Felix Antritter
email: felix.antritter@unibw.de

Jean Lévine
email: jean.levine@mines-paristech.fr

VI Flatness Characterization: Two Approaches

We survey two approaches to flatness necessary and sufficient conditions and compare them on examples.

Introduction

In this survey we consider underdetermined implicit systems of the form

F (x, ẋ) = 0 (1)
with x ∈ X, X being an inifnitely differentiable manifold of dimension n, whose tangent bundle is denoted by TX, and F : TX → R n-m regular in the sense that rk ∂F ∂ ẋ = nm in a suitable open dense subset of TX. Differential flatness, or more shortly, flatness was introduced in 1992 [START_REF] Martin | Contribution à l' Étude des Systèmes Différentiellement Plats[END_REF][START_REF] Fliess | Sur les systèmes non linéaires différentiellement plats[END_REF]. In the setting of implicit control systems it may be roughly described as follows: there exists a smooth mapping x = ϕ(y, ẏ, . . . , y (r) ) with y = (y1, . . . , ym) T of dimension m, r = (r1, . . . , rm) T ∈ N m , such that F (ϕ(y, ẏ, . . . , y (r) ), φ(y, ẏ, . . . , y (r+1) )) ≡ 0 [START_REF] Antritter | Towards a computer algebraic algorithm for flat output determination[END_REF] with ϕ invertible in the sense that there exists a locally defined smooth mapping ψ and a multi-index s such that y = ψ(x, ẋ, . . . , x (s) ). The vector y is called a flat output. This concept has inspired an important literature. See [START_REF] Fliess | Controlling nonlinear systems by flatness[END_REF][START_REF] Martin | Flat systems[END_REF][START_REF] Lévine | Analysis and Control of Nonlinear Systems: A Flatnessbased Approach[END_REF][START_REF] Rudolph | Flatness Based Control of Distributed Parameter Systems[END_REF][START_REF] Rudolph | Flatness Based Control of Distributed Parameter Systems: Examples and Computer Exercises from Various Technological Domains[END_REF][START_REF] Sira-Ramirez | Differentially Flat Systems[END_REF] for surveys on flatness and its applications. Various formalisms have been introduced: finite dimensional differential geometric approaches [START_REF] Charlet | Sufficient conditions for dynamic state feedback linearization[END_REF][START_REF] Franch | Flatness, Tangent Systems and Flat Outputs[END_REF][START_REF] Shadwick | Absolute equivalence and dynamic feedback linearization[END_REF], [START_REF] Sluis | A necessary condition for dynamic feedback linearization[END_REF][START_REF] Schlacher | Construction of flat outputs by reduction and elimination[END_REF], differential algebra and related approaches [START_REF] Fliess | Flatness and defect of nonlinear systems: introductory theory and examples[END_REF][START_REF] Aranda-Bricaire | A linear algebraic framework for dynamic feedback linearization[END_REF][START_REF] Jakubczyk | Invariants of dynamic feedback and free systems[END_REF], infinite dimensional differential geometry of jets and prolongations [START_REF] Fliess | A Lie-Bäcklund approach to equivalence and flatness of nonlinear systems[END_REF][START_REF] Van Nieuwstadt | Differential flatness and absolute equivalence of nonlinear control systems[END_REF][START_REF] Lévine | Analysis and Control of Nonlinear Systems: A Flatnessbased Approach[END_REF][START_REF] Chetverikov | New flatness conditions for control systems[END_REF][START_REF] Chetverikov | Flatness conditions for control systems[END_REF][START_REF] Pomet | A differential geometric setting for dynamic equivalence and dynamic linearization[END_REF], [START_REF] Da Silva | Relative flatness and flatness of implicit systems[END_REF][START_REF] Rathinam | Configuration flatness of Lagrangian systems underactuated by one control[END_REF], which is adopted here. The interested reader may refer to [START_REF] Anderson | Lie-Bäcklund Transformations in Applications[END_REF][START_REF] Fliess | A Lie-Bäcklund approach to equivalence and flatness of nonlinear systems[END_REF][START_REF] Krasil'shchik | Geometry of Jet Spaces and Nonlinear Partial Differential Equations[END_REF], [START_REF] Lévine | Analysis and Control of Nonlinear Systems: A Flatnessbased Approach[END_REF][START_REF] Pomet | A differential geometric setting for dynamic equivalence and dynamic linearization[END_REF][START_REF] Zharinov | Geometrical Aspect of Partial Differential Equations[END_REF] for more details.

The first part of the paper recalls the mathematical setting. In Section 3 the approch introduced in [START_REF] Lévine | Analysis and Control of Nonlinear Systems: A Flatnessbased Approach[END_REF][START_REF] Antritter | Towards a computer algebraic algorithm for flat output determination[END_REF] for the characterization of differentially flat systems is recalled. Then, in Section 4, we introduce a novel characterization using the so-called Generalized Euler-Lagrange Operator. We conclude the paper with examples.

Implicit control systems on manifolds of jets of infinite order

Given an infinitely differentiable manifold X of dimension n, we denote its tangent space at x ∈ X by TxX, and its tangent bundle by TX.

Let F be a meromorphic function from TX to R n-m . We consider an underdetermined implicit system of the form (1) regular in the sense that rk ∂F ∂ ẋ = nm in a suitable open dense subset of TX. Following [START_REF] Lévine | On necessary and sufficient conditions for differential flatness[END_REF][START_REF] Lévine | On necessary and sufficient conditions for differential flatness[END_REF], we consider the infinite dimensional manifold X defined by

X def = X × R n ∞ def = X × R n × R n × .
. ., made of an infinite (but countable) number of copies of R n , with the global infinite set of coordinates3 x = x, ẋ, . . . , x (k) , . . . , , endowed with the product topology. Recall that, in this topology, a function ϕ from X to R is continuous (resp. differentiable) if ϕ depends only on a finite (but otherwise arbitrary) number of variables and is continuous (resp. differentiable) with respect to these variables. C ∞ or analytic or meromorphic functions from X to R are then defined as in the usual finite dimensional case since they only depend on a finite number of variables. We endow X with the so-called trivial Cartan field ( [START_REF] Krasil'shchik | Geometry of Jet Spaces and Nonlinear Partial Differential Equations[END_REF][START_REF] Zharinov | Geometrical Aspect of Partial Differential Equations[END_REF])

τ X = n i=1 j≥0 x (j+1) i ∂ ∂x (j) i . We also denote by Lτ X γ = n i=1 j≥0 x (j+1) i ∂γ ∂x (j) i = dγ
dt the Lie derivative of a differentiable function γ along τ X and L k τ X γ its kth iterate. Since

d dt x (j) i def = ẋ(j) i = x (j+1) i
, the Cartan field acts on coordinates as a shift to the right. X is thus called manifold of jets of infinite order. A regular implicit control system is defined as a triple (X, τ X , F ) with X = X × R n ∞ , τ X its associated trivial Cartan field, and F meromorphic from TX to R n-m ) satisfying rk ∂F ∂ ẋ = nm in a suitable open subset of TX. We next consider the cotangent space T *

x X with dx (j)

i , i = 1, . . . , n, j ≥ 0 as basis, dual to the ∂ ∂x (j) i ´s. 1-forms on X are then defined in the usual way. The set of 1-forms is noted Λ 1 (X). We also denote by Λ p (X) the module of all the p-forms on X.

Flatness

We recall the following definitions and result [START_REF] Lévine | On necessary and sufficient conditions for differential flatness[END_REF][START_REF] Lévine | On necessary and sufficient conditions for differential flatness[END_REF][START_REF] Lévine | Analysis and Control of Nonlinear Systems: A Flatnessbased Approach[END_REF]:

Given two regular implicit control systems (X, τ X , F ), with X = X × R n ∞ , dim X = n and rk ∂F ∂ ẋ = n -m, and (Y, τ Y , G), with Y = Y × R p ∞ , dim Y = p, τ Y its trivial Cartan field, and rk ∂G ∂ ẏ = p-q, we set X0 = {x ∈ X|L k τ X F (x) = 0, ∀k ≥ 0} and Y0 = {y ∈ Y|L k τ Y G(y) = 0, ∀k ≥ 0}.
They are endowed with the topologies and differentiable structures induced by X and Y respectively. Definition 1 The control systems (X, τ X , F ) and (Y, τ Y , G) are said locally Lie-Bäcklund equivalent (or shortly L-B equivalent) in a neighbourhood X0 × Y0 of the pair (x0, y 0 ) ∈ X0 × Y0 if and only if (i) there exists a one-to-one meromorphic mapping Φ = (ϕ, φ, . . .) from Y0 to X0 satisfying Φ(y 0 ) = x0 and such that Φ * τ Y = τ X ; (ii) there exists Ψ one-to-one and meromorphic from X0 to Y0, with Ψ = (ψ, ψ, . . .), such that Ψ (x0) = y 0 and Ψ * τ X = τ Y . The mappings Φ and Ψ are called mutually inverse Lie-Bäcklund isomorphisms at (x0, y 0 ).

Definition 2 The implicit system (X, τ X , F ) is locally flat in a neigh- borhood of (x0, y 0 ) ∈ X0 × R m ∞ if and only if it is locally L-B equivalent around (x0, y 0 ) to the trivial implicit system (R m ∞ , τ R m ∞ , 0).
In this case, the mutually inverse L-B isomorphisms Φ and Ψ are called inverse trivializations.

Theorem 1 The system (X, τ X , F ) is locally flat at (x0, y 0 ) ∈ X0 × R m ∞
if and only if there exists a local meromorphic invertible mapping Φ from R m ∞ to X0, with meromorphic inverse, satisfying Φ(y 0 ) = x0, and such that 4Φ * dF = 0.

(

) 3
3 Necessary and Sufficient Conditions: Generalized Moving Frame Structure Equations

Algebraic characterization of the differential of a trivialization

Consider the following matrix, polynomial with respect to the differential operator d dt (we use indifferently d dt for Lτ X or Lτ R m ∞ , the context being unambiguous):

P (F ) = ∂F ∂x + ∂F ∂ ẋ d dt , P (ϕ) = j≥0 ∂ϕ ∂y (j) d j dt j (4)
with P (F ) (resp. P (ϕ)) of size (nm) × n (resp. n × m). Equation ( 3)) reads:

Φ * dF = P (F )P (ϕ)dy = 0. (5)
Clearly, the entries of the matrices in (4) are polynomials in the differential operator d dt with meromorphic coefficients from X to R. We denote by K the field of meromorphic functions from X to R and by K[

V M U = (∆, 0r,s-r) if r ≤ s, and ∆ 0r-s,s if s ≤ r (6)
with V ∈ Ur[ d dt ] and U ∈ Us[ d dt ] and ∆ diagonal (see e.g. [START_REF] Cohn | Free Rings and Their Relations[END_REF]). U and V are indeed non unique. We say that U ∈ R -Smith (M ) and

V ∈ L -Smith (M ). A matrix M ∈ Mr,s[ d dt ]
is said hyper-regular if and only if its Smith decomposition leads to ∆ = I. An interpretation of this property in terms of controllability in the sense of [START_REF] Fliess | A remark on Willems' trajectory characterization of linear controllability[END_REF], may be found in [START_REF] Lévine | On necessary and sufficient conditions for differential flatness[END_REF]. From now on, we assume that P (F ) is hyper-regular in a neighborhood of x0. In place of (5), we first solve the matrix equation:

P (F )Θ = 0 (7)
where Θ ∈ Mn,m[ d dt ] is not supposed to be of the form P (ϕ). It may be verified that matrices Θ ∈ Mn,m[ d dt ] satisfying [START_REF] Chetverikov | Flatness conditions for control systems[END_REF] have the structure

Θ = U 0n-m,m Im W (8)
with U ∈ R -Smith (P (F )) and W ∈ Um[ d dt ] arbitrary. Clearly Θ is itself hyper-regular and admits the Smith decomposition

QΘZ = QU 0n-m,m Im W Z = Q Û R = Im 0n-m,m (9)
with

Q ∈ Un[ d dt ], Z ∈ Um[ d dt ], R = W Z and Û = U 0n-m,m Im .

Integrability

We denote by ω the m-dimensional vector 1-form defined by

ω(x) =    ω1(x) . . . ωm(x)    = (Im, 0m,n-m) Q(x)dx X 0 (10)
with Q given by ( 9), the restriction to X0 meaning that x ∈ X0 satisfies L k τ X F = 0 for all k and that the dx

(k) j
are such that dL k τ X F = 0 in X0 for all k. Since Q is hyper-regular, the forms ω1, . . . , ωm are independent by construction.

Theorem 2 A necessary and sufficient condition for system (1) to be locally flat around (x0, y 0 ) is that there exist U ∈ R -Smith (P (F )), Q ∈ L -Smith Û , with Û given by ( 9) and a matrix

M ∈ Um[ d dt ] such that d(M τ ) = 0.
We denote by (Λ p (X)) m the space of m-dimensional vector p-forms on X, by (Λ(X)) m the space of m-dimensional vector forms of arbitrary degree on X, and by Lq ((Λ(X)) m ) = p≥1 L (Λ p (X)) m , Λ p+q (X)

m the space of linear operators from (Λ p (X)) m to Λ p+q (X) m for all p ≥ 1, where L (P, Q) denotes the set of linear mappings from a given space P to a given space Q.

In order to develop the expression d(µκ) for µ ∈ Lq ((Λ(X)) m ) and for all κ ∈ (Λ p (X)) m and all p ≥ 1, we define the operator d by:

d (µ) κ = d(µ κ) -(-1) q µ dκ. (11)
Note that [START_REF] Fliess | Sur les systèmes non linéaires différentiellement plats[END_REF] uniquely defines d (µ) as an element of Lq+1 ((Λ(X)) m ).

Theorem 3

The system (X, τ X , F ) is locally flat iff there locally exists µ ∈ L1 ((Λ(X)) m ), and a matrix

M ∈ Um[ d dt ] such that dω = µ ω, d (µ) = µ 2 , d (M ) = -M µ. ( 12
)
with the notation µ 2 = µµ and where ω is defined by [START_REF] Fliess | Controlling nonlinear systems by flatness[END_REF]. In addition, if [START_REF] Fliess | Flatness and defect of nonlinear systems: introductory theory and examples[END_REF] holds true, a flat output y is obtained by integration of dy = M ω.

Remark 1 Note that the two first conditions of ( 12) are comparable to conditions (A) and (B) of [START_REF] Chetverikov | New flatness conditions for control systems[END_REF][START_REF] Chetverikov | Flatness conditions for control systems[END_REF]. However, the last condition of ( 12) is different from condition (C) of [START_REF] Chetverikov | New flatness conditions for control systems[END_REF][START_REF] Chetverikov | Flatness conditions for control systems[END_REF] and is easier to check. Note also that conditions ( 12) may be seen as a generalization in the framework of manifolds of jets of infinite order of Cartan's well-known moving frame structure equations (see e.g. [START_REF] Chern | Lectures on Differential Geometry[END_REF]).

A Sequential Procedure

We start with P (F ) hyper-regular and compute the vector 1-form ω defined by [START_REF] Fliess | Controlling nonlinear systems by flatness[END_REF]. 1. We identify the operator µ such that dω = µω componentwise. It is proven in [START_REF] Lévine | Analysis and Control of Nonlinear Systems: A Flatnessbased Approach[END_REF] that such µ always exists. 2. Among the possible µ's, only those satisfying d (µ) = µ 2 are kept. It is shown in [START_REF] Lévine | Analysis and Control of Nonlinear Systems: A Flatnessbased Approach[END_REF] that such µ always exists. 3. We then identify M such that d (M ) = -M µ componentwise. 4. If, among such M 's, there is a unimodular one, the system is flat and a flat output is obtained by integration of dy = M ω. Otherwise the system is not flat. More details and examples may be found in [START_REF] Lévine | On necessary and sufficient conditions for differential flatness[END_REF][START_REF] Lévine | Analysis and Control of Nonlinear Systems: A Flatnessbased Approach[END_REF]. are independent by assumption, (13) yields, for every j = 1, . . . , m,

Necessary and Sufficient Conditions using the Generalized Euler-Lagrange Operator

                 ∂F ∂ ẋ ∂ϕ ∂y (r j ) j = 0 ∂F ∂x ∂ϕ ∂y (k) j + ∂F ∂ ẋ d dt ∂ϕ ∂y (k) j + ∂F ∂ ẋ ∂ϕ ∂y (k-1) j = 0, ∀k = 1, . . . , rj ∂F ∂x ∂ϕ ∂yj + ∂F ∂ ẋ d dt ∂ϕ ∂yj = 0 (14)
The Generalized Euler-Lagrange operator EF associated to F is defined by

EF = ∂F ∂x - d dt ∂F ∂ ẋ ( 15
)
In the case n-m = 1, it is well-known that the curves that extremize the cost function J = T 0 F (x, ẋ)dt are those satisfying the Euler-Lagrange equation EF = 0, which justifies our terminology. Using [START_REF] Jakubczyk | Invariants of dynamic feedback and free systems[END_REF] and elementary calculus, ( 14) yields: Theorem 4 A necessary and sufficient condition for (1) to be diffferentially flat is that there exist (r1, . . . , rm) with m i=1 ri + m ≥ n and a solution ϕ of the following triangular system of PDEs in an open dense subset of X

                     ∂F ∂ ẋ ∂ϕ ∂y (r j ) j = 0 ∂F ∂ ẋ ∂ϕ ∂y (l) j = r j -l-1 k=0 (-1) k+1 d k dt k EF ∂ϕ ∂y (l+k+1) j
, ∀l = 0, . . . , rj -

1 0 = r j k=0 (-1) k d k dt k EF ∂ϕ ∂y (k) j , (16)
satisfying dϕ1 ∧ . . . ∧ dϕn = 0.

Remark 2 If there exists a coordinate transformation ϕ that satisfies the conditions of Theorem 4 with given r1, . . . , rm, meaning that the system is flat, then gj = n i=1 ∂ϕ i ∂y (r j ) j ∂ ∂ ẋi , if non zero, defines a ruled direction [START_REF] Sluis | A necessary condition for dynamic feedback linearization[END_REF][START_REF] Rouchon | Necessary condition and genericity of dynamic feedback linearization[END_REF][START_REF] Lévine | Analysis and Control of Nonlinear Systems: A Flatnessbased Approach[END_REF].

Examples

An Academic Example: Generalized Moving Frame Approach

We consider the 3-dimensional system with 2 inputs:

ẋ1 = u1, ẋ2 = u2, ẋ3 = sin u1 u2 (17)
or, in implicit form:

F (x1, x2, x3, ẋ1, ẋ2, ẋ3) ẋ3 -sin ẋ1 ẋ2 = 0. (18)
It is readily seen that P (F ) = -cos( ẋ1 ẋ2 ) ẋ-1

2 d dt ẋ1 cos( ẋ1 ẋ2 ) ẋ-2 2 d dt d dt
and that V P (F )U = (1 0 0) with

V = 1, U =     ẋ1 a ẋ2 1 + ẋ1 a( ẋ2 ) 2 cos ẋ1 ẋ2 d dt ẋ1 a ẋ2 d dt 1 a 1 a ẋ2 cos ẋ1 ẋ2 d dt -1 a d dt 0 0 1     (19)
where

a = -1 ẋ2 cos ẋ1 ẋ2 ẍ1 ẋ2 -ẋ1 ẍ2 ( ẋ2 ) 2 . Then, Q Û R =   1 0 0 1 0 0   is com- puted with Q =    1 -ẋ1 ẋ2 0 0 0 1 -1 a ẋ2 cos ẋ1 ẋ2 d dt ẋ1 a( ẋ2 ) 2 cos ẋ1 ẋ2 d dt 1 a d dt    , R = 1 0 0 1 (20) So, (ω1 ω2) T = 1 0 0 0 0 1 Qdx = dx1 -ẋ1 ẋ2 dx2 dx3 T and dω = 1 √ 1-( ẋ3 ) 2 dx2 ∧ dx3 0 T . According to section 3.3, step 1, µ =   0 - ẋ3 (1-( ẋ3 ) 2 ) 3 2 dx2 ∧ d ẋ3 + ηd ẋ3 ∧ d dt 0 0   . (21)
Step 2 yields η = x 2 ẋ3

(1-ẋ3 )

y1 = x1 - ẋ1 ẋ2 x2 + σ2( ẋ3), y2 = x3 (22)
where σ2( ẋ3) is an arbitrary meromorphic function (a primitive of σ1). By inversion of [START_REF] Da Silva | Relative flatness and flatness of implicit systems[END_REF] we get

x1 = y1 -arcsin( ẏ2) 1 -( ẏ2) 2 ÿ2 ( ẏ1 -σ1( ẏ2)ÿ2) -σ2( ẏ2) x2 = - 1 -( ẏ2) 2 ÿ2 ( ẏ1 -σ1( ẏ2)ÿ2) (23) x3 = y2

Academic Example: Euler-Lagrange Operator

We consider once more the example [START_REF] Lévine | On necessary and sufficient conditions for differential flatness[END_REF]. We have

∂F ∂ ẋ = -ẋ-1 2 cos ẋ1 ẋ2 , ẋ1 ẋ-2 2 cos ẋ1 ẋ2
, 1 , EF = (η1, η2, 0) [START_REF] Rathinam | Configuration flatness of Lagrangian systems underactuated by one control[END_REF] with η1 = -ẍ2

+ ẋ1 (ẍ 1 ẋ2 -ẋ1 ẍ2 ) ẋ4 2
sin ẋ1 ẋ2 . The first two equations of ( 16), with r1 = r2 = 2, read

- 1 ẋ2 cos ẋ1 ẋ2 ∂ϕ1 ∂ ẏj - ẋ1 ẋ2 ∂ϕ2 ∂ ẏj + ∂ϕ3 ∂ ẏj = 0, j = 1, 2 (25)
If we assume that ∂ϕ 3 ∂ ÿj = ∂ϕ 3 ∂ ÿj = 0, j = 1, 2 and introduce the variable

ψ = ẋ1 ẋ2 (26)
with ∂ ∂ ÿ ψ = 0 we obtain from ( 25)

∂ϕ1 ∂ ÿj -ψ ∂ϕ2 ∂ ÿj = ∂ ∂ ÿj (ϕ1 -ψϕ2) = 0, j = 1, 2 Setting κ(y, ẏ) = ϕ1 -ψϕ2, we get κ = φ1 -ψ φ2 -ψϕ2 = -ψϕ2 (27)
Using the definition of κ and ( 27) we obtain:

ϕ1 = κ - κ√ 1 -φ3 φ3 arcsin( φ3), ϕ2 = - κ φ3 1 -φ3, ϕ3 = ϕ3(y) (28
) Choosing ϕ3 = y2, κ = y1, we arrive at the invertible transformation

x1 = ϕ1 = y1 - ẏ1 ÿ2 1 -ẏ2 2 arcsin( ẏ2), x2 = ϕ2 = - ẏ1 ÿ2 1 -ẏ2 2 ,
with x3 = ϕ3 = y2, which gives the same formula as [START_REF] Pomet | A differential geometric setting for dynamic equivalence and dynamic linearization[END_REF] with σ1 = σ2 = 0. Hence (y1, y2) is indeed a flat output, which implies that the remaining equations of ( 16) are satisfied.

An Example Proposed by P. Rouchon

Consider the implicit control system

F (x, ẋ) = ẋ1 ẋ3 -( ẋ2) 2 = 0. ( 29
)
We thus have ∂F ∂x = (0 0 0) , ∂F ∂ ẋ = ( ẋ3 -2 ẋ2 ẋ1) and

EF = ∂F ∂x - d dt ∂F ∂ ẋ = - d dt ∂F ∂ ẋ = (-ẍ3 2ẍ2 -ẍ1) .
The lowest possible choice of (r1, r2) in Theorem 4 is r1 = r2 = 1. However, there is no solution of ( 16) for these values, and we choose r1 = r2 = 2. The two first equations of ( 16) read

φ3 ∂ϕ1 ∂ ÿj -2 φ2 ∂ϕ2 ∂ ÿj + φ1 ∂ϕ3 ∂ ÿj = 0, j = 1, 2 (30)
We divide (30) by φ3 to obtain

∂ϕ1 ∂ ÿj -2ψ ∂ϕ2 ∂ ÿj + ψ 2 ∂ϕ3 ∂ ÿj = 0, j = 1, 2 (31)
where, taking account of the system equation ( 29),

ψ = φ2 φ3 = φ1 φ3 . (32)
If we assume that ψ doesn't depend on ÿ1 and ÿ2, equation ( 31) reads ∂ ∂ ÿj ϕ1 -2ψϕ2 + ψ 2 ϕ3 = 0, for j = 1, 2. In other words, there exists a function κ satisfying ∂κ ∂ ÿj = 0 for j = 1, 2, such that

ϕ1 -2ψϕ2 + ψ 2 ϕ3 = κ (33)
Differentiating the latter relation with respect to t, and taking into account the relation φ1 -2ψ φ2 + ψ 2 φ3 = 0 obtained from ( 29) and ( 32), we get

ϕ2 -ψϕ3 = - κ 2 ψ . (34)
We again differentiate the latter relation with respect to t to obtain where κ and ψ are arbitrary functions of y1, y2, ẏ1, ẏ2. Note that choosing κ = y1 and ψ = y2 yields, after inversion of (36) with (32):

y1 = x1 -2x2 ẋ2 ẋ3 + x3 ẋ1 ẋ3 , y2 = ẋ2 ẋ3 ,
which is similar to the solution obtained by F. Ollivier5 . Similarly, the solution of K. Schlacher and M. Schöberl [START_REF] Schlacher | Construction of flat outputs by reduction and elimination[END_REF] may be recovered by posing κ = y1 -y2 ẏ1 ẏ2 and ψ = ẏ1 2 ẏ2 which, again after inversion of (36) with [START_REF] Sluis | A necessary condition for dynamic feedback linearization[END_REF], yields:

y1 = x1 -x3 ẋ1 ẋ3 , y2 = x2 -x3 ẋ2 ẋ3 .

Conclusion

In this survey we presented two dual approaches to flatness necessary and sufficient conditions, one based on the integration of 1-forms and the second based on the integration of a set of PDEs involving a generalized Euler-Lagrange operator. Their complexity is compared on examples.

  Another way of analysing (3) consists in characterizing the change of coordinates corresponding to the mapping Φ in (3). More precisely[START_REF] Aranda-Bricaire | A linear algebraic framework for dynamic feedback linearization[END_REF]

3 2 +m12 d dt 0 1 which yields m12 = - x 2 √ 1 -

 32121 σ( ẋ3). For step 3 we set M = 1 ( ẋ3 ) 2 + σ1( ẋ3) with σ1 a primitive of σ. Thus, d(M ω) = 0 and setting (dy1 dy2) T = M ω, one obtains

  ψ φ3 = 0 from[START_REF] Sluis | A necessary condition for dynamic feedback linearization[END_REF]. Thus, solving the system (33)-(35), we immediately obtainϕ1 = κψ κ ψ + ψ 2 κ ψ -

From now on, x y, . . . stand for the sequences of jets of infinite order of x, y,. . .

Note that if Φ is a meromorphic mapping from Y to X, the (backward) image by Φ of a 1-form is defined in the same way as in the finite dimensional context.

personal communication

Acknowledgements

This work has been partially supported by a PROCOPE program of EGIDE "Algorithmique en Calcul Formel pour les Systèmes Différentiellement Plats", N. 20146UH and DAAD N. 50018800 "Implementierung notwendiger und hinreichender Kriterien für differentielle Flacheit mittels Computer Algebra".


Jérémy Malaizé
email: jeremy.malaize@ifp.fr

Jean Lévine
email: jean.levine@ensmp.fr

An Observer-Based Design for Cogging Forces Cancellation in Permanent Magnet Linear Motors

This paper adresses the high precision positioning issue of permanent magnet (PM) linear motors, in presence of spatially periodic forces, also known as cogging. Using an internal model representation of this perturbation, an observerbased controller only relying on position measurements is derived. The observation error is not autonomous, and the stability analysis of the resulting closed loop system is regarded as the stability of two interconnected systems. For the motor to quickly track a desired trajectory, while being robust to large magnitude cogging forces, a small-gain like theorem is derived and used to tune the gains of the control law in an explicit way. The experimental results obtained through this method are then showed and compared with those of a PID controller.

I. INTRODUCTION

This paper is concerned with motors used in high-end applications, such as wafer steppers central to the lithography process. These machines are expected to carry loads along point-to-point trajectories with stringent positioning accuracy requirements, usually ranging from a micron to a few dozens nanometers. Given this constraint, direct drive solutions are chosen to actuate these machines. They are actually capable of providing the sufficient precision levels and the required smoothness of motion by getting rid of undesirable phenomena such as hysteresis, backlash or mechanical play. In the following, we focus on the use of PM synchronous motors. They are preferred to DC motors for their longer life time, improved cleanliness of operation and better resistance to wear and tear.

Ironless motors feature two magnetic tracks, made up of permanent magnets, while ironcore motors feature only one magnetic track. The excess of ferromagnetic material used for this design allows to trap the magnetic field generated by this only track in the vicinity of the rotor windings. This costeffective design spares the use of expensive and eventually superfluous rare earths magnets and yields motors with better efficiency. However, this design generates perturbation forces, hereafter referred to as cogging that significantly affects their performances, in view of the desired accuracy requirements. Ironless motors, though much more expensive, turn out to be significantly less affected by cogging.

Cogging is due to the interaction of the magnetic field with the ferromagnetic material of the rotor. Given the periodic layouts of magnets and slots (see 1), cogging forces are shown to be spatially periodic perturbations, see [START_REF] Wang | Fringing in tubular permanentmagnet machines: Part I, magnetic field distribution, flux linkage, and thrust force[END_REF], [START_REF]Fringing in tubular permanent-magnet machines: Part II, cogging force and its minimization[END_REF] or [START_REF] Wang | Study of cogging torque in surfacemounted permanent magnet motors with energy method[END_REF] for in depth reviews. Roughly speaking, even if not fed with electrical currents, these motors have a tendency to stabilize at specific positions so as to minimize magnetic energy.

This paper is concerned with getting PM synchronous motors to track reference trajectories with stringent accuracy specifications, related to the aforementioned applications, in spite of the cogging forces. These trajectories can be of any kind, meaning this paper is not limited to constant velocity motion along which these spatially periodic forces would readily come down to temporally periodic perturbations. Position measurements are assumed to be available, while the actual speed of the motor is not measured. The spatial periods of the cogging are known.

Great efforts have been made to minimize this, though natural, limitation of the ironcore motors from a technological point of view. The underlying idea consists in using the free design parameters, like the shapes of the magnets, the spacing in between slots, or the length of the rotor to analytically or numerically find out which configuration actually minimizes cogging forces. All these techniques generally lead to very complex designs, such as skewed magnets layouts, which, in the end, do not manage to completely get rid of cogging forces (see [START_REF] Islam | Issues in reducing the cogging torque of mass-produced permanent-magnet brushless dc motor[END_REF]). The same conclusion applies to ironless motors, but, for the motors considered hereafter, the main spatially perdiodic perturbations are due to unavoidable and slowly time-varying electrical current offsets rather than cogging. For the sake of simplicity, we shall also consider these forces as cogging forces.

Driving a production model motor with a PID controller yields satisfactory results for some applications, but, when stringent accuracy specifications are at stake, a dedicated controller is required to get rid of the oscillations around the desired trajectory induced by the residual cogging forces. It might be tempting to cancel these forces from a preliminary analytical or numerical analysis together with a feedforward compensation scheme, but modeling these perturbations turn out to be tough, see [START_REF] Islam | Issues in reducing the cogging torque of mass-produced permanent-magnet brushless dc motor[END_REF].

The previous analysis makes lean towards an online cogging compensation scheme. For repetitive tasks, the cogging forces affecting the motion during one run may be identified, and, directly compensated for during the next run. This iterative method, known as learning feedforward [START_REF] Otten | Linear motor motion control using a learning feedforward controller[END_REF], can obviously cope with perturbations of arbitrary shape but takes several tries to yield satisfactory results.

Other techniques are dedicated to spatially periodic perturbations and rely on a spatial Fourier series expansion of cogging forces. Adaptive controllers may be designed to estimate both the magnitudes and phases of each of the Fourier series expansion components and achieve position tracking thanks to cogging compensation, see [START_REF] Ahn | State-periodic adaptive compensation of cogging and coulomb friction in permanent-magnet linear motors[END_REF] and [START_REF] Tan | Robust adaptive numerical compensation for friction anf force ripple in permanent-magnet linear motor[END_REF] for instance. Observer-based controllers can also be derived using an internal model by considering an extended system made up of the motor dynamics and the perturbations. In [START_REF] De | Adaptive eccentricity compensation[END_REF], this approach is implemented for mechanical pure sine perturbations but the extension to higher order harmonics does not seem straightforward.

The methods reported so far assume the velocity available, either by direct measurements or numerical differentiation of the position. For high precision positioning applications, assuming the position directly measured is rather sensible and common, but numerical computation of the velocity may generate undesirable noise affecting the overall system performances. In this paper, we propose an observer-based controller only relying on position measurements. Our solution steers the tracking error to zero by cancelling cogging forces defined by an arbitrary number of spatial periods, as well as compensating for Coulomb friction.

This paper is organized as follows. We first model the dynamics of the motors considered so far and give an analytical definition of the cogging forces (section II). An observer structure estimating the states of both the mechanics and the perturbation is proposed together with a controller fed with these estimated values to allow exact reference trajectory tracking despite cogging forces (section III). The corresponding gains have to be tuned to achieve perturbation rejection and for the motor to quickly rally the desired trajectory. The stability analysis (section IV) is mainly based on a small gain result, pointing out the robustness of the proposed observer-based controller with respect to the magnitude of the cogging forces. The performances of the method are then illustrated through experiments carried out on both ironless and ironcore motors (section V).

II. PM SYNCHRONOUS MOTORS

A. Modeling

Figure 1 depicts a two-phase ironcore motor (the modeling turns out to be the same for ironless motors and three-phase windings motors), where permanent magnets, periodically spread out along the stator, create a periodic magnetic field, with spatial period P , trapped in the rotor out of iron. The rotor carries windings and is free to move along the x-axis. The converse design with fixed windings and moving magnetic track might also be conceivable. It is rather classical to assume that a current controller stabilizes the dynamics of the electrical currents. They eventually turn out to be fast in comparison with the mechanical behavior of the motor. When using the field-oriented method to drive the currents while neglecting their dynamics, a force propels the motor ahead, see [START_REF] Malaizé | Active estimation of the initial phase for brushless synchronous motors[END_REF]. The remaining forces acting on the motor are due to cogging and friction. The dynamical behaviour of the motor with speed ν and measured position y = x eventually reads:

ẋ ν = A x ν + B u + Λ(x) -f sgn (ν) y = C x ν (1)
where the notations are given by:

A = 0 1 0 -µ , B = 0 1 , C = 1 0 T .
The parameters µ and f respectively model viscous and dry friction, u is the controlled thrust and Λ(x) models the action of cogging forces on the acceleration of the motor. Though friction is modeled by a differential inclusion, in our case, a simple constant parameter suffices to bring the desired performances to the derived observer-based controller. Moreover, for the considered applications, friction is definitely not the most significant limitation. However, the extension of our work to differential inclusions could be the scope of further research activities. From now on, the modeling (1) , together with assuming friction to be a constant unknown parameter, will be used for either ironcore or ironless motors.

B. Cogging forces

According to [START_REF] Wang | Fringing in tubular permanentmagnet machines: Part I, magnetic field distribution, flux linkage, and thrust force[END_REF], [START_REF]Fringing in tubular permanent-magnet machines: Part II, cogging force and its minimization[END_REF], the finite length of the rotor as well as the windings slots (see figure 1) are some of the design parameters inducing cogging forces. Each of them yields spatially periodic perturbation forces with a specifc period, which in the case of the finite length of the rotor turns out to be P 2 . Current offsets generate cogging-like forces with the same spectral components as the back-emf.

Truncated Fourier series expansion is a rather well-suited tool to model the x-periodic function Λ(x), made up of N sine functions, as given by:

Λ(x) = λ 0 + N n=1 λ n sin 2π P n x + ϕ n , (2)
where the a priori unknown λ n 's and ϕ n 's bear magnitude and phase information of the n-th spectral component with known spatial period P n . This approach copes with cogging forces made up of several fundamental spatial periods, and also with friction through λ 0 .

Adaptive methods mentioned in the introduction mainly cope with the online estimation of the λ n 's and ϕ n 's in order to compensate for Λ(x). In this paper, an observer-based method is proposed, and, to this end, Λ(x) is expressed as the output of a dynamical system. For any n, 1 ≤ n ≤ N , let's note:

ζ n,1 = λ n sin 2π P n x + ϕ n , ζ n,2 = λ n cos 2π P n x + ϕ n and observe: ζn,1 ζn,2 = ν 2π P n 0 1 -1 0 ζ n,1 ζ n,2 := ν M n ζ n,1 ζ n,2 (3)
Moreover, let ζ 0 = λ 0 , with ζ0 = 0, and stacking it up together with the variables ζ n,1 and ζ n,2 in the vector ζ ∈ R 2 N +1 , Λ(x) turns out to be the output of a dynamical system, obtained using an internal model representation:

ζ = ν         0 0 . . . . . . 0 0 M 1 0 . . . 0 . . . 0 . . . . . . . . . . . . . . . . . . . . . 0 0 0 . . . 0 M N         ζ := νM N ζ Λ(x) = 1 1 0 . . . 1 0 ζ := C N ζ (4)
where the matrices M n 's are defined by (3) and 0 stands for null matrices with appropriate dimensions.

III. OBSERVER AND CONTROLLER DESIGN

A. Observer structure

Let χ = (x ν ζ)
T be the whole state of the plant, and, combining (1) together with (4):

χ = A(ν)χ + B u -f sign(ν) y = Cχ (5)
with notations as follows:

A(ν) =     A BC N 0 νM N     , B =     B 0     , C =     C 0     T ,
with 0 standing for null matrices with relevant dimensions. Let's note that, obviously, at standstill, the complete state ζ is not observable. On top of that, as a perturbation, ζ is not controllable but remains bounded according to its initial conditions.

The proposed observer architecture based on the only measurements of the position x is the following:

˙ χ = A(ν * ) χ + Bu + KC (χ -χ) (6)
To some extent, the proposed expression ( 6) is similar to a Luenberger observer in that it is clearly a copy of the dynamics (5) together with a correction term through gain K. However, note that dry friction is not allowed for in [START_REF] Ahn | State-periodic adaptive compensation of cogging and coulomb friction in permanent-magnet linear motors[END_REF], but as a piecewise constant phenomenon, this term is merely estimated along with λ 0 . Cautions have to be taken to properly reset the observer whenever the sign of the desired velocity ν * changes. From now on, for convenience, f is ommited in ( 5), but will still be taken into consideration for the simulation results.

The second point of interest is the use of the reference velocity ν * in [START_REF] Ahn | State-periodic adaptive compensation of cogging and coulomb friction in permanent-magnet linear motors[END_REF]. The observer scheme (6) can be regarded as a kind of linearization of the perturbation, lower block in [START_REF] Otten | Linear motor motion control using a learning feedforward controller[END_REF], around the reference trajectory x * .

B. Closed loop dynamics

Our goal is to use this in parallel with a controller to achieve precise positioning. To this end, observation and positioning errors are denoted by:

x = x -x, ν = ν -ν, ζ = ζ -ζ, ε x = x -x * , ε ν = ν -ν *
and collected in the vectors

χ o = x ν ζ T and χ c = (ε x ε ν ) T .
The following equation concerning the observation error is readily derived from (5) subtracted to [START_REF] Ahn | State-periodic adaptive compensation of cogging and coulomb friction in permanent-magnet linear motors[END_REF]:

χo = (A(ν * ) -KC) χ o + B o ε ν M N ζ, (7)
where the matrix B o reads:

B o =   0 0 I 2N +1  
with 0 null matrices of relevant dimensions, and I 2N +1 the identity matrix of dimenion 2N + 1. Let us move on to the controller design from the previous estimated values. Provided the vector ζ converges to the actual vector ζ, it is sensible to design the input u to compensate for the perturbation Λ = C N ζ. Additional state feedback through the gain matrix L = (L x L ν ) and feedforward terms are added to achieve exact trajectory tracking so that the controller eventually computes u according to:

u = -L x -x * ν -ν * -C N ζ + ν * + µ ν * (8)
In the end, the full state space representation of (5) driven by the observer-based controller made up of ( 7) and ( 8) is given by:

χc = (A -BL) χ c -B C o χ o χo = (A (ν * ) -KC) χ o + ε ν B o M N ζ , (9)
whith C o = (0 L ν C N ). In (9), the evolution of the uncontrollable part ζ of (5) has been ommitted, as we are mainly concerned by the stability of the origin of ( 9) that is equivalent to both observation and positioning errors converging to zero. Looking at [START_REF] Tan | Robust adaptive numerical compensation for friction anf force ripple in permanent-magnet linear motor[END_REF], unlike classical Luenberger observers, the observation error is not completely autonomous, which results from the linearization of Λ(x) around the reference trajectory. Fortunately, as the origin is still an equilibrium point of ( 9) despite the non-controllable part ζ, it might not be hopeless to look for an appropriate tuning of L and K guaranteeing global asymptotic stability of the origin of the closed loop [START_REF] Malaizé | Active estimation of the initial phase for brushless synchronous motors[END_REF].

IV. STABILITY ANALYSIS

In [START_REF] Malaizé | Active estimation of the initial phase for brushless synchronous motors[END_REF], given the non-autonomous observation error, the observation K and positioning L gains tuning may not be decoupled. We shall assume that L is given by some a priori knowledge of the plant and K is derived using some results on the stability of interconnected systems. In doing so, we shall derive an interesting result concerning the robustness of the proposed observer structure versus the magnitude of the perturbation.

A. Small-gain theorem for cogging cancellation

Let us first have a fresh look at [START_REF] Malaizé | Active estimation of the initial phase for brushless synchronous motors[END_REF], and note that this equation can be viewed as the input-output interconnection of:

χc = (A -BL) χ c + Bu c z c = M N ζC c χ c , (10)
with C c = (0 1) and

χo = (A (ν * ) -KC) χ o + B o u o z o = C o χ o , (11)
with u c = -z o and u o = z c . The following small-gain theorem, based on dissipativity, gives sufficient conditions for the observer-based cogging cancellation scheme based on the perturbation linearization to be globally asymptotically stable.

Theorem 1: [Small-gain theorem] Suppose the following properties are fulfilled:

(i) There exist a function V c : R 2 → R, such that V c (0) = 0 and ∀x ∈ R 2 -{0}, V c (x) > 0, a K ∞ function α c and a scalar γ c > 0 such that ∀ (χ c , u c ):

Vc (χ c ) ≤ γ 2 c u T c u c -χ T c C T c C c χ c -α c ( χ c ) (ii) There exist a function V o : R 2N +3 → R, such that V o (0) = 0 and ∀x ∈ R 2N +3 -{0}, V o (x) > 0, a K ∞ function α o and a scalar γ o > 0 such that ∀ (χ o , u o ): Vo (χ o ) ≤ γ 2 o u T o u o -χ T o C T o C o χ o -α o ( χ o ) then the closed loop system is globally asymptotically stable if 2πγ c γ o N n=1 λ n P n 2 < 1, ( 12
)
where the λ n are the coefficients of the series expansion [START_REF]Fringing in tubular permanent-magnet machines: Part II, cogging force and its minimization[END_REF]. Proof: First, denote:

γ N = N n=1 λ n P n 2 . ( 13
)
Let find a scalar a > 0 for V = V c + a V o to be a Lyapunov function of the whole system ( 9) with u c = -z o and u o = z c . Basic requirements concerning V are readily checked, moreover, note that:

z T c z c = 4π 2 γ 2 N × χ T c C T c C c χ c , and compute V (χ c , χ o ): V (χ c , χ o ) ≤ (γ 2 c -a)z T o z o + (aγ 2 o - 1 (2πγ N ) 2 )z T c z c -α c ( χ c ) -aα o ( χ o ) Suppose γ 2 c < a < 1 4π 2 γ 2 o γ 2 N then Vc (χ c ) + a Vo (χ o ) ≤ -α c ( χ c ) -aα o ( χ o )
and global asymptotic stability is showed provided the sufficient condition

2πγ c γ o γ N < 1 is met.
Recalling that the observer is based on the linearization of the perturbation, this theorem states the intrinsic limitation of our approach. It also shows we are not completely helpless before the perturbation, for the gains L and K play a role in the values taken by γ c and γ o .

B. Gain tuning issues

As stated by the following theorem, quadratic Lyapunov functions turn out to be an appropriate tool to translate the conditions (i) and (ii) of theorem 1 into numerically tractable problems where the roles of L and K are clearly identified.

Theorem 2: [Quadratic stabilization] Let α be a given positive real numbers, and suppose there exist:

• two symmetric positive definite matrices P c ∈ R 2×2 and

P o ∈ R (2N +3)×(2N +3) ,
• some gain matrices L and K,

• two positive scalar γ c and γ o that satisfy the following relations where the dependancy of A on ν * is ommitted:

(A-BL) T P c + P c (A-BL) + C T c C c + 2αP c P c B B T P c -γ 2 c ≤0 (14) (A-KC) T P o + P o (A-KC) + C T o C o + 2αP o P o B o B T o P o -γ 2 o I ≤0
(15) then the origin of ( 9) is globally exponentially stable with decay-rate α if the spatially periodic perturbation satisfies 2πγ c γ o γ N < 1, with γ n defined by (13).

Proof:

Let V c (χ c ) = χ T c P c χ c and V o (χ o ) = χ T o P o χ o ,
and differentiating these functions along the trajectories of ( 10) and (11) respectively, together with using the Schurr's complement of ( 14) and (15) yields:

d dt χ T c P c χ c ≤ γ 2 c u T c u c -χ T c C T c C c χ c -2αχ c P c χ c d dt χ T o P o χ o ≤ γ 2 o u T o u o -χ T o C T o C o χ o -2αχ o P o χ o Assuming the small gain condition 2πγ c γ o γ N < 1 is fulfilled, there exists γ 2 c < a < 1 4π 2 γ 2 o γ 2 N so that : d dt χ T c P c χ c + aχ T o P o χ o ≤ -2α χ T c P c χ c + aχ T o P o χ o ,
which ends proving global exponential stability with decayrate α, see for example [START_REF] Boyd | Linear Matrix Inequalities in System and Control Theory[END_REF] for further details.

C. Controller gains tuning

The recommended default PID settings for the motors yield closed loop eigenvalues that may be computed with the modeling [START_REF] Wang | Fringing in tubular permanentmagnet machines: Part I, magnetic field distribution, flux linkage, and thrust force[END_REF]. When omitting the cogging compensation feature of equation ( 8), our controller is an ordinary LTI controller with gain L. We suggest to tune it to have eigenvalues with identical real part as with this PID. Without cogging compensation, we suggest to tune L to yield poles with real parts as negative as with the PID controller. If we rewrite L = ω 2 2ξωµ , it amounts to setting the poles of the transfer H(s):

H(s) = s s 2 + 2ξωs + ω 2 .
H(s) turns out ot be the transfer from u c to C c χ c in [START_REF] Boyd | Linear Matrix Inequalities in System and Control Theory[END_REF]. For L determined this way, we may also compute the smallest γ c fulfilling (14). It turns out to be the H ∞ norm of the transfer H(sα) given by:

H(s -α) = C c ((s -α)I -A + BL) -1 B.
This value is finite, and denoted by γ * c provided the poles of H(s) are located on the left of -α:

γ * c = H(s -α) ∞ .

D. Observer gains tuning

For the previous controller gain L, we propose to compute K in order to minimize the value of γ o for which the condition (15) is met. This problem is not straighforward, especially because A(ν * (t)) is a time-varying system. However, since the velocity along the reference trajectory is bounded, ∀t, ν ≤ ν * (t) ≤ ν, the system (11) is a polytopic system, and we are facing an optimization problem under Linear Matrix Inequalities constraints.

Theorem 3: The minimum value of γ o for which there exist P = P T > 0 and K such that (15) is fulfilled is γ * o , obtained by solving the following optimization problem:

(γ * o ) 2 , P * , Q * = min γ 2 o , P , Q γ 2 o
subject to:

∃ P = P T > 0, P ∈ R (2N +3)×(2N +3) ∃ Q ∈ R 2N +3 A T P + P A -C T Q T -QC + C T o C o + 2αP P B o B T o P -γ 2 o I ≤ 0 A T P + P A -C T Q T -QC + C T o C o + 2αP P B o B T o P -γ 2 o I ≤ 0
where A = A(ν) and A = A(ν) and the corresponding observer gain matrix K * = (P * ) -1 Q * . Proof: Since A(ν * ) is a polytopic system, (15) only has to be enforced on the vertices A and A of A(ν * ). See [START_REF] Boyd | Linear Matrix Inequalities in System and Control Theory[END_REF] for more details.

E. Conclusion

The observer-based controller made up of ( 6) and ( 8) is globally exponentially stable with decay-rate to zero α for perturbations (2) fulfilling the relation:

γ N < γ * N = 1 2π H (s -α) ∞ γ * o , (16)
with γ N given by ( 13), γ * o in the theorem 3 and H(s) in subsection IV-C. This is the largest perturbation for which the stability of the method is proved, and may be regarded as a robustness margin. Computing the numerical value of γ * N allows to estimate the largest perturbations this controller may withstand by computing the maximum admissible value λ * of the λ n . This shows the proposed design can cope with perturbations hardly conceivable from a physical point of view, as illustrated in table I.

V. EXPERIMENTAL RESULTS

We consider two commercial off-the-shelve PM linear motors, both an ironcore and an ironless motor. They are intended to carry loads, for point to point motion, with respective reference speed trajectories given on figure 2. In order to determine the spatial periods P n of the equation (2), we drive each motor with its recommended default PID controller and perform an FFT transform of the recorded tracking error. These results are plotted in blue dotted line on the figures 4 and 3.

At this point, it is noteworthy that the spectral content of the cogging forces is much more intricate in the case of the ironcore motor than with the ironless one. The latter clearly features two spatial periods located at 42mm and 21mm, while the ironcore motor spectrum is much more chaotic. Nevertheless, the spatial periods 24mm, 16mm and 12mm are used to tune the observer. In other words, the assumption about the spatially periodic nature of the cogging forces for the ironcore motor is not as well fulfilled as for the ironless one.

We still implement our observer-based controller for both the motors and get the results presented on figure 4 andtable I. The cogging forces are completely filtered out (see the figure 3 for a quantitative illustration), after a short transient for both motors. As expected, this design is efficient during the constant velocity phases as well as the during acceleration and deceleration phases.

Let's also mention that λ * (the largest admissible λ n for which stability is guaranteed) is given in table I. The nominal acceleration for the reference speed profiles of figure 2 are 4000 mm/s 2 (resp. 2500 mm/s 2 ) for the ironcore (resp. ironless) motor. This means that, if the cogging forces acting on these motors ever had the same magnitude as the nominal thrust, the stability of this controller would not be jeopardized. In view of the efforts made by motors manufacturers to reduce them, cogging forces turn out to be perturbations, whose magnitude actually do not reach such values. We believe our approach allows to cope with most practical use conditions.

In the end, as illustrated in table I, for the ironless motor, the tracking error is 10 times smaller than with a PID, and 20 times smaller for the ironcore motor. One may also note that the ironcore motor driven by the observer-based controller reaches the intrinsic performances of the ironless motor with a PID. In other words, our solution makes ironcore motor perform as well as ironless motors driven by a casual control scheme. On the top of that, ironless motors have their performances clearly improved.

VI. CONCLUSION

A new controller scheme dedicated to cogging forces cancellation has been proposed. This approach only relies on position measurements, is based on physical modeling of these perturbations. It estimates these perturbations online to then get rid of them. As illustrated through the proposed experimental data, the desired trajectory can be of any kind, and, after a short transient, the motor reaches the reference trajectory. This cogging dedicated controller significantly removes unsatisfactory oscillations around the reference trajectory, which makes the proposed controller scheme definitely suited for high accuracy applications, such as in the semiconductor industry. Ironcore motors may thus achieve accuracy usually met with ironless motors, themselves reaching upstream performances.

Fig. 1 .

 1 Fig. 1. Two-phase ironcore linear motor.

Fig. 2 .

 2 Fig. 2. Reference velocity profiles in mm/s.

Fig. 3 .

 3 Fig. 3. FFT of the tracking error wrt. spatial perdiods in mm.

Fig. 4 .

 4 Fig. 4. Experimental tracking error in µm, with a PID controller (--) and the observer-based controller (-).


Ann Zeuner
email: a.zeuner@iss.it

Federica Francescangeli

Michele Signore

Mary Anna Venneri

Francesca Pedini

Concetta Conticello

Alfredo Pagliuca

Nadia Felli

Ruggero De Maria

The Notch2-Jagged1 interaction mediates Stem Cell Factor signaling in erythropoiesis

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

Introduction

Stem Cell Factor (SCF) is a cytokine produced by bone marrow stromal cells that stimulates the expansion of hematopoietic stem and progenitor cells.(1) SCF is specifically essential for erythroid cell development, as mice defective for SCF or its receptor c-kit die before or shortly after birth due to severe macrocytic anemia. [START_REF] Zsebo | Stem cell factor is encoded at the Sl locus of the mouse and is the ligand for the c-kit tyrosine kinase receptor[END_REF][START_REF] Chabot | The proto-oncogene c-kit encoding a transmembrane tyrosine kinase receptor maps to the mouse W locus[END_REF][START_REF] Huang | The hematopoietic growth factor KL is encoded by the Sl locus and is the ligand of the c-kit receptor, the gene product of the W locus[END_REF][START_REF] Geissler | The dominant-white spotting (W) locus of the mouse encodes the c-kit proto-oncogene[END_REF] We and others have shown that SCF promotes erythroblast expansion, retards differentiation and protects erythroid precursors from apoptosis.(5-7) Among the effectors activated by SCF in differentiating erythroid cells, p38, ERK and Bcl-2/Bcl-XL have been implicated in stimulating erythroblast survival and proliferation. [START_REF] Zeuner | Stem cell factor protects erythroid precursor cells from chemotherapeutic agents via up-regulation of BCL-2 family proteins[END_REF][START_REF] Kapur | Role of p38 and ERK MAP kinase in proliferation of erythroid progenitors in response to stimulation by soluble and membrane isoforms of stem cell factor[END_REF] However, the mechanisms through which SCF modulates erythroid differentiation remain mostly unclear.

Notch proteins are a family of highly conserved transmembrane receptors that regulate cell fate decisions, proliferation, differentiation and survival. Notch receptors are synthesized as 300 kDa precursors that are cleaved by furin-like convertases in the trans-Golgi compartment. The resulting N-terminal and C-terminal fragments are assembled into mature heterodimers and exported to the cell surface, where Notch signal transduction is initiated by ligand binding and endocytosis.

Subsequent cleavage by ADAM10/TACE/Kuz/SUP-17 and by the γ-secretase complex lead to the release of the Notch intracellular signal-transducing fragment named Notch IntraCellular Domain (NICD). NICD then translocates to the nucleus where it displaces corepressors associated with the DNA-binding transcription factor CSL (CBF-1/SuH/Lag-1) and forms a transcriptionally active complex with CSL, Mastermind, and coactivators leading to activation of Notch target genes.(9) In the hematopoietic system, Notch family members are expressed on both human and murine hematopoietic stem cells (HSC), where they promote cell expansion, self-renewal and immortalization. [START_REF] Ohishi | Notch signalling in hematopoiesis[END_REF] Although canonical Notch signaling has been shown to be dispensable for HSC maintenance, [START_REF] Maillard | Canonical notch signaling is dispensable for the maintenance of adult hematopoietic stem cells[END_REF] the Notch family still plays a complex role in the hematopoietic system by modulating cell proliferation, survival and differentiation into multiple lineages. The clearest effects of the Notch system can be observed during lymphopoiesis, where Notch activation favours the CD8 and αβ cell fate decisions. [START_REF] Radtke | Deficient T cell fate specification in mice with an induced inactivation of Notch1[END_REF] In the myeloid system, Notch signalling has been reported to modulate differentiation in vitro and in vivo, with cell context-dependent effects that vary between different experimental models and conditions. [START_REF] Schwanbeck | Notch signaling in embryonic and adult myelopoiesis[END_REF] Several in vitro studies have associated Notch signaling with an inhibition of erythroid, granulocytic or megakaryocytic differentiation. [START_REF] Tachikawa | Pivotal role of Notch signaling in regulation of erythroid maturation and proliferation[END_REF][START_REF] Ishiko | Notch signals inhibit the development of erythroid/megakaryocytic cells by suppressing GATA-1 activity through the induction of HES1[END_REF][START_REF] Bigas | Notch1 and Notch2 inhibit myeloid differentiation in response to different cytokines[END_REF] However, Notch has also been reported to increase erythroid maturation or proliferation, to induce monocyte death or to promote megakaryocytic maturation. [START_REF] Walker | The Notch/Jagged pathway inhibits proliferation of human hematopoietic progenitors in vitro[END_REF][START_REF] Ohishi | Monocytes express high amounts of Notch and undergo cytokine specific apoptosis following interaction with the Notch ligand, Delta-1[END_REF][START_REF] Mercher | Notch signaling specifies megakaryocyte development from hematopoietic stem cells[END_REF][START_REF] Henning | mNotch1 signaling and erythropoietin cooperate in erythroid differentiation of multipotent progenitor cells and upregulate beta-globin[END_REF] In vivo studies on Notch function in the hematopoietic system have been precluded for a long time due to the embryonic lethal phenotype of mice homozigously deficient in Notch1 or Notch2. [START_REF] Huppert | Embryonic lethality in mice homozygous for a processing-deficient allele of Notch1[END_REF][START_REF] Hamada | Mutation in ankyrin repeats of the mouse Notch2 gene induces early embryonic lethality[END_REF] Later, an inducible Notch1 knockout mouse revealed an essential role for Notch1 in T cell lineage induction while leaving all the other hematopoietic lineages unaltered. [START_REF] Radtke | Deficient T cell fate specification in mice with an induced inactivation of Notch1[END_REF] In apparent contradiction, mice deficient for the Notch mediator RBP-Jk were found to generate excessive numbers of erythroid cells in the yolk sac, suggesting that the Notch pathway regulates erythroid homeostasis through elimination of developing erythroblasts.( 23) More recently, multilineage effects of Notch signaling on myeloid homeostasis were observed in fucosylation-deficient mice, which display a Notch-dependent increase in granulocytic cells but a decrease of bone marrow TER119 + cells and circulating erythrocytes. [START_REF] Zhou | Notchdependent control of myelopoiesis is regulated by fucosylation[END_REF] Since Notch family members are known to modulate the differentiation of hematopoietic progenitor cells, we investigated whether SCF could modulate the expression of Notch in erythroid progenitors and precursors. We found that SCF induced the expression of the Notch family member Notch2 in differentiating erythroblasts. Interfering with Notch function by gamma secretase inhibitor treatment or expression of a dominant negative Notch2 mutant in primary erythroid precursors resulted in decreased erythropoiesis, indicating a necessary role of Notch2 in mediating SCF effects on erythroblast proliferation and differentiation. Altogether, these results show a new mechanism that involves SCF and Notch to control erythropoiesis, which may contribute to maintain postnatal erythroid homeostasis.

Results

SCF modulates the proliferation and differentiation of primary human erythroblasts.

To investigate the mechanisms responsible for SCF effects on erythroid proliferation and differentiation, we took advantage of a serum-free liquid culture system that allows the production of virtually pure populations of differentiating erythroblasts starting from CD34 + hematopoietic progenitors (Figure 1a). [START_REF] Zeuner | Stem cell factor protects erythroid precursor cells from chemotherapeutic agents via up-regulation of BCL-2 family proteins[END_REF][START_REF] Zeuner | Increased death receptor resistance and FLIPshort expression in polycythemia vera erythroid precursor cells[END_REF][START_REF] Zeuner | Activity of the BH3 mimetic ABT-737 on polycythemia vera erythroid precursor cells[END_REF] Erythroblasts grown in these culture conditions gradually become >90% positive for c-kit expression, while this expression tends to decrease at later stages of maturation (Figure 1b). Conversely, Glycophorin A (GpA) begins to be expressed around day 3 of differentiation reaching 98 ± 2% positivity at late (day 14) stages of culture (Figure 1b). The addition of SCF to erythroblast cultures results in a great increase in cell proliferation (Figure 1c) with a concomitant delay of erythroid differentiation that manifests with the prevalence of immature basophilic erythroblasts at advanced stages of culture and expression of lower levels of GpA (Figure 1d).

SCF induces Notch2 expression in erythroid precursors.

The mechanisms exploited by SCF to increase the proliferation and retard the differentiation of hematopoietic precursors are scarcely understood. Since Notch family members play an important role in maintaining the undifferentiated state of hematopoietic stem cells, we investigated whether SCF could modulate Notch proteins to exert its effects on the immature erythroid compartment. Therefore, we investigated the expression of Notch1 and Notch2, the main members of the Notch family expressed in the hematopoietic system, on cultures of primary erythroblasts untreated or treated with SCF. A short (day 6-day 8) treatment with SCF was chosen to investigate the ability of this cytokine to modify Notch expression in the absence of major alterations of erythroid differentiation. Erythroblast treatment with SCF decreased Notch1 RNA expression while increasing Notch2 RNA (Figure 2a). Notch1 protein levels remained unaltered upon SCF treatment, whereas both the mature and immature forms of Notch2 increased significantly (Figure 2b). In order to verify that the increase in Notch2 expression induced by SCF stimulation in erythroid precursors was not due to the presence of a less mature erythroblast population, we stained with anti-Notch2/anti-GpA antibodies erythroid precursors at day 8 of culture, untreated or treated for 2 days with SCF. Such staining confirmed Notch2 induction by SCF, with the increase in Notch2 expression detected by FACS staining (1 log fluorescence scale) being coherent with that detected by western blot (~10 times-fold). Importantly, however, simultaneous GpA staining did not reveal major differences in GpA expression following the short-term treatment with SCF, indicating that SCF-induced increase in Notch2 was due to protein induction and not to a differentiative shift (Figure 2c). In line with a prevalent role of Notch2, but not of Notch1, in the modulation of erythroid differentiation, we found that Notch2 expression progressively increased, peaking around day 5-day 7 of erythroid unilineage culture (Figure 2d) whereas Notch1 protein expression progressively decreased during erythroid maturation as compared to levels found in CD34 + hematopoietic progenitors (Supplementary Figure 1a).

Erythroblast response to inhibition of the Notch system was subsequently investigated at day 4-day 8 of unilineage culture, when high Notch2 expression reflects the proerythroblast/basophilic erythroblast stage of the vast majority of the cells, which possess a high proliferation potential and whose homeostasis is therefore particularly susceptible by modulation by external stimuli. To investigate whether Notch modulation interfered with the functional effects of SCF stimulation, we treated erythroid precursors with SCF in the presence or in the absence of L-685,458 an inhibitor of gamma secretase which is known to inhibit the production of functional Notch proteins. While a short-term (day 4-day 8) treatment with L-685,458 alone did not significantly interfere with basal erythroid proliferation, gamma-secretase inhibition interfered with SCF-induced cell expansion (Figure 2e). Longer treatments with L-685,458 resulted in cell toxicity (data not shown).

The Notch ligand Jagged1 is expressed during erythropoiesis and contributes to SCFmediated erythroblast expansion. To identify the Notch ligand potentially responsible for Notch2 binding and activation, we investigated the expression of four Notch ligands, Jagged1, Jagged2, Delta-like1 and Delta-like3 in differentiating erythroid cells. We found that only Jagged1 RNA was expressed at detectable levels throughout erythroid maturation (Figure 3a), whereas the other ligands showed very low or absent RNA expression (Supplementary Figure 1b). Jagged1 expression was confirmed at the protein level and appeared to be present during the central phases of erythroid differentiation (Figure 3b). Then, we determined whether SCF was able to increase Jagged1 expression. Erythroid precursors at day 6 of unilineage culture were stimulated for two days with SCF and analyzed for Jagged1 RNA and protein expression. Both Jagged1 RNA and protein remained unvaried upon SCF treatment, suggesting that SCF acts rather through reinforcing Notch2 expression (Figure 3c,d). To rule out a potential role of other Notch ligands in mediating SCF effects, we assessed whether SCF was able to modify the expression of Jagged2, Delta-like1 and Delta-like3, but RNA levels of such factors remained unchanged upon SCF treatment (Supplementary Figure 1c). To understand whether Jagged1 played a role in SCF-mediated modulation of erythropoiesis, we cultivated erythroid precursors for four days (day 4-day 8) in the presence or in the absence of SCF and anti-Jagged1 neutralizing antibodies. We found that blocking Jagged1 receptor-ligand interactions reduced SCF-mediated erythroid cell expansion, suggesting the presence of an autocrine signaling mechanism involving Notch2 and Jagged1 expressed on erythroid precursors (Figure 3e). Even in the absence of increased protein expression, the ability of anti-Jagged1 neutralizing antibodies to inhibit SCF-induced proliferation indicates that basal levels of Jagged1 provide a sufficient stimulus to activate Notch2 and support SCF-mediated erythroid expansion.

SCF modulates the expression of Notch mediators in erythroid precursor cells.

In order to depict a possible mechanism of action downstream of Notch2 and responsible for SCF modulation of erythropoiesis, we assessed RNA and protein expression of two major transducers of Notch signals, Hes-1 and Hey-1. As Notch has been previously been shown to modulate GATA-2 expression in hematopoietic cells to inhibit myeloid differentiation, we also analyzed the expression of GATA-2 and its relative GATA-1 in erythroid precursors at day 8 of differentiation, untreated or previously treated for two days with SCF. We found that Hes-1 RNA and protein levels, but not Hey-1 levels, strongly increased upon SCF stimulation (Figure 4a,b). Likewise, SCF increased RNA and protein levels of the antidifferentiative factor GATA-2, while the pro-erythroid factor GATA-1 remained unvaried (Figure 4a,b). Upregulation of Notch2, Hes-1 and GATA-2 by SCF suggests that this cytokine activates signaling pathways downstream of Notch2 responsible for the modulation of erythropoiesis.

Interfering with Notch2 function inhibits the effects of SCF on erythroblast proliferation and differentiation. In order to confirm Notch2 involvement in SCF signaling we searched for a method to stably interfere with Notch2 activity throughout erythroid cell maturation. To do so, we developed Notch2 mutant molecules based on pioneer studies demonstrating that specific Notch truncations resulted in constitutively active and dominant negative forms of the receptor. [START_REF] Rebay | Specific truncations of Drosophila Notch define dominant activated and dominant negative forms of the receptor[END_REF] The constitutively active Notch2 mutant (Notch2 Intra) was constructed by truncating all the extracellular part of the molecule, whereas a dominant negative Notch2 (Notch2 Extra) was produced by removing the intracellular part of the receptor (Figure 5a). Specifically, the Notch2 Extra mutant was constructed in order to maintain all the extracellular and transmembrane region of Notch2 but excluding the region that interacts with CBF-1, which was demonstrated to encompass a conserved region adjacent to the cdc10/ankyrin repeats. [START_REF] Hsieh | Truncated mammalian Notch1 activates CBF1/RBPJk-repressed genes by a mechanism resembling that of Epstein-Barr virus EBNA2[END_REF] The activity of the two mutants was confirmed by evaluating their ability to modulate the activation of a multimerized CBF-1 binding sequence upstream of the SV40 promoter cloned upstream the luciferase sequence (Figure 5b). The constitutively active and dominant negative Notch2 mutants were cloned in a bicistronic retroviral vector carrying the GFP reporter gene. A full-length Notch2 gene could not be used in this expression system as its large size (~7400 bp) exceeded the packaging threshold of the virus.

Retroviral constructs containing Notch2 mutants were used to transduce cycling CD34 + hematopoietic progenitors, which were subsequently sorted for GFP expression and induced to undergo erythroid differentiation through culture in standard erythroid medium. The expression of the truncated Notch2 proteins was detected in packaging cells and in Notch2 Extra-transduced erythroblasts, whereas sufficient numbers of erythroid precursors for immunoblot analysis could not be collected for the Notch2 Intra sample (Figure 5c). In fact, we observed that upon Annexin V/7-AAD staining, the Notch2 Intra-transduced sample revealed a higher rate of apoptotic erythroblasts as compared to the vector-transduced and Notch2 Extra-transduced samples (Figure 5d). To explain this observation, we hypothesized that an elevated Notch2 activity may be incompatible with the first stages of erythroid differentiation, resulting in the death of cells that incorporated a high number of transgene copies, while only cells with a low copy number of constitutively active Notch2 gene could escape apoptosis and mimic SCF signaling. This hypothesis was confirmed by a transgene dosage experiment revealing a decrease in the number of integrated viral copies at late differentiation stages (day 12) in the Notch2 Intra-transduced population but not in the vectortransduced population (Supplementary Figure 1d). Then, we evaluated the proliferative capacity of erythroblasts transduced with the mutant Notch2 genes. The proliferation of cells expressing Notch2 Extra was evaluated both in the presence and in the absence of SCF, whereas the growth of cells expressing Notch2 Intra was evaluated only in standard erythroid medium as the mutant gene itself was expected to mimic the effects of SCF. Erythroblasts transduced with Notch2 Extra showed a severely impaired rate of proliferation both in the presence and in the absence of SCF, indicating that dominant negative Notch2 interfered with both basal and SCF-induced erythroid expansion (Figure 5e). Erythroblasts expressing the constitutively active Notch2 mutant showed also an impaired proliferation, according to the increased percentage of dying cells in the early maturation phases (Figure 5d,e). When we evaluated the differentiation of erythroblast carrying the Notch2 Extra or the Notch2 Intra genes we found that the Notch2 Intra sample contained increased numbers of immature cells as compared to the vector-transduced sample (Figure 5f, left panel).

Conversely, Notch2 Extra-expressing erythroblasts showed an increased presence of mature cells as compared to the control (Figure 5f, left panel). Accelerated erythroid differentiation of Notch2 Extra-expressing erythroblasts was observed also when cells were cultivated in the presence of SCF, indicating that Notch2 contributes to SCF-mediated delay of erythroid differentiation (Figure 5f, right panel). The presence of mature erythroblasts in culture transduced with the dominant negative Notch2 Extra gene was confirmed by GpA staining, which revealed increased numbers of GpA high cells in the Notch2 Extra sample than in the control (vector-transduced) sample, both in basal conditions and in the presence of SCF (Figure 5g).

Discussion

The Notch signaling system plays a complex role in regulating cellular proliferation, survival, differentiation and fate specification. The context-dependent effects of Notch activation are particularly apparent in the regulation of hematopoiesis, where in different systems the Notch pathway has been reported to promote or suppress apoptosis, proliferation and differentiation or even to be dispensable for hematopoiesis. [START_REF] Maillard | Canonical notch signaling is dispensable for the maintenance of adult hematopoietic stem cells[END_REF][START_REF] Kopan | The canonical Notch signaling pathway: unfolding the activation mechanism[END_REF][START_REF] Mancini | Jagged1dependent Notch signaling is dispensable for hematopoietic stem cell self-renewal and differentiation[END_REF] Redundancy between Notch receptors, complexity of intracellular networks and threshold effects may all contribute to explain the different effects of Notch on hematopoiesis. [START_REF] Schwanbeck | Notch signaling in embryonic and adult myelopoiesis[END_REF] In this scenery, studies on differentiating primary hematopoietic cells or on simple vertebrates may provide an important contribution to clarify the role of Notch in the control of hematopoietic cell production.

The process of erythroid maturation involves sequential waves of proliferation and differentiation that are mainly controlled by erythropoietin. SCF plays a key role in the regulation of erythropoiesis as demonstrated by the impaired development of late erythroid progenitors displayed by mice deficient for SCF or its receptor c-kit.(1-3) Interestingly, c-kit mutation also results in a severe impairment of stress erythropoiesis, underlying the importance of SCF both in basal erythropoiesis and in the recovery from acute anemia. [START_REF] Perry | BMP4, SCF, and hypoxia cooperatively regulate the expansion of murine stress erythroid progenitors[END_REF] The molecular pathways responsible for SCF-mediated erythroid proliferative and antiapoptotic effects have been reported to involve multiple effectors including p38, MAP kinase and Bcl-2/Bcl-XL. [START_REF] Zeuner | Stem cell factor protects erythroid precursor cells from chemotherapeutic agents via up-regulation of BCL-2 family proteins[END_REF][START_REF] Kapur | Role of p38 and ERK MAP kinase in proliferation of erythroid progenitors in response to stimulation by soluble and membrane isoforms of stem cell factor[END_REF] In contrast, pathways activated by SCF that affect erythroid differentiation have been poorly characterized, except for the finding that SCF modulates the activity of cyclin/cyclin dependent kinases to inhibit erythroid cell maturation. [START_REF] Tamir | Stem cell factor inhibits erythroid differentiation by modulating the activity of G1-cyclin-dependent kinase complexes: a role for p27 in erythroid differentiation coupled G1 arrest[END_REF] Since both Notch and SCF are able to delay the differentiation of hematopoietic progenitors, we investigated a possible link between the two systems. We observed that Notch2 was strongly induced upon SCF stimulation and that targeting Notch2 signaling neutralized the effects of SCF on erythroblast expansion and differentiation. The observation that dominant negative Notch2 depresses erythroid proliferation is in agreement with previous reports showing that Notch inhibition results in reduced erythropoiesis. In particular, a 40% decrease of bone marrow erythroid cells was detected in fucosylation deficient mice, which have a defective Notch signaling. [START_REF] Zhou | Notchdependent control of myelopoiesis is regulated by fucosylation[END_REF] Interestingly, studies performed on primary human hematopoietic progenitors reported that the simultaneous presence of SCF and Jagged1 increased erythroid colony formation, [START_REF] Walker | The Notch/Jagged pathway inhibits proliferation of human hematopoietic progenitors in vitro[END_REF] anticipating the link between SCF and the Notch pathway described in the present study. Differently, our observation that Notch inhibition impairs erythropoiesis is apparently in contrast with results obtained in other studies. Mice embryos deficient for the Notch mediator RBP-jk have been reported to display increased numbers of Ter119 + cells at the yolk sac level, due to decreased apoptosis of developing erythroblasts. [START_REF] Robert-Moreno | The notch pathway positively regulates programmed cell death during erythroid differentiation[END_REF] In agreement with this observation, activation of Notch signaling in embryonic stem cells has been recently reported to inhibit primitive erythropoiesis. [START_REF] Cheng | Numb mediates the interaction between Wnt and Notch to modulate primitive erythropoietic specification from the hemangioblast[END_REF] This apparent discrepancy may be explained by hypothesizing distinct roles of Notch signaling in different phases of erythroid development. In early erythroid progenitors as well as during embryonic erythropoiesis, Notch signaling may create a conflict with the process of lineage commitment and result in cell death. Accordingly, we found that CD34 + hematopoietic progenitors transduced with constitutively active Notch2 undergo apoptosis when forced to undergo erythroid differentiation by erythropoietin-containing medium. Differently, in more mature erythroblasts elevated Notch expression can result in increased proliferation and differentiative slowdown.

Notably, mice with a conditional inactivation of Mind bomb-1, which is essential for endocytosis of Notch ligands and subsequent Notch signaling, exhibit expansion of the immature erythroid compartment but reduction of circulating mature erythrocytes. [START_REF] Kim | Defective Notch activation in microenvironment leads to myeloproliferative disease[END_REF] Similarly, mice with defective presenilin activity, which is necessary for gamma-secretase cleavage and generation of intracellular Notch, have decreased numbers of mature erythrocytes. [START_REF] Qyang | Myeloproliferative disease in mice with reduced presenilin gene dosage: effect of gamma-secretase blockage[END_REF] According to these observations, common myeloid progenitors from mice expressing a dominant negative form of Mastermind-like-1 that sequesters the intracellular domains of all mammalian Notch receptors produce only one third of erythroid colonies compared to controls. [START_REF] Mercher | Notch signaling specifies megakaryocyte development from hematopoietic stem cells[END_REF] Altogether, these evidences suggest that the Notch system plays a complex role in regulating the size of the erythroid compartment, possibly by restraining the expansion of immature erythroblasts but at the same time by enhancing the production of more mature erythroid precursors and erythrocytes.

The present study adds a new player in SCF-mediated regulation of hematopoiesis, linking SCFactivated signaling pathways to Notch receptors and intracellular mediators. It may be hypothesized that the links between Notch and SCF are not limited to the hematopoietic system. In fact, a connection between SCF and Notch signaling pathways has been previously identified in neural stem cells, where SCF induces Notch expression possibly contributing to stem cell proliferation. [START_REF] Das | Identification of c-Kit receptor as a regulator of adult neural stem cells in the mammalian eye: interactions with Notch signaling[END_REF] Possible correlations between Notch and SCF may also emerge from the melanocytic compartment, where mutations affecting either the Notch system or the c-kit/SCF system similarly lead to loss of melanocyte precursors and absence of hair pigmentation. [START_REF] Schouwey | Notch1 and Notch2 receptors influence progressive hair graying in a dose-dependent manner[END_REF] Future studies that link different systems regulating cell homeostasis are likely to provide new clues to understand hematopoietic regulation and indicate new potential applications for clinical intervention.

Materials and Methods

Antibodies and reagents

Human recombinant SCF, Epo, IL-3 and GM-SCF were purchased from Peprotech Inc. (Rocky Hill, NJ). Rat monoclonal antibodies against Notch1 (bTAN20), Notch2 (C6516BDHN) and Jagged1 (TS115H) were the supernatants of hybridomas purchased from DSHB Hybridoma Bank (Iowa City, Iowa). Alexa-647-conjugated anti-rat antibodies used for flow cytometry were purchased from Molecular Probes-Invitrogen (Carlsbad, CA). Anti-Jagged1 blocking antibody and anti-c-kit were purchased from R&D systems (Minneapolis, MN). The gamma secretase inhibitor L-685,458 was from Sigma-Aldrich (St. Louis, MO). PE-conjugated anti-Glycophorin A (GpA) was from Pharmingen (San Diego, CA). Annexin V FITC and 7-amino-actinomycin D (7-AAD) were from Invitrogen Molecular Probes (Eugene, OR).

Adult peripheral blood human progenitor cell (HPC) purification and culture

Peripheral blood was obtained from healthy donors after informed consent and approval by the institutional ethical committee (protocol N. CE-ISS 08/207). CD34 + hematopoietic progenitor cells (HPC) were purified using the Midi-MACS separation system (Miltenyi Biotec, Bergisch Gladbach, Germany) and cultured in serum-free medium supplemented with 0.01 U/ml IL-3, 0.001 U/ml GM-CSF and 3U/ml Epo (subsequently referred to as standard erythroid medium) as previously described.(25) These culture conditions routinely yield a progeny of 98 ± 2% GpA-positive cells.

Alternatively, CD34 + cells were kept for two days in serum-free medium supplemented with cycling mixture (see below) for subsequent retroviral infection. In all the experiments, CD34 + cells were obtained from three different healthy donors and pooled. The differentiation stage of erythroblasts was routinely evaluated by May-Grünwald-Giemsa staining and cytologic analysis.

Images were taken with a Nikon Eclipse E1000 microscope equipped with a Nikon Plan Apo 60X/1.4 NA oil immersion objective and a Nikon DXM 1200 digital camera with dedicated acquisition software (Nikon ACT-1 v. 2.1; all from Nikon Instruments, Japan).

Reverse-Transcriptase PCR and Real-time PCR

Total RNA was extracted by the standard guanidium thiocyanate-CsCl method and reversetranscribed with oligo (dT) as a primer. To evaluate Notch1 and Notch2 expression, an aliquot of RT-RNA was amplified within the linear range (30 PCR cycles) with primers indicated in Table 1.

Samples were electrophoresed in 2% agarose, transferred to a nylon membrane and hybridized with probes indicated in Table 1. Sample expression was normalized based on S26 expression. Real-time PCR was performed by TaqMan technology, using the ABI PRISM 7900 DNA Sequence Detection System (Applied Biosystems, Foster City, CA) according to standard procedures. 18S RNA was selected as endogenous control. Commercial ready-to-use primers/probe mixes were used (Applied Biosystems): Hs00164982_m1 (Jagged1), Hs00171432_m1 (Jagged2), Hs00194509_m1 (Delta-like1), Hs00213561_m1 (Delta-like3), Hs00172878_m1 (HES1), Hs00232618_m1 (HEY1), Hs00231112_m1 (GATA-1), Hs00231119_m1 (GATA-2), 4319413E (Eukaryotic 18S rRNA).

To assess the number of vector copies integrated in the genome of retrovirally transduced erythroblasts, quantitative PCR was performed using primers complementary to the vector backbone sequence GFP: forward primer, 5΄-CAGCTCGCCGACCACTA-3΄ at 600 nM final concentration; reverse primer, 5΄-GGGCCGTCGCCGA-3΄ at 600 nM final concentration. As internal reference for normalization of human samples, we amplified a fragment of the human telomerase reverse transcriptase (TERT) gene, using the following set of primers: forward primer, 5΄-GGCACACGTGGCTTTTCG-3΄ at 200 nM final concentration; reverse primer, 5΄-GGTGAACCTCGTAAGTTTATGCAA-3΄ at 600 nM final concentration. A standard curve of genomic DNA carrying 11 retroviral vector (RV) copies, validated by Southern blot analysis, was constructed with DNA extracted from RV-transduced HeLa cells. Reactions were carried out in a total volume of 25 µl, in an ABI PRISM 7700 HT sequence detection system (Applied Biosystems).

Calculation of C/G (copies per genome) was done by using the equation: ((ng retroviral vector)/(ng endogenous DNA)) (n° of RV integrations in the standard curve).

(Notch2 Extra) or XhoI/XhoI (Notch2 Intra) in a modified Pinco retroviral vector for subsequent transduction of CD34 + cells.( 26)

Retroviral infection of primary erythroblasts

Mutant forms of Notch2 (Notch2 Intracellular and Notch2 Extracellular) were cloned into a bicistronic retroviral vector obtained by modification of the Pinco vector under the control of Moloney long terminal repeats together with the GFP reporter gene. [START_REF] Grignani | High-efficiency gene transfer and selection of human hematopoietic progenitor cells with a hybrid EBV/retroviral vector expressing the green fluorescence protein[END_REF] The amphotropic packaging cell line Phoenix was transfected by standard calcium-phosphate/chloroquine method, and culture supernatants containing retroviral particles were collected after 48 hours. HPC infection was performed on CD34 + cells previously kept in serum-free medium supplemented with a SCFfree cycling mixture (100 U/ml IL-3, 100 ng/ml FLT3 Ligand, 100 ng/ml Thrombopoietin) for 48 hours after purification. Cells were suspended at 5x10 4 /ml in the viral supernatant supplemented with cycling mixture and 8 µg/ml polybrene and centrifuged at 1800 rpm for 45 minutes at 32 o C, then placed back in the incubator for 1 hour. Such infection cycle was repeated three times a day for two consecutive days. GFP-positive cells were separated by flow cytometry using a FACSAria (Becton Dickinson, Omaha, CA). Immediately after sorting, HPC were placed in standard erythroid medium to induce erythroid differentiation.

Statistical analysis

Statistical analysis was performed using GraphPad Prism version 4.00 for Windows (GraphPad Software, San Diego, CA; www.graphpad.com). Data are represented as the mean and standard deviation of independent experiments, with the statistical significance expressed by one asterisk (P<0.05), two asterisks (P<0.01) or three asterisks (P<0.001). Results shown in Figure 1d

Figure 1 .

 1 Figure 1. SCF stimulates the proliferation and delays the differentiation of primary erythroblasts in unilineage culture. CD34 + cells derived from the peripheral blood of healthy

Figure 2 .

 2 Figure 2. SCF induces Notch2 expression in erythroid precursors. CD34 + cells were cultivated

Figure 3 .

 3 Figure 3. Jagged1 is expressed during eryhtropoiesis and is involved in SCF-mediated cell

Figure 4 .

 4 Figure 4. Hes-1 and GATA-2 levels increase upon SCF stimulation of differentiating erythroblasts. CD34 + cells were cultivated for 6 days in standard erythroid medium to generate

Figure 5 .

 5 Figure 5. Dominant negative Notch2 inhibits the effects of SCF on erythroblast proliferation

  Fig. 4

  GAG TCA CGC ATA AAC CTG CAT GTT GTT GTG

Acknowledgements

We thank Stefano Guida for excellent technical assistance, Giuseppe Loreto for graphics, Gualtiero Mariani and Mauro Biffoni for flow cytometry, Agnese D'Angiò for cell purification, Ferdinando Pucci for DNA analysis. This work was supported by the Italian Association for Cancer Research (AIRC). M.S. is the recipient of an Italy-USA Collaborative Program fellowship.

This work is dedicated to the memory of Elisa Santolini, our beloved colleague.

Reporter assays

The reporter construct pGL3-N8 was obtained by cloning a multimerized (8-mer) CBF1 binding sequence upstream of the SV40 promoter in the vector pGL3 Promoter (Promega, Madison, WI).

Transfections were performed in the 293T embryonic kidney cell line by lipofection (lipofectamine, Invitrogen, Carlsbad, CA) following the manufacturer's instructions. Briefly, cells were seeded in 12-well plates on day 1. On day 2 cells were transfected with a mixture containing either pGL3 promoter or pGL3-N8, along with pRL-CMV (Promega), and the effector plasmids. On day 4 cells were lysed and the reporter activity was assayed using the Dual Luciferase Reporter Assay (Promega) in a TLX20 luminometer. Relative luciferase activity (RLA) was calculated as the ratio of firefly luciferase activity versus jellyfish activity and relative promoter activation (RPA) was calculated by dividing the RLA of the pGL3-N8 series by the RLA of the pGL3 series and then normalizing by the RPA of the empty vector.

Western blotting

Protein extracts were prepared by resuspending cell pellets in 1% NP40 lysis buffer (20 mM Tris/HCl pH 7.2, 200 mM NaCl, 1% NP40) in the presence of protease inhibitors (Sigma).

Concentration of lysates was determined by the Bradford assay (Bio-Rad Laboratories, Richmond, CA) and equal amounts of proteins were used for SDS-PAGE. Samples were analyzed by standard immunoblot procedure and visualized by chemiluminescence (Super Signal West Pico Pierce, Rockford, IL). The intensity of bands representing relevant proteins was quantified using Scion Image (Scion Corporation, www.scioncorp.com).

Construction of Notch2 mutants

The dominant negative Notch2 mutant (Notch2 Extra) was obtained by amplifying the extracellular and transmembrane portions of full-length Notch2 (5340 nucleotides from the start codon). The constitutively active Notch2 mutant (Notch2 Intra) was obtained by amplifying the intracellular portion of full-length Notch2 starting from nucleotide 5095. Oligonucleotides used for amplification are reported in Table 1. The PCR products were verified by sequencing and cloned XhoI/EcoRI

Conflict of interest statement

The authors have no conflict of interest to declare. Supplementary information is available at Cell Death and Differentiation's website.