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 (H_\infty) control.

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Lecture Outline

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

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References and Further Reading

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Frequently Asked Questions

Q: What did you mean when you wrote V = Rn, 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 = Rn      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 CeA(t − τ)

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 Cn[t0,t1]? A polynomial?

Cn[t0,t1] is the set of all continuous, Rn-value functions defined on the interval [t0,t1]. If n = 1, then a polynomial would be an example of a function that is in the set Cn[t0,t1]. Similarly, the function et is also in Cn[t0,t1]. The function 1 / (t − 0.5) is not in Cn[0,1] since is not continuos at 0.5.
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