CRONE Cruise Control System

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xavier.moreau@ims-bordeaux.fr; pierre[email protected] ). M. Moze and F. Guillemard are with PSA Peugeot Citroën, 78943 Vélizy-. Villacoublay ...
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VT-2014-00452.R2: CRONE Cruise Control System

1

CRONE Cruise Control System A. Morand, X. Moreau, P. Melchior, M. Moze and F. Guillemard  Abstract—This paper deals with vehicle longitudinal control performed by a cruise control (CC) system using CRONE approach. A comparison between this approach, a classical Proportional-Integral (PI) controller and an H-infini (Hinf) controller is presented. Simulations with these controllers are performed considering a complex vehicle model obtained from simulator software taking mass uncertainties, aerodynamic drag, gravity and wind forces into account. This model is also accounting for wheels dynamics with tire-road interaction which allow the analyse of the adherence effect. Results show better robustness to uncertainties with the second generation CRONE controller as already demonstrated with the CRONE suspension system and the CRONE ABS system. Index Terms—CRONE controller, cruise control system, robustness analysis, vehicle dynamics.

I. INTRODUCTION

T

RAFFIC safety is a major concern of automotive research with the aim of always reducing road casualties. In this context “primary safety” refers to systems that prevent crash [1] whereas “secondary safety” refers to systems that minimize the impact of a potential crash on the car occupants [2]. New vehicles tend to be equipped with more Advanced Driver Assistance Systems (ADAS) such as lane keeping systems [3] or Adaptive Cruise Control (ACC) [4]. Until now cost was a limitation and complex systems such as ACC were essentially implemented on premium vehicles. This will fade thanks to the evolutions of EuroNcap norms [5]. These kinds of features will progressively be mandatory to be awarded with five stars. Recent surveys show that most of car makers are working on the autonomous vehicle [6]. Such systems have many advantages not only for safety but also for traffic fluency and for fuel consumption reduction. Some studies [7] [8] are looking to design new ADAS inspired by fish and in particular from their capacity to travel Manuscript received March 14, 2014; revised August 4 14, 2014 and November 10, 2014; accepted December 13, 2014. Date of publication January, 2015; date of current version 2015. The review of this paper was coordinated by Pr. M. Benbouzid. A. Morand is with the Laboratory IMS, UMR 5218 CNRS, University of Bordeaux, 33405 Talence, France, and also with PSA Peugeot Citroën, 78943 Vélizy-Villacoublay, France (e-mail: [email protected]). X. Moreau and P. Melchior are with the Laboratory IMS, UMR 5218 CNRS, University of Bordeaux, 33405 Talence, France, and also with OpenLab PSA-IMS “Electronics and Systems for Automotive” (e-mail: [email protected]; [email protected] ). M. Moze and F. Guillemard are with PSA Peugeot Citroën, 78943 VélizyVillacoublay, France, and also with OpenLab PSA-IMS “Electronics and Systems for Automotive” (e-mail: [email protected]; [email protected])

cooperatively in group without collision and without any leader [9]. The behavior of each individual thus engenders a form of collective behavior promoting smoother travel and greater safety for all. The benefits linked to this type of individual assistance are shown to be greater safety, improved energy performance and shorter travel time. The overall behavior of the shoal obtained through technology is suited to the road environment [10] by: • Avoiding collisions between vehicles and adapting to the road geometry • Making the best use of the space available for efficient traffic management • Minimizing travel time, emissions and consumption by driving at an appropriate speed This fact (time gain) is presented in [11] that shows how the penetration of ACC contributes to driving time reduction. In this paper, focus is made on longitudinal control performed by cruise control system that regulates speed of the vehicle around a reference given by the driver or a more complex system ahead. A first work [12] was already presented by the authors with a very simple vehicle model without the wheels dynamics. However, most of previous works ([13] [14] [15]) on longitudinal control were based on simpler models neglecting important nonlinear aspects of vehicle such as load variation, rolling resistance, road adherence effects. Here the model used is more complex, it is accounting for wheels dynamics with tire-road interaction which allow the analyse of the adherence effect. Different control strategies can easily be found in literature [13] including fuzzy logic, gain scheduling control involving PIDs [14], sliding mode control [15]. The method presented in this paper is based on a CRONE control strategy, already known for its applications [16] to [28] but never applied to cruise control systems. Results are compared to those obtained using a classical proportional-integral (PI) controller because of its wide acceptance across industries and to those obtained using an H-infini (Hinf) controller for its robustness property. This paper is composed of six sections. After this introduction, Section 2 presents the context. Section 3 gives a nonlinear model used for analysis of vehicle dynamic behavior from which a linearized model is deduced for cruise-control design in frequency-domain. Section 4 deals with the design of the PI, Hinf and CRONE controllers, while Section 5 provides a robustness analysis of the stability degree and the simulation results. Finally, conclusion and prospects are presented in Section 6.

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VT-2014-00452.R2: CRONE Cruise Control System II. PROBLEM FORMULATION The car cruise control system design is considered in this section. The road is assumed to be straight and with no overtaking possibility, meaning the vehicles can only drive on one lane. These assumptions are made for the focus on longitudinal dynamics only. The vehicle of interest is a small car designed especially for urban environment. It can take only two persons including the driver. This electrical car is motorized by two in-wheel 30kW motors, similar to the one presented in Fig. 1, in front wheels only [29].

Fig. 1. Active wheel from Michelin

The vehicle is equipped with a cruise control system that the driver can activate with a target velocity from 30 km/h to 110 km/h. Different cases are presented to compare performance of a CRONE controller and a PI one. The driver activates the CC system at a certain velocity, called velocity target in the following, and the vehicle should follow the speed, even when confronted to a wind gust and/or a non-flat road. This study seeks to establish a vehicle model, including wheels dynamics, to obtain a cruise controller taking account any parameter variation. An important care is thus brought to the sensibility analysis. III. SYSTEM MODELLING As the cruise controller is part of a more complex ADAS that will be tested on a driving simulator (Fig. 2), the final model used for this application will be the one from simulator software SCANeR Studio. This simulator is acknowledged to be realistic and to reflect real-life driving situations for both the driver and the vehicle. Feelings of driving are achieved by a real driver’s cab moving with six jacks along rails of 10 m in longitudinal and 15 m in lateral.

2 Remark Regarding all physical quantities, the following notation is adopted in the whole paper: (1) X t   X e  xt  where X(t) is the value of the physical quantity, in an absolute referential, Xe is defined as the reference (constant) value fixed by the steady-state operation around which dynamical behavior is studied, x(t) represents the small variations of X(t) around its equilibrium value Xe. In short, all notations in small letter are variations and all physical quantities without capital notation are quantities with a zero reference. In this framework, Laplace transform of variation x(t) is ~ written X s   Ltxt  . A. Vehicle Model For the purpose of this analysis where the focus is done on longitudinal dynamics, the vehicle representation is the single track one also known as bicycle model. In this approach the forces applied on the vehicle along with its geometric parameters are represented on Fig. 3 [30].

Fig. 3. Vehicle notation

Regarding only longitudinal speed, lateral dynamics are neglected. This model includes longitudinal components of aerodynamic drag Fa, rolling resistance Frr, gravity and wind forces f0x. Longitudinal dynamics can be described by Newton’s second law by (2) M t Vx t   F x t   Fg t   f 0 x t   Fa Vx   Frr Vx  , with

F x t   Fx11t   Fx12 t   Fx 21t   Fx 22 t  ,

(3) where each Fxij is respectively each pneumatic tangential force to the road. For the purpose of clarity, indices are defined for each wheel as a matrix as shown in Fig. 4.

Fig. 4. Wheels notation

Fig. 2. Driving simulator

In this part, the vehicle model is described. This model is not linear and is used for validation purposes only. Some simplifications are thus made in a second time to obtain a linear model that is used for controller design.

Mt is the vehicle mass, f0x(t) is the longitudinal wind force applied at the gravity center, Fg(t) is the gravity induced force according to the road slope αr(t): (4) Fg t   M t g sinr t  , Fa(Vx) is the aerodynamic drag force according to the longitudinal velocity Vx

0018-9545 (c) 2015 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission. See http://www.ieee.org/publications_standards/publications/rights/index.html for more information. Copyright (c) 2015 IEEE. Personal use is permitted. For any other purposes, permission must be obtained from the IEEE by emailing [email protected].

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VT-2014-00452.R2: CRONE Cruise Control System 1  S x C xx Vx2 t  , 2

(5)

in which ρ is the air density, Cxx is the aerodynamic drag coefficient and Sx is the frontal area of the vehicle, and Frr(Vx) is the resultant force of the rolling resistance from the four wheels expressed in the center of gravity : (6) Frr Vx   0  2 Vx2 t  M t g , with λ0 and λ2 are positive constants. Vehicle motion is generated by forces applied at the wheelroad contacts considered as points. In the special case of this study, absence of torque applied at rear wheels leads to negligible forces for Fx21(t) and Fx22(t), hence to the simplification FΣx(t) = Fx11(t) + Fx12(t). Front wheels dynamics are described by:



 J r   J r



Ω11(t )  Cm11(t )  r0 Fx11(t )  C f 11(t ) Ω12 (t )  Cm12 (t )  r0 Fx12 (t )  C f 12 (t )

,

(7)

where Jr is the wheel inertia, r0 is the wheel radius, Ω11/2(t) is the angular velocities of wheels, Cm11/2(t) is the torque applied by the motors at the front wheels and Cf11/2(t) is the viscous friction torque respectively applied at each wheel: (8) C f 11/ 2 (t )  br Ω11/ 2 t  , where br is the viscous friction coefficient. Note that longitudinal efforts Fx11(t) and Fx12(t) can be deduced using the tire model created by Pacejka [31] which takes into account road adherence coefficient μ and slip ratio T11/2(Vx(t),Ω11/2(t)), namely: Fx11/ 2 t    Dx sinC x arctanBx

   , Ex arctanBx Τ11/ 2 t   1  E x  Τ11/ 2 t    Bx   

(9)

where macro-parameters Bx, Cx, Dx and Ex depend on microparameters and on the vertical load Fz on each axle. Moreover, road adherence coefficient μ is considered as an uncertain parameter between 0.4 (slippery road) and 1 (dry road), considered equal for both wheels. In the same way, vertical load Fzij is also an uncertain parameter which equals to the load located at each wheel. The uncertainty comes from the vehicle mass that varies between loaded and unloaded vehicle. Thus, considering symmetrical load on front wheels, Fz is given by: M (10) Fz11  Fz12  1 g cos  r . 2 Motors are brushless and their dynamics are considered to follow those of a direct current motor. Neglecting dynamics of the voltage current amplifier, voltage command, current and torque are thus given by: I (t )  K A Uc t  and Cm11/ 2 (t )  KT I (t ) , (11) where KA is the voltage to current amplifier gain and KT the constant torque associated with each motor. Note that hypothesis of symmetrical load with identical motors assumed in this study the commands are equal for both motors. In this framework, Fig. 5 describes the vehicle from in-wheel motors control voltage to longitudinal speed.

Fig. 5. Block diagram of the validation model

Simulations show that applying a step of the maximum voltage command (11.80 V) with the parameters presented in Table 1 leads to Vx(t) passing from 0 to 100km/h in 9 seconds. This behavior is representative of this kind of vehicles. This model is the one used in the following parts unless otherwise stated. TABLE I MODEL PARAMETERS 600 to 900 Kg

Mt

0.5

Cxx -3

g

10 m/s²

br

10 Nm.s/rad

ρ

1.225 Kg/m3

Jr

1 Kg.m²

Sx

2 m²

KT

25 Nm/A

r0

0.3 m

KA

1 A/V

B. Linear model used for frequency-domain controller design In order to exploit time-frequency duality thus to use a frequency-domain control-system design, the preceding model is linearized around operating points. - Slip ratio T11/ 2 (Vx , 11/ 2 ) varies around the equilibrium point between [-5%; 5%], staying in the linear operating area, as shown by the characteristics function of Fx with respect to T of the Pacejka model in Fig. 6. e

e

e

2500 Fx(T): from Pacejka model Fx(T): approximation at the linear zone 2000

1500

Fx (N)

Fa Vx  

3

1000

500

0 0

10

20

30

40

50 60 Slip ratio (%)

70

80

90

100

Fig. 6. Characteristics Fx with respect to the slip ratio

Longitudinal forces are thus written as: Fx11/ 2 t   Fxe11/ 2  f x11/ 2 t  , where

(12)

e Fxe11/ 2  c x11/ 2 T11 /2

(13)

and

f x11/ 2 t   cx11/ 2 11/ 2 t  , (14) with T11/2(t)=Te11/2 + τ11/2(t) and cx coming from the Pacejka

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VT-2014-00452.R2: CRONE Cruise Control System

4

formula applied for each tire in which Bx, Cx, Dx and Ex are expressed according to the vertical load Fz (numerical indices are deleted for clarity): cx 

Fx Τ

,

(15)

T T e

and in the particular case of the linear operating area: (16) cx   Bx Cx Dx . - Slip ratio needs to be linearized. First order Taylor series expansion gives for the driving phase (index d): Td Vx t , Ωt   Tde  d t   d v x t  ,

(17)

with

(28)

The transfer matrix H is thus given by: H  C s I  A1 B  D  H11( s) H12( s) H13( s) ,

(29)

where

Tde  1 

Vxe r0 Ω

T d  d Ω

e

0,

V x V xe Ω Ωe

Vxe



r0 Ω e

2

 cste  0

(18)

~ V ( s) H11 s   ~x  r ( s)

(19)

and the characteristic polynomial P(s) = det[s I - A] by:

and d 

2c x d  ba  brr 2c    a12  x d  a11   Mt Mt , A   r0c xd  br  a   r0c x d a  22  21  Jr Jr . 1    g 0   Mt , C  1 0 and D  0 B K A KT   0 0  J r 

Td Vx

V x V xe ΩΩe



1 r0 Ω e

 cste  0 .

(20)

- The aerodynamic force is then linearized around the reference velocity fixed by the driver: Fa Vx t 

 Fae

 ba v x t  ,

(21)

with (22)

- In the same way, rolling resistance is linearized: Frr Vx t   Frre  brr v x t  , with

(23)

2   Frre   0  2Vxe  M t g     .  Frr  2 2 M t g Vxe brr  Vx V V e  x x 

(24)

- Moreover, considering the road slope is limited to small angles, (25) sin r t    r t  . A state space representation of this linear model can be established, namely:

uc  0

   (a11a22 )  0  . Ps   s ²   s   , with    a a  a a  0 11 22 12 21 

f 0 x 0

(31)

A numerical analysis shows that P(s) has two negative distinct real roots ω1 and ω2, hence P(s) can be rewritten in the form: 1  2    . Ps   s  1 s  2 , with       1 2



(32)



ba  e 1  f Vx , M t  M t  , with 2  1 .   r0 c x d e 2  g Vx , M t ,   Jr 





(33)

ω1 is thus associated with a slow mode (vehicle dynamics for longitudinal translation) and ω2 is associated with a fast mode (rotation of wheel dynamics). C. System uncertainties Transfer function H13(s) used for frequency-domain design methodology of feedback system is given by: ~ V ( s) G0 H13s   ~x  ,   0   s  U c ( s) r s  1   1 f 0 x  0   1   2 

where

 

2 K A KT . r0 ba

(34)

(26)

G0  h Vxe 

(27)

The numerical values of G0, ω1 and ω2 depend on the operating point. Uncertainties in the vehicle model are associated to mass Mt, reference velocity Vxref and adherence μ of road. In this study, those parameters are considered to be bounded according to:

where

 u1 t    r t      x1 t   v x t   , u   u2 t   f 0 x t  and y  v x t  , x    x2 t   r t   u t   u t   c  3 

uc  0

Numerical and analytical analyses of α and β show that ω1 and ω2 can be approximated by:

2 1  Fae   S x C x Vxe  2  .  Fa   S x C x Vxe  ba  Vx V V e  x x

 x  A x  B u ,  y  C x  D u

~ ~ V ( s) V ( s) (30) , H12 s   ~x , H13 s   ~x f 0 x 0 F0 x ( s)  r  0 U c ( s)  r  0

30 km/h  Vxref  110 km/h   600 kg  M t  900 kg .  0.4    1 

(35)

(36)

0018-9545 (c) 2015 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission. See http://www.ieee.org/publications_standards/publications/rights/index.html for more information. Copyright (c) 2015 IEEE. Personal use is permitted. For any other purposes, permission must be obtained from the IEEE by emailing [email protected].

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VT-2014-00452.R2: CRONE Cruise Control System D. Sensibility analysis The sensibility to the uncertain parameters is observed from the frequency response H13(jω). Fig. 7 to 9 presents the sensibility with respect to mass variation, reference velocity variation and road adherence variation respectively. To summarize, the observed results are in accordance with (33) and (35), namely: - the mass variation leads to variations of ω1 and ω2 (Fig. 7); - the reference velocity variation leads to variations of G0, ω1 and ω2 (Fig. 8); - the road adherence variation leads to variation of ω2 (Fig. 9). Furthermore, we can observe the dynamic decoupling between the two modes.

IV. FREQUENCY-DOMAIN CONTROLLER DESIGN In order to obtain a significant comparison between PI and CRONE controllers, the same rapidity and stability degree are chosen. The specifications are defined for the general case in a first place. Then, considering nominal operating point chosen as minimum mass (600kg), median value for the target velocity (70km/h) and adherence coefficient of 1, those specifications are translated into constraints assigned to each controller design. Fig. 10 presents the diagram used for control design where G(s) = H13(s). Input disturbance Du(s) Vref(s) Reference signal

20 0 Gain (dB)

5

+

Error signal U(s) (s) C(s) +

-

Controller

G(s)

-40

+

-80

-2

10

0

10

2

Phase (deg)

Mt=min

-45

Mt=max

-90 -135 -180

-2

10

0

10 Frequency (rad/s)

2

10

Functional equations associated to Fig. 10 are: - output: ~ ~ Vx s    T s  N m s   GSs  Du s   T s Vref s  ; - error: ~  s    S s  N m s   GS s  Du s   S s  Vref s  ; ~ ~ U s    CS s  N m s   T s  Du s   CS s  Vref s  ,

20 Gain (dB)

0

with

-20 -40 -60 -2

10

0

10

2

10

Phase (deg)

0 Vx=50km/h Vx=70km/h Vx=90km/h

-45 -90 -135 -180

-2

10

0

10 Frequency (rad/s)

2

10

Fig. 8. Sensibility to target velocity variation 20 0 -20 -40 -60 -80

-2

10

0

10

2

10

0 -45 -90

=1 =0.7

-135 -180

=0.4 -2

10

0

10 Frequency (rad/s)

Sensor noise

(37) (38)

- control effort:

Fig. 7. Sensibility to mass variation

-80

Nm(s)

Fig. 10. Closed loop system diagram for controller synthesis

10

0

Gain (dB)

Output

Plant model

-20

-60

Phase (deg)

Vx(s)

2

10

Fig. 9. Sensibility to road adherence variation

1  S s   1   s  : sensitivity function  complementary sensitivity function T s   1  S s  :  GS s   G s  S s  : input disturbance sensitivity function CS s   C s  S s  : input sensitivity function   s   C s  G s  : open - loop transfer function  

(39)

.

A. Performance specifications The performance specifications concern: • the stability degree; • the bandwidth; • the precision in steady state; • the rejection level of the measurement noise; • the rejection level of the input perturbation; • the plant input sensitivity. In the following, the study is performed in the frequency domain. The stability degree can be specified with respect to module margin MM, gain margin MG or phase margin MФ. The last one is used in this article and given by: (40) M     arg   ju  , where ωu is the open-loop crossover frequency defined such that: (41)   ju   1 . The stability degree specification is thus emcompassed within the following constraint: (42) M   M  min , where M  min represents the minimal acceptable value of the

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VT-2014-00452.R2: CRONE Cruise Control System

6

phase margin. Substituting Mϕ from (42) in (40), leads the constraint of the open-loop argument for the crossover frequency ωu to be computed as: (43) arg   ju      M min . The argument of the controller can then be deduced for (43) with respect to: (44) arg   j   arg C j   arg G j  , where G(s) represents the plant transfer function. Hence the reduced form of the controller argument constraint at the crossover frequency is: (45) arg C ju      M min  arg G ju  . The bandwidth specification is also computed to impose the crossover frequency ωu. The aim of this constraint is to fix the closed-loop dynamics. The frequency range that is concerned by this condition is (46) u  u max ,

CS  jCS   D .

(58)

If the frequency ωCS is much bigger than ωu, (57) can be rewritten as: (59)    CS  u , C  j   D . The constraints (52) and (59) represent the same aspect which is the controller gain at high frequencies. Hence, they can be reduced to one constraint by choosing the lowest value of these two relations. Thus, 1 (60)    u , C  j   Min A G jT  , D .





B. User constraints Once the uncertain model transfer function is defined, the user constraints for the design of the three controllers are presented in this section: - the stability degree: M min  45  QT max  3 dB ,

where ωumax represents the maximal acceptable value of ωu. Referring to (41) and knowing that: (47)   j   C j  G j  ,

where QTmax is the maximum of the complementary sensitivity function (resonance peak); - the rejection level of the measurement noise:    T  10u , T  j  dB  AdB  20 dB ;

a controller gain constraint at the open-loop crossover frequency ωu can be deduced from (46) as follow:

- the rejection level of the input perturbation:    GS  0.1u , GS j  dB  BdB  20 dB ;

C  ju   G ju max 

1

.

(48)

Concerning the rejection level of the measurement noise, it is calculated using a specification applied to the complementary sensitivity function module as follow: 1 (49)    T , T  j     j  1    j   A , where A imposes the desired rejection noise level for a given frequency ωT: (50) T  jT   A . If the value ωT is chosen to be much bigger than ωu, (49) can be rewritten as: (51)    T  u , T  j     j   A . Hence, a new controller gain constraint around the frequency ωT is deduced by combining (51) and (47) as follow: 1 (52)    T  u , C  j   A G jT  . The rejection level of the input disturbance is used for low frequencies and is computed using a specification as follow: 1 (53)    GS , GS  j   G j  1    j   B , where B represents the desired rejected input disturbance level for a frequency ωGS: (54) GS  jGS   B . When choosing ωGS ωu) (67)   u , CPI  j   C0PI and arg CPI  j   0 . Considering system (64) and the nominal plant, the constraints of (45), (48), (52), (56) and (59), valid independently of the controller nature, can be rewritten to best suit the PI controller as:

0018-9545 (c) 2015 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission. See http://www.ieee.org/publications_standards/publications/rights/index.html for more information. Copyright (c) 2015 IEEE. Personal use is permitted. For any other purposes, permission must be obtained from the IEEE by emailing [email protected].

This article hasThis beenisaccepted the author's for version publication of aninarticle a future that issue has of been thispublished journal, but in this has journal. not beenChanges fully edited. wereContent made tomay this change versionprior by the to publisher final publication. prior to publication. Citation information: DOI 10.1109/TVT.2015.2392074, The final version of record isIEEE available Transactions at http://dx.doi.org/10.1109/TVT.2015.2392074 on Vehicular Technology

VT-2014-00452.R2: CRONE Cruise Control System arctan b   / 2     M  min  arg G ju max  ,

C0 PI

1  b 2 / b  G ju max 

1

.

C0 PI  A G jT 

   T  u ,

(68) (69)

1

,

G0(j ), namely: Ccrone j   0  j  G01  j  .

(77)

(70)

C0 PI u max / b    B

   GS  u ,

7

1

(71)

and    CS  u ,

C0 PI  D .

(72) The next step consists in applying the user constraints in the preceding relations to determine the optimal values of b, C0PI and ωumax. - From relation (72): (73) max C0PI   D  C0PI  3.16 Vs/m . - From relation (70): G jT   G j10u  

A C0 PI

 G j10u  dB  30dB .

(74)

 10u  10rad/s  u max  1rad/s

- From relation (68):   b  tan M min 

   arg G  ju max   0.933 2 . 

 b  1  i 

u max b

Fig. 11. Illustration of the loop shaping method

(75)

 1rad/s

D. Hinf controller The synthesis of the Hinf controller is made with mixedsensitivity from the user constraints on S(jω), CS(jω), T(jω) and GS(jω). The results obtained with this controller can be seen in the performance analysis, section 5. E. CRONE controller Three generations of CRONE control are available [32] [33] [34] [35] [36]. The fact that the specified crossover frequency u is not in a constant phase frequency range implies the use of the second generation. This controller can ensure robustness when plant reparametrization leads only to gain variation around u. The second generation CRONE approach is based on a regulator design from the open-loop constraints (robust loop shaping) [37]. For 2nd generation, the open-loop transfer function is: s  1 l  s   K0   s   l 

     

nl

s  1 h   s  1  l 

n

  1  nh   1  s    h  

,

(76)

where K0 is a gain,l and h the transitional low and high frequencies, nl the order at low frequencies, n the order around crossover frequency u and nh the order at high frequencies. CRONE controller design is then associated to determination of optimal values of these parameters. Optimality is achieved through minimization of a criterion defining performance robustness in case of nominal plant reparametrization [38]. Nominal open-loop transfer function is denoted β0(s) and is modeled using the loop shaping method (Fig. 11). The frequency response Ccrone(j ) of the fractional controller is obtained from the nominal open-loop frequency response β0(j ) and the nominal plant frequency response

Fractional order n is calculated by taking into account the phase margin MΦ and the phase (-n 90°) of the open-loop around u: 180  M Φ  (78) n  1.5 . 90

Synthesis of the rational controller that can be implemented consists in identifying the frequency response of Ccrone(jω) by the one of a rational transfer function Crat(jω) [39]. Using nl = 2, nh = 3 to meet the steady state specifications considering the plant asymptotic behavior at low and high frequencies and n=1.5, the frequency response becomes:   Crat  j   C0  l   j 

2

 j   j  1  1       1 _ nom   2 _ nom    j  1   h  

N

 i 1

 1   1  

j i' j i

  ,   

(79)

where C0 

K0 G0 _ nom

(80)

and the last product is the approximation of the non-integer part as defined in [40]. In this case, N = 4 is sufficient to obtain adequate performances. Finally, the numerical parameter values are: C0  6.798 Vs/m

l  0.0437 rad/s 1' h  11.45 rad/s 2' 1_nom  0.0383 rad/s 3' 2_nom  277 rad/s 4'

1  0.2412 rad/s 2  0.9703 rad/s 3  3.904 rad/s 4  0.0599 rad/s

 0.1281 rad/s  0.5153 rad/s

. (81)

 2.073 rad/s  8.341 rad/s

V. PERFORMANCE ANALYSIS The performance analysis is performed for all three controllers in frequency domain with the linear model and in time domain with the nonlinear model. Two operating points are selected: the first one is for nominal conditions (Mt = 600kg; μ = 1; Vxref = 70km/h) and the second one for extreme conditions (Mt = 900kg; μ = 0.4; Vxref = 90km/h). Figure 12 presents the Bode plots of the PI controller, the Hinf controller and the CRONE controller. Figure 13 presents

0018-9545 (c) 2015 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission. See http://www.ieee.org/publications_standards/publications/rights/index.html for more information. Copyright (c) 2015 IEEE. Personal use is permitted. For any other purposes, permission must be obtained from the IEEE by emailing [email protected].

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VT-2014-00452.R2: CRONE Cruise Control System

8

the open-loop Bode plots obtained with the PI controller (Fig.13.a), the Hinf controller (Fig.14.b) and the CRONE controller (Fig.13.c) around the first (in dashed blue: nominal) and second (in solid red: extreme) operating points. Figure 14 shows the open-loop Nichols obtained with the PI controller (Fig.14.a), the Hinf controller (Fig.14.b) and with the CRONE controller (Fig.14.c) around the first and second operating points. With the CRONE controller, the phase margin (45°) remains constant for the extreme values, illustrating the stability degree robustness versus gain uncertainties in frequency domain. Figures 15, 16, 17 and 18 present the different sensitivity functions obtained with the PI controller, the Hinf controller and with the CRONE controller around the operating points along with the associated constraints in black dashed lines and blue zones. For the nominal operating point (in dashed blue), all performance bounds are respected with the PI and the CRONE controllers. For both operating points, the Hinf controller cannot respect the constraints fixed for the level of rejection of the input perturbation (Fig 16.b) and neither for the input sensitivity function (Fig 18.b). For the extreme operating point (in solid red), all performance bounds are respected with CRONE controllers, which is not the case for the stability degree obtained with the PI controller (Fig 17.a), neither for the sensitivity function obtained with the Hinf controller (Fig 15.b). Figure 19 shows the closed-loop step responses obtained with the PI controller (Fig 19.a), the Hinf controller (Fig 19.b) and with the CRONE controller (Fig 19.c) around operating points. With the CRONE and Hinf controller, the first overshoot and the damping remain constant for the extreme values, illustration of the stability degree robustness versus gain uncertainties in time domain. The command signals are also presented in Fig. 20. Robustness leads to a constant first overshoot with the CRONE controller in spite of variation of the plant parameters as can be seen in Fig 19.c. 80

Gain (dB)

60 40 20 0 -20 -40 -3 10

-2

10

-1

10

0

10

1

10

2

10

3

10

Phase (deg)

0 -45 -90



1  104







and because 2 >> u 2 u  2 102 , the expression of G(s) can be simplified: V (s) G0 . (82) Gs   x  2

U c (s)  r  0

f 0 x 0

s

1

1

The complementary sensitivity function TPI(s) obtained using the PI controller (61) is then given by: C PI s  G s  TPI s    1  C PI s  G s 

1 1  2 BF

s

i  s     BF

s

 BF

,   

(83)

2

where  BF  C0 PI G0 i 1    1  1  C0 PI G0       BF 2  C0 PI G0    

.

(84)

C0 PI G0 1

i

Replacing 1 by (33), G0 by (35) and considering C0PI G0  1, gives:   BF  C0 PI i      BF  1 C0 PI 2 i  

2K A KT r0 M t

.

(85)

2K A KT r0 M t

In this case, the resulting damping ratio BF is a function of the mass, in such a way that when the mass Mt increases, the damping ratio BF decreases. However, using a second generation CRONE control, the damping ratio depends only on the chosen fractional order n, such that:   (86)  n    cos   0.5 with n  1.5 . n

This characteristic of the CRONE control was already used to design the CRONE suspension [17]. Simulations have been performed using the validation model of the car, with three controllers. Starting at nominal velocity (70km/h), a rectangular pulse of 700N with a duration of two seconds simulating a wind gust is applied at t = 5s. Figure 21 presents the vehicle velocity and the control signal in rejection mode obtained without controller (a, e), with the PI controller (b, f), with the Hinf controller (c, g) and with the CRONE controller (d, h) around the nominal (in dashed blue) and the extreme (in red) operating points. One can note that the control signal remains smaller than the saturation value (+12V). Simulation results are similar to those obtained with linear model, that means the linear model is valid.

PI CRONE Hinf

-135 -180 -3 10

This behavior can also be observed through analysis of the closed-loop damping ratio. By taking into account the dynamic decoupling between the two modes of G(s)

-2

10

-1

10

0

1

10 10 Frequency (rad/s)

2

10

3

10

Fig. 12. Bode plots of the PI (in blue), the Hinf (in black) and the CRONE (in red) controllers

0018-9545 (c) 2015 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission. See http://www.ieee.org/publications_standards/publications/rights/index.html for more information. Copyright (c) 2015 IEEE. Personal use is permitted. For any other purposes, permission must be obtained from the IEEE by emailing [email protected].

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40

20

20

0 -20

Gain (dB)

40

20

0

0 -20

-20

-40 -1 10

0

-40 -1 10

1

10

10

0

0

-40 -1 10

1

10

10

-180 -1 10

0

-90 -135 -180 -1 10

1

10 Frequency (rad/s)

nom ext

-45

10

Phase (deg)

-90 -135

0

1

10

10

0

0

nom ext

-45

Phase (deg)

Phase (deg)

9

40

Gain (dB)

Gain (dB)

VT-2014-00452.R2: CRONE Cruise Control System

0

-90 -135 -180 -1 10

1

10 Frequency (rad/s)

nom ext

-45

10

0

1

10 Frequency (rad/s)

10

(a) (b) (c) Fig. 13. Open-loop Bode plots for the PI controller (a), the Hinf controller (b) and for the CRONE controller (c) Open-loop with PI controller

Open-loop with CRONE controller

Open-loop with Hinf controller

40

40

40

0 dB 30

30

30

20

1 dB

3 dB 6 dB

0

20

1 dB

3 dB

10

Open-Loop Gain (dB)

10

6 dB

0

6 dB

0

-10

-10

-10

-20

-20

-20

-30

-40

-30

-30

-260

-240

-220

-200

-180 -160 Open-Loop Phase (deg)

-140

-120

-100

-40

-80

1 dB

3 dB

10

Open-Loop Gain (dB)

20

Open-Loop Gain (dB)

0 dB

0 dB

-40 -260

-240

-220

-200

-180 -160 Open-Loop Phase (deg)

-140

-120

-100

-80

-260

-240

-220

-200

-180 -160 Open-Loop Phase (deg)

-140

-120

-100

-80

(a) (b) (c) Fig. 14. Open-loop Nichols loci obtained with the PI controller (a), the Hinf controller (b) and the CRONE controller (c) 10

10 10

0

-10

nom ext

-30 -40

-20

Sensitivity function (dB)

-10

-20

Sensitivity function (dB)

Sensitivity function (dB)

0

0

-10

nom ext

-30 -40

-20 -30 -40

-50

-50

-50

-60

-60

-60

-70 -2 10

-1

10

0

10 Frequency (rad/s)

1

-70 -2 10

2

10

10

-1

10

0

10 Frequency (rad/s)

1

-70 -2 10

2

10

10

nom ext

-1

10

0

10 Frequency (rad/s)

1

10

2

10

(a) (b) (c) Fig. 15. Sensitivity functions with: (a) the PI controller, (b) the Hinf controller and (c) the CRONE controller

-20 -25 -30 -35 -40 -45 -50 -55

Input disturbance sensitivity function (dB)

nom ext

-15

Input disturbance sensitivity function (dB)

Input disturbance sensitivity function (dB)

nom ext

-15

-60 -2 10

-10

-10

-10

-20 -25 -30 -35 -40 -45 -50

10

0

10 Frequency (rad/s)

1

10

2

10

-60 -2 10

-20 -25 -30 -35 -40 -45 -50 -55

-55 -1

nom ext

-15

-1

10

0

10 Frequency (rad/s)

1

10

2

10

-60 -2 10

-1

10

0

10 Frequency (rad/s)

1

10

2

10

(a) (b) (c) Fig. 16. Input disturbance sensitivity functions with: (a) the PI controller, (b) the Hinf controller and (c) the CRONE controller

0018-9545 (c) 2015 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission. See http://www.ieee.org/publications_standards/publications/rights/index.html for more information. Copyright (c) 2015 IEEE. Personal use is permitted. For any other purposes, permission must be obtained from the IEEE by emailing [email protected].

This article hasThis beenisaccepted the author's for version publication of aninarticle a future that issue has of been thispublished journal, but in this has journal. not beenChanges fully edited. wereContent made tomay this change versionprior by the to publisher final publication. prior to publication. Citation information: DOI 10.1109/TVT.2015.2392074, The final version of record isIEEE available Transactions at http://dx.doi.org/10.1109/TVT.2015.2392074 on Vehicular Technology

10

5

5

0 -5 -10 -15 -20 -25 nom ext

-30

10

10 5

Complementary sensitivity function (dB)

10

Complementary sensitivity function (dB)

Complementary sensitivity function (dB)

VT-2014-00452.R2: CRONE Cruise Control System

0 -5 -10 -15 -20 -25 nom ext

-30

-40 -2 10

-1

0

10

10 Frequency (rad/s)

1

-40 -2 10

2

10

10

-10 -15 -20 -25 -30

nom ext

-35

-35

-35

0 -5

-1

0

10

10 Frequency (rad/s)

1

-40 -2 10

2

10

10

-1

10

0

10 Frequency (rad/s)

1

2

10

10

20

15

15

10 5 0 -5 -10 nom ext

-15 -20 -2 10

-1

10

20 15

Input sensitivity function (dB)

20

Input sensitivity function (dB)

Input sensitivity function (dB)

(a) (b) (c) Fig. 17. Complementary sensitivity functions with: (a) the PI controller, (b) the Hinf controller and (c) the CRONE controller

10 5 0 -5 -10 nom ext

-15

0

10 Frequency (rad/s)

1

-20 -2 10

2

10

10

-1

10

10 5 0 -5 -10 nom ext

-15

0

10 Frequency (rad/s)

1

10

-20 -2 10

2

10

-1

10

0

10 Frequency (rad/s)

1

10

2

10

(a) (b) (c) Fig. 18. Input sensitivity functions with: (a) the PI controller, (b) the Hinf controller and (c) the CRONE controller 7

6 5 4 3 2 1 0 0

5

10 Time (s)

15

6 5 4 3 2 1 0 0

20

7

nom ext

5

10 Time (s)

15

Normalized speed (km/h)

nom ext

Normalized speed (km/h)

Normalized speed (km/h)

7

5 4 3 2 1 0 0

20

nom ext

6

5

10 Time (s)

15

20

4 3 2 1 0 -1 0

5

10 Time (s)

15

20

nom ext

5 4 3 2 1 0 -1 0

5

10 Time (s)

15

20

Normalized motor voltage (V)

nom ext

5

Normalized motor voltage (V)

Normalized motor voltage (V)

(a) (b) (c) Fig. 19. Normalized step responses with: (a) the PI controller, (b) the Hinf controller and (c) the CRONE controller nom ext

5 4 3 2 1 0 -1 0

5

10 Time (s)

15

20

(a) (b) (c) Fig. 20. Normalized control signal responses in rejection mode with: (a) the PI controller, (b) the Hinf controller and (c) the CRONE controller

0018-9545 (c) 2015 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission. See http://www.ieee.org/publications_standards/publications/rights/index.html for more information. Copyright (c) 2015 IEEE. Personal use is permitted. For any other purposes, permission must be obtained from the IEEE by emailing [email protected].

This article hasThis beenisaccepted the author's for version publication of aninarticle a future that issue has of been thispublished journal, but in this has journal. not beenChanges fully edited. wereContent made tomay this change versionprior by the to publisher final publication. prior to publication. Citation information: DOI 10.1109/TVT.2015.2392074, The final version of record isIEEE available Transactions at http://dx.doi.org/10.1109/TVT.2015.2392074 on Vehicular Technology

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VI. CONCLUSIONS In this paper, a comparison between PI, Hinf and CRONE controllers for the design of a car cruise control system is proposed. Synthesis of the controllers is performed after linearization of the vehicle model presented in Section 3. Due to the system characteristics (almost constant phase around the open-loop crossover frequency) and the desired specifications, the second generation of CRONE control appears to be well suited. Simulations obtained with the nonlinear vehicle model confirm better results in terms of robustness to mass and velocity uncertainties with the CRONE approach. Future work will be oriented to the extension of the approach proposed in this paper to the design of an entire ADAS system taking into account not only longitudinal but also lateral dynamics. REFERENCES [1]

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0018-9545 (c) 2015 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission. See http://www.ieee.org/publications_standards/publications/rights/index.html for more information. Copyright (c) 2015 IEEE. Personal use is permitted. For any other purposes, permission must be obtained from the IEEE by emailing [email protected].

This article hasThis beenisaccepted the author's for version publication of aninarticle a future that issue has of been thispublished journal, but in this has journal. not beenChanges fully edited. wereContent made tomay this change versionprior by the to publisher final publication. prior to publication. Citation information: DOI 10.1109/TVT.2015.2392074, The final version of record isIEEE available Transactions at http://dx.doi.org/10.1109/TVT.2015.2392074 on Vehicular Technology

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Audrey Morand was born in Nantes, France, in 1988. She received the Master degree in intelligent systems from Pierre and Marie Curie University, Paris, France, in 2011 and the Ph.D. degree in control engineering from the University of Bordeaux, France, in 2014. She currently works in the automotive industry as a research engineer specialized in advanced driver assistance systems.

12 applications in engineering sciences and more particularly in vibration control, vehicle dynamics, global chassis control and ADAS. He was awarded the AFCET Trophy in 1995 for the CRONE suspension. Pierre Melchior was born in France in 1961. He is graduated from the Bordeaux I University and obtained his Ph.D. degree in Control Theory from Paul Sabatier University in 1989, and his HDR (Accreditation to supervise research) from the University of Bordeaux in 2014. He is currently Associate Professor in Control Theory of the BordeauxINP/Enseirb-Matmeca engineering school and researcher in the Automatic Control group of the IMS laboratory of Bordeaux University. He is co-manager of the CRONE team. His research interests are in the area of systems, control theory, robotics and bio-mechatronic (fractional systems, fractional path planning, fractional motion planning, muscle modelling). Mathieu Moze was born in Bordeaux, France, in 1979. He received the M.Eng. degree in Mechatronics from the French Ecole Nationale d’Ingenieurs (ENIT), Tarbes, in 2003 and the M.S degree in Control Systems from the Institut National Polytechnique (INP) of Toulouse, France, the same year. He then worked at the IMS Laboratory, Bordeaux, France and was associated with the University of Bordeaux where he received his Ph.D. degree in Control Theory in 2007. In 2008, he became a consultant to a number of industrial firms and was associated with the IMS Laboratory where he studied algebraic approach to fractional order systems analysis and robust control theory. Since 2010 he has been working in the Scientific Department of PSA Peugeot Citroën where he conducts advanced research concerning modeling, design and control of mechatronic systems for automotive. Franck Guillemard was born in France in March 18th, 1968. He obtained his Ph.D. degree in Control Engineering from the University of Lille, France, in 1996. Presently he works in the Scientific Department of PSA Peugeot Citroën where he is in charge of advanced research concerning Computing Science, Electronics, Photonics and Control. He is also expert in the field of modeling, design and control of mechatronic systems for automotive.

Xavier Moreau was born in France in March 23rd, 1966. He obtained his Ph.D. degree in Control Engineering and his HDR (Accreditation to supervise research) from the University of Bordeaux, France, respectively in 1995 and 2003. Presently he is Professor at the Department of Sciences and Technologies, University of Bordeaux. His research focuses on fractional differentiation and its

0018-9545 (c) 2015 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission. See http://www.ieee.org/publications_standards/publications/rights/index.html for more information. Copyright (c) 2015 IEEE. Personal use is permitted. For any other purposes, permission must be obtained from the IEEE by emailing [email protected].