CDS 110b: Introduction to Robust Control

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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 ($H_{\mathrm{\infty }}$) control.

Lecture Outline

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

Lecture Materials

• Blackboard lecture; no slides. MP3 lost (technical error)
• Lecture Notes on system norms
• Reading: DFT, Chapter 2

Frequently Asked Questions

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

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

$V=R^n$		$V=C[-\mathrm{\infty },\mathrm{\infty }]$
$\Vert x{\Vert }_2=\sqrt{\sum _{i=1}^n{x}_i^2}$		$\Vert u{\Vert }_2={(\underset{-\mathrm{\infty }}{\overset{\mathrm{\infty }}{\int }}{u}^2(t)dt)}^{1/2}$

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 $\Vert \cdot \Vert =a$" 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 $\mathrm{\infty }$-norm. For the space of functions $V=C[-\mathrm{\infty },\mathrm{\infty }]$, the $k$ norm is defined as

$\Vert u{\Vert }_k={(\underset{-\mathrm{\infty }}{\overset{\mathrm{\infty }}{\int }}|u(t){|}^{k}dt)}^{1/k}$

Q: How do you find $\underset{\Vert v{\Vert }_a\le 1}{\sup }\Vert w{\Vert }_b$?

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 $\mathrm{\infty }$-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	$$	Not piecewise continuous
$f(x)=\{\begin{array}{c}0ifx<0\\ 1ifx>0\end{array}$		$f(x)=\{\begin{array}{c}0ifx\ne 0\\ 1ifx=0\end{array}$

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 $C{e}^{A(t-\tau )}$

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

Q: Is $\Vert x{\Vert }_{\mathrm{\infty }}$ the same as $\underset{k\to \mathrm{\infty }}{\lim }\Vert x{\Vert }_k$?

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.