PER JOHAN NICKLASSON I_N0~7 NO970521? PASSIVITY-BASED CONTROL OF ELECTRIC MACHINES
RECEIVED
DOKTOR IN GENI0RAVHANDLIN G 1996:21 INSTTTUTT FOR TEKNISK KYBERNETIKK NTM TRONDHEIM UNIVERSITETET I TRONDHEIM NORGES TEKNISKE H0GSKOLE ITK-rapport 1996:17- W
Passivity-Based Control of Electric Machines
Thesis by
Per Johan Nicklasson
Submitted In Partial Fulfillment of the Requirements for the Degree of Dr.ing.
Report 96-17-W Department of Engineering Cybernetics Norwegian University of Science and Technology N-7034 Trondheim, Norway 1996
DiSmaUTTON OF TBS DOCUMENT * WNUM1TED DISCLAIMER
Portions of this document may be illegible in electronic image products. Images are produced from the best available original document Abstract
This thesis presents new results on the design and analysis of controllers for a class of electric machines. Nonlinear controllers are derived from a Lagrangian model representation using passivity techniques, and previous results on induction motors are improved and extended to Blondel-Park transformable machines. The relation to conventional techniques is discussed, and it is shown that the formalism introduced in this work facilitates analysis of conventional methods, so that open questions concerning these methods may be resolved.
In addition to the new controllers derived for a class of electric machines, this work contains the following improvements of previously published results on control of induction motors:
• The improvement of a passivity-based speed/position controller.
• The extension of passivity-based (observer-less and observer-based) con trollers from regulation to tracking of rotor flux norm.
• An extension of the classical indirect FOC scheme to also include global rotor flux norm tracking, instead of only torque tracking and rotor flux norm regulation.
Experimental results from applications of the proposed control schemes to a squirrel- cage induction motor are included to illustrate the design. These results show that the proposed methods have advantages over previous designs with respect to con troller tuning, performance and robustness.
The main results in this thesis have been presented at international conferences, and parts of this work have also been published in international journals.
i Abstract Preface and Acknowledgments
This thesis is submitted in partial fulfillment of the requirements for the degree of Dr.Ing. (Doktor ingeni0r ) at the Norwegian Institute of Technology 1 (NTH), documenting the research part of my work towards this degree. The work has been carried out at Department of Engineering Cybernetics, during the period February 1992 to January 1996. It has been financed first by NTH during the period I worked as a teaching assistant (1992-1993), and then by a scholarship from The Research Council of Norway (February 1993 to January 1996), under grant 100651/410.
I am grateful to my supervisor Professor Dr.Ing. Olav Egeland for encouraging me to work towards a doctoral degree after finishing my M.Sc. in 1991. His enthusiasm has been a valuable source of inspiration and motivation during this work, and he has convinced me that I should always have confidence in what I do, and take a rigorous attitude to any control problem, even if it may seem “obvious ” at the first glance. For increased competence at our department, he has always insisted in contact with leading international researchers and their organizations, and in 1992 he introduced me to CNRS Researcher Dr. Romeo Ortega, Universite de Technologic de Compiegne (UTC), France, who later became my co-supervisor.
I would like to thank Dr. Ortega for his skillful advising during this project. He introduced me to the field of electric machines and passivity-based control, and main parts of the theoretical results in this thesis were initiated during a stay at UTC under his supervision from March to July 1994. This cooperation continued after I returned, and resulted in several joint publications. His optimism and diligent care about every technical difficulty have been especially valuable, and I hope some of it will follow me for use in my future work.
Several people have helped me with this work in various ways — only a few thanks are listed below:
• Professor Gerardo Espinosa-Perez at Universidad National Autonoma de 1From January 1996 this institute has been an integrated part of the Norwegian University of Science and Technology (NTNU).
ill IV Preface and Acknowledgments
Mexico (UNAM), for stimulating discussions and joint work. It has been a pleasure to work with him.
• A. Loria, D. Taoutaou and K. Kim, for stimulating discussions and help during my stay in Compiegne.
• The M.Sc. students J.G. Dyrset, H. Holemark, and L.F. Markussen, for doing their thesis under my guidance. I hope they have benefited as much from this cooperation as I have.
• My colleagues in the Motion Control Group, supervised by the Professors Egeland and Fossen, for creating a very pleasant environment for research. I would especially like to thank Morten Dalsmo, John-Morten Godhavn and Erling Aarsand Johannessen for many useful comments to this work and numerous discussions in general control theory (and theory in general!).
• The staff at our electromechanical workshop, S. Bertelli, T. Haugen, A. Ler- vold, and P.I. Lovold, for helpful comments and suggestions based on years of experience to the practical part of this work — the building of the induction motor setup. The combination of power electronics and signal transmission was a bit tricky, and a considerable amount of work which is difficult to document in a thesis, had do be done. Comments and suggestions from A. Fritzsche at Lust Antriebstechnik GmbH, Lahnau, S. Beineke at Universitat Gesamthochschule Paderborn, and J. Vater at the dSPACE Company, have also been of high importance for the success of this part.
Finally, I would like to express my gratitude to The Center of Maritime Control Systems at NTH and SINTEF, headed by Professor Egeland, for financial support to attend conferences in Orlando (CDC ’94), Seattle (ACC ’95), Rome (ECC ’95) and New Orleans (CDC ’95). These conferences have been important not only for the presentation of this work, but also for making international research contacts.
Trondheim, February 29, 1996 Per J. Nicklasson Contents
Abstract i
Preface and Acknowledgments iii
List of Figures xi
Nomenclature xiii
1 Introduction 1
1.1 Motivation ...... 1
1.2 Previous Work...... 5
1.2.1 Exact Linearization Design ...... 7
1.2.2 Backstepping and Manifold Designs ...... 9
1.2.3 Energy-Shaping Design ...... 13
1.2.4 Other Results...... 15
1.3 Unresolved Problems...... 15
1.4 Contributions of this Thesis...... 17
1.5 Outline of the Thesis ...... 18
2 Control of The Generalized Electric Machine 19
2.1 Introduction ...... 19
2.2 Passive Subsystems Feedback Decomposition ...... 20
v vi ■ Contents
2.3 Generalized Rotating Electric Machine...... 22
2.3.1 Model...... 22
2.3.2 Remarks to the Model...... 24
2.3.3 Examples...... 27
2.4 Problem Formulation and Design Procedure...... 28
2.4.1 Problem Formulation ...... 28
2.4.2 Design Procedure...... 28
2.5 Strict Passifiability via Damping Injection...... 29
2.5.1 Feedback Decomposition ...... 29
2.5.2 Conditions for Damping Injection...... 29
2.5.3 Remarks to Conditions for Damping Injection...... 31
2.6 Current Tracking via Energy-Shaping ...... 32
2.7 From Current Tracking to Torque Tracking ...... 33
2.7.1 Desired Current Behavior ...... 34
2.7.2 Decoupling Conditions ...... 34
2.7.3 Remarks to the BP Transformation ...... 36
2.8 Main Results...... 37
2.8.1 Underactuated Machines, ns 2.8.2 Fully Actuated Machines, ns = ne...... 39 2.8.3 Remarks...... 40 2.9 Examples...... 41 2.9.1 Synchronous Motors ...... 41 2.9.2 PM Stepper Motor ...... 42 2.10 Concluding Remarks...... 43 3 The Induction Motor 45 3.1 Introduction...... 45 3.2 The Squirrel-Cage Induction Motor Model...... 45 Contents vii X 3.2.1 Controlled Outputs ...... 47 3.2.2 Measured Variables...... 48 3.3 The (^-Transformation ...... 49 3.4 Field-Oriented Control...... 50 3.4.1 Background ...... • 50 3.4.2 Rationale Behind FOC...... 51 3.4.3 State Estimation or Reference Values ...... 53 3.4.4 Shortcomings of FOC...... 55 3.5 The o5-Model ...... 57 3.6 Concluding Remarks...... 58 4 Observer-Less Control of The Induction Motor 61 4.1 Introduction ...... 61 4.2 Review of Design Method...... 62 4.3 Definition of Desired Dynamics ...... 63 4.4 Main Result...... 65 4.5 Proof of Main Result...... 66 4.5.1 Remarks to the Observer-Less Controller ...... 69 4.6 Comparison with Indirect Field-Oriented Control ...... 70 4.6.1 Current-Fed Motor ...... 70 4.6.2 An Improvement of Indirect Field-Oriented Control .... 72 4.6.3 Voltage-Fed Motor ...... 72 4.7 Definitions of Desired Rotor Flux Norm...... 73 4.7.1 Minimization of Steady State Losses...... 73 4.7.2 Flux Weakening ...... 75 4.8 Simulation Results: Observer-Less Case...... 75 4.9 Concluding Remarks...... 79 5 Observer-Based Control of The Induction Motor 81 viii Contents 5.1 Introduction...... 81 5.2 Workless Forces...... 82 5.3 Problem Formulation...... 83 5.4 Ideal Case - Full State Feedback...... 84 5.5 Output Feedback Controller...... 85 5.5.1 Remarks to the Controller...... 87 5.6 A (^-Implementation...... 88 5.7 Simulation Results: Observer-Based Case...... 90 5.8 Concluding Remarks...... 92 6 Experimental Results 93 6.1 Introduction...... 93 6.2 Observer-Less Control ...... 94 6.3 Observer-Based Control ...... 100 6.4 Comparison with FOC...... 101 6.5 Conclusions and Suggestions for Future Experimental Work . . . 104 6.5.1 Comments to the Experimental Setup...... 105 7 Concluding Remarks 107 References 109 A Passivity 119 A. l Definition of Passivity ...... 119 B The BP Transformation 121 B. l Proof of Proposition 2.7.1...... 121 B. 2 A Lemma on the BP Transformation...... 122 C Proof of Eqs. (4.21) and (5.19) 125 C. l A Theorem on Positivity of a Block Matrix...... 125 Contents 'ix X C.2 Proof of Eq. (5.19) ...... 126 C. 3 Proof of Eq. (4.21)...... 128 D Derivation of Eqs. (4.38) and (4.39) 129 D. l Derivation of Pioss ...... 129 E Experimental Setup 133 E. l Introduction ...... 133 E.2 Hardware Description...... 133 E.3 Software Description...... 134 E.3.1 Controller Discretization and Speed Estimation...... 135 E.3.2 Phase Transformations...... 136 E.3.3 Pulse-Width Modulation 137 X Contents List of Figures 2.1 Passive subsystem decomposition ...... 30 4.1 References for speed and flux norm...... 77 4.2 Tracking errors for speed and flux norm...... 77 4.3 3 5.1 Tracking errors for flux norm and speed. Observer-based controller. 90 5.2 Components of estimation error gr — qr and error between the real speed g m and the internal speed qmd from (5.11). Observer-based controller...... 91 5.3 3<£ Voltages and currents ua , Ub, ia , h- Observer-based controller. 91 6.1 Speed regulation/flux tracking without integral action in currents. Qmd = 500 rpm...... 97 6.2 Speed regulation/flux tracking with (upper two figures), and without $ in the controller...... 97 6.3 Speed tracking/flux regulation at low speed (±10 rpm)...... 98 6.4 Speed regulation with load torque disturbance, tl « 0.8 Nm for t > 0.58 s. Qmd = 500 rpm. Error between desired torque and measured torque is shown in lower right plot...... 98 6.5 Position control. Passivity-based controller, = 0.2 Wb...... 99 6.6 Effect of desired dynamics in controller for high sampling period, T’sampi = 600 (is. Lower two plots are result from an experiment with only integral action and the nonlinear damping term in the controller...... 99 xi xii List of Figures X 6.7 Speed and flux regulation. Observer-based controller. Estimates of electrical quantities denoted by ...... 100 6.8 Comparison of the passivity-based controller (P-B) with an imple mentation of FOC. Speed regulation/flux tracking...... 101 6.9 Comparison with FOC. Speed tracking/flux regulation ...... 102 6.10 Comparison with FOC. Speed tracking/flux regulation. Rr = 1.572^#.103 6.11 Comparison with FOC. Speed regulation/flux tracking. Rr = 1.572,.#.103 E.l Block diagram of experimental setup...... 138 E.2 Main block diagram for C-code generation from SIMULINK. . . . 139 E.3 Example of SIMULINK block diagram for controller...... 140 Nomenclature The mathematical part of the definitions used in this thesis are adopted from (Desoer and Vidyasagar, 1975) and (Khalil, 1992), while the symbols denoting electromechanical quantities and parameters are adopted from (White and Wood- son, 1959). Abbreviations ASIC Application Specific Integrated Circuit BJT Bipolar Junction Transistor BP Blondel-Park CSI Current Source Inverter DSP Digital Signal Processor FOC Field-Oriented Control MMF Magnetomotive Force PM Permanent Magnet PWM Pulse Width Modulation (-ed) TTL Transistor Transistor Logic VSI Voltage Source Inverter ZOH Zero-Order Hold Mathematical Symbols and Definitions field of real numbers (nonnegative real numbers {i£R:i> 0}) ring of nonnegative integers: {0,1,2,...} N set of natural numbers: {1,2,3,...} ring of proper (i.e. bounded at infinity) rational functions in p with coefficients in R Rn linear space of ordered ro-tuples in R. Rnxm ring of matrices with n rows and m columns, and with elements in R. := “defined as” —» mapping from a domain into a range *-» mapping of elements into their image (•)r transpose operator Xlll XIV N omenclature (•)* conjugate transpose (-)-1 inverse operator ^max {'} minimum/maximum eigenvalue of matrix KOI absolute value of the scalar (•) (Oy submatrix i, j of a block matrix denotes norm, meaning for vectors : x E Rn, ||x|| = QC"=1 | ^i|2)" (2-norm, Euclidean norm) for matrices : X E Rnxm , ||X|| = y/Amax(XTX) (induced 2-norm,maximum singular value) In € RnXn n x n identity matrix Onxm € R"x ™ n x m zero matrix diag{-> diagonal matrix derivative of z = /(?) 19f ° partial derivative of z = /(... ,<;,...) with respect to c time, t € R+ ^(di ) = (0 total time derivative P:=Tt differentiation operator, e.g. pf = %, C°° “smooth ” (k times continuously differentiable in all its arguments (Ck), Vfc € Z+) £noo space of n-dimensional essentially bounded functions: / : R+ -» Rn belongs to ££o iff esssup*> 0 |j/(t)|| < oo a space of n-dimensional square integrable functions: / : 1+ -* Rn belongs to iff (/0°° ||/(<)||2ctt)2 < oo h truncated function: /:«+ -*»*, «*) = { m0] °7>|T rn rn *-'ooeJ *~2e extended n-dimensional function spaces: ££,, = {/: E4 ^ IP |/Tec vt>0} = {/ : 1+ -> ir I /T € CS, VT > 0} SO(n) Special Orthogonal Group of order n (the set of all n x n rotation matrices) SS(n) the set of all n x n skew symmetric matrices 0 -1 J the skew symmetric matrix E SS(2) 1 0 e(") natural exponential function e<‘) matrix exponential function e^() the 2x2 rotation matrix cos(-) - sin(-) E SO(2) sin(-) cos(-) _(•) estimate of (•) denotes error between a quantity and its reference £ “dynamic system ”, i.e. a causal mapping between extended function spaces Nomenclature xv Subscripts a, b a&-quantities (stator-fixed reference frame) e electrical quantities, g e, g e m mechanical quantities, g m, g m d desired function (reference function), e.g. Qed, Qrnd, Superscripts * desired function when subscript d is also used, e g. dg dg-quantity which also has subscript, e.g. qfq (rotor currents in dg-frame) Some Frequently Used Symbols 9, 9 generalized coordinates/velocities g, g 6 En according to the Euler-Lagrange formulation 9e generalized electrical coordinates Ze Blondel-Park transformed electrical coordinates Qm generalized mechanical coordinate (rotor position) Q external forces Me constant matrix L Lagrangian H Hamiltonian 9 e> 9 S1 Qr 9e ~ Qed, Qs ~ Qsdi 9r ~ Qrd Tig total number of windings TI5) 7lr number of actuated-/unactuated windings Tip number of pole pairs ■De(gm) inductance matrix (rotor position dependent) Wl = ^i X9m) flux linkage from permanent magnets W2 W2 = ~$t n torque component due to interactions between magnetics materials Dm moment of inertia Rm mechanical viscous damping constant XVI N omenclature L, Lai Lb i Ldj Lq, Ls , Lr, Lsri Lis inductance parameters Rs) R-r , Ti resistance parameters r 2 a total leakage factor <7=1— Tr time constant of rotor flux dynamics 7 motor parameter 7 - * c + L,aLrTr Pit) desired value of rotor flux norm X$, Ar stator/rotor flux linkages A n A f XT Ardi Xr Xrd 3 TL generated- /desired-/load torque Cl, C2 load torque parameters transfer function, h(p), H{(p) € Ep(p), i € N e, ei, 1£>, -K/, Kjs, fci, A?2 control parameters 7ii,, 72,, 7rz, adaptation gains a, 6 speed filter parameters T time parameter, T € E+ Pi, E7 constant matrices used in the Blondel-Park transformation ee qe - qe, current estimation error N “In reply to Mr. MacMillan ’s question, I would point out that it is al ways necessary to make some assumptions; in fact, even in the simplest problems there invariably exists an enormous number of assumptions most of which are not recognized as such. Therefore a rigorous solution means only ‘rigorous on the basis of the assumptions. ’ The solutions presented in the paper are rigorously correct in this sense. ”. Reply to reviewers in (Park, 1929). xvu xviii Chapter 1 Introduction 1.1 Motivation One of the main driving forces of control theory, has traditionally been the charac terization of classes of systems for which a certain control objective is achievable. This allows for identifying the structural constraints of the system —usually ex pressed in terms of system invariants— which are compatible with the desired per formance. A fairly complete theory along these lines is now available for both linear (Boyd and Barrat, 1991; Kucera, 1991) and nonlinear systems (Isidori, 1995; Ni- jmeijer and van der Schaft, 1990). From a practical viewpoint, the interest in identifying the invariants of the system, is to be able to attach to them some phys ical interpretation that can be used in the controller design. Although in the linear case this characterization is readily available and practically useful, for instance in the form of effective transmission delays or sensor-actuator couplings, their nonlinear counterparts developed to date seem to be too restrictive for practical applications. A good example of this state of affairs concerns the fundamental problem of sta- bilizability. Stabilizable systems have been identified via geometric conditions for linearization (Isidori, 1995; Nijmeijer and van der Schaft, 1990; Marino and Tomei, 1991) or transformation to particular (triangular, pure feedback) forms (Nam and Arapostathis, 1988; Kanellakopoulos, Kokotovic and Morse, 19926), see also (Jiang, 1993; Sontag, 1995) for recent surveys and other characterizations of such systems. Unfortunately, these geometric conditions are hard to verify in practical applications, and usually not related with the physical constraints of the system. Furthermore, since the control laws are derived by neglecting the physical constraints of the system, they are often not well defined in all operating regimes. The rationalization of these controller singularities is an a posteriori step that leaves the designer with only a pious hope that they will not contradict the operation of the system. 1 2 Introduction An alternative approach to systems stabilization that, to a certain extent, over comes the aforementioned shortcomings, has been explored in (Rodriguez and Ortega, 1990; Ortega, 1991; Byrnes, Isidori and Willems, 1991; Seron, Hill and Fradkov, 1995) where the stabilizability of the system is related to the possibility of rendering it passive via feedback1. The relevance of using passivity as a build ing block for control systems design stems, not just from the important role this concept plays in stability analysis, and its close connection with the physics of the system, but also from the invariance of this property vis a vis feedback in terconnection (Desoer and Vidyasagar, 1975). This aspect is particularly relevant in this work, where the key steps in the controller design are the decomposition of the system into passive subsystems, and the choice of a feedback control that preserves the passivity of the closed loop. The idea of passivity-based controller design is to reshape the system ’s natural energy, and inject a required damping in such a way that the control objective is achieved. Evident advantages of this approach are the enhanced robustness prop erties and lack of (controller calculation) singularities, due to the fact that exact cancellation of nonlinearities is avoided. The technique has its roots in classical mechanics (Goldstein, 1980; Arnold, 1989), and was introduced in control theory with the seminal paper (Takegaki and Arimoto, 1981). This method has been instrumental in the solution of several control problems from robotics (Ortega and Spong, 1989; Berghuis and Nijmeijer, 1993; Lanari and Wen, 1992; Ailon and Ortega, 1993), induction motors (Ortega and Espinosa-Perez, 1993; Ortega, Canudas de Wit and Seleme, 1993), power electronics (Sira-Ramirez and Or tega, 1995; Sanders and Verghese, 1992), vibration damping (Kanestrpm and Ege- land, 1994), and surface effect ships (S0rensen and Egeland, 1995). Some of these problems were untractable with other stabilization techniques. The main motivation behind the work presented in this thesis is to contribute, if modestly, to the development of a control theory for physical systems that incorporates at a fundamental level the systems physical structure. To achieve this objective, the work is restricted to a particular, but practically very important class of systems — the electromechanical systems that are included in the model of the generalized rotating electric machine (Meisel, 1966; White and Woodson, 1959) — and conditions for stabilizability are established. These are interpretable in terms of the physics of the system. The main contribution is the definition of a class of machines such that the output feedback torque tracking problem can be solved with passivity-based controllers. Roughly speaking, the class consists of machines whose non-actuated (rotor) dy namics is suitably damped, and whose electrical and mechanical dynamics can be partially decoupled via a coordinate transformation. Machines satisfying the latter condition are known in the electric machines literature as Blondel-Park transformable (Liu, Verghese, Lang and Onder, 1989). In practical terms this re 1Some efforts to reinterpret from a passivity perspective the currently popular “backstepping technique” (Kokotovid, 1992; Kolesnikov, 1987) derived for systems in special forms, are reported in (Lozano, Brogliato and Landau, 1992; Krstid, Kanellakopoulos and Kokotovid, 1994). See also (Sontag, 1995) for some interesting connections between input-to-state stability and dissipativity. 1.1 Motivation 3 quires that the air-gap magnetomotive force can be suitably approximated by the first harmonic in a Fourier expansion. These two conditions, stemming from the construction of the machine, have clear physical interpretations in terms of the couplings between electrical, magnetic and mechanical dynamics, and are satisfied by a large number of practical machines. The importance of the class of physical systems chosen in this thesis can hardly be Overestimated. The field within motion control called mechatronics has become a very important technology for industrial automation. This technology merges mechanics (precision mechanics, coupled electromechanical systems) and electron ics (microelectronics, power electronics), using among other tools control theory, and is of high economic importance. Control of electrical drives holds a central position within this technology. Due to their simplicity from a control point of view, DC motors have been the traditional choice where high dynamic performance, i.e. extremely rapid and ac curate torque/speed/position control in all four quadrants and for a wide speed range (including zero speed), is required (machining tools, robots). These ma chines are expensive and difficult to construct for high power/speed applications, even if slotless armature designs have increased their power range. High torque standstill operation is also difficult, due to brushes being welded to the commuta tor. DC motors are heavy with high rotor inertia and large dimensions, and they have failure prone brushes which are exposed to corrosion and wear, hence regular maintenance is required. Because of the brushes they are not suited for hazardous environments where electric sparks are not allowed (oil and gas industry), unless they are especially encapsulated in material or by inexplosive gases under higher pressure than the surroundings. To summarize, the DC motor can be thought of as an expensive device, but with a cheap controller. AC machinery has been the choice for high power constant speed industry appli cations (compressors, fans, mills, and pumps), or in assembly lines with several machines connected to the same power supply. These machines do not suffer from the typical disadvantages associated with DC motors (no brushes, less complicated rotor construction), but due to their inherent nonlinear dynamics, they have been considered difficult to control and not suited for high dynamic performance appli cations. Compared with DC motors, they can be thought of as cheap devices, but with expensive controllers. The recent years advances in power electronics and microprocessor technology have enabled implementation of advanced nonlinear control schemes using DSPs, and AC machines have replaced DC machines in a large variety of low and medium power applications, leading to higher reliability and lower costs. The new advances in AC motor technology have also led to a re-examination of the control schemes used in the traditional (uncontrolled) constant speed high power industry drives, due to demands on higher product quality (tighter speed control, faster recovery from disturbances). In addition, these advances have led to an increasing number of new applications. Examples of high power, high dynamic 4 Introduction performance applications are ships, where conventional diesel-/turbine-mechanical propulsion has reached the top of its evolution cycle, and is now being replaced with diesel-/turbine-electric propulsion (Ins, 1995). It is even claimed that in future permanent magnet AC motors will take over in application areas where hydraulic and pneumatic actuators, which are bulky and failure prone, have been the traditional choice. Such applications include robotics, aircrafts, and spacecrafts. In many of these applications, and also for ships and vehicles, which carry their own fuel, economy is of high importance, and the concept of power efficiency could be equally important to high dynamic performance. It is estimated that in the United States more than 60% of the generated energy is converted to other forms by electrical drives (Bose, 1993), and only 8% by DC drives. With such amounts of energy being converted, power efficiency of electrical drives in general has become an area of increasing research interest. In fact, it is pointed out in a book from the American Council for an Energy-Efficient Economy, that “adjustable speed drives and other controls are the largest potential source for motor system energy savings ” (Bodson, 1994). This does not only include large industrial drives, where a 2% improvement of speed regulation can give significant long term cost reductions despite higher initial investments. It includes all kinds of electrical drives, in a huge number of applications, including households. To some extent high dynamic performance and power efficiency (typically more than 90% at rated conditions) of a drive can be obtained by the design of the motor itself. However, for motor drives with a highly varying range of operating conditions, motor design alone cannot ensure high performance and efficiency for all conditions, and advanced control schemes must be used together with power electronics. From a control point of view, AC drives can pose the following research problems: • Transfer of electromagnetic into mechanical energy is essentially described by nonlinear models, making standard linear control theory inferior to nonlinear schemes for control of such systems. • They are multivariable systems, with several input voltages or currents and one or more outputs (torque/speed, flux level) to be controlled. • Varying parameters (e.g. temperature dependent resistances and friction parameters, inductances depending on flux level). • Unknown load disturbances. • Only partial state measurement (e.g. unmeasurable fluxes and rotor quanti ties). • In some cases failure prediction and prevention is also needed (supervisory control system to monitor defects in windings or bearings). 1.2 Previous Work 5 Because of these factors AC drives, and especially induction motors, have be come interesting benchmark problems for testing of new nonlinear control schemes (Editorial Board, 1993). The challenging control problems and the rapidly growing number of applications, are also highly motivating factors for working with this class of physical systems. 1.2 Previous Work Due to the importance of electrical drives in industry, thousands of papers and numerous textbooks presenting research in this field, have been published during the last 30 years. It would be a time consuming and difficult task to go into the details of all these approaches, and that is not the intention of this summary. In this section the most important control approaches are explained, with emphasis on recent results within the nonlinear control theory direction this research has taken. For additional information about this direction, the reader should consult (Taylor, 1994) and the references therein. A recent summary of results along the more application oriented branch can be found in (Bose, 1994a). Classical Stationary Control The classical methods for control of AC machin ery have been based on linearizations of the nonlinear equations at steady state operating points. This approach has the advantage that classical lin ear theory can be used for controller design. Typically, this has resulted in schemes where amplitude and frequency of sinusoidal stator voltages or currents are the basic control variables. Such designs give varying dynamic performance when applied to nonlinear systems, depending on to which de gree the underlying small-signal assumption is fulfilled, i.e. depending on how far from a nominal operating point the system is driven. Another dis advantage of applying such methods to multivariable systems, is the problem of coupling between inputs and outputs, making independent control of out puts difficult. For instance, in a voltage fed induction motor, both torque and air gap flux are functions of voltage amplitude and frequency, giving considerable coupling and slow response when trying to control only torque (Bose, 1986). Even with a well tuned scheme it is difficult to match the performance of a DC drive. Vector Control The deficiencies of classical linear methods when applied to con trol of AC machinery, were overcome by the vector control methods intro duced in the period 1969 - 1972. These methods aimed at making the AC motor behave like a DC motor, with asymptotic decoupling of torque and flux control. To achieve this, the nonlinear model of the motor had to be used in the design. The vector methods consist essentially of a nonlinear coordinate transforma tion (a rotation), followed by a nonlinear decoupling feedback. At the time of introduction, implementation of the schemes (especially the rotations) was computationally heavy and difficult with analog electronics. These schemes 6 Introduction were therefore considered to be rather “academic” (Leonhard, 1991), and did not gain their wide popularity until the introduction of the digital micropro cessor around 1980. The basic control variables were now the individual components of the rotated two-dimensional stator current vector , either con trolled directly, or indirectly through a nonlinear feedback voltage control law. Vector control in its various implementations is now the de facto standard for high dynamic performance control of AC drives, and its superior dynamic performance as compared to the use of classical linear methods is widely accepted (Leonhard, 1991). One of the most common implementations of vector control, rotor-flux- oriented control , is discussed in Section 3.4 with application to induction motors. The basic drawbacks of these schemes are the assumption of full state measurement (flux measurement), and the problem of only asymptotic decoupling between torque and flux control. The second problem can be solved with linearizing controllers, while the approach to solve the first has been the use of “a nonlinear separation principle”, for which no rigorous theoretical justification is given. See (Vas, 1990) for the theoretical part of vector control, and (Lorenz, Lipo and Novotny, 1994; Kazmierkowski and Tunia, 1994) for implementation aspects. Modern Nonlinear Control During the last 10 years, there have been signifi cant advances in nonlinear control theory, and among many other applica tions for which linear theory cannot give satisfactory solutions, its application to electric machines has gained widespread interest. Of course, these schemes are also based on nonlinear models and end up with a specification of the current or voltage input vector. However, since they are derived from differ ent and sometimes purely mathematical nonlinear control theories, aiming at formal proofs of stability, they are nemed nonlinear schemes in this thesis instead of vector control schemes. The following review of modern nonlinear control theory applied to electric machines is based on (Taylor, 1994), with the addition of recent results. To limit the number of references, only the recent theoretical and experimental applications of these methods to induction motor control are included here, since this machine will be particularly emphasized in parts of this thesis. According to Taylor ’s recent overview, the methods can be broadly classified into the three following categories: • Exact Linearization Design • Backstepping and Manifold Designs • Energy-Shaping Design Each of these approaches are explained in the following subsections, and some comments are given to methods not belonging to these classes. 1.2 Previous Work 7 1.2.1 Exact Linearization Design Basically, the goal of these schemes is to transform the system into a linear input- output relation between fictitious inputs and outputs, using an inner nonlinear decoupling loop. Controllers can then be designed to ensure stability and per formance of the resulting linear system, by use of conventional linear theory. In strumental to this approach is the choice of coordinates for the representation of the system, together with the design of the inner loop decoupling control in these new coordinates. In contrast to field-oriented control, which also has a coordinate transformation and an inner decoupling loop, the decoupling between outputs is no longer only asymptotic. The coordinate transformations used are also generally more complicated than the rotation used in field-oriented control. Disadvantages of these schemes are that measurement of the full state is needed, exact cancellation of dynamics is necessary, and controller singularities are intro duced, typically for zero rotor flux norm. It also seems to be difficult to apply this method in a general way to a class of machines, and then derive controllers for each machine in particular from a general controller. The dynamic equations for stator and rotor quantities must be transformed to a common frame of reference (typically the stator fixed frame) before the differential geometric tools can be used. Symbolic software is often necessary to answer the question of whether the system, with a given set of inputs and outputs, can be transformed into a linear system or not. For the fifth order induction motor model2 case, recent contributions belonging to this class are: • The exact input-state linearization under the assumption of a slowly vary ing speed presented in (De Luca and Ulivi, 1987), which essentially was a linearizing controller design for the reduced fourth order electromagnetic model. • The exact input-output linearization from stator voltages to torque or rotor speed and square of rotor flux norm proposed in (Krzeminski, 1987), for both the rotor-flux-oriented model and the stator fixed model. This approach was extended in (Marino, Peresada and Valigi, 19936) to handle unknown constant rotor resistance and load torque by the use of adaptation, and local stability results were derived. The resulting zero dynamics with these outputs, is the dynamics of the rotor flux angle, which is bounded. In this paper it was also shown that the fifth order induction motor model is not input-state exactly linearizable, and that the largest input-state feedback linearizable subsystem has dimension four. 2In this model the two rotor currents (or fluxes), two stator currents, and rotor speed are considered to be the states of the system, while the two stator voltages are the inputs. The outputs to be controlled are torque or rotor speed and rotor flux norm. In the results reported here, the stator fixed frame of reference has been used for model representation, unless something else is explicitly stated. The model of the induction motor is also described in Chapter 3 of this thesis. 8 Introduction • In (Chiasson, 1993) it was shown that the extended sixth order system ob tained by augmenting the fifth order model by an integrator in one of the inputs (or the higher order system obtained by one integrator in one input and two in the other), can be exactly input-state linearized, but only for speed, not position control. The result is only locally valid, and the con trol structure is computationally heavy and requires switching between two transformations to avoid singularities. A nonsingular feedback linearizing transformation was shown to exist only for nonzero torque. The result has recently been extended in (Chiasson, 1995) for the case of a third order rotor-flux-oriented reduced dg-model, for which stator dynamics is neglected and stator currents are considered as inputs, while rotor flux angle, amplitude, and rotor speed are states. This model is not input-state exactly feedback linearizable. In the work by Chiasson it was shown that if one integrator is added to the input in the d-axis, a single feedback linearizing transformation and controller exist. The controller has a dynamic singularity condition, and if flux is kept constant, it must be nonzero to avoid this singularity. Otherwise, the dynamic condition sets a limit on how fast it can be decreased. This limits the flux tracking capabilities of the scheme. The controller structure is however computationally feasible. Adding instead one integrator to the g-axis input, results in the fifth order system with position included being feedback linearizable. Unfortunately the required controller is singular for zero torque, and another controller structure is needed when the torque is required to change sign. This limits the practical usefulness of the result. This property of dynamic feedback linearization (Marino and Tomei, 1995) of the induction motor has recently been commented in (Martin and Rou- chon, 1995), where it was shown that it follows from the flatnes£ of the reduced order induction motor model equations, since a flat system can be linearized by dynamic feedback around a regular point. The flat outputs of tb induction motor are the slip angle of the flux and the rotor position, and input-output linearization of the system with these outputs, results in no zero dynamics. These results show how the geometric properties of the induction motor model change with coordinate transformations, and care has to be taken when model representation, inputs and outputs are to be chosen. Otherwise the linearizing approach can result in controllers having singularity condi tions which are difficult to interpret and avoid. • Experiments validating the practical importance of the feedback linearizing approach have been presented in (Kim, Ha and Ko, 1990; Kim, Ha and Ko, 1992) and (Bodson, Chiasson and Novotnak, 1994a). Results from extensions to nonlinear magnetics with saturation and speed observers, have also been presented (Bodson, Chiasson and Novotnak, 19946), (Bodson, Chiasson and Novotnak, 1995). 3 3See (Fliess, L6vine, Martin and Rouchon, 1995) for an explanation of this concept. 1.2 Previous Work 9 Quite a lot of effort has been devoted to the estimation problem, and linear and nonlinear observer theory (Verghese and Sanders, 1988), extended Kalman filters (Cava, Picardi and Ranieri, 1989), and more physically based adaptive observers (Nilsen and Kazmierkowski, 1989) have been proposed as solutions. Recently, an observer which is adaptive with respect to rotor resistance have been proposed in (Marino, Peresada and Tomei, 1994). In this work exponential convergence of flux and rotor resistance errors to zero, is proved using a Lyapunov type argument, under reasonable assumptions on persistency of excitation. The incorporation of estimated states in the control law is however still based on a “nonlinear separation principle”. Even if there are rigorous theoretical proofs of stability in the case of full state measurement, generally no proofs are given for stability of the closed loop system when ad hoc estimates have been substituted for real states in the controller. The use of observers also introduces asymptotic prop erties to the schemes, i.e. there is only exact decoupling after the estimation error has converged to zero. As explained in (Kim, Ortega, Charara and Vilain, 1996b), it is important to consider the effect of the convergence rate of the observation error, on the performance of the total system. For the reduced third order model of the induction motor, an interlaced output feedback controller and observer design has been reported in (Marino, Peresada and Tomei, 1993a). Rotor speed is the only measurement, and the controller provides singularity free (under the assumption of suitable initial conditions for the flux estimates) speed and rotor flux norm tracking. The algorithm is also adaptive with respect to a constant load torque. An implementation taking advantage of using the rotor flux reference frame for digital calculations, has been reported in (Morici, Rossi, Tonielli and Peresada, 1995). 1.2.2 Backstepping and Manifold Designs In the feedback linearization technique, every nonlinearity standing in way for a linear input-output relation is cancelled. Intuitively, it could be expected that not all of the nonlinearities would be harmful to the closed loop dynamics if they were left uncancelled. In fact, by use of Lyapunov theory, it can be shown that some nonlinearities are useful for system stabilization, and hence they should not be cancelled by the controller4. This has now become one of the main points in the philosophy of the backstepping approach (Krstic et al., 1995), which is a recursive Lyapunov procedure for controller design. The first step in this approach is to choose the output to be controlled and derive its dynamic equation. A fictitious control signal is then chosen from this equation. Using a first Lyapunov theory approach (or based on intuition), a desired function for this fictitious control is found, such that the the control objective of the first 4Consider the system x = —xz + cosx + u. For stabilization around zero, it is of interest to cancel the cosine term with the input it, but not the third order nonlinearity, which is helpful. A feedback linearizing controller would typically also cancel out this term. See Example 2.5 in (Krstic, Kanellakopoulos and Kokotovic, 1995). 10 Introduction subsystem can be asymptotically achieved. The control is generally designed with the above points regarding cancellation of dynamics in mind. If the fictitious control is the real input to the system, which can directly be specified to be the desired function, then the design ends here. This is generally not the case, and there will be a deviation between the fictitious control and its desired behavior. The dynamic equation for this error is then derived, and the design above is repeated with the aim of forcing the error to zero by the use of a new fictitious control. Stability and convergence of this new subsystem can be proved by adding a term square in the error to the previous Lyapunov function. The procedure is then repeated until finally the real control can be specified to be a desired function, and the desired control properties can be proved using a final Lyapunov function, which is a sum of the previous functions. The number of steps needed, is equal to the relative degree between the output to be controlled, and the input of the system. For multivariable systems, the design is done separately for each of the outputs. This results in a linear combination of inputs being equal to desired functions, and inversion of a matrix is necessary to specify the real inputs, hence control singularities are likely to occur. There are now several results from the application of backstepping to induction motor control. In all of the results listed below, the fifth order stator fixed model representation has been used, and unless something else is explicitely stated, ob servers have been used to avoid flux measurement. • In (Kanellakopoulos, Krein and Disilvestro, 1992 o) the first application of backstepping to speed and rotor flux norm tracking for induction motors was presented. Exact model knowledge was assumed, and a proof of exponential stability was given for the total system with an observer based on the rotor flux equations. The stability result was only regional in the sense that the invertibility of a matrix needed to calculate the controls, restricted the initial conditions to be in a set which could be estimated a priori. The basic idea used for the interlaced design, was to add nonlinear damping terms in the controller to compensate for interactions due to observation errors. These terms made it possible to dominate cross-terms containing observation errors in the derivative of the Lyapunov function, and it could consequently be made non-positive. The advantage of this scheme, as compared to conventional schemes, which do not compensate for observation errors, was demonstrated by simulations, and showed that not compensating for observation errors could have signifi cant negative impact on transient responses. This result was extended in (Kanellakopoulos and Krein, 1993) to compen sate for unspecified modeling imperfections and external disturbances, by addition of PI-loops for speed and rotor flux norm tracking. The regional property of stability was still present, imposing restrictions on initial condi tions and reference functions. In the observer structure used in that paper, additional nonlinear terms were introduced, as compared to the previous design. In the stability analysis, it was shown that with a proper choice of these terms, they could be used for eliminating instead of dominating cross 1.2 Previous Work 11 X terms stemming from observation errors, in the derivative of the Lyapunov function. The introduction of nonlinear damping terms in the observer/controller, gave a systematic method for handling of estimation errors and other disturbances in stability proofs. Later designs have taken advantages of these ideas. • The above results were changed to semi-global uniform ultimately bounded position tracking error and rotor flux norm regulation in (Hu, Dawson and Qian, 1993). To avoid controller singularities for zero flux estimates, a control parameter had to be made sufficiently large relative to initial conditions. • Departing somewhat from the previous results with interlaced controller and observer designs, adaptive and robust controllers, which could compensate for parametric uncertainty in all parameters (robust case also had additive norm bounded disturbances), were presented in (Hu, Dawson and Qu, 1994) and (Hu, Dawson and Qu, 19956). There were controller singularities for zero rotor flux norm, and proofs of asymptotic position/speed tracking (adaptive case) and uniform ultimate boundedness of position tracking error (robust case), were derived under the assumption of full state measurement. • While all the above approaches required speed measurement, a design with both speed and flux observers was proposed in (Hu, Dawson and Qian, 1995 o). Local exponential rotor flux norm and position tracking was proved, with measurement of only stator currents and position. The localness of this result was again due the the invertibility of a matrix needed for control calculations. To avoid the singularity, certain restrictions had to be imposed on rotor flux norm reference and initial conditions. The result was novel in the way that speed and flux observers were both taken into the stability analysis. • An adaptive controller which could compensate for parameter uncertainty in both rotor resistance and the mechanical subsystem, was presented in (Hu and Dawson, 1995). The observer was based on the adaptive observer from (Marino et oZ., 1994), but to obtain an interlaced design of observer and controller, additional terms were added to the observer. The proof of asymptotic position/speed tracking and norm of estimated fluxes converg ing to a desired function, was only locally valid, due to a singularity in the controller for zero flux. A drawback of this scheme is that asymptotic con vergence of flux estimates to real values was not proved (not even for exactly known rotor resistance), hence rotor flux norm tracking cannot be claimed. However, this was the first asymptotically stable output feedback scheme with adaptation of rotor resistance , which has been reported. • By adaptation of results presented in this thesis and published in (Ortega, Nicklasson and Espinosa-Perez, 19956), globally valid results based on back- stepping designs were derived in (Dawson, Hu and Vedagarbha, 1995) and (Vedagarbha, Dawson, Burg and Ou, 19966). The clue to avoid singularities was to use desired rotor currents and fluxes in the controller design. 12 Introduction X In the first paper a result on global asymptotic position/speed tracking and rotor flux norm regulation was presented. The scheme ii luded adaptation of mechanical parameters. The result in the second paper was a scheme with globally exponential velocity tracking and rotor flux norm regulation. Experimental results were included in both papers. Drawbacks of these schemes are that they do not provide rotor flux norm tracking, electrical parameters are not compensated for, and they are compu tationally heavy. This last point limits the value of the experimental results, since responses are restricted to be very slow because of the high sampling period needed. • Some of the above problems have been solved in (Vedagarbha, Dawson and Burg, 1996a), where an adaptive singularity-free controller for asymptotic rotor position and rotor flux tracking has been proposed. The work was based on (Dawson et al, 1995), but instead of using only desired rotor flux dynamics, rotor flux estimates were also calculated by integration of voltage and current measurements using Stanley ’s equations (Stanley, 1938). This allows for adaptation of rotor flux resistance and mechanical parameters, but relies on zero initial conditions and use of open loop integration for flux calculation. This approach is not numerically robust, and an ad hoc integra tion method with a forgetting factor has to be used for implementation. The performance of the controller was demonstrated by experiments. Unfortu nately the same points as above regarding response times were present. It is believed that this will be solved in future work, either by use of controller simplifications or special hardware (ASICs). It must be pointed out that this scheme (as well as many other backstepping schemes) has been derived with the stator voltages as basic control variables, a; d it is not clear how the controller can be applied to the reduced order model, with currents as inputs. In the above schemes the full order model has been used for controller design, and there are no approximations or implicit assumptions about time scale separations between interacting parts of the dynamics. The rationale behind manifold designs based on singular perturbation or integral manifold theory, is to take advantage of inherent time scale separations between different parts of dynamics in a system. Such a separation exists for instance be tween the high frequency current dynamics and its low frequency average dynam ics, because of small inductances or high-gain current control. Another example is the relatively slow mechanical dynamics for high rotor inertias, as compared to the fast electrical dynamics. These methods essentially consist in first designing a fast control, which steers the fast dynamics to the manifold of the slow dynamics (i.e. makes it attractive), and is inactive once the fast states hit the slow manifold. A second slow control is then designed to give the desired behavior of the slow dynamics, assuming that the reduced order slow model is a sufficiently good approximation of the system 1.2 Previous Work 13 dynamics. These methods can be applied in combination with other approaches, to analyze and implement approximations of for instance schemes based on feedback linearization. Another related approach is to force the systems dynamics to evolve on a manifold called a sliding mode with discontinuous controls, having analogy to classical bang- bang control. The behavior on the sliding mode is specified to be the desired one, giving for instance zero speed and flux tracking error. Once on the sliding mode, the system states will stay on it, due to the controls being discontinuous across it. Any small deviation from the sliding mode will activate controls to force the states back to it. Theoretically this is equivalent to infinitely high gain. The inherently high frequency discontinuous on-off switching of controls have made this approach very appealing for control of power converters and electric machines, where thyristors or other switches are used. Together with PWM, sliding modes is now the de facto standard for converter control. For the design of sliding mode motor controllers, the converter is taken into the model, and the switching pattern follows directly from the controller, not from a PWM block with outputs from the controller scheme as reference inputs. This sets requirements to speed of computation, if high switching frequencies are needed for satisfactory control. Unfortunately, analysis of differential equations with dis continuous right-hand sides is technically difficult, due to the fact that classical theorems of existence and uniqueness are not necessarily satisfied for such systems. This has motivated the development of special tools for rigorous analysis of sliding mode systems. Examples of these schemes and their combinations for motor control can be found in (De-Leon, Alvarez and Castro, 1995; Utkin, 1993), and (Sabonovic, Sabanovic and Ohnishi, 1993). Experimental results have been reported in (Sabanovic and Izosimov, 1981). 1.2.3 Energy-Shaping Design The work in this thesis belongs to this class, and it will be described in detail in the following chapters. As already pointed out in Section 1.1 (see also (Taylor, 1994)), this approach has not evolved from a mathematically motivated approach like feedback linearization, but by consideration of physical properties like energy conservation and passivity. Briefly, it is aimed at reshaping the energy of the system in a way leading to the desired asymptotic output tracking properties. The main goal is to drive the system to a desired dynamics, leaving the closed loop system nonlinear, without cancelling dynamics or introducing controller singularities. There are several results from the application of this approach to induction motor control: 14 Introduction • In (Ortega and Espinosa-P er ez, 1991) the controller design method used in robot motion control to solve the output tracking problem for a class of un deractuated Euler-Lagrange systems, was extended to torque regulation of the induction motor, with all internal states bounded. There were no con troller singularities, but exact model knowledge and full state measurement had to be assumed. It was also indicated how to follow sinusoidally varying torque references. A model representation in a dg-frame of reference was used, and this model became the basis for later designs. • The previous design was extended to a globally stable controller for torque regulation without measurements of rotor variables in (Ortega and Espinosa- Perez, 1993). This globally defined and globally stable interlaced design of controller and observer, was the first such result reported in the control literature (Taylor, 1994). Exact model knowledge was assumed, but it was indicated how to compensate for unknown rotor resistance and load torque, unfortunately under the assumption of full state measurement. The torque reference was restricted to be below a certain upper limit depending on motor and controller parameters, but again it was shown how this could be avoided in the case of full state measurement. This fundamental paper also explained the design method based on shaping the energy of the total electromechanical system. • Torque regulation with a globally defined and stable controller without mea surements of rotor variables, was extended to torque tracking with adapta tion of unknown linearly parameterized load torque in (Ortega et al, 1993). • In (Espinosa-Perez and Ortega, 1995) the passivity-based controllers were extended to include the important case of rotor flux norm regulation with out rotor variable measurements. The coordinate independent properties of this approach were also rigorously explained. I follows that passivity-based controllers can be derived in any frame of reference chosen for model repre sentation. • Recently, a new approach to the induction motor control problem was pre sented in (Espinosa-Perez and Ortega, 1994), where it was shown that global torque tracking and rotor flux norm regulation could be done without flux measurement or estimation. This was accomplished by the fundamental ob servation that the mechanical part of the induction motor dynamics defines a passive feedback around the electrical subsystem, which is also passive. Hence, instead of shaping the energy of the total system as in previous designs, the control goal could be achieved by shaping only the energy of the electrical subsystem, with the mechanical subsystem as a passive distur bance. It was also shown in this paper how to extend the controller for speed tracking with adaptation of a constant load torque. Drawbacks of this scheme are that it is open loop in speed, and that the convergence rate of the speed tracking error is bounded from below by the mechanical time constant, relying on a positive damping of the mechanical system. 1.3 Unresolved Problems 15 However, this paper gave a first rigorous solution to the longstanding prob lem of avoiding rotor flux estimates in induction motor control, still with global stability results, but unfortunately under the assumption of known parameters. Other result along the line of passivity-based control of induction motors have been presented in (Seleme Jr., 1994), where its important application to joint angle tracking for n-link robot manipulators have been addressed. Under the assumption of measurable link positions, link velocities, and stator currents, a local result on convergence of tracking errors to zero was given. The scheme was based on an interlaced design of controller and observer. Another result in this work, is a globally defined and exponentially convergent observer-based scheme for torque tracking of induction motors with magnetic energy minimization. The desired slip speed is used for optimization purposes, hence the optimal rotor flux norm is specified indirectly. 1.2.4 Other Results It might be tempting to include also a fourth class, consisting of all those schemes based on other approaches in nonlinear control theory which do not fit into the above framework. Among these we find the so-called “intelligent ” schemes, based on expert systems, fuzzy logic and neural networks. Common for the schemes are that they do not provide formal proofs of stability for the resulting system, not even under the assumption of full state measurement. Such proofs are important" goals for the other three classes. In lack of theoretical results, the issues of performance and stability are instead demonstrated by simulations or experimental results. An overview of these techniques is given in (Bose, 19946), and examples of recent ap plications can be found in (Theocharis and Petridis, 1994) and (Gorzalczany and Stefanski, 1995). Some of the problems with the derivation of proofs for stability and performance of “intelligent ” schemes, are due to their model free structure, which also seems to be their advantage when the model is “fuzzy ” or missing. It is not yet clear if these nonlinear function approximation techniques have significant advantages over the above approaches, for high-performance control of electrome chanical systems with a structured model but uncertain varying parameters. 1.3 Unresolved Problems To summarize the above review of previous work, it is worthwhile to point out that no systematic comparison of all these approaches has been done for motor control. Intuitively, the schemes seem to have much in common, but to the best of this author’s knowledge, no comparing analysis has yet been reported. Also, the schemes are not general in the sense that estimation and control are not based on a compact and general model. Equations of dynamics are commonly specialized 16 Introduction to a particular machine before any results are derived. A comparison of the above results in a general setting, would be a highly interesting and challenging task. The problem of combined parameter and state estimation is still an unresolved issue in general nonlinear control theory, and a lot of effort has been devoted to this problem of interlacing controller design with the design of adaptive observers. As an example, the main driving force within the field of induction motor research is to provide a satisfactory solution to the problem of rotor resistance adapta tion. Adaptive observers have been designed, and the first results from interlacing these results with output feedback controller design, have been reported for the backstepping scheme. This problem is still open for the other schemes. Another issue to be solved in a general setting, is the problem of incorporating additional nonlinearities arising from inherently nonlinear magnetics and actuator saturation. Some results along this line using nonlinear control theory have been reported in (Bodson et al, 19946), but this is still an active area of research. The problem of digital implementation of controllers for nonlinear systems is yet another problem. Usually nonlinear analysis starts with a continuous model, and stability results are established for the total continuous system. The controllers are however invariably implemented on digital processors. Today this is commonly done by use of an emulation technique with some ad hoc discretization (e.g. ZOH) , under an assumption of “sufficiently short” sampling period. This assumption can be recast in terms of a bound on sampling period relative time constants of the system in the linear case. For nonlinear systems there is no such equivalent, and the performance issues of the resulting systems with nonlinear controllers which have been discretized by use of ad hoc methods, are yet not fully understood. As an example, it is not obvious that exact cancellation of dynamics and stable zero dynamics can be achieved with a discretized feedback linearizing controller, when directly applied to a continuous system. Promising first results towards the rigorous design of discrete-time controllers for electric machines, have been been reported for the CSI induction motor in (Ortega and Taoutaou, 1996), and for the synchronous motor in (Georgiou, Chelouah, Monaco and Normand-Cyrot , 1992). It is expected that the solution of this problem will lead to the design of discrete nonlinear controllers at converter level. This means that the discrete nature of switched converters will be taken advantage of in the controller design, removing the PWM block between controller and converter, and instead directly specifying the switching pattern by the discrete-time controller. Hysteresis controllers for direct control of torque and flux, which specify the converter switching via lookup tables, are now becoming alternatives to field-oriented controllers for industrial use (Kazmierkowski and Kasprowicz, 1995). This principle should be focused on also from the viewpoint of nonlinear control theory. Torque ripple due to current harmonics must be addressed to reduce mechanical vibrations and to obtain higher power efficiency. It is expected that the solu tion of the above points regarding nonlinear magnetics, unknown parameters and converter switching will significantly reduce this problem. 1.4 Contributions of this Thesis 17 There have not been reported many results where both speed and flux observers have been interlaced with controller designs, and rigorously analyzed using non linear theory. Observers for flux and speed are generally designed separately in an ad hoc way, even though a high quality estimate of speed is needed for good flux estimates, and vice versa. This is a problem to be solved for both higher dynamic performance and power efficiency (Bodson et al. , 1995). A natural extension of this problem, is to the design of nonlinear controllers which can be implemented without rotational sensors. Specific to the the energy-shaping method, in addition to the above problems, there were several problems to be solved in a short term setting. For practical usefulness, it was important to extend the result reported in (Espinosa-Perez and Ortega, 1994) so that the rate of speed convergence was independent of mechani cal damping. The extension of this approach and the observer-based approach in (Espinosa-Perez and Ortega, 1995), to handle the case of rotor flux norm tracking, were also problems of high importance, not only for experimental implementation with converter saturations and demands on power efficiency, but also for setting the schemes on equal footing with designs based on feedback linearization. A gen eralization of the results from passivity-based design of induction motor controllers to a larger class of machines, was a problem of interest not only for clarifying the applicability of this design method, but also for building a formal framework for future extensions (nonlinear magnetics, varying parameters). For the passivity- based approach, as for any other design method, experimental results were also of outmost importance for the verification of theoretical results and addressing of implementation aspects. 1.4 Contributions of this Thesis The contribution of this thesis is to extend the line of results in passivity-based control of electric machines as follows: • A novel theory for the control of a large class of Blondel-Park transformable electric machines included in the model of the generalized rotating electric machine is presented. The theory is based on an Euler-Lagrange approach to modeling, and passivity-based controller design. This work has been pub lished in (Nicklasson, Ortega and Espinosa-Perez, 1994) and (Nicklasson, Ortega and Espinosa-Perez, 1996a). • An observer-less passivity-based speed/position controller is derived for the induction motor. The controller allows for independent rotor flux norm and speed/position tracking with global stability results. Parts of this work have been published in (Ortega et al., 19956; Ortega, Nicklasson and Espinosa- Perez, 1996) and (Espinosa-Perez, Nicklasson and Ortega, 19956). • The work on passivity-based controllers for rotor flux norm tracking has also led to an extension of the classical indirect FOG, to allow for global flux 18 Introduction tracking, instead of only flux regulation. • An observer-based controller for induction motors is extended to include rotor flux norm tracking. The design of observer and controller is interlaced, and gives global stability results for the closed loop system. Theoretical results from this work have been published in (Espinosa-Perez, Ortega and Nicklasson, 1995c) and (Espinosa-Perez, Ortega and Nicklasson, 1996). • Experimental result from the application of passivity-based controllers to an induction motor are presented. Parts of this work have also been accepted for publication (Nicklasson, Ortega and Egeland, 19966). 1.5 Outline of the Thesis The outline of the thesis is as follows: • In Chapter 2 the model of the generalized electric machine is presented, and by the use of passivity-based techniques, observer-less controllers for global torque tracking are derived for a class of BP transformable machines. For machines where fluxes are to be controlled, these controllers also provide global flux tracking. • The 2<6-model of the squirrel-cage induction motor is then presented in Chap ter 3, together with a transformation of the dynamic equations to any frame of reference (dq-model, ab-model). This leads to a'presentation and dis cussion of rotor-flux-oriented control. The chapter is a presentation of well known results, and it will hopefully be useful for readers who are not familiar with the induction motor. • In Chapters 4 and 5, observer-less and observer-based controllers for speed (or position) and rotor flux norm tracking of induction motors are presented. The controllers are derived using passivity properties in combination with energy-shaping arguments, and global stability results are given. As a corollary of the results from observer-less control, it is shown in Section 4.6.2 how classical indirect FOC can be extended to include the case of global rotor flux norm tracking. • Experimental results from the application of passivity-based controllers to a small VSI induction motor, are presented in Chapter 6. For purposes of comparison, the chapter also contains results from experiments with an implementation of FOC. • In Chapter 7, concluding remarks to this work are given. • Finally, references followed by appendices with what has been considered lengthy proofs or derivations, and a description of the experimental setup, are given at the end of this thesis. Chapter 2 Control of The Generalized Electric Machine Man mufi immer generalisieren. C. G. J. Jacobi 2.1 Introduction The main objective of this chapter is to characterize a class of machines for which a passivity-based controller solves the output feedback torque tracking problem. Roughly speaking, the class consists of machines whose nonactuated dynamics are well damped and whose electrical and mechanical dynamics can be suitably decoupled via a coordinate transformation. The first condition translates into the requirement of approximate knowledge of the rotor resistances to avoid the need of injecting high gain into the loop. The latter condition is known in the electric machines literature as Blondel-Park transformability, and in practical terms it requires that the air-gap magnetomotive force must be suitably approximated by the first harmonic in its Fourier expansion. These conditions -stemming from the construction of the machine- have a clear physical interpretation in terms of the couplings between its electrical, magnetic and mechanical dynamics, and are satisfied by a large number of practical machines. Since there is no need for flux measurement or estimation in the passivity-based controller, it can be thought of as a generalization of the well known indirect vector controllers, giving new insight and providing stability results to this class of controllers. The passivity-based controller design presented here is an extension of the method used in (Espinosa-Perez and Ortega, 1994) for induction motors, to a larger class of Blondel-Park transformable machines. It proceeds as follows. First a decompo sition of the system dynamics as a feedback interconnection of passive subsystems 19 20 Control of The Generalized Electric Machine is carried out, where the outputs of the forward subsystem are the measurable and controlled outputs. Sufficient conditions for the possibility of this, are established in a lemma on the composition of the system ’s energy function. Second, an inner feedback loop is designed, which via the injection of a nonlinear damping term, ensures that the controlled subsystem defines a strictly passive map from control signals to measurable outputs. Third, the passivity-based technique is applied to this subsystem leaving the feedback subsystem as a “passive perturbation”. As explained in (Ortega and Espinosa-Perez, 1993) this last step involves the defini tion of the desired closed-loop energy function whose associated (target) dynamics evolves on a subspace of the state space ensuring zero tracking error. The overall procedure leads to a nonlinear dynamic output feedback controller that ensures global asymptotic torque tracking with internal stability. In position or speed control applications, an outer loop can be added to the torque controller (Ortega et al, 1995b). As pointed out in (Leonhard, 1985), there ap pears to be universal agreement that this structure, with a fast inner current loop that can be regarded as creating impressed currents to the stator windings, and which is necessary to achieve the desired torque specified by the outer loops, is the most effective control scheme for electrical drives. This procedure naturally leads to the well known cascaded controller structure, which is typically analyzed invoking time-scale separation assumptions. A key feature of the new cascaded control paradigm is that stability is now established using instead energy dissipa tion arguments. The remaining of this chapter is organized as follows. In Section 2.2 a simple, but important, passive systems feedback decomposition lemma is presented. The class of systems considered in this chapter —those which can be described by the model of the generalized electric machine— is given in Section 2.3, and in Section 2.4 the control problem is formulated and the design procedure described. Conditions for strict passifiability of the electrical subsystem are established in Section 2.5. In Section 2.6 and 2.7, it is explained how the passivity-based approach is used to achieve current and torque tracking, and in Section 2.8 the main results of this chapter are presented. Examples and concluding remarks are given in Sections 2.9 and 2.10. 2.2 Passive Subsystems Feedback Decomposition In this section nonlinear electromechanical systems evolving on a smooth (C00) manifold Q called the configuration space of the system are considered. Without loss of generality it is assumed that Q is En with n = ne+nm and nc,nm € N. The tangent bundle of Q is denoted T Q, and the cotangent bundle T* Q. The dynamic behavior of the system is described by the Euler-Lagrange equations of motion (Meisel, 1966): 2.2 Passive Subsystems Feedback Decomposition 21 where C = J*(q,q) - V(g) is the system Lagrangian, J* : TQ —» R (J"* : R2n —» R) is the total system coenergy-state 1, V : Q —► R (V : Rn —*► R) is the total system energy-state function, q € Q (q € Rn) are the generalized coordinates and Q : TQ —> T*Q (Q € Rn) are the external (dissipative and control) forces. It is assumed that the system energy-state function (the potential energy) is bounded from below. Lemma 2.2.1 Assume the Lagrangian of (2.1) can be decomposed in the form T — dl e(qe,qe, 9 m) £m(9 m?9 m) where q := [qj ,qm)T with qe € Rn= and qm € Rn"* . Then, S can be represented as the negative feedback interconnection of two passive2 subsystems Qe qe Ee :Cnc+l /»ne+l 2e -L2e —qm T = ^2” ^2em : (T + Qm) ^ <7m where dC e ggm is the subsystems coupling signal, and Q := [Qj, Qm\T with Qe € , Qm € Rnm. 000 Proof Use of the Euler-Lagrange procedure to derive the equations of motion gives d dC e _ dCe Qe (2.2) dt dq e dq e d .d£.m dC m ]- Qm + r (2.3) dt dq m dqm Evaluating the total time derivative of Ce results in Noting that 1 Coenergy is defined on the tangent bundle of the configuration manifold, as a function in generalized velocities (flows), while kinetic energy is defined on the cotangent bundle in terms of generalized momenta (energy variables). In classical mechanics generally no distinction is made, since energy is usually quadratic in momenta, which are linear in velocities, hence coenergy equals energy (e.g. kinetic (co)energy \mv2 = with momentum p = mv, m -mass, v-velocity) (van der Schaft, 1994). 2 See Appendix A for the definition of passivity. 22 Control of The Generalized Electric Machine inserting this into (2.4), using (2.2) and rearranging the terms, it follows that He = Q^qe-q^r dt where dC T He{qei 9ej 9m) := "TjT 9e — £e Using the arguments of Section 9.1 in (Crandall, Karnopp, Kurtz Jr. and Pridmore- Brown, 1968), it can be shown that He is the total energy of the subsystem Ee. Integrating from 0 to T and setting te := —He(0) proves the passivity of Ee, as defined in Appendix A. A similar procedure can be used to establish the passivity of Em, using the energy function Hm = §^-9 m — £m and (2.3). □□□ Remark 2.2.1 It is clear from the invariance of passivity vis a vis feedback interconnection (Desoer and Vidyasagar, 1975) that the system (2.1) defines a passive operator E : £%e —» L^e '• Q 9 - Of course, this also follows immediately from the well known energy balance equation of Euler-Lagrange systems (Nijmeijer and van der Schaft, 1990) s: H(t,) - H{0) qTQds stored energy dissipated + supplied where H 9 — £is the overall systems total energy . 2.3 Generalized Rotating Electric Machine 2.3.1 Model In this section the generalized rotating machine considered in (Liu et al., 1989), (see also (Youla and Bongiorno, 1980; Willems, 1972)) is described. It consists of in all ne windings on stator and rotor, and ideal symmetrical phases and sinusoidally distributed phase windings are assumed. The permeability of the fully laminated cores is assumed to be infinite, and saturation, iron losses, end winding- and slot effects are neglected. Hence, only linear magnetic materials are considered, and it is further assumed that all parameters are constant and known. Under the assumptions above, application of Gauss’ law and Ampere’s law leads to the following affine relationship between the flux linkage vector A = [Ai,..., AnJT and the current vector qe = [91 ,..., qnJT A — -De(9m)9e + At(9m) (2.5) 2.3 Generalized Rotating Electric Machine 23 where qm € K is the mechanical angular position of the rotor, De = Dj > 0 is the ne x ne multiport inductance matrix of the windings and the vector fi represents the flux linkages due to the possible existence of permanent magnets. Both being bounded and periodic in qm with period 2tt/N, N 6 N. If the generalized coordinates of the system are defined as the electrical charge of each winding qi, i = 1 , ...,ne, and the angular position of the rotor qm, the magnetic-field coenergy (with ' denoting the variable of integration) can be com puted as (Meisel, 1966) / M&)(% = 1 .rn . , t - 29e DeQe + V qe Jo and the mechanical kinetic coenergy as K.m = ^Dmq^, where Dm > 0 is the rotational inertia of the rotor. Neglecting the capacitive effects in the windings of the motor, and considering a rigid shaft, the potential energy V of the system is only due to the interactions between the magnetic materials in stator and rotor, i.e. V = V(qm). This energy contribution is zero if there are magnetic materials in only one part (stator or rotor), and the reluctance properties of the other part is uniform. The Lagrangian is Age, 9m, 9m) = ^9?-De9e + p?qe + ^An9m ~ ^(9m) (2.6) To model the external forces, it will be assumed that the dissipative effects are linear, time-invariant and only due to the resistances in the windings u > 0, i = 1, • • • ,ne, and the mechanical viscous friction coefficient Rm > 0. The control forces are the voltages applied to the windings u € Rn* ,ns < ne. In this work fully actuated as well as underactuated machines, that is, machines where the voltages can be applied only to stator windings (e.g. induction motor), or to both stator and rotor windings (e.g. synchronous motor with field windings) will be considered. Hence, it is convenient to partition the vector of generalized electrical coordinates as qe = [qJ,qJ]T € Rn* , qs € Rn* , qr € Rn”, nr = ne — ns, where the subscripts s,r are used to denote variables related to windings with and without actuation respectively. In the case of underactuated machines the partition coincides with stator and rotor variables as well. Notice however, that there are also machines, like the PM synchronous, PM stepper and variable reluctance motors, where qe consists only of stator variables which are directly actuated by the stator voltages, see the examples in Section 2.3.3. Finally, it is assumed3 that the load torque tl in the mechanical subsystem is of the form (Ortega et al, 1993) TL(qm,qm) = [ci + c2g^]tanh(^-) (2.7) 3The presence of a load torque tl of this form ensures that to every bounded r there exists a bounded qm. Except from this, as will be shown below, the load torque rl plays no role in the torque tracking problem. Also, as shown in (Espinosa-Pdrez and Ortega, 1994) it can be treated as an external disturbance for the speed tracking problem. 24 Control of The Generalized Electric Machine with a scaling parameter c > 0 and c\, C2 € M+. With the considerations above, and by applying the Euler-Lagrange equations to (2.6), the equations of motion of the generalized machine are derived as Deqe + W\ {qrn)QmQe 1^2 (9m ) 9m "4" ReQe — MeU (2.8 ) DmQm 'r(qe, 9m) + RrnQm — T£. (2.9) where dD e(qm) In W2 := Re diag{rsln^TrXnr}, Me dq-m dQm 0 and with r the generated torque T — h{qe, Qra) W1qe + W? qe + r}(qm) (2.10) Vi.Qm) — ~j~ (2.11) dQm which is also bounded and periodic in qm. Notice that the machine is fully characterized by its dynamic functions D := diag{De,Pm}, Me, fi, 77 and the dissipation parameters R := diag{Re,Rm}. Therefore, in the sequel the system described by (2.8), (2.9), (2.10) will be re ferred to as H(D,R,Me,ft,rj). Similarly, the electrical subsystem (2.8 ), (2.10) will be denoted by £e(Z>e,Re,Me, fx,77), and the mechanical subsystem (2.9) by Sm(Hmi Rm)- 2.3.2 Remarks to the Model 1. Lumped System and Euler-Lagrange Formulation Magnetic end electric fields are distributed phenomena, which are naturally modeled with partial differential equations to give a boundary value problem. This is usually done during the construction phase, when effects of different materials and geometric shapes are to be studied. Even in the case of two- dimensional fields, this results in problems which must be solved numerically, for instance by finite-element analysis. A distributed model is too complicated for control purposes, and a lumped model is generally considered to give satisfactory results. In this thesis it is assumed that a lumped model of the system in terms of inductance and resis tance matrices is already available. The lumped parameters are generally de rived as functions of material constants, turns and span of distributed wind ings, air-gap parameters and approximations, by integrating surface-current densities over rotor and stator periphery (Meisel, 1966; Krause, 1986). From 2.3 Generalized Rotating Electric Machine 25 a lumped description, the dynamical equations of motion are derived using the Euler-Lagrange procedure. As discussed in (White and Woodson, 1959), it could be argued that insight into the physical process is lost when this variational approach to modeling is used instead of basic force laws, even though the model equations are equivalent. Against this, it could be argued that physical insight is gained because coupling terms between various subsystems are derived in an an alytically formal way, due to the generality of the method. This property has previously been exploited in numerous examples of controller designs for purely mechanical systems (Murray, 1995), and it is also the main argu ment behind its use here. In particular, the coupling between electrical and mechanical subsystems is highlighted, and storage and dissipation functions for subsequent passivity-based controller design are easily obtained. These properties are obscured if the Euler-Lagrange language is translated into a state-space formulation of the dynamical equations. 2. Voltage Balance Equations If the flux (recall that fluxes and currents are the generalized electrical momenta and velocities of the Hamiltonian formalism (White and Wood- son, 1959)) A as defined by (2.5) is introduced into (2.8), the voltage balance equation becomes A + Reqe = Meu. Particularly useful for further develop ments is the following relationship between rotor fluxes and rotor currents for motors where the rotor windings are short circuited (induction motors) Ar + RrQr = 0 (2.12) where A := [Aj,\J]T and Re := diag{JZs, Rr} with Rs = rsln3 E Rn* xn* , Rr rr2nr € Rn’"xnr . The importance of (2.12) is that it defines a dynamic re lationship qe = /(g e,9 ei9 m,9 m) that is invariant with respect to the control action that will have to taken into account when defining a “desired behav ior” for the machine. 3 4 3. Ignorable Coordinates An interesting property of E(D, R, Me, fi, rj) and other magnetic field devices in which currents are of main interest, is that the electrical charges are ignorable (White and Woodson, 1959), (also known as cyclic in mechanics (Arnold, 1989)). That is, the Lagrangian of the system does not contain qe, although it contains the corresponding currents qe. It must be pointed out that when choosing the form (2.1) of the Euler-Lagrange equations, it is crucial that the currents are expressed in their natural frames, where electrical charges can be obtained by integration of currents, to avoid the introduction of quasi coordinates. See also (White and Woodson, 1959). 4. Mechanical Commutation For a class of machines with mechanical commutation, the relation be tween the physically applied port currents and the rotor currents intro duces non-holonomic constraints (White and Woodson, 1959), and the dy namic equations cannot be obtained directly from (2.1) with the given La grangian. Instead quasi coordinates could be introduced, and the dynamic 26 Control of The Generalized Electric Machine X equations derived by using the Boltzmann-Hamel (White and Woodson, 1959) or Gaponov (Neimark and Fufaev, 1972) form of the Euler-Lagrange equations. These procedures are however quite involved, and the dynamic equation for this class of machines is therefore usually not derived from vari ational principles, but by the use of basic force laws as Faraday ’s law, Ohm’s law and Euler’s law. As pointed out in Section 1.1, motors with mechanical commutation are of less interest for nonlinear control design, and will not be considered in this thesis. 5. Parameters It is well known that the lumped parameters are not constant, but that they vary due to temperature variations, current displacement and magnetic hysteresis and saturation. In some cases, like for the switched reluctance machine, magnetic nonlinearities must be taken into account in the mod eling for satisfactory performance. In other cases the assumption of linear magnetics leads to controller designs which works satisfactory, at least if the machine is forced to operate at flux levels below the saturation limit. A controller design based on a model including magnetic saturation is however highly desirable for performance improvements. For instance, while older uniform air-gap induction motors often were robustly designed to operate under linear magnetic conditions (constant inductances), there is now an in creasing interest of operating these under magnetic saturation (current/flux dependent inductances), saving cost and weight. The Euler-Lagrange procedure is based on energy properties of the system, and the dynamic equations for machines with nonlinear magnetics can also be derived by this procedure. However, it is believed that the simpler prob lems of constant parameters and linear magnetics must first be rigorously solved using passivity-based methods, before the formal framework can be used to incorporate and solve the ore complex problems stated above. Hence, magnetic nonlinearities, time varying resistances, or state dependent inductances will not be considered in this thesis. 6. Complex versus Real Notation In this thesis a real representation of the machine’s model is used, meaning that rotations of vectors will be presented by matrix exponentials, allowing for easy manipulation of the equations. This is a common approach which have been widely used for design of nonlinear controllers, at least within the control theory community. The complex notation, which is closely related to the transfer function language, has been widely used for analysis of sta tionary operation and presentation of two-axis theory using space phasors, but has not become the common approach for the application of recent non linear control theory to electric machines. Of course, the two notations are equivalent4, and nonlinear analysis could be done in either of the two set tings. Recent results from nonlinear control and analysis of induction motors 4 4 To state it more mathematically, this stems from the fact that the metric space of complex numbers is homeomorphic to the metric space of matrices on the form oZg + PJ, where a, p € R, i.e. there exists a continuous and invertible mapping from one space to the other, with a continuous inverse. 2.3 Generalized Rotating Electric Machine 27 \ reported in (Ortega and Taoutaou, 1995) and (Martin and Rouchon, 1995), have taken advantages of using a complex formulation of the induction motor model because of its compactness. 7. Load Torque The model in (2.7) is sufficiently general to include mechanical friction, windage, and pump or compressor loads. The scaling of the hyperbolic tangent function is done to mimic the signum function, replacing the dis continuity in the friction model with a curve of finite slope. This is done to avoid introduction of additional problems with differential equations having discontinuous right hand sides in the stability analysis. However, this model does not provide a true stiction mode, something which may be important if the period of the stick-slip limit cycle is long (Armstrong-Helouvry, Dupont and Canudas de Wit, 1994). The issue of friction compensation and adap tation of other than a constant load torque, is not addressed in this work. 2.3.3 Examples In (Liu et al., 1989) several examples of electric machines described by E(D, R, n, rf) are given. In this section two examples of fully actuated machines, i.e. machines with ns = ne and Me = Tne, are presented. i) The 30 PM synchronous motor (Krause, 1986) has ne = 3 and the parameters De(qm) = Lu + La- Lb cos 2npqm -\La - LB cos 2(npqm - f) —\La — Lb cos 2(npg m — j) Lis + La — Lb cos 2(npqm — ^) — 2La — Lb cos 2(npqm + j) — |La — Lb cos2(ropgm 4- tt) ~\La — Lb cos 2(npqm + j) --^La ~ Lb cos 2(npqm + tt) (2.13) Lu + La — Lb cos 2(npqm + ^) sin Tip (jin P{qm) — sin (npqm — 4^) (2.14) sin(7ipgm + %:) where Lis,La ,Lb are inductance parameters, np is the number of pole pairs and Am is the amplitude of the flux linkage established by the permanent magnet. ii) The 20 PM stepper motor has ne = 2 and the following parameters (Zribi and Chiasson, 1991) L 0 De (2.15) 0 L Km cos(N rqm) HiQrn) r (2.16) Nr sin (Nrqm) 28 Control of The Generalized Electric Machine L is the self-inductance of each winding, and Km is the torque constant. NT is the number of rotor teeth of same polarity. In this case the torque has a term due to the interaction between the permanent magnet and the magnetic material in the stator (detent torque), and 7?(qm) = —Kn sin(4Arrg m), Kd « 5 — 10% of Kmio, where io is a rated current. 2.4 Problem Formulation and Design Procedure 2.4.1 Problem Formulation It will be assumed here that the currents of the actuated windings qs, rotor posi tion qm, and velocity qm are available for measurement. Also, the basic regulated variable is taken to be the generated torque r, which is however unmeasurable since it depends on the variables qT. Notice that the motor speed is related to the latter via a simple linear passive operator (2.9). Thus by controlling r — tl and ensuring passivity in closed loop, as can be done in passivity-based control, a good behavior in the mechanical subsystem can be expected by use of a simple . (PI) outer speed loop. The control problem can therefore be formulated as follows: Definition 2.4.1 (Output Feedback Torque Tracking Problem) Consider the 2ne + 2 dimensional machine model (2.8)-(2.10) S(D,R,Me,p, rf) with state vector [gj, qj, qm, qm]T, inputs u € Rna, regulated output t, measurable outputs qs, qm, qm and smooth disturbance tl £ Coo ■ Find conditions on D, R, fi, p such that, for all continuously differentiable desired output functions ra € Coo with known derivative f<£ € Coo , global torque tracking with internal stability is achieved, i.e. limt—oo |r — ?%| = 0 with all internal signals bounded. Further, for underactuated machines, asymptotic flux amplitude tracking will be required, that is, for a given twice differentiable and bounded function /?(£) > 6 > 0, with known and bounded $(t), $(t), limt—oo j ||Ar|| - (3(t) | = 0 must hold. 000 2.4.2 Design Procedure The rationale of this design stems from the passive subsystems decomposition of Section 2.2 and, “disregarding ” the mechanical dynamics, attempts to control the generated torque r by imposing a desired value to the currents qe. There are therefore three natural steps to follow: 1. Apply the passive subsystems decomposition of Section 2.2 to the machine, to view Ee as the “system to be controlled”, and Em as a passive disturbance. To ensure the latter does not “destroy ” the stability of the loop damping must be injected to Se for strengthening its passivity property to strict passivity. 2.5 Strict Passifiability via Damping Injection 29 2. Define a set of “attainable ” currents qed, i.e., currents for which it is possible to find a control law that ensures Hindoo \\qe — qed\\ = 0. To this end, the energy of the closed loop must be shaped to match a desired energy (storage) function, which is chosen here as Hed := |qe Deqe, with current error defined as Qe := Qe - Qed (2.17) 3. Among the attainable currents choose qed to deliver the desired torque 7%, that is, such that if qe = qtd then r = r^. Finally, give conditions under which limt—oo ||g e - Qed\\ = 0 implies lim(-.oo \T ~ rd\ = 0 with internal stability. 2.5 Strict Passifiability via Damping Injection In this section the first step of our design procedure will be carried out, namely, the decomposition of the model into passive subsystems Ee, Em, and strict passi fiability of Eg. 2.5.1 Feedback Decomposition Proposition 2.5.1 The system H{D,R,Me,p,rj) can be represented as the negative feedback inter connection of two passive subsystems (see Figure 2.1 on page 30) u Qs —Qm r : f~2e —» £>2e ■ (t _ T&) Qm 000 Proof The proof is a corollary of Lemma 2.2.1, noting that the Lagrangian of the electric machine (2.6) can be decomposed into C = Ce + Cm with £e(<7e) Qm) ~ ^eQe + P 9e> Qm) — ~ □□□ 2.5.2 Conditions for Damping Injection Damping is now injected to Ec such that the map from control input to measurable output is output strictly passive. 30 Control of The Generalized Electric Machine X Figure 2.1: Passive subsystem decomposition. Proposition 2.5.2 Consider the subsystem Se(De,Re,p,r}). Assume A.l Re := diag{R s,Rr} with Rs E l71’ Xn% Rr E Rnr X7lr diagonal matrices and Rr> 0 A.2 The nr x nr-dimensional (2,2) block of the matrix W\ = is zero, i.e. (Wa)12 (Wi)21 0nrXnr A.3 The non-actuated rotor components of the vector p are independent of qm, that is dp W2 = , w2s € R71* dqm 0 Under these conditions there is an output feedback of the form u = v + W2sqm - Ki (qm, qm)qs (2.18) such that the map v >qs is output strictly passive for all qm,qm E C2e. 000 Proof The dynamics of Ee is described by (2.8 ), which is repeated here for ease of refer ence Deqe + Wiqmqe + W2qm + Reqe = Meu Closing the loop with (2.18) results in Retie *b Ce (Qm > Qm)<7e “b Res(QmiQm')Qe — MeV (2.19) 2.5 Strict Passifiability via Damping Injection 31 X where the matrices Ce and Res have been defined as Kx 0 Ce -— 2 Res •— Re d" g d* (2.20) 0 0 Taking the time derivative of the total energy of Se, that is He := ^qjDeqe, along the trajectories of Se gives He = qjv - qjResVe Now, let = 4? > sup - ^i)n Then, by using standard matrix results (see Appendix C), the symmetric matrix Res can be shown to be uniformly positive definite in the sense that inf > a >0 (2.22) Qm yQm Integration of He completes the proof. □□□ 2.5.3 Remarks to Conditions for Damping Injection 1. Notice that strict passivity is achieved, via the nonlinear damping term Kxqs, which recovers the positivity of the “damping ” matrix Res. 2. In the case of full actuation, i.e. ns = ne, the damping matrix can be a full matrix, and the required positivity of Res is guaranteed if (see (2.20)) Kx = KT > -Re - \wxqm (2.23) 3. Assumption A.l is a reasonable condition of damping of the nonactuated dy namics which is satisfied in all electric machines. The problem is with condition (2.21), which shows that to overcome the imprecise knowledge of the rotor resis tances, high gains will have to be injected into the loop. 4. A.2 is a decoupled dynamics condition equivalent to requiring that the contri bution to the magnetic coenergy of the terms quadratic in qr must be independent of qm. Physically, this translates into the condition that if there are rotor wind ings, then the rotor flux induced by the rotor currents must be independent of the rotor position. This means that the stator must have uniform reluctance proper ties (non-salient and of uniform magnetic material). This assumption is satisfied by many machines, e.g. classical Park (Youla and Bongiorno, 1980) and polyphase machines. (Willems, 1972) 5. Since the torque (2.10) consists of one component due to the currents, and the other of purely magnetics! origin, and since there is no control on the fields from 32 Control of The Generalized Electric Machine the permanent magnets, it is reasonable to expect that the effect on £e of the flux linkages due to the permanent magnets must be eliminated. This explains the need for assumption A.3. Physically, this assumption also means that if the machine has rotor windings, then the stator must have uniform reluctance properties i.e., if the machine has permanent magnets, then they can only be placed on the rotor. As can be seen from (2.18), the term from the permanent magnets must be cancelled out. The need for this cancellation is a drawback of the scheme. However, the term is generally a vector with periodic functions of a measurable quantity (position), and proportional to a constant which can be precisely identified. 2.6 Current Tracking via Energy-Shaping The “attainable” currents qed in the second step of the design procedure are now to be defined. Proposition 2.6.1 If in (2.18) v, qed € satisfy Mev = Deqed + ceqed + Rested (2.24) then (see (2.17) for the definition of the current error) qe —» 0 as t —► oo indepen dently of qm,qm and the choice of qed . Furthermore, when qed is bounded then qm, qe and r are also bounded. In addition, boundedness of qed ensures that qe and v are also bounded. 000 Proof Rewriting (2.19) in terms of the error signals gives tie + tie + ftie = (2.25) with ^ := Mev - (Deqed + Ceqed + Resqed) (2.26) Then (2.24) implies ij) = 0, and the dynamics of the system is fully described by tie + tie+jtie = 0 (2.27) 4" Rm(lm — T 7% (2.28) These equations are locally Lipschitz in state, and under the assumptions on the desired torque and load torque, they are continuous in t, so there exists t\ > 0 such that in the time interval [0,ti) the solutions exists and are unique. Taking the time derivative of the desired energy function Hed = \q^Deqe along the trajectories of (2.25) results in •JT Red = -qeResqe-, Vt e [0,ti) (2.29) 2.7 From Current Tracking to Torque Tracking 33 It follows from (2.29) and the proof of Proposition 2.5.2 that Hed = -iuReske < -ai&ll2, V*€[0,ti) (2.30) and it can be concluded that IfcWI! < me|l9 %(0)||e-<’-<, V(6[0,ti) (2.31) holds with me = and pe — xma ^(D7i ' w^ere a ^ defined in (2.22). These two constants are independent of ti. From this, and since qed € ££*, it can be deduced that qe is bounded on the open interval [0,*i). Now, it must be proved that it remains bounded also on the closed interval [0,ti]. To this end, notice that the right hand side of (2.28) is also bounded on [0, ti), thus its solution can not grow faster than an exponential, and consequently qm,qm remain bounded W 6 [0,ii]. This in its turn ensures the boundedness of Res, and consequently ||qe|| cannot escape to infinity in this time interval. Since me, pe and a are independent of Urn qe =0 (2.32) and it can be concluded that qe is bounded, which implies that r is also bounded. From this, and the definition of the load torque (2.7), it follows that qm remains bounded. Now, it follows from (2.27) and (2.26) that qe and v will be bounded if qed is also bounded. □□□ 2.7 From Current Tracking to Torque Tracking It is now convenient to take a brief respite and recapitulate the previous deriva tions. The first step of Subsection 2.4.2 was carried out in Section 2.5 where an inner control loop was designed to ensure that (2.19) defines a strictly passive map ping v q8 . In Section 2.6 the second step was carried out, that is, a relationship between the control signals v and qed (2.24) which implies limt_>oo Me - QedW = 0, was established. The third step, to which this section is devoted, demands the definition of qed from the “attainable” set that delivers the desired torque and the establishment of conditions under which current tracking implies torque tracking. Notice that these steps are not straightforward since De, Ce, Res in (2.24) and r in (2.10), depend on qm and qm, thus some additional conditions on the couplings between the subsystems must be satisfied. These conditions are expressed in terms of restrictions on the parameters De,Re, p,,r) of the general electric machine model. 34 Control of The Generalized Electric Machine 2.7.1 Desired Current Behavior Motivated by (2.10) it is proposed to define qed such that, for a given desired torque rd the equation 1 -r • Td 2^ed^l9ed + W2 T-rd - ^eWxqc + q^WiQed + W%'qe with the error signal qe := qe — qed ■ Since W\ and W2 are bounded, it follows that asymptotic torque tracking will be achieved if limi_>co \\qe — qed \\ = 0 with Qed € can be ensured. • It is clear then that to attain the torque tracking objective, bounded qed and v must be defined such that (2.33) and (2.24) both hold. Towards this end, first notice that in the case of fully actuated machines Me = Jn Now, to treat the more difficult case of underactuated machines, when there are not enough control actions to directly set ip = 0 for any given qed , it is convenient to partition ip = [ipj, ipJ]T with ips € Rn* . Since ips = 0 can always be solved with a suitable choice of v, (see the definition of Me), the attention must be focused on the solution of ipr = 0. It will be shown in the sequel that, although (2.24) aii 2.33) are intimately related, additional restrictions on the machine model are required for their simultaneous solution. Interestingly enough, it turns out that these “decoupling ” conditions are (a stronger version of) the well known Blondel- Park (BP) transformation conditions which are fundamental in the analysis of rotating machines (Liu et al., 1989), (Meisel, 1966). 2.7.2 Decoupling Conditions The following definition is in order: Definition 2.7.1 The machine E(D, R, Me, fi, rf) is BP transformable if there exists a current trans formation Ze = P(qm) = Pi e ~u^qe (2.34) such that the dynamics oPE e in these coordinates are independent of qm (but still depend on qm). Pi is any nonsingular constant matrix. If furthermore the matrix 2.7 From Current Tracking to Torque Tracking 35 X U is of the form 0 0 0 0 = -UT = € SS(ne) 0 U22 0 -u£ then the machine is strongly BP transformable. 000 From the structure of the matrix U above it can be seen that strong BP transforma- bility means that the decoupling is achieved by rotating only the rotor variables. As will become clear later, this condition is needed when the rotor circuits are not actuated, as in the induction motor case. In the fundamental paper (Liu et al, 1989) necessary and sufficient conditions for BP transformability are given. Since the definition of the BP transformation given above is slightly different from the one given in (Liu et al ., 1989), and for the sake of self-containment, a simplified version of their theorem is given below. A proof can be found in Appendix B. Proposition 2.7.1 If there exists a constant matrix U € Rn'xn' solution of UDe -DeU = Wi (2.35) ReU = URe (2.36) dW 2 uw2 = (2.37) dq m then B(D,R,Me,p,r]) is BP transformable. In this case the dynamics of Ee (see (2.8)) in the coordinates ze is described by De(0)P^ze + UDMPr^mZe + w2(0)g m + RePr'ze = e ~Uqm Meu = Meu' (2.38) while the dynamics of Em (see (2.9)) is described by DmQm + Rmqm = T - TL (2.39) r = z^P{TUD^)P^ze + W^^P^Ze + r) (2.40) 000 Example 2.7.1 (Park’s Transformation) For a class of electric machines with inductance matrix as in (2.13), the BP trans formation to the dqO-frame is given as (Krause, 1986) cos(npg m) cos(npg m - ^) cos(npg m + ^l) P(Qm) Pi sin(npg m) sin(npqm - %^) sin(npg m + %) (2.41) II 1 2 2 2 where pi = 2/3 (or y/2/3) is a constant. This transformation can also be written as P(qm) = Pi e ~Uq- 36 Control of The Generalized Electric Machine where 1 -i 0 -1 1 1 A—1 71t 10-1 Pi =P1 0 (2.42) V3 i2 -110 i L 2 2 and C7 satisfies (2.35)-(2.37). The inverse transformation for pi = 2/3 is given as cos(npqm) sin(npg m) 1 f-'W = e 1 cos(npqm - x) sin(npg m - 1 (2.43) cos(npg m + x) sin(npg m + x) 1 Several slightly different forms can be found in the literature, (Fitzgerald, Kingsley Jr. and Umans, 1992), (Hemati and Leu, 1992). This stems from the choice of the constant factor pi, which is sometimes chosen to preserve power in the transformed phases (pi = 2/3), or with the objective of making Pi orthogonal (pi = \J%)- 2.7.3 Remarks to the BP Transformation 1. For the purpose of the present work, the key feature of BP transformable machines is that the components of torque that also involves currents become independent of rotor position (see (2.40)) when expressed in suitable coordinates. Notice also that, for constant speed, the electrical subsystem in (2.38) is linear and time-invariant when u' is taken as the new input. This fundamental property has been exploited in the literature to determine stability properties in stationary operation (Verghese, Lang and Casey, 1986). 2. The underlying fundamental assumption for the machine to be BP trans formable, is that the windings are sinusoidally distributed (Youla and Bongiorno, 1980), giving a sinusoidal air-gap magnetomotive force (MMT and sinusoidally varying elements in the inductance matrix De. For a practical machine, this means that the first harmonic in a Fourier approximation of the MMF must give a suf ficiently close approximation of the real MMF. Examples of machines in which higher order harmonics must be taken into account, are the square wave brushless DC motors in (Miller, 1993), and machines with significant saliency in the air gap (Taylor, 1994). The squirrel-cage induction machine is an example of a machine where the squirrel-cage rotor with non-sinusoidally distributed MMF is replaced by an equivalent fictitious sinusoidally wound rotor for analytical purposes, without introducing detrimental effects to controller design. 3. It is interesting to remark that the BP transformation can not be derived from a canonical transformation (Goldstein, 1980) z = Z(q) of the generalized coordinates and momenta. This fact is presented in Appendix B. 4. Since the matrix U is real and skew-symmetric, it follows that e ~u^m is an orthogonal transformation, and the transformation P(qm) is bounded. 5. It is worth to point out that the passivity properties, being input-output properties, are invariant under a change of coordinates on the tangent bundle of 2.8 Main Results 37 the configuration manifold, hence they are preserved for the transformed systems Ee and Em. This can be proved by evaluating the time derivate of along the trajectories of (2.38), and from the fact that the transfer function be tween t — tl and qm is positive real, and the mapping is passive. Thus it is possible to design passivity-based controllers also for the transformed system as done in (Ortega and Espinosa-Perez, 1993) for the case of induction motors. See also (Espinosa-Perez and Ortega, 1995) for further discussion in this respect. 2.8 Main Results The property of BP transformability will now be related with the problem of defining desired currents qed- This is carried out in the following subsections, where underactuated (ns < ne) and fully actuated machines (ns = ne) are discussed separately. 2.8.1 Underactuated Machines, ns < ne Proposition 2.8.1 (Desired Currents for Underactuated Machines) Assume that the machine E(D, R, Me, p, rj) is strongly BP transformable, h = t} = 0, the (2,1) block of De is nonsingular, and that /3(t) is a bounded strictly positive twice differentiable function with known first and second order derivatives. Under these conditions, the following definition of qed satisfies (2.24) and (2.33) for any given Td Qsd iPe)2l [^nr + (De)22 | jpTdU22 + f(7)-^r 1}] Vd Qed — (2.44) Qrd - [^TTdU^ + ATd where Ard is the solution of the differential equation Ard = pffiTdRrU* Xrd + k rd (2.45) with initial conditions such that ||Ard(0)|| = j3(0). Furthermore, l|Ard(*)|| = P(t), Vt > 0 000 Proof The last statement of the proposition follows immediately by taking the time 38 Control of The Generalized Electric Machine derivate of |||Ard||2, substituting (2.45), and using the fact that strong BP trans- formability implies RrU^ + (-Rr = 0. To simplify the notation of the rest of the proof it is convenient to introduce the desired flux Ad = [Ajd, Xjd ]T as Xd := Deqed (2.46) Some simple calculations using (2.26), (2.20) and (2.46) show that5 Ipr = 0 4* XTd + RrQrd = 0 (2.47) Now, notice that BP transformability of the machine implies that (2.33) can be rewritten as (see (2.35)) Td = qJd UDeqed which, in terms of the desired fluxes and currents looks like Td — Qed U\d If further the machine is strongly BP transformable then Td — 9rd^22^rd = —Xjd R~1U22^rd (2.48) where tfr = 0 and (2.47) has been used in the last equation. From this it can be seen that for (2.33) to hold, Xrd must be defined such that (2.48) always holds. It is straightforward to verify that this is the case when Xrd is defined as in (2.45), using = —U22 (see Definition 7.1), the symmetry of Rr and the fact that ||Ard(t)|| = P(t)- The proof is completed using (2.47) to obtain qrd and the definition of Ad to calculate qsd - □□□ The main result for underactuated machines is contained in the proposition below. Proposition 2.8.2 (Underactuated Machines) Consider the machine model (2.8), (2.9), (2.10). Assume that the machine is strongly BP transformable (Definition 2.7.1), ft = 0, rj = 0, (De)2i is nonsingular and A.l - A.3 of Proposition 2.5.2 hold. Under these conditions, there exists a dynamic output feedback controller that ensures global asymptotic torque tracking with internal stability. Furthermore, for all f3(t) (strictly positive bounded twice differ entiable with known bounded first and second order derivatives) the rotor flux Ar satisfies limi_00 | ||Ar|| - j3(t) J = 0. 000 Proof The control law is obtained from (2.18) and (2.26), setting -0s = 0. To this end, 5See (2.5) and the second remark in Subsection 2.3.2 for the physical motivation behind this choice of relations between desired fluxes and currents. 2.8 Main Results 39 the definition of qed in Proposition 2.8.1 is used. Notice that qed is bounded, and can be computed from the available measurements provided fa is known. Convergence of qe —► 0 follows from the arguments of Section 2.6. Boundedness of qed follows from (2.44) and the boundedness of Ard and rd . This establishes asymptotic torque tracking. Electrical rotor flux norm tracking is a consequence of the convergence of the currents to their desired values and the constant norm of Xrd , since Ard — Ar = (De)2l{qsd ~ 2.8.2 Fully Actuated Machines, ns = ne For fully actuated machines Me = Jne, and as previously explained, t/> = 0 can be obtained by a suitable selection of v for given qed and qed - The main difficulty is to find qed such that (2.33) is satisfied. This is done by choosing the desired currents from the BP transformed torque equation, since the matrices relating the transformed currents and the torque are no longer dependent on qm, which considerably simplifies the choice. Proposition 2.8.3 (Fully Actuated Machines) Consider the machine model (2.8),(2.9),(2.10). Assume the machine is BP trans formable (Definition 2.7.1), and A.l - A.3 of Proposition 2.5.2 hold. Let the desired currents and their derivatives be defined as Qed = e^Pf'zed (2.50) Qed = UeUg ’*Pr 1qmzed+ e^P^Zed (2.51) where zed is chosen to satisfy Td -v = ^d P~TUDMP-lzed + W2t (O)Pr^ed (2.52) with Zed, Zed E Under these conditions, use of the dynamic output feedback controller defined in (2.18), with v = Deqed + Ceqed + Resqed (2.53) will ensure global asymptotic torque tracking with internal stability. 000 Proof The expression for the torque in the transformed system is, according to (2.39) r-v = zJP^UDMP-1^ + Wj (0)Pf xie Setting ze = ze — zed and using (2.52) gives r-rd = sfpfrtfDe(0)Pf+ 2?ePiTUDe(0)P-1Zed + W2T(0)Pf(2.54) 40 Control of The Generalized Electric Machine Since ze = P(g m)g c, and P(qm) is a bounded transformation, it follows that lim qe = 0 lim ze = 0 >oo t—>oo limt—>00 T — Tfi It is clear that zed 6 £?e 2.8.3 Remarks 1. Notice that the assumption in Proposition 2.8.1 that the (2,1) block of De is nonsingular implies that the number of actuated and nonactuated windings must be equal. This is the case for typical induction motors, where this matrix is a nonsingular rotation matrix. 3. Equation (2.45) has a solution 6 of the form A,d(t) = M e t'="=,'’-(‘)A,d(0), Ai(t) = ^Td(t),Pd(0) = 0 (2.55) This gives an interpretation of the desired flux in terms of its rotation angle, whose speed {the desired slip) is related to the desired torque. 3. During the derivation of the model and controller, it has been assumed that a VSI has been used to generate the inputs to the actuated windings. It the inverter used is a CSI or a VSI with fast current control, it follows that the inputs to the actuated windings will be the currents qs = qad , where qsd is the vector of desired currents for the actuated windings, as defined in the previous sections. 4. The quadratic form in (2.52) is in general not easy to solve for the components of zed . Examples of solutions for certain machines are given in Section 2.9. 5. For 3 6. The desired transformed currents could be chosen from a similar objective as in the field-oriented approach. The transformed torque equation is generally given as r = c{\d zq ~ Aqzd ] (2.56) 6Recall that if the matrices A(t) and f* A(s)ds commute, then the differential equation x(t) = A(t)x(t), x(0) = xo t > 0, has the solution x(t) = e fo A^ds xo, see Kailath (1980, pp. 595- 596.) 2.9 Examples 41 where c is a constant, and A<*, Ag are d and q components of the transformed flux vector. If AqZd can be made equal to zero and Ad is constant, it will be possible to control the torque by specifying zq. Notice that in this case the angle of the transformation is known, and there is no need to estimate it, as in direct field-oriented control of induction motors. 2.9 Examples In this section controllers are derived for some common fully actuated machines. Controller design for underactuated machines will be the issue of subsequent chap ters, where the squirrel-cage induction motor will be studied. 2.9.1 Synchronous Motors In the last years, synchronous motors, and in particular permanent magnet motors have become attractive alternatives to induction motors in the low to medium power range (Bose, 1993). These machines are generally more expensive than induction motors, but have higher efficiency due to the fact that the rotor losses are negligible. This results in reduced size and cooling problems as compared to induction motors. As low price high-energy permanent magnets become available, the market for these machines will increase even more. The controller given by (2.18), (2.53) with currents satisfying (2.52), can be applied to this type of motor as follows. Using the transformation given in (2.41)-(2.43) with p\ = 2/3, and the model given in (2.13)-(2.14), the torque can be expressed in new coordinates ze — [id, zq, io]r as t — ~2^{(Ld — Lq)z The desired currents are chosen as ' 0 ' ; . - 2li zed — 3np\m 0 from which it follows that ied, ied € £|o, whenever Td, fa € Coo- To satisfy (2.23), taking the uncertainty of the resistances into account, K\ is chosen as Kx — + kTs, k > 0 42 Control of The Generalized Electric Machine The input is then given from (2.53) and (2.18). This approach can also be extended to synchronous reluctance motors 7 with the same inductance matrix as in (2.13). In these machines there are no permanent magnets or windings in the rotor, hence Am = 0 and the torque is given as (Hemati, 1995) T = “rT^d - Lg)ZdZq from which it follows that one of the desired currents should be constant, and the other proportional to desired torque. Also, if the synchronous machine has a field winding on the rotor instead of perma nent magnets, Xm will be proportional to the current in the field winding, which is usually chosen to be constant or varying according to a field weakening objective. The choice of the other desired currents could be done as previously explained. 2.9.2 PM Stepper Motor As another example, in this section it will be shown how to apply the proposed controller (2.18), (2.53) with currents satisfying (2.52), to a PM stepper motor. With the model given in (2.15)-(2.16), the transformation to the dg-frame is (Liu et al., 1989) cos(N rqm) &in(Nrqm) 0 -Nr = e U = P(qm) ~Ug ’ - sin(Nrqm) cos(Nrqm) Nr 0 where U satisfies (2.35)-(2.37). This transformation is orthogonal, and P~l 'g m) = The torque expressed in new coordinates ze = [zd , zq]T is r = Kmzg - Kd sin(4Nrqm) and the desired currents can be chosen as r o * ‘d - i;fo + KDBin(4«,9„)} Notice that (2.23) is satisfied for the choice K\ = kX2 > 0, where k > 0, and r It is worth to point out that the controller in (Bodson, Chiasson, Novotnak and Rekowski, 1993) can be obtained from the passivity-based approach if it is applied 7This motor has been proposed as an alternative to other AC machines, see (Lipo, 1991). 2.10 Concluding Remarks 43 to the full system, without dividing the system into electrical and mechanical parts. As previously pointed out, the underlying assumption of BP-transformability is the sinusoidally distribution of the MMF. It can be discussed whether this is a good approximation in the case of stepper motors, with concentrated windings, significant air gap saliency and often hybrid rotor constructions. The BP trans formation above has however been used in several papers, among them (Zribi and Chiasson, 1991) and (Blanch, Bodson and Chiasson, 1993). 2.10 Concluding Remarks In this chapter the output feedback global tracking problem for a generalized electric machine model has been studied. A passivity-based method was used to design the controller in three steps. First, the dynamics of the machine was de composed as the feedback interconnection of two passive subsystems -electrical and mechanical-. Then, a nonlinear damping was injected to make the electrical subsystem strictly passive. Finally, an energy-shaping controller was designed to make the currents converge exponentially to desired functions, such that the de sired torque is generated. The main contribution is the establishment of physically interpretable conditions on the model, such that the method can be successfully applied. To further relax these conditions, it is believed that passivity ideas must be combined with the powerful new dynamic extension techniques for stabilization of nonlinear systems. Some research along these lines for the robotics problem has been reported in (Brogliato, Ortega and Lozano, 1995). The passivity-based approach gives control schemes which provide global stability results for the closed-loop system. There is no need for observers since unmeasur able states are not used, hence the robustness problems associated with observer- based designs are avoided (e.g. numerical problems from open-loop integrations, unknown parameters). Further, the passivity-based controllers do not introduce singularities, and the need for special precautions to be taken at for example start up is obviated. The performance of the scheme, as measured with the exponential convergence rate of desired currents (and consequently outputs) to their desired values, can be explicitly derived for each machine using the results in Sections 2.5 and 2.6. It follows that the rate of convergence is restricted by the convergence rate of the unactuated dynamics, i.e. the resistance of the unactuated windings. This is a consequence of that additional damping can not be injected into this dy namics, since the involved states are unmeasurable. Since the physical properties of the system are exploited in the controller design and dynamics is not cancelled, the closed-loop system must behave according to the constraints of the system. It must be pointed out that the energy properties of the system are invariant under a change of coordinates, and this gives the possibility of controller implementation in a general dg-frame, chosen from the objectives of minimizing computational burden and increasing numerical robustness. This also allows for implementations 44 Control of The Generalized Electric Machine without measurement of rotor position, if the stator fixed frame is chosen for model representation. To establish a relationship of the controller in this work to existing schemes, it should be noticed that this control input consists of a nonlinear damping term added to the reference dynamics. Henceforth, it can be classified as an indirect vector control scheme, which is the most widely used implementation of field- oriented control (especially well suited for operation close to zero speed (Lorenz et al, 1994)). In particular, for speed control of the induction motor, it is shown in (Ortega, Taoutaou, Rabinovici and Vilain, 1995a) (see also Section 4.6) that the passivity-based controller exactly reduces to the indirect field-oriented con troller under some simplifying assumptions, namely speed regulation with use of a current-fed converter (or high-gain current control), for which the additional problem of stator dynamics is not present. In practice, the assumptions of constant and known parameters will not hold. Re sistances will for instance vary due to temperature changes and the skin effect at high frequencies, and inductances will change when magnetic saturation occurs. To assess the sensitivity of the stability of indirect field-oriented control vis a vis these assumptions, in (de Wit, Ortega and Mareels, 1995) a robustness analysis was carried out. In that paper it was proved that stability is preserved despite large variations in rotor resistance and inductance. In view of the downward compatibility mentioned above, these robustness properties are inherited by the passivity-based controller. It may be possible to enhance performance by identify ing parameters on-line. Globally stable adaptive controllers have been reported in (Espinosa-Perez, 1993), (Ortega and Espinosa-Perez, 1993), both unfortunately re quiring full state measurement. The relaxation of this assumption, together with inclusion of nonlinear magnetics in the model, are challenging topics for future research. Chapter 3 The Induction Motor 3.1 Introduction The induction motor, and especially the squirrel-cage induction motor, has tra ditionally been the workhorse of industry, due to its mechanical robustness and relatively low cost. In a wide range of servo applications with high-performance requirements it has now, due to advances in control theory and power electronics, replaced DC and synchronous drives. In the rest of this chapter the two phase nonlinear squirrel-cage induction motor model1 is first given. The transformation of this model to a general frame of reference is presented in Section 3.3, and leads to the presentation of the principle of field orientation, which has now become the industry standard for high dynamic performance control of these devices. This ap proach for controlling induction motors is explained and discussed in Section 3.4. In Section 3.5 the stator fixed 3.2 The Squirrel-Cage Induction Motor Model For analytical purposes it is common to substitute the squirrel- (single)-cage rotor, which has a uniform conductor distribution, with an equivalent fictitious rotor with the same number of phases as the stator, and sinusoidally distributed conductors. This implies that in the analysis, only the first order harmonic of the rotor MMF is accounted for. Experimental results indicate that analysis and controller designs based on this simplified model will also be valid for the real machine. In cases of 1This is the most commonly used model for control purposes. The derivation of the 2 45 46 The Induction Motor deep bar or double-cage rotors, care should however be taken when modeling these with sinusoidally distributed windings (Vas, 1990; White and Woodson, 1959). Under the same assumptions as in Section 2.3.1, the standard two phase a/? model2 of an np pole pair squirrel-cage induction motor with uniform air-gap has ne = 4, na = nr = 2 and electrical parameters La X2 Lstb ^ •^e(9m) Tj — 0, fl — 0 Lsre~Jn* ,9m Lrl2 RSX2 0 0 -1 Re = -JT 0 RrX2 1 0 cos(n pqm) - sin{npqm) TlpQm 0 * {b ^'y^' sin (npqm) cos (npqm) La ,Lr, Lsr > 0 are the stator, rotor and mutual inductance, Ra ,Rr > 0 are stator and rotor resistances. X2 is the 2x2 identity matrix. The dynamic equations are derived by direct application of the Euler-Lagrange equations (2.2)-(2.3) as in Section 2-3 with the Lagrangian from (2.6). This results in DeiQm^Qe “h Wl{qm^QmQe ReQe Meu (3.1) ^ dDe(q m) _ 0 npLSTjBJnrqm dq m ~ [ -npLar jB-Jnrqm 0 (3.3) qe := [qT,qJ]T = [qs\As2Ar\Ar2]T is the current vector, qm is the rotor angular velocity. Dm > 0 is the rotor inertia. The control signals u = [ui,U2]t are the stator voltages, is the external load torque, and Rm > 0 is the mechanical viscous damping constant. The flux vector A := [Aj\ A;T]r = [Aai, AS2, Ari, Xr2]T is related to the current vector qe via A — De(qm)qe (3.4) from which the second of these vector equations Ar = Lar B-Jn^q3+Lrqr (3.5) is of particular interest for use in the subsequent analysis. Also, notice that due to the short circuited windings of the squirrel-cage rotor, the second of the equations in (3.1) is given by Ar + Rrqr = 0 (3.6) 2In this model the axes for the stator have a fixed position while those corresponding to the rotor are rotating at the rotor (electrical) angular speed. 3.2 The Squirrel-Cage Induction Motor Model 47 3.2.1 Controlled Outputs The outputs to be independently controlled are the norm of the rotor flux ||Ar||, and the torque t. The rotor flux norm needs to be controlled for system optimization (e.g. power efficiency, torque maximization) during changing operating conditions and under inverter limits (Garcia, Luis, Stephan and Watanabe, 1994; Bodson and Chiasson, 1992). Torque control is essential for high dynamic performance. Once torque can be controlled, speed and position can be controlled by simple outer linear loops, at least if the load does not have significantly nonlinear dynamics. The torque can be written as T = = npLsrqJjeJn^mqr (3.7) where the fact that J and e^np9m commute (J'e^Tlpqrn = eJnpqmJ'), and the skew-symmetry of J {JT = —J => xT Jx = 0, V x € R2) has been used. Various other expressions for the torque can be derived from this equation, for instance, solving (3.5) for qr Qr = r-Lsre-Jnpq-qs) (3.8) ±jT and substituting it into (3.7) gives r = np^qJjeJnpq- Xr (3.9) Lr Now, (3.5) can be solved for qa q, = ^-eJn^(XT~LTqr) "ST and then substituted into (3.9) to give t = -np£j\r = —X^JXr (3.10) Kr where qr = — ^Ar from (3.6) has been used. This expression is especially useful for relating the speed of the rotor flux relative to the rotor frame (the slip speed) to the torque. This can be seen by following (Espinosa-Perez and Ortega, 1995) and evaluating | arctan(^) AraArl — ArsArl P A*i 1 \T Af JA, (3.11) II At* 2 p Rr (3.12) np||Ar||: 48 The Induction Motor Hence, torque can be controlled by controlling rotor flux norm and slip speed. This important equation is also of particular interest for passivity-based control designs, for which it is used to define a desired rotor flux reference from given torque and rotor flux norm references. 3.2.2 Measured Variables A common instrumentation of a standard high-performance industrial 3 Currents are often measured using Hall-effect sensors with magnetic compensation, and can give high precision measurements with high bandwidth (DC to 100 kHz) and isolation from measured currents. Velocity measurement can be expensive, and estimation of rotational speed from position measurement with a high resolution digital incremental encoder can give significantly better results than often noisy analog measurements with DC tachome ters (Lorenz and Van Patten, 1991). In some rare cases both velocity and position transducers are used, but the most common approach is to use an encoder, and estimate speed by simple numerical differentiation, or by the use of speed observers driven by reference- or estimated torque and updated from discrete position mea surements (Lorenz et al, 1994). Since rotational transducers and their associated digital or analogue circuits give extra costs and reduce the mechanical robustness of the total system, there has been an increasing interest in schemes without rotational sensors. In some of these cases, speed is estimated by exploiting the influence of the rotational voltages in the dynamic equations. These methods are parameter sensitive with typically low performance at speeds close to zero. Recently, promising results from sen sorless speed and even position estimation have been obtained by modifying rotor slots (introducing magnetic saliencies), and injecting balanced high frequency volt age signals at the terminals (Jansen and Lorentz, 1995). By signal processing of measured voltages and currents in combination with a closed-loop observer, sen sorless control is achieved. This field is still an area of active research, although some successful implementations have already been reported (Ohtani, Takada and Tanaka, 1992; Nilsen and Kasteenpohja, 1995). Additional voltage transducers are also used, not only for some control schemes without rotational sensors, but also in experimental setups for parameter identifi cation (Moons and De Moor, 1995). Signal filtering is then needed, especially with PWM converters. In experimental laboratory setups and in some industrial appli cations (e.g. ships), torque and input/output power are also sometimes measured using current and voltage transducers on the DC-link, and strain gauge rosettes on the motor shaft. Many of the nonlinear control schemes are derived under the assumption of the full state being measured. This is rarely the case, since the rotor currents or the 3.3 The dq -Transformation 49 fluxes, are not directly available for measurement. Measurement of currents in the squirrel-cage rotor is very difficult. Flux sensors (Hall-sensors, extra sensing coils) require expensive modifications of standard motors, and are not robust to mechan ical vibrations and other conditions encountered in rough industrial environments. Also, due to space harmonics, it is very difficult to get good flux estimates by interpolating measurements from a few point sensors. Hence, flux measurement is impractical and against the benefits of the squirrel-cage motor (Leonhard, 1985). It is therefore only used in experimental setups. 3.3 The dg-Transformation As discussed in Section 2.7.2, it is sometimes advantageous to present the equations of an electric machine in another frame of reference than the natural a/3-frame, where the currents can be directly integrated (see the discussion of quasi coordi nates in Section 2.3) to give charges. In this section (Krause, 1986) is followed, and the induction motor model is pre sented in a frame of reference rotating at an arbitrary speed a> a (t). In this model, the natural machine variables (current, voltages, flux linkages) associated with sta tor and rotor windings are substituted with dq -variables associated with fictitious windings. Although this model could be written in terms of new current variables, it is common practice to present the model in terms of fictitious stator currents idq and rotor flux linkages Ad I'd. Ada = e npqm^Xr, idq := = e JBaqs A0 tq (3.13) ud = e ^Ba u Udq •— uq where 9a is the solution of 9a — w*, 0o(O) = 0, with ua a function to be defined later, depending on each particular choice of reference frame. Substituting the expression for qr from (3.8) into (3.6) and multiplying by gives Tr Xr + Xr = Lar e~^np9mqs (3.14) where Tr = jj£ is the time constant of the rotor dynamics. Noting that Ar = e^6a ~npqm^Xdq, computing its derivate, substituting it into (3.14), rearranging terms, multiplying from the left by e~^6a ~npqm^ and using idq = e~J0a qs, finally gives 2"r Adg *t “ TT(jjJa 1lp(lm)\JXdq 4- A dq — ^->sr^dq (3.15) To express the upper two stator equations of (3.1) in this new reference frame, it is started by expressing (3.8) in terms of %d q and A dq- This expression and qs = 50 The Induction Motor eje“idq are then substituted into the stator part of (3.1), written as = u or equivalently as ~ (Lsqs + LsreJn* qmqr) + Rsqs = u Differentiating, multiplying from the left by e~J9a, substituting for -^Xdq from (3.15) and rearranging terms, the stator equations can be written ~^dq + [ua J + 7^2]idq + [npQmJ ~ TjT^Xdq = (3.16) cr = 1 — is the total leakage factor of the motor, and 7 = ■ To express the torque in terms of dq- variables, substitution of Ar = eJ(-da npgm ^Xdq and qs = eJ6aidq into (3.9) gives ±Jsr -T sr \ T = np~jr—‘idq'-'^dq — ■(Xdiq Xq%d ) (3.17) 3.4 Field-Oriented Control 3.4.1 Background The concept of field-oriented control (FOC) was introduced by the German re searchers Hasse and Blaschke about 25 years ago. The ideas presented in (Hasse, 1969) for vector control of a PWM inverter fed induction motor, showed a remark able improvement in response as compared to previous scalar methods, which were based on steady state linear models. The ideas did not seem to be of general inter est, and their realization was quite complex. Some years later a general theory for vector control of AC machinery, based on deep understanding of the physics of the systems and a nonlinear model, was presented in (Blaschke, 1972). This theory is now the de facto standard for high-performance control of AC machinery, due to the strong influence by the pioneering work of Professor Leonhard. The method consists of a nonlinear change of coordinates together with a nonlinear decoupling state feedback. Seen from the viewpoint of modern control theory, FOC was one of the earliest implementations of ideas which are now considered to belong to the more general field of geometric control theory. The fact that it gives superior per formance as compared to methods based on classical linear theory, has motivated and paved the way for other applications of nonlinear control theory. In the rest of this section the rationale behind rotor-flux-oriented vector control3, and the arguments commonly used against this approach are explained. In the following presentation, the definitions of direct and indirect FOC are adopted from Vas (1990, pp. 124-125): 3There are several implementations of FOC, depending on the frame of reference chosen for model representation, e.g. rotor-, magnetizing flux- and stator-oriented. The rotor-oriented implementation is what now commonly is called field-oriented control (Vas, 1990). 3.4 Field-Oriented Control 51 Definition 3.4.1 (Direct versus Indirect FOC) Direct FOC (flux feedback control) refers to an implementation where rotor flux norm and angle are either measured (Hall-effect sensors, search coils, tapped stator windings ) or estimated, while indirect FOC (flux feed-forward control) refers to an implementation where reference values are used instead of measured/estimated values. 000 There is some inconsistency in the literature regarding these definitions, and some times direct FOC refers to the case where flux is measured, while any other ap proach (use of estimated or reference values) is denoted indirect. 3.4.2 Rationale Behind FOC The simplicity of controlling a DC motor is due to mechanical commutation, which ensures that the main flux from the field winding in the stator is always orthogonal to the magnetomotive force created by the current in the armature winding. It follows that the torque, which is proportional to the vector cross product of flux and current, will be proportional to armature current when flux is kept constant. Hence, it can easily be controlled to give high dynamic performance by using two linear loops, controlling flux and armature current. In AC machinery flux and MMF distributions rotate with different speeds, resulting in varying relative angle. This motivates the use of rotating reference frames to analyze the dynamics. Looking at the torque equation for the rotated model of an induction motor in (3.17), T (Adig — A qid) it can be seen that if one of the components of Adq is held constant, while the other is zero, torque can be controlled by controlling one of the rotated stator currents, analogous to the DC motor. To achieve this objective a reference frame in which the rotor flux vector is aligned with one of the axis, should first be chosen. Thus, the angle of the rotor flux vector must be known. Assuming that the d-axis of the reference frame has been aligned with the rotor flux vector, it can be derived from (3.12) by using (3.17), that the rotor flux speed (relative to the rotor frame) can be written in terms of dg-components of flux and currents as Tr Xd Indeed, if it is assumed that Ag = 0, |Aj| > e > 0, and ua is chosen to be the rotational speed of flux relative to the fixed stator frame, Lg r iq Wo flpQm "h (3.18) Tr Xd 52 The Induction Motor the second of the equations in (3.15) gives The first equation in (3.15) becomes TrXd + A d = Lsrid (3.19) Thus, Ad can be controlled by id- For instance, if id = with {3 > 0 a constant, it follows that lim*_oo Ad = ft- Assuming that Ad is held constant, while Xg is still zero, it follows that Lsr r = "'IT and torque can be controlled by the ^-component of the rotated stator currents. The remaining problem is to control the rotated stator currents id q- This can either be done by high-gain current control, or if this does not give satisfactory performance, by exact cancellation of parts of the dynamics in (3.16). Under the assumption of a current-fed machine, the controls are the desired stator currents, which can be written ejea Qsd (3.20) where the d-component of the desired currents in the dg-frame can be defined in a feed-forward way as P C = (3.21) or preferably with feedback from Ad as id = tfi(p)GS-Ad) (3.22) The g-component is defined as ** = n^L^pTd (3.23) where the desired torque 7% for instance can be defined for the purpose of speed4 control as Td = H2{p) (Qrnd - Qm) (3.24) Hi(p) = KiP + Ku^, p := Jj, Kip, Ku > 0, i = 1,2 are Pi-controllers, with control parameters designed to give the desired response of the asymptotically 4 Position control is usually implemented in a cascade manner by adding an outer PI-loop to give the speed reference (Vas, 1990). 3.4 Field-Oriented Control 53 linear dynamics from transformed currents to flux norm and rotor speed in (3.19) and (3.2). The parameters multiplying Td in (3.23) represent only a scaling of the parameters in the Pi-controller, hence they are not explicitly used for implemen tation. Otherwise, with the nonlinear decoupling input Udq — cLs Jidq d" npqmJ — —%2 A dq d- Vdq (3.25) crLsLr equation (3.16) can be written as ~^idq — 'Iidq d" Vdq (3.26) The inputs Vd g can now easily be defined to force id and ig to their desired values. Usually Vdq consist of nested PI-loops as in (Leonhard, 1985) = #,(P)(%-W (3.27) v, = FtWR-W (3.28) where Hi(p), i = 3,4 are also Pi-controllers, and i* d , i* are defined in (3.21) (or (3.22)) and (3.23). Assuming rotor flux is available for measurement, the voltage input for direct FOC may then be implemented as u = e^Udq = &^6a aL s idq + npQmJ — jt22 ^dq d- (3.29) (T LqLj It is also possible to implement high-gain current control by neglecting all terms except Vdq in the equation above. Notice that to compute cv0 from (3.18), the norm of the rotor flux must be strictly greater than zero. This assumption does not hold at startup, giving a controller singularity at this point. To avoid the unwanted controller blow-up for small values of rotor flux norm measurements or estimates, some heuristics is added to the control scheme to make it work, for instance exciting id before iq or setting Ad used in the controller equal to a constant value, when the measured/estimated value is smaller than a certain limit. Reference values have also been used in voltage decouplers aiming at “decoupling ” the stator dynamics in a way analogous to by (3.25) (Vas, 1990). See also Section 4.6.3 for a discussion of this. 3.4.3 State Estimation or Reference Values In the above derivations it has been assumed that all states including rotor flux norm and angle could be measured. In general this assumption does not hold, as 54 The Induction Motor explained in Section 3.2.2. This problem has been a longstanding research topic, and generally there are two ways to solve it. The first one is to estimate the rotor flux angle and amplitude, while the other is to use reference values for these two quantities. As an example of the first method, rotor flux can be estimated in open loop from stator current measurements using the first equation of (3.15), and its angle can be found by integrating (3.18) with the estimated value of A a as Wo = npqm + ^~j- (3.30) 1r Ad where A, = (3.31) The estimated currents id q are computed from measured currents using the esti mated angle Ba and the rotation defined in (3.13). This simple estimation scheme has been used for high-performance control of an induction motor in (Bodson et al.,, 1994a). See (Verghese and Sanders, 1988) for other solutions to the estimation problem. To implement indirect FOC, the same feed-forward way of defining the desired currents as in (3.20) is used, but now the rotor flux speed in (3.18) is also computed using reference values ua = npqm + ~Z-^ (3.32) and 9a is found by integration as before. The indirect approach has become the most popular implementation of FOC since it does not require flux sensors or a flux model, avoiding the need for estimation (Vas, 1990). Also, its performance at low rotor speed is generally better than for direct schemes, for which there are estimation problems present when the Ohmic losses become dominant in the stator equation, and signal integration is problematic (Lorenz et al., 1994). However, all of these schemes are highly sensitive to parameters, especially to changes in the rotor time constant Tr. When this parameter is uncertain, asymptotic decoupling of flux norm and torque is lost, resulting in second order dynamic flux and torque interactions. It is often argued that the indirect approach is much more sensitive to parameter uncertainty than other direct approaches. This issue has recently been addressed in (de Wit et al., 1995), where it is shown that the first requirement of the system, global stability, is preserved for the indirect approach with as much as a 200% error in rotor resistance estimate. This is a remarkably strong result, and to the best of the author’s knowledge, there is no such result for the direct approach, where controllers based on a “nonlinear separation principle” are used. 3.4 Field-Oriented Control 55 3.4.4 Shortcomings of FOC There are mainly two arguments used against the various implementations of FOC: i) It does not give full decoupling: The decoupling between flux and torque (or speed) control is only asymptotic (Marino et ai, 19936). There is only decoupling when the flux has converged to its constant value, giving also a decoupled and linear speed dynamics. This means that simultaneously track ing of both flux amplitude and torque/speed/position using FOC most likely will cause problems, especially if the flux reference is not slowly varying. For instance, as pointed out in (Taylor, 1994), operation in the flux-weakening regime will excite the coupling between flux and speed. This gives undesired speed fluctuations, and could possibly cause instability. The problem is due to the rationale behind FOC, which is to make it behave like a DC motor, where torque is proportional to current when flux is constant. However, this does not give a decoupling between speed/torque and flux norm, which are the outputs a tracking controller must be designed for. For the reasons above it has been natural to operate the machine at maximum constant flux level below rated speed, something which restricts the possibility of power efficient operation. The problem of full (dynamical) decoupling of rotor flux norm and torque (exact input-output linearization), still under the assumption of full state feedback, was solved in (Krzemihski, 1987). The basic idea for the dq- implementation (under the assumption of ideal field orientation, Xg = 0) is to choose Ad and the torque r = np^-Xdig (3.33) L/r as controlled variables, instead of Ad and iq. For a motor with current control loops as previously explained, decoupling can be achieved directly by defining the current reference for iq as i* _ Lr I± ^ 71pLsr Ad Some additional insight into the decoupling problem can be gained if the new nonlinear decoupling terms in the g-direction are found by directly evaluating the dynamic equation for the torque, giving T = 71, A dig + Ad^f Lsr id ~(£+7) Tr Xd T ■Ua A did ^sr'^pL2 n2 . >2 . npL.~p—STsr . . „gmAd+ Adu, (3.35) 56 The Induction Motor where (3.16), (3.19), iq = n^£ 'r yj , and tv0 as defined in (3.18) have been used. From the above it follows that the choice O'LpLs Lar id LepTbp Uo = K + d id + ] (3.36) •Ad Tt A d o L a Lp gives T T + Vq (3.37) and t can be controlled with a linear inner Pi-controller H(p) u, = S"(p)(rd -r) with Td from (3.24). The voltage decoupling term in the d -axis is still given by the first of the equations in (3.25). A linearization from voltage inputs to speed and square of rotor flux norm in the case of the stator fixed a&-model, was also derived in (Krzeminski, 1987), using the new powerful tools adopted from differential geometry (See (Marino et al., 19936) for a clear presentation). It was shown that the implementation complexity of this scheme is no greater than for FOC. Experimental results from various implementations (stator fixed frame of reference, decoupling control for square of rotor flux norm and speed or torque) of this scheme have been presented in (Kim et al., 1990; van Raumer, Dion, Dugard and Thomas, 1994), and (Bodson et al., 1994 o) (dg-frame implementation with high-gain current control). ii) It is based on “a nonlinear separation principle”: It is well known that for linear time-invariant systems, the problem of stabilizability with only partial state measurement can be solved by use of an observer, at least if the system is both stabilizable and detectable. The stability of the total system is only dependent on the observer stability, and the stability of the feedback control system when full state measurement is assumed. This is the so-called (deterministic) separation principle. Motivated by the successful linear controller designs, this principle has been carried over to controller designs for nonlinear and nonautonomous systems, giving a unonlinear separation principle”. Generally no theoretical stabil ity analysis are given for systems resulting from such designs, and the per formance is only verified through simulations or experiments. It can be showed that even for very simple nonlinear systems, with an exponentially convergent state observer5 and known parameters, the approach of separate observer-controller design can lead to explosive instability in terms of finite escape time, when estimated states are used as if they were real states, even if the controller with real states would ensure global exponential stability of the total system (Krstic et al, 1995). Analysis of separate observer-controller 5This is generally the best convergence that can be expected. 3.5 The 06-Model 57 \ design using linearization techniques leads to local stability results, even un der the assumptions of known constant parameters. With only local results, it is very difficult to predict what will happen for certain choices of controller parameters and initial conditions under real operation, and the designer is left with a trial and error approach to controller design, making tuning an often time consuming and difficult task. It must also be pointed out than when observers are used for implementation, asymptotic properties are added to the schemes, i.e. there is only exact decoupling after the estimates have converged to their real values. Even if some heuristics are implemented to avoid instability, estimation er rors can lead to severe effects on system performance and power efficiency, especially during high speed transients. This is why much of the research in nonlinear control of induction motors has aimed at interlacing design of controller and observer, with additional terms in the controller or the ob server to counteract for estimation errors, and obtaining global stability re sults. These schemes have the benefit of guaranteeing stability (under given assumptions), and hence giving a priori information about which modifica tions can reasonably be expected to work and which ones will probably not. In the cases where exponential stability results are established for a nominal system, robustness result for bounded perturbations can also be established (Khalil, 1992). It must be pointed out that even if the robustness and stability issues of the many implementations of FOC are not rigorously established, this scheme has through years of practical experience been developed to a level giving a performance which is difficult to compete with for other nonlinear approaches, at least for nonlinear designs implemented directly from theoretical desk designs. Years of experimental work has resulted in modifications of FOC based on experience and intuition, and problems of parameter uncertainty, flux saturation and other unmodeled dynam ics can be compensated for by several ad hoc methods. Even if these methods are based on “nonlinear certainty equivalence”, using estimated parameters as if they were real parameters in the controller, and proposed without theoretical justifi cation, the most important aspect from an application oriented view is that they work and improve performance. Rigorous analysis of the resulting schemes are left as challenging problems for the academics. 3.5 The afc-Model Another choice of reference frame which is of interest is the stator fixed frame, where rotor variables are associated with fictitious stationary windings (Stanley, 1938). The model is of special interest because it is widely used in the many implementations of controllers based on backstepping (Krstic et at., 1995) and feedback linearization (Marino et ah, 1993 b). This model is denoted the ah-model and can be derived from (3.15) and (3.16) by setting u)a = 0, 6a = 0. Since this 58 The Induction Motor model is usually written in state space form instead of the second order Euler- Lagrange form in (3.1)-(3.2), it is of interest to rewrite it as Xdg = npqmJXdq — jT^dq + -^rUg (3.38) d . \npqmJ — jT^2]Xdg + (3.39) O LgLf (3.40) Instead of subscripts d and q, a and b are used, and by setting x = [qm, Xa , Xb, ia , ib]T [xi,Z2,X3,a:4,Z5]T, ua b = [v >i,U2]t the model can be rewritten as x = f(x) + giUi+g 2U2 (3.41) where Dmll iX2X5 X3X4) DmTL — jrX 2 — UpX\Xz + jtLstX 4 fix) npX\X 2 — ^XZ + -^rLsrX 5 T,b CL*!* X2 + UV cl7l,X iX z ~ -n■PoL.7 lt XlX2 + Tr ffLUr XZ ~ 7 X5 . 0 ' 0 ■ 0 0 0 II 9i = $ 0 1 0 trL, 1 0 . The state space ah-model of the induction motor was used in this work only for implementing a simulation model of the induction motor. It is well suited for this purpose, since it has no rotational transformations. It is also written down for the purpose of comparison with the second order Euler-Lagrange model which has been used in this work, since it is believed that this second order model is rather uncommon within the motor control literature. While the Euler-Lagrange model has a structure which can be used for passivity-based controller design, the model in (3.41) is a set of mathematical equations which are well suited for controller design and analysis using tools from geometric control theory. 3.6 Concluding Remarks In this chapter the dynamic equations for an np pole-pair squirrel-cage induction motor with smooth air-gap have been presented. The second order model structure follows naturally from the Euler-Lagrange approach for modeling, and is well suited for passivity-based analysis. As pointed out in Section 2.3.2, there are parameter variations due to heating and magnetic saturation, and these have not been been specified in the model given in 3.6 Concluding Remarks 59 this chapter. It is well known that the rotor resistance Rr can vary significantly, and ability to compensate for this variation or at least to guarantee stability despite variations, is of outmost importance for any control design. For comparison purposes, the more commonly encountered stator fixed ab-model has also been derived. This model is used extensively for analysis with geometric tools, and in a large number of implementations of controllers based on other designs than the passivity-based. For implementation purposes, this choice of model is sometimes preferred since measured quantities and controls need not be rotated to other frames (Krzeminski, 1987). It is however important to be aware of bandwidth considerations for controllers in such implementations. While currents will be constant in a dg-frame at stationary conditions (constant flux and speed), they will be oscillating with a speed dependent frequency in the stationary frame. The bandwidth must consequently be higher for current controllers if they are implemented in the stationary frame. It is worthwhile to point out the generality of the model presented in this chapter. It could be interpreted as an equivalent model of the usual 3 Chapter 4 Observer-Less Control of The Induction Motor 4.1 Introduction In a recent paper (Espinosa-Perez and Ortega, 1994) an output feedback globally stable speed tracking controller for induction motors was presented. The perfor mance of the scheme is limited by the fact that the convergence rate of the speed tracking errors is determined by the natural mechanical damping of the motor. In this chapter it is shown how to overcome this drawback by using a simple linear filtering of the speed tracking error to inject mechanical damping into the closed loop. This allows to improve the transient performance of the scheme from (Espinosa-Perez and Ortega, 1994) in position and speed tracking applications, without significantly increasing the computational requirements. The global sta bility results are preserved, and can be proved using a Lyapunov-type argument combined with use of the Gronwall’s inequality. The result from (Espinosa-Perez and Ortega, 1994) is also extended from rotor flux norm regulation to tracking. Further, it is shown that if the inverter can be modeled as a current source and the desired speed and rotor flux norm are constant, then the passivity-based scheme exactly reduces to the well known indirect field- oriented control scheme, hence providing a solid theoretical foundation to this popular control strategy. This connection between indirect FOC and passivity- based control is then used to extend classical indirect FOC, which is only valid for rotor flux norm regulation, to include the case of rotor flux norm tracking, with global stability results. For the sake of self-containment, the controller is derived in detail, and of previous results, only the basic model equations from Chapter 3 will be needed. Comments to the relation with the more general results from Chapter 2 will be given at 61 62 Observer-Less Control of The Induction Motor appropriate points. The rest of this chapter is organized as follows. In Section 4.2 a review of the design used in (Espinosa-Perez and Ortega, 1994) is given. Section 4.3 explains the definition of desired dynamics needed to achieve torque and flux tracking. The main result of this chapter, a globally defined observer-less speed and rotor flux norm tracking controller, is then given in Section 4.4, followed by its proof in Sec tion 4.5. In Section 4.6 the proposed controller is compared with classical indirect FOC, and a dg-implementation of the scheme is given. Definitions of the desired rotor flux norm for minimization of power losses and to mimic field weakening are proposed in Section 4.7, and finally simulation results and concluding remarks are given in Sections 4.8 and 4.9. 4.2 Review of Design Method First, to get the necessary skew-symmetric properties of the model which are fundamental for the design, the procedure in (Espinosa-Perez and Ortega, 1994) is followed, and (3.1) (see p. 46) is rewritten as &e(.Qm)Qe 4" Ce{,QmiQm)Qe 4" R(QmiQm)Qe = MeU (4.1) where ’ 0 npLsrJeJn* qm 1 . (/e(9m, 9m) 0 0 j9m [ % o R(Qmi Qm) [ -npLsrJe~Jn* qmqm Rrlz Note that zT j.De(g m) - 2C'e(g m,g rn)| z = 0, Vz € l4. However, the symmetric part of the “damping ” matrix R{qm,Qm)> [R(Qm>Qm)] sy = 2 {-^(9mj Qm) 4* R (9m?9m)} is not positive definite. In (Espinosa-Perez and Ortega, 1994) passivity-based controllers for torque and speed tracking, that do not rely on state reconstruction, were designed for (4.1), (3.2) as follows1. First, it was proved that the system dynamics can be represented as the negative feedback interconnection of the following passive (electrical and mechanical) subsystems (see Fig. 2.1 on p. 30) u Qs £m : (T - tl) w. qm —Qm T 1The model considered in (Espinosa-Perez and Ortega, 1994) is obtained by applying a change of coordinates to (4.1), (3.2). See (Espinosa-P6rez and Ortega, 1995). 4.3 Definition of Desired Dynamics 63 Second, an inner feedback loop which ensures that the electrical subsystem defines a strictly passive map from control signals to stator currents, was designed, This was achieved via the injection of a nonlinear term to the “damping ” matrix. Third, the passivity-based technique was applied to the electrical subsystem leaving the feedback subsystem as a “passive perturbation”. As explained in (Ortega and Espinosa-Perez, 1993) this last step involves the definition of the desired closed- loop energy function whose associated (target) dynamics evolves on a subspace of the state space ensuring zero tracking error. The overall procedure leads to an in ternally stable dynamic output feedback controller that ensures global asymptotic torque or speed tracking. The mechanical part of the system is described by the equation DmQm RmQm — ^iAetQrn) ~ TL — 2^e and unfortunately, to avoid acceleration measurement in speed tracking, the stabil ity proof in (Espinosa-Perez and Ortega, 1994) requires the mechanical damping to be nonzero, i.e., Rm > 0 instead of Rm > 0. Furthermore, it follows from the analysis in (Espinosa-Perez and Ortega, 1994) that the convergence rate of the speed tracking error is bounded from below by the motor time constant Some simulations that reveal the performance degradation for small mechanical damping constants are presented in (Espinosa-Perez, 1993). To improve the re sult from (Espinosa-Perez and Ortega, 1994), Rm is set to zero in the rest of this chapter. The main objective of the work presented in this chapter is to remove this re striction. To this end, it is shown that by simple linear (strictly proper) filtering of the speed tracking error, mechanical damping can be injected into the closed loop without acceleration measurement. This renders the transient behavior of the scheme independent of the natural mechanical damping, which can now be assumed to be zero, thus enhancing the controller performance in position and speed tracking applications. The computational requirements are not significantly increased, and the global stability properties are preserved. Interestingly enough, no conditions on the filter bandwidth are imposed. 4.3 Definition of Desired Dynamics In this section it is explained how torque and rotor flux norm tracking of time- varying references can be achieved by forcing rotor flux and its derivative to track references derived from time-varying torque and flux norm references. The problem is then recast in terms of tracking desired currents qrd and qSd • The result in this section is a direct extension of Corollary 2.1 in (Espinosa-Perez and Ortega, 1995) to the case of a time-varying rotor flux norm reference. Motivated by the relation between torque and rotor flux speed and norm in (3.12), 64 Observer-Less Control of The Induction Motor a desired rotor flux Ard is defined as Rr J Pd Xrd — Pd{ 0) = 0 (4.2) 0 , Pd Tip/?2 which is the solution of (see (2.55) on p. 40) R 3 ^(0) Xrd — '7flT'd($)3X Td + — XTdt Xrd(0) = nPP P 0 The norm reference P(t) = ||Ard|| is a strictly positive twice differentiable and bounded function, with known and bounded first and second order derivatives. Defining Ar := Ar — ATd and using this in (3.10) gives Tip -T t = Xr $X T + Ar 3Xrd + XrdJXr 4- Xrd JArd'} Rr Furthermore, |P{t)pd [ ~ sin(pd) cos(pd) ] +$(t) [ cos(Pd ) sin(pd ) ]|<7 | gfn(prf) cos(pd) [ - sin(pd) cos(pd) ] J sin (pd) Td Hence, noting that if Ard, Ard are bounded, it can be concluded that Ar —» 0 and A,. -+ 0 implies -r —» 7% and | ||Ar|| — /3(t)| -* 0, thus both torque and rotor flux norm tracking follows. If the desired relation between fluxes and currents is defined to be consistent with the motor model, requiring Ad — De{Qrn)Qed (4.3) Xrd RrQrd ~ 0 (4.4) the definitions of desired rotor flux can be translated into desired currents by using (4.4) and the second of the equations in (4.3) Ard = Lsre~Jn* qm qsd + LrC[rd (4.5) which gives £ Td J + S-7 Z2 | Ard = ~eJpd (4.6) * rd - ~KXrd ~'\np^ RrP ^0 4.i = (Xri - Lrgrd ) = -L ((1 + M)l2 + j ) Xrd Li$r J^sr \ RrP TlpP I P + T?j 0 (ftp flfm fid ) (4.7) 4.4 Main Result 65 It then follows by using (3.5) and (3.6) together with (4.5) and (4.4) that Xr — Xrd = Lar e '^npqm (qs — Qsd) + Lr( 4.4 Main Result Proposition 4.4.1 (Speed and Rotor Flux Norm Tracking) Consider the induction motor model (4.1),(3.2). Assume: A.l Stator currents qs, rotor speed qm and position qm are available for measure ment. A.2 All motor parameters are exactly known. A.3 The load torque rz(t) is a known 2 bounded function with known bounded first order derivate, such that |r&(t)| < c% < 00, Vt € [0,00). A.4 The desired rotor speed qmd(t ) is a bounded and twice differentiable function with known bounded first and second order derivatives, such that \qmd(t)\ < c2 < 00, Vt € [0,00). A.5 The desired rotor flux norm 0(t) is a strictly positive bounded twice differen tiable function with known bounded first and second order derivatives, such that 0<6i Let the controller be the nonlinear dynamic output feedback u = Lsqsd + LsreJnp9mqrd + npLar JeJnpqmqmqrd + Ra qsd ~ Ki(qm)qa (4.9) with ’ -t (a+fef ^+i^LJ) ' Qsd 4ed (4.10) Qrd ” + I&Z2) Ard 2 This assumption is made for simplicity, the result can be extended for the case of unknown linearly parameterized load torque (Ortega and Espinosa-Perez, 1993), or in the case of a constant load torque by using a projection to keep the estimate bounded as in (Espinosa-Perez and Ortega, 1994). 66 Observer-Less Control of The Induction Motor where Qs ~ Qsd gr qrd n2L2 Ki(qm) ■= 4"$n + h, 0 < e < Rr, ki > 0 Td(z) = Dmqmd -z + tl (4.11) and controller state equations Rr ^(0) Xrd — rri^d^Zl^\ rd n^rdi Ar Under these conditions the control objective of global asymptotic speed and rotor flux norm tracking is ensured, i.e., lim 'qm = 0, lim | ||Ar|| - j3{t)\ = 0 t—*00 t—*00 with all internal signals uniformly bounded. 000 4.5 Proof of Main Result By substituting (4.9) into (4.1) and using (3.6), it follows that the closed loop system is fully described by De(qm)Qe + Ce(qm, qm)q e + [R(Qm,qm) + lC(qm)] he = 0 (4.14) -Dm9 m = -z + r{qe,qm) - rd (z) (4.15) z = —az + bijm (4.16) ^rd — t^02 'rd(z)^f\rd 4" ^j\rd (4.17) where JC(qm) := diag{Jfi(gm)J2,0}. For later convenience, (4.15) and (4.16) are rewritten as i 9m — Dm (r - Td) i b —a 0 z x = Ax + B(t - Td) (4.18) The matrix A is Hurwitz for all positive values of a and b. Under assumptions A.3 — A.5 the system (4.14)-(4.17) is locally Lipschitz in the state [qe ,qm,z, A^JT and continuous in t. This condition ensures that there exists 4.5 Proof of Main Result 67 a time interval [0,T) where the solutions exist and are unique. First, ||Ar(j|| = (3{t), it 6 [0,T), and it is consequently bounded. Now, consider the following quadratic function Vi = i qcD'(qm)i' whose time derivative along the solutions of (4.14) for all t € [0, T), is given by ^ 9m) + K(Wsy (4-19) where the skew symmetry property mentioned above has been used, and \TJ(n a \ 4- 1C(n )1 — Ra%2 + K\{qrn)^2 2nP^-JarSf e'^ p9m 9m 9m) + K.(9m)J„y - Je-j„ R-I, This matrix will in the rest of this chapter be denoted3 Rea . The matrix is strictly positive definite, uniformly in qm, namely Res > 6X4 > 0 (4.20) where J4 is the 4x4 identity matrix. This can be proved by using standard results from matrix theory and the facts that Rr > 0 and Je~Jnj,qm = B~Jn^qrn J, which leads to (4.20) holding if and only if nlL2sr Rs + Ki(qm) — Qm > f (4.21) 4(Ar-6) See Section C.3 for a detailed derivation of this requirement, which is fulfilled with the chosen definition of Ki(qm). Therefore, from the above and (4.19) it follows that = -qeReake < " inf Amin{Res}||5e||2 Qm iQm and it can be derived that for some constants me > 0 and pe > 0 independent of T HWl < vt € [0,T) (4.22) Notice that, unless qm escapes to infinity in finite time, limt^oo qe = 0. Thus, it must be proved first that this is not the case by showing that the input (t — tj) to the linear filter (4.15)-(4.16) is linearly bounded by the filter state. To this end, notice that the desired torque t& can be written as Td = ^qJdWl(Qm)qed 3Notice that the symmetric matrix above is exactly the matrix Res defined in (2.20). 68 Observer-Less Control of The Induction Motor hence it follows that r-Td = ~^Wi{qm)qe + £Wi(qm)qei (4.23) and the following bound holds TlnL,pJ-'sr /,| ~ m2 |r-rd| < (ll?,lf + 2||e«|||M) (4.24) On the other hand, writing the desired currents in (4.10) as Z% + TL)J + (1 + eJnrq™\rd Qed ~ (j^pziDmQmd + Tl)J + Kd + (4.25) 1^0*^ ^rd := a\(t) + a 2(t)z (4.26) and noting that llaiCOII < {Dmqmd + Tl)J + '^y2'2|| 2)||Xr ll«WII < ^((l^f)2 + <6= <0= yields ||gcd|| < ai +a.2\z\. Replacing this bound, together with ||ge|| < me||ge(0)||, Vt € [0,T) in (4.24) it follows that \r-rd \ < VzhL(m2||£e(0)||2 + 2me||g e(0)||5i) + npLar me||5C(0)||a2|z|, Vt € [0,T) This last inequality proves, via Gronwall’s inequality, that z, and consequently qm, can not grow faster than an exponential in the time interval [0,T). Moreover, since all the constants in the above bound are independent of T, this argument can be repeated to extend the time interval of existence of solutions to the whole real axis. Having proved that (4.22) holds as t —> oo, it must be proved that this implies that limt_>oo x = 0, with qed bounded. Inserting (4.23) into (4.18), and using (4.26) to express qed , results in the system ■JT, x = Ax + 9m o o Z + 2g W,(Sm)ai(t)} + 2D 0 t X Ax + B(t)x + c(t) 4.5 Proof of Main Result 69 Calculation of norms gives ||B(t)|| < -j^npLsrOL2me\\qe($)\\e pct, and ||c(t)|| < nib’J{mell9 e(Q)lle~p oo, it follows have that x —► 0. Further, from (4.26) it is established that qed is bounded. The proof of asymptotic rotor flux norm tracking follows from (4.8) and conver gence of current errors to zero. ODD 4.5.1 Remarks to the Observer-Less Controller 1. Comparison with (Espinosa-Perez and Ortega, 1994) Notice that since qed is required for implementation of the controller, fd is needed. Thus, if Td includes qm, acceleration must be measured. To overcome this problem, in (Espinosa-Perez and Ortega, 1994) it was assumed that Rm > 0, and the speed control strategy 7% = Dmqmd + RmQmd + Tt was proposed. Two drawbacks of this scheme are that it is open loop in the speed tracking error, and that its convergence rate is limited by the mechanical time constant Defining the desired torque Td as in (4.11) with z from (4.13), allows for effectively feeding back the speed tracking error without acceleration measurement. Further, the convergence rate is independent of the natural mechanical damping. 2. Comparison with Results from Chapter 2 Notice that the controller (given the desired torque) follows from the results in Propositions 2.5.2 and 2.6.1. For the completeness of the result in this chapter as an extension to speed/position tracking, the proof was carried out in detail. 3. Position Control It is easy to see that choosing the desired torque in the controller above as Td = Dmqmd - z- fqm + tl (4.27) where qm 9m — Qmd, yields global asymptotic position tracking for all positive values of a, 6, /. In this case, the error equation (4.18) has Qm ' 0 1 0 ' ' 0 X = Qm , A := ~DZ 0 , B:= Z 0 b —CL 0 The matrix A is Hurwitz for all positive values of a, b, /, and the proof of global asymptotic rotor flux norm and position tracking follows verbatim from the proof of the main result. 4. Comment on Integral Action in Stator Currents It is of interest to see if the controller can be robustified to compensate for 70 Observer-Less Control of The Induction Motor unmodeled dynamics by adding an integral term Uj -KIs / qs dt, KIs > 0 to the control in (4.9), with Kjs a scalar or matrix. To study the stability of this system, a term \ qs dt Kjs qs dtj is added to V\. Computing the derivate of this new V\ with the new closed loop system (see (4.14)), De(qm)q e + Ce(qm,qm)q e + [-R(9 m, q-m) + JC{qm)] qe = (4.28) with o results in Vi It follows that qe and the integral term will be bounded for a closed time interval. This is enough to complete the previous details proving that there is no finite escape time, and it follows that qe 6 n L\ and J0* q3 dt 6 C2^. It is however not enough to claim convergence of current errors to zero. For this, the additional requiremer of qe € will be sufficient. From (4.28) it can be seen that this is the -e if the speed is bounded, something which is difficult to establish rigorous without a function V\ which explicitly includes also qm. Even though simulations and experiments indicate that the use of an extra integral term in stator currents will result in global speed convergence with all internal signals bounded, the rigorous proof of this is a technically challenging problem. 4.6 Comparison with Indirect Field-Oriented Con trol 4.6.1 Current-Fed Motor In this section it will be shown that, in speed and rotor flux norm regulation applications, and under some simplifying assumptions on the motor model, the 4.6 Comparison with Indirect Field-Oriented Control 71 controller presented in the previous section exactly reduces to the well known indirect field-oriented control (FOC). In typical applications of FOC it is assumed that the inverter can be modeled as an ideal current source. In this kind of machines the error between stator currents and their references is made singularly perturbed with respect to the rest of the electrical dynamics, and it can be assumed to be arbitrarily fast, i.e., u = j(g s — gs d), for some e —» 0, thus qs actually becomes the control variable. It is also assumed that the norm 0 of the desired rotor flux is constant 0 = 0). Using these assumptions, it follows from (4.10) that the passivity-based control reduces to — CJ npg„ {nppTdJ + l2)Xrd Qs = Qsd 1 (4.29) Lsr 0 Ard (4.30) 0 := Td = DmQmd - GQm + TL (4.31) where fid is the solution of Pd = Tdi Pd(0) = 0 (4.32) npP and \jg := [0 0]r is the desired rotor flux represented in a frame with the d-axis directed along it. Notice that qsd and qrd are no longer needed for controller implementation, and consequently rd must not be used. Thus, there is no need for the filtered speed error in (4.31). Inserting (4.30) into (4.29) results in (see also (4.7)) Jlu eJ(npqm+pd) Qsd (4.33) L,rnvpTd On the other hand, assuming that the desired speed is constant (g m Td = ~(a+~)q m, 7/ > 0 (4.34) The controller (4.32), (4.33), (4.34) exactly coincides with the classical indirect FOC for current fed machines, see (3.32), (3.20), (3.21), (3.23) and (3.24). This result was first rigorously established in (Ortega et al, 1995 o), and that paper should be consulted for further information. See also (Ortega and Taoutaou, 1995) for a proof in terms of complex notation. 72 Observer-Less Control of The Induction Motor 4.6.2 An Improvement of Indirect Field-Oriented Control Classical indirect FOC is only valid for constant rotor flux norm, but using the re sults from passivity-based control, it can now be extended to include the important case of rotor flux norm tracking. Substituting for \rd in (4.10) (with fi(t) a time-varying function) using (4.30) results in ,jea + irf-P Qsd Lar ^ := Lar np0Td i: Oa U)a dt, &a(0) — 'R'pQm(0) U)a Qm *t" Pd with pd from (4.32). This remarkably simple modification of classical indirect FOC to obtain a con troller which ensures also global rotor flux norm tracking, motivates the following corollary. Corollary 4.6.1 Consider the induction motor model (3.1) and (3.2). For current-fed machines the controller of Proposition 4.4.1, which reduces to L- + TCIT 0 Qsd eJ(nPqm+p lim |r - rd| = 0, lim | ||Ar|| - /3(f)| = 0 t--*00 (—*00 holds for all initial conditions with all signals uniformly bounded. 000 4.6.3 Voltage-Fed Motor For the VSI case, it is of interest to rotate the control in (4.9) to a dg-ffame, not only for comparison with indirect FOC, but also for implementation purposes. Using the results from Section 3.3, the equivalent of (4.9) can be derived in the reference frame of the desired rotor flux. As previously pointed out, (4.9) consists of the desired stator dynamics and a nonlinear damping term. By using (3.13) in terms of also desired quantities i* dq = 4.7 Definitions of Desired Rotor Flux Norm 73 e~J9a qa d, X* dq = e~J(6a npQm^Xrd, and comparing (4.9) with (3.16), it can be seen that u = Lsqsd + LsreJnpqm qrd + npLsrJeJnpa J + 7^2}i* dq + ™£~- [npQmJ ~ jT^2]^9 | (4.36) d a — J Wa dt, 8 a( 0) = n.p5’m(0) Jo U)a = Tip 9m "4* Pd where pd is defined by (4.32), and i* dq = T^\P + fcA ^rd ]T,\}q = [/3,0]T Motivated by the successful use of the decoupling terms in (3.25) in combination with estimated states, there have been several attempts to implement equivalents by using reference values for the states, see Vas (1990, 152-158). The first prob lem with these approaches is that the term crLs-^idq is either neglected, resulting in unwanted torque overshoot, or has to be computed in an ad hoc way by nu merical differentiation. The other problem is that these schemes are only suited for constant rotor flux norm. The indirect scheme proposed in this chapter is an alternative method which do not possess these problems, and which also gives a theoretical explanation of why these reference value “decoupling ” approaches actually work. 4.7 Definitions of Desired Rotor Flux Norm In the following two sections examples of how to design the desired rotor flux norm (3 are given. The first result is a direct adaptation of the results in (Vedagarbha et al, 19966) to the passivity-based controller for minimization of steady state losses. The other is an example of how the well known flux weakening approach can be mimicked. 4.7.1 Minimization of Steady State Losses To minimize power losses in the motor, the function Pioss — y 9^ 7"9m (4.37) supplied power mechanical output power is considered. The control u is first eliminated from Pioss by using (3.1) and (3.7), which gives P|o,s = (l, - qjq, + (r, + qjq. - hgztf e (4 3s) 74 Observer-Less Control of The Induction Motor Detailed derivations of this and the following expressions are given in Appendix D. It is assumed that the stator currents qa have converged to their desired values, given in (4.10), and all occurrences of qa and qa can be substituted by their refer ence values. This gives - L;,Z.r j j + -Pi OSS — LlL, L,Lr - Rr L3Rt + 2LrRa , RrLrLs + L2Ra l2 W + . PrLgr RyLs 2 P , Pa p2 nlLL d P3 L2srP vp~sr Lr2 Ls LrL2ar . Lr2 Ra + RrLl 1 + -TdJd. + ' ■m (4.39) IP2 The minimization of the above criterion with respect to a general time-varying strictly positive function4 (3{r,i,fd), is a nontrivial dynamic optimization problem. As a first approach, only the task of minimizing the above expression at stationary conditions with constant torque is considered here. In this case /? can also be considered as a constant, which gives _ Pa 02 , LyPa RrL2r 2 1 vj + n%Vir d lP (4.40) By evaluating and setting = 0, the only extrema is found to be ^ ,Pr Plr kd| (4.41) n2 Rs n\ Evaluation of the second order partial derivative of PiOSs with respect to f32 gives ^PlQSS Ra L2 + RrL2r 2 1 TdW > 0, V/3 > 6 > 0 This implies that the function (4.40) is convex in f32, and y3 2pt is a global minimum. It must be pointed out that the loss model is very simple, with no core losses, current/voltage limits or effects from nonlinear magnetics included. Despite this, the model indicates that to minimize power losses, the flux norm reference should be proportional to the square root of the desired torque, at least as long as the resulting reference is below the maximum allowable value. Other aspects of power efficient control, including effects of core losses, have been considered in (Dyrset, 1995). 4Notice that the derivative of desired torque is also needed for dynamic optimization. 4.8 Simulation Results: Observer-Less Case 75 4.7.2 Flux Weakening In this section it is explained how the controller can be used to mimic the well known flux weakening approach for operation in the constant power region above nominal speed. To get the desired smoothness properties of the norm reference, a linear second order filter is introduced, Xi(t) 0 1 Xi(t) + &ef(() (4.42) . ^2(t) . —2 £u>n x2(t) fl(t) = Xi(t), fi(t) = x2(t) with [xi(0) £2(0)]t = [A-ef(O) 0]T, C = 1 and un > 0. The input signal to the filter (4.42) is given by An > QmN — Qmd — QmN fire (4.43) i |9md| > QmN with ySjv the desired constant nominal value for the rotor flux norm and qmN the base speed. With the given value for the damping constant £, it is ensured that the output of the filter, /?, will always be positive, and singularity points in the controller are avoided. The flux reference has been defined in a feed-forward way from reference speed, instead of by using a memoryless feedback from actual speed. This is due to the additional technicalities involved in stability analysis when an extra feedback loop is used. A possible drawback of this approach is that speed control has to be tight to avoid unwanted saturation and low performance when operating at speeds close to nominal speed and during transients. This implies that the acceleration constraint of the system must be taken into consideration when defining the reference for speed tracking. 4.8 Simulation Results: Observer-Less Case To verify the qualitative behavior of the scheme in Proposition 4.4.1, a simulation study with a SIMULINK5 implementation of the induction motor o6-model was done. This is a stiff system, for which the choice of integration method has to be carefully considered. The Gear-method with small step-size was found to give sat isfactory numerical accuracy. The same model parameters (see Appendix E) as for 5 SIMULINK is a trademark of The Math Works, Inc. 76 Observer-Less Control of The Induction Motor the experimental setup were used, and simulations were done under the assump tions of ideal conditions (zero load torque, known parameters, linear unsaturated amplifier, speed and position measurements). References were generated by use of step and square wave functions which were filtered using third order linear filters of the form The necessary first and second order derivatives were obtained from the state space realizations of the filters. Filtering of flux and speed references was found to be of high importance for avoiding current and voltage saturations, and the values £ = 1, uq — 50, 100 rad/s were chosen for speed and flux norm reference filtering, respectively. The following parameters were used: a = 1000, b = 300, ki = 30, c = 0.45JZr. Integral action was not used in the controller, and all initial conditions were set to zero. References, tracking errors and voltages/currents are shown in Figures 4.1-4.3. As can be seen from these figures, there is no interaction between flux and speed control. This is consistent with the previous analysis. The 3 6 Only two of the 3 Speed Reference 500 - —500 - 0.4 0.5 0.6 Time [s] Flux Norm Reference 0.2 - 0.15 - 0.05 -i Time [s] Figure 4.1: References for speed and flux norm. Speed Tracking Error 0.4 0.5 0.6 Time [s] Flux Tracking Error -0.05 Time [s] Figure 4.2: Tracking errors for speed and flux norm. 78 Observer-Less Control of The Induction Motor X Currents in Phases a, b Time [s] Voltages of Phases a, b Time [sj Figure 4.3: 30 Currents and voltages ia , ib, ua , ub. 4.9 Concluding Remarks 79 \ 4.9 Concluding Remarks In this chapter the results on output feedback globally stable speed tracking con troller for induction motors reported in (Espinosa-Perez and Ortega, 1994) have been extended. The dependence of the convergence rate of speed tracking errors upon the natural mechanical damping of the motor, something that is important for todays low friction motors, has been removed. This has been done by using a linear filtering of the speed error in the controller equations, avoiding the need for acceleration measurement, which is expensive and complicated to implement. The result has also been extended from rotor flux norm regulation to tracking, something which is important for power efficiency and torque optimization. It is worth remarking that the controller is given here in terms of the afi model. However, for the purposes of computation it is more convenient to use an equivalent dq implementation, as explained in (Espinosa-Perez and Ortega, 1995). Especially, using the stator fixed ab frame, the controller can be implemented without position measurement as in (Espinosa-Perez and Ortega, 1994). Implementations of the scheme in the reference frame of the desired rotor flux have been given in Section 4.6 for the cases of current and voltage source inverters. Results from simulations under the assumption of ideal conditions have been given in Section 4.8. The passivity-based scheme does not require the explicit implementation of an observer, has no controller singularities (avoiding the need to take special precau tions at start-up), and gives global stability results. It has further been shown that under some simplifying assumptions on the motor model and the inverter, this scheme exactly reduces to the well known indirect FOC. The extension of this classical scheme to allow for also global rotor flux norm tracking follows in a straightforward way, by use of result from the passivity-based approach. A drawback of the proposed scheme is that known motor parameters have been assumed. This is a well known important problem, especially with respect to the changing rotor resistance. Interestingly enough, it has recently been proved that for current-fed motors in speed regulation, the controller is actually highly insensitive to the rotor time constant Tr. Namely, it has been shown that global stability is preserved for all estimates 0 Chapter 5 Observer-Based Control of The Induction Motor 5.1 Introduction The result in this chapter is decoupled from the previous chapters in that the aim is not to solve the torque and rotor flux tracking problem by an observer-less scheme. In this chapter the problem is solved with an interlaced controller-observer design. To state it in terms of energy-shaping, it is now aimed at shaping the total energy of the system, mechanical and electrical, instead of only shaping the energy of the electrical dynamics, as in previous chapters. The problem of global torque tracking and rotor flux norm regulation of induction motors perturbed by an unknown constant load torque was recently solved with an observer-based controller in (Espinosa-Perez and Ortega, 1995). In this chap ter the result from that publication is extended to treat the practically important case when the rotor flux norm is required to follow a time-varying reference. The controller design follows the passivity-based approach and proceeds in two steps: First, a target closed loop dynamics compatible with the physical model of the mo tor that delivers the desired rotor flux and torque is designed. Second, a nonlinear dynamic output feedback controller is proposed, which ensures that this behavior is asymptotically achieved. A proof of global tracking is given under the assumption of known motor parameters. Some key features of this physically-based design are that the control law does not require measurement of rotor variables, is always well defined and does not rely on (intrinsically nonrobust) nonlinear dynamics cancellation. The controller consists of a globally convergent nonlinear observer and an ob served state feedback control law. This observer-based approach has been pursued in (Ortega and Espinosa-Perez, 1993; Ortega et al., 1993; Espinosa-Perez and Or 81 82 Observer-Based Control of The Induction Motor tega, 1995), and should be contrasted with the recent observer-less schemes, see (Espinosa-Perez and Ortega, 1994) and Section 4.4. The remaining of this chapter is organized as follows. Next section reviews some material from (Espinosa-Perez and Ortega, 1995) pertaining to the design proce dure. The control objective is stated in Section 5.3. In Section 5.4 the problem is first solved under the assumption of full state feedback. This assumption is then removed in Section 5.5, which contains the proposed output feedback observer- based controller. In Section 5.6 a dg-implementation of the scheme is given, and simulations with the proposed scheme are presented in Section 5.7. Finally, some concluding remarks are given in Section 5.8. 5.2 Workless Forces The model (3.1), (3.2), may be rewritten in “Newton’s second law form” as -Wi(qm)qmqe ®(«)sr — 7Zq + Mu + £ \$Wi{q m)qe mass x acceleration —V 1 1 " of forces where V{q) = diag{Be(g m),J9m}, 77, = diag{jRe, Rm}, q = [q£,qm]r, M = The second right hand term corresponds to dissipation force's, while the last two right hand terms constitute the external forces. The fact below, which is the cornerstone of the passivity-based design philosophy, reveals that the the first right hand terms are workless forces. Fact 5.2.1 (Energy Rate of Change) The systems total energy H = ±qTV(q)q has a rate of change (the systems work) H = qT{—lZq + Mu + £) □□□ Workless forces do not affect the systems energy balance, which results from the integration of the equation above [tqTKqds+ f H(t) - H(0) qT(Mu + £)ds Jo Jo stored energy dissipated supplied As a result of this fact the effect of these forces can, roughly speaking, be disre garded in the stability analysis. 5.3 Problem Formulation 83 To carry out the controller design, a suitable linear factorization of these workless forces into a form fUl (9m)9m9e C{q,q)q must be found. Specifically, C(q,q) will be required to be such that: i) V(q) = C{q,q) + CT(q, q)r.hence V{q) - 2C(q, q) = CT(q,q ) - C{q,q), which is skew-symmetric. ii) The third and fourth rows of C(q, q) are independent of qe. These conditions are needed for the following stability analysis. Using (3.3) and the transposed of the last expression in (3.7), it is clear that the objectives can be achieved with the choice 0 0 f(q, q) C(q,q) -npLsrJe~Jnp9mqm 0 0 -fT(q,q) 0 0 with f(q,q) = npLsrJeJnpqmqr (5.1) This factorization leads to the following compact model representation: v(q)q + l(g, q)q + fcq = Mu + £ (5.2) 5.3 Problem Formulation Consider the induction motor model (5.2) with outputs torque r and rotor flux norm ||Ar|| to be controlled. Assume: A.l The load torque tl is an unknown constant. A.2 Stator currents g si,g a2, rotor speed qm and position qm are available for measurement. A.3 All motor parameters are exactly known, and the viscous mechanical damping constant is nonzero, i.e. Rm > 0. Let the desired torque Ta(t ) be a bounded and differentiable function with known bounded first order derivative, and the desired rotor flux norm be a strictly posi tive bounded and twice differentiable function /3(t) with known bounded first and second order derivatives. Under these conditions, design a control law that will ensure internal stability and asymptotic torque and rotor flux norm tracking, that is, the closed loop system must give lim |t - rd| = 0, lim j ||Ar|| - /3{t)| = 0 (5.3) t—►oo t—»oo from all initial conditions and with all signals uniformly bounded. 84 Observer-Based Control of The Induction Motor 5.4 Ideal Case - Full State Feedback For the sake of clarity of presentation, the problem will first be solved under the temporary assumption of full state measurement and known load torque. This is referred to as the ideal case. It will then be explained in the next section how the controller can be modified to remove these assumptions. Following the approach used in (Espinosa-Perez and Ortega, 1995), it can be shown that the control problem can be recast in terms of tracking of the motor currents. To this end, let a vector of desired currents and an internal desired 1 rotor speed be defined as qd := [qjd , qjd , qmd] T, and define the error as q := q — qd = '7 T * i T* [qs, qr , qm\ . Equation (5.2) can then be rewritten as T>(q)q + C{q,q)q+[K +K]q = if; where K. is a positive semidefinite matrix (to be defined below) that injects the required damping in the output feedback case. It is set to zero when the state is measurable. The right-hand side in the equation above is defined as if; := ~'D{q) This is a perturbation term that should be set to zero with a suitable choice of u and qd- If this is possible it follows from Corollary 2.1 of (Espinosa-Perez and Ortega, 1995) that q —* 0 as t —► oo. The problem is henceforth solved if u and qd can be chosen to ensure • if; = 0 • limt-,00 q = 0 => limt_oo |r - rd \ = 0, lim,_cc ||| Ar|| - /3(t)\ = 0 The problem of setting if; = 0 is first solved. It is clear from (5.2) and the definition of M that, for any given q, q, qd , qd , the first two equations of if; = 0 can be satisfied with U = La qsd + Lar eJnpqm qrd + f(q, q)qmd + RsQsd (5.4) It follows from (5.1) that this control law requires the measurement of the rotor currents qr. Also, the fifth equation of if; = 0 is satisfied if qmd is defined as the solution to / (#; 4)4ad "b RmQmd — TL (5.5) The next problem is to solve the third and fourth equations of if; = 0, which for convenience can be rewritten explicitly as Ard + RrC[rd = 0 (5.6) 1 This is an internal reference speed which is generated by the controller from a given desired torque. See also Section 5.5.1. 5.5 Output Feedback Controller 85 X where, motivated by (3.5), Ard has been defined as Ard := Lar erJn* qm qsd + Lrqrd (5.7) From the two equations above it can be seen that the problem is now reduced to the definition of a desired rotor flux (with known derivative) whose norm is /3(t), and such that the machine delivers the desired torque rd . Towards this end, recall from Section 4.3 that this is the case with the choice ' COS(pd) ' ' P(<) ' —— Pd A rd = 0(t) Pd — Uppi Td i Pd(0) — 0 (5.8) . sin (pd) 0 With these choices of u and qd it can be concluded that limt-.oo 9 = 0, and torque and flux tracking follows, as explained in Section 4.3. The above derivations are summarized in the following proposition: Proposition 5.4.1 (Ideal Case) Consider the induction motor model (5.2) in closed loop with (5.4), (5.5), (5.1) where qad, qad , qrd , Qrd are calculated from (5.7) and (5.6) using (5.8). Then, for all initial conditions equation (5.3) holds with all signals uniformly bounded. 000 5.5 Output Feedback Controller The main result of this chapter, a nonlinear observer-based controller, is presented in the proposition below. Proposition 5.5.1 (Main Result) Consider the induction motor model (5.2) with outputs torque r and rotor flux norm ||Ar|| to be controlled, and assumptions A.l - A.3. Let the control law be defined as u - La qsd +Lar eJnPqm qrd + npLar JeJnpqmqrqmd +Ra qa d —Ki(qmd)q a (5.9) where i Qrd —e^Pd 0 1 3 ( Tip 9m Pd ) +&P Qsd and with controller dynamics Pd — np02 Td 5 A*(0) — 0 (5.10) Qmd = zfc (-npLar qTJe~Jn* q™qad - R^qmd - tl + K2(qd)Qm) , 9md(0) = qm (0) (5.11) 86 Observer-Based Control of The Induction Motor The gains Kx (qmd ) and K2(qd ) are given as Ki(qmd) := + h, h > 0 (5.12) 4£i K2{qd ) (9id + gld) + ^2, 0 < ei < k2 > 0 (5.13) while the state estimator and the load adaptation law are -Oe(9m)9e "f" ^1 (9m)9m9e “H -Re9e — MeU L(qm, (jrril&e (5.14) TL = -Trikmi Ttl > 0 (5.15) with ee := qe — qe the observation error and 0 0 T{qmi 9m) — (5.16) npLsrJe * 7np9m 0 9m Under these conditions, the closed loop system achieves global torque and rotor flux norm tracking with all signals uniformly bounded. 000 Proof. Since the control law above uses the estimated instead of the real states, contrary to the ideal case, the error equation in this case takes the following form V{qjii + C{q,q)'q+[n + K,{qd))'q = S(qm,qd )ee +f (5.17) where K(qd ) = diag{#i(qmd)%2,0,0, K2(qd )}, f := [0,0,0,0, fL]T = [0,0,0,0, (fL- tl)]t and 0 npLsrJeJn* qmqmd On the other hand, from (5.14) and (3.1) the observation error ee satisfies the following equation f^e(9m)Ce [Wl (9m)9m "t” ■^'(9m?9m)] &e d" Re&e — 0 (5.18) Now, consider the composite Lyapunov function candidate v = \qV{q)q + \^D,(q^)K + TL whose derivative, taking into account the skew-symmetry of T>{q) — 2C(q,q) and De(qm) - 2 [Wi(9m)9m + L(9m,9m)], yields V" = “9 T + £(9d)] 9 + qTS(qm, 9d)se - i2eee + —+ g T| Ttl Use of (5.15) in the equation above and defining z := [g T,e^]T, results in the following quadratic function V —ztMz 5.5 Output Feedback Controller 87 with w _ 1l + IC(qd) -§<$(Qm,Qd ) — \ST{qmAd) Re Checking that (5.12) and (5.13) ensures strictly positive definiteness of M (see Section C.2), i.e. M > 8 19 > 0 (5.19) it can be concluded that q € , ee € and tl 6 £oo hold (i.e. boundedness) and that ee and q are also square integrable (i.e. belongs to C%, n = 4,5). Since qad,Qrd are bounded by construction, then qe is bounded, which together with ee in its turn implies that qe is bounded. The fact that qmd is bounded follows from boundedness of qr and (5.11) with Rm > 0, and implies qm bounded since qm is bounded. From (5.17) and (5.18) it now follows that q,ee are bounded. Since q, ee are bounded signals with bounded derivatives, they are also uniformly continuous. Together with square integrability, this implies convergence of current errors to zero. Hence, rotor flux norm and torque tracking can be concluded, with all internal signals uniformly bounded. □□□ 5.5.1 Remarks to the Controller 1. Observer Structure Under the assumptions that u € C2^ and q € global exponential convergence of the estimated currents to their real values can be proved, using only the part of V quadratic in ee together with (5.18). To get this result, a speed dependent term (5.16) proportional to deviation in sta tor currents is used to update the estimates in (5.14). This is common to several reduced order observers for which global stability results exists, see (Verghese and Sanders, 1988). The choice of L(qm, qm) follows nat urally from the Euler-Lagrange structure of the model, aiming at getting De(qm)—2 [Wi(qm)qm + L(qm, g m)] skew-symmetric. Unfortunately the con vergence rate of the estimation errors depends on the minimum resistance (i.e. Amin{i2e})- Notice also that even global exponential convergence of current estimation errors to zero is not enough to claim stability of the total system with esti mated states in the controller. Nonlinear damping terms must be introduced in the controller equations to ensure global stability when estimated instead of real states are used. The proof can be adapted to other globally valid and exponentially convergent observers, of which converse theorems guarantee that the part of the Lyapunov function pertaining to the observer has the necessary properties needed for the proof. The gains must then be recalcu- . T lated to ensure that the cross terms in q See of V can be dominated by the square terms, and consequently imply that (5.19) holds. 2. Comparison with Observer-Less Case Comparing (4.9) with (5.9), it can be seen that the difference is in the third 88 Observer-Based Control of The Induction Motor term on the right hand side, where qrqmd is used instead of qrdQm as in (4.9), and in the nonlinear damping term, where qmd is used instead of the real speed qm. In addition qmd is no longer the real reference speed, but an internal speed for the controller defined in (5.11). It depends on the estimated rotor currents and the desired stator currents. After estimates and real currents have converged to their desired values, this speed will indeed be the actual rotor speed. To extend the torque tracking objective above to classical position/speed tracking problems, an outer loop is needed to define the desired torque 7%. Since the derivatives of the desired currents are needed in the control (5.9), fa must also be known. The position/speed control loop in Section 4.4 is an example of how the outer torque generating loop can be defined. Notice that the observed rotor currents and the derivatives of the desired currents are used only in the case of a voltage input (5.9), and hence they are not needed if a CSI or a VSI with current control is used. In these cases the controls will be the stator currents qs = qsd, and the controller above exactly reduces to the controller of Corollary 4.6.1. 5.6 A dg-Implementation For the purpose of implementation it is of interest to formulate the controller in an arbitrary rotating frame of reference. This can be done by using the results from Section 3.3, giving Ad id e-jea Xdq "— = e-J(e*- n*qm )\r, Xg idq '•— ig is ltd Udg ■ — _ U, qf9 := erJ^a ~nv97n)qT Ug 9a is the solution of 6a = ua , 9a(0) = 0, with ua the angular speed of rotation for the reference frame relative to the stator fixed frame. Using these definitions and following the same procedure as for the derivation of (3.15)-(3.16), it follows that the observer in (5.14) and (5.16) can be rewritten &LS ^ ^°^ /y^\idq "h 2 [^p9mv7 — jT^jAdg^ — Udq = 6 ^ a 1l (5.20) TrXdg 4" Tj-{uia m)CfXdg + Adq (5.21) 5.6 A tig-implementation 89 Next, a relation between g^and the other estimated and measured dg-quantities must be derived. Notice from (5.14) and (5.16) that Ar + Rrqr = -npLar qmJe Jnj,qm (qa - q<^ Differentiation of Ar = e^0a ~np9m^Xd q and substitution of this expression in the equation above, gives after a rearrangement of terms qrq — e~J('9a ~nP9m'>qr = -^-[{u)a -npqm)J\dq + \dq -\-TlpLsr>"Jq.m (j'dq ] Substitution of the first two terms in the bracket with terms from (5.21) results in \d.q Qr (5.22) To express the control (5.9) in terms of tig-quantities, notice that if the desired stator and rotor currents are written in terms of rotated quantities i* dq , q* d as qSd = ^J0a i*dq and Q*d = Jt [eJda idq] = * J6a \uaJi*dq + jfiq (5.23) qrd = jt[eJ{6a -n>9m)qti\ = [(ua - npqm) Jqf9 (5.24) From the above and qr = e^^a-np9 m^gf ?, it follows that npLsrJ eJnrgm qrqmd = npLar Jejea q^9qmd (5.25) —npLsrqr Je-Jn* 9- qsd = -npLsr (jff) ^ Ji*dq (5.26) Substitution of the above expressions into (5.9) gives the control in any reference frame. Especially, with the choice oJa = npqm + pd where pd is defined in (5.10), (5.9) can be rewritten using (5.23)-(5.26) as u = ecr(n.,„+..) | + {npim + MJi.^ | + Z,„ ( Jtiil + pd Jq% } +npL„Jq?qmd + R,i‘M - Kidmd) (id , - ij,)] (5.27) where q* 9 is given in (5.22), and qmd is defined by qmd — -Q- npLsr (qr9') -JQq - RmQmd ~ Tl + , The desired stator and rotor currents in the tig-frame are in this case given as (see Proposition 5.5.1) 0 e~Jpdqrd = - Lsr ^pTd The nonlinear gains K\ and Ki can be calculated directly from (5.12) and (5.13), noting that the squared norm of qsd is equal to the squared norm of idq . 5.7 Simulation Results: Observer-Based Case Simulations with the same reference trajectories (see Figure 4.1), filter values and under exactly the same conditions as in Section 4.8 were performed with the observer-based controller in Proposition 5.5.1. The outer speed control loop was chosen as in Proposition 4.4.1 to generate , with filtering of the speed error to allow for speed tracking without acceleration measurement. Control parameters were chosen as: a = 1000, b = 300, ki = 30, k% = 2, €\ = 0.45JfZr, 7T£ = 0. All initial conditions were set to zero. Results from the simulations are shown in Figures 5.1-5.3. Speed Tracking Error 0.5 Time [s] Flux Tracking Error I -0.05 Time [s] Figure 5.1: Tracking errors for flux norm and speed. Observer-based controller. 5.7 Simulation Results: Observer-Based Case 91 X Rotor Currents — Estimation Errors 0.15 0.2 0.25 Time [s] Error of Fictitious Speed o. 0.05 -0.05 Time [s] Figure 5.2: Components of estimation error qr — qr and error between the real speed qm and the internal speed from (5.11). Observer-based controller. Currents in Phases a, b Time [s] Voltages Phases 0.5 Time [s] Figure 5.3: 34> Voltages and currents ua , Ub, ia, h- Observer-based controller. 92 Observer-Based Control of The Induction Motor As can be seen from Figure 5.1, there is no interaction between speed and flux norm control, as predicted from the analysis. Also for this controller filtering of references was found to be of high importance for avoidance of saturations in currents and voltages, and the parameter ki in the additional damping term was important for improving tracking performance for reference speeds close to zero. For small damping Rm, the parameter was also important for the quality of speed tracking. Comparison with Figures 4.2-4.S from the observer-less case shows that there is little difference between the two approaches under ideal conditions. This can be explained as follows: After a short initial transient the internal speed qmd and the rotor current estimates converge to their real values (and consequently also their desired values), and then the two controllers behave exactly similar. 5.8 Concluding Remarks In this chapter a solution to the output feedback torque and rotor flux tracking problem for an induction motor model given in the natural a(3 frame was proposed. The controller is an outgrowth of the work in (Ortega and Espinosa-Perez, 1993; Ortega et al., 1993; Espinosa-Perez and Ortega, 1995). More specifically, it is an extension of the work presented in the last paper to include the important case of rotor flux norm trackin: In this approach the design of controller and observer is interlaced, and glo stability results for the total system can be proved, provided that nonlinear damping terms are added to the controller to compensate for estimation errors. The extension of the previous observer-based results to allow time-varying ref erences for the rotor flux norm is important. From a theoretical perspective it puts this passivity-based approach on equal footing (with respect to the achiev able control objectives) with linearization-based controllers (Marino et al., 1993b). It must be pointed out that the observer used in this approach is simple, with only updating from current error terms in the rotor equations. It would in general be advantageous to have updating also in stator equations for making the observer more robust. The inclusion of error terms with adjustable gains will also give the possibility of specifying a convergence rate less sensitive to resistance parameters. These are possible future extensions of this observer-based result. However, the observer is only needed when a voltage input is used, since the controller proposed in this chapter reduces to the controller of Corollary 4.6.1 for a motor with a CSI or a VSI with current control. Consequently, the robustness results from (de Wit et al, 1995) with respect to uncertainty in rotor flux time constant Tr hold for this controller too in the case of current inputs. Chapter 6 Experimental Results 6.1 Introduction A laboratory setup was built for experimental testing of the controllers proposed in the previous chapters. The motor to be controlled was a 4-pole 3(j> squirrel- cage induction motor with a voltage source switched converter and a standard PWM scheme. Only a brief explanation of the setup is given here, and further implementation details can be found in Appendix E. Position was measured with an incremental encoder and a quadruple counter, giv ing a position resolution of 4Jg§6 « 4.0 • 10-4 rad. Speed was estimated from position measurements using a backward difference approximation, and computed at one third of the main sampling frequency to increase resolution. A dSPACE con troller board with a TMS320C31 main processor and a TMS320P14 slave processor were used for controller implementation. Two of the line currents were measured with LEM transducers, and converted to digital signals using 12 bit (3/zs) A/D converters. The discretized controller was first implemented in SIMULINK, and the design was tested with simulations in the full model. It was then converted to C-code, compiled and down-loaded to the board using dSPACE software. This software also allowed for on-line logging of variables and controller tuning. A simple ZOH approximation was used for the discretization, with a sampling 1 period of Tsampi = 300 /xs. The combination of this sampling period with the number of encoder lines and backward difference estimation, gave a speed estimate resolution of 4.4096 /(3Tsampi) • 60 « 4.1 rpm (Lorenz et al, 1994). Several series of experiments were carried out with this equipment and different controllers, and to limit the number of figures, only plots from what has been considered illustrative experiments are shown. Unless something else is explicitly 1This sampling time has been used in all experiments unless something else is explicitly written. 93 94 Experimental Results stated, the load torque used in the experiments was only due to friction. However, for purposes of regeneration, the converter of the load was turned on during all experiments. Even if the reference for the load torque2 was set to zero, this gave some additional high frequency oscillations of small amplitude in the system. Also, to limit the number of plots, measured currents and reference voltages for PWM have only been included for a few of the reported experiments. In those cases where saturation was experienced, this will be commented. The references were generated by linear first order filtering of step/square-wave signals from real-time implementations of the SIMULINK signal generator. Higher order derivatives of references for position/speed and flux amplitude j3(t) were obtained from state space representations of linear filters, similar to those reported in Section 4.8. For this reason, in the cases where both a desired quantity and an estimated or mea sured quantity are shown in the same plot, the desired value can be distinguished from the other as the smoothest curve. It was aimed at showing that flux norm and speed can be independently controlled, and for this reason either the speed- or flux reference was held constant during each experiment, while the other reference was a varying function. The unfiltered time- varying references were chosen to be square-waves, with a maximum amplitude of the flux reference equal to the nominal flux level of the motor. The outline of the rest of this chapter is as follows. In Section 6.2 some of the results from an implementation of the observer-less scheme in Section 4.4 are reported. Section 6.3 contains experimental results with the observer-based con troller from Section 5.5. In Section 6.4 results from an implementation of the rotor-flux-oriented control scheme in Section 3.4, are given for the purpose of comparison with the passivity-based controllers. Finally, concluding remarks to the experimental work are given in Section 6.5. 6.2 Observer-Less Control The behavior of the controller presented in Section 4.4, and implemented as in Section 4.6.3, was first investigated in a series of experiments. For later convenience (4.9) is rewritten here in terms of its different components u Laq a d + LsreJn* qmqrd + npLar J eJn* 9m qmqrd + Ra qsd '------V------' desired dynamics ‘ 7l2 Tv2 (6.1) 4e nonlinear damping term integral term where a possible integral term in stator currents have been added.. 2The control of the load torque was feed-forward in currents (i.e. torque) for the brushless DC motor. 6.2 Observer-Less Control 95 Since flux measurement was not implemented in the setup, a flux observer had to be run in parallel with the controller for the purpose of verifying flux tracking. For later comparison with an implementation of FOC, the estimation scheme in (3.30)—(3.31) on p. 54 was chosen. To avoid the singularity in the rotor flux speed estimation for zero flux estimate, it was necessary to substitute Xd in the division (see (3.30)) by a small constant c = 0.001 whenever Xd < c. To the desired torque defined in (4.11) and (4.13) (or (4.27) for position control), an integral term was added to compensate for unknown load torque, giving T~d — DjnQmd Z f(.Qm Qmd ) TL (6.2) Z — (LZ + b( The controller was first tested without integral action in stator currents, and a typical response is given in Figure 6.1. As can be seen from the figure, speed regulation is satisfactory, except from some high-frequency ripple, due to a com bination of unknown parameters, PWM, load torque and unmodeled dynamics. There is however an error in flux amplitude, which can be explained as follows: In (4.9) there is only proportional action in stator currents, even if the gain is a nonlinear function of speed. There will always be some unmodeled dynamics in the system, in addition to the introduced discretization effects3 and parameter uncertainty. For this reason real currents deviates from their desired values. In the speed controller there is an outer loop integral action, which forces the speed error to zero, despite error in q-axis current. There is no such feedback in flux control, which is feed-forward. Consequently the error in d —axis current gives an error in flux tracking. For the reasons above, the integral term in (6.1) had to be used for satisfactory performance, in addition to the other terms. The following controller parameters were used in the rest of the experiments reported here: e = 0.5-Rr, ki = 30, Kja = 0.3, a = 1000, b = 300, 7rii = 3.85. In Figure 6.2 the importance of the term in desired currents is shown. This feed-forward term is significant for high-performance flux tracking. As can be seen from Figure 6.3, it was difficult to get good low speed tracking performance. This is due to the resolution of speed estimation together with friction (especially stiction) in the load. Since the speed estimation gives the average speed between the sampling intervals, it is difficult to detect the sign transition precisely, and this gives problems with compensation of stiction terms. In Figure 6.4 an example of load torque rejection is shown, after a step in load 3It can be shown that discretization introduces coupling terms in the dynamic equations which are proportional to sampling period and speed (Vas, 1990). 96 Experimental Results torque of approximately 0.8 Nm. The controller compensates fast for the distur bances, and no steady state error is present. Figure 6.5 shows an example of position tracking for filtered steps in position reference of ±tt rad. The maximum error is approximately 1°. The steady state error is only restricted by the resolution of the position measurement system. In a real implementation, the digital jittering would be eliminated by a dead-zone. The controller parameters e = 0.5Rr, k\ — 30, Kjs = 0.3, a = 1000, b = 95, 7ri = 70, / = 41 were used in this experiment. It is well known that for small and medium size motors with relatively high sam pling frequencies for control calculations and PWM, high-performance control can be achieved with only high-gain current control (Morici et al ., 1995). This was also experienced in this experimental work, were it was found that the desired dy namics in the controller (see (6.1)) had relatively low influence on the performance for the chosen sample period of T = 300 ns, when integral action was also used. However, for a sample period twice this value, the influence of the desired terms was significant, as can be seen from Figure 6.6. It must be pointed out that in all the reported experiments there are interac tions between flux and torque control, resulting in small flux norm peaks during transients. For a real system with unmodeled dynamics and with parameters taken from the data sheet, perfect control can hardly be expected. The peaks are not detrimental for system operation, and result in negligible speed transients. Also, for high speed operation (more than 2000 rpm), the nonlinear damping term in (6.1) became large, and this resulted in amplification of noise from measured currents, which again gave saturation in voltages. For this reason the nonlinear damping term had to be disconnected for high speeds. With integral action in stator currents, the effect of this term was found to be negligible. 6.2 Observer-Less Control 97 X Speed Error Estimated Flux Amplitude and its Reference 0.14 - Time [s] Figure 6.1: Speed regulation/flux tracking without integral action in currents. Qmd = 500 rpm. Estimated Speed and its Reference Estimated Flux Amplitude and its Reference 0.2 r 0.18 j-0.16 =•0.14 0.12 0.1 i 05 1 1.5 2 Time [s] Time [s] Estimated Speed and its Reference Estimated Flux Amplitude and its Reference ¥300 Time [s] Figure 6.2: Speed regulation/dux tracking with (upper two figures), and without /3 in the controller. 98 Experimental Results Est. Speed and its Ref. Est. Flux Ampl. and its Ref. 0.202 0.201 §. 0.2 0.199 0.198. Time [s] Time [s] Meas. Line Curr. Ref. for Stator Volt. Time [s] Time [s] Figure 6.3: Speed tracking/flux regulation at low speed (±10 rpm). Est. Speed and its Ref. Est Flux Ampl. and its Ref. Time [s] Time [s] Ref. for Stator Volt. (Window) Torque Error z-0.2 -100 -150 Figure 6.4: Speed regulation with load torque disturbance, 7% « 0.8 Nm for t > 0.58 s. Qmd = 500 rpm. Error between desired torque and measured torque is shown in lower right plot. 6.2 Observer-Less Control 99 Position Reference Position Tracking Error -0.01 -0.02 Time [s] Time [s] x 1q-4 Position Tracking Error Flux Error 5 -5 1 1.05 1.1 1.15 1.2 Time [s] Time [s] Figure 6.5: Position control. Passivity-based controller. /3 = 0.2 Wb. Est. Speed and its Ref. -100 Time [s] Time [s] Est. Speed and its Ref. Est Flux Ampl. and its Ref. 0.197 Time [s] Time [s] Figure 6.6: Effect of desired dynamics in controller for high sampling period, Tsampi = 600 /Lis. Lower two plots are result from an experiment with only integral action and the nonlinear damping term in the controller. 100 Experimental Results 6.3 Observer-Based Control The controller in Section 5.5 was implemented as given in Section 5.6, with the desired torque as in (6.2). Since this controller exactly reduces to the observer-less controller if stator currents are controlled by high-gain, it was aimed at testing how well it worked without integral action in stator currents. An extensive simulation study was carried out with its discretized version, which work well under ideal conditions. The controller was then tested experimentally. Est. Speed and its Ref. Est. flux ampl. and its ref. /\ 100.5 0.3 / \ / / V'V'X !_0.25 ... __V . i / \ \ 0.2 0.15 0 0.1 0.2 0.3 0.4 Time [s] Time [s] Est. i_q and meas. i_q Est. i_d and meas. i_d Time [s] Figure 6.7: Speed and flux regulation. Observer-based controller. Estimates of electrical quantities denoted by As can be seen from Figure 6.7, except from some ripple, the speed regulation is quite good, but flux regulation is far from good. This was general for all the experiments, and can be explained with both missing integral action in stator currents, and only current error terms in the updating of the rotor quantities of the observer. The effect of the first point has been explained in the previous section. Since there is no updating in the stator terms of (5.14), the estimated stator currents drift off from the measured values due to uncertainty in parameters, noise and other unmodeled dynamics. This again introduces errors in the estimated rotor currents, which are used in the controller. It was possible to reconstruct a similar behavior under simulations, when the parameters used in the observer deviated from the real parameters. 6.4 Comparison with FOC 101 X 6.4 Comparison with FOC For the purpose of comparing the observer-less passivity-based controller with another scheme, the rotor-flux-oriented controller from Section 3.4 was also im plemented. More specifically, the controller in (3.29) was used. Estimated rotor flux amplitude Xa and angle 6a were computed from (3.31) and (3.30) (see p. 54). The Pi-controllers given in (3.27)-(3.28), together with the current references from (3.21), (3.34), were used to give the voltage references vd q- For speed control the desired torque was defined as in (3.24). The controller parameters were the same as in the previous section for the passivity-based controller, and the parameters Kip = 30.2, K\i = 0.3 (PI speed controller), K^p = 0.3, K^i — 3.8 (PI current controllers) were used in the implementation of the FOC scheme. Est. Speed and its Ref. (P-B) Est. Flux Ampl. and its Ref. (P-B) 1 1.5 5 1 1.5 2 Time [s] Time Is] Est. Speed and its Ref. (FOC) Est. Flux Ampl. and its Ref. (FOC) S'0.16 Time [s] Time [s] Figure 6.8 : Comparison of the passivity-based controller (P-B) with an implemen tation of FOC. Speed regulation/flux tracking. Figures (6.8)-(6.9) are representable for the comparison between the two schemes. In both cases the controller parameters were tuned such that currents saturated during transients. The FOC scheme generally gave slower responses, and some what higher maximum tracking errors. This scheme was also more difficult to tune than the passivity-based scheme. In the FOC controller implementation it was ad vantageous to use saturation limits corresponding to the systems constraints both in reference currents and voltages. To investigate the robustness of the schemes, an artificial change in rotor resistance was introduced by using a value different from the nominal value in the controllers. Both controllers were then tuned to give a “best performance” in terms of tran sients and overshoot. The FOC scheme was experienced to give a more oscillatoric 102 Experimental Results behavior than the passivity-based for different values of Rr. Examples are shown in Figures (6.10)-(6.11) for Rr = 1.5RrN- The execution time4 was approximately 150 /xs for the passivity-based scheme (with the flux observer running in parallel), and 145 fis for the FOC scheme. Est. Flux Ampl. and its Ret. (P-B) -100 Time [s] Time [s] Speed Tracking Error (FOC) Time (s) Figure 6.9: Comparison with FOC. Speed tracking/flux regulation. 4Only the execution time was logged, and no signal generators were implemented. 6.4 Comparison with FOC 103 X Est. Speed (P-B) Est. Flux Ampl. and its Ref. (P-B) —500 Time [s] Time [s] Est. Speed (FOC) Est. Flux Ampl. and its Ref. (FOC) | 0.2 -500 Time [s] Figure 6.10: Comparison with FOC. Speed tracking/flux regulation. Rr = l.oRriv- Est. Speed and its Ref.(P-B) Est. Flux Ampl. and its Ref. (P-B) E 502 0.5 1 1.5 Time [s] Time [s] Est. Speed and its Ref. (FOC) Est. Flux Ampl. and its Ref. (FOC) E" 502 0.5 1 1.5 Time [s] Time [s] Figure 6.11: Comparison with FOC. Speed regulation/flux tracking. Rr = 1.5-Rr#. 104 Experimental Results 6.5 Conclusions and Suggestions for Future Ex perimental Work The experimental testing of the proposed controllers can be summarized as follows: The observer-less passivity-based controller was found to give the best dynamic performance and robustness to unmodeled dynamics, as compared to a (direct) FOC scheme and a passivity-based controller with observer. The observer-less controller was easy to tune, and could basically be down-loaded with parameters taken from the simulations, without the need for extensive tuning. Significantly more tuning was necessary to make the FOC scheme give results comparable to the performance of the passivity-based controller. For the passivity-based controller, the feed-forward term from the derivative /? of the flux reference improved flux tracking significantly. The effect of this simple modification of the passivity-based controller to allow for global flux tracking is an interesting result, especially when it is related to conventional implementations of indirect FOC schemes, which only can handle flux regulation. It must be pointed out that high-gain current control was necessary for satisfactory performance of all the controllers which were tested. The nonlinear damping terms which were introduced to prove stability, could not compensate for the unmodeled dynamics of the motor and the converter, and integral action was needed. These damping terms also have certain disadvantages with respect to current noise amplification. Together with the terms in the voltage controller stemming from the reference dynamics, they can be removed when integral action current control is implemented, at least when sampling frequency is high. This experience motivates the derivation of a complete proof for stability when integral action in currents is used. For the case of observer-based control along the direction of the result in Chapter 4, it should be focused on including an observer which is more robust to unmodeled dynamics into the scheme. This could for instance be a scheme which also has updating from stator current error terms in the stator equations, as in (Verghese and Sanders, 1988). An observer-based controller is of interest for future extensions along the line of adaptive observers, and it is believed that it will be possible to include other observers with the desired robustness properties into the scheme, provided they give exponentially convergent estimates. The aspects of discrete implementation and speed estimation also have to be care fully considered for performance improvements. In this experimental work stan dard ad hoc schemes for controller discretization, speed estimation, and generation of switching signals have been used. This is a drawback of the implementation, since there is no theoretical justification for such an approach. Another point that should be tested experimentally, is power efficient operation of the drive, but for this some kind of power measurement is required, something which has not been implemented yet. 6.5 Conclusions and Suggestions for Future Experimental Work 105 The rigorous solutions of the above problems naturally lead to their experimental verification, but it would also be interesting to do experiments with passivity-based controllers in combination with some of the proposed solutions to the following problems: • Parameter identification prior to startup. • Parameter adaptation during operation. • Comparison of different speed estimators and observers updated from posi tion measurements. • Speed control without rotational sensors. • Friction compensation for better position and low speed performance. Other experimental results from the application of the observer-less passivity-based controller to induction motors have recently been reported in (Kim et al., 19966) (comparison with the scheme of (Marino et al., 1993a)), (Kim, Charara, Ortega, Vilain and Loron, 1996a) and (Espinosa-Perez, Campos-Canton, Lara-Reyes and Gomez-Becerril, 1995a) (speed tracking and flux regulation of 2 6.5.1 Comments to the Experimental Setup There were few problems with the dSPACE software, the interface circuits and power electronics in the experimental setup. However, the calibration and nonlin ear characteristics of the torque transducer, together with mechanical vibrations at speeds above 2500 rpm due to poorly balanced elastic couplings, were problems which to some extent hampered the experimental work (torque measurement, high speed responses, testing of field weakening). These problems must be solved before any further experimental work is carried out. Another future modification of the experimental setup required to implement some of the identification schemes, unless satisfactory results can be obtained by the use of reference voltages, is the fitting of voltage transducers at the motor terminals. These are relatively inexpensive, and have interfaces compatible with the con troller board used. Power efficiency could be investigated by adding extra current and voltage transducers at the DC-link side of the inverter. Since there are only 4 A/D interfaces on the board, external electronics for multiplexing combined with software switches is necessary for measurement of all these quantities (2 phase voltages, 2 phase currents, DC-link voltage and current, torque) in the same ex periment. It would also be advantageous to synchronize the current sampling with the PWM, to avoid some of the ripple in the measurements. It is difficult to remove this ripple with only filters without introducing unwanted phase lag. Synchronization of 106 Experimental Results measurement to be in the middle of the PWM period would possibly reduce much of the high frequency ripple, which due to proportional effects in the controller is amplified and added to the voltage reference. To investigate the effects of changing rotor resistance and possible adaptation of this parameter, it would be advantageous to be able to change this parameter physically. This can be done if the motor is exchanged with another having a wound rotor. It should also be pointed out that the experimental setup that has been built is quite general, and with suitable interface circuits it can be used for fast prototype implementation of controllers for a variety of electric machines. The interface with MATLAB and the SIMULINK block library gives the possibility of simulation and implementation of even very complicated controllers in an easy way. Chapter 7 Concluding Remarks In this work new output feedback controllers for a large class of electric machines have been proposed. This has been achieved by extending previous results from passivity-based control of induction motors. The new controllers are observer-less, do not introduce singularities or rely on exact cancellation of unmeasurable states, and global stability results of the total control system can be established under the assumptions of linear magnetics and constant known parameters. Furthermore, the controllers are derived for a general model of an electric machine, and system constraints to be satisfied for the applicability of this method have been identified. These are interpretable in terms of the machine’s physical properties, e.g. place ment and design of windings and permanent magnets. The passivity-based design procedure has been demonstrated for the case of a squirrel-cage induction motor, and controllers for global speed/position and rotor flux norm tracking have been derived. This work also contains an improvement of a previously proposed passivity-based controller for induction motors. An observer-based scheme has been extended from torque tracking with rotor flux norm regulation, to global tracking of both flux norm and torque. The use of reference dynamics in the controllers is instrumental to the passivity- based approach, hence these results are also important for the understanding of commonly used indirect control schemes. The new passivity-based controllers represent alternatives to controllers based on feedback linearization, which have totally dominated the field until now. In con trast to the passivity-based approach, these linearizing designs rely on full state measurement, and generally no a priori stability guarantee can be given when they are used together with state estimators. They also introduce artificial control sin gularities and rely on exact cancellation of unmeasurable dynamics. Furthermore, the applicability of this approach to control of electric machines is answered via a set of mathematical constraints for each machine in particular, and these are very 107 108 Concluding Remarks difficult to give any physical interpretation. Experimental result from an application of the passivity-based design method to a small VSI induction motor have been included. They show promising results for the observer-less controller, even when standard ad hoc discretization and speed estimation schemes are used for the implementation. More specifically, a compari son with an implementation of a direct FOC approach shows that the observer-less passivity-based controller has advantages with respect to simplicity of controller tuning and performance over this classical scheme. This observer-less controller can also be considered as an extension of the classical indirect FOC scheme to allow for global rotor flux norm and torque tracking, instead of only torque track ing and flux norm regulation. The improvement of flux norm tracking resulting from this remarkably simple modification of the classical indirect FOC scheme has been demonstrated by experiments. The experimental work also showed that the observer-based controller was more sensitive to unmodeled dynamics than the observer-less controller, and therefore future work in the direction of passivity- based controllers with observers should focus on interlaced designs utilizing more robust observers. It must be pointed out that relative to the feedback linearizing approach, the application of passivity-based controllers to electric machines is new. While a lot of results extending the first approach in several directions have been proposed, the potential of extensions within passivity-based control is still an open issue. Also, as pointed out in (Taylor, 1994), there is a potential advantage in working with a general model instead of specializing the model equations, and it is hoped that the results in this thesis will motivate for future research in the field of servo applications with utilization of the general machine’s model and its passivity properties. An interesting example of such an application is tracking control for robot manipulators actuated with AC drives. 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IEEE Transactions on Circuits and Systems 27(1), 15-19. Zribi, M. and J. Chiasson (1991). Position control of a PM stepper motor by exact linearization. IEEE Transactions on Automatic Control 36(5), 620-625. Appendix A Passivity A.l Definition of Passivity The following definition of passivity is taken from (Desoer and Vidyasagar, 1975): Definition A. 1.1 (Definition of Passivity) A system E : C\\e —» ££e IS said to be passive (respectively output strictly passive) if 3a > 0 (respectively > 0 ) and 3t E R such that fT rT / ur(t)(£a)(t)dt > a / \\Hu\\2dt + t, Vu E VTE t+. Vo do 000 119 120 Passivity X Appendix B The BP Transformation B.l Proof of Proposition 2.7.1 For the proof of Proposition 2.7.1, the following lemma is needed. Lemma B.1.1 dDe (Qm) ^ q u + uT = 0 (B.l) dQm 000 Proof Note that the differential equation (2.35) has the unique solution De(qm) = e UqmDe(0) e ~Uq™ (Kailath, 1980). e UqmDe{0) e ~Uq™ = e ~uTqmDe(0) e uTqm e uTqm e UqmDe(0) = De(0) e uTqm e Uqm e uTqrn e Uqm = I UT + U = 0 The third implication follows from the fact that U and De do not commute, unless * DdqOr^ = ^ (2-35), and this implies that f(U) = e uTq™ e Uqm and De cannot commute, unless f(U) = I. □□□ Proof of Proposition 2.7.1 From (2.34) it follows that qe = e Uq™ P^1 ze Qe = e Uq-P^ze + Ue Uq~P{lqmzt 121 122 The BP Transformation Inserting these two equations into (2.38), and multiplying from the left by e Uq results in e e ^J^z, + e -^9 "D«(9m)[/ e ^-Pri W, + e ~UqmRe e Uq^P^xze = e ~Uq™Meu (B.2) From De(qm) = e UqmDe(0) e -tZ9m , and since (2.36) implies that e UqmRe = Re e Uqm (Lancaster and Tismenetsky, 1985), notice that e -Uq-De(qm) eUq™ = De(0) (B.3) e -UqmDe(qm)U e Uq™ = De{0)U (B.4) e -^Wiiqm) e Uq™ (2=5) e ~Uqm [U e UqmDe(0) e ~Uqm - e Uq™De(Q)U e ~Uqm] e Uqm = UDe(0)-De(0)U (B.5) e ~UqmRe eUqm = Re (B.6) In addition, it follows from (2.37) that Wziqm) — e t/9m W2(0). This implies that e ~U9mW2(qm) — W2(0), which is constant with respect to qm. Finally, inserting (B.3-B.6) into (B.2), gives De(0)P^xze + L7De(0)Pf1g mie + W2(0)g m + PePf^e = e ~Uq-Meu For the transformed mechanical system Em, it follows that qJWUq^qe = ierPfT e ffT*~Wi( ft») e ^Pf1^ {=1} zeTPfT e "^^(gm) e ^Pf1^ ( =5) ifPfT [Z7Pe(0) - Pe(0)Z7] Pf1^ = 2zeJPfJ[/£>e(0)Pf1ze = 2zJPrTDe(0)UTPrlze W2T(gm ) e ^“Pf1^ W^Pf1^ This completes the proof. □□□ B.2 A Lemma on the BP Transformation Lemma B.2.1 Unless U = 0, the velocities z = [zj, qm]T introduced by the BP transformation cannot be derived from a transformation z = Z(q) of the generalized coordinates Q = [qJ,Qm]T- 000 B.2 A Lemma on the BP Transformation 123 Proof The transformation from the generalized electrical velocities qe and the generalized mechanical velocity qm to z = [zj, qm]T is " ie ' ■ e 0 " o i — l r Qm __ i If z = Z(q), then dZ _ Px e ~Uq™ 0 dq ~ 0 1 since z = Qj^q. From this, it can be seen that ze must be of the form ze = Ze(q) = Pi e Uqmqe 4- c, c € !.nc Taking the total time derivate gives fiZ Ze = -trq = -Pi e ~Uq™ Uqmqe + Pi e ~Uq"qe dq from which it follows that since the BP transformation (see (2.34)) is defined as ze = Px e ~Uqmqe, it must be true that -Pi e ~UqmUqmqe = 0, Vg e € Rn« For this to hold, U must be the zero matrix, since Pi e Uqm is nonsingular, and consequently for U ^ 0 there is no transformation ze = Ze(q) such that ie = fyq. □□□ 124 The BP Transformation X Appendix G Proof of Eqs. (4.21) and (5.19) C.l A Theorem on Positivity of a Block Matrix For use in the following proofs, a theorem on positivity of a block matrix is needed. The results is given in terms of the block elements on the diagonal of the matrix and their corresponding Schur complements. A proof of this theorem can be found in (Kreindler and Jameson, 1972). Theorem C.1.1 An arbitrarily partitioned Hermitian matrix of the form @11 @12 @ = @12 @22 is positive definite (Q > 0) if and only if either f @n > 0 1 @22 — @^@l^@12 > 0 or f @22 > 0 1 @11 — @12@m @12 > 0 000 For a necessary and sufficient condition on the matrix to be positive semidefi- nite when one of the block matrices Qu or Q22 is positive definite (and hence invertible), the requirement to the Schur complements can be relaxed from greater than zero , to greater than or equal to zero. For necessary and sufficient condi tions of positive semidefiniteness in the case where none of the block matrices are invertible, see (Kreindler and Jameson, 1972). 125 126 Proof of Eqs. (4.21) and (5.19) C.2 Proof of Eq. (5.19) Proof It must be shown that 7Z + /C(qd) — | S{qm,qd) Ai = > 61$ >0 (C.l) \ST{qmAd) Re with 'll = diag{fie,-Rm}, = diag{JRri(Qmd)J2,0,0, K2(qd )} and 0 npLSrJeJnrq™qmd S{Qm> qd) 0 0 E R5x4 0 -npLsrqJd JeJn”qm For the use of the theorem in Section C.l it must be checked if there exists a 6 > 0 such that A4 — 61$ > 0 with the given definition of JC(qd ). Since Re > min{Rs, Rr}!?, under the assumption that 0 < 6 < min{Rs,Rr}, which ensures invertibility of Re — 6X4, the theorem in the previous section can be used, and it must only be checked if there exists a 6 within these limits such that n + )C-^S{Re- 61a}-1 St - 61s > 0 Writing out this expression, it follows that {S» 4- Ki(qmd) — 62x2 Oaxi 02x2 {Sr — 6} I2 02x1 0lx2 0ix2 Rm +1^2(g 02X2 02x2 °2X2 r^TsIs 0ix2 -npLarqTdJeJn* qm X 02x2 02x2 O2XI -npLsrJ e Snpqn-md Q2x2 npLaT Je-Jn* qmqad {Rs + Ki (qmd ) — 6} Ig 02x 2 02X1 = O2X2 {Sr — 6}%2 02xl OlX 2 0lx 2 Rm. + K2 (qd ) — 6 " 02x 2 T&rJe*n'q”'qmd ' 1 02X2 02x 2 4A L 0ix 2 -%±fq?d J^n* q~ J 02x 2 02x 2 02xi X -npLsrJe ~Jnpqmqmd 02X2 npLar Je-Jn^mqad {Ra + Ki(qmd ) — 6}%2 02X2 02x 1 = 02x 2 {Sr — 02X1 0 (4 . 0ix 2 X Rm + K2 {qd ) — 6 IgkildZ* 02 2 1 X -R-Zjgmdgjd 02X2 02x 2 02xl 4 nv^'sr ~ ~T L R-r—S Q™dq a d 01X2 . C.2 Proof of Eg. (5.19) 127 X and it must be required that {* n2L2 . . Z2 °2X2 4(H r-4) °2X2 {R-r — 6} %2 °2X21 > 0 T ■ °1 X 2 Rm + R2(9d> “ 4(Hr-’s)«Jd^d ~ 4 Under the assumption that the 4x4 upper left submatrix is positive definite and hence invertible, only the positive semidefiniteness of its Schur complement must be checked according to the theorem. The upper 4x4 matrix is invertible if and only if n2L2 Rs + Ki(qmd) - s This condition can be satisfied by choosing the gain as n2L2 Ki(qmd) := PA SrQmd + h, o < Cl < Rr, fa > 0 (C.2) 4€i For each choice of ei, there will be a corresponding 0 < 8 < min{R3,Rr} such that the requirement above is satisfied, but as —► Rr, 8 —* 0. Calculation of the Schur complement for the upper 4x4 matrix, results in nlLlr 4(Rr - S)9sd9sd > « (C.3) [R, + KMr^) - - f] 16(Rr - Sf Now, since n2L2 8 = 4(Rr - S) tei(R, + h -i)(Sr -j) + njVj'iURr -e-g) 4ei(Rr — 8) (C.3) can be rewritten as Rm+ K^^qd) nlLlr 4ei (Rs + kx - 8) + f? pL2sr(^md qJdQsd > 8 4 4ci(Es + h- 8){Rr -8)+ nlLU2md (Rr -6-d) Choosing Kz(qd) as Kz(qd ) 128 Proof of Eqs. (4.21) and (5.19) gives the requirement ntL: [i~ 4ei(Ra+ki— 6)+n%L2,Tqlnd 7j] fidQsd > 8 Rm + 4ei(Ra+ki—6)(R-r—fi)+n? >L2rq*,(Rr — 8—€ A rearrangement of the terms finally gives P 1 nlLlr- rg)(itr—g-ei)+WpL^rq^1 From this equation it can be seen that there exists a 0 < 8 < mm{Rs, Rr,Rm] such that the above inequality is satisfied, at least for any 0 < e± < |JRr. As ei approaches its upper limit, 8 goes to zero. This bound on e\ becomes the restricting bound. However, 8 goes to zero with the mechanical damping Rm, even if ei can be chosen independent of Rm. This dependence on Rm can be avoided by adding a constant k2 > 0 to K2(qd), giving n2L2 1 Ki(4d) := P^* rqjdtsd + k2, 0 < ei < -Rr, k2>0 □□□ C.3 Proof of Eq. (4.21) Proof For the proof of Eq. (4.21), it must be shown that Qm) + £( Rsl2 + Ki(qm)T2 lnpLa rJeJnr9mqm [■R(?TOJ 9m) + ^C(5m)]es = -\npLsrJe~Jnj,qm qm Rrlz for some 8 > 0, with the given choice of the nonlinear gain Ki(qm). Using the results from Section C.l, the fact that Rr >0 and calculating the Schur complement of the lower 2x 2 matrix, gives the requirement 4(Er-5) c > ^ Using the results from the derivation of (C.2), it follows that the requirement is fulfilled for some 0 < 8 < min{jRs, Rr} if nlLlr + fci, 0 < e < Rr, &i > 0 4e Qm non Appendix D Derivation of Eqs. (4.38) and (4.39) D.l Derivation of Pioss In this section it will be shown how (4.38) and (4.39) are derived. The starting point is (4.37) ■Pioss = uTqs - rqm The control u is first eliminated from the above expressions by using (3.1), which can be rewritten as u = Lsqs + Lsr e Jn* qmqr + npLar qmJ e Jnp9mqr + Ra qa 0 = Lrqr + Lsr e ~Jnp9mqs - nPLar qmJ e ~Jnp9mqs + Rrqr The derivative of the rotor currents, qr, can be eliminated from the stator equation by using the last of the equations above. This results in u Substitution of the expression above together with r = npLar qJ J e 3n* qmqr in 129 130 Derivation of Eqs. (4.38) and (4.39) Pioss, and use of the fact that zT Jz = 0,Vz € R2 (skew-symmetry) gives which is identical to (4.38). Under the assumption of perfect control, i.e. that the stator current tracking error has converged to zero, the desired functions for q3 and qs defined in (4.10) can be substituted into the above expression. For convenience they are rewritten here as 9s — Qsd — % + rd J e JnpqmXrd Jsr ■*M) nP/P := [CI2 + DJ] e JnpQmXrd 1 RrLj/3 - RrLrfi 2LrW n +-kt+d ) j Qs = Qsd ?2 + Lsr np(32 npQm L' - + TdX2 + I 1 + } e Jn* qm\Td np(32 Rr?) Td ^f I Xrd ■LJST 1+M)l2 + 5?Td'7 1 LrP Lrqm _ t /3 LyRf 2 Lar Rrfi L + T Td + n,g m + | J- e jTh,qmXrd nPP “ ' ' flr/3 nPP2 := [AJ2 + BJ] eJn”qmXri where the constants A, B, C and D have been introduced to simplify later calcu lations. In the above calculations Xrd from (4.12) was used. D.l Derivation of PiOSs 131 The above expressions substituted into (D.l) results in Poss = AL(C%2 - DJ) (AZ, + DJ) Ard + (rs + Ar&j \Jd (Cl2 - DJ) (CI2 + DJ) Xrd -^^A^(C%2-DJ)Ard = (i, - ^ (AC + BD) \?d \rd + {C2 + D2) \Jd Xrd — —~^-CX^d Xrd where the skew-symmetry of J has been used in the last transition, and the constants AC, BD, C2 and D2 are given as Lr@ Lrqm /? LrRr Lr0 AC = Td + — ~ 1 + L2r B KB'Ph 1 Lt0 LrQrn t LrRr < + 2- a LI /P 0 nl@4 L2BB L'jqml3 lJ2 l2J Td + Td R202 Rrf33 U i-d +npqm + npLJiT0 + ** BD = Td Td L2r np/32 Rr0 Tip/?2 Tip/?2 LrQm . L2qm0 LrRr JLTdT d + -Td + Td + L2r n%04 RrP* nIB,T“ 2 Lr$ L2J2 C2 = L2r Br/? ^ A2/?2 D2 = llr npP* Use of the above expressions results in 1 LrP 3 L2r00 AC + BD = + Ti + TdTd L\r RrP P RW2 ’ RrP2 712/?5 ' m2/?4 C2 + D2 = 1+?M + !££ + JL^ L2r RrP RIB2 rhB By the use of the above results and the fact that Xjd Xrd = /?2, poss is finally found 132 Derivation of Eqs. (4.38) and (4.39) to be La Lr Pi OSS — ITRr L2LS 1 nn Lr n n . Lr/3 O "^r TdTd - -R-ff - a*# 4- - ^-T,Td np^sr Ra P2 + 2%^-0P + ^LlRt^/3a 2 + Rr Tj — -t} + LIuST TtSJCfr/P ' LlLs - L2L -pp+Ls\ Li-rpp Rr LsRr + 2LrRs ^ RrLrL3 + L2R, Ll-Rr LlRr LrI?„ - L2L, , 0 R* o2 nlL2,r nT2 + W/ LlLs - LrLlfdTd + L2rRs + RrL2r r2 1 + P2 which is identical to (4.39). X Appendix E Experimental Setup E.l Introduction In this chapter the experimental setup is described. The setup was built from scratch with basic ideas for implementation taken from (Schiitte, 1994), but mod ified to fit specific needs, and avoid reported problems with noise and signal trans mission. Additional equipment for load torque generation and torque measurement were built from standard components. For the software interface an integrated sys tem from dSPACE was chosen. This choice allowed for fast prototype implementa tion without extensive C or assembly language coding. A more detailed description of the induction motor part of the setup can be found in (Holemark, 1994). E.2 Hardware Description The controller is implemented on a DS1102 controller board from dSPACE. The board has a 40 MHz TI320C31 32/32 bit floating-point DSP and a 25 MHz TI320P14 32/16 bit micro controller DSP. In addition there are 4 A/D-converters which are used for current (12 bit resolution, 3 /xs conversion time) and torque1 (16 bit resolution, 10 fis conversion time) measurements, 16 bit digital I/O of which 6 bit were used for pulse-width modulation and 1 bit for a converter enable signal, and two 24 bit encoder interfaces, of which one is used for position measurement. The micro controller computes the switching signals for the symmetric carrier- based PWM of the three phases from reference values transferred to it from the main processor at the end of each sampling interval, see Section E.3.3 for a descrip tion. The three signals for the upper transistors in the converter legs are converted to optical signals on an interface card before they are transmitted through optical 1The torque is measured only for illustration purposes. It is not used for feedback in any of the controllers presented in this thesis. 133 134 Experimental Setup fibers to the converter, where complementary switching signals and blanking time of the converter (2 /-is) are generated in hardware using an IXYS IXDP630 digital dead time generator for 3 The converter is a Lust FU2235 l.lkW BJT voltage source converter with a DC- link voltage of Uvc = 300V, and capable of delivering maximum line currents of Imax = 6.8 A. This converter was connected to a 4-pole (np = 2) ASH-11- 10163-00 400W squirrel-cage induction motor from the same company. Nominal2 two phase parameters of the motor given in the data sheet are: Lsn = 99.0 mH, LsrN = 92.3 mH, Zrjv = 97.1 mH, Rsn = 1.8 fl, Rtn = 2.2 ft, Dm = 2.8 kgcm2 , qmN = 3000 rpm, fix = 0.2 Wb, tjv = 1.5 Nm, Rm ~ 0.005 Nms/rad. A Lust BC1200 brake chopper was connected to the DC link for power dissipation. To allow for optical transmission of switching and enable signals from the DSP board to the converter, the standard microprocessor board for voltage/frequency control was removed from the inverter, and replaced by a specially designed inter face card. Over-current protection was implemented both in software and hard ware. Position is measured using an incremental encoder with 4096 lines, and a quadruple counter is used, giving a position measurement resolution of 4 ^9 g rad. The currents are measured with LEM LA 25-NP current transducers and filtered with first order analog anti-aliasing filters, having a cut-off frequency of 1.1 kHz, before they are converted by the 12 bit A/D converters. An offset correction of the current measurements is done at startup, when the inverter switches are disabled and the currents in the motor windings are zero. As a load for the induction motor a current controlled BSM 80A250 brushless DC-motor from Baldor with a BSC1105 driver is used.The motor has a moment of inertia equal to 2.13 kgcm 2, and is capable of producing a nominal torque of 3.2 Nm. A separate PC with a PCL-711B I/O board and external electronics is used to control the load. Torque is measured using a HBM T1 50 Nm strain gauge torque transducer with an AE101 amplifier, and the signal is filtered with a first order low-pass filter. The torque transducer has a moment of inertia equal to 0.6 kgcm 2. An overview of the total system is given in Figure E.l on page 138. E.3 Software Description The DSP board was installed in a 80486/66MHz PC with MATLAB/SIMULINK, and a RTI C code generator which converts the controller graphically described in SIMULINK to C code which can be compiled and run on the DSP. For logging of identification of the 2 Computation of speed, position increment, current and torque measurements and the communication between the main processor and the micro controller, is im plemented in external routines which are linked with the code generated from the block diagram. These routines also provides the mapping between software in puts/outputs specified in the block diagram, and hardware addresses on the board. The code for the PWM running on the micro controller is written in assembly and down-loaded during startup of the processor. In Figure E.2 on page 139 the main block diagram used for code generation is shown, and in Figure E.3 an example of the diagram for the controller block is shown. The controller is implemented using multi rate computation, where the PWM calculation is run relatively fast on the micro controller at a sampling rate of ZpwM = 100/zs independent of control algorithm, with a slower computation of reference voltages and speed control at a rate of Tsampi seconds. These rates depend on the implemented control algorithm. E.3.1 Controller Discretization and Speed Estimation In the derivations below a base sample period of T [s] for controller implementation is assumed, and y(k), k 6 Z+ is used to denote a sample of the signal y(t) at time t = kT. Discretization of controller equations were done using the ZOH (Zero-Order-Hold) approximation of an integration. In this approach the discrete equivalents of transfer functions for linear continuous systems are derived by using the formula hzon(z) = (1 —z 1)Z where z~l is the delay operator, and Z denotes the ^-transform. This gives (Franklin, Powell and Workman, 1990) 1 Linear filter: Tip+1 i y(k) = e~^y(k-l) + (l-e-^)u(k 1 Integration: = jw V i y(k) = y(k — 1 ) + Tu(k - 1) 3MATLAB and SIMULINK are registered trademarks of The Math Works Inc. RTI, COCK PIT and TRACE are registered trademarks of dSPACE GmbH. 136 Experimental Setup To avoid integral wind-up, conditional integration was implemented in all PI- controilers. In this approach, the integral term is held constant when the output of the controller exceeds an adjustable limit. Since the induction motor used in this work has no speed transducer, speed had to be estimated from discrete time position measurement. This was done using the simple backward difference approximation (Lorenz et al, 1994) 9m(fc) Qmjh 3) Q (E.l) m 3T This average speed is a rather rough first estimate with a resolution (assuming quadruple counter) of 2 it 4iVpoa Qm, res 3T iVpOS is the encoder resolution in pulses per rotation. For T = 300 [is and iVpos = 4096, the resolution is g m>res ~ 0.43 rad/s ( « 4.1 rpm ). The backward difference estimate of speed was smoothened using a discrete implementation of the linear filter 1 h{jp) ZTp+1 before it was used for control purposes. E.3.2 Phase Transformations To transform the measured 3 line currents Ia and Ib to equivalent 2 r 3 Qsi ' 2 0 ' 7S1- - 1 fl yS . 9s2 . 1 3 L 2 V3. . VS4 . was used. The 3^-voltages ua , uc used for the PWM calculation were com puted from the voltages uai, uS2 calculated by the controller as V/Q ub Us 2 Uc Note that uc = —(ua + ub), hence only ua and ub are needed. The form of the transformation used here is the so-called power-invariant form of the transformation, see (Vas, 1990). E.3 Software Description 137 E.3.3 Pulse-Width Modulation A standard symmetric carrier-based PWM was chosen for generation of the switch ing signals of the converter. In this method the switching signals for the transistors in the three bridge legs of the converter are generated to be symmetric to the middle of the switching interval, and the on-time for each upper switch t*oN, * E {a,b,c} is computed from \ 0 Ui < -2ns tiON < (§ + i^)rPWM To get the desired symmetry of the digital switching signals, the slave DSP gen erates two signals for each upper switching signal. These are passed through an external XOR gate. Both inputs to the XOR gate are logical high at the begin ning of each switching interval, t = IcTpwm- The first input is set low at time t = fclpwM + and the other at time t = fcJpvvM + The three TTL voltage outputs from the XOR gates are converted to current signals suited for generating optical signals using external electronics, and then transmitted to the converter. Complementary signals for switching of the lower transistors and blanking time is taken care of by a PWM controller in the converter. LOAD 138 /Tpcm 220 RESOLVER SIGNAL 0 Filter VAC DRIVER BLDC SYSTEM I/O INDUCTION 0 BOARD Dept, AT BUS Torque Speed Command Transducers Figure Current of LEM Eng. E.l: Lust INVERTER SWITCHING FU2235 MOTOR dSPACERTl Block SIMUUNK MATLAB Cybernetics, DRIVE SIGNALS diagram _ LOCK Filter SYSTEM CONTROL SIGNAL 220 of AMPLIFIER TRANSMISSION AT VAC FIBEROPTIC e x p e r i m e n t a l NTH, BUS A V NORWAY TI320C31 40 COUNTER PS Experimental SYSTEM Filter setup. Interface Electrical/Optical MHz 1102 ’ dSPACE b Circuits Controller I n D % c I n I/O CONNECTOR TI320P14 25 ADC MHz Board ENCODER SIGNALS Setup E.3 Software Description 139 SIMULM Block Diagram for C-Code Generation Figure E.2: Main block diagram for C-code generation from SIMULINK. 140 Experimental Setup reference transformation rotation to drHfaine Figure E.3: Example of SIMULINK block diagram for controller.