PID tuning — Ziegler-Nichols, loop shaping

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

1. PID controller kya hota hai (WHAT / WHY)

P (proportional): present error par react karta hai. Bada error ⇒ bada push. Akela use karne par steady-state error (offset) chhod deta hai aur zyada bada hone par oscillate kar sakta hai.
I (integral): past error accumulate karta hai. Yeh steady-state error kill karta hai kyunki tab tak push karta rehta hai jab tak accumulated error zero na ho jaye. Lekin yeh phase lag add karta hai ⇒ destabilise kar sakta hai.
D (derivative): slope se future predict karta hai. Yeh damping / phase lead add karta hai, overshoot kam karta hai, lekin noise amplify karta hai.

2. Ziegler-Nichols: ultimate-gain method (HOW, derived)

Procedure (closed-loop / ultimate-gain method):

$K_{i} = 0, K_{d} = 0$ set karo (pure P control).
$K_{p}$ dheere dheere badhao jab tak output sustained, constant-amplitude oscillation na dikhaye.
Uss gain ko ultimate gain $K_{u}$ aur oscillation ki period $T_{u}$ record karo.
Table se gains padhlo.

$K_{u}, T_{u}$ analytically kahan se aate hain?

Stability edge par open loop $L (j ω) = K_{u} G (j ω)$ satisfy karta hai $∠ G (j ω_{u}) = - 180°, K_{u} = \frac{1}{∣ G ( j ω _{u} ) ∣}, T_{u} = \frac{2 π}{ω _{u}} .$

Ek test plant ke liye derivation $G (s) = \frac{1}{( s + 1 ) ^{3}}$ :

Phase: $∠ G (j ω) = - 3 arctan ω$ . Set $= - 180° = - π$ : $arctan ω_{u} = π /3 \Rightarrow ω_{u} = tan 60° = 3$ .
Magnitude: $∣ G (j ω_{u}) ∣ = (ω_{u}^{2} + 1)^{- 3/2} = (3 + 1)^{- 3/2} = 4^{- 3/2} = 1/8$ .
Toh $K_{u} = 1/∣ G ∣ = 8$ aur $T_{u} = 2 π / 3 \approx 3.63$ s.

3. Ziegler-Nichols: open-loop (reaction-curve) method

Step response ko delay $L$ aur slope se approximate karo. Model: $G (s) \approx \frac{K e ^{- L s}}{T s + 1}$ . Maano $R = K / T$ (max slope / step-height). Toh:

Ye kyun? $L$ (dead time) govern karta hai ki tum kitna phase lag tolerate kar sakte ho; bada $L$ ⇒ chhota gain. Ratios $2 L, 0.5 L$ closed-loop $T_{u}$ ratios ko mirror karte hain kyunki aise plants ke liye $T_{u} \sim 4 L$ hota hai.

4. Loop shaping ("samjho" wala method)

PID ek loop-shaping filter ki tarah:

Integral term $K_{i} / s$ low-frequency gain boost karta hai (slope $- 20$ dB/dec, phase $- 90°$ ).
Derivative term $ω_{c}$ ke paas phase lead add karta hai — tum deliberately $C (s)$ ka zero $ω_{c}$ se neechay place karte ho taaki uska $+ 90°$ phase plant lag counter kare aur phase margin raise kare.

5. Common mistakes (Steel-manned)

6. Active recall

Recall Quick self-test (hide karke answer karo)

Closed-loop ZN kaun se do numbers measure karta hai, aur kaise? (→ $K_{u}$ : sustained oscillation par gain; $T_{u}$ : uski period.)
Integral action phase margin kyun reduce karta hai? (→ $- 90°$ lag add karta hai.)
Phase margin define karo aur target range batao. (→ $180° + ∠ L (ω_{c})$ ; $30$ – $60°$ .)
D action loop shaping mein kaise help karta hai? (→ $ω_{c}$ ke paas phase lead add karta hai, PM raise karta hai, damping add karta hai.)

Recall Feynman: ek 12-saal ke bacche ko samjhao

Socho ek toy car ko floor par ek line tak steer karna hai. P: jitna door ho line se, utna zyada wheel ghuma. I: agar hawa baar baar tum ko line se hata rahi hai, thoda extra ghuma te raho jab tak finally line par na aa jao. D: agar tum line ki taraf bahut tezi se aa rahe ho, pehle se ease off karo taaki zyada aage na nikal jao. Tuning mein yeh pick karna hai ki yeh teen reactions kitni strong hongi. Ziegler-Nichols aise hai: "P knob tab tak badhao jab tak car hamesha side-to-side wobble na kare, note karo kitni fast wobble kar rahi hai, phir usi se teeno knobs set karo." Loop shaping hai: "ek picture draw karo ki car fast vs slow wiggles par kaise react karti hai, aur knobs adjust karo taaki slow mistakes strongly fix hon lekin fast jitter ignore ho jaye."

Connections

PID controller basics — teen terms ki origin
Bode plot & frequency response — loop shaping ka canvas
Stability margins (gain & phase margin)
Nyquist stability criterion — kyun $- 180°$ / gain 1 edge hai
Integral windup and anti-windup
State-space control & LQR — hand tuning ka alternative
Root locus method — gain-selection ka ek aur tool

Flashcards

Time domain mein PID control law kya hai?

u (t) = K_{p} e + K_{i} \int e d t + K_{d} \overset{e}{˙}

, jahan

e = r - y

Closed-loop ZN mein

K_{u}

aur

T_{u}

kya hain?

Woh proportional gain jo sustained constant-amplitude oscillation produce karta hai, aur uss oscillation ki period.

K_{u}, T_{u}

se ZN PID gains?

K_{p} = 0.6 K_{u}, T_{i} = T_{u} /2, T_{d} = T_{u} /8

(phir

K_{i} = K_{p} / T_{i}, K_{d} = K_{p} T_{d}

Integral action add karne se stability margin kyun reduce hoti hai?

Har

1/ s

term

- 90°

phase lag add karta hai, crossover ke paas phase margin cut karta hai.

Phase margin define karo.

PM = 180° + ∠ L (j ω_{c})

jahan

ω_{c}

gain-crossover frequency hai (

∣ L ∣ = 1

L (s)

ke liye instability ki edge kaun si condition mark karti hai?

∣ L (j ω) ∣ = 1

aur

∠ L (j ω) = - 180°

simultaneously.

G = 1/ (s + 1)^{3}

ke liye

ω_{u}, K_{u}, T_{u}

nikalo.

ω_{u} = 3

K_{u} = 8

T_{u} = 2 π / 3 \approx 3.63

"Derivative kick" kya hai aur iska fix kya hai?

Reference mein ek step

d e / d t

spike banata hai; fix: measurement ka derivative

- K_{d} \overset{y}{˙}

lo aur filter karo.

Loop shaping mein high open-loop low-frequency gain kya faayda deta hai?

Chhota steady-state error aur achhi disturbance rejection.

Loop shaping mein target gain margin kya hai?

\geq 6

dB (factor of 2).

Classic ZN kaun sa response shape target karta hai?

Quarter-amplitude decay (~25% overshoot) — aggressive, aksar detuning ki zaroorat padti hai.

FOPDT se ZN open-loop PID (

L

=delay,

R

=slope/step)?

K_{p} = 1.2/ (R L), T_{i} = 2 L, T_{d} = 0.5 L