CDS 212, Homework 1, Fall 2010: Difference between revisions

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{{CDS 212 draft HW}}
{{CDS homework
{{CDS homework
  | instructor = J. Doyle
  | instructor = J. Doyle
  | course = CDS 112
  | course = CDS 212
  | semester = Fall 2010
  | semester = Fall 2010
  | title = Problem Set #1
  | title = Problem Set #1
Line 10: Line 9:


=== Reading ===
=== Reading ===
* {{DFT}}, Chapterss 1 and 2
* {{DFT}}, Chapters 1 and 2
* Dullerud and Paganini, Ch 3
* Dullerud and Paganini, Ch 3


Line 16: Line 15:


<ol>
<ol>
<li>DFT 2.1, page 28<br>
<li>[DFT 2.1, page 28<br>
Suppose that $u(t)$ is a continuous signal whose derivative $\dot
Suppose that <amsmath>u(t)</amsmath> is a continuous signal whose derivative <amsmath>\dot
u(t)$ is also continuous.  Which of the following quantities qualifies
u(t)</amsmath> is also continuous.  Which of the following quantities qualifies
as a norm for $u$:
as a norm for <amsmath>u</amsmath>:
* $\sup_t |\dot u(t)|$
<ol type="a">
* $|u(0)| + \sup_t |\dot u(t)|$
<li> <amsmath>\textstyle  \sup_t |\dot u(t)|</amsmath></li>
* $\max \{ \sup_t |u(t)|,\, \sup_t |\dot u(t)| \}$
<li> <amsmath>\textstyle  |u(0)| + \sup_t |\dot u(t)|</amsmath> </li>
* $\sup_t |u(t)| + \sup_t |\dot u(t)|$
<li> <amsmath>\textstyle  \max \{ \sup_t |u(t)|,\, \sup_t |\dot u(t)| \}</amsmath> </li>
<li> <amsmath>\textstyle  \sup_t |u(t)| + \sup_t |\dot u(t)|</amsmath> </li>
</ol>
Make sure to give a thorough answer (not just yes or no).
Make sure to give a thorough answer (not just yes or no).
</li>
</li>


<li> DFT 2.4, page 29] <br>
<li> [DFT 2.4, page 29] <br>
Let $D$ be a pure time delay of $\tau$ seconds with transfer function
Let <amsmath>D</amsmath> be a pure time delay of <amsmath>\tau</amsmath> seconds with transfer function
\begin{displaymath}
<amsmath> \widehat D(s) = e^{-s \tau} </amsmath>. A norm <amsmath>\|\cdot\|</amsmath> on transfer functions is ''time-delay invariant'' if for
  \widehat D(s) = e^{-s \tau}.
every bounded transfer function <amsmath>\widehat G</amsmath> and every <amsmath>\tau > 0</amsmath> we have
\end{displaymath}
<center><amsmath>
A norm $\|\cdot\|$ on transfer functions is {\em time-delay invariant} if for
\| \widehat D \widehat G \| = \| \widehat G \|
every bounded transfer function $\widehat G$ and every $\tau > 0$ we have
</amsmath></center>
\begin{displaymath}
Determine if the 2-norm and <amsmath>\infty</amsmath>-norm are time-delay invariant.
  \| \widehat D \widehat G \| = \| \widehat G \|
\end{displaymath}
Determine if the 2-norm and $\infty$-norm are time-delay invariant.
</li>
</li>


<li> [DFT 2.5, page 30] <br>
<li> [DFT 2.5, page 30] <br>
Compute the 1-norm of the impluse response corresponding to the
Compute the 1-norm of the impluse response corresponding to the
transfer function
transfer function <amsmath> \frac{1}{\tau s + 1}, \quad \tau > 0 </amsmath>.  
\begin{displaymath}
  \fract{1}{\tau s + 1} \qquad \tau > 0.
</li>
</li>


<li> DFT 2.7, page 30] <br> Derive the $\infty$-norm to $\infty$-norm system gain for a stable,
<li> [DFT 2.7, page 30] <br>  
proper plant $\widehat G$.  (Hint: write $\widehat G = c + \widehat G_1$ where $c$ is a constant
Derive the <amsmath>\infty</amsmath>-norm to <amsmath>\infty</amsmath>-norm system gain for a stable,
and $\widehat G_1$ is strictly proper.)
proper plant <amsmath>\widehat G</amsmath>.  (Hint: write <amsmath>\widehat G = c + \widehat G_1</amsmath> where <amsmath>c</amsmath> is a constant
and <amsmath>\widehat G_1</amsmath> is strictly proper.)
</li>
</li>


<li> [DFT 2.8, page 30] <br> Let $\widehat G$ be a stable, proper plant (but not necessarily strictly proper).
<li> [DFT 2.8, page 30] <br>  
# Show that the $\infty$-norm of the output $y$ given an input
Let <amsmath>\widehat G</amsmath> be a stable, proper plant (but not necessarily strictly proper).
   $u(t) = \sin(\omega t)$ is $|\widehat G(jw)|$.
<ol type="a">
# Show that the 2-norm to 2-norm system gain for $\widehat G$ is $\|
<li> Show that the <amsmath>\infty</amsmath>-norm of the output <amsmath>y</amsmath> given an input
   \widehat G \|_\infty$ (just as in the strictly proper case).
   <amsmath>u(t) = \sin(\omega t)</amsmath> is <amsmath>|\widehat G(jw)|</amsmath>.</li>
<li>  Show that the 2-norm to 2-norm system gain for <amsmath>\widehat G</amsmath> is <amsmath>\|
   \widehat G \|_\infty</amsmath> (just as in the strictly proper case).</li>
</ol>
</li>
</li>


<li>[DFT 2.11, page 30] <br>
<li>[DFT 2.11, page 30] <br>
Consider a system with transfer function
Consider a system with transfer function
\begin{displaymath}
<amsmath>\widehat G(s) = \frac{s+2}{4s + 1}</amsmath>
  \widehat G(s) = \fract{s+2}{4s + 1}
and input <amsmath>u</amsmath> and output <amsmath>y</amsmath>.  Compute
\end{displaymath}
<center><amsmath>
and input $u$ and output $y$.  Compute
\begin{displaymath}
   \| G \|_1 = \sup_{\|u\|_\infty = 1} \| y \|_\infty
   \| G \|_1 = \sup_{\|u\|_\infty = 1} \| y \|_\infty
\end{displaymath}
</amsmath></center>
and find an input which achieves the supremum.
and find an input which achieves the supremum.
</li>
</li>


<li> [DFT 2.12, page 30] <br>
<li> [DFT 2.12, page 30] <br>
For a linear system with input $u$ and output $y$, prove that
For a linear system with input <amsmath>u</amsmath> and output <amsmath>y</amsmath>, prove that
\begin{displaymath}
<center><amsmath>
    \sup_{\|u\| \leq 1} \| y \| =
  \sup_{\|u\| \leq 1} \| y \| =
     \sup_{\|u\| = 1} \| y \|
     \sup_{\|u\| = 1} \| y \|
\end{displaymath}
</amsmath></center>
where $\|\cdot\|$ is any norm on signals.
where <amsmath>\|\cdot\|</amsmath> 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>
</li>

Latest revision as of 17:42, 28 September 2010

J. Doyle Issued: 28 Sep 2010
CDS 212, Fall 2010 Due: 7 Oct 2010

Reading

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

Problems

  1. [DFT 2.1, page 28
    Suppose that <amsmath>u(t)</amsmath> is a continuous signal whose derivative <amsmath>\dot u(t)</amsmath> is also continuous. Which of the following quantities qualifies as a norm for <amsmath>u</amsmath>:
    1. <amsmath>\textstyle \sup_t |\dot u(t)|</amsmath>
    2. <amsmath>\textstyle |u(0)| + \sup_t |\dot u(t)|</amsmath>
    3. <amsmath>\textstyle \max \{ \sup_t |u(t)|,\, \sup_t |\dot u(t)| \}</amsmath>
    4. <amsmath>\textstyle \sup_t |u(t)| + \sup_t |\dot u(t)|</amsmath>

    Make sure to give a thorough answer (not just yes or no).

  2. [DFT 2.4, page 29]
    Let <amsmath>D</amsmath> be a pure time delay of <amsmath>\tau</amsmath> seconds with transfer function <amsmath> \widehat D(s) = e^{-s \tau} </amsmath>. A norm <amsmath>\|\cdot\|</amsmath> on transfer functions is time-delay invariant if for every bounded transfer function <amsmath>\widehat G</amsmath> and every <amsmath>\tau > 0</amsmath> we have
    <amsmath> 
    \| \widehat D \widehat G \| = \| \widehat G \|
    </amsmath>

    Determine if the 2-norm and <amsmath>\infty</amsmath>-norm are time-delay invariant.

  3. [DFT 2.5, page 30]
    Compute the 1-norm of the impluse response corresponding to the transfer function <amsmath> \frac{1}{\tau s + 1}, \quad \tau > 0 </amsmath>.
  4. [DFT 2.7, page 30]
    Derive the <amsmath>\infty</amsmath>-norm to <amsmath>\infty</amsmath>-norm system gain for a stable, proper plant <amsmath>\widehat G</amsmath>. (Hint: write <amsmath>\widehat G = c + \widehat G_1</amsmath> where <amsmath>c</amsmath> is a constant and <amsmath>\widehat G_1</amsmath> is strictly proper.)
  5. [DFT 2.8, page 30]
    Let <amsmath>\widehat G</amsmath> be a stable, proper plant (but not necessarily strictly proper).
    1. Show that the <amsmath>\infty</amsmath>-norm of the output <amsmath>y</amsmath> given an input <amsmath>u(t) = \sin(\omega t)</amsmath> is <amsmath>|\widehat G(jw)|</amsmath>.
    2. Show that the 2-norm to 2-norm system gain for <amsmath>\widehat G</amsmath> is <amsmath>\| \widehat G \|_\infty</amsmath> (just as in the strictly proper case).
  6. [DFT 2.11, page 30]
    Consider a system with transfer function <amsmath>\widehat G(s) = \frac{s+2}{4s + 1}</amsmath> and input <amsmath>u</amsmath> and output <amsmath>y</amsmath>. Compute
    <amsmath> 
     \| G \|_1 = \sup_{\|u\|_\infty = 1} \| y \|_\infty
    </amsmath>

    and find an input which achieves the supremum.

  7. [DFT 2.12, page 30]
    For a linear system with input <amsmath>u</amsmath> and output <amsmath>y</amsmath>, prove that
    <amsmath> 
     \sup_{\|u\| \leq 1} \| y \| =
   \sup_{\|u\| = 1} \| y \|
    </amsmath>

    where <amsmath>\|\cdot\|</amsmath> is any norm on signals.

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