Stabilization Of Linear Systems In The Framework Of Nest Algebra

The analysis and synthesis is one of the main content of control theory research. However, the stabilization problem is an important content of synthesis. In this paper, Stabilization of linear systems in the framework of nest algebra is studied by using the operator algebra and operator theory approaches.In Chapter 2, some preliminaries about operator algebra, operator theory and the stabilization of linear time-varying systems in the framework of nest algebras are introduced.In Chapter 3, the concepts of stabilization with internal loop are analyzed for infinite-dimensional discrete time-varying systems. We extend our study of controllers with internal loop to more general use and give a parametrization of all stabilizing controllers with internal loop. Some sufficient and necessary conditions such that a stabilizing controller with internal loop stabilizes plant are obtained. We also analyze two special subclasses of stabilizing controllers with internal loop, called canonical and dual canonical controller and show that all stabilizing controllers can be parameterized by a doubly coprime factorization of the original transfer function. The concept enable a simple Youla parameterization and has some advantages which turn out to be very important for infinite-dimensional systems. It makes the theory of dynamic stabilization simpler and more natural.In Chapter 4, we consider the reliable stabilization of multi-controller systems composed to one plant and two controllers for linear time-varying system. To have the closed loop system stability, which is reliable against a single controller failure, we must ensure that each controller individually stabilizes system as well. Sufficient and necessary conditions for existence of reliable controllers are obtained.In Chapter 5, the stabilization of linear systems in lp(Z)(p? 1) space is studied.Georgiou and Smith have studied the properties such as causality and stabilizability of one-operator model y= Pu over the l2(Z) signal space and they discovered intrinsic difficulties:A causal system could have a noncausal closure and a well-known stabilizable system seemed not to be stabilizable. The two-operator, input-output model Ay= Bu, avoids the intrinsic limitations of the one-operator model on the full time axis.We discuss the properties of causal LTI operators on weighted l2 spaces for different choices of the weighting sequence. Problems of closability of unstable causal LTI convolution operators are also discussed. We shall provide a new type of argument concerning causal LTI operators and robust design that can be applied to a large class of weighted l2 spaces on Z. A large class of weighted l2 spaces on Z have been shown to lead to H? optimization. The unstable causal LTI convolution operators are not closable in weighted l2 spaces.