H Infinity
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## 1 H Infinity ($LaTeX: \mathcal{H}_\infty$)

The H Infinity control paradigm ($LaTeX: \mathcal{H}_\infty$) was developed in the early 1980's by Zames, Doyle and Glover and Kwakernak among others.

The $LaTeX: \mathcal{H}_\infty$ method involves two processes:

1. Definition of specifications based on the $LaTeX: \infty$-norm of the weighted sensitivity function or functions.
2. Controller design and optimisation whereby specifications are met as closely as possible by shaping the weighted sensitivity transfer function.

## 2 Constraints and Specifications: Nominal Stability

Nominal stability is a fundamental requirement of the closed loop system, and simply ensures that under nominal conditions, the system will not be unstable.

## 3 Nominal Performance

For nominal performance, it is required that the $LaTeX: \mathcal{H}_\infty$ norm should conform to the inequality

$LaTeX: \|S(j\omega)W_{s}(j\omega)\|_{\infty}\leq 1$

Where $LaTeX: W_{s}(j\omega})$ is the weighting for the primary sensitivity function $LaTeX: S(j\omega)$