Control system fundamentals — plant, actuator, sensor, controller: Question bank

3.5.26 · D5 · Physics › Guidance, Navigation & Control (GNC) › Control system fundamentals — plant, actuator, sensor, contr

Loop gain ( $G H$ ) ka matlab hai wo total factor jo ek signal loop mein ek baar poora ghoomne par pick up karta hai; characteristic equation ka matlab hai $1 + G (s) H (s) = 0$ , jiske roots closed-loop poles hote hain. Dono parent note mein banaye gaye hain — agar koi bhi unfamiliar lage toh wahan wapas jao.

True or false — justify

A perfect (unity-gain, zero-lag) sensor stability analysis se

H (s)

ko hata deta hai.

False. Sirf tabhi loop gain

G

tak reduce hota hai jab

H (s) = 1

exactly ho;

H

mein koi bhi real dynamics ya scaling

1 + G H

ke andar rehti hai aur poles ko move kar sakti hai — DC par accuracy ka matlab

H (s) = 1

at all frequencies nahi hota. Dekho Poles Zeros and Stability.

Ek integrator add karna hamesha system ko improve karta hai.

False. Ek integrator (

s = 0

par pole) steady-state step error ko khatam karta hai lekin

- 90°

ka phase lag add karta hai, stability margin ko erode karta hai aur oscillation ko invite karta hai — yeh ek trade hai, free win nahi.

Laplace domain mein, cascaded blocks apne transfer functions add karte hain.

False. Cascaded LTI blocks multiply karte hain:

G = C A P

. Addition summing junctions par hoti hai (jaise

r - y_{m}

), chain ke along nahi. Dekho Transfer Functions and Laplace Domain.

Ek stable open-loop plant ek stable closed loop guarantee karta hai.

False. Loop band karne se poles

1 + G H = 0

ke roots par move ho jaate hain; high gain un roots ko right half-plane mein push kar sakta hai, chahe bare plant

P (s)

theek bhi tha.

Reference

r (t)

plant ki dynamics ka hissa hai.

False.

r

Guidance ka ek external command hai; plant

P (s)

wo fixed physics hai jise command ki ja rahi hai. Ye loop mein bilkul alag points par enter karte hain.

Proportional gain

K

badhana hamesha response ko faster aur better banata hai.

False. Yeh

ω_{n}

badhata hai (faster) lekin motor example mein damping

ζ = 1/ K

ko kam karta hai, isliye tum speed khareed rahe ho overshoot aur eventual instability ke saath. Dekho PID Control.

Purely proportional controller ke saath bhi step ke liye steady-state error zero ho sakta hai.

True — lekin sirf tabhi jab plant mein already ek integrator ho (

s = 0

par pole), jisse

G (0) \to \infty

ho; uske bina,

e_{ss} = 1/ (1 + G (0)) > 0

Controller aur actuator ko koi fidelity khoye bina ek block mein merge kiya ja sakta hai.

False. Controller ek abstract number

u

output karta hai; actuator mein hardware limits hoti hain (saturation, rate limit, lag) jo controller math kabhi nahi dekhta. Unhe merge karne se windup chhup jaata hai. Dekho Actuator Saturation and Anti-Windup.

Feedback ek system ko plant ko exactly na jaanne ke liye robust banata hai.

True. Kyunki loop actual output ko measure karta hai aur error ko correct karta hai,

P (s)

mein moderate modelling errors partly suppress ho jaate hain — wahi robustness loop band karne ka pura point hai.

Spot the error

T (s) = G / (1 - G H)

kyunki feedback subtract karta hai."

Sign galat hai: summing junction par subtraction solving

Y = G (R - H Y)

ke baad denominator mein

+ G H

ban jaati hai. Sahi:

T = G / (1 + G H)

"Unity feedback ke liye

H = 1

, toh

G = C A P \cdot 1

aur

T = G

Unity feedback se refer karta hai,

1 + G H

term ko drop karne se nahi.

H = 1

ke saath,

T = G / (1 + G)

, na ki

G

G = C + A + P

forward path ke liye."

Multiply karne ki jagah add kar diya:

G = C A P

. Series blocks

s

-domain mein multiply karte hain.

"Characteristic equation

T (s)

ka numerator hai jo zero ke equal set hota hai."

Nahi — yeh denominator

1 + G H = 0

hai; numerator roots zeros hain (woh response shape karte hain lekin stability decide nahi karte).

"Steady-state error

= 1/ (1 + G (0))

kisi bhi input ke liye kaam karta hai."

Sirf step ke liye. Ramp ke liye ya jab Final Value Theorem ki stability condition fail ho, yeh formula invalid hai.

"Sensor gain badhana harmless hai kyunki yeh sirf reading ko scale karta hai."

Bada

H

loop gain

G H

badhata hai, poles ko shift karta hai aur possibly destabilize karta hai — yeh passive rescaling nahi hai.

"Ek open-loop system wind disturbance ke liye correct kar sakta hai agar pehle se acchi planning ho."

Open loop react nahi kar sakta ek unmeasured disturbance ke liye; sirf feedback resulting error ko measure karta hai aur wapas push karta hai.

Why questions

Loop-gain term sirf

G H

ki jagah

1 + G H

kyun hota hai?

1

" error se output tak direct path hai; loop ki geometric series

1 - G H + (G H)^{2} - \dots

jo apne aap ko chase karti hai,

1/ (1 + G H)

tak sum hoti hai.

Plant woh ek block kyun hai jise tum redesign nahi kar sakte?

Iska dynamics physics se set hoti hai (mass, inertia, aerodynamics); tum sirf woh choose kar sakte ho jo iske around ho — controller, actuator, sensor.

Integral action specifically steady-state step error ko kyun khatam karta hai?

s = 0

par iska pole

G (0) \to \infty

bhejta hai, toh

e_{ss} = 1/ (1 + G (0)) \to 0

— integrator error accumulate karta rehta hai jab tak woh exactly zero na ho.

Loop stability

G (s)

se nahin,

H (s)

ke saath kyun check karni chahiye?

Poles

1 + G (s) H (s) = 0

ke roots par hote hain;

H

ko bahar rakhna ek alag loop ko analyse karta hai, woh nahi jo tumne actually banaya.

Higher gain steady-state error ko kyun kam karta hai lekin saath mein stability ko threaten bhi karta hai?

Bada

G (0)

1/ (1 + G (0))

ko chhota karta hai (kam error) lekin closed-loop poles ko imaginary axis ki taraf aur uske paar bhi kheenchta hai (kam stability). Dekho Second-order System Response.

Hum Laplace

s

-domain mein kaam kyun karte hain?

Kyunki time mein convolution

s

mein multiplication ban jaata hai, dynamic blocks ki chain ko simple algebra

G = C A P

mein badal deta hai. Dekho Transfer Functions and Laplace Domain.

Navigation (state estimation) Control se alag concern kyun hai?

Control estimated state par act karta hai; agar estimate noisy ya biased hai toh loop galat target ko chase karta hai — isliye filtering matter karti hai. Dekho Kalman Filter and Navigation.

Edge cases

T (s) = G / (1 + G H)

ka kya hota hai jab loop gain

∣ G H ∣ \to \infty

T \to 1/ H

: output reference ko sensor gain se divided track karta hai, essentially plant se independent — ideal high-gain limit.

Jab

G H \to 0

(loop gain bahut chhota) ho toh kya hota hai?

T \to G

: feedback almost kuch nahi karta aur system open-loop behave karta hai, apni disturbance rejection khote hue.

Imaginary axis par exactly ek pole (

ζ = 0

) physically kya mean karta hai?

Sustained, undamped oscillation — stable aur unstable ke beech ka boundary; woh marginal case jis par tum practice mein kabhi nahi bethna chahte.

Agar actuator saturate ho, toh kya linear closed-loop formula

T = G / (1 + G H)

still valid hai?

Nahi — saturation nonlinear hai, toh transfer-function analysis toot jaata hai aur integrator windup ho sakta hai. Dekho Actuator Saturation and Anti-Windup.

Agar plant mein already ek pure integrator ho aur input ek step ho toh steady-state error kya hai?

Zero, kyunki

G (0) \to \infty

e_{ss} = 1/ (1 + G (0)) = 0

banata hai — bina integral controller term ke bhi.

Perfectly tracking loop mein steady state par error signal

e (t)

ka kya hota hai?

Yeh zero ki taraf settle hota hai (sufficient system type ke saath step ke liye); ek non-zero residual ek missing integrator ya finite loop gain ka signal deta hai.

Kya zero gain (

C = 0

) wala controller kabhi ek unstable plant ko stabilize kar sakta hai?

Nahi —

C = 0

ke saath forward path dead hai (

G = 0

), koi corrective push plant tak nahi pahunchta, aur iske unstable dynamics freely chalte hain.

Agar sensor fail ho jaaye aur constant

y_{m} = 0

read kare toh closed-loop behaviour kya hoga?

Loop sochta hai error hamesha

r

hai, actuator ko hard drive karta hai chahe true output kuch bhi ho — ek runaway, kyunki feedback effectively broken hai.

Recall Jaane se pehle ek-line self-test

Closed loop ka denominator? ::: $1 + G H$ , jiske roots woh poles hain jo stability decide karte hain. "Bas gain badha do" ke peeche ek trade? ::: Speed ( $ω_{n} ↑$ ) versus damping ( $ζ ↓$ ).