CDS 212, Homework 5, Fall 2010 - Revision history

Sojoudi at 22:07, 26 October 2010

2010-10-26T22:07:13Z

← Older revision		Revision as of 22:07, 26 October 2010
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	~~{{CDS 212 draft HW}}~~
	{{CDS homework		{{CDS homework
	\| instructor = J. Doyle		\| instructor = J. Doyle

Sojoudi: /* Problems */

2010-10-26T19:44:55Z

Problems

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	</amsmath></center>		</amsmath></center>
	</li>		</li>
	<li>We know that the discrete-time system <amsmath>X[k+1]=Ax[k]</amsmath> is stable (i.e, <amsmath>x[k]\rightarrow 0</amsmath> as <amsmath>k\rightarrow 0</amsmath>) if and only if all eigenvalues of <amsmath>A</amsmath> are inside the unit circle. Derive a necessary and sufficient LMI condition for stability of this system.		<li>We know that the discrete-time system <amsmath>x[k+1]=Ax[k]</amsmath> is stable (i.e, <amsmath>x[k]\rightarrow 0</amsmath> as <amsmath>k\rightarrow 0</amsmath>) if and only if all eigenvalues of <amsmath>A</amsmath> are inside the unit circle. Derive a necessary and sufficient LMI condition for stability of this system.
	</li>		</li>
	<li>Consider the following optimization problem:		<li>Consider the following optimization problem:

Sojoudi: /* Problems */

2010-10-26T08:41:38Z

Problems

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	\max_{Q,\alpha}\; \alpha		\max_{Q,\alpha}\; \alpha
	</amsmath></center>		</amsmath></center>
	~~such that~~		subject to
	<center><amsmath>		<center><amsmath>
	AQ+QA^T+\alpha Q< 0, \quad Q>0		AQ+QA^T+\alpha Q< 0, \quad Q>0
	</amsmath></center>		</amsmath></center>
	Find an analytical expression (in terms of <amsmath>A</amsmath>) for the maximum value of <amsmath>\alpha</amsmath>.</li>		Find an analytical expression (in terms of <amsmath>A</amsmath>) for the maximum value of <amsmath>\alpha</amsmath>.</li>

Sojoudi: /* Problems */

2010-10-26T08:40:16Z

Problems

← Older revision		Revision as of 08:40, 26 October 2010
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	<li> Prove the Schur complement inequality		<li> Prove the Schur complement inequality
	<center><amsmath>		<center><amsmath>
	\left[\begin{array}{ccccccc} A&B\\B^T&C \end{array}\right]~~\geq 0, C~~>0 \Longleftrightarrow A-BC^{-1}B^T\~~geq0,~~ C>0:		\left[\begin{array}{ccccccc} A&B\\B^T&C \end{array}\right]> 0 \Longleftrightarrow A-BC^{-1}B^T>0,\quad C>0
	</amsmath></center>		</amsmath></center>
	</li>		</li>
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	such that		such that
	<center><amsmath>		<center><amsmath>
	AQ+QA^T+\alpha Q~~\le~~ 0, \quad Q>0		AQ+QA^T+\alpha Q< 0, \quad Q>0
	</amsmath></center>		</amsmath></center>
	Find an analytical expression ~~for the maximum value~~ of <amsmath>~~\alpha~~</amsmath> ~~in terms~~ of <amsmath>A</amsmath>.		Find an analytical expression (in terms of <amsmath>A</amsmath>) for the maximum value of <amsmath>\alpha</amsmath>.</li>
	</li>

Sojoudi at 08:32, 26 October 2010

2010-10-26T08:32:43Z

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	</ol>		</ol>
	</li>		</li>

	<li>		<li>
	Assume <amsmath>(A,B)</amsmath> is controllable. Show that <amsmath>(F,G)</amsmath> with		Assume <amsmath>(A,B)</amsmath> is controllable. Show that <amsmath>(F,G)</amsmath> with
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	<li>[PD 4.4]<br>		<li>[PD 4.4]<br>
	Controllability gramian vs. controllability matrix. We have seen that the singular values of the controllability gramian <amsmath>X_c</amsmath> can be used to determine "how controllable" the states are. In this problem you will show that the controllability matrix		Controllability gramian vs. controllability matrix. We have seen that the singular values of the controllability gramian <amsmath>X_c</amsmath> can be used to determine "how controllable" the states are. In this problem you will show that the controllability matrix
			</li>
	<center><amsmath>		<center><amsmath>
	M_c=\left[\begin{array}{ccccccc} B&AB&A^2B&\cdots&A^{n-1}B \end{array}\right]		M_c=\left[\begin{array}{ccccccc} B&AB&A^2B&\cdots&A^{n-1}B \end{array}\right]
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	</ol>		</ol>
	</li>		</li>

	<li> Prove the Schur complement inequality		<li> Prove the Schur complement inequality
	<center><amsmath>		<center><amsmath>
	\left[\begin{array}{ccccccc} A&B\\B^T&C \end{array}\right]\geq 0, C>0 \Longleftrightarrow A-BC^{-1}B^T\geq0, C>0:		\left[\begin{array}{ccccccc} A&B\\B^T&C \end{array}\right]\geq 0, C>0 \Longleftrightarrow A-BC^{-1}B^T\geq0, C>0:
	</amsmath></center>		</amsmath></center>
			</li>
			<li>We know that the discrete-time system <amsmath>X[k+1]=Ax[k]</amsmath> is stable (i.e, <amsmath>x[k]\rightarrow 0</amsmath> as <amsmath>k\rightarrow 0</amsmath>) if and only if all eigenvalues of <amsmath>A</amsmath> are inside the unit circle. Derive a necessary and sufficient LMI condition for stability of this system.
			</li>
			<li>Consider the following optimization problem:
			<center><amsmath>
			\max_{Q,\alpha}\; \alpha
			</amsmath></center>
			such that
			<center><amsmath>
			AQ+QA^T+\alpha Q\le 0, \quad Q>0
			</amsmath></center>
			Find an analytical expression for the maximum value of <amsmath>\alpha</amsmath> in terms of <amsmath>A</amsmath>.
			</li>

Sojoudi at 08:07, 26 October 2010

2010-10-26T08:07:41Z

← Older revision		Revision as of 08:07, 26 October 2010
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	</ol>		</ol>
	</li>		</li>

			<li> Prove the Schur complement inequality
			<center><amsmath>
			\left[\begin{array}{ccccccc} A&B\\B^T&C \end{array}\right]\geq 0, C>0 \Longleftrightarrow A-BC^{-1}B^T\geq0, C>0:
			</amsmath></center>

Sojoudi: Created page with '{{CDS 212 draft HW}} {{CDS homework | instructor = J. Doyle | course = CDS 212 | semester = Fall 2010 | title = Problem Set #5 | issued = 26 Oct 2010 | due = 4 Nov 2010 }} …'

2010-10-26T07:55:38Z

Created page with '{{CDS 212 draft HW}} {{CDS homework | instructor = J. Doyle | course = CDS 212 | semester = Fall 2010 | title = Problem Set #5 | issued = 26 Oct 2010 | due = 4 Nov 2010 }} …'

New page

{{CDS 212 draft HW}}
{{CDS homework
| instructor = J. Doyle
| course = CDS 212
| semester = Fall 2010
| title = Problem Set #5
| issued = 26 Oct 2010
| due = 4 Nov 2010
}}

=== Reading ===
* [PD], Chapter 4

=== Problems ===
<ol>
<li>[PD 4.1]<br>
Suppose <amsmath>A, X</amsmath> and <amsmath>C</amsmath> satisfy <amsmath>A^*X+XA+C^*C=0.</amsmath> Show that any two of the following implies the third:
<ol type="i">
<li> <amsmath>A</amsmath> Hurwitz.</li>
<li><amsmath>(C,A)</amsmath> observable.</li>
<li><amsmath>X>0</amsmath></li>
</ol>
</li>

<li>
Assume <amsmath>(A,B)</amsmath> is controllable. Show that <amsmath>(F,G)</amsmath> with
<center><amsmath>
F=\left[\begin{array}{ccc} A&0\\C&0 \end{array}\right], G=\left[\begin{array}{ccc} B\\0 \end{array}\right],
</amsmath></center>
is controllable if and only if
<center><amsmath>
\left[\begin{array}{ccc} A&B\\C&0 \end{array}\right]
</amsmath></center>
is a full row rank matrix.
</li>
<li>[PD 4.4]<br>
Controllability gramian vs. controllability matrix. We have seen that the singular values of the controllability gramian <amsmath>X_c</amsmath> can be used to determine "how controllable" the states are. In this problem you will show that the controllability matrix

<center><amsmath>
M_c=\left[\begin{array}{ccccccc} B&AB&A^2B&\cdots&A^{n-1}B \end{array}\right]
</amsmath></center>
cannot be used for the same purpose, since its singular values are unrelated to those of <amsmath>X_c</amsmath>. In particular, construct examples (<amsmath>A\in\mathcal{C}^{2\times 2}, B\in\mathcal{C}^{2\times 1}</amsmath> suffices) such that
<ol type="a">
<li><amsmath>X_c=I</amsmath>, but <amsmath>\underline{\sigma}(M_c)</amsmath> is arbitrarily small.</li>
<li><amsmath>M_c=I</amsmath>, but <amsmath>\underline{\sigma}(X_c)</amsmath> is arbitrarily small.</li>
</ol>
</li>