State estimation within a guaranteed interval

Automatica, 2013

This paper is devoted to design of interval observers for Linear Time Varying (LTV) systems and a class of nonlinear time-varying systems in the output canonical form. An interval observer design is feasible if it is possible to calculate the observer gains making the estimation error dynamics cooperative and stable. It is shown that under some mild conditions the cooperativity of an LTV system can be ensured by a static linear transformation of coordinates. The efficiency of the proposed approach is demonstrated through numerical simulations.

This paper deals with state estimation and fault detection in the presence of unknown but bounded state perturbations and measurement noise. In this context, most available results are for linear models. Based on interval analysis, a state estimator for nonlinear dynamical systems is presented. Given the perturbation and noise bounds, the proposed method evaluates a set estimate guaranteed to contain all values of the state that are consistent with the available observations. The estimator is then used to regime shift detection. A numerical example is given.

2010

Abstract This paper deals with state estimation and fault detection in the presence of unknown but bounded state perturbations and measurement noise. In this context, most available results are for linear models. Based on interval analysis, a state estimator for nonlinear dynamical systems is presented. Given the perturbation and noise bounds, the proposed method evaluates a set estimate guaranteed to contain all values of the state that are consistent with the available observations.

IEEE Access, 2021

In this paper, we investigate the interval observer problem for a class of discrete-time nonlinear systems, in absence or presence of external disturbances and parametric uncertainties. The interval observers depend on the design of two preserving order observers, providing lower and upper estimations of the state. The main objective is to apply the stability radii notions and cooperativity property in the estimation error systems in order to guarantee that the lower/upper estimation is always below/above the real state trajectory at each time instant from an appropriate initialization, and the estimation errors converge asymptotically towards zero when the disturbances and/or uncertainties are vanishing. For the disturbed case, the estimation errors practically converge to a vicinity of zero, while the lower/upper estimations preserve the partial ordering with respect to the state trajectory. The design conditions, that are valid for Lipschitz nonlinearities, can be expressed as Linear Matrix Inequalities (LMIs). A numerical simulation example is provided to verify the effectiveness of the proposed method.

Automatica, 2016

This paper investigates the interval observer design for a class of nonlinear continuous systems, which can be represented as a superposition of a uniformly observable nominal subsystem with a Lipschitz nonlinear perturbation. It is shown in this case there exists an interval observer for the system that estimates the set of admissible values for the state consistent with the output measurements. An illustrative example of the observer application is given with simulation results.

IEEE Transactions on Automatic Control, 2014

Interval observers are dynamic systems that provide upper and lower bounds of the true state trajectories of systems. In this work we introduce a technique to design interval observers for linear systems affected by state and measurement disturbances, based on the Internal Positive Representations (IPRs) of systems, that exploits the order preserving property of positive systems. The method can be applied to both continuous and discrete time systems.

Symmetry

In this paper, we consider the problem involved when designing the interval observer for the system described by a linear discrete-time model under external disturbances and measurement noises. To solve this problem, we used the reduced order model of the initial system, which is insensitive or has minimal sensitivity to the disturbances. The relations involved in designing the interval observer, which has minimal dimensions and estimates the prescribed linear function of the original system state vector, were obtained. The theoretical results were illustrated by a practical example.

IEEE Access, 2018

This paper presents a new approach to design preserving order and interval observers for a family of nonlinear systems in absence and in presence of parametric uncertainties and exogenous disturbances. A preserving order observer provides an upper/lower estimation that is always above/below the state trajectory, depending on the partial ordering of the initial conditions, and asymptotically converges to its true values in the nominal case. An interval observer is then constituted by means of an upper and a lower preserving order observer. In the uncertain/disturbed case, the estimations preserve the partial ordering with respect to the state trajectory, and practically converge to the true values, despite of the uncertainties/perturbations. The design approach relies on the cooperativity property and the stability radii mathematical tools, both applied to the estimation error systems. The objective is to exploit the stability radii analysis for the family of linear positive systems under the time-varying nonlinear perturbations in order to guarantee the exponential convergence property of the observers, while the cooperativity condition determines the partial ordering between the trajectories of the state and the estimations. The proposed approach, defined for Lipschitz nonlinearities, depends only on two observer matrix gains. The design is reduced to the solution of linear matrix inequalities, which are given by the cooperative condition and convergence constraints. An illustrative example is presented to show the effectiveness of the theoretical results. INDEX TERMS Interval observers, preserving order observers, stability radii, positive systems.

Applied Mathematics and Computation, 2012

The interval Kalman filtering (IKF) can handle parametric interval uncertainties of the system matrices, and it computes the lower and upper boundaries of the estimated states. In this paper, we propose an alternative form of interval Kalman filtering to reduce the conservatism inherent to the existing interval Kalman filtering. First, we address why the existing interval Kalman filtering scheme induces conservatism in the boundary estimation. Then, to remove the conservatism, we derive noise covariance matrices taking into account of interval uncertainties as well as process and measurement noises. Following the typical derivation process of the standard Kalman filtering, a new recursive form of interval Kalman filtering is derived. Through numerical simulations, the superiority of the new algorithm over existing IKF is illustrated.

Asian Journal of Control, 2019

This paper studies the problem of designing interval observers for a family of discrete-time nonlinear systems subject to parametric uncertainties and external disturbances. The design approach states that the interval observers are constituted by a couple of preserving order observers, one providing an upper estimation of the state while the other provides a lower one. The design aim is to apply the cooperative and dissipative properties to the discrete-time estimation error dynamics in order to guarantee that the upper and lower estimations are always above and below the true state trajectory for all times, while both estimations asymptotically converge towards a neighborhood of the true state values. The approach represents an extension to the original method proposed by the authors, which focuses on the continuous-time nonlinear systems. In some situations, the design conditions can be formulated as bilinear matrix inequalities (BMIs) and/or linear matrix inequalities (LMIs). Two simulation examples are provided to show the effectiveness of the design approach.