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Title:  An information theoretic study of modeling and control of dynamical systems 
Author(s):  Sun, Yu 
Director of Research:  Mehta, Prashant G. 
Doctoral Committee Chair(s):  Mehta, Prashant G. 
Doctoral Committee Member(s):  Basar, Tamer; Coleman, Todd P.; Dullerud, Geir E. 
Department / Program:  Mechanical Sci & Engineering 
Discipline:  Mechanical Engineering 
Degree Granting Institution:  University of Illinois at UrbanaChampaign 
Degree:  Ph.D. 
Genre:  Dissertation 
Subject(s):  Bode integral formula
hidden Markov model nonlinear control system KullbackLeibler divergence model reduction distributed optimization dynamical systems 
Abstract:  This dissertation concerns fundamental performance limitation in control of nonlinear systems. It consists of three coherent, closely related studies where the unifying theme is the use of information theoretic tools to investigate modeling and control issues in dynamical systems. The ﬁrst study focuses on entropy based fundamental limitation results for the nonlinear disturbance rejection prob lem. The starting point of our analysis is the socalled KolmogorovBode formula for linear dynamics, which relates the fundamental limitation to certain entropy rates of the input/output signals. We propose a hidden Markov model(HMM) framework for the closedloop system, under which the entropy rate calculations become straight forward. Explicit entropy bounds are thus obtained for both the classical Bode problem(with linear dynamics) as well as certain cases of nonlinear dynamics. An important implication of this study is that the limitations arise due to fundamental issues pertaining to estimation as opposed to the stabilization control problem. The second study is concerned with information theoretic “pseudometrics” for comparing two dynamical systems. It can be regarded as extending the KolmogorovBode formula for model comparison and robustness analysis. Central to the considerations here is the notion of uncertainty in the model: the comparisons are made in terms of additional uncertainty that results for the prediction problem with an incorrect choice of the model. A KullbackLeibler (KL) rate pseudometric is adopted to quantify this additional uncertainty. The utility of the KL pseudometric to a range of model reduction and model selection problems are demonstrated by examples. It is shown that model reduction of nonlinear system using this pseudometric leads to the socalled optimal prediction model. For the particular case of linear systems, an algorithm is provided to obtain optimal prediction auto regressive (AR) models. The third study concerns discrete time nonlinear systems, where the fundamental limitations are expressed in terms of the average cost of an inﬁnite horizon optimal control problem. Unlike usual optimal control problem, the control cost here is deﬁned by a certain KL divergence metric. Under this cost structure, the limitations can be obtained via analysis of a linear eigenvalue problem deﬁned only by the open loop dynamics. The fundamental limitations are investigated for both linear time invariant (LTI) system and nonlinear systems. It is shown that for LTI systems the limitation depend upon the unstable eigenvalues, as in the classical Bode formula. For more general class of nonlinear systems the limitation arise only if the openloop dynamics are nonergodic. Taken together, these studies represent some preliminary effort towards an information theoretical paradigm for study ing control of dynamical systems. The essential interest is to understand the interaction between uncertainties and dynamics, and its implication in closedloop control systems. This thesis also contain my work on two relevant applications, one is about sensor placement design for distributed estimation and the other is about convergence analysis of a distributed optimization algorithm. 
Issue Date:  20120206 
URI:  http://hdl.handle.net/2142/29705 
Rights Information:  Copyright 2011 Yu Sun 
Date Available in IDEALS:  20120206 
Date Deposited:  201112 
This item appears in the following Collection(s)

Dissertations and Theses  Mechanical Science and Engineering

Graduate Dissertations and Theses at Illinois
Graduate Theses and Dissertations at Illinois