Structural deformation reconstruction by the Penrose–Moore pseudo-inverse and singular value decomposition–estimated equivalent force

For structural health monitoring, estimating the external load is a typical ill-posed problem but significant. Because with the external force and the structural finite element model, any required response can be calculated, which is advantageous for further structural health monitoring works. This article first defines an underdetermined equation using a limited number of in-field measurements and the finite element model–calculated influence line matrix, and it proposes a load estimation method using the Penrose–Moore pseudo-inverse (generalized inverse). The objective of the proposed method is to obtain the equivalent nodal force vector with minimum length among all possible force vectors satisfying the deformation constraints. The estimated force is an equivalent nodal force, since it only satisfies limited deformation constraints. With the estimated nodal force, full structure static response can be easily calculated by multiplying the influence line matrix and the equivalent force vector. Besides, the truncated singular value decomposition is used to process the finite element model–calculated influence line matrix to avoid the over-fitting effect due to the measurement noise. The singular values of singular value decomposition represent the significance of the structural deformation modes, and the decreasing rate of the singular values is a good complexity indicator for a structure. The proposed frame can involve any types of static measurements, and it can realize real-time computation because it merely involves the matrix multiplying calculation. Finally, the sensitivity analysis is conducted by numerical simulation, and a large-scale model-based experiment has demonstrated that the algorithm is appropriate for in-field applications.