A Kalman filter estimates a system’s hidden state by predicting how it changes, then correcting that prediction with a noisy measurement. Its central loop is simple: propagate an estimate and its uncertainty, compare the predicted observation with the actual one, and use the resulting error to update the estimate. The standard equations below describe the linear, discrete-time filter.
The linear system the filter estimates
A discrete-time Kalman filter uses a model of how the hidden state evolves and a model of how sensors observe it:
xₖ = Aₖ xₖ₋₁ + Bₖ uₖ + wₖ
zₖ = Hₖ xₖ + vₖ
xₖis the hidden state at stepk, such as position and velocity.uₖis a known control input, such as an applied force;Bₖmaps it into the state.Aₖmaps the previous state to the next state.zₖis the sensor measurement, andHₖmaps the state into measurement space.wₖis process noise: unmodeled variation in how the state evolves. Its covariance isQₖ.vₖis measurement noise. Its covariance isRₖ.
The hats in x̂ mark estimates rather than known true states. The covariance matrix P describes uncertainty in the state estimate. A superscript minus (⁻) means the estimate is before using the current measurement; a plus (⁺) means it is after correction.
Step 1: Predict the state and uncertainty
Start with the previous corrected estimate and propagate it through the motion model:
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x̂ₖ⁻ = Aₖ x̂ₖ₋₁⁺ + Bₖ uₖ
Propagate uncertainty through the same model and add uncertainty from process noise:
Pₖ⁻ = Aₖ Pₖ₋₁⁺ Aₖᵀ + Qₖ
Here Aₖᵀ is the transpose of Aₖ. If process noise enters the state through a separate mapping Γₖ, its contribution is written Γₖ Qₖ Γₖᵀ instead of simply Qₖ. The prediction provides both a best estimate before the next sensor reading and a measure of how uncertain that estimate is.
Step 2: Compare the prediction with the measurement
The model predicts a measurement of Hₖ x̂ₖ⁻. Subtract that from the actual sensor reading to get the innovation, also called the residual:
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yₖ = zₖ − Hₖ x̂ₖ⁻
A positive or negative innovation indicates how far, and in which direction, the observed value differs from the predicted observation. Its covariance is:
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Sₖ = Hₖ Pₖ⁻ Hₖᵀ + Rₖ
Sₖ accounts for uncertainty in both the predicted measurement and the sensor. The Kalman gain is then:
Kₖ = Pₖ⁻ Hₖᵀ Sₖ⁻¹
Kₖ determines how strongly the innovation changes the state estimate. It is calculated from the model and covariances; it is not generally a manually selected, fixed blend.
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Step 3: Correct the estimate
Apply the gain to the innovation to obtain the corrected state estimate:
x̂ₖ⁺ = x̂ₖ⁻ + Kₖ yₖ
Then update the estimate’s covariance using the conventional compact form:
Pₖ⁺ = (I − Kₖ Hₖ)Pₖ⁻
I is the identity matrix. This covariance equation is a standard expression, not the only numerically appropriate implementation: software libraries may use equivalent forms or additional safeguards. Once initialized, the filter repeats prediction and correction as new measurements arrive; MathWorks describes this loop in its Kalman Filtering documentation.
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What the gain means in practice
Consider estimating a vehicle’s position. A motion model predicts where the vehicle should be, while a position sensor reports a noisy location. The innovation is the gap between the reported position and the model’s predicted position. The gain determines how much of that gap to apply to the estimate.
- When predicted state uncertainty is greater, the measurement tends to have more influence, with other factors fixed.
- When measurement uncertainty is greater, the measurement tends to have less influence, with other factors fixed.
Qrepresents uncertainty in state evolution;Rrepresents uncertainty in what the sensor reports.
The filter therefore combines model-based prediction with sensor information according to their represented uncertainties, rather than treating each reading as an independent answer.
Independent reader supportYour contribution helps us test, update, and keep practical guides available for everyone.When the standard Kalman filter fits—and when it does not
The equations above are for linear state and measurement models. MathWorks characterizes the classical Kalman filter as optimal for linear systems with Gaussian process and measurement noise; that claim depends on those assumptions and a suitable model. It should not be extended automatically to nonlinear systems, outliers, changing dynamics, or poorly specified noise covariances. See MathWorks’ Introduction to Estimation Filters for the scope of that characterization.
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For a nonlinear model, extended and unscented Kalman filters are related approaches, but they are not the same equations as the standard linear filter. If the system matrices and noise covariances remain fixed, and design conditions permit convergence, a steady-state gain may be used instead of recalculating a time-varying gain. When model or noise quantities change over time, the filter can retain those changing values. MathWorks discusses steady-state design and time-varying filtering in its Kalman filtering documentation and time-varying Kalman filter example.
Notation varies across references: some use C rather than H for the observation matrix, or include a direct input term such as D uₖ in the measurement equation. Process noise may also be represented through a mapping such as G or Γ. These are notation and modeling choices, not evidence that the prediction-and-correction idea has changed. WPILib’s State Observers and Kalman Filters also gives the discrete predict and update equations.
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