An Encoderless Predictive Torque Control for an ... - IEEE Xplore

IEEE TRANSACTIONS ON INDUSTRIAL ELECTRONICS, VOL. 61, NO. 12, DECEMBER 2014

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An Encoderless Predictive Torque Control for an Induction Machine With a Revised Prediction Model and EFOSMO Fengxiang Wang, Zhenbin Zhang, S. Alireza Davari, Reza Fotouhi, Davood Arab Khaburi, José Rodríguez, Fellow, IEEE, and Ralph Kennel, Senior Member, IEEE

Abstract—The finite control state predictive torque control (FCS-PTC) method is known to produce a fast response. However, when compared with the direct torque control (DTC) method, which is inherently encoderless, the FCS-PTC method has a disadvantage because of its speed dependence. An encoderless FCS-PTC method that is based on a revised prediction model for an induction machine is proposed and experimentally verified in this paper. To observe the unmeasured variables, an encoderless full-order sliding-mode observer (EFOSMO) is applied. Neither the revised prediction model nor the EFOSMO uses the estimated rotor speed value. The disturbance injected by process of speed estimation in the usual encoderless predictive methods is removed from the system. The experimental results show that the proposed algorithm can work well and achieve good performance at a very wide range of speeds. Index Terms—Encoderless control, induction machine (IM), predictive control.

I. I NTRODUCTION

D

IRECT TORQUE control (DTC) is a very important control strategy in the field of electrical drive systems. The DTC method has two main advantages: 1) fast dynamics; and 2) a straightforward algorithm [1]. The DTC method plays a key role in situations where torque control is more favorable, such as traction and paper and steel industries. However, the conventional DTC method has an obvious disadvantage: the torque ripple is considerable. To tackle this drawback, many research efforts have been made. Band-constraining DTC [2], Manuscript received September 30, 2013; revised December 2, 2013 and January 15, 2014; accepted February 20, 2014. Date of publication April 14, 2014; date of current version September 12, 2014. This work was supported by a National Priorities Research Program grant from the Qatar National Research Fund (a member of the Qatar Foundation). (Corresponding author: Z. Zhang.) F. Wang, Z. Zhang, R. Fotouhi, and R. Kennel are with the Institute for Electrical Drive Systems and Power Electronics, Technische Universitaet Muenchen, 80333 Munich, Germany (e-mail: fengxiang.wang1982@gmail. com; [email protected]; [email protected]; [email protected]). S. A. Davari is with the Faculty of Electrical and Computer Engineering, Shahid Rajaee Teacher Training University, Tehran 16788 15811, Iran (e-mail: [email protected]). D. Arab Khaburi is with the Center of Excellence for Power System Automation and Operation, Department of Electrical Engineering, Iran University of Science and Technology, Tehran 13114 16846, Iran (e-mail: khaburi@ iust.ac.ir). J. Rodríguez is with the Department of Electronics Engineering, Universidad Tecnica Federico Santa Maria, 2390123 Valparaiso, Chile (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.2014.2317140

Fig. 1. Simplified block diagrams of the DTC and PTC methods.

DTC by using space vector modulation [3], and predictive DTC [4] are available approaches for the reduction of the torque ripple. Finite control state predictive torque control (FCS-PTC) is a promising method [5]. Aside from reducing torque ripples, the FCS-PTC method also exhibits many other merits; these include easy implementation, a straightforward algorithm, fast dynamics, and the easy inclusion of constraints. The basic idea of the predictive torque control (PTC) method is to calculate the required control signals in advance [6]. In the FCS-PTC method, pulsewidth modulation is unnecessary. The inverter model is considered in the control method. The FCS-PTC method calculates all possible voltage vectors within one sampling interval and selects the best one by using an optimization cost function [7]. To date, the PTC method has been applied in many situations and widely researched, as seen in the articles [8], [9] listed in the reference section. However, compared with the DTC method, the PTC method has two drawbacks: higher calculation time and speed dependence. The PTC method takes more time because of the implementation of the optimization cost function; however, the development of better and faster microprocessing units renders this problem less important [10], [11]. The traditional PTC method for induction machine (IM) applications requires the rotor electrical speed in the prediction steps. The predicted stator current values are dependent on the measured or the estimated speed [5]. However, the DTC method is inherently encoderless. The simplified block diagrams of the DTC and PTC methods are presented in Fig. 1. It is easy to see that the DTC method does not need the rotor speed for the controller; on the other hand, the rotor speed is necessary for PTC of an IM drive. Because of this, the PTC method cannot be applied as flexibly and reliably as the DTC method. The stability of the system is reduced by either the disturbance caused by the speed measurement components or the inaccuracy of the speed estimations [12].

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Because of the merit of low cost and disturbance rejection, many efforts toward achieving encoderless control have been made [13]. For IM drives, the model-based observers are widely used. Luenberger observer [14], model reference adaptive system [15], and extended Kalman filters [16] have been proposed, and they have been verified as good solutions for IM drives. Model predictive control has been recently adapted into encoderless applications [17], [18]. However, in the proposed encoderless FCS-PTC method for IM drives, the model of the prediction is speed dependent, which is a drawback when compared with the DTC method. All mentioned encoderless methods strongly depend on the estimated speed. This means that the estimated speed is a required parameter in the observers. Since speed estimation is usually the last step of the estimation process, the estimated speed is always affected by cumulative errors. When it is fed back to the observer, the flux estimation accuracy will be impacted. The stator voltage model observer is one that does not include rotor speed and is very easy to implement. However, a known problem for such an open-loop observer is weak robustness and low accuracy [17], [19]. This can be compensated for, but the compensation requires the accurate inverter parameters. The slidingmode observer (SMO) is a popular observer, and it has many merits such as disturbances rejection and strong robustness to parameter deviations [20], [21]. More importantly, SMO makes speed-independent design possible [22]. Generally, SMO methods include two categories: reduced-order SMO and full-order SMO (FOSMO). A reduced-order SMO (the voltage model observer with sliding-mode feedback) requires fewer equations and parameters than a FOSMO method; hence, it is easier to implement. However, it reduces the reliability and accuracy of the estimations. In this paper, a FOSMO is used in order to estimate the flux in quality. In [23] and [24], an inherently encoderless PTC that uses a compensated reduced-order observer and SMO method, respectively, is proposed. These works are investigated with simulation results only. As mentioned, although the reducedorder observer is inherently encoderless, it lacks accuracy and reliability [17], [19]. In this paper, a PTC method that uses an encoderless FOSMO (EFOSMO) and an encoderless prediction model is proposed and improved by considering the stator and rotor resistances online identifications and by using a saturation function for reducing the undesirable chattering noise. Using this method, the speed term is omitted from prediction and observer models. Neither the accuracy of the flux estimation nor the prediction of stator current or flux will be influenced by speed calculation errors, which include calculation from synchronous angle to rotor flux speed and slip speed in an IM drive. Like other DTC methods, the system is very flexible. When an adjustable-speed drive is required, it is very easy to add an external speed proportional–integral (PI) controller in order to produce the torque reference for implementing speed control. The proposed system is carried out experimentally, and the performance of the adjustable speed and the load disturbance is thus demonstrated. This paper is structured as follows. In Section II, the models of the IM and the inverter are introduced. In Section III, the proposed encoderless FCS-PTC method is designed. Section IV

Fig. 2.

(Left) Two-level voltage source inverter. (Right) Voltage vectors.

presents the experimental results. This paper is concluded in Section V. II. IM AND I NVERTER M ODEL A squirrel-cage IM can be described by a well-known set of complex equations using a stator reference frame d ψ dt s d ψ − j · ω · ψr 0 = Rr · ir + dt r ψ s = Ls · i s + Lm · i r v s = Rs · is +

ψ r = Lr · i r + Lm · i s 3 T = · p · Im {ψ ∗s · is } 2

(1) (2) (3) (4) (5)

where v s denotes the stator voltage vector; ψ s and ψ r represent the stator flux and the rotor flux, respectively. is and ir are the stator and rotor currents, respectively. Rs and Rr are the stator and rotor resistances, respectively. Ls , Lr , and Lm are the stator, rotor, and mutual inductances, respectively; and ω is the electrical speed. p is the number of pole pairs, and T denotes the electromagnetic torque. The system in stator reference frame αβ can be transformed into rotating synchronous reference frame dq by using Park transformation as follows: d α cos(θ) sin(θ) = (6) − sin(θ) cos(θ) β q where θ is the rotating angle. A two-level voltage source inverter and its voltage vectors are presented in Fig. 2. The switching states S can be expressed by the vector as follows: S=

2 (Sa + aSb + a2 Sc ) 3

(7)

where a = ej2π/3 , Si = 1 means Si is on and S¯i is off, and i = a, b, c. The voltage vector v is related to the switching state S by v = Vdc S

(8)

where Vdc is the dc link voltage. III. E NCODERLESS PTC S YSTEM OF IM Fig. 3 helps to clarify the difference between the proposed system and the usual method. In usual encoderless PTC,

WANG et al.: ENCODERLESS PTC FOR AN INDUCTION MACHINE WITH A PREDICTION MODEL AND EFOSMO

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Lm 1 ˆ d ˆ ˆ r +k2 ·sgn(is −îs ) (11) · is − · ψ ˆ ·ψ ψ = r +j · ω dt r τr τr where τr = Lr /Rr , L2σ = σ · Ls · Lr , and σ = 1 −(L2m /Ls Lr ). Equations (9)–(11) describe the FOSMO in the stator reference frame. The sliding surface is constructed by using the current error between the measured and estimated currents is − îs . The sgn function is applied componentwise for is − îs . To reduce the undesirable chattering problems caused by the sign function, a saturation function is applied and is defined as ⎧ ˆΔ

Fig. 3. Structure of the encoderless PTC. (a) Usual prediction model and FOSMO. (b) Proposed encoderless prediction model and EFOSMO.

FOSMO needs the feedback of the estimated rotor speed ω ˆ. ˆ slip (for This means that the errors generated by either ω ˆ r or ω IM) will influence the accuracy of the estimated stator and rotor ˆ s,r . In addition, the prediction model needs an injection flux ψ of the rotor electrical speed ω ˆ on which the calculations of flux ˆ s,r (k + 1) and stator current prediction îs (k + 1) prediction ψ are based. In the proposed encoderless prediction model and EFOSMO system, the estimated rotor electrical speed ω ˆ is injected neither into the EFOSMO nor into the prediction model. From the diagram, it can be also seen that the estimation of ω ˆ is the last step of the estimation process, which means that ω ˆ includes the accumulative noise and errors. When this inaccurate estimation is fed back to the observer, the accuracy of the flux estimation deteriorates [22]. The same conclusion can be drawn for the prediction model. From this point of view, the proposed encoderless prediction model and EFOSMO system has advantages over the usual one. The system can also work in direct torque mode, which means that torque reference can be set as a fixed value Td∗ . Therefore, the predictive DTC method does not need ω ˆ for speed PI control to generate Tω∗ . With predictive DTC operation, the proposed drive system no longer requires the calculation of ω ˆ , which is only necessary for adjustable-speed drives.

where S = is − îs , x = isα and isβ , and Δ is a tuning parameter. In the applied FOSMO, k1 and k2 are the gains of the observer and shall be selected in such a way that the stability and robustness of the closed-loop control are ensured. In our previous works, pole shifting method [26] and H∞ theory [17] are used to calculate the feedback gains, which leads to a robust stable closed-loop operation in the low speed range (2% of the nominal synchronous speed). The minimum values of normalized gains at this speed that ensure stable operation are k1 = 0.3454+j0.4383 and k2 = −1.95+j1.8631. In order to investigate the stability at higher speeds, the infinity norms of the closed-loop model are calculated for various speeds when gains are kept constant, as calculated for the low speed. In this paper, considering the same approach used in [22] and in order to achieve a better performance in the whole speed range, the gains are introduced with rotor-speed-dependent imaginary components. Considering the fact that (9) is based on the voltage model observer and that, normally, the voltage model observer has flux drift phenomenon and dc offset for removing drifts, compensating the offset, and reducing chattering problems, the PI dynamics are included only in the design for the stator model. The gains of the observer are considered as follows: k1 = k1p +

k1i . s

k2 is set as ˆ k2 = k21 + j · k22 · ω

A. EFOSMO for an IM The aim of this section is to build an encoderless observer to do away with the requirement of the estimated speed in the flux estimations. A FOSMO is selected, and it has the merit of the sliding-mode theory, which is strong robustness [25]. In this work, an SMO is implemented for an IM. According to the IM mathematic model and the SMO theory, a FOSMO could be modeled as d ˆ ψ = v s − Rs · is + k1 · sgn(is − îs ) dt s ˆ s − Lm · ψ ˆ r) îs = 1 · (Lr · ψ L2σ

(9) (10)

(13)

(14)

where ω ˆ is the estimated rotor speed. Considering the noise in this work, reference speed ω ∗ is used instead of the estimated rotor speed. k1p , k1i , k21 , and k22 are SMO constants. To ensure stability, the gains are calculated to be equal to the aforementioned normalized gains at 2% of the nominal synchronous speed. Hence, the normalized gain values corresponding to the minimum pole shift, which is necessary for the stability of the system, are the following: k1p = 0.3454, k1i = 0.0088, k21 = −1.95, and k22 = 93.1550. In addition, the experimental tests at low speed (see Fig. 11) support the closed-loop stability at low speed. The described FOSMO could work for the flux estimation. However, the aim is to design an inherently encoderless

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observer. According to the mentioned model, (9) and (10) are speed independent, but rotor flux model (11) does have a speed term that cannot satisfy the requirement. By defining (11) in synchronous reference frame and taking into account that the imaginary component of (11) is zero, a new rotor flux model, which includes only the real component, can be described, i.e., Lm r 1 ˆr d ˆr îs )·e−j·θˆr ψ rd = ·isd − · ψ +Re k ·Sat(i − 2 s dt τr τr rd (15) where θˆr is the coordinate transformation angle arctan(ψˆrβ / ψˆrα ). In the stator reference frame, the rotor flux ψˆr is

ˆ r = 1 Lr · ψ ˆ s − L2 · i s . ψ σ Lm

(16)

By using (9), (10), and (15), a new EFOSMO is built. Despite the fact that flux estimation is dependent on the rotor flux position, there is no need to use the estimated speed in the observer. The speed estimation procedure involves more error sources than position estimation because the rotor flux is estimated in the observer and no extra observer will be used for position estimation. However, speed estimation requires an extra parameter-dependent observer. Furthermore, the estimation is reinforced by sliding-mode feedbacks. Therefore, the reliable estimated flux will be used for position estimation. B. Revised Prediction Model for IM As an FCS-PTC method, the electromagnetic torque Tˆ(k + ˆ s (k + 1) must be predicted. It must be 1) and the stator flux ψ mentioned that the usual FCS-PTC methods use the electrical speed for state predictions, which means that the accuracy of the prediction is dependent on the measured or the estimated speed. Here, the aim is to build a revised prediction model by removing the rotor speed from the prediction model. To design a predictor, the forward Euler equation is used for discretization, i.e.,

In (19), the predicted stator current irs (k + 1) is necessary for completing the torque prediction. Current predictions are considered in the dq reference frame, where the rotor imaginary component is zero. The current predictions can be described as follows: r îrsd (k + 1) = A · ψˆsd (k + 1) − (B − C · Ts ) r r · ψˆrd (k) − Ts · D · ψˆsd (k)

îr (k sq

+ 1) = A ·

ˆ r (k ψ sq

+ 1)

(20) (21)

where A = Lr /(Ls · Lr − L2m ), B = Lm /(Ls · Lr − L2m ), C = 1/(σ · τr ), and D = B · (Lm /σ · Ls · τr ). The proof of current predictor is presented in the Appendix. With this revised predictor, the predictions of the stator flux, the stator current, and the torque do not use the estimated rotor speed. This means that the prediction model has noise rejection of the speed. The noise generated by the speed estimation, including the calculation of slip speed for an IM and the calculation of the derivative of synchronous angle, will not be injected to this revised prediction model. C. Torque Operation Mode Controller Design The cost function is very flexible and can handle system constraints. It should be designed according to the specific control goals. The cost function includes three items: torque error, stator flux error, and overcurrent protection, i.e., N

∗ ˆ gj = T − T (k + h)j + λ h=1

ˆ · ψ ∗s − ψ s (k + h)j + Im (k + h)j .

(22)

(18)

The torque reference T ∗ is either a constant, as in the case of DTC operation, or is generated by an external speed PI controller for speed-adjustable drives. The coefficient λ denotes the weighting factor, which weights the relative importance of the electromagnetic torque versus flux control. When the same importance is considered, this coefficient should be chosen as λ = Tnom /(ψnom ). The overcurrent protection is activated when the absolute value of the predicted current is higher than its limit. The current limitation is defined as 0 if |i(k + h)| ≤ |imax | (23) Im (k + h) = γ 0 if |i(k + h)| > |imax |

where Ts is the sampling interval. The same sliding-mode function as that introduced in the observer is applied in this prediction model, and the tuning gains are kept the same values as those in the observer; this helps improve the accuracy of the stator flux predictor. To predict the electromagnetic torque Tˆ(k + 1), the IM mathematic model is used, and (5) is considered in the synchronous reference frame, i.e., 3 ˆ sr (k + 1)∗ · îsr (k + 1) . (19) Tˆ(k + 1) = · p · Im ψ 2

where |i(k + h)| = i2α (k + h) + i2β (k + h). If the current is below the maximum current imax , the third part of the cost function does not have any effect on the control procedure. However, if the voltage vector will cause a current beyond the limit, the third part will increase the cost function in a way that the voltage vector will not be chosen. This seems to prevent the control of the motor if the current becomes greater than imax , but the safety of the system is guaranteed. In the cost function, h (h = 1, 2, . . .) is the prediction horizon. j denotes the index of applied voltage vector for the

x(k + 1) − x(k) dx ≈ . dt Ts

(17)

In the next step, which takes place in the synchronous reference frame, stator flux ψ s (k + 1) can be calculated, i.e., ˆ sr (k) ˆ sr (k + 1) = ψ ψ

+ Ts v rs (k) − Rs · irs (k) + k1 · Sat irs (k) − îsr (k)


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prediction. As a two-level voltage source inverter is applied, there are in total eight different switching states, but seven different voltage vectors, as described in Fig. 2. For the FCS-PTC method, all switching states must be considered in one sampling interval. In this case, j = 0, . . . , 6, which means that the cost function must be calculated seven times with a one-step prediction (N = 1) in order to select the best switching state. The zero voltage vector decision is made based on the previous applied switching state. If 011 was the switching state in the previous sample and a zero voltage vector has to be selected, 111 will be chosen. Otherwise, as 010 was the switching state in the previous sampling cycle, 000 will be selected. With this approach, the number of commutations can be reduced. D. Adjustable-Speed Drive In many instances, the adjustable-speed drive is required. Based on the proposed encoderless FCS-PTC method, it is very easy to add a speed control loop. By using the estimated rotor ˆ r (k) and electromagnetic torque Tˆ(k), the estimated flux ψ rotor speed does not feedback into either the EFOSMO nor does it inject into the encoderless prediction model. ω ˆ is calculated as the error between the rotor flux speed and the slip speed, i.e., ω ˆ=

Fig. 4. Simulation results: predicted and measured torque and predicted and measured stator flux magnitude waveforms with an Lm variation (−30%).

ψˆrα (k−1) · ψˆrβ (k)− ψˆrβ (k−1) · ψˆrα (k) 2Rr Tˆ(k) · − . 3p ψˆr2 (k) Ts · ψˆr2 (k) (24)

The torque reference Te∗ is generated by means of a speed PI controller, i.e., T ∗ = kp · (ω ∗ − ω ˆ ) + ki (ω ∗ − ω ˆ ) dt (25) where kp and ki are the gains of the PI modulator. There is no internal current PI control for the system; the inner loop of the PTC method is very fast, and this allows it to increase the bandwidth of the outer speed loop without interference. Therefore, the speed PI controller can reach very fast dynamics. E. Influence of Parameter Variation Parameters of the motor can be measured. However, it is difficult to implement accurate parameters, because they can vary with the motor’s conditions such as motor temperature. From (9), (10), and (16), it can be seen that the accuracy of the observer could be influenced by detuning of magnetizing inductance Lm and resistances Rs and Rr . The sensitivity to parameter detuning is investigated using simulations. Because the torque and the stator flux are controlled in the cost function, the predicted and actual values of the torque and the flux are observed during the parameter variations. In Fig. 4, the predicted and measured torque and the predicted and measured stator flux magnitude over the variations of Lm are observed. The speed reference is 200 r/min, and the motor starts up at time instant (t = 0) with the real value (0.275 mH). The system works in steady state after time instant (about t =

Fig. 5. Simulation results: predicted and measured torque and predicted and measured stator flux magnitude waveforms with a stator resistance variation (+30%).

0.23 s), from when the motor rotates at 200 r/min. At time instant (t = 0.3 s), Lm decreases with a 30.0% magnitude. From the picture, with this variation, the measured torque T is influenced by the variation of Lm . An obvious oscillation can be seen. The predicted torque has a very slow and small change. However, neither the predicted and measured stator flux nor the magnitude is significantly impacted. In Fig. 5, system stability over the variation of stator resistance is observed. At time instant (t = 0.3 s),Rs increases with a 30.0% magnitude. The predicted values are impacted only minimally. Both measured torque and measured stator flux magnitude are influenced by this variation. The reason can be found from the stator flux estimation equation. The stator resistance is an important parameter for the observation, particularly in low speed range. The influence of rotor resistance with a 60.0% variation is also investigated. The simulations confirm that the change in Rr has an almost neglected influence on the system observations. The

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TABLE I PARAMETERS OF THE IM

simulations have also verified that bigger variations of Lm , Rs , and Rr show more influence on the system stability. In this paper, the stator resistance is estimated [27] as follows:

ˆ s = Rs − krs Im ψˆr∗ · Sat irs − îrs dt (26) R

Fig. 6.

Test bench description.

where krs is the gain for the stator resistance estimation. Considering the similar thermal behavior of stator and rotor resistances, it can be assumed that the rotor resistance varies in direct relation to the estimated stator resistance variations [28]. Rˆr can be estimated as ˆ ˆ r = Rr + krr Rr Rs − Rs R Rs

(27)

where Rr is the initial measured value, and krr is a constant for tuning the ratio between the stator and rotor resistances. IV. I MPLEMENTATION AND E XPERIMENTAL R ESULTS A. Test Bench Description The proposed encoderless FCS-PTC method has been tested on an experimental test bench. It consists of two 2.2-kW squirrel-cage IMs. One machine, which is driven by a Danfoss VLT FC-302 3.0-kW inverter, is used as a load machine. The main machine is driven by a modified SERVOSTAR620 14-kVA inverter, which provides full control of the insulatedgate bipolar transistor gates. A self-made 1.4-GHz real-time computer system is used. The rotor position is measured by a 1024-point incremental encoder. The parameters of the main motor are given in Table I. Fig. 6 shows a picture of the test bench. B. Experimental Results In a real implementation, the microprocessor needs time to execute the algorithm. It takes one sampling cycle to generate the control signals. To reach a time-consistent control strategy, the time delay is taken into account and compensated for as discussed in [5]. The first test is to observe the system behavior in steady state. The motor rotates at half speed with a 5-N · m load. Fig. 7 presents the behavior of the stator flux magnitude, the electromagnetic torque, and the stator current. The torque

Fig. 7. Stator flux magnitude, torque, and stator current waveforms at steady state (half speed, 5-N · m load).

ripple is low (less than 1.5 N · m) in this test. The waveform of the stator current is also very good and has a calculated total harmonic distortion of 4.3%. The stator flux magnitude stays almost the same as the reference (0.71 Wb). The second test is developed to observe the system performance in the whole speed range. The flux reference is set to be a constant 0.71 Wb. Fig. 8 shows the measured speed, estimated speed, torque, and stator current waveforms during the full rated speed reversal process. The result shows that the proposed method can work in the full range of speeds. During the transient stage (2772 to −2772 r/min), the electromagnetic torque reaches −7.5 N · m, which is the maximum value generated by the speed PI controller; this is also the rated torque of the motor. To evaluate the accuracy of the speed estimation in the rated speed reversal process, the speed error between the measured speed and the estimated speed is calculated as follows: Δω = (ω − ω ˆ /ω) × 100%. Each group of sampled data from the oscilloscope has 10 k points. In order to clearly express error information, all the presented speed error information in this paper is calculated based on a principle using an average value of every 100 points from the source. Fig. 9 shows the speed error Δω of the full speed process. The


Fig. 8.

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Measured speed, estimated speed, torque, and stator current response during a rated speed reversal maneuver without load (2772 to −2772 r/min).

Fig. 9. Speed error during a rated speed reversal maneuver (2772 to −2772 r/min).

Fig. 11. Measured speed, estimated speed, torque, and stator current response during speed change (1–0.5 Hz).

Fig. 10. Torque response and switching signals during a torque step (2–7.5 N · m).

value is high during the transient stage because it is crossing the low-speed region, including the zero speed. At steady state, the mean value of Δω is about 0.5%. A very important feature of FCS-PTC is its fast torque dynamics. The fast dynamics are due to the absence of both the inner current PI controller and a modulator. To verify this feature, a torque step test is developed. Fig. 10 presents the

torque response and switching signals during this process. In the picture, switching 0 means 000, switching 1 means 001, . . ., switching 7 means 111. The torque step is generated by a sudden change in torque reference. In this test, torque reference changes from 2.0 to 7.5 N · m (rated torque). The torque quasidiscretation phenomenon occurs during the transient state because the torque is estimated based on the sampled currents, which are discreted in time with a constant period of Ts = 62.5 μs. From the picture, it can be seen that the dynamic time is fast: 0.4 ms. Before and after the torque step, the switching changes, among all available switching states, from 0 to 7. During the torque step process, only an active switching signal (signal 6) is selected. This is the reason for a fast settling time. The system performance in low speed range is observed. Fig. 11 presents the measured speed, estimated speed, torque, and stator current waveforms during the low-speed dynamic process. The motor rotates at 60 r/min (1.0 Hz) without load. At time instant (t = 5 s), the speed reference changes to 30 r/min (0.5 Hz). From the presented measured speed and estimated speed waveforms, it can be seen that the system is stable. Fig. 12 shows the speed error Δω. With this low-speed

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Fig. 12. Speed error during a low-speed dynamic process (1–0.5 Hz).

Fig. 14. Speed error during the transient stage with a load torque disturbance (half speed with a 7-N · m load).

V. C ONCLUSION

Fig. 13. Measured speed, estimated speed, and torque response with a load disturbance (half speed with a 7-N · m load).

reference, the absolute mean value of Δω in steady state is 5.0% at 60 r/min and 9.0% at 30 r/min, respectively. The torque response and the stator current are also observed in Fig. 11. The average value of the torque is not zero because of the friction load, which is around 0.6 N · m. During the speed transient (60–30 r/min), there is a very small change in torque due to the small variation of the rotor speed. The stator current behavior is very clear at a different speed reference. With the reduction in the speed, the magnitude of the measured current noticeably decreases. The proposed method results in a better performance than that of the previously discussed encoderless PTC method [17]. The problem of uncertainty injected from calculated speed is solved by this encoderless prediction model and encoderless observer. Therefore, the proposed method is more stable in low-speed performance and shows better torque and current behavior. Finally, the system performance over a full rated load disturbance is observed. The measured speed, the estimated speed, and the electromagnetic torque are shown in Fig. 13. The motor rotates at half rated speed. A 7.0-N · m load is suddenly added to the system. From the picture, it can be concluded that the system has a good response to a load disturbance. At steady state, it works almost the same as shown in Fig. 8. The speed error Δω is presented in Fig. 14. During the transient stage, the value is slightly higher. At steady state, the mean value of Δω is less than 0.5%.

An encoderless PTC method has been proposed in this paper. Neither the prediction model nor the observer uses the estimated rotor speed. An optimization cost function is designed to select the best switching signal. An external speed PI modulator can easily be added to implement the adjustable speed control. To reduce the chattering problem of the EFOSMO, a saturation function is used instead of the sign function. The influence of parameter variations is investigated in simulations. The results show that parameter variation has caused oscillation in the actual flux and torque of the machine in a no-load condition. The impact could be higher with a load. The stator and rotor resistances online identifications are used for reducing this impact and improving the robustness of the system. Experimental results verify that the system works well. Due to the EFOSMO and the encoderless prediction model for an IM, the errors generated by speed calculation process, including slip speed calculation and calculation of derivative of synchronous angle, are neither fed back to the observer nor injected into the prediction model. This advantage further enhances the competitive features of the PTC method. Specifically, it achieves the main advantage of the DTC method, which is an inherently encoderless application. The future work will consider an optimization algorithm to reduce the calculation time for a short sampling interval. Thus, the quality of the measured current will be improved, and the average of switching frequency can be increased; both of these aspects are important for the performance improvement of the encoderless FCS-PTC implementation. Furthermore, a long prediction horizon-based encoderless predictive control would be also an interesting endeavor. A PPENDIX 1) Stator current predictor. To predict the stator current in the synchronous reference frame, the forward Euler discretization is used, i.e., dx x(k + 1) − x(k) ≈ . dt Ts

(28)

By considering the flux (3) and (4) in the dq reference frame, the discretization model can be obtained, i.e.,

ˆ sr (k + 1) − Lm · îr (k + 1) îsr (k + 1) = 1 ψ (29) r Ls


îr (k + 1) = r

r 1 r ˆ ˆ L · ψ (k+1)−L · ψ (k + 1) . s m s r Ls ·Lr −L2m (30)

By using (30) in (29) ˆ sr (k + 1) − B · ψ ˆ rr (k + 1) îsr (k + 1) = A · ψ

(31)

where A = Lr /(Ls · Lr − L2m ), and B = Lm /(Ls · Lr − L2m ). In the dq reference frame, the imaginary part of the r (k + 1) is zero. Thus, the îsr (k + 1) on the rotor flux ψˆrq q-axis can be described as r îrsq (k + 1) = A · ψˆsq (k + 1).

(32)

To reach the predicted îsr (k + 1) on the d-axis, the relationship between rotor flux and stator flux [22] is taken into account, i.e., Ts · Lm ˆr T s ˆr r r (k + 1) = (k) ψˆrd ψ (k) − ψ (k) + ψˆrd Ls · σ sd τr · σ rd (33) With (31) and (33), the predicted îrsd (k + 1) can be calculated as r r r îrsd (k+1) =A· ψˆsd (k+1)−(B −C ·Ts )· ψˆrd (k)−Ts ·D· ψˆsd (k) (34)

where C = 1/(σ · τr ), and D = B · (Lm /σ · Ls · τr ). 2) Rotor speed calculation. ˆ slip For an IM, synchronous speed ωˆr and slip speed ω can be estimated as follows: d ψˆrβ (35) ω ˆr = arctan dt ψˆrα ω ˆ slip =

2Rr Tê (k) · . 3p ψˆr2 (k)

(36)

The rotor speed is an error of the synchronous speed and the slip speed, i.e., ω ˆ=ω ˆr − ω ˆ slip .

(37)

3) Parameters for controllers. 1) The sampling time Ts = 62.5 μs, and the corresponding sampling frequency is 16 kHz. 2) Gains for speed PI controller: kp = 0.21 and ki = 4.01. Parameters are tuned by using MATLAB rltool and slightly revised on the test bench. 3) Parameters in cost function: λ = 32 and imax = 10.0 A. Parameters are decided by obtaining good torque and flux response. R EFERENCES [1] I. Takahashi and T. Noguchi, “A new quick-response and highefficiency control strategy of an induction motor,” IEEE Trans. Ind. Appl., vol. IA-22, no. 5, pp. 820–827, Sep. 1986.

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Fengxiang Wang was born in Jiujiang, China, in 1982. He received the B.S. degree in electronic engineering and the M.S. degree in automation from Nanchang Hangkong University, Nanchang, China, in 2005 and 2008, respectively. From 2008 to 2009, he studied at Tongji University, Shanghai, China. He is currently working toward the Ph.D. degree in the Institute for Electrical Drive Systems and Power Electronics, Technische Universitaet Muenchen, Munich, Germany. His research interests include predictive control and sensorless control for electrical drives.

Zhenbin Zhang was born in Shandong, China, in 1984. He received the B.S. degree in electrical engineering from Harbin Engineering University, Harbin, China, in 2008. Since 2008, he has studied as a combined Master and Ph.D. student in control theory and engineering at Shandong University, Jinan, China. He is currently working toward the Ph.D. degree in electrical engineering at the Institute for Electrical Drive Systems and Power Electronics, Technische Universität München, Munich, Germany. His research interests include multilevel and backto-back power converters, predictive and encoderless control of electrical drives, renewable energy systems, and field-programmable gate array based digital control.

S. Alireza Davari was born in Tehran, Iran, in 1981. He received the M.Sc. and Ph.D. degrees in electrical engineering from the Iran University of Science and Technology, Tehran, in 2006 and 2012, respectively. Between 2010 and 2011, he left for a sabbatical visit at Technische Universitaet Muenchen, Munich, Germany. Since 2013, he has been an Assistant Professor with the Shahid Rajaee Teacher Training University, Tehran. His research interests include encoderless drives, predictive control, power electronics, and renewable energy.

Reza Fotouhi was born in Tehran, Iran, in 1984. He received the M.S. degree in electrical railway engineering from the Iran University of Science and Technology, Tehran, in 2010. From 2010 to 2011, he was with the Engineering Design Team, Transportation Division, MAPNA Company, Tehran. Since 2011, he has been a Research Assistant with the Institute for Electrical Drive Systems and Power Electronics, Technical University of Munich, Munich, Germany. His research interests include power electronics and control of electrical drives.

Davood Arab Khaburi was born in 1965. He received the B.Sc. degree in electronic engineering from the Sharif University of Technology, Tehran, Iran, in 1990, and the M.Sc. and Ph.D. degrees in electrical engineering from école Nationale Supérieure d’électricité et de Mécanique, Institut National Polytechnique de Lorraine, Nancy, France, in 1994 and 1998, respectively. In 1998–1999, he was with University, de Technologie de Compiegne, Compiegne, France. Since January 2000, he has been a Faculty Member with the Electrical Engineering Department, Iran University of Science and Technology, Tehran, where he is currently an Associate Professor, and also a member of the Center of Excellence for Power System Automation and Operation. His research interests include power electronics, motor drives, and digital control. Mr. Arab Khaburi is one of the founders and currently a board member of the Iranian Association of Power Electronics.

José Rodríguez (M’81–SM’94–F10) received the Bachelor’s degree in electrical engineering from Universidad Tecnica Federico Santa Maria (UTFSM), Valparaiso, Chile, in 1977, and the Dr.-Ing. degree in electrical engineering from the University of Erlangen, Erlangen, Germany, in 1985. Since 1977, he has been with the Department of Electronics Engineering, UTFSM, where he is currently a Full Professor and a Rector. He has coauthored over 300 journal and conference papers. His research interests include multilevel inverters, new converter topologies, control of power converters, and adjustable-speed drives. Prof. Rodriguez is a member of the Chilean Academy of Engineering. He has been an Associate Editor of the IEEE T RANSACTIONS ON P OWER E LECTRONICS and the IEEE T RANSACTIONS ON I NDUSTRIAL E LECTRON ICS since 2002. He was a recipient of the Best Paper Award from the IEEE I NDUSTRIAL E LECTRONICS M AGAZINE in 2008, the Best Paper Award from the IEEE T RANSACTIONS ON P OWER E LECTRONICS in 2010, and the Best Paper Award from the IEEE T RANSACTIONS ON I NDUSTRIAL E LECTRONICS in 2007 and 2011.

Ralph Kennel was born in Kaiserslautern, Germany, in 1955. He received the Diploma and Dr.-Ing. degrees in electrical engineering from the University of Kaiserslautern, Kaiserslautern, in 1979 and 1984, respectively. From 1983 to 1999, he worked in several positions with Robert BOSCH GmbH, Stuttgart, Germany. Until 1997, he was responsible for the development of servo drives. From 1994 to 1999, he was a Visiting Professor with the University of Newcastle upon Tyne, Tyne, U.K. From 1999 to 2008, he was a Professor of Electrical Machines and Drives with Wuppertal University, Wuppertal, Germany. Since 2008, he has been a Professor of Electrical Drive Systems and Power Electronics with Technische Universtaet Muenchen, Muenchen, Germany. His research interests include sensorless control of ac drives, predictive control of power electronics, and hardware-in-the-loop systems. Dr. Kennel is a Fellow of the Institution of Electrical Engineers. Within the IEEE, he is the Treasurer of the Germany Section and an Energy Conversion Conference and Exposition Global Partnership Chair of the Power Electronics Society. He is a Chartered Engineer in the U.K.