Control system fundamentals — plant, actuator, sensor, controller: Foundations

3.5.26 · D1 · Physics › Guidance, Navigation & Control (GNC) › Control system fundamentals — plant, actuator, sensor, contr

Parent note the parent topic padhne se pehle, tumhe har letter ka ownership lena hoga jo woh use karta hai. Yeh page unhe bilkul zero se build karta hai, us order mein jisme woh ek dusre par depend karte hain. Koi cheez use nahi hoti jab tak draw na ho.

Part 1 — Signals: woh cheezein jo time ke saath change hoti hain

Picture: ek signal ek curve hai. Horizontal position batata hai kab; height batati hai kitna.

Figure — Control system fundamentals — plant, actuator, sensor, controller

Figure s01 — Ek wiggly curve. Magenta curve (signal) dekho, phir violet dashed drop-line jo ek instant $t$ mark karta hai, aur orange dashed line jo uski height $x (t)$ read off karta hai. Takeaway: curve par ek dot = ek moment par ek number.

Kyun yeh notation chahiye. Control un quantities ke baare mein hai jo evolve hoti hain: rocket ka angle, motor ki position, error. "5°" jaisa ek akela number ek snapshot hai; $x (t)$ poori movie hai. Humein movie ke baare mein baat karni hai, isliye $(t)$ chahiye.

Part 2 — Error: loop ka poora point

Picture: target line $r$ aur measured curve $y_{m}$ ko stack karo; unke beech vertical distance, har instant par, $e$ hai. Target se upar, $e$ negative hai (overshoot); neeche, positive.

Figure s02 — Do curves plus vertical bars. Flat violet line (target $r$ ) dekho, rising magenta curve (measurement $y_{m}$ ), aur unke beech orange vertical bars — har bar ki length us instant par error $e$ hai. Takeaway: error woh shrinking gap hai jo tum band karne ki koshish kar rahe ho.

Kyun subtraction aur ratio nahi? Kyunki ek difference apna sign rakhta hai. $2$ ka ratio nahi bata sakta tum upar ho ya neeche — lekin $e = - 3$ kehta hai "tune 3 se overshoot kiya, doosri taraf push karo." Sign steering direction hai. Isliye loop ek subtraction ke around bana hai.

Part 3 — Chaar organs, signals transform karne wale boxes ki tarah

Har block ek signal andar leta hai aur ek signal bahar deta hai. Har ek ko ek machine samjho: signal left se aata hai, changed signal right se nikalti hai.

Kyun hardware ko software se alag karo? Controller $C$ sirf arithmetic hai — woh koi bhi number output kar sakta hai. Actuator $A$ metal hai — woh sirf itna hi push kar sakta hai (yeh limit saturation kehlati hai, Actuator Saturation and Anti-Windup mein explore ki gayi hai). Donon ko confuse karna real failures ko chhupa deta hai, isliye parent unhe alag letters mein rakhta hai.

Part 4 — Kyun hum Laplace domain ( $s$ ) mein jump karte hain

Yeh tool question hai. Har block apna input delay, smooth, ya amplify karta hai — ek differential equation (rates of change) se describe hota hai. Boxes ko chain karna matlab derivatives ko chain karna, jo time mein horrible algebra hai.

Picture: $s$ ek 2-D map par rehta hai (s-plane). Left half = decaying = safe. Right half = exploding = unstable. Bilkul vertical middle line par = na yeh na woh — sustained oscillation. Signal ka behaviour decide hota hai ki uske special points is map par kahan hain.

Figure s03 — S-plane map. Violet-shaded left half dekho (poles yahan = decaying, STABLE, violet ×'s ki tarah drawn), magenta-shaded right half (poles yahan = growing, UNSTABLE), aur orange dots jo bilkul vertical axis par baithe hain ( $σ = 0$ ) — poles yahan na grow karte hain na decay, forever-ringing oscillation dete hain (marginal stability). Takeaway: horizontal position = stability, height = wobble speed.

Kyun yeh tool aur ODEs solve karna nahi? Kyunki ek baar hum $s$ mein hain, "signal box $C$ se phir box $A$ se phir box $P$ se guzrta hai" simply $C (s) A (s) P (s)$ hai — plain multiplication. Yahi akaula reason hai ki parent itni casually $G = C A P$ likh sakta hai. Full details Transfer Functions and Laplace Domain mein hain.

Part 5 — Transfer functions: ek box ek akeli fraction ki tarah

Upar ke (numerator) roots zeros kehlate hain — $s$ ki woh values jo output ko vanish karti hain.
Neeche ke (denominator) roots poles kehlate hain — $s$ ki woh values jo box ko "blow up" kar deti hain. Poles stability decide karte hain (dekho Poles Zeros and Stability).

Kyun ek fraction? Kyunki ek physical system kuch frequencies par strongly respond karta hai aur kuch par weakly. Do polynomials ka ratio exactly woh shape hai jo "yahan amplify karo, wahan ignore karo" capture karta hai. Example: parent ka motor $P (s) = \frac{1}{s ( s + 2 )}$ mein poles $s = 0$ aur $s = - 2$ par hain.

Part 6 — Loop gain $G H$ aur magic denominator

Forward boxes ko chain karne se $G = C A P$ milta hai. Loop ke puri tarah around jaana (forward through $G$ , back through sensor $H (s)$ ) unhe multiply karta hai: yeh round-trip factor $G H$ loop gain hai.

Intuition Kyun successive laps sign alternate karte hain — derive karo

Comparison node se shuru karo. Error hai reference minus fed-back signal: $E = R - H Y, Y = G E .$ $Y = GE$ ko pehli line mein substitute karo: $E = R - G H E .$ Ab repeated substitution se solve karo — right side ko us $E$ ki jagah wapas plug in karo jo us par appear karta hai, baar baar: $E = R - G H (R - G H E) = R - G H R + (G H)^{2} E = \dots$ Har substitution mein $G H$ ka ek aur factor aata hai, aur kyunki node subtract karta hai, har nayi term sign flip karti hai: $E = R (1 - G H + (G H)^{2} - (G H)^{3} + \dots) .$ Alternating signs literally comparison node ka minus sign hai, ek extra baar har lap mein apply hota hai. Woh geometric series (valid jab $∣ G H ∣ < 1$ ) sum hoti hai $1 - G H + (G H)^{2} - \dots = \frac{1}{1 + G H}, so E = \frac{R}{1 + G H} .$ Denominator mein woh $1 + G H$ loop ka "apne aap ko balance mein chase karna" hai. Iske roots ( $1 + G H = 0$ ) characteristic equation hain — woh poles jo decide karte hain ki poora loop stable hai ya nahi.

Picture: signal loop ke around ghoomta hai, har lap $- G H$ se scale hota hai, laps alternately add aur cancel hote hain jab tak settle na ho. Agar $∣ G H ∣$ chhota hai, woh jaldi mar jaate hain; agar zyada bada, woh badhte hain — instability.

Figure s04 — Bars plus running total. Orange bars dekho (har lap ka contribution $(- G H)^{n}$ — notice karo woh sign flip karte hain aur shrink karte hain), magenta dotted line (running sum jaise laps pile up hoti hain), aur violet dashed line jis par woh settle hoti hai, $1/ (1 + G H)$ . Takeaway: feedback subtraction, repeated, exactly woh geometric series hai jo $1 + G H$ produce karta hai.

Part 7 — Second-order fingerprint: $ζ$ aur $ω_{n}$

Jab loop ka denominator ek quadratic $s^{2} + 2 ζ ω_{n} s + ω_{n}^{2}$ hai, do numbers sab kuch describe karte hain ki woh kaise settle hota hai (dekho Second-order System Response).

Kyun yeh do aur raw coefficients nahi? Kyunki $ζ$ aur $ω_{n}$ directly jo tum feel karte ho se map karte hain: speed aur wobble. Woh parent ko kehne dete hain "bada $K$ $ω_{n}$ badhata hai lekin $ζ$ ghatata hai" — eternal speed-vs-stability trade-off — plain human terms mein. Note karo $ζ = 0$ (no damping) bilkul imaginary axis par baithta hai — Figure s03 se marginal case: forever ringing.

Part 8 — Final Value Theorem: $t = \infty$ par peek karna

Kyun yeh tool? Hum aksar steady-state error chahte hain — sab settle hone ke baad hum kitna off hain — forever simulate kiye bina. $s$ se multiply karke aur $s \to 0$ let karke ( $s$ -plane ka "DC" corner) woh final value seedha transfer function se read ho jaati hai. Isi tarah parent $e_{ss} = 1/ (1 + G (0)) = 3/8$ kisi bhi time-plot ke bina pata karta hai.

Foundations topic ko kaise feed karte hain

Related deeper dives jab yeh tools aa jaayein: PID Control, State-Space Representation, aur Kalman Filter and Navigation.

Equipment checklist

Right side cover karo; kya tum har cheez memory se bata sakte ho?

Signal

x (t)

hai

ek number jo time ke saath change hota hai;

t

= kab,

x

= kitna.

Error

e (t)

equal hai

reference minus measurement,

r (t) - y_{m} (t)

— ek signed gap.

Plant vs. actuator

plant = fixed physics jo tum steer karte ho; actuator = hardware jo push karta hai, real limits ke saath.

Sensor vs. controller

sensor = hardware jo measure karta hai; controller = software jo command decide karta hai.

Laplace transform operator

L {x (t)} = \int_{0}^{\infty} x (t) e^{- s t} d t

, ek time signal ko

X (s)

mein convert karta hai.

Kyun Laplace domain mein jaate hain

derivatives

\times s

ban jaate hain, toh chained boxes sirf multiply karte hain.

Complex variable

s = σ + j ω

σ

= growth/decay rate,

ω

= oscillation speed.

Poles bilkul imaginary axis par matlab

na grow na decay — sustained oscillation, marginal stability.

Transfer function

G (s)

hai

ek box ka output-over-input,

s

mein polynomials ka ratio.

Sensor

H (s)

kya karta hai

true output ko measurement se map karta hai,

Y_{m} (s) = H (s) Y (s)

;

H = 1

unity feedback hai.

Poles hote hain

denominator ke roots; woh stability decide karte hain.

Loop gain

G H

hai

woh factor jo ek signal poore loop ke ek chakkar mein pick up karta hai.

Denominator

1 + G H

kyun hai

subtracting node par repeated substitution se

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

milta hai.

Characteristic equation

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

; iske roots closed-loop poles hain.

s^{2} + a s + b

ω_{n}, ζ

nikalo

ω_{n} = b

ζ = a / (2 b)

Natural frequency

ω_{n}

system kitni tez move karna chahta hai (speed).

Damping ratio

ζ

motion kitna dabaya gaya hai;

< 1

overshoots,

= 1

critical,

> 1

sluggish,

0

forever rings.

Final Value Theorem

e_{ss} = lim_{s \to 0} s E (s)

, valid sirf jab loop stable ho.