New design of linear discrete-time predictor using covariance information

A discrete one-stage predictor algorithm using covariance information in linear systems is derived. The algorithm is obtained for white Gaussian observation noise. The signal is a nonstationary or stationary stochastic process. The auto-covariance function of the signal is expressed using a semi-degenerate kernel of discrete-time systems. The semi-degenerate kernel can represent general covariance functions of random processes by a finite sum of nonrandom functions.     

Initial-value system for linear smoothing problems by covariance information

A Cauchy system suitable for real-time calculation of the solution of integral equations with semi-degenerate kernels of the linear least squares smoothing problems for nonstationary signal process is developed using invariant imbedding theory. Digital simulation results indicate that the algorithms presented are leasible.     

New prediction algorithms by covariance information based on innovation theory

This paper states a new design method of recursive predictor and filter based on the innovations theory, using signal and noise covariance information, for white Gaussian and white Gaussian + coloured observation noises. The derived prediction and filtering algorithms estimate stationary stochastic signal processes.The digital simulation results indicate that the algorithms presented are feasible.     

噪声统计特性未知时的鲁棒卡尔曼滤波算法设计

本文研究了噪声统计特性未知时的鲁棒卡尔曼滤波算法(RKF)设计问题.首先,提出了一种新的RKF算法设计条件,并分析了其合理性;其次,从RKF算法设计条件出发研究了RKF算法的设计问题,把RKF算法的设计过程转化为计算一组线性矩阵不等式(LMI)的可行解;再次,研究了LMI可行解的计算问题,并通过计算该LMI的可行解设计了一种RKF算法;最后,通过仿真验证了所设计的RKF算法的有效性.     

Least squares estimation of nonstationary covariance parameters in linear systems

Least squares estimation techniques are employed to overcome previous difficulties encountered in accurately estimating the state and measurement noise covariance parameters in linear stochastic systems. In the past accurate and rapidly converging covariance parameter estimates have been achieved with complex estimation algorithms only after specifying the statistical nature of the noise in the system and constraining the time variation of the covariance parameters. Weighted least squares estimation allows these restrictions to be removed while achieving near optimal accuracy using a filter on the same order of complexity as a Kalman filter. Allowing the covariance parameters to vary in as general a manner in time as the state in a linear discrete time stochastic system, and assuming that a Kalman filter is applied to this system using incorrect knowledge of the a priori statistics, it is shown how a covariance system is developed similar to the original system. Unbiased least squares estimates of the covariance parameters and of the original state are obtained without the necessity of specifying the distribution on the noise in either system. The accuracy of these estimates approaches optimal accuracy with increasing measurements when adaptive Kalman filters are applied to each system.     

A separation theorem for stochastic control problems with point-process observations

The exact solution is derived for a stochastic optimal control problem involving a linear stochastic plant, quadratic costs, and nonlinear, nongaussian observations. The observations are in the form of a point process in which each point has both a temporal and a spatial coordinate. The state of the stochastic plant influences the intensity of the observed time-space point process. The solution to this dual control problem can be realized with a separated estimator-controller in which the estimator is nonlinear, mean-square optimal, and finite dimensional, and the controller is the certainty equivalent linear controller. Motivation for the stochastic optimal control problem studied here is given in terms of position sensing and tracking for quantum-limited optical communication problems.     

The problem of optimal robust Kalman state estimation via limited capacity digital communication channels

This paper considers a state estimation problem for a continuous-time uncertain system via a digital communication channel with bit-rate constraints. The estimated state must be quantized, coded and transmitted via a limited capacity digital communication channel. Optimal and suboptimal recursive coder–decoder state estimation schemes are proposed and investigated.     

New Kalman filter and smoother consistency tests

We derive three new tests that can be applied to a Kalman filter to check for inconsistencies. The Filter Residual Test can detect observations that are outliers but would be missed by a basic residual test because the uncertainty of the expected observation is large relative to the uncertainty of the observation. The Smoother Residual Test uses the output from a Modified Bryson–Frazier (MBF) smoother to detect observations that are outliers. The Smoother State Test compares the state estimates from the filter and MBF smoother to detect model inconsistencies, in particular insufficient process noise.     

Identification of process and measurement noise covariance for state and parameter estimation using extended Kalman filter

The performance of Bayesian state estimators, such as the extended Kalman filter (EKF), is dependent on the accurate characterisation of the uncertainties in the state dynamics and in the measurements. The parameters of the noise densities associated with these uncertainties are, however, often treated as ‘tuning parameters’ and adjusted in an ad hoc manner while carrying out state and parameter estimation. In this work, two approaches are developed for constructing the maximum likelihood estimates (MLE) of the state and measurement noise covariance matrices from operating input-output data when the states and/or parameters are estimated using the EKF. The unmeasured disturbances affecting the process are either modelled as unstructured noise affecting all the states or as structured noise entering the process predominantly through known, but unmeasured inputs. The first approach is based on direct optimisation of the ML objective function constructed by using the innovation sequence generated from the EKF. The second approach - the extended EM algorithm - is a derivative-free method, that uses the joint likelihood function of the complete data, i.e. states and measurements, to compute the next iterate of the decision variables for the optimisation problem. The efficacy of the proposed approaches is demonstrated on a benchmark continuous fermenter system. The simulation results reveal that both the proposed approaches generate fairly accurate estimates of the noise covariances. Experimental studies on a benchmark laboratory scale heater-mixer setup demonstrate a marked improvement in the predictions of the EKF that uses the covariance estimates obtained from the proposed approaches.     

基于Kalman滤波的统一的Wiener状态滤波器

对于带相关噪声系统 ,基于稳态Kalman滤波器和自回归滑动平均 (ARMA)新息模型 ,提出了统一的渐近稳定的Wiener状态滤波器 ,可统一处理状态滤波 ,平滑和预报问题 .它们构成了一种新的时域Wiener滤波算法 .揭示了Kalman滤波器与Wiener滤波器之间的关系 .一个目标跟踪系统的仿真例子说明了它们的有效性     

Numerical comparison of kalman filter algorithms: Orbit determination case study

Numerical characteristics of various Kalman filter algorithms are illustrated with a realistic orbit determination study. The case study of this paper highlights the numerical deficiencies of the conventional and stabilized Kalman algorithms. Computational errors associated with these algorithms are found to be so large as to obscure important mismodeling effects and thus cause misleading estimates of filter accuracy. The positive result of this study is that the U-D covariance factorization algorithm has excellent numerical properties and is computationally efficient, having CPU costs that differ negligibly from the conventional Kalman costs. Accuracies of the U-D filter using single precision arithmetic consistently match the double precision reference results. Numerical stability of the U-D filter is further demonstrated by its insensitivity to variations in the a priori statistics.     

广义容积卡尔曼滤波

传统CKF采用三阶球面径向容积定律来计算非线性积分,该定律将球面数值积分与径向积分相结合,难以构造高阶CKF算法。此外,CKF在许多非线性问题上表现出估计精度低等问题。为了解决以上问题,提出了一种广义CKF族,所提算法彻底抛弃了球面径向积分定律。进一步指出,传统CKF是这种滤波算法的特殊形式。实验结果表明,高阶CKF比传统的非线性滤波器准确性更高。     

多传感器时滞系统信息融合最优Kalman滤波器

基于线性最小方差最优加权融合估计算法,对多传感器的离散线性状态时滞随机系统,给出了一种非增广分布式加权融合最优Kalman滤波器.推导了状态时滞系统任两个传感器子系统之间的滤波误差互协方差阵的计算公式.它与状态增广加权融合滤波器具有相同的精度.与每个传感器的局部滤波器相比,分布式融合滤波器具有更高的精度.与状态和观测增广最优滤波器相比,具有较小的精度.但避免了增广所带来的高维计算和大的空间存储,可减小计算负担.仿真例子验证了其有效性.     

Nonlinear Bayesian Estimation: From Kalman Filtering to a Broader Horizon

This article presents an up-to-date tutorial review of nonlinear Bayesian estimation. State estimation for nonlinear systems has been a challenge encountered in a wide range of engineering fields, attracting decades of research effort. To date, one of the most promising and popular approaches is to view and address the problem from a Bayesian probabilistic perspective, which enables estimation of the unknown state variables by tracking their probabilistic distribution or statistics (e.g., mean and covariance) conditioned on a system's measurement data. This article offers a systematic introduction to the Bayesian state estimation framework and reviews various Kalman filtering (KF) techniques, progressively from the standard KF for linear systems to extended KF, unscented KF and ensemble KF for nonlinear systems. It also overviews other prominent or emerging Bayesian estimation methods including Gaussian filtering, Gaussian-sum filtering, particle filtering and moving horizon estimation and extends the discussion of state estimation to more complicated problems such as simultaneous state and parameter/input estimation.      

基于渐消卡尔曼滤波算法的航空发动机参数估计方法

针对航空发动自适应模型误差无法完全消除,可能导致参数估计结果严重偏离甚至滤波发散的问题,提出一种带渐消因子的卡尔曼参数估计方法,采用在线调整卡尔曼方程残差的权重、加强现实测量数据在状态估计中作用的策略,保证了发动机性能参数估计的准确性。仿真结果表明,该方法不仅克服了滤波发散现象,具有更优的收敛速度和估计精度,且计算量小,实现简单,便于实际应用     

A novel narrow band digital filter and its application to multivariable system identification

A novel narrow band time-varying digital filter is proposed, which has desirable properties such as global asymptotic stability, asymptotic noise annihilation and asymptotic signal tracking. It is shown that the proposed filter is comparable to the Kalman filter in performance, but with substantial computational simplicity; no Ricatti equation is involved. It is basically a Fourier analysis method but the Fourier coefficients are found recursively. The application of the proposed filter for on-line identification of a linear multivariable system subject to both deterministic and stochastic disturbances is presented; simulation results are given.     

Design and analysis of discrete-time robust Kalman filters

In this paper, the problem of finite and infinite horizon robust Kalman filtering for uncertain discrete-time systems is studied. The system under consideration is subject to time-varying norm-bounded parameter uncertainty in both the state and output matrices. The problem addressed is the design of linear filters having an error variance with an optimized guaranteed upper bound for any allowed uncertainty. A novel technique is developed for robust filter design. This technique gives necessary and sufficient conditions to the design of robust quadratic filters over finite and infinite horizon in terms of a pair of parameterized Riccati equations. Feasibility and convergence properties of the robust quadratic filters are also analyzed.     

A maximum-likelihood Kalman filter for switching discrete-time linear systems

State estimation is addressed for a class of discrete-time systems that may switch among different modes taken from a finite set. The system and measurement equations of each mode are assumed to be linear and perfectly known, but the current mode of the system is unknown. Moreover, additive, independent, normally distributed noises are assumed to affect the dynamics and the measurements. First, relying on a well-established notion of mode observability developed “ad hoc” for switching systems, an approach to system mode estimation based on a maximum-likelihood criterion is proposed. Second, such a mode estimator is embedded in a Kalman filtering framework to estimate the continuous state. Under the unique assumption of mode observability, stability properties in terms of boundedness of the mean square estimation error are proved for the resulting filter. Simulation results showing the effectiveness of the proposed filter are reported.     

Iterative learning Kalman filter for repetitive processes

In this paper, a discrete-time iterative learning Kalman filter scheme is proposed for repetitive processes to reject repeatable disturbances as well as random noises. The proposed state estimator scheme integrates Kalman filter with iterative learning control. The estimation process contains two stages: a conventional Kalman filter is applied in the first stage; the second stage refines the estimates in an iterative learning fashion, leading to a gradual improvement on the estimation performance. According to the estimates that the first stage feeds to the second stage, the optimal design includes two types – posterior type and priori type. In order to reduce the memory and computation load of the optimal design, two suboptimal estimators are provided as well. The stability of the both suboptimal estimators is also studied. Furthermore, a lower bound is given to estimate the ultimate estimation performance before implementing any estimation. Finally, an illustrative example of injection molding is given to verify the performance of the four estimators developed.     

An iterated fuzzy extended Kalman filter for nonlinear systems

The fuzzy extended Kalman filter (FEKF) for state estimation can be used to deal with fuzzy uncertainty effectively. However, the linearisation processing of the FEKF introduces truncation error, which degrades the estimation precision. In order to reduce the error, a new iterated fuzzy extended Kalman filter (IFEKF), based on the FEKF and the maximum a posteriori estimation, is proposed in this article. Compared with the FEKF, the proposed algorithm can be used not only to deal with the fuzzy uncertainty, but also to reduce the truncation error and to estimate the states more accurately. With an algebraic example and a passive location simulation, it is shown that the IFEKF has better estimation precision than that of the FEKF.