Iterative Thresholding and Projection Algorithms and Model-Based Deep Neural Networks for Sparse LQR Control Design

In this article, we consider a linear–quadratic regulator (LQR) design problem for distributed control systems. For large-scale distributed systems, finding a solution might be computationally demanding due to communications among agents. To this aim, we deal with LQR minimization problem with a regularization for sparse feedback matrix, which can lead to achieve the reduction of the communication links in the distributed control systems. For this work, we introduce simple but efficient iterative algorithms—iterative shrinkage-thresholding algorithm and iterative sparse projection algorithm. They can give us a tradeoff solution between LQR cost and sparsity level on feedback matrix. Moreover, in order to improve the speed of the proposed algorithms, we design deep neural network models based on the proposed iterative algorithms. Numerical experiments demonstrate that our algorithms can outperform the previous methods using the alternating direction method of multiplier [Lin et al. (2013)] and the gradient support pursuit [Lian et al (2017)], and their deep neural network models can improve the performance of the proposed algorithms in convergence speed.