A Novel Adaptive Square Root UKF with Forgetting Factor for the Time-Variant Parameter Identification

The unscented Kalman filter (UKF) serves as an efficient estimator widely utilized for the recursive identification of parameters. However, the UKF is not well suited for tracking time-variant parameters. Moreover, the unscented transformation (UT) used in the UKF typically relies on Cholesky decomposition to perform the square root operation of the covariance matrix. This method necessitates the matrix to maintain symmetry and positive definiteness. Due to the adverse influence of rounding error and noise, it becomes challenging to guarantee the positive definiteness of the matrix in each recursive step for practical engineering. The square root UKF (SRUKF) eliminates the need for the square root operation in the UT by directly updating the square root of the covariance matrix during each recursion. However, the SRUKF still relies on the rank 1 update to the Cholesky factorization to perform the recursive process, which also necessitates the matrix to be positive definite. Furthermore, the SRUKF is ineffective in the identification of time-variant parameters. Therefore, this paper proposes a modification to the SRUKF that ensures unconditional numerical stability by utilizing QR decomposition. Subsequently, the modified square root UKF (MSRUKF) method is enhanced by incorporating an adaptive forgetting factor that can be adjusted based on the residual information from each recursive step. This adaptation leads to the development of the adaptive SRUKF with forgetting factor (ASRUKF-FF) method, which significantly improves the tracking capability for time-variant parameters. To validate the effectiveness of the proposed method, this paper demonstrates its application in identifying the time-variant stiffness and damping parameters of a three-story frame structure. In addition, the method is employed to estimate the time-variant stiffness of the bridge excited by vehicles. The simulation results show that the proposed method has the superiority of high accuracy, strong robustness, and widespread applicability, even with incomplete measurements and inappropriate parameter settings.