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403 lines
13 KiB
TeX
403 lines
13 KiB
TeX
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% Beamer Presentation
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% LaTeX Template
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% Version 1.0 (10/11/12)
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%
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% This template has been downloaded from:
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% http://www.LaTeXTemplates.com
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%
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% License:
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% CC BY-NC-SA 3.0 (http://creativecommons.org/licenses/by-nc-sa/3.0/)
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% Changed theme to WSU by William King
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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%----------------------------------------------------------------------------------------
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% PACKAGES AND THEMES
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%----------------------------------------------------------------------------------------
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\documentclass[xcolor=dvipsnames,aspectratio=169]{beamer}
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%Import Preamble bits
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\input{../Assets/preambles/FormattingPreamble.tex}
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\input{../Assets/preambles/TikzitPreamble.tex}
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\input{../Assets/preambles/MathPreamble.tex}
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\input{../Assets/preambles/BibPreamble.tex}
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\input{../Assets/preambles/GeneralPreamble.tex}
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%----------------------------------------------------------------------------------------
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% TITLE PAGE
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%----------------------------------------------------------------------------------------
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\title[MDP Constellations]{Modeling decisions in operating satellite constellations}
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\author{Will King} % Your name
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\institute[WSU] % Your institution as it will appear on the bottom of every slide, may be shorthand to save space
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{
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Washington State University \\ % Your institution for the title page
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\medskip
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\textit{william.f.king@wsu.edu} % Your email address
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}
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\date{\today} % Date, can be changed to a custom date
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\begin{document}
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\begin{frame}
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\titlepage % Print the title page as the first slide
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\end{frame}
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\begin{frame}[allowframebreaks] %Allow frame breaks
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\frametitle{Overview} % Table of contents slide, comment this out to remove it
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\tableofcontents
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%Planned TOC
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% See ../outline2.txt
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\end{frame}
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%-------------------------------------------------------------------------------------
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%%%%%%%%%%%%%%%%%%%% Developing the Model%%%%%%%%%%%%%%%%%%%%%%%%
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\section{Background}
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% Why should we care?
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% Uses of space
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% Pollution in space and it's impacts
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% Kessler Syndrome
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% What is different now
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%
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%
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%
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%-------------------------------------------------------------------------------------
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%-------------------------------------------------------------------------------------
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%%%%%%%%%%%%%%%%%%%% Developing the Model%%%%%%%%%%%%%%%%%%%%%%%%
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\section{Literature}
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%-------------------------------------------------------------------------------------
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\begin{frame}
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\frametitle{Past Literature}
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\begin{enumerate}
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\item Kessler and Cour-Palais: Raised issue of orbital pollution
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\item Adilov et al: Described 2 period salop model of interactions
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\item Adilov et al: Described an infinite period model with symmetric competitive interactions.
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\item Rao and Rondina: Describe a symetric infinite period model (first to do so).
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\end{enumerate}
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\end{frame}
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%-------------------------------
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\begin{frame}
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\frametitle{}
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\end{frame}
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%-------------------------------
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%-------------------------------------------------------------------------------------
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%%%%%%%%%%%%%%%%%%%% Developing the Model%%%%%%%%%%%%%%%%%%%%%%%%
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\section{Model}
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%-------------------------------------------------------------------------------------
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\begin{frame}
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\frametitle{Overview}
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\begin{itemize}
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\item Mathematical Notation
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\item Law of motion for debris
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\item Law of motion for satellite stocks
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\item Kessler Syndrome
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\item Markov Decision Problems
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\end{itemize}
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\end{frame}
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%-------------------------------
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\begin{frame}
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\frametitle{Mathematical Notation}
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\begin{itemize}
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\item $\{s^j_t\}$: The set of constellation satellites stocks
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\item $D_t$: The level of debris
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\item $x^j_t$: The launches from each constellation
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\item
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\end{itemize}
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\end{frame}
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%-------------------------------
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%------------------------------------
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\subsection{Laws of Motion}
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%------------------------------------
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\begin{frame}
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\frametitle{Debris}
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Law of motion for debris
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\begin{align}
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D_{t+1} =& (1-\delta)D_t \tag{Debris decay.} \\
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&+ g\cdot D_t \tag{Debris produced by collision with debris.} \\
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&+ \gamma \sum^N_{i=1} \left(1-R^i(\{s^j_t\},D_t) \right) s^i_t \tag{Debris produced by satellite destruction.} \\
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&+ \Gamma \sum^n_{j=1} \{x^j_t\} \tag{Debris produced by launches.}
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\end{align}
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\end{frame}
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%-------------------------------
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\begin{frame}
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\frametitle{Satellite Stocks}
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\end{frame}
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%-------------------------------
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%------------------------------------
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\subsection{Kessler Syndrome}
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%------------------------------------
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\begin{frame}
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\frametitle{Explanation of Kessler Syndrome}
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\begin{block}{Kessler Syndrome}
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The situation in which collisions between
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objects in orbit produced debris and this debris begins collisions
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with other objects, leading to a runaway growth in debris.
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As debris can persist for millenia, this may make some orbits unusable.
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\autocite{Kessler1978}
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\end{block}
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Often described as a condition with an exponential growth of debris.
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Similar to what is expected with Climate Change
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\end{frame}
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%-------------------------------
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\begin{frame}
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\frametitle{Past approaches to Kessler Syndrome}
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\begin{itemize}
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\item \cite{Adilov2018}:\\
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Develops an analog of kessler syndrome where the condition is met when satellites
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are destroyed immediately after launch by debris.
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\begin{align}
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\left\{ (\{s^j_t\},D_t) : R^i(\{s^j_t\},D_t) = 0 \forall i\right\}
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\end{align}
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\item \cite{RaoRondina2020}:\\
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A working paper in which the authors develop a dynamic model and a definition of
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kessler syndrome that captures all increasing debris levels.
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\begin{align}
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\left\{ (\{s^j_t\},D_t) :
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\lim_{t\rightarrow \infty} D_{t+1}(\{s^j_t\},D_t) = \infty
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\right\}
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\end{align}
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\end{itemize}
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\end{frame}
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%-------------------------------
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\begin{frame}
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\frametitle{My contributions}
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I propose two refinements of these definitions to
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simplify analyzing kessler syndrome in computational models.
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\begin{itemize}
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\item $\epsilon$-Kessler Region
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\item Proto Kessler Region
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\end{itemize}
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\end{frame}
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%-------------------------------------------------------
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\begin{frame}
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\frametitle{$\epsilon$-Kessler Region}
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\begin{block}{$\epsilon$-Kessler Region}
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\begin{align}
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\kappa = \left\{ \left(\{s^j_t\}, D_t \right) : \forall k \geq 0,
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~~ D_{t+k+1} - D_{t+k} \geq \epsilon > 0 \right\}
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\end{align}
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\end{block}
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Notable Features
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\begin{itemize}
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\item $\epsilon$ can be calibrated to capture only economically significant growth.
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\item Requires an explicit description of what is considered economically significant.
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\item Guarantees divergent behavior.
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\item Simulated transition paths can identify the region.
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\end{itemize}
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\end{frame}
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%-------------------------------------------------------
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\begin{frame}
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\frametitle{Proto Kessler Region}
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\begin{block}{Proto Kessler Region}
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\begin{align}
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\left\{
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\left(\{s^j_t\},D_t \right) : ~~ D_{t+1} - D_{t} \geq \epsilon_\text{proto}
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\right\}
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\end{align}
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\end{block}
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Notable Features
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\begin{itemize}
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\item $\epsilon_\text{proto}$ can be calibrated to capture only economically significant growth.
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\item Requires an explicit description of what is considered economically significant.
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\item Does not guarantee divergent behavior.
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\item Easily computable kessler regions.
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\end{itemize}
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\end{frame}
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%-------------------------------------------------------
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\begin{frame}
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\frametitle{Proto Kessler Region}
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With the given law of motion for debris, the proto-kessler region is:
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\begin{align}
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\left\{
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\left(S_t,D_t \right) :
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(g-\delta) D_t
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+ \gamma \sum^n_{i=1} 1-R^i(S_t,D_t)
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+ \Gamma \sum^n_{i=1} x^i_t(S_t,D_t)
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\geq \epsilon_\text{proto}
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\right\}
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\end{align}
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\end{frame}
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%------------------------------------
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\subsection{Markov Decision Problem Formulation}
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%------------------------------------
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\begin{frame}
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\frametitle{Operator's Problem}
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\end{frame}
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%-------------------------------
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\begin{frame}
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\frametitle{Benefit Functions}
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Possible profit functions
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\begin{itemize}
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\item Linear (Currently working on this one)
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\item Cournot Profits
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\item Profits under Partial substitutability
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\item Military capabilities (Keeping up with the Jones')
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\end{itemize}
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\end{frame}
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%-------------------------------
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\begin{frame}
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\frametitle{Planner's Problem}
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\end{frame}
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%-------------------------------
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\begin{frame}
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\frametitle{Planned model expansions}
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Multiple interacting orbital shells and debris terms.
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\end{frame}
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%-------------------------------
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%-------------------------------------------------------------------------------------
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%%%%%%%%%%%%%%%%%%%% Developing the Model%%%%%%%%%%%%%%%%%%%%%%%%
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\section{Analysis}
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%-------------------------------------------------------------------------------------
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\begin{frame}
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\frametitle{Issues}
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The following issues characterize the Operators' or Planner's problem
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\begin{itemize}
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\item Curse of Dimensionality
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\item Unsure of state space, no idea what equilibrium (if it exists) would look like.
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\item Strategic Interaction (Computability issues)
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\end{itemize}
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\end{frame}
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%-------------------------------
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\begin{frame}
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\frametitle{Possible approaches}
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Possible approaches
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\begin{itemize}
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\item Standard VFI/Howards algorithm.
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\item VFI with sparse state space and other computational benefits.
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\item Maliar et al's approachs to supervised learning.
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\item Reinforcement Learning.
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\end{itemize}
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\end{frame}
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%-------------------------------
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\begin{frame}
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\frametitle{Chosen approach}
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% maliar et al - Bellman Residuals
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% present basic approach (using my notation)
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% Discuss basic training loop (use sutton and bartos term Generalized Policy Iteration)
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Bellman Residual minimization \cref{MALIAR2018}
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Use NN to approximate $V(S_t,D_t|\theta_1)$ and $X(X_t,D_t|\theta_2)$.
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The loss function is:
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\begin{align}
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%0 =& \left[
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% V(S_t, D_t) - F(S_t, D_t, X_t)- \beta V(S_{t+1}, D_{t+1})
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% \right]^2
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% - v \left[
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% F(S_t, D_t, X_t) + \beta V(S_{t+1}, D_{t+1})
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% \right] \\
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0 =& \left[
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V(S_t, D_t) - F(S_t, D_t, X_t)- \beta V(S_{t+1}, D_{t+1})
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- \frac{v}{2}
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\right]^2
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- v \left[
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V(S_{t}, D_{t}) + \frac{v}{4}
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\right]
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\end{align}
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Iterate between Value and Policy function estimation.
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\end{frame}
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%-------------------------------
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\begin{frame}
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\frametitle{Training Loop: Planner}
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\end{frame}
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%-------------------------------
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\begin{frame}
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\frametitle{Training Loop: Operators}
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Using a branched policy function and individual value funcitons
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For each training epoch
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Draw random data
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For each operator
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select operator's policy parameters.
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train policy function
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train value function
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\end{frame}
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%-------------------------------
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%------------------------------------
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\subsection{Analysis so far}
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%------------------------------------
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\begin{frame}
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\frametitle{State of the Code}
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Currently functioning
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%Planner training
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%Operator training
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\end{frame}
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%-------------------------------
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\begin{frame}
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\frametitle{Results}
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%simulated debris paths (increase the ammount of debris a couple orders of magnitude)
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%Protokessler region plots?
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\end{frame}
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%-------------------------------
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%-------------------------------------------------------------------------------------
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%%%%%%%%%%%%%%%%%%%% Developing the Model%%%%%%%%%%%%%%%%%%%%%%%%
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\section{Conclusion}
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%-------------------------------------------------------------------------------------
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%-------------------------------
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\begin{frame}
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\frametitle{Summary}
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\end{frame}
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%-------------------------------
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\begin{frame}
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\frametitle{Future Work}
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\end{frame}
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%-------------------------------
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\begin{frame}
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\frametitle{}
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\end{frame}
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%-------------------------------
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\begin{frame}
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\frametitle{Questions?}
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\end{frame}
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%\begin{frame}
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% \frametitle{MarginalRevenue}
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% \begin{figure}
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% \tikzfig{../Assets/owned/ch8_MarginalRevenue}
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% \includegraphics[height=\textheight]{../Assets/copyrighted/KrugmanObsterfeldMeliz_fig8-7.jpg}
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% \label{FIG:costs}
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% \caption{Average Cost Curve as firms enter.}
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% \end{figure}
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%\end{frame}
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\end{document}
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% \begin{frame}
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% \frametitle{Columns}
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% \begin{columns}
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% \begin{column}{0.5\textwidth}
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% \end{column}
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% \begin{column}{0.5\textwidth}
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% \begin{figure}
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% \tikzfig{../Assets/owned/ch7_EstablishedAdvantageExample2}
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% \label{FIG:costs}
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% \caption{Setting the Stage}
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% \end{figure}
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% \end{column}
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% \end{columns}
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% \end{frame}
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% %---------------------------------------------------------------
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