Dynamical Analysis of Three–phase Systems Controlled by

Dynamical Analysis of Three–phaseSystems Controlled by ZAD-FPIC

Techniques.

Nicolás Toro García

Universidad Nacional de ColombiaFaculty of Engineering and Architecture, Department of Electrical, Electronic and Computer Engineering

Manizales, Colombia2012

Análisis Dinámico de Sistemas TrifásicosControlados con Técnicas ZAD–FPIC.


Universidad Nacional de ColombiaFacultad de Ingeniería y Arquitectura, Departamento de Ingenirías Eléctrica, Electrónica y Computación


Dynamical Analysis of Three–phaseSystems Controlled by ZAD-FPIC

Techniques.


A Thesis Presented for the Degree of:Ph. D. Engineering Industrial Automation

Advisors:Ph. D. Fabiola Angulo García

Ph. D. Edgar Nelson Sánchez Camperos

Research Areas:Power Electronics and Control

Research group:Perception and Intelligent Control

Universidad Nacional de ColombiaFaculty of Engineering and Architecture, Department of Electrical, Electronic and Computer Engineering


This work is dedicated...To my wife...

To my daughters...And To my sons...

for time not shared with them.Nicolás.

AcknowledgementsFor the successful completion of this thesis, I am indebted to a number of people andinstitutions who contributed effort and resources.

To my advisors Ph.D. Fabiola Angulo García and Ph.D. Edgar Sánchez Camperos whohave read a large part of an earlier versions of this work and have made many valuablesuggestions and comments.

To my partners Fredy Edimer Hoyos and Yeison A. Garcés for their professional andenthusiastic support to this project.

To Phd. Eduardo Antonio Cano Plata for their support and allow the use of computersand devices in the laboratory of power quality and power electronics.

I would also like to thank the members of search groups PCI, ABCDynamics andGREDyP of the Universidad Nacional de Colombia - sede Manizales, specially toprof. Gerard Olivar Tost and Prof. Gustavo Osorio.

By Last I must to thank the Universidad Nacional and its research direction of Mani-zales(DIMA), for founding my research work; to Department of Electrical, Electronicand Computer Engineering; and CINVESTAV unidad de Guadalajara (México) whereI spent the research stay in the control laboratory with assistance of Prof. SánchezCamperos and Prof. Alexander Loukyanov.

XI

ResumenEl presente trabajo se enfoca en dos aspectos fundamentales en los cuales se han realizado di-ferentes aportes.

El primero de ellos consiste en el modelado de los motores de inducción lineal (LIM), en el cual sehan incluido los efectos de borde en su representación en el espacio de estados. Este fenómeno esaltamente no lineal y resultados numéricos preliminares muestran oscilaciones no deseadas. De-bido a la limitación del recorrido del LIM utilizado en este trabajo (1.5 metros), posteriormente, elcomportamiento del sistema es analizado únicamente a nivel numérico.

Otro aspecto importante desarrollado en la tesis está relacionado con la aplicación de la técnica decontrol ZAD-FPIC a sistemas de alto orden. Hasta ahora la técnica anteriormente propuesta sólohabía sido desarrollada, probada y validada en sistemas de segundo orden. En este trabajo se hageneralizado la técnica ZAD-FPIC, aplicándose a un sistema que consiste en un convertidor con-mutado para el manejo de diferentes cargas trifásicas, lo cual que constituye un sistema de ordensuperior. Las cargas utilizadas en esta investigación son cargas resistivas y cargas inductivas anali-zadas en el documento, como son el motor de inducción de desplazamiento lineal-LIM y el motorde inducción rotativo-RIM. Los motores de inducción de desplazamiento lineal se utilizan en sis-temas de transporte masivo y en aplicaciones especiales en la industria e ingeniería aeroespacial,y los motores de inducción rotativos son utilizados ampliamente a nivel industrial. ZAD-FPIC esuna estrategia de control conmutado que presenta la ventaja de imponer una frecuencia de con-mutación fija. Para el análisis del LIM se usó el modelo obtenido en la primera parte de la tesis;por el contrario, puesto que el modelado del RIM es ampliamente cubierto en la literatura, el con-trol se diseñó usando el clásico circuito equivalente por fase. En la respuesta de estos sistemas sepresentan comportamientos dinámicos complejos debido a su naturaleza no lineal.

Así pues, como parte de los resultados obtenidos de la fase de modelamiento y control, en estetrabajo también se muestra, a nivel experimental y con solución numérica, la posible presencia defenómenos complejos que surgen cuando se alimentan cargas trifásicas con un convertidor conmu-tado trifásico controlado con la técnica ZAD-FPIC. En el documento se presentan los resultadosexperimentales que evidencian la coexistencia de atractores de órbitas periódicas y caóticas comoresultado de las dinámicas no lineales en cada sistema estudiado. La estabilidad del sistema enlazo cerrado fue analizada usando diagramas de bifurcaciones y se mostró en forma numérica laimportancia de los retardos en la estrategia ZAD-FPIC. Se observó que en un amplio rango detrabajo la aplicación de la técnica ZAD-FPIC fue exitosa.

El desarrollo experimental consta de: un inversor conmutado trifásico que usa un módulo de IGBTsintegrado, una tarjeta de control para prototipado rápido (RCP) basado en la plataforma dSPACE,algoritmos de control realizados en MATLAB/SIMULINK y las cargas trifásicas.

Para cada una de las cargas se realizó una descripción del sistema propuesto (arquitectura), se

XII

diseñaron las estrategias de control y se presentaron los resultados numéricos y experimentales.Para el caso de los motores de inducción se llevó a cabo la identificación de parámetros, con elobjetivo de realizar las simulaciones correspondientes. Para cada sistema en estudio se obtuvieronmodelos continuos y discretos (mapas iterativos) del sistema convertidor-carga y se diseñaron lasestrategias de control para el cálculo del ciclo útil del DPWM (Digital Pulse Width Modulation).Debido a que en la práctica no es posible disponer de medidas para las corrientes del rotor en elRIM, se diseñó un observador al cual se le analizó su convergencia con el método de Lyapunov yse comprobó su efecto en el desempeño del controlador. En el documento se muestran diagramasde bifurcaciones en forma numérica y experimental usando Ks, N y frecuencia del voltaje de refe-rencia como parámetros de bifurcación.

Palabras clave: Control ZAD–FPIC, Convertidor trifásico, Diagramas de bifurcación, Control decaos, DPWM, LIM, RIM, Regulación de voltaje, Comportamiento complejo.

XIII

AbstractThis paper focuses on two key aspects which have made different contributions.

The first consists in the modeling of the linear induction motor(LIM), which are included the endeffects in the state space representation. This phenomenon is highly nonlinear and preliminary nu-merical results show undesired oscillations. Due to the limitation of travel of the LIM used in thiswork (1.5 meters), then the system behavior is analyzed only at the numerical level.

Another important aspect in the thesis is related to the implementation of the control ZAD-FPICtechnique to higher order systems. So far the technique previously proposed only had been devel-oped, tested and validated on systems of second order. This paper has generalized the ZAD-FPICtechnique, applied to a system consisting of a switched converter for handling different Three–phase loads, which constitutes a higher order system. The loads used in this research are resistiveand inductive loads analyzed in the document, such as the induction motor of linear movement -LIM and rotary induction motor -RIM. The linear displacement induction motors are used in masstransit systems and special applications in industry and aerospace engineering, and rotary induc-tion motors are widely used in industry. ZAD-FPIC is a switched control strategy which has theadvantage of imposing a fixed switching frequency. For the LIM analysis, the model obtained inthe first part of the thesis was used. On the contrary, because the modeling of the RIM is widelycovered in the literature, its control was designed using the classical equivalent circuit per phase.The response of these systems has complex dynamic behavior due to their nonlinear nature.

Thus, as part of the results obtained from the stage of modeling and control, this paper also shows,on an experimental and numerical solution, the presence of complex phenomena that arise whenfeeding three-phase loads with a controlled three-phase switched converter with ZAD-FPIC tech-nique. The paper presents experimental results that show the coexistence of attractors of periodicorbits and chaotic as a result of nonlinear dynamics in each system studied. The stability of theclosed loop system was analyzed using bifurcation diagrams and numerically showed the impor-tance of delays in the ZAD-FPIC strategy. It was noted that a wide range of work the applicationof the ZAD-FPIC technique was successful.

The experimental development comprises: a switched Three–phase inverter using IGBTs integrat-ed module, a control board for rapid control prototyping (RCP) based on the dSPACE platform,control algorithms performed in MATLAB / SIMULINK and three phase loads.

For each of the loads there was a description of the proposed system (architecture), were designedcontrol strategies and presented the numerical and experimental results. For the case of inductionmotors is carried out to identify parameters in order to perform the corresponding simulations.For each system under study were obtained continuous and discrete models (iterative maps) forconverter-load system and control strategies were designed to calculate the duty cycle of the DP-WM (Digital Pulse Width Modulation). Because in practice it is not possible to have measures for

XIV

the rotor currents in the RIM, we designed an observer which is analyzed for convergence withthe Lyapunov method and tested its effect on driver performance. The document show diagrams ofbifurcations numerically and experimentally using Ks, N and frequency of the reference voltage asbifurcation parameters.

Keywords: Control ZAD–FPIC, Three–phase Converter ,Bifurcation Diagrams, Chaos Control, DP-WM, LIM, RIM, Voltage Regulation, Complex behaviour.

Contents

Resumen XI

Abstract XIII

Contents XV

List of Figures XVIII

List of Tables XXI

List of symbols and abbreviations 1

1. Introduction 21.1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.2. Setting Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.3. Methodology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41.4. Results diffusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

2. Theoretical Framework 72.1. Previous results in modeling, control and nonlinear analysis of RIMs and LIMs . . 72.2. Linear Induction Motor Modeling . . . . . . . . . . . . . . . . . . . . . . . . . . 8

2.2.1. Construction aspects of Linear Induction Motors . . . . . . . . . . . . . . 82.2.2. LIM Model without end effect and considering attraction force . . . . . . . 102.2.3. LIM Model taking into account attraction force and end-effects . . . . . . 162.2.4. LIM behavior . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

2.3. Analysis of Stability and Bifurcation in Power Electronic LIM Drive System . . . 292.3.1. Model of Power Electronic LIM Drive System . . . . . . . . . . . . . . . 302.3.2. Stability Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

2.4. ZAD and FPIC Control Strategies . . . . . . . . . . . . . . . . . . . . . . . . . . 412.4.1. Zero Average Dynamic ZAD . . . . . . . . . . . . . . . . . . . . . . . . . 422.4.2. Control of chaos with FPIC . . . . . . . . . . . . . . . . . . . . . . . . . 43

3. Parameter Estimation of Induction Motors 453.1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 453.2. Parameter Estimation of Linear Induction Motor LabVolt 8228-02 . . . . . . . . . 46

XVI CONTENTS

3.2.1. Experiment Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 483.2.2. Primary Inductance and Resistance Estimation under no-load test. . . . . . 533.2.3. Estimation of Req and Leq . . . . . . . . . . . . . . . . . . . . . . . . . . 573.2.4. Estimation of Lm, Lr and Rr from a Third-Order Polynomial . . . . . . . 593.2.5. Estimation of Lm, Lr, and Rr from the equation system . . . . . . . . . . 613.2.6. Estimation of Lm, Lr, Ls and Rr taking into account end effects . . . . . . 63

3.3. Parameter Estimation of Induction Motor LEESON model C4T34FB5B . . . . . . 663.3.1. Determination of Rs: . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 663.3.2. Determination of Ls: . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 673.3.3. Determination of Lm, Lr and Rr: . . . . . . . . . . . . . . . . . . . . . . 683.3.4. Determination of mechanical parameters: . . . . . . . . . . . . . . . . . . 70

4. Three–Phase Power Converter Controlled With ZAD–FPIC Techniques 734.1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 734.2. Three–phase Power Converter With Resistive Load . . . . . . . . . . . . . . . . . 74

4.2.1. Proposed system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 744.2.2. Mathematical model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 754.2.3. Control strategies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 794.2.4. Numerical and experimental results . . . . . . . . . . . . . . . . . . . . . 814.2.5. Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86

4.3. Three–phase Power Converter With Rotary Induction Motor Load . . . . . . . . . 864.3.1. Proposed system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 884.3.2. Mathematical model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 894.3.3. Control strategies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 914.3.4. Numerical and experimental results . . . . . . . . . . . . . . . . . . . . . 934.3.5. Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105

4.4. Three–phase Power Converter With Linear Induction Motor Load . . . . . . . . . 1074.4.1. Proposed System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1074.4.2. Mathematical Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1084.4.3. Control strategies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1114.4.4. Numerical Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1124.4.5. Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114

5. Main Contributions and Future Work 1185.1. The main contributions of this work are: . . . . . . . . . . . . . . . . . . . . . . . 1185.2. Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119

5.2.1. To adjust the models with appropriate time delay, parameters and no mod-eled dynamics. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119

5.2.2. Stability Analysis of Equilibrium points and Periodic Solutions . . . . . . 1195.2.3. Computer Algorithms for Analysis Bifurcation Phenomena. . . . . . . . . 1205.2.4. Chaotic Behavior Detection . . . . . . . . . . . . . . . . . . . . . . . . . 122

CONTENTS XVII

Bibliography 123

List of Figures

2.1. Cutting and unrolling process to obtain a LIM. . . . . . . . . . . . . . . . . . . . . 92.2. Mover velocity, voltage, currents and fluxes resulting from the model simulation

using ODE45 function of Matlabr, in the continuos case, and discrete Time modelof LIM 2.20 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

2.3. Linor currents at the entry and exit ends for given velocity. . . . . . . . . . . . . . 172.4. (a) Eddy current generation at the entry and exit of the air gap when the primary

coil moves with velocity v. (b) Polarity and decaying profile of the entry and exiteddy currents. (c) Air gap flux profile. Taken from [29] . . . . . . . . . . . . . . . 18

2.5. (a) Effective air gap MMF and (b) eddy current profile in normalized time scale . . 192.6. The LIM equivalent circuits taking into account the end effects. . . . . . . . . . . . 212.7. Mover velocity, velocity differences, currents and fluxes resulting from the model

simulation using ODE45 function of Matlab, taking into account end effects inmodel of LIM 2.40 and without end-effects model 2.3. . . . . . . . . . . . . . . . 28

2.8. LIM behavior with 30 Hz input frequency. Mover velocity and phase portraits ofsome state variables. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29



2.11. Power electronic LIM drive system . . . . . . . . . . . . . . . . . . . . . . . . . . 302.12. Space Vector Modulation (SVM) . . . . . . . . . . . . . . . . . . . . . . . . . . . 352.13. Logical sequence step1...step6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 352.14. Average Space Vector Modulation . . . . . . . . . . . . . . . . . . . . . . . . . . 362.15. PWM for period T . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 362.16. Effective switching Frequency. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 372.17. Modified Inverse Clarke Transform. . . . . . . . . . . . . . . . . . . . . . . . . . 382.18. PWM periods for the sectors of SVM. . . . . . . . . . . . . . . . . . . . . . . . . 392.19. surface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

3.1. Experiment for LIM parameter identification. . . . . . . . . . . . . . . . . . . . . 463.2. Simulink block-set for characterization of a LIM. . . . . . . . . . . . . . . . . . . 483.3. Two pole DC switch-mode power converter. . . . . . . . . . . . . . . . . . . . . . 493.4. Variable magnitude DC voltage source. . . . . . . . . . . . . . . . . . . . . . . . 503.5. Three-phase variable magnitude and variable frequency source. . . . . . . . . . . . 513.6. Sign Control of the LIM mover velocity via its position. . . . . . . . . . . . . . . . 51

LIST OF FIGURES XIX

3.7. first order filter to avoid sudden changes of reference speed in the motor. . . . . . . 523.8. LIM position and velocity measurement. . . . . . . . . . . . . . . . . . . . . . . . 523.9. Control Desk Panel. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 533.10. LIM mover velocity with no-load condition and Vs = 0,549m/s. . . . . . . . . . . 553.11. va and ia (Voltage and current) wave forms with no-load condition. . . . . . . . . . 563.12. Equivalent circuit of a LIM at stanstill. . . . . . . . . . . . . . . . . . . . . . . . . 573.13. va and ia (Voltage and current) wave forms in blocked-mover test. . . . . . . . . . 593.14. Per-phase equivalent circuit of a three-phase LIM with end effects. . . . . . . . . . 633.15. Per-phase equivalent circuit of a three-phase induction motor. . . . . . . . . . . . . 663.16. va and ia (Voltage and current) wave forms with no-load condition. . . . . . . . . . 673.17. Equivalent circuit of a RIM at standstill. . . . . . . . . . . . . . . . . . . . . . . . 683.18. va and ia (Voltage and current) wave forms for rotor-blocked test. . . . . . . . . . . 693.19. Rotor speed for free running of induction motor. . . . . . . . . . . . . . . . . . . . 713.20. Equivalent circuit model per-phase for simulation. . . . . . . . . . . . . . . . . . . 723.21. Stator currents of induction motor. . . . . . . . . . . . . . . . . . . . . . . . . . . 72

4.1. Block diagram of the proposed system . . . . . . . . . . . . . . . . . . . . . . . . 754.2. Electrical circuit for the buck-motor system . . . . . . . . . . . . . . . . . . . . . 764.3. Electrical circuit for the buck converter (equivalent circuit per phase ) . . . . . . . 774.4. Centered PWM. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 784.5. Periodic Solution for the Three–phase converter with resistive load. . . . . . . . . 824.6. Chaotic Solution for the Three–phase converter with resistive load. . . . . . . . . . 834.7. Experimental and simulation results for Three–phase converter with resistive load. 834.8. Error in voltage, with ks andN like bifurcation parameters, and error in duty cycle

for phase a resulting from the model simulation using simulink of Matlabr ofThree–phase converter with resistive load. . . . . . . . . . . . . . . . . . . . . . . 84

4.9. vc voltage, ia current and duty cycle experimental bifurcation diagrams with kslike bifurcation parameter of three phasic converter with resistive load. . . . . . . 84

4.10. vc voltage, ia current and duty cycle experimental bifurcation diagrams with f likebifurcation parameter of three phasic converter with resistive load. . . . . . . . . . 85

4.11. Bifurcation Diagram with ks like bifurcation parameters, taking one sample eachperiod of the reference voltage for phase a during 30 periods considering a T delay. 86

4.12. Bifurcation Diagram with ks like bifurcation parameters, Taking a sample each30 periods of the reference voltage for phase a considering a 3T delay, for thePoincare map 4.6. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87

4.13. Results for Poincare map 4.6 simulation, Taking a sample on each one of last 10periods of the reference voltage for phase a and ia considering a3T delay,varyingfrequency in the reference voltage . . . . . . . . . . . . . . . . . . . . . . . . . . 88

4.14. Block diagram of the proposed system . . . . . . . . . . . . . . . . . . . . . . . . 884.15. Electrical circuit for the converter-motor system . . . . . . . . . . . . . . . . . . . 904.16. Responses of the current rotor observer. . . . . . . . . . . . . . . . . . . . . . . . 93

XX LIST OF FIGURES

4.17. Experimental bifurcation diagrams with f like bifurcation parameter in converter–rotary motor system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94

4.18. Rotor speed Vs Frequency. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 944.19. Results for converter–motor system for 20HZ and 40 Hz of the supply voltage . . . 954.20. Experimental results for coexistence in buck–rotary motor system . . . . . . . . . 954.21. Bifurcation diagrams for duty cycle, converter voltage and converter current of

phase a. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 964.22. Time behaviour for Voltage motor (controlled voltage) of phase a without delay. . . 974.23. Phase portrait for ia and va variables. . . . . . . . . . . . . . . . . . . . . . . . . . 974.24. Bifurcation diagrams for duty cycle, Voltage motor and converter current of phase

a for a time delay of 2T. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 984.25. Time behaviour for Voltage motor (controlled voltage) of phase awith a time delay

of 2T. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 984.26. Phase portrait for ia and va variables with 2T delay in signal control. . . . . . . . . 994.27. Response of the controller with and without rotor currents observer. . . . . . . . . 994.28. Error in output controlled voltages of the controller with and without rotor currents

observer. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 994.29. Experimental results for coexistence in buck–rotary motor controlled system . . . . 1004.30. Time behaviour for Voltage motor (controlled voltage) of phase a for different

initial conditions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1014.31. Phase portrait for motor voltage and current of phase a. . . . . . . . . . . . . . . . 1014.32. Experimental bifurcation results for converter–rotary motor controlled system . . . 1024.33. Chaotic Solution for the Three–phase converter with rotary motor load. . . . . . . 1024.34. Coexistence of Solutions for the Three–phase converter with rotary motor load. . . 1034.35. Periodic Solution for the Three–phase converter with rotary motor load. . . . . . . 1044.36. Chaotic Solution for the Three–phase converter with rotary motor load. . . . . . . 1044.37. Chaotic Solution for the Three–phase converter with rotary motor load. . . . . . . 1054.38. Experimental results for coexistence in buck–rotary motor controlled system . . . . 1054.39. Periodic Solution for the Three–phase converter with rotary motor load. . . . . . . 1064.40. Chaotic Solution for the Three–phase converter with rotary motor load. . . . . . . 1064.41. Block diagram of the proposed system for LIM-Drive system . . . . . . . . . . . . 1074.42. Electric circuit for the converter-motor system . . . . . . . . . . . . . . . . . . . . 1084.43. Response of the secondary current observer. . . . . . . . . . . . . . . . . . . . . . 1124.44. Equivalent circuit model per-phase for simulation. . . . . . . . . . . . . . . . . . . 1134.45. Bifurcation diagrams for duty cycle, converter voltage and converter current of

phase a in the converter with LIM load. . . . . . . . . . . . . . . . . . . . . . . . 1144.46. Time behaviour for Voltage motor (controlled voltage) of phase a. . . . . . . . . . 1154.47. Phase portrait for ia and va variables. . . . . . . . . . . . . . . . . . . . . . . . . . 1154.48. Bifurcation diagrams for duty cycle, converter voltage and converter current of

phase a in the converter with LIM load. . . . . . . . . . . . . . . . . . . . . . . . 1164.49. Time behaviour for Voltage motor (controlled voltage) of phase a without delay. . . 1174.50. Phase portrait for ia and va variables. . . . . . . . . . . . . . . . . . . . . . . . . . 117

List of Tables

2.1. Space Vector Modulation Inverter States . . . . . . . . . . . . . . . . . . . . . . . 342.2. PWMs duty cycle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

4.1. Parameters for simulation and experiment . . . . . . . . . . . . . . . . . . . . . . 81

List of symbols and abbreviations

LIM Linear Induction MotorRIM Rotary Induction MotorZAD Zero Average DynamicFPIC Fixed Point Induced ControlRCP Rapid Control PrototypingDPWM Digital Pulse Width ModulationPID Proportional, Integrative, and Derivative controllerTDAS Time-Delayed Auto-synchronizationETDAS Extended TDASSMC Sliding Mode ControlDSP Digital Signal ProcessorFOC Field Oriented Controlv Linear speedτ Pole pitchλdr, λqr d-axis and q-axis secondary fluxids, iqs d-axis and q-axis primary currentuds, uqs d-axis and q-axis primary voltageTr Secondary time constantρ Leakage coefficientKf Force constantRs Winding resistance per phaseRr Secondary resistance per phase referred primaryLm Magnetizing inductance per phaseLr Secondary inductance per phase referred primaryLs Primary inductance per phaseFL External force disturbanceM Total mass of the moverB Viscous friction coefficientCCM Continuous Conduction Mode

1 Introduction

1.1. Introduction

Basically, there are two kinds of induction motors: the Round-rotor Induction Motor (RIM) andLinear Induction Motor (LIM). Induction motors are widely used as actuators in industrial appli-cations because of the simplicity in construction, lack of maintenance and low cost. Even though,LIM presents an interesting dynamics not always considered for designing devices.

Topics about RIM are quoted broadly in the literature, and a classic technique is the field orientedcontrol [1], which involves nonlinear state transformations and state feedback for the uncoupledasymptotic of magnetic flux and rotor velocity, combined with linear control methods as PID.

Nonlinear methods for the RIM control problem have been applied to improve its performance.An example can be the adaptive control used to deal with uncertainties of parameters in the system[2,3]. It is important to mention some works in this field as: backstepping [4], input/output adaptivelinearization [2], and sliding modes [3, 5, 6], among others. The RIM control by continuous-timesliding mode is found in [3] and [5]. However, this literature is precarious in the nonlinear dynamicanalysis since there are few articles with a deep study about nonlinear behavior that can includelimit cycles, bifurcations and chaos [7–9].

This research is based on a discrete-time control of RIM by sliding modes [10]. Its purpose, inthe RIM and LIM, is to achieve that the mover velocity tracks a reference value via controllingthe supply voltage. The current, flux and force equations are given in a state space representationwhich describe the linear induction motor performance. The discrete time model is obtained todevelop the control law and describe its performance in a computer system.

The novelty on the modelling of the LIM is that the so called end effects are considered in an spacestate representation. This phenomena is highly non linear and preliminary numerical results showundesirable oscillation. On the other hand, the novelty on the analysis will be use the qualitativetheory of non smooth dynamical systems in the controlled LIM via discrete-time quasi slidingmode. As far as we know, the approach taking into account non linear components, parametersvariation and topological changes derived from control actions on RIM and LIM has not beenreported in the literature.

The study of variable structure system uses bifurcation theory in order to determine the conditionsin parameters values which generate stability changes, periodicity and chaotic dynamics in the

1.2 Setting Problem 3

system and allow us to define safe and stable operation zones. The knowledge of these operationranges lets us avoid the presence of non desired phenomena such as auto-sustained oscillations,chaos and evolution to other operation regimes, among others.

In this research,the analysis of controlled loads like resistive, RIM and LIM by quasi sliding modeswas made. This analysis includes: To detect bifurcations, to obtain fixed points and limit cycles,to determine stability and chaos transitions. Moreover, the chaos control algorithms were based onrecent developed technique (ZAD–FPIC [11]) which is being carried out by PCI (Perception andIntelligent Control) and ABCDynamics (Analysis of Bifurcation and Chaos), the research groupson nowadays in Universidad Nacional de Colombia-Manizales.

1.2. Setting Problem

Linear induction machines are being actively investigated for use in high-speed ground transporta-tion. Other applications including liquid metal pumps, magnetohydrodynamic power generation,conveyors, cranes, baggage handling systems, as well as a variety of consumer applications havecontributed to an upsurge in interest in linear induction machines. Unfortunately, analysis of a lin-ear induction machine is complicated by the so-called “end effect”. In a conventional round-rotorinduction motor the behavior of the machine need to be calculated only over one pole pitch. Thesolution for the remaining pole-pitches can then be simply obtained by symmetry. However, thesymmetry argument cannot be used for a linear induction machine since the electrical conditionschange at the entrance and exit (LIM mover ends). A more detailed analysis is required to ade-quately describe its behavior [12].

According to LIM applications, it is possible to choose an operation regime according to the de-veloped task. This operation regime may vary from regulation to tracking. However, the presenceof phenomena such as Magnetic Hysteresis, dead zones (inertia and friction) and saturation (fre-quency limits and mechanic ends) generates unstable behavior in the induction motors driven byfrequency variators at certain operation conditions, which causes unusual vibrations in the sys-tems [13].

Many of studies related to RIM and LIM dynamics are restricted to the stability analysis via smoothdifferential equations [8, 9, 14, 15], which are uncompleted and do not reflect the real dynamicalsystem when it becomes a variable structure system. It is necessary to perform a detailed studyof nonlinear dynamic behavior of the non smooth system bifurcations diagrams in order to definesafety and stable operation zones.

The control action required for Three–phase loads, is implemented usually by power electroniccircuit based on switches. For this reason, controlled commuted system with Three–phase load(Resistive and inductive) becomes a variable structure system defined by non-smooth differentialequations in which a complete theoretical framework does not exist yet that allows its study [16,17]

4 1 Introduction

since its theoretical and numerical analysis represents a extremely difficult problem [18]. In thissense, non-smooth transitions occur when a cycle interact with a boundary of discontinuity in thephase space in a non-generic way, causing periodic additions or sudden chaos transitions [19]. Oneof the most relevant aspects in the bifurcation analysis of non-smooth systems is the absence ofthe period doubling sequences that are observed in smooth systems [18]. Because of characteristicbehavior of non-smooth systems,in many cases, it is no possible to apply analysis techniques forsmooth systems without modifications or adequate adjustments [16].

Undesired chaotic fluctuations appear in non-linear control systems. Several different strategieshave been proposed to control the chaotic dynamics in nonlinear systems based on disturbingsystem parameters. This kind of methods does not adjust for high order systems, since they requirenontrivial computing analysis in each cross of a Poincaré section. In addition, the noise can takethe orbit off the steady state and the controller must remain until the orbit approaches again [20].

The techniques referred before have been applied to nonlinear smooth and non smooth systemsof low order. This proposal searches to show: - The analysis of Three–phase loads controlled byquasi sliding mode via bifurcation detection when the control parameters and the input sourcefrequency vary. - The stability analysis and chaos transition - Determine safe operation regionsfor the controlled loads. To perform these goals, it is necessary to make a high order discrete-time model of LIM and RIM where the parameters that vary in the real system are shown. Usingits analytic or numeric solution and the existent theoretical framework for the study of piecewisesmooth systems we can establish the qualitative dynamics of the system.

1.3. Methodology

To achieve the objectives, we used the experimental and computing techniques defined in the chap-ter 4 of [13], namely:

1. Experimental investigation techniques. Chi K. Tse.

2. Numerical investigation techniques. Soumitro Banerjee - Davic C. Hamill

3. Analytic and computing methods for stability analysis and bifurcation phenomena. YasuakiKuroe.

The designed control strategy was based on quasi sliding mode, then it was necessary to calculatea piecewise smooth flow. For this purpose direct numeric simulation was used.

An equally important research way of nonlinear phenomena consists of beginning with experi-mentation. Experimental study plays the dual role of verifying and establishing certain nonlinearphenomena in physical systems. With the use of oscilloscopes with memory and spectrum analyz-ers we could do phase portraits to identify chaotic behavior and distinguish periodic behavior by

1.4 Results diffusion 5

the frequency spectrum inspection. Chi K. Tse in [21] described experimental techniques whichwere used for calculating Poincaré maps and bifurcation diagrams. These also could be obtainedby Poincaré section method. In the experimental case we used a discrete model as Poincaré map,using an impact map [22], where the states were observed with constant sample frequency. Thestate variables that highlight much more bifurcation phenomena were selected. In [22] a method-ology to obtain not only bifurcation diagrams but also basins of attraction with multiple attractorshas been presented.

For computing bifurcation diagrams, control parameters and voltage reference were varied. Thenumerical analysis was complemented and validated with experiments.

From the pioneering work of E. Ott. C Grebogy and J.A. Yorke (OGY method for chaos con-trol) [23], many strategies have been presented for the chaotic dynamic control in nonlinear sys-tems [24, 25]. Some strategies are based on OGY method, where the control is achieved throughsmall perturbations of accessible parameter. An alternative method called TDAS [26] (Time-Delayed Auto-synchronization), involves a control signal formed with the difference between cur-rent state system and delayed state system by a period of the unstable orbit that is necessary tostabilize. In [22] a TDAS variation named Extended TDAS (ETDAS) uses a linear combination ofsystem signals delayed by several periods of unstable orbit. Likewise, a method is described forthe bifurcation control of boundary collision using the state feedback technique (Pole placement).

In order to chaos control in the Three–phase loads the ZAD–FPIC control strategies were used inthe development of this thesis.

1.4. Results diffusion

The application of control techniques mentioned and some partial results have been presented invarious journals and academic events such as:

1. F. Hoyos, D. Burbano, F. Angulo, G. Olivar, J. Taborda, N. Toro, Effects of Quantization,Delay and Internal Resistances in Digitally ZAD-controlled Buck Converter, InternationalJournal of Bifurcation and Chaos (IJBC), 2011. (Accepted in Sep 22 of 2011). SubmissionIJBC-D-11-00151R1.

2. F. Hoyos, N. Toro, F. Angulo, Rapid Control Prototyping of a Permanent Magnet DC MotorUsing Non-linear Sliding Control ZAD and FPIC, 3rd IEEE Latin American Symposium onCircuits and Systems (LASCAS 2012), Playa del Carmen, México, de Febrero 29 a Marzo2 de 2012.

3. O. Trujillo, F. Hoyos, N. Toro, Design, Simulation and Experiment of a PID Using RapidControl Prototyping Techniques, VI Simposio Internacional sobre Calidad de la EnergíaEléctrica SICEL 2011, Asunción Paraguay, 2, 3 y 4 de Noviembre de 2011.

6 1 Introduction

4. N. Toro, Y. Garcés, E. Sanchez, F. Hoyos, Parameter estimation of linear induction motorlabvolt 8228-02, Congreso Anual, Asociación de México de Control Automático, Sede: In-stituto Tecnológico de Saltillo (ITS), Coahuila, México, del 5 al 7 de Octubre de 2011.

5. F. Hoyos, N. Toro, F. Angulo, Adaptive Control for Permanent Magnet DC Motor UsingZAD-FPIC. In processing for: IEEE Transactions on Power Electronics, 2012.

6. F. Hoyos, F. Angulo, N. Toro, A. Rincón, Adaptive Control for Buck Converter Using ZAD-FPIC: Bifurcations Analysis. In processing for: IEEE Transactions on Circuits and SystemsI-Regular Papers, 2012.

7. F. Hoyos, F. Angulo, N. Toro, ZAD and FPIC Switching mode control Vs SMC and PIDcontrollers in a DC-DC buck converter.In processing for: IEEE Transaction on Power Elec-tronics.In this paper the sliding mode control(SMC), PID and ZAD (Zero Average Dynamic) strate-gies are applied to an electronic DC–DC power converter. Time behaviour for each controlleris shown for numerical solution and experimental realization.The results in SMC and PIDare contrasted with a ZAD controller combined with a recent developed strategy namedFPIC (Fixed Point Induction Control). From a practical point of view, ZAD–FPIC techniquehas advantages no exhibited by PID and SMC working with sample and hold. These ad-vantages have been corroborated experimentally. These designs have been tested in a RCP(Rapid Control Prototyping) system based on DSP from dSPACE platform. Numerical andexperimental performance agree.

.

2 Theoretical Framework

2.1. Previous results in modeling, control and nonlinearanalysis of RIMs and LIMs

The regulation problem for rotary induction motor (RIM) starts from the pioneering work ofBlaschke [27], the field oriented control (FOC) has become a classical technique for inductionmotor control but, more recently, various nonlinear control design approaches have been appliedto induction motors to improve their performance. Among the various approaches, we recall adap-tive input/output linearization [2], adaptive backstepping [4], sliding modes [3,5,6], just to mentiona few. With respect to the regulation theory, there are few works related to the regulation or slidingmode regulation of Linear induction motors . All these approaches require full state measurement.Since secondary (linoric) flux is not usually measurable, some researchers have bypassed the prob-lem using flux observers [14, 28, 29].

With respect to the sampling of the RIM dynamics, Ortega and Taoutaou [30], in an effort forimplementing a FOC with digital devices and to provide stability analysis, they derive an ex-act discrete-time representation of a current-fed rotary induction motor model, that is a third or-der model, using state deffeomorphism. Based on the results presented in [30], Loukianov andRivera [10] derived an approximated voltage-fed sampled-data model (fifth-order) where a slidingmode block control was designed.

There are no works in sampling LIMs dynamics therefore, it is of great significance to investigatean exact sampled-data representation of the full Linear induction motor dynamics (fifth order mod-el) and to design exact discrete-time sliding mode controllers.

Rong-Jong Wai and Wei-Kuo Liu in [14] describe a nonlinear control strategy to control a LIMservo drive for periodic motion based on the concept of the nonlinear state feedback theory andoptimal technique, which is compose of an adaptive optimal control system and sliding-mode fluxobservation system. The control and estimation methodologies are derived in the sense of Lya-punov theorem so that the stability of the control system can be guaranteed. The sliding-mode fluxobservation system is implemented using a digital signal processor with a high sampling rate tomake it possible to achieve good dynamics.

Ezio F. Da Silva et al in [31] and [15] present a mathematical model that describes the dynam-ic behavior of a LIM , divided into two portions. The first part represents the model dynamic of

8 2 Theoretical Framework

conventional induction without the end effects while the second portion describes the attenuationcaused by the end effects on LIM.

In [32] V.H. Benítez et al present a method to control a LIM using dynamic neural networks. Theypropose a neural identifier of triangle form and design a reduced order observer in order to estimatethe secondary fluxes. A sliding mode control is devepoed to track velocity and flux magnitude.

In the nonlinear dynamics study area there are some facts that have contributed in its theoreticaland numeric analysis develop. The history of chaotic dynamics can be traced back to the workof Henri Poincaré on celestial mechanics around 1900. [13]. Chaotic effects in electronic circuitswere first noted by Van der Pol in 1927 [33]. However, the first inkling that chaos might be im-portant in a real physical system was given in 1963 by Lorenz [34], who discovered the extremesesitivity to initial conditions in a simplified computer model of atmospheric convection. At begin-ning this work was unknown, but later it becomes so important like the born of a new science [35].Li and Yorke first used the term chaos in their 1975 paper "Period three implies chaos" [36]. Hamil,Banerjee and Verghese do a complete historical review of nonlinear dynamics and chaos study inpower electronics [13] until 2000 and M. di Bernardo et al. in [16] complete that review and showthe advances in chaotic and bifurcation behavior in piece-wise smooth systems until middle of2006.

Nonlinear dynamics analysis in Rotary induction motors (RIMs) is in I. Nagy and Z. Sütö, [9]work. The main objective of their study is to show the most characteristic bifurcation phenomenain order to prepare the ground for finding ways of locking the trajectory to periodic orbit or in-ducing chaos under certain conditions.Y. Kuroe [37] do an analysis of stability and bifurcation inpower electronic rotary induction motor drive systems. Finally T. Asakura et al in [7] verify theexistence of chaotic motion in the velocity control of RIM and find the generating conditions ofchaotic motion. In order to prevent the chaos occurrence, a chaos control method is proposed byusing the neural network controller.

Some techniques and methods showed before apply to RIMs and they will extend and adjust forfitting to chaotic behavior and bifurcations analysis of LIMs.

2.2. Linear Induction Motor Modeling

2.2.1. Construction aspects of Linear Induction Motors

Linear induction machines are being actively investigated for use in high-speed ground transporta-tion. Other applications including liquid metal pumps, magnetohydrodynamic power generation,conveyors, cranes, baggage handling systems, as well as a variety of consumer applications havecontributed to an upsurge in interest in linear induction machines. Unfortunately, analysis of a lin-ear induction machine (LIM) is complicated by the so-called end effect. In a conventional round-

2.2 Linear Induction Motor Modeling 9

rotor induction motor the behavior of the machine need be calculated only over one pole pitch. Thesolution for the remaining pole-pitches can then be simply obtained by symmetry. However, thesymmetry argument cannot be used for a linear machine since the electrical conditions change atthe entrance and exit. A more detailed analysis is required to adequately describe its behavior [12].

In principle, for each rotary induction motor (RIM) there is a linear motion counter-part. Theimaginary process of cutting and unrolling the rotary machine to obtain the linear induction motor(LIM) is by now classic (Figure 2.1). The primary may now be shorter or larger than the secondary.The shorter component will be the mover. In Figure 2.1 the primary is the mover. The primary maybe double sided (Figure 2.1d) or single sided (Figure 2.1 c, e). The secondary material is copper or

Figure 2.1: Cutting and unrolling process to obtain a LIM.

aluminum for the double-sided LIM and it may be aluminum (copper) on solid iron for the single-sided LIM. Alternatively, a ladder conductor secondary placed in the slots of a laminated core may


be used, as for cage rotor RIMs (Figure 2.1c). This latter case is typical for short travel (up to afew meters) and low speed (below 3 m/s) applications.The primary winding produces an airgap field with a strong travelling component at the linearspeed v.

v = τω1

π= 2τf1 (2.1)

The number of pole pairs does not influence the ideal no-load linear speed. Incidentally, the pe-ripheral ideal no-load speed in RIMs has the same formula 2.1 where τ is the pole pitch (the spatialsemiperiod of the travelling field) [38].

2.2.2. LIM Model without end effect and considering attraction force

The dynamic model of the LIM is modified from the traditional model of three-phase, Y-connectedrotatory induction motor in d − q stationary reference frame and can be described, without endeffect and considering attraction force, by the following differential equations [39] [28]

diqsdt

= −[Rs

ρLs+

1− ρρTr

]iqs −

Lmπ

ρLsLrτvλdr +

LmρLsLrTr

λqr +1

ρLsuqs

didsdt

= −[Rs

ρLs+

1− ρρTr

]ids +

LmρLsLrTr

λdr +Lmπ

ρLsLrτvλqr +

1

ρLsuds

dλqrdt

=LmTriqs +

π

τvλdr −

1

Trλqr

dλdrdt

=LmTrids −

1

Trλdr −

π

τvλqr

dv

dt=Kf

M(λdriqs − λqrids)−

B

Mv − FL

M

(2.2)

where v is the mover linear velocity; λdr and λqr are the d-axis and q-axis secondary flux; ids andiqs are the d-axis and q-axis primary current; uds and uqs are the d-axis and q-axis primary voltage;

Tr = LrRr

is the secondary time constant; ρ = 1 −(

L2m

LsLr

)is the leakage coefficient; Kf = 3

2πLmτLr

is the force constant; Rs is the winding resistance per phase; Rr is the secondary resistance perphase referred primary; Lm is the magnetizing inductance per phase; Lr is the secondary induc-tance per phase referred primary; Ls is the primary inductance per phase; FL is the external forcedisturbance; M is the total mass of the mover; B is the viscous friction and iron-loss coefficientand τ is the pole pitch.

Changing the notation in order to distinguish a stationary reference frame model we obtain an"α−β model". We change the index d and q by α and β respectively, and we omit the primary and


secondary indexes because the voltages and currents are with respect to primary and the fluxes arewith respect to secondary.

diβdt

= −[Rs

ρLs+

1− ρρTr

]iβ −

Lmπ

ρLsLrτvλα +

LmρLsLrTr

λβ +1

ρLsuβ

diαdt

= −[Rs

ρLs+

1− ρρTr

]iα +

LmρLsLrTr

λα +Lmπ

ρLsLrτvλβ +

1

ρLsuα

dλβdt

=LmTriβ +

π

τvλα −

1

Trλβ

dλαdt

=LmTriα −

1

Trλα −

π

τvλβ

dv

dt=Kf

M(λαiβ − λβiα)− B

Mv − FL

M

(2.3)

The discrete time model of LIM is obtained by the continuous time model solution. To over-come this problem the continuous time model is divided in a current-fed linear induction motorthird-order model, where the current inputs are considered as pseudo-inputs, and a second-ordersubsystem that only models the currents of the primary with voltages as inputs. The current-fedmodel is exactly discretized by solving the set of differential equations and the other subsystem isdiscretized by a first-order Taylor series.

The currents subsystem with voltages as inputs is given by 2.4

diβdt

= −[Rs

ρLs+

1− ρρTr

]iβ −

Lmπ

ρLsLrτvλα +

LmρLsLrTr

λβ +1

ρLsuβ

diαdt

= −[Rs

ρLs+

1− ρρTr

]iα +

LmρLsLrTr

λα +Lmπ

ρLsLrτvλβ +

1

ρLsuα

(2.4)

The current-fed induction motor third-order model is given by 2.5

dλαdt

=LmTriα −

1

Trλα −

π

τvλβ

dλβdt

=LmTriβ +

π

τvλα −

1

Trλβ

dv

dt=Kf

M(λαiβ − λβiα)− B

Mv − FL

M

(2.5)

Defining the following matrices, in order to simplify notation

Λ =

[λαλβ

]


I =

[iαiβ

]J =

[0 −11 0

]Then simplified third-order model is 2.6

dΛ

dt=LmTrI − 1

TrΛ +

π

τvJΛ

dv

dt=Kf

MITJΛ− B

Mv − FL

M

(2.6)

Making the following change of variables

Y = e−πτxJΛ

X = e−πτxJI

where x is the mover linear displacement.The exponential factor in the transformation is

e−πτxJ =

[cos(π

τx) sin(π

τx)

−sin(πτx) cos(π

τx)

]applying this transformation to the system 2.6 we obtain the bilinear model 2.7

dY

dt=LmTrX − 1

TrY

dv

dt=Kf

MXTJY − B

Mv − FL

M

(2.7)

The first equation in 2.7 is in form 2.8

dY (t)

dt+ αY = αLmX (2.8)

Multiplying the differential equation 2.8 by the integral factor eαt yields to

eαtdY

dt+ αeαtY = αeαtLmX (2.9)

Integrating 2.9 and considering that input X = X(t0) is always constant during the integrationinterval time [t0, t] it yields to

eαtY (t) = eαt0Y (t0) + LmX(t0)[eαt − eαt0

](2.10)


with α = 1Tr

the first equation in 2.7 solution is

Y (t) = e−1Tr

(t−t0)Y (t0) + Lm(1− e1Tr

(t−t0))X(t0) (2.11)

Turning to electromechanical equation

dv

dt=Kf

MXTJY − B

Mv − FL

Mdv

dt+B

Mv =

Kf

MXTJY − FL

M

(2.12)

Multiplying the differential equation 2.12 by the integral factor eBMt yields to

eBMtdv

dt+B

MeBMtv = e

BMtKf

MXTJY − e

BMtFLM

(2.13)

Integrating 2.13 and considering that input X = X(t0) is always constant during the integrationinterval time [t0, t] it yields to

eBMtv(t)− e

BMt0v(t0) =

Kf

MXT (t0)J

∫ t

t0

eBMhY (h)dh− FL

B

[eBMt − e

BMt0]

(2.14)

To solve the integral in 2.14, where h is a dummy integration variable, it uses 2.10 with α = BM

and XTJX = 0 (skew-symmetry of J) then

Kf

MXT (t0)J

∫ t

t0

eBMhY (h)dh =

Kf

MXT (t0)JY (t0)e

BMt0(t− t0)

The velocity mechanical equation 2.12 solution is

v(t) = e−BM

(t−t0)v(t0) +Kf

MXT (t0)JY (t0)e−

BM

(t−t0)(t− t0)− FLB

[1− e−

BM

(t−t0)]

(2.15)

Solution to bilinear model system 2.7 were found from an initial time t0 to an arbitrary time t.

Y (t) = e−1Tr

(t−t0)Y (t0) + Lm(1− e1Tr

(t−t0))X(t0)

v(t) = e−BM

(t−t0)v(t0) +Kf

MXT (t0)JY (t0)e−

BM

(t−t0)(t− t0)− FLB

[1− e−

BM

(t−t0)] (2.16)


Integrating 2.15 in order to obtain mover position

x(t) =x(t0) +M

B(1− e−

BM

(t−t0))v(t0) +Kf

B(t0 − te−

BM

(t−t0))XT (t0)JY (t0)

+KfM

B2(1− e−

BM

(t−t0))XT (t0)JY (t0)− Kf

Bt0(1− e−

BM

(t−t0))XT (t0)JY (t0)

− FLB

(t− t0) +FLM

B2(1− e−

BM

(t−t0))

(2.17)

In a general form, the initial time is to = kT and states are found in a (k + 1)T time. In discretetime systems T is sample time.Defining a common notation

xk ≡ x(kT )

xk+1 ≡ x((k + 1)T )

Using above notation in 2.16 and 2.17 we have

Yk+1 =e−1TrTYk + Lm(1− e

1TrT )Xk

vk+1 =e−BMTvk +

Kf

MXTk JYke

− BMTT − FL

B

[1− e−

BMT]

xk+1 =xk +M

B(1− e−

BMT )vk +

Kf

B(kT − (k + 1)Te−

BMT )XT

k JYk

+KfM

B2(1− e−

BMT )XT

k JYk −Kf

BkT (1− e−

BMT )XT

k JYk

− FLBT +

FLM

B2(1− e−

BMT )

(2.18)

System 2.18 is a discrete-time version of 2.5 transformed (with position mover added), for come-back to original states, we need to make a backward transformation using following change ofcoordinates

Λ = eπτxJY

I = eπτxJX


to obtain

λβk+1 =sin(πτxk+1

)ρ1 + cos

(πτxk+1

)ρ2

λαk+1 =cos(πτxk+1

)ρ1 − sin

(πτxk+1

)ρ2

vk+1 =e−BMTvk +

Kf

Me−

BMTT (λαk i

βk − λ

βk iαk )− FL

BA

xk+1 =xk +M

BAvk

+

[Kf

B(kT − (k + 1)Te−

BMT ) +

KfM

B2A− Kf

BkTA

](λαk i

βk − λ

βk iαk )

− FLBT +

FLM

B2A

(2.19)

where

A = (1− e−BMT )

ρ1 = e−TTr (cos(

π

τxk)λ

αk + sin(

π

τxk)λ

βk) + (1− e−

TTr )Lm(cos(

π

τxk)i

αk + sin(

π

τxk)i

βk)

ρ2 = e−TTr (cos(

π

τxk)λ

βk − sin(

π

τxk)λ

αk ) + (1− e−

TTr )Lm(cos(

π

τxk)i

βk − sin(

π

τxk)i

αk )

To discretize current’s differential equations we use the backward difference method [40] andfinally we feature the discrete time version of the linear induction motor model 2.42.

iβk+1 =iβk −[Rs

ρLs+

1− ρρTr

]Tiβk −

Lmπ

ρLsLrτTvλαk +

LmρLsLrTr

Tλβk +1

ρLsTuβk

iαk+1 =iαk −[Rs

ρLs+

1− ρρTr

]Tiαk +

LmρLsLrTr

Tλαk +Lmπ

ρLsLrτTvλβk +

1

ρLsTuαk


)ρ1 + cos

(πτxk+1

)ρ2


)ρ1 − sin

(πτxk+1

)ρ2

vk+1 =e−BMTvk +

Kf

Me−

BMTT (λαk i

βk − λ

βk iαk )− FL

BA

xk+1 =xk +M

BAvk +

[M

B2A− 1

BT

]FL

+

[Kf

B(kT − (k + 1)Te−

BMT ) +

KfM

B2A− Kf

BkTA

](λαk i

βk − λ

βk iαk )

(2.20)

Discrete time model Simulation results are shown in Figure 2.2 refereed to axis α.


(a) Mover velocity of LIM (b) β-Axis Voltage input

(c) β-axis Continuos and discrete time currents (d) β-axis Continuos and discrete time fluxes

Figure 2.2: Mover velocity, voltage, currents and fluxes resulting from the model simulation usingODE45 function of Matlabr, in the continuos case, and discrete Time model of LIM2.20

2.2.3. LIM Model taking into account attraction force andend-effects

The dynamic model of the linear induction motor is analyzed by using the d − q model of theequivalent electrical circuit with end effects included [15, 29]. The q-axis equivalent circuit of theLIM is identical to the q-axis equivalent circuit of the induction motor (RIM), i.e. the parametersdo not vary with the end effects. However, the d-axis entry linoric currents affect the air gap fluxby decreasing λdr. Therefore, the d-axis equivalent circuit of the RIM cannot be used in the LIM


analysis when the end effects are considered [29].

Figure 2.3: Linor currents at the entry and exit ends for given velocity.

The end effects are not very noticeable in conventional induction motors. On the other hand, inLIM, these effects become increasingly relevant with the increase in the relative velocity betweenthe primary and the linor. Therefore, the end effects will be analyzed as a function of the LIM’svelocity [41]. The end effects behave differently at the entry and exit ends of the LIM. The currentsinduced in the linor at the entry end decay more slowly than at the exit end, due to its larger timeconstant.

Equivalent Circuit for LIM

Figure 2.4(a) shows a conceptual construccion of LIM. When the primary moves, the linor is con-tinuously replaced by a new linoric region. This new linoric region tends to oppose to the suddenincrease in the penetration of the magnetization flux allowing a gradual accumulation of the mag-netization field density in the air gap. The arising of a new linoric region and its influence in themagnetic field modifies the LIM performance when compared to the traditional rotary inductionmotor (RIM) [29]. Such flux variation along the motor length is illustrated in Figure 2.4(c)

As the primary coil set of the LIM moves, a new field penetrates into the reaction rail in the entryarea, whereas the existing field disappears at the exit area of the primary core. Both generation anddisappearance of the fields create the eddy current in the reaction rail. The eddy current in the entrygrows very rapidly to mirror the primary current, nullifying the primary MMF and reducing theflux to nearly zero at entry [42]. On the other hand, the eddy current at the exit generates a kind ofwake field, dragging the moving motion of the primary core. The rise and decay of the secondaryeddy current are controlled by the sheet leakage time constant Tlr = Llr/Rr and total secondarytime constant Tr = Lr/Rr respectively [42]. Since Tlr is small compared with Tr the eddy currentat the entry grows very rapidly to the primary current level, and then decay slowly (see Fig.2.4(b)).

Hereafter, we consider the problem of incorporating the end effects into an equivalent circuit mod-el. Note from the previous argument that, the secondary eddy current generated by the end effect


Figure 2.4: (a) Eddy current generation at the entry and exit of the air gap when the primary coilmoves with velocity v. (b) Polarity and decaying profile of the entry and exit eddycurrents. (c) Air gap flux profile. Taken from [29]

is in phase opposition to the primary current.

Since we align the reference frame with the reaction linor flux and call it d axis, it results in λqr = 0.It should be noted that as far as λqr = 0 and λdr does not change, the end effect does not play anyrole in equivalent circuit. Since iqε = −iqs the entry q axis eddy current keeps λqr = 0 within smallerror bound. Hence, the q axis equivalent circuit is identical to the case of the RIM. However, the daxis air gap flux is affected much by the eddy current since d axis entry eddy current, idε, reducesλdr. Therefore, the d axis dynamic model of the RIM is not appropriate to that of the LIM.

Magnetizing Inductance Reflecting the End Effects

As the primary moves, the primary MMF seen by the rail will be reduced at the entry and bereflected in the exit rail to maintain the air gap flux (continuity of flux). Specifically, the polarity ofthe entry eddy current is opposite to that of the exit eddy current, since they are, in nature, opposingto the generation and disappearance of the fields, respectively. Note that the entry eddy current hasa longer decaying time constant than the exit eddy current, since inductance is larger in the air gapthan in the open air. The profile of the eddy currents are plotted in Fig. 2.5 based on the normalized


time scale [42].

Figure 2.5: (a) Effective air gap MMF and (b) eddy current profile in normalized time scale

Noting that the d axis entry eddy current decays with the time constant Tr, the average value of thed axis entry eddy current idε over the motor length is given by

iε =idsTv

∫ Tv

0

e−t/Trdt (2.21)

where Tv = D/v, and D, v are the motor length and velocity, respectively. Note that Tv = D/vis the time taken for the motor to traverse a point. Since the travel distance over the period Tr isequal to vTr we can normalize the motor length by vTr such that [42]

Q =vTvvTr

=DRr

(Lm + Llr)v(2.22)

Note that Q is dimensionless but represents the motor length on the normalized time scale. On thisbasis, the motor length is clearly dependent on the motor velocity so that at zero velocity, the motorlength is infinitely long. As the velocity rises, the motor length will effectively shrink. Using 2.22,


equation 2.21 can be rewritten as follows:

iε =idsQ

∫ Q

0

e−xdx = ids1− e−Q

Q(2.23)

Therefore, the effective magnetizing current is reduced such that

ids − iε = ids

[1− 1− e−Q

Q

](2.24)

However, the reduction in the magnetizing current due to the eddy current can be accounted bymodifying the magnetizing inductance such that

L′m = Lm(1− f(Q)) (2.25)

where f(Q) =[1− e−Q

]/Q [42]. As velocity tends to zero, L′m converges to Lm i.e., the LIM

dynamics becomes equivalent to the RIM dynamics as the end effect disappears. Fig. 2.5 showsthe effective air gap MMF and the eddy current profile in normalized time scale.

Equivalent Series Resistor Reflecting Rail Eddy Current Losses

When the entry and exit eddy currents circulate in the rail, ohmic loss will be produced by Rr.Note that the RMS value of the entry eddy current over the motor length is given by

iεrms =

[i2dsQ

∫ Q

0

e−2xdx

]1/2

= ids

[1− e−2Q

2Q

]1/2

(2.26)

Hence, the loss caused by the entry eddy current is evaluated as [42]

Pentry = i2εrmsRr = i2dsRr1− e−2Q

2Q(2.27)

Following the method in [42], we also evaluate the losses due to the exit eddy current by the timerate of change of the magnetic energy as it leaves the motor air gap. Note from 2.23 that the totaleddy current in the air gap is equal to ids(1 − e−Q). This current must be vanished at the exit railduring Tv to satisfy the steady state condition in air gap flux. Hence, the loss due to the exit eddycurrent is given by

Pexit =Lri

2ds(1− e−Q)2

2Tv= i2dsRr

(1− e−Q)2

2Q. (2.28)


Summing 2.27 and 2.28, the total ohmic losses due to eddy current in the rail is given by Thispower loss can be represented as a serially connected resistor Rrf(Q) in the magnetizing currentbranch. Fig. 2.6 shows the final proposed equivalent Duncan’s circuits in which the end effects ofthe LIM are considered.

Figure 2.6: The LIM equivalent circuits taking into account the end effects.

Duncan´s circuit has been developed considering velocity and power loss. It supposes uniformwinding and materials, symmetric impedances per phase and equal mutual inductances. It´s basedon traditional model of three-phase, Y-connected rotatory induction motor whit linear magneticcircuit in a synchronous reference system (superscript “ e ”) aligned with the linor flux. Also onlylongitudinal end effects have been considered.

Duncan´s model has been adopted in order to obtain a space state representation both continuous-time and discrete-time. Several techniques have been developed for non linear dynamics analysisin the state space.


Parameter Q, function f(Q), Magnetizing Inductance Reflecting the End Effects and EquivalentSeries Resistor Reflecting Rail Eddy Current Losses have been derived from circuit theory.

The Q factor is associated with the length of the primary, and to a certain degree, quantifies theend effects as a function of the velocity v as described by equation 2.29

Q =DRr

Lrv(2.29)

Note that the Q factor is inversely dependent on the velocity, i.e., for a zero velocity the Q factormay be considered infinite, and the end effects may be ignored. As the velocity increases the endeffects increases , which causes a reduction of the LIM’s magnetization current. This effect maybe quantified in terms of the magnetization inductance with the equation:

L′

m = Lm(1− f(Q))

where

f(Q) =1− e−Q

Q

The resistance in series with the inductance (L′m) in the magnetization branch of the equivalent

electrical circuit of the d-axis (Figure 2.6(a)), is determined in relation to the increase in losses oc-curring with the increase of the currents induced at the entry and exit ends of the linor. These lossesmay be represented as the product of the linor resistance Rr by the factor f(Q), ie, Rrf(Q). [31]

From the d− q equivalent circuit of the LIM (Figure 2.6), the primary and linor voltage equationsin a stationary reference system aligned with the linor flux are given by:

uds = Rsids +Rrf(Q)(ids + idr) +dλdsdt

uqs = Rsiqs +dλqsdt

udr = Rridr +Rrf(Q)(ids + idr) +dλdrdt

+π

τvλqr

uqr = Rriqr +dλqrdt− π

τvλdr

(2.30)

Due to the short-circuited secondary their voltages are zero, that is, udr = uqr = 0.


The linkage fluxes are given by the following equations:

λds = Lsids + Lmidr − Lmf(Q)(ids + idr)

λqs = Lsiqs + Lmiqr

λdr = Lridr + Lmids − Lmf(Q)(ids + idr)

λqr = Lriqr + Lmiqs

(2.31)

To develop a state space LIM model from 2.30 and 2.31 is necessary to combine both equations.Because q-axis equivalent circuit of the LIM is identical to the q-axis equivalent circuit of theinduction motor (RIM), the parameters do not vary with the end effects and so dλqr

dtand diqs

dtin 2.2

remaind it equals.

diqsdt

= −[Rs

ρLs+

1− ρρTr

]iqs −

Lmπ

ρLsLrτvλdr +

LmρLsLrTr

λqr +1

ρLsuqs

dλqrdt

=LmTriqs +

π

τvλdr −

1

Trλqr

The RIM electrical torque in an arbitrary reference frame is giving by [43], and modifying it withrelation 2.1we obtain following LIM thrust force

Fe =3

2

π

τωr[ω(λdsiqs − λqsids) + (ω − ωr)(λdriqr − λqridr)]

in a stationary reference frame (ω = 0) the thrust force becomes

Fe =3

2

π

τ[λqridr − λdriqr] (2.32)

isolating idr from λdr in 2.31

idr =λdr − Lm(1− f(Q))ids

Lr − Lmf(Q)(2.33)

isolating iqr from λqr in 2.31

iqr =λqr − Lmiqs

Lr(2.34)

Substituting idr and iqr into 2.32 results in:

Fe =3

2

π

τ

LmLr

[λdriqs +

f(Q)

Lr − Lmf(Q)λqrλdr −

1− f(Q)

1− LmLrf(Q)

λqrids

](2.35)


Then space state mechanical equation is giving by 2.36

dv

dt=Kf

M

[λdriqs +

f(Q)

Lr − Lmf(Q)λqrλdr −

1− f(Q)

1− LmLrf(Q)

λqrids

]− B

Mv − FL

M(2.36)

To get the space state equation of λdr we isolate idr from λdr in 2.31and substituting it into udr in2.30 results:

udr =Rr(1 + f(Q))

Lr − Lmf(Q)λdr −

[RrLm(1− f 2(Q))

Lr − Lmf(Q)−Rrf(Q)

]ids +

π

τvλqr +

dλdrdt

Considering short-circuited linor circuit (udr = 0) and solving for dλdrdt

gets

dλdrdt

= −Rr(1 + f(Q))


π

τvλqr +

Rr(Lm − Lrf(Q))

Lr − Lmf(Q)ids (2.37)

Substituting the first equation of 2.31 into first equation of 2.30 results:

uds =

[Rs +Rrf(Q)− Lm

df(Q)

dt

]ids + [Ls − Lmf(Q)]

didsdt

+ Lm[1− f(Q)]didrdt− Lm

df(Q)

dtidr

(2.38)

Isolating idr from λdr in 2.31

idr =1


Lm(1− f(Q))

Lr − Lmf(Q)ids

and substituting into 2.38 results

uds =

[Rs +Rrf(Q)− (Lr − Lm)2

(Lr − Lmf(Q))2Lm

df(Q)

dt

]ids

+

[Ls − Lmf(Q)− L2

m(1− f(Q))2

Lr − Lmf(Q)

]didsdt

+Lm(Lm − Lr)

(Lr − Lmf(Q))2

df(Q)

dtλdr +

Lm(1− f(Q))

Lr − Lmf(Q)

dλdrdt


substituting dλdrdt

(2.37) in last term into above equation we obtain:

uds =

[Rs +Rrf(Q)− (Lr − Lm)2

(Lr − Lmf(Q))2Lm

df(Q)

dt+RrLm(1− f(Q))

Lr − Lmf(Q)

(Lm − Lrf(Q))

Lr − Lmf(Q)

]ids

+

[Ls − Lmf(Q)− L2

m(1− f(Q))2

Lr − Lmf(Q)

]didsdt

+

[Lm(Lm − Lr)

(Lr − Lmf(Q))2

df(Q)

dt− RrLm(1− f 2(Q))

(Lr − Lmf(Q))2

]λdr −

Lm(1− f(Q))

Lr − Lmf(Q)

π

τvλqr

Solving for didsdt

didsdt

=[Rs +Rrf(Q)][Lr − Lmf(Q)]2 − Lm(Lr − Lm)2 df(Q)

dt+RrLm[1− f(Q)][Lm − Lrf(Q)]

[LsLr − LsLmf(Q)− LrLmf(Q)− L2m + 2L2

mf(Q)][Lmf(Q)− Lr]ids

+Lm(Lm − Lr)df(Q)

dt−RrLm[1− f 2(Q)]


mf(Q)][Lmf(Q)− Lr]λdr

+Lm[1− f(Q)]

LsLr − LsLmf(Q)− LrLmf(Q)− L2m + 2L2

mf(Q)

π

τvλqr

+Lr − Lmf(Q)


mf(Q)uds

(2.39)

Grouping the state equations and changing the index d and q by α and β respectively, and omittingthe primary and secondary (linor) indexes because the voltages and currents are with respect to


primary and the fluxes are with respect to secondary, we obtain 2.40:

diβdt

=−[Rs

ρLs+

1− ρρTr

]iβ −

Lmπ

ρLsLrτvλα +

LmρLsLrTr

λβ +1

ρLsuβ

diαdt

=[Rs +Rrf(Q)][Lr − Lmf(Q)]2 − Lm(Lr − Lm)2 df(Q)

dt+RrLm[1− f(Q)][Lm − Lrf(Q)]


mf(Q)][Lmf(Q)− Lr]iα

+Lm(Lm − Lr)df(Q)

dt−RrLm[1− f 2(Q)]


mf(Q)][Lmf(Q)− Lr]λα

+Lm[1− f(Q)]


mf(Q)

π

τvλβ

+Lr − Lmf(Q)


mf(Q)uα

dλβdt

=LmTriβ +

π

τvλα −

1

Trλβ

dλαdt

=− Rr(1 + f(Q))

Lr − Lmf(Q)λα −

π

τvλβ +

Rr(Lm − Lrf(Q))

Lr − Lmf(Q)iα

dv

dt=Kf

M

[λαiβ +

f(Q)

Lr − Lmf(Q)λβλα −

1− f(Q)

1− LmLrf(Q)

λβiα

]− B

Mv − FL

M

dx

dt=v

(2.40)

where v is the mover linear velocity; λα and λβ are the d-axis an q-axis secondary flux; iα andiβ are the d-axis and q-axis primary current; uα and uβ are the d-axis and q-axis primary voltage;

Tr = LrRr

is the secondary time constant; ρ = 1 −(

L2m

LsLr


2πLmτLr

is the force constant; Rs is the winding resistance per phase; Rr is the secondary resistance perphase referred primary; Lm is the magnetizing inductance per phase; Lr is the secondary induc-tance per phase referred primary; Ls is the primary inductance per phase; FL is the external forcedisturbance; M is the total mass of the mover; B is the viscous friction and iron-loss coefficient;τ is the pole pitch; D is the primary length in meters; Q = DRr

Lrvis a factor related to the primary

length, which quatifies the end effects as a function of the speed and f(Q) = 1−e−QQ

is the factorrelated to the losses in the magnetization branch.

To discretize the state LIM model with end effects we use the backward difference method [40] andfinally we obtain an approximate discrete time version of the LIM model 2.43 taking into accountend effects.


iβk+1 =iβk −[Rs

ρLs+

1− ρρTr

]Tiβk −

Lmπ

ρLsLrτTvkλ

αk +

LmρLsLrTr

Tλβk +1

ρLsTuβk

iαk+1 =iαk +[Rs +Rrf(Q)][Lr − Lmf(Q)]2 − Lm(Lr − Lm)2 ∆f(Q)

T+RrLm[1− f(Q)][Lm − Lrf(Q)]


mf(Q)][Lmf(Q)− Lr]Tiαk

+Lm(Lm − Lr)∆f(Q)

T−RrLm[1− f 2(Q)]


mf(Q)][Lmf(Q)− Lr]Tλαk

+Lm[1− f(Q)]


mf(Q)

π

τTvkλ

βk

+Lr − Lmf(Q)


mf(Q)Tuαk

λβk+1 =λβk +LmTrTiβk +

π

τTvkλ

αk −

1

TrTλβk

λαk+1 =λαk −Rr(1 + f(Q))

Lr − Lmf(Q)Tλαk −

π

τTvkλ

βk +

Rr(Lm − Lrf(Q))

Lr − Lmf(Q)Tiαk

vk+1 =vk +Kf

MT

[λαk i

βk +

f(Q)

Lr − Lmf(Q)λβkλ

αk −

1− f(Q)

1− LmLrf(Q)

λβk iαk

]− B

MTvk −

FLMT

xk+1 =xk + vkT

(2.41)

where vk = v(kT ) is the mover linear velocity; λαk = λα(kT ) and λβk = λβ(kT ) are the d-axisan q-axis secondary flux; iαk = iα(kT ) and iβk = iβ(kT ) are the d-axis and q-axis primary current;uαk = uα(kT ) and uβk = uβ(kT ) are the d-axis and q-axis primary voltage; Tr = Lr

Rris the sec-

ondary time constant; ρ = 1−(

L2m

LsLr


2πLmτLr

is the force constant;Rs is the winding resistance per phase; Rr is the secondary resistance per phase referred primary;Lm is the magnetizing inductance per phase; Lr is the secondary inductance per phase referredprimary; Ls is the primary inductance per phase; FL is the external force disturbance; M is thetotal mass of the mover; B is the viscous friction and iron-loss coefficient; τ is the pole pitch; D isthe primary length in meters; Q = DRr

Lrvkis a factor related to the primary length, which quantifies

the end effects as a function of the speed; f(Q) = 1−e−QQ

is the factor related to the losses in the

magnetization branch and ∆f(Q)T

= df(Q)dt|t=kT .

Figure 2.7 shows the end effects on mover velocity, fluxes and currents.


(a) Mover velocity of LIM with and without end-efeccts

(b) Velocity difference Vs mover velocity withoutend-effects

(c) β-axis With and without end-effects currents (d) β-axis With and without end-effects fluxes

Figure 2.7: Mover velocity, velocity differences, currents and fluxes resulting from the modelsimulation using ODE45 function of Matlab, taking into account end effects in modelof LIM 2.40 and without end-effects model 2.3.

2.2.4. LIM behavior

Figures 2.8, 2.9 and 2.10 show the system 2.40 behavior when the frequency of input voltage vary.The steady state velocity is a periodic wave in all cases, but when the fed frequency is lower, higheroutput frequency components appear. Phase portraits in Subfigures (b) and (c) of Figures 2.8,2.9and 2.10 with attractive limit cycles are shown.

2.3 Analysis of Stability and Bifurcation in Power Electronic LIM Drive System 29

(a) Mover velocity of LIM withend-effects

(b) Phase portrait of mover ve-locity Vs iα

(c) Phase portrait of λα Vs iα

Figure 2.8: LIM behavior with 30 Hz input frequency. Mover velocity and phase portraits of somestate variables.




Figure 2.9: LIM behavior with 60 Hz input frequency. Mover velocity and phase portraits of somestate variables.

2.3. Analysis of Stability and Bifurcation in PowerElectronic LIM Drive System

Adjustable speed control of LIM with the use of variable-frequency power source using PWMinverters will be used in the analysis of stability and chaotic behavior in LIM dynamics. We willuse the computer–aided methods which explained in subsection 1.3. First we will derive the math-ematical model of the system and we will investigate the stability properties of their nominalsteady-states solutions. Next the Poincaré map will be defined in terms of the periodicity of thesolutions and their stability will be investigated from the point of view of bifurcation phenome-na. Bifurcations occur in the system depending on the driver and load conditions and bifurcationsvalues will be determined by method described in Section 4.5.6 of [13].





Figure 2.10: LIM behavior with 120 Hz input frequency. Mover velocity and phase portraits ofsome state variables.

2.3.1. Model of Power Electronic LIM Drive SystemFigure 2.11 shows a schematic diagram of the power electronic LIM Drive System that we willstudy. The system comprises a rectifier with filter, an inverter, a three phase LIM, and mechanicalload. The induction motor is supplied power through three symmetrical a, b, c primary windings.The secondary windings are short circuited and only two phases, α and β, are shown for conve-nience.

Figure 2.11: Power electronic LIM drive system

Model of LIM

The discrete time LIM model (2.42) is obtained by the continuous time LIM Model solution,without end effects. For discrete time LIM modeling taking into account end effects we will useRunge-Kutta method [44] (because an analytic solution is not possible). In both cases the model is


a map that describes LIM dynamics.

iβk+1 =iβk −[Rs

ρLs+

1− ρρTr

]Tiβk −

Lmπ

ρLsLrτTvkλ

αk +

LmρLsLrTr

Tλβk +1

ρLsTuβk

iαk+1 =iαk −[Rs

ρLs+

1− ρρTr

]Tiαk +

LmρLsLrTr

Tλαk +Lmπ

ρLsLrτTvkλ

βk +

1

ρLsTuαk


)ρ1 + cos

(πτxk+1

)ρ2


)ρ1 − sin

(πτxk+1

)ρ2

vk+1 =e−BMTvk +

Kf

Me−

BMTT (λαk i

βk − λ

βk iαk )− FL

BA

xk+1 =xk +M

BAvk +

[M

B2A− 1

BT

]FL

+

[Kf

B(kT − (k + 1)Te−

BMT ) +

KfM

B2A− Kf

BkTA

](λαk i

βk − λ

βk iαk )

(2.42)

The LIM model taking into account end-effects discretized by Euler´s method is given by 2.43

iβk+1 =iβk −[Rs

ρLs+

1− ρρTr

]Tiβk −

Lmπ

ρLsLrτTvkλ

αk +

LmρLsLrTr

Tλβk +1

ρLsTuβk

iαk+1 =iαk +[Rs +Rrf(Q)][Lr − Lmf(Q)]2 − Lm(Lr − Lm)2 ∆f(Q)

T+RrLm[1− f(Q)][Lm − Lrf(Q)]


mf(Q)][Lmf(Q)− Lr]Tiαk

+Lm(Lm − Lr)∆f(Q)

T−RrLm[1− f 2(Q)]


mf(Q)][Lmf(Q)− Lr]Tλαk

+Lm[1− f(Q)]


mf(Q)

π

τTvkλ

βk

+Lr − Lmf(Q)


mf(Q)Tuαk

λβk+1 =λβk +LmTrTiβk +

π

τTvkλ

αk −

1

TrTλβk

λαk+1 =λαk −Rr(1 + f(Q))

Lr − Lmf(Q)Tλαk −

π

τTvkλ

βk +

Rr(Lm − Lrf(Q))

Lr − Lmf(Q)Tiαk

vk+1 =vk +Kf

MT

[λαk i

βk +

f(Q)

Lr − Lmf(Q)λβkλ

αk −

1− f(Q)

1− LmLrf(Q)

λβk iαk

]− B

MTvk −

FLMT

xk+1 =xk + vkT

(2.43)


Model of Inverter

We assume that the inverter in Figure 2.11 is the voltage-controlled type and lossless. Such inverteris usually implemented so that their output voltages va, vb, vc, are expressed as va(t)

vb(t)vc(t)

=

sa(t)sb(t)sc(t)

vDC (2.44)

where vDC is the voltage across the capacitor CF of the filter. The functions sa(t), sb(t) and sc(t)represent the characteristics of inverters. They are usually given to be balanced and periodic withperiod Te = 2π/we, and are given by

sa(t) = sa(t+ Te), sb(t) = sb(t+ Te), sc(t) = sc(t+ Te)

sa(t) = sb

(t+

1

3Te

)= sc

(t+

2

3Te

)sa(t) + sb(t) + sc(t) = 0

where fe = 1/Te corresponds to the drive frequency of the inverter.

Then the input voltage of LIM 2.42 or 2.43 is given by uα(t)uβ(t)

0

= C

va(t)vb(t)vc(t)

= C

sa(t)sb(t)sc(t)

vDC =

sα(t)sβ(t)

0

vDC (2.45)

where C is the Park´s transformation,

C =2

3

cosθe cos(θe − 23π) cos(θe + 2

3π)

−sinθe −sin(θe − 23π) −sin(θe + 2

3π)

12

12

12

(2.46)

and its inverse is

C−1 =

cosθe −sinθe 1cos(θe − 2

3π) −sin(θe − 2

3π) 1

cos(θe + 23π) −sin(θe + 2

3π) 1

(2.47)


Line to line voltages are (see Figure 2.11):

Vab = VDC(S1 − S2)Vbc = VDC(S2 − S3)Vca = VDC(S3 − S1)

(2.48)

withS4 = 1− S1

S5 = 1− S2

S6 = 1− S3

andSi ∈ 0, 1

for i = 1 . . . 6

Motor phase voltages can be calculated by:

va(t) = −(1/3)(Vca − Vab)vb(t) = −(1/3)(Vab − Vbc)vc(t) = −(1/3)(Vbc − Vca)

(2.49)

Substituting 2.48 in 2.49 we obtain:

va(t) = −(1/3)VDC(−2S1 + S2 + S3)vb(t) = −(1/3)VDC(S1 − 2S2 + S3)vc(t) = −(1/3)VDC(S1 + S2 − 2S3)

(2.50)

Rearranging before equation

va(t)vb(t)vc(t)

= −1

3

−2 1 11 −2 11 1 −2

S1

S2

S3

VDC =

sa(t)sb(t)sc(t)

VDC (2.51)

The backward transformation can be obtained as:

S1(t) =1

2+va(t)

VDC

S2(t) =1

2+vb(t)

VDC

S3(t) =1

2+vc(t)

VDC

(2.52)

Substituting 2.51 and 2.46 in 2.45, when axis α is aligned with axis a (θe = 0), LIM primary


voltages are: uα(t)uβ(t)

0

= −1

3

2

3

1 −12−1

2

0√

32−√

32

12

12

12

−2 1 11 −2 11 1 −2

S1

S2

S3

VDC (2.53)

then, uα(t)uβ(t)

0

=

23−1

3−1

3

0 1√3− 1√

3

0 0 0

S1

S2

S3

VDC =

sα(t)sβ(t)

0

vDC (2.54)

We will use a pulse-width-modulated inverter. In this inverter, the dc input voltage is essentiallyconstant in magnitude (VDC), therefore the inverter must control the magnitude and frequency ofthe ac output voltages. This is achieved by PWM of the inverter switches [45].

There are various schemes to pulse-width modulate the inverter switches in order to shape the out-put ac voltages to be as close to a sine wave as possible. We will use a SVPWM (Space VectorPulse-Width Modulation) scheme.

Table 2.1: Space Vector Modulation Inverter States

Step S3 S2 S1 Vab Vbc Vca Va Vb Vc Vd Vq Vector

- 0 0 0 0 0 0 0 0 0 0 0 U(000)

1 0 0 1 VDC 0 −VDC 2VDC/3 −VDC/3 −VDC/3 2VDC/3 0 U0

2 0 1 1 0 VDC −VDC VDC/3 VDC/3 −2VDC/3 VDC/3 VDC/√

3 U60

3 0 1 0 −VDC VDC 0 −VDC/3 2VDC/3 −VDC/3 −VDC/3 VDC/√

3 U120

4 1 1 0 −VDC 0 VDC −2VDC/3 VDC/3 VDC/3 −2VDC/3 0 U180

5 1 0 0 0 −VDC VDC −VDC/3 −VDC/3 2VDC/3 −VDC/3 −VDC/√

3 U240

6 1 0 1 VDC −VDC 0 VDC/3 −2VDC/3 VDC/3 VDC/3 −VDC/√

3 U300

- 1 1 1 0 0 0 VDC VDC VDC 0 0 U(111)

Each one of the three inverter outputs can be in one of two states. The inverter output can be eitherconnected to the + bus rail or the – bus rail, which allows for 23 = 8 possible states that the outputcan be in (see Table 2,1). The two states where all three outputs are connected to either the + busor the – bus are considered null states because there is no line-to-line voltage across any of thephases. These are plotted at the origin of the SVM Star (Figure 2.12). The remaining six states are


represented as vectors with 60 degree rotation between each state, as shown in Figure 2.12.

Figure 2.12: Space Vector Modulation (SVM)

To generate a balanced sinusoidal set of three-phase motor voltages the logical sequence step1...step6is cycling. The logical sequence may be started in any step.

Figure 2.13: Logical sequence step1...step6

The process of SVPWM allows the representation of any resultant vector by the sum of the com-ponents of two adjacent vectors. In Figure 2.14 UOUT is the desired resultant. It lies in the sectorbetween U60 and U0. If during a given PWM period T U0 is the output for T1/T and U60 is output


for T2/T, the average for the period will be UOUT.

Figure 2.14: Average Space Vector Modulation

Since T1 + T2 < T (if there is not over modulation), during the remaining time the null vectors areoutput (See Figure 2.15).

Figure 2.15: PWM for period T

The PWM device will be configured for center aligned PWM. Which forces symmetry about thecenter of the period. This configuration produces two pulses line-to-line during each period. The ef-fective switching frequency is doubled, reducing the ripple current while not increasing the switch-ing losses in the power device. The Figure 2.16 shows the effect described before.

The values for T1 and T2 can be extracted with no extra calculations by using a modified inverseClarke transformation, where vd and vq are swaped compared to the normal inverse Clarke.


(a) PWM1 and PWM3 pulses

(b) Line to line pulses

Figure 2.16: Effective switching Frequency.

Vr1 = vq

Vr2 = −vq2

+

√3

2vd

Vr3 = −vq2−√

3

2vd

By reversing vd and vq, a reference axis is generated that is shifted by 30 degrees from the SVMstar. As a result, for each of the six segments one axis is exactly opposite to that segment and theother two axis symmetrically bound the segment (Figure 2.17(a)). The values of the vector com-ponents along those two bounding axis are equal T1 and T2. [46]

SVM sectors can be determinate by the Vr1, Vr2 and Vr3 values. Then dPWM1, dPWM2 and


(a) SVM star and Vr1, Vr2 and Vr3Vectors

(b) Vr1, Vr2 and Vr3 Signals

Figure 2.17: Modified Inverse Clarke Transform.

dPWM3 duty cycles can be calculated for each sector. In table 2.2 T0 = (T − T1 − T2)/2,Tb = T0 + T2 and Ta = Tb + T1.

Table 2.2: PWMs duty cycle

Sector Vr Values T1 T2 dPWM1 dPWM2 dPWM300 − 600 Vr1 ≥ 0 and Vr2 ≥ 0 Vr2T Vr1T Ta Tb T0

600 − 1200 Vr2 < 0 and Vr3 < 0 −Vr2T −Vr3T Tb Ta T01200 − 1800 Vr1 ≥ 0 and Vr3 ≥ 0 Vr1T Vr3T T0 Ta Tb1800 − 2400 Vr1 < 0 and Vr2 < 0 −Vr1 −Vr2 T0 Tb Ta2400 − 3000 Vr2 ≥ 0 and Vr3 ≥ 0 Vr3 Vr2 Tb T0 Ta3000 − 00 Vr1 < 0 and Vr3 < 0 −Vr3 −Vr1 Ta T0 Tb

Let the state variable x be defined

x = [iβk , iαk , λ

βk , λ

αk , vk, xk]


(a) Sector 1 (0 to 60) (b) Sector 2 (60 to 120)

(c) Sector 3 (120 to 180) (d) Sector 4 (180 to 240)

(e) Sector 5 (240 to 300) (f) Sector 6 (300 to 360)

Figure 2.18: PWM periods for the sectors of SVM.

then the LIM drive system can be written in the following nonlinear non autonomous form:


xk+1 = f(xk) + g(xk)uk (2.55)

with

f(xk) =

iβk −[RsρLs

+ 1−ρρTr

]Tiβk − Lmπ

ρLsLrτTvkλ

αk + Lm

ρLsLrTrTλβk

iαk −[RsρLs

+ 1−ρρTr

]Tiαk + Lm

ρLsLrTrTλαk + Lmπ

ρLsLrτTvkλ

βk

sin(πτxk+1

)ρ1 + cos

(πτxk+1

)ρ2

cos(πτxk+1

)ρ1 − sin

(πτxk+1

)ρ2

e−BMTvk +

KfMe−

BMTT (λαk i

βk − λ

βk iαk )

xk + MBAvk +

[KfB

(kT − (k + 1)Te−BMT ) +

KfM

B2 A− KfBkTA

](λαk i

βk − λ

βk iαk )

(2.56)

g(xk) =

1ρLs

T 0 0

0 1ρLs

T 0

0 0 00 0 00 0 −A

B

0 0[MB2A− 1

BT]

(2.57)

uk =

uβkuαkFL

(2.58)

for discrete time model without end effects.

2.4 ZAD and FPIC Control Strategies 41

Or:

f(xk) =

iβk −[RsρLs

+ 1−ρρTr

]Tiβk − Lmπ

ρLsLrτTvkλ

αk + Lm

ρLsLrTrTλβk

iαk +[Rs+Rrf(Q)][Lr−Lmf(Q)]2−Lm(Lr−Lm)2 ∆f(Q)

T+RrLm[1−f(Q)][Lm−Lrf(Q)]

[LsLr−LsLmf(Q)−LrLmf(Q)−L2m+2L2

mf(Q)][Lmf(Q)−Lr] Tiαk

+Lm(Lm−Lr) ∆f(Q)

T−RrLm[1−f2(Q)]

[LsLr−LsLmf(Q)−LrLmf(Q)−L2m+2L2

mf(Q)][Lmf(Q)−Lr]Tλαk

+ Lm[1−f(Q)]LsLr−LsLmf(Q)−LrLmf(Q)−L2

m+2L2mf(Q)

πτTvkλ

βk

λβk + LmTrTiβk + π

τTvkλ

αk − 1

TrTλβk

λαk −Rr(1+f(Q))Lr−Lmf(Q)

Tλαk − πτTvkλ

βk + Rr(Lm−Lrf(Q))

Lr−Lmf(Q)Tiαk

vk +KfMT

[λαk i

βk + f(Q)

Lr−Lmf(Q)λβkλ

αk −

1−f(Q)

1−LmLr

f(Q)λβk i

αk

]− B

MTvk

xk + vkT

(2.59)

and g(x)

g(xk) =

1ρLs

T 0 0

0 Lr−Lmf(Q)LsLr−LsLmf(Q)−LrLmf(Q)−L2

m+2L2mf(Q)

T 0

0 0 00 0 00 0 − T

M

0 0 0

(2.60)

for discrete time model taking into account end-effects.

2.3.2. Stability Analysis

Due to the periodicity of the functions sd(t) and sq(t), and considering load force FL constant thesystem 2.55 has an periodic solution. As stated above, the stability of a periodic solution can beinvestigated by introducing The Poincaré map appropriately and evaluating the eigenvalues of itsJacobian matrix. This is denoted as DP (x∗0). The periodic solution x∗(kT ) is asymptotically sta-ble if all the eigenvalues of the Jacobian matrix DP (x∗0) are inside the unit circle on the complexplane. The Jacobian matrix could be obtained by the use of the Newton-Raphson method (subsec-tion 5.2.2).

2.4. ZAD and FPIC Control Strategies

The controller designed in the present work combines Zero Average Dynamics (ZAD) and FixedPoint Inducting Controller (FPIC) strategies, which have been reported in [47–54].


2.4.1. Zero Average Dynamic ZAD

ZAD control strategy: The application of Centered Pulse Width Modulation (CPWM) generates apiecewise smooth surface.

(a) Centered PWM (b) Surface per phase

Figure 2.19: surface

spwl(t) =

s1 + (t− kT )s+ if kT ≤ t ≤ t1s2 + (t− kT + dk

2)s− if t1 < t < t2

s3 + (t− kT + T + dk2

)s+ if t2 ≤ (k + 1)T(2.61)

The surface per phase can be approximated as a piecewise-linear function given by 2.61, where:

s+ = ((x− xref ) + ks1(x− xref )) + ks2(...x − ...

x ref )) + ks3(....x − ....

x ref ))∣∣∣x=x(kT ),u=1

s− = ((x− xref ) + ks1(x− xref )) + ks2(...x − ...

x ref )) + ks3(....x − ....

x ref )))∣∣∣x=x(kT ),u=−1

s1 = ((x− xref ) + ks1(x− xref )) + ks2((x− xref ) + ks3(...x − ...

x ref ))∣∣∣x=x(kT ),u=1

s2 = dk2s+ + s1

s3 = s1 + (T − dk)s−t1 = kT + dk

2

t2 = kT + (T − dk2

)t3 = (k + 1)T

(2.62)

with ks1 = KS

√LC, ks2 = KS1LC, ks3 = KS2LC

√LC are positive constants. Therefore, the

2.4 ZAD and FPIC Control Strategies 43

zero average condition is

(k+1)T∫kT

spwl(t)dt = 0 (2.63)

To find dk and satisfying zero average requirement, we solve equation (2.63) and obtain

Dk =2s1(x(kT )) + T s−(x(kT ))

s−(x(kT ))− s+(x(kT ))(2.64)

The duty cycle is given by.

dk =

1 if Dk > TDk/T if 0 ≤ Dk ≤ T0 if Dk < 0

(2.65)

We have experimentally measured and noticed that there is a period delay in the control action. Inthis case, the control action is taken from the data acquired in the past sampling time, and then wecompute the duty cycle as:

dk =2s1(x(k − 1)T ) + T s−(x(k − 1)T )

s−(x(k − 1)T )− s+(x(k − 1)T )(2.66)

2.4.2. Control of chaos with FPIC

Control of chaos, meaning suppression of the chaotic regime in a system by means of a small,time-dependent parameter or input perturbation, has been subject of extensive investigation [55].

A method of control chaos was developed in [50]. It is named Fixed Point Induction Control(FPIC). This technique showed great results with systems operating with time-delayed signals.The method starts on previous knowledge of fixed point and based on it the control strategy isdesigned. This control technique was first presented in [50] then tested numerically in [50,51], andthe first experimental results are shown in [54, 56].

FPIC Theorem Let a system described by a set of differential equations:

x (k + 1) = f (x (k)) (2.67)

Where x(t) ∈ <n and f : <n → <n. Suppose that there is a fixed point x∗, which is unstableand corresponds to the orbit to be controlled, ie, x∗ = f (x∗). Let J = ∂f

∂xthe Jacobian of the

system evaluated at the fixed point, then as the system is unstable there is at least one i, suchthat |λi (J)| > 1, where λ corresponds to the eigenvalues of the linearized system. Under these


conditions, the equation:

x (k + 1) =f (x (k)) +Nx∗

N + 1(2.68)

Ensures stabilization of the fixed point for some N positive real.

Proof Initially it is worth to note that the equation (2.68) does not change the fixed point ofequation 2.67. In this case the Jacobian of the new system can be expressed as:

Jc =1

N + 1J (2.69)

Where Jc is the Jacobian of the controlled system and J is the Jacobian of the uncontrolled sys-tem. Therefore, a proper selection of N ensures stabilization of the equilibrium, since the valuesof the controlled system will be the values of the original system, divided by the factor N + 1. Astraightforward way to compute N is through the Jury criterion.

Taking into account the strategies FPIC and ZAD, the new duty cycle is calculated as follow:

dk−FPIC =dk(k) +N · d∗

N + 1(2.70)

Where dk(k) is calculated as (2.66) and d∗ is duty cycle calculate in steady state (x1(kT ) = x1ref ).

The control chaos method described before is applied to Three–phase converter driven Three–phase loads under study in this thesis.

3 Parameter Estimation of InductionMotors

3.1. Introduction

In LIM, the air gap is very large compared with the RIMs. Further, the secondary part normallydoes not have slotted structure. It is just made of aluminum and steel plates. Therefore, the effec-tive air gap is larger than the physical air gap. High air gap makes a larger leakage inductance. Itleads to lower efficiency and lower power factor [57].

The Lab-Volt Model 8228-02 is a Single Side Linear Induction Motor (SLIM). The stator is com-posed of a three-legged laminated iron core upon which are mounted three identical coils A, B, C.Each coil has 500 turns of No. 21 AWG copper wire, with a tap at 300 turns.

The coils produce in each leg and corresponding salient pole, fluxes that are labeled φa, φb andφc. These fluxes are created by the currents ia, ib and ic that flow in the respective windings; con-sequently, the fluxes are 120o out of phase. This phase shift means that the fluxes attain theirmaximum value at different times, separated by intervals of 1

3fwhere f is the frequency of the

source. If the phase sequence is A-B-C, flux φb will attain its maximum value 13f

seconds after φa.Similarly, φc will reach its maximum value 1

3fs after φb. Then, the flux continually shifts from left

to right across the face of the salient poles. If two of the supply lines are interchanged, the phasesequence will reverse, and the flux will shift from right to left across the poles.

Knowing the distance d between the center of the poles, we can calculate the speed at which theflux moves. This is called the synchronous speed vs because it is directly related to the frequencyof the power supply. The synchronous speed is given by

vs = 3df (3.1)

in which,vs = synchronous speed [m/s]d = distance between poles [m]f = frequency [Hz]

in the Model 8228, d = 0,061m.

46 3 Parameter Estimation of Induction Motors

Figure 3.1: Experiment for LIM parameter identification.

3.2. Parameter Estimation of Linear Induction MotorLabVolt 8228-02

The experiment is shown in Figure 3.1, where a DC motor is coupled to the LIM under test. DCresistance test will be done to determine the value of Rs. The primary Inductance Ls will be cal-culated by running the LIM at synchronous speed. The secondary parameters i.e. Llr and R′r willbe calculated by blocked-mover test.There are five major components of the DSP-based electric-drives system, which will be used toperform the experiment. They are as follows: 1) Motor coupling system, 2) Power ElectronicsDrive Board, 3) Adjustable speed driver (ASD), 4) DSP based DS1104 R&D controller card andCP 1104 I/O board and 5) MATLAB Simulink and Control-desk.

1. Motor coupling system: This system contains the LIM that needs to be characterized. Thesystem has a mechanical coupling arrangement to couple LIM and DC motors. The systemalso has a linear encoder mounted which is used to measure the position and the speed ofthe LIM. The motors demand controlled pulse-width-modulated (PWM) voltages to run atcontrolled speed. The PWM voltages are generated by Power Electronics Drive Board andAdjustable speed driver (briefed next); the voltage sources thus generated are connected tothe motor coupling system as shown in Figure 3.1.

3.2 Parameter Estimation of Linear Induction Motor LabVolt 8228-02 47

2. Power Electronics Drive Board: This board has the capability to generate two independentPWM voltage sources (A1B1C1 and A2B2C2) from a constant DC voltage source. Hencetwo machines can be controlled independently for independent control variables, at the sametime. This board also provides the motor phase currents, dc-bus voltage etc. In this experi-ment we will use only one PWM voltage source (A2B2C2) to control the DC motor speed.To generate the controlled PWM voltage source, this board requires various digital controlsignals. These control signals dictates the magnitude of the PWM voltage source. They aregenerated by the DS1104 R&D Controller board inside the computer.

3. Adjustable Speed Driver (ASD): This board has the capability to generate PWM voltagesource (UVW) from a 3-phase voltage source. Hence LIM can be controlled independentlyfor independent control variables. This board also provides the motor phase currents, dc-bus voltage etc. To generate the controlled PWM voltage source, this board requires variousdigital control signals. These control signals dictates the magnitude and phase of the PWMvoltage source. They are generated by the DS1104 R&D Controller board inside the com-puter.

4. DS1104 R&D controller Board and CP 1104 I/O board: In each discrete-time-step, theDS1104 controller board takes some action to generate the digital control signals. The typeof action is governed by what we have programmed in this board with the help of MATLAB-Simulink real-time interface. This board monitors the input (i.e. motor current, speed, voltageetc) with the help of CP1104 I/O board in each discrete-time step. Based on the inputs and thevariables that need to be controlled; it takes the programmed action to generate the controlleddigital signals. The CP1104 I/O board is an input-output interface board among the PowerElectronics Drive Board, Adjustable Speed Driver and DS1104 controller board. It takes themotor current, dc-voltage etc. from the Power Electronics Drive Board and Adjustable SpeedDriver also, speed signal (from linear encoder) from motor coupling system, to the DS1104controller board. In turn, the controlled digital signals supplied by DS1104 controller boardare taken to the Power Electronics Drive Board and Adjustable Speed Driver by CP1104.

5. MATLAB Simulink and Control-desk (Programming DS1104 and control in realtime):Simulink is a software program with which one can do model-based design such as designinga control system for a DC motor speed-control. The I/O ports of CP 1104 are accessible frominside the Simulink library browser. Creating a program in Simulink and procedure to usethe I/O port of CP 1104 will be detailed in this experiment. When you build the Simulinkcontrol-system (CTRL+B) by using real-time option, it implements the whole system insidethe DSP of DS1104 board, i.e. the controlsystem that is in software (Simulink) gets convertedinto a real-time system on hardware (DS1104). Simulink generates a *.sdf file when youbuild (CTRL+B) the control-system. This file gives access to the variables of controlsystem(like reference speed, gain, tuning the controller etc) to separate software called Control-desk. In this software a control panel (see Figure 3.9) can be created that can change thevariables of control-system in real time to communicate with DS1104 and hence change thereference quantities such as the DC motor speed, LIM frequency, V/f ratio and LIM stroke


limit (tope) by sliders or numeric entries.

3.2.1. Experiment DesignThe system for the speed of a DC motor and velocity and position of the LIM is shown in Figure3.1. The encoder signal (speed and position of LIM motor) is fed back to the DS1104 board viaCP 1104. The iv phase-current and dA duty cycle quantities are required to estimation of the LIMparameters.

The real-time Simulink model is shown in Figure 3.2.

Figure 3.2: Simulink block-set for characterization of a LIM.

In this experiment, a Simulink model (*.mdl) of a two pole DC switch-mode power converter willbe built to control the voltage of the DC motor in real-time.

The two-pole switch-mode DC converter voltage is the difference between the individual pole-voltages of the two switching power-poles (Figure 3.3). The average output voltage vab can rangefrom to + VDC and -VDC depending on the individual average pole voltages.

To achieve both positive and negative values of vab, a common-mode voltage equal in magnitudeto VDC/2 is injected in the individual pole-voltages. The pole-voltages are then given by:


Figure 3.3: Two pole DC switch-mode power converter.

dA =1

2+

1

2

vabVDC

dB =1

2− 1

2

vabVDC

The above equations were implemented in Simulink. dA governs the duty cycle of the S1 switchPWM and dB governs the duty cycle of the S2 switch PWM. Like was made earlier the PWMdevice was configured for center aligned PWM which forces symmetry about the center of the pe-riod. This configuration produces two pulses line-to-line during each period. The effective switch-ing frequency is doubled, reducing the ripple current while not increasing the switching losses inthe Power Electronics Drive Board.

A three-phase balanced voltage source of variable-magnitude and frequency is required, to run thelinear induction motor at synchronous frequency. The duty ratios for the three poles A, B and C togenerate this type of voltage source are given by (section 4-6- 2 in [1]):

da(t) =1

2+

1

2

vm,A(t)

Vdlim; vm,A(t) = Vmcos(ωt)

db(t) =1

2+

1

2

vm,B(t)

Vdlim; vm,B(t) = Vmcos(ωt− 2π/3)

dc(t) =1

2+

1

2

vm,C(t)

Vdlim; vm,C(t) = Vmcos(ωt− 4π/3)

(3.2)

Equations 3.3 are modified forms of equations 3.2 given in [1] which are suitable for real-time


Figure 3.4: Variable magnitude DC voltage source.

implementation.

da(t) = 0,5 +1

2u[1]cos(u[2]); u[2] = 2πft = (

1

sf)2π

db(t) = 0,5 +1

2u[1]cos(u[2]− 2π/3); s→ Laplace Operator

dc(t) = 0,5 +1

2u[1]cos(u[2]− 4π/3); u[1] =

VmVdlim

(3.3)

There are various aspects to take into account in the experiment realization. One difficulty associat-ed with the LIM is that due to the limit in the LIM stroke, is not easy to make a high speed no-loadcondition which is needed for primary winding inductance estimation. To avoid such difficulty, weare considering to apply a lower frequency than the nominal frequency, and move the LIM in backand forth by DC motor at corresponding synchronous speed controlling its direction via the LIMposition. The function in Figure 3.6 allows the LIM movement between 0 cm and tope cm only.

To avoid sudden changes of reference speed in the motor a first order filter has been added (Figure3.7). Sudden changes can cause high current peaks.

In order to measure the LIM position and velocity the speedmeasuredc2 simulink block wasmade. The linear encoder has a resolution of 20µm per channel thus the linear position may be inmillimeters dividing the encoder counter value by 50. It has a two quadrature channels, thus theaccuracy is 5µm.

The control-desk panel for run the experiment is shown in Figure 3.9. The control-desk panel allow


Figure 3.5: Three-phase variable magnitude and variable frequency source.

Figure 3.6: Sign Control of the LIM mover velocity via its position.

us to set the DC motor speed, LIM position limits and LIM frequency and monitors the duty cyclesand LIM velocity in real-time.


Figure 3.7: first order filter to avoid sudden changes of reference speed in the motor.

Figure 3.8: LIM position and velocity measurement.

To achieve no-load condition the LIM is feed with a lower frequency than rather frequency and viaDC motor the system is moved at corresponding synchronous velocity.


Figure 3.9: Control Desk Panel.

Since the secondary part of the LIM does not have slotted structure, the secondary leakage in-ductance is much smaller than the primary leakage inductance. Because of this, many parameterestimation methods of the RIM are not applicable to the LIM [58].

The mutual inductance Lm will be calculated by solving a third order polynomial which will bederived from the total equivalent inductance [58]. Such method of obtaining Lm directly allowus to calculate the leakage inductances of the primary and secondary windings separately and thesecondary resistance.

3.2.2. Primary Inductance and Resistance Estimation underno-load test.

Primary resistance was estimated by DC current test. Applying constant line-line voltages Vuv,Vvw, Vwu, we can get generate DC phase currents. Then, we obtain

Rs1 =Ruv

2=Vuv2Iu

= 1,6865Ω

Rs2 =Rvw

2=Vvw2Iv

= 1,6680Ω

Rs3 =Rwu

2=Vwu2Iw

= 1,6900Ω


Primary resistance is the average of Rs1, Rs2 and Rs3

Rs = 1,6875Ω

In estimating the primary inductance Ls, the secondary circuit should be seen as little as possible.To isolate the effects of the secondary circuit it is necessary to minimize the slip moving the LIMat synchronous speed by the DC motor. Then the LIM load is reduced.

To reduce end effect, frequency f = ωe/2π needs to be selected less than 18 Hz, since end effectis negligibly small for f < 18 Hz [59], [60].

Under no load or a low slip condition, we can obtain from the Figure 3.15 an approximate equationsuch that

Vs = Rsis + jωeLsis

Multiplying by is and dividing by 2 both sides of before equation we obtain

Vsis2

=Rsi

2s

2+ j

ωeLsi2s

2Vsrmsisrms = Rsi

2srms + jωeLsi

2srms

S = P + jQ

where Ls = Lls + Lm, ωe is the electric angular frequency, P is the active power and Q is thereactive power.

The experiment was made increasing the reference frequency for the linear induction machine (byref-Vel-LIM/Value slider in the Control-Desk panel) to 3 Hz, corresponding frequency ofthe synchronous velocity, and slowly increasing the DC motor speed up-to LIM mechanical syn-chronous velocity dictated by (3.1) (by ref-Vel-DC/Value slider in the Control-Desk panel),we obtained the LIM performance depicted in Figure 3.10. The V/F ratio and LIM stroke limit wassettled in 300V/20Hz and 100cm respectively.

Taking the waveforms for dA and iv on the oscilloscope (Figure 3.11) we obtain the readings for therms values of these variables. Also we measure the phase difference between the two waveformsusing the cursors.

dA,rms = 0,053033

IA,rms = 0,5356

Phase =35

333, 33360o = θ = 37,8o

The scaling factor for dA and iv are 10 and 0.125 respectively. Actual rms values of the phase‘v’ voltage, phase ‘v’ current and the per phase reactive power drawn by the three-phase linear


Figure 3.10: LIM mover velocity with no-load condition and Vs = 0,549m/s.

induction motor can be calculated as follows:

vv,rms = dA,rms300V =dA,rms

10300V = 15,9099V

iv,rms =iv,rms0,125

= 4,2851A

Q = vv,rmsiv,rmssinθ = 41,7852V AR

P = vv,rmsiv,rmscosθ = 53,8692V A

where cosθ is the displacement power factor, and dA,rms and iv,rms are measured on the scope.

Reactive power is consumed by primary inductance Ls. thus

Q = ωeLsi2vrms

Ls can be calculated from the above equation.

Ls =Q

ωei2vrms

Ls =vv,rmsiv,rmssinθ

ωei2vrms

Ls =vv,rmssinθ

ωeivrms


Figure 3.11: va and ia (Voltage and current) wave forms with no-load condition.

ThenLs = 120,7256mH

Active power is consumed by the primary resistance Rcal. thus

P = Rsi2srms

Rcal can be calculated from above equation.

Rcal =P

i2vrms

Rcal =vv,rmsiv,rmscosθ

i2vrms

Rcal =vv,rmscosθ

ivrms

ThenRcal = 2,93372Ω


3.2.3. Estimation of Req and Leq

For obtaining the mutual Lm and secondary inductances Lr, a large current must flow through thesecondary circuit, i.e., current path through the secondary circuit must be dominant. To provide alarge current flow through the secondary circuit, the mover-locked test is used.

Figure 3.12: Equivalent circuit of a LIM at stanstill.

Since at standstill vx = 0 (mover velocity), the LIM equivalent circuit shown in Fig 3.12 can berepresented as a series – circuit, such that

Zeq = Req + jωeLeq (3.4)

where

Req = Rs +ω2eL

2mRr

R2r + ω2

eL2r

(3.5)

Leq = Lls +Lm(R2

r + ω2eLrLlr)

R2r + ω2

eL2r

(3.6)

Req and Leq denote the total resistance and the total inductance and need to be estimated to obtainthe estimates of the secondary parameters.

The vs(t) voltage is obtained in the steady state, such that

vs(t) = Reqis(t) + jωeLeqis(t) (3.7)


Multiplying by is(t) and dividing by 2 both sides of before equation we obtain

vs(t)is(t)

2=Reqi

2s(t)

2+ j

ωeLeqi2s(t)

2vsrmsisrms = Reqi

2srms + jωeLeqi

2srms

S = P + jQ

where ωe is the exiting angular frequency, P is the active power and Q is the reactive power.

The experiment was made increasing the reference frequency for the linear induction machine (byref-Vel-LIM/Value slider in the Control-Desk panel) to 30 Hz with the LIM mover locked.The V/F ratio and LIM stroke limit was settled in 300V/60Hz and 100cm respectively. In the Figure3.13 the experiment result is shown.

dA,rms = 0,1768

IA,rms = 0,2934

Phase =6

33, 33360o = θ = 64,8o

The vv,rms and iv,rms values are:

vv,rms = dA,rms300V =dA,rms

10300V = 53,04V

iv,rms =IA,rms0,125

= 2,3472A

dA,rms and iv,rms are measured on the scope.

The active power is P = Reqi2srms = vsrmsisrmscosθ = 53V A, and the reactive power Q =

ωeLeqi2srms = vsrmsisrmssinθ = 112,6469V AR. where θ = arctan ωeLeq

Reqthen,

Req =P

i2srms= 9,62Ω (3.8)

and

Leq =Q

ωei2srms= 108,4721mH (3.9)


Figure 3.13: va and ia (Voltage and current) wave forms in blocked-mover test.

3.2.4. Estimation of Lm, Lr and Rr from a Third-Order Polynomial

Utilizing the estimated values, we define δL = Ls − Leq. We let β = Lm/Lr, which is unknown.Choosing ωe high enough so that R2

r ω2eL

2r , one can approximate (3.5) and (3.4) such that

Req ≈ Rs +L2m

L2r

Rr = Rs + β2Rr (3.10)

Leq ≈ Lls +LmLr

Llr = Lls + βLlr (3.11)

Since Ls = Lm + Lls, Lr = Lm + Llr and Leq ≈ Lls + βLlr, it follows that

Lls = Ls − Lm (3.12)

Llr =Lm − δl

β(3.13)


Lr =(1 + β)Lm − δL

β(3.14)

Rr =Req −Rs

β2(3.15)

Note that Lm is the only unknown value in the above definitions. Substituting (3.12), (3.13), (3.14),and (3,15) into (3.6), we obtain a third-order polynomial for Lm, such that

L3m + AL2

m +BLm + C = 0 (3.16)

where A = −(1 + β)δL/β − δL/(1 + β), B = 2δ2L/β, C = −δ3

L/β(1 + β)− βδLR2r/ω

2e(1 + β).

Note again that the coefficients A, B, and C are available with the methods suggested above. Thenumerical solution of (3.16) is found by the solve function of MATLAB R©.

Once Lm is found, the estimates Lls, Llr, Lr, and Rr are obtained directly from (3.12)–(3.15),respectively. Rr is an intermediate estimate needed for deriving polynomial (3.16). Based on theestimates Lm and Lr, we have a more accurate estimation method for Rr than (3.15). Rearranging(3.5), we obtain (3.17)

Rr =(ωeLm)2 −

√(ωeLm)4 − (2ωeLr(Req −Rs))2

2(Req −Rs)(3.17)

MATLAB R© code to solve the third order equation and recalculates Rr:

%Parameter estimation of LIM Lavolt 8228-02%by Nicolás Toro García - Universidad Nacional%de Colombia- Sede Manizales.

f=30;w=2*pi*f;Rs=1.6875Ls=0.1207256Req=9.62;Leq=0.1084721;DeltaL=Ls-Leq;Beta=0.92;Rr=(Req-Rs)/(Beta*Beta)

% Polynomial for Lm


A=-(1+Beta)*DeltaL/Beta-DeltaL/(1+Beta);B=2*DeltaL*DeltaL/Beta;C=-DeltaL*DeltaL*DeltaL/(Beta*(1+Beta))-Beta*DeltaL*Rr*Rr/(w*w*(1*Beta));

Lm=solve(’Lm^3+A*Lm^2+B*Lm+C’);Lm=eval(Lm);Lm=Lm(1)Lls=Ls-LmLlr=(Lm-DeltaL)/Beta Lr=((1+Beta)*Lm-DeltaL)/Beta

Rr=((w*Lm)^2-sqrt((w*Lm)^4-(2*w*Lr*(Req-Rs))^2))/(2*(Req-Rs))

Then the LIM parameters are:

Rs = 1,6875Ω

Ls = 0,1207H

Req = 9,6200Ω

Leq = 0,1085H

Rr = 9,3720Ω

Lm = 0,0420H

Lls = 0,0788H

Llr = 0,0323H

Lr = 0,0743H

Rr, adj = 3,9444Ω

3.2.5. Estimation of Lm, Lr, and Rr from the equation system

Knowing that Ls = Lls + Lm, Lr = Llr + Lm and, substituting Lls and Llr into 3.6, the followequation system is obtained:

Req = Rs +ω2eL

2mRr

R2r + ω2

eL2r

Leq = Ls − Lm +Lm[R2

r + ω2eLr(Lr − Lm)]

R2r + ω2

eL2r

LmLr

= β ⇒ Lr =Lmβ⇒ Llr

Lr= 1− β

(3.18)

Since Lm and Lr are unknown, β is also not known, it is around 0.95 in rotary induction motors,and 0.9 in linear induction motors. Giving a value to β, between 0.9 and 0.95, the unknown vari-


ables are Lm, Rr and Lr.

Substituting Lr in Req and Leq of (3.18) and defining A = Req −Rs and B = Leq −Ls we obtain:

A =ω2eL

2mRr

R2r + ω2

e(Lmβ

)2

B = −Lm +Lm[R2

r + ω2eLmβ

(Lmβ− Lm)]

R2r + ω2

e(Lmβ

)2

(3.19)

Simplifying B

B =−Lmω2

eL2m

β

R2r + ω2

e(Lmβ

)2(3.20)

From before expression of A we can get Lm like:

Lm =

√ARr

ωe√Rr − A

β2

(3.21)

We can calculate Rr Substituting 3.21 into 3.20:

Rr =ω2eA+ A3

B2

ω2β2(3.22)

Lm is obtained putting Rr value in 3.21. Then we can get Lr from the third equation of 3.18 Thenthe LIM parameters are:

Rs = 1,6875Ω

Ls = 0,1207H

Req = 9,6200Ω

Leq = 0,1085H

Rr = 9,1455Ω

Lm = 0,1616H

Lls = −0,0409H

Llr = 0,0050H

Lr = 0,1666H

Rr, adj = 9,1455Ω


3.2.6. Estimation of Lm, Lr, Ls and Rr taking into account endeffects

The per-phase equivalent circuit of a three-phase Linear Induction Motor taking into account endeffects is shown in the Figure 3.14.

Figure 3.14: Per-phase equivalent circuit of a three-phase LIM with end effects.

From the no-load condition test (s = 0) the value of Rs +Rrf(Q) can be obtained as:

Rs +Rrf(Q) =P

i2srms= 2,93372Ω = Rcal

andLls + Lm(1− f(Q)) = 0,12072H = Lcal

From the locked mover and and no-load condition test and replacing Q = DRrLrv

and f(Q) = 1−e−QQ

the system 3.23 is obtained.


Req −Rs =ω2eL

2mRr

R2r + ω2

eL2r



R2r + ω2

eL2r

Rcal −Rs = Rr1− e−

DRrLrv

DRrLrv

Lcal = Ls − Lm + Lm

(1− 1− e−

DRrLrv

DRrLrv

)(3.23)

with Rs = 1,6875Ω, Req = 9,6200Ω, Leq = 0,1084721H , ωe = 2π30, D = 0,061m andv = 0,549m/s.

Combining the second equation with the fourth one of 3.23 and simplifying, the equation system3.24 is obtained.

a =ω2eL

2m

R2r + ω2

eL2r

Rr

b = −aLrRr

+ cLmRr

c =1− e−

DRrLrv

DRrLrv

Rr

(3.24)

where a = Req −Rs, b = Leq − Lcal, c = Rcal −Rs. The unknown variables are Lm, Rr, and Lr.

To solution above equation the fminsearch Mtalb function was used. fminsearch finds the minimumof a scalar function of several variables, starting at an initial estimate. This is generally referred toas unconstrained nonlinear optimization. The Euclidian norm was used like error criterion. fmin-search uses the Nelder-Mead simplex direct search method of [61]. This is a direct search methodthat does not use numerical or analytic gradients.

Code for implicit function.

function y=impl2(x)

w = 30*2*3.141592; a = 7.9325; b = 0.1084721; c =1.246221;d = 0.1207256; k = 0.061/(0.549);

y1=a-w^2*x(1)^2*x(2)/(x(2)^2+w^2*x(3)^2);


y2=b-d +a*x(3)/x(2)-c*x(1)/x(2);y3=c-(1-exp(-k*x(2)/x(3)))*x(3)/(k);

y=norm([y1 y2 y3]);

Commands to solution the implicit function.fx=’impl’x0=[10 10 10];[x, f, EF, out]=fminsearch(fx, x0)

Results:

x =0.137482800713993 75.657648604483427 0.138469000111235

errorf =

5.306191110160208e-009

The current x satisfies the termination criteria using OPTIONS. TolX of 1.000000e-004 andf(X) satisfies the convergence criteria using OPTIONS. TolFun of 1.000000e-004.

Value exit flag

EF =1

Value exitflag (EF) describes the exit condition. which means fminsearch converges to a solution x.

Errors of y1, y2 and y3.

[y1 y2 y3]=1.0e-008*[0.044271431 0.519205586 -0.1001111416]

LIM paramters are Lm=, x(1), Rr= x(2) , and Lr=x(3). Ls can be obtained either second orfourth equations of 3.23.


LIM parameters calculated are:Rs = 1,6875Ω

Ls = 0,1229902H

Req = 9,6200Ω

Leq = 0,1085H

Rr = 75,6576486Ω

Lm = 0,1374828H

Lls = −0,0409H

Llr = 0,0009862H

Lr = 0,138469H.

3.3. Parameter Estimation of Induction Motor LEESONmodel C4T34FB5B

Figure 3.15: Per-phase equivalent circuit of a three-phase induction motor.

In order to construct a simulation model in this section a three-phase RIM will be characterized todetermine the various parameters used in its per-phase equivalent circuit (Figure 3.15) by a DSP-based electric-drives system.

The identification of rotary induction motor parameters will be performed via blocked rotor andno load condition tests. The stator winding resistance will be determined by direct current test.

3.3.1. Determination of Rs:

Measuring the line-line resistances of the induction motor Rs is obtained.

Rs =

(Rab +Rbc +Rca

3

)/2 =

(15,1Ω + 14,9Ω + 15,2Ω

3

)/2 = 7,55Ω

3.3 Parameter Estimation of Induction Motor LEESON model C4T34FB5B 67

3.3.2. Determination of Ls:Under no load or a low slip condition, we can obtain from the Figure 3.15 an approximate equationsuch that

Q = ωeLsi2srms = vsrmsisrmssinθ

where Ls = Lls + Lm, ωe is the electric angular frequency and Q is the reactive power.

Reactive power is consumed for stator inductance Ls. thus Ls can be calculated from the aboveequation.

Ls =Q

ωei2srms

Ls =vs,rmsis,rmssinθ

ωei2vrms

Ls =vs,rmssinθ

ωeisrms

0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1−200

−150

−100

−50

0

50

100

150

200

Pha

se a

Vol

tage

Time (sec)

No load condition test

0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1−2

−1

0

1

2

Pha

se a

Cur

rent

Figure 3.16: va and ia (Voltage and current) wave forms with no-load condition.

Taking the waveforms for vs(t) and is(t) (Figure 3.16) we obtain the readings for the rms valuesof these variables. Also we measure the phase difference between the two waveforms.

vs,rms = 165,2/√

2 V.

Is,rms = 1,06/√

2 A.

Phase =0,0035

0,016667360o = θ = 75,6o

ωe = 377rad/s

ThenLs = 0,4004H


3.3.3. Determination of Lm, Lr and Rr:

For obtaining the magnetizing Lm and rotor Lr inductances and Rr rotor resistance, a large currentmust flow through the rotor circuit, i.e., current path through the rotor circuit must be dominant.To provide a large current flow through the secondary circuit, the rotor-blocked test is used.

Figure 3.17: Equivalent circuit of a RIM at standstill.

Since at standstill wmec = 0 (rotor velocity), the RIM equivalent circuit shown in Fig 3.17 can berepresented as a series – circuit, such that

Zeq = Req + jωeLeq (3.25)

where

Req = Rs +ω2eL

2mRr

R2r + ω2

eL2r

(3.26)

Leq = Lls +Lm(R2

r + ω2eLrLlr)

R2r + ω2

eL2r

(3.27)

Req and Leq denote the total resistance and the total inductance and need to be estimated to obtainthe estimates of the rotor parameters.

For is(t) = Isin(ωet), the vs(t) voltage is obtained in the steady state, such that

vs(t) = Reqis(t) + jωeLeqis(t) (3.28)

Taking the waveforms for vs(t) and is(t) (Figure 3.18) we obtain the readings for the rms valuesof these variables. Also we measure the phase difference between the two waveforms.


0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1−40

−30

−20

−10

0

10

20

30

40

Pha

se a

Vol

tage

0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1−2.5

−1.25

0

1.25

2.5

2

−2

Time (sec)

Pha

se a

Cur

rent

Figure 3.18: va and ia (Voltage and current) wave forms for rotor-blocked test.

vs,rms = 34,9/√

2V.

Is,rms = 2,0678/√

2A.

Phase =0,0022

0,016667360o = θ = 47,52o

ωe = 377rad/s

The power consumed for the induction motor is given by,

vsrmsisrms = Reqi2srms + jωeLeqi

2srms

S = P + jQ

P = vsrmsisrmscosθ = Reqi2srms

Q = vsrmsisrmssinθ = ωeLeqi2srms

where ωe is the electric angular frequency, P is the active power and Q is the reactive power.Then,

Req = 11,3982Ω

Leq = 0,0330H

Knowing that Ls = Lls + Lm, Lr = Llr + Lm and, substituting Lls and Llr into (3.27), the followequation system is obtained:

Req = Rs +ω2eL

2mRr

R2r + ω2

eL2r



R2r + ω2

eL2r

LmLr

= β ⇒ Lr =Lmβ⇒ Llr

Lr= 1− β

(3.29)


β is not known, it is around 0.95 in rotary induction motors, and 0.9 in linear induction motors.Giving a value to β = 0,95, the unknown variables are Lm, Rr and Lr.

Substituting Lr in Req and Leq of (3.29) and defining A = Req −Rs and B = Leq −Ls we obtain:

A =ω2eL

2mRr

R2r + ω2

e(Lmβ

)2

B = −Lm +Lm[R2

r + ω2eLmβ

(Lmβ− Lm)]

R2r + ω2

e(Lmβ

)2

(3.30)

Simplifying B

B =−Lmω2

eL2m

β

R2r + ω2

e(Lmβ

)2(3.31)

From before expression of A we can get Lm like:

Lm =

√ARr

ωe√Rr − A

β2

(3.32)

We can calculate Rr Substituting (3.32) into (3.31):

Rr =ω2eA+ A3

B2

ω2β2(3.33)

Lm is obtained putting Rr value in (3.32). Then we can get Lr from the third equation of (3.29).

3.3.4. Determination of mechanical parameters:

The mechanical equation that governs the speed behaviour is given by the Newton second rota-tional motion law.

Jdw

dt= Tem −Bw − TL

The inertia J = 5,4783e − 0,04Kgm2 is given by the motor manufacturer. Without load TL = 0and without feeding voltage Tem = 0, the before equation solution is:

w = w(0)e−BJt

The Figure 3.19 shows the experimental behaviour and simulated with w(0) = 3500rpm andBJ

= 1/3,2. B is the friction coefficient.


0 1 2 3 4 5 6 7 8−500

0

500

1000

1500

2000

2500

3000

3500

Time (sec)

Spe

ed (

RP

M)

ExperimentalSimulated

Figure 3.19: Rotor speed for free running of induction motor.

Then the RIM parameters are:

Rs = 7,55Ω

Ls = 0,4004H

Req = 11,3982Ω

Leq = 0,0330H

Rr = 4,3227Ω

Lm = 0,3870H

Lls = 0,0134H

Llr = 0,0204H

Lr = 0,4074H

J = 0,00054783Kg −m2

B = 0,000171418N −m− s

A simulation model has been implemented in order to validate the identification process(Figure3.20). In this simulation no load condition and standstill test were executed and the results obtainedare shown in Figure 3.21.


t2

powergui

Continuous

irc

Rr

irb

Rr

ira

Rr

f3

165.2

f2

60

dw3

2*pi

dw1

60/2/pi

Wmreal1

Voltage_G

f

t

Vm

VAN

VBN

VCN

fcn

Vcn

s -+

Vbn

s -+

Van

s -+

Transfer Fcn

1

Jeq.s+B

Tem Model

ira

irb

irc

Wm

Ws

Tem

s_des

fcn

Stator_Currents

Rs2

Rs1

Rs

Ls2

Ls1

Ls

Lr2

Lr1

Lr

Lm2

Lm1

Lm

Goto

[s_desl]

From2

[s_desl]

From1

[s_desl]

From

[s_desl]

Divide2

Divide1

Divide

Constant1

0

isc

i+ -

isb

i+ -

isa

i+ -

irc

i+ -

irb

i+ -

ira

i+ -

s -+

s -+

s -+

Figure 3.20: Equivalent circuit model per-phase for simulation.

0 0.02 0.04 0.06 0.08 0.1−2

−1

0

1

2

Time (sec)

Cur

rent

(A

)

No load condition test experimental results

0 0,02 0,04 0,06 0,08 0,1−2

−1

0

1

2

Time (sec)

Cur

rent

(A

)

No load condition test simulated results

0 0.02 0.04 0.06 0.08 0.1−3

−2

−1

0

1

2

3

Time (sec)

Cur

rent

(A

)

Rotor blocked test experimental results

0 0.02 0.04 0.06 0.08 0.1−3

−2

−1

0

1

2

3

Time (sec)

Cur

rent

(A

)

Rotor blocked test simulated results

Figure 3.21: Stator currents of induction motor.

4 Three–Phase Power ConverterControlled With ZAD–FPICTechniques

4.1. Introduction

This chapter shows the behavior of a Three–phase power converter with resistive load, Rotary in-duction motor (RIM) load and Linear Induction motor (LIM) load using a quasi-sliding controltechnique for output voltage regulation and a novel control technique to control chaos. The con-troller is designed using Zero Average Dynamic (ZAD) and Fixed Point Inducting Control (FPIC)techniques. The designs have been tested in a Rapid Control Prototyping (RCP) system based onDigital Signal Processing (DSP) for dSPACE platform. The results show that the phase voltage inthe load has sinusoidal performance when it is controlled with these techniques. The robustnessof the system is shown by bifurcation diagrams varying a control parameter. Experimental andsimulations results agree only when delay effects are considered in simulations. After this consid-eration, numerical and experimental bifurcations diagrams agree both in stable zone as well as intransition to chaos.

The converters use power electronics for efficient transformation and rational use of electricityfrom generation sources to industrial and commercial use. It has been estimated that 90 % of elec-trical energy is processed through power converters before the final use [62]. The power convertersmust provide a certain level of output voltage, either in tasks regulation or tracking, and be able toreject changes in load and primary supply voltage level. A complete and detailed analysis of theoperation and configuration of the different power converters can be found in [63], [45]. One of themost desirable qualities in these devices is their efficiency in the performance by using switchingdevices generating the desired output with low power consumption.

The use of the Digital Pulse–Width Modulation (DPWM) to control electronic power convert-ers has increased, because of a number of potential advantages. Some of them are: low powerconsumption, immunity to analog component variations, potentially faster design process, lowersensitivity to parameter variations, programmability, reduction or elimination of external passivecomponents, calibration or protection algorithms, ability to interface with digital systems, possibil-ity of implementing nonlinear control techniques and are much easier to implement the advancedcontrol algorithms such as estimation of parameters as mentioned in [64–66]. DPWM has also

74 4 Three–Phase Power Converter Controlled With ZAD–FPIC Techniques

enabled practical applications of high frequency digital controllers for DC–DC converters [67].

Nevertheless, DPWM exhibits the following disadvantages: i). Quantization effects in state vari-ables and in duty cycle can cause undesirable limit-cycle oscillations [68,69] or chaos [70]. Quan-tization is due to Analog–to–Digital (AD) converter. In [68] the presence of steady-state limitcycles in DPWM controlled converters are discussed and conditions on the control law and on thequantization resolution for avoiding these oscillations are imposed. In [64] an approach to improvedynamic responses of digitally controlled DC–DC converters using nonuniform quantization ispresented; ii). Delays in the controller produce instabilities. Indeed, digital controllers generallypresent a poor processing speed and processing delays in the feedback loop, due to sampling fre-quency, calculation time and the real time performance of the controller [71].

Liu et al [65] provides the minimum requirement of digital controller parameters: sampling timeand quantization resolution dimensions.

Parameter estimation is useful to estimate unknown varying parameters of converters [72–74]. FewZAD based controllers have used parameter estimation, e.g. [66, 75].

The controller designed in the present work combines Zero Average Dynamics (ZAD) and FixedPoint Inducting Controller (FPIC) strategies, which have been reported in [47–54]. The designcorresponds to a three phase low power inverter (1500 W) with three phase resistive load usinga dSPACE platform. Numerical and experimental bifurcations are obtained for the ZAD-FPIC-controller, by changing the parameter values. Numerical and experimentally obtained bifurcationdiagrams match. On the other hand, the development and application of the FPIC control tech-nique are presented in [50, 52–54]. This technique allows the stabilization of unstable orbits in asimple way. Recent advances in real-time implementation of ZAD-FPIC control techniques withRecursive Least Square (RLS) estimator are reported in this work.

4.2. Three–phase Power Converter With Resistive Load

This section is organized as follows. Subsection 4.2.1 describes the proposed system. Subsection4.2.2 describes the mathematical model of the system. Subsection 4.2.3 describes the control tech-niques. Subsection 4.2.4 presents the obtained results, and finally, subsection 4.2.5 presents theconclusion.

4.2.1. Proposed system

Figure 4.1 shows the block diagram of the system under study. This system is divided into two ma-jor subgroups: hardware and software. Hardware includes electrical circuits and electronic devices,and software includes signals acquisition and implementation of control techniques. Software is

4.2 Three–phase Power Converter With Resistive Load 75

implemented on a dSPACE platform.

Figure 4.1: Block diagram of the proposed system

The hardware is composed of a Three–phase power converter with resistive load with rated power1500 Watts, rated Volts 600 VDC, rated Amps 20 ADC. In the measure of variables υc (capacitorvoltage) series resistance were used and for measurement of iL (inductor currents) HX10P/SP2current sensors were used. The converter switches were driven by PWM outputs of the controllercard,these signals are coupled via fast optocouplers (6N137).

The digital part is developed in the control and development card dSPACE DS1104, where ZADand FPIC control techniques are implemented. This card is programmed from Matlab/Simulinkplatform and it has a graphical display interface called ControlDesk. The controllers are imple-mented in Simulink and they are downloaded to the DSP. The sampling rate for all variables is setto 6 kHz. The state variables υc, and iL are 12 bits; the duty cycle (d) is 10 bits. Parameters of buckconverter (C, L, rs, rL) and ZAD-FPIC–controller (KS , N , Fs, R) are entered to the control blockby the user, as constant parameters. KS is the bifurcation parameter. At each sampling time thecontroller calculates in real time the duty cycle and the equivalent PWM signal to control switch.

4.2.2. Mathematical model

Figure 4.2 shows a basic diagram of the system. The buck power converter is used to feed theresistive load. Equation (4.1) is obtained for the system model.


Figure 4.2: Electrical circuit for the buck-motor system

vcaiLavcbiLbvcciLc

=

−1RaCa

1Ca

0 0 0 0−1La

−(rs+rLa )

La0 0 0 0

0 0 −1RbCb

1Cb

0 0

0 0 −1Lb

−(rs+rLb )

Lb0 0

0 0 0 0 −1RcCc

1Cc

0 0 0 0 −1Lc

−(rs+rLc )

Lc

vcaiLavcbiLbvcciLc

+

0 0 0ELa

0 0

0 0 00 E

Lb0

0 0 00 0 E

Lc

S1 − S4

S2 − S5

S3 − S6

(4.1)

withS4 = 1− S1

S5 = 1− S2

S6 = 1− S3


andSi ∈ 0, 1

for i = 1 . . . 6

This equation can be expressed in a compact form as:

x = Ax+BU (4.2)

Where the state variables are: vca := x1, iLa := x2, vcb := x3, iLb := x4, vcc := x5, and iLc := x6.

A is a block diagonal matrix, so that the system consists of three uncoupled subsystems that maybe treated independently. Figure 4.3 shows the equivalent circuit per phase.

Figure 4.3: Electrical circuit for the buck converter (equivalent circuit per phase )

Derivation of the discrete time iterative map of the converter

The inherent piecewise switched operation converters implies a multi-topological mode in whichone particular circuit topology describes the system for a particular interval of time. The first stepin the analysis of multi-topological circuit is to write down the state equations which describethe individual switched circuits. For converters operating in continuous conduction mode, twoswitched circuits can be identified. When the switch is ON, it is described by equation (4.3); whenthe switch is OFF, it is described by equation (4.4).[

υciL

]=

[ −1RC

1C

−1L

−(rs+rL )

L

] [υciL

]+

[0EL

](4.3)

[υciL

]=

[ −1RC

1C

−1L

−(rs+rL )

L

] [υciL

]+

[0−EL

](4.4)

The state variables are the capacitor voltage (υc) and the inductor current (iL). These equationscan be expressed in a compact from as x = Ax + Bu with x1 = υc and x2 = iL. The converterpower supply is denoted by E and depending on the control pulse voltage E or −E is injected to


Figure 4.4: Centered PWM.

the system through PWM signal.

Only considering continuous conduction mode (CCM) and according to centered PWM (Figure4.4), the control signal is defined as follows:

u =

1 si kT ≤ t ≤ kT + dT/2−1 si kT + dT/2 < t < kT + T − dT/21 si kT + T − dT/2 < t < kT + T

(4.5)

The solution of the system (4.3) for kT < t < (kT + dT/2) is given by

x(t) = eA(t−kT )x(kT )− A−1[I − eA(t−kT )]B

The solution of the system (4.4) for (kT + dT/2) < t < (kT + T − dT/2) is given by

x(t) = eA(t−(kT+dT/2))x(kT + dT/2) + A−1[I − eA(t−(kT+dT/2))]B

where

x(kT + dT/2) = eA(dT/2)x(kT )− A−1[I − eA(dT/2)]B.

The solution of the system (4.3) for (kT + T − dT/2) < t < (kT + T ) is given by

x(t) = eA(t−(kT+T−dT/2))x(kT + T − dT/2)− A−1[I − eA(t−(kT+T−dT/2))]B

where

x(kT + T − dT/2) = eA(T−dT )x(kT + dT/2) + A−1[I − eA(T−dT )]B.


The general solution of the system for kT < t < (kT + T ) is given by

x((k + 1)T ) = eATx(kT ) + [2eA(dT/2) − 2eA(T−dT/2) + eAT − I]A−1B (4.6)

where k represents the k-th iteration, T is the sampling period and d is the duty cycle.

Equation (4.6) is the discrete-time state equation for the buck converter. In much of the literature,the terms iterative map, iterative function and Poincaré map have been used synonymically withdiscrete-time state equation.

In that follows, the computation of the duty cycle is presented.

4.2.3. Control strategies

The control strategies presented in this section are developed for the per phase equivalent circuit.So for the three phase system the control must be applied for each phase independently takinginto account that the reference voltage will be phase shifted according to the corresponding circuitphase.

ZAD control strategy

As reported in [48,50,51], one of the possibilities for computing the duty cycle is to define a surfaceand to force it to be zero in each iteration. The surface per phase is defined as a piecewise-linearfunction given by

spwl(t) =

s1 + (t− kT )s+ if kT ≤ t ≤ t1s2 + (t− kT + dk

2)s− if t1 < t < t2

s3 + (t− kT + T + dk2

)s+ if t2 ≤ (k + 1)T(4.7)


where

s+ = ((x1 − xref ) + ks(x1 − xref ))∣∣∣x=x(kT ),u=1

s− = ((x1 − xref ) + ks(x1 − xref ))∣∣∣x=x(kT ),u=−1

s1 = ((x1 − xref ) + ks(x1 − xref ))∣∣∣x=x(kT ),u=1

s2 = dk2s1 + s1

s3 = s1 + (T − dk)s2

t1 = kT + dk2

t2 = kT + (T − dk2

)t3 = (k + 1)T

(4.8)

with ks = Ks ∗√LC a positive constant.

The dk satisfying zero average requirement is:

Dk =2s1(x(kT )) + T s−(x(kT ))

s−(x(kT ))− s+(x(kT ))(4.9)

From (4.3), (4.4) and (4.8) we obtain

s1(kT ) = (1 + aks)x1(kT ) + bksx2(kT )− x1ref − ksx1ref

s+(kT ) = (a+ a2ks + bcks)x1(kT ) + (b+ abks + bdks)x2(kT ) + bksEL− x1ref − ksx1ref

s−(kT ) = (a+ a2ks + bcks)x1(kT ) + (b+ abks + bdks)x2(kT )− bks EL − x1ref − ksx1ref

(4.10)

with a = − 1RC

, b = 1C

, c = − 1L

, d = − (rs+rL )

L.

The duty cycle is given by.

dk =

1 if Dk > TDk/T if 0 ≤ Dk ≤ T0 if Dk < 0

(4.11)

We have experimentally measured and noticed that there is a period delay in the control action. Inthis case, the control action is taken from the data acquired in the past sampling time, and then wecompute the duty cycle as:

dk =2s1(x(k − 1)T ) + T s−(x(k − 1)T )

s−(x(k − 1)T )− s+(x(k − 1)T )(4.12)

To apply this technique we need to measure the states at the beginning of each sampling time. For


Table 4.1: Parameters for simulation and experimentParameter Valuers: Internal resistance of the source 4 ΩE: Input voltage 40 VL: Inductance 1,6 mHrL: Internal resistance of the inductor 0,9 ΩC: Capacitance 368 µFN : FPIC control parameter 7Fc: Switching frequency 4 kHzFs: Sampling frequency 4 kHz1T_p: 1 Delay time 0,25 msKS: Control parameters 5

doing this, we carry out a synchronization between the measured signals and the start of the PWM.This synchronization is performed using a trigger signal obtained from the PWM, which gives thecommand to ADC converter for reading υc and iL. On the other hand, we need to know the valuesof the parameters L, C, rs, rL . In this case, we suppose that these parameters will be constant andmeasurable. The load R may be unknown and in this case must be estimated.



N + 1(4.13)

Where dk(k) is calculated as (4.12) and d∗ is duty cycle calculate in steady state (x1(kT ) = x1ref ).From (4.9)

d∗ = Dk

∣∣∣x1(kT )=x1ref

=T

2+T [(1 + rs+rL

R)x1ref + (L

R+ (rs + rL)C)x1ref + LCx1ref ]

2E(4.14)

4.2.4. Numerical and experimental results

In this section numerical and experimental results are shown using KS and N as bifurcation pa-rameters, in addition the system behaviour under frequency and voltage amplitude variations areillustrated graphically. Parameter values used in simulations and experiments are listed in table4.1, initially the reference voltage has a peak of 32V and 40Hz of frequency. For the simulation inSIMULINK model the fixed step size (fundamental sample time) in configuration parameter wassetting in 1/(4Fs).

Figures 4.5 and 4.6 show the experimental behaviour for the Three–phase power converter usingsame parameters and controller but different initial conditions. References Voltages for phases a,


b, c (red signals); phase voltages (va, vb, vc) and phase currents (ia, ib)(blue signals) are shown inthese Figures.

In Figure 4.5 the system exhibits a periodic solution and in Figure 4.6 a chaotic solution. This factshows the coexistence of attractors or solutions in the system. When the solution is periodic thecontrolled voltage follows the voltage reference by the control action unlike the chaotic solutionwhere the output voltage is lower than reference and it has a irregular fashion, in this regime thephase currents have a higher peak. Some times the system toggles between two solutions while itis running, this happens when the solution is near of the border of two regions of attraction.

Figure 4.5: Periodic Solution for the Three–phase converter with resistive load.

The results obtained in simulation and experiment for periodic solution are shown in Figure 4.7.The simulation was executed taking into account a 3T (0,75ms) delay in the duty cycle applica-tion. Under this condition simulation and experimental results match. Controlled voltage vc followsreference voltage for all phases with a maximum error of 2v.

Figure 4.8 shows the bifurcation diagrams for the output error and duty cycle of controlled sys-tem with KS and N like bifurcation parameters, obtained via model simulation using simulink ofMatlabr. ForKS = 1,5 withN = 2 andN = 1,5 withKS = 3 the system presents a qualitativelybehaviour change. Before KS = 1,5 and N = 1,5 the system is chaotic regime and after is stableregime. For constructing these diagrams, the simulation was running for 3 periods of referencevoltage and last 15 samples of output error are taken with initial conditions equal zero and a delayin duty cycle application of 3T.


Figure 4.6: Chaotic Solution for the Three–phase converter with resistive load.

Figure 4.7: Experimental and simulation results for Three–phase converter with resistive load.


(a) Bifurcation Diagram withN = 2

(b) Bifurcation Diagram withKs = 3

(c) Bifurcation Diagram withN = 2

Figure 4.8: Error in voltage, with ks and N like bifurcation parameters, and error in duty cycle forphase a resulting from the model simulation using simulink of Matlabr of Three–phase converter with resistive load.

In next the results of experiments are shown for buck power converter behaviour when the voltagereference and voltage level vary.

(a) vc Bifurcation Diagram (b) ia Bifurcation Diagram (c) Duty cycle Bifurcation Di-agram

Figure 4.9: vc voltage, ia current and duty cycle experimental bifurcation diagrams with ks likebifurcation parameter of three phasic converter with resistive load.

Figure 4.9 shows the bifurcation diagrams for the output error and duty cycle of controlled systemwith KS and N like bifurcation parameters, obtained experimentally. For KS3 the system presentsa qualitatively behaviour change. Before KS = 3 the system is chaotic regime and after is stableregime. For constructing these diagrams, the experiment was running for 2 seconds for each value


of Ks and samples were acquired every 50 milliseconds.

(a) va Bifurcation Diagram (b) ia Bifurcation Diagram (c) Duty cycle Bifurcation Di-agram

Figure 4.10: vc voltage, ia current and duty cycle experimental bifurcation diagrams with f likebifurcation parameter of three phasic converter with resistive load.

Figure 4.10 shows the bifurcation diagrams for the ia current, va voltage and duty cycle of con-trolled system with f (frequency of voltage reference) like bifurcation parameter, obtained ex-perimentally. For f = about 27 Hz the system presents a qualitatively behaviour change. Afterf = 27Hz the system is chaotic regime and before is stable regime. For constructing these dia-grams, the experiment was running for 2 seconds for each value of f and samples were acquiredevery 50 milliseconds. The phenomenon shown in Figure 4.10 is caused by saturation of the induc-tor core in the LC filter, because the core saturation and magnetic hysteresis not were simulatedthese phenomena are not present in simulations.

Figure 4.11(a) shows the bifurcation diagram of duty cycle and Figure 4.11(b) shows bifurcationdiagram of va for the Poincare map 4.6, both of which are numerical. Bifurcation diagrams wereconstructed taking last 30 samples, each one every period of reference voltage for phase a fromthe model 4.6 simulation using Matlabr, while the system was running during 35 periods. Thesimulation was executed taking into account a 1T (0,25ms) delay in the duty cycle application.In these Figures various regimes of operation can be appreciated: periodic windows, Chaos andperiodic solutions.

The Figure 4.12 shows the bifurcation diagram when a 3T delay is considered. Bifurcation dia-grams are quite different for different delay.

Before analysis shows the effects of the delay time in the signal control applied to the power con-verter. Many complex phenomena arise like shown in Figures 4.12(e) and 4.12(f) some of theseare non smooth bifurcations, doubling period bifurcations, chaos and 1-periodic orbits.


(a) Bifurcation Diagram of duty cycle for thePoincare map 4.6

(b) Bifurcation Diagram of va error for the Poincaremap 4.6

Figure 4.11: Bifurcation Diagram with ks like bifurcation parameters, taking one sample eachperiod of the reference voltage for phase a during 30 periods considering a T delay.

In order to analyze the effect of the variation of frequency and voltage level in the reference volt-age, a simulation was carried out.

The results obtained varying the frequency of reference voltage are shown in Figure 4.13. TheFigures 4.13(a), 4.13(b), 4.13(c) show the behaviour of duty cycle, peak voltage and peak currentin C and L respectively when the frequency voltage reference is varied in the Poincare map simu-lation. The amplitude of the output voltage does not undergo significant changes with the variationof frequency but current peak has a linear growth with frequency shown a predominant capacitiveeffect. In this case the delay has not significative impact.

4.2.5. Conclusion

The control strategy ZAD-FPIC was designed and applied to three phase buck converter with resis-tive load. For this system, simulations and experiments were performed. The stability of the closedloop system was analyzed using bifurcation diagrams, and stable and transitions to chaos wereobserved. It was demonstrated in an experimental way, that the delay effects have high importancein ZAD strategy. Simulations and experiments agreed when delay effects were included.

4.3. Three–phase Power Converter With RotaryInduction Motor Load

This section is organized as follows. Subsection 4.3.1 describes the proposed system. Subsection4.3.2 describes the mathematical model of the system. Subsection 4.3.3 describes the control tech-

4.3 Three–phase Power Converter With Rotary Induction Motor Load 87

(a) Bifurcation Diagram of duty cycle. (b) Bifurcation Diagram of va

(c) Bifurcation Diagram of duty cycle. (d) Bifurcation Diagram of va.

(e) Bifurcation Diagram of duty cycle. (f) Bifurcation Diagram of va

Figure 4.12: Bifurcation Diagram with ks like bifurcation parameters, Taking a sample each 30periods of the reference voltage for phase a considering a 3T delay, for the Poincaremap 4.6.

niques. Subsection 4.3.4 presents the obtained results, and finally, subsection 4.3.5 presents theconclusion.


(a) duty cycle. (b) va peak. (c) ia peak.

Figure 4.13: Results for Poincare map 4.6 simulation, Taking a sample on each one of last 10periods of the reference voltage for phase a and ia considering a3T delay,varyingfrequency in the reference voltage

4.3.1. Proposed system

Figure 4.14: Block diagram of the proposed system

Figure 4.14 shows the block diagram of the system under study. This system is divided into twomajor subgroups: hardware and software. The hardware is composed of a Three Phase power con-verter rated power 1500 Watts with RIM load rated power 1/3 HP, voltage 220V and current 1.4 A.

The controllers are implemented in Simulink and they are downloaded to the DSP. Parameters ofconverter (C, L, rs, rL) and ZAD-FPIC–controller (KS , N , Fs, RIM parameters) are entered tothe control block by the user, as constant parameters. KS is the bifurcation parameter. At eachsampling time the controller calculates in real time the duty cycle and the equivalent PWM signalto control the bridge switches.


4.3.2. Mathematical model

A schematic of switched power converter and RIM coupled system is shown in the Figure 4.15,the power converter is used to feed the rotary induction motor. Any time domain analysis of suchsystem normally assumes that the magnetic saturation, hysteresis and eddy current effects are allnegligible. Equation 4.26 is obtained for the system model.

iLavcaisairaiLbvcbisbirbiLcvcciscircwm

=

0 a12a 0 0 0 0 0 0 0 0 0 0 0a21a 0 a23a 0 0 0 0 0 0 0 0 0 0

0 a32a a33a a34a 0 0 0 0 0 0 0 0 00 a42a a43a a44a 0 0 0 0 0 0 0 0 00 0 0 0 0 a12b 0 0 0 0 0 0 00 0 0 0 a21b 0 a23b 0 0 0 0 0 00 0 0 0 0 a32b a33b a34b 0 0 0 0 00 0 0 0 0 a42b a43b a44b 0 0 0 0 00 0 0 0 0 0 0 0 0 a12c 0 0 00 0 0 0 0 0 0 0 a21c 0 a23c 0 00 0 0 0 0 0 0 0 0 a32c a33c a34c 00 0 0 0 0 0 0 0 0 a42c a43c a44c 00 0 0 a51 0 0 0 a52 0 0 0 a53 −B

J

iLavcaisairaiLbvcbisbirbiLcvcciscircwm

+

ELa

0 0

0 0 00 0 00 0 00 E

Lb0

0 0 00 0 00 0 00 0 E

Lc

0 0 00 0 00 0 0

S1 − S4

S2 − S5

S3 − S6

+

00000000000−TL

J

(4.15)

Before equation can be expressed in a compact form as:

x = A(x)x+BU +D (4.16)

The system consists of three subsystems where each phase may be treated independently. The


Figure 4.15: Electrical circuit for the converter-motor system

model per phase, considering TL = 0, is given by the equation (4.17).

iLυcisir

=

0 a12 0 0a21 0 a23 00 a32 a33 a34

0 a42 a43 a44

iLυcisir

+

EL

000

Uc (4.17)

With Uc = −1, 1 according to switches control.

Matrix entries are defined as:

a12 = − 1

L

a21 =1

Ca23 = −a21

a32 =Lm + Llr

LmLls + LlrLm + LlrLlsa33 = −a32Rs

a34 = − LmRr

s(LmLls + LlrLm + LlrLls)

a42 =Lm

LmLls + LlrLm + LlrLlsa43 = −a42Rs

a44 = − (Lm + Lls)Rr

s(LmLls + LlrLm + LlrLls)


s = (ws − wm)/ws

s is the motor slip, wm is the rotor speed and ws is the synchronous speed.

The mechanical equation is given by:

wm =i2raRra

Jsws+i2rbRrb

Jsws+i2rcRrc

Jsws− B

Jwm −

TLJ

(4.18)


The control strategies presented in this section are developed for the per phase equivalent circuit.So for the three phase system the control must be applied for each phase independently takinginto account that the reference voltage will be phase shifted according to the corresponding circuitphase. Because the mechanical dynamic is very slow compared with the electrical dynamic, therotor speed is considered constant in each period of sample.

ZAD–FPIC control strategy

The surface per phase is defined as a piecewise-linear function given by 2.61, where:

s+ = ((vc − vref ) + ks1(vc − vref )) + ks2(...v c −

...v ref )) + ks3(

....v c −

....v ref ))

∣∣∣x=x(kT ),u=1

s− = ((vc − vref ) + ks1(vc − vref )) + ks2(...v c −

...v ref )) + ks3(

....v c −

....v ref )))

∣∣∣x=x(kT ),u=−1

s1 = ((vc − vref ) + ks1(vc − vref )) + ks2((vc − vref ) + ks3(...v c −

...v ref ))

∣∣∣x=x(kT ),u=1

s2 = dk2s+ + s1

s3 = s1 + (T − dk)s−t1 = kT + dk

2

t2 = kT + (T − dk2

)t3 = (k + 1)T

(4.19)



N + 1(4.20)

Where dk(k) is calculated as (2.66) and d∗ is duty cycle calculate in steady state (vc(kT ) = vref ).

d∗ =1

2+vref2E

(4.21)

To apply this technique we need to measure the states at the beginning of each sampling time. For


doing this, we carry out a synchronization between the measured signals and the start of the PWM.This synchronization is performed using a trigger signal obtained from the PWM, which gives thecommand to ADC converter for reading υc, iL and is, per phase. On the other hand, we need toknow the values of the parameters L, C, rs, rL . In this case, we suppose that these parameters willbe constant and measurable.

Rotor current observer

In order to make an experiment for to apply the ZAD–FPIC control strategies, it is necessary obtainthe rotor current in the equivalent circuit per phase model. The original rotor current equation canbe used directly as the observer equation. The implementation of such current observer is the on-line simulation of controlled rotary induction motor. This observer is stable in the large with therate of convergence depending on the rotor time constant. in this case the observer equation isgiven by

dirdt

= LmLmLls+LlrLm+LlrLls

vc − LmRsLmLls+LlrLm+LlrLls

is − (Lm+Lls)Rrs(LmLls+LlrLm+LlrLls)

ir (4.22)

where s = (ws − wm)/ws and ir is the estimate of the per phase rotor current. Defining theestimation error as ir = ir − ir result in the following error dynamic:

dirdt

= − (Lm + Lls)Rr

s(LmLls + LlrLm + LlrLls)ir (4.23)

Select a Lyapunov function candidate as

V =1

2i2r > 0 (4.24)

and calculate the time derivative of V along the solution of (4.31) to yield

V = − (Lm + Lls)Rr

s(LmLls + LlrLm + LlrLls)i2r < 0 (4.25)

This means that if we simply integrate equation 4.22, the mismatch between the real and estimatedrotor current tends to zero asymptotically. The rate of convergence may be improved if properlydesigned observer gain is introduced.

Figure 4.16 shows the simulation results of the currents rotor observer (4.22). To show the con-vergence process, select as initial conditions the observed currents ira = irb = irc = 2A, and thereal rotor currents ira = irb = irc = 0A. Since the observer is just on-line simulation of the motormodel, no observer gain is to be adjusted.


Figure 4.16: Responses of the current rotor observer.

4.3.4. Numerical and experimental results

In this section numerical and experimental results are shown. Initially the converter-motor systembehavior in open loop is illustrated, then the behavior of controlled system is shown via varyingthe parameters of the controller using bifurcation diagrams. KS is used as bifurcation parameter.KS1 and KS2 are fixed to zero.

Figure 4.17 shows the behavior of the current of phase a in the induction motor, the voltage ofphase a and the current in phase a of the LC filter when the reference voltage frequency varies inthe converter. The red graphs show the values of currents and voltages taken each period of thereference signal feed, beginning at the first peak value of each and varying the frequency from 10to 45 HZ.

In Figure 4.17, the blue graphs show the values for the variables mentioned above when the fre-quency is varied from 45 to 10 Hz. By raising the frequency, motor voltage shows a stable behaviorup to the 37 Hz (bifurcation point) and drops abruptly to a chaotic regime. Motor current at thesame frequency point, shows an increase in magnitude as a qualitative change in behavior. Whenthe frequency varies from 45Hz to 10 Hz, the bifurcation point is close to 24 Hz. We observe acoexistence of attractors (periodic and chaotic solution) ranging from 24 to 37 Hz.

In the Figure 4.18 the coexistence of attractors can also be observed. When the frequency is in-creasing speed there is a stable periodic solution, up to 37 Hz where the bifurcation occurs. Whenthe voltage and motor current entering chaotic regime, the motor speed drops to zero. The blue plotshows that the rotor speed is fully recovered when frequency lows below 24 Hz. In what followswe will analyze the system behavior in stable , chaotic and coexistence areas.

In the Figure 4.19 are given the phase diagrams for variables iL versus vC in the converter and ia


(a) Current ib in the adjustablespeed driver.

(b) Phase b motor voltage vb. (c) Current ib in the RIM.

Figure 4.17: Experimental bifurcation diagrams with f like bifurcation parameter in converter–rotary motor system

Figure 4.18: Rotor speed Vs Frequency.

Versus va in the motor obtained experimentally in the Buck–motor system in open loop. The bluegraphs were obtained by applying a three-phase sinusoidal voltage of 20 Hz (stable area) and redgraphics with a frequency in the supply voltage of 40 Hz (zone of chaos).

In both Figures the voltage go down and the current go up with the frequency change from 20Hzto 40 Hz. In 20 Hz the phase portraits show a periodic solution unlike of 40 Hz where a chaoticsolution is shown.

The Figure 4.20 shows the behavior of the system under study for the coexistence area of solutions.


(a) Phase portrait for Voltage motor and con-verter current.

(b) Phase a motor voltage va Vs Phase a mo-tor current ia.

Figure 4.19: Results for converter–motor system for 20HZ and 40 Hz of the supply voltage

(a) Voltage motor and convertercurrent of phase a.

(b) Phase a converter current. (c) Phase a motor voltage va.

Figure 4.20: Experimental results for coexistence in buck–rotary motor system

Again the blue lines illustrate the periodic solution and red lines show the chaotic solution.

For the illustration of numerical results a simulation of the rotary motor drive controlled by ZAD–FPIC control strategies was made. The motor model used in simulation was the per phase equiva-lent circuit.

Figure 4.21 shows the bifurcation diagrams for duty cycle, capacitor voltage vc error , vc andinductor current iL, in phase a when the bifurcation parameter Ks varies from 0 to 1. In thissimulation no delay in signal control is considered and N (FPIC parameter) is zero.

Figure 4.22(a) shows the time behaviour when the bifurcation parameterKs is 0.02. The controlledvoltage has chaotic behaviour and has quite difference respect to the reference voltage for phase a.


(a) Duty cycle of phase a. (b) capacitor voltage vc error for phase a.

(c) capacitor voltage vc in phase a. (d) inductor current iL in phase a.

Figure 4.21: Bifurcation diagrams for duty cycle, converter voltage and converter current of phasea.

In Figure 4.22(b) the tracking error of controlled voltage respect to the reference voltage is morelow, in this case the KS value is 0.7.

Phase portrait for ia and va variables shows a one-periodic solution for Ks = 0,7. When Ks =0,05 the qualitative behaviour is quite different (Figure 4.23). In this case the ZAD control wasused, the effect of the control FPIC was eliminated using N=0.

In the Figure 4.24 the bifurcations diagrams were obtained taking into account a time delay equalto 2T. With the time delay in the signal control, the bifurcation point has suffered a shift from about0.3 to about 0.9 and the variables variation range have been increased.

The Figure 4.25 shows the time behaviour when the signal control has been delayed two sampleperiods. Unlike of the system without delay in signal control the system delayed has the bifurca-tion point shifted. Figure 4.25(a) shows the time behaviour when the bifurcation parameter Ks is0.2. The controlled voltage has oscillatory behaviour around of voltage reference then it has quite


(a) Time behaviour with KS=0.02. (b) Time behaviour with KS=0.7.

Figure 4.22: Time behaviour for Voltage motor (controlled voltage) of phase a without delay.

Figure 4.23: Phase portrait for ia and va variables.

difference respect to the reference voltage for phase a. In Figure 4.25(b) the tracking error of con-trolled voltage respect to reference voltage is smaller because the oscillations amplitude are small,in this case the KS value is 2.

Phase portrait for ia and va variables shows a quasi–periodic solution for Ks = 2 (blue graphic)and Ks = 0,2 (red graphic)(Figure(4.26)).

The effect of time delay was illustrated in simulation. In the experiment the delay is caused byacquisition system and time process on the computer. In the simulation without delay the bifurca-tion parameter KS was varied form 0 to 1, taking into account 2T of time delay. The variation ofKs was from 0 to 2 in order to appreciate the bifurcation point and the effects of the delay on thesystem behaviour.

Before simulations were made without rotor current observer. Figure 4.27 shows the simulationresults taking estimated rotor currents like real currents in the controller. Difference between output


(a) Duty cycle of phase a. (b) capacitor voltage vc error for phase a


Figure 4.24: Bifurcation diagrams for duty cycle, Voltage motor and converter current of phase afor a time delay of 2T.

(a) Time behavior with Ks = 0,2. (b) Time behavior with Ks = 2.

Figure 4.25: Time behaviour for Voltage motor (controlled voltage) of phase a with a time delayof 2T.


Figure 4.26: Phase portrait for ia and va variables with 2T delay in signal control.

voltages is so difficult to see when in the controller is considered the observer and when it is notconsidered.

Figure 4.27: Response of the controller with and without rotor currents observer.

Figure 4.28: Error in output controlled voltages of the controller with and without rotor currentsobserver.


Figure 4.28 shows the effect of the observer on the controller. The error between output controlledvoltages when the currents observer is and is not considered is less than 1.5 %.The response of thesystem, when the controller parameters are varied, is practically the same that the case when theobserver is not included.

The experimental results of controlled system using ZAD and FPIC strategies, including rotor cur-rents observer, are shown in the next Figures. The system behaviour shows coexistence of solutionsfor different initial conditions maintaining the control parameters and operation frequencies.

(a) Voltage motor and convertercurrent of phase a.

(b) Phase a converter current. (c) Phase a motor voltage va.

Figure 4.29: Experimental results for coexistence in buck–rotary motor controlled system

Converter inductors were changed to L = 160µH in this experiment in order to reduce the hys-teresis effects.

The time behaviour of voltage controlled for two considered solutions are illustrated in Figure4.30. The Output voltage follows the reference voltage with a time delay (Figure 4.30(a)).In Figure 4.30(b) the motor voltage can not follow the reference voltage. This phenomenon is pre-sented in Figure 4.31 where the phase portrait presents the stator voltage and current for phase amotor.

In order to do a more complete analysis, graphics bifurcations were made. Figure 4.32 shows thebifurcations diagrams for duty cycle, output converter current (IL) on phase c, and phase c motorvoltage Vc; using Ks1 like bifurcation parameter. In Figure 4.38 the bifurcations diagrams weremade with the FPIC parameter (N ) like bifurcation parameter. The voltage reference for this ex-periment had a voltage peak of 40V and its frequency was settled in 20Hz. When the bifurcationparameter Ks1 was used, the FPIC parameter N was fixed in 1. When the bifurcation parameterN was used, the ZAD parameter Ks1 was fixed in 0.5; in this way the control chaos with FPICtechnique was illustrated.


(a) Time behaviour with no chaos. (b) Time behaviour with chaos.

Figure 4.30: Time behaviour for Voltage motor (controlled voltage) of phase a for different initialconditions.

Figure 4.31: Phase portrait for motor voltage and current of phase a.

Experimental bifurcation results for converter–rotary motor controlled system when Ks1 variesshow three zones where the system behaviour is qualitatively different. For 0 < Ks1 < 0,4the regime is chaotic, Ks1 between 0,4 and 0,7 the dynamic regime exhibits a chaotic and peri-odic solutions (a 1T periodic solutions for vc and 2T periodic solution for ILc), and finally for0,7 < Ks1 < 1 there is a 1T periodic solution.

For Ks1 = 0,3 the system has chaotic behaviour. Figure 4.33 shows the experimental behaviour


(a) Bifurcation diagram for dutycycle of phase c.

(b) Bifurcation diagram forphase c converter current.

(c) Bifurcation diagram forphase c motor voltage vc.

Figure 4.32: Experimental bifurcation results for converter–rotary motor controlled system

Figure 4.33: Chaotic Solution for the Three–phase converter with rotary motor load.

for three phase power converter with rotary motor load. The variables phase voltages ( signals red,green and blue on the left bottom side), duty cycles ( signals red, green and blue on the left upside), inductor converter currents, and motor currents (middle and lower signal on right side re-spectively) have a chaotic pattern.

For Ks1 = 0,5 the system presents both chaotic and periodic behaviour that correspond to the sec-ond zone described before for Figure 4.32. Figures 4.35 and 4.36 show the experimental behaviour


Figure 4.34: Coexistence of Solutions for the Three–phase converter with rotary motor load.

for three phase power converter with rotary motor load when coexistence of solutions is present.The variables phase voltages ( signals red, green and blue on the left bottom side), duty cycles (signals red, green and blue on the left up side), inductor converter currents, and motor currents(middle and lower signal on right side respectively) have a periodic solution in Figure 4.35 whilein Figure 4.36 these signals have a chaotic pattern at same control parameters.

In the zone for 0,7 < Ks1 < 1 where the converter–rotary motor system has stable response thesolution is 1T periodic. This fact is illustrated in Figure 4.37. The voltages, currents, and duty cy-cles signals are periodic and the controlled voltages can follow their respective references.

When the FPIC parameter N vary, arise three zones again. The first one corresponds to 0 < N <0,6 interval, the second one corresponds to 0,6 < N < 1,6 interval and the last one corresponds to1,6 < N < 2,0 interval. The system behaviour is similar to the case analyzed before.

Figures 4.39 and 4.40 show the signals system behavior for chaotic and periodic regimes. The co-existence case considered for the second zone is for N = 1 and Ks1 = 0,5 was described beforein Figures 4.34, 4.35 and 4.36.


Figure 4.35: Periodic Solution for the Three–phase converter with rotary motor load.




(a) Bifurcation diagram for dutycycle of phase c.

(b) Bifurcation diagram forphase c converter current.

(c) Bifurcation diagram forphase c motor voltage vc.

Figure 4.38: Experimental results for coexistence in buck–rotary motor controlled system

4.3.5. Conclusion

The control strategy ZAD-FPIC was designed and applied to three phasic converter with rotaryinduction motor load. Because is not possible to measure the rotor currents, a rotor current observ-er was included in the system. For this system, simulations and experiments were performed. Thestability of the closed loop system was analyzed using bifurcation diagrams, and stable and transi-tions to chaos were observed. In an experimental way the coexistence of attractors were illustrated


Figure 4.39: Periodic Solution for the Three–phase converter with rotary motor load.


in open loop system, the presence of this phenomenon was caused by the magnetic saturation in the

4.4 Three–phase Power Converter With Linear Induction Motor Load 107

core of the converter inductor. When the converter inductors were replaced for other with bettercores this phenomenon disappeared under 80 Hz. It was demonstrated in a simulation way, thatthe delay effects have high importance in ZAD–FPIC control strategy, with no delay the systemhas chaotic behaviour in some range of parameter values and with delay in the control signal thesystem has quasi–periodic solution. Experiments with controller and rotor current observer wereexecuted and the coexistence phenomenon was present. Evidence of a time delay was found in theexperiment realization.

4.4. Three–phase Power Converter With LinearInduction Motor Load

For the system power converter with linear induction motor load were considered its model and itsnumerical solution in the nonlinear analysis behavior. Parameters identification for modeling weredetermined in an experimental way. This section is composed by: Subsection 4.4.1 describes theproposed system, Subsection 4.4.2 describes the mathematical model of the system, Subsection4.4.3 describes the control techniques, Subsection 4.4.4 presents the obtained results, and finally,subsection 4.4.5 presents the conclusion.

4.4.1. Proposed SystemFigure 4.41 shows the system under study. The system is composed for two major parts. The firstone consists in hardware devices: voltage and current sensors, three phase power converter, Lab-Volt 8228-02 linear induction motor and the DSpace based rapid control prototyping. The secondone is the software that include ZAD–FPIC controller and the acquisition data drives for the digitalsignal processing. A more complete description of the LIM motor drive is shown in Figure 3.1.

Figure 4.41: Block diagram of the proposed system for LIM-Drive system

The hardware is composed of a Three–phase power converter with LIM motor load with rated pow-er 1500 Watts, rated Volts 600 VDC, rated Amps 20 ADC. In the measure of variables υc (capaci-


tor voltage) series resistance were used and for measurement of iL (inductor currents) HX10P/SP2current sensors were used. The converter switches were driven by PWM outputs of the controllercard,these signals are coupled via fast optocouplers (6N137).

The digital part is developed in the control and development card dSPACE DS1104, where ZADand FPIC control techniques are implemented. This card is programmed from Matlab/Simulinkplatform and it has a graphical display interface called ControlDesk.

The controllers are implemented in Simulink and they are downloaded to the DSP. The samplingrate for all variables is set to 6 kHz. The state variables υc, and iL per phase are 12 bits; the dutycycle (d) is 10 bits. Parameters of buck converter (C, L, rs, rL) and ZAD-FPIC–controller (KS ,N , Fs, LIM parameters) are entered to the control block by the user, as constant parameters. KS

is the bifurcation parameter. At each sampling time the controller calculates the duty cycle and theequivalent PWM signal to control the bridge switches.

4.4.2. Mathematical Model

A schematic of switched power converter and LIM coupled system taken into account end effectsis shown in the Figure 4.42, the power converter is used to feed the linear induction motor. Anytime domain analysis of such system normally assumes that the magnetic saturation, hysteresis andeddy current effects are all negligible. Equation 4.26 is obtained for the system model.

Figure 4.42: Electric circuit for the converter-motor system


iLavcaisairaiLbvcbisbirbiLcvcciscircvm

=

a11a a12a 0 0 0 0 0 0 0 0 0 0 0a21a 0 a23a 0 0 0 0 0 0 0 0 0 0

0 a32a a33a a34a 0 0 0 0 0 0 0 0 00 a42a a43a a44a 0 0 0 0 0 0 0 0 00 0 0 0 a11b a12b 0 0 0 0 0 0 00 0 0 0 a21b 0 a23b 0 0 0 0 0 00 0 0 0 0 a32b a33b a34b 0 0 0 0 00 0 0 0 0 a42b a43b a44b 0 0 0 0 00 0 0 0 0 0 0 0 a11c a12c 0 0 00 0 0 0 0 0 0 0 a21c 0 a23c 0 00 0 0 0 0 0 0 0 0 a32c a33c a34c 00 0 0 0 0 0 0 0 0 a42c a43c a44c 00 0 0 a51 0 0 0 a52 0 0 0 a53 − B

M

iLavcaisairaiLbvcbisbirbiLcvcciscircvm

+

ELa

0 0

0 0 00 0 00 0 00 E

Lb0

0 0 00 0 00 0 00 0 E

Lc

0 0 00 0 00 0 0

S1 − S4

S2 − S5

S3 − S6

+

00000000000−FL

M

(4.26)

Before equation can be expressed in a compact form as:

x = A(x)x+BU +D (4.27)

The system consists of three subsystems where each phase may be treated independently. Themodel per phase, considering FL = 0, is given by the equation (4.28).

iLυcisir

=

a11 a12 0 0a21 0 a23 00 a32 a33 a34

0 a42 a43 a44

iLυcisir

+

EL

000

Uc (4.28)

With Uc = −1, 1 according to switches control.


Matrix entries are defined as:

a11 = −rs + rLL

a12 = − 1

L

a21 =1

C

a23 = − 1

C

a32 =Lm(1− f(Q)) + Llr

LlsLm(1− f(Q)) + LlrLm(1− f(Q)) + LlrLls

a33 = − Rs(Lm(1− f(Q)) + Llr) +Rrf(Q)LlrLlsLm(1− f(Q)) + LlrLm(1− f(Q)) + LlrLls

a34 = − RrLm(1− f(Q)) + sLlrRrf(Q)

s(LlsLm(1− f(Q)) + LlrLm(1− f(Q)) + LlrLls)

a42 =Lm(1− f(Q))

LlsLm(1− f(Q)) + LlrLm(1− f(Q)) + LlrLls

a43 = − RsLm(1− f(Q)) +Rrf(Q)LlsLlsLm(1− f(Q)) + LlrLm(1− f(Q)) + LlrLls

a44 = − Rr(Lm(1− f(Q)) + Lls + sf(Q)Lls)

s(LlsLm(1− f(Q)) + LlrLm(1− f(Q)) + LlrLls)

s = (vs − vm)/vs

s is the motor slip, vm is the mover speed and vs is the synchronous speed.

The mechanical equation is given by:

vm =i2raRra

Msvs+i2rbRrb

Msvs+i2rcRrc

Msws− B

Mvm −

FLM

(4.29)

then:

a51 =iraRra

Msvs

a52 =irbRrb

Msvs

a53 =ircRrc

Msws



The control strategies presented in this section are developed for the per phase equivalent circuit forthe LIM. So for the three phase system the control must be applied for each phase independentlytaking into account that the reference voltage will be phase shifted according to the correspondingcircuit phase. Because the mechanical dynamic is very slow compared with the electrical dynamic,the mover velocity is considered constant in each period of sample.

ZAD–FPIC control strategy

The surface per phase, the duty cycle dk and the duty cycle in steady state were calculated insimilar way of the rotary induction motor load system.

Secondary current observer

In order to make an experiment for to apply the ZAD–FPIC control strategies, it is necessaryobtain the secondary current in the equivalent circuit per phase model (current flowing throughLlr in Figures 3.14 and 4.42). The original secondary current equation can be used directly asthe observer equation. The implementation of such current observer is the on-line simulation ofcontrolled Linear induction motor. This observer is stable in the large with the rate of convergencedepending on the secondary time constant. in this case the observer equation is given by

dirdt

=Lm(1− f(Q))

LlsLm(1− f(Q)) + LlrLm(1− f(Q)) + LlrLlsvc

− RsLm(1− f(Q)) +Rrf(Q)LlsLlsLm(1− f(Q)) + LlrLm(1− f(Q)) + LlrLls

is

− Rr(Lm(1− f(Q)) + Lls + sf(Q)Lls)

s(LlsLm(1− f(Q)) + LlrLm(1− f(Q)) + LlrLls)ir

(4.30)

where s = (vs − vm)/vs and ir is the estimate of the per phase secondary current. Defining theestimation error as ir = ir − ir results in the following error dynamic:

dirdt

= − Rr(Lm(1− f(Q)) + Lls + sf(Q)Lls)

s(LlsLm(1− f(Q)) + LlrLm(1− f(Q)) + LlrLls)ir (4.31)

Select a Lyapunov function candidate as

V =1

2i2r > 0 (4.32)

and calculate the time derivative of V along the solution of (4.31) to yield

V = − Rr(Lm(1− f(Q)) + Lls + sf(Q)Lls)

s(LlsLm(1− f(Q)) + LlrLm(1− f(Q)) + LlrLls)i2r < 0 (4.33)


This means that if we simply integrate equation 4.30, the mismatch between the real and estimatedsecondary current tends to zero asymptotically. The rate of convergence may be improved if prop-erly designed observer gain is introduced.

4.4.4. Numerical Results

Figure 4.43: Response of the secondary current observer.

Figure 4.43 shows the simulation results of the secondary currents observer (4.30). To show theconvergence process, select as initial conditions the observed currents ira = irb = irc = 3A, andthe real secondary currents ira = irb = irc = 0A. Since the observer is just on-line simulationof the LIM model, no observer gain is to be adjusted. The controlled LIM behavior with observedsecondary currents was very close to behavior of controlled system feeding with the real secondarycurrents in numerical simulation.

For the illustration of numerical results a simulation of the linear induction motor drive controlledby ZAD–FPIC control strategies was made. The motor model used in simulation was the per phaseequivalent circuit (Figure 4.44).

Figure 4.45 shows the bifurcation diagrams for duty cycle, capacitor voltage vc error , vc andinductor current iL, in phase a when the bifurcation parameter Ks varies from 0 to 0,5. In thissimulation a 1T delay in signal control was considered and N(FPIC parameter) was 2.

Figure 4.46(a) shows the time behaviour when the bifurcation parameterKs is 0.25. The controlledvoltage has chaotic behaviour and has quite difference respect to the reference voltage for phase a.In Figure 4.46(b) the tracking error of controlled voltage respect to the reference voltage is more


t

30

3*D

Zero-OrderHold

ZAD-FPIC CONTROL

Variables

Ks

N

Par

Ws

Reference

d_a_b_cf

100

Vcv

+-

A

B

C

+

-

g

A

B

C

+

-

Unit Delay

1/z

Vtri

d

gfcn

1

M.s+B

N

A

B

C

Rs

Rr/sRr*f(Q)

ReferenceVoltage

f

t

Vm

Ws

VAN

VBN

VCN

fcn

Lm(1-f(Q))

Lls Llr

L

Electromganetic Force

ir_a_b_c

Wm

Ws

Femfcn

FL

C

is

i+ -

ir

i+ - iL

i+ -

[CONTROL _PARAMETER]

[PARAMETERS_SYSTEM]

isc

Figure 4.44: Equivalent circuit model per-phase for simulation.

low, in this case the Ks value is 0.5.

Phase portrait for ia and va variables shows a one-periodic solution for Ks = 0,5. When Ks =0,25 the qualitative behaviour is quite different (Figure 4.47).

Figure 4.48 shows the bifurcation diagrams for duty cycle, capacitor voltage vc error , vc and induc-tor current iL, in phase a when the bifurcation parameter N varies from 0 to 3. In this simulation a1T delay in signal control was considered and Ks (ZAD parameter) was settled in 0.35.

Figure 4.49(a) shows the time behaviour when the bifurcation parameter N is 0.1. The controlledvoltage has chaotic behaviour and has quite difference respect to the reference voltage for phase a.In Figure 4.22(b) the tracking error of controlled voltage respect to the reference voltage is morelow, in this case the N value is 3.0.

Phase portrait for ia and va variables shows a chaotic solution for Ks = 0,35 and N = 0,1. WhenN = 3,0 the qualitative behaviour is quite different (Figure 4.50). In this case the ZAD controlwas used keeping Ks constant, the effect of the control FPIC was illustrated changing N = 0,1 toN = 3,0.




Figure 4.45: Bifurcation diagrams for duty cycle, converter voltage and converter current of phasea in the converter with LIM load.

4.4.5. ConclusionThe control strategy ZAD-FPIC was designed and applied to three phasic converter with linear in-duction motor load. Because is not possible to measure the secondary currents, a secondary currentobserver was included in the system. For this system, simulations were performed. The stabilityof the closed loop system was analyzed using bifurcation diagrams, and stable and transitionsto chaos were observed. Due to limitation in the mover path (1.5 meters), experiments with thecontrol system were not made. The bifurcation diagrams like a technique for to adjust controllerparameters in ZAD–FPIC controllers were shown in this section.


0.9 0.92 0.94 0.96 0.98 1

-100

-50

0

50

100

time [s]

Va

[V]

Vout

Vref

(a) Time behaviour with KS=0.25.

0.9 0.92 0.94 0.96 0.98 1-150

-100

-50

0

50

100

150

time [s]

Va

[V]

Vout

Vref

(b) Time behaviour with KS=0.5.

Figure 4.46: Time behaviour for Voltage motor (controlled voltage) of phase a.

-150 -100 -50 0 50 100 150-5

0

5

Va [V]

iLa

[A]

Ks=0.25Ks=0.5





Figure 4.48: Bifurcation diagrams for duty cycle, converter voltage and converter current of phasea in the converter with LIM load.


0.85 0.9 0.95 1

-100

-50

0

50

100

time [s]

Va

[V]

Vout

Vref

(a) Time behaviour with N = 0,1.

0.9 0.92 0.94 0.96 0.98 1-150

-100

-50

0

50

100

150

time [s]

Va

[V]

Vout

Vref

(b) Time behaviour with N = 3,0.

Figure 4.49: Time behaviour for Voltage motor (controlled voltage) of phase a without delay.

-150 -100 -50 0 50 100 150-5

0

5

Va [V]

iLa

[A]

N=0.1N=3.0


5 Main Contributions and Future Work

This chapter presents the different contributions and future work that can be derived from thisresearch.

5.1. The main contributions of this work are:

The first consists in obtain the continuous and discrete model of the linear induction mo-tor(LIM), which include the end effects in the state space representation. This phenomenonis highly nonlinear.

Another important aspect in the thesis is related to the implementation of the control ZAD-FPIC technique to higher order systems. So far the technique previously proposed only hadbeen developed, tested and validated on systems of second order. This paper has generalizedthe ZAD-FPIC technique, applied to a system consisting of a switched converter for handlingdifferent Three–phase loads, which constitutes a higher order system.

Control algorithms with ZAC-FPIC in a RCP system for controlling the tension on resistiveloads and induction motor drive. This system is divided into two major subgroups: hardwareand software. The hardware is composed of a Three Phase power converter rated power 1500Watts,which was designed and implemented in the power electronics laboratory, with RIMload rated power 1/3 HP, voltage 220V and current 1.4 A.

The digital part was developed in the control and development card dSPACE DS1104, whereZAD and FPIC control techniques were implemented. This card is programmed from Mat-lab/Simulink platform and it has a graphical display interface called ControlDesk. The con-trollers are implemented in Simulink and they are downloaded to the DSP.

It was made diagrams of experimental and numerical bifurcation for a switched Three–phaseinverter with different loads. Because is not possible to measure the rotor currents, a rotorcurrent observer was included in the system. For this system, simulations and experimentswere performed.

In an experimental way the coexistence of attractors were illustrated in open loop system,the presence of this phenomenon was caused by the magnetic saturation in the core of theconverter inductor.

5.2 Future Work 119

It was demonstrated in a simulation way, that the delay effects have high importance inZAD–FPIC control strategy, with no delay the system has chaotic behaviour in some range ofparameter values and with delay in the control signal the system has quasi–periodic solution.

Analysis of complex behavior for a Three–phase switched inverter, with different loads,controlled with ZAD-FPIC was reported.

5.2. Future Work

The research presented leaves no completely closed issues in the analysis of complex behavior inmultiphasic switched systems. On the contrary leaves open possibilities for future research mas-ter’s or doctoral degree, being many directions in which this work could continue. This sectionbriefly describes the future work that may result from this research.

In this sense there are some researches of nonlinear phenomena analysis in discrete-time nonlin-ear systems that could be accomplished on Three–phase commuted system. Among them can benamed the next: Stability Analysis of Equilibrium points and Periodic Solutions, Computer Methodto Analyze Bifurcation Phenomena, and Chaotic Behavior Detection.

5.2.1. To adjust the models with appropriate time delay, parametersand no modeled dynamics.

In order to get a better concordance between experimental and numerical results it is necessary toadjust the system parameters and to model dynamics like core´s saturation in inductors.

5.2.2. Stability Analysis of Equilibrium points and PeriodicSolutions

The procedure to check the equilibrium points stability consists in: (I) obtain the equilibriumpoints (x0); (II) compute the Jacobian matrix Df(x0) of vector field f(x0) in each equilibriumpoint x = x0 and finally; (III) evaluate the eigenvalues of Df(x0). The equilibrium point x0 isobtained by solving the nonlinear equation

xk+1 = f(xk) = xk = x0 x0 − f(x0) = 0 (5.1)

The Newton-Raphson method is usually utilized to solve it numerically, by carrying out the itera-tion

xi+10 = xi0 − [I −Df(xi0)]−1[xi0 − f(xi0)]

Where the superscripts denote the iteration index.

120 5 Main Contributions and Future Work

Note that the Jacobian Matrix Df(x0) in (II) is equal to that in the before iteration when it con-verges (xi0 → x0). This means that the equilibrium point and the Jacobian matrix are obtained atsame time by using the Newton-Raphson method.

In order to numerically check the stability of periodic solutions of non autonomous and au-tonomous systems we could use the before procedure. We should compute and test the stabilityof equilibrium points obtaining the periodic solution ϕ(t), computing the Jacobian matrixDP (ϕ0)of the Poincaré map at the periodic solution and evaluating the eigenvalues of DP (ϕ0).

The eigenvalues of the m ∗m Jacobian matrix evaluated at x0, Df(x0), will determine the stabilityof the equilibrium points and period-k orbits. There are many theorems for stability determinationin [37] [44] [76] [77] [18]

5.2.3. Computer Algorithms for Analysis Bifurcation Phenomena.For computing bifurcation diagrams using Monte Carlo method, the initial conditions for each pa-rameter value will be made through a random number generator, in a uniform domain. In this waythe situation of missing attractors with small basins of attraction will be avoided. This is a directsimulation method.

The direct simulation has two disadvantages: the bifurcation points can not be calculated withenough accuracy always, and it only calculates stable invariant sets (attractors). To precise the bi-furcation point localization it will be necessary to calculate unstable invariant sets. Therefore, thedirect methods should be complemented with continuation methods. Continuation methods havebeen a natural way for detection of discontinuity induced bifurcations (when a smooth flow hitsone surface or discontinuity boundary) [17]. The main idea behind numerical continuation appealsto the application of the implicit Function Theorem to compute sequences of points in small inter-vals along the solution curve x(µ) ≈ (xi, µi), i = 0...N .

Classification of Bifurcations: Consider a nonlinear dynamical system described by a discrete-time model depending on a parameter vector µ:

x(k + 1) = fµ(x(k)) (5.2)

with a fixed point x∗,so

x∗ = fµ(x∗) (5.3)

Note also that the fixed points x∗ of 5,2 depend on the parameter µ. There are several kinds ofbifurcations in nonlinear dynamical systems. They are generally classified into three types, de-pending on the conditions that hold at the point of bifurcation.

5.2 Future Work 121

Let Dfµ(x) be the Jacobian matrix of fµ with respect to x. A bifurcation is said to occur in thesystem described by 5.2 at µ = µ∗ if one of the eigenvalues λ of Dfµ(x∗) satisfies one of thefollowing conditions as µ is varied:

1. Type 1: The eigenvalue λ is real and passes through the point (+1,0) in the complex plane atµ = µ∗.

2. Type 2: The eigenvalue λ is real and passes through the point (-1,0) in the complex plane atµ = µ∗.

3. Type 3: The eigenvalue λ is complex and, together with its conjugate λ, passes throug theunit circle at points other than (+1,0) and (-1,0) at µ = µ∗.

A type 1 bifurcation corresponds to a saddle-node bifurcation, a transcritical bifurcation or apitchfork bifurcation, these being differentiated from each other by additional condition. A type2 bifurcation is called a period-doubling or subharmonic bifurcation, in which the stability of x∗

changes at µ = µ∗ and a new orbit which is not a fixed point of fµ bat has period-2 (x∗ 6= fµ(x∗)but x∗ = fµfµ(x∗)) appears. The bifurcation of type 3 is called the Neimark bifurcation, where thestability of x∗ changes and limit cycle surrounding the equilibrium point x∗ emerges.

Method to Determine Bifurcation Values: We now discuss a computer method to determinevalues of system parameters at which bifurcations occur for fixed point or periodic solutions of thesystem (satisfies 5.3). From the above conditions, bifurcations of types 1, 2, and 3 occur at x = x∗and µ = µ∗ if a pair (x∗, µ) satisfies the following equations:

det[I −Dfµ(x∗)] = 0 (5.4)

for type 1

det[−I −Dfµ(x∗)] = 0 (5.5)

for type 2

det[ejβI −Dfµ(x∗)] = 0 (5.6)

for type 3

where β is the angle around the unit circle at which the eigenvalue λ crosses the unit circle. Theproblem of determining bifurcation values of the system parameters is now reduced to determin-ing pair (x∗, µ) that satisfy both the fixed point condition and the appropriate bifurcation condition.

122 5 Main Contributions and Future Work

5.2.4. Chaotic Behavior DetectionChaos is aperiodic long-term behavior in a deterministic system that exhibits sensitive dependenceon initial conditions. This means that two trajectories starting very close together will rapidly di-verge from each other, and thereafter have totally different futures. Sensitive dependence can bequantified by Liapunov exponents λ [78].

Let f be a smooth map on Rm, let Jn = Dfn(v0), and for k = 1, ...,m, let rnk be the lengthof the kth longest orthogonal axis of the ellipsoid JnN for an orbit with initial point v0. Then rnkmeasures the contraction or expansion near the orbit of v0 during the first n iterations. The kthLyapunov number of v0 is defined by

Lk = lımn→∞

(rnk )1/n,

if this limit exists. The kth Lyapunov exponent of v0 is hk = lnLk. If N is the unit sphere in Rm

and A is an m ∗ m matrix, then the orthogonal axes of the ellipsoid AN can be computed in astraightforward way, as it showed in Theorem 2.24 of Chapter 2 of [44]. The lengths of the axesare the square roots of the m eigenvalues of the matrix AAT , and the axis directions are givenby the m corresponding orthonormal eigenvectors. For a map on Rm, each orbit has m Lyapunovnumbers, which measure the rates of separation from the current orbit point along m orthogonaldirections. These directions are determined by the dynamics of the map [44].The natural logarithmof each Lyapunov number is a Lyapunov exponent.

For the chaotic behavior detection we shall analyze the evolution of the probability density func-tion using Frobenius-Perron operator associated to the system map [79].

Using time series we might reconstruct attractors by Takens embedding theorem and we shalldetect chaos through Lyapunov exponents. In [22] and [7] procedures to calculate the Lyapunovexponent maximum and quantify the sensibility to initial condition are shown.Numerical calcula-tions of Lyapunov exponents and Lyapunov dimensions dimension are in [44] [77]

Bibliography[1] D. Taoutaou, R. Puerto, R. Ortega, and L. A. Loron, “New field oriented discrete-time con-

troller for current-fed induction motors,” Commission of European Communities under con-tract ERB CHRX CT 93-0380 and by the DIVA project of the Picardie region, Tech. Rep.,1993.

[2] R. Marino and P. Tomei, Nonlinear Control Design. New York: Prentice Hall, 1995.

[3] G. A. Loukianov, J. M. Cañedo, O. Serrano, I. V. Utkin, and S. Celikovsky, Adative SlidingMode Block Control of Induction Motors. New York: Mc Graw Hill, 1999.

[4] H. Tan and J. Chang, “Adaptive backstepping control of induction motor with uncertainties,”in Proceedings of the American Control Conference, San Diego, California, June 1999.

[5] V. I. Utkin, J. Guldner, and J. Shi, Sliding Mode Control in Electromechanical Systems. Lon-don - Philadelphia: Taylor and Francis, 1999.

[6] J. Vittek, V. A. Utkin, and S. J. Dodds, “Self oscillating, sensorless induction motor drivecontrol system with prescribed closed-loop and rotor flux dynamics,” 1998.

[7] T. Asakura, K. Yoneda, Y. Saito, and M. Shioka, “Chaos detection in velocity control ofinduction motor and its control by using neural network,” in Proceedings of ICSP 2000,2000.

[8] Y. Kuroe, Non Linear Phenomena in Power Electronics. N.Y.: IEEE press, 2001, ch. Com-puter Methods to analyze Stability and Bifurcation Phenomena, pp. 165–177.

[9] I.Ñagy and Z. SüTö, Non Linear Phenomena in Power Electronics. N.Y.: IEEE press, 2001,ch. Non Linear Phenomena in the Current Control of Induction Motors, pp. 328–338.

[10] A. G. Loukianov, J. Rivera, and J. M. Cañedo, “Discrete-time sliding mode control of aninduction motor,” in Proceedings of the 15th World Congress of IFAC.España, vol. 1, no. 1,2002, pp. 106–111.

[11] F. Angulo, G. Olivar, J. A. Taborda, and F. E. Hoyos, “Nonsmooth dynamics and fpic chaoscontrol in a dc-dc zad-strategy power converter,” in ENOC-2008, Saint Petersburg, Russia,June 30 - July 4 2008.

[12] T. A. Lipo and T. A. Nondahl, “Pole-by-pole d-q model of a linear inductuion machine,” IEEETransactions on Power Apparatus and Systems, vol. PAS-98, no. 2, p. 629, March/April 1979.

124 BIBLIOGRAPHY

[13] S. Banerjee and G. C. Verghese, Non linear Phenomena in Power Electronics. New York:IEEE Press, 2001.

[14] R. J. Wai and W. K. Liu, “Non linear control for linear induction motor servo drive,” IEEETransation on Industrial Electronics, vol. 50, no. 5, pp. 920– 935, October 2003.

[15] E. Fernandes, E. B. dosSantos, P. C. M. Machado, and M. A. A. deOliveira, “Dynamic modelfor linear induction motors,” in ICIT 2003, Maribor, Slovenia, 2003, pp. 478–482.

[16] M. diBernardo, C. Budd, A. R. Champaynes, P. Kowalczy, A. B. Nordmark, G. Olivar, andP. T. Piiroinen, “Bifurcatios in nonsmooth dynamical systems,” May 2006, aMS Subject Clas-sifications: 34A36, 34D08, 37D45, 37G15, 37N05, 37N20.

[17] M. diBernardo, C. Budd, A. R. Champaynes, and P. Kowalczyk, Piecewise-smooth Dynam-ical Systems: Theory and Applications. Berlin Heidelberg New York HongKong LondonMilan Paris Tokyo: Springer, 2007.

[18] Z. T. Zhusubaliyev and E. Mosekilde, Bifurcations and Chaos in Piecewise-Smooth Dynami-cal Systems, ser. series on Nonlinear sciencE, A, L. O. Chua, Ed. London: World Scientific,2003, vol. 44.

[19] P. Kowalczyk, M. DiBernardo, A. R. Champneys, S. J. Hogan, M. Hommer, P. T. Phroinen,Y. A. Kuznetsov, and A.Ñordmark, “Two-parameter discontinuity-induced bifurcations ofcycles: Classification and open problems,” International Journal of Bifurcation and Chaos,vol. 16, no. 3, 2006.

[20] M. diBernardo, G. Olivar, and C. Batlle, Non Linear Phenomena in Power Electronics. N.Y.:IEEE press, 2001, ch. Control of Chaos, pp. 393–406.

[21] C. K. Tse, Non Linear Phenomena in Power Electronics. N.Y.: IEEE press, 2001, ch.Techniques of experimental investigation, pp. 111–123.

[22] S. Banerjee and D. C. Hamil, Non Linear Phenomena in Power Electronics. N.Y.: IEEEpress, 2001, ch. Techniques of numerical investigation, pp. 123–129.

[23] E. Ott, C. Grebogi, and J. A. Yorke, “Controlling chaos,” Phys. Rev. Lett., no. 64, pp. 1196–1199, 1990.

[24] G. Chen and X. Dong, From Chaos To Order: Methodologies, Perspectives and Applications.Singapure: World Scientific, 1998.

[25] G. Chen, Controlling Chaos and Bifurcations in Engineering Systems. Boca Raton, FL,USA: CRC Press, 1999.

[26] K. Pyragas, “Control of chaos by self-controlling feedback,” Phys. Lett., vol. 170, pp. 421–428, 1992.

BIBLIOGRAPHY 125

[27] F. Blaschke, “The principle of field orientation applied to the new transvector closed-loopcontrol system for rotating field machines,” Siemens Rev., vol. 39, pp. 217–220, 1971.

[28] R. J. Wai and W. K. Liu, “Nonlinear decoupled control for linear induction motor servo-driveusing the sliding-mode technique,” IEEE Proc-Control Theory Appl., vol. 148, no. 3, pp.217–231, May 2001.

[29] J. Liu, F. Lin, Z. Yang, and T. Q. Zheng, “Fiel oriented control of linear induction motorconsidering attraction force and end-effects,” International Power Electronics and MotionControl Conference IPEMC 2006, 2006.

[30] R. Ortega and D. Taoutaou, “A globally stable discrete-time controller for current-fed induc-tion motors,” Systems and Control letters, vol. 28, pp. 123–128, 1996.

[31] E. Fernandes, C. C. dosSantos, and J. W. Lima, “Field oriented control of linear inductionmotor taking into account end-effects,” in AMC 2004, Kawasaki, Japan, 2004, pp. 689–694.

[32] V. H. Benitez, A. G. Loukianov, and E.Ñ. Sanchez, “Identification and control of a linearinduction motor using an α−β model,” in Proceedings of the American Control Conference,Denver, Colorado, June 4-6 2003.

[33] B. Van-Der-Pol and J. Van-Der-Mark, “Frequency demultiplication,” Nature, 1927.

[34] E.Ñ. Lorenz, “Deterministic nonperiodic flow,” J. Atmos. Sci, vol. 20, no. 130, 1963.

[35] J. Gleick, Chaos: Making a New Science. New York: Viking, 1987.

[36] T. Y. Li and J. A. Yorke, “Period three implies chaos,” American mathematical Monthly,vol. 2, pp. 985–992, 1995.

[37] Y. Kuroe, Non linear Phenomena in Power Electronics. IEEE Press, 2001, ch. 7. Analysis ofStability and Bifurcation in Power Electronic Induction Motor Drive Systems, pp. 338–352.

[38] I. Boldea and S. A. Nasar, The Induction Machine Hanbook, ser. Electric power engineering,L. Grigsby, Ed. Boca Raton: CRC Press, 2002.

[39] V. H. Benitez, E.Ñ. Sanchez, and A. G. Loukianov, “Neural identification and control forlinear induction motors,” Journal of Intelligent and Fuzzy Systems, no. 16, pp. 33–55, 2005,iOS Press.

[40] K. Ogata, Discrete-Time Control Systems. Pretice-Hall, 1987.

[41] A. Boucheta, I. Bousserhane, A. Hazzab, B. Mazari, and M. Fellah, “Linear induction motorcontrol using sliding mode considering the end effects,” Systems, Signals and Devices, 2009.SSD ’09. 6th International Multi-Conference on, pp. 1–6, ’2009’.

126 BIBLIOGRAPHY

[42] J. Duncan, “Linear induction motor-equivalent-circuit model,” Proc. IEE, vol. 130, Pt B,no. 1, pp. 51–57, 1983.

[43] C.-M. Ong, Dynamic Simulation of Electric Machinery Using Matlab/Simulink, R. Hall, Ed.Upper Sadle River, New Jersey 0748: Prentice Hall, 1998.

[44] K. T. Alligood, T. D. Sauer, and J. A. Yorke, CHAOS An Introduction to Dynamical Systems,ser. Textbooks in Mathematical Sciences, T. F. Banchoff, K. Devlin, G. Gonnet, J. Marsden,and StanWagon, Eds. New York: Springer-Verlag, 1996.

[45] N. Mohan, First Course on Power Electronics and drives. Mnpere, USA, 2003.

[46] D. Ross and J. Theys, “Using the dspic30f/dspic33f for vector control of an acim,” Microchip,Tech. Rep., 2007.

[47] E. Fossas, R. Griñó, and D. Biel, “Quasi-sliding control based on pulse width modulation.zero averaged dynamics and the l2 norm,” Advances in Variable Structure Systems. Analy-sis. Integration and Applications (6th International Workshop on Variable Structure Systems(VSS’2000), pp. 335–344, 2000.

[48] F. Angulo, G. Olivar, and J. Taborda, “Continuation of periodic orbits in a zad-strategy con-trolled buck converter,” Chaos. Solitons and Fractals, vol. 38, pp. 348–363, 2008.

[49] F. Angulo, E. Fossas, and G. Olivar, “Transition from periodicity to chaos in a pwm controlledbuck converter with zad strategy,” Int. Journal of Bifurcations and Chaos, vol. 15, pp. 3245–3264, 2005.

[50] F. Angulo, “Análisis de la dinámica de convertidores electrónicos de potencia usando pwmbasado en promediado cero de la dinámica del error (zad),” Ph.D. dissertation, UniversidadPolitécnica de Cataluña, Cataluña, 2004.

[51] J. Taborda, “Análisis de bifurcaciones en sistemas de segundo orden usando pwm y promedi-ado cero de la dinámica del error,” Master’s thesis, Universidad Nacional de Colombia - SedeManizales, Colombia, Mayo 2006.

[52] F. Angulo, E. Fossas, C. Ocampo, and G. Olivar, “Stabilization of chaos with fpic: appli-cation to zad strategy buck converters,” In Proceedings: 16th World Congress InternationalFederation of Automatic Control. Praga (República Checa), july 2005.

[53] F. Angulo, J. Burgos, and G. Olivar, “Chaos stabilization with tdas and fpic in a buck con-verter controlled by lateral pwm and zad,” In Proceedings: Mediterranean Conference onControl and Automation, july 2007.

[54] F. Angulo, G. Olivar, J. Taborda, and F. Hoyos, “Nonsmooth dynamics and fpic chaos controlin a dc-dc zad-strategy power converter,” ENOC-2008, vol. Vol. 55„ pp. 2392–2401, 2008,Saint Petersburg, Russia.

BIBLIOGRAPHY 127

[55] G. Olivar, “Chaos in the buck converter,” Ph.D. dissertation, Universidad Politécnica deCataluña, Cataluña, 1997.

[56] F. Hoyos, “Desarrollo de software y hardware para manejo de un convertidor dc-dc y dc-ac controlado con zad y fpic,” Master’s thesis, Universidad Nacional de Colombia - SedeManizales, Colombia, Julio 2009.

[57] G. Kang, J. Kim, and K.Ñam, “Parameter estimation scheme for low-speed linear inductionmotors having different leakage inductances,” IEEE Trans. on Industrial Electronics, vol. 50,no. 4, pp. 708–716, August 2003.

[58] ——, “Parameter estimation of a linear induction motors with pwm inverter,” The 27th An-nual Conference of the IEEE Industrial Electronics Society, pp. 1321–1326, 2001.

[59] J. F. Gieras, G. E. Dawson, and A. R. Eastham, “new longitudinal end effect factor for linearinduction motors,” IEEE. Trans. Energy Conversion, vol. 22, p. pp. 152–159, Mar. 1987.

[60] J. F. Gieras, “Linear induction drives,” Oxford, U.K.: Clarendon, 1994.

[61] J. Lagarias, J. A. Reeds, M. H. Wright, and P. E. Wright, “Convergence properties of thenelder-mead simplex method in low dimensions,” SIAM Journal of Optimization, vol. 9, no. 1,pp. 112–147, 1998.

[62] S. Banerjee and G. Verghese, “Nonlinear phenomena in power electronics,” IEEE Press. Pis-cataway, 2001.

[63] N. Mohan, T. Undeland, and W. Robbins, Power Electronics: Converters. Aplications andDesign. J. Wiley, 1995.

[64] H. Haitao, V. Yousefzadeh, and D. Maksimovic, “Nonuniform a/d quantization for improveddynamic responses of digitally controlled dc–dc converters,” Power Electronics, IEEE Trans-actions on, vol. 23, no. 4, pp. 1998 – 2005, 2008.

[65] C. Fung, C. Liu, and M. Pong, “A diagrammatic approach to search for minimum samplingfrequency and quantization resolution for digital control of power converters,” Power Elec-tronics Specialists Conference, 2007. IEEE, pp. 826 – 832, 2007.

[66] J. Taborda, F. Angulo, and G. Olivar, “Estimation of parameters in buck converter withdigital-pwm control based on zad strategy,” Circuits and Systems (LASCAS), 2011 IEEE Sec-ond Latin American Symposium on, pp. 1–4, 2011.

[67] D. Maksimovic, R. Zane, and R. Erickson, “Impact of digital control in power electronics,”Power Semiconductor Devices and ICs, 2004. Proceedings. ISPSD ’04. The 16th Internation-al Symposium on,, pp. 13–22, 2004.

128 BIBLIOGRAPHY

[68] A. Peterchev and S. Sanders, “Quantization resolution and limit cycling in digitally controlledpwm converters,” Power Electronics, IEEE Transactions on, vol. 18, no. Issue: 1 , Part: 2, pp.301 – 308, 2003.

[69] P. Hao, D. Maksimovic, A. Prodic, and E. Alarcon, “Modeling of quantization effects indigitally controlled dc-dc converters,” Power Electronics Specialists Conference, 2004. PESC04. 2004 IEEE 35th Annual, vol. 6, pp. 4312 – 4318, 2004.

[70] F. Angulo, F. Hoyos, and G. Olivar, “Experimental results on the quantization in the adc de-vice for a zad-strategy controlled dc-dc buck converter,” 7 th European Nonlinear DynamicsConference, July 2011.

[71] L. Corradini and P. Mattavelli, “Analysis of multiple sampling technique for digitally con-trolled dc-dc converters,” Power Electronics Specialists Conference, 2006. PESC ’06. 37thIEEE, pp. 1–6, 2006.

[72] M. Algreer, M. Armstrong, and D. Giaouris, “Adaptive control of a switch mode dc-dc powerconverter using a recursive fir predictor,” Power Electronics, Machines and Drives (PEMD2010), 5th IET International Conference on, pp. 1–6, 2010.

[73] ——, “Adaptive pd+i control of a switch-mode dc–dc power converter using a recursive firpredictor,” Industry Applications, IEEE Transactions on, vol. 47, no. 5, pp. 2135 – 2144,2011.

[74] M. Shirazi, R. Zane, and D. Maksimovic, “An autotuning digital controller for dc–dc pow-er converters based on online frequency-response measurement,” Power Electronics, IEEETransactions on, vol. 24, no. 11, pp. 2578 – 2588, 2009.

[75] F. Angulo, F. Hoyos, and J. Taborda, Principios de la Estrategia de Control Zero AverageDynamics (ZAD). jmariotta, ISBN: 978-3-8465-7666-3, 2011.

[76] Y. A. Kuznetsov, Elements of Applied Bifurcation Theory, 3rd ed., ser. Applied MathematicalSciences, S. S. Antman, J. E. Marsden, and L. Sirovich, Eds. Berlin Heidelberg New YorkHongKong London Milan Paris Tokyo: Springer, 2004, vol. 112.

[77] T. S. Parker and L. O. Chua, Practical Numerical Algorithms for Chaotic System. NewYork: Springer-Verlag, 1989.

[78] S. H. Strogatz, Nonlinear Dynamics and Chaos With Applications to Physics, Biology, Chem-istry and Engineering. Westview Press-Perseus Books Publishing, 2000.

[79] J. L. Rodríguez, G. C. Verguese, R. Santos, and H. I. Steven, Non Linear Phenomena inPower Electronics. N.Y.: IEEE press, 2001, ch. Computations of averages Under Chaos, pp.129–149.

Documents

Dynamical Analysis of Three–phase Systems Controlled by