Square-Root Cubature Kalman Filter

The Square-Root Cubature Kalman Filter (SR-CKF) is an enhanced nonlinear state estimation algorithm for dynamic systems. It builds upon the standard Cubature Kalman Filter by incorporating square-root decomposition techniques, significantly improving numerical stability during computation.

In MATLAB implementation, this algorithm achieves efficient state estimation through several critical steps. The initialization phase requires setting initial values for the state vector and error covariance matrix. During time update, cubature points propagate through the nonlinear state equation to predict the state at the next time step. The MATLAB implementation typically uses the spherical-radial rule to generate these cubature points, ensuring accurate capture of statistical characteristics in nonlinear systems.

The measurement update forms the core of the algorithm, where cubature rules approximate the probability distribution of nonlinear measurement equations. The square-root decomposition technique plays a vital role here by decomposing covariance matrices, thus preventing potential numerical instability issues from direct computations. This decomposition can be Cholesky decomposition or other matrix factorization methods. In code implementation, maintaining matrix positive definiteness is crucial for algorithm convergence.

Compared to the standard CKF, the SR-CKF version may show slightly lower computational efficiency but offers substantially improved numerical stability. This makes it particularly suitable for handling ill-conditioned matrices or long-term estimation problems. Experimental verification demonstrates that this MATLAB implementation delivers excellent accuracy and robustness across various nonlinear state estimation tasks. The implementation often includes specific functions for covariance matrix updates and numerical integrity checks to ensure reliable performance.