# 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 () control.

## Lecture Outline

- Overview of the robust performance problem
- Linear spaces and norms
- Norm of a linear system

## Lecture Materials

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

## References and Further Reading

## Frequently Asked Questions

**Q: What did you mean when you wrote , ?**

This appeared in a table that listed different norms. The two columns above were showing what the norms were for different . For example,

The first column shows the 2-norm on the set of vectors of length , the second column shows the 2-norm on the set of continuous functions.

**Q: What is a "vector space with norm " for some = 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 -norm. For the space of functions , the norm is defined as

**Q: How do you find ?**

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 -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 The first function is a step function, the second function is identically zero except at the single point (which is not a finite interval). Wikipedia has a pretty good description of this.

**Q: How do ou evaluate **

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

**Q: Is the same as ?**

Yes.

**Q: What is ? A polynomial?**

is the set of all continuous, -value functions defined on the interval . If , then a polynomial would be an example of a function that is in the set . Similarly, the function is also in . The function is

notin since is not continuos at 0.5.