Model Order Reduction of Positive Real Systems Based on Mixed Gramian Balanced Truncation with Error Bounds

In this paper, we discuss the problem of model order reduction for positive real systems based on balancing methods. The mixed gramian balanced truncation (MGBT) method, which is a modification of the positive real balanced truncation (PRBT) method, focuses on solving one Lyapunov equation and one Riccati equation resulting in less computational effort compared to PRBT requiring solving two Riccati equations. One major disadvantage of MGBT is that it cannot provide an error bound in contrast to PRBT. To overcome this issue, we have developed some novel modifications to MGBT which not only work with one Lyapunov and one Riccati equations but also provide error bounds. Thus, we can say that the presented methods take the key features of both MGBT and PRBT. These algorithms are presented with the aid of the new gramians which are extracted from new Lyapunov equations. The second algorithm is the frequency weighted version of the first algorithm. Additionally, it is also observed that the proposed methods can provide better error bounds compared to PRBT. Finally, comprehensive numerical examples are included to figure out the effectiveness of the presented method.