Stéphanie Bay

Benoit Geller

Member, IEEE Alexandre Renaux

Jean-Pierre Barbot

Jean-Marc Brossier

On the Hybrid Cramér-Rao bound and its application to dynamical phase estimation

This letter deals with the Cramér-Rao bound for the estimation of a hybrid vector with both random and deterministic parameters. We point out the specificity of the case when the deterministic and the random vectors of parameters are statistically dependent. The relevance of this expression is illustrated by studying a practical phase estimation problem in a non data-aided communication context.

I. INTRODUCTION

A natural problematic when designing an estimator is the evaluation of its performance. Lower bounds on the Mean Square Error (MSE) mainly answer this question and the well known Cramér-Rao Bound (CRB) is widely used by the signal processing community. Depending on assumptions on the parameters, the CRB has different expressions. When the vector of parameters is assumed to be deterministic, we obtain the standard CRB [START_REF] Cramér | Mathematical Methods of Statistics, ser[END_REF] and when the vector of parameters is assumed to be random with an a priori probability density function (pdf), we obtain the so-called Bayesian CRB [START_REF] Van Trees | Detection, Estimation and Modulation Theory[END_REF].

At the end of the eighties, an extension combining both the standard and the Bayesian CRBs has been proposed [START_REF] Rockah | Array shape calibration using sources in unknown locations-part I: Far-field sources[END_REF]. Indeed, in some practical scenarios, it is natural to represent the parameter vector by a deterministic part and by a random part. This bound has thus been called the Hybrid CRB (HCRB) and a nice tutorial can be found in [START_REF] Messer | The hybrid Cramér-Rao lower bound -from practice to theory[END_REF].

Until now, results available in the literature essentially focussed on the case where the deterministic part and the random part of the parameter vector are assumed to be statistically independent (see, e.g., Eqn. [START_REF] Guarnieri | Hybrid Cramér-Rao bounds for crustal displacement field estimators in SAR interferometry[END_REF] in [START_REF] Rockah | Array shape calibration using sources in unknown locations-part I: Far-field sources[END_REF], Eqn. in [START_REF] Messer | The hybrid Cramér-Rao lower bound -from practice to theory[END_REF] and Eqn. (13) in [START_REF] Guarnieri | Hybrid Cramér-Rao bounds for crustal displacement field estimators in SAR interferometry[END_REF]). To the best of our knowledge, a closed-form expression of the HCRB with a statistical dependence between the deterministic and the random parameters has never been reported in the literature. The goal of this paper is then twofold. First, in Section II, we remind the structure of the HCRB and we point out the specificity of the case when the deterministic part and the random part of the parameter vector are statistically dependent. Second, in Section III, motivated by this analysis we give a closed-form expression of the proposed bound in the practical case of a dynamical phase subject to a linear drift in a non data-aided communication context.

II. THE HYBRID CRAM ÉR-RAO BOUND

A. Background

Let µ = µ T r µ T d T ∈
R n be the parameter vector that we have to estimate. This vector is split into two subvectors µ d and µ r where µ d is assumed to be a (n -m) × 1 deterministic vector and µ r is assumed to be a m × 1 random vector with an a priori pdf p (µ r ). The true value of µ d will be denoted µ d . We consider μ(y) as an estimator of µ where y is the observation vector. The HCRB satisfies the following inequality on the MSE

E y,µ r |µ d ( μ(y) -µ) ( μ(y) -µ) T µ d ≥ H -1 (µ d ) , (1)
where H (µ d ) ∈ R n×n is the so-called Hybrid Information Matrix (HIM) defined as [START_REF] Rockah | Array shape calibration using sources in unknown locations-part I: Far-field sources[END_REF] H

(µ d ) = E y,µ r |µ d -∆ µ µ log p( y, µ r | µ d ) µ d , (2)
where

∆ ν η k,l = ∂ 2 ∂[η] k ∂[ν] l .
When the deterministic and the random parts of the parameter vector are assumed to be independent, and after some algebraic manipulations, the HIM can be rewritten as (see [START_REF] Messer | The hybrid Cramér-Rao lower bound -from practice to theory[END_REF], Eqn. (18))

H (µ d ) = E µ r [F(µ d , µ r )] +   E µ r -∆ µr µ r log p (µ r ) 0 m×(n-m) 0 (n-m)×m 0 (n-m)×(n-m)   , (3)
where

F(µ d , µ r ) = E y|µ d ,µ r -∆ µ µ log p( y| µ d , µ r ) µ d . ( 4
)
With this aforementioned structure, it is straightforward to reobtain the standard and the Bayesian CRBs. Indeed, if µ = µ d , we have

H -1 (µ d ) = E y|µ d -∆ µ d µ d log p( y| µ d ) µ d -1 , ( 5
)
which is the standard CRB, and, if µ = µ r , we have

H -1 = E y,µ r -∆ µ r µ r log p( y| µ r ) + E µ r -∆ µ r µ r log p (µ r ) -1 , ( 6
)
which is the Bayesian CRB.

February 20, 2008 DRAFT

B. Extension when µ r and µ d are statistically dependent

We now assume a possible statistical dependence between µ r and µ d . In other words, µ r is now assumed to be a m × 1 random vector with an a priori pdf p

( µ r | µ d ) = p (µ r ).
Based on the HIM definition given by Eqn. ( 2) and expending the log-likelihood as log p y, µ r |µ d = log p (y|µ d , µ r )+ log p (µ r |µ d ), we obtain the following HIM

H (µ d ) = E µr |µ d [F(µ d , µ r )] + E µr |µd -∆ µ µ log p (µ r |µ d ) µ d , ( 7
)
where F(µ d , µ r ) is given by Eqn. [START_REF] Messer | The hybrid Cramér-Rao lower bound -from practice to theory[END_REF].

In order to explicitly show the modification in comparison with the HIM given by Eqn. ( 3), H (µ d ) can be rewritten as follows

H (µ d ) = E µ r |µ d [F(µ d , µ r )] + E µ r |µ d     -∆ µ r µ r log p (µ r |µ d ) -∆ µ r µ d log p (µ r |µ d ) µ d -∆ µ r µ d log p (µ r |µ d ) µ d T -∆ µ d µ d log p (µ r |µ d ) µ d     . (8)
Obviously, if we assume p ( µ r | µ d ) = p (µ r ) in this expression, we straightforwardly reobtain Eqn. (3).

Based on this structure, one now has to prove that there is still an inequality, i.e., a lower bound on the MSE,

E y,µ r |µ d ( μ(y) -µ) ( μ(y) -µ) T µ d ≥ H -1 (µ d ) , (9)
when H (µ d ) is given by Eqn. [START_REF] Mcneill | Jitter in ring oscillators[END_REF].

Proof: Following the idea of [START_REF] Messer | The hybrid Cramér-Rao lower bound -from practice to theory[END_REF] to prove the inequality (1), one defines a vector h such that

h =   ∇ µ log p (y, µ r |µ d )| µ d μ (y) -µ| µ d   , ( 10
)
where

∇ µ = ∂ ∂[µ] 1 • • • ∂ ∂[µ] n T .
Consequently, the non-negative definite matrix G (µ d ) = E y,µ r |µ d h h T can be decomposed as the following block matrix

G (µ d ) =   H (µ d ) L (µ d ) L T (µ d ) R (µ d )   , ( 11
)
where R (µ d ) is the covariance matrix of μ (y), i.e.,

R (µ d ) = E y,µ r |µ d ( μ(y) -µ) ( μ(y) -µ) T µ d , ( 12
)
and, where L (µ d ) is given by

L (µ d ) = E y,µ r |µ d ∇ µ log p (y, µ r |µ d )| µ d μ (y) -µ| µ d T . ( 13
) Since G (µ d ) ≥ 0, its Schur complement satisfies R (µ d ) ≥ L T (µ d ) H -1 (µ d ) L (µ d ) . ( 14
)
It is straightforward to show that, for an unbiased estimator w.r.t. the pdf p ( y,

µ r | µ d ), L (µ d ) = I n×n .
Consequently, the inequality (9) is proved and H -1 (µ d ) is a lower bound on the MSE.

February 20, 2008 DRAFT

III. HCRB FOR A DYNAMICAL PHASE ESTIMATION PROBLEM

In [START_REF] Bay | Analytic and asymptotic analysis of Bayesian Cramér-Rao bound for dynamical phase offset estimation[END_REF], we have proposed a closed-form expression of the Bayesian CRB for the estimation of the phase offset for a BPSK transmission in a non data-aided context. In this section, we extend these previous results by providing a closed-form expression of the HCRB for the estimation of the phase offset and also of the linear drift. In this more realistic scenario, we show that we have to take into account the statistical dependence between the parameters and, consequently, the HCRB given by Eqn. ( 3) is not adapted to this problem.

A. Observation and state models

We consider a linearly modulated signal, obtained by applying to a square-root Nyquist transmit filter an unknown

symbol sequence a = (a 1 • • • a K )
T taken from a unit energy BPSK constellation. The signal is transmitted over an additive white Gaussian noise channel. The output signal is sampled at the symbol rate which yields to the observations

y k = a k e jθ k + n k with k = 1 . . . K, ( 15
)
where {n k } is a sequence of i.i.d., circular, zero mean complex Gaussian noise variables with variance σ 2 n . We consider that the system operates in a non Data-Aided synchronization mode, i.e., the transmitted symbols are i.i.d.

with P r (a k = ±1) = 1 2 . In practice, several sources of distortions affect the phase. An efficient model representing these effects is the so-called Brownian phase with a linear drift widely studied in the literature (see, e.g., [START_REF] Amblard | Phase tracking: what do we gain from optimality? Particle filtering versus phase-locked loops[END_REF] [8] [9]). This model takes into account a constant frequency shift between the oscillators of the transmitter and of the receiver, the uncertainities due to clocks, and, the jitters of oscillators. The Brownian phase model with a linear drift is given as follows

θ k = θ k-1 + ξ + w k with k = 2 . . . K, ( 16
)
where, for any index k, {θ k } is the sequence of phases to be estimated, ξ represents the deterministic unknown linear drift with true value ξ , and where {w k } is an i.i.d. sequence of centered Gaussian random variables with known variance σ 2 w . The parameter vector of interest is then made up of both random and deterministic parameters µ

= µ T r µ d T where µ r = θ = (θ 1 • • • θ K )
T and µ d = ξ. Moreover, from Eqn. (16), it is clear that p ( θ| ξ ) = p (θ).

B. Derivation of the HCRB

For notational convenience, we drop the dependence of the different matrices on µ d = ξ in the remainder of this paper. From Eqn. ( 8), the HIM H can be rewritten into a block matrix

H =   H 11 h 12 h 21 H 22   , where,              H 11 = E y,θ|ξ -∆ θ θ log p (y|θ, ξ) ξ + E θ|ξ -∆ θ θ log p (θ|ξ ) , h 12 = h T 21 = E y,θ|ξ -∆ ξ θ log p (y|θ, ξ) ξ + E θ|ξ -∆ ξ θ log p (θ|ξ) ξ , H 22 = E y,θ|ξ -∆ ξ ξ log p (y|θ, ξ) ξ + E θ|ξ -∆ ξ ξ log p (θ|ξ) ξ . ( 17
)
February 20, 2008 DRAFT These blocks only depend on the log-likelihoods log p ( y| θ, ξ ) and log p ( θ| ξ ). Let us set y = (y 1 • • • y K )

T and assume that the initial phase θ 1 does not depend on ξ, i.e., p (θ 1 |ξ ) = p (θ 1 ). Using Eqn. ( 15) and ( 16), i.e., the Gaussian nature of the noise and the equiprobability of the symbols, one has

   log p (y|θ, ξ ) = K k=1 -log πσ 2 n -1+ y k 2 σ 2 n + log cosh 2 σ 2 n y k e -jθ k , log p (θ|ξ ) = log p (θ 1 ) + (K -1) log 1 √ 2πσ w - K k=2 (θ k -θ k-1 -ξ ) 2 2σ 2 w . ( 18
)
• Expression of H 11 : assuming that we have no prior knowledge, i.e., E θ 1 ∆ θ 1 θ 1 log p(θ 1 ) = 0, it is shown in [START_REF] Bay | Analytic and asymptotic analysis of Bayesian Cramér-Rao bound for dynamical phase offset estimation[END_REF] (due to the order one Markov structure exhibited by Eqn. ( 16)) that H 11 takes the following tridiagonal structure

H 11 = b             A + 1 1 0 • • • 0 1 A 1 . . . . . . 0 . . . . . . . . . 0 . . . . . . 1 A 1 0 • • • 0 1 A + 1             , ( 19
)
where b = -1/σ 2 w , and, where

A = -σ 2 w J D -2 with J D = E y,θ|ξ -∆ θ k θ k log p (y k |θ k , ξ ) . • Expression of h 12 : since, from Eqn. (18), log p (y|θ, ξ ) is independent of ξ , ∆ ξ θ log p (y|θ, ξ) ξ = 0.
Consequently,

h 12 = E θ|ξ -∆ ξ θ log p (θ|ξ) ξ . ( 20
)
Using the state model, we have

           ∆ θ1 ξ log p (θ|ξ) ξ = -1 σ 2 w , ∆ θ K ξ log p (θ|ξ) ξ = 1 σ 2 w , ∆ θ k ξ log p (θ|ξ) ξ = 0 for k ∈ {2, . . . , K -1} . ( 21
)
Applying the expectation operator E θ|ξ [.], we obtain

h 12 = 1 σ 2 w 0 1×K-2 -1 σ 2 w T . ( 22
)
• Expression of H 22 : since, from Eqn. (18), log p (y|θ, ξ ) is independent of ξ , ∆ ξ ξ log p (y|θ, ξ)

ξ = 0.
Consequently,

H 22 = E θ|ξ -∆ ξ ξ log p (θ|ξ) ξ = K -1 σ 2 w . ( 23
)
• Expression of the HCRB: we now give the expression of H -1 which bounds the MSE. Thanks to the blockmatrix inversion formula, we have

H -1 =   H -1 11 + V K -1 λ H -1 11 h 12 -1 λ h T 12 H -1 11 1 λ   , ( 24
)
where

λ = K-1 σ 2 w -h T 12 H -1 11 h 12 and V K = 1 λ H -1 11 h 12 h T 12 H -1 11 .
February 20, 2008 DRAFT

We start to compute λ corresponding to the inverse of the minimal bound on the MSE of ξ. Due to the particular structure of matrices H 11 and h 12 (Eqn. ( 19) and ( 22)), we obtain

λ = K -1 σ 2 w - 2 σ 4 w H -1 11 1,1 -H -1 11 1,K . (25)
From Eqn. (19), thanks to the cofactor expression in the matrix inversion formula we have for any index k,

H -1 11 1,k = b k-1 |H11| (d K-k + b d K-k-1 ) where d k is the determinant of the following k × k matrix D k D k = b             A 1 0 • • • 0 1 A 1 . . . . . . 0 . . . . . . . . . 0 . . . . . . 1 A 1 0 • • • 0 1 A             . ( 26
)
The sequence {d k } satisfies the following recursive equation

d k = A b d k-1 -b 2 d k-2 with d 0 = 1 and d 1 = bA. d k can thus be written as d k = ρ 1 (r 1 ) k + ρ 2 (r 2 ) k where r 1 , r 2 , ρ 1 and ρ 2 are given by    r 1 = b 2 A + √ A 2 -4 , r 2 = b 2 A - √ A 2 -4 , ρ 1 = √ A 2 -4+A 2 √ A 2 -4 , ρ 2 = √ A 2 -4-A 2 √ A 2 -4 . (27)
Consequently,

H -1 11 1,k = b k-1 |H 11 | ρ 1 r K-k-1 1 (r 1 + b) + ρ 2 r K-k-1 2 (r 2 + b) , (28)
and

λ = K -1 σ 2 w - 2 σ 4 w |H 11 | ρ 1 r K-2 1 (r 1 + b) + ρ 2 r K-2 2 (r 2 + b) -b K-1 . ( 29
)
From the definition of V K , we have

[V K ] k,k = 1 λσ 4 w H -1 11 1,k -H -1 11 1,K+1-k 2 . ( 30
)
Using Eqn. ( 24), (28), and (30), we obtain, for any index k, the analytical expression of the HCRB diagonal elements

H -1 k,k = 1 |H 11 | ρ 2 1 (b + r 1 ) 2 r K-3 1 + ρ 2 2 (b + r 2 ) 2 r K-3 2 - b 2 A -2 (r k-2 1 r K-k-1 2 + r K-k-1 1 r k-2 2 ) + 1 λ σ 4 w |H 11 | 2 b k-1 ρ 1 (r 1 ) K-k-1 (b + r 1 ) + ρ 2 (r 2 ) K-k-1 (b + r 2 ) + b K-k ρ 1 (r 1 ) k-2 (b + r 1 ) + ρ 2 (r 2 ) k-2 (b + r 2 ) 2 . ( 31
)
Remark: Note that, if Eqn. (3) was used instead of Eqn. ( 8), the HIM would not be invertible.

C. Simulation results

We now illustrate the behavior of the HCRB versus the Signal-to-Noise Ratio (SNR) defined by 1 • At high SNR, we first notice that HCRB ξ converges to its horizontal asymptote given by σ 2 w K-1 which is the standard CRB when θ is assumed to be known. The observation noise compared to the phase noise is then not significant enough to disturb the estimation of ξ; consequently HCRB ξ depends only on the phase noise and on the number of observations. Concerning the bounds on θ K , HCRB θ k and BCRB θ k both have the same asymptote given by σ 2 n 2 which is the Modified CRB (MCRB) for one observation (see [10]). It means that, at high SNR, the observation y K is self-sufficient to estimate θ K and the error on ξ does not disturb the performance on θ K . Moreover, the HCRB logically tends to the Data-Aided HCRB.

• For median SNR, HCRB θ K and HCRB ξ leave their respective asymptote. HCRB θ K is still lower bounded by the BCRB and upper bounded by the high-SNR asymptote. This stems from the fact that taking into account a block of observations instead of one observation necessarily improves the performance. However, for large

σ 2
w values (e.g., σ 2 w = 0.1 rad 2 ), HCRB θ K stays close to the MCRB because the correlation between the phase offsets θ k is less significant than the information brought by the observation y K . Moreover, when σ 2 w tends to 0, HCRB θ K is above the BCRB because performance is now limited by the accuracy on the parameter ξ.

• At low SNR, n k is preponderant compared to w k . Both HCRB ξ and HCRB θ K do not depend on σ 2 w : the lack of knowledge on ξ directly affects the estimation on θ K . As expected, the knowledge of the symbols (Data-Aided HCRB) leads to a better estimation of θ and ξ.

IV. CONCLUSION

In this paper, we have studied the hybrid Cramér-Rao bound when the random and the deterministic parts of the parameter vector are statistically dependent. We have applied this bound in order to evaluate the performance of a dynamical phase estimator where the linear drift is unknown in a non data-aided context. In particular, we have illustrated the effect of this unknown linear drift on the phase estimation performance. [9] Demir, A. Mehrotra, and J. Roychowdhury, "Phase noise in oscillators: a unifying theory and numerical methods for characterization," IEEE Trans. on Circuits and Systems, vol. 47, pp. 655-674, May 2000.

[10] M. Moeneclaey, "On the true and the modified Cramér-Rao bounds for the estimation of a scalar parameter in the presence of nuisance parameters," IEEE Transactions on Communications, vol. 46, no. 11, pp. 1536-1544, Nov. 1998.

  (13) Stéphanie Bay and Jean-Pierre Barbot are with the Ecole Normale Supérieure de Cachan, SATIE Laboratory, 61, avenue du President Wilson 94235 Cachan, France. (e-mail: {bay,barbot}@satie.ens-cachan.fr) Benoit Geller is with Ecole Nationale Supérieure des Techniques Avancées, Laboratory of Electronics and Computer Engineering (LEI), 32, Boulevard Victor 75015 Paris, France. (e-mail: geller@ensta.fr) Alexandre Renaux is with University Paris-Sud 11, Laboratory of Signals and Systems, Supélec, 3 rue Joliot-Curie, 91192 Gif-sur-Yvette cedex, France. (e-mail: renaux@lss.supelec.fr) Jean-Marc Brossier is with Grenoble Institute of Technology, GIPSA Laboratory, 961 rue de la Houille Blanche Domaine universitaire -BP 46 38402 Saint Martin d'Hères cedex, France. (e-mail: jean-marc.brossier@gipsa-lab.inpg.fr) This work was supported in part by the French ANR (Agence Nationale de la Recherche), LURGA project and by the European network of excellence NEWCOM++. February 20, 2008 DRAFT

σ 2 n

 2 . We consider a block of K = 40 BPSK transmitted symbols. For two distinct phase-noise variances (σ 2 w = 0.1 rad 2 and σ 2 w → 0 rad 2 ), Figure1superimposes on one side the HCRB (see Eqn. (31)), the Data-Aided HCRB J D = 2 see Eqn. (21) in[START_REF] Bay | Analytic and asymptotic analysis of Bayesian Cramér-Rao bound for dynamical phase offset estimation[END_REF]) on θ K . For the same scenario, Figure2superimposes on one side the HCRB (see Eqn. (25)) and the Data-Aided HCRB on ξ.

Fig. 1 .

 1 Fig. 1. Bounds on θ K versus the SNR (K = 40 observations, σ 2 w = 0.1 rad 2 , and σ 2 w → 0 rad 2 , J D evaluated over 10 8 Monte-Carlo trials).

February 2 Fig. 2 .

 22 Fig. 2. Bounds on ξ versus the SNR (K = 40 observations, σ 2 w = 0.1 rad 2 and σ 2 w → 0 rad 2 , J D evaluated over 10 8 Monte-Carlo trials).


Patrick Maillé

Gwendal Simon

Bruno Tuffin

Toward a Net Neutrality Debate that Conforms to the 2010s

Keywords: FCC, ARCEP, CRTC, OFCOM, ... Service providers Google, Facebook, Yahoo!, .. TV producers, Hollywood, YouTube, Netflix, ... Device builders Samsung, Cisco, Alcatel-Lucent, HTC, ... CDNs/Cloud Akamai, Amazon, Cloud-Front, Limelight, ... Transit networks Cogent, Level 3, CenturyLink, ... ISPs Comcast, Verizon, Orange, ... Users population, companies,

Introduction

Net neutrality is "the principle that Internet service providers should enable access to all content and applications equally, regardless of the source, without favoring or blocking particular online services or websites" (e.g., Oxford Dictionary). This universality principle for the Internet has been present for a long time in the US. In the 1996 Telecommunications Act [3], the Federal Communications Commission (FCC) declared Internet Service Providers (ISPs) as "common carriers" providing a public service. In February 2015, it voted to regulate broadband Internet service as a public utility. The same principle was adopted by the European Parliament in 2014 with a vote that restricts the 1 ability of ISPs to charge specific Service Providers (SPs) [START_REF]pean parliament and of the council laying down measures concerning the european single market for electronic communications and to achieve a connected continent[END_REF]. This principle is often translated as guaranteeing that all packets are treated equally at each intermediary step.

The network neutrality debate started in the 1990s, amongst growing concerns about the business models of network operators, which have to deal with several challenging trends, including the increase of traffic volume, the growing traffic asymmetry between operators, and the fast decrease of transit prices. Recently, spectacular disputes between major Internet actors1 as well as vehement reactions from user associations and governments 2 have drawn attention to that debate. Arguments and discussions on possible regulation of the Internet resulted in refined definitions of network neutrality [START_REF]Net Neutrality or Net Neutering: Should Broadband Internet Services be Regulated[END_REF][START_REF] Odlyzko | Network neutrality, search neutrality, and the never-ending conflict between efficiency and fairness in markets[END_REF][START_REF] Wu | Network neutrality, broadband discrimination[END_REF].

In this paper, we show that the current focus on ISPs behavior is too restrictive in the era of cloud -based content delivery. Indeed, the regulatory bodies in the US and Europe-those most active with regard to neutrality-focus on the ISP/internetwork level, but the value chain in the Internet is not restricted to only ISPs and Content Providers (CPs). In particular, web portals (especially search engines) and Content Delivery Networks (CDNs) have become key components in the delivery chain, while they surprisingly remain absent from the debate. Despite some recurring criticism [START_REF] Pasquale | Federal Search Commission? Access, Fairness and Accountability in the Law of Search[END_REF], search engines have not been forced to conform to any universal policy yet; similarly, to the best of our knowledge, the only mention of CDNs in official net neutrality reports is from the Norwegian regulator, according to which "the ordinary use of CDN servers is not a breach of net neutrality" [9]. Yet, we show in Section 3 that the presence of a CDN, or biases in search listings, can be exploited against the fundamental principles of neutrality.

More generally, we think the net neutrality debate should be extended to all the actors involved in the Internet delivery chain. Our goal is not to discuss the validity of neutrality proponents and opponents, but rather to make a first step toward a global framework that would be more appropriate for the definition of public regulating rules (if any) in the era of the cloud and more generally of information-centric networks. We present our vision of such a framework in Section 4. To foster scientific activities, we highlight in Section 5 some topics, where the rigorous analysis of the supposedly "neutral" network scientists could be especially appreciated by decision makers.

Net neutrality is a highly politicized topic. We have tried to be politically neutral and scientifically objective in this article; 3 however, we realize that some readers may interpret some of our statements as being politicized. If so, we assure that any perceived politicization is unintentional. transit networks often act as intermediaries when users and contents are hosted by different ISPs; CDNs play a key role by storing content closer to users, thereby reducing transit costs and improving performance for that content 4 ; device builders may introduce biases through the features of their products (possibly colluding with some other actors); and search engines (seen as service providers) directly affect the accessibility (visibility) of content. Regulatory bodies therefore face the delicate task of defending fairness and universality principles in this complex ecosystem, where ever-changing technical and business conditions prevent (or considerably complicate) analysis and comparison.

An Intricate Delivery Chain

The success of a given CP partly depends on its interactions with other actors. For example, its visibility is strongly affected by the behavior of search engines, which leaves space for biases that we point out later. Another example, the engagement of end-users for a CP is impacted by the delivery performance [START_REF] Krishnan | Video stream quality impacts viewer behavior: Inferring causality using quasi-experimental designs[END_REF], which is usually under the responsibility of cloud providers and CDNs. Here again, non-neutral behaviors from those actors impact the CP.

The actors that have direct economic relationship negotiate Service Level Agreements (SLAs). In the case of content delivery, the intricate relationships between actors yield chains of SLAs, which are based on network-oriented metrics such as the throughput and the ratio of packet losses. But inferring the Quality of Experience (QoE) from a chain of network metrics, potentially cascading, is a difficult task. Most CPs struggle both to assess the QoE of their customers and to identify the failing actor when the quality of delivery is below expectations.

This complex chain of content delivery and the difficulties to assess the performance of the involved actors potentially leave some room for intermediaries to favor some CPs against others, thereby violating the principles of neutrality. We show later that a packet-level interpretation of neutrality can unfortunately not prevent intermediaries from biasing the competition among CPs.

Existing Protection Tools

Net neutrality opponents often argue that existing legal frameworks already protect fair competition, and thus no additional regulation is needed. This is the case in the US with the antitrust law and the associated Federal Trade Commission (FTC). Such antitrust arsenals do not however address some of the points that are frequently raised by net neutrality proponents.

First, the antitrust framework acts reactively and on a per-case basis although the neutrality regulation looks for pro-active solutions. With the antitrust framework, an actor that reckons to suffer economically from the (nonneutral) behavior of another actor should file a complaint with the FTC to obtain redress. The framework requires lawsuits and judgements once a plaintiff has built a case. On the other hand, neutrality regulation aims at anticipating the possible problems and preventing them by specifying allowed and forbidden behaviors.

Second, focusing on competition corresponds to a partial view-since limited to market aspects-which ignores some other principles raised by net neutrality proponents, such as freedom of speech and innovation fostering (the latter involving equal access to resources regardless of the monetary capacity of CPs).

In our opinion, neutrality rules are essentially complementary to antitrust laws, and setting boundaries between the application field of antitrust laws and that of neutrality rules should be part of the debate as well.

Examples of Neutrality Breaches

As previously stated, the supply chain between content and users includes multiple intermediary actors, which act in a free market with reciprocate engagements based on SLAs. We show in the following that these intermediaries can generate significant biases in the network and create a breach in what represents network neutrality for most people. We focus on CDNs and search engines, which are two key actors in the supply chain.

Competition Biases due to CDNs

First, we consider the impact of a profit-driven CDN provider, by developing two scenarios depicted in Figure 2. Both scenarios model the economic reality of the traditional interactions between ISPs, CDNs, and CPs. : Two cases of neutrality breaches due to the presence of a CDN. On the left, two content providers, each with its own origin server, compete for the storage resources of a regional CDN edge server [START_REF] Maillé | How Neutral is a CDN: An Economic Approach[END_REF]. On the right, two ISPs are served by a shared CDN regional server, but one ISP (here ISP 𝐵 ) installs its own edge server.

𝑂 1 𝑂 2 𝑂 Origin servers 𝐸 𝐸 𝑅 𝑠𝑅 CDN regional server 𝑞 1 𝑞 2 𝑞
A CDN serves a large population of end-users by the means of edge servers, which are usually located near the Point of Peering (PoP) between ISPs. When the CDN receives a request, it has two handling options: either (1) fetch the requested content from the origin server, or (2) fetch the requested content from the edge server. Option 1 incurs transit costs (which can be low if the CDN also owns a transit network, but large otherwise), while Option 2 leads to improved user quality of experience (QoE) 5 , and no transit costs are incurred. Thus, in terms of cost and quality, Option 2 should always be chosen; however, storage/cache and processing resources at 𝐸 are finite. Hence, some requests must be handled by Option 1 (fetch from origin server). A rational CDN will maximize profits by optimizing the balance between Options 1 and 2, playing on caching decisions at 𝐸 (or possibly on the number/capacity of edge servers).

Note that we focus on static content in these models for the sake of clarity. But the CDN models for interactive services, such as live streaming and cloud gaming, are not essentially different: the edge server should still reserve some hardware resources (mostly computing power here) to serve a subset of users. For example in live rate-adaptive video streaming systems, the edge server should both get multiple video representations from the origin server and manage the multiple concurrent requests for the videos.

Biasing the competition among Content Providers

Let us consider the scenario shown in the left part of Figure 2. That scenario is extensively discussed in [START_REF] Maillé | How Neutral is a CDN: An Economic Approach[END_REF]. To reach customers in a given ISP, two competing CPs subscribe to the CDN, which has to share the privileged resources in the edge server 𝐸 among the CPs. The scientific literature comprehensively studies the performances of content replacement strategies in dynamic scenarios, but rarely integrates the economic parameters, namely here: 𝑖) the transit costs 𝑞 1 and 𝑞 2 , which can differ among CPs due to different paths to reach the ISP of interest; 𝑖𝑖) the charging policy of the CDN, with prices per volume possibly differing depending on whether users are served from an edge or an origin server.

A profit-maximizing CDN stores in the edge server the content that yields the largest revenues. If both transit costs 𝑞 1 and 𝑞 2 are equal, and if the CDN charges both CPs the same way, then it should store the most popular content. However, if the transit costs of one CP (say CP 1 ) are higher than the other, then the interest for the CDN of storing CP 1 content in the edge server exceeds that of storing CP 2 content of comparable popularity, which both benefits CP 1 customers and harm CP 2 customers. In other words, the internal costs of a CDN can distort competition.

The situation gets even more problematic when the CDN charges CPs differently. In [START_REF] Maillé | How Neutral is a CDN: An Economic Approach[END_REF] we study a case based on real popularity measures, CP 2 being (on average) approximately five times more popular than CP 1 . Figure 3 displays the "satisfaction" (a normalized customer QoE measure) of both CPs, according to the ratio of content from CP 1 in the edge server.

The "standard scenario" situation corresponds to the "optimal" (for the CDN) sharing of the edge server when both CPs are charged identically. CP 2 being more popular, approximately five-sixths of the edge server is filled with CP 2 content, and the average satisfaction of CP 2 customers exceeds that of CP 1 customers. Hence the CDN favors the dominant player although both content providers pay the same price. Note however that the CDN strategy is the most efficient in terms of traffic reduction and overall satisfaction of endusers.

Now, assume that one of the CPs pays more for its requests fulfilled by the edge server. Two of the vertical lines correspond to two extrema where one CP pays ten times the basic price (see [START_REF] Maillé | How Neutral is a CDN: An Economic Approach[END_REF] for more details). When CP 2 pays more (typically to reinforce its dominant position), the satisfaction gain of CP 2 is small but competitor satisfaction drops more significantly (from 0.94 to 0.89), possibly leading some CP 1 customers to churn. Alternatively, when CP 1 pays more (typically to increase its audience with better quality), the impact is less spectacular. To summarize, the dominant player can leverage the CDN to harm its competitors although challengers cannot.

Biasing the Competition Among ISPs

Consider now the scenario depicted in the right context of Figure 2, where a CDN serves two ISPs. Initially users of both ISPs (labelled 𝐴 and 𝐵) get data either from the origin server 𝑂 or from a regional "shared" edge server 𝐸 𝑅 . The size of the cache in 𝐸 𝑅 is chosen by the CDN according to the transit cost 𝑞 from 𝑂 to 𝐸 𝑅 (based on traffic) and the storage cost 𝑠 𝑅 at 𝐸 𝑅 (based on volume). Now assume one ISP, say ISP 𝐵 , installs an edge server 𝐸 𝐵 within its network 6 with a storage cost 𝑠 𝐵 , in order to improve the QoE for its users through an agreement with the CDN. The profit-maximizing CDN now has four ways to store content: in none of the edge servers (so only in the origin server), in 𝐸 𝑅 only, in 𝐸 𝐵 only, or in both 𝐸 𝑅 and 𝐸 𝐵 .

The appearance of the edge server 𝐸 𝐵 can create conflict, in particular when content previously stored in 𝐸 𝑅 gets only stored in 𝐸 𝐵 (and no longer in 𝐸 𝑅 ): the QoE of ISP 𝐴 users is then degraded. Such a scenario is more likely to occur when the storage cost of 𝐸 𝐵 is significantly lower than 𝐸 𝑅 , or when ISP 𝐵 traffic is significantly bigger ISP 𝐴 traffic. In that case, the content stored in the regional "shared" edge server 𝐸 𝑅 is the "second" most popular in ISP 𝐵 , i.e., the most popular among the content not stored in 𝐸 𝐵 . Thus, agreements between CDNs and a given ISP can degrade the performances for users in another ISP.

Figure 4 illustrates the impact of edge-server 𝐸 𝐵 on the overall satisfaction (still a normalized QoE measure) for users in both networks, using real movie request data over two years obtained from a leading French Video on Demand (VoD) service and including the request origin networks. We set ISP 𝐵 as the network with the highest traffic, and affect to ISP 𝐴 the average traffic from all other networks. The 𝑦-axis shows the ratio between the satisfaction with and without the edge server 𝐸 𝐵 ; the 𝑥-axis corresponds to different storage costs 𝑠 𝐵 . We observe that the installation of an edge server in ISP 𝐵 has an (expected) positive impact on the satisfaction of ISP 𝐵 users, but also an (undesirable) negative impact on the satisfaction of ISP 𝐴 users. Observe that the latter exceeds the former for most of the storage cost prices on 𝐸 𝐵 : a powerful network operator does not need to implement any aggressive pricing policy to degrade the performances of competitors.

Search engines biasing the access to content

Search engines are the preferred way to discover and access content: according to ComScore, in 2014 about 21 billion searches have been treated each month in the US only. Search engines usually present their suggestions corresponding to keywords into two categories: 1) sponsored results, a list of advertisements related to the search and clearly defined as such, from which the engines make money when the links are clicked, and 2) organic results, a ranked list of links, believed to associate the most relevant items to the asked keywords since promotional results are assumed to be within sponsored links.

To increase revenues, search engines can be tempted to include in the organic results some results that are not among the most relevant but that can generate (direct and indirect) short-term revenues. A typical case is that of a content provider (called CP 1 ) vertically integrated with (i.e., owned by) the search engine. The engine is economically incentivized to rank that content higher [START_REF] L'ecuyer | Revenue-Maximizing Rankings for Online Platforms with Quality-Sensitive Consumers[END_REF].

Such temptation has raised a search neutrality debate, parallel to the one on network neutrality [START_REF] Maillé | Telecommunication Network Economics: From Theory to Applications[END_REF]. We do not target here any specific search engine nor claim one is biased: our goal is to show through a model that such biases can be motivated, and to investigate their impact on fairness among CPs.

The model designed in [START_REF] L'ecuyer | Revenue-Maximizing Rankings for Online Platforms with Quality-Sensitive Consumers[END_REF] analyzes the temptation to return "non-neutral" organic results. This model includes a long-term effect: end-users who are unsatisfied with the relevance of the organic results may stop using the search engine. The search engine objective is then to maximize its long-term revenue, which corresponds to the trade-off between the short-term revenue per visit on CP 1 and the longer-term number of visits on the search engine. The model helps to understand revenue-maximizing ranking strategies and to anticipate the impact of regulatory interventions for various scenarios. The principle is illustrated in Figure 5. Table 1 illustrates, for some arbitrarily chosen numerical values, the differences in visit rates to CPs due to the revenue-optimal ranking, the resulting average relevance of the engine's output, and the revenues it can generate, for different values of a parameter 𝛽, which represents the average revenue directly generated per visit to the search engine (via clicks on sponsored links, their assumed single source of revenue); the smaller 𝛽, the larger the bias because of larger (relative) incentives to create revenue through integrated CPs. In this scenario, the impact on the average relevance remains small (around 10%), but the impact on the visibility and the revenues of CPs is substantial, possibly threatening the survivability of non-integrated CPs.

Toward a Global Framework

Net neutrality proponents often put forward that every packet should be treated equally, which misses important aspects and does not prevent unfair situations. Indeed, that proposal focuses on the regulation of the third layer of the OSI model (the so-called network layer) although the end-users get the results of the policies of all intermediaries on the seven layers. As shown in the previous section, intermediaries can introduce significant biases fulfilling this "neutral" condition at the third layer. We therefore call for an extension of the debate to any class of actors having an influence in the way end-users consume content. Among them, CDNs, as well as the providers of data-centers, and search engines.

The need is then for a definition of neutrality that would be applicable to all actors, whose enforcement would prevent the behaviors pointed out in Section 3. Such a definition would need to go beyond the packet level, and may involve a classification of all actors into categories, for each of which specific principles should be followed. The regulation would then be specific to each level; below are a few examples:

• at the ISP level, no differentiation among packets according to the origin, destination, service, or price paid (the common suggestion);

• at the CDN level, caching of only the most popular content, independent of the provider, the type of content, or any monetary aspect;

• at the search engine level, ranking results based only on relevance to the query (and possibly the user), independent of the result page owner, the type of query, or possible payment.

Our purpose in the remaining of this section is not to put forward a clear framework for a new neutrality definition, ideally robust to network evolution, but rather to launch a discussion in the community towards that goal. To start that discussion, we hereafter pose a few questions and provide limited and probably flawed suggestions, which will (hopefully) ignite the debate.

Where should neutrality be applied?

The whole topology of the Internet should be abstracted to capture all the relevant actors that may introduce inappropriate biases. This effort should not be restricted to the physical infrastructure, but it should also consider the actors that contribute to the consumption of content by any means. Typically, search engines are not captured by the current framework targeting packet level and ISPs. This effort should also consider the new actors that are expected to implement and integrate their network management services into the network devices compatible with Network Function Virtualization (NFV) standards. More generally, all actors that transform the content and are part of the delivery should be considered.

A definition of neutrality should study behaviors (actor actions) at each actor level : this would imply defining levels for the different types of actions on content, in a way comparable to the OSI reference model. We would for instance need to characterize (in terms of level) the services provided by CDNs and by search engines. Players at a given level/layer should then be neutral in terms of protocols applied at this level, by avoiding any kind of differentiation through the compliance of level-specific rules, extending the current rules defined for ISPs at the network/transport layer.

Note that, unfortunately it is not easy to identify the level at which an actor plays. For example in France, Skype is considered by the regulator as a virtual network operator, which thus competes with ISPs, and not as a service (or application) provider.

What notion should neutrality be focusing on?

What is the goal of neutrality, what do we want to preserve? The current focus is about equal treatment: no differentiation would be allowed whatever the source, application or service; hence some induced notion of fairness.

Among the underlying principles is the universality of access. This is a baseline (as it only focuses on blocking), but we need stronger constraints in a neutrality definition. Slowing down some applications, or favoring others, is generally refused by neutrality proponents and does not conform to the packetbased equal treatment proposal. On the other hand, the supposed equal treatment at the packet level can be criticized since most Internet applications use TCP at the transport layer, which leads to unfair qualities of service offered since the TCP throughput varies with the user/provider distance (it is indeed inversely proportional to the square root of the round trip time). This characteristic of TCP can harm innovation in regions with bad network quality, even with perfectly equal packet treatment.

Defining the "no-differentiation" principle when talking about CDNs for instance seems subtle, because CDNs precisely differentiate services when selecting what content to cache. The notion of neutrality could be in terms of avoiding money-based differentiation: fairness considerations would help decide what to cache, but there can be several potential interpretations:

• Should any packet have the same probability to be cached? We do not see any reason to do that.

• Should it rather be the most popular content which is cached (what is usually assumed natural to limit the load on the network)? But couldn't we reason differently, by saying that all content providers should be equally cached, or at the end should offer the same quality of experience (something not done for the traditional neutrality principle though)?

It seems reasonable to set a "neutrality rule" requiring CDNs to cache the most popular content, independently of the source identity. But the focus could also be on reducing the overall network load by favoring the caching of distant content (that contributes more to the load since using more links) in addition to considering popularity.

In order to (partially) solve these questions, we suggest to define fairness, like neutrality, at the layer the actor plays (related to the question "where?"). This type of definition could also encompass search engines and ISPs. The basic idea is to have an "equal treatment", but one could also look at other notions, such as social welfare, user welfare, or even service welfare (note however that such efficiency measures are in general incompatible with fairness, an argument often used by non-neutrality defendants). The goal is to determine whether a behavior harms the desired goal at the layer it is operating.

When should it be controlled?

Should we ensure that any newcomer has a particular treatment? Regarding the current strategies of CDNs, the CPs in their initial stages are disadvantaged against the incumbents. However, favoring these companies would be in disagreement with the current neutrality rules of equal treatment. It is therefore a slippery slope, opening the door to "mission creep" situations where new types of biases would defeat the initial goal of achieving neutrality.

With the current focus of the debate, remark that this question is not really a problem, but it is if we try to have a broader view of the Internet network.

Potential next steps

To conclude this article, we would like to present a few issues to be tackled by the community.

Taking into account economics in future Internet studies In general the question of actor profit is rarely central in the literature related to network and service management. Computer science and electrical engineering scientists generally aim at maximizing the efficiency of their proposal regarding technical objectives and not economic ones. Typically, to our knowledge the literature related to Content Centric Network (CCN) does not deal with the economics of actors although, in our opinion, the revamping of the Internet cannot be seriously studied without taking economic factors into account.

Evaluating the impact of possible regulatory policies The role of a regulating agency is to recommend policies that guarantee widely accepted principles. As previously mentioned, the stated principles are not (yet?) well defined when it comes to neutrality. It is up to the scientific community to provide rigorous studies, as unbiased as possible, about potential policies, their impact on the considered actor level, and also the possible impact on the overall delivery chain.

Defining and studying new fairness models Our brief discussion in Section 4 highlights the lack of a better definition of fairness and in particular the subtle equilibrium between the guarantee of fair competition among actors pursuing similar objectives (the notion of actor level ) and the preservation of the motivation for investment toward satisfying end-users better.

Figure 1 :

 1 Figure 1: The main categories of actors in the Internet ecosystem; arrows represent a provider-customer (seller-buyer) relationship.

  Figure2: Two cases of neutrality breaches due to the presence of a CDN. On the left, two content providers, each with its own origin server, compete for the storage resources of a regional CDN edge server[START_REF] Maillé | How Neutral is a CDN: An Economic Approach[END_REF]. On the right, two ISPs are served by a shared CDN regional server, but one ISP (here ISP 𝐵 ) installs its own edge server.

  ratio of content from CP1 to all content in the edge server

Figure 3 :

 3 Figure 3: Satisfaction for both CP 1 and CP 2 customers according to the ratio of the edge server filled with content from CP 1

Figure 4 :

 4 Figure 4: Impact on user satisfaction of the creation of an edge server in ISP 𝐵 , computed as the ratio satisfaction with cache 𝐸 𝐵 satisfaction without 𝐸 𝐵 .

Figure 5 :

 5 Figure 5: Search engine whose ranking policy produces an average relevance of results and an average gain. The number of visits (i.e., popularity of the engine) depends on the average relevance.

Table 1 :

 1 Impacts of ranking when the search engine owns CP 1 .

			CP 1	other CP	CP 1	other CP
		Relevance revenue	revenue	visit rate visit rate
	Neutral	0.635	0.028	0.0283	0.057	0.057
	Non-neutral	0.618	0.066	0.0243	0.112	0.049
	(for 𝛽 = 1)	(-3%)	(+136%)	(-14%)	(+96%)	(-14%)
	Non-neutral	0.592	0.084	0.0215	0.140	0.043
	(for 𝛽 = .5)	(-7%)	(+200%)	(-24%)	(+146%)	(-25%)
	Non-neutral	0.568	0.093	0.0193	0.158	0.039
	(for 𝛽 = .25)	(-11%)	(+232%)	(-32%)	(+177%)	(-32%)

For example, Netflix witnessed a

25% reduction in traffic rates for Comcast users, before a deal was struck[START_REF] Brodkin | Netflix performance on Verizon and Comcast has been dropping for months[END_REF].2 See for example B. Obama's declaration on neutrality in Nov 2014.3 The authors receive funding from Orange (a french ISP) for their research activities, but the content presented in this paper is solely the responsibility of the authors.

The use of CDNs can be seen as against neutrality principles, since they offer an improved delivery for a fee, something new arrivants in the market may not be able to afford. We do not address this question here.

Note that QoE is mainly affected for services requiring high bandwidth or low latency.

In most cases, the ISP owns the edge server, which it rents to the CDN.


Ramdane Tami
email: ramdane.tami@insa-cvl.fr

Driss Boutat
email: driss.boutat@insa-cvl.fr

Didier Aubry
email: didier.aubry@univ-orleans.fr

Haoping Wang
email: hp.wang@njust.edu.cn

Partial observer normal form for nonlinear system $

Keywords: Observer normal form, Partial state observer, Nonlinear systems

In this paper, we investigate the estimation problem for a class of partially observable nonlinear systems. For the proposed Partial Observer Normal Form (PONF), necessary and sufficient conditions are deduced to guarantee the existence of a change of coordinates which can transform the studied system into the proposed PONF. Examples are provided to illustrate the effectiveness of the proposed results.

Introduction

Observability and observer design problem for nonlinear systems have been widely studied during last four decades, and many different methods have been proposed to treat different classes of nonlinear systems, such as adaptive observer, high-gain observer, sliding mode observer and so on Besanc ¸on (2007). In this work, we are interested in the approach based on geometrical transformations to bring the original system into a simple observer normal form. The advantage of this method is that, by well choosing the desired simple observer normal form, we can reuse the existing observers proposed in the literature to estimate the state of the transformed observer normal form, and then obtain the state estimation for the original system by inverting the deduced diffeomorphism. The literature about this technique is vast. Since the pioneer works of [START_REF] Bestle | Canonical form observer design for nonlinear time varying systems[END_REF]; [START_REF] Krener | Linearization by output injection and nonlinear observers[END_REF]) for single output systems and [START_REF] Krener | Nonlinear observers with linearizable error dynamics[END_REF]; [START_REF] Xia | Nonlinear observer with linearizable error dynamics[END_REF]) for the case of MIMO systems, many other results (see [START_REF] Keller | Nonlinear observer design by transformation into a generalized observer canonical form[END_REF]; [START_REF] Marino | Nonlinear control design: geometric, adaptive and robust[END_REF]; [START_REF] Lynch | Nonlinear observers with approximately linear error dynamics: the multivariable case[END_REF]; [START_REF] Phelps | On constructing nonlinear observers[END_REF]; [START_REF] Boutat | New algorithm for observer error linearization with a diffeomorphism on the outputs[END_REF]) were published by following the same idea. However, the solvability of the problem requires the restrictive commutative Lie bracket condition for the deduced vector fields. In order to relax this restriction, we can reconstruct a new family of vector fields which can satisfy the commutative Lie bracket condition. Inspired by this solution, various extensions are developed for output depending nonlinear observer normal form [START_REF] Guay | Observer linearization by output-dependent time-scale transformations[END_REF]; [START_REF] Respondek | Time scaling for observer design with linearizable error dynamics[END_REF]; [START_REF] Zheng | Output dependent observability linear normal form[END_REF][START_REF] Zheng | Multi-output dependent observability normal form[END_REF][START_REF] Zheng | A single output dependent observability normal form[END_REF]; [START_REF] Boutat | Observer error linearization multi-output depending[END_REF]) and for the extended nonlinear observer normal form [START_REF] Jouan | Immersion of nonlinear systems into linear systems modulo output injection[END_REF]; [START_REF] Noh | Nonlinear observer design by dynamic observer error linearization[END_REF]; [START_REF] Back | Dynamic observer error linearization[END_REF]; [START_REF] Boutat | On the transformation of nonlinear dynamical systems into the extended nonlinear observable canonical form[END_REF]; [START_REF] Tami | Extended output depending normal form[END_REF]; [START_REF] Boutat | Extended nonlinear observer normal forms for a class of nonlinear dynamical systems[END_REF]). The related applications can be found as well for the synchronization of nonlinear systems [START_REF] Zheng | Synchronisation of chaotic systems via reduced observers[END_REF], for Dengue epidemic model [START_REF] Tami | Parameters and states estimation for dengue epidemic model[END_REF], for PM synchronous motor Tami et al. (2014a), and so on.

Most of the references cited above are devoted to designing a full-order observer, under the assumption that the whole state of the studied system is observable. Few works have been dedicated to the partial observability which however makes sense in practice when only a part of states are observable or are necessary for the controller design. Among the works on this issue, we can cite the work of [START_REF] Kang | On the observability of nonlinear and switched systems[END_REF] where the authors gave a general definition of observability covering the partial one. Reduced-order observer and LMI technique are proposed in [START_REF] Trinh | Partial-state observers for nonlinear systems[END_REF] to estimate the part of observable states. In [START_REF] Robenack | Observer design using a partial nonlinear observer canonical form[END_REF], the authors proposed a general partial nonlinear observer canonical form and used the geometrical method transforming a nonlinear dynamical system into this observer form. However, the nonlinear term in this canonical form contains as well the unobservable states. Thus, to design an observer for the proposed canonical form, one needs the Lipschitz condition to guarantee the estimation convergence. [START_REF] Jo | Observer design for non-linear systems that are not uniformly observable[END_REF] provided necessary and sufficient geometrical conditions which guarantee the existence of a change of coordinates transforming a nonlinear dynamical system into a special partial observer normal form. This normal form is divided into two dynamical subsystems: the first one contains only a part of states which is of Brunovsky canonical form modulo output injection; the second one is nonlinear containing only the unobservable states and the output. This result can be seen as a direct application of the result of [START_REF] Krener | Linearization by output injection and nonlinear observers[END_REF] to the partial family of vector fields, therefore it suffers from the same restriction on the commutative Lie bracket condition.

In order to relax the restriction of the result presented in [START_REF] Jo | Observer design for non-linear systems that are not uniformly observable[END_REF], this paper proposes a more general PONF for a class of partially observable nonlinear systems. This new form is a generalization of the normal form studied in [START_REF] Jo | Observer design for non-linear systems that are not uniformly observable[END_REF]. The proposed PONF is divided as well into two subsystems, and we relax the form proposed in [START_REF] Jo | Observer design for non-linear systems that are not uniformly observable[END_REF] by involving all states in the second subsystem. To deal with this generalization, we use the notion of commutativity of Lie bracket modulo a distribution. Moreover, our results allow as well to apply additionally a diffeomorphism on the output space. Therefore, the necessary and sufficient geometrical conditions established in this paper are more general and quite different from those stated in [START_REF] Jo | Observer design for non-linear systems that are not uniformly observable[END_REF] because of the introduction of commutativity of Lie bracket modulo a distribution.

The paper is organized as follows. Section 2 is devoted to the technical background and problem statement. In Section 3, some preliminary results are given and then necessary and sufficient conditions allowing the construction of PONF are stated. Section 4 generalizes the result in the previous section by applying a change of coordinates on the output. In the end, an example of Susceptible, Infected and Removed (SIR) epidemic model is presented to highlight the proposed results.

Notation and problem statement

We consider the following nonlinear dynamical system with single output:

ẋ = f (x) + g(x)u = f (x) + m ∑ k=1 g k (x) u k (1) y = h(x) (2)
where x ∈ R n , u ∈ R m , y ∈ R, and the functions f :

R n → R n , g = [g 1 , • • • , g m ] with g i : R n → R n for 1 ≤ i ≤ m, h : R n → R are
supposed to be sufficiently smooth. In this paper, it is assumed that f (0) = 0 and h(0) = 0. Let X ⊂ R n be a neighborhood of 0, for system (1)-( 2), if the pair (h(x), f (x)) locally satisfies the observability rank condition on X , i.e. rank dh, dL f h, • • • , dL n f h (x) = n for x ∈ X , then the following 1-forms:

θ 1 = dh and θ i = dL i-1 f h, for 2 ≤ i ≤ n
are independent on X , where L k f h denotes the k th Lie derivative of h along f . Therefore, there exists a family of vector fields [START_REF] Krener | Linearization by output injection and nonlinear observers[END_REF], where the first vector field τ1 is the solution of the following algebraic equations:

τ = [ τ1 , • • • , τn ] proposed in
θ i ( τ1 ) = 0 for 1 ≤ i ≤ n -1 θ n ( τ1 ) = 1 (3)
and the other vector fields are obtained by induction as τi = where [, ] denotes the Lie bracket. According to [START_REF] Krener | Linearization by output injection and nonlinear observers[END_REF], if

-ad f τi-1 = [ τi-1 , f ] for 2 ≤ i ≤ n,
[ τi , τ j ] = 0 for 1 ≤ i, j ≤ n (4) [ τi , g k ] = 0 for 1 ≤ i ≤ n -1 and 1 ≤ k ≤ m
(5)

then system (1)-( 2) can be locally transformed, by means of a local diffeomorphism ξ = φ (x), into the following nonlinear observer normal form:

   ξ = Aξ + B(y) + m ∑ k=1 α k (y) u k y = Cξ (6)
where A ∈ R n×n is the Brunovsky matrix and

C = [0, • • • , 0, 1] ∈ R 1×n .
Obviously, for the nonlinear dynamical system (1)-( 2), if rank dh, dL f h, • • • , dL n f h (x) = r < n for x ∈ X , which implies that only a part of states of the studied system are observable, the proposed method by [START_REF] Krener | Linearization by output injection and nonlinear observers[END_REF] could not be applied.

To treat this partially observable situation, this paper proposes the following Partial Observer Normal Form (PONF):

         ξ = Aξ + β (y) + m ∑ k=1 α 1 k (y)u k ζ = η(ξ , ζ ) + m ∑ k=1 α 2 k (ξ , ζ )u k y = Cξ (7) where ξ ∈ R r , ζ ∈ R n-r , y ∈ R , β : R → R r , η : R r × R n-r → R n-r , α 1 k : R → R r , α 2 k : R r × R n-r → R n-r , C = (0, • • • , 0, 1) ∈ R 1×r
and A is the r × r Brunovsky matrix:

A =        0 0 • • • 0 0 1 0 . . . . . . . . . 0 1 • • • 0 0 0 • • • • • • 0 0 0 • • • • • • 1 0        ∈ R r×r
For the proposed form (7), one can easily design a reducedorder observer to estimate the part of observable state ξ , by choosing the gain K such that (A -KC) is Hurwitz. Therefore, the rest of this paper focuses only on how to deduce a diffeomorphism which transforms the nonlinear system (1)-(2) into the proposed PONF (7).

Before dealing with this problem, let use highlight the fundamental difference between the proposed PONF with respect to the canonical form proposed in [START_REF] Jo | Observer design for non-linear systems that are not uniformly observable[END_REF].

Remark 1. The normal form described in ( 7) is a generalization of the normal form obtained in [START_REF] Jo | Observer design for non-linear systems that are not uniformly observable[END_REF]. Indeed, in [START_REF] Jo | Observer design for non-linear systems that are not uniformly observable[END_REF] the unobservable dynamics (the second dynamics ζ ) does not contain the observable state ξ . Let us consider the following modified example from [START_REF] Jo | Observer design for non-linear systems that are not uniformly observable[END_REF] to highlight this difference:

       ξ1 = u ξ2 = ξ 1 + y 2 ζ = -ζ + µξ 2 1 + y y = ξ 2 (8)
If µ = 0, then this example is the same one studied in [START_REF] Jo | Observer design for non-linear systems that are not uniformly observable[END_REF] and is of the normal form proposed in [START_REF] Jo | Observer design for non-linear systems that are not uniformly observable[END_REF]. However, if µ = 0, then it is of the proposed normal form (7). Moreover, the conditions stated in [START_REF] Jo | Observer design for non-linear systems that are not uniformly observable[END_REF] are not fulfilled for this example when µ = 0. This fact will be discussed in Example 1.

Geometrical conditions

In this section, we will deduce necessary and sufficient conditions which guarantee the existence of a diffeomorphism to transform the studied partial observer nonlinear system into the proposed PONF. For this, considering the studied system (1)-( 2), it is assumed in this paper that rank dh, dL f h,

• • • , dL n f h (x) = r < n for x ∈ X ⊂ R n
, where X is a neighborhood of 0.

Denote the observability 1-forms for 1 ≤ i ≤ r by θ i = dL i-1 f h and note ∆ = span{θ 1 , θ 2 , • • • , θ r } as the co-distribution spanned by the observability 1-forms. Specifically, thanks to observability rank condition, it is clear that dL i f h ∈ ∆ for i ≥ r + 1. Then one can define the kernel (or the annihilator) of the co-distribution ∆ as follows:

∆ ⊥ = ker∆ = {X : θ k (X) = 0, for 1 ≤ k ≤ r} (9)
for which we have the following properties which will be used in the sequel.

Lemma 1. For the distribution ∆ ⊥ defined in (9), the following properties are satisfied:

P1: ∆ ⊥ is involutive, i.e. for any two vector fields

H 1 ∈ ∆ ⊥ and H 2 ∈ ∆ ⊥ , we have [H 1 , H 2 ] ∈ ∆ ⊥ ; P2: there exist (n -r) vector fields {τ r+1 , • • • , τ n } that span ∆ ⊥ such that [τ i , τ j ] = 0 for r + 1 ≤ i, j ≤ n, i.e. {τ r+1 , • • • , τ n } is the commutative basis of ∆ ⊥ ; P3: ∆ ⊥ is f -invariant, i.e. for any vector field H ∈ ∆ ⊥ , we have [ f , H] ∈ ∆ ⊥ .
Proof. See Appendix A.1. Now, let τ1 be one of the vector field solutions of the following under-determined algebraic equations:

θ k ( τ1 ) = 0 for 1 ≤ k ≤ r -1 θ r ( τ1 ) = 1 ( 10
)
and it is obvious that this solution is not unique, since (10) contains only r algebraic equations. Therefore, for any H ∈ ∆ ⊥ , τ1 + H is also a solution of (10). By following the method proposed in [START_REF] Krener | Linearization by output injection and nonlinear observers[END_REF], we can construct, for any chosen τ1 satisfying (10), the following vector fields:

τi = [ τi-1 , f ] for 2 ≤ i ≤ r (11)
Let τ1 be another solution of (10), which enables us, by following the same method, to construct another family of (r -1) vector fields:

τi

= [ τi-1 , f ] for 2 ≤ i ≤ r (12)
Then the following lemma highlights the relation between these two families of vector fields for the different chosen τ1 and τ1 .

Lemma 2. For any two different solutions τ1 and τ1 of (10), there exist H i ∈ ∆ ⊥ for 1 ≤ i ≤ r such that τi = τi + H i where τi and τi are defined in ( 11) and ( 12), respectively.

Proof. See Appendix A.2. The above lemma reveals an important property: no matter how to choose the first vector field τ1 satisfying (10), the family of vector fields τi for 1 ≤ i ≤ r are defined modulo ∆ ⊥ . Other important properties for this family of vector fields τi are listed in the following lemma.

Lemma 3. For any τ1 satisfying (10) and the associated τi for 2 ≤ i ≤ r deduced from (11), the following properties are satisfied:

1) for any H ∈ ∆ ⊥ , we have [ τi , H] ∈ ∆ ⊥ ; 2) for any H i ∈ ∆ ⊥ and H j ∈ ∆ ⊥ with 1 ≤ i, j ≤ r, we have [ τi + H i , τ j + H j ] = [ τi , τ j ] modulo ∆ ⊥ ; 3) for 1 ≤ i ≤ r and r + 1 ≤ j ≤ n, we have [ τi , τ j ] ∈ ∆ ⊥ (13)
where τ j was defined in P2 of Lemma 1, which is the commutative basis of ∆ ⊥ .

Proof. See Appendix A.3.

Lemma 4. For any τ1 satisfying (10) and the associated τi for 2 ≤ i ≤ r deduced from ( 11), there exists a family of vector fields

τ i = τi modulo ∆ ⊥ such that [τ i , τ j ] = 0 for 1 ≤ i, j ≤ r, if and only if [ τi , τ j ] ∈ ∆ ⊥ .
Proof. See Appendix A.4.

Remark 2. Any family of vector fields τi for 1 ≤ i ≤ r given by ( 10)-( 11) together with the family of vector fields τ j for r + 1 ≤ i ≤ n defined in P2 of Lemma 1, are independent on X . Thus they provide a basis of the tangent fiber bundle T X of X .

The following example is for the purpose of highlighting the above results.

Example 1. Let us consider again system (8). A straightforward calculation shows that θ 1 = dξ 2 and θ

2 = dξ 1 + 2ξ 2 dξ 2 . Then we have ∆ = span{θ 1 , θ 2 }, thus ∆ ⊥ = span ∂ ∂ ζ . Ac- cording to (10), we obtain τ1 = ∂ ∂ ξ 1 + q 1 (ξ 1 , ξ 2 , ζ ) ∂ ∂ ζ where q 1 (ξ 1 , ξ 2 , ζ ) being any function of its arguments. Following (11), we have: τ2 = [ τ1 , f ] = ∂ ∂ ξ 2 + q 2 ∂ ∂ ζ with q 2 = 2µξ 1 - q 1 -∂ q 1 ∂ ξ 2 (ξ 1 + ξ 2 2 ) -∂ q 1 ∂ ζ (-ζ + µξ 2 1 + ξ 2 ).
The Lie bracket of these two vector fields yields:

[ τ1 , τ2 ] = ∂ q 2 ∂ ξ 1 + ∂ q 2 ∂ ζ q 1 - ∂ q 1 ∂ ξ 2 - ∂ q 1 ∂ ζ q 2 ∂ ∂ ζ
which is not equal to 0 if q 1 = 0 and µ = 0. Therefore, the conditions stated in [START_REF] Jo | Observer design for non-linear systems that are not uniformly observable[END_REF] are not fulfilled. However, we have [ τ1 , τ2 ] ∈ ∆ ⊥ . Therefore, thanks to Lemma 4 we can found two other vector fields which are commutative.

It is easy to check that, by choosing

H 1 = -q 1 ∂ ∂ ζ and H 2 = -q 2 ∂ ∂ ζ , we have τ 1 = ∂ ∂ ξ 1 and τ 2 = ∂ ∂ ξ 2 which yields [τ 1 , τ 2 ] = 0.
With the deduced r independent vector fields τi for 1 ≤ i ≤ r, we can then define the following matrix:

Λ 1 = (θ j ( τi )) 1≤i, j≤r =    θ 1 . . . θ r    ( τ1 , • • • , τr ) =      0 • • • 0 1 . . . • • • 1 λ 2,r 0 • • • λ r-1,r-1 λ r-1,r 1 • • • λ r,r-1 λ r,r      (14)
with λ i, j = θ i ( τ j ) for 1 ≤ i, j ≤ r. Since it is invertible, we can define the following 1-forms:

   ω 1 . . . ω r    = Λ -1 1 [θ 1 , • • • , θ r ] T (15)
A straightforward calculation shows that

ω r = θ 1 ω r-k = θ k+1 - n ∑ i=r-k+1 λ k+1,i ω i for 1 ≤ k ≤ r -1 (16)
which are linear combination of the 1-forms θ i for 1 ≤ i ≤ r.

Then it is easy to verify that the deduced 1-forms ω i for 1 ≤ i ≤ r have the following properties.

Lemma 5. 1) the annihilator of 1-forms ω k for 1 ≤ k ≤ r is equal ∆ ⊥ defined in (9). 2) the 1-forms ω k for 1 ≤ k ≤ r are well defined such that they are independent of the choice of τi modulo

∆ ⊥ . 3) for 1 ≤ i, k ≤ r, ω k ( τi ) = δ i k where δ i k represents the Kro- necker delta, i.e. δ i k = 1 if k = i, otherwise δ i k = 0.
Proof. The above properties are just the obvious results of ( 15) and ( 16), thus the proof is omitted.

The above lemma shows that, for any choice solution of (10), the constructed family of vector fields always yields the same 1-form ω k for 1 ≤ k ≤ r, which allows us to state the following theorem.

Theorem 1. There exists a local diffeomorphism (ξ T , ζ T ) T = φ (x) on X which transforms system (1)-( 2) into the proposed normal form (7), if and only if the following conditions are fulfilled:

1) there exists a family of vector fields

τ i = τi modulo ∆ ⊥ such that [τ i , τ l ] = 0 for 1 ≤ i ≤ r and 1 ≤ l ≤ r;
2) [τ i , τ j ] = 0 for 1 ≤ i ≤ r and r + 1 ≤ j ≤ n where τ j was defined in P2 of Lemma 1;

3

) [τ i , g k ] ∈ ∆ ⊥ for 1 ≤ i ≤ n, i = r and 1 ≤ k ≤ m.
If the above conditions are fulfilled, then the diffeomorphism φ is locally determined by its differential φ * := dφ = ω, i.e. φ (x) = γ ω, where γ is any path on X with γ(0) = 0 and γ(1) = x.

Proof. Necessity: To show those conditions are necessary, let us assume that there exists a local diffeomorphism (ξ T , ζ T ) T = φ (x) on X which transforms (1)-( 2) into the proposed form (7).

Note ω = dφ , thus ω i for 1 ≤ i ≤ r are the pullbacks of dξ i , i.e.2 ω i = φ * (dξ ) = dφ i . For the same reason, ω j for r + 1 ≤ j ≤ n are the pullbacks of dζ i for 1 ≤ i ≤ nr with ω j = dφ j .

In the following, we will firstly check that the conditions i), ii) and iii) of Theorem 1 are satisfied for the transformed normal form ( 7), then we will use the property of pullback to prove the necessity for the original system (1)-( 2). In order to avoid the ambiguities, we add the superscript 'nf ' for those calculations made for the transformed normal form (7), otherwise it is for the original system (1)-( 2).

For the transformed normal form (7), following the same procedure presented in Section 3, we have θ n f 1 = dξ r and

θ n f i = dξ r-i+1 + ∑ i-1 j=1 l j θ n f j for 2 ≤ i ≤ r, which yields ∆ n f ⊥ = span { ∂ ∂ ζ 1 , • • • , ∂ ∂ ζ n-r }. Then, we have τn f 1 = ∂ ∂ ξ 1 + H n f 1 with H n f 1 ∈ ∆ n f ⊥
, based on which we can construct the rest vector fields:

τn f i = ∂ ∂ ξ i + H n f i with H n f i ∈ ∆ n f ⊥ for 2 ≤ i ≤ r.
It is clear that there exists a family of vector fields

τ n f i = ∂ ∂ ξ i = τn f i modulo ∆ n f ⊥ such that τ n f i , τ n f l = 0 for 1 ≤ i ≤ r and 1 ≤ l ≤ r.
Moreover, since τ n f j for r + 1 ≤ j ≤ n should be commutative and span ∆ n f ⊥ defined above, without loss of generalities, we can choose

τ n f j = ∂ ∂ ζ j-r
, with which we can prove that τ n f i , τ n f j = 0 for 1 ≤ i ≤ r and r + 1 ≤ j ≤ n. Considering now the term in front of the input in (7). It is

clear that τ n f i , α 1 k (y) α 2 k (ξ , ζ ) ∈ ∆ n f ⊥ for 1 ≤ i ≤ n, i = r and 1 ≤ k ≤ m.
Finally we proved that if there exist a local diffeomorphism (ξ T , ζ T ) T = φ (x) on X which transforms (1)-( 2) into (7), then there exists a family of vector fields τ n f i for 1 ≤ i ≤ n such that the condition i), ii) and iii) are satisfied for the transformed normal form (7).

Since ω = dφ are the pullbacks of [dξ , dζ ] T , due to the fact that

τ n f i = ∂ ∂ ξ i for 1 ≤ i ≤ r and τ n f j = ∂ ∂ ζ j-r for r + 1 ≤ j ≤ n, by defining τ i = φ -1 * τ n f i , for 1 ≤ i ≤ n (17)
we can conclude that if there exists a local diffeomorphism (ξ T , ζ T ) T = φ (x) on X which transforms system (1)-( 2) into the proposed normal form (7), then there exists a family of vector fields τ i for 1 ≤ i ≤ n defined in (17) such that the condition i), ii) and iii) are necessary to be satisfied for the original system (1)-(2).

Sufficiency: Suppose that the conditions 1), 2) and 3) in Theorem 1 are satisfied, and we will prove that there exists a local diffeomorphism.

As [τ i , τ j ] = 0 for 1 ≤ i ≤ n and 1 ≤ j ≤ n, then there exist n independent 1-forms ωi for 1 ≤ i ≤ n such that ωi (τ j ) = δ j i for 1 ≤ j ≤ n. The following shows how to determine those 1-forms.

Firstly, since τ k = τk modulo ∆ ⊥ , according to Lemma 5, we have ω l (τ k ) = δ k l for 1 ≤ l, k ≤ r where ω l was given in ( 15), which implies that ωl = ω l for 1 ≤ l ≤ r. Secondly, for r + 1 ≤ i ≤ n, since we have ωi (τ j ) = δ j i where τ j was defined in P2 of Lemma 1 for 1 ≤ j ≤ n, and this will uniquely determine the rest (nr) 1-forms ωi , noted as

ω i = ωi such that ωi (τ j ) = δ j i for r + 1 ≤ i ≤ n (18)
Finally, we uniquely determine the 1-

forms ω = ω = [ω 1 , • • • , ω n ] T .
For those determined 1-forms, for 1 ≤ k ≤ n we have (see the beginning of the proof for Lemma 1):

dω k (τ i , τ j ) = L τ i ω k (τ j ) -L τ j ω k (τ i ) -ω k ([τ i , τ j ]).
As ω k (τ j ) and ω k (τ i ) are both constant (either 0 or 1 depending on the indices), then one has dω k (τ i , τ j ) = -ω k ([τ i , τ j ]) which implies the equivalence between dω k = 0 and [τ i , τ j ] = 0, since ω k and τ i are dual and independent. Therefore, the condition ii) implies the deduced ω defined in ( 15) and ( 18) is closed, i.e. dω = 0. Then, thanks to Poincar's lemma, there exists a local diffeomorphism φ such that φ * = ω, where φ * represents the Jacobian of φ .

In what follows, we will prove that φ deduced via φ * = ω is exactly the sought diffeomorphism which transforms system (1)-(2) into the proposed normal form (7).

For this, let us firstly show how it transforms f in (1). Setting (ξ T , ζ T ) T = φ (x), since φ * = ω, then we have φ * (τ

i ) = ω(τ i ) = ∂ ∂ ξ i for 1 ≤ i ≤ r and φ * (τ j ) = ω(τ j ) = ∂ ∂ ζ j for r + 1 ≤ i ≤ n.
Moreover, according to the third property of Lemma 5, for any H ∈ ∆ ⊥ , we have

ω i (H) = 0 for 1 ≤ i ≤ r and ω j (H) ∈ span { ∂ ∂ ζ 1 , • • • , ∂ ∂ ζ n-r } for r + 1 ≤ j ≤ n because φ * (τ j ) = ω(τ j ) = ∂ ∂ ζ j and τ j span the distribution ∆ ⊥ . Thus, for any H ∈ ∆ ⊥ we have φ * (H) = ω(H) ∈ span { ∂ ∂ ζ 1 , • • • , ∂ ∂ ζ n-r }.

Now, let us consider the following two cases:

-for 1

≤ i ≤ r -1, we have ∂ ∂ ξ i φ * ( f ) = [ ∂ ∂ ξ i , φ * ( f )] = [φ * (τ i ), φ * ( f )] = φ * [τ i , f ] = φ * (τ i+1 + H i ) for some H i ∈ ∆ ⊥ . Thus we have ∂ ∂ ξ i φ * ( f ) = φ * (τ i+1 ) + φ * (H i ) = ∂ ∂ ξ i+1 + φ * (H i ); (19) -for r + 1 ≤ j ≤ n, we have ∂ ∂ ζ j φ * ( f ) = [ ∂ ∂ ζ j , φ * ( f )] = [φ * (τ j ), φ * ( f )] = φ * [τ j , f ]. Since τ j span the distribution ∆ ⊥ which is f -invariant (see P3 of Lemma 1), then there exist H j ∈ ∆ ⊥ such that H j = [τ j , f ]. Then we have ∂ ∂ ζ j φ * ( f ) = φ * (H j ). (20)
Since for any

H ∈ ∆ ⊥ we have φ * (H) ∈ span { ∂ ∂ ζ 1 , • • • , ∂ ∂ ζ n-r },
( f ) = Aξ + β (y) η(ξ , ζ ) .
Consider now the term g in (1). If the condition iii) of Theorem 1 is satisfied, i.e.

[τ i , g k ] ∈ ∆ ⊥ for 1 ≤ i ≤ n, i = r and 1 ≤ k ≤ m, then we have φ * ([τ i , g k ]) ∈ span { ∂ ∂ ζ 1 , • • • , ∂ ∂ ζ n-r }.
By applying the same argument, for 1 ≤ k ≤ m let us consider the following two cases:

-for 1 ≤ i ≤ r -1, we have ∂ ∂ ξ i φ * (g k ) = [ ∂ ∂ ξ i , φ * (g k )] = [φ * (τ i ), φ * (g k )] = φ * [τ i , g k ]; -for r +1 ≤ j ≤ n, for the same reason we have ∂ ∂ ζ j φ * (g k ) = φ * [τ j , g k ]; Since φ * ([τ i , g k ]) ∈ span { ∂ ∂ ζ 1 , • • • , ∂ ∂ ζ n-r } for 1 ≤ i ≤ n, i = r and 1 ≤ k ≤ m
, the integration of the above two equations yields

φ * (g k ) = α 1 k (y) α 2 k (ξ , ζ ) for 1 ≤ k ≤ m.
Following the same argument, one can prove as well

[τ i , g k ] ∈ modulo ∆ ⊥ for 1 ≤ i ≤ n, i = r and 1 ≤ k ≤ m .
The rest concerns the deduction of the local diffeomorphism. Since ω = dφ on X , then the diffeomorphism can be determined just by integration of ω, i.e. φ (x) = γ ω where γ is any path on X with γ(0) = 0 and γ(1) = x.

For any choice of τ1 satisfying (10) with the associated family of vector fields τi for 2 ≤ i ≤ r, Theorem 1 needs to seek a new family of vector fields τ i = τi modulo ∆ ⊥ such that [τ i , τ l ] = 0 for 1 ≤ i ≤ r and 1 ≤ l ≤ r. Thanks to Lemma 4, we can remove at the beginning all terms in the directions of the basis of ∆ ⊥ in τ1 , noted as τ 1 . It is important to emphasize that this elimination will yield a unique τ 1 for any solution of (10). Then, we can iteratively eliminate all terms in the directions of the basis of ∆ ⊥ in τi , which gives the following simple procedure to calculate τ i for 1 ≤ i ≤ n: Procedure 1. For the nonlinear system (1)-( 2), the family of vector fields τ i for 1 ≤ i ≤ n can be simply computed via the following steps:

Step 1: Compute the observability 1-forms θ i for 1 ≤ i ≤ r;

Step 2: Determine ∆ ⊥ and seek a commutative basis

{τ r+1 , • • • , τ n } that spans ∆ ⊥ ;
Step 3: Choose any solution τ1 of (10), then eliminate all its terms in the directions of

{τ r+1 , • • • , τ n } which yields τ 1 ; Step 4: Iteratively, for 2 ≤ i ≤ r, compute [τ i-1 , f ] in which eliminate all terms in the directions of {τ r+1 , • • • , τ n } which gives τ i .
Remark 3. Like most of existing results by using the differential geometric method, such as the pioneering works in [START_REF] Krener | Linearization by output injection and nonlinear observers[END_REF]; [START_REF] Krener | Nonlinear observers with linearizable error dynamics[END_REF]; [START_REF] Respondek | Time scaling for observer design with linearizable error dynamics[END_REF]; [START_REF] Jo | Observer design for non-linear systems that are not uniformly observable[END_REF], the proposed results in this paper characterize only the solvability of the problem. It is worthy noting that the construction of such family of commutative vector fields is still quite challenging.

The following example is to highlight Theorem 1 by following Procedure 1.

Example 2. Let us consider the following nonlinear system:

       ẋ1 = -x 2 3 + x 3 1 x 3 -1 2 x 3 3 + x 5 2 ẋ2 = x 1 -1 2 x 2 3 ẋ3 = -x 3 + x 3 1 -1 2 x 2 3 y = x 2 (21)
Step 1:

A simple calculation gives rank dh, dL f h, dL 2 f h = 2, thus r = 2, and one has

θ 1 = dx 2 and θ 2 = dx 1 -x 3 dx 3 . since dL 2 f h = 5x 4 2 θ 1 .
Step 2: Then one obtains ∆ = span {θ 1 , θ 2 }, and

∆ ⊥ = span x 3 ∂ ∂ x 1 + ∂ ∂ x 3
Since τ 3 should be a commutative basis of ∆ ⊥ , thus one has

τ 3 = x 3 ∂ ∂ x 1 + ∂ ∂ x 3 .
Step 3: By solving equation ( 10), one obtains τ1 = ∂ ∂ x 1 + q 1 (x)τ 3 . Following Procedure 1, by eliminating all terms in τ1 in the direction of τ 3 , we have

τ 1 = ∂ ∂ x 1 .
Step 4: Then we can calculate:

[τ 1 , f ] = ∂ ∂ x 2 + 3x 2
1 τ 3 , for which the elimination of all terms in the direction of τ 3 yields τ

2 = ∂ ∂ x 2 .
Finally Procedure 1 enables us to easily get the following vector fields:

τ 1 = ∂ ∂ x 1 , τ 2 = ∂ ∂ x 2 , τ 3 = x 3 ∂ ∂ x 1 + ∂ ∂ x 3
with which it is easy to check that all conditions of Theorem 1 are satisfied, thus there exists a diffeomorphism which can transform the studied example into the proposed form (7).

For the deduction of the diffeomorphism, we use the obtained vector field τ = [τ 1 , τ 2 , τ 3 ] which gives:

Λ 1 = θ 1 θ 2 (τ 1 , τ 2 ) = 0 1 1 0 thus one has ω 1 ω 2 = Λ -1 1 θ 1 θ 2 = dξ 1 dξ 2 = d x 1 -1 2 x 2 3 x 2
Following (18), one can solve the following equation:

ω 3 ∂ ∂ x 1 , ∂ ∂ x 2 , ∂ ∂ x 3 + x 3 ∂ ∂ x 1 = (0, 0, 1)
which uniquely determines ω 3 = dx 3 . Finally, one gets the

following diffeomorphism φ (x) =   x 1 -1 2 x 2 3 x 2 x 3
  which transforms the studied system (21) into the following form

         ξ1 = 0 ξ2 = ξ 1 ζ1 = ξ 3 1 -ξ 3 + ξ 2 3 2 3ξ 2 1 + 3 2 ξ 1 ξ 2 3 + 1 4 ξ 4 3 -1 y = ξ 2

Diffeomorphism on the output

This section deals with the case when the conditions of Theorem 1 are not fulfilled. Let us remark that the deduced diffeomorphism φ (x) in Section 3 does not modify the output. This is due to the fact that θ r ( τ1 ) = 1. One way to relax this constraint is to seek a new vector field, noted as σ1 , such that θ r ( σ1 ) becomes a function of the output, and this will introduce a diffeomorphism on the output. For that purpose, this section will modify the new vector field τ 1 obtained in Section 3 by following Procedure 1 and construct a new family of commutative vector fields, with which the deduced diffeomorphism φ (x) will apply as well a change of coordinates on the output (see [START_REF] Krener | Nonlinear observers with linearizable error dynamics[END_REF]; [START_REF] Respondek | Time scaling for observer design with linearizable error dynamics[END_REF]; [START_REF] Boutat | On the transformation of nonlinear dynamical systems into the extended nonlinear observable canonical form[END_REF]). Since this diffeomorphism will modify the output of the studied system (1)-( 2), therefore the PONF ( 7) is adapted as follows:

         ξ = Aξ + β (y) + m ∑ k=1 α 1 k (y)u k ζ = η(ξ , ζ ) + m ∑ k=1 α 2 k (ξ , ζ )u k ȳ = ξ r = ψ(y) (22)
With the same procedure presented in Section 3, we can calculate the observability 1-forms θ i for 1 ≤ i ≤ r where r is the rank of observability matrix, which defines the co-distribution ∆ = span{θ 1 , θ 2 , • • • , θ r }, and the kernel (or the annihilator) of the co-distribution ∆, noted as ∆ ⊥ .

By following Procedure 1 presented in Section 3, it is assumed that we have already obtained a family of vector fields τ j for r + 1 ≤ j ≤ n which are commutative basis of ∆ ⊥ , and τ i for 1 ≤ i ≤ r which do not contain any component in the direction of τ j for r + 1 ≤ j ≤ n. In this section, it is assumed that there exist 1 ≤ i, l ≤ r such that [τ i , τ l ] = 0, thus Theorem 1 cannot be applied. In order to introduce a change of coordinate for the output, let s(y) = 0 be a function of the output which will be determined later. Then one can define a new vector field σ 1 from τ 1 as follows:

σ1 = s(y)τ 1 (23)
By induction, one can define the following new vector fields σi

= [ σi-1 , f ] modulo ∆ ⊥ , for 2 ≤ i ≤ r. A straightforward cal- culation gives σi = i ∑ k=1 (-1) i-k C k-1 i-1 L i-k f s(y)τ k modulo ∆ ⊥ (24)
where C k-1 i-1 is the binomial coefficients and L 0 f s(y) = s(y). For the same reason stated in Procedure 1, by eliminating all terms in the directions of τ j for r + 1 ≤ j ≤ n (the commutative basis of ∆ ⊥ ) we obtain:

σ i = i ∑ k=1 (-1) i-k C k-1 i-1 L i-k f s(y)τ k , for 1 ≤ i ≤ r (25)
For the sake of completeness, we simply note

σ j = τ j , for r + 1 ≤ j ≤ n (26)
where {τ r+1 , • • • , τ n } are commutative basis of ∆ ⊥ , obtained by following Procedure 1 presented in Section 3. We can then define the following matrix:

Λ1 =    θ 1 . . . θ r    (σ 1 , • • • , σ r ) =      0 • • • 0 s(y) . . . • • • s(y) * 0 • • • * * s(y) • • • * *     
(27) Since s(y) = 0, thus it is invertible, and we can define the following 1-forms:

   ω1 . . . ωr    = Λ-1 1 [θ 1 , • • • , θ r ] T (28)
and using the same method in Section 3 we can uniquely determine ωi for r + 1 ≤ i ≤ n by solving the following equations:

ωi (σ j ) = δ j i ( 29
)
for 1 ≤ j ≤ n. Finally we have the following theorem.

Theorem 2. There exists a local diffeomorphism (ξ T , ζ T ) T = φ (x) on X which transforms system (1)-( 2) into the PONF ( 22) if and only if the following conditions are fulfilled:

1) [σ i , σ l ] = 0 for 1 ≤ i ≤ r and 1 ≤ l ≤ r where σ i defined in (25);

2) [σ i , σ j ] = 0 for 1 ≤ i ≤ r and r +1 ≤ j ≤ n where σ j defined in (26);

3

) [σ i , g k ] ∈ ∆ ⊥ for 1 ≤ i ≤ n, i = r and 1 ≤ k ≤ m.
If the above conditions are fulfilled, then the diffeomorphism φ is locally determined by its differential φ * := dφ = ω, i.e. φ (x) = γ ω, where γ is any path on X with γ(0) = 0 and γ(1) = x. Moreover, one has ξ r = ȳ = ψ(y) where ψ(y) = y 0 1 s(c) dc is a change of coordinates on the output.

Proof.

The proof of Theorem 2 is similar to the proof of Theorem 1, thus the following just gives a sketch to determine s(y) and some different elements with respect to the result stated in [START_REF] Boutat | On the transformation of nonlinear dynamical systems into the extended nonlinear observable canonical form[END_REF].

According to (23), we define σ 1 = s(y)τ 1 where τ 1 is obtained in Section 3 by eliminating all terms in the directions of τ j for r + 1 ≤ j ≤ n. By induction, we have σ i = [σ i-1 , f ] modulo ∆ ⊥ , for 2 ≤ i ≤ r. Then one can deduce:

[σ 1 , σ i ] = i ∑ k=1 (-1) i-k C k-1 i-1 L i-k f s(y) [σ 1 , τ k ] = i ∑ k=1 (-1) i-k C k-1 i-1 L i-k f s(y) s(y) [τ 1 , τ k ] -L τ k s(y)τ 1 + i ∑ k=1 (-1) i-k C k-1 i-1 L σ 1 L i-k f s(y)τ k
Imposing [σ 1 , σ i ] = 0, this enables us to determine s(y). More details on this issue can be founded in [START_REF] Boutat | On the transformation of nonlinear dynamical systems into the extended nonlinear observable canonical form[END_REF]. After having deduced s(y), the rest of the proof for Theorem 2 is similar to the proof of Theorem 1.

Example 3. Let us consider the well-known SIR (Susceptible, Infected, Removed) epidemic model describing the contagious disease propagation [START_REF] Raggett | Modeling the eyam plague[END_REF]; [START_REF] Feng | Actuarial applications of epidemiological models[END_REF] as follows:

       Ṡ = -β SI İ = β SI -γI Ṙ = γI y = I
where S denotes the suspected population, I denotes the infected and R denotes the removed population and the total population. By using the same notations introduced in Section 3, a simple calculation gives θ 1 = dI and θ 2 = β IdS + (β Sγ) dI, and we have ∆ = span{dI, β IdS + (β Sγ) dI} which gives

∆ ⊥ = span{ ∂ ∂ R }.

Calculation of τ:

Following the procedure proposed in [START_REF] Krener | Linearization by output injection and nonlinear observers[END_REF], we obtain the following vector fields:

τ1 = 1 β I ∂ ∂ S + q 1 ∂ ∂ R τ2 = [ τ1 , f ] = ∂ ∂ I + β S-γ-β I β I ∂ ∂ S + q 2 ∂ ∂ R
with q 1 (S, I, R) is any function and q 2 = -L f q 1 , which gives [ τ1 , τ2 ] = 0. Therefore the conditions stated in [START_REF] Jo | Observer design for non-linear systems that are not uniformly observable[END_REF] are not fulfilled.

Calculation of τ:

Applying Procedure 1 proposed in Section 3, we obtain the following new vector fields:

τ 1 = 1 β I ∂ ∂ S τ 2 = ∂ ∂ I + β S-γ-β I β I ∂ ∂ S τ 3 = ∂ ∂ R
It is obvious that [τ 1 , τ 2 ] = 0, which implies that the conditions of Theorem 1 are not fulfilled.

Calculation of σ :

In order to construct a new family of commutative vector fields by introducing a change of coordinates on the output, the method proposed in [START_REF] Boutat | On the transformation of nonlinear dynamical systems into the extended nonlinear observable canonical form[END_REF] is used to deduce a non-zero output function s(y). For this, set σ 1 = s(y)τ 1 . According to (25) one has σ 2 = s(y)τ 2s (y) (β SI -γI) τ 1 . A straightforward calculation gives:

[σ 1 , σ 2 ] = 2s
σ 1 = 1 β ∂ ∂ S , σ 2 = -I ∂ ∂ S + I ∂ ∂ I , σ 3 = ∂ ∂ R
with which one has

[σ 1 , σ 2 ] = [σ 1 , σ 3 ] = [σ 2 , σ 3 ] = 0
thus the first two conditions of Theorem 2 are satisfied. As the SIR model does not contain any input, then the third condition of Theorem 2 does not need to be verified. Since

Λ1 = θ 1 θ 2 (σ 1 , σ 2 ) = 0 I I -β I 2 + (-γ + Sβ ) I which yields ω1 ω2 = Λ-1 θ 1 θ 2 = dξ 1 dξ 2 = d β (S + I) ln I
Following (29), one can solve the following equation:

ω3 (σ 1 , σ 2 , σ 3 ) = ω3 1 β ∂ ∂ S , -I ∂ ∂ S + I ∂ ∂ I , ∂ ∂ R = (0, 0, 1)
which uniquely determines ω3 = dR. Therefore, the local diffeomorphism is given as follows φ (x) = [β (S + I), ln I, R] T , which transforms the SIR model into the following form:

       ξ1 = -β γe ȳ ξ2 = ξ 1 -β e ȳ -γ ζ1 = γe ȳ ȳ = Cφ = ln I

Conclusion

This paper deals with partial observability problem of nonlinear system. Necessary and sufficient conditions are established to guarantee the existence of a local diffeomorphism transforming a class of nonlinear systems into the proposed partial observer normal form with and without a change of coordinates on the output. For the transformed system, a simple Luenberger observer can be designed to estimate the part of observable states. Examples are provided to illustrate the feasibility of the proposed results. Proof. Firstly, let us recall that the differential dπ of a differential 1-form π yields a 2-form whose evaluation on two vector fields X and Y is given by:

dπ(X,Y ) = L X π(Y ) -L Y π(X) -π ([X,Y ]) (30)
If π is closed, then dπ = 0 and the above formula becomes:

π ([X,Y ]) = L X π(Y ) -L Y π(X) (31)
Denote ı X π = π(X) as the interior product (or the contraction of a differential form with a vector field) of an 1-form π by a vector field X and recall that by definition the interior product of a function by a vector field is equal 0. Then the interior product relates the exterior derivative and Lie derivative of differential forms by the following so-called Cartan's identity:

L X = dı X + ı X d (32)
Proof of P1: According to the definition of the 1-forms θ i = dL i-1 f h for 1 ≤ i ≤ r, thus θ i are exact (then closed), which implies that for any two vector fields

H 1 ∈ ∆ ⊥ and H 2 ∈ ∆ ⊥ , we have [H 1 , H 2 ] ∈ ∆ ⊥ due to the relation of (31). Thus ∆ ⊥ is involutive.

Proof of P2:

The second property is an obvious consequence of the first one thanks to the well-known Frobenius' Theorem (see [START_REF] Isidori | Nonlinear Control Systems[END_REF] for the details).

Proof of P3:

As θ i for 1 ≤ i ≤ r are exact, thus for any vector field H ∈ ∆ ⊥ , according to the equation (31) we have

θ i ([ f , H]) = L f θ i (H) -L H θ i ( f ) Since H ∈ ∆ ⊥ , then θ i (H) = 0 which implies that L f θ i (H) = 0. Therefore, θ i ([ f , H]) = 0 is equivalent to L H θ i ( f ) = 0 .
Moreover, based on the Cartan's identity (32), we have:

L H θ i ( f ) = dı H θ i ( f ) + ı H dθ i ( f )
Due to the fact that θ i ( f ) is a function, therefore its interior product by a vector field is 0, thus ı H θ i ( f ) = 0, which means dı H θ i ( f ) = 0. So, the above equation becomes

L H θ i ( f ) = ı H dθ i ( f ) It reminds to show that ı H dθ i ( f ) = 0. As dθ i ( f ) = θ i+1 for 1 ≤ i ≤ r -1, then ı H dθ i ( f ) = ı H θ i+1 = θ i+1 (H) = 0. For i = r, we have dθ r ( f ) = dL r+1 f h. Since it is assumed that rank{dh, • • • , dL n f h} = r, it means dθ r ( f ) = r ∑ i=1 γ i θ i which im- plies ı H dθ i ( f ) = 0.
Finally, we proved that, for any vector field H ∈ ∆ ⊥ , one always has θ

i ([ f , H]) = 0, which implies that [ f , H] ∈ ∆ ⊥ . Thus ∆ ⊥ is f -invariant. A.2 Proof of Lemma 2
Proof. Since τ1 and τ1 are both the solution of (10), then there exists an H 1 ∈ ∆ ⊥ such that τ1 = τ1 + H 1 .

According to ( 11) and ( 12), we have

τ2 = [ τ1 , f ] = [ τ1 + H 1 , f ] = [ τ1 , f ] + [H 1 , f ] = τ2 + H 2
where we applied the third property P3 of Lemma 1, i.e.

[H 1 , f ] = H 2 ∈ ∆ ⊥ . By induction, we deduce that for 1 ≤ i ≤ r there exist H i ∈ ∆ ⊥ such that τi = τi + H i .

A.3 Proof of Lemma 3

Proof. 1) For any H ∈ ∆ ⊥ , as dθ k = 0 since θ k = dL k-1 f h for 1 ≤ k ≤ r is closed, we have:

θ k ([ τi , H]) = L τi θ k (H) -L H θ k ( τi ) = -L H θ k ( τi )
where we have applied θ k (H) = 0 for 1 ≤ k ≤ r since H ∈ ∆ ⊥ . When i = 1, then from (10) we have Repeating the above procedure by using the Jacobi's identity, we can prove by induction that, for any H ∈ ∆ ⊥ , we have:

θ k ( τ1 ) = 0, 1 ≤ k ≤ r -1 1, k = r
[ τi , H] ∈ ∆ ⊥ , for 1 ≤ i ≤ r.
2) For any H i ∈ ∆ ⊥ and H j ∈ ∆ ⊥ , it is clear that [H i , H j ] ∈ ∆ ⊥ . Then we have:

[ τi + H i , τ j + H j ] = [ τi , τ j ] + [ τi , H j ] + [H i , τ j ] + [H i , H j ]
Since we have proved in the above that [ τi , H j ] ∈ ∆ ⊥ and [H i , τ j ] ∈ ∆ ⊥ , with the fact that [H i , H j ] ∈ ∆ ⊥ , we get

[ τi + H i , τ j + H j ] = [ τi , τ j ] modulo ∆ ⊥
3) As we have proved that for any H ∈ ∆ ⊥ one has [h, τi ] ∈ ∆ ⊥ for 1 ≤ i ≤ r, due to the fact that τ j for r + 1 ≤ j ≤ n are the basis of ∆ ⊥ , therefore we have: [ τi , τ j ] ∈ ∆ ⊥ .

A.4 Proof of Lemma 4:

Proof. Necessity: Suppose that there exist H i ∈ ∆ ⊥ and H j ∈ ∆ ⊥ , which determine τ i = τi + H i and τ j = τ j + H j for 1 ≤ i, j ≤ r such that [τ i , τ j ] = 0, then according to the second item of Lemma 3 we have 0 = [τ i , τ j ] = [ τi , τ j ] modulo ∆ ⊥ which implies that [ τi , τ j ] ∈ ∆ ⊥ . Sufficiency: Assume that for one choice of τ1 satisfying (10), we have [ τi , τ j ] ∈ ∆ ⊥ for 1 ≤, i, j ≤ r where τ i was defined in (11). Then we need to prove that there exist H i ∈ ∆ ⊥ for 1 ≤ i ≤ r such that [τ i , τ j ] = 0 where τ k = τk + H k . Let {τ r+1 , • • • , τ n } be (nr) determined commutative basis of ∆ ⊥ , it is well-known (see [START_REF] Nijmeijer | Nonlinear dynamical control systems[END_REF]) that there exist r commutative vector fields, noted as {X 1 , • • • , X r }, such that they commute with {τ r+1 , • • • , τ n }, thus form together a basis of the tangent fiber bundle T X of X .

Then, for all 1 ≤ i ≤ r we can rewrite τi as:

τi = r ∑ l=1 γ i,l (x)X l + n ∑ l=r+1 γ i,l (x)τ l
By choosing H i = -∑ n l=r+1 γ i,l (x)τ l ∈ ∆ ⊥ , then we have:

τ i = τi + H i = r ∑ l=1 γ i,l (x)X l
For the same reason, there exist H j = ∑ n l=r+1 γ j,l (x)τ l ∈ ∆ ⊥ yielding:

τ j = τ j + H j = r ∑ l=1
γ j,l (x)X l Moreover, for 1 ≤ i, j ≤ r, due to Lemma 3 we have

[ τi , τ j ] = [τ i , τ j ] modulo ∆ ⊥ . ( 33
)
As [X k , X s ] = 0 for 1 ≤ k, s ≤ r, then (33) becomes

[ τi , τ j ] = r ∑ l,s=1
γ i,l L X l γ j,s X s -r ∑ s,l=1 γ j,s L X s γ i,l X l modulo ∆ ⊥ Thus, if [ τi , τ j ] ∈ ∆ ⊥ , then there exist H i ∈ ∆ ⊥ and H j ∈ ∆ ⊥ defined above, such that

[τ i , τ j ] = r ∑ l,s=1
γ i,l L X l γ j,s X s -r ∑ s,l=1 γ j,s L X s γ i,l X l = 0 due to the fact that X k / ∈ ∆ ⊥ .

Acknowledgement

  2 (y) I -2s(y)s (y) τ 1 thus [σ 1 , σ 2 ] = 0 if and only if the function s(y) fulfills the folto choose s(y) = y = I. Then one has:

Appendix A. 1

 1 Proof of Lemma 1:

  Thus, θ k ( τ1 ) is constant, then its Lie derivative is 0. Therefore, we haveθ k ([ τ1 , H]) = -L H θ k ( τ1 ) = 0 which implies that [ τ1 , H] ∈ ∆ ⊥ .For i = 2, by applying Jacobi's identity we have[ τ2 , H] = [[ τ1 , f ], H] = -[[H, τ1 ], f ] -[[H, f ], τ1 ] Since [ τ1 , H] ∈ ∆ ⊥ , and [H, f ] ∈ ∆ ⊥ (P3 of Lemma 1), then we obtain [ τ2 , H] ∈ ∆ ⊥ .

  thus, by integrating (19) and (20), we obtain φ *

Do not confuse the differential (Jacobian) of φ : φ * where the star is subscript with the pullback: φ * where the star is superscript.

The authors gratefully acknowledge anonymous reviewers for the valuable suggestions and helpful remarks to improve the quality our manuscript.

$ This paper was supported in part by Région Centre France, by Ministry of Higher Education and Research Nord-Pas de Calais Regional Council, by FEDER through the Contrat de Projets Etat Region (CPER), by ARCIR Project ESTIREZ Nord-Pas de Calais Regional Council, by Chinses NSF(61304077), and by International Science Technology Cooperation Program of China (2015DFA01710).