Sojoudi at 22:33, 12 October 2010

2010-10-12T22:33:57Z

← Older revision		Revision as of 22:33, 12 October 2010
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	~~{{CDS 212 draft HW}}~~
	{{CDS homework		{{CDS homework
	\| instructor = J. Doyle		\| instructor = J. Doyle

Sojoudi: /* Problems */

2010-10-12T18:21:42Z

Problems

← Older revision		Revision as of 18:21, 12 October 2010
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	We will assume all plants in the uncertainty set have the same unstable		We will assume all plants in the uncertainty set have the same unstable
	poles.		poles.
			[[Image:HW3-P5.png\|center\|800px]]

	<ol type="a">		<ol type="a">

Sojoudi: /* Reading */

2010-10-12T05:47:26Z

Reading

← Older revision		Revision as of 05:47, 12 October 2010
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	=== Reading ===		=== Reading ===
	* {{DFT}}, ~~Chapters~~ 4		* {{DFT}}, Chapter 4

	=== Problems ===		=== Problems ===

Sojoudi: Created page with '{{CDS 212 draft HW}} {{CDS homework | instructor = J. Doyle | course = CDS 212 | semester = Fall 2010 | title = Problem Set #3 | issued = 12 Oct 2010 | due = 21 Oct 2010 }}…'

2010-10-12T05:46:59Z

Created page with '{{CDS 212 draft HW}} {{CDS homework | instructor = J. Doyle | course = CDS 212 | semester = Fall 2010 | title = Problem Set #3 | issued = 12 Oct 2010 | due = 21 Oct 2010 }}…'

New page

{{CDS 212 draft HW}}
{{CDS homework
| instructor = J. Doyle
| course = CDS 212
| semester = Fall 2010
| title = Problem Set #3
| issued = 12 Oct 2010
| due = 21 Oct 2010
}}

=== Reading ===
* {{DFT}}, Chapters 4

=== Problems ===
<ol>
<li>[DFT 4.1, page 62]<br>
Consider a unity feedback system. True or false: If a controller internally
stabilizes two plants, they have the same number of poles in Re <amsmath> s \geq 0</amsmath>.
</li>

<li> [DFT 4.2, page 62] <br>
Unity-feedback problem: Let <amsmath>P_{\alpha}(s)</amsmath> be a plant depending on a
real parameter <amsmath>\alpha</amsmath>. Suppose that the poles of <amsmath>P_{\alpha}</amsmath> move
continuously as <amsmath>\alpha</amsmath> varies over the interval <amsmath>[0, \ 1]</amsmath>. True or false:
If a controller internally stabilizes <amsmath>P_{\alpha}</amsmath> for every <amsmath>\alpha</amsmath>
in <amsmath>[0, \ 1]</amsmath>, then <amsmath>P_{\alpha}</amsmath> has the same number of poles in Re <amsmath>s \geq 0</amsmath>
for every <amsmath>\alpha</amsmath> in <amsmath>[0, \ 1]</amsmath>.
</li>

<li>[DFT 4.6, page 63]<br>
Consider the unity feedback system with <amsmath>C(s) = 10 </amsmath> and plant
<center><amsmath>
P(s) = \frac{1}{s-a},
</amsmath></center>
where <amsmath>a</amsmath> is real.
<ol type="a">
<li> Find the range of <amsmath>a</amsmath> for the system to be internally stable.</li>
<li> For <amsmath>a = 0</amsmath> the plant is <amsmath>P(s) = 1/s</amsmath>. Regarding <amsmath>a</amsmath> as a perturbations, we can write the plant as
<center><amsmath>
\widetilde P = \frac{P}{1 + \Delta W_2 P}
</amsmath></center> with <amsmath>W_2(s) = -a</amsmath>. Then <amsmath>\widetilde P</amsmath> equals the true plant when <amsmath>\Delta(s) = 1</amsmath>. Apply robust stability theory to see when the feedback system <amsmath>\widetilde{P}</amsmath> is internally stable for all <amsmath>\| \Delta\|_\infty \leq 1.</amsmath> Compare this to your result for part (a).</li>
</ol>
</li>

<li> [DFT 4.10, page 64]<br>
Suppose that the plant transfer function is
<center><amsmath>
\tilde{P}(s) = [1 + \Delta(s)W_2(s)]P(s),
</amsmath></center>
where
<center><amsmath>
W_2(s) = \frac{2}{s + 10}, \ P(s) = \frac{1}{s-1},
</amsmath></center>
and the stable perturbation <amsmath>\Delta</amsmath> satisfies <amsmath>\| \Delta \|_{\infty} \leq 2</amsmath>.
Suppose that the controller is the pure gain <amsmath>C(s) = k</amsmath>. We want the feedback system to be internally stable for all such perturbations. Determine over what range of <amsmath>k</amsmath> this is true.
</li>

<li>
Consider the feedback system in the figure below. Uncertainty in the
plant is described using two stable weighting functions <amsmath>W_1</amsmath> and <amsmath>W_2</amsmath>,
and stable <amsmath>\Delta_1</amsmath> and <amsmath>\Delta_2</amsmath> satisfying
<amsmath>\| \Delta_1 \|_{\infty} \leq 1</amsmath> and <amsmath>\| \Delta_2 \|_{\infty} \leq 1</amsmath>.
We will assume all plants in the uncertainty set have the same unstable
poles.

<ol type="a">
<li> Write down an expression for <amsmath>\tilde{P}</amsmath> (dotted box). If
<amsmath>L = PC</amsmath> and <amsmath>\tilde{L} = \tilde{P}C</amsmath>, write an expression for <amsmath>1 + \tilde{L}</amsmath> with <amsmath>1 + L</amsmath> as a factor.</li>
<li> State and prove a necessary and sufficient condition for robust stability of the closed loop system as a function of <amsmath>P</amsmath>, <amsmath>C</amsmath> and the sensitivity functions.</li>
<li> Apply the condition to the case <amsmath>P=1/(s-1)</amsmath>, <amsmath>C=0.5</amsmath> and <amsmath>W_1 = W_2 = 0.1</amsmath>.</li>
<ol>
</li>

CDS 212, Homework 3, Fall 2010 - Revision history

Sojoudi at 22:33, 12 October 2010

Sojoudi: /* Problems */

Sojoudi: /* Reading */

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