I. INTRODUCTION

Permanent magnet synchronous motors (PMSMs) are widely utilized in servo systems because of their high efficiency, high power density, and high torque current ratio [1], [2]. A high-performance PMSM servo system requires a fast-response current inner loop to ensure the high performance of speed and position loops. The traditional control methods of current loop include current hysteresis control and proportional-integral (PI) control [3], [4]. Current hysteresis control has the problems of variable switching frequency and large steady ripple. PI control is usually accompanied by overshoot because it requires tradeoffs between dynamic and steady-state performances. Both of them hardly meet the high-performance control requirements. With the development of high-speed digital signal processing technology, predictive current control, which requires complex computation, has been the focus of investigation in the high dynamic control of PMSM [5]-[7].

By using a motor model, predictive current control can predict the future current behavior to select a proper voltage vector, under which the current can follow the reference current in an optimal trajectory [8]-[12]. Predictive current controls can be divided into (at least) three classes, namely, direct predictive control (DPC), two-configuration predictive control (2PC), and pulse-width modulation (PWM) predictive control (PPC) [13]. DPC selects a voltage vector that minimizes a cost function and directly applies it to the inverter. Large current and torque ripples exist in DPC because the selected voltage vector is applied to the inverter in the entire sampling interval. A one–zero voltage vector is introduced in 2PC to overcome this major drawback of DPC. An active voltage vector and a zero voltage vector are applied to the inverter in a sampling interval to reduce the current and torque ripples. PPC calculates the ideal voltage vector, which is modulated through space vector PWM, and then applies it to the inverter. Two active voltage vectors and a zero voltage vector are applied to the inverter in one sampling interval to eliminate the current and torque errors. The control block diagrams of PPC and conventional PI control have the same structure; thus, the PPC controller can replace the PI controller. Therefore, PPC is investigated in this study.

Inaccurate model parameters affect the current dynamic response and steady-state error in predictive current control [14]-[21]. Many scholars have conducted extensive research to solve the problem of inaccurate parameters. In [14], a constraint-relaxing deadbeat predictive control strategy was proposed, in which the current offset constraint and the output voltage prediction method were modified to enhance the system stability under the condition of inductance mismatch. A weighting factor was introduced in [15] to improve the system robustness. In [16], two adjacent sampling interval prediction models were subtracted to eliminate constant items to achieve close-loop control, which can avoid the steady-state error and eliminate the influence of flux. In [2], the predictive control method was improved by paralleling with an integrator to control q-axis current to eliminate the static torque current error caused by the flux error. All these methods can improve the performance of predictive current control with inaccurate parameters but cannot eliminate the model parameter error. In [17], the static current error was eliminated by introducing error integration in d-axis current control, and the model flux was dynamically adjusted according to the q-axis current error. However, the authors did not consider the influence of the inductance error on dynamic current performance. The model reference adaptive system (MRAS) was used in [18] and [19] to identify the model parameters of the motor online to eliminate the influence of the parameter error. However, MRAS requires an additional adjustable model, which increased the complexity of the system. In [20] and [21], the function between inductance and flux with id and iq was established and utilized for predictive current control to eliminate the problem of inaccurate parameters. However, inductance-current and flux-current nonlinear maps were required. The maps were measured through repeated experimental procedures or with the help of software, which was not easy to obtain.

In this study, a model parameter correction algorithm for predictive current control of surface-mounted PMSM (SMPMSM) is proposed. The algorithm uses the proportional relation between inductance and d-axis static errors to correct the inductance error. The proportional relation between flux and q-axis static errors is utilized to correct the flux error under the condition that inductance is corrected. Inductance and flux converge to an actual value through the correction algorithm proposed in this study, and the problems of inaccurate inductance and flux are solved.

II. PREDICTIVE CURRENT CONTROL METHOD FOR SMPMSM

The stator voltage and state-space equations of the SMPMSM in the d-q rotor reference frame are provided by Equs. (1) and (2), respectively.

where ud, uq and id, iq are the d-q frame voltages and currents, respectively; L is stator inductance; R is stator resistance; Ψf is the flux established by the permanent magnet; ωe is the electrical angular velocity of the rotor.

Control period T in a servo system is small. Consequently, ωe is considered constant during each sampling period T. Based on the forward Euler approximation method, Equs. (1) and (2) can be discretized into Equs. (3) and (4), respectively.

The PWM predictive current controller is built based on the voltage equation in Equ. (3). We suppose that idr and iqr are the reference currents of d-axis and q-axis, respectively. The goal of predictive current control is for the actual currents to follow the reference currents after one modulation period. Therefore, we suppose that id[k+1] = idr and iq[k+1] = iqr and apply them to Equ. (3). Accordingly, we obtain the following equation.

The ud[k] and uq[k] calculated by Equ. (5) are the required voltage vectors that allow the current vectors to reach the reference currents after one modulation period.

Fig. 1 shows a block diagram of PWM predictive current control.

Fig. 1.Block diagram of PWM predictive current control.

III. PARAMETER SENSITIVITY ANALYSIS

Predictive current control is based on a motor model to calculate the desired voltage vectors. Inaccurate model parameters force the voltage vectors to deviate from the expected ones and thus result in poor control performance.

In this study, we suppose that the actual motor parameters are R0, L0, and Ψf0, and the predictive model parameters are R, L, and Ψf. During one control period, the voltage vectors calculated by Equ. (5) are applied to the actual motor, and the actual motor current response can be presented by Equ. (4). The following equations can be obtained by applying Equ. (5) to Equ. (4).

where ΔR = R − R0, ΔL = L − L0, and ΔΨf =Ψf −Ψf0 are the errors between model and actual motor parameters.

In a practical system, the order of magnitude of T is generally 10−4, R is 10−1, and L is from 10−3 to 10−2. Considering that TΔR is much smaller than ΔL, TΔR is ignored, and Equ. (6) can be simplified as

Considering that id[k+1] = id[k] at steady-state operation, id[k+1] = id[k] is applied to Equ. (7), and the static error of d-axis current response can be obtained.

where Δid= id[k+1] - idr.

Equ. (7) shows that iq[k+1] receives the dual effects of ΔL and ΔΨf. The current response of q-axis is analyzed under the condition of accurate inductance. Considering that iq[k+1] = iq[k] at steady-state operation, ΔL=0 and iq[k+1] = iq[k] are applied to Equ. (7). The static error of q-axis current response can also be obtained.

where Δiq = iq[k+1] - iqr.

Equ. (8) shows that Δid is proportional to ΔL and is not related to flux. Equ. (9) shows that Δiq is proportional to ΔΨf under the condition of accurate inductance. The current responses of different inductance and flux combinations are shown in Table I. A small inductance value causes the static current error of d-axis to be greater than zero. A large inductance value causes the static current error of d-axis to be less than zero. Under the condition of accurate inductance, a small flux value causes the static current error of q-axis to be less than zero. A large flux value causes the static current error of q-axis to be greater than zero.

TABLE ICURRENT RESPONSES OF DIFFERENT INDUCTANCE AND FLUX COMBINATIONS IN THEORY

IV. MODEL PARAMETER CORRECTION ALGORITHM

Based on the analysis in the preceding section, we propose a model parameter correction algorithm for predictive current control.

A. Inductance Correction Algorithm

Equ. (8) shows that the response error of d-axis static current error Δid is proportional to ΔL and is not related to flux. Δid and ΔL have an opposite sign (ωeiq > 0 at steady-state operation). Equ. (10) is defined to correct the inductance error (if Δid > 0, then ΔL < 0; the inductance should be increased and vice versa).

where CL is the inductance constant increment, KIL is the inductance incremental integral coefficient, and KPL is the inductance incremental proportionality coefficient.

In Equ. (10), Equ. (1) is the constant incremental mode, Equ. (2) is the integral incremental mode, and Equ. (3) is the proportional plus integral incremental mode. The first mode is relatively simple, but the second or the third mode is faster by selecting proper proportional and integral coefficients. The mode can be selected according to the control requirement of performance in a practical application.

If and only if ΔL = 0, then Δid = 0. Equ. (10) converges with ΔL = 0 and Δid = 0, which means that inductance L equals actual inductance L0.

B. Flux Correction Algorithm

Equ. (9) shows that the response error of q-axis static current Δiq is proportional to ΔΨf under the condition of accurate inductance, and only one flux variable exists. The signs of Δiq and ΔΨf are identical (at steady-state operation). Equ. (11) is defined to correct the flux error (If Δiq > 0, then ΔΨf > 0; the flux should be decreased and vice versa).

where CΨ is the flux constant increment, KIΨ is the flux incremental integral coefficient, and KPΨ is the flux incremental proportionality coefficient.

In Equ. (11), Equ. (1) is the constant incremental mode, Equ. (2) is the integral incremental mode, and Equ. (3) is the proportional plus integral incremental mode. The mode can be selected according to the control requirement of performance in a practical application.

If and only if ΔΨf = 0, then Δiq = 0. Equ. (11) converges with ΔΨf = 0 and Δiq = 0, which means that flux Ψf equals actual flux ΔΨf0.

C. Flow of the Model Parameter Correction Algorithm

The preceding analysis indicates that the flux is corrected under the condition of accurate inductance. Hence, the flow of the model parameter correction algorithm must correct the inductance first and then the flux. The flow diagram of the model parameter correction algorithm is shown in Fig. 2. At steady-state operation (constant speed), the inductance correction algorithm is enabled first. Then, the flux correction algorithm is enabled after inductance convergence. The entire algorithm is completed when the flux converges.

Fig. 2.Flow diagram of the model-parameter-setting algorithm.

The block diagram of the PWM predictive current control with the model parameter correction algorithm is shown in Fig. 3. A block called parameter correction algorithm is added in Fig. 3 unlike the diagram in Fig. 1. Reference and feedback currents are utilized to correct inductance and flux errors.

Fig. 3.Block diagram of the model parameter correction algorithm for PWM predictive current control.

V. EXPERIMENTAL RESULTS AND ANALYSIS

An experimental platform is established to verify the correctness of the parameter sensitivity analysis for predictive current control and the proposed model parameter correction algorithm. This platform utilizes a Xilinx Spartan-6 field-programmable gate array (FPGA) as the main control chip and two identical SMPMSMs to build a drag system. The specifications of the SMPMSMs utilized in this research are listed in Table II.

TABLE IIMOTOR SPECIFICATION

The sampling period of the current loop is set to 100 μs. The high-speed computing performance of FPGA makes the time delay only 7.4 μs from current sampling to PWM updating, accounting for 7.4% of the sampling period. Therefore, the instant duty cycle update strategy mentioned in [4] is used in this experiment.

The experimental data are sent to the upper computer through the communication module for monitoring and processing. A total of 1000 N (N channels within 0–100 ms) data are stored in the FGPA random-access memory and then read and sent to ensure that the data in each sampling period can be sent without loss.

A. Experiment on Parameter Sensitivity

Experiments are conducted for parameter sensitivity analysis. No speed loop exists in these experiments, and q-axis reference current iqr is set from 0 A to 4 A at 10 ms moment and set from 4 A to 2 A at 20 ms moment. The waveforms during 8–25 ms and the amplified waveform near 10 and 20 ms are recorded. The experimental results of five different conditions (see Table I) are provided (Figs. 4–8).

Fig. 4.Experimental result when L=L0 and Ψf=Ψf0.

Fig. 5.Experiment result when L=0.5L0 and Ψf=Ψf0.

Fig. 6.Experimental result when L=1.5L0 and Ψf=Ψf0.

Fig. 7.Experimental result when L=L0 and Ψf=0.5Ψf0.

Fig. 8.Experimental result when L=L0 and Ψf=1.5Ψf0.

Fig. 4 shows the experimental results when L = L0 and Ψf = Ψf0. The current rise time from 0 A to 4 A is 3T ( three control periods). uq (per-unit value) reaches the limiting value in the first 2T. The falling time from 4 A to 2 A is 1T only, during which uq is not saturated. The static errors of d-axis and q-axis are zero. The predictive current control with accurate parameters thus has a good control performance.

Figs. 5 and 6 show the experimental results when L = 0.5L0 and Ψf = Ψf0 and when L = 1.5L0 and Ψf = Ψf0, respectively. Figs. 7 and 8 show the experimental results when L = L0 and Ψf = 0.5Ψf0 and when L = L0 and Ψf = 1.5Ψf0, respectively. The experimental results in Figs. 4–8 are concluded in Table III.

TABLE IIICURRENT RESPONSES OF DIFFERENT INDUCTANCE AND FLUX COMBINATIONS

Table III verifies that the d-axis static current error is only related to the inductance error. The d-axis static current error is greater than zero with small model inductance (Fig. 5) and less than zero with large model inductance (Fig. 6). Under the condition of accurate inductance, the q-axis static current error is only related to the flux error. The error is greater than zero with large model flux (Fig. 7) and less than zero with small model flux (Fig. 8). The experimental results (Table III) are consistent with the analysis results (Table I), and the correctness of the analysis is verified.

B. Experiment on the Model Parameter Correction Algorithm

The constant incremental mode for the inductance and flux correction algorithm is selected to simplify the system and save on FPGA logic resources. A steady-state operation with a reference speed of 1500 r/min and q-axis current of 4 A by loading is implemented for the experiment. The experimental results of the model parameter correction algorithm are provided (Figs. 9-12).

Fig. 9.Experimental result when L=0.5L0 and Ψf=Ψf0.

Fig. 10.Experimental result when L=1.5L0 and Ψf=Ψf0.

Fig. 11.Experimental result when L=L0 and Ψf=0.5Ψf0.

Fig. 12.Experimental result when L=L0 and Ψf=1.5Ψf0.

Inductance correction is not related to flux. Therefore, only the experimental results when L=0.5L0 and Ψf= Ψf0 and when L=1.5L0 and Ψf= Ψf0 are provided. Figs. 9 and 10 present the experimental results of the inductance correction algorithm when L = 0.5L0 and L = 1.5L0, respectively. The inductance converges to the actual value with a convergence time of approximately 15 ms and a convergence error of approximately 5%. The convergence error is not related to the initial value, and the inductance converges with L = L0 and Δid = 0.

Figs. 11 and 12 show the experimental results of the flux correction algorithm when Ψf = 0.5Ψf0 and Ψf = 1.5Ψf0 after inductance convergence, respectively. The flux converges to the actual value with a convergence time of approximately 12 ms and a convergence error of approximately 1.2%. The convergence error is not related to the initial flux value, and the flux converges with Ψf = Ψf0 and Δiq = 0.

The experimental results (Figs. 9–12) verify the correctness and feasibility of the proposed model parameter correction algorithm.

VI. CONCLUSION

Given its excellent control performance, predictive current control is attractive in the high-performance control of SMPMSM. Predictive current control is sensitive to the errors of model parameters. Based on the analysis of the relations between model parameter and current static errors, a model parameter correction algorithm was developed for predictive current control. Considering that the d-axis static current error is only related to the inductance error, the algorithm corrects the inductance and converges with ΔL = 0 and Δid = 0. Under the condition that the inductance converges, the q-axis static current error is proportional to the flux error. By utilizing this proportional relation, the flux is corrected and converges with ΔΨf = 0 and Δiq = 0. The experimental results show that inductance and flux converge to actual values with convergence times of approximately 15 and 12 ms and with convergence errors of approximately 5% and 1.2%, respectively. These results verify the correctness and feasibility of the proposed algorithm.