Truncation nonlinear filters for state estimation with nonlinear inequality constraints

The paper focuses on the state estimation problem of nonlinear non-Gaussian systems with state subject to a nonlinear inequality constraint. Taking into account the available additional information about the state given by the constraint increases the estimate quality compared to classical state estimation methods which cannot utilize the information. Considering the constraint in the form of an inequality involving a nonlinear function of the state makes the state estimation problem difficult and hence treated only marginally. In this paper, a generic local filter for the inequality constrained estimation problem is proposed. It covers the extended Kalman filter, unscented Kalman filter, and divided difference filter as special cases and enforces the constraint by truncating the conditional density of the state. The truncation is computationally cheap, yet it provides high estimate quality of the constrained estimate. The same idea is then utilized in a truncation Gaussian mixture filter which is also proposed in the paper to increase the estimate quality further by providing a global constrained estimate. Superior estimate quality and computational efficiency of the proposed filters are illustrated in two numerical examples.     

State estimation for linear systems with state equality constraints

This paper deals with the state estimation problem for linear systems with linear state equality constraints. Using noisy measurements which are available from the observable system, we construct the optimal estimate which also satisfies linear equality constraints. For this purpose, after reviewing modeling problems in linear stochastic systems with state equality constraints, we formulate a projected system representation. By using the constrained Kalman filter for the projected system and comparing its filter Riccati equation with those of the unconstrained and the projected Kalman filters, we clearly show, without using optimality, that the constrained estimator outperforms the other filters for estimating the constrained system state. Finally, a numerical example is presented, which demonstrates performance differences among those filters.     

带约束卡尔曼滤波对涡扇发动机状态估计

提出了一种加入线性不等式约束的卡尔曼滤波方法,并用于涡扇发动机的健康状况估计。涡扇发动机数字模型包含10个状态变量、12个量测量、6个控制输入量以及8个健康状况参数。不等式约束不仅保证了状态变量估计在用户自定义的范围内随时间变化平稳缓慢,而且还提高了滤波计算效率,改善了滤波估计精度。同时系统还允许滤波器沿确定的方向修正状态变量估计,以保持状态变量真值恒定。对比传统的无约束卡尔曼滤波,线性化滤波结果显示,该方法对涡扇发动机的健康状况估计尤其行之有效。     

On the choice of importance distributions for unconstrained and constrained state estimation using particle filter

Recursive state estimation of constrained nonlinear dynamical system has attracted the attention of many researchers in recent years. For nonlinear/non-Gaussian state estimation problems, particle filters have been widely used (Arulampalam et al. [1]). As pointed out by Daum [2], particle filters require a proposal distribution and the choice of proposal distribution is the key design issue. In this paper, a novel approach for generating the proposal distribution based on a constrained Extended Kalman filter (C-EKF), Constrained Unscented Kalman filter (C-UKF) and constrained Ensemble Kalman filter (C-EnkF) has been proposed. The efficacy of the proposed state estimation algorithms using a particle filter is illustrated via a successful implementation on a simulated gas-phase reactor, involving constraints on estimated state variables and another example problem, which involves constraints on the process noise (Rao et al. [10]). We also propose a state estimation scheme for estimating state variables in an autonomous hybrid system using particle filter with Unscented Kalman filter as a proposal and unconstrained Ensemble Kalman filter (EnKF) as a proposal. The efficacy of the proposed state estimation scheme for an autonomous hybrid system is demonstrated by conducting simulation studies on a three-tank hybrid system. The simulation studies underline the crucial role played by the choice of proposal distribution in formulation of particle filters.     

一种卡尔曼增益约束滤波算法

在滤波过程中,将卡尔曼增益结合约束条件,可以有效地提高滤波精度.论文推导了含有卡尔曼增益约束条件的代价函数,根据Karush-Kuhn-Tucker（KKT）最小化的一阶必要条件,使用高斯牛顿法求出约束卡尔曼增益的最优约束解.最后对具有约束条件的目标跟踪问题进行了仿真实验,结果表明该算法的跟踪精度要高于普通卡尔曼滤波算法.     

On unscented Kalman filtering with state interval constraints

This paper addresses the state-estimation problem for nonlinear systems for the case in which prior knowledge is available in the form of interval constraints on the states. Approximate solutions to this problem are reviewed and compared with new algorithms. All the algorithms investigated are based on the unscented Kalman filter. Two illustrative examples of chemical processes are discussed. Numerical results suggest that the use of constrained unscented filtering algorithms improves the accuracy of the state estimates compared to the unconstrained unscented filter, especially when a poor initialization is set. Moreover, it is shown that the constrained filters that enforce the state interval constraint on both the state estimate and error covariance yield more accurate state estimates than the methods that enforce such constraint only on the state estimates.     

Some results on linear equality constrained state filtering

This paper addresses the linear equality constrained state filtering for linear dynamic systems from different perspectives. First, by integrating constraint information into the state equation to ensure that the estimates naturally satisfy the constraints, the constrained filtering problem can be transformed into an unconstrained one. Second, according to a linear transformation of the state vector and the linear relationship between different new state components, a reduced-order Kalman filter is developed. Third, adding a projection step after the one-step state prediction in the Kalman filtering algorithm, we present a state prediction projection method. These approaches are mutually equivalent, and the existing null space method proves to be a special case of them. Most of current methods and the proposed approaches can be summed up in a unified framework and boil down to three forms of the projection method. Finally, a vehicle tracking example is provided to compare the performance of the discussed constrained filters.     

一种用于解决非线性滤波问题的新型粒子滤波算法

粒子滤波算法受到许多领域的研究人员的重视,该算法的主要思想是使用一个带有权值的粒子集合来表示系统的后验概率密度.在扩展卡尔曼滤波和Unscented卡尔曼滤波算法的基础上,该文提出一种新型粒子滤波算法.首先用Unscented卡尔曼滤波器产生系统的状态估计,然后用扩展卡尔曼滤波器重复这一过程并产生系统在k时刻的最终状态估计.在实验中,针对非线性程度不同的两种系统,分别采用5种粒子滤波算法进行实验.结果证明,文中所提出的算法的各方面性能都明显优于其他4种粒子滤波算法.     

A recursive state estimator in the presence of state inequality constraints

This paper proposes an optimal recursive estimator to estimate the states of a stochastic discrete time linear dynamic system when the states of the system are constrained with inequality constraints. The case when the constraints are strictly satisfied is treated independently from the case when some of the constraints are violated. For the first case, the well known Kalman filter estimator is used. In the second case, an algorithm which uses a series of successive orthogonalizations on the measurement subspaces is employed to obtain the optimal estimate. It is shown that the proposed estimator has several attractive properties such that it is an unbiased estimator. More importantly, compared to other estimator found in the literature, the proposed estimator needs less computational efforts, is numerically more stable and it leads to a smaller variance. To show the effectiveness of the proposed estimator, several simulation results are presented and discussed.     

Linear minimum mean square error filtering with stochastic linear equality constraints

Constrained filters, through utilising the prior state constraint information, are designed to obtain more accurate state estimates in applications, and most of them deal with the estimation problem of systems with deterministic constraints. In practice, complex environmental disturbance, incomplete information or uncooperative behaviour often brings out uncertainties of the constraints. This paper tackles the filtering problem of dynamic systems subject to the stochastic linear equality constraints expressed by random weighted basis matrices. The corresponding constrained dynamic model is constructed first and the linear-minimum-mean-square-error filter is derived based on the orthogonality principle. Due to the effect of constraint randomness, the resultant filter encounters the problem of nonlinear stochastic calculation of random parameters, which is solved by the Taylor-based and the UT-based schemes, respectively, and the computational complexity as well as the tractability of both schemes are analysed. Finally, a simulation study on a road-constrained vehicle tracking demonstrates that the proposed filter has better performance than the classical estimation projection method in terms of estimation accuracy and computational complexity.     

时域Wiener状态滤波新方法

基于稳态Kalman滤波器和射影理论,提出了统一和通用的时域Wiener状态滤波新方法,用它得到带非零均值相关噪声线性随机系统的渐近稳定的Wiener状态估值器和解耦Wiener状态估值器.它可统一处理状态滤波、预报和平滑问题.发现了Kalman滤波器和Wiener滤波器之间的变换关系,Wiener状态估值器可由Kalman估值器通过自回归滑动平均(ARMA)新息模型得到.一个仿真例子说明了其有效性.     

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     

Dynamic data reconciliation: Alternative to Kalman filter

Process measurements are often corrupted with varying degrees of noise. Measurement noise undermines the performance of process monitoring and control systems. To reduce the impact of measurement noise, exponentially-weighted moving average and moving average filters are commonly used. These filters have good performance for processes under steady state or with slow dynamics. For processes with significant dynamics, more sophisticated filters, such as model-based filters, have to be used. The Kalman filter is a well known model-based filter that has been widely used in the aerospace industry. This paper discusses another model-based filter, the dynamic data reconciliation (DDR) filter. Both the Kalman and the DDR filters adhere to the same basic principle of using information from both measurements and models to provide a more reliable representation of the current state of the process. However, the DDR filter can more easily incorporated in a wide variety of model structures and is easier to understand and implement. Simulation results for a binary distillation column with four controlled variables showed that the DDR filters had equivalent performance to the Kalman filter in dealing with both white and autocorrelated noise.     

Kalman filtering approach to multi-rate information fusion in the presence of irregular sampling rate and variable measurement delay

State estimation for a system with irregular rate and delayed measurements is studied using fusion Kalman filter. Lab data in process plants is usually more accurate compared to other measurements. However, it is often slow rate and subject to variable delay and irregularity in sampling time. Fast rate state estimation can be conducted using fast rate measurement, while the slow rate lab data can be used to improve the accuracy of estimation whenever it is available. For this purpose, two Kalman filters are used to estimate the states based on each type of measurement. The estimates are fused in the next step by considering the correlation between them. An iterative algorithm to obtain the cross-covariance matrix between the estimation errors of the two Kalman filters is presented and employed in the fusion process. The improvement on the accuracy of estimation and comparison with other optimal fusion state estimation techniques are discussed through a simulation example, a pilot-scale experiment and an industrial case study.     

Marginalized adaptive particle filtering for nonlinear models with unknown time-varying noise parameters

Knowledge of the noise distribution is typically crucial for the state estimation of general state-space models. However, properties of the noise process are often unknown in the majority of practical applications. The distribution of the noise may also be non-stationary or state dependent and that prevents the use of off-line tuning methods. For linear Gaussian models, Adaptive Kalman filters (AKF) estimate unknown parameters in the noise distributions jointly with the state. For nonlinear models, we provide a Bayesian solution for the estimation of the noise distributions in the exponential family, leading to a marginalized adaptive particle filter (MAPF) where the noise parameters are updated using finite dimensional sufficient statistics for each particle. The time evolution model for the noise parameters is defined implicitly as a Kullback–Leibler norm constraint on the time variability, leading to an exponential forgetting mechanism operating on the sufficient statistics. Many existing methods are based on the standard approach of augmenting the state with the unknown variables and attempting to solve the resulting filtering problem. The MAPF is significantly more computationally efficient than a comparable particle filter that runs on the full augmented state. Further, the MAPF can handle sensor and actuator offsets as unknown means in the noise distributions, avoiding the standard approach of augmenting the state with such offsets. We illustrate the MAPF on first a standard example, and then on a tire radius estimation problem on real data.     

Estimating Vehicle State by GPS/IMU Fusion with Vehicle Dynamics

In this paper, a low-cost navigation system with high integrity and reliability is proposed. A high-integrity estimation filter is proposed to obtain a high-accuracy state estimate. The filter utilizes a vehicle velocity constraint measurement to enhance the accuracy of the estimate. Two estimation filters, the extended Kalman filter (EKF) and the extended information filter (EIF), are designed and compared to obtain the estimate of the vehicle state. An instrumentation system that consists of a microcontroller, GPS receiver, IMU, velocity encoder, and Zigbee transceiver is used. The microcontroller provides a vehicle navigation solution at 50 Hz by fusing the measurements of the IMU and GPS receiver using the proposed filter design. Extensive experimental tests are conducted to verify the accuracy of the proposed algorithm. These results are processed with and without the velocity constraints. The estimation accuracy improvement with the addition of the velocity constraints is shown. A more than 16 % reduction in the computational time is demonstrated when using the EIF in comparison to the EKF approach.     

Continuous‐time Kalman filtering on the orthogonal group O(n)

This paper presents a continuous‐time O(n)‐constrained Kalman‐like filter. O(n) is the group of n × n orthonormal matrices. The O(n)‐constrained Kalman‐like filter is derived by posing a constrained optimization problem. The solution involves a projection of the unconstrained Kalman state estimate derivative onto the tangent space of O(n). Using this filter, an extended O(n)‐constrained Kalman‐like filter is developed for nonlinear systems where a portion of the states evolve on O(n). A numerical example demonstrates the effectiveness of the extended O(n)‐constrained Kalman‐like filter. Copyright © 2017 John Wiley & Sons, Ltd.     

基于SRCKF 的自适应高斯和状态滤波算法

针对非线性非高斯离散动态系统中的状态估计问题,基于高斯和递推关系,提出一种高斯和状态估计算法GSSRCKF.首先将状态噪声、观测噪声及滤波初值均表示为高斯和的形式,以平方根容积卡尔曼滤波为子滤波器分别估计各高斯子项对应的系统状态;然后结合各子项对应的权值实现全局估计;最后设计高斯子项对应权值的自适应策略,并采用约简控制法降低计算复杂度.仿真结果验证了所提出的算法在滤波稳定性方面的优越性.     

Performance assessment,diagnosis, and optimal selection of non-linear state filters

Non-linear state filters of different approximations and capabilities allow for real-time estimation of unmeasured states in non-linear stochastic processes. It is well known that the performance of non-linear filters depends on the underlying numerical and statistical approximations used in their design. Despite the theoretical and practical interest in evaluating the performance of non-linear filtering methods, it remains one of the most complex problems in the area of state estimation. We propose the use of posterior Cramér–Rao lower bound (PCRLB) or mean square error (MSE) inequality as a filtering performance benchmark. Using the PCRLB inequality, we develop assessment and diagnosis tools for monitoring and evaluating the performance of non-linear filters. Using the PCRLB inequality-based performance assessment tool, an optimal non-linear filter switching strategy is proposed for state estimation in general non-linear systems. The non-linear filter switching strategy is an optimal performance strategy, which maintains high filtering performance under all operating conditions. The complex, high dimensional integrals involved in the computation of the PCRLB inequality-based non-linear filter assessment and diagnosis tools are approximated using sequential Monte-Carlo (SMC) methods. The utility and efficacy of the developed tools are illustrated through a numerical example.