【作者】 王勇；

【导师】 吴俊；

【作者基本信息】 浙江大学 ， 控制理论与控制工程， 2005， 硕士

【摘要】 大系统模型降阶作为一个理论课题，自60年代末至今，受到国内外广大控制界人士的重视，提出了大量模型降阶的方法。然而，1981年，Moore针对线性稳定连续系统提出平衡降阶方法，该方法克服了前述方法的缺点并具有可靠性好、简单易行、降阶效果良好的特点，备受人们的关注。但Moore的这种方法应用于不稳定连续系统时，由于得不到有界正定的能控性与能观性格莱姆矩阵，导致方法失效。Zhou提出可处理不稳定线性连续系统的平衡降阶方法，把Moore方法中能控性和能观性格莱姆阵的定义由时域推广到频域，并且在连续系统中解决了不稳定系统的降阶问题。Al-saggaf将Moore的平衡降阶方法推广到了稳定的线性离散系统，取得良好的降阶效果，但这种方法同样不能应用于不稳定系统。 本论文系统的总结了目前平衡降阶理论的发展与应用，提出一种新的线性离散系统平衡降阶方法，克服了传统Moore方法不能解决不稳定离散系统降阶问题的缺陷。在此基础上对离散系统的平衡降阶问题做了若干频域推广，给出了具体数值算例。主要研究内容： 1．本文阐述了降阶理论的产生及发展，总结了平衡降阶理论的最新进展和传统离散系统平衡降阶理论。 2．提出一种新的频域下的离散能控性矩阵、能观性矩阵的定义，这种定义下能保证对于不稳定离散系统能控性矩阵、能观性矩阵仍然保持正定，从而将平衡降阶方法推广应用到了离散不稳定系统。数值算例表明了这种方法的可靠性和有效性。 3．研究了两种频域平衡降阶方法的推广：离散系统的频域加权平衡降阶方法和离散系统在某一频段上的平衡降阶方法并给出数值算例说明了方法的有效性；阐述了频域平衡降阶方法的时域解释。 最后是全文的总结和展望。更多还原

【Abstract】 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.更多还原