O Gehan

E Pigeon

T Menard
email: tomas.menard@unicaen.fr

GREYC M Pouliquen

H Gualous

Y Slamani

B Tala-Ighil

A nonlinear state feedback for DC/DC boost converters

This paper investigates the control problem for static boost type converters using an high gain state feedback robust controller incorporating an integral action. The robust feature allows to achieve the required performance in presence of parametric uncertainties, while the integral action provides an offset free performance with respect to the desired levels of voltage. The adopted high gain approach is motivated by both fundamental as well as practical considerations, namely the underlying fundamental potential and the design parameter specification simplicity. The stability and convergence analysis has been carried out using an adequate Lyapunov approach, and the control system calibration is achieved throughout a few design parameters which are closely related to the desired dynamical performances. The effectiveness of the proposed control approach has been corroborated by numerical simulations and probing experimental results.

Introduction

The control of static converters has been the subject of an important research activity over the past decades. This interest is mainly due to the emergence of embedded electronics in everyday life, increasing thereby the need for more efficient converters. Indeed, this type of converter is used for many applications such as laptop computers [START_REF] Sahu | A low voltage, dynamic, noninverting, synchronous buck-boost converter for portable applications[END_REF], photo-voltaic [START_REF] Walker | Cascaded dc-dc converter connection of photovoltaic modules[END_REF], vehicular systems [START_REF] Khaligh | Modified pulse-adjustment technique to control dc/dc converters driving variable constant-power loads[END_REF], fuel cell [START_REF] Jin | A hybrid fuel cell power system[END_REF], etc. Furthermore in recent use of electrical devices, the control problem is not the only feature required from the user. There are more and more need of informations on the evolution/aging of the system. In the case of converters, these informations can be obtained by reconstructing voltages and currents thanks to suitable observers.

Of particular interest is the Boost Converter which maintains a low power loss in the conversion process. The counterpart is that the average model presents strong nonlinearities due to the presence of bilinear terms and unstable zero dynamics which reduce the control system performances. These difficulties have given rise to several control solutions involving the converter output voltage. A widely spread method is the current-mode control which consists of two nested closed loop systems where the inner one is a current control system, while the outer one is a voltage control system ( [START_REF] Erickson | Fundamentals of power electronics[END_REF][START_REF] Redl | Near-optimum dynamic regulation of dc-dc converters using feed-forward of output current and input voltage with current-mode control[END_REF][START_REF] Ridley | A new, continuous-time model for current-mode control[END_REF]). The feasibility of the nested control system require the inner control to be relatively faster. Furthermore, it stabilizes the unstable zero dynamics, allowing thereby to apply usual control designs. However the control system performances may be poor when working out of the nominal operating conditions. Even so, this approach, commonly used in the industry, has been developed in many regards since then ( [START_REF] Maksimovic | Impact of digital control in power electronics[END_REF][START_REF] Morales | Average current-mode control scheme for a quadratic buck converter with a single switch[END_REF][START_REF] Olalla | Qft robust control of current-mode converters: application to power conditioning regulators[END_REF][START_REF] Chang | Parameter tuning method for digital power converter with predictive current-mode control[END_REF]).

Several approaches have been recently proposed to circumvent the limitations of the usual methods from both performance improvement and simplicity points of views. They are based on direct output voltage control without resorting to the current control system as well as the stabilization of unstable zeros dynamics. A common technique consists in using the linearized model of the system, which is valid in a neighborhood of the nominal point, with its uncertainties if needed. Some models for these uncertainties have been proposed in [START_REF] Leyva-Ramos | Uncertainty models for switch-mode dc-dc converters[END_REF]. LQR controllers have been used in [START_REF] Leung | The control of switching dc-dc converters-a general lwr problem[END_REF][START_REF] Leung | An improved lqr-based controller for switching dc-dc converters[END_REF] and [START_REF] Garofalo | Control of dc-dc converters with linear optimal feedback and nonlinear feedforward[END_REF] under small signals constraints for construction purposes. In order to cope with large signal disturbances, the linear techniques have been adapted in different ways that can be classified into two main streams. The first approach consists in designing a robust linear controller. Robust linear state feedback have been designed in [START_REF] Leyva | Linear statefeedback control of a boost converter for large-signal stability[END_REF] using passivity and using LMI in [START_REF] Olalla | Robust lqr control for pwm converters: an lmi approach[END_REF][START_REF] Olalla | Lmi robust control design for boost pwm converters[END_REF][START_REF] Olalla | Optimal state-feedback control of bilinear dc-dc converters with guaranteed regions of stability[END_REF][START_REF] Olalla | Robust gain-scheduled control of switched-mode dc-dc converters[END_REF]. These approaches necessarily involve a trade-off between performances and robustness since the structure of the controller is linear. Different works have been done in order to keep the simplicity of the control law, while having better performances. Auto-tunning PID have been considered in [START_REF] Shirazi | An autotuning digital controller for dc-dc power converters based on online frequency-response measurement[END_REF][START_REF] Ebrahimzadeh | Adaptive and fast-response controller for boost pfc converter with wide range of operating conditions[END_REF]. The extended linearization with a PI has been pursued in [START_REF] Sira-Ramirez | Nonlinear pi controller design for switchmode dc-to-dc power converters[END_REF] and the model predictive control has been considered in [START_REF] Beccuti | Explicit model predictive control of dc-dc switched-mode power supplies with extended kalman filtering[END_REF]. Although, these approaches increase the robustness and performances to large disturbances, they would exhibit poor performances on the whole working space and can lead to unexpected behavior.

Given the nonlinear structure of the average model of the boost converter, numerous approaches using nonlinear control law have been proposed in the literature. Early approaches such as passivity based control and Lyapunov based control have been considered in [START_REF] Sira-Ramírez | Adaptive passivity-based control of average dc-to-dc power converter models[END_REF][START_REF] Leyva | Passivity-based integral control of a boost converter for large-signal stability[END_REF][START_REF] Sanders | Lyapunov-based control for switched power converters[END_REF]. These methods have shown to be more robust to large signals disturbances but their performances cannot be compared to the standard techniques. More complicated techniques have then been used in order to achieve better performances using Hammerstein models ( [START_REF] Alonge | Nonlinear modeling of dc/dc converters using the hammerstein's approach[END_REF][START_REF] Alonge | Identification and robust control of dc/dc converter hammerstein model[END_REF]) or time optimal control ( [START_REF] Feng | A new digital control algorithm to achieve optimal dynamic performance in dc-to-dc converters[END_REF][START_REF] Nazarzadeh | Applying bilinear time-optimal control system in boost converters[END_REF]). It is however worth noticing that these techniques need a very precise knowledge of the model which is difficult since the model is usually an average model (this model does not take into account all the uncertainties on the components). Backstepping control has been used in [START_REF] El Fadil | Control of dc-dc power converters in the presence of coil magnetic saturation[END_REF] for current control, the voltage value is indirectly determined from the current using the model knowledge and henceforth the precision on the desired voltage directly relies on the model's accuracy. Fuzzy controllers have been used in [START_REF] Mattavelli | General-purpose fuzzy controller for dc/dc converters[END_REF][START_REF] Perry | A design method for pi-like fuzzy logic controllers for dc-dc converter[END_REF][START_REF] Wai | Total sliding-mode controller for pm synchronous servo motor drive using recurrent fuzzy neural network[END_REF] and [START_REF] Wai | Adaptive fuzzy-neural-network design for voltage tracking control of a dc-dc boost converter[END_REF] without any guarantee of stability and performance. Various contributions have been devoted to the sliding mode control approach for its robustness to bounded uncertainties [START_REF] Oucheriah | Pwm-based adaptive sliding-mode control for boost dc-dc converters[END_REF][START_REF] Sira-Ramirez | Sliding motions in bilinear switched networks[END_REF][START_REF] Tan | A fast-response sliding-mode controller for boost-type converters with a wide range of operating conditions[END_REF][START_REF] Tan | General design issues of sliding-mode controllers in dc-dc converters[END_REF][START_REF] Wai | Design of voltage tracking control for dc-dc boost converter via total sliding-mode technique[END_REF]. They suffer from implementation issues because of the chattering behavior occurring near the sliding surface. The objective of this paper is to propose a state feedback which is easy to calibrate, robust to uncertainties and competitive with respect to classical methods such as current mode control. Of fundamental interest, it performs well on the whole working space and is theoretically guaranteed. In order to cope with the unstable zeros dynamics, only regulation of the output voltage at a constant value is considered, as in most of the applications of this boost converter. The controller consists in combining an integral high-gain state feedback control law with an high-gain observer. Only three design parameters have to be specified and these parameters have physical meaning and are closely related to the underlying control dynamical performance. The admissible assumption of having access to on-line measurements of the current in the inductor and the output voltage is considered here. The current delivered to the load and the input voltage are not measured and are then estimated by a high-gain observer. In most approaches, the robustness toward changes in the current in the inductor and the output voltage is obtained through a degradation of the performances, since they are supposed to be unknown. This is not the case here, since these informations are reconstructed and used in the control law. It should be noted that these informations could be further use for fault detection purposes. This paper is organized as follows. In section 2, the dynamical average model of the boost converter is presented. In the next section, an high gain type control law is synthesized, its construction is split into three parts. First the nominal case is considered and a control law is designed, then an observer reconstruct the current delivered to the load and the output voltage, finally, these two parts are mixed in order to obtain a robust state feedback. Section 4 is dedicated to experimental results. Finally, section 5 concludes the paper.

Modeling of boost converter

The electrical model of the Boost converter considered in this paper is shown on Fig. 1. It consists of an inductor L, a capacitor C and a controlled switch. The output current is denoted i(t) and the control variable u(t) represents the duty cycle of the controlled IGBT. The first objective is to keep the converter's output voltage v c (t) at a constant reference value v * c , independently of the disturbances that could affect the operation of the converter, namely, the variations of the supply voltage of the battery and the load. This control objective is feasible, while tracking a reference sequence voltage is not possible in this form as the zero dynamics are not stable. For cost reasons and simplicity of implementation, the output voltage and the coil current are the unique available measurements. The second objective is to deliver an on line estimation of the unmeasured variables, namely the battery voltage and the output current for process monitoring considerations.

To achieve these objectives, the average model considered in this paper directly includes the losses and can thus be

v loss (t) v ε (t) i l (t) v e (t) u(t) v c (t) i loss (t) i ε (t) i(t) R
Fig. 1: Electric schematic of the Boost converter written into the following form:

vc (t) = 1 C ((1 -u(t))i l (t) -i ε (t)) , il (t) = 1 L (v ε (t) -(1 -u(t))v c (t)) , (1)
where

v ε (t) = v e (t
v c (t) ∈ [v c min , v c max ], i l (t) ∈ [i l min , i l max ], v ε (t) ∈ [v ε min , v ε max ], i ε (t) ∈ [i ε min , i ε max ], u(t) ∈ [u min , u max ]. (2)
3 Control algorithm In this section, the main theoretical development of the paper, namely a robust nonlinear state feedback control law for v c is presented. The proposed algorithm is an integral high gain type controller, which has already been theoretically developed in the literature (see [START_REF] Bédoui | Robust nonlinear controllers for bioprocesses[END_REF]). It should be noted that the explicit incorporation of an integral action, into the synthesis model, is common, whether for linear or nonlinear approaches. Usually, the integral action is introduced after shifting the state space coordinates that leads to the Brunovsky canonical form, because it greatly simplifies the analysis. Indeed, most of the existing nonlinear control are synthesized for systems under this form since every uniformly controllable and observable systems can be put into this form [START_REF] O'reilly | Observers for linear systems[END_REF]. But, in presence of model uncertainties, the inverse transformation that is necessarily done on-line to calculate the value of the control variable, may cause a loss of the integral action effects. To cope with this difficulty and to ensure robust static performances, even in presence of uncertainties, the integral action is directly introduced in the original coordinates. This, then, leads to the following extended synthesis model

     vc (t) = 1 C ((1 -u(t))i l (t) -i ε (t)) , il (t) = 1 L (v ε (t) -(1 -u(t))v c (t)) , u(t) = µ(t), (3)
where µ(t) is the first derivative of the physical control variable u(t), and becomes the virtual input of the system. Another advantage of this structure is that it allows to deal with the negative effects of the actuator saturation on the control variable. This can now be easily solved using a standard anti-windup structure [START_REF] Astrom | Advanced PID control[END_REF]. The presentation of the proposed algorithm is split in three parts. Firstly, a nonlinear control law is designed in the nominal case, that is to say, when the supply voltage and the output current are constant and a priori known. Secondly, an observer is designed in order to deliver an estimation of the variables v ε (t) and i ε (t) using the available on line measurements, namely v c (t) and i l (t). Finally, the two previous parts are combined to obtain both a robust control law that takes into account the changes in the voltage supply and the load and a diagnostic tool for the monitoring of these variables, the proposed control law is illustrated on Fig. 2.

Nominal case

In the nominal case, the supply voltage and the output current are constant and perfectly known. In this configuration, the power losses are equal to zero and system (3) can be written as:

     vc (t) = 1 C ((1 -u(t))i l (t) -i * ε ) , il (t) = 1 L (v * ε -(1 -u(t))v c (t)) , u(t) = µ(t). (4)
where v * ε and i * ε denote the nominal values of v ε (t) and i ε (t) respectively. The Brunovsky canonical form characterizes the class of nonlinear systems for which the nonlinearity has a triangular structure. For such a class of systems, the high-gain control law has been proposed in the literature using the duality with the high-gain type observer. The control law synthesis requires the existence of a diffeomorphism that puts the physical model into the Brunovsky form. For that purpose, let us consider the change of coordinates z = ψ(v c , i l , u; v ε , i ε ) where ψ is defined by

ψ(v c ,i l ,u;v * ε ,i * ε )=    Cv 2 c +Li 2 l 2(i l v * ε -v c i * ε ) 2 v * ε 2 L + i * ε 2 C -2(1-u) v * ε vc L + i * ε i l C    , (5)
Using ( 5), the physical model ( 4) can be put under the following canonical form

     ż1 (t) = z 2 (t), ż2 (t) = z 3 (t), ż3 (t) = f (z(t); v * ε , i * ε )µ(t) + φ(z(t); v * ε , i * ε ), (6)
where

f (z(t); v * ε , i * ε ) and φ(z(t); v * ε , i * ε ) have the following expressions f (z; v * ε , i * ε ) = f ψ -1 (z; v ε , i ε ); v * ε , i * ε , (7)
= f (v c , i l , u; v * ε , i * ε ), φ(z; v * ε , i * ε ) = φ ψ -1 (z; v * ε , i * ε ); v * ε , i * ε , (8)
= φ(v c , i l , u, v * ε , i * ε ), = 2 LC (u -1) 2 (v c i * ε -i l v * ε ). with f (v c , i l , u; v * ε , i * ε ) = 2 v * ε v c L + i * ε i l C , (9)
φ(v c , i l , u, v * ε , i * ε ) = 2 LC (u -1) 2 (v c i * ε -i l v * ε ) , (10)
The nominal values v * ε and i * ε are considered as parameters, hence, the function ψ is seen as a function from R 3 to R 3 , and

ψ -1 (z; v * ε , i * ε )
denotes the function verifying:

ψ -1 (ψ(v c , i l , u; v * ε , i * ε ); v * ε , i * ε ) = (v c , i l , u), (11)
for all

(v c ; i l , u) in R 3 .
The control objective can be expressed in this new basis as: the state vector z(t) must be maintained in the vicinity of the following constant reference vector z * ∈ R 3

z * =   C (v * c ) 2 + L i * l 2 0 0   . ( 12
)
For the class of systems ( 6), the following high-gain type control law is proposed:

µ(t) = - 1 f (v c , i l , u; v * ε , i * ε ) × k c λ 3 , 3λ 2 , 3λ × (z -z * ), (13)
where k c > 0, λ > 0 are synthesis parameters. The control law can now be expressed using the physical variables

µ(t) = - 1 f (v c , i l , u; v * ε , i * ε ) ν, ( 14
)
where ν is given by

ν=k c[ λ 3 , 3λ 2 , 3λ]×(ψ(v c ,i l ,u;v * ε ,i * ε )-ψ(v * c ,i * l ,u * ;v * ε ,i * ε )), (15)
Semicolons are used in the arguments of the different functions to enhance the fact that (v c , i l , u) are the main variables while v * ε and i * ε are only parameters. The nominal values of i l and u, are denoted i * l and u * , and can be obtained by replacing their expressions in (4):

i * l = i * ε v * ε v * c , u * = 1 - v * ε v * c . ( 16
)
One has the following result of convergence.

Proposition 1. If k c and λ are chosen sufficiently large, then the control law ( 14)- [START_REF] Garofalo | Control of dc-dc converters with linear optimal feedback and nonlinear feedforward[END_REF] ensures that the tracking errors

ṽc (t) = v c (t) -v * c , ĩl (t) = i l (t) -i *
l and ũ(t) = u(t) -u * converge to zero where v c (t), i l (t) and u(t) are the solutions of equation ( 4), v * c is the specified output voltage, i * l and u * are given by ( 16).

Remark 1. The control law has been designed for the system in the Brunovsky form and is made up of the product of a linear high-gain state feedback and a nonlinear term. The nonlinear term is used in order to compensate the nonlinearity f and is associated to the parameter k c . When f is perfectly known, it is sufficient to take k c > 1/2. The linear part allows to place all the eigenvalues of the linear part of the closed loop system at -λ. Then λ is closely related to the closed loop time constant. Moreover, λ has to be greater than a lower bound in order to dominate the nonlinearity φ. This lower bound is proportional to the Lipschitz constant of φ. It is worth noting that the different variables are bounded (see ( 2)), thus, if λ is taken too large, the overshoot may lead to a saturation.

Proof. See section A in the appendix. Now let us consider the more realistic case where the converter is not perfect (v loss = 0, i loss = 0) and the supply voltage and the load are unknown but piecewise constant. In that case, the expression of the function f can be deduced from (3) and it depends on the exact values of v ε (t) and i ε (t) instead of their nominal values. As a consequence, the control law [START_REF] Leung | An improved lqr-based controller for switching dc-dc converters[END_REF] will not ensure exact feedback linearization and the closed loop dynamical performances will be degraded. To overcome this problem, a nonlinear observer is synthesized in the next section to give an estimation of v ε (t) and i ε (t) and thus of f . These estimations can also be useful for diagnostic purposes and then achieve the second objective of the proposed monitoring algorithm.

Observer for the perturbation

An observer is designed in this section to estimate v ε (t) and i ε (t) in the case where the supply voltage v e (t) and the output current i(t) changes are piecewise constant. These estimates can be used to get useful informations about the state of charge of the energy storage component which supplies the DC/DC converter [START_REF] Oukaour | Supercapacitors aging diagnosis using least square algorithm[END_REF]. Since the signals i ε (t) and v ε (t) are piecewise constants and the measured signals are v c (t) and i l (t), an observer, based on [START_REF] Farza | Observer design for a class of mimo nonlinear systems[END_REF] is given by

         vc (t) = 1 C (1 -u(t)) îl (t) -îε (t) -2θ ( vc (t) -v c (t)) , il (t) = 1 L ( vε (t) -(1 -u(t)) vc (t)) -2θ îl (t) -i l (t) , iε (t) = Cθ 2 ( vc (t) -v c (t)) , vε (t) = -Lθ 2 îl (t) -i l (t) , (17)
where θ > 1 is the sole observer design parameter.

Proposition 2. Assume that the input u(t) belongs to a bounded set U for all t > 0, then there exists θ 0 > 1 such that for all θ > θ 0 the observation error converges asymptotically to zero, independently of the control signal, as follows:

∃θ 0 > 0, ∀θ > θ 0 , ∃α > 0, ∃µ θ > 0, ∀u ∈ U, ∀ x(0) ∈ R 4 , such that x(t) -x(t) < αe -µ θ t x(0) -x(0) , (18)
where lim

θ→+∞ µ θ = +∞, with x = [v c , i l , i ε , v ε ] T , and x = vc , îl , îε , vε T .
Remark 2. As for the control law, the eigenvalues of the linear part of the observer error system are located at -θ. This design parameter must be greater than a lower bound θ 0 , which is proportional to the Lipschitz constant of the nonlinear part.

Proof. See section B in the appendix.

Now that estimations of v ε and i ε are available, an estimation f e (v c , i l , u; vε , îε ) of f (v c , i l , u; v ε , i ε ) is given as follows:

f e (v c , i l , u; vε , îε ) = 2 v c sat 1 ( vε ) L + i l sat 2 ( îε ) C , (19)
with

sat 1 (σ) =    v ε min if σ < v ε min σ if σ ∈ [v ε min , v ε max ] v ε max if σ > v ε max , sat 2 (σ) =    i ε min if σ < i ε min σ if σ ∈ [i ε min , i ε max ] i ε max if σ > i ε max . ( 20
)
The saturating functions sat 1 (σ) and sat 2 (σ) have been introduced to ensure both that f e (v c , i l , u; vε , îε ) is a positive function, during the observer transient state and that f (v c , i l , u, v ε , i ε ) -f e (v c , i l , u; vε , îε ) converges to zero when v ε (t) and i ε (t) are constant [START_REF] Khalil | Semiglobal stabilization of a class of nonlinear systems using output feedback[END_REF].

Robust control law

In the non-nominal case, all the nominal values used in the control law equations ( 5)-( 14)-( 15)-( 16) are replaced by the estimates of the corresponding variables. The scheme for the implementation of the control law is depicted on Fig. 2. The estimations of v ε (t) and i ε (t) can be directly obtained from observer (17) whereas ( 16) is replaced by :

î * l (t) = îε (t) vε (t) v * c , û * (t) = 1 - vε (t) v * c . (21)
Using, in addition, the estimation f e (v c , i l , u; vε , îε ) delivered by [START_REF] Olalla | Optimal state-feedback control of bilinear dc-dc converters with guaranteed regions of stability[END_REF], the state feedback ( 14)-( 5) can be modified as follows:

µ = - 1 f e (v c , i l , u; vε , îε ) ν, (22)
and

ν=k c[ λ 3 , 3λ 2 , 3λ]× ψ(v c ,i l ,u;sat 1 ( vε ),sat 2 ( îε )) -ψ(v * c , î * l , û * ;sat 1 ( vε ),sat 2 ( îε )) . ( 23
)
The main convergence result of the paper is stated below.

Proposition 3. The control law ( 17), ( 19), ( 22), [START_REF] Sira-Ramirez | Nonlinear pi controller design for switchmode dc-to-dc power converters[END_REF] ensures that v c (t), the solution of system (3), converges exponentially toward v * c provided that θ, k c and λ are chosen sufficiently large.

Remark 3. The sole difference for the tunning of the design parameters compared to the nominal part concerns k c . Indeed, in the non nominal case, the multiplicative nonlinearity f is no longer exactly compensated. The parameter k c is used in order to handle this issue and has to satisfy k c > f min 2 f max where f min and f max are respectively the lower and upper bounds of f .

Proof. See section C in the appendix. Remark 4. The conditions (51) and (56), established in appendix C, which ensure the convergence of the control algorithm may be very restrictive, since they provide a lower bound for the synthesis parameters k c and λ. Nevertheless, many simulations tests have shown that the proposed control law performs well for synthesis parameters values that are lower than the theoretical bounds.

Applications

The performances of the proposed state feedback control law have been firstly validated with simulations on Matlab/Simulink environment and compared to those obtained with a standard current mode controller [START_REF] Ridley | A new, continuous-time model for current-mode control[END_REF]. For that purpose, a realistic boost converter simulator has been created to reproduce as closely as possible the behavior of the continuous system. Several stray inductors, resistors and capacitors associated to the connecting tracks, cables and semiconductor devices have been introduced. The coil is highly nonlinear and displays hysteric behavior due to the magnetic cycle of its core. At usual operating points, effects of these phenomena and of the stray components are however mostly negligible. The IGBT has been considered as an ideal switch. Finally, the imperfections that have been taken into account in our simulation of the average model are the stray resistor of the coil and the saturation of the coil current. Please note that these operating points have usually been chosen in the literature in the more binding cases [START_REF] Olalla | Robust lqr control for pwm converters: an lmi approach[END_REF] and then gave us a satisfying benchmark test for the actual proposed control law. Table 1

Simulation results

First, let us examine the nonlinear control law design. Many design softwares are now available for the synthesis of linear control laws whereas this is not yet the case for nonlinear control schemes where the choice of the design parameters values is often a difficult task. The proposed control law gives a great answer to this challenge because the parameter values are readily determinable. Indeed, as long as k c remains constant, θ and λ directly determine the values of the poles of the linear parts of the observer and of the state feedback respectively. The Fig. 3 shows the transient responses when the load switches from R = 5 Ω to R = 30 Ω at 75 ms and switches back to R = 5 Ω at 150 ms for different values of θ and λ. The relationship between the values of these parameters and the response time is clearly shown there. Subject to compliance with conditions ( 18) and (56) an increase in the values of θ and λ brings a better disturbance rejection but increases the closed loop sensitivity with respect to measurement noise. To determine the best values of the synthesis parameters for the control law [START_REF] Ebrahimzadeh | Adaptive and fast-response controller for boost pfc converter with wide range of operating conditions[END_REF] with regard to the input/output closed loop performances, many simulations have been carried out using the previous simulator. The obtained values are:

θ = 2000, λ = 500, k c = 1. ( 24
)
The performances of the control law [START_REF] Ebrahimzadeh | Adaptive and fast-response controller for boost pfc converter with wide range of operating conditions[END_REF] with the previous design parameters values have been evaluated in many configurations. For comparison purposes, a current mode controller has been designed. It was formed by a peak current modulator associated with a voltage control loop using a PI controller. This control structure is commonly used for static converters in the literature. A comparison of both algorithms allows us to quantify the performances of the nonlinear control law. The PI controller has been designed using the nominal average model and with specifications that guarantee a 60 • phase margin and a 10 dB gain margin which are common specifications in power electronic [START_REF]ECSS : Electrical and Electronic Standart : ECSS-E20A[END_REF].

A first experiment was conducted where the load changes from R = 5 Ω to R = 30 Ω and changes back to R = 5 Ω with two operating point duty cycle values, respectively d = 0.5 (nominal value, Fig. 4a) and d = 0.7 (non-nominal value, Fig. 4b). Then, the performances of the control scheme have been tested in the presence of supply voltage variations (Fig. 4c). The step like corresponding profile consists of a change in level from v e = 12V to v e = 7V at t = 100 ms and again to v e = 12V at t = 200 ms.

The nonlinear control law obviously satisfies the transient and steady state performances requirements for both operating point duty cycles, while the current mode controller does not ensure the stability of the closed loop when working too far from the nominal operating point. Moreover, the nonlinear approach obviously gives the best settling time in the nominal case. It is worth pointing out that the converter, with the nonlinear algorithm, enters in discontinuous mode on figure 4b between t = 0.11 s and t = 0.15 s. The robustness of our algorithm with respect to the change between continuous and discontinuous mode is due to the saturation functions which have been applied directly to the estimates provided by the observer [START_REF] Olalla | Optimal state-feedback control of bilinear dc-dc converters with guaranteed regions of stability[END_REF]. Let us now examine the observer performances. This observer has first been designed to deliver an estimation of the supply voltage and the output current for monitoring purposes using the available measurements, namely, the output voltage and the coil current. An experiment has been conducted where the supply voltage changes from v e = 12V to v e = 7V at t = 250 ms and again to v e = 12V at t = 550 ms (Fig. 5c). Moreover, for each supply voltage level, a load profile has been specified corresponding to a change from R = 5 Ω to R = 30 Ω and again to R = 5 Ω (F 5d). Finally, a step-like output disturbance has been applied at t = 650 ms in order to simulate an offset on the sensor measurement. The observer performances are represented on Fig. 5a, 5b, 5c and5d.

The convergence of the observer is clearly ensured for all the configurations that have been tested thanks to the specified disturbance profiles. The output voltage and the coil current are correctly estimated without any bias during all the steady state phases. A bias appears in the estimation of the supply voltage and the output current which can be easily explained by the model uncertainties that have been taken into account. In fact, the observer delivers estimations of both v ε (t) and i ε (t) instead of v e (t) and i(t) respectively. In the actual simulation case, the difference between the previous signals is directly connected to the coil resistor that has been introduced in the simulator. A particular point needs to be emphasized on Fig. 5c and5d. The sensor offset is responsible for an additional bias on the supply voltage and output current estimates. This test shows that the observer can be used to replace the sensors or to provide an oversight of the converter. Indeed, since v e (t) and i(t) are constants, the differences between the measured and estimated signals are constant. A tabulation of the latter according to the variables can be used to measure the aging deviation of the components constituting the converter.

Current (A) i load (i ε ) est (d) Estimation of i ε (t)
Fig. 5: Simulated state variables (solid lines) and estimated state variables (dashed lines)

Experimental results

The proposed nonlinear control law has been tested on an existing DC/DC Converter. The experimental setup is composed of an Insulate Gate Bipolar Transistor (IGBT) power switches, a capacitor and an inductor. The physical parameter values are given in Table 1. This experiment was conducted with a sampling period of 0.75 ms, which is compatible with the imposed PWM switch frequency (20 kHz), and the reference output voltage v * c has been set to 24V . Starting from the values (24) obtained in simulation with MATLAB/Simulink, the best input/output performances have been obtained with the following values:

θ = 2000, λ = 1500, k c = 1. (25)
The empiric rule for the choice of the parameters λ and θ with respect to the sampling period is to have T s λ max ≈ 1 and T s θ max ≈ 1 [START_REF] Farza | Discrete-time nonlinear observer-based estimators for the on-line estimation of the kinetic rates in bioreactors[END_REF][START_REF] Farza | Simple nonlinear observers for on-line estimation of kinetic rates in bioreactors[END_REF]. Some new approaches allowing to overrule these limitation have recently been proposed in [START_REF] Farza | Continuous-Discrete Time Observer for a class of MIMO Nonlinear Systems[END_REF].

The control algorithm has then been implemented using the 1102 Dspace Environment. One can notice that the voltage and, especially, the current measurements are very noisy, indeed any anti-aliasing filters has been used. It is particularly visible on Fig. 6d, where the stabilized supply voltage measurements are shown. The goal of this experiment is to show that our control law can keep the output voltage to the reference value even in presence of changes in load and supply voltage, all the results are presented on Fig. 6. The changes in the load and the supply voltage have been applied manually to the converter and can be seen on Fig. 6c and6d. The estimates of v ε (t) and i ε (t) are depicted on Fig. 6c and6d. The difference between v e (t) and v ε (t) clearly illustrates the power stage losses. Similarly, the imposed variations of the output current are well estimated by the observer (see Fig. 6c). As shown in Fig. 6a, the regulation is offset position error free despite modeling errors, variations in supply voltage and load. The control objective is achieved with a slightly noisy input signal (Fig. 6e). The nonlinear process behavior is visible with the increase of noise on the output and input during the variation of the supply voltage (3.2 s -4.5 s). The load variations cause a change in the power loss which results in an increase of the average value of the input signal (around 6.2 s). There is no important peak in the value of the control variables to compensate the absence of knowledge of the uncertain parameters. This allows to avoid saturation of the coil which would lead to a magnetic saturation and thus to a nonlinear behavior of the physical system, as shown on Fig. 6b.

Conclusion

In this paper, a nonlinear approach for controlling a static boost converter has been presented, with several improvements with respect to the previous approaches. The underlying control system ensures an offset free performance even in presence of uncertain parameters. Such a feature is achieved by a cancellation of the effect of a nonlinear term acting on the input which is appropriately estimated using a nonlinear high-gain observer. Three features are worth to be mentioned. Firstly, the involved estimates are of importance because they can be used for supervision purposes for batteries or super-capacitors [START_REF] Oukaour | Supercapacitors aging diagnosis using least square algorithm[END_REF]. Secondly, the implementation of the control law has been greatly simplified as the integral action is implemented directly in physical coordinates. And finally, The convergence has been proven and validated by probing experimental results.

A Proof of proposition 1

In the proof, for the sake of clarity, the dependence on the parameters v * ε and i * ε will be omitted, that is, the functions f and φ will be denoted

f (z) = f (z; v * ε , i * ε ) and φ(z) = φ(z; v * ε , i * ε ).
It is of particular importance to notice that the function f (z) is positive on the physical working subspace because of the bound on the physical signals given by [START_REF] Walker | Cascaded dc-dc converter connection of photovoltaic modules[END_REF]. Moreover, the function φ(z) is Lipschitz in its arguments over the domain of interest, since it is continuously differentiable and its arguments stay inside a bounded set, because of (2). The state vector z(t) must be maintained in the vicinity of the constant reference vector v * . This is equivalent to

lim t→+∞ z(t) = 0, (26)
where z(t) is the tracking error defined by

z(t) = z(t) -z * . (27)
Using ( 6), ( 12) and ( 27), the control model can be written in the following Brunovsky triangular canonical form

ż(t) = A c z(t) + B c f (z(t))µ(t) + ϕ(z(t)), (28)
with

A c = 0 2×1 I 2×2 0 0 1×2 , B c = 0 2×1 1 , ϕ(z) = 0 2×1 φ(z) , (29)
where I 2×2 , 0 1×2 and 0 2×1 are respectively the 2 × 2 identity matrix, the zero matrix of dimension 1 × 2 and the zero matrix of dimension 2 × 1.

The control law

µ(t) = - 1 f (z(t)) ν(z(t)), (30)
provides an exponential convergence to zero of the tracking error for a sufficiently high value of the parameter synthesis λ, according to Theorem 3.2 in [START_REF] Bédoui | Robust nonlinear controllers for bioprocesses[END_REF], where

ν(z(t)) = k c B T c SΓ λ z(t), (31)
k c is a synthesis parameter, Γ λ is the following diagonal matrix

Γ λ = diag(λ 3 , λ 2 , λ), (32)
and the matrix S is the positive definite solution of the following algebraic Lyapunov equation:

S + A T c S + SA c = SB c B T c S. (33)
This choice for the gain is a direct transposition of the high-gain observer from [START_REF] Gauthier | A simple observer for nonlinear systems applications to bioreactors[END_REF] exploiting the duality between control and observation. Let . This implies that all the eigenvalue of the matrix A c -B c B T c SΓ λ are equal to -λ, this simplifies the implementation and the proof of convergence without introducing further constraints on the control performance. In the original coordinates, the controller ( 30) is given by ( 14)-( 5), which ends the proof.

B Proof of proposition 2

Our system can be written into the following form (studied in [START_REF] Farza | Observer design for a class of mimo nonlinear systems[END_REF])

ẋ(t) = A o x(t) + ϕ(x(t), u(t)), y = C o x(t), (34)
where u(t) is the control variable, x(t) is the state of system (34), defined by

x(t) = x 1 (t) x 2 (t) ∈ R 4 , (35)
with

x 1 (t) = x 1 1 (t) x 1 2 (t) = v c (t) i l (t) , x 2 (t) = x 2 1 (t) x 2 2 (t) = -i ε (t)/C v ε (t)/L , (36)
and the matrices A o and C o are defined by

A o = 0 2×2 I 2×2 0 2×2 0 2×2 , C o = I 2×2 0 2×2 . (37)
The function ϕ(x(t), u(t)) is globally Lipschitz with respect to x(t) uniformly in u(t) and is given by

ϕ(x(t), u(t)) = ϕ 1 (x(t), u(t)) ϕ 2 (x(t), u(t)) , (38)
with

ϕ 1 (x(t), u(t)) = (1 -u(t))x 1 2 (t)/C -(1 -u(t))x 1 1 (t)/L , ϕ 2 (x(t), u(t)) = 0 2×1 0 2×1 . (39)
Lemma 2.1 in [START_REF] Farza | Observer design for a class of mimo nonlinear systems[END_REF] can be applied, since v ε (t) and i ε (t) are supposed piecewise constant, hence ensuring exponential convergence of the estimation error on every time interval where v ε (t) and i ε (t) are constants.

C Proof of proposition 3

As in the proof of proposition 1, the system (3) is first transformed into the Brunovsky canonical form (6) through the change of coordinates z = ψ(v c , i l , u; v ε , i ε ):

     ż1 (t) = z 2 (t), ż2 (t) = z 3 (t), ż3 (t) = f (z(t); v ε , i ε )µ(t) + φ(z(t); v ε , i ε ), (40)
where f and φ are defined as in ( 7) and ( 9) respectively. For the sake of clarity, the notation f (z) and φ(z) is considered instead of f (z; v ε , i ε ) and φ(z; v ε , i ε ) in the rest of the proof. The reference trajectory, is no longer constant in the non-nominal case, it is given by

z * (t) = ψ(v * c , i * l (t), u * (t), v ε (t), i ε (t)), = ψ v * c , i ε (t) v ε (t) v * c , 1 - v ε (t) v * c , v ε (t), i ε (t) , = κ(v * c , v ε (t), i ε (t)).
Similarly, the estimate of the reference trajectory z * (t) is given by ẑ

* (t) = ψ(v * c , î * l (t), û * (t), sat 1 ( vε (t)), sat 2 ( îε (t))), = κ(v * c , sat 1 ( vε (t)), sat 2 ( îε (t))), =    C (v * c ) 2 + L( sat 2 ( îε (t)) sat 1 ( vε (t)) v * c ) 2 0 0    ,
where the function sat 1 and sat 2 are define in [START_REF] Olalla | Robust gain-scheduled control of switched-mode dc-dc converters[END_REF].

The aim of this proof, is to show that the error z(t) = z(t) -z * (t) converges exponentially to zero on any time interval where z * (t) is constant. Note that saying that z * (t) is constant is equivalent to say that v ε (t) and i ε (t) are constant since z * (t) is a function of v * c , v ε and i ε .

In order to do that, it is sufficient to prove that the next two signals

zc (t) = z(t) -ẑ * (t), (41) zo
(t) = ẑ * (t) -z * (t),
converge to zero, since z(t) = zc (t) + zo (t).

Convergence of zo (t) The signal zo (t) can be seen as the observation error on the reference trajectory. It is directly related to the observation error since :

zo (t) = κ(v * c ,sat 1 ( v * ε (t)),sat 2 (i * ε (t)))-κ(v * c ,v * ε (t),i * ε (t)) . (42)
The function κ is Lipschitz in its arguments over the domain of interest, then one can use the mean value theorem which states that there exists L 1 > 0 such that

zo (t) ≤L 1 ( sat 1 ( v * ε (t))-v * ε (t) + sat 2 ( î * ε (t))-i * ε (t) ), (43)
Since v * ε (t) and i * ε (t) belong to the unsaturated domain and the sat() function is 1-Lipschitz, equation [START_REF] O'reilly | Observers for linear systems[END_REF] becomes

zo (t) ≤ L 1 ( v * ε (t) -v * ε (t) + î * ε (t) -i * ε (t) ). (44)
Using the convergence of the observer, one obtain that if θ is taken sufficiently large, one has

zo (t) ≤L 1 αθe -µ θ t ( v * ε (t)-v * ε (0) + î * ε (t)-i * ε (0) ). (45)
where α and µ θ are given by proposition 2.

Convergence of zc (t) According to [START_REF] Tan | General design issues of sliding-mode controllers in dc-dc converters[END_REF] and ( 41), the error zc (t) is ruled by the following dynamical system: żc (t) = A c zc (t) + B c f (z(t))µ(t) + ϕ(z(t)) -ż * (t),

where A c , B c , f and ϕ are the same as in the proof of proposition 1.

Similarly to the proposition 1, the control law is taken as:

µ(t) = - 1 f e (z(t); vε , îε ) k c B T c SΓ λ (ẑ(t) -ẑ * (t)), (47)
where f e (z; vε , îε ) = f e (ψ -1 (v c ,i l ,u; vε , îε ); vε , îε ), which will be simply denoted f e (z), in the sequel. The matrices Γ λ and S are defined by ( 32) and ( 33) respectively and ẑ by ẑ=ψ(v c ,i l ,u;sat 1 ( vε ),sat 2 ( îε )).

Note that the corrective term in the proposed control law is not zc , because z is not available, indeed, it depends on v ε and i ε .

Let z c = Γ λ zc , then

f max f min , (51)
leads to

V 1 ≤ -λV (z c ) (52)
where f max , f min > 0, are such that f min ≤ f (v c ,i l ,u;v ε ,i ε )≤ f max , these bounds exist since (v c , i l , u, v ε , i ε ) belong to a bounded set given by [START_REF] Walker | Cascaded dc-dc converter connection of photovoltaic modules[END_REF]. Noticing that ϕ(z * ) = 0 when z * is constant, and denoting Using the Cauchy-Schwartz inequality and applying the mean value theorem, since ϕ has a triangular structure and is Lipschitz over its definition range, gives the existence of c 1 , c 2 , c 3 > 0 such that

V 2 ≤c 1 V (z c )+c 2 √ V (z c ) Γ λ zo +c 3 √ V (z c ) ẑ-z (53)
where c 1 and c 2 does not depend on λ.

Denoting W (t) = V (z c (t)), using inequalities (50), ( 52), [START_REF] Khalil | Nonlinear Systems[END_REF] and Cauchy-Schwartz inequality for the term -2z T c SΓ λ ż * yields

Ẇ (t)≤- λ-c 1 2 W (t)+ c 2 2 Γ λ ż * + c 2 2 Γ λ zo + c 3 2 ẑ-z . (54)
It has already been proven that zo (t) converges exponentially toward zero, the same can be done for ż * (t) and (ẑ -z) by using the fact that ż * (t) = żo (t), ẑz = ψ(v c , i l , u; sat 1 ( vε ), sat 2 ( îε )) -ψ(v c , i l , u; v ε , i ε ) ,

≤ L 3 ( v ε -vε + i ε -îε ),
where L 3 > 0 exists since ψ is Lipschitz over its domain of definition. Thus, there exist α, β > 0 such that

Ẇ (t) ≤ - (λ -c 1 ) 2
W (t) + αe -βt . ( 55
)
Note that β does not depend on λ but α may depend on λ. Furthermore, using the comparison lemma 3.4 in [START_REF] Khalil | Nonlinear Systems[END_REF], one obtains :

W (t) ≤ W (0) e -(λ-c 1 )
2 t +W (0) t 0 e - Finally, if λ is chosen so that

λ > c 1 , (56)
then equation (56) implies the exponential convergence of W toward zero and thus of V . If λ is taken sufficiently large zc (t) ≤ α c λ e -min{(λ-c 1 ),β}t

where α c λ > 0. This ends the proof.

Fig. 2 :

 2 Fig. 2: Scheme of the control law implementation presented in section 3.3

  λ= 1000; θ = 3000 λ= 500 ; θ = 2000 λ= 250 ; θ = 1000

Fig. 3 :

 3 Fig. 3: Simulated transient of the boost converter under a load step variation (R = 5 Ω → R = 30 Ω → R = 5 Ω) with different parameter values of the nonlinear controller .

  load step transient (R : 5 Ω → 30 Ω → 5 Ω) in nominal point d = 0.5. load step transient (R : 5 Ω → 30 Ω → 5 Ω) in nominal point d = 0.7. supply voltage step transient (v e : 12V → 7V → 12V ).

Fig. 4 :

 4 Fig. 4: Simulated transient of the boost converter with nonlinear controller (solid line) and current-mode controller (dashed line).

Fig. 6 :

 6 Fig. 6: Experimental results.

  B T c = C c , since the algebraic Lyapunov equation S + A T c S + SA c = C c C T c has a unique symmetric positive definite solution S , one can deduce that equation (33) has a unique symmetric positive definite solution S = T S -1 T with With this choice for the Lyapunov equation, one has B T c S = CS -1 T = [1 3 3]

żc

  =λA c z c -k c f (z) f e (z) λB c B T c S(z c +(ẑ-z))+Γ λ (ϕ(z)-ż * )(48)since Γ λ A c Γ -1 λ = λA c and Γ λ B c = λB c . Let us denote V the following candidate Lyapunov function:V (z c ) = z T c Sz c ,(49)its derivative is given by :V (z c )=2λz T c SA c z c +2z T c SΓ λ ϕ(z)-2k c f (z) f e (z) λz T c SB c B T c Sz c -2z T c SΓ λ ż * -2k c f (z) fe(z) λz T c SB c B T c SΓ λ (ẑ-z). (50)Denoting V 1 =2λz T c SA c z c -2k c f (z) f e (z)λz T c SB c B T c Sz c , using the expression of SA c deduced from the algebraic Lyapunov equation (33) and taking k c such thatk c > 1 2

  V 2 =2z T c SΓ λ ϕ(z)-2k c f (z) fe(z) λz T c SB c B T c SΓ λ (ẑ-z), gives V 2 = -2k c f (z) f e (z) λz T c SB c B T c SΓ λ (ẑ -z) + 2z T c SΓ λ ((ϕ(z) -ϕ(ẑ * )) + (ϕ(ẑ * ) -ϕ(z * ))) .

(λ+c 1 ) 2 (,≤λ-c 1 2

 21 t-s) αe -βs ds,≤ W (0)e -(λ-c 1 ) 2 t +αW (0)e - W (0)e -(λ-c 1 ) 2 t + αW (0)e -(λ+c1 ) 2 t λ-c 1 2 -β e ( -β)t -1 , ≤ W (0)e -(λ-c 1 ) 2 t + αW (0) λ-c 1 2 -β e -βt -e - λ-c 1 2 t .

  ) + v loss (t) and i ε (t) = i(t) + i loss (t). The signals v loss (t) and i loss (t) have been introduced to take into account the modeling errors (stray resistor and voltage drop across the IGBT type switch). Given the sources of the losses, v ε (t), v e (t) and i ε (t), i(t) have the same nature, that is, step-like. The different signals are assumed to belong to the following bounded sets

Table 1 :

 1 reports the simulation parameters values which are the same than the experimental ones, presented next. Experimental parameters values

	Symbol	Quantity	Value Unity
	L	Coil	2	mH
	C	Capacitor	6.8	mF
	R L	Resistor	8	mΩ
	R esr	Resistor	2.5	mΩ
	u max	Maximum control value	0.98	-
	u min	Minimum control value	0.02	-
	v * e	Nominal battery voltage	12	Volts
	v * c	Reference output voltage	24	Volts
		PWM switch frequency	20	kHz


Laurence Cox

L ' Auteur

The multiple traditions of social movement research: theorising intellectual diversity

Keywords: European Movements ; et Marxism and Social Movements social movements, sociology of knowledge, disciplinary norms, interdisciplinarity, normativity, sociology of the global South The multiple traditions of social movement

une des revues internationales les plus connues pour l' étude des mouvements sociaux, et a fondé le réseau de recherches en mouvements sociaux du Council for European Studies.

Les Working Papers et les Position Papers de la Fondation Maison des sciences de l'homme ont pour objectif la diffusion ouverte des travaux en train de se faire dans le cadre des diverses activités scientifiques de la Fondation : Le Collège d' études mondiales, Bourses Fernand Braudel-IFER, Programmes scientifiques, hébergement à la Maison Suger, Séminaires et Centres associés, Directeurs d'études associés... Les opinions exprimées dans cet article n' engagent que leur auteur et ne reflètent pas nécessairement les positions institutionnelles de la Fondation MSH.

The Working Papers and Position Papers of the FMSH are produced in the course of the scientific activities of the FMSH: the chairs of the Institute for Global Studies, Fernand Braudel-IFER grants, the Foundation's scientific programmes, or the scholars hosted at the Maison Suger or as associate research directors. Working Papers may also be produced in partnership with affiliated institutions.

The views expressed in this paper are the author's own and do not necessarily reflect institutional positions from the Foundation MSH.

Les traditions plurielles de recherches en mouvements sociaux : théoriser la diversité intellectuelle

Résumé

Le but du présent article est d' offrir un certain nombre de réflexions sur la diversité des approches actuelles à l'étude des mouvements sociaux. Il propose une esquisse des divergences clé qu' on trouve dans ce domaine, et passe en revue les questions intellectuelles normatives soulevées par la reconnaissance de cette diversité, ainsi que les questions relevant de l'histoire des idées soulevées par toute tentative d'explication. Dans une perspective globale, la question de ce que pourrait constituer une étude des mouvements sociaux du Sud illustre le défi auquel on se trouve confronté. L'auteur s'inspire de la typologie aristotélicienne du savoir pour proposer quelques moyens possibles pour gérer cette situation.

Mots-clefs

mouvements sociaux, sociologie du savoir, normes disciplinaires, interdisciplinarité, normativité, sociologie du Sud Introduction W hy should we think about social movements at all? Or why think about them within the context of global studies? As a topic of academic research, social movements are something of a "poor relation": there is widespread disdain for the subject as being either événementiel or enthusiastic. However, we also find movements regularly used for "kindling" as Borges puts it, for example in academic cover images, introductory stories or discussions of practical implications. This mild schizophrenia could be interpreted as a kind of bad faith: on the one hand academia seeks publicly respectable subjects and even vaunts a certain degree of tedium as a mark of intellectual rigour -but on the other the dramatic figures of movements are used to attract the interests of students, readers and for that matter policy-makers in order to justify particular kinds of study.

This could be read at one level as the challenge raised by a sociology of action to a sociology of structure; but the study of social movements also raises important epistemological questions around knowledge and the place of theory, both in universities and in the wider world. If we turn to face in the other direction, we see that social movement studies experience great difficulty in asserting sole ownership over their object of study. Other academic disciplines, but also movement-derived traditions of thought (notably Marxism, feminism and anarchism), equally claim the right to speak on the subject. They do so more or less effectively in different contexts, but often more effectively than do social movement studies. Scholars in the field can readily observe that an unusually high proportion of public and media commentary on movements is entirely unaware that they are the subject of academic expertise in their own right. This is on the one hand a sign of how much movements matter: as with sociology more generally, it is because they are matters of general concern that others think about them systematically but using their own academic and non-academic categories. On the other hand this situation should propel movement researchers towards a more reflective, and reflexive, understanding which attempts to situate what we do, and these other traditions, in a wider perspective.

The last decade and a half in particular has seen a great enrichment of the space of dialogue in relation to social movements. This derives not so much from any major expansion of third level education in most countries, let alone a rising power of social movement studies within the university. Rather, it is the result of the network of movements broadly discussed as a "movement of movements" or alterglobalisation movement, or more recently as (for example) anti-austerity movements or the mobilisations of 2011 and beyond. These widespread movements have brought questions of collective action much more to the centre of public and hence also academic attention than was the case in the 1990s. At the same time, the multiplicity of movements and sites of conflict and the scale of challenges involved have also contributed to encourage reflection about movements in general, as distinct from the specifics of individual campaigns.

In this context, the phrase "social movements", but also other terms for collective action (e.g. "civil society", "resistance" or "participation"), alongside discussions of the themes which movements organise around, have certainly become much more widespread, including in the form of journals, conferences and academic networks. However, this is far from meaning that "social movement studies" in any simple form have become an uncontested pole of attraction. If anything, we have seen a greater interest in dialogue of various kinds: as we conceptualise this on the journal Interface, this has meant dialogue between different academic disciplines, between different intellectual and political traditions, between movement-based theorising and academic research, between different regions of the world and between reflections around different kinds of movement. This much is familiar and in many ways welcome. It is at one and the same time an assertion of the value of what movement researchers do and an enrichment of our reflections that each new month seems to bring the need to engage with new ways of thinking and talking about collective action, from the most diverse academic and extra-academic sources. I have written about this elsewhere from a normative point of view (Barker C. andCox L. 2011, Cox L. andFlesher Fominaya C. 2009), but here I want to step back from those arguments and note that one tide which these normative arguments about intellectual Living with intellectual diversity: assertion, dismissal, integration

Individual scholars have necessarily learned how to think and write about social movements within one or more concrete social and institutional contexts, through various processes of secondary socialisation. They may for example have first learned to think politically within a movement, completed an undergraduate degree in one discipline in one country and successfully established themselves as graduate or professional researchers within another discipline in another country: such trajectories are common in the post-Seattle generations of researchers. In the process, scholars have of course internalised many different assumptions and invested substantially in what are actually quite situation-specific forms of cultural capital, while also perhaps negotiating the complexities between the different modes of reflecting on movements that they have been trained in.

What I want to suggest is that on the whole our practical ability to shift between registers, or (metaphorically) to creolise and draw on prior learning within our current language, tends to outstrip our explicit reflection on the subject; we are not always necessarily reflexive in our relationship to these different modes of thinking. This is of course particularly important insofar as we are engaged in performative speech (or more often writing) acts, such as peer review of publications or grant proposals, writing professional recommendations or making academic appointments. Such processes require us in a sense to naturalise our own intellectual approach, to take up a position from which we can straightforwardly assess someone else's work in terms of its performance within our own terms of reference. In so doing, of course, we not only naturalise our own approach and ascribe value to it -we also implicitly dismiss other approaches to a greater or lesser extent 1 .

My suggestion, then, is that in those kinds of speech act which are particularly powerful in the construction of academic formations -and hence both for the social realities of intellectual activity and for how we reflect collectively -we are not always as reflective in relation to our own approaches as we might be in other circumstances, or as our own learning processes might suggest. It is perhaps one of the most common experiences in contemporary social movements research -both as subject and as object -to praise or be praised, dismiss or be dismissed, on the basis of (often implicit) assumptions as to what counts as good or bad within a particular approach to movements. Outline of a research project

I
The research project under discussion here draws on a variety of intellectual experiences to approach this subject. Firstly, as noted above, it is something my colleagues and I have been practically exploring within Interface, where we have tried to enable the kinds of dialogue mentioned (across disciplines, across movements, across intellectual traditions, across global regions and between researchers and activist theorists) while simultaneously carrying out effective peer review on a practitioner research model familiar from other fields such as nursing, international relations or architecture (drawing on both academic researchers and movement practitioners) and while recognising the intellectual diversity of different regions of the world (through distinct editorial groups and peer review processes in different contexts). In doing this, we have not attempted to construct a meta-language to enable evaluation of all the different kinds of intellectual approaches in question. Nonetheless, over 16 issues and 8 years, we have built up a substantial body of practical understanding of the nature of the intellectual challenges involved. In other words we have attempted to put B. de Sousa Santos' ( 2006) call for an "ecology of knowledges" into practice.

Secondly, I have paid particular attention to the dialogue between the different forms of knowledge and subject positions of academic researchers and movement theorists, notably in the dialogue between Marxism and social movement studies (which is of course also an encounter between different academic disciplines, or between two multidisciplinary approaches in some cases) and in the participatory action research programmes involving movement participants carrying out research at PhD and MA levels at the National University of Ireland Maynooth (Cox L. 2014(Cox L. , 2015)).

Thirdly, and more recently Here we also find concrete, institutionally-located practices -but now understood explicitly as practices for mediating between different national and disciplinary specificities in the practice of social research (not necessarily of course in terms of relationship to movements). The hope is that these two comparisons will then enable a closer identification of the dimensions along which such approaches differ. Clearly such conditions of existence shape not only how individual researchers arrive at a particular approach to researching movements, but also what kinds of institutional centre are possible: who has to be satisfied of an approach's legitimacy in order to make it take on a lasting shape? In some rare contexts, even in these latter days, it is possible for like-minded researchers to find themselves within the same institution, or to coalesce around a journal or conference -though the like-mindedness then needs to be explained in terms of wider external forces. More commonly it is important to convince powerful others whose support is needed for significant institutional developments (major academic funding bodies, trade unions, university managers, academic barons) of the value of a particular approach in terms of various explicit and implicit criteria. There is also of course a distinction between shorter-lived initiatives tied to project funding and longer-lasting centres, journals and conferences which naturally have a more complex life history as they continue in operation.

Here There are also of course significant material constraints. Cousin and Rui make clear the difficulties faced by PhD students in carrying out sociological intervention, but it is also hard to imagine extensive sociological intervention in a classic sense carried out by British or Irish teacher-researchers under most circumstances (for example, Peillon, M. 1982 adopts Touraine, A.'s (1981) theoretical perspective to analyse Irish society but does so on the basis of secondary literature rather than the methods of sociological intervention).

Lastly it is of course also a question of how social movements themselves impact our fields, for example in terms of student motivations, new researchers' focus of interest, normative concerns etc. but also indirectly, via the effect of Marxism, feminism, Black studies, GLTBQI studies, disability studies, postcolonial studies, subaltern studies and so on. This is of course also true for other political features such as the need to respond to right-wing movements, the concern to defend a liberal public space etc. Less positively, increasingly authoritarian governments and intolerant public spheres can also mean that actual or perceived engagement with movements, or publishing perspectives which are seen to have political implications, can be a very risky activity. Turkey, India, Egypt, Russia and East and Central Europe are all cases where researchers cannot pretend to have the luxury of pursuing research on movements unconstrained by considerations of the potential consequences.

A typology of approaches to social movement research

The following provisional typology may help to clarify some of the dimensions of variation.

One dimension is institutional, on a broadly horizontal vs vertical schema: in one kind of academic system, researchers who share a common approach whatever their level can come together and agree to construct something which then has little difficulty gaining institutional approval. the viewpoint of mainstream media, policy-makers and intermediate spaces such as national academies, middle-brow reviews, etc. Actual movement participation may be more or less welcome and more or less constrained, both from the academic side and in terms of movements' reception of researchers and what is perceived as legitimate.

A further dimension is the extent to which, in particular societies and particular movements, substantial movement-related intellectual institutions (in the sense of not being ultimately reliant on academic authorisation) can be said to exist, and the ways in which these interface with (a) the practice of research, (b) theoretical developments and (c) the social concerns of academic disciplines. Marxism is an obvious case in point here, in that some societies (but not all) have more or less extensive independent Marxist education and research institutions, with a very wide range of relationships to academic activity.

Finally we need to consider the dimension of intellectual competition: how far do multiple disciplines and intellectual traditions (a) lay claim to the field of collective action, more or less successfully, and (b) engage in dialogue with one another in this respect? And how far do they engage in dialogue with movement-based theories and issues (this may of course be a discussion within e.g. Marxism or feminism, ecology or GLTBQI thought as between academic and movement exponents)?

This -very provisional -typology is intended to help a move towards a comparative history of ideas, underlining the social conditions of existence of particular approaches to social movement research within the academy. It also thematises the relationship between academic knowledge, practitioner knowledge and public knowledge of social movements, not only in theoretical terms but perhaps more strongly in terms of methodological assumptions as to the nature of such knowledge (for example: objective, pragmatic and normative).

As I stress in the next section, a sociology of knowledge approach does not exclude recognising the constitutive necessity for active researchers to commit at least provisionally to a specific local approach, at the same time as having relationships both of recognition and of non-recognition with other approaches, other disciplines or other nationally-based models. I want to suggest that this is in fact the key intellectual challenge which we are forced to engage with in practice (to a greater or lesser extent) but which we can benefit from reflecting on more explicitly.

Reflections: normative intellectual questions

The questions raised in 1.2 above are different methodological approaches to the same broad sociology of knowledge question, which can now be stated more precisely: how, in this metaphorically multi-lingual situation, can we (firstly) construct a genuine conversation that is not simply a prisoner of the specific national, disciplinary etc. contexts within which funded research and salaried researchers exist -and (secondly) work towards an acceptance of diversity without (thirdly) abandoning any sense of intellectual value or for that matter normative significance?

Here I am assuming that we do not simply want to allow our intellectual norms to be dictated by what happens to constitute orthodoxy in a particular time and place without reflection, whether that means to abandoning judgement on what constitutes good social movements research either to the globally dominant conversation that is the US subdiscipline of social movement studies or to the locally powerful set of relationships that is the EU's research funding processes in the area. This is, of course, different from consciously judging that either the substance of these approaches or individuals and institutions involved in their social infrastructure are those from which we should take our intellectual lead.

However, if we do not ourselves hold the kind of academic power to be involved in shaping such processes, we are then also by definition in no position to impose our own scholarly norms unilaterally beyond the level of a research programme, a journal or conference which we ourselves control -and which in any case we would typically have to justify in other terms: to funders, colleagues, university authorities and so on. Thus some form of co-existence is a fact of life for those of us who wish to engage beyond our own provincial situation (and here I am reading both the US subdiscipline and EU funding processes as provincial, albeit powerfully so), along with the commitment we necessarily adopt (at least for any given purpose) to a more specific approach. At the same time, I take it that we do not wish to see co-existence mean that we simply interpret what exists as being good by the simple virtue of existing. At some level we seek an intellectual conversation, and a systematisation of how knowledge is generated which can legitimately be called science -indeed, perhaps more legitimately so than one which takes the practice of a particular discipline as it is carried out within a given local institutional setting as ultimately defining of intellectual value. So the attempt to construct such a conversation through the identification of shared points of intellectual reference, even if not central to our own practice; the systematic development of ideas and logical argument, even if we do not always start from the same presuppositions; and the careful attempt to clarify empirical realities, even if those we engage with are not of the same order -these all remain important, and it is here that a more careful specification of the sociology of knowledge shaping our own work is useful.

Furthermore, it becomes empirically evident once one seeks employment outside of the most obvious local possibilities (or supports junior scholars in their own efforts) that social movement studies often has to argue for its right to exist, and that in Europe in particular the argument is far from being accepted. Full-time social movement researchers are a far rarer breed in Europe than many other kinds of researchers, and for this reason we often find ourselves highlighting in different ways the specific real-world significance of what we do, without (hopefully) wishing to reduce our research to what is instrumentally valuable for elites. This is a more sensitive and perhaps "essentially contested" area, and as we shall see different kinds of argument are accepted as being valuable in different national contexts -whether in terms of publishing books whose readership goes beyond a specific academic niche, being in demand from the commercial media or valued online resources such as Open-Democracy, being able to speak to or collaborate with movement actors or for that matter relevant kinds of policy maker, or (for those of us in countries where research institutes such as the FMSH do not exist) showing undergraduates why social movements matter in the first place.

For our own scholarly practice, then, it may be useful to draw on de Sousa Santos' ( 2006) "ecology of knowledges" metaphor. How can we better understand where one another is coming from intellectually and learn from that, while developing (non-managerial) ways of raising the level of the conversation? These are of course classic interdisciplinary questions.

Here I offer the example of the peer review process on Interface, not as a model to be followed but as one which explicitly thematises these various dimensions.

Firstly, we insist on separate, autonomous editorial groups for each region of the world (and one for transnational movements) rather than impose a metropolitan definition of "what matters" for movements globally and how to approach it. Positively, this means actively seeking an openness to the forms of movement theory that are particularly salient in different regions.

If not yet "a sociology of the global South" (Connell R. 2007, Bhambra G. 2007), this is at least an institutional structure intended to enable the articulation of a "social movement studies of Latin America", or Africa, or South Asia.

Secondly, in peer review we seek one researcher to assess a piece from an academic perspective and one movement practitioner to assess its relevance to participants. Of course both of these can have many different dimensions and the corresponding editor is thus the final arbiter. This follows from the journal's self-definition as a practitioner journal on a model familiar from many other disciplines.

Normally we then hope to find both a congruence with the empirical experience of the movement in situ (and not simply its international reception, which can be a very different thing where a movement has become symbolic in international discourse) and that the piece is good of its kind within a particular scholarly or intellectual approach.

Hence it could be said that we are not attempting to prescribe a priori what disciplines, research methods, theoretical approaches or political / intellectual traditions are valid, but rather to create the space for a qualitative judgement on this, taking account of the intellectual landscape of a particular region (or in relation to a given movement etc.) Within this, we are then asking how well the piece meets the internal standards of that approach.

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Lastly we are asking how well the piece is able to speak etically, outside of this particular discourse, to others who are interested in thinking about social movement in whatever form.

As noted, I am not proposing these as universal standards for social movement research, but rather noting that these explicitly thematise some of the dimensions of research noted above.

A social movement studies of the global South? Hence the questions around a social movement studies of the global South, or the post-colonial world, or the non-European world, would have to be "What are the historical presuppositions underlying social movement studies as an institutional reality in different contexts? How far can these presuppositions be usefully relativised by a more global perspective? And what key dimensions of tension would that perspective identify?"

In my attempt to distinguish the primarily US-based history of what I have called canonical social movement studies from the actually-existing history of research on social movements in Europe, one obvious dimension of difference is the extent to which the US-based history is primarily the history of endogenous developments within a small number of academic fields, responding no doubt to the wider world but not, as a discipline, affecting the outer world to anything like the same extent. Conversely, it might be said, a key feature of much European research on movements in the postwar decades, and perhaps again today, is a different kind of relationship between movements, states and universities, in which the relatively weaker field of movement research responds to developments within movements and their interaction with the state.

Ireland, as a post-colonial state, is different again (Cox L. 2016). What we find here is an intellectual tradition within which the disciplines of history and literature were central to the articulation of a narrative of the nation as agent, and hence also of a wide variety of social movements, but within very different intellectual frameworks. Over 90 years after independence this same broad pattern remains as one core thread of research on social movements. Within sociology and associated fields, there is a pattern of research on contemporary social movements of a semi-institutionalised kind (e.g. feminism, GLTBQ movements, environmentalism); often this research has more or less explicitly policy-related interests. Meanwhile, strong Marxist traditions outside the academy associated with left parties, trade unions and independent critical thought contribute to academic research on these same dimensions of working-class movements; while adult education, applied social studies etc. see the presence of substantial bodies of theory tied to community-based activism of various kinds.

In other words, in the Irish context there is usually a relatively clear link between individual disciplines, privileging the study of specific movements, with a more or less visible political purpose, whether in relation to the legitimacy of the nation-state (and Northern Irish politics), social movements as policy actors, the political articulation of left organising or the training of community activists. My assumption in this paper is not that this pattern will prove to hold for other postcolonial states in any simple way, but rather that similar questions as to the political purpose (with a small or large P) of different intellectual / disciplinary projects can usefully be asked. At the broadest level, and now including "Western" approaches, the expectation is that different relationships to social action are more or less embedded within the nature of different research institutions, not least in their different intellectual problematics, their methodological approaches and epistemological presuppositions. We can perhaps use this as a means of grasping the multiplicity of different national situations in terms of the role of the university or of research, types of relationship between academia and social movements, and forms of theory.

This takes us away from a simplistic narrative of "science" and "ideology, "objectivity" and "bias", towards deeper questions of knowledge production and research methodology; in other words questions of what it is we do practically and socially when we act as scientists, how we justify and define this activity (or aspects of it) as being scientific, and how this works differently in different contexts. At another level again, it raises important questions about the nature of knowledge, and here I want to turn briefly to Aristotle.

Aristotle and social movements

Aristotle's conception of knowledge is complex and not entirely consistent (Aristotle Nichomachean Ethics Book VI, Parry R. 2007, Mrvica A. andSides C. 2007). I am not, here, attempting to adopt it for the study of social movements so much as to use it by analogy to reflect on knowledge and movements. In one possible reading, Aristotle distinguishes between true knowledge -not "science" as we now understand it in experimental terms but that which is necessarily true, as with some forms of mathematics; craft (techné), as an instrumental or productive kind of knowledge which involves a general understanding of the objects with which it works in order to effect some sort of change; and phronesis, as a form of self-knowledge which has to do with ethical action and wisdom among other things (excluding an instrumental-productive relationship).

We can, perhaps, glimpse here some of the reasons why the active type of knowledge associated with movements is challenging to what Gramsci would call a purely contemplative type of knowledge that might be considered to be "scientific" and associated with structure alone. Insofar as the study of movements is a sociology of action, we cannot divorce this from actors' own understandings, which -as in sociological intervention but not only there -appear not as fixed (as in some culturalist approaches) but rather subject both to learning and to contestation (de Smet 2015). An adequate understanding of movements cannot avoid engaging with this particular challenge.

However, we might also want to think (perhaps with G. Pleyers' 2010 distinction between instrumental and expressive emphases in social movements) that movement knowledge might in turn vary in its proportions of techné (craft, instrumental knowledge) or phronesis (prudence, action which is good in itself ), and indeed that the tension between these two may be constitutive of certain types of movement, or that particular political paths may result from how this tension is resolved. This is not, I think, to say that the study of social movements must be engaged -although I have argued that on other grounds elsewhere. Here it is to say rather that -just as with anthropology -the credible study of movements must involve a certain capacity to recognise this position of the actor, as craftsperson or prudent, which one masters best by taking up that position oneself, whether in movement engagement or in reflection on one's own instrumental or prudent actor position in other contexts.

Implications

How In each of these cases -and many others like them -the challenge is not so much to bridge the objective and the normative as to bridge the gulf between analysis and action. There is of course a certain rhetorical naïveté in simply attempting to bridge this gulf via the public or academic dissemination of what is in any case already known to those who pay attention to such things. As in the case of climate change, routine debate is not enough (or rather it responds to that form of liberalism for which the means -"a great debate" etc.

-justifies the end).

Social movements play a central role in moving beyond the intellectual full stop of declaring what needs to be done while being unable to articulate any possible agency beyond one's own routine work of dissemination along existing channels. This is not to say that they are the only such forces, but without movements the kinds of normative scholarly arguments identified above fail to identify sociologically credible carriers of their own programmes.

In this respect, social movement studies is about nothing less than the question of the social actors which may be able to offer some hope for the survival of our species, and our planet, in other than disastrous conditions. As social movement researchers we should, perhaps, be less coy about stating this fact -and less narrow in our own concerns.

We are in a period of intensifying conflict, and understanding movements is more important than ever before accessible deposits, while global policy conflicts between fossil and renewable fuels make it clear that the period in which states automatically took the side of the oil industry as the only (strategic, economically viable) game in town is now coming to an end. Conflicts around airport expansion represent another aspect in which the current industrial model is once again (following the conflicts of the late 1970s and early 1980s over nuclear power) increasingly being placed in question by social movements.

Globally, we are seeing an intensification of "anarchy" in international relations, with once-obedient client states pushing authoritarianism as far as they can, including in relation to social movement researchers in countries like Egypt, Turkey and India. We are seeing "bad movements" (such as pro-and anti-Islamist violence, often mediated through states and quasistates) as well as large-scale peace movements setting effective limits to western military adventures in MENA and elsewhere.

Within Europe, we see the juxtaposition of a resurgent right-wing populism, primarily in core states (e.g. AfD, Calais, anti-immigrant violence), with a broadly progressive resistance to the EU and associated austerity programmes, primarily in peripheral countries (Iceland, Spain, Greece, Portugal, Ireland). In the Anglophone world the experiences shaped around Trump and Brexit, Corbyn and Sanders represent similar tendencies of a rapid reshaping of political forces and alliances and a search for stable and effective forms of ensuring popular consent for new kinds of political project.

More broadly, it might be said that there is an ongoing series of uprisings and transformations where it appears that neither the old order can hold nor can something new successfully impose itself (e.g. Black Lives Matter, Trump and women's mobilising in the US; Nuit Debout, the state of emergency, banlieu riots and the presidential election in France; Egypt's political travails since 2011; the Gezi movement and Erdogan's counter-coup…).

In such periods it is precisely through movement and collective action (including collective action from above, or in alliance with the powerful) that the needs of the present are articulated -in deformed or transparent forms -and the potential of the future is articulated. I take it that this is part of the original vision of sociological intervention (that social movements represent something wider than themselves) and indeed part of the intellectual justification for studying movements in general, or for social movement studies to be something more than a minor and self-referential academic field. If there was a falling away from this larger vision in a period when movement seemed off the agenda it is clear that it is now back on the agenda. In this context social movement studies is looking at one of, if not the, most decisive actors in the conflicts currently shaping global futures.

Our potential as social movement researchers

As movement researchers, then, we are engaging with something which can at times represent both a objective referent of what present-day society can be said to need in a general sense and a key actor in the conflicts which express the needs of individuals and shape the future. I take it that some sense of this underlies both our work as researchers and our engagement in movements, when we are also engaged in this way.

I do not think that we should be ashamed of this; in fact we should find ways to articulate this clearly vis-à-vis our academic colleagues who prefer to describe and analyse structures in isolation from collective human action, but also visà-vis our interlocutors in movements who often value our recognition of the significance of what they are doing and the stakes they are fighting for. In fact in doing this we contribute both to a less parochial sense of science in which social movements are somehow unserious flourishes as against the profound theorisations of structure or concept -and to helping movement actors see themselves and their action more clearly.

In our work as researchers, but also as public intellectuals, movement interlocutors, teachers, supervisors and the like we can contribute in what I take to be a Tourainean way to helping movement actors understand more clearly the meanings of their action and reflect more systematically on their strategies. I think this is perhaps better than either finding theoretical ways of eternalising the world as it is (in a period of profound historical change) -or making faint calls for people to abjure sin and behave well without reflection on what that process actually means in

Fondation

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  those of research on social movements in Ireland. G.Accornero and O. Fillieule's (2016) book on social movement studies in Europe asks the same question in a different mode. Even within this supposedly relatively homogenous European space -which according to canonical origin myths has been largely hybridised with the North American model since the later 1980s -diversity is the rule rather than the exception (although, as with linguistic diversity, we do not always acknowledge this).The present project starts from a series of comparisons. The first is perhaps rather grandiose: it is to compare the "Maynooth approach" of participatory action research in social movement practice with the "CADIS approach" of sociological inter-

, I have been exploring the characteristics of European social movement research traditions with C. Flesher Fominaya and scholars in the Council for European Studies (Flesher Fominaya C. and Cox L. 2013), The multiple traditions of social movement research 8/19 Fondation Maison des sciences de l'homme -190 avenue de France -75013 Paris -France http://www.fmsh.fr -FMSH-WP-2017-128 and more specifically vention. I realise that the Maynooth approach is a minnow to CADIS' whale. However both are varieties of what might be imagined as public or engaged sociology with one face, while with the other face they represent varieties of dialogue between researchers and activists at the microlevel of a single institutionalised approach with a clear geographical base. These two approaches can be placed on a spectrum that includes other social movement centres in Europe (e.g. COS-MOS, Göteborg, Bochum, the Wissenschaftszentrum Berlin, Brighton…); national journals (e.g. Forschungsjournal soziale Bewegungen, Partecipazione e Conflitto, Mouvements des idées et des luttes, Interface, Social Movement Studies, Moving the Social…)and perhaps also conferences or mailing lists (Alternative Futures and Popular Protest, bewegungsforschung etc.). The second, equally grandiose perhaps, is to to relate the Interface model to the manifesto of ISA47's current leadership (International Sociological Association 2014a, b; Bringel B. and Pleyers G. 2015) and to the wider project expressed in Penser global (Wieviorka M. et al. 2015; cf Wieviorka M. 2013, Hautcouer P. and Wieviorka M. 2014, Calhoun C. and Wieviorka M. 2015) and the Collège d' études mondiales.

  Thirdly, there are some existing bases for a broader comparison of national approaches to social movements: the previously-mentioned collections by Flesher Fominaya C. and Cox L. (2013) and by Fillieule O. and Accornero G. (2016) do this in relation to Europe, as do Gagyi

	The multiple traditions of social movement research	9/19
	A.'s (2015) reflections on nominally "European" social movement research's difficulties in relation to movements in East and Central Europe (see also Piotrowski G. 2015). These offer some start-ing-points for reflecting more systematically on the different norms which operate at a national and disciplinary level, and to which researchers with allegiance to a particular intellectual trad-ition necessarily need to relate themselves if they are to be able to continue doing so profession-ally within that context, in terms of funding and appointments.
	Lastly, this recognition of geographical diversity may enable us to draw on the current debate on sociologies of the global South, as well as some recent work on the Northern biases of social movement studies (Poulson S. et al. 2014, Mac-Sheoin T. 2016) and my own reflections on Ire-land as a postcolonial state and the implications for movement research (Cox L. 2016), to ask rhetorically what the "social movement studies of the global South" (Fadaee S. 2016; cf Nilsen A. and Roy S. 2015) might be -or more exactly to ask if an intellectual approach which is not centred in the global North and which takes the intellectual lessons of postcolonial studies ser-iously points towards a different kind of social movement studies, or for that matter towards a social movement studies at all.
	Mapping intellectual
	diversity in social
	movement studies
	I want to start by putting slightly more shape on the empirical realities of intellectual divers-ity in social movement studies. In the European context alone we can note a series of centres of social movement research (not all of which would describe themselves in this language). If we take a handful of examples, we can see that while all are very diverse the general understanding of what research is, how it is to be carried out and the relationship between researchers and movements is rather different as between e.g.

• The Collège d' études mondiales;

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  we are not necessarily primarily talking about theoretical orthodoxies in any simple sense; for example,Pleyers G. and Capitaine B. (2016), like Cousin O. and Rui S. (2010) on CADIS, stress the theoretical plurality of even well-established institutions. It is rather a question of the broad problematics, methodologies and epistemologies which are encouraged, as well as of the overall doxa of particular disciplines in terms of what the gatekeepers of funding, PhDs, publications, careers etc. will accept as legitimate.