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

Abstract

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 optimal MOR iterative algorithm to reduce the SISO system. Further, the authors explore the 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 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 optimal MOR problem for general LTI systems. Finally, two numerical examples demonstrate the effectiveness of their algorithms.