Non-convex sparse regularization for impact force identification

Abstract

Many convex regularization methods, such as the classical Tikhonov regularization based on l2-norm penalty and the standard sparse regularization method based on l1-norm penalty, have been widely investigated for impact force identification. However, in many practical applications, these regularization methods commonly underestimate the true solution. In this paper, we propose a non-convex sparse regularization method based on lp-norm (0 < p < 1) penalty, to seek the sufficiently sparse and highly accurate solution of impact force identification. Firstly, a non-convex optimization model based on lp-norm penalty instead of l2-norm penalty or l1-norm penalty is developed for regularizing inverse problems of impact force identification to overcome the mismatch between l0-norm and l1-norm regularizations. Secondly, an iteratively reweighed l1-norm algorithm is introduced to solve such a non-convex model through transforming it into a series of l1-norm regularizations. Finally, numerical simulation and experimental validation are carried out to evaluate the effectiveness and performance of lp-norm regularization when 0 = p ≤ 1. Our numerical results demonstrate that, compared to the state-of-the-art Tikhonov regularization and l1-norm regularization, the solution of lp-norm regularization achieves more stability, sparseness and accuracy when dealing with the heavily noisy response. Additionally, both numerical and experimental results demonstrate that the peak relative error of the identified impact force using lp-norm regularization has a decreasing tendency as p is approaching 0; while the identification of lp-norm regularization with 0 = p ≤ 1/2 bears no significant difference, which always outperforms that the identification with 1/2 = p ≤ 1.