This paper deals with a comparative study of circle criterion based nonlinear observer and H∞ observer for induction motor (IM) drive. The advantage of the circle criterion approach for nonlinear observer design is that it directly handles the nonlinearities of the system with less restriction conditions in contrast of the other methods which attempt to eliminate them. However the H∞ observer guaranteed the stability taking into account disturbance and noise attenuation. Linear matrix inequality (LMI) optimization approach is used to compute the gains matrices for the two observers. The simulation results show the superiority of H∞ observer in the sense that it can achieve convergence to the true state, despite the nonlinearity of model and the presence of disturbance.
Nonlinear Modeling and System Identification of a DC Gear Motor with Unknown ...ijtsrd
Modeling and identification of industrial systems is an essential stage in practical control design and applications. The paper presents linear, state space, nonlinear modeling and identification of a DC gear motor with real time experiments. The main aim of this research is to use the concept of modeling and System Identification method for observing the greater accuracy and better fitness system model, and validate it by applying various data sets of the hardware experiment. System Identification deals with the problem of building mathematical models of dynamical systems based on observed data from the systems. The methodology is based on results obtained from the simulation of theoretical concepts, which are then validated by repeating experiments on the motor. It is very important to do this validation because sometimes these theoretical concepts are not able to fully capture the nature of the physical elements, and both results may differ. Proceeding in this way, it can guarantee a greater extent that the results are correct. Htet Htet Shin | Nay Min Tun "Nonlinear Modeling and System Identification of a DC Gear Motor with Unknown Parameters" Published in International Journal of Trend in Scientific Research and Development (ijtsrd), ISSN: 2456-6470, Volume-3 | Issue-5 , August 2019, URL: https://www.ijtsrd.com/papers/ijtsrd26475.pdfPaper URL: https://www.ijtsrd.com/engineering/electronics-and-communication-engineering/26475/nonlinear-modeling-and-system-identification-of-a-dc-gear-motor-with-unknown-parameters/htet-htet-shin
Robust Exponential Stabilization for a Class of Uncertain Systems via a Singl...ijtsrd
In this paper, the robust stabilization for a class of uncertain chaotic or non chaotic systems with single input is investigated. Based on Lyapunov like Theorem with differential and integral inequalities, a simple linear control is developed to realize the global exponential stabilization of such uncertain systems. In addition, the guaranteed exponential convergence rate can be correctly estimated. Finally, some numerical simulations with circuit realization are provided to show the effectiveness of the obtained result. Yeong-Jeu Sun "Robust Exponential Stabilization for a Class of Uncertain Systems via a Single Input Control and its Circuit Implementation" Published in International Journal of Trend in Scientific Research and Development (ijtsrd), ISSN: 2456-6470, Volume-3 | Issue-6 , October 2019, URL: https://www.ijtsrd.com/papers/ijtsrd29322.pdf Paper URL: https://www.ijtsrd.com/engineering/electrical-engineering/29322/robust-exponential-stabilization-for-a-class-of-uncertain-systems-via-a-single-input-control-and-its-circuit-implementation/yeong-jeu-sun
Evaluation the affects of mimo based rayleigh network cascaded with unstable ...eSAT Publishing House
IJRET : International Journal of Research in Engineering and Technology is an international peer reviewed, online journal published by eSAT Publishing House for the enhancement of research in various disciplines of Engineering and Technology. The aim and scope of the journal is to provide an academic medium and an important reference for the advancement and dissemination of research results that support high-level learning, teaching and research in the fields of Engineering and Technology. We bring together Scientists, Academician, Field Engineers, Scholars and Students of related fields of Engineering and Technology
Nonlinear Modeling and System Identification of a DC Gear Motor with Unknown ...ijtsrd
Modeling and identification of industrial systems is an essential stage in practical control design and applications. The paper presents linear, state space, nonlinear modeling and identification of a DC gear motor with real time experiments. The main aim of this research is to use the concept of modeling and System Identification method for observing the greater accuracy and better fitness system model, and validate it by applying various data sets of the hardware experiment. System Identification deals with the problem of building mathematical models of dynamical systems based on observed data from the systems. The methodology is based on results obtained from the simulation of theoretical concepts, which are then validated by repeating experiments on the motor. It is very important to do this validation because sometimes these theoretical concepts are not able to fully capture the nature of the physical elements, and both results may differ. Proceeding in this way, it can guarantee a greater extent that the results are correct. Htet Htet Shin | Nay Min Tun "Nonlinear Modeling and System Identification of a DC Gear Motor with Unknown Parameters" Published in International Journal of Trend in Scientific Research and Development (ijtsrd), ISSN: 2456-6470, Volume-3 | Issue-5 , August 2019, URL: https://www.ijtsrd.com/papers/ijtsrd26475.pdfPaper URL: https://www.ijtsrd.com/engineering/electronics-and-communication-engineering/26475/nonlinear-modeling-and-system-identification-of-a-dc-gear-motor-with-unknown-parameters/htet-htet-shin
Robust Exponential Stabilization for a Class of Uncertain Systems via a Singl...ijtsrd
In this paper, the robust stabilization for a class of uncertain chaotic or non chaotic systems with single input is investigated. Based on Lyapunov like Theorem with differential and integral inequalities, a simple linear control is developed to realize the global exponential stabilization of such uncertain systems. In addition, the guaranteed exponential convergence rate can be correctly estimated. Finally, some numerical simulations with circuit realization are provided to show the effectiveness of the obtained result. Yeong-Jeu Sun "Robust Exponential Stabilization for a Class of Uncertain Systems via a Single Input Control and its Circuit Implementation" Published in International Journal of Trend in Scientific Research and Development (ijtsrd), ISSN: 2456-6470, Volume-3 | Issue-6 , October 2019, URL: https://www.ijtsrd.com/papers/ijtsrd29322.pdf Paper URL: https://www.ijtsrd.com/engineering/electrical-engineering/29322/robust-exponential-stabilization-for-a-class-of-uncertain-systems-via-a-single-input-control-and-its-circuit-implementation/yeong-jeu-sun
Evaluation the affects of mimo based rayleigh network cascaded with unstable ...eSAT Publishing House
IJRET : International Journal of Research in Engineering and Technology is an international peer reviewed, online journal published by eSAT Publishing House for the enhancement of research in various disciplines of Engineering and Technology. The aim and scope of the journal is to provide an academic medium and an important reference for the advancement and dissemination of research results that support high-level learning, teaching and research in the fields of Engineering and Technology. We bring together Scientists, Academician, Field Engineers, Scholars and Students of related fields of Engineering and Technology
Modeling of Dynamical System Piloted by Discrete Subsystem Based on Bond Grap...IJECEIAES
This paper is a contribution to the analysis and modeling of a mechatronic system with dynamic behavior that is controlled by a digital computer. In this paper, a bibliographic research on mechatronic systems is presented by specifying a case study of the Anti-lock Braking System (ABS). Then, a methodology of systemic modeling of the ABS system based on two methods Structured Analysis Design Technique (SADT) and bond graph (BG) is proposed. The model created is validated with three software programs: CarSim, 20 Sim and Simulink.
International Journal of Computational Engineering Research(IJCER) is an intentional online Journal in English monthly publishing journal. This Journal publish original research work that contributes significantly to further the scientific knowledge in engineering and Technology.
A linear prediction based on logarithmic space (theoretically non-linear) is proposed, which is essentially a geometric series. The number of deaths of coal miners decreases year by year in that proportion.
Neural Network-Based Actuator Fault Diagnosis for a Non-Linear Multi-Tank SystemISA Interchange
The paper is devoted to the problem of the robust actuator fault diagnosis of the dynamic non-linear systems. In the proposed method, it is assumed that the diagnosed system can be modelled by the recurrent neural network, which can be transformed into the linear parameter varying form. Such a system description allows developing the designing scheme of the robust unknown input observer within H1 framework for a class of non-linear systems. The proposed approach is designed in such a way that a prescribed disturbance attenuation level is achieved with respect to the actuator fault estimation error, while guaranteeing the convergence of the observer. The application of the robust unknown input observer enables actuator fault estimation, which allows applying the developed approach to the fault tolerant control tasks.
A New Chaotic System with Line of Equilibria: Dynamics, Passive Control and C...IJECEIAES
A new chaotic system with line equilibrium is introduced in this paper. This system consists of five terms with two transcendental nonlinearities and two quadratic nonlinearities. Various tools of dynamical system such as phase portraits, Lyapunov exponents, Kaplan-Yorke dimension, bifurcation diagram and Poincarè map are used. It is interesting that this system has a line of fixed points and can display chaotic attractors. Next, this paper discusses control using passive control method. One example is given to insure the theoretical analysis. Finally, for the new chaotic system, An electronic circuit for realizing the chaotic system has been implemented. The numerical simulation by using MATLAB 2010 and implementation of circuit simulations by using MultiSIM 10.0 have been performed in this study.
A Fault Detection and Classification Method for SC Transmission Line Using Ph...paperpublications3
Abstract: In this paper, fault detection and classification for Series Compensated Line (SCL) using phasor measurement unit is presented. The algorithm presented in this paper uses the PMU synchronized measurements and not depends on the data to be provided by the electricity utility. The compensated line parameters and Thevenin’s equivalent (TE) of the system at SCL terminals are calculated online, using three independent sets of pre-fault phasor measurements. The accuracy of fault location is performed with respect to fault location/position, types of fault, fault angle. The accuracy of the algorithm is simulated in MATLAB for 9-bus transmission system.
Optimized Parameter of Wavelet Neural Network (WNN) using INGArahulmonikasharma
Genetic algorithm has been one of the most popular methods for many challenging optimization problems. It is a critical problem in which the evacuation time is an important issues. The continuous air traffic growth and limits of resources, there is a need for reducing the congestion of the airspace system. The main objective of this work is to automatically adapt the airspace configurations, according to the evolution of traffic Niche genetic algorithm(INGA) was used in reliability optimization of software system. And also the searching performance of the genetic algorithm was improved by the stochastic tournament model. The multi-module complex software system reliability allocation effectively. Genetic algorithm (GA) and FGA are compared though seven benchmark function. It can be applied to a wider range of problem including multi-level problem. The uniform schema crossover operator and the non-uniform mutation in the genetic algorithm.
Analysis of Four-Bar Linkages Model using Regressioninventionjournals
International Journal of Engineering and Science Invention (IJESI) is an international journal intended for professionals and researchers in all fields of computer science and electronics. IJESI publishes research articles and reviews within the whole field Engineering Science and Technology, new teaching methods, assessment, validation and the impact of new technologies and it will continue to provide information on the latest trends and developments in this ever-expanding subject. The publications of papers are selected through double peer reviewed to ensure originality, relevance, and readability. The articles published in our journal can be accessed online.
Event triggered control design of linear networked systems with quantizationsISA Interchange
This paper is concerned with the control design problem of event-triggered networked systems with both state and control input quantizations. Firstly, an innovative delay system model is proposed that describes the network conditions, state and control input quantizations, and event-triggering mechanism in a unified framework. Secondly, based on this model, the criteria for the asymptotical stability analysis and control synthesis of event-triggered networked control systems are established in terms of linear matrix inequalities (LMIs). Simulation results are given to illustrate the effectiveness of the proposed method.
A New Estimate Sliding Mode Fuzzy Controller for Robotic ManipulatorWaqas Tariq
One of the most active research areas in field of robotics is control of robot manipulator because this system has highly nonlinear dynamic parameters and most of dynamic parameters are unknown so design an acceptable controller is the main goal in this work. To solve this challenge position new estimation sliding mode fuzzy controller is introduced and applied to robot manipulator. This controller can solve to most important challenge in classical sliding mode controller in presence of highly uncertainty, namely; chattering phenomenon based on fuzzy estimator and online tuning and equivalent nonlinear dynamic based on estimation. Proposed method has acceptable performance in presence of uncertainty (e.g., overshoot=0%, rise time=0.8 s, steady state error = 1e-9 and RMS error=0.0001632).
Performance analysis of a liquid column in a chemical plant by using mpceSAT Publishing House
IJRET : International Journal of Research in Engineering and Technology is an international peer reviewed, online journal published by eSAT Publishing House for the enhancement of research in various disciplines of Engineering and Technology. The aim and scope of the journal is to provide an academic medium and an important reference for the advancement and dissemination of research results that support high-level learning, teaching and research in the fields of Engineering and Technology. We bring together Scientists, Academician, Field Engineers, Scholars and Students of related fields of Engineering and Technology
Modeling of Dynamical System Piloted by Discrete Subsystem Based on Bond Grap...IJECEIAES
This paper is a contribution to the analysis and modeling of a mechatronic system with dynamic behavior that is controlled by a digital computer. In this paper, a bibliographic research on mechatronic systems is presented by specifying a case study of the Anti-lock Braking System (ABS). Then, a methodology of systemic modeling of the ABS system based on two methods Structured Analysis Design Technique (SADT) and bond graph (BG) is proposed. The model created is validated with three software programs: CarSim, 20 Sim and Simulink.
International Journal of Computational Engineering Research(IJCER) is an intentional online Journal in English monthly publishing journal. This Journal publish original research work that contributes significantly to further the scientific knowledge in engineering and Technology.
A linear prediction based on logarithmic space (theoretically non-linear) is proposed, which is essentially a geometric series. The number of deaths of coal miners decreases year by year in that proportion.
Neural Network-Based Actuator Fault Diagnosis for a Non-Linear Multi-Tank SystemISA Interchange
The paper is devoted to the problem of the robust actuator fault diagnosis of the dynamic non-linear systems. In the proposed method, it is assumed that the diagnosed system can be modelled by the recurrent neural network, which can be transformed into the linear parameter varying form. Such a system description allows developing the designing scheme of the robust unknown input observer within H1 framework for a class of non-linear systems. The proposed approach is designed in such a way that a prescribed disturbance attenuation level is achieved with respect to the actuator fault estimation error, while guaranteeing the convergence of the observer. The application of the robust unknown input observer enables actuator fault estimation, which allows applying the developed approach to the fault tolerant control tasks.
A New Chaotic System with Line of Equilibria: Dynamics, Passive Control and C...IJECEIAES
A new chaotic system with line equilibrium is introduced in this paper. This system consists of five terms with two transcendental nonlinearities and two quadratic nonlinearities. Various tools of dynamical system such as phase portraits, Lyapunov exponents, Kaplan-Yorke dimension, bifurcation diagram and Poincarè map are used. It is interesting that this system has a line of fixed points and can display chaotic attractors. Next, this paper discusses control using passive control method. One example is given to insure the theoretical analysis. Finally, for the new chaotic system, An electronic circuit for realizing the chaotic system has been implemented. The numerical simulation by using MATLAB 2010 and implementation of circuit simulations by using MultiSIM 10.0 have been performed in this study.
A Fault Detection and Classification Method for SC Transmission Line Using Ph...paperpublications3
Abstract: In this paper, fault detection and classification for Series Compensated Line (SCL) using phasor measurement unit is presented. The algorithm presented in this paper uses the PMU synchronized measurements and not depends on the data to be provided by the electricity utility. The compensated line parameters and Thevenin’s equivalent (TE) of the system at SCL terminals are calculated online, using three independent sets of pre-fault phasor measurements. The accuracy of fault location is performed with respect to fault location/position, types of fault, fault angle. The accuracy of the algorithm is simulated in MATLAB for 9-bus transmission system.
Optimized Parameter of Wavelet Neural Network (WNN) using INGArahulmonikasharma
Genetic algorithm has been one of the most popular methods for many challenging optimization problems. It is a critical problem in which the evacuation time is an important issues. The continuous air traffic growth and limits of resources, there is a need for reducing the congestion of the airspace system. The main objective of this work is to automatically adapt the airspace configurations, according to the evolution of traffic Niche genetic algorithm(INGA) was used in reliability optimization of software system. And also the searching performance of the genetic algorithm was improved by the stochastic tournament model. The multi-module complex software system reliability allocation effectively. Genetic algorithm (GA) and FGA are compared though seven benchmark function. It can be applied to a wider range of problem including multi-level problem. The uniform schema crossover operator and the non-uniform mutation in the genetic algorithm.
Analysis of Four-Bar Linkages Model using Regressioninventionjournals
International Journal of Engineering and Science Invention (IJESI) is an international journal intended for professionals and researchers in all fields of computer science and electronics. IJESI publishes research articles and reviews within the whole field Engineering Science and Technology, new teaching methods, assessment, validation and the impact of new technologies and it will continue to provide information on the latest trends and developments in this ever-expanding subject. The publications of papers are selected through double peer reviewed to ensure originality, relevance, and readability. The articles published in our journal can be accessed online.
Event triggered control design of linear networked systems with quantizationsISA Interchange
This paper is concerned with the control design problem of event-triggered networked systems with both state and control input quantizations. Firstly, an innovative delay system model is proposed that describes the network conditions, state and control input quantizations, and event-triggering mechanism in a unified framework. Secondly, based on this model, the criteria for the asymptotical stability analysis and control synthesis of event-triggered networked control systems are established in terms of linear matrix inequalities (LMIs). Simulation results are given to illustrate the effectiveness of the proposed method.
A New Estimate Sliding Mode Fuzzy Controller for Robotic ManipulatorWaqas Tariq
One of the most active research areas in field of robotics is control of robot manipulator because this system has highly nonlinear dynamic parameters and most of dynamic parameters are unknown so design an acceptable controller is the main goal in this work. To solve this challenge position new estimation sliding mode fuzzy controller is introduced and applied to robot manipulator. This controller can solve to most important challenge in classical sliding mode controller in presence of highly uncertainty, namely; chattering phenomenon based on fuzzy estimator and online tuning and equivalent nonlinear dynamic based on estimation. Proposed method has acceptable performance in presence of uncertainty (e.g., overshoot=0%, rise time=0.8 s, steady state error = 1e-9 and RMS error=0.0001632).
Performance analysis of a liquid column in a chemical plant by using mpceSAT Publishing House
IJRET : International Journal of Research in Engineering and Technology is an international peer reviewed, online journal published by eSAT Publishing House for the enhancement of research in various disciplines of Engineering and Technology. The aim and scope of the journal is to provide an academic medium and an important reference for the advancement and dissemination of research results that support high-level learning, teaching and research in the fields of Engineering and Technology. We bring together Scientists, Academician, Field Engineers, Scholars and Students of related fields of Engineering and Technology
Data-driven adaptive predictive control for an activated sludge processjournalBEEI
Data-driven control requires no information of the mathematical model of the controlled process. This paper proposes the direct identification of controller parameters of activated sludge process. This class of data-driven control calculates the predictive controller parameters directly using subspace identification technique. By updating input-output data using receding window mechanism, the adaptive strategy can be achieved. The robustness test and stability analysis of direct adaptive model predictive control are discussed to realize the effectiveness of this adaptive control scheme. The applicability of the controller algorithm to adapt into varying kinetic parameters and operating conditions is evaluated. Simulation results show that by a proper and effective excitation of direct identification of controller parameters, the convergence and stability of the implicit predictive model can be achieved.
Flatness Based Nonlinear Sensorless Control of Induction Motor SystemsIAES-IJPEDS
This paper deals with the flatness-based approach for sensorless control of
the induction motor systems. Two main features of the proposed flatness
based control are worth to be mentioned. Firstly, the simplicity of
implementation of the flatness approach as a nonlinear feedback linearization
control technique. Secondly, when the chosen flat outputs involve non
available state variable measurements a nonlinear observer is used to
estimate them. The main advantage of the used observer is its ability to
exploite the properties of the system nonlinearties. The simulation results are
presented to illustrate the effectiness of the proposed approach for sensorless
control of the considered induction motor.
This paper presents a study concerning a sensorless vector control of an induction machine fed by a voltage source inverter. The aim is to provide a scheme to control the speed and the rotor flux using a sensorless integral backstepping control approach. The rotor speed estimation is done by an observer using the model reference adaptive system (MRAS) technique whereas the nonlinear backstepping observer is used to get the rotor flux. The main objective is to achieve a robust control, adaptive and efficient, which will allow us to test and evaluate the performance of the proposed observer, combined with a sensorless control of the induction machine. Tests and validation are done using numerical simulations with MATLAB/SIMULINK-PSB (Power System Block set) toolbox. The results show good performance in terms of robustness regarding machine parameter variations and show the excellent quality of the control law associated with the observer, despite the observability problems when the machine operates at low speed.
In this paper, Reduced-Order Observer For Real-Time Implementation Speed Sensorless Control of Induction Using RT-LAB Softwareis presented. Speed estimation is performed through a reduced-order observer. The stability of the proposed observer is proved based on Lyapunov’s theorem. The model is initially built offline using Matlab/Simulink and implemented in real-time environment using RT-LAB package and an OP5600 digital simulator. RT-LAB configuration has two main subsystems master and console subsystems. These two subsystems were coordinated to achieve the real-time simulation. In order to verify the feasibility and effectiveness of proposed method, experimental results are presented over a wide speed range, including zero speed.
Design Nonlinear Model Reference with Fuzzy Controller for Nonlinear SISO Sec...IJECEIAES
Model reference controller is considering as one of the most useful controller to specific performance of systems where the desired output is produced for a given input. This system used the difference between the outputs of the plant and the desired model by comparing them to produce the signals of the control. This paper focus on design a model reference controller (MRC) combined with (type-1 and interval type-2) fuzzy control scheme for single input-single output (SISO) systems under uncertainty and external disturbance. The model reference controller is designed firstly without fuzzy scheme based on an optimal desired model and Lyapunov stability theory. Then a (type-1 and Interval type-2) fuzzy controller Takagi-Sugeno type is combine with the suggested MRC in order to enhance the performer of it, the common parts between the two fuzzy systems such as: fuzzifier, inference engine, fuzzy rule-base and defuzzifier are illustrated. In this paper the proposed controller is applied to controla (SISO) inverted pendulum sustem and the Matlab R2015 software is used to carry out two simulation cases for the overall controlled scheme. The obtained results for the two cases show that the proposed MRC with both fuzzy control schemes have acceptable performance, but it have better performance with the interval type-2 fuzzy scheme.
Steady state stability analysis and enhancement of three machine nine bus pow...eSAT Journals
Abstract
Power System stability study is the important parameter of economic, reliable and secure system planning and operation. Studies
are important during the planning and conceptual design stages of the project as well as during the operating life of the plant
periodically. This paper presents the power system steady state stability analysis for IEEE- 9 bus test system and examines
influence of TCPS FACTS device based controller on test system. It is assumed that system under study has been perturbed from a
steady state equilibrium that prevailed prior to the application of the disturbance. If system is stable, we would expect that for
temporary or permanent disturbance, system will acquire initial or new operating state after a transient period. The stability
study is accessed using Lyapunov’s first method. The effectiveness of damping controller in enhancing the steady state stability is
investigated by incorporating available constraints. For analysis MATLAB software is employed. The conclusions have been
drawn here, based on theoretical and mathematical analysis so as to provide an insight and better understanding of steady state
stability of considered multi machine power system.
Key Words: Lyapunov’s first method, Steady-state stability, Phase portrait, FACTS device, supplementary modulation
controller, eigen value, synchronizing power coefficient, IEEE-9 Bus Test System, Load Flow Study, Differential
algebraic equation.
IJRET : International Journal of Research in Engineering and Technology is an international peer reviewed, online journal published by eSAT Publishing House for the enhancement of research in various disciplines of Engineering and Technology. The aim and scope of the journal is to provide an academic medium and an important reference for the advancement and dissemination of research results that support high-level learning, teaching and research in the fields of Engineering and Technology. We bring together Scientists, Academician, Field Engineers, Scholars and Students of related fields of Engineering and Technology
Autotuning of pid controller for robot arm and magnet levitation planteSAT Journals
Abstract
One of the most essential work of the control engineer is tuning of controller. Majority of the controller used in industry are of the
PID type. An auto tuning is one of the method of controller tuning in which tuning of the parameters of controller is done
automatically and possibly, without any user interaction expect from initiating the operation. Present study emphasis on the relay
based auto tuning of PID controller. An auto-tuning method is implemented based on a relay experiment to determine the ultimate
gain and the ultimate period, with which the PID parameters are obtained using the Ziegler-Nichols tuning rules. An auto tuning
of robot arm model and magnet levitation model are carried out. Performance of relay based auto tuning on the basis of integral
square error is better than artificial neural network.
Keywords: Relay auto tuning, PID, FOPDT, SOPDT, Integral square error.
A novel p q control algorithm for combined active front end converter and shu...eSAT Journals
Abstract
This paper presents a combined active front-end converter and shunt active filter (AFE+SAF) controlled by p-q theory control
algorithm. The Combined AFE+SAF is able to compensate reactive power, harmonic power and unbalanced power produced by
unbalanced linear, non linear and at the same time three phase AC to DC power conversion AFE+SAF will not pollute the source
currents. The p-q theory control algorithm is able to meat the target of load compensation and three phase AC to DC (for drives)
conversion. the AFE+SAF verified by simulation for different cases as unbalanced linear load, non linear load, unbalanced linear
and non linear load, unbalanced linear non linear and drive load. The whole simulation has been carried out on the mat lab/ simulink
software.
Keywords:- active front-end converter, shunt active power filter, p-q theory, power quality, harmonics, reactive power
Feedback linearization and Backstepping controllers for Coupled Tanksieijjournal
This paper investigates the usage of some sophisticated and advanced nonlinear control algorithms inorder
to control a nonlinear Coupled Tanks System. The first control procedure is called the
Feedbacklinearisation control (FLC), this type of control has been found a successful in achieving a global
exponentialasymptotic stability, with very short time response, no significant overshooting is recordedand with a negligible norm of the error. The second control procedure is the approaches of Backsteppingcontrol (BC) which is a recursive procedure that interlaces the choice of a Lyapunov functionwith the design of feedback control, from simulation results it shown that this method preserves tracking, robust control and it can often solve stabilization problems with less restrictive conditions may beencountered in other methods. Finally both of the proposed control schemes guarantee theasymptoticstability of the closed loop system meeting trajectory tracking objectives
FEEDBACK LINEARIZATION AND BACKSTEPPING CONTROLLERS FOR COUPLED TANKSieijjournal
This paper investigates the usage of some sophisticated and advanced nonlinear control algorithms inorder to control a nonlinear Coupled Tanks System. The first control procedure is called the Feedbacklinearisation control (FLC), this type of control has been found a successful in achieving a global exponentialasymptotic stability, with very short time response, no significant overshooting is recordedand with a negligible norm of the error. The second control procedure is the approaches of Backsteppingcontrol (BC) which is a recursive procedure that interlaces the choice of a Lyapunov functionwith the design of feedback control, from simulation results it shown that this method preserves tracking, robust control and it can often solve stabilization problems with less restrictive conditions may beencountered in other methods. Finally both of the proposed control schemes guarantee the asymptoticstability of the closed loop system meeting trajectory tracking objectives.
FEEDBACK LINEARIZATION AND BACKSTEPPING CONTROLLERS FOR COUPLED TANKSieijjournal
This paper investigates the usage of some sophisticated and advanced nonlinear control algorithms in order to control a nonlinear Coupled Tanks System. The first control procedure is called the Feedback linearisation control (FLC), this type of control has been found a successful in achieving a global exponential asymptotic stability, with very short time response, no significant overshooting is recorded and with a negligible norm of the error. The second control procedure is the approaches of Back stepping control (BC) which is a recursive procedure that interlaces the choice of a Lyapunov function with the design of feedback control, from simulation results it shown that this method preserves tracking, robust control and it can often solve stabilization problems with less restrictive conditions may been countered in other methods. Finally both of the proposed control schemes guarantee the asymptoticstability of the closed loop system meeting trajectory tracking objectives.
Modeling control of automatic voltage regulator with proportional integral de...eSAT Journals
Abstract The Automatic Voltage Regulator (AVR) cannot respond generator output voltage quickly resulting in drop voltage and can be affected to the electrical system. AVR controller simulation model created in power system based on real data from National Electricity Company (PLN) ( Tello Power Plant Makassar case). AVR controller response time under normal conditions has the oscillation time of about 9 seconds and response time when loaded is about 15 seconds. Resulting unstable voltage in the system follows load changes. With the addition of Proportional Integral Derivative (PID) controllers acquired a controlling response time to changing loads faster and system voltage will change the voltage stability limit of ± 5% of nominal voltage. AVR controller with PID is very suitable for controlling the voltage of power generation system due to changes in load. Key Words: Automatic Voltage Regulator, Proportional Integral Derivative
Similar to A comparative study of nonlinear circle criterion based observer and H∞ observer for induction motor drive (20)
The aim of this research is the speed tracking of the permanent magnet synchronous motor (PMSM) using an intelligent Neural-Network based adapative backstepping control. First, the model of PMSM in the Park synchronous frame is derived. Then, the PMSM speed regulation is investigated using the classical method utilizing the field oriented control theory. Thereafter, a robust nonlinear controller employing an adaptive backstepping strategy is investigated in order to achieve a good performance tracking objective under motor parameters changing and external load torque application. In the final step, a neural network estimator is integrated with the adaptive controller to estimate the motor parameters values and the load disturbance value for enhancing the effectiveness of the adaptive backstepping controller. The robsutness of the presented control algorithm is demonstrated using simulation tests. The obtained results clearly demonstrate that the presented NN-adaptive control algorithm can provide good trackingperformances for the speed trackingin the presence of motor parameter variation and load application.
This paper presents a fast and accurate fault detection, classification and direction discrimination algorithm of transmission lines using one-dimensional convolutional neural networks (1D-CNNs) that have ingrained adaptive model to avoid the feature extraction difficulties and fault classification into one learning algorithm. A proposed algorithm is directly usable with raw data and this deletes the need of a discrete feature extraction method resulting in more effective protective system. The proposed approach based on the three-phase voltages and currents signals of one end at the relay location in the transmission line system are taken as input to the proposed 1D-CNN algorithm. A 132kV power transmission line is simulated by Matlab simulink to prepare the training and testing data for the proposed 1D- CNN algorithm. The testing accuracy of the proposed algorithm is compared with other two conventional methods which are neural network and fuzzy neural network. The results of test explain that the new proposed detection system is efficient and fast for classifying and direction discrimination of fault in transmission line with high accuracy as compared with other conventional methods under various conditions of faults.
Among the most widespread renewable energy sources is solar energy; Solar panels offer a green, clean, and environmentally friendly source of energy. In the presence of several advantages of the use of photovoltaic systems, the random operation of the photovoltaic generator presents a great challenge, in the presence of a critical load. Among the most used solutions to overcome this problem is the combination of solar panels with generators or with the public grid or both. In this paper, an energy management strategy is proposed with a safety aspect by using artificial neural networks (ANNs), in order to ensure a continuous supply of electricity to consumers with a maximum solicitation of renewable energy.
In this paper, the artificial neural network (ANN) has been utilized for rotating machinery faults detection and classification. First, experiments were performed to measure the lateral vibration signals of laboratory test rigs for rotor-disk-blade when the blades are defective. A rotor-disk-blade system with 6 regular blades and 5 blades with various defects was constructed. Second, the ANN was applied to classify the different x- and y-axis lateral vibrations due to different blade faults. The results based on training and testing with different data samples of the fault types indicate that the ANN is robust and can effectively identify and distinguish different blade faults caused by lateral vibrations in a rotor. As compared to the literature, the present paper presents a novel work of identifying and classifying various rotating blade faults commonly encountered in rotating machines using ANN. Experimental data of lateral vibrations of the rotor-disk-blade system in both x- and y-directions are used for the training and testing of the network.
This paper focuses on the artificial bee colony (ABC) algorithm, which is a nonlinear optimization problem. is proposed to find the optimal power flow (OPF). To solve this problem, we will apply the ABC algorithm to a power system incorporating wind power. The proposed approach is applied on a standard IEEE-30 system with wind farms located on different buses and with different penetration levels to show the impact of wind farms on the system in order to obtain the optimal settings of control variables of the OPF problem. Based on technical results obtained, the ABC algorithm is shown to achieve a lower cost and losses than the other methods applied, while incorporating wind power into the system, high performance would be gained.
The significance of the solar energy is to intensify the effectiveness of the Solar Panel with the use of a primordial solar tracking system. Here we propounded a solar positioning system with the use of the global positioning system (GPS) , artificial neural network (ANN) and image processing (IP) . The azimuth angle of the sun is evaluated using GPS which provide latitude, date, longitude and time. The image processing used to find sun image through which centroid of sun is calculated and finally by comparing the centroid of sun with GPS quadrate to achieve optimum tracking point. Weather conditions and situation observed through AI decision making with the help of IP algorithms. The presented advance adaptation is analyzed and established via experimental effects which might be made available on the memory of the cloud carrier for systematization. The proposed system improve power gain by 59.21% and 10.32% compare to stable system (SS) and two-axis solar following system (TASF) respectively. The reduced tracking error of IoT based Two-axis solar following system (IoT-TASF) reduces their azimuth angle error by 0.20 degree.
Kosovo has limited renewable energy resources and its power generation sector is based on fossil fuels. Such a situation emphasizes the importance of active research and efficient use of renewable energy potential. According to the analysis of meteorological data for Kosovo, it can be concluded that among the most attractive potential wind power sites are the locations known as Kitka (42° 29' 41" N and 21° 36' 45" E) and Koznica (42° 39′ 32″ N, 21° 22′30″E). The two terrains in which the analysis was carried out are mountain areas, with altitudes of 1142 m (Kitka) and 1230 m (Koznica). the same measuring height, about 84 m above the ground, is obtained for these average wind speeds: Kitka 6,667 m/s and Koznica 6,16 m/s. Since the difference in wind speed is quite large versus a difference in altitude that is not being very large, analyses are made regarding the terrain characteristics including the terrain relief features. In this paper it will be studied how much the roughness of the terrain influences the output energy. Also, that the assumption to be taken the same as to how much they will affect the annual energy produced.
Large-scale grid-tied photovoltaic (PV) station are increasing rapidly. However, this large penetration of PV system creates frequency fluctuation in the grid due to the intermittency of solar irradiance. Therefore, in this paper, a robust droop control mechanism of the battery energy storage system (BESS) is developed in order to damp the frequency fluctuation of the multi-machine grid system due to variable active power injected from the PV panel. The proposed droop control strategy incorporates frequency error signal and dead-band for effective minimization of frequency fluctuation. The BESS system is used to consume/inject an effective amount of active power based upon the frequency oscillation of the grid system. The simulation analysis is carried out using PSCAD/EMTDC software to prove the effectiveness of the proposed droop control-based BESS system. The simulation result implies that the proposed scheme can efficiently curtail the frequency oscillation.
This study investigates experimentally the performance of two-dimensional solar tracking systems with reflector using commercial silicon based photovoltaic module, with open and closed loop control systems. Different reflector materials were also investigated. The experiments were performed at the Hashemite University campus in Zarqa at a latitude of 32⁰, in February and March. Photovoltaic output power and performance were analyzed. It was found that the modified photovoltaic module with mirror reflector generated the highest value of power, while the temperature reached a maximum value of 53 ̊ C. The modified module suggested in this study produced 5% more PV power than the two-dimensional solar tracking systems without reflector and produced 12.5% more PV power than the fixed PV module with 26⁰ tilt angle.
This paper focuses on the modeling and control of a wind energy conversion chain using a permanent magnet synchronous machine. This system behaves a turbine, a generator, DC/DC and DC/AC power converters. These are connected on both sides to the DC bus, where the inverter is followed by a filter which is connected to the grid. In this paper, we have been used two types of controllers. For the stator side converter, we consider the Takagi-Sugeno approach where the parameters of controller have been computed by the theory of linear matrix inequalities. The stability synthesis has been checked using the Lyapunov theory. According to the grid side converter, the proportional integral controller is exploited to keep a constant voltage on the DC bus and control both types of powers. The simulation results demonstrate the robustness of the approach used.
The development of modeling wind speed plays a very important in helping to obtain the actual wind speed data for the benefit of the power plant planning in the future. The wind speed in this paper is obtained from a PCE-FWS 20 type measuring instrument with a duration of 30 minutes which is accumulated into monthly data for one year (2019). Despite the many wind speed modeling that has been done by researchers. Modeling wind speeds proposed in this study were obtained from the modified Rayleigh distribution. In this study, the Rayleigh scale factor (Cr) and modified Rayleigh scale factor (Cm) were calculated. The observed wind speed is compared with the predicted wind characteristics. The data fit test used correlation coefficient (R2), root means square error (RMSE), and mean absolute percentage error (MAPE). The results of the proposed modified Rayleigh model provide very good results for users.
This paper deals with an advanced design for a pump powered by solar energyto supply agricultural lands with water and also the maximum power point is used to extract the maximum value of the energy available inside the solar panels and comparing between techniques MPPT such as Incremental conductance, perturb & observe, fractional short current circuit, and fractional open voltage circuit to find the best technique among these. The solar system is designed with main parts: photovoltaic (PV) panel, direct current/direct current (DC/DC) converter, inverter, filter, and in addition, the battery is used to save energy in the event that there is an increased demand for energy and not to provide solar radiation, as well as saving energy in the case of generation more than demand. This work was done using the matrix laboratory (MATLAB) simulink program.
The objective of this paper is to provide an overview of the current state of renewable energy resources in Bangladesh, as well as to examine various forms of renewable energies in order to gain a comprehensive understanding of how to address Bangladesh's power crisis issues in a sustainable manner. Electricity is currently the most useful kind of energy in Bangladesh. It has a substantial influence on a country's socioeconomic standing and living standards. Maintaining a stable source of energy at a cost that is affordable to everyone has been a constant battle for decades. Bangladesh is blessed with a wealth of natural resources. Bangladesh has a huge opportunity to accelerate its economic development while increasing energy access, livelihoods, and health for millions of people in a sustainable way due to the renewable energy system.
When the irradiance distribution over the photovoltaic panels is uniform, the pursuit of the maximum power point is not reached, which has allowed several researchers to use traditional MPPT techniques to solve this problem Among these techniques a PSO algorithm is used to have the maximum global power point (GMPPT) under partial shading. On the other hand, this one is not reliable vis-à-vis the pursuit of the MPPT. Therefore, in this paper we have treated another technique based on a new modified PSO algorithm so that the power can reach its maximum point. The PSO algorithm is based on the heuristic method which guarantees not only the obtaining of MPPT but also the simplicity of control and less expensive of the system. The results are obtained using MATLAB show that the proposed modified PSO algorithm performs better than conventional PSO and is robust to different partial shading models.
A stable operation of wind turbines connected to the grid is an essential requirement to ensure the reliability and stability of the power system. To achieve such operational objective, installing static synchronous compensator static synchronous compensator (STATCOM) as a main compensation device guarantees the voltage stability enhancement of the wind farm connected to distribution network at different operating scenarios. STATCOM either supplies or absorbs reactive power in order to ensure the voltage profile within the standard-margins and to avoid turbine tripping, accordingly. This paper present new study that investigates the most suitable-location to install STATCOM in a distribution system connected wind farm to maintain the voltage-levels within the stability margins. For a large-scale squirrel cage induction generator squirrel-cage induction generator (SCIG-based) wind turbine system, the impact of STATCOM installation was tested in different places and voltage-levels in the distribution system. The proposed method effectiveness in enhancing the voltage profile and balancing the reactive power is validated, the results were repeated for different scenarios of expected contingencies. The voltage profile, power flow, and reactive power balance of the distribution system are observed using MATLAB/Simulink software.
The electrical and environmental parameters of polymer solar cells (PSC) provide important information on their performance. In the present article we study the influence of temperature on the voltage-current (I-V) characteristic at different temperatures from 10 °C to 90 °C, and important parameters like bandgap energy Eg, and the energy conversion efficiency η. The one-diode electrical model, normally used for semiconductor cells, has been tested and validated for the polemeral junction. The PSC used in our study are formed by the poly(3-hexylthiophene) (P3HT) and [6,6]-phenyl C61-butyric acid methyl ester (PCBM). Our technique is based on the combination of two steps; the first use the Least Mean Squares (LMS) method while the second use the Newton-Raphson algorithm. The found results are compared to other recently published works, they show that the developed approach is very accurate. This precision is proved by the minimal values of statistical errors (RMSE) and the good agreement between both the experimental data and the I-V simulated curves. The obtained results show a clear and a monotonic dependence of the cell efficiency on the studied parameters.
The inverter is the principal part of the photovoltaic (PV) systems that assures the direct current/alternating current (DC/AC) conversion (PV array is connected directly to an inverter that converts the DC energy produced by the PV array into AC energy that is directly connected to the electric utility). In this paper, we present a simple method for detecting faults that occurred during the operation of the inverter. These types of faults or faults affect the efficiency and cost-effectiveness of the photovoltaic system, especially the inverter, which is the main component responsible for the conversion. Hence, we have shown first the faults obtained in the case of the short circuit. Second, the open circuit failure is studied. The results demonstrate the efficacy of the proposed method. Good monitoring and detection of faults in the inverter can increase the system's reliability and decrease the undesirable faults that appeared in the PV system. The system behavior is tested under variable parameters and conditions using MATLAB/Simulink.
The electrical distribution network is undergoing tremendous modifications with the introduction of distributed generation technologies which have led to an increase in fault current levels in the distribution network. Fault current limiters have been developed as a promising technology to limit fault current levels in power systems. Though, quite a number of fault current limiters have been developed; the most common are the superconducting fault current limiters, solid-state fault current limiters, and saturated core fault current limiters. These fault current limiters present potential fault current limiting solutions in power systems. Nevertheless, they encounter various challenges hindering their deployment and commercialization. This research aimed at designing a bridge-type nonsuperconducting fault current limiter with a novel topology for distribution network applications. The proposed bridge-type nonsuperconducting fault current limiter was designed and simulated using PSCAD/EMTDC. Simulation results showed the effectiveness of the proposed design in fault current limiting, voltage sag compensation during fault conditions, and its ability not to affect the load voltage and current during normal conditions as well as in suppressing the source powers during fault conditions. Simulation results also showed very minimal power loss by the fault current limiter during normal conditions.
This paper provides a new approach to reducing high-order harmonics in 400 Hz inverter using a three-level neutral-point clamped (NPC) converter. A voltage control loop using the harmonic compensation combined with NPC clamping diode control technology. The capacitor voltage imbalance also causes harmonics in the output voltage. For 400 Hz inverter, maintain a balanced voltage between the two input (direct current) (DC) capacitors is difficult because the pulse width modulation (PWM) modulation frequency ratio is low compared to the frequency of the output voltage. A method of determining the current flowing into the capacitor to control the voltage on the two balanced capacitors to ensure fast response reversal is also given in this paper. The combination of a high-harmonic resonator controller and a neutral-point voltage controller working together on the 400 Hz NPC inverter structure is given in this paper.
Direct current (DC) electronic load is a useful equipment for testing the electrical system. It can emulate various load at a high rating. The electronic load requires a power converter to operate and a linear regulator is a common option. Nonetheless, it is hard to control due to the temperature variation. This paper proposed a DC electronic load using the boost converter. The proposed electronic load operates in the continuous current mode and control using the integral controller. The electronic load using the boost converter is compared with the electronic load using the linear regulator. The results show that the boost converter able to operate as an electronic load with an error lower than 0.5% and response time lower than 13 ms.
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Buying new cosmetic products is difficult. It can even be scary for those who have sensitive skin and are prone to skin trouble. The information needed to alleviate this problem is on the back of each product, but it's thought to interpret those ingredient lists unless you have a background in chemistry.
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In today’s fast-changing business environment, it’s extremely important to be able to respond to client needs in the most effective and timely manner. If your customers wish to see your business online and have instant access to your products or services.
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robustness against nonlinearities assumed to be equivalent to some uncertainties due to internal and extrernal
disturbances and measurement noises [7-12].
To make a comparative study between the circle criterion based observer and H∞ observer design for
induction motor derive, we use one of the standard model of an induction motor. This type of nonlinear
model is generally used for performing nonlinear control strategies, conditions monitoring and faults
diagnosis of electric induction machine systems.
The paper is organized as follows: In the second and third section we present the theory of nonlinear
observer and the nonlinear observer based circle criterion respectively. The robust H∞ observer is presented
in the fourth section. In the fifth section we present the considered nonlinear induction motor model. Finlay,
we present simulation results and comments. A conclusion ends the paper.
2. THE NONLINEAR LUENBERGER OBSERVER DESIGN
We recall that an observer is a dynamical system which uses the available input-output data to
reconstruct the unmeasurable system state variables. It is a “soft sensor” that plays an important role not only
in sensorless control techniques but also in conditions monitoring, fault diagnosis, predictive maintenance
and fault tolerant control techniques [1, 2].
In this paper, we consider a class of nonlinear systems that can be decomposed in linear and
nonlinear parts as:
))
(
),
(
(
)
(
)
( t
u
t
x
F
t
Ax
t
x
(1)
)
(
)
( t
Cx
t
y (2)
Where )
(t
x stand for the system state variable, )
(t
u for system input and )
(t
y for the system output.
The function ))
(
),
(
( t
u
t
x
F
represents the system nonlinearities. For this type of nonlinear system, a general
nonlinear observer (Luenberger) expression is as follows:
))
(
ˆ
)
(
(
)
,
ˆ
(
)
(
ˆ
)
(
ˆ t
y
t
y
L
u
x
F
t
x
A
t
x
(3)
)
(
ˆ
)
(
ˆ t
x
C
t
y (4)
Where )
(
ˆ t
x
stands for the estimated state. We assume that the nonlinear function ))
(
),
(
( t
u
t
x
F
is
locally Lipschitz with respect to the state variable )
(t
x
.
)
(
ˆ
)
(
))
(
),
(
ˆ
(
))
(
),
(
(
/
),
(
ˆ
),
( t
x
t
x
t
u
t
x
F
t
u
t
x
F
t
x
t
x
With 0
as the Lipschitz constant;
Theorem 1 [13]: Consider Lipschitz nonlinear system (1)-(2) along with the observer (3)-(4). The observer
error dynamics are asymptotically stable with maximum admissible Lipschitz constant if there exist scalers
0
and 0
and matrices 0
P and G such that the following LMI optimization problem has a solution
)
min( .
0
2
)
(
)
(
P
P
P
LC
A
P
P
LC
A T
(5)
0
2
1
2
1
I
P
P
I
(6)
With
1
*
)
max(
ˆ
3. Int J Pow Elec & Dri Syst ISSN: 2088-8694
A comparative study of nonlinear circle criterion based observer and H∞ observer….. (Farid Berrezzek)
1231
Proof: Defining the observer error as )
(
ˆ
)
(
)
( t
x
t
x
t
e
, then the observer error dynamics are given by:
))
(
),
(
ˆ
(
))
(
),
(
(
)
(
)
( t
u
t
x
F
t
u
t
x
F
e
LC
A
t
e
(7)
In the following we use notations:
))
(
),
(
( t
u
t
x
F
F and ))
(
),
(
ˆ
(
ˆ t
u
t
x
F
F
The time derivative of the Lyapunov function Pe
e
V T
can be expressed as:
)
ˆ
(
2
)]
(
)
[(
)
( F
F
P
e
e
LC
A
P
P
LC
A
e
t
V T
T
T
(8)
By using the Lipschitz property we have:
e
P
e
F
F
P
e
T
T
2
)
ˆ
(
2
(9)
Then equation (8) becomes:
0
]
2
)
(
)
[(
)
(
e
P
e
LC
A
P
P
LC
A
e
t
V T
T
T
(10)
Those leads to:
0
2
)
(
)
( 1
P
PP
LC
A
P
P
LC
A T
(11)
By using the Schur complement theorem we therefore obtain the following linear matrix inequality (LMI):
0
2
)
(
)
(
P
P
P
LC
A
P
P
LC
A T
(12)
The Lipschitz constant can be computed as follows:
For any Lyapunov function Pe
e
V T
we have:
2
max
2
min )
(
)
( e
P
Pe
e
e
P T
(13)
2
min )
(
2
2 e
P
Pe
eT
(14)
Let: Q
LC
A
P
P
LC
A T
)
(
)
( (15)
Then the time derivative can be written as:
0
2
)
(
2
e
P
Qe
e
t
V T
(16)
0
)
(
2
)
(
)
(
2
max
2
min
e
P
e
Q
t
V
(17)
So:
)
(
)
(
2 min
max Q
P
)
(
)
(
2
1
max
min
P
Q
(18)
In addition, since P is positive definite, suppose
I
Q
.From the equation (18) we can write:
4. ISSN: 2088-8694
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1232
0
)
(
2
1 max
P
(19)
And
)
(
)
( max P
P
So from (19):
2
1
)
(
P
(20)
Which is equivalent to:
0
)
2
1
( 2
P
P
I T
(21)
Using Schur complement lemma
0
2
1
2
1
I
P
P
I
(22)
Defining
1
, (6) is achieved.
3. CIRCLE CRITERION BASED NONLINEAR OBSERVER DESIGN
In contrast of the linearization-based and high-gain approaches which attempt to eliminate the
system nonlinearities using a nonlinear state transformation or to dominate them by a high gain term of
correction, circle-criterion exploits the properties of the system nonlinearities. In its basic form, introduced
by Arcak and Kokotovic [14], the approach is applicable to a class of systems that can be decomposed in
linear and nonlinear parts with a condition that the nonlinearities satisfy the sector property [14-16].
.
3.1. Basic sector properties
A memoryless nonlinear function
p
p
R
R
t
z
F
[
0
[
:
)
,
( is said to belong to the sector [
0
[
if 0
)
,
(
t
z
zF . Let 1
v and 2
v two real positive numbers, by setting 2
1 v
v
z
and
)]
,
(
)
,
(
[
)
,
( 2
1 t
v
F
t
v
F
t
z
F
, the above sector property is equivalent to:
R
v
v
t
v
F
t
v
F
v
v 2
1
2
1
2
1 ,
0
)]
,
(
)
,
(
)[
(
(23)
Relation (23) states that the nonlinear function )
,
( t
z
F is a nondecreasing function. On the other
hand if )
,
( t
z
F is a continuously differentiable function the above relation is equivalent to [7], [8]:
R
z
t
z
F
dz
d
0
)
,
(
(24)
If the nonlinear function )
,
( t
z
F does not satisfy the positivity condition (24) we introduce a function
)
,
( t
z
F such that:
R
z
t
z
F
dz
d
z
t
z
F
t
z
F
,
)
,
(
,
)
,
(
)
,
(
(25)
And:
5. Int J Pow Elec & Dri Syst ISSN: 2088-8694
A comparative study of nonlinear circle criterion based observer and H∞ observer….. (Farid Berrezzek)
1233
R
z
t
z
F
dz
d
t
z
F
dz
d
0
)
,
(
)
,
(
(26)
In the multivariable case the sector property can be written as:
0
)
,
(
t
z
F
zT
. Where z and )
,
( t
z
F
are respectively vectors of an appropriate dimension.
3.2. Nonlinear Observer Design
The circle criterion based nonlinear observer design can be performed for a class of nonlinear
system that the model can be decomposed into linear part and nonlinear part as the following [14-16]:
)]
(
.
[
)]
(
),
(
[
)
(
)
( t
x
H
GF
t
y
t
u
t
Ax
t
x
(27)
)
(
)
( t
Cx
t
y (28)
Where A , C and G are known constant matrices with appropriate dimensions. The pair )
,
( C
A is
assumed to be observable. The term )]
(
),
(
[ t
u
t
y
is an arbitrary real-valued vector that depends only on the
system measured control inputs )
(t
u and outputs )
(t
y . The nonlinear part of the system is included in the
term )]
(
.
[ t
x
H
F
which is a time-varying vector function verifying the sector property. In the following we
recall the main theorem and conditions that are used in this work to study the feasibility of nonlinear observer
design for induction motor sensorless control with respect of circle criterion or sector property. A detailed
proof of the theorem is presented in reference [17].
Theorem 2 [14, 15]: Consider a nonlinear system of the form (27)-(28) with the nonlinear part satisfying the
circle criterion relations (23)-(26). If there exist a symmetric and positive definite matrix
nxn
R
P
and a set
of row vectors
p
R
K
such that the following linear matrix inequalities (LMI) hold:
0
)
(
)
(
Q
LC
A
P
P
LC
A T
(29)
0
)
(
T
KC
H
PG
(30)
The nonlinear observer design refers to the selection of the gain matrices L and K satisfying the LMI
conditions (29)-(30). With n
I
Q
as a defined positive known matrix, n
I is an n-th order unity matrix and
is a small positive real number.
Then a nonlinear observer can be designed as:
))]
(
ˆ
)
(
(
)
(
ˆ
[
)]
(
ˆ
)
(
[
)]
(
),
(
[
)
(
ˆ
)
(
ˆ t
y
t
y
K
t
x
H
F
G
t
y
t
y
L
t
y
t
u
t
x
A
t
x
(31)
)
(
ˆ
)
(
ˆ t
x
C
t
y
(32)
And 0
)
(
ˆ
)
(
)
(
lim
t
x
t
x
t
e
t , where )
(
ˆ t
x is the estimate of the state vector )
(t
x of the
nonlinear system. One can see that the structure of the nonlinear observer is composed of a linear part that is
similar to linear Luenberger observer and a nonlinear part that is an additional term that represents the time-
varying nonlinearities satisfying the sector property.
Proof: The dynamics of the state estimation error )
(
ˆ
)
(
)
( t
x
t
x
t
e
are given by:
))]
(
ˆ
)
(
(
)
(
ˆ
.
(
))
(
.
(
[
)
(
)
(
)
( t
y
t
y
K
t
x
H
F
t
x
H
f
G
t
e
LC
A
t
e
(33)
6. ISSN: 2088-8694
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1234
Let )
(
.
1 t
x
H
v and ))
(
ˆ
)
(
(
)
(
ˆ
.
2 t
y
t
y
K
t
x
H
v
, by setting )
(
)
(
2
1 t
e
KC
H
v
v
z
, the term between
brackets in (33) can be seen as a function of the variable z then: )
,
(
)]
(
)
(
[ 2
1 t
z
f
v
f
v
f
.
The expression )
,
(
)]
(
)
(
)[
( 2
1
2
1 t
z
zF
v
F
v
F
v
v
satisfies the property of the sector.
Taking into account the above result, the error dynamics in (33) can be rewritten as:
)
,
(
.
)
(
)
(
)
( t
z
F
G
t
e
LC
A
t
e
(34)
)
(
)
( t
e
KC
H
z
(35)
Relations (34)-(35) show, once again, that the error dynamics can then be considered as a linear
system controlled by a time-varying nonlinearity function )
,
( t
z
F satisfying the sector property. Circle
criterion establishes that the feedback interconnection of a linear system and a time-varying nonlinearity
satisfying the sector property is globally uniformly asymptotically stable [14, 15].
Based upon the error dynamics, relation (34)-(35), the nonlinear observer design problem is then
equivalent to stabilization of the error dynamics problem. To this end a candidate Lyapunov function
Pe
e
V T
is considered. In order to ensure asymptotic stability of the observer, the derivative of the
candidate Lyapunov function must be negative. With the help of relation (34) and (35) the derivative of the
Lyapunov function becomes:
)
,
(
)
,
(
)
(
)
( t
z
PGF
e
Pe
G
t
z
F
e
LC
A
P
P
LC
A
e
V T
T
T
T
T
(36)
By setting:
Q
LC
A
P
P
LC
A T
)
(
)
(
(37)
And
T
KC
H
PG )
(
(38)
With n
I
Q
and 0
, the derivative of the Lyapunov function can be rewritten as:
)
,
(
.
.
2 t
z
F
z
Qe
e
V T
T
(39)
Thus ends the proof.
Note that the existence of observer (31)-(32) is conditioned by the solution of LMI conditions (29)-(30). By
solving LMI constraints, observer gain matrices L and K that guarantee observer convergence are then
computed. In Ibrir [16], the author has investigated the study of globally Lipschitz systems and bounded-state
nonlinear systems. Bounded-state nonlinear systems constitute a large class of system that includes electric
machine systems. Electric machine models involve the magnetic flux as a key and bounded state variable that
combined with other state variable of the machine, such as rotor angular velocity, leads to the nonlinear part
of the machine model. This is due to the effect of the magnetic material saturation property that is similar to
the sector nonlinearity.
4. NONLINEAR H∞ OBSERVER SYNTHESIS
In this section, we propose to study the robust H∞ observer. This observer is used in the case of
systems involving uncertainties in the model measurements [7, 9, 10, 11, 12, 18].
A literature review shows that many useful H∞ filtering approaches have been developed for several kinds of
systems. One can cite the H∞ observers for one-sided Lipschitz nonlinear systems, the H∞ observer for
singular Lipschitz nonlinear systems and the H∞ control and filtering for uncertain Markovian jump systems
with time-varying delay [9, 10, 11, 19].
The aim of the H∞ approach is to design a full-order filter such that the corresponding filtering error
system is asymptotically stable and satisfies a prescribed H∞ level disturbance attenuation. By using a
Lyapunov function, sufficient conditions are formulated in terms of linear matrix inequalities (LMIs).
7. Int J Pow Elec & Dri Syst ISSN: 2088-8694
A comparative study of nonlinear circle criterion based observer and H∞ observer….. (Farid Berrezzek)
1235
To do this, we extend the result of the previous section to nonlinear robust H∞ observer design method. We
consider the system (1)-(2) with an additional disturbance term as follows:
)
(
))
(
),
(
(
)
(
)
( t
w
B
t
u
t
x
F
t
Ax
t
x d
(40)
)
(
)
( t
Cx
t
y (41)
Where )
,
0
[
2
l
w is an unknown exogenous disturbance.
The goal is to rebuild the state )
(t
x of the system (40)-(41) with some accuracy despite the presence of
disturbance term )
(t
w .
We suppose that:
)
(t
He
z , with H a known constant matrix such that
w
z
, with 0
.
The corresponding observer for the system (40)-(41) is given by:
)]
(
ˆ
)
(
[
))
(
),
(
ˆ
(
)
(
ˆ
)
(
ˆ t
y
t
y
L
t
u
t
x
F
t
x
A
t
x
(42)
)
(
ˆ
)
(
ˆ t
x
C
t
y (43)
Our purpose is to design the observer parameter L such that the observer error dynamics are asymptotically
stable and the following specified norm
H upper bound is simultaneously guaranteed.
Lemma 1: For any
n
R
y
x
, and any positive definite matrix
n
n
R
P
, we have:
y
P
y
Px
x
y
x T
T
T 1
2
Theorem 3 [13]: Consider stochastic Lipschitz nonlinear system (40)-(41), and the corresponding observer
(42)-(43). The observer error dynamics are asymptotically stable with the minimum of the norm 2
l
and , if
there exist scalers s
> 1, 0
and 0
and matrices 0
P and G such that the following LMI
optimization problem has a solution.
0
)
1
(
2
1
I
P
B
PB
I
H
H
T
d
d
s
T
(44)
0
2
)
(
1
2
)
(
1
2
2
H
P
P
I
H
(45)
With G
P
L 1
and
)
min(
ˆ
*
Proof: The observer error dynamics are:
)
(
)
ˆ
(
)
(
)
(
)
( t
w
B
F
F
t
e
LC
A
t
e d
(46)
The time derivative of the Lyapunov function )
(
)
( t
Pe
e
t
V T
is given by:
Pe
B
t
w
t
w
PB
e
F
F
P
e
e
LC
A
P
P
LC
A
e
t
V
T
T
d
T
T
T
T
d
)
(
)
(
)
ˆ
(
2
)]
(
)
[(
)
(
(47)
)
min(
8. ISSN: 2088-8694
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1236
Using Lemma 1 and the Rayleigh leads to the following:
)
ˆ
(
)
ˆ
(
)
ˆ
(
2 1
F
F
P
PP
F
F
Pe
e
F
F
P
e T
T
T
(48)
)
ˆ
(
)
ˆ
(
)
ˆ
(
2 F
F
P
F
F
Pe
e
F
F
P
e T
T
T
(49)
The right term of (49) can be written as:
e
e
P
e
P
Pe
e T
T
)
(
)
( max
2
max
(50)
2
max
2
)
(
)
ˆ
(
)
ˆ
( e
P
F
F
P
F
F
T
(51)
The left term of (49) can be written as:
2
max )
(
2
)
ˆ
(
2 e
P
F
F
P
eT
(52)
Replacing (51)-(52) in equation (49), leads to:
2
max
2
2
max
2
max )
(
)
(
)
(
2 e
P
e
P
e
P
Assume
I
Q
, in the deterministic case we have 0
w and the corresponding time derivative of the
function Lyapunov becomes:
0
)
ˆ
(
2
)
(
F
F
e
Qe
e
t
V T
T
(53)
0
)
(
2
)
(
2
max
e
P
Ie
e
t
V T
(54)
Then:
0
)
(
2
1 max
P
(55)
2
1
)
(
P
(56)
e
e
P
F
F
P
e T
T
)
(
)
1
(
)
ˆ
(
2 max
2
(57)
e
e
e
e
P
F
F
P
e
T
T
T
)
1
(
2
1
)
(
2
1
)
1
(
)
ˆ
(
2 2
(58)
In the stochastic and general case of 0
w and
I
Q s
the above equation of Lyapunov derivative
becomes:
Pe
B
t
w
t
w
PB
e
F
F
P
e
e
LC
A
P
P
LC
A
e
t
V
T
d
T
d
T
T
T
T
)
(
)
(
)
ˆ
(
2
)]
(
)
[(
)
(
(59)
)
(
)
(
)
1
(
2
1
)
(
)
( max
t
w
PB
e
Pe
B
t
w
e
e
e
e
Q
t
V
d
T
T
d
T
T
T
(60)
9. Int J Pow Elec & Dri Syst ISSN: 2088-8694
A comparative study of nonlinear circle criterion based observer and H∞ observer….. (Farid Berrezzek)
1237
)
(
)
(
)
1
(
2
1
)
(
t
w
PB
e
Pe
B
t
w
e
e
e
e
t
V
d
T
T
d
T
T
T
s
(61)
)
(
)
(
)]
)
1
(
2
1
[
)
( 2
t
w
PB
e
Pe
B
t
w
e
e
t
V d
T
T
d
T
T
(62)
)
(t
He
z , H is a known constant with
w
z
Now, we define the following criterion to minimize the difference between the energy of the estimation error
and the energy inducted by the disturbances:
dt
w
w
z
z
J T
T
)
(
0
(63)
Thus:
dt
t
V
w
w
z
z
J T
T
))
(
(
0
(64)
So a sufficient condition for 0
J is that:
This means that:
0
w
w
z
z T
T
(because )
(t
V
is a negative function) hence:
2
2
w
z .
w
z
But we have:
w
w
Pe
B
w
w
PB
e
e
e
He
H
e
t
V
w
w
He
H
e
t
V
w
w
z
z
T
T
d
T
d
T
T
s
T
T
T
T
T
T
T
)]
)
1
(
2
1
[
)
(
)
(
2
(65)
Then, we can write:
w
e
I
P
B
PB
I
H
H
w
e
J
T
d
d
s
T
T
T
]
)
1
(
2
1
[ 2
(66)
So a sufficient condition for 0
J is that the following be negative definite:
0
]
)
1
(
2
1
[ 2
I
P
B
PB
I
H
H
T
d
d
s
T
(67)
According to the Schur’s complement lemma, (67) is equivalent to:
0
I
.
0
1
)
(
2 max
P
B
PB
I
P
H
H T
d
s
T
d
(68)
For equation (68) to be negative, the sum of the first and second terms must be negative because the third
term is always positive:
0
)
(
2 max
I
P
H
H s
T
(69)
0
)
(
,
,
0
t
V
w
w
z
z
t T
T
10. ISSN: 2088-8694
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1238
But as for any other symmetric matrix, for H
H T
, we have:
)
(
)
( max
min H
H
H
H
H
H T
T
T
(70)
I
H
H
H
I
H T
)
(
)
( 2
min
2
max
(71)
Or according to the definition of singular values
0
)
(
2
)
( max
2
s
P
H
(72)
Or:
2
)
(
)
(
2
max
H
P s
(73)
This is equivalent to (45).
5. SIMULATION RESULTS AND COMMENTS
To perform the comparative study between the two nonlinear observers previously view, we use the
model of an induction motor. Described by the following nonlinear differential equations with, the stator
current, rotor flux and rotor angular velocity as selected state variables of the machine [1, 20, 21].
s
s
r
r
r
r
s
s u
l
T
i
i
dt
d 1
s
s
r
r
r
r
s
s u
l
T
i
i
dt
d 1
r
r
r
r
s
r
r
T
i
T
m
dt
d
1
(74)
r
r
r
r
s
r
r
T
i
T
m
dt
d 1
l
l
r
f
s
r
s
r
r T
k
k
i
i
dt
d
)
(
Where: p
r
r
n
, r
p
Jl
m
n2
, r
sl
l
m
m
1
1
1
, r
sl
l
m2
1
, J
f
k r
f
, J
n
k
p
l
, r
r
r
R
L
T
.
The indexes s and r refer to the stator and the rotor components respectively.
i
, and u respectively
denote the stator current, the rotor fluxes, the supplied stator voltage,
R
is the resistance, l is the
inductance, m is the mutual inductance. s
T
and r
T
are the stator and the rotor time constant respectively.
r
is the rotor angular velocity, r
f is the friction coefficient, J is the moment of inertia coefficient, p
n
is
the number of pair poles, r
is the mechanical speed of the rotor and finally l
T
is the mechanical load
torque.
This type of nonlinear model is generally used for performing nonlinear control, conditions
monitoring and faults diagnosis of electric induction machine systems. Performing these techniques requires
estimating unmeasured rotor flux linkage and rotor angular velocity state variables based on the stator current
and voltage measurements [21].
To carry out the comparative study between the two observers previously view, we use the induction
motor with the following characteristics details on Table 1.
11. Int J Pow Elec & Dri Syst ISSN: 2088-8694
A comparative study of nonlinear circle criterion based observer and H∞ observer….. (Farid Berrezzek)
1239
Note that the starting conditions X0= [4 0 0 0 0] for the initial state vector of the machine are taken
into account in simulations of the two observers.
In the application of the nonlinear observer based on the criterion of circle approach, we must write
the model of induction motor (74) in the form (27)-(28), taking into account the properties (29) and (30).
Table 1 Characteristics of the induction motor
Symbol Quantity Numerical value
P Power 1.5 KW
f Supply frequency 50 Hz
U Supply voltage 220 V
p
n Number of pair poles 2
s
R Stator resistance
4.850
r
R Rotor resistance
3.805
s
l Stator inductance 0.274 H
r
l Rotor inductance 0.274 H
m Mutual inductance 0.258 H
r
Rotor angular speed 297.25 rd/s
J Inertia coefficient
0.031
s
kg /
2
r
f Fiction coefficient 0.00114 N.s/rd
l
T Load torque 5 N.m
The nonlinearities of the machine system are function of the flux state variable that is a bounded state
variable. The nonlinearities of the model are of the form rd
r
that can be expressed as:
r
r
rd
r
rd
r
)
( (75)
One can verify that:
0
)
(
rd
r
rd
r
r (76)
With
2
rd
, then one can choose
2
.
The first step of the simulation consists of resolving the LMI conditions, relation (29)-(30), using an
adequate LMI tools such as the LMI tool-box of the Matlab software. The obtained nonlinear observer gain
matrices L and K are the following:
6201
.
1
6201
.
1
7172
.
0
1075
.
0
1075
.
0
7172
.
0
6749
.
1
1188
.
0
1188
.
0
6749
.
1
L
7381
.
0
6037
.
1
1
K
,
6037
.
1
7381
.
0
2
K
,
9193
.
0
3948
.
0
3
K
,
3948
.
0
9193
.
0
4
K
The corresponding Lyapunov matrix for this LMI feasibility test is:
12. ISSN: 2088-8694
Int J Pow Elec & Dri Syst, Vol. 10, No. 3, Sep 2019 : 1229 – 1243
1240
0173
.
0
0505
.
0
0505
.
0
0274
.
0
0274
.
0
0505
.
0
6010
.
5
4659
.
0
0514
.
0
1486
.
0
0505
.
0
4659
.
0
6010
.
5
1486
.
0
0514
.
0
0274
.
0
0514
.
0
1486
.
0
1550
.
0
0710
.
0
0274
.
0
1486
.
0
0514
.
0
0710
.
0
1550
.
0
P
With: 04
.
0
.
The second step of simulation consists of injecting the obtained numerical values of the gain matrix
and the vectors in an S-function-based Matlab program that interacts with the Matlab Simulink software to
simulate the nonlinear system and the nonlinear observer.
The simulation results of the designed nonlinear observer based circle criterion are presented in the
following. Figure 1 and Figure 2 show the measured and estimated stator current and rotor flux components
respectively. The Figure 3 and Figure 4 show the measured and estimated rotor angular velocity and the
corresponding load torque respectively with corresponding estimation error. One can see that the estimated
state variables of the machine follow the desired trajectories.
To highlight these results a load torque is introduced in the simulation at time of sec
5
.
0 , the
simulation results show that all the state variables of the machine are modified accordingly. Thus
demonstrate the effectiveness of the circle criterion based nonlinear observer design for the induction
machine system state estimation.
Figure 1. Measured (blue line) and observed(green
line) d-stator current components and the
corresponding estimation error
Figure 2. Measured (blue line) and observed (green
line) d-rotor flux components and the corresponding
estimation error
Figure 3. Measured (blue line) and observed (green
line ) rotor angular velocity of the (IM) and the
corresponding estimation error
Figure 4. Measured (blue line) and estimated (green
line) electromechanical torque of the
(IM) and the corresponding estimation error
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
-40
-20
0
20
40
d-Stator
current
(A)
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
-2
-1
0
1
2
Time (s)
Estimation
error
(A)
0.25 0.3 0.35
-10
0
10
20
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
-1
-0.5
0
0.5
1
d-Rotor
flux
(wb)
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
-0.5
0
0.5
Time (s)
Estimation
error
(wb)
0.12 0.14 0.16
-0.4
-0.2
0
0.2
0.4
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
-100
0
100
200
300
400
Rotor
anular
velocity
(rad/s)
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
-15
-10
-5
0
5
Time (s)
Estimation
error
(rad/s)
0.22 0.24 0.26 0.28
200
250
300
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
-10
0
10
20
30
40
Electromechanical
torque
(N.m)
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
-5
0
5
Time (s)
Estimation
error
(N.m)
0.120.140.160.180.2
10
15
20
13. Int J Pow Elec & Dri Syst ISSN: 2088-8694
A comparative study of nonlinear circle criterion based observer and H∞ observer….. (Farid Berrezzek)
1241
In simulation experiments of nonlinear
H observer we must write the model of induction motor
(74) in the form (40) - (41).
As in the first observer approach, the simulation of the nonlinear
H approach is performed in two
steps, the first simulation step consists of resolving the LMI conditions, relation (44)-(45), using an adequate
LMI tools such as the LMI tool-box of the Matlab software.
To ensure robustness against nonlinear uncertainty, we use the theorem 1 to maximize the admissible
Lipschitz constant and then the theorem 3, to minimize for the maximized .
7
,
0
*
, 112
,
0
*
, 04
,
0
.
Subsequently, from Theorem 3 and solving the LMI, (40), (41) and (42), then we calculate the nonlinear
observer gain matrices L andG .
In this observer, we apply a disturbance in the power supplyu in order to see these performances.
I
H 10
,
05
.
0
s
, 5
5I
Q , 20
d
B
6924
.
1
0
0
0
0
.
0
4552
.
0
.
0
0143
.
0
0
0
0
4552
.
0
.
0
0143
.
0
0
0143
.
0
0
0005
.
0
0
0
0
0143
.
0
0
0005
.
0
P
0
0
7151
.
1
0
0
7151
.
1
0574
.
0
0
0
0574
.
0
G
0
0
9415
.
2
0
0
9415
.
2
5922
.
213
0
0
5922
.
213
L
The second step of simulation consists of injecting the obtained numerical values of the gain matrix
and the vectors in an S-function-based Matlab program that interacts with the Matlab Simulink software to
simulate the nonlinear system and the nonlinear
H observer.
The simulation results of the designed and nonlinear
H observer are presented in the following.
Figure 5 and Figure 6 show the measured and estimated stator current and rotor flux components
respectively. The Figure 7 and Figure 8 show the measured and estimated rotor angular velocity and the
corresponding load torque respectively with corresponding estimation error. One can see that the estimated
state variables of the machine follow the desired trajectories.
Figure 5. Measured (red line) and observed (blue
line) d-stator current components and the
corresponding estimation error
Figure 6. Measured (red line) and observed (blue
line) d-rotor flux components and the corresponding
estimation error
The simulation results of the designed and nonlinear
H observer are presented in the following.
Figure 5 and Figure 6 show the measured and estimated stator current and rotor flux components
respectively. The Figure 7 and Figure 8 show the measured and estimated rotor angular velocity and the
corresponding load torque respectively with corresponding estimation error. One can see that the estimated
state variables of the machine follow the desired trajectories.
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
-40
-20
0
20
40
d-Stator
current
(A)
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
-2
-1
0
1
2
Time (s)
Estimation
error
(A)
0.28 0.3 0.32
-10
0
10
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
-1
-0.5
0
0.5
1
d-Rotor
flux
(wb)
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
-0.5
0
0.5
Time (s)
Estimation
error
(wb)
0.05
0.06
0.07
0.08
0.09
-0.4
-0.2
0
0.2
0.4
0.6
14. ISSN: 2088-8694
Int J Pow Elec & Dri Syst, Vol. 10, No. 3, Sep 2019 : 1229 – 1243
1242
Figure 7. Measured (blue line) and observed(red
line) rotor angular velocity of the (IM) and the
corresponding estimation error
Figure 8. Measured (red line) and estimated (blue
line) electromechanical torque of the (IM) and the
corresponding estimation error
According to the Figure 7, we note a good trajectory tracking of the speed in the presence of
disturbances and show the high performance of the observer
H . The simulation results demonstrate that,
despite the partly unknown transition probabilities, the designed
H filters are feasible and effective,
ensuring the error systems are stochastically stable.
6. CONCLUSION
This paper gives a comparative study between the nonlinear observer based on circle criterion and
nonlinear
H
observer.
The performance of the nonlinear observers is evaluated in terms of their ability to cope with model
imperfections and process uncertainties such as measurement errors and uncertain initial conditions.
The advantages of the circle-criterion approach are the global Lipschitz restrictions removing and
high gain avoiding. However it introduces linear matrix inequality (LMI) conditions.
From the simulation results, the proposed
H observer has proved to be more robust than the observer based
circle criterion when load variations of the IM occur, in the presence of disturbances.
On the other hand, the results obtained using the nonlinear
H observer show that three
characteristics can be obtained simultaneously. Asymptotic stability, robustness against nonlinear uncertainty
and minimized guaranteed
H
cost.
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0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
-100
0
100
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Rotor
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BIOGRAPHIES OF AUTHORS
Farid Berrezzek was born in Souk ahras , Algeria, in 1968.He received his B.Sc. and M.Sc.
degrees in electrotechnical engineering from the University Badji Mokhtar, Annaba, Algeria, in
1994 and 2006, respectively. Since 2006, he has been with the Department of electrical
engeneering, University of Souk ahras, Algeria, where, since 2016, he has been a Doctorate from
the University Badji Mokhtar. He has published few refereed conference papers. His research
interest covers system control techniques and their application to electric machines. E-mail:
f.berrezzek@univ-soukahras.dz (Corresponding author)
Wafa Bourbia received his B.Sc., M.Sc. and doctorate degrees in electrotechnical engineering
from the University Badji Mokhtar, Annaba, Algeria, in 2004, 2006 and 2016, respectively.
She has published few refereed conference papers. Sher research interest covers system control
techniques and their application to electric machines. E-mail: bourbia_wafa@yahoo.fr
Bachir Bensaker was born in Roknia, Algeria, in 1954. He received the B.Sc. degree in
electronics engineering from the University of Science and Technology of Oran (USTO),
Algeria, in 1979. From 1979 to 1983, he was a Teaching Assistant with the Department of
electronic, University of Annaba, Algeria. He received the M.S. and PhD degrees in
Instrumentation and Control from the University of Le Havre, France, in 1985 and in 1988
respectively. Since 1988, he has been with the Department of electronic, University of Annaba,
Algeria, where, since 2004, he has been a Professor. Prof. Bensaker is IFAC affiliate since 1991.
He has published about 50 refereed journal and conference papers. His research interest covers
system modelling, control, identification, estimation, and system reliability and their applications
in nonlinear control, condition monitoring, fault detection and diagnostics of electrical machines.
Email: bensaker_bachir@yahoo.fr