A framework for globally optimal state estimation for systems with unknown inputs (I): transformation approach

In this paper, a globally optimal state estimation is addressed in light of the conventional Luenberger observer‐type filter. This paper is the first part of a comprehensive extension of an original work by Hsieh, with the main aim being to develop a transformation‐based filtering framework for global unbiased minimum‐variance state estimation (GUMVSE) for systems with unknown inputs that affect both the system and the output. The main contributions of this paper are (i) a complete optimal solution for the GUMVSE is addressed, where both the globally optimal state filter and predictor are presented, and (ii) additional insights for implementing the globally optimal state filter are highlighted via the proposed decorrelation constraint. Compared with existing results, the proposed globally optimal filter has the most general filter form among all transformation‐based globally optimal filters in the sense that it does not use any specific unknown input transformation matrix in the derivation. A simulation example is given to illustrate the usefulness of the proposed results. Copyright © 2011 John Wiley and Sons Asia Pte Ltd and Chinese Automatic Control Society     

Linear dynamic filtering with noisy input and output

State estimation problems for linear time-invariant systems with noisy inputs and outputs are considered. An efficient recursive algorithm for the smoothing problem is presented. The equivalence between the optimal filter and an appropriately modified Kalman filter is established. The optimal estimate of the input signal is derived from the optimal state estimate. The result shows that the noisy input/output filtering problem is not fundamentally different from the classical Kalman filtering problem.     

Robust two-stage Kalman filters for systems with unknown inputs

A method is developed for the state estimation of linear time-varying discrete systems with unknown inputs. By making use of the two-stage Kalman filtering technique and a proposed unknown inputs filtering technique, a robust two-stage Kalman filter which is unaffected by the unknown inputs can be readily derived and serves as an alternative to the Kitanidis' (1987) unbiased minimum-variance filter. The application of this new filter is illustrated by optimal filtering for systems with unknown inputs     

Suboptimal filter for continuous‐time linear systems with unknown parameters

The filtering problem for continuous‐time linear systems with unknown parameters is considered. A new suboptimal filter is herein proposed. It is based on the optimal mean‐square linear combination of the local Kalman filters. In contrast to the optimal weights, the suboptimal weights do not depend on current observations; thus, the proposed filter can easily be implemented in real‐time. Examples demonstrate high accuracy and efficiency of the suboptimal filter. Copyright © 2008 John Wiley and Sons Asia Pte Ltd and Chinese Automatic Control Society     

On the global optimality of unbiased minimum-variance state estimation for systems with unknown inputs

In this paper, a globally optimal filtering framework is developed for unbiased minimum-variance state estimation for systems with unknown inputs that affect both the system state and the output. The resulting optimal filters are globally optimal within the unbiased minimum-variance filtering over all linear unbiased estimators. Globally optimal state estimators with or without output and/or input transformations are derived. Through the global optimality evaluation of this research, the performance degradation of the filter proposed by Darouach, Zasadzinski, and Boutayeb [Darouach, M., Zasadzinski, M., & Boutayeb, M. (2003). Extension of minimum variance estimation for systems with unknown inputs. Automatica, 39, 867-876] is clearly illustrated and the global optimality of the filter proposed by Gillijns and De Moor [Gillijns, S., & De Moor, B. (2007b). Unbiased minimum-variance input and state estimation for linear discrete-time systems with direct feedthrough. Automatica, 43, 934-937] is further verified. The relationship with the existing literature results is addressed. A unified approach to design a specific globally optimal state estimator that is based on the desired form of the distribution matrix from the unknown input to the output is also presented. A simulation example is given to illustrate the proposed results.     

Extension of unbiased minimum-variance input and state estimation for systems with unknown inputs

This paper extends the existing results on joint input and state estimation to systems with arbitrary unknown inputs. The objective is to derive an optimal filter in the general case where not only unknown inputs affect both the system state and the output, but also the direct feedthrough matrix has arbitrary rank. The paper extends both the results of Gillijns and De Moor [Gillijns, S., & De Moor, B. (2007b). Unbiased minimum-variance input and state estimation for linear discrete-time systems with direct feedthrough. Automatica, 43, 934–937] and Darouach, Zasadzinski, and Boutayeb [Darouach, M., Zasadzinski, M., & Boutayeb, M. (2003). Extension of minimum variance estimation for systems with unknown inputs. Automatica, 39, 867–876]. The resulting filter is an extension of the recursive three-step filter (ERTSF) and serves as a unified solution to the addressed unknown input filtering problem. The relationship between the ERTSF and the existing literature results is also addressed.     

自适应Kalman滤波器及其应用

对于带未知噪声统计的单输出系统,本文提出了一种新的自适应Kalman滤波器.应用 现代时间序列分析方法,基于ARMA新息模型的滑动平均(MA)参数的在线辨识,提出了 稳态最优Kalman滤波器增益估计的一种新算法,比Mehra的算法简单.同时还提出了辨 识滑动平均(MA)模型参数的一种新的自适应Kalman滤波算法.此外,给出了在雷达跟 踪系统中的应用,且仿真结果说明了本文算法的有效性.     

Distributed fusion cubature Kalman filters for nonlinear systems

This paper is concerned with the distributed fusion estimation problem for multisensor nonlinear systems. Based on the Kalman filtering framework and the spherical cubature rule, a general method for calculating the cross‐covariance matrices between any two local estimators is presented for multisensor nonlinear systems. In the linear unbiased minimum variance sense, based on the cross‐covariance matrices, a distributed fusion cubature Kalman filter weighted by matrices (MW‐CKF) is presented. The proposed MW‐CKF has better accuracy and robustness. An example verifies the effectiveness of the proposed algorithms.     

An Improved Dual Unscented Kalman Filter for State and Parameter Estimation

A dual unscented Kalman filter (DUKF) is used to estimate the state and the parameter simultaneously via two parallel unscented Kalman filters. The original DUKF usually has performance degradation as a result of assuming the control inputs of each filter are constant, which usually are disturbance inputs or systematic measurement errors in the control system. An improved dual unscented Kalman filter (IDUKF) with random control inputs and sequential dual estimation structure is derived and applicable to the system in which the parameter is linearly observed and uncorrelated with the state. The accuracy, observability, and computational efficiency of the new filter are discussed. Then, the expansibility of the IDUKF for nonlinear parameter observed substructures is investigated. Finally, two simulation experiments about space target tracking and typical time series filtering are shown. The theoretical analyses and simulation results demonstrate the following. (1) the IDUKF can obtain higher accuracy than the original DUKF and a comparative accuracy with the JUKF (joint unscented Kalman filter) when the state and the parameter are not strongly correlated; (2) the IDUKF has better applicability than the DUKF when the state is correlated with the unknown parameter; (3) when the modeling error is not ignorable, the IDUKF is more robust and more accurate than the JUKF due to lower sensitivity to the modeling error.     

Optimal filtering for systems with unknown inputs

An optimal filtering formula is derived for linear time-varying discrete systems with unknown inputs. By making use of the well-known innovations filtering technique, the derivation is an extension of a new observer design method for time-invariant deterministic systems with unknown inputs. The systems under consideration have the most general form. The derived optimal filter has a similar form to the standard Kalman filter with some modified covariance and gain matrices     

干扰解耦滤波器设计及应用研究

提出一种干扰解耦滤波器（ＤＤＦ）设计的新方法。通过一个简单有效的代数变换,将具有未知输入的系统变换成一个不含未知输入的等价系统;利用新息定理得到一种新的干扰解耦滤波器设计方法。针对机动目标跟踪问题,对常规Ｋａｌｍａｎ滤波器、最优干扰解耦观测器和干扰解耦滤波器进行了仿真比较,结果表明ＤＤＦ特别适用于目标高度机动且无有效机动模型的情形。     

线性离散随机系统输入和状态的多步估计方法及应用

具有未知输入的系统的状态估计问题已经在过去几十年里引起了相当的关注.本文对于线性离散随机系统提出了一种基于多步信息的输入和状态同步估计方法.首先,采用多步信息的最小方差方法来获得未知输入.由于引入了包含多个时间步骤的扩张状态和测量向量而计算多步信息,使估计结果与一步估计相比减少了对噪声的敏感性.其次,利用输入估计值和卡尔曼滤波估计过去和当前的状态.该方法在未知输入维数等于状态维数时仍然有良好的估计效果.数值仿真验证了提出的估计方法的有效性.最后,该方法应用于厌氧消化过程反应罐中的溶解甲烷和二氧化碳的浓度估计以验证方法的实用性.     

Connections between H 2 optimal filters and unknown input observers – performance limitations of H 2 optimal filtering

For linear time invariant continuous-time systems with either unknown or white noise input, two well-known filtering problems are considered. These are the unknown input observer problem and the Kalman filtering problem. Most of the available literature on Kalman filtering considers the so-called regular filtering problem. We consider here the general singular filtering problem. We show that such a Kalman filtering problem for a given system can be transformed to the unknown input observer problem for an auxiliary system constructed from the data of the given system. Such transformations between these two filtering problems enable us to study various properties of Kalman filtering, including existence and uniqueness of Kalman filters, computation of performance indices of Kalman filtering, and performance limitations of Kalman filtering as related to the structural properties of the given system.     

Fuzzy model‐based fault detection for Markov jump systems

The robust fault detection filter (RFDF) design problems are studied for nonlinear stochastic time‐delay Markov jump systems. By means of the Takagi–Sugeno fuzzy models, the fuzzy RFDF system and the dynamics of filtering error generator are constructed. Moreover, taking into account the sensitivity to faults while guaranteeing robustness against unknown inputs, the H_∞ filtering scheme is proposed to minimize the influences of the unknown inputs and another new performance index is introduced to enhance the sensitivity to faults. A sufficient condition is first established on the stochastic stability using stochastic Lyapunov–Krasovskii function. Then in terms of linear matrix inequalities techniques, the sufficient conditions on the existence of fuzzy RFDF are presented and proved. Finally, the design problem is formulated as a two‐objective optimization algorithm. Simulation results illustrate that the proposed RFDF can detect the faults shortly after the occurrences. Copyright © 2008 John Wiley & Sons, Ltd.     

广义系统信息融合稳态与自校正满阶Kalman滤波器

基于线性最小方差标量加权融合算法和射影理论,对带多个传感器和带相关噪声的广义系统,提出了分布式标量加权融合稳态满阶Kalman滤波器.推得了任两个传感器子系统之间的稳态满阶滤波误差互协方差阵,其解可任选初值离线迭代计算.所提出的稳态融合滤波器避免了每时刻计算协方差阵和融合权重,减小了在线计算负担.当系统含有未知模型参数时,基于递推增广最小二乘算法和标量加权融合算法,提出了一种两段融合自校正状态滤波器.其中第1段融合获得未知参数的融合估计;第2段融合获得分布式自校正融合状态滤波器.与局部估计和加权平均融合估计相比,所提出的标量加权融合参数估计和自校正状态估计都具有更高的精度.仿真研究验证了其有效性.     

Kalman Filtering for Delayed Singular Systems with Multiplicative Noise

Kalman filtering problem for singular systems is dealt with, where the measurements consist of instantaneous measurements and delayed ones, and the plant includes multiplicative noise. By utilizing standard singular value decomposition, the restricted equivalent delayed system is presented, and the Kalman filters for the restricted equivalent system are given by using the well-known re-organization of innovation analysis lemma. The optimal Kalman filter for the original system is given based on the above Kalman filter by recursive Riccati equations, and a numerical example is presented to show the validity and efficiency of the proposed approach, where the comparison between the filter and predictor is also given.      

基于Huber M估计的鲁棒Cubature 卡尔曼滤波算法

Cubature卡尔曼滤波器(CKF)在非高斯噪声或统计特性未知时滤波精度将会下降甚至发散,为此提出了统计回归估计的鲁棒CKF算法.推导出线性化近似回归和直接非线性回归的鲁棒CKF算法,直接非线性回归克服了观测方程线性化近似带来的不足.具有混合高斯噪声的仿真实例比较了3种Cubature卡尔曼滤波器的滤波性能,结果表明这两种鲁棒CKF滤波精度及估计一致性明显优于CKF,直接非线性回归的CKF的鲁棒性更强,滤波性能更好.     

非高斯噪声下Kalman滤波熵理论算法研究

从一个新的角度结合具体的算法讲述了Kalman滤波器的原理,并对噪声为非高斯情况下结合熵的理论提出了假设,解决了噪声为非高斯情形下的滤波器设计的瓶颈。传统的Kalman滤波器是在噪声为高斯的情形下得出的最优滤波估计,但是现实生活中大多数噪声却是未知的、不确定性并且非高斯的。为了清楚说明熵原理应用于非高斯滤波器的设计结果,运用了数学统计的方法,对比滤波效果,说明了其可行性,证明了这种方法更适应于对噪声情况未知、参数不明确的情况,为研究广义噪声的随机系统提出了一种新的通用的解决途径。