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\begin{document}

\title{ECE 486 (Control Systems) -- Homework 11}
\author{{\bf Due:} December 8}
\date{}

\maketitle

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{problem}

Consider the single-input, single-output transfer function:
\[
Y(s) = \frac{s+1}{s^2 + 2s + 2} U(s)
\]

\begin{enumerate}[(a)]
\item
Find a second-order state-space model that represents this transfer function.
\item
For this state-space model, calculate a state-feedback controller $u = -Kx + r$
that places the closed-loop poles at $-4$ and $-25$.
\item
Construct a stable observer to estimate $x$ based on the known inputs $u$ and
observations $y$. You may use MATLAB for this part.
\item
With the controller and observer from the previous problems in place, calculate
$k_r$ such that $u = -K \hat x + k_r r$ yields a closed-loop system $Y/R$ with
unity gain. You may use MATLAB.
\item
Plot the step response using MATLAB.
\end{enumerate}
\end{problem}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{problem}
Consider the single-input, single-output transfer function:
\[
G_p(s) = \frac{1-s/2}{1+s/2} \frac{1}{s^2}
\]

\begin{enumerate}[(a)]
\item
Find a third-order state-space model that represents this transfer function.
\item
For this state-space model, calculate a state-feedback controller $u = -Kx + r$
that places the closed-loop poles at $-4$, $-13$, and $-25$. You may use MATLAB
to calculate this controller, but not to find the the state-space model.
\item
Construct a stable observer, and put this together to form a compensator of the
form $U = -G_c Y + G_r R$. You may use MATLAB.
\item
Calculate the Nyquist plot of $G_c G_p$. You may use MATLAB to do so. Is the
system stable? If so, calculate the gain and phase margins.
\end{enumerate}
\end{problem}

\begin{problem}
Consider the following nonlinear system
\[
\dot{x}=-4x-2u+u^3
\]
\begin{enumerate}[(a)]
\item
 Find an equilibrium $(\bar{x},\bar{u})$ with $\bar{u}=2$.

\item
Linearize the dynamics around $(\bar{x},\bar{u})$.
\end{enumerate}
\end{problem}

\end{document}
