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

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{{CDS 212 draft HW}}
{{CDS homework
| instructor = J. Doyle
| course = CDS 212
| semester = Fall 2010
| title = Problem Set #1
| issued = 28 Sep 2010
| due = 7 Oct 2010
}}


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


=== Problems ===
=== Problems ===
<ol>
<li>[DFT 2.1, page 28<br>
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>:
<ol type="a">
<li> <amsmath>\textstyle  \sup_t |\dot u(t)|</amsmath></li>
<li> <amsmath>\textstyle  |u(0)| + \sup_t |\dot u(t)|</amsmath> </li>
<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).
</li>
<li> [DFT 2.4, page 29] <br>
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
<center><amsmath>
\| \widehat D \widehat G \| = \| \widehat G \|
</amsmath></center>
Determine if the 2-norm and <amsmath>\infty</amsmath>-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 <amsmath> \frac{1}{\tau s + 1}, \quad \tau > 0 </amsmath>.
</li>
<li> [DFT 2.7, page 30] <br>
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.)
</li>
<li> [DFT 2.8, page 30] <br>
Let <amsmath>\widehat G</amsmath> be a stable, proper plant (but not necessarily strictly proper).
<ol type="a">
<li> 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>.</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>[DFT 2.11, page 30] <br>
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
<center><amsmath>
  \| G \|_1 = \sup_{\|u\|_\infty = 1} \| y \|_\infty
</amsmath></center>
and find an input which achieves the supremum.
</li>
<li> [DFT 2.12, page 30] <br>
For a linear system with input <amsmath>u</amsmath> and output <amsmath>y</amsmath>, prove that
<center><amsmath>
  \sup_{\|u\| \leq 1} \| y \| =
    \sup_{\|u\| = 1} \| y \|
</amsmath></center>
where <amsmath>\|\cdot\|</amsmath> is any norm on signals.
</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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