a user point of view
TRANSCRIPT
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Eperimental Eperimental modeling:modeling:learning models from datalearning models from data
a user point of viewa user point of view
Mario MilaneseDip. di Automatica e Informatica
Politecnico di Torino
The Logic of Modeling
Alta Scuola Politecnica
Milano, April 21, 2006
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OutlineOutline
Models as tools for making inferences from system dataprediction, simulation, control, filtering, fault detection
Model structuresphysical law based, input-output description, linear, nonlinear
Model estimationstatistical/parametric, set membership, structured
Model quality evaluation (vs. model validation)
Application examplesPrediction of atmospheric pollutionSimulation of dam crest dynamicsIdentification of vehicles with controlled suspensions
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Regression form of system representationRegression form of system representation
System So produces output signal y when driven by input signal u :
1 2
1
1 1 2 2
( )
[ ]y u u
t o t
t n t n t nt t t t
y f w
w y y u u u u
+
− − −
=
=
So
u1 y
Output y is related to input u by the regression function f o :
u2
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Regression form of system representationRegression form of system representation
Linear system
If ny=0 : MA (FIR) system
f o is linear in :tw
1 1 11 1
y u
y u
t n t nt t t t to n o ny a y a y a y b u b u b u− −+ − −= + + + + +
If nu=0 : AR system
ARMA system
If f o nonlinear : NARMA, NFIR, NAR systems
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Making inferences from dataMaking inferences from dataIt is desired to make an inference on system So :
prediction, identification, simulation, control, filtering, fault detection
The inference is described by the operator I( f o,wT)
one-step prediction
identification
I(f o,wT)=f o(wT)
I(f o,wT)=f o
The system So is unknown, but a finite number of noise corrupted measurements of yt,wt are available:
1 ( ) , 1, ,t o t ty f w d t T+ = + =dt accounts for errors in data ,t ty w
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Making inferences from dataMaking inferences from data
Problems :for given estimates
evaluate the inference error
The inference error cannot be exactly evaluatedsince f o and wT are not known
ˆ ˆ,o T Tf f w wˆ ˆ( , ) ( , )o T TI f w I f w−
find estimates ˆ ˆ,o T Tf f w w“minimizing” the inference error
Need of prior assumptions on f o and d t for deriving finite bounds on inference error
~~
~ ~
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The model is described by:
type of function ftype of noise d
Model structuresModel structures
which inputs u1, u2,…lag values ny, nu1, nu2,…
1 2
1
1 1 2 2
( )
[ ]y u u
t t t
t n t n t nt t t t
y f w d
w y y u u u u
+
− − −
= +
=
Model structure is defined by:
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Typical assumptions in literature:on system:
known lag values ny, nu1, nu2,…
Functional form of F(θ) required:derived from physical lawsσi : “basis” function (polynomial, sigmoid,..)
1
( ) ( , ) ( , )r
oi i i
i
f f w wθ θ ασ β=
∈ =
∑F =
on noise: iid stochastic noise
Parameters θ are estimated by optimizingLeast Squares (LS) or Maximum Likelihood functionals
Statistical/parametric approachStatistical/parametric approachModel structuresModel structures
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If possible, physical laws are used to obtain theparametric representation of ( ),f w θ
When the physical laws are not well known or too complex, input-output parameterizations are used
“Fixed” basis parametrizationPolinomial, trigonometric, etc.
“Tunable” basis parametrizationNeural networks, wawelets , etc.
often called black-box models
Statistical/parametric approachStatistical/parametric approachModel structuresModel structures
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( ) [ ]
( )
11
( , )
:
r
i i ri
i
f w w
w
θ ασ θ α α
σ=
′= =∑“Basis”
Problem: Can σi ’s be found such that
( )( , ) orf w f wθ →∞→ ?
Statistical/parametric approachStatistical/parametric approachModel structures: Model structures: “fixed” basis“fixed” basis
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For continuous f o, bounded and σi
polynomial of degree i (Weierstrass):
nW ⊂ℜ
( )lim sup ( , ) 0o
r w Wf w f w θ
→∞ ∈− =
Polynomial NARX models
Statistical/parametric approach Statistical/parametric approach Model structures: Model structures: “fixed” basis“fixed” basis
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( )1
1 11
( , ) ,
,
r
i ii
qr rq i
f w wθ ασ β
θ α α β β β
=
=
′ = ∈ℜ
∑
One of the most common “tunable” parameterizationis the one-hidden layer sigmoidal neural network
( ) ( ), Ti i iw w a bσ β σ= +
( )σ •
•
Statistical/parametric approach Statistical/parametric approach Model structures: Model structures: “tunable” basis“tunable” basis
sigmoid
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2 1 1
3 2 2
1
( , )( , )
( , )
o
o
T T o T
y f w dy f w d
y f w d
θ
θ
θ+
= +
= +
= +
( )oY F Dθ= +
Given T noise-corrupted measurements of y t,w t:
Measured output
Known function
( )1
( , ) ,i i
ro o o o
i
f f w wθ α σ β=
= = ∑
Error
Statistical/parametric approach Statistical/parametric approach Model estimationModel estimation
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( )oY F Dθ= + Gaussian pdf
Maximum Likelihood –Least Squares estimate
( )ˆ arg min Rθ
θ θ=
Problem: is in general non-convex
( ) ( ) ( )1 1R D D Y F Y FT T
θ θ θ′
′= = − −
( )R θ
Statistical/parametric approach Statistical/parametric approach Model estimationModel estimation
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“Fixed” basis:“Fixed” basis:
Estimation of θ is a linear problem: oY L Dθ= +
( ) [ ]11
( , )r
i i ri
f w wθ α σ θ α α=
′= =∑
If D is iid gaussian: ( ) 1ˆML L L L Yθ −′ ′=
( ) ( )
( ) ( )
1 1 12 3 1
1
rT
T r T
w wL Y y y y
w w
σ σ
σ σ
+
′ = =
Statistical/parametric approach Statistical/parametric approach Model estimationModel estimation
Statistical/parametric approach Statistical/parametric approach Model estimationModel estimation
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For fixed basis and D iid gaussian:
For tunable basis this results holds asymptotically (T→∞) with:
( ) 1ˆ 2 . . 0.95o MLi i i
iiL L w pϑ θ σ− ′− ≤
Statistical/parametric approach Statistical/parametric approach Estimation accuracyEstimation accuracy
standard deviation of noise component d i
o
FLϑ ϑϑ =
∂ = ∂
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Model structure choice:- “basis” type- Number r of “basis”- Number n of regressors
Problem: “curse of dimensionality”The number r of basis needed to obtain “accurate” approximation of f o may grow exponentiallywith the dimension n of regressor space
More relevant in the case of “fixed” basis
Statistical/parametric approach Statistical/parametric approach Model structures: Model structures: propertiesproperties
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Under suitable regularity conditions on the function to approximate, the number of parameters r required to obtain “accurate” models grows linearly with n
Estimation of θ requires to solve a non-convexminimization problem
Trapping in local minima
Statistical/parametric approach Statistical/parametric approach Model structures: Model structures: propertiesproperties
Using tunable basis:
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Basic to the statistical/parametric approach is the assumption of no modeling error
Statistical/parametric approach Statistical/parametric approach Modeling errorsModeling errors
: ( , )o o of f wϑ ϑ∃ =
stochastic variinde
able pendent of input u
( , )t t od y f w ϑ= −
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Statistical/parametric approach Statistical/parametric approach Modeling errorsModeling errors
Searches for the functional form of unknown f o are time consuming and lead to approximate model structures
d t is no more a stochastic variable independent of u
Statistical estimation in presence of modeling errors is a hard problem
Set Membership approach:no assumption on the functional form of f o
no statistical assumption on d t
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Set Membership approachSet Membership approach
Significant improvements obtained by:
use of “local” bound
bounded set ∈ Rn
{ }1
2
'( ) : ( ) ,of f C f w w Wγ γ∈ ∈ ≤ ∀ ∈F =
scaling of regressors w to adapt to data
, 1 , . . ,t t td t Tε γ δ =≤ +
SM assumptions:
on system:
on noise:
'
2( ) ( )f w wγ≤
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Inference algorithm Φ maps all informationinto estimated inference:
( ) : | ( ) | , 1, ,T t tt tFSS f y f w t Tγ ε γδ ∈ − ≤ =
+= F
All information (prior and data) are summarizedin the Feasible Systems Set:
ˆ ( ) ( , )T o TI FSS I f w=Φ
FSST is the set of all systems ∈F (γ) that could have generated the data
~
Set Membership approachSet Membership approach
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Set Membership approachSet Membership approachPrior assumptions validationPrior assumptions validation
The fact that the priors are validated by using the present data does not exclude that they may beinvalidated by future data (Popper, “Conjectures and Refutations: the Growth of Scientific Knowledge”, 1969)
Prior assumptions are invalidated by dataif FSST is empty
Prior assumptions are considered validatedif FSST ≠ Ø
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Set Membership approach Set Membership approach Prior assumptions validationPrior assumptions validation
Theorem:Conditions for assumptions to be validated are:
necessary:
sufficient:
( ) , 1, ..,t tf w h t T≥ =
( ) , 1, ..,t tf w h t T> =
Define: 21,.., 1( ) min ( || || )t t
t Tf w h w wγ
= −= + −
1 1,t t t t t t t th y h yε γδ ε γδ+ += + + = − −21,.., 1
( ) max ( || || )t t
t Tf w h w wγ
= −= + −
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Set Membership approach Set Membership approach Prior assumptions validationPrior assumptions validation
Used for thechoice of γ,εvalues
In space (γ,ε) the surface * ( ) infTFSS
γ ε γ≠∅
=separates falsified values from validated ones
validated
falsified
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Set Membership approach Set Membership approach Error and optimality conceptsError and optimality concepts
(Local) Inference error:
An algorithm Φ* is optimal if:
| |
ˆ( ) [ ( )] sup sup || ( ) ( , ) ||T T T T T
T T T
f FSS w wE I E FSS FSS I f w
γδε∈ +− ≤= Φ = Φ −
*[ ( )] inf [ ( )]T T TE FSS E FSS r FSSΦ
Φ = Φ = ∀
An algorithm Φα is α-optimal if:
[ ( )] inf [ ( )]T T TE FSS E FSS FSSα αΦ
Φ ≤ Φ ∀
r: (local) radius of information
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InferenceInference
Let || I(f,wT)||=|| f ||p=
Theorem:
1( ) [ ( ) ( )]2
cf w f w f w= +
i) The identification algorithm Φc(FSST)=f c
is optimal for any Lp norm, 1≤ p ≤∞
1[ ] ||2
||cpE f r f f= = −
Define
1/[ | ( ) | d ]p p
W
f w w∫
ii) The radius of information r is:
Identification: Identification: I(I(f,f,wwTT)=)=ff
Set Membership approachSet Membership approach
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InferenceInference
Let: || I(f,wT)||=| f (wT) |
Assume:
Prediction: Prediction: I(I(f,f,wwTT)=)=f f ((wwTT))
Let: { }2( ) :t t tB w w W w wδ δ= ∈ − ≤
t t td ε γδ≤ +
Set Membership approachSet Membership approach
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InferenceInference
Theorem:i) The prediction algorithm
ii) If
Prediction: Prediction: I(I(f,f,wwTT)=)=ff((wwTT))
( ) ( )c T c TFSS f wΦ =
( ) ,T T TB w C Cδ ⊂ ∩ then prediction 1ˆ ( )T c Ty f w+ =is optimal and the radius of information is:
is 2-optimal, with prediction error bounded by:
( ) 1 ( ) ( )2
c T T T TE FSS f w f w γδ Φ ≤ − +
1 ( ) ( )2
c T T TE r f w f w γδ Φ = = − +
Set Membership approachSet Membership approach
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Structured identificationStructured identification
In the case of large dimension of regressor space it is often very hard to obtain satisfactory modeling accuracy.
Structured (block-oriented) identification
The high-dimensional problem is reduced to the identification of lower dimensional subsystems and to the estimation of their interactions
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Structured identificationStructured identification
Not measured
Nonlinear Linear
Typical cases: Wiener, Hammerstein and Lur’e systems
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Iterative identification algorithm:- Initialisation: get an initial guess M2
(0) of M2
- Step k:1) Compute v(k) such that M2
(k-1)[v(k)]=y2) Identify M1
(k) using u and y as inputs, v(k) as output3) Identify M2
(k) using v(k) = M2(k)[u,y] as input, y as output
and return to step 1)
Key feature: The identification error is non-increasing for increasing iteration.
Structured identificationStructured identification
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Model quality evaluationModel quality evaluation
The usual approach is to look for model validity
Model invalidity only can be surely asserted, when the model does not explain the measured data
Even more, infinitely many models exactly explaining the data can be derived
| |t tMy y− > expected noise size
Infinitely many not-invalidated models can be derived
“overfitting” danger
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Model quality evaluationModel quality evaluation
choose #r of basis functions = #T of measured data
( ) ( )
( ) ( )
1 1 1
1
T
T T T
w wL
w w
σ σ
σ σ
=
' 1 'ˆ ( )LL LYϑ −=
invertible
Finding models exactly explaining the data
' 1 'ˆ ( )MY L L LL LY Yϑ −= = =
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Model quality evaluationModel quality evaluation
Example:
1 2 3 42 0.5 0.8 0.5u u u u= =− −= =
1 11 1 1 2( ) t t t
MM y u uϑ ϑ ϑ+ −⇒ = +1 1 2
2 2 1 2( ) ( )t t tMM y u uϑ ϑ ϑ+ −⇒ = +
1 1 33 3 1 2( ) ( )t t t
MM y u uϑ ϑ ϑ+ −⇒ = +
1 2 3 40 1 8 0.125y y y y−= = = =
input
output
candidate model structures
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Model quality evaluationModel quality evaluation
Estimation of M1, M2, M3
1( )M ϑ ⇒ 1
2
8 0.5 20.125 0.8 0.5
ϑϑ
− − =
2t = →Y
3t = →
L= θ
1 1
2
ˆ 2.03ˆ 3.49
L Yϑ
ϑ−
− = =
⇒
2 ( )M ϑ ⇒ 1
2
8 0.5 40.125 0.8 0.25
ϑϑ
− − =
2t = →3t = →
1 1
2
ˆ 0.81ˆ 2.10
L Yϑ
ϑ−
= = −
⇒
3( )M ϑ ⇒ 1
2
8 0.5 80.125 0.8 0.125
ϑϑ
− − =
2t = →3t = →
1 1
2
ˆ 0ˆ 1
L Yϑ
ϑ−
= =
⇒
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y
1 2 3 4 5 6 7 8-8
-6
-4
-2
0
2
4
Model quality evaluationModel quality evaluation
All models M1, M2, M3 explain exactly the given data y
How to choose among them ?
choose the one with the best “predictive ability”
measured by accuracyin simulating data not used for model estimation
estimation data
y,yM1, yM2, yM3
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Model quality evaluationModel quality evaluation
Several indexes have been proposed for estimating the predictive ability of models:
ˆ( )T nFPE RT n
ϑ +=
−2ˆln ( ) nAIC RT
ϑ= +
lnˆln ( ) n TBIC RT
ϑ= +They provide quite crude approximations, especially
for nonlinear systems
A simple but effective approach: splitting of data
• estimation data: estimate candidate models Mi, i=1,..,m• calibration data: choose the best one among Mi
:T numberof data:n numberof parametersϑ
( ) [ ] [ ]1R Y L Y LT
θ ϑ ϑ′= − −
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1 2 3 4 5 6 7 8-8
-6
-4
-2
0
2
4
Model quality evaluationModel quality evaluation
Best model among candidate ones Mi
estimation data
y,yM1, yM2, yM3
calibration data
minimum simulation error on the “calibration” data
Example: M3 is the best one among M1, M2, M3
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ApplicationsApplications
Prediction of atmospheric pollutionSimulation of dam crest dynamicsIdentification of vehicles with controlled suspensions
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Prediction of urban ozone peaks
NOx
SOx
VOCCH4 NH3N2O
COCO2
NO2
NO
O3
O
O2
O2
RH
RO2
smogfotochimico
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Prediction of urban ozone peaks
hCombustion processes and high solar radiation cause high tropospheric ozone concentrationshPrediction of ozone concentrations is important
for authorities in charge of pollution control and preventionhStudies in the literature show that physical models
are not able to reliably forecast the links between precursor emissions (Nox, VOC), methereological conditions and ozone concentrations
Sillman “The relation between ozone, Nox and hydrocarbons”, Atmos. Environ., 1999
Jenkin-Clemitshaw “Ozone and other photochemical polluttants: chemical processes governing their formation”, Atmos. Environ., 1999
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Prediction of urban ozone peaks
Value to bepredicted
[O3]
[µg/m3]
day t-1 day t+1
Used measurements
day t
typical data at Broletto (Bs)
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Prediction of urban ozone peaks
1
1 2 3 4
( )[ ]
t o t
t t t t t t
y f ww y u u u u
+ =
=
• Structure of used models:
- : max O3 concentration at day t- : mean NO2 concentration at 4-8 pm of day t
- : mean O3 concentration at 4-8 pm of day t
- : max temperature at day t
- : forecast of max temperature at day t+14tu
ty1tu
2tu
3tu
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Prediction of urban ozone peaks
• Prediction methods tested:
PERS:
ARCX: periodic ARX
NN: sigmoidal neural net
NF: neuro-fuzzy
NSM: nonlinear set membership
1t ty y+ =
• Hourly data measured at Brescia center:
1995-1998: estimation data set
1999: calibration data set
2000-2001: testing data set
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Prediction of urban ozone peaks
Indexes measuring the ability to predict concentrations exceeding a given threshold:
observed
predicted yes no total
yes a f – a f
no m - a N + a – m – f
N – f
total m N - m N
fraction of Correct Predictions: CP=(a/m)%
fraction of False Alarms: FA=(1-a/f)%
Success index: SI=[(a/m)+((N+a-m-f)/(N-m))-1]%
European Environmental Agency, Tech. Report 9, 1998
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Prediction of urban ozone peaks
PERS ARCX NN NF NSM CP 65.1 61.9 69.8 63.5 71
FA 33.9 25 27.9 25.9 27.4
SI 47.6 51.1 55.7 51.8 51.2
Calibration data set: m=63 exceeded thresholds
PERS ARCX NN NF NSM CP 41.5 35.9 53.8 66.7 71.8
FA 57.5 51.7 40 44.7 44
SI 34.4 31.3 49.6 60.2 63.5
Testing data set: m=39 exceeded thresholds
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Model of Schlegeis Arch Dam
• Model to simulate the crest displacement of the dam as function of:
water level
concrete temperature
air temperature
• Difficulties in deriving reliable physical models
• Models tested: ARX, NN, NSM
• Daily data available in period 1992-2000
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Model of Schlegeis Arch Dam
1
1 1 1 1 11 1 1 2 2 3 3
( )[ ]
t o t
t t t t t t t t t t
y f ww y y u u u u u u u
+
− + − + +
=
=
• Structure of used models:
- : crest displacement at day t
- : concrete temperature at day t
- : mean air temperature at day t
ty1tu
2tu
3tu
- : water level at day t
• Daily data:1992-1996: estimation data set
1997-1998: calibration data set
1999-2000: testing data set
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Model of Schlegeis Arch Dam
ARX model NN modelexperimental datamodel
• Simulation results on the testing data set:
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Model of Schlegeis Arch Dam
NN model NSM modelexperimental datamodel
• Simulation results on the testing data set:
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Identification of vehicles with controlled suspensions
Virtual design and tuning of Continuous Damping Control systems
Derive a model for simulation ofchassis and wheels accelerations as function of road profile and damper control
GOAL:
USE:
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Experimental settinghC-segment prototype vehicle with controlled dampers
and CDC-Skyhook (Continuous Damping Control system).
hMeasurements are performed on a four-postertest bench of FIAT-Elasis Research Center.
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Experimental setting
Road profiles:
hRandom: random road.hEnglish Track: road with irregularly spaced holes and bumps.hShort Back: impulse road.hMotorway: level road.hPavé track: road with small amplitude irregularities.hDrain well: negative impulse road.
Note: The road profiles are symmetric (left=right).
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Experimental setting
Data set: 93184 data, collected with a sampling frequency of 512 Hz, partitioned as follows:
hEstimation data set: 0-5 seconds of each acquisition. hCalibration data set: 5-7 seconds of each acquisition.hTesting set: 7-14 seconds of each acquisition.
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Structure of vehicles vertical dynamics
Since the road profiles are symmetric, a Half-car model has been considered:
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Structured Identification ofvehicles vertical dynamics
Structure decomposition: - CE: chassis + engine- SWT: suspension +
wheel + tire
Measured variables:- prf and prr : front and rear
road profiles. - isf and isr : control currents of
front and rear suspensions. - acf and acr : front and rear
chassis vertical accelerations.
Note: Fcf and Fcr are not measured.
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Results on testing set of NSM modelFront wheel acceleration: english track road
measurements, NSM model
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Results on testing set of NSM modelChassis front accelerations: random road
measurements, NSM model
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Results on testing set of NSM modelChassis rear accelerations: random road
measurements, NSM model.
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Comparison with physical model
Chassis front accelerations: random roadmeasurements, NSM model, physical model