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Practical prescribed-time fuzzy tracking control for uncertain nonlinear systems with time-varying actuators faults
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作者 Shuxing Xuan Hongjing Liang Tingwen Huang 《Journal of Automation and Intelligence》 2024年第1期40-49,共10页
The paper investigates the practical prescribed-time fuzzy tracking control problem for a category of nonlinear system subject to time-varying actuator faults.The presence of unknown nonlinear dynamics and actuator fa... The paper investigates the practical prescribed-time fuzzy tracking control problem for a category of nonlinear system subject to time-varying actuator faults.The presence of unknown nonlinear dynamics and actuator faults makes achieving tracking control within a prescribed-time challenging.To tackle this issue,we propose a novel practical prescribed-time fuzzy tracking control strategy,which is independent of the initial state of the system and does not rely on precise modeling of the system and actuators.We apply the approximation capabilities of fuzzy logic systems to handle the unknown nonlinear functions and unidentified actuator faults in the system.The piecewise controller and adaptive law constructed based on piecewise prescribed time-varying function and backstepping technique method establish the theoretical framework of practical prescribed-time tracking control,and extend the range of prescribed-time tracking control to infinity.Regardless of the initial conditions,the proposed control strategy can guarantee that all signals remain uniformly bounded within the practical prescribed time in the presence of unknown nonlinear item and time-varying actuator faults.Simulation example is presented to demonstrate the effectiveness of the proposed control strategy. 展开更多
关键词 Prescribed-time tracking control Adaptive fuzzy control Actuator faults Uncertain nonlinear system
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Path-Following Control With Obstacle Avoidance of Autonomous Surface Vehicles Subject to Actuator Faults 被引量:1
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作者 Li-Ying Hao Gege Dong +1 位作者 Tieshan Li Zhouhua Peng 《IEEE/CAA Journal of Automatica Sinica》 SCIE EI CSCD 2024年第4期956-964,共9页
This paper investigates the path-following control problem with obstacle avoidance of autonomous surface vehicles in the presence of actuator faults,uncertainty and external disturbances.Autonomous surface vehicles in... This paper investigates the path-following control problem with obstacle avoidance of autonomous surface vehicles in the presence of actuator faults,uncertainty and external disturbances.Autonomous surface vehicles inevitably suffer from actuator faults in complex sea environments,which may cause existing obstacle avoidance strategies to fail.To reduce the influence of actuator faults,an improved artificial potential function is constructed by introducing the lower bound of actuator efficiency factors.The nonlinear state observer,which only depends on measurable position information of the autonomous surface vehicle,is used to address uncertainties and external disturbances.By using a backstepping technique and adaptive mechanism,a path-following control strategy with obstacle avoidance and fault tolerance is designed which can ensure that the tracking errors converge to a small neighborhood of zero.Compared with existing results,the proposed control strategy has the capability of obstacle avoidance and fault tolerance simultaneously.Finally,the comparison results through simulations are given to verify the effectiveness of the proposed method. 展开更多
关键词 Actuator faults autonomous surface vehicle(ASVs) improved artificial potential function nonlinear state observer obstacle avoidance
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Adaptive Fixed-Time Control of Nonlinear MASs With Actuator Faults 被引量:3
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作者 Hongru Ren Hui Ma +1 位作者 Hongyi Li Zhenyou Wang 《IEEE/CAA Journal of Automatica Sinica》 SCIE EI CSCD 2023年第5期1252-1262,共11页
The adaptive fixed-time consensus problem for a class of nonlinear multi-agent systems(MASs)with actuator faults is considered in this paper.To approximate the unknown nonlinear functions in MASs,radial basis function... The adaptive fixed-time consensus problem for a class of nonlinear multi-agent systems(MASs)with actuator faults is considered in this paper.To approximate the unknown nonlinear functions in MASs,radial basis function neural networks are used.In addition,the first order sliding mode differentiator is utilized to solve the“explosion of complexity”problem,and a filter error compensation method is proposed to ensure the convergence of filter error in fixed time.With the help of the Nussbaum function,the actuator failure compensation mechanism is constructed.By designing the adaptive fixed-time controller,all signals in MASs are bounded,and the consensus errors between the leader and all followers converge to a small area of origin.Finally,the effectiveness of the proposed control method is verified by simulation examples. 展开更多
关键词 Actuator faults adaptive fixed-time control multiagent systems(MASs) Nussbaum function
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Distributed fault diagnosis observer for multi-agent system against actuator and sensor faults 被引量:1
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作者 YE Zhengyu JIANG Bin +2 位作者 CHENG Yuehua YU Ziquan YANG Yang 《Journal of Systems Engineering and Electronics》 SCIE EI CSCD 2023年第3期766-774,共9页
Component failures can cause multi-agent system(MAS)performance degradation and even disasters,which provokes the demand of the fault diagnosis method.A distributed sliding mode observer-based fault diagnosis method f... Component failures can cause multi-agent system(MAS)performance degradation and even disasters,which provokes the demand of the fault diagnosis method.A distributed sliding mode observer-based fault diagnosis method for MAS is developed in presence of actuator and sensor faults.Firstly,the actuator and sensor faults are extended to the system state,and the system is transformed into a descriptor system form.Then,a sliding mode-based distributed unknown input observer is proposed to estimate the extended state.Furthermore,adaptive laws are introduced to adjust the observer parameters.Finally,the effectiveness of the proposed method is demonstrated with numerical simulations. 展开更多
关键词 multi-agent system(MAS) sensor fault actuator fault unknown input observer sliding mode fault diagnosis
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Distributed Fault Estimation for Nonlinear Systems With Sensor Saturation and Deception Attacks Using Stochastic Communication Protocols
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作者 Weiwei Sun Xinci Gao +1 位作者 Lusong Ding Xiangyu Chen 《IEEE/CAA Journal of Automatica Sinica》 SCIE EI CSCD 2024年第8期1865-1876,共12页
This paper is aimed at the distributed fault estimation issue associated with the potential loss of actuator efficiency for a type of discrete-time nonlinear systems with sensor saturation.For the distributed estimati... This paper is aimed at the distributed fault estimation issue associated with the potential loss of actuator efficiency for a type of discrete-time nonlinear systems with sensor saturation.For the distributed estimation structure under consideration,an estimation center is not necessary,and the estimator derives its information from itself and neighboring nodes,which fuses the state vector and the measurement vector.In an effort to cut down data conflicts in communication networks,the stochastic communication protocol(SCP)is employed so that the output signals from sensors can be selected.Additionally,a recursive security estimator scheme is created since attackers randomly inject malicious signals into the selected data.On this basis,sufficient conditions for a fault estimator with less conservatism are presented which ensure an upper bound of the estimation error covariance and the mean-square exponential boundedness of the estimating error.Finally,a numerical example is used to show the reliability and effectiveness of the considered distributed estimation algorithm. 展开更多
关键词 Actuator fault deception attacks distributed estimation sensor saturation stochastic communication protocol(SCP).
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A finite-time fuzzy adaptive output-feedback fault-tolerant control for underactuated wheeled mobile robots systems
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作者 Pingfan Liu Shaocheng Tong 《Journal of Automation and Intelligence》 2024年第2期111-118,共8页
This paper investigates the adaptive fuzzy finite-time output-feedback fault-tolerant control (FTC) problemfor a class of nonlinear underactuated wheeled mobile robots (UWMRs) system with intermittent actuatorfaults. ... This paper investigates the adaptive fuzzy finite-time output-feedback fault-tolerant control (FTC) problemfor a class of nonlinear underactuated wheeled mobile robots (UWMRs) system with intermittent actuatorfaults. The UWMR system includes unknown nonlinear dynamics and immeasurable states. Fuzzy logic systems(FLSs) are utilized to work out immeasurable functions. Furthermore, with the support of the backsteppingcontrol technique and adaptive fuzzy state observer, a fuzzy adaptive finite-time output-feedback FTC scheme isdeveloped under the intermittent actuator faults. It is testifying the scheme can ensure the controlled nonlinearUWMRs is stable and the estimation errors are convergent. Finally, the comparison results and simulationvalidate the effectiveness of the proposed fuzzy adaptive finite-time FTC approach. 展开更多
关键词 Underactuated wheeled mobile robots system FINITE-TIME Fuzzy adaptive fault-tolerant control OUTPUT-FEEDBACK Intermittent actuator faults
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Actuator and sensor fault isolation in a class of nonlinear dynamical systems
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作者 Hamed Tirandaz Christodoulos Keliris Marios M.Polycarpou 《Journal of Automation and Intelligence》 2024年第2期57-72,共16页
Fault isolation in dynamical systems is a challenging task due to modeling uncertainty and measurement noise,interactive effects of multiple faults and fault propagation.This paper proposes a unified approach for isol... Fault isolation in dynamical systems is a challenging task due to modeling uncertainty and measurement noise,interactive effects of multiple faults and fault propagation.This paper proposes a unified approach for isolation of multiple actuator or sensor faults in a class of nonlinear uncertain dynamical systems.Actuator and sensor fault isolation are accomplished in two independent modules,that monitor the system and are able to isolate the potential faulty actuator(s)or sensor(s).For the sensor fault isolation(SFI)case,a module is designed which monitors the system and utilizes an adaptive isolation threshold on the output residuals computed via a nonlinear estimation scheme that allows the isolation of single/multiple faulty sensor(s).For the actuator fault isolation(AFI)case,a second module is designed,which utilizes a learning-based scheme for adaptive approximation of faulty actuator(s)and,based on a reasoning decision logic and suitably designed AFI thresholds,the faulty actuator(s)set can be determined.The effectiveness of the proposed fault isolation approach developed in this paper is demonstrated through a simulation example. 展开更多
关键词 Actuator and sensor fault isolation Adaptive approximation Observer-based fault diagnosis Reasoning-based decision logic
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A Novel Robust Attitude Control for Quadrotor Aircraft Subject to Actuator Faults and Wind Gusts 被引量:21
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作者 Yuying Guo Bin Jiang Youmin Zhang 《IEEE/CAA Journal of Automatica Sinica》 SCIE EI CSCD 2018年第1期292-300,共9页
A novel robust fault tolerant controller is developed for the problem of attitude control of a quadrotor aircraft in the presence of actuator faults and wind gusts in this paper.Firstly, a dynamical system of the quad... A novel robust fault tolerant controller is developed for the problem of attitude control of a quadrotor aircraft in the presence of actuator faults and wind gusts in this paper.Firstly, a dynamical system of the quadrotor taking into account aerodynamical effects induced by lateral wind and actuator faults is considered using the Newton-Euler approach. Then,based on active disturbance rejection control(ADRC), the fault tolerant controller is proposed to recover faulty system and reject perturbations. The developed controller takes wind gusts,actuator faults and measurement noises as total perturbations which are estimated by improved extended state observer(ESO)and compensated by nonlinear feedback control law. So, the developed robust fault tolerant controller can successfully accomplish the tracking of the desired output values. Finally, some simulation studies are given to illustrate the effectiveness of fault recovery of the proposed scheme and also its ability to attenuate external disturbances that are introduced from environmental causes such as wind gusts and measurement noises. 展开更多
关键词 Active disturbance rejection control(ADRC) attitude control actuator faults disturbances rejection quadrotor aircraft
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Backstepping adaptive fuzzy control of uncertain nonlinear systems against actuator faults 被引量:13
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作者 Ping LI Guanghong YANG 《控制理论与应用(英文版)》 EI 2009年第3期248-256,共9页
A class of unknown nonlinear systems subject to uncertain actuator faults and external disturbances will be studied in this paper with the help of fuzzy approximation theory. Using backstepping technique, a novel adap... A class of unknown nonlinear systems subject to uncertain actuator faults and external disturbances will be studied in this paper with the help of fuzzy approximation theory. Using backstepping technique, a novel adaptive fuzzy control approach is proposed to accommodate the uncertain actuator faults during operation and deal with the external disturbances though the systems cannot be linearized by feedback. The considered faults are modeled as both loss of effectiveness and lock-in-place (stuck at some unknown place). It is proved that the proposed control scheme can guarantee all signals of the closed-loop system to be semi-globally uniformly ultimately bounded and the tracking error between the system output and the reference signal converge to a small neighborhood of zero, though the nonlinear functions of the controlled system as well as the actuator faults and the external disturbances are all unknown. Simulation results demonstrate the effectiveness of the control approach. 展开更多
关键词 Adaptive control Fuzzy system BACKSTEPPING Uncertain nonlinear system Actuator fault
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Human-in-the-Loop Consensus Control for Nonlinear Multi-Agent Systems With Actuator Faults 被引量:6
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作者 Guohuai Lin Hongyi Li +2 位作者 Hui Ma Deyin Yao Renquan Lu 《IEEE/CAA Journal of Automatica Sinica》 SCIE EI CSCD 2022年第1期111-122,共12页
This paper considers the human-in-the-loop leader-following consensus control problem of multi-agent systems(MASs)with unknown matched nonlinear functions and actuator faults.It is assumed that a human operator contro... This paper considers the human-in-the-loop leader-following consensus control problem of multi-agent systems(MASs)with unknown matched nonlinear functions and actuator faults.It is assumed that a human operator controls the MASs via sending the command signal to a non-autonomous leader which generates the desired trajectory.Moreover,the leader’s input is nonzero and not available to all followers.By using neural networks and fault estimators to approximate unknown nonlinear dynamics and identify the actuator faults,respectively,the neighborhood observer-based neural fault-tolerant controller with dynamic coupling gains is designed.It is proved that the state of each follower can synchronize with the leader’s state under a directed graph and all signals in the closed-loop system are guaranteed to be cooperatively uniformly ultimately bounded.Finally,simulation results are presented for verifying the effectiveness of the proposed control method. 展开更多
关键词 Actuator faults distributed control human-in-the-loop neighborhood observer nonlinear multi-agent systems(MASs)
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Adaptive fault-tolerant control based on boundary estimation for space robot under joint actuator faults and uncertain parameters 被引量:4
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作者 Rong-Hua Lei Li Chen 《Defence Technology(防务技术)》 SCIE EI CAS CSCD 2019年第6期964-971,共8页
Since the joint actuator of the space robot executes the control instructions frequently in the harsh space environment,it is prone to the partial loss of control effectiveness(PLCE)fault.An adaptive fault-tolerant co... Since the joint actuator of the space robot executes the control instructions frequently in the harsh space environment,it is prone to the partial loss of control effectiveness(PLCE)fault.An adaptive fault-tolerant control algorithm is designed for a space robot system with the uncertain parameters and the PLCE actuator faults.The mathematical model of the system is established based on the Lagrange method,and the PLCE actuator fault is described as an effectiveness factor.The lower bound of the effectiveness factors and the upper bound of the uncertain parameters are estimated by an adaptive strategy,and the estimated value is fed back to the control algorithm.Compared with the traditional fault-tolerant algorithms,the proposed algorithm does not need to predetermine the lower bound of the effectiveness factor,hence it is more in line with the actual engineering application.It is proved that the algorithm can guarantee the stability of the closed-loop system based on the Lyapunov function method.The numerical simulation results show that the proposed algorithm can not only compensate for the uncertain parameters,but also can tolerate the PLCE actuator faults effectively,which verifies the effectiveness and superiority of the control scheme. 展开更多
关键词 Space robot Actuator faults Uncertain parameters Effectiveness factor Fault-tolerant control
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Observer-based adaptive control of uncertain time-delay switched systems with stuck actuator faults
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作者 Limin WANG Cheng SHAO 《控制理论与应用(英文版)》 EI 2009年第2期219-223,共5页
This paper concerns the observer-based adaptive control problem of uncertain time-delay switched systems with stuck actuator faults. Under the case where the original controller cannot stabilize the faulty system, mul... This paper concerns the observer-based adaptive control problem of uncertain time-delay switched systems with stuck actuator faults. Under the case where the original controller cannot stabilize the faulty system, multiple adaptive controllers are designed and a suitable switching logic is incorporated to ensure the closed-loop system stability and state tracking. New delay-independent sufficient conditions for asymptotic stability are obtained in terms of linear matrix inequalities based on piecewise Lyapunov stability theory. On the other hand, adaptive laws for on-line updating of some of the controller parameters are also designed to compensate the effect of stuck failures. Finally, simulation results for reference [1] model show that the design is feasible and efficient. 展开更多
关键词 Observer-based adaptive control Tuck actuator faults DELAY-INDEPENDENT Uncertain time-delay switched systems
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Minimum eigenvalue based adaptive fault compensation for hypersonic vehicles
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作者 MA Yajie JIANG Bin REN Hao 《Journal of Systems Engineering and Electronics》 SCIE EI CSCD 2023年第2期492-500,共9页
The attitude tracking control problem is addressed for hypersonic vehicles under actuator faults that may cause an uncertain time-varying control gain matrix.An adaptive compensation scheme is developed to ensure syst... The attitude tracking control problem is addressed for hypersonic vehicles under actuator faults that may cause an uncertain time-varying control gain matrix.An adaptive compensation scheme is developed to ensure system stability and asymptotic tracking properties,including a kinematic control signal and a dynamic control signal.To deal with the uncertainties of the control gain matrix,a new positive definite one is constructed.The minimum eigenvalue of such a new control gain matrix is estimated.Simulation results of application to an X-33 vehicle model verify the effectiveness of the proposed minimum eigenvalue based adaptive fault compensation scheme. 展开更多
关键词 actuator fault adaptive compensation hypersonic vehicle minimum eigenvalue
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Output feedback based adaptive robust fault-tolerant control for a class of uncertain nonlinear systems 被引量:6
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作者 Shreekant Gayaka Bin Yao 《Journal of Systems Engineering and Electronics》 SCIE EI CSCD 2011年第1期38-51,共14页
An adaptive robust approach for actuator fault-tolerant control of a class of uncertain nonlinear systems is proposed.The two chief ways in which the system performance can degrade following an actuator-fault are unde... An adaptive robust approach for actuator fault-tolerant control of a class of uncertain nonlinear systems is proposed.The two chief ways in which the system performance can degrade following an actuator-fault are undesirable transients and unacceptably large steady-state tracking errors.Adaptive control based schemes can achieve good final tracking accuracy in spite of change in system parameters following an actuator fault,and robust control based designs can achieve guaranteed transient response.However,neither adaptive control nor robust control based fault-tolerant designs can address both the issues associated with actuator faults.In the present work,an adaptive robust fault-tolerant control scheme is claimed to solve both the problems,as it seamlessly integrates adaptive and robust control design techniques.Comparative simulation studies are performed using a nonlinear hypersonic aircraft model to show the effectiveness of the proposed scheme over a robust adaptive control based faulttolerant scheme. 展开更多
关键词 fault-tolerant system actuator fault adaptive control robust control.
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LMI approach to reliable H_∞ control of linear systems 被引量:6
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作者 Yao Bo Wang Fuzhong 《Journal of Systems Engineering and Electronics》 SCIE EI CSCD 2006年第2期381-386,共6页
The reliable design problem for linear systems is concerned with. A more practical model of actuator faults than outage is considered. An LMI approach of designing reliable controller is presented for the case of actu... The reliable design problem for linear systems is concerned with. A more practical model of actuator faults than outage is considered. An LMI approach of designing reliable controller is presented for the case of actuator faults that can be modeled by a scaling factor. The resulting control systems are reliable in that they provide guaranteed asymptotic stability and H∞ performance when some control component (actuator) faults occur. A numerical example is also given to illustrate the design procedure and their effectiveness. Furthermore, the optimal standard controller and the optimal reliable controller are compared to show the necessity of reliable control. 展开更多
关键词 reliable control sensor fault actuator fault LMI.
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PID-type fault-tolerant prescribed performance control of fixed-wing UAV 被引量:6
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作者 YU Ziquan ZHANG Youmin JIANG Bin 《Journal of Systems Engineering and Electronics》 SCIE EI CSCD 2021年第5期1053-1061,共9页
This paper introduces a fault-tolerant control(FTC)design for a faulty fixed-wing unmanned aerial vehicle(UAV).To constrain tracking errors against actuator faults,error constraint inequalities are first transformed t... This paper introduces a fault-tolerant control(FTC)design for a faulty fixed-wing unmanned aerial vehicle(UAV).To constrain tracking errors against actuator faults,error constraint inequalities are first transformed to a new set of variables based on prescribed performance functions.Then,the commonly used and powerful proportional-integral-derivative(PID)control concept is employed to filter the transformed error variables.To handle the fault-induced nonlinear terms,a composite learning algorithm consisting of neural network and disturbance observer is incorporated for increasing flight safety.It is shown by Lyapunov stability analysis that the tracking errors are strictly constrained within the specified error bounds.Experimental results are presented to verify the feasibility of the developed FTC scheme. 展开更多
关键词 unmanned aerial vehicle(UAV) fault-tolerant control(FTC) prescribed performance control(PPC) proportional-integral-derivative(PID) composite learning actuator faults
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Fault Tolerant Control for Networked Control Systems with Packet Loss and Time Delay 被引量:5
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作者 Ming-Yue Zhao He-Ping Liu +1 位作者 Zhi-Jun Li De-Hui Sun 《International Journal of Automation and computing》 EI 2011年第2期244-253,共10页
In this paper,a fault tolerant control with the consideration of actuator fault for a networked control system (NCS) with packet loss is addressed.The NCS with data packet loss can be described as a switched system ... In this paper,a fault tolerant control with the consideration of actuator fault for a networked control system (NCS) with packet loss is addressed.The NCS with data packet loss can be described as a switched system model.Packet loss dependent Lyapunov function is used and a fault tolerant controller is proposed respectively for arbitrary packet loss process and Markovian packet loss process.Considering a controlled plant with external energy-bounded disturbance,a robust H ∞ fault tolerant controller is designed for the NCS.These results are also expanded to the NCS with packet loss and networked-induced delay.Numerical examples are given to illustrate the effectiveness of the proposed design method. 展开更多
关键词 Fault tolerant control networked control system (NCS) packet loss actuator fault time delay.
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Multiobjective fault detection and isolation for flexible air-breathing hypersonic vehicle 被引量:4
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作者 Xuejing Cai Fen Wu 《Journal of Systems Engineering and Electronics》 SCIE EI CSCD 2011年第1期52-62,共11页
An application of the multiobjective fault detection and isolation(FDI) approach to an air-breathing hypersonic vehicle(HSV) longitudinal dynamics subject to disturbances is presented.Maintaining sustainable and s... An application of the multiobjective fault detection and isolation(FDI) approach to an air-breathing hypersonic vehicle(HSV) longitudinal dynamics subject to disturbances is presented.Maintaining sustainable and safe flight of HSV is a challenging task due to its strong coupling effects,variable operating conditions and possible failures of system components.A common type of system faults for aircraft including HSV is the loss of effectiveness of its actuators and sensors.To detect and isolate multiple actuator/sensor failures,a faulty linear parameter-varying(LPV) model of HSV is derived by converting actuator/system component faults into equivalent sensor faults.Then a bank of LPV FDI observers is designed to track individual fault with minimum error and suppress the effects of disturbances and other fault signals.The simulation results based on the nonlinear flexible HSV model and a nominal LPV controller demonstrate the effectiveness of the fault estimation technique for HSV. 展开更多
关键词 fault detection and isolation(FDI) hypersonic vehicle(HSV) actuator and sensor faults multiobjective optimization.
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Multimodel-based flight control system reconfiguration control in the presence of input constraints 被引量:2
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作者 Yuying GUO Bin JIANG Yufei XU 《控制理论与应用(英文版)》 EI 2010年第4期418-424,共7页
In this paper,an active fault accommodate strategy is proposed for the plant in the presence of actuator fault and input constraints,which is a combination of a direct adaptive control algorithm with multiple model sw... In this paper,an active fault accommodate strategy is proposed for the plant in the presence of actuator fault and input constraints,which is a combination of a direct adaptive control algorithm with multiple model switching.The μ-modification is introduced in the model reference architecture to construct the adaptive controller.The proof of stability is based on the candidate Lyapunov function,while appropriate switching of multiple models guarantees asymptotic tracking of the system states and the boundedness of all signals.Simulation results illustrate the efficiency of the proposed method. 展开更多
关键词 Actuator fault Adaptive control reconfiguration Multiple model Input constraint RBF neural network
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Reliable Memory Feedback Design for a Class of Nonlinear Fuzzy Systems with Time-varying Delay 被引量:1
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作者 You-Qing Wang Dong-Hua Zhou Li-Heng Liu 《International Journal of Automation and computing》 EI 2007年第2期169-176,共8页
This paper is concerned with the robust reliable memory controller design for a class of fuzzy uncertain systems with timevarying delay. The system under consideration is more general than those in other existent work... This paper is concerned with the robust reliable memory controller design for a class of fuzzy uncertain systems with timevarying delay. The system under consideration is more general than those in other existent works. The controller, which is dependent on the magnitudes and derivative of the delay, is proposed in terms of linear matrix inequality (LMI). The closed-loop system is asymptotically stable for all admissible uncertainties as well as actuator faults. A numerical example is presented for illustration. 展开更多
关键词 Reliable memory control fuzzy uncertain system actuator fault linear matrix inequalities (LMIs).
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