S Riachy
email: samer.riachy@ensea.fr

D Efimov
email: denis.efimov@inria.fr

M Mboup

Universal integral control: An approach based on mollifiers

The Universal Integral Control, introduced in H.K. Khalil [6], is revisited by employing mollifiers instead of a high-gain observer for the differentiation of the output signal. The closed loop system is a classical functional differential equation with distributed delays on which standard Lyapunov arguments are applied to study the stability. Low-pass filtering capability of mollifiers is demonstrated for a high amplitude and rapidly oscillating noise. The controller is supported by numerical simulations.

I. INTRODUCTION

 (see also their bibliography for previous works).

 a universal integral control (UIC) is proposed. It ensures regional and semi-global stabilization of nonlinear systems that can be put in the normal form of an n-integrator with a stable zero dynamics. The UIC is quite interesting since it requires little information about the

system, mainly, the output relative degree and the sign of the control gain. The integral term compensates a priori unknown constant bias. In addition, the UIC reduces to a PID regulator for second order systems and thus it can be seen as a natural extension of a PID to higher order systems. As a consequence, [START_REF] Khalil | Universal integral controllers for minimum-phase nonlinear systems[END_REF] revealed that a PID can be tuned to achieve regional and semi-global stabilization. Note that tuning a PID is an active research domain as attested by the list of 78 patents on tuning rules in [START_REF] Ang | PID control system analysis, design, and technology[END_REF]. The interest in the linear integral control is pursued in [START_REF] Teo | Proportional-Integral Controllers for Minimum-Phase Nonaffine Control Systems[END_REF] where the UIC is extended to non-affine-in-control systems, then it corresponds, for a given choice of initial conditions, to an exact realization of the approximate dynamic inversion (ADI) [START_REF] Hovakimyan | Dynamic Inversion for Nonaffine-in-Control Systems via Time-Scale Separation: Part I & II[END_REF], which is a nonlinear control. In addition the UIC [START_REF] Khalil | Universal integral controllers for minimum-phase nonlinear systems[END_REF] outperforms the ADI [START_REF] Hovakimyan | Dynamic Inversion for Nonaffine-in-Control Systems via Time-Scale Separation: Part I & II[END_REF] in the presence of some perturbations. Moreover, the stabilization of the n-integrator (i.e the special case of input-output linearizable systems) has been studied using sliding mode control/differentiation in [START_REF] Levant | Higher-order sliding modes, differentiation and output-feedback control[END_REF] and via "time delay control" in [START_REF] Youcef-Toumi | Input/output linearization using Time Delay Control[END_REF].

In [START_REF] Khalil | Universal integral controllers for minimum-phase nonlinear systems[END_REF] and [START_REF] Teo | Proportional-Integral Controllers for Minimum-Phase Nonaffine Control Systems[END_REF], a point remained untreated. It is the capability of linear integral control to attenuate a measurement noise which may have a big magnitude. Note that a big magnitude noise can be found in telecommunication systems as well as in power electronic devices such as choppers. This problem is interesting since noises may have undesired effects on the system such as the excitation of hidden modes. It is important to mention that, in a nonlinear context, only noises with small magnitude were considered so far in the literature (see [START_REF] Ahrens | High-gain observers in the presence of measurement noise: A switched-gain approach[END_REF], [START_REF] Levant | Higher-order sliding modes, differentiation and output-feedback control[END_REF]). On the other hand, recall from linear systems theory, that quite basic tools such as the Bode diagram permit to design the cut-off frequency of a linear controller in order to attenuate a high-frequency noise without assumptions on its magnitude. This note revisits the UIC by introducing mollifiers [START_REF] Adams | Sobolev spaces[END_REF] in the feedback loop. A mollifier replaces the high-gain observer used in [START_REF] Khalil | Universal integral controllers for minimum-phase nonlinear systems[END_REF]. It consists of an ideal differentiator cascaded to a low-pass filter. The main features of this note are the following. A representation of a class of big magnitude noise is proposed. The noise consists of functions in a Sobolev space with negative index. A mollifier is introduced as a differentiator whose low-pass filtering capability is demonstrated on the considered noise. The UIC [START_REF] Khalil | Universal integral controllers for minimum-phase nonlinear systems[END_REF] is revisited via mollifiers when the output is corrupted by the considered noise. In a noise free situation, one can consider a limit setting of the mollifier leading to an exact and instantaneous derivative. Compared to the observers in [START_REF] Khalil | Universal integral controllers for minimum-phase nonlinear systems[END_REF] and [START_REF] Levant | Higher-order sliding modes, differentiation and output-feedback control[END_REF], there is no transient time for the differentiator. Numerical simulations show that mollifier based UIC outperforms the high-gain observer based one since better transients and DRAFT noise filtering (on the output and its derivatives) are obtained with mollifiers.

In section II, a class of big magnitude, rapidly oscillating, noises is introduced as well as a lowpass filtering strategy via mollifiers. The stabilization problem of the n-integrator (input-output linearizable systems) through mollifier based integral control is stated in section III. In section IV, the stability analysis of the nonlinear n-integrator is investigated which constitutes the main result of this note. As an auxiliary result, section V proposes an extension to systems involving zero dynamics. Section VI provides numerical simulations and compares the performances of the UIC based on mollifiers and high-gain observers.

II. REPRESENTATION OF A BIG MAGNITUDE NOISE AND IT'S FILTERING

A. Representing big magnitude noise

Let Ω be a connected subset of the real line R and introduce (see [START_REF] Adams | Sobolev spaces[END_REF]) the Sobolev space

W 1,p (Ω) defined by W 1,p (Ω) = {ω ∈ L p (Ω), 9
ω ∈ L p (Ω)} where 9 ω is the weak derivative of ω, L p (Ω) is the space of all measurable functions defined on Ω and satisfying

ω p p △ = ∫ Ω ω(t) p dt < ∞. The functional ω W 1,p (Ω) △ = ( ω p p + 9 ω p p ) 1 p defines a norm on W 1,p (Ω). Introduce W -1,p (Ω)
the dual space of W 1,p (Ω). W -1,p (Ω) is a Banach space under the norm

W -1,p (Ω) = sup ω∈W 1,p (Ω), ω W 1,p (Ω) ≤1 ⟨ω, ⟩ (1)
where ∈ W -1,p (Ω), ⟨ω, ⟩ = ∫ Ω ω(t) (t)dt and ⋅ a standard (finite dimensional) vector norm. W -1,p (Ω) consists of all linear functionals defined on W 1,p (Ω). Elements of W -1,p (Ω)

are functions belonging to L p ′ (Ω) by the Riesz representation theorem where p ′ is dual exponent of p (see [START_REF] Adams | Sobolev spaces[END_REF], page 62). Given a positive constant N , introduce V -1,p N (Ω), a bounded subset of

W -1,p (Ω), defined by V -1,p N (Ω) = { ∈ W -1,p (Ω); W -1,p (Ω) < N }.
Roughly speaking, for a small N , V -1,p N (Ω) contains (but is not restricted to) large magnitude, rapidly oscillating functions, such that their integral is less than N . Indeed, for a bounded Ω, the indicator function

1 Ω of Ω belongs to W 1,p (Ω). Therefore for ∈ V -1,p N (Ω) one has ∫ Ω (t)dt ≤ W -1,p (Ω) ≤ N . It is important to note that N = 0 ⇔ = 0.
As an example, let Ω be bounded and take

N i ∈ R + , i = 1, 2, ⋯, the function h(t) = ∑ i 1 N i cos( t N 2 i ), t ∈ Ω, belongs to V -1,p N (Ω) whenever ∑ i N i ≤ N .
Such an h(t) can represent a (truncated) Fourier series decomposition of some high-frequency noise with big magnitude.

DRAFT

B. Mollifiers

Mollifiers (see [START_REF] Adams | Sobolev spaces[END_REF]) are smooth convolution kernels which are used in functional analysis to construct regularizing (Cauchy) sequences of L p functions. By this procedure, a discontinuous L p function can be represented by a smooth (infinitely differentiable) one which is equal to the original function everywhere on a given domain except on subsets having a zero Lebesgue measure. A standard mollifier is given by J

(τ ) = ke -1 1-τ 2 1 [-, ]
where k and are positive constants and

1 [-, ] denotes the indicator function of [-, ].
In our context, a mollifier is used as a differentiator endowed with a low-pass filter. For the feedback control purpose, we will be using derivative estimates up to a finite order, say, n. Thus, the mollifier is not required to be smooth but, at least, n-times continuously differentiable. The mollifier that we will be using is given by:

ρ ,i (τ ) = (2i + 1)! (i!) 2 2i+1 ( -τ ) i τ i 1 T , i = 0, ⋯, n, (2)
where T = [0, ], 1 T the indicator function of the interval T and is a positive constant.

The mollifier (2) corresponds to the weight function of the Jacobi orthogonal polynomials and also it plays a key in the algebraic numerical differentiators of [START_REF] Mboup | Numerical differentiation with annihilators in noisy environment[END_REF]. Moreover, by taking affine combinations of (2) as indicated in [START_REF] Mboup | Numerical differentiation with annihilators in noisy environment[END_REF], it is possible to synthesize other mollifiers.

Introduce the (causal) convolution product of y ∈ L p (Ω) with (2) by y (t)

△ = [ρ ,i ⋆ y](t) = ∫ +∞ -∞ ρ ,i (
τ )y(t -τ )dτ where y (t) is called the mollified function.

The Young inequality for convolutions will be used. It is given by:

ρ ⋆ y p ≤ ρ r y q , (3)
such that 1 r + 1 q = 1 + 1 p , 1 ≤ p, q, r ≤ ∞, ρ ∈ L r (Ω), y ∈ L q (Ω) and ρ ⋆ y ∈ L p (Ω). Lemma 2.1: Properties of (2): 1) ∫ +∞ -∞ ρ ,i (τ )dτ = ∫ 0 ρ ,i (τ )dτ = 1, i = 0, ⋯, n. 2) If y ∈ L p (Ω) and y is continuous, then y = ρ ,i ⋆ y ∈ L p (Ω), y p ≤ y p and lim →0 + y - y p = 0.
3) For an i-times continuously differentiable function y one has ρ ,i ⋆ y

(i) = d i ρ ,i (τ ) dτ i ⋆ y. Proof: Proof of item 1. The multiplicative constant (i!) 2 2i+1 (2i+1)! = ∫ 0 ( -τ ) i τ i dτ in (2) serves for normalization. DRAFT Proof of item 2.
The proof is adapted from item c, theorem 2.29 of [START_REF] Adams | Sobolev spaces[END_REF]. Firstly, y p ≤ y p follows from Young inequality (3) with q = p and r = 1, item 1 of Lemma 2.1 and that (τ ) i τ i is positive for τ ∈ T. Secondly, the following serie of inequalities is verified y -

y p = ∫ Ω y (t) -y(t) p dt 1 p = ∫ Ω ∫ 0 ρ ,i (τ )[y(t -τ ) -y(t)]dτ p dt 1 p ≤ (sup 0≤τ ≤ y(t -τ ) -y(t) p ) 1 p .
The proof ends by reducing to zero and noticing that y is continuous.

Proof of item 3. It follows by successive integration by parts and noticing that ( -τ ) i τ i vanishes at 0 and and d dt y(tτ ) = -d dτ y(tτ ). The differentiation and low-pass filter aspects of (2) appear explicitly [START_REF] Mboup | A frequency domain interpretation of the algebraic differentiators[END_REF] on the Fourier transform ŷ(i) of y

(i) = ρ ,i ⋆ y (i) : ŷ(i) (w) = (2i + 1)! (i!) 2 2i+1 0 ( -τ ) i τ i e -jwτ dτ low-pass filter (jw) i i th derivative ŷ(w), (4)
where jw denotes the Fourier variable. Note, for example, that with i = 0, the spectrum of the corresponding low-pass filter is the sinc function and ρ ,0 ⋆ y is the convolution of y with the rectangle function which is the simplest low-pass filter.

C. Noise filtering

Let Ω = [t -, t], the noise is any function

∈ V -1,p N ([t -, t]
) such that its i-th derivative (i) is understood in the weak sense. Convolving with (2) leads to:

Lemma 2.2: There exists a positive constant k i such that ρ ,i ⋆ (i) p =

d i ρ ,i (τ ) dτ i ⋆ p ≤ k i N i , i = 0, ⋯, n.
Proof: Let us start with:

[ d i ρ ,i (τ ) dτ i ⋆ ](t) = 0 d i ρ ,i (τ ) dτ i (t -τ )dτ.
With the standard rule of differentiation

d i dτ i ( -τ ) i τ i = i l=0 ⎛ ⎜ ⎝ i l ⎞ ⎟ ⎠ 2 i!(-1) l τ l ( -τ ) i-l ,
one has:

[ d i ρ ,i (τ ) dτ i ⋆ ](t) = (2i + 1)! (i!) 2 2i+1 0 ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ i l=0 ⎛ ⎜ ⎝ i l ⎞ ⎟ ⎠ 2 i!(-1) l τ l ( -τ ) i-l ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ (t -τ )dτ.

DRAFT

The change of variables β △ = τ , 0 ≤ β ≤ 1, leads to:

[ d i ρ ,i (τ ) dτ i ⋆ ](t) = 1 i 1 0 p(β) (t -β)dβ, p(β) = (2i + 1)! (i!) 2 i l=0 ⎛ ⎜ ⎝ i l ⎞ ⎟ ⎠ 2 i!(-1) l β l (1 -β) i-l .
For any given t, one can define a function t (β) = (t -β), 0 ≤ β < 1. This leads to

[ d i ρ ,i (τ ) dτ i ⋆ ](t) = 1 i 1 0 p(β) t (β)dβ
on which (1) will be used. Let k i be such that

(β) ∈ W 1,p ([0, 1]). Finally one has k i ∫ 1 0 p(β) k i t (β)dβ ≤ k i t W -1,p ([0,1]) = k i W -1,p ([t-,t]) ≤ k i N leading to ρ ,i ⋆ (i) p = d i ρ ,i (τ ) dτ i ⋆ p ≤ k i N i
For a small (< 1), one notices that the noise is less attenuated on high-order derivatives.

Moreover, the lower N is (N ≠ 0), the smaller the bound on the output of the convolution is.

Recall that a small N does not mean that the noise magnitude is small. It means that if the magnitude of the noise is big then it is rapidly fluctuating (i.e high-frequency).

III. CONTROL PROBLEM AND BASIC ASSUMPTIONS

In order not to congest the presentation, the main result (Theorem 4.2) concerns input-output linearizable systems (nonlinear n-integrator). Then systems involving zero dynamics are treated in a secondary result (Corollary 5.1). We consider first a SISO nonlinear n-integrator system:

9 X = F (X) + G(X)u, y = x 1 + (5)
where is some noise, X = [x 1 , ⋯, x n ] ′ ∈ R n , ′ denotes matrix transposition and y is the output.

F (X) = [x 2 , ⋯, x n , f (X)] ′ , G(X) = [0, ⋯, 0, g(X)
] ′ are nonlinear vector fields and f (0) is not necessarily equal to zero. Then X = 0 is not necessarily an equilibrium for (5) with u = 0. The problem considered here is to stabilize (5) at X = 0 by output feedback with y perturbed by .

Assumption 1: With Γ ⊂ R n a compact subset, assume that:

1) f (X) is an unknown function with bounded first derivative on compact subsets of R n .

f X (X) < L 1 ∀X ∈ Γ, L 1 is a positive constant and f X (X) △ = Bf (X) BX .
DRAFT 2) g(X) is an unknown function, lower bounded by a positive constant g (g(X) ≥ g) and has bounded first order derivative on compact subsets of R n . g X (X) < L 2 ∀X ∈ Γ, L 2 is a positive constant and g X (X) △ = Bg(X) BX . We will be using the dynamic feedback

α 9 u = -ρ ,n ⋆ y (n) + n i=1 h i ρ ,i-1 ⋆ y (i-1) = - d n ρ ,n (τ ) dτ n + n i=1 h i d i-1 ρ ,i-1 (τ ) dτ i-1 ⋆ y (6)
where the constants h i , i = 1, ⋯, n are chosen such that the characteristic polynomial

s n + ∑ n i=1 h i s i-1 = 0 is Hurwitz and α is a positive constant. Set X = [ρ ,0 ⋆ x 1 , ⋯, ρ ,(n-1) ⋆ x n ] ′ , = - d n ρ ,n(τ ) dτ n + ∑ n i=1 h i d i-1 ρ ,i-1 (τ ) dτ i-1
⋆ and h = [h 1 , ⋯, h n ], the closed loop system is given by:

9 X = F (X) + G(X)u, y = x 1 + (t) (7)
α 9 u = -[ρ ,n ⋆ (f (X) + g(X)u) + hX ] + , (8)
with

ρ ,n ⋆(f (X)+g(X)u) = (2i+1)! (i!) 2 2i+1 ∫ 0 ( -τ ) n τ n [f (X(t-τ )+g(X(t-τ ))u(t-τ )]dτ .
It remains to specify the initial conditions for t = 0. Let C([-, 0]; R) be the space of continuous functions on the interval [-, 0] and take n + 1 bounded functions φ i ∈ C([-, 0]; R), i = 1, ⋯, n + 1. The initial conditions are given by

x i (θ) = φ i (θ), u(θ) = φ n+1 (θ), -≤ θ ≤ 0, (9)
such that sup -≤θ≤0 φ i (θ) < φi where φi are positive constants. The control (8) turns the closed loop system ( 7)-( 9) into a functional differential equation with distributed delays [START_REF] Hale | Theory of functional differential equations[END_REF]. Since (t)

is an exogenous perturbation and thus not state dependent, then according to [START_REF] Hale | Theory of functional differential equations[END_REF], the system ( 7)-( 9) admits, with assumption 1, a unique solution in forward time.

In particular, by assuming that the whole state vector X measured and f (X) and g(X) are perfectly known and in the absence of noise = 0, the dynamic feedback is redefined by:

α 9 u = -( 9 x n + hX) = -(f (X) + g(X)u + hX) ( 10
)
which corresponds to the ADI control [START_REF] Hovakimyan | Dynamic Inversion for Nonaffine-in-Control Systems via Time-Scale Separation: Part I & II[END_REF]. Thus, as demonstrated in [START_REF] Hovakimyan | Dynamic Inversion for Nonaffine-in-Control Systems via Time-Scale Separation: Part I & II[END_REF], the parameter α can be arbitrarily decreased and the separated time scale analysis as well as singular perturbation theory can be applied for the stability analysis of ( 5), [START_REF] Teo | Proportional-Integral Controllers for Minimum-Phase Nonaffine Control Systems[END_REF]. In particular, for n = 2, and with the particular choice of the initial condition -αu(0) = 9 y(0) + h 2 y(0), one obtains a PID regulator (see [START_REF] Teo | Proportional-Integral Controllers for Minimum-Phase Nonaffine Control Systems[END_REF])

u = -1 α 9 y -h 2 α y -h 1 α ∫ y.
DRAFT Remark 1: The control (10) can be derived from ( 6) by a passage to the limit. In fact, by the continuity of X, u, f (X) and g(X), one can use the last equality of item 2 of Lemma 2.1

lim →0 + ρ ,n ⋆ (f (X) + g(X)u) -(f (X) + g(X)u) p = 0 etc.

IV. STABILITY ANALYSIS

The stability analysis is established in three steps. First we show the asymptotic stability of the unperturbed system ( 5), ( 10) with = 0 (Theorem 4.1). In a second step, we show that the mollifier based system ( 7)-( 9) with = 0 is practically stable (Theorem 4.2). That is the trajectories of ( 7)-( 9) converge within a ball centered at X = 0 such that its radius can be rendered arbitrarily small by reducing to zero. Finally, we show that the trajectories of ( 7)-( 9) under noise (with N > 0) converge within a ball whose radius cannot be decreased arbitrarily (Theorem

4.2).

Let us decompose the control (8) into three terms, a nominal term, a mollification error term and a noise term. Since f (X), g(X), X, and u are differentiable, and the partial derivatives, f X (X) and g X (X) are bounded by assumption 1, such a decomposition is possible by Lemma 4.1 and Corollary 4.1. Lemma 4.1 permits to rewrite the control with distributed delays (8) as a control with a variable, pointwise, delay.

Lemma 4.1:

With assumption 1 verified and given continuous and bounded functions X(θ)

and u(θ), θ ∈ [t -, t], there exist (not necessarily unique) n + 1 functions η i (t) ∶ R ≥0 ↦ [t -, t], i = 0, ⋯, n, such that the following is satisfied:

[ρ ,n ⋆ (f (X) + g(X)u)](t) = f (X(η 0 (t))) + g(X(η 0 (t)))u(η 0 (t)) (11)
[ρ ,(i-1)

⋆ x i ](t) = x i (η i (t)), i = 1⋯n. (12)
Proof: With X(θ) and u(θ) being continuous and bounded and since, by assumption 1, f (X) and g(X) are continuous in X and have bounded derivatives, then the composite function

R t (θ) = f (X(θ)) + g(X(θ))u(θ) maps [t -, t] into a convex subset of R, say, Λ t . For any fixed t, notice that the convolution [ρ ,n ⋆ (f (X) + g(X)u)](t) is bounded such that ν 1 (t) × (inf θ∈[t-,t] R t (θ)) ≤ [ρ ,n ⋆ (f (X) + g(X)u)](t) ≤ ν 2 (t) × (sup θ∈[t-,t] R t (θ)), ν 1 (t) ≥ 1 and ν 2 (t) ≤ 1.
Therefore, the result of the convolution belongs to Λ t . Thus for each time instant t, there exists a delay η 0 (t) ∈ [t -, t], such that ( 11) is verified. The proof of the Lemma follows by applying the same reasoning to (12).

DRAFT

As a consequence, the following is satisfied.

Corollary 4.1:

Let assumption 1 and Lemma 4.1 be verified and assume that 9 X(t) and 9 u(t) are bounded. Introduce m(X(t), u(t), X(η 0 ), u(η 0 ), x 1 (η 1 ), ⋯, x n (η n )) given by m(⋅) = f (X(η 0 (t))) + g(X(η 0 (t)))u(η

0 (t)) + ∑ n i=1 h i x i (η i (t)) -[f (X(t)) + g(X(t))u(t) + hX(t)].
Then, there exists a positive constant M such that m(⋅) ≤ M .

Proof: The proof is straightforward. It relies on a first order Taylor expansion. Thus the boundedness of f X (X), g X (X), 9

X and 9 u are needed. In addition, η i (t) -t has to be bounded which is satisfied since η i (t) -t < , i = 0, ⋯, n.

Remark 2: Corollary 4.1 is used in the proof of Theorem 4.2. In fact, if a trajectory (X(t), u(t)) of ( 7)-( 9), initialized within a compact subset of R n+1 , stays in it for future time, it can be shown that 9 X and 9 u will be bounded from ( 7), [START_REF] Mboup | A frequency domain interpretation of the algebraic differentiators[END_REF]. Then Corollary 4.1 is applicable and m(⋅) is bounded.

∎ With Lemma 4.1 and Corollary 4.1, the control can be decomposed into (compare with ( 10)):

α 9 u = -(f (X) + g(X)u + hX) Nominal term + m(⋅) Mollification error + Noise term . (13)
From Lemma 2.2, it is clear that:

∃K ′ > 0 such that ≤ K ′ N n . ( 14
) Set s = x n + αu, ξ = [x 1 , ⋯, x n-1 , s] ′ = [ξ 1 , ⋯, ξ n ] ′ , B = [0, ⋯, 0, -1, h n ] ′ , D = [0, ⋯, 0, 1] ′ and
the matrix A in controllable canonical form:

A = ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ 0 1 ⋯ 0 ⋮ ⋮ ⋱ ⋮ 0 0 ⋯ 1 -h 1 -h 2 ⋯ -h n ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦
.

With the change of variable:

ξ = X + Dαu, (15)
the system ( 7)-( 9) with (13) rewrites

⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩ 9 ξ = Aξ + Bαu + D(m(⋅) + ) α 9 u = -f (ξ -Dαu) -g(ξ -Dαu)u -hξ + h n αu + m(⋅) + .

DRAFT

The initial conditions can be redefined by n+1 continuous and bounded functions ψ i ∈ C([-, 0]; R),

such that ψ i = φ i i = 1, ⋯, n -1, n + 1 and ψ n = φ n + αφ n+1 . Now, with ξ = ξ -ξ, ũ = u -ū, ū = -f 0 g 0 , f 0 = f (0), g 0 = g(0), ξ = [ ξ1 , ⋯, ξn ] ′ = Dαū, the closed loop becomes (Notice that A ξ + Bαū = 0): ⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩ 9 ξ = A ξ + Bαũ + D(m(⋅) + ) α 9 ũ = -f (⋅) -g(⋅)(ũ + ū) -h ξ + h n αũ + m(⋅) + (16)
where

g(⋅) = g( ξ + ξ -Dα(ũ + ū)) and f (⋅) = f ( ξ + ξ -Dα(ũ + ū)).
The initial conditions are once again redefined by:

ϕ i = ψ i -ξi , i = 1, ⋯, n, ϕ n+1 = ψ n+1 -ū. (17)
In particular, if the whole state is measured and f (X) and g(X) are known functions, it is straightforward to rewrite ( 5), [START_REF] Teo | Proportional-Integral Controllers for Minimum-Phase Nonaffine Control Systems[END_REF] as follows:

⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩ 9 ξ = A ξ + Bαũ α 9 ũ = -f (⋅) -g(⋅)(ũ + ū) -h ξ + h n αũ. (18)
Let P = P ′ ⪰ 0 be the solution of the Lyapunov equation A ′ P + P A = -I where I is the identity matrix of dimension n and consider the candidate Lyapunov function:

V ( ξ, ũ) = ξ′ P ξ + 1 2 αũ 2 . ( 19
)
The asymptotic convergence of ( 5), ( 10) is given by the following result.

Theorem 4.1: Let assumption 1 be verified. Set K 1 = 2P B and take α sufficiently small such

that g > α(L 1 + L 2 ū + h n ). Set K 2 = min{1, g -α(L 1 + L 2 ū + h n )} and K 3 = αK 1 +L 1 +L 2 ū+ h 2 .
If K 2 > K 3 , then ξ = 0, ũ = 0 is an asymptotically stable equilibrium for (18). That is X = 0, u = -f 0 g 0 is an asymptotically stable equilibrium for ( 5), [START_REF] Teo | Proportional-Integral Controllers for Minimum-Phase Nonaffine Control Systems[END_REF].

Proof: The derivative of ( 19) is given by:

9 V = ξ′ (A ′ P + P A) ξ + 2αP B ξ ũ + ũ(-f (⋅) -g(⋅)(ũ + ū) -h ξ + h n αũ). (20) Notice that f (X) -f 0 ≤ L 1 X = L 1 ξ -Dαũ and g(X) -g 0 ≤ L 2 X = L 2 ξ -Dαũ . We add ũ(f 0 -f 0 +g 0 ū-g 0 ū) to the right-hand-side of (20) to obtain 9 V ≤ -ξ 2 -[g-α(L 1 +L 2 ū+ h n )]ũ 2 + (αK 1 + L 1 + L 2 ū + h ) ξ ũ . By noticing that ξ ũ ≤ ξ 2 +ũ 2 2 one obtains 9 V ≤ -(K 2 -K 3 )( ξ 2 + ũ2 )
where K 2 and K 3 are defined in theorem statement. This ends the proof. It can be noticed From Fig. 2 that the M-UIC provides better transient response than HGO-UIC but also better filtering of derivatives Fig. 3 and4.

VII. CONCLUSION

The UIC [START_REF] Khalil | Universal integral controllers for minimum-phase nonlinear systems[END_REF] is revisited under noisy measurements where the noise can have a big magnitude.

Mollifiers as differentiators were introduced and their low-pass filtering capability has been highlighted. The stability of the mollifier based feedback loop has been studied using a Lyapunov function. Numerical simulations showed the advantages of the mollifier with respect the highgain observer. Extending this work to non-affine in control systems seems to be an interesting future research direction.

DRAFT

Fig. 1 .Fig. 2 .

 12 Fig. 1. Plot of y(t) for three controllers: Noise-free simulation

Fig. 3 .Fig. 4 .

 34 Fig. 3. Control input, first and second derivative estimates for the M-UIC.

This work (Grant 074-U01) and the Ministry of Education and Science of Russian Federation (Project 14.Z50.31.0031).

Remark 3: The condition K 2 > K 3 can be satisfied by taking α and h sufficiently small. Moreover, g should be sufficiently big and L 1 , L 2 , and ū sufficiently small. ∎

The stability of ( 7)-( 9) is treated in the main result.

Theorem 4.2: Let assumption 1 be verified, take α sufficiently small such that g > α(L 1 + L 2 ū + h n ). Set K 4 = √ 2 max( 2P D , 1) and assume that there exists a positive δ such that δ = K 2 -K 3 where K 2 and K 3 are given in Theorem 4.1. Consider the compact set:

where R B is a positive constant. For any noise ∈ V -1,p N ([t -, t]) with N and small enough, one chooses M sufficiently big such that m(⋅) ≤ M and

Then, the ball:

and attracts any trajectory of ( 16)-( 17) initialized within B B 0 . In particular, if = 0, the radius of B 0 can be arbitrarily decreased by reducing .

Proof: The derivative of (19) along the trajectories of (16) leads to:

As in the proof of Theorem 4.1, we add ũ(f 0 + g 0 ū) = 0: 9

, where the inequality ξ + ũ ≤ √ 2 ξ 2 + ũ2 is used. By definition of the Lyapunov function we have:min{λ

Let us notice that the mapping

then it maps the compact set B into a compact set, say, B 1 . Consider any trajectory initialized DRAFT within B i.e ( ξ0 , ũ0 ) ∈ B. At the initialization instant, we have the following. The variables X 0 and u 0 are bounded. Moreover, f (X 0 ) and g(X 0 ) are bounded on B 1 (due to the boundedness of their gradients by Assumption 1 and since B 1 is compact). Then F (X 0 ) and G(X 0 ) are also bounded. Then from equations ( 7) and ( 8) we conclude that 9 X 0 and 9 u 0 are bounded since ρ ,i , i = 0, ⋯, n, define bounded convolution operators. Therefore, Corollary 4.1 is applicable. Now, for sufficiently small and N , one ensures that R B 0 < R B , then any trajectory initialized within B B 0 satisfies 9 V ( ξ0 , ũ0 ) < 0 therein, which means that the trajectory does not leave B and Corollary 4.1 stays valid for future time with the same constant M . Then, the trajectory ( ξ(t), ũ(t)) reaches B 0 asymptotically. It remains to notice that if = 0, the radius of 2 and it can be arbitrarily decreased by reducing .

V. EXTENSION TO SYSTEMS WITH ZERO DYNAMICS

Under some conditions, the control (8) can be applied to systems involving zero dynamics.

In fact, assume that there exist zero dynamics given by:

Assumption 2: 1) The system 9 Z = F 0 (Z) is globally exponentially stable. That is there exists a positive definite function V 0 (Z) and four positive constants α 1 , α 2 , α 3 and α 4 , such that

With (16), we have the following closed loop system:

with initial conditions (17). The stability is described by the following.

Corollary 5.1:

and suppose, in addition to assumption 2, that α 1 , α 3 , α 4 and γ( X ) satisfies α 4 > α 3 α 1 γ((α 2 + 1)G). Then zero is attractive for Z subsystem and Theorem 4.2 applies to ( ξ, ũ)-subsystem.

Proof: The linear growth condition prevent from peaking (finite escape time of the Z subsystem during the convergence of the other subsystem). In addition note that X = ξ -DRAFT VI. NUMERICAL SIMULATIONS The kinematic car model, borrowed from [START_REF] Levant | Higher-order sliding modes, differentiation and output-feedback control[END_REF], satisfies the following set of equations with

The The first controller proceeds by Non-Linearities Compensation (NLC-UIC) in order to specify a nominal performance. It is given by 0.

The second controller is based on the high-gain observer (HGO-UIC). It is given by 9 u = -(y 4 + h 3 y3 + h 2 y2 + h 1 y1 ) where y1 and the derivatives estimates y2 , y3 and y4 are given by the high-gain observer with χ 1 = 40, χ 2 = 600, χ 3 = 4000, χ 4 = 10 4 :

The third controller is the mollifier based one (M-UIC). It consists of the second equation in


Philippe Fernandez

Clément Morgat

Jürgen Schulz

Delphine Vimont

Joachim Mazere

Frédéric Debordeaux

Elif Hindie

Michèle Allard

Mise

Mise en place d'une plateforme d'imagerie par le Gallium 68 : applications précliniques et cliniques

à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Mise en place d'une plateforme d'imagerie par le Gallium 68 : applications précliniques et cliniques

En effet, des récepteurs de peptides régulateurs «neuropeptides et hormones peptidiques » peuvent être fortement exprimés au niveau de certaines tumeurs, offrant ainsi l'opportunité d'une imagerie moléculaire spécifique grâce à l'utilisation d'analogues radiomarqués de ces peptides. Cette imagerie moléculaire spécifique, outre qu'elle permet une identification de la tumeur et de son extension, permet de définir des groupes de patients répondeurs à une thérapeutique ciblée.

Index terms -Molecular Imaging, Nuclear imaging

L'imagerie par Tomographie par Emission de Positons (TEP) fait l'objet d'un engouement constant notamment en oncologie. Même si la TEP au [ 18 F]-FDG reste aujourd'hui l'examen de référence pour de nombreux cancers, plusieurs pistes sont aujourd'hui explorées pour permettre une imagerie plus spécifique. Parmielles, l'imagerie des récepteurs de peptides régulateurs comme les neuropeptides semble prometteuse. En effet, des récepteurs de peptides régulateurs «neuropeptides et hormones peptidiques » peuvent être fortement exprimés au niveau de certaines tumeurs, offrant ainsi l'opportunité d'une imagerie moléculaire spécifique grâce à l'utilisation d'analogues radiomarqués de ces peptides. Cette imagerie moléculaire spécifique, outre qu'elle permet une identification de la tumeur et de son extension, permet de définir des groupes de patients répondeurs à une thérapeutique ciblée. Un exemple phare a été le développement d'analogues radiomarqués ciblant le récepteur à la somatostatine de sous-type sst2, fréquemment surexprimé dans des tumeurs neuro-endocrines (TNE). Depuis maintenant plus de 10 ans, l'imagerie scintigraphique des TNE avec l'[ 111 In]-DTPA-Octréotide (Octreoscan®) est devenu incontournable. Dans ce contexte, à cause d'une imagerie conventionnelle par l'In111 de qualité médiocre, de nouveaux analogues de l'octréotide marqués au 68 Ga (émetteur ! + de période 68 minutes), permettant une imagerie TEP quantitative et de meilleure résolution en comparaison avec l'imagerie par !-caméra conventionnelle (par exemple [ 68 Ga]-DOTA-TOC) ont récemment augmenté les performances diagnostiques dans ces pathologies. Ces nouvelles molécules ont poussé certains centres à s'équiper des moyens de synthèse nécessaires pour le radiomarquage au gallium, avec certaines difficultés compte tenu de la règlementation. Nous présenterons la plateforme de Bordeaux mise en place dans le cadre du projet Galléa, mais aussi la nécessité de se regrouper en réseaux pour répondre efficacement aux appels à projets mettant en oeuvre ces nouveaux traceurs. Ainsi, nous avons pu mener à bien un 1 er projet clinique d'évaluation du [ 68 Ga]-DOTA-TOC, nous permettant d'être éligible pour de nouveaux projets.

Cette plateforme à visée diagnostique a permis le développement de traceurs dans le domaine des neuropeptides, et parmi les différents récepteurs surexprimés dans les tumeurs humaines, nous nous sommes notamment intéressés aux récepteurs de la bombésine. Concernant plus spécifiquement le cancer du sein, les récepteurs du GRP, les récepteurs à la neurotensine, et ceux au neuropeptide Y, sont parfois fortement exprimés mais de façon hétérogène. Ceci est important car actuellement les choix thérapeutiques offerts dans le cancer du sein sont différents en fonction du sous-type tumoral.

En particulier, l'arsenal thérapeutique étant limité pour les cancers du sein triple-négatif, une forte expression des récepteurs aux neuropeptides pourrait guider le développement de nouvelles thérapeutiques visant ces récepteurs (antagonistes, radiothérapie interne vectorisée). Un analogue a été synthétisé, radiomarqué par le Gallium-68 ( 68 Ga) et évalué dans la perspective d'une imagerie TEP en clinique.

imagerie par Tomographie par Emission de Positons (TEP) fait l'objet d'un engouement constant notamment en oncologie. Même si la TEP au [18F]- FDG reste aujourd'hui l'examen de référence pour de nombreux cancers, plusieurs pistes sont aujourd'hui explorées pour permettre une imagerie plus spécifique. Parmi-elles, l'imagerie des récepteurs de peptides régulateurs comme les neuropeptides semble prometteuse.

  , C Morgat, J Schulz, D Vimont, J Mazere, F Debordeaux, E Hindie, M Allard

	CNRS, INCIA, UMR 5287, F-33076 Bordeaux, France
	Université de Bordeaux, F-33076 Bordeaux, France
	Service de Médecine Nucléaire, Hôpital Pellegrin, CHU Bordeaux, F-33076 Bordeaux, France

P Fernandez * * Corresponding author: philippe.fernandez@chu-bordeaux.fr.

Abstract -L'


Ghislaine Gueudet

Birgit Pepin

Luc Trouche

Textbooks design and digital resources

Keywords: Digital teaching resources, Mathematics teaching resources, Quality process, Textbooks authors intentions, Teacher communities, Textbooks design, Teachers' documentation work, Textbooks use

Introduction: digital means and evolutions of textbooks design

Our contribution relates to theme C: "Design and use of text-based resources". As stated in the ICMI study 22 discussion document, "most teachers use textbooks and/or online packages of materials as their total or main source of tasks." Moreover, the textbook content and structure influence teachers' choices (Pepin, 2009;Haggarty & Pepin 2002), for example in terms of their choice of tasks or sequencing of the topic area; and both content and structure depend on the textbooks design. We investigate the textbook design and its developments brought about by digital means. Textbooks are now often complemented by digital materials: files to be projected during the lesson by the teacher; animated figures and exercises using different software and providing feedbacks to students, available on a CD or on a website. Linking to theme C we focus on the following question:

"How can or should new digital formats influence textbook design: e.g. use of podcasts, twitter, and other social media; implications for design and coherence of materials (either original digital design or transfer from print) if teachers are able to select tasks in varied orders?" (ICMI study 22,discussion document,p.19).

We consider that this question is a very complex one. It encompasses indeed several aspects of the evolutions resulting from digital means, in particular the following: digital means provide new opportunities for the structuring of the textbooks for their use by teachers. They also open up new possibilities for design and further evolution. In previous works we evidenced that digital means foster collective teacher work (Gueudet & Trouche, 2012): teachers discuss by e-mail; or online teacher associations create resources to be shared by all, not only members. Like many other kinds of teaching resources, now the textbook can also be designed by groups of teachers: "bottom-up" designs (in contrast to a traditional "top-down" design) are now developing, in particular in countries where national policies allow it. This constitutes a major evolution, which we retain as a central focus of our study.

Considering these developments, what does the concept of "coherence of a textbook" mean? The discussion document mentions (p.17), "the conceptual coherence". This can be understood as the correctness of the mathematical content; an alignment or 'consistency' with the official curriculum; a sequencing of the notions and properties introduced that permits to avoid gaps in the mathematical progression; a correct articulation between the text of the course and the associated exercises and problems, or indeed to emphasise 'abstraction' as an 'umbrella concept' for the coherence of the textbook. In the case of a collective design, coherence can also mean that the individual mathematical intentions of the individual authors, their epistemological stances, are well coordinated; and that the whole textbook corresponds to the same "mode of address" (defined as the positioning of the user induced by the material, by Remillard, 2012). Moreover, in terms of implications for learning the most important issue linked to coherence is certainly the 'coherence' of what the teacher produces drawing on the textbook. Alike Shield and Dole (to appear), we consider that analysing textbooks can only inform about their potential to assist in teaching and learning, since teachers interact with textbooks in various ways.

Investigating the potential of a resource leads to an exploration of the concept of quality. We consider that the intrinsic quality of a resource has to be distinguished from its adequacy with respect to institutional and users expectations (Trouche, Drijvers, Gueudet & Sacristan to appear). The intrinsic quality encompasses mathematical, didactical, and ergonomic (ease of use) aspects. It also depends on the mathematical topic considered. For example, Trgalová, Soury-Lavergne & Jahn (2011) assessed the quality of dynamic geometry resources, and they differentiated between nine dimensions of quality: mathematical content; technical aspects; instrumental aspects; added-value of dynamic geometry; didactical implementations; pedagogical implementations; the potential of the resource integration into a teaching process/sequence; ergonomic aspects (e.g. presentation and adaptability); and metadata (e.g. accuracy facilitating searchability).

Traditional textbook analysis (e.g. in TIMSS: [START_REF] Valverde | According to the Book-Using TIMSS to investigate the translation of policy into practice through the world of textbooks[END_REF] proposes three aspects: content (e.g. number, measurement, geometry); performance expectation (knowing, using routine procedures, problem solving, mathematical reasoning, and communicating); and perspective (attitudes, careers, participation, interest, and habits of mind). Most textbook analyses focus on tasks (e.g. exercises and working tasks), and many studies have analysed problem-types [START_REF] Zhu | Focus on the presentation of problem types in intended curriculum: a comparison of selected mathematics textbooks from Mainland China and the United States[END_REF], problem solving procedures (Fan & Zhu 2007), procedural complexity [START_REF] Vincent | Do mathematics textbooks cultivate shallow teaching? Applying the TIMSS video study criteria to Australian eighth-grade mathematics textbooks[END_REF], cognitive demand (Jones & Tarr 2007) or concept treatment [START_REF] Cai | Intended treatments of arithmetic average in US and Asian school mathematics[END_REF].

However, how textbooks deal with depth of understanding, for example in terms of mathematical abstraction, is largely left untouched. Without going into deep philosophical discussions, textbooks are without doubt didactical materials, and as such can be seen as providing 'tools and products of abstraction' (e.g. in their tasks, representations, contexts, etc.). Textbooks use different registers of representation [START_REF] Duval | A cognitive analysis of problems of comprehension in a learning of mathematics[END_REF]), e.g. usual language symbols; figures; representations of technological tools. This variety, and the need for conversion between different representations, can be associated with 'depth of understanding'. Another way of analysing depth of understanding might be by conceptualising 'understanding mathematics' in terms of 'making connections' (Pepin 2008): e.g. connections to what the pupils already knows; to authentic situations; across mathematical topics; across other subjects. This is in line with the literature on 'learning mathematics with understanding' (e.g. Hiebert et al. 1997). They contend that students build mathematical understanding by 'reflecting and communicating', and tasks should allow and encourage these processes. This means that such tasks should have the following features:

"First, the tasks must allow the students to treat the situations as problematic, as something they need to think about rather than as a prescription they need to follow. Second, what is problematic about the task should be the mathematics rather than other aspects of the situation. Finally, in order for students to work seriously on the task, it must offer students the chance to use skills and knowledge they already possess. Tasks that fit these criteria are tasks that can leave behind something of mathematical value for students. " (p.18) In summary, and considering the above, the notion of 'quality' is complex: it involves the notion of coherence and depth of understanding in textbook analysis, and this in the light of the evolutions brought on by digital means. We discuss this further in what follows, drawing on the comparison of two differently designed textbooks. The main research questions are: What are the differences between a textbook designed by a team of experts (researchers, teacher trainers etc.); and a textbook designed by a group of teachers using a digital platform? Which are the 'consequences' of these different designs, in particular in terms of coherence and quality of the textbooks produced?

In the next section, we present the study and subsequently the findings and results. The findings are presented under three headings: (1) the two contrasting teams of authors and their different conceptualisations of their respective textbooks; (2) authors' choices concerning content and structure; and (3) authors' intentions concerning the use of their textbooks.

The study

Our study took place in France, where no institutional control of textbooks exists. France has a National Curriculum, which is presented as a text detailing mathematical objectives and accompanied by detailed comments for teachers, in addition to a booklet giving a structured list of pupil competences. For our study we selected two very different textbooks on the basis of contrasting cases. Both textbooks were grade 6 books (first year of lower secondary school, collège, in France). Since no official statistical figures were available on the most commonly bought mathematics textbooks for this grade, we cannot claim that these textbooks were the most used by teachers. However, they were the most commonly used by the teachers we worked with, and most mathematics teachers in the region knew of them. The differences between the two books did not only concern their content or the material they offered: the differences were also linked to the authors' teams and the design processes. Therefore, we designed a questionnaire for each of the authors' teams. The questionnaires were designed by drawing on our knowledge of previous studies on textbook analysis (e.g. Pepin & Haggarty 2001) and on our knowledge of the books, including what teachers told us about their use. The questionnaires included questions on the textbook's design mode; the authors' perspectives on mathematics and their teaching; the design choices, on a general level and on specific aspects of the textbook concerned. The results we present here draw on a cursory analysis of the textbooks, but more importantly on the analysis of the responses to the questionnaires. The analysis involved the identification of similarities and differences, category generation and saturation based on constant comparison as advocated by Glaser and Strauss (1967) and a procedure similar to that described by [START_REF] Woods | Inside schools: Ethnography in educational research[END_REF].

In particular, we have chosen to focus on the differences in this paper.

Different authors, different conceptualizations

In terms of textbook design Helice 6 e [START_REF] Chesné | Hélice 6e[END_REF] has been developed by a team of four "experts": three teacher educators, two of them with a master degree in mathematics education; and a researcher in mathematics education. A grade 6 teacher (not considered an author) 'tested' some of the tasks in his class. Asked about the way they evaluated the relevance of the content, the authors declared that they trusted "research results, or [their] training experience" -we call it 'expert evaluation'. These experts were clearly aware of their expert position.

Sesamath 6 e (Sesamath, 2009) has been developed by a large group of authors: approximately 57 lower secondary school mathematics teachers (being involved in producing both the paper version and the digital complements), their work being coordinated by members of the Sesamath association 1 : teachers who are involved in the design of online resources. The Sesamath association (Gueudet & Trouche, 2012, Sabar & Trouche, to appear) designed many other resources: online exercises; a digital geometry software; and a complete virtual environment, LaboMep 2 . LaboMep allows the co-ordination of various kinds of resources, from Sesamath or from other sources on the web, and their preparation for student use. The "Sesamath 6 e " textbook was published under a free license; it can be downloaded from a website 3 , or used online (figure 1). A paper textbook exists, corresponding to the text files (which are available in .pdf and .odt). Other complementary files (e.g. dynamic geometry files, spreadsheets, slides, online exercises) can only be accessed using the website. In terms of evaluation of the content relevance, the teacher authors and other 1 The name of the association itself, Sesamath, is interesting and linked to « Open sesame », the famous phrase from the Arabian Nights. The motto of the association is "mathematics for everybody". 2 LaboMep-Laboratory for Mathematics in the Pocket 3 http://manuel.sesamath.net/ members of the association used it in their classrooms and observed and evaluated it, in particular in terms of their students' involvement with particular features of the book. Referring to a distinction introduced in the field of computer supported collaborative learning [START_REF] Dillenbourg | The evolution of research on collaborative learning[END_REF], we consider that the collective work of the authors in the case of Helice was collaborative (the authors sharing each part and step of the work). In the case of Sesamath, it consider the authors' collective work as co-operative: different tasks had been assigned to different authors.

Another important difference (we develop it in the discussion part), was that the content of the digital textbook continuously evolved, according to the experiments and contribution of the teacher users. In fact, Sesamath proposed a website, 'Sesaprof', which was open for teachers and comprised of a 'discussion forum' section (discussing the textbook). These online discussion has led to modifications of the textbook's content: for example, when a large number of users asked for the solutions of the textbook's exercises, these were added in the digital textbook.

Beyond these differences, both teams claimed that they were constrained by commercial publishing and user expectations. For example, the Helice team explained that they had wanted to write more guiding comments for the teacher, but had been asked by the publisher to limit these comments. For the Sesamath team, the grade 6 book was the last in a series: they had started with and accomplished textbooks for grades 7, 8 and 9 (in this order) due to a curriculum changes. Thus, the structure of these earlier textbooks gradually became a model familiar to users, and the authors were obliged to keep a similar structure for the grade 6 textbook.

Authors' choices concerning content and structure

The analysis of the textbooks and the authors' answers to the questionnaire showed many differences concerning the textbooks content. We illustrate these statements with examples from the mathematical topic of 'area'.

Organisation of the content

Concerning the textbook's global structure and the organisation of the mathematical content, Sesamath provided a "classical" organisation: chapters were organised according to the headings of the official National curriculum. As an example, the topic of 'area' was in the section entitled "Area and Perimeter", which was the second section of the chapter entitled "Quantities and Measures".

Helice had a very special, spiral organisation, hence the title "Helice" (meaning "Helix"). The book was not structured in chapters, but in "units", working with different 'thematical' lessons, and a 'unit' finished with problems and exercises linking the different notions learnt. The authors of Helice presented this (spiral organisation) as their central and original structure. Indeed, in France Helice was the only textbook retaining this composition. The authors specified that the intention had been that the learner revisited and deepened the same notion at different stages -a spiral curriculum. At every stage the notions were associated with different representations; links between different chapters were frequently made; and differentiation in terms of pupil learning (e.g. pace) was taken into account. The topic 'area' appeared twice in the table of contents: in the unit entitled "Distance and areas" (Unit 4-in a lesson called "Area: comparison and sharing") and in the unit entitled "Division and computation of area" (Unity 5-in a lesson called "division and computation of area"). Moreover, the area of a circle was in fact presented in Unit 7, as an activity demonstrating that the area of a circle and the square of its radius are proportional.

We claim that this difference in terms of structure was influenced by the different design modes and author teams. Following a complex and coherent structure, such as the one retained by Helice, was only possible for a 'steady' author team, and likely to be very difficult for a large and 'variable' author group, such as the Sesamath group. A large 'collective' process, like Sesamath, required splitting the content into different parts, which then were designed by different authors who did not necessarily communicate. This splitting of tasks, we argue, has also influenced the coherence of the textbook. Bringing together the work of more than fifty authors and achieving a coherent didactical structure would require an enormous coordination effort, more than the one organised by the Sesamath team.

Different methods, or a single expert method

Helice authors stated that it was important for them to propose different representations and a rich vocabulary in their book. They also favoured exercises with different possible solutions.

In Sesamath, our analysis showed that some additional research activities suggested and fostered the search for several solutions. However, we contend that most of the exercises led to a single and final solution. In particular, the worked examples were all one-solution exercises-the authors called them 'expert solution' in the questionnaire (figure 2).

Primary-secondary link

Another important aspect, given that the textbooks were for grade 6, was whether the primary-secondary link was explicitly addressed. Some of the Helice authors were teacher educators, both for secondary and primary school level. Thus they had considerable knowledge about the content taught at primary and secondary school. They identified several crucial cognitive and didactical changes in terms of transition from primary to secondary school mathematics. Concerning 'area', one important choice, according to them, was to start with activities on the comparison of areas of two figures, without using any kind of measures. The activity in figure 3 illustrates this choice.

exercises, and of the lessons. The book also offered a large amount of exercises, with the intention of leaving their choice to teachers. According to the Sesamath authors, the digital textbook was a "collection of bricks". Moreover, they stated that it should be thought of as belonging to a more general set of different kinds of resources available in the virtual environment LaboMep. Sesamath authors considered that it was the teachers' responsibility to 'build coherent lessons' and a 'coherent progression'. In their view this was made possible by providing a large range of resources to choose from, the textbook being only one of these. In addition, they wanted to provide 'efficient tools', such as LaboMep, for teachers to build their own teaching from these 'initial bricks'.

Discussion

Going back to our initial questions ("What are the differences between a textbook designed by a team of experts (researchers, teacher trainers etc.) and a textbook designed by a group of teachers using a digital platform? Which are the 'consequences' of these different designs, in particular in terms of coherence and quality of the textbooks produced?), we answer these by comparing the two contrasting textbook cases (Helice and Sesamath).

The investigation of the two textbooks showed that there are differences concerning essentially six levels (at least):

-the modes of design: more collaborative in the first case, more co-operative in the second; -the nature of the structure: the first book is a single whole, with an organised structure (organised by the team of experts); the second is an atomistic system that can be arranged differently by different users; -the organisation of the content: more didactically original, linked to the didactical choices of a 'homogeneous' team in the first case; more aligned with the institutional instructions in the second; -the content: more open to a variety of ways for solving a given problem in the first case; more driven by an expert solution in the second; -the integration into the whole grades 1-9 mathematics curriculum: links with primary school more taken into account in the first case than in the second; and -the links to the users: the textbook provided as a final product given to the teachers in the first case; and as a proposal to be enriched by teachers' contributions in the second. For Helice, the coherence is insured by the authors' didactical expertise, i.e. the mastering of the concepts at stake and of the potential difficulties and misconceptions for learning. It could be said that it is a transcending coherence. For Sesamath, the coherence is insured by the link with the curriculum and the institutional prescriptions; by teacher evaluation in class; and by the discussions among authors faced with the different contributions. Sabra and Trouche (to appear) describe, for example, the discussion in another Sesamath author team (for a grade 10 book) in terms of reaching a coherence and consistency between the introduction of equations (in one chapter), and the introduction of function (in another chapter). We argue that this is a collective and institutional coherence.

Helice was, we claim, of high didactical quality. It offered many rich tasks, organized according to a carefully considered and complex structure. It took into account central aspects of the primary-secondary transition. The Sesamath textbook appeared to be, in its initial version, of a lower intrinsic quality: it offered less problems and less rich tasks. In terms of structure it simply followed the structure of the official National Curriculum.

However, the 'digital additions' and possibilities of Sesamath prompted us to re-consider the notions of 'quality' and 'coherence'. Helice remained the same, in both its paper and pdf versions-we call this static quality. In contrast, the online version of Sesamath had already been modified several times, to take account of 'user comments', i.e. users' experiences and needs. The digital means offered possibilities for modifications, and these were integrated by Sesamath in the process of re-design. The association perceived this as a necessity for meeting users'/teachers' needs in order to insure the quality of the textbook-we call this dynamic quality. Both Helice and Sesamath authors recognized that teachers would select and adapt elements of the textbook for their teaching. However, only Sesamath supported these adaptations and drew on user contributions.

We argue that this was a major development linked to digital means: the involvement of a large group of teachers in the design of resources, which in turn continuously evolved. This also deepens our knowledge of already established phenomena: even teachers who were not involved in the design of textbooks could be considered as 'designers' of their own teaching materials, as teachers selected resources, combined them, set them up in class-a process that Gueudet & Trouche (2009) called teacher documentation work. This leads us to consider that the question raised in the discussion document: "[Which] implications for design and coherence of materials […] if teachers are able to select tasks in varied orders?" does not sufficiently recognize the complex link between 'design' and 'coherence', and the evolving role of teachers. Exploiting the potential of digital means for the design of teaching resources, including textbooks, requires the acknowledgement of this new role.

Drawing on the results presented here, we argue that textbooks can be considered as lived resource (Gueudet, Pepin & Trouche, 2012), or even 'living resources', as they get continuously enriched and renewed by teachers' experience. This new conceptualization of the textbook is likely to be associated with new forms of design, for example, in terms of reflection on meta-design (Fischer and Ostwald 2005): resources which support the design by teachers. It can also lead to favour teams of textbooks designers who combine and involve different 'experts' and expertise (e.g. teachers and other 'experts' such as researchers). From the literature (e.g. Kieran, Tanguay & Solares 2012), it is clear that teacher documentation work and professional knowledge are intrinsically intertwined, one leading to the evolution of the other (e.g. Gueudet & Trouche 2012;Pepin 2012). This has implications, also in terms of policy, for teacher professional development (of 'users' and of 'designers'); mathematical task design and digital means/possibilities; and the reconceptualization of the quality of resources in mathematics education.

Figure 1 .

 1 Figure 1. Sesamath online texbook for grade 6. The text on the screen background corresponds to the content of the .pdf, or paper version. The "complements" window opens when the mouse is placed on a selected activity.

Figure 2 .

 2 Figure 2. Extract of Sesamath 6 e . Method 2: evaluating an area. Example 1. Determine the area of this figure by choosing the yellow triangle as the unit, then calculating the area of this rhombus. To find the area of the figure, you have to count the number of area units of the figure. The purple figure is made of 9 triangles. Its area is thus the area of 9 yellow triangles. The rhombus is made of two yellow triangles. The area of the purple figure, in number of rhombusses, is thus twice as smaller. Hence the area of the figure equals 4.5 rhombuses.

Educational Studies in Mathematics, 61(1-2): 103-131. Fan, L. & Zhu, Y. (2007). Representation of problem-solving procedures: A comparative look at China, Singapore, and US mathematics textbooks. Educational Studies in Mathematics 66, 61-75. Fischer, G., & Ostwald, J. (2005). Knowledge communication in design communities. In R. Bromme, F. Hesse & H. Spada (eds.), Barriers and Biases in computer-mediated knowledge communication -and how they may be overcome (pp 213 -242) . Dordrecht, Netherlands: Kluwer Academic Publishers. Glaser, B.G. & Strauss, A.L. (1967). The discovery of grounded theory. Strategies for qualitative research. Chicago: Aldine. Gueudet, G., & Trouche, L. (2009). Towards new documentation systems for mathematics teachers? Educational Studies in Mathematics 71, 199-218. Gueudet, G., Pepin, B., & Trouche, L. (eds.) (2012). From Textbooks to 'Lived' Resources: Mathematics Curriculum Materials and Teacher Documentation. New York: Springer. Gueudet, G., & Trouche, L. (2012). Communities, documents, and professional geneses:Interrelated stories. In G. Gueudet, B. Pepin, & L. & Trouche (eds.), From Textbooks to 'Lived' Resources: Mathematics Curriculum Materials and Teacher Documentation (pp. 305-322), New York: Springer. Haggarty, L., & Pepin, B. (2002). An investigation of mathematics textbooks and their use in English, French and German Classrooms: who gets an opportunity to learn what? British Educational Research Journal 28 (4): 567-590. Hiebert, J., Carpenter, T., Fennema, E., Fuson, K., Wearne, D., Human, P., Murray, H., & Olivier, A. (1997). Making Sense: teaching and learning mathematics with understanding. Portsmouth, NH: Heinemann. Jones, D.L. & Tarr, J.E. (2007) An examination of the levels of cognitive demand required by probability tasks in middle mathematics textbooks. Statistics Education Research Journal, 6(2): 4-27. Kieran, C., Tanguay, D., & Solares, A. (2012). Researcher-designed Resources and their Adaptation within Classroom Teaching Practice: Shaping both the Implicit and the Explicit. In G. Gueudet, B. Pepin, & L. & Trouche (eds.), From Textbooks to 'Lived' Resources: Mathematics Curriculum Materials and Teacher Documentation (pp. 189-213), New York: Springer. Pepin, B. (2008) "Making connections and seeking understanding: mathematical tasks in English, French and German textbooks", Presentation, Nuffield seminar series on Mathematical Knowledge in Teaching, Loughborough, March 2008. Pepin, B. (2009). The role of textbooks in the 'figured world' of English, French and German classrooms-a comparative perspective. In L. Black, H. Mendick, & Y. Solomon (eds.), Mathematical Relationships: identities and participation (pp. 107-118). London: Routledge. Pepin, B. (2012) 'Working with teachers on curriculum materials to develop mathematical knowledge in/for teaching: task analysis as 'catalytic tool' for feedback and teacher learning'. In G. Gueudet, B. Pepin & L. Trouche (eds) Mathematics Curriculum Material and Teacher Development: from text to 'lived' resources (pp. 123-142). Dordrecht, NL: Springer. Pepin, B., & Haggarty, L. (2001). Mathematics textbooks and their use in English, French and German classrooms: a way to understand teaching and learning cultures, ZDM, The International Journal on Mathematics Education 33 (5), 158-75. Remillard, J. (2012). Modes of Engagement:Understanding Teachers'Transactions with Mathematics Curriculum Resources. In G. Gueudet, B. Pepin, & L. & Trouche (eds.), From Textbooks to 'Lived' Resources: Mathematics Curriculum Materials and Teacher Documentation (pp. 105-122), New York: Springer. Sabra, H., & Trouche, L. (to appear). Designing digital resources in communities of practice: a way to develop mathematics teachers' knowledge. In A. Clarck-Wilson, O. Robutti & A. Sinclair (eds.), The Mathematics Teacher in the Digital Era: An International Perspective on Technology Focused Professional Development. New York: Springer. Sesamath (2009). Le manuel Sésamath 6e. Chambéry: Génération 5. Shield, M., & Dole, S. (to appear). Assessing the potential of mathematics textbooks to promote deep

The official curriculum mentioned "geometrical comparison of areas" as a teaching objective. This aspect was particularly developed in Helice; and this was a deliberate choice of the authors, in order to provide a better link to primary school notions, where comparing areas without using measure was a common task. Sesamath authors declared that they had only very limited knowledge of primary school mathematics curricular aspects, when they started writing the grade 6 book. Concerning 'area', the textbook focused on area computations, whereas the official curriculum also stipulated "geometric comparison of areas". This geometric comparison, without measures, was not evident in the Sesamath textbook. With a better knowledge of primary school, where such tasks were frequent, the authors may have inserted it. However, the Sesamath authors paid attention to one 'classical' difficulty: the potential problem to distinguish between area and perimeter of a figure. This difficulty was known to be important for primary school, and also for grade 6 students, and as teachers of grade 6 students the textbook's authors knew about it.

Writing the book raised the authors' awareness of transition questions (e.g. the importance of mental arithmetic for grade 6.). The paper version of Sesamath did not reflect this, but in the associated online resources, in particular online exercises, the authors attended to this aspect of primary-secondary transition. Whilst the paper book remained the same since 2009, the authors claimed that they had attended to particular 'shortcomings' and that the online complements had considerably developed since (e.g. in LaboMep many online exercises of mental arithmetic for grade 6 had been added).

Authors' intentions concerning the use of their textbooks

In the questionnaire, the Helice designers adopted a general stance about teachers' adaptations of the book to their specific contexts. They said that "it [was] impossible to anticipate all possible adaptations" (in terms of contexts) and that they did not regard it as their "responsibility" to attend to these. This view was reflected in several statements throughout the questionnaire, e.g. the following: "the gap between what is planned and what happens in class is large-the gap between the authors' intentions and the teachers' use is even larger". They also declared that they would anticipate that teachers would combine Helice with the use of other textbooks. However, the spiral progression made such a practice difficult and the very complex spiral structure was clearly an obstacle for the adaptation, or combination of several textbooks.

For Sesamath, the possibility to adapt the content was an important issue. The authors conceptualised the book, from its inception, as a digital textbook, each chapter file in .odt format. This offered opportunities for teachers to modify the texts of the


Elena Nardi

Alejandro S González-Martin

Ghislaine Gueudet

Paola Iannone

Carl Winsløw

WG14: UNIVERSITY MATHEMATICS EDUCATION

Research on university level mathematics education is a relatively young field, which embraces an increasingly wider range of theoretical approaches (e.g. cognitive/developmental, socio-cultural, anthropological and discursive) and methods/methodologies (e.g. quantitative, qualitative and narrative). Variation also characterises research in this area with regard to at least two further issues: the role of the participants, students and university teachers, in the research -from 'just' subjects of the research to fully-fledged co-researchersand the degree of intervention involved in the researchfrom external, non-interventionist research, to developmental/action research in which researchers identify problems and devise, implement and evaluate reforms of practice [START_REF] Artigue | Mathematics Thinking and Learning at Postsecondary Level[END_REF].

2011 marks the 20 th anniversary of the publication of Advanced Mathematical Thinking (Tall, 1991), a volume that is often heralded as a first signal of the emergence of this new area of research. A few years later a second signal was given by the 1998 ICMI study that resulted in The teaching and learning of mathematics at university level [START_REF] Holton | The teaching and learning of mathematics at university level: An ICMI study[END_REF]. In the meantime Advanced Mathematical Thinking (AMT) groups ran both in previous CERME and PME conferences; sessions exclusively on university mathematics education are part of the EMF conferences since 2006 ; the RUME, UMT and Delta conferences emerged in the USA, the UK and South Africa respectively; the International Conferences on the Teaching of Mathematics at University Level were launched in 1998; etc. WG14 (University Mathematics Education, hereafter UME) emerged out of the above developments and out of the realisation that this is a distinct area of mathematics education research. Its distinctiveness can be attributed to several characteristics. Firstly, the classic distinction between 'teacher' and 'researcher' does not always apply in UME as researchers in mathematics education in this area are often university-level teachers of mathematics themselves. In particular there is a growing group of mathematicians specializing in research on mathematics education at university level, where advanced mathematical knowledge and experience is really an asset (if not a necessity). Secondly, mathematics education theories and research methods find new uses, and adaptations, at the university level. These adaptations are often quite radical as the post-compulsory educational context is different in many waysthe voluntary presence of students, the important role of mathematics as a service subject, the predominance of lecturing to large numbers of students, the absence of national programmes for university education, to mention but a few. In this sense UME is a distinct area of mathematics education research, not merely a mirror of mathematics education research at a more advanced educational level. Finally, in recent years, research in this area has been growing in different parts of the world. WG14 is one opportunity to collate evidence of this growing research activity from Europe and beyond.

The WG14 Call for Papers invited contributions from as wide a range of research topics as possible. These included: concept formation, mathematical reasoning and proof in university mathematics; teaching at university level (including the perspectives of university mathematicians); ICT in the teaching and learning of university mathematics (including curriculum design); socio-cultural, anthropological and discursive approaches in university mathematics education; affective and social aspects of the learning and teaching of university mathematics; the transition from secondary to tertiary mathematics; novel approaches to teaching mainstream (e.g. calculus and linear algebra) as well as more advanced topics, both to students of mathematics and other areas (e.g. engineering, sciences, finance); theoretical approaches to the study of teaching and learning mathematics at university (including a focus on specific approaches and on contrasting or combining approaches). This report draws on the presentations, reactions-to and discussions of the 21 accepted papers that met these terms. The number and quality of these papers marks the recent surge in the quality of research outputs and a move away from the earlier days of perhaps more naïve, less rigorous research in this areabrought about partly by the university sector's increasing urge to adapt teaching to changing student cohorts and by a growing, and wider, tendency towards an in-depth probing into traditional teaching practices in higher education.

Across the WG14 discussions certain themes and questions emerged as crucial. These included: exploring whether UME needs to generate new theories or adapt already existing ones; attending to issues of both theory and practice; acknowledging that research on teaching and learning in higher education develops also outside mathematics education, and benefiting from these developments; working towards the generation of new theories while valuing already accumulated knowledge in the field; etc. Colleagues observed that, beyond staple references to classic constructs from the AMT era (such as concept imageconcept definition; APOS theory, process object duality etc.), several works presented in WG14 employ (often in tandem with the above) approaches such as the Anthropological Theory of Didactics (ATD: [START_REF] Chevallard | La transposition didactique: du savoir savant au savoir enseigné[END_REF], and discursive approaches (e.g. [START_REF] Sfard | Thinking as communicating. Human development, the growth of discourse, and mathematizing[END_REF].

Generally speaking papers seem to be classified into those with a focus on the teaching and learning of particular mathematical topics (calculus continues to attract more attention than other topics) or on wider, cross-topical issues such as the transition to university mathematics, use of IT, language, motivation, teacher knowledge and development, curricular, pedagogical and institutional issues, etc.. Furthermore an area of growth is of studies that examine the different role of mathematics in courses towards a mathematics degree, courses for pre-service teachers, as a 'service' subject (physics, biology, economics etc.). While a substantial number of papers remains in the increasingly well-trodden area of students' perceptions of specific mathematical concepts (again calculus prevails in these), a focus on university teachers and teaching is also emerging, if often a little timidly, and diplomatically, resulting in descriptive, openly non-judgemental studies. In conjunction with those studies a genre of collaborative studies, with mathematicians engaged as co-researchers, also seems to be on the rise. In the nutshell descriptions of the WG14 papers that follow the order of presentation is loosely structured around some of the themes mentioned above.

Xhonneux & Henry is one of the papers that employs the ATD framework to distinguish between mathematical and didactic praxeologies in the context of teaching and learning of Lagrange's Theorem in calculus courses to mathematics and economics students. Gyöngyösi, Solovej & Winslow is another: in it a part of a transitional course in Analysis was taught with a combination of Maple and paperbased techniques and resulted in mixed reception and performance by students. A third is Barquero, Bosch & Gascón: from its analyses 'applicationism' emerges as the prevailing epistemology of mathematics in science departments that potentially hinders the teaching of mathematical modelling to science students.

Another set of approaches that was employed by a number of WG14 papers were discursive. Jaworski & Matthews employed such approaches to trace university mathematicians' pedagogical discourse and suggest links of this discourse to their ontological and epistemological perspectives. Biza & Giraldo described how computational inscriptionsin this case differentiabilityhave potentialities and limitations that can be helpful in students' exploration of newly introduced mathematical concepts. Three papers made use of Sfard's commognitive framework. Viirman employed this framework to trace the variation in the pedagogical discourses of mathematics lecturers in the course of their introducing the concept of function. Stadler described students' experience of the transition from school to university mathematics as an often perplexing re-visiting of content and ways of working that seem simultaneously familiar and novel (for example in the case of solving equations). Nardi outlined interviewed mathematicians' perspectives on their newly arriving students' verbalisation skills; and, observed that discourse on verbalisation in mathematics tends to be risk-averse and not as explicit in teaching as necessary.

Several papers focused on the transition from school to university mathematics (including Gyöngyösi, Solovej & Winslow and Stadler mentioned above). Biehler, Fischer, Hochmuth & Wassong proposed that blending traditional course attendance with systematic e-learning study can facilitate the bridging of school and university mathematics. Faulkner, Hannigan & Gill noted the shifting profile of students who take service mathematics courses (in the context of an Irish institution): many more are diagnosed as at risk, fewer have an advanced mathematics secondary qualification and the percentage of non-standard (e.g. mature) students has grown. Zimmermann, Bescherer & Spannagel described MaSE-T, a mathematics self-efficacy test designed to measure the impact that self-efficacy perceptions have on choice of studies.

Vandebrouck noted that in the transition from school to university, mathematics students need to reconceptualise the concept of function in terms of its multiple representations and its process-object duality. Finally, De Vleeschouwer & Gueudet observed that students can learn to appreciate the duality in linear forms (described here in micro-macro terms) if given an appropriate set of tasks that require them to engage with these concepts at both levels.

Many of the papers mentioned above had a clear focus on a specific mathematical concept or issue. In addition to these, Iannone & Inglis discussed a range of weaknesses in Year 1 mathematics students' production of deductive arguments (rather than in the oft-reported perception that a deductive argument was expected of them). Juter reported the highly individual and not easily classifiable character of pre-service secondary mathematics teachers' concept images of elementary Calculus concepts. And Souto-Rubio & Gómez-Chacón mapped out students' difficulty with developing visualisation skills in the context of the Riemann integral. Some papers focused on particular curricular and pedagogical aspects of university mathematics. Agathokleous argued how teaching Abstract Algebra to pre-service primary teachers can facilitate students' appreciation for the connectedness across mathematical domains. Jukic & Dahl, through data collected in the Croatian and Danish context, aimed to illustrate that students taught in different styles are likely to perform differently. Bergsten presented evidence that students tend to find lectures useful and attractive, despite their bad press in some education quarters.

Finally, a few papers addressed theoretical issues directly. Barton outlined efforts to combine the three-fold activity of research, development and theory building into LATUM, a model for learning and teaching university mathematics that is proposed as a model for designing alternative university mathematics delivery. And, Pettersson proposed 'threshold concepts', a theoretical construct from the general education literature, as a means for gaining insight into student learning and engaging teachers in pedagogical discussion.