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\begin{document}

\title{ECE 486 (Control Systems) -- Homework 10}
\author{{\bf Due:} December 1st}
\date{}

\maketitle

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\begin{problem}[25 points]
Calculate the transfer function for the following state-space model.

\[
\dot x = 
\begin{bmatrix}
5 & 2 \\
2 & 1
\end{bmatrix} x + 
\begin{bmatrix}
1 \\
-1
\end{bmatrix} u
\]
\[
y = \begin{bmatrix}1 & 2\end{bmatrix} x + \begin{bmatrix} 3 \end{bmatrix}u
\]

\end{problem}

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\vspace{0.1in}


\begin{problem}(20 points)
For the following transfer function, calculate the controllable canonical form (CCF) state-space model. 
\[
G(s) = \frac{1}{(s+2)(s^2+2s+5)}
\]

\end{problem}


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\begin{problem}
[25 points]
Prove the following:
\renewcommand{\labelenumi}{\roman{enumi})}
\begin{enumerate}
\item $(AB)^\top = B^\top A^\top$
\item $(ABC)^\top = C^\top B^\top A^\top$
\item $(A^\top)^{-1} = (A^{-1})^\top$
\item $(I - TAT^{-1})^{-1} = T(I - A)^{-1} T^{-1}$
\item For any integer $k \geq 0$, $(TAT^{-1})^k = TA^k T^{-1}$
\end{enumerate}

Here, $(\cdot)^\top$ denotes the transpose operator and $(\cdot)^{-1}$ denotes the inverse operator. Assume matrices are invertible whenever needed.

Recall that the definition of $A^{-1}$ is the unique matrix such that $A A^{-1} = A^{-1} A = I$, where $I$ is the identity matrix.

You may use previous parts to prove later parts, e.g. you may invoke Part (i) when proving Part (ii).

\end{problem}


\vspace{0.1in}
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\begin{problem}[30 points]
Determine whether or not the following systems are controllable. If they are controllable, put them in controllable canonical form.

\renewcommand{\labelenumi}{\roman{enumi})}
\begin{enumerate}
\item
\[
\dot x = 
\begin{bmatrix}
1 & 2 & 1 & 0 \\
5 & 1 & 3 & 2 \\
6 & 1 & 3 & 4 \\
1 & 7 & 1 & 0 
\end{bmatrix} x + 
\begin{bmatrix}
0 \\
0 \\
0 \\
0
\end{bmatrix} u
\]
\[
y = 
\begin{bmatrix}
1 & 0 & 0 & 1
\end{bmatrix} x + 
\begin{bmatrix}
1
\end{bmatrix} u
\]

\item 
\[
\dot x = 
\begin{bmatrix}
5 & 2 \\
2 & 1
\end{bmatrix} x + 
\begin{bmatrix}
1 \\
-1
\end{bmatrix} u
\]
\[
y =
\begin{bmatrix}
2 & 1
\end{bmatrix} x + 
\begin{bmatrix}
0
\end{bmatrix} u
\]
\end{enumerate}


\end{problem}


\end{document}
