Information Descriptions And Approaches In Control Systems

Information viewpoint and methods are attracting more and more attentions in the field of control theory. However, most approaches in the literature were to adopt information measures as cost functions, along with some mathematical optimization methods to analyze and design systems. In fact, little has been done on the study of control systems under the framework of information theory, that is, considering control systems as information transmission channels.From the unique viewpoint of Shannon information theory, this dissertation investigates the varying and transmission of information and uncertainty in control systems using measures of entropy, mutual information, Kullback-Leibler information and information rates, in time and frequency domains, respectively. Several problems concerning state estimation, modeling, H∞ control, performance limits and design constraints of control systems are addressed under this framework. The major contributions of this dissertation are specifically stated as follows.Time Domain1) The problem of the estimability of stochastic systems is discussed by analyzing information varying between states and state estimate errors. An information theoretic definition of estimability for general stochastic systems is given in the sense of MinMax entropy estimation. It is concluded that a linear Gaussian stochastic system is estimable if and only if the corresponding linear definite system is observable. Further, a new definition of identifiability of system model is given in the same sense, and an identifiability Gramian matrix with application value is derived.2) By analyzing the information and conditional information description mechanism of system states, the problem of stochastic model reduction is investigated based on state aggregation. The information loss and conditional information loss between the full- and reduced-order models are measured by entropy, while the independence and conditionalindependence within me components of aggregated state are measured by Kullback-Leibler information distance. Several model reduction methods for stable and unstable linear systems are derived by employing two criteria to get aggregation matrices: the minimal information loss and the maximal independence. The properties of reduced-order models, the order selection and the characteristics of system noises are also discussed.Frequency Domain3) The transmissions of information and uncertainty in several kinds of control systems are studied by investigating the relations between entropy rate, mutual information rate, Kullback-Leibler information rate and H∞ entropy. With this analysis, several information theoretic interpretations are given for the H∞ entropy in seeking the connection between information methodology and the H∞ control methodology. In addition, performances of uncertain systems with parameter random perturbation are discussed.4) By using the famous Bode integrals and the derived relations between information rates and H∞ entropy, performance limits of several kinds of linear control systems disturbed by stationary noises are investigated from the viewpoint of information theory. For linear tracking systems with disturbance, design constraints are formulated by two proposed basic information conditions for efficient tracking.5) The variety, a famous concept proposed by the precursor of cybernetics, Ashby, is extended from variable to systems. With the definition of system variety, the gotten conclusions lead to two new laws of variety in control systems: 'the general conservation law of variety' for regulation systems, and 'the law of requisite variety' based on the proposed basic information condition for tracking systems. Our studies connect information transmission to classic control theory, and hence cover some shortages of information theoretic methods.Although research bases of control theory and information theory are not similar, our researches and conclusions demonstrate that, information theoretic viewpoint and methodology lead...