This article addresses state estimation problems with unknown process and measurement noise covariances in both linear and nonlinear systems. By formulating the joint estimation of system state and noise parameters into an optimization problem, a novel adaptive Kalman filter method based on conjugate-computation variational inference, referred to as CVIAKF, is proposed to approximate the joint posterior probability density function of the latent variables. Unlike existing adaptive Kalman filter methods utilizing variational inference in natural parameter space, CVIAKF performs optimization in expectation parameter space, resulting in a faster and simpler solution, particularly for nonlinear state estimation. Meanwhile, CVIAKF divides optimization objectives into conjugate and non-conjugate parts of nonlinear dynamical models, whereas conjugate computations and stochastic mirror-descent are applied, respectively. Notably, the reparameterization trick is employed to minimize the variance of stochastic gradients for the non-conjugate parts. The effectiveness of CVIAKF is validated through synthetic and real-world datasets of maneuvering target tracking.

This paper considers the adaptive Kalman filtering problem with unknown process noise and measurement noise covariances for linear dynamical systems. By formulating the joint estimation of system state and noise parameters as variational optimization problems, the joint posterior probability density function (PDF) of latent variables (i.e., the noise covariance matrices and system state) is approximated by variational Bayesian inference (VB). Different from the existing VB-based adaptive Kalman filtering (VBAKF) methods, which update the variational hyperparameters analytically by constructing conjugate priors, this paper presents a stochastic natural gradient-based VBAKF, referred to as NGAKF-QR, to directly optimize the intractable non-conjugate objectives. By splitting the optimization objective into conjugate and non-conjugate parts, the proposed NGAKF-QR updates the conjugate models of system state and measurement noise covariances with conjugate computations, and the non-conjugate models of process noise covariances with stochastic natural gradient, enabling effective and flexible Bayesian inference. Remarkably, the reparameterization trick for the inverse Wishart distribution is presented to decrease the stochastic gradient variance. Due to the direct estimation of state and noise covariance, the proposed NGAKF-QR has better filtering accuracy than the existing state-of-the-art VBAKF. The effectiveness of NGAKF-QR is validated through maneuvering target tracking scenarios in both simulated and real-world data.

Aiming at tracking sharply maneuvering targets, this paper develops novel variational adaptive state estimators for joint target state and process noise parameter estimation for a class of linear state-space models with abruptly changing parameters. By combining variational inference with change-point detection in an online Bayesian fashion, two adaptive estimators—a change-point-based adaptive Kalman filter (CPAKF) and a change-point-based adaptive Kalman smoother (CPAKS)—are proposed in a recursive detection and estimation process. In each iteration, the run-length probability of the current maneuver mode is first calculated, and then the joint posterior of the target state and process noise parameter conditioned on the run length is approximated by variational inference. Compared with existing variational noise-adaptive Kalman filters, the proposed methods are robust to initial iterative value settings, improving their capability of tracking sharply maneuvering targets. Meanwhile, the change-point detection divides the non-stationary time sequence into several stationary segments, allowing for an adaptive sliding length in the CPAKS method. The tracking performance of the proposed methods is investigated using both synthetic and real-world datasets of maneuvering targets.

This paper presents a unified Lyapunov-based predefined-time stability theorem that includes three sufficient conditions. The standard theoretical analysis method for achieving predefined-time stability of non-linear systems using this theorem is provided within the framework of Lyapunov theory. The developed Lyapunov-based theorem facilitates the establishment of equivalence between the existing Lyapunov theorems concerning predefined-time stability. Furthermore, when the presented sufficient conditions are relaxed, the predefined-time stability conclusion for non-linear systems degenerates into a finite-time one. Consequently, a standard non-singular sliding mode control framework based on the unified Lyapunov-based theorem is developed for a Lagrangian system to ensure its predefined-time stability. Exemplary numerical simulation results are subsequently given, in order to illustrate the convergence behavior of the system states and confirm that the controlled systems are predefined-time stable.

This study is dedicated to address the predefined-time attitude tracking control problem of spacecraft in the presence of actuator fault, unmeasurable angular velocity, system uncertainties, and external disturbance. Invoking the attitude measurement feedback only, a model-free adaptive state observer is designed to estimate the unmeasurable angular velocity. Its estimation error is governed to be stable with predefined-time convergence. Using this estimation, a continuous and non-singular sliding surface-based predefined-time adaptive controller is synthesized. This controller does not have singularity and is free of chattering. It ensures the attitude tracking error to be stable with predefined-time convergence. The proposed controller's effectiveness is finally verified through numerical examples.