Gradient-based neural networks for online solutions of coupled Lyapunov matrix equations

Abstract

This study investigates gradient-based neural networks (GNNs) for solving coupled Lyapunov matrix equations arising in the stability analysis of continuous-time Markovian jump linear systems. First, based on the gradient descent principle, a general framework of GNNs is presented to solve the considered equations by following the ideas in some existing results, which are used to solve the standard Lyapunov matrix equation. Subsequently, a new GNN solver is developed for finding the online solution of the coupled Lyapunov equations. Compared with the general GNN solver, a main advantage of the proposed GNN solver is that its convergence can be directly proven through theoretical analysis. To be specific, according to Lyapunov theory, it is shown that the state of the presented GNN solver with appropriately activation functions can globally converge to the unique positive definite solution of the coupled Lyapunov equations. In addition, to accelerate its convergence rate, an improved version of the proposed GNN is established. Simulation results are given to substantiate the effectiveness of the theoretical results and the superior of the developed GNN solvers compared with the originally obtained general GNN solvers.