SIE550/HW5-ProblemStatements.tex at master · eggsactly/SIE550 · GitHub

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\documentclass[12pt]{article}
\usepackage{lingmacros}
\usepackage{tree-dvips}
\usepackage{amsmath}
\usepackage{accents}
\newcommand{\ubar}[1]{\underaccent{\bar}{#1}}
\usepackage{hyperref}
\hypersetup{
    colorlinks=true,
    linkcolor=blue,
    filecolor=magenta,
    urlcolor=cyan,
}
\begin{document}

{\centering
 \textbf{SIE 550 (Linear) Systems Theory\\Homework \#5 - Due date: Wednesday, April 18, 2017 \newline}\par

}

\noindent
\textbf{Problem 1:} Discuss the observability of system ({\em Linear Systems Theory}) Chapter 6, Problem 1 (page 287)
$$
\dot{\bar{x}}=
\begin{pmatrix}
	\frac{1}{t} & 0 \\
	0 & \frac{1}{t}
\end{pmatrix} \bar{x} +
\begin{pmatrix}
1 \\
1
\end{pmatrix} u
$$
$$y=(1,1)\bar{x}$$

\noindent
\textbf{Problem 2:} Examine the observability of system ({\em Linear System Theory}) Chapter 6, Problem 2 (page 322)
$$
\dot{\bar{x}}=
\begin{pmatrix}
	1 & 1 \\
	2 & 2
\end{pmatrix}
\bar{x}+
\begin{pmatrix}
	1 \\
	0
\end{pmatrix} u
$$
$$y=(1,1)\bar{x}$$
Use Theorem 6.1, and select $t_0=0$.\\

\noindent
\textbf{Problem 3:} Compute matrix $M(t_0,t_1)$ for system({\em Linear System Theory}) Chapter 6, Problem 6 (page 323)
$$ \dot{\bar{x}} =
\begin{pmatrix}
	2 & 1 \\
	0 & 2
\end{pmatrix} \bar{x} +
\begin{pmatrix}
	1 \\
	1
\end{pmatrix} u
$$
$$y=(0,1)\bar{x}$$
and illustrate Properties (i) and (ii) of Theorem 6.2. Select $t_0=0$.\\

\noindent
\textbf{Problem 4:} Examine the observability of system ({\em Linear System Theory}) Chapter 6, Problem 7 (page 323)
$$ \dot{\bar{x}} =
\begin{pmatrix}
	2 & 1 \\
	0 & 2
\end{pmatrix} \bar{x} +
\begin{pmatrix}
	1 \\
	1
\end{pmatrix} u
$$
$$y=(0,1)\bar{x}$$
by using the observability matrix (6.7).\\

\noindent
\textbf{Problem 5:} Find the controllability canonical form (7.14) for the system ({\em Linear System Theory}) Chapter 7, Problem 9 (page 371)
$$\dot{\bar{x}}=\begin{pmatrix}
	1 & 0 & 1 \\
	0 & 1 & 1 \\
	1 & 1 & 1
\end{pmatrix} \bar{x}
+
\begin{pmatrix}
	0\\
	1\\
	0
\end{pmatrix}u$$
$$y=(0,1,0)\bar{x}$$

\noindent
\textbf{Problem 6:} Find the observability canonical form (7.18) for system ({\em Linear System Theory}) Chapter 7, Problem 12 (page 371)
$$\dot{\bar{x}}=\begin{pmatrix}
	1 & 0 & 1 \\
	0 & 1 & 1 \\
	1 & 1 & 1
\end{pmatrix} \bar{x}
+
\begin{pmatrix}
	0\\
	1\\
	0
\end{pmatrix}u$$
$$y=(0,1,0)\bar{x}$$

\end{document}