Florin Doru
email: florin-doru.hutu@insa-lyon.fr

Hut ¸u

Radu Gabriel Bozomitu

Driver's warning notifications by using FM RDS technology

In this paper, the use of Software Defined Radio (SDR) transceivers for broadcasting drivers' warning notifications is tackled. More precisely, by employing the Radio Data System (RDS) protocol of a local Frequency Modulation (FM) transmission, road traffic's dangerous areas (rail crossing, unmarked crossroads, etc.) are announced. The code of a warning message is broadcasted in the close vicinity of the dangerous area and, at the receiver side, a corresponding image is displayed on the car's tablet PC. In order to study the coexistence of such kind of transmissions with the regional or national ones, the performance in terms of bit error rate (BER) with respect to the difference between the two received power levels is studied. Then, based on a simplified channel model, this paper gives a method to estimate the local FM transmission range, able to cover the dangerous area without transmission errors on the RDS part.

I. INTRODUCTION

More than ten years ago, the European Union had the ambition to halve the number of road fatalities by 2020 with respect to the values of 2010. Unfortunately, this target was not reached, and in 2019 the EU average was 51 road fatalities per million inhabitants. In the same time, Romania, with 96 fatalities per million inhabitants, had the most dangerous roads in the European Union. France, with the 50 fatalities per million inhabitants was representative for the EU average and Iceland had the safest roads in 2019 (17 fatalities per million inhabitants [START_REF] Decae | European Commission (2020) Road safety targets -Monitoring report[END_REF].

For Romania, this situation can be explained by the tremendous increase of the car fleet in the last two decades (from 3 million to 8 million vehicles) while the road infrastructure did not keep pace with this evolution. In this context, the implementation of solutions able to increase traffic safety remains a stringent problem.

Many car accidents are due to the occurrence of sudden dangerous situations in traffic for which the drivers are not prepared to react or they are reacting too slowly. In the literature, several solutions, based on the wireless transmission of drivers' notifications, are proposed. Among them, the authors of [START_REF] Bozomitu | Drivers' Warning Application Through Image Notifications on the FM Radio Broadcasting Infrastructure[END_REF] are proposing to broadcast driver's warning notifications through the FM radio infrastructure by the means of pictograms sent through Quadrature Phase Shift Keying (QPSK) modulation on the monophonic component of the multiplex (MPX) signal.

Different other types of radio access technologies, such as cellular vehicle-to-everything (C-V2X) and dedicated short-range communication (DSRC) are used in order to implement vehicular networks for robust communications among vehicles. By using these communication technologies, many applications like road traffic safety together with infotainment related services developed within new intelligent transportation systems (ITS) are envisaged in the near future. DSRC and C-V2X are technologies employed for very high-speed (Mbps) and at high-frequency (5.9 GHz) data exchange and require extensive and very expensive infrastructure deployment. The range of DSRC is typically 300 meters, while C-V2X may have 20-30 % more range than DSRC as well as significant improvement in terms of performance when obstructions between emitters and receivers occurs [START_REF] Gettman | DSRC and C-V2X: Similarities, Differences, and the Future of Connected Vehicles[END_REF]. The classical RDS system provides the TA (traffic announcements) flag which may be set up in order to receive national or regional announcements about traffic events. When TA is set, the radio receiver is tuned to a special FM channel where flash information is broadcasted. Another example of classic traffic broadcasting service is the TMC (Traffic Message Channel). Once again, regional or national traffic information is broadcasted and generally integrated directly into the car's navigation system.

In this context, our paper proposes a local warning system through image notifications based on FM radio transmissions. More precisely, the drivers are receiving notifications from the local FM emitters which are deployed in the close vicinity of an imminent danger (rail crossing, unmarked crossroads, etc.). These notifications are transmitted as codes sent on the RDS (Radio Data System) component and these codes are associated to pictograms which are selected from a database stored on the car's tablet PC. As soon as the code is received, the pictogram is displayed on the tablet PC's screen and an audio notification may be also be played back.

The rest of the paper is organized as follows: in section 2, the scenario of transmitting warning notifications by the means of local FM emitters is detailed. Through simulation results, section 3 presents the impact of the classical FM broadcasting on the local FM transmission. More precisely, the coexistence of the two kind of FM emissions (classical and local ones) is studied and the local RDS signal's transmission quality is tackled. The bit error rate was chosen as indicator of the local FM transmission's communication quality.

II. PROBLEM STATEMENT

In this paper, the scenario of using local FM emitters in order to transmit drivers' warning notifications is studied. These FM transmissions are performed with a relatively high power level in order to overcome the existing broacasting of a classical FM station emitting on the same channel. The range of this FM transmission need to be long enough such that the driver adapts its driving with respect of the imminent danger. More precisely, the code transmitted on the RDS part need be decoded properly and the corresponding image displayed in a short time.

Local FM emitter "Classic" FM emitter SDR Fig. 1. Scenario of a driver's warning notification strategy by using local FM emitters. The local FM emitter is installed at the road sign level and the emission power is high enough to assure a zero BER for the RDS signal in the close vicinity.

For the proposed solution, several advantages can be listed. The main one is that the solution is entirely compatible with the existing car lot. In the absence of a tablet PC, the driver is notified either by the audio signal, either by a text notification, corresponding to the RDS transmission. Another advantage is that the FM radio broadcasting infrastructure is a mature technology, widespread in all European countries and especially in Romania. Consequently, the deployment of the proposed solution is relatively inexpensive.

As can be seen from Figure 2, the classical FM composite signals have two distinct parts. First, the MPX (multiplex component) part is the one generated by a stereo encoder based on the two audio channels. Second, the RDS part is dedicated to broadcast, in a digital form, a certain amount of information. A classical commercial radio receiver, compliant to the RDS standard [START_REF]EN 62106:2001 -Specification of the radio data system (RDS[END_REF], can receive static or dynamic information like the name of the radio station, the name or the type the broadcasted program, etc. On the physical layer, after passing through a differential encoder, the RDS data is transposed around a 57 kHz subcarrier by using the PSK (Phase Shift Keying) modulation. The RDS data is sent with a symbol rate of 1187.5 syms/sec.

According to the recommendations of the ITU (International Telecommunications Union) [START_REF]Transmission standards for FM sound broadcasting at VHF[END_REF], the composite signal The FM emissions are performed with a relatively high power level in order to cover large areas. According to [START_REF]BS.412-9 Planning standards for terrestrial FM soundbroadcasting at VHF[END_REF], the average levels of the received electrical fields in order to obtain a satisfactory FM reception are function of the scenario. Table I is gathering these values together with the corresponding power levels, when the far field condition is satisfied and for a receiving antenna having 5 dBi gain. The considered RF frequency is 100 MHz. As stated, the emitters installed in the close vicinity of the imminent danger are broadcasting warning signals on a FM channel through the RDS component. The power level of the local notification signal can be approximated if the following scenario is considered. The antenna of the local FM emitter is considered placed at a height H = 3.5 meters, an average value of the road sign's height. The antennas installed on the car's rooftop and at the local emitter level are omnidirectional ones. Moreover, a ground plane is considered at the car's aerial level and its height is h = 1.5 meters, an average value for a car's height.

Under these conditions, by employing the Friis formula in free space [START_REF] Balanis | Antenna theory analysis and design[END_REF], the received power level (P R ) dBm can be written as in equation ( 1):

P r = EIRP + G R -22 -20 • log 10 d λ 0 + + 20 • log 10 ∆h 2 + d 2 d 2 cos 2 π 2 • ∆h ∆h 2 + d 2 (1)
where d represents the distance between the two antennas, (G R ) dBi is the gain of the receiving antenna, ∆h = H -h represents the height difference between the two antennas and λ 0 is the free space wavelength. Figure 3 presents the received power level variation with respect to the distance d between the two antennas when the receiving antenna's maximum gain is 5 dBi. The local emitter's EIRP (equivalent isotropic radiated power) is arbitrarily chosen at 10 dBm and the RF frequency f = 100 MHz. By taking into account this variation of the local's FM signal strength and the average received signal strength for the national/regional FM transmissions (Table I), the range of the local FM transmission such that the RDS transmission's BER remains insignificant can be estimated.

III. SIMULATION RESULTS

In order to estimate the BER variation, a simulation model based on the software-defined radio concept was implemented in Matlab -Simulink, including a stereo FM transmitter and receiver using the RDS standard. An additive white Gaussian noise (AWGN) is also considered in order to bring more realism to the communication channel.

In practice, the local FM signal may be obtained by translating the baseband composite part on a carrier frequency (87.5 -108 MHz) that has very good propagation properties by using a professional transmitter as for example an USRP (Universal Software Radio Peripheral) device. In our model, in order to be able to perform the simulations without exceeding the processing capabilities of the computing unit, the carrier frequency was set to 1 MHz.

The simulation setup includes two FM transmitters emitting on the same RF frequency, one for broadcasting the FM high priority local warning signal (s RF L (t)) and having a power level P RF L , and the other one for transmitting the FM signal generated by the national/regional station (s RF N (t) having the power level P RF N ). The sampling frequency used in the proposed system was set to 228 kHz, which is a multiple of the 1187.5 Hz, the data rate of the RDS signal.

The software defined FM receiver uses a non-coherent FM demodulator based on the complex delay line frequency discriminator employed in order to extract the MPX signal. It uses also a stereo FM demultiplexer, a stereo FM decoder to extract the audio stereo part by using blocks from the "RTL-SDR book library" [START_REF] Stewart | Software defined radio using MATLAB and Simulink and the RTL-SDR[END_REF] and a physical layer RDS decoder [START_REF]RDS/RBDS and RadioText Plus (RT+) FM Receiver[END_REF].

As stated, in the vicinity of a traffic dangerous situation, the signal generated by the local FM transmitter need to have a higher power level compared to the signal generated by the national station, so that it can be received by traffic participants without errors. The model proposed in this paper takes into account the distortions introduced by the AWGN communications channel, as well as the distortions generated by the national/regional station, which acts as noise for the local FM transmitter.

The distortions introduced by the AWGN communication channel have been modeled by considering different values for the received RDS signal to noise ratio (S RDS /N ). More precisely, three transmission scenarios, corresponding to different values of the S RDS /N ratio (8 dB, 11 dB and 14 dB) have been analyzed in the paper. In our setup, the transmission of the local FM station without the impact of the national FM one, corresponds to a transmission of the RDS signal without any errors. In this particular case, the signal to noise ration of the RDS component is higher than S RDS /N = 11 dB.

In Fig. 4, the spectrum of the received MPX high priority warning signal for an S RDS /N = 11 dB and a power difference of ∆ P = P RF L -P RF N = 18 dB is presented. In Fig. 5, the spectrum of the received RF signal in which, the high priority local warning signal is predominant. More precisely, this spectrum is obtained in the same conditions as for Fig. 4. In our model, the local warning FM baseband signal was low pass filtered at 100 kHz in order to respect the FM standards. This signal was then translated around 1 MHz carrier. In Fig. 6, the bit error rate of the received RDS signal depending on the difference between the two received RF signal powers, ∆P dB = P RF L -P RF N , for different values of the received RDS signal to noise ratio is plotted. The power of the received high priority local warning signal, P RF L , depends on the distance between the car's aerial and the transmitter's antenna and may be estimated by using the equation (1). From Fig. 3 and Fig. 6, can be estimated the distance between the local FM emitter and the car's aerial at which the RDS component of the local warning signal can be received without errors. Indeed, consider a transmission of the local warning signal with an S RDS /N = 11 dB. According to Fig. 6, in this particular case, the power level of s RF L (t) need to be 17.5 dB higher than the power level of s RF N (t) in order to receive the RDS component without errors. Moreover, by taking into account the average power levels of s RF N (t) (Table I), the following distances can be estimated, as function of the propagation environment (Table II). These estimated values are available for an EIRP = 10 dBm and if no error correction code was implemented after the RDS signal demodulation. In order to increase the transmission range for s RF L (t), either the EIRP can be increased, either error correction codes, decreasing the ∆P value for which no error occur, can be implemented.

In Fig. 7 is given an example of a text message, received by the high priority local warning station in the vicinity of a rail crossing. This message is decoded in Matlab -Simulink from the RDS component of the s RF L (t) signal and may be featured on the auto-radio's display. Moreover, the corresponding warning pictogram, software built at the receiver stage, may be displayed on the car's Tablet PC.

IV. CONCLUSION

The paper presents a strategy to warn drivers from imminent traffic events. By using local FM emitters and the RDS component, codes corresponding to warning pictograms are sent. Software defined radio receivers coupled with tablet PC are demodulating, decoding and then displaying the corresponding pictogram. Moreover, in order to study the coexistence of such kind of local FM transmissions with the national or regional ones, this paper propose a method to estimate the local FM transmission range, exempted from errors. This method employs a simple propagation channel model and is based on the average received power levels of the national/regional FM transmissions. Future work will be dedicated to implement in practice this driver's warning scenario.

Fig. 2 .

 2 Fig. 2. Spectrum of a classical FM composite signal. The percentages are corresponding to the contribution of each component to the composite signal's power level.

Fig. 3 .

 3 Fig. 3. Local FM signal's power variation with respect to the distance d between the car's aerial and the local FM emitter's antenna.

Fig. 4 .

 4 Fig. 4. Example of received high priority warning signal (baseband)

Fig. 5 .

 5 Fig. 5. Spectrum of the received RF signal when the local warning signal is predominant.

Fig. 6 .

 6 Fig. 6. BER of the received RDS signal depending on the difference between the two received RF signal powers ∆P for different values of the received RDS signal to noise ratio.

Fig. 7 .

 7 Fig. 7. Text message extracted from the RDS component of the s RF L (t) signal.

TABLE I MEDIAN

 I FIELD STRENGTH AND THE CORRESPONDING RECEIVED POWER LEVELS FOR DIFFERENT FM RECEPTION SCENARIOS

TABLE II DISTANCE

 II BETWEEN THE LOCAL FM EMITTER AND THE CAR'S AERIAL GUARANTEEING AN RDS TRANSMISSION WITHOUT ERRORS


André Benine-Neto

Saïd Mammar

Piecewise Affine State Feedback Controller for Lane Departure Avoidance

This paper presents the design and simulation tests of a lane keeping assistance system for passenger vehicles based on a piecewise affine state feedback controller. The design of the proposed lane keeping system takes into account the entire domain of lateral tire forces through piecewise affine approximations of the tire forces nonlinear behavior. The computation of the control law is casted as Bilinear Matrix Inequalities optimization procedure which is solved using the V-K-method to find a piecewise quadratic Lyapunov function and the state feedback gain. Simulations show the improved performance of the controller on degraded road adhesion conditions.

I. INTRODUCTION

Lane departure represents a significant fraction of road accidents and in many cases it is due to the driver fatigue or inattentiveness, therefore research on vehicle lateral control has been gained importance over the last 40 years, either for full or partial automation of driving tasks, in order to increase safety by reducing accidents and drivers workload. An overview of the recent research activity is provided in [START_REF] Shladover | Path at 20 -history and major milestones[END_REF]. It can be noted that earlier work was devoted to autonomous vehicles in highway scenarios in which the experienced lateral accelerations are generally low [START_REF] Fenton | On the steering of automated vehicles : Theory and experiment[END_REF], which has also been considered in more recent works for the development of assistance systems which replace the driver during the lane departure avoidance manoeuvres [START_REF] Minoiu-Enache | Driver steering assistance for lane departure avoidance[END_REF]. Nevertheless, in order to provide more safety to the drivers, these Lane Keeping Assistance Systems (LKAS) must be able to operate in the complete domain of tire forces, including the nonlinear behavior [START_REF] Hatipoglu | Automated lane change controller design[END_REF], as many accidents occur due to the vehicle loss of control when the tire forces are saturated.

Vehicle lateral control in saturated tire force conditions has been proposed for autonomous lane keeping system in [START_REF] Falcone | Predictive active steering control for autonomous vehicle systems[END_REF] and in [START_REF] Mammar | Lane keeping automation at tire saturation[END_REF] using a parallel distributed control structure developed on a Takagi-Sugeno model.

Piecewise Affine (PWA) systems [START_REF] Sontag | Nonlinear regulation: The piecewise linear approach[END_REF] provide another possibility to model the nonlinearities. It has been employed in [START_REF] Borrelli | A Hybrid Approach to Traction Control[END_REF] for the design of traction controller and in [START_REF] Bernardini | Driveby-wire vehicle stabilization and yaw regulation: A hybrid model predictive control design[END_REF] for a yaw rate tracking system both using a model predictive control strategy. PWA approximation of the tire forces has also been used in [START_REF] Scalzi | Active steering control based on piecewise affine regions[END_REF] and [START_REF] Benine-Neto | Vehicle yaw rate control based on piecewise affine regions[END_REF] to develop an active steering system to track a yaw rate reference using a PI controller.

In this paper a similar tire force approximation is used for the design of a state-feedback PWA controller for a LKAS, which is able to operate in the entire tire force domain.

Following the results obtained in [START_REF] Rodrigues | Observer based control of piecewise-affine systems[END_REF], the control synthesis is casted as a Bilinear Matrix Inequality (BMI) optimization problem which is solved using the V-K method. This interactive procedure allows the BMI to be transformed into two Linear Matrix Inequalities (LMI) optimizations problems which can be solved more efficiently to find simultaneously the state feedback gain and a Piecewise Quadratic Lyapunov (PWQL) function that ensure stability of the closed loop system. The vehicle model for control synthesis is described in Section II, followed by the controller design shown in Section III. The simulation results obtained on a nonlinear model are presented in Section IV providing a comparison of the PWA controller with a controller designed only for the linear behavior of the tire forces. Section V wraps up the work and provides some perspective for the future work.

II. VEHICLE MODEL

In order to design the control law for the LKAS, a simple nonlinear vehicle model is used, in which the lateral translational and yaw motions are considered and the roll and pitch motions are neglected. The wheels of the front and rear axles are lumped into one located at the axle center leading to a bicycle model. The equations describing this model are given by:

mv( β + r) = f s f + f sr J ṙ = l f f s f -l r f sr , (1)
where the involved variables consist of the vehicle yaw rate, denoted r and the vehicle sideslip angle, β . Concerning the fixed parameters for the model, v represents the longitudinal vehicle speed, m is the vehicle mass, l f (l r ) is the distance from the front (rear) axle to the center of gravity (CG), J is the vehicle inertia with respect to the vertical axle through the CG. All numerical values of these parameters are presented in Table I. The lateral forces f sx , with x = f , r, for the front and rear tires can be modeled according to the Pacejka tire model [START_REF] Pacejka | Tire and Vehicle Dynamics[END_REF]:

f sx (α x ) = D x sin {C x atan [(1 -E x ) B x α x + E x atan (B x α x )]} , (2)
where α x stands for the front (rear) tire sideslip angle. Considering that the angles remain small, the sideslip angles for front and rear tires are given by:

α f = δ f -β - l f r v , α r = -β + l r r v , (3)
where δ f is the steering angle. The Pacjeka tire model is depicted by the solid line in Fig. 1. The simple nonlinear model [START_REF] Ackermann | Robust Control[END_REF] shows, as well known in literature (see for example [START_REF] Ono | Bifurcation in vehicle dynamics and robust front wheel steering control[END_REF]), a limited stability region which also depends on the driver steering wheel angle, two saddle points and a stable one (bifurcation analysis). The causes of the instability are due to the nonlinear behavior of the tire-road forces which are taken into account in the PWA vehicle model described in the next section.

A. Piecewise affine approximation of lateral tire forces

Considering that the front and rear tire forces described by [START_REF] Benine-Neto | Vehicle yaw rate control based on piecewise affine regions[END_REF] are approximated by the following PWA functions:

f s f α f = e f i + d f i α f f sr (α r ) = e ri + d ri α r (4)
where d f i , d ri , e f i and e ri depend on the front and rear tire forces and the index (i) corresponding to the partitioning of the tire force domain, based on the tire sideslip angles, as illustrated by the dotted lines on Fig. 1 for the front wheel.

Replacing the PWA approximation of lateral tire force (4) in the vehicle model (1) leads to the PWA system:

ẋd = A d i x d + B d i u + a d i , (5)
with the corresponding dynamics:

A d i = - d f i +d ri mv -1 - d f i l f -d ri l r mv 2 d ri l r -d f i l f J - d f i l 2 f +d ri l 2 r Jv , (6)
B d i = d f i mv d f i l f J , a d i = e f i +e ri mv e f i l f -e ri l r J . ( 7
)
where the control input is the front wheel steering angle u = δ f and the state variables are the vehicle sideslip angle, β and the vehicle yaw rate, r,

x d = [β , r] T .
Even though this approximation can be refined, it is shown in the simulation section that improved performance can be obtained by simply approximating the tire forces with the proposed PWA functions.

B. Additional dynamics for lane keeping

For lane keeping purposes, the model ( 5) has to be expanded with the dynamics of the relative yaw angle and the lateral displacement with respect to the lane centerline. These measurements are provided by a video sensor. Let ψ L = ψψ d be the yaw angle error which is the angle between the vehicle orientation and the tangent to the road. The road reference curvature ρ re f is defined by ( ψd = vρ re f ), and the following equality can be derived:

ψL = r -vρ re f . (8)
Denoting by l s the look-ahead distance, the equation giving the evolution of the measurement of the lateral offset y L from the centerline at sensor location is obtained by

ẏL = v(β + ψ L ) -l s r. (9)
Equations ( 8) and ( 9) are included in system (5) and the PWA system describing the vehicle dynamics and positioning is described by:

ẋp = A p i x p + B p i u + B p ρ ρ re f + a p i ( 10
)
where

x p = [β , r, ψ L , y L ] T and A p i =   A d i 0 0 0 1 0 0 v l s v 0   , B p i = (B d i ) T 0 0 T , B p ρ = 0 0 -v 0 T , a p i = (a d i ) T 0 0 T . ( 11
)
An illustration of the state variables is provided in Fig. 2 III. CONTROL STRATEGY

The algorithm presented in [START_REF] Rodrigues | Observer based control of piecewise-affine systems[END_REF] is applied for the design of the proposed LKAS. It provides an efficient procedure for control synthesis for PWA systems using a PWA state feedback gain, by means of a BMI optimization procedure, based on the search of a PWQL function.

A. Assumptions for LKAS and PWA system description

As most vehicles have understeering behavior, it is assumed that the front tire forces tend to saturate firstly, therefore only the front tire forces are approximated by PWA functions according to (4), and a single linear approximation is assumed for the rear tire, as follows:

         f s f α f = e f 1 + d f 1 α f , for α f < -ᾱ f , f s f α f = c f α f , for -ᾱ f < α f < ᾱ f , f s f α f = e f 3 + d f 3 α f , for α f > ᾱ f , f sr (α r ) = c r α r ∀α r , ( 12
) where c f = B f C f D f , c r = B r C r D r
and ᾱ f corresponds to the partition shown in Fig. 1.

According to (3), the partitions defined by the front wheel sideslip angle, α f =± ᾱ f , depend on the state variables and control input. As mentioned in [START_REF] Rodrigues | Observer based control of piecewise-affine systems[END_REF] the switching between the regions should depend only on the state x, so models in which there is a feed-through from u to a nonlinearity should be avoided. This can be done by the inclusion of a first order actuator which is also useful to enforce continuity of the control input at the boundaries. Therefore, the steering angle is included as an additional state to the system as follows:

δ f = -τδ f + u c , (13)
where u c becomes the control input of the augmented plant and τ = 10. Disturbances and exogenous inputs, such as the road curvature, are not taken into account in the control synthesis, therefore the model describing the vehicle dynamics, positioning [START_REF] Mammar | Vehicle handling improvement by active steering[END_REF] and first order actuator (13) can be written as:

ẋ = A i x + B i u + a i , (14)
with x = [β , r, ψ L , y L , δ f ] T and

A i = A d i B p i 0 0 0 0 -τ , B i = 0 0 0 0 τ T , a i = (a d i ) T 0 T . ( 15
)
Each of the regions R i , i = 1, 2, 3 can be described as the intersection of the half spaces

R i = {x|H T i x -g i < 0} ( 16
)
Since these regions are slab, it is preferred to express them as degenerated ellipsoids, as suggested in [START_REF] Rodrigues | Piecewise-affine state feedback for piecewise-affine slab systems using convex optimization[END_REF] and [START_REF] Rodrigues | Observer based control of piecewise-affine systems[END_REF], because they describe exactly the slab regions and it often requires fewer parameters than the polytopic description [START_REF] Rodrigues | Observer based control of piecewise-affine systems[END_REF].

The degenerated ellipsoids can be described by:

||E i x + f i || 2 ≤ 1 ∀x ∈ R i , (17)
and following the description in [START_REF] Rodrigues | Piecewise-affine state feedback for piecewise-affine slab systems using convex optimization[END_REF], if R i = {x|d < c T i x < d} then the degenerated ellipsoid is described by:

E i = 2c T i /( d -d) and f i = -( d + d)/( d -d) ( 18
)
For the PWA system ( 14), the ellipsoids describing the slab regions are:

E T 1 = [-1, - l f v , 0, 0, 1] 2 -ᾱ f -d , f 1 = - -ᾱ f +d -ᾱ f -d , E T 2 = [-1, - l f v , 0, 0, 1] 1 ᾱ f , f 2 = -ᾱ f -ᾱ f ᾱ f + ᾱ f = 0, E T 3 = [-1, - l f v , 0, 0, 1] 2 d-ᾱ f , f 3 = - d+ ᾱ f d-ᾱ f , (19)
where d and d can be arbitrarily set, such that d <ᾱ f and d > ᾱ f . The ellipsoidal cell description is used to relax the constraints when searching the PWQL function. In order to ensure the continuity of the PWQL function across the boundaries, a parametric boundary description is needed [START_REF] Hassibi | Quadratic stabilization and control of piecewise-linear systems[END_REF].

Each facet boundary between the neighboring regions R i and R j is contained in the hyperplanes described by:

{x|c T i j x -d i j = 0}. ( 20
)
and the boundary parametric description can be obtained by:

Ri ∩ R j ⊆ {x / l i j + F i j s / s ∈ R n-1 } (21)
where F i j ∈ R n×n-1 (full rank) is the matrix whose columns span the null space of c i j , and l i j ∈ R n is given by l i j = c i j (c T i j c i j ) -1 d i j For the LKAS model ( 14) the parametric description of the boundaries is given by:

F 1,2 =       1 0 0 0 0 1 0 0 0 0 1 0 0 0 0 1 1 l f v 0 0       , F 2,3 =       1 0 0 - l f v 0 0 0 1 0 1 0 0 0 1 0 1 0 0 0       , l 1,2 = ᾱ f 2+l 2 f /v 2 , ᾱ f l f /v 2+l 2 f /v 2 , 0, 0, -ᾱ f 2+l 2 f /v 2 , l 2,3 = -ᾱ f 2+l 2 f /v 2 , -ᾱ f l f /v 2+l 2 f /v 2 , 0, 0, ᾱ f 2+l 2 f /v 2 . ( 22
)

B. PWA State-Feedback control synthesis

Due to the symmetry of the lateral tire force with respect to the origin, only regions R 1 and R 2 are considered in the control synthesis. The same gains obtained for region R 1 can be applied in region R 3 .

The goal is to stabilize ( 14) with a PWA state feedback gain u c = K i x + m i , based on the search of a PWQL function of the form:

V i (x) = x T P i x + 2q T i x + r i . ( 23
)
V i (x) can be a Lyapunov function with a decay rate α i , for the region R i if, for a fixed ε ≥ 0,

x ∈ R i , V i (x) > ε||x -x eq || 2 d dt V i (x) < -α i V i (x), (24)
where x eq is the equilibrium point of the closed loop system.

Rendering the vehicle to the center of the lane is a regulation problem for system [START_REF] Pacejka | Tire and Vehicle Dynamics[END_REF], since in a straight line all state variables should converge to the origin. Therefore the desired equilibrium point of the closed loop system, x eq = 0, is placed at the origin. The equilibrium point must also be the extrema of the Lyapunov function candidate for the corresponding region (see [START_REF] Rodrigues | Observer based control of piecewise-affine systems[END_REF]). In consequence, q 2 = 0 5×1 and r 2 = 0 to ensure that V (0) = 0.

The closed-loop state-space equation is:

ẋ = (A i + B i K i )x + (b i + B i m i ) = Ãi x + bi . ( 25
)
As the closed loop system has a single equilibrium point, the equilibrium points of R 1 , denoted x eq1 and R 3 , x eq3 , must not be contained in its own region, enforcing a transition through the boundaries with R 2 . This can be done using the constraint:

(A i + B i K i )x eq + (b i + B i m i ) = 0. ( 26
)
The stability of the (25) can be guaranteed by S-procedure [START_REF] Boyd | Linear Matrix Inequalties in System and Control Theory[END_REF], using the ellipsoidal cell description [START_REF] Scalzi | Active steering control based on piecewise affine regions[END_REF] and conditions (24), if there exists P i = P T i ≻ 0, and positive constants λ 1 and γ 1 that satisfy:

       P 1 -εI n + λ 1 E T 1 E 1 q 1 + εx eq + λ 1 E T 1 f 1 * r 1 -εx T 1 x 1 + λ 1 ( f T 1 f 1 -1) ≻ 0 P 2 -εI n ≻ 0 (27)                  ÃT 1 P 1 + P 1 + Ã1 -γ 1 E T 1 E 1 + α 1 P 1 P 1 b1 + ÃT 1 q 1 -γ 1 E T 1 f 1 + α 1 q 1 * 2 bT 1 q 1 + α 1 r 1 -γ 1 ( f T 1 f 1 -1)     ≺ 0 ÃT 2 P 2 + P 2 + Ã2 + α 2 P 2 ≺ 0 (28)
Using the parametric description of the boundaries ( 22), the continuity of the PWQL function candidate is ensured by:

     F T 12 (P 1 -P 2 )F 12 = 0 F T 12 (P 1 -P 2 )l 12 + F T 12 q 1 = 0 l T 12 (P 1 -P 2 )l 12 + 2q T 1 l 12 + r 1 = 0 (29)
Ensuring that there is no sliding mode between R 1 and R 2 can be enforced by the equations:

c T 12 ( Ã1 -Ã2 )F 12 = 0 c T 12 ( Ã1 -Ã2 )l 12 + b1 -b2 = 0 (30)
Defining a performance criterion as:

J = min i α i .
The goal is to find the PWA state feedback controller that maximizes the performance criterion J. This optimization problem can be cast as a BMI as: maximise: min

i α i
subjected to: (26), ( 29), ( 27), ( 28), (30),

ε > 0, γ 1 > 0, λ 1 > 0, α 1,2 > l 0 > 0, -l 1 < K i < l 1 , -l 2 < m i < l 2 (31)
where l 0 is a scalar bound, l 1 and l 2 are vector bounds.

The BMI problem (31) can be solved using the V-K method, in which LMIs are solved in an iterative fashion. For that, the equilibrium point of the dynamics corresponding to the affine region x eq1 , is chosen a-priori, such that it is not located in R 1 . For this specific lane keeping problem, the algorithm becomes.

V-step: Given a fixed controller, and a fixed α i , solve:

Find: P 1 , q 1 , r 1 and P 2 , s. t. : (29), ( 27), (28),

ε > 0, γ 1 > 0, λ 1 > 0, (32)
K-step: For P 1 , q 1 , r 1 and P 2 fixed at the previous step, solve: maximise: min (26), ( 27), ( 28), (30),

ε > 0, γ 1 > 0, λ 1 > 0, α 1,2 > l 0 > 0, -l 1 < K i < l 1 , -l 2 < m i < l 2 , (33)
For each iteration of the K-step, the decay rates α 1 and α 2 must be greater than the value computed at the previous iteration. The loop must be repeated until there is no significant improvement on the cost or the LMIs become infeasible.

The gain for the first iteration on the V-step is computed using the algorithm from [START_REF] Minoiu-Enache | Driver steering assistance for lane departure avoidance[END_REF]. The same gain is considered for both regions, and m 1 = m 2 = 0.

IV. SIMULATION RESULTS

The resulting gains from the V-K method are:

K 1 = [ -0.0914 -0.1514 -1.0289 -0.0824 -0.1919], K 2 = [-0.3184 -0.1639 -1.0289 -0.0824 -0.1879], K 3 = K 1 , m 1 = -m 3 = 0.0245, m 2 = 0 (34)
In order to verify the performance of the PWA controller, simulation were carried out on the Matlab/Simulink environment, comparing two vehicles, one equipped with the PWA controller and another equipped with the controller computed for the linear region only (controller K 2 ). The dynamics of the model used for simulation consist of nonlinear 4-wheels vehicle according to [START_REF] Ackermann | Robust Control[END_REF], which is considered to capture the essential vehicle lateral steering dynamics:

m( vy + rv) = f s f cos δ f + f sr J ṙ = l f f s f cos δ f -l r f sr (35)
where v y is the lateral velocity, and the lateral forces are given by ( 2). The sideslip angles α f ,r are given by:

α f (le f t) = δ f -arctan vβ +rl f v-a 2 r α f (right) = δ f -arctan vβ +rl f v+ a 2 r α r(le f t) = -arctan vβ -rl r v-a 2 r α r(right) = -arctan vβ -rl r v+ a 2 r
(36)

As an attempt to make the lane departure avoidance assistance system unobtrusive during normal driving, the activation strategy takes place only in case the driver is inattentive and there is a risk of lane departure. Throughout the simulations the driver is considered inattentive, and analogously to [START_REF] Minoiu-Enache | Driver steering assistance for lane departure avoidance[END_REF] the risk of lane departure is estimated by position of the front wheels with respect to a centered strip on the lane with width 2d. The position of the left (y l ) and right (y r ) front wheels can be described by:

y l = y L + (l f -l s )ψ L + a 2 , y r = y L + (l f -l s )ψ L -a 2 , (37)
where a represents the vehicle width. Enforcing that the front wheels must remain inside the centred lane strip ±d, yields:

- 2d -a 2 ≤ y L + (l f -l s )ψ L ≤ 2d -a 2 (38)
This set, corresponding to the region between two parallel hyperplanes, can be written with respect to the state vector as:

T ∆ = {x ∈ R 5 : |Fx| ≤ 1} (39)
where F = (0, 0,

2(l f -l s )
2d-a , 2 2d-a , 0). The road adhesion is considered degraded, in order to show the performance of the controller in the occurrence of tire forces saturation. The road adhesion coefficient is considered µ = 0.5, which corresponds to wet pavement. This condition is incorporated in the parameters of (2) by changing B to (2 -µ)B, C to ( 5 4 -µ 4 )C and D to µD, [START_REF] Mammar | Vehicle handling improvement by active steering[END_REF]. The simulation scenario consists of an inattentive driver approaching a left-hand side curve. As the right front wheel reaches the centered strip d the LKAS system is activated. While attempting to steer the vehicle to the center of the lane, the front wheel sideslip angle, depicted in Fig. 3 overshoots the limit considered for the linear behavior, corresponding to the saturation of tire forces. The switches between the gains corresponding to R 2 and R 3 are shown in Fig. 3. The corresponding control input and steering angle that can be seen in Fig. 4.

The vehicle sideslip angle and yaw rate for the two vehicles are shown in subplots of Fig. 5. The relative yaw angle and lateral offset at look-ahead distance, representing the vehicle positioning on the road are depicted in Fig 6 . It is interesting to note that the dynamics of the vehicle equipped with the PWA controller are less extreme due to the switching of the controller. This control action to reestablish vehicle stability leads to a slightly larger lateral offset during the maneuver, but it is still within an acceptable range. Moreover it avoids the saturation of the lateral tire forces. In this paper the design and simulation of a PWA controller for LKAS has been described. The nonlinear behavior of the lateral tire forces are approximated by PWA functions. As only understeering vehicles are considered in this work, the regions describing the PWA vehicle model can be exactly described as degenerated ellipsiods which are used to relax the conditions for the existence of a PWQL function. The simultaneous search of a PWQL function and a PWA state feedback gain has been casted as a BMI optimization problem. The solution has been computed using the V-K method in which the BMIs are transformed into two LMI optimization procedures that are solved iteratively. The resulting PWA controller has been tested in degenerated adherence conditions showing its enhanced performance under the effects of tire force saturation and the activation strategy of the assistance system.

Currently commercialized vehicles are not equipped with sensors that are able to measure the vehicle sideslip angles.

Therefore the design of a output feedback controller should be considered in the future work. Taking into account parameter variations as for example vehicle longitudinal speed, and implementation on prototype vehicle should also be included in the future work.

Fig. 1 .

 1 Fig. 1. Front tire forces described by the Pacejka magic formula and corresponding PWA approximations and partitions.

Fig. 2 .

 2 Fig. 2. Single track vehicle model.

Fig. 4 .

 4 Fig. 4. Steering angle and Control input for vehicles with PWA and linear controllers

Fig. 5 .Fig. 6 .

 56 Fig. 5.Vehicle dynamics response for vehicles with PWA and linear controllers

TABLE I VALUES

 I OF THE VEHICLE PARAMETERS.

	Parameter	Value
	c f , front cornering stiffness	40000 N/rad
	c r , rear cornering stiffness	35000 N/rad
	l f , distance form CG to front axle 1.22 m
	l r , distance form CG to rear axle	1.44 m
	l s , look-ahead distance	0.95 m
	m, total mass	1600 kg
	J, vehicle yaw moment of inertia	2454 kgm 2
	v, longitudinal velocity	17 m/s

IFSTTAR -LIVIC, 14 Route de la Minière, 78000 Versailles, France (andre.benine-neto@ifsttar.fr)[START_REF] Bernardini | Driveby-wire vehicle stabilization and yaw regulation: A hybrid model predictive control design[END_REF] IBISC -EA 4526 UEVE, 40 rue du Pelvoux CE1455, 91020, Evry, Cedex, France (e-mail: said.mammar@iup.univ-evry.fr)

ACKNOWLEDGMENT

This work has been partially financed by the FUI French Initiative E-Motive.


Yasmina Bestaoui

Nominal Trajectories of an Autonomous Under-actuated Airship

Keywords: Autonomous airship, trajectory generation, under-actuation

The objective of this paper is to generate a desired flight path to be followed by an autonomous airship. The space is supposed without obstacles. As there are six degrees of freedom and only three inputs for the LSC AS200 airship, three equality constraints appear due to the under-actuation.

INTRODUCTION

Lighter than air vehicles suit a wide range of applications, ranging from advertising, aerial photography and aerial inspection platforms, with a very important application in the areas of environ mental, biodiversity, and climatological research and monitoring. Airships offer the advantage of quiet hover with noise levels much lower than helicopters. Unmanned remotely-operated airships have already proved themselves as camera and TV platforms and for specialized scientific tasks. An actual trend is toward autonomous airships.

Fig. 1 presents the AS200 airship. It is basically a large gas balloon. Its shape is maintained by its internal overpressure. The only solid parts are the gondola, a tail rotor with horizontal axis of rotation, the main rotor with a varying tilt angle and the tail fins. It is actually a remotely piloted airship designed for remote sensing, being transformed to be fully autonomous.

A basic problem to be solved by autonomous vehicles is the problem of trajectory generation. Trajectory generation means the generation and execution of a plan for moving from one location to another location in space to accomplish a desired task. The motion generation module generates a nominal state space trajectory and a nominal control input. Trajectory prediction consists in computing reference values to be given to the controller.

The problem of finding the time-optimal trajectory for a fully actuated robot manipulator along a specified path is a classical one in robotics. This problem has been solved by algorithms proposed by [START_REF] Bestaoui | On line reference trajectory definition with joint torque and velocity constraints[END_REF][START_REF] Bestaoui | Motion generation in Cartesian space for industrial robots with actuators limitations[END_REF][START_REF] Murray | A Mathemati cal Introduction to Robotic Manipulation[END_REF][START_REF] Shin | Minimum time control for robotic manipulators with geometric path constraints[END_REF][START_REF] Thomasson | Equations of motion of a vehicle in a moving fluid[END_REF]. The algorithms find the minimum time scaling of the path with respect to the actuator constraints: dynamic ones [START_REF] Bestaoui | On line reference trajectory definition with joint torque and velocity constraints[END_REF][START_REF] Shin | Minimum time control for robotic manipulators with geometric path constraints[END_REF] and technological ones [START_REF] Bestaoui | Motion generation in Cartesian space for industrial robots with actuators limitations[END_REF][START_REF] Piede | Actuator constraints in optimal motion planning of manipulators[END_REF]. The problem of finding a smooth interpolating curve is well understood in Euclidean spaces, but it is not so clear in curved spaces [START_REF] Belta | An SYD based projection method for interpolation on SE(3)[END_REF]. When planning Cartesian trajectories, it is usually possible to characterize the performance of different trajectories [START_REF] Bestaoui | Motion generation in Cartesian space for industrial robots with actuators limitations[END_REF][START_REF] Shin | Minimum time control for robotic manipulators with geometric path constraints[END_REF][START_REF] Piede | Actuator constraints in optimal motion planning of manipulators[END_REF][START_REF] Belta | An SYD based projection method for interpolation on SE(3)[END_REF][START_REF] Anderson | Constrained Extremal Trajectories and UAV Path Planning[END_REF][START_REF] Avanzini | Frenet based algorithm for trajectory prediction[END_REF][START_REF] Beard | Coordinated target assignment and intercept for unmanned air vehicles[END_REF][START_REF] Bestaoui | Motion planning • of a fully actuated unmanned air vehicle[END_REF][START_REF] Frazzoli | Robust Hybrid Control fo r Autonomous Ve hicle Motion Planning[END_REF][START_REF] Hima | Motion generation on trim trajectories for an autonomous under actuated airship[END_REF][START_REF] Hima | Time optimal paths for lateral navigation of an autonomous underactuated airship[END_REF][START_REF] Olfati-Saber | Nonlinear Control of Under actuated Mechanical Systems with Application to Robotics and Aerospace Ve hicles[END_REF][START_REF] Shiller | Computation of path constrained time optimal motions with dynamic singularities[END_REF][START_REF] Sussmann | Shortest 3 dimensional paths with a prescribed curvature bound[END_REF].

In [START_REF] Bestaoui | Some insights in path planning of small autonomous blimps[END_REF], in the first part, the trajectories considered are trim trajectories. The general condition for trim •requires that the rate of change of the magnitude of the velocity vector is identically zero, in the body fixed frame. The trim problem is generally formulated as a set of non linear algebraic equations. The solution trajectories are helices with constant curvature and torsion. In the second part, variable curvature helices are investigated. In [START_REF] Avanzini | Frenet based algorithm for trajectory prediction[END_REF], a 3rd order expansion is used for transition maneuvering between two helices. [START_REF] Das | Optimal trajectory planning for hot air balloons in linear wind fields[END_REF] investigate optimal trajectory planning for hot air balloons in linear wind fields. The objective function to be minimized is fuel consumption with respect to free end states.

This article is concerned with methods of computing a trajectory in space that describes the desired motion. The contribution of this paper is the characterization of trajectories, considering the under actuation. This paper consists of 6 sections. Section 2 Fig. 1. The AS200 airship.

presents the kinematics while the following one introduces the mechanical system. Section 4 introduces the relationship between trajectory generation algorithms and under-actuation. Simulation results are discussed in Section 5 and finally some concluding remarks are given in section 6.

KINEMATICS

Consider a rigid body moving in free space. Assume any inertial reference frame {F} fixed in space and a frame {M} fixed to the body at the center of gravity. At each instant, the configuration (position and orientation) of the rigid body can be described by a homogeneous transformation matrix correspondinj5 to the displacement from frame {F} to frame {M}.

The origin C of {M} coincides with the center of gravity of the vehicle. Its axes (Xe Y e Ze) are the principal axes of symmetry when available. They must form a right handed orthogonal normed frame. The position of the vehicle C in {F} can be described by 171 = (x y z{ while the orientation is given by '72 = (¢ () If/{ with <j> Roll, () pitch and \jJ Yaw angles. The orientation matrix R is given by (C lj/CB -Slf/C</J + Clj/SBs¢ Slf/S</J + Clj/sBc¢ J R = Slj/CB Clf/C</J + Slj/SBs¢ -Clf/S</J + Slj/SBc¢ ' (

-sB cBs¢ cBc¢

where cB =cos ( B) and se =sin ( B ) . This description .

I'd . h . " "

Is va 1 m t e region --< () < -. A singularity of 2 2

this transformation exists for:

() = " ± ktr, k E Z . 2
The kinematics of the airship can be expressed in the following way: where and 03*3

J ( V J J (11 2 ) Q ' s¢. tan() c¢. s¢/c() c¢. tan B J -s¢ ' c¢/c() (2)
(3)

Fig. 2. The airship frames (cg: center of gravity, cv: center of volume).

w physically corresponds to the angular velocity of the rigid body, while V is the linear velocity of the origin C ofthe frame {M}.

MECHANICAL SYSTEM

. 1 . Dynamics

In this section, analytic expressions for the forces and moments of a system with added mass and inertia such as an airship are introduced [START_REF] Thomasson | Equations of motion of a vehicle in a moving fluid[END_REF][START_REF] Bestaoui | Dynamic modeling of small autonomous blimps[END_REF][START_REF] Fossen | Guidance and Control of Ocean Ve hicle[END_REF][START_REF]Airship Technology[END_REF]]. An airship is a lighter than air vehicle using a lifting gas (helium in this particular case). We will make in the sequel some simplifying assumptions: the earth fixed frame is inertial, the gravitational field is constant, the airship is well inflated, the density of air is uniform. In [START_REF] Azouz | Dynamic analysis of airships with small deformations[END_REF], we considered the case of an airship with small deformations analyzed via the Updated Lagrangian Method.

The translational part being separated from the rotational part [START_REF] Bestaoui | Dynamic modeling of small autonomous blimps[END_REF], the dynamic equations (Euler -Poincare) are given by:

MV =-OJ* MV -b(.) + f(u), J OJ = -OJ * J OJ -V * MV -p ( . ) + r ( u), where b(.) =R T e3(mg-B) + DvV, T - /1(.) = (R e3 * BG)B + DwOJ. (5) (6) (7)
For a system with added masses, the term V * MV is non zero. M and J are respectively the vehicle's mass and rotational tensors and T, � and f, b represent respectively the control and non-conservative torques and forces in body axes. m is the mass of the airship, propellers and actuators. The tensor M includes both the airship's actual mass as well as the virtual mass elements associated with the dynamics of buoyant vehicles. The tensor J includes both the airship ' s actual inertias as well as the virtual inertia elements associated with the dynamics of buoyant vehicles. As the airship displays a very large volume, its added masses and inertias become very signi ficant [START_REF] Fossen | Guidance and Control of Ocean Ve hicle[END_REF]. We will assume that the added mass coefficients are constant. They can be estimated from the inertia ratios and the airship weight and dimension parameters. Added mass should be understood as pressure-induced forces and moments due to a harmonic motion of the body which are proportional to the acceleration of the body [START_REF] Fossen | Guidance and Control of Ocean Ve hicle[END_REF]. Dv is the 3 x3 aerodynamics forces diagonal matrix while Do is the 3x3 aerodynamics moments diagonal matrix. e3 = (0 0 l ) T a unit vector. The 3x1 buoyancy force vector Be3 with:

B= pA g, ( 8
)
where A is the volume of the envelope, p is the difference between the density of the ambient atmosphere P air and the density of the helium Phelium in the envelope, g is the constant gravity acceleration. BG = ( x b Yb zb ) represents the position of the center of buoyancy with respect to the body fixed frame.

The aerodynamic force can be resolved into two component forces, one parallel and the other perpendicular to the direction of motion. Lift is the component of the aerodynamic force perpendicular to the direction of motion and drag is the component opposite to the direction of motion. As the airship is a slow moving object in the air, we can assume a linear relationship between the speed and the drag. Dv = diag(-Xu -Yv -Z w ) , Do.= diag(-L P -Mq -Nr ).

The gravitational force vector is given difference between the airship weight buoyancy acting upwards on it:

[ (mg-B),e ) RT e 3 (mg-B) = -(mg -B)cB. s¢ .
-( mg-B)cB. c</J by and (9) [START_REF] Beard | Coordinated target assignment and intercept for unmanned air vehicles[END_REF] the the [START_REF] Bestaoui | Motion planning • of a fully actuated unmanned air vehicle[END_REF] The gravitational and buoyant moments are given by: (12)

If a system is fully actuated, it can be steered along any given curve on the configuration manifold, i.e it is controllable. This is not true in general for an under actuated system. However, an under-actuated system can be locally controllable if it enjoys the property of nonholonomy. The existence of nonholonomic const-raints translates into the fact that the system can be locally steered along a manifold of dimension larger than the number of independent control inputs [START_REF] Murray | A Mathemati cal Introduction to Robotic Manipulation[END_REF][START_REF] Hima | Motion generation on trim trajectories for an autonomous under actuated airship[END_REF][START_REF] Hima | Time optimal paths for lateral navigation of an autonomous underactuated airship[END_REF][START_REF] Olfati-Saber | Nonlinear Control of Under actuated Mechanical Systems with Application to Robotics and Aerospace Ve hicles[END_REF][START_REF] Bestaoui | Some insights in path planning of small autonomous blimps[END_REF][START_REF] Fossen | Guidance and Control of Ocean Ve hicle[END_REF]. However, practical methods have only been found for simple underactuated systems.

Propulsion

An airship is propelled by thrust. Propellers are designed to exert thrust to drive the airship forward. The most popular propulsion system layout for pressurized non rigid airships is twin ducted propellers mounted either side of the envelope bottom. Another one exists in the tail for torque correction and attitude control. The airship AS200 is an under actuated system with two types of control in a low velocity flight: forces generated by thrusters and angular inputs controlling the direction of the thrusters ( y is the tilt angle of the propellers)

lTM sinr ) Fi= 0 '
TM cosr [START_REF] Hima | Motion generation on trim trajectories for an autonomous under actuated airship[END_REF] where TM and T T represent respectively the main and tail thrusters. Thus in building the non linear six degrees of freedom mathematical model, the additional following assumptions are made: [START_REF] Hima | Time optimal paths for lateral navigation of an autonomous underactuated airship[END_REF] If we consider the plane XZ as a plane of symmetry, the mass and inertia matrices can be written as:

l m+X, M= 0 zx l l,+Lp

TRAJECTORY GENERATION AND UNDERACTUATION

In this paragraph, nominal trajectories are charac terized, the model is supposed perfect and any perturbation such as wind or sensor disturbance is neglected. The three equality constraints deriving from the under-actuation are sought. Considering the dynamics of the airship and its propulsion, the following dynamics equations can be written:

f( u) = M V + w * MV +b(. ) = [ T M ;; nr J • (17)
T M cosy r ( u) = J �+ w * Jw + V * MV + /3(.)

= [fl 3 T ;si nrJ=[� ; ( u)J, ( 18
)
Pz T T Pzfz ( u)
where the kinematics are:

. . p = </J-l/f SB, .

.

q = () C</J + l/f S<jJC(), . . r = -{} S</J + l/f C</JCB. ( 19
)
First equality constraint: The roll moment being zero (see [START_REF] Bestaoui | Some insights in path planning of small autonomous blimps[END_REF]), r 1 = 0 gives J 11 ( ¢-w SB-�BCB J +(J3 3 -J 22 ) ( ec¢+�S¢CB J ( -es¢+�C¢C8 J +D P ( ¢-� S() J + zbBCBS</J = 0.

(

) 20
Second equality constraint: Pi 3 Ji = r 2 gives J 22 x+( M11-M33 )uw

where + (J11 -J 33 ) u -�S() J ( -BS</J +�c¢cB J + Dq ( BC</J +�s¢cB J -zbBS() +Pi 3 M11 �

+ Pi 3 M22 w ( BC</J + � s ¢cB J -Pi 3 M22 v ( -es¢+�c¢cB J +Pi 3 Duu + Pi 3 (mg -B)SB = 0, .. . . (21)
x = () C</J-() </J S</J +If/ S<jJC() +If/ </J C<jJC() -If/() S</JSB.

Third equality constraint: Pilz= r 3 gives J 33X ' + ( Mz2 -M11 ) uv + (J 22 -J11{¢-�sB J ( ec¢+ �s¢cB J c22) +N r ( -es¢+�c¢ce ) +Pi 2 M22� -Pi 2 M22 w ( ¢-�sB ) +Pi 2 M11 u ( -es¢+�c¢cB )

+ fi 2 Dvv -fi 2 (mg-B)S</JC(} = 0, where ..

.. . x' = -BS</J-8</JC</J + l/fC</JCB-l/f </JS</JCB-l/fBC<jJSB .

The following approach is considered. The variations of the roll and pitch angles as well as the longitudinal velocity are imposed and the influence of the under-actuation on the variations of the yaw angle, the lateral and vertical velocities are studied.

The first equality constraint ( 20) is equivalent to the resolution of an ordinary differential equation of the form a(t)�+b(t) ( � r +c(t)�+d(t)�o,

If 2(t)=�(t) then the non autonomous generalized logistic equation must be solved:

a(t)S(t) + b(t)(2(t)) 2 +c(t)2(t) + d(t) = 0. ( 25
)
The third equality constraint ( 22) can be written as: (

w(t) = a0 + a 1 u + a 2 v + a 3 uv + a 4 v, (26)
) (31) 30
By using the method of separation of variables and integration. by partial fractions, in the constant coefficient case, logistic equation can be solved and the behavior of all solutions is analyzed [START_REF] Galanis | Global positive solutions of a generalized logistic equation with bounded and unbounded coefficients[END_REF][START_REF] Grimshaw | Non Linear Ordinary Diff erential Equations[END_REF][START_REF] Jiang | Positive periodic solutions of functional differential equations and population models[END_REF][START_REF] Nkashama | Dynamics of logistic equations with non autonomous bounded coefficients[END_REF].

For ¢ = 0;

Lpt/ lj/ (t) =!f f o e -/Ix.
For B=O;

(32) (33)

For the particular case, where 1ff is constant, classical trim trajectories are encountered. ------'--------' -----2 . Pi v SB M22 x =ax cos( If/ 0 t) +b x sin(lj/ 0 t), .

. . where

A =� B =!2_ x ' x If/ 0 If/ 0 C=� V= lf/o . x v ' v e e
The trajectory is shown in Fig. 3.

-0.5 The trim condition can be a turning (about the vertical axis), descending or climbing (assuming constant air density and temperature), side-slipping maneuver at constant speed. More conventional flight conditions such as hover, cruise, auto-rotation or sustained turns are also trims.

4.1.2 If/ is not constant: Fig. 4 shows the solution If/ ( t) of the differential in (20) while Fig. 5 shows its derivative Vt ( t) .

Even though there is a nonlinear variation of If/ in the beginning of the simulation, the yaw angle has a quasilinear variation after a certain time. A transitional behavior can be recognized before the yaw velocity attains a permanent (constant) value.

Roll and pitch angles linear functions of time

In this paragraph, the roll and pitch angles are assumed to have linear variations:

When the coefficients of the non autonomous logistic equation are no longer constant, no explicit solutions can be found in general and the equilibrium point may become unstable [START_REF] Galanis | Global positive solutions of a generalized logistic equation with bounded and unbounded coefficients[END_REF]. For a study to be complete, the existence of stable periodic or stable bounded solutions is an essential part of qualitative theory of this differential equation, in order to determine non trivial solutions and study their behavior [START_REF] Galanis | Global positive solutions of a generalized logistic equation with bounded and unbounded coefficients[END_REF][START_REF] Grimshaw | Non Linear Ordinary Diff erential Equations[END_REF][START_REF] Jiang | Positive periodic solutions of functional differential equations and population models[END_REF][START_REF] Nkashama | Dynamics of logistic equations with non autonomous bounded coefficients[END_REF]. Nkashama [START_REF] Nkashama | Dynamics of logistic equations with non autonomous bounded coefficients[END_REF] proved that the logistic equation with positive non autonomous bounded coefficients has exactly one bounded solutions that is positive and does not tend to the zero solution.

Solving the first equality constraint, the roll lff SB

(52)
Once the yaw angle is calculated, the linear and angular velocities are determined as well as the 3D path.

SIMUL ATION RESULTS

The In Ta ble 1, fi ve cases are presented for a normalized simulation time = 1. For each case, four subplots are presented: the first one presents the trajectory in space, the second one the variation of the yaw angle 1ff , the linear velocities v and w and finally the angular velocities p, q, r.

The 3D trajectory in Fig. 6 represents a straight line as the angular velocity is null while the linear velocity is constant. The yaw angle is constant. This 3D Table 1. Five cases for a normalized simulation time This methodology can be applied to other types of UAV, taking into account their characteristics. For fixed wing aircraft or helicopter, the added mass and inertia are neglected.

  Manuscript received May 11, 2005; revised April 5, 2006; accepted May 10, 2006. Recommended by Editor Keum-Shik Hong. Yasmina Bestaoui is with the Laboratoire IBISC CNRS FRE2873, University of Evry, 38 rue du Pelvoux, 91025 EVRY, France (e-mail: bestaoui@iup.univ-evry.fr).

  of gravity sits below the center of buoyancy, then BG = ( 0 0 zb ) r . It is important to gain insight into the geometric structure of the equations since this knowledge is useful in motion planning and control.

  where a(t)=J 11( SB), b(t)=(J 33 -J 22 )(C</JS</JC 2 B), c(t)= -S BD P +2(J 33 -J 22 )CBC 2 ¢ -(J 11 +J 33 -J22 )BCB, d (t) = -zbBCBS</J -J , 1 ¢-D P ¢ 2 + ( J 33 -J 22 )B S</JC</J.[START_REF] Azouz | Dynamic analysis of airships with small deformations[END_REF]

1 .

 1 where a'0(J22 -J1i)-lf (mg-B)CBS¢+Dr ( vCBC¢-BS¢ ) ao = -lf M22 (Yrse-¢) J33 (;pcocrp -BSrp -;yBSBC¢-;y¢cBS¢-e¢c¢) ( -v 2 seces¢-vBSBC¢+v¢ces¢+e¢c¢ } The second equality constraint ( 21 ) gives: . . + f36 v + j37 v+ /J s u = 0, where /3 0 = /3'o + /3'3 ao, /32 = /3 15 a1, /31 = /3 '1 + a1f3 '3 + ao /3 's, /33 =/J ' s a2 + a3 /J '3 , (28) /3 4 =/J ' s a3 , /3s = /3 ' s a4 , (29) ' o = -B¢S¢ + lff ¢C¢CB + vB SBS</J-v 2 C q)CBSB, Roll and pitch angles constant The differential equation where a= -J11 (SB), b = (J 33 -J 22 )(S¢CB)( C¢CB), c = D P (-S B ) , d =-zhBCBS¢ admits an analytic general solution.

4. 1 . 1

 11 lf/ is constant: [ </J = c s t =¢ 0 � B = e s t = 80 . lff = l/fo t (34) The first equality constraint becomes a second order polynomial equation: (35) The third equality constraint gives where 2 -v SBCBS¢ (J 22 -J11)+Dr �CBC¢ ao = -

  37 v+ [38 u = 0 with the parameters /Ji given in (29)-(31 ). u = v = w = 0, we retrieve the classical kinematics equations of the trim trajectories.. .

y

  = a y cos( If/ 0 t) +b y sin( If/ 0 t), z = -sin(B0 )u0 + cos(B0 )sin(¢0 )v0 + cos(B0 )cos(¢0 )w0, where ax = cos( B0 )u0 + sin( B0) sin( ¢0 )v0 + sin( B0) cos( ¢o )w0, by= ax, bx =cos( ¢0 )v0 + sin( ¢0 )w0, a y =-bx. (4ll) Integrating these equations, we obtain a x • ( lf/o ) b x ( lf/o ) x=-sm -s --cos -s, lf/o ) ax ( lf/o y=--sm -s --cos -s ), the curvilinear abscissa and we � uppose a uniform motion such that (43) The trajectories represented by these equations ( 44) are classical helices rotated around the vertical axe by an angle of arctan [ �: ) and have a constant curvature and torsion

Fig. 3 .

 3 Fig. 3. Trim trajectory.

  null, Vt, implies L P ¢ 0 = 0 � ¢0 = 0 . Rearranging the first equality constraint with this requirement gives: Bo C¢0S¢0 = 0, three cases are possible l[ Bo = 0 or ¢0 = 0 or ¢0 = -. 2 The first case: trim trajectories has already been studied in paragraph 4.1.1. If the roll angle is null, the following differential equations must be solved: i+Jt( a+b Bo CBI SB J = o, where a= L P !(Ix), h=-( Iz -] y -I z )!(Ix) , The case ¢0 =gives the following differential 2 equations i+ tit( a1 + a2 Bo CB I SB J + a3CB I SB= 0, where a 1 = L P !(Ix), a2 = (I z -I y + I z ) I (Ix ) , a3 = Bzb I Ix. The third equality constraint gives w = 81+82uv+ 83u + 84v+85it.

  lighter than air platform is the AS 200 by Airspeed Airships. It is a remotely piloted airship designed for remote sensing. It is a non rigid 6m long, 1.4m diameter and 8.6m3 volume airship equipped with two vectorable engines on the sides of the gondola and 4 control surfaces at the stern. Envelope pressure is maintained by air fed from the propellers into the two ballonets located inside the central portion of the hull. These ballonets are self regulating and can be fed from either engine. The engines are standard model aircraft type units. The propellers can be rotated through 120 degrees. During flight the rudders and elevators are used for all movements in pitch and yaw.

Fig. 8 .

 8 Figure 6 7 8 9 10

-Fig. 10 .

 10 Fig. 9. Case# 4.


Saïd Mammar

André Benine-Neto

Sébastien Glaser

Naïma Ait Oufroukh

Vehicle Handling Improvement by Fuzzy Explicit Nonlinear Tire Forces Parametrization

Keywords: Vehicle handling, Fuzzy control, Output feedback, LMI, BMI

This paper presents the design and the simulation test of a Takagi-Sugeno (TS) fuzzy output feedback for yaw motion control. The control synthesis is conducted on a nonlinear model in which tire-road interactions are modeled using Pacejka's magic formula. Using sector approximation, a TS fuzzy model is obtained. It is able to handle explicitly the nonlinear Pacejka lateral tire forces including the decreasing or saturated region. The controller acts through the steering of the front wheels and the differential braking torque generation. The computation of the controller takes into account the constraints that the trajectories of the controlled vehicle remain inside an invariant set. This is achieved using quadratic boundedness theory and Lyapunov stability. Some design parameters can be adjusted to handle the trade-off between safety constraints and comfort specifications. The solution to the associated problem is obtained using Linear and Bilinear Matrix Inequalities (LMI-BMI) methods. Simulation tests show the controlled car is able to well achieve standard maneuvers such as the ISO3888-2 transient maneuver and the sine with dwell maneuver.

I. INTRODUCTION

Ground vehicles experience instabilities that are difficult for the driver to control. In fact, bifurcation analysis have shown that the stability region, given for example in the sideslip angle -yaw rate phase plane is limited [START_REF] Tuan | Bifurcation in Vehicle Dynamics and Robust Front Wheel Steering Control[END_REF]. In addition its size is function of the driver input on the steering speed, the road adhesion and the longitudinal speed. Notice that instabilities are mainly due to lateral tire-road forces saturation. It is thus important de help the driver in maintaining control of the vehicle is extreme dynamics and even prevent that the vehicle enters them.

In this context, electronic stability control systems (ESC) is the subject of intense research while solutions are already available and become more and more popular on commercial vehicles in Europe. They have largely contributed during the last decade to accident and death reduction [START_REF]Grands thèmes de la sécurité routière en France[END_REF]. Today's systems act on the vehicle lateral dynamics mainly through independent wheel braking. Recent studies have demonstrated that differential braking may have a better effect on yaw dynamics than independent active wheel braking [START_REF] Zanten | The Vehicle Dynamics Control System of Bosch[END_REF], [START_REF] Hancock | A comparison of braking and differential control of road vehicle yaw-sideslip dynamics[END_REF]. Optimal strategies for braking forces allocation have been explored in [START_REF] Tjoennas | Adaptive Optimizing Dynamic Control Allocation Algorithm for Yaw Stabilization of an Automotive Vehicle using Brakes[END_REF]. In parallel, vehicle handling has been S. Mammar France, (e-mail: naima.aitoufroukh, said.mammar@iup.univ-evry.fr).

A. Benine-Neto and S. Glaser are with IFSTTAR -LIVIC Laboratoire sur les Interactions Véhicule-Infrastructure-Conducteur. 14, route de la Minière, Bât 824, 78000, Versailles, France, e-mail: (sebastien.glaser, andre.benine-neto@ifsttar.fr). also investigated through active steering [START_REF] Gerdes | Rossetter A Unified Approach to Driver Assistance Systems Based on Artificial Potential Fields Authors[END_REF], [START_REF] Mammar | Vehicle Handling improvement by Active Steering[END_REF]. Even if the mechanical linkage between them is still a limiting factor, solutions have been already implemented in series production [START_REF] Kohen | Active Steering -The BMW Approach Towards Modern Steering Technology[END_REF]. In such systems, the additional steering angle is limited and it is expected that real gain from active steering will come with steer-by-wire systems which will offer additional freedom-factors for the controller intervention [START_REF] Ono | Vehicle Dynamics Integrated Control for Four-Wheel-Distributed Steering and Four-Wheel-Distributed Traction/Braking Systems[END_REF].

Using both steering angle rate and differential braking, this paper proposes a dynamic fuzzy output feedback. The Takagi-Sugeno [START_REF] Tanaka | Fuzzy Control Systems Design and Analysis[END_REF] fuzzy formalism allows modeling of the nonlinear behavior of the lateral forces described by the pacejka formula [START_REF] Bakker | A new tire model with an application in vehicle dynamics studies[END_REF]. The nonlinearity includes the decreasing region. The dynamic output feedback formulation considered in this paper presents three main advantages: the use of only the yaw rate and the steering angle as controller input, better flexibility to formulate the stabilization conditions and the ability to handle input or state constraints and bounded disturbances. This controller uses the property of quadratic boundedness and invariant set [START_REF] Ding | Quadratic boundedness via dynamic output feedback for constrained nonlinear systems in Takagi-Sugeno's form[END_REF]. This allows the constraints that the trajectories of the controlled vehicle remain inside an invariant set. In fact, during control intervention, it is important to ensure a good safety level by bounding state variables. The chosen strategy fulfills this requirement and consists of building an invariant set for the system state. It guarantees that each trajectory that starts in the invariant set will not exceed it, hence the trajectories will be bounded inside it [START_REF] Minoiu | Driver steering assistance for lane departure avoidance[END_REF]. Some design parameters can be adjusted to handle the trade-off between safety constraints and comfort specifications. The solution to the associated problem obtained using Linear and Bilinear Matrix Inequalities (LMI-BMI) methods.

The paper is organized as follows: the next Section gives a description of the developed vehicle lateral dynamics Takagi-Sugeno model of the vehicle. The fuzzy output feedback synthesis, including the requirements concerning the quadratic boundedness, the state constraints and control limitation are then presented in Section 3. In Section 4, simulation results for the ISO 3888-2 and the sine with dwell transient maneuvers which excite the nonlinear tire dynamics are provided. The conclusions wrap up the paper.

II. VEHICLE LATERAL DYNAMICS T-S MODEL

As lateral control is concerned, a simple nonlinear model of a vehicle is obtained by neglecting the roll and pitch motions. This model includes the lateral translational motion and the yaw motion (Fig. 1). The two wheels of each axle are lumped into one located at its center. This leads to the vehicle bicycle model. The lateral forces between each tire and the road surface are added at each axle leading to two resulting forces f f ( α f ) = f y1 + f y2 and f r (α r )= f y3 + f y4 at the front and rear wheels of the bicycle model respectively. These forces which will be detailed below are function of the front and rear tires sideslip angle, denoted α f and α r respectively.

The lateral translation and rotational yaw motion equations written in the vehicle fixed frame take the following form [ mv

( β + r ) J ṙ ] = [ 110 l f -l r 1 ] ⎡ ⎣ f f ( α f ) f r (α r ) T z ⎤ ⎦ (1)
where β is the vehicle side slip angle, ψ = r is the yaw rate and T z is the yaw moment input applied by differential wheel braking. m is the vehicle mass while J is the vehicle moment of inertia. The vehicle center of gravity is located at a distance l f from the front axle and a distance l r from the rear axle. The vehicle parameters values are listed in Table I in the Appendix.

Assuming that the angles remain small, the front and the rear sideslip angles are given by:

α f = δ f - ( β + l f v r ) α r = -β + l r v r
(2)

A. Lateral tire forces model

Several types of models of the forces of tire-pavement interaction have been proposed in the literature [START_REF] Bakker | A new tire model with an application in vehicle dynamics studies[END_REF]. They are usually derived from experimental data, as for the Pacejka model, and have as parameters the adhesion, the speed v and the vertical load f ni . The shape of the lateral force is often similar from one model to another. A first linear domain for small sideslip angle allows to define a slope factor called the tire cornering stiffness coefficient. When the sideslip angle increases, the tire enters a nonlinear operating zone where the lateral force saturates. The maximum value defines the limit of the vehicle maneuverability, resulting in a loss of controllability that can cause an understeering phenomenon or an unusual oversteering which may surprise the driver.

Here, the Pacejka magic formula [START_REF] Hans | Tyre and Vehicle Dynamics[END_REF], [START_REF] Versteden | Improving a tyre model for motorcycle simulations[END_REF] is used to represent the efforts exerted on each tire. This model is based on the mathematical representation of the tire dynamic behavior using analytical functions having a particular structure. Lateral forces of front and rear tires are function of the side slip angle α i at the tire-road contact location. The effect of the camber angle is neglected. Here, the index i stands for f (front) or r (rear) :

f i (α i )= d i sin ( c i ⋅ tan -1 (b i (1 -e i )α i + e i ⋅ tan -1 (b i α i )) ) (3)
Notice that the adhesion coefficient and the normal force acting on each tire are embedded inside the parameters b i , c i , d i and e i . See [START_REF] Mammar | Vehicle Handling improvement by Active Steering[END_REF] for further details. The definition and the value of the above parameters are described in the appendix at the end of the paper.

The goal now is to achieve a Takagi-Sugeno fuzzy model which covers the entire operating domain (linear and nonlinear) of the forces [START_REF] Takagi | Fuzzy identification of systems and its application to modeling and control[END_REF].

B. Four rules Takagi-Sugeno vehicle fuzzy model

The nonlinear vehicle model is transformed into a four rules Takagi-Sugeno (T-S) fuzzy model according to the values of the front and rear cornering stiffnesses:

• if α f is m 1 and |α r | is n 1 then { f f = c f 1 α f f r = c r 1 α r • if α f is m 2 and |α r | is n 1 then { f f = c f 2 α f f r = c r 1 α r • if α f is m 1 and |α r | is n 2 then { f f = c f 1 α f f r = c r 2 α r • if α f is m 2 and |α r | is n 2 then { f f = c f 2 α f f r = c r 2 α r
The membership functions m i and n i (i = 1, 2) are determined by the approximation method of nonlinear function by linear sectors. Coefficients c f i and c r i (i = 1, 2) represent the tire cornering stiffnesses associated to each sector. In fact they represent also the slope of the limits of the sectors which include the tire forces (Fig. 2). For example, given two coefficients c f 1 and c f 2 , chosen according to the expected road adhesion and driving conditions, one can determine

the membership functions m 1 ( α f ) and m 2 ( α f )
. The evolution of the two functions m 1 and m 2 as functions of the sideslip angle are shown in Figure 3. They are obtained with numerical values: c f 1 = 1.2c f and c f 2 = 0.6c f .I ti s important to outline that this sector representation is an exact approximation of the nonlinear system. The membership functions n 1 and n 2 for the rear tire forces are obtained by the same procedure. Finally, one can write:

{ f f = [ (h 1 + h 3 ) c f 1 +(h 2 + h 4 ) c f 2 ] α f f r =[(h 1 + h 2 ) c r 1 +(h 3 + h 4 ) c r 2 ] α r ( 4
)
with

h 1 = m 1 × n 1 , h 2 = m 2 × n 1 , h 3 = m 1 × n 2 and h 4 = m 2 × n 2 .
In order to have the front and the rear sideslip angle as state vector components, let us define the state x = [α f , α r , δ f ] T and the control input u =[ δ f , T z ] T , the fuzzy system takes the form:

ẋ = 4 ∑ i=1 h i ( α f , α r ) Āi x + Bu (5)
where

Āi = ⎡ ⎣ a 11i a 12i a 13 a 21i a 22i a 23 000 ⎤ ⎦ , B = ⎡ ⎣ 1 - l f Jv 0 l r Jv 10 ⎤ ⎦ (6)
where

⎧            ⎨            ⎩ a 11i = -v l f +l r -1 v ( 1 m + l r l f J ) c ′ fi , a 12i = v l f +l r -1 v ( 1 m - l f l r J ) c ′ r i , a 21i = -v l f +l r -1 v ( 1 m -l 2 r J ) c ′ fi , a 22i = v l f +l r -1 v ( 1 m + l 2 r J ) c ′ r i , a 13 = v l f +l r , a 23 = v l f +l r .
where

c ′ fi = c f 1 for i = 1, 3 and c ′ fi = c f 2 for i = 2, 4. Similarly, c ′ ri = c r1 for i = 1, 2 and c ′ ri = c r2 for i = 3, 4.

C. Reference yaw rate tracking

Ideally, the vehicle should respond to driver's steering angle δ d as a speed depended yaw rate reference steady state value with almost constant settling time. Let T 0 be the desired transfer function between δ d and r. In order to ensure at nominal speed, the same steady state value for the controlled and the conventional car, the reference model is chosen as a first order transfer function with the same steady state gain as the conventional car. It is of the form r d = K d (v) τs+1 δ d .T h e speed dependent steady state gain is K d (v), derived from the nominal linear bicycle model, and τ = 0.2 sec.

In order to ensure that the yaw rate reference value is achieved in steady state, the integral z of the yaw rate tracking error is added as state a variable:

ż = r -r d = δ f + α r -α f l f + l r v -r d ( 7
)
This variable is thus added to the previous third order model while the desired yaw rate is considered as a disturbance. The fuzzy model is finally discretized at a sample time of T = 0.005 sec. The final fuzzy model is of the form:

x(t + 1)=∑ 4 i=1 h i ( α f , α r ) A i x(t)+Bu(t)+Ew(t) y(t)=Cx(t)+Dw(t) (8)
where

x =[α f , α r , δ f , z] T and y(t)=[r, z] T . The disturbance w(t)=r d (t) ∈ ε Q = { w ∈ R/w T Qw ≤ 1 } is bounded.
Matrices A i and B can be easily derived from equations ( 6) and [START_REF] Mammar | Vehicle Handling improvement by Active Steering[END_REF]. This discrete time fuzzy system is characterized by common B, E and C matrices. This property simplifies drastically the stability and performance conditions as only simple summations are involved.

III. DYNAMIC OUTPUT FEEDBACK FUZZY CONTROLLER

In the following, a dynamic output feedback fuzzy controller is sought. It has the form:

x c (t + 1)=∑ 4 i=1 h i ( α f , α r ) A i c x c (t)+B c y(t) u(t)=C c x c (t)+D c y(t) (9)
where x c ∈ R 4 is the controller state; {A i c , B c ,C c , D c } are matrices to be designed.

This controller uses the parallel distributed compensation (PDC) concept of the fuzzy system control. In this concept, each control rule is distributively designed for the corresponding rule of a T-S fuzzy model. Linear control theory can then be used to design controllers for each of the consequent part of the fuzzy system while ensuring the same properties for the fuzzy system.

As pointed out in [START_REF] Ding | Quadratic boundedness via dynamic output feedback for constrained nonlinear systems in Takagi-Sugeno's form[END_REF], D c is an important parameter for stabilization, and the controller structure is able to handle constraints on the input and the state. By combining ( 8) and ( 9), the augmented closed-loop fuzzy model is given by

x(t + 1)= 4 ∑ i=1 h i ( α f , α r ) Φ i x(t)+Γw(t). ( 10
)
where x =

[ x x c ] , Φ i = [ A i + BD c C BC c B c C A i c ] and Γ = [ BD c D + E B c D ] . Let Φ z = ∑ 4 i=1 h i ( α f , α r )
Φ i , the closed loop system takes the form: x(t + 1)=Φ z x(t)+Γw(t).

A. Invariant set and output feedback PDC control

Assume that there exists a quadratic function V ( x)= xT P x, where P is a symmetric, positive definite matrix that satisfies, for all x, w satisfying (10), w T Qw ≤ 1, V ( x) ≥ 1, the condition [START_REF] Alessandri | Design of state estimators for uncertain linear systems using quadratic boundedness[END_REF]:

V ( x + 1) ≤ V ( x) (11)
Consider the reachable set Λ defined by:

Λ ≜ { x(T )| x, w satisfying (10), x(0)=0, w T Qw ≤ 1, T ≥ 0} (12)
The set ε P defined by:

ε P = { x(t) ∈ R 8 | x(t) T P x(t) ≤ 1}, (13)
is an invariant set for the system (10) with w ∈ R, w T Qw ≤ 1. This means that every trajectory that starts inside ε P remains inside it for t → ∞.

The existence of such a function V ( x) means that the set ε P is an outer approximation of the reachable set Λ.

ε P is also and outer approximation of the reachable set

Λ * ≜ { x(T )| x, w satisfying equation (10), x(0) ∈ ε P , w T Qw ≤ 1, T ≥ 0} (14)
In this section the control law and the invariant set ε P are synthesized. This is achieved using BMI (Bilinear Matrix Inequalities) optimization method such that the system without the disturbance is asymptotically stable and at the same time, the reachable set for an initial state values inside the invariant set is contained in this invariant set.

B. Invariant set -quadratic boundedness

According to the previous considerations, the closed loop linear system x(t + 1)=Φ z x(t)+Γw(t) is strictly quadratically bounded with a common Lyapunov matrix

P > 0f o r all allowable w(t) ∈ ε Q ,f o rt > 0, if x(t) T P x(t) > 1 implies (Φ z x(t)+Γw(t)) T P (Φ z x(t)+Γw(t)) < xT P x, for any w ∈ ε Q .
The corresponding condition is obtained using the S-procedure and invoking the Schur complement, using that the satisfaction of w ∈ ε Q and xT P x ≥ 1 implies w T Qw ≤ xT P x.

Defining P =

[ P 1 P T 2 P 2 P 3 ] and P -1 = [ M 1 M T 2 M 2 M 3 ]
.A ssuming that P 2 and M 2 are full rank matrices and setting:

⎧     ⎨     ⎩ Dc = D c Ĉc = D c CM 1 +C c M 2 Bc = P 1 BD c + P T 2 B c Âi c = P 1 A i M 1 + P 1 BD c CM 1 + P T 2 B c CM 1 +P 1 BC c M 2 + P T 2 A i c M 2 ( 15
)
the existence of the controller is ensured if there exist matrices P 1 , P 2 , M 1 , M 2 and a positive scalar α such that the following condition holds

4 ∑ i=1 h i ( α f , α r ) ϒ i ≥ 0 ( 16
)
where

ϒ i = ⎡ ⎢ ⎢ ⎢ ⎢ ⎣ (1 -α) P 1 * * * * (1 -α)I (1 -α)M 1 * * * 00 αQ * * A i + B Dc C A i M 1 + B Ĉc B Dc D + EM 1 * P 1 A i + Bc C Âi c Bc D + P 1 EIP 1 ⎤ ⎥ ⎥ ⎥ ⎥ ⎦
(17) Notice that the matrices M 1 , M 2 , P 1 and P 2 verify:

M T 2 P 2 = I -M 1 P 1 ( 18
)
In addition, it is possible to handle constraints on the control signal and the state:

-ū ≤ u(t) ≤ ū, -Ψ ≤ Ψx(t + 1) ≤ Ψ, ∀t ≥ 0 ( 19
)
where ū =[ū 1 , ū2 ] T with ū1 > 0, ū2 > 0 and Ψ := [ Ψ1 ,..., Ψq ] T with Ψ j > 0, j = 1,...,q, Ψ ∈ R q×4 and q is the number of imposed constraints. Notice that the bounds are provided separately on each state variables or a combination of state variables. The control input limitation is verified if for a pre-specified scalar η ∈ (0, 1], the additional inequality

⎡ ⎢ ⎢ ⎣ ηP 1 * * * ηI ηM 1 * * 00Q * √ 2 Dc C √ 2 Ĉc √ 2 Dc D Z ⎤ ⎥ ⎥ ⎦ ≥ 0 (20) holds with Z ∈ R 2×2 is such that Z 11 ≤ ū2 1 and Z 22 ≤ ū2 1 .
A similar procedure can be applied for the constraints on the state variables. One can achieve from the convex property conditions (21):

∑ 4 i=1 h i ( α f , α r ) Υi ≥ 0, t ≥ 0, Ξ kk ≤ Ψ2 k , k ∈ {1,...,q} (21)
where Ξ is a symmetric matrix and

Υi = ⎡ ⎢ ⎢ ⎣ ηP 1 * * * ηI ηM 1 * * 00Q * Υi41 Υi 42 Υi 43 Ξ ⎤ ⎥ ⎥ ⎦ (22)
and

Υi 41 = √ 2Ψ ( A i + B Dc C ) Υi 42 = √ 2Ψ ( A i M 1 + B Ĉc ) Υi 43 = √ 2Ψ ( B Dc E + D )

C. Controller synthesis

Under the proposed modeling approach, the desired yaw rate could be seen as an input disturbance under which the closed-loop system should remain stable with bounded values for the state vector components. More generally, the state variables should not exceed the bounds of a "safety zone", namely α f ≤ α M f , |α r | ≤ α M r and δ f ≤ δ M f . Thus, the state vector x has to be confined to a hypercube L(Z M ) defined by the above bounds. Finally, the control input, the steering angle rate and the yaw moment, have to be bounded

δ f ≤ δ M f and |T z | ≤ T M z . According to the equation (19), control limitation is given by ū =[ δ M f , T M z ], while state limitation is given by Ψ = [α M f , α M r , δ M f ] T and Ψ = [ I 3 0
] . The PDC output feedback controller was synthesized with the following numerical values:

α = 0.02, η = 0.02, δ M f = 100 deg /s, T M z = 10KN α M f = α M r = 13 deg δ M f = 6deg
, These design parameters could be adjusted to handle the trade-off between safety constraints and comfort specifications.

The achieved Q is 5, which ensures that the constraints are verified for a disturbance of a magnitude less than 0.447rad/s at the considered longitudinal speed of 20m/s. In fact, the maximum value is constrained by [START_REF] Rajamani | Vehicle Dynamics and Control[END_REF]:

r d max = 0.85 g v .

IV. SIMULATION TESTS

In order to proof the assistance ability to maintain the dynamic vehicle stability in extreme conditions, several type of maneuvers have been defined to test the ESC systems. Among them, double lane-change manoeuvre defined in ISO 3888-2 standard and the sine with dwell transient maneuver. The controller is tested below for the two maneuvers.

A. Testing for the ISO 3888-2 maneuver

The ISO3888-2 double lane-change maneuver setup is depicted in Figure 4-a. The maneuver is carried out with and without the controller at the same speed of 80km/h.D u r i n g the maneuver, the throttle is released.

The driver initiates the maneuver by applying the steering angle shown in dashed line in Figure 4-b. Figures 4-a highlights that the controlled vehicle is able to perform the maneuver (solid line) while the uncontrolled vehicle fails (dashed line). Figure 4-d shows that the controller shares the effort on the steering angle rate an the yaw moment, respectively. In this situation the driver applied steering angle is too high (dashed line in Figure 4

B. Sine with dwell maneuver

The sine with dwell is a transient maneuver considered by NHTSA (National Highway Traffic Safety Administration) for electronic stability control evaluation. Such type of maneuver is suited for the excitation of the vehicle oversteer response. The maneuver is conducted at the same speed of 80km/h. It corresponds to a 0.7Hz frequency sine wave form with dwell steering angle of 500ms. During the maneuver, the throttle is released. Figure 7 shows both the control sharing ISO3888-2 maneuver: Front and rear tire sideslip angle and corresponding lateral forces for the uncontrolled (dashed) and the controlled vehicles (solid), with vertical offset for the uncontrolled one for better display. between the steering and the differential braking. It also shows that the realized relative yaw angle is higher which means that the vehicle is more able to change direction.

V. C ONCLUSION

In this paper the design and the test of an integrated steering and differential braking control for yaw moment generation has been described. Controlled vehicle trajectories are confined inside an invariant set. The nonlinear behavior of the vehicle dynamics are modeled using a fuzzy Takagi-Sugeno approach. An output feedback fuzzy controller constituted by four sub-controllers handles constraints on the state variables and the control inputs. Simulation tests have shown that the controlled vehicle is able to achieve the ISO 3888-2 and the sine with dwell transient maneuvers where the uncontrolled vehicle fails.

APPENDIX

Fig. 1 .

 1 Fig. 1. Vehicle model.

Fig. 2 .

 2 Fig. 2. Tire lateral force given by the pacejka model and sector based approximation.

Fig. 3 .

 3 Fig. 3. Membership functions m 1 and m 2 associated to the front tire contact forces.

Fig. 4 .

 4 Fig. 4. ISO3888-2 maneuver: Yaw rate, steering angle, steering angle rate and trajectory for the uncontrolled and the controlled vehicles.

Fig. 5 .

 5 Fig. 5. ISO3888-2 maneuver: Coefficients h i reflecting the contribution of each sub-controller for the controlled vehicles.

  Fig. 6.ISO3888-2 maneuver: Front and rear tire sideslip angle and corresponding lateral forces for the uncontrolled (dashed) and the controlled vehicles (solid), with vertical offset for the uncontrolled one for better display.

Fig. 7 .

 7 Fig. 7. Sine with dwell maneuver: Steering rate and differential braking control inputs, vehicle absolute yaw angle. Dashed line for the uncontrolled vehicle and solid line for the controlled one.

  and N. Ait Oufroukh are with Universitéd ' Évry Val d'Essonne, France. IBISC: Informatique, Biologie Intégrative et Systèmes Complexes -EA 4526, 40 rue du Pelvoux CE1455, 91020, Evry, Cedex,