Simulation on model predictive control for PMSM drive system based on double extended state observer

CUI Hong-wei, TENG Qing-fang, ZHU Jian-guo, GUO You-guang

(1. Department of Automation & Electrical Engineering, Lanzhou Jiaotong University, Lanzhou 730070, China；2. Key Laboratory of Opto-technology and Intelligent Control (Lanzhou Jiaotong University),Ministry of Education, Lanzhou 730070, China；3. Faculty of Engineering and Information Technology, University of Technology Sydney, Sydney 2007, Australia)

Abstract： A novel double extended state observer (DESO) based on model predictive torque control (MPTC) strategy is developed for three-phase permanent magnet synchronous motor (PMSM) drive system without current sensor. In general, to achieve high-precision control, two-phase current sensors are necessary for successful implementation of MPTC. For this purpose, two ESOs are used to estimate q-axis current and stator resistance respectively, and then based on this, d-axis current is estimated. Moreover, to reduce torque and flux ripple and to improve the performance of the torque and speed, MPTC strategy is designed. The simulation results validate the feasibility and effectiveness of the proposed scheme.

Key words： double extended state observer (DESO); model predictive torque control (MPTC); permanent magnet synchronous motor (PMSM) drive system; without current sensor

0 Introduction

For permanent magnet synchronous motor (PMSM) drive system, real-time measurement of stator currents is required for implementation of model predictive torque control (MPTC). Generally, two-phase current sensors are installed in three-phase voltage source inverters (VSI). During the actual operation, abrupt severe failures of phase current sensors may cause in over-current malfunction of the drive system, which, if there is no protection solution in gate-drive circuit, may lead to unrecoverable fault of power semiconductors in VSI and performance deterioration of motor drive. In addition, some minor failures of phase current sensors, such as gain drift and nonzero offset, may lead to torque ripple synchronized with the VSI output frequency[1]. The larger the offset and the scaling error of phase current sensors, the worse the performance of torque regulation. Furthermore, if the offset and gain drift are beyond a certain level, it may cause over-current trip under high speed and heavy load conditions[2]. Therefore it is indispensable to consider the fault-tolerant control of phase current sensors.

As for current control technique without current sensors, two methods have been reported in the literature. The first method is based on DC-link current[3-4], which belongs to the mainstream method. However, it has unavoidable disadvantages. For example, the cycle of an active switching state is so short that the DC-link current cannot be measurable; there are immeasurable regions in the output voltage hexagon where the DC-link current sampling is limited or impossible to do. To obtain high-precision reconstruction current, many kinds of improved DC-link current reconstruction methods based on pulse width modulation (PWM) strategy have been proposed[5-10]. Although these studies have improved the accuracy of phase current estimation, they are complex. Xu, et al. proposed the vector control of PMSM drive system based on single phase current[11], and an isolated current sensor was used instead of DC bus current sensor. Finally, a new method of PMSM three-phase current reconstruction based on zero voltage vector sampling was proposed. However, the single current sensor sampled twice in each PWM cycle, therefore the dynamic performance of current control was deteriorated[12]. The second method is based on adaptive observer. In Ref.[13], the phase current was reconfigured for induction motor (IM) drive by single phase current sensor; but in Ref.[14], the phase current was estimated for PMSM drive without phase current sensor. The limitation of the Refs.[13-14] is that resistance change is slow.

Apart from above-mentioned two methods, a new extended state observer (ESO)-based method can be employed. By extending a unknown variable as the incremental state of an original system, ESO can estimate the unknown variable without much model information, and consequently it is introduced for PMSM phase current estimation. To improve the system performance and reduce the chattering from the switch function in conventional ESO, a nonlinear exponential function (NEF) can replace such a switch function, making ESO monotonically converge to a steady state value. Compared with the first two methods, the NEF-based ESO has no limitation on applicable scope and constraint condition. Moreover, its operation process is not complicated and algorithm execution is easy. So far there has been no literature on such a current observer.

In this paper, for the PMSM drive system without any phase current sensor, to perform current feedback control, a double ESO (DESO) is introduced to estimate phase currents and time-varying stator resistance. Furthermore, to improve dynamic performance and control accuracy, a novel DESO-based MPTC strategy is put forward.

1 Dynamic model of PMSM drive

For three-phase PMSM drive, the models in rotor synchronous reference frame (dq-frame) can be expressed as

(1)

(2)

ψd=Ldid+ψf,

(3)

ψq=Lqiq,

(4)

whereRsis the stator resistance;Ld,Lq,ud,uqandid,iq,ψd,ψqare thedq-axes stator inductances, voltages, currents and magnet flux, respectively;ψfis the permanent magnet flux;ωris the mechanical rotor speeds;pis the number of pole pairs.

And the mechanical equation is expressed as

(5)

whereJis the moment of inertia;TLis the load torque;Bmis the viscous friction coefficient;Tfis the coulomb friction torque;Teis the electromagnetic torque, which is expressed as

(6)

2 Design of DESO-based MPTC PMSM drive system

A DESO based on MPTC strategy is developed for three-phase PMSM drive system without any current sensor. The proposed DESO can estimate the other two-phase currents and time-varying stator resistance when there are any phase current sensor. The control system block diagram is shown in Fig.1.

Fig.1 Block diagram of DESO-based MPTC PMSM drive system

Our design mission focuses on DESO, PI speed regulator and MPTC. The design idea is as follows: Firstly, one ESO is designed to estimateq-axis current which will be a known variable in the following design process. Secondly, another ESO is designed to estimate stator resistance. Finally,d-axis current is estimated based on above two estimated results and motor model.

2.1 Estimation of q-axis current based on ESO

The design is completed under the following conditions.

1) Due to electric heating during the operation of motor, stator resistanceRsis assumed to be time-varying.

2) Rotor speed and inverter output voltage can be measured. Substituting Eq.(6) into Eq.(5), there is

(7)

Supposingx1=ωr, the state space equation of Eq.(7) can be expressed as

(8)

whereu1andy1are the input and output, respectively;B1is the input matrix; the expressions are

u1=[ωrTL]T,

(9)

(10)

(11)

and let

(12)

|q1(t)|

(13)

Therefore, the state space equation in Eq.(8) can be rewritten as

(14)

which is observable. Now we construct a second-order state observer, denoted as the ESO, in the form of

(15)

whereβ1andβ2are the positive observer gains,a1andδ1are the positive parameters, andf(x1,α,δ) is a nonlinear function which is defined as

(16)

The extended state observer of Eq.(15) is

(17)

(18)

According to Ref.[15], if the parameter value satisfies the following inequality

(19)

2.2 Estimation of stator resistance based on ESO

Supposingv1=iq, the state space equation of Eq.(2) can be expressed as

(20)

whereu2andy2are the input and output, respectively;B2is the input matrix; the expressions are

u2=[uqωr]T,

(21)

(22)

(23)

and let

(24)

|q2(t)|

(25)

Therefore, the state space expression in Eq.(20) can be rewritten as

(26)

Becauseidis an unknown variable,pωridis ignored in the construction of the observer. But the variablepωridactually works, therefore the function ofe2is used to compensate the observer.

Eq.(26) is observable. Now we construct a second-order state observer, denoted as the ESO, in the form of

(27)

whereβ3>0,β4>0,β5>0 and 0≤αi≤1,δi>0, (i=2,3).

The extended state observer of Eq.(27) is

(28)

wherez1andz2are the real-time estimate values forv1andv2, respectively. And the real-time estimation value forRscan be obtained by Eq.(28) as

(29)

According to Ref.[15], if the parameter value satisfies the following inequality

(30)

2.3 Estimation of d-axis current based on ESO

Thed-axis current estimation model is

(31)

By combining Eqs.(15), (27)and (31), we can construct the structure diagram of stator current and stator resistance observer based on DESO, as shown in Fig.2.

Fig.2 DESO-based stator current and stator resistance observer

Remark1 General rules of the parameter selection of the ESO are as follows[15-18]:

1) 0<α≤1.

2)δindicates the width of the linear interval near the original point, to which system steady state error relates.δis selected in the range of[0.000 1 1]. Whenδis less than 0.002 5, it is easy to cause system high frequency pulsation; whileδis too large, the effect of nonlinear feedback control cannot be achieved. In practical engineering application, it is generally set to be 0.01.

3)βi(i=1,2) can be determined on the basis of system tracking performance. Largeβiis helpful to increase tracking speed, however, it may lead to system oscillation and overshoot. In general, the value ofβ1is larger than 1-2 orders ofβ2. There is a compromise betweenβi(i=1,2) andδfor a better noise suppression and tracking performance.

Remark2 To smooth the stator resistance to be estimated, it needs to pass through a low-pass filter (LPF). A first-order inertia link is used as

(32)

whereωcis cut-off frequency.

2.4 Model predictive torque control

The basic idea of MPTC is to predict the future behavior of the variables over a time frame based on the system model. In fact, MPTC is an extension of direct torque control (DTC) as it replaces the look-up table of DTC with an online optimization process in the control of machine torque and flux. Different from the use of hysteresis comparators and switching table in DTC, the principle of vector selection in MPTC is based on evaluating a defined cost function. For six voltage space vectors generated by three-phase six-switch inverter as shown in Fig.3, it is easy to evaluate the effect of each voltage vector and select the one to minimize the cost function in MPTC.

Fig.3 Layout of voltage space vectors and its corresponding switching states

2.4.1 Minimization of cost function

The basic idea of MPTC is to reduce torque and flux pulsations. In general, the cost function is defined as

(33)

2.4.2 Predictive model for stator current

According to Eqs.(1) and (2), the prediction of the stator current at the next sampling instant is expressed as

(34)

2.4.3 Torque and flux estimators

According to Eqs.(3) and (4), the flux-linkageψdandψqcan be defined indq-frame as

(35)

The stator flux linkageψsat (k+1)th instant is

(36)

Electromagnetic torque can be estimated indq-frame as

(37)

3 Simulation result and analysis

In order to verify the validity and correctness of the method proposed in this paper, the simulation is carried out with Matlab. The parameters of the PMSM motor are listd in Table 1.

Table 1 Parameters of PMSM

δ1=0.01,α1=0.5,β1=80 000,β2=1 620,

δ2=0.001,α2=0.5,β4=260,β5=1 000,

δ3=0.001,α3=0.25,β3=2.

The parameters of PI are

kP=1.0,kI=0.08.

The parameter of low pass filter is

ωc=80.

The load torque, rotor speed and the stator resistance are set at 2 N·m, 1 000 rpm and 2.875 Ω, respectively.

3.1 Performance comparison under the variation of stator resistance

Here, the stator resistance varies from 2.875 to 3.5 Ω at 0.3 s. Fig.4 shows the comparison results of two scenarios. From Figs.4(a) and 4(b), it can be seen that the current waveforms inabc-frame for DESO observer of Case 2 coincide exactly with the corresponding ones of Case 1 as is the case indq-frame. Fig.4(c) shows that actual resistance change can be tracked by its estimated resistance value. Figs.4(d) and (e) show that for the MPTC system of Case 2, its ability against resistance variation is weaker than that of Case 1.

Fig.4 Dynamic response comparison between Case 1 and Case 2 under the variation of stator resistance

3.2 Performance comparison under the variation of load torque

Here, the load torque varies from 2 to 3 N·m at 0.2 s. Fig.5 shows the comparison results of two scenarios. From Figs.5(a) and (b), it can be seen that the current waveforms inabc-frame for DESO observer of Case 2 coincide exactly with the corresponding ones of Case 1 as is the case indq-frame. Fig.5(c) shows that actual resistance change can be tracked by its estimated resistance value. Figs.5(d) and (e) show that for the MPTC system of Case 2, the waveforms of the speed and torque are almost as good as the corresponding ones of Case 1, and it can make the MPTC system run stably and reliably.

Fig.5 Dynamic response comparison between Case one and Case 2 under the variation of load torque

4 Conclusion

This paper has put forward a novel DESO-based MPTC scheme for three-phase PMSM drive system with no phase current sensor. The designed DESO is capable of estimating any phase current and time-varying stator resistance based on an assumption that rotor speed and position are available for measurement, and it has strong robustness to stator resistance and load. The designed DESO-based MPTC strategy can ensure that the PMSM drive system achieves satisfactory torque and speed control as well as strong robustness. Comprehensive simulation validates the feasibility and effectiveness of the proposed scheme.