Control system fundamentals — plant, actuator, sensor, controller

3.5.26 · Physics › Guidance, Navigation & Control (GNC)

GNC mein, Guidance decide karta hai kahan jaana hai (reference), Navigation figure out karta hai tum kahan ho (state estimate), aur Control vehicle ko wahan actually pahunchata hai. Ye note us last block ki machinery ke baare mein hai: kisi bhi feedback loop ke chaar organs.

The four building blocks

YE CHAAR KYUN? Kyunki control fundamentally information → action → measurement hai. Tumhe kuch chahiye jo kaam kare (actuator), kuch jo kaam receive kare (plant), kuch jo result dekhe (sensor), aur kuch jo decide kare (controller). Koi ek hata do aur loop toot jaata hai.

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

Open loop vs. closed loop — Feedback KYUN?

Signal KAISE flow karta hai (closed loop):

$r (t) compare e (t) = r (t) - y_{m} (t) controller C u (t) actuator A force plant P y (t) sensor H y_{m} (t)$

$r$ = reference (desired output, Guidance se)
$e$ = error
$u$ = control signal (controller output)
$y$ = true plant output
$y_{m}$ = measured output (fed back)

Closed-loop transfer function pehle principles se derive karna

HAMARA GOAL: $R (s)$ (jo hum maangein) se $Y (s)$ (jo milta hai) tak ek single relation banana, Laplace domain mein jahan cascaded blocks multiply ho jaate hain.

Maano transfer functions hain $C (s)$ (controller), $A (s)$ (actuator), $P (s)$ (plant), $H (s)$ (sensor). Forward path $G (s) = C (s) A (s) P (s)$ mein lump karo.

Step 1 — error likho. Kyun? Error hi controller ko drive karta hai. $E (s) = R (s) - H (s) Y (s)$

Step 2 — forward path output produce karta hai. Kyun? Cascaded LTI blocks se guzarne wale signals $s$ -domain mein multiply hote hain. $Y (s) = G (s) E (s)$

Step 3 — Step 1 ko Step 2 mein substitute karo. Kyun? Internal signal $E$ ko eliminate karna hai aur sirf $R, Y$ rakhne hain. $Y = G (R - H Y) = GR - G H Y$

Step 4 — $Y$ collect karo. Kyun? Algebraically input–output map ke liye solve karo. $Y + G H Y = GR \Rightarrow Y (1 + G H) = GR$

Unity feedback special case ( $H = 1$ , matlab sensor perfect hai aur 1 tak scaled hai): $T (s) = \frac{G ( s )}{1 + G ( s )}$

Worked example 1 — DC motor angular position

Setup: Plant (motor to angle) $P (s) = \frac{1}{s ( s + 2 )}$ , actuator gain $A = 1$ , sensor $H = 1$ , proportional controller $C (s) = K$ .

Step 1 — forward path. Kyun? Cascaded blocks ko multiply karo. $G (s) = K \cdot 1 \cdot \frac{1}{s ( s + 2 )} = \frac{K}{s ( s + 2 )}$

Step 2 — closed loop. Kyun? Boxed formula apply karo. $T (s) = \frac{G}{1 + G} = \frac{K / [ s ( s + 2 )]}{1 + K / [ s ( s + 2 )]} = \frac{K}{s ( s + 2 ) + K} = \frac{K}{s ^{2} + 2 s + K}$

Step 3 — stability & response padho. Kyun? Denominator ko $s^{2} + 2 ζ ω_{n} s + ω_{n}^{2}$ se compare karo. $ω_{n} = K, 2 ζ ω_{n} = 2 \Rightarrow ζ = \frac{1}{K}$ Bada $K$ → faster ( $ω_{n} ↑$ ) lekin kam damped ( $ζ ↓$ ) → zyada overshoot. Yahi eternal control trade-off hai.

Worked example 2 — steady-state error with a step

Setup: unity feedback, $G (s) = \frac{5}{s + 3}$ , input ek unit step $R (s) = 1/ s$ .

Step 1 — error transfer function. Kyun? Hume $E$ chahiye, $Y$ nahi. $\frac{E}{R} = \frac{1}{1 + G} (from E = R - Y, Y = \frac{G}{1 + G} R)$

Step 2 — Final Value Theorem. Kyun? Steady-state error = $lim_{t \to \infty} e (t) = lim_{s \to 0} s E (s)$ (valid kyunki loop stable hai). $e_{ss} = lim_{s \to 0} s \cdot \frac{1}{1 + G ( s )} \cdot \frac{1}{s} = \frac{1}{1 + G ( 0 )} = \frac{1}{1 + 5/3} = \frac{3}{8} = 0.375$

Interpretation: finite error isliye kyunki loop mein koi integrator nahi hai ( $s = 0$ par koi pole nahi). Integral action add karo ( $C = K / s$ ) → $G (0) \to \infty$ → $e_{ss} \to 0$ . Isliye PID ka "I" steady-state error ko khatam karta hai.

Common mistakes (Steel-manned)

Recall Feynman: ek 12-saal ke bache ko explain karo

Socho tum apni aankhon (sensor) se ek mailbox ki taraf cycle chala rahe ho. Tumhara brain (controller) dekhta hai ki tum thoda left ho (error), toh decide karta hai right steer karna. Tumhare haath (actuator) handlebars morate hain, aur bike (plant) move karti hai. Phir tumhari aankhein dobara check karti hain — baar baar — jab tak tum bilkul sahi jagah nahi aa jaate. Agar tumne aankhein band kar li hoti (no feedback) toh shayad crash ho jaate. Yahi baar baar dekho–socho–steer karo–dekho ek control loop hai!

Active recall

Feedback control system ke chaar building blocks kya hain?

Plant, actuator, sensor, controller.

Kaun se block ki dynamics physics se fix hoti hain aur redesign nahi ho sakti?

Plant.

Actuator kya karta hai jo controller nahi karta?

Controller ke command signal ko real physical action mein convert karta hai (hardware limits jaise saturation hote hain).

Closed loop mein error signal define karo.

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

= reference minus measured output.

Closed-loop transfer function batao.

T (s) = \frac{G ( s )}{1 + G ( s ) H ( s )}

with

G = C A P

Characteristic equation kya hai aur yeh kyun matter karta hai?

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

; iske roots closed-loop poles hain jo stability determine karte hain.

Proportional gain badhane se overshoot kyun hota hai?

ω_{n}

badhta hai lekin damping

ζ

ghatta hai, poles oscillatory/unstable region ki taraf jaate hain.

Step ke liye steady-state error kaise eliminate karte hain?

Integral action add karo (

s = 0

par pole) taaki

G (0) \to \infty

aur

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

Sensor

H (s)

ko loop analysis mein kyun rakhte hain chahe accurate ho?

Iski dynamics/gain loop gain

G H

mein appear hoti hai aur loop ko destabilize kar sakti hai.

Term

G H

ka physical meaning kya hai?

Loop gain — woh factor jo ek signal loop mein ek baar poora chakkar lagate hue accumulate karta hai.

Open loop vs closed loop, closed ka key advantage kya hai?

Closed loop feedback use karta hai disturbances aur plant uncertainty ke liye self-correct karne ke liye.

Connections

PID Control — P, I, D kaise $C (s)$ ko shape karte hain
Transfer Functions and Laplace Domain — cascaded blocks kyun multiply hote hain
Poles Zeros and Stability — $1 + G H = 0$ ke roots
Second-order System Response — $ζ, ω_{n}$ trade-offs
State-Space Representation — MIMO GNC ke liye transfer functions ka alternative
Kalman Filter and Navigation — GNC ka sensor/estimator side
Actuator Saturation and Anti-Windup