H∞ control — robust to uncertainty (intro)

3.5.37 · Physics › Guidance, Navigation & Control (GNC)

WHY — woh problem jo hum solve kar rahe hain

Classical/LQR design mein kya galat hota hai? LQR ek quadratic cost ko known model ke liye minimise karta hai. Us model ke liye uske margins bahut achhe hote hain, lekin agar plant thodi alag ho to kuch nahi kehta. H∞ "plant kitni alag ho sakti hai aur phir bhi sab theek rahe?" ko ek solvable optimisation mein badal deta hai.

"Worst case" kaise measure karte hain: signals aur systems pe ek norm se.

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

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

Derivation: peak of frequency response = worst-case energy gain kyun hai

Shuru karte hain wahan se jo hum actually bound karna chahte hain — energy gain: ${\gamma }_{\text{worst}}={\sup }_{w\ne 0}\frac{\Vert z{\Vert }_2}{\Vert w{\Vert }_2}={\sup }_{w\ne 0}\frac{\Vert Gw{\Vert }_2}{\Vert w{\Vert }_2}.$

Yeh step kyun? Yeh literally "unit input energy per worst output energy" hai — robustness margin ka physical meaning.

Parseval's theorem se, energy time aur frequency ke beech preserved rehti hai: $\Vert w{\Vert }_2^2=\frac{1}{2\pi }{\int }_{-\infty }^{\infty }|W(j\omega ){|}^{2}d\omega .$

Yeh step kyun? Yeh hume frequency-by-frequency kaam karne deta hai, jahan $G$ simple multiplication $Z(j\omega )=G(j\omega )W(j\omega )$ ki tarah act karta hai.

Phir (clarity ke liye scalar case): $\Vert z{\Vert }_2^2=\frac{1}{2\pi }\int |G(j\omega ){|}^2|W(j\omega ){|}^{2}d\omega \le ({\sup }_{\omega }|G(j\omega ){|}^2)\frac{1}{2\pi }\int |W{|}^{2}d\omega .$

Yeh step kyun? Sabse bada gain integral se bahar nikaalo — yeh har frequency ke contribution ko upper-bound karta hai.

Toh $\frac{\Vert z{\Vert }_2^2}{\Vert w{\Vert }_2^2}\le {\sup }_{\omega }|G(j\omega ){|}^2$, aur $w$ ki energy ko peak frequency ke paas concentrate karke bound achieve ho jaata hai. Isliye $\boxed{\underset{w}{\sup }\frac{\Vert z{\Vert }_2}{\Vert w{\Vert }_2}=\Vert G{\Vert }_{\infty }}$

Yahi reason hai ki H∞ norm hai worst-case energy amplification. Ise bound karna worst case ko bound karta hai.

Design goal

The Small-Gain Theorem — yeh robustness kyun deta hai

Toh $K$ ko design karna taaki $\Vert M{\Vert }_{\infty }$ 1 se neeche aaye (aur weights ke saath, $\gamma$ se neeche) har uncertainty ke liye stability guarantee karta hai jo us ball mein hai. Yahi saara point hai.

Worked example 1 — Bode plot se H∞ norm padhna

Lo $G(s)=\frac{10}{s^2+0.4s+4}$ (ek lightly damped mode, ${\omega }_n=2$, $\zeta =0.1$).

• Step: $|G(j\omega )|$ ka peak dhundho. Kyun? $\Vert G{\Vert }_{\infty }$ wahi peak hai.
• Resonant peak magnitude $\approx \frac{10/4}{2\zeta \sqrt{1-{\zeta }^2}}=\frac{2.5}{2(0.1)}\approx 12.6$.
• Toh $\Vert G{\Vert }_{\infty }\approx 12.6$: $\omega \approx 2$ rad/s par ek resonant gust ~12.6× amplify hoti hai.
• Interpretation: Agar yeh ek rocket ka bending mode hai, toh H∞ synthesis wahan ek weight insert karega taaki loop use attenuate kare.

Worked example 2 — small-gain sizing

Maano true actuator gain $1\pm 30\%$ hai: $\Delta$ with $\Vert \Delta {\Vert }_{\infty }\le 0.3$.

• Step: Normalise karo: uncertainty ko $0.3\tilde{\Delta }$, $\Vert \tilde{\Delta }{\Vert }_{\infty }\le 1$ likho. Kyun? Small-gain ek unit ball chahta hai.
• Step: Robust stability ke liye chahiye $\Vert 0.3M{\Vert }_{\infty }<1\Rightarrow \Vert M{\Vert }_{\infty }<3.33$.
• Toh jab tak uncertain channel pe nominal loop ki peak gain $3.33$ se neeche rahe, ±30% ki koi bhi actuator error tolerate hogi.
• Yeh powerful kyun hai: ek norm inequality plants ke infinite family ko certify karti hai.

Worked example 3 — weight shaping intuition

Hum achha tracking chahte hain (low frequency pe low error) aur gentle control (high frequency pe roll off).

• Sensitivity $S$ (error/reference) pe $W_S(s)=\frac{s/M+{\omega }_b}{s+{\omega }_{b}A}$ choose karo. Kyun? Low $\omega$ pe bada $W_S$ force karta hai $|S|$ wahan small ho ⇒ achha tracking.
• Require karo $\Vert W_{S}S{\Vert }_{\infty }<1$. Kyun? Kyunki $\Vert W_{S}S{\Vert }_{\infty }<1\Rightarrow |S(j\omega )|<1/|W_S(j\omega )|$ har frequency pe — tum literally ceiling $1/|W_S|$ draw karte ho jiske neeche $|S|$ rehna chahiye.

Active recall

Connections

• LQR and LQG control — known model ke liye optimal; H∞ worst-case robustness add karta hai.
• Sensitivity and Complementary Sensitivity (S+T=1) — woh objects jo H∞ weights shape karte hain.
• Small-Gain Theorem · μ-synthesis (structured uncertainty)
• Singular Value Decomposition — jahan se $\bar{\sigma }$ aata hai (MIMO gain).
• Parseval's Theorem · Bode Sensitivity Integral (waterbed)
• Model Uncertainty in GNC — gusts, flex modes, launch vehicles ke liye mass variation.

Concept Map

Deep Dive

• D1 · Foundations — har symbol zero se
• D2 · Visual walkthrough — derivation pictures mein
• D3 · Worked examples — har scenario
• D4 · Exercises — graded, full solutions
• D5 · Question bank — concept traps