Road Adaptive Semi-Active Suspension in anAutomotive Vehicle using an LPV Controller ∗ ⋆

Juan C. Tudon-Martınez ∗ Soheib Fergani ∗ Sebastien Varrier ∗

Olivier Sename ∗ Luc Dugard ∗ Ruben Morales-Menendez ∗∗

Ricardo Ramırez-Mendoza ∗∗

∗ GIPSA-lab, 11 rue des mathematiques, 38402 Grenoble, France (e-mail:{juan-carlos.tudon, soheib.fergani, sebastien.varrier, olivier.sename,

luc.dugard } @gipsa-lab.grenoble-inp.fr)∗∗ Tecnologico de Monterrey, Av. E. Garza Sada 2501, 64849, Monterrey

N.L., Mexico (e-mail: {rmm, ricardo.ramirez}@itesm.mx)

Abstract: A novel road adaptive Linear Parameter-Varying (LPV) based controller for the semi-activesuspension system of an automotive vehicle is proposed. The analysis is carried on the front-left Quarterof Vehicle (QoV) model generated via CarSimT M vehicle simulator. By using an on-line road roughnessestimation, considered as a scheduling parameter, the proposed LPV/H∞ controller is designed toimprove comfort and road holding. The road profile detector is based on the frequency and amplitudeestimation of the road irregularities by using a Fourier analysis. An H∞ robust observer is designedto estimate the variables related to the QoV vertical dynamics, which are used to compute the roadfrequency and roughness. Different ISO road classes are used to evaluate on-line the proposed roadidentification algorithm. A Receiver Operating Characteristic (ROC) curve is used to monitor theperformance of the roughness estimation; the results show that any road can be identified (at least 70%of success with a false alarm rate lower than 5%). The average error of road identification is 16.2%.Simulation results show that the proposed controller with road adaptation is capable to manage thetrade-off between comfort and road holding. The road adaptive controller increases the comfort (35.8%)when the vehicle is driven on a road of bad quality, by considering an uncontrolled damper (passivesuspension) as a benchmark. If the vehicle is driven on a smooth runway at high velocity, the proposedcontroller improves the road holding around 50%.

Keywords: Road Adaptive Semi-active Suspension, Road detection, LPV Control, H∞ Observer.

1. INTRODUCTION

Since the market demands constantly more safe and comfort-able vehicles, the automobile industry has been concerned ondeveloping intelligent systems in order to increase these fea-tures in modern cars. Moreover, the World Health Organisation(2009) estimates that 2.2% of the global mortality is relatedto vehicle collisions caused by multiple factors, e.g. loss ofstability at high velocities under road irregularities.

Some researches have demonstrated that the roughness esti-mation of the road surface can be used in an active or semi-active suspension control system to improve the comfort andhandling stability of a car, [Hong et al. (2002); Kim et al.(2002); Fialho and Balas (2002)]. However, in these adaptivesuspension systems, there is no integrated approach where theroad roughness estimation and control strategies are developedin a single control objective.

Existing methods for estimating the road roughness are basedeither on visual inspections [Stavens and Thrun (2006)] or onthe use of a fully instrumented vehicle that can take measure-ments from road irregularities, e.g. profilometers [Healy et al.(1977)]; the problem is that both methodologies are extremely

⋆ This work is partially supported by PCP project 03/2010 (France-Mexico).∗ This work is partially supported by the French national project INOVE ANR2010 BLAN 0308.

expensive to be implemented and require a specialized opera-tion, i.e. knowledge for sensors location, signal processing, etc.

Recently, Gonzalez et al. (2008) proposed a road roughness es-timator based on standard vehicle instrumentation (accelerom-eters); however, the road estimation depends on a specific fre-quency, i.e. the approach is designed for a constant vehiclevelocity and the result is not ensured when the velocity changes.In Ngwangwa et al. (2010) the road roughness is estimated atvariable velocity by using different standardized roads (ISO8608), but the ANN-NARX estimator could demand many com-putational resources for an online estimation. In Heyns et al.(2012) a road roughness monitoring system is proposed byusing a Bayesian estimator that performs at variable velocitybut, a priori information of the road is required.

In road adaptive suspension control, an extension to the Sky-Hook controller with adaptation to the road profile is proposedin Hong et al. (2002). The road estimator is based on the Fouriertransform over the road profile signal where the fundamentalfrequency is constant, i.e. an online estimation of the frequencyof motion, which depends on the vehicle velocity, is not used.Moreover, the road estimation is based on conventional filteringand the control strategy does not ensure the continuous andsaturated actuation of the semi-active damper. In Kim et al.(2002) an active suspension controller with road adaptationis proposed, the road-sensing system uses two laser height-

Preprints of the 7th IFAC Symposium on Advances in AutomotiveControl, National Olympics Memorial Youth Center, Tokyo, Japan,September 4-7, 2013

ThC3.3

Copyright © 2013 IFAC 231

meters which are susceptible to be contaminated; additionally,if their location is not good, the controller performance could bedeteriorated. An interesting Linear Parameter-Varying (LPV)controller with road adaptation is proposed in Fialho and Balas(2002), the roughness is used as a scheduling parameter inorder to design the controller; however, it is assumed to have anavailable signal of the road roughness (estimated or measured).

This paper proposes a novel road adaptive semi-active suspen-sion control system for an automotive vehicle based on the LPVapproach. The road roughness is estimated on-line by using aFourier analysis over the road profile signal which is derivedfrom a robust H∞ observer of a Quarter of Vehicle (QoV)model; the fundamental frequency is estimated also online, i.e.the roughness estimation depends on the vehicle velocity and isconsidered as a scheduling parameter in the LPV control designin addition to the semi-active damper constraints. DifferentISO 8608 road profiles are used to analyze the performance ofthe road detector by using a Receiver Operating Characteristic(ROC) curve. Simulations in CarSimT M are used to analyze thecontroller performances in comfort and road holding.

The outline of this paper is as follows: in the next section,the problem formulation is presented. The design of the roadroughness estimator is described in section 3. Section 4 showsthe formulation of the proposed road adaptive LPV controller.Simulation results are discussed in section 5. Finally, conclu-sions are presented in section 6.

2. PROBLEM STATEMENTA customized vehicle model obtained by experimental data isused as a test-bed, whose dynamics is generated via CarSimT M

vehicle simulator. The analysis is carried on the front-left cor-ner, Fig. 1. The system considers a sprung mass (ms = 470 Kg)and an unsprung mass (mus = 110 Kg). A spring with stiffnesscoefficient ks (86,378 N/m) and a Magneto-Rheological (MR)damper model represent the suspension between both masses.The stiffness coefficient kt (270,000 N/m) models the wheeltire. The vertical position of the mass ms (mus) is defined by zs(zus), while zr corresponds to the unknown road profile.

The spring is considered as linear because around 95% of itsoperating zone in an automotive application is linear; however,the semi-active damping force (FMR) depends on an electric cur-rent value and is highly nonlinear with respect to the suspensionmotion. There are different approaches to model semi-activedampers, as presented in Tudon-Martınez et al. (2012). In theparametric model of Guo et al. (2006), the hysteresis loop force-velocity is well modeled with an hyperbolic tangent function.The MR damping force is given by:

FMR = I fc tanh(a1zde f +a2zde f

)+b1zde f +b2zde f (1)

where, the electric current is bounded between 0 ≤ Imin ≤ I ≤Imax ≤ 2.5. Imin and Imax depend on the MR damper specifi-cations. Experimental data obtained from a commercial MR

m s

m us

s

us

z

z

rzkt

ks MR F

Fig. 1. QoV model for a semi-active suspension in a vehicle.

damper are used to model the nonlinearities of this actuator byusing (1). Figure 2 shows the performance of the MR dampermodel used in this analysis, whose parameters are: f c = 600.9,a1 = 37.8, a2 = 22.1, b1 = 2830.8 and b2 =−7897.2.

−0.1 −0.05 0 0.05 0.1−1000−800−600−400−200

0200

400600800

1000

Deflection velocity [m/s]

Fo

rce

[N]

I = 2.0 A

Experimental data(ISO 8608 road profile )

Force-velocity map I = 1.5 AI = 1.0 AI = 0.5 A

I = 0 A

Model (Modeling error = 1.7 %)

Fig. 2. Model of the MR damper for different I values.

The QoV system dynamics, given in a state-space representa-tion, is written as:

zszszuszus

︸ ︷︷ ︸

x

=

0 1 0 0

−ks

ms0

ks

ms0

0 0 0 1ks

mus0

−ks − kt

mus0

︸ ︷︷ ︸

A

zszszuszus

︸ ︷︷ ︸

x

+

0 0−1ms

0

0 01

mus

kt

mus

︸ ︷︷ ︸

B

[FMRzr

]︸ ︷︷ ︸

u

[y1y2

]︸︷︷︸

y

=

−ks

ms0

ks

ms0

ks

mus0

−ks − kt

mus0

︸ ︷︷ ︸

C

zszszuszus

+

−1ms

0

1mus

kt

mus

︸ ︷︷ ︸

D

[FMRzr

]+

[v1v2

]︸︷︷︸

v

(2)

where v is the noise in the accelerometers of the sprung (zs)and unsprung mass (zus). These measurements are related tothe comfort and road holding performances, that depend onthe semi-active damper properties and obviously on the roadirregularities.Problem 1. Design a road adaptive LPV controller for thesemi-active suspension system of an automotive vehicle by in-cluding the non-linearities of the actuator into the QoV model,i.e. (1) in (2), such that:

x = A (ρ) · x+B ·uy = C · x+D ·uu = K (ρ) · x

(3)

whit K (ρ) =∑Ni=1 ξi(ρ)Ki =∑N

i=1 ξi(ρ)[

Aci BciCci Dci

]by appro-

priately choosing the gains Ki, i= 1, . . . ,N such that the closed-loop system (3) be asymptotically stable in all parameter varia-tions. Since N = 8, three varying parameters are used in the LPVcontrol design: ρ1 and ρ2 represent the actuator constraints andρ3 monitors on-line the road quality (roughness estimation).

3. ROAD ROUGHNESS ESTIMATOR

From the practical point of view, variables related to the sus-pension motion (zde f = zs− zus and zde f = zs− zus) are not easyto measure; therefore, an H∞ robust observer is proposed toestimate them.

3.1 Design of the H∞ observerAn H∞ robust observer is designed to be unsensitive to noiseof measurements; also, it includes a weighting function totake into account the unknown road profiles. The performanceis monitored by reducing the estimation error in the statevariables; Wei represents the weighting function in each variableused to minimize the estimation error in the frequency range ofinterest for the suspension motion, while Wzr shapes the roadirregularities in the observer design.

Copyright © 2013 IFAC 232

Wzr =Kzrωzrss+ωzr

Wei =Ke1i

(s2 +2ζe1i

ωe1is+ω2

e1i

)s2 +2ζe2i

ωe2is+ω2

e2i

(4)

By considering the filtering specifications, the generalizedmodel P used for the synthesis of the H∞ observer is givenby (5). Figure 3 shows the structure of its design.

P :=

x = A · x+B ·wy = C2 · x+D2 ·wz = (x− x) ·

[We1We2We3We4

]T(5)

where w =

[FMR

Wzr · zr

], C2 =

[C

01×4

], D2 =

[D

1 0

]and,

˙x = Aobs · x+Bobs · [ zs zus FMR ]T (6)

where Aobs and Bobs are the matrices of the H∞ observer, whichis quadratically stable by solving an optimization problem withLMI techniques, Scherer et al. (1997). The observer reducesthe effect of the measurement noise and avoid drifting in theestimated variables by decreasing asymptomatically the errordynamics, given by: e = x− ˙x.

zr

z

x

,

F

us

..z

MR

^

+x

-

y~ s

..z , FMR

e

~

[ ]A BC D2 2

H Observer

zrW

eiW

Fig. 3. H∞ observer design in a QoV system.

3.2 Road profile identificationBased on the frequency and amplitude estimation of the roadirregularities, it is possible to define the type of road profile onwhich the vehicle is driven.Frequency estimation of the road profile. Since the unsprungmass motion is strongly influenced by the road profile, mea-surements of zus can be used to describe the overall conditionof the road, Gonzalez et al. (2008). By using the estimated statesof the H∞ observer, the road profile can be estimated from thestatic equation of the unsprung mass acceleration, such that:

zr = [muszus − ks(zs − zus)+ kt zus −FMR] · k−1t (7)

where FMR is measured or modeled from experimental data.

According to the ISO 8608, a road profile satisfies a sinusoidalwave whose frequency depends on the vehicle velocity and roadquality. Thus, by assuming an harmonic motion in the road, thisunknown input could be represented by:

zr(t)≈ R · sin(2π · fzr · t) (8)ˆzr(t)≈ 2π · fzr ·R · cos(2π · fzr · t) (9)

where there is no feasible prior information of fzr, whichdepends on: 1) the road surface, 2) tire dynamics and 3) vehiclevelocity. In Cartwright (2007), it is shown that a sum of twoor more sinusoidal waveforms can be obtained by the effectiveRMS value; thus, the road profile frequency fzr can be estimatedby the RMS values of the position and velocity of the road as,Lozoya-Santos et al. (2011):

fzr = zrRMS/(2π · zrRMS

)[Hz] (10)

Equation (11) computes the discrete RMS values of zrRMS andzrRMS ; n is the number of samples in a time window whichguarantees at least 2 cycles of the estimated frequency.

fzr =

√ (z2

r1+ z2

r2+ · · ·+ z2

rn

)(z2

r1+ z2

r2+ · · ·+ z2

rn

)·4π2

(11)

Amplitude estimation of the road profile. By using a Fourieranalysis of the estimated road signal (zr) over a running windowof one cycle of its fundamental frequency, it is possible toestimate the magnitude of the road signal by:

Azr =√

α21 +β 2

1 (12)

where the terms of the Fourier series, related to the fundamentalcomponent, are:

α1 =2T+

∫ t

t−Tzr(t)cos(2π fzr t)dt β1 =

2T+

∫ t

t−Tzr(t)sin(2π fzr t)dt (13)

The on-line estimated frequency fzr is the fundamental fre-quency of the road over the running window, i.e. T = 1/ fzr .Thus, the estimations of the frequency and amplitude of theroad disturbances are used to monitor its roughness. The rough-ness Power Spectral Density (PSD) function, Szr( fzr), is used tocharacterize a road in the frequency domain, Wong (2001):

Szr( fzr) = (Azr)2/(2∆ f ) (14)

where ∆ f is defined by the frequency range of interest. Byusing the limits of roughness for each type of road, definedby the ISO 8608, it is possible to identify the road profile onwhich the vehicle is driven. The thresholds are online computedby using the PSD of each road based on the limit of theroughness coefficient (cr) associated with the pavement quality.The lower/upper control limits for each road are given by:

Szri( fzr) = cr · v−1

x · [ fzr/( f0 · vx)]−nr [m2/Hz] (15)

where fzr is the on-line estimated frequency by using (11), f0is the critical spatial frequency made equal 1 rad/m, vx is thelongitudinal vehicle velocity and nr is a dimensionless constantrelated to the road waves, Robson (1979). Table 1 shows the crcoefficient used to define the thresholds of different ISO roadsin the road identification algorithm.

Table 1. Classification of road profiles (ISO 8608).Type of Road Class Lower cr Upper cr nr

m2/(rad/m) m2/(rad/m)

Smooth runway A 1.6×10−14 3.2×10−7 3.8Smooth highway B 3.2×10−7 1.2×10−6 3.5

Highway with gravel C 1.2×10−6 5.1×10−6 2.1Rough runway D 5.1×10−6 2.0×10−5 2.0

Pasture E 2.0×10−5 8.2×10−5 2.0Plowed field F 8.2×10−5 3.3×10−4 1.6

Since the PSD of the roughness is bounded, it could be usedas a scheduling parameter in order to design a road adaptivesemi-active suspension control system.

3.3 Results of the H∞ observer and road roughness estimatorTable 2 shows the parameters of the weighting functions thatminimize the estimation error of the designed H∞ observer.

Table 2. Parameters of design in the H∞ observer.Filter Wzr Filters Wei

Kzr 3 Ke1 0.5 ωe1 571 rad/sωzr 1 rad/s ζe1,ζe2 1 ωe2 127 rad/s

The effectiveness of the robust H∞ observer is shown in Fig.4.a-b, the estimated states (e.g. deflection velocity, zs − zus) fol-low the transient response of the CarSim model; the noise of theacceleration sensors is random with a distribution N ∼ [0,0.5],Fig. 4.a. Once the states are estimated, the road frequency isestimated by eq. (11) and using zr, Fig. 4.c. Note in Fig. 4.d thatsince the time window for computing the frequency of motionis of 1 s, the estimated frequency has a time delay, which isnegligible during normal driving over a rough road.

Copyright © 2013 IFAC 233

−25

−20

−15

−10

−5

0

5

10

15

20

Sp

run

g m

ass

acce

lera

tio

n [

m/s

]2

Noisy measurement

−2

−1.5

−1

−0.5

0

0.5

1

1.5

2CarSim H Observer

Def

lect

ion

vel

oci

ty [

m/s

]

0 5 10 15 20 25 30−0.08

−0.06

−0.04

−0.02

0

0.02

0.04

0.06

0.08

0.1

Ro

ad p

rofi

le [

m]

Time [s]

CarSim Estimation

0 5 10 15 20 25 30

0

1

2

3

4

5

6

7

Fre

qu

ency

of

the

road

pro

file

[H

z]

Real frequency

Estimated frequency

Time [s]

a

c d

b

Fig. 4. Performance of the H∞ observer and estimator of fzr .

To evaluate the proposed road roughness detector, a sequenceof 120 s with different ISO roads was implemented at vx =30 Km/h, dashed line in Fig. 5.b. Figure 5.a shows the on-line roughness estimation, by considering the log operator;additionally the thresholds, which are computed on-line byusing the limits of cr for each road, are used to obtain theroad classification result, Fig. 5.b. The road type C is morecomplicated to detect because it has a lower operating zonebetween the thresholds.

0 20 40 60 80 100 120A

B

C

D

E

F

Time [s]

Road

Id

enti

fica

tion

−9

−8

−7

−6

−5

−4

−3

−2

log

(

S *

f

)10 zr

z

r

PSD

of

the

rough

nes

s

Estimated roughness Thresholds for ISO roads

Road classification

a

b

^

Fig. 5. On-line roughness estimation (up) and final result in theroad identification algorithm (bottom).

A ROC curve is used to analyze the probability of right clas-sification of each road profile, Pc = T P/(T P + FN), and itsrespective false alarm probability, Pf a = FP/(FP + T N). Inthis case, a True Positive (TP) is when the road class i occursand is well classified and a False Negative (FN) is when thisroad is incorrectly identified as a road class j; inversely, a FalsePositive (FP) is when a road class j occurs but is classified as aroad class i and a True Negative (TN) is when the road class i iscorrectly not classified because really occurs the road class j.

Figure 6 shows the ROC curve obtained by the proposed iden-tification system of road profile. The detection result of the ISOroads A, B, D and E was successful, the probability of correctidentification is greater than 80% with minimal false alarm rate(lower than 5%). To quantify the global performance of the roadestimator, the average error of classification was used:

e = Σri=1Pi

(Σ j =iQi j

)(16)

where, r is the number of classes of road, Pi is the probabilityof each road and Qi j is the number of elements when the classof road i is incorrectly detected as a road j. In the implementedtest, e = 16.2%, i.e. the road was well classified during 100 of120 s.

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.2

0.4

0.6

0.8

1

Pro

bab

ilit

y o

f

cla

ssif

icat

ion

(Pc)

Probability of false alarms (Pfa)

A

BC

Limit of classificationD

E

F

Fig. 6. ROC curve for the classification of ISO road profiles.

4. SEMI-ACTIVE SUSPENSION CONTROL SYNTHESIS

By considering the vertical dynamics in a QoV model, eq. (2),the MR damping force represents the key element to isolatevibrations caused from road irregularities and to ensure goodroad holding. By using the parametric model of Guo et al.(2006), it is possible to represent the non-linear dynamics of thesemi-active damper as a scheduling parameter and to include itin the LPV control design, Do et al. (2012).4.1 LPV model formulation, Do et al. (2012)In this paper, the following QoV LPV model is considered:{

xl pv = Al pv (ρ1,ρ2)xl pv +B1uc +B2wyl pv =C1xl pv

(17)

wherexl pv =

(xsx f

)T

, Al pv (ρ1,ρ2) =

(As +ρ2Bs2Cs2 ρ1BsC f

0 A f

),

B1 =

(0

B f

), B2 =

(Bs10

), C1 =

(Cs0

)T

ρ1 = tanh(Cs2xs) tanh(C f x fF1

) F1C f x f

, ρ2 =tanh(Cs2xs)

Cs2xs

xs, As, Bs, Bs1, Bs2, Cs and Cs2 are state and matrices of a state-space representation of the QoV model by including the MRdamper model in (2) and considering zde f and zde f as output; x f ,A f , B f , C f are state and matrices of a representation of the low-pass filter Wf ilter = ω f /(s+ω f ) which is added to the systemto make the control input matrices parameter independent.Remark 1. This model is obtained, based on the non linearequations (2) and (1), after some mathematical transformationsas in Do et al. (2012). It is a control oriented LPV modelconsidering an input saturation uc provided with two schedulingparameters ρ1 and ρ2.

4.2 LPV/H∞ control synthesisThe main contribution of this paper is the use of a varyingparameter (ρ3) that schedules the suspension actuator workaccording to a new road estimation strategy. Indeed, dependingon the value of the road roughness, the suspension control isadapted to meet the required performances. This schedulingparameter is included in the LPV control synthesis, Fig 7. Theweighting functions are designed as:

Wf ilter = ω f /(s+ω f ), with a large bandwidth to decouple the

input and the varying parameters; Wzs = kzss2+2ζ11ω11s+ω11

2

s2+2ζ12ω12s+ω122 ,to account for passengers comfort at low frequencies; Wzus =

kzuss2+2ζ21ω21s+ω21

2

s2+2ζ22ω22s+ω222 , to account for road holding at high fre-

quencies; Wzr = 5 × ρ3 × 10−2, is the road profile gain alsoscheduled by the road roughness estimation to adapt the controlsynthesis.

Parameters of these weighting functions are obtained by geneticalgorithms, Do et al. (2012). Thus, the corresponding general-ized plant is a 3 linear parameter depending system as follows:

Copyright © 2013 IFAC 234

Σ (ρ1 ,ρ2 )

ρ1, ρ2

RoughnessEstimation

ρ3K( ρ1, ρ2, ρ3)

Wr (ρ3)Wzs

Wzus

zs

zus

zdefzdef

u c

u f

zrz1

z2z3

u

Road

W filter

,ρ3

Fig. 7. Suspension control implementation scheme.

Σgv(ρ1,ρ2,ρ3) :=

ξ = A(ρ1,ρ2,ρ3)ξ +B1u+B2wz = C1ξ +D11u+D12wy = C2ξ +D21u+D22w

(18)

where ξ = [χvert χw]T ; z = [z1 z2 z3]

T ; w = zr; y = [zde f zde f ]T ;

u= uH∞i j ; χvert are the states in the vertical dynamics of the QoV

model and χw are the vertical weighting functions states.

4.3 Scheduling parametersThe control of the vertical dynamics is ensured by the suspen-sion system in order to achieve frequency specification perfor-mances, Savaresi et al. (2010). In this study, both the consideredQoV model for synthesis and the controller are parameters de-pendent; ρ1 and ρ2 allow to ensure that the suspension controlrespects the MR damper constraints and input saturation.

The varying parameter ρ3 allows an on-line adaptation of thesemi-active suspension system to road profiles and is given by:

ρ3 = Kρ3 ·Szr( fzr) ∈ [0,1] (19)

where Kρ3 is used to bound ρ3, such that

I(ρ3) :=

I = Imax i f ρ3 ≥ ρ3Imin < I < Imax i f ρ3 < ρ3 < ρ3I = Imin i f ρ3 ≤ ρ3

(20)

The main idea of this road adaptive control strategy is toprovide a good on-line trade-off between road holding andpassengers comfort, which are conflicting objectives, such that:

• When ρ3 is high, the road roughness is high, and thesemi-active damper is tuned to be “harder” (uH∞ → Imax)in order to improve the car road holding and guaranteethe vehicle safety at high velocities or comfort at lowvelocities.

• Conversely, when ρ3 is low, the road roughness is low, andthe MR damper is set to be “softer” to enhance comfort atlow velocities or road holding at high velocities.

4.4 Problem solutionThe proposed LPV/H∞ robust controller is synthesized by us-ing LMIs solution for polytopic systems; all varying parametersare considered bounded: ρ1 ∈ [−1,1], ρ2 ∈ [0,1] and ρ3 ∈ [0,1].

The global LPV/H∞ controller is a convex combination of thelocal controllers obtained by solving the set of LMIs only ateach vertex of the polytope formed by the limit values of thevarying parameters. Then, the stability will be guaranteed forall trajectories of the varying parameters.Remark 2. In this work, since 3 varying parameters are used,the considered polytope has 8 vertices (8 local controllers, moredetails on LPV robust control design in Scherer et al. (1997)).

5. SIMULATION RESULTSFigure 8 shows the block diagram of the proposed road adaptivesemi-active suspension control system. The estimated statesof the H∞ observer are used in the proposed road estimationalgorithm and are also the controller inputs.

Table 3 shows the parameters used to guarantee the specifica-tions of performance of the designed LPV/H∞ controller. Somesimulation results in the time domain are presented to empha-size the interest of the proposed road adaptive LPV controller.

Table 3. LPV/H∞ controller parameters.kzs 10 ζ12 1 ω11 6.3 rad/s ω22 56.5 rad/skzus 1 ζ21 0.3 ω12 19 rad/s ω f 90 rad/sζ11 0.3 ζ22 1 ω21 56.5 rad/s

Scenario 1: ISO road F at vx = 30 Km/h. In this test,the suspension motion is considerably high, and because thevehicle goes at low velocity, it is primordial to enforce thepassengers comfort. The controller performances for comfort(sprung mass acceleration) and road holding (unsprung massdisplacement) are depicted in Fig. 9. The results show that: 1)the road adaptive controller reduces considerably the magnitudeof oscillations of the sprung mass acceleration in the QoV, Fig.9a, the improvement of comfort according to the RMS valueof zs is 35.8% (reduction of 6.5 m/s2); 2) the road holdingindex (zus) is also reduced with the road adaptive semi-activesuspension, the reduction of motion in the tire is 11.8% (8.1mm), Fig. 9b.

−60−40−20

0

20406080

100

Sp

rung

mas

s

acce

lera

tio

n [

m/s

]

2 LPV controllerPassive Suspension (PS)

−0.25

−0.15

−0.05

0.05

0.15

0.25

Un

spru

ng

mas

s

dis

pla

cem

ent

[m]

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

a

b

02468

1012141618

RM

S [

m/s

]

2

010

20

30

40

50

6070

RM

S [

mm

]

LP

V

PS

Fig. 9. Performance of the road adaptive LPV controller respectto an uncontrolled damper (passive), by using test 1.

Scenario 2: ISO road A at vx = 100 Km/h. When the vehicleis driven at high velocities over highways, minimal road dis-turbances could affect the vehicle stability; thus, it is critical toincrease the road holding index. Figure 10 shows the controllerperformance with respect to the passive suspension system. Theresults show that: 1) the unsprung mass displacement is reducedup to 6 mm by using the road adaptive controller, i.e. the vehiclestability is increased 50% according to the RMS value of zus,Fig. 10b; 2) the comfort performance is practically the same inboth suspension systems, Fig. 10a .

Thus, both tests confirm the efficiency of the proposed roadadaptive LPV controller to manage the trade-off between com-fort and road holding of an automotive suspension systemequipped with an MR damper.

6. CONCLUSIONSA novel road adaptive controller for the semi-active suspensionsystem of an automotive vehicle is proposed. The front-leftQuarter of Vehicle (QoV) model is used to evaluate the con-troller by using the CarSimT M vehicle simulator. The road iden-tification algorithm and road roughness estimation are based on

Copyright © 2013 IFAC 235

zr

QoV

def

.

z

noise

model

F s

..

z

us

..

z

x

noise

,^

Experimental

MR damper

Model

.

sz^us

^s

.

usz

z

z

^

Road profile

estimadorzr^ dt Frequency

estimation

fzr^

Fourier

Analysis

Azr Road profile

identification

Road

typeMR ^

CarSimTM

roughness

H Observer

defz^

ρ3

ρ1

ρ2

Fu=I

,ρ1

ρ2

ρ3,K( )

Fig. 8. Block diagram of the proposed road adaptive semi-active suspension control system.

Spru

ng

mas

s

acce

lera

tio

n [

m/s

]

2

Un

spru

ng

mas

s

dis

pla

cem

ent

[m]

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

RM

S [

m/s

]

2R

MS [

mm

]

LP

V

PS

−3

−2

−1

0

1

2

3

−0.01−0.008−0.006−0.004−0.002

00.0020.0040.0060.0080.01 LPV controllerPassive Suspension

LPV controller

Passive Suspension (PS) a

b

00.2

0.4

0.6

0.81

1.2

1.4

0

0.5

1

1.5

2

2.5

Fig. 10. Performance of the road adaptive LPV controller re-spect to an uncontrolled damper, by using test 2.

the Fourier analysis over the estimated road profile signal byusing an on-line estimation of the frequency. An H∞ robust ob-server is used to estimate the road profile and its frequency. Thecontrol law is scheduled/adapted by using a Linear Parameter-Varying (LPV) based controller. Three scheduling parametersare used in the LPV control design, two for including thenonlinearities and saturation of the actuator and one more forthe road adaptation. Experimental data were used to modelthe actuator dynamics: a Magneto-Rheological (MR) damper.A comparison between the road adaptive controller for theMR damper and an uncontrolled damper (passive suspension)shows the effectiveness of the proposed semi-active suspensioncontrol system to manage the trade-off between comfort androad holding.

A random sequence of 6 different ISO road profiles was usedto evaluate the road identification algorithm, the total errorof identification was 16.2%. Simulation results show that theroad adaptive controller improves the comfort (35.8%) whenthe vehicle is driven on a road of bad quality at 30 Km/h;while, when the vehicle is driven on a smooth runway at 100Km/h, the road holding is improved, i.e. the unsprung massdisplacement is reduced up to 6 mm (the vehicle stability isincreased).

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