A Necessary And Sufficient Condition For The Simultaneous Stabilization Of Three Time-varying Linear Systems

Linear system theory is the most basic and important in modern control theory is a branch of the most mature,is the production process control,information processing and communica-tion system,the basic theory of network system and other aspects.Stability is often necessary for the control system to work properly.In 2011,under the transmission condition,Yu tian qiu proposed the simultaneous stabilization.He gave some sufficient and necessary conditions for the simultaneous stabilization of the three linear systems,but these conditions required at least two systems of prime factorization.This paper considers the simultaneous stabilition problems of three discrete-time time-varying linear systems within the frame wok of nest algebra.Under the transmission condition,a necessary and sufficient condition for the existence of simultaneous stabilizing controllers is given based on the strong representation.Some simultaneous stabiliz-ability criteria are established in terms of single strong representation of only one controller in the transimission condition.The first chapter mainly introduces the development background of control theory,some knowledge of linear system and the problems and research background of this paper.In the second chapter,we give the basic knowledge of Hilbert space,operator theory,set algebra,finiteness and linear system.Third chapter three time-varying linear systems is studied and a sufficient and necessary conditions of calm.This condition only need a controller of single element decomposition.Illustrates the feasibility and practicability of the general theorem.