Souleymen Sahnoun

El-Hadi Djermoune

David Brie

SPARSE MULTIGRID MODAL ESTIMATION: INITIAL GRID SELECTION

Keywords: modal estimation, adaptive sparse approximation, multigrid

Modal retrieval problem can be addressed using sparse estimation techniques coupled with a multigrid approach. Selection of the initial grid, in the multigrid algorithm, is a critical problem that needs satisfactory solutions. In this paper we propose a strategy for selecting a coarse initial grid which guarantees convergence of the algorithm even starting with a single atom in the dictionary. The idea is to define the atoms so as their correlation with the signal modes is greater than a specified threshold. We show through simulations that the proposed solution provides accurate estimations even in presence of noise. Moreover, we investigate the convergence of the multigrid algorithm towards the true parameters.

INTRODUCTION

Consider the modal retrieval problem in which the main aim is to estimate parameters of superimposed damped sinusoids. This topic has versatile applications in engineering, e.g., nuclear magnetic resonance (NMR) spectroscopy, wireless communications, sonar and radar. We will use the following complex modal signal model:

y(m) = F i=1 c i a m i + e(m) (1)
for m = 0, . . . , M -1, where a i = e (-αi+j2πfi) , with {α i } F i=1 the damping factors and {f i } F i=1 the frequencies. {c i } F i=1 are complex amplitudes and e(m) is a white Gaussian noise with variance σ 2 . The problem is to estimate the set of parameters {a i , c i } F i=1 from the observed sequence y(m). This can be done by using three classes of possible approaches. The first class, parametric approaches, such as sub-space based methods [START_REF] Roy | ESPRIT: Estimation of signal parameters via rotational invariance[END_REF][START_REF] Kumaresan | Estimating the parameters of exponentially damped sinusoids and pole-zero modeling in noise[END_REF], whose use requires the choice of the number of components (model order) F before they can be applied to the data vector. The second class, nonparametric approaches, such as the periodograms, which do not require knowledge of F but they suffer from a limited spectral resolution. The third class is that of sparse approximation methods [START_REF] Donoho | Fast solution of 1-norm minimization problems when the solution may be sparse[END_REF][START_REF] Gorodnitsky | Sparse signal reconstruction from limited data using FOCUSS: a re-weighted minimum norm algorithm[END_REF][START_REF] Malioutov | A sparse signal recontruction perspective for source localization with sensor arrays[END_REF] which may be considered as semi-parametric. These later methods, as compared to the previous ones, can greatly enhance the estimation accuracy for noisy signals and their use does not necessarily require a priori knowledge of the model order F [START_REF] Stoica | Sparse estimation of spectral lines: grid selection problems and their solutions[END_REF][START_REF] Sahnoun | Sparse multidimensional modal analysis using a multigrid dictionary refinement[END_REF]. This is why spectral analysis using sparse estimation methods has received considerable interest in the recent years.

Sparse approximation consists in finding a decomposition of a signal y ∈ C M as a linear combination of a limited number of elements from a dictionary Q ∈ C M ×N , i.e., finding a coefficient vector x ∈ C N that satisfies y ≈ Qx, where Q is overcomplete (M < N ). The sparsity condition on x ensures that the underdetermined problem does not have an infinite number of solutions. The dictionary Q can be chosen according to its ability to represent the signal with a limited number of coefficients or it can be imposed by the inverse problem at hand. In the present paper, we consider dictionaries whose atoms are function of some parameters. The different atoms of the dictionary are then formed by evaluating this function over a grid which has to be very fine to achieve a certain degree of resolution. This is the case for the modal estimation problem in which the atoms are formed by discretizing the frequency and damping factor axes. In this situation, the challenge is to get a good approximation without a prohibitive computational cost due to the huge size of the dictionary. To deal with this problem we proposed a multigrid approach [START_REF] Sahnoun | Sparse multiresolution modal estimation[END_REF][START_REF] Sahnoun | Sparse multidimensional modal analysis using a multigrid dictionary refinement[END_REF] which iteratively enhances the set of atoms in the dictionary. The goal of the multigrid algorithm is to improve resolution by avoiding computationally intractable operations. The estimation begins with a coarse dictionary which is then enhanced over several resolution levels according to the activated atoms at each level. However, different numerical simulations suggest that the construction of the initial dictionary (at the first level) should be done carefully in order to ensure convergence of the multigrid algorithm towards the signal modes {a i } F i=1 . In a recent work on grid selection problems, Stoica et al. [START_REF] Stoica | Sparse estimation of spectral lines: grid selection problems and their solutions[END_REF] proposed to use a very fine grid to estimate the parameters of harmonic signals. As a result, the minimum number of atoms in the dictionary have to be greater than a threshold which is generally high. Alternatively, in this paper we propose to investigate the approach which consists of starting with a very coarse dictionary con-taining only a single atom and we derive the condition that ensures convergence to the true modes thanks to the multigrid approach.

The paper is organized as follows. In section 2, we see how the modal retrieval problem may be addressed using sparse approximations. In section 3 we recall the principles of multigrid dictionary refinement. In section 4 we discuss different possible ways and conditions to construct the initial dictionary, then we present an efficient strategy to define the initial grid. In section 5 we present some simulation results to show the effectiveness of the proposed strategy and the convergence of the multigrd algorithm on randomly generated modal signals. Conclusions are drawn in section 6.

PROBLEM FORMULATION

The data model expressed in (1) can be written in a vector form as:

y = F i=1 c i a(α i , f i ) + e (2)
where a(α i , f i ) is a vector of M elements:

a(α i , f i ) = [1, e (-αi+j2πfi) , . . . , e (-αi+j2πfi)(M -1) ] T .
(3) Let α max be an upper bound on {α i } F i=1 and let P the number of points of a uniform grid covering the damping factor interval [0, α max ]. Similarly, let K be the number of points of a uniform grid covering the frequency interval [0, 1[. Finally, let Q = [q(0, 0), . . . , q(0, (K -1)δ f ), q(δ α , 0), . . . ,

q(δ α , (K -1)δ f ), . . . , q((P -1)δ α , (K -1)δ f )] (4)
where q(α, f ) = a(α,f ) ||a(α,f )||2 , δ α = α max /P and δ f = 1/K. The matrix Q, called dictionary, is constructed using N = KP modes derived from the combination of the two uniform grids combining damping factors and frequencies. Each column {q n } N -1 n=0 of Q is called atom. Thus, the dictionary Q is obtained from a 2-D grid. Therefore, we can approximate y in (2) by:

y ≈ Qx + e (5)
where x ∈ C N is a sparse vector, i.e. it contains a few non-zero elements and the rest of elements are equal to zero or negligible. Non-zero elements are nearly equal to {c i } and correspond to columns in Q that are equal (or close) to {a(α i , f i )}. Thus, the modal retrieval problem can be formulated as the sparse estimation of x. It should be emphasized that by doing so, we implicitly assume that the dictionary includes the true signal modes. A first approach to ensure that (at least approximately) is to define Q on a very fine grid resulting in a high dimension dictionary. The main limitation of

Table 1. Sparse multigrid algorithm • Input. A signal y ∈ C M , a matrix Q 0 ∈ C M ×N , a scalar λ and an integer L • Output. A sparse coefficient vector x L-1 ∈ C N .
For l = 0 up to l = L -1

x l = SAM (Q l , y, λ) Q l+1 = ADAPT (Q l , x l ), End For.
such an approach is to drastically increase the computational cost. We have proposed an alternative approach which consists in an adaptive refinement of the dictionary resulting in the so-called sparse multigrid approach [START_REF] Sahnoun | Sparse multidimensional modal analysis using a multigrid dictionary refinement[END_REF]. Its principle is described in the next section.

MULTIGRID DICTIONARY REFINEMENT

To achieve a high-resolution modal estimation, a possible way is to define a high-resolution dictionary often resulting in a prohibitive computational burden. Rather, it is also possible to adaptively refine a coarse one through a multigrid scheme. This results in the algorithm sketched in table 1 where the key step is the adaptation of the dictionary according to the previous one and the estimated vector x. The algorithm amounts to insert (resp. remove) atoms in (resp. from) Q and to rerun the sparse approximation algorithm. We propose two refinement procedures. The first one consists in inserting new atoms in the Q matrix in the neighborhood of active ones. In other words, we first restore the signal x (l) related to the dictionary Q (l) by applying a sparse approximation method (SAM) at level l. Then we refine the dictionary by inserting atoms in between pairs of Q (l) , in the neighborhood of each activated atom and we apply again the SAM at level l + 1 to restore x (l+1) with respect to the refined dictionary Q (l+1) . Thus we refine iteratively the dictionary until the maximum level l = L -1 is reached. This procedure is illustrated in figure 1(a) where the dictionary atoms depend on two parameters f and α. The disadvantage of this procedure is that the size of the dictionary is increasing as new atoms are constantly added between two resolution levels. Hence, the computational cost will be increasing. To cope with this limitation, we propose a second procedure consisting in adding new atoms as in the first procedure and deleting distant non-active ones (fig. 1(b)).

The multigrid dictionary refinement is proposed in the context of modal analysis. However, it is worth noticing that this idea can be straightforwardly extended to any dictionary obtained by sampling a continuous function over a grid. We discuss the selection of the initial grid in the following section.

Add

Q l : level l Q l+1 : level l + 1 f f α Add & remove Q l : level l Q l+1 : level l + 1 f f α

SELECTION OF THE INITIAL GRID

The initial dictionary in the multigrid scheme is of great importance since a bad definition of it may cause completely wrong estimation. Therefore, it should respond to at least two conditions. First, in order to ensure convergence to the true modes, it is necessary that each signal mode has a non negligible correlation with at least one atom q n from Q. Second, to reduce numerical complexity, the dictionary should contain only a few atoms defined from a coarse sampling of the variables (α, f ). These conditions may be fulfilled by defining broadband atoms in the initial dictionary, viz. whatever the number of atoms were included in the dictionary, the spectrum of the atoms have to cover all the spectral range [0, 1[ in order to ensure a certain correlation with the signal modes. To clarify this idea, let us consider a rather elementary dictionary containing only one atom q(α 0 , f 0 ) ∈ C M formed from the signal: q(m) = β 0 e (-α0+j2πf0)m

where β 0 = 1/||a(α 0 , f 0 )|| 2 is a normalization constant. The spectrum of this damped atom is:

Q(f ) = M -1 m=0 q(m)e -j2πf m = β 0 1 -e (-α0+j2π(f0-f ))M 1 -e (-α0+j2π(f0-f ))
It can be shown that the maximum of

|Q(f )| is max f |Q(f )| = Q(f 0 ) = β 0 1 -e -α0M 1 -e -α0
and the minimum is

min f |Q(f )| = Q(f 0 ± 1 2 ) = β 0 1 -(-e -α0 ) M 1 + e -α0
Now, we want Q(f ) to cover all the spectral range in the way that it has some correlation with any possible peaks existing in the signal, as illustrated in figure 2. Hence, given This can be done by keeping the minimum min f |Q(f )| greater than zero. More precisely, the minimum correlation ρ may be controlled using the inequality:

min f |Q(f )| max f |Q(f )| ≥ ρ ( 6
)
where 0 < ρ < 1. It can be verified that the solution of this inequality is given by:

α 0 ≥ ln(1 + ρ) -ln(1 -ρ) (7)
This result will be used in the next section to show the convergence of the multigrid algorithm even starting with a single atom. Note that, by doing so, the limits of the frequency (f min = 0, f max = 1) and the damping factor (α min = 0, α max = α 0 ) are utilized to calculate the position of the new atoms in the second level of the multigrid scheme.

SIMULATION RESULTS

In this section, we present results of computer simulations to illustrate the usefulness of the proposed strategy for the construction of the initial dictionary. First, we present obtained results on a simulated noisy modal signal to prove convergence to true modes. Then we analyze the convergence by estimating parameters of random generated modal signals. Finally, we analyze convergence to true modes with different noise levels. Fig. 3. FFT of y and estimated modes, SNR 1 = 20 dB.

Convergence

In the first example, the modal signal y is composed of M = 50 samples and made up of three modes (superimposed damped sinusoids) having the same amplitudes c i = 1. The variance of the additive Gaussian noise is fixed such that the SNR of mode 1 is SNR 1 = 20 dB. Table 2 shows the parameters of the three simulated modes and their estimates using the multigrid algorithm. The initial grid is defined using the result of the previous section. We choose ρ = 0.635, which gives α 0 = 1.5, and we put f 0 = 0.48 (recall that f 0 can take any value in [0, 1[). For the sparse approximation we use the single best replacement (SBR) method [START_REF] Soussen | From Bernoulli-Gaussian deconvolution to sparse signal restoration[END_REF] because its good performances for modal retrieval [START_REF] Sahnoun | Sparse multiresolution modal estimation[END_REF]. We apply the multigrid algorithm over 50 levels; the algorithm converges before reaching this maximum level but the objective here is to see its behavior with extra levels after convergence. These settings will be used for all remaining simulations. In figure 3 we present the spectrum of the simulated noisy modal signal and the spectrum of each estimated component. Figures 4(a)-(b) present the estimated frequencies and damping factors at each level. It can be seen that the algorithm converges towards the true frequencies. Concerning the damping factors, two of them are very well estimated (α 1 and α 3 ); the third is slightly biased (α 2 = 0.006 versus α 2 = 0). Finally, the algorithm converges at level L = 40.

In order to show the importance of the definition of the initial dictionary, let us consider the case ρ = 0.25 corresponding to α 0 = 0.51. For the same settings as before, the results are shown in figure [START_REF] Malioutov | A sparse signal recontruction perspective for source localization with sensor arrays[END_REF]. We see clearly that the multi- grid algorithm does not converge and fails to detect the first mode.

Resolution

Here we simulate 1000 noise-free signals of 30 samples, each one contains two harmonic sinusoids whose frequencies are generated randomly. The distance between frequencies in the same signal is forced to be greater than or equal to Fourier resolution. Figure 6 presents the estimation error for the two modes in each trial. The standard deviation of the estimation error for the two modes has an order of magnitude of 10 -4 . The mean squared error (MSE) is equal to 8.603 × 10 -8 and 9.758 × 10 -8 for mode 1 and mode 2, respectively. We can conclude that whatever the position of the signal frequencies, the proposed strategy converges to the true modes.

Detection rate

In this simulation, the signal y contains two modes at a 1 = e j2π0.23 and a 2 = e -0.1+j2π0.46 with (c 1 , c 2 ) = (1, 1). The number of samples is M = 30. The parameters of the signal are estimated by the multigrid algorithm at different noise powers. We emphasize here that the mode a 2 as compared to a 1 vanishes rapidly because of its damping factor. We do 100 Monte Carlo trials for each noise level. Then we compute the rate of successful estimations (convergence rate) at each level. The obtained results are presented in figure 7. We can see that, thanks to the proposed strategy, the algorithm converges to true modes with a rate upper than 80% for an SNR 1 more than 9 dB; and the rate of convergence is almost equal to 100% with an SNR 1 upper than 18 dB.

CONCLUSION

We proposed an efficient strategy to select the initial grid in the sparse multigrid modal estimation approach. The proposed solutions consist in starting estimation with a dictionary only containing one atom. Then, it was shown, through simulations, that this strategy provides accurate estimations, guarantees convergence to true damped sinusoids parameters and ensures to have a modal dictionary still be coarse. As Convergence Rate (%) Fig. 7.

Monte Carlo trials: convergence rate to (α 1 , f 1 ) = (0.00, 0.23) and (α 2 , f 2 ) = (0.1, 0.46). future work, we are planing to theoretically study the convergence of the multigrid approach.

Fig. 1 .

 1 Fig. 1. Two multigrid schemes.

Fig. 2 .

 2 Fig. 2. Schematic representation of the correlation between signal modes and a single atom.

  Estimated damping factors. The insert shows details for levels 15 to 50.

Fig. 4 .

 4 Fig. 4. Estimated parameters at each multigrid level: Convergence to true parameters.

Fig. 5 .Fig. 6 .

 56 Fig. 5. Estimated frequencies at each level: the multigrid algorithm fails with ρ = 0.25.

Table 2 .

 2 First example: Simulated and estimated modes

		True		Estimates
	Modes	f	α	f	α
	1	0.100 0.010 0.100 0.010
	2	0.480 0.000 0.480 0.006
	3	0.500 0.000 0.501 0.000


J Alexis
email: joel.alexis@enit.fr

B Etcheverry

J D Beguin

J P Bonino

Structure, morphology and mechanical properties of electrodeposited composite coatings Ni-P/SiC

Keywords:

Physical properties of the NiP/SiC deposits are discussed according to the electroplating parameters and heat treatments. The insertion of silicon carbide in the coatings does not modify their rigidity (E = 230 GPa), increases their hardness slightly (50 Vickers) and decreases the residual stresses in coatings.

The phosphorus content has a major effect on the structure and the physical properties. Indeed, insertion of phosphorus into the deposits generates a grain size reduction or even an amorphisation which results in morphological modifications observed by AFM. In parallel a very important hardening is associated with incorporation of phosphorus: hardness is multiplied by 3 to reach 600 HV0.1.

The deposits tend to become crystalline following the heat treatments. With 420 • C, the precipitation ofaNi 3 P phase which distorts the crystal lattice is observed, increasing the hardness of the deposits.

Introduction

To meet aims of eco-design and use clean technologies, excluding rejections from effluents containing chromium VI, a study on composite coatings NiP/SiC were carried out in order to replace the hard chromium deposits. Indeed, even if no law prohibits the use of baths containing hexavalent chromium, a French ministerial decree (no. 2005-378: 20 April 2005) limits very strongly atmosphere and water pollution by fixing the concentration limits in the effluents at 0.1 mg l -1 and the content limits in gases and vapors rejected to 0.1 mg mm -3 . The insertion of SiC particles should make it possible to increase the mechanical properties of the nickel coatings in order to obtain a hardness comparable with the hard chromium deposits (1000 HV) [1][2][3]. The purpose of the undertaken study is to evaluate the mechanical properties of electrodeposited NiP/SiC coatings. It is based on microhardness and nanohardness tests and analyses by X-ray diffraction in order to determine the nature of the crystalline phases present as well as the rate of internal stresses. To carry out the deposits, we varied the phosphorus and SiC content of the electrolyte and we carried out two post heat treatments: 190 • C under air (amorphous state without residual stress) and 420 • C (crystalline state) under nitrogen. The temperatures of treatment were selected according to the studies undertaken by others authors [4][5][6]. The mechanical properties of the various coatings obtained are discussed according to their morphological and structural characteristics.

Experimental details

NiP/SiC deposits were electrodeposited in a cell whose electrodes are static and vertical, containing 140 cm 3 of electrolyte. The cathode is a copper disk substrate of surface 1.7 cm 2 . The anode was a high-purity nickel ingot with an effective area of 4 cm 2 . The substrates are pickled in a nitric bath of diluted acid and are rinsed with distilled water before deposition. SiC particles (99.9% SiC) come from NEYCO. It has an average diameter of 600 nm. The various deposits are indicated P x S y thereafter with x and y respectively the phosphorous acid H 3 PO 3 contents and the SiC content of the electrolyte.

The temperature of the bath as well as the current density is respectively fixed at 80 • C and 10 A dm -2 . The pH is adjusted to 2 after introduction of SiC particles in suspension in the electrolyte. The bath was stirred by a rotative stirring rod to increase the SiC particles incorporation. Time was fixed to obtain 60 m thick deposits. Four series of samples were carried out by varying the content of SiC particles, the phosphorous acid of the electrolyte and the heat treatments at 190 • C and 420 • C(Table 1). The evolution of the phosphorus content as well as the SiC particles content in the deposits respectively to the phosphorus acid content and the SiC particles content of the electrolyte are presented in Fig. 1.

The topographic observations were carried out using the inductive feeler profilometer SURFASCAN 3D. Scanned surfaces have

Series

Composition of the bath Post-treatment The whole of the analyses by X-ray diffraction was carried out on the diffractometer Philips MRD by using a copper anticathode ( = 0.15415 nm) in order to determine crystalline structure of the deposits in configuration -2 and in addition their residual stress state according to the sin 2 method.

NiSO4•6H2O NiCl2•6H2OH 3 PO4 H3PO3 Na2SO4 SiC 1 210gl -1 60gl -1 50gl -1 0 50gl -1 0to80gl -1 Untreated 2 210gl -1 60gl -1 50gl -1 0to20gl -1 50gl -1 20gl -1 Untreated 3 210gl -1 60gl -1 50gl -1 0to20gl -1 50gl -1 20gl -1 190 • C treated 3 h-air 4 210gl -1 60gl -1 50gl -1 0to20gl -1 50gl -1 20gl -1 420 • C treated 1 h-nitrogen
The microhardness of the various deposits was determined by using the microdurometer Shimadzu HMV-2 under several loads from 25 g to 200 g. The values given are the averages of the hardness's obtained thanks to a minimum of 5 tests for each load and each deposit. The maximum load was given according to the depth ratio of indentation depth/thickness of the deposits which is equal to a maximum of 0.09 in our case in order to remove the influence of the copper substrate on the measurements. The nanohardness tests were carried with the nanoindentor XP from MTS. A matrix of 20 measurements in dynamic mode was carried out on the section of each deposit by imposing a maximum displacement of 500 nm in-depth. Spacing between each indent is of 9 m to avoid the phenomena of work hardening between each test. The Young's modulus of the deposits is calculated starting from the contact stiffness measured at the time of unloading [7]

Se = dP dH = 2 √ Er A(h c ) (1)
where Se the stiffness of contact determined in the higher part of the curve of discharge, P the load applied, H the indentation depth, A (h c ) the contact area function to indentation depth, Er the effective module of the system holding account of the Young's moduli and the Poisson's ratios of indentor (E i , i ) and of the sample (E e , e ). 3. Results

1 E r = 1 -2 i E i + 1 -2 e E e (2)

Influence of the SiC particles content

The evolution of the average roughness R a and the ratio of surface developed/surface scanned Sdr according to the SiC content of the bath is presented Fig. 2. These two parameters decrease slightly then stabilize. This light levelling of the surface of the deposits corresponds to a microstructural refinement observed under the scanning electron microscope and the atomic force microscope. The pyramidal grains are denser and well cuts with a content of SiC increasing. The morphology is typical among that of crystalline deposits (Fig. 3). Moreover, with the increase in SiC content of the bath, phenomena of abrasion on the surface occurs during the elaboration, thus decreasing the roughness of the deposits.

The structure of the various coatings according to the SiC content of the electrolyte is presented Fig. 4. Taking into account the important thickness of the deposits (60 m), no peak of diffraction related to the copper substrate appears. Moreover, SiC is not detected whatever the deposit. Only the Ni phase corresponding to JCPDS card no. 04-0850 is highlighted.

All the diffraction diagrams present well defined and distinct peaks that confirm the crystallinity of the deposits. However, the intensity ratio for each measured peak is different from the intensity ratio of the JCPDS card. It could be due to a preferred crystalline orientation. The peak intensity corresponding to the family of plan (2 0 0) of Ni is more important than that of the peak of the family of plan (1 1 1) which should be most intense according to the JCPDS card. A comparative study of texture has been carried out on copper substrate and on deposits containing different percentage of SiC particles. Other electroplating conditions (pH, temperature, potential difference, etc.) are constant. As Fig. 5 shows, the pole figures of the plans (1 1 1) and (2 2 0) of the P0S0 deposit exhibit a typical 110 texture fibre completely different from the brass texture of Cu substrate. Moreover, this fibre texture seems to be independent of SiC content particles contrary to what is observed by CHOU [8]. The microhardness tests carried out with 100 g and 200 g loads show in all cases a slight increase in the hardness of the deposits with the increase in SiC content of the bath (Fig. 6). The values found for pure Ni deposit are in conformity with those given by Dedeloudis et al. [2]. However, the increase rate in hardness measured by this author with the insertion of SiC particles is not confirmed by our measurements. This difference can be explained by the size of the inserted particles. Indeed, Dedeloudis uses particles of size 150 nm whereas those used for this study have a size around 660 nm. The slight hardness increase in our case seems to be only induced by the intrinsic mechanical properties of the particles which have a high hardness and a high modulus of elasticity. Particles of very low size used by Dedeloudis et al. can take into account, as they showed, the dislocation movement; reinforcing more the nickel matrix. Fig. 7 presents the evolution of the mechanical behaviour of the deposits according to the SiC content of the bath. All the deposits have a virtually identical behaviour; which seems to confirm the preceding observations: the particles of important size would not intercede on the mechanical behaviour of the matrix by blocking any dislocations in particular. They are characterized by a weak elastic return ( ∼ =50 nm). Moreover, the SiC content does not influence the Young modulus value of the deposits which is about 230 GPa (Fig. 8). Nanohardness also remains relatively stable even if a very slight increase for the deposits elaborated with an electrolyte containing 40 g l -1 of SiC particles is observed. Moreover, the nanohardness tests do not highlight any hardness gradient in the thickness of the deposits. This result is to be attributed to the good SiC particles distribution observed in optical microscopy.

Lastly, the internal stresses were determined according to the sin 2 method starting from the variations of the inter-reticular distances of the nickel plans (4 2 0) (2 = 144.66 • ). X-ray crystallographic constants of this plan family, used in the calculation of the stresses, are those given by Casteix et al. [START_REF] Casteix | Publications Scientifiques et Techniques[END_REF]: S1=-1.43 × 10 -6 MPa Fig. 8. Effect of the acid phosphorus content of the electrolyte on the roughness parameters. and 1/2S2 = 6.15 × 10 -6 MPa. In spite of the crystalline anisotropy highlighted previously, the good linearity of the curves D = function (sin 2 ) allowed us to study the residual stresses comparatively.

The results show that the coatings are in tension and the tensile stress decreases with the increase in SiC content in the electrolyte (Table 2). This confirms the results previously shown by Chou [8] and Berkh [1].

Influence of the H 3 PO 3 content

The phosphorus content in the electrolyte very clearly influences the topography of the deposits. Indeed, as the joint reductions in average roughness R a and the ratio Sdr show, the coatings deposited in the presence of acid phosphorous are by far the least disturbed (Fig. 9) and this, just with the incorporation of 2 g l -1 of H 3 PO 3 in the bath.

This levelling according to the H 3 PO 3 content of the bath is explained by a radical change of the morphology of the deposits. The coatings deposited without phosphorus are made up, as we have seen by pyramidal grains, characteristic of a crystalline structure whereas the deposits elaborated in the presence of H 3 PO 3 have a globular morphology which would return rather to an amorphous structure (Fig. 10).

The X-ray diffraction analysis (Fig. 11) confirms the topographic and morphological changes between the coating deposited with or without phosphorus. The transition of the crystalline state towards a less ordered state is characterized by broader and less intense peaks which can continue until disappearing according to increasing phosphorus content. Thus, only deposits of P2S20 containing 7.67 at.% of phosphorus are crystalline, even microcrystalline taking into account the extinction and the widening of the whole of the peaks compared to the P0S20. On the sample without phosphorus, the smoothness of the peaks is the result of strong crystallinity of the deposit for this composition, as has been shown previously. On the other hand, the inversion of the intensities of the peaks (1 1 1) and (2 0 0) for sample P2S20 (7.67 at.% of P) could be explained by a change of crystalline texture with the phosphorus incorporation: the deposit without phosphorus presents a preferential orientation of the (2 0 0) plan parallel to the surface whereas the P2S20 favours the plan (1 1 1). This result was already observed by Lewis and Marshall [START_REF] Lewis | [END_REF]. They assign this privileged orientation of the plans (1 1 1) with increasing phosphorus content in the deposits due to the fact that the deformations of the crystallites are more important according to the crystalline plans (2 0 0) than that of the plans (1 1 1).

Beyond this content of 7.67 at.% of phosphorus, that means for all the other samples, the coating becomes amorphous: the diffraction peaks decrease, widen then disappear. Phosphorus dissolves in solid solution in the nickel mesh and does not appear on the X-ray diffractograms.

The microhardness of the deposits evolves considerably according to the phosphorus content (Fig. 12). It is multiplied by a factor of 3 between coatings deposited without phosphorus and another elaborate with 2 g l -1 in the bath to reach approximately 600 HV0.1. This increase is in conformity with that measured by Yucheng [11] and Cheng [12] and lower than that measured by Apachitei [13]. A slight reduction in hardness is then observed for the high phos- phorus contents of the bath. It is interesting to note that only these deposits tend to fissure with the Vickers tests carried out with a load of 100 g. This can be due to a change of mechanical behaviour; and more particularly to the increased brittleness of the deposits rich in phosphorus. The modifications of the mechanical behaviour with the insertion of phosphorus in the Ni matrix are confirmed by the shape of the loading and unloading curves obtained during nanoindentation tests (Fig. 13). The deposits containing phosphorus are characterized by an elastic return three times larger than the deposit without phosphorus. Moreover, as Weaire [14] underlines it, a reduction of 30% of the value of the Young's modulus is observed for the amorphous deposits (Fig. 14). This fall is explained by local rearrangements within the disordered structure of amorphous alloy under a given pressure [15,16].

Influence of the heat treatments

The study of the NiP-SiC deposits, after a 190 • C heat treatment for 3 h in air, highlights no change of topography or morphology compared to the untreated coatings. According to the X-ray phase's analyses, P2S20 and P6S20 deposits are microcrystalline; their hardness increases by 150-200 Hv (Fig. 15). This increase is due to stress relaxation. The stress rate of the as plated coating without phosphorus is estimated at 69 ± 11.3 MPa and 45.4 ± 7.7 MPa after a 190 • C heat treatment. Moreover, for this temperature former studies made it possible to highlight hydrogen desorption of the Ni-P deposits according to their structural state [4,17].Onthe other hand, amorphous deposits, P12S20 and P20S20, preserve the same hardness than as-plated deposits. However, after this heat treatment, all the deposits are more fragile. Cracks appear after microhardness tests. These observations are confirmed by nanohardness tests (Fig. 16). No gradient of hardness was observed in the thickness of the deposit whatever its composition. For a 420 • C heat treatment of 1 h in nitrogen, no surface modification was highlighted compared to as-plated samples. However, structural changes appear in the deposits (Fig. 17). These give a more stable structure from a thermodynamic point of view. Indeed, as we can see it on the diffraction X-ray diagrams, all the deposits become crystalline again or microcrystalline. Moreover, precipitation of the Ni 3 P phase is observed [18][19][20]. However, this is associated with a very important hardening, in particular for the deposits which were amorphous until then. Thus, an increase in the microhardness (Fig. 15) and nanohardness (Fig. 16) of deposits P12S20 and P20S20 are recorded compared to their untreated state of development. This increase of 500 Hv comes primarily from the presence of the precipitates which distort the network and cause it to become crystalline. We find then that the values of hardness generally agree with the literature [18]. The beginning of the decrease in hardness of the treated P20S20 420 • C comes from the all important phosphorus content: the Ni 3 P phase tends to form larger aggregates which lose their hardening effect. According to Apachitei et al. [18], a spheroidisation of the precipitates can appear, decreasing by the same their hardening effect.

The calculation of the stress state highlights that the deposits are in a state of compression (Table 3). This compression of the matrix is related to the presence of precipitates. The value of the Young's modulus (Fig. 16) increases which once more confirms the return to the crystalline state of these deposits.

Conclusion

This study made it possible to show the small influence of the insertion of SiC particles on the morphology and the structure of the deposits with a nickel-phosphorus matrix. The coatings have a 110 fibre texture independent of that of the substrate and of SiC particles content. We notice a slight increase in the microhardness of the deposits with the insertion of SiC which can be due to the slight reduction in the grain size and to the increased hardness of the SiC particles.

On the other hand, the insertion of phosphorus severely modifies the morphology of the deposits. The grains of pyramidal form become globular with the insertion of phosphorus. These modifications are closely related to the structural variations. Indeed, measurements of X-ray diffraction highlighted an amorphisation of the deposits for the baths whose phosphorus acid content is higher or equal to 6 g l -1 . Consequently, the microhardness also evolves with the insertion of phosphorus. It is maximum for the microcrystalline deposits and decreases slightly for the amorphous deposits. The nanoindentation tests confirm not only the evolution of hardness but make also possible to obtain the evolution of the Young modulus according to the inserted phosphorus content. The amorphisation of the coatings results in a fall of the Young modulus.

The post heat treatment at 190 • C involves a slight increase in hardness which can be correlated to a reduction in the residual tensile stresses. Preceding studies showed a hydrogen desorption for these same deposits at this temperature. The 420 • C treatment allows a return to a much more stable structure. A nickel phosphide precipitation appears, involving a setting in compression of the nickel matrix and by the same process, a significant increase in hardness.

Future research for this study is to look further into our knowledge on the interface particles stamp and to lead, if possible, the same work by using nanometric particles to try to have a "precipitate-matrix" type hardening effect by blocking the propagation of dislocations.

Fig. 1 .

 1 Fig. 1. Effect of SiC content particles of the electrolyte on the evolution of the average roughness Ra and of the developed surface ratio Sdr.

Fig. 2 .

 2 Fig. 2. Effect of SiC particles on the coatings morphologies (a) P0S20 and (b) P0S80.

Fig. 3 .

 3 Fig. 3. Effect of SiC content on the coatings' structure.

Fig. 4 .

 4 Fig. 4. Effect of SiC particles content of the bath on the deposits texture.

Fig. 6 .

 6 Fig. 6. Loading-unloading curves function to SiC particles content of the bath.

Fig. 5 .

 5 Fig. 5. Effect of SiC content of the bath on the microhardness function to the applied load.

Fig. 7 .

 7 Fig. 7. Effect of SiC particles content of the bath on the nanohardness and the Young's modulus.

Fig. 9 .

 9 Fig. 9. Effect of the acid phosphorus content of the bath (a) P0S20, (b) P6S20, and (c) P20S20 on the morphologies of the untreated samples obtained by atomic force microscopy.

Fig. 10 .

 10 Fig. 10. Effect of the acid phosphorus content on the coatings' structure.

Fig. 11 .

 11 Fig. 11. Effect of acid phosphorus content of the bath on the microhardness function to the applied load.

Fig. 12 .

 12 Fig. 12. Loading-unloading curves function to the acid phosphorus content.

Fig. 13 .

 13 Fig. 13. Effect of the acid phosphorus content on the nanohardness and Young's modulus.

Fig. 14 .

 14 Fig. 14. Effect of heat treatments on the microhardness: (a) heat treatment at 190 • C under air and (b) heat treatment at 420 • C under nitrogen.

Fig. 15 .

 15 Fig. 15. Effect of heat treatments on the deposits nanohardness: heat treatment at 190 • C under air and heat treatment at 420 • C under nitrogen.

Fig. 16 .

 16 Fig. 16. Effect of heat treatments on the Young's modulus of the deposits: heat treatment at 190 • C under air and heat treatment at 420 • C under nitrogen.

Fig. 17 .

 17 Fig. 17. Effect of heat treatments on the structure.

Table 1

 1 Elaboration parameters of the coatings.

Table 2

 2 Effect of SiC particles content of the electrolyte on the residual stresses.

	Coatings	Internal stresses (MPa)
		Average	Standard deviation
	P0S0	136.6	13.8
	P0S40	100	10
	P0S80	55.3	7.4

Table 3

 3 Effect of heat treatments on the residual stresses.

	Coatings	Internal stresses (MPa)
		Average	Standard deviation
	P0S20	54.6	8.2
	P2S20 P6S20	-82.9 -66.5	13.1 11.7

Acknowledgement

This study received the financial support of the "Conseil Régional de Midi Pyrénées" under the project heading "multipurpose composite Coatings for the industry of transport".


A Seuret
email: aseuret@laas.fr

F Gouaisbaut

Jensen's and Wirtinger's inequalities for time-delay systems

Keywords: Time-delay systems, stability analysis, Lyapunov

In the huge literature dedicated to stability of time-delay systems, the most popular approach remains the use of Lyapunov-Krasovskii functionals. This framework allows to study a large class of time-delay systems including constant or time-varying delays. Since several years, the main challenge is to propose new functionals and techniques for deriving less and less conservative stability conditions. Nevertheless, all these approaches usually adopt the same procedure which is based on the well-known Jensen's inequality which generally induces some conservatism difficult to overcome. This paper analyses firstly the conservatism induced by the Jensen's inequality and secondly proposes a wide class of new parametrized inequalities. All these are based on an extensive use of Wirtinger inequality which has been recently introduced in Liu and Fridman [2012] and Seuret and Gouaisbaut [2012b] for stability analysis.

INTRODUCTION

Proving stability of time-delay systems is a recurrent problem since delays are inherent in many practical problems in engineering fields and control theory (see [START_REF] Sipahi | Stability and stabilization of systems with time delay[END_REF] for a recent survey). Several methods have been employed, from frequential approaches (see [START_REF] Sipahi | Stability and stabilization of systems with time delay[END_REF] for a recent survey) to state space approaches using Lyapunov functionals [START_REF] Gu | Stability of timedelay systems[END_REF]) along with robust analysis counterpart (like Small gain Theorem [START_REF] Niculescu | Frequency sweeping tests for asymptotic stability: a model transformation for multiple delays[END_REF], [START_REF] Zhang | Stability of time-delay systems: Equivalence between Lyapunov and scaled small-gain conditions[END_REF]), IQC approach [START_REF] Kao | Stability analysis of systems with uncertain time-varying delays[END_REF]) or quadratic separation approach [START_REF] Ariba | Stability interval for time-varying delay systems[END_REF]). In the case of stable linear time-delay system, a Lyapunov-Krasovskii functional can be constructed but it relies on a delay Lyapunov matrix [START_REF] Kharitonov | Lyapunov matrices for time-delay systems[END_REF]). The parameters involved in the so-called complete Lyapunov functional are solutions of a linear differential-difference equation with boundary conditions. As it is therefore quite complicated to solve, many works have been dedicated to the construction of approximate complete Lyapunov-Krasovskii functionals. By definition, the choice of such functionals entails a certain part of conservatism and many efforts have been put on improving such results. Several ways have been therefore explored. One possible solution is to construct more and more complex Lyapunov-Krasovskii functionals by choosing extended state variables, which generally goes along with a certain numerical burden [START_REF] Ariba | An augmented model for robust stability analysis of time-varying delay systems[END_REF], [START_REF] Kim | Note on stability of linear systems with timevarying delay[END_REF]). Often associated with these new functionals, the authors introduce extra variables called slack variables in order to relax the optimization problem, expressed in terms of LMIs. Getting numerically tractable inequalities (generally some LMIs to be optimized) have been then the core of many research since several years. In all these works, Jensen's inequality has been recognized as a powerful tool to obtain efficient results even if it is at a price of an unavoidable conservatism. Surprisingly, this has been barely studied yet except in a few papers [START_REF] Briat | Convergence and equivalence results for the Jensen's inequality -application to time-delay and sampled-data systems[END_REF], Seuret and Gouaisbaut [2012a,b]). In [START_REF] Briat | Convergence and equivalence results for the Jensen's inequality -application to time-delay and sampled-data systems[END_REF], an upperbound of the conservatism induced by Jensen inequality is proposed by using Grüss inequality. This last result is then employed to derive equivalent new inequalities well fitted for an optimization scheme like LMIs. In Seuret and Gouaisbaut [2012a,b], the benefits of using the Wirtinger's inequalities instead of Jensen's one has been shown successfully. In particular it has been proved that the choice of a particular signal (denoted in the sequel by z) which satisfies the necessary assumptions to apply the Wirtinger inequalities, leads to a new inequality which encompasses the well-known Jensen's inequality. The resulting inequality does not only depend on the classical terms x(t) or x(t -h) but also on

t t-h x(s)ds.
This last signal is then directly integrated into a new suitable Lyapunov-Krasovskii functional, highlighting the features of Wirtinger inequality. Notice that this new class of inequalities has been already employed in stability of sampled-data systems by [START_REF] Liu | Wirtinger's inequality and lyapunov-based sampled-data stabilization[END_REF]. A new Lyapunov functional is derived using directly this new inequality.

In this work, the first part is devoted to the construction of a lower bound for Jensen gap. Then, we aim at reducing the conservatism of Lyapunov-Krasovskii techniques by considering an accurate integral inequality which includes the Jensen's one as a special case : the Wirtinger's inequality. We explore the possible choices of signal z which satisfies the assumptions of the Wirtinger's inequality. Based on the general formulation, our objective is to investigate in the optimal choice of the signal z if possible.

Notations: Throughout the paper R n denotes the ndimensional Euclidean space with vector norm | • |, R n×m is the set of all n × m real matrices, and the notation P > 0, for P ∈ R n×n , means that P is symmetric and positive definite. The symmetric matrix A B * C stands for

A B B T C . For any differentiable z : [a, b] → R n , we denote the norm z 2 2 = 1 b-a b a z T (u)z(u)du.

PRELIMINARIES

Using Lyapunov-Krasovskii functionals to derive stability conditions for time-delay systems usually leads to some matrix inequalities quite difficult to optimize. Using Jensen's inequality often allows to derive useful inequality which can be directly incorporated in an optimization setup. For our purpose, let us recall the Jensen's inequality in the following lemma. Lemma 1. For a given matrix R > 0 and for any differentiable signal ω in [a, b] → R n , the following inequality holds:

b a ωT (u)R ω(u)du ≥ (ω(b) -ω(a)) T R(ω(b) -ω(a)) b -a .
(1)

Following [START_REF] Briat | Convergence and equivalence results for the Jensen's inequality -application to time-delay and sampled-data systems[END_REF], the Jensen gap g R (ω) is referred as:

g R ( ω) = b a ωT (u)R ω(u)du - (ω(b) -ω(a)) T R(ω(b) -ω(a)) b -a .
(2)

In the present article, our objectives are twofold.

(a) = z(b) and b a z(u)du = 0. Then for any n × n matrix R > 0, the following inequality holds b a żT (u)R ż(u)du ≥ 4π 2 (b -a) 2 b a z T (u)Rz(u)du. (3)
There exist also other versions of the Wirtinger's inequality, which can be found in [START_REF] Kammler | A first Course in Fourier Analysis[END_REF], and which could be of great interests. For instance the second version of the Wirtinger's inequality proposed anther integral inequality which requires less constraints on the function z to be considered. This correspond to the following lemma. Lemma 3. Let z a differentiable function and z(a) = z(b) = 0. Then for any n × n matrix R > 0, the following inequality holds

b a żT (u)R ż(u)du ≥ π 2 (b -a) 2 b a z T (u)Rz(u)du. (4)
This last inequality have been recently adopted Seuret and Gouaisbaut [2012a,b] to exhibit interesting inequalities which drastically reduce the conservatism of classical Lyapunov-Krasovskii methods. In this paper, we aim at understanding how these inequalities can help to assess stability of time-delay systems. In particular, in the next section, we propose a novel inequality which is proved to be less conservative than the Jensen's one. Remark 1. Another version of the Wirtinger inequality can be found in [START_REF] Liu | Wirtinger's inequality and lyapunov-based sampled-data stabilization[END_REF] but will not be presented here.

The following section constructs a link between the Jensen's and the Wirtinger's inequalities. This is not straightforward, since in the Jensen's Lemma, no assumption on the considered signal are needed contrary to Wirtinger Lemma which requires strong assumptions on the signal z. Recall that the objectives of the present paper is to obtain new lower bounds of the integral b a ω(u)R ω(u)du, in order to be consistent with the Jensen's inequality. Thus a first step consists in defining appropriate function z such that this integral appears naturally in the developments. Thus a necessary condition is that the function z as the following form z(u) = γω(u) -y(u).

(5) where γ is a scalar and where y(•) is a vector function of appropriate dimension.

HOW CONSERVATIVE JENSEN'S INEQUALITY IS?

This section aims at showing how Jensen's inequality could be conservative by providing a lower bound for the Jensen's gap. To this end, we provide the following Corollary: Corollary 4. For a given symmetric positive definite matrix R > 0, any function any differentiable signal ω in [a, b] → R n , then the following inequality holds:

b a ω(u)R ω(u)du - (ω(b) -ω(a)) T R(ω(b) -ω(a)) b -a ≥ 4π 2 (b -a) 2 b a z T (u)Rz(u)du, (6)
where the function z is defined as follows:

z(u) = ω(u) - 1 b -a b a ω(u)du - (u -a) b -a - 1 2 (ω(b) -ω(a)). (7)
Proof : Remark that the function z defined by ( 7) is of the form given in (5) and has been constructed in order to fulfill the assumption on the function z described in Lemma 2. Then, computing the left-hand side of the inequality (3) leads to

b a żT (u)R ż(u)du = b a ωT (u)R ω(u)du - 1 (b -a) (ω(b) -ω(a)) T R(ω(b) -ω(a)).
Hence, applying Lemma 2 allows to conclude. ♦ Compared to Jensen's inequality which simply states that g R (ω) ≥ 0, this previous corollary provides a new interesting lower bound of the Jensen's gap. This bound corresponds to an integral quadratic function and is positive definite. Simple calculations show that this equality can be also rewritten as follows: A simple inspection of this last expression shows that this may be difficult to handle especially if we are looking for numerically tractable inequalities. A naive approach to derive efficient relations could be the use of Jensen lemma on

b a z T (u)Rz(u)du, b a z T (u)Rz(u)du ≥ 1 b -a b a z T (u)du b a z(u)du,
but as b a z(u)du = 0, this last inequality does not lead to any improvements. The constraint on the signal z and its integral are thus very restrictive. They lead to many difficulties when constructing other admissible signals z. Thus in the next section, an analysis of the possible inequalities based on the second version of the Wirtinger's inequality is provided in Lemma 3. Remark 2. We can also draw a parallel with the result of [START_REF] Briat | Convergence and equivalence results for the Jensen's inequality -application to time-delay and sampled-data systems[END_REF], where an upperbound is given using Grüss inequality. Indeed, using Corollary 2.1 from [START_REF] Briat | Convergence and equivalence results for the Jensen's inequality -application to time-delay and sampled-data systems[END_REF], one exhibits the following inequality:

g R ( ω) ≤ (b -a) 3 4 (sup [a,b] |ω|) T R(sup [a,b] |ω|).
As a matter of fact, we get the following result: Corollary 5. The Jensen gap is such that:

4π 2 (b -a) 2 b a z T (u)Rz(u)du ≤ g R ( ω) ≤ (b -a) 3 4 (sup [a,b] |ω|) T R(sup [a,b] |ω|),
where the function z is defined by (7).

APPLICATION OF THE SECOND WIRTINGER'S INEQUALITY

This section focusses on exploiting the second version of the Wirtinger's inequality given in Lemma 3. As it is mentioned above, the use of the second version of the Wirtinger's inequality leads to a additional flexibility in the definition of the function z employed in this lemma. Thus, in the following, we will propose a general method to construct such class of function z of the form (5). In the sequel, the following issues will be addressed.

-For a such class of functions f , is it possible to derive an inequality related to the Jensen Lemma? -Is it possible to find an optimal function f which ensures that the resulting inequality is less conservative than the ones obtained by any other functions f ?

The answers to these questions are given below.

A second inequality

The following lemma is proposed. Lemma 6. For given matrix R > 0 and differentiable

signal ω in [a, b] → R n and a twice differentiable function [a, b] → R such that f (b) = f (a), the following inequality holds: b a ω(u)R ω(u)du ≥ α 1 (f ) η T (a, b)Rη T (a, b) b -a +π 2 ν T (a, b)Rν T (a, b) b -a , (8)
where

η(a, b) = ω(b) -ω(a) ν(a, b) = (1 + α 2 (f ))ω(b) -α 2 (f )ω(a) - b a ω(u) b -a du,
where

α 1 (f ) = b -a f (b) -f (a) ḟ 2 - (b -a) 2 π 2 f 2 , α 2 (f ) = 1 f (b) -f (a)   b a f (u)du b -a + (b -a) π ( ḟ (b) -ḟ (a)) -f (b) .
Proof : Let f be a continuous and twice differentiable scalar function f defined over [a, b] such that f (a) = f (b).

As suggested previously consider the signal z given by z

(u) = (f (b) -f (u))ω(a) +(f (u) -f (a))ω(b) -(f (b) -f (a))ω(u), where u ∈ [a, b].
Since z(a) = z(b) = 0, Lemma 3 can be applied to this signal. This leads to:

b a żT (u)R ż(u)du ≥ 4π 2 (b -a) 2 b a z T (u)Rz(u)du.
The right-hand-side of Lemma 3 is rewritten as:

b a żT (u)R ż(u)du = (f (b) -f (a)) 2 b a ωT (u)R ω(u)du + b a ḟ 2 (u)du (ω(b) -ω(a)) T R(ω(b) -ω(a)) -2(f (b) -f (a))(ω(b) -ω(a)) T R b a ḟ (u) ω(u)du.
+2(f (b) -f (a))(ω(b) -ω(a)) T R× b a f (u)ω(u)du -[ ḟ (b)ω(b) -ḟ (a)ω(a)] .
Introduce the vector

ξ(u) = π (b -a) z(u) + (b -a) π f (u)(ω(b) -ω(a)).
Following this definition, we have

b a ξ T (u)Rξ(u)du = π 2 (b -a) 2 b a z T (u)Rz(u)du + (b -a) 2 b a f 2 (u)du π 2 (ω(b) -ω(a)) T R(ω(b) -ω(a)) +2(ω(b) -ω(a)) T R b a f (u)z(u)du.
(11) The following developments focus on the last term of (11). Performing an integration by parts yields to

a ξ T (u)Rξ(u)du = π 2 (b -a) 2 b a z T (u)Rz(u)du -Ψ - α 1 (f )(f (b) -f (a)) 2 (b -a) (ω(b) -ω(a)) T R(ω(b) -ω(a)).
(12) By combining ( 9) and ( 12) in order to eliminate Ψ, we obtain

(f (b) -f (a)) 2 b a ωT (u)R ω(u)du = α 1 (f ) (b -a) (f (b) -f (a)) 2 (ω(b) -ω(a)) T R(ω(b) -ω(a)) + b a ξ T (u)Rξ(u)du + b a żT (u)R ż(u)du - π 2 (b -a) 2 b a z T (u)Rz(u)du.
(13) Then Lemma 3 and Jensen's inequality ensure that the following inequality holds

a ωT (u)R ω(u)du ≥ 1 (b -a) α 1 (f )(ω(b) -ω(a)) T R(ω(b) -ω(a)) + 1 (f (b) -f (a)) 2 b a ξ T (u)duR b a ξ(u)du . (14)
The last step of the proof consists in the computation of the integral b a ξ T (u)du, which is obtained by performing an integration by parts as follows

a ξ T (u)du = π b -a b a [(f (u) -f (a))ω(b) + (f (b) -f (u))ω(a)] du - π b -a (f (b) -f (a)) b a ω(u)du + (b -a) π b a f (u)du(ω(b) -ω(a)), = π b a f (u)du b -a + (b -a) π 2 ( ḟ (b) -ḟ (a)) (ω(b) -ω(a)) -π (f (b) -f (a)) b -a b a ω(u)du -π(f (a)ω(b) -f (b)ω(a)). (15) Finally we get that b a ξ T (u)du = π(f (b) -f (a))ν(a, b),
where ν(a, b) is defined in the statements of the Lemma. This allows to get the inequality proposed in Lemma 6. ♦

The previous lemma presents a wide class of inequalities. Depending on the choice of the function f , the coefficients α 1 and α 2 may have different values. In the following section, a maximization of the coefficients is proposed.

How to encompass Jensen's inequality?

It is relevant to see if one may find the function which reduces the conservatism of the inequality. As the first step, the proposition is derived. Proposition 7. For any β ∈ R + , γ ∈ R and any function f satisfying the conditions of Lemma 6, then α 1 (βf + γ) = α 1 (f ) and α 2 (βf + γ) = α 2 (f ).

Proof : The proof is performed by simple computations. ♦ Finding the function f which maximizes the right-hand side of ( 8) is quite ambitious and difficult. This is the reason why we propose to investigate only in the maximization of the coefficient α 1 (f ). This is achieved in the following proposition. Proposition 8. For any twice differentiable function f , α 1 (f ) ≤ 1 and the equality holds when f is an affine function.

Proof : Consider a scalar function f which satisfies the conditions of Lemma 6 . Using the Fourier series of the function ḟ , the Parseval identity ensures that b -

a b a ḟ 2 (u)du = a 2 0 + 1 2 i∈N\{0} a 2 i + b 2 i , 1 b -a b a f 2 (u)du = 2π 2 (b -a) 2 i∈N\{0} n 2 a 2 i + b 2 i ,
where

a 0 = 1 b -a b a ḟ (u)du = f (b) -f (a) b -a , a i = 2 b -a b a ḟ (u)c i (u)du, b i = 2 b -a b a ḟ (u)s i (u)du, s i (u) = sin 2iπ b -a (u -a) , c i (u) = cos 2iπ b -a (u -a) .
Hence the coefficient α 1 (f ) can be rewritten as

α 1 (f ) =   1 + i∈N\{0} (1 -4i 2 ) a 2 i + b 2 i 2a 2 0   .
Since 1-4i 2 is always negative for any non negative integer i, the coefficient α 1 (f ) is maximized when the Fourier coefficients a n and b n of ḟ are zero for all integer n > 0. This implies that the function f is an affine function. Moreover, according to Proposition 7, any affine scalar functions yields the same result, i.e. α 1 (f (u) = u). This concludes the proof. ♦

According to the previous proposition, the following corollary is provided.

a ω(u)R ω(u)du ≥ η T (a, b)Rη(a, b) b -a + ν 0 T (a, b)Rν 0 (a, b) b -a , (16)
where η(a, b) is defined in Lemma 6 and

ν 0 (a, b) = ω(b) + ω(a) 2 - 1 b -a b a ω(u)du. Proof : Simple computations yield α 1 (f (u) = u) = 1 and α 2 (f (u) = u) = -1
2 which conclude the proof. ♦ Remark 3. The previous lemma has already been presented in the Seuret and Gouaisbaut [2012b]. However the optimality of the choice of the function f has not been discussed there. Thus the previous contributions aim at showing this first choice was the optimal one regarding the particular definition of the function z. Of course, the present paper does not provide an optimal choice of f for both coefficients α 1 and α 2 but just gives a clue of which function would be suitable to consider. Remark 4. As shown in Corollary 4, the previous corollary provides a lower bound on the Jensen's gap. Indeed the previous inequality can be rewritten as follows

Some additional remarks

As it was mentioned in the previous section, Corollary 9 provides an more accurate inequality than the Jensen inequality. Despite of this main characteristic, several additional comments on the signal z can be added. When f is the identity function, the function z becomes

z(u) = (b -a) u -a b -a ω(b) + b -u b -a ω(a) -ω(u) .
Note that this signal corresponds to the error of an arbitrary signal ω and its Lagrange interpolating polynomial of degree 1 considered over the interval This is represented in Figure 1. Thus it would be interesting, in future works, to investigate other interpolating polynomials with, for instance, a higher degree or considering additional information of the signal ω over the interval [a, b] as, for instance, ω((a + b)/2). This last consideration refers some how to a discretization method. This term corresponds to the approximation error of the integral of an arbitrary signal ω and the area enclosed in the trapeze defined by the ω(a) and ω(b) and the abscissa axis as shown in Figure 2. This last remark is completely in accordance with the trapezoidal rule using Lagrange polynomial of order 1. As a matter of fact, there exists other relevant methods to approximate the integral of a function over an interval of the form [a, b] and it should be interesting to investigate on these other methods which could deliver improved results.

CONCLUSIONS

This paper investigates the use of a new inequality called Wirtinger inequality for stability purpose. Firstly, this inequality is linked to the Jensen inequality and provides some explanations on the conservatism induced by the use of Jensen Lemma. Secondly, we use this new inequality to establish some new numerically tractable inequalities which outperform the well known Jensen inequality often used in the context of time delay systems / sampled data stability.

  b) -ω(a)) T R((ω(b) -ω(a)) dsdu) T R(ω(b)-ω(a)).

  Consider a scalar function f defined over[a, b]. Then define the associated function (5) given byz(u) = (f (b) -f (u))ω(a) +(f (u) -f (a))ω(b) -(f (b) -f (a))ω(u), where u ∈[a, b]. By construction, the function z(u) satisfies the conditions of Lemma 3, i.e. z(a) = z(b) = 0.

  parts of the last term leads to b a ḟ (u) ω(u)du = ḟ (b)ω(b) -ḟ (a)ω(a)u)du (ω(b) -ω(a)) T R(ω(b) -ω(a))

  z(u)du = (f (b) -f (a)) ḟ (b)ω(b) -ḟ (a)ω(a) -b a ḟ 2 (u)du (ω(b) -ω(a)) -(f (b) -f (a)) b a f (u)ω(u)du.Combining the previous expression with (11) yields to

  b

  b

  b

Corollary 9 .

 9 For a given symmetric positive definite matrix R > 0, any function any differentiable signal ω in [a, b] → R n and any twice differentiable function [a, b] → R, then the following inequality holds: b

Fig. 1 .

 1 Fig. 1. Graph representing a signal ω and its interpolating polynomial ω g r ( ω) ≥ ν 0 T (a, b)Rν 0 (a, b) b -a

Following

  this idea, an other important remark concerns the resulting signal ν(a, b). Recall that in Corollary 9

Fig. 2 .

 2 Fig. 2. Graphical representation of ν(a, b) for an arbitrary signal ω.


A Generative Montagovian Lexicon for Polysemous Deverbal Nouns

We propose a computational formalization of some forms of polysemy.

Here we focus on the resultative/processual polysemy of deverbal nouns like assinatura ("signing/signature") or abertura ("opening/aperture") in Portuguese -we also study similar constructs in French, Italian, and English. We follow the Montagovian Generative Lexicon (MGL) introduced in Bassac, Mery & Retoré (2010) based on second-order Girard's F system with several entity types -including at least one type t for propositions and several entity types, as v (event), s (state) ϕ (physical object). Our formalization produces the readings involving one aspect of the polysemous noun, and it also handles properly co-predication phenomena. Indeed, copredications on several aspects of a polysemous term can be correct or incorrect. For instance, one cannot assert a predicate of the resultative meaning and simultaneously a predicate of the processual meaning.

To do so, a lexical entry consists in the "usual" Montagovian λ-term expressing the argument structure (with fine-grained types) and optional modifiers turning the type of an object (e.g. v or s) into another type (e.g. ϕ). Consider the lexical entry to "assinatura" (whose type is v) as the following:

λx v .(assinatura v→t x); id = λx v .x, f v→s r , f v→ϕ ϕ
When there is a type mismatch, one is allowed to apply some optional modifier(s). We thus are able to derive "A assinatura atrasou três dias"1 and "A assinatura estava ilegível"2 . The definite article "a" ("the"), is handled by a typed choice function ι (or ǫ, one of the first smiler approach being [START_REF] Egli | The epsilon operate and E-type pronouns[END_REF] whose type is Λα.(α → t) → α. When this polymorphic ι (Λα...) is specialised to the type v (α := v) it becomes of type (v → t) → v and when applied to assinatura : (v → t) it yields a term ι{v}assinatura of type v whose short hand in the examples is written (sig) v . This term introduces a presupposition: assinatura(ι(assinatura)), saying that the designed event is an "assinatura".

In the examples, let us denote by atra3 : (v → t) the predicate "atrasou três dias" (took three days) which applies to events and by ilg : φ → t the predicate "estava ilegivel" (was illegible) that applies to physical objects. Both predicates are computed from the lexicon, but we cannot include the details.

(1)"A assinatura atrasou três dias" 1 (2)"A assinatura estava ilegível" 2 λy v .(atras3 v→t y)(sig v ) (λy ϕ .ilg ϕ→t y)((sig v )) atras3 v→t (sig v ) (λy ϕ .ilg ϕ→t y)(g v→ϕ (sig v )) ilg ϕ→t (g sig v ) ϕ Now let us show that the co-predication between "took three days" and "was illegible" cannot be derived. Firstly, the conjunction of two predicates that apply to different types (different view of the same object) is depicted using the second order typing. The "and" formalisation is:

ΛαΛβ

λP α→t λQ β→t Λξ λx ξ λf ξ→α λg ξ→β &(P (f x))(Q(g x))
The instantiations for our example should be as follows:

P = atras3, α = v, f = Id v , Q = ilg, β = ϕ, g = f and ξ = v, x = sig v .
This polymorphic "and" takes as arguments two properties P (here: astras3) and Q (here: ilg) which apply to entities of type α (here: v) and β (here: ϕ), returning a predicate that applies to a term x of type ξ. This predicates says that x of type ξ (here sig of type v) which via some f (here Id) can be viewed as an object of type α (here v) enjoying P (here atras3(x)) and that the same x can also be viewed via some g (here f ϕ ) as an object of type β (here φ) enjoying Q (here ilg(f ϕ (x))). -hence x has both properties (here atras3(x)&ilg(f ϕ (x))), provided the proper meanings of x are considered.

The constraint that both the identity and the result are rigid modifiers, means that if one of the occurrences of the argument of the predicate is used via a modifier, so are the others. Here, if one occurrences is the process itself (through the identity) or the result (through the optional modifier) it ought to be used with the same meaning for each of the occurrences -the presence of the identity in the lexical entry allows us to express that the original type itself is incompatible with others that are derived from it. As expected, this flexible/rigid distinction properly blocks the above co-predication that effectively cannot be derived. A less strict rule is possible: such compound infringing the rigid rule are given bad marks.

"The signing was delayed by three days." Example from http://noticias.uol.com. br/inter/efe/2004/03/05/ult1808u6970.jhtm.

"The signature was illegible." Example from http://www.reclameaqui.com.br/

3372739/dix-saude/cancelamento-do-plano-a-mais-de-um-mes-e-nada/.