Search results for: direct field-oriented control

Authors: A. Denker, U. Akıncıoğlu

Abstract:

In this paper a neural adaptive control method has been developed and applied to robot control. Simulation results are presented to verify the effectiveness of the controller. These results show that the performance by using this controller is better than those which just use either direct inverse control or predictive control. In addition, they show that the resulting is a useful method which combines the advantages of both direct inverse control and predictive control.

Keywords: Neural networks, robotics, direct inverse control, predictive control.

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Authors: T. Benmiloud, A. Omari

Abstract:

This paper presents a new technique of compensation of the effect of variation parameters in the direct field oriented control of induction motor. The proposed method uses an adaptive tuning of the value of synchronous speed to obtain the robustness for the field oriented control. We show that this adaptive tuning allows having robustness for direct field oriented control to changes in rotor resistance, load torque and rotational speed. The effectiveness of the proposed control scheme is verified by numerical simulations. The numerical validation results of the proposed scheme have presented good performances compared to the usual direct-field oriented control.

Keywords: Induction motor, direct field-oriented control, compensation of variation parameters, fuzzy logic controller.

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Authors: T. Mohammed Chikouche, K. Hartani

Abstract:

In order to solve the instantaneous power ripple and achieve better performance of direct power control (DPC) for a three-phase PWM rectifier, a control method is proposed in this paper. This control method is applied to overcome the instantaneous power ripple, to eliminate line current harmonics and therefore reduce the total harmonic distortion and to improve the power factor. A switching table is based on the analysis on the change of instantaneous active and reactive power, to select the optimum switching state of the three-phase PWM rectifier. The simulation result shows feasibility of this control method.

Keywords: Power quality, direct power control, power ripple, switching table, unity power factor.

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Authors: Y.Srinivasa Kishore Babu, G.Tulasi Ram Das

Abstract:

This paper presents a comparative study of two most popular control strategies for Induction motor (IM) drives: Field-Oriented Control (FOC) and Direct Torque Control (DTC). The comparison is based on various criteria including basic control characteristics, dynamic performance, and implementation complexity. The study is done by simulation using the Simulink Power System Block set that allows a complete representation of the power section (inverter and IM) and the control system.

Keywords: IM, FOC, DTC, Simulink

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Authors: S. Abouda, F. Nollet, A. Chaari, N. Essounbouli, Y. Koubaa

Abstract:

In this paper we propose the study of a centrifugal pump control system driven by a three-phase induction motor, which is supplied by a PhotoVoltaic PV generator. The system includes solar panel, a DC / DC converter equipped with its MPPT control, a voltage inverter to three-phase Pulse Width Modulation - PWM and a centrifugal pump driven by a three phase induction motor. In order to control the flow of the centrifugal pump, a Direct Torque Control - DTC of the induction machine is used. To illustrate the performances of the control, simulation results are carried out using Matlab/Simulink.

Keywords: Photovoltaic generators, Maximum power point tracking (MPPT), DC/DC converters, Induction motor, Direct torque control (DTC).

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Authors: Jun Liu, Feihang Zhou, Gungyi Wang

Abstract:

This paper expounds on the direct-driven PMSG wind power system control strategy, and analyses the stability conditions of the system. The direct-driven PMSG wind power system may generate the intense mechanical vibration, when wind speed changes dramatically. This paper proposes a new type of torque control strategy, which increases the system damping effectively, mitigates mechanical vibration of the system, and enhances the stability conditions of the system. The simulation results verify the reliability of the new torque control strategy.

Keywords: Damping, direct-driven PMSG wind power system, mechanical vibration, torque control.

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Authors: E. E. El-Kholy, Ahamed Kalas, Mahmoud Fauzy, M. El-Shahat Dessouki, Abdou. M. El-Refay, Mohammed El-Zefery

Abstract:

Direct Torque Control (DTC) is an AC drive control method especially designed to provide fast and robust responses. In this paper a progressive algorithm for direct torque control of threephase induction drive system supplied by photovoltaic arrays using voltage source inverter to control motor torque and flux with maximum power point tracking at different level of insolation is presented. Experimental results of the new DTC method obtained by an experimental rapid prototype system for drives are presented. Simulation and experimental results confirm that the proposed system gives quick, robust torque and speed responses at constant switching frequencies.

Keywords: Photovoltaic (PV) array, direct torque control (DTC), constant switching frequency, induction motor, maximum power point tracking (MPPT).

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Authors: M. S. Merzoug, F. Naceri

Abstract:

This paper presents a comparative study on two most popular control strategies for Permanent Magnet Synchronous Motor (PMSM) drives: field-oriented control (FOC) and direct torque control (DTC). The comparison is based on various criteria including basic control characteristics, dynamic performance, and implementation complexity. The study is done by simulation using the Simulink Power System Blockset that allows a complete representation of the power section (inverter and PMSM) and the control system. The simulation and evaluation of both control strategies are performed using actual parameters of Permanent Magnet Synchronous Motor fed by an IGBT PWM inverter.

Keywords: PMSM, FOC, DTC, hysteresis, PWM.

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Authors: A. Elnady

Abstract:

This paper presents an improved Direct Power Control (DPC) scheme applied to the multilevel inverter that forms a Distributed Generation Unit (DGU). This paper demonstrates the performance of active and reactive power injected by the DGU to the smart grid. The DPC is traditionally operated by the hysteresis controller with the Space Vector Modulation (SVM) which is applied on the 2-level inverters or 3-level inverters. In this paper, the DPC is operated by the PI controller with the Phase-Disposition Pulse Width Modulation (PD-PWM) applied to the 5-level diode clamped inverter. The new combination of the DPC, PI controller, PD-PWM and multilevel inverter proves that its performance is much better than the conventional hysteresis-SVM based DPC. Simulations results have been presented to validate the performance of the suggested control scheme in the grid-connected mode.

Keywords: Direct power control, PI controller, PD-PWM, and power control.

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Authors: L. Moussaoui, L. Rahmani

Abstract:

This paper deals with the direct torque control (DTC) of the induction motor. This type of control allows decoupling control between the flux and the torque without the need for a transformation of coordinates. However, as with other hysteresis-based systems, the classical DTC scheme represents a high ripple, in both the electromagnetic torque and the stator flux and a distortion in the stator current. As well, it suffers from variable switching frequency. To solve these problems various modifications, in conventional DTC scheme, have been made during the last decade. Indeed the DTC based on space vector modulation (SVM) has proved to generate very low ripples in torque and flux with constant switching frequency. It also shows almost the same dynamic performances as the classical DTC system. On the other hand, fuzzy logic is considered as an interesting alternative approach for its advantages: Analysis close to the exigencies of user, ability of nonlinear systems control, best dynamic performances and inherent quality of robustness.

Therefore, two fuzzy direct torque control approaches, for the induction motor fed by SVM-voltage source inverter, are proposed in this paper. By using these two approaches of DTC, the advantages of fuzzy logic control, space vector modulation, and direct torque control method are combined. The performances of these DTC schemes are evaluated through digital simulation using Matlab/Simulink platform and fuzzy logic tools. Simulation results illustrate the effectiveness and the superiority of the proposed Fuzzy DTC-SVM schemes in comparison to the classical DTC.

Keywords: Direct torque control, Fuzzy logic control, Induction motor, Switching frequency, Space vector modulation, Torque and flux ripples.

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Authors: Biao Ma, Jikai Liu, Man Chen, Jianpeng Wu, Liyong Wang, Changsong Zheng

Abstract:

Proposing an appropriate control strategy is an effective and practical way to address the overheat problems of the wet multi-plate clutch in Direct Shift Gearbox under the long-time creeping condition. To do so, the temperature variation of the wet multi-plate clutch is investigated firstly by establishing a thermal resistance model for the gearbox cooling system. To calculate the generated heat flux and predict the clutch temperature precisely, the friction torque model is optimized by introducing an improved friction coefficient, which is related to the pressure, the relative speed and the temperature. After that, the heat transfer model and the reasonable friction torque model are employed by the vehicle powertrain model to construct a comprehensive co-simulation model for the Direct Shift Gearbox (DSG) vehicle. A creeping control strategy is then proposed and, to evaluate the vehicle performance, the safety temperature (250 ℃) is particularly adopted as an important metric. During the creeping process, the temperature of two clutches is always under the safety value (250 ℃), which demonstrates the effectiveness of the proposed control strategy in avoiding the thermal failures of clutches.

Keywords: Creeping control strategy, direct shift gearbox, temperature variation, wet clutch.

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Authors: Akbar Banitalebi, Mohd Ismail Abd Aziz, Rohanin Ahmad

Abstract:

A new stochastic algorithm called Probabilistic Global Search Johor (PGSJ) has recently been established for global optimization of nonconvex real valued problems on finite dimensional Euclidean space. In this paper we present convergence guarantee for this algorithm in probabilistic sense without imposing any more condition. Then, we jointly utilize this algorithm along with control parameterization technique for the solution of constrained optimal control problem. The numerical simulations are also included to illustrate the efficiency and effectiveness of the PGSJ algorithm in the solution of control problems.

Keywords: Optimal Control Problem, Constraints, Direct Methods, Stochastic Algorithm

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Authors: Fayssal Amrane, Azeddine Chaiba

Abstract:

In this paper, type-2 fuzzy logic control (T2FLC) and neuro-fuzzy control (NFC) for a doubly fed induction generator (DFIG) based on direct power control (DPC) with a fixed switching frequency is proposed for wind generation application. First, a mathematical model of the doubly-fed induction generator implemented in d-q reference frame is achieved. Then, a DPC algorithm approach for controlling active and reactive power of DFIG via fixed switching frequency is incorporated using PID. The performance of T2FLC and NFC, which is based on the DPC algorithm, are investigated and compared to those obtained from the PID controller. Finally, simulation results demonstrate that the NFC is more robust, superior dynamic performance for wind power generation system applications.

Keywords: Doubly fed induction generetor, direct power control, space vector modulation, type-2 fuzzy logic control, neuro-fuzzy control, maximum power point tracking.

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Authors: Kalyanasundaram.D., Kavitha. S

Abstract:

Field experiments were conducted at Annamalai University Experimental Farm, Department of Agronomy; to device suitable weed control measures for direct seeded puddled rice and to study the effect of the weed control measures on the soil microbial population. The treatments comprised of incorporation of pressmud @ 6.25 t ha-1 and application of herbicide butachlor @1.5 kg a. i. ha- 1 with and without safener 4 days after sowing (DAS), 8 DAS alone and also in conjunction with hand weeding at 30 DAS. Hand weeding twice and a weedy check were also maintained. At maximum tillering stage, the population of bacteria was significantly reduced by butachlor application. The injury to microbes caused by herbicide disappeared with the advancement of crop's age and at flowering stage of crop, there was no significant difference among the treatments. The fungal and actinomycetes population remained unaltered by weed control treatments at both the stages of observation.

Keywords: Butachlor, Herbicide, Direct sown rice, Microbial population

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Authors: Zundong Zhang, Limin Jia, Yuanyuan Chai

Abstract:

Control of complex systems is one of important files in complex systems, that not only relies on the essence of complex systems which is denoted by the core concept – emergence, but also embodies the elementary concept in control theory. Aiming at giving a clear and self-contained description of emergence, the paper introduces a formal way to completely describe the formation and dynamics of emergence in complex systems. Consequently, this paper indicates the Emergence-Oriented Control methodology that contains three kinds of basic control schemes: the direct control, the system re-structuring and the system calibration. As a universal ontology, the Emergence-Oriented Control provides a powerful tool for identifying and resolving control problems in specific systems.

Keywords: Complex System Control, Emergence, Emergence- Oriented Control Methodology.

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Authors: R. Karbalaei, A. Ghaffari, R. Kazemi, S. H. Tabatabaei

Abstract:

An integrated vehicle dynamics control system is developed in this paper by a combination of active front steering (AFS) and direct yaw-moment control (DYC) based on fuzzy logic control. The control system has a hierarchical structure consisting of two layers. A fuzzy logic controller is used in the upper layer (yaw rate controller) to keep the yaw rate in its desired value. The yaw rate error and its rate of change are applied to the upper controlling layer as inputs, where the direct yaw moment control signal and the steering angle correction of the front wheels are the outputs. In the lower layer (fuzzy integrator), a fuzzy logic controller is designed based on the working region of the lateral tire forces. Depending on the directions of the lateral forces at the front wheels, a switching function is activated to adjust the scaling factor of the fuzzy logic controller. Using a nonlinear seven degrees of freedom vehicle model, the simulation results illustrate considerable improvements which are achieved in vehicle handling through the integrated AFS/DYC control system in comparison with the individual AFS or DYC controllers.

Keywords: Intelligent strategy, integrated control, fuzzy logic, AFS/DYC.

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Authors: Mohd Sazli Saad, Hishamuddin Jamaluddin, Intan Zaurah Mat Darus

Abstract:

This paper presents the development of an active vibration control using direct adaptive controller to suppress the vibration of a flexible beam system. The controller is realized based on linear parametric form. Differential evolution optimisation algorithm is used to optimize the controller using single objective function by minimizing the mean square error of the observed vibration signal. Furthermore, an alternative approach is developed to systematically search for the best controller model structure together with it parameter values. The performance of the control scheme is presented and analysed in both time and frequency domain. Simulation results demonstrate that the proposed scheme is able to suppress the unwanted vibration effectively.

Keywords: flexible beam, finite difference method, active vibration control, differential evolution, direct adaptive controller

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Authors: Abdellatif Kasbi, Abderrafii Rahali

Abstract:

This paper appraises the performances of two control scenarios, for doubly fed induction generator (DFIG) operating in wind generation system (WGS), which are the direct decoupled control (DDC) and indirect decoupled control (IDC). Both control scenarios studied combines vector control and Maximum Power Point Tracking (MPPT) control theory so as to maximize the captured power through wind turbine. Modeling of DFIG based WGS and details of both control scenarios have been presented, a proportional integral controller is employed in the active and reactive power control loops for both control methods. The performance of the both control scenarios in terms of power reference tracking and robustness against machine parameters inconstancy has been shown, analyzed and compared, which can afford a reference to the operators and engineers of a wind farm. All simulations have been implemented via MATLAB/Simulink.

Keywords: DFIG, WGS, DDC, IDC, vector control, MPPT.

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Authors: M. Naserian, M. Jooshaki, M. Fotuhi-Firuzabad, M. Hossein Mohammadi Sanjani, A. Oraee

Abstract:

In recent years, the development of communication infrastructure and smart meters have facilitated the utilization of demand-side resources which can enhance stability and economic efficiency of power systems. Direct load control programs can play an important role in the utilization of demand-side resources in the residential sector. However, investments required for installing control equipment can be a limiting factor in the development of such demand response programs. Thus, selection of consumers with higher potentials is crucial to the success of a direct load control program. Heating, ventilation, and air conditioning (HVAC) systems, which due to the heat capacity of buildings feature relatively high flexibility, make up a major part of household consumption. Considering that the consumption of HVAC systems depends highly on the ambient temperature and bearing in mind the high investments required for control systems enabling direct load control demand response programs, in this paper, a solution is presented to uncover consumers with high air conditioner demand among a large number of consumers and to measure the demand response potential of such consumers. This can pave the way for estimating the investments needed for the implementation of direct load control programs for residential HVAC systems and for estimating the demand response potentials in a distribution system. In doing so, we first cluster consumers into several groups based on the correlation coefficients between hourly consumption data and hourly temperature data using K-means algorithm. Then, by applying a recent algorithm to the hourly consumption and temperature data, consumers with high air conditioner consumption are identified. Finally, demand response potential of such consumers is estimated based on the equivalent desired temperature setpoint changes.

Keywords: Data-driven analysis, demand response, direct load control, HVAC system.

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Authors: Amin Zabihinejad, Philippe Viarouge

Abstract:

In this paper, a direct power control (DPC) strategies have been investigated in order to control a high power AC/DC converter with time variable load. This converter is composed of a three level three phase neutral point clamped (NPC) converter as rectifier and an H-bridge four quadrant current control converter. In the high power application, controller not only must adjust the desire outputs but also decrease the level of distortions which are injected to the network from the converter. Regarding to this reason and nonlinearity of the power electronic converter, the conventional controllers cannot achieve appropriate responses. In this research, the precise mathematical analysis has been employed to design the appropriate controller in order to control the time variable load. A DPC controller has been proposed and simulated using Matlab/ Simulink. In order to verify the simulation result, a real time simulator- OPAL-RT- has been employed. In this paper, the dynamic response and stability of the high power NPC with variable load has been investigated and compared with conventional types using a real time simulator. The results proved that the DPC controller is more stable and has more precise outputs in comparison with conventional controller.

Keywords: Direct Power Control, Three Level Rectifier, Real Time Simulator, High Power Application.

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Authors: A. Ourici

Abstract:

This paper proposes a direct power control for doubly-fed induction machine for variable speed wind power generation. It provides decoupled regulation of the primary side active and reactive power and it is suitable for both electric energy generation and drive applications. In order to control the power flowing between the stator of the DFIG and the network, a decoupled control of active and reactive power is synthesized using PI controllers.The obtained simulation results show the feasibility and the effectiveness of the suggested method

Keywords: Doubly fed induction machine , decoupled power control , vector control , active and reactive power, PWM inverter

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Authors: Naim Suleyman Ting, Yakup Sahin, Ismail Aksoy

Abstract:

This paper presents a zero voltage transition-zero current transition (ZVT-ZCT)-PWM DC-DC boost converter with direct power transfer. In this converter, the main switch turns on with ZVT and turns off with ZCT. The auxiliary switch turns on and off with zero current switching (ZCS). The main diode turns on with ZVS and turns off with ZCS. Besides, the additional current or voltage stress does not occur on the main device. The converter has features as simple structure, fast dynamic response and easy control. Also, the proposed converter has direct power transfer feature as well as excellent soft switching techniques. In this study, the operating principle of the converter is presented and its operation is verified for 1 kW and 100 kHz model.

Keywords: Direct power transfer, boost converter, zero-voltage transition, zero-current transition.

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Authors: P. Subha Karuvelam, M. Rajaram

Abstract:

Brushless DC motors (BLDC) are widely used in industrial areas. The BLDC motors are driven either by indirect ACAC converters or by direct AC-AC converters. Direct AC-AC converters i.e. matrix converters are used in this paper to drive the three phase BLDC motor and it eliminates the bulky DC link energy storage element. A matrix converter converts the AC power supply to an AC voltage of variable amplitude and variable frequency. A control technique is designed to generate the switching pulses for the three phase matrix converter. For the control of speed of the BLDC motor a separate PI controller and Fuzzy Logic Controller (FLC) are designed and a hysteresis current controller is also designed for the control of motor torque. The control schemes are designed and tested separately. The simulation results of both the schemes are compared and contrasted in this paper. The results show that the fuzzy logic control scheme outperforms the PI control scheme in terms of dynamic performance of the BLDC motor. Simulation results are validated with the experimental results.

Keywords: Fuzzy logic controller, matrix converter, permanent magnet brushless DC motor, PI controller.

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Authors: J. Soltani, H. Abootorabi Zarchi, Gh. R. Arab Markadeh

Abstract:

In this paper a sliding-mode torque and flux control is designed for encoderless synchronous reluctance motor drive. The sliding-mode plus PI controllers are designed in the stator-flux field oriented reference frame which is able to track the mentioned reference signals with a minimum pulsations in the state condition. In addition, with these controllers a fast dynamic response is also achieved for the drive system. The proposed control scheme is robust subject to parameters variation except to stator resistance. To solve this problem a simple estimator is used for on-line detecting of this parameter. Moreover, the rotor position and speed are estimated by on-line obtaining of the stator-flux-space vector. The effectiveness and capability of the proposed control approach is verified by both the simulation and experimental results.

Keywords: Synchronous Reluctance Motor, Direct Torque and Flux Control, Sliding Mode, Field-Oriented Frame, Encoderless.

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Authors: Jagadish H. Pujar, S. F. Kodad

Abstract:

Recent developments in Soft computing techniques, power electronic switches and low-cost computational hardware have made it possible to design and implement sophisticated control strategies for sensorless speed control of AC motor drives. Such an attempt has been made in this work, for Sensorless Speed Control of Induction Motor (IM) by means of Direct Torque Fuzzy Control (DTFC), PI-type fuzzy speed regulator and MRAS speed estimator strategy, which is absolutely nonlinear in its nature. Direct torque control is known to produce quick and robust response in AC drive system. However, during steady state, torque, flux and current ripple occurs. So, the performance of conventional DTC with PI speed regulator can be improved by implementing fuzzy logic techniques. Certain important issues in design including the space vector modulated (SVM) 3-Ф voltage source inverter, DTFC design, generation of reference torque using PI-type fuzzy speed regulator and sensor less speed estimator have been resolved. The proposed scheme is validated through extensive numerical simulations on MATLAB. The simulated results indicate the sensor less speed control of IM with DTFC and PI-type fuzzy speed regulator provides satisfactory high dynamic and static performance compare to conventional DTC with PI speed regulator.

Keywords: Sensor-less Speed Estimator, Fuzzy Logic Control(FLC), SVM, DTC, DTFC, IM, fuzzy speed regulator.

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Authors: P.-W. Tsai, C.-Y. Chen, C.-W. Chen

Abstract:

In this paper, the stability analysis of a GA-Based adaptive fuzzy sliding model controller for a nonlinear system is discussed. First, a nonlinear plant is well-approximated and described with a reference model and a fuzzy model, both involving FLC rules. Then, FLC rules and the consequent parameter are decided on via an Evolved Bat Algorithm (EBA). After this, we guarantee a new tracking performance inequality for the control system. The tracking problem is characterized to solve an eigenvalue problem (EVP). Next, an adaptive fuzzy sliding model controller (AFSMC) is proposed to stabilize the system so as to achieve good control performance. Lyapunov’s direct method can be used to ensure the stability of the nonlinear system. It is shown that the stability analysis can reduce nonlinear systems into a linear matrix inequality (LMI) problem. Finally, a numerical simulation is provided to demonstrate the control methodology.

Keywords: Adaptive fuzzy sliding mode control, Lyapunov direct method, swarm intelligence, evolved bat algorithm.

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Authors: K. Kouzi

Abstract:

In order to have stable and high performance of direct torque and flux control (DTFC) of double star induction motor drive (DSIM), proper on-line adaptation of the stator resistance is very important. This is inevitably due to the variation of the stator resistance during operating conditions, which introduces error in estimated flux position and the magnitude of the stator flux. Error in the estimated stator flux deteriorates the performance of the DTFC drive. Also, the effect of error in estimation is very important especially at low speed. Due to this, our aim is to overcome the sensitivity of the DTFC to the stator resistance variation by proposing on-line fuzzy estimation stator resistance. The fuzzy estimation method is based on an on-line stator resistance correction through the variations of the stator current estimation error and its variations. The fuzzy logic controller gives the future stator resistance increment at the output. The main advantage of the suggested algorithm control is to avoid the drive instability that may occur in certain situations and ensure the tracking of the actual stator resistance. The validity of the technique and the improvement of the whole system performance are proved by the results.

Keywords: Direct torque control, dual stator induction motor, fuzzy logic estimation, stator resistance adaptation.

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Authors: Krishna S. Raghuwaiya, Shonal Singh, Jito Vanualailai

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In this paper, we study the formation control problem for car-like mobile robots. A team of nonholonomic mobile robots navigate in a terrain with obstacles, while maintaining a desired formation, using a leader-following strategy. A set of artificial potential field functions is proposed using the direct Lyapunov method for the avoidance of obstacles and attraction to their designated targets. The effectiveness of the proposed control laws to verify the feasibility of the model is demonstrated through computer simulations

Keywords: Control, Formation, Lyapunov, Nonholonomic

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Authors: Tawiwat Veeraklaew, Nathasit Phathana-im, Songkit Heama

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Both the minimum energy consumption and smoothness, which is quantified as a function of jerk, are generally needed in many dynamic systems such as the automobile and the pick-and-place robot manipulator that handles fragile equipments. Nevertheless, many researchers come up with either solely concerning on the minimum energy consumption or minimum jerk trajectory. This research paper proposes a simple yet very interesting relationship between the minimum direct and indirect jerks approaches in designing the time-dependent system yielding an alternative optimal solution. Extremal solutions for the cost functions of direct and indirect jerks are found using the dynamic optimization methods together with the numerical approximation. This is to allow us to simulate and compare visually and statistically the time history of control inputs employed by minimum direct and indirect jerk designs. By considering minimum indirect jerk problem, the numerical solution becomes much easier and yields to the similar results as minimum direct jerk problem.

Keywords: Optimization, Dynamic, Linear Systems, Jerks.

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Authors: Kouamana Bousson, Paulo Machado

Abstract:

The optimization and control problem for 4D trajectories is a subject rarely addressed in literature. In the 4D navigation problem we define waypoints, for each mission, where the arrival time is specified in each of them. One way to design trajectories for achieving this kind of mission is to use the trajectory optimization concepts. To solve a trajectory optimization problem we can use the indirect or direct methods. The indirect methods are based on maximum principle of Pontryagin, on the other hand, in the direct methods it is necessary to transform into a nonlinear programming problem. We propose an approach based on direct methods with a pseudospectral integration scheme built on Chebyshev polynomials.

Keywords: Pseudospectral Methods, Trajectory Optimization, 4DTrajectories

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