This article describes a method of vehicle dynamics estimation for impending rollover detection. We estimate vehicle dynamic states in presence of the road bank angle as a disturbance in the vehicle model using a robu...This article describes a method of vehicle dynamics estimation for impending rollover detection. We estimate vehicle dynamic states in presence of the road bank angle as a disturbance in the vehicle model using a robust observer. The estimated roll angle and roll rate are used to compute the rollover index which is based on the prediction of the lateral load transfer. In order to anticipate rollover detection, a new method is proposed to compute the time to rollover(TTR) using the load transfer ratio(LTR). The nonlinear model, deduced from the vehicle lateral and roll dynamics, is represented by a Takagi-Sugeno(T-S) fuzzy model. This representation is used to account for the nonlinearities of lateral cornering forces. The proposed T-S observer is designed with unmeasurable premise variables to cater for non-availability of the slip angles measurement. The proposed approach is evaluated using Car Sim simulator under different driving scenarios. Simulation results show good efficiency of the proposed T-S observer and the rollover detection method.展开更多
An extended robust model predictive control approach for input constrained discrete uncertain nonlinear systems with time-delay based on a class of uncertain T-S fuzzy models that satisfy sector bound condition is pre...An extended robust model predictive control approach for input constrained discrete uncertain nonlinear systems with time-delay based on a class of uncertain T-S fuzzy models that satisfy sector bound condition is presented. In this approach, the minimization problem of the “worst-case” objective function is converted into the linear objective minimization problem in- volving linear matrix inequalities (LMIs) constraints. The state feedback control law is obtained by solving convex optimization of a set of LMIs. Sufficient condition for stability and a new upper bound on robust performance index are given for these kinds of uncertain fuzzy systems with state time-delay. Simulation results of CSTR process show that the proposed robust predictive control approach is effective and feasible.展开更多
Pressure ripples in electric power steering （EPS） systems can be caused by the phase lag between the driver s steering torque and steer angle, the nonlinear frictions, and the disturbances from road and sensor noise...Pressure ripples in electric power steering （EPS） systems can be caused by the phase lag between the driver s steering torque and steer angle, the nonlinear frictions, and the disturbances from road and sensor noise especially during high-frequency maneuvers. This paper investigates the use of the robust fuzzy control method for actively reducing pressure ripples for EPS systems. Remarkable progress on steering maneuverability is achieved. The EPS dynamics is described with an eight-order nonlinear state-space model and approximated by a Takagi-Sugeno （T-S） fuzzy model with time-varying delays and external disturbances. A stabilization approach is then presented for nonlinear time-delay systems through fuzzy state feedback controller in parallel distributed compensation （PDC） structure. The closed-loop stability conditions of EPS system with the fuzzy controller are parameterized in terms of the linear matrix inequality （LMI） problem. Simulations and experiments using the proposed robust fuzzy controller and traditional PID controller have been carried out for EPS systems. Both the simulation and experiment results show that the proposed fuzzy controller can reduce the torque ripples and allow us to have a good steering feeling and stable driving.展开更多
This paper investigates the problem of finite frequency fuzzy H_∞ control for uncertain active vehicle suspension systems, in which sensor failure is taken into account. TakagiSugeno(T-S) fuzzy model is established f...This paper investigates the problem of finite frequency fuzzy H_∞ control for uncertain active vehicle suspension systems, in which sensor failure is taken into account. TakagiSugeno(T-S) fuzzy model is established for considered suspension systems. In order to describe the sensor fault effectively, a corresponding model is introduced. A vital performance index,H_∞ performance, is utilized to measure the drive comfort. In the framework of Kalman-Yakubovich-Popov theory, the H_∞ norm from external perturbation to controlled output is optimized effectively in the frequency domain of 4 Hz-8 Hz to enhance ride comfort level. Meanwhile, three suspension constrained requirements, i.e., ride comfort level, manipulation stability,suspension deflection are also guaranteed. Furthermore, sufficient conditions are developed to design a fuzzy controller to guarantee the desired performance of active suspension systems. Finally, the proposed control scheme is applied to a quarter-vehicle active suspension, and simulation results are given to illustrate the effectiveness of the proposed approach.展开更多
The aging prediction of railway catenary is of profound significance for ensuring the regular operation of electrified trains.However,in real-world scenarios,accurate predictions are challenging due to various interfe...The aging prediction of railway catenary is of profound significance for ensuring the regular operation of electrified trains.However,in real-world scenarios,accurate predictions are challenging due to various interferences.This paper addresses this challenge by proposing a novel method for predicting the aging of railway catenary based on an improved Kalman filter(KF).The proposed method focuses on modifying the priori state estimate covariance and measurement error covariance of the KF to enhance accuracy in complex environments.By comparing the optimal displacement value with the theoretically calculated value based on the thermal expansion effect of metals,it becomes possible to ascertain the aging status of the catenary.To improve prediction accuracy,a railway catenary aging prediction model is constructed by integrating the Takagi-Sugeno(T-S)fuzzy neural network(FNN)and KF.In this model,an adaptive training method is introduced,allowing the FNN to use fewer fuzzy rules.The inputs of the model include time,temperature,and historical displacement,while the output is the predicted displacement.Furthermore,the KF is enhanced by modifying its prior state estimate covariance and measurement error covariance.These modifications contribute to more accurate predictions.Lastly,a low-power experimental platform based on FPGA is implemented to verify the effectiveness of the proposed method.The test results demonstrate that the proposed method outperforms the compared method,showcasing its superior performance.展开更多
This paper proposes a new method for control of continuous large-scale systems where the measures and control functions are distributed on calculating members which can be shared with other applications and connected ...This paper proposes a new method for control of continuous large-scale systems where the measures and control functions are distributed on calculating members which can be shared with other applications and connected to digital network communications.At first, the nonlinear large-scale system is described by a Takagi-Sugeno(TS) fuzzy model. After that, by using a fuzzy LyapunovKrasovskii functional, sufficient conditions of asymptotic stability of the behavior of the decentralized networked control system(DNCS),are developed in terms of linear matrix inequalities(LMIs). Finally, to illustrate the proposed approach, a numerical example and simulation results are presented.展开更多
By the best approximation theory, it is first proved that the SISO （single-input single-output） linear Takagi-Sugeno （TS） fuzzy systems can approximate an arbitrary polynomial which, according to Weierstrass appro...By the best approximation theory, it is first proved that the SISO （single-input single-output） linear Takagi-Sugeno （TS） fuzzy systems can approximate an arbitrary polynomial which, according to Weierstrass approximation theorem, can uniformly approximate any continuous functions on the compact domain. Then new sufficient conditions for general linear SISO TS fuzzy systems as universal approximators are obtained. Formulae are derived to calculate the number of input fuzzy sets to satisfy the given approximation accuracy. Then the presented result is compared with the existing literature＇s results. The comparison shows that the presented result needs less input fuzzy sets, which can simplify the design of the fuzzy system, and examples are given to show its effectiveness.展开更多
The projection of the chaotic attractor observed from the Lorenz system in the X-Z plane is like a butterfly, hence the classical Lorenz system is widely known as the butterfly attractor, and has served as a prototype...The projection of the chaotic attractor observed from the Lorenz system in the X-Z plane is like a butterfly, hence the classical Lorenz system is widely known as the butterfly attractor, and has served as a prototype model for studying chaotic behaviour since it was coined. In this work we take one step further to investigate some fundamental dynamic behaviours of a novel hybrid Takagi-Sugeno （TS） fuzzy Lorenz-type system, which is essentially derived from the delta-operator-based TS fuzzy modelling for complex nonlinear systems, and contains the original Lorenz system of continuous-time TS fuzzy form as a special case. By simply and appropriately tuning the additional parametric perturbations in the two-rule hybrid TS fuzzy Lorenz-type system, complex （two-wing） butterfly attractors observed from this system in the three dimensional （3D） X-Y-Z space are created, which have not yet been reported in the literature, and the forming mechanism of the compound structures have been numerically investigated.展开更多
In order to obtain accurate prediction model and compensate for the influence of model mismatch on the control performance of the system and avoid solving nonlinear programming problem,an adaptive fuzzy predictive fun...In order to obtain accurate prediction model and compensate for the influence of model mismatch on the control performance of the system and avoid solving nonlinear programming problem,an adaptive fuzzy predictive functional control(AFPFC) scheme for multivariable nonlinear systems was proposed.Firstly,multivariable nonlinear systems were described based on Takagi-Sugeno(T-S) fuzzy models;assuming that the antecedent parameters of T-S models were kept,the consequent parameters were identified on-line by using the weighted recursive least square(WRLS) method.Secondly,the identified T-S models were linearized to be time-varying state space model at each sampling instant.Finally,by using linear predictive control technique the analysis solution of the optimal control law of AFPFC was established.The application results for pH neutralization process show that the absolute error between the identified T-S model output and the process output is smaller than 0.015;the tracking ability of the proposed AFPFC is superior to that of non-AFPFC(NAFPFC) for pH process without disturbances,the overshoot of the effluent pH value of AFPFC with disturbances is decreased by 50% compared with that of NAFPFC;when the process parameters of AFPFC vary with time the integrated absolute error(IAE) performance index still retains to be less than 200 compared with that of NAFPFC.展开更多
This paper is concerned with the problem of stabilization of the Roesser type discrete-time nonlinear 2-D system that plays an important role in many practical applications. First, a discrete-time 2-D T-S fuzzy model ...This paper is concerned with the problem of stabilization of the Roesser type discrete-time nonlinear 2-D system that plays an important role in many practical applications. First, a discrete-time 2-D T-S fuzzy model is proposed to represent the underlying nonlinear 2-D system. Second, new quadratic stabilization conditions are proposed by applying relaxed quadratic stabilization technique for 2-D case. Third, for sake of further reducing conservatism, new non-quadratic stabilization conditions are also proposed by applying a new parameter-dependent Lyapunov function, matrix transformation technique, and relaxed technique for the underlying discrete-time 2-D T-S fuzzy system. Finally, a numerical example is provided to illustrate the effectiveness of the proposed results.展开更多
This paper focuses on the stability analysis of nonlinear networked control system with integral quadratic constraints(IQC) performance, dynamic quantization, variable sampling intervals, and communication delays. By ...This paper focuses on the stability analysis of nonlinear networked control system with integral quadratic constraints(IQC) performance, dynamic quantization, variable sampling intervals, and communication delays. By using input-delay and parallel distributed compensation(PDC) techniques, we establish the Takagi-Sugeno(T-S) fuzzy model for the system, in which the sampling period of the sampler and signal transmission delay are transformed to the refreshing interval of a zero-order holder(ZOH). By the appropriate Lyapunov-Krasovskii-based methods, a delay-dependent criterion is derived to ensure the asymptotic stability for the system with IQC performance via the H∞ state feedback control. The efficiency of the method is illustrated on a simulation exampler.展开更多
The detection and mitigation of cyber-attacks in connected vehicle systems(CVSs)are critical for ensuring the security of intelligent connected vehicles.This paper presents a solution to estimate sensor and actuator c...The detection and mitigation of cyber-attacks in connected vehicle systems(CVSs)are critical for ensuring the security of intelligent connected vehicles.This paper presents a solution to estimate sensor and actuator cyber-attacks in CVSs.A novel method is proposed that utilizes an augmented system representation technique and a nonlinear unknown input observer(UIO)to achieve asymptotic estimation of both CVS dynamics and cyber-attacks.The nonlinear CVS dynamics is represented in a Takagi–Sugeno(TS)fuzzy form with nonlinear consequents,which allows for the effective use of the differential mean value theorem to handle unmeasured premise variables.Furthermore,via Lyapunov stability theory sufficient conditions are proposed,expressed in terms of linear matrix inequalities,to design TS fuzzy UIO.Several test scenarios are performed with high-fidelity Simulink-CarSim co-simulations to show the effectiveness of the proposed cyber-attack estimation method.展开更多
In this article,the problem of state estimation is addressed for discrete-time nonlinear systems subject to additive unknown-but-bounded noises by using fuzzy set-membership filtering.First,an improved T-S fuzzy model...In this article,the problem of state estimation is addressed for discrete-time nonlinear systems subject to additive unknown-but-bounded noises by using fuzzy set-membership filtering.First,an improved T-S fuzzy model is introduced to achieve highly accurate approximation via an affine model under each fuzzy rule.Then,compared to traditional prediction-based ones,two types of fuzzy set-membership filters are proposed to effectively improve filtering performance,where the structure of both filters consists of two parts:prediction and filtering.Under the locally Lipschitz continuous condition of membership functions,unknown membership values in the estimation error system can be treated as multiplicative noises with respect to the estimation error.Real-time recursive algorithms are given to find the minimal ellipsoid containing the true state.Finally,the proposed optimization approaches are validated via numerical simulations of a one-dimensional and a three-dimensional discrete-time nonlinear systems.展开更多
基金supported by the"Conseil Régional de Picardie"and the European Regional Development Fund within the framework of the project"SEDVAC"
文摘This article describes a method of vehicle dynamics estimation for impending rollover detection. We estimate vehicle dynamic states in presence of the road bank angle as a disturbance in the vehicle model using a robust observer. The estimated roll angle and roll rate are used to compute the rollover index which is based on the prediction of the lateral load transfer. In order to anticipate rollover detection, a new method is proposed to compute the time to rollover(TTR) using the load transfer ratio(LTR). The nonlinear model, deduced from the vehicle lateral and roll dynamics, is represented by a Takagi-Sugeno(T-S) fuzzy model. This representation is used to account for the nonlinearities of lateral cornering forces. The proposed T-S observer is designed with unmeasurable premise variables to cater for non-availability of the slip angles measurement. The proposed approach is evaluated using Car Sim simulator under different driving scenarios. Simulation results show good efficiency of the proposed T-S observer and the rollover detection method.
基金Project (No. 60421002) supported by the National Natural ScienceFoundation of China
文摘An extended robust model predictive control approach for input constrained discrete uncertain nonlinear systems with time-delay based on a class of uncertain T-S fuzzy models that satisfy sector bound condition is presented. In this approach, the minimization problem of the “worst-case” objective function is converted into the linear objective minimization problem in- volving linear matrix inequalities (LMIs) constraints. The state feedback control law is obtained by solving convex optimization of a set of LMIs. Sufficient condition for stability and a new upper bound on robust performance index are given for these kinds of uncertain fuzzy systems with state time-delay. Simulation results of CSTR process show that the proposed robust predictive control approach is effective and feasible.
基金supported Foundation of National Development and Reform Commission of China (No. 2040)
文摘Pressure ripples in electric power steering （EPS） systems can be caused by the phase lag between the driver s steering torque and steer angle, the nonlinear frictions, and the disturbances from road and sensor noise especially during high-frequency maneuvers. This paper investigates the use of the robust fuzzy control method for actively reducing pressure ripples for EPS systems. Remarkable progress on steering maneuverability is achieved. The EPS dynamics is described with an eight-order nonlinear state-space model and approximated by a Takagi-Sugeno （T-S） fuzzy model with time-varying delays and external disturbances. A stabilization approach is then presented for nonlinear time-delay systems through fuzzy state feedback controller in parallel distributed compensation （PDC） structure. The closed-loop stability conditions of EPS system with the fuzzy controller are parameterized in terms of the linear matrix inequality （LMI） problem. Simulations and experiments using the proposed robust fuzzy controller and traditional PID controller have been carried out for EPS systems. Both the simulation and experiment results show that the proposed fuzzy controller can reduce the torque ripples and allow us to have a good steering feeling and stable driving.
基金partially supported by the National Natural Science Foundation of China(61622302,61673072,61573070)Guangdong Natural Science Funds for Distinguished Young Scholar(2017A030306014)+1 种基金the Department of Education of Guangdong Province(2016KTSCX030)the Department of Education of Liaoning Province(LZ2017001)
文摘This paper investigates the problem of finite frequency fuzzy H_∞ control for uncertain active vehicle suspension systems, in which sensor failure is taken into account. TakagiSugeno(T-S) fuzzy model is established for considered suspension systems. In order to describe the sensor fault effectively, a corresponding model is introduced. A vital performance index,H_∞ performance, is utilized to measure the drive comfort. In the framework of Kalman-Yakubovich-Popov theory, the H_∞ norm from external perturbation to controlled output is optimized effectively in the frequency domain of 4 Hz-8 Hz to enhance ride comfort level. Meanwhile, three suspension constrained requirements, i.e., ride comfort level, manipulation stability,suspension deflection are also guaranteed. Furthermore, sufficient conditions are developed to design a fuzzy controller to guarantee the desired performance of active suspension systems. Finally, the proposed control scheme is applied to a quarter-vehicle active suspension, and simulation results are given to illustrate the effectiveness of the proposed approach.
基金supported by the Science and Technology Research Project of Henan Province (No.222102210087)the Science and Technology Research Project of Henan Province (No.222102220102).
文摘The aging prediction of railway catenary is of profound significance for ensuring the regular operation of electrified trains.However,in real-world scenarios,accurate predictions are challenging due to various interferences.This paper addresses this challenge by proposing a novel method for predicting the aging of railway catenary based on an improved Kalman filter(KF).The proposed method focuses on modifying the priori state estimate covariance and measurement error covariance of the KF to enhance accuracy in complex environments.By comparing the optimal displacement value with the theoretically calculated value based on the thermal expansion effect of metals,it becomes possible to ascertain the aging status of the catenary.To improve prediction accuracy,a railway catenary aging prediction model is constructed by integrating the Takagi-Sugeno(T-S)fuzzy neural network(FNN)and KF.In this model,an adaptive training method is introduced,allowing the FNN to use fewer fuzzy rules.The inputs of the model include time,temperature,and historical displacement,while the output is the predicted displacement.Furthermore,the KF is enhanced by modifying its prior state estimate covariance and measurement error covariance.These modifications contribute to more accurate predictions.Lastly,a low-power experimental platform based on FPGA is implemented to verify the effectiveness of the proposed method.The test results demonstrate that the proposed method outperforms the compared method,showcasing its superior performance.
文摘This paper proposes a new method for control of continuous large-scale systems where the measures and control functions are distributed on calculating members which can be shared with other applications and connected to digital network communications.At first, the nonlinear large-scale system is described by a Takagi-Sugeno(TS) fuzzy model. After that, by using a fuzzy LyapunovKrasovskii functional, sufficient conditions of asymptotic stability of the behavior of the decentralized networked control system(DNCS),are developed in terms of linear matrix inequalities(LMIs). Finally, to illustrate the proposed approach, a numerical example and simulation results are presented.
文摘By the best approximation theory, it is first proved that the SISO （single-input single-output） linear Takagi-Sugeno （TS） fuzzy systems can approximate an arbitrary polynomial which, according to Weierstrass approximation theorem, can uniformly approximate any continuous functions on the compact domain. Then new sufficient conditions for general linear SISO TS fuzzy systems as universal approximators are obtained. Formulae are derived to calculate the number of input fuzzy sets to satisfy the given approximation accuracy. Then the presented result is compared with the existing literature＇s results. The comparison shows that the presented result needs less input fuzzy sets, which can simplify the design of the fuzzy system, and examples are given to show its effectiveness.
基金Project partially supported by the Natural Science Foundation of Educational Committee of Anhui Province, China （Grant No 2006kj250B）.
文摘The projection of the chaotic attractor observed from the Lorenz system in the X-Z plane is like a butterfly, hence the classical Lorenz system is widely known as the butterfly attractor, and has served as a prototype model for studying chaotic behaviour since it was coined. In this work we take one step further to investigate some fundamental dynamic behaviours of a novel hybrid Takagi-Sugeno （TS） fuzzy Lorenz-type system, which is essentially derived from the delta-operator-based TS fuzzy modelling for complex nonlinear systems, and contains the original Lorenz system of continuous-time TS fuzzy form as a special case. By simply and appropriately tuning the additional parametric perturbations in the two-rule hybrid TS fuzzy Lorenz-type system, complex （two-wing） butterfly attractors observed from this system in the three dimensional （3D） X-Y-Z space are created, which have not yet been reported in the literature, and the forming mechanism of the compound structures have been numerically investigated.
基金Project(2007AA04Z162) supported by the National High-Tech Research and Development Program of ChinaProjects(2006T089, 2009T062) supported by the University Innovation Team in the Educational Department of Liaoning Province, China
文摘In order to obtain accurate prediction model and compensate for the influence of model mismatch on the control performance of the system and avoid solving nonlinear programming problem,an adaptive fuzzy predictive functional control(AFPFC) scheme for multivariable nonlinear systems was proposed.Firstly,multivariable nonlinear systems were described based on Takagi-Sugeno(T-S) fuzzy models;assuming that the antecedent parameters of T-S models were kept,the consequent parameters were identified on-line by using the weighted recursive least square(WRLS) method.Secondly,the identified T-S models were linearized to be time-varying state space model at each sampling instant.Finally,by using linear predictive control technique the analysis solution of the optimal control law of AFPFC was established.The application results for pH neutralization process show that the absolute error between the identified T-S model output and the process output is smaller than 0.015;the tracking ability of the proposed AFPFC is superior to that of non-AFPFC(NAFPFC) for pH process without disturbances,the overshoot of the effluent pH value of AFPFC with disturbances is decreased by 50% compared with that of NAFPFC;when the process parameters of AFPFC vary with time the integrated absolute error(IAE) performance index still retains to be less than 200 compared with that of NAFPFC.
基金Supported by National Natural Science Foundation of China （50977008, 60904017, 60774048, 60728307）, the Funds for Creative Research Groups of China （60521003）, the Program for Cheung Kong Scholars and Innovative Research Team in University （IRT0421）, and the 111 Project （B08015）, National High Technology Research and Development Program of China （863 Program） （2006AA04Z183）
文摘This paper is concerned with the problem of stabilization of the Roesser type discrete-time nonlinear 2-D system that plays an important role in many practical applications. First, a discrete-time 2-D T-S fuzzy model is proposed to represent the underlying nonlinear 2-D system. Second, new quadratic stabilization conditions are proposed by applying relaxed quadratic stabilization technique for 2-D case. Third, for sake of further reducing conservatism, new non-quadratic stabilization conditions are also proposed by applying a new parameter-dependent Lyapunov function, matrix transformation technique, and relaxed technique for the underlying discrete-time 2-D T-S fuzzy system. Finally, a numerical example is provided to illustrate the effectiveness of the proposed results.
基金Supported by the National Natural Science Foundation of China(61472136)the Best Youth of the Education Department of Hunan Province(16B023)
文摘This paper focuses on the stability analysis of nonlinear networked control system with integral quadratic constraints(IQC) performance, dynamic quantization, variable sampling intervals, and communication delays. By using input-delay and parallel distributed compensation(PDC) techniques, we establish the Takagi-Sugeno(T-S) fuzzy model for the system, in which the sampling period of the sampler and signal transmission delay are transformed to the refreshing interval of a zero-order holder(ZOH). By the appropriate Lyapunov-Krasovskii-based methods, a delay-dependent criterion is derived to ensure the asymptotic stability for the system with IQC performance via the H∞ state feedback control. The efficiency of the method is illustrated on a simulation exampler.
基金supported in part by the Key Research Project of North Minzu University under Grant 2021JCYJ09in part by the French Ministry of Higher Education and Research,in part by the National Center for Scientific Research(CNRS)+5 种基金in part by the ANR CoCoVeIA project(ANR-19-CE22-0009)in part by the ANR HM-Science project(ANR-21-CE48-0021)in part by the Hauts-de-France Region under the project RITMEA CPER 2021-2027in part by the National Natural Science Foundation of China under Grant 62163002in part by the Natural Science Foundation of Ningxia Hui Autonomous Region under Grant 2021AAC05011in part by the Advanced Intelligent Perception and Control Technology Innovative Team of Ningxia.
文摘The detection and mitigation of cyber-attacks in connected vehicle systems(CVSs)are critical for ensuring the security of intelligent connected vehicles.This paper presents a solution to estimate sensor and actuator cyber-attacks in CVSs.A novel method is proposed that utilizes an augmented system representation technique and a nonlinear unknown input observer(UIO)to achieve asymptotic estimation of both CVS dynamics and cyber-attacks.The nonlinear CVS dynamics is represented in a Takagi–Sugeno(TS)fuzzy form with nonlinear consequents,which allows for the effective use of the differential mean value theorem to handle unmeasured premise variables.Furthermore,via Lyapunov stability theory sufficient conditions are proposed,expressed in terms of linear matrix inequalities,to design TS fuzzy UIO.Several test scenarios are performed with high-fidelity Simulink-CarSim co-simulations to show the effectiveness of the proposed cyber-attack estimation method.
基金supported in part by the National Natural Science Foundation of China(61973219,61933007,62073158)the China Scholarship Council(201908310148)。
文摘In this article,the problem of state estimation is addressed for discrete-time nonlinear systems subject to additive unknown-but-bounded noises by using fuzzy set-membership filtering.First,an improved T-S fuzzy model is introduced to achieve highly accurate approximation via an affine model under each fuzzy rule.Then,compared to traditional prediction-based ones,two types of fuzzy set-membership filters are proposed to effectively improve filtering performance,where the structure of both filters consists of two parts:prediction and filtering.Under the locally Lipschitz continuous condition of membership functions,unknown membership values in the estimation error system can be treated as multiplicative noises with respect to the estimation error.Real-time recursive algorithms are given to find the minimal ellipsoid containing the true state.Finally,the proposed optimization approaches are validated via numerical simulations of a one-dimensional and a three-dimensional discrete-time nonlinear systems.
