republique algerienne democratique et populaire

REPUBLIQUE ALGERIENNE DEMOCRATIQUE ET POPULAIRE Ministère de l’Enseignement Supérieur et de la Recherche Scientifique Ecole Nationale Polytechnique d’Oran

ENPO - ORAN Département de : Génie Electrique

MEMOIRE POUR L’OBTENTION DU DIPLOME DE MAGISTER EN AUTOMATIQUE OPTION : Commande Avancée des Systèmes Complexes INTITULE :

COMMANDE ADAPTATIVE PAR RETOUR D'ETAT AVEC OBSERVATEUR DU MOTEUR ASYNCHRONE Présenté par : BEKHITI BELKACEM Devant le Jury composé de : o Mr. BOUHENNA.A

MC ‘A’

(ENPO-Oran)

o Mr. MANSOURI.A

Prof.

o Mr. CHENAFA.M

MC ‘A’

(ENPO-Oran)

Exanimateur

o Mr. OMARI A

MC ‘A’

(USTO)

Examinateur

(ENPO-Oran)

Année Universitaire 2013 / 2014

Président Encadreur

ACKNOWLEDGEMENT In the name of Allah, the Most Beneficent and Most Merciful. We thank Allah for all His blessing and strength that He gives us in completing this project. I would like to express my deepest gratitude to my final year project supervisor, Pr: MANSOURI Abdellah for his invaluable advices and guidance throughout this project. His profound knowledge, ideas and support keeps on motivating me to give me all for this project. I wish to thank all jury members, Dr: CHENAFA Mohammed, Dr: BOUHENNA Abderrahmane and Dr: OMARI Abdelhafid our friends and to those whom directly or indirectly guiding and helping me in this project. The knowledge and support that they shared with me will always be remembered. Lastly, and most importantly I wish to dedicate my appreciation to my beloved Parents, brothers and sisters for always be there for me all these year (DIF Ahmed, BELOUDAH Takieddine and ABED_ALKAWI Mohammed). Thanks for their unconditional love, encouragement, and support.

Abstract The theory of hyper-stability as it is developed by Popov was the main subject of researches till nowadays, especially non-linear and/or time variables process control. The more contribution of this concept is in terms of stability and performance of modern control, especially, in adaptive control systems. In this meaning, the actual study is devoted to presenting the general directives for the development of reference type adaptive control by hyper-stability criteria. More applications of this control technique is to the induction motor with its vector control was considered. The adaptive structure obtained is relatively heavy and it’s setting up in real time needs relatively strong calculators, and from other point of view we need sensors for sensing speed which are usually noisy measurement tools. So, sophisticated algorithms have been introduced based on the adaptive state estimation. High performances in terms of speed and torque control can be achieved by using indirect vector control and feedback linearization to decouple the induction motor. However a speed sensor is required to ensure this decoupling. In some industrial applications, the use of speed sensor results in extra cost or at expense of low reliability drive system; consequently replacing the speed sensor by software makes the induction motor more reliable. Many techniques have been developed to meet this objective. In the last part of this thesis three methods are investigated, namely Extended Kalman filter, MRAS observer and adaptive Luenberger observer. A simulation framework is developed in order to compare their performance for different operating conditions. The set of adaptive structures proposed are tested using Matlab and compared to each other.

Key words Hyper-stability, Adaptive control, Adaptation mechanism, Induction motor, Vector control, Extended Kalman filter, Adaptive Luenberger observer, Feedback linearization.

Nomenclature

Nomenclature

SCIM:

Squirrel cage induction motor;

a, b, c :

Subscripts denoting actual three phase axes;

s, m, r:

Subscripts denoting Stator, air- gap and rotor, respectively;

d, q :

Subscripts denoting axes in rotating frame (PARK axes);

α, β :

Subscripts denoting stator-fixed axes (CLARK axes);

S:

Laplace operator

l s, l r :

Apparent self inductance per phase of stator and rotor;

lm :

Apparent mutual inductance;

Rs, Rr :

Resistance of stator and rotor coils per phase;

σ:

Coefficient of dispersion;

j:

Rotor moment of inertia;

f:

Friction coefficient;

p:

Number of pair-poles;

[Vs]:

Stator voltage vector;

[Vr]:

Rotor voltage vector;

[Is ]:

Stator current vector;

[ s]:

Stator flux vector;

[ r]:

Rotor flux vector;

θr :

Rotor electrical angle;

θe :

Stator synchronous electrical angle;

θsl :

Slip electrical angle;

ωe :

Electrical synchronous frequency;

ωr :

Electrical rotor frequency;

ωsl :

Slip frequency;

Ωr :

Rotor angular velocity;

Tr:

Rotor time constant;

Ts :

Stator time constant;

θr :

Rotor flux angle;

θsl :

Slip angle;

θe :

Electrical angle;

ωs :

Electrical Stator frequency;

Nomenclature

Te:

Electromagnetic torque;

TL:

Load torque;

EKF:

Extended kalman Filter;

LO:

Luenberger observer;

[P]:

Park Transformation matrix

P:

Lyapunov matrix

*

:

Reference notation

e:

Nominal value

:

State variable

̇:

Derivative of state variable

BIBO:

Bounded input bounded output stability

PR:

Real positive

SPR:

Strict real positive

CGR:

Constant gain regulator

MRAC:

Model reference adaptive control

MRAS:

Model reference adaptive system

Table of Contents Acknowledgment Abstract Contents General Introduction Chapter 1: Modeling of the Three Phase Induction Machine 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8

Introduction Description of the Cage Induction Machine Advantage of the Asynchronous Machine Problems with the Asynchronous Machine Simplifying Assumptions The Phasors Concepts Park Transformation Model of the Asynchronous Machine 1.8.1 Equations of Bases 1.8.2 The Park Model of the Asynchronous Machine 1.8.3 Definition of the Different Repositories 1.9 Representation of the State Model of the Asynchronous Machine 1.9.1 Modeling the Current Feed Machine 1.9.1.1 Description 1.9.1.2 The State Equations 1.9.1.3 Block Diagram 1.9.2 Modeling the Voltage Feed Machine 1.9.2.1 Description 1.9.2.2 The State Equation 1.9.2.3 Block Diagram 1.10 Simulation of the Asynchronous Machine Model 1.10.1 Overview 1.10.2 Simulation of a Current Feed Machine 1.10.3 Simulation of a Voltage Feed Machine 1.11 Conclusion

1 1 2 2 3 3 3 4 5 6 8 10 10 10 10 11 11 11 11 12 13 13 13 17 20

Chapter 2: Field Oriented Control of Induction Machine 2.1 Introduction 2.2 Control by Flux 2.2.1 Description 2.2.2 Current Control 2.2.3 Voltage Control 2.3 Structure of a Current Driven Motor 2.3.1 Description 2.3.2 Rotor Flux Orientation (Indirect Control) 2.4 Structure of a Voltage Driven Motor 2.4.1 Description 2.4.2 Orientation of the Rotor flux (Indirect Control) 2.5 Simulation and Results Interpretations 2.5 Conclusion

21 22 22 23 24 25 25 25 28 28 29 31 37

Table of Contents Chapter 3: Synthesis of the Adaptive Control via Hyper-stability Criteria 3.1 Introduction 3.2 Hyper-stability Criteria 3.2.1 Basic Principle 3.2.2 Hyper-stability and Positivity: 3.2.2.1 Definition of Hyper-stability 1 3.2.2.2 Definition of Hyper-stability 2 3.2.2.3 Asymptotic Hyper-stability 3.2.2.4 Definition of Weak Hyper-stability 3.2.3 Hyper-stability of Linear Systems 3.2.4 Properties Hyper-stable Systems 3.3 Controllers with Constant Gain 3.4 Adaptive Control with Reference Model (MRAC) 3.6 Model Reference Choice 3.8.1 Reference Model 3.8.2 Choice of Adjustment Gains 3.9 Conclusion

38 39 39 40 40 41 41 41 42 43 45 48 53 53 53 54

Chapter 4: Application to the Control of Asynchronous Machine 4.1 Introduction 4.2 Adaptive control for Current Feed Machine 4.2.1 Description 4.2.2 Reference Model Based on the Flux Orientation 4.2.3 Controller with Constant Gain (CGR) 4.2.4 Adaptive Control with Reference Model (MRAC) 4.2.4.1 Behavior of the Constant Parameters Adaptation Mechanism 4.3 Adaptive Control for Voltage Feed Induction Machine 4.3.1 Description 4.3.2 Behavior of the Constant Parameters Machine 4.4 Behavior of the Adaptation Mechanism with Respect to Parameters Change 4.4.1 Introduction 4.4.2 Influence of Compensating of the Rotor Resistance 4.5 Conclusion

55 56 56 56 57 60 64 71 71 73 75 75 78 89

Chapter 5: Adaptive State Feedback Control via State Reconstruction 5.1 Introduction 5.2 Input Output Linearization 5.3 Asynchronous Machine Control via Input-Output Linearization 5.3.1 Control without Observer 5.4 Kalman Filter 5.4.1 Selection of the Time-Domain Induction Machine Model 5.4.2 Discretized Machine Model 5.4.3 State Estimation Simulation with EKF 5.5 General Theory of Observer 5.5.1 Luenberger Observer 5.5.2 Induction Motor in Stationary Reference Frame (α, β) 5.5.3 Determination of the Gain Matrix K via Adaptation Mechanism 5.5.4 State Representation of Adaptive Luenberger Observer 5.5.5 Adaptive Speed Estimation 5.6. Model Reference Adaptive System Based Observer 5.6.1 MRAS Techniques

80 80 84 84 87 89 90 90 91 91 92 92 93 93 94 94

Table of Contents 5.7 Simulation and Results Interpretation of Adaptive Feedback Control 5.7.1 Adaptive Field Oriented Control a) Field Oriented Control (FOC) with Extended Kalman Filter (EKF) b) Field Oriented Control (FOC) with MRAS Observer c) Comparison Study between MRAS Observer and EKF with IFOC 5.7.2 Adaptive Feedback Linearization Control (AFLC) a) Feedback Linearization Control with Kalman Filter b) Feedback Linearization Control with Adaptive Luenberger Observer c) Comparison Study between Luenberger Observer and EKF with FLC 5.8 Conclusion General Conclusion References Appendices

95 99 99 99 101 105 105 107 109

General Introduction

.

GENERAL INTRODUCTION With the development of power electronics, as well as technologies digital control, wired or programmed, the electric drives for variable speed motors experiencing tremendous progress and that in all sectors of industry. Induction machines (IM) are operated with their advantages of simplicity in construction, and low cost [01-05]. In modern industrial facilities, the asynchronous machine has a particular attention for many reasons: low cost, low mass, robustness and simplicity of construction. However, contrary to the simplifications offered by the collector brushes in the control system of the D.C. machine, the dynamics of the asynchronous machine is not linear, but strongly multivariable coupled system. In Moreover, some of its state variables are inactive transferable to direct (rotor flux) and the machine parameters (in particular, the rotor resistance) are affected by the thermal effect and functioning conditions, which poses a problem to its control. In summary, the key issues affecting asynchronous drives (to vector control, direct torque control ...), available today on the market are: the need for a mechanical sensor and the sensitivity to changes in parameters. Conventional controls performed by the help of standard controllers (PI controller) ensure control with very fast response time. The synthesis and reliability of these depends on the parameters of the machine. Thus an inaccurate identification or a variation of machine parameters affect significantly on the performance brought by these regulators, in addition to these, have difficulty with the dynamic system due to the variation of the load. At this stage, and to find solutions to the problems described above, the adaptive control reference model is a swing. However, most of the works on adaptive control of the asynchronous machine are based on optimizing the performance requirements. This technique command assumes that the machine parameters are accessible and committed to each time using methods of identification, but in practice this is not very realistic, since we usually have access only to system inputs. Furthermore, it has been demonstrated that these methods give only control laws that lead to locally stable systems. Thus recourse to find solutions to these two problems of adaptive control, led us to seek a satisfactory adaptive control structure at time: the development of a stable adaptive control, the adaptation signal additional synthesis. In the majority of the performed works, the stability problem is solved by a synthesis of adaptive controllers using the theory of stability, in particular the second method Lyapunov. This method has the properties of an overall stability without any restrictions to initial condition errors, or to the nature of the inputs or the used references. However, the major drawback of this approach is the need to finding an appropriate Lyapunov function; this function must be defined for all the adaptive laws known. So, there is a problem of the generalization of Lyapunov method. These drawbacks have motivated researchers to move to another concept of stability said hyperstability criterion. In the light of what has been said, the objective of this work is to develop an adaptive reference-model control, with a law synthesized by the hyper-stability concept applied to the induction motor, allowing research for a priori compromise between desired performance and stability of the overall system. In addition, study the robustness of this control vis-a-vis any parametric variations and external disturbances [31, 34]. In this context, the synthesis of adaptive control is company on simplified models of the asynchronous machine in its state form for two power modes (current/voltage driven IM). This structure will be called to control the magnitude of the speed and provided that the electromagnetic torque follows its reference, with no overshoot start times and set-point change. However, the decoupling of the flux and torque is obtained traditionally via vector control (indirect).

General Introduction

.

In a more general way, the objectives can be summarized as following: ■ Develop an adaptive control to ensure the overall stability of the system; ■ Achieve a method of synthesis by a control signal which does not require the added instant identification of parameters of the process to be controlled; ■ Reduce the need for accurate modeling; ■ Improve the robustness of control of the asynchronous machine in the face of parameter variation and model uncertainty; ■ Implement an adaptive faster, simpler, more flexible control, and easier to implement in real time. This thesis is organized as follows: In the first chapter, the modeling of the induction machine will be presented for two power modes (current and/or voltage driven IM). The models will be selected as a functional diagram to be directly usable in Matlab-Simulink. Models of machine have been simulated with the same software and whose results will be presented and commented to validate their accuracies. To highlight the interest of a proposed approach and able to carry out its comparisons with conventional control techniques, chapter two is exposed on one of these techniques. Indeed, the second chapter is devoted to the applications of indirect vector control with oriented rotor fluxes to the two power modes such as simulations were conducted and the responses will be presented and commented to better motivate the use of adaptive control model reference. Chapter three discusses the theoretical question approaches for the synthesis of a control law to adaptive reference model after introducing the hyper-stability concept. Furthermore, guidelines on the choice of the reference model and the gains of the adaptation mechanism drawn from other work in the same axis [24] have been addressed. The fourth chapter is devoted to an application of these techniques to all (vector control + asynchronous machines) for two power modes. Then, an analysis of operation of the machine is performed based on dynamic responses obtained by simulation. In the fifth chapter we present the adaptive state feedback control based on adaptive state reconstruction. However, the (FOC) technique requires knowledge of the rotor flux and the rotor position to control the torque and speed of the machine. For this and for economic reasons and/or of robustness reasons, we will present the estimation technique fluxes and velocity using the Kalman filters for the purpose of direct measurement without using a sensor. Thereafter we present another technique for sensorless control called the adaptive FOC control based on the adaptive observer (MRAS) to cope with the problems brought by the use of conventional controllers (PI). Finally, to address the drawbacks of the vector control we present the basic concepts of linear and nonlinear systems with detailed study of the theory of nonlinear feedback control condition. Then we applied this technique of control of the asynchronous machine to study its operation followed by a numerical simulation whose aim is to validate the proposed control. The last part of this chapter is devoted to the application of nonlinear state feedback control with adaptive rotor flux machine in the presence of asynchronous changes in stator and rotor resistances.

Finally, we give a conclusion to draw up an assessment of simulation results obtained, as well as perspectives to consider regarding potential applications experimental or theoretical research.

Chapter: I

C

hapter

Modeling of the Three Phase Induction Machine

I

Modeling of the Three Phase Induction Machine 1.1 Introduction Industrially, cage asynchronous machine is particularly attractive, by his great electromagnetic robustness, low cost and good standardization. However, its structural simplicity hides great complexity due to the functional absence of independent inducer, nonlinearities, the difficulty of identifying parameters and shifts which are essentially consequences of the rotor cage [1,2]. This resulting in strong contra intestinal control system, therefore an important complexity of control was made [3]. Thus, it is necessary to have a mathematical model accurately representing the dynamic behavior of the asynchronous machine not only steady but also transient. Several modeling methods for the analytical study of the machine are available from the literature. To investigate the control of the machine, the most suitable model of them is the generalized machine model that directly modeled from Park theory. This model presents a good compromise between accuracy and mathematical simplicity and allows to study the behavior of the machine in the way closer to reality [4,12 ]. The complete model of the induction machine taking into account the equations of mechanical behavior, and will be presented in the state space form with the specification of inputs and outputs of the system adopted to promote the use of matrix or operational calculus for the design of the control. To this end, we took as a starting point the different definitions that we consider important as the concept of the rotating field , the transformation in two axes (Park transformation), then we have presented a round turn adapted to the modeling when current controlled and/or voltage controlled motor. These models will be transcribed after predicate model selection in each case in block diagram form in order to be simulated using general-purpose software for the machine control. The simulation of three-phase asynchronous motor associated with its various power supplies (current/voltage) will find the classic question results and validate our models. 1.2 Description of the Cage Induction Machine A cage induction machine is present (figure1.1) in the form of a housing (2) surrounding the magnetic circuit ferromagnetic stator hosting in slots the poly-phase stator winding (usually threephase) in wound insulated copper wire (1). Within this magnetic circuit, which looks like a hollow cylinder, separated by a gap, turn the rotor magnetic circuit (3), which hosts the notches in the bars of the rotor cage, aluminum or copper, each shorted end by rings made of the same material. The rotor magnetic circuit is traversed by the shaft that rests in bearings mounted on the flanges (5), (6) attached to housing.

1

Chapter: I


Fig 1.1 Description of the Induction Machine The asynchronous motor used is characterized: • In the presence of a single poly-phase coil supplied by a source external to the stator, • The presence of a "wound" in solid short circuit rotor. 1.3 Advantage of the Asynchronous Machine Cage asynchronous machine is the machine most answered in the industry: it is robust, reliable, and economical. It is also popular for its good standardization [1-4]. 1.4 Problems with the Asynchronous Machine In the asynchronous machine, the stator current is used to generate both the flux and torque. Natural decoupling DC machine no longer exists. On the other hand, we cannot know the internal variables of the rotor cage (Lr: for example) and through the stator. The inaccessibility of the rotor leads us to change the rotor vector equation to express the rotor greatness through their actions on the stator. Therefore the structural simplicity hides great complexity due to functional characteristics just mentioned but also to nonlinearities, the difficulty in identification and parametric variations (Rr in particular, up to 50%).

2

Chapter: I


1.5 Simplifying Assumptions The modeling is based on a certain number of assumptions that [1.3]: • Symmetrical two-pole, three phases windings, perfect symmetry. • Assimilation of a shorted cage winding with the same number of phases that the stator winding. • The magnetic fields of each winding is sinusoidally distributed along the air gap. • No saturation in the magnetic circuit. 1.6 The Phasor Concept The stator as the rotor, three-phase currents flowing through the phase windings creates pulsating magnetic fields which generate overlapping rotating magnetic fields [6-10].

Fig 1.2 The Principal of Rotating Magnetic Field Creation Given the relationships between different variables, it is possible to extend the phasor concept to seem while three-phase variables: [Iabc_s], [Φabc_s], [Vabc_s], [Φabc_r]… 1.7 Park Transformation Treating a wide range of machines using Park transformation aims unified by reducing them to a single model called primitive machine or IRON machines. This conversion processing often called two axes, matches to the windings of the original machine, similar electrical and magnetic windings (the sensation resulting is that the electric and magnetic fields are energetically instantaneous) arranged in the d and q axes. This transformation aims to independent model making rotation of mutual inductors. In a more general way, the processing of the park transformation which ensures a direct passage between three-phase system with two-phase system ((a b c) → (q d)) is characterized by the transition matrix given by: [ ]

( (

√ [ √

) )

√

( (

) )]

√

Where  is the rotation angle between the (q-d) reference frame with respect to () /(a b c), and the inverse of Park transformation is given by the next transition matrix.

3

Chapter: I


√ [ ]

√ [

(

)

(

)

√

(

)

(

)

√

]

1.8 Model of the Asynchronous Machine The asynchronous machine which we will study is corresponding to the development equation of the basic structure represented by the figure (1.3) in a plane perpendicular to the axis of rotation. In the bipolar case, it is always possible to go to a machine 2p pole by a suitable transformation of the torque equation. The armatures of the stator and rotor are both cylindrical, so separated by a constant gap, and each provided with a three-phase winding [1,4].

Fig 1.3 Schematic Representation of Stator and Rotor of Three Phase Induction Motor We have the following angular definitions, counterclockwise: (

)

(

)

(

)

(

)

(

)

(

)

{(

) 4

Chapter: I


The magnetic axes of three phase windings of the stator "as", "bs", "cs", are presented diagrammatically with respective to the right shift of three rotor windings "ar", "br", "cr". We transform equations by using perpendicular two axes "od" (polar or longitudinal) and "oq" (interpolar a cross), but the position and movement of the axis "od" are not fixed. The angular velocity is obtained by:

1.8.1 Equations of Bases Voltage equations of the stator and rotor phases, represented as matrix under the above simplifying assumptions are written as follows: [

]

[

]

[

]

[

]

[

]

[

]

With [

] [

[

] [

[

] [

] ] ]

Total in the phases of the stator and rotor flux expressed in the form as matrix, [

]

[

][

]

[

][

]

[

]

[

][

]

[

][

]

With [ [ [

] ] ]

Where the matrices are expressed by the following

[

]

[

]

[

]

[

]

( [

( )]

[

( (

) )

) ( )

(

)

5

( (

) )] ( )

Chapter: I


With

The system of differential equation (1.4) and (1.5) is a time-varying coefficient, since they depend on the instantaneous position of the rotor relative to the stator (θ). These equations can be made independent of the position by applying rotor stator Park transformation. 1.8.2 The Park Model of the Asynchronous Machine To simplify the representation of the electrical equations (1.4), (1.5), (1.6) and (1.7). we introduce standardized Park transformation using the matrix passage (1.1). The change of variables on current, the voltage and flux is defined by the transformation (1.1):

(

)

(

)

[ ] [

]

[

]

We use the inverse matrices of the passage given by (1.2), the transformations can be written as follows:

(

)

(

)

[ ] [

]

[

]

Standardized Park transformation comprises the current, the voltage and the flux, a change of variables involving the angle between the axis and the axes of the windings "d" and "q". Equations (1.4), (1.5), (1.6) and (1.7) then give rise to the following systems:

[

[

]

]

[

[

][

][

]

[

]

[

]

[

]

[

6

][

][

]

]

Chapter: I


[

]

[

][

]

[

]

[

][

]

The isotropy of the asynchronous machine that is in the system (1.15) and (1.16) has the same coefficients. The change of variables can be interpreted as replacing windings actual, fictional windings of "ds", "qs", "dr" and "qr" whose magnetic axes are related to the axes "d" and "q" (figure 1.5), but the conductive plates which are to bear. In equations (1.13) to (1.14), we did not include the index components "O", which occur only in the case of the asymmetric feed stator, memory quote for the corresponding equations:

[

]

[

][

]

[

]

[

][

]

[

]

With

Fig 1.4 Physical Interpretation Of The Park Transformation For Induction Motor

7

Chapter: I


Mechanical Equations In the most common case, an asynchronous machine operates as a motor, the stator is powered by a three phase source, and the rotor winding is closed in short-circuit. For a multi-phase machine, if we consider the electrical angle and the electric speed , obtained by multiplying the geometric and mechanical homogues by the number of pole pairs p, the electrical equations are unchanged, and the electromagnetic torque of the equation and the equation of motion can be written: (

)

{ 1.8.3 Definition Of The Different Repositories The repository is the "od"-"oq" system associated with its rotational speed where three types are possible in practice, the choice is made depending on the problem being studied. The slip angular velocity is defined by:

Among the relationships that we have just described, the only transformation which is affected by the choice of reference is equation (1.13) and (1.14).  Fixed Reference With Respect To The Stator It results in the condition:

The electrical s equations take the form:

[

[

]

[

]

[

][

]

][

]

[

]

[

[

]

][

]

This repository will preferably be selected to study large variations in speed, whether or not associated with going variations of the input frequency.

8

Chapter: I


 Fixed Reference With Respect To The Rotor It results in the condition:


[

]

[

[

][

]

]

[

[

]

][

[

][

]

[

]

]

This repository can be interesting in transient problems where speed is considered constant.  Fixed Frame Of Reference With Respect To The Rotating Field It results in the condition:


[

[

]

]

[

[

][

][

]

[

]

[

]

[

]

[

][

][

]

]

This repository is the only one that does not introduce simplification in the transformation equations (1.13) and (1.14), but it is often used for the control problem and vector control. Because the control variables become continuous in this repository.

9

Chapter: I


1.9 Representation Of The State Model Of The Asynchronous Machine This representation depends on a priori several choices: 1. Repository performance, i.e. (frame). 2. State and output variables. 3. Control variables with the type of power. The choice of the representative reference model of the asynchronous machine is conditioned by the objectives that are assigned to achieve. The choice of a marker whose axes are secured to the rotating field is more general, is more complex because it involves the exact knowledge of its position [1,4] . Nevertheless, this choice is still the most suitable for the development of control laws. Several state variables of the machine can be selected. Therefore, more variants for the choice of the state vector are possible, depending on the purpose of the control and observation of state variables. In this section we will present the model has handled the rest of our study, which for the two types of power supplies. In the end, a simulation is performed on these models to validate and verify its correctness which carries a major interest for the command. 1.9.1 Modeling The Current Feed Machine 1.9.1.1 Description For a three-phase asynchronous machine supplied with power in current, the stator currents (ids, iqs) and sliding speed (ωsl) are considered as control variables, the load torque (Tl) as a perturbation. We choose the reference related to the rotating field, the model of three-phase asynchronous machine supplied by current is represented by equations (1.15), (1.16), (1.18) and (1.28), the phases of the rotor are supposed to be short-Circuit: vdr = 0, vqr.= 0. 1.9.1.2 Equations of state Any system can be described by different state variable. For a current fed asynchronous machine, we choose as state variable variant (φdr, φqr, Ωr). While the other variants are presented in Appendix (A). Thus, the model of the asynchronous machine given by the electrical and mechanical equations is obtained after arrangement of equations (1.15), (1.16), (1.18) and (1.29): Electrical Equation:

[

]

] [

[

Mechanical Equation: (

]

[

] )

{ 10

[

]

Chapter: I


1.9.1.3 Block Diagram The system is then represented by the following block diagram:

Figure 1.5 Block Diagram Of The Current Feed Induction Motor (Three Inputs, Three Outputs And Perturbation) 1.9.2 Modeling The Voltage Feed Machine 1.9.2.1 Description For the case when a voltage supplied machine, the stator voltage (vds, vqs) and the speed of the rotating field (ωs) are considered as control variables, the load torque (Tl) as a three-phase asynchronous machine perturbation. We choose the reference related to rotating field, the model of the three-phase induction machine is supplied with voltage represented by the equations (1.15) (1.16) (1.18) (1.28) and (1.29) assuming: vdr = 0, vqr = 0. 1.9.2.2 The State Equation Several state variables of the machine can be chosen, we chose to share our components on the axes (d, q) of the stator current and rotor flux. The currents are easily measurable quantities and are being used to observe the flux. The flux and speed are chosen as variables to be controlled. The flux control in almost can operate the machine at constant level on a fixed state of saturation of the magnetic circuit of the machine and approximate the inductors constants [12]. So our choice is made on the variant (ids, iqs, φdr, φqr, Ωr). Other variants are given in Appendix (A) and detailed in [04]. Electrical and mechanical equations can be drawn after arrangement equations (1.15), (1.16), (1.18), (1.28) and (1.29), we get this: Electrical Equation:

With

[

[

] 11

]

Chapter: I

(


) (

)

[

]

[

Mechanical Equation:

]

(

)

{ Evidently it is nonlinear coupled system due to the interaction between the rotor and stator windings. 1.9.2.3 Block diagram The system is then represented by the following block diagram:

Figure 1.6 Block Diagram Of The Voltage Feed Induction Motor (Three Inputs, Five Outputs And Perturbation) 12

Chapter: I


1.10 Simulation Model Of The Asynchronous Machine 1.10.1 Overview Numerical simulation is well-known today in the field of power machinery, it enables engineers and researchers to model, analyze and simulate systems with great precision, we use Matlab-Simulink for simulation three-phase asynchronous machine and associated controls. 1.10.2 Simulation Of A Current Feed Machine The model is represented by equations (1.29) and (1.30) are used Scheme functional association is given in figure (1.7)

Figure 1.7 Functional Block Diagram Of The Current Feed Induction Motor Other physical quantities are determined by the following equations:

[

]

[

][

And √

13

]

Chapter: I


The machine parameters that can be obtained by identification are given in Appendix (F). 1) The machine is driven by a sinusoidal current system: ( √ [

]

[

)

(

)

(

) ]

Apply the standard Park transformation (1.1) and assume θs = ωs.t, we obtain the two stator currents (ids, iqs) and the pulsation of the following rotor currents (ωsl) √

[ If figure 1.9

(

)

( )

]

[ (

] )

the simulation is given by the

The start time when machine is unloaded is (4.7 s) while the electromagnetic torque reaches the value (4.8 Nm) and stabilizes at an amplitude which compensates for ventilation and friction losses in the steady state (fig. (1.9)). We note that the stator and rotor fluxes are almost equal to their direct component (φds et φdr respectively), while the quadrature components have the shape of the electromagnetic torque up to sign. The machine is assumed linear flux, we do not observe an influence of the saturation of the mutual inductance ring (Φ = 1.3Wbr. and Φ = 1. 45Wb in steady state). Having a current component (iqs=0), the torque considerably lengthens the starting time. 2) The machine is supplied directly by the two stator currents (ids, iqs) and disrupted couple Tl is applied for (5 s) (Fig. (1.8)).

Figure 1.8 The Perturbation Of Resistive Torque For Current Feed Model

14

Chapter: I


In this case (figure (1.10)), the amplitudes of the components of the stator current are set to the nominal values and the startup time is reduced (4.7s) with a maximum electromagnetic torque). (6.8 N.m). Amplitudes of stator and rotor flux are virtually (1.4 Wb) for the system provided no load. When the load is applied to the shaft of the machine, the electromagnetic torque has a virtually instantaneous response with overshoot of 48 % and a rapid payback on asses (0.6 s). This behavior is perfectly symmetrical when the load is deleted. Speed does not vary when the machine is loaded (negligible slip ), but we see a slight decrease in flux and appearance components of rotor currents (-0.8 (A) idr ) and (-3.8A for iqr). The current control therefore gives an answer called " stiff" high overrun for the electromagnetic torque , so it is not recommended for use in open loop system for stability reasons. Electromagnetic Torque (N.m)

200

6

150

4 Torque (N.m)

speed (rad/s)

The mechanical Speed (rad/s)

100 50 0

2 0

0

2

4 time (s)

6

-2

8

0

2

1

0

rotor-current(A)

2

0.5 0 -0.5

0

2

4 time (s)

6

-4 -6

8

0

2

The stator Flux (Wbr)

4 time (s)

1 0.5 0

0

8

-2

1.5

-0.5

6

The rotor current (A)

1.5

stator-flux(Wb)

rotor-flux(Wb)

The rotor Flux (Wbr)

4 time (s)

2

4 time (s)

6

8

Figure 1.9 Unloaded Starting Motor Driven By Current (ids=5(A) iqs=0(A)) 15

6

8

Chapter: I

Modeling of the Three Phase Induction Machine Electromagnetic Torque (N.m)

The mechanical Speed (rad/s) 8

200 150

4

Torque (m/s)

speed (m/s)

6

100

2 0

50 -2

0

0

5

10

-4

15

0

5

time (s)

15

The rotor current (A) 1

1

0 rotor-current(m/s)

1.5

0.5 0 -0.5

-1 -2 -3 -4

0

5

10

-5

15

0

5

time (s)

10

15

time (s)

The rotor Flux (Wbr) 1.5 1

rotor-flux(m/s)

stator-flux(m/s)

The stator Flux (Wbr)

-1

10 time (s)

0.5 0 -0.5 -1

0

5

10

15

time (s)

Figure 1.10 Loaded Starting Motor Driven By Current (ids=5(A) iqs=0(A), Tl=5(N.m))

16

Chapter: I


1.10.3 Simulation Of A Of A Voltage Feed Machine We use the model which is represented by equations (1.30) and (1.31). Other variables (flux and rotor current) are obtained by the following formulas:

[

]

[

][

]

√ The machine is powered by a sinusoidal voltage system, namely: ( √ [

]

[

)

(

)

(

)

]

Apply the standard Park transformation (1.1) and assume θs = ωs.t + θ0s we get the two stator voltages (vds, vqs); Start under full load voltage, allows a rapid establishment (0.28 (s)) of the rotor flux and the stator with maximum instantaneous electromagnetic torque of 45 (N.m) (Fig. (1.12)). The starting time is 0.28 (s) and current ids stabilizes at a value close to zero while the current iqs takes a constant value. The machine not being sought at load, the rotor current component stabilizes at zero values also. The transient current phase has a duration equivalent to the time of starting with a peak of 28 (A) during the first moments of power. The machine is powered by a sinusoidal voltage system (formula 1.42) and a disturbance Tl is applied for 2 (s) (figure (1.11)).

Figure 1.11 The Perturbation Of Resistive Torque For Voltage Feed Model 17

Chapter: I

Modeling of the Three Phase Induction Machine Electromagnetic Torque (N.m)

The mechanical Speed (rad/s) 50

200

40

Torque (N.m)

speed (rad/s)

150

100

30 20 10

50 0

0

0

0.5

1 time (s)

1.5

-10

2

1 time (s)

1.5

2

0.2

30

0 rotor-flux (Wb)

20 10 0 -10 -20

-0.2 -0.4 -0.6 -0.8 -1

0

0.5

1 time (s)

1.5

-1.2

2

The Stator Current

0

(A) in

0.5

1 time (s)

1.5

frame

40 30 20 ia (A)

stator-current (A)

0.5


The stator current (A)

-30

0

10 0 -10 -20 -30

0

0.2

0.4

0.6

0.8

1 time (s)

1.2

1.4

1.6

1.8

2

Figure 1.12 Unloaded Starting Motor Driven By Voltage (vds=380(V) vqs=0(V)) 18

2

Chapter: I



Electromagnetic Torque (N.m)

200

50 40 30

Torque (N.m)

speed (rad/s)

150

100

20 10

50 0

0

0

1

2

3 time (s)

4

5

-10

6

0

2

4

6

time (s)



0.2

30 0

20

rotor-flux (Wb)

0 -10

-0.6 -0.8

-20 -30

-0.4

-1

0

2

4

-1.2

6

0

2

time (s)

4

6

time (s)

40

The Stator Current

(A) in

frame

30

20

10

ia (A)

stator-current (A)

-0.2

10

0

-10

-20

-30

0

1

2

3 time (s)

4

5

6

1.13 Loaded Starting Motor Driven By Voltage (vds=380(V) vqs=0(V) , Tl=5(N.m)) 19

Chapter: I


When the shaft of the machine is loaded by a step load torque (figure (1.12)), the electromagnetic torque instantly compensates for this solicitation with almost optimal response. However, the modules of stator and rotor flux also undergo a significant decline. As against, the quadrature components of the stator and rotor currents remain the same value. While, the direct components are superior to the greedy regime values. We also observe a low growth phase current, but this variation is more difficult to quantify in the case of the components in a reference linked to the rotating field. 1.11 Conclusion This first chapter allowed us to enter a near original formulation of models for three-phase asynchronous motor associated with its power source. To this end, we treated the case of current control and voltage control that we find in most industrial applications. The model used was obtained from the generalized machine and the state equations that result. We also showed the simplifications made when perfect source (voltage or current) is used for supply. The choice of state variables is fundamental for modeling and we were able to propose several interchangeable solutions that enable the user to consider many types of control. To design these controls with a powerful simulation tool, we transcribed our models in block diagram form. This technique is unique for electrical machines due to nonlinearities they put in. However, we were able to present models in functional forms simple enough to be built and operated by specialized software ( Matlab Simulink - in our case ) . This software allowed us to fair simulation conventional operating mode and thus validates our models to go to the next step control.

20

Chapter: II

C

hapter

Field Oriented Control Of Induction Machine

II

Field Oriented Control Of Induction Machine 2.1 Introduction In 1972, Blaschke [06] proposed a new control theory, called flux oriented, which allows assimilating asynchronous machine to a DC machine (Figure 2.1).

Figure 2.1 Principal Of Decoupling Control For Direct Current, Alternative Current Machine Today, this control technique seems quite assimilated several engineers and industrial achievements for the day. However, the theory of directed flux is relatively complex and many questions remain about the choice of the best control strategy for independent adjustments of the flux and torque without using the technique of decoupling directly [07]. The purpose of this technique is to present the possibilities of orientation of the flux of the controlled to the three-phase asynchronous machine by voltage or current with a static converter. Indeed, it is possible to define the components of flux at the stator or the rotor of the gap provided that these quantities are not directly measurable. In all what follows, we assume by definition that control of torque and flux is done by state primary variables namely the component ids for the flux and Component iqs for the couple and we will check with proper simulation that the latter function is performed. We use the results of the first chapter to present the various control algorithms oriented by using a continuous flux formulation. The methodology is to first present the model equations, to its representation in block diagram form and 21

Chapter: II


then add the control to the latter formulation. Complete block diagrams are then simulated in operating mode speed controller for various disturbances (at variable load). Different regulators used in control algorithms will be proportional and integral type (PI), which a priori choice is not restrictive to the extent that the controlled process is not linear and that the first order approximation can be validated in many cases. The two degrees of freedom available in the PI controller will set the dynamic characteristics for an operating point, so we can expect to get the following different temporal responses of different amplitude disturbances. Nevertheless, the performance is sufficient in most books and this type of controller is an industry standard. In the calculation of its parameters, we proposed an analytical approach rather rigorous (Annex C); ranked among the different about technical for synthesis linear systems. This original gold technique has already been validated in industrial studies. We also tested the robustness of the flux oriented control variations of the rotor resistance is a major concern in industrial applications. 2.2 Controls By Flux Orientation 2.2.1 Description In the model of the asynchronous machine represented by the two-phase equations, we choose a related field rotating frame so that the axis "d" coincides with the direction desired flux (rotor flux, stator flux or gap flux: Figure 2.2).

Figure 2.2 Flux Orientations (Stator, Rotor, And Rotating Reference Frame) In figure (2.2) ( )represent the rotor flux( we can orient the three different fluxes

) , the stator flux (

Given rotor flux with the following condition: Given stator flux with the following condition: Given gap flux with the following condition:

22

)and the air gap flux(

). Then

Chapter: II


In this chapter, we present the structure of the orientation of the rotor flux while other types of guidance can be found on [8-10.36]. Thus, we can report that according to the models for the study of the orientation of the stator flux and the orientation of the air gap flux, the stator flux (φs) and the air gap flux (φg) depend on two stator currents (ids, iqs). This coupling effect is due to the stator and rotor leakage for the stator flux and rotor leakage to the air-gap flux. The electromagnetic torque Te depends only on the quadrature component of the stator current (iqs) if the flux (or to stator air gap) is held constant 2.2.2 Current Control Consider the two stator currents (ids, iqs) as control variables, the rotor flux (φr) and the mechanical speed (Ωr) as state variables after arrangement of equations (1.1), (1.29) and (1.30). We obtain:

{ We can observe in the equations (2.4) that only the direct component (axis "d") of the stator current (ids) determines the magnitude of the rotor flux (φr), while the torque depends only on the quadrature component (axis "q") of the stator current (iqs) if the rotor flux is kept constant. Thus is formed a decomposition of stator current into two terms corresponding respectively to flux and torque. The corresponding scheme function is:

Figure 2.3 Current Modeling By The Rotor Flux Orientation

The relations (2.4) constitute the essential of control flow oriented basis.

23

Chapter: II


2.2.3 Voltage Control The ideas are the same as those specified in the preceding paragraph; however we must not only consider the dynamics of rotor but also the dynamics of stator, because the machine is powered by a voltage source. Consider the two stator voltages (vds, vqs) as a control variable, the two stator currents (ids, iqs), the rotor flux (φr) and the mechanical speed (Ωr) as state variables after setting equations (2.1), (1.15), (1.16), (1.31) and (1.35), we obtain the rotor flux orientation model of the three-phase asynchronous machine supplied by voltage. (

) (

*

)

+

{ 2.3 Structure Of a Current Driven Machine 2.3.1 Description For three-phase asynchronous machine limentée running, the two currents stator (ids, iqs) and the pulsation of the current (ωs) or of the sliding speed (ωsl = ωs-ωr) are considered as control variables. We use the model which is represented by the following block diagram

Figure 2.4 Block Diagram Of Three Phase Current Supplied Induction Machine 24

Chapter: II


The other variables are calculated by the formulas (2.6) √

√ [ {

(

)

]

(

[

)

]

√

We present a control which is based on the orientation of the flux, we decouple the torque and flux. In the next section, we only study the indirect vector control which is based on this principle. We seek algorithm to control both stator currents (ids, iqs) and sliding speed (ωsl). 2.3.2 Rotor Flux Orientation (Indirect Control) Consider the rotor flux (φr*) and the torque (Te*) as the command references, we inverse the model via orientation of rotor flux (2.4) and we will get the control equation as: (

)

{ The block diagram of the control structure is the following:

Figure 2.5 Functional Schematic Diagram Of The FOC For The Current Supplied Model

25

Chapter: II


For (φ*= 1 Wbr), and Te* follows the reference figure (2.6), we give the results of simulation in Figure (2.7).

Figure 2.6 The Torque Reference Electromagnetic Torque (N.m)

The rotor Flux (Wbr) 1.2

30

1

20

Torque (N.m)

rotor-flux(wbr)

0.8 0.6 0.4

10 0 -10

0.2

-20

0 -0.2

-30 0

0.5

1

1.5 time (s)

2

2.5

3

0

1

2

3

time (s)

The rotor Flux norm (Wbr) 1.5

f-r(wbr)

1

0.5

0

0

0.5

1

1.5 time (s)

2

2.5

3

Figure 2.7 Simulation Results For Decoupling The Torque And The Rotor Flux In Current Mode

26

Chapter: II


The simulation results (Figure 2.7) clearly show an initialization period of the rotor flux φr that reaches its reference value after 0.01 s. From that moment, the electromagnetic torque of the machine perfectly follows the reference Te* imposed without overshoot or static error. We note that the electromagnetic torque spikes do not affect the rotor flux φr represented by its components and φdr φqr. This last observation allows us to conclude that an automatic decoupling the torque and flux. This remarkable result has led to a systematic use of this scheme control in industrial processes [10], [11]. For this purpose, it is enough to join the open loop control of a flux controller and a torque controller. The speed is measured, it is possible to make its regulation against feedback, in contrary, the flux cannot be measured directly and the simplest control is the reaction (figure 2.8). The diagram thus obtained is called indirect field oriented control. The PI controller (Annex C) is selected for the speed control, while the flux weakening is defined by the following non-linearity:

{ |

|

|

|

|

|

Figure 2.8 Functional Schematic Diagram Of The Speed Control For The Field Orientation Pole (-1/T0) is implanted with the aim of limiting the bandwidth to compensate for the derivation of the flux. A structural change in the control shown may occur considering the rotor flux (φr*) and frequency of the rotor current (ωsl*) as a reference in making a suitable change of variables in the equations (2.7), we obtain notable simplification at block circuit diagram contains a multiplier instead of two divisors involved in the previous scheme [7,36].

27

Chapter: II


2.4 Structure Of A Voltage Supply 2.4.1 Description For dynamic equations of the three-phase voltage-fed induction machine, stator can no longer be neglected in the state representation; the problem of control has become more complicated, because we must not only consider the rotor dynamics but also the stator. Both stator voltages (vds, vqs) and the speed of the rotating field (ωs) are considered as control variables, the stator currents (ids, iqs), the rotor flux (φdr, φqr) and the mechanical speed (Ωr) are considered state variables (equations 1.31-1.34), the model is represented by the following block diagram:

Figure 2.9 Block Diagram Of The Voltage Supplied Three Phase Induction Machine Other variables are calculated by the following formulas: √

{

√

[(

)

(

)

]

[(

)

(

)

]

√

2.4.2 Orientation of the rotor flux (indirect control) Consider the torque (Te*) and the rotor flux (φr*) as a reference control. The inversion of the model (2.5) provides the following control equations:

28

Chapter: II


(

{

)

,(

)

,(

)

(

)

(

( )*

(

)

)

+-

Figure 2.10 Block Diagram Of The FOC Of The Voltage Supplied Induction Machine For (φ* = 1 Wbr, Te*) follows the torque reference figure (2.6), we give the simulation results in Figure (2.11) showing that the couple Te and rotor flux φr are decoupled and the flux φr is oriented in the direction "d" as in the case of indirect current control. However, we note that the flux initialization phase is disturbed and when the regime is completed, the decoupling is not perfect during transient for the step kind of torque. The electromagnetic torque of the machine responds to these levels with a time more significant response. In this case a speed control is considered, it is sufficient to take the principle of indirect control as already shown by adding a proportional integral controller for loop speed and rotor flux control by reaction (Figure 2.12). 29

Chapter: II

Field Oriented Control Of Induction Machine Electromagnetic Torque (N.m)


20

1.2

10

0.8

Torque (N.m)

rotor-flux (Wb)

1

0.6 0.4

0 -10

0.2

-20 0 -0.2

res Torque -30 0

0.5

1

1.5 time (s)

2

2.5

3

0

1

2

3

time (s)

Figure 2.11 Simulation Results Of The Field Oriented Control For The Voltage Supplied Three Phase Induction Machine

2.5 Simulation And Interpretation Of Results To test the behavior of the asynchronous machine with a (PI) controller, the control data structures in figures (2.8) and (2.12) have been simulated for The following operating systems (for parameters of the machine studied, see Appendix F): - Start unloading; - Application and removal of a load of 10 (Nm) to 1.5 seconds (s) and 3 (s), respectively; - Reverse the direction of rotation at time 4 (s). 30

Chapter: II


The responses to the two modes of power are illustrated in Figures (2.13) and (2.14), we find that after a start time of about 0.34 seconds, the curve coincides with its rotor speed reference during the steady state. The application of a load causes a drop in speed to equal 6.5% of the set-point value. Compensation that fall lasts 0.5 seconds. The supply voltage reaches a large rotor flux value (About 1.4Wb) start with more pronounced oscillations. At the start the electromagnetic torque reaches a value equal 2 times that of nominal torque for current supply. While a voltage powered machine admits an electromagnetic torque 3 times its nominal value. When applying a load, the torque developed by the motor attains the value of the shot load after 0.3 seconds. The components of the rotor flux are kept stable after starting the machine. Thus, the control circuit ensures decoupling (between Speed and dynamics of the flux) in the steady state by acting on the control variables. The evolution follows that of the stator currents, magnetic flux and electromagnetic torque; they die at a peak value which is 2 times the nominal for current supply (Figure 2.13) and three times for a voltage supply (Figure 2.14). To test the robustness of vector control to variations of the parameters, we give some variations to the rotor resistance during the simulation. Given the role of the rotor time constant in the decoupling circuit (equation 2.7 and 2.10), the variation of the rotor resistance considerably effect the magnetic flux dynamics for which is expressed by the loss of decoupling (see the evolution of flux components, Figure 2.15 and 2.16). Therefore, risk of magnetic saturating the machine circuit and losing control performance. Furthermore, the compensating the effect of the resistance in the mechanical part (speed and torque) may require current and control voltage higher than the permissible values supported by the machine.

31

Chapter: II



Electromagnetic Torque (N.m) 40

150

real Torque

20 Torque (N.m)

Wr (rad/s)

100 50 0

res Torque

-20 -40

-50 -100

0

-60 0

2

4

6

0

2

6

time (s)

time (s)


The rotor Flux norm (Wbr)

1.5

1.5

1

f-r(wbr)

rotor-flux(wbr)

4

0.5

1

0.5 0

-0.5

0 0

1

2

3 time (s)

4

5

6

0

2

4 time (s)

Figure 2.13 Dynamic Response Of The Current Supplied Induction Machine With Indirect Vector Control

32

6

Chapter: II



150

40

100

20 Torque (N.m)

Wr (rad/s)


50 0

0 res Torque

-20 -40

-50 -100

real Torque

0

2

4

-60

6

0

2

4

6

time (s)

time (s)

The rotor Flux norm (Wbr)

The rotor Flux (Wbr) 2

2

1.5

1 f-r(wbr)

rotor-flux(wbr)

1.5

0.5

1

0 0.5

-0.5 -1

0

2

4

0

6

time (s)

0

1

2

3 time (s)

4

Figure 2.14 Dynamic Response Of The Current Supplied Induction Machine With Indirect Vector Control

33

5

6

Chapter: II



The alpha stator current (A)

200

30

100 speed (rad/s)

ia (A)

20 10 0

0 -100

-10 -20

-200 0

time (s)

4 6 time (s)

The speed error (rad/s)


2

4

6

8

10

0

2

8

10

30

1.5

stator-current (A)

error-speed (rad/s)

1 0.5 0 -0.5

20 10 0

-1 -1.5

-10 0

5 time (s)

10

0


5 time (s)

10

The rotor flux (Wbr)

25

1.5

res Torque 20

rotor-flux (Wb)

Torque (N.m)

15 10 5 0

1

0.5

0

-5 -10

-0.5

0

2

4 6 8 time (s) Figure 2.15 Dynamic Response Of The Current Supplied Induction Machine With Indirect Vector Control 4 6 time (s)

8

10

34

0

2

10

Chapter: II


The mechanical speed (rad/s)

Variation of rotor resistance 5.5

150

5

Rr (hom)

speed (rad/s)

100 50 0

4

-50 -100

4.5

0

2

4

3.5

6

0

2

4

6

time (s)

time (s)

Electromagnetic torque (N.m)

The rotor flux (Wbr) 1.5

40 real Torque rotor-flux (Wb)

torque (N.m)

20 0 -20 res Torque

1

0.5

0

-40 -60

0

2

4

-0.5

6

0

2

time (s)

4

6

time (s)

The speed error (rad/s)

The torque error (rad/s)

4

60 2

error-speed (ras/s)

error-Torque (N.m)

40 20 0

-2

-4

-20 -40

0

0

2

4

6

-6

time (s)

0

2

4 time (s)

Figure 2.15 Dynamic Response Of The Current Supplied Induction Machine With Indirect Vector Control And Under Variation

35

6

Chapter: II


2.5 Conclusion This chapter has allowed us to present the methods of control asynchronous machine by indirect oriented rotor flux associated with the sources of current and/or voltage. Currently, the choice of the power source, and thus the static converter, the technology is related to the power electronics which limits the possibilities for the designer tees. The choice of the power supply leads to the patterns certainly the simplest as defined above control designer in the case of the rotor flux orientation. When the machine is supplied with voltage control schemes lead necessarily to the use of the current loops which allow taking into account the dynamics of the stator. It follows from this state, a great sensitivity to changes machine parameters. Nevertheless, the patterns of indirect control, we obtain comparable results to the indirect current control with response time speed slightly greater. We therefore find, through these investigations, the oriented flux control leads to imply limitations to the associated drive performance. In light of all that is said and in order to overcome the drawbacks of the rotor flux oriented control, we have tried in this present work offer an interesting alternative that fits within the common objectives and it is applied to the asynchronous machine, it is called an adaptive reference model control synthesized by the concept of hyper-stability. The next part will be devoted to the first presentation of this technique.

36

Chapter: III

C

Adaptive Control Analysis Via Hyper-Stability Criteria

hapter

III

Synthesis Of The Adaptive Control Via Hyper-Stability Criteria 3.1 Introduction The previous chapter allows us to show the contribution of the field oriented vector control in different operating regimes. While in the presence of significant variations in the parameters of the machine, the maintenance of a high level of performance may be impossible to reach with a fixed regulator, which is designed beforehand on a so-called ' nominal ', maintaining a decent level of performances, therefore for this objective it is made possible by the development of an adaptive version of the control law [12-14]. In this sense, we present a command that was the concern about several researchers called « model reference adaptive control ». The direct model reference adaptive control (MRAC) was given the first time in 1961 [16] based on the minimization of a performance index, commonly known as the MIT design rule approach. The MRAC strategy specifies the desired shape of the process response to a control signal via a reference model defined parametrically, thus desired future is reached via a closed-loop performance. Then we should develop a mechanism which will deal the output process to the output model, a set of parameters designed such so that the difference between these two outputs goes to zero. This strategy has been extended subsequent to the gradient accelerated method [17]. The stability of these adaptive control systems has been demonstrated regardless of the underlying control law if the parametric adaptation algorithm verifies four well-defined properties. Three former are made by the least square algorithm incorporating a modified regularization of the adaptation gain and ensuring the boundedness of parameters. Last concerning the eligibility of the identified model compared to the control law, adaptation requires persistent excitation condition as soon as input-output information behavior are no longer relevant for estimation. This was achieved in the presence of modeling errors and small parametric variations. Adaptive methods with reference model should be considered from three different approaches. The first, based on state representation described by Landau [18], suppose that state variables are measurable. The second adopts an input-output representation based on the concept of the error signal increased by Monopoli [19]. In this approach, adaptive observers are contained within the controller for counteract the lack of input to the entire state vector. The third concerns the simple adaptive control approach introduced by Kaufman [20]. This approach based on the output feedback method which requires neither state feedback nor adaptive observers. The MRAC approach we propose in this study is different in nature, which can overcome the drawbacks of direct structure described above, is essentially based on the following principles: ■ An indirect adaptation (off line model identified); ■ Adopting the state representation described by Landau; ■ Adaptation of the synthesis signal as an additional control input; ■ A test synthesis based on a stability criterion (avoided the problem of local stability); 38

Chapter: III


The figure (3.1) shows the difference between these two cases.

Figure 3.1 The Structure Of Model Reference Adaptive Control This problem of stability led many researchers in the early 60s to consider the synthesis of adaptive controllers using the stability theory and in particular the second Lyapunov method [21]. This method offers an overall stability without restrictions or conditions on the initial error, or on the nature of the used reference inputs. However, the major drawback in this approach is the need to find an appropriate Lyapunov function and this function must be defined for all adaptive known laws. So there is a problem of generalization of the Lyapunov method. This drawback has motivated in [26] Landau apply a new concept called hyper-stability criterion [27] for the synthesis of model reference adaptive control. To this end, we have presented in this chapter some basic definitions of hyper-stability for linear and nonlinear systems. In addition, we have developed approaches for the synthesis of model reference adaptive control for this concept. 3.2 Hyper-Stability Criteria 3.2.1 Basic Principle Similarity to the concept of absolute stability, the hyper-stability ensures the stability of several classes of system. The theory of hyper-stability can be interpreted in light of the general theory of passive single-input systems. It is assumed that such a system is described by an input signal u(t), an output y(t) and a state vector x(t), then the main power to remember is this: ∫ ( ) ( )

( ) ( )

39

( ) ( )

Chapter: III


The first term on the left side of the inequality represents the energy supplied to the system during the interval t0 ≤ τ ≤ t, the second term represents the energy in the initial state and the right side of the inequality represents the energy instantly stored by the system states. The difference between the two terms is the energy dissipated, so that the energy of a passive system can never be greater than the sum of higher energy supplied from the outside and the initial energy, this type of system is always stable when the output y(t) is a function of the input u(t) (generally nonlinear and time varying). ( )

[ ( ) ]

The equation (3.1) may write in more general form as: ∫ [ ( ) ] ( )

‖ ( )‖

Where √

( ) ( )

The main energy given by (3.1) is chosen not only for "passive linear" systems but also for "real passive" systems, that is to say, as non-linear and/or time varying systems and conversely we can establish that all dynamical systems so please the inequalities (3.1) and (3.2) allowed the same stability properties for passive systems discussed above. The general principle of hyper-stability was developed and demonstrated by Popov (1963.1973). The adaptive control systems and system identification by the adaptive model approach is characterized by a strong non-linearity in the recent years, this theory it has proven that is outstanding compared to other theories in determining the limits of stability. As the method of Lyapunov theory the hyper-stability is sufficient but not necessary for stability conditions. 3.2.2 Hyper-Stability And Positivity The concept of hyper-stability was developed by Popov [26, 28] to study the stability of nonlinear systems, it generalizes the absolute stability. Recall that a system is called absolutely stable if it is stable regardless of the non-linear characteristic located in a region of sector type, that is to say, of the form: ( ) ( ) In other words, it is stable for all gains back between 3.2.2.1 Definition Of Hyper-Stability 1 Let the system described by the state-space representation: ̇( )

[ ( ) ( ) ]

( )

[ ( ) ( ) ] 40

Chapter: III


With x, u, y are respectively the state variable, the input vector and the output vector of dimension (n,1), u and y have the same dimensions (p,1), the a hyper-stability of a system according to Popov the current state describes that: if u satisfies the inequality (3.2), then the states of the system are limited. In the following treatment, we use the inequality in (3.2) the scalar variable: (

)

( ) ( )

∫

The integral term is given in its general form for t0 = 0. While stability is a property of the system in the steady state, so it will be considered as reference stability which is consistent with the theory of Popov. In what follows, several definitions of hyper-stability will be introduced. 3.2.2.2 Definition Of Hyper-Stability 2 The system written by (3.4) and (3.5) is a hyper-stable if there exist following integrality: ( If we set:

)

‖ ( )‖

verify the

‖ ( )‖

and ‖ ( )‖

( ) ( ) , then (3.7) will be equivalent to equation (3.1).

The integrality (3.7) give as the general relation of energy, then there exist more general definition for which is equivalent to that of Popov [23]. 3.2.2.3 Asymptotic Hyper-Stability The system described by (3.4) and (3.5) can be asymptotically hyper-stable if hyper-stable and generally asymptotic for u = 0. Previous definitions of hyper-stability consider the system statements. If we consider that the system is described only by its input-output expressed by: ( )

[ ( ) ]

Similarly, as we have seen in (3.2) provided the hyper-stability is as follows: 3.2.2.4 Definition Of Weak Hyper-Stability The system described by (3.4) and (3.5) is said weakly hyper-stable if there is a finite constant γ02 ≥ 0 which depends on initial conditions, such that the inequality: (

)

It verifies the definition of low hyper-stability and inequality (3.9) are particularly important for adaptive control systems and identification of adaptive models. 41

Chapter: III


3.2.3 Hyper-Stability Of Linear Systems The asymptotic hyper-stability and hyper-stability of linear systems are studied through the transfer function representation according to mono-variable systems and matrix transfer function in the case of multivariable systems [28]. Indeed, the representation function or transfer matrix is independent of the chosen form of state, and the hyper-stability study is reduced to the frequency domain (frequency version of hyper-stability) matching that of positivity, in what follows, the matrices transfer functions are square matrices. A linear time invariant system is hyper-stable, if and only if, it is described by a function (matrix) of strictly positive real transfer (SPR), and is weakly if hyperstable written by a function of (matrix) of transfer only positive real (PR) [34]. Definition 1: A function (matrix) rational transfer G(s) is called positive real (PR) if satisfies the following properties: 1. G(s) is real when s is real, 2. The elements of G (s) have no unstable poles, 3. G(s) + GT(-s) is Hermitian positive semi-definite for Re(s)> 0. Equivalent to that defines the imaginary axis definition, give the in the case of the strict positivity Definition 2: Function (matrix) transfer G (s) is strictly positive real (SPR) if and only if: 1. Elements of G(s) are real for real s. 2. It has no poles for Re(s)> 0. 3. G(s) + GT(-s) is Hermitian positive-defined for all real ω. The concepts of (strict) hyper-stability and (strict) real positivity are equivalent, making a link between temporal (the hyper-stability seen as a restriction of the classical definition of stability) and frequency (properties of the transfer matrix, the restriction conventional frequency stability test). In the invariant case, the positivity is easy to check: a rational function G(s) is positive if and only if it has no poles and unstable if Re(G(s))> 0 when Re(s)> 0 or, equivalently, if it has no unstable poles and the associated Nyquist place is located entirely in the right half-plane. But the concept of positivity as presented above would be unworkable practically in the multivariable case without the following theorem: Theorem: Let G(s) t a rational transfer matrix such that G(∞) = 0 and G has no poles in the plane Re (s)