CDS 212, Homework 1, Fall 2010 - Revision history

Sojoudi at 17:42, 28 September 2010

2010-09-28T17:42:54Z

← Older revision		Revision as of 17:42, 28 September 2010
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	~~{{CDS 212 draft HW}}~~
	{{CDS homework		{{CDS homework
	\| instructor = J. Doyle		\| instructor = J. Doyle

Sojoudi at 05:48, 27 September 2010

2010-09-27T05:48:30Z

← Older revision		Revision as of 05:48, 27 September 2010
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	{{CDS homework		{{CDS homework
	\| instructor = J. Doyle		\| instructor = J. Doyle
	\| course = CDS ~~112~~		\| course = CDS 212
	\| semester = Fall 2010		\| semester = Fall 2010
	\| title = Problem Set #1		\| title = Problem Set #1

Sojoudi: /* Problems */

2010-09-27T05:39:16Z

Problems

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 </amsmath></center>
 where <amsmath>\|\cdot\|</amsmath> is any norm on signals.
 </li>

Murray: /* Problems */

2010-09-26T01:32:52Z

Problems

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	<ol>		<ol>
	<li>DFT 2.1, page 28<br>		<li>[DFT 2.1, page 28<br>
	Suppose that <amsmath>u(t)</amsmath> is a continuous signal whose derivative <amsmath>\dot		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		u(t)</amsmath> is also continuous. Which of the following quantities qualifies
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	</li>		</li>

	<li> DFT 2.4, page 29] <br>		<li> [DFT 2.4, page 29] <br>
	Let <amsmath>D</amsmath> be a pure time delay of <amsmath>\tau</amsmath> seconds with transfer function		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		<amsmath> \widehat D(s) = e^{-s \tau} </amsmath>. A norm <amsmath>\\|\cdot\\|</amsmath> on transfer functions is ''time-delay invariant'' if for
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	</li>		</li>

	<li> DFT 2.7, page 30] <br>		<li> [DFT 2.7, page 30] <br>
	Derive the <amsmath>\infty</amsmath>-norm to <amsmath>\infty</amsmath>-norm system gain for a stable,		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		proper plant <amsmath>\widehat G</amsmath>. (Hint: write <amsmath>\widehat G = c + \widehat G_1</amsmath> where <amsmath>c</amsmath> is a constant

Murray: /* Problems */

2010-09-18T18:59:47Z

Problems

← Older revision		Revision as of 18:59, 18 September 2010
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	<amsmath> \widehat D(s) = e^{-s \tau} </amsmath>. A norm <amsmath>\\|\cdot\\|</amsmath> on transfer functions is ''time-delay invariant'' if for		<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		every bounded transfer function <amsmath>\widehat G</amsmath> and every <amsmath>\tau > 0</amsmath> we have
	<~~ol type=""~~>		<center><amsmath>
	<amsmath>~~\textstyle~~ \\| \widehat D \widehat G \\| = \\| \widehat G \\| </amsmath>		\\| \widehat D \widehat G \\| = \\| \widehat G \\|
	</ol>		</amsmath></center>
	Determine if the 2-norm and <amsmath>\infty</amsmath>-norm are time-delay invariant.		Determine if the 2-norm and <amsmath>\infty</amsmath>-norm are time-delay invariant.
	</li>		</li>

Murray: /* Problems */

2010-09-18T18:58:47Z

Problems

← Older revision		Revision as of 18:58, 18 September 2010
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	<li> DFT 2.4, page 29] <br>		<li> DFT 2.4, page 29] <br>
	Let <amsmath>D</amsmath> be a pure time delay of <amsmath>\tau</amsmath> seconds with transfer function		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 ~~<amsmath> \emph {~~time-delay invariant~~}</amsmath>~~ if for		<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		every bounded transfer function <amsmath>\widehat G</amsmath> and every <amsmath>\tau > 0</amsmath> we have
	<ol type="">		<ol type="">
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	<amsmath>\widehat G(s) = \frac{s+2}{4s + 1}</amsmath>		<amsmath>\widehat G(s) = \frac{s+2}{4s + 1}</amsmath>
	and input <amsmath>u</amsmath> and output <amsmath>y</amsmath>. Compute		and input <amsmath>u</amsmath> and output <amsmath>y</amsmath>. Compute
	<ol>		<center><amsmath>
	<amsmath>\\| G \\|_1 = \sup_{\\|u\\|_\infty = 1} \\| y \\|_\infty</amsmath>		\\| G \\|_1 = \sup_{\\|u\\|_\infty = 1} \\| y \\|_\infty
	</ol>		</amsmath></center>
	and find an input which achieves the supremum.		and find an input which achieves the supremum.
	</li>		</li>
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	<li> [DFT 2.12, page 30] <br>		<li> [DFT 2.12, page 30] <br>
	For a linear system with input <amsmath>u</amsmath> and output <amsmath>y</amsmath>, prove that		For a linear system with input <amsmath>u</amsmath> and output <amsmath>y</amsmath>, prove that
	<ol>		<center><amsmath>
	<amsmath>\sup_{\\|u\\| \leq 1} \\| y \\| =		\sup_{\\|u\\| \leq 1} \\| y \\| =
	\sup_{\\|u\\| = 1} \\| y \\|</amsmath>		\sup_{\\|u\\| = 1} \\| y \\|
	</ol>		</amsmath></center>
	where <amsmath>\\|\cdot\\|</amsmath> is any norm on signals.		where <amsmath>\\|\cdot\\|</amsmath> is any norm on signals.
	</li>		</li>

	<li>Consider a second order mechanical system with transfer function		<li>Consider a second order mechanical system with transfer function
	<ol>		<center><amsmath>
	<amsmath> \widehat G(s) = \frac{1}{s^2 + 2 \omega_n \zeta s + \omega_n^2}</amsmath>		\widehat G(s) = \frac{1}{s^2 + 2 \omega_n \zeta s + \omega_n^2}
	</ol>		</amsmath></center>
	(<amsmath>\omega_n</amsmath> is the natural frequence of the system and <amsmath>\zeta</amsmath> is the		(<amsmath>\omega_n</amsmath> is the natural frequence of the system and <amsmath>\zeta</amsmath> is the
	damping ratio). Setting <amsmath>\omega_n = 1</amsmath>, write a short MATLAB		damping ratio). Setting <amsmath>\omega_n = 1</amsmath>, write a short MATLAB

Sojoudi: /* Reading */

2010-09-18T18:06:46Z

Reading

← Older revision		Revision as of 18:06, 18 September 2010
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	=== Reading ===		=== Reading ===
	* {{DFT}}, ~~Chapterss~~ 1 and 2		* {{DFT}}, Chapters 1 and 2
	* Dullerud and Paganini, Ch 3		* Dullerud and Paganini, Ch 3

Sojoudi: /* Problems */

2010-09-18T18:05:18Z

Problems

← Older revision		Revision as of 18:05, 18 September 2010
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	<li> [DFT 2.5, page 30] <br>		<li> [DFT 2.5, page 30] <br>
	Compute the 1-norm of the impluse response corresponding to the		Compute the 1-norm of the impluse response corresponding to the
	transfer function <amsmath> \frac{1}{\tau s + 1} \~~qquad~~ \tau > 0 </amsmath>.		transfer function <amsmath> \frac{1}{\tau s + 1}, \quad \tau > 0 </amsmath>.
	</li>		</li>

	<li> DFT 2.7, page 30] <br> Derive the <amsmath>\infty</amsmath>-norm to <amsmath>\infty</amsmath>-norm system gain for a stable,		<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		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.)		and <amsmath>\widehat G_1</amsmath> is strictly proper.)
	</li>		</li>

	<li> [DFT 2.8, page 30] <br> Let <amsmath>\widehat G</amsmath> be a stable, proper plant (but not necessarily strictly proper).		<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">		<ol type="a">
	<li> Show that the <amsmath>\infty</amsmath>-norm of the output <amsmath>y</amsmath> given an input		<li> Show that the <amsmath>\infty</amsmath>-norm of the output <amsmath>y</amsmath> given an input
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	</li>		</li>

	<li>		<li>Consider a second order mechanical system with transfer function
	Consider a second order mechanical system with transfer function
	<ol>		<ol>
	<amsmath> \widehat G(s) = \frac{1}{s^2 + 2 \omega_n \zeta s + \omega_n^2}</amsmath>		<amsmath> \widehat G(s) = \frac{1}{s^2 + 2 \omega_n \zeta s + \omega_n^2}</amsmath>

Sojoudi: /* Problems */

2010-09-18T18:01:56Z

Problems

← Older revision		Revision as of 18:01, 18 September 2010
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	as a norm for <amsmath>u</amsmath>:		as a norm for <amsmath>u</amsmath>:
	<ol type="a">		<ol type="a">
	<li> <amsmath>\textstyle \sup_t \|\dot u(t)\|</amsmath></li>		<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 \|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 \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>		<li> <amsmath>\textstyle \sup_t \|u(t)\| + \sup_t \|\dot u(t)\|</amsmath> </li>
	</ol>		</ol>
	Make sure to give a thorough answer (not just yes or no).		Make sure to give a thorough answer (not just yes or no).
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	<li> DFT 2.4, page 29] <br>		<li> DFT 2.4, page 29] <br>
	Let <amsmath>D</amsmath> be a pure time delay of <amsmath>\tau</amsmath> seconds with transfer function		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 \em {time-delay invariant} if for		<amsmath> \widehat D(s) = e^{-s \tau} </amsmath>. A norm <amsmath>\\|\cdot\\|</amsmath> on transfer functions is <amsmath> \emph {time-delay invariant}</amsmath> if for
	every bounded transfer function <amsmath>\widehat G</amsmath> and every <amsmath>\tau > 0</amsmath> we have		every bounded transfer function <amsmath>\widehat G</amsmath> and every <amsmath>\tau > 0</amsmath> we have
	<ol type="">		<ol type="">

Sojoudi: /* Problems */

2010-09-18T17:54:51Z

Problems

← Older revision		Revision as of 17:54, 18 September 2010
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	Suppose that <amsmath>u(t)</amsmath> is a continuous signal whose derivative <amsmath>\dot		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		u(t)</amsmath> is also continuous. Which of the following quantities qualifies
	as a norm for $u$:		as a norm for <amsmath>u</amsmath>:
	<ol type="a">		<ol type="a">
	<li> <amsmath>\textstyle \sup_t \|\dot u(t)\|</amsmath></li>		<li> <amsmath>\textstyle \sup_t \|\dot u(t)\|</amsmath></li>
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	<li> DFT 2.4, page 29] <br>		<li> DFT 2.4, page 29] <br>
	Let $D$ be a pure time delay of $\tau$ seconds with transfer function		Let <amsmath>D</amsmath> be a pure time delay of <amsmath>\tau</amsmath> seconds with transfer function
	~~\begin{displaymath}~~		<amsmath> \widehat D(s) = e^{-s \tau} </amsmath>. A norm <amsmath>\\|\cdot\\|</amsmath> on transfer functions is \em {time-delay invariant} if for
	\widehat D(s) = e^{-s \tau}.		every bounded transfer function <amsmath>\widehat G</amsmath> and every <amsmath>\tau > 0</amsmath> we have
	~~\end{displaymath}~~		<ol type="">
	A norm $\\|\cdot\\|$ on transfer functions is {\em time-delay invariant} if for		<amsmath>\textstyle \\| \widehat D \widehat G \\| = \\| \widehat G \\| </amsmath>
	every bounded transfer function $\widehat G$ and every $\tau > 0$ we have		</ol>
	\~~begin{displaymath}~~		Determine if the 2-norm and <amsmath>\infty</amsmath>-norm are time-delay invariant.
	\\| \widehat D \widehat G \\| = \\| \widehat G \\|
	~~\end{displaymath}~~
	Determine if the 2-norm and $\infty$-norm are time-delay invariant.
	</li>		</li>

	<li> [DFT 2.5, page 30] <br>		<li> [DFT 2.5, page 30] <br>
	Compute the 1-norm of the impluse response corresponding to the		Compute the 1-norm of the impluse response corresponding to the
	transfer function		transfer function <amsmath> \frac{1}{\tau s + 1} \qquad \tau > 0 </amsmath>.
	\~~begin{displaymath}~~
	~~\fract~~{1}{\tau s + 1} \qquad \tau > 0.
	</li>		</li>

	<li> DFT 2.7, page 30] <br> Derive the $\infty$-norm to $\infty$-norm system gain for a stable,		<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 $\widehat G$. (Hint: write $\widehat G = c + \widehat G_1$ where $c$ is a constant		proper plant <amsmath>\widehat G</amsmath>. (Hint: write <amsmath>\widehat G = c + \widehat G_1</amsmath> where <amsmath>c</amsmath> is a constant
	and $\widehat G_1$ is strictly proper.)		and <amsmath>\widehat G_1</amsmath> is strictly proper.)
	</li>		</li>

	<li> [DFT 2.8, page 30] <br> Let $\widehat G$ be a stable, proper plant (but not necessarily strictly proper).		<li> [DFT 2.8, page 30] <br> Let <amsmath>\widehat G</amsmath> be a stable, proper plant (but not necessarily strictly proper).
	# Show that the $\infty$-norm of the output $y$ given an input		<ol type="a">
	$u(t) = \sin(\omega t)$ is $\|\widehat G(jw)\|$.		<li> Show that the <amsmath>\infty</amsmath>-norm of the output <amsmath>y</amsmath> given an input
	# Show that the 2-norm to 2-norm system gain for $\widehat G$ is $\\|		<amsmath>u(t) = \sin(\omega t)</amsmath> is <amsmath>\|\widehat G(jw)\|</amsmath>.</li>
	\widehat G \\|_\infty$ (just as in the strictly proper case).		<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>

	<li>[DFT 2.11, page 30] <br>		<li>[DFT 2.11, page 30] <br>
	Consider a system with transfer function		Consider a system with transfer function
	~~\begin{displaymath}~~		<amsmath>\widehat G(s) = \frac{s+2}{4s + 1}</amsmath>
	\widehat G(s) = \~~fract~~{s+2}{4s + 1}		and input <amsmath>u</amsmath> and output <amsmath>y</amsmath>. Compute
	~~\end{displaymath}~~		<ol>
	and input $u$ and output $y$. Compute		<amsmath>\\| G \\|_1 = \sup_{\\|u\\|_\infty = 1} \\| y \\|_\infty</amsmath>
	~~\begin{displaymath}~~		</ol>
	\\| G \\|_1 = \sup_{\\|u\\|_\infty = 1} \\| y \\|_\infty
	~~\end{displaymath}~~
	and find an input which achieves the supremum.		and find an input which achieves the supremum.
	</li>		</li>

	<li> [DFT 2.12, page 30] <br>		<li> [DFT 2.12, page 30] <br>
	For a linear system with input $u$ and output $y$, prove that		For a linear system with input <amsmath>u</amsmath> and output <amsmath>y</amsmath>, prove that
	~~\begin{displaymath}~~		<ol>
	\sup_{\\|u\\| \leq 1} \\| y \\| =		<amsmath>\sup_{\\|u\\| \leq 1} \\| y \\| =
	\sup_{\\|u\\| = 1} \\| y \\|		\sup_{\\|u\\| = 1} \\| y \\|</amsmath>
	~~\end{displaymath}~~		</ol>
	where $\\|\cdot\\|$ is any norm on signals.		where <amsmath>\\|\cdot\\|</amsmath> is any norm on signals.
	</li>		</li>

	<li>		<li>
	Consider a second order mechanical system with transfer function		Consider a second order mechanical system with transfer function
	~~\begin{displaymath}~~		<ol>
	\widehat G(s) = \~~fract~~{1}{s^2 + 2 \omega_n \zeta s + \omega_n^2}		<amsmath> \widehat G(s) = \frac{1}{s^2 + 2 \omega_n \zeta s + \omega_n^2}</amsmath>
	~~\end{displaymath}~~		</ol>
	($\omega_n$ is the natural frequence of the system and $\zeta$ is the		(<amsmath>\omega_n</amsmath> is the natural frequence of the system and <amsmath>\zeta</amsmath> is the
	damping ratio). Setting $\omega_n = 1$, write a short MATLAB		damping ratio). Setting <amsmath>\omega_n = 1</amsmath>, write a short MATLAB
	program to generate a plot of the $\infty$-norm as a function of the		program to generate a plot of the <amsmath>\infty</amsmath>-norm as a function of the
	damping ratio $\zeta > 0$.		damping ratio <amsmath>\zeta > 0</amsmath>.
	</li>		</li>