Isse pehle ki tum H∞ intro ki ek bhi line padho, tumhe usmein aane wala har symbol earn karna hoga. Yeh page woh toolbox hai. Hum zero se shuru karte hain — ek smart 12-saal ka bachcha jo kabhi sup, jω, ya ∥⋅∥ nahi dekha, woh bhi line one se shuru kar sakta hai.
Figure s01. Poora subject ek picture mein: ek input arrow w(t) (blue) box G mein jaata hai; ek output arrow z=Gw (pink) bahar aata hai. Dhyan do ki yahan output input se lamba hai — box ne ise amplify kiya. Neeche sab kuch in do arrows aur is box ko measure karne ka tarika hai.
Humein "is gust mein kitna stuff hai" ke liye ek single number chahiye. Jo gust tall aur lamba ho woh ek chote blip se zyada punch karta hai.
Figure s02. Blue curve ek decaying gust w(t) hai; yellow curve w(t)2 hai (hamesha positive). Shaded yellow areahi energy ∥w∥22 hai. Kyunki gust khatam ho jaati hai, woh area finite hai — ek legal, "bounded-energy" disturbance. Dhyan do ki curve sirf t≥0 ke liye hai: yahi woh causal window hai jisko integral sum karta hai.
Topic ko yeh kyun chahiye: robustness ka matlab hai "kisi bhi bounded-energy disturbance ke liye chota output." "Chota" tab tak nahi keh sakte jab tak koi ruler na ho — ∥w∥2 woh ruler hai. Yeh formally Parseval's Theorem mein build hota hai.
Kyun bother karein? Kyunki ek LTI system pure sine ke saath kuch beautifully simple karta hai: woh ise sirf scale aur shift karta hai — wahi frequency bahar aati hai, bas badi/choti aur delayed. Yeh mushkil calculus ko simple multiplication mein badal deta hai.
Figure s03. Pink curve ∣G(jω)∣ hai — woh gain jo box har frequency ω par laagoo karta hai (horizontal axis, rad/s mein). Low aur high ω par yeh chota hai lekin ω≈2 ke paas ek sharp peak (yellow dot) par utha jaata hai: woh resonant frequency hai jahan yeh system sabse zyada amplify karta hai. Yeh curve padhna literally padhna hai ki box har possible tone ke saath kya karta hai.
G(jω) ek complex number hai: yeh do facts ek saath store karta hai — sine kitni stretch hui (uska size ∣G(jω)∣) aur kitni delay hui (uska angle). "Worst amplification kitni badi hai" ke liye hum sirf size ki parwah karte hain.
Topic ko yeh kyun chahiye: "sabse buri disturbance jo nature de sakti hai" woh hoti hai jo us frequency par hoti hai jahan system sabse zyada amplify karta hai. Use dhundhne ke liye humein ∣G(jω)∣ ko saare ω par scan karna hoga.
Ab §1 (signals measure karo) ko worst case ki idea ke saath combine karo.
Topic ko yeh kyun chahiye: yeh physical robustness margin hai — "unit-energy disturbance kitna grow kar sakti hai." Poora H∞ theorem kehta hai yeh §3 ke frequency peak ke barabar hai. (Woh identity parent mein Parseval's Theorem se prove hoti hai.)
Rockets ke kai inputs (teen fins, gust) aur kai outputs (pitch, yaw, roll) hote hain. Isliye w aur zvectors hain — numbers ka stack — aur G har frequency par ek matrix hai.
Figure s04. Left (blue): inputs ka unit circle — har direction, har length 1. Box G use right par pink ellipse mein map karta hai. Yellow arrow ellipse ka sabse lamba axis mark karta hai: is pre-image ki direction mein aimed ek input kisi bhi doosre se zyada stretch hota hai. Woh sabse lamba stretch factor exactly wahi hai jo hum aage naam denge.
Right side cover karo aur khud test karo. Agar koi bhi answer fuzzy lage, woh section dobara padho.
Signal w(t) ek phrase mein kya mean karta hai?
Ek number jo time ke saath badlata hai — jaise time ke saath wind gust.
System G ke baare mein "LTI" kya assume karta hai?
Linear (scale karta hai aur jodta hai) aur time-invariant (kisi bhi time par same behaviour) — woh assumption jo frequency methods ko kaam karaati hai.
System box z=Gw kya kehta hai?
Input signal w ko system G mein daalo; output signal z bahar aata hai.
∥w∥22 geometrically kya measure karta hai?
Squared signal ke neeche ka area — uski total energy.
Energy integral t=0 se kyun shuru hota hai?
Causality — clock tab shuru hoti hai jab gust aati hai; us se pehle w=0 hai aur kuch contribute nahi karta.
Finite-energy signal kyun khatam ho jaani chahiye?
Agar woh zero par nahi aati, toh uske square ke neeche ka area infinite ho jaata.
W(jω) kya hai aur kahan se aata hai?
w(t) ka Fourier transform — woh recipe ki signal mein har frequency kitni hai.
Symbol jω kya stand karta hai?
Frequency ω (radians/sec) ki ek pure sine; j imaginary unit hai.
Frequency domain mein G kaise act karta hai?
Multiplication se: Z(jω)=G(jω)W(jω), ek frequency at a time (LTI ki wajah se).
∣G(jω)∣ tumhe kya batata hai?
Woh factor jitna G frequency ω ki ek sine ko amplify karta hai.
supω kis set par scan karta hai?
Saari real frequencies ω∈R par (aur symmetry se sirf ω≥0 scan karna kaafi hai).
Worst-case energy gain γworst kya hai?
Saare nonzero inputs par sabse bada ratio ∥z∥2/∥w∥2.
w2 ki jagah w⊤w kyun use kiya jaata hai?
Jab w channels ki ek list ho, w⊤w saari entries ke squares jodta hai.
σˉ(G) ko ek maximisation ke roop mein define karo.
σˉ(G)=max∥x∥=1∥Gx∥ — saare unit-length inputs par sabse bada output length (induced 2-norm).
H∞ norm likho aur zor se padho.
∥G∥∞=supω∈Rσˉ(G(jω)) — saari real frequencies aur directions par, sabse bade gain ka peak.
Uncertainty block Δ kya hai?
Woh unknown model-vs-reality gap, size-bounded by ∥Δ∥∞≤1.
Feedback loop kab stable rehta hai (intuitively)?
Jab loop ke around har lap signal ko shrink kare — round-trip gain ek se kam ho.