3.5.39 · D1PID tuning — Ziegler-Nichols, loop shaping
Ek controller dekhta hai ki machine abhi kitni galat hai, add karta hai ki woh pehle kitni galat rahi hai, aur notice karta hai ki yeh galti kitni tezi se badal rahi hai — phir woh push karta hai. Tuning yeh choose karna hai ki woh un teeno pieces mein se har ek par kitna hard push kare taaki machine apne target par jaldi settle ho jaye bina zyada wobble kiye.
Yeh page assume karta hai ki aap kuch bhi nahi jaante. Har woh symbol jo parent note ne use kiya tha, woh yahan unpack kiya gaya hai, us order mein jo har ek ko pichle wale par lean karne deta hai.
Kisi bhi symbol se pehle, loop dekho. Jis machine ko hum control karna chahte hain use plant kehte hain. Hum use batate hain kahan jaana hai; woh report karta hai ki woh actually abhi kahan hai; hum compare karte hain; hum correct karte hain. Round and round.
Definition Loop mein char signals
r ( t ) reference (ya setpoint): hum output ko kahan rakhna chahte hain. Dashed target line.y ( t ) measured output : plant abhi actually kahan hai.e ( t ) error : hum kitne galat hain. Neeche define kiya gaya hai.u ( t ) control signal : controller jo push plant ko bhejta hai.
Har letter ke baad chhota sa ( t ) time t hai." t = 0
e ( t ) = r ( t ) − y ( t ) Simple words mein: error = jahan hum jaana chahte hain minus jahan hum hain. Picture: upar ki figure mein dashed target line aur solid output curve ke beech ki vertical gap. Agar output target se neeche hai, toh e upar overshoot kar jaye, toh e Yeh kyun zaroori hai: control ka poora point is gap ko zero par drive karna hai, isliye har controller e
Controller sirf error signal e ( t ) P, I, D sirf har kahani padhne ke naam hain.
Ab teen tools. Har ek ek aisa maths piece hai jo exactly un teeno mein se ek sawaal ka jawaab dene ke liye choose kiya gaya hai.
Definition Proportional termK p e ( t ) Simple words mein: us proportion mein push karo jitna tum abhi galat ho — double error, double push. Picture: orange arrow jo present instant par error curve ki height measure karta hai. Yeh piece kyun: yeh sabse simple possible reaction hai — koi naya symbol nahi, sirf raw error e ( t ) Topic ko yeh kyun chahiye: kuch toh present gap par react karna chahiye, aur P wahi hai; yeh woh backbone hai jis se har loop shuru hoti hai. Yeh kya problem solve karta hai aur iske limits: P akela aapko target tak zyada tar pahuncha deta hai jaldi se, lekin do flaws aate hain. (1) Yeh ek leftover steady-state error chhodta hai — kyunki push karte rehne ke liye, P ko feed karne ke liye ek non-zero error chahiye , isliye yeh target se thoda pahle settle ho jaata hai. (2) Agar aap ise zyada hard scale karo (bada K p oscillate karne lagta hai, stability ko threaten karta hai. Yahi do flaws hain jo exactly isliye hain ki neeche I aur D pieces ko exist karna padta hai.
∫ 0 t e ( τ ) d τ Simple words mein: error ko start (0 t unhe sab add karo . Picture: error curve ke neeche green shaded area. Yeh symbol kyun: lamba pata ∫ τ 0 t t alag letter chahiye kyunki t τ Topic ko yeh kyun chahiye: agar ek steady hawa plant ko target se door dhakelte rehti hai, toh height error choti ho sakti hai lekin area badhta rehta hai — isliye integral tab tak harder push karta rehta hai jab tak gap actually band na ho jaye. Isliye I-action steady-state error ko khatam karta hai , P ke leftover offset ko fix karta hai.
d t d e ( t ) Simple words mein: error per second kitni tezi se change ho raha hai, abhi. Picture: tangent line ki steepness jo error curve ko touch kar rahi hai — figure mein red line. Neeche steep = error tezi se shrink ho raha hai. Yeh symbol kyun: d t d e e d t Topic ko yeh kyun chahiye: agar hum target ki taraf tezi se rush kar rahe hain (steep slope), toh hume pehle se ease off karna chahiye taaki hum zoom past na ho jaayein — yahi damping hai, P ki oscillate karne ki tendency ko tame karna. Dekho PID controller basics ki kaise teeno combine hote hain.
K p , K i , K d K K p K i K d u ( t ) = present K p e ( t ) + past K i ∫ 0 t e ( τ ) d τ + future K d d t d e ( t )
"Tuning" exist kyun karta hai: zyada K p K i K d
Mnemonic Kaun sa knob kya karta haiP = abhi P ower, I = leftover offset ko I ron out karo, D = rush ko D amp karo.
Parent note suddenly likhta hai C ( s ) = K p + s K i + K d s t s
s Time mein integrals aur derivatives ke saath kaam karna clumsy hai. Ek translation trick hai (the Laplace transform) jo "integrate" ko "s s s
Definition Laplace transform khudEk time signal f ( t ) define kiya gaya hai:
L { f ( t )} = F ( s ) = ∫ 0 ∞ f ( t ) e − s t d t Simple words mein: apne signal ko ek decaying probe e − s t s F ( s ) e − s t × s ÷ s Fine print (region of convergence): integral tabhi finite answer tak add hota hai jab s e − s t f ( t ) s region of convergence hai. Control mein well-behaved, bounded signals ke liye yeh s = j ω
Definition Complex frequencys j
s s = j ω system ek pure sine wave ke response mein frequency ω ω angular frequency radians per second mein — wiggle kitni tezi se spin karta hai. Ek full cycle 2 π T ω = T 2 π j = − 1 j i i size aur timing (phase) dono carry karne deta hai.
Definition Transfer functionEk transfer function jaise G ( s ) C ( s ) C ( s ) controller ki recipe hai, G ( s ) plant ki recipe hai. Unhe multiply karna, L ( s ) = C ( s ) G ( s ) open-loop recipe hai — loop ke around signal ka poora safar feedback close hone se pehle. (Heads-up: wahi letter L
Jab hum s = j ω
∣ L ( j ω ) ∣ ∠ L ( j ω )
Magnitude ∣ L ( j ω ) ∣ badi ya choti bahar aati hai — complex plane mein arrow ki length. Value 1 Phase ∠ L ( j ω ) timing mein kitni shift hui hai, angle ke roop mein measure ki gayi. − 180°
Common mistake "Phase sirfarctan ( upar-part / daayein-part ) Yeh sahi kyun lagta hai: arrow picture par, (vertical divided by horizontal) ka arctan sahi bhi hai.
Galti: plain arctan − 90° + 90° real < 0 , imag < 0 − 135° fark nahi kar sakta — woh opposite quadrants ko collapse kar deta hai. Lekin control − 180° Fix: two-argument atan2 ( imag , real ) dono parts ke signs dekhta hai arrow ko correct quadrant mein place karne ke liye aur poore − 180° … + 180° ≈ − 143° Bode plot & frequency response , jo ω
− 180° y r subtract karta hai. Agar ek wave loop travel kare aur dono flipped (phase = − 180° aur full size (magnitude = 1 Nyquist stability criterion aur Stability margins (gain & phase margin) ka seed hai.
Ab parent note ke symbols ka last cluster.
Definition Crossover aur margins
ω c gain crossover frequency : jahan ∣ L ( j ω c ) ∣ = 1 Phase margin PM = 180° + ∠ L ( j ω c ) − 180° 30 60° Gain margin GM = 1/∣ L ( j ω 180 ) ∣ − 180° 1 ≥ 6
Definition Decibels (dB) aur "per decade"
Ek decibel magnitude ko log scale par rewrite karta hai: dB = 20 log 10 ∣ L ∣
Ek decade = frequency mein 10 − 20 ω
L K T Symbol warning: yahan L bilkul naya, unrelated meaning hai — seconds mein ek dead-time delay , na ki §4 ka open-loop transfer L ( s ) L ( s ) ( s ) θ τ d G ( s ) ≈ T s + 1 K e − L s
K T L kuch bhi hone se pehle ek pure delay (e − L s
Neeche ka prerequisite map ek real figure ke roop mein draw kiya gaya hai — colour-coded iss baat se ki teeno "stories" mein se har idea kaun si story se belong karta hai — taaki aap dependency flow dekh sako na ki padho.
Isse bottom-to-top padhein: raw loop signals error ko feed karte hain; error present/past/future pieces mein split hota hai; woh pieces gains se attach hote hain aur control law mein combine hote hain; control law ko s
Aage badhne se pehle right side chhupao aur khud ko test karo.
e ( t ) Error, e ( t ) = r ( t ) − y ( t )
Pure proportional control kaun si problem unsolved chhodta hai? Ek leftover steady-state offset (use push karte rehne ke liye non-zero error chahiye), aur agar K p
Integral ∫ 0 t e d τ Start se ab tak error curve ke neeche accumulated shaded area.
Derivative d e / d t Present instant par error curve ka slope (steepness).
Kaun sa knob past scale karta hai, aur s K i s
Laplace transform ka defining integral likho. F ( s ) = ∫ 0 ∞ f ( t ) e − s t d t
t s Laplace calculus ko algebra mein badal deta hai — integrate ⇒ ÷s s
j ω j = − 1 , imaginary unit;
ω angular frequency hai rad/s mein,
ω = 2 π / T ke saath.
Plain arctan Yeh opposite quadrants collapse kar deta hai (sirf ± 90° atan2 ( imag , real ) − 180° … + 180°
L ( j ω ) Magnitude = wave ka size change; phase = degrees mein timing shift.
Phase = − 180° = 1 Wave flipped aur full-size waapis aati hai, isliye feedback use hamesha ke liye reinforce karta hai.
Phase margin define karo. PM = 180° + ∠ L ( ω c ) 30 60°
K u T u Sustained oscillation par gain, aur us oscillation ka period.
Is page par letter L L ( s ) L
