September 4, 2003 1

Bayesian System Identification and Structural Reliability

Soheil Saadat, Research AssociateMohammad N. Noori, Professor & Head

Department of Mechanical & Aerospace Engineering

September 4, 2003 2

Overview

1. Intelligent Parameter Varying (IPV) Technique

2. Bayesian System Identification (BSI) Technique and Structural Reliability

3. Research Directions

September 4, 2003 3

1. Intelligent Parameter Varying (IPV) Technique

Parametric Non-parametric

“White box” “Black box”

Find “optimal” parameters of a system using “white box” models

Fully derived from the first principles

xy

ux,ufxx,ufx

21

xy

buaxx

Identification Techniques

Modeling Techniques

September 4, 2003 4

1. Intelligent Parameter Varying (IPV) Technique

Find “optimal” functional representation of a system using “black box” models

Solely based on the recorded data

xy

ux,ufxx,ufx

21

Parametric Non-parametric

“White box” “Black box”

Identification Techniques

Modeling Techniques

xy

x,ufx

September 4, 2003 5

Parametric Non-parametric

“White box” “Black box”

Identification Techniques

Modeling Techniques

1. Intelligent Parameter Varying (IPV) Technique

“Gray box”

IPV

Combines the advantages of parametric and non-parametric techniques

A mixture of “white box” and “black box” models

xy

ux,ufxx,ufx

21

xy

uwx,ugxwx,ugx

2211 ,,

September 4, 2003 6

1. Intelligent Parameter Varying (IPV) Technique

Advantages

2. Finds “optimal” functional representation of system constitutive non-linearities

1. Does not require a priori knowledge of system constitutive non-linearities

3. Can detect the presence, location, and time of damage

September 4, 2003 7

1. Intelligent Parameter Varying (IPV) Technique

M 3M 2M 1

xg x1 x2 x3

gxMux,fuCuM

gii xxu

u y

k2=0

RelativeD isplacem ent

Restoring Force

k1

R elativeD isplacem ent

k1

k2=0

R estoring Force

u y

September 4, 2003 8

1. Intelligent Parameter Varying (IPV) Technique

M 3M 2M 1

xg x1 x2 x3

gxMux,fuCuM

gii xxu

u y

k2=0

RelativeD isplacem ent

Restoring Force

k1

R elativeD isplacem ent

k1

k2=0

R estoring Force

u y

September 4, 2003 9

1. Intelligent Parameter Varying (IPV) Technique

Identified restoring forces

September 4, 2003 10

1. Intelligent Parameter Varying (IPV) Technique

Identified restoring forces

September 4, 2003 11

2. Bayesian System Identification (BSI) Technique

Is an statistical approach to system identification that can be applied to a wide range of dynamic systems

NNN

NN

yuyuyu

cppp

,,...,,,, 2211

D

θθDDθ

The unknown model parameters are not “estimated” but their posterior probability distributions are calculated

Thus, the estimated parameters are not point estimates but probability distributions, conditional on the given data

The Baye’s theorem provides the mathematical procedure, where:

September 4, 2003 12

2. Bayesian System Identification (BSI) Technique

cppp NN θθDDθ

The posterior pdf of model parameters,conditional on the given data

The likelihood function, reflects the contributionof the measured data DN in calculating the updatedposterior pdf

The prior pdf of model parameters

Normalizing constant

September 4, 2003 13

2. Bayesian System Identification (BSI)

Procedure

2. Define prior pdf of model parameters

1. Select a model class and structure

3. Define the likelihood function

4. Minimize the posterior pdf with respect to model parameters

September 4, 2003 14

2. Bayesian System Identification (BSI)

2D truss structure

September 4, 2003 15

3. Research Directions

3. Adaptive structural reliability analysis of aerospace structures based on real-time Bayesian system identification

1. Application of Bayesian system identification to aerospace structure

2. Health monitoring and damage detection of aerospace structures based on real-time Bayesian system identification