Some Problems On Linear Time-Varying Systems In The Framework Of Nest Algebra

Control theory is the general theory about modeling,analysis and synthesis for control systems. It is widely used in many different fields such as economics, biology,engineering, medicine and so on. And great effects have been made for the human beings. It is clear that the research on control theory is of invaluable importance. Mathematical theories and methods, such as functional analysis, algebra, topology, geometry and so on, have been introduced into the study of control theory. Lots of powerful, beautiful and elegant techniques and results have been established. This makes that control theory greatly relies on mathematics. The control theory in Hilbert space is right a link between operator theoretic approaches and the study of control theory.In this paper, several problems of linear time-varying systems by using the operator theoretic approaches in the framework of nest algebras are studied.In Chapter2, some known results about the control theory in the Hilbert space are recalled. Moreover, all the causal bounded right (left) inverses of a strong left (right) representation for a plant are fully characterized for the first time. A new criterion for judging the stability of a closed loop system is presented in terms of operator equations.Chapter3is concerned with the relationship between the gap metric and the time-varying gap metric for linear time-varying systems. With the help of the properties of the minimum modulus of a linear operator, we get that when measuring the distance between the orthogonal complement of the inverse graph of a plant and the graph of a controller in a feedback configuration, the gap metric and the time-varying gap metric are in fact identical. The developed criteria are also applied to compute the optimal minimal angles of stabilizable linear time-varying systems. And it is shown that the value of the cosine of the optimal minimal angle is equal to the norm of a time-varying Hankel operator.In Chapter4, the duality theory for the time-varying4-block problem is completely studied for the first time. We express the importance of this block problem for the optimal control theory by converting a general feedback control problem equivalently to the time-varying4-block problem. By using the properties of nest algebra and the notion of M-ideal in the operator theory, we compute the appropriate preannihilator and annihilator. Specific duality theories are established for the general plants and the compact plants, respectively.Furthermore,an example on the measurement feedback control problem is given to show that the optimum obtained by duality theory is allpass.In Chapter5,by using the complete finiteness of a certain discrete nest algebra, we show that a system is stabilizable if and only if it has one kind of strong representation and we also give a parametrization for all the stabilizing controllers in terms of this strong representation. Moreover, a fact is obtained that the other strong representation of a plant can always be derived from the given one and a sufficient condition for the strong stabilization problem is given. These results provide an essential extension to the classical Youla parametrization theorem.