Model Reduction And H_∞Controller Reduction In Linear Systems

Since1960s, many scholars at home and abroad have been interested in the research on the model reduction method for systems as a theoretical problem. A lot of model reduction methods have been presented. However, most of them focus on the normal systems and the reduction methods about the singular systems are very few. Besides, Controllers followed by the H∞theory are usually difficult to realize and maintain, because their orders are too high. It is an obstacle to the application of the H∞theory in the practical engineering.For those two reasons, the main contributions of this dissertation are as follows.(1) The generation and development of model reduction method and the newly development of balanced model reduction method and H∞reduced-order control theory are summarized.(2) A model reduction for singular systems is investigated. First of all, by transforming the original systems, the reduction problem of no impulse models singular systems comes down to a simplification of normal systems reduction. The singular systems of impulse models are decomposed into the casual subsystems and the non-casual subsystems, and the reduction problem becomes merely a simplification of the non-casual subsystems reduction. Based on the Schur decomposition, the controllability and the observability for non-causal subsystems are analyzed. The degree of the systems state controllability and observability is scaled by Hankel singular values, and the results show that the controllability and observability of the states which are corresponding to the smaller Hankel singular values are weaker. Based on it, a new model reduction algorithm of singular systems is given. Finally, these facts are clearly shown by numerical examples.(3) The order-reduced Luenberger observer-based H∞controller for linear systems is studied, and this observer-based controller is with disturbance decoupling. Compared with the previous methods and conclusions, a necessary and sufficient condition for the H∞state-feedback control problem of linear systems is established, which can be solved by LMI toolbox in MATLAB directly. A parameterize functional observer with reduced order is formulated by the explicit solution for Sylvester functions. An order-reduced H∞controller for linear systems is obtained. Finally, numerical example is given to illustrate the validity of the results.(4) We extend the problems, point out the unsolved problems, and propose the perspective on the further more study.