Identification and Adaptive Control for
High-performance AC Drive Systems
by
David M. Reed
A dissertation submitted in partial fulfillment
of the requirements for the degree of
Doctor of Philosophy
(Electrical Engineering: Systems)
in The University of Michigan
2016
Doctoral Committee:
Associate Professor Heath Hofmann, Co-chair
Professor Jing Sun, Co-chair
Professor Jessy Grizzle
Professor Ilya Kolmanovsky
c
David M. Reed
2016
All Rights Reserved
In loving memory of my uncle Davi,
for inspiring my love of science and interest in electrical engineering.
And to my family,
for all of your love, support, and understanding.
ii
ACKNOWLEDGEMENTS
I would like to begin by expressing my deep gratitude to my advisors, Professors
Jing Sun and Heath Hofmann, for their support, guidance, and mentorship over
the past four-plus years. While it took me some time to adapt (pun intended) to
her advising style, I am grateful for Professor Sun’s persistence and patience while
pushing me to realize my potential as an academic researcher more fully. I also greatly
appreciate her use of controls themed analogies in our weekly discussions.
Additionally, I owe a great deal of thanks to Professor Hofmann for bringing me
to Michigan and allowing me the freedom to pursue my research interests, even when
they tilted more towards controls than power and energy. I first starting working
with Heath back when I was a master’s student at Penn State, before I even knew
what a Lyapunov function was, and I am glad that I was able to earn my PhD under
his supervision as well.
I also wish to thank Professor Jessy Grizzle, not only for serving on my committee,
but also for answering my questions and offering advice over the past few years. From
letting me borrow his office phone to call campus security when I locked myself out of
the lab after hours, to engaging in insightful discussions regarding my research (and
sometimes just controls in general). I am fortunate to have had the opportunity to
take his Nonlinear Systems and Control course, and have surely benefited from his
presence here at Michigan.
Additionally, I would like to thank Professor Ilya Kolmanovsky for serving on
my dissertation committee, as well as for his helpful suggestions and discussions
throughout the process. I don’t think I could have found a better cognate member
for my dissertation committee.
I am also grateful to Professors Necmiye Ozay and Ian Hiskens for letting me pick
their brains from time to time. I was fortunate to have had the opportunity to GSI
Linear Systems Theory during my time at Michigan, working with Professor Ozay.
While time consuming, it was a very rewarding experience.
iii
I would also like to express my sincere gratitude to Rebecca “Becky” Turanski
and Michele “Shelly” Feldkamp for all of their help, hard work, and just generally
making the department a more pleasant place to work.
Throughout my studies at Michigan, as well as Penn State, I was lucky to have
had so many excellent teachers - Professors Jim Freudenberg, Jessy Grizzle, Laura
Balzano, Chris Rahn, Jack Langelaan, Jeff Schiano, Jack Mitchell, and Javier GómezCalderón, to name just a few.
I am also grateful for the help of my labmates, particularly Kan Zhou, Abdi Zenyu,
and Jun Hou, who spent a considerable amount of time helping collect data for the
induction machine work presented in this dissertation. And also Vicky Cheung, who
has been the best undergraduate lab assistant one could ask for.
I would be remiss not to acknowledge the many friends and classmates here
at Michigan who have made the process of completing this degree not only more
bearable, but enjoyable - Mai Le, Aaron Stein, Ian Beil, Rob Vandermeulen, Nick
Asendorf, Madison McGaffin, Matt Prelee, JJ Lipor, Mads Almassalkhi, Hamid Ossareh (who was also the best GSI I’ve ever had), Steph Crocker, Greg Ledva, as well
as the members of the RACElab group (for all of their helpful feedback on papers and
presentations). I am also grateful to the “Snarky Comments” crew for welcoming me
into their group and the continuing friendships. My time in the Mitten would not
have been the same without Formal Tailgates, Snarky Christmas, and other gatherings which left me sore from laughter.
Finally, I would like to thank my family for all of their love and support.
*My sincere apologies to anyone I have forgotten!
iv
TABLE OF CONTENTS
DEDICATION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
ii
ACKNOWLEDGEMENTS . . . . . . . . . . . . . . . . . . . . . . . . . .
iii
LIST OF FIGURES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
viii
LIST OF TABLES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
xii
ABSTRACT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
xiii
CHAPTER
I. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.1
1.2
1.3
1.4
Motivation and Overview . . . . . . . . . . . . . . . . . . . .
Background and the State of the Art . . . . . . . . . . . . . .
1.2.1 Background on Electric Machinery and Drive Systems
1.2.2 State-of-the-Art in Offline Identification of Induction
Machine Parameters . . . . . . . . . . . . . . . . . .
1.2.3 Background on Parameter Identification and Adaptive Control . . . . . . . . . . . . . . . . . . . . . .
1.2.4 State-of-the-Art in Simultaneous Identification and
Control . . . . . . . . . . . . . . . . . . . . . . . . .
Open Issues and Contributions . . . . . . . . . . . . . . . . .
1.3.1 Open Issues . . . . . . . . . . . . . . . . . . . . . .
1.3.2 Summary of Contributions and Innovations . . . . .
Reader’s Guide . . . . . . . . . . . . . . . . . . . . . . . . . .
II. Offline Identification of Induction Machine Parameters . . .
2.1
2.2
2.3
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . .
The Steady-State Stator Current Locus . . . . . . . . . . . .
Proposed Parameter Estimation Technique . . . . . . . . . .
2.3.1 Fitting the Parameterized Stator Current Locus Circle to Data . . . . . . . . . . . . . . . . . . . . . . .
v
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26
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III. Adaptive Excitation Decoupling Approach to Simultaneous
Identification and Control of Permanent Magnet Synchronous
Machines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
37
2.4
2.5
2.6
2.7
3.1
3.2
3.3
2.3.2 Procedure for Data Collection . . . .
2.3.3 Dead-time Compensation . . . . . . .
Stator Flux Linkage Estimation . . . . . . . . .
Numerical Analysis . . . . . . . . . . . . . . .
Experimental Results . . . . . . . . . . . . . .
2.6.1 Experimental Setup . . . . . . . . . .
2.6.2 Experimental Results and Discussion
Conclusion . . . . . . . . . . . . . . . . . . . .
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4.4
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IV. Simultaneous Identification and Control of Permanent Magnet Synchronous Machines via Adaptive 2-DOF Lyapunov
Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.1
4.2
4.3
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3.6
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3.4
3.5
Introduction . . . . . . . . . . . . . . . . . . . . . .
Dynamic Model of PMSMs . . . . . . . . . . . . . .
Adaptive Disturbance Decoupling Approach . . . . .
3.3.1 Statement of the Control Objective . . . .
3.3.2 Review of Disturbance Decoupling . . . . .
3.3.3 Excitation Decoupling for PMSMs . . . . .
3.3.4 Gradient-based Parameter Estimation . . .
Selection of Persistently Exciting Inputs . . . . . . .
Simulation Results . . . . . . . . . . . . . . . . . . .
3.5.1 On Conditions for Parameter Convergence
3.5.2 Closed-loop Performance . . . . . . . . . .
Conclusion . . . . . . . . . . . . . . . . . . . . . . .
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Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . .
Two-Phase Equivalent Dynamic Model for PMSMs . . . . . .
Simultaneous Identification and Control Objective and Methodology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Adaptive Control Design . . . . . . . . . . . . . . . . . . . .
Simultaneous Parameter Identification . . . . . . . . . . . . .
4.5.1 Parameter Convergence using Two-Time-Scale Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.5.2 Persistently Exciting Inputs . . . . . . . . . . . . .
Simulation Results . . . . . . . . . . . . . . . . . . . . . . . .
4.6.1 Ideal Case . . . . . . . . . . . . . . . . . . . . . . .
4.6.2 Sampled-data Implementation: Time Delay and Compensation . . . . . . . . . . . . . . . . . . . . . . . .
Experimental Validation . . . . . . . . . . . . . . . . . . . . .
4.7.1 Test Machine Parameters . . . . . . . . . . . . . . .
vi
55
56
57
59
62
63
64
67
67
68
71
71
4.7.2 Description of the Experimental Set-up . . . . . . .
4.7.3 Experimental Results . . . . . . . . . . . . . . . . .
Chapter Conclusion . . . . . . . . . . . . . . . . . . . . . . .
71
72
77
V. Receding Horizon Control Allocation for Simultaneous Identification and Control of PMSMs . . . . . . . . . . . . . . . . .
78
4.8
5.1
5.2
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . .
Proposed Control Architecture . . . . . . . . . . . . . . . . .
5.2.1 Inner-loop Controller . . . . . . . . . . . . . . . . .
5.2.2 Control Allocation . . . . . . . . . . . . . . . . . . .
Receding Horizon Control Allocation for Simultaneous Identification and Control . . . . . . . . . . . . . . . . . . . . . . .
5.3.1 The Fisher Information Matrix and Persistent Excitation . . . . . . . . . . . . . . . . . . . . . . . . . .
5.3.2 Receding Horizon Control Allocation for Simultaneous Identification and Control . . . . . . . . . . . .
5.3.3 The Crucial Role of Past Input and State Data . . .
Simulation Results . . . . . . . . . . . . . . . . . . . . . . . .
5.4.1 Static Control Allocation . . . . . . . . . . . . . . .
5.4.2 RHCA-SIC without Past Input and State Data . . .
5.4.3 RHCA-SIC with Past Input and State Data . . . .
Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . .
83
85
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88
89
90
92
VI. Conclusions and Future Work . . . . . . . . . . . . . . . . . . .
93
5.3
5.4
5.5
6.1
6.2
Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.1.1 Offline Identification of Induction Machine Parameters
6.1.2 Simultaneous Identification and Adaptive Control of
PMSMs . . . . . . . . . . . . . . . . . . . . . . . . .
Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.2.1 Offline Identification of Induction Machine Parameters
6.2.2 Simultaneous Identification and Adaptive Control .
BIBLIOGRAPHY . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
vii
78
79
79
81
82
82
93
93
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95
95
96
98
LIST OF FIGURES
Figure
1.1
Comparison of torque-speed curves for Tesla induction motor and
conventional IC engines [1]. . . . . . . . . . . . . . . . . . . . . . . .
2
Example of steady-state torque errors (ratio) in a PMSM drive for
variations in the permanent magnet flux linkage (left) and quadratureaxis self-inductance (right) for a variety of direct-axis currents. The
following machine parameters were used to generate these plots: Ld =
212.3 µH, Lq = 1.274 mH, and ΛP M = 12.644 m-Vsec. . . . . . . . .
3
1.3
Drive system architecture. . . . . . . . . . . . . . . . . . . . . . . .
5
1.4
Ideal (left) and practical (right) three-phase VSIs. . . . . . . . . . .
7
1.5
Generation of sinusoidal PWM waveform. . . . . . . . . . . . . . . .
7
1.6
Vector diagram depicting the Park transform of an arbitrary vector.
8
1.7
Typical structure of adaptive controllers. . . . . . . . . . . . . . . .
11
2.1
Dynamic 2-phase equivalent circuit model for an induction machine.
21
2.2
Parameterized steady-state stator current locus in the stator flux linkage reference-frame. . . . . . . . . . . . . . . . . . . . . . . . . . . .
23
Data acquisition controller block diagram for the proposed parameter
estimation technique. . . . . . . . . . . . . . . . . . . . . . . . . . .
27
Simulation of the proposed data acquisition procedure depicting the
distortion in the stator current locus due to dead-time effect(left) and
improvement using first-harmonic dead-time compensation (right). .
28
1.2
2.3
2.4
2.5
Timing relationships for sampled-data implementation with unit delay. 29
viii
2.6
Comparison of bilinear and impulse invariance discrete-time second
order integrator approximations with ideal continuous-time integrator. 30
2.7
Simulated parameter errors in the presence of non-ideal effects. . . .
32
2.8
Steady-state induction machine torque error (ratio) due to unmodeled
core loss as a function of slip. This plot was generated using the
following machine parameters: Ls = Lr = 4.4 mH, M = 4.2 mH,
Rr = 23 mΩ, Gc = 30 mΩ−1 , and Ωe = 153.33 Hz. . . . . . . . . . .
33
2.9
Experimental set-up for parameter identification data collection. . .
33
2.10
Experimental estimated machine parameters as a function of stator
flux linkage magnitude. . . . . . . . . . . . . . . . . . . . . . . . . .
34
Estimated core loss power as a function of stator flux linkage magnitude at a electrical base frequency of 153.33 Hz and switching frequency of 10 kHz. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
35
Experimental data (blue circles) with fitted current locus circle (green
dashed line) and estimated locus points (red X’s) for various stator
flux linkage magnitudes. . . . . . . . . . . . . . . . . . . . . . . . .
36
Cross-section of the two-phase equivalent, two-pole smooth airgap
interior-permanent-magnet PMSM machine. . . . . . . . . . . . . .
40
Block diagram of closed-loop system after disturbance decoupling
(adapted from [29]). . . . . . . . . . . . . . . . . . . . . . . . . . . .
43
Block diagram of the closed-loop system with proposed adaptive excitation decoupling controller. . . . . . . . . . . . . . . . . . . . . .
46
Simulations of the closed-loop adaptive system without persistently
exciting input and with zero torque command. . . . . . . . . . . . .
51
2.11
2.12
3.1
3.2
3.3
3.4
3.5
Simulation of closed-loop adaptive system at a fixed rotor velocity of
2000 rpm with excitation input (adaptation turned “on” at t = 1 sec). 52
3.6
Simulation of closed-loop adaptive system at a fixed rotor velocity
of 2000 rpm with zero-mean Gaussian noise added to the current
measurements. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
53
4.1
Depiction of a 1-D manifold in R2 . . . . . . . . . . . . . . . . . . . .
58
4.2
Block diagram of the proposed control law. . . . . . . . . . . . . . .
60
ix
4.3
Simulation result demonstrating state-trajectory convergence to the
desired constant-torque manifolds using the proposed adaptive control design methodology with a step change in the commanded torque
from 0.2 N-m to 0.4 N-m at a fixed rotor speed of 2000 RPM. . . .
67
Simulation of an ideal implementation of the proposed SIC design
for PMSMs demonstrating parameter stagnation due an initial lack
of persistent excitation, and the improvement resulting from the introduction of the excitation signal at 0.75 seconds. . . . . . . . . . .
68
4.5
Timing sequence of digital controller implementation. . . . . . . . .
69
4.6
Simulation of sampled-data system without reference-frame advancing at a speed of 2000 RPM with step changes in command torque
(the same as in Fig. 4.7), leading to poor parameter estimator performance. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
70
Simulation of the proposed adaptive control design in a sampleddata scenario with reference-frame advancing based on (4.27) and
measurement noise at a rotor speed of 2000 RPM. . . . . . . . . . .
70
4.8
Experimental setup. . . . . . . . . . . . . . . . . . . . . . . . . . . .
72
4.9
Experimental torque steps with adaptation on at 2000 rpm. . . . . .
73
4.10
Experimental transient responses of estimated torque (top) and measured quadrature-axis current (bottom) across a wide range of rotor
speeds. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
74
Experimental adaptive parameter estimator for a constant torque
command of 0.2 N-m at a fixed rotor speed of 2000 RPM demonstrating transient characteristics of the parameter estimator as well
as asymptotically vanishing torque perturbation due to the excitation
signal. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
75
Experimental characterization of steady-state parameter estimates
over a wide range of rotor speeds and torque commands. . . . . . .
76
Block diagram of the proposed RHCA-SIC methodology for PMSM
torque regulation. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
79
∗r
Sets of current pairs, (i∗r
d , iq ), which yield various torques for a machine with large saliency (to magnify nonlinearity). . . . . . . . . .
81
4.4
4.7
4.11
4.12
5.1
5.2
x
5.3
5.4
5.5
5.6
Disregard for past input (and state) data leading to a lack of persistent excitation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
86
Simulation of the static control allocation (5.6) without PE maximization. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
89
Simulation of the RHCA with PE maximization and without past
input and state data (5.15). . . . . . . . . . . . . . . . . . . . . . .
90
Simulation of the proposed RHCA-SIC methodology for over-actuated
systems with PE maximization and past data (5.16). . . . . . . . .
91
xi
LIST OF TABLES
Table
2.1
List of induction machine notation. . . . . . . . . . . . . . . . . . .
20
3.1
List of notation for PMSMs. . . . . . . . . . . . . . . . . . . . . . .
39
3.2
Simulation parameters. . . . . . . . . . . . . . . . . . . . . . . . . .
50
4.1
“Nominal” test machine parameters.
. . . . . . . . . . . . . . . . .
71
4.2
Manufacturer machine ratings. . . . . . . . . . . . . . . . . . . . . .
72
5.1
Simulation parameters. . . . . . . . . . . . . . . . . . . . . . . . . .
88
xii
ABSTRACT
Identification and Adaptive Control for High-performance AC Drive Systems
by
David M. Reed
Co-Chairs: Heath Hofmann and Jing Sun
High-performance AC machinery and drive systems can be found in a variety
of applications ranging from motion control to vehicle propulsion. Such applications typically require high-bandwidth and tight regulation of position, speed and/or
torque over a wide range of operating conditions. However, machine parameters can
vary significantly with electrical frequency, flux levels, and temperature, degrading
the performance of the drive system. While adaptive control techniques can be used
to estimate machine parameters online, it is sometimes desirable to estimate certain
parameters offline. Additionally, parameter identification and control are typically
conflicting objectives with identification requiring plant inputs which are rich in harmonics, and control objectives often consisting of regulation to a constant set-point.
In this dissertation, we present research which seeks to address these issues for highperformance AC machinery and drive systems.
The first part of this dissertation concerns the offline identification of induction
machine parameters. Specifically, we have developed a new technique for induction
machine parameter identification which can easily be implemented using a voltagesource inverter. The proposed technique is based on fitting steady-state experimental
data to the circular stator current locus in the stator flux linkage reference-frame
for varying steady-state slip frequencies, and provides accurate estimates of the magnetic parameters, as well as the rotor resistance and core loss conductance. This
approach allows leakage inductance and rotor resistance to be accurately estimated
while avoiding the difficulties associated with inverter-based implementations of the
standard locked-rotor test. Experimental results for a 43 kW induction machine are
xiii
provided which demonstrate the utility of the proposed technique by characterizing
the machine over a wide range of flux levels, including magnetic saturation.
The remainder of this dissertation concerns the development of generalizable design methodologies for Simultaneous Identification and Control (SIC) of overactuated
systems via case studies with Permanent Magnet Synchronous Machines (PMSMs).
Specifically, we present two different approaches to the design of adaptive controllers
for PMSMs which exploit overactuation to achieve identification and control objectives simultaneously. The first approach, termed “Adaptive Excitation Decoupling”,
utilizes a disturbance decoupling control law to prevent the excitation input from
perturbing the regulated output. Machine parameters used in the control law are
updated online via a normalized gradient estimator. The second approach uses a
Lyapunov-based inner-loop adaptive controller to constrain the states to the output
error-zeroing manifold, defined by the torque output mapping, on which they are
varied to provide excitation for parameter identification. Finally, the issue of input
selection (i.e., excitation input design and control allocation) is addressed for the
Lyapunov-based design by incorporating a receding-horizon control allocation which
includes a metric for generating persistently exciting reference trajectories. While
both approaches are shown to achieve the SIC objective, and each hold promise for
generalization, the Lyapunov-based design has robustness and stability advantages
over the Adaptive Excitation Decoupling approach.
xiv
CHAPTER I
Introduction
This dissertation describes a series of related research efforts aimed at advancing
the state-of-the-art in identification and control for high-performance AC motor drive
systems, with the secondary goal of developing control methodologies for simultaneous
identification and control of overactuated systems. Motivated by the rising interest in
electric propulsion for vehicular applications, as well as the desire to fully utilize the
capabilities of the AC machines used in such applications, we consider the problems of
offline identification of induction machine parameters and simultaneous identification
and torque regulation of Permanent Magnet Synchronous Machines (PMSMs). In
particular, Simultaneous Identification and Control (SIC) of PMSMs will serve as a
testbed for SIC methodologies for overactuated systems.
1.1
Motivation and Overview
These days, high-performance AC machinery and drive systems can be found in a
variety of applications ranging from motion control (servo drives) to vehicle propulsion
(traction drives). The distinguishing feature of high-performance drives, as opposed
to lower performance industrial drives (e.g., for pumps and fans), is the need for
high-bandwidth (i.e., fast response times) and tight (i.e., high accuracy) regulation of
position, speed and/or torque [12]. While the low cost of electronic components, particularly powerful microprocessors, continues to make electric drives a cost-effective
alternative for applications once dominated by mechanical systems, there are other
advantages as well. For instance, the dynamics of electrical systems are typically
much faster than those of mechanical systems, and the routing of wiring for electrical power (and control signals) is often easier than lines for fuel and hydraulic fluid.
Furthermore, when properly designed, electrical systems are typically more reliable
than mechanical systems due to a reduction in moving parts which are susceptible
1
to wear-and-tear. Additionally, high-performance AC machinery and drive systems
are capable of delivering impressive performance, particularly in terms of low-speed
torque (see Figure 1.1 for an example), making AC machines an attractive alternative
to SI engines for vehicle propulsion.
Figure 1.1: Comparison of torque-speed curves for Tesla induction motor and conventional IC engines [1].
Interest in hybrid and electric vehicles has increased greatly over the past decade
due to the rising cost of energy along with environmental concerns and government
mandates. While electric drives are a fairly mature technology, their use in vehicle
propulsion applications presents some unique challenges when it comes to maintaining a high level of performance over a very wide speed range, and under a variety
of operating conditions (loads, temperatures, etc.). Machine parameters can vary
significantly [7, 38, 40, 41] with electrical frequency, flux levels, and temperature. For
instance, resistance can increase by as much as 100% with temperature [39], while
inductances vary significantly when high flux levels cause magnetic saturation. Additionally, the sensitivity of the permanent magnet flux linkage to a 100◦ C rise in
temperature in ferrite, neodymium, and samarium cobalt magnets are -19%, -12%,
and -3%, respectively, from nominal [39]. These variations tend to “detune” the electric drive’s control system, degrading its performance. In particular, since torque isn’t
2
-=1
1.5
1.3
1.4
1.25
1.3
1.2
1.2
= /= $
= /= $
0.9
ird = !5A
1.1
1.05
1
0.8
ird = 2.5A
ird = 0A
ird = !2.5A
0.95
0.7
ird
0.6
0.5
ird = 5A
1.15
1.1
1
,=1
0.8
= 5A
0.9
ird = !5A
0.9
1
,
1.1
0.85
0.7
1.2
0.75
0.8
0.85
0.9
0.95
1
-
Figure 1.2: Example of steady-state torque errors (ratio) in a PMSM drive for variations in the permanent magnet flux linkage (left) and quadrature-axis
self-inductance (right) for a variety of direct-axis currents. The following
machine parameters were used to generate these plots: Ld = 212.3 µH,
Lq = 1.274 mH, and ΛP M = 12.644 m-Vsec.
directly measured due the impracticality1 of fielding torque sensors, the accuracy of
the regulated torque is therefore sensitive to variations in parameters which appear
in the torque output mapping (e.g., inductance and permanent magnet flux linkage).
For example [39], the ratio of the regulated torque output, τ , of a PMSM drive to its
reference, τ ∗ , is given by
(Ld − βLq )ird + αΛP M
τ
=
,
τ∗
(Ld − Lq )ird + ΛP M
(1.1)
where ird is the direct-axis current, Ld is the direct-axis inductance, Lq is the quadratureaxis current, and ΛP M is the permanent magnet flux linkage. The scalars α and β are
introduced to represent errors in the permanent magnet flux linkage and quadratureaxis inductance, respectively. This ratio (1.1) is plotted in Figure 1.2 for a range of
uncertainty and direct-axis currents, using machine parameters provided in the caption. Thus, accurate knowledge of machine parameters, including their variations, is
key to maintaining high-performance in electric drive systems. This is particularly
true in all-electric vehicles where it is desirable to run the machine in maximallyefficient operation points which depend upon the parameters of the machine.
1
Torque transducers are expensive and their calibration is sensitive to environmental conditions,
making them unsuitable for use in field applications (e.g., electric vehicles).
3
While parameter identification and adaptive control are mature fields of study,
their application in practice still tends to be challenging. In particular, the need
for inputs to the system under identification and/or control to be persistently exciting fundamentally conflicts with typical control objectives (e.g., set-point regulation), particularly in transportation applications where rider comfort would be negatively impacted by any large perturbations. In applications where accurate parameter
knowledge is important, it is therefore of interest to ensure that the system is persistently excited while eliminating, or at least minimizing, the impact of that excitation
on the regulated outputs. Overactuated systems2 provide an opportunity to achieve
persistent excitation and output regulation objectives simultaneously. In AC machines, the reduced-order two-phase equivalent models have two control inputs, the
direct and quadrature-axis voltages, and a single performance output, electromagnetic torque, to be regulated. Thus, AC machines are an example of an overactuated
system.
In this dissertation, we present research which seeks to address many of these
issues for high-performance AC machinery and drive systems, as well as investigation
of methodologies for simultaneous identification and control of overactuated systems.
Previous research of the authors demonstrated the use of an adaptive rotor resistance
estimator for improving the performance of direct field-oriented torque regulation for
induction machines in the presence of rotor resistance variations [63, 65]. While this
dissertation will not cover field-oriented control of induction machines, the need for
accurate parameter knowledge in such controllers motivates our first research project;
a new offline technique for improved identification of induction machine parameters
over a wide range of operating conditions [64]. In particular, our offline technique
provides accurate estimates of the magnetic induction machine parameters as well
as the rotor resistance and core loss conductance, which may be used in a fieldoriented controller to achieve high-performance torque regulation over a wide operating range that includes magnetic saturation. The remainder of this dissertation covers
research which concerns the development of simultaneous identification and control
methodologies for overactuated systems via case studies with Permanent Magnet Synchronous Machines (PMSMs) [66, 67]. More specifically, we will present two different
approaches, the first of which utilizes a disturbance decoupling control law to prevent
excitation for parameter identification from perturbing the regulated output. The
second approach uses an inner-loop Lyapunov-based adaptive controller to ensure
2
We use the term “overactuated” to refer to systems which have strictly more inputs than outputs
to be controlled.
4
that the states asymptotically track persistently exciting filtered reference commands
generated by a static control allocation based on the torque output mapping, thereby
constraining the states to the manifold described by the output mapping. Finally, the
second approach is modified to use a receding horizon control allocation which generates persistently exciting reference trajectories while also ensuring that the output
regulation objective is achieved.
Since the research to be discussed in this dissertation concerns the application
of system identification and adaptive control techniques to electric machinery and
drive systems, some basic background in these areas will be discussed. Following the
general background discussions, we will discuss the state-of-the-art in offline induction
machine parameter identification, as well as simultaneous identification and control.
Additional literature pertaining to adaptive control of PMSMs in general, will be
discussed at the beginning of Chapter 3. Finally, we will outline the open issues
which will be addressed as well as the specific contributions of this dissertation.
1.2
1.2.1
Background and the State of the Art
Background on Electric Machinery and Drive Systems
Electric Drive
Controller
Command
Values
Electrical
Measurements
Power
Electronics
Mechanical
Measurements
a
b
c
Electric
Machine
Figure 1.3: Drive system architecture.
The term “electric drive” generally refers to the power electronics, controller and
electrical sensors required to operate an electric machine in applications where tight
control over torque and/or speed is desired. The basic motor drive system architecture
is shown in Figure 1.3. It should be noted that, while the focus of this dissertation is
the identification and adaptive control of 3-phase AC machines, the control techniques
used are based on a two-phase equivalent model. The use of two-phase models is
5
common practice, as it leads to reduced order models which are easier to work with,
and may be extended to higher-than-three-phase machines as well.
Assuming that the machine is balanced in its construction, the 3-phase stator
currents sum to zero, and so the machine dynamics are adequately captured by two
electrical states, rather than three. Intuitively, the main principle of operation in
(rotating) electric machines is the generation of a rotating magnetic field, which only
requires two phases to achieve. The reason three phases are used in practice is that
it eliminates the need for a fourth “return” conductor in the AC distribution system.
The mapping from 3-phase (a − b − c) variables to equivalent 2-phase (d − q) is
generally referred to as the Clarke transform, named after Edith Clarke [14]:
 
  
 
xa
xd
xa
2/3 −1/3 −1/3
√  
√
  
 
3/3 − 3/3  xb  = T23  xb  .
 xq  =  0
1/3 1/3
1/3
xc
x0
xc
(1.2)
Likewise, the inverse Clark transform is given by
 
 
 
  
xd
xd
xd
1
0
1
xa
√
 
 
  
−1  
3/2 1 xq  = T23 xq  = T32 xq  .
 xb  = −1/2
√
−1/2 − 3/2 1
xo
xo
xo
xc
(1.3)
However, a simplification can be made by noting that the zero sequence component
(xo ) is equal to zero under our assumption of balanced construction and operation.
This leads to the following mapping referred to as the Modified Clark transform (1.4)
#" #
xa
√
.
2 3
xb
3
" # "
xd
1
= √3
xq
3
0
(1.4)
Additionally, we note that the transform used in this dissertation is scaled such that
the peak values of the 3-phase sinusoidal electrical variables are preserved in the 2phase representation.
In this dissertation, the machines are assumed to be fed by a Voltage Source Inverter (VSI) (see Figure 1.4), which generates Pulse-Width Modulated (PWM) (see
Figure 1.5) versions of the sinusoidal voltages commanded by the control algorithm.
The primary advantage of using PWM voltages is a large reduction in converter losses
since the VSI transistors, which serve as “switches”, are never operated in their “linear” region for an extended period of time, but instead alternate between “ON” and
“OFF” states. The drawbacks are the generation of Electromagnetic Interference
6
“Ideal” Inverter
+
Da+
VBUS
_
Da-
“Practical” Inverter
Db+
Dc+
va
vb
Db-
Dc-
+
vc
Da+
VBUS
_
Da-
Db+
Dc+
va
vb
Db-
Dc-
vc
Figure 1.4: Ideal (left) and practical (right) three-phase VSIs.
(EMI), and that hard-switching (i.e., switching when there is an overlap in voltage
across the device and current through it) increases stress on switching devices, winding insulation, and even machine bearings. By synchronizing the switching of the
transistors and the sampling of the Analog-to-Digital Converters (ADCs), we avoid
any spurious measurements due to EMI; while other design choices can help mitigate
the other issues associated with hard-switching (e.g., using a motor with inverter-duty
rated insulation).
The inverter is typically treated as “ideal” when designing the control algorithm
in that current harmonics generated by switching are neglected, save the desired
fundamental frequency, as well as the dead-time effect and voltage limitations. The
justification for this comes from average-value modeling [69], which holds provided
that the switching frequency, fsw , is sufficiently higher than that of the maximum
fundamental frequency, fmax . A typical rule-of-thumb is that fsw ≥ 21fmax [57]. In
terms of control, the VSI may be viewed as the actuator used to control the AC
1
Mod. Signal
Carrier Wave
0.8
0.6
0.4
0.2
0
0
0.002 0.004 0.006 0.008
0.01 0.012 0.014 0.016 0.018
0.02
0.002 0.004 0.006 0.008 0.01 0.012 0.014 0.016 0.018
Time (sec)
0.02
PWM Waveform
1
0.5
0
−0.5
−1
0
Figure 1.5: Generation of sinusoidal PWM waveform.
7
machine, and our dynamic AC machine models will therefore take the stator voltages
to be the inputs to the dynamics, while currents and/or flux linkages will serve as
the states. Our measured outputs will typically be the stator currents, as well as
the rotor speed and/or position. Electromagnetic torque will generally serve as our
performance (or regulated ) output, which is unmeasured due to the impracticality of
measuring torque.
Finally, it is noted that modern control of AC machines is typically based on what
are referred to as Field-Oriented Control (FOC) techniques. The basic premise of
FOC is to perform the actual control (e.g., current regulation) in a rotating reference
frame. This approach has several advantages:
1. Sinusoidal electrical variables are transformed into constant values in such reference frames under steady-state conditions. This allows the use of conventional
control techniques, such as PI compensation, to regulate the stator currents of
the machine.
2. Further simplification of the AC machine dynamics. For example, in synchronous machines, representing the dynamics in a reference frame aligned with
the rotor position eliminates nonlinear terms associated with the EMF.
3. Additionally, for both synchronous and induction (a.k.a. asynchronous) machines, the expressions relating electrical variables (e.g., currents and flux linkages) to the electromagnetic torque are simplified, which is advantageous as
these expressions are used to generate the command values for regulating the
electrical states in order to achieve a desired torque.
syn
q
syn
syn
q
syn
d
x
syn
d
Figure 1.6: Vector diagram depicting the Park transform of an arbitrary vector.
8
The projection of electrical variables into a rotating reference frame (Figure 1.6) is
typically referred to as the Park transform, named for R.H. Park who published the
first papers in 1929 [59] detailing the application of reference frame theory to the
analysis of AC machines. Starting from two-phase equivalent variables and using
notation from Fig. 1.6, the Park transform is simply given by,
"
~xsyn
# "
#" #
xsyn
cos
(θ
)
sin
(θ
)
xd
syn
syn
= dsyn =
= e−Jθsyn ~x,
xq
−sin (θsyn ) cos (θsyn ) xq
(1.5)
where J is the 90◦ counter-clockwise (CCW) rotation matrix:
"
#
0 −1
J=
.
1 0
For completeness, the inverse Park transform is simply given by,
#
#"
" # "
syn
xd
cos (θsyn ) −sin (θsyn ) xd
~x =
=
= eJθsyn ~xsyn .
syn
xq
xq
sin (θsyn ) cos (θsyn )
1.2.2
(1.6)
State-of-the-Art in Offline Identification of Induction Machine Parameters
Historically, induction machines have been the industrial workhorse while permanent magnet machines have dominated high-performance applications. However,
advancements in their design and control have made induction machines a viable alternative to permanent magnet machines in automotive applications (e.g., the Tesla
Model S) where ruggedness and the absence of expensive rare-earth magnets are desirable characteristics. Nevertheless, the challenge remains that high-performance
control techniques for induction machines, such as field-oriented control, require accurate knowledge of the machine parameters [41].
Over the past few decades, a considerable amount of attention has been given
to the online identification of the rotor time constant and/or rotor resistance, e.g.
[32, 42, 55, 63, 75, 81, 85], as these parameters can vary significantly with temperature,
leading to severe detuning in both direct and indirect field-oriented controllers [7,41].
In addition to the rotor time constant, online techniques have been proposed for other
machine parameters as well, e.g., [25, 50, 61, 73]. However, the added complexity and
design difficulty of adaptive parameter estimation might not be appropriate for some
applications. Furthermore, since the variations in some machine parameters, such as
9
the inductances, may be modeled as functions of known or measurable variables such
as flux linkage magnitude [24], offline identification of such machine parameters for
non-adaptive control methodologies is a viable alternative.
The IEEE standard for induction machine parameter identification [2] uses noload and locked-rotor tests for offline parameter identification. However, accurate
parameter estimation using the standard requires special equipment to conduct these
tests. For example, it is recommended that locked-rotor tests be conducted at electrical frequencies close to typical slip frequencies (e.g., 25% of rated frequency) to obtain
accurate leakage inductance and rotor resistance estimates. While a voltage-source
inverter could be used to generate voltages with such a frequency, the presence of
switching harmonics in the output voltage complicates voltage measurements, unless
the inverter has a significant output filter. An alternative to measuring the voltage
is to calculate the voltage from the inverter duty cycles and bus voltage. However,
transistor voltage drops and the deadtime effect [7] make it difficult to accurately
determine output voltages when they are small, as in the case of the locked-rotor
test.
Alternatives to the IEEE standard for offline identification of induction machine
parameters have been proposed, which can generally be categorized as using either
transient measurements (e.g., [21, 35, 46, 68, 70, 80]) for parameter identification, or
steady-state measurements (e.g., [3,4,43,58]), like the technique proposed in this dissertation. In [3], an adaptive (search boundary) genetic algorithm is used to identify
machine parameters, while a more recent paper [4] has proposed using the LevenbergMarquardt algorithm, commonly used to solve nonlinear least-squares problems, to
estimate induction machine parameters. Other approaches have been proposed as
well [43, 58] which use variable frequency tests at a standstill (i.e., zero rotor speed)
to estimate parameters. However, none of these papers [3,4,21,35,43,46,58,68,70,80]
considers core loss in their parameter identification, which can influence the accuracy
of estimated parameters [8, 79]. Nor is the characterization of magnetic saturation
considered beyond noting that it can have an influence on the accuracy of estimated
parameters [35, 43, 46, 58, 68, 70]. Finally, the value-added by estimating the core loss
conductance and saturation characteristics is that knowledge of these parameters, and
their variations, may be used in the control law (e.g., [24] for inclusion of saturation,
and [79] for core loss) as well as for loss estimation and/or minimization (e.g., [23]).
10
1.2.3
Background on Parameter Identification and Adaptive Control
Model-based control requires reasonably accurate knowledge of the plant parameters. While there are a number of ways to determine the plant parameters, a common
approach is to fit an input-output model of the plant to experimental data obtained by
exciting the plant dynamics with a known input and measuring the output response.
This identification process may be done “offline” when the plant parameters either a)
don’t vary significantly, or b) vary in a known or repeatable manner. As mentioned
in the previous section, the inductances of an AC machine can be modeled as nonlinear functions of flux linkage magnitude to capture saturation effects. The saturation
characteristics don’t vary significantly with temperature or time, and can therefore
be identified offline and “hard-coded” in the controller. The plant parameters may
also be identified “online” in real-time when there is an immediate need or use for
that information, such as condition monitoring and fault detection, for example.
Adjustment
Mechanism
Modeling
Controller
Plant
Design
Figure 1.7: Typical structure of adaptive controllers.
Online identification is typically performed in the context of adaptive control. As
used in this document, the term adaptive control refers to a control methodology in
which control law parameters are updated in real-time via a parameter identification
Permanent Magnet
Synch.typical
Machines structure of an adaptive controller is shown in Figure 1.7. A
algorithm. The
(PMSM)
common design approach for adaptive control, which we have used in our research, is
the so-called certainty equivalence principle in which the control law is designed first
Surface-Mount
Interior Permanent
assuming
that the plant
parameters
are known, and then an adaptive law is designed
Permanent Magnet
Magnet
(SMPM)
(IPM)
to estimate those parameters. Thus, as the parameter estimates converge to their true
values, the performance of the adaptive controller tends to the desired performance
of the certainty equivalence design.
To identify the plant parameters, linear parametric models of the following form
are often used,
z(t) = θ T φ(t),
(1.7)
Implementation
where z and φ consist of measurable signals, and θ consists of the plant parameters
11
we want to identify. Note that in general z is a vector (often a scalar), while φ and θ
are either vectors or matrices, depending on the particular application. Additionally,
it should be noted that z and φ often contain time-differentiated versions of measured
signals, which are typically “estimated” by filtering [28]; this will be discussed in more
detail later when it is needed. Identifying the plant parameters can be thought of as an
optimization problem which seeks to find the estimated parameters, θ̂, which minimize
the difference between the measured signal, z, and its estimated value ẑ = θ̂ T φ
(assuming z and φ are bounded), e.g.:
1
min kz − θ̂ T φk22
θ̂ 2
(1.8)
Parameter estimates are often obtained using the familiar gradient-descent and leastsquares algorithms [28]. However, an important sufficient condition for the estimated
parameters to converge to their true values (i.e., θ̂ → θ as t → ∞) is that the regressor
φ(t) be persistently exciting:
Definition I.1. (Persistence of Excitation (PE) [28]): A piecewise continuous signal
vector φ : R+ 7→ Rn is said to be persistently exciting in Rn with a level of
excitation α0 > 0 if there exist constant scalars α1 , T0 > 0 such that
1
α1 I ≥
T0
Z
t
t+T0
φ(σ)φ> (σ)dσ ≥ α0 I, ∀ t ≥ 0.
(1.9)
While we won’t get into the details here, the persistence of excitation condition on
the regressor is key to proving that the parameter error, θ̃ = θ̂ − θ, goes to zero. A
rule of thumb is that the input to the system under identification must contain at
least one distinct frequency component for every two parameters to be identified; this
condition is referred to as sufficient richness [28].
Analysis of closed-loop adaptive control systems tends to be rather challenging
since even when the plant under control is linear-time invariant (LTI), the closed-loop
system under adaptive control is nonlinear. This is particularly true when the control
law and parameter estimator are designed separately, and then simply “plugged” together to form an adaptive controller. So, even though combining, say, a least-squares
parameter estimator with a particular control law to form an adaptive controller is an
intuitive and easy design approach, it is typically very challenging to prove closed-loop
stability for such a design.
12
An alternative approach is to formulate a control law based on the certainty equivalence principle, and then design the parameter estimator (or update law) such that it
ensures the closed-loop system is stable in the presence of parameter uncertainty, using a Lyapunov stability analysis. Essentially, what happens is that the adaptive law
is designed to cancel the indefinite terms which appear when the Lyapunov function
candidate is differentiated with respect to time, ensuring that the derivative of the
Lyapunov function is negative semi-definite for all time, t, and that the closed-loop
system is therefore stable in the sense of Lyapunov. Barbalat’s lemma [28, 33, 71]
is then typically used to prove asymptotic stability of the closed-loop adaptive control system. While this design approach is generally more involved than the “plugand-play” approach using linear parametric models and least-squares (or gradient
descent) algorithms, it has the advantage of coming with a stability proof. However,
the difficulty associated finding a Lyapunov function, if possible, limits the general
applicability of this approach.
1.2.4
State-of-the-Art in Simultaneous Identification and Control
As noted earlier, identification and control are typically conflicting objectives. For
identification, we need to select plant inputs such that the plant dynamics are persistently excited, which typically involves signals which are rich in harmonic content.
However, for output regulation, we are typically interested in tracking some reference value, often a constant set-point, which may not provide sufficient excitation
for parameter convergence. For single-input single-output (SISO) systems, this is an
unavoidable trade-off between output regulation and parameter identification.
One approach to handling this trade-off is the so-called “dual control”, introduced
by Feldbaum in 1960 [15]. The dual control input is derived by solving a stochastic
optimal control problem which seeks to balance the trade-off between maintaining
tight control and small parameter estimation errors. A key characteristic of dual
control is that the control law is a function of the estimated parameters as well as
their uncertainties, similar to how a Kalman filter estimates states along with the
noise covariance. However, analysis requires nonlinear stochastic control theory, and
solutions to the optimal dual control problem are challenging if not impossible to
find for all but simple problems [84]. For this reason, there has been a fair amount
of interest in finding approximate (or sub-optimal) approaches to achieving the dual
control objective, e.g., [5, 11, 16, 19, 20, 22, 30, 37, 49, 54, 62, 72, 82].
Receding Horizon Control (RHC) (a.k.a. Model Predictive Control (MPC)3 ) [52],
3
In this dissertation, the terms RHC and MPC will be used interchangeably.
13
which has seen a rapid growth in popularity in recent years, provides a natural platform for alternative dual control (i.e., simultaneous identification and control) strategies due in part to its inherent optimization and constraint handling. In fact, most
of the papers cited in the preceding paragraph use MPC as a basis for implementation [5, 11, 19, 20, 22, 37, 54, 62, 72, 82]. Within the MPC framework, a metric for
excitation is incorporated into the optimization problem to ensure the generation of
persistently exciting control signals along with the usual control metrics. The tradeoff between identification and control may then be managed by tuning the weighting
(or penalties) placed on excitation and regulation metrics. For example, in [19, 54]
the metric for persistent excitation is included as an inequality constraint in the MPC
formulation, whereas [37] includes it in the cost function. However, while a tradeoff is unavoidable in SISO systems, overactuated systems provide an opportunity to
circumvent this trade-off.
In an overactuated system, there is no unique input vector which yields a particular
output. Thus, overactuated systems provide an opportunity to achieve persistent
excitation and output regulation objectives simultaneously, potentially without tradeoff, when the excitation is constrained to the “null-space” of the system. While
this idea has been explored for specific applications in recent years [10, 45, 83], the
problem has yet to be treated in a more general framework. In [45], [83] the authors
exploit the redundancy in a spacecraft with an overactuated reaction wheel array by
restricting the optimized excitation signal to the “null-motion” of craft in order to
estimate actuator misalignments. Similarly, in [10], the authors exploit the actuation
redundancy in an electric vehicle with separate drives for the front and rear wheels,
to generate sufficiently rich input signals for road friction coefficient identification
without affecting the vehicle motion.
1.3
1.3.1
Open Issues and Contributions
Open Issues
The open issues in identification and control for high-performance AC drive systems which we will address in this dissertation are as follows:
• Accurate offline identification of induction machine parameters using a voltagesource inverter and over a wide operating range - The IEEE standard for induction machine parameter identification [2] uses no-load and locked-rotor tests for
offline parameter identification. However, accurate parameter estimation us-
14
ing the standard requires special equipment to conduct these tests. While a
voltage-source inverter could be used, the presence of switching harmonics in
the output voltage complicates voltage measurements, unless the inverter has
a significant output filter. Thus, there is a need for an alternative to the IEEE
standard for offline induction machines parameter identification, which can be
implemented in VSI-driven systems and is capable of characterizing machines
over a wide range of operating conditions. While alternatives to the IEEE standard have been proposed in the literature (see Section 1.2.2), these techniques
do not consider core loss or the characterization of magnetic saturation, both
of which can significantly affect the accuracy of the estimated parameters, and
knowledge of which may be used to improve controller performance.
• Development of methodologies for simultaneous identification and control of
overactuated systems - Adaptive control is often used to obtain high-performance
when controlling uncertain systems. However, while steady-state tracking can
typically be guaranteed, regardless of the accuracy of the estimated parameters, transient performance, as well as the regulation of unmeasured outputs,
can suffer in the presence of inaccurate parameter estimates. Furthermore,
accurate parameter knowledge is vital to secondary objectives such as loss minimization (e.g., in AC machines) and condition monitoring, as well as ensuring
that constraints are satisfied in predictive control designs. While parameter
identification and output regulation are typically conflicting objectives, overactuated systems such as AC machines provide an opportunity to achieve these
objectives simultaneously in real-time without compromise. Dual control and
alternatives have been introduced to manage this trade-off, typically for SISO
systems. However, currently no general methodology exists which specifically
exploits overactuation for simultaneous identification and control.
1.3.2
Summary of Contributions and Innovations
This dissertation seeks to address the open issues described in the previous section,
and more, via the following contributions:
• A novel technique for offline identification of induction machine parameters We have developed a new technique for induction machine parameter identification which can easily be implemented using a voltage-source inverter [64]. The
proposed technique is based on fitting steady-state experimental data to the circular stator current locus in the stator flux linkage reference-frame for varying
15
steady-state slip frequencies, and provides accurate estimates of the magnetic
parameters, as well as the rotor resistance and core loss conductance. This approach allows accurate leakage inductance and rotor resistance estimation while
avoiding the difficulties associated with inverter-based implementations of the
locked-rotor test. Experimental results for a 43 kW induction machine are provided which demonstrate the utility of the proposed technique by characterizing
the machine over a wide range of flux levels, including magnetic saturation;
• The development of generalizable design methodologies for simultaneous identification and control of overactuated systems via case studies with permanent magnet synchronous machines - Specifically, this dissertation will present two different approaches to the design of adaptive controllers for PMSMs which exploit
overactuation to achieve identification and control objectives simultaneously by
ensuring that excitation signals introduced for parameter identification have a
minimal impact on the regulated torque output. The first approach, termed
“Adaptive Excitation Decoupling” [67], utilizes a disturbance decoupling control law to prevent the excitation input from perturbing the regulated output,
while machine parameters used in the control law are updated online via a normalized gradient estimator. The second approach [66] uses a Lyapunov-based
inner-loop adaptive controller to constrain the states to the output error-zeroing
manifold, defined by the torque output mapping, on which they are varied to
provide excitation for parameter identification. Finally, the issue of input selection (i.e., excitation input design and control allocation) is addressed for the
Lyapunov-based design by incorporating a receding-horizon control allocation
which includes a metric for generating persistently exciting reference trajectories. While both approaches are shown to achieve the SIC objective, and each
hold promise for generalization, the Lyapunov-based design has robustness and
stability advantages over the Adaptive Excitation Decoupling approach.
• The development of numerical tools for analysis and design of high-performance
R
AC drive systems - Specifically, we have developed Simulink
models for single and three-phase inverters with dead-time effect and center-based PWM,
sampled-data controller models which account for time delays as well as single
and twice-per-period sampling schemes, and finally AC machine models which
allow parameters to be varied;
16
• Construction of a physical testbed for electric drives with hybrid energy storage
- Finally, we have constructed a testbed for the purpose of experimentally validating advanced control algorithms for high-performance AC drive systems, as
well as energy cycling using hybrid energy storage. While it won’t be discussed
in detail here, the development of the testbed has been documented in [26].
1.4
Reader’s Guide
The remainder of this dissertation is organized as follows:
Chapter 2: Offline Identification of Induction Machine Parameters
presents a new technique for offline identification of induction machine parameters using a voltage-source inverter. The stator current locus representation in
the stator flux linkage reference frame is first derived, and next, the estimation
technique is discussed. Following a discussion of the stator flux linkage estimator, simulation and experimental results demonstrating the effectiveness of the
proposed technique are presented.
Chapter 3: Adaptive Excitation Decoupling Approach to Simultaneous Identification and Control of Permanent Magnet Synchronous
Machines begins our discussion of simultaneous identification and control of
permanent magnet synchronous machines. The proposed technique utilizes a
disturbance decoupling control law to prevent the excitation input, introduced
to ensure that conditions for persistency of excitation are satisfied, from perturbing the regulated (i.e., electromagnetic torque) output. A normalized gradientbased identifier is used to estimate the machine parameters and update the
excitation (i.e., disturbance) decoupling control law. Simulations are used to
verify the resulting closed-loop adaptive excitation decoupling controller.
Chapter 4 Simultaneous Identification and Control of Permanent
Magnet Synchronous Machines via Adaptive 2-DOF Lyapunov Design presents an alternative adaptive control design for PMSMs which achieves
the simultaneous identification and control objective using a Lyapunov-based
design. By regulating the states to the output error-zeroing manifold, we ensure
that perturbations to the torque output (due to the presence of an excitation
signal) are minimized while still providing excitation for parameter identification. This approach has the advantages that the stator currents (i.e., the states)
17
are directly regulated, closed-loop stability is proven, and analysis of sufficient
conditions for parameter convergence is tractable.
Chapter 5 Receding Horizon Control Allocation for Simultaneous
Identification and Control of PMSMs extends the approach presented in
Chapter 4 to use an optimization-based control allocation, rather than the fixed
allocation used in Chapter 4. A receding-horizon optimization, which includes a
metric for encouraging the generation of persistently exciting signals, is used to
generate the command current trajectories for a given command torque, which
are fed to the inner-loop adaptive current regulator derived in Chapter 4. This
work address the limitations of the work presented in Chapter 4 in that specialized knowledge (i.e., intuition) of the plant is not needed to design the control
allocation, and the design of persistently exciting signals is guided by a rigorous
metric; these points, the control allocation and design of persistently exciting
signals, are handled automatically by the receding-horizon optimization.
Chapter 6 Conclusions and Future Work summarizes the results of this
research and makes suggestions for future research directions.
18
CHAPTER II
Offline Identification of Induction Machine
Parameters
2.1
Introduction
This chapter presents a new technique for induction machine parameter identification using steady-state current measurements. The proposed technique is based on
fitting experimental data to the circular stator current locus in the stator flux linkage reference-frame for varying steady-state slip frequencies, and provides accurate
estimates of the magnetic parameters, as well as the rotor resistance and core loss
conductance. Numerical simulation results evaluating the accuracy of the estimated
parameters in the presence of non-ideal effects are presented, and experimental results for a 43 kW induction machine are provided which demonstrate the utility of
the proposed technique by characterizing the machine over a wide range of flux levels,
including magnetic saturation.
2.2
The Steady-State Stator Current Locus
In this work, the induction machine is modeled as having a smooth air-gap (i.e.,
slotting effects are neglected) in addition to the following simplifying assumptions:
A1. a quasi-linear magnetics model;
A2. the machine is balanced in its construction with sinusoidally-distributed magnetomotive force (mmf);
This chapter is based on a previously published conference paper [64] which has been expanded and
submitted to a journal and is currently under review:
D. M. Reed, H. F. Hofmann, and J. Sun, “Offline Identification of Induction Machine Parameters
with Core Loss Estimation using the Stator Current Locus,” Under review, 2015.
19
A3. a 1:1 effective turns ratio;
A4. the core loss is modeled as a resistive shunt just after the stator winding resistance (see Figure 2.1).
The first assumption, A1, permits variations in the magnetic parameters with operating conditions, while A2 justifies the use of a 2-phase equivalent model and transformation from 3-phase to 2-phase using the Clarke transform [14]. The third assumption, A3, is common for squirrel-cage induction machines, which do not have
physical rotor windings. Finally, while core loss is typically modeled as a resistance
in parallel with the mutual inductance [17], this placement is somewhat arbitrary as
leakage flux also travels through the machine iron. A4 simplifies the analysis while
still capturing the nature of the core loss (i.e., electrical power which is not converted
into mechanical power). Finally, a list of induction machine notation is provided in
Table 2.1.
Table 2.1: List of induction machine notation.
Symbol
Description
~vs (t) = [vsd (t) vsq (t)]>
Stator Voltage Vector
~is (t) = [isd (t) isq (t)]>
Stator Current Vector
~λs (t) = [λsd (t) λsq (t)]>
Stator Flux Linkage Vector
Rs
Stator Winding Resistance
Ls
Stator Winding Self-Inductance
M
Mutual Inductance
Gc
Core Loss Conductance
ωe
Electrical Frequency
~ir (t) = [ird (t) irq (t)]>
Rotor Current Vector
~λr (t) = [λrd (t) λrq (t)]>
Rotor Flux Linkage Vector
Rr
Rotor Resistance
Lr
Rotor Self-Inductance
ωr
Rotor Angular Velocity
P
ωre =
Number of Poles
P
ω
2 r
ωse = ωe − ωre
Rotor Electrical Angular Velocity
Electrical Slip Frequency
20
Figure 2.1: Dynamic 2-phase equivalent circuit model for an induction machine.
The desired expressions for the steady-state stator currents in the stator flux
linkage reference-frame are developed starting from the flux linkage dynamics in the
stationary reference-frame for the 2-phase equivalent induction machine model:
d~λs
= −Rs~is + ~vs ,
dt
d~λr
= −Rr~ir + ωre J~λr ,
dt
(2.1)
(2.2)
where ~λs = [λsd λsq ]> is the stator flux linkage vector, ~λr = [λrd λrq ]> is the rotor
flux linkage vector, ~is = [isd isq ]> is the stator current vector, ~ir = [ird irq ]> is the
rotor current vector, ~vs = [vsd vsq ]> is the stator voltage vector, and J is the 90◦
CCW rotation matrix. These expressions (2.1)-(2.2) are easily derived by applying
Kirchhoff’s and Faraday’s laws to the equivalent circuit model provided in Fig. 2.1.
Additionally, the following flux linkage/current relationships, which hold for arbitrary
reference-frames (denoted by the superscript x), are needed
~λx = Ls~ix + M~ix ,
s
s
r
~λx = M~ix + Lr~ix ,
r
s
r
(2.3)
(2.4)
where Ls = Lls + M and Lr = Llr + M .
To represent (2.2) in the stator flux linkage reference-frame, we use the Park
transform [59],
~x λs = e−Jθλs ~x,
(2.5)
where the superscript λs is used to designate variables which are being represented in
the stator flux linkage reference-frame, the angle of which is denoted by θλs . Applying
(2.5) to the stationary-frame electrical variables in (2.2) yields
d~λλr s
= −Rr~irλs + ωse J~λλr s ,
dt
21
(2.6)
where ωse = ωe − ωre is the electrical slip frequency. At steady-state1 (2.6) gives us
the following expression for the steady-state rotor currents
Ωse ~ λs
JΛ r ,
I~rλs = −
Rr
(2.7)
where Ωse is the steady-state electrical slip frequency. From (2.3)-(2.4) it can be
shown that
2
~λx = σ ~i x + M ~λx ,
(2.8)
r
Ls r
Ls s
where σ 2 = Ls Lr − M 2 . Plugging (2.8) into (2.7) and solving for the rotor current,
I~rλs , we obtain the following expression for the steady-state rotor currents in the stator
flux linkage reference-frame as a function of slip frequency and stator flux linkage:
I~rλs = −
Ωse M
Rr Ls
1+
Ωse σ 2
Rr Ls
2
Ωse σ 2
~ λs s ,
I+J Λ
Rr Ls
(2.9)
where I is the 2 × 2 identity matrix.
Finally, using the stator current relationship with the core loss conductance (as
defined in Fig. 2.1),
~
~is = Gc dλs + ~i0s ,
(2.10)
dt
which, at steady-state and when represented in the stator flux linkage reference-frame,
is given by
~ λs + I~ λ0s ,
I~sλs = Gc Ωe JΛ
(2.11)
s
s
along with the fact that
1 ~ λs
0
I~sλs =
Λs − M I~rλs ,
Ls
(2.12)
λs
we obtain the desired scalar form expressions for the steady-state direct, Isd
, and
λs
quadrature, Isq
, stator currents in the stator flux linkage reference-frame in which
the direct-axis is aligned with the stator flux linkage vector (i.e., ~λλs s = [||~λs || 0]> ):
"
λs
Isd
= 1+
λs
Isq
1
M
σ
2
s
Ωse /Ωλse,max
2 #
2
~ s ||
||Λ
,
Ls
s
1 + Ωse /Ωλse,max
2
s
~ s ||
Ωse /Ωλse,max
M
||Λ
~ s ||,
=
+ Gc Ωe ||Λ
2
λ
σ
L
s
s
1 + Ωse /Ωse,max
Steady-state variables are denoted by capital letters.
22
(2.13)
(2.14)
Figure 2.2: Parameterized steady-state stator current locus in the stator flux linkage
reference-frame.
~ s || is the stator flux linkage magwhere Ωe is the steady-state electrical frequency, ||Λ
s
nitude and Ωλse,max
= Rσr L2 s is the slip frequency which maximizes torque for a given
stator flux linkage magnitude.
When plotted as a function of slip frequency, (2.13) and (2.14) produce the circular
stator current locus in Figure 2.2. The parameterized stator current locus circle is
given by
2
2
λs
λs
Isd
− xo + Isq
− yo = r 2 ,
(2.15)
where,
Lr
1
+ 2
Ls σ
~ s ||,
yo = Gc Ωe ||Λ
1
xo =
2
r=
M2 ~
||Λs ||.
2σ 2 Ls
~ s ||,
||Λ
(2.16)
(2.17)
(2.18)
The stator current locus in Fig. 2.2 can therefore be used to identify the magnetic
parameters of the induction machine (i.e., Ls ,Lr , and M ), as well as the core loss
conductance, Gc , by fitting the parametric circle (2.15) to experimental data (i.e.,
λs
λs
Isd
(Ωse ) and Isq
(Ωse )) which forms the stator current locus, provided that the stator
flux linkage magnitude is held constant (i.e., regulated) during data collection. Once
estimates of the center location, (x̂o , ŷo ), and radius, r̂, have been computed, the
core loss and magnetic parameters are calculated assuming that the inductance ratio,
Ls
, is known2 . This assumption gives us three equations with three unknowns, Ls,r ,
Lr
2
If the NEMA-design letter is known, this ratio can be found in IEEE Standard 112; otherwise
23
M , and Gc . Once we have estimates of the magnetic parameters and the core loss
conductance, (2.13) and (2.14) along with the corresponding slip frequencies, are used
to estimate the rotor resistance, Rr .
Since the magnetic parameters are obtained from the parameterization of the
fitted locus circle, their estimates are independent of the rotor resistance, which will
vary with temperature. Other benefits of the proposed technique are a reduction in
the dimension of the estimation problem by identifying magnetic parameters (and
core loss conductance) separately from the rotor resistance, the use of multiple data
points in estimating parameters, and the ability to characterize the machine over a
wide range of operating points which include magnetic saturation.
2.3
2.3.1
Proposed Parameter Estimation Technique
Fitting the Parameterized Stator Current Locus Circle to Data
Using the stator current locus presented in the previous section, the magnetic
parameters, as well as the core loss conductance, are identified by fitting the parameterized circle (Fig. 2.2) to experimental data. The fitting is achieved by solving a
reasonably simple minimization problem. According to the theory, the zero-slip data
point and the center of the stator current locus should be aligned horizontally. While
it may seem simplistic to use the zero-slip data point to fix the center of the estimate
stator current locus (circle), it works well in practice, as will be demonstrated in our
numerical analysis.
Enforcing the condition that the zero-slip data point determines the vertical offset
in the locus circle, we see that
ŷo = Isq (Ωse = 0).
(2.19)
The magnetic parameters are then computed by solving the following minimization
problem
(x̂o , r̂) = arg min Jscl (x, r),
(2.20)
(x,r)>0
where the cost function, Jscl (x, r), is given by,
Jscl
N h
X
2
2 i2
λs
λs
− ŷo
,
=
r2 − Isd,n
− x + Isq,n
n=1
assume Ls /Lr = 1.
24
(2.21)
λs
λs
and Isq,n
are the nth measurements of the steady-state direct and quadrawhere Isd,n
ture stator currents in the stator flux linkage reference-frame. This approach (2.20)(2.21) is sometimes referred to as a “pure least-squares” solution [76]. In our work,
we chose to solve this minimization problem numerically using MATLAB’s fmincon
constrained nonlinear programming algorithm.
Given estimates of the center location, (x̂o , ŷo ), and radius, r̂, the stator selfinductance is given by
~ s ||
||Λ
L̂s =
.
(2.22)
x̂o − r̂
Next, we assume that the stator and rotor inductances are the same (i.e. L̂s /L̂r = 1)
and compute the leakage term:
σ̂ 2 =
~ s ||
L̂r L̂s ||Λ
.
~ s ||
2L̂s x̂o − ||Λ
(2.23)
Once the self-inductance and leakage terms are known, the mutual inductance is
calculated:
q
M̂ = L̂s L̂r − σ̂ 2 .
(2.24)
Finally, the estimate of the core loss conductance is given by
Ĝc =
ŷo
.
~ s ||
Ωe ||Λ
(2.25)
To estimate the rotor resistance, we will minimize the sum-of-squares error beλs
λs
tween the experimental data points (i.e., Isd
(Ωse ) and Isq
(Ωse )) and those predicted by
the steady-state model of the stator current locus (in the stator flux linkage referenceframe). Again, the parameter estimation is obtained by solving a constrained minimization problem:
R̂r =
arg min
JRr (Rr )
(2.26)
Rr ∈[Rr,min , Rr,M AX ]
where the cost function, JRr (Rr ), is given by,
JRr (Rr ) =
N X
n=1
λs
Isd,n
λs
− Iˆsd,n
25
2
+
λs
Isq,n
λs
− Iˆsq,n
2 ,
(2.27)
λs
λs
and Iˆsq,n
are functions of Rr derived from (2.13) and (2.14):
where Iˆsd,n


M̂ 2

λs
= 1 + 2 Iˆsd,n
σ̂
M̂ 2
λs
= 2 Iˆsq,n
σ̂
~ s ||
(σ̂ 2 Ωse,n )
 ||Λ
,

2
L̂s
Rr L̂s + (σ̂ 2 Ωse,n )2
2
~ s ||
Rr L̂s σ̂ 2 Ωse,n
||Λ
~ s ||.
+ Ĝc Ωe,n ||Λ
2
2
Rr L̂s + (σ̂ 2 Ωse,n ) L̂s
(2.28)
(2.29)
Note that a non-zero stator flux linkage magnitude and non-zero slips are required
for identification of the rotor resistance. This amounts to a rather intuitive persistent
excitation condition [63], in that it suggests rotor currents must be present in order to
determine the rotor resistance. While this cost function (2.27)-(2.29) is globally nonconvex, it is convex for practical rotor resistance values (e.g., positive values) with
a unique minimum at the true resistance. Therefore, we enforce constraints when
solving the minimization problem, requiring that Rr ∈ [0.1Rs , 10Rs ]. Once again, we
use MATLAB’s fmincon constrained nonlinear programming algorithm to solve the
minimization problem.
2.3.2
Procedure for Data Collection
In order to generate the experimental stator current locus, the measured stator
currents must be projected into the stator flux linkage reference-frame using the Park
transform (2.5). Additionally, the stator flux linkage magnitude must be held constant
while the steady-state direct and quadrature stator currents are recorded for various
steady-state (i.e., constant) slip frequencies, Ωse . To ensure that the flux linkage
magnitude remains constant, a Proportional-Integral (PI) regulator is used to drive
the error between the commanded stator flux magnitude, ||Λ̃s ||, and estimated flux
magnitude, ||Λ̂s ||, to zero. The output of the PI regulator is the stator excitation
voltage magnitude, ||~vs ||, as depicted in Figure 2.3. The slip frequency is varied by
either fixing the electrical frequency, Ωe , of the stator excitation voltage and varying
the regulated rotor speed of the load machine, or vice versa.
Selection of the electrical excitation frequency, Ωe , is somewhat arbitrary. In general, running at higher speeds (and thus, higher electrical frequencies) will reduce
the influence of stator resistance variations, as well as inverter non-ideal effects like
dead-time, by increasing voltage levels in the machine. Additionally, the electrical
frequency should be high enough that the effects of integrator approximations are negligible. Similar to electrical frequency, the use of higher stator flux linkage magnitudes
26

||  s ||
+
_
Σ
P.I.

|| vs ||
ωe
2-Phase
Voltage
Gen.
vsd
Σ
vsq
Σ
vsd*
*
sq
v
Da
Space
Db
Vec.
Dc
Mod.
V.S.I.
a
b
c
I.M.
Dead-Time
Comp.
isds
isqs
isd
Park
Trans.
λs
isq
Clark
Trans.
(ab-to-dq)
isa
isb
cos  s
sin s Stator Flux
Linkage
ˆ
||  s || Estimator
Figure 2.3: Data acquisition controller block diagram for the proposed parameter
estimation technique.
will also help to minimize the influence of stator resistance variations and inverter
non-ideal effects (e.g., dead-time). However, it is advisable to consider multiple flux
linkage magnitudes during data collection, to check at what point the machine (iron)
begins to saturate. The nominal (or rated) specifications from the manufacturer are
a good starting point for selecting the electrical frequency and stator flux linkage
magnitude.
2.3.3
Dead-time Compensation
In practical implementations, it is desirable to avoid the use of stator voltage
measurements due to the added cost and complexity involved in processing the pulsewidth modulated (PWM) voltage waveforms. Instead of measured voltages, our algorithm (Fig. 2.3) uses the commanded stator voltages to estimate the stator flux
linkage. However, use of the commanded voltages requires compensation of non-ideal
inverter characteristics such as the dead-time effect [7], which lead to distortions in
the stator current locus, as depicted in Figure 2.4. In other words, the stator current locus is not circular in the presence of the dead-time effect. For this reason,
first-harmonic dead-time compensation is employed to ensure that the actual voltages applied to the machine terminals closely resemble the commanded values used
to estimate the stator flux linkage. In discrete-time and in the stationary reference∗
∗
∗
frame, the compensated voltage command, ~v˜s,k
= [ṽsd,k
ṽsq,k
]> , at time-step “k” is
27
Without Dead-Time Comp.
100
Simulated Data
Fitted Circle
Est. Locus
Simulated Data
Fitted Circle
Est. Locus
50
λs
Isq
(A)
50
λs
Isq
(A)
With Dead-Time Comp.
100
0
−50
0
−50
−100
0
50
100
150
−100
0
λs
Isd
(A)
50
100
150
λs
Isd
(A)
Figure 2.4: Simulation of the proposed data acquisition procedure depicting the distortion in the stator current locus due to dead-time effect(left) and improvement using first-harmonic dead-time compensation (right).
given by:
∗
~v˜s,k
4
= ~v˜s,k + Vbus td fsw
π
~is,k
eJ(1.5Ts Ωe )
||~is,k ||
!
.
(2.30)
where ~v˜s,k = [ṽsd,k ṽsq,k ]> is the ideal (commanded) stator voltage vector, ~is,k =
[isd,k isq,k ]> is the measured stator current vector, td is the dead-time, fsw is the
switching frequency of the power electronics (in Hertz) and Vbus is the DC bus voltage.
The sinusoidal first-harmonic of the square-wave dead-time voltage is used to avoid
introducing the additional harmonic content associated with the sign function.
Finally, we note that the exponential term in (2.30) is used to compensate for the
time delay present in the experimental sampled-data implementation. While a stator
current predictor could be employed to compensate the time delay, it would require
accurate knowledge of the machine parameters (which we are trying to identify).
Instead, we simply advance the normalized stator current vector by 1.5 times the
angular distance traveled by the stator current vector over one sample period. The
factor of 1.5 is used to center the prediction over the next sample period, which was
found to provide improved performance in numerical simulations.
2.4
Stator Flux Linkage Estimation
Accurate estimation of the stator flux linkage is necessary for the proposed parameter identification technique, as well as for field-oriented control techniques in
28
general. In particular, consideration must be given to the sampled-data nature of
modern controller implementations, which include a time delay between when stator
currents are sampled and when the computed duty cycle is executed.

vs ,k 1

vs ,k


v
s ,k 1
v

vs ,k 3
s ,k  2
k 1

is ,k 1

is ,k

is ,k 1
k 1
k

is ,k 2
k2
t
Figure 2.5: Timing relationships for sampled-data implementation with unit delay.
Typically, the stator flux linkage is estimated by integrating the stator flux linkage
dynamics (2.1). In general, the stator flux linkage vector at time tk is given by
~λs,k = ~λs,k−1 +
Z
tk
tk−1
~
~vs (t) − Rs is (t) dt.
(2.31)
We will assume that the voltage applied to the stator terminals is constant over a
given sample period, Ts , which is true in an average-value sense, and that there is a
one sample-period delay before a computed voltage is applied, as depicted in Figure
2.5. In other words, the voltage/duty cycle computed at time index k is applied at
k + 1. Under these assumptions, the discrete-time estimate of the stator flux linkage
at time k is given by
ˆ
~λ
~ˆ
es,k ,
(2.32)
s,k = λs,k−1 + Ts~
with
Rs ~
~es,k = ~v˜s,k−2 −
is,k + ~is,k−1 ,
2
(2.33)
where ~v˜s,k−2 denotes the commanded voltage computed at time index k − 2 (which
is implemented at k − 1), ~is,k and ~is,k−1 denote the measured stator current at time
index k and k − 1, respectively. In the z-domain, (2.32) may be represented by the
following transfer function
zTs
ˆ
~λ
~es,k .
(2.34)
s,k =
z−1
Note that ~es,k is essentially an input to the discrete-time integrator in (2.34). However,
the use of a pure integrator is undesirable in practice, as it can lead to drift and
instabilities. Instead, we use a discrete-time approximation of a stable second-order
continuous-time integrator approximation.
29
To reject DC biases in the measured currents, and achieve a faster phase transition
(to 90◦ ) we employ a second-order integrator approximation [63],
ˆ
~λ
s =
s
2
s + 2ζωn s + ωn2
~vs − Rs~is ,
(2.35)
where “s” is the Laplace variable, ζ > 0 is the damping constant, ωn sets the corner
frequency of the integrator approximation, and the brackets, e.g., {F (s)}, are used to
indicate a dynamic operator with transfer function F (s). To ensure accurate estimates
of the stator flux linkage, ωn should be set as low as possible3 (i.e., ωn ≤ 0.01 Ωe )
while still providing stable flux linkage estimates. While the discussion of continuoustime representations is conceptually convenient, discrete-time implementations must
be derived for experimental implementation on a microcontroller.
-3
x 10
2
Amplitude
1.5
1
0.5
Impulse Invariance
Bilinear
Ideal Integrator
0
0
1
2
Time (sec)
3
4
-3
x 10
Figure 2.6: Comparison of bilinear and impulse invariance discrete-time second order
integrator approximations with ideal continuous-time integrator.
Two common methods for deriving discrete-time approximations of continuoustime transfer functions are the bilinear transform and the impulse invariance method
[60]. While the bilinear transform is generally favored for filter design, it leads to
a small delay in our application, which is avoided by using the impulse invariance
method, as shown in Figure 2.6. Using the impulse invariance method, the following
discrete-time integrator approximation is obtained for ζ = 0.4 and ωn = 5 rad/sec
ˆ
~λ
s,k =
0.0001z 2 − 0.0001z
z 2 − 2z + 0.9999
~es,k ,
(2.36)
where the coefficients are computed using MATLAB’s c2d command, which converts
3
Note that F (s) ≈
1
s
for ω >> ωn .
30
continuous-time system models to a discrete-time equivalent using the method specified (e.g., ‘impulse’ for the impulse invariance method).
2.5
Numerical Analysis
Numerical simulations implemented in MATLAB/Simulink are used to evaluate
the proposed parameter identification methodology’s accuracy in the presence of nonideal effects which are encountered in experimental implementations. Specifically, our
simulations include non-ideal inverter characteristics such as dead-time, switch resistance, and diode voltage drops, as well as the sampled-data nature of experimental
implementations, which include a one-time-step delay. Additionally, zero-mean Gaussian noise, of amplitude (i.e., variance) comparable to what we have observed in our
experimental test-bed, is added to the three-phase stator current measurements.
To capture the sampled-data nature of the experimental system, our algorithm
is implemented in a triggered subsystem in Simulink, while the machine dynamics
are simulated in a continuous-time environment using MATLAB’s ode45 solver. To
reduce simulation times, an “average-value” inverter model is used. That is, we do
not model the switching nature of the inverter, since the switching frequency is high
enough that its impact on performance is negligible. We do, however, model the
dead-time effect and other non-ideal effects (resistive and diode voltage drops of the
IGBT switches) by appropriately modifying the voltage commands produced by the
identification algorithm before they are fed to the induction machine model.
The methodology for data collection and data acquisition controller (Fig. 2.3)
with stator flux linkage estimation, described in the prequel, were used to generate the
numerical data from MATLAB/Simulink at various slip frequencies. The simulated
data was generated at a variety of flux linkage magnitudes ranging from 0.06 Vsec to 0.14 V-sec, closely mimicking the experimental conditions (same bus voltage,
sampling frequency, etc.) and using machine parameter values similar to those of the
test machine. An electrical base frequency of 153.33 Hz was used, which corresponds
to a zero-slip rotor speed of 4600 rpm (i.e., the rated rotor speed of the experimental
test machine). The simulated data was then used to compute the machine parameters
in the same fashion that the experimental data is processed, using the proposed
technique discussed earlier. The resulting parameter errors are plotted in Figure 2.7
as a function of flux linkage magnitude.
Inspection of the simulation results in Fig. 2.7 reveals that the proposed parameter
identification methodology is capable of estimating the magnetic parameters and rotor
31
% Error
% Error
Self-Inductance: Ls , Lr
Mutual Inductance: M
2
2
1.5
1.5
1
1
0.5
0.5
0
0.06 0.08 0.1 0.12 0.14
0
0.06 0.08 0.1 0.12 0.14
Rotor Resistance: Rr
Core Loss Conductance: Gc
0.4
20
0.3
15
0.2
10
0.1
5
0
0.06 0.08 0.1 0.12 0.14
0
0.06 0.08 0.1 0.12 0.14
||$̀s || (V-sec)
||$̀s || (V-sec)
Figure 2.7: Simulated parameter errors in the presence of non-ideal effects.
resistance with high accuracy in the presence of non-ideal effects. Additionally, while
the core loss conductance proves to be a more challenging parameter to estimate, the
proposed technique provides estimates with reasonable accuracy, and which improve
at higher flux linkage magnitudes. This is due to the fact that the dead-time effect, as
well as the transistor voltage drops, result in a fixed magnitude voltage error. Their
impact on the accuracy of the estimated parameters therefore diminishes as voltage
levels increase with higher flux linkage magnitudes, as well as higher speeds.
Nevertheless, this level of accuracy in the estimated core loss conductance is more
than sufficient for predicting the impact of core loss on torque regulation. This is
verified by plotting the ratio of the “true” (i.e., with core loss) electromagnetic torque
to the “commanded” (i.e., without core loss) electromagnetic torque as a function
of slip frequency, as shown in Figure 2.8. Inspection of this plot reveals that the
impact of even a 20% error in the core loss conductance estimate will have a rather
small influence on our ability to capture the impact of core loss on torque regulation
performance over the slip frequency range of practical interest (i.e., from the minimum
current operating point up to the minimum flux linkage operating point).
32
1
= /= $
0.95
0.9
0.85
0% Error in Ĝc
'20% Error in Ĝc
0.8
0
2
4
6
8
10
Slip Frequency (Hz)
Figure 2.8: Steady-state induction machine torque error (ratio) due to unmodeled
core loss as a function of slip. This plot was generated using the following
machine parameters: Ls = Lr = 4.4 mH, M = 4.2 mH, Rr = 23 mΩ,
Gc = 30 mΩ−1 , and Ωe = 153.33 Hz.
2.6
Experimental Results
2.6.1
Experimental Setup
The experimental parameter identification control algorithm is implemented on a
Speedgoat real-time target machine using auto-generated code from MATLAB/Simulink.
The test motor is a 3-phase, 4,600 rpm (nominal), 43 kW-peak induction machine
from Azure Dynamics, driven by an IGBT inverter with a switching frequency of
10 kHz, bus voltage of 300 V, and dead-time of 2 µs. A center-based pulse-width
modulation technique is employed to synchronize sampling and switching, thereby
r
r
a
a
b
b
BUS
Gate
Drive
Signals
Gate
Drive
Signals
User
Inputs
Data
Out
Figure 2.9: Experimental set-up for parameter identification data collection.
33
avoiding the pickup of electromagnetic interference generated during switching transitions. Thus, the control algorithm for parameter identification is executed at 10
kHz as well, and Space-Vector Modulation (SVM) is used to generate the desired
duty-cycles sent to the inverter. An identical induction machine serves as the load
for the test machine by regulating the rotor speed (Fig. 2.9).
2.6.2
Experimental Results and Discussion
Steady-state data is recorded for several stator flux linkage magnitudes, ranging
from 0.08 V-sec to 0.14 V-sec, at an electrical base frequency of 153.33 Hz. For each
flux linkage magnitude, the direct and quadrature currents (in the stator flux linkage
reference-frame) are recorded for several different slip frequencies, including zeroslip, roughly up to the current limitations of the machines. The resulting machine
parameters, estimated using the proposed technique, are plotted in Figure 2.10 as a
~ s ||.
function of stator flux linkage magnitude, ||Λ
Inspection of the parameter estimates in Fig. 2.10 reveals that the estimated self
and mutual inductance capture the saturation effects in the machine. Additionally,
we have plotted the leakage inductances, Lls and Llr , to confirm our expectation that
these parameters remain constant. As for the estimated rotor resistance, while there
are variations in the estimates, these variations are well within the range expected
Leakage Inductance
Self and Mutual Inductances
L̂s , L̂r , M̂ (mH)
250
200
150
100
0.08
R̂r (m+)
30
0.1
0.12
4.2
4
3.8
3.6
0.08
0.14
Rotor Resistance
60
25
20
15
0.08
4.4
Ĝc (m+!1 )
L̂ls , L̂lr (7H)
300
0.1
0.12
||$̀s || (V-sec)
0.1
0.12
0.14
Core Loss Conductance
40
20
0
0.08
0.14
L̂s , L̂r
M̂
0.1
0.12
0.14
||$̀s || (V-sec)
Figure 2.10: Experimental estimated machine parameters as a function of stator flux
linkage magnitude.
34
due to temperature changes. In fact, the lower resistance estimates at 0.1 and 0.13
V-sec correlate with data collected after allowing the machine time to cool. Furthermore, the expected increase in resistance for a 45◦ C rise in temperature in aluminum
conductors [64] is ∼ 4.07 mΩ, and supports our suspicion that the variations in rotor
resistance observed in Fig. 2.10 are due to temperature changes in the rotor during
data collection. Inspection of Fig. 2.10 also reveals that the core loss conductance
estimates are reasonably consistent across all of the stator flux linkage magnitudes.
However, since core losses are typically thought of in terms of the power they
consume, rather than the equivalent resistance, it is desirable to check the predicted
core loss power using the core loss conductance estimates. The estimated power
dissipated by the core loss conductance is given by
3
~ s ||2 .
P̂core = Ĝc Ω2e ||Λ
2
(2.37)
Estimated Core Loss (W)
Thus, using our experimental data, we obtain the plot provided in Figure 2.11. Inspection of Fig. 2.11 reveals the general trend that core losses increase with stator
flux linkage magnitude, as expected.
1000
800
600
400
200
0
0.08
0.1
0.12
0.14
||$̀s || (V-sec)
Figure 2.11: Estimated core loss power as a function of stator flux linkage magnitude
at a electrical base frequency of 153.33 Hz and switching frequency of
10 kHz.
Finally, the experimental stator current locus plots for several stator flux linkage
magnitudes are provided in Figure 2.12. The estimated locus points are computed
using the estimated machine parameters along with equations (2.13) and (2.14). Inspection of Fig. 2.12 reveals that there is a good consensus between the experimental
and estimated locus points, particularly at high flux linkage magnitudes where the influence of the non-ideal inverter effects and stator flux estimation are less pronounced.
35
6s
Isq
(A)
||$̀s || =0.08V-sec
||$̀s || =0.1V-sec
100
100
50
50
0
0
-50
-50
-100
0
100
200
300
-100
0
6s
Isq
(A)
||$̀s || =0.11V-sec
||$̀s || =0.12V-sec
100
100
50
50
0
0
-50
-50
-100
0
100
6s
Isd
200
Experimental Data
Fitted Circle
Estimated Locus
100
200
300
300
-100
0
100
6s
Isd
(A)
200
300
(A)
Figure 2.12: Experimental data (blue circles) with fitted current locus circle (green
dashed line) and estimated locus points (red X’s) for various stator flux
linkage magnitudes.
2.7
Conclusion
This chapter presented a new technique for offline identification of induction machine parameters, including core loss conductance, using steady-state measurements.
The technique is based on fitting steady-state experimental data to the circular stator
current locus in the stator flux linkage reference-frame for various steady-state slip
frequencies, providing reliable estimates of the magnetic parameters as well as the
rotor resistance and core loss conductance. This approach allows accurate estimation
of leakage inductance and rotor resistance while avoiding the practical challenges of
implementing a locked-rotor test with a voltage-source inverter. Numerical results
verifying the accuracy of estimated parameters in the presence of non-ideal effects
were presented, in addition to experimental results for a 43 kW induction machine,
which demonstrate the proposed technique’s ability to accurately characterize a VSIdriven induction machine over a wide range of operating conditions, including magnetic saturation.
36
CHAPTER III
Adaptive Excitation Decoupling Approach to
Simultaneous Identification and Control of
Permanent Magnet Synchronous Machines
3.1
Introduction
In this chapter, we begin our technical discussion of simultaneous identification
and control for PMSMs and present the first of three approaches to SIC of PMSMs.
As noted in the introduction, parameter identification and output regulation are
typically conflicting objectives. Generally, a trade-off must be made between ensuring
that inputs to the system under control are persistently exciting and maintaining
tight regulation of “performance” (i.e., regulated) outputs. However, in the case
of certain overactuated systems there is an opportunity to achieve these objectives
simultaneously. For example, field-oriented output torque regulation in Permanent
Magnet Synchronous Machines (PMSMs) constitutes an overactuated control problem
in that there are two distinct inputs to the system, the direct-axis voltage input and
the quadrature-axis voltage input, and one regulated output, torque. The direct-axis
voltage is typically used to set magnetic field (flux) levels in the machine by regulating
the direct-axis stator current, while the quadrature-axis voltage is used to regulate
the electromagnetic torque by regulating the quadrature-axis stator current.
The PMSM is a popular choice in high-performance drives, particularly in transportation applications, thanks to its high torque density and high efficiency. However, temperature changes, skin effect, and magnetic saturation lead to changes in
This chapter is based on previously published work:
D. M. Reed, J. Sun, and H. F. Hofmann, “Simultaneous Identification and Torque Regulation
of Permanent Magnet Synchronous Machines via Adaptive Excitation Decoupling,” in American
Control Conference (ACC), 2015, pp.3224-3229, 1-3 July 2015.
37
the machine parameters which in turn detune the drive system, causing performance
degradation. The stator winding resistance is primarily impacted by temperature
variations, which can lead to increases in resistance by as much as 100% [39]. While
the permanent magnet flux magnitude also varies with temperature, the variation
can be small for certain types of magnets. For example, in neodymium (N dF eB)
magnets the variation is around -0.1% per ◦ C [7], which results in a mere 5% error for
a rather large 50◦ C increase in temperature. Finally, while the electrical frequencies
needed to see a significant rise in stator resistance due to skin effect are not typically
encountered, high-speed applications using motors with a high pole-pair count may
see an impact due to skin effect.
Many different approaches to compensating parameter variations in PMSMs have
been proposed by researchers. Steady-state machine models have been used to avoid
the additional complexity that comes with using dynamic models for parameter estimation [34], [44]. Open-loop [13], as well as closed-loop [77], [27] approaches have
been presented which utilize the method of least-squares for PMSM parameter estimation. The gradient method is used in [56] to provide online estimates of the lumped
time-varying disturbances caused by parameter variations. While artificial neural
networks have been proposed for online adaptation [47], Lyapunov-based adaptive
designs provide an attractive alternative, as a stability proof is a byproduct of the design process [48]. However, none of these papers proposes a design which specifically
considers Simultaneous Identification and Control (SIC) in their design.
This chapter presents a simultaneous identification and control methodology for
PMSMs by exploiting the overactuated nature of the machine. Specifically, the parameters to be identified are the direct and quadrature-axis self-inductances, Ld and
Lq , as well as the stator winding resistance, R. And the output to be controlled
is the electromagnetic torque, τ . An indirect adaptive control design using the certainty equivalence principle is proposed in which a disturbance decoupling control
law is utilized to prevent the input selected for excitation from perturbing the regulated output. The machine parameters used in this excitation decoupling control
law are updated via a normalized gradient estimator. Simulation results for a torque
regulating controller for PMSMs confirm the effectiveness of the proposed simultaneous identification and control design methodology. Furthermore, while the focus
of the chapter is on the application of the proposed adaptive excitation decoupling
control methodology to PMSM torque regulation, the prospects of generalizing this
methodology for overactuated systems are promising.
38
Table 3.1: List of notation for PMSMs.
Symbol
Description
Electrical Variables
vdr (t)
Direct-axis Voltage in Rotor Ref. Frame
vqr (t)
Quadrature-axis Voltage in Rotor Ref. Frame
ird (t)
Direct-axis Current in Rotor Ref. Frame
irq (t)
Quadrature-axis Current in Rotor Ref. Frame
R
Stator Winding Resistance
Ld
Direct-axis Stator Self-Inductance
Lq
Quadrature-axis Stator Self-Inductance
ΛP M
Permanent Magnet Flux Linkage
Mechanical Variables
τ
Three-Phase Electromagnetic Torque
ωr
ωre =
Rotor Angular Velocity
P
ω
2 r
Rotor Electrical Angular Velocity
P
3.2
Number of Poles
Dynamic Model of PMSMs
The proposed control algorithm is designed around the standard two-phase equivalent model for permanent-magnet synchronous machines [39], the physical crosssection of which is depicted in Figure 3.1. This model, and the subsequent control
design, are derived under the following assumptions:
A1. The machine to be controlled has a smooth airgap (i.e., slotting effects are
neglected), is fed by an ideal voltage source inverter (VSI), and is balanced
in its construction such that it can be accurately represented by its 2-phase
equivalent;
A2. A linear magnetics model is assumed (i.e., magnetic saturation effects are neglected) and core losses are neglected;
A3. The rotor and mechanical load have enough inertia that a significant time-scale
separation exists between electrical and mechanical dynamics;
39
A4. The sampling frequency of the digital implementation is high enough that a
continuous-time control design can be sufficiently approximated;
A5. The only uncertain parameters are resistance, R, permanent magnet flux linkage,
ΛP M , and the direct and quadrature inductance, Ld and Lq , respectively.
These assumptions are typical and valid under normal operation.
The first three assumptions (A1 - A3) simplify the model by reducing its order and
maintaining linearity. In particular, given that the mechanical dynamics associated
with the rotor velocity are significantly slower than the electrical dynamics and that
measurements of rotor velocity are readily available, ωre may be treated as a known
constant (A3). This simplification allows us to represent the machine as having
linear time-invariant dynamics. The last two assumptions (A4 and A5) pertain to
the control design and methodology.
q
qr
id
N turns
SN
q
θr
q
SN
id
dr
d
Rotor
μ→∞
d
d Stator
μ→∞
Figure 3.1: Cross-section of the two-phase equivalent, two-pole smooth airgap
interior-permanent-magnet PMSM machine.
The dynamic model of a PMSM in the rotor reference frame (denoted by the
superscript r), in which the direct-axis is aligned with the rotor permanent magnet
flux, is given by,
dird
= −Rird + ωre Lq irq + vdr ,
dt
dirq
Lq
= −ωre Ld ird − Rirq + vqr − ωre ΛP M ,
dt
Ld
40
(3.1)
(3.2)
with the unmeasured nonlinear torque output mapping,
τ=
3P
[(Ld − Lq ) ird + ΛP M ] irq .
4
(3.3)
Finally, the mechanical dynamics are given by
1
d
ωr =
[(τ − τl ) − Bωr ] ,
dt
H
(3.4)
where H is the combined moment of inertia of the rotor and load, B is the mechanical
damping, and τl is the load torque. However, because the separation between the
B
,
(dominant) electrical and mechanical time constants (Telec = − LRq and Tmech = − H
respectively) is roughly a factor of 150 for the experimental machine considered in
this work, we will neglect the mechanical dynamics in our design and analysis. We
therefore treat the rotor speed as a known constant with respect to the electrical
dynamics (3.1)-(3.2), as there exists a significant time-scale separation between the
electrical and mechanical dynamics to justify A3.
3.3
Adaptive Disturbance Decoupling Approach
The SIC approach presented in this paper is based upon a certainty equivalence
design in which an excitation input decoupling control law is used to prevent the
excitation input signal from perturbing the regulated output. This excitation decoupling control law is derived by reformulating the problem as a disturbance decoupling
problem [29].
3.3.1
Statement of the Control Objective
The control inputs to the PMSM are the direct and quadrature-axis voltages, urd
and urq , and the (unmeasured) regulated output is electromagnetic torque, τ . Thus,
the PMSM constitutes an overactuated system. Our control objective is to simultaneously achieve parameter identification and asymptotic output regulation in an
overactuated system, namely the PMSM. This is accomplished by using an adaptive
excitation (disturbance) decoupling control design in which one input of the overactuated system is designated as the excitation input used to ensure that the PMSM
dynamics are persistently excited for parameter convergence, and the other input is
used for torque output regulation. The excitation decoupling control law ensures that
the perturbations in the regulated output go to zero asymptotically as the machine
41
parameters converge to their true values. Once identified, the presence of the excitation signal ensures that the parameter estimator will track any changes in the
parameters.
3.3.2
Review of Disturbance Decoupling
To apply this solution to the simultaneous identification and control problem, we
treat the excitation input as a measured disturbance, and derive a state-feedback
controller which decouples the excitation input from the regulated output, provided
that the system parameters are well known. For convenience, we will first review the
general solution for a class of nonlinear systems [29] before applying the result to the
PMSM torque regulation problem.
Consider a nonlinear system of the form
(
Σ:
ẋ = f (x) + g(x)u + p(x)w,
y = h(x),
(3.5)
where x ∈ Rn is the state vector, y ∈ R is the regulated (or “performance”) output,
u ∈ R is the input and w ∈ R is the disturbance input which is to be decoupled.
Given measurements of the full state vector, x, as well as the disturbance, w, it is
possible to decouple the disturbance from the output, y, using a state-feedback law of
the form u = α(x) + β(x)ψ + γ(x)w, where ψ is a control input which will be designed
to yield stable linear closed-loop dynamics provided that,
Lp Lif h(x) = 0 for all 0 ≤ i ≤ ρ − 2,
ρ−1
Lp Lρ−1
f h(x) = −Lg Lf h(x)γ(x),
(3.6)
(3.7)
for all x in the neighborhood of the equilibrium, xo , where Lif h(x) denotes the ith Lie
derivative of h(x) projected along f (x) and ρ is the relative degree of the system, Σ.
Note that the second condition is easily satisfied by solving for γ(x) and including
the term in the feedback law. If these conditions are satisfied for a given plant, a
feedback law which achieves disturbance decoupling, is given by:
u=−
Lρf h(x)
Lg Lρ−1
f h(x)
−
Lp Lρ−1
f h(x)
w
Lg Lρ−1
f h(x)
+
ψ
.
Lg Lρ−1
f h(x)
(3.8)
The design of ψ is best understood by considering the system in the “normal
42
form” [29], which is obtained by defining new coordinates such that,
ż1 = z2
..
.
żρ−1 = zρ
żρ = b(ξ, η) + a(ξ, η)u + d(ξ, η)w
η̇ = q(ξ, η) + k(ξ, η)w
y = z1
where ξ = [z1 · · · zρ ]> and η = [zρ+1 · · · zn ]> . Note that the term η̇ = q(0, η) represents
the zero dynamics of the system, which must be stable (i.e., minimum phase) to ensure
that the closed-loop design is internally stable. In the new coordinate system (i.e.,
normal form) the state-feedback law (3.8) takes the form,
u=−
ψ
b(ξ, η) d(ξ, η)
−
w+
,
a(ξ, η) a(ξ, η)
a(ξ, η)
and the closed-loop system takes on the structure depicted in Figure 3.2, effectively
isolating the output, y, from the disturbance input, w.



Figure 3.2: Block diagram of closed-loop system after disturbance decoupling
(adapted from [29]).
With Figure 3.2 in mind, our choice of the control, v̄, is rather intuitive:
ψ = − (c0 z1 + · · · + cρ−1 zρ ) + ỹ
= − c0 h(x) + · · · + cρ−1 Lρ−1
f h(x) + ỹ,
(3.9)
where ỹ is the reference input (i.e., desired output value) and the coefficients c0 , · · · , cρ−1
are selected to yield the desired linear dynamics with characteristic equation,
sρ + cρ−1 sρ−1 + · · · + c1 s + c0 = 0.
43
(3.10)
It should be noted that, in general, the solution to the disturbance decoupling problem
presented here, and its asymptotic stability properties, are local results (i.e., they hold
only for a neighborhood of the equilibrium point, xo ).
3.3.3
Excitation Decoupling for PMSMs
The dynamic model of the PMSM in the rotor reference frame, (3.1) and (3.2),
may be rewritten in the following form,
 
  
   
Lq
R
1
r
r
−
ω
0
i
i
re
d  d 
d
Ld
Ld
Ld
r


 urd ,




+
uq +
=
L
1
R
dt ir
−ωre Ldq − Lq
irq
0
q
Lq
| {z }
{z
} | {z }
|
,f (x)
,g(x)
(3.11)
,p(x)
with the nonlinear torque output mapping (3.3), repeated for convenience,
τ=
3P
[(Ld − Lq ) ird + ΛP M ] irq ,
4
|
{z
}
(3.12)
,h(x)
where the direct and quadrature currents are the states (i.e., x = [ird irq ]> ) of the
system and the direct and quadrature voltage inputs to the system are assumed to
include a EMF cancellation term, i.e., vqr = urq + ΛP M ωre , and vdr = urd for consistency.
To apply the disturbance decoupling solution to our problem, we first need to
decide which plant inputs will be used for control and which will be used for excitation
(i.e., the “disturbance” to be decoupled). While this can be challenging for general
plant models, many applications lend intuition which may be leveraged to make this
decision. As it concerns our application, we note that the coupling between the direct
and quadrature-axis dynamics is only strong at high speeds (3.11), going to zero at
zero speed (i.e., the system matrix is diagonal at zero speed). Additionally, inspection
of the torque output mapping (3.12) reveals that, because (Ld − Lq ) tends to be much
smaller than ΛP M , the quadrature-axis current, irq , and thus, the quadrature-axis
voltage input, urq , has the most authority over the torque output. Therefore, we take
the quadrature-axis voltage, urq , as the control input and the direct-axis voltage, urd , as
the disturbance input (for excitation) to be decoupled. Thus, the PMSM is treated as
a SISO system with control input, urd , regulated output τ , and measured disturbance
input, urd . While we have selected our control input based on its authority over the
regulated output, for other applications, this decision could also be dictated by which
input provides more excitation for identification.
44
The relative degree of the PMSM system (3.11)-(3.12) from either input (urd or
urq ) to the output, τ , is one (i.e., ρ = 1 for the PMSM), which is easily verified by
differentiating the output (3.12) with respect to time. Therefore, we need only include
the following feed-forward term
γ(x) =
(Ld − Lq )irq
Lq
Ld (Ld − Lq )ird + ΛP M
in the excitation decoupling control law to ensure that the conditions (3.6)-(3.7) are
satisfied. Furthermore, since the PMSM dynamics (3.11)-(3.12) are minimum phase,
we may apply the disturbance decoupling results from the previous section.
While our focus is on the design of the control input, urq , in practice it is beneficial
to cancel the cross coupling term in the direct-axis dynamics as it will lead to resonant
behavior, as well as a large steady-state direct-axis current, at high rotor velocities.
Therefore, we include the following feedback term for the direct-axis,
urd = −ωre L̂q irq + R̂ue ,
(3.13)
where ue is the excitation input which is scaled by the estimated resistance so that
it corresponds to the magnitude of the steady-state direct-axis current generated by
the direct-axis command voltage. Computing the necessary Lie derivatives specified
in (3.8), we obtain the excitation decoupling control law for PMSMs:
urq
=
R̂irq
+
ωre L̂d ird
−
R̂
ˆ L irq
L̂q ∆
ˆ L ir + Λ P M
L̂d ∆
d
(ure − ird ) +
4L̂q
ψ,
r
ˆ
3P ∆L id + ΛP M
(3.14)
where the “hat” ( ˆ· ) designates parameters which will be adaptively estimated, and
ˆ L = L̂d − L̂q (for compactness).
∆
Finally, we design the control input, v, in (3.14) as follows:
ψ = −λh(x) + λτ̃ ,
(3.15)
which yields the following first-order (input-output) closed-loop dynamics,
(
Σcl :
ż = −λz + λτ̃
τ = z = h(x)
(3.16)
where λ > 0 is a control gain which sets the closed-loop bandwidth, and h(x) is
45

uer
vdr
Excitation Decoupling
Control Law

 ˆ
 ˆ
 ˆ
u r   ( x ,  )   ( x ,  )   ( x ,  )uer
Inverse
Park
Transform
(r-to-s)
v qr
vd
vq
a
PWM
&
VSI
PMSM
Machine
b
c
ˆ

Adaptive Estimator

d ˆ
  ΓΦez
dt
idr
iqr
Clarke
&
Park
Transform
(s-to-r)
θre
ωre
ωr
P
2
ia
ib
P
2
θr
Figure 3.3: Block diagram of the closed-loop system with proposed adaptive excitation decoupling controller.
defined by (3.12). Thus, the disturbance and its associated dynamics are decoupled
from the output, z. Finally, we compute the value for λ based on our desired rise-time
of 2 milliseconds and the following relationship for first-order systems [18],
λ=
1.8
,
tr
where tr is our desired rise time (i.e., tr = 2 msec).
Remark : A comprehensive stability analysis of the adaptive excitation decoupling
control will not be pursued in this dissertation. However, once parameter convergence
is obtained, which is guaranteed by the algorithm proposed in the next section, stability of the closed-loop system will follow from the non-adaptive disturbance decoupling
control. Therefore, for the adaptive control problem, stability can be assured if there
is no finite escape time.
Estimates of the machine parameters used in the excitation decoupling control law
(3.14) are provided by a normalized gradient-based algorithm. The resistance, as well
as the direct and quadrature inductances, are directly estimated by the algorithm. A
block diagram of the overall adaptive control system is provided in Figure 3.3.
46
3.3.4
Gradient-based Parameter Estimation
To formulate the parameter estimator, we first construct a linear parameterization
for the system model (3.11),
~
(3.17)
~z = Φ> θ,
where ~z = [urd urq ]> is the observation vector, θ~ = [R Ld Lq ]> is the parameter
vector, and the regressor matrix is given by


d r
r
r
id dt id −ωre iq
.
Φ> =   = 
d r
>
r
r
~
φq
iq ωre id
i
dt q

~>
φ
d

(3.18)
In order to avoid direct computation of derivatives in the regressor matrix, we filter
each side of (3.17) by a stable first-order filter [28], i.e.,
~
{M (s)} ~z = {M (s)} Φ> θ,
where
{M (s)} =
Kf
s + Kf
is the transfer function representation of a stable first-order filter (i.e., Kf > 0) which
operates on the individual elements of ~z and Φ.
The parameter estimates are obtained by integrating the following expression,
ˆ˙
θ~ = ΓΦ~ez
" #
h
i e
zd
~d φ
~q
=Γ φ
ezq
~
~
= Γ φd ezd + φq ezq
(3.19)
where Γ = Γ> > 0 is the adaptation gain matrix and ~ez = [ezd ezq ]> is the normalized
estimator error vector, whose entries are given by,
~ > θ~ˆ
urd − φ
d
,
>~
~
1 + φ φd
(3.20)
~ > θ~ˆ
urq − φ
q
=
.
~ >φ
~q
1+φ
(3.21)
ezd =
d
ezq
q
Finally, we note that the rows of the regressor matrix, Φ, are scaled such that
47
the nominal signal amplitudes are around unity to ensure that the identification
problem is well-conditioned. This is important as the large differences in the orderof-magnitude between resistances and inductances will lead to a numerically poorlyconditioned identification problem if left unscaled.
3.4
Selection of Persistently Exciting Inputs
A disadvantage of the design approach presented in this chapter is that it significantly complicates the analysis of the resulting closed-loop adaptive controller.
However, by making some reasonable simplifying assumptions, we are able to gain
some intuition to guide the design of persistently exciting inputs.
In the following analysis, we will assume that we have accurate knowledge of the
machine parameters, as this represents something of a worst-case scenario since any
error in the estimated parameters is expected to provide additional “information” for
identification. This is because the excitation signal is only perfectly decoupled from
the output when the plant parameters are accurately known. Thus, when there is
parameter uncertainty, the excitation signal will excite additional dynamics in the
plant as compared to when we have accurate parameter knowledge.
To analyze the influence of the reference torque, τ̃ , and the excitation input, ure ,
on the conditioning of the regressor matrix (3.18), we need to relate these inputs to
the direct and quadrature-axis currents, ird and irq . Under the assumption of accurate
parameter knowledge, the closed-loop direct-axis current dynamics are given by
d r
R
R
id = − ird + ure .
dt
Ld
Ld
(3.22)
Additionally, recalling the closed-loop torque regulation dynamics (3.16) and noting
that z = τ in (3.16), we get that
d
τ = −λτ + λτ̃ .
dt
(3.23)
To simplify our analysis of the regressor, we will assume that that Ld ≈ Lq . This is
a conservative assumption, however, as a significant magnetic saliency, i.e., Lq >> Ld ,
will provide additional excitation via the coupling between the direct and quadrature
currents through the torque expression (4.6). Note that for the torque to remain
constant in the presence of excitation introduced via the direct-axis dynamics, the
quadrature-axis current must vary in an inverse manner. Under this assumption, the
48
torque expression is simplified as follows:
3P
[(Ld − Lq ) ird + ΛP M ] irq ,
4
3P
≈
ΛP M irq ,
4
= KΛ irq .
τ=
(3.24)
Next, we relate the reference torque, τ̃ , to the quadrature-axis current, irq , by substituting the relationship in (3.24) into (3.23), to get that
λ
d r
iq = −λirq +
τ̃ .
dt
KΛ
(3.25)
To determine sufficient conditions for persistent excitation on the reference torque,
τ̃ , and the excitation input, ure , we will use the definition of persistent excitation.
Noting that (from (3.18)),
1
T
Z
t
t+T
1
Φ(σ)Φ (σ)dσ =
T
>
Z
t
t+T
~ d (σ)φ
~ > (σ)dσ + 1
φ
d
T
Z
t+T
~ q (σ)φ
~ > (σ)dσ, (3.26)
φ
q
t
~d
it follows that a sufficient condition for Φ to be persistently exciting is that either φ
~ q (or both) be persistently exciting. Assuming the excitation input is sinusoidal,
or φ
i.e., ure = sin(ωt), it follows that the (sinusoidal) steady-state solution to (3.22) is of
the form
ird = a sin(ωt + b).
Additionally, we will assume that the torque reference input is constant. It follows
that the steady-state solution to (3.25) is given by
irq =
τ̃
.
KΛ
Under these assumptions, the direct-axis regressor vector is given by

 

a sin(ωt + b)
ird

 
~ d (t) = 
φ
 dtd ird  = aω cos(ωt + b) .
−ωre Kτ̃Λ
−ωre irq
(3.27)
Finally, using the definition of persistent excitation, and taking the time interval to
49
be [0, 2π
], it can be shown that
ω
det
ω
2π
2π
ω
Z
!
~ d (σ)φ
~ > (σ)dσ
φ
d
0
=
2
τ̃ 2 a4 ω 2 ωre
.
4KΛ2
(3.28)
We conclude that, if τ̃ , a, ω, and ωre 6= 0, then Φ is persistently exciting.
In the next section, we will investigate parameter convergence properties when
the reference torque is equal to zero, i.e., τ̃ = 0, and when the excitation input is
zero, i.e., a = 0, using numerical simulations. Intuitively, we expect these conditions
to cause problems with parameter convergence since τ̃ = 0 and a = 0 (i.e., ird = 0)
lead to a row of zeros in the regressor (3.18). Additionally, it will be verified that, by
leveraging the excitation input, we can ensure complete parameter convergence.
3.5
Simulation Results
We have validated the proposed adaptive excitation decoupling control methodology for SIC in Matlab/Simulink simulations using a dynamic model of the PMSM
and the ode4 Fixed-step Runge-Kutta solver with a step size of 10 µ-sec. Parameters
provided in Table 3.2 were used in all simulations except where noted otherwise.
Table 3.2: Simulation parameters.
Description
Value
Electrical Machine Parameters:
R
102.8 mΩ
Ld
212.3 µH
Lq
424.6 µH
ΛP M
12.644 mV-s
P
10
Control Design Parameters:
λ
900
Γ
diag([16 80 40])
Kf
1000
ωpe
363 rad/sec
50
3.5.1
On Conditions for Parameter Convergence
Currents (A) Torque (N-m)
Currents (A) Torque (N-m)
Due to the overactuated nature of the PMSM, we are able to ensure that the
machine dynamics are persistently excited while minimizing perturbations to the regulated (i.e., torque) output, by utilizing the excitation input. Our only requirement
on the torque reference input to ensure full parameter convergence, is that the torque
command be nonzero, as the analysis in the preceding section suggested. To investigate parameter convergence properties in the closed-loop system, Simulink simulations are run which examine scenarios when the excitation input is set to zero, and
in which the torque (control) command is set to zero.
=
=˜
0.4
0.2
0
0
1
2
3
4
10
0
-10
0
ird
1
2
irq
3
4
0.2
0
0
1
2
3
3
4
ird
irq
-10
0
1
2
3
4
2
L̂q /Lq
1
0
0
2
0
3̂/3 (-)
3̂/3 (-)
L̂d /Ld
1
10
2
R̂/R
=
=˜
0.4
R̂/R
L̂q /Lq
1
0
0
4
time (s)
L̂d /Ld
1
2
3
4
time (s)
(a)
(b)
Figure 3.4: Simulations of the closed-loop adaptive system (adaptation turned “on”
at t = 1 sec): (a) without persistently exciting input (i.e., ure = 0),
leading to partial convergence; and, (b) with zero torque command input
(i.e., τ̃ = 0), again leading to partial convergence.
In Figure 3.4(a) we see that, without the additional information provided by the
excitation input, i.e. ue = 0, the estimate of the direct-axis inductance, L̂d , settles
to an incorrect value. This scenario serves as our baseline adaptive control design
in which the overactuation in the system is not exploited. This essentially gives us
the feedback linearization portion of the controller and a fair basis with which to
compare. Despite the rich harmonic content of the square wave torque command, the
estimated parameters do not fully converge to their true values. However, in Figure
3.4(b) we see that, when the torque command is set to zero, the estimated machine
parameters again fail to converge fully to their true values, despite the presence of
51
a persistently exciting excitation input, ue = 1.5 (sin(ωpe t) + sin(0.5 · ωpe t)). The
zero torque command results in a lack of sufficient richness for the quadrature-axis
inductance estimate, L̂q , to converge to its true value.
3.5.2
Closed-loop Performance
Currents (A) Zoomed Torque (N-m)
The main objective in this work is to demonstrate an adaptive control methodology for overactuated systems which is capable of achieving simultaneous identification
of parameters and control of a regulated output. This is achieved by exploiting the
overactuated nature of a PMSM by designating one input as an excitation input,
which is designed to ensure that the system is persistently excited for parameter
identification, and the other input as the control input used for torque regulation.
Decreasing output ripple
=
=˜
0.4
0.2
0
0
1
2
3
0.4
0.4
0.2
0.2
0
0.25
10
0
3.25
0.26
3.255
3.26
0
-10
0
2
3̂/3 (-)
0.255
4
ird
1
2
irq
3
R̂/R
L̂d /Ld
4
L̂q /Lq
1
Adaptation ON
0
0
1
2
3
4
time (s)
Figure 3.5: Simulation of closed-loop adaptive system at a fixed rotor velocity of 2000
rpm with excitation input (adaptation turned “on” at t = 1 sec).
In Figure 3.4(a) we see that, without leveraging the extra degree of freedom resulting from overactuation (i.e., utilizing the excitation input) the estimated parameters
converge to a set where, despite the fact that the control error goes to zero, the estimate of Ld stagnates at an incorrect value. This is the typical scenario for adaptive
control in which the adaptation drives the control error to zero, but due to a lack of
52
=
=˜
0.4
0.2
0
0
10
1
2
3
4
0
-10
0
2
3̂/3 (-)
Currents (A) Torque (N-m)
persistent excitation, the parameters fail to completely converge to their true values.
However, when we use the overactuated nature of the PMSM to our advantage, we
are able to ensure that the system is persistently excited and so all of the parameters
converge to their true values. This is accomplished by using a disturbance decoupling control law to fix the control allocation such that, given accurate parameter
knowledge, the excitation input is decoupled from the regulated output, allowing us
to introduce excitation for parameter identification while minimizing the impact on
output regulation. Machine parameters used in the control law are then updated by
the online identification, ensuring that any perturbations to the torque output vanish
as the estimated parameters converge to their true values. Inspection of the results in
Figure 3.5 confirm that the closed-loop system performs very well, with torque perturbations due to the excitation input vanishing as the estimated parameters converge
to their true values.
Finally, to demonstrate that the closed-loop adaptive excitation decoupling controller does exhibit robustness to uncertainty as well, simulations are run which include zero-mean Gaussian noise on the stator current measurements. Inspection of
these results, presented in Figure 3.6, indicate that, despite the presence of measurement noise, the parameters converge to their true values. It should be noted that, in
practice, limiting the bandwidth of the adaptive estimator will improve performance
ird
1
2
R̂/R
irq
3
L̂d /Ld
4
L̂q /Lq
1
0
0
1
2
3
4
time (s)
Figure 3.6: Simulation of closed-loop adaptive system at a fixed rotor velocity of 2000
rpm with zero-mean Gaussian noise added to the current measurements.
53
in presence of measurement noise. Additionally, while the simulations presented here
do not include any robustness modification to the adaptive update law (3.19), in an
experimental implementation, the addition of a robustness modification such as a
switching-sigma or projection is advised [28].
3.6
Conclusion
This chapter presented recent research [67] into the application of disturbance decoupling to the development of a simultaneous identification and control methodology
for overactuated systems via a case study with PMSMs. An indirect adaptive control
design using the certainty equivalence principle was proposed in which a disturbance
decoupling control law, termed excitation decoupling, is utilized to prevent the excitation input from perturbing the regulated output. The plant parameters used in
the excitation decoupling control law are updated online via a normalized gradient
estimator. Simulation results for simultaneous identification and torque regulation in
PMSMs confirm the effectiveness of the proposed design methodology. While open
issues remain, such as a comprehensive assessment of closed-loop stability, the “adaptive excitation decoupling” approach shows promise for generalization.
54
CHAPTER IV
Simultaneous Identification and Control of
Permanent Magnet Synchronous Machines via
Adaptive 2-DOF Lyapunov Design
4.1
Introduction
In the previous chapter, an investigation into the application of disturbance decoupling control theory to the simultaneous identification and control of overactuated
systems, dubbed “adaptive excitation decoupling”, was presented. The advantage of
this approach is that it is based upon well established theory and provides a relatively
straightforward design procedure. However, even if the resulting closed-loop adaptive
system is globally asymptotically stable, it will be very challenging to demonstrate
this analytically. Likely, the best that can be argued is that if the closed-loop system
does not posses a finite-escape time, and the system is persistently excited such that
the estimated parameters are converging to their true values, then local stability is
assured by the disturbance decoupling control law.
This chapter presents a Lyapunov-based simultaneous identification and control
design methodology for overactuated systems which is demonstrated on PMSMs and
whose stability properties can be rigorously established. By constraining the states to
a manifold which corresponds to zero regulated-output error, we are able to achieve
excitation and output regulation simultaneously. The controller to be presented for
PMSMs is derived using Lyapunov’s stability theorem, and so the stability of the
closed-loop system is demonstrated in the process of deriving the adaptation law. A
This chapter based on work submitted to a journal and is under review:
D. M. Reed, J. Sun, and H. F. Hofmann, “Simultaneous Identification and Robust Adaptive Torque
Control of Permanent Magnet Synchronous Machines,” Under review, 2015.
55
switching-σ robust modification to the derived adaptive law is used to ensure closedloop stability in the presence of unmodeled disturbances. The control law, designed
using the certainty equivalence principle [28], utilizes a combination of adaptivelytuned feedforward (to achieve zero steady-state error), d − q decoupling (to improve
transient response), and proportional feedback (to add robustness to disturbances)
terms. Overactuation of the system is exploited to simultaneously achieve parameter
convergence and torque regulation. After reviewing the dynamic PMSM machine
model, the derivation and stability proof for the proposed adaptive controller is presented. Necessary conditions for parameter convergence are discussed, and simulation
results verifying the performance of the control design are presented. Remarks specific to experimental implementation challenges, as well as experimental results, are
presented and discussed.
4.2
Two-Phase Equivalent Dynamic Model for PMSMs
For convenience, and continuity with the material to be discussed in this chapter,
presentation of the two-phase equivalent PMSM model is repeated here. This model,
and the subsequent control design, are derived under the following assumptions:
A1. The machine to be controlled has a smooth airgap (i.e., slotting effects are
neglected), is fed by an ideal voltage source inverter (VSI), and is balanced
in its construction such that it can be accurately represented by its 2-phase
equivalent model;
A2. Linear magnetics is assumed (i.e., magnetic saturation effects are neglected),
and core losses are neglected;
A3. The rotor (electrical) velocity, ωre , is a bounded time-varying input which is
known (i.e., measured);
A4. The sampling frequency of the digital implementation is high enough that a
continuous-time control design can be sufficiently approximated;
A5. The only uncertain parameters are resistance, R, permanent magnet flux linkage,
ΛP M , and the direct and quadrature inductances, Ld and Lq , respectively.
Note that these assumptions differ slightly from those in preceding chapter (i.e.,
Chapter 3) in that the assumption of constant rotor velocity is relaxed, as it is not
56
required for the theory to be presented. Additionally, we will consider the permanent
magnet flux linkage as an unknown parameter to be estimated.
The first three assumptions (A1 - A3) simplify the model and reduce its order, while the last two assumptions (A4 and A5) pertain to the control design and
methodology. Under these assumptions, the dynamic model of a PMSM, repeated
here for easy reference, is given by:
dird
= −Rird + ωre Lq irq + vdr ,
Ld
dt
dirq
Lq
= −ωre Ld ird − Rirq + vqr − ωre ΛP M ,
dt
(4.1)
with the unmeasured nonlinear torque output mapping
τ=
4.3
3P
[(Ld − Lq ) ird + ΛP M ] irq .
4
(4.2)
Simultaneous Identification and Control Objective and
Methodology
The SIC design methodology demonstrated in this chapter on torque regulation in
PMSMs is based on constraining the states of the system to a set which corresponds to
a particular desired (regulated) output value, and then varying the state within that
set in order to excite the system dynamics for parameter identification. Specifically,
we are interested in regulating the (unmeasured) electromagnetic torque output of
PMSMs, for which the regulated output error is defined as follows:
eτ = τ̃ − τ,
= τ̃ − h(ird , irq ),
(4.3)
where the “tilde” denotes the reference input and h( · ) : R2 7→ R is the nonlinear
torque output mapping provided in (4.2). Note that eτ describes a 1-D manifold in
the two dimensional (real) state-space (see Figure 4.1). We define this output (error)
zeroing manifold as follows:
(ird , irq ) ∈ M := {(ird , irq ) : eτ = τ̃ − h(ird , irq ) = 0}.
(4.4)
Thus, restricting the system state to this manifold ensures that our regulated output
objective is achieved, while the non-zero dimension of M provides space in which
57
the state may vary for identification purposes. This “wiggle room” for identification
is available provided that the output maps the state-space to a lower-dimensional
output space.
: ̃−ℎ
=0
Δ
Figure 4.1: Depiction of a 1-D manifold in R2 .
However, while it is possible to drive the state to points in the set M with a
single input, provided that M is in the controllable subspace, it is generally not
possible (with a single input) to vary the state within the set M without departing
for a time, which results in a perturbation of the regulated output. Overactuation
provides additional inputs to the system which may be coordinated in such a way that
the state not only converges to M, but also varies within the set without departing.
The torque regulation problem for PMSMs is overactuated since we have two inputs
to the system, vdr and vqr , but only one regulated output, i.e., torque. Thus, for our
application, we wish to find an input pair, (vdr , vqr )(t), such that the states, ird and irq ,
converge asymptotically to the set M, as defined in (4.4).
Assuming that (ird , irq )(t0 ) ∈ M, the following invariance condition must be satisfied to ensure that (ird , irq )(t) ∈ M for all t ≥ t0 ,
ėτ = τ̃˙ − τ̇ ,
d
d
3P
r
r
r
r
∆L
i i + (∆L id + ΛP M )
i
,
= τ̃˙ −
4
dt d q
dt q
= 0,
(4.5)
where ∆L = Ld − Lq . At this point, we could substitute the machine dynamics into
(4.5) and solve for a state-feedback control law satisfying these output-zeroing and
invariance conditions. However, this will result in a highly nonlinear feedback law
which may present robustness challenges and make the design and stability proof of
the closed-loop adaptive controller difficult. We will instead use (4.4) to generate a
reference quadrature stator current, ĩrq , given a desired torque output, τ̃ , and direct
current, ĩrd , which will be treated as our free variable designed to ensure persistent
58
excitation (i.e., ĩrd is our excitation input). If we can find a control law which ensures
that (ird , irq ) → (ĩrd , ĩrq ) and ( dtd ird , dtd irq ) → ( dtd ĩrd , dtd ĩrq ) as t → ∞, then it follows that (4.4)
and (4.5) are satisfied asymptotically. Therefore, by designing a stator current regulator capable of tracking these reference currents, we will ensure that any perturbation
in the torque output due to the excitation input vanishes asymptotically.
Given τ̃ and ĩrd , we solve (4.4) for the reference quadrature current, ĩrq , which is
the primary torque generating component of the stator currents:
ĩrq =
τ̃
3P
4
∆L ĩrd
+ ΛP M
.
(4.6)
The time-derivatives of our reference currents, ĩrq and ĩrd , which are needed for tracking,
are generated using reference models (i.e., filters), M (s). Next, we seek an adaptive
control design which will ensure asymptotic tracking of the reference currents in the
presence of parameter uncertainty.
4.4
Adaptive Control Design
We begin our derivation of the adaptive controller by defining the direct and
quadrature stator current errors:
erid = ĩrd − ird ,
eriq = ĩrq − irq .
(4.7)
The following control law,
dĩrd
− ωre L̂q irq + Kpd erid ,
dt
dĩrq
r
r
vq = R̂ĩq + L̂q
+ ωre L̂d ird + Kpq eriq + ωre Λ̂P M ,
dt
vdr = R̂ĩrd + L̂d
(4.8)
where the “hat” ( ˆ ) is used to denote an estimated value and Kpd , Kpq > 0 are
constant proportional control gains, is formulated using a combination of feedforward,
feedback, and decoupling terms, designed to yield exponentially stable stator current
error dynamics (4.9) under perfect model knowledge (i.e., R̂ = R, L̂d = Ld , L̂q = Lq ,
and Λ̂P M = ΛP M ):
1
ėrid = − (R + Kpd ) erid ,
Ld
(4.9)
1
ėriq = − (R + Kpq ) eriq .
Lq
59
Feedforward
Σ
Σ
_
Σ
Σ
_
Σ
_
Inverse
Park
Transform
(r-to-s)
Σ
PWM
&
VSI
PMSM
Machine
Clarke
&
Park
Transform
(s-to-r)
P
2
P
2
Figure 4.2: Block diagram of the proposed control law.
However, when the parameters R, Ld , Lq , and ΛP M are not well known, one can
show that the closed-loop error dynamics are given by:
~e˙ ir = L−1 ΦT ~eθ − (RI + Kp ) L−1~eir ,
(4.10)
where Kp = diag [Kpd , Kpq ] is a diagonal matrix of the proportional control gains, L =
diag [Ld , Lq ] is a diagonal matrix of the direct and quadrature axis self-inductance,
T
T
~eir = erid eriq
is the stator current error vector, and ~eθ = eR eLd eLq eΛ is
the parameter error vector with eR = R − R̂, eLd = Ld − L̂d , eLq = Lq − L̂q , and
eΛ = ΛP M − Λ̂P M . Finally, the regressor matrix, Φ, in (4.10) is given by
~T
φ
d
ΦT =
~
φT
"
q
#
"
=
ĩrd
d r
ĩ
dt d
ĩrq ωre ird
−ωre irq
d r
ĩ
dt q
0
ωre
#
.
(4.11)
To stabilize (4.10) and ensure that our simultaneous identification and control objectives are achieved in the presence of parameter uncertainty, adaptation is required.
A block diagram of the proposed controller implementation is given in Figure 4.2,
where the crossing arrows behind blocks symbolize portions of the controller which
are tuned by the adaptation. Note that, in practice, implementation of the control
˜
˜
law (4.8) uses filtered commands, i.e., ~ir = {M (s)}~i∗r and dtd ~ir = {sM (s)}~i∗r , as
depicted in Fig. 4.2, where M (s) is a stable, minimum phase, proper, unity dc gain,
first-order transfer function1 , to prevent feeding-forward an unbounded signal during
1
{·} denotes a dynamic operator with transfer function “·”.
60
a step-change in references. Furthermore, the use of a derivative term in the feedforward portion of the control law (4.8) does not amplify noise, as the differentiated
reference signals are free of noise.
To derive the adaptive update law, a Lyapunov stability analysis of the closedloop system is first performed. The adaptive law is then selected such that it makes
the Lyapunov function monotonically decreasing, thereby guaranteeing closed-loop
stability of the controlled system. The following Lyapunov function candidate forms
the basis of the derivation:
V (~eir , ~eθ ) =
1 rT r
~ei L~ei + ~eθT Γ−1~eθ ,
2
(4.12)
where Γ = ΓT > 0 is the adaptation gain matrix. The first derivative of (4.12) with
respect to time is given by
V̇ (~eir , ~eθ ) = ~eirT L~e˙ ir + ~eθT Γ−1~e˙ θ .
(4.13)
Substituting (4.10) into (4.13), with some manipulation, yields:
V̇ = −~eirT [RI + Kp ] ~eir + ~eθT Φ~eir + ~eθT Γ−1~e˙ θ .
(4.14)
It is assumed that the actual machine parameters are changing very slowly, i.e.,
˙
˙ ˆ˙
~ˆ
~e˙ θ = θ~ − θ~ ≈ −θ,
(4.15)
iT
h
ˆ
where θ~ = R̂ L̂d L̂q Λ̂P M . Finally, the adaptive law is selected as,
ˆ˙
θ~ = ΓΦ~eir ,
(4.16)
V̇ (~eir , ~eθ ) = −~eirT [RI + Kp ] ~eir ≤ 0.
(4.17)
and so (4.14) becomes
Therefore, the closed-loop system (4.1), with control law (4.8) and adaptation (4.16),
is stable in the sense of Lyapunov [71].
To establish asymptotic convergence of the stator current error (i.e., ~eir → 0 as
t → ∞), Barbalat’s lemma [71] is used to show that V̇ (~eir , ~eθ ) → 0 as t → ∞.
Note that the preceding Lyapunov stability analysis has established that V (~eir , ~eθ )
61
is differentiable and has a finite limit as t → ∞. To establish uniform continuity of
V̇ (~eir , ~eθ ) we compute:
V̈ (~eir , ~eθ ) = −2~eirT [RI + Kp ] ~e˙ ir ,
(4.18)
and note that:
• ~eir and ~eθ are bounded from (4.12) and (4.17),
˜
˜
• ~ir and dtd ~ir are bounded by design, and so
˜
• ~ir = ~ir − ~eir is bounded,
thus ~e˙ ir is bounded (from inspection of (4.10)), and so V̈ (~eir , ~eθ ) is also bounded.
Therefore, from Barbalat’s lemma we have that V̇ (~eir , ~eθ ) → 0 as t → ∞; and so we
conclude that the control law (4.8) with adaptive law (4.16) renders the system (4.1)
stable in the sense of Lyapunov, with ~eir → 0 as t → ∞.
Lastly, we note that in practice our implementation of the adaptive update law
(4.16) includes a switching σ-modification [28] for robustness, which acts as a “soft
projection”, applying a leakage term, σ, to the adaptive law only when a parameter
is exceeding an expected limit on its range of variation. A benefit of this modification
is that the ideal behavior of the adaptive law is preserved so long as the estimated
ˆ
parameters remain within their acceptable bounds (i.e., |θ~i (t)| < M0,i ).
4.5
Simultaneous Parameter Identification
In the previous section, our analysis established that ~eir → 0 as t → ∞, without
requiring parameter convergence. This is typical of adaptive control designs, as the
estimated parameter set which yields zero steady-state control error is generally not
unique. For the simultaneous identification and control problem, however, parameter
identification is part of the design objective. To establish conditions for parameter
convergence, we will apply two-time-scale theory. While this approach requires the
assumption that the adaptation gain matrix is selected such that the parameter estimates converge at a much slower rate than the control error, it leads to a much more
intuitive and cleaner analysis. Furthermore, in practice it is often desirable to limit
the size of the adaptation gains in order to reduce sensitivity to noise and unmodelled
high-frequency dynamics.
62
4.5.1
Parameter Convergence using Two-Time-Scale Analysis
With the controller and adaptive laws presented in the previous section, the closedloop error dynamics (control and parameter) take the form
L~e˙ ir = ΦT ~eθ − (RI + Kp ) ~eir
~e˙ θ = −ΓΦ~eir .
(4.19)
Note that L and (RI + Kp ) are diagonal matrices, and so their multiplication is
therefore commutative. In practice, the inductances of the machine are relatively
small numbers and can therefore play the role of the “epsilon scalar” commonly used in
singular perturbation analyses to indicate small terms which are to be neglected [36].
Therefore, we may approximate (4.19) as,
0 ≈ ΦT ~eθ − (RI + Kp ) ~eir
~e˙ θ = −ΓΦ~e r .
(4.20)
i
Thus, assuming the fast stator current error dynamics have converged to a quasisteady-state, the slow parameter error dynamics are given by,
~e˙ θ = −ΓΦ [RI + Kp ]−1 ΦT ~eθ ,
1
1
T
T
~
~
~
~
φd φd +
φq φq ~eθ ,
= −Γ
R + Kpd
R + Kpq
(4.21)
~ d and φ
~ q are functions of time as defined in (4.11). Note that the slow
where φ
parameter error dynamics are of the form
~x˙ (t) = A(t)~x(t).
~ d and/or φ
~ q is persistently exciting, that is
It follows that, so long as φ
1
α1 I ≥
T
Z
t
t+T
~ d,q (σ)φ
~ T (σ)dσ ≥ α0 I, ∀ t ≥ 0,
φ
d,q
for some T, α0 , α1 > 0, then ~eθ → 0 as t → ∞.
63
(4.22)
4.5.2
Persistently Exciting Inputs
To determine necessary and sufficient conditions for persistent excitation, and
thus, sufficient conditions for parameter convergence, we will take advantage of the
connection between persistent excitation and linear independence of the functions
which make up the rows of the regressor matrix. The definition for linear independence of vector-valued functions (of time) is similar to that of constant vectors (e.g.,
in Rn ) with the difference being that an interval of interest (i.e., the domain) is specified. The definition for linear independence of vector-valued functions is given here
for convenience:
Definition IV.1. (Linear Independence of Functions [9]): A set of 1 × p real-valued
functions, f~i (t) where i = 1, · · · , n, is said to be linearly dependent on the interval
[t0 , t1 ] over the field of reals if there exist scalars ci , not all zero, such that
c1 f~1 (t) + c2 f~2 (t) + · · · + cn f~n (t) = 0
for all t ∈ [t0 , t1 ]. Otherwise, they are said to be linearly independent on the
interval [t0 , t1 ].
Naturally, there are a number of theorems which may be used to check whether or
not a set of functions is linearly independence. For example, the Grammian matrix
may be used:
Theorem IV.2. (Grammian [9]): Let f~i (t), for i = 1, 2, · · · , n, be 1 × p real-valued
continuous functions defined on the interval [t1 , t2 ]. Let F be the n × p matrix with
f~i (t) as its ith row. Define
Z
t2
W(t1 , t2 ) ,
F(t)F> (t)dt
t1
Then f~1 (t), f~2 (t), · · · , f~n (t) are linearly independent on [t1 , t2 ] if, and only if, the
n × n constant Grammian matrix, W(t1 , t2 ), is positive definite.
At this point, the connections between linear independence of the functions which
comprise the rows of the regressor matrix, and persistent excitation can be made
by noting that the definition of persistent excitation (Definition I.1) is based on the
Grammian matrix. For completeness and easy reference, this connection is summarized in the following theorem:
64
Theorem IV.3. (Linearly Independent Functions and Persistent Excitation): Consider the matrix function Φ(t) : R+ 7→ Rn×m where the elements of Φ(t) are bounded
for all time, t. The regressor matrix Φ(t) is persistently exciting if, and only if,
the rows of Φ(t) are linearly independent on the interval [t, t + T ] for all t ≥ 0
and some T > 0.
Proof: Follows from the Grammian matrix and its properties.
Our regressor matrix (4.11) is a function of the reference signals, ĩrd and ĩrq , as well
as the states of the system, ird and irq . However, we may rewrite the states in terms
of their corresponding reference signals and tracking errors, i.e.,
~T
φ
d
ΦT =
~T
φ
"
q
#
"
=
d r
ĩ
dt d
ĩrd
ĩrq ωre (ĩrd − erid )
−ωre (ĩrq − eriq )
d r
ĩ
dt q
0
ωre
#
.
(4.23)
Recall that our analysis in Section 4.4 established that the stator current error is
bounded (i.e., ~eir ∈ L∞ ) and goes to zero asymptotically (i.e., ~eir → 0 as t → ∞).
˜
˜
From Lemma 4.8.3 in [28], it follows that if ~ir is persistently exciting and ~ir ∈ L∞ ,
˜
then ~ir −~eri is persistently exciting. Therefore, we need only consider the properties of
the reference signals in our analysis of the regressor matrix (i.e., we will take ~eir = 0).
To simplify our analysis, will conservatively assume that the command torque
and rotor electrical velocity are constant, i.e., τ̃ = T̃0 and ωre = Ωre , as well as
Ld ≈ Lq . These assumptions represent something of a “worst-case” scenario since
time-varying torque references and/or rotor electrical velocity, as well as a significant
magnetic saliency, i.e., Lq >> Ld , will aid in parameter identification by providing
additional excitation; either directly, in the case of a varying torque command (and/or
rotor velocity), or indirectly, via coupling between the command currents through the
torque expression (4.6) in the presence of a significant magnetic saliency. With respect
to the latter, note that for the torque to remain constant in the presence of excitation
introduced via the direct-axis dynamics, the quadrature-axis current must vary in an
inverse manner, introducing additional excitation to the quadrature axis dynamics.
Under these assumptions, we can rewrite the regressor matrix (4.11) as follows,

ĩrd (t)
Cτ T̃0



r
 d ĩr (t)

Ω
ĩ
(t)
re d
 dt d

Φ(t) = 
.
−Ωre Cτ T̃0
0 


0
65
Ωre
(4.24)
where Cτ is a positive constant scalar, Cτ = 3P Λ4P M > 0. Additionally, we will neglect
the reference filter M (s) in our analysis as it has no effect on the results2 . Without
loss of generality, we may take ĩrd = sin(ωt):

sin(ωt)
Cτ T̃0



 ω cos(ωt) Ωre sin(ωt)


Φ(t) = 
.
−Ωre Cτ T̃0

0


0
(4.25)
Ωre
From Theorem IV.3, we may establish necessary and sufficient conditions for the
regressor matrix (4.25) to be persistently exciting, by establishing conditions under
which the rows of Φ(t) in (4.25) are linearly independent. To do this, we will use the
following theorem for checking linear independence of functions:
Theorem IV.4. (Derivative Test [9]): Assume that the 1 × p real-valued continuous
functions f~1 (t), f~2 (t), · · · , f~n (t) have continuous derivatives up to order (n − 1) on the
interval [t1 , t2 ]. Let F be the n × p matrix with f~i (t) as its ith row, and let F(k) be the
k th derivative of F. If there exists some t0 in (t1 , t2 ) such that the n × np matrix
.. (1)
.. (2)
..
.. (n−1)
F(t0 ) . F (t0 ) . F (t0 ) . · · · . F
(t0 )
has rank n, then the functions, f~i (t), are linearly independent on the interval [t1 , t2 ]
over the field of reals.
Since we are interested in sinusoidal inputs, we will consider t0 ∈ [0, 2π
]. Applying
ω
π
Theorem IV.4 to (4.25), we take t0 = 2ω and compute:
det
h
i
Φ(t0 ) Φ̇(t0 ) = −Cτ T̃0 ω 2 Ω3re .
(4.26)
We conclude that the rows of Φ(t) are linearly independent on [0, 2π
], and Φ(t) is
ω
therefore persistently exciting, provided that:
1. the direct-axis command current, ĩrd , has at least one sinusoidal component (i.e.,
ω 6= 0),
2. the command torque is non-zero, T̃0 6= 0,
2
Given a persistently exciting u with u̇ bounded, and a stable, minimum phase, proper transfer
function M (s), it follows that y = M (s)u is also persistently exciting [28].
66
3. the rotor (electrical) velocity is non-zero, Ωre 6= 0.
Remark: It should be noted that, in practice, it is important to normalize the rows
of the regressor matrix such that the peak values are all around unity. Otherwise, the
wide range of machine parameters, which are separated by orders of magnitude, will
lead to convergence issues due to poor numerical conditioning. Note that this scales
the corresponding parameter estimates as well.
4.6
4.6.1
Simulation Results
Ideal Case
Simulations using Matlab/Simulink are used to validate the proposed SIC design
for PMSMs. We present results for the ideal case first, which assume a “continuoustime” controller implementation, no time delay, and noise-free stator current measurements. Additionally, the inverter is assumed to be ideal in that the sinusoidal
voltage commands generated by the control algorithm are fed directly into the machine model.
The proposed control methodology is demonstrated in Figure 4.3, specifically constraining the system state (i.e., the stator currents) to manifolds, M, which correspond to a constant torque output (i.e., zero regulated output error). The adaptive
controller is initialized with mismatched parameters and a step-change in the com7
6
5
~ir (t)
Initial State, t = 0
Final State, t = 6 sec
M(τ = 0.4 N-m)
irq
4
3
M(τ = 0.2 N-m)
2
1
0
−10
−5
0
ird
5
10
Figure 4.3: Simulation result demonstrating state-trajectory convergence to the desired constant-torque manifolds using the proposed adaptive control design methodology with a step change in the commanded torque from 0.2
N-m to 0.4 N-m at a fixed rotor speed of 2000 RPM.
67
Torque (N-m)
3̂/3 (-) Currents (A) Zoomed
mand torque occurs 3 seconds into the simulation. Machine parameters which exaggerate the curvature of the manifolds were selected for the purpose of demonstrating
the effectiveness of the proposed methodology.
The simulation results presented in Figure 4.4 demonstrate the stagnation of the
parameter estimates when there is a lack of persistent excitation (t ≤ 0.75 sec). Inspection of Fig. 4.4 reveals that, initially, when the direct-axis current and output
torque commands are zero, the parameters fail to fully converge, as expected. Additionally, while there is partial convergence at 0.25 seconds due to excitation provided
by the step change in command torque, the resistance and direct-axis inductance
estimates do not converge quickly and fully until the excitation signal is added at
0.75 seconds. Finally, the black arrows in the “zoomed” plots in Fig. 4.4 point out
overshoot in the torque resulting from the lack of parameter convergence.
0.4
=
=˜
0.2
0
0
0.5
1
0.4
0.4
0.4
0.4
0.2
0.2
0.2
0.2
0
0.25
5
0.3
0
0.75
0
1.25
0.8
1.5
1.3
0
1.75
1.8
ird
irq
0
-5
0
3
2
0.5
1
R̂/R
L̂d /Ld
1.5
L̂q /Lq
2
$̂P M /$P M
2
1
0
0
0.5
1
1.5
2
time (s)
Figure 4.4: Simulation of an ideal implementation of the proposed SIC design for
PMSMs demonstrating parameter stagnation due an initial lack of persistent excitation, and the improvement resulting from the introduction
of the excitation signal at 0.75 seconds.
4.6.2
Sampled-data Implementation: Time Delay and Compensation
The experimental implementation of the proposed control algorithm must take
into account the sampled-data nature of its execution on a microprocessor. In particular, sampling of stator currents and encoder measurements is synchronized with a
68
center-based pulse-width modulation (PWM) structure to prevent the pickup of electromagnetic interference (EMI) generated by switching transitions during sampling.
A consequence of this synchronization is that it leads to a one-switching-period delay
between sampling measurements and updating duty cycles, as depicted in Figure 4.5.
a,b,c
s
a,b,c
a,b,c
a,b,c
s
s
a,b,c
s
a,b,c
Figure 4.5: Timing sequence of digital controller implementation.
The presence of this time-delay will impose limits on control gains, Kpd and Kpq .
Additionally, the use of reference-frame advancing in the inverse Park transform is
required (see Fig. 4.2), as the rotor angular displacement during the delay interval can
be significant. This discrepancy between the rotor position at the beginning and at
the end of a sample period can lead to instability and parameter drift in the adaptive
controller. To compensate for this angular displacement, the rotor position, θre , at
the center of the next sample period is predicted assuming that the rotor velocity,
ωre , is constant over the sample period, Ts :
3
(4.27)
θ̂re [k + 1] = θre [k] + ωre [k]Ts ,
2
where k = 1, 2, 3... represents the discrete time indices. The predicted rotor position (4.27) is then used to compute the inverse Park transform in the discrete-time
controller implementation.
To demonstrate the impact that this rotor angle discrepancy has on the parameter
estimator, we include the simulation results in Figure 4.6. A “triggered subsystem” is
used in Simulink to capture the sampled-data nature of the experimental implementation. The subsystem is triggered by an inverter model which is using center-based
PWM at a rate of 8 kHz, like the experimental set-up, and includes a one-timestep delay. The simulation which produced Fig. 4.6 did not include reference-frame
advancing based on (4.27). The switching-σ modification bounds the parameter estimates. However, inspection of Fig. 4.6 clearly shows that the parameter estimates
(particularly resistance and quadrature-axis inductance) are sensitive to this rotor
angle error due to the time delay present in the sampled-data implementation.
69
Currents
0
-5
0
ird
irq
1
2
3
3̂/3 (-)
1.5
4
5
L̂d /Ld
R̂/R
6
7
L̂q /Lq
8
$̂P M /$P M
1
0.5
0
0
1
2
3
4
5
6
7
8
time (s)
Figure 4.6: Simulation of sampled-data system without reference-frame advancing at
a speed of 2000 RPM with step changes in command torque (the same as
in Fig. 4.7), leading to poor parameter estimator performance.
τ
τ̃
0.2
0.1
0
−0.1
0
Currents (A)
Torque (N-m)
For comparison (to Fig. 4.6), we include simulation results which include referenceframe advancing based on (4.27) in Figure 4.7. This simulation uses the same Simulink
code that was used to generate the experimental code using Real-time Workshop. It
should be noted that this simulation did not include dead-time effect [7]. The “nominal” parameters provided in Table 4.1 were used in the PMSM model. Inspection of
Fig. 4.7 reveals that the algorithm works as intended under sampled-data conditions,
provided that the time delay is compensated via reference-frame advancing.
2
4
6
8
5
10
ird
irq
0
−5
0
2
4
6
8
10
θ̂/θ (-)
2
R̂/R
L̂d /Ld
L̂q /Lq
Λ̂P M /ΛP M
1
0
0
2
4
6
8
10
time (s)
Figure 4.7: Simulation of the proposed adaptive control design in a sampled-data
scenario with reference-frame advancing based on (4.27) and measurement
noise at a rotor speed of 2000 RPM.
70
4.7
4.7.1
Experimental Validation
Test Machine Parameters
For simulation and comparison purposes, the “nominal” test machine parameters
were determined offline using standard techniques. The (DC) stator resistance was
measured with a Digital Multi-Meter, inductance with an Agilent E4980A LCR meter,
and the permanent magnet flux linkage was identified using an open-circuit test and
a linear regression. These nominal parameters, denoted by an overbar, are provided
in Table 4.1. We must emphasize that we do not expect that the parameter estimates
provided by our adaptive controller will converge to these values, since they are not
necessarily the true physical parameters of the machine. For instance, the resistance
measured with a DMM does not account for skin-effect and inverter losses, while the
formation of eddy currents in the rotor iron can lead to an error in the measured
inductance when using a standard LCR meter.
Table 4.1: “Nominal” test machine parameters.
4.7.2
Parameter
Value
Resistance, R̄
109 mΩ
Direct-axis self-inductance, L̄d
192 µH
Quadrature-axis self-inductance, L̄q
212 µH
PM Flux Linkage, Λ̄P M
12.579 mV-s
No. of Poles, P
10
Description of the Experimental Set-up
The proposed robust adaptive control algorithm has been implemented on experimental hardware using a dSPACE DS1104 controller board, and the test machine
(Table 4.2) is a 3-phase, 10-pole, 250 watt SMPM machine from MOTORSOLVER
with “nominal” parameters (denoted by the over bar) listed in Table 4.1. A 250 watt
DC machine from the same manufacturer serves as the load for the SMPM machine.
A power MOSFET inverter is used to drive the motors with a switching frequency
of 8 kHz and a bus voltage of 42 VDC. First-harmonic dead-time compensation is
used to mitigate the voltage discrepancy resulting from the insertion of dead-time
in the gate-drive signals [64]. Duty cycles are calculated using conventional pulsewidth-modulation, and the ADC sampling is synchronized with, and offset from, the
71
center-based PWM signals to avoid sampling during a switching event (as discussed
in the previous section).
r
r
a
dc
b
BUS
Gate
Drive
Signals
Gate
Drive
Signals
User
Inputs
Figure 4.8: Experimental setup.
Table 4.2: Manufacturer machine ratings.
Test Motor
Load Motor
Type:
PM Brushless
Type:
DC
No. Phases:
3
No. Phases:
N.A.
V/I:
42 V/5.7 A
V/I:
42 V/6 A
Max. Speed: 4000 RPM
Max. Speed: 4000 RPM
Rated Power: 250 W
Rated Power: 250 W
4.7.3
Experimental Results
Since mechanical torque was not measured during these experiments, the quadrature stator current (in the rotor reference frame) is used to evaluate the transient
performance of the proposed torque regulator in addition to the estimated electromagnetic torque (4.28), which can vary with the parameter estimates:
τ̂ =
i
3P h
L̂d − L̂q ird + Λ̂P M irq .
4
(4.28)
It should be noted that, since torque is not measured directly, accurate knowledge
of the permanent magnet flux linkage, as well as the direct and quadrature selfinductance, is required for accurate torque regulation. Torque steps, used to evaluate
72
Torque (N-m)
Est./Nom. (-)
Current (A)
0.2
0.1
0
−0.1
0
τ̂
τ̃
1
2
3
4
5
5
0
−5
0
ird
irq
1
2
3
R̂/R̄
L̂d /L̄d
1
2
4
5
1.5
1
0.5
0
L̂q /L̄q
3
Λ̂P M /Λ̄P M
4
5
time (s)
Figure 4.9: Experimental torque steps with adaptation on at 2000 rpm.
the performance of the proposed adaptive torque regulator, are provided in Figures
4.9 and 4.10.
In Fig. 4.9 we see that the estimated torque tracks the commanded value very well
without any undesirable jumps or drifting in the parameter estimates. A direct-axis
current reference of ĩrd = 1.5 sin(150t) + 1.5 sin(300t) amps provides excitation for parameter estimation. Note that the estimated parameter values have been normalized
with respect to their “nominal” values in Table 4.1, which are not necessarily the true
values (which are unknown), to facilitate plotting on the same axis for comparison.
A feature of the proposed adaptive controller design is that its closed-loop transient
response remains consistent across a wide range of operating points. To demonstrate
this, torque steps from 0 to 0.4 N-m were performed at 2500, 1200, and 0 RPM3 ,
and are plotted in Fig. 4.10. Note that the responses overlay, indicating that the
controller is performing as expected. Additionally, the “ripple” or “noise” which can
be seen in the signals is expected, and is largely due to the non-ideal slotting effects
in the machine.
As discussed earlier in this paper, the proposed adaptive control design achieves
the simultaneous identification and control objective in that it allows excitation sig3
Voltages begin to saturate due to inverter-created voltage constraints above 2500 RPM.
73
τ̂ (N-m)
0.4
0.3
0.2
0.1
0
0
0.01
0.02
0.03
0.04
5
irq (A)
4
3
2
ωr = 2500 RPM
ωr = 1200 RPM
ωr = 0 RPM
1
0
0
0.01
0.02
time (s)
0.03
0.04
Figure 4.10: Experimental transient responses of estimated torque (top) and measured quadrature-axis current (bottom) across a wide range of rotor
speeds.
nals to be introduced for parameter identification whose impact on the output is
minimized (asymptotically, in the case of our design). This property is demonstrated
in Figure 4.11, in which the transient response of the experimental adaptive parameter estimator for a constant torque command of 0.2 N-m at a fixed rotor speed of
2000 RPM is plotted. Initially, the parameter values are intentionally mismatched
such that the excitation signal disturbs the torque output. Inspection of Fig. 4.11 reveals that, as the estimates converge, the disturbance caused by the excitation signal
vanishes, as expected.
To gauge the performance of our parameter identification, we recorded the steadystate values of the estimated parameters over a range of operating points in which
the parameters are all identifiable (i.e., non-zero rotor speed and torque command).
Inspection of the results in Figure 4.12 indicate that the parameter estimation is
working very well overall. The resistance estimate is fairly consistent across rotor
speed, but increases slightly with the torque command, potentially due to temperature rise. The estimated direct-axis inductance and permanent magnet flux linkage
are very consistent across rotor speed and torque, as expected. The drop in the estimated permanent magnet flux linkage at 500 RPM is likely due to the increasing
impact of the dead-time effect at lower speeds (which generally correspond to smaller
74
0.1
0
0
zoomed
Torque (N-m)
0.2
τ̂
τ̃
1
2
3
0.2
0
5
−5
0
2
5
0.2
0.1
Adaptation OFF
0.1
0.2
Adaptation ON
4.5
4.6
0
Est./Nom. (-)
Current (A)
0.1
4
4.7
ird
irq
1
2
3
4
5
1
R̂/R̄
0
0
1
L̂q /L̄q
L̂d /L̄d
2
3
Λ̂P M /Λ̄P M
4
5
time (s)
Figure 4.11: Experimental adaptive parameter estimator for a constant torque command of 0.2 N-m at a fixed rotor speed of 2000 RPM demonstrating
transient characteristics of the parameter estimator as well as asymptotically vanishing torque perturbation due to the excitation signal.
stator voltages). Finally, the wider variation in the quadrature-axis inductance was
anticipated, as this parameter was observed to be particularly sensitive to encoder
misalignment while tuning the experimental controller. This behavior was also observed in simulations which introduced a fixed rotor angle offset error. However,
we have found that the impact of this variation on the controller performance (i.e.,
output regulation) is negligible.
Still, it is worth noting that the quadrature-axis inductance estimate seems to improve at higher speeds, yielding a nearly flat trend at 2000 RPM (see Fig. 4.12), and
estimates around the same value as the direct-axis inductance. This is to be expected
as our test machine was a surface-mount permanent magnet (SMPM) machine which,
characteristically, have a negligible magnetic saliency (i.e., it is commonly assumed
that Ld ≈ Lq for SMPM machines). Recall that under a constant torque command
75
R̂ (m+)
140
0.15
0.2
0.25
0.3
2000 RPM
1500 RPM
1000 RPM
500 RPM
0.35
0.4
0.15
0.2
0.25
0.3
0.35
0.4
0.15
0.2
0.25
0.3
0.35
0.4
0.15
0.2
0.25
=˜ (N-m)
0.3
0.35
0.4
120
100
0.1
L̂d (7H)
400
300
200
0.1
L̂q (7H)
400
300
200
0.1
$̂P M (mV-s)
13
12.5
12
0.1
Figure 4.12: Experimental characterization of steady-state parameter estimates over
a wide range of rotor speeds and torque commands.
and rotor speed, the third row of the regressor, which relates to the quadrature inductance estimate (see equation (4.25)), is dependent on the following term: −Ωre Cτ T̃0 .
At a minimum, a nonzero torque command, T̃0 , and rotor speed, Ωre , are needed
for the regressor to be persistently exciting, otherwise the third row of the regressor
will be all zeros and the estimate of quadrature-axis inductance will stagnate. Practically, it is expected that the estimate of the quadrature-axis inductance, Lq , will
suffer from drifting in the presence of modeling errors, such as encoder misalignments,
at low speeds and/or torque commands. This may explain why the estimate of Lq
seems to improve at high speeds, as well as higher torque commands, as observed in
Fig. 4.12.
76
4.8
Chapter Conclusion
This chapter extended results from [66], which presented a new robust adaptive
torque regulating controller for SMPM machines that estimates resistance, inductance, and permanent magnet flux linkage online. The adaptive controller for PMSMs
presented was derived using Lyapunov’s stability theorem, and a robust modification
to the derived adaptive law is used to ensure closed-loop stability in the presence
of unmodeled disturbances. The control law utilizes a combination of adaptivelytuned feedforward (to achieve zero steady-state error), d − q decoupling (to improve
transient response), and proportional feedback (to add robustness to disturbances)
terms. Overactuation of the system is exploited to simultaneously achieve parameter convergence and torque regulation. Necessary conditions for persistent excitation
were discussed, and simulation results verifying the performance of the control design
were presented. Finally, remarks specific to experimental implementation challenges,
and experimental results validating the performance of the proposed design, were
discussed.
77
CHAPTER V
Receding Horizon Control Allocation for
Simultaneous Identification and Control of
PMSMs
5.1
Introduction
Chapters 3 and 4 explored more traditional control designs for achieving SIC in
PMSMs with emphasis on exploiting overactuation. In this chapter, we present an
optimization-based simultaneous identification and control methodology for achieving the same objective. A receding horizon control allocation (RHCA) is used which
includes a metric for maximizing the excitation characteristics of the generated reference current trajectories. The reference currents produced by the RHCA are fed
to the lower-level adaptive current regulator, presented in Chapter 4, which ensures
asymptotic tracking of a reference model. We begin by discussing the proposed control
architecture and introduce the (static) control allocation problem for PMSM torque
regulation. Metrics for optimizing the conditioning of the Fisher information matrix
and their application to generating persistently exciting inputs are then discussed;
as well as the necessary modifications to the control allocation problem, needed for
excitation maximization, which lead to the RHCA formulation. Finally, the crucial
role of past input and state data in the RHCA-SIC algorithm is discussed, and simulation results demonstrating the effectiveness of the methodology, as well as the need
for past data, are presented.
This chapter based on work submitted to a conference and is under review:
D. M. Reed, J. Sun, and H. F. Hofmann, “A Receding Horizon Approach to Simultaneous Identification and Torque Control of Permanent Magnet Synchronous Machines,” Under review, 2015.
78
Inner-loop Controller
Feedforward
Receding Horizon
Control Allocation
Σ
Σ
_
Σ
Adaptive Current Regulator
_
Σ
Σ
_
Σ
Inverse
Park
Transform
(r-to-s)
PWM
&
VSI
PMSM
Machine
subject to:
Clarke
&
Park
Transform
(s-to-r)
P
2
P
2
Figure 5.1: Block diagram of the proposed RHCA-SIC methodology for PMSM torque
regulation.
5.2
Proposed Control Architecture
The proposed RHCA-SIC design utilizes a two-level structure with reference signals generated by the RHCA being fed to the inner-loop adaptive current regulator, as
depicted in Figure 5.1. The adaptive current regulator ensures fast, accurate tracking
of the filtered1 reference current trajectories, while the “outer-loop” RHCA exploits
the over-actuated nature of the PMSM to generate reference currents which are both
persistently exciting and produce the desired torque. For convenience, we briefly
review the inner-loop adaptive current regulator presented in Chapter 4.
5.2.1
Inner-loop Controller
The inner-loop controller is a Lyapunov-based adaptive current regulator [66]
which has been extended to include PMSMs (i.e., magnetic saliency is considered).
The adaptive current regulator ensures that the 2-phase equivalent stator currents
asymptotically converge to track the trajectories produced by the reference models.
We define the direct and quadrature stator current errors as follows:
erid = ĩrd − ird ,
eriq = ĩrq − irq ,
1
By the reference models, M (s) =
λ
s+λ .
79
(5.1)
where the “tilde” ( ˜· ) denotes filtered reference signals, i.e., the output of M (s).
The control law for our adaptive current regulator uses a mix of feedforward,
feedback decoupling, and proportional feedback terms, and is given by
dĩrd
− ωre L̂q irq + Kpd erid ,
dt
d
ĩrq
r
r
vq = R̂ĩq + L̂q
+ ωre L̂d ird + Kpq eriq + ωre Λ̂P M ,
dt
vdr = R̂ĩrd + L̂d
(5.2)
where the “hat” ( ˆ· ) denotes estimated parameters, Kpd and Kpq are the respective
direct and quadrature-axis proportional gains, and the derivative terms are produced
˜
˜
by the reference model (i.e., ~ir = {M (s)}~i∗r and dtd ~ir = {sM (s)}~i∗r , where M (s) is a
a stable, minimum phase, proper, unity dc gain, first-order transfer function2 ).
The estimated parameters in (5.2) are updated via the following adaptive parameter update law
ˆ˙
θ~ = ΓΦ~e r ,
(5.3)
i
T
where Γ = ΓT > 0 is the adaptation gain matrix, ~eir = erid eriq is the stator current
error vector, and the regressor matrix, Φ, is given by

ĩrd
ĩrq



 d ĩr
ωre ird 

 dt d
Φ=
.
−ωre ir d ĩr 
q
q
dt


0
ωre
(5.4)
It can be shown, using Barbalat’s lemma [71] and the following Lyapunov function
V (~eir , ~eθ ) =
1 rT r
~ei L~ei + ~eθT Γ−1~eθ ,
2
(5.5)
that the control law (5.2) with adaptive update (5.3) renders the PMSM dynamics
(4.1) stable in the sense of Lyapunov with ~eir → 0 as t → ∞, where L = diag [Ld , Lq ]
T
is a diagonal matrix
h of the direct iand quadrature axis self-inductances, and ~eθ =
[R Ld Lq ΛP M ] − R̂ L̂d L̂q Λ̂P M is the parameter error vector. Convergence of
the parameter error follows when the regressor matrix (5.4) is persistently exciting.
Lastly, we note that a “switching σ-modification” [28] is used on (5.3) for robustness.
2
{·} denotes a dynamic operator with transfer function “·”.
80
5.2.2
Control Allocation
The primary objective in any control allocation problem is to find the “best”
distribution of control effort among multiple actuators to achieve a desired effect (e.g.,
generate a “virtual” control input which achieves the desired output). Additionally,
by solving the problem online, the effects of actuator saturation and failures can be
taken into account. Control allocation is particularly well suited to over-actuated
problems which permit the inclusion of secondary objectives, such as control effort
minimization.
8
= $ = 0.4 N-m
6
4
= $ = 0.2 N-m
ir$
q (A)
2
= $ = 0.0 N-m
0
-2
= $ = -0.2 N-m
-4
= $ = -0.4 N-m
-6
-8
-8
-6
-4
-2
0
2
4
6
8
ir$
d (A)
∗r
Figure 5.2: Sets of current pairs, (i∗r
d , iq ), which yield various torques for a machine
with large saliency (to magnify nonlinearity).
Typically, the control allocation problem is treated as a static optimization problem, assuming that the “actuator” response is instantaneous [6, 31]. As it concerns
torque control for the (over-actuated) PMSM, the control allocation problem consists
∗r
∗
of finding a reference current pair, (i∗r
d , iq ), which produce a desired torque, τ . The
inner-loop controller, discussed in the previous subsection, is then tasked with producing the voltage pair, (vdr , vqr ), which generates these reference currents. Since the
problem is over-actuated, there exists an infinite number of reference currents which
yield a given torque. The reference current solution set for some τ ∗ is described by all
∗r
∗r ∗r
∗
~∗r ~
pairs (i∗r
d , iq ) ∈ M := {(id , iq ) : |τ −h(i , θ)| = 0}, and are depicted in Figure 5.2.
In discrete-time, the static control allocation problem for a torque-regulated PMSM
81
can be stated as
T ~∗r
min ~i∗r
k Ri k
~i∗r
k
s.t.
|~i∗r
k | ≤ Imax ,
~ˆ
|τk∗ − h(~i∗r
k , θk )| = 0,
(5.6)
where our secondary objective is the standard weighted quadratic function of the
reference input with R > 0, which minimizes the control effort and, therefore, the
ohmic losses as well. While this problem formulation is sufficient for torque regulation, it doesn’t ensure persistently exciting reference currents without varying the
commanded torque. In the next section, we discuss metrics for persistent excitation
and their inclusion in the control allocation problem.
5.3
Receding Horizon Control Allocation for Simultaneous
Identification and Control
To ensure that the reference currents generated by the control allocation are persistently exciting, we seek a metric which will provide a measure of how persistently
exciting the regressor matrix (5.4) is over some time interval. Such a metric will then
be included in the objective (or cost) function of the control allocation problem to
encourage the generation of reference signals which are persistently exciting.
5.3.1
The Fisher Information Matrix and Persistent Excitation
The identification of parametric models is of interest to a wide variety of disciplines, well beyond that of the control community. In statistics, as well as other fields,
the conditioning of the Fisher information matrix is used to judge how informative
an experiment (i.e., its data) is with respect to the identification of a given parametric
model. Mathematically, given N discrete observations (i.e., measurements) of a single
output3 , y(tk ), at time tk with k ∈ [1 · · · N ], of some process described by
~
y(tk ) = H(t, θ),
(5.7)
the Fisher information matrix is defined as
T N X
∂y(tk )
∂y(tk )
,
F=
∂ θ~
∂ θ~
k=1
3
The extension to multiple outputs is trivial.
82
(5.8)
where θ~ is the parameter vector we are interested in identifying. Note that F is a symmetric positive-semidefinite matrix. When Gaussian noise is considered in the estimation problem formulation, F−1 gives the Cramer-Rao lower bound on the achievable
covariance of an unbiased estimator. Clearly, if the experiments are not informative, the Fisher information matrix (5.8) will be poorly conditioned, leading to high
uncertainty in the parameter estimates.
For processes described by a linear parameterization, e.g.,
~ T (t)θ,
~
y(tk ) = φ
(5.9)
the Fisher information matrix (5.8) simplifies to the familiar form used when defining
persistent excitation
N
X
~ k )φ
~ T (tk ),
φ(t
(5.10)
F=
k=1
~ k ) is the regressor vector. In discrete-time, a bounded vector signal φ(t
~ k ) is
where φ(t
said to be persistently exciting (PE) if there exists N > 0 and α0 > 0 such that
F=
N
X
k=1
~ k )φ
~ T (tk ) ≥ α0 I
φ(t
(5.11)
for all tk ≥ t0 [74].
Finally, we will use the log-determinant of F as our measure of the conditioning
of the Fisher information matrix, i.e.,
JD = log (det (F)) ,
(5.12)
which is often used for the purpose of “optimal experiment design” (sometimes referred to as “D-optimality”) [49, 53].
5.3.2
Receding Horizon Control Allocation for Simultaneous Identification and Control
Since the Fisher information matrix (5.8) becomes singular when evaluated at
any given time instant, the addition of a metric for persistent excitation requires
modifying the control allocation problem (5.6) to consider a finite time horizon in the
optimization, making a receding horizon (or MPC) framework a natural choice. In
the past, researchers have proposed receding horizon control allocation (or MPCA)
83
approaches to account for actuator dynamics, e.g., [51, 78]. In this work, we will
utilize the receding horizon control allocation (RHCA) framework to accommodate
the addition of the persistent excitation metric (5.12).
Implementation of a RHCA requires a dynamic model of the inner-loop system to
predict future state trajectories and evaluate the regressor matrix for optimization.
The prediction model is formulated using the certainty equivalence principle; that is,
assuming that the estimated parameters are equal to their true values. This turns
out to be a minor assumption, however, because we are assured that, as long as the
system is persistently excited, the estimated parameters will converge to their true
˜
values. Augmenting the reference filter states, ~ir , with the PMSM states, ~ir , the
inner-loop dynamics for prediction are therefore given by
~ˆ x + B̄~i∗r ,
~x˙ = Ā(θ)~
(5.13)
~z = C̄~x + D̄~i∗r ,
with,

 −1
−1
L̂
−
λI
R̂I
+
K
L̂
−
R̂I
+
K
p
p
~ˆ = 
,
Ā(θ)
0
−λI
" #
I
B̄ = λ
, C̄ = −λ [0 I] , D̄ = λI,
I
˜
˜
where ~x> = [~ir ~ir ] is the augmented state vector, ~z = dtd ~ir is needed to evaluate the
regressor matrix, and so we treat it as an output of the prediction model, and λ > 0
sets the bandwidth of the (first-order) reference model filters. For the discrete-time
implementation, the prediction model (5.13) is discretized using a zero-order hold.
Since the reference currents, ~i∗r , have no effect on the estimation of the permanent magnet flux linkage4 , ΛP M , we do not include the bottom row of the regressor
matrix (5.4), which corresponds to the ΛP M term, in our optimization. We define the
truncated regressor matrix, used in the optimization, as follows:


Φ̄(~x, ~z) = 

ĩrd
d r
ĩ
dt d
−ωre irq
ĩrq


ωre ird 
.
(5.14)
d r
ĩ
dt q
ˆ
Assuming the estimated parameters, θ~k , torque reference, τk∗ , and rotor electrical
4
It can be shown that identification of ΛP M only requires a non-zero rotor velocity, i.e., ωre 6= 0.
84
velocity, ωre,k , to all be constant over the prediction horizon, the extension of (5.6)
to include a metric (5.12) for persistent excitation is given by
min
~i∗r
j
s.t.
k+N
X−1
j=k
T ~∗r
~i∗r
Rij − ρ log det (F(~x, ~z))
j
ˆ
~xj+1 = Ā(θ~k )~xj + B̄~i∗r
j ,
~zj = C̄~xj + D̄~i∗r
j ,
F(~x, ~z) =
k+N
X−1
(5.15)
Φ̄(~xj , ~zj )Φ̄T (~xj , ~zj ),
j=k
|~i∗r
j |
≤ Imax , ∀ j ∈ [k · · · k + N − 1],
~ˆ
|τk∗ − h(~i∗r
j , θk )| ≤ , ∀ j ∈ [k · · · k + N − 1],
where R ≥ 0 is the input weighting matrix, ρ ≥ 0 is the PE metric weighting,
and > 0 determines the maximum allowable perturbation in the regulated torque
output. While the constraint on the regulated output error could be included in the
objective function and penalized5 , the over-actuated nature of our problem permits
the use of it as a constraint6 . We do, however, include it here as a “relaxed” (i.e.,
inequality) constraint to speed up the numerical optimization, help ensure that a
feasible solution exists, and allow for small perturbations in regulated output if it will
aid the parameter identification.
5.3.3
The Crucial Role of Past Input and State Data
The standard RHCA formulation defined by (5.15), where only future inputs are
considered in calculating the Fisher information matrix, F(~x, ~z), turns out to be
problematic in that it fails to generate persistently exciting trajectories. This is
due to the fact that (1) the effects of past data are not included in evaluating the
Fisher information matrix, and (2) RHCA only implements the first element of the
optimizing sequence.
To highlight the importance of incorporating recent past input and state data
in the calculation of the Fisher information matrix for maximizing excitation in the
receding horizon framework, imagine conditions are such that the optimal predicted
5
This approach was briefly investigated in numerical simulations, but was found to require very
large penalties to achieve reasonable tracking performance which could lead to numerical conditioning
issues.
6
~ˆ
Since we know that, under normal operating conditions, solutions satisfying |τk∗ −h(~i∗r
k+i , θk )| ≤ exist.
85
Past
Future
Figure 5.3: Disregard for past input (and state) data leading to a lack of persistent
excitation.
input trajectory is the same at every subsequent time step. In the receding horizon (or
MPC) framework, only the first step of the optimal sequence is applied at any given
time step. So while the optimal predicted sequence may be persistently exciting, the
actual sequence applied to the system is very much not persistently exciting. This
is depicted graphically in Figure 5.3. When past data is considered, it is clear that
the first time step in each subsequent optimal sequence, which will be applied to the
system, must differ from the previous to ensure that persistently exciting inputs are
indeed generated.
With this issue in mind, we modify the RHCA problem proposed in (5.15) to
include Np points of recent (past) data (i.e., the last Np values of the states and
inputs) in addition to the usual prediction horizon, Nf , in evaluating the Fisher
matrix, F(~x, ~z):
k+Nf −1
min
~i∗r
j
s.t.
X
j=k
T ~∗r
~i∗r
Rij − ρ log det (F(~x, ~z))
j
ˆ
~xj+1 = Ā(θ~k )~xj + B̄~i∗r
j ,
~zj = C̄~xj + D̄~i∗r
j ,
k+Nf −1
F(~x, ~z) =
X
(5.16)
Φ̄(~xj , ~zj )Φ̄T (~xj , ~zj ),
j=k−Np
|~i∗r
j | ≤ Imax , ∀ j ∈ [k · · · k + Nf − 1],
~ˆ
|τk∗ − h(~i∗r
j , θk )| ≤ , ∀ j ∈ [k · · · k + Nf − 1].
The change is subtle, but the effects are profound, as will be demonstrated in the
simulation results to follow.
86
Finally, to reduce the dimension of the numerical optimization problem, a linear
B-spline is used to approximate the control input [37]. For the purpose of trajectory
optimization, the reference currents on the time interval tk ∈ [0, T ] are given by
i∗r
d,q (tk )
=
J
X
j=0
αj B (t̄k ) ,
(5.17)
where t̄k is the normalized time sequence, given by
t̄k =
tk J − 1 j − 1
−
,
T 2
2
and B(t̄k ) are the triangular basis functions,

1 − 2|t̄ | for |t̄ | ≤ 0.5,
k
k
B(t̄k ) =
0
otherwise,
which are precomputed and stored in memory. Thus, we optimize over a vector of
the weighting coefficients, αj , rather than the full resolution time sequence.
Note that a sufficient number of “knot” points (i.e., sufficiently large J) must be
used with respect to the length of the time interval, T , to ensure that signals are
approximated with sufficient fidelity. The advantages of using a linear spline are that
constraints can be enforced simply by looking at the weighting coefficients, αj , which
give the signal value at the knot points7 and a reduction in the dimension of the
optimization problem, speeding up the numerical optimization.
5.4
Simulation Results
Numerical simulations using Matlab/Simulink are used to verify the effectiveness
of the proposed receding horizon control allocation methodology for SIC of PMSMs.
The simulations capture the sampled-data nature of a practical implementation by
implementing the controller in a triggered subsystem which runs at 8 kHz for the
inner-loop (high-bandwidth) adaptive current regulator and a quarter of that (i.e., 2
kHz) for the RHCA, while the machine dynamics are solved using ode45. An ideal
“average-value” inverter model is assumed, that is, the voltage commands generated
by the controller are fed directly into the PMSM model. The optimization problem
is solved using the active-set algorithm in fmincon, and the simulation parameters in
7
Higher-order polynomial basis functions can lead to “peaking” and constraint violation.
87
Table 5.1: Simulation parameters.
Description
Value
Electrical Machine Parameters:
R
109 mΩ
Ld
192 µH
Lq
212 µH
ΛP M
12.579 mV-s
P
10
Control Design Parameters:
Kpd , Kpq
0.2
Γ
diag([30 30 30 30])
λ
225
R
0.1 · I
ρ
10
Prediction Horizon, Nf
25
No. of Recent Data Points, Np
25
Simulation Settings:
Solver
ode45
Max Step Size
25 µ-sec
Table 5.1 were used in all simulations except where otherwise noted.
5.4.1
Static Control Allocation
For completeness, simulation results for the static control allocation problem (5.6)
are provided in Figure 5.4. Inspection of the results in Fig. 5.4 reveals that, without
a metric for excitation, the control allocation algorithm is simply trying to track
the desired torque command using a minimal amount of control effort. Thus, the
commanded direct-axis current is essentially zero for the entirety of the simulation,
corresponding to a minimal current magnitude operating point. The lack of excitation
leads to slow parameter convergence, since excitation is only provided by the step
changes in torque. Additionally, the lack of accurate parameter knowledge leads to a
small but undesirable overshoot in the transient torque responses (see “zoomed” plots
88
=
=˜
0.2
0.4
0.6
0.1
0.1
0
0
-0.1
-0.1
Zoomed
Torque (N-m)
0.2
0.1
0
-0.1
0
Currents (A)
0.36 0.38
0.4
0.42 0.44
0.48
0.8
0.5
1
0.52 0.54 0.56
5
0
-5
0
3̂/3 (-)
3
ird
0.2
R̂/R
0.4
0.6
L̂d /Ld
L̂q /Lq
irq
|Imax |
0.8
1
$̂P M /$P M
2
1
0
0
0.2
0.4
0.6
0.8
1
time (s)
Figure 5.4: Simulation of the static control allocation (5.6) without PE maximization.
in Fig. 5.4). Finally, the steady-state tracking is expected given that the magnetic
parameters more or less converge to their true values, and the inner-loop controller is
designed to guarantee asymptotic convergence of the stator current error regardless
of the accuracy of the parameter estimates.
5.4.2
RHCA-SIC without Past Input and State Data
Again, for completeness, simulation results for the standard RHCA-SIC problem
formulation when past input and state data is not considered (5.15) are provided in
Figure 5.5. Inspection of the results in Fig. 5.5 reveals that, while the controller does
do a good job of tracking the desired torque, the RHCA algorithm fails to generate
persistently exciting signals. The lack of persistently exciting inputs once again leads
to parameter stagnation.
89
0.1
0
-0.1
0
Currents (A)
Torque (N-m)
0.2
=
=˜
0.2
0.4
0.6
0.8
1
5
0
-5
0
ird
0.2
0.4
0.6
irq
|Imax |
0.8
1
3̂/3 (-)
3
L̂q /Lq
L̂d /Ld
R̂/R
$̂P M /$P M
2
1
0
0
0.2
0.4
0.6
0.8
1
time (s)
Figure 5.5: Simulation of the RHCA with PE maximization and without past input
and state data (5.15).
5.4.3
RHCA-SIC with Past Input and State Data
When past input and state data are included in the RHCA (5.16), we see that all
of the parameters converge to their true values, as the simulation results in Figure
5.6 demonstrate. Inspection of the results in Fig. 5.6 reveals that, not only does
the PE metric with past data generate persistently exciting reference currents, but
the overall RHCA-SIC strategy takes advantage of the over-actuated nature of the
plant by utilizing the direct-axis current, which has a small impact on the torque
production, for the majority of the excitation. Meanwhile, the quadrature-axis current
is primarily used to satisfy the torque regulation (i.e., control) objective, agreeing with
our intuition about the SIC problem for PMSMs [66, 67].
Note that, while the torque output is initially perturbed by the additional excitation introduced for parameter identification, this perturbation vanishes asymptotically as the parameter estimates converge to their true values. This happens because
accurate parameter knowledge is needed in order to accurately define the set M in
which the states may vary without perturbing the regulated output. In addition
to the initial perturbations in the regulated output, another trade-off of practical
interest is between the losses incurred due to the excitation, and the identification
objective which favors large signals (which results in faster convergence). This trade-
90
=
=˜
0.2
0.4
0.6
0.1
0.1
0
0
-0.1
-0.1
Zoomed
Torque (N-m)
0.2
0.1
0
-0.1
0
Currents (A)
0.36 0.38
0.4
0.42 0.44
0.48
0.8
0.5
1
0.52 0.54 0.56
5
0
-5
0
ird
0.2
0.4
0.6
irq
|Imax |
0.8
1
3̂/3 (-)
3
L̂q /Lq
L̂d /Ld
R̂/R
$̂P M /$P M
2
1
0
0
0.2
0.4
0.6
0.8
1
time (s)
Figure 5.6: Simulation of the proposed RHCA-SIC methodology for over-actuated
systems with PE maximization and past data (5.16).
off is managed by adjusting the penalty, R, and the generation of exciting signals
can be “turned off” (e.g., after an initial commissioning phase) by setting R to be
the zero matrix. Lastly, while intuition can sometimes be leveraged to decide how
the excitation and control efforts should be allocated (e.g., using the direct-axis for
excitation and the quadrature-axis for control), as was the case in Chapters 3 and 4,
a distinct advantage of the proposed optimization-based RHCA-SIC methodology is
that it automatically determines the optimal allocation strategy for SIC in systems
where intuition is lacking.
91
5.5
Conclusion
In this chapter we presented an optimization-based simultaneous identification
and control methodology for PMSMs which exploits the overactuated nature of the
machine. A receding horizon control allocation (RHCA) framework is used which
includes a metric for maximizing the excitation characteristics of the generated reference current trajectories. The RHCA feeds the computed reference currents to a
lower-level adaptive current regulator which ensures asymptotic tracking of a reference model. The importance of including past input and state data in the RHCA-SIC
algorithm is discussed, and simulation results demonstrating the effectiveness of the
proposed RHCA-SIC methodology with past input and state data are presented, as
well as scenarios without PE maximization and which disregard past data. Finally,
it is worth noting that the optimization problem as posed in this chapter, could lack
a feasible solution in the event that either a exceedingly large torque is commanded;
or if an exceedingly large load torque is preset, requiring stator current magnitudes
beyond the limits of the machine. In this scenario, there is no feasible solution which
satisfies both the current limits and output (i.e., torque) error objective. However,
by introducing a slack variable (or “soft constraint”) [52] on the output error, we can
ensure that feasible solutions exist, at the expense of tracking error performance.
92
CHAPTER VI
Conclusions and Future Work
6.1
6.1.1
Conclusions
Offline Identification of Induction Machine Parameters
In Chapter 2, a new technique for offline identification of induction machine parameters using steady-state data was presented. This work addresses the need for an
alternative to the IEEE standard, and one which is well suited to modern VSI drives
and the characterization of machines over their full range of operation. The proposed
technique is based on fitting experimental data to the circular stator current locus in
the stator flux linkage reference-frame for varying steady-state slip frequencies, and
provides accurate estimates of the magnetic parameters, as well as the rotor resistance and core loss conductance. Implementation issues related to the sampled-data
nature of experimental implementations are considered in the design of the stator
flux linkage estimators, as well the compensation of the dead-time effect. Numerical
simulations evaluating the accuracy of the estimated parameters in the presence of
nonideal effects were presented, and experimental results for a 43 kW induction machine demonstrate the utility of the proposed technique by characterizing the machine
over a wide range of flux levels, including magnetic saturation.
6.1.2
Simultaneous Identification and Adaptive Control of PMSMs
In Chapter 3, an adaptive excitation decoupling approach to achieving simultaneous identification and control was explored via a case study with PMSMs. The
input(s) which have the most authority over the regulated output are designated the
control input(s), while the remaining plant inputs serve as the excitation input(s).
A disturbance decoupling control law is then utilized to prevent the excitation input
from perturbing the regulated output. The machine parameters used in this excitation
93
decoupling control law are updated via an online parameter estimator (normalized
gradient estimator in this work). Simulation results for a torque regulating controller
for PMSMs confirm the effectiveness of the proposed simultaneous identification and
control design methodology. While the focus of the chapter is on the application of
the proposed adaptive excitation decoupling control methodology to PMSM torque
regulation, the prospects of generalizing this methodology for overactuated systems
are promising. However, this approach does have some drawbacks. Like the exact feedback linearization techniques [29,33], disturbance decoupling lacks robustness
to unmodeled uncertainties. Additionally, the indirect design approach used to formulate the closed-loop adaptive excitation decoupling controller, pairing separately
designed control law and parameter identifier, makes proving stability a challenge.
The 2 degree-of-freedom (DOF) Lyapunov design presented in Chapter 4, takes a
different approach to achieving the simultaneous identification and control objective.
By directly regulating the state to the output zeroing manifold, and ensuring that the
invariance condition is satisfied, we are able to achieve the SIC objective by varying
the state within this set (the output zeroing manifold) to ensure the system remains
excited for identification. To do this, we develop an adaptive control law, using a
Lyapunov stability analysis, which ensures that reference trajectories produced by a
reference model are tracked asymptotically. Using the torque output mapping (i.e.,
state to regulated output mapping), we generate the command currents which are
fed to the reference models by solving for the required (i.e., command) quadratureaxis current given torque and excitation input commands. The advantages of this
approach are that closed-loop stability of the adaptive controller is proven in the
design process, as well as the direct regulation of the states.
The SIC approaches presented in Chapters 3 and 4 did not consider optimization of the excitation signal introduced to the system for parameter identification.
Additionally, they fixed the allocation of the applied control inputs based on predetermined control input(s), excitation input(s), and regulated output(s). In Chapter 5,
we addressed these limitations by modifying the Lyapunov-based design in Chapter 4
to used an optimization-based “front-end”. Specifically, the command currents fed to
the inner-loop reference models are generated by a receding-horizon control allocation
(RHCA), which includes in its cost function a metric to promote the generation of
persistently exciting signals. The RHCA considers the output error-zeroing manifold
by including it in the constraints. Additionally, the importance of considering past
data in the generation of persistently exciting signals is discussed and demonstrated
using simulations.
94
6.2
6.2.1
Future Work
Offline Identification of Induction Machine Parameters
Future work specific to offline identification of induction machine parameters
which could be of interest, and potentially improve the performance of the proposed
technique, are as follows:
• Improved inverter models for compensation of additional nonideal effects. While
our use of first-harmonic dead-time compensation partially compensated nonideal inverter behavior, additional efforts to characterize and compensate deadtime effect, as well as other effects such as conduction losses, would improve
the accuracy even further. Additionally, control techniques which require accurate knowledge of the terminal voltages would benefit from such research
efforts. Adaptive parameter estimation and sensorless control techniques are
particularly sensitive to such discrepancies between the actual and command
voltages.
• Investigation into improved algorithms for circular data fits. An issue which
was observed while testing the parameter identification technique on simulation
data was that the sensitivity of the cost function being minimized to variations
in the different free variables (horizontal offset, vertical offset, and radius) is
highly dependent on the distribution of data around the circular locus. The
nature of our application is such that we only get data on the left side of the
circular current locus, and often we are not able to span much of the locus before
we hit the current limits of the machine. As a result, the computed sensitivities
of the horizontal offset and the radius are considerable higher than that of the
vertical offset; meaning that the horizontal offset and radius influence the data
fit more than the vertical offset. While this isn’t a problem for estimating the
magnetic parameters (which are derived from the horizontal offset and radius)
or rotor resistance (which is computed based on the individual locations of locus
points), it did mean that using the estimated center of a fitted circle to compute
the core loss conductance would not yield sufficiently accurate results. While we
found that using the zero-slip data point alone would yield sufficiently accurate
estimates of the core loss conductance, it is possible that introducing weighting
into the cost function for the circular fit could improve the accuracy of the
estimated core loss conductance.
95
6.2.2
Simultaneous Identification and Adaptive Control
Future work specific to simultaneous identification and control, for PMSMs and
in general, which is of interest is as follows:
• A generalized methodology for simultaneous identification and control of overactuated systems. Using the knowledge and intuition gained via case studies,
develop a general simultaneous identification and control methodology for some
class (or classes) of overactuated systems. While the Adaptive Excitation Decoupling approach proposed and explored in Chapter 3 could be promising for
generalization, a closed-loop stability proof remains a challenging result to obtain. Additionally, while we had intuition to guide our selection of excitation
and control inputs for the PMSM, this choice is not as straightforward for more
general plants. The RHCA-SIC approach presented in Chapter 5, inherently
addresses this issue of allocating the excitation (for parameter identification)
and control effort via the optimization problem formulation. It therefore likely,
that the RHCA-SIC approach is the most promising starting point for developing a generalized methodology. Particularly, if an inner-loop adaptive controller
is utilized to ensure closed-loop stability of the plant under control.
• Application to other overactuated plants of interest. In this dissertation, we focused on the application of the proposed simultaneous identification and control
techniques to torque regulation in PMSMs. While these case studies provided
the opportunity to try out ideas on a real system and gain a better understanding of the challenges involved, it would nonetheless be informative to consider other applications as well. For instance, the air-path control problem for
turbocharged diesel engines with variable geometry turbines/compressors [86]
presents a challenging overactuated control problem of practical interest. Simultaneous identification and control could help improve performance over the lifetime of an engine, while also providing more reliable parameter estimates which
could be used for condition monitoring, e.g., indicating mechanical wear in the
turbocharger. Inevitably, the consideration of alternative plants will provide
additional insight that is likely to motivate improvements to the methodology.
• Investigation into the influence of horizon window length and position on the
generation of persistently exciting inputs and parameter identification. In Chapter 5, we used a past-data horizon, Np , which was equal to the prediction horizon, Nf , in our evaluation of the Fisher information matrix. However, this
96
decision was made somewhat arbitrarily. In fact, the Fisher matrix may be
evaluated over any interval of interest, independent of the control allocation
prediction horizon. Intuitively, we expect past data to be more valuable than
future (i.e., predicted) data in our evaluation of the Fisher matrix for the reasons
discussed in Chapter 5. It is therefore of interest, to investigate the influence
that the selection of this time interval, its length and positioning (i.e., how much
past and future data is included), has on the generation of persistently exciting
reference trajectories and parameter identification.
• Experimental verification of the RHCA-SIC approach for PMSMs. To better
understand the challenges of fielding the RHCA-SIC algorithm, experimental
validation is needed. In the simulation results presented in Chapter 5, we had
the luxury of “pausing” to solve the optimization problem at each time step. In
practice, this optimization problem will have to be solved quickly in real-time.
To do so will require an investigation into numerical solution techniques for the
constrained optimization problem at hand.
• Incorporation of voltage constraints in RHCA for PMSMs. While the case study
for PMSMs in Chapter 5 did not include voltage constraints in the problem formulation, it should be straightforward to include them. The voltage constraints
on a voltage-source inverter form a convex set, and including them will ensure that the actuation provided by the inverter is fully utilized. Additionally,
including voltage constraints in the RHCA will lead the controller to automatically perform field weakening at high-speeds when the drive system is voltage
constrained.
97
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