fault detection and diagnosis

# Fault Detection and Diagnosis

## Introduction & Motivation

ML-based fault detection and diagnosis enables early problem identification for preventive maintenance and safety.

Motivation: Detect and diagnose equipment faults.

Applications: Fault detection, diagnostics, root cause analysis, corrective action.

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## Core Concepts & Theory

### Fault Types

Bearing wear, seal failure, cavitation.

### Detection Methods

Threshold-based and model-based.

### Diagnosis Techniques

Classification and reasoning.

### Timetointerventionn

Failure progression.

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## Mathematical Formulation

Residual:
$$r(t) = y(t) - \hat{y}(t)$$

Fault Indicator:
$$f_i = \frac{\sum |r(t)|}{N}$$

Diagnosis:
$$ ext{fault} = \arg\max_c P(c|r)$$

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## Summary & Key Takeaways

ML detects and diagnoses faults efficiently.

Principles: 1. Monitoring, 2. Residuals, 3. Detection, 4. Diagnosis, 5. Action.

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## Appendix: Practical Labs

### Lab 1: Fault Detection

import numpy as np

def detect_fault(measurements, threshold):
 residuals = np.abs(measurements - np.mean(measurements))
 faults = residuals > threshold
 return np.sum(faults) > 0

meas = np.random.normal(100, 5, 100)
fault = detect_fault(meas, 20)
assert isinstance(fault, (bool, np.bool_)), "Detection failed"
print(f"✓ Fault detected: {fault}")

### Lab 2: Fault Classification

import numpy as np

class FaultDiagnosis:
 def __init__(self):
 self.fault_types = ['bearing', 'seal', 'alignment']
 
 def diagnose(self, features):
 scores = np.dot(features, np.random.randn(len(features)))
 return self.fault_types[np.argmax(scores)]

diag = FaultDiagnosis()
feat = np.random.randn(5)
fault_type = diag.diagnose(feat)
assert fault_type in diag.fault_types, "Diagnosis failed"
print(f"✓ Fault type: {fault_type}")

### Lab 3: Root Cause Analysis

import numpy as np

def analyze_root_causes(parameters, baseline):
 deviations = parameters - baseline
 causes = []
 for i, dev in enumerate(deviations):
 if abs(dev) > 2 * np.std(deviations):
 causes.append(i)
 return causes

params = np.array([100, 150, 100, 100, 100])
baseline = np.array([100, 100, 100, 100, 100])
causes = analyze_root_causes(params, baseline)
assert isinstance(causes, list), "Analysis failed"
print(f"✓ Root causes identified at indices: {causes}")

### Lab 4: Fault Severity

import numpy as np

def estimate_fault_severity(fault_indicator, thresholds):
 if fault_indicator < thresholds[0]:
 return 'Normal'
 elif fault_indicator < thresholds[1]:
 return 'Warning'
 else:
 return 'Critical'

severity = estimate_fault_severity(15, [10, 20])
assert severity in ['Normal', 'Warning', 'Critical'], "Severity failed"
print(f"✓ Severity: {severity}")

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