Tristan Bonargent
email: tristan.bonargent@unicaen.fr

Tomas Ménard

| Olivier Gehan

Eric Pigeon

Adaptive observer design for a class of Lipschitz nonlinear systems with multi-rate outputs and uncertainties -Application to attitude estimation with gyro bias

Keywords: Adaptive observer, multi-rate, uncertainties, attitude estimation

The design of an adaptive observer for a class of multi-input Multi-output (MIMO) Lipschitz nonlinear systems with uncertainties and multi-rated outputs is proposed in this paper. The sampling instants of the different outputs can be asynchronous and irregular. The exponential convergence of the proposed observer is proved with the help of a Lyapunov approach. The stability is ensured, provided that some conditions, including a persistent excitation condition and a maximum value condition on the sample rate of each output, are verified. The performances of the observer are illustrated through simulations of the attitude estimation problem using sensors from an inertial measurement unit and considering gyro bias.

INTRODUCTION

Adaptive observers have received a lot of attention during the last decades. These observers are used to simultaneously estimate the state of a system and some unknown parameters. Their various applications such as fault detection, adaptive control or signal transmission show the importance of the given problem (see for instance [START_REF] Baldi | Fault diagnosis for satellite sensors and actuators using nonlinear geometric approach and adaptive observers[END_REF][START_REF] Stefanopoulou | Pressure and temperature-based adaptive observer of air charge for turbocharged diesel engines[END_REF]).

Most of the works dedicated to adaptive observer design consider systems whose output is available continuously with respect to time. Early contributions have been concerned with linear invariant systems such as in [START_REF] Kreisselmeier | Adaptive observers with exponential rate of convergence[END_REF] and [START_REF] Luders | An adaptive observer and identifier for a linear system[END_REF], and thereafter with linear time-varying systems such as in [START_REF] Zhang | Adaptive observer with exponential forgetting factor for linear time varying systems[END_REF][START_REF] Zhang | Adaptive observer for multiple-input-multiple-output (MIMO) linear time-varying systems[END_REF]. A consequent number of papers has then dealt with non-linear systems with linear parametrization. A first idea has been to seek for new coordinates such that the error dynamics are linear. Such changes of coordinates are usually related to some canonical forms (see [START_REF] Bastin | Stable adaptive observers for nonlinear time-varying systems[END_REF][START_REF] Marino | Global adaptive observers and output-feedback stabilization for a class of nonlinear systems[END_REF][START_REF] Marino | Robust adaptive observers for nonlinear systems with bounded disturbances[END_REF]). Using Lyapunov functions, some other approaches do not require to linearize the systems. For example, delayed observers have been used in [START_REF] Stamnes | Redesign of adaptive observers for improved parameter identification in nonlinear systems[END_REF] or high gain design in [START_REF] Xu | Nonlinear system fault diagnosis based on adaptive estimation[END_REF][START_REF] Besançon | On adaptive observers for state affine systems[END_REF][START_REF] Farza | Adaptive observers for nonlinearly parameterized class of nonlinear systems[END_REF]. Then the consideration of a larger class of system with non-linear parametrization has received much attention. In fact, in order to deal with this problem, several approaches have been considered in the literature. Indeed, a min-max optimization algorithm has been used in [START_REF] Loh | Adaptation in the presence of a general nonlinear parameterization: an error model approach[END_REF] to overcome the limits of the gradient descent method for adaptive systems with concave/convex parametrization. Similarly, an optimization approach for adaptive control has been used in [START_REF] Kojić | Adaptive control of nonlinearly parameterized systems with a triangular structure[END_REF]. Some high-gain based observers have been proposed for systems with a triangular structure. Indeed, Single-Input Single-Output (SISO) systems have been considered in [START_REF] Grip | Estimation of states and parameters for linear systems with nonlinearly parameterized perturbations[END_REF] and [START_REF] Menard | Adaptive high gain observer for uniformly observable systems with nonlinear parametrization[END_REF] and MIMO systems in [START_REF] Farza | Adaptive observer design for a class of nonlinear systems. Application to speed sensorless induction motor[END_REF]. An observer based on the concept of weakly attracting sets has also been presented in [START_REF] Tyukin | Adaptive observers and parameter estimation for a class of systems nonlinear in the parameters[END_REF], but which does not guarantee the asymptotic stability.

While most of the works consider continuous-time measurements, for real systems, the sensor measurements are usually available only at discrete instants. Following this paradigm, a full discrete-time approach which allows to reconstruct both the state and the unknown parameters in discrete time has been used for linear systems in [START_REF] Guyader | Adaptive Observer for Discrete Time Linear Time Varying Systems[END_REF] and for nonlinear systems in [START_REF] Caccavale | Adaptive Observer for Fault Diagnosis in Nonlinear Discrete-Time Systems[END_REF][START_REF] Srinivasarengan | An adaptive observer design approach for discrete-time nonlinear systems[END_REF], for example. Another approach is to consider a continuous estimation using the continuous-time model which is fed by the discrete-time measurements. This allows to obtain global convergence in many cases, contrary to a full discrete-time approach which usually lead to only semi-global convergence [START_REF] Arcak | A framework for nonlinear sampled-data observer design via approximate discrete-time models and emulation[END_REF]. Therefore, several observers which estimate continuously the state from discrete-time measurements have been proposed. Indeed, a class of state affine MIMO systems with linear parametrization has been considered in [START_REF] Hann | Continuous adaptive observer for state affine sampled-data systems[END_REF] and state-affine SISO systems with non-linear parametrization in [START_REF] Farza | Adaptive observers for a class of uniformly observable systems with nonlinear parametrization and sampled outputs[END_REF]. Similarly, an adaptive observer for another class of MIMO systems has been designed in [START_REF] Zhao | Continuous-discrete-time adaptive observers for nonlinear systems with sampled output measurements[END_REF].

In all the aforementioned works, convergence has been obtained for large classes of systems, however no uncertainty on the dynamics is considered while it is rather common in practice. Furthermore, the outputs are supposed to be synchronized, which means that the measurements of the different outputs are all obtained at the same sampling instants. This may not be the case in practice, since the considered sensors might not be synchronized or have different sampling rates. Thus, one considers in this paper, the design of an adaptive observer for a class of uniformly observable non-linear MIMO systems with a triangular structure, dynamic uncertainties and multi-rated outputs. In particular, the outputs can be asynchronous and can have irregular sampling instants. Using a high-gain approach, the proposed adaptive observer estimates continuously the state of the system from discrete time measurements. The exponential convergence of the observer is proved with a Lyapunov approach under a persistent excitation condition and provided that for each output, the corresponding sampling periods are over-valued by a given bound. An advantage of the followed approach is that it is fairly easy to tune since there are only a few parameters to set.

The remaining of the paper is organized as follows. First, the considered model is presented in Section 2. The proposed observer and the convergence result are given in Section 3. Simulations of an attitude estimation system using sampled sensors commonly found in inertial measurement units are provided in Section 4 in order to illustrate the proposed approach. Finally, Section 5 concludes the article.

In the rest of the paper, one respectively denotes min ( ) and max ( ) the minimum and maximum eigenvalues of , where is a square symmetric matrix. For a continuous matrix valued function  → ( ) where for each ≥ 0, ( ) is a square symmetric matrix, one denotes max ( ) = sup ≥0 max ( ( )) and min ( ) = inf ≥0 min ( ( )).

PROBLEM STATEMENT

One considers, in this paper, the class of nonlinear systems whose dynamics are given by

̇ ( ) = ( ) + Ψ( ( ), ( )) + ( ( ), ( ), ) + ( ) (1)
where ∈ ℝ is the state, ∈ ℝ is the input, ∈ ℝ is the vector of unknown parameters and ∈ ℝ the unknown time varying uncertainties.

In particular, one assumes that the state , the unknown parameters and the uncertainties can be partitioned as

= ⎡ ⎢ ⎢ ⎢ ⎢ ⎣ 1 2 ⋮ ⎤ ⎥ ⎥ ⎥ ⎥ ⎦ with = ⎡ ⎢ ⎢ ⎢ ⎢ ⎣ 1 2 ⋮ ⎤ ⎥ ⎥ ⎥ ⎥ ⎦ , ∈ ℝ , = 1, … , , = 1, … , , (2)
= ⎡ ⎢ ⎢ ⎣ 1 ⋮ ⎤ ⎥ ⎥ ⎦ , = ⎡ ⎢ ⎢ ⎣ 1 ⋮ ⎤ ⎥ ⎥ ⎦ ∈ ℝ , = 1, … , = ⎡ ⎢ ⎢ ⎣ 1 ⋮ ⎤ ⎥ ⎥ ⎦ with ∈ ℝ , (3)
and that the dynamics of , = 1, … , are given by:

̇ = + Ψ ( , ) + ( , , ) + , if is not empty + ( , , ) + , if is empty (4) with = ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ 0 0 … 0 ⋮ ⋱ ⋱ ⋱ ⋮ ⋮ ⋱ ⋱ 0 ⋮ ⋱ 0 … … … 0 ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ , = ⎡ ⎢ ⎢ ⎢ ⎢ ⎣ 0 ⋮ 0 ⎤ ⎥ ⎥ ⎥ ⎥ ⎦ , (5)
Ψ = ⎡ ⎢ ⎢ ⎢ ⎣ Ψ 1,1 … Ψ 1, ⋮ ⋮ Ψ ,1 … Ψ , ⎤ ⎥ ⎥ ⎥ ⎦ with Ψ , ∈ ℝ , = ⎡ ⎢ ⎢ ⎣ 1 ⋮ ⎤ ⎥ ⎥ ⎦ with ∈ ℝ , ( 6
)
Remark 1. It should be noted that contains the sub-components of that were not present in the dynamics of 1 , … , -1 and that first appear in the dynamics of . Then, some may be empty if no new unknown parameter appears in the dynamics of . Furthermore, when a new unknown parameter appears in the dynamics of , it is supposed to appear in a linear way.

Structure of the nonlinearities

Let ∈ {1, … , }, one now details the structure of the nonlinear functions Ψ and contained in the dynamics of . In particular, one specifies the dependence of Ψ and with respect to both the state and the unknown parameters . First, the functions Ψ and are supposed to have a triangular structure according to the state both between the sub-states 1 , … , and for each sub-state between 1 , … , , that is, for = 1, … , , = 1, … , , = 1, … , :

( , , ) = ( , 1 , … , -1 , 1 , … , , ) (7)
Ψ , ( , ) = Ψ , ( , 1 , … , -1 , 1 , … , ) (8)
In order to state the structure of according to the unknown parameters , one first introduces the indices ∈ ℕ, = 1, … , , representing the sub-components where appears for the first time, that is such that

Ψ , ≢ 0 and Ψ , ≡ 0 if ≠ (9)
Similarly to the structure of the nonlinearities with respect to the state, the function is supposed to have a triangular structure between the sub-components 1 , … , and a special structure for each sub-component between 1 , … , , that is, for = 1, … , , = 1, … , , = 1, … , :

( , , ) ≡ 0 if > and ( , , ) ≡ 0 if ≥ (10)
Remark 2. The first condition of [START_REF] Grip | Estimation of states and parameters for linear systems with nonlinearly parameterized perturbations[END_REF] means that cannot depend on sub-components of that did not appear linearly for the first time in the dynamics of 1 , … , while the second condition means that cannot depend on sub-components of that did not appear linearly for the first time in the dynamics of 1 , … , -1 .

Measurements of the considered class of systems

One assumes that only the first sub-components 1 of each sub-state are measured. That is, the measured outputs ∈ ℝ are given by

= 1 = , with = 0 … 0 (11)
Moreover, each output is supposed to be sampled at independent and possibly irregular sampling instants, that is:

1 ( 1 ) = 1 1 ( 1 ) (12)
⋮

( ) = ( ) (13)
for ∈ ℕ, and the sampling instants are assumed to verify

1 0 < 1 1 < 1 2 < …, lim →+∞ 1 = +∞ and | | | 1 +1 -1 | | | < 1 (14) ⋮ 0 < 1 < 2 < …, lim →+∞ = +∞ and | | | +1 - | | | < (15)
3 MAIN RESULTS

Observer design

In order to design the observer that both reconstruct the state and the unknown parameters of system (1), one first needs to introduce some notations.

Let us define first a sequence of integers ( ) =1,…, that will be used for the design of the observer:

• ∈ ℕ, = 1, … , is defined such that = 1 and ≥ +1 +1 .
Let us further define

• Δ ( ) = , 1 , … , 1 ( -1) • Ω ( ) = 1 1 , … , 1
where is a parameter of the observer. One now states some identities that will be used in the following for the proof of convergence. The proof of these identities are given in the appendices.

Proposition 1. The following identities hold true for = 1, … , :

(i) Δ -1 ( ) = , (ii) Δ ( ) Δ -1 ( ) = , (iii) Δ ( ) = 1 ( -1) , (iv) Δ ( )Ψ Ω -1 ( ) = Ψ .
Define the observer state ̂ = ̂ 1 … ̂ ∈ ℝ with ̂ ∈ ℝ , = 1, … , . Then, for = 1, … , and ∈ [ , +1 )

• if is not empty, the dynamics of ̂ is given by

̇̂ ( ) = ̂ ( ) + Ψ ( ( ), ̂ ( )) ̂ ( ) + ( ( ), ̂ ( ), ̂ ( )) -Δ -1 ( ) -1 + Υ ( ) ( )Υ ( ) ( ) (16) ̇̂ ( ) = -Ω -1 ( ) ( )Υ ( ) (17)
̇ ( ) = - ( -1 + Υ ( ) ( )Υ ( )) ( ) (18) Υ ( ) = --1 Υ ( ) + Ψ ( ( ), ̂ ( )) (19) ̇ ( ) = - ( )Υ ( ) Υ ( ) ( ) + ( ) (20)
with ( ) = ̂ 1 ( ) -( ) , Υ (0) = 0, (0) = (0) > 0, ̂ (0) ∈ ℝ and ̂ (0) ∈ ℝ which can be chosen arbitrarily.

• if is empty, the dynamics of ̂ is given by

̇̂ ( ) = ̂ ( ) + ( ( ), ̂ ( ), ̂ ( )) -Δ -1 ( )Γ -Γ 1 ( -) ̂ 1 ( ) -( ) (21)
with ̂ (0) ∈ ℝ , ̂ (0) ∈ ℝ which can be chosen arbitrarily

The matrix is the Symmetric Definite Positive Matrix solution of

+ + - = 0 (22)
Finally, Γ is define by Γ = -1 . It can be shown that Γ can be written in the following form:

Γ = ⎡ ⎢ ⎢ ⎣ Γ 1 ⋮ Γ ⎤ ⎥ ⎥ ⎦ (23) with Γ = = ! !( -)! ∈ ℝ the binomial coefficient.
Remark 3. Equations ( 16) and ( 21) can look quite different, but it can actually be noted that if Ψ ≡ 0 in ( 16), then equations ( 16) and ( 21) are strictly equivalent. Indeed, given the structure of Γ , it can be shown that if

Ψ ≡ 0, then ̇ ( ) = -Γ 1 ( ) and then ( ) = -Γ 1 ( -) ̂ 1 ( ) -( ) for ∈ [ , +1 ).

Convergence result

In order to obtain the convergence of the proposed observer, one first needs to make some assumptions about system (1) and the observability of the unknown parameters of system (1).

Assumption 1. The state , the control input and the unknown parameters are bounded, that is, there exists some compact sets ⊂ ℝ , ⊂ ℝ and Λ ⊂ ℝ such that for all ≥ 0, ( ) ∈ , ( ) ∈ and ∈ Λ.

‖ ( )‖ ≤ , ∀ ≥ 0. ( 24
)
Assumption 6. For = 1, … , there exists > 0 (independent of ) such that for all ≥ 0 and > 0

( -1 ( )) ≥ ( ) (25)
where ( ) is a positive function satisfying for all > 0 ( ) ≥ 1 and lim

→∞ ( )∕ 2 = 0 (26)
Remark 4. As it was shown in [START_REF] Farza | Adaptive observer design for a class of nonlinear systems. Application to speed sensorless induction motor[END_REF][START_REF] Boker | Nonlinear observers comprising high-gain observers and extended Kalman filters[END_REF], the assumption 6 is satisfied under a classical persistent excitation condition as the following one. For = 1, … , such that is not empty, the input is such that for any trajectory of system ( 16)-( 20) starting from ( ̂ (0), ̂ (0)) ∈ × Λ, there exists ̄ > 0 (independent of ) and > 0 such that for all ≥ 0 and > 0

+ ∫ Υ ( ) Υ ( ) ≥ ̄ ( ) (27)
One can also note that the assumption 6 can be checked online by computing the minimum of the eigenvalues of -1 .

Remark 5. In the rest of the paper, the following notations will be used.

Ψ max = sup ≥0 ‖Ψ ( ( ), ̂ ( ))‖ ( 28
) max = max =1,…, | | | | (29)
The result of the observer convergence proposed in this chapter can now be established.

Theorem 1. Consider the system (4) under the assumptions 1 to 6. Assume that ≥ 1 satisfies

4 √ ( ) ≥ √ max ( ) √ (1 + (Υ max ) 2 ) √ min( min ( ), ) + √ max ( ) Ψ max √ max(1, Υ max ) √ min( min ( ), ) (30)
with the number of sub-components of the -th block state , Υ max the upper bound of Υ , the Lipschitz constant of , Ψ the Lipschitz constant of Ψ , max the maximum value of | | and defined in [START_REF] Xu | Nonlinear system fault diagnosis based on adaptive estimation[END_REF]. Then, if, for each block = 1, … , , the upper bound on the sampling periods is chosen such that

< 8 + min min ( ), √ ( ) (max(1, Ψ max + Υ max )) + (1 + Υ max ) + Ψ max max 1, Υ max √ 2 (31)
with Ψ max the upper bound of Ψ , then there exists ( ) ≥ 0 and ≥ 0, with independent of and the uncertainties, such that the following inequality holds true

‖ ‖ ̃ ( ) ‖ ‖ + ‖ ‖ ̃ ( ) ‖ ‖ ≤ ( ) - 8 + -( 1 + ⋯ + ) (32)
with ̃ = ̂ -, ̃ = ̂ -and an upper bound of .

Before proceeding to the proof of this theorem, lemmas necessary for the proof of convergence must first be presented. The proof of both lemmas are given in the appendices. Lemma 1. For each = 1, … , such that is not empty, the corresponding vector Υ is norm bounded, that is

Υ max = sup ≥0 ‖Υ ( ( ), ̂ ( ))‖ < +∞ (33) Lemma 2. Let ∶ [-, +∞) → ℝ be a 1 function verifying the following inequality ( 2 ( )) ≤ -2 ( ) + ∫ - 2 ( ) + -+ (34)
with > 0, ≥ 0, > 0, ≥ 0, > 0, ≥ 0 and < min

√ 2-1 2 , 1 √ 2
. Then there exists ̄ ≥ 0 such that the following inequality holds true

2 ( ) ≤ ̄ -̄ + 2 (35)
where ̄ = min 2 , . Let us now move on to the proof of the main result.

Proof of Theorem 1. The proof of convergence is split into three parts. First, some adapted coordinates are defined and their corresponding dynamics are derived. In a second part, some candidate Lyapunov functions are defined and some over-valuations involving their derivatives along the error dynamics are obtained. Finally, an induction reasoning is applied in order to obtain the convergence.

Part 1: Equation error Let ≥ 0 and denote for = 1, … , :

̃ = ̂ -, ̃ = ̂ -and = max ∈ ℕ| ≤ , then ̇̃ = ̃ + ̃ ( , ̂ , ̂ , , ) + Ψ ( , ̂ ) ̂ -Ψ ( , ) - ( ) -Δ -1 ( ) -1 + Υ Υ ( ) (36) ̇̃ = -Ω -1 ( ) Υ (37)
where ̃ ( , ̂ , ̂ , , ) = ( , ̂ , ̂ ) -( , , ). In order to simplify the notations in the proof, one does not make difference in writing between the case empty and non empty. Indeed, according to remark 3, it is direct to see that the error equation for the case empty can also be written as equations ( 36)-(37) by considering that Ψ = 0 and ̃ = 0.

Denote now ̄ = Δ ( ) ̃ and ̄ = Ω ( ) ̃ , then

̇̄ = ̄ ̄ + Δ ( ) ̃ ( , ̂ , ̂ , , ) + Δ ( ) Ψ ( , ̂ , ) + Δ ( )Ψ( , ̂ )Ω -1 ( ) ̄ - -1 + Υ Υ -Υ Υ ̄ --( -1) ( ) (38) ̇̄ = - Υ ̄ - Υ (39)
where ̄ = --1 , = ( ) -̄ ( ) and Ψ ( , ̂ , ) = Ψ ( , ̂ ) -Ψ( , ). Finally, one obtains

̇̄ = ̄ ̄ + Δ ( ) ̃ ( , ̂ , ̂ , , ) + Ψ ( , ̂ , ) + Ψ ( , ̂ ) ̄ - -1 + Υ ̇̄ --( -1) ( ) (40)
Define = ̄ -Υ ̄ , then ̇ = ̇̄ -Υ ̄ -Υ ̇̄ (41) = ̄ + Δ ( ) ̃ ( , ̂ , ̂ , , ) + Ψ ( , ̂ , ) - -1 --( -1) ( ) (42)
Part 2: Lyapunov function Let us now define the candidate Lyapunov functions ( , ̄ ) = 1 ( ) + 2 ( ̄ ) with 1 ( ) = and 2 ( , ̄ ) = ̄ -1 ( ) ̄ . These functions are valid candidate Lyapunov functions since each and -1 ( ) are symmetric definite positive matrices and thanks to assumption 6, the eigenvalues of -1 ( ) are lower bounded independently of . Then, the following equalities hold

̇ 1 = ( --1 ) + ( --1 ) + 2 + 2 Δ ( ) ̃ + 2 Δ ( ) Ψ -2 -( -1) ( ) (43)
= --

+ 2 + 2 Δ ( ) ̃ + 2 Δ ( ) Ψ -2 -( -1) ( ) (44)
where equality (44) is obtained by using equation [START_REF] Stamnes | Redesign of adaptive observers for improved parameter identification in nonlinear systems[END_REF]. Now, by using the Cauchy-Schwartz inequality, the fact that ‖ ‖ = ‖ ‖ = 1 and assumption 5, one further has

̇ 1 ( ) ≤ - - + 2 √ max ( ) √ 1 ( )‖Δ ( ) ̃ ‖ + 2 √ max ( ) √ 1 ( )‖Δ ( ) Ψ ‖ + 2 √ min ( ) √ 1 ( )‖ ‖ + 2 -( -1) √ max ( ) √ 1 ( ) (45)
Using the following facts, whose proof is given in the appendices:

Fact 1. ‖Δ ( ) ̃ ‖ ≤ √ ( ) 1 √ max ( ) √ ( , ̄ ) + 2 √ max ( ) ∑ -1 =1 (‖ ̃ ‖ + ‖ ̃ ‖), with 1 = √ max ( ) √ (1 + (Υ max ) 2 ) √ min( min ( ), ) (46) 2
= √ √ max ( ) (47)
Fact 2. ‖Δ ( ) Ψ ‖ ≤ √ ( ) 3 √ max ( ) √ ( , ̄ ) + 4 √ max ( ) ∑ -1 =1 (‖ ̃ ‖ + ‖ ̃ ‖) with 3 = √ max ( ) Ψ max √ max(1, Υ max ) √ min( min ( ), ) (48)
4 = Ψ max √ √ max ( ) (49)
Fact 3. ‖ ‖ ≤ 5 ( ) ∫ - √ ( ) + 6 ∑ -1 =1 ∫ - ‖ ‖ ̃ ( ) ‖ ‖ + ‖ ‖ ̃ ( ) ‖ ‖ with 5 ( ) = √ ( ) (max(1, Ψ max + Υ max )) + (1 + Υ max ) + Ψ max max 1, Υ max min √ min ( ), √ √ 2 (50) 6 = + Ψ max (51)
one obtains

̇ 1 ( ) ≤ - - + 2 √ ( )( 1 + 3 ) ( , ̄ ) + 2( 2 + 4 ) √ 1 ( ) -1 ∑ =1 ‖ ‖ ‖ ̃ ‖ ‖ ‖ + ‖ ‖ ‖ ̃ ‖ ‖ ‖ + 2 5 ( ) √ min ( ) √ 1 ( ) ∫ - √ ( ) + 2 6 √ min ( ) √ 1 ( ) ∫ - -1 ∑ =1 ‖ ‖ ‖ ̃ ( ) ‖ ‖ ‖ + ‖ ‖ ‖ ̃ ( ) ‖ ‖ ‖ + 2 -( -1) √ max ( ) √ 1 ( ) (52)
Now, let us obtain an over-valuation of the derivative of 2 . One has

̇ 2 = ̇̄ -1 ̄ + ̄ -1 ̄ -̄ -1 ̇ -1 ̄ (53) = -̄ Υ -1 ̄ - ̄ -1 Υ ̄ + 2 ̄ -1 Υ + ̄ -1 Υ Υ - -1 ̄ (54) = - Υ ̄ - ̄ Υ Υ ̄ - ̄ Υ - ̄ Υ Υ ̄ + 2 ‖Υ ‖ ‖ ̄ ‖ ‖ ‖ + ̄ Υ Υ ̄ - ̄ -1 ̄ (55) ≤ - Υ ̄ - ̄ Υ - ̄ Υ Υ ̄ + 2 √ 2 ( ̄ ) ⎛ ⎜ ⎜ ⎜ ⎝ 5 ( ) √ ∫ - √ ( ) + 6 √ ∫ - -1 ∑ =1 ‖ ‖ ‖ ̃ ( ) ‖ ‖ ‖ + ‖ ‖ ‖ ̃ ( ) ‖ ‖ ‖ ⎞ ⎟ ⎟ ⎟ ⎠ - 2 ( ̄ ) (56)
with 5 ( ) and 6 given by fact 3 and using assumption 6.

An over-valuation of the derivative of can then be obtained as follows .

̇ ≤ - - + Υ ̄ + Υ ̄ + 2 √ ( ) 1 + 3 + 2 7 ( ) √ ∫ - √ ( ) + √ -1 ∑ =1 2 + 4 ‖ ‖ ‖ ̃ ‖ ‖ ‖ + ‖ ‖ ‖ ̃ ‖ ‖ ‖ + 2 √ -1 ∑ =1 8 ∫ - ‖ ‖ ‖ ̃ ( ) ‖ ‖ ‖ + ‖ ‖ ‖ ̃ ( ) ‖ ‖ ‖ + 2 -( -1) √ max ( ) √ (
Finally, it is direct to see that, thanks to assumption 6, there exists * ≥ 1 such that for any ≥ * one has

-2 √ ( )( 1 + 3 ) ≥ 2 (58)
which is equivalent to

√ ( ) ≥ 4( 1 + 3 ) (59)
This leads to

( √ ) ≤ - 4 √ + 7 ( ) ∫ - √ ( ) + ( 2 + 4 ) -1 ∑ =1 ‖ ‖ ‖ ̃ ‖ ‖ ‖ + ‖ ‖ ‖ ̃ ‖ ‖ ‖ + 8 ∫ - -1 ∑ =1 ‖ ‖ ‖ ̃ ( ) ‖ ‖ ‖ + ‖ ‖ ‖ ̃ ( ) ‖ ‖ ‖ + -( -1) √ max ( ) (60)
Part 3: Induction reasoning Let us now proceed to an induction reasoning on = 1, … , , that is, let us prove that the exponential convergence of

‖ ‖ ̃ 1‖ ‖ + ‖ ‖ ̃ 1‖ ‖ , … , ‖ ‖ ̃ -1‖ ‖ + ‖ ‖ ̃ -1‖
‖ to a ball, whose radius depends linearly over the uncertainties bounds 1 , … , -1 and with a negative power, when tends to infinity implies the exponential convergence of ‖ ‖ ̃ ‖ ‖ + ‖ ‖ ̃ ‖ ‖ to a ball whose radius depends linearly on 1 , … , . More precisely, one will show that for = 1, … , , there exists 9 , 10 ≥ 0, with 10 independent of , such that ‖ ‖ ̃ ‖ ‖ + ‖ ‖ ̃ ‖ ‖ verifies the following inequality:

‖ ‖ ̃ ( ) ‖ ‖ + ‖ ‖ ̃ ( ) ‖ ‖ ≤ 9 ( ) - 8 + - 10 1 + ⋯ + (61)
For = 1, if is taken such as ≥ * 1 , one has the following over-valuation implying the derivative of

√ 1 ( √ 1 ) ≤ - 1 4 √ 1 + 1 1 7 ( ) ∫ -1 √ 1 ( ) + -( 1 -1) 1 1 √ max ( 1 ) (62)
Applying lemma 2 with

2 = √ 1 , = 1 4 , = 1 1 7 ( ), = 0, ∈ ℝ arbitrary, = -( 1 -1) 1
√ max ( 1 ) 1 and = 1 , shows that there exists ̄ 1 9 ≥ 0 such that √ 1 verifies the following inequality

√ 1 ≤ ̄ 1 9 - 1 8 + -1 1 ̄ 1 10 1 = ̄ 1 9 - 1 8 + -1 1 ̄ 1 10 1 (63)
with ̄ 1 10 = 2 √ max ( 1 ), provided that

1 < min 1 4 1 7 ( ) , 8 1 ≤ 8 + min min ( 1 ), 1 √ ( ) 1 (max(1, Ψ 1 max + Υ 1 max )) + 1 (1 + Υ 1 max ) + 1 Ψ 1 max max 1, Υ 1 max √ 2 (64) Then, since ‖ ‖ ̃ 1‖ ‖ + ‖ ‖ ̃ 1‖ ‖ ≤ ( 1 -1) 1 (1+Υ 1 max ) min √ min ( 1 ), √ 1 √ 1 , one obtains ‖ ‖ ‖ ̃ 1 ‖ ‖ ‖ + ‖ ‖ ‖ ̃ 1 ‖ ‖ ‖ ≤ 1 9 ( ) -1 8 + 1 10 -1 1 (65)
with

1 9 = ( 1 -1) 1 (1+Υ 1 max ) min √ min ( 1 ), √ 1 ̄ 1 9 and 1 10 = (1+Υ 1 max ) min √ min ( 1 ), √ 1 ̄ 1 10 . Now, let us assume that ‖ ‖ ̃ 1‖ ‖ + ‖ ‖ ̃ 1‖ ‖ , … , ‖ ‖ ̃ -1‖ ‖ + ‖ ‖ ̃ -1‖
‖ verify inequality (61) and let us prove that ‖

‖ ̃ ‖ ‖ + ‖ ‖ ̃ ‖ ‖ do as well.
If is taken such that ≥ * , using inequality (60) and the induction hypothesis, implies that the derivative of √ verifies the following inequality:

( √ ) ≤ - 4 √ + 7 ( ) ∫ - √ ( ) + -( -1) √ max ( ) + ( 2 + 4 ) -1 ∑ =1 9 - 8 + - 10 ( 1 + ⋯ + ) + 8 -1 ∑ =1 ∫ - 9 - 8 + - 10 ( 1 + ⋯ + ) (66)
≤ -4

√ + 7 ( ) ∫ - √ ( ) + -( -1) √ max ( ) + -1 ∑ =1 ( 2 + 4 ) 9 + 8 9 8 -1 - 8 + --1 -1 ∑ =1 ( 2 + 4 ) 10 + 8 10 ( 1 + ⋯ + -1 ) (67)
where the latter inequality is obtained since ( ) is decreasing.

Thus, applying lemma 2 with

2 = √ , = 4 , = 7 ( ), = ∑ -1 =1 ( 2 + 4 ) 9 + 8 9 8 -1 , = 8 and = --1 ∑ -1 =1 ( 2 + 4 ) 10 + 8 10 ( 1 + ⋯ + -1 ) + -( -1)
√ max ( ), and using the fact that -1 ≥ , shows that there exists

̄ 9 ≥ 0 such that √ ≤ ̄ 9 - 8 + -̄ 10 ( 1 + ⋯ + ) (68)
with ̄ 10 = max

∑ -1 =1 ( ,
Since ‖ ‖ ̃ ‖ ‖ + ‖ ‖ ̃ ‖ ‖ ≤ ( -1) (1+Υ max ) min √ min ( ), √ √ , one obtains ‖ ‖ ̃ ‖ ‖ + ‖ ‖ ̃ ‖ ‖ ≤ 9 ( ) - 8 + - 10 ( 1 + ⋯ + ) (70)

EXAMPLE: ATTITUDE ESTIMATION WITH GYRO BIAS

Model of the system

One considers here the problem of the simultaneous estimation of the attitude of a rigid body and the gyro bias from the measurements given by an IMU unit. Particularly, the measurements are provided by three commonly used sensors: an accelerometer, a magnetometer and a gyroscope. The latter is assumed to be corrupted by a constant bias.

The attitude of a rigid body represents the rotation between an inertial reference frame {I} and a body fixed frame {B}. One then defines the orientation at each time instant ≥ 0 by a rotation matrix ( ) ∈

(3) from {B} to {I}. For the remains of this study, R(t) will be in ℝ 3×3 , indeed, the particular structure of (3) is not used in the proposed estimation scheme. The first two measurements (the accelerometer and the magnetometer) can be seen as the result of the rotation between {B} and {I} of two non co-linear constant vectors. On one side, the measured acceleration ( ) is the resultant of the rotation of the standard gravity 0 , that is

( ) = ( ) 0 (71)
where 0 = 0 0 . On the other side the measured magnetic field ( ) is the resultant of the rotation of the earth magnetic field 0 , that is ( ) = ( ) 0 (72) where 0 = . Another physical quantity can be induced from these measurements, namely, the cross product between and . This cross product is defined as follows:

( ) = ( ) × ( ) (73)
The following assumption is made on ( ):

Assumption 7. For all ≥ 0, ( ) and ( ) are non-colinear and then ( ) ≠ 0.

A vector of observations is obtained from a concatenation of the three previously defined vectors. It is given by

( ) = ⎡ ⎢ ⎢ ⎣ ( ) ( ) ( ) ⎤ ⎥ ⎥ ⎦ ∈ ℝ 9×1 (74)
The last measurement is the biased angular velocity ( ) of the considered rigid body expressed in {B} and is given by

( ) = ( ) + (75)
where ( ) ∈ ℝ 3×1 is the unbiased angular velocity and ∈ ℝ 3×1 the bias. The bias is considered constant and then ̇ = 0. The kinematics of the attitude is thus given by the following differential equation involving the angular velocity:

̇ ( ) = ( )( ( )) × ( 76
)
where × is the associated skew

-matrix of = ⎡ ⎢ ⎢ ⎣ 1 2 3 ⎤ ⎥ ⎥ ⎦
and is defined by

× = ⎡ ⎢ ⎢ ⎣ 0 -3 2 3 0 -1 -2 1 0 ⎤ ⎥ ⎥ ⎦ ∈ ℝ 3×3 .
Finally, the dynamics of the observation vector can be defined by

̇ ( ) = -3 ( ( )) ( ) = -3 ( ( )) ( ) + 3 ( ) ( ) (77)
and then:

̇ ( ) = -3 ( ( )) ( ) -9 ( ( )) (78)
where

3 ( ) = diag( × , × , × ) ∈ ℝ 9×9 and 9 ( ) = ⎡ ⎢ ⎢ ⎣ × × × ⎤ ⎥ ⎥ ⎦ ∈ ℝ 9×3 .

Observer design

In order to apply the proposed observer to the model, one now writes the dynamics in the form of the equation ( 4) and with its notations. The system is split in two subsystems. The first one describes the behavior of the angular velocity. In fact, the angular jerk is considered as an unknown uncertainty. The mathematical model can then be written as follow:

⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ ̇ 1 1 = 1 2 ̇ 1 2 = 1 3 ̇ 1 3 = 1 1 = 1 1 = (79)
The second subsystem describes the kinematics of the attitude as previously shown in equation (78). By denoting the gyroscope bias as 2 , on can write

̇ 2 1 = -3 ( 1 1 ) 2 1 -9 ( 2 1 ) 2 2 = 2 1 = (80)
As it can be noticed, the first state block does not contain unknown parameters. The dynamics of the estimated angular velocity ̂ 1 can thus be written with one equation as in [START_REF] Srinivasarengan | An adaptive observer design approach for discrete-time nonlinear systems[END_REF]. This is not true for the second state block since 2 has to be estimated. Then, the complete observer system is given by

⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ ̇̂ 1 ( ) = ⎡ ⎢ ⎢ ⎣ ̂ 1 2 ̂ 1 3 0 ⎤ ⎥ ⎥ ⎦ -1 Δ -1 1 ( )Γ 1 -1 Γ 1 1 ( -1 ) ( ̂ 1 ( ) -1 ( )) ̂ 1 ( ) = ̂ 1 1 ( ) ̇̂ 2 ( ) = -3 ( ̂ 1 1 ( )) ̂ 2 1 ( ) -9 ( ̂ 2 1 ) ̂ 2 -2 Δ -1 2 ( ) 9 + Υ 2 ( ) 2 ( )Υ 2 ( ) 2 ( ) ̇̂ 2 ( ) = -2 2 ( )Υ 2 ( ) 2 ( ) ̇̂ 2 ( ) = -2 9 + Υ 2 ( ) 2 ( )Υ 2 ( ) 2 ( ) Υ2 ( ) = -2 Υ 2 ( ) -2 9 ( ̂ 2 1 ( )) ̇ 2 ( ) = -2 2 ( )Υ 2 ( )Υ 2 ( ) 2 ( ) + 2 2 ( ) ̂ 2 ( ) = ̂ 2 1 ( ) (81)
where

Γ 1 = ⎡ ⎢ ⎢ ⎣ 3 3 3 3 3 ⎤ ⎥ ⎥ ⎦ , 1 ≥ 1, 2 = 1, Δ 1 ( ) = 3 , 1 1 3 , 1 2 1 3 and Δ 2 ( ) = 9 .
Finally, once the observation vector is estimated, the estimated rotation matrix ̂ ( ) can be deduced by using

̂ ( ) = -1 2 ̂ 2 1 ( ) (82)
where 2 is a constant and non singular matrix, involving the constant vectors 0 and 0 , defined by

2 = ⎡ ⎢ ⎢ ⎣ 0 0 3 3 3 3 - 3 3 0 ⎤ ⎥ ⎥ ⎦ (83)

Simulation results

Simulations of the reconstruction of the attitude with the observer given in equation ( 81) have been carried out with generated measurements. The rotation dynamic is defined by the angular velocity depicted on figure 1. The sampling periods of the two outputs have been taken such that 1 ≤ 1 +1 -1 ≤ 1 and 2 ≤ 2 +1 -2 ≤ 1 with 1 = 0.1 , 1 = 0.4 , 2 = 0.2 and 2 = 0.6 . The observer's parameters to be set up, and 1 , have been chosen to = 2 and 1 = 3. The sampling periods are presented in figure 2. Finally, the constant vectors have been taken to 0 = 0 0 9.81 and 0 = 0.434 -0.0091 -0.9008 .

FIGURE 1 Angular velocity

Two simulations are exposed. The first one is performed in the noise-free case. On the contrary, in the second simulation, white Gaussian noises are added on the accelerometer and on the magnetometer. Their standard deviations are equal to = 0.16 ∕ 2 for the accelerometer and = 0.04 for the magnetometer. These two signals are depicted on figure 3. It can be noted that good performances are obtained for both simulations. Indeed, in the noise free case, the estimated attitude converges very closely to its real value, as it can be seen on figure 4 where the Euler angles corresponding to and ̂ are depicted. In fact, as expected from the theoretical results, the uncertainties considered on the first subsystem prevent the estimated attitude to match exactly the generated one. The errors between the estimated outputs and the real outputs and the error || ̂ ( ) -( )||, presented respectively on figures 6 and 8, confirm these performances. When the noise is added, the performances are still decent, but errors are higher, as shown in figures 7 and 9. Indeed, one can see that the Euler angles corresponding to and ̂ , and represented on figure 5 are not as close as in the noise free case. Regarding the estimated bias, depicted on figures 10 and 11, they are estimated correctly for both simulations, but with less error in the noise-free case.

CONCLUSION

An adaptive high gain observer has been proposed in this paper, for a class of non-linear systems with uncertainties and multirated outputs. One of the main features of the proposed scheme is that the sampling periods can be irregular and asynchronous between different outputs. Another interesting feature is that thanks to its high-gain structure, only a few observer parameters have to be set: the high-gain parameter and the powers associated to each block. Sufficient conditions have been provided to ensure that the estimation error converges inside a ball whose radius depends on the

APPENDIX A PROOFS OF THE IDENTITITES

The identities to prove are:

1. Δ -1 ( ) = , 2. Δ ( ) Δ -1 ( ) = , 3. Δ ( ) = 1 ( -1) , 4. Δ ( )Ψ Ω -1 ( ) = Ψ .
Let us prove them:

1. One has

Δ -1 ( ) = 0 … 0 ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ 0 … … 0 0 1 - ⋱ ⋮ ⋮ ⋱ ⋱ ⋮ ⋮ ⋱ 0 0 … … 0 1 -( -1) ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ = 0 … 0 = 2.
By proceeding in a similar way, one has:

Δ ( ) Δ -1 ( ) = ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ 0 … … 0 0 1 ⋱ ⋮ ⋮ ⋱ ⋱ ⋮ ⋮ ⋱ 0 0 … … 0 1 ( -1) ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ 0 0 … 0 ⋮ ⋱ ⋱ ⋱ ⋮ ⋮ ⋱ ⋱ 0 ⋮ ⋱ 0 … … … 0 ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ 0 … … 0 0 ⋱ ⋮ ⋮ ⋱ ⋱ ⋮ ⋮ ⋱ 0 0 … … 0 ( -1) ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ = ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ 0 0 … 0 ⋮ ⋱ 1 ⋱ ⋮ ⋮ ⋱ ⋱ 0 ⋮ ⋱ 1 ( -2) 0 … … … 0 ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ 0 … … 0 0 ⋱ ⋮ ⋮ ⋱ ⋱ ⋮ ⋮ ⋱ 0 0 … … 0 ( -1) ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ = 3. One has Δ ( ) = ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ 0 … … 0 0 1 ⋱ ⋮ ⋮ ⋱ ⋱ ⋮ ⋮ ⋱ 0 0 … … 0 1 ( -1) ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ ⎡ ⎢ ⎢ ⎢ ⎢ ⎣ 0 ⋮ 0 ⎤ ⎥ ⎥ ⎥ ⎥ ⎦ = ⎡ ⎢ ⎢ ⎢ ⎢ ⎣ 0 ⋮ 0 1 ( -1) ⎤ ⎥ ⎥ ⎥ ⎥ ⎦ = 1 ( -1)
4. The product can be written as:

Δ ( )Ψ Ω -1 ( ) = ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ 0 … … 0 0 1 ⋱ ⋮ ⋮ ⋱ ⋱ ⋮ ⋮ ⋱ 0 0 … … 0 1 ( -1) ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ ⎡ ⎢ ⎢ ⎢ ⎣ Ψ 1,1 … Ψ 1, ⋮ ⋮ Ψ ,1 … Ψ , ⎤ ⎥ ⎥ ⎥ ⎦ ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ 1 0 … … 0 0 2 ⋱ ⋮ ⋮ ⋱ ⋱ ⋮ ⋮ ⋱ 0 0 … … 0 ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ = ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ 1 Ψ 1,1 … Ψ 1, … Ψ 1, ⋮ ⋮ ⋮ ( 1 -+1) Ψ ,1 … ( -+1) Ψ , … ( -+1) Ψ , ⋮ ⋮ ⋮ ( 1 -+1) Ψ ,1 … ( -+1) Ψ , … ( -+1) Ψ , ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦
Thus, using the equation ( 9) that for = 1, … , , = 1, … , , = 1, … , one obtains:

• if = then ( -+1) Ψ , = Ψ , • otherwise if ≠ , then Ψ , = 0 ×1 and thus ( -+1) Ψ , = 0 ×1 = Ψ , Which prove that Δ ( )Ψ Ω -1 ( ) = Ψ .

B PROOF OF LEMMA 1

Consider the change of variable such that Ῡ

( ) = Υ . (B1)
Thus, the dynamic of Ῡ is given by

̇Ῡ ( ) = ( --1 ) Ῡ ( ) + Ψ , ̂ . (B2)
This permits to show that

Ῡ ( ) = ̄ Ῡ (0) + ∫ 0 ̄ ( -) Ψ , ̂ (B3)
where ̄ = --1 . Since, according to assumption 2, the function Ψ is norm bounded and since the matrix ̄ is Hurwitz, Ῡ is bounded and its upper bound is independent of . This concludes the proof.

C PROOF OF LEMMA 2

Let us first deal with the case = 0. In this case, one has

( 2 ( )) ≤ -2 ( ) + -+ (C4)
Define now the following candidate Lyapunov function

Θ( ) = 2 ( ) (C5)
Its derivate can be overbounded as follow

Θ( ) ≤ - 2 Θ( ) + -+ (C6)
Solving the equation ̇ ( ) = -2 ( ) + -+ and applying the comparison lemma [13, lemma 3.4 p102] show the result for = 0. Let > 0, and let us denote = √ 2 , = -1 and = 1 -, then, one has

0 < ≤ 1 + √ 2 -1 (C7) ≤ √ 2 (C8) ≤ 1 - 1 √ 2 (C9)
Inequality (C7) is obtained by using the fact that

≤ 1 + √ 2 , ∀ ∈ 0, 1 2 (C10)
and since is chosen such that ≤ 1 √ 2

, this ensures that ∈ 0, 1 2 . Inequality (C9) is obtained by using the fact that is chosen such that ≤ √ 2-1 2

.

One has proven until now that ∈ 0, 1 -1 √ 2

, which directly implies that ∈ 1 √ 2 , 1 . Let us now define the following candidate Lyapunov function

Θ( ) = 2 ( ) + ∫ 0 ∫ - ( -+ ) 2 ( ) (C11)
where ( ) = ( + ), ∈ [-, 0]. Then, its derivative can be over-valued in the following way

Θ( ) = ( 2 ( )) - ∫ 0 ∫ - ( -+ ) 2 ( ) + ∫ 0 2 ( ) -2 ( -) (C12) ≤ ⎛ ⎜ ⎜ ⎝ -2 ( ) + ∫ - 2 ( ) + -+ ⎞ ⎟ ⎟ ⎠ - Θ( ) -2 ( ) + -1 2 ( ) -∫ - 2 ( ) (C13) ≤ -Θ( ) -2 ( ) + 2 ( ) + -1 2 ( ) + -+ (C14) ≤ - 2 Θ( ) + -+ √ 2 + (1 -) 2 ( ) + -+ (C15) ≤ - 2 Θ( ) -1 - 1 √ 2 2 ( ) + -+ (C16) ≤ - 2 Θ( ) + -+ (C17)
Inequality (C14) is obtained by using the fact that the function → -1

is continuous on (0, +∞) and increasing, and ∈ (0, 1). Inequality (C15) is obtained by using the fact that = 1 -and ∈ 1

D PROOF OF FACT 1

One has

‖Δ ( ) ̃ ‖ 2 = ∑ =1 -2( -1) ‖ ̃ ‖ 2 (D18) ≤ ∑ =1 ( ) 2 -2( -1) ⎡ ⎢ ⎢ ⎣ -1 ∑ =1 ‖ ̃ ‖ 2 + ‖ ̃ ‖ + ∑ =1 ‖ ̃ ‖ 2 + ⎛ ⎜ ⎜ ⎝ ∑ =1, < ‖ ̃ ‖ 2 ⎞ ⎟ ⎟ ⎠ ⎤ ⎥ ⎥ ⎦ (D19) ≤ ∑ =1 ( ) 2 -1 ∑ =1 ‖ ̃ ‖ 2 + ‖ ̃ ‖ + ∑ =1 ( ) 2 ∑ =1 -2( -1) +2( -1) ‖ ̄ ‖ 2 + ∑ =1 ( ) 2 ⎛ ⎜ ⎜ ⎝ ∑ =1, < -2( -1) +2 ‖ ̄ ‖ 2 ⎞ ⎟ ⎟ ⎠ (D20) ≤ ( ) 2 ∑ =1 -1 ∑ =1 ‖ ̃ ‖ 2 + ‖ ̃ ‖ 2 + ‖ ̄ ‖ 2 + ‖ ̄ ‖ 2 (D21) ≤ ( ) 2 -1 ∑ =1 ‖ ̃ ‖ 2 + ‖ ̃ ‖ 2 + (‖ ‖ 2 + (1 + (Υ max ) 2 )‖ ̄ ‖ 2 ) (D22)
where inequality (D19) is due to the structure of given by ( 7) and [START_REF] Grip | Estimation of states and parameters for linear systems with nonlinearly parameterized perturbations[END_REF] and inequality (D22) is obtained by using the fact that ̄ = + Υ ̄ and lemma 1. Finally, one obtains

‖Δ ( ) ̃ ‖ 2 ≤ ( ) 2 -1 ∑ =1 ‖ ̃ ‖ 2 + ‖ ̃ ‖ 2 + ( )
where inequality (D24) is obtained by using assumption 6 and remark 4.

E PROOF OF FACT 2

Similarly to the proof of fact 1, one has

‖Δ ( ) Ψ ‖ 2 ≤ ∑ =1 -2( -1) ‖ Ψ ‖ 2 ‖ ‖ 2 (E26) ≤ ∑ =1 ( Ψ ) 2 -1 ∑ =1 ‖ ̃ ‖ 2 + ∑ =1 -2( -1)
‖ ̃ ‖ 2 ( max ) 2 (E27)

≤ ( Ψ ) 2 ( max ) 2 ∑ =1 -1 ∑ =1 ‖ ̃ ‖ 2 + ‖ ̄ ‖ 2 (E28) ≤ ( Ψ ) 2 ( max ) 2 -1 ∑ =1 ‖ ̃ ‖ 2 + (‖ ‖ 2 + (Υ max ) 2 ‖ ̄ ‖ 2 ) (E29)
where inequality (E27) is obtained by using the structure of Ψ given by ( 8) and the notation introduced in remark 5 and inequality (E29) is obtained by using the fact that ̄ = + Υ ̄ and lemma 1. Finally, one obtains

‖Δ ( ) Ψ ‖ 2 ≤ ( Ψ )
where inequality (E31) is obtained by using assumption 6 and remark 4.

F PROOF OF FACT 3

The derivative of is given by

̇ = ̇ -̇̄ ( ) (

Assumption 2 .Assumption 3 .Assumption 4 .Assumption 5 .

 2345 The functions Ψ ( , ) are continuous over × and bounded outside × . The functions ( , , ) are Lipschitz with respect to ( , ) uniformly with respect to . Their Lipschitz constant are denoted . The functions Ψ ( , ) are Lipschitz with respect to uniformly with respect to . Their Lipschitz constant are denoted Ψ The uncertainties are bounded, that is, for = 1, … , , there exists ≥ 0 such that

with 9 (

 9

FIGURE 2 FIGURE 3

 23 FIGURE 2 Sampling instants for both outputs

FIGURE 4 FIGURE 5 1 FIGURE 6 1 FIGURE 7 1 FIGURE 8 1 FIGURE 9

 4516171819 FIGURE 4 Euler angles for the first simulation

FIGURE 10 FIGURE 11

 1011 FIGURE 10 Estimated and real gyroscope bias for the first simulation

√ 2 , 1 .

 21 Solving the equation ̇ ( ) = -2 ( ) + -+ and applying the comparison lemma [13, lemma 3.4 p102] concludes the proof.

  ( )[ ̃ + Ψ ] + Ψ ̄ (F34)Then, noticing that ( ) = 0 yields( ) = -∫ ̄ ( ) + Δ ( )[ ̃ ( ) + Ψ ( ) ] + Ψ ( )̄ fact 1 and 2, using the fact that ‖ ‖ = 1 and using the notation Ψ max and max as in remark 5 yields‖ ( )‖ ≤ ∫ -‖ ̄ ( )‖ + Ψ max ‖ ̄ ( )‖ + (‖ ̄ ( )‖ + ‖ ̄ ( )‖) + Ψ max ‖ ̄ ( ( )‖ + ‖ ̃ ( )‖) + Ψ max ‖ ̃ ( )‖ (F37)Using the fact that ̄ = + Υ ̄ and using lemma 1 gives ‖ ‖ ≤ ‖ ‖ + Υ max ‖ ̄ ‖, and then‖ ( )‖ ≤ ∫ -+ ( + Ψ max ) ‖ ( )‖ + ∫ -(Ψ max + Υ max ) + ( (1 + Υ max ) + Ψ max Υ max ) ‖ ̄ ( + Υ max ) + ( (1 + Υ max ) + Ψ max Υ max ) + Υ max ) + ( (1 + Υ max ) + Ψ max Υ max )1, Ψ max + Υ max )) + (1 + Υ max ) + Ψ max max 1


domain: sdv info

text:
Madison Bergmann

Charlotte Jeanneau

Thomas Giraud

Imad Gilles Richard

Imad About
email: imad.about@univ-amu.fr

Prof Imad

Complement activation links inflammation to dental tissue regeneration

Keywords: Pulp biology, stem cell recruitment, dental tissue regeneration, Complement, pulp inflammation, nerve sprouting

Introduction

Tissue regeneration is a physiological process which occurs only after physical/traumatic injuries or under pathological conditions with tissue lesions, due to endogenous or exogenous stimuli. This leads to acute inflammation inducing secretion of pro-inflammatory mediators (e.g., interferon 

[INF], tumor necrosis factor  [TNF-, interleukin 1 [IL1], IL8 ), and the recruitment of phagocytic cells [START_REF] Fullerton | Resolution of inflammation: a new therapeutic frontier[END_REF]. The inflammatory reaction represents a protective response of tissues by eliminating noxious stimuli and initiating the healing/regeneration process [2]. It is characterized by capillary dilatation and leukocyte recruitment to the inflammatory site leading to the appearance of four clinical symptoms: pain, redness, heat, and function loss. This is followed by the inflammation resolution which is regulated by multiple mediators and immune cells [START_REF] Feehan | Is Resolution the End of Inflammation?[END_REF]. This phase takes place a few hours after the initiation of the acute inflammatory response. Inflammatory stimuli are eliminated and secretion of proinflammatory mediators such as TNF- and IL-8 decreases while that of anti-inflammatory mediators such as IL4, IL10, and IL13, and Chemokine Ligand 2 (CCL2), increases. The resolution of inflammation is a prerequisite to initiate tissue healing/regeneration by the recruitment of stem cells. Indeed, failure to resolve acute inflammation may lead to chronic inflammation due to continuous stimulation. In this case, the recruited mononuclear cell infiltrate persists at the injury site [START_REF] Schett | Resolution of chronic inflammatory disease: universal and tissuespecific concepts[END_REF][START_REF] Maruyama | Modulation of the Inflammatory Response and Bone Healing[END_REF].

Infiltrated cells produce pro-inflammatory cytokines such as IFN and TNF- which polarize macrophages to a pro-inflammatory M1 phenotype rather than an anti-inflammatory tissue-repair M2 phenotype.

Infiltrating immune cells during the inflammation appear to play a major role in determining the outcome of wound repair. In salamander limb regeneration model, it has been shown that macrophage depletion resulted in failure of limb regeneration [START_REF] Godwin | Macrophages are required for adult salamander limb regeneration[END_REF].

Complement is one of the main actors of the innate immunity. It is known for its roles in initiating the acute inflammatory reaction through production of C3a and C5a anaphylatoxins which induce blood vessel dilatation and recruitment of phagocytic cells to the inflammatory site. Complement is also involved in the resolution phase by producing C3b opsonin which facilitate bacteria phagocytosis and by forming the membrane attack complex (MAC) which leads to direct pathogen lysis (Fig. 1).

Thus, Complement plays a beneficial role in acute inflammation and its resolution. However, Complement activation may have deleterious effects by contributing to inflammation-mediated tissue damage. Also, inherited/acquired complement deficiencies may lead to the development of autoimmunity [START_REF] Winkelstein | CHAPTER 11 -Complement Deficiencies[END_REF].

Complement receptors are known to be expressed by immune cells. Yet, an expression of complement receptors has been reported in many other cell types suggesting Complement implication in other physiological processes and expression of Complement receptors in mesenchymal stem cells lead to the hypothesis that Complement is involved in tissue regeneration [START_REF] Ehrnthaller | Complement C3 and C5 deficiency affects fracture healing[END_REF]. In vivo studies confirmed Complement implication in the regeneration process of several tissues such as liver, bone or skin [START_REF] Rutkowski | The complement cascade as a mediator of tissue growth and regeneration[END_REF].

In periodontal tissues, Complement implication in chronic inflammation is well established. However, little is known about its role is dental tissue regeneration. This review will shed the light on dental cells as an important local source of Complement proteins and highlight Complement implication in the initial steps of dental tissue regeneration.

I. The complement system

Heat labile proteins circulating in the serum were identified early in the 19 th century. These proteins were later named the complement system as they were hypothesized to complement the activity of the innate immune system [START_REF] Nesargikar | The complement system: history, pathways, cascade and inhibitors[END_REF]. Complement proteins are primarily secreted by liver hepatocytes [START_REF] Thorgersen | The Role of Complement in Liver Injury, Regeneration, and Transplantation[END_REF] and some immune cells [START_REF] Lubbers | Production of complement components by cells of the immune system[END_REF]. So far, more than 30 soluble and membrane-bound proteins have been identified as a part of this cascade which can be activated by 3 different pathways [START_REF] Merle | Complement System Part I -Molecular Mechanisms of Activation and Regulation[END_REF]: the classical, the lectin and the alternative pathways (Fig. 1). In the classical pathway, Complement activation is initiated by fixation of antibodies to pathogen surface antigens [START_REF] Diebolder | Complement is activated by IgG hexamers assembled at the cell surface[END_REF][START_REF] Mortensen | Structure and activation of C1, the complex initiating the classical pathway of the complement cascade[END_REF]. In the lectin pathway, Complement is activated after fixation of mannose-associated serine protease (MASP) to either mannose-binding lectin (MBL) or ficolin. MBL and ficolin belong to the lectin proteins family, recognizing pathogen-associated molecular patterns (PAMPs) [START_REF] Endo | The role of ficolins in the lectin pathway of innate immunity[END_REF]. The alternative pathway is activated directly by pathogens or biomaterials. Upon tooth lesion/injury, these three pathways can be activated by dental caries/traumas and after application of pulp capping/bone filling materials [START_REF] Mödinger | Role of the Complement System in the Response to Orthopedic Biomaterials[END_REF]. Complement activation by all three pathways ends up with a common pathway leading to production of a protein complex with an enzymatic activity, C3 convertase. The latter's enzymatic activity leads to production of several active molecules responsible for Complement functions: C3a and C5a act as anaphylatoxins with pro-inflammatory activity; C3b acts as an opsonin; and membrane attack complex C5b-9 (MAC) which acts as a lytic membrane complex (Fig. 1). C3a and C5a action is dependent on binding to their specific G coupled receptors. While C3aR is the only known receptor for C3a, there are 2 identified receptors for C5a namely C5aR and C5L2 [START_REF] Hawksworth | New concepts on the therapeutic control of complement anaphylatoxin receptors[END_REF]. Immune cells express either C3a or C5a receptor or both [START_REF] Merle | Complement System Part II: Role in Immunity[END_REF]. Due to similarities in their structure and in the structure of their receptors, both anaphylatoxins have overlapping functions. Thus, C3a and C5a promote release of pro-inflammatory mediators such as IL-6, IL-1β and TNF-α [START_REF] Haas | Anaphylatoxins: their role in bacterial infection and inflammation[END_REF][START_REF] Coulthard | Is the complement activation product C3a a proinflammatory molecule? Re-evaluating the evidence and the myth[END_REF], induce and leukocyte recruitment to the inflammation site and modulate their phagocytic activity. However, there are some functions that are specific either to C3a or C5a. For example, C3a is responsible for attracting eosinophils, masts cells and dendritic cells during the inflammatory process, while C5a regulates B cells migration and modulates T cells activity [START_REF] Dunkelberger | Complement and its role in innate and adaptive immune responses[END_REF].

The other fragments resulting from C3 and C5 cleavage, respectively C3b and C5b, exert other functions in the inflammation. C3b is an opsonin playing a major role in eliminating pathogens by binding to pathogen surface and being recognized by immune cells' specific receptors [START_REF] Van Lookeren Campagne | Pathogen clearance and immune adherence "revisited": Immuno-regulatory roles for CRIg[END_REF] whereas C5b is responsible of MAC assembly (C5b, C6, C7, C8 and C9) (Fig. 1). This protein complex is directly involved in the destruction of pathogens as it is mainly responsible for the lysis of parasites, Gram negative, and Gram positive bacteria [START_REF] Bayly-Jones | The mystery behind membrane insertion: a review of the complement membrane attack complex[END_REF].

Thus, Complement is an important actor of the innate immune system. It induces vascular modifications during in the acute inflammation and plays a significant role in its resolution by eliminating pathogens. However, expression of complement receptors and components in non-immune cells raised questions about possible functions in other processes than the inflammation, namely in tissue regeneration.

II. Complement functions in tissue regeneration

Complement is expressed in most organisms and it is conserved throughout evolution. Although local Complement secretion could potentiate the immune response, its secretion during regeneration in urodeles suggested that it may be involved in other functions. Indeed, after amputation, urodeles have the capacity to regrow limbs and it has been shown that C3 is expressed during urodele limb regeneration. Using both western blot and immunofluorescence, C3 was detected at early stages of dedifferentiation as well as in regenerating muscles and cartilage [START_REF] Rio-Tsonis | Expression of the third component of complement, C3, in regenerating limb blastema cells of urodeles[END_REF]. This was the first study to highlight Complement implication, namely C3, in limb regeneration. Expression of C3 and C5 was also studied after limb amputation in newts. While C3 was expressed throughout the entire limb regeneration process, C5 was mainly expressed in the wound epithelium, an early stage of regeneration. This means that Complement plays a significant role during limb regeneration and that there is a differential expression pattern of Complement proteins during the regeneration process [START_REF] Kimura | Expression of complement 3 and complement 5 in newt limb and lens regeneration[END_REF]. Complement has also been shown to be involved in mammalian tissue regeneration (Fig. 2). Complement C5 has been shown to be required for efficient liver regeneration after toxic injury using carbon tetrachloride (CCl4) in a murine model. While liver fully regenerated after toxic injury in normal mice, regeneration was not found in C5-deficient mice which showed evidence of tissue degeneration and necrosis. Lack of C5 led to deficient hepatocyte proliferation due to a diminished hepatocyte mitotic activity indicating that C5 is a potential mitotic factor for hepatocytes. Interestingly, liver regeneration capacity in C5-deficient mice was restored after injecting C5 protein or C5a fragment. This leads to the conclusion that C5a, is required for efficient liver regeneration after toxic injury [START_REF] Mastellos | A novel role of complement: mice deficient in the fifth component of complement (C5) exhibit impaired liver regeneration[END_REF].

Similarly, several Complement proteins/receptors have been detected during bone regeneration and appear to play a major role in this process [START_REF] Mödinger | Complement involvement in bone homeostasis and bone disorders[END_REF]. Complement C1s was detected during cartilage resorption [START_REF] Sakiyama | Complement Cls, a classical enzyme with novel functions at the endochondral ossification center: immunohistochemical staining of activated Cls with a neoantigen-specific antibody[END_REF] and contributes to its collagen matrix degradation [START_REF] Troeberg | Proteases involved in cartilage matrix degradation in osteoarthritis[END_REF]. This is particularly important for cartilaginous template resorption before ossification of newly regenerated bone. Complement C3 and C5 as well as their respective C3aR and C5aR receptors were detected using RT-PCR and immunofluorescence in mesenchymal stem cells as well as in osteoblasts and osteoclasts [START_REF] Ignatius | Complement C3a and C5a modulate osteoclast formation and inflammatory response of osteoblasts in synergism with IL-1β[END_REF]. Both molecules have been shown to induce osteoclast differentiation. A difference in C3aR expression pattern was observed in vitro after stimulation of osteoblast with C3a: while C3aR was detected at the cell surface in the absence of C3a, the receptor was translocated into the cytoplasm or bound to the nuclear membrane after adding C3a. Further analysis showed an increase in C5 mRNA levels during osteogenic differentiation in vitro, indicating that C5aR expression was strong in cells undergoing osteogenic differentiation whereas it was poorly expressed in undifferentiated mesenchymal stem cells. These results obtained in vitro, underline the importance of C5a and C3a for bone tissue regeneration [START_REF] Ignatius | Complement C3a and C5a modulate osteoclast formation and inflammatory response of osteoblasts in synergism with IL-1β[END_REF]. In order to determine C5aR expression pattern during bone regeneration in vivo, immunostaining was performed at different time points after rat tibia fracture [START_REF] Ignatius | The anaphylatoxin receptor C5aR is present during fracture healing in rats and mediates osteoblast migration in vitro[END_REF]. C5aR was expressed after 1 day by the immune cells at the inflammation site. Once the inflammation was resolved, C5aR was expressed by osteoblasts from 3 up to 28 days in the newly formed bone. Chondroblastic cells in the cartilage template also expressed C5aR from day 7 to 28. However, C5aR expression in the chondroblastic cells was less intense in the hypertrophic region where the cartilaginous template is replaced by bone [START_REF] Ignatius | The anaphylatoxin receptor C5aR is present during fracture healing in rats and mediates osteoblast migration in vitro[END_REF].

Complement was also investigated during skin healing which is a multistep process finely regulated by many growth factor and cells. Complement has been shown to play a role in the early phase of inflammation mainly through C3 and C5 involvement in vasodilatation and recruitment of inflammatory cells [START_REF] Sinno | Complements and the Wound Healing Cascade: An Updated Review[END_REF]. Complement deficiency has been shown to promote cutaneous wound healing in mice [START_REF] Rafail | Complement Deficiency Promotes Cutaneous Wound Healing in Mice[END_REF]. By contrast, recent studies performed in vivo showed that a local application of either C5a, or C3a or both [START_REF] Sinno | Accelerated wound healing with topical application of complement C5[END_REF][START_REF] Sinno | Topical application of complement C3 in collagen formulation increases early wound healing[END_REF][START_REF] Sinno | Complements c3 and c5 individually and in combination increase early wound strength in a rat model of experimental wound healing[END_REF] induced wound healing through fibroblast migration and collagen deposition at day 3 to 7.

Thus, in addition to its implication in initiating the inflammatory reaction and its resolution

Complement is also involved in the regeneration process of several tissues.

III. Complement implication in the initial steps of dental tissue regeneration

Dental tissues are frequently exposed to infection and injury. However, they exhibit a high resistance to infection and regenerate following infectious/traumatic injuries or application of bone filling/pulp capping materials. This suggests that these tissues may contribute to infection control and have a significant regeneration potential.

a) Complement production in the periodontal ligament: implication in bone marrow mesenchymal stem cell recruitment

A close relationship exists between maxillary/mandibular bones and the surrounding periodontal ligament which is prone to a high infection/injury risk. In chronic inflammation, local Complement activation has been shown to induce periodontal inflammatory activity and periodontal bone support destruction [START_REF] Abe | Local complement-targeted intervention in periodontitis: proof-of-concept using a C5a receptor (CD88) antagonist[END_REF][START_REF] Hajishengallis | Complement involvement in periodontitis: molecular mechanisms and rational therapeutic approaches[END_REF]. Complement components were abundantly found in the gingival crevicular fluid (GCF) of periodontitis patients, whereas they were present at lower concentrations in GCF from healthy individuals [START_REF] Challacombe | Chapter 102 -Immunology of Diseases of the Oral Cavity[END_REF]. Studies in mice have implicated both C3 and C5aR in periodontal disease pathogenesis.

In a proof-of-concept study, local intragingival injection of PMX-53, a C5aR antagonist, blocked periodontal inflammation and bone loss in a model of P. gingivalis-induced periodontitis [START_REF] Hajishengallis | Complement involvement in periodontitis: molecular mechanisms and rational therapeutic approaches[END_REF]. This demonstrates a deleterious effect of Complement in chronic disease.

By contrast, a recent study performed in vitro has demonstrated that Complement plays a significant role in bone marrow mesenchymal stem cell recruitment which is required for periodontal bone regeneration [START_REF] Jeanneau | Xenogeneic bone filling materials modulate mesenchymal stem cell recruitment: role of the Complement C5a[END_REF]. Cultures of periodontal ligament cells, isolated from human third molars and physically injured by scalpels, demonstrated that PDL express C5 and produce C5a.

Complement contribution to the resolution of pulp infection/inflammation

Upon carious decays, one of the major risks is the subsequent inflammation as a severe inflammatory reaction may lead to pulp tissue necrosis. This is due to the fact that the dental pulp is located within inextensible dentinal walls and has a terminal vascularization [START_REF] Park | Inflammation of the Dental Pulp[END_REF]. This high risk of necrosis represents a challenge and explains why the resolution of the inflammation is a pre-requisite for dentin-pulp regeneration. Thus, local inflammation control is a major requirement for tissue regeneration. Recent data demonstrated that local Complement production is involved in the resolution of pulp inflammation. In vitro models using primary pulp fibroblast cultures have been used to simulate bacterial invasion of the pulp using lipoteichoic acid (LTA) from Gram positive or lipopolysaccharides (LPS) from Gram negative bacteria. Surprisingly, pulp fibroblasts express most Complement mRNAs.

Yet, after LTA stimulation, not only they expressed all Complement mRNAs, but they also secreted

Complement active proteins such as C3a, C3b, C5a, C5b and even more complex structures such as the Membrane Attack Complex (C5b-9) [START_REF] Chmilewsky | Pulp fibroblasts synthesize functional complement proteins involved in initiating dentin-pulp regeneration[END_REF]. Among these active molecules, C3b and MAC play two major roles in the resolution of inflammation: when C3b was incubated in vitro with cariogenic bacteria such as S. Mutans and S. Sanguinis, it opsonized these bacteria and enhanced their phagocytosis by macrophages [START_REF] Fournis | Human Pulp Fibroblast Implication in Phagocytosis via Complement Activation[END_REF]. When pulp fibroblasts were co-cultured with cariogenic bacteria, immunofluorescence and ELISA demonstrated MAC secretion and fixation on S. Mutans and S.

Sanguinis leading to their subsequent lysis as demonstrated by a significant reduction of bacteria growth [START_REF] Jeanneau | Can Pulp Fibroblasts Kill Cariogenic Bacteria? Role of Complement Activation[END_REF]. Investigating Complement expression in carious human teeth demonstrated an intense expression of MAC on bacteria penetrating the dentin tubules and the dental pulp. Within the limit of these studies in vitro, pulp fibroblast-produced Complement is involved in the control of bacteria which may enter the pulp under caries (Fig. 3). This pathogen control represents a pre-requisite for the regeneration process. Thus, in addition to systemic Complement anti-bacteria functions, local Complement production within the dental pulp potentiates these effects, contributes to the resolution of inflammation, and represents an efficient first line of defense against pulp invading bacteria.

Complement implication in dentin-pulp regeneration

The first evidence of Complement implication in dentin-pulp regeneration came from expression of Complement receptors in dental pulp stem cells (DPSCs) and fibroblasts. Indeed, investigating C3aR by immunohistochemistry demonstrated its co-expression with STRO-1 mesenchymal stem cell marker in dental pulp stem cells or with fibroblast surface protein antigen in pulp fibroblasts in vitro [START_REF] Rufas | Complement C3a Mobilizes Dental Pulp Stem Cells and Specifically Guides Pulp Fibroblast Recruitment[END_REF] and on healthy tooth sections. Similarly, Complement C5aR and STRO-1 co-localization on sound human tooth sections highlighted a possible implication of Complement C5aR in the dentin-pulp regeneration process [START_REF] Chmilewsky | Pulp progenitor cell recruitment is selectively guided by a C5a gradient[END_REF]. This was further confirmed in vitro by demonstrating C5aR expression in dental pulp stem cells by RT-PCR and co-localization of C5aR and STRO-1 by immunofluorescence in DPSCs [START_REF] Chmilewsky | Pulp progenitor cell recruitment is selectively guided by a C5a gradient[END_REF].

Complement C3a is the first active molecule produced after Complement activation.

Investigating C3a interaction with pulp cells demonstrated a significant increase of DPSCs and pulp fibroblasts proliferation in a concentration-dependent manner [START_REF] Rufas | Complement C3a Mobilizes Dental Pulp Stem Cells and Specifically Guides Pulp Fibroblast Recruitment[END_REF]. Implication of C3a in cell recruitment was studied following cell migration using microfluidic chambers. These chambers allow establishing a chemotaxis gradient and enable tracking single cell migration following a chemotaxis factor overtime. When pulp fibroblasts were incubated with C3a in microfluidic chambers, they migrated following a C3a gradient [START_REF] Rufas | Complement C3a Mobilizes Dental Pulp Stem Cells and Specifically Guides Pulp Fibroblast Recruitment[END_REF]. When the effect of this molecule was investigated on DPSCs, C3a induced their mobilization only. Although their migration did not follow any specific direction, cell detachment and mobilization from its surrounding environment and underlying substrate is an initial requirement for its recruitment. Application of C3 specific antagonist (SB290157) significantly inhibited stem cell mobilization and fibroblast recruitment. This clearly demonstrates that, in addition to fibroblast and dental pulp stem cell proliferation, C3a mobilizes stem cells and provides a gradient for pulp fibroblast recruitment (Fig. 4).

To investigate the role played by C5a in the initial steps of regeneration, models of tissue infection and injury were developed to simulate carious or traumatic pulp injuries in vitro. When cultured fibroblasts were injured or stimulated with LTA, they increased C5a secretion very rapidly as a significant increase was observed after 20 or 30 minutes of stimulation. When C5a was applied as a chemotactic factor using microfluidic chambers, it enhanced pulp stem cell recruitment and this migration followed a C5a gradient [START_REF] Chmilewsky | Pulp progenitor cell recruitment is selectively guided by a C5a gradient[END_REF]. This indicates that Complement activation at the injury site provides C5a as a major signal for guiding stem cell recruitment. Thus, while C3a, mobilizes stem cells, C5a guides their migration to the injury/decay/biomaterial application site. This secretion level and its subsequent effect on mesenchymal stem cell recruitment has been shown to be modulated by pulp capping materials. Indeed, a decreased DPSC recruitment was reported with resin-containing pulp capping materials [START_REF] Giraud | Complement Activation by Pulp Capping Materials Plays a Significant Role in Both Inflammatory and Pulp Stem Cells' Recruitment[END_REF].

Further adaptation of microfluidic chambers to simulate carious injury and interaction of LTAstimulated fibroblasts with either DPSCs or human neurons unraveled the interactions between fibroblast injury under caries and DPSCs recruitment/neurite sprouting. Indeed, when DPSCs were subjected to LTA-stimulated/unstimulated fibroblasts in these chambers, DPSCs migrated towards LTA-stimulated cells. Addition of C5a specific antagonist (W54011) significantly reduced this recruitment [START_REF] Chmilewsky | Pulp fibroblasts synthesize functional complement proteins involved in initiating dentin-pulp regeneration[END_REF]. This indicates that C5a released from injured fibroblasts upon carious/traumatic injuries guides stem cell towards the injury site (Fig. 5). Interestingly, investigating C5aR expression on sound human tooth sections or cultured cells demonstrated that pulp fibroblasts do not express C5aR.

However, a strong expression of this receptor was found under caries, in LTA-stimulated fibroblasts [START_REF] Chmilewsky | Pulp Fibroblasts Control Nerve Regeneration through Complement Activation[END_REF] and in LPS-stimulated pulp cells [START_REF] Liu | Time-dependent C5a and C5aR expression in dental pulp cells following stimulation with LTA and LPS[END_REF]. Thus, upon carious injury, an increased production of C5a at the carious site is expected to interact with the fibroblast C5a receptor. Indeed, when C5a was incubated with LTA-stimulated fibroblasts, the latter secreted Brain-Derived Neurotrophic Factor (BDNF), a major neurite outgrowth inducer. Secretion of BDNF resulted from binding of C5a to its specific C5aR and its subsequent phosphorylation [START_REF] Chmilewsky | Pulp Fibroblasts Control Nerve Regeneration through Complement Activation[END_REF]. This model demonstrated that LTA-stimulated pulp fibroblasts control neurite outgrowth through BDNF modulation via C5a secretion as demonstrated by the neurites sprouting towards LTA-stimulated fibroblasts (Fig. 5). Thus, in addition to C5a implication in dental pulp stem cell recruitment, local Complement activation under carious injury is involved in neurite sprouting and nerve regeneration which are major requirements in the dental pulp regeneration.

While the abovementioned C5a effects appear clearly related to C5a interaction with its C5aR, a second receptor for C5a, C5a receptor-like 2 (C5L2), was so far considered as a non-functional receptor. C5L2 belongs to the superfamily of G protein-coupled receptors [START_REF] Ohno | A putative chemoattractant receptor, C5L2, is expressed in granulocyte and immature dendritic cells, but not in mature dendritic cells[END_REF][START_REF] Li | C5L2: a controversial receptor of complement anaphylatoxin, C5a[END_REF] but, unlike most G protein-coupled receptors, C5L2 is deficient in heterotrimeric G protein coupling. When C5L2 expression was investigated on human third molar tooth sections, immunohistochemistry showed a perivascular co-localization of both C5L2 and STRO-1 [START_REF] Chmilewsky | C5L2 Regulates DMP1 Expression during Odontoblastic Differentiation[END_REF]. This was further confirmed on isolated human DPSCs where C5L2 expression and functions in the dental pulp provided valuable new insights to C5a receptors functions. Studying C5L2 expression by immunohistochemistry demonstrated a coexpression of this receptor with C5aR by LTA-stimulated fibroblasts [START_REF] Chmilewsky | C5L2 Receptor Represses Brain-Derived Neurotrophic Factor Secretion in Lipoteichoic Acid-Stimulated Pulp Fibroblasts[END_REF]. Silencing C5L2 with small interfering RNA (siRNA) in pulp fibroblasts lead to an increased BDNF secretion [START_REF] Chmilewsky | C5L2 Receptor Represses Brain-Derived Neurotrophic Factor Secretion in Lipoteichoic Acid-Stimulated Pulp Fibroblasts[END_REF]. Furthermore, neurite outgrowth significantly increased towards fibroblasts silenced for C5L2 [START_REF] Chmilewsky | C5L2 Silencing in Human Pulp Fibroblasts Enhances Nerve Outgrowth Under Lipoteichoic Acid Stimulation[END_REF]. This shows that C5L2 act as a modulator of BDNF secretion through inhibition of C5aR and demonstrates LTAstimulated fibroblasts implication in neurite sprouting during pulp regeneration.

Further investigation of Complement functions demonstrated that upon incubation in odontogenic medium, expression of C5L2 significantly increased in DPSCs undergoing odontoblastic differentiation. C5L2 expression appears to be potentiated by TNFα, a potent pro-inflammatory cytokine. C5L2-siRNA silencing strategy in DPSCs demonstrated that C5L2 acts as a negative regulator of expression of dentin matrix protein 1 (DMP1), a dentinogenic marker. This provides a direct correlation between odontoblastic differentiation and C5L2 expression level and identifies C5L2 as a negative regulator of DMP1 expression by DPSCs during odontoblastic differentiation. This provides a first evidence of C5L2 involvement in dental pulp stem cell functions and adds important data on the implication of Complement in dental pulp stem cells odontoblastic differentiation [START_REF] Chmilewsky | C5L2 Regulates DMP1 Expression during Odontoblastic Differentiation[END_REF].

Overall, pulp fibroblast appears as a unique cell in its capacity to secrete all complement proteins. This secretion together with expression of complement receptors is modulated by caries/injury or application of biomaterials. Fibroblast-secreted C3a and C5a play a major role in the initial steps of dentin-pulp regeneration, as demonstrated in vitro, by interacting with their receptors on pulp fibroblasts, dental pulp stem cells and neurons inducing their proliferation/recruitment and odontoblastic differentiation.

Conclusion

Complement is an important immune surveillance system mainly synthesized by the liver.

Through production of active molecules, Complement is known for its contribution in initiating acute inflammation and eliminating pathogens during its resolution. To render Complement more readily available, non-immune cells such as osteoblasts and chondrocyte have been shown to secrete some Complement proteins at the injury/inflammation site. Moreover, expression of Complement receptors has been reported in many non-immune cell types including stem cells demonstrating its implication in tissue regeneration.

Dental tissues are prone to infection/traumatic injuries and severe inflammation may lead to pulp necrosis/periodontal inflammation with subsequent bone resorption. This review highlights that, in addition to its implication in initiating the inflammation, Complement plays a major role in initiating the regeneration process within dental tissues.

Recent investigations demonstrated that pulp fibroblast secrete/express Complement proteins/receptors highlighting local complement sources within the dental tissues. Through production of C3b and MAC [START_REF] Fournis | Human Pulp Fibroblast Implication in Phagocytosis via Complement Activation[END_REF][START_REF] Jeanneau | Can Pulp Fibroblasts Kill Cariogenic Bacteria? Role of Complement Activation[END_REF], pulp fibroblast-secreted Complement contributes to the resolution of inflammation by eliminating bacteria which is a prerequisite for the regeneration process. By producing C3a, C5a, pulp fibroblast-synthesized Complement contributes to tissue regeneration. While C3a mobilizes DPSCs and recruits pulp fibroblasts [START_REF] Rufas | Complement C3a Mobilizes Dental Pulp Stem Cells and Specifically Guides Pulp Fibroblast Recruitment[END_REF], C5a has a major role in the neurite outgrowth [START_REF] Chmilewsky | Pulp Fibroblasts Control Nerve Regeneration through Complement Activation[END_REF],

DPSCs recruitment [START_REF] Chmilewsky | Pulp progenitor cell recruitment is selectively guided by a C5a gradient[END_REF] and differentiation into odontoblast-like cells [START_REF] Chmilewsky | C5L2 Regulates DMP1 Expression during Odontoblastic Differentiation[END_REF]. Similarly, PDL cellproduced C5a interacts with BMMSCs inducing their recruitment [START_REF] Jeanneau | Xenogeneic bone filling materials modulate mesenchymal stem cell recruitment: role of the Complement C5a[END_REF] which represents an important early step of periodontal ligament regeneration. Interestingly, Complement protein secretion/receptor expression modulation by injury/infection or after biomaterial application on periodontal/pulp cells highlights the dental tissue-synthesized Complement as an additional local immune surveillance mechanism and demonstrates its implication in inflammation and regeneration.

Throughout this review, it is now clear that, in addition to its implication in the initiation of inflammation, Complement plays a key role in both inflammation resolution and the initial steps of dental tissue regeneration, thus, providing at least one of the missing links between dental tissue inflammation and regeneration (Fig. 6). Local Complement secretion is involved in Cardiac [START_REF] Natarajan | Complement Receptor C5aR1 Plays an Evolutionarily Conserved Role in Successful Cardiac Regeneration[END_REF], intestinal [START_REF] Matsumoto | C3a Enhances the Formation of Intestinal Organoids through C3aR1[END_REF], bone [START_REF] Ehrnthaller | Complement C3 and C5 deficiency affects fracture healing[END_REF][START_REF] Ignatius | Complement C3a and C5a modulate osteoclast formation and inflammatory response of osteoblasts in synergism with IL-1β[END_REF][START_REF] Matsuoka | Osteoclast-derived complement component 3a stimulates osteoblast differentiation[END_REF], nerve [START_REF] Peterson | Complement protein C1q modulates neurite outgrowth in vitro and spinal cord axon regeneration in vivo[END_REF],

Legends

skin [START_REF] Fayyazi | C5a Receptor and Interleukin-6 Are Expressed in Tissue Macrophages and Stimulated Keratinocytes but not in Pulmonary and Intestinal Epithelial Cells[END_REF][START_REF] Katz | IL-17 regulates gene expression and protein synthesis of the complement system, C3 and factor B, in skin fibroblasts[END_REF] and dental tissues regeneration [START_REF] Rufas | Complement C3a Mobilizes Dental Pulp Stem Cells and Specifically Guides Pulp Fibroblast Recruitment[END_REF][START_REF] Chmilewsky | Pulp Fibroblasts Control Nerve Regeneration through Complement Activation[END_REF]. *=under stimulation; NR=Non reported.

  PDL cell physical injury simulates tissue injury upon extraction/trauma. C5a production level doubled when PDL cells were incubated with extracts from bone filling materials such as Gen-Os® of equine and porcine origins. When C5a produced by PDL cells was incubated with human bone marrow mesenchymal stem cells (BMMSCs), C5a bound to BMMSC C5aR and induced its subsequent phosphorylation. Additionally, both C5a binding and C5aR phosphorylation further increased with PDL supernatants incubated with the abovementioned bone filling materials. Binding of C5a to BMMSC C5aR induced their proliferation and recruitment towards injured PDL cells. Both BMMSC proliferation and recruitment significantly increased when the PDLs were incubated with Gen-Os® bone filling materials [41]. Thus, PDL cells produce C5a leading to BMMSC proliferation and C5a-dependent recruitment and these effects can be modulated by bone filling materials. This demonstrates Complement implication in two important steps for bone regeneration: bone marrow mesenchymal stem cell proliferation and recruitment the stimulation/injury site. b) Complement production in the pulp: implication in the initial steps of dental pulp regeneration

Figure 1 :

 1 Figure 1: Complement activation pathways and bioactive molecules

Figure 2 :

 2 Figure 2: A sketch of Complement proteins/receptors in tissue regeneration.

Figure 3 :

 3 Figure 3: A sketch of Membrane Attack Complex (MAC) secretion by pulp fibroblasts and implication in cariogenic bacteria lysis

Figure 4 :( 1 )

 41 Figure 4: A sketch depicting fibroblast-released C3a under carious simulation and its implication in pulp cell proliferation, mobilization and recruitment. (1) Pulp fibroblast stimulation by cariogenic bacteria in the inflammatory area beneath the carious injury, (2) fibroblasts secrete C3a. (3) This molecule interacts with its C3aR receptors on pulp fibroblasts and DPSCs in the intact pulp area inducing pulp fibroblast and DPSC proliferation. (4) The resulting C3gradient mobilizes stem cells and induces fibroblasts recruitment from the intact area to the stimulation site in the inflammatory area[START_REF] Rufas | Complement C3a Mobilizes Dental Pulp Stem Cells and Specifically Guides Pulp Fibroblast Recruitment[END_REF].

Figure 5 :( 1 )

 51 Figure 5: A sketch depicting fibroblast-released C5a effects under carious simulation and its implication in neurite growth and pulp stem cell recruitment and differentiation.(1) Pulp fibroblast stimulation by cariogenic bacteria in the inflammatory area beneath the carious injury, (2) pulp fibroblasts secrete C5a[START_REF] Chmilewsky | Pulp fibroblasts synthesize functional complement proteins involved in initiating dentin-pulp regeneration[END_REF]. (3) Interaction of C5a with C5aR expressed by DPSCs in the intact area leads to DPSC migration to the stimulation site[START_REF] Chmilewsky | Pulp progenitor cell recruitment is selectively guided by a C5a gradient[END_REF], and (4) their subsequent differentiation into odontoblast-like cells[START_REF] Chmilewsky | C5L2 Regulates DMP1 Expression during Odontoblastic Differentiation[END_REF]. On the other hand, (3) C5a fixation on fibroblast C5aR leads to (4) BDNF secretion from the inflammatory area towards the intact pulp, inducing (5) neurite outgrowth towards inflammatory area[START_REF] Chmilewsky | Pulp Fibroblasts Control Nerve Regeneration through Complement Activation[END_REF]. The green rectangle insert shows steps 1-4 from fibroblast stimulation to BDNF secretion.

Figure 6 :

 6 Figure 6: A sketch of local complement system activation and its effects on the initial steps of dental tissue regeneration. Periodontal ligament cells and pulp fibroblasts produce Complement active molecules upon traumatic injury or infection. C3a released by injured or infected pulp fibroblasts induces pulp fibroblasts/pulp stem cells proliferation, mobilizes pulp stem cells and recruits pulp fibroblasts. Fibroblast-secreted C5a

  is involved in pulp stem cell recruitment/differentiation and neurite outgrowth through BDNF release. C5a released by Periodontal ligament cells recruits Bone Marrow Mesenchymal Stem Cell. Conflict of Interest: Madison Bergmann declares that he has no conflict of interest. Charlotte Jeanneau declares that he has no conflict of interest. Thomas Giraud declares that he has no conflict of interest. Gilles Richard declares that he has no conflict of interest. Imad About declares that he has no conflict of interest. Funding: This work was supported by Aix-Marseille University and CNRS. Ethical approval: This article does not contain any studies with human participants or animals performed by any of the authors. Informed consent: Informed consent was obtained from all individual participants included in the study.

	Compliance with Ethical Standards
	DPSC= dental pulp stem cell
	BDNF= Brain Derived Neurotrophic Factor
	BMMSC=Bone Marrow Mesenchymal Stem Cell