Properties of frequency weighted balanced truncation techniques for discrete-time system

The concept of approximating a linear system into a more manageable order has been a constant fascination for many years [1][8][9]. Enns [2] in particular has initiated a method for reducing a stable high order model with frequency weightings based on balanced truncation technique [1]. In Enns' method, when using input or output weighting, the reduced order system will yield stable reduced order model. However, when both weightings are present, the stability of the reduced order system is not guaranteed. Lin and Chiu [3] has since proposed a different method to guarantee stability even when both weightings are present under certain assumptions i.e. using strictly proper functions and no occurence of pole-zero cancellations when forming the augmented systems. Wang et al [5] has also solved the stability problem of Enns' for two-sided case by introducing fictitious input and output matrices