L1-Regularization Technique in System Identification
Based on Bayesian Theory
Jae Hyoung Park
This paper presents a new approach to eliminate effect of modeling errors in system identification scheme. It is well-known that SI problems are a type of ill-posed problems, which suffers from instabilities characterized by non-uniqueness and discontinuity of solutions. The instabilities become severe when measured responses of structures are noisy and incomplete. Regularization techniques have been proven to be very effective in stabilizing the ill-posedness of inverse problems such as SI problems of mechanical systems. Most previous works on the regularization of SI scheme have concerned on the measurement error rather than modeling error.
The modeling error represents the discrepancy between a real structure and its mathematical model employed in the SI. The modeling errors cannot be filtered with regularization techniques used before. Because the measurement errors are random while the modeling errors are systematic in nature. This paper proposes a new error function based on Bayesian theory to reduce the instability caused by modeling error. The L1-regulariztion function is employed to filter out noise in measure responses. The optimization for SI is performed by the PP-TSVD.
The validity of the proposed method is demonstrated through numerical examples on discrete structures under various damage scenarios. The proposed method is able to control the modeling error and measurement error effectively by the combination of the new error function and L1-regulariztion function.
Statistical SI, Bayesian Theory, L1-regularization, L2-regularization, normal distribution, symmetric exponential distribution, measurement error, modeling error, covariance matrix.