H₂ optimal model order reduction by two-sided technique on Grassmann manifold via the cross-gramian of bilinear systems

ABSTRACT

In this paper, we discuss the optimal H₂ model order reduction (MOR) problem for bilinear systems. The H₂ optimal MOR problem of bilinear systems is considered as the minimisation problem on Grassmann manifold, which is stored as a quotient space of the noncompact Stifiel manifold. Grassmann manifold whose tangent space is endowed with a Riemannian metric is a Riemannian manifold. In its tangent space equipped with the Riemannian metric, we derive the negative gradients of the cost function, i.e. the steepest descent direction of the cost function. After that, the formulas of geodesic on Grassmann manifold are given. Then we perform linear searches along geodesics to obtain the optimal solutions. Thereby, a two-sided MOR iterative algorithm is proposed to construct an order-reduced bilinear system, which is used to simulate the output and input responses of the original bilinear system. Numerical examples demonstrate the effectiveness of our algorithm.

KEYWORDS:

• Bilinear systems
• model order reduction
• H₂ optimality
• cross gramian
• Grassmann manifold
• tangent space
• geodesic
• cost function

Acknowledgments

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 authors.

Additional information

Funding

This work was supported by the Natural Science Foundation of China (NSFC) [grant number 11371287], [grant number 11161045]; Xinjiang Introduction Plan Project of High Level Talents and Graduate Innovation Project of Xinjiang Province [grant number XJGRI2015007].