1. Department of Civil and Environmental Engineering
A Probabilistic Optimization Approach Using Bayesian Estimation for Identification of Modal Parameters
Li Li, Faouzi Ghrib
June 11, 2008
Department of Civil and Environmental Engineering
University of Windsor

2. Modal Identification Output/measurement Sensors Input Output Input
System
identification
Identified Parameters

3. Why we do it?
A structure’s health or level of damage can be monitored by identifying changes in modal parameters..
Checking construction quality.
The only reliable way to determine damping in a structure.
The output of identified parameters can be pursued further for model update or validation.

4. What is optimization approach?
Classification of system identification methods (Ljung 1999)
Equation solving approaches
Optimization approaches
Correlation approaches
Subspace approaches

5. Optimization approach
Define an objective function J to be minimized
θ is structural parameter to be identified
Here it includes:

6. Probabilistic optimization approach using Bayesian estimation
We have:
Sensor
Single-output: Y
Y=[y(0),y(1),…,y(N-1)]T
Input (S0)
Assumptions:
Input is Gaussian process, noise is Gaussian process, output is Gaussian process too.
excitation
noise

7. Parameter estimation using Bayesian maximum a posteriori (MAP) estimator
Minimize the Mean-Square-Error between the output process and the measured process:
The Bayesian MMSE estimator (theoretical solution):
The optimal solution: Conditional mean of the parameter given observations Y, or the mean of the a posteriori PDF

8. Parameter estimation using Bayesian maximum a posteriori (MAP) estimator
Using Bayes’ theorem, the posterior PDF of the parameters given the measurement series Y = {y(0),y(1),…,y(N-1)}T is given by
We seek a suboptimal estimate of the parameters by solving a nonlinear optimization problem:
The covariance:

9. Parameter estimation using Bayesian maximum a posteriori (MAP) estimator
Prior PDF P(θ) incorporates any prior knowledge and engineering judgment about the parameters to be identified. If no prior knowledge available, a constant value can be assigned indicating we have no preference over any choice of θ.
In this case, the remaining term is a likelihood function, and the solution is reduced to the maximum likelihood estimation (MLS).
The likelihood function P(Y|θ) can be computed using random vibration theory.

10. Given a set of parameter θ, how to compute the P(Y|θ)
The differential equation governing the motion of MDOF structure:
T is a force distribution matrix, and g(t) is a Gaussian stationary stochastic process with zero mean and spectral density Sg(ω)
Apply modal transform, we obtain a system of uncoupled equations in the modal coordinates. (k =1:N)
we use this model directly

11. Given a set of parameter θ, how to compute the P(Y|θ)
Random vibration theory:

12. Given a set of parameter θ, how to compute the P(Y|θ)
There is measurement noise and modeling errors, i.e. there is a difference between the measured response y(k) and the model response x(k). That we have a Bayesian linear model for MDOF system:
The noise process v is assumed to be white Gaussian noise (WGN) with zero mean and covariance Rv.

13. Given a set of parameter θ, how to compute the P(Y|θ)
The PDF of the measurement {y(k), k=0,…,N-1} for the given parameters θ is also Gaussian:
The number of measurement points is usually a large number,
therefore this formula is computationally prohibitive.

14. Given a set of parameter θ, how to compute the P(Y|θ)
Using Bayes’ rule, the likelihood function can be expanded in terms of the transition probability density as:
NP is chosen as a number much smaller than the total measurement point number N.

16. Given a set of parameter θ, how to compute the P(Y|θ)
Measurement points with a large enough time interval in between can be regarded independent; in this respect the measurement series can be regarded as a Markov process of order k.
k<m
Therefore,

17. Given a set of parameter θ, how to compute the P(Y|θ)
Covariance matrix of Z={y(k-NP),…,y(k)}

18. Given a set of parameter θ, how to compute the P(Y|θ)
According to Bayesian minimum-mean-square-error (MMSE) estimation:
The mean of y(k) given previous NP number of observations {y(k-NP),…,y(k-1)} is
Covariance matrix of the prediction error
YNp is the vector {y(k-NP),…,y(k-1)}

19. Furthermore:
can be computed using a steepest descent algorithm

20. Example: a six-storey building model
This structure is excited at base with Gaussian white noise. Acceleration responses are measured at top floor and the third floor.
10% measurement noise.
NP is chosen as 300, which covers a time lag 6 times the fundamental period. Sampling rate of data is 50Hz.

21. Example: a six-story building model
PSD plot at top floor
PSD plot at third floor

22. Example-2: a six-story building model

23. Example: a six-storey building model
Identification  results from measurement at the top floor  

24. Example: a six-storey building model
Identification  results from measurement at the third floor  

25. Example: a six-storey building model
Frequencies and damping ratios identified by different system identification methods
PEM: Prediction error method implemented in MATLAB system identification toolbox
N4SID: Numerical algorithms for Subspace State Space System Identification method implemented in the MATLAB system identification toolbox
RDM-ERA: Eigensystem realization with random decrement method

26. Conclusions
With a single measurement, this Bayesian identification is able to identify the fundamental frequencies and damping ratios well.
This approach is based on the random vibration analysis of structures and adaptive filtering theory of likelihood computation.
In this work, the Bayesian identification is demonstrated in a transient response problem.
This approach can be extended to other cases: earthquake excitations, dynamic tests using impulse excitations; all that required is the stochastic vibration solution of the problems relating the input and output probabilistic distributions.