Riemannian Gradient-Based Online Identification Method for Linear Systems with Symmetric Positive-Definite Matrix

This paper introduces a stochastic optimization-based approach to the online optimal identification of symmetric linear continuous-time systems. The identification problem is formulated as an optimization problem on a Riemannian manifold, which is a product manifold consisting of three manifolds, i.e., the manifold of symmetric positive definite matrices and two matrix spaces. We specifically address the case where the system matrices to be identified vary over time. We develop a novel algorithm for online identification, called the Riemannian online gradient descent method, in a manner similar to the Riemannian stochastic gradient descent method. Numerical experiments show that the proposed online algorithm considerably decreases the value of the objective function in a practical situation, where time intervals in which the system characteristic does not change significantly are assumed to be small.