State and Unknown Input Observers for Nonlinear Systems With Bounded Exogenous Inputs

International audience

This brief investigates the issue of state estimation and unbounded unknown input reconstruction for a class of one-sided Lipschitz nonlinear interconnected systems with uncertain output and mismatched unknown input distribution matrix. Based on descriptor augmentation strategy and coordinate transformation, the effect of measurement noise and matched unknown input parts are firstly decoupled. Then, a novel multi-intermediate estimator-assisted unknown input observer is constructed to estimate state and mismatched unknown input. In addition, sufficient conditions in linear matrix inequality (LMI) form is derived to guarantee the stability of the observer and to facilitate observer parameter tuning. Finally, a simulation example is provided to illustrate the effectiveness and merits of the proposed methods.

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.

This paper presents a new method to synthesize observers for continuous time nonlinear systems described by Takagi-Sugeno (TS) model with unmeasurable premise variables. First, convergence conditions are established in order to guarantee the convergence of the state estimation error. These conditions are given in linear matrix inequality (LMI) formulation. Secondly, a classical proportional integral observer (PIO) is extended to the considered nonlinear systems in order to estimate the state and the unknown inputs (UI).

Three-stage unscented Kalman filter for state and fault estimation of nonlinear system with unknown input

This paper presents a novel approach for state and unknown input (UI) observer in single-input single-output (SISO) linear time-invariant (LTI) systems. An Orthogonal Neural Network (ONN), which does not require traditional training or offline learning, is incorporated to approximate the unknown input. The proposed method extends the system state to include the ONN weights, creating a new LTI system. Leveraging the structure of orthogonal activation functions, a suitable algorithm for the extended state dynamics is derived. The observer gain is computed using Linear Matrix Inequalities (LMI), ensuring stability and attenuating the approximation error. Unlike existing methods, this approach requires no assumptions about the unknown input, such as matching conditions or boundedness. A numerical example demonstrates the effectiveness of the proposed scheme.

This paper proposes a novel adaptive observer for Lipschitz nonlinear systems and dissipative nonlinear systems in the presence of disturbances and sensor noise. The observer is based on an H∞ observer that can estimate both the system states and unknown parameters by minimising a cost function consisting of the sum of the square integrals of the estimation errors in the states and unknown parameters. The paper presents necessary and sufficient conditions for the existence of the observer, and the equations for determining observer gains are formulated as linear matrix inequalities (LMIs) that can be solved offline using commercially available LMI solvers. The observer design has also been extended to the case of time-varying unknown parameters. The use of the observer is demonstrated through illustrative examples and the performance is compared with extended Kalman filtering. Compared to previous results on nonlinear observers, the proposed observer is more computationally efficient, and guarantees state and parameter estimation for two very broad classes of nonlinear systems (Lipschitz and dissipative nonlinear systems) in the presence of input disturbances and sensor noise. In addition, the proposed observer does not require online computation of the observer gain.

An estimation of the state and the unknown inputs of the reduced nonlinear model of an activated sludge process using the Extended Kalman Filter (EKF) is proposed. First, we present the reduced nonlinear model. This model contained five state variables and four unknown inputs. For satisfying the rank condition for the construction of an EKF, one unknown input has been approximated and the daily mean value of another unknown input has been used. Then, to estimate conjointly the state and the unknown inputs, the reduced nonlinear system is transformed to a nonlinear singular system. High performances of the proposed observer will be shown through the simulation results.

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

This paper adresses the problem of simultaneously estimating the state and the fault of nonlinear discrete-time stochastic systems in light of the unknown input filtering framework. The fault and unknown disturbances which may cause great estimate errors and even divergence of conventional filters, affect both the system state and the measurements. Inspired by the robust two stage Kalman filter for linear discrete-time system, a nonlinear robust state and fault estimator is derived. In order to achieve the aim, the nonlinear system with fault and unknown disturbances is first transformed into an equivalent descriptor system. Next, it is shown that the previously proposed robust two stage Kalman filter can be applied to yield a robust state and fault estimation. Simulation results for a robotic manipulator show the effectiveness of the proposed method.

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

We consider the problem of estimating the state and unknown input for a large class of nonlinear systems subject to unknown exogenous inputs. The exogenous inputs themselves are modeled as being generated by a nonlinear system subject to unknown inputs. The nonlinearities considered in this work are characterized by multiplier matrices that include many commonly encountered nonlinearities. We obtain a linear matrix inequality (LMI), that, if feasible, provides the gains for an observer which results in certified $L$ 2 performance of the error dynamics associated with the observer. We also present conditions which guarantee that the $L$ 2 norm of the error can be made arbitrarily small and investigate conditions for feasibility of the proposed LMIs.

In this paper a new unknown input estimation method is proposed for a class of nonlinear stochastic systems in the presence of time dependent unknown inputs, when the system states and process noises are unknown but bounded. In this study, a new augmented state vector is constructed by augmenting unknown inputs as a new state to the original state vector. Then a recursive algorithm based on unknown but bounded (UBB) uncertainty is developed, that unlike the Bayesian models which consider the state estimate as a single vector, produces a time-varying set of state estimates that contains the system's true state. The particular sets to be discussed, are ellipsoids. The proposed method doesn't need any unknown input detection stage procedure and covariance resetting that are necessary in previous works. At last, efficiency of the proposed method is shown in a numerical simulation for a nonlinear system.

In this paper, we consider the problem of designing a special type of observer to estimate the state and unknown input disturbance for linear time-invariant positive systems. Although positive observers have been designed for positive systems, they are unable to estimate the states when unknown inputs or disturbances are present in the systems. To realize simultaneous estimation of states and unknown inputs, we first design positive unknown input observers (PUIO) for positive linear systems, which is the main contribution of this paper. Then, we provide a procedure to estimate the unknown disturbance. The PUIO design can be performed with the aid of a positive stabilization scheme via LMI. A critical step in the design process is the requirement of imposing a condition on positivity of the generalized inverse of a certain design matrix. We show that this can easily be accomplished with the aid of a unique result, which makes the design of PUIO possible. An alternative LMI with less restrictive assumption is also included to design PUIO. Finally, theoretical results are supported by numerical examples.

For descriptor systems with measurement output noises (input disturbances may exist at the same time), a new descriptor estimator technique is developed. The necessary and sufficient condition for the existence of the present estimator is derived, and a systematic design approach is addressed. The effect of uncertainties is decoupled completely, and the asymptotic estimates of the descriptor system state and the output noise are obtained simultaneously. Furthermore, a normal state/disturbance estimator is also given. For a class of nonlinear descriptor systems with both output noises and input uncertainties, a nonlinear descriptor estimator is derived by using the proposed design approach, together with the linear matrix inequality technique. The existence and convergence of the nonlinear descriptor estimator is proven. The asymptotic estimates of the descriptor nonlinear system state and the output noise are obtained at the same time. The present estimators are applied to the sensor fault diagnosis, and hence the sensor fault can be estimated asymptotically. Finally, two numerical examples are included to illustrate the proposed design procedures and applications.