CDS 140a Winter 2014 Homework 2 - Revision history

Murray at 00:02, 10 February 2014

2014-02-10T00:02:34Z

← Older revision		Revision as of 00:02, 10 February 2014
Line 5:		Line 5:
	\| title = Problem Set #2		\| title = Problem Set #2
	\| issued = 14 Jan 2014 (Tue)		\| issued = 14 Jan 2014 (Tue)
	\| due = 22 Jan ~~2011~~ (Wed) @ noon <br>Turn in to box outside Steele House		\| due = 22 Jan 2014 (Wed) @ noon <br>Turn in to box outside Steele House
	}}		}}

Murray at 03:26, 23 January 2014

2014-01-23T03:26:25Z

← Older revision		Revision as of 03:26, 23 January 2014
Line 7:		Line 7:
	\| due = 22 Jan 2011 (Wed) @ noon <br>Turn in to box outside Steele House		\| due = 22 Jan 2011 (Wed) @ noon <br>Turn in to box outside Steele House
	}}		}}

			<!-- 2014 feedback from TA office hours
			TA1: There were a lot of questions on problem 3. Also, quite a few students either had serious issues with linear algebra or even explicitly said that their linear algebra was poor, so it might be a good idea to do that linear algebra recitation.

			TA2: I also had a lot of questions on problem 3, and also a lot of questions on problem 1a. Two or three persons had some questions on how to compute generalized eigenvectors, and why there are different methods. I gave a brief explanation of how to compute generalized eigenvectors according to the method of Perko (around page 42, depending on the copy of Perko), as one of the students had questions about that, but didn't get into all of the details. That method I believe is a good method and better than other methods I've seen. An example of computing generalized eigenvectors for a 3x3 matrix seemed to be helpful to show some of the students. A lot of people (some in office hours) are not familiar with how to compute e^{Jt} for a Jordan form with off diagonal elements (ones).
			-->

	'''Note:''' In the upper left hand corner of the ''second'' page of your homework set, please put the number of hours that you spent on		'''Note:''' In the upper left hand corner of the ''second'' page of your homework set, please put the number of hours that you spent on

Murray at 04:12, 20 January 2014

2014-01-20T04:12:51Z

← Older revision		Revision as of 04:12, 20 January 2014
Line 1:		Line 1:
	{{CDS homework		{{CDS homework
	\| instructor = R. Murray~~, D. MacMartin~~		\| instructor = R. Murray
	\| course = ACM ~~101~~/AM 125b/CDS 140a		\| course = ACM 101b/AM 125b/CDS 140a
	\| semester = Winter ~~2013~~		\| semester = Winter 2014
	\| title = Problem Set #2		\| title = Problem Set #2
	\| issued = 14 Jan 2014 (Tue)		\| issued = 14 Jan 2014 (Tue)

Murray at 04:11, 20 January 2014

2014-01-20T04:11:43Z

← Older revision		Revision as of 04:11, 20 January 2014
Line 4:		Line 4:
	\| semester = Winter 2013		\| semester = Winter 2013
	\| title = Problem Set #2		\| title = Problem Set #2
	\| issued = 12 Jan ~~2011~~ (Tue)		\| issued = 14 Jan 2014 (Tue)
	\| due = 19 Jan 2011 (~~Tue~~)		\| due = 22 Jan 2011 (Wed) @ noon <br>Turn in to box outside Steele House
	}}		}}

Murray at 05:47, 14 January 2014

2014-01-14T05:47:14Z

← Older revision		Revision as of 05:47, 14 January 2014
Line 7:		Line 7:
	\| due = 19 Jan 2011 (Tue)		\| due = 19 Jan 2011 (Tue)
	}}		}}

	~~{{warning\|This homework set is still in preparation. This banner will be removed when it is in its final form}}~~

	'''Note:''' In the upper left hand corner of the ''second'' page of your homework set, please put the number of hours that you spent on		'''Note:''' In the upper left hand corner of the ''second'' page of your homework set, please put the number of hours that you spent on

Murray at 05:45, 14 January 2014

2014-01-14T05:45:45Z

← Older revision		Revision as of 05:45, 14 January 2014
Line 61:		Line 61:
	</li>		</li>

	<li> '''Perko, Section 1.9, problem 5, parts (c), ~~(d1) and~~ (d2)''': Let <amsmath>A</amsmath> be an <amsmath>n \times n</amsmath> nonsingular matrix and let <amsmath>x(t)</amsmath> be the solution of the initial value problem (1) with <amsmath>x(0) = x_0</amsmath>. Show that<br>		<li> '''Perko, Section 1.9, problem 5, parts (c), (d2)''': Let <amsmath>A</amsmath> be an <amsmath>n \times n</amsmath> nonsingular matrix and let <amsmath>x(t)</amsmath> be the solution of the initial value problem (1) with <amsmath>x(0) = x_0</amsmath>. Show that<br>

	(c) If <amsmath>x_0 \in E^c</amsmath>, <amsmath>x_0 \neq 0</amsmath> and <amsmath>A</amsmath> is semisimple, then there are postive constants <amsmath>m</amsmath> and <amsmath>M</amsmath> such that for all <amsmath>t \in R</amsmath>, <amsmath>m \leq \|x(t)\| \leq M</amsmath>;		(c) If <amsmath>x_0 \in E^c</amsmath>, <amsmath>x_0 \neq 0</amsmath> and <amsmath>A</amsmath> is semisimple, then there are postive constants <amsmath>m</amsmath> and <amsmath>M</amsmath> such that for all <amsmath>t \in R</amsmath>, <amsmath>m \leq \|x(t)\| \leq M</amsmath>;
	:* Note: in the book, Perko defines <amsmath>\sim</amsmath> to mean "set subtraction". So <amsmath>E \sim \{0\}</amsmath> in the book is the set <amsmath>E</amsmath> minus the point 0.		:* Note: in the book, Perko defines <amsmath>\sim</amsmath> to mean "set subtraction". So <amsmath>E \sim \{0\}</amsmath> in the book is the set <amsmath>E</amsmath> minus the point 0.

	(d1) If <amsmath>x_0 \in E^c</amsmath> , <amsmath>x_0 \neq 0</amsmath> and the projection of <amsmath>A</amsmath> onto the center subspace is not semisimple, then there is an <amsmath>x_0 \in {\mathbb R}^n</amsmath> such that <amsmath>\lim_{t \to \pm \infty} \|x(t)\| = \infty;</amsmath>
	~~:* Note: there is a bug in the statement in Perko; use the modified version here~~
	~~:* Hint: put the system into Jordan form; the center subspace is now the set of coordinate directions corresponding to eigenvalues with zero real part.~~

	(d2) If <amsmath>E^s \neq \{0\}</amsmath>, <amsmath>E^u \neq \{0\}</amsmath>, and <amsmath>x_0</amsmath> has nonzero components in both <amsmath>E^s</amsmath> and <amsmath>E^u</amsmath>, then <amsmath>\lim_{t \to \pm \infty} \|x(t)\| = \infty</amsmath>;		(d2) If <amsmath>E^s \neq \{0\}</amsmath>, <amsmath>E^u \neq \{0\}</amsmath>, and <amsmath>x_0</amsmath> has nonzero components in both <amsmath>E^s</amsmath> and <amsmath>E^u</amsmath>, then <amsmath>\lim_{t \to \pm \infty} \|x(t)\| = \infty</amsmath>;
Line 93:		Line 89:
	(a) Show that the unforced system (<amsmath>u = 0</amsmath>) is stable but not asymptotically stable.		(a) Show that the unforced system (<amsmath>u = 0</amsmath>) is stable but not asymptotically stable.

	(b) Given <amsmath>x(0) = x_0</amsmath> and <amsmath>u(t) = cos(\omega*t)</amsmath>, solve for the output <amsmath>y(t)</amsmath>. Show that when <amsmath>\omega=1</amsmath> the output is unbounded.		(b) Given <amsmath>x(0) = x_0</amsmath> and <amsmath>u(t) = \cos(\omega*t)</amsmath>, solve for the output <amsmath>y(t)</amsmath>. Show that when <amsmath>\omega=1</amsmath> the output is unbounded.
	</li>		</li>
	</ol>		</ol>

Murray at 05:43, 14 January 2014

2014-01-14T05:43:22Z

← Older revision		Revision as of 05:43, 14 January 2014
Line 93:		Line 93:
	(a) Show that the unforced system (<amsmath>u = 0</amsmath>) is stable but not asymptotically stable.		(a) Show that the unforced system (<amsmath>u = 0</amsmath>) is stable but not asymptotically stable.

	(b) ~~Show that if the system is driven by the bounded input~~ <amsmath>u = \cos(\omega t)</amsmath> ~~then~~ the output ~~is unbounded for some values of~~ <amsmath>\omega</amsmath>.		(b) Given <amsmath>x(0) = x_0</amsmath> and <amsmath>u(t) = cos(\omega*t)</amsmath>, solve for the output <amsmath>y(t)</amsmath>. Show that when <amsmath>\omega=1</amsmath> the output is unbounded.
	</li>		</li>
	</ol>		</ol>

Murray at 05:36, 14 January 2014

2014-01-14T05:36:26Z

← Older revision		Revision as of 05:36, 14 January 2014
Line 73:		Line 73:
	</li>		</li>

			<!-- Replaced in 2014 with problem below
	<li> '''Perko, Section 1.10, problem 2''': Use Theorem 1 in Section 1.10 to solve the nonhomogeneous linear system		<li> '''Perko, Section 1.10, problem 2''': Use Theorem 1 in Section 1.10 to solve the nonhomogeneous linear system
	<center><amsmath>		<center><amsmath>
Line 82:		Line 83:
	</amsmath></center>		</amsmath></center>
	</li>		</li>
			-->

	<li>		<li>

Murray at 01:33, 13 January 2014

2014-01-13T01:33:28Z

@@ Line 81: / Line 81: @@
 x(0) = \begin{bmatrix} 1 \\ 0 \end{bmatrix}.
 </amsmath></center>
 </li>
 </ol>

Murray at 00:08, 13 January 2014

2014-01-13T00:08:30Z

← Older revision		Revision as of 00:08, 13 January 2014
Line 64:		Line 64:

	(c) If <amsmath>x_0 \in E^c</amsmath>, <amsmath>x_0 \neq 0</amsmath> and <amsmath>A</amsmath> is semisimple, then there are postive constants <amsmath>m</amsmath> and <amsmath>M</amsmath> such that for all <amsmath>t \in R</amsmath>, <amsmath>m \leq \|x(t)\| \leq M</amsmath>;		(c) If <amsmath>x_0 \in E^c</amsmath>, <amsmath>x_0 \neq 0</amsmath> and <amsmath>A</amsmath> is semisimple, then there are postive constants <amsmath>m</amsmath> and <amsmath>M</amsmath> such that for all <amsmath>t \in R</amsmath>, <amsmath>m \leq \|x(t)\| \leq M</amsmath>;
	:* Note: in the book, Perko defines <amsmath>\sim</amsmath> to mean "set subtraction". So <amsmath>E \sim \{0\}</amsmath> in the book is the set <amsmath>E</amsmath> minus the point 0.		:* Note: in the book, Perko defines <amsmath>\sim</amsmath> to mean "set subtraction". So <amsmath>E \sim \{0\}</amsmath> in the book is the set <amsmath>E</amsmath> minus the point 0.

	(d1) If <amsmath>x_0 \in E^c</amsmath> , <amsmath>x_0 \neq 0</amsmath> and the projection of <amsmath>A</amsmath> onto the center subspace is not semisimple, then there is an <amsmath>x_0 \in {\mathbb R}^n</amsmath> such that <amsmath>\lim_{t \to \pm \infty} \|x(t)\| = \infty;</amsmath>		(d1) If <amsmath>x_0 \in E^c</amsmath> , <amsmath>x_0 \neq 0</amsmath> and the projection of <amsmath>A</amsmath> onto the center subspace is not semisimple, then there is an <amsmath>x_0 \in {\mathbb R}^n</amsmath> such that <amsmath>\lim_{t \to \pm \infty} \|x(t)\| = \infty;</amsmath>