An information-theoretic analysis of control and filtering limitations by the I-MMSE relationships

Wan, Neng

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https://hdl.handle.net/2142/124129 Copy

Description

Title

An information-theoretic analysis of control and filtering limitations by the I-MMSE relationships

Author(s)

Wan, Neng

Issue Date

2024-01-09

Director of Research (if dissertation) or Advisor (if thesis)

Hovakimyan, Naira

Doctoral Committee Chair(s)

Hovakimyan, Naira

Committee Member(s)

• Mehta, Prashant
• Shamma, Jeff S.
• Srikant, Rayadurgam

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)

• Control and filtering trade-offs
• Fundamental limitations
• Information-theoretic method
• I-MMSE relationships
• Stochastic control and filtering
• Optimal estimation

Abstract

Fundamental limitations in control and filtering, as the trade-offs between the pursuit of performance and the constraints in the design, have been providing guidelines to the practitioners for the design of controllers and filters in a broad range of applications. Some classical control and filtering trade-offs include Bode's sensitivity integral or the waterbed effect in control and the minimum mean-square error in estimation. By modeling control and filtering systems as noisy communication channels, information-theoretic metrics, such as total information (rate), data transmission rate, and differential entropy (rate) difference, can be also utilized to characterize and quantify the fundamental limitations. Previously, complex analysis and the information-theoretic method based on (differential) entropy are two primal tools for calculating and studying these trade-off metrics. However, both methods can only handle a specific type of systems and are not general enough. In this dissertation, by resorting to the I-MMSE (mutual information - minimum mean-square error) relationships, a comprehensive and estimation-based information-theoretic method is proposed to uniformly calculate and analyze total information (rate), i.e., the mutual information (rate) between transmitted message and channel output, or equivalently, the directed information (rate) from channel input to output, as a trade-off metric that captures the fundamental limitations of numerous control and filtering systems in continuous-time, discrete-time, and infinite-dimensional scenarios. In the continuous-time scenario, with the aid of continuous-time I-MMSE relationships and optimal filtering techniques, we relate total information (rate) to the noise sensitivity trade-offs, (average) performance cost, and lowest achievable mean-square estimation error of various continuous-time control and filtering systems in LTI (linear time-invariant), LTV (linear time-varying), and nonlinear forms. In the discrete-time scenario, we first derive the I-MMSE relationships of discrete-time additive white Gaussian channels with and without feedback. With these relationships and optimal filtering techniques, we then revisit and extend the trade-off properties and interpretations of total information (rate) in discrete-time LTI, LTV, and nonlinear control and filtering systems. Lastly, based on the I-MMSE relationships in abstract Wiener space, a direct and estimation-based scheme is proposed to calculate the total information (rate) of infinite-dimensional control and filtering systems under additive arbitrary Gaussian noise. By applying this scheme to the finite-dimensional systems, we not only recover some well-established fundamental limitations in the control and filtering systems subject to white Gaussian noise, but provide a direct scheme to calculate and study the total information (rate) in some complicated and rarely investigated systems under colored Gaussian noise.

Graduation Semester

2024-05

Type of Resource

Thesis

Copyright and License Information

Copyright 2024 Neng Wan

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