Proposed Interval Observer Research Articles

New structure to design interval observers for linear continuous-time systems

In this contribution, a new approach to design interval observer for continuous-time linear systems is introduced. First, a more general structure than the classical Luenberger observer is proposed and extended to the interval case based on an explicit set integration method. Then, to obtain tight state enclosures, the computation of the design matrices involved in this new structure is formulated as a nonlinear constrained optimisation problem. In addition, using the symmetric property of interval vectors, a less complex form of the proposed interval observer is presented. On the other hand, compared to the conventional interval observers, no preserving order property on the dynamics of the estimation error is required to apply the introduced method. A numerical case study example is considered to illustrate the effectiveness of the proposed method and to compare its performance to that of a former method.

Outlier-resistant interval observer design for multirate time-delayed systems under the adaptive event-triggered protocols

In this paper, an interval observer has been presented for a type of multirate time-delayed systems subject to measurement outliers under the adaptive event-triggered protocol (AETP). First of all, the multirate sampling system is studied to meet the practical requirements. Meanwhile, for relieving the communication burden, an AETP is considered to schedule the measurement information received by the observer. The augmented system is built at a consistent sampling/update rate by using the lifting technique. Moreover, the occurrence of measurement outliers would pose some serious threats to the reliability of the systems. For the addressed issue, an outlier-resistant interval observer is developed by means of saturation function to weaken the impact of the measurement outliers. A sufficient condition is proposed by using the positive system theory and Lyapunov stability theory to ensure that the interval observer is effective and the error system is input-to-state stable. Besides, the gain matrices of the interval observer are obtained by solving a set of linear matrix inequalities (LMIs). Finally, the effectiveness of the proposed interval observer is verified by a numerical simulation.

Limiting State Estimation of Switched Interval Systems with Metzler-Takagi-Sugeno Models

The paper deals with interval observer design for fuzzy switched positive systems. The systems are represented by the Takagi-Sugeno fuzzy models, with premise variables depending on a measurable part of the state vector. Stability conditions for the proposed interval observer structure are formulated via linear matrix inequalities to ensure nonnegative system state estimation. The proposed method allows to compute the lower and upper bounds of the system state under the assumption that the system disturbance are bounded. The properties of proposed approach are illustrated in numerical example.

Interval observer‐based bumpless switching control for switched systems

AbstractAn interval observer‐based bumpless switching (BS) control scheme for switched systems (SSs) is proposed in this article. First, a couple of interval observers are built to capture the system state from its upper and lower bounds, respectively. Then, according to the observers, the controllers with small jumps at switching points and the switching logic are co‐developed to stabilize the SS and reducing the unwanted big steps in the control signal at switching moments. Further, under the proposed interval observers, interval observer‐based controllers, and the switching director, a condition is driven to ensure the solvability of the observer‐based BS control problem, even though every subsystem may not permit a solution to the control problem. Finally, a verification example serving for the established interval observer‐based BS control scheme is offered.

Zonotopic Kalman Filter-Based Interval Estimation for Discrete-Time Linear Systems With Unknown Inputs

This letter proposes an unknown input zonotopic Kalman filter-based interval observer for discrete-time linear time-invariant systems. In such contexts, a change of coordinates decoupling the state and the unknown inputs is often used. Here, the dynamics are rewritten into a discrete-time linear time-invariant descriptor system by augmenting the state vector with the unknown inputs. A zonotopic outer approximation of the feasible state set is then obtained with a prediction-correction strategy using the information from the system dynamics, known inputs and outputs. Bounds for both the state and unknown inputs are obtained from this zonotopic set. The efficiency of the proposed interval observer is assessed with numerical simulations.

Sensor fault detection for switched systems using interval observer with L∞ performance

This paper deals with sensor fault detection (FD) for discrete-time switched systems subject to bounded disturbances. A novel approach is investigated to construct residual framers using interval observer with L∞ performance. The proposed technique is used to provide more degrees of design freedom and to obtain accurate FD results. The design conditions of the FD observer are given in terms of Linear Matrix Inequalities (LMIs) adopting firstly a common quadratic Lyapunov function, under an arbitrary switching signal and secondly multiple quadratic Lyapunov functions, under an Average Dwell Time (ADT) switching signal. Furthermore, the FD decision is based on residual intervals generated by the proposed interval observer. The effectiveness of the proposed approach is highlighted through simulation results of an academic example.

Distributed Fault Detection and Isolation for Multiagent Systems: An Interval Observer Approach

This paper presents a distributed method for detecting and isolating the faults in multiagent systems (MASs). First, for the output-feedback-based closed-loop MAS in normal case and faulty cases, a bank of fault detection and isolation (FDI) interval observers are constructed by taking into account the outputs of local and received from neighbors, the bounds of one node fault signal and the whole disturbances. Then, the observer gains can be determined by solving the disturbance attenuation, fault sensitivity, and nonnegativity conditions, simultaneously. Furthermore, an alarm signal is released when the zero value is excluded from any one of the intervals, a node fault is indicated when the zero value is included in one residual interval and excluded from the surplus intervals. Different from the classical FDI methods, the proposed interval observers are able to both generate the residual signals and imply the thresholds. Under such framework, each agent has the capacity to determine which neighbor agent is faulty. Finally, the developed techniques are demonstrated in a simulation example.

ℓ1-to-ℓ1 interval observation design for discrete-time switched linear systems under dwell time constraint

ABSTRACTIn this brief article, sufficient conditions characterising the -gain of discrete-time switched positive linear systems (DSPLSs) under dwell time constraint are obtained. The dwell time refers to range, minimum and constant dwell time. These conditions are presented in terms of linear programming. Then, convex sufficient conditions on the existence of -to- interval observers for discrete-time switched linear systems (DSLSs) are derived based on the -gain stability analysis results of DSPLSs. By adding some parameter matrices, the proposed interval observer has a new structure that relaxes the design conditions. Suitable observer gains can be extracted from the solution of the dimensional optimisation problem where the -gain of the system mapping the disturbances to the weighted observation errors is minimised. At last, some numerical examples of DSLSs are given for illustration.

Interval estimation for continuous-time LPV switched systems

This paper deals with interval state estimation for continuous-time linear parameter varying (LPV) switched systems with measured polytopic parameter dependence, assuming a known switching signal. Considering that the measurement noise and the state disturbance are unknown but bounded and that the dynamics of the system are described by a convex combination, lower and upper bounds of the state are therefore determined. An interval observer is designed in order to guarantee both stability and cooperativity of the upper and lower observation errors. For the proposed observer, sufficient input-to-state stability conditions are given in terms of linear matrix inequalities, adopting firstly common quadratic Lyapunov functions, under arbitrary switching signals and secondly multiple quadratic Lyapunov functions, under average dwell time (ADT) switching signals. The efficiency of the proposed interval observer is highlighted through simulation results on academic examples.

Interval Observer Design Under Stealthy Attacks and Improved Event-Triggered Protocols

In this article, an interval estimation problem is investigated for a class of discrete-time nonlinear networked systems under stealthy attacks. An improved event-triggered protocol with the time-varying threshold is adopted to govern the received signals of interval observer so as to reduce unnecessary data communication burden. Stealthy attacks occurring in both the sensors and the plant come typically from open networked environments. For the addressed issue, a novel interval observer is developed to estimate the bounds of both system states and stealthy attacks. A sufficient condition is proposed by using the positive system theory and Lyapunov stability theory. Furthermore, the desired observer gain is obtained in terms of the solution to certain matrix inequalities. In comparison with the traditional approach that attacks are regarded as bounded disturbances, the estimated upper and lower bounds of attack signals can effectively reduce the conservatism of estimation performance, and avoid the implementation difficulties of the designed observer in practical engineering. Finally, the effectiveness of the proposed interval observers is verified by a numerical simulation.

Nanoforce Estimation using Interval Observer: Application to Force Sensor based on Diamagnetic Levitation

This paper presents a new design to measure slowly time-varying vertical nanoforces. The proposed sensor is based on diamagnetic levitation. It uses auto-stabilized magnetic springs and relies on a macroscopic seismic mass: a capillary tube used in a vertical configuration. When a vertical external force is applied to it, the capillary tube acts as a transducer that converts this unknown input force into a vertical displacement that is measured. Relying on this measurement, a Generic Linear Extended State Observer based on an interval approach is proposed to reconstruct this unknown input force and the state of the considered system. This is achieved without any a priori knowledge on the vertical force, i.e. magnitude and bounds. The efficiency of the proposed interval observer is illustrated by experimental results.

High-gain interval observer for partially linear systems with bounded disturbances

In this paper, a high-gain interval observer is proposed for a class of partially linear systems affected by unknown but bounded additive disturbances term and measurements noise. The proposed observer is based upon a classical high-gain structure from which an interval observer for the system is designed. The proposed interval observer is designed based on the suitable change of coordinates which ensure the cooperativity of the system. To prove the effectiveness of the proposed approach, two numerical examples are provided and the corresponding simulation results are presented.

Luenberger-like interval observer design for discrete-time descriptor linear system

In this paper, the problem of Luenberger-like interval observer design for discrete-time descriptor linear system is investigated. In the first place, two ways in which the original descriptor system is transformed into general systems are given and the detectability of the equivalent general systems is discussed in detail. A new equivalent general system construction is developed and we prove that the detectability can be kept under this new equivalent transformation. After this, based on the new equivalent general system, a Luenberger-like interval observer is proposed. Finally, two simulation examples are given to demonstrate the correctness and the effectiveness of the proposed interval observer.

Robust fault and icing diagnosis in unmanned aerial vehicles using LPV interval observers

SummaryThis paper proposes a linear parameter varying (LPV) interval unknown input observer for the robust fault diagnosis of actuator faults and ice accretion in unmanned aerial vehicles (UAVs) described by an uncertain model. The proposed interval observer evaluates the set of values for the state, which are compatible with the nominal fault‐free and icing‐free operation and can be designed in such a way that some information about the nature of the unknown inputs affecting the system can be obtained, thus allowing the diagnosis to be performed. The proposed strategy has several advantages. First, the LPV paradigm allows taking into account operating point variations. Second, the noise rejection properties are enhanced by the presence of the integral term. Third, the interval estimation property guarantees the absence of false alarms. Linear matrix inequality–based conditions for the analysis/design of these observers are provided in order to guarantee the interval estimation of the state and the boundedness of the estimation. The developed theory is supported by simulation results, obtained with the uncertain model of a Zagi Flying Wing UAV, which illustrate the strong appeal of the methodology for identifying correctly unexpected changes in the system dynamics due to actuator faults or icing.

Interval observer-based fault detection in finite frequency domain for discrete-time fuzzy systems

Abstract This paper presents a new method for detecting faults in discrete-time fuzzy systems. First, the piecewise interval observers are constructed based on the output-space partition technique. Second, l1 performance is introduced to attenuate the persistent bounded disturbances and H − performance in finite frequency domain is employed to improve fault sensitivity of the residual intervals. Then, the observer gains can be determined by solving the disturbance attenuation, fault sensitivity and non-negativity conditions, simultaneously. Different from the classical fault detection methods with residual evaluation functions and threshold generators, the proposed interval observers are able to generate the natural thresholds. Finally, the developed technique is demonstrated in a simulation example.

Interval observer design for uncertain discrete-time linear systems

This paper proposes a novel interval observer design method for discrete-time linear systems with unknown but bounded disturbance and measurement noise. The proposed interval observer has a new structure that provides more design degrees of freedom. A direct method based on H∞ technique is used to improve the accuracy of interval estimation. The design conditions are formulated into linear matrix inequalities, which can be efficiently solved. Two numerical examples are given to illustrate the design and validate the performance of the interval observers.

Using set invariance to design robust interval observers for discrete‐time linear systems

SummaryBased on interval and invariant set computation, an interval version of the Luenberger state observer for uncertain discrete‐time linear systems is proposed in this work. This new interval observer provides a punctual estimation of the state vector and guaranteed bounds on the estimation error. An off‐line and an on‐line approach to characterize, in a guaranteed way, the estimation error are introduced. Compared with the existing approaches, the proposed interval observer design method is not restrictive in terms of required assumptions, complexity, and on‐line computation time. Furthermore, the convergence issue of the estimation error is well established and to reduce the conservatism of the estimated state enclosure induced by the bounded additive state disturbance and noise measurement, an H∞ method to compute the optimal observer gain is proposed. The performance of the proposed state estimation approach are highlighted on different illustrative examples.

Interval observer for LPV systems with unknown inputs

This study addresses the problem of guaranteed state and unknown input (UI) estimation for a class of linear parameter varying systems affected by UIs and other bounded perturbations. The context of interval observer in the cooperative framework is adopted. Thus, lower and upper bounds of the state and UI are provided. No assumption is made on the UI. The approach is based on a rank condition allowing to decouple the UI from the state estimation errors (so that the state estimation is guaranteed, regardless of the variation of the UI). The convergence of the proposed interval observer is studied by Lyapunov theory and stability conditions are expressed in terms of linear matrix inequalities which ease the design of the gain matrices of the observer. Finally, illustrative examples are given for discussion and comparison with respect to the existing results.

Interval observer fault detection ensuring detectability and isolability by using a set-invariance approach

This paper proposes the design of an interval observer-based Fault Detection and Isolation (FDI) algorithm using the set-invariance approach. Using this approach, both fault detectability and isolability properties of the proposed interval observer based FDI algorithm can be characterized in system steady-state operation. The effect of the uncertainty is taken into account using zonotopic-set representations. Finally, a simulation example based on a two-tanks system is employed to both illustrate and discuss the effectiveness of the proposed approach.

Interval State and Unknown Inputs Estimation for Linear Time-Invariant Systems

This paper deals with a set membership approach to design an Unknown Input Interval Observer for uncertain Linear Time-Invariant (LTI) continuous-time systems. The goal is to compute lower and upper bounds for unmeasured state as well as unknown inputs. The bounds are guaranteed under the assumption that external disturbances and noises are bounded with a priori known bounds. The proposed interval observer structure is based on decoupling the unknown input effect on the state dynamics by solving algebraic constraints on the estimation errors. Numerical simulations on a 5th-order lateral axis model of a fixed-wing aircraft are provided to demonstrate the efficiency of the proposed technique.