book

Electrical Actuators: Applications and Performance

by Bernard de Fornel, Jean-Paul Louis

May 2010

Intermediate to advanced

528 pages

11h 5m

English

Wiley

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Cover
Title Page
Copyright
Introduction
Part I. Measures and Identifications
Chapter 1: Identification of Induction Motor in Sinusoidal Mode
1.1. Introduction1.2. The models1.2.1. Dynamic model of the induction machine1.2.2. Establishment of the four parameter models1.2.2.1. Total rotor leakage model1.2.2.2. Total stator leakage model1.2.2.3. Equivalence of total leakage models1.2.3. Magnetic circuit saturation1.2.4. Iron losses1.2.5. Sinusoidal mode1.2.6. Summary of the different models1.2.7. Measurements1.2.8. Use of the nameplates1.3. Traditional methods from a limited number of measurements1.3.1. Measurement of stator resistance1.3.2. Total rotor leakage model1.3.2.1. Usual method1.3.2.1.1. Simplified calculation1.3.2.1.2. Precise calculation1.3.2.2. Use of any two measurements1.3.3. Total stator leakage models1.3.4. Saturation characteristic1.3.5. Experimental results1.3.5.1. Total rotor leakage model1.3.5.2. Total stator leakage model1.3.5.3. Saturation1.4. Estimation by minimization of a criteria based on admittance1.4.1. Estimation of parameters by minimization of a criterion1.4.2. Choice of criterion1.4.3. Implementation1.4.4. Analysis of estimation errors1.4.4.1. Method for error evaluation1.4.4.2. Experiment design1.4.4.3. Effect of measurement errors1.4.4.3.1. Offset and error of gain on sensors1.4.4.3.2. Stochastic measurement noise1.4.4.4. Effect of model error1.4.4.4.1. Iron losses1.4.4.4.2. Saturation1.4.4.5. Discussion1.4.5. Experimental results1.4.6. Optimal experiment design1.4.6.1. Principle1.4.6.2. Minimization of the effect of measurement noises1.4.6.3. Minimization of the combined effect of measurement noise and iron losses1.4.7. Conclusion on the method1.5. Linear estimation1.5.1. Principle1.5.2. Case of the five parameter model1.5.3. Study of precision1.5.3.1. Measurement errors1.5.3.1.1. Offset1.5.3.1.2. Gain error1.5.3.1.3. Stochastic errors1.5.3.2. Error on stator resistance1.5.3.3. Discussion1.5.4. Experimental results1.5.4.1. Experimental setup1.5.4.2. Non-linear method1.5.4.3. Linear method1.5.4.4. Comparison1.5.5. Conclusion on the “linearizing” method1.6. Conclusion1.7. Appendix1.7.1. Expression of sensitivities1.7.2. Characteristics of the machines used1.8. Bibliography
Chapter 2: Modeling and Parameter Determination of the Saturated Synchronous Machine
2.1. Modeling of the synchronous machine: general theory2.1.1. Description of the machine studied and general modeling hypotheses2.1.2. Fundamental circuit laws for the study of electrical machines2.1.3. Equations of the machine in abc variables2.1.4. Concordia transformation: equations of the machine in 0αβ variables2.1.5. Park transformation: equations of the machine in 0dq variables2.1.6. Connection between the machine and a three-phase link2.1.7. Reduction of rotor circuits to the stator2.1.8. Relative units (per-unit)2.2. Classical models and tests2.2.1. The synchronous non-saturated machine2.2.1.1. Classical linear model (Park model)2.2.1.2. Equivalent diagrams2.2.1.3. Operational reactances in non-saturated mode2.2.1.4. Internal and external parameters in non-saturated mode2.2.2. General classical tests2.2.2.1. Test of the synchronous machine in low-load steady state2.2.2.1.1. No-load test (taking down of the direct-axis magnetic characteristic)2.2.2.1.2. Short-circuit test2.2.2.1.3. Loss determination (ohmic, magnetic, mechanical)2.2.2.1.4. Low slip test (determination of longitudinal and transversal synchronous reactances)2.2.2.2. Synchronous machine tests in transient state2.2.2.2.1. Transient three-phase short-circuit test2.2.2.2.2. Load shedding test2.2.2.3. Transient standstill test2.2.2.3.1. Direct current decay test2.2.2.3.2. Frequency response static tests2.2.3. Potier Method2.2.3.1. Insufficiency of the linear theory2.2.3.2. Potier Model2.2.3.3. Experimental determination of Potier model parameters2.2.3.4. Necessity for a more general theory in the presence of saturation and magnetic saliences2.3. Advanced models: the synchronous machine in saturated mode2.3.1. Elements of the von der Embse theory of saturated electrical machines: inductive circuits in the presence of magnetic saturation2.3.2. General study of magnetic coupling in the presence of saturation2.3.2.1. Transformer effect coupling2.3.2.2. Cross-saturation coupling2.3.2.2.1. General study2.3.2.2.2. The UCL model2.3.2.2.3. Isotropic inductance2.3.3. Implementation of the model2.3.3.1. Leakage flux2.3.3.2. Operational reactances in saturated mode2.3.3.3. Large-amplitude transients2.4. Bibliography
Chapter 3: Real-Time Estimation of the Induction Machine Parameters
3.1. Introduction3.2. Objectives of parameter estimation3.2.1. On control3.2.2. Diagnosis3.3. Fundamental problems3.3.1. Identifiability, parameterization, and validation of the model3.3.1.1. Identifiability and parameterization of the model3.3.1.2. Validation of the model3.3.1.3. Identifiability and parameterization of the induction motor model3.3.2. Choice of the sampling period and digital problems3.3.2.1. General case3.3.2.2. Discretization of the IM dynamic model3.3.2.2.1. Fourth-order state model3.3.2.2.2. Reverse second order model3.3.3. Monitoring and information analysis3.3.3.1. Monitoring problem3.3.3.2. Information analysis and global identification3.3.3.2.1. Distance of structure and distance of state3.3.3.2.2. Connection between distance of structure and distance of state3.3.3.2.3. Hessian conditioning and parametric uncertainty3.4. Least square methods3.4.1. Principle of least squares and instrumental variables3.4.1.1. Least squares and recursive least squares3.4.1.2. Instrumental variables3.4.1.3. Monitoring and parametric uncertainty3.4.2. Application to the induction motor3.4.2.1. Linear model in relation to parameters3.4.2.2. Implementation of least squares3.4.2.3. Experimental results3.5. Extended Kalman filter3.5.1. Principle3.5.2. Tuning of Q and R matrices3.5.3. Application to the induction motor3.5.3.1. Complete order direct model of the induction motor3.5.3.2. Reduced order model of the induction motor3.5.3.3. Covariance matrix adjustment3.5.3.4. Filter initialization and restart3.5.3.5. Results3.5.3.5.1. Simulation results3.5.3.5.2. Experimental results3.6. Extended Luenberger observer3.6.1. Principle3.6.1.1. Discrete time observer3.6.1.2. Extended observer3.6.1.3. Tuning (pole placement)3.6.2. Estimation of induction machine velocity3.6.2.1. Machine model3.6.2.2. Extended Luenberger observer3.6.2.3. Extended Kalman filter3.6.2.4. Comparison between Kalman filter and Luenberger observer3.7. Conclusion3.8. Appendix: machine characteristics3.9. Bibliography
Part II. Observer Examples
Chapter 4: Linear Estimators and Observers for the Induction Machine (IM)
4.1. Introduction4.2. Estimation models for the induction machine4.2.1. Park model for the induction machine4.2.1.1. A dynamic model for observation4.2.1.2. Reference frame changes4.2.1.3. The model in the general reference4.2.2. Different state models for flux estimation4.2.2.1. State representation of a sinusoidal machine4.2.2.2. State vector made up of stator currents and fluxes4.2.2.3. State vector made up of stator currents and rotor fluxes4.2.2.4. State vector made up of stator fluxes and rotor fluxes4.2.3. Different study reference frames for flux estimation4.2.3.1. Different study reference frames4.2.3.2. Reference frames based on prior knowledge of angles4.2.3.3. Reference frames based on subsequent knowledge of angles4.2.3.4. Assessment of the different reference frames appropriate for the estimation of fluxes4.3. Flux estimation4.3.1. Introduction4.3.2. Stator flux estimator4.3.3. Rotor flux estimator4.4. Flux observation4.4.1. Full order deterministic observer4.4.1.1. Principle4.4.1.2. State equation of the observer4.4.1.3. Observability4.4.1.4. Calculation of observer gain4.4.1.5. Example of observer robustness4.5. Linear stochastic observers—Kalman–Bucy filters4.5.1. Introduction4.5.2. Kalman–Bucy filter model4.5.3. Convergence of the Kalman filter4.5.4. Simulation results and experimental results4.6. Separate estimation and observation structures of the rotation speed4.6.1. Introduction4.6.2. General principles4.6.3. Speed estimation and observation methods4.6.3.1. Speed calculation by the self-piloting relation4.6.3.2. Association of a mechanical observer to an estimator by frequency addition4.6.3.3. Adaptive mechanism (MRAS)4.6.3.4. Adaptive mechanism associated with a mechanical observer4.7. Adaptive observer4.7.1. Introduction4.7.2. Determination of observer gains4.7.3. Speed adaptation law4.7.3.1. Verification of the third condition of the hyperstability theory4.7.4. Simulation results and experimental results4.8. Variable structure mechanical observer (VSMO)4.8.1. Basic principle4.8.2. Construction of the VSMO4.8.3. Determination of variable structure observer gains4.8.4. Presentation of observer performances4.8.4.1. The shattering phenomenon4.8.4.2. Rated speed operation4.8.5. Low-speed operation4.8.6. Robustness in relation to parametric variations4.8.6.1. Mechanical parameter variations4.9. Conclusion4.10. Bibliography

Chapter 5: Decomposition of a Determinist Flux Observer for the Induction Machine: Cartesian and Reduced Order Structures
5.1. Introduction5.2. Estimation models for the induction machine5.2.1. Park model of the induction machine5.2.1.1. A dynamic model for control5.2.1.2. Reference changes5.2.1.3. The model in reference (αS, βS)5.2.2. State models for Cartesian and reduced observers5.2.2.1. State representation of an induction machine5.2.2.2. State vector made up of stator currents and fluxes5.2.2.3. State vector made up of stator currents and rotor fluxes5.2.2.4. State vector made up of stator fluxes and rotor fluxes5.2.3. Determination of the flux in the reference used by the control5.2.3.1. Different study references5.2.3.2. Flux estimators5.2.3.3. Flux observers5.3. Cartesian observers5.3.1. Principle and structure of Cartesian observers5.3.1.1. Breakdown of a complete observer5.3.1.2. Coupled sub-observers5.3.1.3. Characteristics of Cartesian observers5.3.2. Different Cartesian observers5.3.2.1. Cartesian observer associated with the stator current and stator flux5.3.2.2. Cartesian observer associated with the stator current and rotor flux5.3.2.3. Cartesian observer associated with the stator flux and rotor flux5.3.3. Synthesis of the Cartesian observer linked to stator and rotor fluxes5.3.3.1. Observability5.3.3.2. Calculation of observer gains5.3.3.3. Equivalence with a complete order observer5.3.4. Discretization of the Cartesian observer linked to stator and rotor fluxes5.3.4.1. Discretization of a complete order observer5.3.4.2. Cartesian observer discretization5.3.5. Validation of the Cartesian observer for stator and rotor fluxes5.3.5.1. Specifications5.3.5.2. Robustness test5.3.6. Assessment on Cartesian observers5.4. Reduced order observers5.4.1. Principle and structure of reduced order observers5.4.1.1. Principle of the reduced order observer5.4.1.2. Breakdown of the complete observer5.4.1.3. Characteristics of reduced order observers5.4.2. Different reduced order observers5.4.2.1. Reduced order stator flux observer5.4.2.2. Reduced order rotor flux observer5.4.3. Synthesis of the reduced order rotor flux observer5.4.3.1. Observability5.4.3.2. Gain calculation5.4.4. Discretization of the reduced order rotor flux observer5.4.5. Validation of the reduced order rotor flux observer5.4.5.1. Specifications5.4.5.2. Robustness test5.4.6. Assessment on reduced order observers5.5. Conclusion on Cartesian and reduced order observers5.6. Appendix: parameters of the study induction machine5.7. Bibliography
Chapter 6: Observer Gain Determination Based on Parameter Sensitivity Analysis
6.1. Introduction6.2. Flux observers6.2.1. Rotor flux estimator6.2.2. Reduced order flux observer6.2.3. Full order flux observer6.2.4. Choice of observer gains6.2.5. Choice of the reference frame6.3. Analysis method of the parametric sensitivity6.3.1. Flux amplitude and phase error estimation6.3.2. Influence of the magnetic saturation6.3.3. Calculation algorithm of errors in the estimated flux6.3.4. Variations of the stator current used6.4. Choice of observer gains6.4.1. Pole placement and parametric sensitivity6.4.2. Optimal observer6.5. Reduced order flux observer6.5.1. Control strategy6.5.2. Error in flux orientation and amplitude6.5.3. Theoretical results6.5.4. Experimental results6.6. Full order flux observer6.6.1. Control strategy6.6.2. Error in flux orientation and amplitude6.6.3. Theoretical results6.6.4. Experimental results6.7. Conclusion6.8. Appendix: parameters of the squirrel-cage induction machine6.9. Bibliography
Chapter 7: Observation of the Load Torque of an Electrical Machine
7.1. Introduction7.2. Characterization of a load torque relative to an axis of rotation7.2.1. Introduction7.2.2. Disruptions of the electrical machine torque7.2.2.1. Generation of the electromagnetic torque7.2.2.2. The cogging torque7.2.2.3. Other disruptive torques non-linked to contact7.2.3. Load torque disruptions by modification of contact actions7.2.3.1. Low-speed friction7.2.3.2. Non-linear friction simulation7.3. Modal control of the actuator with load torque observation7.3.1. Introduction7.3.2. State representation of the actuator7.3.3. Analysis of controllability and observability7.3.4. Control law by state feedback7.3.4.1. Control structure7.3.4.2. Sizing of the control law7.3.4.3. Limitation of the integral action7.4. Observation of load torque7.4.1. Introduction7.4.2. Observer with integration in the loop7.4.2.1. Principle7.4.2.2. Open-loop operation7.4.2.3. Closed-loop operation7.4.2.4. Analytical characterization7.4.3. Complete order observer7.4.3.1. Considerations on observability7.4.3.2. Construction7.4.3.3. Analytical study7.4.4. Reduced order two observer based on the measure of position7.4.4.1. Construction7.4.4.2. Analytical study7.4.4.3. Experimental tests7.4.5. Reduced order one observer based on speed measure7.4.5.1. Construction7.4.5.2. Analytical study7.4.5.3. Experimental results of the reduced order one observer7.4.6. Comparative study of different types of observers7.5. Robustness of control law by state feedback with observation of the resistant torque7.5.1. Introduction7.5.2. Context of the study7.5.3. Robustness of actuator position7.5.3.1. Variations of inertia7.5.3.2. Friction factor variations7.5.4. Robustness of actuator rotation speed7.5.4.1. Variations of inertia7.5.4.2. Friction factor variations7.5.4.3. Torque constant variations7.5.5. Conclusions7.6. Experimental results7.6.1. Introduction7.6.2. Results of modal control with pump torque observer7.6.2.1. Mass lifting7.6.2.2. Variation of the friction factor7.6.2.3. Influence of the inertia variation7.6.3. Non-linear friction influence7.6.3.1. Static friction7.6.3.2. Influence of static friction on control law with resistant torque compensation7.7. Conclusion7.8. Bibliography
Chapter 8: Observation of the Rotor Position to Control the Synchronous Machine without Mechanical Sensor
8.1. State of the art8.2. Reconstruction of the low-resolution position8.2.1. Equivalent electromotive force measure8.2.2. Reconstruction of the sum of electromotive forces using the machine neutral8.2.3. Use of the extended Park reference8.2.4. Use of a two-phase reference8.3. Exact reconstruction by redundant observer8.3.1. Principle and implementation of analytical redundancy8.3.2. Adjustment of correction gains8.3.2.1. First expression of εemfd , εemfq8.3.2.2. Second expression of εemfd, εemfq8.3.2.3. Adjustment of the analytical redundancy observer with filters8.3.2.4. Adjustment of gains and filter in relation to the reference current8.3.3. Sensitivity and robustness8.3.4. Experimental results8.4. Exact reconstruction by Kalman filter8.4.1. Overview8.4.2. Using the Kalman filter for the synchronous machine without mechanical sensor8.4.3. Application for the synchronous machine8.4.4. Gain adjustment8.4.4.1. First tests8.4.4.3. Initial matrix of P0 errors of estimation8.4.4.4. Matrix of state noise Q8.4.5. Assessment on the adjustment of Kalman filter factors8.5. Comparison of reconstructions by Kalman filter or analytical redundancy observer8.5.1. Influence of rating8.5.2. Influence of the initial rotor position8.5.3. Sensitivity to electric parameters8.5.4. Influence and management of load torque8.6. Bibliography
List of Authors
Index

Content preview from Electrical Actuators: Applications and Performance

Chapter 6 Observer Gain Determination Based on Parameter Sensitivity Analysis ¹

6.1. Introduction

The major problem with the implementation of vector control in the induction motor involves controlling machine flux because of the problem in measuring this variable.

To avoid measuring the flux, it is possible to estimate it with the help of models using variables that are easier to acquire (stator current and voltage, mechanical speed). This large dependence on a model leads to major sensitivity problems and uncertainties for the control. These uncertainties are caused by the variations of stator and rotor resistances with temperature and skin effect and the variations of inductances with magnetic saturation.

Rotor resistance is the most difficult parameter to identify with precision, especially in the case of squirrel-cage machines, although it plays an important role in vector control. This parameter can vary by 100% with temperature.

Parameter uncertainties lead to errors of amplitude and flux orientation in the machine with the following consequences:

– the system can become unstable when the error of orientation becomes too large;

– an additional stator current is used to develop a given torque, increasing system losses.

In the first section, we suggest an overview of the principle of the two traditional rotor flux observers: a reduced order observer and a full order observer. The major problem linked to the development of a flux observer is the choice of observer gains that ...

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Electrical Actuators: Applications and Performance

by Bernard de Fornel, Jean-Paul Louis

Chapter 6

Observer Gain Determination Based on Parameter Sensitivity Analysis ¹

6.1. Introduction

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