# 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

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

## 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 $Ce^{{A(t-\tau )}}$

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.