CDS 110b: Introduction to Robust Control

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This lecture provides an introduction to some of the signals and systems concepts required for the study of robust ( $H_\infty$ ) 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 $V = R n$ , $V = C[-\infty, \infty]$ ?

This appeared in a table that listed different norms. The two columns above were showing what the $\|\cdot\|_k$ norms were for different $k$ . For example,

$V = R n$ $V = C[-\infty, \infty]$

$\|x\|_2 = \sqrt{\sum_{i=1}^n x_i^2}$ $\|u\|_2 = \left( \int_{-\infty}^\infty u^2(t)\, dt \right)^{1/2}$

The first column shows the 2-norm on the set of vectors of length $n$ , the second column shows the 2-norm on the set of continuous functions.

Q: What is a "vector space with norm $\|\cdot\| = a$ " for some $a$ = 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 $\infty$ -norm. For the space of functions $V = C[-\infty, \infty]$ , the $k$ norm is defined as $\|u\|_k = \left( \int_{-\infty}^\infty |u(t)|^k\, dt \right)^{1/k}$

Q: How do you find $\sup_{\|v\|_a \leq 1} \|w\|_b$ ?

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 $\infty$ -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 $\qquad\qquad$ Not piecewise continuous

$f(x) = \left\{ \begin{matrix} 0 \quad {if\,\, x < 0} \\ 1 \quad{if\,\, x > 0}\end{matrix} \right.$ $f(x) = \left\{ \begin{matrix} 0 \quad {if\,\, x \neq 0} \\ 1 \quad{if\,\, x = 0}\end{matrix} \right.$

The first function is a step function, the second function is identically zero except at the single point $x = 0$ (which is not a finite interval). Wikipedia has a pretty good description of this.

Q: How do ou evaluate $C e A (t - τ)$

The matrix exponential can be computing using the Jordan form for the matrix. Wikipedia has a pretty good description of this.

Q: Is $\|x\|_\infty$ the same as $\lim_{k \to \infty} \|x\|_k$ ?

Yes.

Q: What is $C n [t 0, t 1]$ ? A polynomial?

$C n [t 0, t 1]$ is the set of all continuous, $R n$ -value functions defined on the interval $[t 0, t 1]$ . If $n = 1$ , then a polynomial would be an example of a function that is in the set $C n [t 0, t 1]$ . Similarly, the function $e t$ is also in $C n [t 0, t 1]$ . The function $1 / (t - 0.5)$ is not in $C n [0,1]$ since is not continuos at 0.5.

CDS 110b: Introduction to Robust Control

From MurrayWiki

Lecture Outline

Lecture Materials

References and Further Reading

Frequently Asked Questions

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$V = R n$		$V = C[-\infty, \infty]$
$\\|x\\|_2 = \sqrt{\sum_{i=1}^n x_i^2}$		$\\|u\\|_2 = \left( \int_{-\infty}^\infty u^2(t)\, dt \right)^{1/2}$

Piecewise continuous	$\qquad\qquad$	Not piecewise continuous
$f(x) = \left\{ \begin{matrix} 0 \quad {if\,\, x < 0} \\ 1 \quad{if\,\, x > 0}\end{matrix} \right.$		$f(x) = \left\{ \begin{matrix} 0 \quad {if\,\, x \neq 0} \\ 1 \quad{if\,\, x = 0}\end{matrix} \right.$