可处理不稳定线性离散系统的平衡降阶方法

针对线性定常离散系统,本文将原定义于时域的能控性矩阵、能观性矩阵推广到频域上.研究表明:对于稳定系统,频域能控、能观矩阵等于时域能控、能观矩阵;但不稳定系统有频域能控、能观矩阵而无时域能控、能观矩阵.在此基础上,从频域角度提出了平衡实现的定义并导出一种新的线性离散系统平衡降阶方法,这种方法克服了传统Moore方法不能解决不稳定系统降阶问题的缺陷.本文详细给出了通过求解离散李亚普诺夫方程来计算频域能控、能观矩阵的过程,深入讨论了通过求解离散黎卡提方程来进行系统平衡降阶的新算法.一个关于离散不稳定系统的降阶算例显示:传统Moore方法完全失效,而本文的方法可行并得到满意的降阶效果.

A balanced model reduction method capable of tackling unstable discrete-time systems

For discrete-time LTI systems, the definition of controllability Gramians and observability Gramians is extended into frequency domain from time domain. It is shown that the frequency domain Gramians of stable systems are equal to the time domain ones, and that unstable systems possess only frequency domain Gramians. Based on the frequency domain Gramians, a new balanced model reduction method is proposed. 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. A calculation approach of frequency domain Gramians is presented by virtue of Lyapunnov equations. The model reduction algorithm is discussed which utilizes Recatti equations. A numerical example of an unstable system shows that the method in the literature is invalid while the proposed method is feasible and satisfying.