Unbiased State Estimation Research Articles

Unbiased minimum-variance disturbance and state estimation for linear systems with both state and output disturbances

This paper addresses the problem of simultaneously estimating the states, the state disturbances and the output disturbances of a linear time variant system. The objective is to develop a simple but optimal filter in the unbiased minimum-variance sense without depending extra complex system decomposition transformation. The derived filter bases on the assumption that no prior knowledge about the dynamical evolution of these disturbances is available.

Unbiased state and fault estimation for discrete-time complex networks with time delays

In this paper, the joint state and fault estimation problem for a class of discrete-time complex networks with time delays is investigated. The information on the dynamics of the fault is not required with an appropriate structure of the proposed fault estimator. Unbiased state and fault estimation can be obtained simultaneously at each node only with locally available information, thereby avoiding the requirement of a centre node which collects information from every node. Considering the effects of time delays and the interconnections between different nodes, an upper bound of the estimation error covariance is calculated, and the desired filter parameters are obtained in a recursive manner such that the trace of the bound is minimised. The existence conditions of the developed filter are provided explicitly as well. Finally, a simulation example is employed to demonstrate the state estimation and fault diagnosis performances of the given strategy.

A linear Bayesian filter for input and state estimation of structural systems

AbstractThis paper proposes a linear recursive Bayesian filter for minimum variance unbiased joint input and state estimation of structural systems. Unlike the augmented Kalman filter (AKF), the proposed filter falls within the category of Bayesian filters in which unknown inputs are estimated without attributing any fictitious input model or statistics. Also, in contrast with the existing algorithms in the latter category, such as the Gillijns and De Moor Filters (GDFs), the developed filter applies to systems with and without direct feedthrough, in particular, systems with a rank‐deficient feedforward matrix. Because of the latter features, the filter is referred to as universal filter (UF) for convenience. The numerical examples show that the UF performs better than the AKF. Due to its structure, the UF does not require the tuning of the hyperparameters for inputs, and therefore the problematic instability of the AKF is not encountered in the case of a large modeling error variance of the input. For systems with direct feedthrough, the error and covariance propagation terms differ due to the distinct state space. Consequently, the UF can enhance estimations due to the well‐conditionedness of the relevant inversion problem. Moreover, the UF can deal with systems with rank‐deficient feedforward matrix where these systems are not covered by GDFs.

Simultaneous seismic input and state estimation with optimal sensor placement for building structures using incomplete acceleration measurements

Simultaneous real-time input and state estimation and optimal sensor placements are investigated in this paper, focusing on systems without direct feedthrough, such as earthquake-excited building structures with absolute floor acceleration measurements. Current studies showed that system properties such as strong observability, strong* detectability (the asterisk distinguishes strong* detectability from strong detectability), and invertibility conditions are crucial to the stability and convergence of unknown input and state estimation, but they can often be violated in practice. Consequently, uncertainties such as modeling errors and measurement noise can greatly degrade the accuracy and stability of the estimations. Estimation in this case has remained challenging due to the above reasons. To fill this gap, this paper develops an optimal sensor placement algorithm (OSPA) for real-time unknown input and state estimation, which ensures the required system conditions are met. The developed OSPA is integrated with two optimal real-time Kalman-based filters, a minimum-variance unbiased input and state estimation filter (MVUIS) and an Augmented State Kalman Filter (ASKF), for simultaneous input and state estimation. In particular, the MVUIS is presented in a recursive three-step structure without using the arbitrary matrix in the gain, which makes no assumptions on the input but requires strong* detectability. To avoid the requirement, ASKF is derived from the MVUIS by incorporating prior knowledge of the input. The developed OSPA along with the MVUIS and ASKF provide the optimal input and state estimation in real-time without incurring low-frequency drift or unstable estimations. Notably, the OSPA improves the performance of MVUIS by reducing amplitude errors and enhances the accuracy of ASKF by reducing phase errors. The developed OSPA integrated with the MVUIS and ASKF are validated through numerical and experimental studies as well as a real-world instrumented building structure under earthquakes using incomplete absolute acceleration measurements.

Robust State and Unknown Input Estimator and Its Application to Robot Localization

The robot localization problem is typically solved using state estimation techniques, where process and sensing inaccuracies are invariably present. Moreover, disturbances in the sensing and actuating mechanisms add to the uncertainties. Any system may degrade over time, and its parameter values may be ambivalently known. Cumulatively, all these sources of errors and uncertainties are considered as unknown inputs. This work aims to address the unknown inputs using a robust state (pose in the robot localization problem) estimator. The proposed robust state estimator deals with the unknown inputs such that the solution of the estimator is constrained in a way that warrants unbiased state estimates in the presence of the unknown inputs. This article explores the formulation of these constraints and the development of a constrained state estimator for a system, where the unknown inputs appear in both the state transition map (i.e., system model) and the state-output map (i.e., measurement model). The theoretical development of such a strategy stems from the localization problem of a wheeled mobile robot. The residuals of the constrained state estimator developed contain information about the unknown inputs. We conceive a recursive least squares strategy to estimate the unknown inputs simultaneously with the states using this information. Using simulations and experimental studies, we demonstrate the adequacy of our strategy for a differential drive robot.

Sequential State and Unknown Parameter Estimation Strategy and its Application to a Sensor Fusion Problem

Soft-sensing and state-space-based methods play crucial roles in fault-tolerant sensor fusion schemes. Kalman filtering is one of the most employed sensor fusion methods. In practice, filtering (or state estimation) is sensitive to initial conditions, variations in model parameters, and faults in sensors and actuators. Cumulatively, these uncertainties that affect the state estimator can be modeled as unknown parameters. In this article, we present the development of a discrete-time minimum variance unbiased estimator (MVUE) that is robust to a class of unknown parameters. The estimation gain of MVUE is designed to facilitate unbiased state estimates in the presence of unknown parameters. We then use the innovations that are generated by the proposed state estimator (MVUE) to estimate the unknown parameters, thus proposing a sequential state and unknown parameter estimation strategy. The development of the decoupled state and unknown parameter estimation strategy is motivated by the fault-tolerant sensor fusion exercise for railway track geometry inspection. This exercise aims to formulate the state-space realization of the track geometry inspection problem to enable sequential state and unknown parameter estimation. The track geometry parameters, initial conditions, and sensor faults are considered unknown parameters for the application. The simulation analysis shows the importance of developing the trinity of the state-space realization for track geometry inspection, the state estimation framework, and the decoupled unknown parameter estimator.

Robust strong tracking unscented Kalman filter for non‐linear systems with unknown inputs

This paper proposes a state estimation approach ‘robust strong tracking unscented Kalman filter with unknown inputs’ that can be applied to non-linear systems with unknown inputs. Specifically, the non-linear state and measurement equations are linearised by statistical linearisation. Then, the estimation equation of the unknown input is derived based on the weighted least squares method. The multiple suboptimal fading factor is introduced into a priori error covariance matrix to improve the tracking ability for the inaccuracy of the system model and the abrupt change of state variables caused by unknown inputs. Finally, based on the unbiased minimum variance estimation, the unbiased state estimation and the error covariance matrix are derived. Singular value decomposition is performed on the error covariance matrix to improve the stability of the algorithm. Simulated results validate the effectiveness of the proposed method.

Proportional integral observer based tracking control design for Markov jump systems

With the use of proportional integral observer, this article intends to discuss an effective output tracking control and disturbance rejection problem for an input time-delayed Markov jump system with state dependent nonlinearities and randomly occurring uncertainties. Notably, the proportional integral observer incorporates both proportional and integral loop, where the presence of additional loop provides unbiased state estimation. Moreover, an improved equivalent input disturbance approach with a gain factor is considered to enhance better disturbance estimation and rejection performance. Naturally, Smith predictor is commenced to handle input time delays. Subsequently, the combination of improved equivalent input disturbance with Smith predictor guarantees the desired tracking performance with input time delays and external disturbances. By endowing Lyapunov stability theory, a set of sufficient conditions is developed in the frame of linear matrix inequalities to promise the asymptotically stability of the closed-loop system. Subsequently, the explicit form of the desired proportional integral gain matrices are parameterized using the matrix inequality techniques. The superiority of our proffered control technique are validated through two numerical examples, which includes a practical model namely RLC circuit.

Air Data Sensor Fault Detection and Diagnosis in the Presence of Atmospheric Turbulence: Theory and Experimental Validation With Real Flight Data

Managing air data sensor fault detection and diagnosis (FDD) in the presence of atmospheric turbulence is challenging since the effects of faults and turbulence are coupled. Existing FDD approaches cannot decouple the faults from the turbulence. To address this challenge, this brief first proposes a novel kinematic model that incorporates the effects of the turbulence. This model is valid inside the entire flight envelope, and there is no need to design a linear parameter varying system. Then, the double-model adaptive estimation algorithm is extended to achieve unbiased state estimation even in the presence of unknown disturbances. The proposed approach is validated using generated turbulence data with various scale lengths and intensities. More importantly, the proposed approach is successfully validated using the real flight test data of a business jet when it is experiencing atmospheric turbulence.

Extended Kalman Filter with Input Detection and Estimation for Tracking Manoeuvring Satellites

The technique of tracking a non-cooperative manoeuvring satellite is important for Space Situation Awareness (SSA). However, the classical extended Kalman filter cannot work successfully in this situation. Motivated by this problem, a novel Extended Kalman Filter with Input Detection and Estimation (EKF/IDE) method is proposed in this paper for tracking a non-cooperative satellite with impulsive manoeuvres. The impulsive manoeuvre is modelled as an unknown acceleration without any prior information. An unbiased minimum-variance input and state estimation method is introduced to estimate the manoeuvre acceleration. An approach based on the Mahalanobis distance of the manoeuvre estimate error is proposed for manoeuvre detection. With the impulsive manoeuvre being detected and estimated accurately, an adaptive extended Kalman filter is proposed to estimate the state of the target. An approach of covariance inflation is proposed to deal with the manoeuvre during the unobserved period. Simulations and Monte Carlo experiments are implemented to demonstrate the feasibility and validity of the proposed method. The results of simulations show that the proposed method can accurately detect and estimate unknown impulsive manoeuvres of a non-cooperative satellite. Through the compensation of the estimated manoeuvre, the estimation of position and velocity after the manoeuvre maintains the accuracy of the pre-manoeuvre period, demonstrating the robustness of the proposed method against unknown manoeuvres.

Augmented unbiased minimum-variance input and state estimation for tracking a maneuvering satellite

The technique of tracking a maneuvering satellite is significantly important for Space Situation Awareness (SSA). Traditional Kalman filters (KF) cannot robustly track unknown maneuvers. Motivated by the problem of tracking a non-cooperative maneuvering satellite, an augmented unbiased minimum-variance input and state estimation (AUMVISE) method is developed for estimating the state and the maneuver acceleration in this study. The maneuver acceleration estimate of the proposed method is proven to be more accurate than the original unbiased minimum-variance input and state estimation (UMVISE) method. Approaches based on the measurement residuals and the acceleration estimates are developed for maneuver start and end detection. The filter is switched between the AUMVISE method and the classical extended Kalman filter (EKF) to obtain an accurate tracking result during both the maneuver period and the non-maneuver period. Simulation results show that the proposed method has a suitable maneuver detection delay and outperforms the UMVISE method in estimating the state and maneuver acceleration.

Optimal Position and Velocity Estimation for Multi-USV Positioning Systems with Range Measurements

This paper investigates the problem on simultaneously estimating the velocity and position of the target for range-based multi-USV positioning systems. According to the range measurement and kinematics model of the target, we formulate this problem in a mixed linear/nonlinear discrete-time system. In this system, the input and state represent the velocity and position of the target, respectively. We divide the system into two components and propose a three-step minimum variance unbiased simultaneous input and state estimation (SISE) algorithm. First, we estimate the velocity in the local level plane and predict the corresponding position. Then, we estimate the velocity in the heave direction. Finally, we estimate the 3-dimensional (3D) velocity and position. We establish the unbiased conditions of the input and state estimation for the MLBL system. Simulation results illustrate the effectiveness of the problem formulation and demonstrate the performance of the proposed algorithm.

Robust extended Kalman filter with input estimation for maneuver tracking

This study investigates the problem of tracking a satellite performing unknown continuous maneuvers. A new method is proposed for estimating both the state and maneuver acceleration of the satellite. The estimation of the maneuver acceleration is obtained by the combination of an unbiased minimum-variance input and state estimation method and a low-pass filter. Then a threshold-based maneuver detection approach is developed to determinate the start and end time of the unknown maneuvers. During the maneuvering period, the estimation error of the maneuver acceleration is modeled as the sum of a fluctuation error and a sudden change error. A robust extended Kalman filter is developed for dealing with the acceleration estimate error and providing state estimation. Simulation results show that, compared with the Unbiased Minimum-variance Input and State Estimation (UMISE) method, the proposed method has the same position estimation accuracy, and the velocity estimation error is reduced by about 5 times during the maneuver period. Besides, the acceleration detection and estimation accuracy of the proposed method is much higher than that of the UMISE method.

Switching and Data Injection Attacks on Stochastic Cyber-Physical Systems

In this article, we consider the problem of attack-resilient state estimation, that is, to reliably estimate the true system states despite two classes of attacks: (i) attacks on the switching mechanisms and (ii) false data injection attacks on actuator and sensor signals, in the presence of stochastic process and measurement noise signals. We model the systems under attack as hidden mode stochastic switched linear systems with unknown inputs and propose the use of a multiple-model inference algorithm to tackle these security issues. Moreover, we characterize fundamental limitations to resilient estimation (e.g., upper bound on the number of tolerable signal attacks) and discuss the topics of attack detection, identification, and mitigation under this framework. Simulation examples of switching and false data injection attacks on a benchmark system and an IEEE 68-bus test system show the efficacy of our approach to recover resilient (i.e., asymptotically unbiased) state estimates as well as to identify and mitigate the attacks.

Minimum-variance unbiased unknown input and state estimation for multi-agent systems with direct feedthrough by using distributed cooperative filters

This paper addresses the problem of simultaneous estimation of unknown inputs and states in multi-agent systems with direct feedthrough. A group of cooperative distributed recursive filters, in the sense of minimum-variance unbiased (MVU), is developed, where the estimations of the unknown input and state are interconnected. Theoretical and numerical analyses show that the existing condition of the proposed filters is significantly relaxed compared with that of the conventional decentralized filters.

Minimum-Variance Unbiased Unknown Input and State Estimation for Multi-Agent Systems by Distributed Cooperative Filters

This paper addressed the problem of the simultaneous estimation of unknown inputs and states in a multi-agent system with time-invariant and time-varying topology. A group of distributed cooperative recursive filters, in the sense of minimum-variance unbiased, was developed, where the estimations of unknown input and state were combined. A necessary and sufficient existing condition is presented and proven for the proposed distributed cooperative filters. Theoretical and numerical analyses demonstrate that the existing condition of the proposed filters is significantly relaxed, in comparison to that of conventional decentralized filters.

Unbiased Minimum Variance Fault and State Estimation for Linear Discrete Time-Varying Two-Dimensional Systems

The fault and state estimation problem is addressed for a class of linear discrete time-varying two-dimensional systems subject to state and measurement noises. Two estimators are proposed to compute the estimation of the system state and/or fault recursively, both of which are unbiased with minimum variance. Through formulating the estimation problem as the solvability problem of the corresponding matrix equations of estimator gains and system constraint, the necessary and sufficient condition of the existence and the solution for the proposed estimators are given. An example is used to demonstrate the effectiveness of the proposed estimators.

Unbiased minimum-variance input and state estimation for systems with unknown inputs: A system reformation approach

This paper presents a system reformation based unbiased minimum-variance input and state estimation for systems with unknown inputs which can be reconstructed with a one-step delay. It is shown that, within this new filtering approach the optimal unknown input and state estimation can be simultaneously achieved through the filter developed by Gillijns and De Moor. An illustrative example is given to show the effectiveness of the proposed results. Moreover, under some additional assumptions the proposed system reformation can be easily extended to consider a multi-step delayed input estimation.

Handling Model Plant Mismatch in State Estimation Using a Multiple-Model-Based Approach

Accurate state estimates are important for the success of model predictive control (MPC). State estimates are obtained using a model, but, in real plants, there will always be model plant mismatch (MPM), which affects these estimates. In this work, we present a multiple-model (MM)-based approach to obtain unbiased state estimates in the presence of MPM. Necessary assumptions on the source of mismatch and models used are presented. It is shown that unbiased output estimates do not guarantee unbiased state estimates. Our approach is shown to provide unbiased state estimates when all the assumptions are met using a froth flotation system. A model-identification-based control approach using our multiple model estimation approach with a conventional MPC was tested on the froth flotation system and was found to successfully provide offset-free reference tracking when all the necessary assumptions for unbiased state estimation were met. A nonlinear offset-free MPC was also tested on the froth flotation system bu...

Iterative unbiased FIR state estimation: a review of algorithms

In this paper, we develop in part and review various iterative unbiased finite impulse response (UFIR) algorithms (both direct and two‐stage) for the filtering, smoothing, and prediction of time‐varying and time‐invariant discrete state‐space models in white Gaussian noise environments. The distinctive property of UFIR algorithms is that noise statistics are completely ignored. Instead, an optimal window size is required for optimal performance. We show that the optimal window size can be determined via measurements with no reference. UFIR algorithms are computationally more demanding than Kalman filters, but this extra computational effort can be alleviated with parallel computing, and the extra memory that is required is not a problem for modern computers. Under real‐world operating conditions with uncertainties, non‐Gaussian noise, and unknown noise statistics, the UFIR estimator generally demonstrates better robustness than the Kalman filter, even with suboptimal window size. In applications requiring large window size, the UFIR estimator is also superior to the best previously known optimal FIR estimators.