readme.md

Kalman Filter Application With Newton's Law of Cooling

This project supposes a (simplified) dynamic system that models the heating of a house as illustrated in the figure below; inspired by [1].

• $T_A$...temperature of the attic
• $T_S$...temperature of the surrounding
• $T_E$...temperature of the earth
• $T_H$...temperature of the heater
• $T_M$...temperature of the main floor
• $k_1,k_2,k_3,k_4,k_5$...coefficients of heat transfer

Based on Newton's law of cooling, this situation can be modeled by a system of two linear ODEs:

$$ \left{\begin{array}{rl} T_M' &= k_1(T_M-T_E) + k_2(T_M-T_S) + k_3(T_M-T_A) + k_5(T_M-T_H),\\ T_A' &= k_3(T_A-T_M) + k_4(T_A-T_S). \end{array}\right. $$

The goal is to design a Kalman filter to accommodate inaccurate temperature measurements.

Kalman Filter

Kalman filter can be summarized using the following equations:

$$ \begin{array}{|l|l|} \hline \text{Prediction} & \text{Update}\\ \hline \hat{\mathbf{x}}=\mathbf{F}\mathbf{x}+\mathbf{B}\mathbf{u} & \mathbf{y}=\mathbf{z}-\mathbf{H}\hat{\mathbf{x}} \\ \hat{\mathbf{P}}=\mathbf{F}\mathbf{P}\mathbf{F}^\mathsf{T}+\mathbf{Q} & \hat{\mathbf{K}}=\hat{\mathbf{P}}\mathbf{H}^\mathsf{T}(\mathbf{H}\hat{\mathbf{P}}\mathbf{H}^\mathsf{T}+\mathbf{R})^{-1} \\ & \mathbf{x}=\hat{\mathbf{x}}+\mathbf{K}\mathbf{y}\\ & \mathbf{P}=(\mathbf{I}-\mathbf{K}\mathbf{H})\hat{\mathbf{P}}\\ \hline \end{array} $$

To tackle the aforementioned problem, I employ Euler's method to approximate the ODEs. Based on these approximations, I generate true and noisy temperature measurements. Subsequently, I apply the Kalman filter. The figure below displays one outcome of the simulation. For further details, please refer to the file newton_cooling.ipynb.