Benjamin Navarro

Aïcha Fonte

Philippe Fraisse

Gérard Poisson

Andrea Cherubini

Physical human-robot interaction with the OpenPHRI library

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Physical human-robot interaction with the OpenPHRI library

Benjamin Navarro [START_REF] Hogan | Impedance control: An approach to manipulation: Part II-Implementation[END_REF][START_REF] Navarro | OpenPHRI: an open-source library for safe physical human-robot interaction[END_REF] , Aïcha Fonte2 , Philippe Fraisse1 , Gérard Poisson 2 , and Andrea Cherubini [START_REF] Hogan | Impedance control: An approach to manipulation: Part II-Implementation[END_REF] 1 LIRMM CNRS-UM, 161 Rue Ada, 34090 Montpellier, France. [START_REF] Navarro | OpenPHRI: an open-source library for safe physical human-robot interaction[END_REF] PRISME Laboratory, University of Orléans, 12, Rue de Blois, 45067 Orleans, France.

OpenPHRI is a C++/Python software with several builtin safety measures, designed to ease robot programming for physical human-robot interaction (pHRI) and collaboration (pHRC). Aside from providing common functionalities, the library can be easily customized and enhanced, thanks to the project's open source nature. The library is distributed online 1 free of charge under the GNU LGPLv3 license [START_REF] Navarro | OpenPHRI: an open-source library for safe physical human-robot interaction[END_REF] . OpenPHRI is written in C++ to maximize efficiency in terms of computation and memory footprint and to easily embed it in existing projects. Python bindings are also provided, since this language is largely used in the robotics community. An interface with the robotics simulator V-REP3 is also provided. The framework consists in a two-layer damping controller depicted in figure 1. Damping control is a particular case of impedance control [1], which makes the robot act as a massspring-damper system. Although impedance control proved useful for complying with interaction forces while following a predefined trajectory, it can be extended to fit many more scenarios. We design a more generic controller that includes real or virtual force inputs and velocity inputs. In this work, the focus has been on :

Two-layer safe damping control framework

• real forces exchanged with the human or environment,

• virtual mass and stiffness forces (generating inertial and elastic effects), • virtual forces repelling away from obstacles,

• velocities generated by a trajectory generator,

• velocities output by a force controller.

To reduce the velocity in unsafe situations, we add constraints, that determine the value of a velocity scaling factor. If multiple constraints are active at the same time, the most restrictive one, i.e. the one leading to the lowest velocity, will be chosen. This makes sure that all the constraints are satisfied at any given time. In less than 35 lines of code one can set up a V-REP scenario where a serial manipulator robot Kuka LWR4+ is moved with an external force applied, while limiting its velocity, reading sensory input, and sending joint commands to the simulator. See Fig. 2 and video available at: http: //bit.do/openphrivideo

For further details on OpenPHRI, the reader should refer to [2].

HOW THE PROPOSED WORK ADDRESSES THE WORKSHOP'S RESEARCH QUESTIONS 1) How can a robot understand operator's actions, intentions and eventually her/his physical and mental status?

In this work, we mainly use force sensing to transmit the operator intentions to the robot. In other works from our group, we have used vision, tact, and muscular (EEG) as well as brain (BCI) activity. 2) What should a robot do to make its behaviour understandable by the operator, therefore enforcing a sense of trust?

The robot motion should be smooth and predictable. For this, OpenPHRI also relies on a trajectory generator, detailed in [2]. 3) How much information should the robot provide to the operator? What means of communication should the robot use? For now, only motion and stiffness are used in Open-PHRI. In our opinion, audio also provides an interesting means of communication, although it has been seldom studied in the pHRI literature. 4) How much information does the robot need from the operator? What are the best ways to represent such knowledge? See reply to 1. 5) How can a non-expert operator communicate with the robot in a natural manner? Audio, vision and touch appear as the most promising means for intuitive human-robot communication.

Fig. 1 :

 1 Fig. 1: Overview of the OpenPHRI framework. Its core components are: force inputs, velocity inputs, and constraints.

Fig. 2 :

 2 Fig. 2: Setup for the experiment.

https://github.com/BenjaminNavarro/OpenPHRI

https://www.gnu.org/licenses/lgpl-3.0.en.html

http://www.coppeliarobotics.com

ACKNOWLEDGMENTS

This work has been supported by the ANR (French National Research Agency) SISCob project ANR-14-CE27-0016.


Kevin Nadaud
email: kevin.nadaud@univ-tours.fr

Caroline Borderon

Raphaël Renoud

Areski Ghalem

Aurelian Crunteanu

Laure Huitema

Frédéric Dumas-Bouchiat

Pascal Marchet

Corinne Champeaux

Hartmut W Gundel

Diffuse phase transition of BST thin films in the microwave domain

Keywords: Ferroelectrics, domain walls, hyperbolic law, microwave frequency, temperature

Ferroelectric materials are widely used in the paraelectric phase in order to realize tunable capacitors with reduced losses and low hysteresis effect. Nevertheless, for polycrystalline thin films, the ferro/paraelectric phase transition can be diffuse and some crystallites can exhibit a ferroelectric nature even if the sample seem to be globally in a paraelectric phase.

In this case, domain wall motions are responsible for the sensitivity of the dielectric properties to the driving field and are very dissipative phenomena. In this paper, we evaluate, in the high-frequencies band, the ferroelectricity impact on the properties of BST thin films near the phase transition, by measuring the film's dielectric properties as a function of the incident RF power and for different temperatures.

To handle the ever-increasing number of different transmission protocols at high transmission data rates, microwave devices should be more and more reconfigurable. The reconfigurability at device or sub-system levels (switches, filters, antennas,...) requires the employment of semiconductor-based or MEMS (MicroElectroMechanical Systems) components or functional materials (such as ferroelectrics) which can conveniently adapt the state of a device to a specific, required state (e.g. operating frequency, polarization ) [START_REF] Nadaud | Loughborough Antennas and Propagation Conference[END_REF][START_REF] Ghalem | Proc. IEEE Int. Symp. Applicat. Ferroelectrics[END_REF][START_REF] Hai | [END_REF] . The advantages of using ferroelectrics for tunable devices come from their large permittivity variation near the transition temperature with an applied external field, their low leakage current (since they behave like insulators) and their very fast response time. Ferroelectric materials can be thus conveniently used to realize variable capacitors for which the capacitance values can be tuned using a static electrical field.

The dielectric characterization of the material is obviously necessary in order to integrate them into a practical device. The non-linear characterizations of permittivity and losses as a function of frequency are usually done using high driving fields [4][5][6] , for which the non-linear phenomena arise by the harmonic generation. Nevertheless, at low driving fields 7,8 ferroelectric materials also exhibit a non-linear behavior due to the domain wall pinning/unpinning 9,10 . These non-linearities need to be taken into account for the integration of the material into a device since they are associated to very dissipative phenomena 11,12 which may considerably decrease the figure of merit of the material 8,13 (FoM, ratio between relative tunability and dielectric losses).

In the present paper we study the influence of the incident power and especially of the temperature on the dielectric properties of a ferroelectric thin film in the microwave range. We focus on moderate power levels (corresponding to a low driving field E AC ) in order to evaluate the influence of the domain wall motion (Rayleigh region) but also on the influence of high power levels in order to study the coalescence of domains.

At low fields, called the Rayleigh region 9 , the polarization state of the material remains identical and thus the hyperbolic law can be applied 10 :

ε r = ε rl + ε 2 r-rev + (α r E AC ) 2 , ( 1
)
where ε rl corresponds to the lattice contribution, ε r-rev to the domain wall vibration and α r to pinning/unpinning of the domain walls.

The present study is done on (100) epitaxial Ba 2/3 Sr 1/3 TiO 3 (BST) thin films deposited by pulsed laser deposition on a (100)-oriented MgO/Ir substrate. More details on the process and the structural characterization are reported elsewhere 7,14 . The film has a thickness of 0.45 µm and it For a single crystal having the studied composition, the Curie temperature is close to 15 • C 15 .

To verify the Curie temperature of our polycrystalline sample, the characteristic ε r (E DC ) loops where measured at 100 MHz at different temperatures (Fig. 1). At 20 17 . For a single crystal, the temperature of the ferro/paraelectric phase transition is well defined but in a polycristal, the size and shape distributions of the crystallites and the stress applied on the thin film can lead to a dispersion of the Curie temperature. If the overall behavior of the material seems clearly paraelectric on the ε r (E DC ) loop, it is necessary to study in more detail the dielectric characteristics of the samples in the transition region.

Metal-insulator-metal (MIM) capacitor has been chosen to perform the dielectric characterizations of the fabricated thin films. The advantage of this geometry is the possibility to perform measurements at low and high frequencies and the facility in extracting the material's permittivity using the well-known parallel plate expression: The complex capacitance, defined as C * = C(1j tan δ) similarly to the complex permittivity 4 , can be obtained using the reflection S -parameter Γ C : 18

C = ε 0 ε r S t BST , (2)
C * = 1 jωZ 0 1 -Γ C 1 + Γ C , (3)
where Z 0 is the reference impedance (50 Ω), ω the angular frequency and j the imaginary unit.

The capacitance and dielectric losses can thus be obtained:

C = Re [C * ] , tan δ = - Im [C * ] Re [C * ] , (4)
The power of the incident high-frequency signal has been varied from -20 dBm to 15 dBm corresponding to a maximum value of the driving field E AC = 80 kV/cm. In addition to the reflection calibration, a power calibration has been performed and the indicated power corresponds to the real power incident on the ferroelectric capacitor.

When using a VNA, only the incident power P RF can be set and a conversion into the actual driving field E AC 7 can be done using:

E AC = V AC t BST = |1 + Γ C | √ 2Z 0 P RF t BST , ( 5
)
with t BST the thickness of the thin film and Γ C the reflection coefficient of the load which can be written as:

Γ C = 1 -jωZ 0 C * 1 + jωZ 0 C * , (6)
where C * (P RF ) is the complex capacitance of the tunable material. For a given frequency ω, the reflection coefficient depends on the capacitance C * . As a consequence, the prefactor |1 + Γ C |, and thus the driving field, decrease when the frequency increases, even if the incident power is kept constant 7 . Fig. 2 shows the relative permittivity and the dielectric losses as a function of the frequency for different incident powers for a temperature of 30 • C. Up to 10 dBm the relative permittivity increases with rising incident power, which is due to the domain wall pinning/unpinning contribution to the permittivity 9,10 . At 100 MHz, the permittivity grows from 955 to 1033, when the incident power increases from -20 dBm to 10 dBm, which represents an increase of 8 %. This indicates that a non-negligible part of the crystallites is in the ferroelectric phase. This behavior has been already shown for this material at lower frequencies 19 . However, the fraction of crystallites in the ferroelectric phase does not seem sufficient in order to have a hysteresis on the ε r (E DC ) loop.

The higher the frequency, the less an increase of the permittivity is visible. This comes from the driving field which stays very low even if the incident power increases, due to the diminution of the prefactor |1 + Γ C | (5). Hence, the increase of the driving field is not sufficient to perceive the presence of the domain wall pinning/unpinning. To be able to observe their presence at frequencies higher than 1 GHz, a smaller device capacitance value would be needed.

The dielectric losses also increase when the incident power increases. Nevertheless, this varia- tion is higher than for the permittivity since at 100 MHz, the losses increase from 0.025 to 0.066, when the incident power varies from -20 dBm to 10 dBm, which represents an increase of 164 % (compared to 8 % for the permittivity). This reveals the very dissipative behavior of the domain wall pinning/unpinning and shows that this contribution should be reduced in order to limit the losses of the material influencing the overall behavior of a practical device.

Above 10 dBm and for frequencies below 200 MHz, the driving field is sufficient to cause the saturation of the material which consists of a decrease of the permittivity. The losses increase since a part of the incident power (at f 0 ) is absorbed and converted into harmonics (at k f 0 ) 4 . This saturation is not visible for higher frequencies since the prefactor |1 + Γ C | becomes small and the resultant driving field stays low.

Fig. 3 shows the real and imaginary parts of the permittivity and the dielectric losses as a function of the driving field for different temperature at a frequency of 100 MHz. The imaginary part of the permittivity has been calculated using ε r = ε r tan δ. The driving field has been calculated using (5) for each temperature. For the three temperatures, the evolutions with the driving field are very similar showing that a non negligible part of the crystallites of the material presents a ferroelectric behavior showing the diffuse nature of the transition.

For driving fields, below 27 kV/cm, the real and imaginary parts of the permittivity increase, due to the domain wall pinning/unpinning contribution, similarly to what have been noticed previously. The dielectric losses also increase showing the increase of the imaginary part is proportionally higher than for the real part, exhibits the dissipative behavior of the domain wall pinning/unpinning [11][12][13] . Again, this is due to the diffuse behavior of the transition. The presence of ferroelectric crystallites remains up to 40 • C and possibly at higher temperature 19 since an increase of the real part of the permittivity is well visible (7.6 %) even at 40 • C.

For driving fields above 27 kV/cm, the real part of the permittivity decreases due to the coalescence of domains. This decrease of the permittivity comes from decrease of the number of domain.

The driving field is sufficient to change the polarization state of the material. In this region, the permittivity diverges from the fit, showing that the hyperbolic law cannot be applied. The imaginary part of the permittivity still increases since a part of the power is absorbed to switch a part of the polarization. Since the imaginary part increases and the real part decreases, the dielectric losses (ratio between imaginary and real parts) become higher.

Rather stable dielectric losses are observed for driving fields higher than 60 kV/cm. Different conflicting phenomena, influencing the real and the imaginary parts of the permittivity, may explain this behavior. As the driving field increases, losses also increase due to enhanced domain wall pinning/unpinning and a part of the power is absorbed to partially switch the polarization. In turn, polarization switching causes a diminution of the number of domains. As the contributions of domain wall vibration and pinning/unpinning to the complex permittivity is proportional to the number of domain walls 20 , a diminution of permittivity and dielectric losses is observed when global switching occurs. The observed "saturation" of the dielectric losses seems to be a compromise between these phenomena. It should be pointed out, that the variation of the permittivity and the losses can not be attributed to a heating effect as in 4 since this would affect the dielectric properties in the whole frequency band, which is not the case here (Fig. 2).

Table I: Hyperbolic coefficients of the real and imaginary parts of the relative permittivity and associated dissipation factor for each contribution to the permittivity at a frequency of 100 MHz. The real and imaginary parts of the permittivity have been fitted using the hyperbolic law and the extracted coefficients are reported in the Table I. The fitting has been made using the Levenberg-Marquat algorithm which also gives the uncertainty on the extracted parameters (ε rl , ε r-rev and α r ). Using real and imaginary coefficients, it is possible to calculate the dissipation factor of all contributions m ε = ε ε with ε corresponding to ε rl , ε r-rev and α r . Since the dissipation factor is the ratio between imaginary and real parts, its uncertainty has been calculated using the following formula:

δm ε = δε ε + δε ε m ε (7)
δε and δε are respectively the uncertainty associated the imaginary and real parts of the considered contribution. The precision on the extraction of the bulk values and the pinning/unpinning coefficients is quite good but the uncertainty is higher for the domain vibration contribution since the value is small compared to the bulk. As a consequence, the uncertainty on the dissipation factor associated to the bulk and pinning/pinning contributions are relatively low.

The real and imaginary parts follow well the hyperbolic law (full line). When the temperature increases, the slope of the real and imaginary parts versus the driving field, represented by the Rayleigh coefficient α r , decrease (Table I). This diminution indicates that the proportion of ferroelectric crystallites in the material decreases when the temperature increases, which has already been reported previously 19 . This decrease is progressive due to the polycristalline structure of the material. The parameters representing the domain vibration ε r-rev are relatively hard to obtain since they are low compared to the lattice parameter (0.5 %). As a consequence, the evolution with the temperature is not easy to determine.

The dissipation factors for the domain wall contributions are one order of magnitude higher than the one associated to the lattice, confirming the extremely dissipative behavior 11,12 . When the temperature increases, the dissipation factor of the lattice contribution decreases which is coherent with a material going to the paraelectric phase. On the contrary, the dissipation factor for the domain wall pinning/unpinning slightly increases. This indicates that when the temperature increases, the domain wall motion is more dissipative. As the temperature increases and the ferroelectric behavior decreases, the polarization in the region surrounding pinning sites can be reoriented at reasonable electric fields. This can be seen on the unpinning threshold field (E th = ε r-rev /α r 10 ) which seems to slightly decrease when the temperature increases, even if the precision on this parameter is quite low (Table I). Therefore, domain walls are more mobile due to unpinning at reasonable applied fields, leading to higher dissipative losses.

In this paper, the dielectric properties of a BST thin film have been studied in the microwave domain, as a function of the incident power and for different temperatures. When the incident power increases, the permittivity increases due to the domain wall pinning/unpinning contribution. A high increase of the dielectric losses is also noticed which is attributed to the very dissipative behavior of the domain wall pinning/unpinning. This proves the diffuse nature of the ferro/paraelectric transition which needs to be considered for the characterization or the use of the material into a device. When the temperature increases, the domain wall pinning/unpinning contribution decreases indicating that the ferroelectric nature of the material "progressively vanishes". This clearly shows that the characterization of functional material needs to be made for the foreseen power since the non-linearities of the ferroelectrics are present even at low and high driving fields.

Real permittivity

Figure 1 :

 1 Figure 1: Relative permittivity of the BST thin film measured at 100 MHz as a function of the bias field and for different temperatures.

with ε 0

 0 = 8.85 × 10 -12 F/m the vacuum permittivity, ε r the relative permittivity of the material, S the area of the electrode and t BST the thickness of the material. The bottom electrode of the MIM structure consists of an iridium layer (100 nm) while the square top electrode (30 × 30 µm 2 ) is made of gold/titanium (200 nm/10 nm). The capacitor is measured using a Cascade ground-signal-ground (GSG) probe of 125 µm pitch and a vector network analyzer (VNA, Rohde & Schwarz ZVA24) in the 50 MHz-2 GHz frequency band. The measurements have been carried out at different temperatures (20 • C, 30 • C and 40 • C). The range of temperature is imposed by our test-bench, consisting of a Peltier heat pump, which is not able to regulate the temperature outside this range.

Figure 2 :

 2 Figure 2: Relative permittivity (a) and dielectric losses (b) of the BST layer integrated in a MIM device, as a function of the frequency for different incident power, at 30 • C.

Figure 3 :

 3 Figure 3: Real (a) and imaginary (b) parts of the relative permittivity and associated hyperbolic fits and the dielectric losses (c) as a function of the driving field for different temperatures, at 100 MHz.

ε

  rl ε r-rev α r (cm/kV) ε rl ε r-rev α r (cm/kV) m ε rl m ε r-rev m α r E th 20 • C 900 ± 2 8 ± 3 4.29 ± 0.08 23.2 ± 0.7 1 ± 1 1.50 ± 0.05 0.027 ± 0.001 0.3 ± 0.4 0.35 ± 0.02 1.9 ± 0.6 30 • C 947 ± 3 6 ± 5 3.3 ± 0.2 22.5 ± 0.3 1 ± 1 1.30 ± 0.02 0.024 ± 0.001 0.3 ± 0.4 0.39 ± 0.03 1.8 ± 0.9 40 • C 946 ± 2 5 ± 3 2.77 ± 0.08 20.5 ± 1.6 1 ± 1 1.13 ± 0.01 0.022 ± 0.002 0.3 ± 0.4 0.42 ± 0.02 1.8 ± 0.9

P

  RF = -20 dBm P RF = -10 dBm P RF = 0 dBm P RF = 10 dBm P RF = 12 dBm P RF

O. Boser, Journal of Applied Physics 62, 1344 (1987).

The authors gracefully acknowledge L. Trupina and L. Nedelcu from the National Institute of Materials Physics (Romania) for the iridium bottom electrode deposition. The authors from XLIM and IRCER acknowledge the financial support from the France Region Limousin under the H2020 European project MASTERS (M-ERA.NET call).


Dimitri Peaucelle

Yoshio Ebihara

Affine Versus Multi-Affine Models for S-Variable LMI Conditions *

Keywords: LMIs, Polytopic uncertainties, S-variables, Robustness, Descriptor systems

This paper discusses and compares LMI results built using the S-variable approach starting from equivalent yet different representations of uncertain systems. Using the fact that S-variable results are well suited to handle descriptor systems and that descriptor system modeling is versatile, we compare results in terms of the impact of modeling on the computational burden and on conservatism. Multi-affine representations allow reduced numerical burden while affine representations lead to less conservative results. Numerical examples show that conservatism reduction is not systematic, but is in some cases quite significant without major increase of the numerical burden.

Introduction

During the past twenty years an efficient method for handling affine polytopic uncertainties which involves additional S-variables has been intensively used for robustness purpose. The method originated in [START_REF] Geromel | LMI characterization of structural and robust stability[END_REF][START_REF] De Oliveira | A new discrete-time stability condition[END_REF][START_REF] Peaucelle | A new robust D-stability condition for real convex polytopic uncertainty[END_REF] and has had many derivations. The book [START_REF] Ebihara | S-Variable Approach to LMI-based Robust Control[END_REF] discusses many of these, including analysis/control design problems, and conservatism/numerical complexity issues. Among the nice features of the approach which is noticed as soon as [START_REF] Trofino | Robust stability and domain of attraction of uncertain nonlinear systems[END_REF][START_REF] Coutinho | Regional stability analysis of implicit polynomial systems[END_REF][START_REF] Trofino | LMI stability conditions for uncertain rational nonlinear systems[END_REF][START_REF] De Oliveira | Perspectives in Robust Control, chapter Stability tests for constrained linear systems[END_REF] is that it is well suited for systems in descriptor form, and this descriptor form is appropriate for manipulating systems rational in the uncertainties (or non-linearities), as if affine [START_REF] Masubuchi | Synthesis of output-feedback gain-scheduling controllers based on descriptor LPV system representation[END_REF][START_REF] Coutinho | Regional stability analysis of implicit polynomial systems[END_REF][START_REF] Trofino | LMI stability conditions for uncertain rational nonlinear systems[END_REF]. Any rational in the uncertainties linear system can be converted to a descriptor linear system with affine dependency in the parameters for which S-variable results apply readily. Moreover, it is shown in [START_REF] Peaucelle | Robust observed-state feedback design for discrete-time systems rational in the uncertainties[END_REF] that S-variable results apply as well to models which are multi-affine in the parameters, a modeling inspired form multi-simplex models from [START_REF] Oliveira | Robust LMIs with parameters in multi-simplex: Existence of solutions and applications[END_REF][START_REF] Morais | H 2 control of discrete-time Markov jump linear systems with uncertain transition probability matrix: Imporved linear matrix inequality relaxations and multisimplex modelling[END_REF], which may as well be represented using tensors as exploited in [START_REF] Baranyi | Tensor Product Model Transformation in Polytopic Model-Based Control[END_REF] that provide related LMI results for Takagi-Sugeno fuzzy systems.

One conclusion drawn from [START_REF] Peaucelle | Robust observed-state feedback design for discrete-time systems rational in the uncertainties[END_REF] is that S-variable results apply readily to affine and to multi-affine descriptor models. But, in case a system has a multi-affine representation it can always be converted to an affine representation. This is usually at the expense of increasing the size of the descriptor model and hence increases the numerical burden for the solvers in the end. If we employ a multi-affine representation, we obtain an LMI that is more conservative but computationally less demanding, whereas if we employ an affine representation, we obtain an LMI that is less conservative but computationally more demanding. The question addressed in this paper is whether the increase of the numerical burden worths the effort in terms of reduced conservatism. We provide notations for handling the affine and multi-affine models. We show the intrinsic source of conservatism in the multi-affine results and we show on examples that it is indeed the case, but not always.

Results are compared numerically to Pólya based results for robust stability analysis ( [START_REF] Scherer | Relaxations for robust linear matrix inequality problems with verifications for exactness[END_REF][START_REF] Scherer | LMI relaxations in robust control[END_REF][START_REF] Oliveira | A simple and less conservative test for D-stability[END_REF]). The comparison reveals the relevance of the proposed methodology in terms of the tradeoff conservatism/complexity. For space limitation reasons we did not perform comparions with sum-of-squares type results that may be applied as well. The reason why we concentrated the comparison with the Pólya approach is that [START_REF] Peaucelle | LMI tests for positive definite polynomials: Slack variable approach[END_REF] shows that in the considered situation S-variable results include those issued from sum-of-squares methodology.

The outline of the paper is a follows. The second section is devoted to multi-affine modeling using multi-simplex representation of uncertainties. The third section recalls the S-variable results for descriptor systems (for the simplest case when the E matrix is full column rank, for more advanced results the reader is invited to consult [START_REF] Ebihara | S-Variable Approach to LMI-based Robust Control[END_REF]). The fourth section is then dedicated to numerical examples which illustrate the impact of modeling on the conservatism of LMI results.

Notation: I n stands for the identity matrix of size n. 0 n,m is the zero matrix with n rows and m columns. When the size can be deduced from the context the subscripts are avoided. A T is the transpose of the matrix A. A B is the matrix inequality stating that A -B is symmetric positive definite. The terminology "congruence operation of A on B" is used to denote A T BA. If A is full column rank, and B 0, the congruence operation of A on B gives a positive definite matrix: A T BA 0. A matrix inequality of the type N (X) 0 is said to be a linear matrix inequality (LMI for short), if N (X) is affine in the decision variables X. In the following, decision variables are highlighted using the blue color.

Ξ v = {ξ v=1...v ≥ 0, v v=1 ξ v = 1} is the unitary simplex in R v .
The elements ξ of unitary simplexes are used to describe polytopic type uncertainties. In the following, uncertainties are highlighted using the red color. The vertices of Ξ v are the v vectors ξ [v] with all zeros coefficients except one equal to 1.

Polytopes and Multi-affine representations

We shall consider uncertain representations depending on parameters θ assumed to lie in a set Θ defined as the cross product of p polytopes:

θ ∈ Θ = {(θ 1 , . . . , θ p) ∈ Θ 1 × • • • × Θ p} . (1)
The p components of θ are independent vectors θ p ∈ R mp . Each set Θ p is assumed to be a polytope with vp vertices from the set

V p = θ [1] p , . . . , θ [vp] p
. Θ p is the convex hull of the vertices, or equivalently, each θ p writes as the weighted sum of vertices with weight from unitary simplexes:

Θ p = Co(V p ) = θ p = vp v=1 ξ p,v θ [v] p : ξ p ∈ Ξ vp . (2)
In the following,

V = V 1 × • • • × V p
is the finite set of all extremal values of the parameters gathered in θ. A generic element of V will be denoted θ [v] with v = (v 1 , . . . , v p) the vector of indices of vertices for each component. I is the set of all vectors of indices v. θ [v] is the one to one mapping from I to V. The cardinality of V is v = Π p p=1 vp . Choosing any ordering of the v components of V the set shall also be described using the notation V = θ [1] , . . . θ [ṽ] , . . . θ [ v] where ṽ is a scalar indexing of the v elements.

A matrix M (θ) is said to be multi-affine in the parameters if it is affine in each θ p taken independently. A multiaffine matrix can be written as the multi-sum of weighted vertices from V denoted as

M (θ) = v∈I ξ 1,v1 • • • ξ p,vp M (θ [v] ) : ξ p ∈ Ξ vp (3)
As we will see on an example, it is quite trivial to notice that any such matrix belongs as well to the polytopic set defined by

M ( θ) = v ṽ=1 ξ ṽ M (θ [ṽ] ) : ξ ∈ Ξv (4)
The simplest case is when the θ p elements are scalars (m p = 1) defined in intervals

θ p ∈ [ θ [1] p , θ [2]
p ], which are polytopes of vp = 2 vertices. For the case when all elements are scalar, the cardinality of V is v = 2 p. In case of two scalar parameters, the multi-sum reads as

M (θ) = v∈{(1,1),(1,2),(2,1),(2,2)} ξ 1,v1 ξ 2,v2 M (θ [v] ) = ξ 1,1 ξ 2,1 M (θ [(1,1)] ) + ξ 1,1 ξ 2,2 M (θ [(1,2)] ) + ξ 1,2 ξ 2,1 M (θ [(2,1)] ) + ξ 1,2 ξ 2,2 M (θ [(2,2)]
).

The following fact

ξ 1,1 ξ 2,1 + ξ 1,1 ξ 2,2 + ξ 1,2 ξ 2,1 + ξ 1,2 ξ 2,2 = ξ 1,1 + ξ 1,2 = 1
allows to conclude that the multi-affine matrix M (θ) is included in the polytope of four vertices M (θ

[(1,1)] ), M (θ [(1,2)] ), M (θ [(2,1)] ), M (θ [(2,2)]
). The converse is not true in general. There are potentially elements in the convex hull of these four vertices that are not in the multi-affine model. An example of this is the matrix

M (θ) = θ 1 θ 1 θ 2 θ 2 with θ 1 ∈ [1, 2] and θ 2 ∈ [1, 2]. The middle of vertices 1 1 1 and 2 4 2 is equal to 3 2 5 2 3 2 . It is inside polytope (4), but since 5 2 = ( 3 2 ) 2 it is not a realization (3) of M (θ).

LMI S-Variable conditions

The previous section allows to conclude about the fact that multi-affine polytopes can be considered as included in polytopes of larger size. This might bring some conservatism. To evaluate this fact we study the LMI conditions for robust analysis of polytopic and multi-polytopic systems. To do so we consider one type of LMI conditions issued from the S-Variable approach.

Let an uncertain descriptor systems described by the following equation:

E xx (θ) ẋ(t) + E πx (θ)π(t) = A(θ)x(t) (5)
where x ∈ R nx is the state and π ∈ R nπ is some auxiliary signal linked to the state via algebraic constraints. Neither E xx (θ) nor A(θ) are supposed to be square and have n rows. For simplicity of the presentation the matrix E xx (θ) E xπ (θ) is assumed to be full column rank for all θ ∈ Θ. More general cases are described in [START_REF] Ebihara | S-Variable Approach to LMI-based Robust Control[END_REF].

For conciseness of notation denote

M (θ) = E xx (θ) E πx (θ) -A(θ)
the uncertain matrix which fully describes the dynamics of the system.

Proposition 1 If the system ( 6) is defined using a multi-affine matrix M (θ) then there exists another augmented representation

Êxx (θ) ẋ(t) + Êπx (θ)π(t) = Â(θ)x(t) (6)
in which the matrix of size n-by-(2n x + nπ ), where n ≥ n and nπ ≥ n π ,

M (θ) = Êxx (θ) Êπx (θ) -Â(θ)
is affine in the parameters.

Rather than proving the proposition for a general representation we shall consider an illustrative example. The proof of the general case follows the same lines. Let the system described by

1 θ 1 0 1 ẋ(t) = -θ 2 0 θ 1 θ 2 -1 x(t). (7)
This multi-affine representation admits the following equivalent affine representation

  1 θ 1 0 1 0 0   ẋ(t) +   0 -θ 1 1   π(t) =   -θ 2 0 0 -1 θ 2 0   x(t). (8)
The proof comes from the fact the last row of the affine representation gives π = θ 2 x 1 , which when included in the second row, gives exactly the multi-affine representation.

Theorem 1 The uncertain system (6) defined by a multi-affine matrix M (θ) is robustly stable if there exist v n x -by-n x symmetric matrices P [v] 0 and a common to all inequalities (2n x + n π )-by-n matrix S such that for all vertices v ∈ I the following LMIs hold

Q(P [v] ) ≺ SM (θ [v] ) + M T (θ [v] )S T (9)
where

Q(P [v] ) =   0 nx,nx 0 P [v] 0 0 nπ,nπ 0 P [v] 0 0 nx,nx   .
Proof If the inequalities of Theorem 1 hold for all vertices v ∈ I, these hold as well for all reordered vertices θ [ṽ] with ṽ = 1 . . . v. By convexity of matrix inequalities this implies that the inequalities hold for the convex hull of the vertices, that is for all ξ ṽ ∈ Ξv one has:

Q(P ( θ)) ≺ SM ( θ) + M T ( θ)S T (10)
for the following parameter-dependent Lyapunov matrix

P ( θ) = v ṽ=1
ξ ṽ P [ṽ] .

Congruence of η = ẋT π T x T T on inequality [START_REF] Masubuchi | Synthesis of output-feedback gain-scheduling controllers based on descriptor LPV system representation[END_REF] gives that ẋT (t)P ( θ)x(t) + x T (t)P ( θ) ẋT (t) < 0 holds along the trajectories of M ( θ)η(t) = 0. Hence robust stability of the embedding polytopic uncertain system is proved by the parameter-dependent Lyapunov function x T P ( θ)x. The multi-affine representation M (θ) being included inside the polytopic embedding M ( θ), stability of the multi-affine model is proved. The proof is well known but is recalled here because it clarifies that these conditions for proving robust stability of multi-affine models are equivalent to proving robust stability of a larger embedding polytopic set of models.

An alternative method to prove robust stability of the multi-affine models is to take advantage of the equivalent augmented representation proposed in Proposition 1.

Corollary 1 The uncertain system (6) defined by a multi-affine matrix M (θ) is robustly stable if there exist v n x -by-n x symmetric matrices P [v] 0 and a common to all inequalities (2n x + nπ )-by-n matrix Ŝ such that for all vertices v ∈ I the following LMIs hold

Q( P [v] ) ≺ Ŝ M (θ [v] ) + M T (θ [v] ) ŜT (11)
where

Q( P [v] ) =   0 nx,nx 0 P [v] 0 0 nπ,nπ 0 P [v] 0 0 nx,nx   .
At this stage here are the characteristics of the two results.

• Theorem 1 is such that:

The decision variables are the v matrices P [v] of size n x -by-n x and a (2n x + n π )-by-n matrix S;

The v constraints (9) have (2n x + n π ) rows (and as many columns); Stability also holds for a polytopic model in which the multi-affine representation is embedded.

• Corollary 1 is such that:

The decision variables are the v matrices P [v] of size n x -by-n x and a (2n x + nπ )-by-n matrix S;

The v constraints [START_REF] Morais | H 2 control of discrete-time Markov jump linear systems with uncertain transition probability matrix: Imporved linear matrix inequality relaxations and multisimplex modelling[END_REF] have (2n x + nπ ) rows (and as many columns).

Theorem 1 is hence potentially more conservative than Corollary 1 but is of smaller size (both in number of decision variables and in size of the constraints). The increase in size of LMIs of Corollary 1 compared to those of Theorem 1 is related to the increase from n π to nπ . This increase is at most equal to the number of products between parameters in the original multi-affine representation.

Since the goal of the paper is to compare results in terms of conservatism and numerical complexity, we shall also compare these to another LMI-based approach. The comparison is done with respect to the Pólya based method [START_REF] Scherer | Relaxations for robust linear matrix inequality problems with verifications for exactness[END_REF][START_REF] Scherer | LMI relaxations in robust control[END_REF][START_REF] Oliveira | A simple and less conservative test for D-stability[END_REF] when applied to inequality

Q(P (θ)) ≺ S(θ)M (θ) + M T (θ)S(θ) T (12)
with parameter dependent S-variables S(θ). The choice of a multi-affine P (θ) and a common to all S(θ) = S gives exactly condition of Theorem 1. Choosing polynomial P (θ) of degree higher than one and polynomial S(θ) leads to LMIs of decreasing conservatism. Conservatism is know to be vanishing as the order grows.

4 Numerical examples

Example 1

The LMI conditions are tested on the example [START_REF] Ebihara | Robust performance analysis of linear time-invariant uncertain systems by taking higher-order time-derivatives of the states[END_REF] For θ 1 = 0 and θ 2 = 0, which is a possible realization when δ = 1, the system is not asymptotically stable. For this example Theorem 1 is highly conservative while Corollary 1 is not. Conservatism reduction thanks to Corollary 1 is not at the expense of a major increase of the computation burden.

These results are now compared to those obtained using Pólya methodology. Table 2 gives the results for different choices of parameter-dependent S-variables. In all cases P (θ) is chosen to be multi-affine in the uncertainties. Four choices are tested: (1a) S(θ) = S is parameter-independent;

(1b) S(θ) = S(θ 1 ) is affine in the first uncertainty;

(1c) S(θ) = S(θ 2 ) is affine in the second uncertainty;

(1d) S(θ) is multi-affine in the two uncertainties. Results of Table 2 illustrate that it is indeed possible to get non-conservative results starting from the multi-affine representation by applying the Pólya methodology with parameter-dependent S-variables. But, in comparison with our proposed strategy consisting in converting the multi-affine problem into an affine representation, the results using parameter-dependent S-variables lead to numerically more demanding LMIs. Our interpretation of this fact is that our proposed methodology exploits more accurately the structure of the problem while systematic methods for handling polynomially parameter-dependent LMIs tend to over parameterize the LMIs.

Example 2

We consider now a more complex example with three uncertain parameters:

    1 θ 1 0 0 0 1 θ 3,1 0 0 0 1 0 0 θ 1 θ 2 θ 2 θ 3,1 1     ẋ =     -θ 2 0 0 0 θ 1 θ 2 -1 0 0 θ 1 θ 3,2 0 -1 0 θ 1 θ 2 θ 3,2 0 0 -1     x (13)
with

θ 1 ∈ Co ( -10δ , +10δ ), θ 2 ∈ Co ( 1 , 1 -δ ) and θ 3 = θ 3,1 θ 3,2 ∈ Co δ 0 , 0 δ , -δ -δ .
Choose π 1 = θ 1 θ 3,2 0 0 0 x -0 θ 1 θ 3,1 0 ẋ to get this other multi-affine model

      1 θ 1 0 0 0 1 θ 3,1 0 0 0 1 0 0 0 0 1 0 θ 1 θ 3,1 0       ẋ +       0 0 0 θ 2 1       π 1 =       -θ 2 0 0 0 θ 1 θ 2 -1 0 0 θ 1 θ 3,2 0 -1 0 0 0 0 -1 θ 1 θ 3,2 0 0 0       x. ( 14
)
Choose π 2 = θ 1 0 0 0 x to get this third multi-affine model

      1 θ 1 0 0 0 1 θ 3,1 0 0 0 1 0 0 θ 1 θ 2 θ 2 θ 3,1 1 0 0 0 0       ẋ +       0 -θ 2 -θ 3,2 -θ 2 θ 3,2 1       π 2 =       -θ 2 0 0 0 0 -1 0 0 0 0 -1 0 0 0 0 -1 θ 1 0 0 0       x. ( 15
)
Combining π 1 and π 2 , the system has the following affine representation

        1 θ 1 0 0 0 1 θ 3,1 0 0 0 1 0 0 0 0 1 0 θ 1 θ 3,1 0 0 0 0 0         ẋ +         0 0 0 -θ 2 0 -θ 3,2 θ 2 0 1 -θ 3,2 0 1         π =         -θ 2 0 0 0 0 -1 0 0 0 0 -1 0 0 0 0 -1 0 0 0 0 θ 1 0 0 0         x. (16)
Models ( 13), ( 14) and ( 15) are multi-affine, and ( 16) is affine. The LMI results are tested for each of these models and results are summarized in Table 3. In this example the LMIs built based on models ( 13) and ( 14) give exactly the same results in terms of conservatism but for an increased numerical burden in case of model [START_REF] Peaucelle | A new robust D-stability condition for real convex polytopic uncertainty[END_REF]. The same comment holds when comparing the LMIs built based on models [START_REF] Peaucelle | Robust observed-state feedback design for discrete-time systems rational in the uncertainties[END_REF] and ( 16). This illustrates the fact that results for multi-affine representations are not necessarily more conservative. But the results for affine representations are, as guaranteed by the theory, the less conservative ones, and the more demanding in terms of numerical complexity.

In order to reduce further the conservatism [START_REF] Ebihara | Robust performance analysis of linear time-invariant uncertain systems by taking higher-order time-derivatives of the states[END_REF] suggests to augment the model representation by including higher derivatives of the states. See also [START_REF] Ebihara | S-Variable Approach to LMI-based Robust Control[END_REF] for discussions about this method. For the treated example, starting from model [START_REF] Peaucelle | A new robust D-stability condition for real convex polytopic uncertainty[END_REF] and including the second derivative of the state, it amounts to considering the augmented system:

  E x(14) (θ) 0 0 E x(14) (θ) 0 I   ẍ ẋ +   E π(14) (θ) 0 0 E π(14) (θ) 0 0   π1 =   A (14) (θ) 0 0 A (14) (θ) I 0   ẋ x
where

E x(14) (θ) =       1 θ 1 0 0 0 1 θ 3,1 0 0 0 1 0 0 0 0 1 0 θ 1 θ 3,1 0       E π(14) (θ) =       0 0 0 θ 2 1       A (14) (θ) =       -θ 2 0 0 0 θ 1 θ 2 -1 0 0 θ 1 θ 3,2 0 -1 0 0 0 0 -1 θ 1 θ 3,2 0 0 0      
We do not provide all the formulas, but the augmentation procedure applies the same way to all four models (13), ( 14), ( 15), ( 16) and we shall denote (13-a), (14-a), (15-a), (16-a) the augmented versions of these. In the augmented models the last rows correspond to the fact that ẋ and ẋ are the same vectors on both sides of the equality constraint. As discussed in [START_REF] Ebihara | S-Variable Approach to LMI-based Robust Control[END_REF] the method hence includes the information that the first derivatives of the uncertain parameters are zero. The rows involved to include this key information are parameter independent and this fact can be used to reduce partly, and without conservatism, the numerical burden (see Chapter 3 in [START_REF] Ebihara | S-Variable Approach to LMI-based Robust Control[END_REF]). Table 4 provides the results for the augmented systems. The size of the LMIs and the number of decision variables are while employing this size reduction technique. Same conclusions apply as for Table 3. Moreover, as expected, the LMIs tested for Table 4 are less conservative than those for Table 3 when comparing results based augmented and non-augmented models. What is more surprising is the conservatism of results based on model (13-a) compared to those based on model [START_REF] Peaucelle | Robust observed-state feedback design for discrete-time systems rational in the uncertainties[END_REF]. With less decision variables, and without introducing the knowledge about derivatives of the parameters, the results based on (15) are less conservative.

For comparison, we apply the Pólya methodology to the parameter-dependent inequality [START_REF] Oliveira | Robust LMIs with parameters in multi-simplex: Existence of solutions and applications[END_REF] for different choices of parameter-dependent variables. Two cases are tested, in both of these P (θ) is taken multi-affine, and (2a) S(θ) = S is common to all uncertainties (result coincides with conditions of Theorem 1);

(2b) multi-affine S(θ).

Results are given in Table 5 and illustrate once again that our proposed strategy has advantages in terms of the numerical burden versus conservatism. To finalize the study of this example it should be said that for δ = 0.644 the vertex defined by the following values θ 1 = 6.44, θ 2 = 0.3560, θ 3 = -0.644 -0.644 gives an unstable system. LMI results based on models (15-a) and (16-a) are hence non-conservative.

Example 3

Now we modify slightly Example 2 and replace the set where the θ 3 parameter lies to the following polytope of four vertices

θ 3 = θ 3,1 θ 3,2 ∈ Co δ 0 , 0 δ , -δ 0 , 0 -δ .
Results are given in Table 6 where (13-0) indicates that the model used is [START_REF] Oliveira | A simple and less conservative test for D-stability[END_REF] and no augmentation is applied; indicates that a first augmentation including ẍ in the equations is applied as in Example 2; indicates that an other augmentation including x (3) in the equations is applied. We do not give results based on equations ( 14) and ( 16) because these are once again strictly the same as for LMIs built based on equations ( 13) and ( 15) respectively. Only the maximal values of δ for which the LMI results are feasible are given.

For the third augmentation that involves derivatives of the state up to x (4) the LMI results provide no measurable improvement compared to the second augmentation (at least with the precision of thee digits that we used). But, applying results from [START_REF] Scherer | Relaxations for robust linear matrix inequality problems with verifications for exactness[END_REF][START_REF] Scherer | LMI relaxations in robust control[END_REF] on the dual variables of the LMIs obtained for (16-3) with δ = 0.850 we were able to find a de-stabilizing worst case:

θ 1 = 8.4995, θ 2 = 0.15, θ 3 = 0.3947 0.4552
for which the poles of the system are 0.0030 ± 3.4839i, -0.0124, -1.0000.

There is clearly an unstable pair of poles which indicates that the upper bound δ = 0.849 obtained at the second relaxation is tight (at the precision of three digits that we have chosen). It should be noticed that without the results of [START_REF] Scherer | Relaxations for robust linear matrix inequality problems with verifications for exactness[END_REF][START_REF] Scherer | LMI relaxations in robust control[END_REF] which were implemented using the methodology described in [START_REF] Ebihara | S-Variable Approach to LMI-based Robust Control[END_REF], it would have been quite complex to find this worst case that is not at one of the v = 16 vertices. We now compare these results with Pólya methodology. 5 cases are tested:

(3a) P ( θ) is multi-affine and S( θ) = S is common to all uncertainties (identical results to (13-0));

(3b) both P ( θ) and S( θ) are multi-affine;

(3c) P ( θ) is polynomial or degree 2 in all the parameters and S( θ) is multi-affine;

(3d) P ( θ) and S( θ) are polynomial or degree 2 in all the parameters;

(3e) P ( θ) is polynomial of degree 3 in all the parameters and S( θ) is polynomial of degree 2.

Results are given in Table 7 and illustrate once again that Pólya based results tend to be numerically more demanding compared to our approach, especially in terms of the size of LMI constraints. In terms of computation, all tests have been made on a MacBook Pro 2.9 GHz Intel Core i5 with Matlab2016b. LMIs were coded using YALMIP (R20141030 release) [9] and solved using SDPT3 (version 4.0) [START_REF] Toh | SDPT3 -a MATLAB software package for semidefinite programming[END_REF]. Solver time for the 4 LMI problems built for models (13-3), (14-3), and for δ = 0.849 is about 70 seconds. The LMIs for the comparison with Pólya based results were build using the ROLMIP toolbox [START_REF] Agulhari | Robust LMI Parser: a computational package to construct LMI conditions for uncertain systems[END_REF].

The code of the examples is available on the web at:

http://homepages.laas.fr/peaucell/papers/rocond18.m

Conclusions

The goal of this paper is to discuss the conservatism of S-variable LMI results when these are applied to multi-affine representations. We have shown that at the expense of a minor augmentation of the numerical burden it is preferable to convert the models into affine representations. Doing so does not increase much the numerical burden, but reduces the conservatism quite significantly on examples. Yet it is not always the case, and we provide examples of the two situations. Methods to detect beforehand if the removal of some of the products between parameters will indeed reduce the conservatism is left as an open question.

Table 1 :

 1 Example 1

	δ	0.556 0.557 0.999 1 nb vars nb rows
	Theorem 1	OK	-	-	-	20	24
	Corollary 1	OK	OK	OK	-	27	28
	last columns indicate the number of decision variables and the total number or rows of the LMI constraints (including
	contraints P [v] 0).

with θ 1 ∈ [ -10δ , 10δ ] and θ 2 ∈ [ 1 , 1 -δ ]. The LMI conditions are tested for different values of δ and are summarized in Table 1. OK indicates that the LMIs are feasible for this value of δ. Corollary 1 is feasible for larger values of δ thus illustrating that it is less conservative than Theorem 1. The

Table 2 :

 2 Example 1 -Pólya method

	δ	0.556 0.557 0.999 1 nb vars nb rows
	(1a)	OK	-	-	-	20	24
	(1b)	OK	-	-	-	28	32
	(1c)	OK	OK	OK	-	28	32
	(1d)	OK	OK	OK	-	44	44

Table 3 :

 3 Example 2

	δ	0.469 0.470 0.602 0.603 nb vars nb rows
	(13)	OK	-	-	-	152	144
	(14)	OK	-	-	-	165	156
	(15)	OK	OK	OK	-	165	156
	(16)	OK	OK	OK	-	180	168

Table 4 :

 4 Example 2 with augmentation

	δ	0.541 0.542 0.643 0.644 nb vars nb rows
	(13-a)	OK	-	-	-	528	240
	(14-a)	OK	-	-	-	572	264
	(15-a)	OK	OK	OK	-	572	264
	(16-a)	OK	OK	OK	-	624	288

Table 5 :

 5 Example 2 -Pólya method

	δ	0.469 0.470 0.643 0.644 nb vars nb rows
	(2a)	OK	-	-	-	152	144
	(2b)	OK	OK	OK	-	504	480

Table 6 :

 6 Example 3 with augmentation

	δ	nb vars nb rows
	(13-0) 0.502	192	192
	(15-0) 0.659	205	208
	(13-1) 0.587	672	320
	(15-1) 0.841	716	352
	(13-2) 0.587	1440	448
	(15-2) 0.849	1533	496
	(13-3) 0.587	2496	576
	(15-3) 0.849	2656	640

Table 7 :

 7

	δ	nb vars nb rows
	(3a) 0.502	192	192
	(3b) 0.752	672	784
	(3c) 0.809	1412	1080
	(3d) 0.847	3780	2920
	(3e) 0.849	6080	3840


Mathieu Avanzi
email: mathieu.avanzi@unine.ch

Lisa Brunetti
email: lisa.brunetti@univ-lyon2.fr

Cédric Gendrot
email: cgendrot@univ-paris3.fr

A Study of Clitic-Left Dislocation in Spontaneous French

Keywords: clitic left dislocation, prosodic phrasing, prosodic restructuring, prominence, topic, salience, spontaneous French

come

Introduction

In the Prosodic Theory framework [START_REF] Selkirk | Phonology and syntax. The relations between sound and structure[END_REF], French clitic-left dislocated elements, as the NP ma mère in the utterance ma mère, elle est institutrice 'My mother, she is a teacher', are predicted to be phrased as Intonational Phrases (IP) [START_REF] Delais-Roussarie | Unités prosodiques et grammaire de l'intonation: vers une nouvelle approche[END_REF], i.e. followed by a strong prosodic break, realized in French by a pitch rise associated to a lengthened syllable, optionally followed by a silent pause (see, among many others [START_REF] Delais-Roussarie | Dislocation[END_REF] and [START_REF] De Cat | French Dislocation. Interpretation, Syntax, Acquisition[END_REF], and the dislocated NP underlined in Fig. 3, §6 below, for an illustration). The phenomenon is also known for other Romance languages such as Italian [START_REF] Frascarelli | The Syntax-Phonology Interface in Focus and Topic Constructions in Italian[END_REF] or Catalan [START_REF] Feldhaussen | Sentential Form and Prosodic Structure in Catalan[END_REF], where it has been formalized with the Align-Topic rule:

(1) Align-Topic: Align the right edge of a leftdislocated element with the right edge of an Intonational Phrase. In recent literature nevertheless, it has been shown that left-dislocated NPs, as well as other extra-sentential elements, can be subjects to restructuring phenomena, and be phrased as Intermediate Phrases (ip) or Accentual Phrases (AP). In other words, they can be ended by a perceptually less strong pitch accent (see [START_REF] Jun | Realization of Accentual Phrase in French[END_REF], [START_REF] Fougeron | Rate effects on French Intonation: prosodic organization and phonetic realization[END_REF] and below, Fig. 4 and Fig. 5 for attested spontaneous data). Avanzi (see [START_REF] Avanzi | L'interface prosodie/syntaxe en français. Dislocations, incises et asyndètes[END_REF]), for instance, says that the alignment rule formulated under [START_REF] Selkirk | Phonology and syntax. The relations between sound and structure[END_REF] is violated in 70% of the cases in his corpus (around 441 clitic-left dislocated constructions extracted from spontaneous speech), and therefore proposes the following restructuring rule:

(2) Align-Topic restructuring: Align the right edge of a left-dislocated NP with the right edge of an Intermediate Phrase or the right edge of an Accentual Phrase. The reasons that motivate prosodic restructuring are numerous. Speech rate and metrical length have been mentioned to be strong factors (see among others [START_REF] Delais-Roussarie | Unités prosodiques et grammaire de l'intonation: vers une nouvelle approche[END_REF] and [START_REF] Fougeron | Rate effects on French Intonation: prosodic organization and phonetic realization[END_REF] for example). As for clitic-left dislocations, some scholars have made the hypothesis that the prosodic variation of the boundary degree of the dislocated NP could be explained in pragmatic terms (see e.g. [START_REF] Delais-Roussarie | Dislocation[END_REF], [START_REF] Lacheret-Dujour | De la notion de détachement topical à celle de constituant thématique extrapropositionnel[END_REF]- [START_REF] Grobet | Constructions à détachement à gauche: les fonctions de la prosodie[END_REF]). According to these authors, the degree of strength of the pitch accent associated with the last syllable of the NP would depend on the activation status of the referent denoted by the NP in discourse. Strongly salient referents would not need to be prosodically prominent, while unactivated or new referents need to be strongly prominent when they are uttered for the first time. The generalization expressed under (3) resumes this hypothesis:

(3) Prosody/Pragmatic Interface Hypothesis: the more the topic is salient in the discourse, the less the right edge of the dislocated constituent is prominent. To our knowledge, the hypothesis formulated within (3) has not received any empirical accreditation yet. In this paper, we provide empirical data in order to test such a hypothesis, by analyzing a set of sentences extracted from spontaneous speech corpora and independently annotated for a prosodic analysis and for a pragmatic analysis.

Material

Data

101 declarative sentences containing a clitic-left dislocated NP subject were extracted from two spontaneous spoken corpora, representing two varieties of French: Northern (Parisian) French [START_REF] Branca-Rosoff | Discours sur la ville. Corpus de Français Parlé Parisien des années 2000[END_REF], and Southern French [START_REF] Bertrand | Le CID -Corpus of Interactional Data -Annotation et Exploitation Multimodale de Parole Conversationnelle[END_REF]. All the NPs of the extracted sentences comprise 2 or 3 syllables only, initiate a new prosodic utterance (they occur after a silent pause or a strong prosodic break) and are coreferent with a 3 rd 2.2.Prosodic processing person subject clitic pronoun (il(s)/elle(s)). In all, 22 speakers (17 females, 5 males), each performing between 1 and 10 utterances, were analyzed.

Once transcribed, each utterance was processed with the EasyAlign script [START_REF] Bertrand | Le CID -Corpus of Interactional Data -Annotation et Exploitation Multimodale de Parole Conversationnelle[END_REF] running under Praat [START_REF] Boersma | Praat: doing phonetics by computer (Version 5.2)[END_REF], which provides a 3-layer automatic segmentation: phone string, syllable string, and graphemic word string. The alignments are manually checked and corrected by one of the authors.

A second tool is then used to categorize the prosodic break associated with the last syllable of each dislocated NP. On the basis of four automatically measured acoustic parameters (relative syllabic duration, relative f0 average, slope contour amplitude and presence of an adjacent silent pause), the software estimates a degree of strength for the last syllable of the NP. Since the system has been described exhaustively in other works (see [START_REF] Avanzi | A corpus-based learning method for prominence detection in spontaneous speech[END_REF] and [START_REF] Avanzi | Toward a Continuous Modeling of French Prosodic Structure: Using Acoustic Features to Predict Prominence Location and Prominence Degree[END_REF]), we will not go further in the presentation of the theoretical and methodological foundations which sustain it. Let's just mention that the calculations rely on two fundamental principles. The first one is a quantity principle: the greater the number of acoustic parameters involved in the identification of a prominence and the distance from predetermined thresholds, the stronger the prominence is perceived. The second one is a compensation principle, which stipulates that if one of the classic parameters involved in the perception of prominence in French presents a low score, and another one a score, there will be the same feeling of prominence as if the two involved parameters presented both medium score. The results of the calculation are provided in a dedicated tier: a mark comprised between 0/10 and 10/10 (from the less prominent to the more prominent) indicates the degree of strength of the break. Fig. 3, Fig. 4 and Fig. 5 in appendix ( §6) give each an illustration of 3 types of prosodic degree boundaries generally recognized as significant in French literature (see [START_REF] D'imperio | Embedded Register Levels in French and Prosodic Phrasing in French[END_REF] for a recent discussion). Fig. 3 presents a case of IP break: the intravocalic rise is not followed by a pause but anchored on a lengthened syllable. We don't see such an increased duration for the final syllable of the NP analyzed in Fig. 4, which represents a typical ip boundary. Moreover, compared with the preceding and following syllables, the average of pitch points on the syllable [] in Fig. 3 are relatively higher than the ones of the syllable les gamins Fig. 4 in . By contrast, the last syllable of the NP included in the utterance transcribed in Fig. 5 (la personne

Pragmatic coding

), which we analyzed as followed by an AP break, does not exhibit a pitch rise, neither bears a static tone or is the object of a lengthening.

In line with a large part of the literature ([20]- [START_REF] Lambrecht | Silence accent in information questions: default and projection[END_REF], etc.), we assume a topic to be an entity that a proposition predicates something about. In [START_REF] Lambrecht | Silence accent in information questions: default and projection[END_REF]'s definition, for example, a topic is "a referent which a proposition is construed to be about in a given discourse situation", and a proposition is about a referent "if it expresses information which is relevant to, and which increases the hearer's knowledge of, this referent" ( [START_REF] Lambrecht | Silence accent in information questions: default and projection[END_REF]: 494).

Speakers generally talk about referents that are already known by them and by the interlocutors, or whose identity the interlocutors are able to recover/infer from the context or their world knowledge. We classified topic subjects into six different types, and subdivided them into three classes depending on their degree of accessibility or salience: class A (active in the speakers' minds), class B (semi-active), class C (not active anymore).

We coded as continuous topic a referent that is already the topic of the current discourse, for instance the topic of an answer to a question (cf. a ratified topic in [START_REF] Lambrecht | Silence accent in information questions: default and projection[END_REF], and a given topic in [START_REF] Lacheret-Dujour | De la notion de détachement topical à celle de constituant thématique extrapropositionnel[END_REF]). We coded as introduced topic a referent that is introduced in the preceding utterance as the upcoming topic (see the discussion on 'topic promotion' in [START_REF] Lambrecht | Information structure and sentence form: Topic focus, and the mental representations of discourse referents[END_REF]). In (4) the presentational construction y a 'there are' introduces the new referent beaucoup de petits chiens 'many small dogs', which becomes the topic of the subsequent utterance (ceux-là, referring to the dogs):

(4) y a beaucoup de petits chiens 'Then there are many small dogs' là ceux-là ils aboient ils font un peu d'bruit mais… those-ones SCL bark SCL make a bit of noise but 'these bark, make a little noise, but…' With both kinds of topic, the referent not only is 'active' in the speakers' minds, but it is also predicted to be the upcoming topic. We therefore grouped them together in the same class A of active topics.

In other utterances of our corpus, the topic referent is inferable from another referent because semantically related to it (cf. an accessible topic in [START_REF] Lambrecht | Information structure and sentence form: Topic focus, and the mental representations of discourse referents[END_REF]'s Topic Accessibility Scale, and a sub-topic in [START_REF] Lacheret-Dujour | De la notion de détachement topical à celle de constituant thématique extrapropositionnel[END_REF]). We call this an inferable topic ((5)).

(5) A : y a Laurence there's Laurence who was of person of mixed race B: oui puis attends surtout son père il venait des… qui était métisse yes then wait above-all his father SCL came from … When the topic is inferable because it is a member of a previously mentioned set, we talk of a partial topic.

The topic referent can also be known to the hearer from world or shared knowledge, as in [START_REF] Feldhaussen | Sentential Form and Prosodic Structure in Catalan[END_REF]. The referent is less predictable than in the previous case, but it is still linked to the current topic: it is for instance an expected entity in a described scene, or it concerns a generic set of entities that are present in the speakers' mind in the current discourse, etc. We coded this as a shared topic.

(6) là Aix-en-Provence, j'aime beaucoup mais euh mes racines elles sont à Grenoble "Aix-en-Provence I like very much but um, as for my roots, they are in Grenoble" These three topic types were grouped in the same class B of semi-active topics (borrowing from [START_REF] Lacheret-Dujour | De la notion de détachement topical à celle de constituant thématique extrapropositionnel[END_REF]'s terminology). They do not have an antecedent with identical reference which is the topic of the current discourse, so they are not active in the speakers' mind at the moment of the utterance; however, they are indirectly accessible from the current discourse, so they are not totally unexpected topics.

The last topic type was coded as a shift topic, and was put in a separate class of salience: class C. Its peculiarity is that it has been functioning as a topic in a previous discourse segment, but not in the immediately preceding one. In other words, the referent is reintroduced as topic after an interruption. This means that the referent has an antecedent, but it is not active anymore in the speakers' mind. As a result, the referent is not anymore the topic of the current discourse nor is predictable as topic of the upcoming discourse, since other referents could as well be.

In [START_REF] Jun | Realization of Accentual Phrase in French[END_REF], speaker B resumes a topic (les coqs) that was introduced about 50 exchanges before, during which speaker A had been talking about other problems related to her apartment.

( We included other cases where the interruption was much shorter (sometimes extremely short) but still, since the topic had been dropped, it was not fully predictable. In [START_REF] Fougeron | Rate effects on French Intonation: prosodic organization and phonetic realization[END_REF], for instance, the speaker stops talking about the old lady -what the discourse was about at the beginning -in order to describe the lady's bedroom, and then resumes the original topic.

(8) A: y avait aussi une vieille personne […] et un matin parce qu'ils la voyaient pas venir donc ils s'étaient dit tiens on va aller réveiller euh la mémé donc la chambre était à moitié remplie de neige!

[…]

'and in fact, there was also an old lady […] and one morning, since they did not see her coming, they said let's go to wake up grandma […] and so the room was half filled with snow!' A: et la mémé elle était dans son lit et elle dormait tranquillement and the grandma SCL was in her bed and she was sleeping quietly 'and grandma, she was in her bed, sleeping quietly' It must be finally pointed out that in our annotation we did not take into account whether the topic had a contrastive interpretation or not, because we considered that contrast is not strictly related to salience. Of course, we do not exclude the possibility that prosodic prominence be influenced by contrast (cf. e.g. [START_REF] Braun | Prosodic Correlates of Contrastive and Non-Contrastive Themes in German[END_REF] for German, [START_REF] Calhoun | Phonetic dimensions of intonational categories: the case of L+H* and H*[END_REF] for English). However, this feature is not much present in our data since only 12 cases of explicit contrast were found (two topics of class A, nine of class B, and one of class C).

Analysis and Results

The comparison of the two coding tasks (degree of prosodic strength and pragmatic activation state) allows us to provide an empirical account of the hypothesis formulated under (3) above.

Degree of prosodic frontier according to the topic type

As a first step, we test whether each topic type could be discriminated according to the degree of strength of the boundaries ending the NPs. ANOVA shows that there is a significant global effect of the topic type on the degree of the prosodic boundary (F=3.66 p=0.00451**), but a post-hoc test shows that there is no significant distinction among all the categories. The only significant contrast concerns the Shift topic category with the Introduced topic, Inferable topic, Partial topic, and Shared topic categories respectively (Fig. 1).

Degree of prosodic frontier according to the salience degree

We then compared the three classes of topic, offering a less fine scale, in order to check whether there was any correlation between the activation status of the referent and its prosodic strength boundary. We ran an ANOVA that shows a significant global effect of the topic class on the degree of the prosodic boundary (F=7.793 p=0.000727***), but again, a post-hoc test shows that there is no significant distinction among the classes A and B, while there is between A and C and between B and C (Fig. 2).

Discussion and Conclusion

In [START_REF] Lacheret-Dujour | De la notion de détachement topical à celle de constituant thématique extrapropositionnel[END_REF], a qualitative study is carried on the relation between prominence and pragmatic salience, based on examples of different types of detachment constructions extracted from spontaneous spoken French corpora. [START_REF] Lacheret-Dujour | De la notion de détachement topical à celle de constituant thématique extrapropositionnel[END_REF]'s classification of topic types approximately corresponds to the three classes of salience presented here. The conclusion drawn in [START_REF] Lacheret-Dujour | De la notion de détachement topical à celle de constituant thématique extrapropositionnel[END_REF] is that a different level of prosodic prominence corresponds to each topic class (the higher the prominence, the less salient the topic). In other words, generalization (3) is fully met in [START_REF] Lacheret-Dujour | De la notion de détachement topical à celle de constituant thématique extrapropositionnel[END_REF]'s study.

Our results, however, lead us to draw a different conclusion. No clear mapping between the degree of salience and the degree of prosodic prominence is given in our data, since only topics of class C, namely shift topics, are significantly more prominent than those of the other two classes, and no significant difference exists between class A and class B.

It is possible that, with a larger set of data, we may also show a significant difference between topics of class A and class B, as predicted by (3) and in compliance with [START_REF] Lacheret-Dujour | De la notion de détachement topical à celle de constituant thématique extrapropositionnel[END_REF]'s work. However, a more interesting explanation can be given to our results. Marking a shift topic does not just mean marking a topic that is not active: it aims at showing that that the topic is not active anymore, namely that it was the topic of a preceding discourse segment, which must be retrieved in order to understand the upcoming discourse. In other words, prominence makes the listener aware of a particular organization of topics in the discourse.

Note that the difference in the activation state of a topic of class B (an inferable topic from previous discourse or from shared knowledge) and one of class C (a shift or reintroduced topic) is sometimes very subtle (shared topics, for instance, can be very weakly accessible to the speakers' mind). Despite this fact, prosodic prominence consistently marks only the latter class. Therefore, it seems to be more important for the interlocutor to understand how the discourse is structured, rather than to simply be aware of the introduction of a new referent.

In conclusion, our data confirm that pragmatic features may be marked by accentual prominence; nevertheless, it is not (or not just) salience that is marked by this prosodic feature, as hypothesized in (3). Rather, prominence seems to signal a more subtle pragmatic phenomenon, namely the way different topics are organized and structured in a discourse.

Annexes

The figures below are ANALOR screenshots (see [START_REF] Avanzi | L'interface prosodie/syntaxe en français. Dislocations, incises et asyndètes[END_REF], [START_REF] Avanzi | A corpus-based learning method for prominence detection in spontaneous speech[END_REF] and [START_REF] Avanzi | Toward a Continuous Modeling of French Prosodic Structure: Using Acoustic Features to Predict Prominence Location and Prominence Degree[END_REF]). In the abscissa, temporal values are given in milliseconds; in the ordinate, the values of F0 in a logarithmic scale can be seen. The transcription tiers are, from top to bottom: phone tier, syllable tier (both in SAMPA alphabet); automatic prominence detection tier and the orthographic word tier. Fig. 3 shows a left-dislocated element followed by an IP boundary, Fig. 4 shows a left-dislocated element followed by an ip boundary and Fig. 5 a left-dislocated element followed by an AP boundary. The final NP syllables are framed in red.

  ) A: le matin […] je suis réveillée par les coqs in the morning […] I am awaken by the roosters […] […] B: et les coqs ils sont dans la pépinière ou…? and the roosters SCL are in the nursery or…? What about the roosters, are they in the nursery?

Fig. 1 :

 1 Fig. 1: Classification of the different topics according to the strength of the prosodic break ending the dislocated NP. The ordinate indicates the prosodic score.

Fig. 2 :

 2 Fig. 2: Classification of the topic classes according to the strength of the prosodic break ending the NPs, with the prosodic score in the ordinate and the three topic classes in the abscissa: A (active), B (semi-active), and C (non active).

Fig. 3 :Fig. 4 :

 34 Fig. 3: ANALOR screen shot. Transcription of the utterance: le chat il était dans l'escalier il a dû le choper [CID]

Fig. 5 :

 5 Fig. 5: ANALOR screen shot. Transcription of the utterance: la personne elle veut plus elle fait plus rien de ses journées [CFPP]


Lisa Brunetti

When generative linguists became interested in structures related to Information Structure (IS), they sought explanations by considering the sentence alone, and tried to explain IS properties by means of computational rules. However, this way of proceeding is problematic: IS properties concern the relation between a sentence and its linguistic and extralinguistic context, and computational mechanisms are not expected to be driven by external (discourse) properties. Chomsky recently proposes (Chomsky 2000 and subsequent work) that multiple spell-outs are possible at fixed points in the derivation, the so-called phases. When a phase is reached, the derivation is visible to the external systems, and interface rules apply. In the present book, Luis López exploits this new theoretical notion to solve the problem of IS-related syntactic phenomena. His proposal is that at phase edges, the derivation is able to access discourse rules, therefore IS properties can be assigned. In other words, in López's proposal, IS features are not assigned by the computational system (C HL ), but by the independent module that he calls pragmatics. The C HL only provides, by feature checking

operations that trigger movement to the edge of a phase, derivations that are readable by the interpretive module. Unlike many former approaches, where IS properties (e.g. Focus and Topic features) are the trigger for movement, this approach exempts the C HL from accounting for discourserelated phenomena. The role of the C HL is only to create derivations that are able to interface with the external systems.

López's theoretical assumptions (mostly argued for in [START_REF] López | Locality and the Architecture of Syntactic dependencies[END_REF] are presented in the introduction of the book. As for phrase structure, he takes the CP to be split into a Finite and a Force Phrase [START_REF] Rizzi | The fine structure of the left periphery[END_REF]), but rejects Topic and Focus Phrases (in fact, he rejects the very notions of topic and focus that are behind these projections). As for syntactic dependencies, he assumes a strictly local probing mechanism of unvalued features, which are carried by THE MOVING ITEM. In this chapter, the constructions that will be object of study are also presented. The ones that López dedicates more attention to are Clitic Left and Right Dislocations (CLLD and CLRD), and Focus Fronting (FF). His data are taken mostly from Catalan, but his proposal is meant to be valid for all southern Romance languages.

López's view of IS is presented in chapter 2, 'Information structure'. IS involves those aspects of the grammar that are relevant for the integration of sentences into discourse. The IS of a syntactic object is that same object augmented with the features assigned by the independent module of pragmatics (cf. the independent module of IS in [START_REF] Vallduví | The Informational Component[END_REF].

It is appreciable that López addresses the question of defining IS categories before proposing any syntactic analysis of IS-related constructions. He first discharges the notions of topic and focus often used in the syntactic literature on IS. By these terms, he actually means 'aboutness' for topic and resolution for a variable left open in the discourse for focus. He then proposes two different notions: anaphoricity and contrast.

For dislocated elements, López makes the original claim that they are strongly anaphoric, namely that they require a local antecedent in the discourse and must be in a relation of structural asymmetry with their antecedent. Structural asymmetry means, within a SDRT framework [START_REF] Asher | Logics of Conversation[END_REF], that dislocations must occur in a subordinated discourse segment, and their antecedent in the superordinate one. So the appropriateness of dislocations in discourse is predictable by discourse structure.

CLLD and CLRD are both assigned the feature [+a(naphoric)] by the interpretive module pragmatics. As for the difference between the two, it lies in their relationship with the antecedent: for CLRD it must be identity, for CLLD it can be a subset-superset, set-membership, or part-whole relationship. In other words, the CLLD opens up a variable and closes it.

That means that a CLLD is assigned an additional feature: [+c(ontrastive)].

A CLRD is instead [-c]. It must be said that in fact it is not so clear that CLLDs always open a quantificational domain (see Brunetti 2009a). In that case, one should doubt of the possibility to assign a unique interpretation to a syntactic derivation, as this book attempts to do.

Focus provides a resolution for a variable left open in previous discourse.

The problem with this definition is that it cannot account for the displacement of focus material to the left periphery (Focus Fronting). FF, for López, BOTH opens up a variable and provides an element to resolve it.

All focus is [-a], but the relevant feature to distinguish FF from in situ focus is then [+c]. In contradiction to these conclusions, in Brunetti 2009b I show that FF may answer a question (resolve a variable in the discourse) if the question is implicit or not salient anymore. This analysis is compatible with López's claim if we specify that FF not only occurs when a variable is not in the discourse, but also when it is not easily or not anymore accessible.

Eventually, my proposal remains different in that I relate the interpretation of FF to the fact that a fronted focus is unambiguously narrow, rather than to a feature [+c] assigned to a particular derivation.

A section at the end of the chapter is dedicated to the relation between stress and focus. López argues that focus projection rules do not always explain deaccenting in English. The corresponding Catalan data, on the contrary, reveal that the syntactic structure always match with IS: the item corresponding to the deaccented one in English is dislocated only when it is anaphoric, otherwise it is not. López concludes that deaccenting (hence intonation) is not always related to anaphoricity and hence to IS.

In chapter 3, 'The syntax of dislocations and focus fronting', López discusses the syntactic position of CLRD, CLLD and FF. He argues for pragmatic features to be assigned at the edges of two phases: v for [±a], and Fin for [±c].

As for CLRD, López shows that the dislocated item undergoes Amovement, is higher than a CLLD, and occupies a position that is between spec,T and the in situ subject. He proposes that such position is a higher spec,v (multiple specifiers are possible in his model). Displacement is related to the presence of the resumptive clitic. The clitic spells out a feature matrix X that merges in v and causes the argument to move to spec,v, where it forms a dependency with v and agrees with X. Since vP is a phase edge, the argument is assigned the interpretive feature [+a]. In sum, [+a] is assigned to a constituent that: (i) it is in spec,v and (ii) it agrees with X. If one of the two conditions is not met, no [+a] feature is assigned (this prevents for instance external arguments to be always assigned [+a]). The complement of v is assigned [-a], as far down as the following vP. In order to explain linear order, López proposes an analysis based on linearization constraints (at the PF interface). Note importantly that pragmatic features come before the PF interface, so PF reads pragmatics and not vice versa.

Assuming that wh-phrases, FF and CLLD all involve contrast, López argues for a unity of all A'-dependencies: these movements are all triggered by the same formal feature and their landing site is always spec,Fin, where the pragmatic feature [+c] is assigned by the interpretive module. CLLD is also [+a], so it has to move to spec,v to get that feature before it moves to spec,Fin. This is shown by the fact that it can strand a floating quantifier in spec,v. Moreover, like a CLRD, a CLLD reconstructs to a position that ccommands the post-verbal subject but is c-commanded by the pre-verbal subject.

As for FF, there is mixed evidence on whether it stops at spec,v or not, but without pragmatic consequences, because the focus constituent does not agree with a clitic. I am wondering though how the French MOI j'ai appelé Pierre would be accounted for, if MOI moves to spec,v. The moved element agrees with the clitic, so it should get [+a].

Finally, López discusses D-linked wh-phrases. They have no resumptive clitic and they can reconstruct. However, they are [+a] (as often claimed in the literature). López proposes that the [+a] feature is in fact assigned in the lexicon in this case to the wh-word.

Occurrence restrictions of A'-moved items are used by Rizzi as evidence for different A' positions. López, on the contrary, argues that they are due to interpretability problems. I find this approach in principle desirable.

Nevertheless, anaphoricity and contrast are not always sufficient or adequate to account for the data. By assuming that CLLD is [+a], it is easy to explain why multiple CLLDs are possible: there is no limit to the number of anaphoric expressions in a sentence. As for the fact that FF must be unique, López argues that it is odd to have a double correction in the same utterance. However, it is not clear why the incompatibility concerns two items that are [-a], but not two that are [+a]. As for the incompatibility between FF and a wh-phrase, it is due to the fact that contrast implies assertion, and it is not possible to make an assertion and a question at the same time. Finally, López claims that ordering of A'-moved items depends on a restriction in the linearization of [+a] and [-a] constituents: the former have to be linearized before the latter.

López eventually gives his contribution to the long debated question whether subjects are always dislocated or not. He argues that they are not, because SVO order is possible in an all-focus sentence, and that is not expected if the subject is dislocated and hence [+a]. But why are subjects preverbal in all focus sentences? López proposes that they check an interpretable feature indicating the beginning of a new discourse. I think here López misses the association between sentence initial position and topic, which is claimed by many scholars to occur in many languages.

Preverbal subjects and CLLDs would be unified by this property. Since topic is not a relevant interpretive notion for López, he cannot pursue this kind of analysis.

In chapter 4, 'The derivation of information structure', López provides further data in support of his phase-based analysis of IS-related phenomena. for Spanish, that it is triggered by the same formal feature as CLRD. The item moves to spec,v, and is assigned the feature [+a]. P-movement occurs in place of CLRD whenever the feature X is not spelled out as a clitic.

Animate accusative complements in Spanish are introduced by the preposition a. López shows that 'accusative A' is a structural phenomenon triggered by a formal feature. Its interpretation can be optionally anaphoric and specific. López proposes that its position is spec,V. This position is not the edge of a phase, so it does not interface with any interpretive module.

That explains why there is no obligatory interpretation. In some varieties of Spanish, accusative A combines with Clitic Doubling (CLD) of the direct object. The position is indeed the same (spec,V), but the interpretation of the object is always specific. López accounts for this difference by proposing that the clitic of CLD is of a special nature, and inherently specific. This explanation is not very convincing, as it messes up the clear opposition between items that are at the edge of the v phase and are always interpreted as [+a], and those that are not and do not receive any obligatory interpretation. These data again make one wonder whether a one-to-one match between syntax and interpretation is possible and even desirable.

Finally, scrambling and object shift in Germanic are also briefly discussed. López shows that the scrambled or shifted object has to be specific and anaphoric, and provides evidence for movement to spec,v. To conclude, López's work contributes to a line of syntax studies that recognizes the importance of context in the interpretation of IS-related constructions and consequently acknowledges that they cannot be explained by internal mechanisms only. What C HL does is just place items to phase edges, where they can be assigned interpretive features. Even this reduced role of C HL , however, seems to be questioned by some data which make wonder on the actual possibility of a one-to-one match between syntax and interpretation.

  The main piece of evidence is sub-extraction. Sub-extraction data provide arguments against two (opposite) alternatives to López's account: one based on a model where interpretation of syntactic structures only takes place when the derivation is completed, and one based on a model in which interpretation takes place each time an operation applies, rather than at specific points. The former model is too rigid and cannot explain why two parts of the same constituent can get two different values of the same feature. The latter falls into the opposite problem: it cannot explain why in certain constructions the two parts of the same constituent cannot get different values. A phase-based model can explain both situations, because it distinguishes between sub-extraction taking place within a phase, when the pragmatic feature has not yet been assigned, and sub-extraction across phase boundary, after the feature has been assigned. López attempts to account for all those constructions that are related to IS and have been object of debate in the literature. Chapter 5, 'Moving objects', is dedicated to P-movement, Accusative A, and Clitic Doubling (CLD), as well as scrambling and object shift in Germanic. P-movement consists of indirect-direct object reordering or subjectobject reordering. This movement was originally proposed by Zubizarreta 1998 as being prosodically motivated. López on the contrary shows, at least

  The proposal made in chapter 3 concerning the syntax of CLLD, CLRD and FF is refined in the last chapter, 'Dislocation debates'. Contra proposals made in the literature, López reaffirms that CLLD and CLRD involve movement. The chapter is also dedicated to the discussion concerning the landing site of CLRD. Many different and often contradictory proposals have been given in the literature concerning the position of CLRD in Romance. López contributes to the discussion by providing new data, but only NPI licensing provides further evidence that the CLRD is located in the middle position.

Author's address: Laboratoire Dynamique du Langage (UMR 5596), Institut des Sciences de l 'Homme,14,avenue Berthelot,69363,Lyon,