On The Simultaneous Stabilization Of Multi-input And Multi-output Linear Time-varying Systems

Simultaneous stabilization problem of the systems, has always been a focus in con-trol theory of linear system, robust control theory and mathematical theory. Just because stabilization of the system is a necessary condition and premise for the safety and smooth work of the system, then simultaneous controller design for several systems gradually be-come a research focus. Different disciplines of the community, include interdisciplinary, start from a different perspective to study simultaneous stabilization problem. The study of stability is very necessary and meaningful. Further, research on criterion of system stability and how to design the controller are also indispensable step in the research of system stability. Therefore, research on the system stability by the scholars in the field also becomes more and more urgent.The problems of stability at the same time main focus on how to design a reasonable controller can calm gens linear system, with the development and evolution of the stabil-ity theory at the same time, many scholars gradually focus on controller design problem of linear system stabilization at the same time. This study will be taken from a new perspective to study the stability at the same time of two discrete time-varying linear systems and deduced three discrete time-varying linear systems exist at the same time, then deduced the stability criterion and the necessary and sufficient conditions of stabi-lization system, then focus on the set of algebraic framework to explore three discrete time-varying linear systems town qualitative theoretical tools and methods at the same time. More specifically, this study based on the research of the stability of two discrete time invariant linear system, time-varying linear systems are given as L0, L1 and L2, if L0 and L1can be calm at the same time by the controller Co, L1 and L2 can be calm at the same time by controller C1, Whether there can L0, L1 and L2 be calm at the same time by a simple controller C? More general, can parametric form of the controller C be deduced?This article studies on the simultaneous stabilization of multi-input and multi-output linear time-varying systems in the frame of Nest Algebras. The main context and structure of the this study are described below:the first part introduces the research background, research significance, both foreign and domestic research state, and structure of this ar-ticle, and put forward the need to solve the problem. The second part mainly covers the basic symbol, basic concept, basic properties, basic theorem, which are used in this study. The third part reviews the stability of the system, the relationship between the stability and the strong representation of the system. Besides, the youla parameterized theorem and youla parametric expansion theorem are introduced. The fourth part based on y-oula parameterized theorem and youla parametric expansion theorem in the third part, the main results of this research are given, that is, a new approach to the simultaneous stabilization of three time-varying linear systems. At the same time, the parametrization of all simultaneously stabilizing controllers for three time-varying linear systems is estab-lished. In addition, the applicability and the feasibility of theorem are testified and proved through an example. The fifth part presents a summary of the research and prospects for the future work.In this article, we study on the simultaneous stabilization of multi-input and multi-output linear time-varying systems in the frame of Nest Algebras. From the perspective of strong transitivity, basing on the strong representation and the coprime factorization principle, from the problem of the simultaneous stabilization of two time-varying discrete-time linear systems to the problem of the simultaneous stabilization of three time-varying discrete-time linear systems, the criterion and necessary and sufficient conditions are de-duced. Besides, give new methods and tools to design controller for time-varying linear systems. In addition, we establish the parametrization of all the simultaneous stabilizing controllers two or three time-varying linear systems.