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An LP V/H∞ integrated Vehicle Dynamic ControllerSoheib Fergani, Olivier Sename, Luc Dugard

To cite this version:Soheib Fergani, Olivier Sename, Luc Dugard. An LP V/H∞ integrated Vehicle Dynamic Controller.IEEE Transactions on Vehicular Technology, Institute of Electrical and Electronics Engineers, 2016,65 (4), pp.1880-1889. 10.1109/TVT.2015.2425299. hal-01230203

1

An LPV/H∞ integrated Vehicle Dynamic Controller

S. Fergani1∗, O. Sename1, L. Dugard1

Abstract—This paper is concerned with the design andanalysis of a new multivariable LPV/H∞ (Linear Pa-rameter Varying) robust control design strategy forGlobal Chassis Control.The main objective of this study is to handle criticaldriving situations by activating several controller subsys-tems in a hierarchical way. The proposed solution consistsindeed in a two-step control strategy that uses semi-active suspensions, active steering and electro-mechanicalbraking actuators.The main idea of the strategy is to schedule the 3 controlactions (braking, steering and suspension) according tothe driving situation evaluated by a specific monitor.Indeed, on one hand, rear braking and front steeringare used to enhance the vehicle yaw stability and lateraldynamics, and on the other hand, the semi-active suspen-sions to improve comfort and car handling performances.Thanks to the LPV/H∞ framework, this new approachallows to reach a smooth coordination between thevarious actuators, to ensure robustness and stability ofthe proposed solution, and to significantly improve thevehicle dynamical behavior.Simulations have been performed on a complex fullvehicle model which has been validated using data ob-tained from experimental tests on a real Renault MéganeCoupé1. Moreover, the suspension system uses Magneto-Rheological dampers whose characteristics have beenobtained through experimental identification tests.A comparison between the proposed LPV/H∞ controlstrategy and a classical LTI/H∞ controller is performedusing the same simulation scenarios and confirms theeffectiveness of this approach.

Index Terms—Global chassis Control, Braking, Steering,semi-active suspension, LPV, monitoring, H∞, LMI.

I. INTRODUCTION

A. Motivations

Road safety has been an international stake in the lastdecades. A close examination of road traffic accidentdata (1.24 million deaths and more than 50 millionsof injuries in 2013, according to the World HealthOrganisation) reveals that the loss of vehicle controlis largely responsible of road accidents. Therefore, en-hancing driving characteristics by ensuring stability incritical situations (i.e. safer vehicles) has been recently

1 GIPSA-lab, Control Systems Department (former LAG), Greno-ble University - Grenoble INP, ENSE3 - Domaine UniversitaireBP46, 38402 Saint Martin d’Hères - Cedex FRANCE∗ corresponding author: soheib.fergani@gmail.com1The authors would like to thank their colleagues of the MIPS

laboratory, Mulhouse, France, especially Prof. Michel Basset forproviding them with experimental data for validating the proposedstrategies.

the main issue for both academical and industrialcommunities.Moreover, the increasing request of the customers interms of driving performances has led several auto-motive manufacturers to seek new efficient strategiesthat prevent vehicles from drifting, spinning or rollingover, and therefore, that improve stability and drivingcharacteristics in critical situations.So, a new trend is to develop multivariable globalchassis control strategies involving several actuatorsand, thus enhancing the car’s dynamics. The purposeof the proposed integrated multivariable vehicle controlis to use, in a collaborative way, the available actua-tors acting on the vehicle dynamics. Such a MIMOcontrollers synthesis allows to adapt the vehicle be-havior to the driving situations. It also focuses hereon improving both comfort and safety objectives bycoordinating the use of the braking, steering and semi-active suspensions subsystems.

B. Related works1) Vehicle stability Steering/Braking control: In thelast decade, lots of works dealing with the control ofautomotive dynamics have been proposed, based onSISO control solutions as in [1], [2] where only thebraking control is used separately to improve lateraland yaw behavior of the vehicle and to tackle criticaldriving situations.More recently, some studies have been carried out onthe multivariable control design for the global chassisstabilization and vehicle dynamics improvement. First,research on vehicle stability and handling has focusedon improving the lateral dynamics behavior usingbraking and steering actuators. Indeed, [3] proposes anoptimal non linear vehicle control based on individualbraking torque and steering angle with an online con-trol allocation to improve vehicle performances.Some of the authors’ results in [4], [5], presentscheduling policies of braking /steering actuators, astep towards a full coordinated control. New strategiesusing steering and braking actuators are detailed in arecently published book chapter [6].However, it has been proven that steering and brakingactuators are not sufficient to improve all the vehicledynamical behaviours. Indeed, road holding character-istics and passenger comfort, which are crucial com-mercial arguments for the automotive manufacturers,can not be handled when using only steering and brak-ing actuators. Both industry and academic communities

have therefore been interested by considering verticaldynamics through suspension systems to improve theoverall vehicle dynamics.2) Road holding and passengers comfort through sus-pension systems control: The importance of suspen-sion systems in the vehicle dynamical behavior hasattracted much interest in the last years. Since thesuspension system ensures the link between the chassisand the wheels, it plays a key role in the automo-tive vertical dynamical attitude. The control strategiesdeveloped for such systems allow to achieve the per-formance objectives concerning the passenger comfortand the car road-holding.Many studies have been dedicated to the suspensioncontrol. A summary of some recent suspension con-trol strategies is presented in [7]. Also, the book [8]presents a detailed description of various suspensionsystems and summarizes several control approachesapplied to semi-active suspensions such as ground-hook, skyhook, ADD (Acceleration Driven Damping)and LPV control strategies. A lot of recent works(see e.g. [9], [10] and references therein) describe newreliable suspension control strategies to enhance bothsafety and comfort of passengers.The authors also developed various control strategiesfor suspension system to enhance passenger comfortand road holding as in [11] where an LPV controlapproach for comfort and suspension travel improve-ments of semi-active suspension systems is presented.The study of the each on of the suspension, steeringand braking systems has shown that an independentdesign for each one of them may lead to performanceconflicts due to the different interactions between thevehicle dynamics. The solutions of this problem isto treat all the vehicle dynamics in the same controlstrategy.3) Global chassis control (GCC) strategy : Thestudy of each of the suspension, steering and brakingsystems has shown that an independent design foreach of them may lead to performance conflictsdue to the different interactions between the vehicledynamics. The solution of this problem is to treat allthe vehicle dynamics in the same control strategy.

Recently, several works have considered integratedcontrol strategies that allow to manage multi-objectiveperformances through all the available actuators andsensors used in these control tasks. The interest forthis type of vehicle control has increased in severalacademic and industrial research centers. As a resultof the interactions between the vertical and lateraldynamics, as previously mentioned, new vehiclecontrol methodologies including suspension andbraking or steering actuators have been presented:in [12] a nonlinear backstepping control designfor anti-lock braking systems assisted by an active

suspension, and a hierarchical fuzzy-neural control ofanti-lock braking system and active suspension in [13].

A detailed survey of all these studies shows theimportance of providing new global chassis controlstrategies. The previously cited approaches yield goodresults, therefore we tried to combine the strength ofthe mutlivariable control for the multiple performanceobjectives with the adaptation of the use of the actua-tors to the driving situations that influence considerablythe dynamical behavior of the vehicle.Furthermore, the authors have developed a new robustcontrol structure to enhance the overall vehicle dy-namics using a coordination approach for the steering,braking and magneto-rheological semi-active dampers.A robust LPV/H∞ based on LMI’s resolution in theLPV framework for those subsystems is developed in[14]. Also, some first results concerning the robustmultivariable control using the three types of actuatorsare established and validated in [15].

C. Paper contributions and structure

In this study, a new LPV/H∞ control strategy(see Fig. 1) provides a hierarchical collaborativecoordination between the actuators of semi-activesuspension, steering and braking subsystems toenhance the vehicle dynamics, and prevents conflictsin terms of performance objectives. On one hand,this GCC (Global Chassis Control) strategy combinesthe monitoring of the driving situation and thecorresponding coordination of the actuators; on theother hand, the LPV/H∞ frame allows a smooth andflexible use of the actuators, with adaptation to thedriving situation, while guaranteeing the robustness ofthe proposed control. The controllers design focuseson enhancing the overall vehicle dynamics, namely,vertical, lateral and longitudinal dynamics.Moreover, in this work, semi-active suspensionsare considered (while in [16] active systems wereused), which suits better industrial requirements(energy saving). This strategy is adapted to driver-aid,depending on the dangerousness of the situation: first,it selects the best actuators coordination to avoidaccidents, and second, it limits the unnecessary use ofthe actuators for energy saving sake.

More precisely, this paper presents, in a unified anddetailed way, some results of previous authors studiesand enhances those studies by adding the main follow-ing contributions:• The use of the real data input collected on a

real car (Renault Mégane Coupé) running on atrack to evaluate the effectiveness of the proposedLPV/H∞ strategy in a more realistic simulationframework.

• The use of the semi-active Magneto-RheologicalDampers (MRD) for the suspension control im-plementation. Such dampers allow to achievegood comfort, to enhance the road holding andto keep a safety suspension deflection.

This strategy is summarized in the following imple-mentation scheme (see Fig. 1) including the vehicle’smodel, the monitoring approach and the subsystemscontrollers.

λrfr

ψ

zdef

v

uij

Tbrj

δSteer Input

zr

v

δ0

λrr

λrl

ψref

+

−

Rs

Rb

Rs

zdefijuij

+

+

eψ

Rs

δ+

v

δ0

ψRb

Tbrj

Σfull

Suspension controller

Steering controller

Braking controllerMonitor

Rs and Rb: scheduling parameters

Road profil

GCC

Clipped Strategy

Fig. 1: Global chassis control implementation scheme.

The paper is organized as follows:section 2 presents the overview of the main contri-bution of the paper which is the coordination betweensemi-active suspension, steering and braking actuators,and the synthesis of different controllers to enhance ve-hicle performances and attitude. The control synthesisfor the braking/steering subsystem (resp. suspensionsubsystem) and the performance analysis through fre-quency domain simulations are detailed in sections 3(resp. 4).In Section 5, time domain simulations are performedon a complex nonlinear full vehicle model equippedwith semi-active suspension MRD. It also emphasizesthe contribution of the proposed LPV strategy bycomparing it to the LTI strategy. Conclusions anddiscussions are given in the last Section.

Paper notations:For modeling and simulation purposes, the follow-ing vehicle parameters and notations are adopted.Throughout the paper: indices i = f, r and j =l, r are used to identify the vehicle front, rear andleft, right positions respectively. Also, since the fullvehicle model is used, some notations will appear.Index s, t holds for suspension and tire forces re-spectively. x, y, z holds for forces and dynamics inthe longitudinal, lateral and vertical axes respectively.

Then let v =√v2x + v2

y denote the vehicle speed,Rij = R − (zusij − zrij ) the effective tire radius,m = ms + musfl + musfr + musrl + musrr thetotal vehicle mass, δ = δd + δ+ is the steering angle(δd, the driver steering input and δ+, the additionalsteering angle provided by steering actuator and Tbijthe braking torque provided by the braking actuator(see Section II).

II. A NEW GLOBAL CHASSIS CONTROL STRATEGY:SUPERVISION AND SYNTHESIS

This section presents the main contribution of thisstudy, namely, the multivariable Global Chassis Con-trol (GCC) involving front active steering, rear brakingand semi-active suspension (see Fig.1). Such a strategy,preliminary introduced by the authors in previousworks (see [16]), involves 2 monitoring parametersRb and Rs, used to evaluate the dangerousness of thedriving situation and to schedule the control actions.

Steering controller

Braking controller

Suspension controller

Coordination strategyMonitoring system

Renault Mégane Coupé

Fig. 2: General structure scheme

The main idea is to synthesize two controllers, onededicated to the lateral dynamics and the other tothe vertical dynamics, that will be coordinated thanksto the scheduling parameters Rb (braking) and Rs(suspension and steering). The controllers synthesis ispresented in the following sections.

A. Driving situation monitoring

The monitoring of the driving situation has been se-lected following [16] from the longitudinal slip ratioof the rear wheels (sij), since it considerably affectsthe yaw stability and the car handling attitude.• Braking monitor:

Rb = minj=l,r

(rbj ), (1)

is a function of the absolute value of the slipratio (|srj |). rbj is defined as a relay (hysteresis

like) function: → 0 when ’on’, → 1 when ’off’.The switch ’on’ (resp. ’off’) threshold is s+ (resps−).When the slipping is low, the vehicle is in anormal situation, hence Rb → 1. When the slipratio raises and becomes greater than s+, a criticalsituation is detected, then Rb → 0. Since Rb isfunction of the slip ratio, s+ and s− are chosenaccording to the tire friction curve. Here (and ina general case), s+ = 9% and s− = 8%, in orderto delimit the linear and peak tire friction forcewith the unstable part of the tire (see [17]).

B. Classification of the driving situations

Based on the previously defined driving situationmonitor Rb, the other varying parameter Rs allowsto classify these driving situations depending on thedangerousness and on the degree of emergency underwhich the vehicle is running. This parameter Rs isdefined as follows:

Rs

= 1 when 1 > Rb > R2

crit

=Rb−R1

crit

R2crit−R1

critwhen R1

crit < Rb < R2crit

= 0 when 0 < Rb < R1crit

(2)

When Rb > R2crit(= 0.9), i.e. when a low slip

(< s−) is detected, the vehicle is not in an emergencysituation and Rs is set to 1. When Rb < R1

crit(= 0.7),i.e. when a high slip occurs (> s+), a critical situationis reached and Rs is set to 0. Intermediate values ofRb correspond to intermediate driving situations.

The Rb and Rs varying parameters are used (as de-tailed later in the design step) to schedule the useof the Active steering, Semi-Active suspension andElectro-Mechanical Braking actuators according to thedriving situation and to optimize their operating rangeas described below, and summarized in Fig. 3.

Rb

NormalSituation

Intermediate

Situation

CriticalSituation

Steering Braking Suspension

Emergency Level

Of

The driving

Situation

Comfort

Objectives

Smooth transition

between performance

Objectives

Roadholding

Objectives

0

Rs = 1Steering action

strongly

RsAdaptation to the driving

Rb

SituationsMore steering Less Braking

Rb → 0

Braking actionAllowed

Rb → 1Braking action

strongly

Rs = 0

Steering action

Fully Allowed penalized

penalized

Fig. 3: Actuators monitoring and scheduling strategy

Normal situation: (Rs = 1, Rb→ 0) the drivingcruise goes smoothly, (no emergency situations).Since there is no risk of wheel locking, as Rb→ 0 therear braking torques is not limited. The semi-active

suspension is tuned in order to preserve the passengerscomfort, without deteriorating the road holding (i.esoft suspension damping), thanks to the schedulingparameter Rs = 1. Also, since the driving situationis safe, no corrective steering action is needed tostabilise the vehicle.

Intermediate situation: (Rs , Rb) As thedriving situation becomes dangerous, the valuesof the scheduling parameters Rs and Rb change.The tire forces approach the non linear zone ofthe tire characteristic. As a result, the value of themonitor Rb starts to rise and the braking torquesare penalized to prevent the wheel locking. At thesame time, the varying parameter Rs decreases andan a corrective steering action is allowed to help thedriver to overcome this situation. Also, the semi-activesuspension characteristics are changed smoothly, fromsoft to hard, depending on the value of Rs, to furtherimprove the car road holding without deteriorating thepassengers comfort.

Critical situation: (Rs = 0, Rb→ 1) When a danger-ous situation is detected through the braking monitorRb = 1 (in terms of longitudinal tire slip), the brakingtorques are limited accordingly in order to bring backthe forces into the linear stable zone of the tire char-acteristic. As Rs reaches zero, the maximum additivesteering angle is generated and the semi-active MRdampers are tuned to be "hard" in order to ensurea good roadholding (small wheel rebound). This willhelp the driver to overcome the critical driving situationand to prevent the vehicle from imminent accidents.

C. Global chassis controllers design synthesis

The scheme in Fig. 1 shows the proposed 2-step GCCLPV/H∞ strategy. The first one is dedicated to thefront steering/ rear braking controller, that aims atimproving the yaw stability and the lateral dynamics.The other one corresponds to the 4 semi-active MRDsuspension system, to enhance the vertical behavior(comfort/roadholding performances of the car). It isworth noting that the coupling effects are handledthrough the scheduling parameter Rs and thanks to an"anti-roll" action of the semi-active suspension.The main strategy is to adapt the control action to thedriving situation as previously presented in Fig.3, usinga self-scheduled controller function of Rb and Rs.This will be achieved thanks to a good coordinationand communication between the actuators of Activefront Steering, Electro-Mechanical rear Braking, andthe Semi-Active MRD Suspension systems.

III. FIRST STEP: THE BRAKING/STEERINGCONTROL PROBLEM FORMULATION

The LPV/H∞ controller synthesis for the brak-ing/steering subsystems is achieved, based on the ex-tended bicycle model.

A. Extended lateral bicycle vehicle model: controloriented

The model describes the lateral dynamics of the ve-hicle. It has been used in many studies in order tosynthesize braking and steering control to enhanceseveral dynamical behaviours such as the yaw rate, thelateral acceleration and the lateral sideslip dynamics(more details see [15]).

B. The LPV/H∞ braking/steering controller synthesismethod:

The following general control configuration (includinggain scheduled weighting functions) is considered:

Extended Bicycle

ModelGCC(Rb,Rs)

Weψ

WTbrj (Rb)

Wδ+(Rs)

ψref +

−

Wvy

Tbrj , δ+

ψ

z1

z2

z3

z4

Fig. 4: Generalized plant for braking/steering controlsynthesis.

where: Weψ= 10 s/500+1

s/50+1 , is used to shape the yawrate error (eψ = ψref − ψ) and Wvy = 10−3,attenuates the lateral acceleration. Also, WTbrj

(Rb) =

Rbs/10$+1s/100$+1 , attenuates the yaw moment control in-

put according to the value of Rb and Wδ+(Rs) =

Rss/κ+1s/10κ+1 , attenuates the steering control input ac-

cording to the value of Rs.where $ (resp. κ) is the braking (resp. steering)actuator cut-off frequency.

The controller is chosen to be scheduled by thevarying parameters Rb and Rs (according to thediagram given in Fig.3) in order to achieve thefollowing objectives:• Normal situation: The tire force is in the linear

friction zone, i.e. there is no risk of wheel locking;so Rb → 0 and the weighting function gain ofWTbrj

is chosen to be low. Therefore, the brakingcontrol is allowed to stabilize the vehicle. At the

same time Rs = 1, the gain of the weightingfunction on the steering control is high and noadditive steering angle is necessary.

• Intermediate situation: When the driving situ-ation becomes more dangerous, the gains of theweighting functions on the braking and steeringactions change to cope with the needs for thevehicle stabilization. Indeed, the braking actionis more and more reduced to avoid the wheelsskidding, while a more corrective steering angleis supplied to help keeping the vehicle stable.

• Critical situation: When a high slip ratio is de-tected, Rb → 1; the gain of the weighting functionis increased to deactivate the braking torques andto prevent the wheels from locking. Then, thevalue of the varying parameter Rs is set to 0,the steering weighting function is not penalizedany more and a maximum corrective action by thesteering actuators is allowed to compensate for thelack of braking, and to preserve the handling andstability of the vehicle. This may help the driverto overcome the critical driving situations.

The corresponding LPV generalized plant is modeledas:

Σ(R(.)) :

xzy

=

A(Rs, Rb) B1(Rs, Rb) B2

C1(Rs, Rb) D11(Rs, Rb) D12

C2 0 0

xwu

(3)

where x includes the state variables of the systemand of the weighing functions, w = Fdy andu = [δ+, Tbrl , Tbrr ] are the exogenous and controlinputs respectively;z = [z1, z2, z3, z4] =[Weψ

eψ,Wvy vy,WTbrj(Rb)Tbrj ,Wδ+(Rs)δ

+] holdsfor the controlled output, and y = ψref (v) − ψ isthe controller input, where ψref (v) is provided by areference bicycle model.Notice that, the LPV model (3) is here affine w.r.tthe parameters Rs and Rb and can be described as apolytopic system, i.e. a convex combination of the sys-tems defined at each vertex formed by PR(.), namelyΣ(R(.)) and Σ(R(.)). The controller is then a convexcombination of 4 vertex controllers obtained at themin/max values of Rb/Rs. From the affine generalizedplant in Fig. 4, an LPV polytopic controller is designedin the framework of the quadratic stabilisation, asexplained for instance in [18].

IV. SECOND STEP: THE SUSPENSION CONTROLPROBLEM FORMULATION

The control of the semi-active suspension system issynthesized using the classical 7-DOF vertical model(see [15]) in order to handle the trade-off betweenthe chassis motion (comfort) and the roll one (han-dling). Here, a vehicle equipped with 4 MR semi-active dampers is considered. As explained in section

4.3, the MR damper is a non-linear component withdissipative capability used in automotive suspensioncontrol systems, where the damping property variesaccording to the applied magnetic field. Such a damperis able to provide adaptive performances in terms ofcomfort and road holding.

A. Full vertical vehicle model: control oriented

This model includes the vertical dynamics of thechassis zs, the vertical motions of the wheels zusij ,the pitch φ and roll θ (more details in [15]).

B. LPV/H∞ suspension controller synthesis:

The control of suspension systems aims at enhancingthe vertical dynamics of the vehicle in order toachieve frequency specification performances, see [8]and [19]. Here the control objectives are orientedtowards bounce and roll motions, characterized bythe frequency-domain weighting functions in the H∞control framework (see Fig. 5).

ΣgvWu

Ks(Rs)uH∞ij

z1

z2

zdefij

z3

Wzs(Rs)

Wθ(1−Rs)

Full vertical

linear model

Fig. 5: Suspension system generalized plant.

where Wzs(Rs) = Rss2+2ξ11Ω11s+Ω11

2

s2+2ξ12Ω12s+Ω122 is shaped

in order to reduce the bounce amplification of thesuspended mass (zs) between [0, 12]Hz, when Rs ishigh.Wθ(Rs) = (1 − Rs)

s2+2ξ21Ω21s+Ω212

s2+2ξ22Ω22s+Ω222 attenuates the

roll amplification in low frequencies, when Rs is low.Wu = 3.10−2 is set to shape the control signal.

This control design schedules the use of the semi-active dampers and the vehicle performance as follows:

In Normal situation: Rs = 1, the semi-activesuspension control enhances the passenger comfortobjectives by using a high gain of the weightingfunction on the chassis displacement zs. Theundesirable vibrations of the chassis are then absorbedby the MR semi-active dampers which are tuned tohave a soft damping characteristic.

Intermediate situation: when the driving situationchanges, the varying parameter Rs decreases, which

increases the weighting on the roll dynamics of thecar caused by the lateral load transfers. Thereforethe suspension control modifies the performanceobjectives from passenger comfort to roadholding.The LPV framework used in the proposed strategyensures a smooth and efficient transition between theseperformance objectives while ensuring the stabilityconditions.

In Critical situation: Rs = 0, the semi-activesuspension control acts to further improve theroadholding. The weighting on the chassis motion isrelaxed since the passenger comfort is no longer thepriority, and a high penalization on the roll motion isset to reduce the load transfer that may lead to vehicleinstability (close to accident).The configuration of the proposed LPV/H∞ ensuresthe appropriate help to the driver by monitoring thedriving situation and the related vehicle dynamics.The main purpose is to preserve the passenger safetyand to help to overcome the different emergencieswhile facing such situations.

Remark 1:The selection of the parameters of the weighting func-

tions is a key step in H∞ control. Usually, they arechosen using empirical rules, thanks to the automotiveengineers experience but it doesn’t guarantee any op-timal value for these parameters. Here, one followsthe methodology described in [20] where a geneticalgorithm was used to optimize the parameter’s valuesthat minimize a criterion representative enough of thevehicle vertical performances in terms of comfort androadholding.According to Fig. 5, the following parameter dependentgeneralized plant (Σgv(Rs)) is obtained:

ξzy

=

A(Rs) B1(Rs) B2

C1(Rs) D11(Rs) D12

C2 0 0

ξwu

(4)

where ξ = [χvert χw]T is the state vector ofthe system plus the state vector of the weightingfunctions; z = [z1 z2 z3]T ; w = [zrij Fdx,y,z Mdx,y]T ;y = zdefij ; u = uH∞

ij , i = f, r and j = l, r; Fdz isthe vertical disturbance and Mdz is the disturbancemoment along the z-axis.

The LPV system (4) includes a single schedulingparameter (Rs) and can be described as a polytopicsystem after some relaxations, i.e, a convex combina-tion of the systems defined at each vertex of a polytopedefined by the bounds of the varying parameter.The LPV/H∞ suspension controller synthesis is ob-tained thanks to LMI’s resolution of the control prob-

lem following the mathematical development given in[21].Remark 2: Since semi-active suspensions are consid-ered, the LPV controllers are clipped in order to copewith the damper constraints

C. The semi-active suspension control implementation

The application of the proposed LPV control to theconsidered semi-active suspension is achieved here,for simplicity, using the clipped strategy (see [8]). TheFig. 6 shows the experimental characteristics of theMR dampers obtained in collaboration with colleaguesfrom ITESM, Monterrey, Mexico (see [22]). Givena deflection speed (zdef ) and a desired controlleddamper force F ∗, the clipped approach consists inprojecting F ∗ onto the admissible force domain, ifnecessary, to get F⊥ .

F ∗1

F⊥1

F ∗2

F⊥2

F ∗3 = F⊥3

F [N ]

zdef

Realistic MR damper force

Cmax = 7282

Cmin = 881

Fig. 6: Illustration of projection principle of the semi-active controlled damper model (F ∗1 and F ∗2 are out ofthe allowed area and F ∗3 is inside)+ the MR damperforce with bi-viscosity "Cmin = 881, Cmax = 7282"(for more details see [23])

V. SIMULATION

In this section, the complete and validated non linearmodel of the vehicle used for simulation purpose isrecalled (a first version of this model is availablein [16]). Then, 2 different simulation scenarios arepresented and the corresponding results are analysed.

All along the paper results, the proposed LPV/H∞VDC, denoted as ’LPV’, will be analyzed and com-pared to the Renault Mégane Coupé car (withoutcontrol denoted "Open Loop") and, for sake of com-pleteness, with the standard LTI/H∞ design of bothactive steering/ braking controller and semi-active sus-pension controller (without scheduled gains), denotedas ’LTI’, which was achieved by solving the previousH∞ problems with the values of the varying parameterfrozen at Rs = 0.1 and Rb = 0.9 (near a criticalsituation).

A. The full non linear vehicle model of a real RenaultMégane Coupé

The parameters used for the simulation were obtainedby experimental identification of the physical param-eters of the "Renault Mégane Coupé" at the MIPSlaboratory in Mulhouse, France.Also, the non linear model used for the simulationspurposes was validated by an experimental proceduremade on a real Renault Mégane Coupé, through aMoose test performed on a real track.

B. Simulation results: a first scenario

In this case, simulation results are presented to em-phasize the improvements of LPV closed-loop control(denoted " CL LPV semi-active") compared to theopen loop results (denoted as "Open Loop") and,for sake of completeness, to the standard LTI/H∞design of both Active Steering/ Braking controller andsemi-active suspension controller (without scheduledgains),(denoted as "CL LTI", which was achieved bysolving the previous H∞ problems with the valuesof the varying parameters frozen at Rs = 0.1 andRb = 0.9.The following scenario is considered. When the vehicleruns at 100km/h in straight line, the following eventsoccur: from t = 0.5s to t = 1s: a 5cm bump on the leftwheels, then the driver perform a double line changefrom t = 2s to t = 6s, and finally another 5cm bumpon the left wheels, during the manoeuvre, t = 3s to t =3.5s. a lateral wind occurs at vehicle’s front, generatingan undesirable yaw moment, is considered t = 2.5s tot = 3s In this scenario, for the robustness analysis, theroad is considered as wet (µ = 0.5, the road adherenceparameter), which reduces the road/tire adhesion andthe lateral tire contact forces.The resulting monitoring signals Rb (see Eq.(1)) andRs (see Eq.(2)) are shown in Fig. ?? and justify theLPV framework of the strategy.

Fig. 7: Monitoring signals

The varying parameters Rb and Rs allow to activate,limit or deactivate the control action when required(for braking and steering actuators). Let recall thatthe Rs scheduling parameter depends on the value ofRb, which itself depends on the slip ratio dynamics.These parameters are very important since they define

the behavior of the vehicle subject to critical drivingsituations. They will be used to provide the driverwith the necessary assistance, through the steering,braking and suspension subsystems.

Fig. 8: Yaw rate

1) Lateral dynamics behavior analysis: It can beensee from Fig. 8 that the proposed LPV/H∞ strategyenhances better the lateral dynamics, here, the vehicleyaw tracking. Compared to the LTI/H∞ controller, itgives good results in terms of vehicle lateral stability.Remark 3: Simulations using an extended bicyclemodel with the driver input have given the "ideal"reference vehicle to be tracked by the vehicle (blackdashed line, see Fig. 8). It helps to compare and toemphasize the improvements brought by the proposedLPV/H∞ strategy

Fig. 9: Vertical chassis displacement zs

2) Vertical dynamics behavior analysis: The verticalmotion of the chassis is shown in Fig. 9. The LPV/H∞controller improves the vertical dynamics better thanthe LTI/H∞ one does. The chassis displacement isconsiderably reduced by the proposed strategy. Thisenhances the passengers comfort while driving onuneven roads.Fig. 10 represents the improvement brought in term

Fig. 10: Roll motion of the chassis θ

of the load transfer mitigation. The roll motion is well

attenuated which, in addition to enhance the vehiclestability, ensures a good road handling of the vehiclerunning in dangerous driving situations. It is seen thatthe use of the semi-active suspension control in thecoordinated "LPV/H∞" strategy (with hierarchicalactivation of the different actuators depending on thedriving situations needs) gives better results than inthe "LTI" case.

3) Actuators dynamics behavior analysis: In additionto enhancing the vehicle various dynamics, theproposed LPV/H∞ improves the use of the actuators(electromechanical braking, active steering and semi-active suspensions) considered for the car under study.The following figures show interesting results for theactuators activation.

Fig. 11 and Fig. 12 show the braking torques

Fig. 11: Rear right Break-ing torque.

Fig. 12: Rear left Breakingtorque.

provided by the vehicle to perform the previouslydefined scenario. The braking torques provided by theLPV/H∞ controller are depicted in red. The torquesare clearly much lower than those provided in the LTIcontroller case (blue curves), that saturate. Moreoverthe use of the LPV/H∞ strategy avoids wheel locking:Fig. 12 shows that for the LTI case, the longitudinalslip ration λrl reaches the 100% value which meansthat the left rear wheel is locked.

Fig. 13: Steer control input

Therefore the "LPV coordination strategy" can helpthe driver to keep the vehicle stable with a minimumeffort. Indeed the steer control considerably decreasesin the "LPV" case, compared to the "LTI" case, and isactivated only when the driving situation is dangerousenough.

Finally, Fig. 14 shows the force/deflection

-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1-8000

-6000

-4000

-2000

0

2000

4000

6000

8000

Time [s]

F damp

erSemi-ActiveCmin=881Cmax=7282

Fig. 14: Damper force/deflection

characteristic of the controlled semi-active suspension.As explained previously, the semi-activeness isobtained by the "Clipped Strategy" that takes intoaccount the min and max damping limits of the MRdampers, namely Cmin = 881 and Cmax = 7282.

Remark 4: In the previous simulations, the LTI controlstrategy gives good results. However, since the rearwheels lock during the manoeuvre (see Fig. 12) leadingto a very high risk of loss of manoeuvrability andsafety degradation, the LPV control appears to be avery efficient way to deal with the braking issues.Moreover, it enhances performances and stability, us-ing the previously presented integrated control strategy.Furthermore, the LPV controller uses the actuators ofbraking, steering and suspension in a coordinated wayto enhance the overall vehicle dynamics and to copebetter with the actuators characteristics and limitations.The improvements brought by the proposed strategycompared to the LTI control case are quantified in Ta-ble. I by calculating the RMS (Real Mean Square valueof the signals) values of the different car dynamicssignals.

Signals improvement % Vehicle dynamicszs 11 Chassis displacementθ 13 Roll motionψ 19 Yaw ratey 32 Lateral accelerationTbrl ;Tbrr 68; 74 Braking torquesδ 86; Steering angleFsij 27, 31, 23; 28 Suspension forces

TABLE I: Performances evaluation

The comparison shown in this Table. I proves theefficiency of the proposed solution for this drivingscenario.

C. Simulation results: a second scenario.

This scenario uses experimental data obtained formodel identification. Indeed, a test (of the realuncontrolled Renault Mégane Coupé car) was firstperformed by a professional driver on a real racetrack. This circuit includes a left bend and then anobstacle avoidance (emergency situation) to determinehow well a vehicle evades a suddenly appearing

obstacle.In the considered simulation case, the focus is puton the "Moose" test only (performed at a velocity of90km.h−1) in order to assess the efficiency of thedesigned controllers for obstacles avoidance.The "driver" inputs (i.e. the steering angle and thelongitudinal speed) are considered as external inputsof the NL closed-loop model for the simulationof the LPV control. The closed-loop simulationresults obtained from real input data (denoted here as"LPV VDC") are then compared with the experimentalones (denoted here as "Measurement passive vehicle").

The resulting varying parameters, Rs and Rb, thatschedule the coordination of the 3 actuator’s con-trollers (Semi-Active Suspension, Active Steering andElectro-Mechanical Braking) are shown on Fig. ??.These parameters have complementary values, whichis coherent with the previously presented monitoringstrategy.

20 25 30 35 40 45 50 55 600

0.2

0.4

0.6

0.8

1

t [S]

Ma

gn

itu

de

Rs

20 25 30 35 40 45 50 55 600

0.2

0.4

0.6

0.8

1

t [S]

Ma

gn

itu

de

Rb

Fig. 15: Monitoring Rs and Rb signals

20 25 30 35 40 45 50 55 60−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

t [S]

ψdt

Yaw rate [rad/s2]

Measurement passive vehicleLPV VDC

Fig. 16: Yaw rate ψ

Remark 5: The passive vehicle dynamical behaviourspresented in this section were measured on the realvehicle (Renault Mégane Coupé) while performingthis scenario on a real circuit path.

Fig. 16 shows the yaw rate behavior of the vehicleusing the proposed LPV/H∞ control compared tothe passive vehicle behavior. One can notice that theyaw rate dynamics of the vehicle are well improvedeven if the vehicle is running with a quite highvelocity (90km.h−1) on the left bend when avoidingthe obstacle.

Fig. 17 shows the improvement of the roll velocity.Indeed, using the designed LPV/H∞ controllers

20 25 30 35 40 45 50 55 60−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

t [S]

θdt

Roll velocity [rad/s]

Measurement passive vehicleLPV VDC

Fig. 17: Roll velocity of the chassis θ

that coordinate the use of the semi-active suspension,steering and braking, the roll motion considerablyreduces (47% less than that of the passive car). It isobvious that the vertical dynamics are better enhancedusing an LPV/H∞ robust controller in emergencysituations.

20 25 30 35 40 45 50 55 60−150

−100

−50

0

50

100

150

t [S]

An

gle

(D

eg

°)

Steering wheel mesurement

20 25 30 35 40 45 50 55 60−3

−2

−1

0

1

2

3

t [S]

An

gle

(D

eg

°)

Additive steering angle from the controller

Fig. 18: Steering wheel angle δ0 (left) and correctivesteering angle from the controller δ+ (right)

Fig. 18 shows the measured rotation angle of thesteering wheel that the driver generates to performthe considered driving scenario (left), and, right, thecorrective steering angle that the controller suppliesto help the driver to ensure the vehicle stability andmanoeuvrability. This corrective steering angle isdirectly applied on the wheel (not on the steeringwheel).Notice that the steering ratio of the car, which isthe rotation angle of a steering wheel divided bythe steer angle of the wheels, is around 10 : 1 to20 : 1 depending on the car’s type (commercial,race, sport...). This means that the corrective steeringangle’s effect is very important.

VI. CONCLUSION

In this paper, a global chassis control strategy has beenproposed, involving active steering, electromechanicalbraking and semi-active suspension. This strategy wasshown to enhance the vehicle dynamical behaviorsubject to critical driving situations. In this framework,the LPV approach plays a major role to efficientlyschedule the use of these actuators. Indeed theoriginality of the proposed approach is first concernedby the coordinated use of these 3 types of actuators,and second by their hierarchical activation, depending

on the driving situations, which allows to reach theperformance objectives.Another advantage of the LPV methodology(compared to classical LTI controllers) is the limitationof the braking actuation in critical situations to avoidwheel locking and skidding, and its coordinationwith active steering and semi-active suspensioncontrollers, leading to vehicle stability and roadhandling improvements.Simulation results, obtained from experimentalinput data, and performed with a validatedcomplex nonlinear vehicle model, have assessedthe performances of the proposed approach. However,a complete control validation step requires a set ofexperiments performed on a test car equipped withthe considered actuators. The real implementation ofthe control algorithm might lead to several problemsthat do not occur in simulation: for instance real-timeconstraints. This could be handled further in anexperimental study.

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