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Design of active disturbance rejection internal model control strategy for SISO system with time delay process 被引量:3
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作者 靳其兵 刘立业 《Journal of Central South University》 SCIE EI CAS CSCD 2015年第5期1725-1736,共12页
A novel control scheme of active disturbance rejection internal model control(ADRIMC) is proposed to improve the anti-interference ability and robustness for the dead-time process. The active anti-interference concept... A novel control scheme of active disturbance rejection internal model control(ADRIMC) is proposed to improve the anti-interference ability and robustness for the dead-time process. The active anti-interference concept is introduced into the internal model control(IMC) by analyzing the relationship between IMC and disturbance observer control(DOB). Further, a design process of disturbance filter is presented to realize the active anti-interference ability for ADRIMC scheme. The disturbance filter is used to estimate an equivalent disturbance consisting of both external disturbances and internal disturbances caused by model mismatches.Simulation results demonstrate that the proposed method possesses a good disturbance rejection performance, though losing some partial dynamic performance. In other words, the proposed method shows a tradeoff between the dynamic performance and the system robust. 展开更多
关键词 internal model control(IMC) disturbance observer(DOB) active disturbance rejection internal model controller(ADRIMC) low pass filter(LPF) ROBUSTNESS
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Modified Two-Degrees-of-Freedom Internal Model Control for Non-Square Systems with Multiple Time Delays 被引量:25
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作者 Jian-Chang Liu Nan Chen Xia Yu 《Journal of Harbin Institute of Technology(New Series)》 EI CAS 2014年第2期122-128,共7页
A modified two-degrees-of-freedom( M-TDOF) internal model control( IMC) method is proposed for non-square systems with multiple time delays and right-half-plane( RHP) zeros. In this method,pseudo-inverse is introduced... A modified two-degrees-of-freedom( M-TDOF) internal model control( IMC) method is proposed for non-square systems with multiple time delays and right-half-plane( RHP) zeros. In this method,pseudo-inverse is introduced to design the internal model controller,and a desired closed-loop transfer function is designed to eliminate the unrealizable factors of the derived controller. In addition,set-point tracking and load-disturbance rejection of each process are separately controlled by two controllers. The simulation results show that in addition to high decoupling performance and robustness,the proposed control method also effectively improves loaddisturbance rejection and simultaneously optimizes the input tracking performance and disturbance rejection performance by selecting the parameters of controllers. Furthermore,the higher tolerance of model mismatch is achieved in this paper. 展开更多
关键词 non-square system two degrees of freedom internal model control
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Vehicle Active Steering Control Research Based on Two-DOF Robust Internal Model Control 被引量:12
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作者 WU Jian LIU Yahui +3 位作者 WANG Fengbo BAO Chunjiang SUN Qun ZHAO Youqun 《Chinese Journal of Mechanical Engineering》 SCIE EI CAS CSCD 2016年第4期739-746,共8页
Because of vehicle's external disturbances and model uncertainties,robust control algorithms have obtained popularity in vehicle stability control.The robust control usually gives up performance in order to guarantee... Because of vehicle's external disturbances and model uncertainties,robust control algorithms have obtained popularity in vehicle stability control.The robust control usually gives up performance in order to guarantee the robustness of the control algorithm,therefore an improved robust internal model control(IMC) algorithm blending model tracking and internal model control is put forward for active steering system in order to reach high performance of yaw rate tracking with certain robustness.The proposed algorithm inherits the good model tracking ability of the IMC control and guarantees robustness to model uncertainties.In order to separate the design process of model tracking from the robustness design process,the improved 2 degree of freedom(DOF) robust internal model controller structure is given from the standard Youla parameterization.Simulations of double lane change maneuver and those of crosswind disturbances are conducted for evaluating the robust control algorithm,on the basis of a nonlinear vehicle simulation model with a magic tyre model.Results show that the established 2-DOF robust IMC method has better model tracking ability and a guaranteed level of robustness and robust performance,which can enhance the vehicle stability and handling,regardless of variations of the vehicle model parameters and the external crosswind interferences.Contradiction between performance and robustness of active steering control algorithm is solved and higher control performance with certain robustness to model uncertainties is obtained. 展开更多
关键词 active steering internal model control model tracking robust performance crosswind disturbances
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Improvement of precision for pendulous integrating gyro accelerometer via adaptive internal model control 被引量:1
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作者 Xu Fengxia Xia Gang +2 位作者 Zeng Ming Sun Baoku Zhao Xuezeng 《Journal of Systems Engineering and Electronics》 SCIE EI CSCD 2007年第4期841-845,共5页
An adaptive internal mode control is proposed to eliminate effectively periodic disturbance with uncertain frequency caused by input error angle of PIGA (Pendulous Integrating Gyro Accelerometer). An adaptive algori... An adaptive internal mode control is proposed to eliminate effectively periodic disturbance with uncertain frequency caused by input error angle of PIGA (Pendulous Integrating Gyro Accelerometer). An adaptive algorithm with periodic disturbance frequency identification on line is applied and the internal model controller parameters are adjusted to eliminate disturbance. Then the convergence of this algorithm and the stability of the system are proved by the averaging method. Simulation results verify the proposed scheme can eliminate periodic disturbance and improve the test precision for PIGA effectively. 展开更多
关键词 PIGA periodic disturbance adaptive algorithm internal model control
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Synthetical Control of AGC/LPC System Based on Neural Networks Internal Model Control
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作者 Hu He, Xiaodong Luan, Yikang Sun Information Engineering School, University of Science and Technology Beijing, Beijing 100083, China 《Journal of University of Science and Technology Beijing》 CSCD 2001年第1期75-77,共3页
One synthetical control method of AGC/LPC system based on intelligence control theory-neural networks internal model control method is presented. Genetic algorithm (GA) is applied to optimize the parameters of the neu... One synthetical control method of AGC/LPC system based on intelligence control theory-neural networks internal model control method is presented. Genetic algorithm (GA) is applied to optimize the parameters of the neural networks. Simulation results prove that this method is effective. 展开更多
关键词 hot strip rolling AGC LOOPER neural networks internal model control GA
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Average Current Control with Internal Model Control and Real-time Frequency Decoupling for Hybrid Energy Storage Systems in Microgrids 被引量:1
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作者 Alejandro Latorre Wilmar Martinez Camilo A.Cortes 《Journal of Modern Power Systems and Clean Energy》 SCIE EI CSCD 2023年第2期511-522,共12页
Among hybrid energy storage systems(HESSs),battery-ultracapacitor systems in active topology use DC/DC power converters for their operations.HESSs are part of the solutions designed to improve the operation of power s... Among hybrid energy storage systems(HESSs),battery-ultracapacitor systems in active topology use DC/DC power converters for their operations.HESSs are part of the solutions designed to improve the operation of power systems in different applications.In the residential microgrid applications,a multilevel control system is required to manage the available energy and interactions among the microgrid components.For this purpose,a rule-based power management system is designed,whose operation is validated in the simulation,and the performances of different controllers are compared to select the best strategy for the DC/DC converters.The average current control with internal model control and real-time frequency decoupling is proposed as the most suitable controller according to the contemplated performance parameters,allowing voltage regulation values close to 1%.The results are validated using real-time hardware-in-the-loop(HIL).These systems can be easily adjusted for other applications such as electric vehicles. 展开更多
关键词 internal model control energy management system(EMS) hybrid energy storage system(HESS) MICROGRIDS real-time frequency decoupling
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Multi-loop adaptive internal model control based on a dynamic partial least squares model 被引量:3
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作者 Zhao ZHAO Bin HU Jun LIANG 《Journal of Zhejiang University-Science A(Applied Physics & Engineering)》 SCIE EI CAS CSCD 2011年第3期190-200,共11页
A multi-loop adaptive internal model control (IMC) strategy based on a dynamic partial least squares (PLS) frame-work is proposed to account for plant model errors caused by slow aging,drift in operational conditions,... A multi-loop adaptive internal model control (IMC) strategy based on a dynamic partial least squares (PLS) frame-work is proposed to account for plant model errors caused by slow aging,drift in operational conditions,or environmental changes.Since PLS decomposition structure enables multi-loop controller design within latent spaces,a multivariable adaptive control scheme can be converted easily into several independent univariable control loops in the PLS space.In each latent subspace,once the model error exceeds a specific threshold,online adaptation rules are implemented separately to correct the plant model mismatch via a recursive least squares (RLS) algorithm.Because the IMC extracts the inverse of the minimum part of the internal model as its structure,the IMC controller is self-tuned by explicitly updating the parameters,which are parts of the internal model.Both parameter convergence and system stability are briefly analyzed,and proved to be effective.Finally,the proposed control scheme is tested and evaluated using a widely-used benchmark of a multi-input multi-output (MIMO) system with pure delay. 展开更多
关键词 Partial least squares (PLS) Adaptive internal model control (IMC) Recursive least squares (RLS)
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Optimized Model Based Controller with Model Plant Mismatch for NMP Mitigation in Boost Converter
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作者 R.Prasanna Uma Govindarajan NSBhuvaneswari 《Computer Systems Science & Engineering》 SCIE EI 2023年第8期1961-1979,共19页
In this paper,an optimized Genetic Algorithm(GA)based internal model controller-proportional integral derivative(IMC-PID)controller has been designed for the control variable to output variable transfer function of dc... In this paper,an optimized Genetic Algorithm(GA)based internal model controller-proportional integral derivative(IMC-PID)controller has been designed for the control variable to output variable transfer function of dc-dc boost converter to mitigate the effect of non-minimum phase(NMP)behavior due to the presence of a right-half plane zero(RHPZ).This RHPZ limits the dynamic performance of the converter and leads to internal instability.The IMC PID is a streamlined counterpart of the standard feedback controller and easily achieves optimal set point and load change performance with a single filter tuning parameterλ.Also,this paper addresses the influences of the model-based controller with model plant mismatch on the closed-loop control.The conventional IMC PID design is realized as an optimization problem with a resilient controller being determined through a genetic algorithm.Computed results suggested that GA–IMC PID coheres to the optimum designs with a fast convergence rate and outperforms conventional IMC PID controllers. 展开更多
关键词 Boost converter non-minimum phase internal model control dynamic controller
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Temperature Control Based on Fuzzy Logic Two-degree-of-freedom Smith Internal Model 被引量:4
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作者 WANG Zhigang HE Meng 《Instrumentation》 2020年第2期1-8,共8页
According to the characteristics of the large time delay,nonlinearity and the great inertia of temperature control system in biomass pyrolysis reactor,a two-degree-of-freedom Smith internal model controller based on f... According to the characteristics of the large time delay,nonlinearity and the great inertia of temperature control system in biomass pyrolysis reactor,a two-degree-of-freedom Smith internal model controller based on fuzzy control is proposed.Firstly,the mathematical model of the temperature control system is established by using the step response method,and then the two-degree-of-freedom Smith internal model controller is designed,and the good tracking performance and disturbance suppression performance can be obtained by designing the set value tracking controller and interference rejection capability.Secondly,the fuzzy control algorithm is used to realize the on-line tuning of the control parameters of the two-degree-of-freedom Smith internal model algorithm.The simulation results show that,compared with the traditional internal model control,fuzzy internal model PID control and two-degree-of-freedom Smith internal model control,the algorithm proposed in this paper improves the influence of lag time on the control system,realizes the separation control of set point tracking and anti-jamming performance and the self-tuning of control parameters,and improves the control performance of the system. 展开更多
关键词 Smith Predictive controller internal model control Two Degrees of Freedom Fuzzy control Algorithm
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Two fuzzy internal model control methods for nonlinear uncertain systems
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作者 Amira Aydi Mohamed Djemel Mohamed Chtourou 《International Journal of Intelligent Computing and Cybernetics》 EI 2017年第2期223-240,共18页
Purpose-The purpose of this paper is to use the internal model control to deal with nonlinear stable systems affected by parametric uncertainties.Design/methodology/approach-The dynamics of a considered system are app... Purpose-The purpose of this paper is to use the internal model control to deal with nonlinear stable systems affected by parametric uncertainties.Design/methodology/approach-The dynamics of a considered system are approximated by a Takagi-Sugeno fuzzy model.The parameters of the fuzzy rules premises are determined manually.However,the parameters of the fuzzy rules conclusions are updated using the descent gradient method under inequality constraints in order to ensure the stability of each local model.In fact,without making these constraints the training algorithm can procure one or several unstable local models even if the desired accuracy in the training step is achieved.The considered robust control approach is the internal model.It is synthesized based on the Takagi-Sugeno fuzzy model.Two control strategies are considered.The first one is based on the parallel distribution compensation principle.It consists in associating an internal model control for each local model.However,for the second strategy,the control law is computed based on the global Takagi-Sugeno fuzzy model.Findings-According to the simulation results,the stability of all local models is obtained and the proposed fuzzy internal model control approaches ensure robustness against parametric uncertainties.Originality/value-This paper introduces a method for the identification of fuzzy model parameters ensuring the stability of all local models.Using the resulting fuzzy model,two fuzzy internal model control designs are presented. 展开更多
关键词 Robust control UNCERTAINTIES Descent gradient method Fuzzy modeling internal model control Nonlinear systems
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Generalized Internal Model Robust Control for Active Front Steering Intervention 被引量:7
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作者 WU Jian ZHAO Youqun +2 位作者 JI Xuewu LIU Yahui ZHANG Lipeng 《Chinese Journal of Mechanical Engineering》 SCIE EI CAS CSCD 2015年第2期285-293,共9页
Because of the tire nonlinearity and vehicle's parameters'uncertainties,robust control methods based on the worst cases,such as H_∞,μsynthesis,have been widely used in active front steering control,however,in orde... Because of the tire nonlinearity and vehicle's parameters'uncertainties,robust control methods based on the worst cases,such as H_∞,μsynthesis,have been widely used in active front steering control,however,in order to guarantee the stability of active front steering system(AFS)controller,the robust control is at the cost of performance so that the robust controller is a little conservative and has low performance for AFS control.In this paper,a generalized internal model robust control(GIMC)that can overcome the contradiction between performance and stability is used in the AFS control.In GIMC,the Youla parameterization is used in an improved way.And GIMC controller includes two sections:a high performance controller designed for the nominal vehicle model and a robust controller compensating the vehicle parameters'uncertainties and some external disturbances.Simulations of double lane change(DLC)maneuver and that of braking on split-μroad are conducted to compare the performance and stability of the GIMC control,the nominal performance PID controller and the H_∞controller.Simulation results show that the high nominal performance PID controller will be unstable under some extreme situations because of large vehicle's parameters variations,H_∞controller is conservative so that the performance is a little low,and only the GIMC controller overcomes the contradiction between performance and robustness,which can both ensure the stability of the AFS controller and guarantee the high performance of the AFS controller.Therefore,the GIMC method proposed for AFS can overcome some disadvantages of control methods used by current AFS system,that is,can solve the instability of PID or LQP control methods and the low performance of the standard H_∞controller. 展开更多
关键词 active front steering system generalized internal model robust control H_∞ optimization PID split-μ road
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Novel power capture optimization based sensorless maximum power point tracking strategy and internal model controller for wind turbines systems driven SCIG
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作者 Ali EL YAAKOUBI Kamal ATTARI +1 位作者 Adel ASSELMAN Abdelouahed DJEBLI 《Frontiers in Energy》 SCIE CSCD 2019年第4期742-756,共15页
Under the trends to using renewable energy sources as alternatives to the traditional ones,it is important to contribute to the fast growing development of these sources by using powerful soft computing methods.In thi... Under the trends to using renewable energy sources as alternatives to the traditional ones,it is important to contribute to the fast growing development of these sources by using powerful soft computing methods.In this context,this paper introduces a novel structure to optimize and control the energy produced from a variable speed wind turbine which is based on a squirrel cage induction generator(SCIG)and connected to the grid.The optimization strategy of the harvested power from the wind is realized by a maximum power point tracking(MPPT)algorithm based on fuzzy logic,and the control strategy of the generator is implemented by means of an internal model(IM)controller.Three IM controllers are incorporated in the vector control technique,as an alternative to the proportional integral(PI)controller,to implement the proposed optimization strategy.The MPPT in conjunction with the IM controller is proposed as an alternative to the traditional tip speed ratio(TSR)technique,to avoid any disturbance such as wind speed measurement and wind turbine(WT)characteristic uncertainties.Based on the simulation results of a six KW-WECS model in Matlab/Simulink,the presented control system topology is reliable and keeps the system operation around the desired response. 展开更多
关键词 power optimization wind energy conversion system maximum power point tracking(MPPT) fuzzy logic internal model(IM)controller
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Integrated Auto-Tuning PID Control of Continuous Casting Process 被引量:1
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作者 GUO Ge WANG Wei CHAI Tian-you 《Journal of Iron and Steel Research(International)》 SCIE EI CAS CSCD 2000年第1期17-21,共5页
PID controllers were used for the hydraulic servo system of sliding gate and the tundish weight control system in continuous caster.These two loops were synthesized in mould level controller based on model reduction a... PID controllers were used for the hydraulic servo system of sliding gate and the tundish weight control system in continuous caster.These two loops were synthesized in mould level controller based on model reduction and internal model control strategy.Satisfactory control performance of this synthetic mould level controller was demonstrated by simulations and on-line experiments. 展开更多
关键词 auto-tuning control model reduction internal model control
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A novel flow control scheme for reciprocating compressor based on adaptive predictive PID control
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作者 刘雯华 Zhang Jinjie +3 位作者 Zhou Chao Wang Yao Sun Xu Hong Huaibing 《High Technology Letters》 EI CAS 2020年第3期243-252,共10页
In the process of capacity regulation of reciprocating compressor, the frequent change of inlet temperature and pressure makes the control of exhaust flow unstable, resulting in the high pressure ratio of the intermed... In the process of capacity regulation of reciprocating compressor, the frequent change of inlet temperature and pressure makes the control of exhaust flow unstable, resulting in the high pressure ratio of the intermediate stage. At last the compressor cannot operate safely. To solve the problem, a novel flow control scheme based on inlet temperature and pressure ratio is proposed. In this scheme, the intake model of the cylinder under the capacity regulation condition is established to calculate the load of the first cylinder. Then, the adaptive predictive PID(APPID) controller is designed to control the pressure ratio of other stages, and the grey prediction model is used to predict the pressure output to overcome the system delay. To solve the problem of control parameters tuning, an improved particle swarm optimization(PSO) algorithm is adopted to obtain the optimal control parameters.The effectiveness of the adaptive predictive PID control method is verified by a two-stage compressor model simulation.Finally, the flow control scheme is applied to the actual four-stage air reciprocating compressor flow control system. Although the temperature difference is greater than 15 ℃, the compressor exhaust flow is maintained at the set value and the pressure ratio is also maintained stable. At the same time, the compressor pressure ratio can be quickly adjusted without overshoot. The application result further verifies the feasibility and effectiveness of the scheme. 展开更多
关键词 reciprocating compressor capacity control predictive PID control internal model control(IMC)
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Start-up current adaptive control for sensorless high-speed brushless DC motors based on inverse system method and internal mode controller 被引量:9
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作者 He Yanzhao Zheng Shiqiang Fang Jiancheng 《Chinese Journal of Aeronautics》 SCIE EI CAS CSCD 2017年第1期358-367,共10页
The start-up current control of the high-speed brushless DC(HS-BLDC) motor is a challenging research topic. To effectively control the start-up current of the sensorless HS-BLDC motor, an adaptive control method is ... The start-up current control of the high-speed brushless DC(HS-BLDC) motor is a challenging research topic. To effectively control the start-up current of the sensorless HS-BLDC motor, an adaptive control method is proposed based on the adaptive neural network(ANN)inverse system and the two degrees of freedom(2-DOF) internal model controller(IMC). The HS-BLDC motor is identified by the online least squares support vector machine(OLS-SVM) algorithm to regulate the ANN inverse controller parameters in real time. A pseudo linear system is developed by introducing the constructed real-time inverse system into the original HS-BLDC motor system. Based on the characteristics of the pseudo linear system, an extra closed-loop feedback control strategy based on the 2-DOF IMC is proposed to improve the transient response performance and enhance the stability of the control system. The simulation and experimental results show that the proposed control method is effective and perfect start-up current tracking performance is achieved. 展开更多
关键词 Adaptive control Brushless DC motors Inverse systems internal model controller Neural networks START-UP Support vector machines
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Model Predictive Control of Resonant Systems Using Kautz Model
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作者 Shamik Misra Rajasekhara Reddy Prabirkumar Saha 《International Journal of Automation and computing》 EI CSCD 2016年第5期501-515,共15页
The scope of this paper broadly spans in two areas: system identification of resonant system and design of an efficient control scheme suitable for resonant systems. Use of filters based on orthogonal basis functions... The scope of this paper broadly spans in two areas: system identification of resonant system and design of an efficient control scheme suitable for resonant systems. Use of filters based on orthogonal basis functions (OBF) have been advocated for modelling of resonant process. Kautz filter has been identified as best suited OBF for this purpose. A state space based system identification technique using Kautz filters, viz. Kautz model, has been demonstrated. Model based controllers are believed to be more efficient than classical controllers because explicit use of process model is essential with these modelling techniques. Extensive literature search concludes that very few reports are available which explore use of the model based control studies on resonant system. Two such model based controllers are considered in this work, viz. model predictive controller and internal model controller. A model predictive control algorithm has been developed using the Kautz model. The efficacy of the model and the controller has been verified by two case studies, viz. linear second order underdamped process and a mildly nonlinear magnetic ball suspension system. Comparative assessment of performances of these controllers in those case studies have been carried out. 展开更多
关键词 model predictive control resonant systems Kautz model orthonormal basis function internal model control
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Output constrained IMC controllers in control systems of electromechanical actuators 被引量:1
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作者 Piotr M.MARUSAK Suwat KUNTANAPREEDA 《Control Theory and Technology》 EI CSCD 2015年第3期245-255,共11页
Electromechanical actuators are widely used in many industrial applications. There are usually some constraints existing in a designed system. This paper proposes a simple method to design constrained controllers for ... Electromechanical actuators are widely used in many industrial applications. There are usually some constraints existing in a designed system. This paper proposes a simple method to design constrained controllers for electromechanical actuators. The controllers merge the ideas exploited in internal model control and model predictive control. They are designed using the standard control system structure with unity negative feedback. The structure of the controllers is relatively simple as well as the design process. The output constraint handling mechanism is based on prediction of the control plant behavior many time steps ahead. The mechanism increases control performance and safety of the control plant. The benefits offered by the proposed controllers have been demonstrated in real-life experiments carried out in control systems of two electromechanical actuators: a DC motor and an electrohydraulic actuator. 展开更多
关键词 Constrained control internal model control (IMC) model predictive control (MPC) electromechanical actuators
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Precise control of a magnetically suspended double-gimbal control moment gyroscope using differential geometry decoupling method 被引量:3
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作者 Chen Xiaocen Chen Maoyin 《Chinese Journal of Aeronautics》 SCIE EI CAS CSCD 2013年第4期1017-1028,共12页
Precise control of a magnetically suspended double-gimbal control moment gyroscope (MSDGCMG) is of vital importance and challenge to the attitude positioning of spacecraft owing to its multivariable, nonlinear and s... Precise control of a magnetically suspended double-gimbal control moment gyroscope (MSDGCMG) is of vital importance and challenge to the attitude positioning of spacecraft owing to its multivariable, nonlinear and strong coupled properties. This paper proposes a novel linearization and decoupling method based on differential geometry theory and combines it with the internal model controller (IMC) to guarantee the system robustness to the external disturbance and parameter uncertainty. Furthermore, by introducing the dynamic compensation for the inner-gimbal rate-servo system and the magnetically suspended rotor (MSR) system only, we can eliminate the influence of the unmodeled dynamics to the decoupling control accuracy as well as save costs and inhibit noises effectively. The simulation results verify the nice decoupling and robustness performance of the system using the proposed method. 展开更多
关键词 Differential geometry decoupling Dynamic compensation internal model controller MSDGCMG Spacecraft control
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A nonsmooth IMC method for mechanical systems with backlash
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作者 Ruili DONG Yonghong TAN David HE 《控制理论与应用(英文版)》 EI CSCD 2013年第4期600-607,共8页
This paper proposes a discrete-time nonsmooth internal model control (NSIMC) approach for mechanical transmission systems described by so-called sandwich system with backlash. In this method, a dynamic compensator i... This paper proposes a discrete-time nonsmooth internal model control (NSIMC) approach for mechanical transmission systems described by so-called sandwich system with backlash. In this method, a dynamic compensator is introduced to compensate for the effect of the input linear subsystem. Thus, the sandwich systems with backlash can be simplified as a pseudo-Hammerstein system with backlash. The corresponding NSIMC strategy is designed to control this system. The design procedure of the controller is presented based on the analysis on the robust stability by considering the model errors involved with the effect of backlash as well as the compensated error of the input linear subsystem. Moreover, as the model is switched among the different operating zones, the robust filters are proposed to guarantee the robust stability and satisfactory control performance of the system. 展开更多
关键词 BACKLASH internal model control Sandwich system Nonsmooth nonlinear systems
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