Héctor Ríos

Roberto Franco

Alejandra Ferreira De Loza

Denis Efimov

A High-Order Sliding-Mode Adaptive Observer for Uncertain Nonlinear Systems

Keywords: Adaptive Observers, Nonlinear Systems, Sliding-Modes

A high-order sliding-mode adaptive observer is proposed to solve the problem of adaptive estimation, i.e., the simultaneous estimation of the state and parameters, for a class of uncertain nonlinear systems in the presence of external disturbances, that does not need to satisfy a relative degree condition equal to one. This approach is based on a highorder sliding-mode observer and a nonlinear parameter identification algorithm. The practical, global and uniform asymptotic stability of the adaptive estimation error, despite the external disturbances, is guaranteed through the small-gain theorem. The convergence proofs are developed based on Lyapunov and inputto-state stability theories. Some simulation results illustrate the performance of the proposed high-order sliding-mode adaptive observer.

I. INTRODUCTION

Control of real systems confronts two shortcomings: first, not all the states are available for measurement; second, the model has discrepancies with respect to the real system (see [START_REF] Ioannou | Robust Adaptive Control[END_REF] and [START_REF] Shtessel | Sliding Mode Control and Observation[END_REF]). Disturbances and uncertainties exist in almost all realworld scenarios in the form of external perturbations, unknown system dynamics and/or unknown parameters. In the last years, the application and the development of adaptive observers have motivated a lot of work, this topic has become a wide and active research field. The adaptive observers estimate, simultaneously, the whole state and the parameters of the system using some online adaptation law (for instance, see [START_REF] Besançon | Nonlinear Observers and Application[END_REF]).

Concerning adaptive observers, early results that solve this problem are presented in [START_REF] Marino | Nonlinear Control Design: Geometric, Adaptive & Robust[END_REF] and [START_REF] Cho | A systematic approach to adaptive observer synthesis for nonlinear systems[END_REF]. Both works guarantee exponential convergence of the estimation error, meanwhile, the convergence of the parameters is exponential. The latter fact is mainly determined by a persistence-of-excitationtype constraint. The problem of adaptive observer synthesis for Lipschitz nonlinear systems with known parameters is developed in [START_REF] Ekramian | Adaptive state observer for Lipschitz nonlinear systems[END_REF]. The sufficient conditions are presented to ensure the state estimation error convergence to zero. Also, an adaptive observer scheme is presented to achieve the adaptive estimation under additional constraints. In [START_REF] Tyukin | Adaptive observers and parameter estimation for a class of systems nonlinear in the parameters[END_REF], based on the concepts of weakly attracting sets and nonuniform convergence, an adaptive observer is proposed for the asymptotic reconstruction of the state and parameter values in a particular class of forward-complete Single-Input-Single-Output nonlinear systems. In [START_REF] Farza | Adaptive observers for nonlinearly parametrized class of nonlinear systems[END_REF], an adaptive observer is proposed to exponentially estimate the state and the unknown parameters under a persistent excitation condition for uniformly observable Multiple-Input-Multiple-Output (MIMO) nonlinear systems. In [START_REF] Ma | Estimation using L 1 adaptive descriptor observer for multivariable systems with nonlinear uncertainties and measurement noises[END_REF], an L 1 adaptive descriptor observer is designed for multi-variable systems with nonlinear uncertainties and measurement noises. The algorithm is constructed to asymptotically estimate states, nonlinear uncertainties, and measurement noise at the same time. In [START_REF] Stepanyan | Robust adaptive observer design for uncertain systems with bounded disturbances[END_REF], a robust adaptive observer design methodology for a class of uncertain nonlinear systems in the presence of time-varying unknown parameters is given ensuring asymptotic convergence of state estimation and boundedness of parameter errors. The above-mentioned works do not consider external disturbances, and the convergence rates are asymptotic or exponential.

In this context, in [START_REF] Vijay | Adaptive observer based approach for the fault diagnosis in solid oxide fuel cells[END_REF], two nonlinear adaptive observers are designed to provide a fault diagnosis of a Solid Oxide Fuel Cell system, ensuring an asymptotic convergence of the fault. In [START_REF] Chen | Fault reconstruction for markovian jump systems with iterative adaptive observer[END_REF], two types of adaptive observer methods are developed to solve the problem of Markovian jump systems with simultaneous time-varying actuator efficiency factors, additive actuator, and sensor faults. The estimation and identification errors for both observers converge asymptotically. In [START_REF] Farza | Adaptive observer design for a class of nonlinear systems: Application to speed sensorlees induction motor[END_REF], the problem of an adaptive high-gain observer for a class of MIMO non uniformly observable systems with unknown constant parameters is addressed. Exponential convergence is established for the state and parameter estimation errors. In [START_REF] Rabiee | Continuous nonsingular terminal sliding-mode control based on adaptive sliding-mode disturbance observer for uncertain nonlinear systems[END_REF], an adaptive sliding-mode disturbance observer is designed to estimate the disturbance in finite time ensuring the convergence to zero of the state estimation error. In [START_REF] Chen | New adaptive sliding observers for position and velocity-sensorless controls of brushless dc motors[END_REF], an adaptive sliding-mode observer is proposed for position and velocity sensorless controls of a cylindrical brushless DC motor. This algorithm guarantees asymptotic convergence of the estimation errors.

On the other hand, in the sliding-mode area, in [START_REF] Angulo | Robust exact uniformly convergent arbitrary order differentiator[END_REF], an arbitrary order differentiator that converges to the true derivatives of the signal after a finite time independent of the initial differentiator error is presented. This approach switches from a uniform differentiator to the classical high-order sliding-mode (HOSM) differentiator providing convergence to zero in finitetime of the differentiation error. In [START_REF] Angulo | On functional observers for linear systems with unknown inputs and HOSM differentiators[END_REF], a functional observer for linear systems with unknown inputs is considered. It is shown that when such an observer is not proper, it is still possible to use an HOSM differentiator to construct a proper quasi-functional unknown input observer.

Regarding finite-time convergence, an adaptive state observer for a class of nonlinear systems with unknown parameters is presented in [START_REF] Pyrkin | Adaptive state observers using dynamic regressor extension and mixing[END_REF]. Under an excitation assumption, the convergence in finite time of the parameter is guaranteed; meanwhile, the state converges asymptotically. In [START_REF] Ríos | An adaptive sliding-mode observer for a class of uncertain nonlinear systems[END_REF], a nonlinear adaptive sliding-mode observer is proposed based on a nonlinear parameter estimation algorithm. The proposed algorithm is included in the structure of a sliding-mode state observer, providing an ultimate bound for the full estimation error and attenuating the effects of the external disturbances concluding finite-time convergence of the identification error and exponential convergence of the estimation error. In [START_REF] Franco | Adaptive estimation for uncertain nonlinear systems with measurement noise: A sliding-mode observer approach[END_REF], an adaptive observer based on [START_REF] Ríos | An adaptive sliding-mode observer for a class of uncertain nonlinear systems[END_REF] is developed, this observer increases the class of systems ensuring finite-time convergence of the parameter identification error and exponential convergence to a neighborhood of the state estimation error.

Most of the above-mentioned works have one of the following shortcomings: either do not consider external disturbances, or the output must satisfy a relative degree condition equal to one with respect to the external disturbances. Moreover, in most cases, the convergence rates are asymptotic or exponential. Motivated for the aforementioned issues, this paper proposes an HOSM adaptive observer for a class of uncertain nonlinear systems affected by external disturbances. In contrast with the previous works, the disturbances can be of a relative degree higher than one. The present approach combines an HOSM observer with a nonlinear parameter identification algorithm. The HOSM observer guarantees finite-time inputto-state stability (FT-ISS) of the state estimation error with respect to the parameter identification error. Whereas, the nonlinear parameter identification algorithm provides FT-ISS of the parameter identification error with respect to the state estimation error. Then, the practical, global, and uniform asymptotic stability of the adaptive estimation error, despite the external disturbances, is guaranteed through the smallgain theorem. The convergence proofs are developed based on Lyapunov and ISS theories.

This manuscript is organized as follows. The problem statement is formulated in section II. The preliminaries are discussed in Section III. The adaptive observer and the main results are presented in Section IV. Simulation results are shown in Section V, followed by the conclusions in Section VI.

A. Notation

The Euclidean norm of a vector q ∈ R n is denoted by q . For a matrix Q ∈ R m×n , denotes its smallest singular value σ min (Q) = λ min (Q T Q) and its induced norm as

Q := λ max (Q T Q) = σ max (Q)
, where λ max is the maximum eigenvalue and λ min is the minimum one, σ max is the largest singular value. For a Lebesgue measurable function u : R ≥0 → R m , define the norm u (t0,t1) := ess sup t∈(t0,t1) u(t) , then u ∞ := u (0,+∞) and the set of functions u with the property u ∞ < +∞ is denoted as L ∞ . For a matrix function Q : R ≥0 → R m×n , denote Q ∞ := Q (0,+∞) . The term ∇V (x)f (x) denotes the directional derivative of a continuously differentiable function V with respect to the vector field f evaluated at any point x. Define the function a γ = |a| γ sign(a), for any γ ∈ [0, 1) and any a ∈ R.

A continuous function α : R ≥0 → R ≥0 belongs to class K if it is strictly increasing and α(0) = 0; it belongs to class K ∞ if it is also unbounded. A continuous function β : R ≥0 ×R ≥0 → R ≥0 belongs to class KL if, for each fixed s, β(r, s) ∈ K with respect to r, and for each fixed r, β(r, s) is decreasing to zero with respect to s. Moreover, the function β is class KL T if β(r, 0) ∈ K ∞ , β is a strictly decreasing function on its second argument s ∈ R + for any fixed first argument r ∈ R + and β(r, T ) = 0 for each fixed r ∈ R + for some 0 ≤ T < +∞.

II. PROBLEM STATEMENT Consider the following class of uncertain nonlinear systems

ẋ = Ax + φ(y, u) + G(t, y, u)θ + Dw(t), (1a) y = Cx, (1b)
where x ∈ R n is the state vector, y ∈ R is the measurable output, u ∈ R m is the control input vector, θ ∈ R is an unknown constant parameter, and w ∈ R represents the external disturbances. The matrices A, C, and D are known, they have corresponding dimensions, and it is assumed that the triple (A, C, D) is strongly observable, which implies that the relative degree of the output y, with respect to the external disturbance w, is equal to n. The functions φ : R × R m → R n and G : R ≥0 × R × R m → R n are also known, and they ensure uniqueness and existence of solutions for system [START_REF] Ioannou | Robust Adaptive Control[END_REF] for all admissible disturbances. The objective is to provide estimations of the state and the unknown parameter, i.e., x and θ, respectively; only using the information of the output y and attenuating the effects of the external disturbances w.

The following assumption is imposed on system (1).

Assumption 1. ||x|| ∞ < +∞, ||u|| ∞ < +∞, ||w|| ∞ < +∞, and
||G(t, y(t), u(t))|| ∞ < +∞ for all t ≥ 0.

III. PRELIMINARIES Consider the following nonlinear system

ẋ = f (t, x(t)), t ≥ t 0 , t 0 ∈ R, (2)
where x ∈ R n is the state vector; f : R + × R n → R n is a continuous function with respect to x and measurable with respect to t, and f (t, 0) = 0. A solution of system (2), for an initial condition x 0 ∈ R n at time instant t 0 ∈ R, is denoted as x(t, t 0 , x 0 ), and it is defined on some finite time interval [t 0 , t 0 + T ], where 0 ≤ T ≤ ∞.

Let Ψ be an open neighborhood of the origin in R n ; then, the following stability properties are introduced for system (2) (for more details see [START_REF] Ríos | An adaptive sliding-mode observer for a class of uncertain nonlinear systems[END_REF], [START_REF] Bernuau | Verification of ISS, iISS and IOSS properties applying weighted homogeneity[END_REF], [START_REF] Jiang | A Lyapunov formulation of the nonlinear small-gain theorem for interconnected ISS systems[END_REF] and [START_REF] Khalil | Nonlinear Systems[END_REF]). Definition 1. [START_REF] Khalil | Nonlinear Systems[END_REF]. At the steady state x = 0, the system (2) is said to be:

1) Uniformly Stable in Ψ if, for any t 0 ∈ R and > 0, there is δ( ) > 0 such that for any x 0 ∈ Ψ , if x 0 < δ( ), then x(t, t 0 , x 0 ) ≤ , for all t ≥ t 0 ; 2) Uniformly Asymptotically Stable if it is Uniformly Stable in Ψ and for any t 0 ∈ R, κ > 0 and > 0, there exists T (κ, ) ≥ 0 such that for any x 0 ∈ Ψ , if x 0 ≤ κ, then x(t, t 0 , x 0 ) ≤ , for all t ≥ t 0 + T (κ, ); 3) Uniformly Finite-Time Stable if it is Uniformly Stable and for any x 0 ∈ Ψ , there exists 0 ≤ T x0 < +∞ such that x(t, t 0 , x 0 ) = 0, for all t ≥ t 0 + T x0 . The function

T (x 0 ) = inf{T x0 ≥ 0 : x(t, t 0 , x 0 ) = 0, ∀t ≥ t 0 +T x0 } is called the settling time of the system (2). If Ψ = R n , at steady-state x = 0, system (
2) is said to be Globally Uniformly Stable (GUS), Globally Uniformly Asymptotically Stable (GUAS) and Globally Uniformly Finite-Time Stable (GUFTS), respectively.

Consider the following nonlinear system

ẋ = f (x, w), (3)
where

x ∈ R n is the state vector, w ∈ R l is the external disturbance; f : R n ×R l → R n is a locally Lipschitz function.
It is assumed that the solution of system (3) for an initial condition x 0 ∈ R n and w ∈ L ∞ is denoted as x(t, x 0 , w) for any t ≥ 0 for which the solution exists.

Definition 2. [START_REF] Bernuau | Verification of ISS, iISS and IOSS properties applying weighted homogeneity[END_REF]. The system (3) is said to be Input-to-State practically Stable (ISpS) if for any w ∈ L ∞ , and any x 0 ∈ R n there exist some functions β ∈ KL, γ ∈ K and a constant κ ∈ R ≥0 such that its solution satisfies

||x(t, x 0 , w)|| ≤ β(||x 0 ||, t) + γ(||w|| ∞ ) + κ, ∀t ≥ 0.
If κ = 0, the system (3) is said to be Input-to-State Stable (ISS). Moreover, if β ∈ KL T and κ = 0, system (3) is said to be Finite-Time Input-to-State Stable (FT-ISS).

If ||w|| ∞ = 0; then, system (3) is said to be practically GUAS, for κ > 0; and GUAS, for κ = 0. Lemma 1. [START_REF] Ríos | An adaptive sliding-mode observer for a class of uncertain nonlinear systems[END_REF]. Let V : R n → R ≥0 be a smooth function. If there exist some positive constants ϕ 1 , ϕ 2 , ϕ 3 , ϕ 4 > 0 and ς ∈ (0, 1] such that

ϕ 1 ||x|| 2 ≤ V (x) ≤ ϕ 2 ||x|| 2 , ∇V (x)f (x, ω) ≤ -ϕ 3 V ς (x), ∀||x|| ≥ µ := ϕ 4 ||w|| ∞ ,
then system (3) is FT-ISS with respect to the input w. Moreover, the following bounds are satisfied:

||x(t, xo, ω)|| ≤ ϕ(t) ϕ 1 , ∀ς ∈ (0, 1), ∀t ≤ T (x0), e -ϕ 3 2 t ϕ 2 ϕ 1 ||x0||, ς = 1, ∀t ≤ T (x0), ||x(t, xo, ω)|| ≤ ϕ2 ϕ1 µ, ∀t > T (x0),
where

ϕ(t) = (ϕ 1-ς 2 ||x 0 || 2(1-ς) -ϕ 3 (1 -ς)t) 1 2(1-ς) and T (x 0 ) ≤      max 0, ϕ 1-ς 2 ||x 0 || 2(1-ς) -ϕ 1-ς 1 µ 2(1-ς) ϕ 3 (1-ς) , ∀ς ∈ (0, 1), max 0, 2[ln(||x 0 ||)-ln(µ)] ϕ 3 , ς = 1.
Let us consider the following interconnected nonlinear system

ẋ1 = f 1 (x 1 , x 2 , w), (4a) ẋ2 = f 2 (x 1 , x 2 , w), (4b)
where

x i ∈ R ni , w ∈ R l , and f i : R n1 × R n2 × R l → R ni
ensures existence of the system solutions at least locally, for i = 1, 2. Consider that both subsystems are ISS with respect to (x 2 , w) and (x 1 , w), respectively, i.e., there exist functions β i ∈ KL, γ j ∈ K, and positive constants κ i with i = 1, 2 and j = 1, 4, such that the conditions

||x 1 (t, x 10 , w)|| ≤ β 1 (||x 10 ||, t) + γ 1 ( x 2 ∞ ) + γ 2 ( w ∞ ) + κ 1 , (5a) ||x 2 (t, x 20 , w)|| ≤ β 2 (||x 20 ||, t) + γ 3 ( x 1 ∞ ) + γ 4 ( w ∞ ) + κ 2 , ( 5b
)
hold for all t ≥ 0. Thus, the following nonlinear smallgain result is introduced, in terms of ISS properties, for the interconnected system (4).

Theorem 1.

[24], [START_REF] Dashkovskiy | An ISS small-gain theorem for general networks[END_REF]. Suppose that the interconnected system (4a)-( 4b) is ISS with respect to (x 2 , w) and (x 1 , w), respectively, satisfying the condition in [START_REF] Cho | A systematic approach to adaptive observer synthesis for nonlinear systems[END_REF]. If there exists a non negative number s 1 satisfying

γ 1 • γ 3 (s) < s, ∀s > s 1 , (6)
then, the system (4) is ISpS with respect to the external input w. If s 1 = 0, the system (4) is ISS with respect to the external input w.

Note that, if the external disturbance w is vanishing, or somehow, it is completely compensated, Theorem 1 implies that the interconnected system (4a)-(4b) is practically GUAS, for s 1 > 0; and GUAS, for s 1 = 0.

IV. HOSM ADAPTIVE OBSERVER

Let us introduce the following adaptive observer

Ω = (A -LC)Ω + G(t, y, u), (7a) θ = ΓΩ T C T y -C x α , (7b) x = z + T -1 v, (7c)
ż = Az + φ(y, u) + G(t, y, u) θ + L(y -Cz) + Ω θ, (7d) v = A 0 v + a(y -Cz) + χ(v 1 , y -Cz). ( 7e
)
where x ∈ R n and θ ∈ R are the estimations for x and θ, respectively; z ∈ R n is a Luenberger state estimation, and Ω ∈ R n is an auxiliary variable. If the signal G is persistently exciting (PE), then, due to the filtering property of the variable Ω, the variable CΩ is also PE. The function y -C x α in (7b) is understood in the component-wise sense. Note that (7b) represents the dynamics of a parameter identification algorithm characterized by α. If α = 0, a discontinuous algorithm is obtained; when α = 1, one has the classic linear algorithm, while for α ∈ (0, 1), a continuous nonlinear algorithm is given. The observer vector gain L ∈ R n has to be selected such that (A -LC) is Hurwitz and

Γ = Γ T > 0. The transformation matrix T ∈ R n×n is given as T -1 = [(A-LC) n-1 ϑ, (A-LC) n-2 ϑ, ..., ϑ]; where ϑ = O -1 h, with h = [0, 0, ..., 1] T ∈ R n ,
and O is the observability matrix of the pair (A -LC, C). The vector v ∈ R n provides an estimation of the error between the system state x and the Luenberger state estimation z. The nonlinear injection χ : R × R → R n takes the following structure:

χ(v 1 , y -Cz) :=       k 1 (y -Cz) -v 1 n-1 n k 2 (y -Cz) -v 1 n-2 n . . . k n (y -Cz) -v 1 0      
, with some positive gains k i , i = 1, n. Finally, the matrix A 0 and the vector a are given as follows

A 0 =        0 1 0 • • • 0 0 0 1 • • • 0 . . . . . . . . . 0 0 0 0 1 0 0 0 0 0        , a =      a 1 a 2 . . . a n     
, where a i , i = 1, n are the vector of coefficients of the characteristic polynomial of the matrix (A -LC). The solutions of the adaptive observer [START_REF] Tyukin | Adaptive observers and parameter estimation for a class of systems nonlinear in the parameters[END_REF] are understood in the Filippov sense [START_REF] Filippov | Differential Equations with Discontinuos Right-hand Sides[END_REF]. Let us introduce the following assumption:

Assumption 2. Let 0 < min ≤ σ min (CΩ(t)) for all t ≥ 0 and CΩ ∞ ≤ max < +∞.
The existence of max is guaranteed by Assumption 1 and the fact that matrix (A -LC) is Hurwitz; the existence of min follows if, for example, the number of sensors is equal to the number of unknown parameters, or CΩ(t) is uniformly injective for all t ≥ 0 (for more details, see [START_REF] Rueda-Escobedo | Discontinuous gradient algorithm for finite-time estimation of time-varying parameters[END_REF]). Moreover, note that Ω(0) can be arbitrarily chosen. Remark 1. Opposite to [START_REF] Ríos | An adaptive sliding-mode observer for a class of uncertain nonlinear systems[END_REF] and [START_REF] Franco | Adaptive estimation for uncertain nonlinear systems with measurement noise: A sliding-mode observer approach[END_REF], the adaptive observer in (7) strikes systems with relative degree larger than one. For instance, mechanical systems, where the relative degree of the output (position), with respect to the external disturbance w(t) (in acceleration), is equal to two.

In the sequel, the stability of the HOSM adaptive observer [START_REF] Tyukin | Adaptive observers and parameter estimation for a class of systems nonlinear in the parameters[END_REF] is analyzed. To this aim, the dynamics (7) can be rewritten as two interconnected subsystems given by the nonlinear parameter identification algorithm and the HOSM observer. Then, the FT-ISS properties of each subsystem are introduced. Finally, the small-gain theorem is used to ensure the stability of the adaptive estimation error.

A. The Nonlinear Parameter Identification Algorithm

Define θ := θ -θ. Then, the error dynamics for θ is given by:

θ = ΓΩ T C T y -C x α . (8)
The FT-ISS convergence properties of the error dynamics (8) with respect to the input η = T δ -v with δ = x -z + Ω θ , for α ∈ (0, 1) and α = 0, are given by the following lemmas. Lemma 2. [START_REF] Franco | Adaptive estimation for uncertain nonlinear systems with measurement noise: A sliding-mode observer approach[END_REF]. Let Assumption 2 be satisfied. Then, the error dynamics (8), with α ∈ (0, 1) and Γ = Γ T > 0, is FT-ISS with respect to η. Moreover, the trajectories satisfy:

θ(t, θ0 ) ≤ √ c 1 c α-1 2 2 θ0 1-α + κ 1 (1 -ψ)(1 -α) α+1 min c α+1 2 2 2 t 1 1-α , ∀t ≤ T θ1 , θ(t, θ0 ) ≤ c 1 c 2 µ θ1 ||η|| ∞ , ∀t > T θ1 ,
where θ0 = θ(0), c 1 := 2λ max (Γ), c 2 := 2λ min (Γ), κ 1 ∈ (0, 1) and

µ θ1 = (2κ 2 ) 1 α+1 ||CT -1 || (ψκ 1 ) 1 α+1 min , T θ1 ≤ max   0, 2(c α-1 2 2 θ0 1-α -c α-1 2 1 µ 1-α θ1 κ 1 (1 -ψ)(1 -α) α+1 min c α+1 2 2   ,
for any θ0 ∈ R, ψ ∈ (0, 1) and

κ 2 = max(κ 1 + 1, κ 1 /(1 - κ 1 α
1 ) α ). Proof. Let us consider the following Lyapunov function:

V θ = 1 2 θT Γ -1 θ, (9)
that satisfies the following inequalities:

c -1 1 θ 2 ≤ V θ ( θ) ≤ c -1 2 θ 2 , (10) c
-α+1 2 1 θ α+1 ≤ V α+1 2 θ ( θ) ≤ c -α+1 2 2 θ α+1 . ( 11
)
Following the procedure given in [START_REF] Franco | Adaptive estimation for uncertain nonlinear systems with measurement noise: A sliding-mode observer approach[END_REF], it can be shown that the time derivative of V θ along the trajectories of the error dynamics [START_REF] Farza | Adaptive observers for nonlinearly parametrized class of nonlinear systems[END_REF], with α ∈ (0, 1), satisfies

Vθ ≤ -κ 1 (1 -ψ) α+1 min c α+1 2 2 V α+1 2 θ , (12a)
∀|| θ|| ≥ (2κ 2 )

1 α+1 ||CT -1 || (ψκ 1 ) 1 α+1 min ||η|| ∞ , (12b)
for any constant ψ ∈ (0, 1). Therefore, the error dynamics (7b) is FT-ISS with respect to η.

Lemma 3. [START_REF] Franco | Adaptive estimation for uncertain nonlinear systems with measurement noise: A sliding-mode observer approach[END_REF]. Let Assumption 2 be satisfied. Then, the error dynamics (8), with α = 0, is FT-ISS with respect to η. Moreover, the trajectories satisfy:

θ(t, θ0 ) ≤ √ c 1 c -1 2 2 θ0 + κ 1 (1 -ψ) min c 1 2 2 2 t , ∀t ≤ T θ2 , θ(t, θ0 ) ≤ c 1 c 2 µ θ2 ||η|| ∞ , ∀t > T θ2 ,
where

µ θ2 = 2κ 2 ||CT -1 || ψκ 1 min , T θ2 ≤ max   0, 2 c -1 2 2 θ0 -c -1 2 1 µ θ2 κ 1 (1 -ψ) min c 1 2 2   ,
for any θ0 ∈ R, ψ ∈ (0, 1) and κ 2 ≥ κ 1 + 1.

Proof. Let us consider the Lyapunov function given in ( 9) that satisfies the inequalities [START_REF] Stepanyan | Robust adaptive observer design for uncertain systems with bounded disturbances[END_REF] and [START_REF] Vijay | Adaptive observer based approach for the fault diagnosis in solid oxide fuel cells[END_REF]. Following the procedure given in [START_REF] Franco | Adaptive estimation for uncertain nonlinear systems with measurement noise: A sliding-mode observer approach[END_REF], the time derivative of V θ along the trajectories of the error dynamics [START_REF] Farza | Adaptive observers for nonlinearly parametrized class of nonlinear systems[END_REF], with α = 0, satisfies

Vθ ≤ -κ 1 (1 -ψ) min c 1 2 2 V 1 2 θ , (13a)
∀|| θ|| ≥ 2κ 2 ||CT -1 || ψκ 1 min ||η|| ∞ , (13b)
for any constant ψ ∈ (0, 1). Therefore, the error dynamics (7b) is FT-ISS with respect to η.

Then, the results of Lemmas 2 and 3 establish that the trajectories of the error dynamics (8), i.e., θ, enter into the bound c 1 /c 2 µ θ1 or c 1 /c 2 µ θ2 , in a finite time for any α ∈ [0, 1). It is clear that the size of such a bound depends on α.

B. The HOSM State Observer

Consider the error dynamics for δ = x -z + Ω θ, i.e.,

δ = (A -LC)δ + Dw(t), (14a)
y δ = y -Cz = Cδ -CΩ θ, (14b)
where y δ is a measurable signal. Then, applying the transformation δ = T δ to system [START_REF] Rabiee | Continuous nonsingular terminal sliding-mode control based on adaptive sliding-mode disturbance observer for uncertain nonlinear systems[END_REF], it follows that

δ = A 0 δ + aCT -1 δ -aCΩ θ + T Dw, (15a)
y δ = δ1 -CΩ θ, (15b)
where δ1 = CT -1 δ and T D = [0, ..., 0, CA n-1 D] T . Then, an observer for system [START_REF] Chen | New adaptive sliding observers for position and velocity-sensorless controls of brushless dc motors[END_REF] is designed as (7e). Therefore, the error dynamics for η = δ -v is given by

η = A 0 η + T Dw -χ(v 1 , y -Cz).
Note that y -Cz -v 1 = δ1 -CΩ θ -v 1 ; therefore, the correction term χ 1 can be rewritten as follows

χ(η 1 , CΩ θ) =          k 1 η 1 -CΩ θ n-1 n k 2 η 1 -CΩ θ n-2 n . . . k n η 1 -CΩ θ 0         
, where η 1 = δ1 -v 1 . Then, the error dynamics is given by

η = A 0 η + T Dw(t) -χ(η 1 , CΩ θ). ( 16
)
Note that dynamics [START_REF] Angulo | Robust exact uniformly convergent arbitrary order differentiator[END_REF] has the same structure as the HOSM differentiator [START_REF] Levant | High-order sliding modes: differentiation and outputfeedback control[END_REF]. The following lemma states the FT-ISS convergence properties of the error dynamics [START_REF] Angulo | Robust exact uniformly convergent arbitrary order differentiator[END_REF] with respect to the input θ. Lemma 4. [START_REF] Levant | High-order sliding modes: differentiation and outputfeedback control[END_REF], [START_REF] Zavala | Levant's arbitrary-order exact differentiator: A Lyapunov approach[END_REF]. Let the observer (7c)-(7e) be applied to the system (1). Let Assumption 1 be satisfied. Then, there exist some positive constants k i , i = 1, n, such that the estimation error dynamics ( 16) is FT-ISS with respect to θ.

According to [START_REF] Levant | High-order sliding modes: differentiation and outputfeedback control[END_REF], if ||w|| ∞ ≤ w, with some known positive constant w; then, the gains k i , i = 1, n, for n = 4, can be chosen as

k 4 = 1.1|CA n-1 D w|, k 3 = 1.5|CA n-1 D w| 1/2 , k 2 = 2|CA n-1 D w| 1/3 and k 1 = 3|CA n-1 D w| 1/4 .
Moreover, the results given by [START_REF] Levant | High-order sliding modes: differentiation and outputfeedback control[END_REF] show that η satisfies the following inequality

η(t, η 0 ) ≤ n i=1 τ i |CA n-1 D w| i-1 n n-i+1 n max || θ|| n-i+1 n ∞ , (17)
for all t ≥ T η > 0, with τ i ≥ 1 depending only on the gains k i . A method to compute the constants τ i has been proposed in [START_REF] Zavala | Levant's arbitrary-order exact differentiator: A Lyapunov approach[END_REF], by means of the following Lyapunov function

V η = n-1 j=1 β j Z j (η j , η j+1 ) + β n 1 p |η n | p , (18)
with some

β i > 0, i = 1, n, p > 1 and Z j (η j , η j+1 ) = rj p |η j | p r j -η j η j+1 p-r j r j+1 + p-rj p |η j+1 | p r j+1 , j = 1, n -1,
and the homogeneity weights

r i = n + 1 -i with i = 1, n.
Hence, note that by definition x = z + T -1 v; thus, since δ = x -z + Ω θ and η = T δ -v, one obtains that

x -x = T -1 η -Ω θ, (19)
then, Lemma 4 implies that the state estimation error dynamics x -x is FT-ISS with respect to θ.

C. Convergence of the Adaptive Observer

In order to establish the convergence properties of the adaptive observer [START_REF] Ekramian | Adaptive state observer for Lipschitz nonlinear systems[END_REF], the statements provided by Lemmas 2, 3, 4 and Theorem 1 are applied. The following result shows that the interconnected error system (8) and ( 16) is GUS for any α ∈ [0, 1). Theorem 2. Let Assumptions 1 and 2 hold. Let L ∈ R n be such that (A -LC) is Hurwitz, α ∈ [0, 1), Γ = Γ T > 0, and k i , i = 1, n, be designed based on the formulas provided by [START_REF] Levant | High-order sliding modes: differentiation and outputfeedback control[END_REF]. If the following conditions

nβµ max ≤ 1, (20a)
[(nβ) n µ max ] 1 n-1 ≤ 1, (20b)
hold with β = max i=1,n [τ i |CA n-1 D w| i-1 n n-i+1 n max ] and µ max = c 1 /c 2 max[µ θ1 , µ θ2
]; then, the interconnected error system (8) and ( 16) is practically GUAS for any α ∈ [0, 1).

Proof. Based on Lemmas 2, 3 and 4 and their proofs, considering (12b), (13b), and [START_REF] Angulo | On functional observers for linear systems with unknown inputs and HOSM differentiators[END_REF], it follows that

θ ≥ µ max η ⇒ Vθ < 0, (21)
n i=1 || θ|| n-i+1 n ≤ β -1 η ⇒ Vη < 0, (22)
with µ max = c 1 /c 2 max[µ θ1 , µ θ2 ] and

µ θ1 = (2κ 2 ) 1 α+1 ||CT -1 || (ψκ 1 ) 1 α+1 min , µ θ2 = 2κ 2 ||CT -1 || ψκ 1 min , β = max i=1,n [τ i |CA n-1 D w| i-1 n n-i+1 n max ].
Note that ( 22) can be rewritten as follows

n|| θ|| ≤ β -1 η ⇒ Vη < 0, for θ > 1, n|| θ|| 1 n ≤ β -1 η ⇒ Vη < 0, for θ ≤ 1.
Therefore, in order to apply the Theorem 1, in terms of the error norms, two cases must be analyzed, i.e., when θ > 1 and when θ ≤ 1.

• Case 1: s > 1

According to ( 21) and ( 22), the small-gain condition is reduced to µ max s ≤ (nβ) -1 s, which is satisfied for any s > 1, the previous condition can be rewritten as in (20a). • Case 2: s ≤ 1

According to ( 21) and ( 22), the small-gain condition is reduced to µ max s ≤ (nβ) -1 s n , which results in the condition given in (20b). Thus, in order to guarantee the practical, global, and uniform asymptotic stability of the interconnected error systems ( 8) and ( 16), conditions (20a) and (20b) are required. This concludes the proof.

Theorem 2 implies that the state estimation error dynamics, x -x; and the parameter identification error dynamics, θ; are practically GUAS for any α ∈ [0, 1).

The procedure to verify the condition (20) includes the following steps:

1) Use (7a) to calculate 0 < min ≤ σ min (CΩ(t)) and CΩ ∞ ≤ max < +∞ offline. 2) Fix the values of κ 1 , ψ ∈ (0, 1), and κ 2 according to Lemma 2 or 3.

3) Define the set

L c = {η ∈ R n |V η ≤ c}, based on V η
given in [START_REF] Pyrkin | Adaptive state observers using dynamic regressor extension and mixing[END_REF], with c such that Vη < 0 in the sublevel set L c ; and then, from the sublevel set L c , the values for τ i can be computed as τ i = max Lc |η i |, for all i = 1, n. 4) Verify the small-gain condition [START_REF] Franco | Adaptive estimation for uncertain nonlinear systems with measurement noise: A sliding-mode observer approach[END_REF]. Note that the values for min and max depend on the excitation given by the term G(t, y, u) in (7a), which depends on the input u. It is well-known, in the adaptive estimation framework, that the input can be modified in order to get more excitation if it is required. Thus, it is possible to modify both min and max by means of the input. Note also that κ 1 , κ 2 and ψ can be adjusted to manipulate the left-hand side of [START_REF] Franco | Adaptive estimation for uncertain nonlinear systems with measurement noise: A sliding-mode observer approach[END_REF].

The following properties for the adaptive observer (7) must be remarked:

1) In the case that inequalities (20a) and (20b) are not satisfied simultaneously, the results from the adaptive observer will be local if (20b) holds. Moreover, if (20b) is not satisfied but (20a) is, the interconnected error dynamics converges to a unitary neighborhood of the origin. Note that, if (20b) holds, the size of the error convergence region is characterized by [(nβ) n µ max ]

1 n-1 . 2) Theorem 2 provides practical, global, and uniform asymptotic stability of the adaptive estimation error, and the rate of convergence given by the HOSM adaptive observer is faster than exponential due to its FT-ISS properties. 3) For the multiple-output and multiple-parameter case, consider system (1) with y ∈ R p , θ ∈ R q and w ∈ R p with q ≤ p, let us assume that the output vector has a vector relative degree [r 1 , r 2 , ..., r p ] with respect to the unknown input. Therefore, if the relative degrees are such that r 1 + r 2 + ... + r p = n; then, there exists a required nonsingular matrix T ∈ R n×n obtained from the rearrangement of the n linearly independent rows of the observability matrix of the pair (A, C). Under this vector relative degree condition, the HOSM state observer (7e) can be designed as follows vj = A j0 v j + A j1 (y j -c j z) + χ j (v j1 , y j -c j z), for j = 1, p, with

A j0 =        0 1 0 • • • 0 0 0 1 • • • 0 . . . . . . . . . 0 0 0 0 1 0 0 0 0 0        rj ×rj , A j1 =      a j1 a j2 . . . a jrj      rj , c j =      1 0 . . . 0      T rj , χ j =        k j1 (y j -c j z) -v j1 r j -1 r j k j2 (y j -c j z) -v j1 r j -2 r j . . . k jrj (y j -c j z) -v j1 0       
,

where v = [v 1 , ..., v p ] ∈ R n , with v j = [v j1 , ...v jrj ] ∈ R rj
, and some constant vectors A j1 , for all j = 1, p.

Then, it is possible to apply the HOSM observers v j by diagonal output blocks, obtaining a small-gain condition for each output block. Thus, if such smallgain conditions hold, it is possible to provide a state estimation and a parameter identification simultaneously (for more details, see [START_REF] Fridman | Observation of linear systems with unknown inputs via high-order sliding-modes[END_REF] and [START_REF] Ríos | Robust output-control for uncertain linear systems: Homogeneous differentiator-based observer approach[END_REF]). 4) For the case of time-varying parameters, i.e., θ(t), the proposed parameter identification algorithm provides the same result, i.e., practically GUAS. However, based on [START_REF] Franco | Adaptive estimation for uncertain nonlinear systems with measurement noise: A sliding-mode observer approach[END_REF], it is possible to prove that the size of the convergence region also depends on the upper bound of the time-varying parameter. 5) The robustness and finite-time properties of the HOSM observer together with the finite-time parameter identification algorithm are illustrated in the next section and, in contrast with [START_REF] Farza | Adaptive observer design for a class of nonlinear systems: Application to speed sensorlees induction motor[END_REF], [START_REF] Chen | New adaptive sliding observers for position and velocity-sensorless controls of brushless dc motors[END_REF], [START_REF] Pyrkin | Adaptive state observers using dynamic regressor extension and mixing[END_REF], [START_REF] Ríos | An adaptive sliding-mode observer for a class of uncertain nonlinear systems[END_REF] and [START_REF] Franco | Adaptive estimation for uncertain nonlinear systems with measurement noise: A sliding-mode observer approach[END_REF], the proposed algorithm covers a broader class of systems.

V. SIMULATION RESULTS The simulations have been done in Matlab Simulink with the Euler explicit discretization method and sampling time equal to 0.001 [s].

Consider the DC motor system [START_REF] Dormin | Controlling mechanical systems with backlash: A survey[END_REF]: T f (q) = α 0q sign(q) + α 1q e -α2qq|q| sign(q), for any q ∈ R and some known coefficients α 0x2 = 3.2, α 1x2 = 2.4 and α 2x2 = 1.1.

ẋ =    -Ra La -Km La -B s B s Km Jm 0 0 0 B s -B l J l -B s    x +   1 La u -T f (x 2 ) 0   +   0 -1 Jm x 2 0   θ +   0 0 1   w(t), y = 0 1 0 x + ς, where x = [x 1 , x 2 , x 3 ] T , x 1 ∈ R is the armature current, x 2 ∈ R is
The external disturbance is given by w(t) = -T d -T f (x 3 ), where T d is the load disturbance torque. For simulation purposes, the load disturbance torque is taken as T d = 0.4 [Nm], and the friction coefficients as α 0x3 = 1.1, α 1x3 = 2.1, and α 2x3 = 0.75. Thus, the external disturbance can be upper bounded by w ∞ ≤ 3.6.

The parameter to be identified is the motor viscous friction given by θ = 4.6 [Nms/rad]. The input is given as u(t) = 3 sin(t) + 1.4 sin(0.45t) + 8 cos(5t).

The proposed algorithm is compared with the linear algorithm, i.e., with α = 1. Note that the adaptive observers [START_REF] Farza | Adaptive observer design for a class of nonlinear systems: Application to speed sensorlees induction motor[END_REF], [START_REF] Chen | New adaptive sliding observers for position and velocity-sensorless controls of brushless dc motors[END_REF], [START_REF] Pyrkin | Adaptive state observers using dynamic regressor extension and mixing[END_REF], [START_REF] Ríos | An adaptive sliding-mode observer for a class of uncertain nonlinear systems[END_REF] and [START_REF] Franco | Adaptive estimation for uncertain nonlinear systems with measurement noise: A sliding-mode observer approach[END_REF] cannot be applied because the relative degree of the output, with respect to the external disturbance, is equal to three. Moreover, measurement noise is considered and is denoted by ς, which is a band limited white noise with power of 1 × 10 -7 .

Consider the initial conditions x(0) = [-0.2, -0.1, 1] T while the observer initial conditions are taken as Ω(0) = 0.1, θ(0) = 0, z(0) = 0 and v(0) = 0. Fixing Γ = 140, k 1 = 12.96, k 2 = 2.84 and k 3 = 3.96 selected as [START_REF] Levant | High-order sliding modes: differentiation and outputfeedback control[END_REF] with w = 3.6. Also, the transformation matrix T and the vector a are given as follows

T =   -304.4 6.3 0 1 0 0 7.1 -15.9 35.1   , a =   -15 -71 -105   .
The Assumption 2 is satisfied with min = 0.024 and max = 0.045. Fixing κ 1 = 0.99, κ 2 = 1.99, ψ = 0.99 and using the method described by [START_REF] Zavala | Levant's arbitrary-order exact differentiator: A Lyapunov approach[END_REF], it can be shown that τ 1 = 0.52, τ 2 = 0.12 and τ 3 = 0.18. Then, it is easy to verify that all the conditions of Theorem 2 are satisfied for all α ∈ [0, 1). Fig. 1 illustrates the convergence of the observer to the states of the system in the presence of measurement noise. Fig. 2 shows the estimation of the parameter for different values of α. It has to be remarked that the nonlinear and the discontinuous adaptive observers, i.e., α ∈ [0, 1), converge to a neighborhood of the origin even in the presence of disturbances, and the convergence rate is faster than the linear algorithm, i.e., α = 1. It is clear that the algorithm with α = 0 provides the best performance.

VI. CONCLUSIONS

This paper contributes with an HOSM adaptive observer for a class of uncertain nonlinear systems with external disturbances. The disturbances may have a relative degree higher than one in contrast with previous works (see, for instance, [START_REF] Ríos | An adaptive sliding-mode observer for a class of uncertain nonlinear systems[END_REF]- [START_REF] Franco | Adaptive estimation for uncertain nonlinear systems with measurement noise: A sliding-mode observer approach[END_REF]). The proposed approach combines an HOSM observer with a nonlinear parameter identification algorithm. The HOSM observer guarantees FT-ISS of the state estimation error with respect to the parameter identification error. The nonlinear parameter identification algorithm provides FT-ISS of the parameter identification error with respect to the state estimation error. Then, the practical, global and uniform asymptotic stability of the adaptive estimation error, despite the external disturbances, is guaranteed through the small-gain theorem. Simulation results illustrate the performance of the proposed HOSM adaptive observer.

  the rotor angular position, and x 3 ∈ R is the rotor angular velocity. The parameters are: L a = 0.21 [H] the armature inductance, R a = 10 [Ω] the armature resistance, K m = 3 the torque constant, B s = 0.18 [Nm/(rad/s)] the inner damping coefficient of the shaft, J m = 19 [Kgm 2 ] the motor inertia moment, J l = 12.5 [Kgm 2 ] the load inertia moment, and B l = 3.4 [Nms/rad] the load viscous friction. The nonlinear function T f represents a mathematical model of the friction-velocity curve representing the transition from static to kinetic friction, including the Coulomb friction, and it is given as

  The state x 1 and the estimated signal x1 . The state x 3 and the estimated signal x3 .

Figure 1 :Figure 2 : 2 ,Figure 3 :

 1223 Figure 1: States behavior and estimated states

This work was supported in part by the SEP-CONACYTECOS-ANUIES Project 315597 and ECOS NORD Project M20M04. The work of Roberto Franco and Héctor Ríos was supported in part by CONACYT CVU 772057, in part by Catedras CONACYT CVU 270504 Project 922, and in part by TecNM Projects. The work of Alejandra Ferreira de Loza was supported by Catedras CONACYT CVU 166403 Project 1537. qq † Tecnológico Nacional de México/I.T. La Laguna,


Christelle Couchoux

François-Xavier Dechaume-Moncharmont

Thierry Rigaud

Loïc Bollache

Couchoux Christelle

Dechaume-Moncharmont François-Xavier

Rigaud Thierry

Bollache Loic
email: loic.bollache@u-bourgogne.fr

Loic Bollache

Male Gammarus roeseli provide smaller ejaculates to females infected with vertically transmitted microsporidian parasites

Keywords: mate choice, mate guarding, sperm investment, Nosema granulosis, Dictyocoela

HAL is

The effects of parasites on the reproduction of their hosts are widespread and diverse [START_REF] Poulin | Evolutionary ecology of parasites (2nd edn)[END_REF]. In particular, the seminal paper by [START_REF] Hamilton | Heritable true fitness and bright birds -a role for parasites[END_REF] drew attention to the fact that parasite infections could be a major evolutionary force driving host's sexual selection and sexual behaviour. Among this broad area of research, studies on effects of parasites on male sperm production and investment are scarce. Most of them have investigated the direct effect of the parasitic status of the male on its sperm production, with evidence of either negative effects (e.g. [START_REF] Yan | Selection by parasites on components of fitness in Tribolium beetles -the effect of intraspecific competition[END_REF]Galipaud, Dechaume-Moncharmont, Oughadou & Bollache, 2011) or positive effects (reflecting an increased investment to compensate negative effects of parasitism on other life-history traits) [START_REF] Figenschou | The relative effect of parasites and social status on sperm traits in Arctic Charr[END_REF][START_REF] Haeussler | Parasitic mites influence intra-and interpopulational variation in sperm length in a simultaneous hermaphrodite land snail (Gastropoda: Helicidae)[END_REF]. Only a few studies have investigated the effect of female parasitic status on sperm allocation in the male [START_REF] Edward | The evolution and significance of male mate choice[END_REF]. Due to intense malemale competition and because males are sperm-limited, strategic sperm investment is predicted to occur at each mating event to maximise the overall fertilisation success [START_REF] Wedell | Sperm competition, male prudence and sperm limited females[END_REF][START_REF] Jarrige | Strategic sperm allocation in response to perceived sperm competition risk in a lekking insect[END_REF]; but see [START_REF] Arundell | Perceived risk of sperm competition affects sperm investment in a mate-guarding amphipod[END_REF]. In particular, males may provide females of low reproductive value with fewer sperm [START_REF] Reinhold | Cryptic male choice: sperm allocation strategies when female quality varies[END_REF]. Parasitic infections decrease host fitness, and are therefore a source of variation in the quality of potential mates. Female parasitic status could therefore promote strategic sperm allocation by males, with greater allocation of sperm to high quality (uninfected) females, and prudent sperm allocation to females of low quality. This is especially true in the eventuality of numerous potential future mates, because these future mates could be of higher quality [START_REF] Reinhold | Cryptic male choice: sperm allocation strategies when female quality varies[END_REF][START_REF] Edward | The evolution and significance of male mate choice[END_REF].

When parasites are vertically transmitted (from mother to offspring via the eggs), they are under strong evolutionary pressure to distort the primary sex ratio of their hosts, through male killing or feminisation [START_REF] Bandi | Inherited microorganisms, sex specific virulence and reproductive parasitism[END_REF].

By reversing genetic males into phenotypic females, feminising microbes increase their transmission efficiency by increasing the frequency of the transmitting sex (females). For instance, feminisation is induced by the bacterium Wolbachia [START_REF] Bouchon | Evidence for widespread Wolbachia infection in isopod crustaceans: molecular identification and host feminization[END_REF][START_REF] Kageyama | Feminizing Wolbachia in an insect, Ostrinia furnacalis (Lepidoptera: Crambidae)[END_REF] and by parasites from the eukaryotic phylum Microspora [START_REF] Terry | Widespread vertical transmission and associated host sex-ratio distortion within the eukaryotic phylum Microspora[END_REF]).

Because they often result in female-biased populations, parasitic sex ratio distorters are a selective force in the evolution of mating behaviour [START_REF] Charlat | Evolutionary consequences of Wolbachia infections[END_REF] and mate choice [START_REF] Jiggins | Sex-ratio-distorting Wolbachia causes sex-role reversal in its butterfly host[END_REF][START_REF] Moreau | Sexual selection in an isopod with Wolbachia-induced sex reversal: males prefer real females[END_REF]). In the crustacean isopod Armadillidium vulgare, not only males prefer to mate with uninfected (real) females vs. infected ones (genetic males reversed by the feminising bacterium Wolbachia) [START_REF] Moreau | Sexual selection in an isopod with Wolbachia-induced sex reversal: males prefer real females[END_REF]), but they also allocate smaller size ejaculates to infected females, resulting in a decrease in fertility [START_REF] Rigaud | A cost of Wolbachia-induced sex reversal and female-biased sex ratios: decrease in female fertility after sperm depletion in a terrestrial isopod[END_REF]). Because on average infected and uninfected females had the same fecundity, this lesser allocation to infected females has been attributed to the abnormal (incomplete) behaviour expressed by the reversed-females during courtship and copulation [START_REF] Moreau | Sexual selection in an isopod with Wolbachia-induced sex reversal: males prefer real females[END_REF][START_REF] Rigaud | A cost of Wolbachia-induced sex reversal and female-biased sex ratios: decrease in female fertility after sperm depletion in a terrestrial isopod[END_REF]).

The crustacean amphipod Gammarus duebeni has been found to be infected with sex ratio distorting parasites: the microsporidian parasites Nosema granulosis and Dictyocoela duebenum [START_REF] Terry | Impact of a novel feminising microsporidian parasite on its crustacean host[END_REF][START_REF] Ironside | Microsporidian parasites feminise hosts without paramyxean co-infection: support for convergent evolution of parasitic feminization[END_REF]. [START_REF] Dunn | Strategic sperm allocation under parasitic sex-ratio distortion[END_REF] showed that males provide smaller size ejaculates to females infected with N. granulosis, relative to uninfected females. As infected females produce fewer eggs than uninfected ones and males are sperm-limited, the smaller ejaculate size has been interpreted as a strategic sperm allocation; the males saving most of their sperm reserve for females of high quality. Assessment of female quality could be favoured in amphipods because of a long-lasting phase of precopulatory mate guarding involving close proximity with the female for up to three weeks [START_REF] Galipaud | Males do not always switch females when presented with a better reproductive option[END_REF].

In most freshwater crustaceans, including Gammarus roeseli, reproduction is characterized by a precopulatory mate guarding phase during which males guard a potential mate by carrying a female beneath their ventral surface for several days before copulation. This behaviour is tightly linked to females' moulting cycle. While males are considered available for mating during most of their moult cycle [START_REF] Sutcliffe | Reproduction in Gammarus (Crustacea: Amphipoda): basic reproductive processes[END_REF], females are only receptive to copulation shortly after moulting and just for a few hours. After copulation, they begin a new moulting cycle, which can last from several days up to several weeks depending on the species [START_REF] Jormalainen | Precopulatory mate guarding in crustaceans: male competitive strategy and intersexual conflict[END_REF].

Gammarus roeseli belong to the group of multivoltine iteroparous annuals (lifetime 12-24 months) and females moult six to eight times, thus potentially producing six to eight broods [START_REF] Pöckl | Reproductive potential and lifetime potential fecundity of the freshwater amphipods Gammarus fossarun and G. roeseli in Austrian streams and rivers[END_REF].

Because females' moulting cycles, and hence receptivity to copulation, are asynchronous, the operational sex ratio is predicted to be strongly biased towards males. Precopulatory mate guarding behaviour is thought to have evolved as a male competitive strategy in response to the brief period of female receptivity [START_REF] Parker | Courtship persistence and female-guarding as male time investment strategies[END_REF][START_REF] Grafen | A model of mate guarding[END_REF]. Males are in competition for access to females, in particular to 'high-quality' females. From a male perspective, the female's quality as a mate strongly depends on fecundity, and the male pairing decision to engage mate guarding is known to be negatively influenced by female parasitism status [START_REF] Bollache | Effects of two acanthocephalan parasites on the fecundity and pairing status of female Gammarus pulex (Crustacea: Amphipoda)[END_REF].

Most populations of the amphipod G. roeseli are infected with microsporidian parasites in the rivers of Western Europe [START_REF] Haine | Coexistence of three microsporidia parasites in populations of the freshwater amphipod Gammarus roeseli, evidence for vertical transmission and positive effect on reproduction[END_REF][START_REF] Gismondi | Microsporidia parasites disrupt the responses to cadmium exposure in a gammarid[END_REF][START_REF] Grabner | Invaders, natives and their enemies: distribution patterns of amphipods and their microsporidian parasites in the Ruhr Metropolis, Germany[END_REF]. Three of these parasite species (Nosema granulosis, Dictyocoela muelleri and Dictyocoela sp. (roeselum)) have been shown to be vertically-transmitted, and associated with female-biased sex ratios [START_REF] Haine | Coexistence of three microsporidia parasites in populations of the freshwater amphipod Gammarus roeseli, evidence for vertical transmission and positive effect on reproduction[END_REF][START_REF] Haine | Infection by a vertically-transmitted microsporidian parasite is associated with a female-biased sex ratio and survival advantage in the amphipod Gammarus roeseli[END_REF]. In contrast with the parasites infecting G. duebeni, the microsporidia in G. roeseli are not associated with a decrease in fecundity [START_REF] Haine | Coexistence of three microsporidia parasites in populations of the freshwater amphipod Gammarus roeseli, evidence for vertical transmission and positive effect on reproduction[END_REF]) and the microsporidia of the genus Dictyocoela had only a slight impact on the host's physiology in absence of other stresses [START_REF] Gismondi | Microsporidia parasites disrupt the responses to cadmium exposure in a gammarid[END_REF]. Nosema granulosis even provides its hosts an advantage in survival relative to uninfected females [START_REF] Haine | Infection by a vertically-transmitted microsporidian parasite is associated with a female-biased sex ratio and survival advantage in the amphipod Gammarus roeseli[END_REF]. Here, we tested the hypothesis of an effect of females' parasitic status on males' reproductive strategy. Male gammarids are sperm-limited, and long lasting precopulatory mate guarding leaves opportunities for an accurate assessment of female quality. We thus predicted that G. roeseli males allocate more sperm to females infected with N. granulosis, but show no difference in sperm allocation between uninfected females and females infected with parasites of the genus Dictyocoela.

Methods

Ethical note

This work followed the ABS/ASAB guidelines for the treatment of animals in behavioural research. Information about individuals' origin, and housing conditions are described below. Transport between sampling site and laboratory, housing conditions, handling, as well as experimental monitoring, have been conducted in order to reduce stress and maximise animals' welfare. We complied with the French regulations for experiments on invertebrates.

Animal collection, maintenance and general procedures

Animals were collected in the river Ouches in Dijon (47°17'51.6"N; 5°02'33.3"E), using kick sampling method with a hand net. Gammarids exhibit a precopulatory mate guarding behaviour (they grasp females several days before egg laying) in order to ensure the reproduction of the future batch of eggs [START_REF] Sutcliffe | Reproduction in Gammarus (Crustacea: Amphipoda): basic reproductive processes[END_REF].

Immediately upon sampling, couples were isolated in individual plastic tubes, and were brought back to the laboratory. Each pair was then housed individually under a 12:12 light:dark cycle regime, at 15°C (± 1), in boxes (h=7cm; Ø=9cm) filled with water from the river mixed with tap water previously dechlorinated, UV-treated and oxygenated.

Couples were then randomly assigned to one of the two following experiments. The first experiment aimed at understanding the effect of female infection status on male ejaculate size. Fertilisation is semi-external in gammarids: after the female has laid eggs, the male deposits sperm in her ventral incubating 'pouch'. As water can flow in the pouch, it is impossible to collect male ejaculates. Sperm investment is therefore estimated by comparing sperm reserves in testes between two treatment groups; males before copulation (during mate guarding), and males after a copulation event (not holding the female anymore, the female having released eggs) [START_REF] Dunn | Strategic sperm allocation under parasitic sex-ratio distortion[END_REF][START_REF] Lemaître | Sperm depletion, male mating behaviour and reproductive 'time-out' in Gammarus pulex (Crustacea, Amphipoda)[END_REF]. Males were anaesthetised using carbonised water and dissected to estimate their sperm reserves. In the first group (n = 64), males were dissected before insemination (within two days after their arrival in the lab). In the second group (n = 61), sperm remaining in testes were counted after copulation, on the same day. Females were also anaesthetised and dissected. The eggs were flushed out from the incubating pouch and counted. To assess the female's infection status by microsporidia, the gonads were dissected and stored in pure ethanol until molecular analysis. Prior to dissection, the size of each animal was estimated by measuring the height of the fourth coxal plate, using a Nikon SMZ 1500 stereoscopic microscope and the Lucia G 4.81 software.

The second experiment aimed at assessing sperm replenishment kinetics after copulation. Sperm were therefore counted in males 2, 4, 8 and 12 days after the insemination (n = 21, 22, 20, 20, respectively). The second group of the first experiment served as reference for sperm reserves just after insemination (0 days after insemination, n = 64). At the end of the experiments, all non-dissected individuals were released back into the river they came from.

Sperm counting

Sperm was counted as described in [START_REF] Lemaître | Sperm depletion, male mating behaviour and reproductive 'time-out' in Gammarus pulex (Crustacea, Amphipoda)[END_REF].

One testis per individual was isolated in a watch glass, in 1000 µL of demineralised water. After isolation, the gonad was dissected under a binocular microscope. The fragments of gonad were submitted to 10 s of ultra-wave treatment to separate the membranes from the sperm (Ultra-waves tank, Branson 2200 Branson Cleaning Equipment Company, Shelton, Co, U.S.A.) and homogenised. For each male, four 10 µL samples were placed on a slide and dried for 10 min at 37°C. All sperm in each drop were counted under an optic microscope Nikon Eclipse E600 (magnification x100). Statistical analyses were carried out using the sum of the number sperms in the four drops, since the counting appeared repeatable between drops (R = 0.964, 95% CI = [0.953; 0.975]).

Infection status

Microsporidia detection and identification were performed using a PCR-RFLP method, following [START_REF] Haine | Coexistence of three microsporidia parasites in populations of the freshwater amphipod Gammarus roeseli, evidence for vertical transmission and positive effect on reproduction[END_REF]. After DNA extraction from the female's gonads, a PCR test was conducted with the primers V1f and 530r, amplifying a fragment of the microsporidian 16S ribosomal gene only in case of parasite infection [START_REF] Haine | Coexistence of three microsporidia parasites in populations of the freshwater amphipod Gammarus roeseli, evidence for vertical transmission and positive effect on reproduction[END_REF]). The size of the amplification product allowed to discriminate between Nosema granulosis and Dictyocoela parasites, and the use of restriction enzymes VspI and Bst1107I (MBI fermentas) allowed to discriminate between the D. muelleri and the D. sp. (roeselum) sequences, respectively [START_REF] Haine | Coexistence of three microsporidia parasites in populations of the freshwater amphipod Gammarus roeseli, evidence for vertical transmission and positive effect on reproduction[END_REF]). VspI enzyme revealed only females infected with Dictyocoela sp. (roeselum) in this study, hereafter referred as "Dictyocoela". Males were not tested for the presence of parasites since prevalence in males in this population is close to zero [START_REF] Haine | Coexistence of three microsporidia parasites in populations of the freshwater amphipod Gammarus roeseli, evidence for vertical transmission and positive effect on reproduction[END_REF]).

Statistical analyses

Female fecundity was analysed with a general linear model including the following factors: female size, insemination status of males (before or after insemination), and infection status of females (uninfected, infected with N. granulosis, infected with Dictyocoela) and their two-order interactions. The repeatability of sperm count was estimated using 'rptR' packages [START_REF] Stoffel | rptR: Repeatability estimation and variance decomposition by generalized linear mixed-effects models[END_REF]. Sperm reserves in males were analysed using a general linear model including the following factors: male size, insemination status (before or after insemination), and infection status of females (uninfected, infected with N. granulosis, infected with Dictyocoela)

and their two-order interactions. Sperm replenishment data were analysed using a general linear model including the following factors: male size, day after insemination (treated as categories, 0, 2, 4, 8 and 12 days after insemination) and their interaction.

We reported Cohen's d with their bootstrapped 95% confidence intervals [START_REF] Nakagawa | Effect size, confidence interval and statistical significance: a practical guide for biologists[END_REF] as measures of effect size for the comparison between mean sperm reserves before and after copulation [START_REF] Garamszegi | Changing philosophies and tools for statistical inferences in behavioral ecology[END_REF]. We reported standardised slopes as a measure of effect size for the relationship between female body length and fecundity [START_REF] Schielzeth | Simple means to improve the interpretability of regression coefficients[END_REF]. Non-parametric post-hoc comparisons after Kruskal-Wallis' tests were performed using Conover's-test implemented in PMCMR packages [START_REF] Pohlert | The pairwise multiple comparison of mean ranks package (PMCMR). R package <URL[END_REF]. The tests were performed using R 3.2.5 (R core team 2016) or JMP 10.0 (SAS institute).

Results

Animal size and female fecundity

A total of 108 pairs (n = 53 before ejaculation, and n = 55 after ejaculation) were measured. Thirty-seven females were infected with N. granulosis, 23 were infected with Dictyocoela, and 48 were uninfected (control). Body length did not differ as a function of the female's parasitic status in either males (F2,105 = 0.43, p = 0.65, supplementary Fig. S1a) or females (F1,106 = 0.03, p = 0.97, supplementary Fig. S1b).

There was no difference in body length between the two groups (before and after ejaculation) in males (F1,106 = 0.42, p = 0.52, Cohen's d = -0.13, 95% CI = [-0.53; 0.22], supplementary Fig. S1c) or in females (F1,106 = 0.18, p = 0.68, Cohen's d = -0.08, 95% CI = [-0.48; 0.30], supplementary Fig. S1d). The mean body size was 3.00 mm (95%IC = [2.94; 3.06]) for males, and 2.70 mm (95% CI = [2.64;2.75]) for females. There was a significant positive relationship between female size and fecundity (F1,106 = 22.3, p < 10 -5 , standardised slope = 0.42, 95%CI = [0.24; 0.59], Fig. 1), but there was no effect of the parasitic status of the female (F2,104 = 0.18, p = 0.82) or the insemination status of the male (F2,104 = 0.0002, p = 0.99) on female size.

Sperm reserves in males before and after insemination

There was no effect of male body size on sperm reserves before (F1,51= 1.89, p = 0.18) or after ejaculation (F1,53 = 0.68, p = 0.41). Sperm reserves were significantly affected by the interaction between the insemination status and the parasitic status (F2,104 = 3.32, p = 0.040, Fig. 2A). The sperm count was significantly larger for males in amplexus (i.e. before copulation) than for males after copulation in each treatment groups: N. granulosis infected females (Cohen'

Sperm replenishment

Sperm reserves increased significantly in males with time spent since copulation (F4,138 = 47.19, P < 10 -5 ) but was not influenced by male size (F1,137 = 2.36, P = 0.140) nor by the interaction between male size and time since copulation (F4,133 = 0.767, P = 0.548). Post hoc comparisons showed no difference in sperm reserves between freshly-mated males and males two days after a copulation event (Fig. 3).

Eight and 12 days after a copulation event, males had more sperm than other males, without any significant difference between them (Fig. 3).

Discussion

We observed a negative effect of females' infection status on males paired with them, in terms of both sperm reserves before ejaculation and ejaculate size. Indeed, males paired with infected females had significantly lower initial sperm reserves than males paired with uninfected females, whereas their final sperm reserves after insemination did not differ, reflecting a lower number of sperm provided during insemination. This could be explained by three non-exclusive hypotheses. First, based on our screening of sperm replenishment dynamics, males paired with infected females could be those that had less time to replenish their reserves between two copulations. Such a pattern could occur if infected females moulted more rapidly than uninfected ones, which would reduce amplexus duration before copulation.

Alternatively, males paired with uninfected females could be in pair for longer prior collection. However, since the dynamics of sperm replenishment is quite rapid (around 8 days), this possible difference was not huge. If such a pairing pattern was true, this would mean that males engaging pairing with uninfected females would be those able to monopolize females earlier.These hypotheses remain to be tested.

Second, when paired with infected females, males could strategically allocate less energy in the production of gametes. As first argued by [START_REF] Dewsbury | Ejaculate cost and male choice[END_REF], a single male gamete may be cheap, but as males transfer large numbers of sperm, gamete production could be energetically expensive. Thus, males could modulate their allocation in sperm production according to female quality by allocating less energy in gamete production when paired with infected females [START_REF] Reinhold | Cryptic male choice: sperm allocation strategies when female quality varies[END_REF]. Third, males paired with infected females could differ in competiveness or body condition from males paired with uninfected females, as suggested by the fact that unpaired males have generally lower sperm reserves than paired males [START_REF] Lemaître | Sperm depletion, male mating behaviour and reproductive 'time-out' in Gammarus pulex (Crustacea, Amphipoda)[END_REF]. Consistent with the major role of male-male competition for access to receptive females in gammarids [START_REF] Ward | The relative roles of habitat structure and male-male competition in the mating system of Gammarus pulex (Crustacea: Amphipoda): a simulation study[END_REF][START_REF] Dick | Effects of natural variation in sex ratio and habitat structure on mate-guarding decisions in amphipods (Crustacea)[END_REF][START_REF] Bollache | Sexual selection on male body size and assortative pairing in Gammarus pulex (Crustacea: Amphipoda): field surveys and laboratory experiments[END_REF], it is possible that those in the best condition have a higher access to uninfected females. If males in poorer condition were less competitive and less efficient in finding or monopolising uninfected females, they might have to accept the second-choice females remaining in the population. Furthermore, the decision to enter into precopula might not solely be under male control, and females could resist male attempts to grasp them [START_REF] Jormalainen | Female resistance and duration of mateguarding in three aquatic peracarids (Crustacea)[END_REF][START_REF] Jormalainen | Precopulatory mate guarding in crustaceans: male competitive strategy and intersexual conflict[END_REF]. If uninfected females were more able to resist attempts from males than infected ones, only the most competitive and in best condition males would be able to pair with them.

Our results are consistent with previous observations in G. roeseli reporting that the parasitic status of females does not appear to affect their fitness, neither in terms of fecundity nor body length [START_REF] Haine | Coexistence of three microsporidia parasites in populations of the freshwater amphipod Gammarus roeseli, evidence for vertical transmission and positive effect on reproduction[END_REF]). This pattern is in contrast with the situation in G. duebeni, where infection confers a fitness cost in terms of fecundity (e.g. [START_REF] Dunn | Strategic sperm allocation under parasitic sex-ratio distortion[END_REF], and therefore, makes it difficult to invoke for G. roeseli the adaptive strategic male sperm allocation based on differences in female fecundity proposed in G. duebeni [START_REF] Dunn | Strategic sperm allocation under parasitic sex-ratio distortion[END_REF]. Differences in sperm reserves and ejaculate size may therefore result from differences in cues used by males to identify infected females. As precopulatory mate guarding is a long-lasting and intimate interaction between the partners, we may propose that the male relies on behavioural cues to assess whether its partner is a genuine female or a feminised male. Nosema granulosis and Dictyocoela sp. have been shown to be sex ratio distorting parasites in gammarids, reversing male hosts into phenotypic females [START_REF] Terry | Widespread vertical transmission and associated host sex-ratio distortion within the eukaryotic phylum Microspora[END_REF][START_REF] Rodgers-Gray | Mechanisms of parasite-induced sex reversal in Gammarus duebeni[END_REF][START_REF] Haine | Infection by a vertically-transmitted microsporidian parasite is associated with a female-biased sex ratio and survival advantage in the amphipod Gammarus roeseli[END_REF][START_REF] Ironside | Microsporidian parasites feminise hosts without paramyxean co-infection: support for convergent evolution of parasitic feminization[END_REF]. A number of infected females are therefore genetic males reversed by parasites. In terrestrial isopod crustaceans infected with the feminising Wolbachia bacteria, such 'false females' do not behave entirely as normal females during the copulation sequence, resulting in higher rates of copulation failure and, in case of successful copulations, lower sperm investment in infected females compared to uninfected ones [START_REF] Moreau | Sexual selection in an isopod with Wolbachia-induced sex reversal: males prefer real females[END_REF][START_REF] Rigaud | A cost of Wolbachia-induced sex reversal and female-biased sex ratios: decrease in female fertility after sperm depletion in a terrestrial isopod[END_REF]).

Since mate recognition in crustaceans is based on contact pheromones present on the cuticle [START_REF] Caskey | Studies on contact sex pheromones of the caridean shrimp Paldemonetes pugio: I. Cuticular hydrocarbons associated with mate recognition[END_REF][START_REF] Zhang | Simultaneously Hermaphroditic Shrimp Use Lipophilic Cuticular Hydrocarbons as Contact Sex Pheromones[END_REF], differences in cuticular compounds between infected and uninfected females may explain the observed pattern. Investigating the behaviour and cuticle compounds of infected vs. uninfected G. roeseli females during mate guarding would therefore be helpful to further understand the sperm investment pattern observed in this study. Another explanation, which does not involve a directional preference for uninfected female, could be that infected females show less resistance to pairing attempts [START_REF] Jormalainen | Female resistance and duration of mateguarding in three aquatic peracarids (Crustacea)[END_REF]. Thus, only the most competitive males (with the largest reserves of sperm) could pair with more resistant uninfected females, the less competitive males only being able to pair with the less resistant infected females.

Gammarus roeseli males showed large sperm investment at each insemination event (between c.a. 50% and 75% of their sperm reserves constitute the ejaculate).

Such a sperm depletion is consistent with the values reported in Gammarus pulex [START_REF] Lemaître | Sperm depletion, male mating behaviour and reproductive 'time-out' in Gammarus pulex (Crustacea, Amphipoda)[END_REF], and confirms that the sperm investment is substantial in Gammarus. However, as also noted by [START_REF] Lemaître | Sperm depletion, male mating behaviour and reproductive 'time-out' in Gammarus pulex (Crustacea, Amphipoda)[END_REF], sperm replenishment was achieved within less than a dozen days. Initial sperm reserves in G. roeseli were double those in G. pulex: around 10 sperm/µL in G. pulex before copulation, and less than 2.5 sperm/µL after copulation, while in G. roeseli they were around 22 sperm/µL before copulation and more than 5 sperm/µL after copulation. As observed in terrestrial isopods [START_REF] Moreau | Variable male potential rate of reproduction: high male mating capacity as an adaptation to parasite-induced excess of females[END_REF], we may propose that these higher sperm reserves could be due to a selection pressure induced by the excess of females in populations of G. roeseli [START_REF] Haine | Coexistence of three microsporidia parasites in populations of the freshwater amphipod Gammarus roeseli, evidence for vertical transmission and positive effect on reproduction[END_REF].

Future studies should carefully assess the influence of parasites on male mate choice and female behaviour in order to understand pairing processes leading to these mating patterns, and the link between male sperm reserves and female infection status.

Caption for figure

  s d = 1.37 with 95% CI = [0.92;1.95], F1,35 = 17.14, p = 0.00021), Dictyocoela infected females (Cohen's d = 1.04 with 95% CI = [0.39;1.84], F1,21 = 6.07, p = 0.022) and uninfected females (control) (Cohen's d = 2.78 with 95% CI = [2.029;4.13], F1,46 = 92.078 p < 10 -5 ). Sperm reserves before ejaculation were significantly different among males, depending on the parasitic status of the female (Kruskal-Wallis test χ2 = 6.7824, df = 2, p = 0.034). Males paired with a female infected with N. granulosis (Conover's post-hoc test, p = 0.048) or Dictyocoela (p = 0.016) had significantly lower sperm reserves than males from the control group. After ejaculation, sperm reserves were not significantly different among males, whether paired with infected females or uninfected females (Kruskal-Wallis test χ2 = 1.77, df = 2, p-value = 0.41). The comparison of sperm investment as a function of the parasitic status of females was based on the effect size (Cohen's d) of the average difference in sperm count before and after ejaculation. For each treatment, the value of the effect size was large (sensu Cohen 1988) but a direct comparison of the 95% CIs (Cumming & Finch 2005, Krzywinsk & Altman 2013) revealed a significantly larger sperm investment (twice larger) for the uninfected group than for the two infected groups (Fig. 2B). We estimated the average sperm number in the ejaculate as the difference between the mean sperm reserves before and after ejaculation (+/-95% confidence interval based on 10 000 bootstraps): n = 532 (95% CI = [415; 639]) for the uninfected females, n = 383 (95% CI = [212; 565]) for females infected with N. granulosis, and n = 234 (95% CI = [49; 427]) for females infected with Dictyocoela. Because we counted the sperm number in 40 µL among the 1000 µL in which one testis was dissected, we can estimate the total ejaculate size as ((532*1000)/40)*2 = 26600 sperm for uninfected females (CI = [20750; 31950]), 19150 for females infected with N. granulosis (CI = [10600; 28250]), and 11700 sperm for females infected with Dictyocoela (CI = [2450; 21350]).

Fig. 1 .

 1 Fig. 1. Female fecundity as a function of female body size (estimated from the height

Fig. 2 .

 2 Fig. 2. (A) Mean sperm number (+/-95% confidence interval) as a function of the

Figure 3 .

 3 Figure 3. Mean sperm number (+/-95% confidence interval) as a function of the time

  Fig. 1

Acknowledgments

This work was supported in part by the Agence Nationale de la Recherche (ANR), grant reference ANR-15-CE32-0006-01 (CytoSexDet).


domain: sde sdu.ocean sdv

text:
Chia-Ting Chen

Daniela Bănaru

François Carlotti

Mireille Harmelin-Vivien

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