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_{\infty }) control.

Lecture Outline

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

Lecture Materials

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 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.