作者

Venkatraman Renganathan

最近上传

5 年前

许可

Creative Commons CC BY 4.0

摘要

Template to write solutions to homework assignments that involve math equations, figures and code. Just one line command to add the Matlab code with its file name.

```
\documentclass[a4paper, 11pt]{article}
\usepackage{comment} % enables the use of multi-line comments (\ifx \fi)
\usepackage{lipsum} %This package just generates Lorem Ipsum filler text.
\usepackage{fullpage} % changes the margin
\usepackage[a4paper, total={7in, 10in}]{geometry}
\usepackage[fleqn]{amsmath}
\usepackage{amssymb,amsthm} % assumes amsmath package installed
\newtheorem{theorem}{Theorem}
\newtheorem{corollary}{Corollary}
\usepackage{graphicx}
\usepackage{tikz}
\usetikzlibrary{arrows}
\usepackage{verbatim}
\usepackage[numbered]{mcode}
\usepackage{float}
\usepackage{tikz}
\usetikzlibrary{shapes,arrows}
\usetikzlibrary{arrows,calc,positioning}
\tikzset{
block/.style = {draw, rectangle,
minimum height=1cm,
minimum width=1.5cm},
input/.style = {coordinate,node distance=1cm},
output/.style = {coordinate,node distance=4cm},
arrow/.style={draw, -latex,node distance=2cm},
pinstyle/.style = {pin edge={latex-, black,node distance=2cm}},
sum/.style = {draw, circle, node distance=1cm},
}
\usepackage{xcolor}
\usepackage{mdframed}
\usepackage[shortlabels]{enumitem}
\usepackage{indentfirst}
\usepackage{hyperref}
\renewcommand{\thesubsection}{\thesection.\alph{subsection}}
\newenvironment{problem}[2][Problem]
{ \begin{mdframed}[backgroundcolor=gray!20] \textbf{#1 #2} \\}
{ \end{mdframed}}
% Define solution environment
\newenvironment{solution}
{\textit{Solution:}}
{}
\renewcommand{\qed}{\quad\qedsymbol}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{document}
%Header-Make sure you update this information!!!!
\noindent
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\large\textbf{Venkatraman Renganathan} \hfill \textbf{Homework - \#} \\
Email: veralevel@gethu.edu \hfill ID: 123456789 \\
\normalsize Course: MECH 6325 - Optimal Estimation \& Kalman Filtering \hfill Term: Fall 2019\\
Instructor: Dr. Sriram \hfill Due Date: $22^{nd}$ November, 2019 \\
\noindent\rule{7in}{2.8pt}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% Problem 1
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{problem}{1}
Consider the scalar system
\begin{align*}
\Dot{x} &= -x + u + w
\end{align*}
$w$ is zero-mean process noise with a variance of $Q$. The control has a mean value of $u_0$, an uncertainty of $2$ (one standard deviation), and is uncorrelated with $w$. Rewrite the system equations to obtain an equivalent system with a normalized control that is perfectly known. What is the variance of the new process noise term in the transformed system equation?
\end{problem}
\begin{solution}
The variance of the new process noise, $w_u$ is $\Sigma_{w_{u}} = Q + \sigma^2_u = Q + 4$.
\begin{align*}
\Dot{x} &= -x + u_0 + \underbrace{w + \Delta u}_{w_{u}}, \quad w_u \sim (0, Q + \sigma^2_u).
\end{align*}
\end{solution}
\noindent\rule{7in}{2.8pt}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% Problem 2
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{problem}{2}
Consider the system
\begin{align*}
x_{k+1} &= \phi x_{k} + w_{k}, \\
y_k &= x_k,
\end{align*}
where $w_k \sim (0, 1)$, and $\phi = 0.9$ is an unknown constant. Design an extended Kalman filter to estimate $\phi$. Simulate the filter for $100$ time steps with $x_0 = 1, P_0 = I , \hat{x}_{0} = 0$, and $\hat{\phi}_{0} = 0$. Hand in your source code and a plot showing $\hat{\phi}$ as a function of time.
\end{problem}
\begin{solution}
Perform the measurement update of the state estimate and estimation error covariance as follows
\begin{align*}
K_k &= P^{-}_k H^{\top}_k (H_k P^{-}_k H^{\top}_k + R_k)^{-1} = P^{-}_k H^{\top}_k (H_k P^{-}_k H^{\top}_k)^{-1}, \quad \text{Since }R_k = 0, \\
\hat{\bar{x}}^{+}_{k} &= \hat{\bar{x}}^{-}_{k} + K_k (y_k - h_k(\hat{\bar{x}}^{-}_{k}, 0)) \\
&= \hat{\bar{x}}^{-}_{k} + K_k (y_k - \hat{x}^{-}_{k}), \quad \text{Since } \hat{\phi}^{-}_{k} = 0, \\
P^{+}_k &= (I - K_k H_k) P^{-}_k
\end{align*}
\begin{figure}[H]
\centering
\includegraphics[scale=0.25]{q2.png}
\caption{Plot showing $\hat{\phi}$ as a function of time.}
\label{fig_q2l}
\end{figure}
\end{solution}
\lstinputlisting{HW6Q2.m}
\noindent\rule{7in}{2.8pt}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\end{document}
```

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