\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}
