Foundations — H∞ control — robust to uncertainty (intro)

Before you can read a single line of the H∞ intro, you need to earn every symbol it throws at you. This page is that toolbox. We go from zero — a smart 12-year-old who has never seen $\sup$, $j\omega$, or $\Vert \cdot \Vert$ can start at line one.

0 — A signal, a system, and a picture

Figure s01. The whole subject in one picture: an input arrow $w(t)$ (blue) enters the box $G$; an output arrow $z=Gw$ (pink) leaves. Notice the output here is taller than the input — the box amplified it. Everything below is a way of measuring those two arrows and that box.

1 — How big is a signal? The 2-norm $\Vert w{\Vert }_2$

We need a single number for "how much stuff is in this gust." A gust that is tall and long carries more punch than a tiny blip.

Figure s02. The blue curve is a decaying gust $w(t)$; the yellow curve is $w(t{)}^2$ (always positive). The shaded yellow area is the energy $\Vert w{\Vert }_2^2$. Because the gust dies away, that area is finite — a legal, "bounded-energy" disturbance. Notice the curve lives only for $t\ge 0$: that is the causal window the integral sums over.

Why the topic needs it: robustness means "small output for any bounded-energy disturbance." You cannot say "small" without a ruler — $\Vert w{\Vert }_2$ is that ruler. This is built formally in Parseval's Theorem.

2 — Frequency: turning a wiggle into a note

Why bother? Because an LTI system does something beautifully simple to a pure sine: it just scales and shifts it — same frequency out, only bigger/smaller and delayed. That turns hard calculus into simple multiplication.

Figure s03. The pink curve is $|G(j\omega )|$ — the gain the box applies at each frequency $\omega$ (horizontal axis, in rad/s). It is small at low and high $\omega$ but rears up to a sharp peak (yellow dot) near $\omega \approx 2$: that resonant frequency is where this system amplifies most. Reading this curve is literally reading how the box treats every possible tone.

$G(j\omega )$ is a complex number: it stores two facts at once — how much the sine is stretched (its size $|G(j\omega )|$) and how much it is delayed (its angle). For "how big is the worst amplification" we only care about the size.

Why the topic needs it: the "nastiest disturbance nature can throw" is the one at the frequency where the system amplifies most. To find it we must scan $|G(j\omega )|$ across all $\omega$.

3 — "The biggest possible": $\sup$

4 — Worst over all inputs: the energy gain

Now combine §1 (measure signals) with the idea of a worst case.

Why the topic needs it: this is the physical robustness margin — "how much can a unit-energy disturbance grow." The whole H∞ theorem says this equals the frequency peak of §3. (You'll see that identity proved in the parent using Parseval's Theorem.)

5 — When signals are lists: vectors, $\top$, and singular values

Rockets have many inputs (three fins, gust) and many outputs (pitch, yaw, roll). So $w$ and $z$ are vectors — stacks of numbers — and $G$ is a matrix at each frequency.

Figure s04. Left (blue): the unit circle of inputs — all directions, each of length 1. The box $G$ maps it to the pink ellipse on the right. The yellow arrow marks the ellipse's longest axis: an input aimed along its pre-image is stretched more than any other. That longest stretch factor is exactly what we name next.

6 — Assembling the H∞ norm $\Vert G{\Vert }_{\infty }$

Every ingredient is now earned: $\sup$ over $\omega \in \mathbb{R}$ (§3), $\bar{\sigma }$ as induced 2-norm (§5), $G(j\omega )$ via Fourier/Laplace (§2). Nothing is a mystery symbol.

7 — The loop, uncertainty $\Delta$, and $\Vert \cdot \Vert <1$

The parent then wires two systems in a feedback loop and asks when it stays stable.

Prerequisite map

Equipment checklist

Cover the right side and test yourself. If any answer is fuzzy, reread that section.