Variational Bayesian adaptation of process noise covariance matrix in Kalman filtering

Adaptive Kalman filtering with unknown constant or varying process noise covariance matrix is studied. A resolution is proposed to directly estimate or tune the process noise covariance matrix in Kalman filtering using variational Bayesian technique. By state augmentation, conjugacy of the process noise covariance matrix's inverse-Wishart distribution is realized in the estimation at each time instant. The methodological development is given. Illustration examples are presented to demonstrate the improved state filtering performance and the process noise covariance tracking performance of the new method.     

Combined state and parameter estimation for a bilinear state space system with moving average noise

This paper considers the identification problem of bilinear systems with measurement noise in the form of the moving average model. In particular, we present an interactive estimation algorithm for unmeasurable states and parameters based on the hierarchical identification principle. For unknown states, we formulate a novel bilinear state observer from input-output measurements using the Kalman filter. Then a bilinear state observer based multi-innovation extended stochastic gradient (BSO-MI-ESG) algorithm is proposed to estimate the unknown system parameters. A linear filter is utilized to improve the parameter estimation accuracy and a filtering based BSO-MI-ESG algorithm is presented using the data filtering technique. In the numerical example, we illustrate the effectiveness of the proposed identification methods.     

Kalman filtering based gradient estimation algorithms for observer canonical state-space systems with moving average noises

This paper focuses on the joint parameter and state estimation issue for observer canonical state-space systems with white noises in state equations and moving average noises in output equations. By means of the Kalman filtering and the gradient search, we derive a Kalman filtering based extended stochastic gradient algorithm. For purpose of achieving the higher parameter estimation accuracy, a Kalman filtering based multi-innovation extended stochastic gradient algorithm is proposed on the basis of the multi-innovation identification theory. Finally, the effectiveness of the proposed algorithms is validated through a numerical example.     

Sliding mode filtering for stochastic systems with polynomial state and observation equations

In this paper, the mean-square and mean-module filtering problems for polynomial system states over polynomial observations are studied proceeding from the general expression for the stochastic Ito differentials of the estimate and the error variance. The paper deals with the general case of nonlinear polynomial states and observations. As a result, the Ito differentials for the estimates and error variances corresponding to the stated filtering problems are first derived. The procedure for obtaining an approximate closed-form finite-dimensional system of the sliding mode filtering equations for any polynomial state over observations with any polynomial drift is then established. In the examples, the obtained sliding mode filters are applied to solve the third-order sensor filtering problems for a quadratic state, assuming a conditionally Gaussian initial condition for the extended second-order state vector. The simulation results show that the designed sliding mode filters yield reliable and rapidly converging estimates.     

Maximum correntropy criterion variational Bayesian adaptive Kalman filter based on strong tracking with unknown noise covariances

The performance of the current state estimation will degrade in the existence of slow-varying noise statistics. To solve the aforementioned issues, an improved strong tracking maximum correntropy criterion variational-Bayesian adaptive Kalman filter is presented in this paper. First of all, the inverse-Wishart distribution, as the conjugate-prior, is adopted to model the unknown and time-varying measurement and process noise covariances, then the noise covariances and system state are estimated via the variational Bayesian method. Secondly, the multiple fading-factors are obtained and evaluated to modify the prediction error covariance matrix to address the problems associated with inaccurate error estimation. Finally, the maximum correntropy criterion is employed to correct the filtering gain, which improves the filtering performance of the proposed algorithm. Simulation results show that the proposed filter exhibits better accuracy and convergence performance compared to other existing algorithms.     

Distributed Kalman filtering over sensor networks with fading measurements and random link failures

This paper investigates the distributed state estimation problem for a linear time-invariant system characterized by fading measurements and random link failures. We assume that the fading effect of the measurements occurs slowly. Additionally, communication failures between sensors can affect the state estimation performance. To this end, we propose a Kalman filtering algorithm composed of a structural data fusion stage and a signal date fusion stage. The number of communications can be decreased by executing signal data fusion when a global estimate is required. Then, we investigate the stability conditions for the proposed distributed approach. Furthermore, we analyze the mismatch between the estimation generated by the proposed distributed algorithm and that obtained by the centralized Kalman filter. Lastly, numerical results verify the feasibility of the proposed distributed method.     

Three-stage Kalman filter for state and fault estimation of linear stochastic systems with unknown inputs

The paper studies the problem of simultaneously estimating the state and the fault of linear stochastic discrete-time varying systems with unknown inputs. The fault and the unknown inputs affect both the state and the output. However, if the dynamical evolution models of the fault and the unknown inputs are available the filtering problem will be solved by the Optimal three-stage Kalman Filter (OThSKF). The OThSKF is obtained after decoupling the covariance matrices of the Augmented state Kalman Filter (ASKF) using a three-stage U–V transformation. Nevertheless, if the fault and the unknown inputs models are not perfectly known the Robust three-stage Kalman Filter (RThSKF) will be applied to give an unbiased minimum-variance estimation. Finally, a numerical example is given in order to illustrate the proposed filters.     

Reduction of prediction error sensitivity to parameters in Kalman filter

The desensitized Kalman filter Karlgaard and Shen (2013)[1] is a practical and intuitive robust filtering method. However, a thorough analysis of its stability and impact of assumptions is missing. This paper expands the theory of desensitized Kalman filtering by proposing a stochastic approach to reduce estimation error sensitivity to parameters. The novel approach leads to the exact desensitized Kalman filter that does not neglect the gain sensitivity to a parameter. The suboptimal form equivalent to the original desensitized Kalman filter in a special form is proposed. The stability analysis and the definition of stability conditions are possible due to the proposed form that can be interpreted as the Kalman filter with correlated process and measurement noise with time-variant statistics. Furthermore, adaptive normalization of objectives is introduced, which improves the desensitizing performance.     

Widely linear estimation for multisensor quaternion systems with mixed uncertainties in the observations

The optimal widely linear state estimation problem for quaternion systems with multiple sensors and mixed uncertainties in the observations is solved in a unified framework. For that, we devise a unified model to describe the mixed uncertainties of sensor delays, packet dropouts and uncertain observations by using three Bernoulli distributed quaternion random processes. The proposed model is valid for linear discrete-time quaternion stochastic systems measured by multiple sensors and it allows us to provide filtering, prediction and smoothing algorithms for estimating the quaternion state through a widely linear processing. Simulation results are employed to show the superior performance of such algorithms in comparison to standard widely linear methods when mixed uncertainties are present in the observations.     

Data filtering based maximum likelihood extended gradient method for multivariable systems with autoregressive moving average noise

For multivariable systems with autoregressive moving average noises, we decompose the multivariable system into m subsystems (m denotes the number of outputs) and present a maximum likelihood generalized extended gradient algorithm and a data filtering based maximum likelihood extended gradient algorithm to estimate the parameter vectors of these subsystems. By combining the maximum likelihood principle and the data filtering technique, the proposed algorithms are effective and have computational advantages over existing estimation algorithms. Finally, a numerical simulation example is given to support the developed methods and to show their effectiveness.     

Fractional-order Kalman filters for continuous-time fractional-order systems involving colored process and measurement noises

This study proposes fractional-order Kalman filers using Tustin generating function and the average value of fractional-order derivative to estimate the state of fractional-order systems involving colored process and measurement noises. By Tustin generating function, a fractional-order differential equation is provided to approximate the dynamics of a continuous-time fractional-order system and colored process and measurement noises. By constructing an augmented system with respect to state, the process noise and the measurement noise to deal with colored noises, the fractional-order Kalman filter using Tustin generating function is proposed to improve the estimation accuracy. Besides, the average value of fractional-order derivative is proposed, and the corresponding fractional-order Kalman filter by the augmented system method is presented to reduce estimation error. Finally, three illustrative examples are given to illustrate that the proposed two kinds of Kalman filters are more effective than fractional-order Kalman filter based on Gr$\ddot{\mathrm{u}}$nwald–Letnikov definition.     

Fusion estimation for stochastic uncertain systems with time-correlated rician fading channels

In this article, a fusion estimation scheme is proposed for stochastic uncertain systems with time-correlated fading channels (TFCs). A batch of random variables obeying Gaussian distributions is employed to describe the parameter uncertainties. The sensor communicates with the local filter through a TFC where the evolution of the channel coefficient is characterized by a certain dynamic process with one-step correlated noises. For further analyzing the effects of TFCs, a class of additional variables is first introduced by augmenting the dynamics of channel coefficients and the concerned system. Then, a new group of modified local filters is developed and the unbiasedness of local filters is examined by means of inductive method. Furthermore, the filter gains which minimize the local filtering error covariances are designed for the modified local filters in the simultaneous presence of stochastic uncertainties and TFCs. Subsequently, the cross-covariances among local estimates are computed iteratively and, based on the obtained cross-covariances as well as the unbiased local estimates and their corresponding filtering error covariances, a fusion estimate is obtained by using weighted least square fusion method. Finally, the effectiveness of the proposed fusion estimation scheme is verified by two examples.     

Robust extended fractional Kalman filter for nonlinear fractional system with missing measurements

Accurate and effective state estimation is essential for nonlinear fractional system, since it can provide some vital operation information about the system. However, inevitably missing measurements and additive uncertainty in the gain will affect the performance of estimation result. Thus, in this paper, in order to deal with these problems, a novel robust extended fractional Kalman filter (REFKF) is developed for states estimation of nonlinear fractional system, by which the states can be estimated accurately even with missing measurements. Finally, simulation results are provided to demonstrate that the proposed method can achieve much better estimation performance than the conventional extended fractional Kalman filter (EFKF).     

A continuous finite-time convergence fixed-lag FIR smoother using multiple IIR filters

In this paper, we propose a continuous finite-time convergence finite impulse response (FIR) fixed-lag smoother using multiple, or more than two, computationally efficient IIR filters. We describe the optimal design to improve and further optimize an existing scheme based on two IIR filters. Multiple IIR filters are utilized to minimize the estimation error variance of the proposed smoother under the condition that its estimate converges to a real state in a finite time. As the number of adopted IIR filters increases, the proposed smoother improves and its performance approaches that of the heavy computational fixed-lag minimum variance FIR smoother. By choosing the appropriate number of IIR filters, we can balance the trade-off between improved accuracy and increased implementation costs. To realize the optimal design of IIR filters with the limited number of IIR filters, their gains are determined using a particle swarm optimization scheme. Numerical examples are used to show that with an increasing number of IIR filters, the estimation error variance decreases monotonically while guaranteeing finite-time convergence.     

A new distributed Kalman filtering based on means-quare estimation upper bounds

In this paper, a novel distributed Kalman filter consisting of a bank of interlaced filters is proposed for a signal model whose dynamic equation and measurement equation are coupled. Each of the interlaced filters estimates a part of state rather than the global state using its and its neighbor information, which is different from other distributed filters already existed (e.g., distributed Kalman filter based on diffusion strategy or consensus strategy, distributed fuzzy filter and distributed particle filter with Gaussian mixer approximation, etc). This relieves the calculation and communication burden in networks. In addition, the proposed distributed Kalman filtering contains no consensus strategies, which is useful in some cases since consensus usually requires an infinite number of iterations.     

Recursive distributed fusion estimation for nonlinear stochastic systems with event-triggered feedback

This paper focus on the distributed fusion estimation problem for a multi-sensor nonlinear stochastic system by considering feedback fusion estimation with its variance. For any of the feedback channels, an event-triggered scheduling mechanism is developed to decide whether the fusion estimation is needed to broadcast to local sensors. Then event-triggered unscented Kalman filters are designed to provide local estimations for fusion. Further, a recursive distributed fusion estimation algorithm related with the trigger threshold is proposed, and sufficient conditions are builded for boundedness of the fusion estimation error covariance. Moreover, an ideal compromise between fusion center-to-sensors communication rate and estimation performance is achieved. Finally, validity of the proposed method is confirmed by a numerical simulation.     

Globally exponentially stable single beacon underwater navigation with unknown sound velocity estimation

Single beacon navigation methods with unknown effective sound velocity (ESV) have recently been proposed to solve the performance degeneration induced by ESV setting error. In these methods, a local linearization-based state estimator, which only exhibits local convergence, is adopted to estimate the navigation state. When the initial ESV setting error or vehicle initial position error is large, the local linearization-based state estimators have difficulty guaranteeing the filtering convergence. With this background, this paper proposes a linear time-varying single beacon navigation model with an unknown ESV that can realize global convergence under the condition of system observability. A Kalman filter is adopted to estimate the model state, and the corresponding stochastic model is inferred for the application of the Kalman filter. Numerical simulation confirms that the proposed linear time-varying single beacon navigation model can realize fast convergence in the case of a large initial error, and has superior steady-state performance compared with the existing methods.     

Aitken based modified Kalman filtering stochastic gradient algorithm for dual-rate nonlinear models

This paper develops an Aitken based modified Kalman filtering stochastic gradient algorithm for dual-rate nonlinear models. The Aitken based method can increase the convergence rate and the modified Kalman filter can improve the estimation accuracy. Thus compared to the traditional auxiliary model based stochastic gradient algorithm, the proposed algorithm in this paper is more effective, and this is proved by the convergence analysis. Furthermore, two simulated examples are given to illustrate the effectiveness of the proposed algorithm.     

Secure distributed Kalman filter using partially homomorphic encryption

In this paper, we present a secure distributed estimation strategy in networked systems. In particular, we consider distributed Kalman filtering as the estimation method and Paillier encryption, which is a partially homomorphic encryption scheme. The proposed strategy protects the confidentiality of the transmitted data within a network. Moreover, it also secures the state estimation computation process. To this end, all the algebraic calculations needed for state estimation in a distributed Kalman filter are performed over the encrypted data. As Paillier encryption only deals with integer data, in general, this, in turn, provides significant quantization error in the computation process associated with the Kalman filter. However, the proposed estimation approach handles quantized data in an efficient way. We provide an optimality and convergence analysis of our proposed method. It is shown that state estimation and a covariance matrix associated with the proposed method remain with a certain small radius of those of a conventional centralized Kalman filter. Simulation results are given to further demonstrate the effectiveness of the proposed scheme.     

Event-based state and fault estimation for stochastic nonlinear system with Markov packet dropout

This paper investigates the event-based state and fault estimation problem for stochastic nonlinear system with Markov packet dropout. By introducing the fictitious noise, the fault is augmented to the system state. Then combining the unscented Kalman filter (UKF) with event-triggered and Markov packet dropout, the modified UKF is proposed to estimate the state and fault. Meanwhile, the stochastic stability of the proposed filter is also discussed. Finally, two simulation results illustrate the performance of the proposed method.