H∞ control — robust to uncertainty (intro)

WHY — the problem we are solving

WHAT goes wrong with classical/LQR design? LQR minimises a quadratic cost for the known model. It has great margins for that model, but says nothing about a plant that differs. H∞ turns "how different can the plant be, and still be OK?" into a solvable optimisation.

HOW we measure "worst case": with a norm on signals and systems.

Building block 1 — signal size (the 2-norm)

Building block 2 — the H∞ norm (system "gain")

Derivation: why the peak of the frequency response = worst-case energy gain

Start from what we actually want to bound — the energy gain: $\gamma_{\text{worst}} = \sup_{w\neq 0}\frac{\|z\|_2}{\|w\|_2}=\sup_{w\neq 0}\frac{\|Gw\|_2}{\|w\|_2}.$

Why this step? This is literally "worst output energy per unit input energy" — the physical meaning of robustness margin.

By Parseval's theorem, energy is preserved between time and frequency: $\|w\|_2^2 = \frac{1}{2\pi}\int_{-\infty}^{\infty}|W(j\omega)|^2\,d\omega.$

Why this step? It lets us work frequency-by-frequency, where $G$ acts as simple multiplication $Z(j\omega)=G(j\omega)W(j\omega)$ .

Then (scalar case for clarity): $\|z\|_2^2=\frac{1}{2\pi}\int|G(j\omega)|^2|W(j\omega)|^2 d\omega \le \Big(\sup_\omega |G(j\omega)|^2\Big)\frac{1}{2\pi}\int|W|^2 d\omega.$

Why this step? Pull the biggest gain out of the integral — it upper-bounds every frequency's contribution.

So $\dfrac{\|z\|_2^2}{\|w\|_2^2}\le \sup_\omega|G(j\omega)|^2$ , and by concentrating $w$ 's energy near the peak frequency the bound is achieved. Hence $\boxed{\ \sup_w \frac{\|z\|_2}{\|w\|_2}=\|G\|_\infty\ }$

That is why the H∞ norm is the worst-case energy amplification. Bounding it bounds the worst case.

The design goal

The Small-Gain Theorem — why this gives robustness

So designing $K$ to push $\|M\|_\infty$ below 1 (and, with weights, below $\gamma$ ) guarantees stability for every uncertainty in the ball. That is the whole point.

Worked example 1 — reading an H∞ norm from a Bode plot

Take $G(s)=\dfrac{10}{s^2+0.4s+4}$ (a lightly damped mode, $\omega_n=2$ , $\zeta=0.1$ ).

Step: Find peak of $|G(j\omega)|$ . Why? $\|G\|_\infty$ is that peak.
Resonant peak magnitude $\approx \dfrac{10/4}{2\zeta\sqrt{1-\zeta^2}}=\dfrac{2.5}{2(0.1)}\approx 12.6$ .
So $\|G\|_\infty\approx 12.6$ : a resonant gust at $\omega\approx2$ rad/s is amplified ~12.6×.
Interpretation: If this is a bending mode of a rocket, H∞ synthesis will insert a weight there to force the loop to attenuate it.

Worked example 2 — small-gain sizing

Suppose the true actuator gain is $1\pm30\%$ : $\Delta$ with $\|\Delta\|_\infty\le 0.3$ .

Step: Normalise: write uncertainty as $0.3\,\tilde\Delta$ , $\|\tilde\Delta\|_\infty\le 1$ . Why? Small-gain wants a unit ball.
Step: Robust stability requires $\|0.3\,M\|_\infty<1\Rightarrow \|M\|_\infty<3.33$ .
So as long as the nominal loop's peak gain at the uncertain channel stays below $3.33$ , any actuator error in ±30% is tolerated.
Why this is powerful: one norm inequality certifies an infinite family of plants.

Worked example 3 — weight shaping intuition

We want good tracking (low error at low frequency) and gentle control (roll off high frequency).

Choose $W_S(s)=\frac{s/M+\omega_b}{s+\omega_b A}$ on the sensitivity $S$ (error/reference). Why? Large $W_S$ at low $\omega$ forces $|S|$ small there ⇒ good tracking.
Require $\|W_S S\|_\infty<1$ . Why? Since $\|W_S S\|_\infty<1\Rightarrow |S(j\omega)|<1/|W_S(j\omega)|$ at every frequency — you literally draw the ceiling $1/|W_S|$ that $|S|$ must stay under.

Common mistake Steel-manning the wrong ideas

"H∞ just makes gains huge to crush disturbances." Why it feels right: disturbance rejection = high loop gain, so more gain seems better. The fix: you cannot have $|S|$ small at all frequencies — the waterbed / Bode integral law $\int \ln|S|\,d\omega = 0$ (for typical plants) means pushing $S$ down here pushes it up elsewhere. H∞ trades via weights; it finds the best-shaped compromise, not brute force.

"H∞ norm is the DC gain / the average gain." Why it feels right: people quote "the gain" of a system. The fix: it's the peak over all frequencies (largest singular value), because the worst disturbance concentrates energy at the worst frequency.

"Small-gain is conservative, so it's useless." Why it feels right: $\|M\|_\infty<1$ ignores $\Delta$ 's structure and can reject safe plants. The fix: it's sufficient, and structured versions (μ-synthesis) reduce the conservatism while keeping the guarantee.

Recall Explain it to a 12-year-old (Feynman)

Imagine you build a robot to balance a broom. You practised with one broom, but tomorrow the broom might be heavier, or a fan might blow on it. A normal robot only knows its practice broom. The H∞ robot asks: "What is the meanest push the wind could give, and can I still stay standing?" It designs itself so that even the worst push only tips the broom a little. The "H∞ number" is just the biggest wobble the worst wind can cause — and we make that number small.

Active recall

What does the H∞ norm

\|G\|_\infty

represent physically?

The worst-case energy gain

\sup_w \|z\|_2/\|w\|_2

, equal to the peak over frequency of the largest singular value

\sup_\omega \bar\sigma(G(j\omega))

Which theorem links H∞ synthesis to robustness?

The Small-Gain Theorem:

M

–

\Delta

loop is stable for all

\|\Delta\|_\infty\le1

iff

\|M\|_\infty<1

Why does bounding

\|T_{zw}\|_\infty<\gamma

help?

It guarantees no exogenous disturbance is amplified into the performance outputs by more than

\gamma

, for the whole uncertainty set.

What mathematical tool converts time-domain energy to frequency domain in the derivation?

Parseval's theorem,

\|w\|_2^2=\frac{1}{2\pi}\int|W(j\omega)|^2 d\omega

What do weighting functions

W_i

encode?

The designer's frequency-dependent specs (tracking, control effort, noise rejection); e.g.

\|W_S S\|_\infty<1

forces

|S|<1/|W_S|

Why can't sensitivity

|S|

be small at all frequencies?

Bode/waterbed integral

\int\ln|S|\,d\omega=0

: reducing it in one band raises it in another.

For a plant with resonance

\zeta=0.1

, roughly how big is the resonant peak factor?

About

1/(2\zeta)=5

times the low-frequency gain, dominating

\|G\|_\infty

How is a ±30% gain uncertainty turned into a small-gain requirement?

Write it as

0.3\tilde\Delta

\|\tilde\Delta\|_\infty\le1

; robust stability needs

\|0.3 M\|_\infty<1

, i.e.

\|M\|_\infty<3.33

Connections

LQR and LQG control — optimal for the known model; H∞ adds worst-case robustness.
Sensitivity and Complementary Sensitivity (S+T=1) — the objects H∞ weights shape.
Small-Gain Theorem · μ-synthesis (structured uncertainty)
Singular Value Decomposition — where $\bar\sigma$ comes from (MIMO gain).
Parseval's Theorem · Bode Sensitivity Integral (waterbed)
Model Uncertainty in GNC — gusts, flex modes, mass variation for launch vehicles.

Concept Map

Hinglish (regional understanding)

Intuition Hinglish mein samjho

Dekho, real rocket ya drone kabhi bhi apne math model ke exactly barabar nahi hota. Wind gust aayega, mass change hoga, kuch bending modes model me the hi nahi. Agar tumne controller sirf "nominal" (perfect) model pe tune kiya, toh real hardware pe woh unstable ho sakta hai. H∞ control ka idea simple hai: sirf average case mat sochoin, balki worst-case — sabse kameena disturbance — pe bhi system safe aur accha rahe. Yehi robustness hai.

Core quantity hai H∞ norm, $\|G\|_\infty$ . Yeh koi average gain nahi hai — yeh saare frequencies me se sabse bada amplification hai (peak of the Bode magnitude, MIMO me largest singular value). Kyun? Kyunki worst disturbance apni saari energy us frequency pe daal degi jahan system ka gain sabse zyada hai. Parseval theorem se hum time-domain energy ko frequency me le jaate hain, aur prove karte hain ki worst-case energy gain exactly $\|G\|_\infty$ ke barabar hai.

Robustness ka guarantee Small-Gain Theorem deta hai: agar uncertainty ko ek bounded block $\Delta$ ( $\|\Delta\|_\infty\le1$ ) maano jo loop me lagi hai, toh loop tabhi tak stable rahega jab tak $\|M\|_\infty<1$ . Loop ke around signal har baar $M\Delta$ se multiply hota hai; agar yeh product 1 se chhota hai toh geometric series converge kar jaati hai, matlab bounded, matlab stable. Isliye H∞ design ka goal: $\|T_{zw}\|_\infty$ ko $\gamma$ ke neeche push karo — tab koi bhi disturbance $\gamma$ se zyada amplify nahi hogi, aur infinite plants ka poora family ek hi inequality se certify ho jaata hai.

Practical me hum weighting functions lagate hain — yeh batate hain ki kaunsi frequency pe kya important hai (tracking low frequency pe, control effort high pe). $\|W_S S\|_\infty<1$ ka matlab hai $|S| < 1/|W_S|$ — matlab tum ek ceiling draw karte ho jiske neeche error ko rehna padega. Bas yaad rakho: gain infinite nahi bana sakte, kyunki waterbed effect (Bode integral) hai — ek jagah dabaoge toh doosri jagah uthega. H∞ isi trade-off ka best solution nikaalta hai.

H∞ control — robust to uncertainty (intro)

WHY — the problem we are solving

Building block 1 — signal size (the 2-norm)

Building block 2 — the H∞ norm (system "gain")

Derivation: why the peak of the frequency response = worst-case energy gain

The design goal

The Small-Gain Theorem — why this gives robustness

Worked example 1 — reading an H∞ norm from a Bode plot

Worked example 2 — small-gain sizing

Worked example 3 — weight shaping intuition

Active recall

Connections

Concept Map

Hinglish (regional understanding)

Go deeper — visual, from zero

Test yourself — Guidance, Navigation & Control (GNC)

Connections