D Efimov

S Aranovskiy

A A Bobtsov

T Raïssi

On fixed-time parameter estimation under interval excitation

The problem of estimation in the linear regression model is studied under the hypothesis that the regressor may be excited on a limited initial interval of time only. Then the estimation solution is based on the framework of finite-time or fixed-time converging dynamical systems. Two estimation algorithms are proposed. The robustness is analyzed using the notion of short-time input-to-state stability property with the use of a Lyapunov function. The performance of the estimators is demonstrated in numerical experiments.

I. INTRODUCTION

One of the basic and the most popular problems in the theory of identification and estimation is the parameter estimation in a linear regression model:

y(t) = ω (t)θ + w(t), t ∈ R,
where θ ∈ R n is the vector of unknown constant parameters that is necessary to find, ω : R → R n is the regressor function (usually assumed to be bounded and known), y(t) ∈ R is the signal available for measurements with a measurement noise w : R → R (here R denotes the set of real numbers). There are plenty of methods to solve this problem that need a complete statistics on the process (in other words these tools are mainly oriented on estimation offline): the linear least squares, the maximum-likelihood estimation, the Bayesian linear regression, the principal component regression [START_REF] Yan | Linear Regression Analysis: Theory and Computing[END_REF], [START_REF] Rencher | ser. Wiley Series in Probability and Statistics[END_REF], to mention a few. In the theory of adaptive control and identification there exist also many methods for adaptive and online estimation [START_REF] Sastry | Adaptive Control: Stability, Convergence and Robustness[END_REF], [START_REF] Khalil | Nonlinear Systems[END_REF], and applicability of many of them is based on the condition of persistence of excitation [START_REF] Narendra | Persistent excitation in adaptive systems[END_REF], [START_REF] Efimov | Design of impulsive adaptive observers for improvement of persistency of excitation[END_REF]. Consequently, these approaches are also implicitly based on the asymptotic statistics. Recently, several concepts have been proposed to relax the requirement on the excitation [START_REF] Praly | Convergence of the gradient algorithm for linear regression models in the continuous and discrete time cases[END_REF], [START_REF] Barabanov | On global asymptotic stability of ẋ = -φ(t)φ (t)x with φ(t) bounded and not persistently exciting[END_REF], [START_REF] Efimov | Robust stability under relaxed persistent excitation conditions[END_REF], with improved estimation algorithms [START_REF] Chowdhary | Concurrent learning adaptive control of linear systems with exponentially convergent bounds[END_REF], [START_REF] Kamalapurkar | Model-based reinforcement learning for approximate optimal regulation[END_REF], [START_REF] Wang | On robust parameter estimation in finite-time without persistence of excitation[END_REF], [START_REF] Belov | Enhanced parameter convergence for linear systems identification: The DREM approach[END_REF], [START_REF] Wang | Fixed-time estimation of parameters for non-persistent excitation[END_REF], which require only an interval estimation of the regressor ω(t).

Considering convergence on a finite interval, the amplitude of the initial error becomes of great importance, since if this deviation is not bounded, then it is complicated to ensure global convergence of the estimates to their ideal values in a limited time. A notion that overcomes this drawback has been proposed recently, and it is called fixed-time or predefined-time stability/convergence [START_REF] Andrieu | Homogeneous Approximation, Recursive Observer Design, and Output Feedback[END_REF], [START_REF] Cruz-Zavala | Uniform robust exact differentiator[END_REF], [START_REF] Polyakov | Nonlinear feedback design for fixed-time stabilization of linear control systems[END_REF], [START_REF] Song | Time-varying feedback for regulation of normal-form nonlinear systems in prescribed finite time[END_REF]. In the present work we discuss several fixed-time convergent algorithms solving the parameter estimation problem in a linear regression model, which are independent in the initial guesses for the values of parameters and regressor excitation 1 , and This work was supported by the Ministry of Science and Higher Education of Russian Federation, passport of goszadanie no. 2019-0898. 1 In one of our previous studies [START_REF] Wang | On robust parameter estimation in finite-time without persistence of excitation[END_REF] only a finite-time convergence has been guaranteed, therefore, the synthesized algorithms use an upper bound on the initial error.

whose robustness against the measurement noise is assessed using the input-to-state stability (ISS) theory. Following [START_REF] Wang | Fixed-time estimation of parameters for non-persistent excitation[END_REF], a notion of fixed-time input-to-state stability on a short interval of time is applied together with a Lyapunov function analysis. The considered approach is based on the dynamic regressor extension and mixing (DREM) method [START_REF] Aranovskiy | Performance enhancement of parameter estimators via dynamic regressor extension and mixing[END_REF], [START_REF] Belov | Enhanced parameter convergence for linear systems identification: The DREM approach[END_REF], which allow the vector estimation problem to be decoupled on a series of scalar ones. The obtained solutions are illustrated by computer simulations.

Notation

• R+ = {x ∈ R : x ≥ 0}, where R is the set of real numbers.

• |x| denotes the absolute value for x ∈ R or a vector norm for x ∈ R n , and the corresponding induced matrix norm for a matrix A ∈ R n×n is denoted by A . • Define e = exp [START_REF] Yan | Linear Regression Analysis: Theory and Computing[END_REF]. Define the Lambert function Lambert : R → R, also called the omega function or product logarithm, as the branches of the inverse relation of the function f (z) = ze z for z ∈ R (in general it is introduced for z ∈ C, but it has also two brances of solutions for z ∈ R). • Denote s α = |s| α sign(s) for any s ∈ R and α ∈ R+.

II. PRELIMINARIES

Consider a time-dependent differential equation:

dx(t)/dt = f (t, x(t), d(t)), t ≥ t0, t0 ∈ R, (1)
where x(t) ∈ R n is the state vector, d(t) ∈ R m is the vector of external inputs and d ∈ L∞(R, R m ); f : R n+m+1 → R n is a continuous function with respect to x, d and piece-wise continuous with respect to t, f (t, 0, 0) = 0 for all t ∈ R. A solution of the system (1) for an initial condition x0 ∈ R n at time instant t0 ∈ R and some d ∈ L∞(R, R m ) is denoted as X(t, t0, x0, d), and we assume that f ensures definiteness and uniqueness of solutions X(t, t0, x0, d) in forward time at least on some finite time interval [t0, t0 +T ), where T > 0 may be dependent on the initial condition x0, the input d and the initial time t0.

A. Stability definitions

Let Ω, Ξ be open neighborhoods of the origin in R n , 0 ∈ Ω ⊂ Ξ, then following [START_REF] Ríos | Homogeneous time-varying systems: Robustness analysis[END_REF], [START_REF] Wang | On robust parameter estimation in finite-time without persistence of excitation[END_REF] introduce several stability notions: Definition 1. At the steady state x = 0 the system (1) with d = 0 is said to be (a) short-time stable with respect to (Ω, Ξ, T 0 , T f ) if for any x0 ∈ Ω and t0 ∈ [-T 0 , T 0 ], X(t, t0, x0, 0) ∈ Ξ for all t ∈ [t0, t0 + T f ];

(b) short-finite-time stable with respect to (Ω, Ξ, T 0 , T f ) if it is short-time stable with respect to (Ω, Ξ, T 0 , T f ) and finite-time converging from Ω with the convergence time T t 0 ,x 0 ≤ t0 + T f :

|X(t, t0, x0, 0)| = 0 ∀t ≥ T t 0 ,x 0 , for all x0 ∈ Ω and t0 ∈ [-T 0 , T 0 ];
(c) globally short-finite-time stable for T 0 > 0 if for any bounded set Ω ⊂ R n containing the origin there exist a bounded set Ξ ⊂ R n , Ω ⊂ Ξ and T f > 0 such that the system is shortfinite-time stable with respect to (Ω, Ξ, T 0 , T f );

(d) short-fixed-time stable for T 0 > 0 and T f > 0, if for any bounded set Ω ⊂ R n containing the origin there exists a bounded set Ξ ⊂ R n , Ω ⊂ Ξ such that the system is short-finite-time stable with respect to (Ω, Ξ, T 0 , T f ).

B. Robust stability definitions

Consider the following definition of robust stability for (1) with d = 0 [START_REF] Wang | On robust parameter estimation in finite-time without persistence of excitation[END_REF], [START_REF] Wang | Fixed-time estimation of parameters for non-persistent excitation[END_REF]:

Definition 2. The system (1) is said to be (a) short-finite-time ISS with respect to (Ω, T 0 , T f , D) if there exist β ∈ GKL and γ ∈ K such that for all x0 ∈ Ω, all d ∈ L∞(R, R m ) with d ∞ < D and t0 ∈ [-T 0 , T 0 ]: |X(t, t0, x0, d)| ≤ β(|x0|, t -t0) + γ( d ∞) ∀t ∈ [t0, t0 + T f ] and β(|x0|, T f ) = 0;
(b) globally short-finite-time ISS for T 0 > 0 if there exist β ∈ GKL and γ ∈ K such that for any bounded set Ω ⊂ R n containing the origin there is

T f > 0 such that for all x0 ∈ Ω, all d ∈ L∞(R, R m ) and t0 ∈ [-T 0 , T 0 ]: |X(t, t0, x0, d)| ≤ β(|x0|, t -t0) + γ( d ∞) ∀t ∈ [t0, t0 + T f ] and β(|x0|, T f ) = 0 (the system is short-finite-time ISS with respect to (Ω, T 0 , T f , +∞)); (c) short-fixed-time ISS for T 0 > 0 and T f > 0, if there exist β ∈ GKL and γ ∈ K such that for all x0 ∈ R n , all d ∈ L∞(R, R m ) and t0 ∈ [-T 0 , T 0 ]: |X(t, t0, x0, d)| ≤ β(|x0|, t -t0) + γ( d ∞) ∀t ∈ [t0, t0 + T f ]
and β(|x0|, T f ) = 0 (for any bounded set Ω ⊂ R n containing the origin the system is short-finite-time ISS with respect to (Ω, T 0 , T f , +∞)).

The difference of global short-finite-time or short-fixed-time ISS and a conventional (finite-time or fixed-time) ISS [START_REF] Hong | H∞ control, stabilization, and input-output stability of nonlinear systems with homogeneous properties[END_REF], [START_REF] Bernuau | Verification of ISS, iISS and IOSS properties applying weighted homogeneity[END_REF] is that in the former case the stability property is considered on a finite interval of time [t0, t0 + T f ] only. Theorem 1. [START_REF] Wang | Fixed-time estimation of parameters for non-persistent excitation[END_REF] Let the constants T 0 ≥ 0 and T f > 0 be given. Let the system in (1) possess a Lyapunov function V :

R×R n → R+ such that for all x ∈ R n , d ∈ R m and t ∈ [-T 0 , T 0 + T f ]: α1(|x|) ≤ V (t, x) ≤ α2(|x|) α1, α2 ∈ K∞; (2) V (t, x) ≤ -u(t) V 1-η (t, x) + V 1+η (t, x) + κ(|d|)
for κ ∈ K, η ∈ (0, 1) and a function u : R → R+ satisfying t+ t u(s)ds ≥ υ > 0, > 0 for all t ∈ [-T 0 , T 0 + T f ] . Then the system (1) is short-fixed-time ISS for T 0 and T f with

β(s, t) = α -1 1 (2 η -1 min{max{0, η α η 2 (s) -η υ 4 (t -)}, 1 η α -η 2 (s) + η υ 4 (t -) }), γ(s) = α -1 1 [2 η -1 ( κ(s) + max{( 4 υ κ(s)) 1 1-η , ( 4 υ κ(s)) 1 1+η })]
provided that

T f ≥ 2(1 + 4 ηυ ) .
Finally, let us formulate a useful lemma:

Lemma 1.
[23] Let x, y ∈ R and p > 0, then for any κ1 ∈ (0, 1) there exists κ2 > 0 such that

x x + y p ≥ κ1|x| p+1 -κ2|y| p+1 .
In particular, κ2 = max{1 + κ1,

κ 1 (1-κ 1/p 1 ) p }.

C. Dynamic regressor extension and mixing method

Consider the estimation problem as above:

x(t) = ω (t)θ, t ∈ R, (3)
y(t) = x(t) + w(t).
Introduce the following hypothesis:

Assumption 1. Let ω ∈ L∞(R, R n ) and w ∈ L∞(R, R).
As it has been proposed in [START_REF] Aranovskiy | Performance enhancement of parameter estimators via dynamic regressor extension and mixing[END_REF], in order to overcome the limitations imposed by the condition that ω is PE and also to improve the transient performance, the DREM procedure transforms (3) to n new one-dimensional regression models, which allows the decoupled estimates of θi, i = 1, n to be computed under a condition on the regressor ω that differs from the persistent excitation.

For this purpose n -1 linear operators Hj : L∞(R, R) → L∞(R, R) are introduced for j = 1, n -1 (for instance an operator Hj can be chosen as a stable linear time-invariant filter with the transfer function Wj(s) = α j s+β j , where s ∈ C is a complex variable and αj = 0, βj > 0 are selected to filter the noise w in (3); or it can realize the delay operation with the transfer function Wj(s) = e -τ j s for τj > 0). Note that y ∈ L∞(R, R) under Assumption 1, then these operators are applied to the measured output y(t) of (3), and using the superposition principles (the operators Hj are linear) we obtain:

ỹj(t) = Hj(y(t)) = ω j (t)θ + wj(t), j = 1, n -1,
where ỹj(t) ∈ R is the j th operator output, ωj : R → R n is the j th filtered regressor function and wj(t) : R → R is the new j th noise signal, which is composed by the transformation of the noise w(t) by Hj and other exponentially converging components related to the initial conditions of the filters. By construction ωj ∈ L∞(R, R n ) and wj ∈ L∞(R, R) for all j = 1, n -1. Define new vector variables

Ỹ (t) = [y(t) ỹ1(t) . . . ỹn-1(t)] ∈ R n , W (t) = [w(t) w1(t) . . . wn-1(t)] ∈ R n
and a time-varying matrix

M (t) = [ω(t) ω1(t) . . . ωn-1(t)] ∈ R n×n ,
then stacking the original equation ( 3) with the n -1 filtered regressor models we construct an extended regressor system:

Ỹ (t) = M (t)θ + W (t).
For any matrix M (t) ∈ R n×n the following equality is true:

adj (M (t)) M (t) = det (M (t)) In, even if M (t)
is singular, where adj (M (t)) is the adjugate matrix of M (t) and det (M (t)) is its determinant. Recall that each element of the matrix adj (M (t)),

adj (M (t)) k,s = (-1) k+s M k,s (t)
for all k, s = 1, n, where M k,s (t) is the (k, s) minor of M (t), i.e., it is the determinant of the (n -1) × (n -1) matrix that results from deleting the k th row and the s th column of M (t). Define

Y (t) = adj (M (t)) Ỹ (t), W (t) = adj (M (t)) W (t), φ(t) = det (M (t)) ,
then multiplying from the left the extended regressor system by the adjugate matrix adj (M (t)) we get n scalar regressor models of the form:

Yi(t) = φ(t)θi + Wi(t) (4)
for i = 1, n. Again, by construction Y ∈ L∞(R, R n ), W ∈ L∞(R, R n ) and φ ∈ L∞(R, R).
For the scalar linear regression model (4) the conventional gradient estimation algorithm takes the form:

θi(t) = γiφ(t) Yi(t) -φ(t) θi(t) , γi > 0 (5)
for all i = 1, n, where now the estimation processes for all components of θ are decoupled, and the adaptation gain γi can be adjusted separately for each element of θ. However, all these estimation algorithms are dependent on the same regressor φ(t) (determinant of M (t)).

Remark 1. The choice of the operators Hj is a degree of freedom of the DREM procedure, where a poor choice may compromise the excitation level of the original regressor ω and can yield a singular matrix M . In [START_REF] Korotina | On parameter tuning and convergence properties of the DREM procedure[END_REF] it is proposed to apply the Kreisselmeier's regressor extension of the form

Ṁ (t) = -λM (t) + ω(t)ω (t)
and

Ẏ (t) = -λ Ỹ (t) + ω(t)y(t),
where λ > 0, and it is shown that under this choice the excitation of ω is always preserved.

III. PROBLEM STATEMENT

Consider the static linear regression model (3) under Assumption 1, and suppose that the DREM method has been applied in order reduce the initial problem of vector estimation to n scalar regressor models in the form [START_REF] Khalil | Nonlinear Systems[END_REF].

It is necessary to propose an algorithm generating an estimate θ(t) ∈ R n of the vector of unknown parameters θ ∈ R n , and for W ∞ = 0 providing the property of short-fixed-time stability (see Definition 1) of the estimation error e(t) = θ -θ(t) dynamics under assumptions 1 for some given T 0 and T f . If W ∞ = 0 then short-fixed-time ISS for T 0 and T f (see Definition 2) has to be guaranteed.

Since by applying DREM method the problem is decoupled on n independent ones, for brevity of notation, we will further omit the index i in (4) by assuming that n = 1:

Y (t) = φ(t)θ + W (t), (6)
then θ ∈ R, Y ∈ L∞(R, R), W ∈ L∞(R, R) and φ ∈ L∞(R, R).

IV. DESIGN OF ESTIMATION ALGORITHMS CONVERGING IN SHORT-FIXED-TIME

Two different solutions to the posed estimation problem are discussed in this section, whose difference consists in the requirements imposed on excitation of φ(t) and on the guaranteed robustness abilities with respect to W (t).

A. Algorithm 1

Consider an adaptive estimation algorithm proposed in [START_REF] Ríos | Time-varying parameter identification algorithms: Finite and fixed-time convergence[END_REF], [START_REF] Ríos | An adaptive sliding-mode observer for a class of uncertain nonlinear systems[END_REF]:

θ(t) = φ(t){γ1 Y (t) -φ(t) θ(t) 1-α (7) +γ2 Y (t) -φ(t) θ(t) 1+α }
for γ1 > 0, γ2 > 0 and α ∈ [0, 1), with θ(t0) ∈ R, which admits the following properties: Proposition 1. [START_REF] Wang | Fixed-time estimation of parameters for non-persistent excitation[END_REF] Let Assumption 1 be satisfied, and for given T 0 > 0 and T f > 0,

t+ t min{|φ(s)| 2-α , |φ(s)| 2+α }ds ≥ υ > 0 ( 8
)
for all t ∈ [-T 0 , T 0 + T f ] and some ∈ 0,

T f 2 . Take min{γ1, γ2} > 2 2+ α 2 αυ T f 2 -1 ,
then the estimation error e(t) = θ -θ(t) dynamics of ( 7) is shortfixed-time ISS for T 0 and T f .

Proof. The error dynamics for the estimation algorithm ( 7) can be written as follows:

ė(t) = -φ(t){γ1 φ(t)e(t) + W (t) 1-α +γ2 φ(t)e(t) + W (t) 1+α }.
Consider a Lyapunov function candidate V (e) = 0.5e 2 , whose derivative has an upper estimate for some κ11, κ12 ∈ (0, 1) and κ21, κ22 > 0 coming from Lemma 1:

V (t) = -γ1e(t)φ(t) φ(t)e(t) + W (t) 1-α -γ2e(t)φ(t) φ(t)e(t) + W (t) 1+α ≤ -γ1κ11|e(t)φ(t)| 2-α -γ2κ12|e(t)φ(t)| 2+α +γ1κ21|W (t)| 2-α + γ2κ22|W (t)| 2+α ≤ -u(t) V 1-α 2 (t) + V 1+ α 2 (t) + σ(|W (t)|)
for any e(t) ∈ R and W (t) ∈ R, where

u(t) = min{2 1-α 2 γ1κ11|φ(t)| 2-α , 2 1+ α 2 γ2κ12|φ(t)| 2+α }, σ(s) = γ1κ21s 2-α + γ2κ22s 2+α is a function from class K∞. Note that u(t) ≥ 2 1-α 2 min{γ1κ11, γ2κ12} min{|φ(t)| 2-α , |φ(t)| 2+α },
then under the imposed restrictions for φ, the system is shortfixed-time ISS for T 0 and T f due to Theorem 1 provided that the constraint

T f ≥ 2(1 + 2 2+ α 2 α min{γ1κ11, γ2κ12}υ )
is satisfied. The imposed restriction on min{γ1, γ2} guarantees that there exist κ11, κ12 ∈ (0, 1) such that all trajectories converge to the origin faster than T f if W ∞ = 0.

B. Algorithm 2

And, finally, let us introduce a version of the algorithm [START_REF] Praly | Convergence of the gradient algorithm for linear regression models in the continuous and discrete time cases[END_REF], which extends the nonlinear paradigm of the former by time-varying powers:

θ(t) = sign(φ(t)){γ1 Y (t) -φ(t) θ(t) α(t) +γ2 Y (t) -φ(t) θ(t) ς+α(t) } (9) γ1 > 0, γ2 > 0, ς > 1, α(t) = |φ(t)| 1 + |φ(t)| ,
where θ(t0) ∈ R. The idea of this design is that the power α(t) is approaching zero together with the regressor φ(t), then the contribution of the regressor in the adaptation rate is proportional to |φ(t)| α(t) , which is strictly separated with zero even for a convergent regressor, the detailed proof of this fact is given below.

Proposition 2. [14]

Let Assumption 1 be satisfied, and ϑ ∈ L∞(R, R+) where ϑ(t) = W (t) φ(t) . Then the estimation error e(t) = θ -θ(t) dynamics of (9) is short-finite-time ISS for any T 0 ≥ 0 with the input ϑ.

Let additionally for given T 0 > 0 and T f > 0,

t+ t |φ(s)| ς ds ≥ υ > 0 (10)
for all t ∈ [-T 0 , T 0 + T f ] and some ∈ (0, T f ), and

min{γ1, γ2} > √ 2 1 + φmax + 4 (ς-1)υ (T f -)g(xmin)
,

where φmax = max t∈[-T 0 ,T 0 +T f ] |φ(t)|, g(x) = x
x 1+x and xmin = Lambert(e -1 ), then the estimation error e(t) = θ -θ(t) dynamics of (9) is short-fixed-time ISS for T 0 and T f with the input ϑ.

Proof. The error dynamics for the estimation algorithm (9) can be written as follows:

ė(t) = -sign(φ(t)){γ1 φ(t)e(t) + W (t) α(t) +γ2 φ(t)e(t) + W (t) ς+α(t) }.
Consider a Lyapunov function V (e) = 0.5e 2 and observe that

r a(t) (t) ≥ r a min (t) r(t) ≥ 1 r amax (t) r(t) < 1 ≥ min{1, r amax min }, r a(t) (t) ≤ r amax (t) r(t) ≥ 1 r a min (t) r(t) < 1 ≤ max{1, r amax max }
for any r : R → R+ and a : R → R+ such that rmin = inf t∈R r(t), rmax = sup t∈R r(t), amin = inf t∈R a(t) and amax = sup t∈R a(t) for some rmin, rmax, amin, amax ∈ R+, then the time derivative of V for the estimation error dynamics admits an upper estimate:

V (t) = -sign(φ(t))e(t){γ1 φ(t)e(t) + W (t) α(t) +γ2 φ(t)e(t) + W (t) ς+α(t) } = -γ1|φ(t)| α(t) e(t) e(t) + ϑ(t) α(t) -γ2|φ(t)| ς+α(t) e(t) e(t) + ϑ(t) ς+α(t) ≤ -γ1κ11|φ(t)| α(t) |e(t)| 1+α(t) -γ2κ12|φ(t)| ς+α(t) |e(t)| 1+ς+α(t) +γ1κ21|φ(t)| α(t) |ϑ(t)| 1+α(t) + γ2κ22|φ(t)| ς+α(t) |ϑ(t)| 1+ς+α(t) ≤ - √ 2 min{γ1κ11, γ2κ12}g(|φ(t)|)[ξ1 (V (t)) +|φ(t)| ς ξ2 (V (t))] + σ( ϑ ∞), where
ξ1(s) = s 0.5 s ≥ 1 s 0.5+φmax 1+φmax s < 1 , ξ2(s) =    s 1+ς 2 s ≥ 1 s 1+ς+ φmax 1+φmax 2 s < 1 , σ(s) = [γ1κ21 max{1, φ φmax 1+φmax max } +γ2κ22 max{1, φ ς+ φmax 1+φmax max }s ς ] × s 1+2φmax 1+φmax s ≥ 1 s s < 1
and κ11, κ12 ∈ (0, 1), κ21, κ22 > 0 are from Lemma 1 (since φ(t) is upper bounded in amplitude due to Assumption 1 such constants exist). Note that g(0) = g(1) = 1 and

∂g(x) ∂x = 1 + ln(x) 1 1 + x g(x) 1 + x , ∂ 2 g(x) ∂x 2 = x 2 ln 2 (x) + x(x + 1) 2 xg(x)(x + 1) 4 , then ∂ 2 g(x)
∂x 2

≥ 0 for x ≥ 0, and the function g(x) has the only minimum xmin ∈ [0, 1] that satisfies the equality:

1 + ln(xmin) 1 1 + xmin = 0.
Solving this equation we obtain xmin = Lambert(e -1 ), then g(xmin) 0.757. Therefore, the estimate for the Lyapunov function can be represented as follows: By repeating the arguments of Theorem 1, the short-fixed-time and the short-finite-time ISS [START_REF] Wang | On robust parameter estimation in finite-time without persistence of excitation[END_REF] can be established (in the latter case there is no restriction on excitation of φ(t) and even the conventional results [START_REF] Sontag | On characterizations of the input-to-state stability property[END_REF] can be used).

V (t) ≤ -[ξ1 (V (t)) + |φ(t)| ς ξ2 (V (t))] + σ( ϑ ∞) ≤ -min{1, |φ(t)| ς }ξ (V (t)) + σ( ϑ ∞),
In order to check the restrictions on T f for the case W ∞ = 0, assume that V (t0) > 1, denote by T1, T2 > 0 the instants of time such that V (t0 + T1) = 1 and V (t0 + T1 + T2) = 0, and use the estimate

V (t) ≤ -max{V 0.5+φmax 1+φmax (t), |φ(t)| ς V 1+ς 2 (t)}, then since 0.5+φmax
1+φmax ∈ (0, 1) and 1+ς 2 > 1 we obtain

V ς-1 2 (t) ≤ 1 V -ς-1 2 (t0) + ς-1 2 υ 4 (t -t0 -)
for all t ∈ [t0, t0 + T1], and

V 0.5 1+φmax (t) ≤ 1 - 0.5 1 + φmax (t -t0 -T1)
for all t ∈ [t0 + T1, t0 + T1 + T2]. Hence, the upper bounds on T1 and T2 follow (due to the properties of α(t) the time T2 is independent in the excitation):

T1 ≤ [1 + 8 (ς -1) υ ], T2 ≤ 2 1 + φmax ,
which give by resolving the inequality T1 + T2 ≤ T f the required restriction on γ1 and γ2.

Thus, the idea of the algorithm (9) consists in the utilization of a nonlinearity such that the function g(|φ(t)|) becomes separated with zero overcoming the absence of excitation in the system. The price for that is the robustness with respect to a noise W with a well-defined ratio W (t) φ(t) . Remark 2. One of the most important features of estimation algorithms, after estimation error convergence in the ideal case, is their robustness with respect to measurement noises. In our case, since the regressor φ(t) may converge to zero, the appearance of W (t) = 0 additionally limit the time of convergence, since it is reasonable to use the output Y (t) for estimation with t ∈ [t0, t0+T ] only while

|Y (t)| > |W (t)| + ε for some ε > 0. If Y (t)| ≤ |W (t)| + ε (or |Y (t)| is almost equal to |W (t)
| for a sufficiently small ε), then the measured output mainly contains the measurement noise, and it is ambiguous to ask an algorithm to estimate θ due to a bad ratio between the signal and the noise. In this sense, the requirement that the signal W (t)

φ(t)
is well-defined is not much restrictive. Roughly speaking, it merely assumes that the ratio between the useful signal and the noise lies in some reasonable limits.

If the regressor φ(t) is just asymptotically converging without crossing zero, then the algorithms ( 7) and ( 9) can be applied for any finite T > 0 and t0 ∈ R. V. EXAMPLE Consider a measured scalar signal y(t) that consists of two terms. The first term is a harmonic oscillation, and the second term is a decaying process:

y(t) = θ1 sin(t) + θ2 1 t + 1 + w(t),
where w(t) is a measurement noise, and θ1, θ2 are unknown constant parameters. Since the second term decays, the parameter θ1 can be estimated asymptotically with a standard gradient estimator; however, estimation of θ2 is more challenging. The signal y(t) can be rewritten as [START_REF] Sastry | Adaptive Control: Stability, Convergence and Robustness[END_REF], where the regressor ω(t) = [sin(t) 1 t+1 ] , obviously, is not persistently exciting. Thus, standard gradient or least-square approaches cannot guarantee an asymptotic estimation. This section illustrates how θ2 can be estimated by the means of the DREM procedure and the proposed fixed-time parameter estimation algorithms.

First, the DREM procedure is applied to get a scalar linear regression equation for θ2. To this end, choose the delay operator H(u(t)) = u(t -τ d ), where τ d > 0 is the tuning parameter, and define ω1(t) = H(ω(t)) = ω(t -τ d ), ỹ1(t) = H(y(t)) = y(t -τ d ).

Then the procedure described in subsection II-C yields Y2(t) = φ(t)θ2 + W2, where Y2, W2, and φ are computed as it has been explained above, and

φ(t) = sin(t) 1 + t -τ d - sin(t -τ d ) 1 + t
for t ≥ τ d . Thus, the system is not persistently excited (the signal φ(t) converges to zero) and the algorithm (5) cannot be applied. However, since φ(t) decays only asymptotically, there exist and υ such that (8) and ( 10) are satisfied on any finite interval of time.

For simulations, the algorithms (7) and ( 9) are tuned as follows: γ1 = γ2 = 10, α = 0.25, and ς = 1.25. All algorithms are initialized with θ2(0) = 0, while the true value is θ2 = -2. For all algorithms no estimation is performed for t ∈ [0, τ d ], τ d = 1 since φ(t) is identically zero on this interval due to the DREM procedure.

Simulation results for the case when w(t) is a uniformly distributed noise, |w(t)| ≤ 0.2, are given in Fig. 1. As expected, all algorithms provide finite-time parameter estimation. The algorithm [START_REF] Praly | Convergence of the gradient algorithm for linear regression models in the continuous and discrete time cases[END_REF] has both small mean error and small error oscillations, where the trade-off is the largest transient time for similar tuning parameters.

VI. CONCLUSIONS

The problem of estimation in the linear regression model has been considered on a bounded interval of time. Two fixed-time converging estimators are presented. In order to analyze the robustness of these estimation algorithms, a short-time fixed-time input-to-state stability property has been used. The obtained estimation algorithms are compared in numerical experiments.

Figure 1 .

 1 Figure 1. Simulation results for the algorithms (7) and (9), estimation error θ 2 -θ2 (t) vs. time, seconds. The case of uniformly distributed measurement noise

identity matrix of dimension n × n is denoted as In.

	decreasing
	and limt→+∞ β(s, t) = 0 for each fixed s ∈ R+; a function
	β : R+ × R+ → R+ belongs to the class GKL if β(s, 0) ∈ K,
	β(s, •) is decreasing and for each s ∈ R+ there is Ts ∈ R+
	such that β(s, t) = 0 for all t ≥ Ts.

• For a Lebesgue measurable and essentially bounded function x : R → R n denote x ∞ = ess sup t∈R |x(t)|, and define by L∞(R, R n ) the set of all such functions with finite norms

• ∞; if +∞ -∞ |x(t)| 2 dt < +∞ then this class of functions is denoted by L2(R, R n ). • A

continuous function α : R+ → R+ belongs to the class K if α(0) = 0 and the function is strictly increasing, a function α ∈ K belongs to the class K∞ if it is increasing to infinity. A function β : R+ × R+ → R+ belongs to the class KL if β(•, t) ∈ K for each fixed t ∈ R+ and β(s, •) is • The


Keywords: Implicit Lyapunov-Krasovskii Functional (ILKF), nonlinear control, delay-dependent output feedback, Linear Matrix Inequalities (LMIs), practical stability

The problem of output accelerated stabilization of a chain of integrators is considered. Proposed control law nonlinearly depends on the output and its delayed values, and it does not use an observer to estimate the unmeasured components of the state. It is proven that such a nonlinear delayed control law ensures practical output stabilization with rates of convergence faster than exponential. The effective way of computation of feedback gains is given. It is shown that closed-loop system stability does not depend on the value of artificial delay, but the maximum value of delay determines the width of stability zone. The efficiency of the proposed control is demonstrated in simulations.

On output-based accelerated stabilization of a chain of integrators: Implicit Lyapunov-Krasovskii functional approach Artem N. Nekhoroshikh * , * * , * * * Denis Efimov * , * * , * * * Andrey Polyakov * , * * Wilfrid Perruquetti * * Igor B. Furtat * * * , * * * * Emilia Fridman †

INTRODUCTION

Output feedback stabilization of dynamical systems is one of the central problems in the control theory and application. For linear systems with relative degrees greater than one, output derivatives are necessary to construct a control law. Since sometimes they cannot be measured directly, their estimates should be obtained, and usually for this goal the controller dynamics has to be augmented by an observer [START_REF] Khalil | Nonlinear systems; 3rd ed[END_REF]. In some cases, if our control has to be embedded in an actuator with a low computational capacity, any decreasing the control low complexity and dimension is appreciated. To this end one can use finite time approximations of velocities [START_REF] Selivanov | An improved timedelay implementation of derivative-dependent feedback[END_REF]), which can be reconstructed based on the value of position in the current instant of time and a delayed one.

It has been shown that delayed-induced feedback preserves the stability for small enough values of artificial delay [START_REF] Borne | Stabilization of inverted pendulum by control with delay[END_REF]; [START_REF] Fridman | Delay-induced stability of vector second-order systems via simple lyapunov functionals[END_REF]).

This work was partially supported by the Ministry of Science and Higher Education of Russian Federation, passport of goszadanie no. 2019-0898, by Government of Russian Federation (Grant 08-08), by the grant of Russian Federation President (No. MD-1054.2020.8, agreement No. 075-15-2020-184).

It is worth to mention that inducing artificial delays has other benefits also. For instance, in [START_REF] Mazenc | New control design for bounded backstepping under input delays[END_REF]; [START_REF] Mazenc | Stabilization and robustness analysis for a chain of saturating integrators with imprecise measurements[END_REF] this technique allows one to relax the smoothness requirements imposed on the fictitious controls during backstepping design.

In addition, in many applications convergence rate and the settling time are the main optimization criteria. In such cases the problem of non-asymptotic (finite-time and fixed-time) stable system design arises. For example, a convergence rate of homogeneous systems could be changed significantly by modifying only their degree of homogeneity. Another conventional solution for acceleration consists in feedback gains increasing. Nevertheless, for the timedelayed systems such a method has a limited use: for any given delay h sufficiently large gains make the closed-loop system unstable.

Non-asymptotic stability analysis could be performed by using different methods. For instance, homogeneous systems described by ordinary differential equation can be investigated in a simple way. The main feature of such systems is that it is necessary and sufficient to study their behavior on a sphere in state space. However, time-delayed systems in general do not inherit such properties [START_REF] Efimov | Comments on finite-time stability of time-delay systems[END_REF]).

One of the main theoretical tools for stability analysis and control synthesis of nonlinear systems is the Lyapunov function method [START_REF] Khalil | Nonlinear systems; 3rd ed[END_REF]). In [START_REF] Polyakov | Finite-time stabilization using implicit lyapunov function technique[END_REF] the Implicit Lyapunov Function (ILF) method has been extended to non-asymptotic analysis. Theorems on Implicit Lyapunov-Krasovskii Functional (ILKF) are presented in [START_REF] Polyakov | Implicit lyapunov-krasovski functionals for stability analysis and control design of time-delay systems[END_REF]. Both implicit methods allow to check all stability conditions by analyzing the algebraic equations. Therefore, a simple and effective procedure for parameter tuning based on Linear Matrix Inequalities (LMIs) can be obtained (see, for example, [START_REF] Lopez-Ramirez | Fixed-time output stabilization and fixed-time estimation of a chain of integrators[END_REF]).

Nevertheless, stability analysis of time-delay systems can be done not only in the time domain. For example, in [START_REF] Kharitonov | Static output feedback stabilization nec-essary conditions for multiple delay controllers[END_REF] necessary conditions for the existence of multiple delay controllers are presented in terms of Hurwitz stability of some polynomials. However, feedback gains and admissible delays could be found only by solving transcendental equations.

The present work studies the problem of nonlinear output delay-dependent feedback design providing a practical accelerated (faster than exponential) stabilization of a chain of integrators. Although the chain of integrators is a quite simple model, it is also a very useful benchworking tool since all linear controllable systems and many nonlinear ones can be transformed into this particular form.

The goal of this work is to extend the results obtained in [START_REF] Efimov | On hyper-exponential output-feedback stabilization of a double integrator by using artificial delay[END_REF] for the case of a second order system to the chain of n ≥ 2 integrators. However, in this paper the closed-loop system is not homogeneous, therefore, for stability analysis Implicit Lyapunov-Krasovskii functional is introduced. It is proven that the proposed control law ensures practical output stabilization with the rates of convergence faster than exponential. In contrast to [START_REF] Efimov | On hyper-exponential output-feedback stabilization of a double integrator by using artificial delay[END_REF] the effective way of computation of feedback gains is given. Moreover, it is shown that closed-loop system stability does not depend on the value of artificial delay. Nevertheless, maximum value of delay determines the width of stability zone.

The outline of this work is as follows. The notations and auxiliary lemmas are given in Section 2. The Implicit Lyapunov-Krasovskii approach for stability analysis of time-delay systems is introduced in Section 3. The finitedifferences approximation of derivatives is touched upon in Section 4. The problem statement and the control design with stability analysis are considered in section 5. An example is presented in Section 6.

PRELIMINARIES

Notations

Through the paper the following notations will be used:

• N is the field of natural numbers; • a series of integers 1, 2, . . . , n is denoted by 1, n; • R is the field of real numbers, R >0 := {x ∈ R : x > 0} and R ≥0 := R >0 ∪ 0; • C i is a class of i times continuously differentiable functions R >0 → R; • • α := sign(•)| • | α ; • • is the Euclidean norm in R n ; • C 1 h is the space of continuously differentiable func- tions [-(n -1)h, 0] → R n with the norm • h defined as follows Φ h := max τ ∈[-(n-1)h,0] Φ(τ ) for Φ ∈ C 1 h ; • C 1,0 h = {Φ ∈ C 1 h : Φ(0) = 0} is a subspace of C 1 h ; • diag{λ i } n
j=1 is the diagonal matrix with the elements λ j ∈ R, j = 1, n on the main diagonal;

• if P ∈ R n×n is symmetric, then the inequalities P > 0 (P < 0) and P ≥ 0 (P ≤ 0) mean that P is positive (negative) definite and semidefinite, respectively; • λ min (P ) and λ max (P ) are the minimal and maximal eigenvalues of a symmetric matrix P ∈ R n×n , respectively;

• I n ∈ R n×n is the identity matrix and O n ∈ R n×1 is the zero column; • j i := j! i!(j-i)! is the binomial coefficient; • | | A := inf η∈A -η is the distance between ∈ R n and A ⊆ R n .

Auxiliary lemmas

Lemma 1. [START_REF] Solomon | New stability conditions for systems with distributed delays[END_REF]

). Let f, κ : [a, b] → [0, ∞) and φ : [a, b] → R be such that integration concerned is well defined. Then: b a κ(s)φ(s)ds 2 ≤ b a f -1 (s)κ(s)ds b a f (s)κ(s)φ 2 (s)ds. Lemma 2. (Lopez-Ramirez et al. (2018)). For ∀s ∈ [0, 1], β ∈ R >0 \ {1} the function g β (s) := |s β -s| admits the following estimate max s∈[0,1] g β (s) ≤ g β (β 1/(1-β) ).

IMPLICIT LYAPUNOV-KRASOVKSII FUNCTIONAL APPROACH

Consider the system of the form ẋ

(t) = f (x t ), t ∈ R >0 , x 0 = Φ ∈ D ⊆ C 1 h , (1) where x(t) ∈ R n , x t ∈ C 1 h is the state function defined by x t (τ ) := x(t + τ ) with τ ∈ [-(n -1)h, 0] with h > 0 (time delay) and f : C 1 h → R n is a continuous operator.
Assume that the origin is an equilibrium point of the system (1), i.e. f (0) = 0. A solution of the system (1) with the initial function Φ ∈ D is denoted by x(t, Φ).

In applications frequently it is only required to stabilize the origin of the system in some zone, the width of which is determined by technical requirements. The next definition present an asymptotically stability with respect to the set. Definition 3. The system (1) is said to be asymptotically stable with respect to the set A, if it is: a) Lyapunov stable with respect to the set A:

for any ε > 0 there exists ∆(ε

) > 0 such that |x(t, Φ)| A ≤ ε for all t ≥ 0 and Φ ∈ D : Φ h ≤ ∆; b) attractive with respect to the set A: |x(t, Φ)| A → 0 as t → ∞ for any Φ ∈ D.
If the set could be chosen as A = 0, then the origin of the system (1) is asymptotically stable. The set D is called the domain of attraction of the system (1). If D = C 1 h , then the corresponding stability becomes global.

In this paper only asymptotic stability is considered. Therefore, hereafter stability relates to asymptotic one.

Before formulating the theorem concerning stability analysis of time-delay systems using ILKF, a special class of comparison functions introduced by the following definition. Definition 4. [START_REF] Polyakov | Implicit lyapunov-krasovski functionals for stability analysis and control design of time-delay systems[END_REF]). The function q : R 2 >0 → R, (ρ, s) → q(ρ, s) is said to be of the class IK ∞ if and only if:

1) q is continuous on R 2 >0 ; 2) for any s ∈ R >0 there exists ρ ∈ R >0 such that q(ρ, s) = 0; 3) for any fixed s ∈ R >0 the function q(•, s) is strictly decreasing on R >0 ; 4) for any fixed ρ ∈ R >0 the function q(ρ, •) is strictly increasing on R >0 ; 5) lim

s→0 + (ρ,s)∈Γ ρ = lim ρ→0 + (ρ,s)∈Γ s = 0, lim s→∞ (ρ,s)∈Γ ρ = ∞,
where Γ = {(ρ, s) ∈ R 2 >0 : q(ρ, s) = 0}. The next theorem on Implicit Lyapunov-Krasovskii functional presents conditions that guarantee stability of the nonlinear time-delayed system.

Denote partial derivatives of functional Q(V, χ) at time instant t as follows:

Q t (V, x t ) := ∂Q(V,χ) ∂χ χ=xt f (x t ) and Q V (V, x t ) := ∂Q(V,χ) ∂V χ=xt
.

Theorem 5. If there exists a continuous functional Q :

R >0 × C 1 h → R such that: C1) for all χ ∈ C 1 h the function (V, τ ) → Q(V, χ(τ )) is continuously differentiable; C2) for any χ ∈ C 1 h there exists V ∈ R >0 such that Q(V, χ) = 0; C3) there exist q i ∈ IK ∞ , i = 1, 2 such that for all V ∈ R >0 q 1 (V, χ(0) ) ≤ Q(V, χ), ∀χ ∈ C 1 h \ C 1,0 h , Q(V, χ) ≤ q 2 (V, χ h ), ∀χ ∈ C 1 h \ {0}; C4) Q V (V, χ) < 0 for all V ∈ R >0 and χ ∈ C 1 h such that Q(V, χ) = 0; C5) for all (V, x t ) ∈ Ω = {(V, x t ) ∈ (V min , V max ) × C 1 h : Q(V, x t ) = 0} such that x t satisfies (1), we have Q t (V, x t ) ≤ θV 1+µ Q V (V, x t ), ∀t ∈ R >0 , where µ ∈ (-1, 0) ∪ R >0 , θ > 0, V min , V max ∈ R ≥0 .
Then the system is stable with the domain of attraction

D := {Ψ ∈ C 1 h : Q(V max , Ψ) ≤ 0} (2) (globally stable if V max = +∞) with respect to the set A := {Ψ ∈ C 1 h : Q(V min , Ψ) ≤ 0} (3) (at the origin if V min = 0).

The transition between sets S

1 := {Ψ ∈ C 1 h : Q(V 1 , Ψ) = 0} and B 2 := {Ψ ∈ C 1 h : Q(V 2 , Ψ) ≤ 0} is bounded as: T (µ, S 1 , B 2 ) ≤ (V -µ 2 -V -µ 1 )/(θµ), (4) where V min ≤ V 2 < V 1 ≤ V max .
Proof. This Theorem can be proven in the same way as Theorems 2 and 3 in [START_REF] Polyakov | Implicit lyapunov-krasovski functionals for stability analysis and control design of time-delay systems[END_REF] and for brevity the proof is omitted. Nevertheless, it is worth to note that condition C5) provides V ≤ -θV 1+µ , since Q := Q t (V, x t )+Q V (V, χ) V = 0 for all (V, x t ) ∈ Ω. Hence, Theorem 5 implies the stability and the transition time estimate (4). 2

FINITE-DIFFERENCES APPROXIMATION OF DERIVATIVES

In this section the exact form of finite-differences approximation error δ i (t) := ỹi (t) -y (i) (t) is presented. Proposition 6. [START_REF] Selivanov | An improved timedelay implementation of derivative-dependent feedback[END_REF]). If y ∈ C i and y (i) is absolutely continuous with i ∈ N, then the following relation holds:

δ i (t) = ỹi-1 (t) -ỹi-1 (t -h) h -y (i) (t) = - 0 -ih ϕ i (-τ )y (i+1) (t + τ )dτ , (5)
where ỹ0 (t) := y(t) and

ϕ 1 (ξ) := 1 -ξ/h, ξ ∈ [0, h] ϕ i+1 (ξ) :=                  ξ 0 ϕ i (λ) h dλ + h -ξ h , ξ ∈ [0, h], ξ ξ-h ϕ i (λ) h dλ, ξ ∈ (h, ih), ih ξ-h ϕ i (λ) h dλ, ξ ∈ [ih, ih + h]. (6)
For further stability analysis some characteristics of the functions ϕ i are introduced. Proposition 7. The functions ϕ i (ξ) defined in ( 6) and ψ i (ξ) := ih ξ ϕ i (λ)dλ possess the following properties: P1)

ϕ i ≤ 0; P2) ϕ i (ξ) + ϕ i (ih -ξ) = 1; P3) ψ i (0) = ih/2 and ψ i (ih) = 0; P4) ψ i (ξ) = -ϕ i (ξ); P5) ϕ i (ξ) is concave on ξ ∈ [0, ih/2] and convex on ξ ∈ [ih/2, ih] for i ≥ 2; P6) ψ i (ξ) ≤ (ih/2)ϕ i (ξ); P7) ζ i := ih 0 ϕ 2 i (ξ)
ψi(ξ) dξ does not depend on h; P8) for all x i+1 (t) = y (i) (t) it holds:

0 -ih ϕ i (-τ ) ẋ2 i+1 (t + τ )dτ ≥ 2 ih δ 2 i (t), ( 7a
) 0 -ih ψ i (-τ ) ẋ2 i+1 (t + τ )dτ ≥ 1 ζ i δ 2 i (t). ( 7b
)
Proof of the properties P1)-P4) can be found in [START_REF] Selivanov | An improved timedelay implementation of derivative-dependent feedback[END_REF]. Other properties can be proven using P1)-P4) and Lemma 1, but due to space limitation are not presented in this paper.

It is worth to note that ζ i is well-defined. Indeed, function φi (ξ) := ϕ 2 i (ξ)/ψ i (ξ) is continuous on ξ ∈ [0, ih) and φi (ih-) = 0. Therefore, function φi (ξ) is integrable.

NONLINEAR DELAY-DEPENDENT CONTROL

Problem statement

Consider a chain of n ≥ 2 integrators:

ẋ(t) = Ax(t) + Bu(t), y(t) = Cx(t), (8)
where x(t) ∈ R n is the state vector, u(t) ∈ R is the control input, y(t) ∈ R is the output available for measurements and

A = O n-1 I n-1 0 O n-1 , B = [0 . . . 0 1] , C = [1 0 . . . 0] .
The goal is to design an output-feedback control practically stabilizing the system (8) with the rate of convergence faster than exponential.

Main result

Define nonlinear delay-dependent control algorithm as follows:

u = K x α (9) where K = [K 1 . . . K n ] , x α = [ x1 α1 . . . xn αn ] , x1 (t) := x 1 (t), xi+1 (t) := xi (t) -xi (t -h) h , i = 1, n -1, α j := 1/r j , r j (µ) := 1 -(n + 1 -j)µ, j = 1, n.
(10) Let us rewrite control law (9) in a form suitable for practical implementation:

u(t) = n j=1 K j j-1 i=0 (-1) i h j-1 j -1 i z i (t) αj , (11)
where Therefore,control (11) depends on only n + 2 parameters: h > 0 is the artificial delay, µ ∈ (-1, 0) ∪ (0, 1/n) is the degree of nonlinearity and K j < 0, j = 1, n are feedback gains. The restrictions on effective selection of these parameters and the conditions to check are given in the following theorem.

z i (t) := y(t -ih), t > ih, y(0), t ≤ ih.
Before formulating the main result, let us introduce an Implicit Lyapunov-Krasovskii functional (ILKF) by the equality:

Q(V, χ) := -1 + χ (0)Λ -r V P Λ -r V χ(0) +V µ n-1 i=1 0 -ih ψ i (-τ ) 2ζ ihS V -ri+2 χi+1 (τ ) 2 dτ , (12)
where P = P > 0 and Λ -r

V := diag{V -rj } n j=1 . Theorem 8. Let for some µ ∈ (-1, 0) ∪ (0, 1/n) the system of LMIs: 0 < XH r + H r X ≤ 2ω(µ)X (13a) X Y Y 1 ≥ 0, X I n I n SI n ≥ 0, aI n X X SI n ≥ 0, b S S 1 ≥ 0, X ≤ 1 2 I n , (13b)
   Φ 11 b σ B B Y * -S 0 b σ * * -4ζ(1 -a) 1 * * * -S/ζ    ≤ 0, (13c)
where

H r := diag{r j } n j=1 , ω(µ) := 1 -(n + 1)µ, µ < 0, 1, µ > 0, Φ 11 := XA + Y B + AX + BY + (ζ + 2)aI n , σ j := g αj (α 1/(1-αj ) j ), ζ := max i=1,n-1 i 0 ϕ 2 i ( ξ) i ξ ϕ i ( λ)d λ d ξ be feasible for some S, a, b > 0, X ∈ R n×n , X = X > 0, Y ∈ R 1×n , with j = 1, n.
Then for any h > 0 the system (8), ( 11) with K := Y X -1 : a) is globally stable with respect to the set A (3), where V min := ((n -1)hS) -1/µ , if µ < 0; b) is stable at the origin with the domain of attraction D (2), where V max := ((n -1)hS) -1/µ , if µ > 0; c) converges from the set S 1 to the set B 2 with time estimate (4), where 1/θ := 2ω(µ)S 2 .

The proof of Theorem 8 is given in Appendix A.

Corollary 9. If LMIs (13) from Theorem 8 are feasible for some µ max , then they also hold for all µ = γµ max , 0 ≤ γ ≤ 1.

Proof. Indeed, r j (µ) could be rewritten as:

r j (µ) = (1-γ)+γ -(n+1-j)γµ max = (1-γ)+γr j (µ max ).
Therefore, condition (13a) can be expressed as follows

0 < 2(1 -γ)X + γ(XH rmax + H rmax X) ≤ 2ω(µ)X, where r max = r(µ max ).
Clearly, this condition holds for all µ = γµ max , 0

≤ γ ≤ 1, since 0 < XH rmax + H rmax X ≤ 2ω(µ max )X and ω(µ) = (1 -γ) + γω(µ max ).
And taking into account that: µ ∂σ j ∂µ = µr r j 1-r j j n + 1 -j 1 -r j ln (r j ) sign(g αj (α 1/(1-αj ) j

)) > 0, one can deduce that max µ σ = σ(µ max ) . Hence, inequality (13c) also holds for all µ = γµ max . 2

Therefore, Corollary 9 postulates that sufficiently small µ, for which LMIs (13) are feasible, always could be found.

The following proposition claims that for any feedback gains K, obtained as the solution of LMIs (13) from Theorem 8, and fixed delay h, the nonlinear closed-loop system (8), ( 11) with µ = 0 is always converging faster than its linear analog with µ = 0 between properly selected sets S 1 and B 2 , which depend on µ and h. Proposition 10. Let conditions of Theorem 8 hold for some µ ∈ (-1, 0) ∪ (0, 1/n). Then there exist two sets

S 1 := {V 1 ≥ 1, Ψ ∈ C 1 h : Q(V 1 , Ψ) = 0} and B 2 := {V 2 ≤ 1, Ψ ∈ C 1 h : Q(V 2 , Ψ) ≤ 0}
such that in the system (8) and the control (11):

T (µ, S 1 , B 2 ) < T (0, S 1 , B 2 ) (14) for fixed K and h > 0 satisfying h ≤ V -µ 2 /((n -1)S) for µ < 0, V -µ 1 /((n -1)S) for µ > 0.

A procedure for selection of sets S 1 and B 2 is given in the proof below.

Proof. For linear system µ = 0 the time of convergence between sets S 1 and B 2 is lower-bounded:

T (0, S 1 , B 2 ) ≥ T 0 (h)(ln V 1 -ln V 2 ),
where T 0 (h) is the fastest time of transition between levels V 1 = e and V 2 = 1 for µ = 0, given delay h and fixed K.

From (4) it follows that condition ( 14) is verified if:

(V -µ 2 -V -µ 1 )/(θµ) ≤ T 0 (h)(ln V 1 -ln V 2 ) or taking into account property c) from Theorem 8 V -µ 1 -V -µ 2 ln V -µ 1 -ln V -µ 2 < T 0 (h) 2(1 -(n + 1)µ)S 2 for µ < 0, V -µ 2 -V -µ 1 ln V -µ 2 -ln V -µ 1 < T 0 (h) 2S 2 for µ > 0, which for any µ ∈ (-1, 0) (µ ∈ (0, 1/n)), h > 0 and V 2 ∈ [V min , 1] (V 1 ∈ [1, V max ]) could be guaranteed by decreasing V 1 (increasing V 2 ).
Conditions on h guarantee that A ⊂ B 2 for µ < 0 and S 1 ⊂ D for µ > 0. 2 Remark 11. Since the system (8), ( 11) keeps a constant value of µ for all t ≥ 0, then it is clear that the set S 1 represents the set of initial conditions.

Remark 12. Taking into account that V -µ 2 ≤ 1 for µ < 0 and V -µ 1 ≤ 1 for µ > 0, it follows that the maximum value of artificial delay is bounded as:

h max := 1/((n -1)S). (15)
On the one hand, the greater values of h leads to the bigger stable set for µ < 0 and smaller domain of attraction for µ > 0. On the other hand, the smaller values of h could make the control system (11) unimplementable. Therefore, there is the minimum value of artificial delay h min depending on technical realization.

EXAMPLE

Consider a chain of three integrators to show the main advantages of the proposed control law. Selecting µ = -0.01 and µ = 0.01 one can obtain, respectively, the following feedback gains K -= [-0.9754, -2.4398, -1.9303], K + = [-0.9667, -2.4473, -1.9459] and S -= 11.4824, S + = 11.5149.

According to (15), the maximum value of artificial delay is h max = 0.0434 s. Therefore, let us choose the nominal value as h = 0.025 s to enlarge the stability zone and to keep the control system implementable.

The norm of the state x(t) obtained in simulation of the system (8) governed by the proposed nonlinear output delay-induced feedback (11) (blue solid line) in comparison with linear one (red solid line), as well as nonlinear (blue dashed line) and linear (red dashed line) state feedback, are depicted in figures 1 and 2 in logarithmic scale. For the sake of clarity, the feedback gains K are chosen the same for all control schemes as well as parameter µ for both nonlinear feedback and artificial delay for delay-dependent control laws.

Obviously, in the nonlinear case the rate of convergence is faster than in the linear one close to the origin for µ < 0 and far outside for µ > 0. Moreover, proposed control law and nonlinear state feedback show the same performance. The paper addresses the problem of output feedback stabilization of a chain of integrators using a nonlinear delay-dependent control law that achieves the rates of convergence faster than exponential. The effective way of computation of feedback gains is presented. It is shown that the value of artificial delay does not affect closedloop system stability, but determines the width of stability zone. Simulation results show feasibility of the proposed control scheme. The efficiency of the proposed approach is demonstrated in simulations, and a comparison with a linear analogue as well as nonlinear and linear state feedback is carried out.

  Fig. 1. Trajectories of stabilized system with µ = -0.01

Let us show that ILKF (12) satisfies all conditions from Theorem 5. Since proof of conditions C1)-C3) can be done in a straightforward way (see, for example, [START_REF] Polyakov | Implicit lyapunov-krasovski functionals for stability analysis and control design of time-delay systems[END_REF]), we just focus on conditions C4) and C5).

A. Proof of condition C4) of Theorem 5

The derivative of Q(V, χ) with respect to V is:

Taking into account (13a), one can see that:

So we finally conclude that

is the solution of the system (8), (9), then using properties P3) and P4), we obtain

BK, the first term in (A.2) could be rewritten as follows:

where

Obviously, disturbance terms d 1 and d 2 represent nonlinear deviation of feedback and finite-differences approximation error respectively.

The second term in (A.2) either can be upper-bounded by using property P6) and supposing that

or by using (7a):

It is easy to see that the term Σ 2 could compensate crossterms in (A.3) for V ≥ V min if µ < 0 or V ≤ V max if µ > 0. Therefore, only global stability with respect to the set A (3) and stability at the origin with the domain of attraction D (2) could be obtained for µ < 0 and µ > 0 respectively. Hence, we deduce:

Taking into account that from (13b) it follows that: K K ≤ P ≤ SI n ≤ aS 2 P 2 ⇒ 1/S 3 ≤ a, S 2 ≤ b, the term Σ 2 could be finally estimated as follows:

(A.4) Since S -1 I n ≤ aP 2 , the third term in (A.2) is bounded: .5), applying the Schur complement with respect to the vector [x Λ -r V , (b σ ) -1 Kd 1 , Kd 2 ] and using (13c), we deduce that Q t (V, x t ) ≤ -S -2 V µ . Taking into account (A.1), we conclude the proof. 2

The nonlinear disturbance term d 1 d 1 can be rewritten as:

(A.6) Applying (7b) to (12) for all (V, x t ) ∈ Ω, we deduce that:

Since ζ ≥ ζ i for any h > 0 due to P7) and V -µ ≥ (n-1)hS, the following inequality holds for P ≥ 2I n :

Hence, it follows that |V -rj xj | ≤ 1 and, applying Lemma 2 to (A.6), one can finally conclude that d 1 d 1 ≤ σ 2 . 2


Axel Touchard

Alain Dejean

Pierre Escoubas

Jérôme Orivel

Jack Neff

Intraspecific variations in the venom

Keywords: Ant venoms, MALDI-TOF MS, Odontomachus haematodus, peptidome, polyethism

HAL is

Introduction

Due to their ubiquity in terrestrial environments, ants are amongst the most abundant venomous animals on Earth; for instance, they represent 15-20% of the animal biomass in tropical forests [START_REF] Hölldobler | The Ants[END_REF]). Among ants, ca. 9100 species belonging to 16 subfamilies possess a stinging apparatus, while other subfamilies have lost their ability to sting [START_REF] Hölldobler | The Ants[END_REF]. Ant venoms contain a variety of toxins that paralyze prey, ward off predators and protect the colony against microbial pathogens [START_REF] Orivel | Ponericins, new antibacterial and insecticidal peptides from the venom of the ant Pachycondyla goeldii[END_REF][START_REF] Schmidt | Biochemistry of insect venoms[END_REF]). As they have been little studied, ant venom peptides represent a potentially promising source of bioactive molecules with novel scaffolds and original pharmacological activities. Previous studies [START_REF] Aili | Diversity of peptide toxins from stinging ant venoms[END_REF][START_REF] Touchard | Elucidation of the unexplored biodiversity of ant venom peptidomes via MALDI-TOF mass spectrometry and its application for chemotaxonomy[END_REF][START_REF] Touchard | The complexity and structural diversity of ant venom peptidomes is revealed by mass spectrometry profiling[END_REF] have demonstrated that the venoms of stinging ants are mostly comprised of small peptides, similarly to spider, scorpion and cone snail venoms. A limited number of peptidic toxins from several ant subfamilies such as the Ponerinae [START_REF] Cologna | Peptidomic comparison and characterization of the major components of the venom of the giant ant Dinoponera quadriceps collected in four different areas of Brazil[END_REF][START_REF] Johnson | A biochemical characterization of the major peptides from the venom of the giant Neotropical hunting ant Dinoponera australis[END_REF][START_REF] Orivel | Ponericins, new antibacterial and insecticidal peptides from the venom of the ant Pachycondyla goeldii[END_REF], Paraponerinae [START_REF] Piek | Poneratoxin, a novel peptide neurotoxin from the venom of the ant, Paraponera clavata[END_REF], Ectatomminae [START_REF] Arseniev | Toxic principle of selva ant venom is a pore-forming protein transformer[END_REF][START_REF] Pluzhnikov | Methods and tools in biosciences and medicine: animal toxins[END_REF], Myrmicinae [START_REF] Rifflet | Identification and characterization of a novel antimicrobial peptide from the venom of the ant Tetramorium bicarinatum[END_REF], Myrmeciinae [START_REF] Inagaki | Molecular cloning and biological characterization of novel antimicrobial peptides, pilosulin 3 and pilosulin 4, from a species of the Australian ant genus Myrmecia[END_REF][START_REF] Inagaki | Pilosulin 5, a novel histamine-releasing peptide of the Australian ant, Myrmecia pilosula (Jack Jumper Ant)[END_REF] and Pseudomyrmecinae [START_REF] Pan | Isolation and characterization of myrmexins, six isoforms of venom proteins with anti-inflammatory activity from the tropical ant, Pseudomyrmex triplarinus[END_REF] have been characterized.

One of the major issues in the biochemical and pharmacological study of venoms is the reproducibility of studies conducted on field-collected samples, which requires accurate species identification. We have previously demonstrated that, at the species level, the peptidic fingerprints of ant venoms are reliable chemotaxonomic markers for species determination and may possibly allow the discrimination of unresolved species complexes [START_REF] Touchard | Elucidation of the unexplored biodiversity of ant venom peptidomes via MALDI-TOF mass spectrometry and its application for chemotaxonomy[END_REF]. However, intraspecific variations can also occur as shown recently in Dinoponera quadriceps for which only 48 peptides were shared between colonies out of the more than 300 peptides found in total [START_REF] Cologna | Peptidomic comparison and characterization of the major components of the venom of the giant ant Dinoponera quadriceps collected in four different areas of Brazil[END_REF]. Also, intraspecific variations in venom composition have been observed in snakes, scorpions, tarantulas and cone snails, this variation being linked to geographical distribution [START_REF] Núñez | Snake venomics and antivenomics of Bothrops atrox venoms from Colombia and the Amazon regions of Brazil, Perú and Ecuador suggest the occurrence of geographic variation of venom phenotype by a trend towards paedomorphism[END_REF][START_REF] Shashidharamurthy | Variation in biochemical and pharmacological properties of Indian cobra (Naja naja naja) venom due to geographical distribution[END_REF], age [START_REF] Escoubas | High-performance liquid chromatography matrix-assisted laser desorption/ionization time-of-flight mass spectrometry peptide fingerprinting of tarantula venoms in the genus Brachypelma: chemotaxonomic and biochemical applications[END_REF] or sex [START_REF] Escoubas | High-performance liquid chromatography matrix-assisted laser desorption/ionization time-of-flight mass spectrometry peptide fingerprinting of tarantula venoms in the genus Brachypelma: chemotaxonomic and biochemical applications[END_REF][START_REF] Herzig | Intersexual variations in the pharmacological properties of Coremiocnemis tropix (Araneae, Theraphosidae) spider venom[END_REF][START_REF] Herzig | Intersexual variations in the venom of the Brazilian 'armed'spider Phoneutria nigriventer (Keyserling, 1891)[END_REF][START_REF] Herzig | Intersexual variations in northern (Missulena pruinosa) and eastern (M. bradleyi) mouse spider venom[END_REF]). However, it remains unclear whether such variation is a common denominator for all venomous animals or is restricted to some taxa or species. This key point is still a subject of debate amongst specialists, since venom sampling conditions are often limiting and broad species-wide surveys of venom composition have been difficult to conduct.

In ants as in all hymenopterans, only females are venomous, so that sex cannot account for venom variation. Therefore, intraspecific variations in venom composition could be related to geographical distribution, diet, age or division of labor (polyethism). In most ant species, reproduction is carried out by the queen(s), while all other tasks are performed by the workers for whom the division of labor is based on physical caste (there is polymorphism in the worker caste) or, most often, age [START_REF] Fresneau | Biologie et comportement social d'une fourmi néotropicale (Pachycondyla apicalis)[END_REF]. Usually, the youngest workers are involved in intranidal activities, whereas older workers are assigned to tasks outside the nest such as defense and foraging (da [START_REF] Da Silva-Melo | Division of Labor in Pachycondyla striata Fr. Smith, 1858 (Hymenoptera: Formicidae: Ponerinae)[END_REF][START_REF] Sendova-Franks | Self-assembly, self-organization and division of labour[END_REF][START_REF] Wilson | The social biology of ants[END_REF]).

As venom is mostly used by workers performing extranidal activities, one can hypothesize that polyethism could affect venom composition. To test this hypothesis, we investigated both intracolonial and intercolonial variations in venom composition in the Neotropical ponerine species, Odontomachus haematodus. The monomorphic workers of this species possess a peptide-rich venom [START_REF] Touchard | Elucidation of the unexplored biodiversity of ant venom peptidomes via MALDI-TOF mass spectrometry and its application for chemotaxonomy[END_REF]) used in colony defense and prey capture. To assess possible venom variations, we characterized the entire venom peptidome by combining HPLC chromatographic separation with offline MALDI-TOF mass spectrometry analysis and explored the putative differences in venom composition between castes and type of activity (i.e., queens, and worker nurses and foragers) as well as colonies from different geographical locations via the comparison of their venom peptidic fingerprints.

Materials and methods

Ant collection and taxonomy

Odontomachus haematodus colonies were collected from three different areas in French Guiana: six colonies were collected on the Campus Agronomique, Kourou; three in Sinnamary; and one in Angoulême (Fig. 1). In the laboratory, the ant colonies were conserved in artificial nests made of plastic boxes (11 cm × 11 cm × 6 cm) filled with 2 cm of molded plaster to create chambers and covered by a plate of red glass. These boxes were connected to a foraging arena consisting of a second, similar plastic box without plaster. The colonies were kept at 25 °C and provided with dead mealworms and honey twice a week.

Voucher specimens wer1e deposited in the Laboratorio de Mirmecologia, Cocoa Research Centre, Ilhéus, Bahia, Brazil.

Behavioral observations and ant groups

To investigate the division of labor, workers were individually marked with different colored dots of paint on their thoraxes and gasters. Worker tasks were determined by scan sampling their behavior (three scans per day at 9:00 am, 2:00 pm and 5:00 pm; 5 days per week over 3 weeks). The percentage of presence in the foraging area for each individual over the 3-week period was calculated to define the behavioral groups. Workers that had either never been seen in the foraging area or were there between 0% to 25% of the time were considered nurses ([group 0%] and [group 25%], respectively). Those observed between 25% to 50% of the time in the foraging area were considered intermediates [group 50%], and those observed between 50% to 75% or between 75% to 100% of the time in the foraging area were considered foragers ([group 75%] and [group 100%], respectively). Moreover, winged females present in the colonies were named "virgin queens" in order to differentiate them from "reproductive queens" devoid of wings (see Fig. 1).

Venom analysis

Ants were killed by freezing at -20 °C prior to dissecting their venom glands. The dissected venom glands were placed in 10% acetonitrile (ACN)/water (v/v), centrifuged for 5 min at 14,400 rpm and the supernatant was collected and lyophilized prior to storage at -20 °C for subsequent biochemical analysis. To study intra-colonial variations and the influence of the role of individual ants in the colony, five venom glands from each behavioral group (cf. Fig. 1) were dissected and pooled for each colony. Furthermore, 100 venom glands from the workers of one colony (Kou06; 3.55 mg of dry crude venom in total) were dissected to carry out an in-depth exploration of the whole venom by LC-MS. panels show information about each colony, including GPS coordinates, colony code and the different behavioral groups. One hundred dissected workers from colony Kou06 were used for LC-MS investigation.

Chromatographic separation

In order to fully explore the Odontomachus haematodus peptidome, a venom sample pooled from 100 workers was fractionated by reversed-phase high performance liquid chromatography (RP-HPLC) on a Waters Xterra-C18 5µm, 2.1 × 100 mm column with an Agilent HP 1100 HPLC system. Fractionation was achieved using a gradient of solvent A (water / 0.1% trifluoroacetic acid TFA) and solvent B (ACN / 0.1% TFA). The percentage of solvent B was modified at a flow rate of 0.3 mL/min as follows: 0% for 5 min, 0-60% for 60 min, 60-90% for 10 min and 90-0% for 15 min. The absorbance of the column effluent was monitored at 215 nm on a diode-array detector. The signal was monitored in real time, and fractions were collected manually for each eluting peak. Individual fractions were then dried and reconstituted in 50µL of water/0.1% TFA for subsequent off-line MALDI-TOF MS analysis and disulfide bond reduction.

Chemical reduction of disulfide bonds

To map the distribution of disulfide-linked peptides in the venom, 5µL of each fraction were incubated in 10µL of a reducing buffer (100 mM Tris, pH 8, 6M guanidine) with 10 mM dithiothreitol (DTT) for 1h at 56 °C in the dark. The reaction was stopped by the addition of 5µL of water / TFA 0.1%. Prior to mass spectrometry analysis, reduced fractions were desalted using Ziptip® C18 (Millipore) pipette tips. As the chemical reduction of disulfide bonds results in a mass increase of 2 Da for each bond, comparing mass shifts between native and reduced venom fractions allowed the presence of disulfide-linked peptides and the number of disulfide bonds for each to be compared.

Mass spectrometry analysis

Mass spectrometry analyses were performed on a Voyager DE-Pro MALDI-TOF mass spectrometer (Applied Biosystems; CA, USA) using α-cyano-4-hydroxycinnamic acid (CHCA) matrix dissolved at 5 mg/mL in water/ACN/TFA (50/50/0.1 v/v/v). Prior to MS analysis, crude venoms were desalted using Ziptip® C18 (Millipore) pipette tips. Then, 1µL of each reconstituted HPLC fraction or the desalted crude venom was deposited on the MALDI target plate followed by 1 µL of the matrix. Each spectrum was calibrated externally using a mixture of peptides of known molecular masses in the same m/z range (Peptide calibration Mix 4, LaserBio Labs, Sophia-Antipolis, France). External calibration was performed by depositing, adjacent to each sample, 0.5 µL of the calibration mixture co-crystallised with 0.5 µL of the CHCA matrix. All spectra were acquired in reflector mode to maximize the accuracy of the mass determination. Spectra were collected over the m/z 500-10,000 range in positive ion mode (200 shots per spectrum) and were automatically calibrated using the sequence module of the Voyager® control software (Applied Biosystems, USA).

Data analysis

The mass spectra were subjected to a baseline correction (0.7 correlation factor) and Gaussian smoothing (5-point filter width) using Data Explorer® 4.11 software. Potential sodium and potassium adducts were manually removed from all mass lists. Masses matching within ± 1.0 Da were defined as identical peptides in this study. Identical masses in adjoining HPLC fractions, which were interpreted as reflecting an incomplete separation, were also removed. Two-dimensional scatter plots, termed "2D venom landscapes", were constructed using SigmaPlot 12.0 software. All peptide masses detected in the HPLC fraction spectra were plotted as dot graphs with m/z values s on the y-axis and RP-HPLC elution time on the x-axis. A Principal Component Analysis (PCA) of the relative abundance of peptides in the mass spectra was performed using PAST 3.02 software.

Results

Venom peptidome analysis

The LC-MS analysis of Odontomachus haematodus venom revealed the presence of 105 peptides (Table 1). All of the peptides are small, falling within a narrow mass range of m/z 777.49 to 2978.5 (M+H + ) (Fig. 2A-B). We estimated that the number of residues varied between 7 and 27 based on a theoretical estimate of MW av of 111.1254 Da for an average amino acid (Averagine). The value is derived from the statistical occurrence of amino acids in proteins [START_REF] Senko | Determination of monoisotopic masses and ion populations for large biomolecules from resolved isotopic distributions[END_REF], and calculated with the formula C 4.9384 H 7.7583 N 1.3577 S 0.0417 . All of the peptides eluted between 15% and 45% ACN (retention time: 20-50 min), with the most abundant peptides in the venom eluting between 35% and 45% ACN (retention time: 40-50 min) (Fig. 2C). The mass analysis of the chemically reduced HPLC fractions did not show any mass shift between native and reduced fractions, demonstrating that Odontomachus haematodus venom is exclusively composed of linear peptides (i.e., devoid of disulfide bonds).

Venom peptidome variations

We collected 43 venom samples from the nine Odontomachus haematodus colonies monitored: 18 venoms from nurses; eight from intermediates, seven from foragers, six from fertilized queens and four from virgin queens. The MALDI-TOF MS peptidic mass fingerprinting of these 43 crude venoms resulted in the selection of the 20 characteristic peptides masses (i.e., showing the most abundant signals) which constituted the matrix used for the principal component analysis (PCA) (m/z 1842.6, 1861.1, 1916.91, 1962.89, 2019.91, 2044.69, 2062.27, 2086.03, 2095.89, 2117.03, 2219.16, 2245.32, 2387.28, 2473.22, 2515.32, 2590.18, 2679.79, 2784.34, 2789.25, 2802.39). A PCA based on the relative abundance of the selected peptides revealed that the first two principal components accounted for 68.1% of the variance (Fig. 3). The PCA showed that the venom composition was not related to caste or type of activity (Fig. 3). Indeed, venoms from different behavioral groups showed similar patterns, indicating that polyethism and reproductive status did not affect the peptidic composition of the venoms (Fig. 4). Yet, some qualitative intracolonial differences were noted in the behavioral groups as illustrated by the nurse group (Fig. 5). The venoms from Angoulême, which contained two specific peptides (m/z 1861.1 and 2062.27), were separated from those from the two other localities and differed only by the relative proportions of the mass 2019.91 m/z (Fig. 3).

Discussion

Ant venoms are complex cocktails of peptides which have evolved to act on multiple biological targets. By combining MALDI-TOF MS with chromatographic separation, we have shown that the Odontomachus haematodus venom peptidome is composed of more than 100 small and linear peptides in the 700-3000 m/z mass range. This feature is consistent with a previous study on the venoms of five Neotropical Odontomachus species [START_REF] Touchard | Elucidation of the unexplored biodiversity of ant venom peptidomes via MALDI-TOF mass spectrometry and its application for chemotaxonomy[END_REF] as well as wasp and cone snail venoms which are usually comprised of peptides with fewer than 35 residues (Baptista-Saidemberg et al. 2011; [START_REF] Cologna | Peptidomic comparison and characterization of the major components of the venom of the giant ant Dinoponera quadriceps collected in four different areas of Brazil[END_REF][START_REF] De Souza | Mass spectrometric characterization of two novel inflammatory peptides from the venom of the social wasp Polybia paulista[END_REF][START_REF] Gomes | Structure-function relationships of the peptide Paulistine: A novel toxin from the venom of the social wasp Polybia paulista[END_REF][START_REF] Johnson | A biochemical characterization of the major peptides from the venom of the giant Neotropical hunting ant Dinoponera australis[END_REF][START_REF] Lewis | Conus venom peptide pharmacology[END_REF][START_REF] Orivel | Ponericins, new antibacterial and insecticidal peptides from the venom of the ant Pachycondyla goeldii[END_REF]. In contrast, spider, scorpion and snake venoms may also contain larger peptides with typically from 30 to 100 amino acids [START_REF] Olivera | Diversity of Conus neuropeptides[END_REF].

Because the presence of peptides and proteins in venoms is associated with the metabolic cost of venom production, we hypothesized that ants dedicated to tasks within the nest, typically nurses and queens, may possess less complex venoms than foragers, the latter using their venom to subdue prey and deter enemies and therefore needing venoms with a higher level of efficacy. Yet, our results show that the venom composition does not differ between nurses, intermediates, foragers or even queens in Odontomachus haematodus. By comparison, the toxicity of Neoponera commutata (Ponerinae) worker venoms was not related to age or task specialization, but the workers from different behavioral castes possess different amounts of venom in their reservoir [START_REF] Schmidt | Venom and task specialization in Termitopone commutata (Hymenoptera: Formicidae)[END_REF]. Also, callow Harpegnathos saltator (Ponerinae) workers have empty venom sacks, and workers dedicated to tasks inside the nest have lower amounts of venom than do older ones [START_REF] Haight | Patterns of venom production and temporal polyethism in workers of Jerdon's jumping ant, Harpegnathos saltator[END_REF]. Therefore, we can hypothesize that, in most Ponerinae, nurses limit the metabolic cost of venom by producing lower amounts, but with the same peptidic composition as the foragers. Interestingly, these results contrast with the case of fire ants whose alkaloid venom composition changes with the size and age of the workers [START_REF] Deslippe | Venom alkaloids of fire ants in relation to worker size and age[END_REF][START_REF] Haight | Patterns of venom synthesis and use in the fire ant, Solenopsis invicta[END_REF]. The venom composition of all fire ant queens is also presumably different from workers which assumes different biological effects [START_REF] Eliyahu | Venom alkaloid and cuticular hydrocarbon profiles are associated with social organization, queen fertility status, and queen genotype in the fire ant Solenopsis invicta[END_REF][START_REF] Fox | Intraspecific and intracolonial variation in the profile of venom alkaloids and cuticular hydrocarbons of the fire Ant Solenopsis saevissima Smith (Hymenoptera: Formicidae)[END_REF]. Also, Neoponera commutata (Ponerinae) and Pogonomyrmex spp. (Myrmicinae) queens produce less venom than do workers and their venom is significantly less lethal and paralytic than that of the workers, suggesting differences in venom composition [START_REF] Schmidt | Venom and task specialization in Termitopone commutata (Hymenoptera: Formicidae)[END_REF][START_REF] Schmidt | Queen versus worker venoms: are they equally lethal?[END_REF].

In Odontomachus haematodus, differences in venom composition seem rather associated with geographic variations as the venom peptidic fingerprints clearly differed between colonies, particularly if they came from locations separated by large distances. Such inter-colonial variations have previously been reported for Dinoponera quadriceps (Ponerinae) collected from different areas in Brazil [START_REF] Cologna | Peptidomic comparison and characterization of the major components of the venom of the giant ant Dinoponera quadriceps collected in four different areas of Brazil[END_REF]). Among animal venoms, intraspecific variations related to geography are a common phenomenon and have been reported in snakes [START_REF] Shashidharamurthy | Variation in biochemical and pharmacological properties of Indian cobra (Naja naja naja) venom due to geographical distribution[END_REF], cone snails [START_REF] Duda Tf | Geographic variation in venom allelic composition and diets of the widespread predatory marine gastropod Conus ebraeus[END_REF], scorpions [START_REF] Omran | Intraspecific variation in scorpion Leiurus quinquestriatus venom collected from Egypt (Sinai and Aswan deserts)[END_REF], spiders [START_REF] Escoubas | Composition and neurotoxicity of tarantula venoms: biogeographic and taxonomic variations[END_REF]) and both social [START_REF] Dias | Peptide diversity in the venom of the social wasp Polybia paulista (Hymenoptera): A comparison of the intra-and inter-colony compositions[END_REF]) and parasitoid wasps [START_REF] Poirié | Insights into function and evolution of parasitoid wasp venoms[END_REF]. In snakes, intraspecific variations have been shown to exhibit a differential venom effectiveness towards different prey [START_REF] Casewell | Complex cocktails: the evolutionary novelty of venoms[END_REF]). This may be the result of allelic variations in the genes coding the peptides as shown for Conus ebraeus venom [START_REF] Duda Tf | Geographic variation in venom allelic composition and diets of the widespread predatory marine gastropod Conus ebraeus[END_REF], increasing the venom peptidic diversity in this species. This intraspecific diversity is essential for natural selection and ant venom diversification.

Previous studies have demonstrated that toxins (peptide and alkaloids) in ant venoms can be used as chemotaxonomic markers in order to identify species but also to reveal cryptic ant species [START_REF] Fox | Intraspecific and intracolonial variation in the profile of venom alkaloids and cuticular hydrocarbons of the fire Ant Solenopsis saevissima Smith (Hymenoptera: Formicidae)[END_REF][START_REF] Touchard | Elucidation of the unexplored biodiversity of ant venom peptidomes via MALDI-TOF mass spectrometry and its application for chemotaxonomy[END_REF]. Thus, the observed intraspecific variation in the venom composition of Odontomachus haematodus might also result from the presence of cryptic species. It would be interesting in the future to extend this study through further genetic analysis to assess the possible presence of different cryptic species.

Conclusion

The present study constitutes the first exploration of the Odontomachus haematodus venom peptidome, revealing that this venom is comprised of more than 100 small linear peptides. Also, the peptidic diversity in this species is amplified due to intraspecific variations. The present results show that these venom variations are not related to caste or type of activity, but seem to be related to the geographical location of the ant colonies or to a hypothetical complex of cryptic species. It would be interesting in the future to analyze whether such variations can affect the effectiveness of the venom in prey capture. It will also be necessary to consider such intercolonial variations in peptidic composition to ensure the reproducibility of further biochemical and pharmacological studies on ant venoms.

Figure 1 .

 1 Figure 1. Sites where the 10 Odontomachus haematodus colonies were collected in French Guiana. Tablepanelsshow information about each colony, including GPS coordinates, colony code and the different behavioral groups. One hundred dissected workers from colony Kou06 were used for LC-MS investigation.

Figure 2 .

 2 Figure 2. Investigation of the whole Odontomachus haematodus venom peptidome by LC-MS. (A) Twodimensional landscape of the venom. Dots indicate peptides. (B) Box-and-whisker plot of the peptide mass distribution presented in the 2D venom landscape. The bottom and top ends of the box represent the first and third quartiles, respectively, while the line inside each box represents the median mass. The ends of the whiskers represent the 5-95 percentile range while the dots represent masses outside the 5-95 percentile range. (C) C18 RP-HPLC chromatogram of the venom. The dashed line shows the slope of the ACN gradient.

Figure 3 .

 3 Figure 3. Ordination diagram based on the principal components of the relative abundance of peptides from 43 Odontomachus haematodus venoms. Solid symbols represent ant venoms from Kourou, empty symbols represent ant venoms from Sinnamary and grey symbols are venoms from Angoulême. The 95% confidence ellipses are displayed. The different behavioral groups and castes are shown by the following shapes: circle [nurses]; diamond [intermediates]; square [foragers]; triangle [reproductive queens] and inversed triangle [virgin queens].

Figure 4 .

 4 Figure 4. Intracolonial mass spectra variations among the different behavioral groups of colony Kou05. Few qualitative variations can be observed and many dominant peptides were present in all groups, particularly the shaded masses.

Figure 5 .

 5 Figure 5. Intercolonial mass spectra variation among the nurses [0% group]. Shaded areas highlight major mass variations among the different Odontomachus haematodus colonies.

Table 1 .

 1 Mass list of peptides (m/z) from O. haematodus venom collected in Kourou (Kou06).

Acknowledgments

We are grateful to Andrea Yockey-Dejean for proofreading the manuscript. Financial support for this study was provided by the programme Convergence 2007-2013, Région Guyane from the European Community (Bi-Appli, 115/SGAR-DE-2011/052274) and the BIOPEPMED grant from the Programme Amazonie II of the French Centre National de la Recherche Scientifique. This study has also benefited from a ''Investissement d'Avenir'' grant managed by the Agence Nationale de la Recherche (CEBA, ref.

ANR-10-LABX-25-01).