l_p-norm regularization for impact force identification from highly incomplete measurements

The standard l1-norm regularization is recently introduced for impact force identification, but generally underestimates the peak force. Compared to l1-norm regularization, lp-norm (0 ≤ p < 1) regularization, with a nonconvex penalty function, has some promising properties such as enforcing sparsity. In the framework of sparse regularization, if the desired solution is sparse in the time domain or other domains, the under-determined problem with fewer measurements than candidate excitations may obtain the unique solution, i.e., the sparsest solution. Considering the joint sparse structure of impact force in temporal and spatial domains, we propose a general lp-norm (0 ≤ p < 1) regularization methodology for simultaneous identification of the impact location and force time-history from highly incomplete measurements. Firstly, a nonconvex optimization model based on lp-norm penalty is developed for regularizing the highly under-determined problem of impact force identification. Secondly, an iteratively reweighed l1-norm algorithm is introduced to solve such an under-determined and unconditioned regularization model through transforming it into a series of l1-norm regularization problems. Finally, numerical simulation and experimental validation including single-source and two-source cases of impact force identification are conducted on plate structures to evaluate the performance of lp-norm (0 ≤ p < 1) regularization. Both numerical and experimental results demonstrate that the proposed lp-norm regularization method, merely using a single accelerometer, can locate the actual impacts from nine fixed candidate sources and simultaneously reconstruct the impact force time-history; compared to the state-of-the-art l1-norm regularization, lp-norm (0 ≤ p < 1) regularization procures sufficiently sparse and more accurate estimates; although the peak relative error of the identified impact force using lp-norm regularization has a decreasing tendency as p is approaching 0, the results of lp-norm regularization with 0 ≤ p ≤ 1/2 have no significant differences.