Damage Detection Scheme Based on Bayesian Theory Man Woo Park
ABSTRACT 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 illposedness of deterministic inverse problems. The regularizing effects of L_{1} and L_{2}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. Lognormal 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.
Key Word Bayesian Theory, L_{2}regularization, L_{1}regularization, normal distribution, symmetry exponential distribution, lognormal distribution, modeling error, covariance.
