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Robust yaw stability controller design and hardware-in- the-loop testing for a road vehicle

Citation Guvenc, B.A., L. Guvenc, and S. Karaman. Robust Yaw StabilityController Design and Hardware-in-the-Loop Testing for a RoadVehicle. Vehicular Technology, IEEE Transactions on 58.2(2009): 555-571. 2009 IEEE

As Published http://dx.doi.org/10.1109/TVT.2008.925312

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IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL. 58, NO. 2, FEBRUARY 2009 555

Robust Yaw Stability Controller Design andHardware-in-the-Loop Testing for a Road Vehicle

Bilin Aksun Gven, Member, IEEE , Levent Gven, Member, IEEE , and Serta Karaman

Abstract Unsymmetrical loading on a car like -split brak-ing, side wind forces, or unilateral loss of tire pressure resultsin unexpected yaw disturbances that require yaw stabilizationeither by the driver or by an automatic driver-assist system.The use of two-degrees-of-freedom control architecture knownas the model regulator is investigated here as a robust steeringcontroller for such yaw stabilization tasks in a driver-assist system.The yaw stability-enhancing steering controller is designed in theparameter space to satisfy a frequency-domain mixed sensitivityconstraint. To evaluate the resulting controller design, a real-timehardware-in-the-loop simulator is developed. Steering tests withand without the controller in this hardware-in-the-loop setup

allow the driver to see the effect of the proposed controller toimprove vehicle-handling quality. The hardware-in-the-loop sim-ulation setup can also be used for real-time driver-in-the-loopsimulation of other vehicle control systems.

Index Terms Hardware-in-the-loop simulation, model regula-tor, robust control, yaw stability control.

I. INTRODUCTION

D ANGEROUS yaw motions of an automobile result fromunexpected yaw disturbances that are caused by unsym-metrical car dynamic perturbations like the unilateral loss of tire pressure or braking on a unilaterally icy road called -split

braking. Safe driving requires a driver to react extremelyquickly in such dangerous situations. This is not possible, as thedriver who can be modeled as a high gain control system withdead time overreacts, resulting in instability. Consequently, theimprovement of automobile yaw dynamics by active control toavoid such catastrophic situations has been and is continuing tobe a subject of active research.

The commercially available solution to this problem usesindividual wheel braking [1] due to the ease of implementationthrough the antilock braking system (ABS) actuators. This easeof implementation is enhanced with the recent commercialavailability of brake-by-wire systems [2]. Designing individual

Manuscript received February 4, 2007; revised July 2, 2007, October 21,2007, February 21, 2008, and April 6, 2008. First published May 14, 2008;current version published February 17, 2009. This work was supported inpart by the European Commission Framework Program 6 (Project Autocom,INCO-16426) and in part by the Turkish State Planning Organization(Project DriveSafe, DPT). The review of this paper was coordinated byProf. M. Benbouzid.

B. Aksun Gven and L. Gven are with the Department of Mechan-ical Engineering, Istanbul Technical University, Istanbul, Turkey (e-mail:[email protected]; [email protected]).

S. Karaman is with the Laboratory for Information and Decision Systems,Massachusetts Institute of Technology, Cambridge, MA 02139 USA (e-mail:[email protected]).

Color versions of one or more of the gures in this paper are available onlineat http://ieeexplore.ieee.org.

Digital Object Identier 10.1109/TVT.2008.925312

wheel braking controllers that accommodate tire force satura-tion and coupling between braking and steering in corneringmaneuvers improves the capabilities of yaw stability control[3]. During a yaw stabilization maneuver, however, the driverof a car not only applies the brakes but also tries to steer thecar. This second approach of using steering for yaw distur-bance rejection is treated in this paper. Steering control hasbeen treated in a large number of references, including [4]and [5]. Integrated control of steering and braking has alsoreceived a lot of attention in recent years [6][10]. Hardware

implementation of steering controllers will be much easierwhen steer-by-wire systems become commercially available.In steer-by-wire systems, the steering wheel is no longermechanically connected to the tires [11], [12]. The steering-wheel commands are electrically transmitted to the steer-by-wire actuator, which laterally moves the tires. Only thesteer-by-wire controller code has to be changed to modify asteering controller. The presence of a steer-by-wire system isassumed here.

The typical presence of large amounts of uncertainty invehicle steering control for yaw stability enhancement man-dates the use of a robust controller. Uncertainty, for example,occurs in the roadtire friction coefcient and the vehiclemass. The large variations in the longitudinal velocity andthe dependence of the models that are used on this parameterresult in a parameter-varying model that requires a velocityscheduled implementation. It is also desired to use a steering-based yaw stability controller that is easy to tune. Consequently,a robust steering controller based on the model regulator isproposed here, as it has been seen to successfully fulll similarrequirements in earlier work by the authors [13], [14]. A designapproach based on mapping frequency-domain performancecriteria to model the regulator parameter space is introduced inthis paper. Explicit formulas are obtained for basic choices of model regulator lters. An interactive program is used to obtain

multiobjective solution regions for the chosen lter parameters.The single-track model of a vehicle models lateral dynamicsquite accurately for moderate levels of lateral acceleration.However, the single-track model cannot capture extreme ma-neuver dynamics [15]. Such extreme maneuvers are importantfor testing yaw stability controllers. Therefore, a full-vehiclemodel is also introduced and used in this paper, along witha single-track model. The model regulator-based yaw stabilitycontroller developed here is applied to the full-vehicle model toforce it to behave like the steady-state of the single-track model,whose handling properties drivers are used to.

The ultimate and nal test for a vehicle stability controlleris road testing on a test track. Road tests are expensive and

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556 IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL. 58, NO. 2, FEBRUARY 200

Fig. 1. Single-track model for car steering.

dangerous. Therefore, all major modications to a vehicle yawstability controller should be taken care of before road testing,leaving only nal checks and ne-tuning. The approach pro-posed here is to use ofine simulations followed by hardware-in-the-loop testing for thorough evaluation before road tests.A hardware-in-the-loop test setup developed for this purposeis introduced in this paper. This hardware-in-the-loop setupallows the evaluation of the control algorithm with the driverin the loop.

The organization of the rest of this paper is as follows.The single-track and full-vehicle models that are used in thispaper are given in Section II. The steering actuated yaw sta-bility controller design specications are given in Section III.The model regulator steering control architecture being usedis presented in Section IV. Robust steering controller designis introduced in Section V. A simulation study is given inSection VI. The hardware-in-the-loop test setup that is usedfor driver-in-the-loop testing is presented in Section VII, wherereal-time simulation results are also given. This paper ends withconclusions.

II. VEHICLE MODELS USED

A. Single-Track Model

The single-track model, which is also called the bicyclemodel, is shown in Fig. 1 [16], [17]. The nomenclature usedin dening the major variables and geometric parameters of thesingle-track and full-vehicle models is given in Table I. Thenonlinear single-trackmodel is characterized by the steering an-gle projection (also called the force coordinate transformation)

F x

F yM z =sin df sin drcos df cosdrlf cos df lr cos dr

F f F r (1)

the dynamics equations

mv( + r )mvI z r

= sin cos 0

cos sin 00 0 1

F xF yM z

(2)

and the equations of kinematics/geometry

tan f = tan + lf rv cos

(3)

tan r = tan lr r

v cos . (4)

TABLE INOMENCLATURE

The tire longitudinal forces F f and F r are nonlinear func-tions of the corresponding sideslip anglesa f and a r . F f andF ralso depend on the friction characteristics between the road andthe tires. The nonlinear single-track model is shown in the toppart of the block diagram of Fig. 2. A vehicle with front-wheelsteering is considered here. Thus, r = 0 in (1).

The variable exhibiting the largest variation in the single-track model is longitudinal velocity v at the vehicle centerof mass. It is customary to linearize the single-track modelwhile keeping v as a variable parameter. The resulting modelis a linear-parameter-varying one whose operating conditiondepends on vehicle velocity. It is also customary to design con-trollers for several velocity values and to use a gain scheduling


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AKSUN GVEN et al. : ROBUST YAW STABILITY CONTROLLER DESIGN AND HARDWARE-IN-THE-LOOP TESTING 557

Fig. 2. (Top) Nonlinear and (bottom) linearized single-track model block diagrams.

controller. This approach is used here as well. The nonlinearsingle-track model is linearized assuming small steering angles f and small sideslip angle . The tire force characteristics arelinearized as

F f ( f ) = cf 0 f = cf f F r ( r ) = cr 0 r = cr r (5)

wherecf and cr are the tire cornering stiffnesses, is the roadtire friction coefcient, and the tire sideslip angles are given by

f = f + lf v

r (6)

r = lrv

r . (7)

Note that cf 0 and cr 0 in (5) are the nominal values for = 1 of the tire cornering stiffnesses. The transfer functionfrom front-wheel steering angle f to yaw rate r is given by

Gr f (s, v ) = r(s) f (s)

= b1 (v)s + b0 (v)

a 2 (v)s2 + a 1 (v)s + a 0 (v) (8)

with

b0 = cf cr (lf + lr )vb1 = cf lf mv

2

a 0 = cf cr (lf + lr )2 + ( cr lr cf lf )mv

2

a 1 = cf I z + l2f m + cr I z + l

2r m v

a 2 = I z mv2 .

Gr f (s, v ) in (8) is also called the steering-wheel inputresponse transfer function here. The dc gain of the nominalsingle-track model (8) is given by

K n (v) = lims0

Gr f (s, v ) = n =1

(9)

at chosen longitudinal speed v and at nominal frictioncoefcient = n , which is taken as unity here.

The yaw moment disturbance input response is given by

GrM z (s, v ) = r(s)M z (s)

= mv2 s + ( cf + cr )v

a 2 (v)s2 + a 1 (v)s + a 0 (v). (10)

The block diagram of the linearized single-track model is

shown at the bottom of Fig. 2.

B. Full-Vehicle Model

The full-vehicle model consists of the vehicle body rep-resented as a sprung mass with three degrees of free-dom, connected to the tires through suspensions modeled asmassspringdamper systems. The tires are connected to eachother by a double-track model. The full-vehicle model is illus-trated in Fig. 3. The equations of motions governing longitudi-nal and lateral dynamics of the double-track base are given by

m(ax rv y ) = F x =4

i =1(F xi cos i F yi sin i ) (11)

m(ay + rv x ) = F y =4

i =1(F xi sin i + F yi cos i ) (12)

where i = 1 , 2, 3, and 4 refers to individual tires.The equation of motion around the yaw axis is

I z r = M z =4

i =1(lxi (F xi cos i + F yi sin i )

+ lyi (F xi sin i + F yi cos i )) + M d . (13)

A vehicle with front-wheel steering is considered here. Thus, 1 = 2 = f , and 3 = 4 = r in (11)(13).


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Fig. 3. Full-vehicle model.

The tire rotational dynamics are given by

I t i = T i R e F xi (14)and the tire center velocities are given by

v_1 = ( vx r (lw / 2)) i + ( vy + rl f ) jv2 = ( vx + r (lw / 2)) i + ( vy + rl f ) j

v3 = ( vx r (lw / 2)) i + ( vy rl f ) jv4 = ( vx + r (lw / 2)) i + ( vy rl f ) j (15)

where i and j are the unit vectors in the x- and y-directions,respectively. The tire sideslip angles are

1 = f + tan 1 vy + rl f

vx r (lw / 2) 2 = f + tan

1 vy + rl f vx + r (lw / 2)

3 = tan 1 vy rl rvx r (lw / 2)

4 = tan 1 vy rl rvx + r (lw / 2) (16)

and the longitudinal wheel slip ratio is dened as

s i = R e i V xiV xi , Re i < V xi (braking)R e i V xiR e i , Re i > V xi (traction).

(17)

The Dugoff tire model with

F xi = f i C xi s i

F yi = f i C yi i (18)

as the longitudinal and lateral tire forces, respectively, is used.

The coefcients f i are calculated using

f i =1, F Ri i F zi2

2 i F zi2 F Ri i F zi2 F Ri , F Ri > i F zi2(19)

F Ri = (C xi i )2 + ( C xi i )2 . (20)The equation of motion for the vertical dynamics of the

sprung mass is

m s az = F z

az = z x + y . (21)


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The forces in thez-direction consist of a summationof springand damper forces and the weight of the sprung mass given by

F z = ms g (F S 1 + F S 2 + F S 3 + F S 4+ F D 1 + F D 2 + F D 3 + F D 4 ). (22)

The spring and damper forces are given by

F S 1 = K RF f tF

2 lf + z + U RF

F S 2 = K LF f tF

2 lf + z + U LF

F S 3 = K RR tR2

lr + z + U RR

F S 4 = K LR tR2

lr + z + U LR (23)

F D 1 = BRF t

F 2 lf + z + U RF

F D 2 = BLF tF 2

lf + z + U LF

F D 3 = BRRtR2

lr + z + U RR

F S 4 = BLR tR2

lr + z + U LR . (24)Then, the equations of motion regarding the roll and pitchdynamics are, respectively, given as

T RC = I x

= m s ghCG m s (y + x + z )hCG

tF 2

(F S 1 + F D 1 ) + tF

2 (F S 2 + F D 2 )

tR

2 (F S 3 + F D 3 ) +

tR2

(F S 4 + F D 4 ) (25)

T P C = I y

= m s ghCG m s (x y + z )hCG+ ( F

S 1 + F

S 2 + F

D1 + F

D2 )l

f

(F S 3 + F S 4 + F D 3 + F D 4 )lr . (26)Finally, the forces in the z-direction for each tire can becalculated as

F zi = F Si + F Di (27)

which is fed back to (19) of the tire model, coupling thedouble-track base and the vertical suspension dynamics.

III. MODEL UNCERTAINTY AND DESIGN SPECIFICATIONS

The two variables exhibiting the largest variation duringoperation are longitudinal speed v and mass m of the vehicle.

Fig. 4. Uncertainty specications.

The tire cornering stiffnesses can also exhibit large variationsdue to variations in the tireroad friction coefcient. In additionto the vehicle model dynamics, there is also the steer-by-wire actuator, whose dynamics should be considered in thesimulations. The steering actuator model that is assumed to beunder closed-loop position control is modeled as a second-ordersystem with a bandwidth of a = 215 rad/s (15 Hz) and adamping ratio of a = 0 .7 as

Gsa = f ref

= 2a

s 2 + 2 a a s + 2a. (28)

The uncertain parameters and their uncertainty ranges willrst be given before the design specication. Longitudinalspeedv is assumed to vary between a minimum value of 10 m/sand a maximum value of 50 m/s during operation. Note that

longitudinal speed v of the vehicle is a measurable quantity.It will be assumed to be measured and will be used in theyaw stability controller implementation introduced later in thispaper. The yaw dynamics compensator is assumed to be shut off at speeds below 10 m/s since the driver is easily able to rejectyaw disturbances at these speeds without the need for steeringcontroller assistance. Road adhesion factor is assumed toexhibit the characteristics displayed in Fig. 4 with velocity. Themaximum valueof is unity at all speeds in this gure, whereasits minimum possible value linearly varies between 0.30 at lowspeeds and 0.80 at high speeds.

In this study, the operating regions for the vehicle are

based on the values of the vehicle longitudinal speed between10 and 50 m/s. Six exemplary operating points have beenchosen and are shown with large dots having separate labels andcolor coding in Fig. 4. The aim in steering compensator designis to make sure that, rst, stable operation, and then improvedyaw dynamics, are achieved for all operating regions and allpossible values of the uncertain parameters. The improved yawdynamics corresponds to good disturbance rejection properties,where the possible disturbances include the effect of side windforces, tire rupture, unsymmetrical brake wear, and -splitbraking. The steering-wheel input response in the absence of disturbances should also be well damped, regardless of thespeed of operation. A model regulator-based steering controlleris designed and shown to effectively achieve the desired aimsin the following sections.


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Fig. 5. Steer-by-wire system with a model regulator.

IV. MODEL REGULATOR ARCHITECTURE

A review of model regulator basics is given in this section(see [18][21]). Referring to Fig. 5, the vehicle model can alsobe expressed as

r = Gr f f + d = ( Gn (1 + m )) f + d (29)

where Gr f = Gn (1 + m ) is the actual inputoutput relationbetween front-wheel steering angle f and vehicle yaw rate r . m is the multiplicative model uncertainty in our knowledgeof nominal model Gn , and d is the external disturbance. Thelocation of steering actuator transfer function Gsa in Fig. 5corresponds to a steer-by-wire system. The aim in the modelregulator steering controller design is to approximately obtain

r s

= Gn Gsa (30)

as the inputoutput relation (the steering transfer function)despite the presence of model uncertainty m and externaldisturbanced. s is the steering-wheel angle input by the driver.Rewrite (29) as

r = Gn f + ( Gn m f + d) = Gn f + e (31)

by dening the extended disturbance e and solving for it as

e = r Gn f . (32)The control law

f = Gsa s G1

n e = Gsa s G1

n r + f (33)

where the expression in (32) for e has been substituted for onthe right-hand side will result in aim (30) when substituted inthe dynamics given by (31). The control law in (33) cannot beimplemented, as it is not causal since f appears on both sidesand since G1n will not be a proper transfer function. Feedbacksignals also have to go through steering actuator dynamics Gsain a real implementation. All of these problems are solved bymultiplying the feedback signals in (33) by unity dc gain low-pass lterQ and steering actuator dynamics lterGsa , resultingin the following model regulator implementation equation:

f = Gsa s QG1

n r + Q f (34)

which is shown in Fig. 5 within the dashed rectangle. Therelative degree of unity dc gain low-pass lter Q is chosen tobe at least equal to the relative degree of Gn for causality of Q/G n . The structure of Q is assumed to be

Q = 1

Q s + 1. (35)

Note that this choice will result in an integrator within theloop in Fig. 5 and, hence, zero steady-state error in response tostep steering commands and yaw torque disturbances [14].

The nominal (or desired) yaw dynamic model is chosen as arst-order system given by

Gn (s, v ) = K n (v) n s + 1

(36)

where K n (v) is the dc gain of the nominal single-track modelat the chosen longitudinal speed v. The aim in the choice of (36) for desired steering dynamics Gn is to force the controlledsystem to possess the behavior of the single-track model withinthe bandwidth of Q. A nonovershooting rst-order dynamics isalso imposed for faster response. Vehicle velocity v is a mea-surable variable. It is, therefore, used in the controller in (36),as K n (v) in that equation is the velocity-dependent static gainof the bicycle or single-track model. The dependence of Gn onvehicle velocity v makes the controller continuously velocity-scheduled. Note that the yaw dynamics of a vehicle, as repre-sented by its bicycle or single-track model, change with vehiclevelocity v. The continuous gain scheduling approach that isused allows the adaptation of the controller to these changes.

According to (29) and (36), model uncertainty m isgiven by

m = Gr f Gn

Gn=

Gr f (s, v )( n s + 1)K n (v) 1 (37)

and is parameter varying, as it depends on the measurable speedof the vehicle.

The open-loop gain of the model regulator-compensated yawdynamics model is given by

L = Gr f Gsa QGn (1 Gsa Q)

(38)

where the model regulation, disturbance rejection, and sensornoise rejection transfer functions are given by

r s

= Gn Gsa Gr f Gn (1 Gsa Q) + Gsa Gr f Q

(39)

rd

= 11 + L

:= S = Gn (1 Gsa Q)

Gn (1 Gsa Q) + Gsa Gr f Q (40)

rn

= L1 + L

:= T = Gsa Gr f Q

Gn (1 Gsa Q) + Gsa Gr f Q. (41)

S in (40) is the sensitivity function, and T in (41) is thecomplementary sensitivity function of the controlled system. Itis obvious from (39)(41) that for good performance, Q mustbe a unity gain low-pass lter (Gsa is a unity gain low-passlter as well). This choice will result in r/ s

Gn (model

regulation),r/d 0 (disturbance rejection) at low frequencies,where Q 1, and r/n 0 (sensor noise rejection) at high


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Fig. 6. Steering model regulator design requirements.

frequencies, where Q

0, as is desired. The model regulator

design is, thus, shaping lter Q to satisfy the design objectives.The rst limitation on the bandwidth of Q comes from thesensor noise rejection at sensor noise frequencies. The secondlimitation is that the bandwidth of Q should not be larger thanthe bandwidth of the actuator used, as it makes no sense tocommand what cannot be achieved.

A. Stability and Stability Robustness Analysis

Checking the stability of the model regulator control systemwhen there is no uncertainty is a simple stability check, where

Re

{roots(1 + L = 0)

} < 0 or

Re roots Gn (1 Gsa Q) + Gsa Gr f Q = 0 < 0 (42)needs to be satised; that is, all roots of the characteristicequation must lie in the left half-plane. Robustness of stabilityis more important for the model regulator control system andresults in another bandwidth limitation for the Q lter. Thecharacteristic equation for the model regulator control loop isgiven by

Gn [(1 Gsa Q) + Gsa (1 + m )Q]= Gn (1 + Gsa m Q) = 0 (43)

which is equivalent to

(1 + Gsa m Q) = (1 + Q) = 0 , Gsa m (44)since Gn is stable, or

| | |Gsa m |

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