An improved deadbeat predictive current control for permanent magnet linear synchronous motor

Transcription

1 Indian Journal of Engineering & Materials Sciences Vol. 22, June 2015, pp An improved deadbeat predictive current control for permanent magnet linear synchronous motor Mingyi Wang, iyi i, Donghua Pan & Qingbo Guo Department of Electrical Engineering, Harbin Institute of Technology, China Received 3 March 2014; Accepted 10 March 2015 For the permanent magnet linear synchronous motor (PMSM) applied in the high-precision equipment, motor control system should have characteristics of high dynamic response and strong robustness. Predictive current control is an effective method to improve the dynamic response of the system, but it requires accurate motor parameters. This paper presents an improved deadbeat predictive current control for PMSM based on the space vector pulse width modulation (SVPWM) technology, a correction factor is introduced to reduce the response time on the basis of ensuring the system stability, and in order to overcome the parameter mismatch between the linear motor and the controller, a parameter disturbance observer is added, this method improves the robustness of the current loop. Simulation results show the correctness and effectiveness of the proposed strategy, and experiment platform based on the field programmable gates array (FPGA) is constructed, experimental results demonstrate that the current loop has high bandwidth and good robustness. Keywords: Permanent magnet linear synchronous motor, Deadbeat predictive current control, Space vector pulse width modulation, High bandwidth, Robustness Compared with the traditional model of the rotating motor with the transmission mechanism, the linear motor has advantages of fast response, high acceleration and control precision, therefore, the linear motor drive system has an important application value in high-precision equipment, such as high-end computerized numerical control (CNC) machine tool and lithography 1, 2. High-precision PMSM control system mainly includes position loop, velocity loop and current loop. Current loop as the inner loop is the core part, it links up the control system with the motor. Especially in the PMSM control system of the lithography equipment, high position loop bandwidth is the necessary condition to improve the tracking accuracy of the system, therefore, current loop needs higher dynamic performance. In the current control strategy, there are three major types: the hysteresis control, the proportional plus integral (PI) control and the predictive current control 3. The hysteresis control has the advantages of fast transient response and simple application. However, switching frequency is not fixed, which degrades the system steady-state performance 4,5. PI control has been acknowledged as being superior, unlike the hysteresis control, it has a Corresponding author ( liliyi.hit@gmail.com) fixed switching frequency and yields zero control error, nevertheless, its dynamic performance is not optimal 6. Combined with SVPWM technology 7, the predictive current control results in better dynamic performance and less current harmonics. Theoretically, the motor current reaches its command value during two control periods by this control algorithm, and it is very suitable for applying in high dynamic performance occasions The conventional predictive current control utilizes reference and actual currents at the beginning of the switching period to calculate the required direct and quadrature voltages, but calculated voltages are applied to the motor through the inverter in the next period, it leads to a large overshoot and oscillation. In Ref. 12, a predictive current control scheme based on a discretized model can ensure the system stability, but it reduces the transient performance. Meanwhile, this control scheme is highly dependent on the motor and inverter parameters, and the parameter mismatch between the motor and the controller degrades the system performance. The aim of this paper is to improve the dynamic and steady state performance of the deadbeat predictive current control for PMSM, the transient response time is reduced by regulating the correction factor, while retaining its good steady-state

2 274 INDIAN J ENG. MATER. SCI., JUNE 2015 performance is retained, compared with the reported results 12, the proposed control scheme improves the current loop bandwidth. The disturbance caused by parameter variations can be estimated by a parameter disturbance compensator, thus, the steady state response can be significantly improved, and the current loop has strong robustness. The proposed predictive current control is realized by XC3S400 FPGA, simulation and experimental results are shown to demonstrate the feasibility and effectiveness of the proposed scheme. System Descriptions A reliable model is an indispensable part of the successful implementation of a control algorithm. Any discrepancy between the real system and the model may result in a failure to meet the design speciation and may al cause instability in the closed loop system. Therefore, modeling is usually the first step in the progress of controller design. Modeling of PMSM In this study, the stator voltages of PMSM can be described by using Eqs (1) and (2) in synchronous rotating reference frame. v R i p d (1) q o q qo e do q v R i p d (2) d o d do e qo d where qo qoiq (3) i (4) do do d fo n (5) e p r (6) r v / v n v 2 f (7) e p e In the equations, v q, v d are the q-axis and d-axis voltages, i q, i d are the q-axis and d-axis currents, R is the phase winding resistance, q, d are the q-axis and d-axis inductances, λ f is the permanent magnet flux linkage, ω r is the angular velocity of the mover, ω e is the electrical angular velocity, n p is the number of pole pairs, τ is the pole pitch, v is the linear velocity of the mover, v e is the electric linear velocity, f e is the electric frequency, p is the differential operator. subscript o denotes the nominal value, and the lump of uncertainties caused by parameter variations can be represented by d q, d d, they are expressed as: diq dq Riq q dide e f (8) dt did dd Rid d qiqe (9) dt where ΔR=R-R o, Δ q = q - qo, Δ d = d - do, Δλ f =λ f -λ fo, the adopted PMSM is the surface mounted motor, q = d = s, and ΔR, Δ q, Δ d, Δλ f are disturbance variations of motor parameters. The discrete-time equations of Eqs (1) and (2) can be described as: v k R i k i k i k 1 q o q q q Ts i k d k e d e fo q v k R i k i k i k 1 d o d d d Ts i k d k e q d (10) (11) T s is the sampling period. Utilizing nominal parameters, Eqs (10) and (11) can be given as: Ι k 1 G I k H V k λ M D k (12) Where I V k i k i k T q k v k v k T λ D q T efo 0 k d k d k T q 1 Ts Ro Ts e G T 1 T R Ts H 0 d d d s e s o Ts M 0 T s 0 T s 0 SVPWM Technology The drive system in this paper consisting of threephase voltage urce inverter (VSI) and PMSM is shown in Fig. 1. Three-phase VSI has eight different switching states, each state corresponds to a basic voltage space vector, including six active-voltage vectors and two identical zero-voltage vectors, voltage space vectors are shown in Fig. 2, and any

3 WANG et al: PERMANENT MAGNET INEAR SYNCHRONOUS MOTOR 275 calculated command voltage space vector can be synthesized by those basic voltage space vectors. SVPWM technology has the advantages of higher utilization of the DC voltage, lower harmonic distortion and fast transient response. Proposed Current Control Algorithm In order to realize a high dynamic response and strong robustness in the current control structure, an improved predictive current control combined with the parameter disturbance observer is proposed in this paper. Figure 3 depicts the simplified block diagram of the proposed current control scheme, the actual current can follow the current command quickly, and the parameter disturbance observer can overcome the parameter mismatch between the linear motor and controller. Current controller According to the deadbeat predictive current control, the purpose of the scheme is to calculate the command voltage V k by the command current I k 1 and actual current k I at the beginning of the k th switching period, and reduces the error between the command and actual currents to zero at the end of the switching period. The equation can be written as Eq. (13). I k 1 G I k H V k λ (13) M D k In practical applications, however, the command voltage V k doesn t execute immediately, it loads to three-phase VSI at the beginning of the (k+1) th switching period, as written in Eq. (14), and the time sequence of predictive current control is shown as Fig. 4. Ignoring the disturbance D k temporarily, it Fig. 1 Three-phase voltage urce inverter Fig. 3 Overall block diagram for the proposed control scheme Fig. 2 Basic voltage space vectors Fig. 4 Time sequence of predictive current control

4 276 INDIAN J ENG. MATER. SCI., JUNE 2015 is noted that the actual current I k 2 at the end of the (k+1) th switching period is equal to the command I k 1, it can be expressed as Eq. (15). current V k V k 1 (14) I k 2 I k 1 G I k 1 H k V λ (15) If the actual current I k 1 is not optimized, a large disturbance can be brought in the system. This problem can be lved by the predictive current, written as Eq. (16). In order to improve the dynamic performance, the correction factor η is introduced, Eq. (16) can be adjusted as Eq. (17). k k k I p 1 G I H V λ (16) k 1 k 1 1 k I I p I (17) ~ Where I ( k 1) is the predictive current with the n correction factor. By substituting Eq. (16), Eq. (17) into Eq. (15), the equation can be given as Eq. (18): k 1 1 k G H V k H V k I G G I I G I H λ (18) where I is an identity matrix. The adoptive algorithm requests a proper choice of the correction factor η to ensure the dynamic and steady-state performance. When η = 0, ~ ~ I n ( k 1) I ( k), Eq. (18) is equal to Eq. (13), the current has a significant overshoot and oscillation, it is similar to a kind of high gain proportional ~ ~ controller; when η = 1, I ( k 1) I ( k), this approach is the same as given by Moon 12, it improves the steady-state performance of the current loop, but reduces the dynamic performance; when 0 < η < 1, the current loop synthesizes the features of the first two cases, especially compared with Moon 12, the dynamic performance is enhanced. Parameter disturbance observer The predictive current control scheme is sensitive to motor parameter variations, and the performance degradation is yielded by the n p parameter mismatch between the motor and controller. According to Eq. (13), the current control robustness can be effectively improved by estimating parameter variation value D ˆ k, and the voltage disturbance is added to the current closed loop control system in the form of the feedforward 14. If the sampling frequency is high enough, it is considered that the disturbance value remains constant in one sampling period, which can be obtained as Eq. (19). D k 1 D k (19) In order to establish a relationship between the estimated disturbance D ˆ k 1 and D ˆ k, in Eq. (20), an observer gain matrix γ is introduced. Dˆ k 1 Dˆ k γ M Dˆ k I k 1 k k G I H V λ To facilitate the calculation, the matrix (20) Q k is defined as Eq. (21). Q k D ˆ k γ I k (21) Then, Q k 1 Dˆ k γ M Dˆ k k k G I H V λ (22) By using Eq. (22) and choosing the appropriate observer gain matrix γ, the estimated disturbance D ˆ k gradually converges to the actual disturbance D k, the error between the actual disturbance D k and the estimated disturbance k k ˆ k D ˆ k is defined as Eq. (23): E D D (23) By using Eqs (13), (21) and (22), the estimated error of the adjacent sampling period is written as Eq. (24): E k 1 I γ M E k (24) The dynamic characteristic of the observer is determined by the eigenvalue of the matrix I γ M, the appropriate observer gain matrix γ can be achieved by utilizing the eigenvalue of the matrix I γ M, thus, the proposed predictive current control scheme can be easily tuned to obtain adequate convergence properties. Therefore, the current loop has enough robustness to copy with the parameter variation.

5 WANG et al: PERMANENT MAGNET INEAR SYNCHRONOUS MOTOR 277 Results and Discussion Simulation study To evaluate the performance of the proposed control scheme, this paper establishes the simulation model in the Matlab/Simulink environment, Table 1 shows parameters of PMSM model. Improved predictive current control In this study, the correction factor is introduced to improve the dynamic performance, while retaining the steady-state performance. The correlative simulation study proves the validity of the proposed algorithm. The DC link voltage is 380 V, the sampling period T s is 100 μs and the dead time is 2 μs. The q-axis and d-axis command currents are given as 10 A and zero, respectively, and the load thrust is 775 N. Figure 5 shows the q-axis command current I q and actual current I q with different η values. Figure 5(a) shows when η = 0, I q has a significant oscillation; In Fig. 5(d), there is no overshoot and oscillation at η = 1, but the rise time is 2.5 ms, the dynamic performance is poor; Fig. 5(b) or Fig. 5(c) shows the dynamic performance at η = 0.4 or η = 0.6 is better than that at η = 1, but when η = 0.4, I q has a small overshoot. In summary, with the correction factor value increasing, the steady-state performance of the current loop is enhanced, but the dynamic performance is degraded, when the correction factor equals 0.6, the proposed controller provides an ideal performance. Robust to parameter variations In this section, PMSM is starting with parameter variations (R o = 3R, s = 2, λ f = 0.75λ fo ). The q-axis and d-axis command currents are given as 5 A and zero, respectively, and the load thrust is 0 N. Figure 6 shows the predictive current control under matching conditions. The actual currents can track quickly the command currents, and the current adjustment time is less than 1 ms. Figure 7 shows the control performance without a disturbance compensator under the resistance mismatch (R o = 3R ). As shown in Fig. 7(b), a Table 1 Parameters of PMSM model R o 3.9 Ω 26.8 mh λ fo 0.2 Wb n p 10 τ 12 mm mover mass 50 kg Fig. 5 Simulation results of the proposed current controller with different η values: (a) η = 0, (b) η = 0.4, (c) η = 0.6 and (d) η = 1

6 278 INDIAN J ENG. MATER. SCI., JUNE 2015 steady-state error of 0.15 A is observed due to the resistance mismatch. Figure 8 shows the control performance of the proposed scheme under the resistance mismatch. In Fig. 8(b), it is obvious that the steady-state error is zero, and Fig. 8(c) shows the estimated disturbances, and the q-axis disturbance is 39 V. Therefore, the simulation is consistent with the theoretical analysis. Figure 9 shows the control performance without a disturbance compensator under the inductance mismatch ( s = 2 ). It is clear that inductance mismatch mainly affects d-axis performance in Fig. 9(a), which will lead to the phase delay between the actual phase current and the command phase current. Figure 10 shows the control performance of the proposed scheme under the inductance mismatch. With the estimated disturbances in Fig. 10(c), d-axis current and q-axis current control both have a precise current tracking performance, as shown in Figs 10(a) and 10(b), respectively. Figure 11 shows the control performance without a disturbance compensator under the flux linkage mismatch (λ f = 0.75λ fo ). In Fig. 11(b), as the speed increases, the q-axis actual current is larger than the command value gradually. Figure 12 shows the control performance of the proposed scheme under the flux linkage mismatch. In Fig. 12(b), it is obvious that the dynamic tracking errors are zero, and the q-axis estimated disturbance increases with the increase of speed, where Δλ f = -0.25λ fo. Experimental study The experimental system based on XC3S400 FPGA is used to verify the correctness and feasibility of the proposed scheme. The test motor parameters are given in Table 2, the position feedback adopts the optical encoder with 0.1 μm relution, and aerostatic slideway is used to suspend the mover. The DC link voltage is 310 V, the sampling period T s is 100 μs and the dead time is 2 μs. The digital signals are converted to analog signals by a 12 [b] D/A convert. Fig. 6 Control performance under matching conditions: (a) d-axis current component and (b) q-axis current component Fig. 7 Control performance without a disturbance compensator under resistance mismatch: (a) d-axis current component and (b) q-axis current component

7 WANG et al: PERMANENT MAGNET INEAR SYNCHRONOUS MOTOR 279 Fig. 8 Control performance with a disturbance compensator under resistance mismatch: (a) d-axis current component, (b) q-axis current component and (c) estimated disturbances Fig. 9 Control performance without a disturbance compensator under inductance mismatch: (a) d-axis current component and (b) q-axis current component Fig. 10 Control performance with a disturbance compensator under inductance mismatch: (a) d-axis current component, (b) q-axis current component and (c) estimated disturbances

8 280 INDIAN J ENG. MATER. SCI., JUNE 2015 Fig. 11 Control performance without a disturbance compensator under flux linkage mismatch: (a) d-axis current component and (b) q-axis current component Fig. 12 Control performance with a disturbance compensator under flux linkage mismatch: (a) d-axis current component, (b) q-axis current component and (c) estimated disturbance Table 2 Parameters of the test motor R o 1.8 Ω] 2.2 mh λ fo Wb n p 4 τ 85.4 mm mover mass 50 kg Figure 13 shows experimental results of the proposed algorithm with different η values. The q-axis command current is changed form 0 to 5 A at 0.25 ms, and the mover of PMSM is fixed, it can be seen that experimental results are generally consistent with simulation results, the rise time is 0.5 ms at η = 0.6, and the actual current can quickly and smoothly track the command current, therefore, the experimental results further confirm the correctness of the proposed scheme. Figure 14 shows experimental results of control performance under parameters mismatch. In this experiment, the controller parameters cannot match the motor s (R = 0.9 Ω, = 1.5 mh, λ fo = 0.12 Wb), and Fig. 14(a) shows the performance of d-axis and q-axis current without a disturbance compensator, it is clear that the tracking error of the q-axis current increases gradually, and in Fig. 14(b), with a disturbance compensator, the q-axis actual current

9 WANG et al: PERMANENT MAGNET INEAR SYNCHRONOUS MOTOR 281 Fig. 13 Experimental results of the proposed algorithm with different η values: (a) η = 0, (b) η = 0.4, (c) η = 0.6 and (d) η = 1 quickly track the command current, and the estimated disturbance is shown as Fig. 14(c). Therefore, well robustness under the situation of disturbance and model mismatch can be obtained by using a disturbance compensator. Conclusions In this paper, an improved deadbeat predictive current control for PMSM with the correction factor and the disturbance compensator has been presented. The conventional method lves the overshoot and oscillation of the predictive current control, however, it weakens the transient response. Compared with the traditional control strategy, the proposed scheme not only maintains the stability of the system, but al improves the dynamic performance with the suitable factor. By introducing the disturbance compensator, the parameter mismatch problem between the controller and the motor is lved, the control system has well robustness. The proposed algorithm is implemented on the experiment platform based on XC3S400 FPGA, and simulation and experimental results verify the correctness. Thus, the control performance can be significantly improved with a relatively simple control scheme. Fig. 14 Experimental results of control performance under parameters mismatch: (a) performance of d-axis and q-axis current without a disturbance compensator, (b) performance of d-axis and q-axis current with a disturbance compensator and (c) estimated disturbances Acknowledgments This work was supported by the National Science Foundation for Distinguished Young Scholars of China (Grant No ).

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