Comparison with KF and EKF

Kalman Filter (KF)

• Linear Systems: The Kalman Filter is designed for linear systems. It assumes that the state transition and observation models are linear.

• Equations: The state transition and observation models are represented as: where is the state vector, is the control input, is the observation, and and are process and observation noise, respectively.

• Optimal Estimation: It provides optimal state estimation for linear systems with Gaussian noise.

Extended Kalman Filter (EKF)

• Nonlinear Systems: The EKF extends the KF to handle nonlinear systems by linearizing the nonlinear models around the current estimate.

• Equations: The state transition and observation models are represented as: where and are nonlinear functions.

• Linearization: The EKF linearizes these functions using a first-order Taylor series expansion around the current estimate. This involves calculating the Jacobian matrices and of the functions and , respectively.

• Approximate Estimation: While the EKF is not optimal for nonlinear systems, it provides a practical solution by approximating the nonlinear models with linear ones12.

Key Differences

1. System Type: KF is for linear systems, while EKF is for nonlinear systems.

2. Model Representation: KF uses linear models, whereas EKF uses nonlinear models and linearizes them.

3. Optimality: KF provides optimal estimates for linear systems, while EKF provides approximate estimates for nonlinear systems.