Matrix regularization-based method for large-scale inverse problem of force identification

Abstract Identification of force via structural responses has been widely studied due to the force is always difficult or even impossible to be measured directly. Force identification is a typical ill-posed problem. To overcome this drawback, regularization methods have been widely studied. However, existing regularization methods used for force identification belong to the vector-based method, in which unknown force and structural responses are organized in two vectors, respectively. This characteristic decides that the identified process of a long-duration problem will be time-consuming because the size of system matrix is large. In view of this, a novel matrix regularization-based method is proposed for force identification in this paper. Combing with moving time windows, the structural responses are extracted and organized in a form of matrix. A system matrix is formulated by considering both unknown force and unknown initial conditions. Then a governing equation is established, in which structural excitation sources such as force and initial condition are orderly stored in a form of matrix. To obtain a stable solution, matrix regularization is introduced for improving the matrix-based governing equation. Herein, the sparse penalty term is considered as the sum of absolute values of elements in the matrix of excitation sources. Fast iterative shrinkage-thresholding algorithm (FISTA) is applied for solving the matrix regularization model. The regularization parameter is selected according to Bayesian information criterion (BIC). Finally, numerical simulations and experimental studies are carried out for assessing the feasibility and effectiveness of the proposed method. Illustrated results show that the proposed method is effective and time-saving. It can use a system matrix with a relatively small size to dealing with a large-scale problem of force identification. Some related issues are discussed as well.