Problems On Linear Systems In Hilbert Space

As known to all,control theory is the theoretical fundament of systematic engineering. It is in the development foreland of mathematics,computer science and engineering.And it plays an important role in the new technical revolution,which representative is automatization, computer,robot and etc.At present,it has been widely applied in many fields such as biology,ecology,medicine,economy,finance and sociology.System is the research object of the control theory.In this doctorial dissertation,we study the stabilization,strong stabilization and estimation problem in the framework of nest algebras in Hilbert space. Stabilization of linear causal systems is stated in Chapter 3;Strong stabilization problem is investigated in Chapter 4;and the last Chapter is devoted to the general estimation problems.In Chapter 2,we simply recall some preliminaries about operator theory,and describe the fundamental conceptions and theorems.In Chapter 3,we firstly introduce the control theory in Hilbert space.The physical notions of causality are presented in the framework of extended spaces.Secondly,we present internal stability of feedback systems.The fundamental idea is to represent a linear system as the range of a 2×1 operator matrix with causal entries.Stabilization is seen to be equivalent to left causal invertibility of such a matrix.This is an appropriate formulation of what is usually called coprime factorization.The classical Youla-Ku(?)era parametrization theorem is presented in this fashion.Finally,we give a necessary and sufficient condition for simultaneous stabilization of several systems by using Youla-Ku(?)era parametrization theorem.In Chapter 4,we mainly study the strong stabilization of linear time-varying systems. We know that practicing engineers prefer using stable compensators for the purpose of stability.This gives us a motivation for considering whether or not there exists a stable compensator for a given plant.Firstly,we obtain parametrization of the stable controllers for a linear time-varying system L which admits a coprime factorization B^-1A.Also we study the simultaneous and robust stabilization problems for this class of systems.In particular,these results hold if A is compact.Secondly,we present a new approach to solve the strong stabilization problem in the context of nest algebras.A necessary and sufficient condition for the solution of strong stabilization problems is used as a basis to develop theoretical results leading new insight into the solution of strong stabilization problems.By using inner-outer factorization in nest algebras,this problem is related to a minimization problem between a certain operator and the nest algebra.The minimum is solved and is shown to be equal to the norm of a Hankel type operator.Finally,we consider the weighted sensitivity problem with stable controllers.We reduce it to a minimization problem using the parametrization of stable controllers that obtained in our paper.And the optimal solutions are given in the cases of time-invariant and time-varying,respectively.In Chapter 5,we discuss a general estimation problem and multiobjective H²/H^∞estimation problem.A.Feintuch had considered the problem,and gave formulas that depend only on the given data using inner-outer factorization for operators in nest algebras.In the front of this chapter,using operator inner-outer faetorization,we reduce the estimation problem to a distance minimization between a certain operator and the nest algebra in question. In the same time,the existence of an optimal time-varying estimator is given.Also we relate our problem to a linear time-varying operator analogous to the Hankel operator, which is known to solve the standard optimal H^∞problem in the linear time-invariant case. At last,we point to a necessary and sufficient condition which guarantees compactness of the Hanke]operator,and in which case the optimal estimator can be computed.Our approach is purely input-output and does not use any state space realization,therefore the results derived here apply to infinite dimensional LTV systems.In the last of this chapter, we introduce multiobjective H²/H^∞estimation problem.We use the Banach space duality theory and operator theory to give an exact solution,also the solution is shown to satisfy an allpass property.