On Powertrain Oscillation Damping using Feedforward and LQFeedback Control

Maria Bruce, Bo Egardt and Stefan Pettersson

Abstract— Torsional resonances in the vehicle powertrain area problem in the automotive industry, since they affect thedriveability in a negative way. This paper aims at modelling,control design and verification of a heavy duty powertrain.A model based control system with the aim of damping thepowertrain oscillations has been developed, using feedforwardand feedback control. The control system has been testedin simulations in Matlab/Simulink and experimental tests areunderway.

I. INTRODUCTION

Longitudinal oscillations in vehicles often come fromtorsional resonances in the powertrain. This is known asshuffle, which is an unwanted effect in many powertrainapplications. The torsional oscillations occur in particularadjacent to gearshifts and during tip-in and tip-out (when thedriver pushes and releases the accelerator pedal). The firstoscillation frequency for a heavy duty vehicle is between1 and 7 Hz depending on the gear ratio. The oscillationsaffect the driveability of the vehicle in a negative way.The shuffle also causes wear of the mechanical parts inthe powertrain. Shuffle can also impair hill climbing andacceleration performance, since the minimum time for torqueremoval and torque application is limited by the shuffleproperties.

A driveline consists of clutch, gearbox, propeller shaft,final drive, drive shafts and wheels, [1], [2]. When an engineis connected to the driveline the configuration is called apowertrain. The vehicle mass, acting on the wheel radiusthrough the tire friction surface contact, constitutes a largemass moment of inertia at the end of the powertrain. It is thetorsional flexibility between the different mechanical parts,the mass moment of inertia of these parts, the backlash andthe low mechanical damping that make the powertrain proneto oscillate.

It is of interest to develop a model-based controller thatuses the engine as an actuator to damp out powertrainoscillations. To be able to develop such a controller, a reliableand simple powertrain model is of importance. The controllermust be able to work under strict real time constraints andit must also be able to handle limitations of the powertrainsystem. For example, the size of the control signal is limited,due to how much torque the engine can produce.

This work was supported by Volvo Powertrain Corporation and ChalmersUniversity of Technology. The project has been financed by VINNOVA (theSwedish Agency for Innovation Systems) and Volvo Powertrain Corporation.

M. Bruce is with Volvo Powertrain Corporation, Volvo Powertrain Swe-den, Goteborg, Sweden maria.bruce@volvo.com

B. Egardt and S. Pettersson are with the Department of Signals andSystems, Automatic Control, Chalmers University of Technology, Goteborg,Sweden egardt,stp@s2.chalmers.se

A. Previous work

Several powertrain models have already been developedand explained in different papers. The complexity of themodels vary but it is often shown that a simple two-inertiamodel with one flexibility is enough when it comes toexplaining the first oscillation mode in the powertrain [3],[4], [5]. The two-inertia model is derived from the lawsof motions. The drive shaft is considered to be the mainflexibility that connects the lumped engine and transmissioninertias with the wheel inertia. In the powertrain model usedin the research for this paper, the propeller shaft and theclutch are considered to be stiff. The clutch is also assumedto be engaged. Three state equations describe the two-inertiamodel. The three states are the drive shaft torsion, the enginespeed and the wheel speed.

Different approaches to develop controllers to damp outpowertrain oscillations have also been applied. The mostpromising controllers seem to be either LQG/LTR controllersor pole placement controllers, [4], [5], [6]. Also other con-troller designs like PID, backstepping and H∞ have beenstudied, [7], [8], [9]. The majority of these controllers havebeen tested in a simulation environment with promisingresults. All the controllers mentioned above are one-degree-of-freedom feedback controllers. In this paper a controlstrategy based on both feedforward and feedback control ispresented. The aim with this two-degree-of-freedom systemis to damp out the powertrain oscillations more effectively.

The research work in this paper is based on [7], where apowertrain model was developed, parameterized and verifiedin a heavy duty vehicle. In the same article a PD controllerwas developed and used in experiments in a heavy duty truck.It was shown that the PD controller had a damping effect,although it could not damp out the powertrain oscillationscompletely. The conclusion of [7] was that a more sophisti-cated controller solution is needed.

B. Present work

The aim of this work is to develop a model based controlsystem that damps out powertrain oscillations arising duringa tip-in on gear seven, see Fig. 1. The case study is thepowertrain of a heavy duty truck (Volvo FH12).

The control system is a two-degree-of-freedom controlsystem with a feedforward part based on a filter and aninverse model, and with a feedback part based on LQ designwith a state observer. The control loops are based on a loworder powertrain model [7] that is parameterized for gearseven and for a vehicle with a weight of 22.5 tons. Thecontrol system has been simulated in Matlab/Simulink.

Proceedings of the2005 IEEE Conference on Control ApplicationsToronto, Canada, August 28-31, 2005

WB2.4

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8

10

12

14

16

Time, [s]

Whe

el S

peed

, [ra

d/s]

Fig. 1. Measurement of wheel speed signal in open loop during a tip-inon gear seven at 2.5 s and a tip-out at 6 s.

C. Paper outline

A powertrain model that has been verified in a heavyduty truck is presented in Section II. This model is usedin the controller design that is based on both feedforwardand feedback control. This controller design is described inSection III. Simulation results obtained in Matlab/Simulinkare presented in Section IV. Finally, Section V contains theconclusions.

II. POWERTRAIN MODEL

A two-inertia powertrain model has been derived from thelaws of motion that also can be found in the literature, e.g.,[4], [1], [2]. The three states are the drive shaft torsion angle(x1), the engine speed (x2) and the wheel speed (x3). Thestate dynamics become:

x1 =1

it · ifx2 − x3 (1)

x2 =

(Te −

k

it · ifx1 −

−

(bt

i2t+

bf + c

i2t · i2

f

)x2 +

c

it · ifx3

)·

·

(1

Jm + Jt

i2t

+Jf

i2t·i2

f

)(2)

x3 =

(1

Jw + m · r2w

)(kx1 +

c

it · ifx2 −

−

(c + bw + m · fs · r

2

w

)x3 −

− rw · m ·

(f0 + g · sin(α)

))(3)

In the two-inertia model presented in this paper it isassumed that the main flexibility is in the drive shaft. Otherstiffness and damping coefficients than those related to thedrive shaft are neglected. The model has been experimentallyverified for all gears in a heavy duty truck [7].

Several of the model parameters can be determined frommechanical design data. Examples are the moment of inertiasfor the engine, transmission, final drive and wheels (Jm,

Jt, Jf , Jw), the vehicle mass (m), the wheel radius (r),the transmission ratio (it) and the final drive ratio (if ). Theinput to the model is the calculated engine torque (Te). Inthe experiments, Te is estimated mainly from the amount offuel that is injected. The estimated Te can always be loggedduring an experiment and is therefore a realistic input tothe powertrain model. The road inclination (α) is measuredduring the experiments and this value is also used as a modelinput.

Parameters that have to be set and that are not known ornot exactly known are the viscous damping in the transmis-sion (bt), the spring stiffness (k) and the damping coefficient(c). These three parameters have also turned out to be tuningparameters that have to be changed for every gear. In apowertrain the viscous damping arises in the transmission,final drive and wheels where the different mechanical partsare moving in oil.

III. POWERTRAIN CONTROL

The objective is to control the engine torque to reduce theoscillations that occur in the wheel speed. A reduction of os-cillations in the wheel speed should improve the driveabilityand response of the vehicle. The powertrain control systemhas been developed and simulated in Matlab/Simulink.

A. Control strategy and system structure

The total powertrain control system is seen in Fig. 2.

Accelerator

pedal

Reference

Filter &

Inverse

Powertrain

Model

Powertrain

ModelLQ

Power-

train

ObserverTorsion Angle

Engine Speed

Wheel Speed

Fig. 2. Overview of the control system with a feedforward and a feedbackloop.

The powertrain is controlled by two model based controlloops, one feedforward loop and one feedback loop. Thefeedforward loop is based on an approximate inverse plantmodel in combination with a filter that mimics the desiredclosed loop dynamics. The aim of the feedforward loop is toprovide a fast control signal compensating for the oscillatorymodes of the plant. The feedback loop contains an LQcontroller, whose main task is to compensate for disturbancesand deviations in the model compared to the real powertrain.All blocks in the control system are discrete time except thereal powertrain itself, which is regarded to be continuoustime. It is the position of the accelerator pedal that gives arequested wheel speed setpoint to the control system.

Since the powertrain model contains of three states, the LQcontroller also needs three reference signals to follow. Thesereference states are created with the powertrain model whoseinput is the control signal from the feedforward loop. The

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latter can be regarded as the nominal engine torque, neededto damp out the oscillations in the wheel speed. The threestates of the powertrain model form the reference trajectoriesfor the LQ controller.

The LQ controller also needs to have three feedbacksignals, one for every state. Since the drive shaft torsion anglestate cannot be measured it is estimated with an observer. Theother two states are measured and directly fed back to theLQ controller.

The two control signals from the feedforward loop andthe feedback loop are added and sent as a torque requestto the engine that responds to the request and produces atorque that affects the driveline. There is a saturation in thetorque request that initially is set to 1000 Nm which thenis increased along a positive slope as soon as the enginetorque starts to increase. This is because the engine has aphysical limit of how much torque it can produce, but alsodue to engine software limiting functions. An example of thelatter is the smoke limiter, which limits the torque request,i.e. amount of fuel, in relation to the boost pressure.

B. Design Considerations

1) Choice of reference signal: In this work two referencesignals were considered, the engine speed and the wheelspeed. The only block that is affected in Fig. 2 is the inversemodel. It turned out that the inverse model of the enginespeed in combination with the model gave an oscillatingsystem. The reason for this is that there are two poles veryclose to the imaginary axis that lessen the robustness of thesystem. The system was much more robust in the case whenthe inverse model of the wheel speed was used. That is whythe wheel speed is used as reference signal in the system.

2) Closed loop specification: The amount of torque thatthe engine can deliver is limited, which limits the bandwidthof the closed loop system. The role of the filter in the feedfor-ward loop is to define the desired closed-loop dynamics fromcommand signal to wheel speed. A fast filter will require alarge control signal. If the filter is too slow, the response ofthe accelerator pedal will be sluggish and this will lessen thedriveability of the vehicle. In this work the filter is chosenslightly faster than what the engine physically can manage.

3) Control signal saturation: Occasionally, the feedfor-ward control signal may violate the engine torque limits. Oneway to compensate for this is to apply a rate limitation tothe reference signal according to the following pseudo code:

if ((refStep - wheelSpeed) > threshold) thennewReference := wheelSpeed + threshold;

elsenewReference := refStep

end

This ”stepwise” reference signal, calculated on-line, resultsin a control signal that stays within bounds, if the thresholdis given an appropriate value.

An alternative way to force the feedforward control signalto stay within the torque constraints is to use a referencegovernor, see [10], [11] and [12]. The idea is that, at each

sampling interval, the rate of change of the reference signalis constrained in such a way that the resulting control signalwill stay within bounds, assuming the reference signal wouldremain constant at the current level. The principle will beexplained with reference to a close-up of the two leftmostblocks in Fig. 2, shown in Fig. 3.

RG Gff

Saturationr v y u

Fig. 3. The reference governor and filter/inverse model block.

The command signal v is calculated from the setpoint r

as

v(k + 1) = v(k) + α(r(k) − v(k)) (4)

where α is to be chosen as the maximal value, subject to0 ≤ α ≤ 1, such that if v where “frozen” at the new value,the control signal y would stay within the allowed range[0, ymax]. The correct assignment of α can be found usingthe state realization (F, g, h) of the feedforward filter Gff .Let σ(·) denote the step response of Gff , and let y0

k(·) bethe “free response” of Gff , assuming v is frozen at time k

to the value v(k). Then α should be chosen according to

α = min

(infi≥1

{ymax − y0

k(k + i)

σ(i)(r(k) − v(k))

}, 1

)(5)

where we have assumed that the denominator is positive;minor modifications are needed if this is not the case. Itshould be noted that in practice the minimum need only betaken over a finite range of indices i, so that the calculationsare indeed not extensive.

4) LQ design: The state penalty matrix of the LQ con-troller has been designed so that the biggest penalty willbe on the wheel speed compared to the engine speed andthe torsion angle. This is because the aim with the controlsystem mainly is to damp out the oscillations that occur inthe wheel speed.

5) Observer estimates: The observer estimates all threestates in the model even though it is just the drive shafttorsion angle that is not measurable. The estimated signals ofthe wheel speed and the engine speed were compared to themeasured signals. There were no obvious quality differencesbetween the signals, which lead to the decision that themeasured engine speed and wheel speed are used directlyby the LQ controller and it is just the torsion angle that isestimated.

IV. RESULTS

Parameter settings for the powertrain model are chosen tofit a Volvo FH12, which is a heavy duty truck with a 12-litreengine and in this case also equipped with an automatedmechanical transmission with twelve gears. The weight ofthe rigid truck is assumed to be 22.5 tons.

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A. Simulation results

In the simulations, a step in the wheel speed referenceof 10 rad/s has been applied. The control signal is definedas requested engine torque. A saturation that will mimic thebehaviour of the smoke limiter has also been added to thesystem. The smoke limiter will initially allow an enginetorque of 1000 Nm and will then increase along a slopeas soon as the engine torque starts to increase. This willbe referred to as maximum available engine torque in thesimulation figures.

1) Ideal powertrain model: In the first simulations thepowertrain model is the same as the virtual ”real” powertrain,Fig. 2. This means that it is the feedforward control signalthat does all work to damp the oscillations in the powertrain.The LQ control signal from the feedback loop is zero, sincethere are no disturbances or model errors.

The first simulation shows the comparison between therate limiter and the reference governor. The threshold stepsize in the rate limiter has been adjusted to a limit thatwill keep the feedforward control signal below the maximumavailable engine torque. The feedforward control signals areseen in Fig. 4 and the three states are seen in Fig. 5, Fig. 6and Fig. 7.

10.5 11 11.5 12 12.5 13

500

1000

1500

2000

2500

Time, [s]

Fee

dfor

war

d co

ntro

l sig

nal,

[Nm

]

reference governormaximum available engine torquerate limiter

Fig. 4. Feedforward control signal during simulations with an idealpowertrain model.

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0.02

0.04

0.06

0.08

0.1

0.12

Time, [s]

Tor

sion

ang

le, [

rad]

reference governorrate limiter

Fig. 5. Drive shaft torsion angle during simulations with an ideal powertrainmodel.

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70

80

90

100

110

120

130

Time, [s]

Eng

ine

spee

d, [r

ad/s

]

reference governorrate limiter

Fig. 6. Engine speed during simulations with an ideal powertrain model.

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6

7

8

9

10

11

12

Time, [s]W

heel

spe

ed, [

rad/

s]

reference governorrate limiter

Fig. 7. Wheel speed during simulations with an ideal powertrain model.

As can be seen in Fig. 4, both the control signals staybelow the maximum available engine torque. The referencegovernor is however using almost all available engine torquewhich makes the vehicle response much faster. The statesfollow the references perfectly in Fig. 5, Fig. 6 and Fig. 7.It is impossible to distinguish between the state signals andtheir respective reference signals in the figures. Since thefeedforward control signal is smaller in the rate limiter case,the corresponding reference trajectories from the powertrainmodel become slower. There are no oscillations in the wheelspeed which means that the goal with the control system isreached. There are however a slight oscillation tendency inthe engine speed mainly due to the engine’s effort to work inanti-phase with the powertrain oscillations. This oscillationis small and would probably not affect the driver physicallyeven though it might be audible to the driver.

2) Powertrain model differs from powertrain: In thissimulation the mass of the vehicle was set to 20 tons in the”real” powertrain while the powertrain model kept its originalvalue of 22.5 tons. The model error makes the feedback loopwith the LQ controller become active. The simulations havebeen performed for both the rate limiter and the referencegovernor cases.

The feedforward control signal and the total control signalare seen in Fig. 8. The feedforward control signal looksapproximately the same for both the rate limiter and thereference governor as in the ideal simulation case see, Fig4. The big difference lies in the LQ control signal, whichnow has a non-zero, negative contribution to the total controlsignal. The total control signal stays below the maximum

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10.5 11 11.5 12 12.5 13

500

1000

1500

2000

2500

Time, [s]

Con

trol

sig

nal,

[Nm

]tot ctrl, reference governormaximum available engine torquefeedforward, reference governortot ctrl, rate limiterfeedforward, rate limiter

Fig. 8. Feedforward control signal and total control signal duringsimulations with a parameterisation error in the powertrain model. The LQcontroller has a negative contribution to the total control signal.

10 10.5 11 11.5 12 12.5 130

0.02

0.04

0.06

0.08

0.1

0.12

Time, [s]

Tor

sion

ang

le, [

rad]

torsion angle reference, rate limitertorsion angle, rate limitertorsion angle reference, reference governortorsion angle, reference governor

Fig. 9. Drive shaft torsion angle during simulations with a parameteri-sation error in the powertrain model. This gives a negative control signalcontribution from the LQ controller.

10 10.5 11 11.5 12 12.5 1360

70

80

90

100

110

120

130

Time, [s]

Eng

ine

spee

d, [r

ad/s

]

engine speed reference, rate limiterengine speed, rate limiterengine speed reference, reference governorengine speed, reference governor

Fig. 10. Engine speed during simulations with a parameterisation error inthe powertrain model.This gives a negative control signal contribution fromthe LQ controller.

available engine torque and the states are well damped, seeFig 9, Fig. 10 and Fig. 11.

If instead the ”real” powertrain is parameterised with avehicle mass of 25 tons while the powertrain model is un-changed, the LQ control signal gives a positive contributionto the total control signal, as seen in Fig. 12. As seen, thetotal control signal exceeds the maximum available torquelimit. The total control signal for the rate limiter is toobig just for a short moment while the control signal forthe reference governor constantly stays over the limit. Thereason for the bad behaviour in the reference governor caseis that it is only in the feedforward loop that the reference

10 10.5 11 11.5 12 12.5 13

6

7

8

9

10

11

12

Time, [s]

Whe

el s

peed

, [ra

d/s]

wheel speed reference, rate limiterwheel speed, rate limiterwheel speed reference, reference governorwheel speed, reference governor

Fig. 11. Wheel speed during simulations with a parameterisation error inthe powertrain model. This gives a negative control signal contribution fromthe LQ controller.

governor has been introduced. The reference governor doesnot know anything about the dynamics of the feedback loop.Since the reference governor works according to the specifiedmaximum engine torque limit, any positive contribution fromthe feedback controller will make the total control signalexceed the torque limit.

10 10.5 11 11.5 12 12.5 13

500

1000

1500

2000

2500

Time, [s]

Con

trol

sig

nal,

[Nm

]tot ctrl, reference governormaximum available engine torquefeedforward, reference governortot ctrl, rate limiterfeedforward, rate limiter

Fig. 12. Feedforward control signal and total control signal duringsimulations with a parameterisation error in the powertrain model. The LQcontroller has a positive contribution to the total control signal.

10 10.5 11 11.5 12 12.50

0.02

0.04

0.06

0.08

0.1

0.12

0.14

Time, [s]

Tor

sion

ang

le, [

rad]

torsion angle reference, rate limitertorsion angle, rate limiter torsion angle reference, reference governortorsion angle, reference governor

Fig. 13. Drive shaft torsion angle during simulations with a parameter-isation error in the powertrain model. This gives a positive control signalcontribution from the LQ controller.

Even though the total control signal exceeds the torquelimit, that also exists in the ”real” powertrain, the systemseems to be well damped as seen in the states in Fig. 13,Fig. 14 and Fig. 15. Since there are no oscillations in thewheel speed and the engine speed the conclusion is that this

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70

80

90

100

110

120

130

140

Time, [s]

Eng

ine

spee

d, [r

ad/s

]engine speed reference, rate limiterengine speed, rate limiterengine speed reference, reference governorengine speed, reference governor

Fig. 14. Engine speed during simulations with a parameterisation error inthe powertrain model. This gives a positive control signal contribution fromthe LQ controller.

10 10.5 11 11.5 12 12.5 13

6

7

8

9

10

11

12

Time, [s]

Whe

el s

peed

, [ra

d/s]

wheel speed reference, rate limiterwheel speed, rate limiterwheel speed reference, reference governorwheel speed, reference governor

Fig. 15. Wheel speed during simulations with a parameterisation error inthe powertrain model. This gives a positive control signal contribution fromthe LQ controller.

system will feel well damped for a driver.

V. CONCLUSIONS

This paper describes the concept of using a feedforwardcontroller in combination with an LQ feedback in the aimof damping out powertrain oscillations by using the engineas an actuator. Simulations in Matlab/Simulink have beenperformed with good results both in the case with an idealmodel and when a parameter error has been introduced inthe model.

One major challenge in the control system is that thereare limitations in how much engine torque that actuallycan be produced by the engine. The calculation of thereference signal to the feedforward controller is therefore

very important, since the step size in the reference signal hasa major impact on the magnitude of the feedforward controlsignal. In this paper two methods of calculating the referencesignal have been presented, the rate limiting method and thereference governor method.

The reference governor has only been introduced in thefeedforward control loop. This means that a large positivecontribution from the feedback loop might make the totalcontrol signal for the whole system too large for the engineto handle. In the future, the reference governor shouldbe extended to handle also the control signal contributionfrom the feedback loop. The described control system hasbeen implemented and is currently undergoing experimentalvalidation.

REFERENCES

[1] T. N. Gillespie, Fundamentals of Vehicle Dynamics. Society ofAutomotive Engineers, 1992.

[2] U. Kiencke and L. Nielsen, Automotive Control Systems For Engine,Driveline and Vehicle. Society of Automotive Engineers, 2000.

[3] I. Arsie, C. Pianese, G. Rizzo, and G. Serra, “A dynamic model forpowertrain simulation and engine control design,” Consiglio Nazionaledelle Ricerche, SAE International, no. 2001-01-17, 2001.

[4] M. Pettersson, “Driveline modeling and control,” Ph.D. dissertation,Department of Electrical Engineering, Linkoping University, Linkop-ing, Sweden., 1997.

[5] S. Richard, P. Chevrel, and B. Maillard, “Active control of future ve-hicles drivelines,” in Proceedings of the IEEE Conference on Decisionand Control, vol. 4. IEEE/CSS, December 1999, pp. 3752–3757.

[6] J. Fredriksson, H. Weiefors, and B. Egardt, “Powertrain control foractive damping of driveline oscillations,” in Vehicle Systems Dynamics,no. 37(5), 2002, pp. 359–376.

[7] M. Bruce, “Powertrain modelling and experimental validation in aheavy duty vehicle,” in AVEC ’04, 7th International Symposium onAdvanced Vehicle Control, vol. 1. AVEC, August 2004, pp. 71–76.

[8] J. Fredriksson, “Nonlinear model-based control of automotive pow-ertrains,” Ph.D. dissertation, Department of Signals and Systems,Chalmers University of Technology, Gothenburg, Sweden., 2002.

[9] D. Lefebvre, P. Chevrel, and S. Richard, “Control analysis tools foractive attenuation of vehicle longitudinal oscillations,” in Proceedingsof the 2001 IEEE International Conference on Control Applications,vol. 01CH37204. IEEE, September 2001, pp. 811–816.

[10] E. G. Gilbert and I. Kolmanovsky, “Discrete-time reference governorsfor systems with state and control constraints and disturbance inputs,”in Conference on Decision and Control, vol. 43. IEEE, December1995, pp. 1189–1194.

[11] E. G. Gilbert and K. T. Tan, “Linear systems with state and controlconstraints: The theory and application of maximal output admissiblesets,” in IEEE Transactions on Automatic Control, vol. 36. IEEE,September 1991, pp. 1008–1020.

[12] A. Bemporad, “Reference governor for constrained nonlinear systems,”in IEEE Tractions on Automatic Control, vol. 43. IEEE, March 1998,pp. 415–419.

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