\subsection{Homework 9 - Due: 04/02}
We plan to release the solutions the day the homework is due so that you
may prepare for the midterm; therefore (a) seek help early if you are
stuck and (b) absolutely no extensions.
\subsubsection{Problem 1}
Construct minimal realizations of the following transfer functions: \[
G_1(s) = \dfrac{s-3}{s^2-5s+6}\,,\qquad\qquad G_2(s) = \dfrac{s^2+1}{s^3-2s^2+s}
\]
\subsubsection{Problem 2}
Consider the system \[
\dot x=\begin{bmatrix} 0 & 1 & 2\\
0 & 0 & 3\\
0 & 0 & -1\end{bmatrix}x+
\begin{bmatrix} 0\\1\\0\end{bmatrix}u
\]
\begin{enumerate}
\def\labelenumi{\alph{enumi}.}
\item
Is this system asymptotically stabilizable by a state feedback
\(u=Kx\)? If yes, find such a \(K\); if not, explain why not.
\item
Consider the output \[y= \begin{bmatrix} 1 &0 &0 \end{bmatrix}x\]
Determine the transfer function of the resulting (open-loop) system
\ital{without directly computing it}. Explain your answer.
\end{enumerate}
\subsubsection{Problem 3}
Consider the system \[
\dot x=\begin{bmatrix} 0 & -1 & 1\\
1 & 2 & -1\\
0 & 5 & -3\end{bmatrix}x, \qquad
y=
\begin{bmatrix} 1 &-1&1\end{bmatrix}x
\]
\begin{enumerate}
\def\labelenumi{\alph{enumi}.}
\item
Design an asymptotic observer for this system such that the
eigenvalues of the error dynamics are \(-1\) and \(-2\pm i\). (Hint:
You can \emph{carefully} use the result of Problem 8 of
\href{hw08.html\#problem-8}{previous homework}).
\item
What would happen to the entries of the output injection matrix \(L\)
if the desired observer eigenvalues were moved to \(-10\) and
\(-20\pm i\), or to \(-100\) and \(-200\pm i\)? Explain why.
\end{enumerate}
\subsubsection{Problem 4}
Consider the system \[
\dot x=\begin{bmatrix} -1 & 0 & 4\\
2 & -2 & 3\\
0 & 1 & 1\end{bmatrix}x, \qquad
y=
\begin{bmatrix} 0 &0&1\end{bmatrix}x
\]
Design an Luenberger (i.e.~a \emph{reduced order}) observer for
\(x_1,x_2\) in the form of a dynamical system with state dimension 2.
\subsubsection{Problem 5}
Consider the harmonic oscillator with position measurements: \[
\ddot x+x=u, \qquad y=x
\]
\begin{enumerate}
\def\labelenumi{\alph{enumi}.}
\item
Show that it cannot be asymptotically stabilized by static output
feedback of the form \(u=ky\).
\item
Find a dynamic output feedback law that asymptotically stabilizes this
system.
\end{enumerate}
\subsubsection{Problem 6}
Consider the system \[
\dot x = Ax+Bu+d_1 \qquad y =Cx +d_2
\]
Suppose the objective is to make \(y(t)\) asymptotically approach a
reference signal \(r(t)\), in spite of the disturbances. In this
problem, take \(r(t)\) to a sinusoidal signal:
\[r(t)=\sin(t), \qquad t\ge 0 \] Solve the problem by following these
steps, along the lines of the construction given in class for tracking
constants \& ramps.
\begin{enumerate}
\def\labelenumi{\alph{enumi}.}
\item
Reduce the problem to asymptotic stabilization of an auxiliary system
whose state contains the tracking error \(e:=y-r\). You can assume
here that the disturbances belong to the same class as \(r\), but make
this assumption precise and explain how it helps your analysis.
\item
Assuming controllability of this auxiliary system, state what type of
control law can be used to stabilize it (you don't need to investigate
conditions for controllability in terms of the original data \(A\),
\(B\), \(C\) like we did in class). Your final answer for the
controller should be a state-space dynamical system (don't leave it in
the form containing integrators).
\item
Comment in what sense your results reflect the internal model
principle.
\end{enumerate}