G Zheng
email: gang.zheng@inria.fr

F J Bejarano
email: javbejarano@yahoo.com.mx

Observer design for linear singular time-delay systems $

Keywords: Delayed singular systems, commensurate delays, observer

This paper investigates the problem of observer design for a general class of linear singular time-delay systems, in which the time delays are involved in the state and its derivatives, the output and the known input (if there exists). The involvement of the delay could be multiple which however is rarely studied in the literature. Sufficient conditions are proposed which guarantees the existence of a Luenberger-like observer for the general system.

Introduction

Singular systems (known also as Descriptor systems/Algebraic-Differential systems) enable us to model many physical, biological and economical systems [START_REF] Campbell | Singular systems of differential equations II[END_REF], which has been well discussed in [START_REF] Wang | Analysis and Design of Singular Markovian Jump Systems[END_REF]. For such systems, time delay can be also encountered [START_REF] Niculescu | Delay-independent stability in lossless propagation models with applications[END_REF]. Thus, the singular time-delay systems have been extensively analyzed during last decade. Lots of researches are focused on the stabilization of singular time-delay systems, such as H ∞ control, observer-based control and so on [START_REF] Gu | Analysis and Synthesis of Singular Systems with Time-Delays[END_REF].

Concerning the concepts of observability and observer for the singular time-delay systems, a direct adaptation of existing results on observability and observers from regular systems to singular systems is not immediate due to the fact that they involve both differential and algebraic equations. Up to now, most of the existing works are based on the simple case with only one delay in the state, i.e. E ẋ(t) = A 0 x(t) + A 1 x(th) where the input could be also involved. For this simple case, a general solution was derived in [START_REF] Wei | General solution and observability of singular differential[END_REF], based on which a sufficient condition for exact observability in finite time was deduced. When designing an observer for the mentioned simple linear singular time-delay systems, a few results can be found in the literature. In [START_REF] Feng | Observer design for linear singular timedelay systems[END_REF], three kinds of observers (functional observer, reduced-order observer and full-order observer) are studied for the above simple case. In [START_REF] Ezzine | Unknown inputs functional observers designs for descriptor systems with constant time delay[END_REF], a functional observer for singular time-delay systems with unknown inputs was presented, and the existence condition of such observer and the gain implemented in the design are obtained by solving LMIs. A Luenberger-like observer is proposed in [START_REF] Khadhraoui | Design of full order unknown input observers for delayed singular systems with state variable time delay[END_REF] for the linear singular time-delay systems with unknown inputs not affected by time delays.

In this paper, we deal with a quite general linear singular time-delay system of the form: ∑ l i=0 E i ẋ(tih) = ∑ k i=0 A i x(tih), which in fact covers different types of time-delay system (singular or not, with or without neutral term). For such a presentation, there exist lots of applications, such as LC electrical lines [START_REF] Brayton | Small-signal stability criterion for electrical networks containing lossless transmission lines[END_REF] and so on. More concrete applications can be found in [START_REF] Niculescu | Delay effects on stability: a robust control approach[END_REF]. For such a general form, observability analysis and observer design become more difficult. Recently, inspired by the well-known Silverman and Molinari algorithm (see [START_REF] Silverman | Inversion of multivariable linear systems[END_REF]; [START_REF] Molinari | A strong controllability and observability in linear multivariable control[END_REF]) to analyze the observability for linear systems (with or without unknown input), a similar and checkable sufficient condition was proposed in Bejarano and Zheng (2016a) to analyze the observability for this general singular time-delay systems. As far as we know, the only work on the observer design for this general case is [START_REF] Perdon | An unknown input observer for singular time-delay systems[END_REF], where a Luenberger-like observer is proposed by using a virtual discrete time system with the matrices of the originally considered continuous-time system. In this method, a geometric notion of conditioned invariant submodule is introduced for this class of systems over a ring and a design procedure is presented. However, the sufficient condition deduced in that paper is difficult to be checked, since it highly depends on the assumption of the existence of Hurwitz polynomial. The contributions of this paper are as follows. Firstly, the class of the studied systems is quite general (we allow multiple delays both in the x(t) and its derivative), which in fact can cover four different classes of systems. As far as we known, there exist some methods to eliminate (or reduce the degree of) the delay, such as [START_REF] Lee | Canonical forms for time delay system[END_REF][START_REF] Germani | An asymptotic state observer for a class of nonlinear delay systems[END_REF] and [START_REF] Garate-Garcia | Equivalence of linear time-delay systems[END_REF]. It has been proven in [START_REF] Garate-Garcia | Equivalence of linear time-delay systems[END_REF] that the elimination or the reduction of delay degree via a bicausal transformation with the same dimension is possible if some conditions on A(δ ) and B(δ ) are satisfied. In other words, the elimination or the reduction of delay degree is not always possible. This issue has been highlighted in [START_REF] Zheng | Unknown input observer for linear time-delay systems[END_REF]; Bejarano and Zheng (2016c, 2014, 2016b). Therefore, allowing multiple delays in the state, in the derivative, and in the output is an essential generalization. The second contribution of this paper is to deduce a checkable sufficient condition such that a Luenberger-like observer exists for such a general linear singular time-delay system. Moreover, it provides as well a constructive way to synthesize the proposed observer.

Notations and problem statement

In this paper, we consider the following class of linear systems with commensurate delays:

       k e ∑ i=0 Ẽi ẋ (t -ih) = k a ∑ i=0 Ãi x (t -ih) ỹ (t) = k c ∑ i=0 Ci x (t -ih) (1)
where the vector x (t) ∈ R n , the system output vector ỹ (t) ∈ R p , h represents the basic delay, the initial condition ϕ (t) is a piece-wise continuous function ϕ

(t) : [-kh, 0] → R n (k = max {k a , k c , k e }); thereby x (t) = ϕ (t) on [-kh, 0]. Ãi , Bi and
Ẽi are matrices of appropriate dimension with entries in R. It is worth noting that the studied system of the form (1) is quite general, and it covers different types of time-delay systems. More precisely:

• if k e = 0 and Ẽ0 = I, then system (1) becomes the classical linear time-delay system of the form ẋ =

k a ∑ i=0 Ãi x (t -ih);
• if k e = 0 and the rank of Ẽi is not full, then system (1) is equivalent to a linear singular time-delay systems of the

form Ẽ0 ẋ (t) = k a ∑ i=0 Ãi x (t -ih);
• when k e > 0 and Ẽ0 = I, then system (1) can be written as

ẋ(t) = k a ∑ i=0 Ãi x (t -ih) - k e ∑ i=1
Ẽi ẋ (tih) which is typically a linear time-delay neutral system;

• moreover, if k e > 0 and the rank of Ẽ0 is not full, system (1) represents a general linear singular time-delay neutral sys-

tem of the form Ẽ0 ẋ(t) = k a ∑ i=0 Ãi x (t -ih) - k e ∑ i=1
Ẽi ẋ (tih).

Nevertheless, the form (1) is so general that it might include as well advanced systems for which the existence of the solution cannot be always guaranteed. In order to exclude such ill-conditioned systems, it is assumed in this paper that system (1) admits at least one solution. It is worth noting that the existence of a unique solution is not necessary for the observability analysis and the observer design. Take any system with several solutions as an example, this issue can be easily identified if the output of this system is all the state. In this case, the system is always observable even if several solutions exist.

Remark 1. For the special class of systems mentioned above, there exist some results on observer design in the literature, for example [START_REF] Campbell | Singular systems of differential equations II[END_REF], [START_REF] Hou | An observer design for linear timedelay systems[END_REF] and [START_REF] Ezzine | Unknown inputs functional observers designs for descriptor systems with constant time delay[END_REF], and lots of the existing works consider only one delay in x(t), i.e. k e = k c = 0 and k a = 1 in (1). However, for a linear system with commensurate delays of the general form (1) covering a large class of linear singular (or not) time-delay neutral (or not) systems, to the best of our knowledge, rare results on observer design have been reported in the literature. Therefore, the problem of designing an observer is still an open problem for the general form (1).

Motivated by this fact, this paper proposes a Luenberger-like observer and sufficient conditions are deduced which guarantee the existence of such an observer.

In the following, for the purpose of simplifying the analysis, let us introduce the delay operator δ :

x (t) → x (t -h) with δ k x(t) = x(t -kh), k ∈ N 0 .
Then the following notations will be used in this paper. R is the field of real numbers. The set of positive integers is denoted by N. I n for n ∈ N means the identity matrix of order n. R [δ ] is the polynomial ring over the field R. R n [δ ] is the R [δ ]-module whose elements are the vectors of dimension n and whose entries are polynomials. By R q×s [δ ] we denote the set of matrices of dimension q × s, whose entries are in R

[δ ]. For a matrix M (δ ), rank R[δ ] M (δ ) means the rank of the matrix M (δ ) over R [δ ]. M(δ ) ∼ N(δ ) means the sim- ilarity between two polynomial matrices M(δ ) and N(δ ) over R[δ ], i.e. there exist two unimodular matrices U 1 (δ ) and U 2 (δ ) over R[δ ] such that M(δ ) = U 1 (δ )N(δ )U 2 (δ ).
After having introduced the delay operator δ , system (1) may be then represented in the following compact form:

Ẽ(δ ) ẋ (t) = Ã (δ ) x (t) ỹ (t) = C (δ ) x (t) (2)
where

à (δ ) ∈ R n×n [δ ], C (δ ) ∈ R p×n [δ ] and Ẽ (δ ) ∈ R n×n [δ ]
are matrices over the polynomial ring R [δ ], defined as à (δ ) :=

k a ∑ i=0 Ãi δ i , C (δ ) := k c ∑ i=0 Ci δ i and Ẽ (δ ) := k e ∑ i=0 Ẽi δ i . Without loss of generality, it is thus assumed in this paper that n ≤ n, rank R[δ ] Ẽ(δ ) = q ≤ n and rank R[δ ] C(δ ) = p.
Remark 2. The above assumptions are not restrictive. rank R[δ ] Ẽ(δ ) = q ≤ n ≤ n is due to the fact that the studied system might be singular or neutral.

Moreover, if rank R[δ ] C(δ ) = p < p, we can always eliminate the dependent outputs of (2). More precisely, if rank R[δ ] C(δ ) = p < p, then there exists a polynomial matrix U(δ ) ∈ R p×p [δ ] such that U(δ ) C(δ ) ∈ R p×n [δ ] is of full row rank. In this case, we can rewrite the output as U(δ ) ỹ = U(δ ) C(δ )x with rank R[δ ] U(δ ) C(δ ) = q.

Definition, assumptions and preliminary result

For each instant t, the available measurements are only y(t) and its delayed values which can be used to estimate x(t). We cannot in fact utilize the future value of y(t), otherwise it is not causal. Therefore, it is desired to use only the actual and the past information (not the future information) of the measurement to design an observer for time-delay systems due to the requirement of the causality. Thus the following definition of backward observability is given. 2) is said to be backward observable on [t 1 ,t 2 ] if andonly if, for each τ ∈ [t 1 ,t 2 ] there exist t1 andt2

Definition 1. System (
≤ τ such that y(t) = 0 for all t ∈ [t 1 , t2 ] implies x(τ) = 0.
The above definition of backward observability is related to the final observability given [START_REF] Lee | Observability and related structural results for linear hereditary systems[END_REF], and it can be interpreted that x(t) depends only on the previous values of y(t), its time delayed values, and its derivatives.

Since we are going to analyze system (2) which is described by the polynomial matrices over R[δ ], let us firstly give the following definition of unimodularity over R[δ ].

Definition 2. [START_REF] Antsaklis | A Linear Systems Primer, 1st Edition[END_REF]

) A polynomial matrix A(δ ) ∈ R n×q [δ ] is said to be left (or right) unimodular (or in- vertible) over R[δ ] if there exists A -1 L (δ ) ∈ R q×n [δ ] with n ≥ q (or A -1 R (δ ) ∈ R q×n [δ ] with n ≤ q), such that A -1 L (δ )A(δ ) = I q×q (or A(δ )A -1 R (δ ) = I n×n ). A square matrix A(δ ) ∈ R n×n [δ ] is said to be unimodular (or invertible) over R[δ ] if A -1 L (δ ) = A -1 R (δ ) = A -1 (δ ).
It is well known that for any polynomial matrix

D(δ ) ∈ R p×m [δ ] with rank R[δ ] D(δ ) = r ≤ min{p, m}, there exists a unimodular matrix U(δ ) over R [δ ] such that U (δ ) D (δ ) = D(δ ) 0 where rank R[δ ] D(δ ) = rank R[δ ] D(δ ) and D(δ ) is of full row rank over R [δ ]
. Also, there exist two unimodular matrices U (δ ) and V (δ ) over R [δ ] such that D (δ ) is reduced to its Smith form, i.e.,

U (δ ) D (δ )V (δ ) = S D(δ ) 0 0 0 (3) with S D(δ ) being a diagonal matrix as S D(δ ) = diag{ψ 1 D(δ ) , • • • , ψ r D(δ ) } where ψ i D(δ )
are the invariant factors of D (δ ).

For simplicity, let us denote

Inv S [D(δ )] = ψ i D(δ ) 1≤i≤r
as the set of its invariant factors of the Smith form of D (δ ), the following result, adapted from the result on the left unimodularity stated in [START_REF] Hou | An observer design for linear timedelay systems[END_REF], is obvious.

Lemma 1. For any polynomial matrix D (δ ) ∈ R p×m [δ ], it is left (or right) unimodular over R[δ ] if and only if the following conditions are satisfied:

1. rank R[δ ] D(δ ) = m ≤ p (or rank R[δ ] D(δ ) = p ≤ m ); 2. Inv S [D(δ )] ⊂ R.
Concerning the definition of backward observability, and following the ideas of [START_REF] Silverman | Inversion of multivariable linear systems[END_REF] and [START_REF] Molinari | A strong controllability and observability in linear multivariable control[END_REF], define Ñk (δ ) the matrices generated by the following algorithm for the triple Ẽ(δ ), Ã(δ ), C(δ ) , proposed in Bejarano and Zheng (2016a):

• Initialization: Set ∆1 (δ ) C(δ ) and Ñ1 (δ ) ∆1 (δ );

• Iteration:

For k ≥ 1, there exists a unimodu-

lar matrix Sk (δ ) = Sk,1 (δ ) Sk,2 (δ ) Sk,3 (δ ) Sk,4 (δ ) such that Sk (δ ) I n Ẽ (δ ) 0 Ñk (δ ) Sk,1 (δ ) Λk (δ ) Sk,3 (δ ) 0 with rank R[δ ] Λk (δ ) = rank R[δ ] Ẽ(δ ) Ñk (δ ) ; then set ∆k+1 (δ ) = Sk,3 (δ ) Ã(δ ) and define Ñk+1 (δ ) = Ñk (δ ) ∆k+1 (δ ) (4)
With the above algorithm, the following result was stated in Bejarano and Zheng (2016a).

Lemma 2. Bejarano and Zheng (2016a) If there exists a least integer k * such that Ñk * +1 (δ ) is left unimodular over R [δ ], then system (2) is backward observable (or the triple

Ẽ(δ ), Ã(δ ), C(δ ) is backward observable).
Due to the above result, the following assumption is imposed.

Assumption 1. For system (2) with the triple Ẽ(δ ), Ã(δ ), C(δ ) , there exists a least integer k * such that Ñk * +1 (δ ) defined in (4) is left unimodular over R [δ ].
Let us highlight that the condition imposed in Assumption 1 is only sufficient but not necessary for system (2) to be backward observable. However, it becomes to be necessary and sufficient condition of observability for system (2) without delays.

Since rank R[δ ] Ẽ(δ ) = q ≤ n, then there exists a unimodular matrix

Π(δ ) = Π 1 (δ ) Π 2 (δ ) over R [δ ] such that Π(δ ) Ẽ(δ ) = E(δ ) 0 Π(δ ) Ã(δ ) = A(δ ) A(δ ) (5) with E(δ ) ∈ R q×n [δ ] of full row rank over R[δ ] which is equal to q. It is worth noting that the choice of Π(δ ) is not unique, thus different choices of Π(δ ) will give different E(δ ) and A(δ ).
The influence of different choices of Π(δ ) will be analyzed in Section 4. It is obvious that if q = n, then we can just choose Π(δ ) = I n and E(δ

) = Ẽ(δ ). By noting C(δ ) = A(δ ) C(δ ) and y(t) = 0 ỹ(t)
, system (2) can be rewritten as follows:

E(δ ) ẋ (t) = A (δ ) x (t) y (t) = C (δ ) x (t) (6) with E(δ ) ∈ R q×n [δ ], A(δ ) ∈ R q×n [δ ] and C(δ ) ∈ R ( n+p-q)×n [δ ].
Lemma 3. If Assumption 1 is satisfied, then the triple (E(δ ), A(δ ),C(δ )) for system (6) is backward observable.

Proof. Suppose that there exists a least integer k * such that Ñk * +1 (δ ) defined in ( 4) is left unimodular over R [δ ]. Applying the algorithm for system (2) of the triple Ẽ(δ ), Ã(δ ), C(δ ) , when k = 1, we have Ñ1 (δ ) = ∆1 (δ ) = C(δ ). Due to the fact that

I n Ẽ (δ ) 0 Ñ1 (δ ) = Π -1 (δ ) 0 0 I p Π(δ ) Π(δ ) Ẽ (δ ) 0 Ñ1 (δ ) = Π -1 (δ ) 0 0 I p   Π 1 (δ ) E (δ ) Π 2 (δ ) 0 0 Ñ1 (δ )  
then there exists a unimodular matrix

S1 (δ ) =   S1,1 (δ ) 0 S1,2 (δ ) 0 I n-q 0 S1,3 (δ ) 0 S1,4 (δ )   such that S1 (δ )   Π 1 (δ ) E (δ ) Π 2 (δ ) 0 0 Ñ1 (δ )   =   S1,1 (δ )Π 1 (δ ) Λ1 (δ ) Π 2 (δ ) 0 S1,3 (δ )Π 1 (δ ) 0   with rank R[δ ] Λ1 (δ ) = rank R[δ ] Ẽ(δ ) Ñ1 (δ )
. Thus we have

Ñ2 (δ ) =   ∆1 (δ ) Π 2 (δ ) Ã(δ ) S1,3 (δ )Π 1 (δ ) Ã(δ )   =   C(δ ) A(δ ) S1,3 (δ )A(δ )   .
Applying the same algorithm for system (6) of the triple

(E(δ ), A(δ ),C(δ )), when k = 1, we have N 1 (δ ) = ∆ 1 (δ ) = C(δ ) = A(δ ) C(δ ) thus there exists a unimodular matrix ϒ 1 (δ ) over R [δ ] such that N 1 (δ ) = ϒ 1 (δ ) Ñ1 (δ ) A(δ )
. Moreover, by applying the same matrix S1 (δ ) obtained above, we have

S1 (δ ) I q E (δ ) 0 C(δ ) = S1 (δ )   I q E (δ ) 0 A(δ ) 0 C(δ )   =   S1,1 (δ ) Λ1 (δ ) 0 A(δ ) S1,3 (δ ) 0   with rank R[δ ] Λ1 (δ ) = rank R[δ ] E(δ ) C(δ )
. Thus there exists a

unimodular matrix S 1 (δ ) over R[δ ] such that S 1 (δ ) I q E (δ ) 0 C(δ ) = S 1,1 (δ ) Λ 1 (δ ) S 1,3 (δ ) 0 =   S 1,1 (δ ) Λ 1 (δ ) S A 1 (δ ) 0 S1,3 (δ ) 0   with rank R[δ ] Λ 1 (δ ) = rank R[δ ]   E(δ ) A(δ ) C(δ )   , which gives N 2 (δ ) =   C(δ ) S A 1 (δ )A(δ ) S1,3 (δ )A(δ )   .
This means that there ex-

ists a unimodular matrix ϒ 2 (δ ) over R [δ ] such that N 2 (δ ) = ϒ 2 (δ ) Ñ2 (δ ) S A 1 (δ )A(δ )
.

By iteration, we can find the following relation:

N k+1 (δ ) = ϒ k+1 (δ ) Ñk+1 (δ ) S A k (δ )A(δ )
where ϒ k+1 (δ ) is a unimodular matrix over R [δ ]. If there exists a least integer k * such that Ñk * +1 (δ ) is left unimodular over R [δ ], then N k * +1 (δ ) is also left unimodular over R [δ ], which implies that the triple (E(δ ), A(δ ),C(δ )) for system (6) is backward observable.

Assumption 2. For the triple Ẽ(δ ), Ã(δ ), C(δ ) of system (2), it is assumed that the following condition:

Inv S   Ẽ(δ ) Ã(δ ) 0 Ẽ(δ ) 0 C(δ )   = Inv S Ẽ(δ ) 0 0 I n (7)
is satisfied.

Remark 3. When the treated system (2) does not contain any delay, i.e. the triple Ẽ(δ ), Ã(δ ), C(δ ) becomes a constant triple Ẽ, Ã, C . The set of invariant factors can be characterized by using the rank condition over R. Therefore, the condition (7) imposed in the above assumption is equivalent to the following one:

rank R   Ẽ Ã 0 Ẽ 0 C   = rank R Ẽ 0 0 I n = n + rank R Ẽ
It has been shown in [START_REF] Campbell | Singular systems of differential equations II[END_REF] that the above condition is necessary for the existence of a Luenbergerlike observer. In this sense, the required condition (7) in Assumption 2 is not restrictive.

Lemma 4. The condition (7) in Assumption 2 is equivalent to the following one:

Inv S   E(δ ) A(δ ) 0 E(δ ) 0 C(δ )   = Inv S E(δ ) 0 0 I n (8)
Proof. With the decomposition by the unimodular matrix Π(δ ) over R [δ ] defined in (5), we have the following similarity property:

  Ẽ(δ ) Ã(δ ) 0 Ẽ(δ ) 0 C(δ )   ∼     E(δ ) A(δ ) 0 A(δ ) 0 E(δ ) 0 C(δ )     ∼   E(δ ) A(δ ) 0 E(δ ) 0 C(δ )  
Following the same argument, we have:

Ẽ(δ ) 0 0 I n ∼ E(δ ) 0 0 I n
Therefore, the condition (7) in Assumption 2 is equivalent to (8).

Lemma 5. For system (6), the matrix E(δ ) C(δ ) is left unimodular over R[δ ] if and only if Assumption 2 is satisfied for system (2).

Proof. According to Lemma 4, it is equivalent to prove that

E(δ ) C(δ ) is left unimodular over R[δ ] if and only if (8) is sat- isfied. Since E(δ ) is full row rank over R[δ ] which is equal to q, de- noting by S E(δ ) = diag{ψ 1 E(δ ) , • • • , ψ q E(δ )
} the diagonal matrix of the invariant factors of E (δ ), then (8) can be rewritten as:

Inv S   S E(δ ) A(δ ) 0 E(δ ) 0 C(δ )   = Inv S S E(δ ) 0 0 I n
The above equation implies that

rank R[δ ]   S E(δ ) A(δ ) 0 E(δ ) 0 C(δ )   = rank R[δ ] S E(δ ) 0 0 I n (9)
Therefore, we have:

rank R[δ ] E(δ ) C(δ ) = n. Denote S EC(δ ) = diag{ψ 1 EC(δ ) , • • • , ψ n EC(δ ) } as the diagonal matrix of the invariant factors of E(δ ) C(δ ) , then (9) becomes rank R[δ ] S E(δ ) A(δ ) 0 S EC(δ ) = rank R[δ ] S E(δ ) 0 0 I n
The above equation implies that

Inv S E(δ ) C(δ ) ⊂ R.
According to Lemma 1, we can then state that

E(δ ) C(δ ) is left unimodular over R[δ ] if and only if Assumption 2 is satisfied.
Corollary 1. If Assumption 2 is satisfied, then there exists a unimodular matrix P(δ

) ∈ R ( n+p)×( n+p) [δ ] over R [δ ] such that P(δ ) E(δ ) C(δ ) = I n 0 .
Proof. As stated in Lemma 5, if Assumption 2 is satisfied, then

we have rank R[δ ] E(δ ) C(δ ) = n. and Inv S E(δ ) C(δ ) ⊂ R. Thus,
there exists a unimodular matrix S EC (δ ) over R [δ ] that transforms E(δ ) C(δ ) into the following Hermite form:

S EC (δ ) E(δ ) C(δ ) = Q(δ ) 0 where rank R[δ ] E(δ ) C(δ ) = n and Q(δ ) ∈ R n×n [δ ] is of full rank over R[δ ]. Since E(δ ) C(δ ) is left unimod- ular over R[δ ], then Q(δ ) is unimodular over R[δ ] as
well, which implies that there exists Q-1 (δ ) such that Q-1 (δ ) Q(δ ) = I n . Therefore, for the left unimodular matrix E(δ ) C(δ ) over R[δ ], there exists a unimodular matrix P(δ ) = P 1 (δ ) P 2 (δ )

P 3 (δ ) P 4 (δ ) = Q-1 (δ ) 0 0 I n+p-n S EC (δ ) over R [δ ],
with -q) [δ ], such that

P 1 (δ ) ∈ R n×q [δ ], P 2 (δ ) ∈ R n×( n+p-q) [δ ], P 3 (δ ) ∈ R ( n+p-n)×q [δ ] and P 4 (δ ) ∈ R ( n+p-n)×( n+p
P(δ ) E(δ ) C(δ ) = I n 0 , i.e.
, the following equations:

P 1 (δ )E(δ ) + P 2 (δ )C(δ ) = I n P 3 (δ )E(δ ) + P 4 (δ )C(δ ) = 0 (10)
are always satisfied.

In the following, let us denote

Ā(δ ) = P 1 (δ )A(δ ) ∈ R n×n [δ ] (11)
and C(δ

) = P 3 (δ )A(δ ) C(δ ) ∈ R (2 n-n-q+2p)×n [δ ] (12)
Then define the following polynomial matrix over R[δ ]:

Ōl (δ ) =      C(δ ) C(δ ) Ā(δ ) . . . C(δ ) Āl-1 (δ )      ∈ R (2 n-n-q+2p)l×n [δ ] (13)
where l ∈ N, and we arrive to the following lemma.

Lemma 6. Suppose Assumptions 1 and 2 are both satisfied for system (2), then there exists a least integer l * ∈ N such that Ōl * (δ ) defined in ( 13) is left unimodular over R [δ ].

Proof. If Assumption 2 is satisfied, then according to Lemma 5

the matrix

E(δ ) C(δ ) is left unimodular over R[δ ]. Moreover,
if Assumption 1 is satisfied, i.e. there exists a least integer k * such that Ñk * +1 (δ ) for ( 2) is left unimodular over R [δ ], then Lemma 3 ensures that there exists a least integer k * such that N k * +1 (δ ) for ( 6) is left unimodular over R [δ ] when applying the proposed algorithm.

For system (6), note now

∆ 1 (δ ) = C(δ ), N 1 (δ ) = ∆ 1 (δ ). Since E(δ ) C(δ ) is left unimodular over R [δ ]
, then there exists a unimodular matrix P(δ

) = P 1 (δ ) P 2 (δ ) P 3 (δ ) P 4 (δ ) over R [δ ]
such that (10) is satisfied. Set S 1 (δ ) = P(δ ), we have

S 1 (δ ) I q E(δ ) 0 N 1 (δ ) = S 1 (δ ) I q E(δ ) 0 C(δ ) = P 1 (δ ) I n P 3 (δ ) 0
Then we have ∆ 2 (δ ) = P 3 (δ )A(δ ), and

N 2 (δ ) = ∆ 1 (δ ) ∆ 2 (δ ) = C(δ ) P 3 (δ )A(δ ) Since E(δ ) C(δ ) is left unimodular over R [δ ], then   E(δ ) C(δ ) P 3 (δ )A(δ )   = E(δ ) N 2 (δ )
is also left unimodular over [δ ], thus there exists a unimodular matrix S 2 (δ ) = 

R
 P 1 (δ ) P 2 (δ ) 0 P 3 (δ ) P 4 (δ ) 0 P 3 (δ )A(δ )P 1 (δ ) P 3 (δ )A(δ )P 2 (δ ) -I n+p-n   over R [δ ] such that S 2 (δ ) I q E(δ ) 0 N 2 (δ ) =   P 1 (δ ), P 2 (δ ) 0 P 3 (δ ), P 4 (δ ) 0 P 3 (δ )A(δ ) [P 1 (δ ), P 2 (δ )] -I n+p-n     I q E(δ ) 0 C(δ ) 0 P 3 (δ )A(δ )   =   P 1 (δ ) I n P 3 (δ ) 0 P 3 (δ )A(δ )P 1 (δ ) 0   thus we have ∆ 3 (δ ) = P 3 (δ )A(δ ) P 3 (δ )A(δ )P 1 (δ )A(δ )
, and

N 3 (δ ) = N 2 (δ ) ∆ 3 (δ ) =   C(δ ) P 3 (δ )A(δ ) P 3 (δ )A(δ )P 1 (δ )A(δ )   . By induc- tion, for k ≥ 2, if Assumption 2 is satisfied we have N k+1 (δ ) =      C(δ ) P 3 (δ )A(δ )
. . .

P 3 (δ )A(δ ) [P 1 (δ )A(δ )] k-1     
Now, if there exists a least integer k * such that Ñk * +1 (δ ) defined in (4) for ( 2) is left unimodular over R [δ ], it implies that, according to Lemma 3,

N k * +1 (δ ) =      C(δ ) P 3 (δ )A(δ )
. . .

P 3 (δ )A(δ ) [P 1 (δ )A(δ )] k * -1      is left unimodular over R [δ ]. Since Ōk * (δ ) =      C(δ ) C(δ ) Ā(δ ) . . . C(δ ) Āk * -1 (δ )      ∼      N k * (δ ) C(δ )P 3 (δ )A(δ ) . . . C(δ ) [P 3 (δ )A(δ )] k *      thus Ōk * (δ ) is left unimodular over R [δ ].
With the above lemma, we can state the following result.

Lemma 7. If there exists a least integer l * ∈ N such that Ōl * (δ ) defined in ( 13) is left unimodular over R [δ ] , then there exists a matrix Γ(δ ) and a left unimodular matrix T (δ

) over R [δ ] such that T (δ ) Ā(δ ) -Γ(δ ) C(δ ) T -1 L (δ ) is constant (independent on δ ) and Hurwitz.
Proof. The proof is based on the results stated in [START_REF] Hou | An observer design for linear timedelay systems[END_REF] in which it has been proven that there exists a left unimodular matrix T (δ ) over R [δ ] such that

T (δ ) Ā(δ )T -1 L (δ ) = A 0 + F(δ )C 0 C(δ )T -1 L (δ ) = C 0 (14)
where

F(δ ) = F T 1 (δ ), • • • , F T l * (δ ) T and A 0 =        0 I 2 n-n-q+2p 0 • • • 0 0 • • • I 2 n-n-q+2p • • • 0 . . . . . . . . . . . . . . . 0 • • • 0 • • • I 2 n-n-q+2p 0 0 0 • • • 0        C 0 = I 2 n-n-q+2p 0 • • • 0 ( 15
) if and only if Assumption 2 is satisfied. Moreover, the matrix T (δ ) is given by: 16) with F i (δ ) being determined through the following equation:

T 1 (δ ) = C(δ ) T i+1 (δ ) = T i (δ ) Ā(δ ) -F i (δ ) C(δ ), for 1 ≤ i ≤ l * -1 (
[F l * (δ ), • • • , F 1 (δ )] = C(δ ) Āl * (δ ) Ōl * (δ ) -1 L (17)
Obviously, the pair (A 0 ,C 0 ) is observable, thus there exists a constant matrix Γ 0 such that (A 0 -Γ 0 C 0 ) is Hurwitz. Therefore, if Assumption 2 is satisfied, we have

T (δ ) Ā(δ ) -Γ(δ ) C(δ ) T -1 L (δ ) = A 0 + F(δ )C 0 -T (δ )Γ(δ )C 0 By choosing Γ(δ ) = T -1 L (δ ) [F(δ ) + Γ 0 ], we obtain T (δ ) Ā(δ ) -Γ(δ ) C(δ ) T -1 L (δ ) = A 0 -Γ 0 C 0 ( 18
)
which is Hurwitz, independent of the time-delay.

Observer and design procedure

Based on the deduced result, we are ready to present our main result.

Theorem 1. Suppose Assumptions 1 and 2 are both satisfied for system (2), then there exist the following matrices:

N(δ ) = P 1 (δ )A(δ ) -M(δ )P 3 (δ )A(δ ) -H(δ )C(δ ) K(δ ) = P 2 (δ ) -M(δ )P 4 (δ ) L(δ ) = H(δ ) + N(δ )K(δ ) (19)
where P i (δ ) for 1 ≤ i ≤ 4 are defined in (10) and

H(δ ) M(δ ) = T -1 L (δ ) [F(δ ) + Γ 0 ] (20)
with T -1 L (δ ) and F(δ ) being defined in ( 16) and ( 17), and Γ 0 being a constant matrix which makes (A 0 -Γ 0 C 0 ) Hurwitz, such that the following dynamical system:

ẇ = N(δ )w + L(δ )y x = w + K(δ )y (21)
is an exponential observer for system (2).

Proof. As we have proved that, if Assumption 2 is satisfied, then there exists a matrix P(δ ) such that ( 10 20), then we have

I n -K(δ )C(δ ) = [P 1 (δ ) -M(δ )P 3 (δ )] E(δ )
Therefore we obtain

ė = [P 1 (δ ) -M(δ )P 3 (δ )] E(δ ) ẋ - ẇ = [P 1 (δ ) -M(δ )P 3 (δ )] A(δ )x -N(δ )w -L(δ )y = [P 1 (δ )A(δ ) -M(δ )P 3 (δ )A(δ ) -L(δ )C(δ ) + N(δ )K(δ )C(δ )] x -N(δ ) x
By choosing

L(δ ) = H(δ ) + N(δ )K(δ )
and

N(δ ) = P 1 (δ )A(δ ) -M(δ )P 3 (δ )A(δ ) -H(δ )C(δ ) we have ė = [P 1 (δ )A(δ ) -M(δ )P 3 (δ )A(δ ) -H(δ )C(δ )] x -N(δ ) x = Ā(δ ) -[H(δ ), M(δ )] C(δ ) e = N(δ )e (22)
According to Lemma 7, if Assumptions 1 and 2 are both satisfied for system (2), then there exists a left unimodular matrix T (δ ) defined in ( 16) and a gain matrix

Γ(δ ) = T -1 L (δ ) [F(δ ) + Γ 0 ] such that T (δ ) Ā(δ ) -Γ(δ ) C(δ ) T -1 L (δ ) is Hurwitz. By setting Γ(δ ) = H(δ ), M(δ ) = T -1 L (δ ) [F(δ ) + Γ 0 ]
then we obtain that ė = N(δ )e with N(δ

) = Ā(δ ) -Γ(δ ) C(δ ).
By noting z = T (δ )x and e z = T (δ )e, we arrive at

ėz = T (δ ) ė = T (δ )N(δ )e = T (δ )N(δ )T -1 L (δ )T (δ )e = [A 0 -Γ 0 C 0 ] T (δ )e = [A 0 -Γ 0 C 0 ] e z
where we used the equality (18). Since [A 0 -Γ 0 C 0 ] is Hurwitz and independent of δ , thus we proved that e z = T (δ )e is exponentially stable for any value of delay. Due to the fact that T (δ ) is left unimodular over R[δ ], we can conclude that the observation error e = T -1 L (δ )e z exponentially tends to zero.

Remark 4. When designing observers for time-delay systems, the convergence of observation error dynamics might either depend or not depend on the delays. For the case depending on the delays, the most existing methods need to impose some conditions (like boundedness of time delay and its derivative) on the delay in order to prove the stability of the observation error dynamics [START_REF] Bhat | An observer theory for time delay systems[END_REF]; [START_REF] Fattouh | An unknown input observer design for linear time-delay systems[END_REF]; [START_REF] Sename | New trends in design of observers for time-delay systems[END_REF]; [START_REF] Fridman | A descriptor system approach to h control of linear time-delay systems[END_REF]; [START_REF] Richard | Time-delay systems: an overview of some recent advances and open problems[END_REF]; [START_REF] Nguyen | Non-linear observer design for a class of singular time-delay systems with lipschitz non-linearities[END_REF]. However, the proposed method is based on the output injection (delayed) technique. Since the basic delay h is constant, thus all the delays involved in the studied system are constant. It can be seen that the observation error dynamics ( 22) is equivalent to ėz = [A 0 -Γ 0 C 0 ]e z . It is obvious that the convergence speed of e z is independent of the delay, which implies that this method can be applied to any commensurate and constant delay. In other words, no limitations of delays are required for such a method, the only limitations are Assumptions 1 and 2, and these assumptions impose structural conditions which are independent of the delay size.

Remark 5. It is worth noticing that the stability of the observation error depends on neither the choice of the matrix Π nor that of the matrix P. This is clear since the previous theorem is valid for any selection of such matrices. Moreover, the convergence rate depends only on the choice of the matrix Γ 0 , which modifies the eigenvalues of the closed-loop matrix A 0 -Γ 0 C 0 .

In this sense, we may say that matrices Π and P do not play an important role in the asymptotic behavior of the observation dynamics.

Remark 6. The proposed method can be easily extended to treat the linear singular time-delay system with known input of the following form:

E(δ ) ẋ (t) = A (δ ) x (t) + B (δ ) u (t) y (t) = C (δ ) x (t) + D (δ ) u(t) (23)
By applying the same procedure, it can be proven that, if Assumptions 1 and 2 are both satisfied for system (2), then the following dynamics:

ẇ = N(δ )w + L(δ )y + Q(δ )u x = w + K(δ )y + R(δ )u (24)
with N(δ ), K(δ ) and K(δ ) defined in (19), and

R(δ ) = -K(δ )D(δ ) Q(δ ) = P 1 (δ )B(δ ) -M(δ )P 3 (δ )B(δ ) -L(δ )D(δ )
is an exponential observer for system (2).

If all conditions of Theorem 1 are satisfied for (2), then the following summarizes the procedure to design the proposed observer for system (2):

Step 1: For the triple Ẽ(δ ), Ã(δ ), C(δ ) , calculate a unimodular matrix Π(δ

) over R [δ ] such that Π(δ ) Ẽ(δ ) = E(δ ) 0 and Π(δ ) Ã(δ ) = A(δ ) A(δ ) . Then define C(δ ) = A(δ ) C(δ )
;

Step 2: For the deduced triple (E(δ ), A(δ ),C(δ )), determine the unimodular matrix P(δ ) such that P(δ )

E(δ ) C(δ ) = I n 0 ;
Step 3: With the deduced P(δ ), compute Ā (δ ) = P 1 (δ )A(δ ) and C(δ ) = P 3 (δ )A(δ ) C(δ ) ;

Step 4: After having obtained Ā(δ ) and C(δ ), deduce T (δ ) and F(δ ) defined in ( 16) and ( 17) respectively; Step 5: With A 0 and C 0 defined in (15), calculate Γ 0 such that (A 0 -Γ 0 C 0 ) is Hurwitz, and get M(δ ) and H(δ ) according to (20); Step 6: Following (19), compute N(δ ), L(δ ) and K(δ ) based on which the observer of the form (24) can be achieved.

Illustrative example

Consider the following linear singular time-delay neutral system of form (1):

               ẋ1 (t) = -ẋ2 (t -h) + x 1 (t -h) + x 4 (t -h) ẋ2 (t) = -ẋ3 (t -h) + x 1 (t) + x 4 (t) ẋ3 (t) = -ẋ1 (t -h) + x 1 (t) + x 2 (t) 0 = x 1 (t) + x 2 (t) + x 3 (t) y 1 (t) = x 3 (t -h) y 2 (t) = x 4 (t)
which can be written into the form (2) with

Ẽ(δ ) =     1 δ 0 0 δ 0 1 0 0 1 δ 0 0 0 0 0     , Ã(δ ) =     δ 0 0 δ 1 1 0 0 1 0 0 1 1 1 1 0     and C(δ ) = 0 0 δ 0 0 0 0 1
Step 1:

It is clear that we can choose Π(δ ) = I 4 such that

E(δ ) =   1 δ 0 0 δ 0 1 0 0 1 δ 0   , A(δ ) =   δ 0 0 δ 1 1 0 0 1 0 0 1   and C(δ ) =   1 1 1 0 0 0 δ 0 0 0 0 1  
Step 2: It is easy to check that for the given triple

( Ẽ(δ ), Ã(δ ), C(δ )) Assumption 2 is satisfied, thus E(δ ) C(δ ) is left unimodular over R[δ ].
In fact we can find P(δ ) = P 1 (δ ) P 2 (δ )

P 3 (δ ) P 4 (δ ) =         1 0 -δ 0 δ 0 0 0 1 0 -1 0 -δ 1 δ 2 0 -δ 2 0 0 0 0 0 0 1 δ -1 -1 δ -1 -δ 2 1 1 -δ + δ 2 0 δ 2 -δ -δ 3 0 1 + δ 3 0        
Step 3:

We have Step 4: Then according to (17) we have

[F 2 (δ ), F 1 (δ )] =      
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -1 0 0 0 0 0 0 0 0 0 -δ 0 0 0 0 0 0 0 0 0 0 0 0 0 0

     
which yields the following transformation

T (δ ) =    
-2 -δ 1 0 0 -1 -δ 0 -δ 0 -1 -δ 1 0 0 0 0 0 0 0 0 0 1 δ 0 0 0 0 0 0 -1 0 0 0 1 -1 -δ 0 -δ 0 T we assign (-5, -5, -10, -10, -20, -20, -25, -25, -50, -50) as the eigenvalues to the Hurwitz matrix A 0 -Γ 0 C 0 . After having determined Γ 0 , we can then follow ( 19) to calculate all necessary matrices N(δ ), K(δ ) and L(δ ) for the observer:

Conclusion

For linear singular time-delay systems, the most existing results focus on the simple case, i.e. E ẋ(t) = A 0 x(t) + A 1 x(tτ). Few results have been stated for the general linear singular time-delay systems of the form ∑ l i=0 E i ẋ(tiτ) = ∑ k i=0 A i x(tiτ), which covers also neutral delay systems. For such a general case, this paper deduced sufficient conditions, with which a simple Luenberger-like observer can be designed in order to exponentially estimate the states.

  ) is satisfied. Denote e = xx, then e = [I n -K(δ )C(δ )] xw which gives: ė = [I n -K(δ )C(δ )] ẋẇBy choosing K(δ ) = P 2 (δ ) -M(δ )P 4 (δ ) where M(δ ) is given in (

  unimodular over R[δ ].

  C(δ )T -1 L (δ ) = I 5 0 5×5 It is clear that the pair (A 0 ,C 0 ) is observable. By choosing

  The corresponding simulation results are depicted in the following figure.

Figure 1 :

 1 Figure 1: Estimation errors for x 1 , x 2 , x 3 and x 4 with h = 0.5s.

Preprint submitted to ElsevierDecember 27, 2016

$ This paper was supported in part 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. F.J. Bejarnano acknowledges as well the support of Proyecto SIP 20161492, Mexico.


Gang Zheng

Driss Boutat

Synchronization of chaotic systems via reduced observers

This paper treats synchronization problem of chaotic systems from a novel point of view, by using a change of coordinates to transform chaotic systems into a common canonical form, for which the synchronization problem can be easily studied via reduced observer. Sufficient and necessary conditions are given and the proposed method is illustrated by the synchronization of Rössler chaotic system.

I. INTRODUCTION

Since chaotic system is quite sensitive to its parameters and initial conditions, it was thought to be excluded in practice for a long time. After [16] successfully synchronized two identical chaotic systems with different initial conditions, chaos synchronization has been intensively studied in various fields. Since the work of [14], unidirectional synchronization can be viewed as a special case of observer design problem. Many techniques arising from observation theory have been applied to the problem of synchronization, such as observers with linearizable dynamics [9], adaptive [7], generalized hamiltonian form based observers [18], algebraic method [19], [2], and so on [6], [5], [25].

However, most of proposed methods are based on the full order observer, which needs to estimate all states of the system, including the measurable states which are the outputs. In order to reduce the complexity of observer design, the so-called reduced observer was proposed, which needs only estimate unmeasurable states of the studied system. It was firstly introduced for linear systems to reduce the number of dynamical equations by estimating only the unmeasurable states.

Then it is generalized for nonlinear dynamical systems by imposing the Lipschitz conditions for

nonlinear terms [START_REF] Haddad | Optimal reduced-order observer-estimators[END_REF], [START_REF] Xu | Reduced-order observer design for one-sided lipschitz nonlinear systems[END_REF], [START_REF] Sundarpandian | Reduced order observer design for nonlinear systems[END_REF] and invariant manifold [START_REF] Karagiannis | Invariant manifold based reduced-order observer design for nonlinear system[END_REF]. It is known that the formalism to design nonlinear observer, including reduced observer, for general nonlinear systems is still missing, hence some researchers tried to solve this problem by introducing the concept of normal form, i.e. transform the nonlinear systems via a diffeomorphism into the normal form, which is easy to design an observer, and then estimate the original states by the inverse of the diffeomorphism. For this technique, reader can refer to [START_REF] Xia | Nonlinear observer with linearizable error dynamics[END_REF], [START_REF] Respondek | Time scaling for observer design with linearization error dynamics[END_REF], [START_REF] Noh | Nonlinear observer design by dynamic observer error linearization[END_REF], [START_REF] Krener | Nonlinear observer with linearizable error dynamics[END_REF], [START_REF] Krener | Linearization by output injection and nonlinear observer[END_REF], [START_REF] Bestle | Canonical form observer design for nonlinear time variable systems[END_REF], [START_REF] Boutat | New algorithm for observer error linearization with a diffeomorphism on the outputs[END_REF], [START_REF] Zheng | Signal output dependent observability normal form[END_REF], [START_REF] Zheng | Secure data transmission based-on multi-input multi-output delayed chaotic system[END_REF] and the references therein. Moreover, in [START_REF] Nijmeijer | An observer looks at synchronization[END_REF], authors pointed out that sufficient and necessary conditions to guarantee the existence of a reduced observer are not reported, thus need to be studied.

Inspired by the concept of normal form, for the synchronization problem of chaotic systems, this paper proposes a nonlinear canonical form which permits to designe a reduced observer.

Sufficient and necessary geometrical conditions are also deduced to determine whether nonlinear systems can be transformed into such a form thought a diffeomorphism. After that a reduced observer is designed for the proposed canonical form.

The paper is organized as follows. In section II we give notations, the definition of reduced observer and also the problem statement. Then a nonlinear canonical form is proposed in Section III and Section IV gives necessary and sufficient geometric conditions which allow us to transform nonlinear systems into the proposed nonlinear canonical form. The corresponding reduced observer is studied in Section V and an illustrative example is presented in Section VI.

II. NOTATIONS, DEFINITIONS AND PROBLEM STATEMENT

Without loss of generality, we assume that the studied chaotic system can be written as follows:

         ẋ1 = F 1 (x) ẋ2 = F 2 (x) y = x 2 (1)
where x 1 ∈ R r and x 2 ∈ R p are the states and y ∈ R p is the measured output, F 1 : R r+p → R r and F 2 : R r+p → R p are smooth vector functions.

For [START_REF] Abraham | Foundation of mechanics[END_REF], the following gives the definition of the reduced observer.

Definition 1: The dynamical system defined as follows:

.

x 1 = F 1 (x 1 , x 2 )
July 27, 2010 DRAFT where x 2 is the output of (1), is an asymptotically reduced observer for (1) if

lim t→∞ x1 (t) -x 1 (t) = 0.
Moreover, it is said to be an exponentially reduced observer if

x1 (t) -x 1 (t) ≤ ae -bt x1 (0) -x 1 (0)
for t > 0, where a, b are both positive constants.

Let recall the famous result of reduced observer for linear systems. For the following system:

         ẋ1 = A 11 x 1 + A 12 x 2 ẋ2 = A 21 x 1 + A 22 x 2 y = x 2 (2)
where A ij are decomposition matrices for 1 ≤ i, j ≤ 2, then it is easy to see [START_REF] Leunberger | An introduction to the observers[END_REF] the following dynamic is a reduced observer for (2): which can be stabilized by the choice of K, provided that the pair (A 11 , A 21 ) is observable.

ẋ1 = A 11 x1 + A 12 x 2 + K(A 21 x 1 -A 21 x1 ), (3)
In fact, the reduced observer can be easily extended for a more generic linear system than [START_REF] Barbot | An algebraic framework for the design of nonlinear observers with unknown inputs[END_REF] in the following form: • What is the canonical form which enables us to design a reduced observer?

             ẋ = A 11 x 1 + A 12 ζ + A 13 ϑ ζ = A 21 x 1 + A 22 ζ + A 23 ϑ θ = µ (x, ζ, ϑ) y = y T 1 , y T 2 T = (ζ T , ϑ
• What are the sufficient and necessary conditions to transform nonlinear systems (chaotic systems) to such a canonical form?

• How to design a reduced observer for the proposed canonical form?

III. CANONICAL FORM FOR REDUCED OBSERVER

In what follows, we will present first the nonlinear canonical form which will be studied in this paper, then the sufficient and necessary geometric conditions to transform a generic chaotic system to such an observable nonlinear canonical form will be analyzed. An exponentially reduced observer can be easily designed for the proposed nonlinear canonical form, which will be detailed in the next section.

A. Nonlinear canonical form

Assume that the studied chaotic system can be decomposed into the following form:

ẋ = F 1 (x, ζ, ϑ) = f (x, ζ, ϑ) (4)
ζ = F 21 (x, ζ, ϑ) = γ 1 (ζ)H(x) + γ 2 (ζ, ϑ) (5)
θ = F 22 (x, ζ, ϑ) = ε (x, ζ, ϑ) (6)
y = ζ T , ϑ T T (7)
where

x ∈ R r , ζ ∈ R, ϑ ∈ R p , y ∈ R p+1 , f : R r+p+1 → R r , H ∈ R, γ 1 ∈ R and γ 2 ∈ R.
Moreover, it is assumed that (4-7) is observable and ( 4) is observable with respective to H (x), which implies that the following 1-forms:

θ i = dL i-1 f H (8)
for 1 ≤ i ≤ r are independent.

July 27, 2010 DRAFT Then, consider the following observable nonlinear canonical form:

             ż = Az + β(y 1 )z r + ρ(y) ξ = α 1 (y 1 )z r + α 2 (y) η = µ (z, y) y = y T 1 , y T 2 T = ξ T , η T T (9)
where

z = (z 1 , • • • , z r ) T ∈ R r β = (β 1 , • • • , β r ) T ∈ R r ρ = (ρ 1 , • • • , ρ r ) T ∈ R r η = (η 1 , • • • , η p ) T ∈ R p ξ ∈ R, α 1 ∈ R, α 2 ∈ R and A =           0 • • • 0 0 0 1 • • • 0 0 0 . . . . . . . . . • • • . . . 0 • • • 1 0 0 0 • • • 0 1 0           with z r = Cz and C = (0, • • • , 0, 1).
Remark 1: Since the nonlinear canonical form ( 9) is supposed to be observable, thus z r in (9) can be observed from the output y 1 , which implies α 1 (y) = 0.

In section V we will see that a reduced observer can be designed for the proposed canonical form [START_REF] Huijberts | Nonlinear discrete-time synchronization via extended observers[END_REF]. Thus the following sub section concerns the deduction of sufficient and necessary conditions in order to transform (4-7) into (9).

IV. SUFFICIENT AND NECESSARY CONDITIONS FOR THE EXISTENCE OF A DIFFEOMORPHISM

This section is devoted to seeking necessary and sufficient geometric conditions which allow us to transform (4-7) into the proposed nonlinear canonical form [START_REF] Huijberts | Nonlinear discrete-time synchronization via extended observers[END_REF].

For (4-7), denote

F = f T , F T 2 T , F 2 = F T 21 , F T 22 T
and the decomposition of F 2 into F 21 and F 22 makes γ 1 (ζ) = 0.

July 27, 2010 DRAFT After having defined θ i for 1 ≤ i ≤ r in (8), then define the vector field τ 1 according to the following equations:

θ j (τ 1 ) = 0, for 1 ≤ j ≤ r -1 θ r (τ 1 ) = 1 (10)
By induction, we can define the family of vector fields τ j as follows:

τ j = [τ j-1 , f ] for 2 ≤ j ≤ r.
As we will see, the existence of a diffeomorphism necessarily implies that all τ i commutes with each other, i.e. [τ i , τ j ] = 0 for 1 ≤ i ≤ r and 1 ≤ j ≤ r. We suppose that this condition is satisfied, and assume that σ, ν 1 , • • • , ν p are the vector fields determined by the following equations:

1)

[τ i , τ j ] = [τ i , σ] = [τ i , ν l ] = [σ, ν l ] = [ν l , ν s ] = 0 for 1 ≤ i, j ≤ r and 1 ≤ l, s ≤ p;
2) dζ(σ) = 1 and dϑ(σ) = 0;

3) dζ(ν j ) = 0 and dϑ i (ν j ) = δ j i for 1 ≤ i, j ≤ p where δ j i represents Kronecker delta, i.e.

δ k i = 1 if i = k, otherwise δ k i = 0. Set θ = (θ 1 , • • • , θ r , dζ, dϑ 1 , • • • , dϑ p ) T τ = (τ 1 , • • • , τ r , σ, ν 1 , • • • , ν p ) T
and denote Λ = θ(τ i , σ, ν j ) for 1 ≤ i ≤ r and 1 ≤ j ≤ p the evaluation of θ over τ . Due to the observability property, Λ is invertible, hence we can define the following multi 1-forms

ω = Λ -1 θ =   ω 1 ω 2   (11)
where

ω 2 = (dζ, dϑ 1 , • • • , dϑ p ) T
and ω 1 is the rest of ω. Then we are ready to state our main result.

Theorem 1: There exists a diffeomorphism (z T , ξ T , η T ) = φ(x, ζ, ϑ) which transforms the dynamical system (4-7) into the nonlinear canonical form (9) if and only if the following conditions are satisfied:

1) [τ i , τ j ] = 0, for 1 ≤ i ≤ r and 1 ≤ j ≤ r;

2) [τ r , F ] = V (y 1 ) is a vector field which only depends on y 1 modulo the sub space spanned by {τ i , σ} for 1 ≤ i ≤ r;

July 27, 2010 DRAFT

3) [τ j , F 2 ] ∈ ker ω 1 for 1 ≤ j ≤ r -1.

Proof: Necessity: Indeed, if (4-7) can be transformed into (9) via the diffeomorphism

(z T , ξ T , η T ) = φ(x, ζ, ϑ), then τ i = ∂ ∂z i for 1 ≤ i ≤ r, σ = ∂ ∂ξ and υ l = ∂ ∂η l for 1 ≤ l ≤ p.
And it is easy to check that all conditions of Theorem 1 are satisfied.

Sufficiency: Consider the multi 1-forms ω defined in [START_REF] Krener | Linearization by output injection and nonlinear observer[END_REF], we have ω(τ ) = I (r+p+1)×(r+p+1) , which implies ω(τ i ) for 1 ≤ i ≤ r, ω(σ) and ω(υ l ) for 1 ≤ l ≤ p are constant. Therefore,

dω(τ i , τ k ) = L τ i ω(τ k ) -L τ k ω(τ i ) -ω([τ i , τ k ]) = -ω([τ i , τ k ])
thus, we can calculate vector fields σ and υ 1 , • • • , υ p , such that {τ i , σ, υ l } forms a basis, satisfying the equations mentioned above. Following the same principle, we have

dω(τ i , σ k ) = -ω([τ i , σ k ]), dω(τ i , υ l ) = -ω([τ i , υ l ]) dω(υ l , υ t ) = -ω([υ l , υ t ])
Since ω is an isomorphism, this implies the equivalence between

[τ i , τ k ] = [τ i , σ] = [τ i , ν l ] = [σ, ν l ] = [ν l , ν t ] = 0 and dω = 0.
According to theorem of Poincaré [START_REF] Abraham | Foundation of mechanics[END_REF], dω = 0 implies that there exists a local diffeomorphism

(z T , ξ T , η T ) = φ(x, ζ, ϑ) such that ω = dφ. We note ω i = dφ i for 1 ≤ i ≤ 2.
Since condition (1) in Theorem 1 is satisfied and τ is a basis, it implies φ * (τ i ) = ∂ ∂z i , φ * (σ) = ∂ ∂ξ and φ * (υ l ) = ∂ ∂η l for 1 ≤ i ≤ r and 1 ≤ l ≤ p. Now let us clarify the affect of this transformation on f (x, ζ, ϑ) defined in [START_REF] Boutat | New algorithm for observer error linearization with a diffeomorphism on the outputs[END_REF]. By the diffeomorphism φ(x, ζ, ϑ), we have

     ż ξ η      = φ * (F ) where F =   f F 2   . It is easy to see that ω   f F 2   =   ω 1 (f + F 2 ) ω 2 (f + F 2 )   =   ω 1 (f ) + ω 1 (F 2 ) ω 2 (f ) + ω 2 (F 2 )  
July 27, 2010 DRAFT Then, for 1 ≤ i ≤ r, we get

∂ (φ * (F )) ∂z i =   [ω 1 (τ i ) , ω 1 (f ) + ω 1 (F 2 )] [ω 2 (τ i ) , ω 2 (f ) + ω 2 (F 2 )]   =   ω 1 [τ i , f ] + ω 1 [τ i , F 2 ] [ω 2 τ i , ω 2 (f ) + ω 2 (F 2 )]   =   ∂ ∂z i+1 [ω 2 (τ i ) , ω 2 (f ) + ω 2 (F 2 )]   since condition (3) [τ i , F 2 ] ∈ ker ω 1 for 1 ≤ i ≤ r -1 implies ω 1 [τ i , F 2 ] = 0.
By integrating we obtain: ω 1 (F ) = Az + ̺(y, z r ).

Moreover, as we know

∂ ∂z k H • φ -1 = dH(τ k ) = θ 1 (τ k )
then according to the definition of τ 1 in (10), we get

∂ ∂z r H • φ -1 = 1
which implies (5) can be written as

ζ = γ 1 (ζ)z r + γ 2 (ζ) (12)
Hence, by setting ω 2 = dφ 2 where φ 2 = I (p+1)×(p+1) , then we get

φ * (F ) =      Az + ̺(y, z r ) γ 1 (y 1 )z r + γ 2 (y) µ (z, ξ, η)      (13)
Finally, by the condition (2), we have

∂φ * ( F ) ∂zr = φ * τ r , F = φ * r j=1 V j (y 1 )τ j + W (y 1 )σ = r j=1 V j (y 1 ) ∂ ∂z j + W (y 1 ) ∂ ∂ξ
which means W (y 1 ) = α 1 (y 1 ) and ̺(y, z r ) in ( 12) can be decomposed as:

̺(y, z r ) = β(y 1 )z r + ρ(y)
Thus we proved that (4-7) can be transformed to [START_REF] Huijberts | Nonlinear discrete-time synchronization via extended observers[END_REF].

July 27, 2010 DRAFT

V. REDUCED OBSERVER DESIGN FOR THE CANONICAL FORM

If Theorem 1 is satisfied, then (4-7) can be transformed into (9) by a diffeomorphism (z T , ξ T , η T ) = φ(x, ζ, ϑ). This section is devoted to designing a reduced observer for the deduced nonlinear canonical form. Once the states of the canonical form have been estimated, by the inverse of the diffeomorphism, we can then recover the states of original chaotic system in the form of (4-7).

First of all, for (9) if we can accurately measure y 1 , y 2 and calculate ẏ1 , this allows us to define a "new" output Y being a function of known output y and the derivative of y 1 in ( 9):

Y = α -1 1 (y 1 ) ( ẏ1 -α 2 (y)) (14)
Then we have the following preliminary result.

Proposition 1: The following dynamical system:

ż = Aẑ + β(y 1 )C ẑ + ρ(y) -K(y 1 )(Y -C ẑ) (15)
where K(y 1 ) = -β(y 1 ) + κ and Y is defined in [START_REF] Nijmeijer | An observer looks at synchronization[END_REF], is an exponentially reduced observer for [START_REF] Huijberts | Nonlinear discrete-time synchronization via extended observers[END_REF], if the chosen κ makes (A + κC) Hurwitz.

Proof: Let e = ẑ -z be the estimation error. Since z r = Cz, then we can easily derive the dynamic of observation error from ( 9) and ( 15) as follows

ė = [A + (β(y 1 ) + K(y 1 ))C]e (16)
Since the gain matrix K(y 1 ) can be freely chosen, hence without loss of generality we set K(y 1 ) = -β(y 1 ) + κ which makes ( 16) become

ė = (A + κC)e. ( 17
)
Consequently, if κ is chosen in such a way that matrix (A+κC) is Hurwitz, then the exponential convergence of ẑ to z can be guaranteed.

Let us remark that the proposed reduced observer ( 15) is based on the "new" output Y defined in [START_REF] Nijmeijer | An observer looks at synchronization[END_REF], which clearly shows that the derivative of the real output y 1 should be calculated according to [START_REF] Nijmeijer | An observer looks at synchronization[END_REF]. However, it is well-known that the derivative of noisy signal should be avoided if possible in practice, since derivative operation will amplify the influence of noise. Hence, in order to overcome this problem, we introduced an algebraic estimator as follows:

ς = ẑ + Γ(y 1 ) (18)
July 27, 2010 DRAFT where Γ(y 1 ) = K(y 1 )α -1 1 (y 1 )dy 1 .

Remark 2:

The term Γ(y) always exists since y 1 ∈ R and α(y 1 ) is invertible, and it can be obtained by the following procedure:

1) being given A and C, determine κ such that (A + κC) is Hurwitz;

2) being given β(y 1 ) and determined κ, calculate K(y 1 ) = -β(y 1 ) + κ;

3) being given the invertible α(y 1 ) and calculated K(y 1 ), compute the integral Γ(y 1 ) = K(y 1 )α -1 1 (y 1 )dy 1 .

Remark 3: It should be noted that Γ(y 1 ) defined above can be considered as a signal of filtered y 1 , and thus limit the influence of noise on y 1 .

Inserting Y defined in ( 14) into [START_REF] Noh | Nonlinear observer design by dynamic observer error linearization[END_REF], it leads:

ż + K(y 1 )α -1 1 (y 1 ) ẏ1 = (A + β(y 1 )C + K(y 1 )C)ẑ +ρ(y) + K(y 1 )α -1 1 (y 1 )α 2 (y) (19)
In order to avoid the derivative of y 1 , we take the new variable ς into account, then a more practical reduced observer can be derived from [START_REF] Sira-Ramirez | An algebraic state estimation approach for the recovery of chaotically encrypted messages[END_REF] as follows

ς = (A + β(y 1 )C + K(y 1 )C) (ς -Γ(y 1 )) +ρ(y) + K(y 1 )α -1 1 (y 1 )α 2 (y 1 ) = (A + κC)ς + ρ(y) -(A + κC)Γ(y 1 ) +K(y 1 )α -1 1 (y 1 )α 2 (y) (20)
with Γ(y 1 ) defined in [START_REF] Sira-Ramirez | Synchronization of chaotic systems: a generalized hamiltonian approach[END_REF].

The proof of the convergence of ς -Γ(y 1 ) in [START_REF] Sundarpandian | Reduced order observer design for nonlinear systems[END_REF] to z of the canonical form ( 9) is evident, and then one can estimate x of (4-7) by φ -1 . The next section gives an example to illustrate the feasibility of the proposed method.

VI. ILLUSTRATIVE EXAMPLE

In order to highlight the proposed method in this paper, let consider the well-known Rössler chaotic system as follows:

             ẋ1 = x 2 + ax 1 ẋ2 = -x 1 -x 3 ẋ3 = c + x 3 (x 2 -b) y = x 3 (21)
July 27, 2010 DRAFT with a = c = 0.2, b = 5.7.

Fig. 1 depicts the chaotic attractor of Rössler system when a = c = 0.2, b = 5.7, from which we can see that x 3 is always positive, thus there is on observation singularities of x 1 and x 2 . By setting x = (x 1 , x 2 ) T and ζ = x 3 , we can rewrite [START_REF] Xia | Nonlinear observer with linearizable error dynamics[END_REF] into the following form:

               ẋ =   ẋ1 ẋ2   =   x 2 + ax 1 -x 1 -ζ   ζ = c + ζ (x 2 -b) y = ζ (22)
which is of the form (4-6) with

H (x) = x 2 , f =   x 2 + ax 1 -x 1 -ζ   F 2 = c + ζ (x 2 -b) , F = f T , F T 2 T
July 27, 2010 DRAFT

Then we can define the following 1-forms:

θ 1 = dx 2 , θ 2 = -dx 1 -dx 3 and dζ = dx 3
which yields

τ 1 = - ∂ ∂x 1 , τ 2 = -a ∂ ∂x 1 + ∂ ∂x 2 and σ = ∂ ∂x 3
It is easy to check that [τ 1 , τ 2 ] = 0 and [τ i , σ] = 0 for 1 ≤ i ≤ 2. Moreover we have:

[τ 2 , F ] = 1 -a 2 ∂ ∂x 1 + a ∂ ∂x 2 + x 3 ∂ ∂x 3 = -τ 1 + aτ 2 + x 3 σ 1
In order to calculate the diffeomorphism, let compute:

Λ = θτ =      0 1 0 1 a -1 0 0 1      and it yields ω = Λ -1 θ =      -1 -a 0 0 1 0 0 0 1      which gives ω 1 = (-x 1 -ax 2 , x 2 ) T .
Then one can check that ω 1 [τ 1 , F 2 ] = 0, implying [τ 1 , F 2 ] ∈ ker ω 1 . Thus all conditions of Theorem 1 are satisfied, and one deduces the following diffeomorphism:

φ = (z 1 , z 2 , ξ) T = (-x 1 -ax 2 , x 2 , ζ) T
which is a global change of coordinates and can transform [START_REF] Xia | Nonlinear observer with linearizable error dynamics[END_REF] into

         ż1 = -z 2 + aξ, ż2 = z 1 + az 2 -ξ ξ = z 2 ξ + c -bξ y = ξ (23)
which is in the proposed normal form [START_REF] Huijberts | Nonlinear discrete-time synchronization via extended observers[END_REF] with One can then design a reduced observer for [START_REF] Zheng | Secure data transmission based-on multi-input multi-output delayed chaotic system[END_REF]. Being given A and C defined above, it is easy to find κ to make (A + κC) Hurwitz. In our simulation, we chose κ = (-4, -4) T such that (A + κC) has two equal eigenvalues -2. With the above given β and the chosen κ, one obtains

A =   0 0 1 0   , β =   -1 a   , ρ =   aξ -ξ   C = (0, 1) , α 1 = y, α 2 = c -by July
K (y) = -β(y) + κ =   -3 -a -4  
Then, according to the definition of Γ(y), one gets

Γ (y) = K(y)α -1 (y)dy =   -3 -a -4   1 y dy =   -3 -a -4   ln |y|
As shown in Fig. 1, y is always positive (if y = 0, then system ( 21) is not observable), thus the deduced Γ(y) can be simply written as

Γ(y) =   -3 -a -4   ln y
Thus, according to Proposition 1, the reduced observer for (23) can be designed as follows:

.

ẑ1 = -ẑ 2 + az 3 + 3 (z 2 -ẑ2 ) . ẑ2 = ẑ1 + aẑ 2 -z 3 + (a + 4) (z 2 -ẑ2 )
where z 2 = ( ẏ + by -c) /y, which explicitly depends on the derivative of y.

Since the derivative of the output will amplify the influence of noise, so in order to overcome this problem, let introduce the following algebraic estimator:

ς = ẑ + Γ(y) = ẑ +   -3 -a -4   ln y
Then, after a straightforward calculation, one has the following reduced observer for [START_REF] Zheng | Secure data transmission based-on multi-input multi-output delayed chaotic system[END_REF] [START_REF] Zheng | Secure data transmission based-on multi-input multi-output delayed chaotic system[END_REF], and it is shown that the deduced observer can perfectly estimate the unmeasurable states without estimating the output.

VII. CONCLUSION

The synchronization problem of chaotic systems via reduced observer is studied in this paper.

We proposed a new canonical form which enables us to design a reduced observer. Sufficient and necessary conditions are given to determine whether a chaotic system can be transformed into such a form. Then a reduced observer is studied for the proposed canonical form, and the feasibility is highlighted by an illustrative example.

with A 21 x 1

 1 = ẏ -A 22 y. The observation error e = x1 -x 1 is determined by the following linear system: ė = (A 11 -KA 21 )e

3 Fig. 1 .

 31 Fig. 1. Phase portrait of Rössler chaotic system with initial conditions x1(0) = -0.8, x2(0) = -1 and x3(0) = 1.

1 Fig. 2 .

 12 Fig. 2. z1 and the estimate of ς1 with initial conditions z1(0) = 1 and ς1(0) = 0.

2 Fig. 3 .

 23 Fig.3. z2 and the estimate of ς2 with initial conditions z2(0) = -1 and ς2(0) = 0.

Fig. 4 .

 4 Fig.[START_REF] Boutat | New algorithm for observer error linearization with a diffeomorphism on the outputs[END_REF]. z1 and its estimate with initial conditions z1(0) = 1 and ẑ1(0) = ς1(0) = 0.

Fig. 5 .Fig. 6 .Fig. 7 .

 567 Fig.5. z2 and its estimate with initial conditions z2(0) = -1 and ẑ2(0) = ς2(0) = 0.


Frederic Jamme
email: jamme@synchrotron-soleil.fr

Slávka Kaščáková

Sandrine Villette

Fatma Allouche

Stéphane Pallu

Valerie Rouam

Matthieu M Refregiers

Frédéric ; Jamme

Frédéric Jamme

Slavka Kascakova

Valérie Rouam

Matthieu Réfrégiers

Frédéric Jamme

Keywords: Cellular imaging, Plants, Oncology/cancer, Metabolism Autofluorescence, UV, DUV, synchrotron, microspectroscopy VUV, vacuum ultraviolet, DUV, deep ultraviolet, HeLa, Human cervix epithelial cell line, FAD, Flavin adenine dinucleotide, FPs, Fluorescent proteins, BPA, Bisphenol A, PVC, Polyvinyl chloride

teaching and research institutions in France or abroad, or from public or private research centers.

Introduction

The ultraviolet autofluorescence of cells and tissues has long been evaluated for its diagnosis potential. However, since sources and microscopes that allow obtaining reliable information in the DUV (bellow 350 nm) are difficult to obtain and couple for medical diagnosis imaging of tissues, most of the medical diagnosis concentrated on the autofluorescence spectroscopy and microscopy of endogenous probes such as porphyrins which have emission in the visible range.

Synchrotron radiation is a broadband light that can be monochromatized in almost any energy range. The DISCO beamline at the synchrotron SOLEIL optimizes the vacuum ultraviolet (VUV) to the visible range of the spectrum (Giuliani et al., 2009).

The deep ultraviolet (DUV) energies accessible on two microscopes at atmospheric pressure are defined by quartz cut-off, starting from 200 nm to longer wavelengths (600 nm) (Jamme et al., 2010;[START_REF] Tawil | In situ tracking of enzymatic breakdown of starch granules by synchrotron UV fluorescence microscopy[END_REF]. For biological relevance, we specifically focus on the range between 200 and 350 nm that is very rarely covered especially under 250 nm.

The deep ultraviolet range is of particular interest for studying biomacromolecules due to their absorption in this energy range. While spectrometers can often go down to this range to study isolated biomacromolecules, microscopes are usually very limited in this range with either 350 nm cut-off or access to only a limited number of excitation wavelengths.

Köhler developed deep ultraviolet microscopy in 1904 as a bright field transmission technique [START_REF] Kohler | A microphotographic device for ultraviolet light (lambda=275 mu) and investigations into organic tissues using this device[END_REF]. This was implemented for one main reason: to improve the spatial resolution of cellular imaging. It passed through the century allowing seeing the unseen, namely nucleic acids in cells [START_REF] Mellors | Ultraviolet Microscopy and Microspectroscopy of Resting and Dividing Cells: Studies with a Reflecting Microscope[END_REF], and to observe colour [START_REF] Ludford | Ultra-Violet microscopy of living malignant cells[END_REF]. Thanks to developments in detectors, it became a quantitative method to observe protein and DNA contrast at 280 and 260 nm [START_REF] Zeskind | Nucleic acid and protein mass mapping by live-cell deep-ultraviolet microscopy[END_REF].

However, due to differences of optical density in deep UV transmission images, scattering errors are difficult to avoid and interpreting DUV images are more challenging than spectral imaging using our confocal DUV fluorescence microscope.

Nonlinear microscopy is an alternative approach for live tissue visualization of autofluorescent compounds [START_REF] Zipfel | Live tissue intrinsic emission microscopy using multiphoton-excited native fluorescence and second harmonic generation[END_REF]Diaspro et al., 2005). The infrared photons exciting the sample are an asset, allowing deep tissue penetration in the optical window of living tissue. However this methodology present two inconvenient as compared to DUV monophotonic excitation; being a non-linear process relying onto infrared photons, its resolution remains in the µm range, second, the two photon action spectrum are very broad and do not permit as fine selectivity as DUV microscopy.

While most biomolecules do present a contrast in DUV transmission microscopy, few of them will reemit fluorescence. This autofluorescence is very important in a sense that it permits better discrimination of molecules. Moreover, following autofluorescence opens label free studies of molecules of interest without any external probes or radiolabelling that could impair activity of the molecule of interest [START_REF] Tawil | In situ tracking of enzymatic breakdown of starch granules by synchrotron UV fluorescence microscopy[END_REF][START_REF] Batard | Diffusion of ofloxacin in the endocarditis vegetation assessed with synchrotron radiation UV fluorescence microspectrocopy[END_REF].

While many studies were conducted in DUV fluorescence spectroscopy as a diagnosis tool, mainly for cancer [ [START_REF] Wagnieres | In vivo fluorescence spectroscopy and imaging for oncological applications[END_REF][START_REF] Pavlova | Understanding the biological basis of autofluorescence imaging for oral Cancer detection: high-resolution fluorescence microscopy in viable tissue[END_REF], very few were conducted on DUV fluorescence microscopy or microspectrofluorimetry on biopsies and tissues. Most of them were using excitation wavelength longer than 350 6 nm, mostly due to the lack of laser lines below this value, despite very promising sensibility and sensitivity using DUV excitation [START_REF] Ramanujam | In vivo diagnosis of cervical intraepithelial neoplasia using 337-nm-excited laser-induced fluorescence[END_REF]. Indeed, many endogenous fluorophores have an absorption / excitability maximum in DUV below 350 nm, especially tryptophan [START_REF] Wagnieres | In vivo fluorescence spectroscopy and imaging for oncological applications[END_REF]. Signal attributed to tryptophan typically exhibits fluorescence intensities orders of magnitude greater than those from other endogenous fluorophores. And although its fluorescence can serve as an additional marker for monitoring cellular status [START_REF] Ramanujam | Fluorescence spectroscopy of neoplastic and non-neoplastic tissues[END_REF] due to lack of DUV fluorescence microscopy, this fluorophore is often ignored.

In addition, considering the recent work by Cox et al, no specific labels are needed for super resolution microscopy techniques (Cox et al., 2011). Therefore, aufluorescent (free labeling) molecules blinking and bleaching could be use to improve resolution.

In this study, we first present the main autofluorescent biomolecules spectra detected in cells or tissues. Then, we demonstrate the usefulness of DUV microscopy for in situ studies of cell biology and tissue histology. We choose the most occurring natural fluorochromes and representative samples of cells either isolated or in various tissular surrounding, namely cultured cancerous cells, osteocytes in a bone

Results

Isolated auto fluorescent compounds

When measuring endogenous signals, no additional dyes or contrast agents are necessary to determine native characteristics of cells or tissues. However, when considering an autofluorescence signal from biologic material, all the contributing fluorophores and chromophores must be accounted for (due to excitation, emission and absorption spectra overlapping). To identify the major biological autofluorescence compounds in DUV, we selected and recorded their fluorescence emission and excitation spectra (Figures 1234). Most of them have a maximum excitation peak below 350 nm. One can notice that each of them have distinct spectrum (maximum peak). However, several chromophores, such as pyridoxine, collagen and elastin, do present very close emission spectra that become difficult to discriminate without spectral detection. In those cases, fine-tuning of the excitation (Figure 1) may permit better separation of the species by altering their relative intensities.

The fluorophores were measured in PBS at pH = 7 (Figure 1), which does not represent a realistic in vitro and/or in vivo situation. Indeed, the spectroscopic characteristics of fluorophores can change due to environment-related effects and spectral shifts may occur. A bathochromic shift of absorption maxima of bilirubin is observed when bound to HSA, compared to free (and unbound) bilirubin in PBS [START_REF] Kanick | Integration of single-fiber reflectance spectroscopy into ultrasound-guided endoscopic lung cancer staging of mediastinal lymph nodes[END_REF]. Also coenzymes such as NAD(P)H reveals distinct spectroscopic characteristics, as shown on Figure 2. Free NADH in PBS has an emission maximum located at 460 nm, while, a hypsochromic shift of 10 nm in the position of its fluorescence maximum is observed when it is bound to alcohol dehydrogenase. The knowledge of those spectral differences of NAD(P)H allowed previously to envisage the monitoring and quantification of total NAD(P)H concentration (free and bound NAD(P)H), which is an important parameter for the interpretation of cellular metabolism [START_REF] Paul | Oxygen concentration and the oxidation-24 reduction state of yeast: determination of free/bound NADH and flavins by time-resolved spectroscopy[END_REF][START_REF] Villette | Ultraviolet-induced autofluorescence characterization of normal and tumoral esophageal epithelium cells with quantitation of NAD(P)H[END_REF]). However, the example of bilirubin or NAD(P)H shows also the difficulty of the identification of spectral components within the global autofluorescence signal of cell or tissue. It becomes obvious that the detailed knowledge of the influence of factors such as pH, binding, hydrophilicity/lipophilicity on the spectral properties of fluorophores is necessary to identify the chemical composition of biological material under investigation.

Grass lignocelluloses are a major resource in the emerging cellulose to ethanol strategy for biofuels [START_REF] Anderson | Structural and chemical properties of grass lignocelluloses related to conversion for biofuels[END_REF]). However, the potential bioconversion of carbohydrates is limited by the associated aromatic constituents within the grass fibre. These aromatics include both lignin and low-molecular weight phenolic acids. The two main ester linked phenolics (ferulic and para-coumaric acids) can be differentiated by DUV absorption maximums near 326 nm and 314 nm, respectively. As shown in Figure 4, ferulic acid and para-coumaric acid have an emission peak centred at 415 and 400 nm, respectively while lignin shows a maximum of autofluorescence around 475 nm. Ferulic acid and para-coumaric acid can hardly be distinguished from each other using conventional fluorescence microscope, infrared or Raman spectroscopies [START_REF] Saadi | Spatial Confocal Fluorescence Spectral Imaging[END_REF][START_REF] Robert | Change in wall composition of transfer and aleurone cells during wheat grain development[END_REF]. The flux was measured (Ph/s) at the entrance of the microscope using a calibrated photodiode (AXUV 100, IRD, CA, USA) coupled to a current amplifier (DLPCA-200, laser Components, UK). As described previously (Jamme et al., 2010), the excitation light is provided by a bending magnet emission on the DISCO beamline at synchrotron SOLEIL (Giuliani et al., 2009). White beam from 180 to 600 nm is monochromatized by an iHR320 (Jobin-Yvon, FR) equipped with a 100 grooves per millimetre grating with a 240 nm peak efficiency (Spectrum Scientific, Inc, Irvine, CA, USA). Thus the flux variation observed, shown in Figure 5, is mainly attributed to the excitation monochromator grating response convoluted by the transmission view port (UV fused silica, MPF, SC, USA).

Synchrotron flux profile and microscope spectral response

The spectral response of the microscope was measured using a calibrated lamp (QTH 45W, Microcontrole, FR) through the 40X objective (Ultrafluar, Zeiss, D) and calculating the ratio between the detected signal and the known emittance of the lamp (Figure 5). The response variation observed in Figure 5 is mostly attributed to the emission spectrometer gratings (T64000, Horiba Jobin-Yvon, FR).

Applications to cell and tissues

Isolated living cells

In order to assess the distribution of endogenous fluorophore in single cell, the whole living cell was scanned under 275 nm excitation with 1s exposure time and the fluorescence emission spectra of the cell were recorded at each pixel. The Figure 6 represents the spectra of autofluorescence obtained from the highlighted pixels. As it can be seen from Figure 6, in a simplified model, the emission spectrum of a living cell is mainly tryptophan and tyrosine fluorescence arising from the proteins. In a [START_REF] Villette | Ultraviolet-induced autofluorescence characterization of normal and tumoral esophageal epithelium cells with quantitation of NAD(P)H[END_REF][START_REF] Lakowicz | Principles of Fluorescence Spectroscopy[END_REF].

Maize stem cell walls

Improving the ease and yield of cell wall saccharification represents the major technological hurdle that must be overcome before the full vision of the plant-fuelled biorefinery can be realized. Thus, mapping of the spatial distribution of the fluorescent species at a micrometric resolution as a function of growth stage allows the assessment of the best enzymatic treatments and high yield of recovery of useful chemical for biofuel or biomaterials productions can be foreseen.

As shown in Figure 7, the spectral images studied recorded using a 275 nm excitation, covered a region in the vascular bundle of the maize internodes crosssection that contained mainly three cell types: parenchyma, sclerenchyma and phloem cells. For clarity, average spectra were calculated for each cell type region as shown in Figure 7 (bottom left). In the case of the phloem, a peak around 415 nm was observed in the spectra while, in the case of sclerenchyma, the peak was centred on 420 nm and fluorescence was also detected from 440 nm to 540 nm. This confirmed that sclerenchyma cell walls contained lignin while phloem cell walls did not [START_REF] Allouche | Coupling hyperspectral image data having different spatial resolutions by extending multivariate inter-battery Tucker analysis[END_REF]. In addition, multivariate data analysis reveals that paracoumaric acid was found negatively correlated to lignin [START_REF] Allouche | Coupling hyperspectral image data having different spatial resolutions by extending multivariate inter-battery Tucker analysis[END_REF].

Biopsies

The main features observable on tissues under 280 nm excitation are related to tryptophan, tyrosine and in a lesser extent to constitutive proteins [START_REF] Petit | Multimodal spectroscopy combining time-of-flight-secondary ion mass spectrometry, synchrotron-FT-IR, and synchrotron-UV microspectroscopies on the same tissue section[END_REF].

Bones

Osteoporotic fractures represent a significant public health concern. Bone quality (bone microarchitecture, collagen content) was investigated at subcellular resolution.

Synchrotron DUV spectroscopy allows to characterize and distinguish biochemical content of the osteocytes and their surrounding matrix, since osteocytes are considered as the orchestrator of the bone remodelling [START_REF] Rochefort | Osteocyte: the unrecognized side of bone tissue[END_REF]. In a rat experimental model of alcoholic induced osteoporosis, sections of cortical bone were investigated and osteocytes autofluorescence images are presented in Figure 8. The fluorescence emission spectra present three peaks situated at 305 nm, 333 nm and 385 nm, attributed to tyrosine, tryptophan and collagen respectively. The ratio between tyrosine and tryptophan is dependent of the localisation in the bone tissue. Osteocytes do present a higher tyrosine to tryptophan ratio than the matrix. Moreover, from our knowledge and personal experience, no other tissue does present a similar high level of tyrosine fluorescence. Many probes that are usually excited in the visible may be also excited in the UV range. For example, fluorescent proteins (FPs) can be excited either directly through their chromophore or by exciting the tryptophan residues that transfer their energy to the chromophore [START_REF] Visser | Direct observation of resonance tryptophan-to-chromophore energy transfer in visible fluorescent proteins[END_REF]. This could lead to higher resolution studies of FPs by lowering their excitation wavelength. Moreover, this opens the possibility to Of note, the ideal drug to be followed in UV shall have an emission shifted from the tryptophan fluorescence (since tryptophan has strong contribution to autofluorescence signal), or to bypass the tryptophan fluorescence by an excitation outside the 240 to 300 nm range, or if not, then a drug with high quantum yield of fluorescence is required (order of higher than the autofluorescence signal).

Otherwise, multivariate data treatment is required to separate the molecule of interest from the proteins fluorescence (Batard et al., 2001).

From the examples discussed above, we clearly showed that DUV microspectroscopy could monitor the numerous endogenous fluorophores and their Although, the autofluorescence of cells and tissues is often (almost ever) considered as a parasitic signal that must be minimized; here, we demonstrate, that using the right experimental conditions, this so-called background signal, is rich in information and can become useful. Absorption: To avoid inner filter effect in our fluorescence emission experiment, for each sample absorption spectrum was also measured. Absorption spectra were recorded at 20° C using UV-VIS spectrophotometer (Specord 210, Analytic Jena AG, Jena, Germany) in the 190-900 nm range with slits width corresponding to a resolution of 2 nm. For measurements, we used a 1 cm path length quartz cell.

Fluorescence: Fluorescence spectra were measured on optically diluted samples (i.e. absorption < 0.1 at the excitation wavelength) recorded at 20° C using a FluoroMax-4 (HORIBA Jobin Yvon INC, Chilly Mazarin, France) spectrofluorimeter.

To record the excitation and emission spectra of individual compounds, the best excitation and emission wavelengths were chosen after optimisation (see Figure 2).

Para-coumaric acid, ferulic acid and lignin

Para-coumaric acid and ferulic acid were purchased from Sigma-Aldrich (Lyon, France). Powder was first dissolved in water at a concentration of 3.2 mg/mL and 2.6 Lwoff-CNRS, Villejuif Cedex, France. Cells were routinely cultured as monolayer and were grown in Dulbecco's modified Eagle's medium (DMEM) containing L-glutamine (862 mg L -1 ), sodium pyruvate (110 mg L -1 ) and glucose (4500 mg L -1 ), supplemented with 10% fetal calf serum (FCS), penicillin (50 µg mL -1 ) and streptomycin (50 µg mL -1 ). All chemicals were obtained from Gibco Invitrogen (Villebon sur Yvette, France). The cells were maintained at 37 °C in a humidified atmosphere of 5% CO 2 . For the microscopic detection of endogenous fluorophores from HeLa cells, the cells were plated in plastic Petri dishes (35x10 mm) containing 25 mm round quartz coverslip (Cening, Cn) and incubated for 2 days. Cell attachment and growth was monitored by visible inspection using a microscope.

Coverslip with desired cell growth confluence was rinsed in DMEM in absence of FCS or antibiotics for 1-2 minutes. Thereafter, the living cells were directly mounted on the microscope and spectra of cells in PBS were taken.

Maize stem cell walls

Maize from a reference genotype F2 were grown at INRA Lusignan and stems were harvested at the female flowering stage. The internodes under the ear were collected and stored in 70% (v/v) ethanol/water.

Samples were taken from the middle of maize internodes, embedded into paraffin and sectioned using a microtome (Jamme et al., 2008). Proteins were removed using a protease enzyme named subtilisin A type VIII, bacterial, from bacillus licheniformis.

Preliminary results showed that 10 µm thick sections were adapted for fluorescence corresponding to a vascular bundle near the epidermis was selected as showing a chemical variability according to cell types (Figure 7) [START_REF] Jung | Maize Stem Tissues[END_REF].

Bones

These studies were approved by the National Institute for Health and Medical

Research (INSERM). The procedure for the care and killing of the animals was in accordance with the European Community standards on the care and use of laboratory animals (Ministère de l'agriculture, France, Authorisation INSERM45-001).

Wistar male rats (8 weeks old at baseline) were chosen at random and assigned to one of the three following groups: controls and two different percentages ethanol beverage (25% v/v and 35% v/v) during 19 weeks. After the sacrifice, tibias were dissected free of connective and fat tissue. They were fixed in a 4% v/v formalin solution and kept at + 4°C. As required the tibias were cut transversally in slices (thickness 300 µm) in the superior third part of the diaphysis, with a high speed rotary tool (Dremel 300, Dremel, USA).

UV monochromatised light (typically between 270 and 330 nm) was used to excite cortical bone sections through a 40 ultrafluar objective (Zeiss, Germany). The fluorescence emission spectrum arising from each excited pixel is recorded.

Rastering of the sample allows one to record x, y, lambda, I maps of interest [2].

Mapping of 20  20 µm 2 was performed with a 2•2 µm step size, chosen by considering the relative size of the cells and the area to be scanned, with a 5 s acquisition time per spectrum. The Region of Interest (ROI) was centred on the osteocyte. For each rat sample, 3 acquisitions were achieved both in osteocytes ROI and surrounding matrix ROI.

  under UV excitation for cancer diagnosis

  due to environmental influence, tryptophan fluoresces at 335 nm and tyrosine at 306 nm

  shifting the excitation wavelength to 340 nm, the collagen fluorescence increases significantly.

  can be observed inside isolated cells or in a tissue. During physiological and/or pathological processes the changes occurring in the cell and tissue result in modifications of the amount and distribution of those endogenous fluorophores and chemical-physical properties of their microenvironment. Therefore, monitoring the endogenous fluorophores distribution and the intensity of their fluorescence emission can give diagnosis insights and can help to understand biochemical transformations. While the spectral characteristics of isolated molecules can differ from physiological conditions, it is possible to identify them inside our chosen examples.As for plant tissues, during plant-pathogen infection, the physiological state of the invaded tissues is altered. This can be reflected in changes in photosynthesis, transpiration or both. Fluorescence imaging is thus useful for visualization of emerging biotic stresses while detecting the chlorophyll fluorescence (endogenous fluorochrome, which mediates photosynthesis) or monitoring metabolic cell activity through detection of fluorescent coenzymes, such as NAD(P)H (Chaerle andVan der Straeten, 2001;[START_REF] Kasimova | The free NADH concentration is kept constant in plant mitochondria under different metabolic conditions[END_REF]. In addition, the possibility to follow the spectroscopic state at micrometer scale of the major compounds without any markers is of importance for high yield recovery of vegetal wastes.Nicotinamide adenine dinucleotide (NAD(P)H) and flavin adenine dinucleotide (FAD) are metabolic cofactors that act as electron donors and acceptors in the electron transport chain of the mitochondria, thus having a major role in metabolism pathways of eukaryotic cells. While the reduced form, NAD(P)H, is fluorescent and has an excitation maximum in the region of 350 nm with emission at 450 nm (Figure1), the + , does not produce fluorescence[START_REF] Paul | Oxygen concentration and the oxidation-24 reduction state of yeast: determination of free/bound NADH and flavins by time-resolved spectroscopy[END_REF][START_REF] Kirkpatrick | Endogenous Fluorescence Spectroscopy of Cell Suspensions for Chemopreventive Drug Monitoring[END_REF][START_REF] Wang | The role of mitochondria in apoptosis[END_REF]. Conversely, in the case of flavins, the fluorescent form is the oxidized form (excitation maximum at 450 nm with emission at 535 nm) and the reduced form does not fluoresce. Figure6shows the possibility of spatial detection of tryptophan and tyrosine-related fluorescence. One can envisage following the changes of tyrosine and tryptophan signal to monitor the proteins synthesis, degradation or both. Thus, synchrotron DUV microspectroscopy opens boundaries, since understanding of the spatial tryptophan and tyrosine signal changes, in relation to metabolic changes, may have both diagnostic as well as prognostic potential when treatment is concerned.As for bone tissue, synchrotron UV microspectroscopy may give quantitative information on the cortical bone content of some autofluorescent molecules both in bone cells and their surrounding extracellular matrix. Results have shown that strong chronic alcohol consumption induced a decrease of the ratio tyrosine/tryptophan both in cell and surrounding matrix. This new methodology warrants a cell characterization in situ that allows evaluating osteocyte metabolism and gives a better understanding of the key role of this cell in the bone remodelling regulation. This local measurement of tyrosine and tryptophan levels may give a better understanding of cellular metabolism according to different physio-pathological conditions such as osteoporosis.

  different FPs with one single excitation of their respective tryptophan.Moreover, many drugs and toxins do present fluorescence when excited in the DUV; it is then possible to follow their pharmacokinetic at tissular, cellular or subcellular level by directly following their fluorescence distribution without adding external probes or radiolabels(Diaspro et al., 2005). As an example in the area of material science, phenol-or diphenol-molecules, like Bisphenol A (BPA), exhibit absorption and fluorescence in DUV region(Del Olmo et al., 1999). BPA is used as a polymerization inhibitor in PVC while epoxy resins containing BPA are used as coatings on the inside of almost all food and beverage cans. As such, BPA leaching can occur when they are being treated with high temperature or extreme pH[START_REF] Brede | Increased migration levels of bisphenol A from polycarbonate baby bottles after dishwashing, boiling and brushing[END_REF][START_REF] Munguía-López | Migration of bisphenol A (BPA) from can coatings into a fatty-food simulant and tuna fish[END_REF]. Because of health risk, one can envisage to use synchrotron DUV microspectroscopy to detect its presence at very low concentrations in food constituents and/or to use this technique to develop and validate sample preparations methods, which would lead to heath safe requirements.

  tissular as well as cellular level. Note, working in monochromatic excitation mode, the light power is very low (µW). This allowed us to record the spectral map of living cells without detecting any changes in cell morphology during or after the measurement. As discussed above, measuring of live cell opens up possibility to follow the dynamic changes within the specimen. However, in experiments were dynamic processes are not issue; one can work with fixed specimens. Indeed, fixation with formaldehyde did not reveal any changes in autofluorescence of single cell when excited with 275 nm; e.g. for fixed as for live cells tryptophan fluoresced at 335 nm and tyrosine at 306 nm (data not shown).

  Pyridoxin, N-acetyl-L-tryptophanamide, N-acetyl-L-tyrosinamide, •-Nicotinamide adenine dinucleotide, reduced dipotassium salt (NADH), Alcohol dehydrogenase from Saccharomyces cerevisiae (ADH), Collagen type I, Elastin, phosphate-buffered saline (PBS), sodium hydroxide (NaOH) and TRIS were purchased from Sigma-Aldrich (Lyon, France). Acetic acid was purchased from Merck (Fontenay sous-Bois, France) and hydrogen chloride (HCl) from Carl Roth GmbH (Germany). Preparation of solutions: Stock solution of Pyridoxine (c = 8.75x10 -3 M) was prepared by dissolving pyridoxine powder in 1 M HCl. For recording the pyridoxine absorption and fluorescence spectra, the stock solution was diluted with PBS pH = 7.4 to a final concentration of 5x10 -6 M. Collagen type I was first dissolved in 0.1 Macetic acid to a concentration of 1 mg/ml. The stock of collagen was then diluted with PBS pH = 7.4 to a final concentration of 0.074 mg/ml. Stock solution of Elastin (c = 1 mg/ml) was prepared by dissolving elastin powder in 1M NaOH, followed by dilution in PBS pH = 7.4 to a final concentration of 0.027 mg/ml. The N-acetyl-Ltryptophanamide and N-acetyl-L-tyrosinamide were both dissolved in 25x10 -3 M Tris-HCl pH = 7.8 to obtain stock solutions concentrations of 8.15x10 -3 M for N-acetyl-Ltryptophanamide and 6.39x10 -3 M for N-acetyl-L-tyrosinamide respectively. Both stock solutions were then diluted to a concentration of 10x10 -6 M for N-acetyl-Ltryptophanamide and 21.3x10 -6 M for N-acetyl-L-tyrosinamide. Absorption and fluorescence spectra were measured immediately after dilution.

  ADH complexes: Stock solutions of NADH (c = 1.85x10 -2 M) and stock solution of ADH (c = 2.83x10 -3 M) were prepared in 0.1M Tris-HCl pH = 8. Solutions of complexes were obtained by dilution of stock solutions of NADH and ADH with 0.1M Tris-HCl pH = 8. Three different complexes of NADH:ADH have been prepared at the following concentrations for both components: NADH (10x10 -6 M) :ADH (1x10 -4 M); NADH (8.2x10 -6 M) : ADH (5x10 -4 M) and NADH (6.5x10 -6 M) : ADH (1x10 -3 M). The absorption and fluorescence spectra of thus prepared solutions were measured immediately after mixing. In order to distinguish between the spectral characteristics of these components, the NADH and ADH were measured also individually: NADH in 0.1M Tris-HCl pH = 8 at 10x10 -6 M concentration and ADH at 1.62x10 -4 M concentration.

  respectively. Samples were further diluted in water in order to obtain an absorbance value of about 0.03 when measured in 1 cm path length cuvettes.Fluorescence excitation and emission spectra were recorded on a Fluorolog-3 spectrophotometer (HORIBA Jobin Yvon INC, Chilly Mazarin, France), using 3 mm path length cuvettes (Hellma France, Paris). Lignin was observed as a powder in a special holder (HORIBA Jobin Yvon INC, Chilly Mazarin, France) in a front-face configuration. A long pass filter with a cut-off at 351 nm was inserted between sample and detector for ferulic acid measurement.DUV microspectrometerMicrospectrofluorescence spectra were recorded on Polypheme, the DUV inverted microspectrofluorimeter installed at DISCO Beamline(Jamme et al., 2010). Excitation was provided by the continuous emittance from the DISCO beamline bending magnet at Synchrotron SOLEIL(Giuliani et al., 2009). While the system was already described elsewhere(Jamme et al., 2010), it is constructed around an Olympus IX71 inverted microscope stand with homemade replacement of the intermediate lenses that were not transparent in UV. Light detection is collected through a DUV lens and an adjustable pinhole. In order to suppress the Rayleigh band whatever the excitation wavelength, the detection system (T6400, Jobin-Yvon, Fr) uses a triple monochromator in a substractive mode. Thereafter, the fluorescence emission spectrum of the studied voxel is projected onto a -70°C peltier cooled iDus CCD (Andor) of 1024x256 pixels with a 26x26 µm pixel size and a 26.6x6.7 mm detector size and a dark current of less than 0.002e/pixel/sec. Spectra are recorded at a 1024 pixels depth and a 16 bits dynamic range. culture condition: HeLa cells, a human cervix epithelial cell line, were a generous gift from Dr. Denis Biard from CEA-DSV-iRCM/INSERM U935, Institut A.

  Sections were collected on ZnS windows. A region

Figure 1 :

 1 Figure 1: Fluorescence excitation spectra of the main autofluorescent contributors found in mammalian cells and tissues. Maxima of excitation are presented following the name of the compound. Within brackets is indicated the emission wavelength at which was recorded the excitation spectrum.

Figure 2 :

 2 Figure 2: Fluorescence emission spectra of the main autofluorescent contributors observable in mammalian cells and tissues. Maxima of emission are presented following the name of the compound. Within brackets is indicated the excitation wavelength at which was recorded the emission spectrum.

Figure 3 :

 3 Figure 3: Fluorescence excitation spectra of the main autofluorescent contributors found in plant tissues: lignin, para-coumaric and ferulic acids.

Figure 4 :

 4 Figure 4: Fluorescence emission spectra of the main autofluorescent contributors found in plant tissues: lignin, para-coumaric and ferulic acids.

Figure 5 :

 5 Figure 5: (a) Spectral distribution of the photon flux delivered to the sample and (b) relative response of the microscope over the whole spectral range.

Figure 6 :Figure 7 :Figure 8 :Figure 1 :Figure 2 :Figure 3 :Figure 4 :Figure 5 :Figure 6 :Figure 7 :Figure 8 :

 67812345678 Figure 6: (a) Transmission image of a living single Hela cell, (b) corresponding sum intensity spectral image and (c) spectrum of the grey pixel. The gaussian deconvolution presents two main components: tryrosine and tryptophan. Pixel size 1.5 x 1.5 µm 2

  context, vegetal cells as appropriate illustrative examples.

	F o
	r
	P
	e e r
	R e v i e w
	7

Part of this work was supported by the Région Centre (FRANCE). Synchrotron SOLEIL support through projects #20100064, 201000181, 20100949 and 20110131 is acknowledged.

Biology of the Cell Biology of the Cell Biology of the Cell Biology of the Cell Biology of the Cell Biology of the Cell Biology of the Cell Biology of the Cell Biology of the Cell Biology of the Cell Biology of the Cell Biology of the Cell Biology of the Cell Biology of the Cell Biology of the Cell Biology of the Cell Biology of the Cell Biology of the Cell Biology of the Cell Biology of the Cell Biology of the Cell Biology of the Cell Biology of the Cell Biology of the Cell Biology of the Cell Biology of the Cell Biology of the Cell Biology of the Cell Biology of the Cell Biology of the Cell Biology of the Cell Biology of the Cell Biology of the Cell Biology of the Cell Biology of the Cell Biology of the Cell Biology of the Cell Biology of the Cell Biology of the Cell Biology of the Cell Biology of the Cell Biology of the Cell

Author contributions

F.J., S.V. and M.R. conceived and designed the research; F.J., S.K., S.V., F.A. and V.R. carried out the experiments; F.A. and S.P. provided some of the samples; F.J., S.K, S.V. and M.R analysed the data; F.J., S.K., S.V. and M.R. wrote the article.