Distributed Algorithm for Solving Sylvester Matrix Equation via Iterative Learning Control

As the most fundamental type of matrix equation, the Sylvester equation has been widely applied in control theory, signal processing, and many other scientific fields in recent years. The traditional method for solving the Sylvester equation is to transform it into a linear algebraic equation (LAE). However, this method will lead to an increase in the dimension of the coefficient matrix, which makes it difficult to solve the LAE. To alleviate the above problem, a distributed algorithm for solving the Sylvester equation is presented in this paper. Firstly, we obtain a LAE equivalent to the Sylvester equation by utilizing vectorization operation and Kronecker product. Then, a group of agents in the multi-agent system is considered to implement the distributed solution for LAE, where each agent only solves its local task by constantly exchanging information with its neighbors. By constructing the iterative learning control system, a discrete linear system about the tracking error of the agent is obtained. Based on the average neighbor information and the feedback control design, an updating rule for each agent iteratively updating its state is obtained. It is shown that all agents converge to the vectorization solution of the Sylvester equation when the communication topology between agents is undirected complete graph. Finally, a simulation example is provided to demonstrate the effectiveness of the proposed distributed algorithm.