Estimation Error Covariance Matrices Research Articles

Optimal Distributed Kalman Filtering Fusion With Singular Covariances of Filtering Errors and Measurement Noises

In this paper, we present the globally optimal distributed Kalman filtering fusion with singular covariances of filtering errors and measurement noises. The following facts motivate us to consider the problem. First, the invertibility of estimation error covariance matrices is a necessary condition for most of the existing distributed fusion algorithms. However, it can not be guaranteed to exist in practice. For example, when state estimation for a given dynamic system is subject to state equality constraints, the estimation error covariance matrices must be singular. Second, the proposed fused state estimate is still exactly the same as the centralized Kalman filtering using all sensor raw measurements. Moreover, the existing performance analysis results on the distributed Kalman filtering fusion for the multisensor system with feedback are also extended to the singular covariance matrices of filtering error. The final numerical examples support the theoretical results and show an advantage of less computational burden.

Kalman Filter Riccati Equation for the Prediction, Estimation, and Smoothing Error Covariance Matrices

The classical Riccati equation for the prediction error covariance arises in linear estimation and is derived by the discrete time Kalman filter equations. New Riccati equations for the estimation error covariance as well as for the smoothing error covariance are presented. These equations have the same structure as the classical Riccati equation. The three equations are computationally equivalent. It is pointed out that the new equations can be solved via the solution algorithms for the classical Riccati equation using other well-defined parameters instead of the original Kalman filter parameters.

Unscented Kalman Filter Over Unreliable Communication Networks With Markovian Packet Dropouts

The paper investigates the stability of the discrete-time modified unscented Kalman filter (MUKF) in transmitting measurement outputs to the filter via a network. Since the communication channel states do not always vary in time, the arrival of the observations is modeled as a two state time-homogeneous Markov process γk. The stability of the estimation error covariance matrices at packet reception times is analyzed. Sufficient conditions (related to Hk and the initial conditions) for the peak covariance stability and the usual covariance stability are given. Also the relationship between the different types of stability notions is illustrated for systems with i.i.d observation dropouts. Numerical example is given to illustrate the effectiveness of the techniques developed.

Stochastic stability of extended filtering for non‐linear systems with measurement packet losses

This study is concerned with stochastic stability of a new extended filtering for non-linear systems subject to measurement packet losses. The measurements sensored are transmitted to the estimator through a packet-dropping network. By introducing a time-stamped packet arrival indicator sequence, the measurement loss process is modelled as an independent, identically distributed (i.i.d.) and therefore a Bernoulli process. The boundedness of estimation error covariance matrices is proved by showing the existence of a critical threshold for measurement packet arrival probability. It is also shown that, under appropriate assumptions, the estimation error remains bounded as long as the noise covariance matrices and the initial estimation error can be ensured small enough. Finally, simulation results validating the effectiveness of this proposed filtering framework are also presented.

Mean square stability for Kalman filtering with Markovian packet losses

This paper studies the stability of Kalman filtering over a network subject to random packet losses, which are modeled by a time-homogeneous ergodic Markov process. For second-order systems, necessary and sufficient conditions for stability of the mean estimation error covariance matrices are derived by taking into account the system structure. While for certain classes of higher-order systems, necessary and sufficient conditions are also provided to ensure stability of the mean estimation error covariance matrices. All stability criteria are expressed by simple inequalities in terms of the largest eigenvalue of the open loop matrix and transition probabilities of the Markov process. Their implications and relationships with related results in the literature are discussed.

Lossless Linear Transformation of Sensor Data for Distributed Estimation Fusion

In distributed estimation fusion, processed data from each sensor is sent to the fusion center. By taking linear transformation of the raw measurements of each sensor, two optimal distributed fusion algorithms are proposed in this paper. Compared with existing fusion algorithms, they have three nice properties. First, they are optimal in the sense that they are equivalent to the optimal centralized fusion. Second, their communication requirements from each sensor to the fusion center are equal to or less than those of the centralized and most existing distributed fusion algorithms. Third, they do not need the inverses of estimation error covariance matrices, which are assumed to exist in most existing algorithms but can not be guaranteed to exist. So the proposed algorithms can be applied in more cases. Pros and cons of these two new algorithms are analyzed. A possible way to reduce the computational complexity of the new algorithms, an extension to the case of a singular covariance matrix of measurement noise, and an extension to the reduced-rate communication case for some simple systems are also discussed.

Signal estimation based on covariance information from observations featuring correlated uncertainty and coming from multiple sensors

The linear least-squares estimation problem of signals from observations coming from multiple sensors is addressed when there is a non-zero probability that each observation does not contain the signal to be estimated ( uncertain observations). At each sensor, this uncertainty in the observations is modeled by a sequence of Bernoulli random variables correlated at consecutive sampling times. To estimate the signal, recursive filtering and (fixed-point and fixed-interval) smoothing algorithms are derived without requiring the knowledge of the signal state-space model but only the means and covariance functions of the processes involved in the observation equations, the uncertainty probabilities and the correlation between the variables modeling the uncertainty. To measure the estimation accuracy, recursive expressions for the estimation error covariance matrices are also proposed. The theoretical results are illustrated by a numerical simulation example where a signal is estimated from observations featuring correlated uncertainty and coming from two sensors with different uncertainty characteristics.

Linear least-squares estimation based on covariances from multiple correlated uncertain observations

In this paper, the linear least-squares estimation problem of signals from correlated uncertain observations coming from multiple sensors is addressed. It is assumed that, at each sensor, the signal is measured in the presence of additive white noise and that the uncertainty in the observations is characterized by a set of Bernoulli random variables which are only correlated at consecutive time instants. Assuming that the probability and correlation of such variables are not necessarily the same for all the sensors, a recursive filtering and fixed-point smoothing algorithm is proposed. The derivation of such algorithm does not require the knowledge of the signal state-space model, but only the covariance functions of the processes involved in the observation equation of each sensor, as well as the probability and correlation of the Bernoulli variables modeling the uncertainty. Recursive expressions for the estimation error covariance matrices are also provided, and the performance of the estimators is illustrated by a numerical simulation example wherein a signal is estimated from correlated uncertain observations coming from two sensors with different uncertainty characteristics.

Linear and quadratic least-squares estimation using measurements with correlated one-step random delay

This paper considers the linear and quadratic least-squares estimation of a discrete-time signal from observations randomly delayed by one sampling time such that the delay at a given time instant depends on a previous delay. It is assumed that the signal is measured with an additive white noise and that the delay in the observations is characterized by a set of Bernoulli variables which are correlated when the difference between times is equal to a certain value m. Linear and quadratic recursive filtering and fixed-point smoothing algorithms for such a class of models are constructed using an innovation approach; they do not require full knowledge of the state-space model for the signal process, but just the moments up to the fourth order of the signal (admitting a separable form) and the observation noise, as well as the probability and correlation of the Bernoulli variables modelling the delay. Recursive expressions for the estimation error covariance matrices are also given, and the performance of the different estimators is illustrated by means of a numerical example.

Optimal filtering and smoothing for discrete-time stochastic singular systems

A stochastic singular system with correlated noises at the same time is transferred to the equivalent nonsingular system with correlated noises at the same and neighboring time. Applying time-domain innovation analysis method, the recursive full-order predictor, filter and smoother are presented for this nonsingular system. Further, the full-order filter and smoother are given for original stochastic singular linear systems with correlated noises. Recursive and nonrecursive computational formulas for estimation error covariance matrices are given. Furthermore, the steady-state filter and smoother are also investigated. The asymptotic stability is proved. All results generalize the standard Kalman filtering. A simulation example shows the effectiveness.

A quantization architecture for track fusion

Many practical multi-sensor tracking systems are based on some form of track fusion, in which local track estimates and their associated covariances are shared among sensors. Communication load is a significant concern, and the goal of this paper is to propose an architecture for low-bandwidth track fusion. The scheme involves intelligent scalar and vector quantization of the local state estimates and of the associated estimation error covariance matrices. Simulation studies indicate that the communication saving can be quite significant, with only minor degradation of track accuracy.

On the robustness of the finite-length MMSE-DFE with respect to channel and second-order statistics estimation errors

The finite-length minimum mean square error decision-feedback equalizer (MMSE-DFE) is an efficient structure mitigating intersymbol interference (ISI) introduced by practically all communication channels at high-enough symbol rates. The filters constituting the MMSE-DFE, as well as related performance measures, can be computed by assuming perfect knowledge of the channel impulse response and the input and noise second-order statistics (SOS). In practice, we estimate the unknown quantities, and thus, inevitable estimation errors arise. We model the estimation errors as small perturbations, and we derive a second-order approximation to the excess MSE. Furthermore, we derive second-order approximations to the mean excess MSE in terms of the parameter estimation error covariance matrices and simple and informative bounds, revealing the factors that govern the behavior of MMSE-DFE under mismatch. Simulations confirm that the derived second-order approximations provide accurate estimates of the MMSE-DFE performance degradation due to mismatch.

On the robustness of the finite-length MMSE-DFE with respect to channel and second-order statistics estimation errors

The finite-length minimum mean square error decision-feedback equalizer (MMSE-DFE) is an efficient structure mitigating intersymbol interference (ISI) introduced by practically all communication channels at high-enough symbol rates. The filters constituting the MMSE-DFE, as well as related performance measures, can be computed by assuming perfect knowledge of the channel impulse response and the input and noise second-order statistics (SOS). In practice, we estimate the unknown quantities, and thus, inevitable estimation errors arise. We model the estimation errors as small perturbations, and we derive a second-order approximation to the excess MSE. Furthermore, we derive second-order approximations to the mean excess MSE in terms of the parameter estimation error covariance matrices and simple and informative bounds, revealing the factors that govern the behavior of MMSE-DFE under mismatch. Simulations confirm that the derived second-order approximations provide accurate estimates of the MMSE-DFE performance degradation due to mismatch.

Forward-only and forward-backward sample covariances – A comparative study

In some applications the covariance matrix of the observations enjoys a particular symmetry: it is not only symmetric with respect to its main diagonal but also with respect to the anti-diagonal. The standard forward-only sample covariance estimate does not impose this extra symmetry. In such cases one often uses the so-called forward-backward sample covariance estimate. In this paper, a direct comparative study of the relative accuracy of the two sample covariance estimates is performed. An explicit expression for the difference between the estimation error covariance matrices of the two sample covariance estimates is given. This expression shows quantitatively the gain of using the forward-backward estimate compared to the forward-only estimate. The presented results are also useful in the analysis of estimators based on either of the two sample covariances. As an example, spatial power estimation by means of the Capon method is considered. Using a second-order approximation, it is shown that Capon based on the forward-only sample covariance (F-Capon) underestimates the power spectrum, and also that the bias for Capon based on the forward-backward sample covariance is half that of F-Capon.

A contribution to performance prediction for probabilistic data association tracking filters

The probabilistic data association (PDA) algorithm for tracking in clutter contains a stochastic (data-dependent) Riccati equation for updating the estimation error covariance matrix. This note details a simple analytic approximation to the stochastic Riccati equation that allows precomputation of the estimation error covariance matrices. The potential of the approximation for performance analysis of PDA-based tracking algorithm is demonstrated using a simple example.

Assimilation of TOPEX sea level measurements with a reduced‐gravity, shallow water model of the tropical Pacific Ocean

Sea surface height variability measured by TOPEX is analyzed in the tropical Pacific Ocean by way of assimilation into a wind‐driven, reduced‐gravity, shallow water model using an approximate Kalman filter and smoother. The analysis results in an optimal fit of the dynamic model to the observations, providing a dynamically consistent interpolation of sea level and estimation of the circulation. Nearly 80% of the expected signal variance is accounted for by the model within 20° of the equator, and estimation uncertainty is substantially reduced by the voluminous observation. Notable features resolved by the analysis include seasonal changes associated with the North Equatorial Countercurrent and equatorial Kelvin and Rossby waves. Significant discrepancies are also found between the estimate and TOPEX measurements, especially near the eastern boundary. Improvements in the estimate made by the assimilation are validated by comparisons with independent tide gauge and current meter observations. The employed filter and smoother are based on approximately computed estimation error covariance matrices, utilizing a spatial transformation and an asymptotic approximation. The analysis demonstrates the practical utility of a quasi‐optimal filter and smoother.

Optimal measurement scheduling for state estimation

We consider the problem of optimal allocation of measurement resources, when: (1) the total measurement budget and time duration of measurements are fixed, and (2) the cost of an individual measurement varies inversely with the (controllable) measurement accuracy. The objective is to determine the time-distribution of measurement variances that minimizes a measure of error in estimating a discrete-time, vector stochastic process with known auto-correlation matrix using a linear estimator. The metric of estimation error is the trace of weighted sum of estimation error covariance matrices at various time indices. We show that this problem reduces to a nonlinear optimization problem with linear equality and inequality constraints. The solution to this problem is obtained via a variation of the projected Newton method. For the special case when the vector stochastic process is the state of a linear, finite-dimensional stochastic system, the problem reduces to the solution of a nonlinear optimal control problem. In this case, the gradient and Hessian with respect to the measurement costs are obtained via the solution of a two-point boundary value problem and the resulting optimization problem is solved via a variation of the projected Newton method. The proposed method is illustrated using four examples. >

Second-order state estimation experiments using acceleration measurements

The estimation of dynamic states for feedback control of structural systems using second-order differential equations and acceleration measurements is described. The formulation of the observer model and the design of the observer gains are discussed in detail. It is shown that the second-order observer is highly stable because the stability constraints on the observer gains are model independent. The limitation of the proposed observer is the need for nearly collocated actuators and accelerometers. Experimental results using a control-struct ure interaction testbed are presented that show that the second-order observer provided more stability than a Kalman filter estimator without decreasing closed-loop performance. F LEXIBLE-BODY vibration control of large structural systems often requires estimation of dynamic states that cannot be measured directly. The state estimator or observer plays a key role in the overall stability of the control system. Reference 1 has shown that the linear quadratic regulator (LQR) technique provides 60 deg of phase margin and infinite gain margin when the full state is available. However, when an observer must be used in conjunction with LQR, no guaran- tees can be made on the robustness properties due to state reconstruction errors.2'3 Observer state reconstruction suffers from model errors, measurement noise, and unmodeled dy- namics.4 Although much work has been done on optimal state estimation using first-order Kalman filters, the closed-loop stability remains a strong function of the observer model accuracy. Although there is more design freedom when using a first- order model, second-order observers provide a great deal of physical insight when scientists attempt to perform state esti- mation of a second-order system. Second-order observer mod- els have received recent attention in the literature. Reference 5 utilized a Kalman filter for discretized (time and space) sec- ond-order structure models. In addition, robust computa- tional procedures for solution of the Kalman filter estimation error covariance matrices have been developed for second-or- der models in Ref. 6. A dissipative observer for use with velocity measurements that possessed the second-order form was introduced in Ref. 7. This observer was analogous to the collocated controller whereby symmetric positive definite feedback gains were designed to insure stability. Recently, the authors of Ref. 8 developed the conditions for optimal estima- tion using the second-order form assuming velocity measure- ments were available. Accelerometers are frequently used to measure the dynamic response of structural systems because of their low cost and their light weight and because they provide an inertial mea- surement. The first-order observer filters acceleration mea- surements in both space and time to derive the proper phase information to reconstruct the unmeasured states. Spatial fil- tering, which projects the measurement data onto the system coordinate basis, requires accurate dynamic models. The re- duced design freedom available with the second-order ob- server has usually required velocity measurements since nei- ther spatial nor temporal filtering of the measurement is performed. In this work, collocated acceleration measurements and control actuators are used for state estimation by augmenting the second-order observer with additional states. (The addi- tional states provide temporal filtering of the acceleration measurements to insure asymptotic convergence of the esti- mated state error.) By using collocated sensors and actuators, the observer does not require spatial filtering, which signifi- cantly reduces the sensitivity to model errors. Equations for the typical first-order observer and a logical form for the second-order observer are presented. Stability and performance considerations for the first- and second-or- der observers are discussed. A design method for second-order observer gains with acceleration measurements is then de- scribed. Practical aspects of the observers are examined, and both numerical and experimental results are presented.

Use of approximate discrete-time models in filtering and regulation of continuous-time processes

A commonly used approximate discrete-time model for the estimation of the states of linear, continuous-time invariant processes is analysed. This analysis is performed by carefully defining and evaluating the true-error-covariance matrix of the estimate and by comparing it with what has been commonly believed to be the error covariance matrix. It is shown that there exists a class of unstable processes for which the former error-covariance matrix attains an unbounded norm, in spite of the fact that the norm of the other error-covariance matrix remains bounded. It is also shown that for another class of processes, the approximate model provides a particularly good estimation for short sampling periods. Based on the equations that govern the behaviour of the estimation-error covariance matrices, an alternative improved approximation is proposed that always leads to a true-error-covariance matrix with a bounded norm. Finally, following similar lines of analysis, it is shown that the approximate discrete-time...