Kalman filters

What Are Kalman Filters?

Kalman filters are recursive algorithms for estimating the state of a dynamic system from a sequence of incomplete and noisy measurements. Introduced by Rudolf E. Kálmán in 1960, the algorithm produces optimal state estimates under the assumption that system dynamics are linear and that both process noise and measurement noise are Gaussian. The filter operates in two alternating steps: a prediction step that projects the current state estimate forward in time using a mathematical model of the system, and an update step that corrects that prediction using newly arrived sensor data. This predict-correct cycle runs continuously, making the Kalman filter suited to real-time applications where measurements arrive sequentially.

The filter draws on probability theory, linear algebra, and control theory. Its central quantity is the covariance matrix, which encodes uncertainty about the current state. Each update step shrinks that uncertainty by an amount proportional to the Kalman gain, a weighting factor that balances trust in the model prediction against trust in the incoming measurement. When measurements are highly accurate relative to the model, the gain drives the estimate toward the measurement. When the model is reliable and measurements are noisy, the estimate stays closer to the prediction.

State Estimation

State estimation is the core problem the Kalman filter solves: given a model of how a system evolves and a stream of sensor readings, compute the best estimate of the system's internal state at each moment. The state vector can represent position, velocity, temperature, voltage, or any other set of quantities that evolve according to known dynamics. The IEEE Transactions on Signal Processing has documented data-driven extensions of the classical estimator that replace hand-crafted noise parameters with learned ones, broadening applicability to systems whose statistical properties are difficult to characterize analytically. Variants such as the extended Kalman filter and the unscented Kalman filter handle moderately nonlinear systems by linearizing the dynamics around the current estimate or by propagating a carefully chosen set of representative points through the nonlinear model.

Sensor Fusion

Sensor fusion is one of the most widely used settings for Kalman filters. When a system is observed by multiple sensors, each with its own noise characteristics, the filter can combine their readings into a single coherent state estimate that is more accurate than any individual sensor alone. Inertial measurement units, GPS receivers, magnetometers, and barometers are commonly fused using this approach in navigation systems. A distributed formulation of the Kalman filter, described in research on large-scale estimation with sensor networks, decomposes the fusion problem across a network of local filters that communicate only with neighbors, enabling scalable estimation over systems with thousands of state variables.

Extended and Adaptive Variants

The original Kalman filter applies strictly to linear, time-invariant systems with known noise statistics. Practical systems rarely satisfy all these conditions, motivating a family of extensions. The extended Kalman filter approximates nonlinear models through first-order Taylor expansion. The unscented Kalman filter uses a deterministic sampling scheme called the unscented transform to propagate uncertainty through nonlinear functions more accurately than linearization allows. Adaptive Kalman filters adjust their noise covariance estimates online when true noise statistics are unknown or time-varying. Soft sensors, which infer unmeasured process variables from secondary measurements using a model, frequently incorporate Kalman filtering to handle measurement uncertainty and model mismatch. The tutorial introduction to Kalman filtering published by the IET provides a systematic treatment of these extensions and their design trade-offs.

Applications

Kalman filters have applications in a wide range of fields, including: