TVC dynamics — gimbal servo bandwidth, time delay: Question bank

3.5.45 · D5 · Physics › Guidance, Navigation & Control (GNC) › TVC dynamics — gimbal servo bandwidth, time delay

Questions se pehle, har symbol ko pin down karte hain jinpar ye traps rely karte hain, kyunki concept trap ka poora point yahi hai ki wo ek undefined letter mein chhupta hai.

Definition Nomenclature — pehle yeh padho

$ω$ (Greek "omega") ::: ek frequency — kitne radians per second ek oscillating command wiggle karta hai. Bada $ω$ = fast wiggling. Neeche har " $ω$ -something" ek specific frequency hai jis par hum dhyan dete hain.
$s$ ::: Laplace variable, "time mein differentiate karne ke operation" ka ek bookkeeping stand-in. $s$ mein likha hua transfer function hume ek differential equation ko ordinary algebra ki tarah treat karne deta hai. Jab hum frequency $ω$ ki steady oscillation test karte hain, hum $s = j ω$ set karte hain.
$j$ ::: imaginary unit, $j = - 1$ (engineers $j$ likhte hain, $i$ nahi). Ye woh machinery hai jo " $s$ " ko ek rotation mein badalta hai, isliye $e^{- j ω τ}$ size mein koi badalav ke bina $ω τ$ size ka ek pure phase-shift hai.
$ω_{n}$ ::: natural frequency — servo kitni tezi se nozzle ko swing kar sakta hai; $ω_{n} = (k + K) / J$ .
$ω_{B}$ ::: bandwidth — wo sabse zyada frequency jisko servo abhi bhi faithfully follow kar sakta hai, define kiya jata hai jahan iski gain DC value ki $1/ 2$ ( $- 3$ dB) tak gir jaati hai.
$ω_{c}$ ::: loop ki gain-crossover frequency — jahan open-loop gain 1 se guzarti hai; yahi wo frequency hai jahan stability decide hoti hai (dekho Bode plots & phase margin).
$ζ$ ::: damping ratio — rings ( $ζ < 1$ ), cleanly settle karta hai ( $ζ \approx 0.7$ ), slowly move karta hai ( $ζ > 1$ ).
$τ$ ::: pure time delay — computer ke angle decide karne aur nozzle ke move shuru karne ke beech ke seconds.

Neeche di gayi figure is poore page ki mental picture hai — jab bhi koi trap "delay," "phase," ya "magnitude" mention kare, ise dekh lo.

Figure — TVC dynamics — gimbal servo bandwidth, time delay

Recall Figure se kya read karna hai (alt text / caption)

Left panel ::: Nozzle angle mein ek yellow commanded step; blue actual response red band (delay $τ$ ) ke baad hi shuru hoti hai, phir settle hone se pehle rings karti hai — "commanded vs actual" gap dikhata hai. Right panel ::: Pure delay ke liye, green magnitude curve sabhi frequencies par 1 par pinned rehti hai jabki red phase curve steadily neeche slide karti hai ( $- ω τ$ ) — visual proof ki delay sirf phase ko touch karta hai, amplitude ko kabhi nahi.

True or false — justify karo

True or false: Ek pure delay

e^{- s τ}

high frequency par servo ke output ko chhota kar deta hai.

False. Iski magnitude

∣ e^{- j ω τ} ∣ = 1

har frequency par hoti hai — yeh sirf phase ko touch karta hai, amplitude ko kabhi nahi (figure ka right panel).

True or false: Delay ke liye, phase lag

ω τ

frequency badhne ke saath unbounded grow karta hai.

True.

∠ e^{- j ω τ} = - ω τ

ω

mein linear hai, isliye yeh hamesha aur negative hota jaata hai — unlike second-order lag jo

- 180°

par level off hota hai.

True or false: Bandwidth

ω_{B}

hamesha natural frequency

ω_{n}

ke barabar hoti hai.

False. Ye sirf

ζ = 1/ 2 \approx 0.707

par coincide karte hain.

ζ = 0.5

ke liye,

ω_{B} \approx 1.27 ω_{n}

;

ζ = 1

ke liye,

ω_{B} \approx 0.64 ω_{n}

True or false: Loop gain badhana delay-induced stability problem ka valid ilaaj hai.

False. Gain crossover frequency

ω_{c}

ko lift karta hai, aur phase lost

ω_{c} τ

hai — ek higher

ω_{c}

zyada phase spend karta hai, ise aur worse banata hai.

True or false: Zyada servo bandwidth vehicle ke liye unconditionally achhi hai.

False. Agar

ω_{n}

structural bending ya slosh frequencies mein climb kare, to servo flexible modes ko excite karne lagta hai aur TVC ek oscillator ban jaata hai — dekho Structural bending modes & notch filters.

True or false: Delay ki first-order Padé approximation mein ek right-half-plane zero hota hai.

True. Yeh

s = + 2/ τ

par ek zero place karta hai; woh RHP zero "output pehle galat direction mein move karta hai" ka mathematical fingerprint hai, delay ka destabilising essence — dekho Padé approximation of transport delay.

True or false: Ek under-damped servo (

ζ < 0.707

) ki bandwidth iski natural frequency se neeche hoti hai.

False. Under-damped systems roll off hone se pehle thoda resonate karte hain, isliye woh

ω_{n}

se aage bhi faithfully track karte hain; isliye

ζ < 0.707

ke liye

ω_{B} > ω_{n}

hota hai.

True or false:

ζ = 0.707

par servo ke frequency response mein ek flat resonant peak hota hai (magnitude mein koi overshoot nahi).

True.

ζ = 1/ 2

maximally-flat (Butterworth) condition hai jahan magnitude curve bina kisi peak ke roll off hoti hai, exactly isliye yeh design sweet spot hai.

True or false: Agar hum actuator authority

K

ko double kar dein (with

K ≫ k

), to DC gain

K / (k + K)

unity se aur door ho jaata hai.

False. Bada

K

K / (k + K)

ko 1 ki taraf drive karta hai, isliye ek strong inner loop ko unit steady gain wala assume kiya jaata hai.

Error dhundho

"Delay

ω_{c} τ

radians ka gain margin kha jaata hai." — kya galat hai?

Yeh phase margin khaata hai, gain margin nahi. Delay ki magnitude 1 hai, isliye yeh gain ko touch nahi kar sakta;

ω_{c} τ

ek phase angle hai.

"Kyunki

ζ = 0.7

ω_{B} = ω_{n}

hota hai, main hamesha assume karunga ki woh equal hain." — kya galat hai?

Equality sirf us ek

ζ

value par hoti hai. Pehle actual damping ratio check karo, kyunki

ζ = 0.5

par bandwidth

ω_{n}

se 27% zyada hai.

"20 ms ki delay theek hai kyunki 20 ms chhota hai." — kya missing consideration hai?

Jo matter karta hai woh

ω_{c} τ

hai, sirf

τ

nahi.

ω_{c} = 15

rad/s par woh "tiny" delay

0.30

rad

\approx 17°

phase cost karti hai — ek

45°

margin se ek bada bite.

"Servo ko fast banane ke liye mujhe sirf iski inertia

J

reduce karni hai." — yeh kya overlook kiya?

ω_{n} = (k + K) / J

, isliye stiffness

k + K

badhana bhi kaam karta hai; aur speed akeli goal nahi hai — tumhe

ω_{n}

ko first bending mode se neeche rakhna hoga.

e^{- s τ}

ko root-locus analysis se drop kiya ja sakta hai kyunki yeh sirf ek shift hai." — kya galat hai?

Root-locus ko ek rational transfer function chahiye; transcendental

e^{- s τ}

ko pehle Padé approximation se replace karna hoga, jo real poles/zeros introduce karta hai jo stability ko sach mein alter karte hain.

"Hamne ek snappy servo ke liye

ζ

ko 0.3 set kiya." — ek designer objection kyun kar sakta hai?

ζ = 0.3

lightly damped hai: yeh overshoot aur ring karta hai, aur iski resonant peak

ω_{n}

ke paas commands ko amplify kar sakti hai, flexible modes mein energy feed karke.

\approx 0.7

sweet spot peak se bachta hai.

Why questions

Ek pure delay stability ke liye ek equivalent-looking slow lag se zyada dangerous kyun maana jaata hai?

Kyunki iski phase lag

ω τ

bina limit ke grow karti hai, isliye high frequency par yeh total phase ko

- 180°

se aage push kar sakti hai jahan ek bounded lag kabhi nahi pahunch sakta, silently instability threshold cross karte hue.

Bandwidth factors jaise

1.27

(

ζ = 0.5

) aur

0.64

(

ζ = 1

) kahan se aate hain?

Closed form

ω_{B} = ω_{n} (1 - 2 ζ^{2}) + (1 - 2 ζ^{2})^{2} + 1

se:

ζ = 0.5

plug karne par factor

1.27

milta hai, aur

ζ = 1

par

0.64

milta hai. Intuitively, kam damping servo ko

ω_{n}

se aage resonate karne deta hai (factor

> 1

); heavy damping ise pehle roll off karta hai (factor

< 1

Designers

ω_{n} ≳ 5 ω_{c}

(servo bandwidth loop crossover se kaafi upar) kyun target karte hain?

Crossover ke paas second-order servo ka apna phase lag

arctan (\frac{2 ζ ( ω _{c} / ω _{n} )}{1 - ( ω _{c} / ω _{n} ) ^{2}})

hai, jo chhote ratio

ω_{c} / ω_{n}

ke liye approximately

2 ζ (ω_{c} / ω_{n})

rad tak reduce hota hai.

ζ \approx 0.7

aur ratio

0.2

ke saath yeh roughly

16°

hai — itna chhota ki servo rigid-body loop ko "instantaneous" lagta hai.

Servo ko second order kyun model kiya jaata hai na ki first order?

Yeh physically ek motor torque hai jo damping

c

aur stiffness

k

ke against ek nozzle inertia

J

drive karta hai — ek mass–spring–damper, jo inherently second order hai (dekho Second-order systems — natural frequency & damping).

Delay se ladne ke liye gain add karna backfire kyun karta hai?

Gain crossover frequency

ω_{c}

badhata hai, aur phase lost

ω_{c} τ

hai; bada

ω_{c}

usi

τ

ko ek bade phase loss mein multiply karta hai, margin aur erode karta hai.

Padé form

\frac{1 - \frac{τ}{2} s}{1 + \frac{τ}{2} s}

use karta hai na ki simple pole

\frac{1}{1 + τ s}

— kyun?

Dono

e^{- s τ}

ki Taylor series ko first order tak match karte hain, lekin sirf Padé form right-half-plane zero reproduce karta hai — "pehle galat direction mein jaata hai" wala behaviour jo akela pole completely miss kar deta.

Servo bandwidth ko first structural bending frequency se neeche kyun rakhna chahiye?

Kyunki ek fast servo apni passband ke andar frequencies par respond karta hai; agar bending resonances us band mein padein, to actuator unhe drive karta hai, aur TVC flexible mode ke saath couple ho jaata hai steering ki jagah — ek margin-destroying feedback loop.

Edge cases

Edge case: Jab

ζ \to 0

(no damping), bandwidth ka kya hota hai?

Formula deta hai

ω_{B} = ω_{n} 1 + 2 \approx 1.55 ω_{n}

, aur resonant peak infinity ki taraf grow karta hai — "bandwidth" number misleading ho jaata hai kyunki response

ω_{n}

ke paas wildly amplify karta hai.

Edge case: Zero frequency (

ω \to 0

) par delay dwara khaaya gaya phase kitna hai?

Zero —

ω τ \to 0

, isliye ek steady (DC) command par koi phase penalty nahi hoti; delay sirf fast, oscillatory commands ko bite karta hai.

Edge case: Agar

τ = 0

(koi delay nahi), to kya Padé zero phir bhi exist karta hai?

Nahi — zero

s = 2/ τ

par hota hai, jo

τ \to 0

ke saath infinity ki taraf chala jaata hai, isliye wrong-way behaviour vanish ho jaata hai aur delay term

1

ban jaata hai.

Edge case: Ek over-damped servo (

ζ > 1

) ke liye,

ω_{n}

ke relative bandwidth ka kya hota hai?

Yeh

ω_{n}

se neeche gir jaata hai (jaise

ζ = 1

par

ω_{B} \approx 0.64 ω_{n}

); servo sluggish hai aur apni natural frequency se pehle roll off karta hai, isliye yeh

ω_{n}

tak bhi track nahi kar sakta.

Edge case: Agar digital sampling rate dominant delay source ho to?

Tab

τ

roughly ek sample period hai (plus computation), isliye faster sampling directly

τ

shrink karta hai aur phase margin wapas dilata hai — dekho Digital control — sampling & computational delay.

Edge case: Limit

K ≫ k

mein, servo DC gain kya ban jaata hai aur yeh kyun matter karta hai?

Yeh

K / (k + K) \to 1

approach karta hai, matlab nozzle steady state mein exactly commanded angle par settle karta hai — unit-gain assumption jo har jagah downstream Thrust Vector Control geometry & torque mein use hoti hai.

Recall Lock karne ke liye ek-line summary

Ek pure delay kaun si quantity change karta hai, aur kaun si untouched chodta hai? ::: Yeh phase change karta hai (by $- ω τ$ ) aur magnitude untouched chodta hai — yahi poori reason hai ki yeh ek silent margin-eater hai.