CDS 110b: Linear Quadratic Optimal Control - Revision history

Carson: /* Course Materials */

2007-02-13T04:40:02Z

Course Materials

← Older revision		Revision as of 04:40, 13 February 2007
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	* {{cds110b-wi07 pdfs\|optimal.pdf\|Notes on optimal control}}		* {{cds110b-wi07 pdfs\|optimal.pdf\|Notes on optimal control}}
	* {{cds110b-wi07 pdfs\|lqr.pdf\|Notes on linear quadratic regulators}}		* {{cds110b-wi07 pdfs\|lqr.pdf\|Notes on linear quadratic regulators}}
			* [http://www.cds.caltech.edu/~murray/courses/cds110/wi07/mp3/7Feb07.mp3 MP3] of Wednesday lecture, 7 Feb 2007
			* [http://www.cds.caltech.edu/~murray/courses/cds110/wi07/mp3/9Feb07.mp3 MP3] of Friday lecture, 9 Feb 2007
	* {{cds110b-wi07 pdfs\|dfan_lqr.m\|dfan_lqr.m}} - Ducted fan LQR example		* {{cds110b-wi07 pdfs\|dfan_lqr.m\|dfan_lqr.m}} - Ducted fan LQR example
	* {{cds110b-wi07 pdfs\|hw5.pdf\|Homework #5 (due 14 Feb @ 5 pm)}}		* {{cds110b-wi07 pdfs\|hw5.pdf\|Homework #5 (due 14 Feb @ 5 pm)}}

Murray: /* Course Materials */

2007-02-09T12:32:42Z

Course Materials

← Older revision		Revision as of 12:32, 9 February 2007
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	* {{cds110b-wi07 pdfs\|optimal.pdf\|Notes on optimal control}}		* {{cds110b-wi07 pdfs\|optimal.pdf\|Notes on optimal control}}
	* {{cds110b-wi07 pdfs\|lqr.pdf\|Notes on linear quadratic regulators}}		* {{cds110b-wi07 pdfs\|lqr.pdf\|Notes on linear quadratic regulators}}
			* {{cds110b-wi07 pdfs\|dfan_lqr.m\|dfan_lqr.m}} - Ducted fan LQR example
	* {{cds110b-wi07 pdfs\|hw5.pdf\|Homework #5 (due 14 Feb @ 5 pm)}}		* {{cds110b-wi07 pdfs\|hw5.pdf\|Homework #5 (due 14 Feb @ 5 pm)}}

Murray at 12:31, 9 February 2007

2007-02-09T12:31:06Z

← Older revision		Revision as of 12:31, 9 February 2007
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	== Course Materials ==		== Course Materials ==
	* ~~[[Media:~~optimal.pdf\|Notes on optimal control]]		* {{cds110b-wi07 pdfs\|optimal.pdf\|Notes on optimal control}}
	* Notes on linear quadratic regulators		* {{cds110b-wi07 pdfs\|lqr.pdf\|Notes on linear quadratic regulators}}
	* {{cds110b-wi07 pdfs\|hw5.pdf\|Homework #5 (due 14 Feb @ 5 pm)}}		* {{cds110b-wi07 pdfs\|hw5.pdf\|Homework #5 (due 14 Feb @ 5 pm)}}

Franco: /* Course Materials */

2007-02-09T04:22:22Z

Course Materials

← Older revision		Revision as of 04:22, 9 February 2007
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	== Course Materials ==		== Course Materials ==
	* ~~{{cds110b-wi07 pdfs\|~~optimal.pdf\|Notes on optimal control}}		* [[Media:optimal.pdf\|Notes on optimal control]]
	* Notes on linear quadratic regulators		* Notes on linear quadratic regulators
	* {{cds110b-wi07 pdfs\|hw5.pdf\|Homework #5 (due 14 Feb @ 5 pm)}}		* {{cds110b-wi07 pdfs\|hw5.pdf\|Homework #5 (due 14 Feb @ 5 pm)}}

Franco: /* Course Materials */

2007-02-09T04:16:52Z

Course Materials

← Older revision		Revision as of 04:16, 9 February 2007
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	== Course Materials ==		== Course Materials ==
	* {{cds110b-~~wi06~~ pdfs\|optimal.pdf\|Notes on optimal control}}		* {{cds110b-wi07 pdfs\|optimal.pdf\|Notes on optimal control}}
	* Notes on linear quadratic regulators		* Notes on linear quadratic regulators
	* {{cds110b-wi07 pdfs\|hw5.pdf\|Homework #5 (due 14 Feb @ 5 pm)}}		* {{cds110b-wi07 pdfs\|hw5.pdf\|Homework #5 (due 14 Feb @ 5 pm)}}

Franco: /* Course Materials */

2007-02-09T04:16:15Z

Course Materials

← Older revision		Revision as of 04:16, 9 February 2007
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	== Course Materials ==		== Course Materials ==
	* {{cds110b-~~wi07~~ pdfs\|optimal.pdf\|Notes on optimal control}}		* {{cds110b-wi06 pdfs\|optimal.pdf\|Notes on optimal control}}
	* Notes on linear quadratic regulators		* Notes on linear quadratic regulators
	* {{cds110b-~~wi06~~ pdfs\|hw5.pdf\|Homework #5 (due 14 Feb @ 5 pm)}}		* {{cds110b-wi07 pdfs\|hw5.pdf\|Homework #5 (due 14 Feb @ 5 pm)}}

	== References and Further Reading ==		== References and Further Reading ==

Franco: /* Course Materials */

2007-02-09T04:15:32Z

Course Materials

← Older revision		Revision as of 04:15, 9 February 2007
Line 5:		Line 5:
	* {{cds110b-wi07 pdfs\|optimal.pdf\|Notes on optimal control}}		* {{cds110b-wi07 pdfs\|optimal.pdf\|Notes on optimal control}}
	* Notes on linear quadratic regulators		* Notes on linear quadratic regulators
	* {{cds110b-~~wi07~~ pdfs\|hw5.pdf\|Homework #5 (due 14 Feb @ 5 pm)}}		* {{cds110b-wi06 pdfs\|hw5.pdf\|Homework #5 (due 14 Feb @ 5 pm)}}

	== References and Further Reading ==		== References and Further Reading ==

Murray at 05:09, 7 February 2007

2007-02-07T05:09:02Z

New page

{{cds110b-wi07}}
This Wednesday lecture provides an overview of optimal control theory. Beginning with a review of optimization, we introduce the notion of Lagrange multipliers and provide a summary of the Pontryagin's maximum principle.

== Course Materials ==
* {{cds110b-wi07 pdfs|optimal.pdf|Notes on optimal control}}
* Notes on linear quadratic regulators
* {{cds110b-wi07 pdfs|hw5.pdf|Homework #5 (due 14 Feb @ 5 pm)}}

== References and Further Reading ==
* {{cds110b-pdfs|LS95-optimal.pdf|Excerpt from LS95 on optimal control}} - This excerpt is from [http://books.google.com/books?ie=UTF-8&hl=en&vid=ISBN0471033782&id=jkD37elP6NIC Lewis and Syrmos, 1995] and gives a derivation of the necessary conditions for optimaliity. A few pages have been left out from the middle that contained some additional examples (which you can find in similar books in the library, if you are interested). Other parts of the book can be searched via [http://books.google.com Google Books] and purchased online.
* [http://www.statslab.cam.ac.uk/~rrw1/oc/L13.pdf Notes on Pontryagin's Maximum Principle] - these come from a set of [http://www.statslab.cam.ac.uk/~rrw1/oc/index.html lecture notes on optimization and control] by [http://www.statslab.cam.ac.uk/~rrw1/ Richard Weber] at Cambridge University. The notes are based on dynamic programming (DP) and uses a slightly different notation than we used in class.

== Frequently Asked Questions ==
'''Q: In the example on Bang-Bang control discussed in the lecture, how is the control law for <math>u</math> obtained?'''

<blockquote>
Pontryagin's Maximum Principle says that <math>u</math> has to be chosen to minimise the Hamiltonian <math>H(x,u,\lambda)</math> for given values of <math>x</math> and <math>\lambda</math>. In the example, <math>H = 1 + ({\lambda}^TA)x + ({\lambda}^TB)u</math>. At first glance, it seems that the more negative <math>u</math> is the more <math>H</math> will be minimised. And since the most negative value of <math>u</math> allowed is <math>-1</math>, <math>u=-1</math>. However, the co-efficient of <math>u</math> may be of either sign. Therefore, the sign of <math>u</math> has to be chosen such that the sign of the term <math>({\lambda}^TB)u</math> is negative. That's how we come up with <math>u = -sign({\lambda}^TB)</math>.

Shaunak Sen, 12 Jan 06
</blockquote>

'''Q: Notation question for you: In the Lecture notes from Wednesday, I'm assuming that <math>T</math> is the final time and <math>T</math> (superscript T) is a transpose operation. Am I correct in my assumption?'''

<blockquote>
Yes, you are correct.

Jeremy Gillula, 07 Jan 05
</blockquote>

'''Q: What do you mean by penalizing something, from Q>=0 "penalizes" state error?'''

<blockquote>
According to the form of the quadratic cost function <math>J</math>, there are three quadratic terms such
as <math>x^T Q x</math>, <math>u^T R u</math>, and <math>x(T)^T P_1 x(T)</math>. When <math>Q \geq 0</math> and if <math>Q</math> is relative big, the value of <math>x</math> will have bigger contribution to the value of <math>J</math>. In order to keep <math>J</math> small, <math>x</math> must be relatively small. So selecting a big <math>Q</math> can keep <math>x</math> in small value regions. This is what the "penalizing" means.

So in the optimal control design, the relative values of <math>Q</math>, <math>R</math>, and <math>P_1</math> represent how important <math>X</math>, <math>U</math>, and <math>X(T)</math> are in the designer's concerns.

Zhipu Jin,13 Jan 03
</blockquote>