CDS 110b: Two Degree of Freedom Control Design - Revision history

Murray at 03:31, 2 March 2008

2008-03-02T03:31:03Z

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	{{cds110b-wi08}}		{{cds110b-wi08 lecture\|prev=Main Page\|next=Optimal Control}}
	In this set of lectures we describe the problem of trajectory generation and tracking. We use differential flatness to generate feasible trajectories for the system, which are then tracked by a local (gain-scheduled) controller.		In this set of lectures we describe the problem of trajectory generation and tracking. We use differential flatness to generate feasible trajectories for the system, which are then tracked by a local (gain-scheduled) controller.

Murray: /* Frequently Asked Questions */

2008-01-10T15:55:44Z

Frequently Asked Questions

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	'''What's an example of a system that ''isn't'' differentially flat?'''		'''What's an example of a system that ''isn't'' differentially flat?'''
	: While many systems are differentially flat, there are many systems that aren't. One example is given by the following set of differential equations
			While many systems are differentially flat, there are many systems that aren't. One example is given by the following set of differential equations
	<center><amsmath>		<center><amsmath>
	\aligned		\aligned

Murray: /* Frequently Asked Questions */

2008-01-10T15:55:18Z

Frequently Asked Questions

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	\dot x_2 &= u_2 \\		\dot x_2 &= u_2 \\
	\dot x_3 &= x_1 u_2 - x_2 u_1 \\		\dot x_3 &= x_1 u_2 - x_2 u_1 \\
	\dot x_4 &= x_3 u_1		\dot x_4 &= x_3 u_1 \\
	\dot x_5 &= x_3 u_2.		\dot x_5 &= x_3 u_2.
			\endaligned
	</amsmath></center>		</amsmath></center>
	Showing that this system isn't differentially flat is complicated and relies on mathematical tools that are beyond those that we present in the class. If you are interested in learning more, take a look at a [http://www.cds.caltech.edu/~murray/papers/2003d_mmr03-cds.html survey article] on differential flatness by Martin, Murray and Rouchon.		Showing that this system isn't differentially flat is complicated and relies on mathematical tools that are beyond those that we present in the class. If you are interested in learning more, take a look at a [http://www.cds.caltech.edu/~murray/papers/2003d_mmr03-cds.html survey article] on differential flatness by Martin, Murray and Rouchon.

Murray: /* Frequently Asked Questions */

2008-01-10T15:54:48Z

Frequently Asked Questions

← Older revision		Revision as of 15:54, 10 January 2008
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	== Frequently Asked Questions ==		== Frequently Asked Questions ==

			'''What's an example of a system that ''isn't'' differentially flat?'''
			: While many systems are differentially flat, there are many systems that aren't. One example is given by the following set of differential equations
			<center><amsmath>
			\aligned
			\dot x_1 &= u_1 \\
			\dot x_2 &= u_2 \\
			\dot x_3 &= x_1 u_2 - x_2 u_1 \\
			\dot x_4 &= x_3 u_1
			\dot x_5 &= x_3 u_2.
			</amsmath></center>
			Showing that this system isn't differentially flat is complicated and relies on mathematical tools that are beyond those that we present in the class. If you are interested in learning more, take a look at a [http://www.cds.caltech.edu/~murray/papers/2003d_mmr03-cds.html survey article] on differential flatness by Martin, Murray and Rouchon.

Murray at 18:29, 9 January 2008

2008-01-09T18:29:49Z

← Older revision		Revision as of 18:29, 9 January 2008
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	* {{cds110b-wi08 pdfs\|L1-1_intro.pdf\|Lecture presentation: course overview}}		* {{cds110b-wi08 pdfs\|L1-1_intro.pdf\|Lecture presentation: course overview}}
	* {{cds110b-wi08 pdfs\|L1-1_tracking.pdf\|Lecture notes: trajectory tracking and gain scheduling}}		* {{cds110b-wi08 pdfs\|L1-1_tracking.pdf\|Lecture notes: trajectory tracking and gain scheduling}}
	* {{cds110b-wi08 pdfs ~~placeholder~~\|L1-2_flatness.pdf\|Lecture notes: trajectory generation and differential flatness}}		* {{cds110b-wi08 pdfs\|L1-2_flatness.pdf\|Lecture notes: trajectory generation and differential flatness}}
	* Homework 1 (due 14 Jan @ 5 pm): {{obc08\|problems 1.2, 1.3, 1.4 and 1.5}}		* Homework 1 (due 14 Jan @ 5 pm): {{obc08\|problems 1.2, 1.3, 1.4 and 1.5}}
	** {{cds110b-wi07 pdfs\|normsteer.m\|normsteer.m}} - Normalized model for steering control system		** {{cds110b-wi07 pdfs\|normsteer.m\|normsteer.m}} - Normalized model for steering control system

Murray at 03:11, 8 January 2008

2008-01-08T03:11:59Z

← Older revision		Revision as of 03:11, 8 January 2008
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	* {{cds110b-wi08 pdfs\|L1-1_tracking.pdf\|Lecture notes: trajectory tracking and gain scheduling}}		* {{cds110b-wi08 pdfs\|L1-1_tracking.pdf\|Lecture notes: trajectory tracking and gain scheduling}}
	* {{cds110b-wi08 pdfs placeholder\|L1-2_flatness.pdf\|Lecture notes: trajectory generation and differential flatness}}		* {{cds110b-wi08 pdfs placeholder\|L1-2_flatness.pdf\|Lecture notes: trajectory generation and differential flatness}}
	* Homework 1 (due 14 Jan @ 5 pm): ~~Murray,~~ problems 1.2, 1.3, 1.4 and 1.5		* Homework 1 (due 14 Jan @ 5 pm): {{obc08\|problems 1.2, 1.3, 1.4 and 1.5}}
	** {{cds110b-wi07 pdfs\|normsteer.m\|normsteer.m}} - Normalized model for steering control system		** {{cds110b-wi07 pdfs\|normsteer.m\|normsteer.m}} - Normalized model for steering control system

Murray at 02:49, 8 January 2008

2008-01-08T02:49:08Z

← Older revision		Revision as of 02:49, 8 January 2008
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	* {{cds110b-wi08 pdfs\|L1-1_tracking.pdf\|Lecture notes: trajectory tracking and gain scheduling}}		* {{cds110b-wi08 pdfs\|L1-1_tracking.pdf\|Lecture notes: trajectory tracking and gain scheduling}}
	* {{cds110b-wi08 pdfs placeholder\|L1-2_flatness.pdf\|Lecture notes: trajectory generation and differential flatness}}		* {{cds110b-wi08 pdfs placeholder\|L1-2_flatness.pdf\|Lecture notes: trajectory generation and differential flatness}}
	* ~~{{cds110b-wi08 pdfs\|hw1.pdf\|~~Homework 1}} (due 14 Jan @ 5 pm)		* Homework 1 (due 14 Jan @ 5 pm): Murray, problems 1.2, 1.3, 1.4 and 1.5
	** {{cds110b-wi07 pdfs\|normsteer.m\|normsteer.m}} - Normalized model for steering control system		** {{cds110b-wi07 pdfs\|normsteer.m\|normsteer.m}} - Normalized model for steering control system

Murray at 02:45, 8 January 2008

2008-01-08T02:45:27Z

← Older revision		Revision as of 02:45, 8 January 2008
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	* {{cds110b-wi08 pdfs\|L1-1_intro.pdf\|Lecture presentation: course overview}}		* {{cds110b-wi08 pdfs\|L1-1_intro.pdf\|Lecture presentation: course overview}}
	* {{cds110b-wi08 pdfs\|L1-1_tracking.pdf\|Lecture notes: trajectory tracking and gain scheduling}}		* {{cds110b-wi08 pdfs\|L1-1_tracking.pdf\|Lecture notes: trajectory tracking and gain scheduling}}
	* {{cds110b-wi08 pdfs\|L1-2_flatness.pdf\|Lecture notes: trajectory generation and differential flatness}}		* {{cds110b-wi08 pdfs placeholder\|L1-2_flatness.pdf\|Lecture notes: trajectory generation and differential flatness}}
	* {{cds110b-wi08 pdfs\|hw1.pdf\|Homework 1}} (due 14 Jan @ 5 pm)		* {{cds110b-wi08 pdfs\|hw1.pdf\|Homework 1}} (due 14 Jan @ 5 pm)
	** {{cds110b-wi07 pdfs\|normsteer.m\|normsteer.m}} - Normalized model for steering control system		** {{cds110b-wi07 pdfs\|normsteer.m\|normsteer.m}} - Normalized model for steering control system

Murray at 02:43, 8 January 2008

2008-01-08T02:43:37Z

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	== References and Further Reading ==		== References and Further Reading ==
	* {{~~AM07 supplement~~\|Trajectory Generation and Tracking}}		* R. M. Murray, ''Optimization-Based Control''. Preprint, 2008: {{cds110b-wi08 pdfs\|trajgen_07Jan08.pdf\|Chapter 1 - Trajectory Generation and Tracking}}
	* {{AM07\|Chapter 7 - Output Feedback}} Section 7.5		* {{AM07\|Chapter 7 - Output Feedback}} Section 7.5
	* [http://www.cds.caltech.edu/~murray/papers/1996l_nm96-ijrnc.html Real Time Trajectory Generation for Differentially Flat Systems], M. J. van Nieuwstadt and R. M. Murray, Int'l. J. Robust & Nonlinear Control 8:(11) 995-1020, 1998.		* [http://www.cds.caltech.edu/~murray/papers/1996l_nm96-ijrnc.html Real Time Trajectory Generation for Differentially Flat Systems], M. J. van Nieuwstadt and R. M. Murray, Int'l. J. Robust & Nonlinear Control 8:(11) 995-1020, 1998.

	== Frequently Asked Questions ==		== Frequently Asked Questions ==

Murray: /* Course Materials */

2008-01-07T07:28:42Z

Course Materials

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	== Course Materials ==		== Course Materials ==
	* {{cds110b-wi08 pdfs\|L1-1_intro.pdf\|~~Course~~ overview}}		* {{cds110b-wi08 pdfs\|L1-1_intro.pdf\|Lecture presentation: course overview}}
	* {{cds110b-wi08 pdfs\|L1-1_tracking.pdf\|Lecture notes: trajectory tracking and gain scheduling}}		* {{cds110b-wi08 pdfs\|L1-1_tracking.pdf\|Lecture notes: trajectory tracking and gain scheduling}}
	* {{cds110b-wi08 pdfs\|L1-2_flatness.pdf\|Lecture notes: trajectory generation and differential flatness}}		* {{cds110b-wi08 pdfs\|L1-2_flatness.pdf\|Lecture notes: trajectory generation and differential flatness}}

← Older revision		Revision as of 18:29, 9 January 2008
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	* {{cds110b-wi08 pdfs\|L1-1_intro.pdf\|Lecture presentation: course overview}}		* {{cds110b-wi08 pdfs\|L1-1_intro.pdf\|Lecture presentation: course overview}}
	* {{cds110b-wi08 pdfs\|L1-1_tracking.pdf\|Lecture notes: trajectory tracking and gain scheduling}}		* {{cds110b-wi08 pdfs\|L1-1_tracking.pdf\|Lecture notes: trajectory tracking and gain scheduling}}
	* {{cds110b-wi08 pdfs ~~placeholder~~\|L1-2_flatness.pdf\|Lecture notes: trajectory generation and differential flatness}}		* {{cds110b-wi08 pdfs\|L1-2_flatness.pdf\|Lecture notes: trajectory generation and differential flatness}}
	* Homework 1 (due 14 Jan @ 5 pm): {{obc08\|problems 1.2, 1.3, 1.4 and 1.5}}		* Homework 1 (due 14 Jan @ 5 pm): {{obc08\|problems 1.2, 1.3, 1.4 and 1.5}}
	** {{cds110b-wi07 pdfs\|normsteer.m\|normsteer.m}} - Normalized model for steering control system		** {{cds110b-wi07 pdfs\|normsteer.m\|normsteer.m}} - Normalized model for steering control system

← Older revision		Revision as of 03:11, 8 January 2008
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	* {{cds110b-wi08 pdfs\|L1-1_tracking.pdf\|Lecture notes: trajectory tracking and gain scheduling}}		* {{cds110b-wi08 pdfs\|L1-1_tracking.pdf\|Lecture notes: trajectory tracking and gain scheduling}}
	* {{cds110b-wi08 pdfs placeholder\|L1-2_flatness.pdf\|Lecture notes: trajectory generation and differential flatness}}		* {{cds110b-wi08 pdfs placeholder\|L1-2_flatness.pdf\|Lecture notes: trajectory generation and differential flatness}}
	* Homework 1 (due 14 Jan @ 5 pm): ~~Murray,~~ problems 1.2, 1.3, 1.4 and 1.5		* Homework 1 (due 14 Jan @ 5 pm): {{obc08\|problems 1.2, 1.3, 1.4 and 1.5}}
	** {{cds110b-wi07 pdfs\|normsteer.m\|normsteer.m}} - Normalized model for steering control system		** {{cds110b-wi07 pdfs\|normsteer.m\|normsteer.m}} - Normalized model for steering control system

← Older revision		Revision as of 02:49, 8 January 2008
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	* {{cds110b-wi08 pdfs\|L1-1_tracking.pdf\|Lecture notes: trajectory tracking and gain scheduling}}		* {{cds110b-wi08 pdfs\|L1-1_tracking.pdf\|Lecture notes: trajectory tracking and gain scheduling}}
	* {{cds110b-wi08 pdfs placeholder\|L1-2_flatness.pdf\|Lecture notes: trajectory generation and differential flatness}}		* {{cds110b-wi08 pdfs placeholder\|L1-2_flatness.pdf\|Lecture notes: trajectory generation and differential flatness}}
	* ~~{{cds110b-wi08 pdfs\|hw1.pdf\|~~Homework 1}} (due 14 Jan @ 5 pm)		* Homework 1 (due 14 Jan @ 5 pm): Murray, problems 1.2, 1.3, 1.4 and 1.5
	** {{cds110b-wi07 pdfs\|normsteer.m\|normsteer.m}} - Normalized model for steering control system		** {{cds110b-wi07 pdfs\|normsteer.m\|normsteer.m}} - Normalized model for steering control system

← Older revision		Revision as of 02:45, 8 January 2008
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	* {{cds110b-wi08 pdfs\|L1-1_intro.pdf\|Lecture presentation: course overview}}		* {{cds110b-wi08 pdfs\|L1-1_intro.pdf\|Lecture presentation: course overview}}
	* {{cds110b-wi08 pdfs\|L1-1_tracking.pdf\|Lecture notes: trajectory tracking and gain scheduling}}		* {{cds110b-wi08 pdfs\|L1-1_tracking.pdf\|Lecture notes: trajectory tracking and gain scheduling}}
	* {{cds110b-wi08 pdfs\|L1-2_flatness.pdf\|Lecture notes: trajectory generation and differential flatness}}		* {{cds110b-wi08 pdfs placeholder\|L1-2_flatness.pdf\|Lecture notes: trajectory generation and differential flatness}}
	* {{cds110b-wi08 pdfs\|hw1.pdf\|Homework 1}} (due 14 Jan @ 5 pm)		* {{cds110b-wi08 pdfs\|hw1.pdf\|Homework 1}} (due 14 Jan @ 5 pm)
	** {{cds110b-wi07 pdfs\|normsteer.m\|normsteer.m}} - Normalized model for steering control system		** {{cds110b-wi07 pdfs\|normsteer.m\|normsteer.m}} - Normalized model for steering control system