Sparsity-constrained Extended Kalman Filter concept for damage localization and identification in mechanical structures

Structural health monitoring (SHM) systems are necessary to achieve smart predictive maintenance and repair planning as well as they lead to a safe operation of mechanical structures. In the context of vibration-based SHM the measured structural responses are employed to draw conclusions about the structural integrity. This usually leads to a mathematically illposed inverse problem which needs regularization. The restriction of the solution set of this inverse problem by using prior information about the damage properties is advisable to obtain meaningful solutions. Compared to the undamaged state typically only a few local stiffness changes occur while the other areas remain unchanged. This change can be described by a sparse damage parameter vector. Such a sparse vector can be identified by employing $L_1$-regularization techniques. This paper presents a novel framework for damage parameter identification by combining sparse solution techniques with an Extended Kalman Filter. In order to ensure sparsity of the damage parameter vector the measurement equation is expanded by an additional nonlinear $L_1$-minimizing observation. This fictive measurement equation accomplishes stability of the Extended Kalman Filter and leads to a sparse estimation. For verification, a proof-of-concept example on a quadratic aluminum plate is presented.