Kalman filter

Kalman filter. The Kalman filter is a recursive algorithm for estimating the state of a dynamic system from a series of noisy measurements. It is a central tool in control theory and signal processing, providing an optimal estimate under the assumption of Gaussian noise. The filter's efficiency and versatility have made it fundamental to modern engineering, from NASA's Apollo program to contemporary GPS receivers and autonomous vehicle systems.

Overview

The algorithm operates by predicting a system's future state and then updating this prediction with new measurement data, a process known as predictor-corrector. It assumes the system can be described by a linear dynamical system driven by Gaussian noise. The filter's optimality is proven within the framework of minimum mean square error estimation. Its development is closely associated with the work of Rudolf E. Kálmán, though similar concepts appeared in the work of Peter Swerling and Thorvald Nicolai Thiele. Key properties include its recursive nature, which avoids the need to store a complete history of data, and its provision of a real-time estimate of the estimation error covariance.

Mathematical description

The filter addresses a discrete-time system defined by a state-space representation. The state evolution is governed by the state-transition model, represented by the matrix $\backslash mathbf\{F\}\_k$, and is subject to process noise $\backslash mathbf\{w\}\_k$ with covariance $\backslash mathbf\{Q\}\_k$. Observations are made via a measurement model matrix $\backslash mathbf\{H\}\_k$, corrupted by measurement noise $\backslash mathbf\{v\}\_k$ with covariance $\backslash mathbf\{R\}\_k$. The core algorithm consists of two steps: the predict step, which projects the state estimate ($\backslash hat\{\backslash mathbf\{x\}\}\_\{k|k-1\}$) and its error covariance ($\backslash mathbf\{P\}\_\{k|k-1\}$) forward, and the update step, which computes the Kalman gain ($\backslash mathbf\{K\}\_k$) to blend the prediction with the new measurement $\backslash mathbf\{z\}\_k$. This yields the updated estimate $\backslash hat\{\backslash mathbf\{x\}\}\_\{k|k\}$ and covariance $\backslash mathbf\{P\}\_\{k|k\}$.

Derivation

The derivation stems from probability theory and the goal of finding the estimator that minimizes the posterior mean squared error. A foundational approach uses the concept of orthogonal projection onto the subspace of linear measurements, a principle from Hilbert space theory. Alternatively, it can be derived via Bayesian inference by treating the state as a random variable with a Gaussian distribution; the predict step applies the Chapman-Kolmogorov equation, while the update step is an application of Bayes' theorem. The solution can also be obtained through maximum a posteriori estimation or by solving the Wiener–Hopf equation for the non-stationary case. The elegant result is that the optimal estimate is a linear combination of the prior prediction and the new measurement.

Extensions and variants

For nonlinear systems, the Extended Kalman filter linearizes the dynamics around the current estimate. The Unscented Kalman filter, developed by researchers like Simon J. Julier, uses a deterministic sampling technique to better approximate nonlinear transformations. The Ensemble Kalman filter, used extensively in numerical weather prediction by institutions like the European Centre for Medium-Range Weather Forecasts, employs a Monte Carlo method. Other important variants include the Information filter, which is algebraically equivalent but uses an information matrix, and the Kalman–Bucy filter for continuous-time systems. For multi-sensor fusion, the Federated Kalman filter architecture is often employed.

Applications

The filter was crucial for the navigation systems of the Apollo Command Module and Lunar Module. It is ubiquitous in modern inertial navigation systems found in aircraft like the Boeing 777 and missiles such as the Tomahawk (missile). In robotics, it is used for simultaneous localization and mapping and sensor fusion in platforms from Boston Dynamics. Consumer electronics, including smartphones and wearable technology, use it for tracking orientation and position. It is also applied in econometrics for time series analysis, in electrical engineering for power system state estimation, and in computational finance for tracking volatile financial market parameters. Category:Estimation theory Category:Digital signal processing Category:Control theory Category:Algorithms