Regularization Techniques in System Identification
for Damage Assessment of Structures
Hyun Woo Park
Regularization techniques in system identification (SI) for damage assessment of structure are proposed. This study adopts an SI scheme based on the minimization of the least squared error between measured and calculated responses, which is a nonlinear inverse problem.
A general concept of the regularity condition of the system property is presented. By imposing a proper regularity condition, inherent ill-posedness of the SI scheme is alleviated satisfactorily. It is shown that the regularity condition for elastic continua is defined by the L2-norm of the system
properties. Tikhonov regularization technique is employed to impose the regularity condition on the error function. The characteristics of nonlinear inverse problems and the role of the regularization are investigated by the singular value decomposition of a sensitivity matrix of response. It is shown that the regularization results in a solution of a generalized average between the a priori estimates and the a posteriori solution. Based on this observation, a geometric mean scheme (GMS) is proposed. In the GMS, the optimal regularization factor is defined as the geometric mean between the maximum singular value and the minimum singular value of the sensitivity matrix of responses. The validity of the GMS is demonstrated through numerical examples with measurement errors and modeling errors.
It is shown that a solution space defined by the L2-norm of system property is not appropriate for framed structures unlike elastic continua. The L1--norm of the system property is introduced as new regularization function for framed structures. The truncated singular value decomposition (TSVD) is employed to filter out noise-polluted solution components in quadratic sub-problems of the error function. The discretized regularity condition defined by the L1--norm of the stiffness parameter vector is imposed as a separate optimization problem in each quadratic sub-problem. The optimization of the L1--norm is performed by the simplex method. The optimal truncation number is determined by the cross validation. The final damage status of a framed structure is assessed by the statistical approach based on the data perturbation and the hypothesis test. The validity of the proposed regularity condition for framed structures is presented by detecting damage of a two-span continuous truss with different damage cases with measurement errors.
system identification, regularization technique, damage assessment, ill-posedness,
regularity condition, geometric mean scheme