3.5.43
Ek feedback loop tab unstable ho jaata hai jab returned signal khud ko reinforce karta hai instead of die out karne ke. Woh magic point − 1 sign flip hokar aur same size mein — perfect self-reinforcement. Nyquist plot bas loop ki response ka ek map hai har frequency par , aur hum poochhte hain: kitni baar, aur kis direction mein, woh map danger point − 1 Woh winding number humein stability batata hai bina roots solve kiye.
Definition Closed-loop stabilityUnity-feedback system ke liye jisme open-loop transfer function L ( s ) = G ( s ) H ( s ) T ( s ) = 1 + L ( s ) G ( s ) . 1 + L ( s ) 1 + L ( s ) = 0 koi bhi RHP mein nahi hai (right-half plane).
Definition Nyquist criterion (encirclement form)Maano
P L ( s ) N L ( j ω ) − 1 Z closed-loop poles in the RHP ka number.
Toh
Z = N + P iff Z = 0 , yaani iff N = − P − 1 exactly P karna chahiye).
Characteristic equation hai 1 + L ( s ) = 0 L ( s ) = − 1 closed-loop pole wahan hota hai jahan open-loop L − 1 Yeh study karna ki L − 1 1 + L 0
Ab F ( s ) = 1 + L ( s ) C Nyquist contour banaao — poora imaginary axis s = j ω poore RHP ko enclose karta hai.
F = Z F L = P Origin ke around 1 + L = N
Lekin 1 + L 0 L − 1 N = Z − P ⇒ Z = N + P .
Yahi poora criterion hai — seedha angle changes count karne se nikal aata hai.
L ( j ω ) ω : 0 + → ∞ ω : − ∞ → 0 − L ( − j ω ) = L ( j ω ) Contour close karo (big semicircle ka map — usually strictly proper L
− 1 N L P Z = N + P Z = 0
Worked example Example 1 — Stable, no RHP open-loop polesL ( s ) = ( s + 1 ) ( s + 2 ) 2
P − 1 , − 2 ⇒ P = 0 Low freq: L ( 0 ) = 2/2 = 1 Kyun? ω = 0
High freq: magnitude → 0 → − 180° × − 90° Kyun? Har pole asymptotically − 90°
Plot + 1 woh kabhi − 1 , isliye N = 0
Z = N + P = 0 + 0 = 0 stable. ✓
Worked example Example 3 — Unstable open-loop plant made stableL ( s ) = s − 1 K s = + 1 P = 1
Stability ke liye hum chahte hain Z = 0 ⇒ N = Z − P = − 1 − 1 ek counter-clockwise encirclement chahiye.
L ( j ω ) = j ω − 1 K = 1 + ω 2 K ( − 1 − j ω ) ω = 0 L = − K ω → ∞ → 0 Plot ek circle hai jo negative real axis par − K K > 1 − 1 andar hota hai aur ek baar counter-clockwise encircle hota hai ⇒ N = − 1 Z = 0 stable. Agar K < 1 Z = 1
Yeh ulta kyun lagta hai lekin sahi hai: ek unstable plant ko itna strong feedback chahiye jo curve ko − 1
Common mistake "Encirclements of the origin"Kyun sahi lagta hai: humne ise 1 + L Fix: hum L 1 + L − 1 − 1
Common mistake "Any encirclement means unstable"Kyun sahi lagta hai: P = 0 N > 0 hai unstable. Fix: stability hai Z = 0 N = − P P > 0 chahiye . Sign aur count dono matter karte hain.
Common mistake CW / CCW sign ofN Fix: Yahan standard convention — Nyquist contour clockwise traverse hota hai, aur N clockwise encirclements ko positive count karta hai. Ek CCW wrap N < 0
Common mistake Imaginary axis par poles bhool jaanaKyun sahi lagta hai: tum ω = 0 Fix: contour ko j ω right ki taraf ek tiny semicircle se indent karo taaki woh RHP ke bahar rahe; woh arc plot par ek large sweep mein map hoti hai.
Recall Test yourself (answers chhupao)
Closed-loop poles kaunsi equation define karti hai? → 1 + L ( s ) = 0
Point − 1 L = − 1
Count formula batao. → Z = N + P
Ise kaunsa principle underlie karta hai? → Cauchy's argument principle.
N , P N = − P Z = 0
Recall Feynman: ek 12-saal ke bachche ko explain karoSocho tum ek canyon mein chilla rahe ho. Agar echo wapas kamzor aati hai, awaaz mar jaati hai — safe. Lekin agar echo bilkul utni hi tej aur usi direction mein wapas aati hai jis direction mein tumne push kiya tha , toh woh stack up hoti hai aur louder aur louder hoti jaati hai — yahi instability hai. Nyquist plot ek treasure-map hai isi baat ka ki echo tumhari awaaz ki har pitch par kaise behave karti hai. Map par ek "danger X" hai jise − 1
"Z = N + P" → "Z ombies N eed P oles" : unstable (zombie) closed-loop poles Z N yquist wraps plus P lant ke RHP poles se. Aur danger point hai minus one — "echo flipped aur full-strength."
Argument Principle (Cauchy) — mathematical engine.Bode Plot & Gain/Phase Margins — margins, plot ke along − 1 Routh–Hurwitz Criterion — RHP roots count karne ka algebraic alternative.Root Locus — closed-loop poles ko gain K Feedback Control Basics — jahan se T = G / ( 1 + L ) Stability Margins in GNC Loops — attitude/autopilot loops mein application.
Nyquist: hum encirclements kis point ke count karte hain? Point − 1 + 0 j
Nyquist count formula? Z = N + P Z N − 1 P
− 1 Characteristic eq 1 + L = 0 L = − 1 L − 1
Nyquist kaunsa theorem deta hai? Cauchy's argument principle: closed contour ki image, origin ke around Z − P
N , P N = − P Z = 0
Agar P = 0 − 1 Haan: N = 0 ⇒ Z = 0
Unstable plant P = 1 − 1 N = − 1
1 + L L 1 + L 0 L − 1 L − 1
j ω Contour ko right ki taraf ek small semicircle se indent karo (pole ko RHP ke bahar rakho).
− 1 Ek closed-loop pole RHP mein cross ho gaya → instability.
Closed-loop T(s) = G / (1+L)
Characteristic eqn 1+L(s)=0
Poles in RHP means unstable
Cauchy Argument Principle
Nyquist contour encloses RHP
N = CW encirclements of -1
Z = closed-loop RHP poles
