Reducing computational complexity in nonlinear model input design via sparse estimation

The probability density function of the input plays a crucial role in the process of identifying nonlinear systems, with a finite representation commonly employed in the process. However, input design for nonlinear models is a challenging task because it usually involves optimizing a problem with a large number of free variables, which is computationally heavy. The first contribution of this paper is to demonstrate that the majority of these free variables are zero. Consequently, there is no necessity to optimize all of them. The second contribution is to identify the non-zero variables within this set of free variables associated with input design. To address this, we propose an alternating minimization approach. In the first step, we compute the per-sample Fisher Information Matrix (FIM). Then, in the second phase, we estimate the positions of the non-zero elements within the vector of free variables using the previously derived per-sample FIM. Additionally, in the later phase, we calculate the Lagrangian multipliers in our optimization problem using the Karush–Kuhn–Tucker conditions and derive an upper bound for the hyperparameter, which promotes sparsity. This bound ensures the maximum required number of non-zero variables to represent the per-sample FIM. Following this, the original input design problem is streamlined to optimize the cost function only with respect to the non-zero elements, resulting in a significant reduction in computational time. To assess the effectiveness of our proposed method, we conduct a comprehensive numerical performance evaluation by comparing it to state-of-the-art input design algorithms.