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.

在本文中，我们讨论了基于平衡方法的正实系统的模型降阶问题。混合gramian平衡截断（MGBT）方法是对正实数平衡截断（PRBT）方法的改进，其重点是解决一个Lyapunov方程和一个Riccati方程，与需要解决两个Riccati方程的PRBT相比，其计算量较小。MGBT的一个主要缺点是，与PRBT相比，它无法提供错误界限。为了克服这个问题，我们对MGBT进行了一些新颖的修改，不仅可以处理一个Lyapunov和一个Riccati方程，而且可以提供误差范围。因此，可以说，所提出的方法具有MGBT和PRBT的关键特征。这些算法是借助从新Lyapunov方程中提取的新gramians提出的。第二种算法是第一种算法的频率加权版本。另外，还观察到，与PRBT相比，所提出的方法可以提供更好的误差范围。最后，通过综合数值算例，验证了所提出方法的有效性。