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

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

Q: Is \(\|x\|_\infty\) 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.