A control system viewpoint for manipulating heat equation and maximum entropy principle for design of experiments

This thesis has two parts where we present our solutions to two different problems. In the first part, we estimate and manipulate the effective thermal properties of a one-dimensional rod through feedback control methods. In the second part of this work, we consider an unrelated problem of the design of experiments with imputable features. In particular, in the first part of this work, we use the central difference method and basis function approach to discretize heat equation into a linear time invariant (LTI) system. This aids in borrowing control theoretic approaches to manipulate the thermal properties of materials. In this regard, we demonstrate a robust optimal control approach to modify the temperature profile of a one-dimensional rod to behave as another reference material. We propose the utilization of closed-loop feedback control techniques leading to robust matching compared to traditionally used open-loop methods such as 3-omega. Further, we present conditions necessary for the estimation of multiple thermal conductivities of a non-homogeneous material. Simulations show that we could successfully increase the bandwidth of the system by 100% through optimal robust control and under full state feedback, we could estimate the spatially varying thermal conductivities with an average of 99.73% accuracy. In the second part of this work, we also explore the maximum entropy-based approach for optimum feature data imputation and selection of experiments. Tactical selection of experiments to estimate an underlying model is an innate task across various fields. Since each experiment has costs associated with it, selecting statistically significant experiments becomes necessary. Classic linear experimental design deals with experiment selection so as to minimize (functions of) variance in the estimation of regression parameters. Typically, standard algorithms for solving this problem assume that data associated with each experiment is fully known. This is not often true since missing data is a common problem. For instance, remote sensors often miss data due to poor connection. Hence experiment selection under such scenarios is a widespread but challenging task. Though decoupling the tasks and using standard data imputation methods like matrix completion followed by experiment selection might seem a way forward, they perform sub-optimally since the tasks are naturally interdependent. The standard design of experiments is an NP hard problem, and the additional objective of imputing for missing data amplifies the computational complexity. In this paper, we propose a maximum-entropy-principle-based framework that simultaneously addresses the problem of the design of experiments as well as the imputation of missing data. Our algorithm exploits homotopy from a suitably chosen convex function to the non-convex cost function; hence avoiding poor local minima. Further, our proposed framework is flexible to incorporate additional application-specific constraints. Simulations on various datasets show improvement in the cost value by over 60% in comparison to benchmark algorithms applied sequentially to the imputation and experiment selection problems.

Name thesis:degree_name

M.S.

Level thesis:degree_level

Thesis

Discipline thesis:degree_discipline

Mechanical Engineering

Grantor

University of Illinois at Urbana-Champaign

Year dc:date

2021

Author dc:creator

• Velicheti, Raj Kiriti

Contributors dc:contributor

• Salapaka, Srinivasa

• Robust optimal control
• Combinatorial optimization
• Maximum entropy principle
• Parameter estimation
• heat equation

Statement dc:rights

• Copyright 2021 Raj Kiriti Velicheti

Language dc:language

en

Handle dc:identifier

http://hdl.handle.net/2142/110745

OAI identifier oai:identifier

oai:www.ideals.illinois.edu:2142/110745