On The Transitivity In Robust Stabilisation Of Time-varying Systems

Robust stability of the linear system is an important research topic in the control theory.It plays an important role in different fields,including engineering,medicine,economics and so on,which has great significance for practical development.Many scholars used different methods to study,such as algebra,topology,geometry,and so on.And some achievements have been made.In this paper,we mainly study the robust stabilisation problem of infinite dimensional time-varying linear systems using operator theory and operator algebra theory in the framework of nest algebra.And we focus on the problem of transitivity of robust stabilisation of three uncertain sets.This kind of problem is the extension of the transitivity problem of simultaneous stability.The first part mainly introduces the background of this paper,the research status at home and abroad,as well as the problems and basic structure of this paper.In the second part,we first briefly review the knowledge of operator theory and introduce the basic concepts of the nest algebra.Then give the standard model of the research object feedback system,and the sufficient conditions for the robust stabilisation of the system.Finally we focus on the main tool gap metric of this paper,give the concept and the nature of the gap metric,and describe the relationship between gap metric and system stabilisation.The third part is to study the transitivity of robust stabilisation of the spherical uncertain set defined by the gap metric.Specifically,the robust stabilisation of three uncertain sets is discussed on the premise that there are two uncertain sets are known to be stable,and the sufficient conditions for robust stabilisation of the system family are obtained.It is still an open question to calculate the prime factorization of systems in the framework of nest algebra.Therefore,all the conclusions in this paper avoid the use of prime factorization.The fourth part discusses the transitivity of robust stabilisation when the system or controller is stable.When the controller is stable,it is the strong robust stabilisation problem;When the system is stable,the perturbation of the system is defined as a norm ball,and the sufficient conditions for transitivity and strong transitivity of robust stabilisation of a set of stable spherical uncertainty are given.