ON-LINE PARAMETER ESTIMATION AND ADAPTIVE CONTROL OF PERMANENT MAGNET SYNCHRONOUS MACHINES. A Dissertation. Presented to

Transcription

1 ON-LINE PARAMETER ESTIMATION AND ADAPTIVE CONTROL OF PERMANENT MAGNET SYNCHRONOUS MACHINES A Dissertation Presente to The Grauate Faculty of the University of Akron In Partial Fulfillment Of the Reuirements for the Degree Doctor of Philosophy Samuel J. Unerwoo May, 2006 i

2 ON-LINE PARAMETER ESTIMATION AND ADAPTIVE CONTROL OF PERMANENT MAGNET SYNCHRONOUS MACHINES Samuel J. Unerwoo Dissertation Approve: Accepte: Avisor Department Chair Dr. Ibal Husain Dr. Alexis De Abreu-Garcia Committee Member Dean of the College Dr. Robert Veillette Dr. George K. Haritos Committee Member Dean of the Grauate School Dr. Joan Carletta Dr. George R. Newkome Committee Member Date Dr. Graham Kelly Committee Member Dr. Kevin Kreier ii

3 ABSTRACT High performance control of permanent magnet machines (PMSM) reuires accurate knowlege of the parameters that escribe their mathematical moels. This parameter information enables the controller to optimize the rive performance an efficiency, an to react to possible changes in the machine moel. Several methos have been teste in orer to get motor parameter estimates. Most of them are base on off-line measurements or estimates, which are then store in the controller. Other methos publishe are compatible with on-line implementation, but these are usually restricte to a subset of the machine parameters. This issertation proposes a solution to the problem of on-line estimation of PMSM stator resistance, torue constant an - inuctances. An analysis of the machine parameters an their effects on motor rive performance that motivates the evelopment of a new parameter estimation algorithm is presente. This algorithm combines two instances of the recursive least suares metho, which interact in orer to account for ifferent machine parameter ynamics. As a conseuence, the presente metho is able to provie the controller with parameter estimates even when suen changes in operation take place. The effectiveness of the propose parameter estimation algorithm is valiate using both a computer simulation an an experimental motor rive. This simulation moel was evelope specifically for this project an inclues accurate inverter an iii

4 controller moeling, in aition to a machine moel that features parameter variation. The simulation moel is use in both the algorithm evelopment stages an its valiation. The experimental setup provies aitional verification of the effectiveness of the propose algorithm. It is base on the use of a igital signal processor for the controller algorithm implementation, an inclues motor rive classical algorithms as well as the propose parameter estimation program. Both simulation an experimental results emonstrate the performance of the parameter estimation algorithm, both in transient an steay state operations. iv

5 DEDICATION To my fiancée, Coral, an my family. v

6 ACKNOWLEDGEMENTS First, I woul like to thank Dr. I. Husain, who offere me the opportunity to come to Akron an gave me the motivation an assistance I neee to accomplish this. I am eeply grateful for his help an support. I woul also like to thank the members in my Ph.D. committee, for their avice an their help uring the project. I i not get the chance to take classes with all of them, but I woul like to a that the classes I took with Dr. Husain an Dr. Veillette have improve my skills an unerstaning of motor rives an control systems greatly. My thanks also go to Dr. Mir an Dr. Islam, with whom I worke for two summers at Delphi Saginaw Steering Systems. This experience was extremely beneficial for me. Finally, I woul like to express my gratitue to the staff of the epartment of electrical an computer engineering, an particularly to Mrs. Boen. vi

7 TABLE OF CONTENTS Page LIST OF TABLES...xiii LIST OF FIGURES..xiv CHAPTER I. INTRODUCTION Synchronous Machines Types of PM Synchronous Machines Research Objective Dissertation Organization..8 II. PM SYNCHRONOUS MACHINES PMSM Drive Structure PMSM Moeling PMSM Control PMSM Torue Controller Zero -axis Current Control Maximum Torue per Ampere Control Maximum Torue per Voltage Control Loss Minimization Control Flux Weakening Control 22 vii

8 Summary of Current Vector Control Schemes PMSM Current Controller Existing Achievements for Parameter Estimation of PMSM Stator Resistance an Torue Constant Estimation Inuctance Estimation Offline Inuctance Estimation On-line Inuctance Estimation Shortcomings in Existing Research.29 III. PARAMETER VARIATION PROBLEM ANALYSIS Machine Parameter Sensitivities Parameter Sensitivities to Temperature Stator Resistance Sensitivity to Temperature Torue Constant Sensitivity to Temperature Inuctance Sensitivity to Temperature Parameter Sensitivities to Magnetic Saturation Stuy on the Effects of Parameter Variation on Controller Performance Impact of Parameter Variation on Torue Controller Torue Constant Variation Effects of Saturation Impact of Parameter Variation on Current Controller Research Objectives Tracking of Parameter Variations ue to Temperature Tracking of Inuctance Variation ue to Saturation.51 viii

9 3.3.3 Steay-State Detection Capability Conclusion...52 IV. ON-LINE PARAMETER ESTIMATION ALGORITHM Least Suares Algorithms Introuction to the LS Algorithm RLS Algorithm Definition of the RLS Algorithm Moifie RLS Algorithm Estimation Algorithms Overview Fast Estimation Algorithm Slow Estimation Algorithm Conclusion...69 V. PARAMETER ESTIMATION SIMULATION MODEL Machine Moel Inverter Moel Basic Inverter Operation Inverter Nonlinearities Deatime Switch Conuction Transients Switch Steay State Voltage Drops Simulation Moel..80 ix

10 5.3 PWM Algorithm Current Controller Spee Estimation Fast Ientification Algorithm Low freuency controller Conclusion...92 VI. PARAMETER ESTIMATION SIMULATION RESULTS Machine Moels Machine A Inuctances Machine B Inuctances Novel Parameter Estimation Algorithm Algorithm Structure Supervising Program Effects of Neglecting Differential Terms Sensitivity Analysis of Fast Algorithm Estimates Algorithm Initialization Effect of Cross-Saturation Tracking of Temperature Effects Effect of Back-Emf Harmonics Conclusion.125 VII. PMSM DRIVE EXPERIMENTAL DESIGN Inverter Inverter Choice x

11 7.1.2 Inverter Voltage Compensation Deatime Compensation Switch Drop Compensation Compensation Verification Digital Signal Processor PWM Generation Analog to Digital Converters Encoer Interface DSP Central Processing Unit Introuction to Fixe-Point Mathematics Known Limitations Program Execution Times Feeback Circuits Current Sensing Voltage Sensing Feeback Filtering Conclusion.149 VIII. PARAMETER ESTIMATION EXPERIMENTAL RESULTS Time Scale Uncertainty Current Controller Performance Inuctance Estimation Steay State Operation Initial Convergence.158 xi

12 8.3.3 Tracking Ability Complete Algorithm Tests Current Waveforms axis Inuctance Estimation Initial Convergence Steay State Operation Operation During Transient Conclusions 168 IX. CONCLUSIONS AND FUTURE WORK Introuction Research Contributions Limitations in Experimental Setup Suggeste Future Work REFERENCES 174 APPENDIX xii

13 LIST OF TABLES Table Page 3.1 Machine B parameters Operation of ieal inverter leg Steay state operation of an actual inverter leg Simulation moel parameters Machine A -axis inuctance in mh Machine A -axis inuctance in mh Performance of parameter estimation in Torue-spee regions Effects of controller voltage compensation axis inuctance estimate by fast algorithm xiii

14 LIST OF FIGURES Figure Page 1.1 PM machine construction example Different rotor configurations for PMSM PMSM motor rive PM machine reference frames PMSM euivalent circuits incluing iron losses PMSM controller ata flow Summary of current vector control schemes Observer base on-line inuctance estimation Saturation in iron material Inuctance variation in IPM ue to saturation [9] Saturation effects in surface mount PMSM Torue constant change in max Torue / Ampere controller Inuctance waveforms of machine B Saturation effect on max Torue / Ampere Response of current controller to sinusoial spee perturbation Current controller response with parameter error to spee perturbation Comparison between RLS an LS algorithm structures...54 xiv

15 4.2 Propose algorithm structure Fast estimation program structure Slow estimation program structure Estimation algorithm behavior uring current transient Global controller moel structure Machine simulation moel Three phase brige inverter Inverter leg Inverter IGBT turn-on transient (Junction Temperature = 150 C) [24] DSP Deatime insertion Inverter switch voltage rops Simulate ieal inverter response to constant - voltage inputs Simulate real inverter response to constant - voltage inputs Experimental inverter response with locke rotor Machine A -axis inuctance Machine A -axis inuctance Example of interaction between elements of algorithm Simulation example of inuctance step change Inuctance step change without fast inuctance copy Comparison between inclusion an exclusion of ifferential term Torue-spee regions for parameter estimation Initial parameter estimation convergence Initial parameter convergence with 20% initial error xv

16 6.10 Cross-saturation effect on machine inuctances Parameter estimation results with cross-saturation Parameter estimation results with no cross-saturation Parameter estimation with +1 C/s ramp temperature on machine A Parameter estimation with +2 C/s ramp temperature on machine B Effect of 7 th harmonic on - moel ((a) Phase a / (b) - euivalent) Effect of 7 th harmonic on parameter estimation ((a) non filtere / (b) filtere) Deatime error as a function of phase current Switch voltage rop error as a function of phase current Experimental inverter response with locke rotor an voltage compensation Machine bench test result Program executions as a function of time Analog current feeback circuit Analog voltage feeback circuit Transmission of one ata byte Current controller response without feeforwar term Current controller response with feeforwar Current controller response with feeforwar (30% K T error) Initial convergence of fast algorithm Fast algorithm step response Typical current waveforms axis inuctance estimation problem Initial convergence of complete algorithm xvi

17 8.10 Algorithm steay state operation Operation uring transient xvii

18 CHAPTER I INTRODUCTION The evelopment of igital electronics an the recent technological avancements in the fiel of power electronics have cause major changes in the inustry relate to electrical machines. The range of possible applications for such evices has expane tremenously because of their ease of use an their excellent efficiency. Those technological changes have also le to the increase use of new types of electrical machines. For instance, the very popular DC motor is now being challenge for servo applications by Permanent Magnet (PM) motors, Switche Reluctance (SR) motors, an even inuction motors. The evelopment of vector control theory has also allowe improvements in terms of control for existing motor technologies, such as inuction machines. The introuction of Digital Signal Processors (DSP) in motor control applications has allowe electrical machines to reach their full potential, in terms of spee range an ynamic behavior. Complex control algorithms can now be implemente an the motor rive can perform a wier range of operations with optimization of algorithms with regar to efficiency, robustness or ynamic response. For example, in permanent magnet machines the controller can optimize the machine output torue in orer to minimize the reuire current, the reuire voltage, or the power losses. 1

19 The performance of electric motor rives now relies as much on software as on harware configuration. Numerous algorithms have been evelope that can now substitute estimations for measurements, reucing the rive cost an increasing its robustness. The most popular of these inirect parameter estimations are relate to rotor position estimation because of the price an bulkiness of position sensors. These estimators mostly use measure machine currents an electrical parameters to extract position information. On the other han, efforts have been mae in orer to reuce the number of current sensors by reconstructing the three phase currents from DC bus current rather than from measurement of phase currents. A property that most avance control algorithms have in common is their nee for accurate knowlege of the machine analytical moel. A control system esigne for a plant that is ifferent from the one it was intene for is likely to have poor performance. This is why the focus of this issertation will be the estimation of plant parameters. 1.1 Synchronous Machines Permanent magnet (PM) machines are electromechanical energy conversion evices that mainly use the interaction of the stator electromagnetic an rotor magnetic fiels to prouce torue. Most of these machines are non-salient, but epening on the mounting of the rotor magnets, they can also present magnetic saliency that can be use for torue prouction. In their operation an even construction, the PM machines are very similar to the woun rotor (WR) AC synchronous machines. The ifference resies in the fact that the rotor excitation is fixe an provie by permanent magnets instea of coming from an 2

20 external circuit through slip rings an brushes. The stator construction can be the same for both types of machines. In the past, AC synchronous machines were use mostly for generator applications. Their use as a motor was limite ue to the ifficulty of controlling the freuency of their supply voltages. The introuction of power electronics PWM inverters has allowe the motor rive to have complete control over the magnitue an freuency of machine phase to phase voltages. Another factor that helpe the evelopment of PM synchronous machines is the expansion of inustrial prouction of permanent magnets. The first magnet type to be prouce on an inustrial scale was the Alnico in the early twentieth century. As a result, S. Evershe [1][2] in 1920 mae some important contributions to principles of PM torue prouction. At first, PM machines receive severe criticism because of the large tolerance they have in terms of control parameters. Permanent magnet materials exhibit important nonlinearities an are sensitive to temperature an operating point. In 1946, W. Kober first mentione using PM synchronous machines for alternator applications [3]; in 1951, R. M. Sauners an R. H. Weakley significantly contribute to their esign consierations [4]. Most of these first approaches to PM machine esign consiere only Alnico type magnets. Rare earth magnets, which are significantly superior to the Alnico type, appeare in the 1970s. While at first very expensive, these materials have foun an increasing interest in the last few years an are now commonly use in PM machines. PM synchronous machines present several avantages when compare to the WR type machines. First, the PM machines present a much larger energy ensity an can therefore be of smaller size for a given power. They also have much lower rotor inertia, 3

21 which is an important avantage for applications where a fast response is neee. Finally, the absence of brushes to supply the rotor circuit makes them much more mechanically robust. On the other han, the price of PM materials is uite high an PM machines are not economically interesting above a certain power rating (about 20 kw). WR machines are conseuently still use, typically for electrical energy prouction. 1.2 Types of PM Synchronous Machines At this point it is necessary to mention the existence of two families of PM machines, epening on their stator construction. The first one, which will be referre to as PM synchronous machines an which will be the focus of this research, involves sinusoially istribute winings on the stator sie. It is, in that regar, essentially euivalent to a WR synchronous machine with no amper winings. The secon family correspons to a case where the stator winings are concentrate, so that the electromotive force generate by rotor movement is generally trapezoial. These machines are usually calle brushless DC machines, because their operation is very similar to that of stanar DC machines. The focus of this research is not irectly applicable to this type of machine. PM synchronous machines can be further ecompose into two main categories, epening on their rotor construction. While the stator remains essentially the same, the machine rotor can present varying magnetic properties epening on how the permanent magnets are attache to the rotor. Here one nees to familiarize oneself with the general structure of a PM machine. 4

22 Fig. 1.1: PM machine construction example. Figure 1.1 shows a PM machine as an entity compose of two main parts, a stator an a rotor. The stator is a part that is mechanically fixe an connecte to external circuitry. It can be broken own into an iron part, which is magnetically conuctive, an wining slots, which contain electrical winings that generate the stator magnetic flux. The rotor, on the other han, is the part that is mechanically free to rotate an is attache to the stator only with bearings (mechanical, sometimes magnetic). The rotor is also mae of two parts: iron that conucts the magnetic flux, an permanent magnets that prouce the rotor magnetic flux. The interaction between stator an rotor fluxes is what generates the main part of the machine electromagnetic torue. The istinction between the ifferent types of PM machines is mae essentially from the arrangement an location of the rotor permanent magnets. One important fact is that the permeability of the permanent magnets, which can be seen as the magnetic euivalent of the electrical conuctivity, is almost the same as that of air. Depening on how the magnets are mounte on the rotor, there can be a large ifference between a 5

23 magnetic path which inclues magnets an one that oes not. Figure 1.2 shows three examples of PMSM construction. Fig. 1.2: Different rotor configurations for PMSM. These three rotor configurations are the three most commonly foun in the inustry. The rotor on the left has internal permanent magnets, an these are magnetize tangentially, with alternating irections. The mile example also has internal PM, but is magnetize raially. The rotor on the right is ifferent from the two others in that the magnets are mounte on the external surface of the rotor iron. The interesting point about this thir one is that the magnets are completely transparent to the stator magnetic flux, because from a magnetic stanpoint they are euivalent to air. This thir example in Figure 1.2 is also the most popular configuration because it is the easiest one to manufacture. This esign is calle a Surface Mounte PMSM. The magnets are glue to the iron, an a magnetically neutral wrap is also typically place aroun them. It is interesting to note that this configuration oes not show any magnetic saliency, an therefore cannot exploit any form of reluctance torue. On the other han, the left two configurations o present magnetic saliency, as a ifference in magnetic paths between incluing the magnets an not oing so. This particularity allows them to offer both magnetic interaction an reluctance torue 6

24 prouction capabilities to the user. Usually these machines are use for higher spee operation than surface mounte PMSM because they can still prouce reluctance torue in fiel weakening operation. These machines are usually calle Interior or Internal PMSM, or IPM machines. They are naturally more expensive to manufacture, an slightly more complex to control, because of the nee to optimize the combination of two methos of torue prouction. 1.3 Research Objective The main objective of this research is to aress the issue of on-line parameter estimation for IPM machines. The analysis presente shoul also be compatible with a surface mounte configuration an shoul provie performance improvements for that motor rive as well. This research will focus on the machine parameters that are the most relevant to a control system esign for a PMSM. On-line parameter estimation for a PMSM controller is particularly relevant because of the time varying nature of these parameters. The most common perturbation factor to consier is the change in temperature, which cannot be measure in most controllers because of a lack of temperature feeback. Another factor which is often omitte in controller esign is magnetic saturation, which has a very noticeable effect in most IPM machines. Several attempts have been mae to solve this problem in the past. However, most only focuse on a subset of the parameters, relying on the assumption that the rest of the parameters were sufficiently known. Even for such a case, only a few methos were suitable for on-line parameter estimation. Other algorithms relie on the use of offline measurements that are introuce in the control algorithm using look-up tables or 7

25 interpolating functions. These methos present the important rawback of being unable to eal with conitions that cannot be easily accounte for, such as machine aging. The main problem that all these methos avoi or try to overcome is the poor mathematical conitioning of on-line parameter estimation for PM machines. One paper [5] presente a proceure for on-line parameter estimation of a WR synchronous machine, which has a more complex moel than PM machines, but also relie on fixing a small subset of machine parameters an use a non linear metho. The research presente here aims at surmounting the numerical ifficulties associate with the problem of IPM parameter estimation by using specific properties of parameter subsets to ease computation. Even though the focus of this research is on IPM machines, it will be shown that some surface mounte PMSM can also have significant saturation-relate parameter epenency, an coul potentially take avantage of the presente algorithm. 1.4 Dissertation Organization This issertation began with an introuction to the focus of this research. A brief history of permanent magnet machines was presente, followe by a presentation of the ifferent types of such machines, an then the current research objectives were explaine. Chapter II will go further into etails with a more thorough presentation of PM machines moeling an control. A literature review will follow to show the prior research efforts in the area of PM machines parameter estimation. 8

26 Chapter III will focus on the parameter set that is of interest for this research. It will analyze how they can be moifie as the machine operates, an will then stuy the possible conseuences of these changes on controller performance. Chapter IV will introuce the reaer with the solution to the problem of on-line parameter estimation for PM machines that is the subject of this research. The recursive least suares algorithm, which is the basis for this project, will be presente first, an then the propose algorithm structure use in this research will be given. Chapters V an VI will be eicate to the simulation moel that was evelope for this research in orer to esign an first valiate the propose algorithm. Chapter V will focus on the simulation moel itself, escribing how it was esigne, whereas Chapter VI will present the simulation results relevant to this research. Chapters VII an VIII will have a similar structure, but they will be eicate to the experimental setup. Chapter VII will present the ifferent elements of the experimental setup an analyze features that are relevant to this research, while Chapter VIII will show experimental results obtaine with this setup. Finally, Chapter IX will conclue this issertation an will present possible future research topics relate to this control area of PM machine rives. 9

27 CHAPTER II PM SYNCHRONOUS MACHINES 2.1 PMSM Drive Structure A PM synchronous machine rive inclues several elements in aition to the machine itself. The complete motor rive is a structure that inclues the machine, its associate sensors, a power electronics converter, an the controller. The latter processes sensor feebacks an controls the converter for the esire operation. Figure 2.1 shows the motor rive structure that is use in most PMSM applications, an will also be use in this research. Inverter PMSM V c a b c Switch commans Voltage feeback Current feebacks Machine Controller Position feeback Fig. 2.1: PMSM motor rive. 10

28 In the rive structure shown, the position feeback is use to synchronize the stator flux with the rotor one. Position feeback is also use for spee estimation, an for spee or position control. The two current sensors allow the controller to reconstruct the three phase currents an to perform current control, which in turn allow torue control. The DC bus voltage feeback is use in the PWM controller to translate machine voltage commans into switch uty cycles. These feebacks will also be use to achieve the goals of this research, an more emphasis will be place on this aspect in later chapters. 2.2 PMSM Moeling PM synchronous machines are three phase AC machines that involve the interaction of the stator flux, which is controlle by the motor rive, an both the rotor PM flux an the reluctance flux path. The rotor has no winings or electrical connections to the stator. In orer to operate the machine properly, the rotor position has to be known, either from the feeback given by a position sensor, or from a position estimation algorithm. The following assumptions will be mae for this research: Saturation will be taken into account through parameter changes The machine inuce Electro-Motive Force (EMF) is sinusoial Ey currents an hysteresis losses are negligible There are no fiel current ynamics. The approach that is commonly use in orer to moel three phase machines is the one base on the Park transformation [6]. This metho transforms a three phase balance system into a two-imensional one. The transformation changes a complex non 11

29 linear moel into a much simpler one, where machine variables are reference to a rotating reference frame attache to the rotor magnetic axis. Figure 2.2 illustrates this for a two-pole machine. Fig. 2.2: PM machine reference frames. In Figure 2.2, the magnetic axes of the three stator winings are labele a, b an c. From this three imensional coorinate base system two possible results for the Park transformation are commonly use. The first one is labele α-β an is attache to the a phase axis. It is conseuently calle the fixe reference frame. The corresponing machine moel is rather complex, but can be useful in applications where the rotor angle is unknown, such as position estimators. The other case, which will be use in the analysis of this research, is attache to the rotating magnetic axis of the rotor, an is usually calle the - rotor flux reference frame. With this base, the machine moel becomes uite simple, an this makes it easier to evelop control algorithms. The machine moeling starts in the abc reference frame with the following set of euations: V V V a b c R = R 0 0 i 0 i R i a b c + t ϕ a ϕb ϕ c (2.1) 12

30 where V abc are the machine voltages reference to the groun, i abc are the machine phase currents, R is the machine phase resistance, an φ abc are the magnetic fluxes associate with each phase. The Park transformation is a matrix transformation which converts the threephase abc system to the - reference frame. A thir component calle 0 is also present in orer to have a bijective transformation. This 0 or homopolar component is eual to zero in balance three-phase systems an will be omitte later in the chapter. The matrices for the magnitue invariant Park transformation an its inverse are: ( ) ( ) ( ) ( ) ( ) ( ) + + = c b a r r r r r r f f f f f f cos 3 2 cos cos 3 2 sin 3 2 sin sin π θ π θ θ π θ π θ θ (2.2) an ( ) ( ) ( ) ( ) ( ) ( ) + + = cos 3 2 sin cos 3 2 sin 1 cos sin f f f f f f r r r r r r c b a π θ π θ π θ π θ θ θ (2.3) When the Park transformation is applie to euation (2.1) with θ r being the rotor position an taking into account the previous assumptions, we obtain + + = + + = e e p i R V p i R V ϕ ω ϕ ϕ ω ϕ (2.4) where + = = mag i L i L ϕ ϕ ϕ In the above euations, the -axis variables are the ones that are aligne with the permanent magnet position, whereas the -axis correspons to an axis 90 egrees ahea. 13

31 In these euations, p is the Laplace ifferential operatot. It can be note that the permanent magnet flux φ mag only appears on the -axis. In terms of notations, V are the - an -axes stator voltages, which are the results of the Park transformation applie to V abc, an the same conclusion applies to i an φ. L are the inuctances associate with the - an -axes, ω e is the electrical spee of the motor, which is eual to the number of machine pole pairs times the mechanical spee ω r, an p is the Laplace ifferential operator. The machine torue is obtaine from the erivative of the magnetic energy with respect to the rotor position an is given as T e [ mag i + ( L L ) i i 3 P = ϕ ] (2.5) 2 In euation (2.5), P is the number of rotor pole pairs in the machine. There are two torue proucing terms present in euation (2.5). The first one involves the interaction of the magnet flux an the -axis current an is the main machine torue. The secon is base on the ifference between the - an -axes inuctances, an is therefore calle the reluctance torue. This latter component is almost non-existent in surface mount PM synchronous machines but is a particularly interesting feature of IPM machines, giving them extene spee range capabilities. For convenience, a state space representation of the machine moel can be obtaine as p i = p ( V R i ωe L i KT ωr ) i = ( V R i + ω L i ) L p ω = r ( T T ) e e loa J L (2.6) 14

32 In the previous set of euations, K T is the torue or back-emf constant of the machine an J is the moment of inertia of the rotor an its loa. These euations can be use to buil a moel of the machine when couple with euations (2.2) an (2.3). It is necessary at this point to emphasize the fact that the moel evelope from euations (2.6) gives a somewhat simplifie moel for a PMSM. The exclusion of iron losses (ey current an hysteresis losses) has a small impact on the accuracy of simulate results. The reason why these are not inclue here is that the controller complexity reuire for them to be taken into account generally is not justifie by the small error introuce by neglecting them. The moel escribe here is suitable for control oriente problems. However, in a machine esign problem the emphasis woul certainly be ifferent an iron losses shoul be inclue. A simple way to visualize the impact of incluing iron losses in the moel is to raw the machine euivalent circuit shown in Figure 2.3. Fig. 2.3: PMSM euivalent circuits incluing iron losses. 15

33 Usually, the core losses resistance R c is not a constant, but a function of the operating freuency given as 1 R c = 1 R c0 + R c1 1 ω r (2.7) In this research, however, this resistance will be consiere infinite, an conseuently the core losses will be neglecte. 2.3 PMSM Control The machine moel that has been presente can be use for controller esign. Since the focus of this research is on the electrical parameters of PM synchronous machines, our emphasis will be mostly on the current an torue controllers. If an outer loop were to be implemente for spee or position control, the ynamics involve woul mostly rely on the mechanical parameters. Fig. 2.4: PMSM controller ata flow. 16

34 Figure 2.4 shows the ifferent blocks that are commonly foun in PMSM controllers. In this section, the operation of the torue an current controllers will be iscusse, an the other blocks will be escribe in etail later in a harware relate part. The PWM controller oes not reuire motor parameters because the operation it performs scales an shifts the voltage references in orer to convert them into uty cycles for the inverter switches. In the following sections, the euations relating the machine torue to its currents will be iscusse along with the escription of how to control current by acting on the voltages. This will then allow us to visualize more effectively the importance of machine parameters in such controllers PMSM Torue Controller This section will present the various algorithms that can be use with a PMSM in orer to relate its machine currents with its electromagnetic torue. The motivation behin the choice of one algorithm over another is usually a function of complexity an the operating point of the machine at a given time. Each of the presente algorithms relies on euation (2.5) which is repeate here for convenience: T e [ mag i + ( L L ) i i 3 P = ϕ ]. (2.5) Zero -axis Current Control For a surface mounte PMSM (L = L ) or if the -axis current is set to zero, euation (2.5) becomes: 17

35 T e 3 P = ϕ 2 mag i (2.8) Euation (2.8) shows that the machine electromagnetic torue is completely inepenent of the -axis current an is proportional to the -axis current. In such a case, the -axis current is usually controlle to remain at zero, so that the current vector magnitue is minimize. This operation is typically calle i = 0 control [6] an the reference currents are obtaine from i 2 Te = 3 P ϕ mag (2.9) The algorithm is also compatible with IPM machines an is attractive ue to its simplicity, but it completely nullifies the possible contribution of the reluctance torue. It is euivalent to using an IPM as a SM PMSM, which is not esirable ue to the consierable price ifference between the two types of machines. Euation (2.9) shows that this simple algorithm also has the attractive feature of using only one machine parameter φ mag, since P can be regare as a known constant Maximum Torue per Ampere Control Another possible algorithm in the case of IPM machines is the one that minimizes the input current to the machine for a given output torue. This type of operation is usually referre to as maximum torue per ampere (MTPA). The current to minimize is I = i + i (2.10) s

36 T The algorithm is base on fining the point where e = 0. Euation (2.10) I substitute into euation (2.5) gives [8]: s T e 2 2 [ + ( L L ) i ] I i 3 P = ϕ mag s. 2 T An e = 0 I s gives ( ) mag + 8 L L I s ( L ) L I s ϕmag + ϕ β = arcsin (2.11) 4 where β is the angle between I s an i. From this relation the controller can obtain the - an -axes currents that give the maximum torue for a given current magnitue. The euations associate with this algorithm are nonlinear an reuire much more computational power than in the case of euation (2.9). However, this operation is very interesting because it maximizes the motor rive s torue capability when the machine operates below its rate spee. It also minimizes copper losses, which are proportional to the suare of the stator currents. In this case, the -axis current is likely to be ifferent from zero, which implies that the controller takes avantage of both the main machine torue an the reluctance torue capabilities of the machine to minimize the current use. With the MTPA algorithm, however, a goo knowlege of three machine parameters is reuire: both the - an -axes inuctances in aition to the permanent magnet flux. Unlike in the i = 0 control, the relation between the esire torue an the machine currents is not linear an an error in machine parameters can have significant conseuences. 19

37 Maximum Torue per Voltage Control The maximum torue per voltage (MTPV) control for the PM machine, involves the optimization of the current vector to minimize the reuire input voltage to the machine. This is euivalent to minimizing the machine flux linkage between stator an rotor, an that is why this algorithm is also calle maximum torue per flux (MTPF) control. The motor rive operates with both current an voltage constraints an it may happen in an application that the voltage constraint is the harer one to satisfy. This occurs typically at higher spees, when the motor back-emf becomes so ominating that it leaves only small freeom in terms of voltage available. At low spees, the MTPA algorithm is usually preferre because it yiels a higher efficiency. In the case of MTPV, the variable to minimize in the torue euation (2.5) is the flux linkage, which is given in euation (2.12) 2 2 ( L i + ) + ( L i ) ϕ = ϕ. (2.12) 0 mag When this term is combine with euation (2.5) an the resulting euation is ifferentiate with respect to the flux linkage, the following result is obtaine [8] i i ϕ = = mag + Δϕ 2 2 ϕ 0 Δϕ L L (2.13) where L Δϕ = ϕ mag + 2 ( L ϕ mag ) + 8 ( L L ) 4 ( L L ) 2 ϕ

38 This last term can be regare as the amount of flux weakening necessary for the algorithm to achieve optimum operation. The MTPV algorithm reuires the same parameters as the MTPA one. The MTPV is also an interesting control metho since it minimizes the iron losses through the minimization of stator flux. These losses can be significant at high spees an are irectly relate to the flux linkage of the machine Loss Minimization Control It has been mentione in the two previous sections that the maximum torue per ampere minimizes the copper losses in the machine an the maximum torue per flux minimizes the iron losses. If the controller objective is to maximize the efficiency of the machine, then the resulting optimal control will be a combination of these algorithms. This combine control known as loss minimization control. In this case the current vector will be both a function of the machine spee an its torue. For a given torue, at low spees the current vector loci will be close to the maximum torue per Ampere curve, an will graually shift towars the maximum torue per flux trajectory as the spee increases. It is ifficult to obtain an analytical function giving the current vector as a function of the torue an spee, an one way of oing it involves using the circuit of Figure 2.4 with a moel for R c. Common implementations of this algorithm involve a two-imensional look-up table of results that are calculate offline. The problem with this metho is that it fixes the algorithm for the set of machine parameters it was calculate for, which can vary, as will be shown. 21

39 Flux Weakening Control The flux linkage ue to the permanent magnet cannot be controlle by the user, but euation (2.12) shows that its contribution can be minimize by injecting a negative -axis current. This feature becomes interesting as machine spee increases because it can allow the rive to reach spees that woul otherwise cause the rive to excee its voltage capability limit. The flux or fiel weakening metho of control is one that will follow the voltage limit trajectory. This metho is attractive because it is simple to implement an uite robust with regars to parameter changes since it relies on the rive electrical limits. With V ' max = V max r s I max we get i ϕ = L mag ± 1 L V ω r ' max 2 ( L i ) 2 (2.14) Even though this metho reuires the same three machine parameters, its goal is not to achieve optimum performance, but rather to achieve operation at a esire operating point. In this regar, its operation at a point that woul not be the optimum but woul give the esire torue an spee coul be eeme acceptable Summary of Current Vector Control Schemes The algorithms presente up to this point are summe up in Figure 2.5 for a 0.7 kw IPM machine [9]. For a given torue, there is a multitue of possibilities of current vectors. The one that has the smallest magnitue correspons to the maximum torue per ampere algorithm (point A). The other algorithms reuire larger current vectors, but can minimize losses (point D), or the voltage reuire from the inverter (point B). The flux 22

40 weakening trajectory lies on a constant voltage ellipse, an the zero -axis current control lies on the -axis (point E). Fig. 2.5: Summary of current vector control schemes PMSM Current Controller In the block iagram of Figure 2.4, the purpose of the current controller block is to fin the - axis voltages reuire from the inverter in orer to establish the esire currents in the machine. Two methos can be use for that purpose. The first one is calle hysteresis control [7]; it forces the machine phase currents to remain within a preefine range of their reference by switching inverter configurations whenever the error gets too large. This techniue has the important rawback of forcing a variable switching freuency, which can get very large; it also reuires external circuitry an makes it harer for the controller to track the voltage commans sent to the inverter. 23

41 The secon metho is the one that will be use in this research; it uses Pulse With Moulation (PWM) at a fixe freuency to control the voltage vectors sent to the machine. This techniue reuires the calculation of the voltage vector to be sent to the inverter, an is therefore mae reaily accessible by the controller [6][7]. Let us start from the machine electrical euations + = = e s T r e s i L i L p i r V K i L i L p i r V ω ω ω (2.15) A common metho use to make the controller simpler is to use a fee forwar compensator, which ecouples the - an -axes, an cancels the back-emf term. The previous euations then become: + = + = + = = s e s T r e i L p i r i L V V i L p i r K i L V V ω ω ω ' ' (2.16) This in turn gives + = + = s s L p r V i L p r V i 1 1 ' ' The above two transfer functions are simple an linear, an a PI controller is usually sufficient to achieve high performance current control. However, in orer to reach this stage, a goo knowlege of the machine parameters K T, L an L is reuire. If the values use in the controller iffer significantly from the actual machine parameters, the system will present cross coupling an the perturbation of the back-emf, which woul normally reuire a more complex type of control. 24

42 2.4 Existing Achievements for Parameter Estimation of PMSM Several methos exist in the literature that try to compensate for parameter variation in PM machines. These methos can be classifie into offline ones an on-line ones. Offline algorithms use measurements taken at a certain point in time to buil lookup tables or interpolating functions to get parameter estimates as a function of external factors. These techniues are easy to implement, but are ifficult to implement with parameters that can be functions of several variables. They will also fail to account for any change in the machine parameters ue to factors unknown at the time of measurement, such as ageing factors. On the other han, on-line methos use measurements provie by the controller while the machine is in operation in orer to get their parameter estimates. The great avantage of these methos is that they can track parameter variations almost inepenently of their sources. Whether the stator resistance changes ue to temperature or mechanical amage will not matter from the point of view of an on-line metho. Nevertheless, these methos reuire complex numerical algorithms, which may sometimes become unstable an lea to rive malfunction. Particular care must be given to their esign Stator Resistance an Torue Constant Estimation These two parameters nee to be treate somewhat separately from the inuctances, because most controllers for surface mounte PMSM will not even consier inuctance variations. As a conseuence, the problem of estimating r s an K T is common to surface mount an IPM machines. Offline ientification of these two parameters as a separate subset is ifficult to implement even if the assumption is mae that they are only 25

43 temperature epenent. The reason is that temperature feeback is generally unavailable in common PMSM rives. As a conseuence, the only viable approach to this problem is on-line parameter estimation. Papers on this subject usually take the approach of consiering only the steay state moel of the machine [10]. By setting the ifferential terms in euations (2.15) to zero, one obtains V = R i V + ω L = R i e i ω L e + ω K i r T (2.17) The above euations will be referre to as steay state euations. Ientification of the resistance an torue constant is usually one by consiering the inuctances to be constant; then the problem becomes well conitione for either an observer structure or an algorithm like the least suares metho. This approach is vali because the temperature change ynamics are much slower than the electrical ones. This techniue is uite simple an is commonly use for surface mounte PMSM that o not have much saturation sensitivity. On the other han, if saturation occurs, the errors in inuctance will have a large impact on the estimation error. This approach was use in [10] with a position sensorless algorithm for a surface mount PMSM Inuctance Estimation The problem of estimating the inuctance for the controller is uite ifferent from the previous one. In this case, it is possible to measure these two functions offline since each inuctance can be consiere as a function of the current in its axis, if crosssaturation is neglecte. The inuctance in the controller is then obtaine from the measure currents in the machine as it runs. 26

44 Offline Inuctance Estimation Several methos of parameter estimation in PMSM have focuse on offline inuctance measurement. The methos vary since the - an - axes inuctances are not reaily available for measurement, but are somewhat fictitious parameters obtaine through the Park transformation. A first metho consists of getting inuctance estimates from a finite element analysis. This techniue reuires the user to know precisely the geometric an material properties of the machine, an then to run a series of computationally intensive simulations. However, the ata necessary to have accurate simulation results is not usually available from the manufacturer, an these simulation results often nee to be ouble checke with experimental ata for possible numerical problems. Another problem is that this metho oesn t solve the issue of fining estimates for the resistance an torue constant. An example of this techniue has been presente in [11] an [12]. In both cases the results were verifie experimentally: in [12], the inuctances were obtaine from locke-rotor measurements. Other methos for offline measurements are possible. Some use the ecay time of a phase current following a voltage pulse for ifferent locke rotor positions an initial current levels [13]. Offline measurements are usually more convenient than on-line ones because they can force situations that are inaccessible to a non-intrusive on-line algorithm. In the case of a locke rotor situation, the motor - euations become much simpler: V V = R i = R i + + p L p L i i 27

45 Another techniue to extract the machine inuctances consists in running the machine in a succession of steay states an calculating inuctance estimates from voltage, current an spee measurements. To get to this point, some methos irectly work with the Park transformation of the measurement, like in [14], while others work with the Fourier transform of motor phase variables [15]. When it comes to using measurements obtaine offline, a common metho was given in [16] an [17], that use piecewise linear functions to approximate the inuctances as a function of their respective currents. This approach of course neglecte the impact of cross saturation On-line Inuctance Estimation On-line parameter estimation methos have also been evelope to ientify the machine inuctances. One approach foun in [18] an [19] is to consier the stator resistance an torue constant to be fixe, an to run an observer base on the machine moel. Aitional information from offline inuctance measurements was also injecte in orer to improve ynamic behavior an numerical stability. Figure 2.6 shows the implementation structure for this metho. Three main rawbacks can be foun for this techniue. The first one is that errors in resistance an torue constant will have a negative impact on the estimation. Another one is that compensation for cross saturation can harly be mae. Finally, the metho reuires offline measurements, which take away some of the flexibility that coul be expecte from an on-line algorithm. 28

46 Fig. 2.6: Observer base on-line inuctance estimation. A secon approach was foun in [5] where the parameters of a woun rotor synchronous generator are estimate using a non-linear version of least suares estimation. The moel use in this paper is ifferent from the one use in this research (ifferent machine structure), an the metho consiere a small subset of the machine parameters to be constant in orer to obtain ecent numerical stability. This paper is mentione here because it can be consiere to be the closest one available in the literature to provie a solution to the problem of on-line parameter estimation of electrical machines. 2.5 Shortcomings in Existing Research The previous section has shown the various methos that have been trie to overcome the problem of parameter variation in PMSM. Off-line methos have been implemente base on both simulation results (FEA) or experimental measurements, but prove to be limite when it comes to having estimates for all four parameters. These methos also make it ifficult for the controller to compensate for the effects that were 29

47 not foreseen at the time results were taken. These methos also lack compensation of temperature effects, since temperature sensors are usually not present in motor rives. On the other han there have been attempts to evelop on-line parameter estimation, but the examples foun limite themselves to a subset of the parameters an/or use a combination of off-line an on-line results to operate properly. The avantages of having an on-line parameter estimation algorithm for PMSM are numerous. These inclue the fact that parameter estimates track machine parameters regarless of the causes of parameter changes. From the point of view of the algorithm, it is euivalent to estimate parameter variations ue to cross saturation or temperature changes, as oppose to an off-line metho. If the controller is constantly supplie with accurate parameter estimates, it can operate in an optimal way, epening on the control algorithm it uses. This allows the motor rive to reach its full potential, in terms of efficiency, spee range or ynamic response. 30

48 CHAPTER III PARAMETER VARIATION PROBLEM ANALYSIS Base on the literature review one in the previous chapter, the nee for the following have been ientifie: An analysis of how an why the machine parameters change, the factors that affect them an their properties. A complete stuy of the effects of electrical parameter variation on the performance of the torue an current controllers of PMSM. These elements will serve as a basis for this research project an will show how on-line parameter estimation can improve controller performance. 3.1 Machine Parameter Sensitivities It has been shown in the previous chapter how machine parameters play an important role in etermining the location of the esire current vector for a given torue level, machine spee an metho of optimization. In this section we will focus on the ifferent factors which can affect these parameters. The set of machine parameters that will be the subject of this research is efine in euation (3.1). 31

49 L L θ = (3.1) r s K T These four parameters are the ones foun in the euivalent circuit of Figure 2.3, once core losses have been neglecte. L is the inuctance associate with a magnetic path that links stator an rotor going between magnet poles L is the inuctance of a magnetic path linking stator an rotor that goes through the permanent magnets r s is the electrical resistance associate with one stator phase K T is the torue constant of the machine, an is irectly proportional to φ mag The torue constant K T is given by K T P = ϕ (3.2) 2 mag The objective of this research is to estimate these parameters an to track their possible variations on-line uring machine operation. The following sections focus on the factors that affect them, an how they affect them Parameter Sensitivities to Temperature The main external factor that can cause the machine moel to vary is the machine temperature since it affects both electrical an magnetic material properties. Most motor rives o not inclue temperature sensors an it is usually ifficult to compensate for temperature variations. These variations can be cause by either the machine s external 32

50 environment or the machine itself. Both copper an iron losses contribute to a rise in machine s internal temperature, an mechanical losses ue to friction in the bearings may a to this effect. It is ifficult to uantify the way in which these temperature changes affect the ifferent machine parameters, because these are essentially relate to the types of material use in the motor construction. It is necessary to keep in min that in PM machines, temperature changes are consiere to have extremely slow ynamics when compare with the electrical or mechanical ynamics Stator Resistance Sensitivity to Temperature In the case of the stator resistance r s, the problem is not too complex because most electrical machines have copper winings. The general euation that relates resistance changes in conuctors as a function of temperature is R T ( T 0 ( ) R + ( )) = 0 1 α T (3.3) where R 0 is the resistance of the conuctor measure at the reference temperature T 0, an α is the temperature coefficient of the conuctor material. This relation is of course an approximation, but is suitable for most applications. Most of the time α is given for a temperature T 0 of 20 C or sometimes 0 C. For copper conuctors, α = C -1 an for aluminum conuctors α = C -1, both measure at 20 C. This means that for a copper conuctor, a change of 20 C from the initial point leas to a variation of 8 % in its resistance value. PM machines usually have an operating temperature range of about 100 C, an the associate resistance range (about 40%) shoul not be neglecte in controller esign euations. 33

51 Torue Constant Sensitivity to Temperature This machine parameter is irectly associate with the intensity of the magnetic flux inuce by the machine permanent magnets. This flux is temperature epenent, but the way it is affecte is a function of the permanent magnet material, its shape, an the magnetic circuit attache to it. It conseuently is very ifficult to erive a simple moel for the effects that temperature changes will have on the magnetic flux. A rough estimate for a temperature coefficient can however be given by permanent magnet manufacturers an was also foun in [20]. For example, the N3571 Neoimium Iron Boron (NFeB) will have a coefficient of -0.11%/ C for its resiual magnetism B r. As a conseuence, an euation similar to euation (3.3) coul be erive for this parameter. A ecrease in B r is also likely to affect the magnet s intrinsic coercive force, which is its resistance to emagnetization. Both these effects are likely to aversely affect the magnetic flux inuce by the material in a given circuit. In aition to the thermal coefficient, most permanent magnet materials will have an approximate Curie temperature [20], which is also function of magnet material, shape an circuit. This parameter is the temperature at which the permanent magnet becomes permanently emagnetize, an shoul conseuently be avoie at all costs in permanent magnet machines. One of the few isavantages of NFeB when compare to other types of permanent magnet material such as Samarium Cobalt (SmCo) is its much lower Curie temperature an its higher sensitivity to temperature changes. For example, the temperature coefficient for the B r of a SmCo magnet is aroun -0.03%/ C. However, SmCo magnets are generally more expensive than NFeB an have a lower (B.H) max energy prouct. Both these magnet types are use in high performance PM machines. 34

52 Inuctance Sensitivity to Temperature Temperature changes also affect the - an - axes inuctances of the machine. Most of the magnetic paths associate with these inuctances take place in a ferromagnetic or ferrimagnetic material. Such types of materials also present certain sensitivity to temperature changes. For example, an increase in temperature in a typical ferrite of 80 C can result in a permeability rop of about 25%. This, in turn, results in a ecrease of about 25% in its inuctance Parameter Sensitivities to Magnetic Saturation This section only concerns the changes in the machine - an - axes inuctances ue to iron saturation. Magnetic saturation is a phenomenon that occurs in the magnetic iron parts of PM machines. The electrical resistance is not affecte by this effect, an the torue constant can be affecte, but in a negligible way. On the other han, machine inuctances will irectly be affecte. Figure 3.1 provies a better unerstaning of the problem. Figure 3.1 shows a typical flux response in an inuctor with an iron core when subjecte to an increasing current. One can see that past a certain point (P1), the curve ceases to be linear an magnetic flux tens to increase in a slower way. This is what is calle magnetic saturation. The inuctance is the slope of the curve at a given point. The inuctance will remain constant at its maximum value for the linear portion of the curve in Figure 3.1; then, its value rops as saturation comes into play. Electrical machines are much more complicate to moel, but this example can serve as a basis for unerstaning the phenomenon. Another thing to notice here is that inuctance changes 35

53 in response to current changes in the machine have much faster time scales than temperature relate parameter variations. Flux (Wb) L 1 P2 L 2 <L 1 P1 Electrical Current (A) Fig. 3.1: Saturation in iron material. When it comes to PM synchronous machines, both the - an - axis inuctances are affecte by saturation. However, it is necessary to consier important ifferences between the behaviors of these two parameters: The -axis of the machine inclues the permanent magnet, an the corresponing iron path is subjecte to the magnetic flux. This important magnetic flux can be the source of a certain level of saturation even when the machine is not excite (as in point P2 of Figure 3.1). The electrical currents associate with this axis are usually oriente to oppose the magnetic flux, either to exploit reluctance torue, or to achieve flux weakening. Depening on the intensity of these currents, a noticeable change may occur in the machine inuctance. However, the artificially increase air gap thickness ue to the magnets also reuces the sensitivity of L towars i. 36

54 The -axis inuctance oes not inclue the permanent magnets, an is not excite when the machine is at rest (except for remanent flux). For small -axis currents, one can consier that the -axis magnetic circuit operates in the linear region on the iron material. However, for large currents (i.e. large torues), the magnetic path may become saturate an the -axis inuctance will conseuently rop. Surface mounte machines iffer from IPM machines because when the machine is at rest, their - an - axes inuctances are almost the same. A conseuence of their construction is also that they generally have larger air gaps than IPM, in orer to accommoate the permanent magnets. However, when a surface mounte machine operates at heavy loa, saturation effects can still appear in the -axis an these effects will be iscusse later. An assumption is usually mae that the - an - axes inuctances in PM machines are ecouple, which implies that i will not affect L an vice versa. Figure 3.2 shows the inuctances of an IPM machine [9] with low saliency ratio, which is the kin of machine that was use in the experiments for this research project. Fig. 3.2: Inuctance variation in IPM ue to saturation [9]. 37

55 This IPM machine has a small sensitivity towars saturation, an its saliency ratio L /L is not significantly affecte by it. It is also possible to see that the -axis inuctance oes not seem to be affecte much by the -axis current. Similar stuies have also been conucte for surface mounte PMSM [21], as shown in Figure 3.3. Fig. 3.3: Saturation effects in surface mount PMSM. In Figure 3.3, the currents are measure in Amperes an the inuctances in Henries. One can notice large changes in inuctances, with L ecreasing with increasing -axis currents, an L increasing with large emagnetizing currents (negative -axis). Saturation effects in surface mounte machines are often neglecte in controller esign but the results of this research show that a high performance controller coul probably be improve from consiering these changes. 3.2 Stuy on the Effects of Parameter Variation on Controller Performance The previous chapter has shown that the issue of on-line parameter estimation for a permanent magnet synchronous machine has not been aeuately aresse in previous research. The solutions that have been foun to compensate for parameter variation 38

56 usually fail to correct what was not accounte for at the time of controller esign. Temperature effects, cross-saturation an ageing effects, for example, will in most cases have an impact on controller performance. This section focuses on the impact of parameter variation on the controller, which serve as the primary motivation for this research. The parameters of interest for this research have an impact on both the torue an current controller of PM synchronous machines, an the stuy on the effect of parameter variation is presente in the following sections. The stuy was restricte to the operation of a maximum torue per ampere controller in orer to show the results. However, a similar analysis coul be conucte base on another algorithm, which woul lea to similar conclusions Impact of Parameter Variation on Torue Controller It has been seen in section that for all torue controller esigns that were presente, only the torue constant an - inuctances were important. The stator resistance i not have an impact on the esign euations of torue controllers. The results presente in the following sections were obtaine with the Matlab / Simulink moel evelope for this project. They aim at simulating results that are achievable with a machine referre to as machine B, whose parameters are given in Table 3.1. Since etaile information was not available about the machine construction materials, it was assume that the stator winings were mae of copper an the permanent magnets were NFeB for the temperature effects. 39

57 Table 3.1: Parameter name Machine B parameters Machine B Stator resistance, r s (Ω) 1.45 Torue constant, K T (V.s) L (mh) no saturation 18 L (mh) no saturation 6 Rotor inertia, J (N.m.s 2 ) 99.6e-6 Number of pole pairs, P 2 Rate current, I max (A) Torue Constant Variation The results shown in this section are the (i, i ) loci for a torue reference that changes linearly from zero to almost the machine rate torue. Two sets of results were obtaine for this section corresponing to two values of the torue constant or temperature. In this type of problem it is necessary to consier both the machine an controller sies to stuy the effect of temperature. Most controllers are esigne by consiering K T to be constant an use the value measure at room temperature. In these tests, first, the controller was assume to be running in ieal conitions (room temperature), an then the rive was simulate to be operating at a higher temperature, inucing a 30% change in K T, both for the same controller, an for one that woul have a feeback in K T. These results were extracte from a spee control test at low spee, with a ramp loa torue. The curve on the left sie, labele ML-CL correspons to an initial test at 40

58 low temperature, where controller an machine have matching parameters. On the righthan sie, the two loci were taken from simulations where the machine was at high temperature (lower K T ), one where the controller an machine parameter matche (MH- CH) an one where they i not (MH-CL). Fig. 3.4: Torue constant change in max Torue / Ampere controller. A conclusion that can be rawn from this test is that the machine will reuire larger currents to operate at the same torue level for operation at higher temperature. As an example, the points that are the farthest from origin correspon to the same torue level. However, the irect corresponence between torue level an current magnitue was omitte here for clarity. This is ue to the fact that φ mag rops with rising temperatures an so oes the torue prouction capability of the machine for the interaction between stator an rotor magnetic fluxes. For a given torue level, the 41

59 machine will reuire larger currents at higher temperature to compensate for this rop. One can notice though, that the curve on the left (Figure 3.4 (a)), can almost be superimpose to the MH-CL one on the right (Figure 3.4 (b)), which is why it was left alone. The reason for this is that in both cases the controller operates on the same parameters, giving the same current loci. For high temperature operation, the two curves in Figure 3.4(b) show that whether the controller has perfect knowlege of the torue constant or not will not make a large ifference in the maximum Torue per Ampere locus. The sensitivity of this type of controller is therefore low to changes in temperature. This behavior is nonetheless a function of the machine, an a larger change coul be seen on a machine with lower saliency. There is however one interesting comment one can make about the two curves in Figure 3.4(b). The one that has the parameter mismatch (K T larger than it is in the machine) uses larger -axis currents than the other one. In this particular case, the controller that uses the accurate parameter values appears to be relying more on the saliency component of the torue euation (2.5). The reason for this is that as the torue constant rops (or as the temperature increases), it becomes more an more efficient to use the saliency component rather than the permanent magnet one for a given current magnitue to maximize the output torue. The following euations may help visualize the problem. T T e e K = A = A Tlow [ KThigh i + B i i ] ' ' [ K i + B i i > K Tlow Thigh ' ] (3.4) 42

60 The same torue T e is obtaine with ifferent - an -axes currents at two ifferent temperatures. The controller that has the larger torue constant will try to use a larger -axis current because it thinks that the machine s permanent magnet torue capability is higher than it actually is Effects of Saturation The simulations realize for this section are base on the machine B saturation characteristics shown in Figure 3.5. The loa torue of the machine is varie linearly from zero to a value close to the rate one, an the machine is operate uner spee control moe. As the loa torue increases, the machine torue also increases (to maintain constant spee), an the - an - axis currents escribe the maximum torue per ampere trajectory. However, as the currents increase, saturation starts to play its role as escribe in Figure 3.5, affecting machine performance. The left sie of Figure 3.6 (Fig. 3.6(a)) shows the maximum torue per Ampere loci in two ifferent cases in terms of controller parameters, an the right sie shows the corresponing stator current magnitue as a function of machine torue, which is suppose to be minimize for a given torue. Relating Figure 3.6 with Figure 3.5, it can be observe that the -axis inuctance remaine constant for the values of currents that were use in this test. On the other han, the -axis current went as high as about 18 A, which represents a ecrease of about 70% of the value of -axis inuctance. The large increase in -axis current affecte the saliency ratio of the machine in a significant way. 43

61 L = f(i) L = f(i) L, L (H) I, -I (A) Fig. 3.5: Inuctance waveforms of machine B. Fig. 3.6: Saturation effect on max Torue / Ampere. 44

62 Three tests were conucte to stuy the effects of saturation. The first one correspons to the ieal case where there is no saturation (Mnosat-Cnosat). In that case the machine has constant - an -axes inuctances, an the controller values match them. The secon test (Msat-Cnosat) was essentially the same, except that the machine use the inuctance waveforms of Figure 3.5, an the controller remaine unchange. In this case the current vector locus is almost the same as in the ieal case, although larger in magnitue ( Mnosat-Cnosat stops at point A in Figure 3.6). The reason for this is that the controller operates on the same characteristic, even though it oes not correspon to the optimum trajectory. The ifference between these two cases shows on the right plot, where one can see that for the rate torue there is a 50% ifference in motor current magnitue. The thir test is what is more important an really shows the importance of having accurate parameter feeback in the controller. In this last case, the machine presente saturation, an the controller use the saturate values of inuctance to calculate the maximum Torue per Ampere trajectory. The ifference between this locus an the others is very large. As the machine torue an currents increase, the saliency ratio iminishes an the machine begins to operate somewhat like a surface-mount machine. The contribution of the saliency torue becomes small an the torue controller fins it more efficient to use more of the magnetic flux interaction for the same torue level. T T e e B ' = A = A [ K T i + B i i ] ' ' ' ' [ K i + B i i ] T << B << K T 45

63 From the above euations it is clear that the reuire optimal -axis current will ' ' have to be larger for the controller that takes saturation into account ( i i ), for the same level of torue as in the initial one. However, the operating point that woul be given by the maximum Torue per Ampere algorithm with parameter feeback is not the optimal operating point. The euations use to erive this algorithm in section make the assumption that the - inuctances in the machine are constant. To get the optimal point, it woul be necessary to repeat the erivation an consier the inuctances to be functions of the machine currents. This approach is however incompatible with a basic on-line parameter estimation algorithm. The reason for this is that optimum operation woul reuire algorithm erivation with complete knowlege of machine parameters, an the parameter estimation algorithm only provies the controller with the inuctance parameters that correspon to the current operating point. These results also show that for an interior PM synchronous machine with high saturation it is not efficient to operate at high levels of saturation. When these machines are operate uner flux weakening or maximum torue per flux, the controller takes avantage of the saliency torue an reuces the -axis current, thereby reucing the effect of saturation., Impact of Parameter Variation on Current Controller It has been shown in section how machine parameters play a certain role in the performance an esign of the current controller. The feeforwar compensator which rejects the isturbances create by the cross coupling between the - an - axes an the back-electromotive force takes avantage of parameter feeback. The terms 46

64 introuce by that compensator are all proportional to the motor spee, an o not make a ifference at stanstill. The results to be presente in this section show the effect of spee as a isturbance in the complete current controller, i.e. one that inclues feeforwar. These results were obtaine with a machine moel that matches the one use in the experimental setup (machine A ). The simulate machine was run at a spee of about 1500 RPM with a sinusoial variation of 500 RPM aroun that. The - an - axes currents were to be maintaine constant by the current controller. Current controller with no error in feeforwar an sinusoial spee Currents (A) Iref Iref I I wr (r/s) time (s) Fig. 3.7: Response of current controller to sinusoial spee perturbation. In Figure 3.7, the response of the current controller to a step in the - an -axes reference currents as the machine spee varies sinusoially is shown. The current responses track the references very effectively, an the spee perturbation oesn t seem to affect the performance. 47

65 Figure 3.8 on the other han shows the response of the current controller to the same spee perturbation, but with errors in its torue constant an -axis inuctance. Just as in section 3.1.1, a variation of 30% in torue constant an 40% in -axis inuctance were use. 30% error in Kt an sinusoial spee Currents (A) Iref Iref I I % error in L an sinusoial spee Currents (A) Iref Iref I I Time (s) Fig. 3.8: Current controller response with parameter error to spee perturbation. The first plot in Figure 3.8 shows that a variation in the torue constant translates into a poor compensation of the spee variation in the -axis current. In contrast, a variation of the -axis inuctance has no effect on the -axis current, but shows the sinusoial perturbation on the -axis current. This can be explaine by looking at the esign euations of the feeforwar current controller (2.16). 48

66 V ' = V V ' ω L = V e i + ω L e ω K r i T = r s = r i s i + p L + p L i i Due to the cross-coupling effect between - an -axes of the machine, the conseuences of a poor estimation of L will lea to errors in the -axis current control an vice versa. On the other han, errors in the torue constant will lea to problems on the -axis. In aition to this, for the machine uner stuy in this research, we have K T >> L or. max i max L max i max This makes the current controller much more sensitive to errors in the torue constant than in inuctance, which can be observe from Figure 3.5. The effects of stator resistance change only the ynamics of the current controller an affect only milly the esign of the PI controller. Resistance feeback woul be useful if the current controller was esigne to account for such changes. In the present case, ynamic performance is satisfactory for the purpose of this research. The possible role an importance of stator resistance estimation will be further analyze later. 3.3 Research Objectives The objectives for the research presente in this paper are multiple, but its general goal is to fin a solution to the problem of on-line parameter estimation of PM machines for the parameters outline in section 3.1. The algorithm to be foun shoul have the capabilities of tracking parameter variations ue to temperature an saturation, an parameter estimation at steay state. Further etails on the objectives of the algorithm are presente in the following subsections. 49

67 3.3.1 Tracking of Parameter Variations ue to Temperature The metho to be foun shoul be able to track parameter variations ue to temperature, which are slow compare to the ynamics involve in the machine controller. These temperature effects can affect all four parameters, but they are likely to affect the stator resistance an torue constant most. Interest in tracking the stator resistance changes has not been emphasize so far, because it can be seen as seconary from the point of view of controller esign. The resistance only comes into play in the esign of the compensator for the current controller, which is important for very high performance controllers, but reuires aitional computations as the machine is running. On the other han, the material properties of electrical conuctors are very well known, an it coul be possible to use an on-line resistance estimate to get an iea of the machine operating temperature. Monitoring resistance changes coul also be interesting for the purpose of machine or inverter iagnostics. This coul be one as a system self check before startup. Finally, the most important role of knowlege of stator resistance for this project is that it will allow a much easier operation for the ientification of other machine parameters, as it will be shown later. A reliable resistance estimate will likely be crucial for the numerical stability of the entire algorithm. Existing algorithms [18] [19] acknowlege the importance of having goo resistance an torue constant estimates for the purpose of ientifying the - an -axes inuctances. The machine torue constant is another parameter that changes mostly because of temperature variation. Having a goo estimate for this parameter is very important for the operation of the current controller an is appreciable when it comes to the torue 50

68 controller, as it has been mentione in sections an Depening on the magnet material use in the rotor, the torue constant may present a high sensitivity to temperature. As for the stator resistance, the torue constant will also be of importance for the overall stability of the parameter estimation algorithm. Ientification of parameter variation ue to temperature changes shoul be realize with the steay-state machine moel in orer to maintain numerical simplicity an to avoi the inclusion of current erivative terms that are especially ifficult to estimate in motor rive applications. This shoul not be an issue because of the relative slowness of temperature variation when compare to the motor rive ynamics Tracking of Inuctance Variation ue to Saturation The problem relate to saturation is of a ifferent nature than the one relate to temperature changes. As oppose to the latter, saturation effects are irectly relate to the current levels flowing in the machine, an their ynamics are as fast as the currents. As a conseuence, an on-line estimation algorithm for inuctances must operate at a much faster rate than one esigne for temperature relate phenomena. The algorithm may have to use the machine moel incluing current ynamics, or operate at a rate that makes it possible to neglect them. Another solution coul be to use the results given by the algorithm in steay state operation an upate a look-up table use by the controller. An avantage in the use of an on-line estimation algorithm is that it can track the effects of cross-saturation transparently from the user point of view. The metho shoul also be able to operate so as not to isturb the one relate to temperature changes if they are istinct. These issues will be iscusse further in chapters IV an VI. 51

69 3.3.3 Steay-State Detection Capability In orer to best use an coorinate the estimation of all electrical machine parameters, an algorithm will have to be evelope that can operate when the machine is in steay state. For example, a metho base on the steay-state moel of the machine is boun to show errors uring a change of operating point because the moel it uses oes not represent the controlle system uring that time. This steay-state etection algorithm shoul then be able to enable or isable the operation of the parameter estimation algorithms, an even iscar some estimation results. This metho will be base on the current or voltage commans that the ifferent controllers issue. 3.4 Conclusion This chapter presente the phenomena that affect the main machine parameters, an the subseuent effect of parameter variation on controller performance. This analysis serve as a basis for the etermination of precise objectives for this research project. 52

70 CHAPTER IV ON-LINE PARAMETER ESTIMATION ALGORITHM The objective of this research is to provie the machine controller with parameter estimates that are calculate as the machine is operate. The Recursive Least Suares (RLS) algorithm was chosen as a basis for this purpose. It is a relatively fast algorithm that provies reliable parameter estimations, even in the presence of white noise. Since the structure of the machine moel was known beforehan, this algorithm is suitable for this research. 4.1 Least Suares algorithms The RLS algorithm is base on the Least Suares (LS) algorithm. When applie to system parameter estimation, the latter simply gives an estimate of unknown moel parameters base on a given number of input an output system measurements. These measurements can be one at any point in time prior to the execution of the LS algorithm. On the other han, the RLS algorithm is one that can be irectly aapte to on-line parameter estimation because its structure allows easier expressions for new parameter estimates each time it is supplie with new input an output ata. 53

71 Input/Output (t) Input/Output (t-1) Least Suares Algorithm Parameter Estimates (t) Input/Output (1) Input/Output (0) Eual Input/Output (t) Parameter Estimates (t-1) Recursive Least Suares Algorithm Parameter Estimates (t) Fig. 4.1: Comparison between RLS an LS algorithm structures. Figure 4.1 shows the funamental structural ifference between the RLS an LS algorithms. A simple techniue to obtain an on-line estimation algorithm from the LS algorithm woul be to execute it each time new input/output ata is available. Although this coul be possible in theory, the numerical computation power reuire for that woul uickly become an impeiment for an on-line implementation. One major constraint that one encounters when confronte with real-time numerical control techniues is that the various programs associate with an application have to be execute fast enough in orer to be able to maintain the esire sampling rate. In applications, the sampling rate is set to a value that allows the control system to take into account the main ynamics of the system. The sampling rate is usually euce from Shannon s law, which states that the sampling rate has to be at least twice that of the highest freuency to be consiere. Usually, the sampling rate choice of a control system is a compromise between software complexity an system stability. Control algorithms 54

72 are typically execute at the sampling rate in orer to account for the complete system ynamics. For that to be possible, the various algorithms associate with the application have to be execute in a time that is shorter than the sampling perio. If this conition is not realize, the program can easily become unstable. The inherent structure of the LS algorithm makes it unsuitable for on-line applications. The RLS algorithm was evelope to overcome this limitation an still give the same estimates for a given set of inputs. 4.2 Introuction to the LS algorithm For a system escribe by y ( ) = ϕ ( T k k ) θ (4.1) where y is the system output, φ represents a measurement vector an θ the system parameter vector that is to be estimate. If we have N euations like euation (4.1), we can write them in a combine form as Y = Φ θ where [ ϕ,..., ] T, 2 an [ ] T, 2 Φ ( k ) = ( 1) ϕ ( ) ϕ ( k ) ( k ) y( 1) y( ) y( k ) Y =,...,. The Least Suares algorithm calculates an estimate of the system parameters such that V θ est = 1 = 2 argθ ( min( V( θ ))) N T ( y k ϕ k θ ) ( θ ) ( ) ( ) k = 1 The solution to this problem is as follows [22]: est T 1 T ( Φ Φ) Φ Y 2 θ = (4.2) 55

73 One can see that this solution reuires the inversion of a matrix that can be large, epening on the number N of input/output euations consiere. The RLS algorithm is one that woul only use ata available at a particular point in time an its previous estimate to provie the user with the same result as the LS algorithm. 4.3 RLS algorithm Definition of the RLS algorithm The approach taken to get the RLS algorithm from the regular LS algorithm is interesting because it is one that coul be applie to other offline algorithms. To obtain a formulation of the RLS algorithm, we start from the result that the LS algorithm woul give at the k th sample (E (4.2)) as: est T 1 T ( Φ Φ ) Φ ( k ) Y( k ) θ = (4.3) ( k ) ( k ) ( k ) [ ] T T Introucing F ( k ) = Φ ( k ) Φ ( k ), leas to F( k ) θ est( k ) = Φ ( k ) Y( k ). By ecomposing the matrices at the k th step, one can get F T ( k 1) [ Φ ϕ ] T T = T 1 ( k ) ( k 1 ) ( k ) Φ ϕ ( k ) = Φ Φ + ϕ ϕ = F + ϕ T ϕ ( k 1) ( k 1) ( k ) ( k ) ( k ) ( k ) ( k ) The F matrix can be obtaine recursively using the above euation. Going back to the LS algorithm, it is possible to get: θ est = F 1 Φ T Y = F 1 Y T ( k 1) [ Φ ϕ ] ( k ) ( k ) ( k ) ( k ) ( k ) ( k 1) ( k ) An then ( k ) ( k ) ( k 1) ( k 1) ( k ) ( k ) y( k ) T 1 [ Φ Y + ϕ y ] = F( k ) [ F( k 1) θ est( k ) + ϕ ( k ) y( k )] 1 θ est = F 1 56

74 This expression can be further simplifie as follows: ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) [ ] ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) [ ] k k k est T k k k est k k k est k k k est T k k k k k est y F F y F F + = + = ϕ θ ϕ ϕ θ θ ϕ θ ϕ ϕ θ An from this last euation, the first version of the RLS algorithm can be euce: ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) + = + = T k k k k k est T k k k k k est k est F F y F ϕ ϕ θ ϕ ϕ θ θ This algorithm can be practically implemente as it is, but it presents the problem of reuiring the inversion of a possibly large matrix at each step. Matrix inversions are extremely time consuming operations an are to be avoie as much as possible in realtime algorithms. As a conseuence, this version of the RLS algorithm ha to be further moifie in orer to make it more practical. A simplifie recursive expression for the inverte matrix ha to be foun. For this purpose, the following matrix is introuce: ( ) ( ) 1 = k k F P An from ( ) ( ) ( ) ( ) T k k k F k F ϕ + ϕ = 1 one can obtain ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) [ ] = + = T k k k k T k k k k P P P P ϕ ϕ ϕ ϕ At this point, a matrix inversion lemma can be use: [ ] [ ] = + A D C B A D B A A D C B A with where I n is the ientity matrix. ( ) ( ) ( ) T k n k k D I C B P A ϕ ϕ = = = =,,,

75 Therefore, P ( k ) ( k ) ( k 1) ( k ) ( k ) ( k 1) ( k ) T 1 T [ ϕ P ϕ + I n ] ϕ( k ) P( 1) = P 1 P k Introucing, K ( k ) P( k ) ( k ) ϕ. T = ϕ an ε k = y k ϕ k θ est k 1, ( ) ( ) ( ) ( ) Rewriting the previous algorithm an obtain the final implementation of the RLS algorithm [22] [23] can be obtaine as: K θ est( k ) est( k 1) ( k ) ( k ) T 1 ( k ) = P( k 1) ϕ( k ) [ I n + ϕ( k ) P( k 1) ϕ( k )] (4.4) P = = θ + K T [ I K ϕ ] P( k 1) ( k ) n ( k ) ( k ) ε One can notice that this algorithm may also reuire a matrix inversion, but this time the matrix to be inverte is a function of the number of system outputs. In the case of a Single-Input-Single-Output (SISO) system, this matrix is scalar. In the case of interest, the matrix to be inverte will be one with a imension of 2, which is uite simple to invert Moifie RLS algorithm The RLS algorithm as it is presente here is still not suitable for the kin of parameter estimation reuire by this project. The algorithm gives eual weight to all measurements, just like in the original LS algorithm, an the aaptation gain matrix K tens towars the zero matrix as k increases. This comes from the assumption that the parameters are constant for all measurements. The aim of this project is to have an algorithm that can estimate motor parameters an track their variations. The RLS 58

76 algorithm has to be moifie in orer to give higher importance to the most recent measurements, since they correspon to the most recent motor moel parameters. A solution to this problem can be obtaine by changing the initial cost function an introucing an exponential forgetting factor : V θ est 1 = 2 = argθ ( min( V( θ ) ) N T 2 ( y k ϕ k θ ) ( θ ) ( ) ( ) k = 1 λ N k The introuction of λ, which is a real number that has a value between 0 an 1, allows the algorithm to weigh the measurements an give more importance to the most recent ones. If the forgetting factor is eual to 1, we will get the regular RLS algorithm, an if it is set at zero, we will get a simplifie algorithm calle graient algorithm. Usually, values of the forgetting factor range from 0.95 to If the same approach that gave the RLS algorithm (E. (4.4)) is use with this moifie cost function, the following algorithm is obtaine [23]: θ ε = θ ( k ) est( k 1) ( k ) ( k ) = y T ϕ θ ( k ) ( k ) ( k ) est( k 1) K P est = P ϕ [ λ I + T [ I K ϕ ] P( k 1) ( k ) ( k 1) ( k ) n ( k ) ( k 1) ( k ) = + K ( k ) n ( k ) ( k ) ε T ϕ λ P ϕ ] 1 (4.5) The above algorithm is the one that will serve as a basis for parameter estimation for the research project. 59

77 4.4 Estimation Algorithms Overview The parameter vector to be estimate in this project was given in E. (3.1). L L θ = (3.1) r s K T The inuctances can vary because of saturation at a rate that is comparable to that of the machine currents. On the other han, the most important factor that affects the motor phase resistance an the torue constant is the motor temperature, which varies at a much slower rate than the other electromechanical variables in the machine. A ecoupling between the inuctances an the latter two parameters was necessary in orer to minimize the program execution time. A simple approach to the problem of estimating all four parameters in the machine woul be to have a RLS algorithm which woul aim at estimating all four parameters at a fast rate, possibly at the sampling rate of the application. This approach is not possible for a few reasons, one of which is that it woul be a heavy buren on the control program, an another one will be mentione later in the chapter. A more simple approach to the problem woul be to split the estimation algorithm as a function of the ifferent execution rates reuire. A simpler algorithm woul focus only on the estimation of the - an -axes inuctances an run at the sampling rate, with the assumption that the resistance an torue constant remain constant at this rate. The estimation of the remaining two parameters can be one in a separate program that woul be run at a lower freuency. 60

78 Fig. 4.2: Propose algorithm structure. The algorithm chosen for the PM machine parameter estimation has the structure shown in Figure 4.2. It is ecompose into two subroutines, a fast one that estimates the inuctances, an a slower one that estimates all four parameters. A metho where the two pairs of parameters woul be completely ecouple was teste but prove to be unstable; an error in any of the four parameters woul get amplifie by the other estimation program an buil up Fast Estimation Algorithm The goal for this portion of the program is to estimate the machine - an -axes inuctances, assuming the motor phase resistance an torue constant are known. As mentione earlier, this routine will run at the fastest rate possible in the program, which is the sampling rate. The sampling freuency was set to 20 khz for the experimental an simulation setups; this value is commonly use in high performance motor control applications. Because of measurement limitations, the algorithm was evelope from the steay state moel of the machine (the euations are restate here for clarity): 61

79 = + + = e s r T e s i L i r V K i L i r V ω ω ω (2.17) This moel oes not inclue the current ifferential terms because of the limite precision, the signal to noise ratio that was available for the experimental esign, an the high sampling rate. These terms are of the following form: t i L A = For a small t, a small error in i woul lea to a large error in A. On the other han, a larger t woul introuce an unesirable lag in the treatment of the measurements. If they ha been inclue in the experimental algorithm, they woul have introuce a level of error that woul probably outweigh the motivations for their inclusion. Excluing these terms will cause the moel to misrepresent the machine behavior whenever there is a change in machine current. These changes however are usually relatively short but epen on the application. The benefits of incluing these terms will be stuie through simulation. The RLS algorithm is set up as follows. The voltage euations are rewritten in orer to isolate the parameters to be ientifie: = = e s e r T s i L i r V i L K i r V ω ω ω This can then be written as the RLS matrices: = e e s r T s L L i i i r V K i r V 0 0 ω ω ω = s r T s i r V K i r V y ω, an = 0 0 e e T i i ω ω ϕ = L L θ 62

80 The RLS algorithm can be implemente with the above euations an give estimates for the motor inuctances as a function of the inputs an measurements to an from the motor. The only constraint is that the phase resistance an motor torue constant have to be supplie externally. When the machine is at room temperature, these parameters can be set to constant values that are measure offline, but for the final algorithm implementation, these will be supplie from another ientification routine. The program structure is given in Figure 4.3. Fig. 4.3: Fast estimation program structure. The primary concern in terms of numerical implementation was the size of the matrix to be inverte in euation (4.5): λ T I n + ϕ P 1 ϕ ( k ) ( k ) ( k ) In this case, both ϕ an T ϕ are 2x2 matrices, an from this we can euce that P is of imension 2x2. Finally, the ientity matrix must have the same size in orer to be ae properly. The matrix that has to be inverte is of size 2x2, an its inversion is a simple operation: 63

81 1 a b 1 = c a b c c b a This ientification program is suitable to be run continuously an will converge even if the machine stays in one set operating point in terms of currents, voltages an spee. Usually, for the RLS algorithm to converge properly, it nees to be supplie with input an output ata that is rich enough in information. In the present case, the vector that has to be estimate is of size 2, an one operating point gives two euations, one for the -axis, an another for the -axis. There is therefore enough information in one operating point for the RLS algorithm to converge properly. This woul not be true if the vector to be estimate was any larger in size, as it is the case with the secon part of the estimation algorithm Slow Estimation Algorithm The secon part of the complete estimation metho is one that was esigne to be run at a much slower freuency than the sampling rate (500 to 2000 Hz in simulation an experiments). The purpose of this program is to estimate all four parameters using the RLS algorithm. However, the inuctance estimate given by this algorithm is not use in other parts of the program an was introuce in orer to avoi isolating the two sets of machine parameters an improve algorithm stability. This algorithm is also base on the steay state - moel of the machine, but as oppose to the fast ientification program, it will only be run uring steay states. This time the machine moel euations are written as: 64

82 = T s e r e K r L L i i i i V V ω ω ω An the RLS matrices are: = V V y, an = e r e T i i i i ω ω ω ϕ = T s K r L L θ In this case, ϕ is a 4x2 matrix an is a 2x4 matrix, an from this we can euce that P is of imension 2x2. The matrix that has to be inverte for this algorithm will be a matrix of size two, just like in the case of the fast algorithm. On the other han, the algorithm will reuire matrix proucts an sums that will hanle larger matrices than in the case of the fast algorithm, an will conseuently take a longer time to execute. T ϕ The primary constraint associate with the implementation of this RLS algorithm came from the fact that it woul reuire ata from more than one operating point in orer to be stable. Since the parameter vector is of size four an the system is of size two, this algorithm nees at the very least ata from two ifferent operating points in orer to operate properly. On the other han, a esign constraint associate with a practical implementation of the algorithm is that it shoul be unintrusive as possible. In other wors, the estimation algorithm oes not set the machine operating points; they are efine by the application (spee, torue or position control). The solution that was chosen was a compromise: the estimation algorithm will a a perturbation to the -axis 65

83 of the machine an get its ata from the machine feeback. This perturbation was chosen to be on the -axis because it usually is the least torue proucing one, especially in a surface mounte machine or an IPM with low reluctance. In the case of an IPM with very high reluctance, or a synchronous reluctance machine, it may become more interesting to use the -axis. The -axis perturbation will have to be as small as possible, in orer to minimize the torue ripple in the machine, but will have to be large enough to have its effects felt in the controller feeback. It is therefore mostly a function of the harware in terms of the uality of the machine current feeback. The perturbation was chosen to be a succession of small steps at a rather low freuency, leaing to a succession of electrical steay states in the machine. The excitation freuency was chosen to be much lower than the electrical ynamics of the machine, so that the current transients woul not ominate a time step. On the other han, it also ha to be chosen high enough in orer to be able to fee the RLS algorithm with ata that woul allow it to track parameter variations. This minimum magnitue is a irect function of the sensing capabilities of the system (resolution). The perturbation magnitue also has to be fast an small enough in orer to minimize the machine torue ripple an the perturbation in mechanical ynamics. Another esign constraint associate with the implementation of this algorithm is that the injecte current perturbation may also result in an inuctance perturbation, ue to saturation an/or cross saturation. The solution to this problem was to isable the algorithm uring current transients, an to copy the estimate inuctances coming from the fast algorithm when the slow algorithm woul be enable again. 66

84 The program structure associate with the implementation of this algorithm is shown on Figure 4.4. Fig. 4.4: Slow estimation program structure. The algorithm uses the same inputs as the fast estimation algorithm, except for the fact that it reuires upates in its estimate inuctances at the en of current transients. Figure 4.4 shows in etails the interaction between the two algorithms uring a transient. When a current transition occurs in the controller, either ue to a change in controller operating point, or to the perturbation necessary to excite the slow algorithm, the estimation algorithms have to respon accoringly. In the example shown in Figure 4.5, a change in the -axis current was introuce by the controller. The first plot shows how the -axis current reference changes from one level to another, an how the actual - axis current tracks it, this being the result of the operation of the current controller. The secon plot shows the operation of the fast estimation algorithm; it is assume that -axis inuctance value is being effectively tracke initially. When the transient occurs, the ifferential terms start to appear in the machine euations, but are 67

85 not taken into account by the estimation algorithm. This explains why the estimation cannot track the inuctance uring the current transient. Once the current has reache steay state again, the estimation algorithm converges again. During that time however, the slow estimation algorithm was isable, an the estimate resistance an torue constant remaine unchange. This approach is vali because current transients are much faster than temperature transients. Fig. 4.5: Estimation algorithm behavior uring current transient. However, when the slow estimation algorithm resumes its operation, it still has the ol -axis inuctance value in memory, which oes not match the actual machine inuctance anymore. The inuctance coming from the fast algorithm is therefore copie in the slow algorithm when it resumes its operation. This allows the current transient to be almost transparent from the point of view of the slow algorithm. 68

86 4.5 Conclusion The recursive least suares algorithm an its erivation were presente in this chapter. An on-line parameter estimation algorithm was then introuce, base on this algorithm. The propose structure takes avantage of the ifferent ynamics of the parameters that ha to be estimate an is ecompose into three main blocks. One will be eicate to machine inuctance estimation, another one will try to estimate all four machine parameters, an the last block will supervise their operations. In the following chapters, the effectiveness of this propose algorithm will be investigate an verifie both through simulation an experiments. 69

87 CHAPTER V PARAMETER ESTIMATION SIMULATION MODEL The first step in eveloping an valiating an algorithm for on-line parameter estimation in PMSM is to evelop a machine an controller moel that woul be use for simulation. This moel woul serve the purpose of verifying the effectiveness of the propose algorithm in a simulate environment that is as close as possible to the experimental esign. In aition to this, the moel coul also be use to verify an measure variables that may not be easily accessible in the experimental esign. The platform that was chosen to evelop the simulation moel was Matlab, an its extension Simulink. They were chosen because of the simple visual representation of the system they provie, an also because they provie easy access to any moel variable. Designing a moel with Simulink also enables the programmer to visualize the program in ways that are helpful for the final experimental implementation. The simulation moel is ecompose into blocks that can be separate or couple in ways that can reflect the DSP program structure. Newer versions of Simulink can also prouce coe from the moel that is irectly ownloaable into a DSP, making software evelopment much faster. The global moel structure in Matlab/Simulink environment for this project is given in Figure

88 Fig. 5.1: Global controller moel structure. Figure 5.1 shows the ecomposition of the moel into important subsystems. These will be presente in etails next. 5.1 Machine Moel The first step in creating a simulation moel for the whole controller is to esign a moel for the machine behavior. This moel is base on the euations given by the - transformation of a three phase PMSM. This part of the moel calculates the phase currents, the machine torue, spee an rotor position for a given input voltages an loa torue. It is also necessary for the purpose of this project to be able to have the machine parameters be functions of external or internal factors. The phase resistance an torue constant are conseuently chosen to be linear functions of the temperature, an the - an -axes inuctances were mae functions of the machine currents. The temperature is an external variable, an for most of the experiments, it was set to be constant or a slowly increasing linear function of time. 71

89 Fig. 5.2: Machine simulation moel. The machine moel takes three phase voltages as its inputs, performs the Park transformation on them using the rotor position, an calculates the - input voltages. The electromechanical moel block is the one that performs the integration of the - currents as a function all the inputs. Euations (2.6) were aapte for that purpose as: i i = = V V R i R i ω L + ω L L e e L i i K t T ω r t (5.1) In numerical system moeling, it is usually preferable to avoi ifferentiations, which can cause instability an accuracy errors, an hence the electrical euations were written in an integral form. The currents are then use in the torue euation to calculate the machine electromechanical torue. Euation (2.5) was use for that purpose an is given here for the sake of clarity. T e [ mag i + ( L L ) i i 3 P = ϕ ] (2.5) 2 72

90 Both the torue an currents are outputs to this block. The currents are transforme back to the abc omain using the Park inverse transformation an are sent out, because they are measure in the experimental esign an available to the experimental controller. The ifference between machine an loa torue is integrate to get machine spee (Euation (2.6)), which is then integrate to obtain the rotor position, neglecting rotational losses like bearing friction. This position is also use as an output because of the presence of a position sensor in the experimental esign. On the other han, the machine spee is not set to be an output, because there is no tachometer in the experimental esign. The machine spee is estimate from the rotor position feeback information available in the controller. One can also notice the presence of blocks that calculate the values of the machine moel parameters as functions of temperature an currents. The structure of the machine moel is shown in Figure Inverter Moel Basic Inverter Operation The power inverter is the element that converts the small power Pulse With Moulation (PWM) outputs of the microcontroller into higher power signals that are use by the machine. It is also the source of non-iealities an iscrepancies between the esire machine voltages an their actual value. The inverter is a evice that is mae of six switching cells, each of them being subivie into two semiconuctor switches. One of them is a controllable switch, which will turn on an off as a function of its input voltage. These controllable switches can be from ifferent families, like the Insulate Gate Bipolar Transistors (IGBT), the Metal Oxie Semiconuctor Fiel Effect 73

91 Transistors (MOSFET), or the Bipolar Junction Transistors (BJT). They also are usually limite in their operation to one voltage-current uarant: they are uniirectional in both voltage an current. To overcome this limitation, common inverter esigns attach a ioe in parallel to the controllable switch. This ioe is not controllable by any external signal, an its state will be ictate by the external circuit. Fig. 5.3: Three phase brige inverter. Fig. 5.3 shows the circuit use to esign a classical brige inverter structure. The six switching cells (labele 1 to 6) are arrange into three legs (1 an 2, 3 an 4, 5 an 6) an each of these legs is connecte to a machine phase in the experimental esign. Fig. 5.4: Inverter leg. Figure 5.4 can be use to escribe the operation of an ieal inverter leg. The same analysis can be conucte on the complete three phase brige inverter. If we assume the 74

92 current labele I has a return path through other parts of the circuit an that the inverter switches (transistor an ioe) are ieal, the voltage V ph at the output will take values as escribe in Table 5.1. It is necessary here to point out that in such an inverter leg, the top an bottom switching cells are never orere to be on at the same time, since that woul result in a source short circuit. In aition to this, in a typical PMSM application, a configuration where both switching cells are orere off oes not occur. Table 5.1: Operation of ieal inverter leg. Switch commane on Sign(I) V ph 1 + V c 1 - V c One can see from Table 5.1 that if the top cell is commane to be on, the inverter leg will act as if the cell was a close switch; the current sign (irection) only etermines which switch (transistor or ioe) is conucting. Likewise, when the bottom cell is commane to be on, the inverter acts as if the bottom cell was close Inverter Nonlinearities A real inverter, like the one use in the experimental esign associate with this project, iffers to some extent in its operation from its ieal moel. In most motor control applications, these ifferences o not affect the motor control much an they are ignore. However, in the present application, any error between the voltages esire in the 75

93 controller an the voltages applie to the machine will have a irect impact on the parameter estimation performance. This impact will be the subject of a later chapter. The main errors introuce by the inverter will be stuie in etail in the following sections Deatime The switches that compose the inverter cannot change state instantaneously, an in orer to avoi DC bus short circuits, it is necessary to introuce a eatime in the PWM algorithm uring which both switches in transition are orere to be open. This eatime is a elay that is introuce whenever there is a transition between commaning top an bottom switch conuctions. The simultaneous turn-on of both the switches in one leg of the inverter ue to switch non-iealities is prevente by the insertion of this eatime perio, whenever a switch transition comman comes from the PWM algorithm. In the experimental esign, the DSP controls the eatime perio by orering both the switches off for a short preetermine uration. Figure 5.5 shows the transient associate with the turn-on of one of the experimental esign IGBTs. The inverter circuit use in this project was a single integrate circuit, incluing both switches an gate rivers (part #IRAMX16UP60A). It is possible to see that it takes approximately 200 μs before the switch can be consiere on. Dioe conuction transients an IGBT turn-off characteristics also present nonlinear responses. The inverter manufacturer recommens the introuction of a 300 μs 76

94 eatime for safe inverter operation which is fast for an IGBT base inverter. These types of inverters usually reuire eatime elays ranging up to the orer of a microsecon. Fig. 5.5: Inverter IGBT turn-on transient (Junction Temperature = 150 C) [24]. Figure 5.6 shows an example of eatime insertion by the DSP. The ratio between PWM perio an eatime was mae larger in the plots than in the experimental case to be able to visualize it. In the experimental esign the eatime is 160 times smaller than the PWM perio T PWM. The elay t 2 t 1 is eual to t 4 t 3 an they are both eual to the esire eatime. It is also possible to see from Figure 5.6 that the time when the switch 1 is on is ifferent from the time in the ieal case. This error is small in most cases but can become significant when the esire switch on time is small. An important thing to notice for the effect of the eatime elay is that its effect on the output voltage is a function of the phase current. If, in Figure 5.6, the phase current is sufficiently large an positive (flowing towars the machine), the ioe at the bottom of the inverter leg will conuct uring the eatime. On the other han, if the current is 77

95 sufficiently large an negative, the top ioe will conuct uring the eatime. Having a current that is sufficiently large in these cases means that the current is large enough to maintain its polarity throughout the PWM perio. Fig. 5.6: DSP Deatime insertion Switch Conuction Transients Another problem introuce by the real inverter as oppose to the ieal one is also shown in Fig The switch transient behavior iffers from an ieal one where it is assume that the currents an voltages woul instantaneously assume their new values. The values taken by the switch current an voltage uring transient are also sources of iscrepancies between the esire voltage in the controller an the voltage at the output of the inverter. This error is very small an woul reuire extensive knowlege of the inverter switches for any type of compensation to be implemente. It woul also reuire 78

96 an extremely small simulation time step, which woul lea to extremely long simulation times. It is conseuently neglecte in this project Switch Steay State Voltage Drops In the experimental esign for this project, the inverter transistors are IGBTs. This type of transistor presents a voltage rop when it conucts current, an so oes the ioe connecte in parallel with it. These voltage rops change slightly the applie voltage to the machine from the ieal situation. The voltage rops introuce by the inverter switches are not constant, but are functions of the current flowing through the switch at a given time. The voltages applie to the motor phase winings after taking the noniealities into account can be erive accoring to the logic outline in Table 5.2. In the table, V IGBT represents the voltage rop in the IGBT an V D represents the voltage rop in the ioe. Table 5.2: Steay state operation of an actual inverter leg. Switch commane on Sign(I) V ph 1 + V c V IGBT(I) 1 - V c + V D(I) V D(I) V IGBT(I) The switch voltage characteristics were measure experimentally an are compile in Figure

97 Dioe Voltage rop (V) IGBT Voltage rop (V) Dioe current (A) IGBT current (A) Fig. 5.7: Inverter switch voltage rops Simulation Moel The inverter moel use for the simulation moel for this project inclues the eatime elay generator block, an also the steay state switch characteristics shown in Figure 5.7. However, it oes not inclue the switch transient behavior, because it woul reuire a much longer simulation time for results that woul not be affecte noticeably. On the same note, eatime was not always inclue in the simulations, because it was effectively compensate for in the experimental esign, an it woul also reuire a smaller than practical time step in orer to be simulate effectively. The eatime impose in the experimental esign was 300 ns, an the time step that was esire for the simulations was 400 ns, which is a compromise between simulation spee an result precision. From the point of view of the controller, the existence of the eatime is 80

98 transparent, since it was compensate for in the PWM block, for the controller the eatime it was as if it i not exist. The switch voltage rops were also compensate for in the experimental esign, an hence, were not inclue in simulation results shown in Chapter VI. Both the eatime an switch voltage rops compensation algorithms will be explaine in etails in Chapter VII. Initial simulations for this project were one with an ieal inverter because the effect of the inverter nonlinearities was assume to be negligible. However, experiments showe that the errors introuce were uite large, sometimes even comparable to values of certain terms in the voltage euations that the estimation algorithms ha to estimate. The effect of the inverter non-linearities on the machine applie voltage were simulate using the evelope simulation moel. Figures 5.8 an 5.9 show the result of a low spee test over one mechanical revolution where the control algorithm fixe the - input voltages of the inverter. The DC bus voltage was set at 50 V, because the experimental results that were obtaine for similar results were one with a voltage aroun this value. Figure 5.8 shows the results obtaine with an ieal inverter configuration, where the eatime elay is set to zero, an where the switch voltage rops are also set to zero. The results seen in that figure are the ones that we woul like to achieve in the experimental esign, because they correspon to a case where the inverter is almost transparent from the point of view of the controller. It is necessary to bear in min however, that even an ieal inverter will introuce perturbations in the machine phase voltages, because of the lagging response of the 81

99 PWM, an because of the high freuency content of the suare wave signals that are applie to the machine. In most cases however these affects can be neglecte. 10 V (A) 5 0 V V I (A) 0 I I Ia (A) Time (s) Fig. 5.8: Simulate ieal inverter response to constant - voltage inputs. Figure 5.8 shows the - voltages an currents in the machine, as well as the phase a current. The - voltages were not irectly accessible as a continuous signal because of the nature of the inverter output voltages. They were obtaine by filtering the output of the Park transformation at the input of the machine moel. The plot that shows the - voltages of the machine also shows the reference voltages at the input of the controller, which were constant at: 82

100 V = 8V V = 4V However, it is ifficult to istinguish them in Figure 5.8 because they match closely. Fig. 5.9: Simulate real inverter response to constant - voltage inputs. In Figure 5.9, the simulation moel of the inverter inclue both the eatime elay, an the steay state switch voltage rops. One can see significant changes on each of the three plots from the ieal case shown in Figure 5.8. In terms of the - voltages of the machine, it is possible to see both a istortion of the signals from their reference, an a change in their average level. Aitional simulations were run in orer to isolate the 83

101 problems, an showe that most of the istortions were ue to the eatime, because the switch voltage rops ten to cause a constant rop in the - voltages. These results show a significant problem from the controller point of view: while the controller thinks it is sening the reference - voltages (V = -4V, V = 8V), the inverter is only applying (V = -3.5V, V = 7V), an the machine feeback is a response to this latter pair. This error, if not compensate or taken into account by the estimation algorithms, can have an enormous impact on the performance of the controller. In the case of the - currents, the values obtaine are slightly ifferent from the ones in the ieal case, an the oscillations in the machine voltages have an impact on the current waveforms. Another way to look at the istortion cause in the current waveforms is to look irectly at the phase currents; phase a current was shown in both figures 5.8 an 5.9. With the realistic inverter results, we can see some small istortions in the current waveform, mostly aroun zero current, from the waveform obtaine in the ieal case, which was sinusoial. The istortion aroun zero comes from the fact that in the case of small currents, the effect of the eatime on the controller voltage changes because the phase current get small enough to allow the ioe to reach an open state or even change the conucting ioe uring the PWM perio. On the other han, when the current is sufficiently large, the effect of eatime is easy to estimate because the ioe remains in a known state of conuction. The effect of the eatime for large currents is to introuce a voltage error whose polarity is a function of the sign of the current, because of the ioe that conucts uring the eatime elay. This is not the case when the current gets smaller an the istortion comes from the change of behavior of the perturbation. 84

102 It is also necessary to note that whereas the voltage isturbance ue to the switch voltage rops is a function of the current flowing through them, the eatime effect, except for smaller currents, is a function of the switch uty cycles. Since the eatime is a fixe elay, the ratio between the on time of the switch an the esire on time is a function of the latter. The impact of the eatime elay for larger currents will be felt mostly in the case of smaller uty cycles. A series of tests was one in orer to estimate an moel the error in the controller voltage introuce by the inverter. The experimental approach chosen was esigne to use the experimental machine in the simplest way. The machine was connecte to the inverter an the controller applie ifferent sets of constant -axis input voltages (the -axis voltage remaining at zero). The machine rotor was locke an current measurements were taken in steay state. In these conitions, the machine electrical moel reuces to: V V = rs i = r i s The tests were conucte with the machine at room temperature an were complete within a short time, so that we coul assume the machine temperature i not change significantly. As a result, it was possible to assume the stator resistance remaine constant at a value that was measure offline. The error introuce by the inverter was then estimate by taking the ifference between the controller voltage an the estimate resistive rop. The results shown in the lower part of Figure 5.10 (Figure 5.10(b)) show that the error introuce by the inverter for the locke position an the type of voltage use in the 85

103 experiment is pretty much constant. Theoretically, there shoul be a small increase in the error as the current gets larger with the switch characteristics shown in Figure 5.7, but the noise level was too high for it to appear clearly. These results show, however, that the error introuce by the inverter is uite significant an reuires compensation. A similar plot was obtaine in the case of the -axis. Fig. 5.10: Experimental inverter response with locke rotor. 5.3 PWM Algorithm The PWM algorithm block performs the transformation from the esire - machine voltages to the inverter switch uty cycles. The uty cycle for a power electronics switch is the ratio between its on time an the switching perio. This is the first block of the simulation moel that is part of the machine controller, which is not run 86

104 continuously at the simulation time step. In the experimental esign, the various segments of the control program are either run perioically or on the occurrence of certain events. The controller simulation moel is execute at a freuency matching the experimental esign, in orer to represent the real situation more accurately. In the project, the PWM freuency was set at 20 khz, an the simulate PWM was execute at the same rate. For comparison, the simulation freuency (inverse of time step), which is use by the machine an inverter blocks as real time, was 2.5 MHz. In aition to being more realistic, having portions of the simulation moel execute at lower freuencies allows the complete moel to run much faster than it woul if everything was execute at each time step. The inverse Park transformation is the first step in the implementation of the conversion - reference voltages into uty cycles. The transformation is obtaine from euation (2.3): V V V a b c = sin sin sin( θ r ) cos( θ r ) ( θ r 2 π 3) cos( θ r 2 π 3) ( θ + 2 π 3) cos( θ + 2 π 3) r r V V (5.2) The next step is to scale the three phase voltages in orer to obtain the correct uty cycles at the inverter input. The solution chosen for this project consists in scaling the three esire phase voltages with respect to the available DC voltage at the source of the inverter, an then creating a virtual zero for a uty cycle of 50%. As a conseuence, negative voltages will result in uty cycles that are smaller than 50%, an positive voltages give uty cycles larger than this value. There are two immeiate conseuences of using this algorithm: First, a strong homopolar voltage component is applie to the 87

105 machine stator (which oes not affect operation or performance), an the secon is that the maximum achievable phase voltages are half the DC bus voltage. On the other han, using this metho makes it easier to compensate switch non-iealities. Another metho was also consiere, which was base on the use of the smallest of the three phase voltages as a fictitious reference an expresses the other two phase voltages as ifferences between their initial values an the new reference. This algorithm uses phase to phase voltages instea of iniviual phase voltages. Let us see on an example how both these algorithms woul operate on a set of input voltages to obtain the uty cycles abc. For Va = 10V Vb = 5V V c = 15V an V DC = 100V The first algorithm gives a b c = = = ( Va VDC ) ( Vb VDC ) ( 0.5 V V ) c DC an finally a = b = c = ( ) ( ) ( ) = 60% = 55% = 35% An the secon PWM algorithm gives V min = V a = a = a = c = 15V ( Va Vmin ) VDC = ( 10 ( 15) ) ( Vb Vmin ) VDC = ( 5 ( 15) ) ( V V ) V = ( 15 ( 15) ) c min DC 100 = 25% 100 = 20% 100 = 0% The immeiate avantage that one can see in using the secon algorithm is that it always keeps one of the three phases at a zero uty cycle, which means the corresponing phase voltage remains unchange for the PWM perio. This helps minimizing the 88

106 switching losses in the inverter. The switching losses are a conseuence of the non-ieal behavior of the inverter switches epicte in Figure 5.5: uring a switching transient, the prouct between voltage an current in the switch becomes nonzero an translates into heat losses. By switching only two phases out of the three, one reuces significantly these losses. On the other han, maintaining a phase at a zero uty cycle makes it impossible for the voltage compensation algorithm to operate optimally. The first algorithm has been chosen, since aeuation between machine moel an machine response is one of the primary concerns of this project. The inverter compensation algorithms have been implemente insie the PWM moel block to achieve a proper match between the controller voltages an the inverter outputs. 5.4 Current Controller The current controller moel block calculates the - voltages reuire to obtain the esire levels in - currents an is similar to the experimental DSP program. This segment of the program is execute at the PWM freuency of 20 khz which is the same as in the PWM block. The first step in achieving current control is to have aeuate feeback. The currents that are measure in the experimental esign are not the - currents, but usually two of the abc phase currents. It is therefore necessary to perform the Park transformation on them in orer to obtain the corresponing - currents. We can use euations (2.2) for this purpose: 89

107 ( ) ( ) ( ) ( ) ( ) ( ) + + = c b a r r r r r r i i i i i 3 2 cos 3 2 cos cos 3 2 sin 3 2 sin sin 3 2 π θ π θ θ π θ π θ θ (5.3) The simulation moel also incorporates the precision of the Analog-to-Digital Converter (ADC) of the DSP at this point, which is 12 bits in theory. However, measurement noise an the real precision of the ADC reuce the available resolution to about 9-10 bits. This effect was not simulate except through a ecrease in simulate ADC resolution. The next task for the current controller is to extract the comman voltages reuire to establish the esire machine currents. The controller algorithm uses the machine electrical moel along with the current feebacks. The transfer functions that escribe the operation of the current controller with feeforwar have alreay been presente an are repeate here for the sake of convenience: + = + = s s L p r V i L p r V i 1 1 ' ' where + = + = + = = s e s T r e i L p i r i L V V i L p i r K i L V V ω ω ω ' ' The error between the - reference currents an the feeback is sent to a Proportional an Integral (PI) controller, whose output is shifte by the feeforwar terms so that the system is linear from the controller s perspective. 90

108 5.5 Spee Estimation The machine spee is an important variable in the program, but is not reaily available to the controller. The position signal given by the optical encoer nees to be integrate to obtain the spee information. This sensor provies a relative rotor position with a precision of 4096 counts per mechanical revolution, or 2048 counts per electrical revolution, since the machine has four rotor poles. To obtain the motor spee, the program subtracts the current position with the one it ha uring its last execution, an applies a low-pass filter to the result. The filter pole is ajuste to get a compromise between ripple an response spee. 5.6 Fast Ientification Algorithm Another program that is execute at the sampling rate is the fast parameter estimation algorithm. The algorithm uses the - current feeback calculate by the Park transformation an the output voltages sent to the inverter to estimate the - machine inuctances. Two other parameters, K T an r s are reuire in this fast algorithm. The operation of this algorithm an the euations on which it is base have alreay been presente in Chapter IV. The fast ientification algorithm is inclue in the current control block of the global simulation moel for convenience. 5.7 Low Freuency Controller The last block that is a part of both the simulation moel an the experimental program runs non time-critical tasks an executes at a slower rate than the sampling freuency. The cycle freuency for this block was chosen to be 2 khz. 91

109 The spee controller has been inclue in this block, because the mechanical ynamics are much slower than electrical ynamics. The spee controller uses a PI compensator on the error between the estimate spee feeback an the esire reference spee, giving the reference magnitue of the stator current. This magnitue is then converte into the - currents by using either the maximum Torue per Ampere algorithm, or the zero -axis current algorithm, both of which were presente in Chapter II. It is to be note that the spee controller is to be use only when the application emans it. Other cases exist where the parameter to be controlle is the machine torue or rotor position. Another program that was execute at the lower freuency is the slow ientification algorithm, which is mostly use to track the variations in the stator resistance an the machine torue constant. The euations an operation of this slow ientification algorithm were iscusse in Chapter IV. 5.8 Conclusion This chapter has introuce the reaer with the simulation moel evelope for this project. The various blocks of the moel, an the constraints present an necessary to be moele were iscusse. Chapter VI will focus on how this simulation moel was use to evelop an verify the parameter estimation algorithm in PMSM that is the primary objective of this research. 92

110 CHAPTER VI PARAMETER ESTIMATION SIMULATION RESULTS The first step in valiating the propose algorithm was to verify its effectiveness using the simulation moel presente in the previous chapter. For this purpose, the algorithm was teste on two ifferent machine moels. One of these moels matches the motor that was use experimentally for this project. A problem that was encountere with this machine is that it oes not show significant saturation characteristics, an as a conseuence it coul not be use to show some of the results that motivate this research. The secon motor moel chosen isplaye significant inuctance saturation at higher torue levels; it was obtaine from the motor ata use in [19]. This latter motor was selecte for simulation to show the possible effects of saturation on the maximum Torue per Ampere algorithm in Chapter III. 6.1 Machine Moels In this section the machine moel parameters that were use for simulation will be presente. The parameters of the two machines are shown in Table 6.1. The machine A has been use experimentally to emonstrate the functionality of the evelope algorithm. Parameter variations ue to temperature were introuce as variations from the nominal values given in Table

111 Parameter variation ue to saturation was introuce by changing the - inuctance values in the program as a function of the machine currents. Table 6.1: Simulation moel parameters. Parameter name Machine A Machine B Stator resistance, r s (Ω) Torue constant, K T (V.s) L (mh) no saturation L (mh) no saturation Rotor inertia, J (N.m.s 2 ) 46.1e e-6 Number of magnet pole 2 2 pairs, P Rate current, I max (A) Machine A Inuctances In the case of the experimental machine, the inuctance waveforms were measure offline by taking steay state measurements for a variety of operating points an extracting the corresponing inuctance values from the moel euations. These tests were one with the machine at room temperature so that the resistance an torue constant coul be assume to be at their nominal values: These parameter values were also measure with other tests. The phase resistance was measure irectly with a multimeter, an the torue constant was obtaine from an open-circuit test escribe in section of Chapter VII. The mechanical loa in the experimental esign was a controlle brake, that was not able to prouce motoring torue on the motor shaft. Because of this it was not possible to o the tests with the machine -axis current set to 94

112 zero since they woul not prouce any motoring torue from the machine, an the motor spee woul be zero. A non-zero spee is reuire for the inuctance terms to appear in the steay state moel euations. The -axis current is reuire to get motoring torue (see euation (2.5)). For a given electrical operating point, the machine was run at a constant spee in steay state an the controller recore a large number of ata points (compensate comman voltages an measure currents). An average of these points was one in orer to improve precision, an the steay state moel of the machine was use to extract the machine inuctances from euation (2.17), which is state again below for convenience: V = r s V i = r + K s i T ω + L r L ω i e ω i e (2.17) Which gives L L V = rs i V = ωe i r i K s ω i e T ω r. Results were only obtaine for a negative -axis an a positive -axis current because positive -axis currents are never use in practice for motoring applications (they generate negative torue) an the uarant that correspons to the negative -axis current is symmetrical to the positive sie. The values of the inuctance for the applie - axes currents are given in Tables 6.2 an

113 Table 6.2: Machine A -axis inuctance in mh. i \ -i (A) Table 6.3: Machine A -axis inuctance in mh. i \ -i (A) In tables 6.2 an 6.3, the values in bol correspon to inuctance values that were extrapolate (clampe) from neighboring results. The reason for extrapolation is that for small values of current for both the - an - axes, the inuctance term is too small to overcome the feeback noise. Other measurements that correspon to the largest values of - an - axes currents were not obtaine because they were too close to the machine ratings. The corresponing surface plots are shown in figures 6.1 an

114 Fig. 6.1: Machine A -axis inuctance. Fig. 6.2: Machine A -axis inuctance. 97

115 The measurements one to get the - an -axes inuctances show the presence of cross-coupling between the two axes. Each of the two inuctances is a function of both the -axis an the -axis currents. For a constant -axis current, the -axis inuctance is ecreasing as a function of the -axis current because of magnetic saturation in the corresponing iron path. In the case of a constant -axis current, the -axis inuctance is increasing as a function of the -axis current magnitue. The -axis flux works against the magnet flux as the -axis current becomes negative, thereby reucing the global amount of flux in the machine an saturation, an hence the inuctance increases. This epenency of the -axis inuctance on -axis current is referre to as cross-coupling phenomenon. The -axis inuctance plots can also be explaine similarly accounting for the cross-coupling effects. Increasing the -axis current magnitue reuces the -axis flux for a constant -axis current, an conseuently the corresponing inuctance gets closer to its linear region value an increases in value. Reucing the magnitue of the -axis current for a constant -axis current reuces the -axis flux an the cross coupling flux, which increases the -axis inuctance Machine B Inuctances In the case of machine B, the ata available provie only the -axis inuctance as a function of the -axis current. The -axis inuctance waveform was provie only for positive -axis currents, which have not been use in this project. The author of the paper [19] state that the -axis inuctance i not vary much an we mae the 98

116 assumption that the -axis inuctance was constant at the value given for a zero -axis current. The inuctance characteristics of machine B are shown in Chapter III, in Figure 3.5. This machine shows a more significant saliency ratio than machine A. As a conseuence, the saliency torue proucing term will be more important to the controller an the machine will be more suitable for high spee operation. However, this motor also shows a more significant saturation characteristic than the other one an a conseuence of this is that motor rive performance may get egrae if it is not taken into account. Figure 3.2 also shows a region, where the -axis inuctance coul get smaller than the - axis inuctance, making the reluctance torue zero or even negative. 6.2 Novel Parameter Estimation Algorithm Algorithm Structure The novel parameter estimation algorithm propose for this research has a moular structure. The algorithm can be ivie into three main components: The first one is the fast ientification algorithm. It is a part of the program that runs continuously an provies estimates of the machine inuctances from the other two parameters an the machine inputs an feeback. This program is nonintrusive, an coul also be use inepenently of the rest of the program if it were provie with estimates of the stator resistance an the torue constant. The secon part is the one referre to as the slow ientification algorithm. As oppose to the previous program, this one is more emaning in terms of inputs. The primary factor for its effective execution is that it nees to be supplie with 99

117 ata that correspons to steay state operating points. An error in its inputs coul have important repercussions consiering its slow rate of execution. Another constraint associate with this program is the necessity to provie it with ata that is rich enough with information about the system. The ata for the slow algorithm was enriche with a small perturbation signal was introuce on the - axis. This program can therefore be seen as intrusive in the controller operation. However, consiering the very slow rate of change of the stator resistance an torue constant, it is conceivable to isable the algorithm for long perios of time an only activate it when new estimates for these two parameters are reuire. The thir an last component of the propose parameter estimation algorithm is one that controls the interactions between the first two programs an also between the motor rive an them. This block etects when the motor rive is operating in steay state an enables or isables the slow ientification program. It also controls the times at which the fast program will upate the parameters in the slow one. This block also generates the small -axis perturbation use by the slow algorithm. Figure 6.3 shows a clearer picture of the interaction between the three blocks. Figure 6.3 shows a typical machine startup operation with machine B. The first plot shows the machine - currents as a function of time for a perio of one secon. The -axis current is set at a constant value, which is the case in most current or spee control applications in steay state. The -axis current, on the other han, shows small step variations aroun the set point. These are the conseuence of the small perturbation 100

118 injecte in the -axis of the controller. The secon plot shows a closer look on the -axis current, an also shows the signal that enables or isables the slow estimation algorithm. When this signal is at 1, the slow algorithm is enable an will run continuously, though at a slow rate. Fig. 6.3: Example of interaction between elements of algorithm. It can be observe that this Enable signal oes not really correspon to times where the machine is in steay state. For example, between the time 0.82 an the time at which the enable signal turns to 1, the machine currents can be consiere constant an the machine coul be consiere to be in steay state. The reason for this is that enabling the slow algorithm for a too long perio of time on the information of a single operating point will ecrease the richness of its input an may lea to instability of the 101

119 algorithm. In this example, the slow algorithm is execute about 10 times uring an enable perio (for a 500 Hz execution loop) Supervising Program The program that ecies whether or not to enable the slow ientification program operates by using both the current references in the controller an the current feeback. In all cases, if the current feeback shows a change in the - currents that is larger than a threshol, the algorithm will be isable. The avantage in using the current references is that they usually lea the actual machine currents an allow the controller to isable the slow algorithm before it is too late. Another avantage is that the perturbation that is use in the -axis is known an its transitions can be monitore precisely. Another feature that is inclue in this portion of the program is the introuction of a elay when the isabling of the parameter estimation algorithm is triggere by the reference current. This elay allows the user to control the winow uring which the ientification program is active. The appropriate length of this winow an the freuency of the -axis perturbation will vary from motor rive to motor rive, mainly as a function of the RLS forgetting factor. Another feature of the supervising program concerns the monitoring of times when the parameter estimates coming from the fast algorithm are copie into the slow one. This aspect was introuce to solve the following problem: when there is a large change in the operating point of the machine, the inuctances may suenly vary epening on the amount of saturation. The fast algorithm will follow accurately this possible change, an the slow algorithm is isable uring that time. However, when 102

120 the latter program resumes its operation, a large error woul have been introuce in its estimation an the program will have to reconverge, possibly causing transient errors in the other parameters. Such transients are usually fast compare to the mechanical an temperature ynamics in the machine. It is conseuently a vali assumption to assume the stator resistance an the torue constant o not change uring a current transient. The fast algorithm shoul operate properly given the set of machine parameters at the beginning of the transient an for its uration. On the other han, from the slow algorithm s point of view, its parameter estimates after resuming operation shoul be a closer match than if its execution ha not been isable. Combining the two algorithms in the way escribe above allows the parameter estimation algorithm to track fast inuctance changes without affecting the estimation of the other two machine parameters. An example of the operation uring an inuctance step change with machine B can be seen in Figure 6.4. In this figure, the top an mile plots show the -axis inuctance an its controller estimate. The top plot shows a larger time scale than the mile, which focuses on a step change in the inuctance, which was itself cause by a step change in the -axis current from 12 A to 8 A an back. In the mile plot, the machine inuctance matches the controller fast estimate, labele Lf, an the slow algorithm estimate is labele Ls. This nomenclature will appear throughout this chapter an is also use for the -axis inuctances. 103

121 Fig. 6.4: Simulation example of inuctance step change. The bottom plot of Fig. 6.4 shows the machine torue constant ( Kt ) an its estimate, labele Kt est. This plot emonstrates that the estimation of the torue constant is not affecte by the transient. The operation of the supervising algorithm can be seen from the mile plot. At the beginning of the transient, the slow parameter estimation is isable an the corresponing parameter estimates are frozen. The fast estimation algorithm follows the variation in inuctance effectively, an at the en of the transition (6.4s), the slow algorithm is enable again an uses the new fast inuctance estimates instea of the values it was using before the transient. For comparison, Figure 6.5 shows simulation for the same conitions as in Figure 6.4, but with the fast inuctance copy to the slow algorithm isable. Before the transient, the results in both figures match closely. When the inuctance step occurs, the slow ientification algorithm is isable an operation is the same as in the previous 104

122 case, but as soon as the algorithm resumes its estimation (at about 6.4s), the error cause by the mismatch between the slow algorithm inuctance estimate an the correct estimate provie by the fast algorithm causes a large error in the torue constant estimate. It can also be observe that the error in the resistance an torue constant estimates affects the fast algorithm by looking at the fast estimate after the problem starts. The algorithms are able to converge after some time, but such an operation an parameter error has averse effects on the torue an current controllers. Fig. 6.5: Inuctance step change without fast inuctance copy. 6.3 Effects of Neglecting Differential Terms It has been mentione in Chapter IV that the fast RLS algorithm implemente in the experimental esign associate with this project oes not inclue the ifferential terms from the electrical moel euations. 105

123 These terms are of the following form: i A = L t an i B = L t These terms are present on both the - an -axes. This omission is euivalent to making the assumption that the algorithm is execute at steay state. However, whether it is cause by the -axis perturbation or a fast change in operating point, the fast ientification algorithm will at times be execute in a case where the ifferential terms are not zero. The effect of such an omission is presente in Figure 6.6, where results from the fast algorithm are obtaine in the case where the ifferential terms are neglecte an in the case where they are not. These terms coul easily be accounte for with a ifferent an improve experimental setup, but in the present implementation the noise level increases the errors in estimation when these terms are inclue. The conclusion that can be rawn from Figure 6.6 is that the fast algorithm is affecte by the exclusion of the ifferential terms. The small pikes observe in the estimate -axis inuctance show this. On the other han, as soon as the transient ens, the ientification algorithm converges again to the same value as in the case where the ifferential terms were inclue. Special care shoul be taken uring transient operation an when results are copie from the fast algorithm into the slow one in the case of an implementation without the ifferential terms. The effect of ifferential terms coul also be low-pass filtere to extract the esire signal. 106

124 Fig. 6.6: Comparison between inclusion an exclusion of ifferential term. 6.4 Sensitivity Analysis of Fast Algorithm Estimates The performance of the fast ientification algorithm is iscusse in this section. In contrast to the slow algorithm, if the fast algorithm is provie with accurate estimates of the stator resistance an the torue constant, it will be able to provie accurate estimates of the machine inuctances. The analysis in this section is presente with the assumption that there are no errors in the machine currents. This is a vali assumption since in the experimental esign the currents are irectly measure an except for their average signal to noise ratio, these measurements are accurate. 107

125 V = r Let us consier the machine electrical euations in steay state: s V i = r + K s i T ω + L r L ω i e ω i e The following expressions can be extracte from the above euation L L V = rs i V = ωe i r i K s ω i e T ω r In the context of the parameter estimation algorithm, conitions are sought for a set of four parameters that will verify these euations. If a fixe operating point is consiere such that the inuctances are constant, the estimate inuctances will satisfy the following relationship L est L est V = ctrl r = sest r i V ωe i i K sest ω i e ctrl T est ω r (6.1) In these euations, the subscript est enotes an estimate of a parameter, an the subscript ctrl refers to variables as they are thought to be by the controller. As shown in Chapter IV, there can be a mismatch between the esire machine voltages in the controller an the voltages at the output of the inverter because of its non-iealities. In the experimental program, this error is compensate using the metho that will be escribe in Chapter VII, but its effects will be investigate here. For the operating point uner consieration one can obtain L est L = r est sest r sest L + K est T est K T est L + L est est L est L + V est ctrl V ctrl L + V est ctrl V ctrl 108

126 an L est L = r est sest r sest L + K est T est K T est L + L est est L est L + V est ctrl V ctrl L + V est ctrl V ctrl E. (6.1) gives L est i = ω i e r sest 1 ω i e V ctrl an L est i = ω i e r sest 1 i K T est 1 + ω i e V ctrl In the experimental case, the inverter voltage error is compensate for an there is an accurate match between the controller an machine voltages. In the en, the previous sensitivity euations reuce to: L est i = ω i e r sest L est i = ω i e r sest 1 i K T est Both the inuctance estimates will be affecte by an error in the stator resistance. The impact of error on the -axis inuctance will be smaller for operating points that present a large -axis current, high spee an low -axis current. Usually, in IPMs the high spee operation translates into flux weakening operation, which reuces the -axis current an increases the -axis one. In the case of a surface mount machine, the -axis current is set to zero for operation below base spee, which minimizes the resistance error impact. On the other han, the -axis inuctance estimation reuires low -axis an 109

127 high -axis currents an spee to minimize the error. Operation in the flux weakening region will conseuently be best suite for this estimate. An important problem associate with the -axis inuctance estimation is the impact that an error in the torue constant can have on its accuracy. The corresponing sensitivity term is only a function of the -axis current an can be uite large. The conclusion of this analysis is that the estimation of the -axis inuctance will be more sensitive to errors in the other parameter estimates than the estimation of the - axis inuctance. Conseuently, this estimation will probably be of poorer uality than the latter. However, it is common in permanent magnet machines to have a much lower - axis inuctance variation ue to saturation than on the -axis an the -axis in such cases coul be consiere constant for this problem. Such a solution can also be consiere for cases where the -axis current is too small to extract the -axis inuctance, in low torue an spee for an IPM or below base spee operation for a surface mount machine. In this case, the -axis inuctance is consiere constant at a value that can be measure offline, an the fast estimation algorithm can be rewritten to only estimate the -axis inuctance. Figure 6.7 shows a simple representation of the torue-spee characteristic of a PM machine. The space within the envelope of the characteristic can be ivie into four regions epening on the performance of the parameter estimation algorithm. The primary concern in estimating the machine inuctance is that the corresponing term in the electrical euation, L ω i has to be measurable. For this e reason, if the currents or the machine spee are too small, the inuctance cannot be extracte from the electrical moel. In the case of low torue operation, the controller can 110

128 use the inuctance values given by the manufacturer or values measure offline. On the other han, operation at low spee, whether it is at low or high torue, is not suitable for this algorithm. Fig. 6.7: Torue-spee regions for parameter estimation. A ifferent metho that relies on the ifferential terms in the electrical euation coul be implemente, but woul be very ifficult to implement experimentally in a nonintrusive manner because of the noise associate with estimation of the ifferential terms. Table 6.4: Performance of parameter estimation in Torue-spee regions. Region Surface mount PMSM IPM 1 Spee is too low for operation Spee is too low for operation 2 L unimportant Average performance for L est Goo performance for L est 3 Average performance for L est Goo performance for L est Better performance for L est Average performance for L est Average performance for L est 4 - currents are too small - currents are too small 111

129 The ifference between surface mount an IPM machines in region 3 of Figure 6.7 is ue to the fact that usually IPM machines reuire a larger -axis current than surface mount PM machines. 6.5 Algorithm Initialization The RLS algorithm nees to be properly initialize when the machine starts to achieve maximum stability of the system. At machine startup, the assumption is that the machine is at room temperature, so that the stator resistance an torue constant are at their nominal values. This oes not necessarily reuire offline measurements, because these values are usually given by the manufacturer. Offline measurements woul be more accurate since these parameters can change slightly from one machine to another. On the other han, the information of the - inuctances is selom available an it was assume that only a rough estimate was available. The first step in starting up the parameter estimation algorithm is to make sure that the fast ientification program has converge before the other algorithm is starte. Initially, only the fast algorithm will be run, after startup, an will use the room temperature estimates of the resistance an torue constant. After the fast algorithm has converge an steay state has been reache, the slow algorithm begins its operation using the estimates from the fast algorithm as its initial values for inuctance, an the complete algorithm gets unerway. An example of initial convergence of the algorithm can be seen in Figure 6.8. The top plot shows the - inuctances an their estimates. The fast estimation algorithm is 112

130 enable at 0.2s an converges towars the correct value for the -axis inuctance very uickly. Fig. 6.8: Initial parameter estimation convergence. The slow algorithm starts its operation at 0.5s, an uses the results from the fast estimation algorithm. After that point, the three waveforms are iniscernible. In the case of the -axis inuctance, the same conclusions apply (the result from the slow estimation program was omitte for clarity) except for the fact that the estimate appears to be more noisy an has a small error until the slow algorithm starts. This is consistent with our previous analysis, which state that the -axis inuctance estimate was more likely to be poor than the -axis inuctance estimate. In this simulation, the inuctances were initialize with values matching the machine inuctances when there is no current 113

131 (nominal values). There was therefore an error present when the fast algorithm was first enable, which was reuce in a few cycles. One can also notice the small step waveform that the -axis inuctance has; this is ue to the introuction of the -axis perturbation necessary for the slow algorithm to converge. The mile an bottom plots of Figure 6.8 show the stator resistance an the torue constant along with their estimates. Just like in the first plot, the execution of the parameter ientification program that estimates them oes not start until 0.5s. A small steay state error is present after the initial transient. The small oscillations before the final convergence at 3 s o not affect the algorithm stability. These are ue to the limite simulate precision in controller measurements. Another test was conucte with the same conitions as that of the first test except that the initial conitions on the stator resistance an the torue constant were off by 20%. These results are shown on Figure 6.9. The objective of these results was to show that the algorithm coul converge even if the initial guesses for the machine parameters were wrong. This coul happen if the machine was starte without being at room temperature, for example after a long high torue operation. The parameters take a longer time than in Figure 6.8 to converge, but they o so uite effectively. It is possible to note that the error in the stator resistance reflects irectly in the error in -axis inuctance (similar waveform). On the other han, both the error in resistance an torue constant affect the -axis inuctance, with the initial 20% error causing more than 300% error in the inuctance estimate. This numerical property of the -axis inuctance estimate will efinitely be a problem in some cases. Note that parameter estimates are 114

132 clampe for increase stability an values that woul threaten system stability shoul not be allowe. The results shown in Figure 6.9 also inicate that the algorithm may be able to run intermittently. In other wors, to minimize the intrusive aspect of the slow ientification program knowing that the stator resistance an the torue constant change slowly as a function of time, it may be possible to run this algorithm for limite perios in time. This allows the algorithm to reconverge to possibly new parameter values in a manner such as the one seen in Figure 6.9 if parameters neee to be upate significantly. On the other han, the execution of the fast algorithm is non-intrusive an shoul be execute at all times. Fig. 6.9: Initial parameter convergence with 20% initial error. 115

133 6.6 Effect of Cross-Saturation Cross-saturation is a phenomenon that makes both the - an -axes inuctances functions of both - an -axes currents. In other wors, the magnetic flux associate with magnetic paths aligne with the rotor permanent magnets affects the magnetic flux that is in uarature with it, an vice versa. Cross-saturation surfaces were measure experimentally for machine a an the results were shown in figures 6.1 an 6.2. The cross-saturation has one averse effect with regars to the task of parameter estimation: the -axis perturbation, which in the case where no cross saturation is present affects only the -axis inuctance, may now affect both - an -axes inuctances. In most cases, however, the -axis inuctance variation is small, but the introuction of cross-saturation may affect the -axis inuctance significantly. Fig. 6.10: Cross-saturation effect on machine inuctances. 116

134 Figure 6.10 shows the effect of cross-saturation on both the - an -axes inuctances in the context of execution of the parameter estimation algorithm. A variation of about 25% in the -axis current is the cause of a variation of 2% in -axis inuctance, an about 4% in -axis inuctance. Although these changes are not significant, they cause the system to have ifferent inuctance parameters at each step of the perturbation, as oppose to the normal case, where only the -axis inuctance is affecte. Fig. 6.11: Parameter estimation results with cross-saturation. Parameter estimation results are shown in Figure 6.11 in the case where saturation is present in machine A. The -axis inuctance, stator resistance an torue constant have fairly accurate estimates, but the -axis inuctance estimate is of poor uality. This is because the -axis inuctance is also a function of the -axis current, which is uite 117

135 large in this case. Having a large -axis current increases the amount of flux present in the stator iron an conseuently increases saturation. A conseuence of this is that the - axis inuctance becomes smaller than in the case where the -axis current is zero (no cross saturation case). Having a smaller -axis inuctance term will make the corresponing inuctive rop smaller in the electrical moel euations, an more ifficult to estimate. Another reason for the poor estimate is that the small inuctance changes measure by the fast inuctance algorithm are not copie into the slow one at every cycle, since it woul nullify the inuctance estimation capability of the latter program, an make the overall algorithm less stable. The solution aopte in the experimental case is to set the -axis inuctance to be constant, since small errors in its estimation have negligible effect on the other parameter estimates. Fig. 6.12: Parameter estimation results with no cross-saturation. 118

136 For comparison purposes, simulation results obtaine without cross-saturation in machine A are shown in Figure In this case, the -axis inuctance estimate is uite accurate, as well as the other three parameter estimates. 6.7 Tracking of Temperature Effects In this section, the simulation results that correspon to cases where the machine temperature increases linearly as a function of time are shown. The temperature change affects the stator resistance an the machine torue constant irectly. It is important for the parameter estimation algorithm to be able to track these changes. Fig. 6.13: Parameter estimation with +1 C/s ramp temperature on machine A. In Figure 6.13, parameter estimation results are shown in the case of a ramp temperature increase in machine A. The rate of change was set to be uite fast 119

137 compare to common temperature ynamics in electrical machines. The three plots show that the ifferent parameter estimates converge within acceptable range of the real machine parameters. The fact that the algorithm converges with such a temperature rate of change is a goo inication that the algorithm will be able to track parameter variations ue to temperature changes in any experimental case. Fig. 6.14: Parameter estimation with +2 C/s ramp temperature on machine B. Similar results an conclusion were obtaine with machine B an are shown in Figure The temperature rate of change in this simulation was set to be twice as large (2 C/s) as in Figure 6.12, an the results were still satisfactory. 120

138 6.8 Effect of Back-Emf Harmonics A common non-ieality associate with permanent magnet machines an the - moel is that the back electromotive force is not a perfect sine wave. The presence of higher orer harmonics is sometimes felt, especially in the case of IPMs. The purpose of this section is to investigate the effect that such harmonic perturbations can have on the parameter estimation algorithm. The analysis can easily be one with the three phase machine moel expresse in the - omain. To begin this analysis, it is easier to go back to the three phase omain. Going back to the Park transform euations, presente in Chapter II, we have f f f a b c = sin sin sin( θ r ) cos( θ r ) 1 f ( θ ) ( ) r 2 π 3 cos θ r 2 π 3 1 f ( ) ( ) θ r + 2 π 3 cos θ r + 2 π 3 1 f 0 Consiering the back emf term K T ω term in the -axis electrical euation, the r inverse Park transform of this term will only have a component on the -axis. This component is constant at constant spees, neglecting signal harmonics. After we apply the inverse Park transform to this -axis term, we can see that it correspons to a threephase signal as follows: E E E a b c = K T sin( θ r ) ( θ 2 π 3) ω r sin r ( ) sin θ r + 2 π 3 In this above euation, E x (x =a, b or c) symbolizes the phase back-emf. The introuction of an aitional term which correspons to the k th harmonic will conseuently lea to the following form: 121

139 ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) = 3 2 sin 3 2 sin sin 3 2 sin 3 2 sin sin π θ π θ θ π θ π θ θ ω r r r Tk r r r r T c b a k k k K K E E E where K Tk is the magnitue coefficient of the k th harmonic. It is now necessary to go back to the - omain, using the Park transform euations, to see the effects of the introuction of harmonics on the - moel of the machine. The - transformation euations are repeate here for convenience. ( ) ( ) ( ) ( ) ( ) ( ) + + = c b a r r r r r r f f f f f f cos 3 2 cos cos 3 2 sin 3 2 sin sin π θ π θ θ π θ π θ θ Using the istributive property of the Park transform, we can separate the Park transform of the funamental back-emf ( r T K ω in the -axis) from the Park transform of the k th harmonic. An interesting property arises in the case of harmonics of ranks that are multiples of 3 (k = 3.p, for some integer p): ( ) ( ) = sin 3 2 sin sin 3 2 sin 3 sin sin π θ π θ π θ π θ θ θ p p p K E p T ( ) ( ) sin 3 2 sin sin sin = = π θ π θ θ θ p T K E An in the case of the -axis: ( ) ( ) = cos 3 2 cos cos 3 2 cos 3 cos cos π θ π θ π θ π θ θ θ p p p K E p T 122

140 E = 2 K T 3 p 3 cos 2π 3 2π 3 ( θ ) cos( θ ) + cos θ + cos θ + = 0 A conseuence of these results is that harmonics of orers that are multiples of 3 will not affect the result of the Park transformation. As a result, the - moel will not be affecte by such harmonics, an the results obtaine in this chapter woul not be affecte by their introuction. However, harmonics that are not of such orers will introuce a perturbation voltage on both the - an -axes of the machine electrical moel. As an example, we chose to stuy the effect of the 7 th harmonic, which is often present in the back-emf of PM machines. The effect of introucing this harmonic on the - moel of the machine can be seen in Figure Fig. 6.15: Effect of 7 th harmonic on - moel ((a) Phase a / (b) - euivalent). 123

141 In Figure 6.15, the effect of a 10% seventh harmonic is shown when applie to a funamental back-emf of unity magnitue. The left han sie of the Figure (Fig (a)) shows the phase back-emf waveform, with an without the aitional harmonic, an the right han sie (Fig (b)) shows the corresponing - voltages. One can see that the perturbation associate with the seventh harmonic irectly affects both the - an -axes back emfs. The effect of this aitional term will conseuently be a perturbation on both axes of the - moel of the machine. This perturbation is not accounte for by the controller an the parameter estimation algorithm. In orer to simulate the effects of such a perturbation on the performance of parameter estimation algorithm, the perturbation was introuce on both the - an -axes in the form of a sine function (cosine for the - axis) of the position having a magnitue proportional to the motor spee. A test similar to the one that gave Figure 6.12 was conucte with the introuction of this perturbation an the results are shown in Figure Fig. 6.16: Effect of 7 th harmonic on parameter estimation ((a) non filtere / (b) filtere). 124

142 Figure 6.16 has two sets of results: Figure 6.16 (a) shows the results that woul be obtaine irectly from the parameter estimation algorithm, an Figure 6.16 (b) comes from estimation results that have been filtere. The motivation for filtering is apparent from the noise present in the results of Figure 6.16 (a), which is a irect conseuence of the perturbation. It can be observe that the perturbation affects the -axis inuctance more that the -axis one, although the perturbation is of same magnitue on both axes an the term corresponing to the -axis inuctance rop is smaller than the -axis one. One can see that once the perturbation has been filtere, the results are acceptable. The lag introuce by the filtering is acceptable an has no effect on the slow parameter estimation program as long as it is not enable uring an inuctance transient, which is usually not the case. 6.9 Conclusion This chapter was eicate to the presentation of simulation results that were a first step in valiating the propose algorithm. These results were either relate to ifficulties or concerns at the algorithm evelopment stage, or were relate to actual results obtaine with the propose parameter estimation algorithm in various situations. Two machine moels were use to valiate the algorithm performance, an the results were satisfactory. The following chapters will focus on the experimental verification of the effectiveness of the parameter estimation algorithm. 125

143 CHAPTER VII PMSM DRIVE EXPERIMENTAL DESIGN This chapter introuces the ifferent elements of the experimental esign create an use to verify the effectiveness of the propose algorithm. Each of these elements is introuce in etails, with special emphasis on features that coul affect the project. The motor moel parameters were presente in Chapter VI an the inverter was introuce in Chapter V. This chapter follows with the etails of the feeback circuit esign an the igital processor. 7.1 Inverter Inverter Choice The inverter chosen for this project has the classical three-phase brige structure presente in the previous chapters. The package chosen inclue all six switches an the reuire river in one integrate circuit. In aition to saving space, this structure also allows a better match between switch characteristics an propagation elays. However, the problem with this structure is that the whole integrate circuit has to be replace even if only one switch is amage. The inverter ratings were chosen to largely excee the machine ratings in terms of voltage an current, in orer to minimize amage. 126

144 The switch river in the package inclue a temperature an a current monitor that coul be use for shutown in the event of a problem. These features were isable for this project. Another interesting feature is the insertion of a force eatime an the prevention of inverter short circuit even when a faulty controller woul comman it Inverter Voltage Compensation The non-linear characteristics of the experimental inverter were presente in Chapter V. Their effects were investigate an it was shown that they woul nee to be minimize in the context of this project. The compensation algorithm use for this project treate two problems. The first one being the eatime compensation an the secon is the switch voltage rop. The goal of this algorithm is to get a better match between the controller voltage an the inverter output Deatime Compensation The metho chosen for the purpose of compensating the eatime elay in the controller is a simple one. It was erive from Figure 5.6 in Chapter V an observations mae about it. The introuction of a eatime elay changes the voltage applie to the machine by changing the effective input uty cycles of the inverter. Figure 7.1 shows the DSP outputs that correspon to a case of eatime introuction, an the corresponing phase voltage at the machine terminal. Two assumptions were mae in this compensation techniue; the first one is that the phase current oes not change sign uring the switch eatime, an the secon one is that the switch voltage rops were negligible. The secon assumption is vali because the switch 127

145 voltage rops are compensate for by another algorithm. The assumption that the phase current oes not change sign may in some cases not be vali, but even if the current changes sign, the ioe reverse recovery time (the time it takes for a ioe to switch off ) will maintain the voltage error constant longer. There will be cases, however, where this assumption is not vali, but they will introuce a very small error, for a very short time. Fig. 7.1: Deatime error as a function of phase current. The error introuce by the eatime elay T ea is a function of the sign of the phase current. The explanation for this is that when both switches are commane to open, the inuctive nature of the motor circuit will prevent the current from switching to zero an will force one of the leg ioes to turn on. The ioe that turns on epens on the sign of the current. If it is positive (going towar the machine), the bottom ioe will conuct an the phase voltage will be zero, an in the other case the top ioe will conuct an the phase voltage will be high, as shown in Fig If we refer back to the 128

146 ieal case, shown in Figure 5.6 of Chapter V, we know that the controller comman voltage woul have the inverter switch at times t 1 an t 3. It follows that t = 2 t1 = t4 t3 T ea. The voltage error can be expresse as: V err ea = V ctrl V inverter = V c T T ea PWM sign ( i ) ph where T PWM is the PWM perio, V ctrl is the controller comman voltage, V inverter is the actual inverter output voltage an i ph is the phase current. The voltage error term is introuce in the controller at the PWM stage to compensate the error given by the above euation. This way, the compensation is transparent from the controller s point of view Switch Drop Compensation As mentione in Chapter V, the inverter switches o not have ieal characteristics, an when they conuct, both ioes an IGBTs introuce a voltage rop. The metho presente in this chapter is specific to the harware that was use in this project. Other power electronics evices, such as MOSFETs, woul not present the same characteristics, since when they are on they act like a small resistance. In the case of IGBTs an ioes, a voltage rop is present uring conuction that is a non-linear function of the current going through the switch. The voltage rop was recore as a function of this current an the results were shown in Chapter V for both types of switch. For the analysis, it will be assume that eatime is not present (or compensate), an that the phase current oes not change sign uring the PWM perio. The effect of the 129

147 switch voltage rops is a function of the phase current sign, similar to the case of eatime compensation as shown in Table 5.2. Fig. 7.2: Switch voltage rop error as a function of phase current. Figure 7.2 shows the two possible cases that can be obtaine from the inverter as a function of the sign of the phase current in the case of phase a. Both the ioe an IGBT voltage rops are also function of the magnitue of the phase current, as shown in Chapter V. In an ieal case, both the ioe an IGBT voltage rops woul be zero, an the phase uty cycle coul be calculate as: V a = V a ctrl c where V a ctrl is the phase a voltage esire by the controller. If the switch voltage rops are taken into consieration, this euation is not vali anymore, an compensation is necessary to match the controller an inverter voltages. To obtain an expression for the 130

148 compensation voltages, or the uty cycle, it is necessary to consier two cases separately. In the case of a positive phase current, we have: V a ctrl ( Vc VIGBT ) a ( a ) VD = 1 where the controller is assume to have the same voltage comman as the inverter output voltage. The phase uty cycle can then be extracte as: a V = V + V c a ctrl D + V D V IGBT In this euation, the ioe an IGBT voltage rops are positive an uite close to each other consiering the results shown in Figure 5.7. These two variables can also be consiere to be much smaller than V c, which is at least larger than 50 V in this project. On the other han, the phase a controller voltage is a variable an may not be consiere to be much larger than the ioe forwar voltage rop. As a conseuence, the compensate uty cycle can be rewritten as a Va ctrl + VD = (7.1) V c In the case of a negative phase current, a similar analysis can be conucte. The controller phase voltage can be expresse as V a ctrl ( Vc + VD ) a ( a ) VIGBT = 1 An the compensate uty cycle can be extracte as a V = V c a ctrl + V V D IGBT V IGBT In this case the enominator is the same as in the previous case, an the same conclusion applies. The final compensate uty cycle can be obtaine as 131

149 a Va ctrl VIGBT = (7.2) V c One can notice that in the case of a positive phase current, the error inuce by the switch voltage rops is ominate by the ioe rop, an in the case of a negative current the IGBT rop ominates. The compensation algorithm uses look-up tables to store the results shown in Figure 5.7. Interpolation functions coul also be use an this choice epens on a compromise between calculation complexity an program memory reuirement. In the PWM program, the measure currents are use to obtain the switch voltage rops as a function of each phase current. The corresponing three phase uty cycles are calculate from euations (7.1) an (7.2) an the eatime compensation algorithm is applie to the result Compensation Verification To verify the effectiveness of the combine compensation algorithms, an experiment similar to the one conucte to obtain Figure 5.10 was one. The -axis voltage was varie in steps for a locke rotor conition, an the inverter voltage error was measure by taking the ifference between the resistive voltage rop, an the controller voltage. These results are shown in Figure 7.3. If the results on this plot are compare with the ones shown on Figure 5.10, one can see a substantial improvement. The voltage error in the case that ha no compensation was aroun 1 V for the consiere conitions, an is now aroun zero. As a conseuence, the controller voltage matches the inverter one much more accurately. 132

150 Fig. 7.3: Experimental inverter response with locke rotor an voltage compensation. Another way of showing the effect of the voltage an eatime compensations is to look at the controller voltages for a given operating point for ifferent cases of compensation. Table 7.1: Effects of controller voltage compensation. Types of compensation V (V) V (V) L est (mh) None Deatime Deatime an switch voltage Table 7.1 shows results that were obtaine in the case of a current-control experiment at constant spee. The results shown correspon to an operating point that has the -axis current at 1.5 A, an the -axis current at -0.5 A at a spee of 1000 RPM. The 133

151 controller voltages are shown, as well as the estimate -axis inuctance, in orer to show the effect of the voltage error on parameter estimation. Since the operating point is the same in the three cases, we can euce that the inverter output voltage remains constant too. If we refer to the inuctance table shown in Chapter VI (Table 6.2), we see that the -axis inuctance for the consiere operating point is between 9.6 an 9.9 mh. The case that inclues both voltage compensations matches this value closely, but this is not the case with the other experiments. The omission of the voltage compensation algorithms has a significant effect on the inuctance estimation for this operating point. 7.2 Digital Signal Processor The DSP that was chosen for this project is the TMS320F2812 from Texas Instruments. Its esign is optimize for motor control operations. Conseuently, it offers many interesting features for this project which will be highlighte in the following sections PWM Generation The DSP features a number of igital PWM outputs, with associate timers that can generate PWM signals with minimal CPU supervision. In other wors, after the eicate registers are initialize, the only operation neee from the CPU is to upate the uty cycle. The way PWM is generate is by having a timer counting either upwars, or upwars an ownwars (symmetric or asymmetric PWM) up to a value chosen by the user, at a rate chosen by the user. Conseuently, both PWM precision an freuency can 134

152 be accurately selecte. The timer value is then compare at each step with a variable containe in another register, calle the compare register. Depening on the esire output logic, the PWM output will have a state that will epen on whether or not the timer value is larger than the compare value. It is also possible to associate two PWM outputs with one timer an compare value, in orer to control the two switches that form an inverter leg. Various interrupts can be generate to synchronize the programs with PWM. An interrupt function is a program that is execute when a preprogramme event occurs, such as PWM perio or unerflow. This can be useful for example to synchronize the various program measurements (analog to igital conversions) with the PWM. An aitional feature that the DSP offers in terms of PWM generation is the automatic introuction of a user-specifie eatime elay. This feature is necessary for a PMSM rive an its effects, an the compensation techniues for them have been iscusse in previous sections Analog to Digital Converters The 2812 DSP has 16 possible analog to igital (AD) inputs that are multiplexe with two sample an hol units an one analog to igital converter (ADC). Two inputs can be sample simultaneously, an then converte seuentially by the ADC. This is convenient for applications where synchronization is important. For example, the machine phase currents reuire at least the measurements of two phase currents simultaneously for optimal precision. If the conversions are not taken at the same time, it may have an effect on the measurement uality. This is however a minor concern an 135

153 there are structures where a single current sensor is use in combination with a variable elay to reconstruct the three phase currents. Another very interesting aspect of the ADC in the 2812 DSP is the spee at which they operate. They are able to provie a precision of up to 12 bits for a conversion time of 60 to 200ns. This allows very high sampling freuencies an the possibility of oversampling, in orer to improve measurement uality an precision. Twelve bits of precision give a maximum accuracy of 2.5 ma for a maximum current of 10 A. The ADC can also be automatically synchronize with the PWM circuits so that conversions can be starte when specific events occur, without any CPU supervision. This can be helpful in some current sampling cases, like the one that was mentione previously (single shunt current sensor for three phase machine). However, in reality, for the present project an its associate harware, the effective precision of the ADC was closer to 10 bits. The ADC input voltage range is 3 V, which was mae to correspon to 20 A an this gives about ma for maximum accuracy. Another problem introuce by the ADC was the introuction of an offset an ramp error. These errors were specific to each DSP chip an ha to be compensate. The response of the ADC can be escribe by the following euation ADC 4096 = ( G Vin Off ) (7.3) 3 out + where ADC out is the igital output of the ADC (between 0 an ), G is the ADC gain, V in is the ADC input voltage, an Off is the ADC offset. The 4096/3 gain comes from the 12 bit precision of the ADC an its maximum input voltage, which is 3 V. 136

154 Ieally, G woul be eual to 1 an Off woul be zero, but this was not the case in practice. For example, the last DSP use for this project ha a 32% full range gain error an a 40 mv offset. These were measure with known fixe voltages, which were compare with the conversion result. Each time the DSP ha to be change, the gain an offset parameters ha to be measure an were introuce in the program, so that their effect coul be compensate. If these errors were not compensate, the ADC woul introuce significant errors when measuring sinusoial signals, because its error woul be largely ifferent from the maximum of the signal to its minimum, introucing significant istortion. Another constraint that was associate with the ADC was the relatively low maximum input voltage. In orer to get maximum precision from the DSP, it was necessary to scale all input signals so that they woul strictly remain within the 0-3 V region. If an ADC input signal were to be outsie of this region, even for a very short time, the ADC circuits woul most likely be amage. The level of noise introuce by the inverter was important; an ADC amage was efinitely an inconvenience, since the whole processor ha to be change Encoer Interface The DSP has eicate circuits that can be interface with a igital encoer, which is the type of position sensor use for this project. The encoer outputs, which are two suare-waves in uarature, are ecoe so that they either increment or ecrement the counter value in a timer. The DSP etects the rotor movement irection from the leaing of the two signals an the timer s value is upate accoringly. In aition to 137

155 this, the encoer ha an inex output, which is a igital output that gives a small pulse once per revolution. This allows the program to get absolute rotor position. This reuire an aitional offline test where the motor was spun with open terminals. The machine phase to phase back-emf was measure with an oscilloscope an was isplaye with the encoer inex output. The result is shown in Figure 7.4. Fig. 7.4: Machine bench test result. It is possible to see from Figure 7.4 that the inex pulse of the encoer was synchronize with the zero crossing point of the phase to phase (a-b) back-emf of the machine. The zero position chosen for the Park transform is the zero crossing point of the phase a back-emf, which coul not be irectly measure (neutral point not accessible), but is 30 electrical egrees away from the phase to phase zero crossing. As a result, when the DSP receives a pulse from the associate encoer output, it executes an interrupt 138

156 function that resets the controller position at 30. This allows the controller to avoi accumulation errors. The test that was conucte in orer to get Figure 7.4 was also use to get an estimate of the torue constant. The peak value of the phase to phase back emf is irectly relate to the torue constant value by euation (7.4). K T V peak = (7.4) 3 ω r where V peak is the peak value of the phase to phase back-emf, an the motor spee is the one that was measure for the test (which can also be obtaine from the back-emf freuency) DSP Central Processing Unit The 2812 DSP is a 32-bit fixe point processor that is able to process up to 150 millions of instructions per secon (MIPS). It was chosen for this project because it is a processor that is becoming increasingly popular in the inustry an it meets the performance reuirements. However, working with a fixe point DSP makes it more ifficult to calculate mathematical expressions that involve non-integer variables. A floating point DSP on the other han woul have solve this issue, but these are more expensive processors, an are conseuently less popular Introuction to Fixe-Point Mathematics In the software program that was written for this project, ecimal numbers ha to be represente in a special way because only integers can be irectly represente in the 139

157 CPU. Depening on its range an the precision we woul want with it, each variable was assigne a certain number of bits (binary igits) after their ecimal point. A simple example given below explains how this was one. Let us consier the calculation where the prouct of A an B for A =1. 25 an B = 0.04 is esire. If a precision of 4 bits after the ecimal point for A, an 8 bits for B is chosen, the prouct can be written as A = = an B = = = = The number A can irectly be represente by the integer 20 if we keep in min that it has four bits after the ecimal point. On the other han, B can be roune to 10 an we woul have eight bits after the ecimal point. A simple notation for that was introuce: A4 = 20 an B 8 = 10, where the subscript marks the precision associate with each number. It is possible to see that in the case of B, an error was introuce by this metho. This error coul be reuce by introucing more bits after the ecimal point. 5 For example, B B an B B 4. The prouct of the two variables can be obtaine as follows: A = B 8 = = The same example coul be followe for the prouct of a x variable an a y one, which woul always give a (x + y) result. From the processor s point of view in this example, the only operation necessary is the prouct of 10 an 20, but the user has to take into account the precision of the result. For comparison purposes, = 0.05, an = If the user wante the result to be of any 140

158 ifferent precision, all that is neee is to multiply or ivie by the appropriate power of two. Divisions are a very time consuming operation for fixe point processors an have to be avoie at all costs. Fortunately, multiplying or iviing by a power of two is the same as oing a register shift left or right by the corresponing number of bits, which comes from the fact that the CPU operates in base two. An analogy can be mae with iviing by 10 in base 10, which amounts to shifting the ecimal point to the left. It was possible to see from this example that the error introuce by the rouning of B ha an important conseuence on the result of the prouct. It is a ifficult problem to fin the optimal precision to associate with each variable. One coul think that having more than enough precision for each variable woul be a solution, but this can also introuce problems. To illustrate the ifficulty, let us consier we are working with an eight bit CPU, which can only work with integers that are between 0 an 255 (or -128 to +127 if signe variables are use). If one attempte to increase the accuracy by increasing the precision on B, the result woul be A = 4 B 10 = The above result cannot be easily represente in the CPU because it is outsie of the range of integers that can be represente, as oppose to the result obtaine when B was a 8. Choosing the precision for each variable is conseuently a compromise between the actual precision esire an the capacity of the CPU. 141

159 Known Limitations Fortunately, the 2812 DSP has a 32 bit CPU which can effectively multiply 32 bit variables, because it has a 64 bit register that can be use to extract a result if it excees 32 bits. However, one very important limitation introuce in the program was from the interfaces with external harware, the ADC an the position sensor. The analog to igital converter has an effective precision that ranges from 9 to 12 bits, an the position sensor has a precision of 11 bits per electrical cycle. If we take the case of the ADC an current sampling, the worst case scenario gives us a precision of about 0.02 A. For such a case, it is useless to represent the associate integer variables with a precision greater than 9. In the case of the rotor position, the maximum precision is aroun r an anything larger than a 9 woul also be useless. These precision limitations may have an effect on the precision of calculations that are present in the motor control programs Program Execution Times A constraint associate with real time implementation of a program is that its ifferent elements have to execute fast enough to sustain the sampling rate an not interfere with each other. In the final program implementation, there were two sorts of programs that were implemente: Programs that woul be synchronize with the PWM or another event, such as ata acuisition an treatment, current control, PWM an parameter estimation algorithms. The encoer inex coul also trigger a program to reset the rotor position. These programs are execute from an interrupt routine which is 142

160 triggere when specific events happen. In the case of PWM synchronization, timer perio match an unerflow were use, an for the encoer inex, an input capture interrupt was use. Programs that woul not reuire execution at specific times. The main example for this is the serial communication interface that was implemente to transmit ata to a computer (to isplay results an ebug). The communication program woul run with a low priority an its execution woul be stoppe by any interrupt event an resume when the DSP woul have execute the interrupt program. DSP iagnostics an other non time critical programs coul be execute like that. The way various program sections were execute as a function of time is explaine in Figure 7.5. Fig. 7.5: Program executions as a function of time. 143

161 In Figure 7.5, ifferent letters correspon to ifferent types of programs: A: This type of program inclues high priority time critical applications. They are execute on a timer unerflow (zero value) interrupt, at the beginning of the PWM perio. These programs inclue: ADC result treatment, Park transformation of currents, position an spee upate, current control, PWM algorithm, voltage compensation an fast parameter estimation algorithm. The total maximum execution time for this block is aroun 19 µs. B/C: The slow parameter estimation algorithm was split into two blocks because of its high execution time, so that time was available for D programs to execute in the PWM perio. In aition to the parameter estimation algorithm, the spee control program coul also be inclue in these blocks. These programs were chosen to be synchronize with the timer perio interrupt an conseuently have a high priority. Execution times are aroun 7.9 µs for B an 11.4 µs for C. D: Most of the programs inclue in this block have a low priority an stop their execution whenever an interrupt occurs. There are two exceptions for that. The first one was with the interrupt associate with the encoer inex, which resets the rotor position an has a very small execution time. The secon one is the interrupt associate with the reloaing of the serial communication transmit register, whenever its buffer is empty. The other programs in this category are not time critical 144

162 an inclue ata conversion an serial transmission to the computer. They coul also inclue iagnostics an a supervising program. Figure 7.5 contains a lot of information about how the ifferent portions of the program were arrange together in orer to provie smooth execution. The top iagram shows the value that the PWM counter takes as a function of time. A symmetric PWM carrier was chosen for this project, an the PWM timer counts up an own from zero to a chosen perio value that sets the PWM freuency. The other two iagrams show how programs are execute in two cases. Case 1 correspons to program execution that occurs most of the time, almost every 50 µs. Case 2 happens when a program nees to execute at a specific freuency lower than the PWM freuency, like the slow ientification algorithm. For example, if a freuency of 2 khz was chosen for the slow ientification program, case 2 woul take place two times (one for B an another for C) every ten PWM cycles. The time note t ADC in Figure 7.5 correspons to the time at which ADC conversions are triggere. This time was chosen so that the ata use in the A programs woul use measurements that woul be as recent as possible. The software program was written with special care so that they woul execute as fast as possible. Time consuming operations, such as ivisions, trigonometric functions an repetitions were avoie as much as possible. For example, the inversion of a 2x2 matrix was calculate by multiplying each element of the transpose comatrix by the inverse of the original matrix s eterminant (calculate once), instea of iviing each transpose comatrix element by the eterminant (four ivisions). It is very important that 145

163 the various program elements o not overlap because that coul lea to program instability an the loss of synchronization with PWM. 7.3 Feeback Circuits The last experimental esign part that nees to be mentione in this chapter is the analog feeback circuits. As state earlier in the chapter, the analog inputs to the DSP have to be strictly within the 0 3 V range. The sensor outputs of the analog values that were measure were not in that range an aitional circuits ha to be esigne to aapt them Current Sensing In the case of current sensing, Hall effect sensors were use an their output is a biirectional current that is proportional to the current they are sensing. This current ha to be converte into a voltage that woul be within the ADC input range. Figure 7.6 shows the analog circuit that was esigne for this purpose. The output of the Hall effect sensor ha to be converte first into a voltage. This was one with a simple resistor R 1 whose value was esigne to introuce a gain such that the largest possible phase current woul not inuce a voltage larger (in magnitue) than 1.5 V. This voltage was conseuently proportional to the machine phase current an coul be negative, because of the sinusoial nature of the machine currents. It was then necessary to shift this voltage by 1.5 V an create a virtual zero for the DSP analog to igital converter. 146

164 Fig. 7.6: Analog current feeback circuit. The OA1 an OA3 operational amplifiers have a follower function, which means their output is eual to their input, but they loa their input in a minimal way. The voltage ivier at the input of OA3 is set to provie a 1.5 V reference. The resistors R 3 an R 4 were chosen so that R4 15 R + R 3 4 = 1.5 The last operational amplifier, OA2, is an inverting aer whose gain is simply -1 in this case. It is important to note that for a sinusoial input current, the output will be a sinusoial voltage oscillating aroun 1.5 V, with a 180 egrees phase shift from the original current. 147

165 Two of the circuits shown in Figure 7.6 were use for this project. Upon initialization, before the machine is starte, the DSP program measures the input voltages coming from these circuits an saves them. These values, which are aroun 1.5 V, correspon to a zero current an will be subtracte from the later measurements in orer to obtain current samples oscillating aroun zero. In aition to this, the DSP will also have to invert the obtaine waveform to compensate the inverting gain of OA Voltage Sensing In the case of the DC bus voltage feeback, the problem was simpler, because it is a strictly positive voltage. The only necessary operation was to scale own the high voltage such that ADC ratings are never exceee. This was one with a combination of a voltage ivier an a voltage follower, as shown in Figure 7.7. Fig. 7.7: Analog voltage feeback circuit. The resistors in this circuit (Figure 7.7) were chosen accoring to the following ineuality V DC Rb max < 3 R + R a b 148

166 7.3.3 Feeback Filtering The presence of the capacitor C in Figure 7.7 is the only ifference of this circuit when compare with the current feeback circuits. It was mentione throughout this issertation that noise was an important issue for this project. The capacitor, combine with the two resistors, provies a low-pass filter that helps improve the feeback uality. On the other han, such an approach coul not be taken with the current feeback circuits, because the machine currents are sinusoial waveforms, an woul be affecte by a phase shift (lag) if a low pass filter was applie to them, even in the steay state. A software filter was chosen in the case of the currents. However, it i not act on the three phase currents but rather on the - currents obtaine from them with the Park transform. In steay state, the effect of this filter is to reuce the noise level in the current feeback. It only has a small averse effect uring transients. 7.4 Conclusion This chapter has introuce the reaer with the experimental esign an the various issues that ha to be resolve for this project. The introuction of the inverter mae it necessary for some of its non-linearities to be compensate. The DSP an its features were presente, as well as the feeback harware. The next chapter will focus on experimental results that were obtaine with this experimental setup. 149

167 CHAPTER VIII PARAMETER ESTIMATION EXPERIMENTAL RESULTS This chapter presents the results that were obtaine with the experimental esign in orer to verify the effectiveness of the propose parameter estimation algorithm. These results inclue ata from the fast algorithm either alone or combine with the slow one. Current controller performance improvements are also shown. 8.1 Time Scale Uncertainty It was mentione in Chapter VII that the serial communication program that was use to retrieve ata from the DSP was execute in an asynchronous way. This means that epening on the tasks with a higher priority that the DSP has to execute, the transfer rate will not be constant. The ata transmission software was programme in two parts: The first one accepte a number of integer inputs, corresponing to the value of variables to be transmitte (in a preset orer), an converte them into ASCII (American Stanar Coe for Information Interchange) coe. For example, the number 73, which is coe by in the DSP, will be coe as two characters, 7 an 3, which are represente respectively by an This conversion mae it easier on the PC sie to retrieve ata an store it with a minimum processing, but put an aitional buren on the DSP. 150

168 Different variables coul be separate by a space character an ata corresponing to ifferent times coul be separate by a carriage return. The program use on the computer sie shoul use the same norms. The secon part configure the DSP serial communication harware so that it woul sen the characters that ha to be transmitte. After the transmission of the first characters, it uses an interrupt to reloa the transmission buffer register until the ata corresponing to a full message has been sent. Once this has been one, it goes back to the first program which converts the next message ata. The serial ata transmission spee was set to bau, or bits per secon, but with an aitional start bit, a stop bit, an a parity bit. Fig. 8.1: Transmission of one ata byte. Figure 8.1 shows the transmission of one byte of ata through the serial communication harware. When the serial line is at rest, it is at a high state. The start bit notifies the receiver that transmission is starting, an the eight bits that follow are the actual ata to be transmitte, one at a time, where LSB is Least Significant Bit an MSB is Most Significant Bit. The parity bit is an interesting feature that prevents some transmission errors. Both transmitter an receiver count the number of high states in the ata byte, an if for example, an even parity was chosen, the parity bit will be at zero if that number is even. The transmitter sens this bit, an the receiver compares it to what it counte; if there is a mismatch, the ata byte is iscare. The stop bit notifies the receiver of the en of transmission of the current ata byte. One can see that it takes

169 bits to sen an 8 bit piece of ata. With a transmission rate of bau, the maximum byte transmission rate is aroun 10 khz. The number 73 takes two characters an will conseuently take at least 0.2 ms to sen. A conseuence of this structure is that epening on how busy the CPU is an how long the variables to be sent are, the effective transmission rate will be affecte. A larger number will take longer to convert than a smaller one, an will also reuire more characters (bytes) to be sent. Conseuently, it was ifficult to get an exact estimate of the times to which each ata sample correspone on the PC sie. A solution to this problem woul be to inclue a variable in the transmitte ata that woul inicate the exact time at which the sample was taken, but this woul further ecrease the transmission spee. The solution chosen was to have only a rough estimate of the transmission rate as a function of the number of integer variables sent. Time was not a critical piece of information for this project an an estimate of it was enough for most results. These variables were scale to a reuce precision so that only the most important bits woul be sent. For most of the results shown in this chapter, the transmission spee measure was aroun 500 Hz. The variables sent through serial communication are in integer format, an were then scale to ecimal numbers epening on their precision. The low transmission spee an time uncertainty was an important harware limitation for this project. However, if the time scale is not exactly accurate, results shown in this chapter were synchronize together. In other wors, for each of the plots that follow, all the ata was taken with the same time scale. 152

170 8.2 Current Controller Performance This section presents the results obtaine with the current controller in orer to verify the nee for accurate parameter feeback for optimal performance. These results can be relate with the simulation results shown in section of Chapter III. The machine was run in current control moe (i = 1.5 V an i = -0.5 V) with a loa that limite its spee at 1000 RPM. After steay state was reache, a perturbation was introuce in the loa that affecte the machine spee. This perturbation was introuce in the form of friction an was not exactly repeatable. However, the conclusions that will be rawn are not affecte by this. The effects of this perturbation were then stuie in ifferent cases of current controller parameter feeforwar. The euations relate to these terms are shown here for the sake of clarity: V ' = V V ' ω L = V e i + ω L e ω K r i T = r s = r i s i + p L + p L i i (2.16) The following are referre to as feeforwar terms V ff = L V ff ω i e = L + K ω i e T ω r Figure 8.2 shows the results obtaine in the case of a current control that oes not ' inclue any feeforwar term ( V = V ). The error obtaine from the ifference between reference an feeback current went through a PI controller an its output was use as a voltage comman for both - an - axes. The spee perturbation was of fairly small magnitue in this experiment, but one can see that the -axis current control was irectly affecte by it. 153

171 Fig. 8.2: Current controller response without feeforwar term. In Figure 8.2, the -axis current shows small variations ue to the perturbation, an they are negligible. The ifference in - an - axes currents for the chosen operation arises from the ominating back-emf term in the electrical moel of the machine compare to the other terms. This back-emf term only affects the -axis an is irectly proportional to the rotor spee. For example, if the spee ecreases suenly, the backemf will ecrease, an the current controller PI will have to compensate that voltage to maintain the -axis current constant. However, its response is not instantaneous an this is why we can see perturbations in the -axis control. For example, at 2.2 s, the machine shows a fast acceleration an the current ecreases from its set point. In this Figure, at 2.2 s in Figure 8.2, a variation of 27% in spee inuces an error of 11% in current control. 154

172 In the case of the -axis current, the only term that coul affect current control is the inuctive rop L ω i, but it is small an its variations are uickly compensate e for by the PI controller. To prove the effectiveness of a current controller with accurate feeforwar compensation, another test was conucte in similar conitions but with feeforwar compensation an its results are shown in Figure wr (r/s) I (A) 1 I -I Time (s) Fig. 8.3: Current controller response with feeforwar. In the plots shown in Figure 8.3, a small spee perturbation was followe by a very large one (70%). The - currents were not affecte at all by this perturbation an remaine constant an eual to the reference currents. The reason is the spee perturbation oes not change anything from the PI compensator s point of view. The ifference in voltage reuire by a change in spee is irectly taken into account by the 155

173 feeforwar terms an is transparent from the compensator s viewpoint. In this last case the perturbation was uite large, an one can imagine that a controller without feeforwar terms woul have been strongly affecte. One last test was then conucte to check the effects of poor parameter estimation on controller performance. In this test, a 30% error on the torue constant was introuce in the feeforwar compensation. The corresponing results are shown in Figure wr (r/s) I (A) I -I Time (s) Fig. 8.4: Current controller response with feeforwar (30% K T error). Figure 8.4 shows that the torue constant error makes the controller sensitive to changes in spee. However, when these results are compare with those of Figure 8.2, one can consier that the problem has been compensate by at least by 70% (70% from torue constant plus the inuctive rops). It is also possible to see that the error seems to be more important when the acceleration or eceleration is most important. For example, 156

174 the error introuce in Figure 8.4 from 1.2s to 2s is smaller than the one introuce from 2.3s to 2.8s. The reason for this is that in the case of a slow variation, it will be easier for the PI controller to compensate the perturbation. The test results shown in figures 8.2, 8.3 an 8.4 show that for optimum current controller robustness, it is important to have accurate parameter estimates with a feeforwar compensator for optimum current controller robustness. For applications where spee an torue transients are not important, such as pumps an fans, this might not be necessary. On the other han, high performance servo controllers can take avantage of this feature. 8.3 Inuctance Estimation Steay State Operation Several results that were use to buil the inuctance tables shown in Chapter VI (tables 6.2 an 6.3) were compare with results obtaine using the fast ientification algorithm with the experimental harware. A similar table was built to verify the effectiveness of this part of the algorithm in the harware. Table 8.1 shows results obtaine from the fast estimation algorithm for the same tests that were use to buil tables 6.2 an 6.3. The currents are shown in Amperes, an the internal cells give the estimate inuctances in mh, an the percentage of error between these results an the ones obtaine in Table 6.2. This table shows a goo match between estimate an calculate inuctances, especially for larger currents, where the signal-to-noise ratio gets larger. The cells that were not fille correspon to results that were extrapolate in Table

175 Table 8.1: -axis inuctance estimate by fast algorithm. i \ i % N/A 10.4 N/A N/A % % % % % % % % % % N/A % 9.5 5% N/A N/A N/A Initial Convergence An experimental test was conucte to show how the fast estimation algorithm converges towars a satisfactory estimate for machine inuctance. The machine was starte in current control moe with its spee set at 1000 RPM, an the parameter estimation program starte after the initial transient. The -axis current was set at a 2.5 A reference an the -axis current was set at -0.5 A. Figure 8.5 shows the associate results. In Figure 8.5, the top plot shows estimation results obtaine from this initial convergence test for a s winow, an the bottom plot is a zoom on the first part of the top plot. In this test, the serial communication program ha to be change in orer to be able to see the algorithm operate at the PWM freuency. The samples that ha to be transmitte were store in a memory table in the DSP at the PWM freuency, starting from the time when the fast algorithm was activate. Once the table was fille, the communication program was starte an transmitte the ata at low spee. This way, it was possible to extract a short winow of high freuency ata with an exact time scale. 158

176 The DSP memory was too small to be able to use this metho for an extene amount of time, an woul be a limitation for other experimental results shown in this chapter. Fig. 8.5: Initial convergence of fast algorithm. It is possible to see from Figure 8.5 that algorithm convergence was very fast, taking less than ten PWM cycles to converge towars its final value from initial values of 15 mh for the -axis inuctance an 10 mh for the -axis inuctance. The algorithm was able to converge even in the case of a large initial error, because of the inherent robustness of the fast algorithm. It is also possible to see the high level of noise present in the estimation, which is mostly ue to the noise level in the current feeback. A low pass filter was later use when the fast estimation algorithm was interface with other parts of the controller program. 159