Micky Rakotondrabe

Cédric Clévy

Philippe Lutz

Modelling and Robust Position/Force Control of a Piezoelectric Microgripper

This paper deals with the control of a piezoelectric microgripper based on two piezocantilevers. To avoid the destruction of the manipulated micro-object and to permit a high accurate positioning, the microgripper is controlled on position and on force. Each piezocantilever is separately modelled and controlled: while the one is controlled on position, the second is controlled on force. Because the models are subjected to uncertainties and the micromanipulation requires good performances, a H∞ robust controller is designed for each system. The experiments end the paper and show that good performances are obtained.

I. Introduction

In micromanipulation, i.e. manipulation of object from 1µm to 1mm sizes, the required accuracy is generally sub-micrometric. Instead of hinges, active materials are used to design systems for micromanipulation. In fact, hinges are characterized by frictions and may decrease the performances (accuracy) of the micromanipulation. Among these active materials, piezoelectric materials are widespread because of their fast response time and their high resolution. One of the main applications of piezoelectric materials in microsystems is piezoelectric microgrippers. A piezoelectric microgripper is based on two piezoelectric cantilevers (piezocantilevers). It is used to pick a micro-object, transport and place it with a high positioning accuracy. Nevertheless, to avoid the destruction of the micro-object, a control of the manipulation force is necessary.

In the litterature, many studies have been done on the modelling and control of a piezocantilever but few concern a whole microgripper. The majority of these studies especially refer to the deflection of the cantilever: [START_REF] Haddab | Design and development of a micromanipulation system with force/position control[END_REF] is an example in the linear approach while [START_REF] Ge | Tracking control of a piezoceramic actuator[END_REF][3][4] [START_REF] Rakotondrabe | Plurilinear modeling and discrete µ-synthesis control of a hysteretic and creeped unimorph piezoelectric cantilever[END_REF] takes into account the nonlinearities of the material. On the other hand, the control of the force is today a partially solved problematic because it requires the integration of a very small sensor, the use of the piezoelectric properties or a compliant structure. Up to now, very few solutions are proposed: [START_REF] Carrozza | Towards a force-controlled microgripper for assembling biomedical microdevices[END_REF][7] [START_REF] Liguo | Hybrid control of vision and force for MEMS assembly system[END_REF].

In this paper, we propose to model and control a piezoelectric microgripper. In order to ensure good performances for the micromanipulation, a H ∞ robust con-Laboratoire d'Automatique de Besançon LAB CNRS UMR6596 ENSMM -Université de Franche-Comté 24, rue Alain Savary 25000 Besançon -France {mrakoton,cclevy,plutz}@ens2m.fr troller is used. The position (directly measured) and the manipulation force (estimated) are considered as the references. The paper is organized as follows. First, the modelling of the microgripper is presented. Then, the design of the H ∞ controller for each piezocantilever is presented. Finally, the experimental results are presented and discussed.

II. Modelling

Let the Fig. 1 present the piezoelectric microgripper manipulating a micro-object. In this figure, F m is the manipulation force applied by the two piezocantilevers to the micro-object. Our objective is to control the position of the micro-object by means of one piezocantilever and to maintain a constant value of F m by means of the second piezocantilever. Because the adhesion forces [START_REF] Gauthier | Forces analysis for micromanipulation in dry and liquid medi[END_REF] are insignificant relative to the manipulation force range of our concern (more than millinewton), and because we experiment objects from 500µm to 1.5mm in size, they will not be taken into account in this paper. The relation between the applied voltage U i , the force F i applied to the piezocantilever at its tip and the resulting deflection δ i , with i ∈ {l, r}, in the static mode is [START_REF] Smits | The constituent equations of piezoelectric heterogeneous bimorphs[END_REF]:

δ i = d i • U i + s i • F i (1)
where d i > 0 is the equivalent piezoelectric constant of the piezocantilever and s i > 0 is the equivalent elastic constant.

However, it has been shown that the transient part of the (U i , δ i )-transfer and the transient part of the (F i , δ i )transfer of a piezocantilever are similar [START_REF] Haddab | A microgripper using smart piezoelectric actuators[END_REF]. Thus, we have:

δ i = (d i • U i + s i • F i ) • D i (s) (2)
where D i (s) (such as D i (0) = 1) represents the dynamic part and s the Laplace variable.

On the other hand, the relation between the manipulation force F m and the contraction of the micro-object is:

(δ l -δ l0 ) -(δ r -δ r0 ) = s o • D o (s) • F m (3)
where s o > 0 is the elastic constant of the micro-object, D o (s) (such as D i (0) = 1) is its dynamic characteristic and δ l0 and δ r0 are the deflections of the left and right piezocantilevers before touching the object (Fig. 2). Noting ∆δ = δ l0 -δ r0 and replacing F l = F r = -F m in the (Eq. 2), we have:

   δ l = (d l • U l -s l • F m ) • D l (s) δ r = (d r • U r -s r • F m ) • D r (s) (δ l -δ r ) -∆δ = s o • D o (s) • F m (4)
Without loss of generality, we choose the left piezocantilever for the force F m actuation while the right one for the micro-object position actuation. From the set of equations (Eq. 4), two approaches are possible to model the microgripper:

• the first approach uses one bivariable system where the inputs are the two voltages and the outputs are the deflection δ r and the force F m . It takes into account the effects of the piezocantilevers deflections to each other. So, it is possible to design a multivariable controller leading to very good performances. However, this approach needs a precise model, • the second approach consists in modelling independently the two piezocantilevers. While the one is modelled on force, the second is modelled on deflection. The advantage is that the two models are easier than of the first approach. However, since the deflection of one cantilever disturbs that of the other, and vice versa, the design of each feedback controller should takes into account such disturbances.

In both approaches, the models are dependent on the characteristics of the micro-object. In this paper, we choose the second approach because of the simplicity of the models and of the issued controllers.

A. Model of the voltage/deflection transfer

Here, we model the right piezocantilever. From the second equation of the set (Eq. 4), we have the nominal model:

δ r = d r • D r (s) • (U r + b r ) (5)
where b r = -sr•Fm dr is an input disturbance to be rejected. Fig. 3 shows the corresponding scheme.

B. Model of the voltage/force transfer

The left piezocantilever is modelled in this part. From the first and last equations of the set (Eq. 4), we have:

F m = 1 (s o + s l ) • D k (s) • (d l • D l (s) • U l -δ r -∆δ) (6)
with:

D k (s) = (s o + s l ) (D o (s) + D l (s)) (7)
and D k (0) = 1. The (Eq. 6) clearly shows that the model depends on the characteristics of the micro-object. Nevertheless, it is not practical to identify the model and to synthesize a controller at each change of manipulated micro-object. Thus, we propose to have a nominal model independent of the micro-object characteristics, i.e. s o = 0 and D k (s) = 1, and use a robust controller to ensure the stability and the performances. Such hypothesis will be verified in the experimental results. Finally, we have the following nominal model:

F m = d l s l • D l (s) • U l + F pert (8)
where

F pert = -(δr+∆δ)
s l is an output disturbance to be rejected. Fig. 4 shows the corresponding scheme.

C. Identification

For our experiments, we use two unimorph piezocantilevers based on a P IC151 piezolayer [12] and a Copper layer. The sizes of each piezocantilever are: 15mm × 2mm × 0.3mm (length, width and thickness) where the thickness of the piezolayer is 0.2mm . The experimental setup (Fig. 5) is made up of:

• the microgripper,

• two laser sensors (Keyence sensor with 500nm of accuracy) to measure the deflections of the piezocantilevers, • an amplifier with two lines for U l and U r , • a computer-DSpace material to acquire the measurements and to generate the input voltages U l and U r . For each piezocantilever, a harmonic experiment was performed. Afterwards, the (U i , δ i )-transfer function is identified in the frequenty domain. A second order model is assumed to be sufficient for piezocantilevers [START_REF] Low | Modeling of a three-layer piezoelectric bimorph beam with hysteresis[END_REF]. We have:

   d r = 0.545 µm V D r (s) = 1 1.86×10 -8 •s 2 +4.1×10 -6 •s+1 (9) and    d l = 0.525 µm V D l (s) = 1 1.85×10 -8 •s 2 +2.72×10 -6 •s+1 (10)
Fig. 6 shows the results of (U i , δ i )-transfer function for the right piezocantilever. Finally, to identify the elastic constant s i , a mass is hung on at the tip of each piezocantilever and the resulting deflection is measured. We have s r = s l = 1.931 × 10 -3 m N . Because the characteristics of the micro-object have been neglected in the nominal model, the latter is subjected to uncertainties. Such approach let us avoid the model identification and the controller synthesis of at each change of micro-object. Therefore, we choose a robust controller to ensure the stability and performances robustness face to the uncertainties, we use the H ∞ controllers. Moreover, the disturbance rejection will be taken into account during the controller design.

III. Control of the deflection δ r

Let the Fig. 7 be the closed-loop scheme. In the figure, δ r r indicates the reference input. Two weighting functions are used: W r 1 for the closed-loop performances and W r 2 for the disturbance rejection.

A. Standard form

Let P r (s) be the augmented system including the nominal system and the weighting functions. Fig. 8 shows the corresponding standard scheme.

The standard H ∞ problem consists in finding an optimal value γ > 0 and a controller K i (s) stabilizing the closed-loop scheme of the Fig. 8 and guaranteeing the following inequality [START_REF] Balas | ţ-analysis and synthesis toolbox[END_REF]:

+ - + + r b r U r i r o r ε r δ r r δ r K 1 r W 2 r W ( ) r r r G d D s = ⋅
F low (P i (s), K i (s)) ∞ < γ (11)
where F low (., .) is the lower Linear Fractionar Transformation and is defined here by F low (P r (s), K r (s)) = or(s) er(s) . From the Fig. 7, we have:

o r = W r 1 • S r • δ r r -W r 1 • S r • G r • W r 2 • i r (12)
where S r = (1 + K r • G r ) -1 is the sensitivity function.

Using the condition (Ineq. 11) and the (Eq. 12), we infer:

W r 1 • S r ∞ < γ W r 1 • S r • G r • W r 2 ∞ < γ ⇔ |S r | < γ |W r 1 | |S r • G r | < γ |W r 1 •W r 2 | (13)
To solve the problem (Ineq. 13), we use the Glover-Doyle algorithm which is based on the Riccati equations [START_REF] Glover | State-space formulae for all stabilizing controllers that satisfy an H∞-norm bound and relations to risk sensivity[END_REF] [START_REF] Doyle | State-space solutions to standard H 2 and H∞ control problems[END_REF]. The issued controller K r is robust in the fact that it ensures the stability and the performances even if the nominal system G r has an uncertainty relative to the real plant. The wanted performances are introduced through the weighting functions.

B. Choice of the weighting functions

The transfer functions 1

W r 1 and 1 W r 1 •W r 2
are chosen from the specifications respectively on the tracking performances and on the disturbance rejection. Thus, the weighting functions W r 1 and W r 2 can be automatically deduced. The specifications are:

• the maximal response time is 100ms, • the overshoot is null,

• the maximal statical error is 0.1%, • finally, the rejection of the disturbance is a highpass filter with a cutting frequency more than 50Hz.

When the disturbance is a static force, its influence on the deflection has been chosen to be lower than 1.7 µm 10mN . From the performances specifications, we choose:

1 W r 1 = 10 -3 • 3 • s + 1 30 × 10 -3 • s + 1 (14)
Using the specifications on the disturbance rejection and using the equivalence b r = -sr•Fm dr , we choose:

1 W r 1 • W r 2 = 0.5 × 10 -7 • 3 • s + 1 3 × 10 -3 • s + 1 (15)

C. Calculation of the controller

The computed controller has an order of 5. To minimize the memory and time consumptions in the computer, the controller order has been reduced to 2 using the balanced realization technique [START_REF] Moore | Principal component analysis in linear systems: controllability, observability and model reduction[END_REF]. We obtain:

     γ opt = 10.1 K r = -15810•(s+4×10 6 )•(s-1.4×10 4 ) (s+1.5×10 7 )•(s+0.3) (16)
IV. Control of the force F m for the output disturbance rejection. 9. The closed-loop scheme with the weighting functions.

+ - + + ( ) l l l l d G D s s = ⋅ l K 1 l W 2 l W l ε l U r m F m F l i l o Fig.
From the Fig. 9, we have:

o l = W l 1 • S l • F r m -W l 1 • S l • W l 2 • i l (17)
where

S l = (1 + K l • G l ) -1
is the sensitivity function.

Using the condition (Ineq. 11) and the (Eq. 17), we infer:

W l 1 • S l ∞ < γ W l 1 • S l • W l 2 ∞ < γ ⇔ |S l | < γ |W l 1 | |S l | < γ |W l 1 •W l 2 | (18)

A. Choice of the weighting functions

The following specifications are used:

• in order to maintain the micro-object during the positioning, the response time in force should be faster than the one of the deflection. For that, we choose a response time equal to 10ms, • the overshoot is null,

• the maximal statical error is 1%,

• finally, the rejection of the disturbance is a highpass filter with a cutting frequency is 5Hz. Such a value is chosen because the disturbance, which is the deflection of the other piezocantilever, has a low bandwidth (response time). When the disturbance is a static deflection, its influence on the output must be lower than 0.2 mN 40µm . From the performances specifications, we choose:

1 W r 1 = 10 -2 • 3 • s + 1 3 × 10 -3 • s + 1 (19)
Using the specifications of the disturbance rejection and using the equivalence F pert = - (δr+∆δ) s l , we choose:

1 W r 1 • W r 2 = 10 -2 • 3 • s + 1 30 × 10 -3 • s + 1 (20)

B. Calculation of the controller

The computed controller has an order of 5. Once again, the controller order has been reduced. We obtain:

     γ opt = 1.02 K l = 0.06•(s+7.3×10 8 )•(s 2 +277•s+8.2×10 7 ) (s+1.2×10 6 )•(s+1.9×10 4 )•(s+0.3) (21)
V. Experimental results The two controllers given by the (Eq. 16) and the (Eq. 21) have been implemented to a computer-DSpace real-time material. The Matlab-Simulink and the Con-trolDesk softwares were used for that. Since there is no adequate force sensor, a force estimator has been used [START_REF] Haddab | Design and development of a micromanipulation system with force/position control[END_REF]. Fig. 10 shows the principle scheme of the experiments.

Experiments with two types of micro-object were performed (Fig. 11): a flexible material (polystyrene) and a rigid material (rigid plastic). The sizes of the microobjects were chosen so that ∆δ ≈ 0. Its width, corresponding to the gap between the two piezocantilevers, is nearly 1mm.

The results show that the same performances are obtained for both. Fig. 12 presents the experimental results with the rigid plastic mini gear. Initially, the deflection and the force are null. Then, a manipulation force reference of 5mN is applied. A small effect is seen on the deflection (t = 1.3s) but is quickly rejected. Between t = 2s and t = 5s, a series of positive and negative steps (amplitude: 20µm) in position reference is applied. As we can see, its influence on the force is also quickly rejected despite the high amplitude of the The microgripper manipulating a mini gear (in rigid plastic).

diturbance effect. Finally (t = 5.8s), the manipulation force reference is set to zero. Since the size of the microobject is equal to the gap of the microgripper, a negative deflection reference should be applied to release it. On the other hand, we can remark that the accuracy is very high for the deflection and for the force. In fact, the static errors are very small and are not visible by the sensors.

In order to evaluate the response time, we zoom in the step reponse of each output (Fig. 13). It can be seen that the response time is exceeded: t resp ≈ 250ms for the deflection and t resp ≈ 150ms for the force. This is explained in the fact that the optimal value of γ is high for the deflection. For all that, the force response time still stays lower than the one of the deflection and the general performances are always suitable for the micromanipulation task requirements. In addition, ramp signals are often used as references in micromanipulation instead of step signals. So, such response times are largely sufficient.

VI. Conclusion

Piezoelectric materials are very prized in micromanipulation due to their rapidity and high resolution. One of their application is piezoelectric microgrippers which are based on two piezocantilevers. A microgripper permits the pick-transport-and place of micro-objects with a high positioning accuracy. This paper has presented the modelling and the control of a piezoelectric microgripper dedicated to manipulate micro-objects. Both the positioning and the manipulation force were taken into account. For that, two separated models have been proposed: while one piezocantilever is modelled on deflection, i.e. position, the second is modelled on force. Because the characteristics of the manipulated micro-objects are neglected in the model, the nominal model is subjected to uncertainty. To ensure stability and performances, we have proposed a H ∞ controller for each piezocantilever. Whatever the manipulated micro-object characteristics are, good performances suitable for the micromanipulation (disturbances rejection, micrometric positioning accuracy, submillinewton of force accuracy) were obtained in the experiments.

Fig. 1 .

 1 Fig. 1. A piezoelectric microgripper manipulating a micropart.

Fig. 2 .

 2 Fig. 2. Initial situation: the micro-object is not in contact with the piezocantilevers.

Fig. 3 .

 3 Fig. 3. Scheme of the nominal model of the right piezocantilever.

Fig. 4 .

 4 Fig. 4. Scheme of the nominal model of the left piezocantilever.

Fig. 5 .

 5 Fig. 5. Experimental setup.

Fig. 6 .

 6 Fig. 6. Magnitude of the right piezocantilever.

Fig. 7 .

 7 Fig. 7. The closed-loop scheme with the weighting functions.

Fig. 8 .

 8 Fig. 8. The standard form.

Fig. 9

 9 Fig. 9 shows the closed-loop scheme. In the figure, F r m indicates the reference input. Two weighting functions are used: W l 1 for the closed-loop performances and W l 2

Fig. 10 .

 10 Fig. 10. Principle scheme of the microgripper control.

Fig. 12 .Fig. 13 .

 1213 Fig. 12. Experimental results with a rigid plastic mini gear.

Acknowledgment

This work is partially supported by the European Project EUPASS (http://www.eupass.org/).


Catherine Muller-Vibes
email: catherine.muller@tbs-education.fr

Chantal Roucolle
email: chantal.latge-roucolle@enac.fr

Miguel Urdanoz
email: m.urdanoz@tbs-education.fr

Dynamic Price Competition in the Air Transport Market: An Analysis on Long-Haul Routes

Keywords: Air transportation, Price discrimination, Revenue management, Panel data; JEL codes: L93, L110, L130, C33

The pricing policy of airlines is based on revenue management. Revenue management analysts daily observe competitive prices and strategically adjust their own tariffs. One could expect this behavior to lead to a sound homogenization of airline prices evolution while competing on a market. We test empirically whether airline pricing strategies evolve on a similar manner, on a particular set of long-haul routes. Using new and original data including information on ticket prices paid, purchasing and departure dates, we estimate a model for the effect of dynamic factors on the evolution of ticket prices, based on economic theory. We use a 3 rd degree polynomial regression between prices and number of days to departure for each airline operating on the routes, and control for key revenue management variables, competition factors and individual effects. Our results show that competing airlines pricing strategies are statistically distinct during their ticket sale period. Airlines maximize their profits by sequentially increasing or decreasing their prices, but they do so in a non-synchronized fashion, and with different magnitudes.

Introduction

Airlines tend to differentiate their products and use complex pricing discriminatory practices on the basis of their Revenue Management (RM) process. Revenue managers open or close tariff classes during a flight booking period, taking into account the load factor and the level of demand. Also, to address competitive pressure during this optimization process, they observe daily prices of competitors and adjust their tariffs accordingly. 1 This practice should lead to similar pricing strategies across time for similar products, however we show that airlines choose to significantly differentiate their tariffs and sometimes even alternate high and low prices in an opposite way.

The literature tries to explain the evolution of observed prices and their dispersion over time. However, most studies are built on databases that either lack information on dates of purchase and dates of departure (primarily articles based on the DB1B database from the US government) or are built according to data on posted fares (usually collected through spider programs 2 ) rather than actual prices paid. In this article we have access to a unique and confidential database gathering information on the tickets purchased on four long-haul routes for which sales are registered since the moment they are proposed, i.e. one year in advance. This distinctive database allows us to analyze how pricing behavior differs across airlines in some competitive markets. However its confidentiality rules prevent us from giving any specific information on the markets and competing airlines.

To our knowledge, there is a limited number of articles that use observed purchased tickets; [START_REF] Sengupta | Comparing Price Dispersion on and off the Internet Using Airline Transaction Data[END_REF]2014), [START_REF] Hernandez | Nonlinear Pricing Strategies and Competitive Conditions in the Airline Industry[END_REF], [START_REF] Puller | Testing Theories of Scarcity Pricing in the Airline Industry[END_REF] or [START_REF] Puller | Price Discrimination by Day-of-Week of Purchase: Evidence from the U.S. Airline Industry[END_REF]. They all have access to tickets bought between June and December 2004 for flights in the fourth quarter of the same year for the US domestic market. They use fixed effects or airlines market shares to control for airlines specificities, however none of these studies differentiates pricing strategies among airlines.

With our dataset we observe distinct patterns among airlines in the evolution of their prices over time. Figure 1 depicts the average of the prices paid for all economy tickets sold a specific number of days prior to the departure of any flights in our sample. 3 We observe individual changes in trends and magnitudes. In this paper, we empirically show that airlines, while dynamically taking pricing decisions of their competitors into account: i) internalize the number of days prior to departure to increase their revenue; but ii) do not engage in similar price patterns over a booking period. More specifically, not only carriers differentiate the products they are offering on the market but they also adopt different pricing strategies, even if punctually adjusting for competing tariffs (thanks to RM technics). These findings could also explain some of the contradictory results observed in the literature. For example [START_REF] Borenstein | Competition and Price Dispersion in the U.S. Airline Industry[END_REF] show that price dispersion increases on more competitive routes, using 1989 cross section data from the US DB1B database. [START_REF] Gerardi | Does Competition Reduce Price Dispersion? New Evidence from the Airline Industry[END_REF] find the opposite result: competition has a negative impact on dispersion. They use panel data, rather than cross-sectional data, from 1 For instance Amadeus announces in its website: "Based on real-time shopping session information (e.g. trip context), market information (e.g. competitor offers), and the airline's revenue management and pricing strategy (e.g. product consistency), Amadeus Dynamic Pricing is modelling customer behavior and decision, considering his willingness to pay, to calculate the optimum price and instantly adjusts the available fares and taxes" ('Amadeus Dynamic Pricing' 2017). All these practices are described in Talluri and Van Ryzin (2004) among others. 2 Spider programs, also called website crawlers or 'bots' are software programs created to gather information from all the websites they visit. 3 See Section 3 for more details on the average price computation.

the same US DB1B database on the period 1993-2006. Our approach illustrates that price dispersion measures could be biased if pricing strategies are not well specified.

The paper is organized as follows. In Section 2 we present the data, then we describe the model and estimation method in Section 3. Section 4 summarizes the results and Section 5 concludes.

A Rich Dataset

We use a database gathering information on purchased tickets for four airlines, all operating on the same four long-haul routes. We observe individual tickets from the Computer Reservation System, purchased between September 1, 2012 and February 1, 2014.4 This period of observed booking corresponds to 2541 flights departing from September 1, 2013 to February 1, 2014. A route is a one way Origin-Destination market at the airport level. This ticket-level information is provided for all four airlines operating on those markets. Three of them are Full Cost Carriers (FCC), and one is a Low Cost Carrier (LCC). 5The tickets in our sample are collected and observed daily. Such a disaggregated database opens a large scope of research. To begin with, we observe the date of purchase of each ticket and the date of the flight departure. Second, the dataset avoids the restrictions imposed by the usual US DB1B database, which restricts the empirical analysis to the domestic US market and more importantly presents the limitation of quarterly aggregated data. Third, our observations contain information on the ticket characteristics6 and the flight arguments. 7 In particular we observe ticket classes. 8 The classes are used by revenue managers to determine the price to offer according to their revenue maximization objective. .

The date of purchase and the date of departure allow us to recover the number of days prior to departure which is crucial information for the analysis of dynamic price competition. We restrict the analysis to direct flights and to a booking period of 200 days before departure. Before these 200 days the number of tickets purchased represents only 2.5% of the total tickets sold. We focus on economy cabin tariffs only, to derive consistent results based on a homogenous service quality. The strength of our dataset lies in the observation of the two above mentioned dates, jointly with ticket classes. Some competition mechanisms on the market can be partly inferred from these observations. Each observation in our dataset is defined as a combination of a route, an airline and a number of days prior to departure (DTD). We compute an average price per observation, weighted by the corresponding number of passengers: For flights departing between days 1 and T, we compute the average of all the transactions realized at dates 1-DTD, 2-DTD,…and T-DTD. Our dataset contains 3198 observations for the booking period under scrutiny. Our sample covers 30% of the offered seats.

The model

The equation we fit on this data is a reduced form based on an economic model, even though it does not explain the full behavior of airlines. The theoretical foundations of our model are that airlines offer varying prices over the booking period for a flight, to maximize their profit by trading off between attracting more passengers and generating a higher margin. Airlines take into account the dynamics of their cumulative revenue on a flight as well as their competitors' prices. We control for these dynamics to better identify and measure the impact of the number of days prior to departure for the different airlines, and consequently better distinguish their differentiated strategic pricing behavior.

First, as displayed in Figure 1, the dynamic evolution of prices according to the days to departure seems to follow a non-linear relationship. We introduce in our model the days to departure as a third-order polynomial to match this behavior as in [START_REF] Escobari | Systematic Peak-Load Pricing, Congestion Premia and Demand Diverting: Empirical Evidence[END_REF]. Second, in order to test whether airlines implement different pricing strategies across time, we interact the number of days to departure with each airline, allowing for a different effect of time on each competitor's decision, as in [START_REF] Bilotkach | Are Airlines' Price-Setting Strategies Different?[END_REF]. Third, to capture the effect of competition on prices, we build a variable measuring the average competitors' price for a ticket, for the day preceding the observed booking. We are then able to address the main (but not only) component of RM, which is the permanent adjustment of prices to the competitors'. Obviously matching closely competitors' prices is only optimal for an airline when it also takes into account its cumulated revenue up until the previous day. Indeed, the urge to fill an aircraft close to the departure date might drive airlines to lower their prices, or on the contrary to maintain a high level of prices if revenue is already significant. We introduce a variable measuring the sum of all past revenues for a flight (the cumulative revenue), until the previous day of booking. We finally control for seasonal effects, as the markets under consideration are likely to be sensitive to national holidays, implying different demands, and therefore different prices.

To control for unobserved route and airline individual effects that could affect prices (like reputation, costs specificities…), we perform a standard within group estimation. This technique, described in details in [START_REF] Wooldridge | Econometric Analysis of Cross Section and Panel Data[END_REF] allows solving for potential endogeneity issues when no good instrumental variables are available..

Our model is presented in the equation below. The subscripts i, j and DTD refer to an airline, an OD pair and a number of days prior to departure, respectively:

        
                       
where 𝑃 𝑖𝑗𝐷𝑇𝐷 is the average price paid for flights operated by airline i, on the route j, at a certain number of days prior to departure DTD. 9 DTD belongs to the [0; 200] interval. It is equal to 0 when customers purchase tickets on the flight departure day. It is equal to 200 when tickets are purchased 200 days before the departure of the flight. 𝑃 -𝑖𝑗𝐷𝑇𝐷+1 and 𝐶𝑅 𝑖𝑗𝐷𝑇𝐷+1 are respectively the average price of competing airlines' tickets and airline i's cumulated revenue on route j at DTD+1 (that is to say the cumulative revenue up until the day before). The variables 𝑃 -𝑖𝑗𝐷𝑇𝐷+1 and 𝐶𝑅 𝑖𝑗𝐷𝑇𝐷+1 are interacted with dummy variables capturing each airline's business model, characterizing respectively FCCs and LCCs (the dummy takes the value one if the airline is an LCC). 𝐿𝑜𝑤𝑆𝑒𝑎𝑠𝑜𝑛 𝑖𝑗𝐷𝑇𝐷 is a variable for low season periods. It indicates the percentage of tickets, for a given combination of airline, route and DTD, for which the corresponding departure date does not fall into a holiday period. This exogenous variable controls for price adjustments relative to exogenous demand patterns. We estimate the model with the Generalized Least Squares approach, which aims at correcting for potential heteroskedasticity. This is the appropriate estimation method with the panel nature of our dataset. We use the SAS software for data treatment and estimation.

Empirical Results

The results of our estimation are presented in Table 1. The global fit of our model to the data is good, with an R-square of 0.54, and most parameters significant at the 1% level. 9 Let t=1…T be the date for departing flights. The first parameters of interest are those capturing the effect of days prior to departure on prices, for each airline. We find that the polynomial specification fits the behavior of three airlines in the list, as most of the parameters are strongly significant, although interestingly they present opposite signs or different magnitudes. For the third full cost carrier, no particular trend is observed. Wald tests were conducted and statistically confirm these different price dynamics across airlines. These results highlight a significantly distinct pricing behavior among airlines, over a booking period. Airlines do not all choose to apply the same strategy over the period, some starting to sell with lower prices than others, then making prices vary in different or even opposite directions, when getting closer to the departure date. This result is crucial in better understanding airlines' pricing decisions, which are not solely based on short term RM adjustments. We believe this result has not been consistently estimated in the literature so far, and also calls for a more structural model including a pricing equation for airlines, derived from game theory models. This is left for further research.

Our specification is reinforced by the additional variables we introduce. The two variables capturing the typical strategic dimensions of RM are interacted with the airlines' business model and exhibit reasonable and informative effects. Indeed, the parameter for the cumulative revenue 𝐶𝑅 𝑖𝑗𝐷𝑇𝐷+1 is only significant for LCCs and indicates that if the average cumulative revenue of an LCC for a flight increases, the price for this flight the next day will increase as well. Simultaneously, the parameter for competing price 𝑃 -𝑖𝑗𝐷𝑇𝐷+1 is only significant for FCCs, and has a positive sign. These findings are in line with RM objectives applied to the distinct business models. Because of their market positioning implying relatively constant low prices, LCCs have lower incentives to fill an aircraft at any cost when they already have generated certain revenue. They would rather increase their margin on the remaining seats, taking advantage of passengers' lower sensitivity to price as the departure date gets closer. For instance [START_REF] Bachis | Low-Cost Airlines and Online Price Dispersion[END_REF] show that LCC's fares do not necessarily increase monotonically over time, peaking a few days before departure. Due to their market positioning, LCCs pricing is barely affected by the fares set by FCCs. Indeed the prices set in the previous period by the FCCs do not have a significant impact on the LCCs pricing strategy.

Oppositely FCCs adjust their prices in the same direction as their competitors' in the previous period, always trading off the sale of an additional seat and the highest possible margin: If the competing prices increase, no need to decrease their own price to sell a ticket as increasing the price a little might still make them sell the seat and generate a higher marginal profit at the same time. The intense price competition between the FCCs could explain why the cumulated revenue does not have a significant impact over the FCCs pricing strategy.

Finally, prices decrease during low season, which is intuitive and consistent with empirical observation.

Conclusion

In this article we estimate the direct effect of competition on the daily prices for economy tickets supplied by four airlines, three Full Cost Carriers and one Low Cost Carrier. We focus our analysis on measuring the impact of the number of days prior to departure on prices, while controlling for airlines, routes, seasonality and competitive pressure. The days to departure are introduced with a 3 rd degree polynomial form, for each airline. We show that the polynomial form between the days to departure and the price level is statistically significant and that the coefficients of the polynomial differ from one airline to the other. These results confirm the differences in pricing strategies implemented by airlines during the booking period for a flight. Moreover we also observe differences while identifying the RM analysts' behavior. We show that the FCCs adapt their current prices to competitors' prices charged during the previous period, while LCCs would rather focus on their cumulative revenue.

The available data open a large scope of extensions for further analysis. First, our model, based on average price per route, airline and days to departure, fails to account for the airlines' price dispersion for a given day prior to departure. The next step of our analysis will consist in using price dispersion per airline as an endogenous variable and better assessing discriminatory practices linked to the booking day. This will lead to the estimation of a more sophisticated system of equations where the endogenous variables would be the average price and the price dispersion. The different strategies adopted by the airlines could explain the controversial results found in the literature with respect to price dispersion.

Second, given the limits of a reduced form model, we will use structural models to better identify the type of competition between airlines. In particular we will use game theory models to study airline's strategic interactions and coordinated effects.

Figure 1 .

 1 Figure 1. Average Price Evolution per Day Prior to Departure, on a Given Market

   represents the number of tickets bought date t-DTD at price ijt DTD P  .

  

	ln	ijDTD P	1 i	DTD	2	i	DTD	23 3 i DTD	4	FCC	ln	P	ijDTD	1	FCC	4	LCC	ln	P	ijDTD	1	LCC
		5	FCC	ln	ijDTD CR	1	FCC	5	LCC	ln	ijDTD CR	1	LCC		6	ijDTD LowSeason	ijDTD u

Table 1 . Estimation Results

 1

						Parameter
	Variable			Estimate	t Value (1)
	Intercept			3.563***	3.95
	Days to Departure (2)
	11  DTD			-0.016***	-5.23
	21  DTD 2			2e -4 ***	9.74
	31  DTD 3			-5.78e -7 ***	-9.34
	12  DTD			-0.004	-1.44
	22  DTD 2			7.35e -5 ***	2.95
	32  DTD 3			-2.15e -7 ***	-3.09
	13  DTD			0.018	0.87
	23  DTD 2			-9.43e -6	-0.91
	33  DTD 3			3.14e -8	0.87
	14  DTD			0.015***	16.08
	24  DTD 2			-1.02e -4 ***	-10.82
	34  DTD 3			2.33e -7 ***	5.94
	Competitors' prices on the route -previous period
		4LCC	 ln ijDTD P  1		CC L	0.124	1.55
		4FCC	 ln ijDTD P  1		CC F	0.171***	10.18

Airline Cumulative Revenue -previous period

		5LCC	ln		ijDTD CR	1 		LCC	0.304***	11.41
		5FCC	ln		ijDTD CR	1 		FCC	0.102	1.42
	6  Low Season				-1.233***	-11.74
	N=3198						R²=0.5392
	Note: *** significant at the 1% level
	(1) Computed with Robust GLS
	(2) Airlines 1, 2, 3 are FCCs and Airline 4 is an LCC.

Our dataset does not include tickets bought directly from the airlines websites.

The Low Cost business model for airlines focuses on strongly reducing costs in order to offer significantly lower prices to customers.

More precisely: Hour, date and city of purchase, city and country of departure and arrival, day and hour of departure and arrival.

More precisely: Flight number, number and date of stops, marketing and operating airlines, aircraft type.

Using RM wording, the distinction between Economy and Business tickets corresponds to a distinction in terms of cabin layout and quality. Several fare classes are distinguished within each Economy or Business. RM pricers open or close classes during the booking period according to their objectives in terms of revenue while taking into account the behavior of competitors on the market.