Aerospace ML applications — fault detection, system identification

5.6.15 · Coding › Machine Learning (Aerospace Applications)

WHY aerospace ko ML ki zaroorat hai yahan?

Intuition Fixed rules aur physics equations kyun nahi use karte?

Faults rare, subtle, aur expensive hote hain. Ek cracked turbine blade vibration spectrum ko bahut thoda shift karta hai — kaafi pehle se — jab koi human notice bhi nahi karta. ML normal pattern seekhta hai aur deviations flag karta hai.
Real systems apne textbook models se drift kar jaate hain. Wind-tunnel se derive kiya hua UAV ka model kabhi bhi real airframe se match nahi karta (manufacturing tolerances, wear, icing). System identification ek model measured flight data se fit karta hai taaki control laws accurate rahen.
80/20: Do techniques cover karti hain aerospace-ML ki zyaadatar value — (1) faults ke liye residual-based anomaly detection, (2) system ID ke liye least-squares regression. In dono ko master kar lo aur tum field ka 80% samajh jaoge.

Part 1 — Fault Detection

WHAT hota hai ek residual? (derive karo)

Maano ek healthy system ko predictor $\overset{y}{^}$ se model kiya gaya hai. True measurement $y$ ke liye:

$r = y - \overset{y}{^}$

Ye kaam kyun karta hai, first principles se: agar model accha hai aur system healthy hai, toh $\overset{y}{^} \approx y$ , isliye $r \approx 0$ (sirf sensor noise bachta hai). Ek fault physics change karta hai, toh $\overset{y}{^}$ (jo healthy physics pe built hai) $y$ se match nahi karta, aur $r$ badh jaata hai. Hum $r$ monitor karte hain, $y$ ko directly nahi — ye known dynamics cancel out kar deta hai aur sirf "surprise" bacha rehta hai.

HOW karte hain hum noise vs fault threshold?

Noise random hota hai. Healthy conditions mein, model residuals roughly $r \sim N (0, σ^{2})$ hote hain. Hum standardise karte hain:

$z = \frac{r - μ}{σ}$

Fault declare karo jab $∣ z ∣ > τ$ . $τ = 3$ choose karna (normal CDF se) ek false-alarm rate deta hai approximately $0.27%$ per sample.

Figure — Aerospace ML applications — fault detection, system identification

Part 2 — System Identification

Least-squares SysID scratch se derive karna

Ek discrete linear system model karo (ek ARX model — AutoRegressive with eXogenous input):

$y_{k} = a_{1} y_{k - 1} + \dots + a_{n} y_{k - n} + b_{1} u_{k - 1} + \dots + b_{m} u_{k - m}$

Past values ko ek regressor row $φ_{k}^{⊤} = [y_{k - 1}, \dots, u_{k - m}]$ mein stack karo aur unknowns $θ = [a_{1}, \dots, b_{m}]^{⊤}$ . Phir $y_{k} = φ_{k}^{⊤} θ$ . $N$ samples pe:

$y = Φ θ, Φ = φ_{1}^{⊤} ⋮ φ_{N}^{⊤}$

Least squares kyun? Data noisy hai, toh koi exact $θ$ ye solve nahi karta. Hum woh $θ$ choose karte hain jo squared prediction error minimize kare:

$J (θ) = \sum_{k} (y_{k} - φ_{k}^{⊤} θ)^{2} = ∥ y - Φ θ ∥^{2}$

HOW minimize karte hain — normal equations derive karo. Expand karo aur differentiate karo: $J = y^{⊤} y - 2 θ^{⊤} Φ^{⊤} y + θ^{⊤} Φ^{⊤} Φ θ$ $\frac{\partial J}{\partial θ} = - 2 Φ^{⊤} y + 2 Φ^{⊤} Φ θ = 0$

Common mistakes (steel-manned)

Feynman

Recall Ek 12-saal ke bachche ko explain karo

Socho tumhare dost ko school jaane mein hamesha lagbhag 20 minute lagte hain. Wo "20 minute" tumhara model hai. Ek din unhe 40 lagte hain — woh fark (residual) ek surprise hai, toh tum guess karte ho kuch galat hua (fault detection). Aur agar tumne unhe kaafi dino dekha aur rule figure out kiya "time = distance ÷ speed," tum ne abhi data se system identify kiya, bajaye rule bataye jaane ke. Airplanes pe computers exactly yahi karte hain vibrations aur temperatures ke saath.

Flashcards

Fault detection mein residual kya hota hai?

Fark

r = y - \overset{y}{^}

measured output aur model prediction ke beech; healthy hone pe near zero, faults mein badh jaata hai.

Raw sensor values ki jagah residuals kyun monitor karte hain?

Residuals known/expected dynamics subtract kar dete hain, sirf "surprise" bacha lete hain, taaki normal operating variation ko fault na samjha jaaye.

Mahalanobis distance

Σ^{- 1}

kyun use karta hai?

Ye har direction ki variability se rescale karta hai aur sensors ko de-correlate karta hai, woh faults pakadta hai jo inter-sensor relationships tod dete hain chahe har channel in-range ho.

SysID ke liye least-squares (normal-equation) estimate bolo.

\hat{θ} = (Φ^{⊤} Φ)^{- 1} Φ^{⊤} y

Normal equations ek line mein derive karo.

∥ y - Φ θ ∥^{2}

minimize karo;

\partial J / \partial θ = - 2 Φ^{⊤} y + 2 Φ^{⊤} Φ θ = 0 \Rightarrow Φ^{⊤} Φ θ = Φ^{⊤} y

set karo.

ARX model kya hota hai?

AutoRegressive with eXogenous input:

y_{k} = \sum a_{i} y_{k - i} + \sum b_{j} u_{k - j}

, output ka ek linear model past outputs aur inputs se.

Zyada ARX terms kyun hurt kar sakte hain?

Overfitting — extra parameters noise fit karte hain, training error kam karte hain lekin generalisation kharab karte hain; validate karo / AIC/BIC use karo.

z \sim N (0, 1)

ke liye,

τ = 3

common kyun hai?

P (∣ z ∣ > 3) \approx 0.27%

, ek low per-sample false-alarm rate jo real deviations pakadta bhi rehta hai.

Fault detection aur SysID kaise connect hote hain?

SysID healthy predictor

\overset{y}{^} = φ^{⊤} \hat{θ}

deta hai; usse nikle residuals fault detection ke liye threshold kiye jaate hain.

Connections

Least Squares Regression
Kalman Filter — dynamic systems ke liye model-based residual generation
Anomaly Detection
Normal Distribution and Z-scores
Covariance Matrix aur Mahalanobis Distance
ARX and State-Space Models
Overfitting and Model Selection (AIC/BIC)
Digital Twins in Aerospace