Class Of Lipschitz Nonlinear Systems Research Articles

Stabilization of a Class of Lipschitz Nonlinear Systems Through an Event-Triggered Impulsive Controller

Stabilization of a Class of Lipschitz Nonlinear Systems Through an Event-Triggered Impulsive Controller

Estimation of Multiple Unknown Parameters Using Adaptive Observers for Lipschitz Nonlinear Systems

This work presents the design of an adaptive observer for a class of Lipschitz nonlinear systems, where the observer is able to estimate more than one unknown parameter. Thanks to this, it is possible to estimate the system states, which facilitates the monitoring of nonlinear Lipschitz systems. The proposed observer is tested for monitoring a semiactive suspension, in order to know the stiffness parameters of the tire and chassis, allowing the detection of possible anomalies in the system. To ensure observer stability, the gains are calculated by LMI obtained by Lyapunov analysis. The proposed observer is evaluated by estimating the unknown states and parameters of a mechanical system allowing monitoring of its performances.

Sensor fault estimation based on L∞ unknown input observer

Abstract This study investigates the fault estimation problem for a class of Lipschitz non-linear systems with sensor faults and disturbances. An augmented system consisting of system states and sensor faults is constructed to estimate sensor faults by designing an augmented state. A novel L∞ performance index is presented to reduce the influence of disturbances on fault estimation results. An unknown input observer (UIO) is proposed based on the L∞ performance index to achieve an asymptotic estimation of sensor faults. The design conditions of the L∞ UIO are deduced and solved using linear matrix inequalities. The effectiveness of the proposed L∞ UIO and the fault estimation performance are demonstrated via numerical simulations of a single-link flexible joint robot.

A robust distributed observer design for Lipschitz nonlinear systems with time-varying switching topology

This paper deals with state estimation for a class of Lipschitz nonlinear systems under a time-varying disconnected communication network. A distributed observer consists of some local observers that are connected to each other through a communication network. We consider a situation where a communication network does not remain connected all the time, and the network may be caused by intermittent communication link failure. Moreover, each local observer has access to a local measurement, which may be insufficient to ensure the system’s observability, but the collection of all measurements in the network ensures observability. In this condition, the purpose is to design a distributed observer where the estimated state vectors of all local observers converge to the state vector of the system asymptotically, while local observers exchange estimated state vectors through a communication network and use their local measurements. According to theoretical analysis, a nonlinear and a robust nonlinear distributed observer exist when in addition to the union of all communication topologies being strongly connected during a time interval, the component of each communication graph is also strongly connected during each subinterval. The existence conditions of the distributed observers are derived in terms of a set of linear matrix inequalities (LMIs). Finally, the effectiveness of the presented method is numerically verified using some simulation examples.

Finite-time observer–based robust fault estimation for nonlinear systems

This article investigates the finite-time fault-estimation problem for a class of Lipschitz nonlinear systems with actuator faults and unknown input disturbances. Specifically, the original system is preliminarily decoupled into two reduced-order subsystems using a linear nonsingular transformation. A novel finite-time observer is proposed to estimate actuator faults in finite time. Subsequently, the proposed finite-time observer is extended to nonlinear systems with unknown input disturbances to investigate the robust fault-estimation problem. The actuator fault and unknown input disturbance are estimated simultaneously using two finite-time observations. Furthermore, a fault-estimation scheme is designed based on finite-time observers. The finite-time stability of the observers is then analysed using the Moreno–Lyapunov function method. The analytical proofs presented herein demonstrate that estimation errors can converge to a region of zero in finite time. Finally, the effectiveness of the developed finite-time observers and fault-estimation scheme was validated through simulations of a sample single-link flexible-joint robot.

Intermediate variable‐based adaptive fault estimation for a class of Lipschitz nonlinear systems

SummaryIn this paper, the problem of fault estimation based on intermediate variable‐based adaptive observer for a class of nonlinear systems with Lipschitzian nonlinearities and faults is studied. An adaptive observer based on lower triangular matrix transformation is proposed, which comprise the output derivatives information so as to enhance the accuracy and speed of fault estimation effectually. The designed observer can be simplified to the lower triangular estimation observer, and is superior to the comprehensive estimation observer in the related literature. In addition, a more general error dynamics is derived based on the improved adaptive estimation observer, which makes it less conservative to obtain stability conditions. The uniformly ultimately bounded stability conditions of the estimated error dynamics are given by linear matrix inequalities (LMIs). Numerical examples are given to illustrate the effectiveness and merits of the proposed method.

Adaptive delay dependent sliding mode fault-tolerant controller design for nonlinear systems with unknown time-varying input and state delays

This paper presents a novel adaptive delay-dependent fault-tolerant sliding mode control (SMC) strategy for the class of nonlinear Lipschitz systems with time-varying unknown delays in system states and inputs. A modified integral sliding surface (ISS) based on which the SMC approach is developed using a linear matrix inequality. Lyapunov-Krasovskii stability theory is employed to guarantee asymptotic stability of the closed-loop system such that system states starting from any arbitrary initial conditions with time-varying delays reach the predefined sliding surface in a finite time. It is also proved that states stay on the surface for all subsequent time, and the effects of the actuator’s faults are simultaneously attenuated. Adaptive tuning laws are used to adjust controller parameters and estimate actuators’ faults. The controllers’ structures are more straightforward than the most existing recent fault-tolerant control methods. Simulation results of practical nonlinear inverted pendulum system verified the outstanding merits and capabilities of the proposed scheme in the presence of actuators’ faults and multiple delays in the system states and inputs.

Robust sliding‐mode observer‐based multiple‐fault diagnosis scheme

AbstractIn this study, we simultaneously evaluate the multiple‐fault diagnosis problem of a class of Lipschitz nonlinear systems with actuator and sensor faults and unknown input disturbances. A nonsingular system transformation is used to transform the original system into two subsystems for multiple‐fault diagnosis: subsystems 1 and 2. At the system level, two robust sliding‐mode observers (RSMOs) are proposed. An RSMO is designed for subsystem 1 to detect actuator faults subjected to unknown input disturbances, and another RSMO is designed for subsystem 2 to detect sensor faults subjected to actuator faults. At the component level, a bank of RSMOs is proposed to detect and isolate actuators (sensors) with faults using a dedicated observer scheme. The reachability of RSMOs is comprehensively investigated in the estimation error space. Accordingly, the proposed observer parameters are designed as an optimization problem and solved using the linear matrix inequality (LMI) optimization technique. The effectiveness of the proposed multiple‐fault diagnosis scheme was validated through simulations of a modified seventh‐order aircraft system.

H−/L∞Fault Detection Observer Design for Uncertain Lipschitz Nonlinear Systems

This paper deals with the robust fault detection problem in the finite frequency domain for a class of Lipschitz nonlinear systems with uncertain parameters. The design conditions ofH−/L∞fault detection observers are proposed. These conditions make the error estimation system stable, and the residual hasL∞disturbance attenuation performance for external disturbances and uncertain parameters. In addition, considering that fault occurs in the finite frequency domain, the residual also hasH−fault sensitivity performance. To ease of solution, the observer design conditions are further transformed into linear matrix inequalities. On this basis, a dynamic threshold design method based onL∞analysis is presented. Finally, simulation results demonstrate the superiority of the proposed method.

Event-triggered dynamic output-feedback control for a class of Lipschitz nonlinear systems

Event-triggered dynamic output-feedback control for a class of Lipschitz nonlinear systems

Fast actuator and sensor fault estimation based on adaptive unknown input observer

This study evaluates the robust fault estimation problem of systems with actuator and sensor faults though the simultaneous use of unknown input disturbances and measurement noise. Specifically, an augmented descriptor system is preliminarily developed by creating an augmented state consisting of system states and sensor faults. Next, a novel fast adaptive unknown input observer (FAUIO) is proposed for the system to enhance its fault estimation performance. The existence condition of the novel FAUIO is then introduced for linear time-invariant systems with unknown input disturbances. Furthermore, the proposed FAUIO is extended to a class of Lipschitz nonlinear systems with unknown input disturbances and measurement noise to investigate the robust fault estimation problem. Accordingly, an H∞ performance index is employed to attenuate the influence of disturbances on fault estimation. Moreover, the linear matrix inequality (LMI) technique is applied to solve the designed FAUIO. Finally, the effectiveness of the developed FAUIO is validated via the simulation of two examples.

Robust Quantized Sampled–Data Stabilization for a Class of Lipschitz Nonlinear Systems With Time–Varying Uncertainties

In this letter, a robust quantized sampled-data controller is provided for a class of nonlinear systems affected by time-varying uncertainties, actuation disturbances and measurement noises. Sufficient conditions based on linear matrix inequalities and ensuring the existence of the proposed robust quantized sampled-data controller are given. Quantization of both state measurements and input signals is simultaneously considered. Input-to-state stability redesign technique is used in order to attenuate the effects of bounded actuation disturbances and of bounded observation errors. It is proved that, under suitably fast sampling and accurate quantization of the input/output channels, the proposed controller achieves the semi-global practical stability, with arbitrarily small final target ball, of the related quantized sampled-data closed-loop system provided that the observation errors do not affect (or affect marginally) the robustification term added in the controller and, that the bounds of the actuation disturbances as well as of the observation errors are a priori known. The theory here developed includes also the cases of time-varying sampling intervals and of non-uniform quantization of the input/output channels as well as the stability analysis of the inter-sampling system behavior. The provided results are validated through an example of one-link manipulator.

Observer for a class of Lipschitz nonlinear systems with multiple time‐varying delays in the nonlinear measured outputs

AbstractThis paper provides a novel chain observer for a class of Lipschitz nonlinear systems where the nonlinear measured outputs are corrupted by multiple time‐varying long delays. The main contribution is to synthesize a chain observer for a wider class of Lipschitz nonlinear systems with multiple outputs (described by Lipschitz nonlinear functions) with long time‐varying delays. In the design of the proposed observation scheme, the provided observer is comprised of a chain of state observers, where each one has a similar structure. Each observer is tasked to estimate the state over a short time horizon, whereas the first item of the chain provides the actual state estimation. Moreover, each item of the chain observer furnishes a proportional‐integral dynamical term which allows to compensate for the time‐delay effect. The observer gains are obtained from the solution of a set of delay‐dependent Linear Matrix Inequalities (LMIs), which is based on less restrictive LMI synthesis conditions. The convergence analysis relies on a Lyapunov–Krasovskii functional, which demonstrates that the observation error decays to zero. Numerical examples are given, in order to highlight the effectiveness of the provided observer.

Adaptive observer design for a class of Lipschitz nonlinear systems with multirate outputs and uncertainties: Application to attitude estimation with gyro bias

AbstractThe design of an adaptive observer for a class of multi‐input multi‐output Lipschitz nonlinear systems with uncertainties and multirated outputs is proposed in this paper. The sampling instants of the different outputs can be asynchronous and irregular. The exponential convergence of the proposed observer is proved with the help of a Lyapunov approach. The stability is ensured, provided that some conditions, including a persistent excitation condition and a maximum value condition on the sample rate of each output, are verified. The performances of the observer are illustrated through simulations of the attitude estimation problem using sensors from an inertial measurement unit and considering gyro bias.

A new actuator fault‐tolerant control for Lipschitz nonlinear system using adaptive sliding mode control strategy

AbstractThis article proposed a new adaptive integral sliding mode (ISM) based fault‐tolerant control (FTC) strategy to solve the actuator's faults and failures compensation problem for the class of Lipschitz nonlinear systems. A nominal state feedback virtual control law is designed first to stabilize the Lipschitz nonlinear system and to attain the desired nominal performance. To cater for the effect of faults and failures, the control allocation (CA) scheme reorganizes the virtual input signals among the healthy redundant actuators based on actuator's effectiveness level. Then, a nonlinear adaptive integral sliding mode controller (ISMC) is incorporated with the CA scheme to compensate for unknown disturbance and uncertainties effect that arises in the system by virtue of actuator faults/failures and error in fault estimation . An effective synthesis procedure is adopted to ensure the closed‐loop stability condition using linear matrix inequality (LMI) optimization. Finally, the FTC scheme is tested to achieve maneuvering control of coaxial octorotor unmanned aerial vehicle (UAV) system. Simulations are performed on the nonlinear system at different actuator faults/failures combinations. Moreover, the effect of wind‐gust, parameter variations, estimated state feedback and sensor noise are considered in nonlinear simulations. Finally, the results obtained during faults/failures are compared with the nominal condition and fixed ISMC based CA scheme.

State and fault estimation of a class of nonlinear systems with slow internal dynamics: Asymptotic Decoupling approach

Abstract This paper presents an Unknown Input Observer (UIO) for a class of Lipschitz nonlinear systems affected by disturbances/faults and having slow internal dynamics after the decoupling of the Unknown Input (UI). Thanks to the Asymptotic Decoupling notion, the constraint related to the internal dynamics is bypassed. The nonlinear UIO presents time varying dynamics that allows to obtain the Asymptotic Decoupling notion. It will be shown that the proposed approach allows to enhance the convergence rate of the UIO. The convergence enhancement of the state estimation error is demonstrated via a Lyapunov analysis and the established conditions are expressed in terms of Linear Matrix Inequalities. Finally, simulations will be given in order to illustrate the proposed approach with some comparisons to existing techniques.

An Enhanced Observer for Nonlinear Systems With Time-Varying Measurement Delays

In this article, we propose an observer for a class of Lipschitz nonlinear systems affected by time-varying and known measurement delays, which is an improvement of the one presented by Cacace <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">et al</i> ., 2014. Under the assumption that the delay function is piecewise continuous and differentiable, we prove that exponential convergence to zero of the observation error can be achieved with any desired decay rate, by suitably tuning a gain vector. The delay bound achieved with the observer proposed here is less conservative than the one obtained by Cacace <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">et al.</i> , 2014, as confirmed by numerical tests. For the sake of brevity, in this article, only one-step observers are considered. However, a cascade observer can be arranged to cope with arbitrarily long delays.

An integral sliding mode fault tolerant control for a class of non‐linear Lipschitz systems

This paper proposes an active fault-tolerant control (FTC) strategy for a class of non-linear Lipschitz systems. The proposed FTC approach employs the integral sliding mode control (ISMC) technique due to its inherent capability of dealing with system uncertainties. First, under the nominal fault-free condition, the linear matrix inequality technique is introduced to design the primary controller for the non-linear Lipschitz system. To accommodate the actuator faults/failures, the ISMC law is combined with a control allocation scheme that distributes the control signals to the redundant actuators. A non-linear octorotor system is then used as a test bench to validate the tolerance performance of the proposed FTC strategy. In particular, the proposed FTC strategy is applied for the inner-loop control, while in the outer loop, a fractional-order control approach is used to achieve the precise longitude and latitude control. Finally, various simulations are performed to justify the effectiveness of the proposed control scheme.

Robust Adaptive Stabilization by Delay Under State Parametric Uncertainty and Measurement Bias

An output robust adaptive control is designed for a class of Lipschitz nonlinear systems under assumption that the measurements are available with a constant bias and the state equations linearly parameterized by unknown parameters and external disturbances. A dynamic state reconstruction (synthesis of an observer) is avoided by using delayed values of the output in the feedback and adaptation laws. The analysis of robust stability for the resulted time-delay system is performed by using the Lyapunov–Krasovskii approach. The control and adaptation gains can be selected as a solution of the proposed linear matrix inequalities. This research is motivated by a nonlinear pendulum control problem, and the efficacy of the developed control is demonstrated on this application through experiments.

Fault Estimation for Discrete-Time Systems With Lipschitz Perturbation and Time-Variant Coefficients

This brief investigates the H <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">∞</sub> fault estimation problem for a class of Lipschitz nonlinear systems with timevariant coefficient matrices in discrete-time settings. By introducing an auxiliary unknown input based on the nonlinear term, a quasi-linear model and its corresponding indefinite quadratic performance function for fault estimation are respectively given in lieu of the original nonlinear dynamics and the H <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">∞</sub> performance metric, such that the estimation problem is converted as an indefinite optimization problem. By artificially constructing a Krein-space based dynamic model, the classical linear estimation technique in H <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sub> sense is employed to seek a suitable choice of the estimation of the fault. A condition that ensures the existence of the estimator is derived analytically. A Kalman-filter-like estimator recursion is proposed simultaneously.