A Riemannian conjugate gradient method for discrete algebraic Lyapunov equations with simulation-based validation

The discrete algebraic Lyapunov equation (DALE) plays a central role in control theory, signal processing, and the analysis of dynamical systems. Traditional approaches for solving the DALE primarily rely on Euclidean optimization methods, often neglecting the intrinsic geometric structure of the solution space, namely the manifold of symmetric positive definite matrices. In this study, we propose a geometric framework for solving the DALE using a Riemannian conjugate gradient (RCG) algorithm defined over matrix manifolds. By leveraging the Riemannian structure and employing Cayley-type retraction for updates, the proposed method maintains feasibility and accelerates convergence compared to classical methods. We present a comprehensive comparison of RCG with standard solvers, including the Euclidean gradient descent algorithm (GDA), natural gradient descent algorithm (NGDA), and Fisher-preconditioned iteration method (FPIM), both in terms of theoretical properties and empirical performance. Extensive numerical experiments are conducted using synthetic datasets and real data obtained from the Robotarium, a multi-robot research platform. Our results demonstrate that RCG-DALE consistently achieves faster convergence, improved numerical stability, and enhanced scalability in high-dimensional settings. The findings underscore the advantages of exploiting the geometry of positive definite matrices for solving matrix equations, offering a robust and efficient alternative for real-time control and estimation tasks.