In this paper, we present a new nonlinear filter for high-dimensional state estimation, which we have named the cubature Kalman filter (CKF) The heart of the CKF is a spherical-radial cubature rule, which makes it possible to numerically compute multivariate moment integrals encountered in the nonlinear Bayesian filter Specifically, we derive a third-degree spherical-radial cubature rule that provides a set of cubature points scaling linearly with the state-vector dimension The CKF may therefore provide a systematic solution for high-dimensional nonlinear filtering problems The paper also includes the derivation of a square-root version of the CKF for improved numerical stability The CKF is tested experimentally in two nonlinear state estimation problems In the first problem, the proposed cubature rule is used to compute the second-order statistics of a nonlinearly transformed Gaussian random variable The second problem addresses the use of the CKF for tracking a maneuvering aircraft The results of both experiments demonstrate the improved performance of the CKF over conventional nonlinear filters

/pdf/cubature-kalman-filters-13gbzxhdp7.pdf

Cubature Kalman Filters

We explore in this chapter questions of existence and uniqueness for solutions to stochastic differential equations and offer a study of their properties. This endeavor is really a study of diffusion processes. Loosely speaking, the term diffusion is attributed to a Markov process which has continuous sample paths and can be characterized in terms of its infinitesimal generator.

Stochastic Differential Equations

Now in its second edition, this accessible text presents a unified Bayesian treatment of state-of-the-art filtering, smoothing, and parameter estimation algorithms for non-linear state space models. The book focuses on discrete-time state space models and carefully introduces fundamental aspects related to optimal filtering and smoothing. In particular, it covers a range of efficient non-linear Gaussian filtering and smoothing algorithms, as well as Monte Carlo-based algorithms. This updated edition features new chapters on constructing state space models of practical systems, the discretization of continuous-time state space models, Gaussian filtering by enabling approximations, posterior linearization filtering, and the corresponding smoothers. Coverage of key topics is expanded, including extended Kalman filtering and smoothing, and parameter estimation. The book's practical, algorithmic approach assumes only modest mathematical prerequisites, suitable for graduate and advanced undergraduate students. Many examples are included, with Matlab and Python code available online, enabling readers to implement algorithms in their own projects.

Bayesian Filtering and Smoothing

Filtering and smoothing methods are used to produce an accurate estimate of the state of a time-varying system based on multiple observational inputs (data). Interest in these methods has exploded in recent years, with numerous applications emerging in fields such as navigation, aerospace engineering, telecommunications, and medicine. This compact, informal introduction for graduate students and advanced undergraduates presents the current state-of-the-art filtering and smoothing methods in a unified Bayesian framework. Readers learn what non-linear Kalman filters and particle filters are, how they are related, and their relative advantages and disadvantages. They also discover how state-of-the-art Bayesian parameter estimation methods can be combined with state-of-the-art filtering and smoothing algorithms. The book’s practical and algorithmic approach assumes only modest mathematical prerequisites. Examples include MATLAB computations, and the numerous end-of-chapter exercises include computational assignments. MATLAB/GNU Octave source code is available for download at www.cambridge.org/sarkka, promoting hands-on work with the methods.

贝叶斯滤波与平滑 (Bayesian filtering and smoothing)

With the rapid development of new energy electric vehicles and smart grids, the demand for batteries is increasing. The battery management system (BMS) plays a crucial role in the battery-powered energy storage system. This paper presents a systematic review of the most commonly used battery modeling and state estimation approaches for BMSs. The models include the physics-based electrochemical models, the integral and fractional order equivalent circuit models, and data-driven models. The state estimation approaches are analyzed from the perspectives of remaining capacity and energy estimation, power capability prediction, lifespan and health prognoses, and other crucial indexes in BMS. This present paper, through the analysis of literature, includes almost all states in the BMS. The estimation approaches of state-of-charge (SOC), state-of-energy (SOE), state-of-power (SOP), state-of-function (SOF), state-of-health (SOH), remaining useful life (RUL), remaining discharge time (RDT), state-of-balance (SOB), and state-of-temperature (SOT) are reviewed and discussed in a systematical way. Moreover, the challenges and outlooks of the research on future battery management are disclosed, in the hope of providing some inspirations to the development and design of the next-generation BMSs.

A comprehensive review of battery modeling and state estimation approaches for advanced battery management systems

In this paper, a new version of the quadrature Kalman filter (QKF) is developed theoretically and tested experimentally. We first derive the new QKF for nonlinear systems with additive Gaussian noise by linearizing the process and measurement functions using statistical linear regression (SLR) through a set of Gauss-Hermite quadrature points that parameterize the Gaussian density. Moreover, we discuss how the new QKF can be extended and modified to take into account specific details of a given application. We then go on to extend the use of the new QKF to discrete-time, nonlinear systems with additive, possibly non-Gaussian noise. A bank of parallel QKFs, called the Gaussian sum-quadrature Kalman filter (GS-QKF) approximates the predicted and posterior densities as a finite number of weighted sums of Gaussian densities. The weights are obtained from the residuals of the QKFs. Three different Gaussian mixture reduction techniques are presented to alleviate the growing number of the Gaussian sum terms inherent to the GS-QKFs. Simulation results exhibit a significant improvement of the GS-QKFs over other nonlinear filtering approaches, namely, the basic bootstrap (particle) filters and Gaussian-sum extended Kalman filters, to solve nonlinear non- Gaussian filtering problems.

http://soma.mcmaster.ca/papers/Paper_2.pdf

Discrete-Time Nonlinear Filtering Algorithms Using Gauss–Hermite Quadrature

In this paper, we extend the cubature Kalman filter (CKF) to deal with nonlinear state-space models of the continuous-discrete kind. To be consistent with the literature, the resulting nonlinear filter is referred to as the continuous-discrete cubature Kalman filter (CD-CKF). We use the Ito-Taylor expansion of order 1.5 to transform the process equation, modeled in the form of stochastic ordinary differential equations, into a set of stochastic difference equations. Building on this transformation and assuming that all conditional densities are Gaussian-distributed, the solution to the Bayesian filter reduces to the problem of how to compute Gaussian-weighted integrals. To numerically compute the integrals, we use the third-degree cubature rule. For a reliable implementation of the CD-CKF in a finite word-length machine, it is structurally modified to propagate the square-roots of the covariance matrices. The reliability and accuracy of the square-root version of the CD-CKF are tested in a case study that involves the use of a radar problem of practical significance; the problem considered herein is challenging in the context of radar in two respects- high dimensionality of the state and increasing degree of nonlinearity. The results, presented herein, indicate that the CD-CKF markedly outperforms existing continuous-discrete filters.

/pdf/cubature-kalman-filtering-for-continuous-discrete-systems-1gyt9q5czx.pdf

Cubature Kalman Filtering for Continuous-Discrete Systems: Theory and Simulations

Discrete-Time Nonlinear Filtering Algorithms Using Gauss-Hermite Quadrature New computationally efficient methods are proposed for more accurately analyzing and modeling dynamic processes that are nonlinear and subject to non-Gaussian noise.

The quadrature Kalman filter (QKF) is a recursive, nonlinear filtering algorithm developed in the Kalman filtering framework It computes the mean and covariance of all conditional densities using the Gauss-Hermite quadrature rule In this correspondence, we develop a square-root extension of the quadrature Kalman filter using matrix triangularizations The square-root quadrature Kalman filter (SQKF) propagates the mean and the square root of the covariance Although equivalent to the QKF algebraically, the SQKF exhibits excellent numerical characteristics, but at the expense of increased computational complexity We also present possible refinements of the generic SQKF

Square-Root Quadrature Kalman Filtering

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