Hanyu Liu

Gaussian filters with deterministic sample points, such as the Unscented Kalman Filter (UKF), Cubature Kalman Filter (CKF), Gauss– Hermite Quadrature Filter (GHQF), etc., have been widely employed for nonlinear state estimation. However, these filters utilize a fixed set of sample points irrespective of the system's nonlinearity. While various studies have explored data-driven approaches to optimize the three parameters of the UKF, these methods do not generalize well to other Gaussian filters with a greater number of sample points. In this letter, we propose a novel neural network-based unified framework for finetuning sample points across all Gaussian filters with deterministic sample points. Specifically, we first pretrain a Multi-Layer Perceptron (MLP) to approximate the mapping from the state's mean and covariance to the sample points of the original Gaussian filter. The MLP then replaces the sample points generation strategy in the Gaussian filter and is further refined by maximizing the marginal likelihood of the observed measurement data. Simulation results demonstrate that the Gaussian filters leveraging the well-trained MLP as the sample point generation strategy achieve higher filtering accuracy compared to their original counterparts.

State estimation for nonlinear models has been a longstanding challenge in the field of signal processing. Classical nonlinear filters, such as the extended Kalman filter (EKF), unscented Kalman filter (UKF), and particle filter (PF), often struggle to balance accuracy and computational cost in highly nonlinear models. Inspired by the property that data-driven methods trained offline in advance are generally efficient during inference phrase, we propose a physics-informed Data-driven Autoregressive nonlinear Filter (DAF) based on the Transformer architecture. During inference phrase, the DAF relies entirely on the neural network and does not retain any structure from classical nonlinear filters. Simulation results demonstrate that the DAF not only achieves higher accuracy than classical nonlinear filters without requiring access to ground-truth state values but also offers competitive computational efficiency.

The unscented Kalman filter (UKF) has demonstrated its effectiveness for state estimation in highly nonlinear systems over the past two decades. However, the UKF's assumption of a single Gaussian probability density function (PDF) adversely affects its filtering accuracy. To address this limitation, this paper proposes a variant of the UKF, referred to as MaxUKF, which enhances estimation accuracy by incorporating the maximum point of the PDF, denoting the point getting maximum value of the PDF. First, two mathematical theorems are presented, providing direct ways to accurately predict the maximum point of the PDF of the state after nonlinear transformation. Subsequently, a Gaussian sum PDF is constructed to accurately match the mean, covariance, and maximum point of the PDF after nonlinear transformation. Then, the MaxUKF algorithm is presented in a manner analogous to the UKF, following a two-step process: time-update step and measurement-update step, during which the non-Gaussian PDF of the state is represented by the constructed Gaussian sum PDF. Finally, the effectiveness of the proposed MaxUKF is verified through a series of numerical simulations. The results demonstrate that the MaxUKF algorithm offers higher estimation accuracy compared to the conventional UKF while requiring similar computational time.

The unscented Kalman filter (UKF) finds extensive application in the state estimation of systems characterized by significant nonlinearity. A constraint of the UKF is its presumption that the probability density function (PDF) of the states maintains Gaussian distribution throughout the filter recursion, which restricts the estimation accuracy even though the UKF may still be applicable when the Gaussian assumption is not strictly met. To overcome this problem, a skewed unscented Kalman filter (SUKF) based on Gaussian sum is presented in this paper. First, as the theoretical basis, the conventional UKF is reviewed. Then, the SUKF algorithm is presented in a manner analogous to the UKF, following a two-step process: In time-update step, an approximation method, employing no additional sigma points compared to the UKF algorithm, has been proposed to obtain the skewness of the random variable after nonlinear transformation, which approximates the Taylor series of skewness up to fourth order terms. In measurement-update step, a Gaussian sum PDF matching the known mean, covariance, and skewness is constructed to represent the non-Gaussian joint PDF of states and measurements. Finally, taking one nonlinear transformation and three nonlinear systems with different dimensions as examples, the effectiveness of the proposed SUKF is verified through numerical simulations The results demonstrate that the SUKF algorithm offers higher estimation accuracy compared to the conventional UKF while requiring similar computational time, which can provide a practical option for state estimation for highly nonlinear systems.

In this work, an efficient multi-stage constant-vector thrust control algorithm is proposed for spacecraft to achieve finite-thrust Lambert transfer considering J ₂ perturbation. In contrast to most designs that apply continuous-thrust requiring time-varying direction and the magnitude of the thrust, the proposed control scheme utilizes a multi-stage constant-vector thrust which means the modulus and direction of the thrust are constant in each stage determined by a given set of time nodes dividing the whole transfer time, which reduces the control complexity of the engine. Based on the multi-stage sensitivity matrix, which describes the first-order relationship between the thrust and the change of the orbital elements, a rapid algorithm is presented to obtain the multi-stage constant-vector thrust solution with the required accuracy. Furthermore, the optimization problem of the multi-stage constant-vector thrust solution is established, and the direct optimization method is proposed to obtain an optimal multi-stage constant-vector thrust solution for a given set of time nodes. Based on this approach, the node-sequence optimization method is further proposed to obtain a series of feasible optimized solutions rapidly corresponding to a monotonically increasing time-node sets. Compared with the existing finite-thrust methods, the presented methods balance the fuel consumption and control requirements of the engine because the situation that the engine always needs to change the direction and the magnitude of the thrust could be avoided, which are practical options for engineering applications.

Compared with a conventional unscented Kalman filter (UKF), the recently proposed marginalized unscented Kalman filter (MUKF) uses a partially sampling strategy to achieve similar filter accuracy with fewer sigma points, demonstrating its powerful ability to deal with state estimation with mixed linearity and nonlinearity. However, the hypothesis that the accuracy of MUKF is equivalent to that of UKF is not true for all systems. In this paper, it is proved that when the state equation is expressed as a differential equation, the accuracy of MUKF is equivalent to that of UKF only when the propagations of the states to be sampled and the states not to be sampled in MUKF are independent of each other. The above condition corresponds to a common problem in engineering, which is a system with measurement biases. Through theoretical proof and numerical simulation, it is verified in this paper that for the systems with measurement biases, the differences of the computed means and covariances in filtering recursions between MUKF and UKF are restricted to the same order infinitesimal as the square of the scaling parameter. To sum up, this paper evaluates the accuracy of MUKF, providing references for engineers to choose MUKF or UKF in different systems.