An enhanced sparse regularization method for impact force identification

The standard sparse regularization method based on l ₁ -norm minimization for impact force identification has already proved to be an interesting alternative to the classical regularization method based on l ₂ -norm minimization. However, choosing the l ₁ -norm as a convex relaxation of the l ₀ -norm, the corresponding sparse regularization model generally offers a sparse but underestimated solution. In this paper, considering the sparsity of impact force, an enhanced sparse regularization method based on reweighted l ₁ -norm minimization is developed for reducing the peak force error and improving the identification accuracy of impact force. First, a weighted l ₁ -norm convex optimization model is presented to overcome the ill-posed nature of the inverse problem of impact force identification. Second, to solve such a regularized model efficiently, an iteratively reweighted l ₁ -norm minimization algorithm is introduced, where the weights are adaptively updated from the previous solution. The application of the iteratively reweighted scheme is to overcome the mismatch between l ₁ -norm minimization and l ₀ -norm minimization, while keeping the enhanced sparse regularization problem solvable and convex. Finally, numerical simulation and experimental verification including the single and double impact force identification on a plate structure are presented to illustrate the superior performance of the enhanced sparse regularization method compared to classical regularization approaches. Effects of reweighting iteration number, tuning parameters, initial conditions and response locations are successfully investigated in detail. Results demonstrate that compared with the standard l ₁ -norm regularization method and the classical l ₂ -norm regularization method, the enhanced sparse regularization method based on reweighted l ₁ -norm minimization whose solution is much sparser, can greatly improve the identification accuracy of impact force. Moreover, the proposed method is much more robust to the choice of tuning parameters and noisy measurements.