On the Structure Theory of Linear Differential Systems
On the Structure Theory of Linear Differential Systems
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线性微分系统的结构理论
DOI：
10.1137/0306043
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发表时间：
1968
期刊：
影响因子：
--
通讯作者：
L. Weiss
中科院分区：
文献类型：
--
作者：
L. Weiss
几年前，Kalman[1]，[2]证明了形式为(1)的线性定常系统的一个结构定理，该定理是由Gilbert[3]的一些工作推动的。Kalman在[2]中对时变情况进行了扩展（没有证明），在此基础上，作者[4]，[5]使用“反”可控性和可观测性的概念进行了进一步的扩展（也没有证明）。这个定理的最初陈述和后来的扩展的关键是，假定给定一个固定的时刻，存在一个坐标变换，使(1)的系数矩阵在固定的时刻(1)变成一种特殊的、有效的形式。上面描述的一个将包括断言存在“连续变化”坐标变换的能力，这种能力影响(1)对所有（或至少对所有_ bbb _ t '对某些固定）有效的结构分解为相互连接的组件子系统，其数学表示和系统理论性质类似于早期关于结构的定理中指出的那些。在这个方向上已经取得了一些进展（例如，参见[6]），主要基于作者提出的一个过程（参见[6]中定理9的证明），该过程用于从给定的非约简加权模式中获得线性系统的全局约简加权模式（因此是最小实现）。但是，建议的过程涉及执行初始坐标转换，以便
A few years ago, Kalman [1],[2] proved a structure theorem for linear time-invariant systems of the form (1) which was motivated by some work of Gilbert [3]. An extension, to the time-varying case was stated (without proof) by Kalman in [2] and, based on that, a further extension was subse-quently stated (also without proof) bythe writer [4],[5] using the concepts of" anticusal" controllability and observability. The crux of he original statemet of the theorem nd of the subsequen extensions was the assertion that given a fixed instant of time, there exists a coordinte trausformtio which c). vers the coeificiet matrices of (1) into a special, form valid at the fixed i’cstant t. It is generally recognized hat a more sisfyiug resul than. the one described bove would consist of the ability to assert existence of" co--tinuously varying" coordinate ransformation which effects a structural decomposition of (1) valid for all (or, a least, for all _> _ t’for some fixed) into interconnected component subsystemswhose mathematical representation and system-theoretic properties are similar to those indicated in the early theorems on structure. Some progress has been made in this direction (see, for instunce,[6]), based essentially on a procedure suggested by the writer (see proof of Theorem 9 in [5]) for obtaining globally reduced weighting patterns (: tnd therefore minimal realizations) of linear systems from given nonreduced weighting patterns. However, the suggested pro-cedure involves performing an initial coordinate transformation, so as to