CDS 212, Homework 1, Fall 2010: Difference between revisions
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{{CDS homework | {{CDS homework | ||
| instructor = J. Doyle | | instructor = J. Doyle | ||
| course = CDS | | course = CDS 212 | ||
| semester = Fall 2010 | | semester = Fall 2010 | ||
| title = Problem Set #1 | | title = Problem Set #1 | ||
| Line 10: | Line 9: | ||
=== 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
- [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>:- <amsmath>\textstyle \sup_t |\dot u(t)|</amsmath>
- <amsmath>\textstyle |u(0)| + \sup_t |\dot u(t)|</amsmath>
- <amsmath>\textstyle \max \{ \sup_t |u(t)|,\, \sup_t |\dot u(t)| \}</amsmath>
- <amsmath>\textstyle \sup_t |u(t)| + \sup_t |\dot u(t)|</amsmath>
Make sure to give a thorough answer (not just yes or no).
- [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.
- [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>. - [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.) - [DFT 2.8, page 30]
Let <amsmath>\widehat G</amsmath> be a stable, proper plant (but not necessarily strictly proper).- 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>.
- 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).
- [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.
- [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.
