# 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 (${\displaystyle H_{\infty }}$) control.

## Lecture Outline

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

## Lecture Materials

Q: What did you mean when you wrote ${\displaystyle V=R^{n}}$, ${\displaystyle V=C[-\infty ,\infty ]}$?

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

 ${\displaystyle V=R^{n}}$ ${\displaystyle V=C[-\infty ,\infty ]}$ ${\displaystyle \|x\|_{2}={\sqrt {\sum _{i=1}^{n}x_{i}^{2}}}}$ ${\displaystyle \|u\|_{2}=\left(\int _{-\infty }^{\infty }u^{2}(t)\,dt\right)^{1/2}}$

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

Q: What is a "vector space with norm ${\displaystyle \|\cdot \|=a}$" for some ${\displaystyle a}$ = 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 ${\displaystyle \infty }$-norm. For the space of functions ${\displaystyle V=C[-\infty ,\infty ]}$, the ${\displaystyle k}$ norm is defined as

${\displaystyle \|u\|_{k}=\left(\int _{-\infty }^{\infty }|u(t)|^{k}\,dt\right)^{1/k}}$

Q: How do you find ${\displaystyle \sup _{\|v\|_{a}\leq 1}\|w\|_{b}}$?

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

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

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

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

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

Yes.

Q: What is ${\displaystyle C^{n}[t_{0},t_{1}]}$? A polynomial?

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