Author's reply to “Comments on ‘Performance evaluation of UKF-based nonlinear filtering”’

It is stated in the above-mentioned comment that the main result of the paper Xiong, Zhang, et al. [(2006). Performance evaluation of UKF-based nonlinear filtering. Automatica 42(2), 261-270] can be extended to a class of filters, such as the extended Kalman filter (EKF). As we show here, this belief can be justified in a rigorous way, even for the nonlinear stochastic system with a nonlinear measurement equation.     

Comments on “Performance evaluation of UKF-based nonlinear filtering”

This note points out that the main result of the paper [Xiong, K., Zhang, H. Y., & Chan, C. W. (2006). Performance evaluation of UKF-based nonlinear filtering. Automatica, 42(2), 261-270] can be extended to a class of so-called Gaussian filters. It justifies a practical countermeasure of the divergence, i.e., adding small quantities to the noise covariance matrix to stabilize the filter.     

Freeway traffic estimation within particle filtering framework

This paper formulates the problem of real-time estimation of traffic state in freeway networks by means of the particle filtering framework. A particle filter (PF) is developed based on a recently proposed speed-extended cell-transmission model of freeway traffic. The freeway is considered as a network of components representing different freeway stretches called segments. The evolution of the traffic in a segment is modelled as a dynamic stochastic system, influenced by states of neighbour segments. Measurements are received only at boundaries between some segments and averaged within possibly irregular time intervals. This limits the measurement update in the PF to only these time instants when a new measurement arrives, while in between measurement updates any simulation model can be used to describe the evolution of the particles. The PF performance is validated and evaluated using synthetic and real traffic data from a Belgian freeway. An unscented Kalman filter is also presented. A comparison of the PF with the unscented Kalman filter is performed with respect to accuracy and complexity.     

数值鲁棒的非线性滤波新方法

应用多维情形的二阶插值公式构造新型非线性滤波器。该滤波器不需非线性函数的偏导计算，便能代替常规的扩展卡尔曼滤波器，并有滤波精度高、数值计算稳定和适用范围宽等优点。仿真实例表明新滤波器具有较高的性能。     

无迹卡尔曼滤波算法在污水处理BSM1过程中的应用研究

污水处理过程具有多变量、强非线性和强扰动等特性,且系统输入具有随机性,不同天气状况和不同时间段的污水的排量不同.扩展卡尔曼滤波存在估计精度低和鲁棒性差等缺陷,当系统模型参数变化和外界环境噪声较大时,扩展卡尔曼滤波估计性能下滑.将无迹卡尔曼滤波算法应用到污水处理系统中,并与扩展卡尔曼滤波算法相比较,结果显示,无迹卡尔曼滤...     

Unscented Kalman filter with advanced adaptation of scaling parameter

The paper deals with state estimation of the nonlinear stochastic systems by means of the unscented Kalman filter with a focus on specification of the σ$\sigma$-points. Their position is influenced by two design parameters—the scaling parameter determining the spread of the σ$\sigma$-points and a covariance matrix decomposition determining rotation of the σ$\sigma$-points. In this paper, a choice of the scaling parameter is analyzed. It is shown that considering other values than the standard choice may lead to increased quality of the estimate, especially if the scaling parameter is adapted. Several different criteria for the adaptation are proposed and techniques to reduce computational costs of the adaptation are developed. The proposed algorithm of the unscented Kalman filter with advanced adaptation of the scaling parameter is illustrated in a numerical example.     

Constrained output feedback model predictive control for nonlinear systems

A constrained output feedback model predictive control approach for nonlinear systems is presented in this paper. The state variables are observed using an unscented Kalman filter, which offers some advantages over an extended Kalman filter. A nonlinear dynamic model of the system, considered in this investigation, is developed considering all possible effective elements. The model is then adaptively linearized along the prediction horizon using a state-dependent state space representation. In order to improve the performance of the control system as many linearized models as the number of prediction horizons are obtained at each sample time. The optimum results of the previous sample time are utilized for linearization at the current sample time. Subsequently, a linear quadratic objective function with constraints is formulated using the developed governing equations of the plant. The performance and effectiveness of the proposed control approach is validated both in simulation and through real-time experimentation using a constrained highly nonlinear aerodynamic test rig, a twin rotor MIMO system (TRMS).     

A state prediction scheme for discrete time nonlinear dynamic systems

A state prediction scheme is proposed for discrete time nonlinear dynamic systems with non-Gaussian disturbance and observation noises. This scheme is based upon quantization, multiple hypothesis testing, and dynamic programming. Dynamic models of the proposed scheme are as general as dynamic models of particle predictors, whereas the nonlinear models of the extended Kalman (EK) predictor are linear with respect to the disturbance and observation noises. The performance of the proposed scheme is compared with both the EK predictor and sampling importance resampling (SIR) particle predictor. Monte Carlo simulations have shown that the performances of the proposed scheme, EK predictor, and SIR particle predictor are all model-dependent, that is, one performs better than the others for a given example. Some examples, for which the proposed scheme performs better than the others do, are also given in the paper.     

A separation theorem for nonlinear systems

In this paper we have obtained a nonlinear separation result for controlled stochastic systems. The result is based on a sequential technique, introduced by the second author, which has been applied with significant success for nonlinear deterministic systems.     

Approximate conditional least squares estimation of a nonlinear state-space model via an unscented Kalman filter

The problem of estimating a nonlinear state-space model whose state process is driven by an ordinary differential equation (ODE) or a stochastic differential equation (SDE), with discrete-time data is studied. A new estimation method is proposed based on minimizing the conditional least squares (CLS) with the conditional mean function computed approximately via the unscented Kalman filter (UKF). Conditions are derived for the UKF–CLS estimator to preserve the limiting properties of the exact CLS estimator, namely, consistency and asymptotic normality, under the framework of infill asymptotics, i.e. sampling is increasingly dense over a fixed domain. The efficacy of the proposed method is demonstrated by simulation and a real application.     

UKF与EKF在卫星姿态估计应用中的比较

针对卫星的姿态和角速度估计问题,分别给出基于Unscented卡尔曼滤波(UKF)与推广卡尔曼滤波(EKF)的估计算法,并做了相应比较.为了避免欧拉角带来的奇异问题,UKF选用Rodrigues参数而EKF选用四元数参数法来描述姿态误差.考虑卫星的非线性模型,UKF采用Unscented变换而EKF采用线性化方法对姿态误差进行估计.利用陀螺和磁强计的测量信息,KF和EKF都可得到三轴稳定卫星的姿态估计值,但UKF的收敛速度高于EKF.数值仿真结果表明,当初始姿态存在大偏差时,所给出的UKF的滤波算法性能明显优于EKF.     

State estimation of nonlinear dynamical systems using nonlinear update based Unscented Gaussian Sum Filter

Two attractive features of Unscented Kalman Filter (UKF) are: (1) use of deterministically chosen points (called sigma points), and (2) only a linear dependence of the number of sigma points on the number of states. However, an implicit assumption in UKF is that the prior conditional state probability density and the state and measurement noise densities are Gaussian. To avoid the restrictive Gaussianity assumption, Gaussian Sum-UKF (GS-UKF) has been proposed in literature that approximates all the underlying densities using a sum of Gaussians. However, the number of sigma points required in this approach is significantly higher than in UKF, thereby making GS-UKF computationally intensive. In this work, we propose an alternate approach, labeled Unscented Gaussian Sum Filter (UGSF), for state estimation of nonlinear dynamical systems, corrupted by Gaussian state and measurement noises. Our approach uses a Sum of Gaussians to approximate the non-Gaussian prior density. A key feature of this approximation is that it is based on the same number of sigma points as used in UKF, thereby resulting in similar computational complexity as UKF. We implement the proposed approach on two nonlinear state estimation case studies and demonstrate its utility by comparing its performance with UKF and GS-UKF.     

Unscented Kalman filter based nonlinear model predictive control of a LDPE autoclave reactor

This paper presents a multivariable nonlinear model predictive control (NMPC) scheme for the regulation of a low-density polyethylene (LDPE) autoclave reactor. A detailed mechanistic process model developed previously was used to describe the dynamics of the LDPE reactor and the properties of the polymer product. Closed-loop simulations are used to demonstrate the disturbance rejection and tracking performance of the NMPC algorithm for control of reactor temperature and weight-averaged molecular weight (WAMW). In addition, the effect of parametric uncertainty in the kinetic rate constants of the LDPE reactor model on closed-loop performance is discussed. The unscented Kalman filtering (UKF) algorithm is employed to estimate plant states and disturbances. All control simulations were performed under conditions of noisy process measurements and structural plant–model mismatch. Where appropriate, the performance of the NMPC algorithm is contrasted with that of linear model predictive control (LMPC). It is shown that for this application the closed-loop performance of the UKF based NMPC scheme is very good and is superior to that of the linear predictive controller.     

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.     

Computing arrival cost parameters in moving horizon estimation using sampling based filters

Moving horizon estimation (MHE) is a numerical optimization based approach to state estimation, where the joint probability density function (pdf) of a finite state trajectory is sought, which is conditioned on a moving horizon of measurements. The joint conditional pdf depends on the a priori state pdf at the start of the horizon, which is a prediction pdf based on historical data outside the horizon. When the joint pdf is maximized, the arrival cost is a penalty term based on the a priori pdf in the MHE objective function. Traditionally, the a priori pdf is assumed as a multivariate Gaussian pdf and the extended Kalman filter (EKF) and smoother are used to recursively update the mean and covariance. However, transformation of moments through nonlinearity is poorly approximated by linearization, which can result in poor initialization of MHE. Sampling based nonlinear filters completely avoid Taylor series approximations of nonlinearities and attempt to approximate the non-Gaussian state pdf using samples and associated weights or probability mass points. The performance gains of sampling based filters over EKF motivate their use to formulate the arrival cost in MHE. The a priori mean and covariance are more effectively propagated through nonlinearities and the resulting arrival cost term can help to keep the horizon small. It is also possible to find closed-form approximations to the non-Gaussian a priori pdf from the sampling based filters. Thus, more realistic nonparametric arrival cost terms can be included by avoiding the Gaussian assumption. In this paper the use of the deterministic sampling based unscented Kalman filter, the class of random sampling based particle filter and the aggregate Markov chain based cell filter are discussed for initializing MHE. Two simulation examples are included to demonstrate the benefits of these methods over the traditional EKF approach.     

H∞ nonlinear filtering

This paper investigates the problem of H_∞ estimation of nonlinear processes. An estimator, which may be nonlinear, is looked for so that a given bound on the ratio between the energy of the estimation error and the energy of the oxogeneous inputs to the estimated process is achieved. Conditions for the existence of such an estimator and formulas for its derivation are obtained using both the game theory approach and the theory of dissipative systems. The results of the paper extend the recent results on H_∞ nonlinear control. They are demonstrated by a simple example of a linear system with a nonlinear measurement rule and compared with corresponding results that are obtained by the extended Kalman filter.     

Unscented Kalman滤波的流量测量研究

将UKF滤波用于超声波流量测量,使用UKF滤波算法来处理超声波回波信号,得到回波信号的包络线,并且将包络模型的参数作为UKF处理的状态向量。根据流量测量的特点改进了UKF滤波运算过程,给出了UKF迭代开始和结束的条件。最后在Matlab上仿真UKF的性能及收敛速度,证明UKF是有效的和容易实现的。     

带非线性等式约束无迹卡尔曼滤波方法

针对带非线性等式约束的非线性系统的状态估计问题,给出了一种新形式的基于无迹卡尔曼滤波及伪观测手段的处理约束的状态估计方法（SPUKF）。在该方法中原动态系统被虚拟地分离成两个并行的子系统,各时刻的状态估计由基于这两个子系统构建的两套滤波链交替得到。相对于伪观测法中的序贯形式估计器,SPUKF无需事先确定观测及约束的处理次序且能获得更好的估计结果,故可以用来解决序贯方法中观测与约束的处理次序问题。由钟摆运动的实例仿真结果看到,SPUKF不仅有好于序贯形式无迹卡尔曼滤波的估计效果,误差改善比达到22%左右,而且算法运行时间与序贯形式估计器相近。此外,其估计效果还与批处理无迹卡尔曼滤波相当。     

Exponential nonlinear observer for parametric identification and synchronization of chaotic systems

This work proposes the use of a new exponential nonlinear observer for the purpose of parametric identification and synchronization of chaotic systems. The exponential convergence of the observer is guaranteed by a persistent excitation condition. This approach is shown to be suitable for a wide variety of chaotic systems. In order to illustrate the observer design procedure, several examples with simulation results are presented.