IEEE TRANSACTIONS ON CONTROL SYSTEMS TECHNOLOGY, VOL. 15, NO. 6, NOVEMBER 2007
1049
Speed Sensorless Control of Induction Motors Based
on a Reduced-Order Adaptive Observer
Marcello Montanari, Sergei M. Peresada, Carlo Rossi, and Andrea Tilli
Abstract—A novel speed sensorless indirect field-oriented control for the full-order model of the induction motor is presented.
It provides local exponential tracking of smooth speed and flux
amplitude reference signals together with local exponential field
orientation, on the basis of stator current measurements only and
under assumption of unknown constant load torque. Speed estimation is performed through a reduced-order adaptive observer
based on the torque current dynamics, while no flux estimate is
required for both observation and control purposes. The absence
of the flux model in the proposed algorithm allows for simple and
effective time-scale separation between the speed–flux tracking
error dynamics (slow subsystem) and the estimation error dynamics (fast subsystem). This property is exploited to obtain a
high performance sensorless controller, with features similar to
those of standard field-oriented induction motor drives. Moreover,
time-scale separation and physically-based decomposition into
speed and flux subsystems allow for a simple and constructive
tuning procedure. The theoretical analysis based on the singular
perturbation method enlightens that a persistency of excitation
condition is necessary for the asymptotic stability. From a practical viewpoint, it is related to the well-known observability and
instability issues due to a lack of back-emf signal at zero-frequency
excitation. A flux reference selection strategy has been developed
to guarantee Persistency of excitation in every operating condition. Extensive simulation and experimental tests confirm the
effectiveness of the proposed approach.
Index Terms—Adaptive observers, indirect field oriented control, induction motor (IM) drives, sensorless, velocity control.
I. INTRODUCTION
ECTOR-CONTROLLED induction motors (IMs) [1]
without speed sensor have received wide attention in
the field of low and medium performance electrical drives.
Industrial applications where sensorless IM drives are widely
spread are manufacturing machines, belt conveyors, cranes,
lifts, compressors, trolleys, electric vehicles, pumps, and fans.
With respect to standard vector controlled IM drives, absence
of speed/position sensor reduces cost and size and increases
the reliability of the overall system. On the other hand, performance degradation strongly limits the safe applicability of
sensorless controllers. As a matter of fact, not only reduced
V
Manuscript received November 7, 2005; revised August 23, 2006. Manuscript
received in final form January 12, 2007. Recommended by Associate Editor
A. Bazanella.
M. Montanari, C. Rossi, and A. Tilli are with the Center for Research on
Complex Automated Systems “G. Evangelisti” (CASY), Department of Electronics, Computer, and System Sciences (DEIS), University of Bologna, 40123
Bologna, Italy (e-mail: mmontanari@deis.unibo.it; crossi@deis.unibo.it; atilli
@deis.unibo.it).
S. M. Peresada is with the Department of Electrical Engineering, National
Technical University of Ukraine “Kiev Polytechnic Institute,” Kiev 252056,
Ukraine (e-mail: peresada@i.com.ua).
Digital Object Identifier 10.1109/TCST.2007.899714
speed tracking accuracy and load torque rejection capability
affect the sensorless approach, but instability phenomena can
also occur in the low-speed region and in the regenerating mode
when synchronous frequency approaches to zero [2].
Many researchers have focused on the design of sensorless
control algorithms for IMs. Nevertheless, no control method has
been clearly established as the ultimate solution nor theoretical
questions related to the stability of the sensorless controlled IM
have been completely and rigorously solved. The reader could
refer to [2]–[6] for a detailed overview of various methods. From
the control system perspective, the problem is usually faced with
field oriented control strategy (direct or indirect) or direct torque
control method [4], combined with a speed–flux estimator. Various methodologies have been exploited for speed–flux estimation: MRAS observers [7], [8], extended Kalman filter [9], adaptive observers [10], [11], sliding-mode technique [12].
In more recent papers, the composite analysis of speed estimation combined with the speed–flux controller is carried out.
In [13], a sliding mode rotor flux, stator current, and speed observer together with a sliding mode torque-flux controller were
designed. Stability analysis of the overall dynamics has been
carried out using model-order reduction related to sliding mode
technique and applying linearization method. The sensorless
controller [14] is based on a velocity observer which exploits
a back-emf term in the stator current dynamics. It guarantees
semiglobal speed–flux tracking under assumption of known and
constant load torque and stator flux measurement (obtained by
integration of the stator voltage equations with zero initial conditions). Under the same assumptions a methodology for the
combined design of an adaptive flux-speed observer and controller has been presented in [15]. The proposed controller ensures global asymptotic tracking of smooth speed and flux references. In [16], Montanari et al. designed an adaptive speed observer exploiting stator current regulation errors and using rotor
fluxes, obtained by stator model integration. Local exponential
speed and flux tracking is formally proven under condition of
unknown, but constant, load torque. The algorithm in [17] is
designed under assumptions of unknown rotor/stator fluxes but
with known and smooth load torque. It guarantees local exponential speed–flux tracking, provided that persistency of excitation related to IM observability properties is ensured. The same
control problem with explicitly computable domain of attraction has been solved in [18].
These theoretical results represent a significant contribution
to the theory of sensorless vector control of IMs. Nevertheless,
all of them are based on strong simplifying assumptions, such
as required flux measurement and/or known load torque, and
therefore, they cannot be considered as “true” speed sensorless
solutions suitable for industrial implementation.
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The aim of this paper is to design and test a new speed sensorless controller, which provides local exponential speed/flux
tracking and field orientation, on the basis of stator current
measurement only, under assumption of unknown constant load
torque. The concept of improved indirect field orientation (see
[19] and [20]), combined with a novel reduced-order adaptive
observer, based on the torque current dynamics (see [21] for a
preliminary version with no formal stability proof), are used for
the control algorithm development. Speed is estimated through
the speed-dependent back-emf signal perturbing the torque
current dynamics. The control algorithm generates estimates
for time-varying motor speed and constant load torque, that
converge to the corresponding true values under persistency
of excitation conditions. For controller performances and stability of the overall error dynamics, meaningful properties are a
suitable decoupling between the electromechanical, electromagnetic, and estimation subsystems and the possibility to design the
dynamics of the speed observer arbitrarily fast with respect to
the dynamics of the field-oriented speed–flux controller. On the
basis of system structure and fundamental properties of the IM
electromagnetic subsystem under persistency of excitation, the
stability analysis of the proposed solution is performed using
the singular perturbation method [22], [23].
It results that persistency of excitation, which is related to
observability properties of the IM model, is satisfied if the dc
excitation condition, i.e., null synchronous speed, is avoided.
Exploiting the dependence of slip frequency on rotor flux, flux
reference may be varied in order to avoid this situation, thus
guaranteeing correct speed estimation and stability of the electromagnetic subsystem in every working conditions, defined by
the imposed speed reference and applied load torque. In [24], an
algorithm for flux reference selection in order to avoid the critical region around zero excitation frequency has been combined
with a speed sensorless torque control. In [25], a flux selection
strategy is developed and combined with an adaptive full-order
speed–flux observer and a field oriented controller.
In this paper, a new flux reference selection algorithm is developed, with the aim of imposing the optimal flux which maximizes the modulus of the synchronous speed, considering constraints on the minimum and maximum flux level.
Experimental tests demonstrate that the achievable performances are similar to those of standard vector controlled IM
drives with speed measurement.
The main advantages and novelty of the proposed solution are
as follows:
• common simplifying assumptions are avoided: no flux
measurement or pure integration of stator flux model is
required, and load torque is assumed unknown (but constant, according to standard adaptive control approach) for
both control and observation purposes;
• the controller has a physically based structure and no flux
estimation is required;
• time-scale separation and system decomposition allows for
simple and constructive tuning procedure for controller
gains;
• observability properties of the IM are guaranteed by
the flux reference selection algorithm in any operating
conditions.
According to the basic sensorless control problem formulation, the controller is designed under assumptions of constant
and known motor parameters, while, in real-world applications,
these parameters could be affected by significant uncertainties
and could vary during motor operations, e.g., thermal drift of
stator and rotor resistance. Nevertheless, the proposed solution
is suitable for real applications, since its asymptotic stability
property ensures intrinsic robustness to parameter uncertainties
[23]. Deep analysis of the robustness properties of the proposed
controller is beyond the purpose of this paper. By the way, it
is well known that the stator resistance is one of the most critical parameters affecting performance of sensorless controllers,
mostly in the low speed and regenerating region, when excitation frequency is close to zero [26], [27]. In the proposed controller, the stator resistance variation can be compensated or by
measuring the stator temperature, or by means of an online stator
resistance estimation algorithm embedded in the sensorless controller, as in [12], [24], [26], and [28].
This paper is organized as follows. The IM model and control problem formulation are given in Section II. The proposed
solution is presented in Section III. In Section IV, the stability
analysis and the proposed flux selection algorithm are reported.
Details on the stability proof based on the singular perturbation
technique are given in Appendixes I and II. In Section V, results of simulation and experimental tests are reported. Finally,
Section VI draws conclusions.
II. INDUCTION MOTOR MODEL AND CONTROL
PROBLEM STATEMENT
A. Induction Motor Model
The equivalent two-phase model of the symmetrical IM,
under assumptions of linear magnetic circuits and balanced
operating conditions, is expressed in an arbitrary rotating
reference frame ( - ) [1] as
(1)
where
denote
stator current, rotor flux, and stator voltage vectors [subscripts
and stand for vector components in the ( - ) reference frame],
is the rotor speed,
is the load torque,
and
are
the angular speed and position of the reference frame ( - ) with
respect to a fixed stator reference frame ( - ), where the physical
variables are defined. Transformed variables in (1) are given by
with
(2)
where
stands for any 2-D vector of IM model.
MONTANARI et al.: SPEED SENSORLESS CONTROL OF INDUCTION MOTORS
Positive constants related to electrical and mechanical parameters of the IM are defined as
, where is the total rotor inertia, is the
are stator/rotor resistances and
friction coefficient,
is the magnetizing inductance,
inductances, respectively,
and is the number of pole pairs.
B. Control Objectives
General specifications for speed-sensorless controlled electric drives require to control the two IM outputs, speed, and rotor
flux modulus, defined as
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and -axis current estimate
, estimation errors are defined as
and the estimated
. The
speed tracking error is defined as
.
slip frequency is
A. Speed-Flux Controller
In the framework of improved indirect field oriented control
[19], replacing measured speed with the estimated one, the following speed–flux tracking sensorless controller is defined:
Flux controller
(6)
using the 2-D stator voltage vector , on the basis of measured
. Defining
,
variables vector
and
are speed and flux reference trajectories,
where
the speed and flux modulus tracking errors are
(3)
Speed controller
(7)
Current controller
(8)
Following the concept of indirect field orientation [1], [19] the
- and -axis flux tracking errors are defined as
with
(4)
is the condition of asymptotic field
Note that
orientation. From (3) and (4), it follows that the condition
implies
.
The speed–flux tracking control problem is formulated as follows. Consider the IM model (1), (2) and assume the following.
A.1. Stator currents are available from measurements.
A.2. Motor parameters are exactly known and constant.
is unknown, bounded, and constant.
A.3. Load torque
A.4. Speed and flux reference trajectories
are
smooth functions with known and bounded first and
.
second time derivatives, and
Under these assumptions, it is required to design an output feedback controller which guarantees local asymptotic rotor speed
and flux amplitude tracking together with asymptotic field orientation, i.e.,
(5)
(9)
where
are proportional and integral gains of the
are proportional gains of the
speed controller,
are additional correction terms
current controller, and
designed according to Lyapunov-like technique as shown in
is a constant tuning gain.
Section IV, and
B. Reduced-Order Speed Observer
The proposed speed observer is based on the torque current
dynamics only, therefore, it is a reduced-order solution with
respect to the IM electromechanical model. According to the
adaptive control approach, an integral component is adopted to
estimate the speed tracking error, while no information on the
mechanical model are required. The observer equations are the
following:
with all signals bounded.
III. SPEED-FLUX SENSORLESS CONTROL ALGORITHM
The proposed solution exploits the concept of improved indirect field-oriented control presented by Peresada and Tonielli
[19] and an original reduced-order observer based on the torque
current dynamics. Controller and observer designs are presented
in Sections III-A and III-B, respectively, while properties of the
overall controller structure are discussed in Section III-C.
as refThe following notations are introduced. Defining
currents, respectively, current tracking errors
erences for
are defined as
. Introducing the
of load constant
,
speed estimation , the estimation
(10)
where
are the observer tuning gains.
Remark 1: The main idea for the speed observer design recurrent dylies on the influence of the rotor speed on the
, which is estimated
namics through the back-emf term
as
, assuming that the flux vector converges to its reference. Since no flux estimation is performed for the speed reconstruction, the performances are strictly related to flux control
properties. This basic consideration will be confirmed by the
theoretical analysis in Section IV.
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IEEE TRANSACTIONS ON CONTROL SYSTEMS TECHNOLOGY, VOL. 15, NO. 6, NOVEMBER 2007
Fig. 1. Block diagram of the equivalent error dynamics of the mechanical and estimation subsystems.
C. Controller and Observer Analysis
From IM model (1), control algorithm (6)–(8), and observer
(10), the speed–flux tracking and the estimation error dynamics
are
(11)
(12)
(13)
with
(14)
(15)
The error dynamics (11)–(13) is given by the tracking error dynamics (11) and (12) and the observer error dynamics (13). The
former is composed by the nonlinear electromagnetic dynamics
in (11) and the linear mechanical dynamics in (12) in feedback
with linear/biinterconnection through the coupling term
linear properties, while is the flux-error-dependent perturbation for speed estimation.
The mechanical and estimation subsystems (12) and (13) are
shown in the block diagram of Fig. 1. Two linear second-order
dynamics related to PI speed controller and speed estimator can
, defined in (14) and
be observed, while nonlinear terms
(15), establish the interconnection between mechanical/estimation dynamics and electromagnetic subsystem (11). From the
scheme reported in Fig. 1, it is natural to define the nominal
dynamics of the observer and speed subsystems as the one obtained when no perturbation comes from the electromagnetic
subsystem. Hence, the nominal dynamics are given by (12) and
and
. From its structure, it follows that
(13) with
of the speed tracking error is the output of
the estimation
a second-order linear filter with unitary static gain, which can
be viewed as sensor subsystem in the negative-feedback speed
control loop. Therefore, it is straightforward imposing the nominal estimation dynamics much faster than the nominal speed
subsystem. This is a key element of the controller tuning as explained in Section IV.
The high-gain observer design relies on adaptive control
theory. As shown in Fig. 1, the integral action with gain
in the
dynamics in (10) imposes negative
feedback with constant static gain to the speed estimation
loop, thus achieving linear exponentially stable estimation
dynamics (33). In
dynamics, as also enlightened by the
Appendix I, convergence properties are formally derived.
Remark 2: The speed estimation dynamics can be designed
, since no flux model
arbitrarily fast by selection of gains
is embedded in the observer. Nevertheless, independently of
how fast the speed observer (13) is, cannot be directly approxowing to the perturbation coming from the flux
imated by
tracking error dynamics. The flux information obtained by the
integration of the stator flux equations with known initial conditions [7], [14]–[16], [26], [27] could be attractive for compensation of the perturbation in the speed observer (13). However,
it is well known that this kind of flux estimator is not reliable in
actual implementation, owing to effects of imperfect knowledge
of stator resistance, nonidealities in stator voltage actuation, and
measurement offsets.
IV. STABILITY ANALYSIS
The stability of the overall error dynamics (11)–(13) is based
on structural properties of the feedback interconnected electromagnetic, mechanical, and speed estimation subsystems, generated by the field oriented speed–flux controller together with
the speed observer, whose dynamics can be designed arbitrarily
fast with respect to the control one. On the basis of considerations in Section III-C, time-scale separation is imposed between the speed–flux tracking error dynamics (slow-subsystem)
and the speed observation error dynamics (fast-subsystem). In
particular, estimation error dynamics (13), dependent on tuning
, is imposed to be faster than electromagnetic (11)
gains
and mechanical (12) dynamics. The mechanical dynamics can
, while current-flux
be tuned through selection of gains
, due
dynamics are characterized by the rotor time-constant
to IFO controller structure.
MONTANARI et al.: SPEED SENSORLESS CONTROL OF INDUCTION MOTORS
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According to the singular perturbation method [22], [23], the
analysis is focused first on the reduced-order system, given by
the slow-subsystem with the fast dynamics approximated by
its quasi-steady state, as long as exponential stability of the
fast-subsystem is guaranteed. Details on the application of the
singular perturbation method for systems (11)–(13) are given
in Appendix I. The resulting quasi-steady state for the observer
error dynamics (13) is the following:
(16)
Remark 3: From (16), it can be noted the effect of flux amplitude regulation and orientation errors on the speed estimation,
as previously mentioned in Remark 1.
The reduced-order system is composed by the electromagnetic error dynamics (11) and the mechanical error
with their
dynamics (12) replacing the fast variables
quasi-steady-state expression (16). Defining new state variables
, which are proportional to
stator flux tracking errors, and substituting (16) into (11) and
(12), the reduced-order system expressed with state variables
, and
is given by
(19)
First, local exponential stability of the reduced-order dynamics
(17) and (18) is proven, while the full-order system is considered
in Appendix II. Stability properties of the electromagnetic (17)
and mechanical dynamics (18) are analyzed separately, and then
composite reduced-order dynamics is investigated, exploiting
the linear-bilinear (and higher order) properties of [see (14)].
In particular, the nominal mechanical subsystem is linear and
the stability analysis is straightforward, while the Lyapunov-like
method is used for the nonlinear time-varying electromagnetic
dynamics.
A. Electromagnetic Subsystem
In order to analyze the stability properties of electromagnetic
dynamics (17) the following candidate Lyapunov function is defined:
(17)
(20)
where
is a constant positive tuning gain. Its time derivative
along trajectories of (17) is
(18)
where
are the quasi-steady-state values of
,
respectively, i.e., obtained from (6) and (9) replacing
with
from (16), and
. Partitioning
and
into
time-dependent and state-dependent parts, their expressions are
given by
Selecting
(21)
and substituting expressions of
defined in (9) and evaluated on the slow manifold as in (19), becomes
(22)
.
with
Remark 4: Note that
have been suitably selected to
compensate for the quadratic cross terms in the time derivative
of .
No result can be directly obtained from properties of and
in (20) and (22), but the previous considerations will be instrumental for the following analysis. Substituting (9) in (17),
and defining
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IEEE TRANSACTIONS ON CONTROL SYSTEMS TECHNOLOGY, VOL. 15, NO. 6, NOVEMBER 2007
the flux subsystem dynamics can be rewritten as follows:
is differentiable and
is bounded. If there exist two pos, such that the persistency of excitation condition
itive reals
for all
(23)
(27)
is satisfied, then from [30, Lemma 1 (extended persistency of
excitation lemma)] it follows that the origin of the linear timevarying system (24), which represents the linearization of (23),
is exponentially stable.
B. Reduced-Order Dynamics
Consider, first, the linearization of the reduced-order system
(17) and (18), which is compactly expressed as
where
represents the linear part and
includes all the higher order terms.
,
Remark 5: The interconnection matrices between
in
[see (23)] are not skew-symmetric, owing
and
to presence of flux tracking errors in the quasi-steady-state estimated speed (16).
As a first step, the stability properties of the linearized dynamics of (23)
(28)
, and
where
, while
are defined as
(24)
with
are analyzed. Considering the
time-varying coordinate transformation
(25)
with
, dynamics (24) in new coordinates become
(26)
where
are defined as
Note that (26) is in standard form of adaptive systems [29, Sec.
[defined in
2.8]. Evaluating the time derivative of
with
] along the linearized dynamics (26),
(20) replacing
, hence,
are bounded.
it follows that
From (26), it follows that
are bounded, and
therefore,
are uniformly continuous. Since
, it follows that
are square
integrable signals and, therefore, according to Barbalat’s
lemma [23, Lemma 4.2] it results that
.
Since
, and
are bounded by assumption, it follows
that the dynamical matrix of (26) is continuous and bounded,
Note that the linearized electromagnetic dynamics has been decontains only the
rived in Section IV-A, while the matrix
reference-dependent parts of
.
The main feature of the linear approximation (28) is that
current-flux and speed dynamics are in series interconnection.
is
Since the LTV current-flux subsystem
globally exponentially stable if PE is satisfied, matrix
is
and
is bounded, it follows that
Hurwitz with
(28) is GES. Hence, from standard nonlinear control results [23,
Th. 3.11], it follows that the origin of the nonlinear reducedorder system (17) and (18) is locally exponentially stable.
C. Full-Order Dynamics
Standard singular perturbation technique can be directly applied to prove stability of the full-order error dynamics (11)–(13)
(see Appendix II). Briefly, the full-order system given by (11)–
(13) is presented as feedback interconnected reduced-order
system (32) and boundary-layer estimation dynamics (33). The
reduced-order system is LES if PE is satisfied, and the boundary
layer system (33) is linear and exponentially stable, therefore, the
full-order dynamics is LES, provided that parameter is sufficiently small [22, Sec. 7.5]. Hence, it has been proven that local
exponential speed–flux tracking together with field orientation
are achieved, provided that PE condition (27) is satisfied.
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D. Persistency of Excitation Condition and Flux Reference
Selection
It is important to get a physically meaningful comprehension
of the persistency of excitation result (27), in order to understand
in which IM operating conditions the proposed controller fails
to be asymptotically stable due to lack of PE. Then, in order
to avoid these conditions, an algorithm for the reference flux
selection is proposed.
A theoretical analysis of PE condition (27) is complex considering generic operating conditions of the IM. Assuming conand load torque
, it
stant speed and flux references
in (19) is constant, hence,
. When
follows that
, i.e., with dc excitation, it results
hence, the matrix in (27) is positive semi-definite and the PE
, choosing
, it
condition fails. When
results
Fig. 2. Selected flux reference
estimated torque J 1 T^ .
as a function of speed reference
!
and
TABLE I
INDUCTION MOTOR DATA
for all
Hence, it follows that the PE condition is satisfied under all constant operating conditions apart from zero frequency.
depends on
, and , the flux reference can
Since
be used as an extra degree of freedom to avoid lack of PE, i.e.,
. To this purpose a simple strategy for the flux reference
selection which maximizes synchronous frequency taking into
account physical flux bounds is proposed. In particular, considering the steady-state contributions only, the following auxiliary
synchronous frequency can be defined:
(29)
Hence, for given
the flux reference
can be selected
is maximum, while satisfying the constraint
such that
. Flux bound
represents the minimum
admissible flux not impairing the IM performances, while
is defined according to IM magnetic saturation. The following
flux reference selection algorithm solves the proposed maximization problem:
if
if
(30)
with
. In Fig. 2 it is
shown how the flux reference is selected depending on
and
. Notice that the minimum flux level is selected during motoring mode, while the maximum flux level is imposed only
in some regenerating conditions. Solution (30) guarantees that
dc-excitation is always avoided.
The flux selection strategy is enabled only at low excitation
frequency and with high torque current. In fact, at high excitation frequency, no observability issues arise, hence, the flux reference is imposed to the rated value, or a different flux selection
technique, e.g., for power efficiency optimization or flux weakening, can be utilized. Moreover, operation with low torque current and low excitation frequency is not critical from a practical
viewpoint, since the small motor torque is comparable to friction
torque; on the other hand, at low motor torque, the variation of
excitation frequency achieved by changing the flux level is less
relevant [see (29)].
Additionally, in order to reduce sensitivity with respect to
measurement noise and estimation errors, signal filtering and
hysteresis are introduced to avoid spurious commutations of the
flux reference. Moreover, in the implementation the flux reference (30) is filtered, in order to provide bounded time-derivatives and to avoid quick flux changing, which may induce large
flux errors and torque oscillations.
V. SIMULATION AND EXPERIMENTAL RESULTS
The proposed speed sensorless control algorithm has been
tested by means of simulations and experiments using a 1.1-kW
induction motor whose rated data are reported in Table I.
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Fig. 3. Block diagram of the simulated IM model with the speed sensorless controller.
A. Controller Tuning
Controller parameters have been selected according to the
following tuning rules, which derive from the structure of the
error dynamics (11)–(13) and according to the analysis reported
in Appendix I. Some implementation aspects, such as robustness with respect to parameter uncertainties and unmodelled dynamics, sensitivity with respect to measurement noise, issues
related to discrete-time implementation of the algorithm, have
been taken into account in a heuristic way, performing some
simulations and/or experiments in order to refine the controller
tuning.
Selecting gains
and
, the speed
controller is tuned imposing a time constant
and damping factor
for the secondorder nominal mechanical error dynamics (12) (see Fig. 1). The
second-order speed observer dynamics (13) is designed imposing a time constant
and damping factor
by selection of
and
.
From a practical viewpoint, time-scale separation between
dynamics (12) and (13) is achieved with
.
The bandwidth of the current error dynamics is imposed by
selecting the proportional -axis and -axis current controller
gains
. Selection of large gains
, which leads to
the current-fed condition, ensures fast current dynamics, and reduced sensitivity to imperfect knowledge of IM electrical parameters, such as stator resistance. Nevertheless, large gain
implies a high value of gain , which is computed according
to (21), thus, leading to high sensitivity to current measurement
noise superimposed on .
The following controller gains have been chosen for all simulations and experiments. The speed controller gains are set at
120 s
7440 s , imposing a mechanical time
constant
11.6 ms. The tuning gains of the speed observer
are selected as
240 s
93400 s , imposing an
3.3 ms. Current controller gains
observer time constant
have been set at
150 s
300 s
.
B. Simulation and Experimental Set-Up
The continuous-time version of the algorithm is considered in
the simulations. Ideal stator voltage actuation is assumed, i.e.,
no pulse-width modulation (PWM) switching of the inverter is
considered and no noise is superimposed on the stator currents.
A detailed block diagram of the controller (6)–(10) together
with the IM model (1) is shown in Fig. 3. The IM model is expressed in the stationary reference frame ( ), while the controller is implemented in the rotating reference frame ( ). In
the controller, direct Park transformation (2) is utilized to transform stator currents
into
, while the inverse Park
transformation is utilized to compute stator voltages
from
. In the controller equations,
are analytically
computed from expressions of
in (6) and (7).
The experimental tests have been carried out using a rapid
prototyping station, which includes the following:
• a personal computer acting as the operator interface;
• a custom floating-point digital signal processor (DSP)
board (based on TMS320C32) connected to the ISA PC
bus; the DSP board performs data acquisition (eight 12-bit
A/D data channels and two interfaces for incremental
MONTANARI et al.: SPEED SENSORLESS CONTROL OF INDUCTION MOTORS
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Fig. 4. Speed, flux references, and load torque profile (dashed line).
encoder), implements the control algorithm and generates
the PWM signals (two symmetrical three-phase PWM
modulators with programmable dead-time);
• a 50 A/400 V
three-phase inverter, operated at 10-kHz
switching frequency; dead-times of the inverter have been
set to 1.5 s;
• a 4-pole, 50-Hz, 1.1-kW standard induction motor, whose
data are listed in Table I;
• a vector-controlled permanent magnet synchronous motor
used to provide the load torque.
In order to filter out the modulation ripple, two stator phase currents, measured by Hall-effect zero-field sensors, are simultaneously and synchronously sampled at the symmetry point of the
PWM signals. Only for monitoring purposes, the motor speed
is measured by means of a 512 pulse/revolution incremental encoder. Dead-time effects have been compensated with a simple
method based on current sign [5], [31].
The simple Euler method has been used in order to get the
discrete time version of the control algorithm, with 200- s sampling time.
C. Operating Sequences
The following flux and speed references and load torque profiles are used during tests, as shown in Fig. 4.
1) The machine is excited during the initial time interval
0–0.12 s applying a flux reference trajectory starting at
0.01 Wb and reaching the motor rated value of
0.86 Wb with maximum first and second time derivatives
equal to 8 Wb/s and 500 Wb/s , correspondingly.
2) The unloaded motor is required to track the speed reference
trajectory characterized by the following phases: starting at
0.4 s from zero initial value, speed reference trajectory
reaches 100 rad/s at
0.45 s; from this time to
1.3 s
1.3 s to
1.35 s
a constant speed is imposed; from
the motor is required to stop at zero speed reference. Maximum absolute values of the first and second derivatives
of the speed reference trajectory are equal to 2200 rad/s
and 20 000 rad/s , correspondingly. Speed tracking during
reference trajectory variations requires a dynamic torque
equal to the motor rated one.
3) From time
0.7 s to
1.0 s a constant load torque,
equal to 100% of the motor rated value (7.0 Nm), is
applied.
D. Simulation and Experimental Results
Dynamic performance of the controller during speed tracking
and load torque rejection has been tested in various speed-load
operating conditions.
Fig. 5. Simulation results: dynamic behavior of the sensorless controller:
Speed errors and q -axis current estimation error; stator currents; stator voltages;
rotor flux errors.
In the first test, simulations and experiments during transient
with 100-rad/s maximum speed and 7.0-Nm constant load
torque have been performed. In Fig. 5 simulation results are
shown. Speed estimation and tracking errors are negligibly
small during speed trajectory tracking, while a maximum speed
tracking error of about 14 rad/s is present during load torque
rejection. Time-scale separation between speed estimation and
regulation dynamics guarantees accurate speed estimation with
maximum estimation error of about 6 rad/s. As a result, the
speed, current, and flux tracking errors caused by the unknown
load torque perturbation are compensated by the load estimation mechanism based on the estimated speed tracking error.
Results related to experimental tests performed under the
same operating conditions of Fig. 4 are shown in Fig. 6. From
Figs. 5 and 6, it can be noted good similarity between simulation
and experimental results. Zero steady-state speed estimation
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IEEE TRANSACTIONS ON CONTROL SYSTEMS TECHNOLOGY, VOL. 15, NO. 6, NOVEMBER 2007
Fig. 7. Experimental results: dynamic behavior of the improved IFO controller.
Fig. 6. Experimental results: dynamic behavior of the sensorless controller
with maximum speed reference equal to 100 rad/s and applied load torque equal
to 7.0 Nm: Speed errors and q-axis current estimation error; stator currents;
stator voltages.
and tracking errors are achieved when constant speed reference
and load torque are applied. Similar dynamic performance and
amplitude of speed tracking and estimation errors are obtained
during load torque rejection, with maximum speed tracking
error of about 13 rad/s. It is worth noting that, in actual implementation, maximum speed tracking and estimation errors are
about 4 rad/s during speed reference variation. This is mainly
due to the imposed fast speed transient (50 ms), IM parameter
uncertainties and nonidealities such as dead time effects and
stator voltage inverter distortion. Stator currents are quite close
to the reference ones.
The proposed sensorless controller is compared with the improved indirect field oriented (IFO) control algorithm [19], [20],
which is based on speed measurement, under the same operating conditions of Fig. 4. IFO controller parameters are selected as follows. The proportional and integral speed controller
gains have been set as in the sensorless controller, i.e.,
, proportional and integral current controller
gains are
, and
. The
experimental results reported in Fig. 7 demonstrate the dynamic
performance of the improved IFO controller. Maximum amplitude of the speed tracking error is 11 rad/s during load torque
rejection transient and it is almost null during speed tracking.
The settling time achieved during load torque rejection is the
same of the sensorless controller case. Comparison of the experiments performed with the sensorless controller (see Fig. 6)
and with the improved IFO controller (see Fig. 7) demonstrates
that the achievable performances are similar to those obtained
with speed feedback, under the tested operating conditions.
The second set of experiments have been performed to test the
performance of the proposed controller under low-speed operating conditions. These regimes are very critical since they are
characterized by reduced index of PE ( can be almost zero
in regenerating mode), increasing sensitivity to IM parameter
uncertainties (e.g., stator and rotor resistance variations) and to
inverter nonidealities (e.g., dead-time and resistive losses) [26].
The same sequence of motor operations (see Fig. 4) with maximum speed of 10 rad/s and maximum acceleration of 550 rad/s
is used to test the dynamic behavior of the controller under
constant rated load (7 Nm) and regenerating ( 7 Nm) applied
torques. Note that in the regenerating case synchronous speed
is
8.5 rad/s. Transients reported in Figs. 8 and 9 confirm
that no speed tracking performance degradation is present with
respect to the high-speed case for both motoring and regenerating modes. In Fig. 10 rejection of constant rated load torque
with zero speed reference is considered. During load torque application from
0.7 s to
1.0 s synchronous speed is
11.5 rad/s, as confirmed also from the stator current profiles,
hence, PE condition is verified.
An important observation can be made from all previous experiments (see Figs. 6 and 8–10): the final test conditions (from
1.35 s) are characterized by lack of PE, in fact
, and
. Nevertheless, the proposed solution guarantees
negligible speed error for this particular condition.
The third set of experiments was carried out to test the
behavior of the speed controller when PE condition is not
satisfied. The performances of controller with and without
the flux reference selection algorithm of Section IV-D are
MONTANARI et al.: SPEED SENSORLESS CONTROL OF INDUCTION MOTORS
Fig. 8. Experimental results: dynamic behavior of the sensorless controller
with maximum speed reference equal to 10 rad/s and applied load torque equal
to 7.0 Nm: speed errors and stator currents.
1059
Fig. 10. Experimental results: dynamic behavior of the sensorless controller
with speed reference equal to 0 rad/s and applied load torque equal to 7.0 Nm:
speed errors and stator currents.
Fig. 11. Speed reference (solid line) and load torque profile (dashed line).
Fig. 9. Experimental results: dynamic behavior of the sensorless controller
with maximum speed reference equal to 10 rad/s and applied regenerating torque
equal to 7.0 Nm: speed errors and stator currents.
0
comparatively evaluated when motor operates in regenerating
mode. In these tests, parameters of the flux reference selection
algorithm have been chosen as follows: minimum and maximum fluxes are 10% of the rated flux, i.e.,
0.77 Wb,
0.95 Wb, while the resulting flux reference is filtered by
the nonlinear filter [32] with maximum first and second time
derivatives equal to 2 Wb/s and 50 Wb/s . According to Fig. 11,
a speed reference profile with steady-state value
7.5 rad/s
is imposed and constant rated regenerating torque ( 7 Nm) is
applied from
0.6 s to
1.3 s. In the first test, reported
in Fig. 12, a constant flux reference
is adopted,
leading to
(i.e., lack of PE) when the regenerating
torque is applied. From this test it is clear that speed is not
correctly estimated, hence, nonnull speed tracking error is
obtained. Differently, in the second test, reported in Fig. 13,
the flux reference is selected according to (30). In this way,
the synchronous speed is not equal to zero and, as a result,
correct speed estimation and almost null speed tracking error
are achieved.
In order to test the behavior of the controller during motoring,
plugging, regenerating, zero speed, and zero frequency conditions, a slow speed reference from 20 to 20 rad/s and back
with acceleration equal to 10 rad/s is imposed to the IM, with
constant rated torque equal to 7.0 Nm. During tests, minimum
and maximum flux levels are imposed at
0.74 Wb and
1.11 Wb and flux reference is filtered, such that maximum time derivative of the flux reference is equal to 10 Wb/s.
The flux selection strategy (30) is enabled only if excitation frequency
becomes lower than 15 rad/s (in modulus) and estimated torque is greater than 2.2 Nm, while it is disabled for
becoming greater than 25 rad/s. In other operating conditions, flux reference is imposed to
0.86 Wb. Experimental
results are reported in Fig. 14. Flux reference is maintained at
its constant rated value
0.86 Wb till 2.45 s (when
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IEEE TRANSACTIONS ON CONTROL SYSTEMS TECHNOLOGY, VOL. 15, NO. 6, NOVEMBER 2007
0
Fig. 14. Experimental results: speed transient from 20 to 20 rad/s and back,
with applied torque equal to 7 Nm, with optimal flux reference selection: speed,
synchronous speed, and flux reference; stator currents, speed estimation error.
Fig. 12. Experimental results: dynamic behavior of the sensorless controller
with speed reference equal to 7.5 rad/s and applied regenerating torque equal
7.0 Nm, with constant flux reference: speed errors, flux reference, and
to
synchronous speed; stator currents.
0
15 rad/s); then, flux level is decreased at
0.74 Wb, hence,
-axis current decreases, while excitation frequency and torque
current increase. Induction motor works in motoring mode till
2.4 s (
0 rad/s), then it starts operating in plugging mode
. At
3.2 s, the flux level is increased at
1.11 Wb, thus avoiding operation close to zero excitation frequency. The -axis current increases, while slip frequency dechanges sign, thus IM operates in regenerating
creases and
mode. At 4.2 s, when
25 rad/s, the flux level is imposed to its rated value. Speed transient from 20 to 20 rad/s is
performed in a similar way. During operations, estimation and
tracking error with maximum amplitude of about 3 rad/s are
present during flux change, due to magnetic saturation effect.
The presented tests confirm that the proposed flux reference
selection algorithm combined with the speed sensorless control
are profitable to overcome the problem of zero frequency excitation, when PE is not satisfied or is close to fail.
VI. CONCLUSION
Fig. 13. Experimental results: dynamic behavior of the sensorless controller
with speed reference equal to 7.5 rad/s and applied regenerating torque equal
to 7.0 Nm, with optimal flux reference selection: speed errors, flux reference,
and synchronous speed; stator currents.
0
In this paper, a novel speed sensorless controller for IM,
which guarantees local exponential tracking of smooth speed–
flux references together with field orientation, has been
presented. In the proposed method, no flux estimation or reconstruction through integration of the stator voltage model is
needed, while load torque is assumed constant but unknown. A
persistency of excitation condition, which fails to be guaranteed
at dc excitation, is required for stability. Physical understanding
of such result has been exploited to develop an effective flux
reference selection procedure, in order to avoid instability
issues.
Experiments and simulations show that the proposed controller combined with the flux selection algorithm is suitable for
high-performance sensorless controlled IM drives.
The proposed controller has been designed under assumption
of known and constant motor parameters; nevertheless, electrical parameters may vary during motor operations and it is
MONTANARI et al.: SPEED SENSORLESS CONTROL OF INDUCTION MOTORS
well known that parameter uncertainties affect the stability and
transient performance of sensorless controllers. The design of
adaptive version of the controller with respect to motor parameters will be the subject of further works.
APPENDIX I
ANALYSIS OF THE FULL-ORDER ERROR DYNAMICS USING
SINGULAR PERTURBATION TECHNIQUE
Methodology and formalism of singular perturbation technique [22], [23] have been exploited in order to give a formal
approach to time-scale separation of the full-order error model
and to perform the stability analysis. Introducing the following
, and
state variables:
, selecting
tuning parameters according to
(31)
with
and defining the small parameter
, error dynamics (11)–(13) is expressed using as
as
state variables
1061
, the quasi-steady state
From (33) with
is obtained as the unique isolated solution of the alge. It is given by
braic equation
(36)
Note that from definition of
, the quasi-steady state
for corresponds to the quasi-steady state for and given
in (16).
Defining the boundary-layer variable
and
, the so-called boundary layer dynamics is defined as
(37)
where and are considered as fixed parameters [see (33)].
According to Tikhonov’s Theorem for nonlinear time-varying
systems, if the origin of the boundary layer system is asymptotically stable and the small parameter is sufficiently small,
the behavior of the full-order dynamics (32) and (33) can be approximated by the so-called reduced-order system
APPENDIX II
STABILITY OF THE FULL-ORDER DYNAMICS
(32)
Stability proof is carried out transforming the full-order error
dynamics (11)–(13) into standard form for the singular perturbation method and exploiting the following properties: series
interconnection of the linearized reduced-order flux and speed
subsystems, exponential stability of the flux dynamics provided
that PE condition is satisfied, and exponential stability of the
speed and estimation isolated subsystems.
Imposing tuning relations given in Appendix I, from (32) and
(33) the full-order error dynamics is rewritten with state variables
[see (9), (21), and Appendix I for definitions] as
(33)
(38)
Systems (32) and (33) are in standard singular perturbation
form
(34)
(35)
where
state vector,
vector,
parameter, and
that
denotes the “slow”
denotes the “fast” state
represents the small positive perturbation
are smooth and bounded functions, such
.
(39)
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IEEE TRANSACTIONS ON CONTROL SYSTEMS TECHNOLOGY, VOL. 15, NO. 6, NOVEMBER 2007
(40)
Decomposing signals
into time-varying state-independent, fast-variable-independent, and fast-variable-dependent
terms, respectively, i.e., expressing the generic signal as
, the full-order
error dynamics can be rewritten enlightening decomposition
into “nominal” dynamics, slow-variable-dependent and fastvariable-dependent coupling terms as in (41)–(43), shown at the
bottom of the page. Additional signals are defined in (19) and
The following properties of (41)–(43) are meaningful for the
stability analysis.
is
1) If PE condition is satisfied, system
exponentially stable, hence, from the converse Lyapunov
function theorem for LTV systems [23, Th. 3.10] there
exist a continuously differentiable symmetric matrix
and a continuous symmetric matrix
such
that
.
, and
as in Appendix I, matrices
2) Selecting
are Hurwitz, hence, there exist symmetric matrices
such that
and
.
3) Interconnection terms in (41)–(43) can be expressed as
linear and higher order polynomial functions of state
variables. Bearing in mind that the smallest powers are
dominant nearby the origin, interconnection terms can be
bounded by linear or bilinear bounds, which hold uniformly in time within given compact sets of state variables.
,
Assuming that
(41)
(42)
(43)
MONTANARI et al.: SPEED SENSORLESS CONTROL OF INDUCTION MOTORS
with
compact sets containing the origin, denoting
, the following holds.
the Euclidean norm with
• There exist bounded functions
(where dependence on time is due to reference signals
, and load torque
), and there exist positive
, such that
constants
.
• There exist bounded functions
and there exists a positive constant
,
, such that
.
• There exist bounded functions
and there exists a positive constant
,
, such that
.
• There exist bounded functions
and there exist a positive constant
that
,
, such
1063
Remark 6: It is worth noting that since the existence of the
Lyapunov function for the flux subsystem relies on converse
is not
Lyapunov theorem and PE condition, expression of
explicitly known. Moreover, intensive use of Young inequalities
and comparison functions lead to conservative evaluation of the
domain of attraction and tuning rules. Additionally, choice of
comparison functions is not unique, therefore, different evaluations are possible. Anyhow, performed stability result is significant from the theoretical viewpoint.
for the
Considering the Lyapunov function
boundary layer dynamics, its time derivative along trajectories
of (43) is
where Young inequalities have been applied to terms dependent
, and
. Introducing
on
the composite Lyapunov function
and applying Young inequalities to bilinear terms, it
holds
, with
.
• There exist bounded functions
,
. It is possible to select coefficients
with sufficient degrees of freedom and
(enlightening
to choose sufficiently small parameter
and
dynamics), in
time-scale separation between
, thus obtaining
order to impose
. Therefore, the
of the full-order
equilibrium point
error dynamics is locally exponentially stable.
and
there exist positive constants
and
, such that
.
contains linear terms,
Previous inequalities enlighten that
while
does not. Consider first the
dynamics. Exploiting properties stated before, the time derivative of the
(with
Lyapunov function
) along trajectories of (41) and (42) is
where the Young inequality
has been applied to the term dependent on
. Assuming
and selecting sufficiently small
such
,
that
, in the preliminary case of
and
it holds
, thus proving local exponential stability of the
reduced-order error dynamics (41) and (42) (see also results in
Section IV-B).
ACKNOWLEDGMENT
The authors would like to thank Prof. A. Tonielli from CASYDEIS, University of Bologna, for his help and support during the
preparation of this paper.
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Marcello Montanari was born in Ravenna, Italy,
in 1974. He received the Dr. Ing. degree in computer science engineering and the Ph.D. degree in
automatic control from the University of Bologna,
Bologna, Italy, in 1999 and 2003, respectively.
Since 2003, he has held a Postdoctoral position
with the Department of Electronics, Computer, and
System Science, University of Bologna. He is a
cofounder of ARCA Tecnologie srl, Bologna, Italy,
a spin-off company of the University of Bologna,
started in 2004. His current research interests include
applied nonlinear and adaptive control, in the field of electric drives, automotive
applications, and electro-mechanical and electro-hydraulic systems.
Sergei M. Peresada was born in 1952, in Donetsk,
USSR. He received the Diploma of electrical engineering from the Donetsk Polytechnic Institute,
Donetsk, Ukraine, in 1974 and the Candidate of
Technical Sciences degree (corresponding to the
Ph.D. degree) in control and automation from the
Kiev Polytechnic Institute, Kiev, Ukraine, in 1983.
Since 1977, he has been with the Department of
Electrical Engineering and Automation, National
Technical University of Ukraine “Kiev Polytechnic
Institute,” Kiev, Ukraine, where he is currently
a Professor of control and automation. He has been a Visiting Professor
with the University of Illinois (Urbana-Champaign), Urbana, Department
of Electronics, Computer, and System Science (DEIS), University of Rome
Tor Vergata, Rome, Italy, and the Institute of Advanced Study University of
Bologna, Bologna, Italy. He is the author of about 150 scientific publications
and a co-author of the volumes: Theory and Control of Electrical Drives and
Control of Electromechanical Systems. His research interests include nonlinear
and adaptive control of electromechanical systems based on ac motors, control
of power converters.
Dr. Peresada currently serves as an Associate Editor of the IEEE
TRANSACTIONS ON CONTROL SYSTEMS TECHNOLOGY.
Carlo Rossi received the Dr.Ing. degree in electronic
engineering and the Ph.D. degree in system science
and engineering from the University of Bologna,
Bologna, Italy, in 1989 and 1993, respectively.
In 1989, he joined the Department of Electronics,
Computer, and System Science (DEIS) of the University of Bologna. From 1994 to 2001, he was with
Magneti Marelli Powertrain, Bologna, Italy, where
he had the responsibility of the Control Design and
System Analysis Groups. Since 2001, he has been an
Associate Professor with the Automatic Control Department, University of Bologna. He is a cofounder and CEO of ARCA Tecnologie srl, Bologna, Italy, a spin-off company of the University of Bologna
started in 2004. He is also a member of the executive board of the Centre for
Research on Complex Automated Systems (CASY), University of Bologna. His
research interests include electric motor drives, nonlinear control, internal combustion engines, and vehicle powertrain modeling and control.
Dr. Rossi was a recipient of the Best Paper Award from the IEEE
TRANSACTIONS ON INDUSTRIAL ELECTRONICS in 1994.
Andrea Tilli was born in Bologna, Italy, on April
4, 1971. He received the Dr.Ing. degree in electronic
engineering and the Ph.D. degree in system science
and engineering from the University of Bologna,
Bologna, Italy, in 1996 and 2000, respectively, where
his thesis was based on nonlinear control of standard
and special asynchronous electric machines.
Since 1997, he has been with the Department of
Electronics, Computer, and System Science (DEIS),
University of Bologna, where, in 2001, he became a
Research Associate. His current research interests include applied nonlinear control techniques, adaptive observers, electric drives,
automotive systems, power electronics equipments, active power filters, and
DSP-based control architectures.
Dr. Tilli was a recipient of a research grant from DEIS on modeling and control of complex electromechanical systems in 2000.