Robust Finite Control Set Model Predictive Current Control for Induction Motor Using Deadbeat Approach in Stationary Frame

The model predictive control increased its prominence in the field of induction machine drives. However, the performance of this strategy depends on the accuracy of the machine model parameters. In order to overcome this deficiency, this paper proposes a robust model predictive current control employing indirect field-oriented control in stationary reference frame using the stator current and the rotor flux vectors as state and stator voltage vector as the input. The control algorithm combines the classical model predictive control with the deadbeat approach in order to calculate the applied stator voltage vector in two components: one element considers the stator current reference, and another employs the disturbances caused due to the parameter errors, which allows to compensate the parameter mismatches in the plant model. The minimized cost function employs the predicted stator voltage vector to select the voltage vector to be applied to the stator terminals of the motor. The control method performance was verified using an experimental test bench analyzing the system steady-state and dynamic actions. In this way, the results corroborate the proposed controller robustness against parametric variations.


I. INTRODUCTION
The induction motor (IM) has been widely used in applications that demand high torque control accuracy over a wide speed range such as in electric vehicles [1], [2], [3], [4], [5]. Considering the control strategies for IM, one solution that has recently been used is the model predictive control (MPC) due to its high flexibility and simplicity [6], [7]. The basic idea of MPC is to use a model of a plant to predict the behavior of system variables within a time horizon. In this way, the minimization of a cost function using the idealized and the predicted behavior, an optimized control The associate editor coordinating the review of this manuscript and approving it for publication was Jiefeng Hu . performance can be obtained selecting the control input to the system [8], [9]. However, the classical MPC employs the model of the IM to design the controllers. So, it can degrade the performance of the controller due to the parameter errors.
Recently, some MPC strategies have been proposed in order to increase the robustness against parameters mismatch of the machines as presented in [10], [11], [12], and [13]. In [14], the ordinary MPC employs the nonlinear model considering the taylor representation of IM for each equilibrium point. A predictive torque control for IM employing a combined integral sliding mode observer and an adaptive Luenberger observer was presented in [15].
The work [16] proposes a model-free predictive current control (MFPCC) based on the ultra-local model. The whole controller only needs the input and output data without the need for the motor parameters. In [17], a model-free control is also proposed and the control signal is determined by the sampled current and the current error. In [18] and [19], linear extended state observers are employed to this task. The robust MPC considering a generalized proportional-integral observer is presented in [20]. The reported results are satisfactory during the IM parameter errors.
In [21], it is proposed a robust deadbeat predictive current control (DPCC), which employs the error between the measured currents and the predicted currents to compensate the influence of machine parameter mismatches. This method presents a good robustness against parameter mismatches.
Aiming to decrease the error of the stator current prediction, [22] and [23] employ the Luenberger observer to achieve this goal. The robust deadbeat-based current control presented by [22] has satisfactory dynamic performance and strong robustness in the experimental tests compared to PI current control and traditional deadbeat. The experimental results of the MPC strategy proposed by [23] have superior performance against the traditional MPC current control.
This paper proposes a robust finite control set model predictive current control for IM in stationary reference frame using the stator current and the rotor flux vectors as state and stator voltage vector as the input variables. The proposal employs indirect field-oriented control (IFOC) and presents its structural robustness properties [24]. In this case, the proposed solution computes two stator voltage vectors in which one predicts the voltage vector considering the classical MPC and another stator voltage vector element considering the deadbeat approach to compensate the parameter errors of the IM. Hence, the minimized cost function considering the cited voltage elements selects the stator voltage vector to be applied to the stator terminals of the IM. The results endorse the performance of the proposed robust model predictive current control in stationary frame for IM in normal operation, during the presence of the parameter errors and against the classical predictive current control.

II. ROBUST FINITE CONTROL SET MODEL PREDICTIVE CURRENT CONTROL FOR IM
The proposed robust finite control set model predictive current control of IM employs IFOC. In this way, the proposal predicts the future actions of the stator voltage vector of IM considering the parameter mismatches, selecting the voltage vector to be applied to the stator of the IM considering the minimization of the cost function. The mentioned cost function was computed by using the predicted and the reference voltage vectors.

A. INDUCTION MOTOR MODELLING
The model of IM can be expressed using the vector notation and it can be represented in the stationary frame according to [25]: (4) where L r and L s are the self inductance of the rotor and stator and L m corresponds to the mutual inductance. R r and R s are the rotor, stator resistances. J m is the total inertia, T load is the load torque, p is the number of pole pairs and ω mec is the rotor mechanical speed. The vectors i αβs = i α + ji β ; v αβs = v α + jv β and ψ αβr = ψ αr + jψ βr are from the stator current, the stator voltage, and the rotor magnetic flux, respectively. The symbol ''*'' expresses the conjugate of the complex number.

B. ROBUST FINITE CONTROL SET CURRENT CONTROL
This proposal of predictive current control considers the errors in the estimation of the machine parameters in the controller for the prediction of the stator voltage vector (5). The predicted voltage vector v p αβs (k) is composed of two components: one for the voltage calculated considering the stator current reference v ff (k), and another for the voltage that takes into account the parametric errors of the IM ( v fb ). The parametric mismatches are modeled around variations in stator currents.
The selection of the voltage vector to be applied to the stator terminals is performed by using the minimization of the cost function g (6), in which it is limited to the voltage levels that can be applied by the inverter ( v ref αβs ).
This minimization ensures that the behavior of the controlled variable reaches the reference signal.
The presented control method uses a two-level voltage source inverter (VSI) which enable the application of eight voltage vectors as presented in Table 1.
where S i represents the switching state of each i arm of the inverter, a = e j 2π 3 and V dc is the dc-link voltage. The relationship between the stator current, the rotor flux and the voltage vectors is shown in (1). Hence, it can be employed to represent v ff (k). The starting point for calculating v ff (k) is the discretization of (1) using the Euler's method for a sampling time T s [11], [26]. In this way, the discretization of (1) is expressed by: The calculation of v ff (k) can be performed using (8). In this is the stator current vector reference. So, it can be expressed as: The calculation of v fb (k) can be performed using (8) delayed one sampling time as presented in (10).
The assumption ( i αβs (k) = 0) is performed to ensure that there is no increment due to parameter mismatches of the IM [11], [27].
The (11) can be rearranged in the form of: Equation (12) permits to calculate the stator voltage vector increment when there is increment in stator current vector due to the parameter mismatch of the plant. In this case, the idea behind the definition of the component v fb is to express it as a function of the variation of the stator current vector due to the errors associated with the plant parameters. So, v fb using (12) is defined in (13).
where G represents the gain defined as: Which permits the calculation of v fb (k) and it can be expressed as: The calculation of v p αβs (k) can be described in (5); the voltage vectors v ff (k) and v fb (k) are obtained employing the Eqs. (9) and (15), respectively. Therefore, the minimized cost function in (6) is computed using the predicted values of v p αβs (k) and the eight voltage levels of the inverter. Hence, the proposed solution permits to control the stator current vector of the IM.  (16) and (17) The variables presented in Fig. 1, w s and w ele represent the slip speed and the electrical synchronous speed, respectively.

III. EXPERIMENTAL RESULTS
The presented control method was implemented on an experimental test bench shown in Fig. 2. The test bench is composed by an 1.1 kW squirrel-cage IM driven by a two?level voltage source inverter with a dc-link voltage controlled by a 3 kVA variac. The load torque applied on the motor shaft is done by an eddy current brake. The control system is composed by Texas Instruments development board LAUNCHXL-F28379D with TMS320F28379D microprocessor incorporated, using auxiliary circuits built in the laboratory for signal conditioning. A 3600 pulse per revolution incremental encoder measures the rotor position.
The IM parameters are: nominal electromagnetic torque is 6.18 N·m; nominal speed is 1700 rpm; nominal frequency is 60 Hz; pole pairs = 2; R s = 7.1 ; R r = 3.98 ; L s = 545 mH; L r = 545 mH and L m = 526 mH. The inverter was operated with constant dc-link voltage, so V dc = 412 V. The control algorithm runs at a sampling frequency of 20 kHz and uses one-step prediction horizon. The PI controller used in speed control loop was tuned in a process of trial and error.

A. DYNAMIC BEHAVIOR WITHOUT SPEED CONTROL LOOP
The first test was conducted with no load to verify the performance of the proposed control method using the i αs and i sβ as control commands (Fig. 3) The results for i αβs components allow us to observe that the elements of the stator currents reaches the references. It is possible to observe in steady state periods that as rotor speed raises the current frequency also increases. In the interval of 0.65 s to 0.775 s the mean absolute error (MAE) for i αs is 0.074 A and for i βs is 0.058 A. The values of root mean square error (RMSE) are 0.09 A for alpha component and 0.072 A for beta component of the stator current. The calculated current total harmonic distortion (THD) registered for the currents is 7.48%.
The action of the elements is depicted in details in Fig. 4(a) and (b). The equivalent step response of the stator current vector magnitude | i sαβ | is plotted in Fig. 4(c) and its setting time is 0.5 ms with no overshoot. After the settling time period, the peak to peak ripple for the measured values was estimated in 0.1858 A. The MAE is 0.0285 A and the mean relative error (MRE) is 1.76%.

B. DYNAMIC BEHAVIOR WITH SPEED CONTROL LOOP
The performance of the control method with speed control loop was evaluated by two tests. In the first one the speed range control was evaluated varying the rotor speed from −570 rpm to 570 rpm with no load applied on the motor shaft. The signals of speed, torque and one phase of stator current measured during the test are depicted in Fig. 6. The proportional and integral gains of the PI controller were set at 0.3 and 0.1, respectively.
The settling time is 80 ms for the reversing process in which the maximum torque applied was limited to 6 N·m and it is nearly its nominal value. After the setting time, the MAE  is 9.4 rpm and the MRE is 1.7%. The control system followed the speed reference and presented low torque ripples. The stator current has an amplitude near 1.3 A in steady-state and, during the reversion of the speed, a peak of −3.5 A during the test.
Another test was carried out considering the rated mechanical speed. In this case, it was applied a speed step from −1700 rpm to 1700 rpm with no load on the motor shaft. The speed, torque and stator phase current measured during the test are presented in Fig. 6. The proportional and integral gains of the PI controller were set at 0.1 and 0.3, respectively. In this test, the setting time is 270 ms, the MAE is 35.8 rpm, and the MRE is 2.1%.
In a third test, a load torque (T load = 4.6 N·m) is applied to the motor shaft while the speed is maintained constant and it is depicted in Fig. 7. The reference speed was set at 570 rpm and the eddy current brake was switched on for a short period of time changing the load torque from 0.3 N·m to 4.6 N·m until it is switched off again. The proportional and integral gains of the PI controller were set at 0.1 and 0.3, respectively. Thus, the electromagnetic torque reference calculated by the PI controller could be properly attended for the torque reference as long as the rotor speed reached its reference.

C. STEADY-STATE BEHAVIOR UNDER PARAMETER VARIATION CONDITIONS AND COMPARISON WITH CLASSICAL FINITE CONTROL SET
In order to verify the robustness of the proposed control method against parameter mismatches, its steady-state performance was analyzed when the parameters of the IM are changed. The IM was kept at constant speed of 850 rpm and the load torque is 3.8 N·m for all tests.
In the first experiment, the stator resistance value, R s , was changed multiplying it by (1 + σ R s ) for the calculation of the reference voltage, v ff (k), in the proposal. Additionally, the variation σ R is virtually inserted in the control algorithm [28] to allow a comparison between the strategies.
The rotor speed response for the proposal is shown in Fig. 8 (a). In Figs. 8 (b) and (c) was ploted the behavior of the stator current components, i αs and i βs , and the vector magnitude, | i sαβ |, is presented in Fig. 8(d). The values of σ R s were changed in regular steps along time and they can be seen in Fig. 8 (e).
Next, R s was changed dividing it by (1 + δ R s ) for the calculation of the reference voltage, maintaining the same conditions from the previous test. The experimental results VOLUME 11, 2023  are presented in Figs.9(a), (b), (c), (d), and (e), for rotor speed, i αs , i βs , | i sαβ | σ R s , respectively.
The variation of R s values did not affect the performance of the controller significantly as it can be seen in the data presented in the Tables 2 and 3. Under the maximum variation of 800%, the MAE and the RMSE did not ultrapass 0.14A. Only in | i αβs | an increase of MRE from 2.9% to 3.7% could be seen.
The impact of the variation of R r values presented in Figs. 10(a), (b), (c), (d), (e) when R r is multiplied by (1 + σ R r ) for rotor speed, i αs , i βs , | i sαβ | σ R r , respectively. And Figs.11(a), (b), (c), (d), and (e) when R r is divided by (1+σ R r ) for rotor speed, i αs , i βs , | i sαβ | σ R r , respectively. Therefore,  the data presented in Tables 4 and 5 confirm the absence of variations visualized in the figures. In the same way as the tests presented when variations occur in R s value.
Other tests were conducted for simultaneous variations in both resistances, and this time the proposed robust finite control set model predictive control was compared to the classical current finite control set [28] with the same rotor field orientation depicted in Fig. 1.
The classical predictive current control [28] was tested under the same conditions described to the proposal. The resistances R r and R s were changed multiplying them by (1 + σ R ). The signals of speed, current components, current vector magnitude, and imposed variations are presented in   Both strategies were able to maintain the rotor speed next to 850 rpm. However, the proposed robust finite control set demonstrates a superior performance when it is considered the MAE, RMSE, MRE, and THD values for the stator current components, as presented in Table 6, considering the max (800%) variation.
Keeping the same boundary conditions, R r and R s were also changed dividing them by (1 + δ R ). The speed, the stator current components, the current vector magnitude and the variation δ signals are presented in   The proposed robust finite control set current controller was able to keep the current elements close to their references under all range of variations but with smaller error ranges when compared to the classical approach. The MAE, RMSE, MRE, and THD values for the stator current components during parameters variations test can be observed in Table 7.
The effect of variation of inductances values was also verified in another test. The most significant impact occurred VOLUME 11, 2023    The classical predictive current control [28] demonstrated poor responses for i αs and | i αβs | when δ L increases. According to the Table 8, the RMSEs of i αs for the classical predictive current control were more than two times greater than

D. IMPACTS OF PARAMETERS VARIATION OVER FLUX ESTIMATION
Due to the importance of the rotor flux angle estimation in the proposed control algorithm, its sensitivity to variations in the parameter R r was also investigated. These alterations can affect the dynamic response of the control, which could be evaluated through the execution of step response tests for rotor speed as depicted in Figs. 18(a), (b), (c), (d), and (e), respectively. The speed was varied from −570 rpm to 570 rpm keeping 6 N·m as maximum torque reference generated by the speed PI control where the gains are 0.1 for the proportional gain and 0.3 for the integral gain. The value of R r was changed only for the calculation of θ(k) depicted in Fig. 1. It can be observed that the torque reaches the references. However, there is a speed error of more than 5% in Fig. 18(a). It can be corrected by considering other values of the PI gains.
Although outside the scope of this work, there are new observer alternatives for estimating rotor flux as presented in [29], [30], and [31]. So that, they can be incorporated into the proposed control, improving its performance.

IV. CONCLUSION
This paper proposes a robust finite control set model predictive current control for IM using IFOC in stationary reference frame. The proposal calculates the applied stator voltage vector as a sum of two voltages elements: one employs the classical MPC and another one considering the deadbeat approach to compensate parameter mismatch. Hence, the stator voltage vector is selected using the minimized cost function.
The proposal demonstrated strong robustness considering the variations in the IM resistance values. Even variations of 800% of the nominal values did not compromise the steadystate performance, indicating potential for applications that require stability in the control of IM. The experimental results endorse dynamic-state and steady-state performances for the proposed control and its superiority over classical predictive current control. The impact of parametric mismatches on the rotor flux orientation was also analyzed and it can be observed that the strategy works satisfactorily even when the rotor resistance varies up to 300%.
THIAGO BALDIM DOS SANTOS received the bachelor's degree in energy engineering from the Maua Institute of Technology, in 2019. He is currently pursuing the Ph.D. degree in energy with the Federal University of ABC. His research interests include electrical systems and power electronics applied to electric machine drives and electric vehicles. VOLUME 11, 2023 IGOR OLIANI received the bachelor's degree in science and technology from the Federal University of ABC, in 2019, where he is currently pursuing the master's degree in energy engineering and electrical engineering. His research interests include electrical systems and power electronics applied to electric motor drives and electric vehicles. He is currently a Professor of agricultural machine design and robotics with the Faculty of Agricultural Engineering (FEAGRI), UNICAMP, and an Extension Coordinator of FEAGRI/UNICAMP. He is experienced in agricultural engineering, with emphasis on soil dynamics, design and evaluation of agricultural machinery, agroecological machinery, equipment for family farming, optics applied to agricultural engineering, rural buildings with alternative material, physical and mechanical properties of biological material, agricultural aviation, quality management in agriculture, energy in agriculture (biomass and wind), and robotics in agriculture.