Riemannian optimal identification method for linear systems with symmetric positive-definite matrix

TL;DR

This paper introduces Riemannian conjugate gradient methods for identifying linear systems with symmetric positive-definite matrices, improving efficiency and accuracy over traditional approaches through geometric optimization on specialized manifolds.

Contribution

The study formulates three novel Riemannian optimization problems for symmetric system identification and proposes conjugate gradient algorithms tailored to these manifold structures.

Findings

Proposed methods outperform Gauss-Newton in accuracy.

Effective initial point selection via subspace methods.

Numerical simulations validate the approach's efficiency.

Abstract

This study develops identification methods for linear continuous-time symmetric systems, such as electrical network systems, multi-agent network systems, and temperature dynamics in buildings. To this end, we formulate three system identification problems for the corresponding discrete-time systems. The first is a least-squares problem in which we wish to minimize the sum of squared errors between the true and model outputs on the product manifold of the manifold of symmetric positive-definite matrices and two Euclidean spaces. In the second problem, to reduce the search dimensions, the product manifold is replaced with the quotient set under a specified group action by the orthogonal group. In the third problem, the manifold of symmetric positive-definite matrices in the first problem is replaced by the manifold of matrices with only positive diagonal elements. In particular, we examine the quotient geometry in the second problem. We propose Riemannian conjugate gradient methods for the three problems, and select initial points using a popular subspace method. The effectiveness of our proposed methods is demonstrated through numerical simulations and comparisons with the Gauss--Newton method, which is one of the most popular approach for solving least-squares problems.