CDS 212, Homework 4, Fall 2010 - Revision history

Sojoudi at 21:39, 19 October 2010

2010-10-19T21:39:26Z

← Older revision		Revision as of 21:39, 19 October 2010
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	~~{{CDS 212 draft HW}}~~
	{{CDS homework		{{CDS homework
	\| instructor = J. Doyle		\| instructor = J. Doyle

Sojoudi: /* Problems */

2010-10-19T20:04:19Z

Problems

← Older revision		Revision as of 20:04, 19 October 2010
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	=== Problems ===		=== Problems ===
	<ol>		<ol>
	<li>[DFT 6.1~~, page 103~~] <br>		<li>[DFT 6.1] <br>
	Show that any stable transfer function can be uniquely factored as the product of an all pass function and a minimum phase function, up to a choice of sign.		Show that any stable transfer function can be uniquely factored as the product of an all pass function and a minimum phase function, up to a choice of sign.
	</li>		</li>

	<li>[DFT 6.4~~, page 103~~] <br>		<li>[DFT 6.4] <br>
	Let		Let
	<center><amsmath>		<center><amsmath>
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	</li>		</li>

	<li>[DFT 6.5~~, page 104~~] <br>		<li>[DFT 6.5] <br>
	Define <amsmath>\epsilon = \\|W_1 S\\|_\infty</amsmath> and <amsmath>\delta = \\| C S \\|_\infty</amsmath>, so that <amsmath>\epsilon</amsmath> is a measure of tracking performance and <amsmath>\delta</amsmath> measures control effort. Show that for every point <amsmath>s_0</amsmath> with Re <amsmath>s_0 \geq 0</amsmath>,		Define <amsmath>\epsilon = \\|W_1 S\\|_\infty</amsmath> and <amsmath>\delta = \\| C S \\|_\infty</amsmath>, so that <amsmath>\epsilon</amsmath> is a measure of tracking performance and <amsmath>\delta</amsmath> measures control effort. Show that for every point <amsmath>s_0</amsmath> with Re <amsmath>s_0 \geq 0</amsmath>,
	<center><amsmath>		<center><amsmath>
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	</li>		</li>

	<li>[DFT 6.6~~, page 104~~] <br>		<li>[DFT 6.6] <br>
	Let <amsmath>\omega</amsmath> be a frequency such that <amsmath>j \omega</amsmath> is not a pole of <amsmath>P</amsmath>. Suppose that		Let <amsmath>\omega</amsmath> be a frequency such that <amsmath>j \omega</amsmath> is not a pole of <amsmath>P</amsmath>. Suppose that
	<center><amsmath>		<center><amsmath>
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	Derive a lower bound for <amsmath>C(j\omega)</amsmath> that blows up as <amsmath>\epsilon \to 0</amsmath>. Hence good tracking at a particular frequency requires large controller gain at this frequency.		Derive a lower bound for <amsmath>C(j\omega)</amsmath> that blows up as <amsmath>\epsilon \to 0</amsmath>. Hence good tracking at a particular frequency requires large controller gain at this frequency.

	<li>[DFT 6.7~~, page 104~~]<br>		<li>[DFT 6.7]<br>
	Consider a plant with transfer function		Consider a plant with transfer function
	<center><amsmath>		<center><amsmath>

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2010-10-19T06:19:09Z

Created page with '{{CDS 212 draft HW}} {{CDS homework | instructor = J. Doyle | course = CDS 212 | semester = Fall 2010 | title = Problem Set #4 | issued = 19 Oct 2010 | due = 28 Oct 2010 }}…'

New page

{{CDS 212 draft HW}}
{{CDS homework
| instructor = J. Doyle
| course = CDS 212
| semester = Fall 2010
| title = Problem Set #4
| issued = 19 Oct 2010
| due = 28 Oct 2010
}}

=== Reading ===
* {{DFT}}, Chapter 6

=== Problems ===
<ol>
<li>[DFT 6.1, page 103] 
Show that any stable transfer function can be uniquely factored as the product of an all pass function and a minimum phase function, up to a choice of sign.
</li>

<li>[DFT 6.4, page 103] 
Let
<center><amsmath>
P(s) = 4\frac{s-2}{(s+1)^2}
</amsmath></center>
and suppose that <amsmath>C</amsmath> is an internally stabilizing controller such that <amsmath>\| S \|_\infty = 1.5.</amsmath> Give a positive lower bound for
<center><amsmath>
\max_{0 \leq \omega \leq 0.1} |S(j\omega)|.
</amsmath></center>
</li>

<li>[DFT 6.5, page 104] 
Define <amsmath>\epsilon = \|W_1 S\|_\infty</amsmath> and <amsmath>\delta = \| C S \|_\infty</amsmath>, so that <amsmath>\epsilon</amsmath> is a measure of tracking performance and <amsmath>\delta</amsmath> measures control effort. Show that for every point <amsmath>s_0</amsmath> with Re <amsmath>s_0 \geq 0</amsmath>,
<center><amsmath>
|W_1(s_0)| \leq \epsilon + |W_1 (s_0) P(s_0)|\, \delta.
</amsmath></center>
Hence <amsmath>\epsilon</amsmath> and <amsmath>\delta</amsmath> cannot both be very small and so we cannot get good tracking without exerting some control effort.
</li>

<li>[DFT 6.6, page 104] 
Let <amsmath>\omega</amsmath> be a frequency such that <amsmath>j \omega</amsmath> is not a pole of <amsmath>P</amsmath>. Suppose that
<center><amsmath>
\epsilon := |S(j\omega)| < 1.
</amsmath></center>
Derive a lower bound for <amsmath>C(j\omega)</amsmath> that blows up as <amsmath>\epsilon \to 0</amsmath>. Hence good tracking at a particular frequency requires large controller gain at this frequency.

<li>[DFT 6.7, page 104] 
Consider a plant with transfer function
<center><amsmath>
P(s) = \frac{1}{s^2 - s + 4}
</amsmath></center>
and suppose we want to design an internally stabilizing controller
such that
<ol type="a">
<li><amsmath>|S(j\omega| \leq \epsilon</amsmath> for <amsmath>0 \leq \omega \leq 0.1</amsmath> </li>
<li> <amsmath>|S(j\omega| \leq 2</amsmath> for <amsmath>0.1 \leq \omega \leq 5</amsmath> </li>
<li> <amsmath>|S(j\omega| \leq 1</amsmath> for <amsmath>5 \leq \omega \leq \infty</amsmath></li>
</ol>
Find a (positive) lower bound on the achievable <amsmath>\epsilon</amsmath>.
</li>