HW4-ProblemStatements.tex

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{\centering
 \textbf{SIE 550 (Linear) Systems Theory\\Homework \#4 - Due date: Wednesday, April 5, 2017 \newline}\par
 
} 

\noindent
\textbf{Problem 1:} Consider the following non-linear autonomous system

$$
\begin{cases} 
	\dot{x_1} = - |x_1|^{0.5} sgn(x_1)- x_1^5+x_2 \\
	\dot{x_2}=-|x_2|^{0.5} sgn(x_2)-x_2^5-x_1
\end{cases}
$$
Study the Finite-Time (FTS) of the origin and answer the following questions:
\begin{enumerate}
	\item Is the origin FTS?
	\item Is the origin Globally FTS?
	\item Find, if possible, the settling time
\end{enumerate}

\noindent
\textbf{Problem 2:} Consider the following non-linear non-autonomous first-order system
$$
\dot{x}=-\frac{|x|}{1+(cos(t))^2}
$$
Study the Finite-Time Stability (FTS) of the system \\

\noindent
\textbf{Problem 3:} Examine the controllability of the system ({\em Linear Systems Theory} Chapter 5, Problem 1 (page 287)
$$
\dot{x}=
\begin{pmatrix} 
	\frac{1}{t} & 0 \\
	0 & \frac{1}{t}
\end{pmatrix} x + 
\begin{pmatrix} 
1 \\
1
\end{pmatrix} u
$$

\noindent
\textbf{Problem 4:} Compute matrix $W(t_0,t_1)$ for the system described below, and illustrate Properties (i) and (ii) of Theorem 5.2. Select the [0,1] interval. ({\em Linear Systems Theory} Chapter 5, Problem 3 (page 288)
$$
\dot{x}=
\begin{pmatrix}
	1 & 1 \\
	2 & 2
\end{pmatrix}
x + 
\begin{pmatrix}
	1 \\
	0
\end{pmatrix} u
$$

\noindent
\textbf{Problem 5:} Determine if system
$$ \dot{x} = 
\begin{pmatrix}
	2 & 1 \\
	0 & 2
\end{pmatrix} x +
\begin{pmatrix} 
	1 \\
	1
\end{pmatrix} u
$$
is completely controllable by using the controllability matrix (5.10). ({\em Linear Systems Theory} Chapter 5, Problem 7 (page 288)\\

\noindent
\textbf{Problem 6:} Is there any input for system 
$$ \dot{x}=
\begin{pmatrix}
1 & 1 \\
2 & 2
\end{pmatrix} x + 
\begin{pmatrix}
1 \\
1
\end{pmatrix}u,\quad x(0)=
\begin{pmatrix}
1 \\
1
\end{pmatrix},$$
which controls the trajectory to 
$$x(t)=\begin{pmatrix}
t + 1 \\
1
\end{pmatrix}
$$
in the interval [0,1]? ({\em Linear Systems Theory} Chapter 5, Problem 8 (page 288)\\

\noindent
\textbf{Problem 7:} Discuss the controllability of the mechanical system
$$\dot{x}=
\begin{pmatrix}
	0 & 1 \\
	0 & -6
\end{pmatrix}	
x + 
\begin{pmatrix}
0 \\
2
\end{pmatrix} u
$$
introduced in Problem 3.7. ({\em Linear Systems Theory} Chapter 5, Problem 11 (page 289)

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