Glossary term
Kalman Filter
A recursive estimator that combines a dynamic model with noisy measurements to estimate the state of a system.
Definition
methodA Kalman filter is a recursive state estimator that fuses a system model with noisy measurements while tracking estimation uncertainty.
For linear systems with Gaussian noise, the Kalman filter gives the minimum-variance estimate of the system state. It alternates between prediction, using the state-space model, and correction, using incoming measurements. The filter maintains both a state estimate and an error covariance matrix. Variants such as the extended Kalman filter and unscented Kalman filter are used for nonlinear systems.
A Kalman filter estimates the internal state of a dynamic system when measurements are noisy, incomplete, or indirect. It combines two sources of information: a model that predicts how the state evolves and measurements that partially observe the state. The filter also tracks uncertainty, so it can weigh model prediction against measurement correction.
Prediction and correction
In the prediction step, the filter propagates the current state estimate through the state-space model. It also propagates the covariance matrix, increasing uncertainty according to process noise. In the correction step, the filter compares the predicted measurement with the actual measurement. The difference is the innovation. The Kalman gain determines how strongly the estimate should move toward the measurement.
If measurement noise is high, the filter trusts the model more. If model uncertainty is high, it trusts the measurement more. This balance is controlled by process-noise covariance Q and measurement-noise covariance R.
Applications
Kalman filters are used in inertial navigation, aircraft attitude estimation, vehicle tracking, robotics, GPS fusion, motor control, battery state estimation, radar tracking, sensor calibration, and process monitoring. A common example is combining gyroscope data with accelerometer, magnetometer, encoder, or GPS measurements. The gyroscope gives smooth short-term motion information but drifts over time; other sensors can correct the drift.
Variants
The classical Kalman filter assumes linear dynamics and measurement equations with Gaussian noise. The extended Kalman filter linearizes nonlinear models using Jacobian matrices. The unscented Kalman filter propagates selected sample points through nonlinear models. Particle filters are used when distributions are strongly non-Gaussian or multimodal, but they are computationally heavier.
Common mistakes
A common mistake is treating the filter as a magic smoother. If the model is wrong, sensors are biased, timestamps are inconsistent, or noise covariances are unrealistic, the estimate can be confidently wrong. Another mistake is tuning only until the output looks smooth. Excessive smoothing can add delay and degrade control. Good filter design checks observability, sampling rate, sensor calibration, units, time synchronization, innovation statistics, covariance consistency, and behaviour during sensor faults.