L2Observers for a Class of Nonlinear Systems with Unknown 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.

This paper considers the reduced-order observer design problem for one-sided Lipschitz nonlinear systems with unknown inputs. The systems under consideration are a larger class of nonlinear systems than the well-studied Lipschitz systems and have inherent advantages with respect to conservativeness. For such systems, we propose a reduced-order unknown input observer (UIO) by using the linear matrix inequality (LMI) technique. Sufficient conditions that guarantee the existence of the reduced-order UIO are established and transformed into the format of LMIs via using the matrix generalized inverse technique. Simulation results are finally given to illustrate the effectiveness of the proposed design.

A new method to design a fuzzy bilinear observer (FBO) with unknown inputs is developed for a class of nonlinear systems. The nonlinear system is modeled as a fuzzy bilinear model (FBM). This kind of T‐S fuzzy model is especially suitable for a nonlinear system with a bilinear term. The proposed fuzzy bilinear observer subject to unknown inputs is developed to ensure the asymptotic convergence of the error dynamic using the Lyapunov method. The proposed design conditions are given in linear matrix inequality (LMI) formulation. The paper studies also the problem of fault detection and isolation. An unknown input fuzzy bilinear fault diagnosis observer design is proposed. This work is given for both continuous and discrete cases of fuzzy bilinear models. Illustrative examples are chosen to provide the effectiveness of the given methodology.

In this paper, a step by step algorithm is given to design functional unknown input observer for nonlinear discrete time systems under the Polytopic Takagi-Sugeno framework (also known as multiple models systems). Applying the Lyapunov theory and the ℒ 2 attenuation, Linear Matrix Inequalities (LMI)s conditions are deduced which are solved for feasibility to obtain observer design matrices. To reject the effect of unknown input, classical approach of decoupling the unknown input for the linear case is used. A solver based solution is proposed. The novelty of the proposed approach consists in solving simultaneously both structural constraints and LMIs, which ensure a mean for the efficient design of the gains of the observers. To illustrate the proposed theoretical results, an application example of model reference tracking control applied to an electro-mechanical model of a motor with a time varying parameter is discussed.

A systematic design methodology for state observers for a large class of nonlinear systems with bounded exogenous inputs (disturbance inputs and sensor noise) is proposed. The nonlinearities under consideration are characterized by an incremental quadratic constraint parameterized by a set of multiplier matrices. Linear matrix inequalities are developed to construct observer gains, which ensure that a performance output based on the state estimation error satisfies a prescribed degree of accuracy. Furthermore, conditions guaranteeing estimation of the unknown inputs to arbitrary degrees of accuracy are provided. The proposed scheme is illustrated with a numerical example, which does not satisfy the so-called “matching conditions.”

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

For the direct feedthrough system where unknown input affects both process equation and measurement equation, an unscented recursive three-step filter based unbiased minimum-variance (URTSF-UMV) estimation algorithm is proposed. The algorithm is based on the unbiased minimum-variance (UMV) method and uses statistical linearisation technology to approximate the nonlinear system and the nonlinear measurement function to linear regression form. The estimation of unknown input is obtained from innovation through weighted least-squares (WLS) estimation. The approximate linear regression form allows us to process the nonlinear system with unknown input direct feedthrough by using the UMV state estimation framework, and derive the filter by minimising the trace of the state error covariance of the measurement update under the unbiased condition. The URTSF-UMV requires that the unknown input distribution matrix is full column rank. We use the full rank decomposition method to deal with the nonlinear system containing the unknown input rank-deficient distribution matrix. The effectiveness of the URTSF-UMV algorithm is demonstrated through an illustrative simulation example.

Designing delay-dependent functional observers for LTI systems with multiple known time-varying state delays and unknown time-varying input delays is studied. The input delays are arbitrary, but the state delays should be upper-bounded. In addition, two scenarios of slow-varying and fast-varying state delays are investigated. The results of the paper can also be considered as one of the first contributions considering unknown-input functional observer design for linear systems with multiple time-varying state delays. Based on the Lyapunov Krasovskii approach, delay-dependent sufficient conditions of the exponential stability of the observer in each scenario are established in terms of linear matrix inequalities. Because of using effective techniques, such as the descriptor transformation and an advanced weighted integral inequality, the proposed stability criteria can result in larger stability regions compared with the other papers that study functional observers for time-varying delay systems. Furthermore, to help with the design procedure, a genetic algorithm-based scheme is proposed to adjust a weighting matrix in the established linear matrix inequalities. Two numerical examples illustrate the design procedure and demonstrate the efficacy of the proposed observer in each scenario. Copyright © 2016 John Wiley & Sons, Ltd.

This paper treats the design problem of full-order observers for nonlinear descriptor systems with unknown input (UI). Depending on the available knowledge on the UI dynamics, two cases are considered. First, a UI proportional observer (UIPO) is proposed when the spectral domain of the UI is unknown. Second, a PIO is proposed when the spectral domain of the UI is in the low frequency range. Sufficient conditions for the existence and stability of such observers are given and proved. Based on the linear matrix inequality (LMI) approach, an algorithm is presented to compute the observer gain matrix that achieves the asymptotic stability objective. An example is included to illustrate the method.

Robust fault detection for a class of nonlinear time-delay systems

In this paper, a differential neural network (DNN) implemented as a robust observer estimates the dynamics of perturbed uncertain nonlinear systems affected by exogenous unknown inputs. In the first stage, the identification error converges into a neighborhood around the origin. Then, the second-order sliding mode supertwisting algorithm implemented as a robust exact differentiator reconstructed the unknown inputs. The approach proposed in this paper can be applied in the case of full access to the state vector (identification problem) and in the case of partial access to the state vector (estimation problem). In the second case, the nonlinear system under study must have well-defined full relative degree with respect to the unknown input. Numerical examples showed the effectiveness of the proposed algorithm. The first example tested the DNN working as an identifier into a mathematical model describing the dynamics of a spatial minisatellite. The second example (with a DNN implemented as an observer) tested the methodology of this paper over a single link flexible robot manipulator represented in a canonical (Brunovsky) form. In both examples, the mathematical models served as data generators in the testing of the neural networks. Even when not exact mathematical description of both models was used in the input estimation, the accuracy obtained with the DNN is comparable with the case of applying a high-order differentiator with complete knowledge of the plant.

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.

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.

A Polytopic Observer Design Approach for Landing Control of a Quadrotor UAV