CDS 110b: Introduction to Robust Control - Revision history

Murray: /* Lecture Materials */

2006-02-21T19:52:13Z

Lecture Materials

← Older revision		Revision as of 19:52, 21 February 2006
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	== Lecture Materials ==		== Lecture Materials ==
	* ~~{{cds110b-pdfs\|L6-2_robust.mp3\|MP3 of lecture}} (blackboard~~ lecture; no slides)		* Blackboard lecture; no slides. MP3 lost (technical error)
	* {{cds110b-pdfs\|sysnorm.pdf\|Lecture Notes on system norms}}		* {{cds110b-pdfs\|sysnorm.pdf\|Lecture Notes on system norms}}
	* Reading: DFT, Chapter 2		* Reading: DFT, Chapter 2

Murray: /* Lecture Materials */

2006-02-21T19:48:45Z

Lecture Materials

← Older revision		Revision as of 19:48, 21 February 2006
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	== Lecture Materials ==		== Lecture Materials ==
	* {{cds110b-pdfs\|L6-~~2_robudst~~.mp3\|MP3 of lecture}} (blackboard lecture; no slides)		* {{cds110b-pdfs\|L6-2_robust.mp3\|MP3 of lecture}} (blackboard lecture; no slides)
	* {{cds110b-pdfs\|sysnorm.pdf\|Lecture Notes on system norms}}		* {{cds110b-pdfs\|sysnorm.pdf\|Lecture Notes on system norms}}
	* Reading: DFT, Chapter 2		* Reading: DFT, Chapter 2

Murray: /* Lecture Materials */

2006-02-21T19:48:36Z

Lecture Materials

← Older revision		Revision as of 19:48, 21 February 2006
Line 10:		Line 10:

	== Lecture Materials ==		== Lecture Materials ==
	* {{cds110b-pdfs\|L6-~~2_sysnorm~~.mp3\|MP3 of lecture}} (blackboard lecture; no slides)		* {{cds110b-pdfs\|L6-2_robudst.mp3\|MP3 of lecture}} (blackboard lecture; no slides)
	* {{cds110b-pdfs\|sysnorm.pdf\|Lecture Notes on system norms}}		* {{cds110b-pdfs\|sysnorm.pdf\|Lecture Notes on system norms}}
	* Reading: DFT, Chapter 2		* Reading: DFT, Chapter 2

Murray: /* Frequently Asked Questions */

2006-02-14T15:10:27Z

Frequently Asked Questions

@@ Line 49: / Line 49: @@
 '''Q: What are the criteria for a function to be piecewise continuous?'''
 <blockquote>
-A function is piecewise continuous if it is continuous in intervals.  [http://en.wikipedia.org/wiki/Piecewise Wikipedia] has a pretty good description of this.
+A function is piecewise continuous if it is continuous in finite length intervalsintervals.
 </blockquote>

Murray: /* Frequently Asked Questions */

2006-02-14T04:54:51Z

Frequently Asked Questions

@@ Line 55: / Line 55: @@
 <blockquote>
 The matrix exponential can be computing using the Jordan form for the matrix.  [http://en.wikipedia.org/wiki/Matrix_exponential Wikipedia] has a pretty good description of this.
 </blockquote>

Murray: /* Frequently Asked Questions */

2006-02-12T18:59:05Z

Frequently Asked Questions

← Older revision		Revision as of 18:59, 12 February 2006
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	== Frequently Asked Questions ==		== Frequently Asked Questions ==
			'''Q: What did you mean when you wrote <math>V = R^n</math>, <math>V = C[-\infty, \infty]</math>?'''
			<blockquote>
			This appeared in a table that listed different norms. The two columns above were showing what the <math>\\|\cdot\\|_k</math> norms were for different <math>k</math>. For example,
			<center>
			{\|
			\|-
			\| align=center \| <math>V = R^n</math>
			\|
			\| align = center \| <math>V = C[-\infty, \infty]</math>
			\|-
			\| <math> \\|x\\|_2 = \sqrt{\sum_{i=1}^n x_i^2} </math> \|\| \|\| <math> \\|u\\|_2 = \left( \int_{-\infty}^\infty u^2(t)\, dt \right)^{1/2} </math>
			\|-
			\|}
			</center>
			The first column shows the 2-norm on the set of vectors of length <math>n</math>, the second column shows the 2-norm on the set of continuous functions.
			</blockquote>

			'''Q: What is a "vector space with norm <math>\\|\cdot\\| = a</math>" for some <math>a</math> = some number?'''
			<blockquote>
			A vector space specifies the operations of additional and (scalar) multiplication. We can also put a norm on a vector space, but there are different norms for a given vector space (for example, the 2-norm and the <math>\infty</math>-norm. For the space of functions <math>V = C[-\infty, \infty]</math>, the <math>k</math> norm is defined as
			<center><math>
			\\|u\\|_k = \left( \int_{-\infty}^\infty \|u(t)\|^k\, dt \right)^{1/k}
			</math></center>
			</blockquote>

			'''Q: How do you find <math>\sup_{\\|v\\|_a \leq 1} \\|w\\|_b</math>?'''
			<blockquote>
			It can be very difficult to compute the induced norm for a general function. As we shall see, the induced 2-norm for a linear system turns out to be the <math>\infty</math>-norm of the corresponding transfer functions, which is just the maximum gain as a function of the frequency. Hence for this particular norm, it can be computed by looking at the magnitude portion of a Bode plot (more on this in the next lecture).
			</blockquote>

			'''Q: What are the criteria for a function to be piecewise continuous?'''
			<blockquote>
			A function is piecewise continuous if it is continuous in intervals. [http://en.wikipedia.org/wiki/Piecewise Wikipedia] has a pretty good description of this.
			</blockquote>

			'''Q: How do ou evaluate <math>C e^{A(t - \tau)}</math>'''
			<blockquote>
			The matrix exponential can be computing using the Jordan form for the matrix. [http://en.wikipedia.org/wiki/Matrix_exponential Wikipedia] has a pretty good description of this.
			</blockquote>

Murray: /* Lecture Materials */

2006-02-10T06:19:35Z

Lecture Materials

← Older revision		Revision as of 06:19, 10 February 2006
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	== Lecture Materials ==		== Lecture Materials ==
	* {{cds110b-pdfs\|L6-2_sysnorm.mp3\|MP3 of lecture}})		* {{cds110b-pdfs\|L6-2_sysnorm.mp3\|MP3 of lecture}} (blackboard lecture; no slides)
	* {{cds110b-pdfs\|sysnorm.pdf\|Lecture Notes on system norms}}		* {{cds110b-pdfs\|sysnorm.pdf\|Lecture Notes on system norms}}
	* Reading: DFT, Chapter 2		* Reading: DFT, Chapter 2

Murray at 06:19, 10 February 2006

2006-02-10T06:19:10Z

New page

{{cds110b-wi06}}
This lecture provides an introduction to some of the signals and systems concepts required for the study of robust (<math>H_\infty</math>) control. __NOTOC__

== Lecture Outline ==
<ol type=I>
<li> Overview of the robust performance problem
<li> Linear spaces and norms
<li> Norm of a linear system
</ol>

== Lecture Materials ==
* {{cds110b-pdfs|L6-2_sysnorm.mp3|MP3 of lecture}})
* {{cds110b-pdfs|sysnorm.pdf|Lecture Notes on system norms}}
* Reading: DFT, Chapter 2

== References and Further Reading ==

== Frequently Asked Questions ==