Stochastic stability of the discrete-time extended Kalman filter

The authors analyze the error behavior for the discrete-time extended Kalman filter for general nonlinear systems in a stochastic framework. In particular, it is shown that the estimation error remains bounded if the system satisfies the nonlinear observability rank condition and the initial estimation error as well as the disturbing noise terms are small enough. This result is verified by numerical simulations for an example system     

Relation between filter using covariance information and Kalman filter

This paper presents sequential algorithms for the optimal impulse function, Kalman gain and the error variance in linear least squares filtering problems, when the autocovariance function of the signal is given in the form of a semi-degenerate kernel, and the additive observation noise in white Gaussian. A digital simulation result indicates that the algorithms presented in this paper are feasible, and that the values of Kalman gain and the error variance calculated by these algorithms approach to those obtained by the Kalman filter theory, for time sufficiently large.     

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     

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.     

基于渐消记忆自适应Kalman滤波的GPS/DR数据融合

针对标准的卡尔曼滤波器对系统模型依赖性强、鲁棒性差,而GPs/DR系统的精确系统模型难以建立的问题,提出了一种渐消记忆自适应联邦卡尔曼滤波器.融合了自适应联邦滤波算法和SageHusa自适应滤波算法,估计变化的系统观测噪声方差阵,使之更符合真实的模型,并有效对GPS的定位数据的传统算法的发散得到收敛,提高组合定位的精度.计算机仿真结果表明了该算法的可行性和有效性.     

Polynomial extended Kalman filter

This work presents a polynomial version of the well-known extended Kalman filter (EKF) for the state estimation of nonlinear discrete-time stochastic systems. The proposed filter, denoted polynomial EKF (PEKF), consists in the application of the optimal polynomial filter of a chosen degree /spl mu/ to the Carleman approximation of a nonlinear system. When /spl mu/=1 the PEKF algorithm coincides with the standard EKF. For the filter implementation the moments of the state and output noises up to order 2/spl mu/ are required. Numerical simulations compare the performances of the PEKF with those of some other existing filters, showing significant improvements.     

Decomposition of the extended Kalman filter

The use of the extended Kalman filter as an approximate estimator for the states and parameters of nonlinear systems is well known. A decomposition is pointed out in this letter, which is possible with the usual augumentations of the state space by parameters, and which may lead to a more efficient computing algorithm.     

An interlaced extended Kalman filter

An estimation algorithm for a class of discrete time nonlinear systems is proposed. The system structure we deal with is partitionable into in subsystems, each affine w.r.t. the corresponding part of the state vector. The algorithm consists of a bank of m interlaced Kalman filters, and each of them estimates a part of the state, considering the remaining parts as known time-varying parameters whose values are evaluated by the other filters at the previous step. The procedure neglects the subsystem coupling terms in the covariance matrix of the state estimation error and counteracts the errors so introduced by suitably “increasing” the noise covariance matrices. Comparisons through numerical simulations with the extended Kalman filter and its modified versions proposed in the literature illustrate the good trade-off provided by the algorithm between the reduction of the computational load and the estimation accuracy     

Robust stability of nonlinear model predictive control based on extended Kalman filter

This work deals with state estimation and process control for nonlinear systems, especially when nonlinear model predictive control (NMPC) is integrated with extended Kalman filter (EKF) as the state estimator. In particular, we focus on the robust stability of NMPC and EKF in the presence of plant-model mismatch. The convergence property of the estimation error from the EKF in the presence of non-vanishing perturbations is established based on our previous work [1]. In addition, a so-called one way interaction is shown that the EKF error is not influenced by control action from the NMPC. Hence, the EKF analysis is still valid in the output-feedback NMPC framework, even though there is no separation principle for general nonlinear systems. With this result, we study the robust stability of the output-feedback NMPC under the impact of the estimation error. It turns out the output-feedback NMPC with EKF is Input-to-State practical Stable (ISpS). Finally, two offset-free strategies of output-feedback NMPC are presented and illustrated through a simulation example.     

Online tuning fuzzy PID controller using robust extended Kalman filter

Fuzzy PID controllers have been developed and applied to many fields for over a period of 30 years. However, there is no systematic method to design membership functions (MFs) for inputs and outputs of a fuzzy system. Then optimizing the MFs is considered as a system identification problem for a nonlinear dynamic system which makes control challenges. This paper presents a novel online method using a robust extended Kalman filter to optimize a Mamdani fuzzy PID controller. The robust extended Kalman filter (REKF) is used to adjust the controller parameters automatically during the operation process of any system applying the controller to minimize the control error. The fuzzy PID controller is tuned about the shape of MFs and rules to adapt with the working conditions and the control performance is improved significantly. The proposed method in this research is verified by its application to the force control problem of an electro-hydraulic actuator. Simulations and experimental results show that proposed method is effective for the online optimization of the fuzzy PID controller.     

Adaptive Kalman filter for MEMS IMU data fusion using enhanced covariance scaling

MEMS (micro-electro-mechanical-system) IMU (inertial measurement unit) sensors are characteristically noisy and this presents a serious problem to their effective use. The Kalman filter assumes zero-mean Gaussian process and measurement noise variables, and then recursively computes optimal state estimates. However, establishing the exact noise statistics is a non-trivial task. Additionally, this noise often varies widely in operation. Addressing this challenge is the focus of adaptive Kalman filtering techniques. In the covariance scaling method, the process and measurement noise covariance matrices Q and R are uniformly scaled by a scalar-quantity attenuating window. This study proposes a new approach where individual elements of Q and R are scaled element-wise to ensure more granular adaptation of noise components and hence improve accuracy. In addition, the scaling is performed over a smoothly decreasing window to balance aggressiveness of response and stability in steady state. Experimental results show that the root mean square errors for both pith and roll axes are significantly reduced compared to the conventional noise adaptation method, albeit at a slightly higher computational cost. Specifically, the root mean square pitch errors are 1.1? under acceleration and 2.1? under rotation, which are significantly less than the corresponding errors of the adaptive complementary filter and conventional covariance scaling-based adaptive Kalman filter tested under the same conditions.     

Asymptotic convergence properties of the extended Kalman filter using filtered state estimates

In a recent paper, Ljung has given a convergence analysis of the extended Kalman filter (EKF) as a parameter estimator for linear systems. The analysis is done for a version of the EKF using predicted values of the state vector. In this note a similar convergence analysis is done for the EKF using filtered values of the state vector. The convergence properties of the two algorithms are similar, but not identical. The recalculation of a simple example given by Ljung indicates that using the filtered estimate of the state vector gives improved convergence properties of the algorithm.     

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.     

Mobile-robot pose estimation and environment mapping using an extended Kalman filter

In this paper an extended Kalman filter (EKF) is used in the simultaneous localisation and mapping (SLAM) of a four-wheeled mobile robot in an indoor environment. The robot’s pose and environment map are estimated from incremental encoders and from laser-range-finder (LRF) sensor readings. The map of the environment consists of line segments, which are estimated from the LRF’s scans. A good state convergence of the EKF is obtained using the proposed methods for the input- and output-noise covariance matrices’ estimation. The output-noise covariance matrix, consisting of the observed-line-features’ covariances, is estimated from the LRF’s measurements using the least-squares method. The experimental results from the localisation and SLAM experiments in the indoor environment show the applicability of the proposed approach. The main paper contribution is the improvement of the SLAM algorithm convergence due to the noise covariance matrices’ estimation.     

On innovation sequence testing of the Kalman filter

SequentiaUy proven statements are given showing that the whiteness of the innovation sequence of a steady-state Kalman filter is not a sufficient condition for the optimality of the filter. Simulation results are given which verify each of the statements. Definite conclusions are reached concerning the identification of a class of systems by using the output sequence.     

Conditions for stability of the extended Kalman filter and their application to the frequency tracking problem

The error dynamics of the extended Kalman filter (EKF), employed as an observer for a general nonlinear, stochastic discrete time system, are analyzed. Sufficient conditions for the boundedness of the errors of the EKF are determined. An expression for the bound on the errors is given in terms of the size of the nonlinearities of the system and the error covariance matrices used in the design of the EKF. The results are applied to the design of a stable EKF frequency tracker for a signal with time-varying frequency.This research was supported by the Co-operative Research Centre for Robust and Adaptive Systems ((CR)² ASys). The authors wish to acknowledge the funding of the activities of (CR)² ASys by the Australian Commonwealth Government under the Co-operative Research Centre Program.     

Robust stability of nonlinear model predictive control with extended Kalman filter and target setting

This work deals with the closed‐loop robust stability of nonlinear model predictive control (NMPC) coupled with an extended Kalman filter (EKF). First, we point out the gaps between the practical formulations and theoretical research. Then, we show that the estimation error dynamics of an EKF are input‐to‐state stable (ISS) in the presence of nonvanishing perturbations. Moreover, a target setting optimization problem is proposed to solve the target state corresponding to the desired set points, which are used in the objective function in NMPC formulation. Thus, the objective function is a Lyapunov function candidate, and the input‐to‐state practical stability (ISpS) of the closed‐loop system can be established. Moreover, we see that the stability property deteriorates because of the estimation error. Simulation results of the proposed scheme are presented.Copyright © 2012 John Wiley & Sons, Ltd.     

基于广义卡尔曼滤波器的路面辨识

为防止车辆驱动轮打滑、提高制动效能,并使车辆在转向工况下获得稳定的行驶特性,对路面情况的实时辨识进行研究.利用μ-s曲线的斜率特性,采用广义卡尔曼滤波器结合路面变化探测模块的方法估算车轮纵滑刚度;利用仿真实验验证该方法的准确性和有效性,路面辨识方法的响应速度得到明显提高.该方法有利于整车控制策略的开发.     

A Zakai equation derivation of the extended Kalman filter

A discrete time filter is considered where both the observation and signal process have non-linear dynamics with additive Gaussian noise. Using the reference probability framework a convolution type Zakai equation is obtained which updates the unnormalized conditional density. Using first order approximations this equation can be solved recursively and the extended Kalman filter can be derived.