同时稳定化的强传递性及同时控制器的设计

The simultaneous stabilization problem is concerned with the design of a single controller that stabilizes a finite family of linear system. This has been widely used in robust stabilization. In 1982, Sakes and Murray proved that the simultaneous stabilization of two plants is equivalent to strong stabilization.The simultaneous stabilization problem gets more complicated for three(or more) linear systems. Many results have been obtained for simultaneous stabilization in various frameworks. Despite all these efforts, at present no tractable necessary and sufficient conditions exist for simultaneous stabilization except for the case of two systems. In this topic, in view of the strong transitivity, using operator theoretic methods, we consider the simultaneous stabilization and simultaneous controller design of three (or more) discrete time-varying linear systems within the framework of nest algebras in terms of doubly coprime factorizations.

同时稳定化问题是寻找一个控制器稳定多个系统，它在鲁棒控制领域中有着广泛的应用。1982年，Saeks和Murray证明了两个系统可同时稳定化当且仅当某一构造的系统是强稳定化的。而三个及三个以上系统的同时稳定化问题就变得复杂得多。尽管多年来，这方面出现了大量的研究成果，但是人们仍无法找到同两个系统同时稳定化类似的易于判定的充分必要条件。在本课题中，我们将应用算子理论中所形成的经典方法和技巧从同时稳定化的强传递性的角度入手，利用互素因子分解的语言研究套代数框架下三个及三个以上系统的同时稳定化问题以及同时稳定多个系统的控制器设计。

本项目旨在利用算子理论中所形成的方法从同时稳定化的强传递性的角度入手，研究套代数框架下三个及三个以上系统的同时稳定化问题以及同时稳定系统的控制器设 计。我们按照项目计划书开展研究，在若干关键问题上均取得了突破和进展，基本完成了项目的研究任务。代表性的成果有：1. 针对于某类系统，刻画了同时稳定化的强传递性及其控制器的参数化表示。 2. 基于强传递性，得到了三个线性时不变系统同时稳定化较为简洁的充分条件。3. 基于传递性，刻画了套代数框架下线性时变系统的同时稳定化。