Updating Nonlinear Updating Nonlinear Dynamical Models Using Dynamical Models Using Response Measurements OnlyResponse Measurements Only
Reference: Yuen and Beck (2003), “Updating properties of nonlinear dynamical systems with uncertain input”, J. Engng Mech. (at website)
ObjectiveObjective
Identification of nonlinear dynamical systems using only incomplete noisy response measurements– Quantify model and input uncertainties
– Perform Bayesian updating in frequency domain since FFT of stationary response is nearly Gaussian discrete white noise
System IdentificationSystem Identification
Linear or NonlinearDynamical SystemInput Output
??Model uncertain input by stationary stochastic process
Specification of Model ClassSpecification of Model Class
Equation of motion
Observation/Output Equation
Model parameters to be identified
)()|,( ts FθxxGxM =+)|( FF θS ω
{ }, 0 ,s F n=a θ θ Σ
( ) ( ) ( ), 0,1, ...n n n nη= + =y q0
0
: model response at ( ) observed DOF: Gaussian prediction error with zero mean and
covariance matrix (max. entropy PDF)
d
n
N Nη
≤q
Σ
model) DOF-( dN
Bayesian Approach to System IDBayesian Approach to System ID
Bayes Theorem:
- conditioning on class of models left implicit
- use to update PDF for parameters a based on measured response:
- likelihood difficult to evaluate for nonlinear systems subject to stochastic excitation
)|()()|( 1 aYaYa NN ppcp =
[ ])1(),...,0( −= NN yyY
Bayesian Spectral Density Approach Bayesian Spectral Density Approach
Key ideas:- FFT is nearly (complex) Gaussian white
noise for any stationary stochastic process- Use likelihood function for spectral density
estimates rather than response history
Reference Yuen and Beck (2003), “Updating properties of nonlinear dynamical systems with uncertain input”, J. Engng Mech. (at website)
Bayesian Spectral Density Approach Bayesian Spectral Density Approach
[ ])1(),...,0( −= NyyNY
( ) ( )21
,0
exp ( )2
N
y N k kn
t i n t y nN
ω ωπ
−
=
=
∆− ∆∑S
Illustrate with only one DOF observed:
Spectral Density Estimator from FFT:
where ( ) ( ) ( )i.e. observed model prediction error
y n q n nη= += +
Bayesian Spectral Density ApproachBayesian Spectral Density Approach
Asymptotic results for :
Good approximation for large N-Estimate mean by simulation or sometimes by
( ) mean with ddistributelly exponentia is - , kNyS ω
( ) ( ) lkSS lNykNy ≠ ift independen are & - ,, ωω
∞→N
( )[ ] ( )[ ] 0,, nkNqkNy SSESE += ωω
( )[ ] ∑ ∆∆∆
=−
=
1
0, )cos()]([
2N
mkqmkNq tmtmR
NtSE ωγ
πω
Bayesian Bayesian SpectralSpectral DensityDensity ApproachApproach
Single set of data using K frequencies
Bayes’ Theorem:
[ ] )](),...,([)1(),...,0( ,,, ωω ∆∆=⇒−= KSSNyy NyNyK
NyN SY
∏= ⎪⎭
⎪⎬⎫
⎪⎩
⎪⎨⎧−≈
=
K
k kNy
kNy
kNy
KNy
KNy
KNy
SES
SEp
ppcp
1 ,
,
,,
,2,
]|)([)(
exp]|)([
1)|(
)|()()|(
aaaS
aSaSa
ωω
ω
Bayesian Spectral Density ApproachBayesian Spectral Density Approach
Multiple non-overlapping sets of data)(,
,)1(,
,)()1( ,...,,..., MK
NyK
NyM
NN SSYY ⇒
Updated PDF for parameters
∏==
M
m
mKNy
MKNy ppcp
1
)(,,3
,, )|()()|( aSaSa
Bayesian Approach to System IDBayesian Approach to System ID
Parameter estimation and uncertainty
data spectral given the valuesprobablemost i.e.
)|( maximizingby ˆparameter Optimal- ,,
MKNyp Saa
,,- ( | ) locally approximated by a Gaussian
PDF with optimal parameters as mean and withcovariance matrix:
K My Np a S
1,, )}]|ˆ(lnHessian{[)(Cov −−= MKNyp Saa
Example 1: Example 1: DuffingDuffing OscillatorOscillator
Duffing oscillator
)(3 tfxkxxcxm =+++ µGaussian white noise 0)( :)( ff SStf =ω
Good approx. for auto-correlation function from
0)0( ,)0( ,0)()3()()( 22 ===+++ xxxxxxx RRRkRcRm στµσττ
• Expected value of the spectral density estimator
( )[ ] 0
1
0, )cos()]([
2 n
N
mkxmkNy StmtmR
NtSE +∑ ∆∆
∆=
−
=ωγ
πω
Example 1: Example 1: DuffingDuffing OscillatorOscillator
noise) (20% 0.0526m~ and sN01.0~N/m0.1~ 4.0N/m,~
0.1kg/sc ,1sec,1.0 sec,1000
)1(0
2)1(0
3
=
=
==
===∆=
n
fS
k
kgmtT
σ
µ
µk
Example 1: Example 1: DuffingDuffing OscillatorOscillator
noise) (20% 0.1092m~ and sN04.0~
but with before As
)1(0
2)1(0
=
=
n
fS
σ
µk
Example 1: Example 1: DuffingDuffing OscillatorOscillator
Example 1: Example 1: DuffingDuffing OscillatorOscillator
k
µ
Example 1: Example 1: DuffingDuffing OscillatorOscillator
µ
k
x Exact
Gaussian
x Exact
Gaussian
Example 1: Example 1: DuffingDuffing OscillatorOscillator
0.10250.1092
0.05140.0526
0.04540.0400
0.00980.0100
0.98681.0000
3.94204.0000
0.10210.1000
OptimalActualParameter
c
)2(0fS)1(
0nσ
)1(0fS
)2(0nσ
kµ
1.490.0410.0045
0.550.0420.0022
2.640.0510.0020
0.410.0460.0005
0.100.1300.1295
1.250.0120.0463
0.200.1080.0108
Error/S.D.COVS.D.
Example 2: 4Example 2: 4--DOF DOF ElastoElasto--plastic Systemplastic System
yx
ik
~ and ~ Scale yyyiii xxkk θθ ==
4m
3m
2m
1m
deflection-forceInterstory
0fS
Example 2: 4Example 2: 4--DOF DOF ElastoElasto--plastic Systemplastic System
Example 2: 4Example 2: 4--DOF DOF ElastoElasto--plastic Systemplastic System
hysteresisstory Top
Example 2: 4Example 2: 4--DOF DOF ElastoElasto--plastic Systemplastic System
0.00620.0063
0.00220.0022
0.00760.0060
0.95771.0000
0.99471.0000
0.99031.0000
0.99071.0000
1.01221.0000
OptimalActualParameter
0fS
1nσ
2nσ
yθ4θ3θ2θ1θ
0.410.0400.0002
0.030.0470.0001
2.030.1320.0008
0.790.0530.0533
0.690.0080.0078
0.950.0100.0103
1.040.0090.0089
1.260.0100.0097
Error/S.D.COVS.D.
Concluding RemarksConcluding Remarks
A Bayesian spectral density approach is available for system identification of nonlinear systems with uncertain input. It is based on the FFT of any stationary response being nearly Gaussian discrete white noise.
The Bayesian probabilistic framework explicitly treats both model uncertainty and excitation uncertainty.
Concluding RemarksConcluding Remarks
The spectral density matrix estimator for a stationary vector process follows a central complex Wishart distribution while it reduces to an exponential distribution for a stationary scalar process.
The updated (posterior) PDF is usually well approximated by a Gaussian distribution centered at the optimal parameters, so parameter uncertainty may be conveniently and efficiently characterized.
Appendix: Multi-DOF Case
Bayesian Spectral Density ApproachBayesian Spectral Density Approach
Spectral density matrix estimator
( ) ( )∑−
=
= ∆−−∆ 1
0,
)()(),(, )(exp)()(
2
N
pmk
ljk
ljNy tmpipymy
Nt ω
πωS
( ) ( )
( )[ ])}exp()]([
)exp()]({[4
][][
),(
1
0
),(),(,
0,,,
timtmR
timtmRNtSE
EE
krs
qm
N
mk
srqmk
srNq
nkNqkNy
∆∆+
∑ ∆−∆∆
=
+=−
=
ωγ
ωγπ
ω
ωω SSSExpected value of the estimator
Bayesian Spectral Density ApproachBayesian Spectral Density Approach
M sets of time histories
osetNNyNy
NNN NNsetset ≥⇒ ,,...,,..., )(
,)1(,
)()1( SSYY
∑=
=set
setN
m
mNy
set
NNy N 1
)(,,
1 SS
Bayesian Spectral Density ApproachBayesian Spectral Density Approach
Asymptotic results
1. ,
0
,
As , ( ( ) | ) asymptotically tends to a
central complex Wishart distribution of dimension with DOFs and mean [ ( ) | ] :
setNy N k
set y N k
N p
NN E
ω
ω
→ ∞ S a
S a
2. ( ) ( ) . ift independen are and ,, lkSS lNykNy ≠ωω
( )
( ))](]|)([[exp
|]|)([||)(|
|)(
,1
,
,
,0,
kN
NykNy
Nsetk
NNy
NNk
NNy
kN
Ny
set
set
osetset
set
Etr
Ecp
ωω
ωω
ω
SaS
aSS
aS
−
−
−×
≈
Bayesian Spectral Density ApproachBayesian Spectral Density Approach
Finite N
The above two properties hold approximately
Bayes’ Theorem
∏=
≈
=K
kk
NNy
KNNy
KNNy
KNNy
setset
setset
pp
ppcp
1,
,,
,,4
,,
)|)(()|(
)|()()|(
aSaS
aSaSa
ω