Amina Benarab

Alejandro Apaza-Perez

Global observers with finite-time and fixed-time convergence for a class of uncertain mechanical systems

This paper proposes global sliding mode observers for a class of one-degree-of-freedom mechanical systems. For the observer design, besides the usual Coriolis and centrifugal forces, we consider uncertain dry frictions and disturbances. Moreover, the system is not required to be bounded-input bounded-state stable, rendering the observer design problem challenging. The observer design exploits the specific relationships between the inertia and Coriolis terms providing a sliding-mode observer, with global theoretically exact finite-time and fixed-time convergence to the velocities of the mechanical system. The efficiency of the proposed observer is validated through simulations on an inverted pendulum.

I. INTRODUCTION

Control of mechanical systems usually requires information about position and velocity. Often only the position is available for which an estimation of the velocity is required, this has given rise to interesting challenges in the area of control and observation [START_REF] Spong | Robot Modeling and Control. Wiley select coursepack[END_REF], [START_REF] Ortega | Passivity-based Control of Euler-Lagrange Systems: mechanical, electrical and Electromechanical applications[END_REF]. The main challenge in building observers to estimate the velocity in mechanical systems is the presence of nonlinear terms (e.g. Coriolis and centrifugal forces) and disturbances (e.g. unmodeled dynamics). The sliding-mode observers/differentiators [START_REF] Levant | Robust exact differentiation via sliding mode technique[END_REF], [START_REF] Davila | Second-order sliding-mode observer for mechanical systems[END_REF], [START_REF] Fridman | Higher-order sliding-mode observer for state estimation and input reconstruction in nonlinear systems[END_REF], [START_REF] Cruz-Zavala | Uniform robust exact differentiator[END_REF], [START_REF] Oliveira | Global and exact hosm differentiator with dynamic gains for output-feedback sliding mode control[END_REF] provide theoretically exact finite-time and fixed-time convergence despite the presence of bounded disturbances when the system has bounded-input boundedstate (BIBS) stability, therefore they are semi-global. If the BIBS property is not satisfied, then finite-time/fixed-time convergence is not guaranteed, see an illustrative example in [START_REF] Apaza-Perez | Dissipative approach to sliding mode observers design for uncertain mechanical systems[END_REF].

In this paper, we will consider mechanical systems of one degree of freedom (1-DOF) with disturbances, which could not have the BIBS property.

State of art and motivation: 1) If the dynamical systems with disturbances satisfying the conditions for the existence of an observer, among them the relative-degree-one condition [START_REF] Hautus | Strong detectability and observers[END_REF], [START_REF] Rocha-Cózat | Dissipativity and design of unknown input observers for nonlinear systems[END_REF], then dissipative observers [START_REF] Moreno | Approximate Observer Error Linearization by Dissipativity Methods[END_REF], [START_REF] Rocha-Cózatl | Dissipative design of unknown input observers for systems with sector nonlinearities[END_REF], which include high-gain/Lipschitz observers, result to be efficient to deal with BIBS restriction. They are able to estimate globally and exponentially the real states using dissipative properties which could have the nonlinear terms. But mechanical systems with disturbances have the relative degree greater than one, with respect to A. Benarab is with the Université Paris-Saclay, CNRS, CentraleSupélec, Inria, Laboratoire des signaux et systèmes, 91190, Gif-sur-Yvette, France, with Institut Polytechnique des Sciences Avancées, 94200 Ivry-sur-Seine, France. amina.benarab@centralesupelec.fr W.A. Apaza-Perez is with the Université Paris-Saclay, CNRS, Centrale-Supélec, Laboratoire des signaux et systèmes, 91190, Gif-sur-Yvette, France willy-alejandro.apaza-perez@l2s.centralesupel ec.fr (w.r.t.) the measured position.

2) There are many works (e.g. see [START_REF] Besanc ¸on | Global output feedback tracking control for a class of lagrangian systems[END_REF], [START_REF] Mabrouk | On global observers for some mechanical systems[END_REF], [START_REF] Astolfi | A globally exponentially convergent immersion and invariance speed observer for mechanical systems with non-holonomic constraints[END_REF], [START_REF] Stamnes | A constructive speed observer design for general euler-lagrange systems[END_REF]) dealing with the Coriolis and centrifugal forces, which provide global observers when the model of the mechanical system is completely known. Through state transformations, [START_REF] Astolfi | A globally exponentially convergent immersion and invariance speed observer for mechanical systems with non-holonomic constraints[END_REF], [START_REF] Stamnes | A constructive speed observer design for general euler-lagrange systems[END_REF] propose observers with fairly high dimension, namely 3n + 1 and 2n + 2 respectively, where n is the dimension of the unmeasured velocity. [START_REF] Besanc ¸on | Global output feedback tracking control for a class of lagrangian systems[END_REF], [START_REF] Mabrouk | On global observers for some mechanical systems[END_REF] propose observers with the same dimension of the system for a class of mechanical systems. However, the challenge of dealing with frictions, disturbances, and obtaining an estimation theoretically exact of velocity was not considered.

3) To overcome the BIBS restriction and the presence of Coriolis and centrifugal forces some recent works have been presented. [START_REF] Apaza-Perez | Dissipative approach to sliding mode observers design for uncertain mechanical systems[END_REF] and [START_REF] Edwards | Super-twisting observation for a class of lagrangian systems[END_REF] propose a finite-time sliding-mode observer for 1-DOF mechanical systems with disturbances and uncertain inertial matrix, respectively, using state transformations inspired by [START_REF] Besanc ¸on | Global output feedback tracking control for a class of lagrangian systems[END_REF]. An extension to the 2-DOF was proposed in [START_REF] Apaza-Perez | Global sliding mode observers for some uncertain mechanical systems[END_REF] where some assumptions in the inertia matrix are required. But, global observers with convergence in fixed time have not been considered by previous works.

Main contribution. A global sliding mode observer for a class of mechanical systems with Coriolis and centrifugal forces, dry frictions, and bounded disturbances is proposed. The proposed observer does not require a prior state transformation and provides finite-time and fixed-time convergence to the position and velocity of the mechanical system. BIBS stability of the mechanical system is not required, and its gains can be obtained from a feasible matrix inequality. Some definitions and preliminary results are described in Section II. The problem statement is presented in Section III. In Section IV, the observer design for the finite-time/fixedtime convergences is described, with the main results. Finally, to illustrate our results, numerical simulations for an inverted pendulum on a cart are shown in Section V.

II. PRELIMINARIES

Notations. Symbols R, R ≥0 , R >0 denote the set of real, of nonnegative real numbers, of positive real numbers, respectively. | • | denotes the absolute value in R and • denotes the Euclidean norm on R n . Let λ min (P) and λ max (P) be respectively the minimum and the maximum eigenvalues of the positive definite symmetric matrix P ∈ R n×n and P T defines the transposed matrix of P.

Define the signed power function x r , for any x ∈ R and r ∈ R ≥0 , as an increasing odd function defined as x r = |x| r sign(x), where sign(x) is the standard sign function. The function x r satisfies following properties:

• For each r ∈ [0, 1), x r is differentiable for all x = 0, x r ∈ C 1 if r ∈ [1, 2] and x r ∈ C 2 if r ∈ (2, ∞).
• d dx x r = r |x| r-1 and d dx |x| r = r x r-1 , for all x = 0 and r = 0.

• For any r, s ∈ R ≥0 , x r x s = |x| r+s and |x| r x s =

x r+s and x 0 = sign(x).

A. Finite-time and fixed-time stability Consider the following system

ẋ(t) = f (t, x(t)), t > t 0 x(t 0 ) = x 0 , (1)
where x ∈ R n is the state vector and f : R + × R n → R n is a nonlinear continuous vector field. Assume that the origin is an equilibrium point of (1), i.e. f (0) = 0. Definition 1 ( [START_REF] Sanjay | Finite-time stability of continuous autonomous systems[END_REF]): The origin x = 0 of system ( 1) is said to be globally Finite-Time Stable if it is globally asymptotically stable and any solution x(t, x 0 ) of ( 1) reaches the equilibria at some finite-time moment, i.e. x(t, x 0 ) = 0 ∀t ≥ T (x 0 ) where T : R n → R ≥0 is the so-called the settlingtime function.

Definition 2 ([20]):

The origin x = 0 of system ( 1) is said to be globally Fixed-Time Stable if it is globally finite-time stable and the settling time function T is bounded by some positive number T max > 0, i.e. T (x 0 ) ≤ T max for ∀x 0 ∈ R n .

We recall some technical results and well known properties of continuous homogeneous functions.

Lemma 3 (Young's inequality [START_REF] Hardy | Inequalities[END_REF]): For any positive real numbers a > 0, b > 0, c > 0, p > 1 and q > 1, with 1 p + 1 q = 1, the following inequality holds

a b ≤ c p p a p + c -q q b q
and equality holds if and only if a p = b q . Lemma 4 (Jensen's inequality [START_REF] Hardy | Inequalities[END_REF]): For any real numbers a 1 , ..., a n , if 0

< p < q n ∑ i=1 |a i | p 1 p ≤ n ∑ i=1 |a i | q 1 q .
For convenience notations, in the following we will consider x, q, e, u, δ instead of x(t), q(t), e(t), u(t), δ (t), respectively.

III. PROBLEM STATEMENT

Consider the following 1-DOF mechanical systems

m(q) q + c(q) q2 + • sign( q) + g(q) = u + δ (2)
where q ∈ R denotes the (measured) generalized position, q is the generalized velocity, m(q) denotes the inertia term, c(q) q2 is Coriolis and centrifugal force, sign( q) is a dry friction, u is the measured torque, g(q) denotes gravitational forces and δ represents the disturbance in the system. The objective is to design an observer for the system (2) with theoretically exact global finite-time/fixed-time convergence to the real values of position and velocity.

Consider the system (2), which additionally satisfies the following assumptions:

(A-1) There exist constants ς m > 0 and ς M > 0 such that 0 < ς m ≤ m(q) ≤ ς M , for all q, is satisfied. (A-2) The inertia and Coriolis functions satisfy ṁ(q) = 2 c(q) q (A-3) The disturbance δ (t) is bounded, i.e. there exists a constant L δ > 0 such that δ (t) ≤ L δ is satisfied. Assumptions (A-1) and (A-2) are assumed in most of the mechanical systems [START_REF] Spong | Robot Modeling and Control. Wiley select coursepack[END_REF]. Assumption (A-3) is a standard condition for the construction of the observer due to the fact that the system has the relative degree greater than one [START_REF] Hautus | Strong detectability and observers[END_REF] IV. OBSERVER DESIGN AND RESULTS

Considering q 1 = q, q 2 = q, the state space representation of ( 2) is given by

     q1 = q 2 q2 = 1 m(q 1 ) (u -g(q 1 ) -c(q 1 )q 2 2 + δ ) y = q 1 (3) where δ = δ -• sign( q).
We propose the following observer 1 for system (3):

         q1 = 1 √ m(q 1 ) m( q1 ) q2 + k 1 ψ 1 (q 1 -q1 ) q2 = 1 √ m( q1 ) m(q 1 ) (u -g(q 1 ) + k 2 ψ 2 (q 1 -q1 )) -1 m( q1 ) c( q1 ) q2 2 (4)
where the correction functions ψ 1 and ψ 2 are defined by

ψ 1 (•) =α • 1 2 + β • + γ • 3 2 , ( 5
)
ψ 2 (•) = α 2 2 • 0 + 3α β 2 • 1 2 + β 2 • + (6) 2 α γ • + 5 2 β γ • 3 2 + 3 2 γ 2 • 2 ,
with α, β , γ ≥ 0. Note that when γ = 0 the structure of the generalized super-twisting algorithm [START_REF] Moreno | A linear framework for the robust stability analysis of a generalized super-twisting algorithm[END_REF] is recovered, and when β = 0 and γ = 0 the standard super-twisting algorithm [START_REF] Levant | Robust exact differentiation via sliding mode technique[END_REF]. The higher-order terms corresponding to γ are introduced to improve the convergence properties of the observer and to obtain the uniformity of the convergence of the observer w.r.t. the initial conditions.

A. Finite-time observer

In this section, consider the observer (4), and the nonlinear functions ψ 1 and ψ 2 given in ( 5) and ( 6), respectively, with α > 0 and β , γ ≥ 0.

Lemma 5:

Let C = [ 1 0 ], B = [ 0 1 ] T and L δ > 0. Then, there exist ρ > 0, α > 0, a constant matrix A = -k 1 1
-k 2 0 and a symmetric and positive definite matrix P ∈ R 2×2 , such that the following nonlinear matrix inequality holds.

A T P + P A +C T L 2 δ C + ρ P P B B T P -α 4 4 < 0. ( 7
)
1 The solutions of the system (3) are defined in Filippov's sense (see [START_REF] Filippov | Differential Equations with Discontinuous Righthand Sides: Control Systems (Mathematics and its Applications)[END_REF]).

Proof: For the symmetric matrix C T L 2 δ C + I (I = I 2×2 ), there is a positive definite symmetric solution P = P T > 0 for the next inequality (quadratic Lyapunov inequality)

A T P + P A +C T L 2 δ C + I < 0. (8)
In other hand, one has 0 < λ 2 min (P) I ≤ P ≤ λ 2 max (P) I

which implies that

I ≥ P λ 2 max (P) . ( 10
)
Substituting inequality [START_REF] Rocha-Cózat | Dissipativity and design of unknown input observers for nonlinear systems[END_REF] in ( 8), one gets

P A + A T P +C T L 2 δ C + ρ P < 0 where ρ = 1 λ 2
max (P) . For α > 0 sufficiently large the following inequality is satisfied

- α 4 4 + B T P [P A + A T P +C T L 2 δ C + ρ P] -1 P B < 0.
Finally, by Schur complement one can deduce that ( 7) holds.

Theorem 6: If the parameters β , γ are not negative and k 1 , k 2 , α are selected such that ( 7) is satisfied, then the system (4) is a global finite-time observer for the system (3) in the sense that the estimation states q1 and q2 converge exactly in finite time to the states q 1 and q 2 of system (3).

Proof: Define the error estimation of the systems (3) and (4) as:

e 1 = q 1 -q1 e 2 = m(q 1 ) q 2 -m( q1 ) q2
and its dynamic error is given by

   ė1 = e 2 -k 1 ψ 1 (e 1 ) √ m(q 1 ) ė2 = 1 √ m(q 1 ) (-k 2 ψ 2 (e 1 ) + δ ) (11)
where ṁ(q 1 ) = 2 c(q 1 ) q 2 and ṁ( q1 ) = 2 c( q1 ) q2 are satisfied by Assumption (A-2). Consider a homeomorphism η (from [ e 1 e 2 ]) defined as η = [ ψ 1 (e 1 ) e 2 ] T and a quadratic Lyapunov function V 1 (η) = η T P η where P = P T > 0 is obtained from the matrix inequality [START_REF] Oliveira | Global and exact hosm differentiator with dynamic gains for output-feedback sliding mode control[END_REF]. The time derivative of η with e 1 = 0 is given by η = ψ 1 (e 1 ) ė1 ė2

T = ψ 1 (e 1 ) m(q 1 ) A η + B δ ψ 1 (e 1 ) (12)
where ψ 1 (e 1 ) denotes the derivative with respect to e 1 , ψ 2 (e 1 ) = ψ 1 (e 1 ) ψ 1 (e 1 ), A = -k 1 1

-k 2 0 and B = 0 1 .

The time derivative of V 1 along trajectories [START_REF] Moreno | Approximate Observer Error Linearization by Dissipativity Methods[END_REF]:

V1 (η) = ηT P η + η T P η = ψ 1 (e 1 ) m(q 1 ) η T (A T P + P A) η + η T P B δ ψ 1 (e 1 ) + δ ψ 1 (e 1 )
B T Pη = ψ 1 (e 1 ) m(q 1 ) η δ ψ 1 (e 1 ) T A T P + P A P B B T P 0

η δ ψ 1 (e 1 ) (13)
Note that nonlinear correction term ψ 2 defined in (6) satisfies

|ψ 2 (e 1 )| = e 1 0 α 2 2 + 3 α β 2 |e 1 | 1 2 + β 2 |e 1 | + 3 2 γ 2 |e 1 | 2 +2 α γ|e 1 | + 5 2 β γ|e 1 | 3 2 ≥ α 2 2 which implies δ ψ 1 (e 1 ) 2 = δ ψ 2 (e 1 ) ψ 1 (e 1 ) 2 ≤ 2 L δ α 2 2 ψ 2 1 (e 1 ) (14)
with |δ | ≤ L δ . The inequality ( 14) can be written as a matrix inequality in terms of η and δ

ψ 1 (e 1 ) 0 ≤ η δ ψ 1 (e 1 ) T C T L 2 δ C 0 0 -α 2 2 2 η δ ψ 1 (e 1 ) . (15)
Adding the inequality [START_REF] Astolfi | A globally exponentially convergent immersion and invariance speed observer for mechanical systems with non-holonomic constraints[END_REF] in the time derivative of Lyapunov function V 1 (η), we obtain

V1 (η) ≤ ψ 1 (e 1 ) m(q 1 ) η δ ψ 1 (e 1 ) T A T P+P A+C T L 2 δ C P B B T P -α 2 2 2 η δ ψ 1 (e 1 ) ≤ -ρ ψ 1 (e 1 ) m(q 1 ) V 1 (η) (from (7)) ≤ - ρ m(q 1 ) α 2 |e 1 | -1 2 + β V 1 (η). (16)
Note that

α |e 1 | 1 2 ≤| e 1 0 (α |e 1 | 1 2 + β |e 1 | + γ |e 1 | 3 2 )| ≤|ψ 1 (e 1 )| ≤ η ≤ V 1 2 1 (η) λ min (P) (17)
holds, which is given from [START_REF] Hautus | Strong detectability and observers[END_REF] and implies in ( 16) the next inequality holds

V1 (η) ≤ - ρ α 2 λ min (P) 2 m(q 1 ) V 1 2 1 (η) - ρ β m(q 1 ) V 1 (η) (18)
Note that trajectories of the estimation error dynamics cannot stay on the set

S = {(e 1 , e 2 ) ∈ R 2 \ {0}|e 1 = 0}. This means V 1 is
v = -γ 1 v 1/2 -γ 2 v is given by v(t) = exp(-γ 2 t) v 1/2 0 + γ 1 γ 2 (1 -exp( γ 2 2 t
)) , we can conclude that the equilibrium point (e 1 , e 2 ) = 0 is globally finite-time stable.

B. Fixed-time observer

In this section, consider the observer (4) with the nonlinear functions ψ 1 and ψ 2 given in ( 5) and ( 6) respectively, where α, γ > 0 and β = 0 for simplicity of the proof, which is inspired in the Lyapunov function proposed in [START_REF] Cruz-Zavala | Uniform robust exact differentiator[END_REF].

Theorem 7: If γ > 0 and k 1 , k 2 , α are selected such that ( 7) is satisfied, the system ( 4) is a global fixed-time observer for the system (3) in the sense that the estimation states q1 and q2 converge uniformly w.r.t. the initial condition and in fixed-time to the states q 1 and q 2 of system [START_REF] Levant | Robust exact differentiation via sliding mode technique[END_REF].

Proof: Consider a robust Lyapunov-Like function defined as

V 2 (e) = a |e 1 | 3 -b e 1 e 2 4 3 + c |e 2 | 2 .
where a, b and c are positive.

It is necessary to show that V 2 is positive definite and V2 is negative definite. For this, we use the following form inequalities, derived from the Young's inequality, Lemma 3:

(i) |e 1 ||e 2 | 4 3 ≤ 1 3 ω 3 0 |e 1 | 3 + 2 3 ω -3 2 0 |e 2 | 2 , ∀ω 0 > 0, (ii) |e 1 | 1 2 |e 2 | 4 3 ≤ 1 5 ω 5 1 |e 1 | 5 2 + 4 5 ω -5 4 1 |e 2 | 5 3 , ∀ω 1 > 0, (iii) |e 1 | 3 2 |e 2 | 4 3 ≤ 3 7 ω 7 3 2 |e 1 | 7 2 + 4 7 ω -7 4 2 |e 2 | 7 3 , ∀ω 2 > 0, (iv) |e 1 | |e 2 | 1 3 ≤ 2 3 ω 3 2 3 |e 1 | 3 2 + 1 3 ω -3 4 |e 2 |, ∀ω 3 > 0, (v) |e 1 | 2 |e 2 | 1 3 ≤ 4 5 ω 5 4 4 |e 1 | 5 2 + 1 5 ω -5 4 |e 2 | 5 3 , ∀ω 4 > 0, (vi) |e 1 | 3 |e 2 | 1 3 ≤ 6 7 ω 7 6 5 |e 1 | 7 2 + 1 7 ω -7 5 |e 2 | 7 3 , ∀ω 5 > 0, (vii) |e 1 | |e 2 | ≤ 2 5 ω 5 2 6 |e 1 | 5 2 + 3 5 ω -5 3 6 |e 2 | 5 3 , ∀ω 6 > 0.
From inequality (i), the function V 2 satisfies

ã1 |e 1 | 3 + ã2 |e 2 | 2 ≤ V 2 (e) ≤ b1 |e 1 | 3 + b2 |e 1 | 2 (19)

3

. Therefore,

V 2 > 0 if a > b 3 ω 3 0 and c > 2 3 bω -3 2
0 . The derivative of V 2 along the trajectories of ( 11) is given by

V2 (e) = 1 m(q 1 ) -3 a k 1 α 2 |e 1 | 5/2 -3 a α k 1 γ |e 1 | 7/2 -b |e 2 | 7 3 + 3 a α e 1 2 e 2 + bk 1 α e 1 1 2 e 2 4 3 + b k 1 γ e 1 3 2 e 2 4 3 + 2 3 b k 2 α 2 |e 1 | |e 2 | 1 3 + 8 3 b k 2 α γ |e 1 | 2 |e 2 | 1 3 + 2 b k 2 γ 2 |e 1 | 3 |e 2 | 1 3 -c k 2 α 2 e 1 0 e 2 -4 c k 2 α γ e 1 e 2 -3 c k 2 γ 2 e 1 2 e 2 - 4 3 b δ e 1 e 2 1 3 + 2 c δ e 2
which can be bounded using absolute value as follows, where a is selected as a = c k 2 γ 2 α for canceling the term e 1 2 e 2 and c is selected sufficiently large: where

V2 (e) ≤ 1 m(q 1 ) -3 c k 1 k 2 αγ 2 |e 1 | 5 2 -3 c k 1 k 2 γ 3 |e 1 | 7 2 -b |e 2 | 7 3 + b k 1 α |e 1 | 1 2 |e 2 | 4 3 + b k 1 γ |e 1 | 3 2 |e 2 | 4 3 + 2 3 b k 2 α 2 + 4 3 b L δ |e 1 | |e 2 | 1 3 + 8 3 b k 2 α γ |e 1 | 2 |e 2 | 1 3 + 2 b k 2 γ 2 |e 1 | 3 |e 2 | 1 3 + (c k 2 α 2 + 2 c L δ )|e 2 | +4 c k 2 α γ |e 1 | |e 2 |} .
ϑ 1 =3 c k 1 k 2 γ 3 - 3 7 b k 1 γ ω 7 3 2 - 12 7 b k 2 γ 2 ω 7 6
5 , ,

ϑ 2 =b - 4 7 b k 1 γ ω -7 4 2 - 2 7 b k 2 γ 2 ω -7 5 , ϑ 3 = max{ς 3 , ς3 }, ϑ 4 = max{ς 4 , ς4 }, ς 3 = -c k 2 α γ 3 k 1 γ - 4 5 ω 5 2 6 + 1 5 b k 1 α ω 5 1 + 16
ς 4 = 4 9 b k 2 α 2 + 2 b L δ ω 3 2 3 , ς4 =c(k 2 α 2 + 2 L δ ) + 2 9 b(k 2 α 2 + 2 L δ )ω -3 4 .
We recall the fundamental inequality proposed in [25, Thm. 16, Section 2.9] 2 (e),

( σ |x 1 | s 1 + (1 -σ )|x 2 | s 1 ) 1 s 1 ≤ ( σ |x 1 | s 2 + (1 -σ )|x 2 | s 2 ) 1 s 2 (20) 0 < σ < 1, ∀x ∈ R 2 , 0 < s 1 ≤ s 2 .
for all e

∈ C = e ∈ R 2 | e h ≥ ϑ 6 + ϑ 2 6 +4(1-σ )ϑ 7 2(1-σ )
and 6 , which can be bounded independently of the initial condition V 2 (e 2 (0)), i.e. T k ≤ 6 ϑ 8 k- 1 6 . This implies that the trajectories of the system (11) reach the compact set C = e ∈ R 2 | e h ≥ k in a time bounded by 6 ϑ 8 k- 1 6 independently of the initial condition. The time to reach e = 0 with initial conditions in C can be bounded by the compactness of C . Thus, we can conclude that the equilibrium point (e 1 , e 2 ) = 0 is globally fixed-time stable.

ϑ 8 = 2 -1 6 σ ϑ 5 c-7 6 with c = max{ b1 , b2 }. Since the solution of the differential equation v = -γ v 7 6 is given by v(t) = (v -1 6 0 + γ 6 t) -6 , then, from (21) and compari- son principle, V 2 (e(t)) ≤ (V 2 (e(0)) -1 6 + ϑ 8 6 t) -6 which implies V 2 (e(t)) ≤ V 2 (e(0)). Put V 2 (e(t)) = k ≥ ϑ 6 + ϑ 2 6 +4(1-σ )ϑ 7 2(1-σ ) , the time T k to go from V 2 (e(0)) to k is bounded by 6 ϑ 8 k-1 6 -V 2 (e(0)) - 1

V. SIMULATION EXAMPLE

Consider an inverted pendulum on a cart which is adapted from [START_REF] Sieber | Extending the permissible control loop latency for the controlled inverted pendulum[END_REF] 1 -3 ε 4 cos 2 (q) q + 3 ε 8 q2 sin(2 q)

sin(q)u cos(q) + δ = 0.

(

) 22
where m and M are the pendulum mass and cart mass respectively, and ε = m m+M is the relative mass, u is the control law that is the horizontal driving force and δ represents the disturbance. Assumption (A-1) is satisfied with ς m = 1 -3 ε and ς M = 1, (A-2) can be straightforwardly verified from trigonometric properties.

For the numerical results reported bellow, the following parameters were chosen solving the matrix inequality [START_REF] Oliveira | Global and exact hosm differentiator with dynamic gains for output-feedback sliding mode control[END_REF] with disturbance bound L δ = 0.5: P = 0.78 -0.4 -0.4 0.37 , A = (ug(q 1 ) + 1.87 ψ 2 (q 1 -q1 ))

-1 m( q1 ) c( q1 ) q2 2
where m(q 1 ) = 1 -3 ε 4 cos 2 (q 1 ), m( q1 ) = 1 -3 ε 4 cos 2 ( q1 ), c( q1 ) = 3 ε 8 sin(2 q 1 ), ψ 1 (e 1 ) = 1.1 e 1 2 + e 1 . Figure 2. presents the simulation results for finite-time observer for initial conditions q(0) = 0.5 and q(0) = 5. Note that the considered mechanical system is unstable, but the convergence of the observed state to the real state of the mechanical system is guaranteed in a finite-time (less than 2 seconds).

VI. CONCLUSION

In this paper, we propose a global sliding mode observer for a class of mechanical systems with Coriolis and centrifugal forces and disturbances. The proposed observer does not require of a prior transformation and provides finite-time and fixed-time convergence to the position and velocity of the mechanical system. BIBS stability of mechanical system is not required for the observer design, and its gains can be obtained from a feasible matrix inequality. Future work will focus on extending the approach to other types of specifications and to other types scenarios such as time-delay in the framework of Multiplicity-Induced-Dominancy (MID) properties [START_REF] Benarab | The mid property for a second-order neutral time-delay differential equation[END_REF].

1 m(q 1 ) -ϑ 1 |e 1 | 7 2 -ϑ 2 |e 2 | 7 3 + ϑ 3 (|e 1 | 5 2 + |e 2 | 5 3 ) +ϑ 4 (|e 1 | 3 2+

 117735343 From inequalities (ii)-(vii), the previous inequality leads to the next inequality V2 (e) ≤ |e 2 |)

15 b k 2 α γ ω 5

 5

h + ϑ 6 e 5 2 h + ϑ 7 e 3 2 h= -σ ϑ 5 e 7 2 h -e 3 2 h ( 1 -σ )ϑ 5 e 2 h -ϑ 6 e h -ϑ 7 where 0 7 2 + |e 2 | 7 3 27 and ϑ 5 = min{ϑ 1 , ϑ 2 }, ϑ 6 = 2 2 7 ϑ 3 , ϑ 7 = 2 4 7 ϑ 4 .

 22221270735122344 < σ < 1, e h = |e 1 | We can ensure that: V2 (e) ≤ -σ ϑ 5 e

-1.956 1 - 1 .Fig. 1 . 1 √

 1111 Fig. 1. Inverted pendulum on a cart.

1 2 + e 1 and ψ 2 (e 1

 121

Fig. 2 .

 2 Fig.2. Position, Velocity and Errors of system[START_REF] Filippov | Differential Equations with Discontinuous Righthand Sides: Control Systems (Mathematics and its Applications)[END_REF]


Thomas Orriere
email: thomas.orriere@univ-poitiers.fr

Darwin Kurniawan

Yi-Chen Chang

David Z Pai
email: david.pai@univ-poitiers.fr

Wei-Hung Chiang
email: whchiang@mail.ntust.edu.tw

Effect of

come

Introduction

As a zero-dimensional carbon nanomaterial, graphene quantum dots (GQDs) have gained enormous popularity owing to their tunable and stable photoluminescence properties, biocompatibility, and chemical stability.

Numerous GQD-based applications have been reported, including bioimaging, chemical and biological sensing, drug delivery, catalysis, LEDs, and energy-related applications (Zheng et al. 2015, Yan et al. 2019, Shen et al. 2012, Kumar et al. 2020). The synthesis of GQDs is classified into top-down and bottom-up methods. The former involves exfoliation from bulk carbon materials into GQDs which is excellent for large scale production. However, it is difficult to obtain a well-controlled structure and requires the usage of toxic and hazardous chemicals, along with long reaction time and harsh conditions (Kumar et al. 2020, Pan et al. 2010, He et al. 2018, Valappil et al. 2017). Meanwhile, the latter provides ease of tunability though carbonization, pyrolysis or step-wise chemical reaction pathways, but usually suffers from low production yield, long reaction time, and tedious synthesis procedures (Kumar et al. 2020, Valappil et al. 2017, Dong et al. 2012). Among these methods, bottom-up synthesis atmospheric microplasma jets is promising because of its scalability and simplicity [START_REF] Chiang | [END_REF]. Usually, an aqueous solution containing the precursors is exposed to a plasma jet supplied with an inert gas such as argon or helium. The plasma is generated between the jet and the liquid by applying high voltage.

When the plasma is geometrically confined under the mm-scale, it obtains useful properties such as low energy consumption, low gas heating, stability and higher reactive species densities. This special class of discharge is called the microplasmas [START_REF] Chiang | Advanced Materials[END_REF]. The ignited plasmas help accelerate chemical reactions and promote processes which are usually difficult to achieve [START_REF] Chiang | Advanced Materials[END_REF].

With these advantages, microplasmas have been exploited to synthesized GQDs or carbon dots (CDs) with tunable properties using simple carbon-containing precursors (Kurniawan & Chiang 2020, Yang, Pai & Chiang 2019, Wang et al. 2015, Joffrion et al. 2019, Ma et al. 2019). GQDs with a 4.9 nm size have been produced by treating an organosulfate aqueous solution by an atmospheric microplasma jet in ambient air (Yang, Pai & Chiang 2019). DC negative voltage was applied to the jet. By adding H 2 O 2 , which is an electron scavenger (Rumbach et al. 2015), the authors showed that solvated electrons play a major part in GQD formation and the possible mechanism was discussed. During the synthesis, initially activated dodecanol molecules and related moieties were formed due to the SDS molecules interacting with the electrons and the radicals produced by the plasma. Thereafter, the activated species were cyclized and subsequently reconstructed to form nanographenelike structures, yielding to GQDs. However, faster production of carbon dots from ethylenediamine occurs when applying positive rather than negative DC voltage to the jet (Wang et al. 2015). In electrochemistry, the term "plasma anode" often refers to applying a positive polarity on the jet (as in (Wang et al. 2015)). In this paper, we use the term positive discharge (PD) when applying a positive voltage and negative discharge (ND) when applying a negative voltage to avoid any confusion with the plasma cathode or plasma anode. As pointed by Wang et al. (Wang et al. 2015), comparisons of PD and ND are rarely reported, even though in some cases the PD shows faster and more energy efficient nanoparticle synthesis. For silver nanoparticle synthesis, Shirai et al. (Shirai et al. 2014) showed a faster synthesis by using the ND whereas gold nanoparticles were more easily produced by the PD. The aim of this paper is to compare the production of GQDs from Sodium Dodecyl Sulfate (SDS) aqueous solution with PD and ND. In the PD experiment, the metal tube is the anode of the discharge. In the ND experiment, the solution is the anode. We used electrical measurements with passive probes and optical emission spectroscopy to monitor the deposited power and the emitted light from the discharge during the experiment. By examining photoluminescence and UV-visible absorption spectra of our samples, we show a faster production of the GQDs with PD. Additionally, the GQD structure was studied by employing transmission electron microscopy (TEM) and atomic force microscopy (AFM).

Experimental setup

The experimental setup is shown on figure 1. A DC power supply (Stanford Research Systems Model PS 350) is connected via a ballast resistor R 1 = 250 kΩ to a capillary tube with an inner diameter of 180 µm. Argon gas was flow through the tube at a flow rate of 25 sccm. The 10 mL solution contained DI Water and SDS (Sigma Aldrich, CAS number 151-21-3) at a concentration of 16 mM. The 10 mL solution was placed into a 20 mL glass beaker and grounded via a shunt resistor R shunt = 1 kΩ in series with a Pt electrode. For synthesis experiments, the grounded electrode was a DC plate immersed 1 mm in depth into the solution, but for the electrical measurements, we used a Pt wire with a similar cross-sectional area of immersion. The gap distance between the liquid and the tube was about 1 mm. For the power supply, the voltage was set to ± 2 kV while the current was maintained at 5 mA for both PD and ND. These are averaged values, but time-resolved voltage and current waveforms were recorded with an oscilloscope (Lecroy Waverunner 204Mxi 2 GHz bandwidth) using a passive high voltage probe (Lecroy 400 MHz bandwidth) and a low-inductance currentsensing resistor (Caddock), respectively. Transmission electron microscopy (TEM) measurements were performed using a field emission gun TEM (FEI TecnaiTM G2 F-20 S-TWIN) with an accelerating voltage of 200 kV. TEM samples were prepared from GQDs solution dry-casting onto formvar/carbon-coated copper grids (300 mesh, EM Resolutions Ltd.). Digital Micrograph 3.7 software was utilized to obtain the particles size distributions. The topographic heights of the synthesized GQDs were measured by an Atomic Force Microscope (Ardic Instrument P-100 AFM) in tapping mode. AFM samples were prepared from GQDs solution dry-casting onto a mica disc (9.9 mm diameter, Ted Pella Inc.). Ardic Instrument image processing software was used to obtain the particles height distributions.

Results

Electrical characterization

The voltage was measured at the steel capillary tube, representing the potential over the plasma and the solution. For both PD and ND, the current supply was maintained at 5 mA during at least 50 minutes and seemed stable to the naked eye and on the DC power supply current monitor. Occasionally the PD extinguished and needed to be reignited manually. We report on figure 2a an example of the time resolved voltage, current and power at t = 30 min after initiating the discharge, for both polarities. The main discrepancy between the the two discharges is that the current, voltage and power fluctuate more for the PD. The voltage oscillates mostly between 0.4 and 0.7 kV, the current between 2 and 8 mA and the power between 0.7 and 4 W. We also observed brief extinctions of the discharge where the current drops to zero and the voltage can rise up to 1.4 kV as shown in figure 2a between t = 7 µs and t = 13 µs. When the discharge reignites itself, current and power peaks reach amplitudes of 50 mA and 35 mW respectively. These strong fluctuations are limited by the discharge circuit including the DC power supply.

We recorded the electrical signals during time windows of 0.5 ms duration every 20 s to monitor the fluctuations over the course of synthesis. The standard deviation of the current, shown on figure 2b, increases with time up to values on the order of the averaged current for the PD, but never exceeds 1% of the average for the ND. The height of the solution changes with time. Therefore, we monitored this level and added DI water to maintain a gap distance of about 1 mm. This issue is more severe in the PD case. As a result, the voltage slowly increases with experiment time and quickly decreases when DI water is added (figure 2b). The rise of the standard deviation indicates that the discharge evolves from a DC glow like discharge to a pulsed glow like regime and/or a self pulsing regime. The ND, however, is a true DC discharge. For the same gap distance, higher applied voltage is required for the PD to maintain the same current. Therefore, a higher total power is dissipated in the discharge and the liquid for the PD ( 2.5 W) than for the ND ( 1.8 W).

Discharge emission

In this section, we describe the light emission of the ND and the PD. Imaging of the PD and ND enabled control of the solution level in the beaker. The PD is always brighter than the ND. The integrated emission intensities have been computed from the images of the ND and PD and are shown on figure 3a relative to its initial value. The intensity of the ND decays significantly with time, unlike the PD.

The emission has also been monitored by optical emission spectroscopy (OES). As mentioned previously, the PD requires manual re-ignition at later times, probably when the solution evaporates or becomes too conductive. To mitigate this problem, for OES measurements H 2 O 2 (an electron scavenger) was added in the solution at a concentration low enough (0.005 mM) not to affect the UV-vis spectra. Timeintegrated spectra in both cases, shown in figure 3b, are largely similar but exhibit several discrepancies. For the PD, the atomic argon lines have higher relative intensities, and hydrogen atom balmer line H α can be observed. On the other hand, the relative intensity of molecular bands such as OH(A → X), N O(A → X) and N 2 (B → A) are higher for the ND. For the ND, the total intensity as well as the individual intensities of the N 2 bands and the Ar lines start to decrease after 15 minutes. The emission spectra exhibits the same decay discrepancy as the integrated emission, shown on figure 3a. The decay is slower for the PD, and the balance between the Ar and N 2 emission changes. The OES data were obtained without control of the gap distance, showing that the time evolution of discharge emission is robust. The discharge emission and the electrical characterization showed that the PD is unstable and needs to be reignited. Furthermore, the reignition is harder with the PD than for the ND. Increasing the resistance between the jet and the power supply can make ignition of the PD easier. Also, the addition of H 2 O 2 stabilizes the discharge while HCl destabilizes it.

GQDs properties

We used UV-Vis absorbance to characterize the effect of polarity as well as the addition of H 2 O 2 at a concentration of 0.05 mM on GQD production for 1 hour of synthesis time. The absorbance spectra for PD and ND, shown on figure 4(a), both indicate similar profiles with strong UV absorption and a shoulder positioned between 250 nm and 300 nm. These features are usually associated with the π → π* and n → π* transitions of aromatic C=C structures. The highest production yield was obtained with the PD without H 2 O 2 . Both the polarity and the presence of H 2 O 2 have an influence. The yield of the GQDs synthesized from both cases were quantified by drying the purified GQDs to obtain GQD powder. Then, different amounts of GQD powder were weighed and dissolved in 1 mL of DI water, and the absorbance of the solution with different GQD contents was measured using a UV-Vis spectrometer. A standard concentration calibration line was constructed by plotting the concentration versus absorbance , from which the GQD concentration can be quantified and the production yield can be estimated. The production yield of the ND is only 20.37% while the PD can offer almost 4 times higher, 79.32%. The deposited energy per unit mass m (J.g -1 ) produced can be calculated by equation 1 with the averaged power P (W ) (see section 3.1), the synthesis time T (s), the volume of the solution V s (L), the molarity c s (M ) and the molar mass of SDS M SDS (g.mol -1 ). The PD offers an efficiency almost three times higher (246 J.mg -1 ) than the ND (691 J.mg -1 ). The energy cost using our microplasma reactor appears low compared to graphene synthesis using an Ar plasma torch (Dato et al. 2008), where 250 W of microwave power produced 2 mg/min of carbon products, equivalent to an energy cost of 7500 J/mg.

m = T × P yield × V s × c s × M SDS (1)
Raman spectra summarized on figure 4(b) confirmed the synthesis of GQDs. The three large peaks positioned at 1350 cm -1 , 1590 cm -1 and 2700 cm -1 could be assigned respectively to the D, G and 2D bands of graphene (Yang, Li, Lu, Yu, Huo, Man, Pan, Si, Jiang & Zhang 2019, Kim et al. 2013). Also plotted on the figure is the Raman spectrum of the precursor (SDS). Therefore, the peaks positioned at 1290 cm -1 , 1440 cm -1 , 2840 cm -1 and 2880 cm -1 found in the spectra of the synthesized products belong to the precursor. For both PD and ND, the aliphatic hydrocarbon of SDS has been transformed into graphene-like structures. For the ND, the SDS remains on the sample while it is barely observable for the PD, indicating a higher conversion rate. Two types of spectra were identified from the sample produced by the PD. The first contains broad D, G and 2D peaks as observed with the ND (dark grey on figure 4(b)). The intensity ratio is I D /I G = 1.22 compared to I D /I G = 1.27 for the ND. The second, occurring more rarely, possesses sharp D and G bands (green on figure 4(b)). According to Ferrari et al. (Ferrari et al. 2006), the latter can be attributed to a structure of several layers of graphene due to the shoulder observable on the 2D peak at 2700 cm -1 . The inhomogeneous result of the Raman data could be due to the sample preparation.

Representative photoluminescence (PL) spectra of the product from the PD and the ND, respectively illustrated by figures 4(c) and (d), indicate a broadband emission when the samples were excited in the 300 to 500 nm range. The maximum PL intensity of the ND sample is found at excitation and emission wavelengths of 330 nm and 422 nm, respectively. A similar result is obtained for the PD but the PL center is red shifted to 380 nm excitation and 470 nm emission, indicating larger produced GQDs compared to the ND experiment. The quantum yield of the as-produced GQDs was estimated by collecting the PL emission spectra of various GQD dilutions. Identical measurements were performed using a calibration standard of quinine sulfate in 0.5M H 2 SO 4 with a known efficiency of 54%. We estimate the quantum yield of the GQDs to be around 1%, which is comparable to that of carbon-based dots prepared by plasma methods (Wang et al. 2015, Huang et al. 2015).

TEM measurements were performed to understand the morphologies of the GQDs synthesized from both PD and ND. Particle-like nanostructures with no aggregation nor amorphous coatings can be observed in the PD case, as shown in figure 5(a). In contrast, most of the particles in the ND case are aggregated and enveloped by amorphous coatings (figure 5(a)). We attribute this amorphous coating to unreacted SDS or some intermediate by-products that bond with the functional groups of GQDs. These observations strongly support our findings where the plasma is brighter in the PD case (figure 3), providing higher energy to break almost all of the SDS moieties to form GQDs, as supported by the Raman measurements in figure 4. The crystalline structure of the synthesized GQDs from both cases was confirmed from the fast Fourier transform (FFT) images (figure 5c). High-resolution TEM images reveal GQDs lattice spacings of 0.28 and 0.24 nm, corresponding to the ( 002) and ( 1120) planes of graphite (Kumar et al. 2018) and graphene (Kurniawan & Chiang 2020) respectively for the PD case, as depicted in figure 5(c). It is difficult to obtain an accurate lattice spacing for the ND case due to particle agglomeration. The corresponding size distribution histogram in figure 5(d) shows average sizes of 3.79±0.73 nm and 4.32±0.81 nm based on standard normal distribution fitting over 40 and 25 different particles for PD and ND cases, respectively. The larger GQD size for the ND case may be due to particle aggregation, and the actual particle size can be much smaller since the maximum emission of PL is blue-shifted relative to the PD case. Moreover, Raman spectroscopy was also performed to study the product quality. In general, for carbon materials, the D and G bands are respectively assigned to the presence of disorders in carbon systems and to the inplane vibration of the CC bond, respectively (Dresselhaus et al. 2005). The Raman spectra of both the ND and PD cases show broad D and G peaks, similar to those found by (Kim et al. 2013) for GQDs. However, the PD cases also show sharp peaks suggesting the presence of an additional species of GQD with higher crystallinity, similar to the Raman spectra of GQDs shown by (Pan et al. 2010, Peng et al. 2012, Zhuo et al. 2012). Generally, the value of the intensity ratio of D band to G band (I D /I G ) is used to estimate the degree of disorder and defects of carbon materials (Dresselhaus et al. 2005). The lower I D /I G in the green curve suggests product with higher crystallinity or less defects are existed in the product. Overall, the TEM and Raman data suggest good quality of the GQD for PD case.

AFM was conducted to identify the topographic height of GQDs synthesized from both cases. The AFM images (figure 6 corresponding histogram (figure 6(c)) was fitted with a normal distribution, yielding average heights of 4.49±2.21 nm and 6.00±2.35 nm to as-synthesized GQDs produced by PD and ND, respectively. These results indicate that the structure of synthesized GQDs for both cases consists of a few layers of graphene, consistent with our previous result (Kurniawan & Chiang 2020). As with TEM measurements (figure 5(d)), it is also important to note that some aggregation and particle envelopment are observed in the AFM image for the ND case (figure 6c). Therefore, the actual value of the average topographic height of individual GQDs synthesized by ND may be shorter than in the PD case.

Discussion

In this section, we discuss all the results we presented in the previous sections in order to understand the mechanism for GQD production from the SDS precursor, particularly with regards to the voltage polarity. We discuss now all the results in order to understand the mechanism for GQD production from the SDS precursor particularly with regards to the voltage polarity. We first consider a straightforward explanation for the higher production by the PD discharge based on our electrical measurements in section 3.1. At equivalent gap distance, the power and the voltage applied on steel capillary are higher for the PD discharge. However, our UV absorption measurements showed an absorbance nearly 3 time higher for the PD than for the ND, whereas the power was only 40% higher. Therefore, increased power by itself cannot explain the higher production rate of the PD. Furthermore Yang et al. (Yang, Pai & Chiang 2019) showed a linearly increasing production rate with increasing current or synthesis time using ND. This points to total charge injection as a key parameter in growth. Yet, in our case PD and ND production rates are quite different despite identical currents. However charge injection remains relevant, as demonstrated by the effect of adding H 2 O 2 . The suppression of GQD growth by using H 2 O 2 as an electron scavenger for a ND has been demonstrated previously (Yang, Pai & Chiang 2019).

In both cases H 2 O 2 scavenging decreases GQD production by similar amounts according to the decrease in absorbance observed upon adding H 2 O 2 . Thus, even when high-energy plasma electrons are injected into the liquid (in the case of ND), their contribution to synthesis is in the form of charge rather than energy. Their energy is lost to some process that is not that critical to synthesis. However charge injection alone does not produce GQDs. We performed additional experiments for both PD and ND in which the plasma is replaced by a Pt wire just touching the solution, i.e. conventional electrolysis. Applying voltage to the Pt wire such that 5 mA ran through the solution did not result in GQD synthesis.

Next, we consider how the experiment evolves with time. The time-resolved emission of the ND during 60 minutes, presented previously, shows a sharp decrease of the emitted light from N 2 (C → B) and the Ar lines, implying that the fluxes of the charged species, radicals, and photons also decrease with time. However, the input power, voltage and current are stable. It is also known that GQD production scales linearly with time and current (Yang, Pai & Chiang 2019). As mentioned previously, current generation alone without plasma does not result in synthesis. Therefore, the variation of the plasma properties for the ND have a minor effect on the growth rate, but the plasma is needed to start the reaction by fragmenting or transforming the SDS with radicals, UV photons or charged species which are known to interact with the liquid (Bruggeman et al. 2016). The reaction is completed by a phenomenon dependent on charge injection, perhaps related to electrolysis.

The sustained strength of the PD over time compared to the ND would also imply more conversion of SDS. The Raman spectra indicate that the PD completely destroyed the SDS in the solution because no peaks from SDS remain after synthesis. The Raman spectrum obtained from the sample treated by the ND clearly shows some SDS peaks. The enhanced plasma properties of the PD may more effectively fragment SDS, as indicated by its more complete destruction, leading to the higher GQD production yield observed.

The PD may also create more favorable transport in the liquid. The PD may electrostatically attract the SDS to the area affected by the plasma and thus easily destroy the SDS. Visual inspection of bubble movement in the liquid suggests greater convection in PD than ND, possibly due to electrohydrodynamic forces generated by the plasma and applied field. Shirai et al.(Shirai et al. 2014) suspected the influence of ion drift in the liquid, when using a dual plasma reactor (with both polarities) to produce either silver or gold nanoparticles.

Conclusion

We have reported on the synthesis of GQDs from aqueous solution by a DC argon plasma jet at atmospheric pressure, focusing on the effect of the voltage polarity on the jet. The positive discharge (PD) is more effective for producing the GQDs, though less stable. PD stability can be improved by controlling the gap distance during the synthesis, and by adding H 2 O 2 at concentrations as low as 0.005 mM. The light intensity decays during the first 30 minutes of synthesis with the ND, while this decay is rather slow for the PD. The first step for GQD synthesis is the fragmentation of the SDS, and the positive polarity has a greater ability for this purpose. Greater fragmentation of the PD may be due to more favorable plasma parameters (higher fluxes of photons, reactive species and ions, higher power and voltage) or indirectly via transport (electrohydrodynamic phenomenon, electrostatic drift). The second step appears to involve charge injection. The production yield increases with time and current yet is suppressed with the addition of H 2 O 2 , indicating that this electrolysis-related second step remains essential for both polarities. Finally, this study demonstrates that solvated electrons generated during the plasma interaction with liquid are not the only reactive species for GQD synthesis for the PD case because the electron flux for the PD is direct away from the liquid. Further experimental and modeling studies are necessary to provide more fundamental understanding.

Figure 1 :

 1 Figure 1: Experimental setup of the microplasma jet (PD). The DC power supply is reversed for the ND.

Figure 2 :Figure 3 :

 23 Figure 2: (a) Typical voltage, current and power waveforms for t = 30 min and (b) Evolution of the mean (darker shades) and the standard deviation (lighter shades) of the voltage V and the current I as a function of the experiment time.

Figure 4 :

 4 Figure 4: (a) Comparison of the UV/vis absorption spectra of the product obtained with the PD and the ND in pure SDS solution (darker shade) and with the addition of H 2 O 2 (0.05 mM, dashed lighter shade). (b) Raman spectra of the SDS solution, as well as the products from ND and PD synthesis. (c) Normalized photoluminescence colormap of the product after synthesis by using the PD and (d) the ND.

Figure 5 :

 5 Figure 5: TEM images of the synthesized GQDs (a) by the PD (low resolution), (b) by the ND (low and high resolution) and (c) high resolution images of the PD product with the highlighted lattice spacing and corresponding fast Fourier transform. (d) The corresponding histogram obtained with 40 and 25 samples for the PD and the ND, respectively.

  (a) for the PD and figure 6(b)) indicate quite uniform GQDs heights have been synthesized by atmospheric microplasma. The

Figure 6 :

 6 Figure 6: AFM images of the synthesized GQDs (a) by the PD and (b) by the ND with (c) the corresponding histogram obtained with 100 and 45 samples for the PD and the ND respectively.

Acknowledgements

This work was supported by the Partenariat Hubert Curien program Orchid (40938YL), the Ministry of Science and Technology of Taiwan (MOST Grant no. MOST 107-2628-E-011-002-MY3), National Taiwan University of Science and Technology (NTUST), the Agence Nationale de la Recherche program JCJC PLASMAFACE (ANR-15-CE06-0007-01), the Investissements dAvenir program LABEX INTERACTIFS (ANR-11-LABX-0017-01) of the French government, and the CPER-FEDER program of the Rgion Nouvelle Aquitaine.

The data that support the findings of this study are available from the corresponding author upon reasonable request.