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Effect of fractional order damping control on braking performance for electric vehicles ⁎
9th IFAC International Symposium on Advances in Automotive 9th IFAC International Symposium on Advances in Automotive Control 9th IFAC International International Symposium Symposium on on Advances Advances in in Automotive Automotive 9th IFAC Control Available online at www.sciencedirect.com Orléans, June Symposium 23-27, 2019 Control Control 9th IFAC France, International Orléans, France, June 23-27, 2019 on Advances in Automotive Orléans, France, France, June June 23-27, 23-27, 2019 2019 Orléans, Control Orléans, France, June 23-27, 2019
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IFAC PapersOnLine 52-5 (2019) 231–236
Effect of fractional order damping control Effect Effect of of fractional fractional order order damping damping control control on braking performance for electric on braking performance for electric Effect of fractional order damping control on braking performance for electric vehicles vehicles for electric on braking performance vehicles ∗,∗∗ vehicles ∗∗ ∗
Hussein Xavier Moreau ∗∗ Clovis Hussein Termous Termous ∗,∗∗ Clovis Francis Francis ∗∗∗ ∗,∗∗ Xavier Moreau ∗ ∗∗ Hussein Xavier Moreau Francis Hassan ∗ ∗∗ Clovis Hussein Termous Termous ∗,∗∗ XavierShraim Moreau Clovis Francis Hassan Shraim ∗ ∗ ∗∗ Clovis Francis ∗ Hassan Shraim Hussein Termous ∗,∗∗ XavierShraim Moreau Hassan ∗ ∗ HassanofShraim Engineering, Scientific Research ∗ Lebanese University, Faculty ∗ Lebanese University, Faculty of Engineering, Scientific Research ∗ Lebanese University, Faculty of Engineering, Scientific Research Center in Engineering (CRSI), Campus Rafic Hariri, Beirut, Lebanon, Lebanese University, Faculty of Engineering, Scientific Research Center in Engineering (CRSI), Campus Rafic Hariri, Beirut, Lebanon, ∗ Center in Engineering (CRSI), Campus Rafic Hariri, Beirut, Lebanon, (e-mail: [email protected]; [email protected]; Lebanese University, Faculty of Engineering, Scientific Research Center in Engineering (CRSI), Campus Rafic Hariri, Beirut, Lebanon, (e-mail: [email protected]; [email protected]; (e-mail: [email protected]; [email protected]; [email protected]) Center in Engineering (CRSI), Campus Rafic Hariri, Beirut, Lebanon, (e-mail: [email protected]; [email protected]; [email protected]) ∗∗ [email protected]) of Bordeaux, Lab IMS CNRS 5218, Talence, Bordeaux, (e-mail: [email protected]; [email protected]; ∗∗ University [email protected]) ∗∗ University of Bordeaux, Lab IMS CNRS 5218, Talence, Bordeaux, ∗∗ University of Lab France (e-mail: [email protected]) [email protected]) University of Bordeaux, Bordeaux, Lab IMS IMS CNRS CNRS 5218, 5218, Talence, Talence, Bordeaux, Bordeaux, France (e-mail: [email protected]) ∗∗ France (e-mail: [email protected]) University of Bordeaux, Lab IMS CNRS 5218, Talence, Bordeaux, France (e-mail: [email protected]) France (e-mail: [email protected]) Abstract: Abstract: Active Active suspensions suspensions are are an an object object of of interest interest nowadays, nowadays, they they have have been been aa topic topic of of Abstract: Active suspensions are an object of interest nowadays, they have aa topic of research decades due in Global Chassis Control (GCC). Indeed, aa wellAbstract: suspensions arekey an role object interest nowadays, they have been been topic of research for forActive decades due to to their their key role in of Global Chassis Control (GCC). Indeed, wellresearch for decades due to their key role in Global Chassis Control (GCC). Indeed, a wellcontrolled active suspension system may considerably improve not only the passenger comfort Abstract: Active suspensions are an object of interest nowadays, they have been a topic of research for decades due to system their key role in Global Chassis Indeed,comfort a wellcontrolled active suspension may considerably improveControl not only(GCC). the passenger controlled active suspension system may considerably improve not only(GCC). thethe passenger comfort by compensating the chassis dynamics but also road handling by minimizing dynamics of the research for decades due to their key role in Global Chassis Control Indeed, a wellcontrolled active suspension system may considerably improve not only the passenger comfort by compensating the chassis dynamics but also road handling by minimizing the dynamics of the by the chassis dynamics but also road by minimizing dynamics of wheels normal forces. It enhance vehicle stability by controlling the load distribution, controlled active suspension system may only thethe passenger comfort by compensating compensating the chassis butconsiderably also road handling handling by not minimizing the dynamics of the the wheels normal forces. It may may dynamics enhance vehicle stability byimprove controlling the wheel wheel load distribution, wheels normal forces. It may enhance vehicle stability by controlling the wheel load distribution, which influences the lateral and longitudinal dynamics to provide better steering performance, by compensating chassis dynamics but also road handling by minimizing the dynamics of the wheels normal forces. It mayand enhance vehicle stability bytocontrolling the wheel loadperformance, distribution, which influences the lateral longitudinal dynamics provide better steering which influences the lateral lateral and longitudinal dynamics tocontrolling provide better better steering performance, and safety by the distance. The contribution this paper is wheels normal forces. It may enhance vehicle stability byto theof loadperformance, distribution, which influences the and longitudinal dynamics provide steering and safety by reducing reducing the braking braking distance. The main main contribution ofwheel this paper is to to study study and impact safety by reducing the braking distance. The main mainin contribution ofofthis this paper is to study study the of fractional order (FO) control methods the context an active suspension which influences the lateral and longitudinal dynamics to provide better steering performance, and safety by reducing the braking distance. The contribution of paper is to the impact of fractional order (FO) control methods in the context of an active suspension the impact of fractional order (FO) control methods in the context of an active suspension system and in particular fractional order damping control. We derive two optimal damping and safety by reducing the braking distance. The main contribution of this paper is to study the impact of fractional order (FO) control methods in the context of an active suspension system and in particular fractional order damping control. We derive two optimal damping system and in particular fractional control. derive optimal damping control strategies: aa comfort-oriented and aa damping road holding strategy, based on the impact order (FO) order control methods inoriented the We context an active suspension system and of in fractional particular fractional order damping control. We deriveof two two optimal damping control strategies: comfort-oriented and road holding oriented strategy, based on non-integer non-integer control strategies: strategies: a comfort-oriented comfort-oriented andrelevant road holding holding oriented strategy, based on non-integer non-integer derivation deflection. benefits of methods on vehicle response system andof particular fractionalThe order control. We strategy, derive two damping control a and aa damping road oriented based on derivation ofinsuspension suspension deflection. The relevant benefits of both both methods onoptimal vehicle response derivation of suspension suspension deflection. The relevant benefits of both bothstrategy, methods on vehicle vehicle response while driving on rough roads and during braking are evaluated using computer simulations control strategies: a comfort-oriented and a road holding oriented based on non-integer derivation of deflection. The relevant benefits of methods on response while driving on rough roads and during braking are evaluated using computer simulations while driving on rough roads and during braking are evaluated using computer simulations applied on a two-wheel vehicle model. derivation of suspension deflection. The relevant benefits of both methods on vehicle response while driving on rough roads and during braking are evaluated using computer simulations applied on a two-wheel vehicle model. applied on model. while on rough vehicle roads and during braking are evaluated using computer simulations applieddriving on a a two-wheel two-wheel vehicle model. © 2019, IFAC (International Federation of Automatic Control) Hosting by Elsevier Ltd. All rights reserved. applied on a two-wheel vehicle model. Keywords: Keywords: Active Active suspension, suspension, fractional fractional order, order, braking, braking, comfort, comfort, road road holding, holding, damping damping Keywords: Active suspension, fractional order, braking, comfort, road holding, control. Keywords: Active suspension, fractional order, braking, comfort, road holding, damping damping control. control. Keywords: Active suspension, fractional order, braking, comfort, road holding, damping control. control. 1. INTRODUCTION INTRODUCTION and 1. and longitudinal longitudinal dynamics dynamics (Gaspar (Gaspar et et al. al. (2007))(Sammier (2007))(Sammier 1. and longitudinal dynamics (Gaspar et et al. (2003)). In this domain, fractional order 1. INTRODUCTION INTRODUCTION and longitudinal dynamics (Gaspar et al. al. (2007))(Sammier (2007))(Sammier et al. (2003)). In this domain, fractional order control control et al. al. (2003)). In this prevalence domain, fractional order control laws have gained their where they show their 1. INTRODUCTION Active suspension systems are proving their importance and longitudinal dynamics (Gaspar et al. (2007))(Sammier et (2003)). In this domain, fractional order control Active suspension systems are proving their importance laws have gained their prevalence where they show their laws have gained their prevalence where they show their Active suspension systems are proving their importance ability in solving different control problems (Moreau et al. in the field of vehicle dynamics. Many studies in the et al. (2003)). In this domain, fractional order control laws have gained their prevalence where they show their Active suspension systems are proving their importance in the field of vehicle dynamics. Many studies in the ability in solving different control problems (Moreau et al. ability in solving different control problems (Moreau et al. in the field of vehicle dynamics. Many studies in the (2001)). literature show their ability to enhance comfort for paslaws have gained their prevalence where they show their Active suspension systems are proving their importance ability in solving different control problems (Moreau et al. in the field of vehicle dynamics. Many studies in the (2001)). literature show their ability to enhance comfort for pas(2001)). literature show their ability to enhance comfort for passengers (Moreau et al. (2009)), vehicle stability (Altet ability in solving different control problems (Moreau et al. in the field of vehicle dynamics. Many studies in the literature(Moreau show their ability to enhance pas- (2001)). sengers et al. (2009)), vehicle comfort stabilityfor(Altet In this work, the impact of fractional order control methsengers (Moreau et al. (2009)), vehicle stability (Altet In this work, the impact of fractional order control methet al. (2003)) and driving performance (Termous et al. (2001)). literature show their ability to enhance comfort for passengers (Moreau et al. (2009)), vehicle stability (Altet et al. (2003)) and driving performance (Termous et al. ods In work, the of order control methhas been where we two In this this the impact impact of fractional fractional control dampmethhaswork, been studied, studied, where we derive derive order two optimal optimal dampet al. and performance et (2018b)) compared to passive systems. One important important sengers (Moreau al. passive (2009)), vehicle (Termous stability (Altet et al. (2003)) (2003)) andetdriving driving performance (Termous et al. al. ods (2018b)) compared to systems. One odsthis haswork, been studied, where we derive derive order twoand optimal damping control strategies, a comfort-oriented a road holdIn the impact of fractional control methods has been studied, where we two optimal damp(2018b)) compared to passive systems. One important ing control strategies, a comfort-oriented and a road holdsafety requirement of any vehicle is the braking distance et al. (2003)) and of driving performance (Termous et al. ing control (2018b)) compared toany passive systems. One important safety requirement vehicle is the braking distance strategies, a comfort-oriented comfort-oriented and a road road holdoriented strategy, based on the non-integer derivation ods has been studied, where we derive two optimal damping control strategies, a and a holdsafety requirement of any vehicle is the braking distance ing oriented strategy, based on the non-integer derivation in order to prevent accidents or at least lessen the impact. (2018b)) compared to passive systems. One important safety of any vehicle thelessen braking in orderrequirement to prevent accidents or at is least thedistance impact. ing oriented strategy, based on the non-integer derivation of the suspension deflection using the active suspension control strategies, a comfort-oriented and a road holding oriented strategy, based on the non-integer derivation in to prevent or least lessen the impact. the suspension deflection using the active suspension For this purpose, rapid advance has been made in this this of safety requirement of any advance vehicle thebeen braking in order order to preventaaaccidents accidents or at at is least lessen thedistance impact. For this purpose, rapid has made in of the suspension deflection using the active suspension system introduced in the Active Wheel (Laurent et al. ing oriented strategy, based on the non-integer derivation of the suspension deflection using the active suspension system introduced in the Active Wheel (Laurent et al. For this purpose, a rapid advance has been made in this field, most notably the development of the the anti-lock brake in to preventathe accidents or at least lessen the impact. Fororder this purpose, rapid advance has been made inbrake this of field, most notably development of anti-lock system introduced inkind the of Active Wheel (Laurent etused al. (2000)) (Fig.1). This active wheels is mostly the suspension deflection using the active suspension system introduced in the Active Wheel (Laurent et al. field, most notably the development of the anti-lock brake (2000)) (Fig.1). This kind of active wheels is mostly used systems (ABS). For purpose, rapid advance has been made inbrake this in field,this most notablyathe development of the anti-lock systems (ABS). (2000)) (Fig.1). This kind of active wheels is mostly mostly used light electrical cars, where two in-wheel 30-kW mosystem introduced in the Active Wheel (Laurent et al. (2000)) (Fig.1). This kind of active wheels is used systems (ABS). light electrical cars, where two in-wheel 30-kW mofield, most notably the development of the anti-lock brake in systems (ABS). in light electrical cars, where two in-wheel 30-kW motors motorize it. The suspension system is composed However, due to the coupling effects between the suspen(2000)) (Fig.1). This kind of active wheels is mostly used in light electrical cars, where two in-wheel 30-kW moHowever, due to the coupling effects between the suspen- tors motorize it. The suspension system is composed of of systems (ABS). motorize it. The system is of However, due the coupling effects between the ators and engine. This system offers sion dynamics and the braking system (Hamersma and in light electrical cars,suspension where in-wheel 30-kW motors motorize it. electric The suspension system is composed composed of However, due to toand thethe coupling effects between the suspensuspension dynamics braking system (Hamersma and a spring spring and an an electric engine.two This system offers new new a spring and an electric engine. This system offers new sion dynamics and the braking system (Hamersma and perspectives in the suspension control, which indeed offers Els (2014)), active suspension control laws, where the tors motorize it. The suspension system is composed of However, due to the coupling effects between the suspena spring and an electric engine. This system offers new sion (2014)), dynamicsactive and the braking control system laws, (Hamersma Els suspension where and the perspectives in the suspension control, which indeed offers perspectives in the suspension control, which indeed offers Els (2014)), active suspension control laws, where the important opportunities in global chassis control. It proves strategy is comfort-oriented, may degrade the performance a spring and an electric engine. This system offers new sion dynamics and the braking system (Hamersma and perspectives in the suspension control, which indeed offers Els (2014)), active suspension control laws, where the strategy is comfort-oriented, may degrade the performance important opportunities in global chassis control. It proves important opportunities in chassis control. It strategy is may degrade the in safer steering using the active of braking onactive rough suspension roads. Indeed, Indeed, the active active suspension perspectives in the suspension control, which indeed offers Els (2014)), laws, where the its important opportunities in global global chassis control. It proves proves strategy is comfort-oriented, comfort-oriented, may control degrade the performance performance of braking on rough roads. the suspension its effectiveness effectiveness in providing providing safer steering using the active its effectiveness in providing safer steering using the of braking on rough roads. Indeed, the active suspension roll control system by compensating roll dynamics when forces introduced by the electric actuator affect the wheel important opportunities in global chassis control. It proves strategy is comfort-oriented, may degrade the performance in providing safer steering using the active active of braking on rough roads. Indeed, the active forces introduced by the electric actuator affectsuspension the wheel its rolleffectiveness control system by compensating roll dynamics when rolleffectiveness control system system by compensating compensating roll dynamics dynamics when forces introduced by the electric actuator affect the wheel manoeuvring (Termous et al. (2018a)). normal forces, that in turn impact the wheel longitudinal its in providing safer steering using the active of braking on rough roads. Indeed, the active suspension roll control by roll when forces introduced by the electric actuator affect the wheel normal forces, that in turn impact the wheel longitudinal manoeuvring (Termous et al. (2018a)). manoeuvring (Termous et al. (2018a)). normal forces, that in turn impact the wheel longitudinal contact forces that are directly responsible for the braking roll control system by compensating roll dynamics when forces introduced by the electric actuator affect the wheel manoeuvring (Termous et al. (2018a)). normal forces, that in turn impact the wheel longitudinal contact forces that are directly responsible for the braking The optimized control concept is done to utilize the availcontact forces that are directly responsible for the braking The optimized control concept is done to utilize the availbehaviour of a vehicle. For this purpose, searching for new manoeuvring (Termous et al. (2018a)). normal forces, that in turn impact the wheel longitudinal contact forces that are directly responsible for the braking behaviour of a vehicle. For this purpose, searching for new The The optimized optimized control the concept is done to utilize the under available adhesion tires and the road surface control concept done utilize the availbehaviour of vehicle. this purpose, for adhesion between between the tires is and the to road surface under design control strategies using suspension control becomes contact forces are For directly responsible for thebecomes braking behaviour of aa that vehicle. For thissuspension purpose, searching searching for new new able design control strategies using control able adhesion between the tires and the road surface under braking to the maximum possible degree, to minimize The optimized control concept is done to utilize the available adhesion between the tires and the road surface under design using suspension control becomes to the maximum possible degree, to minimize topiccontrol of of research nowadays order enhance lateral a strategies vehicle. For thisin purpose, for new braking design control strategies using suspension control aabehaviour topic of research nowadays in order to tosearching enhancebecomes lateral to the maximum possible degree, to minimize braking distance. This control strategy will offer a better able adhesion between the tires and the road surface under to the maximum possible degree, to minimize braking distance. This control strategy will offer a better adesign of nowadays order to enhance lateral control strategies using in suspension becomes a topic topicwork of research research nowadays order research to control enhance lateral distance. This control strategy will offer a better was supported by the in Lebanese program and riding and reductions. A two-wheel braking to the maximum possible degree, to minimize This distance. Thiscost control strategy offer a vehicle better This work was supported by the in Lebanese research program and riding experience experience and cost reductions. A will two-wheel vehicle a topic of research nowadays order to enhance lateral theThis AUF-CNRSL-UL program. This work was was supported supported by the the Lebanese Lebanese research research program program and riding experience and cost reductions. A two-wheel vehicle braking distance. This control strategy will offer a better work by and riding experience and cost reductions. A two-wheel vehicle the AUF-CNRSL-UL program. the AUF-CNRSL-UL program. theThis AUF-CNRSL-UL program. work was supported by the Lebanese research program and riding experience and cost reductions. A two-wheel vehicle the AUF-CNRSL-UL program. Copyright © 231 Hosting by Elsevier Ltd. All rights reserved. 2405-8963 © 2019 2019, IFAC IFAC (International Federation of Automatic Control) Copyright © 2019 IFAC 231 Peer review under responsibility of International Federation of Automatic Copyright © 2019 IFAC 231 Copyright © 2019 IFAC 231 Control. 10.1016/j.ifacol.2019.09.037 Copyright © 2019 IFAC 231
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- sprung masses motion: m2i z¨2i = f0i + k2 (z1i − z2i ) + b2i (z˙1i − z˙2i ) + uai ,
(3)
where f0i are the front and rear load shift due to the longitudinal acceleration or deceleration, formulated as: f01 =
h M ax (t), and a+b
(4) h f02 = − M ax (t). a+b The heave and pitch motion of the chassis could be calculated using the relation between zG , θG and the dynamics of front and rear sprung masses, by:
Fig. 1. Active wheel from Michelin.
z21 = zG − a sin(θG ), and
(5)
z22 = zG + b sin(θG ). Supposing that θG has a small value, and using the inverse of (5), we get the following relation: −1 z21 1 −a zG . (6) = 1 b θG z 22 2.2 Nonlinear model of the longitudinal dynamics
Fig. 2. Two-wheel, 7 degrees of freedom, car model. model is used to test the proposed control strategy for the vehicle chassis control. This paper is organized as follows: the next section presents the two-wheel vehicle model, then in section 3 we introduce the different suspension systems used. Section 4 illustrates the optimization method and the control strategy. Section 5 shows the results obtained and finally a conclusion and further prospects are given.
where vx (t) is the vehicle longitudinal velocity, Fa (t) is the aerodynamics force described by: Fa (t) =
1 Sx ρ Cx vx (t)2 , 2 2
The motion of the two-wheels car model, as shown in Fig.2, is described by seven differential equations. There are four equations to derive a linear model for the vertical dynamics and three equations to derive a nonlinear model for the longitudinal dynamics. 2.1 Linear model of the vertical dynamics The linear model of the vertical dynamics can be described by the second order differential equations as follows: - unsprung masses motion: (1)
where uai are the forces developed by the electric actuators embedded in the active wheel, with i = 1 for front and i = 2 for the rear axles. Denote by fzi , the wheel ground contact forces, that is defined by: fzi = (m1i + m2i )g + k1i (z1 − z0 ) + b1i (z˙1 − z˙0 ). (2) 232
(8)
Frr (t) is the rolling resistance force: Frr (t) = (σ0 + σ2 vx (t)2 ) M g,
2. DYNAMIC MODEL
m1i z¨1i = k1i (z1 − z0 ) + b1i (z˙1 − z˙0 ) − k2i (z1i − z2i ) − b2 (z˙1i − z˙2i ) − uai ,
In this paper, only the longitudinal dynamics were considered and the lateral dynamics were neglected, and it is defined by: (7) M v˙ x (t) = Fx1 (t) + Fx2 (t) − Fa (t) − Frr (t),
(9)
Fxi (t) are the tyre longitudinal forces generated according to Pacejka wheel model, one of the most complete studies of tire force generation (Pacejka (2005)), where more than 30 empirical parameters are needed to fully describe experimental measurements. The calculation of the tire longitudinal forces depends on the value of the road adherence coefficient, supposed to be µ = 1 for dry pavement, and on the value of tire slip rate λi that is defined by: vx (t) λ =1− , if Ωi (t) r ≥ vx (t) (accelerating) i Ωi (t) r , Ω λi = i (t) r − 1, if Ωi (t) r < vx (t) (braking) vx (t) (10) where Ωi (t) is the wheel angular velocity. The dynamics of the wheel angular velocity is given by: ˙ Jω Ω(t) = Tmi (t) − Tbi (t) − r Fxi (t) − br Ω(t),
(11)
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where Tmi (t) represents the traction torque generated by the electric motor on each wheel and Tbi (t) is the braking torque applied.
V1 (s) =
• isolate the vehicle from road surface irregularities; • control variations in vertical load at the road/tire interface; • control body motion caused by road inputs, inertia forces, and aerodynamic forces; • maintain vehicle directional stability during maneuvers. In fact, the main problem is the conflict between these various aspects of vehicle behavior. 3.1 Traditional suspension A passive vibration control unit consists of a spring and an energy dissipator. The suspension is modeled as a linear spring and a damper. The expression of the force fsi developed by the suspension is as follows: fsi (t) = k2i (z2i − z 1i ) + b2i (z˙2i − z˙ 1i ).
(12)
3.2 Active wheel suspension system The active suspension system embedded in the Active Wheels can be modeled by a spring, an electric actuator that acts as a force generator and a small frictional damping ratio that results from the mechanical links of the system. The idea is to use the force generator as a continuously controllable damper that develops a force ua that is proportional to the fractional derivative of the suspension deflection, namely: uai (t) = ba
d dt
ni
(z2i (t) − z1i (t)) ,
(13)
with 0 < n < 2 is the non-integer derivative order, and ba is a constant gain factor assumed equal to the damping coefficient of the passive system. The Laplace transform of relation (13) leads to the following transfer function: 1 Uai (s) = −ba 1−ni , (14) (V2i (s) − V1i (s)) s The force fsi developed by the active suspension is then given by: ni d fsi (t) = k2i (z2i − z 1i ) + ba (z2i (t) − z1i (t)) . (15) dt The Laplace transform of the vertical dynamics of the quarter vehicle system with fractional order damping control leads to the following transfer functions:
233
(m2 s2 + b2 s + ba sn + k2 )(b1 s + k1 ) V0 (s) den(s) s(b2 + ba sn + k2 ) + F0 (s), den(s)
3. SUSPENSION MODEL The vehicle suspension system is required to fulfill a number of functions. These can be stated as follows:
233
(16)
(b2 s + ba sn + k2 )(b1 s + k1 ) V0 (s) + den(s) (17) s m1 s2 + (b2 + b1 )s + ba sn + (k2 + k1 ) F0 (s), den(s)
V2 (s) =
with den(s) = m1 m2 s4 + (m2 (b1 + b2 ) + m1 b2 ) s3 + (m2 (k1 + k2 ) + m1 k2 + b1 b2 ) s2 + (k1 b2 + b1 k2 ) s +k1 k2 + ba (m1 + m2 ) s2+n + ba b1 s1+n + ba k1 sn . (18) V2 (s), V1 (s), V0 (s) and F0 (s) are the Laplace transform of Z˙ 2 (t), Z˙ 1 (t), Z˙ 0 (t) and f0 (t)respectively. It can be noticed that for ba = 0 is the case of passive suspension, and when b2 = 0 is the case of active suspension. For the particular case, n = 1 and ba = b2 , the active suspension system acts as the passive system.
4. CONTROL STRUCTURE The proposed strategy for the active suspension is shown in Fig. 3. It consists of a feedback control law used to regulate the suspension deflection versus road input disturbances. Supposing “ba ” is constant, the only controlled parameter is the fractional order “n”. The value of n should be chosen according to the selected objectives. In order to find these values of n, three sensitivity transfer function were formulated: A2 (s) sV2 (s) = , Ha (s) = V0 (s) F0 V0 (s) F0 =0
(19)
where Ha (s) represents the sensitivity of the vehicle’s body vertical acceleration to road disturbances. It evaluates the vibration isolation, where small gains results in better ride comfort. Z1 (s) − Z2 (s) , H12 (s) = V0 (s) F0 =0
(20)
H12 (s) is a constrain for the suspension deflection, and
H01 (s) =
Z1 (s) − Z0 (s) , V0 (s) F0 =0
(21)
where H01 (s) is a road holding indicator, and a constrain for wheel normal forces. For optimization purposes, three criteria were formed:
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2 Ja J12 J01
1.8 1.6 1.4
Criteria
1.2 1 0.8 0.6 0.4 0.2 0 0
(22)
0.6
0.8
1
1.2
1.4
1.6
1.8
2
1.5 1 0.5 0 -0.5 -1 -1.5
According to these criteria, when the value is less than 1 it indicates an improvement over the passive system. The optimal control problem can thus be formulated as finding the minimal values of ni for each criterion. Here, two strategies are derived (Fig.4). The comfort-oriented strategy where: (23)
and for the road holding oriented strategy we have: n1 = 1.1 and n2 = 1,
0.4
Fig. 4. Optimized damping control normalized criteria, ( - - ) Rear model, (—) Front model.
where η1 ,η2 , and η3 represent the values of the calculated criteria for the passive suspension system (ba = 0). The range of integration is between ωmin = 2π0.1rad/s and ωmax = 2π30rad/s.
n1 = 0.67 and n2 = 0.64,
0.2
Order n
Elevation (cm)
Fig. 3. Block diagram of the control strategy. ω max 1 2 |Ha (jω)| dω (n) = J (n) = J a 1 η 1 ω min ω max 1 2 J2 (n) = J12 (n) = |H12 (jω)| dω , η 2 ω min ω max 1 2 J3 (n) = J01 (n) = |H01 (jω)| dω η3 ω min
(24)
The suspension control strategy must be changed according to the road state of surface. In case of a smooth road, the road holding strategy is then preferred. It is the same case also when a critical braking situation revealed where comfort-oriented strategy is no longer a priority, here the suspension must be switched to road holding oriented strategy in order to minimize the variation of the wheels normal forces to reduce the braking distance. These considerations are assigned to the “supervisor”, the high-level control unit in the global chassis control, that takes decisions based on all available measurements and estimated variables. 5. PERFORMANCE In order to show the effectiveness of each strategy, two different scenarios were done. First scenario chosen to be a vehicle driven on rough surface with constant speed of 50km/h without a braking phase. The road surface unevenness is chosen to include high frequency components so that it affects ride comfort and it is shown in Fig. 5. 234
0
20
40
60
80
100
120
140
160
180
200
Distance (m)
Fig. 5. Road profile. The optimal values of n for front and rear axles, in the two strategies, are very near to each other. For this purpose, and in order to reduce the complexity of the design strategy, one value for “n” is chosen to each strategy for both the front and the rear model. For comfort-oriented strategy, the chosen fractional order is n = 0.65, that is the average of the two obtained values, and for road holding oriented strategy n = 1 is considered. Fig. 6 shows the bode response of the 2nd generation CRONE controller Ua (s), where the fractional form of the transfer function (red) is compared to its rational form (blue) obtained using N recursive zeros and N recursive poles. It was calculated by the help of a powerful method developed by Oustaloup (Oustaloup (1995)). It is important to consider the identical behavior for both transfer functions in the range of [0.1-30] Hz, in order to well present the calculated fractional controller in the time domain for comfort and road holing band frequencies. Two simulations were done, the first one with the comfortoriented strategy and the other one is with the road holding oriented strategy. It should be noted that all values are normalized with respect to the values obtained from passive system. For this reason, it will be noticed in the following bar graphs that the values of the road holding strategy (blue) are almost equal to one since the derivative order was chosen n=1. The passive system and the system with road holding strategy have the same frequency response in the range of [0.1 - 30 ]Hz, as a result they have very similar response in the time domain. Fig.7 and 8 show the comparison of the results obtained with the first scenario. Fig. 7 shows the maximum values
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235
Scenario one: Without Braking 1.4
90
Comfort- oriented Road holding oriented
Gain (dB)
80
1.2
70
1 60
0.8
50 40 10-3
10-2
10-1
100
101
102
103
0.6 0.4
0 Fractional Rational
Phase (deg)
-10
0.2
-20
0
max(zG )
-30
max( G )
rms(zG )
rms( G )
-40 10-3
10-2
10-1
100
101
102
Fig. 7. RMS values of scenario one: without braking (1).
103
Frequency (rad/s)
Scenario one: Without Braking
Fig. 6. Fractional and rational transfer functions.
1.5
and the RMS values of heave and pitch angle. Fig. 8 shows the RMS values of the heave acceleration, pitch acceleration, front tire normal force and rear tire normal force. The comfort-oriented strategy results in the reduction of the RMS values of the vertical dynamics of the chassis, which is clearly presented in the heave motion, heave acceleration and pitch acceleration. However, the tire normal forces are lower for road holding oriented strategy. The second scenario depicts a sharp deceleration phase to stop the vehicle in a critical situation with an initial speed of 50 km/h. For comparison purposes, the same rough surface used for scenario one is used here. Fig.9 and 10 show a comparison between the results obtained with the two suspension control strategies. For instance, Fig.9 shows the maximum and RMS values for heave and pitch dynamics, while Fig 10 shows the comparison of the braking distance, the RMS values of heave acceleration, pitch acceleration, front tire normal force and rear tire normal force. In comfort-oriented strategy, the enhancement is obvious for heave and pitch dynamics where it leads to higher vibration isolation of the chassis at the expense of increasing the braking distance. In road holding strategy, the values of tire normal forces are lower which leads to decreasing the braking distance by 80cm. Consequently, this provides more safer drive performance in critical situations. 6. CONCLUSION
Comfort- oriented Comfort- oriented Road holding oriented Road holding oriented
1
0.5
0
rms( G )
rms(zG )
rms(Fzf )
rms(Fzr )
Fig. 8. RMS values of scenario one: without braking (2).
1.2
Scenario two: With Emergency Braking Comfort- oriented Road holding oriented
1
0.8
0.6
0.4
0.2
In this paper we emphasize the effect of coupling between the braking system and the active suspension system. We propose two control strategies for the active suspension system that could affect the performance of the braking system in order to lower the braking distance. For this purpose, two-wheel vehicle model, with seven degrees of freedom that includes most of the important characteristics of the longitudinal and vertical dynamics of the vehicle, is used for testing the performance of the two control strategies. On the other hand, the developed control strategy can provide a comfort ride for passengers where there are no critical situations. These control strategies 235
0
max(zG )
max( G )
rms(zG )
rms( G )
Fig. 9. RMS values of scenario two: with braking (1). will then be added to the global chassis control with a convenient cooperation with others strategies in order to improve the chassis dynamics of an electric vehicle.
2019 IFAC AAC 236 Orléans, France, June 23-27, 2019
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Termous, H., Shraim, H., Talj, R., Francis, C., and Charara, A. (2018b). Coordinated control strategies for active steering, differential braking and active suspension for vehicle stability, handling and safety improvement. Vehicle System Dynamics, 1–36.
Scenario two: With Emergency Braking
1.4
Comfort- oriented Road holding oriented
1.2
80cm 1 0.8
Table 1. Nomenclature.
0.6
Parameter
0.4
i
0.2
fxi
longitudinal forces generated at the tires
fzi
normal forces generated at the tires
Tb
braking torques
Tm
Motor torques
Vx
Longitudinal Velocity
ax
Longitudinal acceleration
0
Braking Distance
rms(zG )
rms( G )
rms(Fzf )
Description Front: i=1, rear: i=2
rms(Fzr )
Fig. 10. RMS values of scenario two: with braking (2). ACKNOWLEDGEMENTS This work was supported by the National Council for Scientific Research of Lebanon (CNRS-L), ”L’Agence universitaire de la Francophoniethe” (AUF), the lebanese University (UL) program and by the Lebanese research program.
m2 , m 1 , M
REFERENCES
a,b
Altet, O., Nouillant, C., Buisson, J., Moreau, X., and Oustaloup, A. (2003). Stability of hydractive crone suspension. In Analysis and Design of Hybrid Systems 2003 (ADHS 03): A Proceedings Volume from the IFAC Conference, St. Malo, Brittany, France, 16-18 June 2003, 153. Elsevier. Gaspar, P., Szabo, Z., Bokor, J., Poussot-Vassal, C., Sename, O., and Dugard, L. (2007). Toward global chassis control by integrating the brake and suspension systems. In 5th IFAC Symposium on Advances in Automotive Control, IFAC AAC 2007, 6. IFAC. Hamersma, H.A. and Els, P.S. (2014). Improving the braking performance of a vehicle with abs and a semiactive suspension system on a rough road. Journal of Terramechanics, 56, 91–101. Laurent, D., Sebe, M., and Walser, D. (2000). Assembly comprising a wheel and a suspension integrated into the wheel. US Patent 6,113,119. Moreau, X., Nouillant, C., and Oustaloup, A. (2001). Effect of the crone suspension control system on braking. IFAC Proceedings Volumes, 34(1), 53–58. Moreau, X., Rizzo, A., and Oustaloup, A. (2009). Application of the crone control-design method to a lowfrequency active suspension system. International Journal of Vehicle Autonomous Systems, 7(3-4), 172–200. Oustaloup, A. (1995). La d´erivation non enti`ere. BOOK. Hermes. Pacejka, H. (2005). Tire and vehicle dynamics. Elsevier. Sammier, D., Sename, O., and Dugard, L. (2003). Skyhook and h8 control of semi-active suspensions: some practical aspects. Vehicle System Dynamics, 39(4), 279–308. Termous, H., Moreau, X., Francis, C., and Shraim, H. (2018a). From the standard pid to the crone first generation controller: Application to an anti-roll system for electric vehicles. IFAC-PapersOnLine, 51(4), 733– 738. 236
r
h k1 , k2
Sprung mass, unsprung mass, total mass Wheel radius longitudinal distance from the vehicle’s front axle and rear axle to the centre of gravity height of the centre of gravity tire stiffness, suspension stiffness
b2
suspension damping
b1
tire damping
λ
longitudinal tire slip
σ0 , σ 2
rolling resistance coefficient
Sx, Cx
vehicle fonrtal area, drag coefficient
ρ
air density
Jω
wheel inertia
br
wheel viscous friction coefficient
ScienceDirect
IFAC PapersOnLine 52-5 (2019) 231–236
Effect of fractional order damping control Effect Effect of of fractional fractional order order damping damping control control on braking performance for electric on braking performance for electric Effect of fractional order damping control on braking performance for electric vehicles vehicles for electric on braking performance vehicles ∗,∗∗ vehicles ∗∗ ∗
Hussein Xavier Moreau ∗∗ Clovis Hussein Termous Termous ∗,∗∗ Clovis Francis Francis ∗∗∗ ∗,∗∗ Xavier Moreau ∗ ∗∗ Hussein Xavier Moreau Francis Hassan ∗ ∗∗ Clovis Hussein Termous Termous ∗,∗∗ XavierShraim Moreau Clovis Francis Hassan Shraim ∗ ∗ ∗∗ Clovis Francis ∗ Hassan Shraim Hussein Termous ∗,∗∗ XavierShraim Moreau Hassan ∗ ∗ HassanofShraim Engineering, Scientific Research ∗ Lebanese University, Faculty ∗ Lebanese University, Faculty of Engineering, Scientific Research ∗ Lebanese University, Faculty of Engineering, Scientific Research Center in Engineering (CRSI), Campus Rafic Hariri, Beirut, Lebanon, Lebanese University, Faculty of Engineering, Scientific Research Center in Engineering (CRSI), Campus Rafic Hariri, Beirut, Lebanon, ∗ Center in Engineering (CRSI), Campus Rafic Hariri, Beirut, Lebanon, (e-mail: [email protected]; [email protected]; Lebanese University, Faculty of Engineering, Scientific Research Center in Engineering (CRSI), Campus Rafic Hariri, Beirut, Lebanon, (e-mail: [email protected]; [email protected]; (e-mail: [email protected]; [email protected]; [email protected]) Center in Engineering (CRSI), Campus Rafic Hariri, Beirut, Lebanon, (e-mail: [email protected]; [email protected]; [email protected]) ∗∗ [email protected]) of Bordeaux, Lab IMS CNRS 5218, Talence, Bordeaux, (e-mail: [email protected]; [email protected]; ∗∗ University [email protected]) ∗∗ University of Bordeaux, Lab IMS CNRS 5218, Talence, Bordeaux, ∗∗ University of Lab France (e-mail: [email protected]) [email protected]) University of Bordeaux, Bordeaux, Lab IMS IMS CNRS CNRS 5218, 5218, Talence, Talence, Bordeaux, Bordeaux, France (e-mail: [email protected]) ∗∗ France (e-mail: [email protected]) University of Bordeaux, Lab IMS CNRS 5218, Talence, Bordeaux, France (e-mail: [email protected]) France (e-mail: [email protected]) Abstract: Abstract: Active Active suspensions suspensions are are an an object object of of interest interest nowadays, nowadays, they they have have been been aa topic topic of of Abstract: Active suspensions are an object of interest nowadays, they have aa topic of research decades due in Global Chassis Control (GCC). Indeed, aa wellAbstract: suspensions arekey an role object interest nowadays, they have been been topic of research for forActive decades due to to their their key role in of Global Chassis Control (GCC). Indeed, wellresearch for decades due to their key role in Global Chassis Control (GCC). Indeed, a wellcontrolled active suspension system may considerably improve not only the passenger comfort Abstract: Active suspensions are an object of interest nowadays, they have been a topic of research for decades due to system their key role in Global Chassis Indeed,comfort a wellcontrolled active suspension may considerably improveControl not only(GCC). the passenger controlled active suspension system may considerably improve not only(GCC). thethe passenger comfort by compensating the chassis dynamics but also road handling by minimizing dynamics of the research for decades due to their key role in Global Chassis Control Indeed, a wellcontrolled active suspension system may considerably improve not only the passenger comfort by compensating the chassis dynamics but also road handling by minimizing the dynamics of the by the chassis dynamics but also road by minimizing dynamics of wheels normal forces. It enhance vehicle stability by controlling the load distribution, controlled active suspension system may only thethe passenger comfort by compensating compensating the chassis butconsiderably also road handling handling by not minimizing the dynamics of the the wheels normal forces. It may may dynamics enhance vehicle stability byimprove controlling the wheel wheel load distribution, wheels normal forces. It may enhance vehicle stability by controlling the wheel load distribution, which influences the lateral and longitudinal dynamics to provide better steering performance, by compensating chassis dynamics but also road handling by minimizing the dynamics of the wheels normal forces. It mayand enhance vehicle stability bytocontrolling the wheel loadperformance, distribution, which influences the lateral longitudinal dynamics provide better steering which influences the lateral lateral and longitudinal dynamics tocontrolling provide better better steering performance, and safety by the distance. The contribution this paper is wheels normal forces. It may enhance vehicle stability byto theof loadperformance, distribution, which influences the and longitudinal dynamics provide steering and safety by reducing reducing the braking braking distance. The main main contribution ofwheel this paper is to to study study and impact safety by reducing the braking distance. The main mainin contribution ofofthis this paper is to study study the of fractional order (FO) control methods the context an active suspension which influences the lateral and longitudinal dynamics to provide better steering performance, and safety by reducing the braking distance. The contribution of paper is to the impact of fractional order (FO) control methods in the context of an active suspension the impact of fractional order (FO) control methods in the context of an active suspension system and in particular fractional order damping control. We derive two optimal damping and safety by reducing the braking distance. The main contribution of this paper is to study the impact of fractional order (FO) control methods in the context of an active suspension system and in particular fractional order damping control. We derive two optimal damping system and in particular fractional control. derive optimal damping control strategies: aa comfort-oriented and aa damping road holding strategy, based on the impact order (FO) order control methods inoriented the We context an active suspension system and of in fractional particular fractional order damping control. We deriveof two two optimal damping control strategies: comfort-oriented and road holding oriented strategy, based on non-integer non-integer control strategies: strategies: a comfort-oriented comfort-oriented andrelevant road holding holding oriented strategy, based on non-integer non-integer derivation deflection. benefits of methods on vehicle response system andof particular fractionalThe order control. We strategy, derive two damping control a and aa damping road oriented based on derivation ofinsuspension suspension deflection. The relevant benefits of both both methods onoptimal vehicle response derivation of suspension suspension deflection. The relevant benefits of both bothstrategy, methods on vehicle vehicle response while driving on rough roads and during braking are evaluated using computer simulations control strategies: a comfort-oriented and a road holding oriented based on non-integer derivation of deflection. The relevant benefits of methods on response while driving on rough roads and during braking are evaluated using computer simulations while driving on rough roads and during braking are evaluated using computer simulations applied on a two-wheel vehicle model. derivation of suspension deflection. The relevant benefits of both methods on vehicle response while driving on rough roads and during braking are evaluated using computer simulations applied on a two-wheel vehicle model. applied on model. while on rough vehicle roads and during braking are evaluated using computer simulations applieddriving on a a two-wheel two-wheel vehicle model. © 2019, IFAC (International Federation of Automatic Control) Hosting by Elsevier Ltd. All rights reserved. applied on a two-wheel vehicle model. Keywords: Keywords: Active Active suspension, suspension, fractional fractional order, order, braking, braking, comfort, comfort, road road holding, holding, damping damping Keywords: Active suspension, fractional order, braking, comfort, road holding, control. Keywords: Active suspension, fractional order, braking, comfort, road holding, damping damping control. control. Keywords: Active suspension, fractional order, braking, comfort, road holding, damping control. control. 1. INTRODUCTION INTRODUCTION and 1. and longitudinal longitudinal dynamics dynamics (Gaspar (Gaspar et et al. al. (2007))(Sammier (2007))(Sammier 1. and longitudinal dynamics (Gaspar et et al. (2003)). In this domain, fractional order 1. INTRODUCTION INTRODUCTION and longitudinal dynamics (Gaspar et al. al. (2007))(Sammier (2007))(Sammier et al. (2003)). In this domain, fractional order control control et al. al. (2003)). In this prevalence domain, fractional order control laws have gained their where they show their 1. INTRODUCTION Active suspension systems are proving their importance and longitudinal dynamics (Gaspar et al. (2007))(Sammier et (2003)). In this domain, fractional order control Active suspension systems are proving their importance laws have gained their prevalence where they show their laws have gained their prevalence where they show their Active suspension systems are proving their importance ability in solving different control problems (Moreau et al. in the field of vehicle dynamics. Many studies in the et al. (2003)). In this domain, fractional order control laws have gained their prevalence where they show their Active suspension systems are proving their importance in the field of vehicle dynamics. Many studies in the ability in solving different control problems (Moreau et al. ability in solving different control problems (Moreau et al. in the field of vehicle dynamics. Many studies in the (2001)). literature show their ability to enhance comfort for paslaws have gained their prevalence where they show their Active suspension systems are proving their importance ability in solving different control problems (Moreau et al. in the field of vehicle dynamics. Many studies in the (2001)). literature show their ability to enhance comfort for pas(2001)). literature show their ability to enhance comfort for passengers (Moreau et al. (2009)), vehicle stability (Altet ability in solving different control problems (Moreau et al. in the field of vehicle dynamics. Many studies in the literature(Moreau show their ability to enhance pas- (2001)). sengers et al. (2009)), vehicle comfort stabilityfor(Altet In this work, the impact of fractional order control methsengers (Moreau et al. (2009)), vehicle stability (Altet In this work, the impact of fractional order control methet al. (2003)) and driving performance (Termous et al. (2001)). literature show their ability to enhance comfort for passengers (Moreau et al. (2009)), vehicle stability (Altet et al. (2003)) and driving performance (Termous et al. ods In work, the of order control methhas been where we two In this this the impact impact of fractional fractional control dampmethhaswork, been studied, studied, where we derive derive order two optimal optimal dampet al. and performance et (2018b)) compared to passive systems. One important important sengers (Moreau al. passive (2009)), vehicle (Termous stability (Altet et al. (2003)) (2003)) andetdriving driving performance (Termous et al. al. ods (2018b)) compared to systems. One odsthis haswork, been studied, where we derive derive order twoand optimal damping control strategies, a comfort-oriented a road holdIn the impact of fractional control methods has been studied, where we two optimal damp(2018b)) compared to passive systems. One important ing control strategies, a comfort-oriented and a road holdsafety requirement of any vehicle is the braking distance et al. (2003)) and of driving performance (Termous et al. ing control (2018b)) compared toany passive systems. One important safety requirement vehicle is the braking distance strategies, a comfort-oriented comfort-oriented and a road road holdoriented strategy, based on the non-integer derivation ods has been studied, where we derive two optimal damping control strategies, a and a holdsafety requirement of any vehicle is the braking distance ing oriented strategy, based on the non-integer derivation in order to prevent accidents or at least lessen the impact. (2018b)) compared to passive systems. One important safety of any vehicle thelessen braking in orderrequirement to prevent accidents or at is least thedistance impact. ing oriented strategy, based on the non-integer derivation of the suspension deflection using the active suspension control strategies, a comfort-oriented and a road holding oriented strategy, based on the non-integer derivation in to prevent or least lessen the impact. the suspension deflection using the active suspension For this purpose, rapid advance has been made in this this of safety requirement of any advance vehicle thebeen braking in order order to preventaaaccidents accidents or at at is least lessen thedistance impact. For this purpose, rapid has made in of the suspension deflection using the active suspension system introduced in the Active Wheel (Laurent et al. ing oriented strategy, based on the non-integer derivation of the suspension deflection using the active suspension system introduced in the Active Wheel (Laurent et al. For this purpose, a rapid advance has been made in this field, most notably the development of the the anti-lock brake in to preventathe accidents or at least lessen the impact. Fororder this purpose, rapid advance has been made inbrake this of field, most notably development of anti-lock system introduced inkind the of Active Wheel (Laurent etused al. (2000)) (Fig.1). This active wheels is mostly the suspension deflection using the active suspension system introduced in the Active Wheel (Laurent et al. field, most notably the development of the anti-lock brake (2000)) (Fig.1). This kind of active wheels is mostly used systems (ABS). For purpose, rapid advance has been made inbrake this in field,this most notablyathe development of the anti-lock systems (ABS). (2000)) (Fig.1). This kind of active wheels is mostly mostly used light electrical cars, where two in-wheel 30-kW mosystem introduced in the Active Wheel (Laurent et al. (2000)) (Fig.1). This kind of active wheels is used systems (ABS). light electrical cars, where two in-wheel 30-kW mofield, most notably the development of the anti-lock brake in systems (ABS). in light electrical cars, where two in-wheel 30-kW motors motorize it. The suspension system is composed However, due to the coupling effects between the suspen(2000)) (Fig.1). This kind of active wheels is mostly used in light electrical cars, where two in-wheel 30-kW moHowever, due to the coupling effects between the suspen- tors motorize it. The suspension system is composed of of systems (ABS). motorize it. The system is of However, due the coupling effects between the ators and engine. This system offers sion dynamics and the braking system (Hamersma and in light electrical cars,suspension where in-wheel 30-kW motors motorize it. electric The suspension system is composed composed of However, due to toand thethe coupling effects between the suspensuspension dynamics braking system (Hamersma and a spring spring and an an electric engine.two This system offers new new a spring and an electric engine. This system offers new sion dynamics and the braking system (Hamersma and perspectives in the suspension control, which indeed offers Els (2014)), active suspension control laws, where the tors motorize it. The suspension system is composed of However, due to the coupling effects between the suspena spring and an electric engine. This system offers new sion (2014)), dynamicsactive and the braking control system laws, (Hamersma Els suspension where and the perspectives in the suspension control, which indeed offers perspectives in the suspension control, which indeed offers Els (2014)), active suspension control laws, where the important opportunities in global chassis control. It proves strategy is comfort-oriented, may degrade the performance a spring and an electric engine. This system offers new sion dynamics and the braking system (Hamersma and perspectives in the suspension control, which indeed offers Els (2014)), active suspension control laws, where the strategy is comfort-oriented, may degrade the performance important opportunities in global chassis control. It proves important opportunities in chassis control. It strategy is may degrade the in safer steering using the active of braking onactive rough suspension roads. Indeed, Indeed, the active active suspension perspectives in the suspension control, which indeed offers Els (2014)), laws, where the its important opportunities in global global chassis control. It proves proves strategy is comfort-oriented, comfort-oriented, may control degrade the performance performance of braking on rough roads. the suspension its effectiveness effectiveness in providing providing safer steering using the active its effectiveness in providing safer steering using the of braking on rough roads. Indeed, the active suspension roll control system by compensating roll dynamics when forces introduced by the electric actuator affect the wheel important opportunities in global chassis control. It proves strategy is comfort-oriented, may degrade the performance in providing safer steering using the active active of braking on rough roads. Indeed, the active forces introduced by the electric actuator affectsuspension the wheel its rolleffectiveness control system by compensating roll dynamics when rolleffectiveness control system system by compensating compensating roll dynamics dynamics when forces introduced by the electric actuator affect the wheel manoeuvring (Termous et al. (2018a)). normal forces, that in turn impact the wheel longitudinal its in providing safer steering using the active of braking on rough roads. Indeed, the active suspension roll control by roll when forces introduced by the electric actuator affect the wheel normal forces, that in turn impact the wheel longitudinal manoeuvring (Termous et al. (2018a)). manoeuvring (Termous et al. (2018a)). normal forces, that in turn impact the wheel longitudinal contact forces that are directly responsible for the braking roll control system by compensating roll dynamics when forces introduced by the electric actuator affect the wheel manoeuvring (Termous et al. (2018a)). normal forces, that in turn impact the wheel longitudinal contact forces that are directly responsible for the braking The optimized control concept is done to utilize the availcontact forces that are directly responsible for the braking The optimized control concept is done to utilize the availbehaviour of a vehicle. For this purpose, searching for new manoeuvring (Termous et al. (2018a)). normal forces, that in turn impact the wheel longitudinal contact forces that are directly responsible for the braking behaviour of a vehicle. For this purpose, searching for new The The optimized optimized control the concept is done to utilize the under available adhesion tires and the road surface control concept done utilize the availbehaviour of vehicle. this purpose, for adhesion between between the tires is and the to road surface under design control strategies using suspension control becomes contact forces are For directly responsible for thebecomes braking behaviour of aa that vehicle. For thissuspension purpose, searching searching for new new able design control strategies using control able adhesion between the tires and the road surface under braking to the maximum possible degree, to minimize The optimized control concept is done to utilize the available adhesion between the tires and the road surface under design using suspension control becomes to the maximum possible degree, to minimize topiccontrol of of research nowadays order enhance lateral a strategies vehicle. For thisin purpose, for new braking design control strategies using suspension control aabehaviour topic of research nowadays in order to tosearching enhancebecomes lateral to the maximum possible degree, to minimize braking distance. This control strategy will offer a better able adhesion between the tires and the road surface under to the maximum possible degree, to minimize braking distance. This control strategy will offer a better adesign of nowadays order to enhance lateral control strategies using in suspension becomes a topic topicwork of research research nowadays order research to control enhance lateral distance. This control strategy will offer a better was supported by the in Lebanese program and riding and reductions. A two-wheel braking to the maximum possible degree, to minimize This distance. Thiscost control strategy offer a vehicle better This work was supported by the in Lebanese research program and riding experience experience and cost reductions. A will two-wheel vehicle a topic of research nowadays order to enhance lateral theThis AUF-CNRSL-UL program. This work was was supported supported by the the Lebanese Lebanese research research program program and riding experience and cost reductions. A two-wheel vehicle braking distance. This control strategy will offer a better work by and riding experience and cost reductions. A two-wheel vehicle the AUF-CNRSL-UL program. the AUF-CNRSL-UL program. theThis AUF-CNRSL-UL program. work was supported by the Lebanese research program and riding experience and cost reductions. A two-wheel vehicle the AUF-CNRSL-UL program. Copyright © 231 Hosting by Elsevier Ltd. All rights reserved. 2405-8963 © 2019 2019, IFAC IFAC (International Federation of Automatic Control) Copyright © 2019 IFAC 231 Peer review under responsibility of International Federation of Automatic Copyright © 2019 IFAC 231 Copyright © 2019 IFAC 231 Control. 10.1016/j.ifacol.2019.09.037 Copyright © 2019 IFAC 231
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- sprung masses motion: m2i z¨2i = f0i + k2 (z1i − z2i ) + b2i (z˙1i − z˙2i ) + uai ,
(3)
where f0i are the front and rear load shift due to the longitudinal acceleration or deceleration, formulated as: f01 =
h M ax (t), and a+b
(4) h f02 = − M ax (t). a+b The heave and pitch motion of the chassis could be calculated using the relation between zG , θG and the dynamics of front and rear sprung masses, by:
Fig. 1. Active wheel from Michelin.
z21 = zG − a sin(θG ), and
(5)
z22 = zG + b sin(θG ). Supposing that θG has a small value, and using the inverse of (5), we get the following relation: −1 z21 1 −a zG . (6) = 1 b θG z 22 2.2 Nonlinear model of the longitudinal dynamics
Fig. 2. Two-wheel, 7 degrees of freedom, car model. model is used to test the proposed control strategy for the vehicle chassis control. This paper is organized as follows: the next section presents the two-wheel vehicle model, then in section 3 we introduce the different suspension systems used. Section 4 illustrates the optimization method and the control strategy. Section 5 shows the results obtained and finally a conclusion and further prospects are given.
where vx (t) is the vehicle longitudinal velocity, Fa (t) is the aerodynamics force described by: Fa (t) =
1 Sx ρ Cx vx (t)2 , 2 2
The motion of the two-wheels car model, as shown in Fig.2, is described by seven differential equations. There are four equations to derive a linear model for the vertical dynamics and three equations to derive a nonlinear model for the longitudinal dynamics. 2.1 Linear model of the vertical dynamics The linear model of the vertical dynamics can be described by the second order differential equations as follows: - unsprung masses motion: (1)
where uai are the forces developed by the electric actuators embedded in the active wheel, with i = 1 for front and i = 2 for the rear axles. Denote by fzi , the wheel ground contact forces, that is defined by: fzi = (m1i + m2i )g + k1i (z1 − z0 ) + b1i (z˙1 − z˙0 ). (2) 232
(8)
Frr (t) is the rolling resistance force: Frr (t) = (σ0 + σ2 vx (t)2 ) M g,
2. DYNAMIC MODEL
m1i z¨1i = k1i (z1 − z0 ) + b1i (z˙1 − z˙0 ) − k2i (z1i − z2i ) − b2 (z˙1i − z˙2i ) − uai ,
In this paper, only the longitudinal dynamics were considered and the lateral dynamics were neglected, and it is defined by: (7) M v˙ x (t) = Fx1 (t) + Fx2 (t) − Fa (t) − Frr (t),
(9)
Fxi (t) are the tyre longitudinal forces generated according to Pacejka wheel model, one of the most complete studies of tire force generation (Pacejka (2005)), where more than 30 empirical parameters are needed to fully describe experimental measurements. The calculation of the tire longitudinal forces depends on the value of the road adherence coefficient, supposed to be µ = 1 for dry pavement, and on the value of tire slip rate λi that is defined by: vx (t) λ =1− , if Ωi (t) r ≥ vx (t) (accelerating) i Ωi (t) r , Ω λi = i (t) r − 1, if Ωi (t) r < vx (t) (braking) vx (t) (10) where Ωi (t) is the wheel angular velocity. The dynamics of the wheel angular velocity is given by: ˙ Jω Ω(t) = Tmi (t) − Tbi (t) − r Fxi (t) − br Ω(t),
(11)
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where Tmi (t) represents the traction torque generated by the electric motor on each wheel and Tbi (t) is the braking torque applied.
V1 (s) =
• isolate the vehicle from road surface irregularities; • control variations in vertical load at the road/tire interface; • control body motion caused by road inputs, inertia forces, and aerodynamic forces; • maintain vehicle directional stability during maneuvers. In fact, the main problem is the conflict between these various aspects of vehicle behavior. 3.1 Traditional suspension A passive vibration control unit consists of a spring and an energy dissipator. The suspension is modeled as a linear spring and a damper. The expression of the force fsi developed by the suspension is as follows: fsi (t) = k2i (z2i − z 1i ) + b2i (z˙2i − z˙ 1i ).
(12)
3.2 Active wheel suspension system The active suspension system embedded in the Active Wheels can be modeled by a spring, an electric actuator that acts as a force generator and a small frictional damping ratio that results from the mechanical links of the system. The idea is to use the force generator as a continuously controllable damper that develops a force ua that is proportional to the fractional derivative of the suspension deflection, namely: uai (t) = ba
d dt
ni
(z2i (t) − z1i (t)) ,
(13)
with 0 < n < 2 is the non-integer derivative order, and ba is a constant gain factor assumed equal to the damping coefficient of the passive system. The Laplace transform of relation (13) leads to the following transfer function: 1 Uai (s) = −ba 1−ni , (14) (V2i (s) − V1i (s)) s The force fsi developed by the active suspension is then given by: ni d fsi (t) = k2i (z2i − z 1i ) + ba (z2i (t) − z1i (t)) . (15) dt The Laplace transform of the vertical dynamics of the quarter vehicle system with fractional order damping control leads to the following transfer functions:
233
(m2 s2 + b2 s + ba sn + k2 )(b1 s + k1 ) V0 (s) den(s) s(b2 + ba sn + k2 ) + F0 (s), den(s)
3. SUSPENSION MODEL The vehicle suspension system is required to fulfill a number of functions. These can be stated as follows:
233
(16)
(b2 s + ba sn + k2 )(b1 s + k1 ) V0 (s) + den(s) (17) s m1 s2 + (b2 + b1 )s + ba sn + (k2 + k1 ) F0 (s), den(s)
V2 (s) =
with den(s) = m1 m2 s4 + (m2 (b1 + b2 ) + m1 b2 ) s3 + (m2 (k1 + k2 ) + m1 k2 + b1 b2 ) s2 + (k1 b2 + b1 k2 ) s +k1 k2 + ba (m1 + m2 ) s2+n + ba b1 s1+n + ba k1 sn . (18) V2 (s), V1 (s), V0 (s) and F0 (s) are the Laplace transform of Z˙ 2 (t), Z˙ 1 (t), Z˙ 0 (t) and f0 (t)respectively. It can be noticed that for ba = 0 is the case of passive suspension, and when b2 = 0 is the case of active suspension. For the particular case, n = 1 and ba = b2 , the active suspension system acts as the passive system.
4. CONTROL STRUCTURE The proposed strategy for the active suspension is shown in Fig. 3. It consists of a feedback control law used to regulate the suspension deflection versus road input disturbances. Supposing “ba ” is constant, the only controlled parameter is the fractional order “n”. The value of n should be chosen according to the selected objectives. In order to find these values of n, three sensitivity transfer function were formulated: A2 (s) sV2 (s) = , Ha (s) = V0 (s) F0 V0 (s) F0 =0
(19)
where Ha (s) represents the sensitivity of the vehicle’s body vertical acceleration to road disturbances. It evaluates the vibration isolation, where small gains results in better ride comfort. Z1 (s) − Z2 (s) , H12 (s) = V0 (s) F0 =0
(20)
H12 (s) is a constrain for the suspension deflection, and
H01 (s) =
Z1 (s) − Z0 (s) , V0 (s) F0 =0
(21)
where H01 (s) is a road holding indicator, and a constrain for wheel normal forces. For optimization purposes, three criteria were formed:
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2 Ja J12 J01
1.8 1.6 1.4
Criteria
1.2 1 0.8 0.6 0.4 0.2 0 0
(22)
0.6
0.8
1
1.2
1.4
1.6
1.8
2
1.5 1 0.5 0 -0.5 -1 -1.5
According to these criteria, when the value is less than 1 it indicates an improvement over the passive system. The optimal control problem can thus be formulated as finding the minimal values of ni for each criterion. Here, two strategies are derived (Fig.4). The comfort-oriented strategy where: (23)
and for the road holding oriented strategy we have: n1 = 1.1 and n2 = 1,
0.4
Fig. 4. Optimized damping control normalized criteria, ( - - ) Rear model, (—) Front model.
where η1 ,η2 , and η3 represent the values of the calculated criteria for the passive suspension system (ba = 0). The range of integration is between ωmin = 2π0.1rad/s and ωmax = 2π30rad/s.
n1 = 0.67 and n2 = 0.64,
0.2
Order n
Elevation (cm)
Fig. 3. Block diagram of the control strategy. ω max 1 2 |Ha (jω)| dω (n) = J (n) = J a 1 η 1 ω min ω max 1 2 J2 (n) = J12 (n) = |H12 (jω)| dω , η 2 ω min ω max 1 2 J3 (n) = J01 (n) = |H01 (jω)| dω η3 ω min
(24)
The suspension control strategy must be changed according to the road state of surface. In case of a smooth road, the road holding strategy is then preferred. It is the same case also when a critical braking situation revealed where comfort-oriented strategy is no longer a priority, here the suspension must be switched to road holding oriented strategy in order to minimize the variation of the wheels normal forces to reduce the braking distance. These considerations are assigned to the “supervisor”, the high-level control unit in the global chassis control, that takes decisions based on all available measurements and estimated variables. 5. PERFORMANCE In order to show the effectiveness of each strategy, two different scenarios were done. First scenario chosen to be a vehicle driven on rough surface with constant speed of 50km/h without a braking phase. The road surface unevenness is chosen to include high frequency components so that it affects ride comfort and it is shown in Fig. 5. 234
0
20
40
60
80
100
120
140
160
180
200
Distance (m)
Fig. 5. Road profile. The optimal values of n for front and rear axles, in the two strategies, are very near to each other. For this purpose, and in order to reduce the complexity of the design strategy, one value for “n” is chosen to each strategy for both the front and the rear model. For comfort-oriented strategy, the chosen fractional order is n = 0.65, that is the average of the two obtained values, and for road holding oriented strategy n = 1 is considered. Fig. 6 shows the bode response of the 2nd generation CRONE controller Ua (s), where the fractional form of the transfer function (red) is compared to its rational form (blue) obtained using N recursive zeros and N recursive poles. It was calculated by the help of a powerful method developed by Oustaloup (Oustaloup (1995)). It is important to consider the identical behavior for both transfer functions in the range of [0.1-30] Hz, in order to well present the calculated fractional controller in the time domain for comfort and road holing band frequencies. Two simulations were done, the first one with the comfortoriented strategy and the other one is with the road holding oriented strategy. It should be noted that all values are normalized with respect to the values obtained from passive system. For this reason, it will be noticed in the following bar graphs that the values of the road holding strategy (blue) are almost equal to one since the derivative order was chosen n=1. The passive system and the system with road holding strategy have the same frequency response in the range of [0.1 - 30 ]Hz, as a result they have very similar response in the time domain. Fig.7 and 8 show the comparison of the results obtained with the first scenario. Fig. 7 shows the maximum values
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235
Scenario one: Without Braking 1.4
90
Comfort- oriented Road holding oriented
Gain (dB)
80
1.2
70
1 60
0.8
50 40 10-3
10-2
10-1
100
101
102
103
0.6 0.4
0 Fractional Rational
Phase (deg)
-10
0.2
-20
0
max(zG )
-30
max( G )
rms(zG )
rms( G )
-40 10-3
10-2
10-1
100
101
102
Fig. 7. RMS values of scenario one: without braking (1).
103
Frequency (rad/s)
Scenario one: Without Braking
Fig. 6. Fractional and rational transfer functions.
1.5
and the RMS values of heave and pitch angle. Fig. 8 shows the RMS values of the heave acceleration, pitch acceleration, front tire normal force and rear tire normal force. The comfort-oriented strategy results in the reduction of the RMS values of the vertical dynamics of the chassis, which is clearly presented in the heave motion, heave acceleration and pitch acceleration. However, the tire normal forces are lower for road holding oriented strategy. The second scenario depicts a sharp deceleration phase to stop the vehicle in a critical situation with an initial speed of 50 km/h. For comparison purposes, the same rough surface used for scenario one is used here. Fig.9 and 10 show a comparison between the results obtained with the two suspension control strategies. For instance, Fig.9 shows the maximum and RMS values for heave and pitch dynamics, while Fig 10 shows the comparison of the braking distance, the RMS values of heave acceleration, pitch acceleration, front tire normal force and rear tire normal force. In comfort-oriented strategy, the enhancement is obvious for heave and pitch dynamics where it leads to higher vibration isolation of the chassis at the expense of increasing the braking distance. In road holding strategy, the values of tire normal forces are lower which leads to decreasing the braking distance by 80cm. Consequently, this provides more safer drive performance in critical situations. 6. CONCLUSION
Comfort- oriented Comfort- oriented Road holding oriented Road holding oriented
1
0.5
0
rms( G )
rms(zG )
rms(Fzf )
rms(Fzr )
Fig. 8. RMS values of scenario one: without braking (2).
1.2
Scenario two: With Emergency Braking Comfort- oriented Road holding oriented
1
0.8
0.6
0.4
0.2
In this paper we emphasize the effect of coupling between the braking system and the active suspension system. We propose two control strategies for the active suspension system that could affect the performance of the braking system in order to lower the braking distance. For this purpose, two-wheel vehicle model, with seven degrees of freedom that includes most of the important characteristics of the longitudinal and vertical dynamics of the vehicle, is used for testing the performance of the two control strategies. On the other hand, the developed control strategy can provide a comfort ride for passengers where there are no critical situations. These control strategies 235
0
max(zG )
max( G )
rms(zG )
rms( G )
Fig. 9. RMS values of scenario two: with braking (1). will then be added to the global chassis control with a convenient cooperation with others strategies in order to improve the chassis dynamics of an electric vehicle.
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Termous, H., Shraim, H., Talj, R., Francis, C., and Charara, A. (2018b). Coordinated control strategies for active steering, differential braking and active suspension for vehicle stability, handling and safety improvement. Vehicle System Dynamics, 1–36.
Scenario two: With Emergency Braking
1.4
Comfort- oriented Road holding oriented
1.2
80cm 1 0.8
Table 1. Nomenclature.
0.6
Parameter
0.4
i
0.2
fxi
longitudinal forces generated at the tires
fzi
normal forces generated at the tires
Tb
braking torques
Tm
Motor torques
Vx
Longitudinal Velocity
ax
Longitudinal acceleration
0
Braking Distance
rms(zG )
rms( G )
rms(Fzf )
Description Front: i=1, rear: i=2
rms(Fzr )
Fig. 10. RMS values of scenario two: with braking (2). ACKNOWLEDGEMENTS This work was supported by the National Council for Scientific Research of Lebanon (CNRS-L), ”L’Agence universitaire de la Francophoniethe” (AUF), the lebanese University (UL) program and by the Lebanese research program.
m2 , m 1 , M
REFERENCES
a,b
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r
h k1 , k2
Sprung mass, unsprung mass, total mass Wheel radius longitudinal distance from the vehicle’s front axle and rear axle to the centre of gravity height of the centre of gravity tire stiffness, suspension stiffness
b2
suspension damping
b1
tire damping
λ
longitudinal tire slip
σ0 , σ 2
rolling resistance coefficient
Sx, Cx
vehicle fonrtal area, drag coefficient
ρ
air density
Jω
wheel inertia
br
wheel viscous friction coefficient