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

  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 , ?

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 ?


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 not in since is not continuos at 0.5.