Adaptive Optimal Control for Suppressing Vehicle LongitudinalVibrations

Donghao Hao1, Changlu Zhao1, Guoming G. Zhu2, Ying Huang1, and Long Yang1

Abstract— Sudden torque change under the tip-in operationoften causes driveline low-frequency torsional vibrations, whichseriously impacts vehicle drivability. Typical driveline resonancefrequency is under 10Hz in the longitudinal direction and itcannot be eliminated through mechanical design optimization.To provide a smooth acceleration with minimal vibrations,an adaptive optimal tracking controller of engine torque isdesigned in this paper. A nonlinear model, elaborating thedriveline and vehicle longitudinal dynamics, is developed. Basedon the linearized control-oriented model, a receding horizonlinear quadratic tracking (RHLQT) controller is designed alongwith the Kalman optimal state estimation. The optimal controldesign parameters (weightings) are tuned under different roadconditions. In addition, the road surface contact friction coef-ficient is estimated using the recursive Least-Squares method.The RHLQT adapts to the estimated road condition (surfacefriction). The control performance of the adaptive RHLQT isstudied under different road conditions, compared with fixedcontrol parameters LQT controllers. The simulation resultsconfirm the effectiveness of the proposed control scheme.

I. INTRODUCTION

This paper focuses on the longitudinal low-frequencyvibrations that leads to unpleasant oscillations and reducesdriving comfortability. Typical resonance frequency is below10Hz occurring under tip-in and tip-out operations [1], [2].These oscillations are mainly due to the vehicle drivelineelasticity. In addition, the gear backlash and high frictionroad surface also increase vehicle oscillation intensity [3].

To reduce vehicle oscillations, several control methodsare used widely in automotive industry. The easiest way toreduce the low-frequency oscillations is through open-loopcontrol. One commonly used method is rate shaping by filter-ing desired torque signal [4]. The main disadvantage is thatsignificant calibration time is required to generate calibrationlookup tables as functions of gear-ratio, engine speed andpedal position. Proportional-Integral-Derivative (PID) con-troller is widely used to damp out the longitudinal vibrations[5]. Templin developed an LQR based driveline anti-jerkcontroller used to generate engine compensation torque [6].Fredriksson studied different linear controllers such as PID,pole placement and linear quadratic Gaussian/loop transferrecovery (LQG/LTR) controllers [5], which shows that the

*This work was supported by the National Natural Science Foundationof China.

1Hao, Zhao, Huang and Yang are with the School of Mechani-cal Engineering, Beijing Institute of Technology, Beijing, China. Thiswork was completed when Hao is a visiting scholar at the MichiganState University. hdh@bit.edu.cn, clzhao@bit.edu.cn,hy111@bit.edu.cn, yanglong3011@163.com

2Zhu is with the Department of Mechanical Engineering, Michigan StateUniversity, East Lansing, MI 48824, USA and he is the correspondingauthor. zhug@egr.msu.edu

proposed LQG/LTR controller is the best in terms of controltuning and showed performance improvement in simulationsand experiments. However, the transient performance im-provement reduces as the LQG/LTR weighting on the torsionvibrations increases. Bruce combined a feedforward control,based on an approximate inverse plant model to provide fasttransient control compensation, with a linear quadratic (LQ)feedback control [7]. Lagerberg proposed an MPC controllerwith constraints on input torque and its rate [8]. It providesgood performance for suppressing powertrain vibrations butthe proposed MPC controller has very high computationalload, making practical implementation impossible.

As a summary, it can be concluded that the open-loopcontrol, PID control, LQ control and MPC all have goodperformance for suppressing longitudinal vibrations withpenalty in vehicle acceleration performance. In addition, allcontrollers above are evaluated under a given road conditionand they are not able to compensate for the change ofroad friction condition. Reference [9] shows that the vehicledriveline vibrations are heavily affected by the road frictioncondition. As a result, in this paper, the negative effect ofthe road condition change to the control performance will bereduced by the proposed adaptive parameters switch strategy.

The rest of paper is organized as follows. Section IIprovides the overview and control problem formulation forthe target vehicle vibration system. Section III presentsthe control-oriented model, model linearization and its dis-cretization. The receding horizon LQT controller is designedbased on the discrete-time model in Section IV. The adaptiveoptimal control based on the estimated road friction conditionis also presented in this section. The simulation verificationof the proposed control method is described in Section V.Conclusions are drawn in Section VI.

II. SYSTEM OVERVIEW AND CONTROL OBJECTA. Low-frequency vibration phenomenon

Fig. 1 shows the schematic of the vehicle driveline sys-tem including engine, gearbox, differential, wheel/tire andsuspension systems. The torsional vibration of the drivelinesystem results in fluctuation of the wheel driving torque,which in turn affects the vehicle longitudinal acceleration.The resonant frequency is between 1 and 10Hz; see Fig.2. Torsional and longitudinal vibrations are mainly causedby system elasticity such as clutch system stiffness kc,halfshaft stiffness khs and tire rotational stiffness kt. Theadhesion condition between the driving tire and road alsohas a huge influence on the vibration characteristics. As asummary, vehicle longitudinal vibration is a combination of

2019 American Control Conference (ACC)Philadelphia, PA, USA, July 10-12, 2019

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e

c1g

eJ

cT

eT1gJ

2g2gJ

dfJ df

hsrThslT

gi

dfi

ck

cc

hsk

hscriml

rimr

hsk

hsc

tl tr

cJ

(a) Driveline

2z2x

1x

1z

hsT

sfksfc

blk

blc

tfk

t

bM

rim

rimJtireJ

tk

xF

tc

Longitudinal Low-frequency Vibration

Vehicle Body

Road

(b) Tire, suspension and body

Fig. 1. Vehicle schematic

the transmission system torsional vibration, tire nonlinearvibration, and suspension vertical and body pitch vibration[10].

B. Control problem description

1) Anti shuffle: Fig. 2 shows the vehicle system tip-in re-sponses, where large longitudinal oscillations (acceleration)can be observed and an anti-shuffle controller is needed tocompensate the demand torque for suppressing oscillations.Therefore, the control objective is to generate the enginecompensation torque to improve vehicle comfort by reducingoscillations occurring during the vehicle acceleration process,where the engine speed and vehicle wheel speed are used asfeedback signals so that the vehicle have a smooth acceler-ation. That is, the vehicle longitudinal vibration is reduced.The following are a few key performance requirements to beconsidered.

• The oscillation reduction shall be realized without pe-nalizing the response performance. This means thatthe engine, wheel and vehicle speeds should have thesame response performance with or without anti shufflecontroller.

• Only engine, wheel speed measurements and enginedemand torque can be used for the controller.

• Minimal tuning parameters are required to adjust con-troller on-line.

0 0.5 1 1.5 2 2.5 3Time (s)

0

1

2

3

4

5

6

Veh

icle

long

itudi

nal a

ccel

erat

ion

(m/s

2 )

Acc at Dry asphaltAcc at Wet asphaltAcc at Cobble dryEngine torque/25 [Nm]Engine speed/1000 [rpm] at Dry asphalt

Fig. 2. Acceleration in different road conditions

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Slip ratio [%]

-0.1

0.1

0.3

0.5

0.7

0.9

1.1

1.3

Fric

tion

coef

fici

ent

Dry asphaltWet asphaltCobble dry

Fig. 3. Friction coefficient-slip ratio curves in different roads

2) Reduce the effects of road conditions on the vibration:Fig. 2 shows the longitudinal vibration under different roadadhesion conditions. The corresponding friction coefficientand slip ratio curves of the three roads are shown in Fig. 3.Due to the different peak values and slope ratios, the vehiclewill have different slip ratio ranges for a given driving torqueunder different road conditions. It will affect the amplitude oflongitudinal force directly and so does the longitudinal vibra-tion. Therefore, the designed vibration suppression controllershould adapt to different road conditions. In order to do so, aroad condition parameter estimator is required to obtain thereal-time adhesion characteristics. As a result, the designedanti shuffle controller will be adapt to the road conditionbased on the estimated road condition parameter.

III. SYSTEM MODEL

A. System dynamics

The detailed dynamics of the vehicle longitudinal vi-bration is derived in [10]. In order to design a model-based controller, the system is simplified to a three-DOF(degree of freedom) system, as shown in Fig. 4, where theflywheel, clutch, gearbox input shaft, gears, output shaft, anddifferential output shaft are lumped together as one momentof inertia J1; J2 is the wheel rotational moment inertia; Mis the vehicle mass; Fr is the resistance force consisting ofwind drag, slope resistance and tire rolling resistance; Fx

is the driving force for one driving wheel; kr and cr arethe equivalent stiffness and damping of driveline; and i is

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the overall transmission ratio equal to the product of ratios of gearbox and differential. System dynamic equations areshown below (1).

x1 = f1(x) =−2cr−i2b1

J1i2· x1 +

2crJ1i

· x2 +−2kr

J1i· x3 +

1J1Te

x2 = f2(x) =crJ2i

· x1 +−cr−b2

J2· x2 +

kr

J2· x3 +

−rFzd·sin(c·arctan(b· rx2−x4rx2

−e(b· rx2−x4rx2

−arctan(b· rx2−x4rx2

))))

J2

x3 = f3(x) =1i · x1 − x2

x4 = f4(x) =2Fz

M d · sin(c · arctan(b · rx2−x4

rx2− e(b · rx2−x4

rx2− arctan(b · rx2−x4

rx2))))− 0.5·x4

2cdAcρ+Mcr1+Mcr2x4

M

y = Cx =

1i 0 0 00 1 0 01i −1 0 0

x =

x1

ix2

x1

i − x2

(1)

i

1 1J

eT 2 2J

M

xFrk

rcrF

Fig. 4. Control oriented 3-dof model

where x = [x1 x2 x3 x4]T is the system state vector

and y is the output. State x1 is the engine speed; x2 isthe wheel speed; x3 is the angle difference between theengine flywheel divided by i and wheel speed; and x4 is thevehicle longitudinal speed. Coefficients b, c, d, and e are thetire magic formula parameters that reflect the road adhesionconditions. b1 and b2 are the friction coefficients in engineflywheel side and wheel side, respectively. R is the wheeldynamic effective radius. Fz , equal to Mg

4 , is the verticalforce in the driving wheel, ignoring vehicle pitch movementand gravity center transfer. cd is the drag coefficient. Ac isthe cross sectional area. ρ is the density of the air. cr1 and cr2are the rolling force coefficients. Te is the effective enginetorque which is calculated by the proposed controller in thisstudy. Parameter values are shown in the Appendix.

B. Model linearization

It can be seen that Model (1) is nonlinear due to thenonlinear items in f2(x) and f4(x), such as slip ratio rx2−x4

rx2

and tire longitudinal force Fzdsin(·). A linearized model isneeded for designing the optimal linear quadratic recedinghorizon controller. At each operational condition, the sys-tem model can be linearized using the classical Jacobianlinearization. Therefore, model (1) can be transformed intothe following linearized model.

{∆x = A∆x+B∆u∆y = C∆x

(2)

where ∆x = x−x0, ∆y = y−y0, ∆u = u−u0 and systemmatrices are given below.

A =∂f

∂x

∣∣∣∣(x0,u0)

=

−2cr−i2b1

J1i22crJ1i

−2krJ1i 0

crJ2i

−cr−b3J2

+Fzdµ·x40−x2

20·J2

krJ2

Fzdµx20·J2

1i −1 0 0

0gdµx402rx2

200 −gdµ

2rx20−CdAcρ

M −cr2

B =

∂f

∂u

∣∣∣∣(x0,u0)

=[

1J1

0 0 0]T

(3)

Note that (x0, u0, y0) denotes the equilibrium point thatcan be solved using (1) at the steady state by setting x = 0.The determined equilibrium point is used to calculate thesystem coefficient matrices, and u0 is a known value as afunction of driving pedal position and used as the nominal(feedforward) control. The compensation torque, denoted by∆u, is calculated by the designed controller used to reducethe vibrations. Therefore, the control input for the systemcan be written as

u = u0 +∆u (4)

C. Discrete-time model

The linearized continuous-time model (2) can be dis-cretized using the forward Euler approximation, resulting thefollowing discrete-time model.{

∆x(k + 1) = A(k)∆x(k) +B(k)∆u(k)∆y(k) = C(k)∆x(k)

(5)

where {A(k) = eATs

B(k) = A−1(eATs − I)

and Ts is the sample period which is set to 0.02s in ourstudy.

IV. LINEAR QUADRATIC CONTROL

In this section, a receding horizon linear quadratic trackingcontroller is designed to make the system output y(k) trackthe reference r(k). The reference r(k) is calculated by areference vehicle model for a flexible driveline with a rigidaxle. Since the control design will be based on the deviation

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model (5) to provide an incremental control input ∆u(k)at each operational point, the control target is to make thedeviation output ∆y(k) track the deviation reference ∆r(k).From (2) it can be seen that the actual engine and wheelspeed will follow the reference trajectory without vibrationswhen ∆y(k) is regulated down to 0 during the tip-in process.

A. Receding horizon linear quadratic control

The control objective of the receding horizon LQT controlis to minimize the tracking error e(k) defined in (6) withfeasible control effort ∆Te, where tracking error e(k) isdefined as

e(k) = ∆y(k)−∆r(k) = C(k)∆x(k)−∆r(k). (6)

To simplify notations, ∆xk, ∆yk, ∆uk, ∆rk, ∆ek, Ak,Bk and Ck are used to denote ∆x(k), ∆y(k), ∆u(k), ∆r(k),∆e(k), A(k), B(k) and C(k) at current time step k in therest of the paper. The performance cost function of the LQTcontroller at each control step is defined as

J(k) =1

2eTkf

Fekf+

1

2

kf−1∑k=k0

[eTkQek +∆uTkR∆uk] (7)

where F = FT ≥ 0, Q = QT ≥ 0, and R = RT > 0 aregiven weighting matrices. [k0 ∼ kf ] is the receding horizonmoving window at each control step k for predefined N -step(N = kf −k0) tracking reference. That is, an N-step optimalcontroller is designed at each control step for the given N -step tracking trajectory and only the first control step will beused, which is the so-called moving optimization problem inMPC (model predictive control). Since the LQT controller isdesigned based on the deviation model (5) linearized at thecurrent equilibrium point (at k = k0) and the control objectis to set ∆y to 0, the tracking reference for the deviationoutput ∆yk is

∆rk = 0, k = k0, k1, ..., kf . (8)

The optimal tracking problem can be solved by following theminimum principle approach in [11]. The deviation control∆uk can be obtained as

∆uk = −∆uFBk +∆uFFrk = −LFBk∆xk + LFFkgk+1

(9)where

LFBk = [R+BTk Kk+1Bk]

−1BTk Kk+1Ak

LFFk = [R+BTk Kk+1Bk]

−1BTk

gk = ATk {I −Kk+1[I +BkR

−1BTk Kk+1]

−1

BkR−1BT

k }gk+1 + CTk Q∆rk.

∆uFBk is the state feedback control. Matrix K in the controlgains LFBk can be obtained by solving the Riccati (10)backwards online using the boundary conditions in (11).

Kk = ATkKk+1[I+BkR

−1BTk Kk+1]

−1Ak+CTk QCk (10)

Kkf= CT

k FCk, gkf= 0 (11)

Note that in (9) the system state ∆x3k in ∆xk used forfeedback control cannot be measured and is estimated onlineusing the Kalman filter addressed in the next subsection.

B. Kalman State Estimation

The Kalman state estimation is a stochastic filter thatminimizes the covariance of the estimation error to providean optimal state estimation subject to Gaussian noise inputs.For a given initial state ∆x0, it uses the current outputmeasurement ∆yk and control ∆uk to estimate the next state∆xk+1 in the following form.

∆xk+1 = Ak∆xk +Bk∆uk +Hk(∆yk − Ck∆xk) (12)

Note that the initial state is calculated as follows

∆x0 = xk−1 − xk,0 (13)

The subscript ”k, 0” denotes the equilibrium point at time k.It shows that the initial state is updated once the equilibriumpoint xk,0 is switched. The Kalman filter gain Hk is obtainedas follows

Hk = AkΣkCTk (CkΣkC

Tk + V )−1 (14)

where the estimation error covariance matrix Σk is calculatedrecursively by the following difference Riccati equation usingthe initial condition Σ0 = E(∆x0∆xT

0 ).

Σk+1 = AkΣkATk −HkCkΣkA

Tk (15)

Using the estimated state vector in (9) and substituting it into(4), the system control input can be obtained as

uk = ∆uk + uk,0 = −LFBk∆xk + uk,0 (16)

C. Adaptive optimal control design

From (3) it can be seen that some entries of system matrixA are related to the road condition, e.g., dµ, which is theslope ratio of the friction coefficient-slip curve. It shows thatthe road adhesion condition affects the system dynamics.To maintain the same vibration reduction performance, thesecontrol parameters should be modified under different roadconditions. Three typical roads are chosen in this study asshown in Fig. 3. Since the longitudinal vibration occursmainly under the rising phase of the friction coefficient-slipcurve, the slope ratio at each slip ratio is selected as theindicator of the road condition. The adaptive switch strategyis based on the slope ratio of the friction-coefficient-slipcurve. Optimal control parameters found under different roadconditions are shown in Table I. Note that, when the sloperatio is less than 3.9, e.g. ice road, the longitudinal vibrationswill not occur. Therefore, no more control is needed.

TABLE ICONTROL PARAMETERS

Road type dµ F Q R

Dry asphalt ≥ 23.6[500 0 00 1000 00 0 500

] [25 0 00 200 00 0 50

]0.008

Wet asphalt ≥ 6.4[80 0 00 300 00 0 3000

] [10 0 00 100 00 0 200

]0.04

Cobble dry ≥ 3.9[10 0 00 100 00 0 5000

] [5 0 00 5 00 0 500

]0.1

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D. Road Condition Estimation

The basic idea of Least Squares (LS) method is fitting amathematical model using a sequence of observed data byminimizing the sum of the squares of the difference betweenthe observed and computed data. By doing so, the effect ofnoise in the data can be minimized. Therefore

y(k) = φT (k)θ + v

V (θ, k) =1

2

k∑i=1

λk−i(y(i)− φ(i)T θ)2.(17)

The cost function V (θ, k) in (17) can be minimized byselecting proper parameters [12]. To get the solution, therecursive form is given as follows

θ(k) = θ(k − 1) + L(k)(y(k)− φT (i)θ(k − 1)) (18)

whereL(k) = P (k)φ(k)

= P (k − 1)φ(k)(λ+ φT (k)P (k − 1)φ(k))−1

P (k) = (I − L(k)φT (k))λ−1P (k − 1).

P (k) is referred to as the error covariance matrix and λ isa forgetting factor. The parameter to be identified is definedas follows

θ(k) = dµ (19)

where dµ is the slope ratio of the friction coefficient-slipratio curve. y(k) is updated whenever newly observed Fx,i

from the tire force estimator is available (For more detailsabout the tire force estimator, see [13]). φ(k) is updated asthe real-time slip ratio.

V. SIMULATION VALIDATION

The discrete-time model and the receding horizon LQTcontroller are implemented in Simulink and validated insimulation studies. Fig. 5 illustrates the control scheme. Themodel is verified by experiment data. The vehicle is runningunder three different road conditions sequentially and theroad condition change time is marked using the gray verticaldashed-lines in Figs. 6 and 7.

Vehicle

Matrix A updateDiscretize

Receding

horizon

Rigid vehicle

reference model

Kalman

Filter

Road

condition

estimator

Adaptive Switch

kA kB

, ,F Q R

, ,i i iF Q R

d

,0ˆ ˆk k kx x x ˆ

kx

FBkL

k 1g

1, 2,3i

k 1

ˆ

L g

FBk k

FFk

L x

ku,0k k ku u u

ky

,0kx,0ku

,0ku

FFkL

Fig. 5. Implemented control scheme

Figs. 6 and 7 shows the simulation results with a steptorque input. The vibration suppression performance, accel-eration performance and the controller output u for the three

control schemes are presented. Vehicle longitudinal vibrationhappens without any active torque control is represented bythe blue solid line. Under dry asphalt road, the fixed LQT1(FLQT1) controller (red dot line) has a good vibration sup-press performance and speed tracking performance, whichindicates that it is able to provide good drivability withoutsacrificing acceleration performance. However, FLQT1 is notable to eliminate the vibrations under the wet asphalt andcobble dry roads. For the fixed LQT 2(FLQT2) controller(black dot-dashed-line), it is able to reduce the vibrationunder all three road conditions whereas it is at the costof speed tracking performance under both dry asphalt andcobble dry roads as shown in Fig. 6 (a) and (b). Theperformance of the adaptive LQT(ALQT) is represented bythe green dashed-line in Figs. 6 and 7. It can be seen that theALQT controller is able to provide consistent performanceunder three different road conditions. The drivability andspeed tracking performances are guaranteed by adapting tothe estimated road condition.

0 1 2 3 4 5 6 7 8 9 10Time [s]

-100

102030405060708090

100110120130140

Eff

ectiv

e E

ngin

e T

orqu

e [N

m]

Dry asphalt Wet asphalt Cobble dry

w/o controlw/i FLQT 1w/i FLQT 2w/i ALQT

(a) Effective engine torque

(b) Longitudinal acceleration

Fig. 6. Validation results

Fig. 8 shows the road condition estimation results. Itcan be seen obviously that the estimated value has a goodagreement with the actual value, which guarantees the correctcontrol parameters used under different road conditions.

VI. CONCLUSIONSIn this paper, an adaptive receding horizon LQT controller

with Kalman state estimation and road condition estimator

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(a) Engine speed

(b) Driving wheel speed

Fig. 7. Validation results

0 1 2 3 4 5 6 7 8 9 10Time [s]

0

5

10

15

20

25

30

Slop

e ra

tio o

f fr

ictio

n co

effi

cien

t-sl

ip c

urve

Dry asphalt Wet asphalt Cobble dry

Engine demand torque/4 [Nm]Real slope ratioEstimated by RLS

Fig. 8. Road friction coefficient-slip curve slope ratio comparison

is designed based on a linearized model at the currentoperational condition from the developed nonlinear modelof vehicle longitudinal vibrations. The optimal LQT controlparameters are adapted based on the estimated road condi-tion parameter. Both vehicle drivability and speed trackingperformances are guaranteed under various road conditions.Future work is to validate the controller experimentally.

APPENDIX

A. Vehicle parameters

A list of the main parameters of the vehicle is given inTable II.

TABLE IIMAIN PARAMETERS OF THE VEHICLE

Symbol Value Symbol Value

M 1420 kg cd 0.12A 2.2 m2 ρ 1.225 kg/m3

cr1 0.057 cr2 0.052J1 0.2 kgm2 J2 1.53 kgm2

kr 4650 Nm/rad cr 4.76 Nm/(rad/s)i 13.75

ACKNOWLEDGMENT

The authors would like to acknowledge the NationalNatural Science Foundation of China for the financial supportunder Project numbers 51475043 and 50975026, and also theChina Scholarship Council (CSC) for supporting DonghaoHao’s study at Michigan State University as a joint PhDstudent.

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