Necessary and sufficient conditions for strict causal detectability and Luenberger observer for DAEs with general unknown inputs

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.

State estimation of stochastic discrete-time linear systems subject to persistent unknown inputs has been widely studied but only few works have been dedicated to the case where unknown inputs may be simultaneously or sequentially active or inactive. In this technical note, a Kalman filter approach is proposed for state estimation of systems with unknown intermittent inputs. The design is based on the minimisation of the trace of the state estimation error covariance matrix under the constraint that the state estimation error is decoupled from the unknown inputs corrupting the system at the current time. The necessary and sufficient stability conditions are established considering the upper bound of the prediction error covariance matrix.

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.

ABSTRACTThe paper considers the issues of state estimation and output disturbance reconstruction for a class of switched linear systems with unknown inputs. A singular switched system is derived from the original switched system by taking the output disturbance as a part of a new extended state vector. For the constructed singular switched system, a robust sliding-mode switched observer which can not only estimate the states of original switched system but also reconstruct output disturbances is proposed, where the switching of the observer is synchronous with that of the switched system. A sufficient condition is provided to guarantee the existence of the switched observer by the feasibility of an optimisation problem with linear matrix inequality constraint, and the corresponding switching signal with average dwell time is designed such that the convergence of the estimation error system is proven to be exponential. Based on the state estimation of singular switched system, the methods of state estimation and output disturbance reconstruction of original switched system are proposed. Finally, the simulation results confirm the predicted performance of the proposed methods.

The use of networked sensors or humans-in-the-loop for measuring outputs of dynamical systems inevitably leads to the introduction of time-varying measurement delays. The presence of such delays can lead to instability or severe degradation of system performance. In this paper, linear matrix inequality-based sufficient conditions are proposed for the design of state and unknown input observers informed by delayed measurements for a class of nonlinear systems, where the nonlinearities are characterized by incremental multiplier matrices. The proposed observer is guaranteed to perform at specified operational levels in the presence of unknown exogenous inputs acting on the states and measurement outputs. Sufficient conditions are also provided for the estimation of these unknown inputs to a specified degree of accuracy. The effectiveness of the proposed approach is demonstrated via a numerical example.

This brief is aimed at estimating the state and unknown input of a linear system in the multi-sensor network. First, a distributed filter consisting of the unknown input estimator and the state estimator is proposed, in which the unknown input estimator only uses its own information while the state estimator uses its in-neighbors’ information. Second, the optimal gain matrices are obtained by employing the weighted least squares method and minimizing the performance function, which allow to realize the optimal unbiased estimation of the proposed filter. Third, a sufficient condition about the given filter’s convergence is derived. Finally, a simulation is given for validating the given methodologies.

State and unknown input observers for nonlinear systems with delayed measurements

The problem of state estimation for discrete-time stochastic time-varying systems in the presence of unknown process inputs or disturbances is addressed in this paper. A Kalman-type filter is proposed, and the optimal oracle filter gain in the sense of minimizing the mean squared error of the state estimate is obtained. To tackle the unknown quantities in the gain matrix, a nonlinear equation is introduced and its solution is taken as the estimate of unknown inputs, and then, a novel nonlinear equation-based unknown input filtering (NEUIF) is proposed. A scalar-based iterative algorithm for related fixed point problem is developed so that the dichotomy method is employed to solve the above nonlinear equation very efficiently. Adopting the same strategy for the dynamic systems with unknown inputs or disturbances, we provide two applications of the proposed state estimation algorithm. One is for a class of nonlinear dynamic systems with linear observations by taking the residual term in linearizing the transition function as an unknown input in the derived linear system. The other is for tracking maneuvering targets in which the bias between the real motion and modeled motions is regarded as an unknown input in the state transition equation. Some numerical simulations demonstrate the effectiveness of the proposed NEUIF method for tackling various uncertainties in complicated dynamic systems.

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.

International audience

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.

In rollover prevention systems, a traditional rollover index can detect only un-tripped rollovers that happen due to high lateral acceleration from sharp turns. It fails to detect tripped rollovers that happen due to tripping from external inputs such as forces when a vehicle strikes a curb or a road bump. In order to develop a new rollover index that can detect both tripped and un-tripped rollovers, state estimation in the presence of unknown disturbance inputs is required. Therefore, this paper develops a methodology for estimation of unknown inputs in a class of nonlinear systems. The methodology is based on nonlinear observer design and dynamic model inversion to compute the unknown inputs from output measurements. The observer design utilizes the mean value theorem to express the nonlinear estimation error dynamics as a convex combination of known matrices with time varying coefficients. The observer gains are then obtained by solving linear matrix inequalities (LMIs). The developed approach can enable observer design for a large class of bounded Jacobian nonlinear systems with unknown inputs. The developed nonlinear observer is then applied for rollover index estimation. The developed rollover index is evaluated through simulations with an industry standard software, CARSIM, and with experimental tests on a 1/8 <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">th</sup> scaled vehicle. The simulation and experimental results show that the developed nonlinear observer can reliably estimate vehicle states, unknown normal tire forces, and rollover index for predicting both un-tripped and tripped rollovers.

This paper proposes a robust adaptive decentralized dynamic state estimation method for power system with unknown inputs of the highly detailed synchronous machine model. The temporal and spatial correlations among the unknown inputs are used to derive a vector auto-regressive model. The latter is further integrated together with state transition and measurement models for joint state and unknown inputs estimation. Thanks to the consideration of implicit cross-correlations between the states and the unknown inputs, only generator terminal voltage and current phasors are needed. Test results on a realistic hydro power plant using the field PMU measurements show that the proposed method is able to track both the system dynamic states and unknown controller inputs. These information could significantly benefit the validation and calibration of generator controller parameters.

This paper deals with the problem of state and faults estimation for nonlinear uncertain systems described by Takagi–Sugeno fuzzy structures (called also multiple models). In this work, actuator faults are considered as unknown inputs. The state and faults estimation is made using a structure of sliding mode observer where an integral term is added. This new structure of observer is called proportional integral sliding mode observer. The added integral term permits the unknown input estimation. For the sensor faults estimation, a mathematical transformation is used. The application of this mathematical transformation to the initial system output let to conceive an augmented system where the initial sensor fault appears as an unknown input. The observer convergence conditions are formulated in the form of Linear Matrix Inequalities allowing computing the observer gains. The proposed proportional integral sliding mode observer is applied to a numerical example showing the efficiency of the fault and the state estimation. In order to show the efficiency of the proposed method, it is applied to a turbo-reactor system.

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.