基于无味卡尔曼滤波的多雷达方位配准算法

将无味卡尔曼滤波（Unscented Kalman filter，UKF）应用于雷达配准，提出一种新的多雷达方位配准算法。在该算法中，目标的运动状态和方位误差由选定的采样点来近似，在每个更新过程中，采样点随着状态方程传播并随非线性测量方程变换，得到目标的运动状态和方位误差的均值，避免了对非线性方程的线性化，且具有较高的计算精度。与传统的扩展卡尔曼滤波（Extended Kalman filter，EKF）方法进行了仿真比较，结果表明UKF方法能有效地克服非线性跟踪问题中很容易出现的滤波发散问题，且估计精度高于UKF方法。     

Extended state space recursive least squares

A new extended state space recursive least squares (ESSRLS) algorithm is proposed for state estimation of nonlinear systems. It is based on state space recursive least squares (SSRLS) approach and uses first order linearization of the system. It inherits the capability of obtaining state estimate without knowledge of process and measurement noise covariance matrices (Q and R respectively). The proposed approach is considered to provide new design option for scenarios where noise statistics and system dynamics vary. ESSRLS is initialized using delayed recursion method and a forgetting factor λ is employed to optimize the performance. The selection of λ can be problem specific as shown through experimental validations. However a value closer to and less than unity is generally recommended. Theoretical bases are validated by applying this algorithm to problems of tracking a non-conservative oscillator, a damped system with amplitude death and a signal modeled by mixture of Gaussian kernels. Simulation results show an MSE performance gain of 20 dB and 23 dB over extended Kalman filter (EKF) and unscented Kalman filter (UKF) while tracking van der Pol oscillator without knowledge about noise variances. The computational complexity of ESSRLS falls within that of EKF and UKF.     

Unscented卡尔曼滤波在状态估计中的应用

针对非线形系统的滤波问题，无法使用卡尔曼滤波器（KF），扩展卡尔曼滤波（EKF）方法虽能应用于非线形系统，但给出的是状态的有偏估计，并且对模型误差的鲁棒性较差。为了给出更好的状态估计值，该文介绍了Unscented卡尔曼滤波（UKF）的基本原理。其思想是：基于unscented变换，UKF滤波算法能够给出更精确的均值和协方差的估计，从而带来更高的精度。最后通过Mackey—Glass模型时间序列的状态估计仿真实侧说明：同EKF相比，UKF的滤波精度和稳定性都显著提高了，还可避免计算烦琐的Jacobi矩阵，是一种良好的非线性滤波方法。     

Performance evaluation of UKF-based nonlinear filtering

The performance of the modified unscented Kalman filter (UKF) for nonlinear stochastic discrete-time system with linear measurement equation is investigated. It is proved that under certain conditions, the estimation error of the UKF remains bounded. Furthermore, it is shown that the design of noise covariance matrix plays an important role in improving the stability of the algorithm. Error behavior of the UKF is then derived in terms of mean square error (MSE), and the Cramér-Rao lower bound (CRLB) is introduced as a performance measure. The modified UKF is found to approach the CRLB if the difference between the real noise covariance matrix and the selected one is small enough. These results are verified by using Monte Carlo simulations on two example systems.     

Robust Kalman filtering for nonlinear multivariable stochastic systems in the presence of non‐Gaussian noise

The presence of outliers can considerably degrade the performance of linear recursive algorithms based on the assumptions that measurements have a Gaussian distribution. Namely, in measurements there are rare, inconsistent observations with the largest part of population of observations (outliers). Therefore, synthesis of robust algorithms is of primary interest. The Masreliez–Martin filter is used as a natural frame for realization of the state estimation algorithm of linear systems. Improvement of performances and practical values of the Masreliez‐Martin filter as well as the tendency to expand its application to nonlinear systems represent motives to design the modified extended Masreliez–Martin filter. The behaviour of the new approach to nonlinear filtering, in the case when measurements have non‐Gaussian distributions, is illustrated by intensive simulations. Copyright © 2015 John Wiley & Sons, Ltd.     

Joint state and parameter robust estimation of stochastic nonlinear systems

Successful implementation of many control strategies is mainly based on accurate knowledge of the system and its parameters. Besides the stochastic nature of the systems, nonlinearity is one more feature that may be found in almost all physical systems. The application of extended Kalman filter for the joint state and parameter estimation of stochastic nonlinear systems is well known and widely spread. It is a known fact that in measurements, there are inconsistent observations with the largest part of population of observations (outliers). The presence of outliers can significantly reduce the efficiency of linear estimation algorithms derived on the assumptions that observations have Gaussian distributions. Hence, synthesis of robust algorithms is very important. Because of increased practical value in robust filtering as well as the rate of convergence, the modified extended Masreliez–Martin filter presents the natural frame for realization of the joint state and parameter estimator of nonlinear stochastic systems. The strong consistency is proved using the methodology of an associated ODE system. The behaviour of the new approach to joint estimation of states and unknown parameters of nonlinear systems in the case when measurements have non‐Gaussian distributions is illustrated by intensive simulations. Copyright © 2015 John Wiley & Sons, Ltd.     

On the iterated forms of Kalman filters using statistical linearization

The extended Kalman filter (EKF) is a suboptimal estimator of the conditional mean and covariance for nonlinear state estimation. It is based on first order Taylor series approximation of nonlinear state functions. The unscented Kalman filter (UKF) and the ensemble Kalman filter (EnKF) are suboptimal estimators that are termed as Jacobian free because they do not require the existence of the Jacobian of the nonlinearity. The iterated form of EKF is an estimator of the conditional mode that employs an approximate Newton–Raphson iterative scheme to solve the maximization of the conditional probability density function. In this paper, the iterated forms of UKF and EnKF are presented that perform Newton–Raphson iteration without explicitly differentiating the nonlinear functions. The use of statistical linearization in iterated UKF and EnKF is a nondifferentiable optimization method when the measurement function is nonsmooth or discontinuous. All three iterated forms can be shown to be conditional mean estimators after the first iteration. A simple numerical example involving continuous and discontinuous measurment functions is included to evaluate the performance of the algorithms for the estimation of conditional mean, covariance and mode. A batch reactor simulation is shown for estimating both the states and unknown parameters.     

带有色量测噪声的非线性系统 Unscented 卡尔曼滤波器

传统Unscented卡尔曼滤波器(Unscented Kalman filter, UKF)要求噪声必须为高斯白噪声, 无法解 决带有色噪声的非线性系统滤波问题. 为此, 本文提出了一种带有色量测噪声的UKF滤 波新算法. 首先,基于量测信息增广和最小方差估计, 推导出一类带有色量测噪声的非 线性离散系统状态的最优滤波框架, 接着采用Unscented变换(Unscented transformation, UT)来计算最优框架中的 非线性状态后验均值和协方差, 进而得到有色量测噪声下UKF滤波递推公式. 所设 计的UKF新方法能有效地解决传统UKF在量测噪声有色情况下非线性滤波失效的问题, 数 值仿真实例验证了其可行性和有效性.     

A comparison of simultaneous state and parameter estimation schemes for a continuous fermentor reactor

This article proposes a maximum likelihood algorithm for simultaneous estimation of state and parameter values in nonlinear stochastic state-space models. The proposed algorithm uses a combination of expectation maximization, nonlinear filtering and smoothing algorithms. The algorithm is tested with three popular techniques for filtering namely particle filter (PF), unscented Kalman filter (UKF) and extended Kalman filter (EKF). It is shown that the proposed algorithm when used in conjunction with UKF is computationally more efficient and provides better estimates. An online recursive algorithm based on nonlinear filtering theory is also derived and is shown to perform equally well with UKF and ensemble Kalman filter (EnKF) algorithms. A continuous fermentation reactor is used to illustrate the efficacy of batch and online versions of the proposed algorithms.     

Noise covariance identification for time-varying and nonlinear systems

Kalman-based state estimators assume a priori knowledge of the covariance matrices of the process and observation noise. However, in most practical situations, noise statistics and initial conditions are often unknown and need to be estimated from measurement data. This paper presents an auto-covariance least-squares-based algorithm for noise and initial state error covariance estimation of large-scale linear time-varying (LTV) and nonlinear systems. Compared to existing auto-covariance least-squares based-algorithms, our method does not involve any approximations for LTV systems, has fewer parameters to determine and is more memory/computationally efficient for large-scale systems. For nonlinear systems, our algorithm uses full information estimation/moving horizon estimation instead of the extended Kalman filter, so that the stability and accuracy of noise covariance estimation for nonlinear systems can be guaranteed or improved, respectively.     

基于UKF的闪烁噪声机动目标跟踪

闪烁噪声下的机动目标跟踪是一个非线性非高斯系统滤波问题,传统的卡尔曼理论很难保证其跟踪精度.文中提出了一种基于UKF的闪烁噪声机动目标跟踪算法,首先对目标系统的状态方程进行无味变换,然后再进行滤波估计,以减小跟踪误差.UKF不需要求导,它能比EKF更好地迫近目标运动模型的非线性特性,具有更高的估计精度,计算量却与EKF同阶.在仿真实验中采用"协同转弯模型"作为机动目标的运动模型,雷达的量测方程也是非线性的,分别应用UKF和EKF跟踪闪烁噪声下的机动目标,结果表明,UKF能够较好地解决闪烁噪声下跟踪机动目标的难题,其跟踪精度要远远高于EKF.     

Maximum correntropy unscented filter

The unscented transformation (UT) is an efficient method to solve the state estimation problem for a non-linear dynamic system, utilising a derivative-free higher-order approximation by approximating a Gaussian distribution rather than approximating a non-linear function. Applying the UT to a Kalman filter type estimator leads to the well-known unscented Kalman filter (UKF). Although the UKF works very well in Gaussian noises, its performance may deteriorate significantly when the noises are non-Gaussian, especially when the system is disturbed by some heavy-tailed impulsive noises. To improve the robustness of the UKF against impulsive noises, a new filter for non-linear systems is proposed in this work, namely the maximum correntropy unscented filter (MCUF). In MCUF, the UT is applied to obtain the prior estimates of the state and covariance matrix, and a robust statistical linearisation regression based on the maximum correntropy criterion is then used to obtain the posterior estimates of the state and covariance matrix. The satisfying performance of the new algorithm is confirmed by two illustrative examples.     

Variants of extended Kalman filtering approaches for Bayesian tracking

We provide a tutorial for a number of variants of the extended Kalman filter (EKF). In these methods, so called, sigma points are employed to tackle the nonlinearity of problems. The sigma points exactly represent the mean and the variance of the state distribution function in a dynamic state equation. The initially developed EKF variant, that is, unscented Kalman filter (UKF) (also called sigma point Kalman filter) shows enhanced performance compared with that of conventional EKF in the literature. Another variant, which is not well known, is central difference Kalman filter (CDKF) whose way to approximate the nonlinearity is based on the Sterling's polynomial interpolation formula instead of the Taylor series. Endeavor to reduce the computational load resulted in the development of square root versions of both UKF and CDKF, that is, square root unscented Kalman filter and square root central difference Kalman filter (SR‐CDKF). These SR‐versions are supposed to be numerically more stable than their original versions because the state covariance is guaranteed to be positive definite by avoiding the step of matrix decomposition. In this paper, we provide the step‐by‐step algorithms of above‐mentioned EKF variants with their pros and cons. We apply these filtering methods to a number of problems in various disciplines for performance assessment in terms of both mean squared error (MSE) and processing speed. Furthermore, we show how to optimize the filters in terms of MSE performance depending on diverse scenarios. According to simulation results, CDKF and SR‐CDKF show the best MSE performance in most scenarios; particularly, SR‐CDKF shows faster processing speed than that of CDKF. Therefore, we justify that SR‐CDKF is the most efficient and the best approach among the Kalman variants including the EKF for various nonlinear problems. The motivation of this paper targets at the contribution to the disseminative usage of the Kalman variants approaches, particularly, SR‐CDKF taking advantage of its estimating performance and high processing speed. Copyright © 2016 John Wiley & Sons, Ltd.     

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.     

An adaptive unscented Kalman filter approach to secure state estimation for wireless sensor networks

Wireless sensor networks are vulnerable to false data injection attacks, which may mislead the state estimation. To solve this problem, this paper presents a chi-square test-based adaptive secure state estimation (CTASSE) algorithm for state estimation and attack detection. Taking advantage of Kalman filters, attack signal together with process noise or measurement noise are described as total white Gaussian noise with uncertain covariance matrix. The chi-square test method is used in the adaptation of the total noise covariance and attack detection. Then, a standard adaptive unscented Kalman filter (UKF) is used for the state estimation. Finally, simulation results show that the proposed CTASSE algorithm performs better than other UKFs in state estimation and is also effective in real-time attack detection.     

Online State–Space Modeling Using Recurrent Multilayer Perceptrons with Unscented Kalman Filter

A nonlinear black-box modeling approach using a state–space recurrent multilayer perceptron (RMLP) is considered in this paper. The unscented Kalman filter (UKF), which was proposed recently and is appropriate for state–space representation, is employed to train the RMLP. The UKF offers a derivative-free computation and an easy implementation, compared to the extended Kalman filter (EKF) widely used for training neural networks. In addition, the UKF has a fast convergence rate and an excellent capability of parameter estimation which are appropriate for online learning. Through modeling experiments of nonlinear systems, the effectiveness of the RMLP trained with the UKF is demonstrated.     

随机逼近自适应滤波在捷联惯导系统初始对准中的应用

对于测量噪声方差未知的捷联惯导系统(SINS),采用常规Kalman滤波进行初始对准会造成较大状态估计误差,甚至使滤波器发散。为了解决系统测量噪声方差未知或不确切知道时SINS的误差估计问题,提出一种基于随机逼近的自适应滤波方法。该方法将Robbins-Monro算法与Kalman滤波相结合,通过简化求逆运算,解决了系统观测噪声特性未知情况下SINS的误差估计问题,并提高了算法的数值稳定性。仿真结果表明,该方法能在系统测量噪声方差未知情况下有效实现SINS初始对准。     

基于SVD 的改进抗差UKF 算法及在组合导航中的应用

针对GPS/INS组合导航中因观测异常导致系统状态先验信息矩阵失去对称正定性,及传统等价权函数抗差算法易遇到病态矩阵,引起滤波性能下降的问题,提出一种基于奇异值分解的改进抗差UKF算法。该算法克服了先验协方差矩阵负定性变化,通过判断矩阵病态性实现智能选取抗差策略。最后利用车载实测数据进行验证,所得结果表明, SVD-UKF导航解精度稍优于EKF算法,改进的抗差策略能够极大减弱单独、连续以及混合的观测异常对导航解的影响,提高了导航解精度和可靠性。     

The Cramér-Rao estimation error lower bound computation for deterministic nonlinear systems

For continuous-time nonlinear deterministic system models with discrete nonlinear measurements in additive Ganssian white noise, the extended Kalman filter (EKF) convariance propagation equations linearized about the true unknown trajectory provide the Cramér-Rao lower bound to the estimation error covariance matrix. A useful application is establishing the optimum filter performance for a given nonlinear estimation problem by developing a simulation of the nonlinear system and an EKF linearized about the true trajectory.     

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.