Difference between revisions of "CDS 212, Homework 1, Fall 2010"

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=== Problems ===
=== Problems ===
<ol>
<li>DFT 2.1, page 28<br>
Suppose that $u(t)$ is a continuous signal whose derivative $\dot
u(t)$ is also continuous.  Which of the following quantities qualifies
as a norm for $u$:
* $\sup_t |\dot u(t)|$
* $|u(0)| + \sup_t |\dot u(t)|$
* $\max \{ \sup_t |u(t)|,\, \sup_t |\dot u(t)| \}$
* $\sup_t |u(t)| + \sup_t |\dot u(t)|$
Make sure to give a thorough answer (not just yes or no).
</li>
<li> DFT 2.4, page 29] <br>
Let $D$ be a pure time delay of $\tau$ seconds with transfer function
\begin{displaymath}
  \widehat D(s) = e^{-s \tau}.
\end{displaymath}
A norm $\|\cdot\|$ on transfer functions is {\em time-delay invariant} if for
every bounded transfer function $\widehat G$ and every $\tau > 0$ we have
\begin{displaymath}
  \| \widehat D \widehat G \| = \| \widehat G \|
\end{displaymath}
Determine if the 2-norm and $\infty$-norm are time-delay invariant.
</li>
<li> [DFT 2.5, page 30] <br>
Compute the 1-norm of the impluse response corresponding to the
transfer function
\begin{displaymath}
  \fract{1}{\tau s + 1} \qquad \tau > 0.
</li>
<li> DFT 2.7, page 30] <br> Derive the $\infty$-norm to $\infty$-norm system gain for a stable,
proper plant $\widehat G$.  (Hint: write $\widehat G = c + \widehat G_1$ where $c$ is a constant
and $\widehat G_1$ is strictly proper.)
</li>
<li> [DFT 2.8, page 30] <br> Let $\widehat G$ be a stable, proper plant (but not necessarily strictly proper).
# Show that the $\infty$-norm of the output $y$ given an input
  $u(t) = \sin(\omega t)$ is $|\widehat G(jw)|$.
# Show that the 2-norm to 2-norm system gain for $\widehat G$ is $\|
  \widehat G \|_\infty$ (just as in the strictly proper case).
</li>
<li>[DFT 2.11, page 30] <br>
Consider a system with transfer function
\begin{displaymath}
  \widehat G(s) = \fract{s+2}{4s + 1}
\end{displaymath}
and input $u$ and output $y$.  Compute
\begin{displaymath}
  \| G \|_1 = \sup_{\|u\|_\infty = 1} \| y \|_\infty
\end{displaymath}
and find an input which achieves the supremum.
</li>
<li> [DFT 2.12, page 30] <br>
For a linear system with input $u$ and output $y$, prove that
\begin{displaymath}
    \sup_{\|u\| \leq 1} \| y \| =
    \sup_{\|u\| = 1} \| y \|
\end{displaymath}
where $\|\cdot\|$ is any norm on signals.
</li>
<li>
Consider a second order mechanical system with transfer function
\begin{displaymath}
  \widehat G(s) = \fract{1}{s^2 + 2 \omega_n \zeta s + \omega_n^2}
\end{displaymath}
($\omega_n$ is the natural frequence of the system and $\zeta$ is the
damping ratio).  Setting $\omega_n = 1$, write a short MATLAB
program to generate a plot of the $\infty$-norm as a function of the
damping ratio $\zeta > 0$.
</li>

Revision as of 16:28, 18 September 2010

  1. REDIRECT HW draft
J. Doyle Issued: 28 Sep 2010
CDS 112, Fall 2010 Due: 7 Oct 2010

Reading

  • DFT, Chapterss 1 and 2
  • Dullerud and Paganini, Ch 3

Problems

  1. DFT 2.1, page 28
    Suppose that $u(t)$ is a continuous signal whose derivative $\dot u(t)$ is also continuous. Which of the following quantities qualifies as a norm for $u$:
    • $\sup_t |\dot u(t)|$
    • $|u(0)| + \sup_t |\dot u(t)|$
    • $\max \{ \sup_t |u(t)|,\, \sup_t |\dot u(t)| \}$
    • $\sup_t |u(t)| + \sup_t |\dot u(t)|$
    Make sure to give a thorough answer (not just yes or no).
  2. DFT 2.4, page 29]
    Let $D$ be a pure time delay of $\tau$ seconds with transfer function \begin{displaymath} \widehat D(s) = e^{-s \tau}. \end{displaymath} A norm $\|\cdot\|$ on transfer functions is {\em time-delay invariant} if for every bounded transfer function $\widehat G$ and every $\tau > 0$ we have \begin{displaymath} \| \widehat D \widehat G \| = \| \widehat G \| \end{displaymath} Determine if the 2-norm and $\infty$-norm are time-delay invariant.
  3. [DFT 2.5, page 30]
    Compute the 1-norm of the impluse response corresponding to the transfer function \begin{displaymath} \fract{1}{\tau s + 1} \qquad \tau > 0.
  4. DFT 2.7, page 30]
    Derive the $\infty$-norm to $\infty$-norm system gain for a stable, proper plant $\widehat G$. (Hint: write $\widehat G = c + \widehat G_1$ where $c$ is a constant and $\widehat G_1$ is strictly proper.)
  5. [DFT 2.8, page 30]
    Let $\widehat G$ be a stable, proper plant (but not necessarily strictly proper).
    1. Show that the $\infty$-norm of the output $y$ given an input
    $u(t) = \sin(\omega t)$ is $|\widehat G(jw)|$.
    1. Show that the 2-norm to 2-norm system gain for $\widehat G$ is $\|
    \widehat G \|_\infty$ (just as in the strictly proper case).
  6. [DFT 2.11, page 30]
    Consider a system with transfer function \begin{displaymath} \widehat G(s) = \fract{s+2}{4s + 1} \end{displaymath} and input $u$ and output $y$. Compute \begin{displaymath} \| G \|_1 = \sup_{\|u\|_\infty = 1} \| y \|_\infty \end{displaymath} and find an input which achieves the supremum.
  7. [DFT 2.12, page 30]
    For a linear system with input $u$ and output $y$, prove that \begin{displaymath} \sup_{\|u\| \leq 1} \| y \| = \sup_{\|u\| = 1} \| y \| \end{displaymath} where $\|\cdot\|$ is any norm on signals.
  8. Consider a second order mechanical system with transfer function \begin{displaymath} \widehat G(s) = \fract{1}{s^2 + 2 \omega_n \zeta s + \omega_n^2} \end{displaymath} ($\omega_n$ is the natural frequence of the system and $\zeta$ is the damping ratio). Setting $\omega_n = 1$, write a short MATLAB program to generate a plot of the $\infty$-norm as a function of the damping ratio $\zeta > 0$.