An autoregressive (AR) model based stochastic unknown input realization and filtering technique

A solution to the state estimation problem of systems with unmeasurable non-zero mean inputs/disturbances, which do not satisfy the disturbance decoupling conditions, is given using the Kalman filtering and Bayesian estimation theory. The proposed estimation algorithm, named Supervisory Kalman Filter (SKF), consists of a Kalman filter with an extra update step which is inspired by the particle filtering technique. The extra step, called supervisory layer, numerically solves the measurement equations for the portion of the state vector that cannot be estimated by the Kalman filter. First, it produces N randomly generated state vectors, the particles, which are distributed based on the Kalman filter’s last updated estimate. Then, the estimated measurement vector associated with each particle is compared to the actual measurement vector to identify the particle’s probability to be a solution. Finally, a so-called resampling stage is implemented to refine the particles with higher likelihoods. The effectiveness of the SKF is demonstrated by comparing its estimation performance with that of the standard Kalman Filter and the particle filter for a vehicle state estimation problem. The estimation results confirm that the SKF precisely estimates those states of the vehicle that cannot be estimated by either the Kalman filter or the particle filter, regardless of the unknown disturbances from the road. The filtering methodology offered in the article has a potential to improve performance of the systems presented in the patents WO2011115960, WO2010024751, US8073528, and US20110299730. Keywords: Bayesian state estimation, disturbance decoupling conditions, inputs/disturbances, particle filtering, state estimation, systems with unknown kalman filtering, unscented kalman filtering

In this paper, a robust state estimation problem for discrete-time systems with unknown exogenous inputs is investigated utilizing a risk-sensitive filtering approach. By proposing a Radon-Nikodym derivative, we introduce a reference measure under which the measurement and system state become independent. Based on this independent property and by treating the unknown inputs as a process modeled by a non-informative prior, we derive the reformulated risk-sensitive cost criterion under the reference measure and further propose a recursive algorithm for the risk-sensitive state estimate. A simulation example is provided to validate the theoretical results, where the proposed estimator is shown to outperform the MMSE estimator for unknown input case under the scenario subject to system parameter uncertainties.

In the paper, a practical unknown input filtering problem, whether or not the unknown input model is given, is explored. Through the existing various two-stage Kalman filters, a unified two-stage Kalman filter based on the previously proposed optimal two-stage Kalman filter is developed to implement different unknown input estimators. Moreover, a multiple unknown input model-based system transformation is proposed to transform the original system into an augmented state system. Then, a cost-effective multi-stage Kalman filter is developed to implement the augmented state filter within a parameterized unknown input model-based filtering approach, which is suitable for parallel computing. An illustrative example is given to show the effectiveness of the proposed results.

SummaryThis paper considers the state estimation problem of bilinear systems in the presence of disturbances. The standard Kalman filter is recognized as the best state estimator for linear systems, but it is not applicable for bilinear systems. It is well known that the extended Kalman filter (EKF) is proposed based on the Taylor expansion to linearize the nonlinear model. In this paper, we show that the EKF method is not suitable for bilinear systems because the linearization method for bilinear systems cannot describe the behavior of the considered system. Therefore, this paper proposes a state filtering method for the single‐input–single‐output bilinear systems by minimizing the covariance matrix of the state estimation errors. Moreover, the state estimation algorithm is extended to multiple‐input–multiple‐output bilinear systems. The performance analysis indicates that the state estimates can track the true states. Finally, the numerical examples illustrate the specific performance of the proposed method.

This paper has studied the identification problem of linear mechanical systems where inputs are unknown and only displacement data are accessible for measurement. Eigensystem Realization Algorithm (ERA) has been used along with physical constraints considerations in time domain to simultaneously identify two separate models for the physical system and the unknown inputs. Inputs are assumed to be an arbitrary combination of harmonic signals with frequencies higher than natural frequencies of the physical system by which a linear mechanical system is meant in this paper. Adding physical constraints and utilizing canonical real Jordan form of the identified system leads to a unique analytical solution. To validate the theory part, a set of simulations has been run that demonstrates the physical parameters and input model can be estimated accurately.

Formation-containment tracking for general linear multi-agent systems with a tracking-leader of unknown control input

In this paper, the state estimation for dynamic system with unknown inputs modeled as an autoregressive AR (1) process is considered. We propose an optimal algorithm in mean square error sense by using difference method to eliminate the unknown inputs. Moreover, we consider the state estimation for multisensor dynamic systems with unknown inputs. It is proved that the distributed fused state estimate is equivalent to the centralized Kalman filtering using all sensor measurement; therefore, it achieves the best performance. The computation complexity of the traditional augmented state algorithm increases with the augmented state dimension. While, the new algorithm shows good performance with much less computations compared to that of the traditional augmented state algorithms. Moreover, numerical examples show that the performances of the traditional algorithms greatly depend on the initial value of the unknown inputs, if the estimation of initial value of the unknown input is largely biased, the performances of the traditional algorithms become quite worse. However, the new algorithm still works well because it is independent of the initial value of the unknown input.

This paper focuses on the joint input and state estimation for linear stochastic discrete-time systems. To obtain unique estimators that are optimal in the sense of being both least-squares and minimum-variance unbiased (MVU) and their recursive solution, an objective function of the sum of squared errors of outputs is first minimized with respect to an extended state vector including states and unknown inputs. A recursive solution of the extended state vector is then derived with the aid of matrix manipulations. Finally, appropriate matrix decomposition and transformation are used to extract the recursive solutions only involving the current states and unknown inputs from the recursive solution obtained in the previous step. This approach avoids excessive computation due to the dimensions of the matrices that increase with time. As a result, the resulting recursive solution is referred to as a general Kalman filter with unknown inputs (GKF-UI), which covers the most general case of unknown inputs so far in the literature without resorting to transforming the outputs and/or unknown inputs. The unknown inputs to be estimated by the proposed GKF-UI could be arbitrary signals without any prior information and the properties of the proposed GKF-UI are also examined. A numerical example with partial feedthrough from three unknown inputs is used to demonstrate the effectiveness and accuracy of the proposed algorithm.

This paper describes an efficient reduced order model (ROM) applied in the aerodynamic analysis of bluff bodies. The proposed method, which is based on eigensystem realization algorithm (ERA), uses the impulse response of the system obtained by computational fluid dynamics (CFD) analysis to construct a ROM that can accurately predict the response of the system to any arbitrary input. In order to study the performance of the proposed technique, three different geometries including elliptical and rectangular sections as well as the deck cross section of Great Belt Bridge (GBB) were considered. The aerodynamic coefficients of the impulse responses of the three sections are used to construct the corresponding ROM for each section. Then, the aerodynamic coefficients from an arbitrary sinusoidal input obtained by CFD are compared with the predicted one using the ROM. The results presented illustrate the ability of the proposed technique to predict responses of the systems to arbitrary sinusoidal and other generic inputs, with significant savings in terms of CPU time when compared with most CFD codes. The methodology described in this paper has wide application in many offshore engineering problems where flexible structures interact with unsteady fluid flow, and should be useful in preliminary design, in design optimization, and in control algorithm development.

The proper selection of the model error covariance matrix and the measurement noise covariance matrix of Kalman filter is an optimization problem. Some scholars have studied this, but there is relatively little research on the selection of the two covariance matrices for Kalman filters with an unknown input. Recently, the authors proposed a modified directed bat algorithm (MDBA) which introduces the best historical location of individuals and the elimination strategy to effectively prevent falling into local optimal solution. So, two methods are proposed in this paper to optimize the model error covariance matrix and measurement noise covariance matrix of Kalman filter with unknown inputs (KF-UI) and extended Kalman filter with unknown inputs (EKF-UI) by MDBA, respectively. The objective functions are constructed using the measurement vectors and the corresponding estimated values, and MDBA is adopted to optimize the two covariance matrices of KF-UI and EKF-UI. To validate the effectiveness of proposed methods, two simple structure examples and a benchmark example are adopted. The influence of structural parameter uncertainties on KF-UI is also considered. The result shows that the MDBA-optimized KF-UI has a strong convergence and can take into account the effect of parameter uncertainties. Then, the effectiveness of the proposed MDBA-optimized EKF-UI method is validated by comparing it with EKF-UI with empirically selected covariance values through trial-and-error. The identification results showed that the proposed methods achieved better identification accuracy and enhanced convergence compared to KF-UI and EKF-UI with empirical covariance values.

T HIS Note describes a robust modification of the divided difference filtering technique. The robust technique relies on Huber’s generalized maximum likelihood approach to estimation [1]. Specifically,Huber’smethod is a combinedminimum ‘1and ‘2norm estimation technique, which exhibits robustnesswith respect to deviations from the commonly assumed Gaussian error probability density functions, for which the least–squares or minimum ‘2-norm technique exhibits a severe degradation in estimation accuracy [2]. TheHuber-based estimates are robust in the sense that theyminimize the maximum asymptotic estimation variance when applied to contaminated Gaussian densities. The Huber technique was originally developed as a generalization of maximum likelihood estimation, applied first to estimating the center of a probability distribution in [1] and generalized tomultiple linear regression in [2– 4]. The Kalman filter is a recursive minimum ‘2-norm technique and therefore exhibits sensitivity to deviations in the true underlying error probability distributions [5]. For this reason, theHuber technique has been further extended to dynamic estimation problems. Boncelet and Dickinson [6] first proposed to solve the robust filtering problem by means of the Huber technique at each measurement, by expressing the discrete-time Kalman filter as a sequence of linear regression problems. The authors do not provide any simulation results to validate the proposed technique. Kovacevic et al. [7] follow thework of [6] and develop a robust Kalman filter using the Huber technique applied to a linear regression problem at each measurement update. References [8–10] express the dynamic filtering problem as a sequential linear regression to be solved by the Huber technique and apply the filter to underwater vehicle tracking, power system state estimation, and spacecraft rendezvous navigation, respectively. The increase in computation due to the use of the Huber technique was found in [10] to be small. It should be noted that [6–10] apply the Huber methodology to linearized filters. The divided difference filter is one of several new estimation techniques that are collectively known as sigma-point Kalman filters (SPKF). The first-order (DD1) and second-order (DD2) divided difference filters [11,12] are generalizations of the filter introduced by Schei [13] and are two examples of SPKF-class estimators; other examples can be found in [14–16]. Like the basic Kalman filter, the SPKFs seek to determine a state estimate that minimizes the ‘2 norm of the residuals. The SPKF technique differs from the standard Kalmanfilter in the sense that the SPKFs do not linearize the dynamic system for the propagation, but instead propagate a cluster of points centered around the current estimate to form improved approximations of the conditional mean and covariance. Specifically, the divided difference filters make use of multidimensional interpolation formulas to approximate the nonlinear transformations. As a result of this approach, the filter does not require knowledge or existence of the partial derivatives of the system dynamics and measurement equations. SPKFs have the additional advantage over the basic Kalman filter in that they can easily be extended to determine second-order solutions to the minimum ‘2-norm filtering problem, which increases the estimation accuracywhen the system andmeasurement equations are nonlinear. It is important to note that the SPKFs use a minimum ‘2-norm measurement update and are therefore subject to the same sensitivity to non-Gaussian measurement errors as the Kalman filter. Therefore, the purpose of this Note is tomodify the DD1 andDD2measurement update equations by making use of the Huber technique to provide robustness against deviations from Gaussianity without a large increase in computation. This Note first provides a short review of the DD1 and DD2 filters and then shows how the measurement update can be expressed in terms of a standard regression problem, which can be solved using the robust Huber technique. The filtering techniques are then applied to a benchmark problem that involves estimating the trajectory of an entry body from discrete-time range data measured by a radar tracking station. The simulation is conducted using Monte Carlo techniques for both Gaussian and non-Gaussian cases. The computational cost associated with each filter is provided.

A robust three stage Kalman filtering problem is investigated in this article for non‐linear systems with stochastic non‐linearities, random faults and intermittent missing measurements. Therefore, a more general system model is governed in which unknown stochastic non‐linearities are considered in both the system state and measurement equations. The fault terms are included by stochastic coefficients. Also, the sensor measurements occur in a random behaviour that encounters stochastic missing of data for each sensor. Both faults and unknown inputs are covered in the proposed filter. Moreover, the robust features of the developed estimator resolve the problems arising from the need for accurate models of faults and unknown inputs. Furthermore, any predetermined knowledge about the statistical characteristics of the above factors is not required. The augmented filters are mostly used to provide the fault diagnosis features, however, in the current work, this filter is decoupled to reduce the computational volume attributed to this approach. It is proved that the estimation error is ultimately bounded despite different stochastic terms and inaccurate information about the fault and unknown inputs. Finally, an illustrative example is proposed to demonstrate the effectiveness of the developed method.

Input and state estimation for linear systems with a rank-deficient direct feedthrough matrix

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Sequential state estimation with interrupted observation