Optimal Distributed Filter for State and Unknown Input Estimations in Multi-Sensor Networks

Adaptive schemes for unknown input and state estimation are proposed for a class of uncertain systems with bounded unknown inputs. First, using a Lyapunov approach, conditions are derived that ensure the state and unknown input estimation errors converge to zero for a constant unknown input. Next, combining a Lyapunov approach and linear matrix inequalities, conditions are given that guarantee a prescribed performance level for state and unknown input estimation for a bounded not necessarily constant unknown input.

For nonlinear discrete systems with dual unknown inputs, there are many limitations regarding previous nonlinear filters. This paper proposes two new, improved square-root cubature Kalman filtering (ISRCKF) algorithms to estimate system states and dual unknown inputs. Improved square-root cubature Kalman filtering 1 (ISRCKF1) introduces an innovation that first obtains the unknown input estimates from the measurement equation, then updates the innovation to derive the unknown input estimates from the state equation, then uses the already obtained estimates of the dual unknown inputs to correct the one-step estimate of the state, and finally the minimum variance unbiased estimate of the state is obtained. Improved square-root cubature Kalman filtering 2 (ISRCKF2) builds a unified innovation feedback model, then applies the minimum variance unbiased estimation (MVUE) criterion to obtain the estimates of system states and dual unknown inputs, refining a more concise recursive filter but requiring stronger assumptions. Finally, simulation results demonstrate that the above two algorithms can achieve the optimal estimates of system states and dual unknown inputs simultaneously, and ISRCKF2 further enhances the accuracy of both state and dual unknown inputs estimation, which verifies the validity of the proposed algorithms.

Problem statement: The estimation of states and the unknown inputs of a nonlinear system described by a multimodel are done by a multiobserver. The stabilization of the multiobserver calls upon uses both quadratic and no quadratic functions of Lyapunov. Although the stabilization using the quadratic approach is interesting from the point of view implementation, the step showed its limits for the multimodel. However, the problem paused by the quadratic method lies in the obligation to satisfy several LMI with respect to the same Lyapunov matrix P, these results are shown very conservative. Approach: To reduce the conservatism of the quadratic approach we propose another approach which is exclusively based on Lyapunov piecewise quadratic functions. The conditions obtained by the stabilization of the multiobserver are expressed in term of matrix inequalities with constraints on the matrices rank. Results: The estimation of both states and unknown inputs of a multimodel using the quadratic approach per pieces leads to results less conservative than the quadratic approach. Academic examples illustrate the robustness of the piecewise quadratic approach. Conclusion: In this article we proposed new sufficient conditions of stability of a multiobserver able to the estimation of states and unknown inputs of a nonlinear system describes by a multimodel subjected to the influence of the unknown inputs. The study in was carried out by considering two approaches. The first approach is based on Lyapunov quadratic functions; it is significant to note the great difficulty in finding satisfying results by this approach for the multimodel systems. For this reason we proposed an approach based on piecewise quadratic functions which led to interesting results (proposition 1) and less conservative than the quadratic approach. The conditions suggested in this article concern both the multiobserver stabilization and the estimation of states and the unknown inputs of a multimodel with measurable variables of decision (μξ(k)). The synthesis of a multiobserver with no measurable variables of decision is not approached. This point can constitute an interesting prospect for this study.

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.

This study addresses the problem of the estimation of state when heterogeneous multiagent systems are affected by homologous unknown inputs (UIs). Homologous UIs refer to identical UIs affecting different agents. An improved semidistributed filter based on previous research is proposed. The improved filter uses neighbors’ information for UI estimation but not state estimation. A necessary and sufficient condition for the proposed filter to achieve minimum-variance unbiased estimation is presented and proven. Moreover, the asymptotic stability of the filter is analyzed. A sufficient condition of the asymptotic stability is presented and proven. The theoretical and numerical analyses indicate that the proposed filter has less communication pressure, fewer calculation requirements, and better estimation performance compared with the existing solutions. <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">Note to Practitioners</i> —In the industry, homologous unknown inputs (UIs) exist in many different systems. For example, the same ambient temperature affects the performance of every battery in a battery pack. Similarly, the same wind power can affect different aircrafts flying in the same region. Temperature and wind power can be considered the homologous UIs of a multiagent system. Estimation of homologous UIs is important because of the latter’s massive impact on the system. In this study, data transmission delay and packet loss are ignored. Hence, the study is limited to low-rate systems. Moreover, nonlinear filters must be studied further in future work.

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

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

This paper proposes a novel filtering algorithm for simultaneous estimation of states and unknown inputs of a class of nonlinear discrete-time heterogeneous multi-agent systems. Based on the Taylor approximation of the nonlinear multiagent system, a distributed semi-cooperative switch-mode filter is developed to get the minimum-variance unbiased (MVU) estimation of the unknown inputs and states. Compared with the conventional decentralized EKF-based unknown input filter, the proposed distributed filter has a more relaxed existence condition, which makes it more applicable in reality. This new type of filter is then successfully applied to the simultaneous estimation of state of charge (SOC) and temperature of a battery pack for battery management of electric vehicles and grid-tied energy storage systems.

The paper addresses a systematic procedure to deal with the state, unknown input and parameter uncertainty estimation for nonlinear time-varying systems. This is realized by designing a robust observer for dynamic nonlinear systems using a Takagi-Sugeno (T-S) multi-model (MM) approach with nonlinear outputs. The method applies the technique of descriptor systems by considering unknown inputs and parameter uncertainty as auxiliary state variables. This approach allows to apply the tools of the linear automatic to dynamic nonlinear systems by using the Linear Matrix Inequalities (LMI) optimization. The observer estimates the previous mentioned variables and minimizes the effect of external disturbances on the estimation error. The model uncertainties are included in the model in a polynomial way which allows to consider the model uncertainty estimation as a fault detection problem. The residual sensitivity to faults while maintaining robustness according to a noise signal is handled by ‛ ∞ /ℋ − approach.

Infinity augmented state Kalman filter and its application in unknown input and state estimation

Novel discrete-time (DT) state observers and unknown input estimators are proposed for a class of DT nonlinear systems whose nonlinearity can be characterized by incremental multiplier matrices. The inputs to the nonlinear systems are allowed to have an unknown component. These nonlinear models are represented as linear models by treating the nonlinearity as a nonlinear input with a known structure. A novel DT state observer is proposed and a condition for its existence is given in terms of a linear matrix inequality (LMI). The unknown input estimator estimates the system unknown input with one sampling period time-delay. Unknown input estimators for continuous-time (CT) systems using their DT plant models are proposed. The proposed state observer and unknown input estimator are tested on numerical examples. An application of the proposed unknown input estimator to reconstruct malicious packet drops during the control signal transmission is given.

International audience

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 multi-step delay. It is shown that, within this new system reformation 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 approach.

This paper presents a system augmentation-based unbiased minimum-variance input and state estimation for systems with unknown inputs which can be reconstructed with a multi-step delay. An estimable input generating model (EIGM)-based system augmented approach is proposed to facilitate the filter design. It is shown that, via this new filtering approach the optimal unknown input and state estimation can be simultaneously achieved through the previously proposed robust two-stage Kalman filter (RTSKF). An illustrative example is given to show the effectiveness of the proposed results.

This article addresses the problem of state and unknown inputs (UIs) estimation for nonlinear systems with arbitrary relative degree with respect to the UIs. For this purpose, a novel nonlinear unknown input observer (UIO) is proposed, which is able to decouple the UIs by using the derivatives of the output signal. The error dynamics is attained by an exact handling and a factorization of its gradient to obtain a local polytopic representation suitable for input‐affine nonlinear systems. For that representation, a novel design condition based on convex optimization and linear matrix inequalities is proposed to exponentially stabilize the estimation error and to guarantee the validity of the proposed nonlinear UIO. Numerical simulations indicate the effectiveness of the proposed approach for different classes of nonlinear systems, for which the UIs could be totally decoupled from the state estimation.