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IEEE TRANSACTIONS ON INDUSTRIAL ELECTRONICS, VOL. 53, NO. 1, FEBRUARY 2006

Low-Speed Sensorless Control of Induction Machine

Gregor Edelbaher, Karel Jezernik, Senior Member, IEEE, and Evgen Urlep

Abstract—Induction motor (IM) speed sensorless control, allowing operation at low and zero speed, optimizing torque response and ef?ciency, will be presented in this paper. The magnitude and the orientation angle of the rotor ?ux of the IM are determined by the output of the closed-loop rotor-?ux observer based on the calculation of the extended electromotive force of the machine. The proposed rotor-?ux-oriented control scheme is robust to parameter variations and external disturbances. Both observer and controller utilize the continuous sliding mode and Lyapunov theory. A smooth transition into the ?eld-weakening region and the full utilization of the inverter current and voltage capability are thus possible. The produced torque is a continuous output variable of control. The performance of the proposed method is investigated and veri?ed experimentally on a digital signal processor. Index Terms—Drives, induction motors (IMs), observers, sensorless control, variable-structure systems (VSSs).

mismatched operating variations. A uni?ed control approach for the stator currents and torque control based on the discretetime chattering-free vector control for an IM drive is used [5]. In this way, the chattering-control input has been eliminated and the excitation of the dynamic system without high-frequency oscillations has been achieved. The proposed method is investigated and veri?ed experimentally.

II. M ACHINE D YNAMICS AND P ROBLEM S TATEMENT A. Mathematical Model The state equation of the IM viewed from the stator frame (a–b frame) driven by a voltage source inverter is given by di s 1 s = dt σLs

s dψr =? dt s us s ? Rs is ?

I. I NTRODUCTION ECTOR CONTROL has been widely used for the highperformance drive of an induction motor (IM). Recently, sensorless vector control without the speed sensor is much more focused and has progressed [1]. Though speed sensorless control has a high performance in the high-speed range, lowspeed (including zero speed) control is very dif?cult. If the zero-speed estimation is achieved, sensorless control will be spread for many applications, i.e., for electric vehicle, elevator, crane, and so on. In this paper, by combining variable-structure systems (VSSs) and Lyapunov designs, a new sliding-mode-observer algorithm for the IM is developed. A Lyapunov function is chosen to determine the speed and rotor resistance of the IM simultaneously based on the assumption that the speed is an unknown constant parameter. The proposed method has no integration problem. This method uses measurements of the stator currents only to estimate speed and rotor resistance of the motor, and thus, does not need differentiation of measured states variables, while the inverter output voltage is compensated for nonlinearities using the switching instant detection and nonsymmetrical pulsewidth modulation (PWM) unit realized in a ?eld-programmable gate array (FPGA). It is well known that the conventional vector-control scheme is sensitive to IM parameter variation [3], [4]. We have introduced a discretized current control in the continuous sliding mode with PWM to assure fast dynamic responses in transient states, a minimal current variation for the sinusoidal form in the steady state, and a stable behavior in case of parameterManuscript received June 30, 2004; revised December 10, 2004. Abstract published on the Internet November 25, 2005. The authors are with the Institute of Robotics, Faculty of Electrical Engineering and Computer Science, University of Maribor, Maribor, Slovenia. Digital Object Identi?er 10.1109/TIE.2005.862307

V

Lm dψ s r Lr dt

(1) (2) (3) (4)

Rr Lm s ? jpωr ψ s i r + Rr Lr Lr s

2 Lm s s s s Te = p (ψra isb ? ψrb isa ) 3 Lr J dωr = Te ? Tl dt

where ωr is the mechanical rotor angle velocity, twos s s dimensional complex space vectors ψ s s = ψsa + jψsb , ψ r = s s s s s s s s ψra + jψrb , us = usa + jusb , is = isa + jisb are the stator and rotor ?ux, stator voltage and current, respectively, Te is the motor torque and Tl is the load torque, J is the inertia of the rotor, and p is the number of pole pairs.

B. Torque and Flux Variation In order to illustrate the nonlinear behavior of inductionmotor control, a theoretical deduction of the torque derivative was carried out. This yields, for torque variation Rs dTe Rr 2 pLm s s =? + ( ψ s × us Te + s ? pωr ψ s · ψ r ) dt σLs σLr 3 σLs Lr r (5) where × indicates cross product (active term), de?ned as ψ s r × s s s s = ψ u ? ψ u and · indicates dot product (residual us s ra sb rb sa s s s s s term), de?ned as ψ s r · us = ψra usa + ψrb usb . It can be deduced from (5) that torque variation is the sum of two terms. The ?rst term depends on the stator (Rs ) and rotor (Rr ) resistances and reduces the absolute value of the torque (Te ). It is independent from applied voltage (us s ) and rotor speed (ωr ).

0278-0046/$20.00 ? 2006 IEEE

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The second term represents the effect of the applied voltage vector (us s ) on the torque and is dependent on the operating condition of the machine. It can be noted that some PWM voltage vectors may cause positive torque variation at low dynamic electromotive force (EMF) magnitude value and negative torque variation at high value of ωr ψ s r. Similarly, the rotor-?ux variation can be described with d Lr 1 |ψ s r| = dt Lm |ψ s r|

s s s ψs r · us ? Rs is · ψ r ? σLs

design approach to the torque and ?ux control of the IM to achieve the following. 1) The generated torque becomes a linear output with respect to control states. 2) Regulating the rotor-?ux amplitude can increase the power ef?ciency and make it possible to operate in the ?ux-weakening region. 3) Employing the nonlinear continuous sliding-mode control on the torque and ?ux possesses the robustness to the matched and mismatched uncertainties. Therefore, we have the following problem statement: Design a ?ux/speed observer to estimate the ?ux and speed simultaneously based on the measurement of the stator currents and voltages and then design a corresponding controller to guarantee that the real torque tracks the desired one. III. P ROPOSED C ONTINUOUS VSS S LIDING -M ODE C ONTROL S CHEME A. Design Procedure The so-called sliding-mode motion is represented by the state trajectory being forced to stay in the selected state-space manifold (sliding-mode manifold). The convergence to the sliding-mode manifold takes ?nite time. In continuous time, the control that may guaranty the above properties happens to be discontinuous with high-frequency switching, while in the discrete time, the control that guarantees the motion in the sliding-mode manifold may be continuous [5], [10], [11]. The design of the sliding-mode system generally consists of two procedures: design of the switching surface and design of the sliding-mode controller [2]. The switching surface is designed to obtain a design performance for the system output variables. The VSS theory [12]–[14] has been applied to nonlinear processes. One of the main features of this approach is that one only needs to drive the error to a “switching surface,” after which the system is in “sliding-mode” and will not be affected by any modeling uncertainties and/or disturbances. Let us consider the plant ˙ = f (x, t) + B (x)u x (9)

dis s · ψs r . dt (6)

The variation of rotor ?ux is determined with the dot product between rotor ?ux and applied voltage vector and depends mostly on stator parameters Rs and σLs . C. Control Objective The most popular control method, known as ?eld-oriented control (FOC), transformed the motor equations in a coordinate system that rotates in synchronism with the rotor-?ux vector [6], [7]. These new coordinates are called ?eld coordinates ψ r = ψrd + j 0! The right-hand side of control-system equation for torque variation (5) and rotor-?ux variation (6) will be simpli?ed as follows: dTe + dt = d |ψ | dt r = Lr 1 Lm |ψ r | ψrd usd ? Rs isd ψrd ? σLs disd ψrd . dt (8) Rs Rr + σLs σLr Te (7)

2 p Lm (ψrd usq ? pωr ψsd ψrd ) 3 σLs Lr

In real induction-motor control, rotor ?ux in the q -axis will not be 0 and the FOC method is due mainly to variations of parameters that are inaccurate in sensorless and high dynamic drive applications. The second most popular method, which departs from the idea of coordinate transformation and with analogy with dc motor control, is direct torque control (DTC), which originated from studies of Depenbrock [8] and Takahashi and Noguchi [9]. These innovators propose to replace motor decoupling with hysteresis control, which is in accordance with work in ON – OFF operation of inverter switching elements. In conventional DTC, torque and ?ux errors are used as inputs to hysteresis controllers, which determine the necessary space voltage vectors to reduce these errors. The control method is nonlinear but during steady-state high torque, ?ux and current pulsations occur. In principle, the DTC method is speed sensorless, but due to the use of the voltage stator model of the IM and approximation of stator resistance Rs ? 0 in the ?ux model, ?ux, current, and torque variation by low speed are slightly higher than in the case of nominal speed. The main purpose of this paper is to apply the nonlinear continuous sliding-mode control combined with the Lyapunov

with rank(B ) = m, x ∈ Rn , u ∈ Rm . In VSS control, the goal is to keep the system motion on the manifold S , which is de?ned as S = {x : σ (x, t) = 0}. The solution to achieve this goal can be calculated from the requirement that σ (x, t) = 0 is stable. The control should be chosen such that the candidate Lyapunov function satis?es the Lyapunov stability criteria. The aim is to force the system states to the sliding surface de?ned by σ = G(xd ? x). Firstly, a candidate Lyapunov function is selected as v= σT σ ˙ < 0. > 0 and v ˙ = σT σ 2 (11) (10)

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Fig. 1. Proposed control scheme with the included closed-loop observer.

It is aimed that the derivative of the Lyapunov function is negative de?nite. This can be assured if we can somehow make sure that ˙ < 0. v ˙ = ?σ T D σ D is always positive de?nite. From (11) and (12) ˙ = ?Dσ σ (13) (12)

Using (15) for the equivalent control can be written as u(t) = ueq (t) + (GB )?1 Dσ . (20)

By looking at (17), an estimation for ueq can be made using the property that u(t) is continuous and cannot change too much in a short time as follows: ? eq (t) = u ? (t ? ?t) + (GB )?1 u dσ dt (21)

can be written. Equalizing (13) to 0 results in what is known as “equivalent control” [2]. In other words, the control that makes the derivative of the sliding function equal to 0 is called equivalent control. The derivative of (10) is ˙ d ? G (f (x, t) + Bueq ) = 0. Gx (14)

where ?t is a short delay time. This estimation is also consistent with the logic that ueq is selected as the average of u. By putting (21) into (20), we get the last form for the controller ˙ ). u(t) = u(t ? ?t) + (GB )?1 (Dσ + σ By using Euler interpolation u(t) = u(t ? ?t) + (GB )?1 ((D ?t + 1)σ (t) ? σ (t ? ?t)). ?t (23) (22)

As a result, the equivalent control can be written in the following form: ueq = (GB ) G f (x, t) ? x ˙ From derivative of (10) and using (15) dσ = (GB )(ueq ? u) dt (16)

?1 d

.

(15)

The next step in controller design is the discretization of (23) into the state equation on the assumption that the voltage vectors us (k ) and us (k ? 1) are constant within small kT th and (k + 1)T th sampling intervals. B. Discretized Current Control of IM In this paper, a novel method is being proposed for IM sensorless control based on estimation of EMF es = ((Rr /σLr ) ? jpωr )ψr in which motor mechanical speed information is included (Fig. 1). In order to express the EMF es in the motion equation of an IM in stator reference frames (1) and (2), the following relation is used: is = 1 σLs ψs ? Lm ψ Lr r (24)

is obtained. Then, another equation for equivalent control can be written as follows: ueq (t) = u(t) + (GB )

?1 d σ

dt

.

(17)

By using the de?nition given by (9) and (10) in (13) ˙d ?x ˙ ) = G(x ˙ d ? f (x, t) ? B (x)u) = ?Dσ G(x the control is obtained as ˙ d ? f (x, t) + Dσ . u = (GB (x))?1 G x (19) (18)

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so that the rotor model (2) can be rewritten in the following form: dψ r Lm Rr L2 m = ?es + Rr is + ψr . dt Lr σLs L2 r The stator model of the IM can now be expressed as us = Rs + Rr L2 m L2 r is + σLs Lm dis Rr L3 m ? es + ψr . dt Lr σLs L3 r (26) (25)

The proposed discrete current control is designed to assure fast dynamic responses in transient states, a minimal current variation in the steady state, and a stable behavior in the case of R ? L parameter operating variations (Fig. 1). Let us use (d ) to denote the desired value of the stator current that determines the desired voltage ud s = Rs + Rr L2 m L2 r id s + σLs Lm d did Rr L3 m d s ? es + ψr . dt Lr σLs L3 r (27)

term with the continuous one with the discrete-time equation. The feedback gain matrix Ks is chosen in such a way, that the eigenvalues of the system lie in the unity circle of the z plane, even by R?L parameter variation. The asymptotic stable transient phenomenon is obtained and near “ideal” sliding mode of the system’s operation is realized. Now, the vector of the complete supply voltage remains to be found. It is composed of the feed-forward part in which the EMF es s of the IM is taken into account, as well as the feedback part from the current controller (Cs ). The unknown real EMF ?s is replaced with the estimated one from rotor-?ux observer e s as follows:

s ?s ud s [k ] + ?us [k ]. s [k + 1] = e

(32)

If (26) is subtracted from (27), the difference is obtained as ?us s + Lm d (e ? es s) = Lr s Rs + Rr + σLs L2 m L2 r ?is s

Equation (32) gives the full expression of the total desired stator voltage used to control the PWM, composed of the feed-forward part, which takes into account the back induced voltage es s , depending on the rotor’s ?ux and the rotor speed ω . The feedback part of the current controller compensates for voltage drops in stator windings. To compensate time delay due to measurement and computation time, we introduce the rotation matrix in the control scheme to affect the algorithm predictability C = ej ?θ r ?θr = θr [k ] ? θr [k ? 1]. (33)

d Rr L3 m ?is ?ψ s s + r . (28) dt σLs L3 r

The ac phase voltage of the PWM inverter is uniquely determined from ud s (k + 1) using the coordinate transformation

j ud s = u1 + u2 e

2π 3

On the left-hand side of (28) is the vector of unknown statord s voltage error ?us s = us ? us along with back induced volts age feed-forward compensation error (ed s ? es ), which will be determined by the rotor-?ux observer, and on the right-hand d side of (28) is the vector of the stator-current error ?is s = is ? s is . The current controller is made robust to R?L parameter variations by introducing the negative feedback loop. Due to its outstanding robustness, a discrete variable-structure controller (VSC) will be used. Consequently, the following switching function is chosen: dσ s s + Dσ s = 0 σ s = id s ? is . dt (29)

+ u 3 ej

4π 3

d = ud a + jub .

(34)

From sinusoidal PWM, which constitutes a suitable sequence of active and zero inverter output vectors uk ? π ? 2 Ud ej (k?1) 3 , for k = 1, . . . , 6 uk = (35) 3 ? 0, for k = 0, 7 the stator ?ux moves along a track resembling a circle. The rotor ?ux, however, rotates continuously with the actual synchronous speed along a near-circular path, because its components are sinusoidal. C. Torque and Flux Control The bene?ts of the robust control in the sliding mode of the proposed IM vector scheme are insensitivity to parameter variations and excellent decoupling properties. The sinusoidal form of stator currents is assured with the proposed discretized current control. The actual current tracks the desired value without delay, it is robust to the variations of rotor resistance Rr and mutual inductance Lm , and it decouples in?uences interacting between stator windings. We assume that the rotor back induced voltage es s behaves like disturbance acting in a stator plant. The time constant that determines the transients of stator current (Ts ) is some order smaller than the rotor time constant (Tr ). This physical property of the time-dependent

In the discrete form, the continuous signal is replaced by its difference in the ?rst step of approximation as follows dσ s σ s [k ] ? σ s [k ? 1] ≈ . dt T (30)

The feedback part of nonlinear sliding mode is generated in the discrete form as

s ?us s [k +1] = ?us [k ]+ Ks

σLs [(1+ T · D)σ s [k ] ? σ s [k ? 1]] . T (31)

The advantage of introducing the discrete control law in the sliding mode is twofold. Firstly, the chattering of the control function is eliminated by replacing the discontinuous hysteresis

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IEEE TRANSACTIONS ON INDUSTRIAL ELECTRONICS, VOL. 53, NO. 1, FEBRUARY 2006

the stator-current observer with the combination of voltage and current models

s 1 d? is = dt σLs

?s Lm us s ? Rs is ? Lr

?s ?e s +

Rr Lm ? s ψ σLs Lr s (37)

? r is the adaptation signal, ?s = ((Rr /σLr ) ? jpω where e ?r )ψ which has the meaning of EMF. Estimation error is determined as 1 dεi = dt σLs Lm s Rr L2 m ?s ?s (es ? e ψs s ? ψs s ) ? Rs εi + 2 Lr σLs Lr (38)

?s with εi = is s ? is . Equation (38) could also be rewritten in the following discrete form: ?s [k ] = es [k ] ? e σLs Lr Ts Lm 1 + Ts Rs σLs +

Fig. 2. Closed-loop rotor-?ux observer.

εi [k ] ? εi [k ? 1] Rr L2 m ?r ?ψ σLs L2 r (39)

variability of stator current and rotor-?ux transients can be pro?tably used to express the relation between the rotor ?ux and stator current. The direct axis of magnetizing current Isd and the quadrature axis of the torque producing stator current Isq in the reference frame tied to the rotor-?ux vector can be expressed. Thus, the vector orientation of the rotor ?ux is de?ned as ψ r = ψrd + j 0 and I s = Isd + jIsq . In this way, the desired components of stator current in the stationary reference frame are determined as id sa = and id sb =

d d ψr 3 Lr Te cos θr . sin θr + d| Lm 2 Lm |ψr d d ψr 3 Lr Te sin θr cos θr ? d Lm 2 Lm |ψr |

2 where Rs = Rs + Rr (L2 m /Lr ). The sliding-mode algorithm could be used to calculate new value of the adaptation ?s signal e s as follows:

?s ?s e s [k ] = e s [k ? 1] +

Ke ((I + T D i )εi [k ] ? εi [k ? 1]) . T (40)

The process of zeroing the current error utilizing ?ux-regulated PWM is the essence of sliding-mode control. A rotor-?ux observer could be selected having the structure of the rotor model ? Rr L2 dψ r m ?s ?s = ?e ψ r + Kψ ?ψ r s + dt σLs L2 r

s

(41)

(36)

IV. C LOSED -L OOP R OTOR -F LUX O BSERVER It is known that the rotor ?ux is needed for the implementation of torque or speed control. Since measurement of rotor ?ux is intrusive and degrades mechanical instability, it is common practice to estimate rotor ?ux rather than measure it. If the angle speed is available, the rotor speed can be estimated with an observer (Fig. 2). However, if no information about the mechanical variables is acquired, the design of the observer is no more a trivial problem [15]. Conventional methods for the estimation are the use of current and voltage model ?ux observers. In this paper, the used rotor-?ux observer is based on the estimation of EMF ?s e s in which motor mechanical speed information is included (Fig. 1). This leads to the following selection of the structure of

where Kψ ?ψ r is the correction factor. To calculate the rotor ?uxes from (41), an integrating process is needed. However, it is dif?cult to maintain the system stability of a pure integrator due to motor-parameter variation. In order to overcome this problem, the model (41) is modi?ed by the use of the correction ? r , where ψ d could be calcu? r = ψd ? ψ factor such that ?ψ r r lated from the current model ? dψ r = dt

d

?

Rr ? d + Rr Lm is . + jpω ?r ψ r Lr Lr s

(42)

Since the adaptation signal has the meaning of EMF, mechanical rotor speed can be estimated from ?s ?s e r × ψr ?s ψ r

2

ω ?r =

.

(43)

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Using the EMF (40), an adaptation algorithm for rotorresistance estimation can also be obtained as follows: ?s · e ?r ?s R ψ = r 2s . ?r σL ?s ψ r (44)

V. F LUX C ONTROL FOR E FFICIENCY I MPROVEMENT The maximum output torque developed by the machine is dependent on the allowable current rating and the maximum voltage that the inverter can apply to the machine. Therefore, to use the inverter capacity fully, it is desirable to use the control scheme considering the voltage and current limit condition, which can yield the maximum torque per ampere over the entire speed range. The maximum stator voltage is determined by the available dc link voltage Udc and PWM strategy [16]. The IM will operate at the highest possible rotor ?ux maintained by the dc link voltage Udc in the entire speed range. Operation at the highest ?ux assures a maximum motor torque at the lowest slip frequency. As the motor losses are proportional to the slip frequency, we may expect the proposed control method to yield a highly ef?cient electromechanical energy conversion of the drive. The ef?ciency of IM drive is improved by adjusting the rotor-?ux level automatically in accordance with the torque command, as shown with (45). This will be especially true at light loads when low ?ux level is used. The maximum allowed rotor ?ux will depend on actual stator frequency and dc link voltage Udc Udc d ψr max = kv √ , 3ωs and torque-dependent optimal rotor ?ux

d ψr opt =

Fig. 3. Bridge branch output voltage; reference ud i , output ui .

A. Compensation of Power-Inverter Nonlinearity The task of the power inverter is to produce the desired voltage on the stator winding with the switching of switch elements controlled by the PWM modulator. Since the switching times of the existing transistors are not in?nitely short, the necessary blanking time to avoid short circuiting the dc link during commutations must be introduced, which is also known as dead time. This small time delay is the most important cause of the inverter nonlinearity and introduces a magnitude and phase error in the output-voltage vector. In addition to the dead time, there is also the ?nite voltage drop across the switch during the ON state, which introduces an additional error in the magnitude of the output voltage. The dead time introduced by the inverter causes serious waveform distortion and fundamental voltage drop when the switching frequency is high compared to the fundamental output frequency. One way of dealing with inverter imperfections is to use a nonlinear inverter model to compensate for the dead time [17], which gives quite a good voltage estimation. However, when exact knowledge is needed, such models often are not suf?cient. Due to the bridge nonlinearities, e.g., delays of the inverter driver and the bridge switching elements, actual switching of the bridge branch (i = 1, 2, 3) voltage is delayed for ?t according to the reference voltage ud i , as shown in Fig. 3. The switching period is represented with T . The mean value of branch voltage can be approximated as

T

kv ? 1

(45)

2 3p

2 L2 m Rr + Lr Rs d T . Rs

(46)

d d The effective command will be used as ψr opt ≤ ψr max .

VI. C OMPENSATION OF P OWER -I NVERTER N ONLINEARITY AND S TATOR -R ESISTANCE A DAPTATION In speed and voltage sensorless control, the output-voltage compensation is needed because the output-voltage error is not negligible in the very-low-speed region. The following estimations and compensation are required for the low- and zero-speed operation: 1) rotor-resistance estimation presented in (44); 2) stator-resistance estimation; 3) output-voltage compensation. In this paper, a new algorithm for the Rs and Rr estimation under the speed transient state is proposed. The adaptive algorithm is used for the Rs estimation. The simultaneous estimation of the output voltage and Rs is very dif?cult without the voltage sensor, because each piece of information is in the same stator equation.

1 ui = T

0

ui (t)dt =

Udc ti . T

(47)

The goal of the compensation is to achieve the branch voltage equal to its reference, ui = ud i . Both time delays, delay at the rising edge and delay at the falling edge of the bridge branch voltage waveform, are measured and used to compensate the switching delays using the reference voltage. The resulting reference is uc i = Udc c Udc d t = ti + ?tia ? ?tib T i T (48)

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robust property to stator-resistance variations is proposed. First, we de?ne a new quantity q as a cross product

s s s s q = is s × (us ? Rs is ) = is × us

(51)

q represents the instantaneous reactive power maintaining the magnetizing current. From (51), it is evident that this quantity is independent from stator resistance due to the property of the cross product. The estimated EMF from the sliding-mode ?ux ? s dependent observer is resistance R ?s q ? = is s. s ×e

Fig. 4. Closed-loop compensation diagram.

(52)

By using discrete-time control, the reference voltage is compensated using the measured delays, and delayed for one sampling time due to the discretization. The resulting reference voltage in discrete form is uc i [k ] = Udc d ti [k ] + ?tia [k ? 1] ? ?tib [k ? 1] . T (49)

Using (51) as the reference model and (52) as the adjustable model, respectively, a model reference adaptive system (MRAS) can be used, where proportional and integral operation are utilized as the adaptation mechanism

t

? s [k + 1] = Kp (q ? q R ?) + Ki

0

(q ? q ?)dτ.

(53)

The stator-voltage estimation is based on the compensated stator voltage and on the measured error at the same discrete time u ?i [k ] = uc i [k ] ? Udc (?tia [k ] ? ?tib [k ]) . T (50)

? s from (53) is then used The estimated stator-resistance value R as an input signal to the sliding-mode ?ux and speed observer (Fig. 2). VII. P ARAMETER S ENSITIVITY OF THE P ROPOSED O BSERVER The accuracy of the estimated state variables is dependent not only on observer inputs, but also on the parameters used in the observer. There are two types of parameters in the observer: model parameters and observer own parameters. It is obvious that model parameters must be known exactly; but unfortunately, all parameters are not perfectly known, since some of them are changing during the operation. From the structure of the observer as well from the observer own parameters depends observer sensitivity to the parameter variations. For the evaluation of observers, estimation accuracy frequency response (EAFR) is primarily used [18]. Ideal EAFR amplitude of an observer would be 1 at all frequencies and EAFR phase of 0 rad. For the presented rotor-?ux observer, the EAFR is shown in Fig. 5. The presented EAFR is shown for the observer with mismatched stator resistance and different values of correction factor Kψ . It can be seen from Fig. 5 that a high value of Kψ is required at very low frequencies and low Kψ is required at high frequencies. This way, the observer with minimal phase and magnitude errors at all frequencies can be achieved. VIII. R ESULTS The IM used for simulations and experiments is type T100L6, 380 V three phase, Y connection, 50 Hz, 1.5 kW, three pole, 920 r/min. Parameters in the per-phase steadystate equivalent circuit are: stator resistance Rs = 5.1 ?, rotor resistance Rr = 5.28 ?, stator and rotor inductance Ls = Lr = 0.218 H, mutual inductance Lm = 0.196 H, and motor and load inertia J = 0.0046 kg · m2 .

The advantage of (50) instead of u ? i = ud s is in using the accurate measured signal. The second advantage is that there is no sampling delay between the reference signal and the measured value. Conversion to the vector form is achieved by using the known 3 → 2 transformation. The implemented compensation is shown in Fig. 4. The measurement of the branch switching delays is implemented by using the analog comparator between the bridge branch voltage and the half of the dc link voltage. The discrete measurement of the time delay between reference and branch voltages is implemented in the FPGA, since a pulse-generation unit is also implemented in the same FPGA. For that purpose, the pulse-generation unit must support the nonsymmetric pulse generation, which is more complicated to implement, but gives better results considering the phase-to-phase voltages. B. Stator-Resistance Adaptation The dynamic performance of the IM control strongly depends on motor-parameter accuracy. A parameter mismatch produces an error in ?eld orientation and undesirable coupling between the ?ux and torque control. Although it is possible to determine the motor parameters in advance, signi?cant changes occur during the operation. Among these parameters, the exact knowledge of the stator resistance is especially an important issue. The problem is present in the case of the use of the speed sensor, but its importance increases when in sensorless applications. Here, a new approach to speed estimation with

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Fig. 5. Estimation accuracy frequency response of the presented rotor? s ). ?ux observer with mismatched parameter of stator resistance (Rs = 1.1R (a) Kψ = 122. (b) Kψ = 40. (c) Kψ = 2.

? r ). (a) Rotor-?uxFig. 7. In?uence of rotor-resistance mismatch (Rr = 1.2R estimation magnitude error. (b) Rotor-?ux-estimation phase error.

? r ). (a) Desired, Fig. 6. In?uence of rotor-resistance mismatch (Rr = 1.2R actual, and estimated speed. (b) Actual and load torque.

A. Simulation Results In addition to the EAFR, simulation can also be used for the parameter sensitivity analysis of the proposed observer. The presented simulation results in Figs. 6 and 7 were made using MATLAB and Simulink software package to show the in?uence of inaccurate value of rotor resistance in the observer, and on the estimation of rotor ?ux and speed. Therefore, simulation test consists of several parts: starting of motor at 0.5 s, lowspeed no-load slow reversing starting at the 1-s time mark, operation at constant speed with load change starting at 7 s, and speed change at constant load starting at 8.5 s. Desired, actual, and estimated speeds during the test are presented in Fig. 6(a), whereas actual and load torque are presented on Fig. 6(b). At no load of the machine, speed estimation error is negligible and the actual speed follows the desired one closely, as can be observed from Fig. 6(a). Transition through zero speed as well as zero frequency succeeds without any problem. The speed estimation error is load dependent, but the system still remains stable.

Fig. 8. Block diagram of the DSP-2 board.

In Fig. 7, the in?uence of rotor-resistance mismatch during the simulation test on the rotor-?ux-estimation accuracy is shown. Fig. 7(a) presents rotor-?ux-estimation magnitude error, whereas rotor-?ux-estimation phase error is shown in Fig. 7(b). It is evident that the rotor resistance has a very small in?uence on both estimated rotor-?ux amplitude as well as for FOC, more important rotor-?ux phase. B. Experimental Results Experiments were obtained using a DSP-2 [19] motor controller board (Fig. 8), based on ?oating-point DSP TMS320C32 from Texas Instruments Incorporated. Connection to the peripherals and PWM unit is made with an FPGA. All measured and controller internal variables are accessible through the serial link to the PC, where graphical data-analysis software can be run. The sampling time of the measurements and computation of control algorithm are both 200 ?s. A brushless ac servomotor

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Fig. 11. Zero rotor speed operation. (a) Desired, actual, and estimated rotor speed. (b) Desired and load torque.

Fig. 9. Experimental system.

stator frequency, but the system remains stable even in such conditions. The low-speed performance can further be evaluated at zerospeed operation with fast load changes. Desired, actual, and estimated speeds are presented in Fig. 11(a). Load changes are presented in Fig. 11(b). Estimation error is very small in steady state and becomes higher at points of load changes, which occur very fast. Otherwise, zero speed is holding satisfactorily. The applied torque of the load machine is measured, and thus, the value of the torque applied by the drive machine is obtained. Actual values of load torque as well as desired values of torque obtained by the speed controller are presented in Fig. 11. It can be seen that the desired torque matches the load torque in steady state. IX. C ONCLUSION In this paper, an advanced discrete-time chattering-free control scheme of the IM has been presented. The distinctive features of the scheme are its robustness to different initialcondition values and parameter mismatch of nonlinear behavior of induction-motor control. Nonlinear control principles have been used, namely the combined feed forward and the robust negative feedback path based on the VSS-control approach. In the used discrete-time expression, the discontinuous operation of VSS controllers in the sliding mode, which was the main obstacle in the past for the use of these kind of techniques in practical applications, has been replaced by the continuous one. In this way, the chattering of the control input signal has been eliminated and the excitation of the dynamic system, without high-frequency oscillation, has been achieved. Unlike other VSC algorithms intended to avoid chattering, the presented approach uses the information about the distance from the sliding-mode manifold to derive the control. R EFERENCES

[1] K. Rajashekara, Sensorless Control of AC Motors. Piscataway, NJ: IEEE Press, 1996. [2] V. I. Utkin, “Sliding mode control design principles and applications to electric drives,” IEEE Trans. Ind. Electron., vol. 40, no. 1, pp. 23–36, Feb. 1993.

Fig. 10. Slow reversing. (a) Desired, actual, and estimated rotor speed. (b) Components of stator current in the stationary reference frame.

mechanically connected to the IM under test was used as the load machine. The control of both speed and applied torque is possible, thus, a hardware-in-the-loop operation can also be performed (Fig. 9). The performance of the presented control system with sensorless sliding-mode rotor-?ux observer is shown with two experiments; low-speed no-load slow reversing (Fig. 10) and zero-speed operation with fast load changes (Fig. 11). For the speed sensorless systems, no-load low-speed slow reversing is a very tough task because of longer operation near zero stator frequency as well as inevitable at zero frequency during the speed reversal. Desired, actual, and estimated values of speed at low-speed reversing from ?5 to 5 rad/s are presented in Fig. 10(a), whereas in Fig. 10(b), components of stator current in the stationary reference frame are presented. As can be seen from Fig. 10, transition through zero speed succeeds, and some problems occur during the transition through the zero

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[3] K. Ohnishi et al., “Model reference adaptive system against rotor resistance variation in induction motor drives,” IEEE Trans. Ind. Electron., vol. IE-33, no. 3, pp. 217–223, Jul. 1986. [4] R. D. Lorenz and D. W. Novotny, “A control system perspective of ?eld oriented control for AC servo drives,” in Proc. Contr. Eng. Conf., Chicago, IL, 1988, pp. XVIII-1–XVIII-11. [5] S. V. Drakunov and V. I. Utkin, “On discrete-time sliding modes,” in Proc. Nonlinear Contr. Syst. Design Conf., Capri, Italy, 1989, pp. 273–278. [6] P. Vas, Vector Control of AC Machines. Oxford, U.K.: Clarendon, 1990. [7] W. Leonhard, Control of Electric Drives, 3rd ed. Berlin, Germany: Springer-Verlag, 2001. [8] M. Depenbrock, “Direct self control (DSC) of inverter-fed induction motors,” IEEE Trans. Power Electron., vol. 3, no. 4, pp. 420–429, Oct. 1988. [9] I. Takahashi and T. Noguchi, “A new quick response and high ef?ciency control strategies of an induction motor,” IEEE Trans. Ind. Appl., vol. IA-22, no. 5, pp. 820–827, Sep./Oct. 1986. [10] K. Furuta, “Sliding mode control of a discrete system,” Syst. Contr. Lett., vol. 14, no. 2, pp. 145–152, Feb. 1990. [11] V. I. Utkin, “Sliding mode control in discrete time and difference systems,” in Variable Structure and Lyapunov Control, A. S. Zinober, Ed. London, U.K.: Springer-Verlag, 1993. ? [12] A. Sabanovi? c, “Sliding mode in robotic manipulators control systems,” Electrotech. Rev., vol. 60, no. 2–3, pp. 99–107, 1993. ? ? [13] A. Sabanovi? c, K. Jezernik, N. Sabanovi, and K. Wada, “Chattering free sliding mode,” in Proc. Workshop Robust Contr. Variable Structure Lyapunov Tech. (VSLT), Benevento, Italy, 1994, pp. 315–318. [14] V. I. Utkin, Sliding Modes in Control and Optimization. Berlin, Germany: Springer-Verlag, 1992. [15] P. Vas, Sensorless Vector and Direct Torque Control. London, U.K.: Oxford Univ. Press, 1998. [16] J. Holtz, “Pulsewidth modulation for electronic power conversion,” Proc. IEEE, vol. 82, no. 8, pp. 1194–1214, Aug. 1994. [17] ——, “Sensorless vector control of induction motors at very low speed using a nonlinear inverter model and parameter identi?cation,” in Proc. IEEE IAS Annu. Meeting, Chicago, IL, 2001, pp. 2614–2621. [18] R. D. Lorenz, “Observers and state ?lters in drives and power electronics,” in Proc. IEEE IAS OPTIM . Brasov, Romania, 2002, CD-ROM. ? [19] D. Hercog, M. Curkovi? c, G. Edelbaher, and E. Urlep, “Programming of the DSP2 board with the Matlab/Simulink,” in Proc. IEEE Int. Conf. Industrial Technology (ICIT), Maribor, Slovenia, Dec. 2003, pp. 709–713.

Gregor Edelbaher received the B.S. and Ph.D. degrees in electrical engineering from the Faculty of Electrical Engineering and Computer Science, University of Maribor, Maribor, Slovenia, in 1999 and 2004, respectively. He is currently a Teaching Assistant in the Institute of Robotics, University of Maribor, where, since 1999, he has been a Faculty Member. His research interests include electrical drives, especially control of ac drives.

Karel Jezernik (M’77–SM’04) received the B.S., M.S., and Ph.D. degrees in electrical engineering from the University of Ljubljana, Ljubljana, Slovenia, in 1968, 1974, and 1976, respectively. He was a Visiting Research Fellow at the Institute of Control, TU Braunschweig, during 1974–1975. In 1976, he joined the University of Maribor, Maribor, Slovenia, where, since 1985, he has been a Full Professor and Head of the Institute of Robotics. His research and teaching interests include automatic control, robotics, power electronics, mechatronics, and electrical drives. Prof. Jezernik is the Vice President for Technical Activities of the IEEE Industrial Electronics Society.

Evgen Urlep received the B.S. degree in mechatronics from the Faculty of Electrical Engineering and Computer Science, University of Maribor, Maribor, Slovenia, in 2003, and is currently working toward the Ph.D. degree in electrical engineering and computer science at the University of Maribor. His ?eld of interest is mostly on the control of servo drives, especially speed sensorless control of synchronous machines.

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