Reduced H₂ optimal models via cross Gramian for continuous linear time-invariant systems

In this study, the authors focus on the problem of model order reduction (MOR) suitable for continuous linear time-invariant (LTI) systems. Specifically, for single-input single-output (SISO) LTI systems, a couple of MOR algorithms via the cross Gramian are presented. The authors first give a cross Gramian-based H₂ optimal MOR iterative algorithm to reduce the SISO system. Further, the authors explore the H₂ optimisation problem on the Stiefel manifold. Based on the geometric notions on this manifold, a conjugate gradient iterative algorithm (CGIAHM) is proposed. The conjugate gradient direction used to search for the H₂ minimiser is derived by applying the notion of vector transport. It is worth mentioning that the conjugate gradient used in the CGIAHM algorithm is a decent direction for the cost function due to the ingenious construction of this algorithm. In addition, the authors' algorithms are extended to solve the H₂ optimal MOR problem for general LTI systems. Finally, two numerical examples demonstrate the effectiveness of their algorithms.