CDS 110b: Introduction to Robust Control

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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 ($H_{\infty }$) control.

Lecture Outline

1. Overview of the robust performance problem
2. Linear spaces and norms
3. Norm of a linear system

Lecture Materials

• MP3 of lecture (blackboard lecture; no slides)
• Lecture Notes on system norms
• Reading: DFT, Chapter 2

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 intervals. Wikipedia has a pretty good description of this.

Q: How do ou evaluate $Ce^{A(t-\tau )}$

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

Q: Is $\|x\|_{\infty }$ 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.