An information theoretic study of modeling and control of dynamical systems

Bode integral formula;hidden Markov model;nonlinear control system;Kullback-Leibler divergence;model reduction;distributed optimization;dynamical systems

Sun, Yu

关键词: Bode integral formula;    hidden Markov model;    nonlinear control system;    Kullback-Leibler divergence;    model reduction;    distributed optimization;    dynamical systems;   

Others  :  https://www.ideals.illinois.edu/bitstream/handle/2142/29705/Sun_Yu.pdf?sequence=1&isAllowed=y

美国|英语

来源: The Illinois Digital Environment for Access to Learning and Scholarship

【 摘 要 】

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 relatesthe 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 certaincases of nonlinear dynamics. An important implication of this study is that the limitations arise due to fundamentalissues 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. Itcan be regarded as extending the Kolmogorov-Bode formula for model comparison and robustness analysis. Centralto the considerations here is the notion of uncertainty in the model: the comparisons are made in terms of additionaluncertainty 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 rangeof model reduction and model selection problems are demonstrated by examples. It is shown that model reduction ofnonlinear system using this pseudo-metric leads to the so-called optimal prediction model. For the particular case oflinear 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 termsof the average cost of an infinite horizon optimal control problem. Unlike usual optimal control problem, the controlcost here is defined by a certain K-L divergence metric. Under this cost structure, the limitations can be obtainedvia 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 thelimitation depend upon the unstable eigenvalues, as in the classical Bode formula. For more general class of nonlinearsystems 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 anddynamics, 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 distributedestimation and the other is about convergence analysis of a distributed optimization algorithm.