Cédric Legout

Guilhem Freche

Romain Biron

Michel Esteves

Oldrich Navratil

Guillaume Nord

Magdalena Uber

Thomas Grangeon

Nico Hachgenei

Brice Boudevillain

A critical zone observatory dedicated to suspended sediment transport: The meso-scale Galabre catchment (southern French Alps)

Keywords: Hydrology, meteorology, soil erosion, sediment transport, sediment fingerprinting, hydrochemistry, critical zone, observatory

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

Summary Paragraph

The 20 km² Galabre catchment belongs to the French network of critical zone observatories. It is representative of the sedimentary geology and meteorological forcing found in Mediterranean and mountainous areas. Due to the presence of highly erodible and sloping badlands of various lithologies, the site was instrumented in 2007 to understand the dynamics of suspended sediments (SS) in such areas. Two meteorological stations including measurements of air temperature, wind speed and direction, air moisture, rainfall intensity, raindrop size and velocity distribution are installed both in the upper and lower part of the catchment. At the catchment outlet, a gauging station records the water level, temperature and the turbidity (10 min. timestep). Water and sediment samples are collected automatically to estimate SS concentrationturbidity relationships, providing SS fluxes quantifications with known uncertainties. The sediment samples are further characterized by measuring their particle size distributions (PSD) and by applying a low-cost sediment fingerprinting approach using spectrocolorimetric tracers.

Thus, the contributions of badlands on different lithologies to total SS flux are quantified at a high temporal resolution providing the opportunity to better analyze the links between meteorological forcing variability and watershed hydrosedimentary response. The set of measurements was extended to the dissolved phase in 2017. Both the river electrical conductivity and its major ion concentrations are measured each week and every three hours during storm events. This allows progress in understanding both the origin of the water during the events and the partitioning between particulate and dissolved fluxes in the critical zone.

Site Descriptions and Measurements

The 20 km² Galabre catchment belongs to the Draix-Bléone Observatory (https://oredraixbleone.inrae.fr/en/) which is also part of the research infrastructure OZCAR [START_REF] Gaillardet | and co-authors: OZCAR: the French network of Critical Zone Observatories[END_REF]. It is located in the southern French pre-Alps (Figure 1a) in the Bléone catchment (Figure 1b). The Bléone river is a tributary of the Durance river which delivers 18% of the total sediment fluxes released by the Rhône river to the Mediterranean Sea [START_REF] Sadaoui | Controls, budgets and variability of riverine sediment fluxes to the Gulf of Lions (NW Mediterranean Sea)[END_REF]. The elevations of the Galabre catchment range from 735 to 1909 m a.s.l. (above sea level) with mean slopes of 54%, 19% and 6% for hillslopes, intermittent rivers and the main river, respectively. The geology is composed of sedimentary rocks overlain by shallow soils (depth < 50 cm). The different geology types include limestones (34 %), marls (30 %), gypsum (9 %), molasses (9%) and quaternary deposits (18%) [START_REF] Esteves | Medium term high frequency observation of discharges and suspended sediment in a Mediterranean mountainous catchment[END_REF]. Unvegetated badlands covering 9% of the total area are distributed throughout the catchment on all types of geologies (Figure 1c). Grasslands cover 40% of the total area while forests cover more than half of the catchment (34% deciduous and 17% coniferous). The climate is both mountainous and Mediterranean, characterized by freezing periods in winter and both high temperatures and high rainfall intensities in summer.

Meteorology

Two meteorological stations are installed at la Robine sur Galabre (since 2014) and at Ainac (since 2019) with similar sensors and configurations (Table 1)

Hydrology

The hydrological station was installed at la Robine sur Galabre in 2007 under a bridge. The river Galabre flows on limestone slabs at this location ensuring a high stability of the cross section without sedimentation or bed scouring. The configuration of this station is similar to that of the meteorological stations (i.e. data logger, energy supply and remote data transfer) with measurements performed every 10 minutes. Water temperature is measured with a Decagon ES-2 sensor. Water level is measured with a 24-GHz radar (Paratronic Cruzoe).

Thirty-five discharge measurements were performed, either using a current flow meter or dilution methods to build a stage-discharge rating curve. As these manual measurements ranged from 0.017 m 3 s -1 to 0.948 m 3 s -1 , an extrapolation was performed for higher water levels with a 1-D hydraulic model (HEC-RAS software; [START_REF] Esteves | Medium term high frequency observation of discharges and suspended sediment in a Mediterranean mountainous catchment[END_REF].

Particulate matter

The hydrological station also includes a turbidimeter (WTW Visolid 700-IQ) connected to a transmitter (IQ182) performing low concentration measurements by 90° diffusion and high concentrations by backscattering. When turbidity thresholds of 5 and 20 g L -1 SiO2 are exceeded, the data logger triggers sample collection by an automatic sequential water sampler (Teledyne ISCO 3700) every 3 and 1 hour, respectively. The 500 ml water and sediment samples are processed in the laboratory to measure the suspended sediment concentration (SSC), particle size distribution (PSD) and spectral reflectance. Most SSC measurements are obtained by weighing the sample after drying for 24 h at 105°C. When SSC are visually assessed to be small (< 2 g L -1 ) measurements are performed by filtration through pre-weighed 0,45µm fibreglass filters and then weighed after drying. Continuous time series of SSC are obtained with turbidity-SSC relations either general (whole period) or specific to a given flood event. [START_REF] Navratil | Global uncertainty analysis of suspended sediment monitoring using turbidimeter in a small mountainous river catchment[END_REF] assessed the uncertainties at the storm event scale to be, on average 20% and 30% for SSC and suspended sediment yield (SSY), respectively. PSD were measured on samples collected during five events using a laser diffraction sizer (Malvern Mastersizer 2000). The protocol detailed in [START_REF] Grangeon | Variability of suspended particles size during highly concentrated flood events in a small mountainous catchment[END_REF] provides, for each measured sample, eleven PSD measurements, the first one being the effective PSD, while the others provide the disaggregation dynamics of suspended particles and an absolute PSD. An aggregation degree was deduced from these measurements. The dried sediment samples were also measured with a portable spectrocolorimeter (Konica Minolta 2600d). Similar measurements were also carried out on source soil samples collected in badland areas of different geological nature. From the diffuse reflectance spectra, a low-cost sediment fingerprinting approach was developed [START_REF] Legout | Quantifying suspended sediment sources during floods in headwater catchments by spectrocolorimetry[END_REF]. It allows to quantify with a maximum uncertainty of 20% the contribution of primary geological sources molasse, marl and limestone to each SS sample.

Dissolved matter

Data characterizing the dissolved phase are available since 2017. Measurements are performed in samples collected automatically by the sequential sampler at a weekly frequency and a time step of a few hours during runoff events. Electrical conductivity is measured with a portable sensor (WTW 340i) regularly calibrated with commercial standards. Major ion concentrations (F -, Cl -, NO2 -, Br -, NO3 -, SO4 2-, Na + , NH4 + , K + , Ca 2+ , Mg 2+ ) are measured by ionic chromatography (Metrohm modular systems 732-733).

Results and interest of the data set

The Galabre catchment is characterised by high SSC with 50% of the total SSY transported at SSC exceeding 13g L -1 . It is highly intermittent with 50% of the total SSY transported in less than 0.1% of time and with less than 5% of the water volume. SS transport occurs mainly during two periods associated to two distinct rainfall regimes [START_REF] Esteves | Medium term high frequency observation of discharges and suspended sediment in a Mediterranean mountainous catchment[END_REF]. In early winter (November and December) widespread rainfall events associated to low rainfall intensities and large durations (>5 h) lead to a high hydrological connectivity within the catchment generating high liquid discharges (e.g. max. 34 m 3 s -1 on 24/12/09). Even though the SSC are usually low during these events (max. 16 g L -1 on 24/12/09), SS fluxes remain high. In summer (June to September), convective storms are usually brief (1-2 h) over limited areas (few square km) leading to high SSC (e.g. max. 360 g L -1 on 9/9/12) with low flow discharge (max. 0.3 m 3 s -1 on 9/9/12) due to a low hydrological connectivity. While the median effective diameters of SS are small (10µm), they are aggregated at various levels and exhibit either a stability or variability at the event scale [START_REF] Grangeon | Variability of suspended particles size during highly concentrated flood events in a small mountainous catchment[END_REF]. The sediment fingerprinting also revealed a stability of the geological source contribution for some events while others exhibit high temporal variations [START_REF] Legout | Quantifying suspended sediment sources during floods in headwater catchments by spectrocolorimetry[END_REF].

The overall data set allows to address process-based understanding of the transfer of water and matter in the critical zone, particularly the links between sediment export from meso-scale watershed and meteorological forcing. It was already used in studies on sediment budgets at larger scales in the Bléone catchment in order to assess the spatial variations of erosion intensity [START_REF] Evrard | Combining suspended sediment monitoring and fingerprinting to determine the spatial origin of fine sediment in a mountainous river catchment[END_REF] or the drivers of the temporal variability of SS fluxes [START_REF] Navratil | Temporal variability of suspended sediment sources in an alpine catchment combining river/rainfall monitoring and sediment fingerprinting[END_REF].

It also allows to test conceptual or physically based models investigating assumptions such as the respective contribution of badlands and the riverbed to SSY [START_REF] Park | Influences of catchment and river channel characteristics on the magnitude and dynamics of storage and re-suspension of fine sediments in river beds[END_REF][START_REF] Misset | An attempt to Link suspended load hysteresis patterns and sediment sources configuration in alpine catchments[END_REF] or the role of the catchment configuration on SS connectivity [START_REF] Uber | How do modeling choices impact the representation of structural connectivity and the dynamics of suspended sediment fluxes in distributed soil erosion models? Earth Surface Dynamics Discussion[END_REF].

  . From 2008 to 2019 Ainac was equipped with an autonomous tipping bucket raingauge (Rainew 111). The common set of measurements at both stations are air temperature and moisture (Campbell Sc. CS215 sensors), wind speed and direction measured with a 2D ultrasonic anemometer (WindSonic4), precipitation measured with a tipping bucket raingauge (Precis Mecanique 3039) and drop size distributions (DSD) measured with an optical disdrometer (OTT Parsivel2). All sensors are supplied by a combination of battery and photovoltaic panel and controlled by a data logger (Campbell CR800) triggering measurement cycles every 15 min. (1 min. for DSD). Data are transmitted each day to a remote ftp server by a 2G/3G modem (Sierra Fastrack FXT009) through the LoggerNet software (Campbell Sc.).

Acknowledgements

The data acquired in the Galabre catchment is funded by the Institut National des Sciences de l'Univers (INSU/CNRS) and the Observatoire des Sciences de l'Univers de Grenoble (OSUG / Université Grenoble Alpes). The authors are grateful to Laurent Bourgès and OSUG for the publication of the DOI of datasets.

Data availability

The data set summarised in Table 1 can be accessed through the OSUG data center at https://doi.osug.fr/public/DRAIXBLEONE_GAL/index.html. A brief description and the data policy is provided for the overall data set (https://doi.org/10.17178/DRAIXBLEONE_GAL.all) and for each sub data set. Alternatively, some data sets can be visualized directly on the BDOH database (https://bdoh.irstea.fr/DRAIX/nos-donnees?site=Galabre). BDOH also enables the interpolation of time steps for export in multiple formats after creating an account.


Valentin Alleaume
email: valentin.alleaume@minesparis.psl.eu

Pauline Bernard
email: pauline.bernard@minesparis.psl.eu

KKL set-valued observers for non-observable systems

Keywords:

KKL observer design consists in transforming the system dynamics into a filter of the output, which admits a trivial observer, and left-inverting the transformation to recover an estimate of the state in the system coordinates. This left-inversion is typically guaranteed under a backward-distinguishability condition. In this paper, instead, we demonstrate how this KKL approach may also be applied without any such observability assumption. We show that there exist appropriate choices of the filter such that any filter solution asymptotically contains the full information about the state indistinguishable class, namely the set of points generating the same output. Then, we investigate the existence of a set-valued left-inverse allowing to estimate asymptotically this indistinguishable class, in the Hausdorff sense. We prove that the estimate tends to be included asymptotically in the indistinguishable classes of the limit points of the system solution. Finally, we provide a numerical example illustrating this convergence.

INTRODUCTION

We consider an autonomous dynamical system ẋ = f (x) , y = h(x)

(1) with state x ∈ R nx , output y ∈ R ny , f : R nx → R nx locally Lipschitz and h : R nx → R ny continuous. We assume that the trajectories of interest remain in a compact subset X ⊂ R nx .

A standard question is the so-called observation problem: at each time t ≥ 0, knowing the past history of the output y on [0, t], can we produce an estimate x(t) of the state x(t) such that the error x(t) -x(t) asymptotically goes to zero? Among many possible methods, a possible path is the design of an observer, namely a dynamical system ż = F (z, y) , x = T (z, y)

(2) fed with the known signal y that asymptotically estimates the real system state. Such designs are reviewed in [START_REF] Bernard | Observer design for continuous-time dynamical systems[END_REF]. However, for an observer to exist, detectability or observability properties are typically required to ensure that the output carries enough information to determine the system state uniquely. In this paper, instead, we are interested in building an algorithm extracting from the output y all the possible information about the state x, which may not be uniquely determined without observability properties.

KKL observer design with distinguishability

When the system is observable, one of the possible routes towards an observer is the nonlinear Luenberger or Kazantzis-Kravaris-Luenberger (KKL) design. Its idea originates from [START_REF] Luenberger | Observing the State of a Linear System[END_REF] where he first introduced his observer for single-output linear systems ẋ = F x , y = Hx with observable pair (F, H) ∈ R nx×nx × R 1×nx . Indeed, he showed that, for any controllable pair (A, B) ∈ R nx×nx × R nx with A Hurwitz and having no common eigenvalue with F , there exists a linear invertible change of coordinates T ∈ R nx×nx such that z = T x follows the dynamics ż = Az + By .

(3) Then, since A is Hurwitz, implementing (3) for any initial condition asymptotically provides an estimate of T x; thus, an estimate of x can be recovered through x = T -1 z.

It turns out that this procedure extends to nonlinear systems. Indeed, the idea is to look for a (nonlinear) change of coordinates T : R nx → R nz with n z ≥ n x such that z = T (x) is governed by (3) in the new coordinates. In other words, T is chosen such that

∂T ∂x (x)f (x) = AT (x) + Bh(x) ∀x ∈ X . (4)
Then, if the map T is injective and admits a (uniformly) continuous left-inverse T inv : R nz → R nx , an estimate of x is obtained with x = T inv (z) for any solution z to (3).

The existence of the map T was first considered in [START_REF] Shoshitaishvili | Singularities for projections of integral manifolds with applications to control and observation problems[END_REF], [START_REF] Kazantzis | Nonlinear observer design using Lyapunov's auxiliary theorem[END_REF] and [START_REF] Krener | Nonlinear observer design in the siegel domain through coordinate changes[END_REF] in the analytic context and around an equilibrium point. Then, the localness was dropped following another perspective in [START_REF] Kreisselmeier | Nonlinear observers for autonomous Lipschitz continuous systems[END_REF], where a global existence result was proposed based on a strong observability assumption which unfortunately did not provide an indication on the necessary dimension n z of the pair (A, B). This problem was solved in [START_REF] Andrieu | On the Existence of a Kazantzis-Kravaris/Luenberger Observer[END_REF] by proving the existence of the injective map T under a weak backward-distinguishability condition, for A complex diagonal of dimension n x + 1, with a generic choice of n x + 1 distinct complex eigenvalues and recently in [START_REF] Brivadis | Further remarks on KKL[END_REF] for almost any real controllable pair (A, B) of dimension n z = 2n x + 1 with A diagonalizable. In this latter paper, an existence result was also provided in a different paradigm, where, instead of picking a sufficient number of distinct eigenvalues, the filter (3) is picked triangular with a single eigenvalue of sufficiently large multiplicity, but without any indication about the necessary dimension n z .

The distinguishability property assumed in [START_REF] Andrieu | On the Existence of a Kazantzis-Kravaris/Luenberger Observer[END_REF]; [START_REF] Brivadis | Further remarks on KKL[END_REF] and guaranteeing the existence of an observer basically says that any distinct states x a , x b in X can be distinguished from their respective past output. In this paper, we are interested in the state estimation problem for (1) when no such distinguishability is assumed.

KKL observer design without distinguishability

If distinct solutions to (1) produce the same output, namely, they are indistinguishable, without asymptotically tending to each other, there is no hope to design an observer producing an asymptotic estimate of x. However, we could imagine to have an algorithm producing at each time a set guaranteed to converge asymptotically -for a certain distance to be defined -to the indistinguishable class of x. In other words, build a set-valued observer.

The interest for such an algorithm comes from the application of electrical machines where nonlinear models producing finite numbers of indistinguishable states have been exhibited (see, e.g., Moreno et al. (2017); [START_REF] Verrelli | Steady-state speed sensor fault detection in induction motors with uncertain parameters: A matter of algebraic equations[END_REF] or [START_REF] Bernard | Estimation of position and resistance of a sensorless PMSM : a nonlinear luenberger approach for a non-observable system[END_REF] in the context of induction motors and permanent magnet synchronous motors (PMSM) respectively) Note that this phenomenon could not appear in a linear context where non observability necessarily implies an infinite number of indistinguishable states.

Indeed, when the number of possible indistinguishable states is finite at each time, one could hope to design an observer estimating this finite number of possibilities. This was done in Moreno et al. (2017); [START_REF] Moreno | Multivalued finitetime observers for a class of nonlinear systems[END_REF] through sliding mode tools for systems that can be written in an "observable-like" form, but which, due to unobservability, involves a set valued map that is considered as an unknown input in the design. Instead, in [START_REF] Bernard | Estimation of position and resistance of a sensorless PMSM : a nonlinear luenberger approach for a non-observable system[END_REF], the KKL route is employed on a particular application featuring a PMSM with unknown resistance. Indeed, it is shown that undistinguishable trajectories exist, but always less than six, and that there exists a map T : R nx → R nz transforming the dynamics into (3) whose inversion enables to reconstruct all the possible states.

In this paper, we theoretize this approach in a general context and investigate whether the KKL paradigm may be used for non observable systems. We show in Section 2 that there exists appropriate choices of the pair (A, B) such that any solution t → z(t) to (3) asymptotically contains the full information about the indistinguishable class of x, namely the set of points generating the same output as x. Then, after studying continuity and convergence of set-valued maps in Section 3, we investigate in Section 4 the existence of a set-valued map T inv : R nz ⇒ R nx such that T inv (z) converges to the indistinguishable class of x. Finally, a numerical example is provided in Section 5. Remark 1. Another paradigm exploiting KKL design for non observable systems is the so-called functional observer problem in [START_REF] Spirito | On the existence of KKL functional observers[END_REF], where the full state is not necessarily observable but only a function of the state q(x) is estimated. This does not lead to set-valued observers since q(x) is assumed observable and is thus reconstructed uniquely by the observer.

Notations

We denote X(x, t) the solution of (1) initialized at x at time 0 and evaluated at time t, and (σ -(x), σ + (x)) its maximal domain of definition. For x ∈ R nx and ε > 0, we denote B(x, ε) the open ball centered at x and with radius ε. The identity matrix of dimension m is denoted I m .

INJECTIVITY WITH RESPECT TO INDISTINGUISHABLE CLASS

Following the KKL methodology, we propose to look for a map T : R nx → R nz transforming the dynamics (1) into (3), namely solution to (4). This first step is the same as in [START_REF] Andrieu | On the Existence of a Kazantzis-Kravaris/Luenberger Observer[END_REF], Theorem 1) in the observable context and can be simplified as follows when X is compact. Lemma 1. Pick n z ∈ N. There exists ρ > 0 such that for any Hurwitz matrix A ∈ R nz×nz and for any B ∈ R nz×ny with A + ρI nz Hurwitz, there exists a C 1 map T : R nx → R nz such that (4) holds.

But then, without observability/distinguishability we cannot hope to prove injectivity of T on X , namely the fact that for all (x a ,

x b ) ∈ X × X , T (x a ) = T (x b ) =⇒ x a = x b .
(5) Instead, we would like to prove that T (x) contains all the possible distinguishable information about x. Definition 1. Two states x a , x b ∈ R nx are said backwardindistinguishable, which we denote

x a ∼ I x b , if h(X(x a , t)) = h(X(x b , t)) ∀t ∈ (max{σ -(x a ), σ -(x b )}, 0] . Given O ⊆ R nx , for a state x ∈ R nx , we then denote I O (x) = {x ′ ∈ O, x ′ ∼ I x} (6) its backward indistinguishable class in O.
In other words, I O (x) contains all the states in O that cannot be distinguished from x based on the knowledge of the past output. To ease the notation, we will omit the mention of O when O = R nx .

Note that for analytic systems, t → X(x, t) is analytic in time, and therefore, equality of outputs during an arbitrarily short amount of time implies equality of outputs on the whole interval of definition. In other words, two solutions t → X(x a , t) and X(x b , t) are indistinguishable, i.e., have the same output, on some interval of time is equivalent to them being indistinguishable on their whole interval of definition, and equivalently, X(x a , t) ∼ I X(x b , t) at all times.

The following result shows the existence of a map T : R nx → R nz transforming the dynamics (1) into (3) such that T distinguishes the distinguishable states. Theorem 1. Assume there exists an open bounded set O, containing X , that is backward invariant by f . Denote n 0 = 2n x + 1 and n z = (2n x + 1)n y . There exists ρ > 0 such that for almost any pair (A 0 , B 0 ) ∈ R n0×n0 ×R n0 with A 0 + ρI n0 Hurwitz, there exists T : R nx → R nz verifying (4) with

A = I ny ⊗ A 0 , B = I ny ⊗ B 0 and for all (x a , x b ) ∈ O × O, T (x a ) = T (x b ) ⇐⇒ x a ∼ I x b . (7)
Remark 2. The "almost any" comes from the fact that the pair (A, B) is picked controllable, with A diagonalizable, and with almost any p C eigenvalues in C and p R eigenvalues in R, all distinct with real part smaller than -ρ, such that 2p C + p R ≥ 2n + 1. Remark 3. It is always possible to make the set O backward invariant by modifying f outside of O. However, then, (7) holds for the modified system. It follows that T (x a ) = T (x b ) ensures equality of the outputs of the original system only as long as X(x a , t) and X(x b , t) remain in the set where f has not been modified. Note however that for any t → x(t) evolving in the compact set X , its omega-limit set is compact and backward invariant. It means that the values taken by T at those limit points are not impacted by the modification of f .

Proof. We adapt the proof of Theorem 3.4 in [START_REF] Brivadis | Further remarks on KKL[END_REF] by removing the "backward-distinguishability" assumption, and replacing the injectivity of T by ( 7). Indeed, we first note that the proof follows in the same way if Proposition D.1 holds with T diag verifying (7) instead of being injective. Digging into the proof of Proposition D.1, we see that T diag is of the form T diag (x) = (T 0 (λ 1 , x), . . . , T 0 (λ 2nx-ℓ+1 , x)) where ℓ is the number of independent complex eigenvalues of A 0 , (λ

i ) ∈ C ℓ ρ × R 2(nx-ℓ)+1 ρ
where

R ρ = {λ ∈ R : λ < -ρ} , C ρ = {λ ∈ C : Re(λ) < -ρ} , and by backward-invariance of O, for (λ, x) ∈ C ρ × O, T 0 (λ, x) = 0 -∞ e -λs h(X(x, s))ds . From this expression, it is clear that for all (x a , x b ) ∈ O × O, x a ∼ I x b implies T diag (x a ) = T diag (x b ). Then, the converse of this implication is obtained by following Section D.2.3, but applying Lemma D.3 to Υ = {(x a , x b ) ∈ O 2 , ∃t < 0, h(X(x a , t) ̸ = h(X(x b , t))} instead of Υ old = {(x a , x b ) ∈ O 2 , x a ̸ = x b } .
Indeed, by continuity of h and of (x, s) → X(x, s), Υ is open and by definition, for all (x a , x b ) ∈ Υ, there exists s < 0 such that ∆(x a , x b , s) ̸ = 0, where ∆(x a , x b , s) = exp(ρs)(h(X(x a , s) -h(X(x b , s)) . Lemma D.3 then allows to say that for almost any (λ i ) ∈

C ℓ ρ × R 2(nx-ℓ)+1 ρ
, and for all (

x a , x b ) ∈ O 2 , T diag (x a ) = T diag (x b ) implies (x a , x b ) / ∈ Υ, which is equivalent to x a ∼ I x b .
In other words, (7) holds for T diag . ■

Thanks to (4), we know that for any initial condition, solutions to (3) converge to T (x). Since T is not injective we cannot deduce a unique x converging to x by leftinversion of T . However, exploiting (7), we may hope to estimate the class of indistinguishability of x, or some elements of it, by looking for the set of pre-images of z by T . Indeed, (7) says that for all x ∈ O and for any subset S of O, the pre-image set in S, corresponds to the set of indistinguishable states in S, i.e, T -1 (T (x)) ∩ S = I S (x) .

(8) In particular, for the given compact set X , this allows to consider a set-valued map

T inv : R nz ⇒ R nx verifying T inv (T (x)) = I X (x) ∀x ∈ X , (9) for instance, T inv (z) = argmin xs∈X ∥T (x s ) -z∥ . ( 10
)
Since z converges to T (x), it is natural to study the convergence properties of T inv (z) towards I X (x).

CONTINUITY AND CONVERGENCE OF SET-VALUED MAPS

Definitions

In order to compare T inv (z) and I X (x), we consider the Hausdorff distance d H defined as follows (see for instance [START_REF] Aubin | Set-Valued Analysis[END_REF]). Note that, for closed sets, we have

δ(S a , S b ) = 0 ⇐⇒ S a ⊆ S b (11a) d H (S a , S b ) = 0 ⇐⇒ S a = S b (11b)
Considering now a set-valued map T inv : R nz ⇒ R nx , evaluating limits of quantities of the type d H (T inv (z a ), T inv (z b )) typically requires (Hausdorff ) continuity, which contains two concepts : upper semicontinuity (usc) and lower semicontinuity (lsc) (see [START_REF] Aubin | Differential Inclusions: Set-Valued Maps and Viability Theory[END_REF], Chapter 1)). Definition 3. A set-valued function T inv is said to be upper semicontinuous at z ⋆ if for any open neighbourhood

V containing T inv (z ⋆ ) there exists a neighbourhood W of z ⋆ such that for all z ∈ W , T inv (z) ⊆ V . This is equivalent to lim z→z ⋆ δ(T inv (z), T inv (z ⋆ )) = 0 . ( 12
)
Definition 4. A set-valued function is said to be lower semicontinuous at z ⋆ if for any x ∈ T inv (z ⋆ ) and any neighborhood V of x, there exists a neighborhood W of z ⋆ such that for all z ∈ W ,

V ∩T inv (z) ̸ = ∅. This is equivalent to lim z→z ⋆ δ(T inv (z ⋆ ), T inv (z)) = 0 . ( 13
)
The following result is a direct consequence of the Maximum Theorem (see [START_REF] Aubin | Differential Inclusions: Set-Valued Maps and Viability Theory[END_REF], Chap 1, Sec 2, Theorem 6)).

Lemma 2. The map T inv defined in ( 10) is upper semicontinuous.

However, T inv is a priori not lower-semicontinuous, so that Hausdorff continuity and Hausdorff convergence of T inv (z) to I X (x) is not guaranteed. However, exploiting upper-semicontinuity, i.e., (12), and (11a), one could hope to have asymptotically at least some inclusion of T inv (z) into I X (x), or the reverse. The following counter-examples show that this is not even guaranteed when both z and x move with time.

Counter-examples

Consider the map T (x) = -x 5 + 2x 3 -x (14) on the domain X = [-1.5, 1.5]. Its plot is given on Figure 1. Imagine that the system solution t → x(t) tends to x * = 1 asymptotically. Then, any solution t → z(t) converges to T (x * ) = 0 marked by the orange dashed line on Figure 1. We have I X (x * ) = T inv (T (x * )) = {-1, 0, 1} marked by the red dots. Now let us compare T inv (z(t)) and I X (x(t)) = T inv (T (x(t))). On the one hand, T inv (T (x(t)) equals after a certain time a set with three elements marked by the three green dots below the orange dashed line, which asymptotically converge to the set {0, 1}. On the other hand, if z converges to zero from above, T inv (z(t)) equals after a certain time a set with three elements marked by the three green dots above the orange dashed line, which asymptotically converge to the set {-1, 0}. It follows that T inv (z(t)) tends to be included asymptotically in I X (x * ), i.e., lim t→∞ δ(T inv (z(t)), I X (x * )) = 0 which illustrates the upper semicontinuity of T inv . This asymptotic inclusion in the limit indistinguishable set of x * is formalized and extended in Theorem 2 below. However, we do not have the Hausdorff convergence, i.e., lim t→∞ d H (T inv (z(t)), I X (x * )) ̸ = 0 since 1 ∈ I X (x * ) will never be "visible" in T inv (z(t)).

But now if we compare T inv (z(t)) and I X (x(t)) dynamically throughout time, we neither have lim

t→∞ δ(T inv (z(t)), I X (x(t)) = 0 nor lim t→∞ δ(I X (x(t)), T inv (z(t))) = 0
and therefore even less lim

t→∞ d H (T inv (z(t)), I X (x(t)) = 0 .
In other words, we can state a convergence result only with respect to the limit indistinguishable set I X (x * ).

Note that this example seems to suggest at least some asymptotic "intersection" between T inv (z(t)) and I X (x(t)), namely the fact that lim

t→∞ inf x ′ ∈T inv (z(t)) d(x ′ , I X (x(t))) = 0 ,
but this is not even ensured as illustrated on Figure 2 for a particular map T exhibiting a plateau. Indeed, if t → x(t) tends to -0.5 from the left, t → T (x(t)) tends to zero from below so that t → I X (x(t)) tends to {-0.5}, while if t → z(t) tends to zero from above T inv (z(t)) tends to {0.5}.

CONVERGENCE OF KKL OBSERVER

As we have seen, the fact that T inv may not be Hausdorff continuous heavily restricts the kind of convergence result we can state in the KKL context. However, we can still state an interesting asymptotic property if we no longer consider the distance of T inv (z(t)) to the current indistinguishable set I X (x(t)), but rather to the indistinguishable set of a limit point of x, as suggested by the examples above.

Theorem 2. Assume there exist a C 1 map T : R nx → R nz verifying (4) and an upper semicontinuous map T inv : R nz → R nx verifying (9). Consider a trajectory t → x(t) of (1) in the compact set X and x ⋆ a limit-point of t → x(t).

Then, for every time sequence (t k ) k∈N such that

lim k→∞ t k = +∞ , lim k→∞ x(t k ) = x ⋆
and any solution t → z(t) of (3), we have

lim k→∞ δ(T inv (z(t k )), I X (x ⋆ )) = 0 (15)
In other words, T inv (z(t)) tends to be included asymptotically in the indistinguishable sets of the limit points of x. In particular, if the system solution t → x(t) converges to a periodic limit-cycle t → x ⋆ (t) with period τ , then we have for any solution t → z(t) of ( 4) and for any Proof. First, x ⋆ exists and is in X since X is compact. Then, according to (4), t → T (x(t)) is solution to (3). Since A is Hurwitz, z(t) -T (x(t)) tends to zero asymptotically, and therefore, by continuity of T and definition of (t k ), lim k→∞ z(t k ) = T (x ⋆ ). Let α = T (x ⋆ ), ε > 0 and the open set

t ⋆ ∈ [0, τ ], lim n→∞ δ(T inv (z(t ⋆ + nτ )), I X (x ⋆ (t ⋆ ))) = 0 .
V ε = x∈I X (x ⋆ ) B(x, ε)
By upper semicontinuity of T inv , since T inv (α) = I X (x ⋆ ) according to ( 9), there exists a neighborhood W of α such that for all z ∈ W , T inv (z) ⊆ V ε . Since (z(t k )) converges to α, we know that there exists

k ε ∈ N such that for k > k ε , z(t k ) ∈ W . Therefore, for any k > k ε , T inv (z(t k )) ⊆ V ε and, by definition of V ε , δ(T inv (z(t k )), T inv (α)) < ε.
The result follows by making ε go to zero. ■

NUMERICAL EXAMPLE

Consider

f (x) = x 2 + x 1 (1 -(x 2 1 + x 2 2 )) -x 1 + x 2 (1 -(x 2 1 + x 2 2 )) h(x) = x 2 1 -x 2 2 2x 1 x 2 (16)
and a set of interest X = B(0, 1.7), where the solutions to be estimated evolve. The phase portrait of the corresponding dynamics are given in Figure 3. It is easily seen that if t → x(t) is a solution of ẋ = f (x), then -x is too. Since h(x) = h(-x), x and -x are thus indistinguishable and we actually have for any x ∈ R 2 , I X (x) = {x, -x}.

In order to fall into the scope of Theorem 1, we need to make an open set O, containing X , backward invariant for the dynamics, while preserving the dynamics on X and the indistinguishable sets. This is done by considering f (x) = χ(∥x∥)f (x), where χ : R → R is a C 1 map, equal to 1 when ∥x∥ ≤ 2 and 0 when ∥x∥ ≥ 10.

Inspired by Theorem 1, we should pick a pair (A, B) of dimension (2n + 1)n y = 10. However, to reduce dimension, following Remark 2, we may try to pick p C = n + 1 eigenvalues in R (included in C) leading to a real pair (A, B) of dimension (n + 1)n y = 6, aware that this choice might be in the zero-measure set that does not ensure (7).

For the simulations, we choose A = -diag (3,4,5,6,7,8) and

B = I 2 ⊗ 1 1 1
, which is not exactly the form recommended by Theorem 1 and shows the flexibility of the result.

Numerical computation of T

In order to compute numerically a map T satisfying (4), we use the algorithm developed in [START_REF] Ramos | Numerical design of Luenberger observers for nonlinear systems[END_REF] and the associated toolbox Buisson-Fenet et al. ( 2022):

(1) Create a grid of x 0 points covering X .

(2) Simulate ẋ = f (x) backward initialized with x 0 during a time t b sufficiently large compared to the slowest eigenvalue of A. This gives a new grid of points x 0,b . Note here the interest of using f instead of f , which explodes in finite backward time.

(3) Simulate forward during t b units of time the augmented system (x, z) made of ( 1)-( 3) and initialized at (x 0,b , 0). ( 4) We obtain a grid of regression points (x 0 , z 0 ), where z 0 is an approximation of T (x 0 ) if t b is sufficiently large for (3) to "forget" its initial condition. (5) Fit a neural network model of T .

Applying this method, we obtain an approximation of a map T : R 2 → R 6 such that for any solution t → x(t) to (1) remaining in X , any solution t → z(t) of (3) converges to T (x) as illustrated on Figure 4. We use a data set with 10 5 points uniformly chosen in X and a neural network with 5 hidden layers of 50 neurons, ReLU activation function, learning rate 0.001, weight decay 10 -6 , and scheduler with factor 0.1, patience 3 and threshold 0.0001. Note that, as expected from Theorem 1, the mean relative error between T (x) and T (-x) is low (under 3%). In [START_REF] Buisson-Fenet | Learning to observe : neural network-based KKL observers[END_REF], written in the spirit of observer design where T is injective, this is done by learning a neural model of a left-inverse T inv of T such that x 0 ≈ T inv (z 0 ) on the previously obtained grid. However, due to the non-injectivity of T , we cannot exploit such a method here. Indeed, the minimization cost will be flawed by the presence of distinct targets for the same input. For instance, if a given z 0 should be mapped both to x 0 and -x 0 , then, the algorithm typically finds T inv (z 0 ) = argmin everywhere. An alternative would be to learn T inv on a reduced data set where only one pre-image of each z 0 is kept. But this raises regularity and threshold issues. Instead, we implement a gradient descent algorithm online, solving (10) at each time step. Results of simulations appear on Figure 5.

As expected, once z has converged to T (x), the algorithm gives as estimate either x or -x, with possible jumps depending on the initialization of the optimization (see Simulation 1). A "warm start" of the optimization around the previously found estimate allows to follow a continuous selection in T inv (z(t)) and avoid those jumps (see Simulation 2). This does not ensure to follow the trajectory x, but a potential indistinguishable trajectory, namely -x here. Note that more efficient algorithms could be implemented to follow the optimum through time, or better yet, reconstruct both candidates x and -x.

CONCLUSION

Even without any observability assumption, the KKL approach is well-suited to estimate information about the state, up to its indistinguishable class. We studied which kind of convergence could be expected from the estimator.

The advantage of such an approach is that no particular normal form is needed unlike high-gain or slidingmode based methods in Moreno et al. (2017); [START_REF] Moreno | Multivalued finitetime observers for a class of nonlinear systems[END_REF]. Further work includes improving the obtained convergence theorem and finding better sufficient conditions guaranteeing Hausdorff continuity. In particular, the apparent link between critical points of T and lack of lower semicontinuity of T inv could be studied.

Definition 2 .

 2 Given two subsets S a and S b of R nx , the Hausdorff distance is defined as d H (S a , S b ) := max δ(S a , S b ), δ(S b , S a ) where δ(S a , S b ) := sup xa∈Sa d(x a , S b ) = sup xa∈Sa inf x b ∈S b d(x a , x b ) .

Fig. 1 .

 1 Fig. 1. Counter-example to asymptotic inclusion betweenT inv (z(t)) and I(x(t)) = T inv (T (x(t)))

Fig. 3 .

 3 Fig. 3. Phase portrait of ẋ = f (x) with f defined in (16).

Fig. 4 .

 4 Fig. 4. Convergence of the error z -T (x)

xFig. 5 .

 5 Fig. 5. Simulation results in the x-coordinates via gradient descent algorithm.