Function Implementation of Extended Kalman Filter Algorithm

The Extended Kalman Filter (EKF) is a widely used algorithm for state estimation in nonlinear systems, demonstrating exceptional performance particularly in target tracking applications. Unlike the standard Kalman filter, EKF handles nonlinearity through local linearization, enabling adaptation to more complex dynamic models.

The algorithm's core logic consists of two main phases: prediction and update. During the prediction phase, the system calculates the prior state estimate and its covariance matrix for the next time step based on the current state and system model. Since the system is nonlinear, first-order Taylor expansion is employed to approximate nonlinear functions, producing Jacobian matrices for linearization. In code implementation, this typically involves calculating partial derivatives of state transition functions with respect to state variables.

The update phase utilizes actual measurements to correct the predictions. Similarly, if the observation equation is nonlinear, it undergoes linearization treatment. By computing the Kalman gain, the algorithm balances the reliability between predicted values and measured values, ultimately yielding posterior state estimates and updated covariance matrices. Programmatically, this involves matrix operations including measurement residual calculation, innovation covariance computation, and optimal gain derivation.

In target tracking scenarios, EKF effectively handles trajectory prediction for non-uniform motion targets, such as aircraft performing turning maneuvers. Its advantages include relatively low computational complexity and satisfactory estimation accuracy for weakly nonlinear systems. However, it's important to note that for highly nonlinear systems, linearization errors may lead to degraded filter performance. Code implementations often incorporate sanity checks for Jacobian matrix conditioning and covariance matrix positive-definiteness to maintain numerical stability.