MATLAB Code Implementation of Extended Kalman Filter (EKF)

The Extended Kalman Filter (EKF) is a widely used algorithm for state estimation in nonlinear systems. In nonlinear problems such as target tracking, EKF effectively addresses the limitation of the standard Kalman filter (which only applies to linear systems) by performing local linearization of nonlinear systems at each time step.

The core concept of EKF involves performing first-order Taylor expansion of nonlinear functions at each time step, using Jacobian matrices to replace the state transition matrix and observation matrix of linear systems. A typical MATLAB implementation includes several key components: state prediction, covariance prediction, Kalman gain calculation, state update, and covariance update. In code implementation, these correspond to discrete steps where system dynamics and measurement models are linearized using Jacobian matrices computed at the current state estimate.

For target tracking applications, EKF can effectively handle nonlinear dynamic models of moving targets (such as turning maneuvers) and nonlinear observation models (such as polar coordinate measurements from radar). Important implementation considerations include: 1) Proper selection of process noise and measurement noise covariance matrices (Q and R matrices); 2) Ensuring accurate computation of Jacobian matrices, which can be done numerically or symbolically; 3) Handling potential numerical stability issues through techniques like square-root filtering or regularization.

MATLAB's powerful matrix operation capabilities and rich mathematical function library facilitate EKF implementation. Complex dynamic models can be processed using numerical integration methods like ode45, while the Symbolic Math Toolbox can automatically derive Jacobian matrices, significantly simplifying the implementation difficulty. The implementation typically involves creating functions for state transition (f), measurement (h), and their corresponding Jacobian calculations (F and H), followed by the standard EKF prediction and update cycles implemented through matrix operations.