Iterative Shrinkage-Thresholding Algorithm and Model-Based Neural Network for Sparse LQR Control Design

This paper considers an Linear Quadratic Regulator (LQR) design problem for multi-agent distributed control systems where designing an optimal feedback controller by considering communications among agents is desired for the reduction of communication burden in a network. To aim this, we deal with an LQR minimization problem with a regularization for sparse feedback matrix, where the sparsity in the feedback matrix is related to the reduction of the communication links in the multi-agent distributed control systems. We propose a simple but efficient iterative algorithm, so-called Iterative Shrinkage-Thresholding Algorithm (ISTA) for sparse LQR optimal control design. The proposed method can provide a trade-off solution between LQR cost and sparsity level on feedback matrix. Through various numerical experiments, we demonstrate that our proposed method can outperform the previous work using the Alternating Direction Method of Multiplier (ADMM) in terms of computational speed. Additionally, based on our proposed method, we introduce its deep neural network model, which can further improve the performance of the proposed algorithm in convergence speed.