TENG Qing-fang, LUO Wei-duo

(1. School of Automation and 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)

Abstract： Based on the fractional order theory and sliding mode control theory, a model prediction current control (MPCC) strategy based on fractional observer is proposed for the permanent magnet synchronous motor (PMSM) driven by three-level inverter.Compared with the traditional sliding mode speed observer, the observer is very simple and eases to implement. Moreover, the observer reduces the ripple of the motor speed in high frequency range in an efficient way. To reduce the stator current ripple and improve the control performance of the torque and speed, the MPCC strategy is put forward, which can make PMSM MPCC system have better control performance, stronger robustness and good dynamic performance.The simulation results validate the feasibility and effectiveness of the proposed scheme.

Key words： permanent magnet synchronous motor (PMSM); three-level inverter; fractional sliding mode speed observer; model predictive current control (MPCC)

0 Introduction

In recent years, in the motor system,multi-level inverters have attracted a lot of attention. The three-level neutral-point-clamped (NPC) inverter is one of the most commonly used multi-level inverter topologies in high-power drives.Three-level inverter has the advantages of high efficiency and low loss. More output levels provide more freedom in vector selection and it is possible to synthesize waveforms that are more sinusoidal in shape[1-5].

Permanent magnet synchronous motor (PMSM) drive nowadays is widely used in the industry applications due to their high efficiency and high power/torque density. As for high performance PMSM drive systems, the popular control strategies include two types: torque-based control and current-based control. The most established torque-based control is direct torque control (DTC). It directly switches the inverter to regulate torque without explicit stator current regulation. However, it suffers from variable switching frequency and more challenges to implementation in digital controllers. The current-based control uses the measured stator current to track the required values accurately with short transient interval as much as possible, which thus plays an important role in high-performance PMSM drive and ensures the quality of the torque and speed control[6-7].

Over past years, four main kinds of current-based control schemes have developed for an inverter-fed PMSM: the hysteresis control,the ramp comparison control, the PI control with space vector modulation (SVM), and the model predictive current control (MPCC)[8-9].

In recent years, scholars have conducted a lot of research on the motor system driven by three-level inverter. Speed sensor research has attracted widespread attention since it was proposed[10-18]. At present, the commonly used methods include model reference adaptive system (MARS)[14], sliding mode(SM)[15], etc. The MRAS is greatly influenced by the change of parameters at low speed. The structure of sliding mode is simple and robust, but the design of sliding mode surface has chattering phenomenon. In Ref.[14], sliding mode reference adaptive observer was proposed. In Ref.[15] a full order observer method was proposed and applied to PMSMs.

For three-phase three-level inverter in PMSM drive systems, in order to improve dynamic performance and control accuracy and system robustness, a novel MPCC strategy based on fractional sliding mode observer is proposed in our research.The fractional sliding mode observer can increase the control redundancy,and also to a large extent, solve the inherent defects of traditional sliding mode control.

1 Topology and mathematical model of PMSM system driven by three-level inverter

1.1 Topology of PMSM

For three-phase PMSM, the schematic diagram of three-level inverter topology is shown in Fig.1.

Fig.1 Topology diagram of PMSM diven by three-level inverter

Each phase of the bridge of the inverter is composed of two clamp diodes, four power switches and four freewheeling diodes. Taking phaseaas an example, by analyzing the power switching sequence and corresponding phase voltage condition (phasesb,candaare the same), we can know that whenSa1andSa2are on whileSa3andSa4are off, the output voltage of phaseais +Vdc/2; whenSa2andSa3are on whileSa1andSa4are off, the output voltage of phaseais 0; whenSa4turns on whileSa1andSa2turn off, the output voltage of phaseais -Vdc/ 2.Therefore, each phase of three-level inverter bridge arm outputs -Vdc/2,0 andVdc/ 2 (denoted by “-”, “0” and “+”), respectively. And 12 power switch tubes on the three-phase bridge arm can be combined into 27 switch states, correspondingly, 27 space voltage vectors are generated. The relationship between the three-level inverter space voltage vector and the different combinations ofa-phase,b-phase andc-phase outputs is shown in Fig.2, which is defined as

Vi∈{V0,V1,…,V25,V26},i=0,…,26,

(1)

Fig.2 Relationship of switching state and space voltage vector of NPC three-level inverter

1.2 Mathematical model of PMSM

As for three-phase PMSM drive, as shown in Fig.3, the model in rotor synchronous reference frame (dq-frame) expressed as

(2)

(3)

whereudanduqare the stator voltages,idandiqare the stator currents,EdandEqare the back electromotive forces;φfis the permanent magnet flux linkage,Rsis the stator resistance,Lis the stator inductance,ωris the rotor angular velocity, andpis the motor pole pairs.

Ignoring Coulomb friction torque, the PMSM mechanical rotation equation is expressed as

(4)

whereJis the moment of inertia,T1is the load torque,Bmis the viscosity friction coefficient, andTeis the electromagnetic torque. In the two-phasedqrotating coordinate system,Tecan be expressed as

Te=1.5pφfiq.

(5)

Fig.3 Block diagram of MPCC system based on fractional sliding mode observer

2 Design of MPCC based on fractional sliding mode speed observer

2.1 Design of fractional sliding mode speed observer

The observer equations for current and back electromotive can be obtained from Eqs.(2) and (3) as

(6)

(7)

(8)

whereD-μis a fractional differential operator,λ1∈R+,λ2∈R+, andμ∈(0,1].

According to Eq.(8), there is

(9)

2.2 Stability of fractional slide speed observer

In order to verify the stability of the designed fractional sliding mode observer, the Lyapunov function is defined as

(10)

Taking the derivative of Eq.(10), there is

(11)

Assuming that

(12)

then

(13)

(14)

Under Lyapunov stability conditions, we can get

(15)

Assuming thatg>max[Ud,Uq], we can get

-[g×sgn(sd)(g-Ud)+g×sgn(sq)(g-Uq)]).

(16)

Then there is the steady motion of the system, and the fractional sliding mode observer asymptotically converges. Here function sgn(si) is defined as

(17)

2.3 Speed estimation

According to

(18)

the permanent magnet synchronous motor speed and angular velocity observation variables can be expressed as

(19)

(20)

3 Design of model predictive current

The basic idea of MPC is based on the fact that a finite number of possible switching states can be generated by a power unit and the mathematical model of the system can be used to predict the future behavior of the variables over a time frame for each switching state. For the selection of an appropriate voltage space vector to be applied, a cost function containing control objective is defined. By means of evaluating the defined cost function for each possible voltage vector, the corresponding switching state that minimizes the cost function is selected. As shown in Fig.3, MPCC mainly includes two parts: current prediction model and minimum cost function.

3.1 Predictive model for stator current

The prediction of the stator current at the next sampling instant can be expressed as

(21)

3.2 Minimum cost function

For conventional MPCC, the minimum cost function is such chosen that bothidandiqindq-system at the end of the cycle are as close to their reference values as possible. Its definition is

s.t.Vi∈{V0,…,V26},i=0,…,26,

(22)

In order to compensate for the computation delay in the digital implementation of the MPCC algorithm, the cost function for the control of thedqsystem stator current is changed to

s.t.Vi∈{V0,…,V26},i=0,…,26,

(23)

4 Simulation and analysis

In order to verify the correctness and validity of the designed system, the simulation was carried out in Matlab / Simulink. The parameters of PMSM are shown in Table 1.

Table 1 Parameters of PMSM

In order to verify the correctness and validity of the fractional speed observer, four systems are compared, which corresponds to the same MPCC PMSM system fed by three-level inverter. The first is based on traditional sliding mode observer, the second is based on model reference adaptive system(MRAS) observer, the third is based on sliding mode variable structure observer, and the fourth is based on fractional sliding mode observer corresponding to the MPCC PMSM system fed by three-level inverter based on fractional sliding mode observer.For convenience, the first scenario is marked as case 1; the second, case 2; the third, case 3; the fourth, case 4. The sampling period is set to be 26 μs.

Figs.4 and 5 show their comparison results in terms of rotor speed, torque, and stator current when the reference speednis set to be 1 000 r/min,the load torque is increased from 0 to 2 N·m at 0.2 s. The parameters of the fractional slide velocity observer areλ1=2.5,λ2=4 andμ=0.153.

(a) Comparison of actual and estimated speeds of case 1

(b) Comparison of actual and estimated speeds of case 2

(c) Comparison of actual and estimated speeds of case 3

(d) Comparison of actual and estimated speeds of case 4

(e) Speed error of case 1

(f) Speed error of case 2

(g) Speed error of case 3

(h) Speed error of case 4

It can be seen from Figs.4(a)-4(h) that for the MPCC PMSM system fed by three-level inverter based on fractional sliding mode observer, its speed can sharply track reference speed in a satisfactory manner, and its speed error is more smaller than that of other cases. From Figs.5(a)-5(f), it can be observed that for the MPCC PMSM system fed by three-level inverter based on fractional sliding mode observer, the torques anda-axis currents inabc-frame anddq-frame response are better than that of case 1.

(a) Torque response of case 1

(b) Torque response of case 4

(c) dq-axis current response of case 1

(d) dq-axis current response of case 4

(e) a-axis stator current in abc-frame of case 1

(f) a-axis stator current in abc-frame of case 4

5 Conclusion

For the PMSM system driven by three-level interver, this paper presents an MPCC strategy based on fractional sliding mode speed observer. Firstly, an fractional sliding mode observer is designed, which can track the rotational speed in real time. Secondly, the MPCC strategy is employed to reduce the stator current ripple and improve the control precision of current control. The proposed MPCC strategy based on fractional sliding mode observer can achieve satisfactory dynamical performance and strong robustness. Comparative simulation demonstrates the feasibility and effectiveness of the proposed scheme.