Gain-scheduled LPV/Hinf controller based on direct yaw moment and active steering for vehicle handling improvements

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1 Gain-scheule LPV/Hinf controller base on irect yaw moment an active steering for vehicle hanling improvements Moustapha Doumiati, Olivier Sename, John Jairo Martinez Molina, Luc Dugar, Charles Poussot-Vassal To cite this version: Moustapha Doumiati, Olivier Sename, John Jairo Martinez Molina, Luc Dugar, Charles Poussot- Vassal Gain-scheule LPV/Hinf controller base on irect yaw moment an active steering for vehicle hanling improvements 49th IEEE Conference on Decision an Control, CDC 2, Dec 2, Atlanta, Georgie, Unite States pp6, 2 <hal-5362> HAL I: hal Submitte on 5 Nov 2 HAL is a multi-isciplinary open access archive for the eposit an issemination of scientific research ocuments, whether they are publishe or not The ocuments may come from teaching an research institutions in France or abroa, or from public or private research centers L archive ouverte pluriisciplinaire HAL, est estinée au épôt et à la iffusion e ocuments scientifiques e niveau recherche, publiés ou non, émanant es établissements enseignement et e recherche français ou étrangers, es laboratoires publics ou privés

2 Gain-scheule LPV/H controller base on irect yaw moment an active steering for vehicle hanling improvements M Doumiati*, O Sename, J Martinez, L Dugar an C Poussot-Vassal Abstract This paper eals with the esign of a control scheme that integrates braking an front steering to enhance the vehicle yaw stability an the lateral vehicle ynamics The propose VDSC (Vehicle Dynamic Stability Controller) allows control of the yaw rate an obtains goo response for the sieslip angle Besies, this controller takes into account the braking actuator limitations (ie braking only the rear wheels) an limits the use of the steering actuator only in the linear vehicle hanling region (stability region) To reach these objectives, an original parameter epenent LPV controller structure with consistent performances weights is esigne The solution of the problem is obtaine within the LMI framework, while warranting H performances To prevent tires longituinal slip ue to brake forces generate by the controller, an ABS strategy is inclue in the control scheme Computer simulations, carrie out on a complex full vehicle moel subject to critical riving situations, confirm the effectiveness of the propose control system an the overall improvements in vehicle hanling an stability I INTRODUCTION A close examination of accient ata reveals that losing the vehicle control is responsible for a huge proportion of car accients Uner critical riving circumstances, such as emergency cornering, it is usually ifficult for a river to stabilize the vehicle, an angerous accients coul happen To ensure vehicle stability an hanling, many avance active chassis control systems base on active yaw moment control have been evelope an brought into the market, like the conventional ESP (Electronic Stability program) an the 4WS (4-wheel steering) systems Safety of groun vehicles may be greatly improve through active yaw/sieslip control The basic iea of the active vehicle stability control is to keep the vehicle within the linear or stability region that is familiar to the river One approach for yaw an lateral vehicle ynamics improvement is to use ifferential braking, thereby creating the moment that is necessary to counteract the unesire yaw motion This technique is referre to as Direct Yaw moment Control (DYC) Some researchers, like in [] an [2], emphasize the DYC concept to improve the vehicle stability, especially in severe maneuvers However, this metho is not esirable in normal riving situations because of the irect influence of the control action on the longituinal vehicle ynamics (ie it causes the vehicle to slow own significantly) An alternative approach is to comman aitional steering angle to create the counteracting moment This technique is referre to as Active Steering (AS), an is mainly effective when the M Doumiati, O Sename, J Martinez an L Dugar are with GIPSA-lab UMR CNRS 526, Control Systems epartment, Grenoble INP-ENSE3 University, BP 46, 3842 Saint Martin Hères, France *moustaphaoumiati@gmailcom C Poussot-Vassal is with ONERA DCSD epartment, 2 avenue Eouar Belin, 355 Toulouse, France charlespoussot-vassal@onerafr Symbol Value Unit Signification m 535 kg vehicle mass m r 648 kg vehicle rear mass I z 249 kgm 2 vehicle yaw inertia C f 8 N/r Cornering stiffness of front tires C r 8 N/r Cornering stiffness of rear tires l f 4 m istance COG - front axle l r m istance COG - rear axle t r 4 m rear axle length R 3 m tire raius µ [2/5; ] tire/roa contact friction interval v [5; 3] km/h vehicle velocity interval TABLE I NOTATIONS AND VEHICLE PARAMETERS lateral tire forces linearly epen on the sieslip angles [3], [4] AS control collapses when the vehicle reaches the hanling limit ue to the tire saturations Consequently, these ifferent control methos are optimize iniviually in specific hanling regions, an the maximum benefit can be gaine through the coorinate an combine use of both methos in the control strategy On this topic, some relevant results in the literature coul be foun in [5]-[9] In this paper, a new VDSC (Vehicle Dynamic Stability Controller) system is evelope The propose VDSC is a unifie controller that coorinates AS (for front tires) an DYC in orer to preserve the vehicle stability in extreme hanling situations, while achieving a goo rie comfort More precisely, this stuy enhances the existing one propose by the authors in [], bringing the following main contributions: The VDSC system tracks a esire vehicle behavior, while controlling both the yaw rate an the sieslip angle ynamics Both AS an DYC are activate in the linear region of vehicle hanling However, AS is involve only in a frequency range where the river is not able to act Applying this strategy ensures that the ae corrective steering angle is not inconvenient for the river, an that the vehicle spee oes not slow own consierably The AS rolls off in severe maneuvers that lea to instability an only DYC (braking only one rear wheel at a time) remains on The bounary of juging the vehicle stability is euce from the phase-plane of the sieslip angle an its time erivative The propose VDSC is base on a 2-DOF linear vehicle moel an synthetize as a parameter epenent controller, gain-scheule MIMO (Multi-Input Multi Output) system This controller activates the require actuator(s) epening on the riving conitions The overall VSDC is built in the LPV (Linear Parameter Varying) framework with an LMI

3 (Linear Matrix Inequalities) solution that warrants the H performances The response of the vehicle with the propose control scheme has been evaluate via computer simulations using a full vehicle moel valiate on a real car The rest of this paper is structure as follows Section II first introuces the global control scheme, an then eveloppe an synthetize the MIMO vehicle ynamical stability controller Performance analysis is one in Section III through time simulations performe on a complex nonlinear full vehicle moel Conclusions an iscussions are given in Section IV Paper notations: Throughout the paper, the following notation will be aopte: inex i = {f, r} an j = {l, r} are use to ientify vehicle front, rear an left, right positions respectively Table I summarizes the notations an values use in the paper II REALIZATION OF THE CONTROL SCHEME Figure represents the total control scheme This architecture inclues an estimator an a controller Signals such as steering wheel angle, wheel spees, yaw rate, lateral acceleration are available in reasonable costs or alreay exist on vehicle equippe with ESP system The sieslip angle is a ifficult an an expensive measurement to achieve in practice Thus, it must be estimate The estimator use here (EKF moel-base observer) was propose by the authors in a previous stuy [] Moel-following technique is use in vehicle ynamic control The yaw rate an the sieslip angle, respectively an β, of a reference moel base on the river s steering input an the vehicle velocity, are the esire responses tracke by the actual vehicle The reference moel is aopte to provie vehicle stability As seen in figure, both inputs of the propose controller are the yaw rate an slip angle errors, an the outputs are the active steer angle an the brake torque applie at only one rear wheel at a time epening on the riving situation It is worthwhile to note that the steering angle applie to the vehicle is =, where is the angle provie by the river an is the aitive steering angle commane by the controller an generate by the AS actuator In the following, each block of the propose control system is escribe in etails A Reference moel The aim of the evelope VDSC uring cornering is twofol: tracking a reference yaw rate an a reference boy sieslip angle In this work, these references values are obtaine as the outputs of a 2-DOF (Degree Of Freeom) classical linear bicycle moel The use of this moel is explaine in etail by Dugoff, Francher, an Segel [2] Note that roll, pitch, an longituinal ynamics are neglecte to simplify the vehicle ynamics The equations governing the lateral an yaw motions in this linear moel can be expresse as: V Reference moel β Fig β VDSC controller Sieslip angle estimator Tb * Tb * ABS EMB * AS Global control scheme Equation of lateral motion: ( ) ( mv β = C f β l f v Driver comman Vehicle Mz ) ( ) C r β l r v () Equation of yaw motion: ( ) ( ) I z ψ = Cf l f v β l f C r β l r l r (2) v Besies, for a safe rive, these references must be saturate by the physical limit impose by the current roa ahesion coefficient µ Since the lateral acceleration of the vehicle cannot excee the maximum friction coefficient µ, the esire yaw rate must be limite by the following value: max µg/v (3) The upper limit β max is obtaine by ensuring that the slip angle oes not become too large Thus, the tires are prevente to approach their limits of ahesion It was foun that this upper limit correspone to β max = arctan(2µg) (empirical relation) For more etails concerning the upper limits formulation, one may refer to [3] B VDSC Design ) Vehicle moel for synthesis: The LPV/H controller is obtaine base on a linear bicycle moel which represents well the lateral an yaw motions of the vehicle (see equations () an (2)) However, taking into account the controller structure, the classical bicycle moel is extene to inclue the rear brake torques as inputs: β = (F tyf F tyr )/(mv) ψ = [ l f F tyf l r F tyr F txr t r M z ] /Iz (4) where F tyf an F tyr are the front/rear lateral tire forces respectively, F txr is the ifferential rear braking force, which epens on the applie braking torques, an M z enotes the yaw moment isturbance (ie effects of the win, ) Assuming that low slip value are preserve, F txr may be written as: F txr = F txrl F txrr (5) = Rm rg (T brl T brr ) (6) 2

4 β β e e β W W2 K(ρ,ρ2) LPV/H Fig 2 g * Tb,rj * W3 W4 EMB AFS Generalize plant moel z z2 z3 z4 Vehicle system (2-DOF linear moel) Consequently, the extene linear bicycle moel is given by: [ ] [ Cf ] C r mv = β ψ lrcr l fc f [ ] mv β 2 l rc r l f C f l 2 f C f l 2 r Cr I z I zv [ ] Cf mv M z l f C f I z I z a a 2 T brl T brr where: a = a 2 = mrgrtr 2I z 2) Generalize plant: VDSC is the propose controller that provies the esire braking torque (Tb rj ) an the aitive steering angle ( ) to maintain the vehicle control VDSC is feee by the yaw rate an sieslip angle errors, e an e β respectively, an scheule by two parameters, ρ an ρ 2, that evolve accoring to the riving situations To synthesize the so calle VDSC, the H control performance is use For more information about the H an LMI theories, reaer can refer to [4], [5] In the following, we present the generalize synthesis plant, calle g an illustrate in figure 2, together with the performance weighting functions an the actuator ynamics g is given thereafter: Σ g : ẋ z y = A C B D B 2 D 2 C 2 x w u Mz (7) (8) where w = [, β, M z ] T is the exogenous input vector, u = [, Tb rl, Tb rr ] T represents the control input signals, y = [, β] T is the measurement vector, an z = [z, z 2, z 3, z 4 ] T contains the weighte controlle outputs which have to be as small as possible Weighting functions: In orer to formulate the stanar structure for the H controller efine in figure 2, the weighting functions W, W 2, W 3, an W 4 are efine to characterize the performance objectives an the actuator limitations The reason behin the weights selection is summarize below: z is the weighte yaw rate error output signal It represents the yaw rate tracking performance The corresponing weight W is: W = sg e /2πf 2G e s/2πf where f = Hz is the cut-off frequency of the high pass filter an where G e = is the attenuation level for low frequencies (f < f ) In this case means that the static error must be lower than % z 2 is the weighte sieslip angle error output signal It represents the sieslip tracking performance The corresponing weight W 2 is: W 2 = 4 sg e /2πf 2G e s/2πf (9) () z 3 is the braking control signal attenuation Its associate weight W 3 is: W 3 = 4 s/2πf 2 s/α2πf 2 () where f 2 = Hz is the braking actuator banwith z 4 is the steering control signal attenuation Its associate weight W 4 is: W 4 = G (s/2πf 3 )(s/2πf 4 ) (s/α2πf 4 ) 2 G ( f /α2πf 4 ) 2 = ( f /2πf 3 )( f /2πf 4 ) f = 2π(f 4 f 3 )/2 (2) where f 3 = Hz is lower limit of the actuator intervention an f 4 = Hz is the steering actuator banwith This filter is esigne in orer to allow the steering system to act only in [f 3, f 4 ] frequency range Outsie of this frequency range, the filter rolls off Between the frequency, an more specifically, at f /2, the filter gain is unitary This filter esign is inspire from [3] These weighting functions are recalle in the sensitivity function plots as upper bouns limits, /W i (see figure 3) More etails are provie in the next subsection 3) LPV controller structure an LMI solution: The controller structure is fixe, but we introuce two parameters epenency, ρ an ρ 2, on the control output matrix The controller has the following structure: ẋ c = A c(ρ, ρ 2)x c B c(ρ, ρ [ 2)e ] [ ] ρ K(ρ) : Tb rl = ρ 2 Cc(ρ, ρ 2) Tb rr ρ 2 }{{} C c(ρ,ρ 2 ) (3) Consequently, accoring to ρ an ρ 2 epenency parameters, a particular controller output will be use More specifically: Steering action is use only if the vehicle is in the stability region The bounary for juging the vehicle stability is erive from the phase plane of the sieslip angle an its time erivative A stability boun efine in [8] is use here, which is formulate as: x c SI < 8 (4)

5 where SI = 249 β 955β is the "Stability Inex" Thus, ρ is chosen as: { if SI < 8 ρ = (5) if SI > 8 The braking torque generate by the controller is always positive an is applie at one wheel at each time It is worthwhile to note that, besies its effectiveness in generating a yaw moment, another avantage of the scheme to apply the brake torque only at one wheel at each time is that the vehicle is not so much ecelerate as when brake torque is applie at more than one wheel to generate the same amount of yaw moment Consequently: when ρ 2 =, the T b rr signal is set to zero when ρ 2 =, the T b rl signal is set to zero Then, by choosing: when we have, e > e ρ 2 = sat [,] [sign(e )] (6) ρ 2 = (only rear left brake is activate) ρ 2 = (only rear right brake is activate) (7) which is consistent with the braking torques practical behavior ( > in the couterclockwise irection) The interest of this original LPV structure is that uring the synthesis step, the controller knows which actuator(s) to activate at each time Consiering the structure iscusse above, it is obvious that the system moel an actuators are LTI, but the controller is LPV The stabilizing controller, ensuring H performances while minimizing the attenuation level γ for ρ {, } an ρ 2 {, }, is obtaine using the LMI tools The polytopic approach to this problem consists on fining a solution at each vertex of the polytope escribe by ρ i = [ρ, ρ 2, ρ 2 ], by using a common Lyapunov function For more etails on the computation solution, reaer is invite to rea [] an [5] By solving the LMI problem using Yalmip interface [6] an SeDumi solver [7], one obtains the suboptimal value γ opt = 669 Remark: It is crucial to note that an LTI controller structure synthetize on the same plant Σ g with the same weighte filters, results in a controller which may provie a negative torque (equivalent to an acceleration), which is, practically impossible [] Accoring to the sensitivity functions Boe iagrams illustrate in figure 3, it is interesting to make the following euctions: The yaw rate error signal, e, is well attenuate for the LPV controller (see figure 3(a)) The sieslip angle error, e β, is not attenuate so much However, we note that thanks to the LPV esign the close-loop stability of the system is ensure, an the sieslip angle is suppose to follow its target (see figure 3(b)) Figure 3(c) shows that if the steering is activate (ρ = ), it ecreases the use of braking for controlling the yaw rate For the LPV control strategy, when the steering control is activate (ρ = ), it acts on the specifie frequency range as illustrate in figure 3() Moreover, this figure eluciates the contribution of the steering in controlling the yaw rate For ρ =, steering is forbien C ABS To prevent tires longituinal slip ue to brake forces generate by the controller, an ABS strategy is inclue in the control scheme The local ABS is implemente on each of the rear wheels, an it is activate only when high slipping occurs It provies T b rj, the braking torque, accoring to the set point T b rj provie by the VDSC control bloc (see figure 2) This ABS system is recently evelope in [8] D Actuator moels The control input signals use are the steering angle an the rear braking torques Let consier the following actuators: As braking system, we consier an EMB (Electro Mechanical Braking) actuators, proviing a continuously variable braking torque The moel is given by: T brj = 2π ( T b rj T brj ) (8) where, = Hz is the actuator cut-off frequency, T b rj an T bij are the local braking controller an actuator outputs respectively Note that in this paper, only the rear braking system is use to avoi coupling phenomena occurring with the steering system This actuator control is limite between [, 2] Nm As Active Steering (AS) system, we consier an active actuator proviing an aitional steering angle Such actuator is moele as: = 2πκ( ) (9) where, κ = Hz is the actuator cut-off frequency, an are the steering controller an actuator outputs respectively III SIMULATIONS Simulations from nominal as well as averse riving conitions were carrie out on ifferent roa conitions in orer to assess the performance of the propose control scheme Simulations are performe using a full vehicle moel valiate on a real french car: (Renault Mégane Coupé) In this paper, we report a ouble-lane-change maneuver on a ry roa maneuver (one of a number of simulations that we carrie out), where the ynamic contributions play an important role In the following, on each plot, the uncontrolle Mégane is plotte in blue ot, the LPV control in re ashe an the yaw rate an sieslip angle references in black soli Scenario escription: In this critical test, the vehicle is riven at very high spee 5 km/h The yaw rate, the sieslip angle, an the trajectory of the vehicle are shown in figure 4 Figure 4 confirms that the vehicle with the propose control task is superior to the uncontrolle vehicle in terms

6 5 5 e / e / β e / M z 5 2 /W /W /W (a) Close loop transfer functions between e an exogenous inputs e / /M β e β e β β z 9 ρ= /W / /W /W (b) Close loop transfer functions between e β an exogenous inputs * Tb / /W3 ρ= * Tb / β 5 /W3 /W3 ρ= (c) Close loop transfer functions between T b * Tb /M z ρ= an exogenous inputs * / / β / M * * z 5 5 /W4 /W4 /W () Close loop transfer functions between an exogenous inputs Fig 3 Close loop transfer functions: LPV (re ashe (ρ = ), or green soli (ρ = )) synthesis results; Weighting functions (black ashe) of /W, /W 2, /W 3, an /W 4 of following the linear moel behavior The sieslip angle of the controlle vehicle remains close to its target all over the vehicle trajectory, which ensures the vehicle stability Figure 5(a) shows the generate corrective steering angle an the brake torques to enhance the lateral vehicle control It is obvious that the LPV/H controller only provies positive braking torques, which are achievable by the consiere actuators Therefore, the controller fits to the actuator constraints For this test, ue to braking, the vehicle spee is reuce to 45 km/h, which is not much compare to 5 km/h Figure 5(b) illustrates how the stability inex an the epenancy parameters ρ an ρ 2 evolve accoring to the riving situations: ρ = (SI < 8) Steering is activate ρ 2 = (e > ) Left brake is activate, otherwise, the right brake is activate Note that, even when scheuling, the close-loop stability of the system is ensure thanks to the LPV esign IV CONCLUSION Vehicle hanling an stability can be effectively improve using steering an braking systems, a new LPV/H controller, that coorinates between these two actuators, is esigne in this paper The propose LPV controller is esigne in an original way an ensures that: The steering action is activate only in normal riving conition, an in a specifie range of frequency where the river coul not act The braking torque is always positive by selecting the appropriate rear wheel Since, the general structure of the propose control scheme oes not involve any online optimization process, it shows to be easy to function in real-time Simulation of a critical riving situations that compare the responses of a controlle vehicle with respect to a passive vehicle show the valiation of the propose control esign Future work consists to implement the controller on a real car, an to test its robustness with respect to real riving conitions REFERENCES [] J H Park, H irect yaw-moment control with brakes for robust performance an stability of vehicles, JSME International Journal, Series C, Vol 44, No 2, 2 [2] M Canale, L Fagiano, M Milanese, an P Boroani, Robust vehicle yaw control using an active ifferential an IMC techniques, Control Engineering Practice, Vol 5, pages , 27 [3] B A Guven, T Bunte, D Oenthal, an L Guven, Robust two egreeof-freeom vehicle steering controller esign, IEEE Transaction on Control System Technology, Vol 5, pages , 27 [4] S Mammar an D Koenig, Vehicle hanling improvement by active steering, Vehicle System Dynamics, Vol 38, No 3, pages 2-242, 22 [5] M Nagai, Y Hirano, an S Yamanaka, Integrate control of active rear wheel steering an irect yaw moment control, Vehicle System Dynamics, Vol 27, pages , 997 [6] B L Boaa, M J L Boaa, an V Diaz,Fuzzy-logic applie to yaw moment control for vehicle stability, Vehicle System Dynamics, Vol 43, No, pages , 25 [7] G Burgio an P Zegelaar, Integrate vehicle control using steering an brakes, International Journal of Control, Vol 79, No 5, pages , 26 [8] J He, D A Crolla, M C Levesley, an WJ Manning, Coorination of active steering, riveline, an braking for integrate vehicle ynamics control, Proc IMechE, Vol 22, PartD: Automobile Engineering, 26

7 [eg/s] β[eg/s] 2 Yaw rate (v =5km/h an roa type:dry) Reference Uncontrolle Controlle Sieslip angle (v =5km/h an roa type:dry) y [m] Vehicle path (v =5km/h an roa type:dry) Controlle vehicle Uncontrolle vehicle 5 x [m] Fig 4 Vehicle responses in a ouble lane change maneuver (ry roa): with LPV (re ashe), Uncontrolle (blue ot), Reference moel (black soli) [eg] Steer an Brakes control (v =5km/h an roa type:dry) T b rl [Nm] T b rr [Nm] 5 5 SI ρ ρ (a) Control signals for the ouble lane change maneuvre Stability inex(v =5km/h an roa type:dry) ρ parameter variations (v =5km/h an roa type:dry) 2 3 ρ 2 parameter variations (v =5km/h an roa type:dry) 2 3 [9] X Yang, Z Wang, an W Peng, Coorinate control of AFS an DYC for vehicle hanling an stability base on optimal guarantee cost theory, Vehicle System Dynamics, Vol 47, No, pages 57-79, 29 [] C Poussot-Vassal, O Sename, an L Dugar, Robust vehicle ynamic stability controller involving steering an braking, Proceeings of the European Control Conference, Buapest, Hungary, 29 [] M Doumiati, A Victorino, A Charara, an D Lechner, A metho to estimate the lateral tire force an the sieslip angle of a vehicle: Experimental valiation, Proceeings of the American Control Conference, Baltimore, USA, 2 [2] H Dugoff, PS francher, an L Segel, An analysis of tire traction properties an their influence on vehicle ynamic performance, SAE transactions, vol 79, pp , 97 [3] R Rajamani, Vehicle ynamics an control, Springer, 26 [4] S Skogesta an I Postlethwaite, Multivariable feeback control, analysis an esign, Wiley, 27 [5] C Scherer, P Gahinet, an M Chilali, Multiobjective output-feeback control via LMI optimization, IEEE Transaction on Automatic Control, Vol 4, pages 896-9, 997 [6] J Lofberg, YALMIP: a toolbox for moeling an optimization in MATLAB, Proceeings of the CACSD Conference, Taipei, Taiwan, 24 [7] J F Sturm,Using sedumi 2, a Matlab toolbox for optimization over (b) ρ an ρ 2 variations: ρ = Steering is activate, ρ 2 = Left brake torque is activate, otherwise, right brake torque is activate Fig 5 Control signals accoring to ρ an ρ 2 variations symmetric cones, Optimization Methos an Software -2, pages , Special issue on Interior Point Methos, 999 [8] M Tanelli, R Sartori, an S Savaresi, Combining Slip an eceleration control for brake-by-wire control systems: a sliing-moe approach, European Journal of Control, Vol 3, pages 593-6, 27 [9] P Apkarian an P Gahinet, A convex characterization of gain scheule H controllers, IEEE Transaction on Automatic Control, Vol 4, pages , 995