Automotive driveline modelling, inverse-simulationand compensation

A. AbassDepartment of EngineeringUniversity of Liverpool

Liverpool, UKEmail: a.m.f.abass@liv.ac.uk

A.T. ShentonDepartment of EngineeringUniversity of Liverpool

Liverpool, UKEmail: shenton@liv.ac.uk

Abstract—A nonlinear automotive driveline model and its in-verse simulation are developed. The model contains four inertiascomprising engine, clutch, transmission, coupling, and wheels.In this paper both nonlinear effects arising from the clutchand backlash are considered. The backlash in the proposedmodel is sandwiched between the compliance of the clutchand the compliance of the drive shafts. A basic simulationapproach is outlined for obtaining the Horowitz linear time-invariant equivalent (LTIE) of the automotive driveline modelby a nonparametric identification using a Windowed FourierTransformation (WFT) method. The inverse simulation modelis used as a compensator and is shown to achieve calibrateduncertainty reductions over the crossover frequency range. Thereduction in the equivalent linear uncertainty of the nonlinearityavailable from the inverse compensation is demonstrated by theWFT analysis.Index Terms—automotive; driveline; inverse compensator; un-

certainty reduction;

I. INTRODUCTIONGenerally, the vehicular drive line consists of four main

parts: engine, clutch, transmission, shafts and wheels. As thedriveline is a fundamental part of any vehicle its dynamicsimpact on many aspects of the vehicle development, andconsequently it has been modelled in many different waysdepending on purpose [14]. In driveline control problems, thenonlinear effect arising from backlash and clutch, limits theperformance of speed and jerk control in vehicle drives. In1997, Nordin et. al. [1] presented a simple backlash modelbased on phase plane analysis for an elastic shaft with internaldamping connected to a backlash. Nordin’s model convergesto a dead zone model when damping tends to zero. The sameauthor in [5] presented an historical survey about controllingmechanical systems with backlash. For the purpose of studyingthe torsional vibrations of a driveline under static torque andexcited by engine torque fluctuations, Couderc et. al. [2]used a finite element model to study the driveline transientperformance. To design a feedback controller for damping thevehicle shuffle, DeLaSalle et. al. [3] extended a model from[11] to a four mass model to add rattle and tyre spin modesto the basic shuffle modeand also include the key nonlineardriveline backlash element. The controllers presented in [3]were verified experimentally.A QFT approach to limit-cycle control with backlash using

a switch-mode method was presented in [4], but this is limited

to a particular nonlinear system structure which is not directlyapplicable to the driveline problem.In [6] a model-based scheme is described and experimen-

tally tested. A parametric continuous-time linear phenomeno-logical model is used and its parameters are identified fromexperimental data. The parameters were found to be highlydependent on engine speed, and this dependency was attributedmainly to the system backlash. An observer scheme was usedin conjunction with a Smith predictor. The continuous observermodel and controller were discretized and their parameterswere scheduled in a look-up table. An attempt to obtainrobustness was made by a Root-Locus synthesis, however itwas found necessary at the end of the design process to tunethe controller on the vehicle, which thereby compromised theactual controller robustness and made the final key design stepunsystematic.Grotjahn et. al. [7] started with a driveline two mass model

and extended it to a three mass model to apply a model basedanti-jerk controller design approach. The model parameterswere estimated by means of a nonlinear least squares opti-mization procedure and validated on different diesel cars.A nonlinear estimator for backlash size and state were

developed for the driveline in [8] using Kalman filter theory.The desired estimators were validated experimentally.

II. DRIVELINE MODELLINGIn this section, a four inertia mathematical model for the

vehicle is developed. The model contains two nonlinearities,clutch nonlinearity and backlash nonlinearity. Both nonlineari-ties are represented by a discontinuity of the elastic coefficientsas in Fig. 1. Crucially, for both realism and also to allow testingof our methodology with an arbitrary driveline nonlinearity,the backlash in the proposed model is sandwiched between thecompliant clutch and the compliance of the drive shafts. Themodel contains four inertias which are engine, transmission,coupling, and an equivalent inertia representing the vehiclechassis and wheels. A physical representation of the vehicledriveline is shown in Fig. 2. The constitutive equations of thesystem are obtained as follows:

2010 International Conference on Intelligent Systems, Modelling and Simulation

978-0-7695-3973-7/10 $26.00 © 2010 IEEE

DOI 10.1109/ISMS.2010.53

244

2010 International Conference on Intelligent Systems, Modelling and Simulation

978-0-7695-3973-7/10 $26.00 © 2010 IEEE

DOI 10.1109/ISMS.2010.53

244

Fig. 1. Nonlinear elastic torque characteristics.

Fig. 2. Driveline mathematical model.

1) Engine: The output torque of the engine Tcl, the drivingtorque Te, the engine inertia Je and the engine damping be arerelated by

Te − Tcl = Je

d2θe

dt2+ be

dθe

dt(1)

2) Clutch: The clutch torque Tcl is the sum of frictiontorque Tclf and the clutch spring torque Tcls are related by

Tcl = Tclf + Tcls ; Tclf = Ccl

dΔθcl

dt(2)

and the friction gives a damping rate Ccl. The characteristicof the clutch compliance is a result of arranging low andhigh stiffness springs in series. The overall two-stage gradedstiffness for Tcl is given as a piecewise increasing linearfunction of change in angle Δθcl = θe − θte where M1cl

is the initial stiffness and M2clis the final stiffness after a

change in angle of magnitude more than αcl, such that

Tcls =

⎧⎨⎩

M1clΔθcl ;−αcl ≤ Δθcl ≤ αcl

M2cl(Δθcl − αcl) + M1cl

αcl;Δθcl > αcl

M2cl(Δθcl − αcl)−M1cl

αcl;Δθcl < −αcl

(3)

3) Transmission: Taking the combined gear ratio of thegearbox and final drive as tr, the difference of torque on theengine side Tte and on the shaft side Tts is determined from

Tcl − Tte = Jt

d2θte

dt2+ bt

dθte

dt(4)

where Jt is the transmission inertia and bt the transmissiondamping. The transmission torque on the shaft side Tts andthe transmission angle of rotation θts are obtained from

Tts = Ttetr ; θtstr = θte (5)

4) Coupling: Assuming that the couplings are frictionlessthe difference between the transmission torque Tts and thedriveshaft torque Tw is

Tts − Tw = Jco

d2θco

dt2+ bco

dθco

dt(6)

where inertia and damping constant of the coupling are Jco,b

co and the coupling rotation is θco. The wheels are assumedcompliantly coupled to the transmission through the backlashelement of the coupling with backlash gap 2αbl. Taking Δθbl

as the angular difference in the coupling shaft ends, Δθbl =θ

ts−θco the transmission torque at the shaft end is then takenas

Tts =

⎧⎨⎩

M1blΔθbl ;−αbl ≤ Δθbl ≤ αbl

M2bl(Δθbl − αbl) + M1bl

αbl;Δθbl > αbl

M2bl(Δθbl + αbl)−M1bl

αbl;Δθbl < −αbl

(7)

whereM1blis the initial stiffness andM2bl

is the final stiffnessafter a change in angle of magnitude of more than αbl. Thetorque in final drive shaft to the wheel Tw is determined by

Tw = Cs

(dθco

dt−

dθw

dt

)+ Ks (θco − θw) (8)

where Ks is the shaft stiffness and Cs is the shaft damping.5) Wheels: Following the usual automotive modelling as-

sumptions, the torque in final drive shaft to the wheel Tw andthe aerodynamic drag load torque at the wheel Tl are thenrelated to the wheel rotation θw by

Tw − Tl = Jeq

d2θw

dt2+ beq

dθw

dt(9)

where

Jeq = Jw + (mv + mw)r2w, (10)

and

Tl = 0.5CdρAr3w

(dθw

dt

)2

, (11)

Jeq is the equivalent inertia of both wheel pair and vehiclechassis, beq the equivalent viscous damping due to wheelsand chassis, Cd is the areodynamic drag coefficient, ρ is themass density of air, and A is the effective frontal vehicle crosssectional area.6) Linearised model: For the purpose of comparing the

nonlinear model with its linearised equivalent, equations (2),(3), and (7) can be combined and presented in linear form as

Tcl = Ccl

(dθe

dt−

dθte

dt

)+ Kcl (θe − θte) (12)

and

Tts = Kbl (θts − θco) (13)

where Kcl and Kbl are the clutch and backlash linearisedstiffnesses.

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III. DRIVELINE SIMULATIONTo investigate the effect of clutch and backlash nonlin-

earities on the system response, simulation is carried outboth with and without these nonlinearities. The model blockdiagram used in the simulation process is shown in Fig. 6.Fig.3 shows that the effect of backlash can be seen clearlyto a step input representing a slow ’tip-in’, as it causes amuch higher amplitude oscillation in wheel acceleration. Theeffect of backlash dramatically decreases when the appliedstep torque increases as can be seen from Fig. 4 and Fig. 5.The system response in these light and heavy tip-ins and themagnitude of the effect of the backlash are consistent withthose reported in [3].

0 1 2 3 4 5−0.5

0

0.5

1

1.5

2

2.5

Time [s]

Whe

el a

ccel

erat

ion

[m/s

2 ]

without lash and clutch nonlinearitieswith lash and clutch nonlinearities

Fig. 3. Nonlinear effect in light tip-in (Te = 50Nm).

0 1 2 3 4 5−0.5

0

0.5

1

1.5

2

2.5

3

3.5

4

Time [s]

Whe

el a

ccel

erat

ion

[m/s

2 ]

without lash and clutch nonlinearitieswith lash and clutch nonlinearities

Fig. 4. Nonlinear effect in medium tip-in (Te = 100Nm).

IV. INVERSE DRIVELINE COMPENSATOR DESIGNAs the nonlinear effect increases the required model uncer-

tainty in the LTIE plant makes it more difficult to control.

0 1 2 3 4 5−2

0

2

4

6

8

10

12

14

Time [s]

Whe

el a

ccel

erat

ion

[m/s

2 ]

without lash and clutch nonlinearitieswith lash and clutch nonlinearities

Fig. 5. Nonlinear effect in heavy tip-in (Te = 400Nm).

To address this, an inverse compensator can be designed andincorporated in series with the plant to reduce the systemuncertainty. Such an inverse compensator design techniquecould be obtained experimentally on the real plant by col-lecting appropriate input-output data and by removing thetime delay off-line and then by using system identification,with the output data used as an input and vice versa. Thesystem time delay can usually be estimated by applying a stepinput and directly measuring the response delay. In simulation,The inverse compensator is the inverse simulation system,constructed from the original system equations but with aninverse causality and time-delays removed. The resulting in-verse compensator is shown in Fig. 7. It is established thatany approximation in the inverse compensator arising fromfiltering to achieve the required causality, can result in thereduction in system uncertainty being achieved only a limitedfrequency range. To be effective as a control compensatorhowever, it is only required to reduce the uncertainty over thesystem gain-crossover frequency to phase-crossover frequencyrange, which range is typically found to be 1 rad/s to 15 rad/sin the driveline.

V. DRIVELINE UNCERTAINTY REDUCTION

Following Horowitz [9], [10], a set of typical input signalswhich are sufficient to cover the full input operating rangeis selected. The typical input signals, which in the realdriveline case are the throttle (accelerator) pedal inputs fora wide range of different driver types, are represented forthe simulation study by the output signals from a secondorder filter of the form ω2

n

s2+2ζωns+ω2n; ζ ∈ [0.5912, 1.2] and

ωn ∈ [2.6667, 3.3832] subject to a set of step inputs rangedfrom 100N/m to 400N/m. Such inputs are found typical ofdriver inputs, however the use of a wider range of inputsrecorded experimentally from a range of drivers would bethe natural way to enhance the reliability of the proposedscheme. The WFT is used to estimate the frequency response

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Fig. 6. Driveline block digram.

Fig. 7. Driveline inverse compensator.

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0 0.5 1 1.5 2 2.5 3 3.5 40

50

100

150

200

250

300

350

400

450

Time [s]

Eng

ine

torq

ue [N

m]

Fig. 8. Typical input signals set.

for each input-output pair with 4096 frequency samples takenin a sampling rate of 50Hz. The Hann window, given by [13]as:

wN

(τ) =1

2

(1− cos

(2πτ

N

))(14)

is used to smooth the frequency response function obtainedby the Fourier transform, where τ = 0, 1, 2, .., N − 1 and Nis the number of time samples (2500 in this case). Since areduction in the area of the structured uncertainty templatesis required within the frequency range 1 rad/s to 15 rad/s, arange of frequencies extending over this range was chosen toinvestigate the reduction actually achieved. Fig. 9 shows theresulting uncertainty templates found both with and withoutthe inverse compensator for the selected frequencies, 0.5, 1,2, 5, 18, 20 rad/s.

VI. IMPROVING THE SIMULATION RESULTSA significant problem when derivatives are used in the

inverse compensator is that the accuracy of the output result isvery sensitive to the time step size. With the derivative blockin the Simulink software, unlike blocks that have continuousstate, the solver does not take smaller steps when the inputchanges rapidly. The result is the appearance of additionalnoise on the computed signal. To handle this problem, thederivative du

dtcan be approximated at low frequencies by a

transfer function of the form ωbss+ωb

. where the filter breakfrequency ωb is chosen to be close to the system bandwidth.In this paper, the scheme is implemented by use of a low-pass filter of the form 1

1

ωbs+1

immediately after each Simulinkderivative block..

VII. CONCLUSIONS• A nonlinear model of the driveline was presented whichincludes the main nonlinearities of clutch nonlinearity andcoupling backlash. The backlash was embedded betweeninertial and compliant elements in contrast to the less

−0.2 −0.1 0 0.1

−0.4−0.2

00.2

Real

Imag

inar

y

ω=0.5rad/s

0 0.1 0.2 0.3

−0.2

0

0.2

0.4

Real

Imag

inar

y

ω=1rad/s

0 0.1 0.2 0.3

−0.1

−0.05

0

Real

Imag

inar

y

ω=2rad/s

−5 0 5

x 10−3

−0.02

−0.01

0

Real

Imag

inar

y

ω=5rad/s

−8 −6 −4 −2 0 2

x 10−3

−15−10−5

0

x 10−3

Real

Imag

inar

y

ω=12rad/s

−15 −10 −5 0

x 10−3

0

2

4x 10−3

Real

Imag

inar

y

ω=15rad/s

−4 −3 −2 −1 0

x 10−3

−2024

x 10−3

Real

Imag

inar

y

ω=18rad/s

−2 −1 0

x 10−3

0123

x 10−3

Real

Imag

inar

y

ω=20rad/s

original modelwith inverse compensator

Fig. 9. Uncertainty boundaries.

realistic but analytically convenient approach of backlashconnected to the end inertial element.

• The model used typical vehicle parameters. A block-diagram for simulation of the driveline model was pre-sented. The model gave similar results to those reportedby [3], with accentuated oscillation due to the vehiclebacklash.

• A WFT method was presented for obtaining Horowitz’sLTIE model from input-output data obtainable directlyfrom the simulation.

• A block-diagram for inverse-simulation of the drivelinewas presented. A nonlinear compensator based on theinverse simulation model was proposed as a means ofreducing the uncertainty in Horowitz’s LTIE model re-quired for QFT control design.

• A reduction in the uncertainty as measured by the areaand minimum covering radius of the QFT templates wasshown to be gained through using the nonlinear inversecompensator in series with the driveline plant.

• The reduced uncertainty with the inverse compensatorshould allow the design in the future, of robust controllerswith higher performance.

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VIII. SYSTEM PARAMETERSSymbol ValueA 3 m2

be 0.05 N.m.s/radbt 0.8 N.m.s/radbeq 5.6 N.m.s/radCcl 0.2 N.m.s/radCbl 1 N.m.s/radCs 120 N.m.s/radJe 0.3 kg.m2

Jt 0.01 kg.m2

Jco 0.01 kg.m2

Jeq 140 kg.m2

Kcl 527.12 N.m/radKbl 10000 N.m/radKs 10000 N.m/radM1cl

10 N.m/radM2cl

527.12 N.m/radM1bl

0.0001 N.m/radM2bl

10000 N.m/radmv 1200 kgmw 60 kgrw 0.304 mtr 13.873αcl 0.001 radαbl 0.0873 radρ 1.2 kg/m3

REFERENCES

[1] Nordin, M, Galic, J and Gutman, P. (1997). New models for backlashand gear play. Int. J. of Adaptive Control and Signal Processing, 11,49–63.

[2] Couderc, P, Callenaere, J, Hagopian, J. and Ferraris, G. (1998). Vehicledriveline dynamics behaiour: expermental and simulation. J. of Soundand Vibratian, 218(1), 133–157.

[3] DeLaSalle, S., Jansz, M. and Light, D. (1999). Design of a feedbackcontrol system for damping of vehicle shuffle. EAEC Conference,Barcelona, 278–284.

[4] Boneh, R. and Yaniv, O. (1999). Reduction of limit cycle amplitude inthe presence of backlash. Transactions of ASME, 121, 278–284.

[5] Nordin, M. and Gutman, P. (2002). Controlling mechanical systems withbacklash - a survey. Automatica, 38, 1633–1649.

[6] Baumann, J., Torkzadeh, D.D., Ramstein, A., Kienke, U. and Schelegl,T. (2006). Model-based predictive anti-jerk control. Control EngineeringPractice, 14, 259–266.

[7] Grotjahn, M, Quernheim, L. and Zemke, S. (2006). Modelling andidentification of car driveline dynamics for anti-jerk controller design.IEEE 3rd Internatonal Conference on Mechartonics, ICM, 131–136.

[8] Lagerberg, A and Egardt, B. (2007). Backlash estimation with applica-tion to automotive powertrains. IEEE Transactions On Control SystemsTechnology. 15(3), 483–493.

[9] Horowitz, I. (1981). Improvement in nonlinear feedback design bycancellation. Int. J. of Control, 34, 547–560.

[10] Horowitz, I. (1993). Quantative Feedback Design Theory. QFTPublications, Boulder, Colorado 80303.

[11] Hrovat, D. and Tobler, W.E. (1991). Bond graph modelling of automo-tive power trains. J. of the Franklin Institute, 328, 623–662.

[12] Karnopp, D., Margolis, D.L., and Rosenberg, R.C. (2006). Systemdynamics: modelling and simulation Of mechatronic Systems. Wiley.

[13] Ljung, L. (1999). System identification, theory for the user. PrenticeHall PTR, New Jersey.

[14] Kiencke, U and Nielsen, L. (2000). Automotive control systems forengine, driveline, and vehicle. Springer-Verlag Berlin Heidelberg NewYourk.

IX. NOMENCLATUREA effective frontal vehicle cross sectional area (m2)be engine viscus damping coefficient (N.m.s/rad)bt transmission viscus damping Coeff. (N.m.s/rad)bw wheels viscus damping coefficient (N.m.s/rad)Cd aerodynamic drag coefficient (N.s2/kg.m)Ccl clutch coulomb damping coefficient (N.m.s/rad)Cs shaft viscus damping coefficient (N.m.s/rad)Je engine inertia (kg.m2)Jt transmission inertia (kg.m2)Jco coupling inertia (kg.m2)Jw wheel inertia (kg.m2)Kcl clutch stiffness (N.m/rad)Kbl lash linear equivalent stiffness (N.m/rad)Ks shaft stiffness (N.m/rad)M1cl

clutch initial stiffness (N.m/rad)M2cl

clutch final stiffness(N.m/rad)M1bl

lash non contact mode stiffness (N.m/rad)M2bl

lash contact mode stiffness (N.m/rad)mv vehicle mass (kg)mw wheels mass (kg)rw wheel radius (m)Te engine torque (N.m)Tcl clutch torque (N.m)Tte transmission torque at engine side (N.m)Tts transmission torque at shaft side (N.m)Tw wheel torque (N.m)Tl aerodynamic drag torque load(N.m)tr transmission ratiov vehicle longitudinal speed (m/s)αcl max clutch initial slip angle (rad)αbl lash gab (rad)ρ mass density of air (kg/m3)θe engine angular displacement. (rad)θte transmission displacement at engine side (rad)θts transmission displacement at shaft side (rad)θco coupling angular displacement (rad)θw wheel angular displacement (rad)

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