Chaotic Fractional Bayesian Filtering for Nonlinear Dynamic Systems with Random Delays and Uncertain Measurements

To address the state estimation problem of nonlinear dynamic systems exhibiting fractional memory characteristics, chaotic dynamic behavior, random time delay, and observation uncertainty, this study proposes a unified Chaotic Fractional Bayesian Filtering (CFBF) framework. At the modeling level, this framework introduces the Caputo fractional derivative to characterize the system's long-term memory and nonlocal evolution properties. By segmenting the continuous random time delay interval and introducing Bernoulli random variables, a structured transformation from random time delay to an equivalent random parameter system is achieved. This transformation ensures the causality and resolvability of the model. At the inference level, this paper derives a prediction mechanism based on the fractional Chapman-Kolmogorov equation. To address the prevalent non-Gaussian chaotic perturbations during observation, a chaos-sensitive weighted update strategy is introduced. The resulting CFBF algorithm integrates a fractional memory kernel function in the prediction phase. This function introduces chaotic robust weights in the update phase, achieving collaborative modeling and inference of complex dynamic uncertainties. In numerical experiments, a three-dimensional nonlinear fractional-order chaotic system was selected as the test object. It was methodically compared with EKF, UKF, PF, and fractional-order Kalman filter (FO-KF) under random time delay and uncertain observation conditions. In comparison to EKF, UKF, PF, and FO-KF, the average RMSE of CFBF in three-dimensional state estimation was lowered by roughly <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$\mathbf{3 2. 6 \%, ~} \mathbf{2 4. 1 \%} \boldsymbol{,} \mathbf{1 8. 3 \%} \boldsymbol{,}$</tex> and <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$\mathbf{1 1. 7 \%}$</tex> in <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$\mathbf{1 0 0}$</tex> Monte Carlo simulations.