Preserving-periodic Riemannian descent model reduction of linear discrete-time periodic systems with isometric vector transport on product manifolds

Abstract

In this paper, we propose a Riemannian descent model reduction iterative method for linear discrete-time periodic (LDTP) systems. The preserving-periodic $H_2$ optimal problem for LDTP systems is posed on a product of Stiefel manifolds. The key feature of the proposed method is the use of the Riemannian geometry, including the orthonormal tangent bases, the intrinsic representation of tangent vectors, and the isometric vector transport by parallelization. By deriving the intrinsic representation of tangent vectors in orthonormal tangent bases, the vector transport by parallelization on the product manifold is given and then we present a Riemannian descent search direction in conjugate gradient scheme to construct the desired reduced systems. The main advantages of our method are that it not only preserves the periodic time-varying structure during the reduction process, but also guarantees the convergence to stationary points. Finally, a numerical example is tested to show that the proposed method has good convergence performance in constructing $H_2$ optimal reduced systems.