Indirect Temperature Estimation Using Kalman Filter (Two Sensors)

Problem Overview

In Computational Fluid Dynamics (CFD) simulations of building environments, accurate boundary conditions are critical. Dr. Nguyen Hop Minh's experimental setup requires precise knowledge of a hot bulb's surface temperature (200–300°C) to validate CFD models. However, direct measurement is impractical—sensors would melt at such temperatures.

We developed an indirect measurement approach using two air temperature sensors positioned at different distances from the heat source, combining their readings through a Kalman Filter to estimate the bulb's temperature.

Methodology

Experimental Setup

Figure 1a: Real experimental setup showing the heat source box with two ambient temperature sensors at different distances.

Figure 1b: Schematic diagram - Hot bulb (center) radiates heat to Sensor A (close, d_A = 5 cm) and Sensor B (far, d_B = 15 cm).

Mathematical Framework: Bayesian Inference

The Kalman Filter implements optimal Bayesian estimation using Gaussian distributions:

The Kalman Filter computes the posterior mean (μ_post) and variance (σ²_post) in closed form through two steps:

Heat Diffusion Model

We model heat diffusion from the bulb using a distance-based attenuation formula with characteristic length ℓ = 10 cm:

This leads to the observation model z = Hx + v where:

where v_A ~ N(0, R_A = 2.0°C²) and v_B ~ N(0, R_B = 0.5°C²) represent Gaussian measurement noise. The process noise Q = 0.1°C² accounts for natural bulb temperature fluctuations.

CSV Data Input: The raw measurements come from temperature_data.csv with columns:

Figure: Schematic diagram showing sensor geometry and distances from the heat source.

Kalman Filter Algorithm

Prediction Step (Prior Propagation)

Purpose: Propagate previous estimate forward in time, accounting for process uncertainty.

Interpretation: Since we have no control input, the best prediction is the previous estimate. However, uncertainty increases by Q = 0.1°C² due to natural temperature fluctuations. This gives the prior p(x) = N(x; x̂_pred, P_pred) before incorporating new measurements.

Update Step (Posterior via Bayes)

Purpose: Fuse prediction with sensor measurements to reduce uncertainty.

Sensor A Update: Combine prior p(x) = N(x; x̂_pred, P_pred) with likelihood p(z_A|x) = N(z_A; H_Ax, R_A)

Key Insight: K_A balances trust in prediction vs measurement:

• If P_pred ≫ R_A (prediction uncertain, sensor reliable) ⇒ K_A ≈ 1 (trust sensor)
• If P_pred ≪ R_A (prediction confident, sensor noisy) ⇒ K_A ≈ 0 (trust prediction)

Sensor B Update: Sequentially update with second sensor p(z_B|x) = N(z_B; H_Bx, R_B)

Result: Final estimate x̂_new optimally fuses both sensors. Since R_B < R_A (Sensor B more reliable), it receives higher weight in fusion.

Experimental Results

Figure 2: Real experimental data over 600 seconds showing Sensor A (orange), Sensor B (blue), and Kalman Filter estimate (green).

Data Statistics

Key Observations

• Sensor A exhibits higher variance due to proximity to heat source and convection effects
• Sensor B provides more stable baseline measurements
• The Kalman Filter estimate smoothly tracks the bulb temperature while rejecting sensor noise
• At t=100s: z_A=40.5°C, z_B=31.5°C, KF estimate=69.0°C

Understanding Sensor Noise Impact

What Does High Noise Mean?

The noise parameters R_A and R_B represent the variance of the measurement error. High noise means the sensor readings have large random fluctuations around the true value.

Why This Matters for Our Application

In the building environment experiment:

• Physical Reality: Sensor A (close to bulb) experiences more turbulence and thermal convection → higher R_A
• Stability Trade-off: Sensor B (far from bulb) is more stable but has weaker signal → lower R_B
• Optimal Estimation: Kalman Filter weighs both sensors according to their reliability, achieving better accuracy than either sensor alone
• Uncertainty Quantification: The variance P tells us how confident the estimate is — crucial for CFD model validation

Interactive Simulation Tool

Try the Kalman Filter with your own parameters or simulate custom sensor data:

Python Implementation

Below is the complete Python code used to analyze the experimental data. This implementation matches the mathematical framework described above.

import numpy as np
import pandas as pd

# Load data
data = pd.read_csv('temperature_data.csv')
time = data['Time'].values
sensor_a = data['Middle_Heat_Source'].values  # Close to heat source
sensor_b = data['Air_Tube_Output'].values     # Farther from heat source

# Kalman Filter parameters (simulating bulb temperature estimation)
# True bulb temperature would be higher than both measurements
Q = 0.1  # Process noise
R_A = 2.0  # Sensor A noise (more noisy, closer)
R_B = 0.5  # Sensor B noise (less noisy, farther)

# Distance factors (heat diffusion model)
d_A = 5.0  # cm, close to bulb
d_B = 15.0  # cm, farther from bulb

# Initialize
x_est = 45.0  # Initial bulb temperature estimate
P = 10.0  # Initial uncertainty

# Store results
bulb_estimates = []

# Kalman Filter loop
for i in range(len(time)):
    # Prediction step
    x_pred = x_est
    P_pred = P + Q
    
    # Update with Sensor A (accounting for distance)
    H_A = 1.0 / (1 + d_A/10)  # Observation model (heat decreases with distance)
    innovation_A = sensor_a[i] - H_A * x_pred
    S_A = H_A * P_pred * H_A + R_A
    K_A = P_pred * H_A / S_A
    x_est = x_pred + K_A * innovation_A
    P = (1 - K_A * H_A) * P_pred
    
    # Update with Sensor B
    H_B = 1.0 / (1 + d_B/10)  # Heat decreases more with distance
    innovation_B = sensor_b[i] - H_B * x_est
    S_B = H_B * P * H_B + R_B
    K_B = P * H_B / S_B
    x_est = x_est + K_B * innovation_B
    P = (1 - K_B * H_B) * P
    
    bulb_estimates.append(x_est)

# Save results
results = pd.DataFrame({
    'Time': time,
    'Sensor_A_Close': sensor_a,
    'Sensor_B_Far': sensor_b,
    'Bulb_Estimate': bulb_estimates
})
results.to_csv('kalman_results.csv', index=False)

# Calculate statistics
print('=== Temperature Statistics ===')
print(f'Sensor A (Close): Mean={sensor_a.mean():.2f}C, Std={sensor_a.std():.2f}C')
print(f'Sensor B (Far): Mean={sensor_b.mean():.2f}C, Std={sensor_b.std():.2f}C')
print(f'Bulb Estimate: Mean={np.mean(bulb_estimates):.2f}C, Std={np.std(bulb_estimates):.2f}C')
print(f'\nAt t=100s: Sensor A={sensor_a[10]:.1f}C, Sensor B={sensor_b[10]:.1f}C, Estimate={bulb_estimates[10]:.1f}C')
print(f'At t=300s: Sensor A={sensor_a[30]:.1f}C, Sensor B={sensor_b[30]:.1f}C, Estimate={bulb_estimates[30]:.1f}C')
print(f'At t=600s: Sensor A={sensor_a[60]:.1f}C, Sensor B={sensor_b[60]:.1f}C, Estimate={bulb_estimates[60]:.1f}C')

Code Explanation

Discussion

This project demonstrates three key concepts:

• Indirect Measurement: When direct observation is impossible, we can infer hidden states from related observable quantities using domain knowledge (heat diffusion physics).
• Sensor Fusion: Multiple imperfect sensors provide complementary information. The Kalman Filter optimally combines them, outperforming any single sensor.
• Probabilistic Approach: By modeling uncertainty explicitly (measurement noise, process noise), we obtain not just point estimates but confidence intervals for decision-making.

Future work could incorporate thermal dynamics modeling (heat capacity, cooling rates) and extend to multi-sensor arrays for spatial temperature field reconstruction.

References