Control system fundamentals — plant, actuator, sensor, controller: Visual walkthrough

3.5.26 · D2 · Physics › Guidance, Navigation & Control (GNC) › Control system fundamentals — plant, actuator, sensor, contr

Hum yeh maanke chalte hain ki aap sirf yeh jaante hain: ek signal ek number hai jo time ke saath badalta hai (aap kitna off hain, aap kitni takkat lagate hain). Baaki sab hum khud banate hain.

Step 0 — Ek "block" kya hota hai, aur cheezein multiply kyun hoti hain?

KYA. Ek box banao. Ek wire signal ko andar laati hai; box kuch karta hai uske saath; ek wire signal ko bahar le jaati hai. Box ke upar $G$ jaisa koi letter likha hota hai. Box ka rule hai: output = (box ka factor) $\times$ (input).

KYUN yeh matter karta hai. Real world mein signals time $t$ ke wiggly functions hote hain, aur unhe combine karna messy hota hai (uske liye convolution chahiye — ek smearing operation). Engineers ek trick use karte hain jise Laplace transform kehte hain: yeh har signal ko ek nayi variable $s$ se re-label karta hai, aur us nayi language mein cascaded boxes sirf multiply hoti hain. Toh teen boxes ki chain ek multiplication ban jaati hai. Yahi ek reason hai ki hum $s$ mein kaam karte hain — yeh "smear" ko "times" mein badal deta hai.

PICTURE. Do boxes ek ke baad ek: signal ek baar multiply hota hai, phir dobara.

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

Step 1 — Chaar blocks ko ek wire-loop mein lagao

KYA. Hum parent note ke chaar organs ko us order mein rakhte hain jis order mein signal actually travel karta hai: controller $C$ → actuator $A$ → plant $P$ (yeh teeno vehicle ko push karte hain), aur sensor $H$ (yeh dekhta hai aur report karta hai).

KYUN. Control ka matlab hai decide → push → watch → report → phir decide. Ise ek loop ke roop mein draw karna (line nahi) poora point hai: output curling back karta hai input se milne ke liye. Baaki sab is ring mein rehta hai.

PICTURE. Arrow ko ek baar poora ghoomte hue follow karo. Chaar named signals wires par sawaar hain:

$r$ — reference: Guidance chahta hai ki hum kahaan hon.
$e$ — error: reference minus jo sensor ne report kiya.
$u$ — control signal: controller ka decision.
$y$ — true output: vehicle actually kahaan hai.
$y_{m}$ — measured output: sensor kya kehta hai $y$ hai.

Step 2 — Error likhein (woh subtraction jo sab kuch shuru karta hai)

KYA. Sab se aage ek comparator baitha hai: ek circle jo reference se fed-back measurement ko subtract karta hai.

E (s) = jo hum maangte hain R (s) - jo sensor report karta hai H (s) Y (s)

KYUN. Controller ko tabhi kaam karna chahiye jab hum target se off hon. Agar measurement reference ke barabar ho, toh $E = 0$ aur controller relax karta hai. Isliye error literally hai "wish aur reality ke beech ka gap." Note karein ki $Y$ sensor box $H$ se guzarta hai pehle subtract hone se — loop reported value se compare karta hai, raw truth se nahi.

Term by term: $R (s)$ target signal hai; $Y (s)$ true output hai; $H (s)$ , $Y$ ko report $y_{m} = H Y$ mein reshape karta hai; minus sign hi hai jo ise negative feedback banata hai — correction hamesha gap ko oppose karta hai.

PICTURE. Comparator circle reference wire par $+$ ke saath aur feedback wire par $-$ ke saath.

Step 3 — Error ko forward path se push karein

KYA. Error bade forward box $G$ ka input hai. Iska output true position hai:

Y (s) = controller \cdot actuator \cdot plant G (s) \cdot hum kitna off hain E (s)

KYUN. Step 0 ke hisaab se, ek signal ko cascaded blocks mein bhejna ek multiplication hai. Error decide karta hai ki correction kitni hard karni hai; $G$ us decision ko ek actual movement $y$ mein badle deta hai.

PICTURE. Seedha run: $e$ $G$ box mein jaata hai, $y$ usse nikalata hai — loop ka "action" waala aadha hissa, feedback se alag draw kiya gaya clarity ke liye.

Step 4 — Internal signal $E$ ko khatam karne ke liye substitute karein

KYA. Hamare paas $E$ share karte hue do equations hain. Step 2 ko Step 3 ke andar rakhein:

Y = G E = G (R - H Y) = GR - G H Y

KYUN. $E$ ek internal wire hai — hum ise bahar se measure nahi kar sakte aur humein iske baare mein parwaah nahi. Hum ek clean statement chahte hain jo sirf jo hum maangte hain ( $R$ ) aur jo hume milta hai ( $Y$ ) ko link kare. Substitution middle-man ko mita deta hai.

Term by term: $GR$ = "agar feedback na hoti" response (reference par pure forward push). $- G H Y$ = woh correction jo loop feedback karta hai — output ring mein ek baar travel karta hai ( $H$ se guzar ke, $G$ se guzar ke) aur wapas aata hai apne aap se fight karne ke liye. Woh extra $G H Y$ term feedback ko visible banata hai.

PICTURE. Output $y$ ko $H$ se peeche aur $G$ se aage trace kiya gaya — same signal equation ke dono sides par appear kar raha hai.

Step 5 — $Y$ collect karein aur solve karein

KYA. Har $Y$ ko ek side par gather karein:

Y + G H Y = GR ⟹ Y (1 + G H) = GR

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

KYUN. Yeh ab sirf algebra hai — $Y$ factor out karein, divide karein. Inaaam ek single box $T (s)$ hai jo poore loop ko niga laata hai: ise $R$ pucho, yeh aapko $Y$ deta hai.

Term by term:

Numerator $G$ = forward push (controller·actuator·plant). Bada $G$ → strong action.
Denominator $1 + G H$ = ek, plus loop gain. " $1$ " aap ka maangna hai; " $G H$ " loop ka pushback hai.

PICTURE. Poori ring ek equivalent box mein collapse ho jaati hai jis par $T (s)$ label hai.

Step 6 — KYUN bilkul " $1 + G H$ "? (chasing picture)

KYA. Ek signal ko ring mein step by step follow karein ek baar mein solve karne ki jagah. $R$ bhejon. Loop last output ka $G H$ subtract karta hai, phir phir subtract hota hai, aur phir:

\frac{1}{1 + G H} = 1 - G H + (G H)^{2} - (G H)^{3} + \dots

KYUN. Yeh ek geometric series hai — loop mein har trip bacha hua leftover $- G H$ se multiply karta hai (minus negative feedback se, $G H$ ek poore lap se). Har correction thoda kam overshoot karta hai pichle se, aur infinite sum $\frac{1}{1 + G H}$ par settle hoti hai. Formula loop ka balance par converge karna hai.

PICTURE. Shrinking corrections ki ek staircase, har ek pichle se $- G H$ guna, final value par pile up hoti hue.

Step 7 — Edge cases (koi bhi scenario unseen mat chhodon)

"Loop gain kitna bada hai" ka har combination alag kahani sunata hai. Chaaon padhein.

Case A — $H = 1$ (perfect unity sensor). Report hi truth hai, toh $T (s) = \frac{G}{1 + G} .$ Yeh parent note ka "unity feedback" hai. Kuch special nahi — bas $H$ ko $1$ set kar diya.

Case B — huge loop gain, $∣ G H ∣ ≫ 1$ . " $1$ " chhota ho jaata hai: $T \approx \frac{G}{G H} = \frac{1}{H} .$ Output ab messy plant $G$ par depend hi nahi karta — sirf sensor $H$ par! KYUN yeh matter karta hai: strong feedback loop ko plant errors se immune banata hai. Yahi deep reason hai ki feedback use ki jaati hai.

Case C — tiny loop gain, $∣ G H ∣ ≪ 1$ . Ab $1 + G H \approx 1$ , toh $T \approx G$ — loop aise behave karta hai jaise open ho, koi self-correction nahi. Bahut weak feedback kuch nahi karta.

Case D — degenerate/dangerous: $1 + G H = 0$ . Denominator zero hit karta hai aur $T$ blow up karta hai. KYUN yeh poora game hai: $s$ -values jo solve karte hain $1 + G (s) H (s) = 0$ woh closed-loop poles hain. Agar koi $s$ -plane ke right half mein baithta hai, toh loop unstable hai — Step 6 ki staircase shrink karne ki jagah grow karti hai (kyunki ab $∣ G H ∣ > 1$ hai, geometric series diverge karti hai). Yahi ek equation hai jiski wajah se hum gain sirf "crank up" nahi kar sakte: $G$ ko bahut aage push karo aur ek root cross over kar jaata hai.

PICTURE. Loop gain ke ek single map par chaar regimes side by side, "kuch nahi karta" se "blow up" tak.

Ek-picture summary

Upar sab kuch, ek canvas par: ring apne chaar blocks aur paanch signals ke saath, woh subtraction jo error banata hai, forward push $G$ , feedback path $H$ , aur $T = \frac{G}{1 + G H}$ mein collapse, denominator ke liye shrinking-staircase reason ke saath.

Recall Feynman retelling — poora walkthrough simple words mein

Ek wire ko ring mein bend hua imagine karo. Upar ek subtracting circle baitha hai: yeh leta hai aap kahaan jaana chahte hain ( $r$ ) aur subtract karta hai aapki aankhon ne kya report kiya ( $y_{m}$ ), deta hai gap ( $e$ ). Woh gap ek bade pusher-box $G$ mein jaata hai — yeh aapka brain, aapke muscles, aur vehicle sab ek mein — aur nikalta hai aap actually kahaan hain ( $y$ ). Lekin $y$ sirf chala nahi jaata; yeh aapki aankhon se (sensor box $H$ ) loop back karta hai aur subtracting circle mein wapas aata hai, toh loop hamesha apne aap se compare kar raha hai. Jab hum algebra karte hain sirf "maanga" aur "mila" ko link karne ke liye, loop ka apna pushback $G H$ ke roop mein dikhta hai fraction ke neeche $1$ ke saath. " $1$ " aap ka ek baar maangna hai; " $G H$ " loop ka jawab dena hai, baar baar, har jawab thoda chhota — shrinking corrections ki staircase jo $\frac{1}{1 + G H}$ par settle hoti hai. Aur woh ek jagah jahan yeh settle nahi hoti — jahan $1 + G H$ zero hit karta hai — exactly wahi hai jahan loop unstable hota hai. Ek bent wire mein feedback ki poori kahani yahi hai.

Recall Quick self-test

Cascaded blocks multiply kyun hote hain add ki jagah? ::: Laplace ( $s$ ) domain mein, ek signal ko series mein blocks se bhejna ek multiplication hai — yahi poora reason hai ki hum $s$ par switch karte hain. $E = R - H Y$ mein minus sign kya create karta hai? ::: Negative feedback — correction hamesha error ko oppose karta hai, toh corrections ki staircase shrink hoti hai. Ek line mein, denominator $1 + G H$ kyun hai? ::: Yeh loop ka apne aap par feedback karne ki geometric series hai: $1 - G H + (G H)^{2} - \dots = \frac{1}{1 + G H}$ . Bahut large loop gain ke saath, $T \approx$ kya hai, aur yeh kyun accha hai? ::: $T \approx 1/ H$ — output plant par depend karna band kar deta hai, toh loop plant errors se immune ho jaata hai. Closed-loop poles konsi equation deti hai? ::: $1 + G (s) H (s) = 0$ — iske roots stability decide karte hain.

Yeh bhi dekhein: State-Space Representation aur Kalman Filter and Navigation, jab sensors noisy hon toh "main kahaan hoon" ( $y$ , $y_{m}$ ) kaise estimate kiya jaata hai.