Outlined presentation and began work on it.
youainti
4 years ago
parent
56a89717ab
commit
f5447b116c
@ -0,0 +1,10 @@
|
|||||||
|
%----------------------------------------------------------------------------------------
|
||||||
|
% BIBLIOGRAPHY SETUP
|
||||||
|
%----------------------------------------------------------------------------------------
|
||||||
|
|
||||||
|
|
||||||
|
%%% Setup Bibliography
|
||||||
|
\usepackage[backend=biber,style=apa,autocite=inline]{biblatex}
|
||||||
|
\addbibresource{../Assets/preambles/References.bib}
|
||||||
|
|
||||||
|
|
||||||
@ -0,0 +1,38 @@
|
|||||||
|
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
||||||
|
%
|
||||||
|
% Based on a template downloaded from:
|
||||||
|
% http://www.LaTeXTemplates.com
|
||||||
|
%
|
||||||
|
% License:
|
||||||
|
% CC BY-NC-SA 3.0 (http://creativecommons.org/licenses/by-nc-sa/3.0/)
|
||||||
|
%
|
||||||
|
% Changed theme to WSU by William King
|
||||||
|
%
|
||||||
|
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
||||||
|
|
||||||
|
%----------------------------------------------------------------------------------------
|
||||||
|
% COLORS AND THEMES
|
||||||
|
%----------------------------------------------------------------------------------------
|
||||||
|
%%% Setup color
|
||||||
|
\usetheme{Berkeley}
|
||||||
|
\definecolor{WSUred}{RGB}{152,30,50}
|
||||||
|
\definecolor{WSUgrey}{RGB}{94,106,113}
|
||||||
|
\setbeamercolor{palette primary}{bg=WSUred,fg=white}
|
||||||
|
\setbeamercolor{palette secondary}{bg=WSUred,fg=white}
|
||||||
|
\setbeamercolor{palette tertiary}{bg=WSUred,fg=white}
|
||||||
|
\setbeamercolor{palette quaternary}{bg=WSUred,fg=white}
|
||||||
|
\setbeamercolor{structure}{fg=WSUgrey} % itemize, enumerate, etc
|
||||||
|
\setbeamercolor{section in toc}{fg=WSUred} % TOC sections
|
||||||
|
\setbeamercolor{block body}{fg=WSUred} % block body sections
|
||||||
|
|
||||||
|
|
||||||
|
%\setbeamertemplate{footline} % To remove the footer line in all slides uncomment this line
|
||||||
|
\setbeamertemplate{footline}[page number] % To replace the footer line in all slides with a simple slide count uncomment this line
|
||||||
|
\setbeamertemplate{navigation symbols}{} % To remove the navigation symbols from the bottom of all slides uncomment this line
|
||||||
|
|
||||||
|
|
||||||
|
%%% setup packages
|
||||||
|
\usepackage{graphicx} % Allows including images
|
||||||
|
\graphicspath{{./img/}} %Set a graphics path
|
||||||
|
\usepackage{booktabs} % Allows the use of \toprule, \midrule and \bottomrule in tables
|
||||||
|
|
||||||
@ -0,0 +1,16 @@
|
|||||||
|
%Include other generally important packages
|
||||||
|
|
||||||
|
\usepackage{hyperref} % Allows for weblinks
|
||||||
|
\hypersetup{
|
||||||
|
colorlinks=true,
|
||||||
|
citebordercolor=WSUgrey,
|
||||||
|
citecolor=WSUred,
|
||||||
|
linkcolor=WSUred,
|
||||||
|
urlcolor=Blue
|
||||||
|
}
|
||||||
|
|
||||||
|
|
||||||
|
\usepackage{cleveref}
|
||||||
|
|
||||||
|
%Add algorithms
|
||||||
|
\usepackage{algorithm,algpseudocode}
|
||||||
@ -0,0 +1,20 @@
|
|||||||
|
|
||||||
|
%%%%%%%%%Packages%%%%%%%%%%%%%%%
|
||||||
|
\usepackage{amsmath}
|
||||||
|
\usepackage{mathtools}
|
||||||
|
\usepackage{amsthm}
|
||||||
|
\usepackage{amssymb}
|
||||||
|
\usepackage{thmtools, thm-restate}
|
||||||
|
|
||||||
|
|
||||||
|
%%%%%%%%%%%% MATH FORMATTING %%%%%%%%%%%%%%%%%%%%%
|
||||||
|
%Helpful bits
|
||||||
|
\newcommand{\bb}[1]{\mathbb{#1}}
|
||||||
|
|
||||||
|
%Derivatives etc.
|
||||||
|
\newcommand{\parder}[3]{\ensuremath{ \frac{\partial^{#3} #1}{\partial #2~^{#3}}}}
|
||||||
|
\newcommand{\der}[3]{\ensuremath{ \frac{d^{#3} #1}{d #2~^{#3}}}}
|
||||||
|
|
||||||
|
%Math Operators
|
||||||
|
\DeclareMathOperator{\argmax}{argmax}
|
||||||
|
\DeclareMathOperator{\argmin}{argmin}
|
||||||
@ -0,0 +1,12 @@
|
|||||||
|
%-----------------------------------------------------------
|
||||||
|
% Tikzit Preamble
|
||||||
|
%-----------------------------------------------------------
|
||||||
|
|
||||||
|
%Setup graphing
|
||||||
|
\usepackage{tikz}
|
||||||
|
\usepackage{tikzit}
|
||||||
|
\input{../Assets/preambles/WSU_Econ.tikzstyles}
|
||||||
|
%\input{../Assets/preambles/tikzit.sty}
|
||||||
|
|
||||||
|
|
||||||
|
|
||||||
@ -0,0 +1,32 @@
|
|||||||
|
% TiKZ style file generated by TikZiT. You may edit this file manually,
|
||||||
|
% but some things (e.g. comments) may be overwritten. To be readable in
|
||||||
|
% TikZiT, the only non-comment lines must be of the form:
|
||||||
|
% \tikzstyle{NAME}=[PROPERTY LIST]
|
||||||
|
|
||||||
|
% Node styles
|
||||||
|
\tikzstyle{CrimsonNode}=[fill={rgb,255: red,152; green,30; blue,50}, draw={rgb,255: red,152; green,30; blue,50}, shape=circle, tikzit category=WSU, tikzit draw={rgb,255: red,152; green,30; blue,50}, tikzit fill={rgb,255: red,152; green,30; blue,50}]
|
||||||
|
\tikzstyle{GreyNode}=[fill={rgb,255: red,94; green,106; blue,113}, draw={rgb,255: red,94; green,106; blue,113}, shape=circle, tikzit category=WSU, tikzit draw={rgb,255: red,94; green,106; blue,113}, tikzit fill={rgb,255: red,94; green,106; blue,113}]
|
||||||
|
\tikzstyle{Box}=[fill={rgb,255: red,94; green,106; blue,113}, draw={rgb,255: red,94; green,106; blue,113}, shape=rectangle, tikzit draw={rgb,255: red,94; green,106; blue,113}, tikzit fill={rgb,255: red,94; green,106; blue,113}]
|
||||||
|
\tikzstyle{Red Box}=[fill={rgb,255: red,152; green,30; blue,50}, draw={rgb,255: red,152; green,30; blue,50}, shape=rectangle]
|
||||||
|
\tikzstyle{new style 0}=[fill=white, draw=black, shape=circle, tikzit draw=black]
|
||||||
|
\tikzstyle{new style 1}=[fill={rgb,255: red,128; green,0; blue,128}, draw=black, shape=circle]
|
||||||
|
\tikzstyle{Box}=[fill=white, draw=black, shape=rectangle]
|
||||||
|
\tikzstyle{rotated text}=[fill=none, draw=none, shape=circle, rotate=270, tikzit draw={rgb,255: red,191; green,191; blue,191}]
|
||||||
|
|
||||||
|
% Edge styles
|
||||||
|
\tikzstyle{RightArrow}=[->]
|
||||||
|
\tikzstyle{LeftRightArrow}=[<->]
|
||||||
|
\tikzstyle{CrimsonBar}=[-, draw={rgb,255: red,152; green,30; blue,50}]
|
||||||
|
\tikzstyle{GreyBar}=[-, draw={rgb,255: red,94; green,106; blue,113}, tikzit draw={rgb,255: red,94; green,106; blue,113}]
|
||||||
|
\tikzstyle{divider}=[draw={rgb,255: red,64; green,64; blue,64}, dashed, dash pattern=on 2mm off 1, -]
|
||||||
|
\tikzstyle{bars}=[{|-|}]
|
||||||
|
\tikzstyle{Dashed}=[-, dashed, dash pattern=on 1mm off 2mm, tikzit draw={rgb,255: red,128; green,128; blue,128}]
|
||||||
|
\tikzstyle{Light Arrow}=[->, draw={rgb,255: red,191; green,191; blue,191}]
|
||||||
|
\tikzstyle{lightgreybar}=[-, draw={rgb,255: red,191; green,191; blue,191}]
|
||||||
|
\tikzstyle{lightred}=[-, draw={rgb,255: red,222; green,148; blue,178}]
|
||||||
|
\tikzstyle{Purple}=[-, draw={rgb,255: red,128; green,0; blue,128}, tikzit draw={rgb,255: red,128; green,0; blue,128}, line width=1mm]
|
||||||
|
\tikzstyle{new edge style 1}=[draw={rgb,255: red,121; green,23; blue,40}, ->]
|
||||||
|
\tikzstyle{filled2}=[-, fill={rgb,255: red,255; green,191; blue,191}, draw=black, tikzit draw=black, tikzit fill={rgb,255: red,255; green,191; blue,191}, opacity=0.5]
|
||||||
|
\tikzstyle{filled1}=[-, fill={rgb,255: red,191; green,191; blue,191}, draw=black, tikzit draw=black, opacity=0.5, tikzit fill={rgb,255: red,191; green,191; blue,191}]
|
||||||
|
\tikzstyle{emptyFill1}=[-, fill={rgb,255: red,255; green,191; blue,191}, draw=none, tikzit draw=blue, opacity=0.3]
|
||||||
|
\tikzstyle{new edge style 0}=[-, draw=none, fill={rgb,255: red,191; green,191; blue,191}, tikzit draw=green, opacity=0.3, tikzit fill={rgb,255: red,191; green,191; blue,191}]
|
||||||
@ -0,0 +1,42 @@
|
|||||||
|
\usepackage{tikz}
|
||||||
|
\usetikzlibrary{backgrounds}
|
||||||
|
\usetikzlibrary{arrows}
|
||||||
|
\usetikzlibrary{shapes,shapes.geometric,shapes.misc}
|
||||||
|
|
||||||
|
% this style is applied by default to any tikzpicture included via \tikzfig
|
||||||
|
\tikzstyle{tikzfig}=[baseline=-0.25em,scale=0.5]
|
||||||
|
|
||||||
|
% these are dummy properties used by TikZiT, but ignored by LaTex
|
||||||
|
\pgfkeys{/tikz/tikzit fill/.initial=0}
|
||||||
|
\pgfkeys{/tikz/tikzit draw/.initial=0}
|
||||||
|
\pgfkeys{/tikz/tikzit shape/.initial=0}
|
||||||
|
\pgfkeys{/tikz/tikzit category/.initial=0}
|
||||||
|
|
||||||
|
% standard layers used in .tikz files
|
||||||
|
\pgfdeclarelayer{edgelayer}
|
||||||
|
\pgfdeclarelayer{nodelayer}
|
||||||
|
\pgfsetlayers{background,edgelayer,nodelayer,main}
|
||||||
|
|
||||||
|
% style for blank nodes
|
||||||
|
\tikzstyle{none}=[inner sep=0mm]
|
||||||
|
|
||||||
|
% include a .tikz file
|
||||||
|
\newcommand{\tikzfig}[1]{%
|
||||||
|
{\tikzstyle{every picture}=[tikzfig]
|
||||||
|
\IfFileExists{#1.tikz}
|
||||||
|
{\input{#1.tikz}}
|
||||||
|
{%
|
||||||
|
\IfFileExists{./figures/#1.tikz}
|
||||||
|
{\input{./figures/#1.tikz}}
|
||||||
|
{\tikz[baseline=-0.5em]{\node[draw=red,font=\color{red},fill=red!10!white] {\textit{#1}};}}%
|
||||||
|
}}%
|
||||||
|
}
|
||||||
|
|
||||||
|
% the same as \tikzfig, but in a {center} environment
|
||||||
|
\newcommand{\ctikzfig}[1]{%
|
||||||
|
\begin{center}\rm
|
||||||
|
\tikzfig{#1}
|
||||||
|
\end{center}}
|
||||||
|
|
||||||
|
% fix strange self-loops, which are PGF/TikZ default
|
||||||
|
\tikzstyle{every loop}=[]
|
||||||
@ -0,0 +1,402 @@
|
|||||||
|
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
||||||
|
% Beamer Presentation
|
||||||
|
% LaTeX Template
|
||||||
|
% Version 1.0 (10/11/12)
|
||||||
|
%
|
||||||
|
% This template has been downloaded from:
|
||||||
|
% http://www.LaTeXTemplates.com
|
||||||
|
%
|
||||||
|
% License:
|
||||||
|
% CC BY-NC-SA 3.0 (http://creativecommons.org/licenses/by-nc-sa/3.0/)
|
||||||
|
%
|
||||||
|
% Changed theme to WSU by William King
|
||||||
|
%
|
||||||
|
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
||||||
|
|
||||||
|
%----------------------------------------------------------------------------------------
|
||||||
|
% PACKAGES AND THEMES
|
||||||
|
%----------------------------------------------------------------------------------------
|
||||||
|
|
||||||
|
\documentclass[xcolor=dvipsnames,aspectratio=169]{beamer}
|
||||||
|
|
||||||
|
|
||||||
|
%Import Preamble bits
|
||||||
|
\input{../Assets/preambles/FormattingPreamble.tex}
|
||||||
|
\input{../Assets/preambles/TikzitPreamble.tex}
|
||||||
|
\input{../Assets/preambles/MathPreamble.tex}
|
||||||
|
\input{../Assets/preambles/BibPreamble.tex}
|
||||||
|
\input{../Assets/preambles/GeneralPreamble.tex}
|
||||||
|
|
||||||
|
|
||||||
|
|
||||||
|
|
||||||
|
|
||||||
|
%----------------------------------------------------------------------------------------
|
||||||
|
% TITLE PAGE
|
||||||
|
%----------------------------------------------------------------------------------------
|
||||||
|
|
||||||
|
\title[MDP Constellations]{Modeling decisions in operating satellite constellations}
|
||||||
|
|
||||||
|
\author{Will King} % Your name
|
||||||
|
\institute[WSU] % Your institution as it will appear on the bottom of every slide, may be shorthand to save space
|
||||||
|
{
|
||||||
|
Washington State University \\ % Your institution for the title page
|
||||||
|
\medskip
|
||||||
|
\textit{william.f.king@wsu.edu} % Your email address
|
||||||
|
}
|
||||||
|
\date{\today} % Date, can be changed to a custom date
|
||||||
|
|
||||||
|
|
||||||
|
|
||||||
|
|
||||||
|
|
||||||
|
\begin{document}
|
||||||
|
\begin{frame}
|
||||||
|
\titlepage % Print the title page as the first slide
|
||||||
|
\end{frame}
|
||||||
|
|
||||||
|
|
||||||
|
\begin{frame}[allowframebreaks] %Allow frame breaks
|
||||||
|
\frametitle{Overview} % Table of contents slide, comment this out to remove it
|
||||||
|
\tableofcontents
|
||||||
|
%Planned TOC
|
||||||
|
% See ../outline2.txt
|
||||||
|
\end{frame}
|
||||||
|
|
||||||
|
%-------------------------------------------------------------------------------------
|
||||||
|
%%%%%%%%%%%%%%%%%%%% Developing the Model%%%%%%%%%%%%%%%%%%%%%%%%
|
||||||
|
\section{Background}
|
||||||
|
% Why should we care?
|
||||||
|
% Uses of space
|
||||||
|
% Pollution in space and it's impacts
|
||||||
|
% Kessler Syndrome
|
||||||
|
% What is different now
|
||||||
|
%
|
||||||
|
%
|
||||||
|
%
|
||||||
|
%-------------------------------------------------------------------------------------
|
||||||
|
|
||||||
|
%-------------------------------------------------------------------------------------
|
||||||
|
%%%%%%%%%%%%%%%%%%%% Developing the Model%%%%%%%%%%%%%%%%%%%%%%%%
|
||||||
|
\section{Literature}
|
||||||
|
%-------------------------------------------------------------------------------------
|
||||||
|
\begin{frame}
|
||||||
|
\frametitle{Past Literature}
|
||||||
|
\begin{enumerate}
|
||||||
|
\item Kessler and Cour-Palais: Raised issue of orbital pollution
|
||||||
|
\item Adilov et al: Described 2 period salop model of interactions
|
||||||
|
\item Adilov et al: Described an infinite period model with symmetric competitive interactions.
|
||||||
|
\item Rao and Rondina: Describe a symetric infinite period model (first to do so).
|
||||||
|
\end{enumerate}
|
||||||
|
\end{frame}
|
||||||
|
%-------------------------------
|
||||||
|
\begin{frame}
|
||||||
|
\frametitle{}
|
||||||
|
\end{frame}
|
||||||
|
%-------------------------------
|
||||||
|
%-------------------------------------------------------------------------------------
|
||||||
|
%%%%%%%%%%%%%%%%%%%% Developing the Model%%%%%%%%%%%%%%%%%%%%%%%%
|
||||||
|
\section{Model}
|
||||||
|
%-------------------------------------------------------------------------------------
|
||||||
|
\begin{frame}
|
||||||
|
\frametitle{Overview}
|
||||||
|
|
||||||
|
\begin{itemize}
|
||||||
|
\item Mathematical Notation
|
||||||
|
\item Law of motion for debris
|
||||||
|
\item Law of motion for satellite stocks
|
||||||
|
\item Kessler Syndrome
|
||||||
|
\item Markov Decision Problems
|
||||||
|
\end{itemize}
|
||||||
|
\end{frame}
|
||||||
|
%-------------------------------
|
||||||
|
\begin{frame}
|
||||||
|
\frametitle{Mathematical Notation}
|
||||||
|
\begin{itemize}
|
||||||
|
\item $\{s^j_t\}$: The set of constellation satellites stocks
|
||||||
|
\item $D_t$: The level of debris
|
||||||
|
\item $x^j_t$: The launches from each constellation
|
||||||
|
\item
|
||||||
|
\end{itemize}
|
||||||
|
\end{frame}
|
||||||
|
%-------------------------------
|
||||||
|
%------------------------------------
|
||||||
|
\subsection{Laws of Motion}
|
||||||
|
%------------------------------------
|
||||||
|
\begin{frame}
|
||||||
|
\frametitle{Debris}
|
||||||
|
Law of motion for debris
|
||||||
|
\begin{align}
|
||||||
|
D_{t+1} =& (1-\delta)D_t \tag{Debris decay.} \\
|
||||||
|
&+ g\cdot D_t \tag{Debris produced by collision with debris.} \\
|
||||||
|
&+ \gamma \sum^N_{i=1} \left(1-R^i(\{s^j_t\},D_t) \right) s^i_t \tag{Debris produced by satellite destruction.} \\
|
||||||
|
&+ \Gamma \sum^n_{j=1} \{x^j_t\} \tag{Debris produced by launches.}
|
||||||
|
\end{align}
|
||||||
|
\end{frame}
|
||||||
|
%-------------------------------
|
||||||
|
\begin{frame}
|
||||||
|
\frametitle{Satellite Stocks}
|
||||||
|
\end{frame}
|
||||||
|
%-------------------------------
|
||||||
|
%------------------------------------
|
||||||
|
\subsection{Kessler Syndrome}
|
||||||
|
%------------------------------------
|
||||||
|
\begin{frame}
|
||||||
|
\frametitle{Explanation of Kessler Syndrome}
|
||||||
|
\begin{block}{Kessler Syndrome}
|
||||||
|
The situation in which collisions between
|
||||||
|
objects in orbit produced debris and this debris begins collisions
|
||||||
|
with other objects, leading to a runaway growth in debris.
|
||||||
|
As debris can persist for millenia, this may make some orbits unusable.
|
||||||
|
|
||||||
|
\autocite{Kessler1978}
|
||||||
|
\end{block}
|
||||||
|
Often described as a condition with an exponential growth of debris.
|
||||||
|
|
||||||
|
Similar to what is expected with Climate Change
|
||||||
|
\end{frame}
|
||||||
|
%-------------------------------
|
||||||
|
\begin{frame}
|
||||||
|
\frametitle{Past approaches to Kessler Syndrome}
|
||||||
|
\begin{itemize}
|
||||||
|
\item \cite{Adilov2018}:\\
|
||||||
|
Develops an analog of kessler syndrome where the condition is met when satellites
|
||||||
|
are destroyed immediately after launch by debris.
|
||||||
|
\begin{align}
|
||||||
|
\left\{ (\{s^j_t\},D_t) : R^i(\{s^j_t\},D_t) = 0 \forall i\right\}
|
||||||
|
\end{align}
|
||||||
|
\item \cite{RaoRondina2020}:\\
|
||||||
|
A working paper in which the authors develop a dynamic model and a definition of
|
||||||
|
kessler syndrome that captures all increasing debris levels.
|
||||||
|
\begin{align}
|
||||||
|
\left\{ (\{s^j_t\},D_t) :
|
||||||
|
\lim_{t\rightarrow \infty} D_{t+1}(\{s^j_t\},D_t) = \infty
|
||||||
|
\right\}
|
||||||
|
\end{align}
|
||||||
|
\end{itemize}
|
||||||
|
\end{frame}
|
||||||
|
%-------------------------------
|
||||||
|
\begin{frame}
|
||||||
|
\frametitle{My contributions}
|
||||||
|
I propose two refinements of these definitions to
|
||||||
|
simplify analyzing kessler syndrome in computational models.
|
||||||
|
\begin{itemize}
|
||||||
|
\item $\epsilon$-Kessler Region
|
||||||
|
\item Proto Kessler Region
|
||||||
|
\end{itemize}
|
||||||
|
\end{frame}
|
||||||
|
%-------------------------------------------------------
|
||||||
|
\begin{frame}
|
||||||
|
\frametitle{$\epsilon$-Kessler Region}
|
||||||
|
\begin{block}{$\epsilon$-Kessler Region}
|
||||||
|
\begin{align}
|
||||||
|
\kappa = \left\{ \left(\{s^j_t\}, D_t \right) : \forall k \geq 0,
|
||||||
|
~~ D_{t+k+1} - D_{t+k} \geq \epsilon > 0 \right\}
|
||||||
|
\end{align}
|
||||||
|
\end{block}
|
||||||
|
|
||||||
|
Notable Features
|
||||||
|
\begin{itemize}
|
||||||
|
\item $\epsilon$ can be calibrated to capture only economically significant growth.
|
||||||
|
\item Requires an explicit description of what is considered economically significant.
|
||||||
|
\item Guarantees divergent behavior.
|
||||||
|
\item Simulated transition paths can identify the region.
|
||||||
|
\end{itemize}
|
||||||
|
\end{frame}
|
||||||
|
%-------------------------------------------------------
|
||||||
|
\begin{frame}
|
||||||
|
\frametitle{Proto Kessler Region}
|
||||||
|
|
||||||
|
\begin{block}{Proto Kessler Region}
|
||||||
|
\begin{align}
|
||||||
|
\left\{
|
||||||
|
\left(\{s^j_t\},D_t \right) : ~~ D_{t+1} - D_{t} \geq \epsilon_\text{proto}
|
||||||
|
\right\}
|
||||||
|
\end{align}
|
||||||
|
\end{block}
|
||||||
|
|
||||||
|
Notable Features
|
||||||
|
\begin{itemize}
|
||||||
|
\item $\epsilon_\text{proto}$ can be calibrated to capture only economically significant growth.
|
||||||
|
\item Requires an explicit description of what is considered economically significant.
|
||||||
|
\item Does not guarantee divergent behavior.
|
||||||
|
\item Easily computable kessler regions.
|
||||||
|
\end{itemize}
|
||||||
|
\end{frame}
|
||||||
|
%-------------------------------------------------------
|
||||||
|
\begin{frame}
|
||||||
|
\frametitle{Proto Kessler Region}
|
||||||
|
With the given law of motion for debris, the proto-kessler region is:
|
||||||
|
\begin{align}
|
||||||
|
\left\{
|
||||||
|
\left(S_t,D_t \right) :
|
||||||
|
(g-\delta) D_t
|
||||||
|
+ \gamma \sum^n_{i=1} 1-R^i(S_t,D_t)
|
||||||
|
+ \Gamma \sum^n_{i=1} x^i_t(S_t,D_t)
|
||||||
|
\geq \epsilon_\text{proto}
|
||||||
|
\right\}
|
||||||
|
\end{align}
|
||||||
|
\end{frame}
|
||||||
|
%------------------------------------
|
||||||
|
\subsection{Markov Decision Problem Formulation}
|
||||||
|
%------------------------------------
|
||||||
|
\begin{frame}
|
||||||
|
\frametitle{Operator's Problem}
|
||||||
|
\end{frame}
|
||||||
|
%-------------------------------
|
||||||
|
\begin{frame}
|
||||||
|
\frametitle{Benefit Functions}
|
||||||
|
|
||||||
|
Possible profit functions
|
||||||
|
\begin{itemize}
|
||||||
|
\item Linear (Currently working on this one)
|
||||||
|
\item Cournot Profits
|
||||||
|
\item Profits under Partial substitutability
|
||||||
|
\item Military capabilities (Keeping up with the Jones')
|
||||||
|
\end{itemize}
|
||||||
|
|
||||||
|
\end{frame}
|
||||||
|
%-------------------------------
|
||||||
|
\begin{frame}
|
||||||
|
\frametitle{Planner's Problem}
|
||||||
|
\end{frame}
|
||||||
|
%-------------------------------
|
||||||
|
\begin{frame}
|
||||||
|
\frametitle{Planned model expansions}
|
||||||
|
Multiple interacting orbital shells and debris terms.
|
||||||
|
\end{frame}
|
||||||
|
%-------------------------------
|
||||||
|
%-------------------------------------------------------------------------------------
|
||||||
|
%%%%%%%%%%%%%%%%%%%% Developing the Model%%%%%%%%%%%%%%%%%%%%%%%%
|
||||||
|
\section{Analysis}
|
||||||
|
%-------------------------------------------------------------------------------------
|
||||||
|
\begin{frame}
|
||||||
|
\frametitle{Issues}
|
||||||
|
The following issues characterize the Operators' or Planner's problem
|
||||||
|
\begin{itemize}
|
||||||
|
\item Curse of Dimensionality
|
||||||
|
\item Unsure of state space, no idea what equilibrium (if it exists) would look like.
|
||||||
|
\item Strategic Interaction (Computability issues)
|
||||||
|
\end{itemize}
|
||||||
|
\end{frame}
|
||||||
|
%-------------------------------
|
||||||
|
\begin{frame}
|
||||||
|
\frametitle{Possible approaches}
|
||||||
|
|
||||||
|
Possible approaches
|
||||||
|
\begin{itemize}
|
||||||
|
\item Standard VFI/Howards algorithm.
|
||||||
|
\item VFI with sparse state space and other computational benefits.
|
||||||
|
\item Maliar et al's approachs to supervised learning.
|
||||||
|
\item Reinforcement Learning.
|
||||||
|
\end{itemize}
|
||||||
|
\end{frame}
|
||||||
|
%-------------------------------
|
||||||
|
\begin{frame}
|
||||||
|
\frametitle{Chosen approach}
|
||||||
|
% maliar et al - Bellman Residuals
|
||||||
|
% present basic approach (using my notation)
|
||||||
|
% Discuss basic training loop (use sutton and bartos term Generalized Policy Iteration)
|
||||||
|
|
||||||
|
Bellman Residual minimization \cref{MALIAR2018}
|
||||||
|
|
||||||
|
Use NN to approximate $V(S_t,D_t|\theta_1)$ and $X(X_t,D_t|\theta_2)$.
|
||||||
|
|
||||||
|
The loss function is:
|
||||||
|
\begin{align}
|
||||||
|
%0 =& \left[
|
||||||
|
% V(S_t, D_t) - F(S_t, D_t, X_t)- \beta V(S_{t+1}, D_{t+1})
|
||||||
|
% \right]^2
|
||||||
|
% - v \left[
|
||||||
|
% F(S_t, D_t, X_t) + \beta V(S_{t+1}, D_{t+1})
|
||||||
|
% \right] \\
|
||||||
|
0 =& \left[
|
||||||
|
V(S_t, D_t) - F(S_t, D_t, X_t)- \beta V(S_{t+1}, D_{t+1})
|
||||||
|
- \frac{v}{2}
|
||||||
|
\right]^2
|
||||||
|
- v \left[
|
||||||
|
V(S_{t}, D_{t}) + \frac{v}{4}
|
||||||
|
\right]
|
||||||
|
\end{align}
|
||||||
|
|
||||||
|
Iterate between Value and Policy function estimation.
|
||||||
|
\end{frame}
|
||||||
|
%-------------------------------
|
||||||
|
\begin{frame}
|
||||||
|
\frametitle{Training Loop: Planner}
|
||||||
|
|
||||||
|
\end{frame}
|
||||||
|
%-------------------------------
|
||||||
|
\begin{frame}
|
||||||
|
\frametitle{Training Loop: Operators}
|
||||||
|
Using a branched policy function and individual value funcitons
|
||||||
|
|
||||||
|
For each training epoch
|
||||||
|
Draw random data
|
||||||
|
For each operator
|
||||||
|
select operator's policy parameters.
|
||||||
|
train policy function
|
||||||
|
train value function
|
||||||
|
\end{frame}
|
||||||
|
%-------------------------------
|
||||||
|
%------------------------------------
|
||||||
|
\subsection{Analysis so far}
|
||||||
|
%------------------------------------
|
||||||
|
\begin{frame}
|
||||||
|
\frametitle{State of the Code}
|
||||||
|
Currently functioning
|
||||||
|
%Planner training
|
||||||
|
%Operator training
|
||||||
|
\end{frame}
|
||||||
|
%-------------------------------
|
||||||
|
\begin{frame}
|
||||||
|
\frametitle{Results}
|
||||||
|
%simulated debris paths (increase the ammount of debris a couple orders of magnitude)
|
||||||
|
%Protokessler region plots?
|
||||||
|
\end{frame}
|
||||||
|
%-------------------------------
|
||||||
|
%-------------------------------------------------------------------------------------
|
||||||
|
%%%%%%%%%%%%%%%%%%%% Developing the Model%%%%%%%%%%%%%%%%%%%%%%%%
|
||||||
|
\section{Conclusion}
|
||||||
|
%-------------------------------------------------------------------------------------
|
||||||
|
%-------------------------------
|
||||||
|
\begin{frame}
|
||||||
|
\frametitle{Summary}
|
||||||
|
\end{frame}
|
||||||
|
%-------------------------------
|
||||||
|
\begin{frame}
|
||||||
|
\frametitle{Future Work}
|
||||||
|
\end{frame}
|
||||||
|
%-------------------------------
|
||||||
|
\begin{frame}
|
||||||
|
\frametitle{}
|
||||||
|
\end{frame}
|
||||||
|
%-------------------------------
|
||||||
|
\begin{frame}
|
||||||
|
\frametitle{Questions?}
|
||||||
|
\end{frame}
|
||||||
|
%\begin{frame}
|
||||||
|
% \frametitle{MarginalRevenue}
|
||||||
|
% \begin{figure}
|
||||||
|
% \tikzfig{../Assets/owned/ch8_MarginalRevenue}
|
||||||
|
% \includegraphics[height=\textheight]{../Assets/copyrighted/KrugmanObsterfeldMeliz_fig8-7.jpg}
|
||||||
|
% \label{FIG:costs}
|
||||||
|
% \caption{Average Cost Curve as firms enter.}
|
||||||
|
% \end{figure}
|
||||||
|
%\end{frame}
|
||||||
|
\end{document}
|
||||||
|
% \begin{frame}
|
||||||
|
% \frametitle{Columns}
|
||||||
|
% \begin{columns}
|
||||||
|
% \begin{column}{0.5\textwidth}
|
||||||
|
% \end{column}
|
||||||
|
% \begin{column}{0.5\textwidth}
|
||||||
|
% \begin{figure}
|
||||||
|
% \tikzfig{../Assets/owned/ch7_EstablishedAdvantageExample2}
|
||||||
|
% \label{FIG:costs}
|
||||||
|
% \caption{Setting the Stage}
|
||||||
|
% \end{figure}
|
||||||
|
% \end{column}
|
||||||
|
% \end{columns}
|
||||||
|
% \end{frame}
|
||||||
|
% %---------------------------------------------------------------
|
||||||
@ -0,0 +1,42 @@
|
|||||||
|
\usepackage{tikz}
|
||||||
|
\usetikzlibrary{backgrounds}
|
||||||
|
\usetikzlibrary{arrows}
|
||||||
|
\usetikzlibrary{shapes,shapes.geometric,shapes.misc}
|
||||||
|
|
||||||
|
% this style is applied by default to any tikzpicture included via \tikzfig
|
||||||
|
\tikzstyle{tikzfig}=[baseline=-0.25em,scale=0.5]
|
||||||
|
|
||||||
|
% these are dummy properties used by TikZiT, but ignored by LaTex
|
||||||
|
\pgfkeys{/tikz/tikzit fill/.initial=0}
|
||||||
|
\pgfkeys{/tikz/tikzit draw/.initial=0}
|
||||||
|
\pgfkeys{/tikz/tikzit shape/.initial=0}
|
||||||
|
\pgfkeys{/tikz/tikzit category/.initial=0}
|
||||||
|
|
||||||
|
% standard layers used in .tikz files
|
||||||
|
\pgfdeclarelayer{edgelayer}
|
||||||
|
\pgfdeclarelayer{nodelayer}
|
||||||
|
\pgfsetlayers{background,edgelayer,nodelayer,main}
|
||||||
|
|
||||||
|
% style for blank nodes
|
||||||
|
\tikzstyle{none}=[inner sep=0mm]
|
||||||
|
|
||||||
|
% include a .tikz file
|
||||||
|
\newcommand{\tikzfig}[1]{%
|
||||||
|
{\tikzstyle{every picture}=[tikzfig]
|
||||||
|
\IfFileExists{#1.tikz}
|
||||||
|
{\input{#1.tikz}}
|
||||||
|
{%
|
||||||
|
\IfFileExists{./figures/#1.tikz}
|
||||||
|
{\input{./figures/#1.tikz}}
|
||||||
|
{\tikz[baseline=-0.5em]{\node[draw=red,font=\color{red},fill=red!10!white] {\textit{#1}};}}%
|
||||||
|
}}%
|
||||||
|
}
|
||||||
|
|
||||||
|
% the same as \tikzfig, but in a {center} environment
|
||||||
|
\newcommand{\ctikzfig}[1]{%
|
||||||
|
\begin{center}\rm
|
||||||
|
\tikzfig{#1}
|
||||||
|
\end{center}}
|
||||||
|
|
||||||
|
% fix strange self-loops, which are PGF/TikZ default
|
||||||
|
\tikzstyle{every loop}=[]
|
||||||
@ -0,0 +1,27 @@
|
|||||||
|
|
||||||
|
utline
|
||||||
|
|
||||||
|
Tell story
|
||||||
|
- Why do we care? Kessler syndrome.
|
||||||
|
- Why is space access needed? (satellite internet, GPS, military, hobbies, science, R&D, etc.)
|
||||||
|
- Why now? (cubesats, lower launch costs, massive expansion in who launches, etc.)
|
||||||
|
|
||||||
|
|
||||||
|
Present past models
|
||||||
|
-
|
||||||
|
-
|
||||||
|
-
|
||||||
|
|
||||||
|
Present Current model
|
||||||
|
-
|
||||||
|
-
|
||||||
|
-
|
||||||
|
|
||||||
|
|
||||||
|
Present work on solving models.
|
||||||
|
-
|
||||||
|
-
|
||||||
|
-
|
||||||
|
|
||||||
|
|
||||||
|
Future work
|
||||||
@ -0,0 +1,117 @@
|
|||||||
|
\documentclass{article}
|
||||||
|
|
||||||
|
%Setup Subfiles
|
||||||
|
\usepackage{subfiles}
|
||||||
|
|
||||||
|
|
||||||
|
|
||||||
|
%Include preambles
|
||||||
|
\input{assets/preambles/MathPreamble}
|
||||||
|
%\input{assets/preambles/TikzitPreamble}
|
||||||
|
\input{assets/preambles/BibPreamble}
|
||||||
|
\input{assets/preambles/GeneralPreamble}
|
||||||
|
|
||||||
|
\title{Summary of my work on Satellite Constellations}
|
||||||
|
\author{William King}
|
||||||
|
|
||||||
|
\begin{document}
|
||||||
|
\maketitle
|
||||||
|
|
||||||
|
\section{Introduction}
|
||||||
|
\subfile{sections/00_Introduction} %Currently from past semesters.
|
||||||
|
%roughly done 2021-07-15
|
||||||
|
|
||||||
|
%Describe sections
|
||||||
|
The paper is organized as follows.
|
||||||
|
Section \ref{SEC:Models}, describes the laws of motion
|
||||||
|
governing satellites and debris(\ref{SEC:Laws})%,
|
||||||
|
%places limits on various measures of risk (\ref{SEC:Survival}),
|
||||||
|
and reviews various definitions of kessler syndrome in
|
||||||
|
the context of dynamic modeling, their advantages, and their disadvanteges (\ref{SEC:Kessler}).
|
||||||
|
It then describes the dynamic problem faced by constellation operators
|
||||||
|
(\ref{SEC:Operator}) and social planners (\ref{SEC:Planner}).
|
||||||
|
Section \ref{SEC:Computation} describes the computational approach and the
|
||||||
|
results are reported in \cref{SEC:Results}.
|
||||||
|
Section \ref{SEC:Conclusion} concludes with a discussion of limitations, concerns,
|
||||||
|
and remaining policy questions.
|
||||||
|
|
||||||
|
%\section{Modeling the Environment}\label{SEC:Environment}
|
||||||
|
\section{Model}\label{SEC:Models}
|
||||||
|
\subsection{Laws of motion}\label{SEC:Laws}
|
||||||
|
\subfile{sections/01_LawsOfMotion} %Roughly done 2021-07-15
|
||||||
|
|
||||||
|
%\subsection{Marginal survival rates}\label{SEC:Survival}
|
||||||
|
%\subfile{sections/03_SurvivalAnalysis} %roughly done 2021-07-14
|
||||||
|
% Thoughts on removal: This doesn't add much to the actual questions.
|
||||||
|
% It is interesting, but in a paper is just too much.
|
||||||
|
% I'll keep it here to add it back easily.
|
||||||
|
|
||||||
|
\subsection{Kessler Syndrome}\label{SEC:Kessler}
|
||||||
|
% Kessler syndrome follows laws of motion because it is the main
|
||||||
|
% threat of orbital pollution and needs to be included.
|
||||||
|
% Also, there is not really a better place to place it.
|
||||||
|
\subfile{sections/02_KesslerSyndrome} %roughly done before 2021-07-14
|
||||||
|
\subfile{sections/06_KesslerRegion} %roughly done before 2021-07-14
|
||||||
|
|
||||||
|
\subsection{Constellation Operator's Program}\label{SEC:Operator}
|
||||||
|
\subfile{sections/04_ConstellationOperator} %Reasonably done.
|
||||||
|
|
||||||
|
\subsection{Social Planner's Program}\label{SEC:Planner}
|
||||||
|
\subfile{sections/05_SocialPlanner} %Reasonably done?
|
||||||
|
|
||||||
|
\section{Computation}\label{SEC:Computation}
|
||||||
|
\subfile{sections/07_ComputationalApproach} %needs some clarifications.
|
||||||
|
|
||||||
|
\section{Results}\label{SEC:Results}
|
||||||
|
\subfile{sections/09_Results} %TODO
|
||||||
|
|
||||||
|
\section{Conclusion}\label{SEC:Conclusion}
|
||||||
|
\subfile{sections/08_Conclusion} %TODO
|
||||||
|
|
||||||
|
|
||||||
|
\newpage
|
||||||
|
\section{References}
|
||||||
|
\printbibliography
|
||||||
|
\newpage
|
||||||
|
\section{Appedicies}
|
||||||
|
\subsection{Mathematical Notation}
|
||||||
|
Needs completed.
|
||||||
|
%\subsection{Deriving Marginal Survival Rates}\label{APX:Derivations:SurvivalRates}
|
||||||
|
%\subfile{sections/apx_01_MarginalSurvivalRates}
|
||||||
|
|
||||||
|
\subsection{Deriving Euler Equations}\label{APX:Derivations:EulerEquations}
|
||||||
|
\subfile{sections/apx_02_GeneralizedEuEqSteps}
|
||||||
|
|
||||||
|
|
||||||
|
\subsection{Collected Assumptions and Caveats}\label{APX:CollectedAssumptions}
|
||||||
|
I hope to write a section clearly explaining assumptions, caveats, and shortcomings here.
|
||||||
|
These will later get written back into the other sections, but I want to collect them
|
||||||
|
in a single place first.
|
||||||
|
%time periods are long enough for debris to disperse after collisions.
|
||||||
|
%Only a single type of debris
|
||||||
|
%With my current computational idea; each constellation provides the same risk to each other constellation
|
||||||
|
% That can be easily adjusted in the computational models.
|
||||||
|
|
||||||
|
\newpage
|
||||||
|
%Just for simplicity, remove later
|
||||||
|
\tableofcontents
|
||||||
|
\end{document}
|
||||||
|
|
||||||
|
%%% Notes to keep track of
|
||||||
|
% Possible other things to investigate
|
||||||
|
% - Free-entry conditions: which of the following?
|
||||||
|
% - When for every operator, the current stocks imply no more launches(x = 0).
|
||||||
|
% - When for every type of operator, the current stocks plus an own stock of 0 imply no more launches(x = 0).
|
||||||
|
%
|
||||||
|
%
|
||||||
|
%
|
||||||
|
%
|
||||||
|
%
|
||||||
|
%
|
||||||
|
%
|
||||||
|
%
|
||||||
|
%
|
||||||
|
%
|
||||||
|
%
|
||||||
|
%
|
||||||
|
%
|
||||||
Loading…
Reference in New Issue