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IEEE TRANSACTIONS ON CONTROL SYSTEMS TECHNOLOGY, VOL. 11, NO. 1, JANUARY 2003
Brief Papers_______________________________________________________________________________
Robust Speed and Torque Estimation in Electrical Drives
by Second-Order Sliding Modes
Giorgio Bartolini, Alfonso Damiano, Gianluca Gatto, Ignazio Marongiu, Alessandro Pisano, and Elio Usai
Abstract—This paper presents the synthesis and practical implementation of a robust digital differentiator that provides the
first and second derivative of a sampled smooth signal. The robustness of the proposed digital device, based on second-order slidingmodes (2-SMs), is analyzed with respect to measurement errors.
Experimental results on an induction motor (IM) drive show that
fast and accurate estimates of speed and torque can be obtained
in several operating conditions by double differentiation of the encoder position measurement.
Index Terms—Electrical drives, induction motor (IM), nonlinear
estimation, numerical differentiators, second-order sliding mode
(2-SM).
I. INTRODUCTION
IGH-PERFORMANCE control of induction motor (IM)
drives requires the knowledge of rotor flux, electromagnetic torque, and shaft speed [23]. Because of the difficulty
to directly measure these quantities the use of nonlinear observers [16], [18], especially those based on high-gain, adaptive, or sliding-mode control [22], [20], [19], [9] appear to be a
convenient choice when uncertainties and disturbances must be
faced, and is attracting a number of research efforts [15], [12].
In this paper, we propose the substitution of the speed and
torque measurement (or estimation) devices by a real-time numerical differentiator that processes the shaft position signal.
DC or brushless tachometers, resolvers, and encoders are
the usual devices for measuring shaft position/speed of motor
drives. In particular, digital incremental encoders are widely
used due to their low cost and to the digital nature of the output
signal. However, in the low-speed region they suffer from some
drawbacks that may cause a speed ripple with unpredictable
frequency and magnitude.
The simplest approach to speed estimation using sampled
position measurements provided by an encoder is the classical
backward-difference method (BDM), where the number of
pulses generated by the encoder during each sampling interval
is divided by the sampling period. Near the zero speed a little
H
Manuscript received February 2, 1999; revised November 28, 2000. Manuscript received in final form September 3, 2002. Recommended by Associate
Editor S. Bortoff.
The authors are with the Department of Electrical and Electronic Engineering,
University of Cagliari, 09123 Cagliari, Italy (e-mail: giob@diee.unica.it;
alfio@diee.unica.it;
gatto@diee.unica.it;
marongiu@diee.unica.it;
pisano@diee.unica.it; eusai@diee.unica.it).
Digital Object Identifier 10.1109/TCST.2002.806441
increment of pulses is detected at each sampling instant, and
BDM may experience unacceptable errors that can largely
overcome 100%, the higher the sampling rate the larger the
error. Increasing the encoder resolution improves the accuracy
but does not remove completely the low-speed error, and
suitable techniques have been proposed to face the difficulties
associated to the low-speed measurement [17], [11].
In [11], an adaptive speed/torque observer for ac drives was
proposed which on-line identifies the mechanical inertia. Near
the zero velocity, the observed torque gives the speed estimation
scheme the lacking information.
The idea of using observers as “aiding sources” when a direct
speed measurement is not effective (e.g., at low speed) is certainly interesting, but at the same time it motivates a research
activity aimed to improve the basic differentiation algorithms,
and this is the scope of this paper. Moreover, as many motor
drives are controlled by means of a DSP, the discrete-time implementation of real-time differentiators must be considered.
In [6] it has been shown that the bilinear discrete implementation of the continuous-time high-gain observer encompasses
other classes of numerical differentiators: the classical BDM,
the inverse-integrator model [1], and that based on spline interpolation [7]. It was claimed in [6] that the spline interpolation
method should be preferred in the presence of significant noise.
Sliding-mode differentiators (SMDs) [24], [14], [2] offer an
interesting tradeoff between accuracy and noise-immunity. A
systematic comparison among existing numerical differentiators is out from the scope of this paper. We only analyze the possibility of using an SMD based on the suboptimal second-order
sliding mode (2-SM) control algorithm [5], [2] to estimate the
shaft speed and acceleration on the basis of the sampled encoder
measurements.
The proposed device requires a very small computational effort, and its dynamic features can be defined by means of few
constant tuning parameters. In particular, it appears well suited
to be used at low speed, since the associated error is mainly located at the high frequencies.
This paper is organized as follows: In Section II, the 2-SM
approach to real-time differentiation is detailed. In Section III,
the proposed differentiator scheme is completed in order to provide also an estimate of the second derivative. In Section IV, the
overall scheme is summarized and some implementation issues
are discussed. Finally, some experimental results are given in
Section V.
1063-6536/03$17.00 © 2003 IEEE
IEEE TRANSACTIONS ON CONTROL SYSTEMS TECHNOLOGY, VOL. 11, NO. 1, JANUARY 2003
85
provided that the control amplitude
satisfies the relationship
(7)
are suitable positive constants.
where , , , and
Proof: See the Appendix, within which the procedure to
is described.
compute the constants , , , and
If the sampled input sequence
is not affected by measurement errors, Proposition 1 ensures that after a finite transient
Fig. 1. Differentiator scheme.
(8)
II. APPROXIMATE DIGITAL DIFFERENTIATION VIA 2-SMs
The 2-SM methodology can be applied to design a digital
device that gives an estimate of the first derivative of a sampled
signal having bounded second derivative [2]. The structure of
the proposed differentiator consists of a double integrator driven
by a digital variable-structure controller (VSC) (see Fig. 1).
The design objective is to constrain the outputs of the integraand
, to track the input signal
and its derivators,
is the piecewise-contive, respectively. The driving signal
denote the
stant output of the digital VSC. Let
sampled value of a generic signal
at the sampling instant
,
, being the sampling period. From
the scheme in Fig. 1, it results
(1)
, and put
. The control
Define
and
, with
task is fulfilled by steering to zero
unmeasurable. The associated dynamics is represented by
(2)
Provided that a positive constant
is known such that
(3)
a number of 2-SM control algorithms [5] can be applied to stabilize system (2). The theoretical properties, and the actual performance, are comparable from almost all points of view. The
“suboptimal” 2-SM control algorithm will be considered in the
present treatment.
The following proposition is proved.
Proposition 1: Given (2)–(3), then the application of the digital control signal
(4)
where
is evaluated as follows:
gives an estimate of the derivative of the input signal
that is
affected by an
high-frequency chattering error component.
Note that due to the piece-wise constant form of the differthe overall scheme can be implemented
entiator control
without any discretization error by means of the following
simple digital algorithm:
(9)
The effect of unknown-but-bounded (UBB) measurement noise
with maximum amplitude affecting the input sequence
deserves to be investigated. We shall demonstrate that the proposed differentiator is regular with respect to the presence of
noise, i.e., the differentiation error is bounded and tends to zero
if both and tend to zero.
If the digital peak-detector (5) receives as input a noisy sequence it may detect fictitious singular points with a potentially
destabilizing effect. In order to avoid this drawback, one can
as follows:
prefilter the sequence
set
set
if
otherwise.
(10)
The nonlinear filter (10) performs a piecewise constant approx. If the peak-detector (5) is
imation of the input sequence
applied to the filtered sequence
, the estimated singular
will differ from the actual ones by an error that depoints
pends not only on the sampling period [as in (21)] but also on
the noise magnitude . More specifically, if the presence of an
UBB measurement noise is taken into account, and the prefilter
(10) is used, then (21) turns out to be modified as follows:
(11)
if
If the proof of Proposition 1 is rewritten using (11) in place of
(21), it yields a boundary layer of size
otherwise
(5)
guarantees the finite-time attainment of the following conditions:
(6)
(12)
which means that the “noise-free” differentiation accuracy (8)
turns out to be worsened, due to the measurement noise, in accordance with
(13)
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IEEE TRANSACTIONS ON CONTROL SYSTEMS TECHNOLOGY, VOL. 11, NO. 1, JANUARY 2003
where
and
are proper positive constants. The evaluation
and
follows the same procedure as in the proof of
of
Proposition 1 but is slightly more involved since system (26) is
affected by , and the solution to the numerical optimization
problem for evaluating and requires the actual noise magnitude to be prespecified. Note that in the actual case coin, where is the number
cides with the encoder resolution
of ppr (pulses per revolution) of the considered device.
These robustness issues, which are a common feature of
sliding differentiators based on 2-SMs, have been thoroughly
analyzed in [14] making reference to a family of differentiators
based on a different 2-SM controller, the so-called “supertwisting” algorithm.
III. ESTIMATING THE SECOND DERIVATIVE
Fig. 2.
It is well known that during a sliding mode behavior the
so-called “equivalent control” (which is the nonswitching
control action that would maintain the system exactly on the
actual sliding manifold) can be estimated by linear low-pass
filtering of the discontinuous switching control [22].
The definition of the equivalent control has been recently generalized to the 2-SM setting [4], and it was also shown that a
proper choice of the filter time-constant can minimize the difference between the equivalent control and its estimate.
As far as the differentiator system (2) is concerned, the equivis the second derivative of
, i.e.,
alent control
(14)
Under the further assumption that the equivalent control is globally Lipschitz, i.e.,
(15)
it has been shown in [4] that after a finite transient, the signal
defined by
(16)
is an
approximation of
, i.e.,
(17)
and
are proper constants, depending on
and
where
, which can be computed according to the formulas given in
[4, Th. 2].
is piecewise constant, filter (16) can be
As the control
exactly discretized as follows:
(18)
The effect of UBB measurement noise traduces in worsening
the accuracy (17). In particular, if one rewrites the proof of Theorem 2 in [4] with (12) used instead of (6), it can be computed
, which depends on
and
, such that
a positive constant
the accuracy of the second derivative estimate is upperbounded
as follows:
(19)
Overall scheme for estimating speed and acceleration.
IV. VARIABLE STRUCTURE SYSTEM (VSS)-BASED ROTOR
SPEED AND ACCELERATION ESTIMATION
The encoder signal consists of two square-waves whose frequency and reciprocal phase displacement indicates the speed
magnitude and the direction of rotation, respectively. An highfrequency counter is properly updated according to the rising
and/or falling edges of the encoder signals.
Near zero speed the number of pulses detected during each
sampling period is close to the unity, and even an error of 1
pulse can lead to unacceptable errors. In this condition, the frequency of the speed ripple is strictly linked to the ratio between
the frequency of the square wave and the sampling frequency.
By these considerations, it is clear that the effectiveness of
the linear filtering of the estimated speed depends on the actual
operating conditions, and that the low-speed ripple cannot be filtered out without unacceptable degradation of the measurement
bandwidth.
The proposed SMD attenuates the low-speed measurement
drawback since the switching nature of the differentiator concauses the resulting error to be mainly located at high
trol
frequencies, and thus it is possible to filter it out efficiently.
In Fig. 2, the overall structure of the proposed digital device is summarized in terms of discrete transfer functions.
and
represent the estimated velocity and acceleration, respectively, and the constant depends on the time constant
of the digital low-pass filter [
, see (18)].
is expressed according to (4), and
The switching control
it commutes at high frequency between the constant, opposite,
.
values
value directly affects the dynamic performances of
The
the differentiator, because it actually sets the range for the estimated acceleration (and, at the same time, the slew rate of the
should
estimated speed). To achieve the best performance
be chosen taking into account the actual sampling period, the
drive parameters and the characteristics of the hardware setup.
will result in a larger bandwidth, but it will cause
Increasing
larger oscillations to affect the estimated speed and acceleration.
is defined according to the required dynamic perforOnce
mance, the measurement accuracy defines a lower threshold
under which decreasing of the sampling time does not guarantee
any improvement [see relationships (13) and (19)].
IEEE TRANSACTIONS ON CONTROL SYSTEMS TECHNOLOGY, VOL. 11, NO. 1, JANUARY 2003
87
(a)
6
Fig. 3.
1500 r/min 2-Hz square-wave reference speed. Speed detected by
BDM and by the VSS.
V. EXPERIMENTAL RESULTS
The proposed estimation algorithm has been implemented on a
commercial DSP board (TMS320C31, 40 MHz clock frequency),
and the shaft velocity and acceleration of a 15-Nm FO-controlled
IM drive have been estimated in the open loop. The position signal
was provided by a 5000-ppr incremental encoder.
Different operating conditions have been analyzed. All the reported tests have been performed at the sampling frequency of
value has been set at 12 000 rad s , and the
7.5 kHz. The
acceleration filter time constant has been set at 4.53 ms. To further reduce the high-frequency residual chattering, the estimated
speed has been filtered with time constant of 3.46 ms. As for this
latter filter, it can be inserted without compromising the performance since the filtered signal is not fed back to the differenof the switching signal
tiator. Remember that the amplitude
must be sufficiently large to provide the desired bandwidth, and
which propathe residual high-frequency components of
gates through the discrete integrator (see Fig. 2) can be attenuated by filtering. Direct comparison with the “unfiltered” curves
clearly indicates the positive effect of post-filtering.
The estimated velocity has been compared with that obtained
byBDM.Twotestshavebeenperformed,imposingasquare-wave
reference speed with amplitudes 1500 r/min and 4 r/min, respectively. It can be noted the perfect matching between VSS and
BDM estimates when the speed is large (Fig. 3), while the behaviors are much different at low speed (Fig. 4).
In the absence of external load torque, the estimated speed
and acceleration can be used to estimate the electromagnetic
, in accordance with
torque
(20)
where is the shaft velocity estimate, is the estimated acceleration, is the total inertia, and is the damping factor.
The electromagnetic torque observed by the sensorless
(analog) FO IM drive has been compared with that obtained by
(20) using the VSS estimates. Three experiments at high speed,
(b)
6
Fig. 4.
4 r/min 2-Hz square-wave reference speed. Speed detected by (a)
BDM and (b) VSS.
with different wave-shape for the reference velocity, have been
made (see Figs. 5–7).
The torque observed by the VSS has smaller peaking errors,
and the ripple is located at higher frequencies, as compared with
those obtained by the servo drive.
In the presented tests, the estimated signals are not used
for feedback, and the proposed differentiator is actually in the
open-loop. It must be underlined that the reduced computational burden allows the VSS algorithm to compute both speed
and torque estimates in less than 4 s.
VI. CONCLUSION
This paper dealt with the robust speed and torque estimation in
electrical drives by VSS approach. A digital double differentiator,
based on 2-SMs, has been proposed. The encoder signal has been
processed by the VSS algorithm in order to provide robust estimates of the shaft speed and acceleration, from which torque can
be estimated in turns. Good performance in the low-speed region,
robustness against measurement errors and small computational
demand are the main features of the proposed method. Experimental tests have been conducted on an FO IM drive.
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(a)
(a)
(b)
(b)
6
6
Fig. 5.
1500 r/min 2-Hz square-wave reference speed. Torque observed by
(a) the servo drive and (b) VSS.
Fig. 6.
1500 r/min 2-Hz sinusoidal reference speed. Torque observed by (a)
the servo drive and (b) VSS.
APPENDIX
PROOF OF PROPOSITION 1
condition”
then a first singular point
is attained
after a finite time.
Our objective is to show that, from this point on, a sequence
of singular values is generated, featuring the convergence
toward a vicinity of zero, i.e.,
The convergence of the state trajectories to the origin of the
plane is demonstrated by showing that the feedback law
(4)–(5) causes subsequent crossings of the state trajectory with
the abscissa axis (singular points of ), and that the control
amplitude
can be set in order to ensure that these subsequent
crossings are nearer and nearer to the origin of the state plane.
The actual system trajectory lies between limiting arcs deof the uncertain
fined by the maximum constant bounds
dynamics (2)–(3) (see Fig. 8).
is an estimate of the last singular point of
,
,
the last crossing point of the state trajectory with the abscissa
axis. It is estimated by means of the peak-holder defined by (5),
error such that
which introduces a
(22)
(23)
,
, be the actual th singular value of
. AsLet
. Analogous consume, for the sake of simplicity, that
siderations are also valid if
.
Taking into account the approximate detection of the singular
points (21), and the switching delay due to sample-and-hold effect, it is possible to show that the subsequent singular point
will be such that
(21)
It is not difficult to verify that, independently from the initial
conditions, if the control amplitude satisfies the “dominance
(24)
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89
(a)
Fig. 8.
Limit trajectories in the lower half-plane.
, and it was
then the first inequality of (6) holds with
.
shown also that the second of (6) is verified with
In general, if the requirement of minimizing is dispensed
the problem admits a solution prowith, for any
vided that
belongs to a suitable open set whose upper and
lower bounds depend on
(28)
(b)
6
Fig. 7.
1500 r/min 2-Hz triangular reference speed. Torque observed by (a)
the servo drive and (b) VSS.
Let us introduce the normalized nonnegative variables defined
as follows:
(25)
Considering (24), the contraction condition (22) is equivalent to
the following system of inequalities:
and
are constants which do not depend on
where
the sampling period [3]. The precise expression of and as
functions of the system parameters is quite complex, and it is not
reported since it is not useful for practical design purposes. Relationship (28) means that to achieve convergence the control
must be set sufficiently higher than the motor
magnitude
maximum acceleration but not too high in order to avoid destabilizing effects due to discretization.
As the time interval between two subsequent singular values
is finite, the finite time convergence of the system to
of
the residual set (6) is a straightforward consequence of the contraction condition.
REFERENCES
(26)
variable represents the normalized size of the
Since the
boundary layer, it should be desirable to solve system (26)
is minimized. This
under the additional constraint that
problem has been solved in [3], leading to the approximate
,
. This means that if the control
solution
amplitude is chosen as
(27)
[1] M. A. Al-Aloui, “A class of second-order integrators and low-pass differentiators,” IEEE Trans. Circuits Syst. I, vol. 42, pp. 220–223, Apr.
1995.
[2] G. Bartolini, A. Ferrara, and E. Usai, “Real time output derivative estimation by means of higher order sliding modes,” in Proc. IMACS Multiconf. CESA ’98, Hammameth, Tunisia, Apr. 1998.
[3] G. Bartolini, A. Pisano, and E. Usai, “Variable structure control of nonlinear sampled data systems by second order sliding modes,” in Variable
Structure Syst., Sliding Mode Nonlinear Contr., K. D. Young and U. Ozguner, Eds. New York: Springer-Verlag, 1999, vol. 247, Lecture Notes
in Control and Information Sciences, pp. 44–67.
[4] G. Bartolini, A. Ferrara, A. Pisano, and E. Usai, “Adaptive reduction
of the control effort in chattering free sliding mode control of uncertain
nonlinear systems,” J. Appl. Math. Comput. Sci., vol. 8, pp. 51–71, 1998.
90
IEEE TRANSACTIONS ON CONTROL SYSTEMS TECHNOLOGY, VOL. 11, NO. 1, JANUARY 2003
[5] G. Bartolini, A. Ferrara, A. Levant, and E. Usai, “On second order
sliding mode controllers,” in Variable Structure Systems, Sliding Mode
and Nonlinear Control, K. D. Young and U. Ozguner, Eds. New
York: Springer-Verlag, 1999, vol. 247, Lecture Notes in Control and
Information Sciences, pp. 329–350.
[6] A. Dabroom and H. K. Khalil, “Numerical differentiation using
high-gain observers,” in Proc. 37th Conf. Decision Control, San Diego,
CA, Dec. 1997, pp. 4790–4795.
[7] S. Diop, J. W. Grizzle, P. E. Moraal, and A. Stefanopoulou, “Interpolation and numerical differentiation for observer design,” in Proc. 1994
Amer. Contr. Conf., Baltimore, MD, June 1994, pp. 1329–1333.
[8] S. Diop, J. W. Grizzle, and S. Ibrir, “On regularized numerical observers,” in Proc. 38th Conf. Decision Contr., Phoenix, AZ, Dec. 1999,
pp. 2902–2903.
[9] S. Drakunov, “Sliding mode observers based on equivalent control
method,” in Proc. 31st Conf. Decision Contr. (CDC’92), Tucson, AZ,
December 1992, pp. 2368–2369.
[10] M. Djemai, T. Boukhobza, J. P. Barbot, J. L. Thomas, and S. Poullain,
“Rotor speed and flux nonlinear observer for speed sensorless induction
motors,” in Proc. 1998 Int. Conf. Contr. Applicat., Trieste, Italy, Sept.
1998, pp. 484–852.
[11] K. Fujita and K. Sado, “Instantaneous speed detection with parameter
identification for ac servo systems,” IEEE Trans. Ind. Electron., vol. 28,
pp. 864–872, July 1992.
[12] H. K. Khalil and E. G. Strangas, “Robust speed control of induction
motors using position and current measurements,” IEEE Trans. Automat.
Contr., vol. 41, pp. 1216–1220, Aug. 1996.
[13] A. Levant, “Sliding order and sliding accuracy in sliding mode control,”
Int. J. Contr., vol. 58, pp. 1247–1263, 1993.
[14]
[15]
[16]
[17]
[18]
[19]
[20]
[21]
[22]
[23]
[24]
, “Robust exact differentiation via sliding mode technique,” Automatica, vol. 34, pp. 379–384, 1998.
R. Marino, S. Pesada, and P. Tomei, “Output feedback control of current-fed induction motors with unknown rotor resistance,” IEEE Trans.
Contr. Syst. Technol., vol. 4, pp. 336–347, July 1996.
E. A. Misawa and J. K. Hedrick, “Nonlinear observers—A
state-of-the-art survey,” J. Dyn. Syst., Meas., Contr., vol. 111, pp.
344–352, 1989.
A. H. Khadim, T. K. M. Babu, and D. OḰelly, “Measurement of steadystate and transient load angle, angular velocity and acceleration using
an optical encoder,” IEEE Trans. Instrum. Meas., vol. 41, pp. 486–489,
July 1993.
R. Rajamani, “Observers for Lipschitz nonlinear systems,” IEEE Trans.
Automat. Contr., vol. 43, pp. 397–401, Mar. 1998.
J.-J-E. Slotine, J. K. Hedrick, and E. A. Misawa, “On sliding observers
for nonlinear systems,” J. Dyn. Syst., Meas., Contr., vol. 109, pp.
245–252, 1987.
A. Tornambé, “High-gain observers for nonlinear systems,” Int. J. Syst.
Sci., vol. 23, pp. 1475–1489, 1992.
M. Tursini, R. Petrella, and F. Parasiliti, “Adaptive sliding-mode observer for speed-sensorless control of induction motors,” IEEE Trans.
Ind. Applicat., vol. 36, pp. 1380–1387, Sept. 2000.
V. I. Utkin, Sliding Modes in Control and Optimization. Berlin, Germany: Springer-Verlag, 1992.
P. Vas, Vector Control of AC Machines. Oxford, U.K.: Oxford Sci.,
1992.
X. Yu and J. X. Xu, “Nonlinear derivative estimator,” Electron. Lett.,
vol. 32, p. 16, 1992.