Damage Detection Scheme Based on Bayesian Theory
Man Woo Park
Deterministic SI based on a least square method is investigated from the viewpoint of Bayesian theory. SI based on a least square method is interpreted in terms of the maximum likelihood of the conditional probability of system parameters under given measurements. The conditional probability distribution is calculated by combining three probability distributions associated with measurement error, modeling error and system parameter. The probability distribution of system parameter in Bayesian theory is equivalent to the regularization function which is usually adopted to alleviate ill-posedness of deterministic inverse problems. The regularizing effects of L1 and L2-regularization functions are realized by applying normal distribution and symmetry exponential distribution to the probability distribution of system parameters, respectively. The ratio of standard deviation of measurement error to standard deviation of system parameter is represented by the regularization factor of regularization technique.
Stiffness properties are positive in any case and assumed to have asymmetric probability distribution. Log-normal distribution representing the asymmetric probability is proposed to impose the distribution characteristics of stiffness properties upon SI. The method of filtering modeling error that deterministic SI scheme couldn¡¯t consider appropriately is proposed introducing probability concepts. Modeling errors are represented by the difference between actual displacements and calculated displacements and assumed to be normal. Modeling errors are filtered out through the covariance matrix of the normal distribution. The validity of the proposed method is demonstrated through some examples.
Bayesian Theory, L2-regularization, L1-regularization, normal distribution, symmetry exponential distribution, log-normal distribution, modeling error, covariance.