The Convergence of the Extended Kalman Filter

Convergence analysis of a family of robust Kalman filters based on the contraction principle

In this paper we analyze the convergence of a family of robust Kalman filters. For each filter of this family the model uncertainty is tuned according to the so called tolerance parameter. Assuming that the corresponding state-space model is reachable and observable, we show that the corresponding Riccati-like mapping is strictly contractive provided that the tolerance is sufficiently small, accordingly the filter converges.

Low-rank Kalman filtering under model uncertainty

We consider a robust filtering problem where the nominal state space model is not reachable and different from the actual one. We propose a robust Kalman filter which solves a dynamic game: one player selects the least-favorable model in a given ambiguity set, while the other player designs the optimum filter for the least-favorable model. It turns out that the robust filter is governed by a low-rank risk sensitive-like Riccati equation. Finally, simulation results show the effectiveness of the proposed filter.

Application of Extended Kalman Filter to Tactical Ballistic Missile Re-entry Problem

The objective is to investigate the advantages and performance of Extended Kalman Filter for the estimation of non-linear system where linearization takes place about a trajectory that was continually updated with the state estimates resulting from the measurement. Here tactile ballistic missile Re-entry problem is taken as a nonlinear system model and Extended Kalman Filter technique is used to estimate the positions and velocities at the X and Y direction at different values of ballistic coefficients. The result shows that the method gives better estimation with the increase of ballistic coe…

一种更新鲁棒的卡尔曼滤波方法

An update-resilient Kalman filtering approach

针对观测中存在模型不确定性的情况，提出一种新的鲁棒滤波框架，推导出更新鲁棒卡尔曼滤波器，并分析其稳定性和有效性。

Robust Kalman Filtering Under Model Uncertainty: the Case of Degenerate Densities

We consider a robust state space filtering problem in the case that the transition probability density is unknown and possibly degenerate. The resulting robust filter has a Kalman-like structure and solves a minimax game: the nature selects the least favorable model in a prescribed ambiguity set which also contains non-Gaussian probability densities, while the other player designs the optimum filter for the least favorable model. It turns out that the resulting robust filter is characterized by a Riccati-like iteration evolving on the cone of the positive semidefinite matrices. Moreover, we st…

Convergence of discrete time Kalman filter estimate to continuous time estimate

This article is concerned with the convergence of the state estimate obtained from the discrete time Kalman filter to the continuous time estimate as the temporal discretization is refined. We derive convergence rate estimates for different systems, first finite dimensional and then infinite dimensional with bounded or unbounded observation operators. Finally, we derive the convergence rate in the case where the system dynamics is governed by an analytic semigroup. The proofs are based on applying the discrete time Kalman filter on a dense numerable subset of a certain time interval $[0,T]$.

Distributed Estimation of Oscillations in Power Systems: an Extended Kalman Filtering Approach

Online estimation of electromechanical oscillation parameters provides essential information to prevent system instability and blackout and helps to identify event categories and locations. We formulate the problem as a state space model and employ the extended Kalman filter to estimate oscillation frequencies and damping factors directly based on data from phasor measurement units. Due to considerations of communication burdens and privacy concerns, a fully distributed algorithm is proposed using diffusion extended Kalman filter. The effectiveness of proposed algorithms is confirmed by both s…

使用卡尔曼滤波器进行状态估计的几何视角

A geometric perspective of state estimation using Kalman filters

本文探讨了卡尔曼滤波器中李群与黎曼方法的关系，基于仿射联络开发了一种泛化两者的新描述，并通过新库GeometricKalman.jl验证了其在状态估计中提供更大自由度的优势。

A New Kalman Filter Model for Nonlinear Systems Based on Ellipsoidal Bounding

In this paper, a new filter model called set-membership Kalman filter for nonlinear state estimation problems was designed, where both random and unknown but bounded uncertainties were considered simultaneously in the discrete-time system. The main loop of this algorithm includes one prediction step and one correction step with measurement information, and the key part in each loop is to solve an optimization problem. The solution of the optimization problem produces the optimal estimation for the state, which is bounded by ellipsoids. The new filter was applied on a highly nonlinear benchmark…

未知噪声协方差下有限时域卡尔曼滤波器的策略优化

Policy Optimization of Finite-Horizon Kalman Filter with Unknown Noise Covariance

本文将有限时域卡尔曼滤波转化为对偶系统策略优化问题，提出梯度估计与随机梯度下降法，并证明其在未知噪声协方差下的全局线性收敛性和样本复杂度。

The Extended Kalman Filter is a Natural Gradient Descent in Trajectory Space

The extended Kalman filter is perhaps the most standard tool to estimate in real time the state of a dynamical system from noisy measurements of some function of the system, with extensive practical applications (such as position tracking via GPS). While the plain Kalman filter for linear systems is well-understood, the extended Kalman filter relies on linearizations which have been debated. We recover the exact extended Kalman filter equations from first principles in statistical learning: the extended Kalman filter is equal to Amari's online natural gradient, applied in the space of trajecto…

Online Natural Gradient as a Kalman Filter

We cast Amari's natural gradient in statistical learning as a specific case of Kalman filtering. Namely, applying an extended Kalman filter to estimate a fixed unknown parameter of a probabilistic model from a series of observations, is rigorously equivalent to estimating this parameter via an online stochastic natural gradient descent on the log-likelihood of the observations. In the i.i.d. case, this relation is a consequence of the "information filter" phrasing of the extended Kalman filter. In the recurrent (state space, non-i.i.d.) case, we prove that the joint Kalman filter over states a…

On the Projective Geometry of Kalman Filter

Convergence of the Kalman filter is best analyzed by studying the contraction of the Riccati map in the space of positive definite (covariance) matrices. In this paper, we explore how this contraction property relates to a more fundamental non-expansiveness property of filtering maps in the space of probability distributions endowed with the Hilbert metric. This is viewed as a preliminary step towards improving the convergence analysis of filtering algorithms over general graphical models.

Eigen Value Analysis in Lower Bounding Uncertainty of Kalman Filter Estimates

In this paper we are concerned with the error-covariance lower-bounding problem in Kalman filtering: a sensor releases a set of measurements to the data fusion/estimation center, which has a perfect knowledge of the dynamic model, to allow it to estimate the states, while preventing it to estimate the states beyond a given accuracy. We propose a measurement noise manipulation scheme to ensure lower-bound on the estimation accuracy of states. Our proposed method ensures lower-bound on the steady state estimation error of Kalman filter, using mathematical tools from eigen value analysis.

当代立方体求积法和无迹卡尔曼滤波方法是否总是优于传统扩展卡尔曼滤波？

Do the Contemporary Cubature and Unscented Kalman Filtering Methods Outperform Always the Traditional Extended Kalman Filter?

本文探讨了在存在刚性行为的随机系统中，传统扩展卡尔曼滤波可能优于当代方法的问题。

比较卡尔曼滤波器与观测器用于存在模型不确定性及恶意网络攻击的电力系统动态状态估计

Comparing Kalman Filters and Observers for Power System Dynamic State Estimation with Model Uncertainty and Malicious Cyber Attacks

本文比较了卡尔曼滤波器与观测器在处理电力系统动态状态估计中的模型不确定性及网络攻击问题，通过测试发现CKF和观测器在鲁棒性上更优。

Optimal Sensing Precision in Ensemble and Unscented Kalman Filtering

We consider the problem of selecting an optimal set of sensor precisions to estimate the states of a non-linear dynamical system using an Ensemble Kalman filter and an Unscented Kalman filter, which uses random and deterministic ensembles respectively. Specifically, the goal is to choose at run-time, a sparse set of sensor precisions for active-sensing that satisfies certain constraints on the estimated state covariance. In this paper, we show that this sensor precision selection problem is a semidefinite programming problem when we use l1 norm over precision vector as the surrogate measure to…

面向不确定动力系统的风险规避确定性卡尔曼滤波器

Risk averse deterministic Kalman filters for uncertain dynamical systems

针对含参数不确定性的动力系统，从确定性视角扩展Kalman-Bucy滤波器，提出可降低大重构误差概率的风险规避设计，推导误差界并通过数值算例对比两类估计器性能。

Utility and Privacy in Object Tracking from Video Stream using Kalman Filter

Tracking objects in Computer Vision is a hard problem. Privacy and utility concerns adds an extra layer of complexity over this problem. In this work we consider the problem of maintaining privacy and utility while tracking an object in a video stream using Kalman filtering. Our first proposed method ensures that the localization accuracy of this object will not improve beyond a certain level. Our second method ensures that the localization accuracy of the same object will always remain under a certain threshold.

可逆卡尔曼滤波器在真实数据上验证方法的实证研究

Validation methodology on real data of reversible Kalman Filter for state estimation with Manifold

本文提出了一种在真实数据中验证可逆卡尔曼滤波器准确性的方法，通过事件检测步骤切换滤波器类型以提高性能。

Power System Dynamic State Estimation by Unscented Kalman Filter with Guaranteed Positive Semidefinite State Covariance

In this paper an unscented Kalman filter with guaranteed positive semidefinite state covariance is proposed by calculating the nearest symmetric positive definite matrix in Frobenius norm and is applied to power system dynamic state estimation. The proposed method is tested on NPCC 48-machine 140-bus system and the results validate its effectiveness.

Invariant extended Kalman filter on matrix Lie groups

We derive symmetry preserving invariant extended Kalman filters (IEKF) on matrix Lie groups. These Kalman filters have an advantage over conventional extended Kalman filters as the error dynamics for such filters are independent of the group configuration which, in turn, provides a uniform estimate of the region of convergence. The proposed IEKF differs from existing techniques in literature on the account that it is derived using minimal tools from differential geometry that simplifies its representation and derivation to a large extent. The filter error dynamics is defined on the Lie algebra…

使用Oja主成分流的低秩近似卡尔曼滤波器用于离散时间线性系统

Low-rank approximated Kalman filter using Oja's principal component flow for discrete-time linear systems

本文针对离散时间线性系统提出基于Oja主成分流的低秩近似卡尔曼滤波器，给出有界均方误差估计的系统理论条件，并通过数值仿真验证其有效性。

Kalman Filter Meets Subjective Logic: A Self-Assessing Kalman Filter Using Subjective Logic

Self-assessment is a key to safety and robustness in automated driving. In order to design safer and more robust automated driving functions, the goal is to self-assess the performance of each module in a whole automated driving system. One crucial component in automated driving systems is the tracking of surrounding objects, where the Kalman filter is the most fundamental tracking algorithm. For Kalman filters, some classical online consistency measures exist for self-assessment, which are based on classical probability theory. However, these classical approaches lack the ability to measure t…

低秩卡尔曼滤波器的几何结构

The geometry of low-rank Kalman filters

研究低秩卡尔曼滤波器的几何性质，通过合适的度量分析低秩Riccati流的收缩特性。

Two novel costs for determining the tuning parameters of the Kalman Filter

The Kalman filter (KF) and the extended Kalman filter (EKF) are well established techniques for state estimation. However, the choice of the filter tuning parameters still poses a major challenge for the engineers [1]. In the present work, two new costs have been proposed for determining the filter tuning parameters on the basis of the innovation covariance. This provides a cost function based method for the selection of suitable combination(s) of filter tuning parameters in order to ensure the design of a KF or an EKF having an optimally balanced RMSE performance. Index Terms-Kalman filter, t…

Probabilistic Performance Bounds for Randomized Sensor Selection in Kalman Filtering

We consider the problem of randomly choosing the sensors of a linear time-invariant dynamical system subject to process and measurement noise. We sample the sensors independently and from the same distribution. We measure the performance of a Kalman filter by its estimation error covariance. Using tools from random matrix theory, we derive probabilistic bounds on the estimation error covariance in the semi-definite sense. We indirectly improve the performance of our Kalman filter for the maximum eigenvalue metric and show that under certain conditions the optimal sampling distribution that min…

The Kalman Like Particle Filter : Optimal Estimation With Quantized Innovations/Measurements

We study the problem of optimal estimation and control of linear systems using quantized measurements, with a focus on applications over sensor networks. We show that the state conditioned on a causal quantization of the measurements can be expressed as the sum of a Gaussian random vector and a certain truncated Gaussian vector. This structure bears close resemblance to the full information Kalman filter and so allows us to effectively combine the Kalman structure with a particle filter to recursively compute the state estimate. We call the resulting filter the Kalman like particle filter (KLP…

An Explicit Probabilistic Derivation of Inflation in a Scalar Ensemble Kalman Filter for Finite Step, Finite Ensemble Convergence

This paper uses a probabilistic approach to analyze the converge of an ensemble Kalman filter solution to an exact Kalman filter solution in the simplest possible setting, the scalar case, as it allows us to build upon a rich literature of scalar probability distributions and non-elementary functions. To this end we introduce the bare-bones Scalar Pedagogical Ensemble Kalman Filter (SPEnKF). We show that in the asymptotic case of ensemble size, the expected value of both the analysis mean and variance estimate of the SPEnKF converges to that of the true Kalman filter, and that the variances of…

平均场集合卡尔曼滤波器：近高斯设定

The Mean Field Ensemble Kalman Filter: Near-Gaussian Setting

本文首次分析了平均场集合卡尔曼滤波器在高斯设定之外的精度，通过引入加权全变差度量证明其稳定性，并证明在高斯问题邻域内误差很小。

一种用于计算 Kalman 滤波器鲁棒裕度的凸优化框架

A Convex Optimization Framework for Computing Robustness Margins of Kalman Filters

该研究提出联合确定 Kalman 增益与噪声鲁棒裕度的凸优化框架，可在保证指定稳态误差的同时最大化过程和传感器噪声，并通过航天器与 F-16 模型验证。

集合卡尔曼滤波器的贝叶斯递归更新

Bayesian Recursive Update for Ensemble Kalman Filters

本文提出贝叶斯递归更新滤波器（BRUF）及其集合版本 BRUEnKF，通过将非线性测量更新划分为多步递归过程，放宽 EKF 的线性假设，并在高维系统中验证其优于传统 EnKF。

On Computational Complexity Reduction Methods for Kalman Filter Extensions

The Kalman filter and its extensions are used in a vast number of aerospace and navigation applications for nonlinear state estimation of time series. In the literature, different approaches have been proposed to exploit the structure of the state and measurement models to reduce the computational demand of the algorithms. In this tutorial, we survey existing code optimization methods and present them using unified notation that allows them to be used with various Kalman filter extensions. We develop the optimization methods to cover a wider range of models, show how different structural optim…

On-Line Learning of Linear Dynamical Systems: Exponential Forgetting in Kalman Filters

Kalman filter is a key tool for time-series forecasting and analysis. We show that the dependence of a prediction of Kalman filter on the past is decaying exponentially, whenever the process noise is non-degenerate. Therefore, Kalman filter may be approximated by regression on a few recent observations. Surprisingly, we also show that having some process noise is essential for the exponential decay. With no process noise, it may happen that the forecast depends on all of the past uniformly, which makes forecasting more difficult. Based on this insight, we devise an on-line algorithm for improp…

A Quantum Extended Kalman Filter

A stochastic filter uses a series of measurements over time to produce estimates of unknown variables based on a dynamic model. For a quantum system, such an algorithm is provided by a quantum filter, which is also known as a stochastic master equation (SME). For a linear quantum system subject to linear measurements and Gaussian noise, the quantum filter reduces to a quantum Kalman filter. In this article, we introduce a quantum extended Kalman filter (quantum EKF), which applies a commutative approximation and a time-varying linearization to non-commutative quantum stochastic differential eq…

Learning the Kalman Filter with Fine-Grained Sample Complexity

We develop the first end-to-end sample complexity of model-free policy gradient (PG) methods in discrete-time infinite-horizon Kalman filtering. Specifically, we introduce the receding-horizon policy gradient (RHPG-KF) framework and demonstrate $\tilde{\mathcal{O}}(ε^{-2})$ sample complexity for RHPG-KF in learning a stabilizing filter that is $ε$-close to the optimal Kalman filter. Notably, the proposed RHPG-KF framework does not require the system to be open-loop stable nor assume any prior knowledge of a stabilizing filter. Our results shed light on applying model-free PG methods to control…

卡尔曼滤波器的一种简单推导

A simple derivation of the Kalman filter

本讲义基于最小二乘估计基础，给出离散时间卡尔曼滤波器方程的简洁自包含推导，兼顾严谨性与通用性，减少数学开销。