An unconstrained H2 model order reduction optimisation algorithm based on the Stiefel manifold for bilinear systems

In this paper, the optimal H-2 model order reduction (MOR) problem for bilinear systems is explored. The orthogonality constraint of the cost function generated by the H-2 MOR error makes it is posed not on the Euclidean space, but can be discussed on the Stiefel manifold. Then, the H-2 optimal MOR problem of bilinear systems is turned into the unconstrained optimisation on the Stiefel manifold. The explicit expression of the gradient for the cost function on this manifold is derived. Full use of the geometry properties of this Stiefiel manifold, we propose a feasible and effective iterative algorithm to solve the unconstrained H-2 minimisation problem. Moreover, the convergence of our algorithm is rigorously proved. Finally, two practical examples related to bilinear systems demonstrate the effectiveness of our algorithm.