LMI-based minimax estimation and filtering under unknown covariances

In this paper, we consider a minimax approach to the estimation and filtering problems in the stochastic framework, where covariances of the random factors are completely unknown. The term ‘random factors’ refers either to unknown parameters and measurement noise in the estimation problem or to disturbance process and the initial state of a linear discrete-time dynamic system in the filtering problem. We introduce a notion of the attenuation level of random factors as a performance measure for both a linear unbiased estimate and a filter. This is the worst-case variance of the estimation error normalised by the sum of variances of all random factors over all nonzero covariance matrices. It is shown that this performance measure is equal to the spectral norm of the ‘transfer matrix’ and therefore the minimax estimate and filter can be computed in terms of linear matrix inequalities (LMIs). Moreover, the explicit formulae for both the minimax estimate and the minimal value of the attenuation level are presented in the estimation problem. It turns out that the above attenuation level of random factors coincides with the attenuation level of deterministic factors that is the worst-case normalised squared Euclidian norm of the estimation error over all nonzero sample values of random factors. In addition, we demonstrate that the LMI technique can be applied to derive the optimal robust estimator and filter, when there is a priori information about convex polyhedral sets which unknown covariance matrices of random factors belong to. Two illustrative examples show advantages of the minimax approach proposed.     

Minimax estimation by probabilistic criterion

A minimax estimation problem in multidimensional linear regression model containing uncertain parameters and random quantities is considered. Simultaneous distribution of random quantities that are a part of the observation model is not prescribed exactly; however, it has a fixed mean and a covariance matrix from the given set. For estimation algorithm optimization, we applied a minimax approach with the risk measure in the form of the exceedance probability of the estimate of a prescribed level by an error. It was shown that a linear estimation problem is equivalent to the minimax problem with the mean-square criterion. In addition, the corresponding linear estimate will be the best (in the minimax sense) by the probabilistic criterion at the class of all unbiased estimates. The least favorable distribution of random model parameters is also constructed. Several partial cases and a numerical example are considered.     

The Optimality of Linear Estimation Algorithms in Minimax Identification

The optimality of linear estimates in minimax estimation of a stochastically uncertain vector in a linear observation model by mean-square criterion is studied. In the Gaussian case, a uniformly optimal linear estimate is shown to exist in the class of all unbiased estimates. Moreover, it is minimax in the class of all nonlinear estimates if the nonrandom parameters of the observation model are unbounded. If the a priori information on random parameters are given as constraints on the covariance matrix, linear estimates are shown to be minimax.     

Comments on ‘ Dynamic feedback methodology for interconnected control systems ’ and ‘ Dynamic multilevel optimization for a class of non-linear systems ’

The paper considers the Kalman-Bucy filter for a linear system when the measurement noise covariance matrix is singular. It is shown that the problem of infimizing the square of a linear functional of the state estimation error is the dual of the optimal singular linear regulator problem. Furthermore there is an optimal reduced-order Kalman-Bucy filter for minimization of the trace of the state error covariance matrix, when all extremal controls for a dual regulator have finite order of singularity, and no Luenberger observer is needed. The proof is constructive. Necessary and sufficient conditions for the existence of a reduced-order optimal estimator are derived.     

Estimation of the matrices of parameters and covariations of the perturbation vectors in multidimensional discrete-time dynamic systems under special structure of the unknown covariance matrices

For the multidimensional linear dynamic system obeying a difference equation with an unknown covariance matrix of the vector of random perturbations having dependent components, consideration was given to estimation of the matrix of system parameters and the covariance matrix represented by a linear combination of the given symmetrical matrices. The family of joint probability densities of the observation vector was factorized, and the sufficient statistics was determined. For the estimates of the maximum likelihood of the system parameter matrices and the estimates of the coefficients of expansion of the covariance matrix, equations were presented. Developed was a recurrent procedure for joint estimation of the system parameter matrices and the covariance matrix with arrival of observations.     

Robust filtering of process in the stationary difference stochastic system

A technique to construct the robust Kalman filter for process estimation in the difference linear stationary stochastic system with an unknown covariance observation error matrix was developed. Consideration was given to the algorithm of constructing the set of permissible covariance matrices from a priori statistical data. A numerical method for solution of the general minimax optimization problem was proposed; and on its basis an iterative algorithm to calculate the robust filter parameters was developed, and its convergence was proved. Results of the numerical experiment were presented.     

Optimal-observer design for linear dynamical systems with uncertain parameters

This paper presents a design technique for the synthesis of robust observers for linear dynamical systems with uncertain parameters. The perturbations under consideration are modelled as unknown but bounded disturbances and an ellipsoidal set-theoretic approach is used to formulate the optimal-observer design problem. The optimal criterion introduced here is the minimization of the ‘size’ of the bounding ellipsoid of the estimation error. A necessary and sufficient condition for this optimal design problem is presented. The results are stated in terms of a reduced-order observer with constant gain matrix, which is then determined by solving a matrix Riccati-type equation. Furthermore, a gradient-search algorithm is presented to find the optimal solution when the free parameter that enters in the construction of the bounding ellipsoids of the estimation error is considered as a design parameter. The effectiveness of the proposed approach is illustrated through a numerical example.     

Linear filtering with adaptive adjustment of the disturbance covariation matrices in the plant and measurement noise

For filtering a nonstationary linear plant under the unknown intensities of input signals such as plant disturbances and measurement noise, a new algorithm was presented. It is based on selecting the vectors of values of these signals compatible with the observed plant output and minimizing the error variances of the last predicted measurement. The measurement prediction is determined from the Kalman filter where the input signals are assumed to be white noise and the covariance matrix coincides with the empirical covariance matrix of the selected vectors. Numerical modeling demonstrated that the so-calculated filter coefficients are close to the optimal ones constructed from the true covariance matrices of plant disturbances and measurement noise. The approximate Newton method for minimization of the prediction error variance was shown to agree with the solution of the auxiliary optimal control problem, which allows to make one or some few iterations to find the point of minimum.     

The optimal filtering problem for a discrete-time distributed parameter system

In this paper, the optimal filtering problem for a discrete-time linear distributed parameter system is considered. Using the least squares estimation error criterion, the Wiener-Hopf equation for the discrete-time distributed parameter system is derived. Based on the Wiener-Hopf equation, the equations satisfied by the optimal filtering estimate and the minimum error covariance matrix function are derived by using the matrix inversion lemma for a distributed parameter system. Finally, we show that the approximation of the results obtained for a distributed parameter system by using the Fourier expansion method produces those of the Kalman filtering problem for the lumped parameter system.     

A decision theoretic approach to parameter estimation

A decision theoretic approach to estimation of unknown random and nonrandom parameters from a linear measurements model is proposed, when the a priori statistics are incomplete and only a small number of data points are available. The unknown statistics are partially characterized by considering two regions in the measurement space, namely, good and bad data regions and constraining the partial probability, the partial covariance, or the combination thereof of the measurements. The random parameter is assumed to be Gaussian variable with known mean and known covariance. Choosing the minimum covariance criterion, the min-max estimator is found to be soft-limiter or tangent type nonlinear function depending upon the a priori statistic available. The estimator for the unknown nonrandom parameter is obtained from the root of some function of the residuals, the function being obtained by minimizing the error covariance. The estimator obtained is similar to a random parameter case.     

随机逼近自适应滤波在捷联惯导系统初始对准中的应用

对于测量噪声方差未知的捷联惯导系统(SINS),采用常规Kalman滤波进行初始对准会造成较大状态估计误差,甚至使滤波器发散。为了解决系统测量噪声方差未知或不确切知道时SINS的误差估计问题,提出一种基于随机逼近的自适应滤波方法。该方法将Robbins-Monro算法与Kalman滤波相结合,通过简化求逆运算,解决了系统观测噪声特性未知情况下SINS的误差估计问题,并提高了算法的数值稳定性。仿真结果表明,该方法能在系统测量噪声方差未知情况下有效实现SINS初始对准。     

带未知通信干扰和丢包补偿的多传感器网络化不确定系统的分布式融合滤波

研究了带有未知通信干扰、观测丢失和乘性噪声不确定性的多传感器网络化系统的状态估计问题.通过白色乘性噪声描述系统状态和观测中的随机不确定性,采用一组服从Bernoulli分布的随机变量描述网络传输过程中存在的观测丢失现象,且数据传输中存在未知的网络通信干扰.当发生丢包时,以当前丢失观测的预报值进行补偿.对每个单传感器子系统,应用线性无偏最小方差估计准则设计了不依赖于未知通信干扰的最优线性滤波器.推导了任两个局部滤波误差之间的互协方差阵.进而,应用矩阵加权融合估计算法给出了分布式融合状态滤波器.仿真例子验证了算法的有效性.     

Decision theoretic approach to real-time robust identification

A decision-theoretic approach to the estimation of unknown parameters from a linear discrete-time dynamic measurement model in the presence of disturbance uncertainty is considered. The unknown disturbance statistics are characterized by a certain class of distributions to which the real disturbance distribution is confined. Using game theory and the asymptotic estimation error covariance matrix as the criteria of how good an estimator is, the stochastic gradient-type algorithm is shown to be optimal in the min-max sense. Since the optimal solution is not tractable in practice, several suboptimal procedures are derived on the basis of suitable approximations. The convergence of the derived algorithms is established theoretically using the ordinary differential equation approach. Monte Carlo simulation results are presented for the quantitative performance evaluation of the algorithms.     

Z基于传感器网络的远程状态估计

针对传感器网络中的远程状态估计, 提出一种多传感器切换的卡尔曼滤波器. 通过分析估计误差的统计特性, 证明估计误差的协方差具有边界, 采用线性矩阵不等式的形式给出了边界的收敛条件. 研究测量数据丢失对估计器性能的影响, 使用临界到达概率作为估计器的稳定性判据, 得到采用线性矩阵不等式求解临界到达概率的方法. 数值仿真证实了结论的正确性.     

Linear filtering for time-varying systems using measurements containing colored noise

The Kalman-Bucy filter for continuous linear dynamic systems assumes all measurements contain "white" noise, i.e. noise with correlation times short compared to times of interest in the system. It is shown here that if correlation times are not short, or if some measurements are free of noise, the optimal filter is a modification of the Kalman-Bucy filter which, in general, contains differentiators as well as integrators. It is also shown for this case that the estimate and its covariance matrix are, in general, discontinuous at the time when measurements are begun. The case of random bias errors in the measurements is shown by example to be a limiting case of colored noise.     

Adaptive divided difference filtering for simultaneous state and parameter estimation

A novel adaptive version of the divided difference filter (DDF) applicable to non-linear systems with a linear output equation is presented in this work. In order to make the filter robust to modeling errors, upper bounds on the state covariance matrix are derived. The parameters of this upper bound are then estimated using a combination of offline tuning and online optimization with a linear matrix inequality (LMI) constraint, which ensures that the predicted output error covariance is larger than the observed output error covariance. The resulting sub-optimal, high-gain filter is applied to the problem of joint state and parameter estimation. Simulation results demonstrate the superior performance of the proposed filter as compared to the standard DDF.     

The accuracy comparison of multisensor covariance intersection fuser and three weighting fusers

For multisensor systems with exactly known local filtering error variances and cross-covariances, a covariance intersection (CI) fusion steady-state Kalman filter without cross-covariances is presented. It is rigorously proved that it has consistency, and its accuracy is higher than that of each local Kalman filter and is lower than that of the optimal Kalman fuser with matrix weights. Under the unbiased linear minimum variance (ULMV) criterion, it is proved that the accuracy of the fuser with matrix weights is higher than that of the fuser with scalar weights, and the accuracy of the fuser with diagonal matrix weights is in between both of them, and the accuracies of all three weighting fusers and the CI fuser are lower than that of centralized Kalman fuser, and are higher than that of each local Kalman filter. The geometric interpretations of the above accuracy relations are given based on the covariance ellipsoids. A Monte-Carlo simulation example for tracking system verifies correctiveness of the proposed theoretical accuracy relations, and shows that the actual accuracy of the CI Kalman fuser is close to that of the optimal Kalman fuser, so that it has higher accuracy and good performance. When the actual local filtering error variances and cross-covariances are unknown, if the local filtering estimates are consistent, then the corresponding robust CI fuser is also consistent, and its robust accuracy is higher than that of each local filter.     

Active fault tolerant control using state-space self-tuning control approach

This paper presents a new fault tolerant control scheme for unknown multivariable stochastic systems by modifying the conventional state-space self-tuning control approach. For the detection of faults, a quantitative criterion is developed by comparing the innovation process errors occurring in the Kalman filter estimation algorithm, which, for faulty system recovery, a weighting matrix resetting technique is developed by adjusting and resetting the covariance matrices of the parameter estimate obtained in the Kalman filter estimation algorithm to improve the parameter estimation of the faulty systems. The proposed method can effectively cope with partially abrupt and/or gradual system faults and/or input failures with fault detection. The modified state-space self-tuning control scheme can be applied to the multivariable stochastic faulty system without requiring prior knowledge of system parameters and noise properties.     

Optimal location of process measurements

The problem of optimally locating a given number of Bensors for observing a general linear distributed parameter system is considered. Measurements at the sensors are assumed to be available continuously in time, and the design criterion is minimization of a scalar measure of the covariance of the estimate error in the optimal linear filter. Necessary conditions for optimality are derived based on the formulation of a distributed parameter matrix minimum principle. A computational algorithm is developed for determining the optimum set of measurement locations. The algorithm is applied to the problem of optimally locating temperature sensors in a solid undergoing transient heat conduction.     

A note on the nature of optimality in the discrete Kalman filter

A derivation is presented in which it is shown that the optimal gain sequence of the discrete Kalman filter minimizes not only the trace of the estimation covariance matrix, but any linear combination of the main diagonal elements of that matrix.