HSH-norm optimal MOR for the MIMO linear time-invariant systems on the Stiefel manifold

Abstract

This paper focuses on the Hilbert–Schmidt-Hankel-norm ($\mathrm{H}\mathrm{S}\mathrm{H}$ norm) optimal model order reduction (MOR) of large-scale multiple-input multiple-output (MIMO) linear time-invariant (LTI) systems on the Stiefel manifold. First, a cost function is constructed in regard to the $\mathrm{H}\mathrm{S}\mathrm{H}$ norm. By introducing the orthogonality constraints, the $\mathrm{H}\mathrm{S}\mathrm{H}$ norm optimal MOR problem is converted into an unconstrained minimisation problem on the Stiefel manifold. We derive the Riemannian gradient of the cost function on the Stiefel manifold. The MOR algorithm is developed associated with the Riemannian conjugate gradient method. Global convergence is guaranteed with mild conditions. Finally, the effectiveness of the proposed method is illustrated by two numerical examples.

Keywords:

• Model order reduction
• linear time-invariant systems
• Stiefel manifold
• optimisation

Acknowledgements

The authors would like to thank the editors and the reviewers for their valuable comments and constructive suggestions that have helped us improve the quality of the paper.

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Funding

The work of the authors was supported by the National Natural Science Foundation of China [grant number 11871393].