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Description  |
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FIELD OF THE INVENTION
This invention relates generally to permanent-magnet synchronous motors
and, more particularly, to a method and apparatus for estimating the rotor
position and rotor velocity of a permanent-magnet synchronous motor
operating without shaft-mounted motion sensors.
BACKGROUND OF THE INVENTION
Permanent-magnet synchronous motors are brushless motors characterized by
low cost, physical ruggedness, and simple construction. There are
essentially three types of construction for such motors. The
"surface-magnet" type has radially magnetized arc-shaped magnets attached
to the surface of a smooth rotor and is widely available with either
sinusoidal (due to distributed phase windings) or trapezoidal (due to
concentrated phase windings) back-EMF voltage characteristics. The
"interior-magnet" type has alternately poled, radially or
circumferentially magnetized, rectangular magnets embedded in a smooth
rotor, with distributed or concentrated phase windings. The
"hybrid-stepper" type has a single axially magnetized cylindrical rotor
magnet enclosed by a two-piece rotor shell having projecting rotor teeth
(with teeth offset between the two pieces), and the stator has
concentrated phase windings on pairs of projecting poles. In the absence
of magnetic saturation, the surface-magnet and hybrid-stepper types
generally do not exhibit angle-dependent phase inductance, whereas the
interior magnet type generally does exhibit this characteristic.
Regardless of their specific construction features, permanent-magnet
synchronous motors are attractive as servo drives because of their high
power densities. The stator phases of such motors may be electrically
excited to produce a controlled torque on the rotor, the torque being
proportional to the field intensity of the rotor magnet(s) and the
amplitude of the stator phase excitation, thus permitting control of the
rotor motion. When such motors are employed for high performance servo
drive applications which require precise control of rotor motion, the use
of feedback signals representing rotor position and rotor velocity becomes
necessary. The most common method for obtaining sufficiently accurate
feedback signals is to mount a high-resolution magnetic resolver or
optical encoder to the rotor shaft, in order to directly measure the rotor
position with sufficient accuracy, and then to electronically process this
direct rotor position measurement to obtain an indirect measurement of
rotor velocity. There are several disadvantages associated with
shaft-mounted rotor position sensors, including their cost, size, mass,
and potential unreliability.
Another approach for obtaining the feedback signals needed to control
permanent-magnet synchronous motors is to install Hall-effect position
sensing devices inside the stator housing. Such Hall-effect devices detect
only the polarity of the rotor's magnetic field and, hence, provide only a
coarsely quantized measure of rotor position. In general, the quantization
factor (i.e., resolution) associated with Hall-effect devices is
determined by the number of rotor poles and stator phases, and cannot be
improved upon for a given motor. By contrast, shaft-mounted position
sensors have quantization factors that depend only on the precision of the
sensor construction, not on the construction details of the motor to which
it is attached. The coarse quantization of the rotor position signal
obtained from Hall-effect devices also leads to difficulties in the
determination of the corresponding rotor velocity signal. At low velocity,
the rotor passes from one quantization interval to another infrequently
and, hence, the indications of rotor velocity cannot be updated at a
sufficiently high rate to be accurate.
A motor equipped with a shaft-mounted high accuracy position sensor can be
controlled in servo fashion, with the stator phase excitation continuously
modulated substantially in response to the accurately measured values of
rotor position and rotor velocity. Due to the high accuracy of the
feedback signals in such a servo system, it is possible to control the
instantaneous value of rotor torque and thus to achieve precise control of
rotor motion. The motion control goals that can be achieved in such a
servo system include velocity control, in which the rotor is commanded to
regulate to a fixed desired velocity or to track a time-trajectory of
desired velocities, and position control, in which the rotor is commanded
to regulate to a fixed desired position or to track a time-trajectory of
desired positions. Since the phase excitation is determined by the
feedback signals, the accuracy of the motion control critically depends on
the accuracy of the sensor measurements. If Hall-effect devices are used
as the only means of measuring rotor position and rotor velocity, the
quantization of the feedback signals limits the possible control actions
to selection of commutation instants or phase firing angles. Since phase
excitation is not continously modulated using Hall-effect devices, only
the average value (rather than instantaneous value) of rotor torque can be
controlled, and consequently the accuracy of rotor motion is rather
limited.
As disclosed in prior art, permanent-magnet synchronous motors can be used
simultaneously as actuators and sensors of motion. For high performance
servo drive applications, this combined actuator sensor mode of operation
requires, at minimum, the highly accurate estimation of rotor position
from purely electrical measurements taken at the stator terminals (with
highly accurate estimation of rotor velocity achieved by processing the
rotor position estimates in the traditional way). Alternatively, rotor
position and rotor velocity may be estimated simultaneously from stator
terminal measurements, instead of using the traditional sequential
processing. Measurable stator terminal signals are limited to the phase
currents (the currents flowing through the phase windings), and either the
applied phase voltages (in case the phase is receiving excitation from the
power source and hence has a nonzero current flowing through its winding)
or the open-circuit phase voltages (in case the phase is unexcited and
hence is disconnected from the power source with no current flowing
through its winding).
It is well known that if the rotor of a permanent-magnet synchronous motor
is rotating with significant velocity, then the rotating magnetic field
set up by the rotor magnet(s) will induce an electromotive force, or
back-EMF voltage, on the stator phase windings. The back-EMF voltage is
dependent upon both rotor position and velocity and, when it is present,
it influences the stator phase dynamics. Consequently, the back-EMF
voltage, when it is present, can potentially play a useful role in rotor
motion estimation schemes. However, the back-EMF voltage is periodic with
respect to rotor position (with an integer number of cycles per revolution
determined by the construction of the rotor) and, more significantly, is
linearly proportional to the rotor velocity. Hence, the back-EMF voltage
is not present on any stator phase if the rotor is not rotating. If the
rotor is rotating but with negligible velocity, then the back-EMF voltage
will be contaminated by noise. The back-EMF voltage therefore possesses no
direct utility for the estimation of rotor position when the rotor is
completely or practically motionless. Even when the back-EMF voltage is
present, it cannot be directly measured at the stator terminals of a given
stator phase unless this same stator phase is unexcited (open-circuited
with no current flowing through it). Due to the periodicity of the
back-EMF voltage with respect to rotor position, schemes using this signal
cannot estimate the rotor position in an absolute sense (at least without
including some heuristic procedures), but instead can only estimate rotor
position relative to the electrical cycle.
Subject to the limitations discussed above, the back-EMF voltage has been
used in prior art to estimate rotor motion (i.e., rotor position and rotor
velocity). Other rotor motion estimation schemes, such as those relying on
naturally present variable phase inductance or saturation-induced variable
phase inductance, are not suitable for all types of permanent-magnet
synchronous motors. Most of the existing rotor motion estimation schemes
based on the back-EMF voltage do not have the objective of estimating
rotor position with accuracies typical of traditional shaft mounted
sensors, such as magnetic resolvers or optical encoders. Instead, the goal
of most existing rotor motion estimation schemes is simply to eliminate
the need for Hall-effect position sensing devices mounted inside the
stator housing. Since quantization can be tolerated in this case, these
rotor motion estimation schemes seek to detect events, measurable at the
stator terminals, which are expected to occur once each step within the
commutation sequence. For example, a detectable event directly related to
the back-EMF voltage is the zero-crossing of an open-circuit phase
voltage. All so-called event detection methods for estimating rotor motion
have the disadvantage of coarse feedback quantization, as well as the
corresponding disadvantage of limited motion control accuracy. Thus, there
is a general need for a more precise method of estimating rotor motion,
not limited by the quantization effects of event detection schemes.
Prior art also discloses rotor motion estimation schemes with the potential
to serve as replacements for traditional high accuracy shaft mounted
sensors, subject to certain limitations. For example, U.S. Pat. No.
5,134,349 to Kruse discloses a sensorless controller for permanent-magnet
synchronous motors which continuously modulates the phase excitation to
achieve instantaneous torque control, with the feedback signals obtained
by processing the back-EMF voltage in a continuous fashion. However, this
technique is disclosed only for motors with two stator phases (or motors
with three interconnected stator phases), with sinusoidal back-EMF voltage
characteristics, and with sensing coils mounted inside the stator. The
most appropriate motors for many applications have more than two or three
phases, and have non-sinusoidal back-EMF voltage characteristics.
Moreover, the use of internal sensing coils is a disadvantage because such
coils add to the cost and size of the motor, reduce the power density of
the motor, and decrease reliability due to the additional wiring
connections between the motor and the control electronics. Therefore, a
need still exists for a rotor motion estimation method for motors with any
number of stator phases, with any periodic back-EMF voltage shape, and
without sensing coils mounted inside the stator.
A more elaborate method for potentially replacing high accuracy shaft
mounted sensors is disclosed in the article "Real-Time Observer-Based
(Adaptive) Control of a Permanent-Magnet Synchronous Motor Without
Mechanical Sensors," by R. B. Sepe and J. H. Lang, 1991. A sensorless
controller is described which continuously modulates the phase excitation
to achieve instantaneous torque control, with feedback signals obtained by
simulating (solving forward in time from assumed initial conditions) a
mathematical model of the motor and its load, augmented with a correction
term used to compensate for errors between the values of stator current
predicted by the simulated model and the measured values of stator
current. However, this technique is disclosed only for motors with two
stator phases (or motors with three interconnected stator phases), with
sinusoidal back-EMF voltage characteristics, and with a known model for
the rotor load. The disadvantages associated with the required number of
phases and required back-EMF voltage shape have already been set forth.
Furthermore, rotor load parameters such as friction coefficients, load
torque, and load inertia often are difficult or impossible to measure or
approximate accurately, and consequently this method of estimation is
adversely affected by such unavoidable parametric errors. There still
exists a need for a rotor motion estimation method for motors with any
number of stator phases, with any periodic back-EMF voltage shape, and
which does not need explicit knowledge of rotor load parameters.
Prior art rotor motion estimation methods fail to function as desired at
low velocities and at zero velocity. In the method of Kruse, for example,
the loss of the back-EMF voltage at low and zero velocities requires an
abrupt transition to a hold-mode wherein large currents are applied to the
phases in order to hold the rotor in place. Also, in the method of Sepe
and Lang, the error between the estimated and actual rotor positions does
not converge to zero at a reasonable rate at low velocities, and may
actually diverge at low velocities for a motor wherein the phase
inductance is independent of rotor position. Selection of gains for the
correction term is not systematic. Hence, there still exists a need for a
technique which is capable of estimating rotor position at standstill. It
is to the provision of this need and the additional needs identified above
that the present invention is primarily directed.
SUMMARY OF THE INVENTION
Briefly described, the present invention is a method and apparatus for
simultaneously estimating instantaneous rotor position and instantaneous
rotor velocity for a multiphase permanent-magnet synchronous motor. The
method is implemented by measuring the phase currents and (if necessary)
the phase voltages, processing these measurements to provide an
approximation of the phase back-EMF voltages, and producing an indication
of instantaneous rotor position and instantaneous rotor velocity from the
back-EMF voltage approximations. The final step of the method, in which
position and velocity are ascertained from back-EMF approximations, may be
carried out on the basis of a single point in time formulation or a
multiple points in time formulation.
The apparatus includes means for measuring the phase currents and phase
voltages, means for processing the phase currents and phase voltages to
provide an approximation of the phase back-EMF voltages, and means for
producing an indication of instantaneous rotor position and instantaneous
rotor velocity from the back-EMF voltage approximations. The apparatus
also includes means for determining rotor position and rotor velocity from
the back-EMF approximations based on either a single point in time
formulation or a multiple points in time formulation.
The apparatus may preferably be embodied in a single microprocessor. In
addition, the means for estimating the instantaneous rotor position and
instantaneous rotor velocity may be combined with a controller and
inverter to form a complete closed-loop control system for a
permanent-magnet synchronous motor.
Accordingly it is an object of the present invention to provide a rotor
motion estimation scheme for permanent-magnet synchronous motors which
achieves high resolution estimation of rotor motion regardless of the
number of stator phases or the shape of the back-EMF voltage.
Another object of the present invention is to provide a rotor motion
estimation scheme which is highly accurate and reliable at low velocities
and at zero velocity.
Another object of the present invention is to provide a rotor motion
estimation scheme which is insensitive to measurement noise present in the
phase current and phase voltage measurements.
It is another object of the present invention to provide a motion control
scheme which uses estimates of rotor position and rotor velocity, instead
of motion measurements obtained from shaft-mounted sensors or Hall-effect
sensors, for the control of rotor motion.
It is yet another object of the present invention to provide sensorless
control schemes for the control of either rotor position or rotor
velocity.
These and other objects, features and advantages of the present invention
will become more apparent upon reading the following description in
conjunction with the accompanying drawings.
BRIEF DESCRIPTION OF THE DRAWINGS
FIG. 1 is a schematic block diagram of a system for implementing the
methods of the present invention;
FIG. 2 is a schematic illustration of one embodiment of the system of FIG.
1 shown in conjunction with a device for determining the performance of
the system; and
FIG. 3 is a schematic illustration of another embodiment of the system of
FIG. 1.
DETAILED DESCRIPTION OF THE INVENTION
Referring now to FIGS. 1-3 wherein like reference numerals represent like
parts, a complete closed-loop control system 10 including the motion
estimation method and apparatus of the present invention is shown in
conjunction with a permanent-magnet synchronous motor 12. Motor 12 is not
equipped with a shaft-mounted position sensor nor with Hall-effect
position sensors. The phase windings of motor 12 are electrically
connected to an inverter 32. A current sensor 24 measures phase currents
22 which are sensed as phase current measurements 44 by a microprocessor
26 via analog-to-digital converter 34. Microprocessor 26 estimates rotor
position 20 and rotor velocity 18 and controls rotor position 20 and rotor
velocity 18 based on the motion estimates. Both features of estimating and
controlling rotor position 20 and rotor velocity 18 are embedded within
the hardware and software of microprocessor 26. Microprocessor 26 receives
input command 36 via analog-to-digital converter 28. Microprocessor 26 is
capable of estimation-based control of either rotor position or rotor
velocity, depending on the type of command presented at input command 36.
Microprocessor 26 also commands inverter 32 via digital-to-analog
converter 30 and inverter command 46 to apply specified phase voltages 38
to motor 12.
Referring now more specifically to FIG. 2, an embodiment of the present
invention is shown in conjunction with a device useful for determining the
performance of motion estimation and control. Motor 12 is equipped with
rotor motion sensor 40 mounted on rotor shaft 42. Motion sensor 40
provides microprocessor 26 with direct measurements of rotor position 20
and rotor velocity 18 for comparison with estimates of rotor position 20
and rotor velocity 18. Current sensor 24 provides phase current
measurement 44 to microprocessor 26 via analog-to-digital converter 28.
Microprocessor 26 estimates rotor motion and commands inverter 32 via
inverter command 46 obtained from digital-to-analog converter 30, and
inverter 32 applies phase voltages 38 to the phase windings.
Referring now more specifically to FIG. 3, an alternate embodiment of the
present invention is shown comprising microprocessor 26, inverter 32,
current sensor 24, and motor 12. In this embodiment, analog or digital
position or velocity commands may be fed to microprocessor 26 at input
command 36. The phase current measurement 44 is used by microprocessor 26
to estimate the rotor motion, and microprocessor 26 determines inverter
command 46 for purposes of controlling the rotor motion, resulting in the
application of phase voltage 38.
Having above described the general sensorless control scheme, including
both the rotor motion estimation scheme and the feedback controller which
uses the rotor motion estimates to guide the rotor motion, attention is
now turned to more detailed descriptions of the present invention. The
present invention estimates rotor motion by considering the back-EMF
voltage, either at a single point in time or at multiple points in time,
and these two cases are described separately.
Prior to any discussion of how back-EMF voltage can be used to estimate
rotor motion, it is first necessary to consider techniques for extracting
the back-EMF voltage from the stator terminals of a permanent-magnet
synchronous motor. If all stator phases of the motor are open-circuited,
then the voltage directly measured at each of the stator terminals is
equal to the back-EMF voltage of the corresponding stator phase. However,
motion control applications require that at least one of the stator phases
be driven from an excitation source at any given time. Moreover, the most
efficient mode of operation, which achieves the highest possible power
density for a given motor, requires that all stator phases be connected to
the excitation source at all times. In this preferred operating mode, no
open-circuited stator phases are available, so direct measurement of the
back-EMF voltage is impossible.
One aspect of the present invention is to avoid the need for open-circuited
stator phases altogether, by reconstructing (instead of directly
measuring) the back-EMF voltage from stator terminal measurements known to
be available under all circumstances. The stator currents (the currents
flowing through the stator phase windings) are always available for
measurement, and the stator voltages (the voltages across the stator phase
windings) are always either known, because they are commanded by a
controller, or else can always be measured. With knowledge of stator
currents and stator voltages, the back-EMF voltage v.sub.emf can be
reconstructed according to
##EQU1##
where v denotes the vector of stator voltages, i denotes the vector of
stator currents, R denotes the diagonal matrix of phase winding
resistances, L denotes the positive-definite symmetric matrix of phase
winding inductances, and t denotes time. Note that equation (1) holds true
whether a stator phase is energized or open-circuited, the latter case
implying the substitution of i=0 into the last two terms. Provided that
the motor is capable of producing torque at every rotor position, the
signal v.sub.emf is zero if and only if the rotor velocity is zero. Since
the time derivative of stator current is not directly measurable, use of
equation (1) generally requires some approximations, even if R and L are
known precisely.
A point to consider when reconstructing the back-EMF voltage from equation
(1) is that the motor controller typically commands either the stator
voltages or the stator currents. If the controller is a digital
controller, then the voltage or current command signals will typically be
piecewise constant signals, i.e. signals which are constant over each
sampling interval and which discontinuously change their values at the
sampling instants. Regardless of which command mode is used by the
controller, certain simplifying approximations to equation (1) may be made
to account for the abovementioned features, as will be clear to those
skilled in the art. Another approximation which is often acceptable in
practice is to neglect the effect of inductance, by assuming that
L.apprxeq.0. Such an approximation is convenient, since it removes the
need to approximate the time derivative of stator current. When the
influence of inductance is deemed to be critical, approximation schemes
for time derivatives may be used. One such approximation would be to
divide the difference between two consecutive values of stator current by
the difference between the corresponding two time instants. Regardless of
the specific details behind the simplifications and approximations
introduced, equation (1) forms the basis for reconstructing the back-EMF
voltage without requiring any stator phases to be open-circuited.
Perhaps the greatest difficulty faced in the implementation of rotor motion
estimation schemes relates to measurement noise, parametric errors, etc.
The influence of these undesirable effects may lead to incorrect estimates
of rotor position and rotor velocity. Moreover, if the motor has more than
two stator phases, then the number of back-EMF voltage constraints at any
single point in time will always be greater than the number of unknowns
(rotor position and rotor velocity). When considering a history of
back-EMF voltage at multiple points in time, the number of constraints on
possible rotor positions and rotor velocities grows. The back-EMF voltage
constraints thus generally form an overdetermined system of algebraic
equations. Overdetermined systems of equations are usually inconsistent,
meaning in this case that there do not exist values of rotor position and
rotor velocity that solve each of the equations simultaneously. Hence, one
way to proceed in this case would be to select only two of the back-EMF
voltage constraints and to solve them uniquely, if possible, for rotor
position and rotor velocity. The drawback of such an approach is that the
two selected constraint equations may be the ones most affected by noise
and other error sources, and the resulting estimates of rotor position and
rotor velocity will contain the worst possible errors. Uniquely solving
for the rotor variables from the minimum number of back-EMF voltage
constraints is undesirable, because the estimates are extremely sensitive
to noise and other error sources.
A better, alternate method for estimation can be employed, which is
generally described as follows. Given a nonlinear system of m equations
and n unknowns with m>n, say f(x)=y where x is the vector of unknowns, it
is highly unlikely that an x can be found for which f(x) equals y.
Instead, it makes sense to look for a vector x for which f(x) is closest
to y. For each x, there is an associated residual r(x)=f(x)-y. The
distance between f(x) and y is given by the Euclidean norm of this
residual, namely .parallel.r(x).parallel.. The nonlinear least squares
problem is to find a vector x for which .parallel.r(x).parallel. will be a
minimum. The problem takes its name from the fact that minimizing
.parallel.r(x).parallel. is equivalent to minimizing
.parallel.r(x).parallel..sup.2, which is equal to the sum of the squares
of the m residual components. A vector x that minimizes the norm, or the
norm squared, of the residual is said to be a least squares solution to
the system of nonlinear equations f(x)=y. For the special case where the
equations depend on x in a linear way, i.e. f(x)=Ax for some constant
matrix A, then the nonlinear least squares problem reduces to a linear
least squares problem which is particularly simple to solve using standard
techniques from linear algebra. Solution of a nonlinear least squares
problem is much more challenging, typically requiring iterative methods
such as the Gauss-Newton and Levenberg-Marquardt algorithms. The existence
of a unique solution to a linear least squares problem is guaranteed if
the matrix A has full column rank. The existence of a locally unique
solution to a nonlinear least squares problem, near some current estimate
x.sub.c of the problem's true solution x.sub.*, is guaranteed if the
Jacobian matrix of
##EQU2##
has full column rank. If f(x) is a one-to-one function on its domain, then
this Jacobian rank condition guarantees a unique global solution, on the
domain of f(x), to the nonlinear least squares problem.
The method according to one form of the present invention is essentially
the application of the least squares data fitting technique, as described
above, to the problem of rotor motion estimation from reconstructed
back-EMF voltage. Consider first the case in which the back-EMF voltage at
just a single point in time is to be used for estimation. In this case
there are two unknowns, rotor position .theta. and rotor velocity .omega.,
which play the role of x in the above general description of least
squares. The function which depends on these unknowns is the back-EMF
voltage function, which will be denoted by H.sub.1, with the subscript I
intended to indicate the single point in time formulation. The function
H.sub.1 may be predetermined and completely characterized for any given
motor, using standard measurement methods. Playing the role of f(x) in the
above general description of least squares, the function is given by
H.sub.1 (.theta.,.omega.)=.omega.K(.theta.) (2)
where K(.theta.) is a periodic function. Although H.sub.1 depends on
.omega. in a linear way, it depends on .music-flat. in a nonlinear way. It
is desired to fit the function H.sub.1, at each time t, to the
reconstructed back-EMF voltage data, obtained by approximating the signal
v.sub.emf defined in equation (1). This approximation step yields from
equation (1) an approximate back EMF voltage
e.sub.1 (t)=v.sub.emf (t) (3)
where v.sub.emf denotes any approximation of v.sub.emf and the subscript 1
indicates the single point in time formulation. The approximate back-EMF
voltage e.sub.1 plays the role of y in the above general description of
least squares. The available data e.sub.1 and the function H.sub.1 ideally
would be consistent, such that the set of constraint equations
H.sub.1 (.theta.(t),.omega.(t))=e.sub.1 (t) (4)
could be satisfied by a unique pair of unknowns (.theta.,.omega.).
Naturally though, equation (4) is expected to be inconsistent in practice,
due to errors introduced in the characterization of H.sub.1, errors in the
approximation of v.sub.emf, and errors in the measurement of stator
voltage and stator current. Moreover, if the motor has more than two
phases, then there may be no pair of unknowns capable of simultaneously
satisfying all constraint equations, even if the abovementioned error
sources are practically not present. Hence, a nonlinear least squares data
fitting problem is set up, and the estimates of rotor position and rotor
velocity at time t are taken to be
##EQU3##
From the above general description of least squares, it is known that the
Jacobian matrix of H.sub.1 is useful in ascertaining the existence of a
unique local minimizer. It is easy to verify from equation (2) that the
Jacobian of H.sub.1 has full rank (equal to 2) if .omega..noteq.0, for
typical K(.theta.) characteristics. It follows that estimation of rotor
position and rotor velocity when .omega..noteq.0 is feasible, using the
single point in time formulation. However, regardless of K(.theta.), the
Jacobian of H.sub.1 does not have full rank if .omega.=0. Hence,
estimation of rotor position at zero velocity is not feasible, because the
rotor position is not (locally) uniquely determined when .omega.=0, using
the single point in time formulation.
Since the method described above is not feasible if .omega.=0, it is
natural to augment the least squares data fitting problem to include
back-EMF voltage values at two or more points in time. The difficulty at
zero velocity is due to a loss of independent constraints. Therefore, by
adding additional constraints by including more of the available data,
there will be a greater opportunity for a (locally) unique least squares
solution to exist. Consider specifically the case in which the back-EMF
voltage at two points in time is to be used for estimation. The two
unknowns, as before, are the rotor position .theta. and the rotor velocity
.omega., so the interpretation of x from the general least squares problem
is unchanged. The back-EMF voltage function of interest here will be
denoted by H.sub.2, with the subscript 2 intended to indicate the two
points in time formulation. The function H.sub.2 may be predetermined and
completely characterized for any given motor and load, using measurement
methods known to those skilled in the art, provided however that the rotor
load, once characterized, is not subject to change. Playing the role of
f(x) for the general least squares problem, the function is given by
##EQU4##
where F.sub..theta. and F.sub..omega. model the mechanical dynamics of the
rotor and load, as defined by
##EQU5##
The time arguments appearing in equation (7) indicate a discrete sampling
process, a common form of sampling being periodic sampling with period T
(in which case the nth sampling instant would be t.sub.n =nT). The
mechanical model of the rotor and load generally depends on parameters
such as rotor and load inertias, viscous and Coulomb friction
coefficients, and load torque. Note that H.sub.2 has twice as many terms
as H.sub.1, that both components of H.sub.2 represent back-EMF voltages,
and that the upper and lower components differ by one sampling instant.
Perhaps the most striking feature of H.sub.2 in comparison with H.sub.1 is
that H.sub.2 explicitly depends on v, the stator phase voltages. The
general idea is to fit the function H.sub.2, at each time t.sub.n-1, to
the reconstructed back-EMF voltage data, obtained by approximating the
signal v.sub.emf defined in equation (1) at times t.sub.n-1 and t.sub.n.
The two approximate back-EMF voltages are grouped according to
##EQU6##
where v.sub.emf denotes any approximation of v.sub.emf and the subscript 2
indicates the two points in time formulation. The approximate back-EMF
voltages of e.sub.2 play the role of y from the general least squares
problem. If the characterization of H.sub.2, the approximation of
v.sub.emf, and the measurements of stator current and stator voltage were
error free, then the set of constraint equations
H.sub.2 (.theta.(t.sub.n-1),.omega.(t.sub.n-1), v(t.sub.n-1))=e.sub.2
(t.sub.n) (9)
would be simultaneously satisfied by a (locally) unique pair of unknowns
(.theta.,.omega.). Since equation (9) is expected to be inconsistent, the
estimates of rotor position and rotor velocity at time t.sub.n-1 are taken
from the nonlinear least squares problem
##EQU7##
and the estimate values at the present time t.sub.n are determined by
propagating the least squares estimates from equation (10) through the
mechanical model of the rotor and load, i.e.
##EQU8##
Due to the fact that H.sub.2 depends explicitly on the stator phase
voltage v, the Jacobian matrix of H.sub.2 can have full rank (equal to 2),
even when .omega.=0. This desirable situation requires appropriate choices
of stator excitation v. With appropriate choices of v, the full rank
Jacobian of H.sub.2 implies the existence of a unique local minimizer. It
follows that estimation of rotor position at zero velocity is achieved
using the two points in time formulation.
The differences between the two formulations disclosed above can be
illustrated using physical insight. The methodology based on the back-EMF
voltage at a single point in time fails for the stationary rotor case
because, even though a measurement of v.sub.emf (t)=0 directly implies
.omega.(t)=0, there is simply no information in equation (4) from which to
infer a value for .theta.(t). If the rotor is stationary, then equation
(4) will hold true regardless of the assumed value of rotor position, and
hence there is no mechanism for reducing any rotor position estimation
error which may be present. It should also be clear that the choice of
stator excitation has no influence on this issue. On the other hand,
consider a reconstruction of the back-EMF voltage at two points in time,
say t.sub.1 and t.sub.2, which for a stationary rotor would yield
v.sub.emf (t.sub.1)=v.sub.emf (t.sub.2)=0. Assuming that sampling instants
t.sub.1 and t.sub.2 are spaced sufficiently close together, several
logical conclusions may be made. Since the two back-EMF voltages are both
equal to zero, it follows that .omega.(t.sub.1)=.omega.(t.sub.2)=0, that
no acceleration or deceleration of the rotor has occured over the sampling
interval, and thus that the torque produced on the rotor by the stator
excitation has exactly balanced the load torque over the sampling
interval. Hence, it follows that
.theta.(t.sub.1)=.theta.(t.sub.2)=.theta..sub.* where .theta..sub.* is any
rotor position at which the known stator excitation would produce a
torque, according to a known torque model, which would balance the load
torque.
Determination of .theta..sub.* in this fashion clearly requires a knowledge
of load torque. This requirement can be viewed either as an advantage, due
to the additional constraints on possible rotor positions that are
introduced, or as a disadvantage, due to the fact that in some
applications the load torque cannot be accurately modeled. The important
point is that, with a load torque model, it is possible to limit possible
rotor positions to those for which a torque balance is achieved. For the
stationary rotor case, no such limitation on possible rotor positions is
available when considering only a single point in time. In contrast, when
two or more reconstructions of the v.sub.emf signal indicate that the
rotor is stationary, any potential rotor position estimation error present
will be eliminated by requiring that the torque balance equation hold
true.
Note that this concept places constraints on the type of stator excitation
which must be present at a zero velocity steady state. For example, if the
rotor is unloaded, then the torque balance will be achieved by ensuring
that zero torque is produced on the rotor. The catch is that even an
absence of stator excitation would suffice in this case to achieve the
torque balance. However, if no stator excitation were applied, then the
torque produced on the rotor would be zero regardless of the rotor
position, so there would be no hope to isolate the true rotor position. To
overcome this difficulty, it is necessary to intentionally apply a nonzero
stator excitation, even though the excitation is not actually needed to
maintain the torque balance, so that rotor position estimation errors will
not persist.
In the zero load torque case, it is possible to choose the stator
excitation on the basis of the commanded rotor position or the estimated
rotor position. Using the commanded rotor position, it is known that zero
torque is produced by the chosen stator excitation only when the actual
rotor position matches the commanded rotor position (assuming zero load
torque). Hence, the possible rotor positions are limited and must be equal
to the commanded rotor position, relative to the electrical cycle. If a
non-persisting disturbance occurs, resulting in a change in the rotor
position (but with zero load torque assumed), then the estimated rotor
position will become correct again, even though the estimate does not
change value, due to the fact that the stator excitation will return the
rotor to the commanded rotor position once the disturbance has been
removed. Using the estimated rotor position, the stator excitation is
selected in response to the rotor position estimate. If the estimate is
correct, then the choice of stator excitation will indeed produce zero
torque, the rotor will stay stationary, the next reconstruction of
back-EMF voltage will indicate zero rotor velocity, and thus the same
stator excitation will be applied again as the entire process repeats. If
the estimate is not correct, then necessarily the stator excitation will
produce a nonzero torque, resulting in a motion which, by proper selection
of the excitation polarities, will be in the direction of the originally
estimated rotor position, such that the estimation process is
self-correcting.
The present invention may be implemented on the basis of either on-line or
off-line solution of the least-squares problem. The on-line approach
involves real-time computation of the least-squares solution on a
sufficiently fast microprocessor. By presolving the least-squares problem
for all possible values of back-EMF and storing the solutions in a memory
chip, the need for a microprocessor is eliminated in the off-line
approach.
From the above, it will be apparent that a new and improved method and
apparatus for accurately estimating the instantaneous rotor position and
instantaneous rotor velocity, in simultaneous fashion, from measurements
available at the stator terminals of a permanent-magnet synchronous motor,
has been developed. The present invention applies to a large class of
motors, including those with any number of phases and those with any
periodic back-EMF voltage shape. The present invention does not require
the presence of Hall-effect position sensors, sensing coils inside the
stator, or shaft-mounted motion sensors. The accuracy of the present
invention is not affected by the number of poles or the number of phases.
Although a specific embodiment has been described and depicted herein, it
will be appreciated by those skilled in the art that various
modifications, substitutions, deletions and additions may be made, without
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