IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL. 64, NO. 10, OCTOBER 2015 4461

Simultaneous Estimation of Road Profile and TireRoad Friction for Automotive Vehicle

J. J. Rath, Student Member, IEEE, K. C. Veluvolu, Senior Member, IEEE, and M. Defoort

Abstract—The longitudinal motion control of automotive vehi-cles is heavily reliant on information about the time-varying tireroad friction coefficient. In the presence of varying road roughnessprofiles, the effective vertical load on each wheel varies dynam-ically, influencing the tire friction. In this paper, we integratedthe vertical and longitudinal dynamics of a quarter wheel toform an integrated nonlinear model. In the modeled dynamics,the time-varying random road profile and the tire friction aretreated as unknown inputs. To estimate these unknown inputsand states simultaneously, a combination of nonlinear Lipschitzobserver and modified super-twisting algorithm (STA) observer isdeveloped. Under Lipschitz conditions for the nonlinear functions,the convergence of the estimation error is established. Simula-tion results performed with the high-fidelity vehicle simulationsoftware CarSim demonstrate the effectiveness of the proposedscheme in the estimation of states and unknown inputs.

Index Terms—Automotive vehicle, higher order sliding-modeobserver (SMO), road profile, tire friction, unknown inputs.

I. INTRODUCTION

AUTOMOTIVE vehicle control systems have witnessedsignificant contributions in various active safety control

systems such as antilock braking systems, traction control sys-tems, active cruise control [1], [2], etc. The precise and accurateimplementation of these control systems requires informationabout various factors affecting the dynamic performance of thevehicle. The automotive vehicle is a complex nonlinear dy-namic system whose performance is affected by environmentalfactors such as aerodynamic drag, tire friction, road profile,etc. These factors act as unknown inputs/disturbances for thevehicle system and thus affect the performance of various safetyprotocols implemented.

Tire road friction is a rapidly varying dynamic phenomenonthat affects the longitudinal motion control of the vehicle. Itis heavily dependent on tire parameters such as pressure and

Manuscript received December 30, 2013; revised May 22, 2014 andAugust 29, 2014; accepted October 28, 2014. Date of publication November 24,2014; date of current version October 13, 2015. This work was supported bythe Basic Science Research Program through National Research Foundationof Korea funded by the Ministry of Education, Science and Technology underGrant 2014R1A1A2A10056145. The review of this paper was coordinated byProf. J. Wang.

J. J. Rath and K. C. Veluvolu are with the School of Electronics Engineering,Kyungpook National University, Daegu 702-701, Korea (e-mail: veluvolu@ee.knu.ac.kr).

M. Defoort is with LAMIH, CNRS UMR 8201, UVHC, 59314 Valenciennes,France.

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

Digital Object Identifier 10.1109/TVT.2014.2373434

temperature, along with environmental factors such as roadsurface type. To replicate the effect of friction, various static anddynamic friction models such as Magic Formula tire [1], Dugoffmodel [1], LuGre model [3], etc., have been developed. Thevariations in tire friction are more prominent in the presenceof dynamic vertical load on the wheel. The effective verticalload on each wheel is determined by the vertical dynamics ofthe vehicle governed by the suspension system. The verticalload is influenced by the presence of suspension nonlinearitiesand, in particular, by the varying road profile. Road profilecharacterizes the vertical height of the ground below the wheel,determining the condition of the road for longitudinal vehiclemotion and is generally measured by expensive instrumentscalled profilographs [4]. This entity is replicated on the Inter-national Roughness Index (IRI), which is the widely used roadroughness indicator proposed by the International Organizationfor Standardization. The IRI is a scale that characterizes differ-ent road profiles in terms of its roughness, ranging from verygood to unpaved rough roads. Under the influence of varyingroad surface and random roughness profile, tire friction esti-mation, and consequently tractive control, is severely affected.The estimation of tire friction and road profile simultaneouslyfor effective implementation of various safety control schemesis a very important problem in research for autonomous vehicledynamics.

In [5], the robust nonlinear estimator for tire forces was de-signed, and consequently, an effective traction control schemewas proposed for different road surface conditions. However,this method did not consider the dynamic effects of road profileon the vertical loading effect for the wheels. In [6], an adaptivespeed control technique for longitudinal motion of the vehiclewas designed where a nonlinear observer was designed toestimate the internal friction state of the LuGre friction model.Although in [6] the dynamic vertical load on each wheel wasevaluated, the effect of road profile was not yet considered.The real-time estimation of tire friction for a nonlinear longi-tudinal tire force model of the vehicle was performed in [7].Similarly, the estimation of friction in real time by consideringboth the lateral and longitudinal dynamics was performed in[8]. The effect of vertical loading on the real-time estimationof tire friction by use of sensors such as accelerometers, GlobalPositioning System, torque sensors, etc., was discussed in [9].However, these works did not consider the effect of road profileor the suspension nonlinearities on the vertical loading whileestimating the tire friction. Similarly in [10], neural-network-based approaches were designed to estimate the random roadprofile by considering the vertical dynamics of the vehicle only.

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4462 IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL. 64, NO. 10, OCTOBER 2015

In all the approaches discussed above, where the estimation ofroad profile was performed, the nonlinearities in the suspensionsystem were not considered.

Sliding-mode theory has evolved over time for effectiveestimation of unknown inputs [11]–[14] for various classes ofsystems. In [15], first-order SMO (FOSMO)-based observerswere designed for estimation of faults/unknown inputs fornonlinear systems. To overcome the issues of chattering duringestimation without the use of filtering, higher order sliding-mode (HOSM) observers were designed. The use of HOSMobservers for state and unknown input estimation in uncertainnonlinear systems has been discussed in [16]. There have beensignificant works contributing to estimation of tire friction androad profile using sliding-mode observers (SMOs). In [2] and[19], FOSMOs were designed to estimate the tire friction fornonlinear longitudinal vehicle dynamics. These approaches,however, did not consider the effect of road profile on thevertical load that affects the tire friction significantly. Similarlyin [17], a FOSMO was designed to estimate the states of thesuspension system under the influence of different road profiles.In [18], the super-twisting algorithm (STA) observer was usedto estimate the road profile acting as an unknown input to thesystem. This work considered the nonlinear vertical dynamicsof the vehicle but did not consider the nonlinearities of thesuspension system during the estimation. Recently, in [19], aHOSM observer based on the STA observer was developedfor estimation of tire road friction coefficient. However, thismethod only employed the longitudinal dynamics of the vehi-cle. The necessity of designing an approach where the nonlin-earities of the vehicle system can be considered for estimatingthe road profile and tire friction simultaneously with no filteringis apparent.

To address this issue by estimating the tire friction underthe effect of dynamic vertical load, a nonlinear vehicle modelthat integrates the vertical–longitudinal dynamics has beenemployed for simultaneous estimation of the tire friction andthe road profile, which is a first attempt in this direction tothe best of the authors’ knowledge. The longitudinal frictionforce as a function of the tire friction and the road profilebased on road roughness values [4] and the engine frictionwith uncertainties are time-varying entities that serve as threeunknown inputs to the coupled dynamics of the quarter vehiclesystem. Under rank conditions for the output matrix, a trans-formation is developed that transforms the developed systeminto two subsystems. For the system affected by unknowninputs, modified STA-based HOSM observers are designed toensure stability of error dynamics in finite time. Once slidingmode is attained, the unknown inputs are then simultaneouslyreconstructed. For a subsystem without unknown inputs, a non-linear observer under Lipschitz conditions is designed to ensurestability.

The rest of this paper is organized as follows: In Section II,the dynamics of the quarter wheel vehicle is discussed. Theintegrated nonlinear vertical–longitudinal model for a quarterwheel is developed. The conceptual design of the proposedobservers is discussed in Section III. The simulation results fordifferent driving scenarios performed in CarSim are presentedin Section IV. Section V concludes this paper.

II. VEHICLE DYNAMIC MODEL

The effect of environmental factors such as road surface, roadprofile, road bank angle, road grade, etc., affects the vehiclemotion prominently. To analyze the effect of road surface androad profile variations on vehicle motion, an integrated vehiclemodel is developed. For developing such a model, longitudinaland vertical dynamics of motion without any steering action areconsidered. The incorporation of independent axle suspensionsystems and independently driven wheels in vehicle systemsfacilitates the modeling of a quarter wheel vehicle, affected byroad surface and road profile. It is assumed that the governingdynamics for each quarter wheel can be replicated for the otherwheels [1].

A. Vehicle Vertical Dynamics

The suspension dynamics of a vehicle govern the verticalstability aspects of motion for the vehicle. The suspensionsystem can be designed as independent wheel suspension oras dependent front axle/rear axle wheel suspension. The utilityof independent suspension systems is critical for analysis ofquarter car models as vertical deflections of each wheel areindependent. The nonlinearities of the suspension system arecontributed by the spring, and damper deflections are modeledas follows [25]:⎧⎪⎨

⎪⎩fk = ksnl(zs − zu)

3

fb = bsnl√|zu − zs|sign(zu − zs)

Ψ = fk + fb

(1)

where zs and zu are the sprung and unsprung mass displace-ments, respectively; zs and zu are the sprung and unsprungmass velocities, respectively; ksnl and bsnl are the nonlinearfactors of the spring and damper coefficients of the suspensionsystem, respectively; and Ψ represents the total nonlinearitiesof the suspension dynamics. The motion of the vehicle over anirregular surface such as a bump or a ditch restricts wheel travelin the given range and prevents contact between the tire andthe ground. These nonlinear spring characteristics effectivelyreplicate the suspension deflection under such irregular mo-tion. In the presence of nonlinearities and disturbance inputs,the qualitative performance of suspension systems relative topassenger comfort, road handling, road holding, etc., degrades.To effectively improve the performance of the suspension,active feedback control is provided. The dynamics of an activesuspension system (with feedback control) under the influenceof nonlinearities (1) and disturbance corresponding to a quartercar model is given by [1]

muzu = bszsu + kszsu − kr(zu − ζ) + Ψ− Fa

mszs = −bszsu − kszsu −Ψ+ Fa (2)

where ms is the sprung mass, mu is the unsprung mass,ks is the spring stiffness, bs is the damping constant, kr isthe tire stiffness, Fa is the controlled actuator force, zsu =(zs − zu) is the suspension deflection, zsu = (zs − zu) is therate of suspension deflection, and ζ(t) is the road excitation

RATH et al.: ESTIMATION OF ROAD PROFILE AND TIRE ROAD FRICTION FOR AUTOMOTIVE VEHICLE 4463

profile. The variations in vertical height of the road act as adisturbance input for the suspension dynamics of the vehicle.For longitudinal motion of the vehicle, this vertical height ofthe road can be characterized by roughness patterns of the roadsurface represented as the road profile [10]. The IRI-based roadclassification can be employed to analyze the road roughness ofvarious road classes from very good road (Class A) to very poorroad conditions (Class E). To generate the road profile as perthis standard, a differential-equation-based model was modeledin [20]

ζ(t) = −2πn0vζ(t) + 2π√σvw0 (3)

where v is the vehicle longitudinal velocity, σ is the road rough-ness coefficient, n0 is the reference space frequency, and w0

is the Gaussian white noise. For the case of active suspensionsystems, in order to provide compensation for the road profile,the control effort Fa is employed. Under the assumptions thatthe vertical load of the vehicle is distributed equally on eachtire, the dynamics of the vertical load Fn can be given by [1] as

Fn = (ms +mu)g − kr (zu − ζ(t)) (4)

where the vertical load on each wheel is contributed by thesuspension dynamics and the road profile.

B. Longitudinal Dynamics

The longitudinal dynamics of the vehicle is responsible forthe forward motion of the vehicle. The governing nonlineardynamics for the longitudinal motion of the vehicle is givenby [1] as F = Ffriction − Froll, i.e.,

Mv = Fx − CrFn (5)

where F is the effective longitudinal force, M is the quartermass of the vehicle, Fx is the longitudinal friction force, Cr

is the rolling resistance coefficient, and Fn is the verticalload acting on the wheel. The rolling resistance force Froll

is dependent on a variety of factors such as tire inflationpressure, temperature, vehicle velocity, and road surface type[21]. However, its longitudinal component is a function of thevertical load and affects the vehicle longitudinal motion. Byintegrating the vehicle dynamics with the rotational dynamicsof the wheel, the effective dynamics for the longitudinal motioncan be obtained. The equations that govern the dynamics of thetire are given by [1]

Jwr = −rFx ± Tw (6)

where J is the wheel inertia, wr is the angular velocity of thewheel, r is the radius of the wheel, and Tw is the braking/drivingtorque. The relative velocity between the vehicle longitudinalspeed and the effective speed of the wheel is given as vr =rwr − v. The phenomenon that translates the motion of thewheel into effective tractive force for the vehicle is the frictionbetween the tires and the ground. The coefficient of frictiondepends on many factors that influence the interaction betweenthe surface and the tire. Many methods, such as the MagicFormula [1], the Dugoff model [1], and the LuGre friction

Fig. 1. Integrated vertical–longitudinal dynamics.

model [3], have been proposed to model the tire road frictioncoefficient. All these models describe the relationship betweenthe friction force and the vehicle vertical load in terms of thefriction coefficient. The relationship between the longitudinalforce and the vertical load can be expressed as Fx = μFn,where μ represents the dynamic friction coefficient. As shownin (4), the vertical load is a function of suspension dynamics andis highly dynamic in nature owing to variations in road profile.Thus, coupling exists between the vertical and longitudinal mo-tion for the wheel. This affects the friction coefficient betweenthe tires and the ground, which is a ratio of the longitudinalforce to the vertical load. Hence, the tire friction estimationis now affected by the variations in the road roughness profileleading to estimation complexities. A schematic representationof coupled dynamics for the quarter vehicle in the presence ofdisturbance inputs is shown in Fig. 1.

C. Engine Dynamics

The engine dynamics for the vehicle is given by [1]

Jewe = Te + [Tfric(we, t)−ΔTengine]−Tw

N(7)

where we is the engine crankshaft speed of rotation, Te isthe engine torque due to combustion, Je is the moment ofinertia of the engine, ds is the damping in the drive shaft, Nis the gear conversion ratio, Tfric is the engine internal friction,and ΔTengine is the lumped uncertainties and losses presentin the transmission and drive shaft dynamics representing thevariation in stiffness and damping parameters form the engineto the drive shaft. Engine internal friction is a significantparameter in modeling of drive-line dynamics as it representsthe losses incurred during combustion. It is generally modeledas a function of the engine speed [1], [25] and the enginetemperature. By deducting these losses and the uncertaintiesfrom the engine combustion torque, the effective engine torquetransferred to the drive shaft can be obtained.

4464 IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL. 64, NO. 10, OCTOBER 2015

D. Integrated Quarter Vehicle Model

The nonlinear longitudinal and vertical dynamics of thequarter wheel vehicle are considered to develop an integratedmodel. It is assumed that aerodynamic drag is negligible. Forthe suspension dynamics, the damping effect provided by thetire is neglected as it does not significantly affect the suspensionsystem and is complex to model. For the dynamics (1)–(7), byconsidering the following state vector:

x = [x1 x2 x3 x4 x5 x6]T

= [zu zs zs zu vr we]T (8)

the integrated longitudinal and vertical dynamics can beformulated as{

x = Ax + Γ(u, t) +GΦ(x, t) + Ef(x, t)

y = Cx(9)

where

A =

⎡⎢⎢⎢⎢⎢⎢⎣

0 0 0 1 0 00 0 1 0 0 0ks

ms− ks

ms− bs

ms

bsms

0 0−kr+ks

mu

ks

mu

bsmu

− bsmu

0 0−Crkr

M 0 0 0 0 00 0 0 0 0 0

⎤⎥⎥⎥⎥⎥⎥⎦

Γ(u, t) =[0 0 Fa

ms− Fa

mub51 b61

]TG = [02×1 g21 02×1]

T , E = [03×3 E11]T

f(x, t) =

⎡⎣f1(x, t)f2(x, t)f3(x, t)

⎤⎦ , E11 =

⎡⎣

kr

mu0 0

Crkr

M d22 00 0 1

Je

⎤⎦

b51 = Crg + (rTw/J), b61 = (Te/Je)− (Tw/JeN), d22 =−(r2M + J)/(JM), and g21 = [−(1/ms) 1/mu].

The nonlinearity affecting the system is given by Φ(x, t) =Ψ(x, t) and the unknown inputs by f(x, t). The nonlinearfunction affecting the system model is given as

Ψ(x, t) = ksnlx321 + bsnl

√|x34|sign(x34)

where x21 = (x2 − x1) and x34 = (x3 − x4). This nonlin-ear function replicates the nonlinearities of the suspensiondynamics. The unknown inputs affecting the system are⎧⎪⎨

⎪⎩f1(x, t) = ζ(t)

f2(x, t) = Fx

f3(x, t) = Tfric(x, t) −ΔTengine.

(10)

The unknown input f1(x, t) characterizes the road profile,whereas the unknown input f2(x, t) represents the longitudinalfriction force. The effect of the engine friction torque andthe uncertainties affecting the drive shaft dynamics has beenconsidered as the third unknown input f3(x, t). In the modeledsystem dynamics (9), the control inputs for the system Fa, Tw,and Te represent the active controlled actuator force for the

suspension system, the driving torque for the wheel, and theengine combustion torque, respectively. The displacement ofsprung mass and velocities of sprung mass and unsprung mass,relative velocity, and engine speed are considered as outputs.The output matrix can be thus defined as

C =

⎡⎢⎢⎢⎢⎣

0 1 0 0 0 00 0 1 0 0 00 0 0 1 0 00 0 0 0 1 00 0 0 0 0 1

⎤⎥⎥⎥⎥⎦ =

[05×1 I5×5

]5×6

.

The objective is now to estimate the states “x” and the unknowninputs f1(x, t), f2(x, t), and f3(x, t).

Remark 1: The measurements (relative velocity and enginespeed) are usually considered measured in automotive vehicleresearch literature [1], [2]. Various sensors such as encoders,linear position sensors, etc., are available for these measure-ments directly. The other considered outputs (displacement ofsprung mass, velocity of sprung mass, and velocity of unsprungmass) cannot be measured directly. However, acceleration ofunsprung mass and sprung mass are easily measurable withaccelerometers [22], [23]. Estimation of position and veloc-ity from acceleration suffers from notorious integration drift.Recently, new methods have been developed to provide drift-free displacement and velocity estimation from acceleration forvarious applications [24].

III. OBSERVER DESIGN

Here, the design of observers to estimate states of system (9)and the unknown inputs f1(x, t), f2(x, t), and f3(x, t) is dis-cussed. To estimate the states and unknown inputs, a combina-tion of a nonlinear Lipschitz observer (NLO) and a modifiedSTA observer based on HOSM is employed. For design of theobservers, the following assumptions are required:

Assumption 3.1: rank(C [G E]) = rank([G E]), where[G, E] is full rank.

Assumption 3.2: The nonlinear function Φ(x, u) : R6+2 �→R6 satisfies the Lipschitz condition

‖Φ(x, t)− Φ(x, t)‖ ≤ lα‖x− x‖.

Assumption 3.3: System (9) is uniformly observable.Assumption 3.4: The unknown functions f(x, t) and its first

derivative are bounded.Assumption 3.5: The control inputs are bounded, and the

system is assumed to be bounded-input bounded-state stable.Remark 2: For the system defined in (9), it is clear that the

system is uniformly observable except when x321=ks/ksnl and√

|x34|sign(x34)=bs/kbnl. As the nonlinearities ksnl and bsnlare a fraction of the original values (ks and bs, respectively),this causes the suspension deflection x21 and the rate of sus-pension deflection |x34| to attain very high values. However,in practical driving scenarios, these conditions are not satisfiedwith the proper selection of Fa. Hence, the observability of thesystem can be guaranteed.

RATH et al.: ESTIMATION OF ROAD PROFILE AND TIRE ROAD FRICTION FOR AUTOMOTIVE VEHICLE 4465

For the system dynamics (9), it can be seen thatAssumption 3.1 is satisfied. The following lemma discussesthe Lipschitz continuity for the nonlinear function Φ(x, t).

Lemma 1: For system (9), satisfying Assumptions 3.1–3.5,the local Lipschitz continuity of the nonlinear function Φ(x, t)can be established with a Lipschitz constant lα.

Proof: See Appendix A. �In the system dynamics (9), the unknown input f1(x, t)

represents the road profile. Its upper bound depends on thecomplexity and unpredictability of the road roughness profile.However, for general road conditions or surfaces, the unknowninput ζ(t) and its derivative are bounded by an a priori knownconstant. Similarly, the upper bounds of the unknown inputfunctions f2(x, t) and f3(x, t) together with their first deriv-atives can be shown to be bounded as these are functions ofthe system states that are bounded. Hence, Assumption 3.4is satisfied. With the control inputs Fa and Tw and Te welldefined, Assumption 3.5 is also satisfied.

To facilitate the design of the observer, a transformation isrequired to decouple the nonlinearities and the unknown inputsof the original system (9) into two subsystems, namely, S1

and S2, where the unknown inputs appear in the subsystem S2

and the system nonlinearities appear in the subsystem S1. Forease of exposition, the transformation matrix is formulated byvisual inspection. The formulation of the transformation for ageneral class of system similar to the automotive vehicle modelis discussed in Appendix B. The transformation matrix can begiven by

S =

⎡⎢⎢⎢⎢⎢⎢⎣

1 0 0 0 0 00 1 0 0 0 00 0 1 0 0 00 0 ms

kr

mu

kr0 0

0 0 CrJms

Mr2+JCrJmu

Mr2+J − MJr

Mr2+J 00 0 0 0 0 Je

⎤⎥⎥⎥⎥⎥⎥⎦. (11)

By employing this transformation, where S is invertible, onecan obtain

SAS−1 = A =

[A13×3

A23×3

A33×3A43×3

], CS−1 = C = [C1 C2]

SΓ = Γ =[Γ1 Γ2

]T, S[G E] =

[G13×1

03×3

03×1 E223×3

]

where

A1 =

⎡⎣ 0 0 −ms

mu

0 0 1ks

ms− ks

ms− bsM

msmu

⎤⎦ , A2 =

⎡⎣

kr

mu0 0

0 0 0bskr

msmu0 0

⎤⎦

A3 =[A31 03×2

], A4 = [03×3], Γ1 =

[01×2 − Fa

ms

]TΓ2 =

[0 Γ22 Γ23

], G1 =

[01×2 − 1

ms

]T, E22 = [I3]

C1 =

[0 1 00 0 c13

], C2 =

⎡⎣c21 0 0

0 c22 00 0 c23

⎤⎦ , A31 =

⎡⎣ 0

0ks

ms

⎤⎦

with Γ22 = −(M(JCrg + rTw)/(Mr2 + J)), Γ23 =(Te − (Tw/N)), c13 = (1 − (ms/mu)), c21 = ((kr/mu) +(Crkr/M)), c22 = −(Mr2 + J)/(MJ), and c23 = 1/Je.It can be seen that C1 and C2 are full rank, where C2 isinvertible. With the transformation (11), system (9) can be nowtransformed as

S1 :

{z1 = A1z1 +A2z2 + Γ1(u, t) +G1Φ(x, t)

y1 = C1z1(12)

S2 :

{lz2 = A3z1 +A4z2 + Γ2(u, t) + E22f(x, t)

y2 = C2z2(13)

where the transformed states are given by

z =Sx = [x1 x2 x3 x4 x5 x6]T

z1 = [x1 x2 x3]T , z2 = [x4 x5 x6]

T . (14)

A. HOSM Observer Based on Modified STA

Here, the HOSM observer based on modified STA [19], [28]is designed for the S2 subsystem. With Assumptions 3.1–3.5satisfied, the HOSM observer can be designed as follows:

˙z2 = A3z1 +A4z2 + Γ2(u, t) + E22ν(t) (15)

where ν(t) = [ν1 ν2 ν3]T represents the matrix of the ro-

bust sliding mode terms designed according to modified STA[19], [28] as

νi(t) = −Ki1φ1 (e2i(t)) −Ki2

t∫0

φ2 (e2i(t)) dt (16)

φ1 (e2i(t)) = e2i +Ki3|e2i|12 sign(e2i) (17)

φ2 (e2i(t)) = e2i +K2

i3

2sign(e2i) +

32Ki3|e2i|

12 sign(e2i)

(18)

Ki = diag(Ki1,Ki2,Ki3) represents the designed diagonalpositive gain matrices with i = 1, 2, 3. The estimation errordynamics e2 = z2 − z2 = z2 − C−1

2 y2 can be computed from(13) and (15) as

e2 = A3e1 +A4e2 + ν(t)− f(x, t) (19)

with, e2 = [e21 e22 e23]T . Similarly, the error dynamics

for the subsystem S1 can be defined as e1 = z1 − z1 =[e11 e12 e13]

T . Further details on the convergence of errordynamics for the subsystem S1 is discussed in the next section.The following theorem establishes the convergence of the errordynamics for the subsystem S2.

Theorem 1: For system (9) satisfying Assumptions 3.1–3.5,the observer system (15) with the robust term (16) will ensurethat error dynamics {e2 = 0} for the subsystem S2 will con-verge to zero in finite time.

Proof: For the simplicity of exposition, we only con-sider the first state error dynamics of the S2 subsystem and

4466 IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL. 64, NO. 10, OCTOBER 2015

prove the convergence. The error dynamics for the first stateof the S2 subsystem can be written from (19) as

e21 = −e11 + ν1(t)− f1(x, t)

= ν1(t) + δ2(e, t) (20)

where δ2(e, t) = δ21(e, t) + δ22(e, t) with δ21(e, t) = 0 andδ22(e, t) = −e11 − f1(x, t). The unknown input f1(x, t) isthe road profile ζ(t) that is bounded by a priori knownconstant. The perturbation δ22(e, t) can now be shown tobe bounded by constant ρ22. With the boundedness condi-tion satisfied, the convergence of the error dynamics cannow be proved. Similarly, the convergence of the error dy-namics e22 and e23 can be proven. For more details, seeAppendix B. �

B. NLO

The S1 subsystem (13) after transformation is free fromunknown inputs. To estimate the states, an NLO [15], [26], [27]can be designed as

˙z1 = A1z1 +A2C−12 y2 + Γ1(u, t)

+G1Φ(x, t) + L(y1 − C1z1) (21)

where L =

[l11 l12 l13l21 l22 l23

]Tis the feedback gain to be dis-

cussed later. By defining the error dynamics as e1 = z1 − z1 =[e11 e12 e13]

T , one can obtain

e1 = (A1 − LC1)e1 +G1 (Φ(x, t)− Φ(x, t)) . (22)

The following theorem establishes the stability of the S1 sub-system.

Theorem 2: For system (9) satisfying Assumptions 3.1–3.5,observer (21) ensures that for the subsystem S1, the stateestimation error (e1) is asymptotically stable provided the gainL satisfies

R(A1 − LC1) + (A1 − LC1)TR+ l2αRR+ I < 0 (23)

where lα is the Lipschitz constant for Φ(x, t), and R is apositive definite matrix.

Proof: With the choice of the Lyapunov functionas V (e1) = eT1 Re1 and differentiating with respect to time,one has

V (e1) = eT1[(A1 − LC1)

TR+R(A1 − LC1)]e1

+ 2eT1 G1R (Φ(x, t)− Φ(x, t)) .

By denoting (A1 − LC1)TR+R(A1 − LC1) = P and under

the Lipschitz condition for Φ(x, t) from Assumption 3.2, onecan deduce

V (e1) ≤ eT1 Pe1 + 2lα∥∥G1R[e1 e2]

T∥∥ ‖e1‖

It has been shown earlier in Theorem 1 that in sliding mode, theerror dynamics e2 = z2 − z2 = 0. Hence, we have

e = [e1 e2]T = [e1 0]T .

It can thus be written as

V (e1) ≤ eT1 Pe1 + 2lα‖G1Re1‖‖e1‖.

As the inequality 2lα‖G1Re1‖e1‖ ≤ (lα)2eT1 G1RRe1 + eT1 e1

is satisfied, one can obtain

V (e1) ≤ eT1 Pe1 + eT1(l2αG1RR+ I

)e1.

If the feedback gain L is designed such that (23) is satisfied,then V (e1) < 0. The error dynamics will thus be asymptoti-cally stable. �

The above condition (23) can be written in the form of analgebraic Riccati inequality as

R(A1−LC1)+(A1−LC1)TR+l2αRR+I+εI<0 (24)

for some ε > 0. If the above inequality is satisfied and ifthere exists a stable (A1 − LC1) matrix, then there exists asymmetric positive definite (SPD) solution R = RT for Riccatiequation (23). The gain L can also be computed by using thedistance to observability of the pair (A1, C11) as discussedin [26].

Remark 3: The stability of the error dynamics is establishedin two stages. During the initial stage when e2 → 0, e2 re-mains bounded. As e2 remains bounded during the transient,the boundedness of e1 can be easily established similar toTheorem 2. The presence of term e2 in the dynamics of e1due to coupling will ensure the ultimate boundedness for theerror dynamics e1. In the second stage, after e2 → 0 in finitetime, we have e2 = 0. The subsystem S1 error dynamics e1 isasymptotically stable in the sliding mode of e2 = 0, as shown inTheorem 2. The proposed design guarantees finite-time stabilityfor e2 and asymptotic stability for e1.

C. Reconstruction of Unknown Inputs

Once the sliding mode is established, the equivalent outputerror injection signal νeq can be obtained as

νeq = −A3e1 + f(x, t). (25)

Theorem 2 ensures the ultimate boundedness of error dynamicsto a small residual set containing the origin. Hence, as t →∞, we have e1 ≈ 0. The unknown inputs signals can be thenreconstructed from the sliding mode. The estimated road profilecan be obtained as

ζ(t) ≈ f1(x, t) = K12

t∫0

φ2(e21, t) dt. (26)

Similarly, the longitudinal friction force can be estimated as

Fx(t) ≈ f2(x, t) = K22

t∫0

φ2(e22, t) dt. (27)

RATH et al.: ESTIMATION OF ROAD PROFILE AND TIRE ROAD FRICTION FOR AUTOMOTIVE VEHICLE 4467

TABLE ICLASS-D SEDAN PARAMETERS

The estimation of the uncertain engine friction Tfric(x, t) andthe engine uncertainties ΔTengine affecting the engine dynam-ics can be obtained as

f3(x, t) = K32

t∫0

φ2(e23, t) dt. (28)

IV. RESULTS AND DISCUSSION

The simulation tests were performed on a high-fidelity ve-hicle model in CarSim. To perform the simulation, a class-Dsedan was considered with no steering.

A. Parameter Selection

The vehicle parameters that were considered for the simu-lation are shown in Table I. In this paper, the class-D sedanvehicle from CarSim environment was used. In CarSim, inde-pendent suspension systems for all wheels were considered. Forexcitation purposes, the four-wheel drive was considered, andall estimations were performed corresponding to the rear leftwheel of the vehicle. The initial conditions for the plant and theobserver were chosen as x(0) = [0 0 0 0.001 0.01 0]and x(0) = [0 0 0 0 0 0]. In the following, we firstdiscuss the gain selection for the HOSM observers. To obtainthe bound ρ22, the rate of change of the perturbation δ22(e, t)can be obtained from (20) and (33) as

δ2(e, t) = −e11 − f1(x, t). (29)

The unknown input f1(x, t) represents the road profile ζ(t) thatcorresponds to the known road roughness levels (3). In thispaper, we consider a class-D road profile whose derivative canbe shown to be upper bounded by 0.3195, which is verifieda posteriori through numerical simulations. The boundednessof e11 has been established in Remark 3, and its value isobtained through numerical simulations as 0.01. This procedureis usually employed in HOSM applications for evaluation ofbounds [29]. Based on these bounds, the gains K11, K12, andK13 are selected as 0.009, 0.018, and 8, respectively. It can beclearly seen that these gains satisfy condition (35). Similarly,the gains for the other states of the subsystem S2 can be selectedas K21 = 10, K22 = 60, K23 = 25, K31 = 3, K32 = 20, andK33 = 10.

For the gain design of the NLO, the Jacobian of the non-linear function Φ(x, t) and its bound are evaluated through

numerical simulations. The Lipschitz constant is obtained aslα = 45. In the linear matrix inequality (LMI) (24) as lα isknown, the problem of finding L and the SPD matrix is astandard LMI feasibility problem. Alternatively, we first select

L =

[0 10 1

−46.3293 77.2154 −24.0767

]Tsuch that A1 − LC11 is

stable. The SPD matrixR that satisfies the ARE (24) is obtained

as R =

⎡⎣ 0.00499 −0.000019 0.000193−0.000019 0.004894 0.000069

0.00193 0.000069 0.004434

⎤⎦ .

B. Simulation

To generate the road profile, the mathematical model pro-vided in (3) is employed. Depending on the class of roadroughness [4], different profiles can be generated using thismathematical model (3). CarSim employs these road profiledata to generate an input for the vehicle simulation. Similarly,to consider the effect of dynamic friction, the LuGre frictionmodel [3] has been employed. This dynamic friction modelreplicates the tire friction as the surface conditions varies dueto variation in the road adhesion coefficient [19]. By employingthis friction model, the generated dynamic friction is then pro-vided as input to CarSim which employs the same to generatethe longitudinal force Fx. Similarly, for the engine friction, themathematical model [1] that is dependent on engine crankshaftspeed of rotation has been considered as input to the CarSimmodel. In the presence of transmission losses and uncertaintiestogether with the engine friction considered, the third unknowninput f3(x, t) is suitably determined. The nonlinearities of thesystem also play a critical role in the analysis of vehicle dy-namics. The influence of the suspension nonlinearity is shownin Fig. 3(b).

It shows that the suspension nonlinearity warrants the pres-ence of an active control force to negate its effect. This controleffort can be provided in the active suspension system as afunction of the suspension deflection (sky hook control) orthe tire deflection (ground hook control). As shown earlier in(4), the vertical load is influenced by the road profile. Thetire friction obtained as a variation of vertical load to thelongitudinal force is thus affected by the road profile implicitly.It can also be seen that the model of road profile considered in(3) depends on the vehicle velocity, as discussed in [30].

C. Vehicle Accelerating Motion: No Speed Controller

In this driving scenario, the vehicle was allowed to accelerateunder open-loop throttle input in the absence of a predefinedspeed controller. The estimation of the states of the systemunder this driving condition is shown in Fig. 2. To furthersupplement this, the norm of the estimation error ‖e‖ of allthe states is shown in Fig. 3(a). It can be clearly seen that theestimation error is confined to a very small bound. This showsthat the estimation of the states is accurately obtained underthe influence of unknown inputs and system nonlinearitiesusing the proposed observer. The unknown inputs were thenreconstructed using (26) and (27).

4468 IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL. 64, NO. 10, OCTOBER 2015

Fig. 2. Estimation of states (a) zs, (b) zu, (c) zs, (d) zu, (e) vr , and (f) we x.

Fig. 3. (a) Norm of the estimation error. (b) Effect of suspension nonlinearity.

The reconstruction of the road profile and longitudinal fric-tion force is shown in Fig. 4(a) and (b), respectively. Thereconstruction of the unknown input f3(x, t) composed of theengine friction and the uncertainties is shown in Fig. 4(c). It canbe seen that the reconstruction is accurate even in the presenceof system nonlinearities. Now, the coefficient of friction can be

Fig. 4. Reconstruction of (a) ζ, (b) Fx, (c) f3(x, t), and (d) μ.

reconstructed from the estimated longitudinal force, as shownin Fig. 4(d). It can be seen that the estimation remains accurateeven when the vehicle is faced with varying road surfaceand a class-D (poor) road profile in the presence of systemnonlinearities.

D. Vehicle Motion: Proportional–Integral Speed Controller

In this driving scenario, a reference speed profile was pro-vided from the CarSim simulation environment, as shown inFig. 5(b). The main criterion for selection of this velocity profilewas to replicate the low slip conditions [see Fig. 5(a)]. Thefocus in this driving scenario is on the changes in vehiclevelocity profile and its subsequent effects on the estimationprocedure. To perform this analysis, the road surface is variedfrom dry to snow conditions over time. Furthermore, a class-Droad profile is selected under the effect of system nonlinearities.The norm of the estimation error shown in Fig. 5(c) shows theeffectiveness of the state estimation with the proposed method.It can be further concluded from Fig. 6(a) that the estimationroad profile is accurate. Furthermore, the estimation of thelongitudinal friction force f2(x, t) is accurate, as shown inFig. 6(b). The estimation of the friction coefficient is shownin Fig. 6(c). By increasing the throttle input continuously, aslip ratio greater than 30% can be achieved such that the tireroad force becomes saturated. In such a case, the longitudinalfriction force can be easily determined using the proposedobserver.

Although the estimation results are accurate, the unknowninput estimation is prone to parametric uncertainties andmodeling mismatch. The existence of model mismatch between

RATH et al.: ESTIMATION OF ROAD PROFILE AND TIRE ROAD FRICTION FOR AUTOMOTIVE VEHICLE 4469

Fig. 5. Depiction of (a) SlipL2, (b) v, and (c) ‖e‖.

Fig. 6. Reconstruction of (a) ζ , (b) Fx, and (c) μ.

the CarSim and the observer affects the estimation of unknowninputs, and it can be observed in the zoom portion of Fig. 6(c).

V. CONCLUSION

In this paper, a HOSM observer based on modified STAin combination with NLO has been developed for state andunknown input estimation for the integrated nonlinear verticaland longitudinal quarter vehicle model. Under the Lipschitzcondition for the nonlinear functions, the convergence of theestimation error is proved. The proposed scheme accuratelyestimates the states and the unknown inputs even in the pres-ence of uncertainties. The random road profile, the longitudinalfriction force, and the engine friction with uncertainties weretreated as unknown inputs and were accurately estimated fromthe sliding mode without the use of a low-pass filter. With

the estimation of longitudinal friction force, the dynamic tirefriction was consequently estimated. The simulations for dif-ferent driving scenarios are performed on high-fidelity vehiclesimulation software CarSim to demonstrate the effectiveness ofthe proposed scheme.

APPENDIX ALIPSCHITZ CONTINUITY FOR Φ(x, t)

The nonlinear function Φ(x, t) in system dynamics (9) con-sists of the nonlinearity Ψ(x, t). To analyze the Lipschitzcontinuity, the nonlinear function Ψ(x, t) can be divided intoΨ1 and Ψ2 as follows:{

Ψ = Ψ1 +Ψ2

Ψ1 = ksnlx321, Ψ2 = bsnl

√|x34|sign(x34).

In Ψ1 and Ψ2, the physical states (zs, zu, zs, and zu) ofthe automotive suspension system are bounded and satisfyAssumption 3.5. For the nonlinear function Ψ2, the continuitycan be established at every point except at the origin, where|x34| = 0. To evaluate the Lipschitz constant, the Jacobian ofthe nonlinear functions Ψ1 and Ψ2 can be computed as

JΨ1=

[3ksnlx2

21 −3ksnlx221

]

JΨ2=

⎡⎣ 2bsnlδ(x34)

√(|x34|) + bsnlsign(x34)

2√

|x34|

−2bsnlδ(x34)√(|x34|)− bsnlsign(x34)

2√

|x34|

⎤⎦T

where δ represents the Dirac function. It can be seen that theJacobian JΨ2

is not computable when |x34| = 0. Thus, theLipschitz continuity for the nonlinear function Φ(x, t) can beestablished locally except at the origin. We can show that

‖JΨ1‖∞ = max

{∣∣3ksnlx221

∣∣} ≤ a1

‖JΨ2‖∞ = max

{∣∣∣∣∣2bsnlδ(x34)√

(|x34|)

+bsnlsign(x34)

2√|x34|

∣∣∣∣∣}

≤ a2

where a1 and a2 are bounds computed through numericalsimulations. Thus, the norm of the Jacobian JΦ can be shownto be bounded as ‖JΦ‖∞ ≤ a1 + a2, and the Lipschitz constantfor Φ(x, t) is given as lα = a1 + a2.

APPENDIX BTRANSFORMATION

For the general automotive vehicle system considered in (9),without loss of generality, it is assumed that the output matrixhas the form C = [0 I]. It is assumed that the number ofoutputs is the same or more than the total number of unknowninputs and nonlinearities. Thus, we obtain p ≥ r + q and q =p− (q + r). Partition [G E] such that

[G E] =

[G1 E1

G2 E2

]

4470 IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL. 64, NO. 10, OCTOBER 2015

where G1 ∈ R(n−p)×r , and E1 ∈ R(n−p)×q . From above, wehave C[G E] = [G2 E2]. As ([G2 E2]) is full rank, thereexist a matrix T1 and a nonsingular matrix T2 ∈ Rp×p such that[G1 E1] = [T1G2 T1E2]

T2[G2 E2] =

⎡⎣G21 0r×q

0q×r 0q×q

0q×r E22

⎤⎦

where G21 ∈ Rr×r and E22 ∈ Rq×q are of full rank. Now,we can construct a nonsingular transformation matrix S thattransforms

S =

[In−p −T1

0 T2

]

S[G E] =

⎡⎣0(n−p)×r 0(n−p)×q

G21 0(p−q)×q

0(p−r)×r E22

⎤⎦ =

[G1 0(n−q)×q

0q×r E22

].

APPENDIX CMODIFIED SUPER-TWISTING ALGORITHM

Let us consider the following uncertain system:{es(t) = ν(t) + δ(es, t)

es(t0) = es0(30)

where δ(es, t) is the unknown input/ perturbation, and

ν(t) = −K1φ1 (es(t))−K2

t∫0

φ2(es(t)) dt (31)

where φ1(es(t)) and φ2(es(t)) are defined in (17) and (18), andK1, K2, and K3 are appropriately designed positive constants.It is assumed that the perturbation δ(es, t) can be written as

δ(es, t) = δ1(es, t) + δ2(es, t) (32)

with the bounded terms{‖δ1(es, t)‖ ≤ ρ1‖es‖∥∥∥δ2(es, t)∥∥∥ ≤ ρ2

(33)

for some positive constants ρ1 and ρ2.Remark 4: The sliding dynamics in (19) can be expressed in

the form of (30). Furthermore, condition (20) is similar to (32).Proposition 1: Under Assumption 2, the origin of system

(30) is a finite-time stable equilibrium point. Furthermore, theterm K2

∫ t

0 φ2(es(t)) dt provides a smooth estimation of theunknown perturbation δ2(es, t) in finite time.

Proof: The term K3 is tuned in order to withstand per-sistently exciting perturbation terms. Indeed, since ‖φ2(e)‖ ≥K2

3/2, one gets ‖δ2(es, t)‖ ≤ ‖φ2(es)‖. If K3 ≥√

2ρ2, let usselect a Hurwitz matrix A0, i.e.,

A0 =

[−K1 1−K2 0

]

whereK1 > 0 andK2 > 0 are appropriately designed. Systems(30) and (31) can be equivalently represented by the system oftwo first-order equations, i.e.,{

es1 = es2 −K1φ1(es1) + δ1

es2 = −K2φ2(es1) + δ2(34)

with es1 = es and es2 = δ2 −K2

∫ t

0 φ2(es1) dt. Consideringnew states Υ = [Υ1 Υ2]

T = [φ1(es1) es2 ]T , system (34)

can be rewritten as

Υ =

(1 +

K3

2|es1 |

−12

)(A0ξ +

[δ10

])+

[0δ2

].

Employing the Lyapunov function V (Υ) = ΥTQΥ, the stabil-

ity analysis can be performed with Q = QT =

[λ+ 4ε2 −2ε−2ε 1

],

λ > 0, and ε > 0. It is worth noting that the matrix Q is positivedefinite if λ and ε are any real number. To guarantee the positivedefiniteness of matrix Δ similar to [19] and [28], one choosesK2 = λ+ 4ε2 + 2εK1. Matrix Δ is positive definite if

K1 >4ε+ 2ελ+ 8ε3 + λρ1 + 4ε2ρ1

λ+

(1 + 2ερ1)2

4ελ. (35)

With K3 > 0, it can be shown that

V ≤ −λmin(Δ)

λ12max(Q)

K23

2V

12 − λmin(Δ)

λmax(Q)V.

The closed-loop system (34) is practically stabilized in finitetime. With ‖Υ‖ converging to zero in finite time, es1 and es2converge to zero. Therefore, the term K2

∫ t

0 φ2(es(t)) dt pro-vides a smooth estimation of the unknown perturbation δ2(es, t)in finite time. �

REFERENCES

[1] R. Rajamani, Vehicle Dynamics and Control. ser. Mechanical EngineeringSeries, New York, NY, USA: Springer-Verlag GMBH, 2006.

[2] G. A. Magallan, “Maximization of the traction forces in a 2WD elec-tric vehicle,” IEEE Trans. Veh. Technol., vol. 60, no. 2, pp. 369–380,Feb. 2011.

[3] C. Canudas de Wit, “Comments on “A new model for control ofsystems with friction,” IEEE Trans. Autom. Control, vol. 43, no. 8,pp. 1189–1190, Aug. 1998.

[4] ASTM E1274-03, “Standard test method for measuring pavement rough-ness using a profilograph,” ASTM Int., West Conshohocken, PA, USA,2003. [Online]. Available: www.astm.org

[5] T. Hsiao, “Robust estimation and control of tire traction forces,” IEEETrans. Veh. Technol., vol. 62, no. 3, pp. 1378–1383, Mar. 2013.

[6] Y. Chen and J. Wang, “Adaptive vehicle speed control with input in-jections for longitudinal motion independent road frictional conditionestimation,” IEEE Trans. Veh. Technol., vol. 60, no. 3, pp. 839–848,Mar. 2011.

[7] J. Wang, L. Alexander, and R. Rajamani, “Friction estimation on high-way vehicle using longitudinal measurements,” J. Dynamic Syst., Meas.,Control, vol. 126, no. 2, pp. 256–275, Aug. 2004.

[8] M. Choi, J. J. Oh, and S. B. Choi, “Linearized recursive least squaresmethods for real-time identification of tire–road friction coefficient,”IEEE Trans. Veh. Technol., vol. 62, no. 7, pp. 2906–2918, Sep. 2013.

[9] R. Rajamani, G. Phanomchoeng, D. Piyabongkarn, and J. Y. Lew,“Algorithms for real-time estimation of individual wheel, tire–road fric-tion coefficients,” IEEE/ASME Trans. Mechatronics, vol. 17, no. 6,pp. 1183–1195, Dec. 2012.

[10] H. M. Ngwangwa, P. S. Heyns, F. J. J. Labuschange, andG. K. Kululanga, “Reconstruction of road defects and road roughnessclassification using vehicle responses with artificial neural networkssimulation,” J. Terramech., vol. 47, no. 2, pp. 97–111, Apr. 2007.

RATH et al.: ESTIMATION OF ROAD PROFILE AND TIRE ROAD FRICTION FOR AUTOMOTIVE VEHICLE 4471

[11] C. Edwards, S. K. Spurgeon, and R. J. Patton, “Sliding mode observersfor fault detection and isolation,” Automatica, vol. 36, no. 4, pp. 541–553,Apr. 2000.

[12] K. C. Veluvolu and Y. C. Soh, “High gain observer with sliding mode forstate and unknown input estimation,” IEEE Trans. Ind. Electron., vol. 56,no. 9, pp. 3386–3393, Sep. 2009.

[13] K. C. Veluvolu and Y. C. Soh, “Multiple sliding mode observers andunknown input estimations for Lipschitz nonlinear systems,” Int. J. RobustNonlinear Control, vol. 21, no. 11, pp. 1322–1340, Jul. 2011.

[14] K. C. Veluvolu, M. Y. Kim, and D. Lee,“Nonlinear sliding mode highgain observers for fault estimation,” Int. J. Syst. Sci., vol. 42, no. 7,pp. 1065–1074, 2011.

[15] K. C. Veluvolu and Y. C. Soh, “Fault reconstruction and state estima-tion with sliding mode observers for Lipschitz non-linear systems,” IETControl Theory Appl., vol. 5, no. 11, pp. 1255–1263, Jul. 2011.

[16] H. Ros, J. Davila, and L. Fridman, “High-order sliding mode observersfor nonlinear autonomous switched systems with unknown inputs,”J. Franklin Inst., vol. 349, no. 10, pp. 2975–3002, Dec. 2012.

[17] R. K. Dixit and G. D. Buckner, “Sliding mode observation and con-trol for semiactive vehicle suspensions,” Veh. Syst. Dyn., vol. 43, no. 2,pp. 83–105, 2005.

[18] H. Imine and V. Dolcemascolo, “Sliding mode observers to heavy vehi-cle vertical forces estimation,” Int. J. Heavy Veh. Syst., vol. 15, no. 1,pp. 53–64, 2008.

[19] J. J. Rath, K. C. Veluvolu, M. Defoort, and Y. C. Soh, “Higher-ordersliding mode observer for estimation of tyre friction in ground vehicles,”IET Control Theory Appl., vol. 8, no. 6, pp 399-408, Apr. 2014.

[20] J. Cao, H. Liu, P. Li, and D. J. Brown, “Sate of the art in vehicle activesuspension adaptive control systems based on intelligent methodologies,”IEEE Trans. Intell. Transp. Syst., vol. 9, no. 3, pp. 392–404, Sep. 2008.

[21] C. E. Tannoury, S. Moussaoui, F. Plestan, N. Romani, and G. P. Gil,“Synthesis and application of nonlinear observers for the estimation oftire radius and rolling resistance of an automotive vehicle,” IEEE Trans.Control Syst. Technol., vol. 21, no. 6, pp. 2408–2416, Nov. 2013.

[22] B. L. Pence, H. K. Fathy, and J. L. Stein, “Sprung mass estimation of offroad vehicles via base excitation suspension dynamics and recursive leastsquares,” in Proc. Amer. Control Conf., 2009, pp. 5043–5048.

[23] D. Hugo, P. S. Heyns, R. J. Thompson, and A. T. Visser,“ Haul roadcondition monitoring using vehicle response measurements,” in Proc.12th ICSV , 11–14 Jul., Lisbon, Portugal, 2005.

[24] K. C. Veluvolu and W. T. Ang, “Physiological tremor estimation fromaccelerometers for real time applications,” Sensors, vol. 11, no. 3,pp. 3020–3036, 2011.

[25] U. Kiencke and L. Nielsen, Automotive Control Systems: For Engine,Driveline, and Vehicle. New York, NY, USA: Springer-Verlag, 2005.

[26] R. Rajamani, “Observers for Lipschitz nonlinear systems,” IEEE Trans.Autom. Control, vol. 43, no. 3, pp. 397–401, 1998.

[27] A. J. Koshkoueki and A. S. I. Zinober, “Sliding mode state observation fornonlinear systems,” Int. J. Control, vol. 77, no. 2, pp. 118–127, 2004.

[28] J. A. Moreno, “Strict Lyapunov functions for the super-twisting algo-rithm,” IEEE Trans. Autom. Control, vol. 57, no. 4, pp. 1035–1040,Apr. 2012.

[29] T. Gonzalez, J. A. Moreno, and L. Fridman, “Variable gain super twist-ing sliding mode control,” IEEE Trans. Autom. Control, vol. 57, no. 8,pp. 2100–2105, Aug. 2012.

[30] P. Gaspar and L. Nadai, “Parameter estimation of coupled road–vehiclesystems,” in Proc. 2nd Int. Workshop Soft Comput. Appl., pp. 199–204,2007.

J. J. Rath (S’13) received the B.Tech degree inelectrical engineering from KIIT University, India,in 2010 and the M.Tech degree in mechatronicsfrom AcSIR, India, in 2012. He is currently workingtoward the Ph.D. degree with Kyungpook NationalUniversity, Daegu, Korea.

From 2010 to 2012, he was with Central Me-chanical Engineering Research Institute, India, as aScientist Trainee while pursuing the M.Tech degree.His research interests include electric drives control,nonlinear systems modelling, and estimation of un-

known inputs/faults with specific focus on automotive applications.

K. C. Veluvolu (S’03–M’06–SM’13) received theB.Tech. degree in electrical and electronic engi-neering from Acharya Nagarjuna University, Guntur,India, in 2002 and the Ph.D. degree in electrical en-gineering from Nanyang Technological University,Singapore, in 2006.

Since 2009, he has been with the College ofIT Engineering, Kyungpook National University,Daegu, Korea, where he is currently an AssociateProfessor. During 2006–2009, he was a ResearchFellow with the Biorobotics Group, Robotics Re-

search Center, Nanyang Technological University. His current research in-terests include nonlinear estimation and filtering, sliding mode control,brain–computer interface, biomedical signal processing, and surgical robotics.

M. Defoort received the Ph.D. degree in 2007 fromEcole Centrale de Lille, Lille, France.

Since 2009, he has been active as an Asso-ciate Professor with the University of Valenciennes,France. He was a Research Fellow with the De-partment of System Design Engineering, Keio Uni-versity, Japan, from 2007 to 2008. His researchinterests include nonlinear control with applica-tions to power systems, multi-agent systems, andvehicles. He has published more than 60 papers(23 journal papers) in his area of research.