Monika Dörfler

Peter Balazs

Florent Jaillet

Removing Components from a Time-Frequency Representation

ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Introduction

Time-frequency representations such as the spectrogram or short-time Fourier transform seem to be well suited to the task of removing certain components with approximately disjoint support in the time-frequency domain. For example, one might be interested in suppressing a certain instrument's contribution from a music signal. Such approaches are used in Computational Auditory Scene Analysis by the name of Time-Frequency masks. However, the trivial approach of just deleting the corresponding component in the time-frequency representation leads to artifacts such as "ghost-tones", musical noise and "phasing effects". We compare different approaches to tackle this problem: To gain some insight in the nature of the problem, we consider the optimal separating mask for known signal components. A softthresholding procedure is applied and compared to a model promoting sparsity in the representation. The latter can lead to favorable results by yielding a feasible approximation of the optimal mask. The results are presented visually and acoustically during the presentation.

Time-frequency representations

Time-frequency analysis aims at providing simultaneous information on a signal's time-and frequency-content. In order to clarify this idea, time-frequency representations are often compared to a music-score, which, in fact, very efficiently conveys the information which frequency, or rather pitch, should sound at which instant. For the mathematical understanding, however, this comparison may be misleading, as, according to Heisenberg's uncertainty principle, an exact separation of signal components in the time-frequency domain ia not possible. In particular, when it comes to removing certain components from a signal's time-frequency representation, it is usually not easy to delete the components' contribution to the timefrequency coefficients, for several reasons. First of all, no window function can be band limited and have a compact support at the same time. This statement is equivalent to saying that no ideal low-pass FIR filter can exist. Secondly, the phase information of the time-frequency representation seems to be of fundamental importance, yet is not fully understood, also see [START_REF] Jaillet | On the structure of the phase around the zeros of the short-time Fourier transform[END_REF] in this volume. We next proceed to define the basic mathematical notions. Given a discrete sequence of real or complex numbers, x[n], n ∈ Z, as well as a, usually compactly supported, window function ϕ[n], n ∈ Z, the short-time Fourier transform (or STFT) of x[n] is given, for k ∈ Z and ω ∈ [-0.5, 0.5] by

V ϕ x(k, ω) = ∞ n=-∞ x[n]ϕ[n -k]e -2πiωn . (1)
Now, in practice, a subsampled version of (1) will usually be applied. Also, since the window ϕ has finite length l, we deal with a finite number of frequency bins. Hence, the result of the sampled STFT, also called Gabor transform, [START_REF] Feichtinger | Gabor Analysis and Algorithms[END_REF], is a matrix of size N × M , where N is the number of time shifts by a timeconstant, or hop-size, a considered. M is the number of frequency bins, hence the length of the FFT, given by l/b, b being the frequency-shift constant. Under certain conditions, usually fulfilled in practice, Gabor theory yields a convenient reconstruction method by using either a dual window in the synthesis step or a tight window for both analysis and synthesis, see [START_REF] Dörfler | Time-frequency Analysis for Music Signals. A Mathematical Approach[END_REF] for the corresponding formulas. Writing φ for the dual window, we obtain the following perfect reconstruction formula:

x[n] = N k=1 M m=1 V ϕ x(ka, mb) φ[n -ka]e 2πinm M . (2)
In fact, the Gabor transform is commonly used in acoustics, but often in a disguised form, called sliding window transform, which, in fact, equals the classical Gabor transform up to a phase factor, see [START_REF] Dörfler | Time-frequency Analysis for Music Signals. A Mathematical Approach[END_REF]. In applications, the obtained coefficients are usually modified before reconstruction, yielding an output of the form:

Gx[n] = N k=1 M m=1 m(m, k)V ϕ x(ka, mb) φ[n -ka]e 2πinm M , (3)
where m denotes the mask which is applied to the coefficients.

From a mathematical point of view, this approach generates the interesting situation, that we reconstruct a signal from coefficients which are not the canonical coefficients of any existing original signal. Figure 1 illustrates this effect. It is easily seen that even in this fairly simple synthesized example, the separation of signal (chirp) and noise is not as easy as it might seem at first sight. The next section shows how optimal reconstruction is obtained in this situation.

The optimal mask

When trying to design an appropriate mask for signal separation, we have to keep in mind that phase plays a central role in the interpretation of (sampled) short-time Fourier transforms. In particular, if we know the signals at hand, we may calculate an optimal, though highly signal dependent mask as follows. Let us assume that we know (a good estimate of) our noise signal n, the signal of interest is denoted by x and x = x + n is the observed signal. Then, if we use, whenever

|V ϕ x(ka, mb)| 2 > 0 for all k, m m opt (k, m) = V ϕ x(ka, mb) • V ϕ x(ka, mb) |V ϕ x(ka, mb)| 2 , (4)
we obtain, by a straight-forward multiplication

V ϕ x(ka, mb) = m opt (m, k) • V ϕ x(ka, mb). (5)
Obviously, this observation is of minor practical interest, as the signal x may not be observed directly. However, we may take a look at the behavior of m opt in order to gain a better understanding of the process of masking in the time-frequency domain.

We consider the synthesized signal from Figure 1 again, this time calculating the optimal mask according to (4) and we look at the inverse effect, see Figure 2.

Cutting out the chirp according to the mask depicted in Figure 1, the "common" approach to masking out components, we look at the Gabor coefficient of the resulting reconstruction, second display. The third display, then, shows the modulus of the optimal mask and the Gabor coefficients of the resulting signal, pure noise, can be seen in the last display. Interestingly, if the specific realization of the noise changes, the resulting noise-signal still does not reveal the original (chirp) signal, as opposed to the result from applying a "common" masking procedure. 1 As stated above, using an "optimal" mask, can hardly be an approach of practical interest. However, it seems reasonable to reconsider the classical approach of changing only the modulus of the Gabor coefficients without touching the phase. As a starting point, we suggest extensive studies on the combination and separation of simple signals in order to achieve a better

1 Audio examples will be provided during the presentation. understanding of masking of time-frequency components.

It is interesting to note, that the coefficients obtained by optimal masking according to (4) yield the Gabor coefficients of some signal. 2

Soft thresholding

For denoising, i.e. the separation of noise and signal components, in the frequency domain, various models based on a statistical approach have been suggested, compare [START_REF] Godsill | Digital Audio Restoration[END_REF]. Here, by using an estimate of the power spectrum S n (k, m) of the noise for each frequency bin k and every time-point m, different soft-thresholding procedures are realized. For simplicity of notation, let us write, V ϕ x(k, m) = V ϕ x(ka, mb). A suppression rule based on a "pseudo-Wiener" noise reduction is given by:

V ϕ x(k, m) = ρ(k, m) • V ϕ x(k, m), (6)
where

ρ(k, m) = |Vϕ x(k,m)| 2 -Sn(k,m) |Vϕ x(k,m)| 2 , |V ϕ x(k, m)| 2 > S n (k, m) 0 else (7)
Variants of the above formula exist, leading, e.g., to the power subtraction method, if ρ(k, m) is used in place of ρ(k, m). We display the result of denoising according to ( 6) and ( 7) in Figure 3. In this experiment, we consider a music signal (piano, bass, drums), synthetically corrupted by Gaussian white noise. We estimate the average noise level S n from the Gabor coefficients of n and ρ(k, m) is then obtained directly from the Gabor coefficients of the noisy signal. The first display of Figure 3 shows the Gabor coefficients of the noisy signal (redundancy is 4), while the coefficients selected by the thresholding procedure can be seen in the second display. The last display points out once more, that the Gabor coefficients of the reconstructed signal are not equal to the masked coefficients. It is also obvious and well-known, that one of the most common problems related to thresholding methods is the regular occurrence of artifacts, most notably, of musical noise, which arises from the randomness inherent in the estimation of the signal power spectrum used. Several advanced suppression rules have been suggested to tackle this problem, see [START_REF] Wolfe | Efficient alternatives to the Ephraim and Malah suppression rule for audio signal enhancement[END_REF][START_REF] Godsill | Digital Audio Restoration[END_REF] and references therein.

Sparse representations

We next suggest a novel approach to the separation of various signal components in the time-frequency domain. Partly, the problems arise from the fact that coefficients belonging to a particular, possibly isolated or well-concentrated signal component is smeared as a consequence of the analysis window's properties. in this way, components which are expected to be separated in time-frequency may have essentially over-lapping Gabor coefficients. On the other hand, if we are convinced, that the signal components of interest have a sparse, at least approximative, representation in the atomic systems we use, then we may avoid the problem by looking for relevant coefficients only. A sparse approximation has a small number of nonzero elements, while still giving a satisfying representation and reconstruction of a signal or a certain signal component. One way to enforce sparsity is to choose an expansion of x such that as many coefficients as possible are zero. Mathematically, however, minimization of an ℓ 1 -constraint on the coefficients yields explicit solutions and fast algorithms as well as similar solutions. 3 In the present situation, we are going to minimize the following expression:

∆(x) = k,m c k,m φkm -x 2 2 + µ c ℓ 1 (8)
where φkm = φ[n -ka]e 2πinm M and c ℓ 1 = k,m |c k,m | is the ℓ 1 -norm of the coefficient sequence. The solution can be found by an iterative algorithm called Landweber iterations [START_REF] Daubechies | An iterative thresholding algorithm for linear inverse problems with a sparsity constraint[END_REF]. We next present the results of two experiments. First, we consider a synthetic signal comprised of two sinusoids with frequencies 1300Hz and 1400Hz, given a sampling rate of 8192. We use a Gaussian window of 400 samples length and calculate the Gabor coefficients, shown in the first display of Figure 4. The second display, then, shows the coefficients resulting from ℓ 1 -penalization on the expansion coefficients according to [START_REF] Jaillet | On the structure of the phase around the zeros of the short-time Fourier transform[END_REF]. It is immediately obvious, that the algorithm visually separates the two signal components. Note that reconstruction from these coefficients is possible, but not perfect, as may be seen in Figure 5. In fact, we face a trade-off between accuracy of reconstruction and sparsity of representation, which is reflected in the choice of µ in [START_REF] Jaillet | On the structure of the phase around the zeros of the short-time Fourier transform[END_REF]. However, the sparse representation allows the separation of signal components by designing a mask that suppresses undesired signal components more easily than the usual, more redundant representation. For comparison, we also apply the Landweber iteration algorithm to the audio signal considered in the previous section. The results are displayed in Figure 6. Quite obviously, we encounter problems similar to those resulting from soft-thresholding. However, in the present approach, we have not made any a priori assumptions about the signal, in particular, about the statistical properties of the noise. We point out, that sparsity approaches bear the potential of more sophisticated

Figure 1 :

 1 Figure 1: Gabor coefficients of a synthesized signal (chirp + noise). It can be seen that the reconstruction introduces smearing due to the spread of the window's FFT

Figure 2 :

 2 Figure 2: Gabor coefficients of a synthesized signal (chirp + noise), top left, of a signal obtained from commonly used masking as in Figure 1, top right, (modulus of) optimal mask, bottom left, and Gabor coefficients of resulting signal, bottom right.

FrequencyFigure 3 :

 3 Figure 3: Denoising by soft thresholding.

Figure 4 :

 4 Figure 4: Gabor coefficients and sparse representation of two close sinusoids

As mentioned before, this is by no means obvious for a general mask applied to Gabor coefficients, as the coefficients have to satisfy a reproducing kernel equation, see[START_REF] Dörfler | Representation of operators in the time-frequency domain and generalized Gabor multipliers[END_REF] for details.

Note, that it has been proved that certain situations ℓ 1minimization in fact yields the optimally sparse solution, see[START_REF] Chen | Atomic decomposition by Basis Pursuit[END_REF].

Acknowledgments

M.Dörfler and F.Jaillet were funded by project MA07-025 of WWTF Austria. We wish to thank Bruno Torrésani for fruitful discussions on the present topic.

penalization terms leading to results better adapted to the signals at hand, compare [START_REF] Kowalski | Sparsity and persistence: mixed norms provide simple signal models with dependent coefficients[END_REF]. Also note that the resulting coefficients heavily depend on the choice of µ, which is usually a difficult task.

Gabor transform of noisy signal

Summary and Outlook

We summarized and compared various approaches to the problem of separating possibly over-lapping components in the time-frequency domain. While (soft) thresholding is a commonly used approach to tackle this task, we also mentioned signal separation by means of a sparsity approach. We believe, that the latter bears the potential to perform better than thresholding procedures in certain situations. Note that so far we have not performed any (pre-)denoising on the signal. Also, in order to achieve favorable performance, the procedure has to be further refined. An additional problem connected to sparsity methods is the slow convergence of the iterative algorithms involved. Alleviations have been suggested, see, e.g. [START_REF] Daubechies | Accelerated projected gradient methods for linear inverse problems with sparsity constraints[END_REF]. We also suggested to look at the optimal mask for separation of signal components, in order to gain insight in the nature of masking procedures in the timefrequency domain. We pointed out that, in contrast to common practice, approaches involving complex-valued masks may be well worth considering.


Samir Adly
email: samir.adly@unilim.fr

Ba Khiet Le

Sliding Mode Observer for Set-valued Lur'e Systems and Chattering Removing

Keywords: Sliding mode observer, set-valued Lur'e systems, chattering effect AMS Subject Classification. 28B05, 34A36, 34A60, 49J52, 49J53, 93D20

In this paper, we study a sliding mode observer for a class of set-valued Lur'e systems subject to uncertainties. We show the well-posedness of the problem and highlight the clear advantages of our approach over the existing Luenberger-like observers. Furthermore, we provide a new continuous approximation to remove the chattering effect in the sliding mode technique. Some numerical examples are given to illustrate our theoretical approach.

Introduction

Hybrid systems are a class of dynamic systems that may present both continuous and discrete behavior. They are characterized by the presence of continuous state variables, inputs/outputs, as well as discrete state variables, inputs/outputs. On the other hand, Lur'e-type nonlinear systems [START_REF] Lur'e | On the theory of stability of control systems[END_REF] are represented by the combination of a linear time-invariant system and a memoryless nonlinearity through a feedback connection. The link between Lur'e systems and hybrid systems lies in the coupling of the continuous and discrete dynamics through the nonlinear state equation and the linear output equation. This coupling makes Lur'e systems well-suited for modeling complex physical, biological, or social systems that exhibit both continuous and discrete behavior, and have found applications in areas such as control systems, signal processing, and system identification. For a comprehensive guide to hybrid dynamical systems from the modeling, stability and robustness point of view, we refer to [START_REF] Goebel | Hybrid dynamical systems. Modeling, stability, and robustness[END_REF]. In this paper we focus on Lur'e set-valued dynamical systems where the nonlinear feedback is given by a set-valued relation. The Lur'e set-valued dynamical system is a widely studied model in applied mathematics and control theory. In recent decades, it has received significant attention, as evidenced by the numerous studies found in references such as [START_REF] Acary | Nonsmooth Modeling and Simulation for Switched Circuits[END_REF][START_REF] Adly | Nonsmooth Lur'e Dynamical Systems in Hilbert Spaces[END_REF][START_REF] Adly | Maximal Monotonicity and Cyclic-Monotonicity Arising in Nonsmooth Lur'e Dynamical Systems[END_REF][START_REF] Brogliato | Absolute stability and the Lagrange-Dirichlet theorem with monotone multivalued mappings[END_REF][START_REF] Brogliato | Well-posedness, stability and invariance results for a class of multivalued Lur'e dynamical systems[END_REF][START_REF] Brogliato | Dissipative Systems Analysis and Control[END_REF][START_REF] Brogliato | Observer Design for Lur'e Systems With Multivalued Mappings: A Passivity Approach[END_REF][START_REF] Brogliato | Dynamical Systems Coupled with Monotone Set-Valued Operators: Formalisms, Applications, Well-Posedness, and Stability[END_REF][START_REF] Camlibel | Linear passive systems and maximal monotone mappings[END_REF][START_REF] Huang | Adaptive full-order and reduced-order observers for the Lur'e differential inclusion system[END_REF][START_REF] Huang | Actuator fault detection and estimation for the Lur'e differential inclusion system[END_REF][START_REF] Huang | H ∞ Observer design for singular one-sided Lur'e differential inclusion system[END_REF][START_REF] Le | On a class of Lur'e dynamical systems with state-dependent set-valued feedback[END_REF][START_REF] Le | Well-posedness and nonsmooth Lyapunov pairs for state-dependent maximal monotone differential inclusions[END_REF][START_REF] Le | Sliding Mode Observers for Time-Dependent Set-Valued Lur'e Systems Subject to Uncertainties[END_REF][START_REF] Tanwani | Stability and observer design for Lur'e systems with multivalued, non-monotone, time-varying nonlinearities and state jumps[END_REF]. Despite advancements in the understanding of the system's existence and uniqueness, stability and asymptotic analysis, the design of control and observers still poses many interesting open questions. Specifically, most observers for Lur'e set-valued systems follow the Luenberger design, which has limitations when the system is subjected to uncertainty, as discussed in references such as [START_REF] Huang | Adaptive full-order and reduced-order observers for the Lur'e differential inclusion system[END_REF][START_REF] Huang | Actuator fault detection and estimation for the Lur'e differential inclusion system[END_REF][START_REF] Huang | H ∞ Observer design for singular one-sided Lur'e differential inclusion system[END_REF][START_REF] Tanwani | Stability and observer design for Lur'e systems with multivalued, non-monotone, time-varying nonlinearities and state jumps[END_REF]. Recently, using the powerful sliding mode technique, B. K. Le in [START_REF] Le | Sliding Mode Observers for Time-Dependent Set-Valued Lur'e Systems Subject to Uncertainties[END_REF] proposed a sliding mode observer for a general class of set-valued Lur'e systems subject to uncertainties as follows

                          
ẋ(t) = Ax(t) + Bλ(t) + Eu(t) + Gξ(t, u, x) for a.e. t ∈ [0, +∞), w(t) = Cx(t) + Dλ(t), λ(t) ∈ -F t (w(t)), t ≥ 0,

y(t) = F x(t), x(0) = x 0 , (1)
where

A ∈ R n×n , B ∈ R n×m , C ∈ R m×n , D ∈ R m×m , E ∈ R n×l , F ∈ R p×n , G ∈ R n×k
are given matrices, λ and w are two connecting variables, u is the control input, y is the physically measurable output and ξ is some uncertainty. The set-valued operator F t : R m ⇒ R m is time-dependent and assumed to be maximal monotone. Note that (1) can be rewritten into a first order time-dependent differential inclusion as follows

ẋ ∈ Ax -B(F -1 t + D) -1 Cx + Eu + Gξ, x(0) = x 0 . (2)
If D = 0 and F t = F, we obtain the following classical case ẋ ∈ Ax -BFCx + Eu + Gξ.

The proposed sliding mode observer for (1) is

                     ẋ ∈ Ax + B λ -Le y + Eu -P -1 F T Ψ(e y ), w = C x + D λ, λ ∈ -F( w + Ke y ), ỹ = F x, (3)
where e = xx, e y = ỹy = F e, Ψ(e y ) = σ 1 e y + ( J ρ(t, u, y) + σ 2 )Sign(e y ) and ρ is a bound of the uncertainty for some suitable matrix J and real number σ 1 ≥ 0, σ 2 > 0.

Then under some mild conditions, the observer state converges to the original state asymptotically and the observation error converges to zero in finite time (we refer to [START_REF] Le | Sliding Mode Observers for Time-Dependent Set-Valued Lur'e Systems Subject to Uncertainties[END_REF] for more details).

In this paper, our first contribution is to provide a sliding mode observer for the system considered in [START_REF] Huang | Adaptive full-order and reduced-order observers for the Lur'e differential inclusion system[END_REF] as follows

           ẋ = Ax + Bω + f 1 (x, u) + f 2 (x, u)θ(t), ω ∈ -F(Cx), y = F x, (4)
where

A ∈ R n×n , B ∈ R n×m , C ∈ R m×n , F ∈ R p×n are given matrix, x ∈ R n is the state, F : R m ⇒
R m is a maximal monotone operator, u ∈ R r is the control input and y ∈ R p is the measurable output. The functions f 1 , f 2 are known smooth while θ ∈ R l is an unknown constant parameter vector. An adaptive Luenberger-like observer was proposed by Huang et al in [START_REF] Huang | Adaptive full-order and reduced-order observers for the Lur'e differential inclusion system[END_REF]. In our paper, the parameter θ may not always be constant, such as in the case of perturbations. Our sliding mode approach is efficient and straightforward, without the need to solve an additional ordinary differential equation, as seen in [START_REF] Huang | Adaptive full-order and reduced-order observers for the Lur'e differential inclusion system[END_REF]. Our results show that the convergence of the observer state to the actual state is exponential, while the rate of convergence in [START_REF] Huang | Adaptive full-order and reduced-order observers for the Lur'e differential inclusion system[END_REF] is unknown. Additionally, if the matrix function f 2 is bounded, the conditions for solvability of the associated LMI are significantly improved, leading to finite time convergence of the observation error under more favorable assumptions than in [START_REF] Le | Sliding Mode Observers for Time-Dependent Set-Valued Lur'e Systems Subject to Uncertainties[END_REF]. Our contributions also extend to the reduced-order case, where we have improved conditions. Lastly, we propose a new smooth approximation of the sliding mode technique that removes the chattering effect while still effectively managing uncertainty.

The paper is organized as follows. In Section 2, we revisit some key concepts. In Section 3, we establish the well-posedness of the problem, addressing the existence and uniqueness of a solution. Additionally, we propose a sliding mode observer for (4) and show its significant advantages over the Luenberger-like observer discussed in [START_REF] Huang | Adaptive full-order and reduced-order observers for the Lur'e differential inclusion system[END_REF]. Section 4 focuses on the reduced-order observer. In Section 5, we present a new and refined version of the sliding mode technique. Numerical examples to support the results are provided in Section 6. Finally, Section 7 concludes the paper and highlights future possibilities.

Notations and mathematical background

We denote the scalar product and the corresponding norm of Euclidean spaces by •, • and • respectively. A matrix P ∈ R n×n is called positive definite, written P > 0, if there exists a > 0 such that

P x, x ≥ a x 2 , ∀ x ∈ R n .
The sign and Sign functions in R n are defined by sign(x) = (sign(x 1 ), sign(x 2 ), . . . , sign(x n ))

and

Sign(x) =    x x if x = 0 B if x = 0,
where B denotes the unit ball in R n . Both functions coincides in R, which are widely used in sliding mode technique (see, e.g., [START_REF] Orlov | Nonsmooth Lyapunov Analysis in Finite and Infinite Dimensions[END_REF][START_REF] Shtessel | Sliding Mode Control and Observation[END_REF] and the references therein). Sign function is usually used in Walcott and Zak observer (see, e.g., [START_REF] Spurgeon | Sliding mode observers -a survey[END_REF][START_REF] Walcott | State observation of nonlinear uncertain dynamical systems[END_REF][START_REF] Xiang | On the design of Walcott-Zak sliding mode observer[END_REF]). It seems to be inherited from the Filippov approach to the meaning of differential equations with discontinuous right-hand side.

x Sign(x)

1 1 0 Figure 1: Sign function in R A set-valued mapping F : R m ⇒ R m is called monotone if for all x, y ∈ R m and x * ∈ F(x), y * ∈ F(y), one has x * -y * , x -y ≥ 0.
Furthermore, F is called maximal monotone if there is no monotone operator G such that the graph of F is contained strictly in the graph of G.

Lemma 1. [START_REF] Le | Sliding Mode Observers for Time-Dependent Set-Valued Lur'e Systems Subject to Uncertainties[END_REF] Let F ∈ R p×n be a full row rank matrix (p ≤ n) and P ∈ R n×n be a symmetric positive definite matrix. If x ∈ im(P -1 F T ) then

F T (F P -1 F T ) -1 F x = P x, (5)
where im(A) denotes the range of A.

Finally, let us recall the concept of orbital derivative. Let x : [0, +∞) → R n be an absolutely continuous function, V : R n → R and W (t) := V (x(t)). Then the orbital derivative of V along x(•) is defined by: V = dW dt .

3 Well-posedness and the convergence analysis

In this section, we will propose a sliding mode observer for the system (4) under the following assumptions:

Assumption 1: The set-valued operator F : R n ⇒ R n is a monotone, upper semi-continuous with non-empty, closed convex and bounded values.

Assumption 2: The continuous functions f 1 : R n × R r → R n and f 2 : R n × R r → R n×l are Lipschitz continuous w.r.t x, i. e., there exist L 1 > 0 and L 2 > 0 such that for all x 1 , x 2 ∈ R n , u ∈ R r , we have

f 1 (x 1 , u) -f 1 (x 2 , u) ≤ L 1 x 1 -x 2 and f 2 (x 1 , u) -f 2 (x 2 , u) ≤ L 2 x 1 -x 2 .
Assumption 3: The unknown θ(t) is continuous and bounded by L 3 > 0.

Assumption 4:

Let γ = L 1 + L 2 L 3 . There exist > 0, P ∈ R n×n > 0, L ∈ R n×p , K ∈ R m×p and the matrix function h : R n × R r → R l×p such that P (A -LF ) + (A -LF ) T P + γP 2 + γI + I ≤ 0, (6)
B T P = C -KF , (7)
f T 2 (x, u)P = h(x, u)F . ( 8
)
Assumption 4': the same to Assumption 4 but with γ is replaced by γ = L 1 .

Remark 1. i) From ( 8), we have

f 2 (x, u) = P -1 F T h T (x, u). It means that im(f 2 (x, u)) ∈ im(P -1 F T ) for all (x, u).
ii) Assumption 4 is widely used in the Control literature, which is based on the passivity and the matching property of disturbances (see, e.g., [START_REF] Brogliato | Observer Design for Lur'e Systems With Multivalued Mappings: A Passivity Approach[END_REF][START_REF] Huang | Adaptive full-order and reduced-order observers for the Lur'e differential inclusion system[END_REF][START_REF] Le | Sliding Mode Observers for Time-Dependent Set-Valued Lur'e Systems Subject to Uncertainties[END_REF]).

Lemma 2. Let F ∈ R p×n be a matrix with full row rank. If f 2 is bounded, then h is also bounded. Furthermore, if f 2 is Lipschitz continuous with respect to x then h is also Lipschitz continuous with respect to x.

Proof. From Remark 1, we imply that h(x, u) = (F P -1 F T ) -1 F f 2 (x, u) and the condition follows.

The proposed sliding mode observer for (4) is

           ẋ = Ax + B ω -Le y + f 1 (x, u) -βP -1 F T h(x, u) Sign(e y ), ω ∈ -F(C x -Ke y ), ỹ = F x, (9)
where β ≥ L 3 . In the following theorem, we establish the well-posedness of the problem by proving the existence and uniqueness of solutions for both the original system (4) and the sliding mode observer (9), under Assumptions 1-4.

Theorem 3. Suppose Assumptions 1-4 are satisfied. Then, the following statements hold: (i) The original system given by ( 4) and the sliding mode observer given by ( 9) both possess solutions.

(ii) If F is a matrix of full row rank, then the solution to the sliding mode observer ( 9) is unique.

(iii) If there exists a symmetric matrix Q > 0 such that B T Q = C, then the solutions of ( 4) is unique.

Proof. (i) The existence of solutions of both systems follows immediately since both systems can be reduced into first order differential inclusions where the right-hand side operators are upper semi-continuous and have non-empty, convex and compact values (see for instance [START_REF] Aubin | Differential Inclusions: Set-valued Maps and Viability Theory[END_REF][START_REF] Brezis | Opérateurs Maximaux Monotones et Semi-groupes de Contractions dans les Espaces de Hilbert[END_REF][START_REF] Huang | Adaptive full-order and reduced-order observers for the Lur'e differential inclusion system[END_REF]).

(ii) Let x1 , x2 be two solutions of ( 9) and x is a solution of (4). Let ẽ(t

) := x1 (t) -x2 (t), ẽy1 := F (x 1 -x), ẽy2 := F (x 2 -x). Consider the Lyapunov function V 1 (ẽ) = 1
2 P ẽ, ẽ and compute its orbital derivative

V1 = P ẽ, ė = P ẽ, Aẽ + B(ω 1 -ω2 ) + f 1 (x 1 , u) -f 1 (x 2 , u) -βP -1 F T ( h(x 1 , u) Sign(ẽ y1 ) -h(x 2 , u) Sign(ẽ y2 ))
The following inequalities hold true

P ẽ, B(ω 1 -ω2 ) = B T P ẽ, ω1 -ω2 = (C -KF )ẽ, ω1 -ω2 ≤ 0, P ẽ, Aẽ + f 1 (x 1 , u) -f 1 (x 2 , u) ≤ ( P A + L 1 ) ẽ 2 .
Further calculations yield

Γ := -P ẽ, P -1 F T ( h(x 1 , u) Sign(ẽ y1 ) -h(x 2 , u) Sign(ẽ y2 )) = -F ẽ, h(x 1 , u) Sign(ẽ y1 ) -h(x 2 , u) Sign(ẽ y2 ) = -h(x 1 , u) F ẽ, Sign(ẽ y1 ) -Sign(ẽ y2 ) -F ẽ, ( h(x 1 , u) -h(x 2 , u) )Sign(ẽ y2 ) ≤ F L h ẽ 2 ,
where L h is the Lipschitz constant of h (Lemma 2) and we have used the monotonicity of the Sign function. Therefore V1 ≤ ρ 1 ẽ 2 where ρ 1 := P A + L 1 + L h F , allowing us to draw the conclusion by using the Gronwall's inequality.

(iii) Let x 1 and x 2 be two solutions of (4). Define the error e(t) as e(t) := x 1 (t)x 2 (t). We choose the Lyapunov function V 2 (e) = 1 2 Qe, e and compute its orbital derivative

V2 = Qe, ė = Qe, Ae + B(ω 1 -ω 2 ) + f 1 (x 1 , u) -f 1 (x 2 , u) + f 2 (x 1 , u)θ(t) -f 1 (x 2 , u)θ(t) .
By using the monotonicity of F, we obtain

Qe, B(ω 1 -ω 2 ) = B T Qe, ω 1 -ω 2 = Ce, ω 1 -ω 2 ≤ 0.
Additionally, we have the following inequalities

f 1 (x 1 , u) -f 1 (x 2 , u) ≤ L 1 e . f 2 (x 1 , u)θ(t) -f 2 (x 2 , u)θ(t) ≤ L 2 L 3 e .
Therefore, V2 ≤ ρ 2 e 2 where ρ 2 := QA + L 1 + L 2 L 3 and the conclusion follows.

The following result confirms the exponential convergence of the observer state to the original state.

Theorem 4. Let Assumptions 1-4 hold. Then the observer state x of (9) exponentially converges to the original state x of (4).

Proof. Let e = xx, e y = ỹy = F e. From ( 4) and ( 9), we have

ė ∈ (A -LF )e -B(ω -ω) + f 1 (x, u) -f 1 (x, u) -βP -1 F T h(x, u) Sign(e y ) -f 2 (x, u)θ. ( 10
)
Consider the Lyapunov function V (e) = P e, e . Then the orbital derivative of V is V (e) = 2 P ė, e = 2 P (A -LF )e -P B(ωω), e

+ 2 P (f 1 (x, u) -f 1 (x, u)), e -2β h(x, u) e y -2 P f 2 (x, u)θ, e . (11)
From ( 7) and the monotonicity of F, we have

P B(ω -ω), e = ω -ω), B T P e = ω -ω, (C -KF )e ≤ 0. ( 12
)
On the other hand

2 P (f 1 (x, u) -f 1 (x, u)), e ≤ 2L 1 e P e ≤ L 1 ( e 2 + P e 2 ) (13)
and

Y := -2β h(x, u) e y -2 P f 2 (x, u)θ, e = -2β h(x, u) e y -2 P (f 2 (x, u) -f 2 (x, u))θ, e -2 P f 2 (x, u)θ, e ≤ -2β h(x, u) e y + 2L 2 L 3 e P e -2 θ, h(x, u)e y ≤ L 2 L 3 ( e 2 + P e 2 ). (14)
Let α max and α min be the largest eigenvalue and the smallest eigenvalues of P , respectively. From ( 6), ( 11), ( 12), ( 13) and ( 14), we have

dV dt ≤ -e 2 ≤ -α max V (t).
Using Gronwall's inequality, we obtain

α min e 2 ≤ V (t) ≤ exp( -t α max )V (t 0 ).
Hence, e ≤ exp( -t 2αmax ) V (t 0 ) α min and the conclusion follows.

Remark 2. (i) According to [START_REF] Huang | Adaptive full-order and reduced-order observers for the Lur'e differential inclusion system[END_REF], under similar conditions, our implementation of a sliding mode observer leads to an exponential convergence of the observer state, whereas the adaptive observer in the same study only achieves convergence without a defined rate. Furthermore, our method avoids the need to solve a high-dimensional additional ODE, which can be computationally expensive, as seen in [START_REF] Huang | Adaptive full-order and reduced-order observers for the Lur'e differential inclusion system[END_REF].

(ii) In Assumption 4, if h is bounded, then γ = L 1 + L 2 L 3 can be replaced by just L 1 , which is usually much smaller than γ. This makes it easier to satisfy ( 6)-( 8) and enables us to solve a wider class of set-valued Lur'e systems that cannot be tackled using the method in [START_REF] Huang | Adaptive full-order and reduced-order observers for the Lur'e differential inclusion system[END_REF]. Furthermore, our technique also ensures finite time convergence of the observation error e y to zero. A bounded matrix function h can be guaranteed if f 2 is bounded and F is full row rank (Lemma 2), as one can always choose a full row rank matrix F for the output y = F x without losing any information.

Theorem 5. Let Assumptions 1, 2, 3, 4' hold and suppose that h(x, u) is upper-bounded by some constant L 4 > 0. Then we can still ensure the exponential convergence to the original state by using the observer

           x = Ax + B ω -Le y + f 1 (x, u) + βP -1 F T Sign(e y ), ω ∈ -F(C x -Ke y ), ỹ = F x ( 15
)
where β > L 3 L 4 . Furthermore, if the matrix F ∈ R p×n has full row rank and im(B) ⊂ im(P -1 F ) then the observation error e y converges in explicitly finite time.

Proof. Under the new assumption, in [START_REF] Camlibel | Linear passive systems and maximal monotone mappings[END_REF] we have

-β e y -P f 2 (x, u)θ, e = -β e y -θ, h(x, u)e y ≤ -(β -L 3 L 4 ) e y ≤ 0
and similarly one obtains the exponential convergence of the observer state.

To attain the finite time convergence of the observation error, we consider the new Lyapunov function W = 1 2 e y , (F P -1 F T ) -1 e y . Then

dW dt = (F P -1 F T ) -1 ėy , e y .
From [START_REF] Brogliato | Dissipative Systems Analysis and Control[END_REF], we have

ėy = F (A -LF )e -F B(ω -ω) + F f 1 (x, u) -f 1 (x, u) -βF P -1 F T Sign(e y ) -F f 2 (x, u)θ. (16)
Since im(B) ⊂ im(P -1 F ), using Lemma 1, we have

F T (F P -1 F T ) -1 F B = P B
and hence

(F P -1 F T ) -1 F B(ω -ω), e y = F T (F P -1 F T ) -1 F B(ω -ω), e = P B(ω -ω), e = ω -ω, B T P e = ω -ω, (C -KF )e ≤ 0.
On the other hand, from [START_REF] Brogliato | Absolute stability and the Lagrange-Dirichlet theorem with monotone multivalued mappings[END_REF] we deduce that f 2 = P -1 F T h T and thus

(F P -1 F T ) -1 F f 2 (x, u)θ = (F P -1 F T ) -1 (F P -1 F T )h T (x, u)θ = h T (x, u)θ. (17)
Since e ≤ exp( -t 2αmax ) V (t 0 ) α min as in the proof of Theorem 4, we can find some t 1 > 0 such that for all t ≥ t 1 , one has

(F P -1 F T ) -1 F (A -LF )e + F f 1 (x, u) -f 1 (x, u) ≤ σ 2 where σ = β -L 3 L 4 > 0. Indeed, we can choose t 1 := 2α max ln 2 (F P -1 F T ) -1 F ( A -LF + L 1 ) σ V (t 0 ) α min .
Then for all t ≥ t 1 , we have

dW dt ≤ σ 2 e y -(β -h T (x, u)θ ) e y ≤ σ 2 e y -(β -L 3 L 4 ) e y = - σ 2 e y ≤ -κ W (t),
where κ := σ/ √ 2γ max and γ max is the largest eigenvalue of (F P -1 F T ) -1 since e y ≥

2W (t)

γmax . Suppose that W (t) > 0 for all t ≥ t 1 then we have

W 2 √ W ≤ - κ 2
and thus

W (t) -W (t 1 ) ≤ - κ 2 (t -t 1 ) → -∞ as t → ∞,
a contradiction. Let t f ≥ t 1 be the first time such that W (t f ) = 0, then we deduce that W (t) = 0 for all t ≥ t f since W is non-negative and decreasing. It means that e y converges to 0 in finite time. Similarly as above we have

-W (t 1 ) = W (t f ) -W (t 1 ) ≤ - κ 2 (t f -t 1 )
and hence t f ≤ t 1 + 2 W (t 1 )/κ, which completes the proof of the theorem.

Remark 3. (i) Based on the observation (17), we can have a significantly better estimation for the gain β than in [START_REF] Le | Sliding Mode Observers for Time-Dependent Set-Valued Lur'e Systems Subject to Uncertainties[END_REF]. Indeed, in the current paper the gain β does not change to obtain the finite time convergence of the observation error from the exponential convergence of the observer state.

In addition, we can also provide an explicit estimation for t f . (ii) If Assumption 1 is substituted by the maximal monotone property of the set-valued F, similar results can still be obtained. The existence and uniqueness of solutions for the original system (4) remains guaranteed, as shown in references such as [START_REF] Aubin | Differential Inclusions: Set-valued Maps and Viability Theory[END_REF][START_REF] Brezis | Opérateurs Maximaux Monotones et Semi-groupes de Contractions dans les Espaces de Hilbert[END_REF]. Existence of solutions to the observer systems ( 9), ( 15) can be also obtained by rewriting these systems into the form ẋ ∈ -Ax + G(t, x) where A is a maximal monotone operator and G is a upper semi-continuous set-valued function w.r.t x with non-empty convex compact values (see, e.g., [START_REF] Azzam-Laouir | Multi-Valued Perturbations to a Couple of Differential Inclusions Governed by Maximal Monotone Operators[END_REF][START_REF] Brogliato | Observer Design for Lur'e Systems With Multivalued Mappings: A Passivity Approach[END_REF]). The uniqueness can be proved similarly as the proof of Theorem 3. It's worth mentioning that the normal cone operator N C , which is associated to a nonempty closed convex set C, is an important maximal monotone setvalued mapping in mechanical and electrical engineering that does not satisfy the boundedness requirement in Assumption 1.

Reduced-order observer

Suppose that the given matrices and matrix-functions can be decomposed as follows

x =   x 1 x 2   , A =   A 11 A 12 A 21 A 22   , B =   B 1 B 2   , C = (C 1 C 2 ), F = (F q 0) P =   P 11 P 12 P 21 P 22   , f 1 (x, u) =   f 11 (x, u) f 12 (x, u)   , f 2 (x, u) =   f 21 (x, u) f 22 (x, u)   ,
where F q ∈ R q×q is an invertible matrix and the following is satisfied:

Assumption 4": There exist > 0, Q ∈ R (n-q)×(n-q) > 0, P 22 ∈ R (n-q)×(n-q) invertible, P 21 ∈ R (n-q)×q such that

Q(A 22 + KA 12 ) + (A 22 + KA 12 ) T Q + L 1 Q(KK T + I n-q )Q +(L 1 + )I n-q ≤ 0, ( 18
) (B 2 + KB 1 ) T Q = C 2 , ( 19
)
P 21 P 22 f 2 (x, u) = 0, (20)
where K = P -1 22 P 21 .

Note that (4) can be rewritten as follows

                               ẋ1 = A 11 x 1 + A 12 x 2 + B 1 ω + f 11   x 1 x 2   , u + f 21   x 1 x 2   , u θ ẋ2 = A 21 x 1 + A 22 x 2 + B 2 ω + f 12   x 1 x 2   , u + f 22   x 1 x 2   , u θ ω ∈ -F(C 1 x 1 + C 2 x 2 ) y = F q x 1 . (21)
Using ( 20), we have

                       ż = (A 22 + KA 12 )z + (B 2 + KB 1 )ω + [(A 21 + KA 11 ) -(A 22 + KA 12 )K]x 1 + (K I n-q )f 1   x 1 z -Kx 1   , u ω ∈ -F(C 2 z + (C 1 -C 2 K)x 1 ), x 2 = z -Kx 1 . ( 22
)
The adaptive observer is

                       ż = (A 22 + KA 12 )z + (B 2 + KB 1 )ω + [(A 21 + KA 11 ) -(A 22 + KA 12 )K]x 1 + (K I n-q )f 1   x 1 z -Kx 1   , u ω ∈ -F(C 2 z + (C 1 -C 2 K)x 1 ), x2 = z -Kx 1 . ( 23
)
Theorem 6. Let Assumptions 1, 2, 3, 4" hold. Then ( 23) is a reduced-order observer of (4), i.e., lim

t→∞ (x 2 (t) -x2 (t)) = 0.
Proof. Let e z = zz. Then we have

                       ėz = (A 22 + KA 12 )e z + (B 2 + KB 1 )(ω -ω) + (K I n-q ) f 1   x 1 z -Kx 1   , u -f 1   x 1 z -Kx 1   , u ω ∈ -F(C 2 z + (C 1 -C 2 K)x 1 ). ω ∈ -F(C 2 z + (C 1 -C 2 K)x 1 ). ( 24
)
Let us consider the Lyapunov function W (e z ) = Qe z , e z , then Ẇ (e z ) = 2 Q ėz , e z . From [START_REF] Le | On a class of Lur'e dynamical systems with state-dependent set-valued feedback[END_REF] and the monotonicity of F, we have

Q(B 2 + KB 1 )(ω -ω), e z = ω -ω, (B 2 + KB 1 ) T Qe z = ω -ω, C 2 e z ≤ 0.
On the other hand

2Q(K I n-q ) f 1   x 1 z -Kx 1   , u -f 1   x 1 z -Kx 1   , u , e z ≤ 2L 1 (K I n-q ) T Qe z e z ≤ L 1 e T z Q(K I n-q )(K I n-q ) T Qe z + L 1 e T z e z ≤ L 1 e T z Q(KK T + I n-q )Qe z + L 1 e T z e z .
Combining with [START_REF] Huang | H ∞ Observer design for singular one-sided Lur'e differential inclusion system[END_REF], similarly as in the proof of Theorem 4, we have Ẇ (e z ) ≤e z 2 . It deduces that e z converges to zero exponentially and the conclusion follows.

Remark 4. (i) Note that if we have [START_REF] Brogliato | Absolute stability and the Lagrange-Dirichlet theorem with monotone multivalued mappings[END_REF], then P f 2 (x, u) = F T h(x, u) and thus P 21 P 22 f 2 (x, u) = 0. It would be interesting to improve or remove the condition [START_REF] Le | Well-posedness and nonsmooth Lyapunov pairs for state-dependent maximal monotone differential inclusions[END_REF]. (ii) When the matrices are decomposable, it is more effective to provide assumptions directly with the new lower-dimension matrices. Note that Assumption 4" is strictly weaker than Assumption 4 even when Q = P 22 [START_REF] Huang | Adaptive full-order and reduced-order observers for the Lur'e differential inclusion system[END_REF]. It is remarkable that Q > 0 can be different from P 22 and it is unnecessary to require that P > 0 but only invertible P 22 . This enhancement significantly broadens the potential applications. For simplicity one can choose P 21 = 0, which would consequently result in K = 0. It's noteworthy that in Assumption 4", only L 1 is involved, just as in Assumption 4' used for sliding mode observer [START_REF] Goebel | Hybrid dynamical systems. Modeling, stability, and robustness[END_REF]. However, one of the drawbacks of the reduced-order observer is the necessity to perform some linear transformations if F is given in a general form. Additionally, despite satisfying Assumptions 1-3 and 4", the numerical convergence of the reduced-order observer (23) may fail in certain sensitive cases, as illustrated in Example 3, due to its reliance on the approximate output x 1 from the original system. In contrast, the sliding mode observer is known for its enhanced robustness.

A new continuous approximate of the sliding mode technique

Although the sliding mode method is effective, it has a persistent issue: the chattering effect caused by the discontinuity of the Sign function. To eliminate this issue, the Sign function in R is typically approximated by the "sigmoid function" (as noted in [START_REF] Shtessel | Sliding Mode Control and Observation[END_REF])

Sign(x) ≈ x |x| +
or by [START_REF] Shokouhi | A new continuous approximation of sign function for sliding mode control[END_REF] Sign(x) ≈ x √

x 2 + for some small fixed δ > 0. Other continuous approximates can be also found in [START_REF] Shokouhi | A new continuous approximation of sign function for sliding mode control[END_REF].

× R n → R n as follows Sign δ (t, x) =              x x -1-x /δ(t) (1+M x ) N x x if x = 0 and x ≤ δ(t), x x if x > δ(t), 0 if x = 0, (25)
for some M , N > 0 and the function δ(•) is non-increasing. For example, we can choose δ(t) = exp(-k 1 tk 2 ) for some k 1 > 0, k 2 > 0. It can be seen that the Sign δ (t, x) function is a continuous function with respect to both time t and state x. When the magnitude of x is greater than δ(t), the function Sign δ (t, x) becomes equal to Sign(x). These properties make it effective in reducing the chattering effect while still handling uncertainty. In comparison, the norm of the sigmoid function

x |x|+ is always less than 1 which does not deal with the uncertainty entirely. The sigmoid function only leads to convergence of the state to an approximate region of the sliding surface, as reported, e.g., in [START_REF] Shtessel | Sliding Mode Control and Observation[END_REF]. The suitable choice of the guiding function δ can remarkably reduce the chattering effect. In practice, one can choose δ(t) = max{10 -3 , exp(-k 1 tk 2 )} to avoid very small values when t is large. This can be seen in Example 1, Figures 34567.

Numerical Examples

In this section we provide some numerical examples to show the effectiveness of our approach.

Example 1. To test the effectiveness of our new approximation method, we will analyze a simple one-dimensional system: ẋ ∈ µ -LSign(x), where µ is an uncertainty with a constraint |µ| < L.

As an example, let us set µ = 3 sin x and L = 4. It is known that the state variable x will converge to zero in a finite time. To evaluate the numerical simulations, we use the explicit scheme with an initial value of x 0 = 0.1 (see Figure 3). The Sign function creates a chattering effect (as seen in the state x 1 ). However, by replacing the Sign function with Sign δ (t, x) defined in equation ( 25), using a guiding function δ(t) = e -0.5t and M = 1, N = 3, the chattering effect can be eliminated and the convergence to zero is achieved (as seen in the state x 3 ). The performance of this approximation is better than using the sigmoid function with = 10 -3 (as seen in the state x 2 ). Similarly, we can consider the same system ẋ ∈ µ -Lu in R 3 where the control u can be sign(x), Sign(x) or certain continuous approximation. Let us consider µ = (3 2 -1) T cos t, L = 40 with the initial condition x 0 = (0.01 0.02 0.01) T . One can see in Figures 4 and5 that the Sign function reduces chattering better than the sign function since the Sign function is only discontinuous at zero. On the other hand, Sign δ (t, x) has the best performance in reducing chattering (Figure 7). It confirms that our time-dependent approximation worths considering as a good alternative besides existing fixed approximations and merits further investigations.

Example 2. Next we consider the system (4) with Suppose that the unknown θ = 3, the control input u = 8 cos t and The well-posedness of the original system and the sliding mode observer system follows (Theorem 3) since we have

A =       -11 5 0 9 -10 0 0 0 -11       , B =       2 -3 4       , f 1 (x, u) =       3u + 0.
f 2 (x, u) =       3 sin x 2 0 0       , C = 5 -3 4 , F = 1 0 0 .
F(x) =    sign(x)(2|x| + 5) if x = 0, [-5, 5] if x = 0.
B T Q = C where Q =       5/2 0 0 0 1 0 0 0 1      
Then L 1 = 0.9, L 2 = 3, L 3 = 3 and γ = 9.9. Then Assumptions 1-4 are satisfied with

P =       1 0 0 0 1 0 0 0 1       , L =       0 14 0       , = 0.2, K = 3.
We have a comparison between a standard Luenberger observer and the sliding mode observers (9), ( 15) with x 0 = (3 2 1) T , x0 = (15 -20 11) T . Both sliding mode observers has quite same performance and converge faster than the Luenberger observer obviously (Figure 8).

Example 3. Finally, we present an example to show the applicability of our approach, which cannot be achieved using the Luenberger observer mentioned in [START_REF] Huang | Adaptive full-order and reduced-order observers for the Lur'e differential inclusion system[END_REF]. Specifically, we consider the identical data as in Example 2, with the only difference being that Let us assign the following values: L 1 = 0.9, L 2 = 3, L 3 = 3, γ = 9.9 and γ = 0.9. It becomes apparent that, for any matrix L, it is impossible to satisfy the condition A -LF ≤ -1.1I. Basic calculations show that Assumption 4 cannot be fulfilled, thereby rendering the application of the Luenberger observer in [START_REF] Huang | Adaptive full-order and reduced-order observers for the Lur'e differential inclusion system[END_REF] unfeasible. However, by using the same values of P , L, , K as in Example 2, it is straightforward to verify Assumptions 1, 2, 3, 4' of Theorem 5. By using the sliding mode observer [START_REF] Goebel | Hybrid dynamical systems. Modeling, stability, and robustness[END_REF] with the gain β = 10, initializing the system with x 0 = (3 2 1) T , x0 = (15 27 16) T and employing an explicit scheme for F, Figure 9 shows the convergence of the error e = xx to zero.

Note that, Theorem 6 can be also applied with K = 0, Q = I 2 , = 0.2. The reduced-order observer [START_REF] Orlov | Nonsmooth Lyapunov Analysis in Finite and Infinite Dimensions[END_REF] becomes However, the numerical convergence of the reduced-order observer (23) fails and is explosive in this sensitive case as it relies on an approximation of x 1 from the original system. Conversely, the reduced-order observer ( 23) can be successfully applied to Example 2, primarily due to the highly negative definiteness of A.

                      

Conclusion and perspectives

In this paper, the advantages of sliding mode observers are demonstrated for set-valued Lur'e dynamical systems that are faced with uncertainties. The robustness of this observer technique is a significant factor in the analysis and control of such systems. We also present a new continuous approximation of the sliding mode technique, which provides improved performance compared to conventional methods. However, the traditional condition ( 8) is a limiting factor in the applicability. Further research is needed to explore the possibility of relaxing or removing this condition to increase the range of systems that can be analyzed. This is an area that merits further investigations and has the potential to enhance the performance of sliding mode observers for set-valued Lur'e dynamical systems.

Figure 2 :

 2 Figure 2: Sign function in R and its approximations

Figure 3 : 3 Figure 4 :

 334 Figure 3: An effective continuous approximation of Sign function in R

3 ×10 - 3 Figure 5 :

 335 Figure 5: The sytem state in R 3 using Sign function

Figure 6 :

 6 Figure 6: The sytem state in R 3 using sigmoid function with = 10 -3

3 Figure 7 :

 37 Figure 7: The sytem state in R 3 using Sign δ (t, x) with δ(t) = max{10 -3 , e -5.5t }, M = 1, N = 3

3 Figure 8 :

 38 Figure 8: Errors using a Luenberger observer (in red) and the sliding mode observer (9) (in blue)

  ω ∈ -F( -3 4 z + 5x 1 ).

3 Figure 9 :

 39 Figure9: The convergence of the sliding mode observer[START_REF] Goebel | Hybrid dynamical systems. Modeling, stability, and robustness[END_REF]

Acknowledgement. The authors would like to show their gratitude to the handling Editor and the Reviewers for their careful reading of our manuscript and for raising many interesting questions, which helps us to improve the paper significantly.


J Y Maradj

J-Y P Hoarau

B Bibérian

Aufray

H Jamgotchian
email: jamgotchian@cinam.univ-mrs.fr

B Ealet

Y Colignon

H Maradj

J-P Biberian

B Aufray

A comprehensive study of the (2 √ 3

Keywords: Silicene, Ag(111), superstructure, Moiré pattern, periodic deformation, strain relaxation PACS numbers: 62.23.Kn, 68.35.Gy, 68.37.Nq, 68.37.Ef

diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Introduction

The silicene, the counterpart of graphene for silicon is mainly obtained by epitaxial growth of silicon on oriented metallic substrates [START_REF] Kara | A review on silicene: New candidate for electronics[END_REF][START_REF] Liu | Silicene on substrates: interaction mechanism and growth behavior[END_REF]. Today, silver is still the most widely used substrate even though silicene has been also synthetized with success on Ir(111) [START_REF] Meng | Buckled silicene formation on Ir(111)[END_REF], ZrB2(111) [START_REF] Fleurence | Experimental Evidence for Epitaxial Silicene on Diboride Thin Films[END_REF] and MoS2 [START_REF] Chiappe | Two-dimensional Si nanosheets with local hexagonal structure on a MoS(2) surface[END_REF].

Silver is an appropriate substrate due to its strong tendency to undergo phase separation (no silicide formation) and to the quasi-perfect match between four silver inter-atomic distances (1.156 nm), and three silicon inter-hexagons distances (1.152 nm). On a silver substrate, the first observations of "graphite-like honeycomb structure" were obtained on Ag(100) (nano-stripes) [START_REF] Léandri | Growth of Si nanostructures on Ag(001[END_REF], then on Ag(110) (nano-ribbons) [START_REF] Aufray | Graphenelike silicon nanoribbons on Ag(110): A possible formation of silicene[END_REF] and finally on Ag(111) (silicene sheet) which is today the most studied orientation [START_REF] Lalmi | Epitaxial growth of a silicene sheet[END_REF][START_REF] Enriquez | Silicene structures on silver surfaces[END_REF][START_REF] Lin | Structure of Silicene Grown on Ag[END_REF][START_REF] Gao | Initial geometries, interaction mechanism and high stability of silicene on Ag(111) surface[END_REF][START_REF] Jamgotchian | Growth of silicene layers on Ag(111): unexpected effect of the substrate temperature[END_REF][START_REF] Vogt | Silicene: compelling experimental evidence for graphenelike twodimensional silicon[END_REF][START_REF] Chen | Spontaneous Symmetry Breaking and Dynamic Phase Transition in Monolayer Silicene Physical[END_REF][START_REF] Acun | The instability of silicene on Ag(111[END_REF][START_REF] Arafune | Structural transition of silicene on Ag(111[END_REF][START_REF] Moras | Coexistence of multiple silicene phases in silicon grown on Ag(1 1 1)[END_REF][START_REF] Liu | Various atomic structures of monolayer silicene fabricated on Ag(111[END_REF][START_REF] Sone | Epitaxial growth of silicene on ultra-thin Ag(111) films[END_REF].

With the (111) orientation, four structures are generally observed depending on growth temperature and deposition rate. We have shown that all these structures can be explained by different orientations of the silicene layer with respect to the silver substrate [START_REF] Jamgotchian | Growth of silicene layers on Ag(111): unexpected effect of the substrate temperature[END_REF]. These orientations of the silicene layer are characterized by the rotation angle (α) between the Si[110] and the Ag [110] directions. Starting from a (4x4) structure, a rotation of the silicene layer relative to the silver substrate gives rise successively to the following structures: a (√13x√13)R13.9°, a (2√3x2√3)R30°, a (√7x√7)R19.1° (which is rarely observed) [START_REF] Chiappe | Local electronic properties of corrugated silicene phases[END_REF] and finally a second (√13x√13)R13.9°. a Bulk Si (dHSi = 0.384 nm) and bulk Ag (dAg = 0.289 nm) distances are used as references.

b Length of the unit cells based on the silver structure.

c Length of the unit cells based on the bulk silicon structure.

d Strain factor.

In order to know if the silicene layer is compressed or expanded, we have calculated the strain factor LSi/LAg as shown in table 1. The ratio shows that the silicene layer should be expanded for the (4x4) and (√13x√13)R13.9°, and contracted for the (2√3x2√3)R30° structures. The formation and the existence of all these structures are strongly dependent on substrate temperature [START_REF] Jamgotchian | Growth of silicene layers on Ag(111): unexpected effect of the substrate temperature[END_REF]. The domain of existence of quasi-pure (4x4) is between ~180°C and ~220°C. In the 220°C-280°C range, there is a mix of (4x4), (√13x√13)R13.9° and (2√3x2√3)R30° structures. Above 280°C a quasi-pure (2√3x2√3)R30° is obtained. In this temperature domain, some studies have shown perfect LEED patterns of (2√3x2√3)R30° with large spots [START_REF] Jamgotchian | Growth of silicene layers on Ag(111): unexpected effect of the substrate temperature[END_REF][START_REF] Moras | Coexistence of multiple silicene phases in silicon grown on Ag(1 1 1)[END_REF]. Others have shown sharp LEED patterns of the (2√3x2√3)R30° with some additional spots which have been interpreted as a mixture of (√19x√19)R23.4° and (3.5x3.5)R26° structures [START_REF] Arafune | Structural transition of silicene on Ag(111[END_REF][START_REF] Moras | Coexistence of multiple silicene phases in silicon grown on Ag(1 1 1)[END_REF]. Recently the LEED pattern with extra spots has been interpreted as (√133x√133)R4.3° and the formation of a surface alloy [START_REF] Rahman | Growth of Si on Ag(111) and determination of large commensurate unit cell of high-temperature phase Japanese[END_REF]. However, under the same conditions, low resolution STM images show apparent ordered (2√3x2√3)R30° structures but with an average angle which can be different from the expected 30° angle [START_REF] Jamgotchian | Silicene on Ag(111) : domains and local defects of the observed superstructures[END_REF] and, sometimes, associated with a Moiré pattern [START_REF] Feng | Evidence of Silicene in Honeycomb Structures of Silicon on Ag(111[END_REF]. On high resolution STM images, the structures appear locally ordered with the expected morphology [START_REF] Enriquez | Silicene structures on silver surfaces[END_REF], but surrounded by many defects [START_REF] Jamgotchian | Silicene on Ag(111) : domains and local defects of the observed superstructures[END_REF]. One paper claims that this structure is not ordered enough to exist and then "die suddenly" [START_REF] Liu | The fate of the (2√3 × 2√3)R30° silicene phase on Ag(111)[END_REF]. However, a more recent study has shown that this structure can be used to build a Field Effect Transistor [START_REF] Tao | Silicene field-effect transistors operating at room temperature[END_REF].

The present paper is focused on the (2√3x2√3)R30° structures obtained after deposition of silicon with substrate temperatures at 390°C, 370°C and 270°C. From a detailed analysis of STM images (low and high resolutions) and from LEED observations, we propose in this paper a comprehensive atomic model of the silicene layer forming the (2√3x2√3)R30° structure. This atomic model is based on a periodic arrangement of perfect areas of (2√3x2√3)R30° surrounded by defect areas. A generalization of this model explains remarkably well the experimental observations: deviations of the average directions of the structure, Moiré patterns, and apparent global disorders.

Experimental set-up

The experiments were performed in an ultra-high vacuum (UHV) system equipped with Low Energy Diffraction (LEED), Auger Electron Spectroscopy (AES) and Room Temperature Scanning Tunneling Microscopy (Omicron RT-STM 1). The protocol used for the Ag(111) surface preparation is the usual one [START_REF] Jamgotchian | Growth of silicene layers on Ag(111): unexpected effect of the substrate temperature[END_REF]. Silicon deposition is performed by thermal evaporation from a silicon wafer heated by the Joule effect. The monolayer (ML) coverage is checked by AES. The sample is heated by a tungsten resistor located under the sample holder. Because the observed structures are very sensitive to the substrate temperature and since the thermocouple is located close to the sample holder, the temperature has been recalibrated using the melting temperature of two metals (Zn and Pb), positioned in place of the Ag sample. The possible drift of the STM images is corrected using the hexagonal symmetry of the surface checked by FFT. Such a correction increases the precision of the angle measurements.

Experimental results

The first silicon deposition has been performed with a substrate temperature at 390°C. The low resolution STM image is shown in figure 1 direction of silver and the white lines to the direction of the perfect (2√3x2√3)R30° structure. The average orientation (22±1°) of the structure (red line) is far from the expected directions of either the (√13x√13)R13.9° or the (2√3x2√3)R30° structures (blue and white lines respectively). From this type of image, it is not possible to identify the structure. Furthermore, the periodicity i.e. the distance between two black holes, is smaller (0.96 ± 0.05 nm) than the expected ones (1.001 nm and 1.042 nm for (2√3x2√3)R30° and (√13x√13)R13.9° respectively -table 1). Interestingly, the FFT of the STM

Periodic local relaxation model

These three experiments show that a small variation of growth temperature has a large effect on the observed superstructures of the silicene layer characterized by LEED and STM. In this section, we show that all these structures are constituted of perfect (2√3x2√3)R30° domains separated by defect areas. Only the average size and the size distribution of the ordered domains depend on growth temperature. We propose an atomic model, which explains in a coherent manner our experimental observations as well as those of other experimentalists.

Presentation of the model

The perfect (2√3x2√3)R30° structure presents two domains (α = ±10.9°). These two domains exist on the silver surface but cannot be distinguished by STM since only the top silicon atoms are visible [START_REF] Jamgotchian | Growth of silicene layers on Ag(111): unexpected effect of the substrate temperature[END_REF][START_REF] Liu | Various atomic structures of monolayer silicene fabricated on Ag(111[END_REF][START_REF] Feng | Evidence of Silicene in Honeycomb Structures of Silicon on Ag(111[END_REF][START_REF] Pflugradt | Silicene-derived phases on Ag(111) substrate versus coverage: Ab initio studies[END_REF]. Nevertheless, as shown above in the previous section, on the observed structures, the angles are different from the expected ones. In the following, we detail the defects at the atomic scale, interpreting them by local deformation/relaxation of the silicene layer.

The geometrical atomic model proposed to explain the above results is based on the fact that on silver substrates, only the silicon atoms on top sites are visible in STM images and on the hypotheses that the silicene layer keeps a honeycomb structure. In addition, we assume that the silver substrate remains rigid under the silicene layer (no deformation). To build up the model, we start from the high resolution STM image of the figure 2(a). The large regular hexagons are perfect (2√3x2√3)R30° areas, in agreement with theoretical studies [START_REF] Enriquez | Silicene structures on silver surfaces[END_REF][START_REF] Pflugradt | Silicene-derived phases on Ag(111) substrate versus coverage: Ab initio studies[END_REF][START_REF] Guo | Absence and presence of Dirac electrons in silicene on substrates[END_REF][START_REF] Kaltsas | Structural evolution of single-layer films during deposition of silicon on silver: a first-principles study[END_REF]. Each of them is constituted of seven silicene hexagons.

In defect areas, the large hexagons, which are highlighted in green and in pink in Note that the atomic model proposed does not predict that the defect areas must appear darker than the perfect areas as observed on the STM image of figure 1. This suggests that the silicon atoms in the defect areas are not located exactly on top of silver atoms but close to them, inducing darker areas.

A dynamic analysis of the LEED patterns for such large superstructures being almost impossible to calculate, we used a simple kinematic approach, equivalent to a Patterson function [START_REF] Patterson | A Fourier series method for the determination of the components of interatomic distances in crystals[END_REF][START_REF] Cunningham | Determining surface relaxation from lowenergy electron diffraction via a transform method[END_REF], which is a convolution of the (133x133)R4.3° large superstructure unit cell with the perfect small local (2√3x2√3)R30° structure. As a consequence, on the LEED patterns, only the spots of the large superstructure close to the (2√3x2√3)R30° spots will be visible [START_REF] Perdereau | LEED studies of adsorption on vicinal copper surfaces Surface[END_REF]. In fact both, the (2√3x2√3)R30°

and the (133x133)R4.3° structures are commensurable with Ag(111), but not commensurable together.

Generalization of the model

This model shows that the periodic deviation creates an average rotation of the structure which value depends directly on the size of the perfect areas. The sign of the angle of this average deviation (angle shift) and the sign of  are the same.

By changing the size of the perfect (2√3x2√3)R30° areas, the size of the new superstructure and the angle shift will change: the larger the perfect domains, the smaller the angle shift and the structure will go towards a perfect (2√3x2√3)R30° structure. In other words, the smaller the number of defects, the smaller will be the angle shift, which is always easier to measure on low-resolution STM images. On the contrary, for high-resolution STM images, it is possible to measure the quantity of defects but not the angle shift. Experimentally, the angle shift measurement allows estimating the average size of the perfect domains, which is equivalent to a statistical determination of the defects.

The model has been developed for the (133x133)R4.3° superstructure with hexagonal symmetry, constituted by seven large perfect hexagons of (2√3x2√3)R30° structure. The simplest superstructure with hexagonal symmetry is the (31x31)R8.9° superstructure constituted by one large perfect hexagon of (2√3x2√3)R30° structure. A generalization of the model allows us to predict the average angle (i.e. the angle shift) of the (2√3x2√3)R30° structure versus the distance between two perfect domains (i.e. the periodicity of the Moiré) for  < 0, and for  > 0 as it is shown on figure 7(a) (continuous lines). The (31x31)R8.9° and (133x133)R4.3° superstructures are indicated by vertical dashed lines. For an infinite number of hexagons a complete perfect (2√3x2√3)R30° structure would be formed. The sign of the angle shift is directly relative to the sign of . When two domains with < 0, and > 0 are simultaneously observed on the same STM image, the angle between both domains will be equal to twice the angle shift as it has been shown in the figure 9 of [START_REF] Jamgotchian | Silicene on Ag(111) : domains and local defects of the observed superstructures[END_REF].

Note that experimentally, for a given temperature, we observe a dispersion of the sizes of perfect areas as shown on figure 1(a) and figure 2(a). On this calculated curve, we have reported our experimental data and those issued from other studies (see discussion section). In the case of figure 3, where the Moiré pattern is not observable on the STM image, we have used the FFT, which reveals the periodicity. It is remarkable that this curve fits well all the experimental data.

With the same approach, figure 7(b) shows the expected variations of the average size of the structure (which is measured on low resolution STM images) as a function of the angle shift (continuous line). This curve shows that starting from a perfect (2√3x2√3)R30° structure the average distance should continuously decrease with the angle shift. On this calculated curve, we also report our experimental data and those from other studies (discussed in the following). Note again that the theoretical curve fits well our three experimental results and not very well the value issue from [START_REF] Liu | The fate of the (2√3 × 2√3)R30° silicene phase on Ag(111)[END_REF].

Discussion

The geometrical model that we propose allows a better understanding of the variety of (2√3x2√3)R30° structures observed by us and various groups [START_REF] Jamgotchian | Growth of silicene layers on Ag(111): unexpected effect of the substrate temperature[END_REF][START_REF] Arafune | Structural transition of silicene on Ag(111[END_REF][START_REF] Moras | Coexistence of multiple silicene phases in silicon grown on Ag(1 1 1)[END_REF][START_REF] Chiappe | Local electronic properties of corrugated silicene phases[END_REF][START_REF] Jamgotchian | Silicene on Ag(111) : domains and local defects of the observed superstructures[END_REF][START_REF] Feng | Evidence of Silicene in Honeycomb Structures of Silicon on Ag(111[END_REF][START_REF] Liu | The fate of the (2√3 × 2√3)R30° silicene phase on Ag(111)[END_REF]. In the following, we analyze LEED and STM results of other groups.

A LEED pattern, showing extra spots close to the (2√3x2√3)R30° structure, has also been observed by Arafune et al. (figure 3(c) in [START_REF] Arafune | Structural transition of silicene on Ag(111[END_REF]), Moras et al. [START_REF] Moras | Coexistence of multiple silicene phases in silicon grown on Ag(1 1 1)[END_REF] and recently by Rahman et al. [START_REF] Rahman | Growth of Si on Ag(111) and determination of large commensurate unit cell of high-temperature phase Japanese[END_REF]. Arafune et al. [START_REF] Arafune | Structural transition of silicene on Ag(111[END_REF] and Moras et al. [START_REF] Moras | Coexistence of multiple silicene phases in silicon grown on Ag(1 1 1)[END_REF] interpret the LEED patterns as a mixture of various structures, including (19x19)R23.4° and (3.5x3.5)R26°. We propose that the diffraction spots of these LEED patterns are from (133x133)R4.3° (or close to) structures. Finally, Rahman et al. [START_REF] Rahman | Growth of Si on Ag(111) and determination of large commensurate unit cell of high-temperature phase Japanese[END_REF] have indexed a similar LEED pattern with a same large (133x133)R4.3° superstructure. Without STM observations, they suggest from Auger Electron Spectroscopy the formation of a surface alloy. Knowing that at high temperature deposition, there is dissolution of silicon in the silver substrate [START_REF] Liu | The fate of the (2√3 × 2√3)R30° silicene phase on Ag(111)[END_REF], the saturation of the Auger silicon signal is probably not the signature of a surface alloy formation but more likely due to a partial dissolution of silicon in silver. Finally, Liu et al. [START_REF] Liu | The fate of the (2√3 × 2√3)R30° silicene phase on Ag(111)[END_REF] report the growth of a highly disordered (2√3x2√3)R30° structure. The authors interpret the STM image (figure 2 in [START_REF] Liu | The fate of the (2√3 × 2√3)R30° silicene phase on Ag(111)[END_REF]) as a mixture of (2√3x2√3)R30° and (√19x√19)R23.4° structures. Figure 8 is a copy of their STM image and its FFT. The FFT presents triplets, similar to the ones of our experiments (figure 1(c), figure 2(d) and figure 3(c)), that we surrounded by white circles. We have drawn a white line corresponding to the perfect (2√3x2√3)R30°

An equivalent

direction. Using the model, if we assume that the triplets come from a Moiré pattern, the extra spots give its periodicity (~2.5 nm). When we superimpose a hexagonal lattice (yellow) with periodicity of 2.5 nm (pink diamond), all the perfect areas of (2√3x2√3)R30° structures (surrounded by white circles by authors) coincide with this lattice. This reveals a quasi-periodic large structure composed of small perfect areas of (2√3x2√3)R30° structure surrounded by defect areas as proposed in the model. From figure 7(a) we deduce an angle shift of 24°, in perfect agreement with the angle given by the authors of 23°. The apparent disordered structure obtained by Liu et al. [START_REF] Liu | The fate of the (2√3 × 2√3)R30° silicene phase on Ag(111)[END_REF] is not so surprising since they have used growth conditions at a temperature such that large Moiré structures cannot be formed.

In our model a linear defect corresponds to a uniaxial stain and a point defect corresponds to a biaxial one. Recent DFT calculations on similar homogeneous strains [START_REF] Qin | First-principles calculations of mechanical and electronic properties of silicene under strain[END_REF][START_REF] Qin | Uniaxial strain-induced mechanical and electronic property modulation of silicene[END_REF] show strong modifications of the electronic structure of standalone silicene layer.

Finally, Tao et al. [START_REF] Tao | Silicene field-effect transistors operating at room temperature[END_REF] have built a field effect transistor with a silicene layer having a (2√3x2√3)R30° structure. Their STM image of the (2√3x2√3)R30° (figure 2(e) of [START_REF] Tao | Silicene field-effect transistors operating at room temperature[END_REF]) shows many apparent defects close to the one shown in figure 8a. Nevertheless, Raman spectroscopy signatures of this structure are characteristic of a continuous silicene layer as expected from theoretical calculations [START_REF] Houssa | Theoretical aspects of graphene-like group IV semiconductors[END_REF]. Furthermore, the electrical characterization of this silicene transistor device displays ambipolar electron-hole symmetry as expected from a silicene layer [START_REF] Cahangirov | Twoand One-Dimensional Honeycomb Structures of Silicon and Germanium[END_REF]. These experimental and theoretical results are in line with our model of a continuous silicene layer with only topological defects.

Conclusion

With the help of various STM images recorded at different magnifications, we propose a geometrical model allowing a better understanding of the (2√3x2√3)R30° structures obtained by us and by other groups. This new atomic model of the silicene layer is based on periodic arrangements of perfect areas of (2√3x2√3)R30° surrounded by defect areas on a rigid lattice of silver. In the model, the perfect areas of silicene layers are slightly contracted due to a strain epitaxy, whereas the defect areas are attributed to a local relaxation of this strain. It is important to note that a small expansion of the silicene layer induces a contraction in the STM images, therefore the defect areas appear as shrunken hexagons. Such a model is in line with a strong localized interaction between silver and silicon atoms in agreement with recent theoretical calculations [START_REF] Pflugradt | Silicene-derived phases on Ag(111) substrate versus coverage: Ab initio studies[END_REF][START_REF] Guo | Absence and presence of Dirac electrons in silicene on substrates[END_REF]. This shows that despite the impression of disorder given by the STM images, the silicene film would be a continuous honeycomb layer with only local and periodic deformations.

  (a) and the corresponding LEED pattern in figure 1(b). The FFT of the STM image is shown on figure 1(c). On the STM image, the Ag[110] direction is identified by a yellow line. The average direction of the structure (red line) is at 25.5 ± 1° with respect to the Ag[110] direction. This is different from the 30° expected from a perfect (2√3x2√3)R30° structure direction (white line). On this image, bright and dark areas create a Moiré pattern. Bright areas, which correspond to perfect (2√3x2√3)R30° areas, are separated by defect areas (darker areas). A pink diamond (3.7 ± 0.1nm) highlights the unit cell of the Moiré pattern. The LEED pattern (figure 1(b)) shows sharp spots characteristic of a well-ordered structure. Some doublets and triplets (inside white circles) close to the expected spots of a perfect (2√3x2√3)R30° structure are visible. Since those doublets and triplets are also visible on the FFT of the STM image of figure 1(c), the LEED pattern is indeed a signature of the Moiré. The white line (figure 1(c)) crosses the middle of the triplets and the red line is exactly on one of a triplet spots. The discussion section presents a detailed analysis of the LEED.

Figure 1 .

 1 Figure 1. (2√3x2√3)R30° structure obtained after deposition of one monolayer of silicon on Ag(111) at TSub = 390°C. (a) Typical large view STM image of the structure showing a Moiré pattern (V=-1.4V; I=0.1nA). Pink diamond is the unit cell of the Moiré pattern. White-blue zigzag line connects the different domains (perfect and defect areas). White lines (# 1) are the direction of a perfect (2√3x2√3)R30° structure. Red lines (#2) correspond to the average direction of the observed structure. (b) Experimental LEED pattern (E = 78 eV). (c) FFT of the STM image shown in (a).

Figure 2 (

 2 Figure 2(a) shows a high resolution STM image after deposition of one monolayer of silicon at a slightly lower temperature of 370°C. Figure 2(b) is a blow-up of the blue rectangle of figure 2(a). The corresponding LEED and FFT patterns are presented in figure 2(c) and 2(d) respectively. The LEED

Figure 2 .

 2 Figure 2. (2√3x2√3)R30° structure obtained after deposition of one monolayer of silicon on Ag(111) at TSub = 370°C. (a) High resolution STM image (V = -1.7V, I = 1.1nA) showing the different defects areas: linear defects in green and point defects in pink. White lines (# 1) are the direction of a perfect (2√3x2√3)R30° structure. Red lines (#2) correspond to the average direction of the observed structure. (b) Blow-up of the blue rectangle of (a). (c) Corresponding LEED pattern (E = 58eV). (d) FFT of the STM image shown in (a).

Figure 3 (

 3 Figure 3(a) shows a low resolution STM image obtained after deposition of a Si monolayer at 280°C. Such an image is not representative of the whole surface which contains different structures as

  image (figure3(c)) shows extra spots (inside the white circle) evidencing the presence of a superstructure not directly seen on the STM image. On the FFT, the extra spots inside the white circles are similar to those observed on figure1(a) and 2(a). This confirms that a similar superstructure exists related to a (2√3x2√3)R30° structure, but not visible on the STM images. From the FFT, we deduce a size of the unit cell of this superstructure (1.8 nm).

Figure 3 .

 3 Figure 3. (2√3x2√3)R30° structure obtained after deposition of one monolayer of silicon on Ag(111) at TSub = 280°C. (a) Large view STM image (V = -1.1V, I = 0.8nA). White lines (# 1) are the direction of a perfect (2√3x2√3)R30° structure. Red lines (#2) correspond to the average direction of the observed structure. Blue lines (# 3) correspond to the direction of a perfect (√13x√13)R13.9° structure. (b) Corresponding LEED pattern (E = 58eV). (c) FFT of the STM image shown in (a).

Figure 4 .

 4 Figure 4. Schematic presentation of a possible formation of local defects in the (2√3x2√3)R30° structure for α = -10.9°. (a) Perfect (2√3x2√3)R30° structure. The green area highlights where the relaxation will take place. (b) Linear defects obtained by dilatation in one direction (blue arrow) of a perfect silicene layer ( dAg /3). The atoms visible by STM (brighter and larger) appear as shrunken hexagons. (c) Point defects (pink) obtained by the combination of three linear defects (green) which correspond to a small local rotation of the silicene layer around a central silicon atom. The atoms visible by STM form a deformed hexagon (pink).

Figure 5 .

 5 Figure 5. Large scale ball model of the (2√3x2√3)R30° structure ( = +10.9°) with perfect periodic areas surrounded by defects areas. The upper left part shows the expected STM image. The white-blue zigzag line connects the different domains (perfect and defect areas). The white line corresponds to the orientation of a perfect (2√3x2√3)R30° structure and the red line to the average angle which should be observed on STM images at large scale. The unit cell pattern (pink diamond) is a (√133x√133)R4.3° superstructure.

Figure 6 (

 6 Figure 6(a) is the expected LEED pattern for one domain of the (133x133)R4.3° superstructure  = +10.9°). The yellow dots correspond to the silver substrate and the blue dots to the (133x133)R4.3° extra spots. Experimentally, the LEED pattern will be a combination of the two domains  = ±10.9°), as shown in figure 6(b). Red dots evidence the agreement with the experimental LEED pattern of figure 1(b).

Figure 6 .

 6 Figure 6. Expected LEED patterns of the (√133x√133)R4.3° superstructure (a) LEED pattern for α = 10.9°, where only the dots near the (2√3x2√3)R30° structure are shown (blue dots). Yellow dots belong to the Ag(111) substrate; a large white circle allows easy comparison with FFT of the STM images. The white and red lines are equivalent to those of figure 1(c). (b) LEED pattern for two domains, α = ±10.9°. Red dots correspond to the more intense spots visible in the experimental LEED patterns.

Figure 7 .

 7 Figure 7. Calculated variations of the apparent (2√3x2√3)R30° structure parameters. (a) Average angle of the structure relative to Ag[110] (left axis) and angle shift (right axis) with respect to a perfect (2√3x2√3)R30° structure as a function of the Moiré periodicity (top curve  = +10.9° , bottom curve  = -10.9°). The size of the error bars is of the order of the symbols dimensions. (b) Variation of the average size of the (2√3x2√3)R30° structure as a function of the angle shift.

  Concerning our own experiments, for three different deposition temperatures, we interpret the STM images with the same model. However, the LEED patterns show extra spots only at the highest growth temperature. On the LEED pattern of figure 2(c) the extra spots are not visible due to the likely dispersion of the size of the perfect domains, as shown on the STM image (figure 2(a)). Probably for the same reason and the presence of the two other structures, (4x4) and (13x13)R13.9°, the extra spots are not visible on the LEED pattern of figure 3(b). Note that on the contrary, the FFT characteristic of welldefined local areas (figure 2(c) and figure 3(b)) are in good agreement with the model of figure 6(a).

  Moiré pattern is observed by Feng et al. (figure 1(b) in[START_REF] Feng | Evidence of Silicene in Honeycomb Structures of Silicon on Ag(111[END_REF]). The measured Moiré periodicity (3.8 nm) is very close to ours (3.7 nm). They interpret this Moiré by a periodic succession of perfect and defect areas. Unfortunately, they do not show a complete high resolution STM image of the defect areas and, they do not mention the average deviation of the angle of the structure: "the angle between the direction of Moiré pattern and honeycomb structure is about 30°"[START_REF] Feng | Evidence of Silicene in Honeycomb Structures of Silicon on Ag(111[END_REF]. Concerning the periodicity of the structure they mention "lattice period about 1.0 nm are observed at the bright part of the Moiré pattern"[START_REF] Feng | Evidence of Silicene in Honeycomb Structures of Silicon on Ag(111[END_REF]. Therefore, we could not plot their values on the curve.

Figure 8 .

 8 Figure 8. (2√3x2√3)R30° structure reproduced and modified from Liu et al. [24]. (a) We have superimposed on the STM image a hexagonal periodic lattice (yellow tram) showing a long distance periodicity given by pink diamond, as well as a red line corresponding to the average structure

  Moreover, this model allows a comprehensive interpretation of LEED patterns observed by us and by other groups. A generalization of this model explains the main observations of this structure: deviation of its average direction, Moiré patterns and apparent global disorder. The same approach will be used in the interpretation of STM images for (√13x√13)R13.9° structure in a forthcoming paper[37].

Table 1 .

 1 Parameters of different unit cell structures a

	Structures relative to Ag	α (°)	LAg b (nm)	LSi c (nm)	LSi/LAg d
	(4x4)	0	1.156	1.152	0.997
	(2√3x2√3)R30°	-10.9	1.001	1.016	1.015
	(2√3x2√3)R30°	+10.9	1.001	1.016	1.015
	(√13x√13)R+13.9° Type I	-27	1.042	1.016	0.975
	(√13x√13)R+13.9° Type II	-5.2	1.042	1.016	0.975
	(√13x√13)R-13.9° Type I	+27	1.042	1.016	0.975
	(√13x√13)R-13.9° Type II	+5.2	1.042	1.016	0.975

Acknowledgments

This work was partially supported within the project SyProSi of the Provence-Alpes-Côte-d'Azur region.


Aline Roc
email: rocaline@proton.me

Fabien Lotte

What can we learn from user interviews in BCI sessions?

Introduction: What factors contribute to the success of motor imagery training for novice users of EEG-based Brain-Computer Interfaces (BCI)? How do novice users experience and adapt to Kinesthetic Motor Imagery (KMI) training? One way to find out more is to ask the users themselves and to collect qualitative data with interviews during BCI sessions. However, the practice is not common, although it tends to be more democratized in recent literature, mostly with patients [1--10].

Methods: Semi-structured interviews were conducted at the end of each of three BCI training sessions for 24 individuals, for a total of 72 interviews. The interview questions covered performances, difficulties and strategies, i.e. movements that users chose to imagine for KMI. The questions were designed to clarify users' choices, the reasoning behind them, and identify any barriers they encountered during the training. Interviews were analysed to identify any training parameters that users found difficult or distracting, which is a lever to better understand users' needs and to ultimately design better exercises for the future of BCI user training [START_REF] Roc | A review of user training methods in BCIs based on mental tasks[END_REF].

Results: Thematic analysis showed important inter-individual discrepancies. While some user struggled to find strategies, others found it easy. Similarly, KMI was difficult for some users, but many others explained that the KMI was not difficult per se. Still, users tend to report quick improvements in their KMI auto-evaluation within and between sessions. Overall, this series of interviews highlighted various issues with common training protocols, some well-known, but that may not receive enough attention within the community. This includes:

-Users tend to be prone to overthinking, and they struggle to go or stay in a relaxed state.

-BCI requires focus, fighting fatigue, and coping with bad performances.

-Handling artefacts and avoiding movements is difficult and consumes attention.

-Eyes-open imagination is hard, especially with continuous feedback.

-Inter-trial duration can feel too short, and MI can be hard to maintain over the trial duration.

-Some MI tasks may implicitly involve other body parts and interfere with right/left separation.

-Users do not always contain their curiosity and sometimes can show unexpected behaviours such as disobeying instructions. -Understanding feedback is hard, notably identifying min/max and assimilating sham feedback.

-Contextual factors can strongly affect the KMI experience, e.g. room temperature.

Discussion/Significance: Interviews were conducted with healthy users over their three first BCI sessions. Although results might not apply well to all populations, they suggest overall that BCI training is highly demanding and causes users to struggle despite high motivation. This emphasizes the need for the development of more engaging and effective training approaches that help novice BCI users explore the interaction, understand instructions and feedback, and maintain focus during training.

What can we learn from user interviews in BCI sessions?

Research questions

➔How do novice users experience and adapt to KMI / EEG-BCI? ➔What factors contribute to success for novice users?

Method

Post-session interviews were conducted by experimenters, covering performances, difficulties encountered, and movements that users chose to imagine for KMI. Recordings were transcribed for a systematic thematic analysis.

Main results

-= dominant themes, i.e. widely reported trends common in 5+ users. -= lesser-common themes, i.e. valuable insights but mentioned by < 5 users.

• High motivation and shared perception of skill improvement.

• Long and repetitive sessions cause boredom, an increasing feeling of fatigue, and a struggle to focus over time (intrusive thoughts).

• Users are prone to overthinking the task and struggle to be or stay in a relaxed state, especially when facing low performances.

• Difficulty to stand still during KMI worsened by prolonged immobility.

• Eyes-open imagination is hard, especially with continuous visual feedback.

• Limited grasp of feedback, notably in identifying min/max, understanding there is no idle class, and assimilating what sham feedback is.

• Inter-trial duration can feel too short, as users need time to feel ready.

• KMI can be hard to maintain over time for the trial duration.

• Some MI tasks may implicitly involve other body parts and interfere with right/left separation. KMI is easier with the dominant hand.

• Users do not always contain their curiosity and sometimes can show unexpected behaviours such as disobeying instructions.

• Contextual factors strongly affect KMI experience, e.g. room temperature.

• Tensions and artefacts can be caused by important imagined muscular effort.

• Trials switch too often and too randomly. Users attempt to anticipate the next cue and are disturbed by wrong prediction.

• Users struggle with inter-trial consistency (stability within task).

• Some users felt not enough support to find the right strategy.

Electroencephalography (EEG-)based BCI Kinesthetic Motor Imagery (KMI) training 24 healthy users 3 sessions 72 interviews

Contact info

rocaline@proton.me As the runs go on, it's more and more tiring, it's more difficult because it's hard to keep your attention. Two or three times, I think I forgot to be really focused.

What is difficult is to stay focused when you see that it doesn't work. It takes me out of my routine, I have parasitic thoughts, and it distracts me, so I get frustrated.

As soon as I see the blue bar going up and down, I lose my focus.

I think that maybe they are trying to frustrate me and that the blue bar is not moving forward on purpose, aiming to frustrate me. Maybe they don't tell me and make me believe that it is not moving forward, and in fact it is moving forward very well. ➔ Provide engaging approaches and opportunities to experiment without cues during training or at the end of the session.

Conclusion

  Aline ROC * , Fabien LOTTE * * Inria, France (Potioc team) & LaBRI (CNRS / Bordeaux INP / Univ. Bordeaux), FranceRelated workWorking with qualitative data is not a common practice in the BCI field, although it tends to be more democratized in recent literature, mostly with patients [1--10].Dataset 24 participants took part in three training sessions on different days with a common protocol (Graz feedback, 2-class CSP/LDA classification). Semi-structured interviews were conducted at the end of each session, for a total of 72 interviews.

  https://team.inria.fr/ References We're not used to imagining a movement without actually doing it. And so, to maintain it, that feeling of maintaining the movement, I think that's what's hard, because there's no haptic feedback, it's just… it's hard to keep imagining, actually. I was like "uuugh" and I felt like even my breathing was changing at the same time, and I was trying not to make any movements and stuff, but I couldn't stop holding my breath like I was lifting something real! Sample quotes I did a test where… maybe I'm a bad subject but… when it told me to do right, I did left! But I didn't do that often!!! Just that one time. Besides, I knew it wasn't right, but I told myself: I'll try anyway.

  Everyone knows that BCI training is highly demanding and causes users to struggle despite high motivation. And still, this may not receive enough attention within the community. Exploring the training experience with real interview data can help to identify appropriate solutions connected to actual user needs, and ultimately improve user training protocols. [11] Practical takeaways ➔ Help users understand KMI and feedback with tutorials and demonstrations. ➔ Offer real breaks and longer inter-trial duration to help maintain energy and focus. ➔ Present some exercises with less information density for beginners (e.g. open loop exercises, only post-trial feedback, eye-close KMI or non-visual feedback).

Acknowledgements: This work is supported by the grant ERC-2016-STG-714567.

Special thanks

Dreyer, P., Kolodzienski, L., Appriou, A., Monseigne, T.

Research supported by the grant ERC-2016-STG-714567