphericity assumptions that can be used in modelling the relationship between structural and textural properties.

As LD measurements are the most widely used in the dairy field, but have limitations, the first objective of this study was to analyze its suitability for heterogeneous (in composition) and irregularly shaped systems like microgels. This analyze of reliability was made by comparing the LD results with the ones obtained with 2D image analysis and 3D reconstruction (from confocal images). The other objective was to compare the ability of the three different techniques to provide information on the size, shape and fractal dimension of stirred yogurt microgels. For this purpose, a fat free and a high fat commercial yogurts were selected and diluted in purified water. The size distributions and the fractal dimensions of the microgels were determined using all three techniques, whereas their shape factors (length and roughness index distributions) were determined only using 2D and 3D image analyses. The different results obtained were then compared and analyzed as a function of the technique.

Materials and methods

Stirred yogurt sampling

Two types of plain stirred yogurts from different commercial brands were purchased in the market.

Perle de Lait (Yoplait, France) was chosen for its high fat (F) content and its classic protein (P) content (9.3 g/100g fat, 3.2 g/100g protein). Taillefine Le Brassé 0% (Danone, France) was selected because it is fat free and has a quite high protein content (0.1 g/100g fat, 4.5 g/100g protein). For the rest of the study, the stirred yogurt samples are referred as FP3 for Perle de Lait and P4.5 for Taillefine Le Brassé 0%. FP3 and P4.5 were chosen to have a similar aging time (based on their similar expiration dates). They were stored in the same conditions (i.e. at 4 °C). All the measurements were performed on two consecutive days. Purified water used for the dilutions was obtained using a Milli-Q purification system (Millipore, Merck, Germany). It was checked and proved that the level of dilution did not have a significant impact on the results. To achieve good sampling and homogeneity, each yogurt was gently mixed using a small spoon rotated 4 times from the bottom of the pot towards the top, with a quarter turn between each movement. For this study, three dilutions were performed from different pots of a same batch of FP3 and of P4.5.

Laser diffraction analysis

Stirred yogurts were diluted 1:10 (w/w) with purified water in a 100 mL pot and the microgels were dispersed by reversing the pot several times. Size distributions were measured by laser diffraction with a MasterSizer 2000 (Malvern Instruments, UK). To achieve a constant level of obscuration, only some drops of 1:10 diluted stirred yogurts were poured in dispersant tank for the measurement (three repetitions), resulting in a total dilution of 1:100. A refractive index of 1.33 for water and 1.46 for the microgels (refractive index of milk proteins), and an absorption index of 0.01 for the microgels were used [START_REF] Huc | Design of a multi-scale texture study of yoghurts using rheology, and tribology mimicking the eating process and microstructure characterisation[END_REF]. Several data were deduced from the PSD (Malvern Instruments Ltd., 2007): size volume distribution, particle sizes representing less than 10% (d(0.1), µm), 50% (median diameter d(0.5), µm) and 90% (d(0.9), µm) of the sample, volume (D[4,3], µm) and surface (Sauter mean diameter D [3,2], µm) weighted mean diameters (D[m,n], Eq. 1) and width of the distribution (span, Eq. 2).

[ , ] = ∑ ×

∑ ×

Eq. 1 = 0.9 -0.1 0.5 Eq. 2

In addition, it was also possible to extract the fractal dimension of the microgel aggregates from the scattering data. The light scattered by porous aggregated structures entails more modeling complexity than the scattering of solid homogenous spheres. One way to overcome this problem is to use the Rayleigh-Gans-Debye theory [START_REF] Gregory | Monitoring particle aggregation processes[END_REF][START_REF] Sorensen | Light scattering by fractal aggregates : A review[END_REF]. Assuming the primary particles that comprise the aggregate behave like Rayleigh scatterers (i.e. the diameter of the initial particles is much smaller than the wavelength of the incident beam λ), it is possible to introduce a structure factor S(q) in the expression of the light scattered intensity I(q) so that (Eq. 3):

# $ ∝ & $ * ( $ Eq. 3
where P(q) is the form factor and is due to primary particles. q (m -1 ) is the scattering vector and is expressed by Eq. 4, where θ is the scattering angle and n the refractive index of the dispersing medium.

$ = 4* + , - . 2 0
Eq. 4

As q -1 represents the characteristic length probed with the light scattering measurement, information on the aggregate structure can only reasonably be extracted for q -1 values so that r0<<q -1 <<Rag, where r0

denotes the characteristic size of the primary particles, and Rag (m) the characteristic size of the aggregates.

Under this condition, the structure factor depends on the fractal dimension, and it is thus possible to write the proportionality relation between the intensity of the light scattered and the structure factor as stated by Eq. 5.

# $ ∝ $ 1 2 Eq. 5

Using a Log-Log scale plot, it was thus possible to access the mean mass fractal dimension of the sample by simply determining the slope of the scattering plot in the above-mentionned q -1 region (see Fig. 5 in Supplementary material). This theory has been successfully applied in several studies involving colloidal suspensions, particularly latexes, well calibrated in size and shape [START_REF] Burns | A light scattering study of the fractal aggregation behavior of a model colloidal system[END_REF][START_REF] Lachin | Aggregation and breakup of acrylic latex particles inside millimetric scale reactors[END_REF][START_REF] Selomulya | Evidence of shear rate dependence on restructuring and breakup of latex aggregates[END_REF]. More closely connected with the food and dairy industries, some successes have been achieved in the light scattering study of model casein and micellar casein aggregates [START_REF] Chardot | Growth kinetics and fractal dimensions of casein particles during acidification[END_REF][START_REF] Panouillé | Aggregation and gelation of micellar casein particles[END_REF][START_REF] Vétier | Effect of Temperature and Aggregation Rate on the Fractal Dimension of Renneted Casein Aggregates[END_REF].

2D dynamic image analysis

Dynamic image analysis was performed using a QICPIC/R modular particle size and shape analyzer and a LIXELL wet dispersing unit (Sympatec GmbH, DE). A precision M4 lens measuring from 1 to 750 µm with a 0.5 mm cuvette was used. Stirred yogurts were diluted 1:2000 (w/w) with purified water in a 1000 mL beaker to disperse the microgels and the dispersed microgels were then stirred at 100 rpm for 1 min and pumped into the dispersing unit with a peristaltic pump (Masterflex L/S Model 77201-60, Cole-Parmer, FR) at a flow rate of 25 mL/min. For each dilution, two 30-second image acquisitions were performed at 10 Hz.

The images were processed using PAQXOS application software (PAQXOS, Version 2.2.2, Sympatec GmbH, DE). Size measurement data such as volume distribution, d(0.1), d (0.5), d(0.9), D[4,3] and D [3,2] were retrieved from the image analysis. The diameters of the equivalent surface circle of microgels and maximum (Fmax, µm) and minimum (Fmin, µm) Feret diameters, derived respectively using the maximum and minimum distance between two tangents of the contour of the particle, were determined by the software. The width of the distribution (span) was calculated by Eq. 2. The software was also able to determine shape factors including the roughness index (Eq. 6).

The results of the dynamic image analysis made it possible to measure fractal dimensions. Some studies have already proposed methods of calculating two-dimensional fractal dimensions (D2) from image analysis [START_REF] Jiang | Fractal Dimensions of Aggregates Determined from Steady-State Size Distributions[END_REF][START_REF] Serra | Structure of the aggregates during the process of aggregation and breakup under a shear flow[END_REF]. The two-dimensional fractal dimension was determined by the relationship between the area (A) of the microgels and their maximum Feret diameter (Fmax) (Eq. 7). In the specific case of the calculation of the two-dimensional fractal dimensions, the microgels below 10 µm in diameter were not selected due to their low image resolution (1 µm = 1 pixel). For each yogurt analysis, 7,000 microgel images were randomly selected and classified according to their roughness index. For each class of roughness, a plot Log(A) vs. Log(Fmax) was performed. A weighted average of these classes was performed to determine a representative D2 value of all measured stirred yogurts.

Using simulated aggregates, [START_REF] Lee | Prediction of three-dimensional fractal dimensions using the twodimensional properties of fractal aggregates[END_REF] found a relationship between the two-dimensional fractal dimension (D2) obtained from image analysis and the three-dimensional fractal dimension (D3) from the laser diffraction results (Eq. 8). The equation was validated by comparing experimental D3 (laser diffraction and electrical sensing) with simulated D3 on different particles, particularly spherical ones [START_REF] Baalousha | Aggregation and surface properties of iron oxide nanoparticles: influence of pH and natural organic matter[END_REF][START_REF] Lee | Prediction of three-dimensional fractal dimensions using the twodimensional properties of fractal aggregates[END_REF].

3D reconstruction from confocal images

Acquisition by confocal microscopy and 3D processing

The stirred yogurts were first diluted 1:100 (w/w) with purified water in a 100 mL pot, and the microgels were then gently dispersed by reversing the pot several times. The proteins that made up the microgels were then stained by mixing 250 µL of this solution with 2.5 µL of DyLight 488 nm (Thermo Fisher Scientific, Waltham, MA, USA) (one repetition per dilution). Confocal images were acquired with a TCS SP8 AOBS inversed confocal laser scanning microscope (CLSM) (Leica, Solms, Germany) equipped with a Helium-Neon laser (458 nm excitation wavelength) and an Argon laser (633 nm excitation wavelength). From 93 to 195 images (x,y) were acquired by z-scan (0.8 µm steps) with a magnification ×40.

For each sample, the z-stacks obtained were combined and processed to reconstitute the 3D microgels using Scan IP™ software (version 7.0, build 2656, © 2000-2014 Simpleware Ltd.). The different processing steps are based on the work of [START_REF] Leverrier | Assessment of deformability of soft plant cells by 3D imaging[END_REF] and are illustrated in Fig. 1. The 2D confocal images (x, y) of 3 4ℎ 2 = 3, 7 3 8 $ , 7 9,39 3 3, 7 3 Eq. 6

: ∝ ; < 1 = Eq. 7 = 1.391 + 0.01 @. AB1 = Eq. 8 each z-series were first combined into a 3D reconstruction. A median filter was then applied to the background of the images (neighborhood radius of 1 × 1 × 1 pixel) to eliminate noise. By comparison with the initial confocal images, a threshold was eventually chosen to select the level of grey that differentiated the stained microgels from the background. The 3D reconstitutions shown here were chosen as being representative of the replications.

Fig. 1 Processing steps used to reconstitute the 3D microgels of stirred yogurt: 1) z-acquisition of 2D confocal images (x, y) (proteins in green); 2) 3D reconstruction and application of a median filter (a → b); 3) 3D identification of the microgels.

Data computation from 3D reconstruction

From the 3D reconstructions, Scan IP TM software provided several data on both size and shape, some of which were either recovered or processed in this study. First, the software provided the number of individual microgels identified in the 3D reconstruction and their corresponding volume (µm 3 ). In order to obtain the size distribution of the microgels in equivalent sphere, their volumes were discretized (logarithmic scale). To ensure good quality discretization, at least eight classes were required to plot each distribution (i.e.

with volume fractions greater than 0%), with a minimum of two classes per decade. Like with laser diffraction, d(0.1) (µm), d(0.5) (median diameter, µm), d(0.9) (µm), D [3,2], D [4,3] and span were retrieved from the reconstituted size distribution. From the volume of each microgel, their equivalent sphere diameter (Eq. 9) then the surface of their equivalent sphere (Eq. 10) were calculated. A roughness index of the microgels was calculated by dividing the surface of an equivalent sphere by the real surface (µm 2 ) given by the software (Eq. 11). For each microgel, the volumes of the oriented bounding ellipsoid and the corresponding minor, medial and major lengths (µm) were also obtained using the same software. The distributions of the roughness index, minor length and major length were plotted by discretizing the data (using the volume of equivalent sphere). It was also possible to use the data provided by the 3D processing to estimate the mean fractal dimension of the sample concerned. By definition of the fractal scaling, the mass of a fractal aggregate mag (kg) composed of initial particles of radius r0 (m) and mass m0 (kg) can be linked to the characteristic cluster size Rag so that [START_REF] Bushell | On techniques for the measurement of the mass fractal dimension of aggregates[END_REF][START_REF] Gregory | Monitoring particle aggregation processes[END_REF][START_REF] Lazzari | Fractal-like structures in colloid science[END_REF]) (Eq. 12):

J = KL M = N M . - O KL 3 M 0 1 2
Eq. 12 where np stands for the original number of particles in the aggregate. The radius of gyration is often taken as the characteristic aggregate size. However, as mentioned by [START_REF] Lazzari | Fractal-like structures in colloid science[END_REF], any characteristic length of the aggregate can be used instead. The shape of the relation remains identical, but the effective value of k0

changes. The effective density ρe of a fractal aggregate (taking its porosity into account) is proportional to Rag as presented by Eq. 13 [START_REF] Gregory | Monitoring particle aggregation processes[END_REF]:

P Q ∝ O KL 1 2
Eq. 13

The 3D reconstruction did not allow the determination of the mass of each single aggregate. However, it provided values for the volume of each aggregate Vag (m 3 ) and its surface envelope Sag (m 2 ). In this study, it was chosen to use the ratio Vag/ Sag as the characteristic length of the aggregates. By combining the two last mentioned relations (Eq. 12 and Eq. 13), it was then possible to find a proportionality relation between the volume of the aggregates and its volume-over-surface ratio so that (Eq. 14):

R KL ∝ S R KL & KL T 1 2 1 2 U
Eq. 14 Thus, by plotting Log(Vag) vs. Log(Vag/Sag) (see Fig. 5 in Supplementary material) for all the stirred yogurt microgels, and extracting the slope of the linear correlation, it was possible to estimate the average mass fractal dimension of the microgels.

Statistical analysis

Statistical analyses were performed using XLSTAT 2015.1 software (Addinsoft, Paris, France).

Analysis of variance (ANOVA) was used to evaluate differences between values using Tuckey's test. A significance level of p < 0.05 was used.

Results and discussion

The size distribution, shape and fractal dimension of the stirred yogurt microgels were measured using the three techniques (laser diffraction and/or 2D image analysis and 3D reconstruction) and are reported in the following tables and figures in order to evaluate the suitability, advantages and limitations of the three techniques. The two stirred yogurts (FP3 and P4.5) are rarely compared since they are intentionally chosen as being different to compare techniques in two systems representative of the variety of stirred yogurt microstructures.

Comparison of microgel size distributions (LD, 2D, 3D)

Fig. 2 shows the size distributions obtained using the three measurement techniques and, below, some data that are characteristic of these distributions. For both FP3 (Fig. 2 A) and P4.5 stirred yogurts (Fig. 2 B), the distributions obtained by laser diffraction, 2D image analysis and 3D reconstruction were all unimodal and rather overlapped for a given stirred yogurt. The FP3 microgels were smaller than those of the P4.5 stirred yogurt, with a median size between 10 and 16 µm for FP3 and between 17 and 24 µm for P4.5. These results mainly indicate that the three techniques are consistent. Moreover, the orders of magnitude of the obtained sizes are in accordance with measurements made by some authors who used laser diffraction or CLSM for different stirred yogurts [START_REF] Cayot | Creaminess in relation to consistency and particle size in stirred fat-free yogurt[END_REF][START_REF] Hahn | Adjusting rheological properties of concentrated microgel suspensions by particle size distribution[END_REF][START_REF] Huc | Design of a multi-scale texture study of yoghurts using rheology, and tribology mimicking the eating process and microstructure characterisation[END_REF]. The differences between the two stirred yogurts (Fig. 2 A andB) were certainly mainly due to their composition and their stirring process, which are known to have the most impact on microgel size [START_REF] Mokoonlall | Post-processing of fermented milk to stirred products: Reviewing the effects on gel structure[END_REF]; van Marle, van den Ende, de Kruif, & Mellema, 1999).

Fig. 2 Size distributions obtained using laser diffraction (dotted lines), 2D dynamic image analysis (dashed lines) and 3D reconstruction (solid line). The tables give the diameters and descriptive parameters corresponding to the different size distributions. The table on the left shows data for the FP3 microgels (A) and the table on the right shows data for the P4.5 microgels (B). Values with different letters in the same row differ significantly at p < 0.05.

Although unimodal and in the same size ranges (similar order of magnitude), the distributions obtained also showed some differences depending on the measurement technique used, mainly for bigger sizes. For the FP3 stirred yogurt (Fig. 2 A), the LD measurement displayed the broadest distribution resulting in a significantly higher span. The size distributions obtained from 2D image analysis and 3D reconstruction had similar spans, but the 2D sizes were significantly bigger (d(0.5), d(0.9), D [4,3]). For the P4.5 stirred yogurt (Fig. 2 B), the 3D distribution differed from that of the LD and 2D distributions, in particular by being significantly narrower (smaller span) and by displaying fewer big microgels (smaller d(0.9) and D [4,3]). Several authors also reported that the size distributions differed with the technique used when the particles were non-spherical particles. [START_REF] Yu | Evaluation of dynamic image analysis for characterizing pharmaceutical excipient particles[END_REF] showed that the LD size distributions of elongated microcrystalline cellulose particles (150-250 µm) were wider than their 2D distributions measured by dynamic image analysis. [START_REF] Califice | Influence of particle shape on size distribution measurements by 3D and 2D image analyses and laser diffraction[END_REF] demonstrated that 2D dynamic image analysis tended to overestimate/underestimate the size of non-spherical particles (50-500 µm elongated metallic particles) compared to 3D reconstruction values obtained from X-ray microtomography images. The literature explained the differences in size distributions by both the measurement technique and the method of calculation used [START_REF] Califice | Influence of particle shape on size distribution measurements by 3D and 2D image analyses and laser diffraction[END_REF][START_REF] Köhler | Investigations on non-Spherical Reference Material Using Laser Diffraction and Dynamic Image Analysis[END_REF][START_REF] Tinke | Particle shape and orientation in laser diffraction and static image analysis size distribution analysis of micrometer sized rectangular particles[END_REF][START_REF] Yu | Evaluation of dynamic image analysis for characterizing pharmaceutical excipient particles[END_REF]. In the present study, LD hypothesized that the particles analyzed were spherical. The calculation of the 2D diameter corresponded to the diameter of a circle of equal projection area (EQPC) and depended on the orientation of the microgel when measured. With 3D reconstruction, the measurement was protein-specific (CLSM staining) and the calculated diameter corresponded to the diameter of the equivalent sphere in volume (and did not depend on the orientation of the microgel). All these differences between the techniques likely explain the slight discrepancies shown in Fig. 2 for each of the stirred yogurts and suggest their microgels were not spherical. Further analysis of the microgel shape was thus performed to better understand the differences in size distribution, to compare the techniques and to characterize the stirred yogurt microgels more precisely.

Comparison of the shape of the microgels (2D, 3D)

2D image analysis and 3D reconstruction were both used to determine the microgel shape. Fig. 3 A illustrates how the characteristic lengths were obtained from 2D (Fmax and Fmin) and 3D (major and minor) analyses. Fig. 3 B(a) and C(a) below show the distributions of the different lengths for FP3 and P4.5 stirred yogurts, respectively. These length distributions are classically used to provide information about the shape (spherical or elongated) of the particles [START_REF] Califice | Influence of particle shape on size distribution measurements by 3D and 2D image analyses and laser diffraction[END_REF][START_REF] Yu | Evaluation of dynamic image analysis for characterizing pharmaceutical excipient particles[END_REF]. When microgels are spherical, the maximum length is obviously the same as the minimum length [START_REF] Yu | Evaluation of dynamic image analysis for characterizing pharmaceutical excipient particles[END_REF]. Here, minor and Fmin length distributions were smaller than major and Fmax distributions for FP3 and P4.5, indicating that stirred yogurt microgels are not spherical, as previously suspected based on differences in size distributions obtained with the LD, 2D dynamic image analysis and 3D reconstruction. These results are in agreement with the fresh cheese microgels observed by [START_REF] Hahn | Control of microgel particle growth in fresh cheese (concentrated fermented milk) with an exopolysaccharide-producing starter culture[END_REF] using CLSM, which were also irregular in shape. In addition, the P4.5 length distributions obtained from the 2D image analysis were broader than those obtained from 3D reconstruction. The differences between the 3D lengths (i.e. between minor and major) were more important than the differences between the 2D lengths (i.e. between Fmin and Fmax). These results reveal some differences between the 2D and 3D distributions that can mainly be explained by the way the lengths were obtained with each technique (Fig. 3 A). From the 2D image analysis, Fmin and Fmax lengths could be biased by the orientation of microgels when measured (orientated lengthwise due to the flow). A similar concern has been expressed for irregular concrete aggregates [START_REF] Cepuritis | Comparison of 2-D and 3-D shape analysis of concrete aggregate fines from VSI crushing[END_REF]. With 3D reconstruction, the microgel may not be in direct contact with the ellipsoid edge to encompass the entire microgel (in length, width and thickness) (Fig. 3 (A)). This technique may therefore overestimate the minor and major lengths. Based on X-ray microcomputed tomography, [START_REF] Cepuritis | Comparison of 2-D and 3-D shape analysis of concrete aggregate fines from VSI crushing[END_REF] reported that 3D minor and major lengths depended on the dimension of the rectangular box enclosing the particle. In addition, in the present study, there were more differences between the two techniques for the P4.5 stirred yogurt. This result showed that P4.5 stirred yogurt microgels are more heterogeneous in shape (with more different types of elongation) than FP3 ones.

Images (b) and (c) in Fig. 3 B andC, show the stirred yogurt microgels obtained using 3D reconstruction (from the z-stack confocal images) and 2D images analysis, respectively, confirming that the microgels were very heterogeneous in size and shape. This is in agreement with the results of [START_REF] Hahn | Adjusting rheological properties of concentrated microgel suspensions by particle size distribution[END_REF], who observed CLSM images of fresh cheese under different processing conditions. 2D images of FP3 and P4.5 stirred yogurts also showed different degrees of microgel compactness (Fig. 3 (c)). For example, the enlarged #1 microgels obtained from the screenshots (2D image analysis) appear to be more compact than the #2 ones (Fig. 3 B (c) and Fig. 3 C (c)). Fig. 4 presents the roughness index distributions obtained for FP3 (A) and P4.5 (B) stirred yogurts.

The roughness index value ranges from 0 to 1 and describes the surface unevenness on the microgels. The index tends towards 1 for microgels with no unevenness (i.e. a smooth circle (2D) or sphere (3D)).

For each technique considered independently, the roughness distributions of the two stirred yogurts were globally similar, even if that of P4.5 was slightly broader. The differences in the yogurt compositions and stirring processes could explain this slight difference in roughness distributions. However, there were bigger differences between the 2D and 3D roughness distributions. Using 3D reconstruction, the distributions were narrow and unimodal, with a median roughness of 0.8, whereas using 2D images analysis, they displayed a main peak with a shoulder, with a first peak at 0.6 and a second one at 0.8-0.9. The 2D distributions were also broader (from 0.2-0.3 to 1) than 3D ones (0.4-0.5 to 1). These results indicate that the stirred yogurt microgels appears less uniform in roughness with 2D images analysis. The difference could be explained by the processing steps used to reconstitute the 3D microgels. The application of a median filter and the selection of a threshold (subsection 2.4.1 and Fig. 1) could smooth the microgel surfaces (i.e. the boundary between the background and the microgels) and therefore underestimate the width of the roughness distributions.

Although these microgels tended towards a smooth surface (roughness mostly between 0.7 and 0.8), the range of widths of the distribution underlined the heterogeneity of the stirred yogurt microgels that can be linked to microgel size. Some studies already linked the shape of the particles such as the roughness index, to their size [START_REF] Yan | Evolution of grain grading and characteristics in repeatedly reconstituted assemblages subject to one-dimensional compression[END_REF][START_REF] Zhou | Generation of a realistic 3D sand assembly using X-ray micro-computed tomography and spherical harmonic-based principal component analysis[END_REF]. In the present study, the roughness index decreased (i.e. surface unevenness was greater) in bigger microgels (data not shown). Rougher microgels are probably due to the bigger size (> 30 µm) of microgels that were mostly measured using 2D analysis rather than 3D reconstruction (subsection 3.1).

Fig. 4 Weighted average curves (three repetitions) of the roughness index distributions obtained from 2D image analysis (dashed lines) and 3D reconstruction (solid lines) for FP3 (A) and P4.5 (B) stirred yogurts.

Comparison of the microgel fractal dimension (LD, 2D, 3D)

All three techniques were used to estimate the microgel fractal dimension of the two stirred yogurts (FP3 and P4.5). For the LD measurements, the slopes were extracted with very high regression coefficients (higher than 0.99). For 3D reconstruction, the linear regression also proved to be very high, with values systematically higher than 0.96. With 2D dynamic image analysis, the two-dimensional fractal dimensions (D2) were also extracted with very high regression coefficients, i.e. higher than 0.91. These high values indicated excellent fitting, thus allowing high confidence in the results obtained using these techniques (Table 1). This regime is called "reaction limited aggregation" (RLA) and leads to denser aggregates, with Df around 2.1. In the present study, the volume fractions of milk proteins (before any dilution) were higher than 1% in both stirred yogurts (FP3 and P4.5), which explains why the values obtained were significantly higher [START_REF] Bremer | Theoretical and experimental study of the fractal nature of the structure of casein gels[END_REF]. In addition, the colloidal calcium phosphate, which ensures the structure integrity of the casein micelles, dissolves during acidification. This dissolution results in the loosening of the micelles (increasing their volume), which likely promotes the compaction of the protein aggregates due to loss of repulsive interactions and thus leads to denser structures [START_REF] Andoyo | Spatial arrangement of casein micelles and whey protein aggregate in acid gels : Insight on mechanisms[END_REF].

The Df value calculated from 3D reconstruction could be considered as the most accurate of the three techniques, because it relies on direct visualization of the aggregates and assumes no strong assumption.

However, the LD technique proved to be a very good alternative technique to obtain Df as the differences between LD and 3D were very small. However, the values obtained using 2D image analysis and the equation proposed by [START_REF] Lee | Prediction of three-dimensional fractal dimensions using the twodimensional properties of fractal aggregates[END_REF] differed more from 3D measurements. Estimating Df from 2D image analysis using this equation thus appears to be questionable in the case of stirred yogurt microgels. [START_REF] Lee | Prediction of three-dimensional fractal dimensions using the twodimensional properties of fractal aggregates[END_REF] reported underestimation of Df in the case of E. coli aggregates and explained that it was partly because E. coli were not spherical, which could also be the case of the stirred yogurt microgels.

The 3D and LD techniques are thus recommended over 2D analysis.

Comparison of the advantages and limitations of LD, 2D and 3D

To complete the comparison of the performances of the three techniques, Table 2 summarizes the size distribution range, the measurement conditions, the time needed for measurement and data treatment per sample, the properties obtained directly or calculated from the data as well as the assumptions and weaknesses.

While LD, 2D image analysis and 3D reconstruction proved to be quite consistent in characterizing stirred yogurt microgels, Table 2 shows that they each had their advantages and limitations. The LD technique mainly assumes that the analyzed particles are homogeneous and spherical, which has been shown (subsection 3.1) to lead to overestimation of the bigger particles and/or underestimation of the smaller particles when measuring the microgel sizes of the stirred yogurts. Moreover, this technique requires refractive and adsorption indexes, which can be difficult to estimate for complex systems composed of different ingredients. However, in the case of the stirred yogurt microgels, these indexes were not problematic since no variation in the size distribution was observed when their values varied (due to the sufficiently large size of the microgels). Although LD obviously does not allow access to shape factors, it is quick and user-friendly for accessing the size distribution and the fractal dimension. It also makes it possible to measure particles less than a micron in size, which is not the case of the 2D and 3D techniques presented here (limited by their optical geometry characteristic). As mentioned above, 2D dynamic image analysis cannot reasonably measure sizes smaller than 1 µm, and this needs to be taken into account when studying food structures that can be below this threshold (colloidal systems, for example). The first limitation is that although the mass fractal dimension can be estimated using a specific relation reported in the literature, (subsection 3.3) its use was shown to be questionable in the case of the stirred yogurt microgels. Moreover, the time required to process the data is quite long, and this technique analyzes the projected areas (2D) of the measured particles, which may depend on their orientation during measurement. On the other hand, it has the advantage of allowing a very large number of particles to be analyzed, which should offsets the orientation bias. Moreover, it enables relatively rapid measurement and direct access to the size and shape properties. It provided a 2D view of the particles that revealed that the microgels were not spherical, but showed varying degrees of roughness, and were sometimes porous (fractal dimension) in stirred yogurt.

The smallest size that can be measured with 3D reconstruction depends on the resolution of the microscope and may be high (i.e. allowing to observe small sizes) in food structure analysis. Data acquisition is time consuming and the analysis of the properties of size, shape and fractal dimension requires complete data processing. Moreover, a threshold has to be chosen to select pixels of interest (identification of the stained microgels). The results showed in subsections 3.1, 3.3 and 3.3 demonstrated that the choice made for this study was appropriate in the case of the stirred yogurt microgels studied here. One of the advantages of the 3D technique (using CLSM) is the limited shear undergone by the particles. This is particularly relevant for the study of brittle systems such as stirred yogurt microgels and most food matrices. Based on molecule staining, it also allows the selection of specific compounds within the particles and tailored measurement of the structure. The main strength of 3D reconstruction is that it enables full visualization of the particles, with no orientation bias or sphericity assumption. This specificity was particularly useful in the present study since it offered the opportunity to clearly observe the diverse sizes and shapes of the yogurt microgels.

Conclusions

Laser diffraction, 2D dynamic image analysis and 3D reconstruction were shown to be relevant and complementary for the characterization of the size (through PSD), shape and fractal dimension of heterogeneous (in composition) and irregularly shaped systems like stirred yogurt microgels. By comparing LD with 2D image analysis and 3D reconstruction on two different stirred yogurts, we showed that LD was fully relevant to access the size distribution and the mean mass fractal dimension of non-spherical yogurt microgels. The use of 2D dynamic image analysis and 3D reconstruction also raised the question of the characterization of the shape of the stirred yogurt microgels. While rarely used for food systems, 2D dynamic image analysis proved to be advantageous to visualize the microgels and quickly estimate their morphological parameters. 3D reconstruction also has very useful features as it enables access to shape factors while avoiding the possible bias resulting from particle orientation using 2D analysis. However, the 3D technique usually entails time consuming sample preparation and analysis, and is thus not really to be recommended for routine analysis. This comparison of the three techniques provides useful guidelines for studying complex food systems. Moreover, these techniques can offer new perspectives to accurately explain the relationship between the microstructure and the macro-scale properties (such as flow properties) of a food system at each step of its processing chain.

Fig. 3

 3 Fig. 3 Details of microgel lengths (major, Fmax, minor, Fmin) (A) studied for FP3 (B) and P4.5 (C) stirred yogurts through: (a) the length distributions (average curves) obtained from 2D image analysis (dashed lines) and 3D reconstruction (solid lines), (b) the 3D reconstructions and (c) a screenshot of the movie processed throughout 2D image analysis.

Table 1

 1 Average fractal dimensions obtained from laser diffraction, 2D image analysis and 3D reconstruction. Values with different letters in the same column differ significantly at p < 0.05.The average of the mass fractal dimensions (Df) obtained from LD and 3D reconstruction were similar even if significantly different, with values around 2.3 for the two samples. These values are in good agreement with values reported in the literature for fermented stirred milk gels[START_REF] Van Marle | Steady-shear viscosity of stirred yogurts with varying ropiness[END_REF]. In conventional studies on Brownian aggregation of particles (generally latex suspensions), the obtained Df values are discussed in the frame of two limiting regimes. Such studies are conducted under low volume concentrations during aggregation (typically 10 -3 -10 -4 %), ensuring the validity of the theory. When there is no energy barrier between the colliding particles, each collision leads to aggregation. This regime is called "diffusion limited aggregation" (DLA) and results in loose open structures with Df around 1.7 -1.8. When the repulsion forces are still significant, the particles can penetrate the aggregate structure before adhering.

	Technique	FP3	P4.5
	Laser diffraction	2.31 ± 0.01 a	2.37 ± 0.03 a
	2D image analysis	2.05 ± 0.02 c	2.08 ± 0.02 c
	3D reconstruction	2.26 ± 0.01 b	2.27 ± 0.03 b

Table 2

 2 Comparison of laser diffraction, 2D image analysis and 3D reconstruction. Information in bold indicates the advantages of each technique.

	Technique	Laser diffraction	2D image analysis	3D reconstruction
	Equipment	MasterSizer 2000 (Malvern)	QICPIC/R and LIXELL (Sympatec)	CLSM (Leica) and Scan IP TM (Simpleware)
	Size range	0.02 to 2,000 µm	1 to 750 µm (M4 lens)	0.532 µm (i.e. pixel) to a few millimeters
	Measurement conditions	Dilution 1:100, Agitation, Pumping	Dilution 1:2000, Agitation, Pumping	Dilution 1:100, Staining
	Measuring time per sample	10 min	30 sec	30 min
	Time needed for data treatment per sample	10 min	30 min	1 h
	Properties obtained directly	Size	Size, Shape factors, projection (2D) Visualization of microgel	-
				Size, Shape factors, Fractal
	Calculated properties	Fractal dimension	Fractal dimension	dimension, realistic visualization of the microgels
				(3D)
	Assumptions and limitations	(i) Particles considered as homogeneous and spherical (ii) Need for refractive and absorption indexes (iii) No access to particle shape	(i) Data based on projected areas of the particles (depending on their orientation) (ii) Low camera resolution (iii) Need for low concentrations of particles	(i) Threshold to select pixels of interest (identification of the stained particles) (ii) Small number of particles (iii) Time consuming data processing

Acknowledgements

This work was funded by a single interministerial fund (France) through a collaborative R&D project certified by Vitagora and Valorial competitiveness clusters as well as by the research programs of the Fonds de recherche du Québec -Nature et technologies, Novalait Inc., the Ministère de l'Agriculture, des Pêcheries et de l'Alimentation du Québec (MAPAQ), and the Fondation Famille-Choquette. The authors would like to thank Philippe Violle, sales manager of Sympatec France, for his help and advice with the dynamic image analysis processing. The authors also thank Paul Menut and Artemio Plana-Fattori for sharing their experience and contributing to the development and improvement of this paper.

Supplementary material

Fig. 5 Graph of principle used to obtain the fractal dimension from the laser diffraction data (A) and the 3D reconstruction data (B) where q (m -1 ) is the scattering vector, I(q) is the light scattered intensity, Vag (m 3 ) is the volume and Sag (m 2 ) the surface envelope of each aggregate.


Nicolas Ellero

David Gucik-Derigny

David Henry

An unknown input interval observer for LPV systems under L2-gain and L∞-gain criteria

Keywords: Interval observer, monotone systems, unknown input observer, LPV systems, H∞ performance

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

Introduction

Several set membership/interval observer-based solutions have been proposed in the literature to solve the so-called "guaranteed" state estimation problem for Linear Parameter Varying (LPV)/quasi-LPV systems [START_REF] Efimov | Interval estimation for LPV systems applying high order sliding mode techniques[END_REF]; [START_REF] Chebotarev | Interval observers for continuous-time LPV systems with L 1 /L 2 performance[END_REF]; [START_REF] Wang | Interval observer design for LPV systems with parametric uncertainty[END_REF].The design of an interval observer is generally based on the monotone system theory [START_REF] Farina | Positive linear systems: theory and applications[END_REF] to get the cooperativity property of the observation error dynamics. The difficulty to obtain this property was relaxed for a large class of systems using a time invariant state transformation [START_REF] Raissi | Interval state estimation for a class of nonlinear systems[END_REF] and a time varying state transformation [START_REF] Mazenc | Interval observers for linear time-invariant systems with disturbances[END_REF]; [START_REF] Thabet | An effective method to interval observer design for time-varying systems[END_REF]. Another approach consists in applying a transformation to put the system into a positive form before designing an interval observer [START_REF] Cacace | A new approach to design interval observers for linear systems[END_REF]. However, it has been proposed only for Linear Time Invariant (LTI) and Linear Time Variant (LTV) systems. A last approach has been proposed based on Müller's theorem and interval analysis for nonlinear system [START_REF] Kieffer | Guaranteed nonlinear state estimation for continuous-time dynamical models from discrete-time measurements[END_REF]; [START_REF] Meslem | Interval observer design based on nonlinear hybridization and practical stability analysis[END_REF]. This approach also exploits the monotone property. The main limitation of the previous mentionned ap-Email addresses: nicolas.ellero@ims-bordeaux.fr (Nicolas Ellero), david.gucik-derigny@ims-bordeaux.fr (David Gucik-Derigny), david.henry@ims-bordeaux.fr (David Henry).

proaches is their ability to simultaneously satisfy the conditions of the existence of the interval observer (especially cooperativity property) and a certain level of a priori given performance like the interval length. The work reported in [START_REF] Chebotarev | Interval observers for continuous-time LPV systems with L 1 /L 2 performance[END_REF] aims at proposing a solution to this problem within the L 1 /L 2 framework. Unfortunately, the problem of considering the Meztler property as a part of the design process, is not addressed. Furthermore, it should be pointed out that the proposed approach is based on a LTI interval observer (to deal with LPV systems) and this leads, obviously, to conservative results. The work introduced in [START_REF] Briat | Interval peak-topeak observers for continuous-and discrete-time systems with persistent inputs and delays[END_REF] proposes an interval observer design method based on L ∞ performance criterion, but it is limited to LTI systems. The contribution of this paper must be thought in this context. The goal is to develop an interval observer for a general class of LPV systems, under unknown input decoupling, L 2 -gain and L ∞ -gain criteria (and then a mixed L 2 /L ∞ -gain criterion), considering the cooperativity property of the observation error dynamics as a part of the design process. To this end, a new structure of an interval observer is proposed. The major advantages with regards to existing solutions is that it does not require the explicit knowledge of a (time-varying) state transformation, for the estimation error dynamics to satisfy the Meztler property. interval communities: R and R + := [0, ∞) ⊂ R are the set of real numbers and the set of nonnegative real numbers, respectively. R n is a n-dimensional real space and R n×m is the set of real n × m matrices. I, 0 denote respectively the identity and the null matrices of appropriate dimensions. For a matrix A ∈ R n×m , ≤, ≥, <, > refer to component-wise. Denote

A + = max(A, 0), A -= A + -A, |A| = A + + A -and A, A = A ∈ R n×m | A ≤ A ≤ A .
The same notations are adopted for vectors. For a vector v ∈ R n , the no-

tation v * refers to v * = (v + ) T (v -) T (v + ) T (v -) T T , ṽ = v -v, v = v -+ v + . For a matrix A, A 0 (A ≺ 0)
defines a strictly positive (negative) definite matrix. A † refers to the Moore-Penrose inverse of a full column matrix rank A and He{A} = A + A T . In large symmetric matrix expressions, terms denoted refer to terms induced by symmetry.

Preliminaries

This section is devoted to definitions and lemmas that will be later used in the paper.

Definition 1 [START_REF] Farina | Positive linear systems: theory and applications[END_REF] (Theorem 2 p.14). A square matrix is said to be Metzler if all its off-diagonal elements are non-negative.

Definition 2 [START_REF] Poole | A survey on Mmatrices[END_REF] (Theorem 2.1 p.420). Let a square matrix W = [w i,j ] satisfying w ij ≤ 0, ∀i = j and w ij > 0 ∀i = j then W is said to be a M-matrix if W -1 exists and W -1 ≥ 0.

The following lemmas provide useful results for Metzler matrices and cooperative systems.

Lemma 3 Consider a (known) parameter-dependent matrix

L(ρ) ∈ R n×n where ρ ∈ Ψ ⊂ R s , an invertible M-matrix W ∈ R n×n and a real scalar µ ≥ 0. Then L(ρ) is a Metzler matrix if W L(ρ) + µW ≥ 0, ∀ρ ∈ Ψ. PROOF. Since W is a M-matrix then W -1 ≥ 0. Mul- tiplying W L(ρ) + µW by W -1 leads to L(ρ) + µI n ≥ 0, which terminates the proof.
.

Lemma 4 Farina & Rinaldi (2000) (Theorem 2 p.14).
Let a non-autonomous system given by ė ||Σ ρ u|| ∞ defines the L ∞ -gain.

(t) = Ae(t)+b(t) where e ∈ R n , the matrix A ∈ R n×n is Metzler and b(t) ∈ R n ≥ 0, ∀t ≥ 0 . If e(0) ≥ 0 then e(t) ≥ 0, ∀t ≥ 0. Such systems are called cooperative. Definition 5 (L 2 -norm, L ∞ -norm) Let u : R + → R n .The L 2 -norm of u (often called simply the 2-norm of u(t)) is defined as u 2 := ∞ 0 u(t) T u(t)dt 1/2 . The L ∞ -norm of u is defined as u ∞ := sup t≥0 u(t) T u(t) 1/2 . The shorthand u ∈ L 2 (u ∈ L ∞ ) where L 2 (L ∞ ) refers to the space of signals mapping R + to R n with finite L 2 -norm (L ∞ -norm), is equivalent to ||u|| 2 < ∞ (||u|| ∞ < ∞).

Problem statement

Consider the following class of LPV systems, driven by unknown inputs w(t) ∈ R m and d(t

) ∈ R q ẋ(t) = A(θ(t))x(t) + B(θ(t))w(t) + E(θ(t))d(t) y(t) = C(θ(t))x(t) + D(θ(t))w(t) (1a) (1b)
Here, x(t) ∈ R n and y(t) ∈ R p denote state and measured output vectors, respectively. Both d(t) and w(t) are unknown input signals. However, w(t) is assumed to be bounded with known bounds w(t), w(t), whereas there is no assumption on d(t). The scheduling parameter vector θ(t) ∈ Θ ⊂ R l and its time derivative θ(t) ∈ Γ θ are assumed to be bounded with a priori known values.

Assumption 7 It is assumed that θ m (t) = [θ 1 (t) θ 2 (t) • • • θ s (t)] ∈ Θ m , 0 ≤ s ≤ l, is measured through ρ(t) = [ρ 1 (t) ρ 2 (t) • • • ρ s (t)] ∈ Ψ where ρ i (t) = λ i (t)θ i (t) + δ i (t) with λ i = 0,δ i are elements of λ ∈ ∆ λ ⊂ R s , δ ∈ ∆ δ ⊂ R s . Each vector λ, δ are continuously differentiable such that λ ∈ Γ λ ⊂ R s , δ ∈ Γ δ ⊂ R s .
It is also assumed that ρ and ρ evolve in hyper-rectangles so that

V = (α 1 , α 2 , • • • , α s ) | α i ∈ ρ i , ρ i (2) R = (β 1 , β 2 , • • • , β s ) | β i ∈ ρi , ρi (3)
Here, ρ i , ρ i , ρi , ρi are the (known) extremal values. The domain of ρ is denoted Γ ρ .

In order to improve the readability of the following developments, the time dependency of signals and parameters is omitted from now on. The following assumption is made:

Assumption 8 E(θ) is assumed to be of full column rank and rank(C(θ)E(θ)) = rank(E(θ)) = q, ∀θ ∈ Θ with q ≤ p.

Note that assumption 8 can be done without loss of generality since, in (1a), d refers to the maxiumum unknwon inputs the interval unknown input observer can decouple. That's why it is a classical assumption in Unknown Input Observer (UIO) theory, [START_REF] Darouach | Fullorder observers for linear systems with unknown inputs[END_REF].

The objective is to design the unknown input interval observer (UIIO) that admits the following structure r, r, w, w, ρ) r = z+N y+ϕ(w, w, ρ) x =g (r, r, ρ)

                 ż = L(ρ)z + (M (ρ) + L(ρ)N )y + η(r, r, w, w, ρ) r = z + N y + ϕ(w, w, ρ) x =g (r, r, ρ) ż = L(ρ)z + (M (ρ) + L(ρ)N )y + η(
(4a) (4b) (4c) (4d) (4e) (4f)
so that its dynamics is decoupled from the unknown inputs d and the effect of w on the size of the total state estimate envelop x = x -x : x ∈ R n is minimized, in the L 2 -gain and L ∞ -gain sense.

In Eqs. (4a)-(4f), η, η : R 2n+2m+s → R n , ϕ, ϕ :

R 2m+s → R n , g, g : R 2n+s → R n , L(ρ) ∈ R n×n , M (ρ), N ∈ R n×p . z, z ∈ R n are
, respectively, the upper and lower bounds of the state z ∈ R n that we define as a parameter dependent linear combination of x, so that:

z = R(θ)x s.t. R(θ)E(θ) = 0 (5)
Eq. ( 5) guarantees that z is decoupled from the unknown inputs d, and so the envelope tight x is. By extending the results presented in [START_REF] Darouach | Fullorder observers for linear systems with unknown inputs[END_REF] to Eq. ( 5), it can be shown that Eq. ( 5) is fulfilled, if and only if R(θ) = K(θ) (I n -S(θ)C(θ)) where S(θ) = E(θ)(C(θ)E(θ)) † and where K(θ) ∈ R n×n is an arbitrary matrix. Note that the classical choice in the UIO theory for K(θ) is K(θ) = I n . Here, K(θ) is a degree of freedom that will be exploited later. Additionally, we enforce the following relation 4e),(4c),(4f) are respectively the upper and lower bounds of r and x.

x = Q(θ)r Q(θ) ∈ R n×n (6) so that r, r ∈ R n , x, x ∈ R n in (4b),(
Assumption 9 It is assumed that the matrices K(θ) and N are known matrices such that

Q(θ) in (6) is in- vertible and satisfies Q(θ) -1 = R(θ) + N C(θ).
Since C(θ) is of full row rank ∀θ ∈ Θ, it is always possible to find a matrix N that compensates the rank deficiency of S(θ) since it is guaranteed by assumption 2 that q ≤ p. Note that the particular case p = q is covered by the definition of R(θ) since it corresponds to particular parametrization of K(θ), see [START_REF] Darouach | Fullorder observers for linear systems with unknown inputs[END_REF].

The objective is now to design the functions η,η,ϕ,ϕ,g,g and the matrices K(θ), N , L(ρ), M (ρ) so that the effect of w on the size of the total state estimate envelop x is minimized, in the L 2 -gain and L ∞ -gain sense, x being decoupled from d, thanks to (5). The following section states the solution to this problem. Section 4.2 is devoted to the derivation of necessary and sufficient conditions for (4a)-(4f) to be an UIIO for the system (1a)-(1b). Section 4.3.1 proposes a method to derive η,η,ϕ,ϕ,g,g in a non conservative way by keeping the dependency of the variables entering in η and ϕ. while section 4.3.2 proposed a Semi Definite Programming (SDP) formulation to optimally compute L(ρ), M (ρ). With regards to K(θ) and N , guidelines are also provided in section 5.

Solution to the UIIO design problem

Preliminary results

The following lemma provides a constructive method to design the lower and upper bounds of a product of a vector by a parameter dependent matrix in the setmembership context. The core element is that any matrix can always be decomposed into two (non unique) matrices which elements are positives.

Lemma 10 Let a vector v ∈ v, v ⊂ R n and some matrices X k (θ) ∈ R m×n , θ ∈ Θ, k ∈ {a, b}, such that X (θ) = X a (θ) -X b (θ). Then: 1) ∀θ ∈ Θ, assume that ∃X k (θ) ∈ X k , X k ⊂ R m×n with X k ≥ 0. Then, with X • = X a -X a -X b X b , X • = -X b X b X a -X a , it follows X • v * ≤ X (θ)v ≤ X • v * (7) 2) Assume that a subset of θ ∈ Θ is known through ρ ∈ Ψ, 0 < s ≤ l, suppose there exist X k (θ) ∈ X k (ρ), X k (ρ) with X k (ρ) ≥ 0, it follows X • (ρ)v * ≤ X (θ)v ≤ X • (ρ)v * (8) PROOF. The proof for 1) is addressed. Given that v ∈ [v, v] then 0 ≤ v + ≤ v + ≤ v + , 0 ≤ v -≤ v -≤ v -. Now considering that 0 ≤ X k ≤ X k (θ) ≤ X k then X k v + ≤ X k (θ)v + ≤ X k v + and X k v -≤ X k (θ)v -≤ X k v -. Since X (θ)v = (X a (θ) -X b (θ)) (v + -v -), we deduce that X • v * ≤ X (θ)v ≤ X • v * .
The proof for 2) is similar to the proof for 1) by a slight adaptation considering the parameter dependence on ρ.

Note that, Lemma 10 generalizes Lemma 2 in [START_REF] Efimov | Interval estimation for LPV systems applying high order sliding mode techniques[END_REF]. Indeed, in the particular case where matrix part is not anymore unique and do not necessarily depend on the non-linear operators (.) + and (.) -for the matrix X (θ).

The following notations are now adopted in the following

X (ρ) = X a (ρ) -X b (ρ) X (ρ) = X a (ρ) -X b (ρ) X (ρ) = X (ρ) -X (ρ) X (ρ) = X a (ρ) + X b (ρ) (9)

Necessary and sufficient conditions

The following theorem gives the existence conditions for Eqs. ( 4a) -( 4f) to be an UIIO for the system (1a)-(1b).

Theorem 11 Under the assumptions 8 and 9, the state x involved in Eqs. (1a)-( 1b) is bounded by x, x defined by Eqs. (4c), (4f) i.e.

x(t) ∈ [x(t), x(t)] ∀t ≥ 0 (10)
iff the following conditions hold for the system (4a)-( 4f):

(i) ∀ρ ∈ Ψ, there exists a matrix L(ρ) such that L(ρ) is Metzler (11a) (ii) ∀ρ ∈ Ψ, ∀θ ∈ Θ, ∀ θ ∈ Γ θ , ∃ η, η, ϕ, ϕ, g, g so that η(r, r, w, w, ρ) ≤ η(r, w, ρ, θ, θ) ≤ η(r, r, w, w, ρ) (11b) ϕ(w, w, ρ) ≤ ϕ(w, θ) ≤ ϕ(w, w, ρ) (11c) g(r, r, ρ) ≤ g(r, θ) ≤ g(r, r, ρ) (11d)
where η(r, w, ρ, θ,

θ) = F (ρ, θ, θ)r + G(ρ, θ)w, ϕ(w, θ) = H(θ)w, g(r, θ) = Q(θ)r, with F (ρ, θ, θ) = T (ρ)Y (θ, θ) -L(ρ), G(ρ, θ) = T (ρ)Z(θ), H(θ) = -N D(θ) and T (ρ) = L(ρ) M (ρ) I , Y (θ, θ)=    0 -C(θ)Q(θ) R(θ)A(θ) + Ṙ(θ) Q(θ)   , Z(θ)=    -N D(θ) -D(θ) R(θ)B(θ)    (iii) the initial conditions for z, z are given by z(0) ≤ R • (ρ(0))x(0) * z(0) ≥ R • (ρ(0))x(0) * (11e)
where the notations (•)

• , (•)

• are defined in lemma 10.

PROOF. Consider the two following observation errors

e z = z -z e z = z -z (12)
Computing the time derivative of e z , e z and considering the decoupling condition (5) (i.e. R(θ)E(θ) = 0) then :

ėz = L(ρ)e z + η(r, r, w, w, ρ) -η(r, w, ρ, θ, θ) (13a) ėz = L(ρ)e z + η(r, w, ρ, θ, θ) -η(r, r, w, w, ρ) (13b)
Under the condition (11b), the Metzler property (11a) and the initial conditions (11e), it can be verified that (13a)-(13b) fulfil the cooperativity property and then e z (t), e z (t) ≥ 0 ∀t ≥ 0 (14) Now, combining ( 5) and ( 12) , it is deduced that z, z are equivalent to

z = e z + R(θ)x z = R(θ)x -e z ( 15
)
Using ( 15) and (1b) to replace z, z, y into (4b),(4e) and considering the inverse of (6

) (i.e. r = (R(θ) + N C(θ)) x) then r -r = e z + ϕ(w, w, ρ) -ϕ(w, θ) (16a) r -r = e z + ϕ(w, θ) -ϕ(w, w, ρ) (16b)
Under the conditions (11c) and ( 14), it is deduced that r(t) ∈ [r(t), r(t)] , ∀t ≥ 0. Finally, considering (6) and recalling that g(r, θ) = Q(θ)r in (11d) leads to x(t) ∈ [x(t), x(t)] , ∀t ≥ 0 where the bounds are defined in (4c),(4f), which concludes the proof.

SDP formulation of L 2 /L ∞ -gain performance

By virtue of (16a)-( 16b), the following equation yields

r = z + ϕ(w, w, ρ) -ϕ(w, w, ρ) (17)
where r = r -r and z = e z + e z . The following lemma gives the solution to determine the functions g, g.

Lemma 12 Consider Q(θ) defined in (6) satisfying Q(θ) = Q a (θ) -Q b (θ) such that, for k ∈ {a, b}, Q k (θ) ∈ Q k (ρ), Q k (ρ) with Q k (ρ) ≥ 0, ∀ρ ∈ Ψ. Then, for r ∈ [r, r], the inequality (11d) of Theorem 11 holds with g(r, r, ρ) = Q • (ρ)r * and g(r, r, ρ) = Q • (ρ)r * .
PROOF. Direct application of lemma 10 on (6).

With lemma 12, (4c) and (4f), it can be verified that the following equation yields,

x = Q(ρ)r + Q(ρ)r ( 18
)
where the notations Q(ρ), Q(ρ) are defined in (9). Then, using (13a)-(13b), it follows that the time derivative of z = e z + e z is given by

ż = L(ρ)z + η(r, r, w, w, ρ) -η(r, r, w, w, ρ) (19)
Finally, injecting (17) into (18) and combining the result with (19), it follows that:

ż = L(ρ)z + η(r, r, w, w, ρ) -η(r, r, w, w, ρ) x = Q(ρ) z + ϕ(w, w, ρ) -ϕ(w, w, ρ) + Q(ρ)r (20a) (20b)
Thus, the problem turns out to be the determination of η, η, ϕ, ϕ. Note that it is always possible to derive bounds for η and ϕ ignoring the coupling of the variables they depend on, but it seems obvious that it may lead to a conservative solution and thus may increase the tight of the interval x. Towards this end, it is preferred here to keep the dependency of variables entering in η and ϕ.

Determination of η, η, ϕ, ϕ

The determination of η, η, ϕ, ϕ is based on an adaptation of Lemma 10 applied on η, ϕ. Its application on each element of η, ϕ is based on the following decompositions:

F (ρ, θ, θ) = F a (ρ, θ, θ) -F b (ρ, θ, θ) (21) G(ρ, θ) = G a (ρ, θ) -G b (ρ, θ) (22) H(ρ, θ) = H a (ρ, θ) -H b (ρ, θ) ( 23
)
Following the materials discussed in lemma 10, the matrices F k (ρ, θ, θ), G k (ρ, θ), H k (ρ, θ), k ∈ {a, b}, must be defined so that each component of the lower bound is positive. Possible candidates for these matrices can be proposed as follows: Consider the decomposition

T (ρ) = T a (ρ) -T b (ρ) (24)
with

T a (ρ)= L a (ρ) M a (ρ) I , T b (ρ)= L b (ρ) M b (ρ) 0 . such that L a (ρ), L b (ρ), M a (ρ), M b (ρ) ≥ 0, ∀ρ ∈ Ψ.
• With regards to the matrix F (ρ, θ, θ), the following form for F a (ρ, θ, θ) and F b (ρ, θ, θ) is retained

F a (ρ, θ, θ) = L b (ρ) + (T a (ρ) -T b (ρ)) Y (θ, θ) F b (ρ, θ, θ) = F b (ρ) = L a (ρ), (25)
where

Y (θ, θ) satisfies the decomposition Y (θ, θ) = Y a (θ, θ) -Y b (θ, θ) such that, for k ∈ {a, b}, Y k (ρ, θ, θ) ∈ Y k (ρ), Y k (ρ) with Y k (ρ) ≥ 0, ∀ρ ∈ Ψ, ∀ θ ∈ Γ θ and the constraint L b (ρ) + T a (ρ)Y (ρ) -T b (ρ)Y (ρ) ≥ 0 holds, for all ρ ∈ Ψ. The notations Y (ρ), Y (ρ) are defined in (9).
• With regards to the matrix G(ρ, θ), the following form for G a (ρ, θ) and

G b (ρ, θ) is retained G a (ρ, θ) = T a (ρ)Z a (θ) + T b (ρ)Z b (θ) G b (ρ, θ) = T a (ρ)Z b (θ) + T b (ρ)Z a (θ) ( 26
)
where the matrices Z a (θ), Z b (θ) are given by the decomposition

Z(θ) = Z a (θ) -Z b (θ) such that, for k ∈ {a, b}, Z k (θ) ∈ Z k (ρ), Z k (ρ) with Z k (ρ) ≥ 0 for all ρ ∈ Ψ.
• Finally, with regards to H(θ), the following conditions for H k (θ), k ∈ {a, b} are considered

H k (θ) ∈ H k (ρ), H k (ρ) with H k (ρ) ≥ 0, ∀ρ ∈ Ψ (27)
The following lemma provides the solution to the design of η, η, ϕ, ϕ.

Lemma 13 Let us consider the matrices

F k (ρ, θ, θ), G k (ρ, θ), H k (ρ, θ), k ∈ {a,
η(r, r, τ ) = F • (ρ)r * + G • (ρ)w * (28) η(r, r, τ ) = F • (ρ)r * + G • (ρ)w * (29) ϕ(τ ) = H • (ρ)w * (30) ϕ(τ ) = H • (ρ)w * (31)
where

F a (ρ) = L b (ρ) + T a (ρ)Y (ρ) -T b (ρ)Y (ρ), F a (ρ) = L b (ρ) + T a (ρ)Y (ρ) -T b (ρ)Y (ρ), F b (ρ) = F b (ρ) = L a (ρ), G a (ρ) = T a (ρ)Z a (ρ) + T b (ρ)Z b (ρ), G a (ρ) = T a (ρ)Z a (ρ)+T b (ρ)Z b (ρ), G b (ρ) = T a (ρ)Z b (ρ)+ T b (ρ)Z a (ρ), G b (ρ) = T a (ρ)Z b (ρ) + T b (ρ)Z a (ρ).
PROOF. First, note that the bounds of

F k (•), G k (•), H k (•)
, k ∈ {a, b} satisfy the conditions of lemma 10.

Then, applying lemma 10 on the parts F (ρ, θ, θ)r, G(ρ, θ)w and H(θ)w, see the inequalities (11b) and (11c) respectively, leads to ( 28)-( 29) and ( 30)-(31).

Sufficient LMIs for L(ρ), M (ρ)

Using the above derived expressions of η, η, ϕ, ϕ, (20a)-(20b) can be rewritten as equations depending linearly on L(ρ), M (ρ). To proceed, and similarly to (18), we have

η(r, r, τ ) -η(r, r, τ ) = F (ρ)r + (T a (ρ) + T b (ρ)) Y (ρ)r + (T a (ρ) + T b (ρ)) Z(ρ) w + (T a (ρ) + T b (ρ)) Z(ρ) ŵ (32) ϕ(τ ) -ϕ(τ ) = H(ρ) w + H(ρ) ŵ (33)
with the notations (•), (•) defined in ( 9) and where τ = [w T w T ρ T ]. Injecting ( 32)-(33) into the system (20a)-(20b) with r defined in (17), it follows that

               ż = A(ρ)z + f (z) + F (ρ) H(ρ) w + H(ρ) ŵ + (T a (ρ) + T b (ρ)) Z(ρ) w + Z(ρ) ŵ x = Q(ρ)z + g(z) + Q(ρ) H(ρ) w + H(ρ) ŵ z(0) = R • (ρ(0))x(0) * -R • (ρ(0))x(0) * (34a) (34b) (34c)
where

f (z) = (T a (ρ) + T b (ρ)) Y (ρ)r(z, z), g(z) = Q(ρ)r(z, z) and A(ρ) = L(ρ) + F (ρ).
The problem remains now to design L(ρ) and M (ρ) so that the effect of the envelop size w = w-w and the magnitude ŵ on the length of the total envelop x = x -x is minimized, in the L 2 -gain and/or in the L ∞ -gain sense.

The used method is based on the cooperative property to demonstrate that both the state z and the output x of the system (34a)-(34c) can be bounded by the state and the output of a fictitious stable (pure) LPV system, scheduled by the parameter vector ρ and driven by an augmented vector that contains w and ŵ. Then, by proving that the L 2 -gain (L ∞ -gain) of the fictitious LPV system is an upper bound for the L 2 -gain (L ∞ -gain) of the system (34a)-(34c), it follows that the system (34a)-( 34c) is stable with a finite L 2 -gain (L ∞ -gain) performance.

For that purpose, the proposed theory uses the slack matrix method initially proposed in [START_REF] Bara | Parameter-dependent state observer design for affine LPV systems[END_REF] with a parameter-dependent Lyapunov matrix through the concept of Affine Quadratic Stability (AQS).

To proceed, we first derive the following lemma that provides a procedure to get the decomposition K(θ, θ) = K a (θ, θ) -K b (θ, θ) for the matrix K ∈ {Q, Y, Z, H} defined in lemmas 12 and 13.

Lemma 14 Considering that a part of the vector θ, denoted as θ m ∈ R s is measured through ρ ∈ R s (see section 3) and for i = {1, • • • , s}, let some functions n i such that

h i = ρi λi -n i ≥ 0, ∀ρ i ∈ Ψ i . For K(θ, θ) such that K(θ, θ) = K 0 (θ, θ) + s i=1 θ i K i (θ, θ) (35)
then, under assumption 7 i.e θ i = ρi-δi λi , the decomposition K(θ, θ) = K a (θ, θ) -K b (θ, θ) is ensured for (35) by adopting, for k ∈ {a, b},

K k (θ, θ) = K k0 (θ, θ) + s i=1 h i K ki (θ, θ) ( 36
)
where

K a0 (θ, θ) = Σ(θ, θ) + , K ai (θ, θ) = K i (θ, θ) + , K b0 (θ, θ) = Σ(θ, θ) -, K bi (θ, θ) = K i (θ, θ) -, with Σ(θ, θ) = K 0 (θ, θ) + s i=1 ε i K i (θ, θ) and h i = ρi λi -n i , ε i = n i -δi λi . For K j (θ, θ) ∈ K j , K j ⊂ R m×n , j ∈ {0, • • • , s}, upper and lower bounds of K k (θ, θ) are K a (ρ) K a (ρ) K b (ρ) K b (ρ) = Σ + Σ + Σ -Σ -+ s i=1 h i K i + K i + K i -K i - ( 37
)
where

Σ = K 0 + s i=1 K i • ε * i , Σ = K 0 + s i=1 K i • ε * i . PROOF. It is obvious that (35) is equivalent to the form K(θ, θ) = K a (θ, θ) -K b (θ, θ)
where, for k ∈ {a, b}, the matrices K k (θ, θ) are defined by Eq. ( 36). Designing term by term the bounds of Eq. (36) leads to (37) and

K k (ρ) ≥ 0, ∀ρ ∈ Ψ.
Note that the construction of the bounds K j , K j for j ∈ {0, • • • , s} is possible since K(θ) and N are assumed to be a priori known, see assumption 9.

The following theorems, i.e. theorem 15 and theorem 17, give the solutions to the L ∞ -gain and L 2 -gain performance problems, respectively.

Theorem 15 Consider the assumptions 7-9 and the following LPV model

   ν = A(ρ)ν + B(ρ)ψ ξ = C(ρ)ν + D(ρ)ψ ν(0) ≥ z(0). (38a) (38b) (38c) where ν, ξ ∈ R n , ψ = wT ŵT |r| T T
and the matrices

C(ρ) = Q(ρ), D(ρ) = Q(ρ) H(ρ) Q(ρ) H(ρ) Q(ρ) , A(ρ) = L(ρ) + F (ρ), B(ρ) = (T a (ρ) + T b (ρ)) Ω(ρ) + F (ρ)Φ(ρ) with Ω(ρ) = Z(ρ) Z(ρ) Y (ρ) , Φ(ρ) = H(ρ) H(ρ) 0 n,n
, and the notations (•), (•) are defined in (9). Consider also matrices P j = P T j ∈ R n×n and

I j = L aj L bj M aj M bj where L aj , L bj ∈ R n×n , M aj , M bj ∈ R n×p , for j ∈ {0, • • • , s}, such that P (α) = P 0 + α 1 P 1 + • • • + α s P s 0, ∀α ∈ V (39) I (α) = I 0 + α 1 I 1 + • • • + α s I s > 0, ∀α ∈ V (40)
and matrices

E ii = E T ii 0, F ii = F T ii 0, H ii > 0 so that E(α) = α 2 1 E 11 + • • • + α 2 s E ss 0, ∀α ∈ V (41) F(α) = α 2 1 F 11 + • • • + α 2 s F ss 0, ∀α ∈ V (42) H(α) = α 2 1 H 11 + • • • + α 2 s H ss > 0, ∀α ∈ V . ( 43
)
If there exist some real scalars µ ≥ 0, γ 1 , γ 2 , γ 3 , γ 4 > 0 and a M-matrix W ∈ R n×n such that the following constraints hold:

(i) ∀(α, β) ∈ V × R, Υ(α, β, γ 1 , γ 2 )+E(α) ≺ 0 (44) Ξ(α, γ 3 , γ 4 ) + F(α) ≺ 0 (45) -(L a (α) -L b (α)) -µW < 0 (46) -L b (α) + T a (α)Y (α) -T b (α)Y (α) + H(α) < 0 (47) with Υ(α, β, γ 1 , γ 2 ) =      -He{W } Υ 2,1 (α) T Υ 2,2 (α, γ 2 ) Υ 3,1 (α) T 0 Υ 3,3 (α, β, γ 1 ) W T 0 0 -P (α)      , Υ 2,1 (α) = (T a (α) + T b (α)) Ω(α) J (α) , Υ 3,1 (α) = P (α) + L a (α) -L b (α) + J (α), J (α) = L b (α) + T a (α)Y (α) - T b (α)Y (α) + L a (α), T a (α) = L a (α) M a (α) W , T b (α) = L b (α) M b (α) 0 , Υ 2,2 (α, γ 2 ) = -γ 2 I Φ(α) -I , Υ 3,3 (α, β, γ 1 ) = (γ 1 -1)P (α)+P (β)-P 0 , Ξ(α, γ 3 , γ 4 ) =     -γ 3 P (α) 0 -γ 4 I C(α) D(α) -I     . (ii) for i ∈ {1, • • • , s}, Υ ii + E ii 0 (48) Ξ ii + F ii ≺ 0 (49) -T ai Y i -T bi Y i + H ii > 0 ( 50
)
where

Υ ii =      0 (Υ ii ) 2,1 T 0 J ii 0 0 0 0 0 0      , Ξ(α) =     0 0 0 0 D ii 0     , (Υ ii ) 2,1 = (T ai + T bi ) Ω i J ii , J ii = T ai Y i -T bi Y i , Ω i = Zi Z i Y i , T ai = L ai M ai 0 , T bi = L bi M bi 0 , D ii = Qi Hi Qi H i 0 , where K i , Ki , K i , K i for K ∈ {Q, Y, Z, H} deduced from the decompositions K(ρ) K(ρ) K(ρ) K(ρ) = K 0 K0 K 0 K 0 + s i=1 ρ i K i Ki K i K i .
Then, with the UIIO matrices

L(ρ) = L a (ρ) -L b (ρ) and M (ρ) = M a (ρ) -M b (ρ) defined according to L k (ρ) M k (ρ) = W -1 L k (ρ) M k (ρ) k ∈ {a, b} (51) the system (34a)-(34c) is stable with its asymptotic L ∞ - gain performance level, less to γ L∞ = γ3γ2 γ1 + γ 4 .
PROOF. The proof is organized into two main steps: a) First, it is shown that the LPV model ( 38a)-( 38c) is stable ∀ρ ∈ Ψ and ∀ ρ ∈ Γ ρ with asymptotic L ∞gain less or equal to γ L∞ (sufficient condition). b) Second, it is shown that this is a sufficient condition for the system (34a)-(34c) to be stable with an asymptotic L ∞ -gain performance level less to γ L∞ .

Step a): Let us first consider E(α) = 0, F(α) = 0, E ii = 0 and F ii = 0 in Eqs. ( 44), ( 45), ( 48) and ( 49). This will be justified later, see remark 16. Since in theorem 15,

L k (ρ) = W L k (ρ) and M k (ρ) = W M k (ρ) for k ∈ {a, b}
are LMI variables, it follows that the set of constraints ( 39)-( 50) implies ∀ρ ∈ Ψ, ∀ ρ ∈ Γ ρ (see [START_REF] Gahinet | Affine parameter-dependent lyapunov functions and real parametric uncertainty[END_REF]) 54)-( 55) ensure that L(ρ) is Metzler and that L(ρ), M (ρ) satisfy (24). Eq. ( 56) guarantees the decomposition (25) and the conditions ( 52)-( 53) guarantee L ∞ -gain performance for (38a)-(38c). Now, consider the condition (52). Since by definition,

Υ(ρ, ρ) ≺ 0 (52) Ξ(ρ) ≺ 0 (53) W (L a (ρ) -L b (ρ)) + µW > 0 (54) W L a (ρ) L b (ρ) M a (ρ) M b (ρ) > 0 (55) W L b (ρ) + T a (ρ)Y (ρ) -T b (ρ)Y (ρ) -H(ρ) > 0 (56) Since W -1 ≥ 0, Eqs. (
Υ 2,1 (ρ) = W G(ρ) and Υ 3,1 (ρ) = P (ρ) + W A(ρ) with G(ρ) = (T a (ρ) + T b (ρ)) Ω(ρ) F (ρ) then applying the Schur complement to (52) leads to Λ(ρ, ρ) + Q(ρ) T W T I + I T W Q(ρ) ≺ 0 ( 57
)
where

I = I 0 Q(ρ) = -I G(ρ) A(ρ) I and Λ(ρ, ρ) =      0 0 Υ 2,2 (ρ) P (ρ) 0 Υ 3,3 (ρ, ρ) 0 0 0 -P (ρ)     
. By virtue of the projection lemma, (57) has a solution for some matrix W if and only if

N Q (ρ) T Λ(ρ)N Q (ρ) ≺ 0 N I (ρ) T Λ(ρ)N I (ρ) ≺ 0 (58)
where the null-spaces N Q (ρ) and N I (ρ) are given by

N Q (ρ) =    A(ρ) T 0 I 0 G(ρ) T I 0 0 I 0 0 I    T , N I (ρ) =    0 I 0 0 0 0 I 0 0 0 0 I    T .
Applying the Schur complement on the left inequality of (58) leads to A (ρ, ρ) + B(ρ) ≺ 0 (59)

A (ρ, ρ) = He{P (ρ)A(ρ)} + Ṗ (ρ) + γ 1 P (ρ) B T P (ρ) -γ 2 I B(ρ) = P (ρ) F (ρ) F (ρ) T P (ρ) T 0 0 Φ(ρ) T Φ(ρ) Since B(ρ) is strictly positive definite, it follows that A (ρ, ρ) ≺ 0 (sufficient condition). Since A (ρ, ρ) is at- tached to the Lyapunov function V (ρ, ν) = ν T P (ρ)ν characterized by V (ρ, ν) + γ 1 V (ρ, ν) -γ 2 ψ T ψ < 0, it is deduced that V (ρ, ν) < -γ 1 V (ρ, ν) + γ 2 ψ 2 ∞ . Then V (ρ(t), ν(t)) < V (ρ(0), ν(0))e -γ1t + γ2 γ1 ψ 2 ∞ (60) Noting that (53) is equivalent to ξ T ξ + γ 3 V (ρ, ν) - γ 4 ψ T ψ < 0, then ξ T ξ < γ 3 V (ρ, ν) + γ 4 ψ 2 ∞ , which leads with (60) to ξ(t) T ξ(t) < γ 2 L∞ ψ 2 ∞ + γ 3 V (ρ(0), ν(0))e -γ1t (61)
which is the characterisation of the L ∞ -gain for the LPV system (38a)-(38c). It follows that the LPV model (38a)-( 38c) is stable ∀ρ ∈ Ψ and ∀ ρ ∈ Γ ρ with asymptotic L ∞ -gain performance less or equal to γ L∞ . Now coming back to the right inequality of (58), since it is nothing else than an additional constraint to the L ∞gain inequality, it follows that LMIs (39)-( 50) provides only a sufficient condition. Unfortunately, there still exists a coupling between the parameter-dependent matrices P (ρ) and L a (ρ), L b (ρ), M a (ρ), M b (ρ).To overcome this drawback, the slack matrix technique is applied to (59), leading to (52).

Remark 16 As it is outlined in [START_REF] Gahinet | Affine parameter-dependent lyapunov functions and real parametric uncertainty[END_REF], it is possible to reduce the conservatism of ( 52)-( 53) by somewhat relaxing the multiconvexity requirement (48)-( 49). This is the role of the matrices E(α), F(α), E ii and F ii in Eqs. ( 44)-( 45) and ( 48)-( 49). The resulting inequalities are generally less conservative since (52)-( 53) corresponds to E 1 = ... = E s = 0 and F 1 = ... = F s = 0. However, this improvement is at the expense of a higher computational overhead, since the number of optimization variables increases.

Step b): We first demonstrate that

0 ≤ x(t) ≤ ξ(t) ∀t ≥ 0 (62)
Under ( 54) and Lemma 3, the condition (i) of Theorem 11 holds. Since W is a M-matrix, we have W -1 ≥ 0, then Eqs. ( 55)-( 56) ensure the existence conditions of Lemma 13. Combining the expressions of η, η, ϕ, ϕ given in Lemma 13 and the initial conditions (34c), it is deduced that Theorem 11 holds. A direct consequence is 0 ≤ x(t), ∀t ≥ 0. So the left part of the inequality of ( 62) is proved. Now, let us consider the right part of (62). Subtracting (34a)-( 34c) and ( 38a)-(38c), leads to

     ν -ż = A(ρ) (ν -z) + F (ρ) (|r| -r) ξ -x = Q(ρ) (ν -z) + Q(ρ) (|r| -r) ν(0) -z(0) ≥ 0 (63a) (63b) (63c)
where |r| = r + + r -, r = r -+ r + , A(ρ) = L(ρ) + F (ρ). Indeed, for the vector r ∈ [r, r], we have 0 ≤ r + ≤ r + , 0 ≤ r -≤ r -and, by definition, F (ρ) ≥ 0, ∀ρ ∈ Ψ, it is obvious that F (ρ)(|r| -r) ≥ 0. Since L(ρ) is Metzler and F (ρ) ≥ 0, ∀ρ ∈ Ψ then A(ρ) is Metzler. Then with (63c) and by virtue of the cooperative system theory, it follows that ν -z ≥ 0, ∀t ≥ 0. Since by definition Q(ρ) ≥ 0, Q(ρ) ≥ 0, ∀ρ ∈ Ψ, it follows that (63b) is positive, and then x(t) ≤ ξ(t), ∀t ≥ 0.

Combining the aforementioned results, i.e. 0 ≤ x(t) ≤ ξ(t), ∀t ≥ 0 and

ξ ∞ ≤ γ L∞ ψ ∞ ∀ρ ∈ Ψ, ∀ ρ ∈ Γ ρ , it
follows that γ L∞ is an upper bound of the asymptotic L ∞ -gain performance level of the system (34a)-(34c). Furthermore, since 0 ≤ z(t) ≤ ν(t), ∀t ≥ 0 where 0 ≤ z(t) is proven by applying the cooperative system theory on (34a) and since it has been proven in the previous step that ν(t) is stable, (34a)-( 34c) is stable.

The following theorem gives the solution for the L 2 -gain performance problem.

Theorem 17 Consider the assumptions 7-9, the LPV model (38a)-(38c), the inequalities (39)-( 40), ( 43) and matrices G ii = G T ii 0, so that

G(α) = α 2 1 G 11 + • • • + α 2 s G ss 0, ∀α ∈ V (64)
If there exist scalars µ ≥ 0, γ > 0 and a M-matrix W ∈ R n×n such that the following constraints hold: (i) ∀(α, β) ∈ V × R, let consider the inequalities (46)-( 47) in Theorem 15 and

Π(α, β, γ) + G(α) ≺ 0 ( 65
)
where Π(α, β, γ) = Υ(α, β, 0, γ)

Π 2,1 (α) -I with Υ defined in (44) and Π 2,1 (α) = 0 D(α) 0 C(α) 0 (ii) for i ∈ {1, • • • , s}, let consider the inequality (50) and Π ii + G ii 0 ( 66
)
where

Π ii = Υ ii (Π ii ) 2,1 0 , (Π ii ) 2,1 = 0 D ii 0 with Υ ii ,
D ii defined in (48), (49).

Then, with the UIIO matrices

L(ρ) = L a (ρ) -L b (ρ) and M (ρ) = M a (ρ) -M b (ρ) with L k (ρ), M k (ρ) k ∈ {a, b}
defined by (51), the system (34a)-( 34c) is stable with its L 2 -gain performance level, less or equal to γ L2 = √ γ.

PROOF. The proof is similar to the previous one.

Theorems 15 and 17 being posed as SDP problems, it is straightforward to combine them into a mixed problem, by using a weighted convex optimization formulation.

Academic example

Consider the following model with the same notations than used in the previous sections:

A(θ) =    θ 3 -2 -0.5 0 θ 4 θ 1 -1 -0.2 0.1 0 θ 2 -2   , C(θ) = 1 0 0 0 1 + θ 1 0 , B(θ) =    0 1 0.2 θ 1 -0.1 0 0 2 θ 2 0   , E(θ) =    θ 3 + 2 0 θ 1   , D = 0 1 0 0 0 1 .
The parameter vector θ and its rate of variations θ are unknown but assumed to be bounded by the sets Following the method described in the previous sections, K(θ) and N must be chosen adequately in order to derive a not too conservative solution for the UIIO.

     θ 1 θ 2 θ 3 θ 4      ∈      -0.
For K(θ), it is proposed to retain a parameter independent matrix so that K(θ) = κI n with κ = 0. Then a valid definition for

N is N = K(θ)S = κ    1 0 0 0 0.081 0   
where S is the upper bound of the matrix S(θ), see (5). As rank(C(θ)E(θ)) = rank(E(θ)) = 1, ∀θ ∈ Θ then assumption 8 holds, which leads to

R(θ) = κ    0 0 0 0 1 0 -θ1 θ3+2 0 1    , X(θ) = κ    1 0 0 0 1 0 0.081 -θ1 θ3+2 0 1    .
κ being non null, it is obvious that X(θ) is invertible ∀θ ∈ Θ and thus, that assumption 9 holds.

The next step is to design A(ρ), B(ρ), C(ρ), D(ρ) involved in (38a)-(38c) using the method proposed in Lemma 14 with n i = -ρ i , i ∈ {1, 2} for all matrices. Furthermore, and as it is usually done in the L 2 -gain setting, the design problem is weighted by introducing a constant weighting function W ψ = diag(w ψi ), i = 1, ..., 2m + n so that ψ = W ψ b ψ where b ψ is a fictitious vector. The scalars w ψi , i = 1, ..., 2m + n are introduced to manage the robustness level of each channel, separately. Noting that γ, γ 2 , γ 4 enters linearly in ( 39)-( 50), ( 64)-( 66), we can address the following optimization problem min γ n = γ + γ 2 + γ 4 s.t. ( 39)-( 50), ( 64)-( 66)

Since ( 39)-( 50), ( 64)-(66) depend nonlinearly on κ, γ 1 , γ 3 and the LMI variables, this optimization problem is not convex. A simple remedy consists of a grid of κ, γ 1 , γ 3 with a sufficiently dense grid and to minimize γ n at each griding point. Here, by setting γ 1 = 0.7, γ 3 = 1, the result, presented in figure 1, is a κ -γ opt graph where it can be observed the value(s) of κ that lead(s) to a minimal value of γ n (γ n ≈ 65.7 for κ ≈ ±1.4). The functions η, η, ϕ, ϕ are then deduced from lemma 13, which leads to the complete definition of the UIIO.

Simulation results :

Simulations are next performed to appreciate the performance of the designed UIIO. The time behaviour of the parameters are chosen according to θ 1 (t) = 0.15sin(2t + π 2 ), θ 2 (t) = 0.25sin(t + π 4 ), θ 3 (t) = 0.15sin(5t + π 3 ) and θ 4 = 0.5. The unknown input d(t) is fixed to d(t) = 2cos(2t) and the vector w(t) is chosen to be composed of the signal w 1 (t) = -2sin( t 2 ) and two noises w 2 (t), w 3 (t) bounded by known values w 2 = w 3 = 0.1, w 2 = w 3 = -0.1. The results of the simulations are presented in figures 2-4. As expected, x(t) and x(t) are lower and upper bounds of x(t) ∀t ≥ 0 and the distance x -x converges with a reasonable transient behaviour. Finally, a tight interval can be observed, thanks to the L 2 /L ∞ -gain performance. x 1 (t)

x 1 (t)

x 1 (t)

Fig. 2. Interval estimation of the state x1

Note that it can be observed a high frequency behaviour on x 1 (t)/x 1 (t) that does not exist on x k (t)/x k (t), k = 2, 3. This is due to the disturbance w(t) and its second component w 2 (t) to be more precise. This can be appreciated by looking at the principal gains of the transfers between the different component of the exogenous input w and the estimated upper/lower bounds x k /x k , i = 1, 2, 3, for frozen parameters θ(t). These plots are not presented x 2 (t)

x 2 (t)

x 2 (t) x 3 (t)

x 3 (t)

x 3 (t) here due to page limitation, but they reveal that the high frequency content of w 2 (t) is not filtered on x 1 (t)/x 1 (t), whereas it is on x k (t)/x k (t), k = 2, 3.

Conclusion and future works

This paper investigated the design of an unknown input interval observer, for linear parameter varying systems, to robustly estimate the state of linear parameter varying systems, in a guaranteed way under L 2 and L ∞ -gain performance. A SDP formulation is derived to design all the observer parameters. The major advantages with regards to existing theories for interval observer design is that it does not require the explicit knowledge of a (timevarying) state transformation, for the estimation error dynamics to satisfy the Meztler property. Furthermore the unknwon input decoupling and L 2 /L ∞ constraints are formulated on the thickness of the interval length in spite of half the interval length. Finally, we would like to conclude by a remark about the conservativeness of the affine parameter dependency of the Lyapunov matrix and the matrices L k , M k , k ∈ {a, b} of theorems 15 and 17. This choice can be relaxed by using the Sum-of-Square (SoS) theory [START_REF] Parrilo | Structured Semidefinite Programs and Semialgebraic Geometry Methods in Robustness and Optimization[END_REF].

The SoS theory enables to use a general polynomial structure for these matrices and thus offers the potential to derive a solution in the general case but there exist some practical limitations such as the a priori choice of the degree and the structure of the polynomials. Nevertheless, its investigation is under current topic.

  b} defined by Eqs. (25), (26), (27). Then, for r ∈ [r, r], the inequality (11b) and (11c) of Theorem 11 holds with

  Fig. 1. κ -γ graph With κ ≈ 1.4, we obtain γ L∞ = 7.15, γ L2 = 5.81 and the matrices L k (ρ), M k (ρ), k ∈ {a, b} follow from (51).The functions η, η, ϕ, ϕ are then deduced from lemma 13, which leads to the complete definition of the UIIO.

Fig. 3 .

 3 Fig. 3. Interval estimation of the state x2

Fig. 4 .

 4 Fig. 4. Interval estimation of the state x3

  The parameters θ 1 and θ 2 are respectively measured through ρ 1 = θ 1 + δ 1 , ρ 2 = θ 2 + δ 2 where δ 1 ∈ -0.03, 0.03 and δ 2 ∈ -0.03, 0.03 are bounded noises such that δ1 ∈ -10 3 , 10 3 , δ2 ∈ -10 3 , 10 3 .

			15, 0.15 			θ1		 -0.30, 0.30 
			-0.25, 0.25 -0.15, 0.15 0.2, 0.8    	,	   	θ2 θ3 θ4	   	∈	    -0.25, 0.25 0, 0  -0.75, 0.75   	.
	It follows that ρ and ρ satisfy		ρ 1 ρ 2	∈	-0.18, 0.18 -0.28, 0.28	,
	ρ1 ρ2	∈	-1000.30, 1000.30 -1000.25, 1000.25	. Hence, assumption 7 is
	satisfied.

X a (θ) = X (θ) + , X b (θ) = X (θ) -and X (θ) ∈ X , X then Lemma 2 in[START_REF] Efimov | Interval estimation for LPV systems applying high order sliding mode techniques[END_REF] is equivalent to Lemma 10. It allows that the bounds structure of the