H∞ control — robust to uncertainty (intro): Visual walkthrough

3.5.37 · D2 · Physics › Guidance, Navigation & Control (GNC) › H∞ control — robust to uncertainty (intro)

Neeche sab kuch yeh assume karke hai ki aapne kabhi koi norm, frequency response, ya integral is tarah use nahi dekha. Har symbol ko use karne se pehle samjhaya gaya hai.

Step 1 — Signal kya hota hai, aur yeh kitna "bada" hai?

KYA HAI. Ek signal $w (t)$ bas ek aisa number hai jo time ke saath badalta rehta hai — jaise rocket par wind gust ki taakat, second by second. Hume "is signal mein kitna stuff hai" ke liye ek single number chahiye, kyunki baad mein hum output stuff ko input stuff se compare karte hain.

YEH TOOL KYU (square ka integral). Hum sabse bade spike ko measure kar sakte the, lekin ek brief tall spike mein zyada "punch" nahi hota, jabki ek lambi gentle push mein bahut zyada ho sakta hai. Total punch ka sahi measure hai energy: signal ko square karo (taaki upar-neeche dono positive count hon), phir un saare squares ko time ke across jod do. "Continuous time mein add up karna" exactly yahi hai jo integral $\int \cdot d t$ karta hai — ek smooth sum. Isliye hum plain maximum ki bajay yahi use karte hain.

PICTURE. Squared curve ke neeche ka area energy hai. Ek tall-lekin-brief blip aur ek low-lekin-long push ka same shaded area ho sakta hai.

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

Term-by-term: $w (t)$ = signal, $w (t)^{2}$ = uska instantaneous punch (hamesha $\geq 0$ ), $\int d t$ = total accumulated punch = shaded area.

Step 2 — Ek system frequency-by-frequency amplifier hota hai

KYA HAI. Ek stable linear system $G$ input signal $w$ leta hai aur output $z = Gw$ deta hai. Jadui fact yeh hai: agar aap $G$ ko frequency $ω$ (radians per second) ka pure sine wave feed karo, toh output mein same frequency ka sine wave aata hai, bas rescaled (aur shifted). Yeh rescaling factor ek number hai jise hum $∣ G (j ω) ∣$ kehte hain — frequency $ω$ par gain.

YEH KYUN MATTER KARTA HAI. Kyunki har signal secretly sine waves ka sum hota hai (frequency ka poora idea yahi hai), agar hume pata hai $G$ har single sine ke saath kya karta hai, toh hume pata hai yeh sab ke saath kya karta hai. Symbol $j$ bas yeh flag hai "hum frequency $ω$ ke sine wave ke baare mein pooch rahe hain"; $∣ G (j ω) ∣$ hai "yeh sine kitne times taller hokar bahar aata hai."

PICTURE. Low-frequency sines shayad gently pass hon; ek resonance ke paas ek particular sine enormously blow up ho jaati hai; bahut high frequencies squash ho jaati hain.

Woh tall red spike peak gain hai — isse yaad rakho, yeh show ka star ban jaata hai.

Step 3 — Woh exact sawaal jo hum answer karna chahte hain

KYA HAI. Robustness poochti hai: un sab disturbances mein se jo nature humpe throw kar sakti hai, kaun sabse zyada amplify hota hai, aur kitna? Energy terms mein:

$sup$ KYU, "average" KYUN NAHI. Robustness ek promise hai nastiest case ke against, typical ke against nahi. Average ek catastrophic single disturbance ko chhupa sakta hai. Hume ek aisi guarantee chahiye jo no matter what hold kare, isliye hum range ka top lete hain.

PICTURE. Imagine karo hazaaron input shapes try kar rahe ho; har ek ek ratio-axis par ek dot deta hai. $γ_{worst}$ sabse oopar wala dot hai.

Mushkil yeh hai: infinitely many possible $w$ hain. Hum inhe haath se try nahi kar sakte — isliye hume is supremum ko compute karne ki ek trick chahiye. Woh trick agla step hai.

Step 4 — Parseval: energy time aur frequency dono mein same hoti hai

KYA HAI. Parseval's Theorem kehta hai time mein compute ki gayi signal ki energy frequency mein compute ki gayi energy ke barabar hoti hai. $W (j ω)$ likho $w$ ke andar chhupe frequency- $ω$ sine ki quantity ke liye (uska spectrum). Tab

YEH TOOL KYU AUR KOI KYU NAHI. Time domain mein, $z = Gw$ ek messy convolution hai (ek smearing). Lekin frequency domain mein, Step 2 ne bataya ki $G$ , simple multiplication se act karta hai: $Z (j ω) = G (j ω) W (j ω)$ . Parseval woh bridge hai jo humein convolution se ladna band karwata hai aur frequency by frequency multiply karne deta hai. Yahi exact reason hai ki hum ise yahan invoke karte hain.

PICTURE. Do histograms jinki total area equal hai: ek hai "energy vs time," doosra "energy vs frequency." Parseval kehta hai totals match karte hain.

Step 5 — Output energy ko frequency sum mein convert karo

KYA HAI. Output $z$ par Parseval apply karo, phir Step 2 se $Z = G W$ use karo:

=\frac{1}{2\pi}\int_{-\infty}^{\infty}\underbrace{|G(j\omega)|^2}_{\text{gain}^2\text{ at }\omega}\ \underbrace{|W(j\omega)|^2}_{\text{input energy at }\omega}\,d\omega.$$ **KYU.** Yeh key rewrite hai: output energy, input ki energy spectrum hai, **har frequency par gain squared se weighted**. Frequency ka har slice $|G|^2 \cdot |W|^2$ contribute karta hai. **PICTURE.** Step 4 ke input spectrum bars lo aur har bar ki height ko gain-squared curve se multiply karo. Resonant peak ke paas ke bars tall ho jaate hain; jahan $G$ small hai wahan ke bars shrink ho jaate hain. ![[deepdives/dd-physics-3.5.37-d2-s05.png]] --- ## Step 6 — Peak bahar nikalo (inequality) **KYA HAI.** **Har** frequency par, local gain $|G(j\omega)|^2$ kahin bhi sabse **bade** gain se zyada nahi hota, jise hum name karte hain $$\|G\|_\infty^2=\sup_\omega |G(j\omega)|^2 \quad(\text{Bode curve ka peak}).$$ Toh har $|G(j\omega)|^2$ ki jagah woh single sabse badi value rakh do — isse sum sirf bada ya equal ho sakta hai: $$\|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)\underbrace{\frac{1}{2\pi}\int |W(j\omega)|^2 d\omega}_{=\ \|w\|_2^2\ \text{(Parseval again)}}.$$ Dono sides ko $\|w\|_2^2$ se divide karo: $$\boxed{\ \frac{\|z\|_2^2}{\|w\|_2^2}\ \le\ \|G\|_\infty^2\ \Longrightarrow\ \frac{\|z\|_2}{\|w\|_2}\le\|G\|_\infty\ }$$ **KYU.** Integral mein se maximum bahar nikalna ek clean ceiling paane ka standard tarika hai: "koi frequency apne worst gain par apne share se zyada contribute nahi karti." Yeh prove karta hai ki ratio **kabhi** $\|G\|_\infty$ se exceed nahi kar sakta. **PICTURE.** Wiggly gain-squared curve ko peak height par ek horizontal ceiling tak flatten karo — flat ceiling ke neeche ka area (times input spectrum) true area se kabhi chhota nahi hota. ![[deepdives/dd-physics-3.5.37-d2-s06.png]] --- ## Step 7 — Ceiling actually reach hoti hai (yeh sirf $\le$ kyun nahi, equality kyun hai) **KYA HAI.** Upper bound tab tak bekar hai jab tak koi use achieve na kare. Lekin hum ek aisi disturbance bana sakte hain jo ceiling **hit** kare: aisa $w$ choose karo jiska energy *peak $\omega^\star$ ke aaspaas frequencies ki narrow band mein* concentrated ho (jahan $|G|$ sabse bada ho). Tab $w$ ki almost saari energy exactly wahan hoti hai jahan $G$ sabse hard amplify karta hai, aur $$\frac{\|z\|_2}{\|w\|_2}\ \longrightarrow\ \|G\|_\infty.$$ **YEH ARGUMENT KYUN CLOSE KARTA HAI.** Step 6 ne dikhaya ki ratio $\|G\|_\infty$ se *exceed nahi kar sakta*. Yeh step dikhata hai ki yeh $\|G\|_\infty$ ke *arbitrarily close ja sakta hai*. Saath mein: supremum $\|G\|_\infty$ ke **barabar hai**. Yahi parent note ka boxed result hai, ab earn kiya gaya. > [!formula] Result, fully earned > $$\sup_{w\neq 0}\frac{\|z\|_2}{\|w\|_2}=\|G\|_\infty=\sup_\omega|G(j\omega)|.$$ > Worst-case **energy** amplification frequency response ke **peak** ke barabar hai. **PICTURE.** Ek spectrum jo resonant peak ke upar seedha ek thin spike jaisa shaped hai — "meanest" disturbance. Uska output woh spike hai jo peak gain se scale hua hai. ![[deepdives/dd-physics-3.5.37-d2-s07.png]] --- ## Step 8 — Edge aur degenerate cases (koi scenario unshown mat chhodna) **KYA HAI.** Hume corners check karne chahiye taaki reader ko kuch surprise na kare. - **Zero input, $w=0$.** Ratio $\|z\|_2/\|w\|_2$ hai $0/0$ — undefined, yahi exact reason hai ki supremum $w\neq 0$ par liya jaata hai. Koi information lost nahi hoti; peak *tiny but non-zero* inputs se approach hoti rehti hai. - **$G$ mein koi resonant peak nahi (monotone roll-off).** Tab sabse bada gain $\omega=0$ (DC) par hai. Formula phir bhi hold karta hai — "peak" bas left edge par hai. *Resonance mein kuch special nahi; yeh bas wahan hoti hai jahan peak sit karti hai.* - **Unstable $G$.** Tab kuch sine unbounded grow karti hai, $\sup_\omega|G(j\omega)|=\infty$, isliye $\|G\|_\infty=\infty$. H∞ norm sahi tarah se unstable systems ke liye *exist karne se mana kar deta hai* — yahi reason hai ki H∞ synthesis pehle ek **stabilising** controller demand karta hai. - **Vector / matrix $G$ (MIMO).** Kai inputs aur outputs ke liye, "$\omega$ par gain" ek number nahi balki ek matrix hai. $|G(j\omega)|$ ki sahi jagah **largest singular value** $\bar\sigma(G(j\omega))$ hai [[Singular Value Decomposition]] se — woh sabse bada stretch factor jo matrix kisi bhi input direction par apply karta hai. Isliye $\|G\|_\infty=\sup_\omega\bar\sigma(G(j\omega))$, jo ek input aur output hone par $|G(j\omega)|$ reduce ho jaata hai. **PICTURE.** Teen mini-panels: (a) ek monotone curve apne peak ke saath DC par, (b) ek unstable pole curve ko infinity par bhejta hua, (c) ek matrix input directions ke unit circle ko ek ellipse mein stretch karta hua jiska longest axis $\bar\sigma$ hai. ![[deepdives/dd-physics-3.5.37-d2-s08.png]] > [!mistake] Woh trap jo yeh walkthrough khatam karta hai > **"$\|G\|_\infty$ average ya DC gain hai."** Nahi — upar ka har step **peak** ki taraf point karta raha. Worst disturbance (Step 7) apni energy deliberately wahan dump karta hai jahan $G$ *sabse bada* hai, average nahi. Averaging ek single resonant gust ko aapki safety promise se slip through karne deta. --- ## Ek-picture summary Sab kuch ek flow mein collapse ho jaata hai: **signal energy → frequencies mein split karo → har frequency gain se scale hoti hai → sabse bada gain worst case set karta hai → woh sabse bada gain *hi* $\|G\|_\infty$ hai.** ![[deepdives/dd-physics-3.5.37-d2-s09.png]] > [!recall]- Feynman retelling (plain words) > Apni disturbance ko sunlight samjho aur apne system ko ek magnifying glass. Light ki "energy" hai total glow kitna hai (Step 1). Magnifying glass alag *colours* ko alag amounts se bend karta hai — yahi frequency response hai (Step 2). Hum jaanna chahte hain: fixed amount of light se yeh glass *sabse hot* spot kaisa bana sakta hai (Step 3)? Trick: light ko uske colours mein sort karo (Parseval, Step 4), aur note karo glass bas har colour ki brightness ko apne factor se multiply karta hai (Step 5). Koi single colour glass ki *strongest* focusing power se zyada boost nahi hoti, isliye total heat us strongest factor ko beat nahi kar sakti (Step 6). Aur agar aap sirf *us ek best colour* ki light shine karo, toh aap exactly maximum heat reach kar lete ho (Step 7). Woh maximum focusing factor — peak — H∞ norm hai. Use small karo, aur koi light, koi gust, koi disturbance tumhe kabhi zyada nahi jala sakti. Corner cases: zero light se koi ratio nahi milta, ek glass jo sab kuch infinity tak focus kare woh ek broken (unstable) glass hai, aur ek fancy multi-colour multi-spot glass bas sabse bade "stretch" (largest singular value) ko apni focusing power ki tarah use karta hai. --- ## Active recall > [!recall] Quick self-test > 1. Hum signal ka size $\int w^2\,dt$ kyun measure karte hain, uski tallest spike kyun nahi? > 2. Woh kaun sa single fact hai jo humein messy time-convolution ko clean frequency-multiplication se swap karne deta hai? > 3. Step 6 mein, $|G(j\omega)|$ ko uske peak se replace karna valid *upper* bound kyun hai? > 4. Step 7 ne ek special disturbance build ki — uski shape kya thi, aur yeh equality kyun prove karta hai? > 5. Agar $G$ unstable ho toh $\|G\|_\infty$ ka kya hota hai, aur H∞ kyun care karta hai? H∞ norm frequency ke upar *peak* kyun hai, average kyun nahi? ::: Worst-case disturbance (Step 7) apni energy maximum gain ki single frequency par concentrate karta hai, isliye sirf peak — average nahi — sabse badi possible energy amplification determine karta hai. Which theorem bridges time-domain energy aur frequency-domain gain ko derivation mein? ::: [[Parseval's Theorem]] — yeh $\|w\|_2^2$ ko dono domains mein equal banata hai taaki $G$ ko frequency-by-frequency multiplication ki tarah treat kiya ja sake. Ek matrix (MIMO) system ke liye, "$\omega$ par gain" ki jagah kya aata hai? ::: Largest singular value $\bar\sigma(G(j\omega))$ [[Singular Value Decomposition]] se.