CDS 110b: Optimal Control - Revision history

Murray at 03:30, 2 March 2008

2008-03-02T03:30:38Z

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	{{cds110b-wi08}}		{{cds110b-wi08 lecture\|prev=Two Degree of Freedom Control Design\|next=Linear Quadratic Regulators}}
	This 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. __NOTOC__		This 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. __NOTOC__

Soto: /* Frequently Asked Questions */

2008-01-23T04:07:45Z

Frequently Asked Questions

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	<blockquote>		<blockquote>
	<p>The psi function represents a general form of terminal constraint for the state variables. It gives a way of indicating which states have a terminal cost attached to them. For example, by defining psi_i(x(T))=x_i(T)-x_i,f for i=1,2,...n, we can impose terminal costs on all states (a fully constrained case) by letting p=n (n being the # of states). When we optimize over time and want x(T)=x_f, then x(T)-x_f=0, and so psi(x(T))=0.</p>		<p>The psi function represents a general form of terminal constraint for the state variables. It gives a way of indicating which states have a terminal cost attached to them. For example, by defining psi_i(x(T))=x_i(T)-x_i,f for i=1,2,...n, we can impose terminal costs on all states (a fully constrained case) by letting p=n (n being the # of states). When we optimize over time and want x(T)=x_f, then x(T)-x_f=0, and so psi(x(T))=0. If we take x_f=0, then psi(x(T))=x(T).</p>

	<p>Luis Soto, 22 Jan 08</p>		<p>Luis Soto, 22 Jan 08</p>

Soto: /* Frequently Asked Questions */

2008-01-23T04:02:57Z

Frequently Asked Questions

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	== Frequently Asked Questions ==		== Frequently Asked Questions ==
			'''Q: Could you please explain what the psi function is or what it means if psi(x(T))=0 versus what it means if psi(x(T))=x(T)?'''

			<blockquote>
			<p>The psi function represents a general form of terminal constraint for the state variables. It gives a way of indicating which states have a terminal cost attached to them. For example, by defining psi_i(x(T))=x_i(T)-x_i,f for i=1,2,...n, we can impose terminal costs on all states (a fully constrained case) by letting p=n (n being the # of states). When we optimize over time and want x(T)=x_f, then x(T)-x_f=0, and so psi(x(T))=0.</p>

			<p>Luis Soto, 22 Jan 08</p>
			</blockquote>
	'''Q: In Problem 2.4d, are the boundary conditions for the differentially-flat trajectory correct?'''		'''Q: In Problem 2.4d, are the boundary conditions for the differentially-flat trajectory correct?'''

Soto: /* Frequently Asked Questions */

2008-01-21T23:23:47Z

Frequently Asked Questions

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	<blockquote>		<blockquote>
	<p>Please ignore the boundary conditions given in part 2.4d for the differentially-flat trajectory and instead use x(0)=1 for the initial condition and x(1)=0 for the condition at final time t=1. Moreover, use c=100 instead of c=1. Note: the x(tf) of the optimal solution won't be exactly 0,		<p>Please ignore the boundary conditions given in part 2.4d for the differentially-flat trajectory and instead use x(0)=1 for the initial condition and x(1)=0 for the condition at final time t=1. Moreover, use c=100 instead of c=1. Note: the x(t_f) of the optimal solution won't be exactly 0,
	but will be close enough for the intent of this problem.</p>		but will be close enough for the intent of this problem.</p>

Soto: /* Frequently Asked Questions */

2008-01-21T23:22:51Z

Frequently Asked Questions

← Older revision		Revision as of 23:22, 21 January 2008
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	== Frequently Asked Questions ==		== Frequently Asked Questions ==
			'''Q: In Problem 2.4d, are the boundary conditions for the differentially-flat trajectory correct?'''

			<blockquote>
			<p>Please ignore the boundary conditions given in part 2.4d for the differentially-flat trajectory and instead use x(0)=1 for the initial condition and x(1)=0 for the condition at final time t=1. Moreover, use c=100 instead of c=1. Note: the x(tf) of the optimal solution won't be exactly 0,
			but will be close enough for the intent of this problem.</p>

			<p>Luis Soto, 21 Jan 08</p>
			</blockquote>

	'''Q: In problem 2.4(d) of the homework, to what positive value should the parameter b be set?'''		'''Q: In problem 2.4(d) of the homework, to what positive value should the parameter b be set?'''

Braman: /* Frequently Asked Questions */

2008-01-19T01:04:23Z

Frequently Asked Questions

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

			'''Q: In problem 2.4(d) of the homework, to what positive value should the parameter b be set?'''

			<blockquote>
			<p>Use b = 1 for part d when solving for and comparing the two trajectories found symbolically in previous parts. </p>

			<p>Julia Braman, 18 Jan 08</p>
			</blockquote>

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

Murray: /* Frequently Asked Questions */

2008-01-15T06:23:57Z

Frequently Asked Questions

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	<p>Jeremy Gillula, 07 Jan 05</p>		<p>Jeremy Gillula, 07 Jan 05</p>
	~~</blockquote>~~

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

	~~<blockquote>~~
	~~<p>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.</p>

	<p>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.</p>

	~~<p>Zhipu Jin,13 Jan 03</p>~~
	</blockquote>		</blockquote>

Murray at 06:23, 15 January 2008

2008-01-15T06:23:39Z

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	* {{cds110b-wi08 pdfs\|L2-1_optimal.pdf\|Lecture notes: optimal control}}		* {{cds110b-wi08 pdfs\|L2-1_optimal.pdf\|Lecture notes: optimal control}}
			* Homework 2 (due 22 Jan @ 5 pm): {{obc08\|problems 2.3, 2.4a-d, 2.6}}

	== References and Further Reading ==		== References and Further Reading ==
			* R. M. Murray, ''Optimization-Based Control''. Preprint, 2008: {{cds110b-wi08 pdfs\|optimal_14Jan08.pdf\|Chapter 2 - Optimal Control}}
	* {{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.		* {{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.		* [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.

Murray at 21:28, 14 January 2008

2008-01-14T21:28:36Z

← Older revision		Revision as of 21:28, 14 January 2008
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	This 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. __NOTOC__		This 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. __NOTOC__

	* {{cds110b-wi08 pdfs\|~~optimal~~.pdf\|Lecture notes on optimal control}}		* {{cds110b-wi08 pdfs\|L2-1_optimal.pdf\|Lecture notes: optimal control}}

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

Murray at 21:21, 14 January 2008

2008-01-14T21:21:20Z

← Older revision		Revision as of 21:21, 14 January 2008
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	This 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. __NOTOC__		This 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. __NOTOC__

	* {{cds110b-~~pdfs\|L1-2_Optimal.pdf\|Lecture Presentation}}~~		* {{cds110b-wi08 pdfs\|optimal.pdf\|Lecture notes on optimal control}}
	* {{cds110b-pdfs\|optimal.pdf\|Lecture notes on optimal control}}

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