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     

Unbiased minimum-variance state estimation for linear systems with unknown input

The problem of state estimation for a linear system with unknown input, which affects both the system and the output, is discussed in this paper. A recursive optimal filter with global optimality in the sense of unbiased minimum variance over all linear unbiased estimators, is provided. The necessary and sufficient condition for the convergence and stability is also given, which is milder than existing approaches.     

Unscented Kalman filter for continuous‐time nonlinear fractional‐order systems with process and measurement noises

This study proposes the design of unscented Kalman filter for a continuous‐time nonlinear fractional‐order system involving the process noise and the measurement noise. The nonlinear fractional‐order system is discretized to get the difference equation. According to the unscented transformation, the design method of unscented Kalman filter for a continuous‐time nonlinear fractional‐order system is provided. Compared with the extended Kalman filter, the proposed method can obtain a more accurate estimation effect. For fractional‐order systems containing non‐differentiable nonlinear functions, the method proposed in this paper is still effective. The unknown parameters are also discussed by the augmented vector method to achieve the state estimation and parameter identification. Finally, two examples are offered to verify the effectiveness of the proposed unscented Kalman filter for nonlinear fractional‐order systems.     

On Kalman filtering for conditionally Gaussian systems with random matrices

We consider linear stochastic systems with additive white Gaussian noise, with the added generality that the system matrices are random and adapted to the observation process. The main result of this paper is that in order for the standard Kalman filter to generate the conditional mean and conditional covariance of the conditionally Gaussian distributed state, it is sufficient for the random matrices to be finite with probability one at each time instant. This generalizes the best previous results available to date, to our knowledge, which require the more stringent hypothesis that the entries of the random matrices should possess finite second moments at each time instant.

A significant application of the results of this paper is to the problem of recursive identification of the unknown parameters of a controlled linear stochastic system. In such problems, the observation matrix is typically generated by complicated nonlinear feedback, as for example in adaptive control, and the finiteness of the second moments is difficult, if not impossible, to establish, while the finiteness with probability one has been established in many applications.     

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.     

Finite‐horizon robust Kalman filtering for uncertain discrete time‐varying systems with state‐delay

A finite‐horizon robust estimator design approach is developed for a class of discrete time‐varying uncertain systems with state‐delay. It extends the Kalman filter to the case in which the considered system is subject to norm‐bounded uncertainties in both state and output matrices. The state and gain matrices of the designed filter are optimized to give a minimal upper bound such that the estimation error variance is guaranteed to lie within a certain bound for all admissible uncertainties. A simulation example is presented to show the effectiveness of the proposed approach by comparing to the traditional Kalman filtering method. Copyright © 2011 John Wiley and Sons Asia Pte Ltd and Chinese Automatic Control Society     

Decomposition of meta‐frontier inefficiency in the two‐stage network directional distance function with quasi‐fixed inputs

Few studies have considered the quasi‐fixed inputs that have an impact on hospitality. The number of guest rooms and the area of the catering department of international tourist hotels are shown to be constant in Taiwan from 2008 to 2009. Therefore, this paper proposes the use of the two‐stage network directional distance function with quasi‐fixed inputs to explore the productive and service efficiencies of tourist hotel formats (chained versus independent). In addition, a new approach is developed to decompose the technical inefficiency of the hotels within the meta‐frontier; this approach helps to identify the source of the meta‐frontier inefficiency. Demonstration of this new approach with practical data reveals that the average productive efficiency is greater than the average service efficiency for Taiwanese tourist hotels. Additionally, the causes of the overall productive inefficiency and the overall service inefficiency are mainly derived from the input excess of the productive process and the output shortfall of the service process, respectively.     

带未知有色观测噪声的自校正融合Kalman滤波器

对于带未知有色观测噪声的多传感器线性离散定常随机系统, 未知模型参数和噪声方差的一致的融合估值器用递推增广最小二乘法(RELS)和求解相关函数方程得到. 将这些估值器代入到最优解耦融合Kalman滤波器中, 得出了自校正解耦融合Kalman滤波器, 并用动态方差误差系统分析(DVESA)和动态误差分析(DESA)方法证明了它收敛于最优解耦融合Kalman滤波器, 因而具有渐近最优性. 一个带3传感器跟踪系统的仿真例子说明了其有效 性.     

Full-order observer design for linear systems with unknown inputs

In this article, a full-order observer without unknown inputs reconstruction is suggested in order to achieve finite-time reconstruction of the state vector for a class of linear systems with unknown inputs. The observer is a simple one, its derivation being direct and easy. It will be shown that the problem of full-order observers for linear systems with unknown inputs can be reduced in this case to a standard one (the unknown input vector will not interfere in the observer equations). The effectiveness of the suggested design algorithm is illustrated by a numerical example (aircraft longitudinal motion), and, for the same aircraft dynamics, we make a comparison between our new observer and other already existing observers from the existence conditions and dynamic characteristics’ point of view; the superiority of the new designed observer is demonstrated.     

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.     

A Framework for Globally Optimal State Estimation for Systems with Unknown Inputs (II): Untrammeled Approach

In this paper, a globally optimal state estimation, in light of unbiased minimum‐variance filtering over all linear unbiased estimators, is addressed. This paper is the second part of a comprehensive extension of original work by Hsieh, with the main aim being to develop an untrammeled filtering framework that does not need any transformations 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 more general derivation of the globally optimal state estimator (GOSE) for the GUMVSE is presented; (ii) a more direct proof, verifying the global optimality of the GOSE by finding the global minimum of the trace of the estimation error covariance, is constructed; and (iii) an application of the proposed result to re‐derive a specific transformation‐based GOSE, i.e., the recursive optimal filter proposed by Cheng et al., is illustrated. The relationship with previously proposed results is also addressed. 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     

Unscented and square‐root unscented Kalman filters for quaternionic systems

Proper construction of an unscented Kalman filter (UKF) for unit quaternionic systems is not straightforward due to the incompatibility between the algebraic properties of the unit quaternions and the common real vector space operations (additions and scalar multiplications) needed in the steps of a filter algorithm. This work studies, in detail, all UKFs and square‐root UKFs for quaternionic systems proposed in the literature. First, we classify the algorithms according to the preservation of the unity norm of the quaternion variables. Second, we propose two new algorithms: the quaternionic additive unscented Kalman filter (QuAdUKF) and a square‐root variant of it. The QuAdUKF encompasses all known UKFs for quaternionic systems of the literature preserving, in all steps, the norm of the unit quaternion variables. Besides, it can also yield new UKFs with this norm preservation property. The QuAdUKF's square‐root variant has better properties in comparison with all the square‐root UKFs for quaternionic systems of the literature. Numerical experiments for a spacecraft attitude estimation problem illustrate the theoretical results.     

含未知输入的广义系统的状态与输入估计

研究了含未知输入的广义系统的输入解耦观测器设计问题. 在系统脉冲能控的条件下通过系统输入-状态对的非奇异变换, 把此问题等价地转化为正常状态空间系统的相应问题. 用大家熟知的方法设计正常状态空间系统的观测器, 从而得到广义系统的输入解耦观测器. 然后用广义系统的观测器状态和系统输出的线性组合渐近估计系统的状态与未知输入.     

Extended Kalman filters for fractional‐order nonlinear continuous‐time systems containing unknown parameters with correlated colored noises

By using the Grünwald‐Letnikov (G‐L) difference method and the Tustin generating function method, this study presents extended Kalman filters to achieve satisfactory state estimation for fractional‐order nonlinear continuous‐time systems that containing some unknown parameters with the correlated fractional‐order colored noises. Based on the G‐L difference method and the Tustin generating function method, the difference equations corresponding to fractional‐order nonlinear continuous‐time systems are constructed respectively. The first‐order Taylor expansion is used to linearize the nonlinear functions in the estimated system, which provides the system model for extended Kalman filters. Using the augmented vector method, the unknown parameters are regarded as new state vectors, and the augmented difference equation is constructed. Based on the augmented difference equation, extended Kalman filters are designed to estimate the state of fractional‐order nonlinear systems with process noise as fractional‐order colored noise or measurement noise as fractional‐order colored noise. Meanwhile, the extended Kalman filters proposed in this paper can also estimate the unknown parameters effectively. Finally, the effectiveness of the proposed extended Kalman filters is validated in simulation with two examples.     

Robust Kalman filtering for uncertain discrete-time systems with probabilistic parameters bounded within a polytope

We consider the problem of finite horizon discrete-time Kalman filtering for systems with parametric uncertainties. Specifically, we consider unknown but deterministic uncertainties where the uncertain parameters are assumed to lie in a convex polyhedron with uniform probability density. The condition and a procedure for the construction of a suboptimal filter that minimizes an expected error covariance over-bound are derived.     

Designing fault detection observers for linear systems with mismatched unknown inputs

Algebraic unknown input observers (UIOs) that have been previously reported in the literature can be constructed under the assumption that linear systems with unknown inputs satisfy the so-called observer matching condition. This condition restricts practical applications of UIOs for fault detection and isolation (FDI). We present an algebraic design for fault detection observers (FDOs) for the case in which the observer matching condition is not satisfied. To loosen the restriction imposed by the observer matching condition, the UIO design method combined with the unknown input modeling technique is proposed to design an FDO that decouples the effect of mismatched unknown inputs. To do this, first, unknown inputs that denote the faults of no interest and process disturbances are decomposed into algebraically rejectable unknown inputs and modeled unknown inputs such that the observer matching condition is satisfied. Under the assumption that mismatched unknown inputs are deterministic and can be expressed as the responses of fictitious autonomous dynamical systems, an augmented system is obtained by combining the original system model with the unknown input model. Finally, through the design technique of a UIO for the augmented system, a reduced-order FDO is constructed to estimate an augmented state vector that consists of both the original state variables and the augmentative state variables. The estimated state is then used to generate the residual, which should be designed to be insensitive to unknown inputs while being sensitive to the faults of interest. Two numerical examples are provided to show the usefulness and the feasibility of the presented approach.     

Distributed adaptive high‐gain extended Kalman filtering for nonlinear systems

In this work, we propose a distributed adaptive high‐gain extended Kalman filtering approach for nonlinear systems. Specifically, we consider a class of nonlinear systems that are composed of several subsystems interacting with each other via their states. In the proposed approach, an adaptive high‐gain extended Kalman filter is designed for each subsystem. The distributed Kalman filters communicate with each other to exchange estimated subsystem state information. First, assuming continuous communication among the distributed filters within deterministic form of subsystems, an implementation strategy that specifies how the distributed filters should communicate is designed and the detailed design of the subsystem filter is described. Second, we consider the case of stochastic subsystems for which the designed subsystem filters communicate to exchange information at discrete‐time instants. A state predictor in each subsystem filter is used to provide predictions of states of other subsystems. The stability properties of the proposed distributed estimation schemes with both continuous and discrete communications are analyzed. Finally, the effectiveness and applicability of the proposed schemes are illustrated via the application to a chemical process example. Copyright © 2017 John Wiley & Sons, Ltd.     

Partial state observers for linear systems with unknown inputs

We consider the problem of constructing partial state observers for discrete-time linear systems with unknown inputs. Specifically, for any given system, we develop a design procedure that characterizes the set of all linear functionals of the system state that can be reconstructed through a linear observer with a given delay. By treating the delay as a design parameter, we allow greater flexibility in estimating state functionals, and are able to obtain a procedure that directly produces the corresponding observer parameters. Our technique is also applicable to continuous-time systems by replacing delayed outputs with differentiated outputs.     

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.