Kalman Filter

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tags: - colorclass/bifurcation theory ---Kalman filters are a powerful set of mathematical equations providing an efficient computational means to estimate the state of a linear dynamic system from a series of noisy measurements. They are widely used in applications requiring real-time estimation and prediction, such as in navigation systems (GPS and INS), robotics, and signal processing, to name a few. The Kalman filter has the unique advantage of being recursive, meaning it can process measurements as they arrive, one at a time, without the need for the entire data set to be available from the start.

Overview

Developed by Rudolf E. Kalman in 1960, the Kalman filter addresses two fundamental problems: 1. Prediction: Estimating the future state of a process based on a mathematical model of the physical system. 2. Update: Refining these estimates as new measurements become available.

How It Works

The Kalman filter operates in two steps: predict and update.

- Prediction Step: The filter predicts the current state and error covariance estimates to obtain the prior estimated state. This step uses the process model of the system.

- Update Step: Once a new measurement is available, the filter updates its predictions based on the measurement to minimize the error in the estimated state. This step incorporates the measurement model of the system.

Mathematical Formulation

The essence of the Kalman filter can be captured in the following set of equations, which are executed in a loop as new data becomes available:

1. Prediction: - Predicted State Estimate: ${\hat{x}}_{k|k-1}=F_k{\hat{x}}_{k-1|k-1}+B_k{u}_k$ - Predicted Covariance Estimate: $P_{k|k-1}=F_k{P}_{k-1|k-1}{F}_k^T+Q_k$

Where ${\hat{x}}_{k|k-1}$ is the predicted state estimate at time $k$ given the information up to time $k-1$, $F_k$ is the state transition model, $B_k$ is the control-input model, $u_k$ is the control vector, $P_{k|k-1}$ is the predicted estimate covariance, and $Q_k$ is the process noise covariance matrix.

2. Update: - Kalman Gain: $K_k=P_{k|k-1}{H}_k^T(H_k{P}_{k|k-1}{H}_k^T+R_k{)}^{-1}$ - Updated State Estimate: ${\hat{x}}_{k|k}={\hat{x}}_{k|k-1}+K_k(z_k-H_k{\hat{x}}_{k|k-1})$ - Updated Covariance Estimate: $P_{k|k}=(I-K_k{H}_k)P_{k|k-1}$

Here, $K_k$ is the Kalman gain, $H_k$ is the measurement model, $z_k$ is the measurement at time $k$, $R_k$ is the measurement noise covariance matrix, and $I$ is the identity matrix.

Applications

- Navigation and Tracking: Kalman filters are extensively used in GPS and INS systems for accurate positioning and navigation. - Robotics: For sensor fusion, where data from multiple sensors are combined to estimate the robot’s position and orientation accurately. - Economics and Finance: To estimate variables of interest, like GDP or stock prices, from noisy observations. - Control Systems: For controlling the state of a process in real-time, adjusting the inputs based on the estimated state.

Extensions and Variations

The original Kalman filter is designed for linear systems with Gaussian noise. Several extensions have been developed for more complex scenarios: - Extended Kalman Filter (EKF): Approximates non-linear systems by linearizing about the current mean and covariance. - Unscented Kalman Filter (UKF): Uses a deterministic sampling technique to capture the mean and covariance of a non-linear transformation of a probability distribution. - Particle Filter: A more general approach that uses a set of particles (samples) to represent probability distributions, capable of dealing with non-linear and non-Gaussian problems.

The Kalman filter’s elegance lies in its simplicity, efficiency, and wide applicability, making it a cornerstone of modern control theory and signal processing.