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# An information theoretic study of modeling and control of dynamical systems

#### Sun, Yu

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https://hdl.handle.net/2142/29705

## Description

- Title
- An information theoretic study of modeling and control of dynamical systems
- Author(s)
- Sun, Yu
- Issue Date
- 2012-02-06T20:12:04Z
- Director of Research (if dissertation) or Advisor (if thesis)
- Mehta, Prashant G.
- Doctoral Committee Chair(s)
- Mehta, Prashant G.
- Committee Member(s)
- Basar, Tamer
- Coleman, Todd P.
- Dullerud, Geir E.

- Department of Study
- Mechanical Sci & Engineering
- Discipline
- Mechanical Engineering
- Degree Granting Institution
- University of Illinois at Urbana-Champaign
- Degree Name
- Ph.D.
- Degree Level
- Dissertation
- Keyword(s)
- Bode integral formula
- hidden Markov model
- nonlinear control system
- Kullback-Leibler 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 so-called Kolmogorov-Bode 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 closed-loop 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 “pseudo-metrics” for comparing two dynamical systems. It can be regarded as extending the Kolmogorov-Bode 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 Kullback-Leibler (K-L) rate pseudo-metric is adopted to quantify this additional uncertainty. The utility of the K-L pseudo-metric 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 pseudo-metric leads to the so-called 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 K-L 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 open-loop dynamics are non-ergodic. 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 closed-loop 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.
- Graduation Semester
- 2011-12
- Permalink
- http://hdl.handle.net/2142/29705
- Copyright and License Information
- Copyright 2011 Yu Sun

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