J P Barbot
email: barbot@ensea.fr.

M Djemai
email: djemai@ensea.fr.

N Manamanni

M Djemaï

State Observer and Observability Conditions for a Class of Hybrid Continuous-Discrete Dynamic System

This paper deals with observability conditions and state observer design for a class of hybrid systems combining continuous and discrete dynamics. The main contribution of the work lies in the performed observability conditions for this class of systems and a design of hybrid observer to reconstruct both continuous and discrete states starting only from the knowledge of a continuous output. An illustrative example is presented showing the efficiency of the proposed observer.

Introduction

Hybrid dynamical systems (HDS) was widely studied in literature these last decades in science and engineering, (see for example [1]). A common definition is that they are modelled or defined as a combination of both continuous and discrete event subsystems. This paper addresses observability conditions and state observer design for a particular class of hybrid systems with combined continuous and discrete state dynamics and called hybrid systems in state. Such models can represent a very broad range of systems for which the control and/or observer design problems remain still open. Indeed, some systems may be controlled by discrete system when computers are used in the control procedure. Also, in some applications in electrical drive, this type of representation can be met after discretization of a part of the dynamics which will be controlled by a discrete system (system with commutation). In the case of the electric motors, the use of the PWM (Plush Width Modulation) is generally used. The PWM being discrete by definition, since the system works by commutation and makes it possible to control the electrical part of the engine (see for example [START_REF] Djemai | Singularly Perturbed Method for the Control Design of a Synchronous Motor with its PWM Inverter[END_REF][START_REF] Vas | Sensorless Vector and Direct Torque Control[END_REF]). In this case, the control strategy consists in discretizing the electrical part of the dynamics (which will be controlled by the orders of commutations) and to keep the mechanical part in continuous time. Note that, in the literature, several papers have defined observability for different classes of hybrid systems ([2, [START_REF] Baglietto | Active mode observability of switched linear systems[END_REF][START_REF] Balluchi | A Hybrid Observer for the Driveline Dynamics[END_REF][START_REF] Balluchi | Observability for Hybrid Systems, 42nd[END_REF][START_REF] Bemporad | Observability and controllability of piecewise affine and hybrid systems[END_REF][START_REF] De Santis | Observability of internal variables in interconnected switching systems[END_REF][START_REF] Saadaoui | Exact differentiation and Sliding mode observer for switched Lagrangian systems[END_REF][START_REF] Vidal | Observability of linear hybrid systems[END_REF]). This paper concerns hybrid observer design for the following particular class of HDS:

ẋ = F (x, Z(k)) (1)
Z(k + 1) = A d Z(k) + β d (v(k), y(k)) (2)
y = y c = H(x) = [h 1 (x), . . . , h p (x)] (3)
where x ∈ N , with N an open set of ℜ n , is the continuous state vector, Z = (z 1 , ..., z m ) T ∈ M with M an open set of ℜ m , is the discrete state vector with

Z(t) = Z(t k ) = Z(k) for t ∈ [t k , t k+1 [ ∀ k > 0. A d ∈ ℜ m×m
, is constant matrix with appropriate dimension and F (x, Z(k)), H(x) and β d are a vectors functions of appropriate dimensions, y(k) = y(t = t k ) is the output and v(k) ∈ ℜ ν a known input. This work deals with the hybrid observer design in the case where the continuous output subsystem is available and all the switching instants t k i.e. the moments when the discrete system switches, are also available and supposed to be known throughout this paper. Nevertheless, it is possible to overcome the previous assumption by using some technics (see for example ([12])).

Remark 1

The considered class of HDS is supposed without jumps for the continuous subsystem i.e., x(t + k ) = x(t - k ) where t k is the time when the discrete state switches from Z(k -1) to Z(k) and t + k and t - k are the instants just before and just after the switching time respectively.

Hybrid observability conditions

Throughout this paper, one considers a class of HDS of form (1) combining continuous and discrete dynamics and supposed to be bounded for both continuous and discrete states and verifies the following assumptions: Assumption 1 The period T k = t k+1t k for the dynamics (2) is supposed to check the following condition:

T k > τ m > 0 ∀ k ≥ 0 (4)
This in order to avoid the particular problem of systems with Zeno phenomena.

Let us now introduce some observability definitions for the considered subsystems of (1).

Definition 1 A pair of points x 0 and x1 in N are uniformly distinguishable if for all input sequences Z(k) ∀ k ≥ 0, generated by (2), the system (1) generates solutions x 0 (t) and x 1 (t)

satisfying x i (0) = x i such that x i (t) ∈ N ∀ t ∈ [t k , t k+1 [, k ≥ 0 and h(x 0 (t)) = h(x 1 (t)) for some t ∈ [t k , t k+1 [, k ≥ 0.
Remark 2 The subsystem [START_REF] Djemai | Singularly Perturbed Method for the Control Design of a Synchronous Motor with its PWM Inverter[END_REF] given in OCF is U-observable with respect to Z(k), but the original subsystem (1) needs the knowledge of Z(k) and Φ -1 Z (ζ) to be able to estimate x.

We denote by I(x 0 ; U ⊆ N ) the set of all points x 1 ∈ U,an open neighborhood of x 0 , that are not uniformly distinguishable from x 0 . Definition 2 The system (1) is uniformly observable at x 0 ∈ N if I(x 0 ; N ) = x 0 and is (Locally Uniformly) LU-observable at x 0 ∈ N if, there exist an open neighborhood U of x 0 , such that I(x 0 ; U ) = x 0 . Definition 3 The dynamics (1) is (Locally Regularly Weakly Uniformly) 1 LRWU-observable2 at x 0 , for all t ∈ [t k , t k+1 [ ∀ k > 0, and Z(k) is constant during the same interval if there exist U (x 0 ), a neighborhood of x 0 and p integers (k 1 , ..., k p ), independent from Z(k) that form, after a possible reordering of the outputs, the smallest p-tuple with respect to the lexicographic ordering, such that for all x ∈ U (x 0 ) and Z(k

) constant for t ∈ [t k , t k+1 [ ∀ k ≥ 0. i) k 1 ≥ k 2 ≥ ... ≥ k p ≥ 0, ii) p i k i = n,(k i are observability indices of (1))
iii) The rank observability condition is verified ∀ Z(k) and x(t

) with t ∈ [t k , t k+1 [, k ≥ 0 rank                 dh 1 dL F h 1 . . . dL (k 1 -1) F h 1 . . . dh p . . . dL (kp-1) F h p                 |x(t) = n (5)
Remark 3 The choice of k i is not unique and the assumption that the observability indices k i are not affected by Z(k) is a strong one.

Under condition ( 5), ( 1)-( 3) may be written in Observability Canonical Form (OCF) ( [START_REF] Fliess | Generalized Controller Canonical Forms for Linear and Nonlinear Dynamics[END_REF]), ∀ t ∈ [t k , t k+1 [ and ∀ k thanks to the diffeomorphism ζ = Φ Z (x) defined as:

ζ = (ζ T 1 , .., ζ T i , .., ζ T p ) T = h 1 (x), L F h 1 (x), ..., L k 1 -1 F h 1 (x), ..., h p (x), L F h p (x), ..., L kp-1 F h p (x) T (6) with ζ i = (ζ 1,i , ζ 2,i , ..., ζ k i ,i ) T , for i = 1, .., p.
Remark 4 Diffeomorphism Φ z(k) (x) is parameterized by Z(k) and is discontinuous at each switching instant t k .

To guarantee the linearity of the OCF, with respect to Z(k), the following assumption is needed Assumption 2 The continuous dynamics (1) should verify the following conditions :

i) ∃ at least j ∈ {1, ..., p} such that ∂ L k j F (x=Φ -1 Z (ζ),Z) h j ζ ∂Z = 0, (7)
ii) For all i = 1, .., p, we have

∂ 2 L k i F (x=Φ -1 Z (ζ),Z) h i ζ ∂Z 2 = 0, (8)
Obviously, assumption 2 is sufficient for the considered problem and can be more relaxed with respect to the left invertibility problem ( [START_REF] Barbot | State and unknown input estimation for linear discrete-time systems[END_REF][START_REF] Hirschorn | Invertibilty of multivariable nonlinear control systems[END_REF]). Hence, subsystem (1)-( 3), under conditions ( 5)-( 7)-( 8) and diffeomorphism ( 6) can be rewritten in OCF linear with respect to Z(k) as:

                         ζ1,i = ζ 2,i ζ2,i = ζ 3,i . . . ζk i ,i = L k i F (x,Z) h i (x) x=Φ -1 Z (ζ) = Γ i (ζ) + Λ i (ζ)Z(k) = Γ i (ζ) + m j=1 Λ ij (ζ)z j (k) y i = ζ 1,i i = 1, . . . , p (9)
where :

Γ i (ζ) = L k i F (x,0) h i x=Φ -1 0 (ζ) i = 1, ..p Λ i,j (ζ) = ∂ L k i F (x=Φ -1 Z (ζ),Z) h i ζ ∂z j , j = 1, ..m
We note that the system (9) can present jumps even if it ont the case for the original one. In fact,

x(t + k ) = x(t - k ) = x(t k ) (10) using (6), for t ∈ [t k-1 , t k [ and t ∈ [t k , t k+1 [, one has ζ(t) = Φ z(k-1) (x(t)) ∀ t ∈ [t k-1 , t k [ (11)
ζ(t) = Φ z(k) (x(t)) ∀ t ∈ [t k , t k+1 [ (12)
Now, starting from these equations ( 11)-( 12) and ( 10), at the sampling instants t k one obtains:

ζ(t + k ) = Φ z(k) Φ -1 z(k-1) (ζ(t - k )) = R [z(k-1),z(k)] (ζ(t - k )) (13)
This relation gives the reset equation of ζ(t) at time t + k just after the switching time.

The following proposition summarize the previous result : Proposition 1 Under conditions ( 5)-( 7)-( 8), the subsystem ( 1) is transformable by diffeomorphism parametrized by Z(k) into the OCF (9).

Remark 5 As the subsystem ( 9) is given in OCF, it is U-observable with respect to Z(k) ( [START_REF] Gauthier | Observability for any u(t) of a class of nonlinear systems[END_REF]). Nevertheless, in order to reconstruct the original state x of (1), the knowledge of

Φ -1 Z (ζ) is necessary and consequently Z(k) ∀ k.
To overcome this difficulty, we will use a software sensor from the continuous dynamics (written in OCF), in order to obtain information with respect to the discrete states Z(k).

Thus, from ( 9), all the last row of the sub-dynamics of ζ i (ζ k i ,i for i = 1, .., p), may be written as:

θ = Γ(ζ) + Λ(ζ)Z(k) (14)
with θ = ζ k 1 ,1, , ..., ζ kp,p, T , and Γ(ζ) = [Γ 1 (ζ), ..., Γ p (ζ)] T is a column vector of dimension p, and
Λ(ζ)| x=Φ -1 Z (ζ) = Λ T 1 (ζ), ..., Λ T m (ζ)
T is a p × m matrix and Λ i (ζ) is a line vector of dimension m defined as:

Λ i (ζ) = [Λ i,1 (ζ), ..., Λ i,m (ζ)] , i = 1, ..., p
For the sake of simplicity, and because the continuous states in ζ coordinates are also known after the convergence of the continuous observer, we choose to

consider each Λ i,j (ζ) at time t - k , i.e., Λ i,j (ζ(t k )) = Λ i,j (ζ(t - k )
),to guarantee the finite time convergence of the continuous observer.

The discrete dynamics (2) remains linear in Z(k), and will be completed by an output

Y d (k) = Λ(ζ(k))Z(k): Z(k + 1) = A d Z(k) + β d (v(k), y(k)) (15)
Y d (k) = Λ(ζ(k))Z(k) (16)
Where

Λ(ζ(k)) = Λ(ζ(t = t - k )
). The discrete subsystem ( 15)-( 16) will now be used to design a discrete observer. As Λ(ζ(k)) is a (p × m) constant matrix only at time t ∈ [t k , t k+1 [ but may change at each t k ∀ k > 0, we need to define the distinguishability and observability of ( 15)- [START_REF] Hirschorn | Invertibilty of multivariable nonlinear control systems[END_REF].

Definition 4 A pair of points Z 0 and Z 1 in M is uniformly distinguishable with respect to v(k) ∀ k > 0, and ∀ ζ(0) if the system (15)-( 16) generates, ∀ ζ(0) and v(k), solutions Z 0 (k) and Z 1 (k) satisfying Z i (0) = Z i with Z i (k) ∈ M, and there exists k such that

Λ(ζ(k))Z 0 (k) = Λ(ζ(k))Z 1 (k).
We denote by J (Z 0 ; M) all points Z 1 ∈ M that are not uniformly distinguishable from Z 0 . Definition 5 The system ( 15)-( 16) is uniformly observable at

Z 0 ∈ M if J (Z 0 ; M) = Z 0 .

Definition 6

The system ( 15)-( 16) is, in a delayed way, U-observable at Z 0 ∈ M, if there exist a bounded µ ≥ 0 such that, for all ζ(0) and v(k), ∀ k < µ the system is uniformly observable at Z(µ).

The above definition means that, even if the reconstruction of Z(0) is not guaranteed, then the reconstruction of Z(µ) may be possible. In fact, to reconstruct a state vector Z at time µ starting from equation ( 15)-( 16

Y µ d = Λ(ζ µ )Z µ Y µ+1 d = Λ(ζ µ+1 ) [A d Z µ + β d,µ ]
. . .

Y µ+N d = Λ(ζ µ+N )A N d Z µ + Λ(ζ µ+N ) N i=1 A N -i d β d,µ+i-1
which can be rewritten as:

     Y µ d Y µ+1 d . . . Y µ+N d      =      Λ(ζ µ ) Λ(ζ µ+1 )A d . . . Λ(ζ µ+N )A N d      Z µ +      D µ D µ+1 . . . D µ+N      (17)
where

D µ+j = Λ(ζ µ+j ) j i=1
A j-i d β d,µ+i-1 for j = 1, .., N, is a known expression depending only on (v(.), y(.)).

Proposition 2 (Observability of Z(µ)) The discrete system (15)-( 16) is observable at Z(µ) ∈ M for a fixed ζ(µ) ∈ N and for a fixed set of input v(k), k > µ if and only if (Λ(.), A d ), verifies the following observability conditions:

There exists a bounded integer Ñ such that:

rank          Λµ =           Λ(ζ µ ) Λ(ζ µ+1 )A d . . . Λ(ζ µ+ Ñ )A Ñ d                    = m (18)
Proof.

• Sufficiency: From (17), verifying [START_REF] Menold | The peaking Finite time convergent observers for linear time-varying systems[END_REF] and using the Left Pseudo Inverse, one obtains Z(k).

• Necessary: If for all integer Ñ , rank Λµ < m, then Ker Λµ is at least of dimension 1, and consequently the unobservable space is at least of dimension one.

The construction of the matrix Λµ is based on FIFO (First Input First Output) principle.

Corollary 1 (Uniform Observability of Z(µ)) The discrete system (15)-( 16) is

U-observable at Z(µ) ∈ M, ∀ ζ(0) ∈ U and ∀ v(k) for k > µ, if and only if (Λ(.), A d ),

verifies the following observability conditions:

There exists a bounded integer N (ζ(0), v(k)), noted N such that:

rank          Λµ =           Λ(ζ µ ) Λ(ζ µ+1 )A d . . . Λ(ζ µ+N )A N d                    = m (19)
Remark 6 If µ = 0, the previous corollary leads to the U-Observability

Obviously, for causality reasons, µ + N is the present instant usually noted k and we don't know Z at time k but k -N and then equation [START_REF] Levant | Robust Exact Differentiation via sliding mode technique[END_REF] will be rewritten as

     Y k-N d Y k-N +1 d . . . Y k d      =      Λ(ζ k-N ) Λ(ζ k-N +1 )A d . . . Λ(ζ k )A N d      Z k-N +      D k-N D k-N +1 . . . D k      (20)
Now, from definitions (2)-( 3) and ( 5)-( 6), we are able to propose an observability definition for the whole HDS system defined in (1).

Definition 7 A pair of continuous and discrete state x 0 (0), z 0 (0) and x 1 (0), z 1 (0) in N × M are uniformly distinguishable if for all input v(k), the system (1) generates solutions x 0 (t), z 0 (t) and x 1 (1), z 1 (1) such that h(x 0 (t)) = h(x 1 (t)) for some t = +∞.

We denote by £(x 0 , z 0 ; U × M) all sets of couple x 1 , z 1 ∈ U × M that are not distinguishable from x 0 , z 0 . Definition 8 The class of HDS ( 1) is (Locally Uniformly) LU-observable

at x 0 , z 0 ∈ N × M if £(x 0 , z 0 ; U × M) = x 0 , z 0 .
Definition 9 The hybrid system (1) is in delayed way (Locally Uniformly) LU-Observable at (x 0 , Z 0 ) if there exists a bounded integer k > 0, such that, the system (1) is LU-Observable at (x(k), Z(k))

Proposition 3 If the continuous dynamics (1) verify conditions ( 5)-( 7)-( 8) and it is (Locally Regularly Weakly) LRW-Observable at x 0 , for all t ∈ [t k , t k+1 [ ∀ k and the subsystems ( 15)-( 16) is uniformly observable at z 0 ∈ M , then the whole HDS defined in ( 1) is (Locally Uniformly) LUobservable at x 0 , z 0 and it is LU-observable, in delayed way, at x tµ , z µ for µ > 0, if the subsystems ( 15)-( 16) is uniformly observable in delayed way at z µ ∈ M.

In the next, a high order sliding modes observer combined with discrete state reconstructor will be designed for this class of HDS which is LU-Observable (or in a delayed way LU-Observable). Advantages of such observer, with respect to other ones which may be used, are its finite time convergence, and its ability to consider the variable structure systems.

Hybrid Observer Design

The hybrid observer design is carried out in the following way : (figure 1). Once the diffeomorphism carried out, a sliding mode based observer (Super Twisting Algorithm), is firstly applied to the continuous system transformed in the OCF (9). Secondly, after the estimation of all continuous states of the system in the ζ coordinates in n -1 first steps, the last n th step allows reconstructing the whole dynamics and by consequence obtaining

Y d (k) = Λ( ζ(t - k ))Z(k) = Λ( ζ(k -))Z(k).
The discrete observer consists in using Y d (k) like an output for the discrete dynamics, in order to design a discrete observer to estimate Ẑ(k).

Z c = Ẑ, ζ c = ζ et x c = x

Super Twisting Algorithm

In [START_REF] Saadaoui | Exact differentiation and Sliding mode observer for switched Lagrangian systems[END_REF], a Super Twisting Algorithm (STA) was used for switched system. In our case, a STA is used for the continuous part of the system. In fact, the continuous dynamics will be supposed as a forced commutation, since it is the discrete dynamics which makes the difference while Z(k) change its value.

The STA is given by the following structure [START_REF] Levant | Robust Exact Differentiation via sliding mode technique[END_REF]:

obs = u(e 1 ) = u 1 + λ 1 |e 1 | 1 2 sign(e 1 ) u1 = α 1 sign(e 1 ) λ 1 , α 1 > 0 (21)
where e 1 = ζ 1,i -ζ1,i , λ 1 , α 1 are positive parameters, and u 1 is the differentiator output where:

sign(.) =    +1 if (.) > 0; -1 if (.) < 0; ∈ [-1, 1] if (.) = 0
The step by step exact differentiator applied to each subsystem (9), leads to the following form:

ζ1,i = ζ2,i + λ 1,i |ẽ 1,i | 1/2 sign(ẽ 1,i ) ζ2,i = α 1,i sign(e 1,i ) ζ2,i = E 1,i ζ3,i + λ 2,i |ẽ 2,i | 1/2 sign(ẽ 2,i ) . . . ζkp-1,i = E kp-2,i ζkp,i + ... +λ kp-1,i ẽkp-1,i 1/2 sign(ẽ kp-1,i ) ζkp,i = E kp-2,i α kp-1,i sign(ẽ kp-1,i ) ζkp,i = E kp-1,i θi + λ kp,i ẽkp,i 1/2 sign(ẽ kp,i ) θi = E kp-1,i α kp,i sign(ẽ kp,i ) (22)
for i = 1, . . . , p, where ẽj,i = ζj,iζj,i , with ζ1,i = ζ 1,i for j = 1, .., k p , and the vector θi represent the estimation of the dynamic Γ i (ζ) + Λ i (ζ)Z(k) and E j,i for j = 1, ...k i -1 are defined as

E j,i = 0 if |ẽ j,i | ≥ ε > 0, else E j,i = 1 (23)
See [START_REF] Floquet | Super twisting algorithm based step-bystep sliding mode observers for nonlinear systems with unknown inputs[END_REF] for more details concerning the convergence proof and the definition of E j,i . We also have to take into account the jumps (13) due to the diffeomorphism and use the reinitialisation procedure

∀ t = t k , k > 0. ζ+ (t k ) = Φ ẑ(k) Φ -1 ẑ(k-1) ( ζ(t - k )) (24)
The convergence of the observation error for all the continuous states is obtained in (n -1) steps in finite time and allows to reconstruct the state vector. To obtain the discrete output

Y d = Λ( ζ)Z(k), let us consider: θ(k) = Γ( ζ(k -)) + Λ( ζ(k -))Z(k)
where k -is the time t = t - k (just before switching instant) then at time t = t k one obtains:

Y d (k) = Λ( ζ(k -))Z(k) = [ θ(k) -Γ( ζ(k -))] (25)
Thanks to the canonical form, the observation of ζ doesn't need the knowledge of Z(k). In fact, the sliding mode observer based on the STA needs only to know y i = ζ i,1 . This is why the hybrid observer design is carried out in two sccessives stages, starting with a continuous observer. The hybrid observer design requires the following condition: Condition 1 The sliding mode observer is designed such that the convergence time τ k c during the interval [t k , t k+1 [ ∀ k ≥ 0 verifies the following condition:

τ m > max τ k c > 0 ∀ k ≥ 0 (26)
with τ m is defined in (4).

Condition ( 4) is justified by the fact that the convergence time τ k c of the designed observer for each constant value of Z(k) is not the same and also ensures that the designed observer has the time to estimate the continuous state during [t k , t k+1 [ ∀ k ≥ 0. The following proposition summarizes the properties of the observer [START_REF] Vas | Sensorless Vector and Direct Torque Control[END_REF] (see [START_REF] Floquet | Super twisting algorithm based step-bystep sliding mode observers for nonlinear systems with unknown inputs[END_REF] for more details).

Proposition 4 Let us consider the system (9), supposed to be bounded for all t ∈ [t k , t k+1 [ ∀ k > 0, and a sliding mode observer [START_REF] Vas | Sensorless Vector and Direct Torque Control[END_REF] All switching instants are known by both observers and the reset information (24) is transmitted from the discrete observer to the continuous one.

Discrete state observer

In this section, we will design the observer of the discrete subsystem to estimate the discrete state. Let us consider the system (1) and the observation structure given in the previous section, the discrete observer receives as input : the observed state ζ, the output y(k), the information E k i -1,i for i = 1, .., p and v(k). Its task is to provide an estimation Ẑ of the discrete state Z of the hybrid plant starting from both the reconstructed discrete output Y d (k) given in (25). and the sampled continuous state ζ(k). To design the discrete observer, let us consider that (1) verifies the observability condition [START_REF] Menold | The peaking Finite time convergent observers for linear time-varying systems[END_REF]. After the estimation of the continuous state ζ and a discrete output Y d = Λ( ζ(k))Z(k), the following system will be used to estimate Ẑ:

Z(k + 1) = AZ(k) + β d (v(k), y(k)) Y d = Λ( ζ(k))Z (27)
If condition ( 18) is verified, it is possible to design an observer for the time varying system (27) to obtain Ẑ(k) which converges to Z(k). Note that the designed method performed in [2, 18], for finite time convergent observers for linear time-varying systems, can be used in our case.

The estimation Ẑ(k -N ) can also be obtained by using left invertibility on the equation ( 20), one obtains:

Ẑk-N =    Λ(ζ k-N ) . . . Λ(ζ k )A N d    -1 *    Y k-N d -D k-N . . . Y k d -D k    (28)
where (.) -1 * represents a pseudo inverse of (.). Obviously, as it is mentioned before, for causality reasons and in order to estimate the discrete state at time k a discrete time predictor of N -steps will be used as follows:

Starting from (27) and denoting

β d,k-N = β d (v(k -N ), y(k -N )), and Z k = Z(k), one obtains: Ẑk-N+1 = A d Ẑk-N + β d,k-N Ẑk-N+2 = A 2 d Ẑk-N + A d β d,k-N +1 + β d,k-N . . . Ẑk = A N d Ẑk-N + N i=1 A N -i d β d,k-N +i-1 (29)
which gives the prediction of the discrete state Z at time k starting from the knowledge of Z(k -N ).

where using (24), the reinitialization procedure is given by

x = (x 1 , x 2 , x 3 , x 4 ) T , Z = (z 1 , z 2 , z 3 ) T , y 1 = x 1 , y 2 = x 3 F (x, Z(k)) =     x 2 + z 1 (k) -10x 2 -5 sin(0.5x 3 ) -x 2 1 z 1 (k) x 4 + z 1 (k) -x 2 3 + 5x 1 + x 1 z 2 (k)     and A d =   0.50 0 1 -0.2 0.1 0 -0.20 0 -0.1   , β d =   y 1 (k) 0 y 2 (k)   Using the following diffeomorphism ζ = Φ Z (x) as ζ 1 = x 1 ; ζ 2 = x 2 + z 1 (k); ζ 3 = x 3 and ζ 4 = x 4 + z 1 (k), one obtains ζ1 = ζ 2 ζ2 = -10ζ 2 -5 sin(0.5ζ 3 ) -ζ 2 1 z 1 (k) + 10z 1 (k) ζ3 = ζ 4 ; ζ4 = -ζ 2 3 + 5ζ 1 + ζ 1 z 2 (k) -z 1 (k) Z(k + 1) = A d Z(k) + β d (y(k))
ζ+ 1 (t k ) = ζ1 (t - k ); ζ+ 2 (t k ) = ζ2 (t - k ) -z 1 (k -1) + z 1 (k) ζ+ 3 (t k ) = ζ3 (t - k ); ζ+ 4 (t k ) = ζ4 (t - k ) -z 1 (k -1) + z 1 (k)
Now, Ẑ(k -1) is given by equation ( 28) and to obtain Ẑ(k), one uses (29) to realize the predictor with N = 1:

Ẑk = A d Ẑk-1 + β d , k-1
Finally, using Φ -1 Z , one obtains the estimation of the continuous state in the original coordinates. The simulation results are depicted in (figures 2-4)

  ) with the following notations Y µ d := Y d (µ), ζ µ := ζ(µ), Z µ := Z(µ), and β d,µ := β d (v(µ), y(µ)), one obtains:

Fig. 1 :

 1 Fig. 1 : Structure of the Hybrid Observer Continuous and Discrete States with : Z c = Ẑ, ζ c = ζ et x c = x

  For any initial condition ζ(k), ζ(k), and for all Z(k), there exist bounded α i,j and λ i,j such that ζ converges in finite time (smaller than τ m ) towards ζ, i.e. ∃ t = τ k c > 0 such that ζ = ζ, and θ converges towards Γ( ζ) + Λ( ζ)Z(k), and by consequence reconstruct a discrete output Y d (k) = Λ( ζ)Z(k) as in (25).

y 1 =

 1 ζ 1 ; and y 2 = ζ 3 with Z(t) = Z(k) for all t ∈ [T k, T (k + 1)[. To simplify the study, we consider a constant sampling period T = t k+1t k = 0.5s. The associated continuous observer is then given as in (22): After a finite time, one obtains ζ and then Y d = Λ(ζ)Z(k) in the following way:-ζ 2 1 (k)z 1 (k) = ( θ1 (k) -10ζ 2 (k) -0.5 sin(ζ 3 (k))) ζ 1 (k)z 2 (k) = ( θ2 (k) + ζ 2 3 (k) -5ζ 1 (k)) and thus, if the output Y d (k) is considered, one can use a pseudo inverse or left inversion to obtain Ẑ(k).

Fig. 2 :Fig. 3 :Fig. 4 :

 234 Fig. 2 : Real x and observed x continuous states

The LRWU-observability leads to the LU-observability.

This is a modified version of observability conditions given in[START_REF] Hermann | Nonlinear controllability and Observability[END_REF].


Marine Pouget

Alex Baumel

Katia Diadema

Fre ´de ´ric Me ´dail

Daniel Sanchez

Conservation unit allows assessing vulnerability and setting conservation priorities for a Mediterranean endemic plant within the context of extreme urbanization

Keywords: Conservation priorities, Ecology, Endemism, Habitat fragmentation, Phylogeography, Vulnerability, Maritime Alps

Current biodiversity patterns are a temporary state in a continuum of ecological and evolutionary changes. Conservation policies must incorporate this dynamic to ensure the long-term conservation of biodiversity which is particularly challenging in a context of extreme urbanization. An original approach, rarely used for plant conservation, is to define conservation units to set conservation priorities within species by combining ecological and evolutionary divergences. In the Maritime Alps (southern France), the ecological and evolutionary divergences between the populations of Acis nicaeensis has allowed us the determination of conservation units of this endemic plant threatened by severe urbanization and land-use changes. Phylogeographical data (cpDNA haplotype) were considered as a proxy for evolutionary legacy, and ecological data (multivariate analysis of habitat) were used as a proxy for ecological distinctiveness. Our goal was to explore the potential of this approach to assess vulnerability and set conservation priorities for narrow endemic species in the context of conflict between biodiversity and human activities. The results highlight five different conservation units within A. nicaeensis distribution. Genetic and ecological divergences are present at fine-scale. This pattern is highly endangered by urbanisation.

Introduction

Faced with the challenge of reducing the loss of biodiversity, conservation biologists and ecologists strive to develop efficient strategies and methods for biodiversity conservation [START_REF] Margules | Systematic conservation planning[END_REF][START_REF] ¨rtel | Grouping and prioritization of vascular plant species for conservation: combining natural rarity and management need[END_REF]. Conservation biology aims to preserve current diversity and the ongoing diversification processes [START_REF] Mace | Preserving the tree of life[END_REF][START_REF] Forest | Preserving the evolutionary potential of floras in biodiversity hotspots[END_REF]. Therefore, identifying spatial distribution of evolutionarily lineages is a priority to highlights geographic areas which have played an important role in the past in terms of emergence and persistence of biodiversity [START_REF] Becerra | Sources and sinks of diversification and conservation priorities for the Mexican tropical dry forest[END_REF][START_REF] Davis | The California Hotspots Project: identifying regions of rapid diversification of mammals[END_REF][START_REF] Rosauer | Phylogenetic endemism: a new approach for identifying geographical concentrations of evolutionary history[END_REF][START_REF] Sauquet | Contrasted patterns of hyperdiversification in Mediterranean hotspots[END_REF]) and to ensure that the evolutionary legacy is recognized and protected [START_REF] Moritz | The importance of an evolutionary perspective in conservation policy planning[END_REF].

The Evolutionarily Significant Unit (ESU) was conceptualized by [START_REF] Ryder | Species conservation and systematics: the dilemma of subspecies[END_REF]. [START_REF] Ryder | Species conservation and systematics: the dilemma of subspecies[END_REF] defined an ESU as a population presenting significant adaptive variation based on concordance between sets of data related to life history information, morphometric, range and distribution, and genetic structure. The concept has since gained support, various definitions and criteria have been proposed [START_REF] Ryder | Species conservation and systematics: the dilemma of subspecies[END_REF][START_REF] Waples | Pacific Salmon, Oncorhynchus spp., and the definition of ''Species'' under the Endangered Species Act[END_REF][START_REF] Dizon | Rethinking the stock concept: a phylogeographic apporach[END_REF][START_REF] Moritz | Defining ''Evolutionarily Significant Units'' for conservation[END_REF][START_REF] Vogler | Diagnosing units of conservation management[END_REF][START_REF] Crandall | Considering evolutionary processes in conservation biology[END_REF][START_REF] Fraser | Adaptive evolutionary conservation: towards a unified concept for defining conservation units[END_REF]. For [START_REF] Moritz | Defining ''Evolutionarily Significant Units'' for conservation[END_REF], the concept of ESU should be based upon historical biogeography rather than on adaptive diversity. The focus on evolutionary legacy is linked to the postulate that a set of populations, historically isolated and genetically different, are likely to be divergent lineages and will contribute to future diversification of the species [START_REF] Moritz | Defining ''Evolutionarily Significant Units'' for conservation[END_REF][START_REF] Moritz | Strategies to protect biological diversity and the evolutionary processes that sustain it[END_REF]. Therefore, according to [START_REF] Moritz | Defining ''Evolutionarily Significant Units'' for conservation[END_REF][START_REF] Moritz | Strategies to protect biological diversity and the evolutionary processes that sustain it[END_REF] and [START_REF] Moritz | The importance of an evolutionary perspective in conservation policy planning[END_REF] the function of the ESU is to prioritize conservation efforts according to an historic biogeographical structure [START_REF] Davis | The California Hotspots Project: identifying regions of rapid diversification of mammals[END_REF][START_REF] Kraft | Range size, taxon age and hotspots of neoendemism in the California flora[END_REF][START_REF] Hendry | Evolutionary biology in biodiversity science, conservation, and policy: a call to action[END_REF][START_REF] Hendry | Evolutionary principles and their practical application[END_REF][START_REF] Isaac | Mammals on the EDGE: conservation priorities based on threat and phylogeny[END_REF][START_REF] Malaney | Using biogeographical history to inform conservation: the case of Preble' s meadow jumping mouse[END_REF][START_REF] Geeta | Biodiversity only makes sense in the light of evolution[END_REF]. However, the use of the term ESU alone to set operational conservation units has been debated [START_REF] Fraser | Adaptive evolutionary conservation: towards a unified concept for defining conservation units[END_REF][START_REF] Guia | The gap between the concept and definitions in the Evolutionarily Significant Unit: the need to integrate neutral genetic variation and adaptive variation[END_REF]. [START_REF] Crandall | Considering evolutionary processes in conservation biology[END_REF] suggested focusing on the identification of distinct populations through ecological and genetic exchangeability tests over historical and recent times. Exchangeability, in essence, means that populations are replaceable, which means that these replaceable populations are lower in population prioritization with regard to conservation.

This theoretical debate appeared far from the reality of conservation issues [START_REF] Fraser | Adaptive evolutionary conservation: towards a unified concept for defining conservation units[END_REF]. It appeared indeed that ESU-related research papers are rarely used to improve conservation policies or assess threats to biodiversity [START_REF] Lopez | A multi-faceted approach for assessing evolutionary significant conservation units in the endangered Omphalodes littoralis subsp. gallaecica (Boraginaceae)[END_REF]. Furthermore the concept of the ESU has been used almost exclusively for mammals or birds species [START_REF] Guia | The gap between the concept and definitions in the Evolutionarily Significant Unit: the need to integrate neutral genetic variation and adaptive variation[END_REF] but it has almost never been used for plants [START_REF] Newton | Molecular phylogeography, intraspecific variation and the conservation of tree species[END_REF][START_REF] Lopez | A multi-faceted approach for assessing evolutionary significant conservation units in the endangered Omphalodes littoralis subsp. gallaecica (Boraginaceae)[END_REF].

Here, we are attempting to examine an approach related to the ESU concept to delineate conservation units and set conservation priorities for a Mediterranean narrow endemic plant. Mediterranean plant populations are the results of long term accumulation of diversity in very heterogeneous landscapes leading eventually to a strong evolutionary legacy on a very fine scale [START_REF] Tatoni | The impact of changes in land use on ecological patterns in Provence (Mediterranean France)[END_REF]Me ´dail and Diadema 2009;[START_REF] Pouget | Phylogeography sheds light on the central-marginal hypothesis in a Mediterranean narrow endemic plant[END_REF]. Today, this legacy is endangered by severe anthropogenic pressures (e.g. [START_REF] Blondel | Biodiversity and conservation[END_REF], especially in lowland areas and near the coasts where urbanization is increasing at an unprecedented rate of ?17% from 1990 to 2000 [START_REF] Underwood | Threats and biodiversity in the mediterranean biome[END_REF]. In this context, we combined phylogeographical data considered as a proxy for evolutionary legacy, to ecological data, used as a proxy for ecological distinctiveness, to define conservation units and to set conservation priorities. We apply this approach to delineate conservation units within the rare geophyte plant species Acis nicaeensis (Ardoino) Lledo

´, A.P. Davis and M.B. Crespo (Amaryllidaceae), an endangered narrow endemic, restricted to a small area in the Maritime Alps, including the major cities of Nice and Monaco within the severely impacted French Riviera [START_REF] Diadema | Ecologie, distribution et morphologie compare ´es des nive ´oles de Nice (Acis nicaeensis) et de Fabre (Acis fabrei), Alliaceae ende ´miques des Alpes maritimes et de la Nesque (Vaucluse)[END_REF].

The objectives of this study are (i) to infer the phylogeography of A. nicaeensis from the distribution of maternal lineages in twenty two populations (ii) which will be combined with ecological distinctiveness of populations, (iii) to delineate conservation units and then to assess the vulnerability of each unit in relation with urban sprawl.

Materials and methods

Study system

The studied species is the Nice's snowflake A. nicaeensis, a narrow endemic geophyte plant from the southern part of the Maritime Alps, with a patchy distribution on calcareous rocky grasslands. The distribution of A. nicaeensis covers approximately 12 9 35 km between the Pre-Alps of Grasse (col de Vence) to the west, and the only Italian population near the French-Italian border (Grimaldi, Ventimiglia) to the east. The highest population densities are found between Nice and Menton with c. 50 patches or populations. The delimitation of populations is described below (''Assessment of vulnerability (or threats)'' section). This snowflake has quite a large ecological amplitude, and the three main habitats of A. nicaeensis are structured along an altitudinal gradient from sea level to mid-altitude mountains: (i) a thermophilous matorral (n = 19 populations) with Euphorbia dendroides, Arisarum vulgare, Brachypodium retusum, Asparagus acutifolius and Smilax aspera evolving towards a thermo-Mediterranean pre-forest with Pinus halepensis and Quercus ilex; (ii) a mid-altitude situation (n = 11 populations), characterized, more or less, by disturbed xerophytic grassland with annual species and the presence of several ruderal perennial plants (Artemisia verlotiorum, Centaurea paniculata, and Lobularia maritima); (iii) a higher altitude (600-1000 m) xerophytic grasslands with perennial species (Bromus sp., Lavandula angustifolia, Genista hispanica, n = 17) on limestone rocks [START_REF] Diadema | Ecologie, distribution et morphologie compare ´es des nive ´oles de Nice (Acis nicaeensis) et de Fabre (Acis fabrei), Alliaceae ende ´miques des Alpes maritimes et de la Nesque (Vaucluse)[END_REF][START_REF] Noble | La flore des Alpes-Maritimes et de la Principaute ´de Monaco: originalite ´et diversite[END_REF]. During the early XXth century, some populations of A. nicaeensis were severely damaged by massive sampling for herbarium collections and several populations have disappeared since the middle of the XXth century as a result of urbanization and land use changes [START_REF] Diadema | Ecologie, distribution et morphologie compare ´es des nive ´oles de Nice (Acis nicaeensis) et de Fabre (Acis fabrei), Alliaceae ende ´miques des Alpes maritimes et de la Nesque (Vaucluse)[END_REF]. Several factors are threatening this species, including the spatial increase of trees and shrubs covers induced by the collapse of traditional land-use, and above all the huge development of urbanization along the Riviera coast and the increase of tourism related activities [START_REF] Diadema | Apport de la phyloge ´ographie, de la dynamique et de la structure des populations pour la conservation des ve ´ge ´taux ende ´miques me ´diterrane[END_REF]. For these reasons this snowflake is recorded in the annexes II and IV of the European Habitats Directive, and included in the 'Livre rouge de la flore menace ´e de France, espe `ces prioritaires' [START_REF] Olivier | Livre rouge de la flore menace ´e de France. Tome 1: Espe `ces prioritaires. Collection Muse ´um national d'Histoire naturelle[END_REF].

Sampling, DNA extraction and sequence analysis

The study area is bounded by the distribution range limit of A. nicaeensis. Twenty-two populations were sampled throughout their entire geographical range in April 2012 (see Fig. 1). Leaves were dried with silica gel and then stored at -20 °C. Total DNA extraction is following the CTAB protocol of [START_REF] Doyle | A rapid DNA isolation procedure for small quantities of fresh leaf tissue[END_REF]. The polymorphism of the chloroplast genome was investigated in one to three randomly sampled individuals per population (n = 63) on five plastid regions: the internal part of the matK gene; trnL-trnF, trnT-trnL and trnT-trnD intergenic spacer, trnH intron plus 5 0 part psbA (abbreviate respectively matK, LF, TL, TD and psba). PCRs were performed in a 50 ll volume reaction that contained 2 ll of each primer (5 mM), 1.25 units of Taq polymerase (Q-Biogen), 5 ll of incubation buffer, 6 ll of MgCl2 (10 mM), 4 ll of dNTP (25 mM) and 5 ll of DNA (50 ng/ll). Amplification was performed using a PTC-200 thermocycler (MJ Research). Direct sequencing (ABI 3730 xl, Applied Biosystems) was carried out by Eurofins sequencing services (Germany). After quality checking and alignment, the cpDNA data set contained 63 sequences.

Phylogenetic and ecological distinctiveness to delineate conservation units

A median-joining network [START_REF] Bandelt | Median-joining networks for inferring intraspecific phylogenies[END_REF]) has been constructed using SplitsTree4 software [START_REF] Huson | Application of phylogenetic networks in evolutionary studies[END_REF] including both substitution and insertion/deletion (indel). Indels sites were reduced, considered as a fifth state and weighted as a substitution.

Twenty-one populations of A. nicaeensis were surveyed for environmental and vegetation characteristics (Conservatoire Botanique National Me ´diterrane ´en de Porquerolles (CBNMed) and Conservatoire Botanique National Alpin (CBNA) (2015). SILENE-Habitats. http://habitats.silene.eu), since the twenty-second population was destructed by urbanisation. Releve ´s were performed on 100 m 2 homogeneous vegetation. For each releve ´, 15 environmental and ecological variables were taken into account: average annual temperature (worldclim data base), average annual precipitation (worldclim data base), altitude, exposition, slope, proportion of naked ground, pebbles, rocks and blocks, proportion of cover by herbaceous plants, by small shrubs (\1 m), by high shrubs (between 1 and 4 m), by low trees (10 m) and by high trees ([10 m), as well as the total vegetation cover (RcvtTotal). A principal component analysis (PCA function ''dudi.pca'', package ''ade4'', R core team 2015) has been built for the 15 mesologic and environmental variables. Then ecological groups were delineated by a kmeans classification (function ''kmeans'', R core team 2015) based on PCA component coordinates of each releve ´s to set groups of releve ´s that describe comprehensively the two first PCA components. On the basis of our knowledge a design according to three ecological groups was better to describe the habitat diversity of A. nicaeensis than with two or more. Kmeans method was performed on PCA components rather than on raw data to avoid bias due to variable collinearity.

Finally, each cpDNA haplotype was considered as a distinct evolutionary group. Ecological groups have been merged with haplotypes information in order to delineate conservation units. Both information were considered as equal, two populations being assigned to the same conservation unit if bearing the same haplotype and living in the same ecological group. The map representing the different elements (haplotypes, ecological groups and conservation units) has been displayed using the GIS software QGIS (QGIS Development Team 2015).

Assessment of vulnerability (or threats)

Several field trips were conducted for the data collection of 662 points of occurrence of A. nicaeensis, databased in SILENE-Flore (http://flore.silene.eu; CBNMed and CBNA 2015). They have been grouped by location, for a total of 52 populations. The delimitation of populations is based on net geographical discontinuities between the occurrences (more than 500 m around occurrence). Most of the population structures have been extracted from the thesis of [START_REF] Diadema | Apport de la phyloge ´ographie, de la dynamique et de la structure des populations pour la conservation des ve ´ge ´taux ende ´miques me ´diterrane[END_REF] and completed by several field trips [START_REF] Diadema | Apport de la phyloge ´ographie, de la dynamique et de la structure des populations pour la conservation des ve ´ge ´taux ende ´miques me ´diterrane[END_REF]Me ´dail from 2007 to 2014). For each population studied, the total number of individuals observed has been counted. The minimum and maximum number of individual per population is recorded for each conservation units (Table 1). Population areas have been calculated as the surface of a polygon including the area of the population in two dimensions (altitude has not been taken into account) using the GIS software QGIS (QGIS Development Team 2015). Conservation unit range size, calculated as the area of the unit, is the sum of all populations' superficies in km 2 (Table 1).

The isolation of the conservation units represents the mean distance separating the population of the unit from all the others population. The distance is calculated using the centroid of the population and the function ''rdist.earth'' (package ''fields'', R core team 2015). The value is given in km.

Because the main threat for A. nicaeensis is the urban sprawl along the Riviera coast [START_REF] Diadema | Apport de la phyloge ´ographie, de la dynamique et de la structure des populations pour la conservation des ve ´ge ´taux ende ´miques me ´diterrane[END_REF] we quantified it, other threats are existing but they are more spatially restricted and they will be discussed.

As a proxy of A. nicaeensis threats, the actual urbanization cover has been estimated per conservation unit using the IGN (National Institute of Geography, France) map of 2006. It has been included inside the category ''urbanization'' all possible artificialized structures such as, buildings of all types, sport centres and fields, roads, staircases, parking, water tank, etc. But the use of the IGN map systematically underestimates our perception of the artificialisation by masking some features (like pedestrian road, artificial green area, forest plantation, etc.). A percentage of urbanization cover has been given for every 10,000 m 2mesh. The mesh size was defined in order to facilitate the visual assessment of the urbanization and to allow for the following spatial analysis (using the GIS software QGIS). 0% represents no urbanization at all, 100% representing a strictly urbanized mesh. The final percentage value of a unit is the mean percent of all meshes that intersect the populations of a conservation unit (Table 1).

Results

Phylogeography as a proxy of evolutionary legacy

The molecular study of A. nicaeensis based on five cpDNA spacers (matK, LF, TL, TD and psba), giving a combined sequence length of 3797 bp, revealed five polymorphic sites (three substitutions and three indels) with the presence of three haplotypes (Fig. 1a), all well supported by at least one substitution and one indel. MatK presents two substitutions, LF presents one indel, TL presents one substitution, and psba presents two indels (Genbank accession number: Haplotype A: KX018277, KX018280, KX018282, KX018284, KX018285; Haplotype B: KX018278, KX018281, KX018283, KX018284, KX018286; Haplotype C: KX018279, KX018280, KX018282, KX018284, KX018287). Up to three individuals per population has been sampled to verify the monotypic identity of population.

No diversity has been found within populations. Each cpDNA haplotype was considered as a distinct evolutionary unit. The haplotype A occurs from the West (CVE, col de Vence) to the East (CAS, Castillon) of A. nicaeensis the distribution (Fig. 1b). The haplotype B is present in seven studied populations and is mainly distributed in the central and eastern parts of the distribution. The haplotype C is the most restricted as it occurs only in the south-western part of the distribution, near Nice and the Cap-Ferrat peninsula, including only three populations (Fig. 1b).

Multivariate analysis of habitat as a proxy of ecological distinctiveness

Environmental and vegetation data characterizing the 21 releve ´s allow the distinction of 3 main ecological groups for A. nicaeensis (Fig. 2a,b), with the two main axes of the ACP representing 35% of inertia. The first group (A) corresponds to the rocky lawns populations growing in the highest altitude (e.g. the population of Castillon CAS or Col de Vence CVE), with a high cover of low shrubs (e.g. Thymus vulgaris, Euphorbia spinosa) or herbaceous species and a reduced cover of high trees and shrubs. This group of releve ´s is typical of the meso-Mediterranean vegetation belt in the Maritime Alps [START_REF] Noble | La flore des Alpes-Maritimes et de la Principaute ´de Monaco: originalite ´et diversite[END_REF]. It is mainly located in the northern range of the distribution, with the largest range from east to west of the distribution, between altitudes 400 m and 960 m a.s.l. (Fig. 2c). The second and third groups of releve ´s are located within the thermo-Mediterranean vegetation belt, defined by several thermophilous plant species. The second group (B) represents populations growing on habitats with a high percentage of rock cover and high slope; it corresponds to rocky crevice and steep scree habitats, characterized by rocky dry lawns with Brachypodium retusum associated with annual and bulbous plants such as Allium acutiflorum. The B group encompasses populations occurring in the centre of the distribution, from 250 m of altitude to around 500 m a.s.l. The third group (C) includes populations whose habitat is determined by a quite important cover of high trees, shrubs and litter. It corresponds to the coastal thermophilous matorral with Euphorbia dendroides, Pistacia lentiscus, Olea europaea subsp. europaea and Ceratonia siliqua. This ecological group is located close to the coast in two areas (Nice, Cap-Ferrat and Monaco), from 10 m altitude to less than 200 m a.s.l.

Delimitation of the conservation units and assessment of vulnerability

Phylogenetic and ecological data have been merged to delineate the conservation units. Both data were considered as equal, two populations being assigned to the same conservation unit if bearing the same haplotype and living in the same ecological group. Among the nine possible combinations between phylogenetic and ecological data, five units were observed and distinguished as conservation units for A. nicaeensis. They have been named firstly by the haplotype number, then by ecological group, accordingly AA, AC, BA, BB and CB units. There is a large difference between units in terms of spatial isolation and geographical distribution of populations (Fig. 3), but also in term of size (range size or individual number) and urbanisation threats (Table 1). It is interesting to note that the most geographically isolated populations (such as CVE and PSL, respectively the westernmost and easternmost populations) are included in conservation units that are also present in the centre of the species distribution (AA and BA) (Table 1). These latter units exhibit the largest distributions, the largest range size but also a lower level of urbanisation cover. The AC units possess a smaller distribution in the southern range but with equivalent number of populations than BA (5 populations) and a low isolation. These units include the largest populations of A. nicaeensis in terms of individual number. The CB and BB units are the smallest ones in terms of distribution range and number of populations (respectively only three and two populations), a very restricted range size especially for the BB unit restricted to Monaco, and have very few individuals per population. These two units are included in the same ecological group, and the CB represents by itself one single haplotype revealing a unique evolutionary distinctiveness. These last two units are also the most threatened, because of a very high proportion of urban cover and the lowest number of individuals per population (Table 1).

Discussion

Assessment of evolutionary and ecological distinctiveness

In the context of an extreme environmental change induced by urbanization (Arau ´jo 2003; [START_REF] Aronson | A global analysis of the impacts of urbanization on bird and plant diversity reveals key anthropogenic drivers A global analysis of the impacts of urbanization on bird and plant diversity reveals key anthropogenic drivers[END_REF][START_REF] Mccleery | Urban wildlife conservation. Theory and practice[END_REF] protected species or habitats are continuously being degraded along the Mediterranean coastline [START_REF] Sirami | Long-term anthropogenic and ecological dynamics of a Mediterranean landscape: impacts on multiple taxa[END_REF][START_REF] Lhotte | Are there contrasted impacts of urbanization and land uses on population persistence? The case of Teucrium pseudochamaepitys, an endangered species in Southern France[END_REF]). The consequence is an extreme fragmentation of populations, which persist in patchy and highly urbanised areas [START_REF] Mckinney | Urbanization, biodiversity, and conservation[END_REF][START_REF] Iucn | Effects of habitat fragmentation on biodiversity[END_REF]. To limit the vulnerability of the populations, it is essential to define conservation units for the rarest and most endangered plant species, notably the endemics species located in areas of high urban sprawl. The French Riviera represents unfortunately an ''emblematic case'' of this worrying socioecological trend where refugial areas of unique biodiversity are threatened by direct human impact ( Me ´dail and Diadema 2006). It is therefore important to preserve the evolutionary legacy of narrow endemics like A. nicaeensis whose populations were moulded by the complex historical biogeography of the Maritime Alps hotspot (Me ´dail and Verlaque 1997; [START_REF] Casazza | Dispersal mechanisms in some representatives of the genus Moehringia L. (Caryophyllaceae)[END_REF].

Our approach is based on a heuristic comparative analysis, able to gather information from various sources and quality. It is essential to establish a fast diagnostic and to provide strong arguments to help conservationists to focus their actions in the field. Our study detected five conservation units within A. nicaeensis by using simple molecular tools (PCR and direct Sanger sequencing for a total cost \1000 €) combined with analysis of habitat structure and threats. Our capacity to design discrete conservation units in the case of A. nicaeensis is particularly valuable because (i) these units are under different pressures of urban sprawl and (ii) their loss would severely decrease the intraspecific diversity of A. nicaeensis.

Three haplotypes are found among the populations of A. nicaeensis, and this genetic structure is well structured and supported despite the very restricted distribution range of the species. Variation from cpDNA genome, maternally inherited in Angiosperm, is highly affected by demographic history [START_REF] Avise | Phylogeography: the history and formation of species[END_REF]. Studying the demographic history through phylogeography is crucial to identify geographic areas having played a role in terms of persistence or migration of populations [START_REF] Avise | Phylogeography: retrospect and prospect[END_REF][START_REF] Hewitt | The genetic legacy of the quaternary ice ages[END_REF]. CpDNA markers often present a lack of variation [START_REF] Schaal | Phylogeographic studies in plants: problems and prospects[END_REF], but several recent studies revealed that it is not always the case for Mediterranean plants (e.g. [START_REF] Molins | Molecular phylogeography of Thymus herba-barona (Lamiaceae): insight into the evolutionary history of the flora of the western Mediterranean islands[END_REF][START_REF] Migliore | Genetic diversity and structure of a Mediterranean endemic plant in Corsica (Mercurialis corsica, Euphorbiaceae)[END_REF]2012;[START_REF] Mayol | Patterns of genetic variability and habitat occupancy in Crepis triasii (Asteraceae) at different spatial scales: Insights on evolutionary processes leading to diversification in continental islands[END_REF][START_REF] Pouget | Phylogeography sheds light on the central-marginal hypothesis in a Mediterranean narrow endemic plant[END_REF]. The phylogeographic data is structured and supported. This information is relevant from the historical perspective and can help to preserve the lineages and biogeographical pattern of the species [START_REF] Ladle | Conservation biogeography[END_REF].

These three haplotypes differ in terms of spatial distribution and abundance (Fig. 1). The first conservation issue is that haplotype C is restricted to a small area neighbouring Nice city in the southern part of the distribution, the second issue is the presence of the three haplotypes in the fringe coastal zone. The coastal and low altitude populations are therefore crucial for the preservation of the evolutionary legacy of A. nicaeensis. In parallel, multivariate analysis of habitat as a proxy of ecological distinctiveness highlight the ecological originality of the coastal area: the ecological group C is narrowly restricted to the thermophilous matorral distributed from sea level to 200 m a.s.l., and its area of occupancy is very close to, or within major cities of the coast (Nice and Monaco-city). Two ecological groups are present near the coast, so, two-thirds of the ecological diversity of A. nicaeensis population is situated near the coastline. The coastal fringe exhibits most of the biogeographical originality of this plant as it encompasses geographically restricted ecological groups and unique evolutionary legacy.

Delineation of conservation units allows us to summarise threats

The use of both genetic and ecological information to delineate conservation units has rarely been done among plant species (e1.g. [START_REF] Lopez | A multi-faceted approach for assessing evolutionary significant conservation units in the endangered Omphalodes littoralis subsp. gallaecica (Boraginaceae)[END_REF]. This lack of studies is quite surprising because it denies the consideration of evolutionary legacy in conservation. Our results demonstrate that endemic species, even with extremely narrow ranges, can contain multiple conservation units, they are potentially genetically diverse and range along ecological gradients (e.g. [START_REF] Pouget | Phylogeography sheds light on the central-marginal hypothesis in a Mediterranean narrow endemic plant[END_REF].

Three main results are emerging from the delineation of the conservation units. One major result is the presence of different units separated by short distances (Fig. 3), a second one is that by combining the two types of information it emphasized the role of the coastal fringe from where two conservation units are very restricted; one around Nice (CB) and one in Monaco (BB). The third result is that gathering populations in conservation units allows us to assess information about threats and enables us to summarize them (Table 1).

AA and BA conservation units cover large areas with populations often outnumbering 100 individuals, but they contains the more isolated populations, these are mainly threatened by land-use changes (see [START_REF] Diadema | Apport de la phyloge ´ographie, de la dynamique et de la structure des populations pour la conservation des ve ´ge ´taux ende ´miques me ´diterrane[END_REF], i.e. land-abandonment due to the collapse of agro-sylvo-pastoral practices (see [START_REF] Tatoni | The impact of changes in land use on ecological patterns in Provence (Mediterranean France)[END_REF]San Roman Sanz et al. 2013;[START_REF] Otero | Land abandonment, landscape, and biodiversity: questioning the restorative character of the forest transition in the Mediterranean[END_REF] but not by urbanization. Indeed they present low urbanisation cover (Table 1). The AC conservation unit represents an intermediary situation which stretches along an important area at the margin of the major cities of Nice and Monaco. As a consequence AC units are under high disturbance pressure, easily seen in the field (trampling, paths, waste and ruderalisation) and its populations are close to areas of ongoing urban sprawl (and a urban cover 40%, Table 1). Most of the populations of AA, BA and AC conservation units are exempt from direct, major threats, but regular monitoring of AC unit populations is necessary in order to estimate their dynamic trends.

As human population continue to grow, urban areas will continue to expand [START_REF] Meyer | Human population growth and global land-use[END_REF]Me ´dail and Diadema 2006;[START_REF] Underwood | Threats and biodiversity in the mediterranean biome[END_REF]). The regression of the ''natural habitats'' due to urban sprawl constitutes a major threat to biodiversity worldwide [START_REF] Wilcove | Threats to imperiled quantifying species in the United States[END_REF][START_REF] Chapin | Consequences of changing biodiversity[END_REF][START_REF] Mcdonald | The implications of current and future urbanization for global protected areas and biodiversity conservation[END_REF]) and particularly in the Mediterranean area and the French Riviera [START_REF] Hoekstra | Confronting a biome crisis: global disparities of habitat loss and protection[END_REF]Me ´dail and Diadema 2006;[START_REF] Cuttelod | The mediterranean: a biodiversity hotspot under threat[END_REF]. Furthermore, the urban ecological footprint extends beyond city boundaries and drives environmental change at local to global scales [START_REF] Grimm | Global change and the ecology of cities global change and the ecology of cities[END_REF]. Indeed, urban areas are threatening ecosystems through both direct habitat conversion [START_REF] Clergeau | Biodiversite dans les paysages urbains: des concepts aux applications[END_REF][START_REF] Mckinney | Urbanization, biodiversity, and conservation[END_REF] and various indirect effects, e.g. land use change in the periphery of urban areas, fragmentation by linear infrastructures such as transport road among urban areas, waste generation, water pollution, disturbances associated with recreational activities [START_REF] Defries | Land use change around protected areas: management to balance human needs and ecological function[END_REF]). In some areas, such as the coast of the French Riviera, there is a strong probability of invasion by exotic species, also linked to human pressures. In our case, the threats caused by major plant invasions (Opuntia div. sp. or Freesia alba) jeopardize the future of these snowflake populations [START_REF] Diadema | Ecologie, distribution et morphologie compare ´es des nive ´oles de Nice (Acis nicaeensis) et de Fabre (Acis fabrei), Alliaceae ende ´miques des Alpes maritimes et de la Nesque (Vaucluse)[END_REF].

Thus, the CB unit (near Nice and cap Ferrat) and BB unit (within the Principaute ´de Monaco) are in imminent danger of extinction. They represent unique evolutionary lineages and ecological group that make them original, but at the same time they present a restricted range, a small number of populations and a low number of individuals per population (Table 1). All these combined features make them highly vulnerable. These units are threatened by a very high level of urbanization as in BB (Monaco) with more than 90% of urban cover, and this unit occurs within one of the most populated area of the world (16,000 inhabitant per km 2 ). It is not surprising that the population of Monaco St-Devote (MSD) became extinct in situ in 2006 (CBNMED/IMBE unpubl., [START_REF] Pouget | Comment pre ´server l'he ´ritage e ´volutif singulier des ve ´ge ´taux ende ´miques me ´diterrane ´ens[END_REF]. Moreover the areas of these units (CB and BB) are not included in a high level protection area; sensu IUCN such as a nature reserve or national park [START_REF] Dudley | Guidelines for applying protected area management categories[END_REF][START_REF] Lefebvre | Les espaces prote ´ge ´s franc ¸ais: une pluralite ´d'outils au service de la conservation de la biodiversite[END_REF]. Even if management priorities are currently directed toward in situ conservation, ex situ conservation should also be required, to restore or reinforce populations of BB and CB conservation units. Indeed, at least 10 years monitoring, (Diadema and Me ´dail, unpublished data) shows that the demographic regeneration is limited and a probable extinction of these units should not be excluded.

Conclusions

In most cases characterized by intensive human impacts, there is no choice but to accept the challenge of managing the future of the local biodiversity in extreme urbanization area. Overall, the results of this study highlight the vulnerability of units near the coast whose originality and restricted distribution call for rapid conservation management to avoid the loss of an evolutionary legacy. The presence of different units within a species has direct implication for the management of the species, for instance in suggesting a sampling strategy for an efficient ex situ conservation sampling (e.g. [START_REF] Ferguson | Mapping the geographical distribution of genetic variation in the genus Lens for the enhanced conservation of plant genetic diversity[END_REF], or in deciding which populations should be enhanced and/or preserved (e.g. [START_REF] Fischer | Genetic Allee effects on performance, plasticity and developmental stability in a clonal plant[END_REF]. Another interesting perspective in the case of severe habitat fragmentation and urbanization, but also strong floristic diversity, would be to target conservation action on a network of small areas. In the Mediterranean region, the Plant Micro-Reserves network (PMR) represents a valuable practical conservation framework supported by the European Union [START_REF] Laguna | The role of small reserves in plant conservation in a region of high diversity in eastern Spain[END_REF][START_REF] Lumbreras | The micro-reserves as a tool for conservation of threatened plants in Europe[END_REF][START_REF] Kadis | Plant micro-reserves: from theory to practice. Experiences gained from EU LIFE and other related projects[END_REF]. This approach, first tested in Spain and then extended to other Mediterranean areas (Cyprus, Crete, Croatia), seems promising, especially because it allows closer monitoring of the most endangered plant populations, implementing action, appropriate to the particular needs of species.

Fig. 1 a

 1 Fig. 1 a Location of the study site (red star); b network of the three cpDNA haplotypes detected for the 63 individuals of A. nicaeensis by combining the polymorphisms of five loci. Each haplotype is associated with a colour and a sphere size proportional to the number of individuals. Two mutations were observed between A and C, three between A and B; c distribution of the three cpDNA haplotypes among the A. nicaeensis populations. Populations without genetic information are indicated in white. (Color figure online)

Fig. 2

 2 Fig. 2 Ecological distinctiveness of A. nicaeensis populations based on habitat variables assessed by field releve ´s and climatic data. a Principal components analysis (PCA) of the 21 studied populations of A. nicaeensis, showing the importance of each variable within the scattergram delimited by the axis 1 and 2. b Structure and distribution of the three ecological groups obtained by a kmean clustering method applied to PCA coordinates of populations. c Geographical distribution of the three ecological groups among the A. nicaeensis populations (A in yellow B in blue and C in light brown). (Color figure online)

Fig. 3

 3 Fig. 3 Distribution of the five conservation units among the A. nicaeensis populations. Conservation units are designed according to their cpDNA haplotype (first letter, cf. Fig. 2) and then to their ecological group (second letter). (Color figure online)

Table 1

 1 Summary of population characteristics and main threats for each conservation unit of A. nicaeensis

	Range size	Populations		Threats
	Area sum	Nb	Nb indiv/	Isolation Urbanisation	Main threats
	(km 2 )	pop	pop		(%)
	AA 4.672	6	31-2000	8.45	28.3	Low to urbanisation, high to land-use
						change
	BA 4.266	5	85-414	9.59	21.0	Low to urbanisation, high to land-use
						change
	AC 3.366	5	148-460	8.08	40.3	At the margin of major city, high
						disturbance
	BB 0.252	2	10-50	8.37	92.5	High to urbanisation, high
						competition with exotic
	CB 1.947	3	2-94	7.93	48.3	High to urbanisation, high
						competition with exotic

Biodivers Conserv (2017) 26:293-307

Acknowledgements This research was part of the Ph.D. thesis of Marine Pouget supported by the Re ´gion Provence-Alpes-Co ˆte d'Azur (PACA) and performed at the IMBE/Aix-Marseille University. Molecular screening was realized by the molecular biology service of IMBE and we thanks Nathalie Duong for her help. For their great contribution in the record of Acis nicaeensis populations, we would like to thanks Julien Baret (BIODIV) and Je ´ro ˆme Volant (ECOMED). Field and molecular works were partly funded by the Prince's Government of the Principality of Monaco (Department of the Environment), the Prince Albert II of Monaco Foundation (project MONACOBIODIV) and the Provence-Alpes-Co ˆte d'Azur regional environmental management centre (DREAL-PACA).