Nyquist stability criterion — encirclements of −1: Foundations

3.5.43 · D1 · Physics › Guidance, Navigation & Control (GNC) › Nyquist stability criterion — encirclements of −1

Yeh page har symbol build karta hai jo parent note use karta hai, un cheezoon se shuru karke jo ek 12-saal-ka bacha already jaanta hai. Ise top to bottom padho: har brick apne neeche waali par tiki hai.

1. $s$ kya hai? The complex-frequency plane

Kisi bhi transfer function ke parentheses mein kya hota hai, yeh samajhne se pehle aapko $L (s)$ ke andar ki cheez jaanni hogi.

Picture. Ek flat map banao. Left–right hai $σ$ , upar–neeche hai $ω$ . Us map ka har point $s$ ki ek value hai. Yeh map $s$ -plane kehlata hai.

Figure — Nyquist stability criterion — encirclements of −1

Figure 1 — $s$ -plane. Vertical (bold) line imaginary axis $σ = 0$ hai. Mint-shaded left half ( $σ < 0$ ) woh jagah hai jahan signals decay karte hain (safe); coral-shaded right half ( $σ > 0$ ) RHP hai jahan signals grow karte hain (unstable). Axis par lavender dot ek pure wiggle $s = j ω$ mark karta hai.

Right half ( $σ > 0$ ) par koi point matlab "time ke saath barhta hai" → yeh Right-Half Plane (RHP) hai, danger zone.
Left half ( $σ < 0$ ) par koi point matlab "time ke saath ghatta hai" → safe.
Exactly vertical axis par ( $σ = 0$ , toh $s = j ω$ ) koi point matlab "pure oscillation, na growing na shrinking" — frequency $ω$ par ek steady wiggle.

Yeh topic isko kyun needs karta hai. Poora Nyquist question hai "kya koi poles RHP mein hain?" Aap yeh tab tak nahi pooch sakte jab tak aap map par RHP point nahi kar sakte. $s$ -plane wahi map hai.

2. Transfer function $L (s)$ , poles, aur zeros

Picture. $s$ -plane par, har pole ko ek chota $\times$ se aur har zero ko ek chota $\circ$ se mark karo. Poles "spikes" hain jahan function infinity par scream karta hai; zeros "holes" hain jahan yeh nothing par drop karta hai.

Figure 2 — Poles ( $\times$ ) aur zeros ( $\circ$ ) $s$ -plane par plot kiye gaye. $- 1$ aur $- 2$ par do lavender $\times$ safe LHP poles hain; $s = + 1$ par coral $\times$ ek RHP pole hai (ek explosive natural motion); teal $\circ$ ek zero hai jahan $L = 0$ hai.

Yeh topic isko kyun needs karta hai. Parent ke letters sab pole-counts hain: $P$ = open loop $L$ ke RHP poles; $Z$ = closed loop ke RHP poles. Agar aap ek pole ko ek spike ki tarah nahi dekh sakte jiska position stability decide karta hai, toh $Z = N + P$ ka kuch matlab nahi hai.

3. Feedback: open loop → closed loop, aur $1 + L$ kahan se aata hai

Formula se pehle, iske do ingredients se milo:

Definition Forward path

$G (s)$ aur feedback path $H (s)$ Ek feedback loop ke do blocks hain jinse signal guzarta hai:

$G (s)$ = forward path — controller-plus-plant jo aapka command ek action mein convert karta hai (e.g. "spacecraft tilt karo"). Yeh woh block hai jo kaam karta hai.
$H (s)$ = feedback path — sensor/measurement jo output read karta hai aur use command se compare karne ke liye wapas bhejta hai (e.g. ek gyro jo actual tilt report karta hai). Yeh woh block hai jo dekhta aur report karta hai.

Picture. Signal andar aata hai → $G$ se guzarta hai → bahar; output ki ek copy tap ki jaati hai, $H$ se wapas bheji jaati hai, aur incoming command se subtract ki jaati hai. Product $L (s) = G (s) H (s)$ woh hai jisse us loop ke around ek full trip aapke signal ko multiply karta hai — open-loop gain.

Ek cheez jo aapko ZAROOR samajhni chahiye:

Picture. $s$ ki jo bhi value open-loop $L$ ko exactly $- 1$ banaye woh closed-loop pole ban jaati hai. Toh number $- 1$ arbitrary nahi hai — yeh exact numerical fingerprint hai "is frequency ne closed loop ki natural motion le li."

Yeh topic isko kyun needs karta hai. Yeh single line, $L = - 1$ , yehi reason hai ki hum kisi aur point ki jagah $- 1$ point hunt karte hain.

4. Complex number ki size aur angle (magnitude & argument)

Nyquist plot $L (j ω)$ ko ek moving point ki tarah draw karta hai. Har point ek complex number hai, aur har complex number do cheezoon se completely describe hota hai — aapko dono words chahiye.

Picture. $0$ se $w$ tak ka arrow banao. Uski length $∣ w ∣$ hai; woh angle jo yeh rightward horizontal se kholti hai woh $ar g w$ hai. Yeh same right-triangle idea hai jaise $θ = arctan (y / x)$ — arrow hypotenuse hai, $x$ adjacent hai, $y$ opposite hai.

Figure 3 — Complex number $w = x + j y$ lavender arrow ki tarah drawn. Uski length magnitude $∣ w ∣ = x^{2} + y^{2}$ hai (mint horizontal = adjacent $x$ , coral vertical = opposite $y$ ), aur origin par teal wedge uska argument $ar g w$ hai.

Counting formula kahan se aati hai (Argument Principle, words mein)

$F = 1 + L$ set karo aur $C$ ko poora RHP fence karne do: enclosed zeros $= Z$ (unstable closed-loop poles), enclosed poles $= P$ (open-loop RHP poles), aur $1 + L$ ke origin ke around turns = $L$ ke $- 1$ ke around turns (ek se shift). Yahi exactly $Z = N + P$ hai.

Yeh topic isko kyun needs karta hai. Bode Plot & Gain/Phase Margins same do ingredients split karta hai: gain = magnitude $∣ L ∣$ , phase = argument $ar g L$ .

5. Frequency response aur FULL Nyquist contour

Figure 4 — $s$ -plane mein Nyquist contour. Lavender arrows positive imaginary axis par upar jaate hain (leg 1); coral dashed arc bada $R \to \infty$ semicircle hai jo RHP sweep karta hai (leg 2); mint arrows negative imaginary axis par neeche aate hain (leg 3, mirror). Pura loop clockwise traverse kiya jaata hai aur shaded RHP fence karta hai.

Definition

$j ω$ -axis par zeros — mirror problem $1 + L$ ka ek zero (ek closed-loop pole) exactly $j ω$ -axis par matlab closed loop mein ek pure, undamped oscillation hai — marginally stable, knife's edge par. Do cheezein follow hoti hain: (1) Plot $- 1$ se guzarta hai. Axis par $1 + L$ ka ek zero matlab $1 + L (j ω_{0}) = 0$ hai, i.e. $L (j ω_{0}) = - 1$ exactly — Nyquist curve danger point ko touch karti hai. (2) Argument Principle wahan toot jaata hai, kyunki $F = 1 + L$ contour par $0$ hit karta hai, aur formula sirf zeros count karta hai strictly andar. Fix / reading: axis pole ke saath bilkul waisa hi, contour ko $s = j ω_{0}$ ke around tiny semicircle se indent karo; aap left ya right bulge karo yeh decide karta hai ki woh borderline root inside (RHP) count hota hai ya outside. Practically, ek curve jo exactly $- 1$ par land karti hai woh ek marginally stable design ki definition hai — encirclement count ambiguous hai exactly isliye kyunki system boundary par hai, aur aapko gain nudge karna hoga decide karne ke liye woh kis side par hai.

6. Nyquist contour direction aur encirclements ( $N$ )

Wrap kaise padhen. Point $- 1$ par khade ho. Arrow dekho jo aapse moving curve ki taraf point karta hai jab $ω$ poore contour par run karta hai. Agar woh arrow ek full clockwise spin complete kare (odometer $- 360°$ ), $N$ ek se badh gaya.

Figure 5 — Encirclement count karna. Lavender loop Nyquist curve hai; chote lavender arrows dikhate hain ki yeh clockwise traverse kiya gaya hai. Coral $\times$ danger point $- 1$ hai. $- 1$ se curve tak teal arrow dekho: poore loop par yeh ek baar clockwise spin karta hai, toh yahan $N = + 1$ hai.

Yeh topic isko kyun needs karta hai. $N$ $Z = N + P$ ka middle letter hai. Yeh wahi quantity hai jo aapko picture dekh ke milti hai; $P$ aapko plant se milta hai, aur $Z$ aap deduce karte ho.

7. Letters ko saath rakhna

Symbol	Seedhe words mein	Picture	Kahan se aata hai
$s = σ + j ω$	complex frequency	map par point	§1
RHP	$σ > 0$ region	map ka right half	§1
$L (s)$	open-loop response	polynomials ka fraction	§2
pole $\times$	jahan $L = \infty$	spike = natural motion	§2
proper	$de g N \leq de g D$	bottom top se chota nahi	§2
$G, H$	forward / feedback path	loop ke do blocks	§3
$1 + L = 0$	characteristic eq.	closed-loop poles	§3
$- 1$	"flipped, full-strength"	danger point	§3
$	L	,\arg L$	size & angle
$Δ_{C} (ar g F)$	total angle turned	arrow ka odometer	§4
contour $C$	RHP-fencing loop	3 legs + detours	§5
$L (j ω)$	frequency response	Nyquist curve	§5
$P$	RHP open-loop poles	fence mein trapped spikes	§2,§6
$N$	wraps of $- 1$	arrow spins	§6
$Z$	RHP closed-loop poles	jawab	§3,§6

Sab kuch parent ke ek boxed line mein culminate hota hai: $Z = N + P .$ Compare karo Routh–Hurwitz Criterion se ( $Z$ paane ka ek algebra-only tarika) aur Root Locus se (jo track karta hai ki gain change hone par closed-loop poles kahan move karte hain).

Prerequisite map

Neeche diagram Obsidian mein top-down flow ki tarah render hota hai; ise padho "har box us arrow ko feed karta hai jo woh point karta hai, saari streams $Z = N + P$ par converge ho rahi hain."

Equipment checklist

Right side cover karo aur dekho ki aap reveal karne se pehle har jawab de sakte ho.

s = σ + j ω

ke do parts kya hain aur har ek kya measure karta hai?

σ

= growth/decay (real part),

ω

= oscillation speed (imaginary part).

s

-plane par unstable region kahan hai?

Right-Half Plane,

σ > 0

(vertical axis ke right).

Ek pole kya hai, ek phrase mein?

s

ki woh value jahan

L

infinity tak blow up karta hai — system ki ek natural motion.

RHP pole "unstable" kyun hai?

Uski natural motion

e^{σ t}

ki tarah grow karti hai jab

σ > 0

, time ke saath explode karti hai.

"Strictly proper" ka matlab kya hai aur yeh kyun matter karta hai?

de g N < de g D

; tab

∣ L ∣ \to 0

infinity par toh bada closing semicircle ek point par collapse hota hai aur koi encirclements add nahi karta.

G (s)

aur

H (s)

kya represent karte hain?

G

= forward path (controller+plant jo act karta hai);

H

= feedback path (sensor jo output measure karta hai aur wapas bhejta hai);

L = G H

Closed-loop poles define karne waali equation kaunsi hai?

1 + L (s) = 0

, i.e.

L (s) = - 1

Number

- 1

special kyun hai?

Uski size

1

hai (full-strength) aur angle

180°

hai (flipped) — woh echo jo khud reinforce karta hai.

Complex number

w = x + j y

ki magnitude aur argument kya hain?

∣ w ∣ = x^{2} + y^{2}

aur

ar g w =

origin se

w

tak arrow ka angle.

ar g w

ko continuously track kyun karna chahiye, calculator se read kyun nahi?

Principal argument branch cut par

360°

jump karta hai; sirf continuous (odometer) total encirclements sahi count karta hai.

Cauchy ka CCW baseline Nyquist ke CW walk se kaise reconcile hota hai?

CW par reverse karna turn-sign

P - Z

kar deta hai;

N

ko CW wraps define karna (

N = Z - P

)

Z = N + P

restore karta hai.

Nyquist contour ke teen legs kaunse hain?

Positive

j ω

-axis upar, bada RHP semicircle, aur negative

j ω

-axis neeche (pehle ka mirror).

Negative-frequency half draw kyun karte hain?

Contour closed hona chahiye Argument Principle ke liye;

L (- j ω) = \overline{L (j ω)}

toh yeh positive half ka sirf mirror hai.

j ω

-axis par baitha hua pole kaise handle karte hain?

Tiny semicircle se detour karo jo RHP mein bulge kare taki pole fence ke bahar rahe (uncounted); yeh plot par ek large arc par map karta hai.

Nyquist curve ka exactly

- 1

se guzarna kya signify karta hai?

1 + L

ka axis par ek zero — ek undamped closed-loop oscillation, i.e. boundary par ek marginally stable design.

Criterion aur uska sign convention state karo.

Z = N + P

contour clockwise aur

N

- 1

ke CW wraps ke saath; stable iff

Z = 0