5126
IEEE TRANSACTIONS ON INDUSTRIAL ELECTRONICS, VOL. 61, NO. 9, SEPTEMBER 2014
Efficient Multivariable Generalized Predictive Control for Sensorless Induction Motor Drives Patxi Alkorta, Member, IEEE, Oscar Barambones, Senior Member, IEEE, José Antonio Cortajarena, Member, IEEE, and Asier Zubizarrreta, Member, IEEE
Abstract—This paper presents the design and the experimental validation of a new linear multivariable generalized predictive control for speed and rotor flux of induction motor. This control approach has been designed in the d–q rotating reference frame, and the indirect vector control has been employed. Load and flux observers, as well as the possibility of including a modelreference-adaptive-system speed estimator, have been considered in the implementation. The proposed controller not only provides enhanced dynamic performance but also guarantees compliance with physical voltage and current constraints. Hence, it ensures that the space vector pulsewidth modulation (SVPWM) always operates in the linear area and that the stator windings are not damaged due to overcurrent. Moreover, the controller includes a novel torque current tracker that allows obtaining an effective electromagnetic torque without a chattering phenomenon. Several simulation and experimental tests have been carried out, both in suitable and adverse conditions, even at zero speed zone, demonstrating that the proposed controller provides an efficient speed tracking and suggesting its use in industry. Index Terms—Induction motor, multivariable generalized predictive control (GPC), sensorless vector indirect control, speed and rotor flux control.
N OMENCLATURE Symbols for Induction Motor Viscous friction coefficient. Bv J Moment of inertia. Magnetizing inductance. Lm Stator inductance. Ls Rotor inductance. Lr Rr Rotor resistance. Stator resistance. Rs Manuscript received February 28, 2013; revised June 8, 2013 and July 28, 2013; accepted July 29, 2013. Date of publication September 9, 2013; date of current version March 21, 2014. This work was supported in part by the Basque Government (BG) and the University of the Basque Country (UPV/EHU) for the support of the S-PE12UN015 (BG), GUI10/01, and UFI11/07 (UPV/EHU) projects. P. Alkorta is with the Department of Systems Engineering and Automatics, School of Engineering of Eibar, University of the Basque Country (UPV/EHU), 20600 Eibar, Spain (e-mail:
[email protected]). O. Barambones is with the Department of Systems Engineering and Automatics, School of Engineering of Vitoria, University of the Basque Country (UPV/EHU), 01006 Vitoria-Gasteiz, Spain (e-mail:
[email protected]). J. A. Cortajarena is with the Department of Electronic Technology, School of Engineering of Eibar, University of the Basque Country (UPV/EHU), 20600 Eibar, Spain (e-mail:
[email protected]). A. Zubizarrreta is with the Department of Systems Engineering and Automatics, Faculty of Engineering of Bilbao, University of the Basque Country (UPV/EHU), 48013 Bilbao, Spain (e-mail:
[email protected]). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TIE.2013.2281172
p Te TL ωm ωr Ψr Imax
Number of poles. Electromagnetic torque. Load or disturbance torque. Mechanical rotor speed. Electrical rotor speed. Rotor flux. Stator rated current.
Symbols for GPC N Future horizon. Control horizon. Nu Rotor flux voltage (vsd ) control weighting factor. λ1 Torque voltage (vsq ) control weighting factor. λ2 Rotor flux current (isd ) tracking weighting factor. δ1 Torque current (isq ) tracking weighting factor. δ2 δ3 Mechanical speed (ωm ) tracking weighting factor. Constraint of the control voltage module. |vs |max |vsd |max Constraint of the rotor flux control voltage module. |vsq |max Constraint of the torque control voltage module. Constraint of the torque control current. Isq max ΔIsd max Differential constraint of the rotor flux control current. I. I NTRODUCTION
G
ENERALIZED PREDICTIVE CONTROL (GPC) is the most popular formulation of model-based-predictivecontrol algorithms [1], which have been successfully implemented in the control of many single-input single-output (SISO) and multiple-input multiple-output (MIMO) processes in industry. The operational principle of predictive control is to calculate in advance the control signal to be applied to a given system, in order to track a known future reference, for which the model of the system is required. This presents an advantage over the rest of the control algorithms, which only consider the past and present tracking errors, but not the future ones. Moreover, predictive algorithms allow the integration of system constraints in the controller design and minimize a cost function considering these, ensuring near-optimal performance of the system. However, the computational cost of predictive control algorithms is higher than the one of conventional control approaches [1]. Hence, predictive controllers have traditionally been implemented in slow processes. Nevertheless, recent technological advances in the performance and execution speed of microprocessors, particularly the digital signal process, have allowed to implement predictive algorithms in faster processes such
0278-0046 © 2013 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission. See http://www.ieee.org/publications_standards/publications/rights/index.html for more information.
ALKORTA et al.: EFFICIENT MULTIVARIABLE GPC FOR SENSORLESS INDUCTION MOTOR DRIVES
as power electronics. The predictive control of electric motor drives is one of the main research areas in the power electronics field. Several research works in this area were based on using the GPC as the main controller for the position or speed [2] of the motor. However, authors included cascade GPC controllers to control rotor flux, stator currents, or rotor flux and torque [3] in addition to the position/speed of the motor shaft. Recent works have focused on motor current control or multivariable control of the motor, where the use of the sensorless concept is of great importance. The predictive current control approach is enhanced in [4] by compensating the time delay in the actuation on switching frequency for the voltage-sourced inverter (VSI) due to large amount of calculations. The combination of this controller in the current loop and pulsewidth modulation (PWM) is employed also in a sensorless speed and rotor flux control of induction motor [5]. Relevant comparative work between some predictive current controls with respect to traditional proportional–integral-PWM current control is presented in [6]. Regarding multivariable predictive control, a multivariable predictive sensorless speed and rotor flux control of induction motor is presented in [7], using the analytical solution and without considering the constraints in the design of the predictive controller. Also, the multivariable predictive scheme for the permanent-magnet synchronous motor (PMSM) is employed in several research works such as [8]–[10], where the results are very satisfactory. In [11], the computational burden of predictive control is emphasized. In recent years, there has been a tendency for control of ac machines such as PMSM and switched reluctance motor [12]–[14]. However, the induction motors are used today due to their low moment of inertia, low ripple of torque, high starting torque, robust architecture, and low cost. The most popular technique employed in the induction motor position and speed control is the indirect field-oriented control (FOC) method [15]. The FOC technique guarantees the decoupling of torque and flux control commands of the induction motor so that the induction motor can be controlled linearly as a separately excited dc motor. Major induction machine modeling research works [16], [17] and comparative studies [18], [19] are still carried out. Nowadays, the model reference adaptive system (MRAS) [20], [21] and the Kalman filter [22], [23] are the most commonly used speed estimators in electrical machines. Although the Kalman filter provides good performance, its high computational cost makes the use of other estimators such as the MRAS preferable. In this paper, a novel GPC approach is proposed for the control of sensorless induction motor drives. The control structure is based on a multivariable GPC controller that allows rotor flux and speed tracking while considering the physical voltage and current constraints of the motor’s stator. The presented controller includes a novel current tracker to avoid the chattering phenomenon, and combining with the MRAS speed estimator, it allows obtaining excellent results in the rotor flux and speed tracking, even at zero speed zone. This paper is organized as follows. Section II describes the dynamics of the induction motor, the design of the multivariable GPC speed and rotor flux regulator, and the VSI voltage and
5127
stator current constraints. In Section III, the description of the MRAS speed estimator and its stability is presented. Section IV contains a brief description of the used experimental platform and the simulation and experimental tests carried out implementing the proposed regulator with and without the speed sensor. Comparative results are given of the performance of the GPC speed controller and the proportional–integral (PI) and sliding mode (SM) speed regulators. Finally, the conclusions are addressed in Section V. II. M ULTIVARIABLE GPC R EGULATOR D ESIGN A. Induction Motor Dynamics The dynamics of the motor can be described by the stator voltage equations and the rotor flux equation, assuming that ψrq ≈ 0 and ψr ≈ ψrd , expressed all in the d–q synchronous rotating reference frame [15]. These sets of equations can be formulated in state-space form, considering the load of the motor TL and the synchronous speed ωe disturbances ⎡ disd ⎤ dt
⎢ disq ⎥ ⎢ dt ⎥ ⎣ dψrd ⎦ dt dωm dt
⎡
⎢ =⎢ ⎣
−
(Rs +Rr L2m /L2r ) σLs
0 Lm Rr /Lr 0 ⎤ ⎡ 1 ⎡ isd σLs ⎢ isq ⎥ ⎢ 0 ⎢ ×⎣ ⎦+⎣ ψrd 0 ωm 0 ⎡ isq ⎢ −isd − Lm ψrd σLs Lr +⎢ ⎣ 0 0
2
Lm Rr /Lr 0 σLs −Rs /(σLs ) 0 −Rr /Lr 0 −KT ψrd /J ⎤ 0 1 ⎥ v sd σLs ⎥ 0 ⎦ vsd 0 ⎤ 0 0 ⎥ ⎥ ωe 0 ⎦ TL −1/J
⎤ 0 0 ⎥ ⎥ 0 ⎦ −Bv /J
and the output equation of the motor ⎤ ⎡ ⎤⎡ 0 0 0 0 isd ψrd ⎢ 0 0 0 0 ⎥ ⎢ isq ⎥ =⎣ ⎦. ⎦⎣ ωm ψrd 0 0 1 0 ωm 0 0 0 1
(1)
(2)
B. Multivariable GPC Speed and Rotor Flux Regulator The typical multivariable GPC scheme for induction motor indirect vector control is based on the use of voltage control signals (the stator voltage linked to the rotor flux, vsd , and the voltage linked to the electromagnetic torque, vsq ). However, the vector control techniques used in the induction motor control employ current control signals, i.e., isd and isq [15]. Although it is theoretically possible to control the induction motor using the voltage control signals, in this case, it should be ensured that the values of these currents are limited in order to guarantee that the stator windings are not damaged. For this reason, isd and isq current constraints should be considered in the design of the GPC regulator.
5128
IEEE TRANSACTIONS ON INDUSTRIAL ELECTRONICS, VOL. 61, NO. 9, SEPTEMBER 2014
predicts the future dynamic behavior of the motor along the horizon N : y = Gu + f .
(5)
Based on this output equation, the control action u∗ to be applied for each time step is calculated by solving the following constrained quadratic minimization problem Z: Z = min (y − w)T Q(y − w) + uT Ru u,x,y
Au u ≤ bu (6) s.t. Ay y ≤ by
Fig. 1. Block diagram of the proposed multivariable predictive regulator for induction motor speed and rotor flux control.
where w contains the future references associated to the outputs y for a horizon of N steps and Q and R are the weighting matrices for tracking error and control input [1]. The constraints of the voltages and currents, corresponding to the control inputs and outputs of the GCP and characterized by Au , bu , Ay , and by , are calculated next. C. Constraints on VSI Voltages
In order to consider all constraints in the GPC regulator ∗ ∗ , vsq , isd , isq ), a novel GPC control scheme is considered (vsd (see Fig. 1). This approach considers the reference signals of ∗ , and both currents the controller, the mechanical speed ωm ∗ ∗ isd and isq . This last one is considered a virtual reference ∗ , obtained from the fourth equainput, as it depends on ωm tion of (1) ∗
i∗sq
∗ + TL J dωdtm + Bv ωm = . KT ψrd
(3)
The proposed GPC regulator is based on the following state-space model that includes measurable and nonmeasurable (noise term η(k)) disturbances [1]:
x(k + 1) = Ad x(k) + Bd u(k) + Dd d(k) (4) y(k + 1) = Cd x(k + 1) + η(k + 1) where the discretized dynamics of the motor are derived from matrices Ac , Bc , and Dc (1) and are calculated for each time step, being constant in the prediction horizon. Similarly, the output matrix Cc is ⎡ ⎤ 1 0 0 0 ⎢0 1 0 0⎥ Cd = ⎣ ⎦ 0 0 0 0 0 0 0 1
In electric drives employing vector control techniques with space vector pulsewidth modulation (SVPWM) with VSI, the module of voltage reference vector vs∗ must be limited in order to ensure that the modulator works in the linear area [15]. Hence, as the voltage vector is formed by the direct and quadrature components
VDC DC 2 + v 2 ≤ V√ . (7) |vs∗ |max = √ → vsd sq 3 3 Operating in (8) and introducing a factor 0 ≤ ε ≤ 1 [8], the constraints of the voltages used to define Au and bu [1] for each time step k can be defined Δvsd max Δvsq max Δvsd min Δvsq min
= vsd max − vsd (k − 1) = vsq max − vsq (k − 1) = vsd min − vsd (k − 1) = vsq min − vsq (k − 1)
(8)
where VDC vsq max = ε √ 3 VDC vsd max = (1 − ε2 ) √ 3 vsq min = − vsq max vsd min = − vsd max .
and the state, input, disturbance, and output variables are x(k) = [isd (k)
isq (k)
u(k) = [vsd (k)
vsq (k)]T
d(k) = [ωe (k)
TL (k)]T
y(k) = [isd (k)
isq (k)
ψrd (k)
ωm (k)]T
ωm (k)]T .
Using (4) and following the procedure detailed in [1], the vector output prediction equation can be calculated, which
D. Constraints on Induction Motor Currents The output constraints of the GPC are those limiting the current in the windings of the motor and avoiding overcurrents that could damage it permanently. These constraints are characterized by the following expression from which Ay and by [1] can be extracted: ymax > Gu + f ymin < Gu + f
(9)
ALKORTA et al.: EFFICIENT MULTIVARIABLE GPC FOR SENSORLESS INDUCTION MOTOR DRIVES
where
⎤ isd max = ⎣ isq max ⎦ 0 ⎡
y max
⎤ isd min = ⎣ isq min ⎦ . 0
5129
⎡
y min
Fig. 2. MRAS equivalent diagram for stability analysis.
The limit values of the direct isd and quadrature isq currents are calculated from the electric equations of the electric motor
εωr = (isα − ˆisα )ψˆrβ − (isβ − ˆisβ )ψˆrα .
isd max = isd + Δisd max isd min = isd − Δisd max
in the α − β stationary reference frame [20]. The MRAS error to estimate the rotor flux and speed is
(10)
where isd is responsible for the rotor flux of the induction motor and is calculated considering its steady-state value, ψr = Lm isd , and a margin of error, i.e., Δisd max . Regarding the isq current, its limit values are calculated considering both the rotor flux and the electromagnetic torque generated by the motor
2 = −I 2 − Isd (11) Isq max = Imax sq min with Isd = ψr_rated /Lm and considering that the rotor flux takes the rated value.
(15)
The rotor speed will be estimated using a PI controller as ω ˆ r = KP εωr + KI εωr dt. (16)
B. MRAS Stability Analysis The MRAS equivalent scheme for its stability analysis is shown in Fig. 2. Considering that the rotor flux q component is zero and the H(s) transfer function H(s) =
εωr ψˆ2 (c11 a3 − c13 ) = rd0 Δωr |sI − A|
2 2 a3 a5 − ωr0 a1 + a2 a5 a4 c11 = − a1 a24 + ωr0
E. ωe , θe , and Load Torque Calculation
c13 = a25 a2 − a5 a4 a1
In order to calculate the θe angle, which is necessary to implement Park’s transformation, the indirect vector control method, obtained by integrating the ωe synchronous speed, is used. As it is known, the synchronous speed has the following expression: ωe = ωs + ω r
(12)
where ωs is the slip speed and ωr is the rotor electrical speed. The relation between ωr and the mechanical speed is p ωr = ωm . 2
(13)
As it is assumed that the d direct and q quadrature components of the rotor flux are decoupled, then, Ψrq ≈ 0, and dΨrq /dt ≈ 0; consequently, the rotor flux is formed only by the direct component [15]. In this context, the slip speed is obtained from (14). The load torque, as ωe , is a part of the measurable disturbance that is calculated using the electromechanical equation of the motor, the fourth equation of (1) Lm R r isq . ωs = ψ r Lr
(14)
III. S TATOR C URRENT MRAS S PEED AND ROTOR F LUX E STIMATOR A. MRAS Rotor Speed and Flux Estimator The MRAS estimator is based on the comparison between the measured stator current of the machine and the estimated current obtained from the stator current model, both expressed
a1 =
Rs L2r + Rr L2m σLs L2r
a4 =
Rr Lr
a5 =
a2 =
Lm R r σLs L2r
a3 =
Lm σLs Lr
Lm R r . Lr
(17)
The characteristic equation of the closed-loop transfer function for the stability analysis is expressed as follows: KI 1 + H(s) KP + = 0. (18) s If the state-space equation of the machine is combined with the mechanical equation and the small signal analysis is carried out, the unstable regions in the torque–speed plane are obtained for the machine in Table I. Fig. 3 shows the stability limits and the no observability line in the speed–torque plane. IV. S IMULATION AND E XPERIMENTAL R ESULTS A. Experiment Platform The control platform is formed by a PC with MatLab7/ Simulink R2007a, dsControl 3.2.1, and the DS1103 real-time interface of dSpace, with a floating point PowerPC processor to 1 GHz. The electric motor used to implement the proposed controller is a M2AA 132M4 ABB commercial 7.5-kW 1445-r/min squirrel-cage induction motor (see Table I), connected to a 540-V dc bus and controlled by a VSI. The load torque is generated by a 10.6-kW 190U2 Unimotor synchronous ac servo motor. The mechanical speed is measured using the G1BWGLDBI LTN incremental encoder of 4096 square impulses per revolution.
5130
IEEE TRANSACTIONS ON INDUSTRIAL ELECTRONICS, VOL. 61, NO. 9, SEPTEMBER 2014
TABLE I PARAMETERS OF THE M2AA 132M4 ABB 7.5-kW 1445-r/min I NDUCTION M OTOR
Fig. 3. Stability limits in the speed–torque plane for the induction machine in Table I.
The electromagnetic torque current command, i∗sq , has been limited to ±20 A (11), to protect against overcurrents in the induction motor’s stator. The SVPWM modulator frequency is fixed at 10 kHz for the speed sensor case and 7 kHz for the speed sensorless case. The sample time employed for the GPC regulator and MRAS estimator is 100 μs, where the estimator constants are Kp = 7.3 and Ki = 630. Regarding the minimization cost function (6), the simulation tests have been realized using MatLab’s quadprog() function, while the experimental tests have been realized using the tnc() function. This routine is written in C programming language [24] and has been specially adapted for the proposed multivariable GPC regulator in an S-Builder Function block of Simulink. B. Rotor Speed and Flux Tracking With Speed Sensor In this section, the simulation and experimental performance results of the proposed D1 multivariable GPC regulator design based on the speed sensor (see Table II) are shown. In order to evaluate the controller, a trapezoidal reference speed of 1000 r/min with a period of 2 s is used. The rotor flux current reference is defined in two stages: The first is generated by a 6.61-A-amplitude 3-s-period sinusoidal signal, while the second provides a constant value of 8.61 A, associated to the rated rotor flux. The load disturbance is 0 during the first 3 s of the test and is constant at 10 N · m between 3 and 4.5 s. A square load torque of ±30 N · m is applied (4.5–7.5 s), and a
TABLE II GPC R EGULATOR D ESIGNS
last stage of ±45 N · m is introduced. Figs. 4 and 5 show the results of the tests. The data show that the speed tracking is very satisfactory, as shown in Figs. 4(a) and 5(a), obtaining an error lower than 5 r/min (0.5%) in steady state for the worst experimental case, i.e., when the load disturbance is greatest (between 4.5 and 10 s), as seen in Figs. 4(b) and 5(b). Although the rotor flux is quite weak and the load torque takes a value of 10 N · m in t = 4 s, it can be seen that the controller is able to provide very good speed tracking. It is important to note that, in this interval of time, as seen in Figs. 4(e) and 5(e), the torque current reaches its maximum value (20 A) but does not exceed it, the same as the case for the stator current, as shown in Figs. 4(g) and 5(g). Later, at t = 4.5 s, an additional torque of 20 N · m is applied, but as the rotor flux is sufficiently large, the motor overcomes its effect and offers an excellent speed tracking. Regarding the isd rotor flux current tracking, Figs. 4(c) and 5(c) show that flux current tracking, and, consequently, also the rotor flux tracking, is very satisfactory, as seen in Figs. 4(d) and 5(d). The rotor flux tracking offers worse results than speed tracking because the proposed regulator has been adjusted in favor of the electromagnetic torque [see (8) and (9) and ε in Table II]. However, this design is effective as it allows arbitrarily generating a weaker or stronger flux, in order to suit the needs of each moment. Figs. 4(f) and 5(f) show the electromagnetic torque generated by the motor and the applied and estimated load disturbance. In these graphs, it can be observed that the electromagnetic torque is smooth but very effective and the estimated load torque is very similar to the real load. Figs. 4(h) and 5(h) show that the electromagnetic torque voltage does not exceed the voltage constraint imposed (7), which is 311 V (see Table II). Figs. 4(i) and 5(i) show the two voltages generated by the GPC regulator. In these graphs, the relevance of the electromagnetic torque voltage with respect to the rotor flux voltage is observed, and also, it is observed that these signals do not exceed the values imposed as constraints. Finally, Figs. 4(j) and 5(j) show the number of iterations used for the minimization functions. In the simulation case, it takes the value between 1 and 2, and in the experimental case, it takes the value between 2 and 10, where the employed execution time is around 60 μs.
ALKORTA et al.: EFFICIENT MULTIVARIABLE GPC FOR SENSORLESS INDUCTION MOTOR DRIVES
Fig. 4. Simulation test of the induction motor controlled with the multivariable predictive regulator D1 design, with speed sensor.
C. Rotor Speed and Flux Tracking Without Speed Sensor The proposed D2 design is based on the previously detailed D1 design for the sensorless case, where the vsd and vsq torque voltage weighting factors (λ2 = 1e − 4 and λ2 = 5e − 3) and the isq and ωm tracking weighting factors (δ2 = 6e1 and δ3 = 25e4) are relaxed for the use of the MRAS speed estimator (see Table II). Moreover, in order to demonstrate the robustness to the parameter variations due to the increase of winding temperature and the moment of inertia, the D2 design considers two cases of uncertainties in the motor parameters. First, the Rs and Rr resistances employed by the controller and the estimators (rotor flux, load torque, and speed) are 30% smaller
5131
Fig. 5. Experimental test of the induction motor controlled with the multivariable predictive regulator D1 design, with speed sensor.
than their respective real values, and J is 50% smaller than its real moment of inertia (see Figs. 6 and 7). Second, the low and zero speed tests, as shown in Figs. 8 and 9, respectively, correspond to Rs , Rr , and J which are 10% smaller than their respective real values. A comparative experimental test set has been carried out to demonstrate the effectiveness of the proposed GPC controller under the first uncertainty case described previously (Rs and Rr + 30%, and J + 50%). The experimental test is similar to the one detailed in the previous section, but using the MRAS speed estimator instead of the speed sensor and employing a rated rotor flux. The test has been also carried out implementing
5132
IEEE TRANSACTIONS ON INDUSTRIAL ELECTRONICS, VOL. 61, NO. 9, SEPTEMBER 2014
Fig. 8. Experimental test of the induction motor controlled with the multivariable predictive regulator D2 design, with parameter uncertainties (Rs Rr and J + 10%) and MRAS speed estimator.
Fig. 6. Experimental test of the induction motor controlled with the multivariable predictive regulator D2 design, with parameter uncertainties (Rs and Rr + 30%, and J + 50%) and MRAS speed estimator.
Fig. 7. Comparative experimental tests: Speed responses of GPC (D2), SM, and PI.
a PI linear speed controller and an SM nonlinear controller. The PI linear speed controller has been tuned with Kp = 1.8 and Ki = 32.94 [15], while the SM nonlinear speed controller has the following parameters: a = 0.3947, b = 77.601, f = 17.544 ∗ TL , k = 400, γ = 1.3, and ζ = 2.5 [25]. Both use the PI current regulators for the isd (Kp = 3.61 and Ki = 395.45) and isq (Kp = 3.15 and Ki = 244) current loops and also the SVPWM modulation [15].
Fig. 9. Experimental test of the induction motor controlled with the multivariable predictive regulator D2 design, with parameter uncertainties (Rs Rr and J + 10%) and MRAS speed estimator.
In Fig. 6, it can be observed that the speed tracking is very satisfactory [see Fig. 6(a)] without and with load torque, where, in the worst case (the end of the test), the speed error in the steady state is around 30 r/min (3%) [see Fig. 6(b)].
ALKORTA et al.: EFFICIENT MULTIVARIABLE GPC FOR SENSORLESS INDUCTION MOTOR DRIVES
Also, it is noted that the rotor flux remains satisfactorily at the imposed reference rated value [see Fig. 6(c)]. Regarding the electromagnetic torque, it can be seen that the motor torque evolution is smooth and it does not chatter [see Fig. 6(d)]. It should be noted that the estimated load torque uses the estimated speed. Fig. 6(e) shows the stator current, where it is observed that the maximum value is limited to 20 A due to the fixed current constraints imposed in the multivariable GPC algorithm. Fig. 6(f) and (g) shows the stator voltage module and its two components, respectively, where their values are into the bounds defined by voltage constraints and are very similar with respect the speed sensor case, as shown in Fig. 5(h) and (i), respectively. Finally, it is observed that the number of iterations is similar to the speed sensor case, and the computational cost is a little greater: around 70 μs (GPC: 33.8 μs, MRAS: 13 μs , Calc. ωe , θe , TL : 5.1 μs , Cal. ABC → dq, dq → ABC, SVPWM: 18.1 μs). Fig. 7 shows the experimental results of the comparative analysis of the three controllers (GPC, PI, and SM), under the same test setup detailed in Fig. 6 for the GPC regulator. The employed execution time by the PI and SM speed controllers is around 23 μs in each case. The comparison of the performance of the three control approaches can be seen in Fig. 7, where it is illustrated that the GPC controller is faster than the PI and variable structure control (VSC), as shown in Fig. 7(a), and also that the GPG controller presents better performance when there is high load effect on the speed tracking, as shown in Fig. 7(b). Fig. 8 shows the performance test of the GPC regulator at low speed. For this test, a square reference speed of ±10 r/min with a period of 5 s is used. The rotor flux current reference is fixed in its rated value, and the load torque takes values into the stable region for this speed, as shown in Fig. 3. Taking into account that it is working at low speed with +10% parameter uncertainties and supporting load torque, from the results shown in Fig. 8(a) and(b), it is observed that the speed tracking is satisfactory, obtaining an error lower than 3 r/min (33%) in steady state for the favorable case (without load) and around 5 r/min (50%) for the worst case (with load, 3.5–4.75, 6–7, and 8–10 s). Regarding the rotor flux, Fig. 8(c) shows that the rotor flux keeps efficiently its rated value. From the result shown in Fig. 8(d), it is observed that the load torque is estimated adequately and that the electromagnetic torque is smooth and effective. Finally, Fig. 8(e) and (f) shows that the performance of the three control approaches can be seen in stator current and stator voltage module values, respectively, which are lower than the corresponding constraints (see Table II). Fig. 9 shows the performance test of the GPC regulator at zero speed, where the rotor flux current reference is fixed in its rated value and the load torque takes values into the stable region for this speed, as shown in Fig. 3. From the results shown in Fig. 9(a) and (b), it is observed that the speed tracking is satisfactory, obtaining an error lower than 2 r/min in steady state for the favorable case (without load) and around 3 r/min for the worst case (with load, between 6.5 and 13 s). Moreover, Fig. 9(c) shows that the rotor flux keeps efficiently its rated value. From the result shown in Fig. 9(d), it is observed that the electromagnetic torque is smooth and, consequently, it does not generate the chattering phenomenon.
5133
V. C ONCLUSION In this paper, a new linear multivariable GPC regulator for induction motor is presented, which controls the mechanical speed and rotor flux. The design uses the linear state equations of the machine, and the nonlinear terms are considered as measurable disturbance terms. The proposed regulator incorporates an electromagnetic current tracker providing the system with very satisfactory speed and rotor flux tracking both in suitable and adverse conditions (load disturbance and parameter uncertainties). The design of the regulator also takes into account the practical voltage and current constraints. Except for stator voltages and currents, the regulator uses estimated variables such as rotor flux, mechanical speed, and load torque. The proposed approach can work with both a speed sensor or a sensorless approach by using a stable MRAS speed estimator. Making use of the proposed multivariable GPC regulator, the induction motor can work with different speed references and amplitudes, even at zero speed zone, with it being robust against important disturbances and parameter uncertainties. The experimental tests carried out provide the experimental validation of the proposed multivariable GPC regulator, presenting faster dynamical response and overcoming high load disturbances quicker than other speed controllers such as proportional–integral and SM. Moreover, experimental tests of the controller ensure its implementability in real time in industrial applications, due to the fact that its computational cost is medium–low.
R EFERENCES [1] E. F. Camacho and C. Bordons, Model Predictive Control, 2nd ed. London, U.K.: Springer-Verlag, 2004, ch. 6. [2] P. A. Egiguren, O. B. Caramazana, A. J. Garrido, and I. Garrido, “SVPWM linear generalized predictive control of induction motor drives,” in Proc. IEEE Int. Symp. Ind. Electron., Cambridge, U.K., 2008, pp. 588–593. [3] R. Hedjar, P. Toumi, P. Boucher, and D. Dumur, “Cascaded nonlinear predictive control of induction motor,” Eur. J. Control, vol. 10, no. 1, pp. 65–80, 2004. [4] P. Cortés, J. Rodríquez, C. Silva, and A. Flores, “Delay compensation in model predictive current control of a three-phase inverter,” IEEE Trans. Ind. Electron., vol. 59, no. 2, pp. 1323–1325, Feb. 2012. [5] J. Guzinski and H. Abu-Rub, “Speed sensorless induction motor drive with predictive current control,” IEEE Trans. Ind. Electron., vol. 60, no. 2, pp. 699–709, Feb. 2013. [6] C. S. Lim, E. Levi, M. Jones, N. A. Rahim, and W. P. Hew, “FCSMPC based current control of five-phase induction motor and its comparison with PI-PWM control,” IEEE Trans. Ind. Electron., vol. 61, no. 1, pp. 149–163, Jan. 2014. [7] E. Souza de Santana, E. Bim, and W. Caradori do Amaral, “A predictive algorithm for controlling speed and rotor flux of induction motor,” IEEE Trans. Ind. Electron., vol. 55, no. 12, pp. 4398–4407, Dec. 2008. [8] S. Chai, L. Wang, and E. Rogers, “Model predictive control of a permanent magnet synchronous motor,” in Proc. Annu. Conf. IEEE Ind. Electorn. Soc., Melbourne, Vic., Australia, 2011, pp. 1928–1933. [9] H. Liu and S. Li, “Speed control for PMSM servo system using predictive functional control and extended state observer,” IEEE Trans. Ind. Electron., vol. 59, no. 2, pp. 1171–1183, Feb. 2012. [10] S. Chai, L. Wang, and E. Rogers, “A cascade MPC control structure for a PMSM with speed ripple minimization,” IEEE Trans. Ind. Electron., vol. 60, no. 8, pp. 2978–2987, Aug. 2013. [11] R. Errouisi, M. Ouhrouche, W.-H. Chen, and A. M. Trynadlowski, “Robust nonlinear predictive controller for permanent-magnet synchronous motors with an optimized cost function,” IEEE Trans. Ind. Electron., vol. 59, no. 7, pp. 2849–2858, Jul. 2012. [12] M.-Y. Wei and T.-H. Liu, “A high-performance sensorless position control system of a synchronous reluctance motor using dual current-slope
5134
[13]
[14]
[15] [16]
[17]
[18]
[19] [20] [21]
[22]
[23] [24] [25]
IEEE TRANSACTIONS ON INDUSTRIAL ELECTRONICS, VOL. 61, NO. 9, SEPTEMBER 2014
estimating technique,” IEEE Trans. Ind. Electron., vol. 59, no. 9, pp. 3411–3426, Sep. 2012. J. Castro, P. Andrada, and B. Blanqué, “Minimization of torque ripple in switched reluctance motor drives using an enhanced direct instantaneous torque control,” in Proc. XXth Int. Conf. Elect. Mach., Marseille, France, 2012, pp. 1021–1026. F. L. M. dos Santos, J. Anthonis, F. Naclerio, J. C. Gyselinck, H. Van der Auweraer, and L. C. S. Góes, “Multiphysiscs NVH modeling: Simulation of a switched reluctance motor for an electric vehicle,” IEEE Trans. Ind. Electron., vol. 61, no. 1, pp. 469–476, Jan. 2014. N. Mohan, Advanced Electric Drives. Mineapolis, MN, USA: MNPERE, 2001, ch. 5. M. Degano, P. Zanchetta, L. Empringham, E. Lavopa, and J. Clare, “HF induction motor modelling using automated experimental impedance measurement matching,” IEEE Trans. Ind. Electron., vol. 59, no. 10, pp. 3789–3796, Oct. 2012. Y. Kawase, T. Yamaguchi, K. Tanaka, M. Otsubo, N. Toida, and K. Sato, “Characteristics analysis of matrix-rotor induction motor using parallel computing system,” in Proc. XXth Int. Conf. Elect. Mach., Marseille, France, 2012, pp. 98–102. P. E. Kakosimos, E. M. Tsamporius, A. G. Kladas, and C. Gerada, “Aerospace actuator design: A comparative analysis of permanent magnet and induction motor configurations,” in Proc. XXth Int. Conf. Elect. Mach., Marseille, France, 2012, pp. 2538–2544. P. Vaclavek, P. Blaha, and I. Herman, “AC drive observability analysis,” IEEE Trans. Ind. Electron., vol. 60, no. 8, pp. 3047–3059, Aug. 2013. T. Orlowska-Kowalska and M. Dybkowski, “Stator-current-based MRAS estimator for a wide range speed-sensorless induction-motor drive,” IEEE Trans. Ind. Electron., vol. 57, no. 4, pp. 1296–1308, Apr. 2010. A. V. Ravi Teja, C. Chakraborty, S. Maiti, and Y. Hori, “A new model reference adaptive controller for four quadrant vector controlled induction motor drives,” IEEE Trans. Ind. Electron., vol. 59, no. 10, pp. 3757–3767, Oct. 2012. M. Butcher, A. Masi, M. Martino, and A. Tacchetti, “Implementation and tuning of the extended Kalman filter for a sensorless drive working with arbitrary stepper motors and cable lengths,” in Proc. XXth Int. Conf. Elect. Mach., Marseille, France, 2012, pp. 2216–2222. S. Jafarzadeh, C. Lascu, and M. S. Fadali, “State estimation of induction motor drives using the unscented Kalman filter,” IEEE Trans. Ind. Electron., vol. 59, no. 1, pp. 4207–4216, Nov. 2012. J.-S. Roy, “TNC: A non linear optimization package,” 2004. [Online]. Available: http://js2007.free.fr/code/index.html O. Barambones and P. Alkorta, “A robust vector control for induction motor drives with an adaptive sliding-mode contro law,” J. Franklin Inst., vol. 348, no. 2, pp. 300–314, Mar. 2011.
Patxi Alkorta (M’12) received the B.S. degree in electronics engineering (specialty in automatic control) from the Faculty of Science and Technology of Leioa, University of the Basque Country (UPV/EHU), Leioa, Spain, in 2004 and the Ph.D. degree in robotics and automatic control systems from UPV/EHU, Faculty of Engineering of Bilbao, in 2011. He currently teaches in the Department of Systems Engineering and Automatics, School of Engineering of Eibar, UPV/EHU, Eibar. His main research interests are induction motor drives, speed and position control, variable structure control, and predictive control.
Oscar Barambones (M’07–SM’13) received the M.Sc. degree in applied physics (specialty in electronics and automatic control) and the Ph.D. degree from the Faculty of Science and Technology of Leioa, University of the Basque Country (UPV/ EHU), Leioa, Spain, in 1996 and 2000, respectively. He currently teaches in the Department of Systems Engineering and Automatics, School of Engineering of Vitoria, UPV/EHU, Vitoria-Gasteiz. His main research interests are induction machine drives, speed control, position control, variable structure control, and predictive control.
José Antonio Cortajarena (M’12) received the B.S. degree in electronics engineering (specialty in automatic control) from the Faculty of Science and Technology of Leioa, University of the Basque Country (UPV/EHU), Leioa, Spain, in 2003. He is currently teaching in the Department of Electronic Technology, School of Engineering of Eibar, UPV/EHU, Eibar. His main research interests are electric machines, torque, speed and position control, sensorless control, and intelligent control, in addition to renewable energy and grid connection and control.
Asier Zubizarrreta (M’10) received the M.Eng. degree in automatics and industrial electronics, and the Ph.D. degree in robotics and automatic control systems from the University of the Basque Country (UPV/EHU), Faculty of Engineering of Bilbao, in 2006 and 2010, respectively. He currently teaches in the Department of Systems Engineering and Automatics, Faculty of Engineering of Bilbao, UPV/EHU, Bilbao. His main research interests are mechatronics, parallel robots, modelbased advanced control, real-time applications, and electric vehicle control.
