Stability Theory And Corresponding Problems Of Time-Varying Systems In The Framework Of Nest Algebra

In this dissertation, we use the complete finiteness of discrete nest algebra, the commutant lifting theorem for representations of the nest algebra, Douglas’s Range Inclusion Theorem and some operator theory and operator algebra theory tools to deal with the stability theory and some correspoding control problems of discrete linear time-varying systems over single and double-infinite time axis in the frame work of nest algebra.The main research work consists of the following subjects:1. The stability analysis via quadratic constraints;2. Strong stabilization, simultaneous stabilization, strongly simultaneous stabilization and the application to the simultaneous stabilization problems using the multi-controller configura-tion;3. Optimal control corresponding with four-block problem and suboptimal model-matching problem.4. The stability analysis of discrete linear time-varying systems over double-sided signal space12(-∞,+∞).In Chapter1, we give a review about the linear system control theory from point of view of mathematics. More precisely, we introduce the development of H∞control theory and then introduce the main achievements obtained by some mathematicians in the research field of time-varying system theory in the framework of nest algebra.In Chapter2, we give some preparations with an emphasis on the basic knowledge of mathematics and control theory, such as the background of discrete linear time-varying systems, nest algebra and complete finiteness, the tools for the stability analysis and some classical results about stability.In Chapter3, we introduce the concept of "quadratic constrain" to deal with various stability problems for infinite-dimensional discrete linear time-varying systems in the framework of nest algebra. First, we derive necessary and sufficient conditions for the close-loop stability, simultaneous stabilization and strong stabilization based on quadratic constraints. An equivalent condition of this stability criterion presents the relationship between the stabilization of each finite dimensional truncation system and that of the whole system. This stability criterion can be realized by the computer. Moreover, applying this stability criterion and the complete finiteness of the nest algebra, we show that the plant is stabilizable if and only if it has a single strong representation. This is an extension of the classical Youla parametrization theorem. We also use quadratic constraints to cope with a type of robust stabilization problem, in which the uncertainty set is a connected set defined by gap metric. At last, we characterize some stabilization problems of unstable, but densely defined, linear systems based on the quadratic constraint approach. Comparing with the obtained results, these criterions are more clear and distinct associated to the input-output signal set of the system and controller.Chapter4deals with the strong stabilization, simultaneous stabilization, strongly simul-taneous stabilization problems and the applications to simultaneous stabilization using multi-controller. First, some necessary and sufficient conditions for the simultaneous stabilizability and strong stabilizability are derived and then the parameterizations of simultaneously stabi-lizing controllers and strongly stabilizing controllers are geiven, respectively. Moreover, the further result to the strongly simultaneous stabilization problem based on the obtained results is presented. In particular, we give a simple and effective parametrization of all simultaneously strong stabilizing controllers for two linear time-varying systems. As an application, we study the simultaneous stabilization problems using two-parameter compensator:the tracking config-uration using two controllers and the simultaneously reliable stabilization problem for discrete linear time-varying systems in the framework of nest algebra.In Chapter5, the optimal control and suboptimal control for linear time-varying systems are considered in the framework of nest algebra. Applying the commutant lifting theorem for representations of the nest algebra, the existence of optimal solutions to the model matching problem and a certain optimal feedback control problem, each of which corresponds with one type of four-block problem, is demonstrated. Furthermore, it is shown that the optimum is equal to the norm of a certain time-varying Hankel-Toeplitz operator. For the suboptimal control, we concentrate on the suboptimal model-matching problem. It is shown that the existence of two coupled J-spectral factorizations determined by the "models "implies the existence of suboptimal solutions to model-matching problem by applying Douglas’s Range Inclusion Theorem, and derive a parametrization of all suboptimal solutions.The principal aim of Chapter6is to study the stabilization problem of discrete linear time-varying systems over double-sided signal space l2(-∞,+∞) in the frame work of nest algebra. First, we illustrate the Georgiou-Smith paradox by considering a linear time-varying system, which is a generalization of the result associated with linear time invariant systems. Next, applying the complete finiteness of nest algebra and coprime factorization approach, we present that there is no problem with stabilization of the time-varying system l2(-∞,+∞) when it is closed. In particular, it is shown that the stabilizability of time-varying system over l2(-∞,+∞) is equivalent to the existence of a single coprime factorization. In the last, Youla parametrization is established based on coprime factorization approach.