Updating Nonlinear Dynamical Models Using Response...

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