This paper proposes a novel realization approach to generate a low-order Fornasini–Marchesini (F-M) (second) model for multidimensional (n-D) systems. First, based on two developed matrix relation properties, the F-M model realization problem is converted to establishing a desired standard matrix by performing elementary operations on an associated matrix initialized from the given transfer matrix. Then, a general constructive procedure and two corresponding techniques to compute the F-M models are established for both the left and right matrix fractional descriptions (MFDs) of the given transfer matrix. It turns out that compared with the existing methods the proposed approach can produce an F-M model with lower order, and furthermore, this is the first method that can treat both the left and right MFDs for obtaining an F-M model realization in a unified manner. Non-trivial examples are given to illustrate the details and effectiveness of the proposed approach.

本文提出了一种新颖的实现方法，该方法可生成多维（n）的低阶Fornasini-Marchesini（FM）（第二）模型-D）系统。首先，基于两个已开发的矩阵关系属性，通过对从给定传递矩阵初始化的关联矩阵执行基本运算，将FM模型实现问题转换为建立所需的标准矩阵。然后，为给定的传输矩阵的左矩阵分数描述和右矩阵分数描述（MFD）建立了通用的构造过程和两种相应的技术来计算FM模型。事实证明，与现有方法相比，该方法可以生成较低阶的FM模型，而且，这是第一种可以同时处理左右MFD的方法，以统一的方式实现FM模型的实现。给出了一些简单的例子来说明所提出方法的细节和有效性。