Aerospace ML applications — fault detection, system identification

WHY does aerospace need ML here?

Part 1 — Fault Detection

WHAT is a residual? (derive it)

Let a healthy system be modelled by predictor $\hat{y}$ . For a true measurement $y$ :

$r = y - \hat{y}$

Why this works from first principles: if the model is good and the system is healthy, $\hat y \approx y$ , so $r \approx 0$ (only sensor noise remains). A fault changes the physics, so $\hat y$ (built on healthy physics) no longer matches $y$ , and $r$ grows. We monitor $r$ , not $y$ directly — this cancels out the known dynamics and leaves only the "surprise."

HOW do we threshold noise vs fault?

Noise is random. Under healthy conditions, model residuals are roughly $r \sim \mathcal{N}(0,\sigma^2)$ . We standardise:

$z = \frac{r - \mu}{\sigma}$

Declare a fault when $|z| > \tau$ . Choosing $\tau = 3$ gives (from the normal CDF) a false-alarm rate of about $0.27\%$ per sample.

Figure — Aerospace ML applications — fault detection, system identification

Part 2 — System Identification

Deriving least-squares SysID from scratch

Model a discrete linear system (an ARX model — AutoRegressive with eXogenous input):

$y_k = a_1 y_{k-1} + \dots + a_n y_{k-n} + b_1 u_{k-1} + \dots + b_m u_{k-m}$

Stack the past values into a regressor row $\varphi_k^{\top} = [\,y_{k-1},\dots,u_{k-m}\,]$ and unknowns $\theta = [a_1,\dots,b_m]^{\top}$ . Then $y_k = \varphi_k^{\top}\theta$ . Over $N$ samples:

$\mathbf y = \Phi\,\theta, \qquad \Phi = \begin{bmatrix}\varphi_1^\top\\ \vdots \\ \varphi_N^\top\end{bmatrix}$

WHY least squares? Data is noisy, so no exact $\theta$ solves this. We choose $\theta$ minimising squared prediction error:

$J(\theta) = \sum_k (y_k - \varphi_k^\top\theta)^2 = \|\mathbf y - \Phi\theta\|^2$

HOW to minimise — derive the normal equations. Expand and differentiate: $J = \mathbf y^\top\mathbf y - 2\theta^\top\Phi^\top\mathbf y + \theta^\top\Phi^\top\Phi\,\theta$ $\frac{\partial J}{\partial \theta} = -2\Phi^\top\mathbf y + 2\Phi^\top\Phi\,\theta = 0$

Worked example Identify a first-order UAV pitch-rate response

Suppose $y_k = a\,y_{k-1} + b\,u_{k-1}$ . Data (3 points): $(y_{k-1},u_{k-1})\to y_k$ : $(0,1)\to 0.5$ , $(0.5,1)\to 0.9$ , $(0.9,0)\to 0.72$ .

Regressor rows: $[0,1],[0.5,1],[0.9,0]$ . Why? Each row = "inputs that produced this output."
Form $\Phi$ and $\mathbf y=[0.5,0.9,0.72]^\top$ , solve $\hat\theta=(\Phi^\top\Phi)^{-1}\Phi^\top\mathbf y$ .
Result $\approx (a,b)\approx(0.8,0.5)$ — meaning ~80% of pitch-rate carries over each step (inertia), gain 0.5 per control unit. Why care? This $a$ tells you the time constant for the autopilot.

Common mistakes (steel-manned)

Feynman

Recall Explain to a 12-year-old

Imagine you know your friend always walks to school in about 20 minutes. That "20 minutes" is your model. One day they take 40 — the difference (the residual) is a surprise, so you guess something went wrong (fault detection). And if you watched them many days and figured out the rule "time = distance ÷ speed," you just identified the system from data instead of being told the rule. Computers on airplanes do exactly this with vibrations and temperatures.

Flashcards

What is a residual in fault detection?

The difference

r=y-\hat y

between measured output and model prediction; near zero when healthy, grows under faults.

Why monitor residuals instead of raw sensor values?

Residuals subtract out known/expected dynamics, leaving only the "surprise," so normal operating variation isn't mistaken for a fault.

Why does the Mahalanobis distance use

\Sigma^{-1}

It rescales by each direction's variability and de-correlates sensors, catching faults that break inter-sensor relationships even if each channel is in-range.

State the least-squares (normal-equation) estimate for SysID.

\hat\theta=(\Phi^\top\Phi)^{-1}\Phi^\top\mathbf y

Derive the normal equations in one line.

Minimise

\|\mathbf y-\Phi\theta\|^2

; set

\partial J/\partial\theta=-2\Phi^\top\mathbf y+2\Phi^\top\Phi\theta=0\Rightarrow\Phi^\top\Phi\,\theta=\Phi^\top\mathbf y

What is an ARX model?

AutoRegressive with eXogenous input:

y_k=\sum a_i y_{k-i}+\sum b_j u_{k-j}

, a linear model of output from past outputs and inputs.

Why can too many ARX terms hurt?

Overfitting — extra parameters fit noise, lowering training error but worsening generalisation; validate / use AIC/BIC.

For

z\sim\mathcal N(0,1)

, why is

\tau=3

common?

P(|z|>3)\approx0.27\%

, a low per-sample false-alarm rate while still catching real deviations.

How do fault detection and SysID connect?

SysID gives the healthy predictor

\hat y=\varphi^\top\hat\theta

; residuals from it are thresholded for fault detection.

Connections

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

Concept Map

Hinglish (regional understanding)

Intuition Hinglish mein samjho

Dekho, aircraft ek aisi machine hai jo udte waqt lagataar data banati rehti hai — vibration, temperature, pressure sab. Fault detection ka core idea simple hai: pehle hum ek "healthy model" banate hain jo predict karta hai ki normal condition me sensor ka reading kya hona chahiye ( $\hat y$ ). Phir actual reading $y$ me se prediction ghata dete hain — isko residual $r = y - \hat y$ bolte hain. Agar sab theek hai to $r$ almost zero rehta hai (sirf noise), lekin jab koi part kharab hota hai to physics change ho jaati hai aur $r$ badhne lagta hai. Bas usko threshold se compare karo — z-score $|z|>3$ hua to alarm bajao.

System identification ka matlab hai data se system ka equation nikalna, textbook physics se nahi. Real UAV kabhi bhi wind-tunnel model se exactly match nahi karta, isliye hum flight ka input-output data lekar model fit karte hain. ARX model use karte hain: aaj ka output = past outputs + past inputs ka linear combination. Noise ki wajah se exact solution nahi milta, isliye hum squared error minimise karte hain aur seedha normal equation $\hat\theta=(\Phi^\top\Phi)^{-1}\Phi^\top\mathbf y$ mil jaata hai.

Sabse important baat — ye dono ek hi pipeline ke do hisse hain. Pehle least squares se healthy model identify karo, phir usi model se residuals nikaal kar fault detect karo. Yaad rakhne ka trick: I-R-T (Identify, Residual, Threshold).

Aur ek galti mat karna: bada raw reading dekh ke ghabra mat jaana. High thrust pe temperature high hona to normal hai! Fault tab hai jab relationship toote — isiliye multi-sensor case me Mahalanobis distance ( $\Sigma^{-1}$ ke saath) use karte hain jo sensors ke beech ke correlation ko bhi check karta hai.

Test yourself — Machine Learning (Aerospace Applications)

Connections