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.     

Unbiased minimum-variance input and state estimation for linear discrete-time systems with direct feedthrough

This paper extends previous work on joint input and state estimation to systems with direct feedthrough of the unknown input to the output. Using linear minimum-variance unbiased estimation, a recursive filter is derived where the estimation of the state and the input are interconnected. The derivation is based on the assumption that no prior knowledge about the dynamical evolution of the unknown input is available. The resulting filter has the structure of the Kalman filter, except that the true value of the input is replaced by an optimal estimate.     

Extension of minimum variance estimation for systems with unknown inputs

In this paper, we address the problem of minimum variance estimation for discrete-time time-varying stochastic systems with unknown inputs. The objective is to construct an optimal filter in the general case where the unknown inputs affect both the stochastic model and the outputs. It extends the results of Darouach and Zasadzinski (Automatica 33 (1997) 717) where the unknown inputs are only present in the model. The main difficulty in treating this problem lies in the fact that the estimation error is correlated with the systems noises, this fact leads generally to suboptimal filters. Necessary and sufficient conditions for the unbiasedness of this filter are established. Then conditions under which the estimation error and the system noises are uncorrelated are presented, and an optimal estimator and a predictor filters are derived. Sufficient conditions for the existence of these filters are given and sufficient conditions for their stability are obtained for the time-invariant case. A numerical example is given in order to illustrate the proposed method.     

Joint attack detection and secure state estimation of cyber‐physical systems

This paper deals with secure state estimation of cyber‐physical systems subject to switching (on/off) attack signals and injection of fake packets (via either packet substitution or insertion of extra packets). The random set paradigm is adopted in order to model, via random finite sets (RFSs), the switching nature of both system attacks and the injection of fake measurements. The problem of detecting an attack on the system and jointly estimating its state, possibly in the presence of fake measurements, is then formulated and solved in the Bayesian framework for systems with and without direct feedthrough of the attack input to the output. This leads to the analytical derivation of a hybrid Bernoulli filter (HBF) that updates in real time the joint posterior density of a Bernoulli attack RFS and of the state vector. A closed‐form Gaussian mixture implementation of the proposed HBF is fully derived in the case of invertible direct feedthrough. Finally, the effectiveness of the developed tools for joint attack detection and secure state estimation is tested on two case studies concerning a benchmark system for unknown input estimation and a standard IEEE power network application.     

Secure estimation for cyber‐physical systems with adversarial attacks and unknown inputs: An L2‐gain method

This paper investigates the attack‐resilient state estimation problem for linear systems with adversarial attacks and unknown inputs, where the upper bound of the unknown inputs is unknown. It is assumed that the attacker has limited resources and can only manipulate a certain number of sensors. In most of the existing observer design approaches for the systems with unknown inputs even in the absence of attacks, the observer matching condition should be satisfied. To overcome this restriction, a novel switched observer is proposed, where the matched unknown inputs will be completely compensated by means of the outputs and the mismatched part will be suppressed in terms of L₂‐gain rejection property. Meanwhile, the observer can provide an attack‐resilient state estimation. Compared with the existing results, the proposed observer can guarantee that the resulting observer error system is stable with unknown input attenuation level γ that can be optimized. Finally, a simulation example of an unmanned ground vehicle is provided to show the effectiveness of the proposed approach.     

Pre‐filter design for tracking error specifications in QFT

This note shows how the pre‐filter may be designed in quantitative feedback design of single‐input, single‐output systems with tracking error specifications (Eitelberg, Automatica, 2000; 36(2):319). The method uses gain and phase information for the pre‐filter design. The design is conveniently performed on the log polar complex plane using standard CAD tools. Copyright © 2003 John Wiley & Sons, Ltd.     

Applying a novel extended Kalman filter to missile–target interception with APN guidance law: A benchmark case study

This paper considers the estimation of the target acceleration with unknown dynamics along with other states of a benchmark example of a nonlinear 2D missile–target engagement system in presence of model uncertainties and measurement noises. The objective is to implement the augmented proportional navigation (APN) guidance law for the missile–target interception to minimize the distance between the missile and the target. The estimated target acceleration can be treated as an unknown input to the nonlinear 2D missile–target engagement system. A novel analytical recursive approach referred to as extended Kalman filter with unknown inputs without direct feedthrough (EKF-UI-WDF) is derived with the weighted least squares estimation for an extended state vector including states and unknown inputs which can be any type of signals without prior information. By applying the proposed EKF-UI-WDF approach to a 2D missile–target interception control system, simulation results demonstrate that this approach is capable of (i) estimating the states and unknown input (target acceleration) well, and (ii) achieving more reasonable interception performance comparing with the traditional extended Kalman filter (EKF) approach.     

Estimation of the disturbance structure from data using semidefinite programming and optimal weighting

Designing a state estimator for a linear state-space model requires knowledge of the characteristics of the disturbances entering the states and the measurements. In [Odelson, B. J., Rajamani, M. R., & Rawlings, J. B. (2006). A new autocovariance least squares method for estimating noise covariances. Automatica, 42(2), 303-308], the correlations between the innovations data were used to form a least-squares problem to determine the covariances for the disturbances. In this paper we present new and simpler necessary and sufficient conditions for the uniqueness of the covariance estimates. We also formulate the optimal weighting to be used in the least-squares objective in the covariance estimation problem to ensure minimum variance in the estimates. A modification to the above technique is then presented to estimate the number of independent stochastic disturbances affecting the states. This minimum number of disturbances is usually unknown and must be determined from data. A semidefinite optimization problem is solved to estimate the number of independent disturbances entering the system and their covariances.     

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.     

Author's reply: “A comment on optimal control of SISO continuous-time systems”

In the above-mentioned comment, the author points out a technical problem with the paper (Wang, Z. Q., & Sznaier, M. (1997). Automatica, 33(1), 85–90). As we show here, this technical problem can be easily solved. Moreover, it affects neither the main formulation nor the results, which remain valid.     

Improved Input and State Estimation for Linear Stochastic Systems with Direct Feedthrough

This paper is concerned with the problem of joint input and state estimation for linear stochastic systems with direct feedthrough. Based on the fact that each unknown input between any two time steps is always bounded, a novel improved algorithm is proposed. Compared with existing results, this algorithm can effectively enhance estimation accuracy. Moreover, the stability of the algorithm is also discussed. Finally, an illustrative example is given to demonstrate the effectiveness of the proposed approach.     

基于条件高斯分布的未知输入与状态估计算法

针对未知输入同时存在于系统方程和测量方程的直接馈通线性随机系统, 提出了一种同时估计未知输入 和状态的算法. 首先, 通过将未知输入模型描述为有限方差的高斯分布, 利用条件高斯分布的性质, 推导出新的滤波 算法, 以同时得到未知输入估计和状态估计. 其次, 证明了当未知输入的方差趋于无穷大时, 本文提出的算法等价于 已有的递归三步滤波算法. 最后, 分析了本文算法的渐进稳定性条件, 结果表明, 与已有算法相比, 本文的算法适用 范围更广.     

On the optimality of two‐stage Kalman filtering for systems with unknown inputs

This paper is concerned with the optimal solution of two‐stage Kalman filtering for linear discrete‐time stochastic time‐varying systems with unknown inputs affecting both the system state and the outputs. By means of a newly‐presented modified unbiased minimum‐variance filter (MUMVF), which appears to be the optimal solution to the addressed problem, the optimality of two‐stage Kalman filtering for systems with unknown inputs is defined and explored. Two extended versions of the previously proposed robust two‐stage Kalman filter (RTSKF), augmented‐unknown‐input RTSKF (ARTSKF) and decoupled‐unknown‐input RTSKF (DRTSKF), are presented to solve the general unknown input filtering problem. It is shown that under less restricted conditions, the proposed ARTSKF and DRTSKF are equivalent to the corresponding MUMVFs. An example is given to illustrate the usefulness of the proposed results. Copyright © 2010 John Wiley and Sons Asia Pte Ltd and Chinese Automatic Control Society     

Simultaneous state and input estimation of hybrid systems with unknown inputs

This paper addresses the problem of the simultaneous state and input estimation for hybrid systems when subject to input disturbances. The proposed algorithm is based on the moving horizon estimation (MHE) method and uses mixed logical dynamical (MLD) systems as equivalent representations of piecewise affine (PWA) systems. So far the MHE method has been successfully applied for the state estimation of linear, hybrid, and nonlinear systems. The proposed extension of the MHE algorithm enables the estimation of unknown inputs, or disturbances, acting on the hybrid system. The new algorithm is shown to improve the convergence characteristics of the MHE method by reducing the delay of convergent estimates, while assuring convergence for every possible sequence of input disturbances. To ensure convergence the system is required to be incrementally input observable, which is an extension to the classical incremental observability property.     

状态估计与未知输入和可测噪声同时重构方法

针对一类不确定线性系统,讨论了状态估计及未知输入和可测噪声重构方法。首先,对仅具有未知输入的线性系统,讨论了观测器匹配条件不满足前提下的状态估计和未知输入重构问题;通过设计降维观测器和高阶滑模观测器,提出一种未知输入代数重构方法;然后,将以上结论上升到具有未知输入和可测噪声的线性系统,以此提出了状态估计及未知输入和可测噪声同时重构的方法;最后,通过对飞行器模型进行仿真,验证了所提出方法的有效性。     

离散Lipschitz非线性系统状态和未知输入估计

针对含有未知输入和可测噪声的离散Lipschitz非线性系统,研究了状态估计、未知输入以及可测噪声同时估计的问题.首先,对含有未知输入的系统,设计了比例积分观测器,达到同时估计系统状态和未知输入之目的.分析了残差系统的观测性和稳定性,利用H_∞实现该观测器对时变未知输入的有效估计;其次,将观测器增益矩阵的求解转化为求解线性矩阵不等式的形式;进一步地,基于系统状态扩展方法,将所提方法推广至同时含有未知输入和可测噪声的系统;最后,通过两个仿真算例说明了所提方法的正确性和有效性.     

Multiple sliding mode observers and unknown input estimations for Lipschitz nonlinear systems

Multiple sliding mode observers for state and unknown input estimations of a class of MIMO nonlinear systems are systematically developed in this paper. A new nonlinear transformation is formulated to divide the original system into two interconnected subsystems. The unknown inputs are assumed to be bounded and not necessarily Lipschitz, and do not require any matching condition. Under structural assumptions for the unknown input distribution matrix, the sliding mode terms of the nonlinear observer are designed to track their respective unknown inputs. Also, the unknown inputs can be reconstructed from the multiple sliding mode structurally. The conditions for asymptotic stability of estimation error dynamics are derived. Finally, simulation results are given to demonstrate the effectiveness of the proposed method. Copyright © 2011 John Wiley & Sons, Ltd.     

Recursive Filter with Partial Knowledge on Inputs and Outputs

 This paper investigates the problem of state estimation for discrete-time stochastic linear systems, where additional knowledge on the unknown inputs is available at an aggregate level and the knowledge on the missing measurements can be described by a known stochastic distribution. Firstly, the available knowledge on the unknown inputs and the state equation is used to form the prior distribution of the state vector at each time step. Secondly, to obtain an analytically tractable likelihood function, the effect of missing measurements is broken down into a systematic part and a random part, and the latter is modeled as part of the observation noise. Then, a recursive filter is obtained based on Bayesian inference. Finally, a numerical example is provided to evaluate the performance of the proposed methods.     

A note on interval observer design for unknown input estimation

This paper addresses the problem of interval observer design for unknown input estimation in linear time-invariant systems. Although the problem of unknown input estimation has been widely studied in the literature, the design of joint state and unknown input observers has not been considered within a set-membership context. While conventional interval observers could be used to propagate with some additional conservatism, unknown inputs by considering them as disturbances, the proposed approach allows their estimation. Under the assumption that the measurement noise and the disturbances are bounded, lower and upper bounds for the unmeasured state and unknown inputs are computed. Numerical simulations are presented to show the efficiency of the proposed approach.     

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