Xavier Margueron

Jean-Pierre Keradec
email: jean-pierre.keradec@g2elab.inpg.fr

J-P Keradec

A Besri

Current Sharing Between Parallel Turns of a Planar Transformer: Prediction and Improvement Using a Circuit Simulation Software

Keywords: planar transformers, PCB parallel windings, circulation current, model, analytical calculation. I

Current sharing between parallel turns of a planar transformer: prediction and improvement using a circuit simulation software

INTRODUCTION

Frequency rise in power electronic converters has induced a diminution of active and passive component sizes. Above 100 kHz, traditional round winding transformers are often replaced by planar ones which use low profile magnetic cores and printed circuit board (PCB) for turns.

For actual applications which need high output currents, the use of PCB conductors 50-150µm thick is problematic. Even if it is some cm wide, cross section of one layer remains small so several layers have to be wired in parallel to sustain several hundredths of Amps [START_REF] Wei | Model and design of PCB parallel winding for planar transformer[END_REF]. Unfortunately, paralleling several layers doesn't result automatically in a balanced current sharing. Indeed, flux flowing through these parallel turns is slightly different and so are induced voltages. As a result, additional currents, here called "circulation currents", are created by these low differential voltages. These internal extra currents are only limited by turn impedances so they seriously impact component losses and reliability. Consequently, accounting for them is not optional for the designer.

At first sight, to design a transformer and predict the current distribution between parallel layers, fem simulations can be used [START_REF] Shangyang | On winding design of planar transformer for miniaturized megahertz DC-DC converters[END_REF]. In practice, planar transformers are quite complex-we studied one including 44 conductive layers-so detailed description of shape and physical properties is time consuming. Worst, to correctly account for eddy currents, meshing must be fine enough so a very large memory computer is needed to carry out a simulation at around 1 MHz and such a simulation lasts several hours. In order to short the "engineer-time" needed to study such a planar transformer intended for an industrial application and to ease optimization process, we developed a method, based on an analytical approach, which enables the designer to obtain the needed results by using any circuit simulation software [START_REF]PSIM 6.0, Power Sim Inc[END_REF]. Such software is among the familiar tools of the designer and related simulations last seconds rather hours, even for complex multi-layer planar transformer.

II. PROBLEM DUE TO PARALELL WINDINGS

Impact of eddy currents on copper losses of a high frequency has been widely studied. Several analytical methods have been proposed to account for these effects, especially when windings can be represented as stacks of conductive layers insulated from each other [START_REF] Reatti | Comparison of various methods for calculating the AC resistance of inductors[END_REF][START_REF] Prieto | 1D magnetic component model for planar structures[END_REF]. They all start from the solution of propagation equation in an infinite plan layer made of conductive linear homogenous and isotropic material. For one layer, total losses split into skin and proximity losses.

In a transformer, when wires are connected in parallel, adding skin and proximity losses of all wires leads to total copper losses only if currents flowing in all wires connected in parallel have not been supposed equal from the beginning. Indeed, as explained above, circulation currents add to the mean current per wire so skin losses must be calculated with the real current that flows in it.

Recently we presented the effect of circulating currents into a 9 kVA, 125-kHz, planar transformer (Fig. 1) [START_REF] Margueron | Les courants de circulation dans les conducteurs en parallèle : Influence dans un transformateur planar[END_REF]. This industrial transformer includes 44 conductive layers. Its PQ core [7] is made of ferrite PC40. This component has two secondaries, each made of 11 paralleled layers. Its primary is made of 11 pairs of layers. Each pair is wired in parallel, then, all these pairs are connected in series. Due to external circuit, while one secondary is loaded (output current : 275 A eff ), the other is in open circuit. The loaded is modeled by a 0.12 Ω resistance. Eleven primary layers are located above and eleven under the secondaries. The current calculated in the different layers, as it is obtained with 2D fem. simulation software [START_REF] Flux | 10 Chemin de Pré Carré-ZIRST[END_REF], is presented in Fig. 2. For the loaded secondary, nominal current should be of 25 A eff per layer. The maximal current value (80 A eff ), which appears for the external layer of the loaded winding, is more than three times higher so, power losses in this layer is ten times than expected. For the open circuit winding, despite nominal current is null (Fig. 2 shows only modulus!), currents in layers 12 and 22 reach 40 A eff .

Even if it doesn't impact on transformer efficiency (still above 98.5 %), this catastrophic current sharing results in strongly localized extra losses (Fig. 3), even in the open circuit secondary. That creates hot points into PCB and, sometimes, destroys the transformer. The impact of circulating currents may be destructive for transformer. Designers have to predict them while they are dimensioning their components.

A. Limits of fem Simulations

Traditional method for determining circulation current is based on fem simulation. Introducing planar transformer geometry in software is quite easy but it is a long task. Besides, meshing of such a component (for example the 44 layers one) must place 2 mashes in a skin depth to properly account for eddy currents. As a consequence, calculation requires a large memory size computer. In practice, the major problem arises when geometrical optimizations are carried out. In order to find the best topology for loss reduction, numerous simulations have to be run [START_REF] Prieto | Using parallel Windings in Planar Magnetic Components[END_REF][START_REF] Skutt | Analysis and measurement of highfrequency effects in high-current tranformers[END_REF] and both description and computation times are often prohibitive.

In order to reduce transformer's development and optimization time, we looked for an analytical method convenient to easily and quickly check different topologies.

B. Analytical Method : 1-D propagation assumption 1) Principe

Looking at its 2D cross section, a planar transformer can be seen as a stack of different types of layers made of conducting, insulating or magnetic material (Fig. 4). Each material is considered as linear, homogeneous, isotropic and it is described by its complex permeability and its complex permittivity. Assuming energy propagation is 1-D (normal to the layers) [START_REF] Keradec | Power transfer in a two winding transformer: From 1D propagation to an equivalent circuit[END_REF][START_REF] Schellmanns | Equivalent circuits for transformers based on one-dimensional propagation: Accouting for multilayer structure of windings and ferrite losses[END_REF], power exchange of a layer is defined, by 6 parameters linearly linked: electric and magnetic fields on each faces, and supplied current and voltage (if the layer is conductive). Such an approach leads to a 3-port equivalent circuit that synthetically represents all energy transfers of a layer.

Magnetic

In order to develop this unidirectional approach, let's consider an infinite plane layer of thickness a , perpendicular to Oz axis. A portion of this layer (height b and depth c ) is taken into account (Fig. 5). This plate is exposed to plane waves (incident and reflected) on its two faces. Magnetic field H is parallel to Oy while electric field E is directed toward Ox . If this plate is a conductive one, a conduction courant can flow inside. This layer is submitted to three power flux: one at the input, one at the output and the third is injected.

Relations between the six external variables ( g E and g H

at the input, d E and d H at the output, V and I) can be written as a 3x3 matrix which is associated to a passive 3-port circuit. However, Ampere's theorem shows that current I is linked to magnetic excitation ( g H and d H ) on both faces [START_REF] Wei | Model and design of PCB parallel winding for planar transformer[END_REF].

J H H d g = - ( 1
)
As a consequence, matrix relation is reduced to a couple of equations ( 2) [START_REF] Keradec | Power transfer in a two winding transformer: From 1D propagation to an equivalent circuit[END_REF] that can be represented by a 2-port circuit (Fig. 6).

J Z H Z E E g s g ⋅ + ⋅ = - 2 1 J Z H Z E E d s d ⋅ + ⋅ - = - 2 1
(2)

with b I J - = and c V E s - = b c J E I V Z s = = ⇔ (3) 1 Z 1 Z 2 Z s E J g E d E g H d H Figure 6. Equivalent 2-port circuit (quadripolar form)
Owing to relation [START_REF]PSIM 6.0, Power Sim Inc[END_REF], this circuit can be modified to introduce potential V and current I (Fig. 7)

1 Z 1 Z 2 Z V b c I 1 Z 1 Z 2 Z V b c I Figure 7. Equivalent circuit for a conductive layer

2) Impedance formulation

Analytical expressions of impedances 1 Z and 2 Z (Fig. 7) depend on complex phase φ (4) and on attenuation A (5) introduced by the plate of thickness a . They depend also on characteristic impedance c Z of this media [START_REF] Margueron | Les courants de circulation dans les conducteurs en parallèle : Influence dans un transformateur planar[END_REF].

a µ ⋅ = ε ω φ (4) φ j e A - = (5)
ε

µ Z c = (6)
Z1 and Z2 can then simply be written with ( 7) and [START_REF] Flux | 10 Chemin de Pré Carré-ZIRST[END_REF].

      ⋅ ⋅ = + - = 2 tan 1 1 1 φ j Z A A Z Z c c (7) ( ) φ sin 1 1 2 2 2 ⋅ = - = j Z A A Z Z c c (8)
These formulas are general and they need to be personalized to any type of material present in transformers. For example, only conductive layers need an electrical supply. Other types of material, not supplied, are represented by a serial impedance ( 12Z ).

To study a transformer, three different types of material need to be described: conductive, insulating and magnetic ones.

a) Conductive layer (copper…)

A conductive layer is defined by its resistivity (or its conductivity σ) σ ρ

1 =
and its thickness ep . For a conductor, permittivity ε is directly linked to conductivity σ (9) so φ and c Z are expressed as [START_REF] Skutt | Analysis and measurement of highfrequency effects in high-current tranformers[END_REF] and [START_REF] Keradec | Power transfer in a two winding transformer: From 1D propagation to an equivalent circuit[END_REF].

ω σ ε ⋅ = j (9) ( ) δ ω σ ω φ ep j ep µ j - = ⋅ ⋅ = 1 0 (10)
( ) σδ ρ ω

1 1 0 j µ j Z c + = ⋅ ⋅ ⋅ = ( 11
)
With

δ ep x = , 1 Z and 2 Z become : ( ) ( ) ( ) ( ) [ ] ( ) ( ) x x x x j x x x r Z cos cosh sin sinh sin sinh 0 1 + + ⋅ + - ⋅ ⋅ = (12) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) [ ] ( ) ( ) x x x x x x j x x x x x x x r Z 2 cos 2 cosh sin cosh cos sinh 2 cos 2 cosh sin cosh cos sinh 2 0 2 - ⋅ - ⋅ ⋅ + - ⋅ + ⋅ ⋅ ⋅ = (13) with a r ⋅ = σ 1 0 b) Air layer (or insulating)
An insulating layer (thickness : a ep _ ) is modelled by a simple inductance [START_REF] Yuequan | Problems of paralleling windings for planar transformer and solutions[END_REF].

a ep µ i Z a _ 0 ⋅ ⋅ ⋅ = ω L a ep µ = ⇔ _ 0 (14)

c) Magnetic layer (ferrite)

A magnetic layer is characterized by its thickness f ep _ and its permeability r µ . The simplest model for such a layer is an inductance (15). If necessary, losses can be taken into account by adding a parallel resistance and even more sophisticated models have been developed [START_REF] Keradec | Accounting for Resistivity and Permeability in High Frequency Permeability measurements: Application to MnZn Ferrites[END_REF]. However, at long as we focus essentially on copper losses such sophisticated models are useless. For the same reasons, fem simulations will be carried out assuming permeability is real.

f ep µ µ i Z r f _ 0 ⋅ ⋅ ⋅ ⋅ = ω L f ep µ µ r = ⇔ _ 0 (15)

3) Example

Let's take a simple two windings EP transformer as an example. Each winding is assumed to be made of only one layer. Axial symmetry of this device enables to obtain a layer stack after unrolling the transformer (Fig. 8). Depth c is taken equal to average turn length. Height b is equal to layer's width. Each layer is separated from the others by an insulating zone which has the same permeability air. Turns number of primary and secondary are introduced in the equivalent circuit by the mean of coupler ratios [START_REF] Schellmanns | Equivalent circuits for transformers based on one-dimensional propagation: Accouting for multilayer structure of windings and ferrite losses[END_REF] which links the layer to its power supply. Finally, the equivalent circuit obtained this way is presented on Fig. 8. After some elementary transformations, this equivalent circuit looks more familiar, as one can see on Fig. 9. All impedances depend on frequency parameter. Open circuit impedance of this transformer depends essentially of magnetic impedance ( ( )

Secondary Primary

Insulator insulator Insulator Ferrite Ferrite p Z 1 p Z 1 s Z 1 s Z 1 p Z 2 s Z 2 1 a Z 2 a Z 3 a Z 1 f Z 2 f Z p V s V b c nt1 b c nt2 Secondary Primary Insulator insulator Insulator Ferrite Ferrite p Z 1 p Z 1 s Z 1 s Z 1 p Z 2 s Z 2 1 a Z 2 a Z 3 a Z 1 f Z 2 f Z p V s V b c nt1 b c nt2 p Z 1 p Z 1 p Z 1 p Z 1 s Z 1s Z 1 s Z 1s Z 1 p Z 2 p Z 2 s Z 2s Z 2 1 a Z 1 a Z 2 a Z 2 a Z 3 a Z 3 a Z 1 f Z 1 f Z 2 f Z 2 f Z p V p V s V s V b c nt1 b c nt1 b c nt2 b c nt2
1 3 1 2 1 f a p Z Z Z + + ⋅ η ). In this expression, ferrite impedance 1 f Z is dominating. p Z 2 2 1 ⋅ η p V s V p Z 1 2 1 ⋅ η p Z 1 2 1 ⋅ η 3 2 1 a Z ⋅ η 1 2 1 f Z ⋅ η 1 2 1 a Z ⋅ η s Z 1 2 1 ⋅ η s Z 1 2 1 ⋅ η 2 2 1 a Z ⋅ η 2 2 1 f Z ⋅ η s Z 2 2 1 ⋅ η 1 2 t t n n b c n t1 1 = η Impedances that can
be neglected in comparison with magnetizing impedance

Figure 9. Classical model (similar to Fig. 8)

4) Application to 44-layers transformer

Geometry of 44-layers transformer (Fig. 3) seems very interesting to check the equivalent circuit representation introduced above. In fact, all conductors have the same width and they are separated by identical insulating layers.

In order to implement this model in circuit simulation software [START_REF]PSIM 6.0, Power Sim Inc[END_REF], 309 impedances (3 resistances and 3 reactances per layer) and 44 couplers must be introduced. Results obtained with this model are presented in Fig. 10. This simulation gives very good results. Current in the load is equal to 275 A eff and the repartition obtained with fem software is quite similar to that deduced from our equivalent circuit representation introduced in circuit simulation software (difference less than 5%). The most interesting aspect of this "analytical simulation" is that it is very fast. Computing time is less than 20 s and this can be advantageously compared to the 20 min necessary to run an fem simulation. Introduction of such model into a circuit software is also simpler (and faster) because numerous identical cells can be repeated.

In the final part of this paper, different layer stacks will be compared in order to reduce circulation currents and to improve transformer reliability.

IV. PLANAR TRANSFORMER STUDIES

A. Reduction of circulation currents

Circulation currents are due to leakage flux enclosed by parallel layers and these flux increase with the distance separating two conductive layers. So, best solution appears to be the use only one thick layer with the total copper thickness! In standard transformers, one alternative solution consists in twisting wires such as in Litz wires. This does reduce the flux embraced by each pair of wires but it is not applicable for PCB windings.

The second solution consists in modifying winding arrangement to reduce the leakage inductance of the transformer [START_REF] Yuequan | Problems of paralleling windings for planar transformer and solutions[END_REF]. This is achieved by interleaving primary and secondary to locally cancel Ampere-turns. This solution, which reduces rms induction in the transformer window when one winding is shorted, decreases stored energy and leakage inductance. It also reduces eddy current losses which are roughly proportional to this rms induction.

B. Layer stack improvements

Impact of interleaving is now tested using our equivalent circuit method. All layers stay at the location occupied in the 44-layer transformer but they are connected differently. This leads to the three different layer stacks shown in Fig. 11. The first one (Fig. 11-a) describes the initial planar device. Each primary layer in the figure is composed of two parallel layers. Secondaries are both placed between the two halfprimaries. The second design (Fig. 11-b) is obtained by interleaving 2 primary layers, 2 "A" secondary layers and 2 "B" secondary layers. The third one (Fig. 11-c) consists in interleaving the three winding layers one by one. According to our guiding idea, this last arrangement seems to be the ideal one but can not always be done because of manufacturing reasons.

P1 P2 P3 P4 P5 SA1 SA2 SA3 SA4 SA5 SA6 SA7 SA8 SA9 SA10 SA11 SB1 SB2 SB3 SB4 SB5 SB6 SB7 SB8 SB9 SB10 SB11 P6 P7 P8 P9 P10 P11 P1 P2 SA1 SA2 SB1 SB2 P3 P4 SA3 SA4 SB3 SB4 P5 P6 SA5 SA6 SB5 SB6 P7 P8 SA7 SA8 SB7 SB8 P9 P10 SA9 SB9 SB10 P11 SA11 SB11 P1 SA1 SB1 P2 SA2 SB2 P3 SA3 SB3 P4 SA4 SB4 P5 SA5 SB5 P6 SA6 SB6 P7 SA7 SB7 P8 SA8 SB8 P9 SA9 SB9 P10 SA10 SB10 P11 SA11 SB11 (a) (b) (c)
Comparison between theses different arrangements can be realized very quickly with the use of circuit simulation software. Only connections between plates must be modified to achieve different arrangements. For the open circuit winding (Secondary B), currents values obtained with circuit simulation are presented on Fig. 13. Circulations currents are reduced to 5 A eff with the arrangement 1. This value can be compare to the 45 A eff reached in the initial device. Arrangement 2 enables to still reduce currents to 1 A eff (nominal values may be null). Total power dissipated in copper can also be compared: for the initial design, it is evaluated to 70 W, for arrangement 1, losses are reduced to 19 W and, for arrangement 2, losses still decrease slightly to 17.8 W. This values doesn't include losses in the PCB vias and in the magnetic core.

C. Final optimized transformer

Owing to our equivalent circuit method, connection between layers as well as thickness of insulating and conductive layers can be changed easily and quickly. However, the solution showing lowest losses is not necessary the one that engineers adopt. Other industrial considerations are taken into account such as volume, weight, cost, …. The study of this 44layers planar transformer has been proposed by Thales and it gave us the opportunity of working with Thales engineers to improve their component. After balancing best efficiency solution with cost and ease of manufacturing, they built the component described below (Fig. 14). Currents repartitions in secondary layers are presented in Fig. 15 for the loaded secondary, and in Fig. 16 for the open circuit secondary. On these figures, results obtained with fem software are compared with those deduced using circuit simulation software. Results are in very good agreement. Current in loaded windings is included between 30 A eff and 55 A eff . This values must be compared to the nominal value : 45.8 A eff . In the open circuit secondary, current is included between 2 A eff and 13 A eff . Copper losses are close to 20.6 W and, according to measurements, power efficiency seems to be slightly above 99 %.

Compared to the first prototype, better performances and reliability are obtained (lower copper losses, reduce hot spots) and this component is easier to manufacture. V. CONCLUSION

In this paper, we investigated the impact of circulation currents on high frequency transformers. These currents appear as soon as at least one winding is made of strands connected in parallel. They are never taken into account in analytical approaches. They can be studied by f.e.m. simulations providing equality of currents circulating in parallel strands is not imposed from the beginning.

Despite paralleling winding seems to be a good solution to reduce eddy current effects while sustaining high frequency strong currents, impact of circulation currents must be evaluated to avoid destructive hot spots. This evaluation is long and quite incompatible with an optimization process when it is done using traditional approach (analytical and f.e.m. simulation).

In this paper, an exact analytical approach has been formatted to appear as a circuit problem that can be solved using standard circuit simulation software. This method, based on the 1-D propagation assumption, leads to an equivalent circuit suitable to determine all layer currents, including circulation ones. Owing to this tool, checking concurrent topologies for a planar transformers with given specification is easy and fast. Using it, engineers can save a lot of time.

Since this paper have been proposed, equivalent circuit method has been extended and it is now usable with non harmonic waveforms. This will be published soon.

Figure 1 .

 1 Figure 1. 44-conductive layers, 3 windings planar transformer

Figure 2 .

 2 Figure 2. Current repartition in secondary layers. Current modulus is plotted versus secondary layer number

Figure 3 .

 3 Figure 3. Power repartition in secondary layers. The Clearer the color is, the higher the power density is

Figure 4 .

 4 Figure 4. Example of layer stack

Figure 5 .

 5 Figure 5. One layer

Figure 8 .

 8 Figure 8. Equivalent circuit for a two winding transformer

Figure 10 .

 10 Figure 10. Comparison of current repartition in secondary layers

Figure 11 .

 11 Figure 11. Connections of the 44 conductive layers of the 3-winding transformer. (a) : initial design | (b) : arrangement 1 | (c) : arrangement 2

Fig. 12 Figure 12 .

 1212 Fig.12presents the comparison of currents in secondary A (loaded winding). Both arrangements 1 and 2 lead to quite equal currents in parallel layers.

Figure 13 .

 13 Figure 13. Current repartition in open circuit winding (secondary B)

  This component is made of a smaller number of layers. Both secondary are made of 6 parallel layers. Thickness of PCB layers have been increased (surface x1.5) to sustain 275/6 = 45.8 A eff . Arrangement of primary and both secondary is a mix (2 primaries-1 secondary A-1 secondary B) between Fig.11-band Fig.11-c. Final performances are presented below.

Figure 14 .

 14 Figure 14. 24-layers optimized transformer

Figure 15 .

 15 Figure 15. Currents repartition in optimized 24-layer planar transformer (Secondary A)

Figure 16 .

 16 Figure 16. Currents repartition in optimized 24-layer planar transformer (Secondary B)


Assia Daid

Eric Busvelle

Mohamed Aidene

On the convergence of the unscented Kalman filter

Keywords: Nonlinear observers, Unscented Kalman filter, Unscented Kalman Observer

A convergence analysis of the modified unscented Kalman filter (UKF), used as an observer for a class of nonlinear deterministic continuous time systems, is presented. Under certain conditions, the extended Kalman filter (EKF) is an exponential observer for non-linear systems, i.e., the dynamics of the estimation error is exponentially stable. It is shown that unlike the EKF, the UKF is not an exponentially converging observer. A modification of the UKF -the unscented Kalman observer -is proposed, which is a better candidate for an observer. This paper is a first step towards a proof of the global convergence of the high-gain version of the UKO.

Introduction

y(t) = AK(1 -e -t
τ H(t) For technological reasons (material, feasibility), reliability (failure of measurements) or economic (cost of sensors), in many applications the measurement of the whole state of a system is not possible, but the knowledge of the state at any moment is essential, for example: the synthesis of a command [START_REF] Anderson | Optimal control: linear quadratic methods[END_REF] or the diagnosis [START_REF] Simani | Model-based fault diagnosis in dynamic systems using identification techniques[END_REF] and detection of defects in an industrial process [START_REF] Zou | Automatic weld defect detection method based on Kalman filtering for real-time radiographic inspection of spiral pipe[END_REF].

State observers (sometimes called soft sensors or intelligent sensors) are a way to handle this problem. The task of an observer is to provide an estimate (in some sense, asymptotic or exponential) of the state of a studied system according to the information available about this system (the input control and output measurements and the dynamic model of the process, [START_REF] Yacine | Observateurs pour l'Estimation de la Dynamique Latérale du véhicule et Application à la Détection de Situations Critiques[END_REF]).

The first observers dedicated to the estimation of the state of a linear system that are characterized by necessary and sufficient conditions, were published around the 1960s by Kalman (Kalman filter, [START_REF] Kalman | A new approach to linear filtering and prediction problems[END_REF]) in a stochastic framework and subsequently by Luenberger in a deterministic framework (Luenberger observer, [START_REF] Luenberger | Observers for multivariable linear systems[END_REF][START_REF] Luenberger | An Introduction to Observer[END_REF]). These two observers are widely used today, but linear systems cover only a small percentage of industrial processes. Indeed, most real applications have non-linear behavior, which has prompted the development of non-linear filters and observers ( [START_REF] Busvelle | Observation and identification tools for non-linear systems: application to a fluid catalytic cracker[END_REF][START_REF] Deza | Exponentially converging observers for distillation columns and internal stability of the dynamic output feedback[END_REF][START_REF] Viel | Stability of polymerization reactors using I/O linearization and a high-gain observer[END_REF]).

Among these non-linear systems, we consider those non-linear dynamic systems in continuous time which have the form dx(t) dt = f (x(t), t)

y(t) = h(x(t), t) (1)
x(t) ∈ R n , y(t) ∈ R p . Since f and h depend explicitly on the time t, we omit writing explicitly the control variable u(t) without loss of generality. The extended Kalman filter (EKF) is an important and widely used tool for state estimation in such non-linear dynamic systems [START_REF] Chui | Kalman Filtering with Real-Time Applications[END_REF][START_REF] Jazwinski | Stochastic Processes and Filtering Theory[END_REF]. It is based on a linearization of the system along the estimated trajectory. However, this linearization can lead to poor performance (and even the divergence of the filter for highly non-linear problems), and the required derivation of Jacobian matrices may not be easy in many applications (see, for instance, [START_REF] Lafont | An adaptive high-gain observer for wastewater treatment systems[END_REF] and a way to work around the problem). Although the linearization around the estimated trajectory of the system usually prevents analytically proving the observer's convergence for any initial error, such proofs exist in the deterministic and stochastic cases when the initial estimation errors and disturbing noise are small enough (see [START_REF] Baras | Dynamic observers as asymptotic limits of recursive filters: Special cases[END_REF][START_REF] Boutayeb | Convergence analysis of the extended Kalman filter used as an observer for nonlinear deterministic discrete-time systems[END_REF][START_REF] Reif | Stochastic stability of the discrete-time extended Kalman filter[END_REF][START_REF] Reif | Unbehauen Modification of the extended Kalman filter with an additive term of instability[END_REF][START_REF] Reif | An EKF-Based Nonlinear Observer with a Prescribed Degree of Stability[END_REF][START_REF] Song | The Extended Kalman Filter as a Local Asymptotic Observer for Nonlinear Discrete-Time Systems[END_REF][START_REF] Reif | Stochastic stability of the continuous-time extended Kalman filter[END_REF][START_REF] Reif | The extended Kalman filter as an exponential observer for nonlinear systems[END_REF]).

A recent improvement to the EKF is the unscented Kalman filter (UKF) [START_REF] Julier | A new approach for filtering nonlinear systems[END_REF][START_REF] Julier | A new method for the nonlinear transformation of means and covariances in filters and estimators[END_REF]. The UKF approximates the probability density resulting from the non-linear transformation of a random variable instead of approximating the non-linear functions with a Taylor series expansion. The approximation is done by evaluating the non-linear function on a minimal set of carefully chosen sample points [START_REF] Särkkä | On Unscented Kalman Filtering for State Estimation of Continuous-Time Nonlinear Systems[END_REF][START_REF] Särkkä | Bayesian Filtering and Smoothing[END_REF][START_REF] Julier | New extension of the Kalman filter to nonlinear systems[END_REF][START_REF] Julier | A general method for approximating nonlinear transformations of probability distributions[END_REF]. Consequently, the UKF has received great attention in many areas such as control theory, aerospace navigation, eye tracking, and information fusion [START_REF] Zu-Tao | Sampling strong tracking nonlinear unscented Kalman filter and its application in eye tracking[END_REF][START_REF] Liu | Control of epileptiform spikes based on nonlinear unscented Kalman filter[END_REF]. This approach may be seen as a Monte Carlo approach, although the number of points used to propagate the law of probability remains small (see [START_REF] Gustafsson | Some relations between extended and unscented Kalman filters[END_REF]).

To summarize, there exist two approaches to non-linear state estimation.

• The first one is non-linear filtering, which consists in studying the probability law of the state of a stochastic dynamic system conditionally on the observations. The extended Kalman filter is based upon an approximation by a normal distribution of a pair of random vectors X ∈ R n and Y ∈ R p such that X is a Gaussian random vector obeying the law N (m, P ) and Y = g(X) where g is a non-linear transformation (see [START_REF] Picard | Efficiency of the extended Kalman filter for nonlinear systems with small noise[END_REF][START_REF] Pardoux | Filtrage non linéaire et équations aux dérivées partielles stochastiques associées[END_REF]). The usual approximation of the law of the pair (X, Y ) uses the first-order approximation

X Y ∼ N m g(m) , P P A t AP AP A t ( 2
)
where A is the Jacobian matrix of g at the point m. To better take into account the non-linear transformation of a Gaussian vector, the unscented transformation has been developed, which consists of writing

X Y ∼ N m µ U , P C U C t U S U (3)
where the vector µ U and the matrices C U and S U are approximated from g(m) and from the image by g of 2n + 1 σ-points, (2n points are placed around m, we will explain further this construction in Section 2.1, (8), ( 9) and [START_REF] Del Moral | On the stability and the concentration of extended Kalman-Bucy filters[END_REF], see also [START_REF] Julier | New extension of the Kalman filter to nonlinear systems[END_REF][START_REF] Wan | The unscented Kalman filter for nonlinear estimation[END_REF][START_REF] Julier | Unscented Filtering and Nonlinear Estimation[END_REF][START_REF] Almeida | State estimation of nonlinear systems using the Unscented Kalman Filter[END_REF]). This non-linear version of the Kalman filter has several advantages, the first being a better consideration of the propagation of a Gaussian noise in a non-linear system. The second advantage is that this non-linear version does not require calculating two Jacobian matrices, which are sometimes complex and sources of numerical instabilities (see [START_REF] Doumiati | Unscented Kalman filter for real-time vehicle lateral tire forces and sideslip angle estimation[END_REF] for a concrete example).

• The second approach to the problem of estimating the state of a system is purely deterministic. It consists in constructing another dynamic system ('the observer'), which uses as input the available measurements and whose state converges asymptotically (and generally exponentially) to the state of the system [START_REF] Boizot | Adaptive-gain observers and applications[END_REF][START_REF] Gauthier | Deterministic Observation Theory and Applications[END_REF]. A classical way to construct an observer is to apply a stochastic filter in the deterministic framework. In this approach, the matrices Q and R that represent the covariance matrices of the state noise and the measurement noise are interpreted as tuning parameters (and corresponds to quadratic cost matrices in a quadratic cost minimization framework).

There are some techniques that show that the observer/filter (EKF, UKF), converges exponentially locally to the state of the system under certain conditions (observability). Some authors use the Lyapunov method [START_REF] Boutayeb | Convergence analysis of the extended Kalman filter used as an observer for nonlinear deterministic discrete-time systems[END_REF][START_REF] Reif | The extended Kalman filter as an exponential observer for nonlinear systems[END_REF], or the contraction theory, developed by Lohmiller and Slotine [START_REF] Lohmiller | On Contraction Analysis for Non-linear Systems[END_REF][START_REF] Maree | On convergence of the unscented Kalman-Bucy filter using contraction theory[END_REF] for estimating the state of the stochastic or deterministic non-linear systems. In the stochastic framework, a novel stochastic analysis based on exponential concentration inequalities and uniform χ-square type estimates for stochastic quadratic type systems, has been adapted to complement the above-mentioned techniques. For more details, see [START_REF] Del Moral | On the stability and the concentration of extended Kalman-Bucy filters[END_REF][START_REF] Karvonen | On stability of a class of filters for non-linear stochastic systems[END_REF].

Compared to the extended Kalman filter, little research has been done to analyse the convergence of the UKF. Some proofs have been established for non-linear stochastic discrete time systems (see [START_REF] Xiong | Performance evaluation of UKF-based nonlinear filtering[END_REF][START_REF] Li | Stochastic stability of the unscented Kalman filter with intermittent observations[END_REF][START_REF] Dymirkovsky | New results on stochastic stability of discrete-time Unscented Kalman Filter[END_REF][START_REF] Hu | A derivative UKF for tightly coupled INS/GPS integrated navigation[END_REF][START_REF] Hu | Stochastic stability of the derivative unscented Kalman filter[END_REF]) and in the stochastic continuous-time case (see [START_REF] Xu | Stochastic stability of the continuous-time unscented Kalman filter[END_REF]). However, in the case of nonlinear stochastic systems with linear state equation, such as the tightly coupled INS/GPS integrated system [START_REF] Hu | A derivative UKF for tightly coupled INS/GPS integrated navigation[END_REF], the use of the UKF causes a great amount of redundant computation in the prediction process, because the system state equation is used to propagate 2n + 1 σ-points. For remedy this the authors have proposed a derivative unscented Kalman filter (DUKF) to reduce the computational complexity by uses the original Kalman filter (KF) in the prediction process and they have analyse the stochastic stability of the DUKF in [START_REF] Hu | Stochastic stability of the derivative unscented Kalman filter[END_REF].

Despite its interest (due to the fact that it does not require a priori Jacobian computation), UKF convergence has not been studied in a deterministic setting. Two reasons may explain this lack: the first one is that the UKF is an improvement of EKF only because it tackles Gaussian noise more efficiently. The second one is because without modification, it is not convergent, as we will see in Section 2.3. We claim that our study explains theoretically several experimental studies comparing EKF and UKF (see, e.g. [START_REF] Rhudy | Evaluation of Matrix Square Root Operations for UKF within a UAV GPS/INS Sensor Fusion Application[END_REF][START_REF] Stastny | Comparison of the extended and unscented Kalman filters for angles based relative navigation[END_REF]) as well as what can be expected from the UKF when the noise is small (and how to tune the UKF in such cases).

The structure of this paper is as follows. In Section 2, we recall the unscented transformation and the unscented Kalman filter, and we show that the unscented Kalman filter does not converge in a deterministic setting, as soon as the system is non-linear. Therefore, in Section 3, we propose a modification which avoids the bias of the UKF. This new algorithm is called the unscented Kalman observer (UKO). In Section 4 we prove the convergence of the UKO, as a deterministic observer for continuous-time systems, when the initial estimation errors are small enough. The proof relies heavily on the existence of bounds for the solution of the Riccati-like observer equation. The bounds on the solution of this Riccati-like equation are obtained by small perturbation of a Riccati equation. For the sake of clarity, the proof of this technical result is relegated to Appendix A. It basically follows the ideas developed in [START_REF] Gauthier | Deterministic Observation Theory and Applications[END_REF] (to prove that the solution of the disturbed Riccati equation is bounded, we followed the ideas developed in [START_REF] Gauthier | Deterministic Observation Theory and Applications[END_REF] which require several lemmas. In our article, only the significantly different proofs are detailed). A simulation has been done to illustrate the UKO's performance against the UKF and EKF in Section 5.2.

Continuous-Time Unscented Kalman Filter

In this section, we present the unscented Kalman filter in matrix form (see [START_REF] Särkkä | On Unscented Kalman Filtering for State Estimation of Continuous-Time Nonlinear Systems[END_REF], it having been first introduced by Julier and Uhlmann in [START_REF] Julier | A new approach for filtering nonlinear systems[END_REF]). It uses the unscented transformation which propagates σ-points, (denoted by X) through the non-linear equation and uses the results to capture the means and covariance of the posterior distribution [START_REF] Julier | Unscented Filtering and Nonlinear Estimation[END_REF][START_REF] Wan | The unscented Kalman filter for nonlinear estimation[END_REF]. In the following, we will recall the unscented transformation algorithm, tuning parameters, and weights.

The unscented transformation

We now introduce some notation that will be convenient for the calculations and proofs.

Notation 1. Let 1 n and 0 n denote, respectively, the row vector The procedure to implement the UKF in continuous-time is summarized as follows [START_REF] Särkkä | On Unscented Kalman Filtering for State Estimation of Continuous-Time Nonlinear Systems[END_REF] 1. Choose 2n + 1 points in R n called σ-points:

1 n = (1, 1, • • • , 1) ∈ R n
X(t) = m(t) u + √ c P (t) U (4)
• X(t) is the matrix of σ-points (written in columns), X(t) ∈ M n,2n+1 (R), the set of matrices of dimension n × (2n + 1)1 ,

• k ≥ 0 and c ∈]0, n + k] are parameters to be set,

In the proofs, the parameter c will play a crucial role and will be used as a tuning parameter. P (t) denotes the principal square root of P (t), that is, the symmetric definite positive matrix such that P (t) = P (t)

2

. We remark that P (t) is not the Cholesky decomposition of P (t), as is often the case in the UKF literature (see, e.g. [START_REF] Särkkä | On Unscented Kalman Filtering for State Estimation of Continuous-Time Nonlinear Systems[END_REF]). This does not change any basic results (since P (t) = P (t) P (t) t as usual).

Remark 1. In [START_REF] Rhudy | Evaluation of Matrix Square Root Operations for UKF within a UAV GPS/INS Sensor Fusion Application[END_REF], several alternatives for the square root in the UKF are considered, without any clear empirical or theoretical reasons for making a choice between the Cholesky or the principal square root (or anything else). In [START_REF] Stastny | Comparison of the extended and unscented Kalman filters for angles based relative navigation[END_REF], a divergence problem is noted when using the Cholesky decomposition, and the principal square root was chosen, which performed well. In the present paper, we reach the same conclusion from purely theoretical considerations.

Calculate the weights W

m = (W (0) m , W (1) m , . . . W (2n) m ) t ∈ M 2n+1,1 (R) associated with the σ-points: W (0) m = λ n + λ ; W (i) m = 1 2(n + λ) , i = 1, . . . , 2n (5)
and

W c = (W (0) c , W (1)
c , . . . W (2n) c

) t ∈ M 2n+1,1 (R) :

W (0) c = λ n + λ + 1 + β - c n + k W (i) c = 1 2(n + λ) , i = 1, . . . , 2n (6)
where λ is a scalar parameter defined by λ = c -n. Some authors set k = 0. β is a non negative weighting term which can be used to incorporate knowledge of the higher order moments of the distribution. For a Gaussian distribution the optimal choice is β = 2 [START_REF] Julier | The Scaled Unscented Transformation[END_REF].

3. Propagate each σ-points via the non-linear transformation g

Y = g(X) (7)
where

X ∈ M n,2n+1 (R), Y ∈ M p,2n+1 (
R), and g : R n → R p , [START_REF] Busvelle | Observation and identification tools for non-linear systems: application to a fluid catalytic cracker[END_REF] means that the i th column Y i of the matrix Y is calculated from the i th column of X, as follows:

Y i = g(X i ), i.e. g(X) = [g(X 0 ) • • • g(X 2n )].
4. The mean and the covariance of g(X) are given by

E[g(X)] ≈ µ u def. = g(X)W m = 2n i=0 W (i) m g(X i ) (8) Cov(g(X)) ≈ S u def. = 2n i=0 W (i) c (g(X i ) -m)(g(X i ) -m) t . (9)
5. The cross-covariance of X and g(X) can estimated as

C u def. = 2n i=0 W (i) c (X i -m)(g(X i ) -µ u ) t . (10)
Finally, we define the matrix W ∈ M 2n+1,2n+1 (R) by

W = I -W m • • • W m × diag W (0) c • • • W (2n) c × I -W m • • • W m t . (11)

The unscented Kalman filter algorithm

The equations for the UKF in the continuous case for the system (1) are derived from the unscented transformation and given in [START_REF] Särkkä | On Unscented Kalman Filtering for State Estimation of Continuous-Time Nonlinear Systems[END_REF][START_REF] Särkkä | Bayesian Filtering and Smoothing[END_REF]:

           K(t) = X(t)W h(X(t), t) t R -1 (t) dm(t) dt = f (X(t), t)W m + K(t) (y(t) -h(X(t), t)W m ) dP (t) dt = X(t)W f (X(t), t) t + f (X(t), t)W X(t) t + Q(t) -K(t)R(t)K(t) t . (12)
In this algorithm, Q(t) and R(t) are the covariance matrices of the state measurements and the noise, respectively. They are symmetric and positive definite. In the deterministic case, these two matrices will be considered as parameters to be set.

The non-convergence of UKF as an observer

The unscented Kalman filter is not a good candidate for an exponential observer. Indeed, assume that the function h (in (1)) is linear, and suppose that x(t) = 0 is an equilibrium state. If m(t) = 0, one expects that the observer remains at the equilibrium state 0 (since the innovation term y(t) -h(X(t), t)W m is equal to zero). The state equation in [START_REF] Doumiati | Unscented Kalman filter for real-time vehicle lateral tire forces and sideslip angle estimation[END_REF] becomes

dm(t) dt = f (X(t), t)W m = 1 2(n + λ) n i=1 f ( √ c P (t) i , t) + f (- √ c P (t) i , t). (13)
The right-hand side of ( 13) is equal to zero if f is linear (or simply odd): if it were different from zero, 0 would not be an equilibrium state of [START_REF] Dymirkovsky | New results on stochastic stability of discrete-time Unscented Kalman Filter[END_REF].

For instance, and to be more concrete, set f (x, t) = -x(1 + (2x -1) 2 ) and h(x, t) = x with x(0) = 0. This example is constructed so that 0 is a globally asymptotically stable point of the system, in which x -→ f (x, t) is not odd function (obviously non-linear), as mentioned above. Indeed, if m(t) = 0, i.e. if the observer perfectly estimated the state, then (13) would become

dm(t) dt = f (X(t), t)W m = 1 2(n + λ) 4c P (t) 2 = 2P (t) ( 14
)
where P (t) > 0. Thus, this result contradicts the hypothesis that m(t) converges asymptotically to the solution. The result of a numerical simulation of the example above is shown in Figure 1. A logarithmic scale was used for the time axis to deal with the exponential convergence rate. Figure 1 shows that the UKF (dashed red curve) does not converge (unlike the EKF, the dash-dotted green curve), and that there remains an estimation bias, consistent with Eq. ( 14). Figure 1 also shows the performance of the UKO (dotted blue curve), which will be explained below.

Unscented Kalman Observer

Equations for the UKO

The UKF does not converge as an observer because of the differential equation for m(t) in [START_REF] Doumiati | Unscented Kalman filter for real-time vehicle lateral tire forces and sideslip angle estimation[END_REF]. To resolve this, we introduce a variant using both the approximation of the first order (2) and the unscented approximation (3):

X Y ∼ N m g(m) , P C U C t U S U . ( 15
)
Like the two previous ones, this transformation makes it possible to establish the equations of an observer for system (1), which we will call the unscented Kalman observer (UKO) (for which a simulation has been shown in Figure 1):

           K(t) = X(t)W h t (X(t), t)R -1 (t) dm(t) dt = f (m(t), t) + K(t) (y(t) -h(m(t), t)) dP (t) dt = X(t)W f t (X(t), t) + f (X(t), t)W X t (t) + Q(t) -K(t)R(t)K t (t). (16)

Perturbed Riccati equation

In this section, we will prove that, unlike the UKF used as an observer, the UKO is a locally exponentially converging observer.

We first give a sketch of the proof. First of all, we will develop each term of the Riccati equation with respect to c. Recall that c is a scaling parameter representing the dispersion of σ-points. We will see that the limit when c goes to 0 of the matrix equation in ( 16) is the classical Kalman's Riccati equation. We will then prove, using a perturbation argument, that P (t) remains bounded if c is small enough (Appendix A).

Remark 2. The fact that c should be small in our result may be considered as a strong hypothesis regarding the meaning of c in the UKF. However, it should be recalled that • if c is large and non-linearities are too important, the quality of the unscented transformation may be very poor;

• we want to establish the exponential convergence of an observer, so we need to use an observability hypothesis to ensure convergence without noise. If c is not small enough, the solution of the Riccatilike equation may even be undefined.

We will then conclude, using the classical Lyapunov approach, that the UKO converges exponentially.

Before calculating f (X(t), t)W X t (t) (and h(X(t), t)W X t (t), which is similar), we remark that uW m = 1, U W m = 0 n and U U t = 2Id n (see Notation 1 in section 2.1). We omit the time variable t in the equations below for lighten the notations in calculations.

W X t = (Id 2n+1 -W m u) diag(W c )(Id 2n+1 -W m u) t (mu + √ c √ P U ) t = (Id 2n+1 -W m u) diag(W c )(mu + √ c √ P U -m uW m =1 u - √ c √ P U W m =0n u) t = (Id 2n+1 -W m u) diag(W c )U t √ c √ P = W (1) c (Id 2n+1 -W m u)U t √ c √ P = 1 2c U t √ c √ P since diag(W c )U t =    W 0 c . . . W 2n+1 c      0 t n Id 2n+1 -Id 2n+1   = W (1) c U t .
We will write, temporarily, B = √ P to lighten the notation:

f (X, t)W X t = 1 2 √ c f (mu + √ cBU, t)U t B = 1 2 √ c f (m, t), f (m1 n + √ cB, t), f (m1 n - √ cB, t) U t B = 1 2 √ c f (m1 n + √ cB, t) -f (m1 n - √ cB, t) B.
We now remark that setting f = Id n , we see that XW X t = P and hence the UKO is, in the linear case, just a Kalman filter. Using Taylor's formula, develop each coefficient of the n × (2n + 1) matrix up to second order f (m1

n + √ cB, t) -f (m1 n - √ cB, t) f (m1 n + √ cB, t) -f (m1 n - √ cB, t) i,j = 2 √ c n k=1 ∂f i ∂x k (m, t)B k,j + c n k,l=1 ∂ 2 f i (m + τ i,j √ cB •,j , t) ∂x k ∂x l B k,j B l,j (17)
where -1 ≤ τ i,j ≤ 1 and B •,j denotes the j th column of B. The approximation of f in matrix form is then

f (X, t)W X t = 1 2 √ c f (m1 n + √ cB, t) -f (m1 n - √ cB, t) B = AP + √ c 2 H f √ P (18)
and similarly for h:

h(X, t)W X t = 1 2 √ c h(m1 n + √ cB, t) -h(m1 n - √ cB, t) B = CP + √ c 2 H h √ P (19)
where A = A(t) and C = C(t) are time-varying matrices (the Jacobian of f and h respectively)

A(t) = ∂f ∂x (m, t) and C(t) = ∂h ∂x (m, t), (20)
H f ∈ M n,n (R) (and H h ∈ M n,n (R) which is similar) being defined as (H f ) i,j = n k,l=1 ∂ 2 f i (m + τ i,j √ cB •,j , t) ∂x k ∂x l B k,j B l,j = B j,• ∂ 2 f i (m + τ i,j √ cB •,j , t)B •,j . (21)
We assumed that each coefficient of the Hessian matrices2 are bounded from above by M f > 0 (respectively, M h > 0), that is to say,

M f = max 1≤i,k,l≤n sup x∈R n ,t≥0 ∂ 2 f i (x, t) ∂x k ∂x l ( 22
)
and the same with M h :

M h = max 1≤i,k,l≤n sup x∈R n ,t≥0 ∂ 2 h i (x, t) ∂x k ∂x l . ( 23
)
Using the expressions in ( 18) and ( 19), the system (16) becomes

                                             K(t) = P (t)C(t) t R -1 (t) + √ c 2 P (t)H h (t) t R -1 (t) dm(t) dt = f (m(t), t) + K(t) (y(t) -h(m(t), t)) dP (t) dt = P (t)A(t) t + A(t)P (t) + Q(t) + √ c 2 ( P (t)H f (t) t + H f (t) P (t) t ) -P (t)C(t) t R -1 (t)C(t)P (t) - √ c 2 (P (t)C(t) t R -1 (t)H h (t) P (t) t + P (t)H h (t) t R -1 (t)C(t)P (t)) - c 4 P (t)H h (t) t R -1 (t)H h (t) P (t) t . ( 24
)
Rewriting the system [START_REF] Julier | A general method for approximating nonlinear transformations of probability distributions[END_REF] with S(t) = P (t) -1 , we obtain

                                       K(t) = S -1 (t)C(t) t R -1 (t) + √ c 2 P (t)H h (t) t R -1 (t) dm(t) dt = f (m(t), t) + K(t) (y(t) -h(m(t), t)) dS(t) dt = -A(t) t S(t) -S(t)A(t) -S(t)Q(t)S(t) - √ c 2 (S(t) P (t)H f (t) t S(t) + S(t)H f (t) P (t) t S(t)) + C(t) t R -1 (t)C(t) + √ c 2 (C(t) t R -1 (t)H h (t) P (t) t S(t) + S(t) P (t)H h (t) t R -1 (t)C(t)) + c 4 S(t) P (t)H h (t) t R -1 (t)H h (t) P (t) t S(t). (25)

Convergence analysis

First, we will consider the matrix equation in [START_REF] Kalman | A new approach to linear filtering and prediction problems[END_REF]. We will prove the following result, which is a crucial result for the proof of the convergence of the UKO. This result applies to the matrix equation of the UKO, which is the same matrix equation as for the UKF. Therefore, this result may be applied to establish a priori bounds on S(t) (or, equivalently, on P (t)), as required in Theorem 3.1 in [START_REF] Xu | Stochastic stability of the continuous-time unscented Kalman filter[END_REF].

Definition 2. (See, for instance, [START_REF] Bornard | Observability and observers[END_REF]). We define the observability Gramian as

WT (t) = t t-T Ψ t (τ, t)C t (τ )R -1 (τ )C(τ )Ψ(τ, t)dτ
for T > 0 and t ≥ T , where Ψ(τ, t) is the state-transition matrix associated to the linear system defined by

A(t), that is,    d dt Ψ(τ, t) = A(τ )Ψ(τ, t) Ψ(t, t) = Idn. (26)
If WT (t) is symmetric positive definite and uniformly bounded for any t ≥ 0, then the system characterized by the pair (A(t), C(t)) and written as dx(t) dt = A(t)x(t)

y(t) = C(t)x(t) (27)
is said to be completely uniformly observable.

Before proving that the proposed observer is an exponential observer, we need to expand the functions f and h of the system (1) up to first order, and make the following assumptions (see [START_REF] Julier | A new approach for filtering nonlinear systems[END_REF] for notation):

f (x(t), t) -f (m(t), t) = A(t)(x(t) -m(t)) + ψ(x(t), m(t)), h(x(t), t) -h(m(t), t) = C(t)(x(t) -m(t)) + φ(x(t), m(t)) (28)
where A(t) and C(t) have been defined in [START_REF] Julier | A new approach for filtering nonlinear systems[END_REF], and ψ(x(t), m(t)), and φ(x(t), m(t)) are the remaining non-linear terms.

Assumption 1. There are positive real numbers κ ψ , κ φ , ψ , φ > 0 such that the non-linearities are bounded by

ψ(x1(t), x2(t)) ≤ κ ψ (x1(t) -x2(t)) 2 , φ(x1(t), x2(t)) ≤ κ φ (x1(t) -x2(t)) 2 . for x1(t), x2(t) ∈ R n such that x1(t) -x2(t) ≤ ψ and x1(t) -x2(t) ≤ φ , respectively.
Assumption 2. There are positive real numbers c, rmin and qmin > 0 such that Q(t), R(t) and C(t) are timevarying matrices bounded by

C(t) ≤ c √ nrmin ≤ R(t) qminIdn ≤ Q(t).
with c is the lower bound of the matrix C(t), rmin and qmin are the smallest eigenvalue of Q(t) and R(t) respectively.

Theorem 3. We now consider the UKO in continuous time, and the following Riccati-like equation,

   dS(t) dt = -A(t) t S(t) -S(t)A(t) -S(t)(Q(t) + Ω(S(t)))S(t) + C(t) t R -1 (t)C(t) S(0) = S0, (29)
where

Ω(S(t)) = - √ c 2 ( P (t)H f (t) t + H f (t) P (t) t ) + √ c 2 P (t)C(t) t R -1 (t)H h (t) P (t) t + P (t)H h (t) t R -1 (t)C(t)P (t) + c 4 P (t)H h (t) t R -1 (t)H h (t) P (t) t (30)
and where (A(t), C(t)) is a time-dependent observable pair (Definition 2). We assume that all the elements of the matrices A(t) and C(t) are uniformly bounded, Q(t) and R(t) are symmetric positive definite matrices and S(0) is a symmetric positive definite matrix taken in the compact set S = {S ∈ S + n , aIdn ≤ S(0) ≤ bIdn}, with 0 < a < b. Assuming ( 22) and (23) hold for f and h, under Assumptions 1 and 2, there exist c0 and 0 < α < β such that for any c ≤ c0, αIdn ≤ S(t) ≤ βIdn.

α and β are called the lower and upper bounds of S(t) respectively.

Proof. The proof is detailed in the Appendix.

Let us define

ε(t) = x(t) -m(t), then ε(t) = ẋ(t) -ṁ(t) = f (x(t), t) -f (m(t), t) -K(t) (h(x(t), t) -h(m(t), t)) . (31)
Definition 3. The equilibrium point ε(t) = 0 is an exponentially stable equilibrium point of Eq. ( 31) if there exist constants m, α > 0 and such that

ε(t) ≤ η ε(0) e -αt (32)
holds for every t ≥ 0 and for all ε(0) ≤ .

Theorem 4. Consider the non-linear system (1) and the observer (UKO) given by system [START_REF] Kalman | A new approach to linear filtering and prediction problems[END_REF] and suppose that all the assumptions of Theorem 3 are fulfilled, so that αIdn ≤ S(t) ≤ βIdn.

Then, for all t ≥ 0, there are positive constants k,L,δ and η, hence the following inequality is satisfied:

ε(t) 2 ≤ β α ε(0) 2 e -αq min + (2 k+ √ cβL) 0 α + √ c α (βδ+ √ cβη) t (33)
Remark 5. As can be seen from [START_REF] Maree | On convergence of the unscented Kalman-Bucy filter using contraction theory[END_REF], c is defined as in Section 2.1 and it must be small enough to ensure the negativity of -αqmin

+ (2 k+ √ cβL) 0 α + √ c α (βδ + √ cβη)
. Exact expressions for k,L,δ and η appear at the end of the proof.

Proof. The proof of this theorem relies on an analysis of the dynamics (31) of the estimation error ε(t).

Using the development [START_REF] Kalman | A new approach to linear filtering and prediction problems[END_REF] :

ε(t) = [A(t) -K(t)C(t)]ε(t) + ψ(x(t), m(t)) -K(t)φ(x(t), m(t)) = A(t)ε(t) -P (t)C(t) t R(t) -1 C(t)ε(t) -P (t)C(t) t R(t) -1 φ(x(t), m(t)) + ψ(x(t), m(t)) - √ c 2 P (t)H h (t) t R(t) -1 C(t)ε(t) - √ c 2 P (t)H h (t) t R(t) -1 φ(x(t), m(t)). (34)
We choose the classical quadratic Lyapunov function

V (ε(t)) = ε(t) t S(t)ε(t), (35)
where S(t) is the solution of the Riccati equation ( 25), and we will compute its time derivative

V (ε(t)) = 2 ε(t) t S(t)ε(t) + ε t Ṡ(t)ε(t). ( 36
)
In the following, to ease the notation and as long as there is no ambiguity, we omit again the time variable t in the equations. Inserting ε, Ṡ in [START_REF] Picard | Efficiency of the extended Kalman filter for nonlinear systems with small noise[END_REF], we obtain

V (ε) = 2[Aε -P C t R -1 Cε - √ c 2
√ P H t h R -1 Cε + ψ(x, m) -P C t R -1 φ(x, m) - √ c 2
√ P H t h R -1 φ(x, m)] t Sε + ε t [-A t S -SA -SQS - √ c 2 (S √ P H t f S + SH f √ P t S) + C t R -1 C + √ c 2 (C t R -1 H h √ P t S + S √ P H t h R -1 C) + c 4 S √ P H t h R -1 H h √ P t S]ε. (37)
With a few rearrangements, V (ε) becomes

V (ε) = -ε t SQSε -ε t C t R -1 Cε + 2ε t Sψ(x, m) -2ε t C t R -1 φ(x, m) - √ cε t S[ √ P H t h R -1 φ(x, m) + √ P H t f Sε - √ c 4 √ P H t h R -1 H h √ P t Sε]. (38)
Using Theorem 3 and Assumptions 1 and 2 we have

V (ε) ≤ -αqminε t Sε + (2 k + √ cβL) ε + ( √ cβδ + cβη) ε 2 (39) with k = (βκ ψ + c √ nκ φ r min ), L = n 2 M h κ φ n 3/4 r min α 3/2 , δ = n 3/2 M f βn 3/4 α 3/2 and η = n 5 M 2 h β 4α 3 r min . For ε ≤ 0 with 0 ≤ min( ψ , φ ), V (ε) ≤ -αqminε t Sε + (2 k + √ cβL) 0 + ( √ cβδ + cβη) ε 2 . ( 40
)
From ( 40) and

ε 2 ≤ 1 α ε t Sε we obtain V (ε) ≤ -αqmin + (2 k + √ cβL) 0 α + √ c α (βδ + √ cβη) ε t Sε (41)
and hence

V (ε(t)) ≤ V (ε(0))e -αq min + (2 k+ √ cβL) 0 α + √ c α (βδ+ √ cβη) t . ( 42
)
Since the Lyapunov function is bounded by

α ε(t) 2 ≤ V (ε(t)) ≤ β ε(t) 2 (43)
then

ε(t) ≤ β α ε(0) e 1 2 -αq min + (2 k+ √ cβL) 0 α + √ c α (βδ+ √ cβη) t (44)
Let 0 and c be small enough such that

-αqmin + (2 k + √ cβL) 0 α + √ c α (βδ + √ cβη) < 0 and let ε(0) ≤ β α 0 such that ε(t) ≤ 0e 1 2 -αq min + (2 k+ √ cβL) 0 α + √ c α (βδ+ √ cβη) t . (45)
This ensure that ε(t) ≤ 0, ∀t ≥ 0 and that the inequality (45) holds, which proves Theorem 4.

Applications and simulations

Distillation column : the binary distillation process

To illustrate the theorem, let us consider the classical CMO (constant molar overflow) model of a binary distillation column. In the following, we will only consider a three plates model, which is very simple and easy to describe (see [START_REF] Deza | Exponentially converging observers for distillation columns and internal stability of the dynamic output feedback[END_REF] for a more general presentation). The difference between this low-dimension model with the CMO model of a depropanizer or debutanizer is only quantitative. In particular, nonlinearities are exactly the same. The CMO model is only a material balance of the lightest component on each plate of the column:

             H1 dx1 dt = V (t)(k(x2) -x1) H2 dx2 dt = F (ZF -x2) + L(t)(x1 -x2) + V (t)(k(x3) -k(x2)) H3 dx3 dt = (L(t) + F )(x2 -x3) + V (t)(x3 -k(x3)) (46)
where H1, H2, H3 (liquid holdups), F (feed flow) and ZF (feed composition) are positive constants, k(x) = αx 1+(α-1)x with α > 1, and L (liquid flow) and V (vapor flow) are smoothly varying positive functions of time (control variables). The state is (x1, x2, x3) (composition in liquid phase) and the output is x3 (bottom composition). This system has been extensively used to illustrates high-gain observer approach because it is already in a canonical form of observability, see [START_REF] Deza | Exponentially converging observers for distillation columns and internal stability of the dynamic output feedback[END_REF][START_REF] Gauthier | Deterministic Observation Theory and Applications[END_REF]). The state of this system belongs to [0, 1] 3 but the state of the filter may leave the state space so one should extend k such that it remains twice differentiable and Lipschitz. It is an easy task, see for instance Figure 2 where the function have been extended between 1 and 1 + ε using a polynomial function of degree 3 (a spline), and extended linearly beyond 1 + ε such that k remains Lipschitz. For negative values, k is extended so as to be an odd function. Therefore, k(x) is clearly Lipschitz (linear outside a compact interval). We have

A(t) =    -V (t) H 1 V (t) H 1 dk dx (m1(t)) 0 L(t) H 2 1 H 2 (-F -L(t) -V (t) dk dx (m2(t))) V (t) H 2 dk dx (m3(t)) 0 L(t)+F H 3 1 H 3 (-L(t) -F + V (t)(1 -dk dx (m3(t))))   
and C(t) = 0 0 1 . Therefore, matrix are bounded and second derivatives also (since k is Lipschitz twice differentiable on R). Moreover, the system is clearly uniformly observable since the following matrix is triangular and its diagonal is bounded from below by a positive constant: (The coefficients below the diagonal do not have to be calculated) The non-linearities of the system being only on k, it is clear that Assumption 1 is satisfied, as well as ( 22) and ( 23), and since C is constant, Assumption 2 is also verified.

Numerical Simulations

Now, we simulate the model of the binary distillation column [START_REF] Stastny | Comparison of the extended and unscented Kalman filters for angles based relative navigation[END_REF]. For that we choose H1 = 40, H2 = 10, H3 = 80, F = 10, ZF = 0.4, L = 13, V = 17 and α, the parameter of the function k(x), is set to 2. The initial conditions are x0 = [0.5, 0.5, 0.5] t for the system and m0 = [1, 0.6, 0.3] t for the observer. The initial condition of the Riccati differential equation ( 16) is set to P (0) = Id3, while the positive definite matrix Q is set to Q = Id3 and R = 1. We have set β = 0, k = 0 and c = 0.03 in UKO (see (4), ( 5) and ( 6)).

Figure 3 shows the evolution of the state and estimates of the system as a function of time for both algorithms UKO and EKF. We did not add noise, our objective being to compare the convergence rates.

Observers UKO and EKF converge locally and perform similarly. They only differ in the calculation made: EKF requires the calculation of the Jacobian of the non-linear functions f (.) and h(.), rather than the propagation of the σ-points, as does the UKO.

Conclusion

In this paper, we have shown that the unscented Kalman filter is not an exponentially convergent observer, and we have proposed a slight modification of it which allows building an exponentially convergent observer, called the unscented Kalman observer. The convergence of this observer has been proved locally if the tuning parameter c of the observer is small enough.

In [START_REF] Daid | Unscented Kalman Observer[END_REF], we built a high-gain version of the unscented Kalman observer and performed some comparison of the performance of each observer ( the unscented Kalman filter, the unscented Kalman observer, and the high-gain unscented Kalman filter). Moreover, if we assume that the solution of the Riccati equation is bounded, then the unscented Kalman observer becomes a globally converging observer. Using Theorem 3 of this paper and the proof in Appendix A, we should be able to prove the global exponential convergence of the high-gain unscented Kalman observer under the same observability conditions.

  and the column vector 0 n = (0, 0, . . . , 0) t ∈ R n . Let u denote the row vector u = [1, 1 n , 1 n ] ∈ R 2n+1 and U the n × (2n + 1) matrix U = [0 n , Id n , -Id n ] (The brackets [ ] denote matrix concatenation).

Figure 1 :

 1 Figure 1: The biased estimation of UKF

Figure 2 :

 2 Figure2: k (in red, between 0 and 1) and its odd (black) C 2 -continuation, spline prolongation between 1 and 1 + ε (blue), linear beyond (green).

  0

Figure 3 :

 3 Figure 3: State estimation with UKO and EKF.

More generally, Mq,p(R) is the set of real-valued matrices of dimension q × p.

H f and H h are not strictly speaking Hessian matrices, and can not be written as the product of Hessian matrices with B, since the coefficients are second derivatives of f (respectively, h) but are not evaluated at the same points.

This work is carried out with the contribution of the Hubert Curien Tassili Research Program (PHC)

Appendix A. Bounds on the solution of the Riccati equation

In this appendix, we prove Theorem 3, following [START_REF] Gauthier | Deterministic Observation Theory and Applications[END_REF]. However, the Riccati-like equation of UKO is different from the classical Kalman Riccati equation, and we interpret the difference as a perturbation of the Kalman Riccati equation. Only proofs notably different from those of [START_REF] Gauthier | Deterministic Observation Theory and Applications[END_REF] will be given in detail.

First of all, we introduce some notation and recall some basic properties.

Notation 4.

• Mn is the set of (n × n) matrices having real entries;

• Sn is the set of symmetric matrices of Mn;

• S + n is the set of positive definite matrices of Sn; • Tr(A) denotes the trace of the matrix A ∈ Mn;

• For any matrix A, A 2 denote the matrix norm induced from the Euclidean vector norm : A 2 = sup x 2 =1 Ax 2;

• For any matrix A, A F denote the Frobenius norm: A F = Tr(A t A);

• • denotes any matrix norm;

• For any matrix Q ∈ Sn, qmin (respectively, qmax) denotes the smallest (respectively, largest) eigenvalue of Q.

We now recall some properties.

Properties 6.

• d dt Tr(M ) = Tr dM dt for any M ∈ Mn; • let A, B ∈ Mn, | Tr(AB) |≤ Tr(A t A) Tr(B t B); 2 , and Tr(SQS) ≥ q min n Tr(S) 2 ;

We consider the Riccati equation with a disturbance term depending on time as follows.

In addition, we assume that ω(t) is symmetric and ω(t) F ≤ ω0 so that (Q(t) + ω(t)) remains symmetric positive definite. Since qminIdn ≤ Q(t) ≤ qmaxIdn, it is sufficient that ω0 < qmin.

The proof is divided into three parts:

• In the first part, we prove the existence and the positive definiteness of S(t) and the existence of an upper bound on S(t); • In the second part, we prove the existence of a lower bound on S(t);

• Finally, we deduce the result when ω(t) is replaced by Ω(S(t)) as in Theorem 3.

Appendix A.1. Upper bound with perturbation ω(t) Lemma 7. (Lemma 2.12 in [START_REF] Gauthier | Deterministic Observation Theory and Applications[END_REF] For any µ ∈ R * and any solution S : [0, T [→ Sn of (A.1) (possibly T = +∞), we have, for all t ∈ [0, T [

for t ∈ [0, T [. Applying the variation of constants formula to

and writing Φ(t, τ ) for the state-transition matrix defined by dΦ(t,τ ) dτ = -A(t) t Φ(t, τ ), Φ(τ, τ ) = Idn, we get the relation (A.2), due to the fact that Φ(t, τ ) = Ψ t (τ, t). Where Ψ(τ, t) is defined in (26) (Definition 2). Moreover, Tr(S(t)) ≤ max(Tr(S(0)), s1) where s1 is the unique positive solution of -as 2 + 2bs + c = 0.

As a consequence,

hence, setting β = max(Tr(S(0)), s1), for all t < T , S(t) ≤ βIdn.

Proof.

The matrix is bounded due to Assumption 2, so using the properties of the trace (Properties 6),

Tr(S(t))

and

Therefore, (A.3) holds. The fact is that Tr(S(t)) ≤ max(Tr(S(0)), s1) is independent of the perturbation as soon as ω(t) F ≤ ω0. Roughly speaking, this is a direct consequence of (A.3), by noting that • -as 2 + 2bs + c = 0 has two roots s0 and s1 s.t. 0 < s0 < s1, • d Tr(S(t)) dt is negative once S(t) > s1.

Lemma 9. (Lemma 2.15 in [START_REF] Gauthier | Deterministic Observation Theory and Applications[END_REF]). Let S : [0, e(S)[→ S + n be a maximal semi-solution of (A.1). If S(0) = S0 is positive definite, then e(S) = +∞ and S(t) is positive definite for all t ≥ 0 (and therefore, T = +∞ in Lemma 8).

Proof. Suppose that S is not always positive definite. Let θ = inf t|S(t) / ∈ S + n . Then S(t) ∈ S + n for all t ∈ [0, θ[. Using lemma 2.13 in [START_REF] Gauthier | Deterministic Observation Theory and Applications[END_REF] and Lemma 8, we see that Tr(S(t)) ≤ max(Tr(S0), s1). Then, S(t) = Tr(S(t) 2 ) ≤ Tr(S(t)) ≤ max(Tr(S0), s1). Choose µ >| (Q(t) + ω(t)) | max(Tr(S0), s1). Apply Lemma 7 with t = θ. S(θ) = (I) + (II) + (III), (I) = e -µθ Φ(θ, 0)S0Φ t (θ, 0),

• (I) is obviously positive definite, hence S(0) is positive definite.

• (II) R(t) is symmetric positive definite and C(t) is bounded by c > 0 hence (II) is symmetric positive definite.

• (III) depends on S(τ ) -S(τ )(Q(τ )+ω(τ ))S(τ ) µ , which can be rewritten

Since S(t) is positive definite for 0 < t < θ. Therefore the positiveness of (III) depends on Idn -√

. From the definition of µ we have

Hence S(τ )-S(τ )(Q(τ )+ω(τ ))S(τ ) µ is positive definite. So is (III). This is in contradiction with the supposition that S(θ) is not positive definite. Now by Lemma 8, Tr(S(t)) ≤ max(Tr(S0), s1) for all t ∈ [0, e(S)[ and e(s) = +∞.

Appendix A.2. Lower bound with perturbation ω(t) Lemma 10. For each time T0 > 0, there exists a constant γ0 > 0 such that if S : [0, +∞[→ Sn is a solution of (A.1) such that S(0) = S0 ∈ S + n , then S(t) ≥ γ0Idn, for t ≥ T0. Moreover, if S(0) > 0, there exists α such that S(t) ≥ αIdn, ∀t ≥ 0.

Proof. We use formula (A.2) from Lemma 7. The first and the third terms are positive (as soon as µ is large enough, see the proof of Lemma 9). Now consider the second term,

≥ e -µT 0 K(T0) for any t ≥ T0, with K(T0) > 0 by definition of persistent observability (see [START_REF] Bornard | Observability and observers[END_REF]). We get the lemma taking γ0 = e -µT 0 K(T0).

Therefore, S(t) > 0 for all t ≥ 0 (Lemma 9) and S(t) ≥ γ0Idn ∀t ≥ T0. So, α0 = inf 0≤t≤T 0 S(t) is strictly positive and α = inf(α0, γ0) is such that S(t) ≥ αIdn ∀t ≥ 0, which proves the result.

Appendix A.3. Bounds on the Riccati-like equation

Let us come back to the Riccati-like equation with perturbation Ω(S(t)) :

Lemma 11. Suppose there exists α1 and β1 such that α1Idn ≤ S(t) ≤ β1Idn. Then there exists Ω0 > 0, and c0 > 0 small enough such that for any 0 < c < c0,, we have

Proof. From (30), we have

Under Assumption 1 on the second order derivatives of f and h, and Assumption 2, and the bound on S(t)

F , it follows that

for c small enough and Ω(S(t)) To prove the Theorem 3, which bounds the solution S(t) of the Riccati-like equation,

• We first choose ω0 such that if ω(t) F ≤ ω0 then αIdn ≤ S(t) ≤ βIdn (Lemmas 8, 9 and 10);

• Then we choose c such that if α 2 Idn ≤ S(t) ≤ 2βIdn then Ω(S(t)) F ≤ √ cΩ0 ≤ ω0 (Lemma 11).

We have to prove that ∀t ≥ 0, αIdn ≤ S(t) ≤ βIdn.

Using a reductio ad absurdum, let t0 be the largest time value such that, ∀t ≤ t0, αIdn ≤ S(t) ≤ βIdn. Then there exist t1 > t0 (see Figure A.4) such that α 2 Idn ≤ S(t) ≤ 2βIdn because S(t) is continuous, and so Ω(S(t)) F ≤ ω0 for t ∈ [t0, t1], and therefore for any t0 < t ≤ t1, αIdn ≤ S(t) ≤ βIdn. This contradicts the existence of t0.