application of adaptive kalman filter for estimation of ...titan.fsb.hr/~bskugor/neizrazito i...
TRANSCRIPT
2008-01-0585
Application of Adaptive Kalman Filter for Estimation of Power Train Variables
Danijel Pavković and Joško Deur University of Zagreb
Ilya Kolmanovsky and Davor Hrovat Ford Research and Advanced Engineering
Copyright © 2008 SAE International
ABSTRACT
The paper presents the estimator design procedures for automotive power train systems based on the adaptive Kalman filter. The Kalman filter adaptation is based on a simple and robust algorithm that detects sudden changes of power train variables. The adaptive Kalman filter has been used to estimate the SI engine load torque and air mass flow, and also the tire traction force and road condition. The presented experimental results indicate that proposed estimators are characterized by favorable response speeds and good noise suppression abilities.
INTRODUCTION
The estimation of power train variables is important for power train/vehicle monitoring and diagnostics, and various control applications. For example, the load torque estimate can be effectively used for load torque compensation, and tire force/road condition estimation can improve traction control and vehicle dynamics control performance. The power train/vehicle estimators are based on appropriate power train/vehicle models and readily available measurements. In order to facilitate accurate and robust power train estimation, advanced estimators structures should be considered.
The application of adaptive Kalman filter methodology, proposed in [1], has been recently investigated by the authors for the purpose of load torque estimation [2, 3]. The basis of the adaptive Kalman filter is the first-order integrator-type model (i.e. engine inertia model) with load torque (input variable) treated as a stochastic disturbance variable at the process input, which is described by a second-order stochastic disturbance model. The proposed adaptive load torque estimator comprises a stationary (non-adaptive) Kalman filter tuned to obtain favorable noise rejection in the estimated load torque, and an adaptation mechanism to increase the Kalman filter feedback gains over a relatively short period of time if a sudden change of load torque has been detected. The adaptive Kalman filter tuning is based on a straightforward multi-step tuning procedure. The experimental results in [3] have indicated that the
utilization of the proposed adaptive Kalman filter approach yields a fast response of estimated load torque (i.e. good tracking ability) while maintaining favorable noise levels. The proposed adaptive Kalman filter estimator has also been effectively applied as a key part of an adaptive load torque compensator within idle speed control system (ISC system) [3].
It can be anticipated that, if some other power train variable can be described as an input disturbance variable, then the adaptive Kalman filter approach from [1-3] can be applied to those power train systems as well. References [4] and [5] have suggested that the first-order input disturbance model could be utilized as a basis for the design of input disturbance observers of exhaust gas recirculation (EGR) flow, and tire traction force, respectively.
As shown in [6], the adaptive Kalman filter methodology can also be attractive for the road condition estimation based on the estimation of tire static curve gradient for normal (low-slip) driving, which is directly related to the friction potential (i.e. road condition) [7]. The adaptive Kalman filter design in [6] is based on the linear regression model of tire static curve in low-slip region. However, the proposed road condition estimator requires the measurement (estimation) of applied wheel torque and wheel slip, which may not be available in all driving conditions. Another road condition estimation concept presented in [8] is based on the wheel speed resonant mode at approximately 40 Hz, which is caused by tire sidewall compliance. Since the damping ratio of the 40 Hz tire mode is inversely proportional to the low-slip tire static curve gradient, the information about the tire static curve gradient can be obtained from the properties of wheel speed signal only [8]. In order to estimate the 40 Hz mode damping ratio, a second-order autoregressive model (AR model) of tire vibrations is used as a basis for the estimator design [8]. The adaptive Kalman filter may be worth investigating as an alternative approach to the instrumental variable estimation method presented in [8].
This paper presents an overview of the theory underlying the adaptive Kalman filter approach, and design procedures of adaptive Kalman filters utilized for the
estimation of SI engine load torque and air mass flow, and tire traction force and road condition. For the purpose of adaptive Kalman filter design, appropriate stochastic process models are established. The proposed adaptive estimators are experimentally verified by utilizing an SI engine setup, a test vehicle equipped with different automotive sensors, and an experimental electrical vehicle equipped with an in-wheel high-bandwidth servomotor and high-precision wheel speed sensors.
PROCESS MODELS
This section presents process models of different automotive power train subsystems which are used as a basis for the estimator design.
MODEL OF SI ENGINE ROTATIONAL DYNAMICS
The design of adaptive Kalman-filter estimator of SI engine load torque in [2, 3] was based on the simple first-order inertia-type model of rotational dynamics =ω&J
bMM − (where ω is the engine speed, J is the engine inertia, while M and Mb are the engine torque and load torque respectively). The load torque Mb changes are not known in advance, so it is treated as a disturbance variable (modeled by a second order linear stochastic model). Since the manifold air mass flow (EGR flow) and tire friction may also be treated as disturbance variables, this model structure may be used as a basis for the design of respective Kalman filter-based estimators.
SI ENGINE INTAKE MANIFOLD
The estimation of EGR flow is usually based on the isothermal mean-value manifold model [4]:
)( oEGRi WWWV
RTdtdp
−+= , (1)
where p is the manifold air pressure, T is the air temperature, V is the manifold volume, R is the gas constant and Wi, WEGR, and Wo are intake air mass flow, port flow, and EGR flow respectively.
EGR flow can be estimated from the measurement of manifold air pressure p, intake air mass flow Wi, and reconstruction of port flow Wo (Eq. (1)). Since the particular engine setup does not include EGR (WEGR = 0), the effectiveness of the air mass flow estimator is illustrated for the analogous problem of estimation of intake air mass flow Wi based on port flow reconstruction. Note also that the intake air mass flow estimation can be beneficial in practical applications (for diesel engines) in order to avoid the influence of air mass flow sensor drift, and low sensor accuracy at low flow rates [3].
The port air mass flow Wo is reconstructed by utilizing the so-called speed density equation (see [9] and references therein):
ω)(120
ˆ1 o
do sps
RTVW += , (2)
where Vd is the engine displacement volume, ω is the engine speed, and s1 and s0 are the coefficients of the speed density equation for the particular engine.
DRIVEN WHEEL MODEL
The estimation of tire traction force can be based on a simple dynamic model of the driven wheel. It is assumed that the wheel driving torque Mm is known directly in the case of utilization of an in-wheel motor, or through the power train torque estimate in traditional engine-based approaches. The tire force can be reconstructed according to the following expression [5]:
w
wapp
w
wm
rJ
Fr
JMF
ωω &&−=
−= , (3)
where ω& w is the driven wheel angular acceleration, F and Fapp = Mm/rw are the tire friction force and applied tire force, respectively, rw is the driven wheel radius, and J is the equivalent wheel inertia.
LOW-SLIP TIRE-ROAD FRICTION EFFECTS
Low-slip tire static characteristics
The tire friction force is usually described by the tire static characteristic (tire static curve), i.e. the friction coefficient µ = F/Fz (Fz – wheel normal force) versus wheel slip s dependence. The longitudinal wheel slip s for the case of front wheel drive and no braking is defined as:
FF
RR
FF
F
rr
rv
sωω
ω−=−= 11 , (4)
where vF is the front wheel center speed (vehicle speed), ωF and ωR are front wheel and rear wheel speed, respectively, and rR and rF are the effective rear and front wheel radii.
If the rear wheel and front wheel radii are equal (rR = rF), the wheel slip equation (4) is reduced to the equation for practical slip sm (measured slip):
F
Rms
ωω
−= 1 . (5)
However, the wheel radii rR and rF may vary due to the difference in front and rear wheel pressure, normal load, or tire wear [6, 7]. Hence, the wheel slip calculated by Eq. (5) may be characterized by an error, which affects the accuracy of tire static curve reconstruction. This is especially important for the low-slip region (i.e. normal driving), where the wheel radii error is manifested in the form of slip bias [6, 7]:
00 ≠==µ
δ ms . (6)
For the purpose of characterization of low-slip tire-road friction, the tire static curve can be approximated by a linear µ-slip relation [6, 7]:
)( δµ −= mg sk , (7)
where kg is tire static curve gradient at low wheel slips.
The experimental analysis of low-slip friction effects in [6, 7] has shown that the gradient of tire static curve can be ten times lower for icy road than for dry or wet asphalt. Fig. 1 shows the results of experimental estimation of low-slip tire static curves µ(sm) from [10]. The tire force and wheel slip data are reconstructed based on Eqs. (3) and (5) respectively. The experimental static curve data for wet snow and wet ice are approximated in the least-squares sense by second-order polynomials (p0, p1 and p2 are the interpolation coefficients):
012
2)( pspsps mmm ++=µ , (8)
while in the case of dry concrete data, a simpler straight-line fit (p2 = 0) is found to be more appropriate. The interpolating curves are also shown in Fig. 1. The dissipation of tire static curve data points µ(sm) is not pronounced, which points out to good accuracy of tire static curve reconstruction in the critical low-slip region.
Based on Eq. (8) the low-slip tire static curve gradients kg(sm) are readily calculated as:
122)(
)( pspds
sdsk m
m
mmg +==
µ . (9)
Note that the zero-slip gradient kg(0) = dµ(0)/dsm is equal to the interpolation coefficient p1.
The static curve gradients for dry concrete, wet snow and wet ice are shown in Fig. 2. The results in Fig. 2 indicate that the zero-slip tire force gradient for dry concrete is 5-8 times larger for dry concrete compared to wet ice, and 3-5 times larger compared to wet or dry snow [10]. Hence, the information on low-slip tire static curve gradient can indeed be used for road condition estimation [6,7]. Note
also that the gradient values do not overlap in a relatively wide range of low wheel slips sm.
Tire sidewall torsional vibrations
Tire torsional vibrations are caused by the tire sidewall compliance effects, and can be excited by uneven road surface (so-called road noise). Fig. 3 shows the mass-spring model of torsional vibrations from [8]. It assumes that the wheel inertia can be separated into wheel rim inertia J1 and tire belt inertia J2, linked by the tire sidewall compliance (characterized by its stiffness coefficient c). The road noise effect can be modeled as stochastic, white noise-like torque component ∆Md [8]. Typical resonance frequency of tire torsional vibrations is around 40 Hz (the tire stiffness coefficient c varies with tire pressure [8]).
The tire torsional vibrations ∆ω1 referred to the wheel rim side can be approximately described by the second-order oscillator model [8]:
221
22
)(nn
no
d ssK
MsG
Ω+Ω+
Ω≈
∆∆
=ζζω
. (10)
where the natural frequency of sidewall torsional vibrations fn = Ωn/2π, the damping ratio ζ, and gain Ko are given in as follows (rw - wheel radius):
12
1Jcfn π
= , 1
221
21
Jc
rJJ
wαζ
+= ,
21
1JJ
Ko += .
The parameter α is the so-called longitudinal stiffness coefficient related to the tire static curve gradient kg = ∆µ /∆s as follows [8] (v – wheel longitudinal speed):
vkg /=α . (11)
Namely, the damping ratio ζ of the tire torsional vibrations is inversely proportional to the tire static curve gradient. As a consequence, the wheel speed amplitude spectra around 40 Hz should exhibit different shapes of resonance peaks depending on the road condition. This has been confirmed in [8, 10] through estimation of wheel speed amplitude spectra and tire vibration damping ratio ζ.
The estimation of tire vibration damping ratio ζ is based in [8] on the second-order auto-regressive model (AR model) of band-pass filtered wheel speed signal ωbp, which is given in the following form (k – sampling step):
( ) )()(1 22
11 kkqaqa cbp ηω =++ −− , (12)
where q-1 is the discrete-time unit-delay operator, and ηc represents colored road noise.
cJ1
rw
J2
Road
Tire belt
Wheelrim
∆Md
sθ
1ω
2ω
Fig. 3. Model of tire torsional vibrations
0 0.4 0.8 1.2 1.6-0.05
0
0.05
0.1
0.15
0.2
0.25
0.3
µ =
F/F z
sm [%]
-0.4
MeasuredInterpolatedMeasuredInterpolatedMeasuredInterpolated
Wet ice
Wet snow
Dryconcrete
+
Fig. 1. Comparative experimental tire static curves for low wheel slipsand straight-line driving on wet ice, wet snow and dry concrete.
k g =
dµ/d
s m
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.80
12
24
36
48
60
72
sm [%]
Wet iceWet snowDry concrete
Fig. 2. Tire static curve gradient vs. wheel slip plots for dry concrete, wet snow and wet ice.
The wheel speed signal needs to be band-pass filtered to extract the interesting 40 Hz tire mode. Since the band-pass filtering of measured wheel speed signal ωm effectively changes the properties of the stochastic perturbations at the process input, the colored road noise source ηc needs to be assumed instead of white road noise in Eq. (10) [8]. The parameters of AR model in Eq. (12) should be estimated by means of the instrumental variable identification method (IV method) [11].
Based on the analysis in [10], the following analytical relationships between the parameters of tire torsional vibration model fn and ζ, and AR model parameters a1 and a2 are valid:
.|)ln(|
1/4arctan41|)ln(|
41
,
|)ln(|1/4arctan
41
1
2
2
212
2
2
2
212
−+=
−+
=
aaa
aT
f
aaa
sn π
ζ
(13)
The results of estimation of the wheel speed amplitude spectra and 40 Hz mode damping ratios from [10] are shown in Fig. 4. The amplitude spectra in Fig. 4a clearly indicate the tire vibration mode resonant peaks located at approximately 40 Hz. The narrowest resonant peak (i.e. bandwidth) of the 40 Hz tire vibration mode is associated with the dry concrete surface. However, note that the wheel speed spectra comprise other resonant modes as well. These modes are also excited by road noise, and may notably interfere with the 40 Hz tire vibration mode.
The corresponding results of 40 Hz mode damping ratio estimation are shown in Fig. 4b. These results indicate that larger values of damping ratio ζ are obtained for road conditions with smaller values of low-slip tire static curve gradients (cf. Fig. 2). However, the estimated damping ratio ζ is not ideally inversely proportional to the low-slip tire static curve gradient kg. Namely, the estimated damping ratio ζ for dry concrete is only about 50% smaller compared to wet ice, while the low-slip static curve gradient kg for dry concrete is up to 8 times larger compared to wet ice (cf. Fig. 2). This could be caused by the relatively low road noise excitation, and the overlap distortion of 40 Hz mode due to its overlap with other tire
modes ([10, 14], see also Fig. 4a).
ADAPTIVE KALMAN FILTER
Design and tuning procedures of different automotive power train estimators based on the adaptive Kalman filter are presented in this section.
BASIC KALMAN FILTER
It is assumed that the process is described by the linear, discrete-time SISO model, given in the following state-space form (k – sampling step) [12]:
)1()1()1()1()1()1()( −−+−−+−−= kkkukkkk νΩGxFx ,
)()()()( kekkky += xH , (14)
where F, G, H, Ω are the system matrix, input matrix, output matrix and state perturbation matrix, respectively, x is the process state vector, u and y are system scalar input and output signals respectively, v is the state perturbation vector, and e is the measurement noise (scalar-valued).
The state perturbation vector v and the measurement noise e are assumed to be mutually independent zero-mean Gaussian noise components, characterized by the state covariance matrix Q(k-1), and measurement noise variance r(k), respectively.
The Kalman filter (estimator) for the process described by Eq. (14) is given by the following set of equations [12]:
,)1|()()()1|()|(
,)1|(ˆ)()1|(ˆ)|(ˆ
,)()()1|()(
)()1|()(
,)1()1()1(
)1()1|1()1()1|(,)1|(ˆ)1()()1|(ˆ
,)1()1()1|1(ˆ)1()1|(ˆ
−−−=
−+−=+−
−=
−−−+
−−−−=−−−−=−
−−+−−−=−
kkkkkkkk
kkkkkkkkrkkkk
kkkk
kkk
kkkkkkkkkkykk
kukkkkkk
T
T
T
T
PHKPP
KxxHPHHPK
ΩQΩFPFP
xH
GxFx
ε
ε
(15)
where:
)1|(ˆ −kkx , )|(ˆ kkx – a-priori and a-posteriori state vector updates, P(k|k-1), P(k|k) – a-priori and a-posteriori error covariance matrix updates,
K(k) – Kalman gain matrix, )1|(ˆ −kkε – a-priori prediction error.
By assuming time-invariant properties of the stochastic perturbations v and measurement noise e, the state covariance matrix Q and noise variance are time-invariant. Due to the fact that the individual components of v vector are mutually independent, the Q matrix also has a diagonal form.
The state covariance matrix Q and the measurement noise variance r are the design parameters of the Kalman
Fig. 4. Estimated wheel speed amplitude spectra for driving at 40 km/hon wet ice, wet snow and dry concrete (a), and corresponding estimatedvalues of 40 Hz mode damping ratio (b).
filter. Since the measurement noise variance r can usually be estimated fairly well from the measurements, the covariance matrix Q turns out to be the only tuning parameter of the Kalman filter. According to (15), if large state perturbations are expected (large Q), the estimator gains K will take on large absolute values. Thus, the state estimate )|(ˆ kkx relies then more on the measurements, and can track fast perturbations of system states. On the other hand, the state covariance matrix elements should be set to small values, in order to improve the Kalman filter noise rejection ability. Therefore, the choice of Q matrix elements is usually a trade-off between favorable suppression of system state perturbations and good estimator tracking ability.
PROCESS MODELS FOR ESTIMATOR DESIGN
Estimation of manifold intake air mass flow
The dynamic model of intake manifold of an SI engine without EGR is based on the isothermal manifold model given by Eq. (1). The intake air mass flow Wi is treated as a disturbance variable described by a second-order model, thus facilitating accurate tracking of ramp changes of disturbance variable [2]. The stochastic part of the process model comprises two mutually independent Gaussian perturbation sources, vo and vi, which are added to the reconstructed port flow signal Wo and the second time-derivative of the intake flow iW&& . Taking into account the above assumptions, and replacing the port flow Wo with its reconstruction oW (Eq. (2)), the following linear process model is obtained [14]:
.
,
),ˆ()ˆ(
ii
ii
ooiooi
dtWd
Wdt
dW
vWWavWWV
RTdtdp
ν=
=
+−=+−=
&
& , (16)
Since the air pressure and air mass flow are usually expressed in [bar] and [g/s], the first expression in model (16) is can be conveniently normalized by the following normalizing factor kpW = 105Pa/bar⋅103g/kg [14]. Note also that the actual manifold air temperature can vary in practical applications. Since the manifold air temperature T is not a state variable in the process model (16) it is treated as a time-varying parameter, which is updated on-line within the estimator based on the manifold air temperature measurement [14].
Based on Eq. (16), the discrete-time manifold model can be given in the state-space form (14), with the state vector x, and perturbation vector v given as follows:
[ ]Tii WWp &=x , [ ]Tio vv=v , (17)
while the model matrices F, G, Ω and Q read as follows (Ts – sample time):
=
100
102
12
s
ss
T
aTaT
F ,
−
=
0
0
saT
G ,
=
s
s
ss
T
T
aTaT
02
0
62
3
Ω ,
=
i
o
q
q
0
0Q (18)
The model of air pressure measurement pm is given as (e - measurement noise with variance r):
,]001[,)()()( =+= HHx kekkpm (19)
Estimation of tire traction force
The design of adaptive Kalman filter for tire force estimation is based on the simple inertia model in Eq. (3), rewritten in the following form:
)( FFJr
appw
w −=ω& (20)
The tire traction force F (disturbance variable) is modeled by a second-order linear stochastic disturbance model (see previous subsection). The stochastic perturbations vF and vm (with time-invariant properties) are added to the tire traction force second time-derivative F&& , and applied wheel torque Mm:
.
,
,)(1
F
mwmw
vdtFd
FdtdF
vFrMJdt
d
=
=
+−=
&
&
ω
(21)
The state vector x, state perturbation vector v and the matrices F, G, Ω and Q of the corresponding discrete-time state-space are given as follows:
[ ]Tw FF &ω=x , [ ]TFm vv=v ,
=
F
m
q
q
0
0Q , (22)
,
100
102
12
−−
= s
swsw
TJTr
JTr
F ,
0
0
=JTs
G ,
02
0
62
3
−
=
s
s
sws
T
TJTr
JT
Ω (23)
while the wheel speed measurement model reads:
]001[,)()()( =+= HHx kekkmω . (24)
Estimation of tire static curve parameters
For the purpose of estimator design, the linear tire static curve model (7) is transformed in the following form [6]:
δµ += mg
m ks 1 . (25)
The application of above model results in separation of static curve parameters kg and δ, which is motivated in [6] by the following assumptions: (i) these parameters vary independently, and (ii) the gradient kg changes are much more emphasized than changes of slip bias parameter δ. Furthermore, the noise variance in the reconstructed slip signal sm is typically much larger than the variance of friction coefficient reconstruction µm [6]. In that case, the noise and possible uncertainties of the µm reconstruction should have less effect on the estimation accuracy. The measurement noise e can then be regarded as an additive component to the slip reconstruction signal sm, and the measurement model can be given in the following linear regression form:
[ ] )()()()()(
)(/11)()( kekkke
k
kkkks
gmm +=+
= xH
δµ . (26)
The unknown low-slip tire static curve parameters kg and δ can be regarded as stochastic state variables of the so-called random-walk model [6] (see also [12, 13]):
)1()1()( −+−= kkk νxx , Tk vv ][ δ=ν , (27)
where vk and vδ are independent Gaussian perturbations in the gradient parameter kg and slip bias δ, respectively.
Due to F and Ω being unit matrices, the Kalman filter is simplified to the following form [7]:
),1|()()()1|()|()],1|(ˆ)()[()1|1(ˆ)|(ˆ
),1|1(ˆ)()1|(ˆ
,)()()1|()(
)()1|()(
),1()1|1()1|(
−−−=−−+−−=
−−=−
+−
−=
−+−−=−
kkkkkkkkkkskskkkkk
kkkkkskrkkkk
kkkk
kkkkk
m
T
T
PHKPPKxx
xHHPHHPK
QPP
(28)
with the covariance matrix Q defined as follows:
=
δq
qk
0
0Q . (29)
Estimation of tire vibrations damping ratio
The adaptive Kalman filter design is based on the AR model of torsional vibrations excited by the “colored” road noise signal ηc (Eq. (12)), which is the consequence of band-pass filtering of wheel speed signal ωm in order to extract the interesting tire mode at 40 Hz. The unknown AR model parameters a1 and a2 are modeled as stochastic state variables within a random-walk model, excited by the corresponding Gaussian stochastic perturbation sources v1 and v2. Based on the above assumptions, the following state-space model is obtained:
434214342143421
)1(
2
1
)1(
2
1
)(
2
1
)1(
)1(
)1(
)1(
)(
)(
−−
−
−+
−
−=
kkk ARARAR
k
k
ka
ka
ka
ka
νxx
ν
ν, (30)
).()(
)(])2()1([)(
)(
2
1
)(
kka
kakkk c
kk
bpbpbp
AR
AR
ηωωω +
−−−−=
4342144444 344444 21
xH
(31)
The adaptive Kalman filter is fed by the band-pass filtered wheel speed signal. Consequently, the band-pass filter needs to be incorporated into the estimator within the model of “colored” road noise ηc (i.e. as band-pass filtered “white” road noise vη [14]). Based on the above assumptions, the final process model is given as follows (I – unit matrix, 0 – zero matrix):
)1(0)1(
)1(
0
0
)(
)( 33
)1(
22
)(
−
+
−
−
=
×
−
×k
k
k
k
k
k
AR
bp
k
ARν
I
x
x
F
I
x
x
ΩxFx32144344214342143421
ηη
, (32)
[ ]43421
44 344 21
)()(
)(
)()()(
k
AR
k
bpARbpk
kkk
xH
x
xHH
=
η
ω , (33)
where v =[v1 v2 vη]T is the state perturbation vector, xη is the band-pass filter state vector, and Fbp and Hbp are the system and output matrices of the colored road noise model (i.e. wheel speed band-pass filter model), respectively. Note that the order of the “colored” road noise vector and the band-pass filter applied to the measured wheel speed ωm signal are equal.
Assuming that state perturbation components v1, v2 and vη are independent, the state perturbation matrix Q has the following diagonal form:
=
ηq
q
q
00
00
00
2
1
Q , (34)
BASIC KALMAN FILTER TUNING
The basic (non-adaptive) Kalman filter is tuned as a trade-off between fast response and low noise sensitivity. The tuning is primarily carried out by means of state perturbation matrix Q (measurement variance r can be estimated directly from the measurements).
Estimation of air mass flow and tire traction force
The Kalman filter-based estimators of air mass flow and tire traction force have similar structures as the load torque estimator. Thus, the tuning of non-adaptive Kalman filter is carried out according to the procedure derived in [3]:
• The response speed is adjusted by means of dominant Q matrix parameter, which is the variance of stochastic perturbations in the disturbance variable
second derivative (parameter qi for air mass flow estimator, and qF parameter for tire force estimator). Larger values of dominant parameter result in faster estimator response, and vice versa.
• The less dominant tuning parameter is the variance of the input variable (i.e. the qo parameter for air mass flow estimation, and qm parameter for tire force estimation). The response damping can be improved to some extent by increasing the less-dominant parameter value. However, it should be kept relatively small compared to the dominant parameter in order to preserve favorable estimator response speed [3].
Estimation of tire static curve parameters
Due to relatively simple static curve process model structure (matrices F and Ω being unit matrices), and the assumption of time-invariant properties of stochastic perturbations v and e, the Kalman filter gain values K(k) are determined by the normalized state covariance matrix Qnorm = Q/r [6]. In that case, the variance parameter r can be set to the unit value (r = 1) without any loss of generality, and the Kalman filter is designed based on the normalized matrix Qnorm only [6].
The parameters qk and qδ of the Qnorm matrix are naturally chosen as a compromise between the response speed and the noise in the estimated state variables. However, the variance of the slip bias parameter is usually set to a much smaller value compared to the variance of gradient parameter qk, because bias parameter δ is characterized by small variations (i.e. changes of wheel radii due to tire wear or pressure changes are relatively slow) [6].
Estimation of tire vibrations damping ratio
The choice of tuning parameters q1 and q2 (i.e. variances of AR model parameters a1 and a2) is a trade off between the response speed and the magnitude of variations of the estimated tire vibration model parameters ζ and Ωn in the estimator steady state. On the other hand, the parameter qη represents the variance of white road noise signal vη at the model input, which may significantly vary with the road surface type. Therefore, the qη parameter needs to be corrected on-line based on the variance σω
2 of the band-pass filtered wheel speed signal ωbp (whose magnitude is proportional to road noise [14]). The block diagram of the on-line correction of the parameter the qη is shown in Fig. 5. The initial value of wheel speed signal variance is determined from the speed signal ωbp, while the initial road noise variance qη0 is set empirically to the value which ensures favorable suppression of noise in the estimated AR model parameters [14].
ADAPTATION MECHANISM
In order to improve the estimator response speed with respect to sudden changes of system state variables, while retaining low noise levels in the estimator steady state, the Kalman filter is extended by a change detection adaptation mechanism. It is based on the calculation of
the so-called cumulative sum g(k) of the a-posteriori prediction error )|(ˆ)()()|(ˆ kkkkykk xH−=ε [6]:
.)|(ˆ)1()(
,0)0(kkkgkg
gε+−=
= (35)
Since the a-posteriori error )|(ˆ kkε is close to the white noise in the estimator steady state, the steady-state cumulative sum g(k) is close to zero. Therefore, if the cumulative sum absolute value |g(k)| exceeds the predefined threshold value, then a sudden change of state variables x has most likely occurred. In that case, the absolute values of Kalman filter gains K should be increased in order to facilitate fast adaptation (i.e. good tracking ability). This is done by increasing the value of the so-called adaptation matrix Q* = Ω Q ΩT in Eq. (15).
In the sampling step following the change detection, the adaptation matrix Q* is reset to its initial value (value prior to the change), and the cumulative sum g(k) is reset to zero and restarted. Even though the elements of Q* matrix are increased during one sampling step only, their change is propagated by the dynamics of error covariance matrix P (Eq. (15)). Thus, the coefficients of the Kalman estimator gain matrix K retain large absolute values over a somewhat longer period of time.
For the case of manifold intake air mass flow estimation and tire traction force estimation, the adaptation matrix Q* is a symmetrical 3x3 matrix, whose second and third diagonal elements Q*(2,2) and Q*(3,3) are typically much larger than any other Q* matrix elements [14]. This is favorable from the standpoint of adaptation, because these diagonal elements directly affect the Kalman filter gains responsible for the estimation of the disturbance variables (i.e. intake air mass flow, and tire traction force). Thus, instead of increasing the overall adaptation matrix Q* (which is computationally more demanding), the adaptation can be carried out by increasing the values of aforementioned matrix elements Q*(2,2) and Q*(3,3). The magnitudes of Q*(2,2) and Q*(3,3) changes are chosen such to obtain favorable Kalman filter response speed while maintaining low noise levels in the estimator steady state. The tuning can be carried out by means of simulation analysis, as illustrated in [3].
In the case of tire static curve parameter estimation and the estimation of tire vibration model parameters, the process model is of a random-walk type where the process states (i.e. estimated parameters) can vary independently. In those cases the adaptation is carried out by increasing the overall adaptation matrix Q* to such a value which will ensure fast estimator response while maintaining low levels of steady-state noise [6, 14].
2)(⋅Fs
Fs
TT
TT
ez
e/
/1-
-
-
-20ωσ
Estimation of speed signal variance
0ηqηqbpω
Low-pass filter2ωσ
Fig. 5. On-line modification of road noise variance parameter qη.
The change detection threshold parameter gt should be set to a value which guarantees robust change detection triggering. This can be done by choosing the threshold value 50% to 100% larger than the expected magnitude of steady-state perturbations in the signal of prediction error cumulative sum [3, 14].
EXPERIMENTAL VERIFICATION
This section presents the results of experimental verification of the proposed adaptive Kalman filter-based estimators of automotive power train variables. Experimental setups used for estimator verification are also described.
EXPERIMENTAL SETUPS
SI engine setup
The SI engine experimental setup, described in detail in [15], is shown in Fig. 6. It comprises a carburetor-based Briggs & Stratton 14 HP V2 SI engine without EGR, an electronic throttle, an AC servomotor as a high-bandwidth dynamometer (fBW ≈ 80 Hz), and a variety of engine sensors. The setup is controlled and monitored by a Pentium III-based PC computer equipped with data acquisition cards.
Experimental electrical vehicle with in-wheel servomotor
The electrical test vehicle, equipped with GoodYearTM Ultra Grip 6 W195/60R15 M+S tires, is shown in Fig. 7. The vehicle control system is built around a Pentium III-based industrial PC equipped with appropriate data acquisition cards. The vehicle comprises a high-bandwidth electrical servomotor built directly into the front wheel, which can develop a torque of up to 880 Nm in
only 2 ms and transfer it directly to the wheel. The driven wheel and one of the non-driven (rear) wheels are equipped with high-precision incremental encoders with sinusoidal outputs whose effective resolution is more than 2.106 pulses per revolution. The measurement data is transferred from the control computer to the operator by means of wireless communication. A detailed description of the test vehicle can be found in [16].
Ford Focus test vehicle
The principal schematic of Ford Focus test vehicle is shown in Fig. 8. The vehicle is equipped with Mastercraft Glacier Grip W195/60R15 M+S tires, four standard ABS sensors with the resolution of 44 pulses/rev, two precise (HiRes) sensors with the resolution of 512 pulses/rev, and a high-bandwidth half shaft torque sensor mounted at the right front wheel [10]. The vehicle sensor positions are shown schematically in Fig. 8.
SI ENGINE LOAD TORQUE ESTIMATION AND COMPENSATION
The adaptive Kalman filter-based estimator of input disturbance variable has been extensively investigated in [3] for the case of SI engine load torque estimation. The benefits of application of adaptive load torque estimator are illustrated in Fig. 9 for the PI controller-based idle speed control system with or without load torque compensation. The load torque changes in the stepwise manner with the magnitude of 4 Nm (approximately 12%
Fig. 6. Photograph of SI engine setup.
Fig. 7. Photograph of electrical experimental vehicle.
ABS wheelspeed sensors
(44 pulsesper rev.) Half-shaft
torquesensor
"Dummy"torquesensor
Externally mounted high-resolution wheel speed sensors
(512 pulses per rev.)
Front side
Rear side
RR RF
LR
FW drivedifferential
LF
Fig. 8. Locations of Ford Focus vehicle sensors – top view.
800900
1000110012001300
t [s]ω [r
pm]
7 8 9 10 11 12 13 14 15 16
ωR
012345
-202468
t [s]
t [s]
θ R [d
eg]
]N
m[
ˆ bM
7 8 9 10 11 12 13 14 15 16
7 8 9 10 11 12 13 14 15 16
Mb
PI ctrl. onlyPI ctrl. w/ adaptive KF
Fig. 9. Comparative responses of PI controller-based ISC systems without and with adaptive Kalman filter-based load torque compensator.
of the engine rated torque). Fig. 9 illustrates that the ISC system with adaptive Kalman filter is able to detect fast load torque changes (bottom plot in Fig. 9), and is thus able to provide fast control action (middle plot in Fig. 9). This results in improved disturbance (load torque) rejection performance compared to the case when only PI controller is used (top plot in Fig. 9).
AIR MASS FLOW ESTIMATION
The adaptive Kalman filter-based air mass flow estimator is validated for the SI engine operation within the electronic throttle control loop [17].
Fig. 10 shows the experimental SI engine responses for the cases of abrupt 20 deg throttle angle changes (throttle tip-in/tip-out experiments), and constant engine load torque (approximately 20% of rated engine torque). The experimental responses of engine speed ω, manifold pressure p, air mass flow measurement Wi (from the intake air mass flow sensor), and port flow reconstruction Wo are characterized by high levels of noise predominantly caused by the engine combustion pulsations.
The comparative responses of different types of Kalman filters (estimators) for throttle tip-in and throttle tip-out intervals are shown in Fig. 11. The air mass flow estimators are verified against the intake air mass flow measurement signal. If the stationary (non-adaptive)
Kalman filter is tuned for the fast response, it can track fast changes of intake air mass flow measurement signal Wi. However, the high gain values of fast stationary Kalman filter also result in high levels of noise in the estimated air mass flow. By tuning the non-adaptive Kalman filter for slower response, the noise issues are effectively avoided, but the estimator is not able to track fast changes of air mass flow. The application of adaptive Kalman filter based on the slow stationary Kalman filter results in favorable response time of approximately 50 ms. Note that the initial part of response (after the abrupt throttle angle change) is identical to the slow stationary Kalman filter response due to inherent delay of the change detection-based adaptation triggering. The adaptation is triggered only immediately after the sudden change of air mass flow is detected, which ensures good noise suppression ability of the adaptive Kalman filter, i.e. noise levels comparable to slow stationary Kalman filter.
TIRE TRACTION FORCE ESTIMATION
The verification of Kalman filter-based estimators of tire traction force has been carried out for straight-line driving in an ice arena by utilizing the experimental electrical vehicle [16]. The vehicle data which are used in the analysis correspond to a closed-loop traction control experiment carried out by the application of traditional PI controller of front wheel slip [14].
The results of tire traction force estimation within the traditional PI controller-based traction control system, and 10 % wheel slip target are shown in Fig. 12. Initial parts of responses in Fig. 12a correspond to the vehicle launch, which is characterized by rather abrupt changes of tire force and wheel slip, caused by the so-called dynamic tire friction potential (see e.g. [16]). Estimator responses in Fig. 12b further illustrate the advantages of adaptive
p [b
ar]
ω [1
03 rpm
]W
i, W
o [g
/s]
a
00.40.81.2
1.21.82.4
33.6
-137
11
0
0 0.2 0.4 0.6 0.8 1 1.2 1.4
0 0.2 0.4 0.6 0.8 1 1.2 1.4
t [s]
t [s]
t [s]
0.2 0.4 0.6 0.8 1 1.2 1.4
Wi
Wo^
Fig. 10. SI engine experimental responses for throttle angle changes of 20 deg.
0.3 0.4 0.5 0.6 0.7 0.8-1258
11a Throttle tip-in (∆θR = 20 deg)
t [s]
b Throttle tip-out (∆θR = -20 deg)
Measurement
0.8 0.9 1 1.1 1.2 1.3-1258
11
t [s]Measurement
Wi [
g/s]
Wi [
g/s]
b
Fast stationary K.F.Slow stationary K.F.Adaptive K.F.
^^
Fig. 11. Comparative responses of different air mass flow estimators for throttle tip-in (a) and throttle tip-out (b).
s m [%
]F ap
p [N
]
0 0.8 1.6 2.4 3.2 4 4.8 5.6-20
0
20
40
0 0.8 1.6 2.4 3.2 4 4.8 5.6300500700900
1100
t [s]
t [s]
a
Vehicle launch
300500700900
1100
b
0.1 0.2 0.3 0.4 0.5 0.6t [s]
Fast stationary K.F.Slow stationary K.F.Adaptive K.F.F
[N]
1.8 1.9 2 2.1 2.2 2.3t [s]500
600
700
800
F [N
]
Reconstruction:F = Fapp - Jω&
Fig. 12. Comparative experimental responses of traction control system (a), and details of estimated tire force responses for different estimators (b).
Kalman filter for tire traction force estimation. Namely, the adaptive Kalman filter is able to capture fast changes of tire friction force with the response speeds similar to fast stationary Kalman filter, while maintaining good suppression noise (comparable to slow stationary Kalman filter).
ESTIMATION OF LOW-SLIP TIRE CURVE PARAMETERS
Fig. 13 shows the wheel slip signal s and friction coefficient signal µ reconstructed from the Ford Focus vehicle ABS and half-shaft torque sensor measurements during transitions from dry concrete to wet ice, and dry concrete to wet snow, for vehicle speed of 40 km/h. The friction coefficient signals µ are characterized by low-frequency pulsations caused by the wheel imbalance. The wheel slip signals s are characterized by noise caused by the reconstruction of wheel speed based on the time-differentiation of wheel position signal [10]. Since the transition between two road surface types is uneven (bumpy), it causes the perturbations of the reconstructed signals µ and s to temporarily increase.
The comparative experimental responses of non-adaptive Kalman filter tuned for good suppression of noise and corresponding adaptive Kalman filter are shown in Fig. 14. The experimental responses for concrete-ice and concrete-snow transitions show that the application of adaptive Kalman filter yields much faster response of estimated gradient parameter kg compared to the non-adaptive Kalman filter. Furthermore, the adaptive Kalman filter utilization results in very accurate steady state values of gradient parameter kg (cf. results in Fig 2). The effective response delay of the adaptive Kalman filter is approximately 250 ms for concrete-ice transition (Fig. 14a), and 300 ms for concrete-snow transition (Fig. 14b). These delays are caused by the inherent delay of change detection-based adaptation mechanism, the non-homogenous transition between two road surfaces, and finite time required for the tire contact patch to pass from one surface type to another [14].
The slip bias parameter δ does not change significantly for the case of adaptive Kalman filter (somewhat larger changes of δ parameter are observed for the case non-adaptive Kalman filter). The adaptation mechanism is robust, i.e. the adaptation is triggered only after a sudden change of road condition is detected, which results in relatively smooth (noise-free) steady-state estimates.
ESTIMATION OF TIRE VIBRATION MODEL PARAMETERS
The proposed Kalman filter-based estimators of tire vibration model parameters are verified by utilizing the experimental wheel speed signals from Ford Focus vehicle HiRes sensors. The wheel speed data is band-pass filtered to extract the interesting 40 Hz tire vibration mode by utilizing the fourth-order Chebyshev band-pass filter (defined by matrices Fbp and Hbp in Eqs. (32) and (33)) with the pass-band located between 20 and 60 Hz. The experimental data correspond to driving at approximately 40 km/h.
Fig. 15 shows band-pass filtered wheel speed signals ωbp (estimator input signals) for the cases of transitions from dry concrete to wet ice, and from dry concrete to wet snow. The magnitudes of wheel speed signals ωbp indicate that driving on wet snow is characterized by significantly larger wheel speed variations compared to driving on dry concrete and wet ice. Furthermore, the transition details of wheel speed signals in Fig. 15 show that a notable increase of 40 Hz mode magnitude occurs, which is caused by uneven (“bumpy”) transition between two road surfaces.
6 6.5 7 7.5 8 8.5 90.050.1
0.150.2
t [s]
µ
Dry concrete Wet ice
a 6 6.5 7 7.5 8 8.5 9-2.5
02.5
57.5
t [s]
s [%
]
7 7.5 8 8.5 9 9.5 10-6-226
10
t [s]
7 7.5 8 8.5 9 9.5 100.050.1
0.150.2
t [s]
µs
[%]
b
Dry concrete Wet snow
Fig. 13. Friction coefficient and wheel slip signals for vehicle speed v ≈ 40 km/h and transition from dry concrete to ice (a), and from dry concrete to wet snow (b).
6 6.5 7 7.5 8 8.5 90
1530456075
6 6.5 7 7.5 8 8.5 9-0.50
0.51
1.52
t [s]
t [s]
k gδ
[x 1
0-3]
Dry concrete Wet ice
≈ 250 ms
Non-adaptive K.F.Adaptive K.F.
a
7 7.5 8 8.5 9 9.5 10
7 7.5 8 8.5 9 9.5 10
Dry concrete Wet snow
≈ 300 ms t [s]
t [s]
01530456075
-0.50
0.51
1.52
k gδ
[x 1
0-3]
Non-adaptive K.F.Adaptive K.F.
b
Fig. 14. Comparative experimental responses of Kalman filter-based estimators of tire static curve parameters for dry concrete to wet icetransition (a), dry concrete to wet snow transition (b).
The results of estimation of tire vibration model parameters are shown in Fig. 16. They show that the aforementioned increase of 40 Hz mode magnitude caused by bumpy transition between two road surfaces has unexpectedly positive effect regarding the response speed of proposed estimators (i.e. the estimation of road condition-related damping ratio parameter ζ). Namely, the responses in Fig. 16 show that due to relatively intense short-term excitation of tire vibration mode, the application of non-adaptive Kalman filter results in practically the same response speed of the estimated damping ratio ζ (approximately 50 ms response time is obtained), compared to the more complex adaptive Kalman filter for the case of concrete-ice transition. This effect is somewhat less emphasized for the transition from concrete to snow (i.e. somewhat faster response is obtained by utilizing the adaptive Kalman filter). The results in Fig. 16 also suggest that the adaptation mechanism is robust, i.e. low levels of noise are obtained
in the estimator steady state for all road condition. The steady-state values of estimated 40 Hz mode damping ratio correlate fairly well with the results obtained by off-line identification shown in Fig. 5. The estimated natural frequency of tire vibrations is somewhat changing with the road condition (which is in agreement with the results of amplitude spectra estimation in Fig. 5a).
It should be noted that the presented results are obtained for driving on regular roads without bumps or potholes (except for the “bumpy” transition itself), so that the 40 Hz tire mode is easily extracted (while other tire resonant modes are effectively attenuated) by means of band-pass filter. In practical applications, however, the reliability of estimation of 40 Hz mode damping ratio may depend on the properties of road noise excitation which excites all tire resonant modes (i.e. other tire modes may interfere with the 40 Hz mode estimation).
CONCLUSION
The paper has presented an overview of design procedures of adaptive Kalman filter-based estimators of different power-train variables, which include SI engine load torque and air mass flow, and tire traction force and road condition-related low-slip tire-road effects. Estimators are based on two distinctive linear stochastic model types. For the case of air mass flow and tire traction force estimation, a first-order linear model with one input variable treated as a disturbance variable (modeled by a second-order linear stochastic model) is used. On the other hand, the estimation of road condition-related effects is based on the so-called random walk model where the low-slip tire static curve parameters and the parameters of tire torsional vibration model are treated as stochastic variables.
The adaptive Kalman filter comprises a stationary (non-adaptive) Kalman filter tuned to obtain favorable response speed while maintaining low levels of noise in the estimated system states. It is extended by a change detection adaptation mechanism which increases the Kalman filter feedback gains over a relatively short period of time if a sudden change of process variables is detected, thus improving the estimator response speed. The change in process variables is detected based on the monitoring of the estimator prediction error cumulative sum.
The proposed adaptive Kalman filter concepts have been verified experimentally. The adaptive load torque estimator has been used as a basis for an adaptive load torque compensator, and verified within the PI controller-based idle speed control system (ISC system). The results of experimental verification have shown that the utilization of the adaptive load torque compensator notably improves the ISC system disturbance rejection performance. The verification of air mass flow estimator has shown that the noise sensitivity of the adaptive Kalman filter can be notably reduced when compared to the non-adaptive Kalman filter tuned for fast response, while preserving its ability to accurately track fast changes of air mass flow (the estimator response time is approximately 50 ms). The adaptive Kalman filter-based
3.5 4 4.5 5 5.5 6 6.5-2-1012
ωbp
[rad
/s]
b
-2-1012
1.5 2 2.5 3 3.5 4 4.5
ωbp
[rad
/s]
a
Dry concrete Wet ice
Dry concrete Wet snow
Bump
Bump
Fig. 15. Band-pass filtered wheel speed signals from Ford Focus HiRessensor for vehicle speed v ≈ 40 km/h: transition from dry concrete to wetice (a), and from dry concrete to wet snow (b).
4143454749
00.020.040.060.08
0.1
1.5 2 2.5 3 3.5 4 4.5
1.5 2 2.5 3 3.5 4 4.5
Dry concrete Wet ice
Non-adaptive K.F.Adaptive K.F.ζf n [
Hz]
a
3.5 4 4.5 5 5.5 6 6.50
4143454749
0.020.040.060.08
0.1
3.5 4 4.5 5 5.5 6 6.5
Non-adaptive K.F.Adaptive K.F.
Dry concrete Wet snow
b
ζf n [
Hz]
Fig. 16. Comparative experimental responses of Kalman filter-based estimators of tire vibration model parameters for dry concrete to wet icetransition (a), dry concrete to wet snow transition (b).
tire traction force estimator is also characterized by good tracking ability and low noise levels.
The adaptive Kalman filter approach has also been successfully applied for the estimation of road condition-related low-slip tire static curve gradient and the damping of the 40 Hz tire vibration mode. The application of proposed static curve gradient estimator facilitates accurate detection of different road surface types (i.e. dry concrete, wet snow and wet ice), while the effective time delay of road condition transition is less than 300 ms. For the case of 40 Hz tire mode damping estimation, relatively good discrimination between dry concrete and wet ice and snow has been obtained. The estimator response times are only about 50 ms for both the adaptive and non-adaptive Kalman filter, which is the consequence of ample excitation of 40 Hz tire vibration mode at the uneven transition between two road surfaces. The above results indicate that the tire vibration approach is generally more favorable in terms of response speed. Furthermore, the tire vibration approach only requires the measurement of wheel speed, which makes it attractive for applications in 4WD. However, it should be noted that the tire vibration-based approach may be generally more sensitive to other tire vibration sources (e.g. tire vertical tire mode, tire imbalance mode and suspension mode), which may interfere (overlap) with the tire torsional vibrations mode at approximately 40 Hz.
ACKNOWLEDGMENTS It is gratefully acknowledged that this work has been supported by the Ford Motor Company and partly by the Ministry of Science, Education and Sports of the Republic of Croatia.
REFERENCES
1. F. Gustafsson, “Adaptive Filtering and Change Detection”, John Wiley & Sons, 2000.
2. J. Deur, D. Pavković, & D. Hrovat, “Estimation of SI Engine Load Torque: Adaptive Kalman Filter vs. Luenberger Estimator”, IMECE 2004, November, 2004, Anaheim, CA, USA.
3. D. Pavković, J. Deur, V. Ivanović & D. Hrovat, “SI Engine Load Torque Estimator based on Adaptive Kalman Filter and Its Application to Idle Speed Control”, SAE paper No. 2005-01-0036.
4. A. Stotsky & I. Kolmanovsky, “Application of Input Estimation Techniques to Charge Estimation and Control in Automotive Engines”, Control Engineering Practice, Vol. 10, Issue 12, pp. 1371-1383, December 2002.
5. M. R. Uchanski, “Road Friction Estimation for Automobiles Using Digital Signal Processing Methods”, Ph. D. Thesis, University of California, Berkley, 2001.
6. F. Gustafsson, “Slip-based Tire-Road Friction Estimation”, Automatica, vol. 33, No. 6, pp 1087-1099, 1997.
7. T. Dieckmann, “Assessment of Road Grip by Way of Measured Wheel Variables”, Proceedings of XXIV FISITA Congress, pp. 75-81, London, June 1992.
8. T. Umeno, E. Ono, K. Asano, S. Ito, A. Tanaka. Y. Yasui, M. Sawada, “Estimation of Tire-Road Friction using Tire Vibration Model”, SAE paper No. 2002-01-1183.
9. J. Deur, D. Hrovat, J. Petrić, & Ž. Šitum, "A Control-Oriented Polytropic Model of SI Engine Intake Manifold", CD-ROM Proceedings of IMECE 2003, Vol. 2, Washington, D.C., USA, 2003.
10. D. Pavković, J. Deur, J. Asgari & D. Hrovat, “Experimental Analysis of Potentials for Tire Friction Estimation in Low-Slip Operating Mode”, SAE paper No. 2006-01-0556, 2006.
11. Ljung, L., “System Identification, Theory for the User”, Prentice-Hall, Inc., 1987.
12. M. S. Grewal, & A. P. Andrews, “Kalman Filtering – Theory and Practice”, John Wiley and Sons, New York, USA, 2001.
13. F. Gustafsson, “Adaptive Filtering and Change Detection”, John Wiley & Sons, Chichester, UK, 2000.
14. D. Pavković, Automotive Powertrain State Estimation with Control Applications”, Ph. D. thesis (in Croatian), University of Zagreb, 2007.
15. J. Petrić, J. Deur, D. Pavković, I. Mahalec & Z. Herold, “Experimental Setup for SI-Engine Modeling and Control Research”, Strojarstvo, Vol. 46, No 1-3, pp. 39-50, 2004.
16. J. Deur, M. Kostelac, Z. Herold, V. Ivanović, D. Pavković, J. Asgari, C. Miano, D. Hrovat, "An In-Wheel Motor-Based Tyre Test Vehicle", International Journal of Vehicle System Modelling and Testing, in press.
17. J. Deur, D. Pavković, N. Perić, M. Jansz & D. Hrovat, “An Electronic Throttle Control Strategy Including Compensation of Friction and Limp-Home Effects”, IEEE Transactions on Industry Applications. Vol. 40, No. 3, pp. 821-834, May/June 2004.
CONTACT Dr. Danijel Pavković, is a senior teaching/research assistant, while Dr. Joško Deur is an Associate Professor at the Faculty of Mechanical Engineering and Naval Architecture of the University of Zagreb, I. Lučića 5, HR-10000, Zagreb, Croatia; e-mails: [email protected], [email protected]; Dr. Ilya Kolmanovsky is a Technical Leader Powertrain Control R&A, while Dr. Davor Hrovat is a Henry Ford Technical Fellow at the Ford Research and Advanced Engineering; MD 2036, P.O. Box 2053, Dearborn, MI 48121, USA; e-mails: [email protected], [email protected]