CDS 110b: Introduction to Robust Control: Difference between revisions
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== Lecture Materials == | == Lecture Materials == | ||
* | * 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 | ||
| Line 17: | Line 17: | ||
== 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 finite length intervalsintervals. | |||
<center> | |||
{| | |||
|- | |||
| Piecewise continuous | |||
| align=center | <math>\qquad\qquad</math> | |||
| align=center | Not piecewise continuous | |||
|- | |||
| <math>f(x) = \left\{ \begin{matrix} 0 \quad {if\,\, x < 0} \\ 1 \quad{if\,\, x > 0}\end{matrix} \right.</math> | |||
| | |||
| <math>f(x) = \left\{ \begin{matrix} 0 \quad {if\,\, x \neq 0} \\ 1 \quad{if\,\, x = 0}\end{matrix} \right.</math> | |||
|} | |||
</center> | |||
The first function is a step function, the second function is identically zero except at the single point <math>x = 0</math> (which is not a finite interval). [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> | |||
'''Q: Is <math>\|x\|_\infty</math> the same as <math>\lim_{k \to \infty} \|x\|_k</math>?''' | |||
<blockquote> | |||
Yes. | |||
</blockquote> | |||
'''Q: What is <math>C^n[t_0, t_1]</math>? A polynomial?''' | |||
<blockquote> | |||
<math>C^n[t_0, t_1]</math> is the set of all continuous, <math>R^n</math>-value functions defined on the interval <math>[t_0, t_1]</math>. If <math>n = 1</math>, then a polynomial would be an example of a function that is in the set <math>C^n[t_0, t_1]</math>. Similarly, the function <math>e^t</math> is also in <math>C^n[t_0, t_1]</math>. The function <math>1/(t-0.5)</math> is ''not'' in <math>C^n[0, 1]</math> since is not continuos at 0.5. | |||
</blockquote> | |||
Latest revision as of 19:52, 21 February 2006
See current course homepage to find most recent page available. |
| Course Home | L7-2: Sensitivity | L8-1: Robust Stability | L9-1: Robust Perf | Schedule |
This lecture provides an introduction to some of the signals and systems concepts required for the study of robust (Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle H_\infty} ) control.
Lecture Outline
- Overview of the robust performance problem
- Linear spaces and norms
- Norm of a linear system
Lecture Materials
- Blackboard lecture; no slides. MP3 lost (technical error)
- Lecture Notes on system norms
- Reading: DFT, Chapter 2
References and Further Reading
Frequently Asked Questions
Q: What did you mean when you wrote Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle V = R^n} , Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle V = C[-\infty, \infty]} ?
This appeared in a table that listed different norms. The two columns above were showing what the Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \|\cdot\|_k} norms were for different Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle k} . For example,
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle V = R^n} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle V = C[-\infty, \infty]} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \|x\|_2 = \sqrt{\sum_{i=1}^n x_i^2} } Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \|u\|_2 = \left( \int_{-\infty}^\infty u^2(t)\, dt \right)^{1/2} } The first column shows the 2-norm on the set of vectors of length Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle n} , the second column shows the 2-norm on the set of continuous functions.
Q: What is a "vector space with norm Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \|\cdot\| = a} " for some Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a} = some number?
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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \infty} -norm. For the space of functions Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle V = C[-\infty, \infty]} , the Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle k} norm is defined as
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \|u\|_k = \left( \int_{-\infty}^\infty |u(t)|^k\, dt \right)^{1/k} }
Q: How do you find Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \sup_{\|v\|_a \leq 1} \|w\|_b} ?
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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \infty} -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).
Q: What are the criteria for a function to be piecewise continuous?
A function is piecewise continuous if it is continuous in finite length intervalsintervals.
Piecewise continuous Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \qquad\qquad} Not piecewise continuous Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f(x) = \left\{ \begin{matrix} 0 \quad {if\,\, x < 0} \\ 1 \quad{if\,\, x > 0}\end{matrix} \right.} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f(x) = \left\{ \begin{matrix} 0 \quad {if\,\, x \neq 0} \\ 1 \quad{if\,\, x = 0}\end{matrix} \right.} The first function is a step function, the second function is identically zero except at the single point Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x = 0} (which is not a finite interval). Wikipedia has a pretty good description of this.
Q: How do ou evaluate Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle C e^{A(t - \tau)}}
The matrix exponential can be computing using the Jordan form for the matrix. Wikipedia has a pretty good description of this.
Q: Is Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \|x\|_\infty} the same as Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \lim_{k \to \infty} \|x\|_k} ?
Yes.
Q: What is Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle C^n[t_0, t_1]} ? A polynomial?
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle C^n[t_0, t_1]} is the set of all continuous, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle R^n} -value functions defined on the interval Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle [t_0, t_1]} . If Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle n = 1} , then a polynomial would be an example of a function that is in the set Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle C^n[t_0, t_1]} . Similarly, the function Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle e^t} is also in Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle C^n[t_0, t_1]} . The function Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 1/(t-0.5)} is not in Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle C^n[0, 1]} since is not continuos at 0.5.
