A Study On Balanced Model Reduction Theory And Algorithms Of Linear Discrete-Time Systems

Since 1960s many scholars at home and abroad have been following with interest the research on the model reduction method for large-scale systems as a theoretical problem. A lot of model reduction methods have been presented . In 1981, Moore developed a new internal Balanced Realization theory that has renovated these model reduction methods and this method is feasible, reliable and satisfying. It takes people great concern, and More practical reduced methods are formed once many classical methods are combined with balanced theory. Because there are none boundary and positive Gramians, this method can't be applied for unstable systems. Kemin Zhou proposed a new method which resolved the model reduction problems for continuous unstable systems by introducing some frequency controllability and observability Gramians. Al-saggaf proposed a discrete version of Moore's method and achieved good results for asymptotically stable discrete systems, but the method can not be applied for unstable systems, too.This paper has summarized available literatures in this area, and proposed a new balanced model reduction method. The new method can solve the model reduction problem of unstable discrete-time systems, while the existing ones can not deal with unstable discrete-time systems. Based on this balanced method, the model reduction methods has been generalized in frequency domain, the numerical examples are given. My main results are obtained as follows:1. The generation and development of model reduction method and the newly development of balanced model reduction method are summarized. Moore's method is introduced on discrete systems in detail.2. A frequency version definition of controllability Gramian, observability Gramian is introduced for discrete-time systems. The Gramians remain positive even though the system is not stable and so the balanced modelreduction method can be applied for unstable systems, too. A numerical example shows that the proposed method is reliable and effective. 3. Two model reduction methods via frequency-domain balanced structure are discussed: frequency weighted balanced model reduction method and balanced model reduction method for a section of frequency-domain, the digital realization process for each typical method is given. Some time domain interpretation of the frequency balanced model reduction is given. The conclusion and perspective are given in the end of the paper.