In practice, simultaneous impact localization and time history reconstruction can hardly be achieved, due to the ill-posed and under-determined problems induced by the constrained and harsh measuring conditions. Although \(\ell_{1}\) regularization can be used to obtain sparse solutions, it tends to underestimate solution amplitudes as a biased estimator. To address this issue, a novel impact force identification method with \(\ell_{p}\) regularization is proposed in this paper, using the alternating direction method of multipliers (ADMM). By decomposing the complex primal problem into sub-problems solvable in parallel via proximal operators, ADMM can address the challenge effectively. To mitigate the sensitivity to regularization parameters, an adaptive regularization parameter is derived based on the K-sparsity strategy. Then, an ADMM-based sparse regularization method is developed, which is capable of handling \(\ell_{p}\) regularization with arbitrary p values using adaptively-updated parameters. The effectiveness and performance of the proposed method are validated on an aircraft skin-like composite structure. Additionally, an investigation into the optimal p value for achieving high-accuracy solutions via \(\ell_{p}\) regularization is conducted. It turns out that \(\ell_{0.6}\) regularization consistently yields sparser and more accurate solutions for impact force identification compared to the classic \(\ell_{1}\) regularization method. The impact force identification method proposed in this paper can simultaneously reconstruct impact time history with high accuracy and accurately localize the impact using an under-determined sensor configuration.
