CDS 140a Winter 2011 Homework 8 - Revision history

Murray at 01:03, 6 March 2011

2011-03-06T01:03:22Z

← Older revision		Revision as of 01:03, 6 March 2011
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	<nowiki>*</nowiki> Part c is ''optional'', not extra credit. If you attempt this problem, one of the instructors will look at your solution and give you feedback. This problem is quite difficult, so you should only do this if you complete the rest of the set quickly and don't have any other homework or studying that is a higher priority before the exam is due. This problem is related to a patent that was filed by Caltech on how to mitigate magnitude and rate limits in active control of rotating stall.		<nowiki>*</nowiki> Part c is ''optional'', not extra credit. If you attempt this problem, one of the instructors will look at your solution and give you feedback. This problem is quite difficult, so you should only do this if you complete the rest of the set quickly and don't have any other homework or studying that is a higher priority before the exam is due. This problem is related to a patent that was filed by Caltech on how to mitigate magnitude and rate limits in active control of rotating stall.
	</li></ol>		</li></ol>

	'''Note:''' this homework set is due at 5 pm on the last day of classes. You can turn in during class on Tue (8 Mar) or put in the box outside Richard Murray's door (109 Steele) on Wed by 5 pm. Homeworks will ''not'' be graded before the final is due, so please make a copy of your solutions if you wish to have them available for the exam.

Murray at 01:02, 6 March 2011

2011-03-06T01:02:47Z

← Older revision		Revision as of 01:02, 6 March 2011
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	the system avoids hysteresis.		the system avoids hysteresis.
	</li></ol>		</li></ol>
	* ~~This problem~~ is ''optional'', not extra credit. If you attempt this problem, one of the instructors will look at your solution and give you feedback. This problem is quite difficult, so you should only do this if you complete the rest of the set quickly and don't have any other homework or studying that is a higher priority before the exam is due. This problem is related to a patent that was filed by Caltech on how to mitigate magnitude and rate limits in active control of rotating stall.		<nowiki>*</nowiki> Part c is ''optional'', not extra credit. If you attempt this problem, one of the instructors will look at your solution and give you feedback. This problem is quite difficult, so you should only do this if you complete the rest of the set quickly and don't have any other homework or studying that is a higher priority before the exam is due. This problem is related to a patent that was filed by Caltech on how to mitigate magnitude and rate limits in active control of rotating stall.
	</li></ol>		</li></ol>

	'''Note:''' this homework set is due at 5 pm on the last day of classes. You can turn in during class on Tue (8 Mar) or put in the box outside Richard Murray's door (109 Steele) on Wed by 5 pm. Homeworks will ''not'' be graded before the final is due, so please make a copy of your solutions if you wish to have them available for the exam.		'''Note:''' this homework set is due at 5 pm on the last day of classes. You can turn in during class on Tue (8 Mar) or put in the box outside Richard Murray's door (109 Steele) on Wed by 5 pm. Homeworks will ''not'' be graded before the final is due, so please make a copy of your solutions if you wish to have them available for the exam.

Murray at 01:02, 6 March 2011

2011-03-06T01:02:02Z

← Older revision		Revision as of 01:02, 6 March 2011
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	<li> Suppose that we can modulate the throttle, so that <amsmath>\gamma = \gamma_0 + u</amsmath>. Analyze the performance of the system using the Liaw-Abed control law <amsmath>u = k J</amsmath>. Show that if we choose <amsmath>k</amsmath> sufficiently large to cause the bifurcation to stall to be supercritical.</li>		<li> Suppose that we can modulate the throttle, so that <amsmath>\gamma = \gamma_0 + u</amsmath>. Analyze the performance of the system using the Liaw-Abed control law <amsmath>u = k J</amsmath>. Show that if we choose <amsmath>k</amsmath> sufficiently large to cause the bifurcation to stall to be supercritical.</li>

	<li> (Optional) Suppose that we impose magnitude and rate limits on <amsmath>u</amsmath>:		<li> (Optional*) Suppose that we impose magnitude and rate limits on <amsmath>u</amsmath>:
	<center><amsmath>		<center><amsmath>
	\|u\| \leq 1, \qquad \|\dot u\| \leq 1.		\|u\| \leq 1, \qquad \|\dot u\| \leq 1.
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	the system avoids hysteresis.		the system avoids hysteresis.
	</li></ol>		</li></ol>
			* This problem is ''optional'', not extra credit. If you attempt this problem, one of the instructors will look at your solution and give you feedback. This problem is quite difficult, so you should only do this if you complete the rest of the set quickly and don't have any other homework or studying that is a higher priority before the exam is due. This problem is related to a patent that was filed by Caltech on how to mitigate magnitude and rate limits in active control of rotating stall.
	</li></ol>		</li></ol>


	'''Note:''' this homework set is due at 5 pm on the last day of classes. You can turn in during class on Tue (8 Mar) or put in the box outside Richard Murray's door (109 Steele) on Wed by 5 pm. Homeworks will ''not'' be graded before the final is due, so please make a copy of your solutions if you wish to have them available for the exam.		'''Note:''' this homework set is due at 5 pm on the last day of classes. You can turn in during class on Tue (8 Mar) or put in the box outside Richard Murray's door (109 Steele) on Wed by 5 pm. Homeworks will ''not'' be graded before the final is due, so please make a copy of your solutions if you wish to have them available for the exam.

Murray at 23:53, 4 March 2011

2011-03-04T23:53:56Z

← Older revision		Revision as of 23:53, 4 March 2011
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	\| title = Problem Set #8		\| title = Problem Set #8
	\| issued = 26 Feb (Sat)		\| issued = 26 Feb (Sat)
	\| due = 9 Mar 2011 (Tue)		\| due = 8 Mar 2011 (Tue)
	}}		}}

Murray at 06:23, 4 March 2011

2011-03-04T06:23:30Z

← Older revision		Revision as of 06:23, 4 March 2011
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	~~{{HW draft}}~~
	{{CDS homework		{{CDS homework
	\| instructor = R. Murray, G. Buzi		\| instructor = R. Murray, G. Buzi

Murray at 06:18, 4 March 2011

2011-03-04T06:18:33Z

← Older revision		Revision as of 06:18, 4 March 2011
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	</ol>		</ol>
	</li>		</li>
	<li> Perko, Section 4.3, Exercise 7</li>		<li> Perko, Section 4.2, Exercise 7</li>
	<li> Perko, Section 4.4, Exercise 1</li>		<li> Perko, Section 4.4, Exercise 1</li>
	<li> The Moore-Greitzer equations model rotating stall and surge in gas turbine engines describe the dynamics of a compression system, such as those in gas turbine engines. The three-state "MG3" equations have the form:		<li> The Moore-Greitzer equations model rotating stall and surge in gas turbine engines describe the dynamics of a compression system, such as those in gas turbine engines. The three-state "MG3" equations have the form:

Murray at 19:04, 27 February 2011

2011-02-27T19:04:32Z

← Older revision		Revision as of 19:04, 27 February 2011
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	\aligned		\aligned
	\frac{d\psi}{dt} &= \frac{1}{4 B^2 l_c}\left(\phi - \Phi_T(\psi) \right), \\		\frac{d\psi}{dt} &= \frac{1}{4 B^2 l_c}\left(\phi - \Phi_T(\psi) \right), \\
	\frac{d\phi}{dt} &= \frac{1}{l_c}\left( \~~Phi_c~~(\phi) - \psi + \frac{J}{8}		\frac{d\phi}{dt} &= \frac{1}{l_c}\left( \Psi_c(\phi) - \psi + \frac{J}{8}
	\frac{\partial^2 \Psi_c}{\partial \phi^2} \right), \\		\frac{\partial^2 \Psi_c}{\partial \phi^2} \right), \\
	\frac{dJ}{dt} &= \frac{2}{\mu + m} \left(		\frac{dJ}{dt} &= \frac{2}{\mu + m} \left(
	\frac{\partial \~~Phi_c~~}{\partial \phi} + \frac{J}{8}		\frac{\partial \Psi_c}{\partial \phi} + \frac{J}{8}
	\frac{\partial^3 \~~Phi_c~~}{\partial \phi^3} \right) J,		\frac{\partial^3 \Psi_c}{\partial \phi^3} \right) J,
	\endaligned		\endaligned
	</amsmath></center>		</amsmath></center>

Murray at 18:38, 27 February 2011

2011-02-27T18:38:46Z

@@ Line 14: / Line 14: @@
 <ol>
 <li> Perko, Section 4.1, Exercise 1</li>
-<li> TBD (simple sensitivity analysis)</li>
+<li> Consider the dynamical system
 <li> Perko, Section 4.3, Exercise 7</li>
 <li> Perko, Section 4.4, Exercise 1</li>

Murray at 16:03, 27 February 2011

2011-02-27T16:03:44Z

← Older revision		Revision as of 16:03, 27 February 2011
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	\| title = Problem Set #8		\| title = Problem Set #8
	\| issued = 26 Feb (Sat)		\| issued = 26 Feb (Sat)
	\| due = 9 Mar 2011 (~~Wed~~)~~, 5 pm~~		\| due = 9 Mar 2011 (Tue)
	}}		}}

Murray at 02:53, 27 February 2011

2011-02-27T02:53:35Z

← Older revision		Revision as of 02:53, 27 February 2011
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	\| semester = Winter 2011		\| semester = Winter 2011
	\| title = Problem Set #8		\| title = Problem Set #8
	\| issued = ~~1 Mar 2011~~ (~~Tue~~)		\| issued = 26 Feb (Sat)
	\| due = 9 Mar 2011 (Wed), 5 pm		\| due = 9 Mar 2011 (Wed), 5 pm
	}}		}}