domain: phys spi

text:
M Khodsiani

F Beyrau

B Fond

Flame-particle interaction inside a packed bed of cooled particles

Flame-particle interaction inside a packed bed of cooled particles

Mohammadhassan Khodsiani, Frank Beyrau, Benoît

Introduction

Many chemical processes are designed to transform material in the form of large particles (larger than a few millimeters in size) that travel at a slow pace through a reactor. These reactors are known as packed bed reactors and they are used for particle transformation process such heating, drying, pyrolysis, calcination, and gasification. Those reactors significantly contribute to large CO 2 emission of process industries.

The temperature distribution within the bulk solid, as well as the gas composition surrounding it, are directly determined by the extent and location of the flame within the void space between the particles [START_REF] Krause | Coupled three dimensional DEM-CFD simulation of a lime shaft kiln-Calcination, particle movement and gas phase flow field[END_REF]. Both parameters play a crucial role in the heat and mass transfer between the bulk and the flame, which ultimately impacts the quality of the final product and the energy efficiency of the process.

Flame-wall interactions (FWI) in combustion engines and porous inert media (PIM) combustion are typically two separately treated area of combustion, but combustion in packed bed reactors directly relate to both of them. In nearly all configurations with a flame, walls play a major role in flame propagation by means of radical exchange, heat exchange and also by inducing a boundary layer. This is a two-may coupling, as the wall affects the propagation of the flame by means of inducing a boundary layer, exchanging radicals and dissipating heat, and the flame in return heats the particle up. The specific wall area per unit volume dictates the strength of this coupling.

In case of low specific area per volume, as in the internal combustion engines and gas turbines, thor-ough experimental and numerical investigation have addressed various aspects of the flame-wall interaction (FWI). [START_REF] Poinsot | Direct simulation and modeling of flame-wall interaction for premixed turbulent combustion[END_REF][START_REF] Jainski | Sidewall quenching of atmospheric laminar premixed flames studied by laserbased diagnostics[END_REF].

On the other hand, combustion in porous inert media (PIM), which has gained attention as a new low NO x [START_REF] Trimis | Combustion in a Porous Medium-Advances and Applications[END_REF][START_REF] Wood | Porous burners for leanburn applications[END_REF], lies on the "high coupling" side of the interaction as there is a high specific area of particle surface per volume. There the fuel-air mixture combusts in the interstices and pores of a porous inert media such as ceramic foams, pellets or packed beds.

In the context of the interaction of a flame and solid surfaces, a Peclet number is defined as the ratio of the distance from the wall at which the flame quenches (δ Q ), to the Zeldovich premixed laminar flame thickness (δ F ), that is, Pe=δ Q /δ F . For porous media burner it is the ratio of the pore size to the laminar thickness. A minimum Pe for the flame to propagate inside a porous medium is reported to be 65 [START_REF] Trimis | Combustion in a Porous Medium-Advances and Applications[END_REF]. Considering a regular spherical packed bed of particles and a methane air mixture, this translates into a minimum sphere diameter of 10 mm diameter with a pore size of 2.5 mm for the sphere to propagate inside the packing. A few experimental studies describe the propagation of the bulk flame zone in the interstices of the packed bed placed in a quartz glass tube [START_REF] Okuyama | Turbulent combustion characteristics of premixed gases in a packed pebble bed at high pressure[END_REF][START_REF] Wu | Experimental investigation on low velocity filtration combustion in porous packed bed using gaseous and liquid fuels[END_REF]. Nevertheless, the harsh scattering of light inside the packed bed and 3D-flame front were among the obstacles to an accurate and highlyresolved detection of the flame front.

In short, there is a need for an accurate description of the systems in which a reactive gas flows through a packing of particles. Also, CFD simulations must be validated as the high volume specific surface ratio means that the thermodiffusivity and wall chemistry play an important role. Obtaining unambiguous data in a setup with well defined boundary conditions is central to the validation of numerical simulations . Existing set of of experimental data do not allow such validation, as a) their flame image recordings incorporate complex 3Deffects b) the temperature of the particles is now known c) the geometry of the bed can not be fully reconstructed in simulations.

In this work we elaborate on the design of a cylindrical packed bed assembly in which a 2D flame can be stabilized between the cylinders. The cylindrical geometry also allows for a line-of-sight optical access. The first version of this assembly includes cooled cylinders to accurately compare with simulations and to replicate cold start-up conditions. Wall chemistry is expected to have less impact on flame propagation under these conditions than uncooled particles, which will be implemented in the next phase of the work. Prior to the flame studies, the uniformity of the temperature along the cylinder and the inflow velocity are verified to ensure twodimensionality of the flow field and thermal boundary conditions. The impact of flame position and extent on flame to particle heat transfer is studied using telecentric imaging of the reaction and a double calorimetric technique. The experiments are performed under different equivalence ratio and inflow velocity conditions.

Two dimensional flame in a packed bed setup

In the design of the two-dimensional packed bed burner, three criteria are taken into account, a) transferabiliy and reproducibility of the experimental results, b) relevance of both geometry and boundary conditions to the packed beds in the industry and their flow specifications, c) allowing for measurements of quantities such as wall temperature, gas velocity, temperature and also species concentration. The geometry of the experimental setup comprises a slit burner and a packed bed of cylindrical particles. A cross sectional view of the cylinder arrangement in the central region is given in Fig. 1. As shown, rows of cylinders are offsetted by half the center to center distance c, imposing turns to the flow similar to spherical packed beds. The slit burner is designed to supply a uni-form two-dimensional flow all along the cylinder length axis. It is composed of an inner duct delivering fuel/air mixture and an outer duct shielding the mixture by a coflow of air. The mixture of fuel and air flows between the two cylinders in the center of the first row.

To set the porosity, taken as void fraction here, the distance c can be adjusted. Eq. ( 1) is used to calculate the porosity within a unit cell defined as the dashed triangle which connects the particle centers in Fig. 1.

1 - π 2 √ 3 ( D c ) 2 (1)
As described in the introduction, flame-particle interaction is largely affected by the porosity. To have a porosity of 40%, relevant to the typical spherical packed beds, the distance c was chosen to be 1.23 D. Thus, the reactive flow inside the cylindrical packed bed is exposed to a similar level of porosity and tortuosity. The specific area of the solid/gas boundary is only 33% lower than a spherical packed bed. The cylinder diameter D is 10 mm, leading to a minimum gap of 2.3 mm between the particles, corresponding to the limit Peclet number of porous media burner and and therefore is the case of intense interaction between the flame and the particles.

The main advantage of this geometry is its through optical access which enables the implementation of lineof-sight optical diagnostics. The two-dimensionality of the flow and of the gas thermochemical state remains a significant consideration for the following reasons: 1) It simplified the flame behaviour description in both experiments and numerical simulations. 2) It avoids the signal averaging error induced when implementing diagnostics such as chemiluminescence imaging and schlieren, reliant on the integration of light all along the line-of-sight.

A three dimensional view of the assembly of the packed bed assembly and burner is shown in Fig. 2a To keep the flow two-dimensional, end walls are placed in the cylinder axes normal direction to confine the flow between the cylinders and avoid the apparition of a third flow velocity component which would be otherwise be caused by the escape of the flow through the open ends of the cylinders. For this reason, the uncooled cylinders are also extended past the holding frames to maintain similar pressure drop in the end region than through the cylinder array. A pair of optical windows on front and back side of the packed bed is used to keep the optical access. Also, side walls are also used to position the aluminum holding plates and to confine the co-flow and central jet and facilitate comparison with numerical simulations. The cylinder holder plates were machined as to provide an extended optical access to a large connected region, shown in Fig. 2b. This optical access covees the inflow region between the duct exit region below the first row, the region of the reactive flow between the three cooled cylinders, and also a large region of burnt gases. The two dimensionality of the inflow conditions was verified by particle image velocimetry measurements, described in [START_REF] Khodsiani | Spatially resolved experimental investigation of particle-flame interaction in a twodimensional model packed bed[END_REF]. The inflow velocity variation along the cylinder axis was found to vary by less than 5%.

Flame front imaging

A telecentric imaging system was employed to both detect the two dimensional flame reaction zone position, and to measure the real geometry of the rig compared to ideally conceived geometry. Chemiluminescence emission from the intermediate radicals is used to mark the position and shape of the flame front through the lineof-sight optical access in between the cooled cylindrical particles. CH* chemiluminescnece at 431 nm and C 2 * at 471 nm are recorded to mark the reaction zone. The error in the marked flame front using visible emission should be limited to around a few percent of the laminar flame thickness [START_REF] Bellenoue | Direct measuremrment of laminar flame quenching distance in a closed vessel[END_REF]. That the maximum intensity position of C 2 * and CH* emission are pretty similar is also experimentally demonstrated in [START_REF] Kojima | Spatially resolved measurement of OH, CH, and C2 chemiluminescence in the reaction zone of laminar methane/air premixed flames[END_REF]. Otherwise integration of the emitted light from the two radicals would have led to the detection of an artificially thicker flame front. However, imaging by conventional lenses would cause perspective distortion along the line-of-sight, so that the flame zone closer to the lens would be imaged with a larger magnification compared to the regions further from the lens. This finally leads to a loss of resolution when collecting all emitted light along the line of sight. To remedy this problem, a telecentric lens (Edmund Optics #62-912) was employed to maintain the same magnification all along the line of sight over which the chemiluminescence is emitted, as illustrated in Fig. 3(a). The telecentric lens axis is aligned with the cylinders axes and mounted on a sCMOS camera (PCO Edge 5.5) to record the chemiluminescence emanating from the reaction front over an exposure of 100 ms. The telecentric lense was equiped with a 500 nm short pass filter (Edmund Optics #84-719) to distinct the blue chemiluminescence from stray light and thermal radia-tion.

Image processing

An image processing algorithm is developed in MATLAB to accurately detect the cylinder boundaries as well as the location of the two dimensional flame front. This is both to assess the real geometry of the bed, and to reveal the location of the reaction front with respect to this geometry. A gradient-based algorithm detects the points lying on the edges of the top, right and left cylinders. Fitting a circle to those points, the center coordinates of cylinders is found. Blue, green and red circles of Fig. 3(c) are the fitted circles and delineate the detected edges of the top, right and bottom cylinders, respectively. Images of the flame luminos-ity are then recorded as shown in Fig. 3(b). Nevertheless, the accurate detection of the reaction front requires the extraction of the high intensity core of the flame in Fig. 3(b). In doing so, an algorithm is developed to access the centreline of the reaction zone by a combination of a Canny edge detection algorithm [START_REF] Canny | A Computational Approach to Edge Detection[END_REF], to isolate the reaction zone, and then a subsequent skeletonization to extract the high intensity core of that the reaction zone. The extracted centerline, being a one pixel thick line is superimposed on the blue reaction front in Fig. 3(b). In order to mark the location of the methane-air flame relative the cylinders, the detected cylinder walls and the extracted flame centerline are put together on a grid in Fig. 3 (c). This forms the premise for the discussion of flame front location under different operating conditions in section .

Internal cylinder cooling and heat transfer measurements

Besides the velocity uniformity, the temperature of the aluminum cylinders must be kept uniform along the cylinder length to make sure that flame stabilizes at the same height and lateral position all along the cylinder length, forming a two dimensional flame surface. Therefore, a sufficiently high flow of coolant must flow inside the hollow aluminum cylinders to prevent an oil temperature increase along the cylinder direction. Two cooling circuits are designed to feed a high flow of silicon oil into the cylinders cooling channels. Based on the flame power, oil coolers were selected to provide a high enough oil flow rate to reduce the cylinders temperature down to about 100 °C while minimizing the surface temperature variations along the cylinder axis to below 1°C. Surface temperature measurements are described in [START_REF] Khodsiani | Spatially resolved experimental investigation of particle-flame interaction in a twodimensional model packed bed[END_REF].

Heat transfer rates from the flame to to each individual cylinder are important quantities in the investigation of the interaction between the flame and the particles. To measure the heat transfer to each cylinder, we resort to a double calorimetric technique. First, through Eq. 2, the heat transfer rate is related to the oil temperature increase and its mass flow rate as it flows through the cylinder. T i and T o are the oil temperature measured at the inlet and outlet of the top cylinder. The same equation holds for the side cylinders.

QTop cylinder = ṁoil c p oil (T o -T i ) (2)
To measure the oil temperature rise in the presence of the flame, Resistance Temperature Detection (RTD) sensors with a diameter of 1 mm were inserted to the inlet and outlet of the cylinder. Probes are four-wired platinum resistance thermometers with class A accuracy, corresponding to a ±0.15 °C absolute temperature accuracy and an excellent capability to measure relative temperature variations due to very low drift. Such temperature resolution was required as a high silicon oil flow rate is used to keep the oil temperature rise below one degree, ensuring the uniformity of the temperature uniformity all along the cylinder. A second calorimetric technique was used to measure the oil mass flow rate. This technique relates the oil mass flow rate to another temperature difference. These in-house designed calorimetric flowmeters were implemented in the cooling circuit. More details on the method can be found in Ref. [START_REF] Khodsiani | Spatially resolved experimental investigation of particle-flame interaction in a twodimensional model packed bed[END_REF] Having measured the oil mass flow rate and the inlet to outlet oil temperature rise, Eq. 2 yields the heat transfer to the cylinder which is discussed in conjunction with the flame chemiluminescence imaging measurements later in section .

Results

In this section, two parameters, namely, the inflow velocity (u) and the equivalence ratio (φ ) are altered, and the flame position and geometry is discussed together with the heat transfer to the cylinders.

Chemiluminescence images recorded between the cylinders under three different operating conditions are shown in Figure . 4(a,b,c). Blue region in between them delineates the methane-air reaction front, while black areas mark the cylinders. The extracted flame skeletons are shown in Fig. 4(d) to better discern the flame position and geometry under three different conditions. A stable flame is recorded as shown in Fig. 4 at the reference condition, an inflow velocity of 0.3 m/s and stoichiometric condition. A high chemiluminescence intensity region, with a thickness of about 600 µm, marks the location of the reaction zone in between the cylinders. The extracted flame skeleton of Fig. 4(d) mark the exact geometry and extent of the flame front. The uppermost flame skeleton point is located at a distance of around 3 mm from the top cylinder, and the flame skeleton horizontally extends over a length of about 3.5 mm. A high pressure region, due to the stagnation of the fast burnt gases downstream of the flame front, is induces near the top cylinder. This stagnation region pushes the flame front further upstream. The flame front has a negative curvature, flattening on the side, which may be caused by the presence of a low velocity recirculation zone combined with the slower flame speed due to flame to wall heat losses.

Maintaining φ at 1 and slowing u down to 0.25 m/s, the flame front contracts and moves upstream considerably to the location observed in Fig. 4(b); as also observed in the extracted skeleton in Fig. 4(d). With the equivalence ratio and the laminar flame speed remaining untouched, the flame front descends by one cm upstream to a region where the gas speed under the current inflow velocity (u=0.25 m/s) is as fast as it was for inflow velocity (u) of 0.3 m/s. In addition, compared to the reference condition, the flame front is considerably flatter and the curvature is even slightly positive. This positive curvature is probably due to the thinner boundary layer and the smaller size of the recirculation region, which leads to a top hat reactant velocity profile. In this case, the positive flame curvature near the wall might have been induced by the slowed flame speed due to the heat losses when approaching the walls. Transiting to the last operating condition, u is matintained at 0.25 m/s while φ is decreased to 0.95. A lean mixture slows the laminar flame speed down, and therefore the reaction front ascends half a centimeter downstream to settle in a region of lower gas velocity. As shown in Fig. 4c and also the flame skeleton of Fig. 4d, the reaction front expands, its curvature becomes more pronounced compared to the stoichiometric condition, but remains still a lower than the the curvature of the flame at the reference condition.

The rate of flame to particle heat transfer is measured for the top and bottom cylinders three operating conditions. In order to take the variation of the total flame power into account, when comparing the heat transfer rates under different operating conditions, a parameter named heat transfer efficiency (η) is introduced. η is defined as the ratio of the heat transferred to the cylinder divided by the theoretically calculated flame power, being the product of the flow rate of the fuel and its lower heating value.

Having detected the flame skeletons and measured the flame to particle heat transfer rate, the influence of the flame stabilization position on the heat transfer to the cylindrical particles can be quantified. To do so, the distance between the tip of the flame skeleton to the cylinder surface is evaluated for both top and bottom cylinders and plotted against the heat transfer efficiency (η) in Fig. 5. The heat transfer efficiency of both top and bottom cylinder diminishes as the flame front moves away from the cylinder. For the top cylinder, this might be attributed to a lower temperature gradients over the cylinder surface as the flame front moves further up-stream away from the cylinder. In case of the bottom cylinders, the flame front moves downstream away from the cylinders, exposing a lower portion of the cylinder surface to the hot and fast burnt gas. This in turn leads to a weaker heat transfer.

Conclusion

Spatially-resolved experimental data on the flameparticle interaction in a packed bed model reactor are essential to develop and validate numerical models. Therefore a two-dimensional packed bed of cooled cylinders was conceived and operated. Telecentric imaging of the CH* chemiluminescence emanating from the reaction zone, in conjunction with flame-toparticle heat transfer measurements, revealed the effect of the flame extent and position on the individual flameto-particle heat transfer. A flame was stabilised above the reactant inflow in the void space between the first and second row of cylinder at a Reynolds number of 250. Due to the closeness of the flame to the cylinders, a significant flame to wall heat transfer was measured, being 25 % ± 2.5% of the total flame thermal power. By varying the inflow velocity and equivalence ratio, the reaction front was stablized at different locations downstream of the narrow gap between the cylinders in the bottom row. These changes also led to different heat transfer distribution between the upstream and downstream cylinders. Also, a correlation was found between the flame tip to cylinder stand-off distance and the heat transfer rate efficiency to that cylinder. These data provide the heat transfer rate, thermal boundary condition and reaction front stabilization position over a fully characterised geometry precluding any spatial ambiguity. They can therefore be directly exploited to validate the implementation of wall heat transfer and chemistry models in direct numerical simulations. In future works, flow velocity measurements by laser Doppler anemometry will allow to interpret the flame position and shape with respect to complex geometry of the flow and to determine the local flame consumption speed. Moreover, we will also investigate the case of hot cylinder temperature, replacing the three cooled cylinders by ceramic cylinders.

Figure 1 :

 1 Figure 1: a) A cross-section of the packed bed assembly in the central region. Here the cylinder-to-cylinder distance (c) is 1.23 diameter D, resulting in a porosity of 40% packed bed porosity.

Figure 2 :

 2 Figure 2: a) Finally manufactured assembly of the cylindrical packed bed and burner b) front view of the assembly; white area delineates the region with optical access)

Figure 3 :

 3 Figure 3: a) Telecentric lens mounted on the sCMOS camera and aligned with through optical access; b) Recorded methane-air flame image and superimposed extracted skeleton

Figure 4 :

 4 Figure 4: a to c) CH* chemiluminescence recorded under different operating conditions; d) detected flame skeletons. Black lines on the top, left and right are the detected edge of the particles

Figure 5 :

 5 Figure 5: Variation of the heat transfer efficiency (η) with the stand-off distance of the flame skeleton tip from the top and bottom cylinders. The corresponding operating condition of each data point is annotated.

Acknowledgment

The authors gratefully acknowledge funding by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) -Project-ID 422037413 -TRR 287.


Sijia Kong

Delphine Bresch-Pietri

Probabilistic Sufficient Conditions for Prediction-based Stabilization of Linear Systems with Random Input Delay

Keywords: Delay systems, prediction-based controller, distributed parameter systems

This paper focuses on the prediction-based stabilization of a linear system subject to a random input delay. Modeling the delay as a finite-state Markov process, it proves that a constant time-horizon prediction enables robust compensation of the delay, provided the horizon prediction is sufficiently close to the delay values in average. Simulation results emphasize the practical relevance of this condition.

I. INTRODUCTION

Delays are among the most common phenomena in engineering practice [START_REF] Palmor | Time-delay compensation-Smith predictor and its modifications[END_REF] as they range from control or sensing processing time, to transport delays. Lately, with the rise of communication and information technologies, communication delays have become a major concern for multiple research areas, such as multi-agent systems coordination or traffic estimation and control based on Vehicle-to-Vehicle transmissions. In such network controlled systems, transmitted information often suffers from lag, data reordering, packets dropouts or quantization [START_REF] Simon | Real-time control systems: feedback, scheduling and robustness[END_REF]. These phenomena can be accounted for by a random delay model.

This paper considers the case where such a random delay affects the input of a dynamical system and investigates the design of a prediction-based controller to compensate for this delay. This control technique is well-known for constant delays [START_REF] Kwon | Feedback stabilization of linear systems with delayed control[END_REF], [START_REF] Manitius | Finite spectrum assignment problem for systems with delays[END_REF], [START_REF] Smith | A controller to overcome dead time[END_REF], and has since then been extended to various contexts including time-varying delays [START_REF] Artstein | Linear systems with delayed controls: a reduction[END_REF], uncertain input delays or disturbances [START_REF] Mondié | Delay robustness of closed loop finite assignment for input delay systems[END_REF], and nonlinear systems [START_REF] Bekiaris-Liberis | Nonlinear control under nonconstant delays[END_REF], [START_REF] Karafyllis | Predictor feedback for delay systems: Implementations and approximations[END_REF]. It consists in computing a state prediction over a time window of the length of the (current or future) delay, and using this prediction in the feedback loop to eliminate or mitigate the effect of the delay in the closed-loop dynamics. Yet, while this technique has been recently applied to linear random Delay Differential Equations in [START_REF] Cacace | Predictor based outputfeedback control of linear stochastic systems with large i/o delays[END_REF] and for a deterministic delay term multiplied by a random variable in [START_REF] Li | Predictor-based h∞ leader-following consensus of stochastic multi-agent systems with random input delay[END_REF], its extension to the case where the delay itself is a random variable remained to be carried out.

Recently, in our preliminary work [START_REF] Kong | Prediction-based controller for linear systems with stochastic input delay[END_REF], we studied for the first time the problem of prediction-based control of dynamical systems subject to random input delays and proposed to use a constant time-horizon prediction. The present paper pursues this study and extends significantly its scope of application. Modeling the delay as a Markov process with a finite number of states as in [START_REF] Kolmanovsky | Mean-square stability of nonlinear systems with time-varying, random delay[END_REF], we reformulate the dynamics as a Partial Tel: +33140519330 Emails: sijia.kong@mines-paristech.fr, delphine.bresch-pietri@mines-paristech.fr Differential Equation-Ordinary Differential Equation (PDE-ODE) system as in the now standard methodology for stability analysis of input-delay systems proposed by Krstic and coworkers [START_REF] Bekiaris-Liberis | Nonlinear control under nonconstant delays[END_REF], [START_REF] Krstic | Boundary control of PDEs: A course on backstepping designs[END_REF], but applied to our random context. Using the so-called technique of probabilistic delay averaging [START_REF] Kolmanovsky | Mean-square stability of nonlinear systems with time-varying, random delay[END_REF] to a new Lyapunov functional, we prove that mean-square exponential stabilization of the closed-loop system is obtained provided the prediction horizon is sufficiently close to the delay values in average, in the sense of the expected value. This considerably generalizes the condition of [START_REF] Kong | Prediction-based controller for linear systems with stochastic input delay[END_REF] which was deterministic and thus quite restrictive, and constitutes the main contribution of the paper.

The paper is organized as follows. In Section II, we formulate the problem under consideration, design a constanthorizon prediction-based controller and formulate our stabilization result. In Section III, we propose a backstepping reformulation of the system, which then enables us to analyze the stability of the closed-loop system in Section IV and prove the paper main result. Simulation results are then provided in Section V to illustrate the practical relevance of the proposed sufficient condition for stabilization.

Notations. In the following sections, for a signal v : (x, t) ∈ [0, 1]×R → v(x, t) ∈ R, we denote v(t) its spatial L 2 -norm with respect to x. λ(A) denotes the spectrum of a square matrix A, while min(λ(A)) and max(λ(A)) are its minimum and maximum eigenvalues, respectively. Additionally, |A| denotes its Euclidean norm |A| = max(λ(A T A)) in which A T denotes the transpose of A.

E(x) denotes the expectation of a random variable x. For a random signal x(t) (t ∈ T ⊂ R), the conditional expectation of x(t) at the instant t knowing that x(s) = x 0 at the instant s ≤ t is denoted E [s,x0] (x(t)). Finally, e i denotes the ith standard basis vector of R r (r ∈ N + and i ∈ {1, ..., r}).

II. PROBLEM STATEMENT AND MAIN RESULT

We consider a controllable linear system with random input delay of the form

Ẋ(t) = AX(t) + BU (t -D(t)) , (1)
in which X ∈ R n and U ∈ R are the state and control input, respectively. The random delay D is a Markov process with the following properties:

(P1) D(t) ∈ {D i , i ∈ {1, . . . , r}}, r ∈ N with 0 < D ≤ D 1 < D 2 < • • • < D r ≤ D. (
i = j , (2)
in which τ ij and c j = r k=1 τ jk are positivevalued functions such that τ ii (t) = 0. In addition, we assume that the functions τ ij are bounded by a constant τ ⋆ > 0. (P3) The realizations of D are right-continuous.

Considering a finite number of delay values in (P1) is a common assumption considered, e.g., in [START_REF] Kolmanovsky | Mean-square stability of nonlinear systems with time-varying, random delay[END_REF], [START_REF] Sadeghpour | Stability of linear continuoustime systems with stochastically switching delays[END_REF]. In the case of network systems, these discrete values can be seen as a measure of the congestion state of the network. Likewise, Property (P3) is standard for the modeling of continuous-time Markov Chain and to assess the system well-posedness.

Furthermore, it is important to emphasize that the two properties (P1) and (P3), along with the Markov property, guarantee that P ij satisfies the Kolmogorov Equation ( 2) for certain positive-valued functions τ ij , c j (see [START_REF] Rausand | System reliability theory: models, statistical methods, and applications[END_REF], [START_REF] Ross | Introduction to probability models[END_REF]). In that sense, Property (P2) is only requiring the functions τ ij to be bounded, which constitutes a mild modeling assumption.

Finally, its is worth observing that the parameter τ ij ∆t is approximately the probability of transition from D i to D j on the interval [t, t + ∆t). Similarly, 1 -c j (t)∆t represents the probability of staying at D j during the time interval [t, t+∆t).

We aim at controlling the random-delay system (1) with a prediction-based controller. As the delay may not be known 1and, in any case, varies abruptly, using the current delay value as prediction horizon would in all likelihood result into a chattering control law and an inaccurate prediction 2 . Thus, we propose to use the following prediction-based controller with constant time horizon

D 0 (D 0 ∈ [D, D]) U (t) = K e AD0 X(t) + t t-D0 e A(t-s) BU (s)ds , (3)
in which K is a feedback gain such that A + BK is Hurwitz. Obviously, contrary to the deterministic case [START_REF] Artstein | Linear systems with delayed controls: a reduction[END_REF], such a predictor can only robustly compensate for the random input delay. We now provide the main result of the paper: a sufficient condition for such a robust compensation.

Theorem 1: Consider the closed-loop system consisting of the system (1) and the control law (3). There exists a positive constant ǫ ⋆ (K), such that if, for all time t ≥ 0,

E [0,D(0)] (|D 0 -D(t)|) ≤ ǫ ⋆ (K) , (4)
then, the closed-loop system is mean-square exponentially stable, that is,

E [0,(Υ(0),D(0))] (Υ(t)) ≤ RΥ(0)e -γt , (5)
for certain positive constants R and γ and with

Υ(t) = |X(t)| 2 + t t-3D U (s) 2 ds . (6)
Theorem 1 requires the prediction horizon to be sufficiently close in average to the delay values. This is in accordance with the constant-horizon feature of the prediction used in (3) and generalizes the restrictive deterministic condition obtained in our previous work [START_REF] Kong | Prediction-based controller for linear systems with stochastic input delay[END_REF] and bearing on |D 0 -D l | for all l ∈ {1, . . . , r}. Indeed, requiring |D 0 -D(t)| to be small enough for all time implies that the delay values are themselves close enough, otherwise such a choice of D 0 cannot be achieved. In that sense, Condition (4) represents a considerable relaxation, by distinguishing among the delay distributions.

Besides, it is worth mentioning that an expression for the positive constant ǫ ⋆ is provided3 in the proof of Theorem 1 detailed in the sequel. However, this value is likely to be of very little practical use, due to the conservativeness of the Lyapunov analysis carried out. Nevertheless, thanks to this expression, one can observe that the positive constant ǫ ⋆ depends on the feedback gain and that this dependence is likely to be considerable. Providing a quantitatively meaningful bound and studying its relation with K in view of increasing the closed-loop robustness is out of the scope of the present paper, but should be the focus of future works.

Finally, note that the interval of definition of the integral in ( 6) is [t-3D, t] a technical reason, namely, for Lemma 3 used in the stability analysis to hold. We now provide the proof of this theorem in the following sections.

III. BACKSTEPPING TRANSFORMATION

In order to prove Theorem 1, we follow the now standard stability analysis methodology for input-delay systems of [START_REF] Bekiaris-Liberis | Nonlinear control under nonconstant delays[END_REF], [START_REF] Krstic | Boundary control of PDEs: A course on backstepping designs[END_REF]. Consequently, we reformulate (1) as a PDE-ODE cascade and introduce additional distributed variables to better account for the effect of the prediction-based control law [START_REF] Cacace | Predictor based outputfeedback control of linear stochastic systems with large i/o delays[END_REF].

We denote v(x,

t) = v 1 (x, t) . . . v k (x, t) . . . v r (x, t) T the vector of distributed actuators v k (x, t) = U (t + D k (x -1)).
Then, one can rewrite the linear system (1) as the following PDE-ODE cascade with random parameter δ

     Ẋ(t) = AX(t) + Bδ(t) T v(0, t) Λ D v t (x, t) = v x (x, t) v(1, t) = 1U (t) , (7)
in which Λ D = diag(D 1 , . . . , D r ), 1 is a r-by-1 all-ones vector and δ(t) ∈ R r is such that, if D(t) = D j , δ(t) = e j , the j th vector of the standard basis of R r . Hence, δ(t) is a Markov process with the same transition probabilities as the process D(t), but with the finite number of states (e i ) instead of (D i ).

we introduce several distributed variables

v(x, t) = U (t + D 0 (x -1)) , (8) ṽ
(x, t) = v(x, t) -1v(x, t) , (9) µ(x, t) = U (t -D 0 + (3D -D 0 )(x -1)) . (10)
In details, v represents the control input U (t) within the interval [t-D 0 , t], ṽ the corresponding input estimation error while µ represents the controller within the interval [t -3D, t -D 0 ]. The extended state (X(t), v(x, t), ṽ(x, t), µ(x, t)) then satisfies

                         Ẋ(t) = AX(t) + Bv(0, t) + Bδ(t) T ṽ(0, t) D 0 vt (x, t) = vx (x, t) v(1, t) = U (t) Λ D ṽt (x, t) = ṽx (x, t) -Σ D vx (x, t) ṽ(1, t) = 0 (3D -D 0 )µ t (x, t) = µ x (x, t) µ(1, t) = v(0, t) , (11)
in which Σ D = ( D1-D0 D0 , . . . , Dr-D0 D0 ) T and 0 is a r-by-1 all-zeros vector.

Finally, in view of stability analysis, we introduce the backstepping transformation (see [START_REF] Bekiaris-Liberis | Nonlinear control under nonconstant delays[END_REF], [START_REF] Krstic | Boundary control of PDEs: A course on backstepping designs[END_REF])

w(x, t) = v(x, t) (12)
-K e AD0x X(t) + D 0

x 0 e AD0(x-y) Bv(y, t)dy .

Lemma 1: The backstepping transformation [START_REF] Li | Predictor-based h∞ leader-following consensus of stochastic multi-agent systems with random input delay[END_REF], jointly with the control law (3), transform the plant (11) into the target system (X, w, ṽ, µ)

                             Ẋ(t) = (A + BK)X(t) + B δ(t) T ṽ(0, t) + w(0, t) D 0 w t (x, t) = w x (x, t) -D 0 Ke AD0x Bδ(t) T ṽ(0, t) w(1, t) = 0 Λ D (x, t) = ṽx (x, t) -Σ D h(t + D 0 (x -1)) ṽ(1, t) = 0 µ t (x, t) = 1 (3D -D 0 ) µ x (x, t) µ(1, t) = KX(t) + w(0, t) , (13)
in which, h is defined for t ≥ 0 as h(t) = D 0 K (A + BK)e AD0 X(t) + e AD0 Bδ(t) T ṽ(0, t)

+ D 0 (A + BK) 1 0 e AD0(1-x) B w(x, t) + Ke (A+BK)D0x × X(t) + x 0 KD 0 e (A+BK)D0(x-y) Bw(y, t)dy dx . ( 14
)
Proof: The proof is similar to the one of [8, Lemma 2].

With this new set of coordinates, we are now ready to analyze the exponential stabilization of the closed-loop system.

IV. STABILITY ANALYSIS

A. Definition of the infinitesimal generator

Let us define the state of the target system (13) as

Ψ = (X, w, ṽ, µ) ∈ R n × L 2 ([0, 1], R) × L 2 ([0, 1], R r ) × L 2 ([0, 1], R) D Ψ .
As the solution to (13) is unique (see [START_REF] Kong | Prediction-based controller for linear systems with stochastic input delay[END_REF] for further details), (Ψ, δ) defines a continuous-time Markov process and we can therefore introduce the following elements for stability analysis.

Define the infinitesimal generator L (see [START_REF] Kolmanovskii | Introduction to the theory and applications of functional differential equations[END_REF] and [START_REF] Kolmanovsky | Mean-square stability of nonlinear systems with time-varying, random delay[END_REF]) acting on a functional V : D Ψ × {e 1 , . . . , e r } → R as

LV (Ψ, δ) = (15) lim sup ∆t→0 + 1 ∆t E [t,(Ψ,δ)] (V (Ψ(t + ∆t), δ(t + ∆t)) -V (Ψ, δ) .
We also define L j , the infinitesimal generator of the Markov process (Ψ, δ) obtained from the target system (13) by fixing δ(t) = e j , as

L j V (Ψ) = dV dΨ (Ψ, e j )f j (Ψ) + r l=1 (V l (Ψ) -V j (Ψ)) τ jl (t) , (16)
in which V l (Ψ) = V (ψ, e l ) and f j denotes the operator corresponding to the dynamics of the target system (13) with the fixed value δ(t) = e j , that is, for Ψ = (X, w, ṽ, µ),

f j (Ψ)(x) =     (A + BK)X + Be T j ṽ(0) + Bw(0) 1 D0 w x (x) -D 0 Ke AD0x Be T j ṽ(0) Λ -1 D ṽx (x) -Σ D h(• + D 0 (x -1)) µ x (x)/D     . ( 17
)
For the sake of conciseness, in the sequel, we denote V (t), LV (t), V j (t) and L j V (t), for short, instead of V (Ψ(t), δ(t)), LV (Ψ(t), δ(t)), V (Ψ(t), e j ) and L j V (Ψ(t)), respectively.

Due to the dynamics (2) of the transition probabilities, the infinitesimal generators ( 15) and ( 16) are related as follows

r j=1 P ij (0, t) dV j dΨ (Ψ(t))f j (Ψ(t)) + r j=1 ∂P ij ∂t (0, t)V j (t) = r j=1 P ij (0, t)L j V (t) = LV (t) . (18)
Therefore, for stability analysis, one can first focus on the constant delay functional L j V . This is the probabilistic delay averaging approach [START_REF] Kolmanovsky | Mean-square stability of nonlinear systems with time-varying, random delay[END_REF], which we follow in the sequel.

B. Lyapunov analysis

Consider the following Lyapunov functional candidate

V (Ψ, δ) = X T P X + c 1 0 (1 + x)(δ • D) T ṽ(x) 2 dx ( 19
)
+ bD 0 1 0 (1 + x)w(x) 2 dx + d(3D -D 0 ) 1 0 (1 + x)µ(x) 2 dx ,
with b, c, d > 0, P the symmetric positive definite solution of the equation P (A + BK) + (A + BK) T P = -Q, for a given symmetric positive definite matrix Q, and D = (D 1 . . . D r ) T and where • denotes the Hadamard multiplication and the square in ṽ(x) 2 should be understood component-wise. Note that, contrary to [START_REF] Kong | Prediction-based controller for linear systems with stochastic input delay[END_REF], the functional (19) explicitly depends on δ. We can then get the following result. Lemma 2: There exist (b, c, d) ∈ (R * + ) 3 such that the Lyapunov functional V defined in [START_REF] Simon | Real-time control systems: feedback, scheduling and robustness[END_REF] satisfies, for t ≥ 2D,

LV (t) ≤ -η -M E [0,D(0)] (|D 0 -D(t)|) -N g(t) V (t) , (20)
with η, M, N > 0 positive constants and the function g defined as g(t)

r j=1 |D j -D 0 | 2 ∂Pij (0,t) ∂t + c j (t)P ij (0, t) .
Proof: According to [START_REF] Sadeghpour | Stability of linear continuoustime systems with stochastically switching delays[END_REF], we first consider L j V defined in [START_REF] Rausand | System reliability theory: models, statistical methods, and applications[END_REF]. For the first term in [START_REF] Rausand | System reliability theory: models, statistical methods, and applications[END_REF], from [START_REF] Ross | Introduction to probability models[END_REF], applying integration by parts and Young's inequality, one obtains

dV j dΨ (Ψ)f j (Ψ) ≤ - min(λ(Q)) 2 -4d|K| 2 |X(t)| 2 (21) -b(1 -2D 0 |K||B|e |A|D0 γ 1 ) w(t) 2 -c 1 - 2 D 0 |D 0 -D j |γ 2 ṽj (t) 2 -d µ(t) 2 -b -4d - 4|P B| 2 min(λ(Q)) w(0, t) 2 -c - 4|P B| 2 min(λ(Q)) -2bD 0 |K||B|e |A|D0 1 γ 1 ṽj (0, t) 2 -dµ(0, t) 2 + 2c D 0 γ 2 |D 0 -D j | h(t + D 0 (• -1)) 2 , for any γ 1 , γ 2 ≥ 0. Observing that D 0 ∈ [D, D], let us choose (b, c, d, γ 1 , γ 2 ) ∈ (R * + ) 5 as follows (a) d < min(λ(Q)) 8|K| 2 , (b) b ≥ 4d + 4|P B| 2 min(λ(Q)) , (c) γ 1 < 1 2D|K|e |A|D |B| , (d) γ 2 < 1 4 min D D-D1 , D D r -D , (e) c ≥ 4|P B| 2 min(λ(Q)) + 2bD|K||B|e |A|D 1 γ1 , and define η 0 = min{min(λ(Q))/2 -4d|K| 2 , b(1 - 2D 0 |K||B|e |A|D0 γ 1 ), d} , which implies dV j dΨ (Ψ)f j (Ψ) ≤ -η 0 |X(t)| 2 + w(t) 2 + µ(t) 2 + 2c γ 2 D 0 |D 0 -D j | h(t + D 0 (• -1)) 2 . ( 22
)
Using Lemmas 3 and 4 for the index j 0 ∈ {1, . . . , r} such that e j0 = δ(t), this finally gives

dV j dΨ (Ψ)f j (Ψ) ≤ -ηV (t) + 2cM 1 γ 2 D 0 |D 0 -D j |V (t) , (23)
with η = η0 2 max{max(λ(P )),2bD,2cDr max{NX ,Nw,Nµ},2d(3D-D)} , in which N X , N w , N µ are defined in Lemma 4.

In addition, for the second term in [START_REF] Rausand | System reliability theory: models, statistical methods, and applications[END_REF], by definition of the Lyapunov function [START_REF] Simon | Real-time control systems: feedback, scheduling and robustness[END_REF], one obtains

r l=1 (V l (Ψ) -V j (Ψ))τ jl (t) (24) = r l=1 c 1 0 (1 + x)τ jl (t) D l ṽl (x, t) 2 -D j ṽj (x, t) 2 dx ,
in which, from the definition (9) of the input estimation error,

D l ṽl (x, t) 2 -D j ṽj (x, t) 2 (25) = D l t+D l (x-1) t+D0(x-1) U (s)ds -D j t+Dj (x-1) t+D0(x-1) U (s)ds × D l t+D l (x-1) t+D0(x-1) U (s)ds + D j t+Dj (x-1) t+D0(x-1)
U (s)ds

≤ (1 -x) 2 | D l -D j ||D j -D 0 | + D|D l -D j | × D |D l -D 0 | + |D j -D 0 | max s∈[-D,0] U (t + s) 2 ≤ M 2 M 3 |D j -D 0 | + |D l -D j ||D l -D 0 | V (t) ,
in which we used Lemma 3 in the last inequality and with M 3 = max 3D -2(DD) 1/2 |D -D|, D . Therefore, gathering (23), ( 24) and (25), one gets

L j V (t) ≤ -ηV (t) + M 4 |D 0 -D j |V (t) + N r l=1 τ jl (t)|D l -D j ||D l -D 0 |V (t) , (26)
with M 4 = 2c M1 γ2D0 + M 2 M 3 rτ ⋆ and N = 2cM 2 M 3 . Then, from [START_REF] Sadeghpour | Stability of linear continuoustime systems with stochastically switching delays[END_REF] and as

r j=1 P ij (0, t)|D 0 -D j | = E [0,D(0)] (|D 0 -D(t)|), the following inequality holds LV (t) ≤ -η -M 4 E [0,D(0)] (|D 0 -D(t)|) V (t) + N r j=1 P ij (0, t) r l=1 τ jl (t)|D l -D j ||D l -D 0 |V (t) .
Hence, applying the triangle inequality and using (2), one finally gets

LV (t) ≤ -(η -M 4 E [0,D(0)] (|D(t) -D 0 |))V (t) (27)
+ N r l=1 r j=1 P ij (0, t)τ jl (t)|D l -D 0 | 2 V (t) + N τ ⋆ E [0,D(0)] (|D(t) -D 0 |) r l=1 |D l -D 0 |V (t) ≤ -η -(M 4 + N τ ⋆ r|D -D|)E [0,D(0)] (|D(t) -D 0 |) V (t) + N r j=1 |D j -D 0 | 2 ∂P ij (0, t) ∂t + c j (t)P ij (0, t) V (t) .
Lemma 2 is then proved with M = M 4 + N τ ⋆ r|D -D|.

C. Conclusion of the stability analysis

With Lemma 2, we are now ready to conclude the proof of Theorem 1. Let us denote γ 0 (t) = η -M E [0,D(0)] (|D(t) -D 0 |) -N g(t), in which η, M, N and g are defined in Lemma 2 and introduce the functional Z as Z(t) = exp ( t 0 γ 0 (s)ds)V (t). Applying Lemma 2, we obtain LZ(t) = γ 0 (t)Z(t) + exp t 0 γ 0 (s)ds LV (t) ≤ 0 . Therefore, for t ≥ 3D, according to Dynkin's formula [4, Theorem 5.1, p. 133],

E [3D,(Ψ,D)(3D)] (Z(t)) -Z(3D) = E [3D,(Ψ,D)(3D)] t 3D LZ(s)ds ≤ 0 , (28)
from which, using standard conditional expectation properties, one deduces E [0,(Ψ,D)(0)] (Z(t)) ≤ E [0,(Ψ,D)(0)] (Z(3D)) . In addition, observe that

E [0,(Ψ,D)(0)] (Z(t)) ≥ E [0,(Ψ,D)(0)] V (t) (30) × exp(-N (D -D)E [0,D(0)] (|D 0 -D(t)|) + t 0 (η -(M + N c ⋆ (D -D))E [0,D(0)] (|D 0 -D(s)|))ds) .
Thus, if (4) holds with

ǫ ⋆ ∆ = η 2(M + N c ⋆ (D -D)) , (31)
one obtains from ( 28) and ( 30)

E [0,(Ψ,D)(0)] e -N (D-D)ǫ ⋆ + η 2 t V (t) ≤ E [0,(Ψ,D)(0)] (Z(t)) ≤ E [0,(Ψ,D)(0)] (Z(3D)) ≤ e 2Dη E [0,(Ψ,D)(0)] (V (3D)) , (32)
which implies, with γ = η 2 ,

E [0,(Ψ,D)(0)] (V (t)) ≤ E [0,(Ψ,D)(0)] (V (3D))e 3Dη+N (D-D)ǫ ⋆ -γt . (33)
Notice that V and Υ are equivalent, that is, there exist positive constants q 1 and q 2 such that for ∀t ≥ 0, q 1 V (t) ≤ Υ(t) ≤ q 2 V (t) (see [START_REF] Kong | Constant time horizon prediction-based control for linear systems with time-varying input delay[END_REF]Lemma 4] for a proof of this fact in a similar case). It thus follows that E [0,(Υ(0),D(0))] (Υ(t)) ≤ q2 q1 e 3Dη+N (D-D)ǫ ⋆ e -γt . In addition, as the dynamics (1) is linear, there exists a constant R 0 > 0 (see [START_REF] Kong | Prediction-based controller for linear systems with stochastic input delay[END_REF]Lemma 5]) such that Υ(t) ≤ R 0 Υ(0), t ∈ [0, 3D]. Consequently, [START_REF] Karafyllis | Predictor feedback for delay systems: Implementations and approximations[END_REF] follows with R = R 0 e 3Dη+N (D-D)ǫ ⋆ q 2 /q 1 .

V. SIMULATIONS

To illustrate Theorem 1 and in particular the role played by the condition (4), we consider the following toy example

Ẋ(t) = 0 1 -1 1 X(t) + 0 1 U (t -D(t)) . (34)
The control law (3) is applied with the feedback gain K = -1 2 resulting in conjugate closed-loop eigenvalues λ(A + BK) = {-0.5000 + 1.3229i, -0.5000 -1.3229i}. The initial conditions are chosen as X(0) = [1 0] T and U (t) = 0, for t ≤ 0. Simulations are carried out with a fixed-step solver in Matlab-Simulink and a sampling time ∆t = 0.01 s. Finally, the integral in (3) is discretized using its zero-order hold approximation, in line with a suggestion in [START_REF] Manitius | Finite spectrum assignment problem for systems with delays[END_REF]. We consider 3 different delay values (D 1 , D 2 , D 3 ) = (0.1, 2.0, 2.1). The initial transition probabilities are taken as4 P 1 (0, 0 + ) = 0.02, P 2 (0, 0 + ) = 0.69 and P 3 (0, 0 + ) = 0.29, which means that the delay values are initially concentrated in D 2 , and D 3 . We pick the prediction horizon as D 0 = 2. Notice that the delay margin of the closed-loop system (34) and (3) with constant delay D 0 is ∆D = 0.096 (see [START_REF] Kong | Constant time horizon prediction-based control for linear systems with time-varying input delay[END_REF] for details on the computation of this quantity). Thus, the realizations of both D 1 and D 3 lead to a delay difference which is beyond the robustness margin of the closed-loop system, resulting in a challenging set-up as prediction-based controllers are wellknown to be sensitive to delay mismatch [START_REF] Mondié | Delay robustness of closed loop finite assignment for input delay systems[END_REF].

Simulations performed for constant transition probabilities equal to the above initial conditions (i.e., τ ij = 0) resulted VI. CONCLUSION In this paper, we proposed a constant horizon predictionbased controller to compensate for a random input delay modeled as a Markov process with a finite number of values. We proved the exponential mean-square stability of the closedloop control system provided that the chosen prediction horizon is in average close enough to the delay value. Simulations illustrated the relevance of this condition and the interest of this prediction-based control law.

Future works will focus on the adaptation of the predictionhorizon to the current delay distribution, as it is likely to increase the closed-loop delay robustness, and thus represents an interesting design feature to explore. APPENDIX Lemma 3: Consider the control law defined in (3) and the function h defined in [START_REF] Mondié | Delay robustness of closed loop finite assignment for input delay systems[END_REF], there exist M 1 , M 2 > 0 such that

h(t + D 0 (• -1)) 2 ≤ M 1 V (t), t ≥ D 0 , (36)
U (t + s) 2 ≤ M 2 V (t), t ≥ 2D . ( 37
)
Proof: (36) is proved in [START_REF] Kong | Prediction-based controller for linear systems with stochastic input delay[END_REF]. Observing that h(t) = D 0 U (t), (37) is obtained with similar arguments.

Lemma 4: There exist N X , N w , N µ > 0 such that, for all j ∈ {1, . . . , r} and t ≥ D,

ṽj (t) 2 ≤ N X |X(t)| 2 + N w w(t) 2 + N µ µ(t) 2 . (38)
Proof: From the definition of the input estimation error [START_REF] Kong | Constant time horizon prediction-based control for linear systems with time-varying input delay[END_REF], it follows that ṽj (t) 2 =

0 E

 0 ds ≤(D -D) E [0,D(0)] (|D 0 -D(t)|) (29) + c ⋆ t [0,D(0)] (|D 0 -D(s)|)ds , as c j = r k=1 τ jk ≤ rτ ⋆ c ⋆ . Hence, it follows that

  Dynamic of transition probabilities P 1 , P 2 and P 3 . (c) Monte Carlo simulation of log X and the closed-loop input U (100 trials), in which the means and the standard deviations are highlighted by the colored lines.

Fig. 2 :

 2 Fig. 2: Simulation results of the closed-loop system (34) and (3) for D = (0.1, 2.0, 2.1) T , X(0) = [1 0] T and U (t) = 0 for t ≤ 0. The prediction horizon is D 0 = 2.0. The transition probabilities follow (2) with τ (t) = τ ⋆ e -kt (1 3×3 -I 3 ) (τ ⋆ = 0.2 and k = 0.1).VI. CONCLUSION In this paper, we proposed a constant horizon predictionbased controller to compensate for a random input delay modeled as a Markov process with a finite number of values. We proved the exponential mean-square stability of the closedloop control system provided that the chosen prediction horizon is in average close enough to the delay value. Simulations illustrated the relevance of this condition and the interest of this prediction-based control law.Future works will focus on the adaptation of the predictionhorizon to the current delay distribution, as it is likely to increase the closed-loop delay robustness, and thus represents an interesting design feature to explore.

  max s∈[-D,0]

1 0(U

 1 U (t + D j (x -1)) -U (t + D 0 (x -1))) (s) 2 ds ≤ 4 D + D 0 D v(t) 2 + µ(t) 2 .Besides, from the inverse of the backstepping transformation[START_REF] Li | Predictor-based h∞ leader-following consensus of stochastic multi-agent systems with random input delay[END_REF], which is v(x, t) =w(x, t) + Ke (A+BK)D0x X(t)+ x 0KD 0 e (A+BK)D0(x-y) Bw(y, t)dy ,(40)it follows, using Young's and Cauchy-Schwarz inequalities that v(t) 2 ≤ N 1 |X(t)| 2 + N 2 w(t) 2 with the positive constants N 1 = 3|K| 2 e 2|A+BK|D0 and N 2 = 3(1 + |K| 2 D 2 0 e 2|A+BK|D0 |B| 2 ). Hence, (38) follows with N X = 4N 1 (D + D 0 )/D, N w = 4N 2 (D + D 0 )/D and N µ = 4(D + D 0 )/D.

  P2) The transition probabilities P ij (t 1 , t 2 ), which quantify the probability to switch from D i at time t 1 to D j at time t 2

	b) P ij is a differentiable function which, for s < t,
	follows the Kolmogorov equation
	∂P ij (s, t) ∂t	= -c j (t)P ij (s, t) +	r k=1	P ik (s, t)τ kj (t) ,
	P ii (s, s) =1 and P ij (s, s) = 0 for

((i, j) ∈ {1, . . . , r} 2 , t 2 ≥ t 1 ≥ 0) satisfy a) P ij : R 2 → [0, 1] with r j=1 P ij (t 1 , t 2 ) = 1.

Time stamping of the exchange data can be used, but requires the controller internal clock and the system one to be synchronized, which could be difficult to guarantee in practice.

For instance, if the current delay value is much larger than its average, leading to an over-estimation of the prediction horizon.

Namely, in Equation (31), involving itself various other parameters such as the intermediate constants chosen in (a)-(e) or introduced in Lemmas 3 and

The subscript i is omitted in this section, as the probability transitions do not depend on the initial delay value. This is consistent with the fact that the expectation in (5) is conditioned by the initial delay value. Besides, to avoid a conflict between the initial condition in[START_REF] Bekiaris-Liberis | Nonlinear control under nonconstant delays[END_REF] and their discretized version used in simulation, we denote their initial conditions as P j (0, 0 + ).


Affective Guidance in the Creative Process: Ritual Anthropology as a Model -The Case of Jackson Pollock

Keywords: Affects, Creative Process, Ritual Anthropology, Jackson Pollock, Methodology

INTRODUCTION

"If we were to ask [an artist] why he did this or changed that, he might not be able to tell us. He does not follow any fixed rules. He just feels his way" (Gombrich 1951: 15). Ernst dealing with an anthropological reading of art. In Gell's account, artists can be described as having an "active" or "passive" position relative to their artworks, but the transition between the two states is left unclear.

Finally, the model is compared to existing ones in the psychology of creativity to assess the place given to affects in relation to agency.

FROM PASSIVITY TO AGENCY

Gell's model aims to encompass the sociology of people and artworks acting upon each other.

In this article, my primary focus is the creative process leading to the artwork's completion rather than its reception by the viewers. From that standpoint, we will examine his distinction between active and passive positions.

Gell states, in a paragraph actually referencing Pollock's art, that the artist is active when the passive artwork is but "a congealed 'trace' of the artist's creative performance (Gell 1998, 33). At this point, a problem of terminology arises. For Gell, the artist is active at that moment because his artwork can act upon the viewers. However, in our case, the first viewer of the completed artwork is Pollock himself. If viewers attribute agency to the artist via the artwork, whose agency is Pollock perceiving when looking at his own works? The agency perceived is that of Pollock in the recent past, during his actual creation process. Thus, the artist is passive as a viewer of his own completed works, and active while producing them.

Here is how Pollock phrasing: When I am in my painting, I'm not aware of what I'm doing. It is only after a sort of "get acquainted" period that I see what I have been about. I have no fears about making changes, destroying the image, etc., because the painting has a life of its own. I try to let it come through. It is only when I lose contact with the painting that the result is a mess.

Otherwise there is pure harmony, an easy give and take, and the painting comes out well. (Karmel 1999, 18;original emphasis).

For the artist, the artwork appears to be more than just a material "index" as suggested by Gell's Peircean terminology. Pollock's engagement with the painting during the process leads him to perceive it as manifesting a form of life There is a way to engage with the painting during the process that makes Pollock describe it as a form of life. This engagement is discussed extensively in Tim Ingold's Making, where this author advocates for a neverending process (Ingold 2013, 96), an idea which I reject partially as we will see later. For now, let us examine how, from Pollock's point of view, his actions during the creative process lead to the feeling of agency: it appears less as mere object-making and more like an exchange taking place between two agents. Critically though, this "life-giving process" is guided by a specific type of affect Pollock identifies as "harmony". The painting is felt as complete when this harmony is achieved during the act of painting, provided that the artist maintains contact with it. Feeling harmony, in other words, is a sign that the process is going in the right direction, and amplifying this harmony to a certain level becomes the goal of the process. When the right level of harmony is felt by Pollock, as we will see, this marks the switch from an active to a passive position: he can then cease making changes to the painting's surface. He then becomes a passive observer, affected by the achieved level of harmony in the painting, an affect that eliminates the need for further work on the artwork. This aligns with one of Gell's definitions of an artist's passivity in relation to an artwork: when the artwork "possesses the characteristics which motivate its selection by the artist" (Gell 1998, 30). In fact, there are two moments in the creative process where this type of passivity occurs. In the first moment, as we've just seen, the artwork is "selected" as complete when it displays qualities that make Pollock feel the right level of harmony. The second moment takes place at the very beginning of the creative process, before any actual work has begun. This is illustrated by the long-running competition Pollock felt he was in with Picasso, as reported by artist Lee Krasner, Pollock's wife: ...there's no question that he admired Picasso and at the same time competed with him, wanted to go past him… I remember one time I heard something fall and then Jackson yelling, 'God damn it, that guy missed nothing!' I went to see what had happened. Jackson was sitting, staring; and on the floor, where he had thrown it, was a book of Picasso's work... (Karmel 1999, 36).

Here, Pollock appears deeply frustrated by Picasso's art, given that Picasso has presumably materialized many ideas into finished artworks, ideas Pollock would have liked to realize himself. Frustration, by definition, implies a strong limitation of agency. If Pollock is frustrated due to his inability to produce artworks comparable to Picasso's, this moment aligns with Gell's definition above. It is the particular characteristics of Picasso's artworks that lead Pollock to "select" them. And it is Picasso's artworks in particular that trigger the most frustration in Pollock among the many works by numerous artists he saw.

Pollock experiences the full affective cycle as follows: 1) passivity, sparked by seeing someone else's art; 2) agency, during the hands-on work of making his own artwork and aiming for a balance of harmony; 3) returning to passivity, once that harmony is achieved, signaling that the artwork is complete. Gell's analysis doesn't account for this chronological transition from one position to another. Although the switch from agency (artist as creator) to passivity (artist as first viewer of the completed artwork) can be understood from these two vignettes, the switch from initial passivity to creative agency is less obvious. This is where Favret-Saada's ethnographic study of the dynamic of affects provides a useful model.

RITUAL DYNAMIC AS A MODEL

Gell makes several references to sorcery in his book, specifically Volt, to illustrate two points. First, a "prototype" is distinct from the artwork that it shapes. As an example, the drawing of an apple, the artwork, is different from the apple that forms its prototype. Then, sorcery provides insight into how a prototype "acts" upon the other agents within his system.

We will come back later at what serves as Pollock's prototype. In the meantime, I will continue to examine the sorcery analogy to uncover its inherent affective dynamics. I'll then adapt these dynamics to our understanding of the creative process, which could give us a fresh perspective on the entire artistic procedure.

Favret-Saada's 1970s ethnography of sorcery in rural France identifies three positions that exist, like Gell's, within the active-passive dichotomy. But unlike the latter, her ethnography provides a description of the shift from passivity to agency.

According to Favret-Saada, rural sorcery is a farmer's last recourse to end a series of adverse events (unexpected livestock deaths or failed crops). These events are said to be the result of a sorcerer's curse. The farmer is the victim, thus in a passive position. To regain agency, the victim undergoes a ritual. A ritual specialist, or "unsorcerer", performs a visually striking ritual action (e.g., inserting rusty nails in a boiled ox's heart) in front of the victim. This violent ritual action is perceived as the means to fight the sorcerer's own violence inflicted upon the victim. The ritual action has to be as violent as the sorcerer's action.

Favret-Saada shows that the sorcerer position is actually always a fictive one: the victim identifies someone as their sorcerer without the latter being aware. It is the victim who, in a paranoid reflex, interprets events as being aggressively directed against them from a common origin. Thus Favret-Saada's understanding is that the specialists ritual action is a pragmatic example of the affect the victim has to harness to escape their paranoid tendencies, a "shifter" in Favret-Saada's parlance (2009,88). In other words, the specialist provides an affective guidance to the victim. Violence is the specific affect that the farmer has to summon in order to regain agency against their identified source of the curse. After the specialist's ritual, smaller daily rituals, prescribed by the specialist, are performed by the farmer, relying on the violent affect experienced during the main ritual. The mobilization of this violent affect by the farmer positions them as an agent. These smaller rituals are prescribed as a way to avoid any relapse in paranoia, the farmer becoming hereby their own private ritual specialist.

We can now interpret Pollock's reaction to Picasso's art in the same fashion. The farmer's adverse events reduce or halt his agency, like Picasso's artworks do for Pollock. The farmer is then regaining agency through the violence of the visual ritual performed by the specialist. Here is the main difference between rural sorcery and the creative process: Pollock is his own specialist, performing the necessary actions to regain agency. There are exceptions to this difference though. Pollock, as we will see, has been helped to refine his creative process by two types of specialists, art teachers, and therapists. In both the sorcery and the creative cases, through different means, specialists aim to foster further autonomy of their (Lewison 1999, 22) since the artworks share four elements of composition (fig. 5, bottom). In Gell's terminology, Picasso's painting is the prototype for Pollock's.

We can now imagine Pollock flipping through the book on Picasso's art and "selecting" this painting, for it exhibits specific characteristics. regain agency then? What are the pseudo-ritual actions to take? In this example, the first step seems to imitate the general composition from which, for Pollock, most of the harmonious feeling seems to come. This is a way to tap into that something felt in Picasso's art. But then Pollock introduces differences seemingly aimed at affirming his own agency (for simple imitation would leave Picasso's unchallenged): Picasso's composition is inverted on the vertical axis, and the palette is shifted to a colder hue.

The eponymous girl in Picasso's painting is peacefully drawing. His execution exhibits mastery. In contrast, Pollock's rework of the main character appears more nervous.

However, the main difference between the two artworks doesn't lie in the way similar elements are drawn. Pollock's painting displays many black lines, the "stenographic figures" of its title. Since those added lines are absent in Picasso's version, their origin has to be clarified. In a note written in 1950, Pollock might lead us on a path: Technic is the result of a need-/ new needs demand new technics-/ total controldenial of / the accident-/ States of order-/ organic intensity-/ energy and motion / made visible-/ memories arrested in space, / human needs and motive-/ acceptance- (Karmel 1999, 24) Mentions of "control and the denial of the accident" are probably a response to an art critic condemning Pollock's later abstract paintings for their apparent simplistic technique.

Pollock's defense rests on the notion of "need", a term he uses here three times. It is not just the artist's will that dictates the creative process. There is a deeper need to fulfill with the help of a particular technique. The technique then allows the capture of "memories in space", a formulation echoing Gell's "congealed trace" seen earlier. Yet, contrary to the latter, Pollock's formulation seems spoken from the perspective of the artist during the creative process. The memories in question could be the ones of the "give and take" process, the dialogue taking place with the painting in the making. For the right harmony to come about, this dialogue needs to happen, motivated by Pollock's affects, translated into movements of applying paint on the canvas. This idea is condensed in the expression "make energy and motion visible". The painting could be thus seen as the evidence of a fight rooted in his need to recover agency over, in our example, Picasso.

In this light, the additional lines we see in Stenographic figure are ways to render the fight against Picasso. To be more precise, they are ways to enhance Pollock's first rework of Picasso's composition. On the far right of his composition, for instance, one element painted in white, blue, and ochre is overpainted with black lines that highlight and reinforce the impression of movement (fig. 6). We can deduce that these lines are added after Pollock completes the first version of the composition. Conventionally, an artist signs his work after completing it. In this case, Pollock's signature is partially covered with the same black lines as the element above. These added lines can thus be viewed as a way to "make energy and motion visible". They enable Pollock to witness in his painting the same kind of harmony he witnessed in Picasso's, despite the poor execution of his first rework. Picasso's painting is thus the "prototype" of Pollock's in two ways described by Gell. First in terms of "appearance", they share compositional elements. But then and foremost, Picasso's artwork is a prototype in terms of "presence" (Gell 1998, 26). Pollock's first rework was perhaps a good start (appearance), but to achieve the proper level of harmony experienced in front of Picasso's painting, extra movements had to be suggested. Although these lines partly mask the initial composition, they are the price Pollock pays to infuse in his artwork the same "energy" as Picasso's original (presence).

This outlines what the visually striking pseudo-ritual Pollock performs would look like. Pollock's initial passivity in front of Picasso's art is actually his frustration with the absence of his own agency, all the while feeling the right level of harmony triggered by the Spaniard's artwork. The frustration of lost agency creates a need to regain that capacity.

Pollock starts by imitating the elements causing this harmonious feeling for him. During the creation of the painting, the successive traces that he paints change his overall impression of the whole canvas, an impression that guides him as to how close he is to achieving the right level of harmony. As that right level begins to affect him, the initial need to fight Picasso disappears. He can now witness a novel artwork, the result of his own agency, which allows him to experience the specific harmony he perceived while discovering Picasso's cubist painting. In this view, the creation of artworks is for artists foremost a fight over existing artifacts from their surrounding material culture that affect them in peculiar ways, but without the experience of their own agency.

AFFECTIVE GUIDANCE OVER A LIFETIME

This model reflects three phases in the creative process of producing a novel artifact. It starts with the initial loss of agency from an artwork displaying the right level of harmony. Pollock then engages in the pseudo-ritual as a sequence of actions aimed at regaining agency, guided by the increasing level of the right harmony displayed by the artwork in the making. Finally, a feeling of completion, once that right level is reached, pushes the artist into another kind of passivity: witnessing his now completed work. What this model doesn't do is clarify why artists engage in the process multiple times. As we have seen, the primary difference with Favret-Saada's ethnography is that the artist assumes the position of the specialist after being affected by someone else's artifact. One reason for the cyclic creative engagement, unlike the farmer's "unsorcering" ritual, might be that the initial trigger for the artist and the result of the artist's efforts to regain agency are of the same kind: artifacts. Indeed, following the regaining of agency through the creative process, the artist relapses into a kind of passivitywitnessing an artwork displaying the right level of harmony. There are then limited possibilities to get that feeling of agency outside of engaging in artwork-making again.

Let us now examine some elements of Pollock's career to see that dynamic unfold throughout his lifetime. This dynamic is not the mere repetition of the model above for a single artwork. Through the iterations, Pollock learns to make "energy and motion visible" in a unique way, fine-tuning his creations to more accurately grasp the right harmony he seeks. The genesis of Pollock's famous "dripping" technique, where paint is poured on the canvas on the floor, has been tentatively traced to many places: a house painter splashing clean his brushes on a board; Pollock's possible ocular migraine producing visual hallucinations; a memory of his father drawing patterns on the ground by urinating (Naifeh and White Smith 2007, 488). More probable prototypes are the works of Janet Sobel (1894-1968) "a self-taught, fifty-two-year-old grandmother from Brooklyn" (ibid., 475) which Pollock sees prior to his abstract period. Sobel produces small paintings made of curvy splashes on the whole surface of the support (fig. 1). According to the model, Sobel's artworks affect Pollock to such a degree that he feels the need to surpass their "energy" by creating similar but much larger compositions. Pollock's well-known abstract paintings occupy roughly 10,000 times more surface than Sobel's. The way they affect him thus clearly matches or surpasses Sobel's.

But if we only consider the two types of passivities, the affecting prototypes (right harmony, without agency) on one side, and completed artworks (right harmony, with past agency) on the other side, we return to where Gell was, not taking into account the transition from one to the other. So we need not only to consider prototypes of composition (Picasso, Sobel) but prototypes of action. It is during the process of making the artwork-the pseudoritual-that energy and motion are made visible, and that presence-the "easy give and take" exchange with the artwork-can be experienced by Pollock. Let us examine some of these possible action prototypes.

Charles and Sanford Pollock, two of his four older brothers, have an artistic practice.

Compared to theirs, Jackson's technique might be referred to as "pictorial dyslexia" (Karmel and Varnedoe 1999, 55): a frustrating incapacity to master figurative painting techniques, against which Pollock has to fight. An action prototype that might have helped him overcome this struggle can be found in an anecdote reporting how Pollock's teacher asks his teenage students to mix color paints in a liquid (Naifeh and White Smith 2007, 117).

After moving to New York in the 1930s, Pollock enrolls in Thomas Hart Benton's courses at an art school. Benton is one of the rare artists that Pollock names publicly: "my work with Benton was important as something against which to react very strongly, later on; in this, it was better to have worked with him than with a less resistant personality who would have provided a much less strong opposition. At the same time, Benton introduced me to Renaissance art" (Karmel 1999, 15). Benton's work is fairly figurative. As a "regionalist" artist, he developed his style depicting farm work and American landscapes with exaggerated curves (fig. XXX top and middle). He masters traditional techniques of figuration, a source of frustration of agency for a "dyslexic" Pollock. Nevertheless, Pollock tries, according to one classmate, "to replicate Benton's technique and subject matter 'down to the last brush stroke'", but only achieves "jagged imitations of Benton's sure, undulating lines" (Naifeh and White Smith 2007, 173, 202). Even if jagged, Pollock does imitate the gestures, as action prototypes, in order to understand the movements in the paintings: in contact with technically perfect artworks (appearances), Pollock needs to appropriate their "energy" (presence) to regain some proper agency in his own paintings (fig. XXX bottom).

In 1936, Pollock participates in the Experimental Workshop led by Mexican muralist artist David Alfaro Siqueiros, who aims at making explicitly political figurative artworks fighting American imperialism. In his workshop, Siqueiros asks the participants to prepare painted backgrounds on panels for his personal artworks. With panels placed on the ground, paint is splashed and poured directly from the can. Pollock displays little interest in the figurative aspect of Siqueiros's works, yet is quite attracted by that preparatory phase for which the muralist artist developed a theory of "controlled accidents" (Karmel and Varnedoe 1999, 46)-wording similar to Pollock's note above. Pollock thus begin experimenting with the fluidity of paint to cover surfaces well before his abstract period. He is in the active position of the specialist looking to solve his pictorial dyslexia through the action prototypes of others. Several more years will pass before he puts this solution into practice for himself, and as a self-sufficient way of making art, independent of traditional drawing techniques.

Pollock pursues therapy for his chronic drinking problem. In 1939, he begins working with Doctor Joseph L. Henderson, a direct disciple of Jung. Henderson is also very interested in the work of Picasso. As is usual for a Jungian therapist, Henderson engages with Pollock's drawings, still figurative at the time, blending Picassoian motifs with Jungian imagery.

Henderson confesses that he did not seek to treat Pollock's alcoholism, but rather he "offered Left : Pablo Picasso, Guernica, 1937 (detail). Right : Jackson Pollock, plate 40 in Psychoanalytical drawings, (Wysuph 1970, 72).

Horse's head by Picasso and Pollock

occasional criticisms which [he] thought might be helpful in order the free him from his influences (mainly Picasso) that seemed to inhibit his own native ability" (Leja 1993, 139).

Henderson does not assist Pollock with his alcoholism but contributes to the artist's fight against Picasso. Henderson occupies the typical position of the ritual specialist in Favret-Saada's ethnography: helping the victim escape the sorcerer. Yet, Henderson does not produce the visually striking ritual to free Pollock. It is Pollock who performs the necessary actions in order to free himself. If Pollock, through therapy, cannot entirely control his alcoholism, he does, however, further develop his practice based on Picasso's works, and regains some agency. In a drawing resulting from the interaction with Henderson, for instance, the horse's head in Picasso's Guernica is reworked. He adds a floating mane, with Bentonian-like curves as action prototypes, to his version (absent in Picasso's), thereby increasing the head's "energy and motion," which he would have felt was missing from his

drawing (fig. XXX).

Stenographic figure is made in 1942, and Mural, probably the gateway to the abstract Pollock we know, is produced in 1944, the same year Pollock reportedly sees Sobel's work (Zalman 2015, 20). Mural, a 6-meter-long painting, is a commission from the gallerist and collector Peggy Guggenheim, which pushes Pollock to change his process. Accustomed to the easel, he now has to walk along the canvas-a new action prototype for him-to apply the layers of paint. While often considered abstract, the painting is actually built on a base of "stick figures" (Karmel and Vernadoe 1999, 90), partly covered with other colors, following their rhythm. Phosphorescence (1947), entirely abstract to the spectator's eyes, is revealed on X-Rays to contain an early sketch made of these figures, which Pollock "veils" (ibid,104).

Krasner relates: "many of the most abstract [artworks] began with more or less recognizable imagery … Once I asked Jackson why he didn't stop the painting when a given image was exposed. He said 'I choose to veil the imagery'" (Karmel 1999, 36). This shows the evolution of Pollock's practice. At first, there is the need to replicate the "energy" of Picasso's art. Then, work after work, as he acquires confidence, the addition of lines, the veiling, becomes his specific way of making art, his action prototype. He fully becomes a specialist able to regain agency during creation, and witness afterward the "life of its own" (presence) the painting has gained.

In August 1949, Life magazine publishes an article on Pollock, illustrated with his abstract artworks, titled "Jackson Pollock: Is he the greatest living painter in the United States?" This is a consecration of sorts for it gives his work visibility in mainstream media.

The tone of the article, though, might seem mocking at times. The apparent ease of making, which his paintings exhibit, suggests that they are "simple" (Friedman 1995, 140). Pollock has to prove to others and to himself that this is not the case, that his paintings are not, as a critic writes at the 1950 Venice Biennale, "chaotic" or "lacking any technique" (Karmel 1999, 68).

Pollock has to find a way out. In 1951, he notes in a letter that he is running tests on Japanese papers (Friedman 1995, 171) given to him by his artist friend Tony Smith. Pollock pours lines of black ink on their surface (Delahunty 2015, 17). The paper absorbs and diffuses the ink in peculiar ways, another type of "controlled accident" as an action prototype. This reaction of the paper might have echoed Pollock's use of unprimed canvases at times since 1947-the canvas fibers spread the paint in similar ways-for the 1951 series of black paintings that directly succeed his abstract period. He explains: "I've had a period of drawing on canvas in black-with some of my early images coming thru-[I] think the nonobjectivists [abstract expressionist artists] will find them disturbing-and the kids who think it simple to splash a Pollock out" (Friedman 1995, 174). The paint he uses is more fluid, employing at times a turkey baster, for greater precision while drawing. He can now show that he controls what he is doing while still achieving results as harmonious to his eyes as the abstract works (fig. 7). This line no longer "veils" a weak copy of a Picasso, or covers the initial rhythmic stick figures, but is a drawing technique, an action prototype, that belongs to him. This is his line. It "makes the energy and motion visible" even with simpler designs. Whether figuration reappears or not, this is his mastery of the line which is at stake. With it, comes a certain maturity of the specialist position. This maturity is the confidence to be able to create visually striking artworks to fight the energy of the works of Picasso, Sobel, and others along the way. However, in 1951, he also fights his own abstract paintings from the previous years. Indeed, the abstract Pollock became internationally famous (Zimmer 2016, 221), but was criticized for the apparent ease of his production. For the 1951 Pollock, the abstract Pollock took the sorcerer's position. After showing these new works, one critic writes that Pollock "found his own way of dealing with human experience". Yet, the global audience is now used to the abstract works and finds it difficult to see the black paintings as continuous with the earlier works (Delahunty 2015, 19). In 1952 and 1953 Pollock uses colors again, contrasting with the black from the preceding year. Those years, however, are dominated by depression and the return of alcoholism (Zimmer 2016, 239). Despite these detours and the criticism, Pollock seems to have never given up his path of "making energy and motion visible" through his line, building confidence, painting after painting. However, this confident position is fragile. It can be occupied at times during the creative process but never secured. Indeed, when a painting approaches completion, Pollock reverts to the passive position we've seen earlier, being a witness of his own art, seeing but "memories arrested in space" of a time when he is about to feel the specific harmony he is seeking.

CONCLUSION

With the model we have discussed, each painting is the result of affective guidance aimed at overcoming a felt passivity, defined as the feeling of the right harmony displayed in an artifact, but without the accompanying feeling of agency. This process occurs for artifacts made by others, but along the way also for the ones made by oneself. The critical point here is that agency, by definition, can only be felt by the artist through is creative actions. The artist's double transition from passive to active, at the start of the process, and from active to passive, at the end, forms the basis of the dynamic by which a body of works is slowly constituted. Artworks, in this view, are but means for the artist to experience, during the act of creation, a specific type of affect, a specific harmony in the case of Pollock. Competition, initially against the work of others and then against one's own, sustains the activity throughout a lifetime. Since each successive work has to match or surpass the previous one's "energy" or presence, artists are incrementally guided throughout their careers to perfect the means, the action prototypes, to feel their specific affect.

By focusing on affects as the main motivation of the artist's action, the gap between the passive and active positions in Gell's argument seems bridged. Affects are specific to each person. What affects me might not affect you, and the way I can produce an artifact that can affect me in the way I need will not be the same as yours. In other words, this is a subjective matter. Yet, the expression of this subjectivity is related to Edward Morphy's critique, in the pages of this very journal, of Gell's lack of consideration for the artifacts' "meaning" beyond their semiotic content. If art is a "mode of action", it must integrate how "people who…interact with works of art…do so with some background of knowledge and experience…of the artwork itself and of innumerable previous interactions involving it, or similar objects" (Morphy 2009, 14). Here, I propose that affects participate in the meaning-making process one goes through, since background knowledge and experience can be expressed in the harmony Pollock finds during creation: his paintings feel harmonious to him because they fit the previous experience of harmonious art Pollock had, with his own works or those of others. To Pollock's eyes, his works are meaningful because they make him feel that specific affect he has been after his whole life. Or as Pollock states: "painting is selfdiscovery. Every good artist paints what he is." (Leja 1993, 186). Through art-making, as a mode of action, one experiences the self as an agent capable of producing that specific affect, rooted in subjectivity.

The model I propose builds on Gell's focus on agency, enriched by the dynamic of affects found in Favret-Saada's ethnography of sorcery. It is built on historical rather than ethnographic data, for this article's aim is to provide a perspective on a long-term practice, challenging to study through ethnography alone. Applications of this model are currently tested in close collaboration with artists, in particular regarding its limitations, such as the somewhat linear description of the creative process. For instance, many compositional and action prototypes are involved in the creation of new artworks.

The development of an apt model for the creative process may require an interdisciplinary effort. In conclusion, I will therefore compare my model to one elaborated by a succession of psychologists. In 1926, Graham Wallas identifies, after a speech by Hermann von Helmholtz in 1891, four stages in the "art of thought", that we can assimilate to the creative process at large: preparation ("hard, conscious, systematic and fruitless analysis"), incubation ("unconscious mental exploration"), illumination ("appearance of a happy idea"), and verification (testing and shaping of the idea) (Wallas 1926, 80-1). In 1961, Mel Rhodes proposes the "four P's of creativity". For our purpose, the Person refers to the artist, the Process encapsulates Wallas' four stages above, the Product means the artwork, and the Press is the global environment of creation. A person creates a product by going through a process surrounded by a pressing environment, made of "sensations and perceptions from both internal and external sources [reflecting] uniquely upon the originator's self" (Rhodes 1961, 308). Yet Rhodes advocates for the study of each "P" in relative isolation from the others to dissipate the "fog" surrounding what creativity actually is. This then results in a lack of dynamism in the model. Vlad [START_REF] Glăveanu | Rewriting the Language of Creativity: The Five A's Framework[END_REF] thus proposes to expand these individualfocused definitions and proposes the "five A's model", a more dynamic way of linking Actor (a person performing social roles), Action (psychological and behavioral dimension of the process), Artifacts (as cultural objects, broader than bare material products), Audiences (the social aspect of Rhodes' Press), and Affordances (physical aspects of the Press). This change of terminology widens our understanding of each component. For instance, "actor" implies the existence of an audience and an action to be performed for it, whereas "person" tends to be understood in narrower terms: a person doesn't necessarily need an audience and action to be defined. Thus, creativity, in this view, takes place in the interrelation between the five A's.

Glăveanu's model is for now proposed as a framework for future research and not as the result of a study. The five A's could also be mapped onto Gell's terminology: actor corresponds to the artist; artifact to the index or artwork; audience to the recipients; and affordances to a form of prototypes. After appreciating the similarities and without stressing further here the differences between these psychological and anthropological models, it seems interesting to underline that in both cases the vector of the interrelations between the terms can be identified as affects. From what we have seen with the Pollock case, I would argue that affect is the vector. Pollock, as actor, is affected by the affordances of artifacts made by others or by himself. He performs his art -the aptly named "action painting" to affect an audience, in the middle of which he stands as the first viewer of his finished paintings. His actions are guided by the quest for an affect, toward completion.

Positioning affect at the center of these models opens interesting research possibilities across various academic fields. As I have tried here, examples drawn from art history can be examined for the affective connections that artworks establish with each other, following Aby Warburg's notion of "Pathosformeln". This enables anthropology to link these longitudinal lines of creative processes to other forms of affect-driven, meaning-making actions such as rituals in non-Western contexts, thus touching on processes that might be universally human.

Furthermore, methodologies from experimental psychology could be adapted to conduct fieldwork with artists in their environment, to examine precisely how affects are driving meaningful actions. Such an interdisciplinary effort would broaden our understanding of art and creativity.

  patients, another way to regain agency. Under that model, what would Pollock's visually striking pseudo-ritual look like? Following Krasner's indication of Pollock's perceived rivalry with Picasso, we will examine a pre-abstract painting of the former where the link between the two artists is easily understandable. Stenographic Figure (fig. 5, top right), is a mostly figurative painting from 1942. The art historian Jeremy Lewison proposes to see it as an inverted reworking of Picasso's cubist Interior with a girl drawing (fig. 5, top left), created seven years earlier

  Pollock might not be able to express what those characteristics are, yet he might have felt affected by this painting more than by others. Just as the farmer does, Pollock then makes Picasso his sorcerer of sorts since the painting that affects Pollock has been made by Picasso. But instead of adverse events for the farmer, Pollock perceives in Picasso's painting the very specific level of harmony seen earlier, and his frustration comes from not being the source of that felt harmony. How to 8 Top left: Pablo Picasso, Interior with a girl drawing, 1935, oil on canvas, 130x195 cm, MoMA, New York. Top middle: Jackson Pollock, Stenographic Figure, 1942, oil on canvas, 102x142 cm, MoMA, New York. Bottom middle: Pollock's painting inverted on the horizontal axis and compared to similar elements in Picasso's (author's modifications). Right: detail of Stenographic Figure Stenographic Figure as an inverted Picasso

A

  common entry point in Pollock's work is his renowned abstract paintings (fig. XXX). Their 1946, oil and enamel on cardboard, 45 x 35 cm, MoMa, New York. 1. Janet Sobel, untitled lack of figurative elements helps us focus on the presence felt by the artist rather than their mere appearance (as is partly the case for Stenographic figure). It can thus be said that Pollock is searching for an inner prototype guiding his actions. 13 Top: Benton, Chilmark landscape, 1922. Middle: with underlined curves. Bottom: Pollock, ca. 1935, oil on fiberboard, Smithsonian American Art Museum, Washington DC.

  1951, 143x167cm, enamel on canvas, National Gallery of Art, Washington DC.7. Jackson Pollock, Number 7Even though the two Pollocks are the same person, in terms of his experience of creation, the abstract Pollock is someone else. The abstract paintings are material proof for the 1951 Pollock that there was then a Pollock able to produce artworks which now affect him in a very specific way. Furthermore, his abstract paintings probably have a more precise effect on the 1951 Pollock than Picasso's artworks had on the abstract Pollock. The abstract works were made by the abstract Pollock to surpass the Spaniard's works' presence. In turn, the 1951 Pollock feels the need to escape the abstract Pollock paintings to gain his agency back. This is the moment in Pollock's career that encapsulates the best the self-competition artists entertain. It is probably present earlier in the development of the artist, but oftentimes obscured by the competition against others.