Speed sensorless FCS-MPTC based on FOSM-MRAS five-phase permanent magnet synchronous motor

ZHANG Haiming， WANG Yunkun， MIAO Zhongcui， WANG Zhihao

(1. School of Mechanical and Electrical Engineering, Lanzhou Jiaotong University, Lanzhou 730070， China；2. School of Automation and Electrical Engineering, Lanzhou Jiaotong University, Lanzhou 730070， China)

Abstract： For the two-level five-phase permanent magnet synchronous motor (FP-PMSM) drive system, an improved finite-control-set model predictive torque control (MPTC) strategy is adopted to reduce torque ripple and improve the control performance of the system. The mathematical model of model reference adaptive system (MRAS) of FP-PMSM is derived and a method based on fractional order sliding mode (FOSM) is proposed to construct the model reference adaptive system (FOSM-MRAS) to improve the motor speed estimation accuracy and eliminate the sliding mode integral saturation effect. The simulation results show that the FP-PMSM speed sensorless FCS-MPTC system based on FOSM-MRAS has strong robustness, good dynamic performance and static performance, and high reliability.

Key words： model reference adaptive system (MRAS)； finite-control-set model predictive torque control (FCS-MPTC)； fractional order sliding mode (FOSM)； speed sensorless； five-phase permanent magnet synchronous motor (FP-PMSM)

0 Introduction

Compared with the traditional three-phase motor the five-phase permanent magnet synchronous motor (FP-PMSM) has the advantages of large capacity, more control freedom, and good fault tolerance due to its redundant phase number. It is widely used in occasions with high reliability requirements, such as new energy vehicles, ship propulsion system, etc.

Direct torque control (DTC) is a high-performance control method developed after vector control. It has been widely used in the FP-PMSM control system. However, the traditional DTC has some problems, such as low equivalent switching frequency, poor steady-state performance and large torque ripple[6-9]. These problems hinder its application in high-precision control situations. In recent years, finite set model predictive torque control has received widespread attention in the field of motor control due to its easy online implementation, fast dynamic response, and flexible control[10-13]. Since this method directly considers the constraints of the input, output and state variables of the controlled process, and directly deals with the multi-variable coupling system, it can overcome the shortcomings of the above-mentioned DTC and does not require a modulator[14-16]. In Ref.[17], FCS-MPTC replaces the traditional DTC and significantly improved the motor control performance. In Refs.[18-19], the predictive current algorithm based on FP-PMSM control system is studied, and 25 basic voltage vectors are classified according to the magnitude. In Ref.[20], the harmonic subspace current under vector space decomposition (VSD) coordinate transformation is considered in the traditional FCS-MPC objective function, and the third harmonic of multi-phase motor is suppressed. In Ref.[21], an optimal deadbeat method based on torque and flux linkage is proposed to estimate the voltage vector in each control cycle and to reduce the computation burden of FCS-MPTC algorithm on line.

In the FP-PMSM control system, it is necessary to obtain feedback information of the motor speed through a sensor. However, a high-resolution speed sensor will increase the complexity of the system and may cause problems such as measurement noise quantization, which will reduce the reliability of the system. Speed sensorless control technology can reduce the cost of control system, improve the stability of FP-PMSM system, and broaden the application of motor control systems. In Refs.[22-23], the expansion Kalman filter is used to realize the speed identification. This method does not require the initial position information and mechanical parameters of the motor, and has strong robustness. But its calculation is complicated and the parameter selection is difficult. In Refs.[24-26], the speed estimation is achieved by superimposing a high frequency signal on the fundamental signal. But the injected high frequency signal affects the dynamic performance of the system. In Refs.[27-29], the sliding mode control algorithm is used to identify the motor parameters, which enhances the robustness of the system to motor parameter changes and load disturbances. However, the discontinuous switching variables make the estimated variables contain high-frequency distortion caused by switching. Also, it causes phase deviation. In Refs.[30-31], it is easy to realize a digital control system by using the observer based on model reference adaptive system (MRAS) to identify motor parameters. However, the precision of the parameters of the reference model directly affects the precision of the speed identification, and the dynamic response is affected in many aspects. In Refs.[32-33], sliding mode control (SMC) and MRAS are combined to construct an SM-MRAS observer, which can obtain better dynamic response and robustness. However, the sliding mode surface based on the integer-order integral of the state variable has an integral saturation effect when the initial error is relatively large or the given signal changes suddenly, which will cause the transient performance of the system to decrease and even become unstable. Fractional order sliding mode (FOSM) control algorithm can effectively improve the dynamic and static performance of the control system[34-36]. If the fractional calculus is applied to the SM-MRAS observer to design an FOSM-MRAS observer, it can accurately estimate the rotor position and speed, eliminate integral saturation and reduce sliding mode chattering.

Taking the surface-mounted FP-PMSM as the control object, a speed sensorless FCS-MPTC system based on the FP-PMSM is designed in this study. By combining the fractional order theory with SM-MRAS, a new type of FOSM-MRAS speed observer is designed to improve the precision of rotational speed observation. The nonlinear exponential function is used to reduce the chattering of the sliding mode and carry on deduction and demonstration. Finally, its validity is verified by simulation.

1 Mathematical model of FP-PMSM

1.1 Generalized Clark transformation matrix

The five-phase motor model contains four degrees of freedom and zero sequence components. Under normal operating conditions, according to the generalized Clark transformation matrix, the symmetric physical quantities in the natural coordinate system can be mapped to the double synchronous rotation coordinate systemsd1-q1andd2-q2, The mathematical model of FP-PMSM is as follows:

1) Voltage vector equation

(1)

whereUdq=[ud1uq1ud2uq2u0]TandIdq=[id1iq1id2iq2i0]Tare the stator phase voltage and phase current, respectively；Rdq=diag[RsRsRsRsRs] is stator resistance matrix；Ldq=diag[LdLqLlsLlsLls]; andΓ=[(-Lqiq) (Lqiq+ψf) 0 0 0]T. Among them,LdandLqrepresent the inductances of the orthogonal and direct axes in thed-qcoordinate system, respectively;Llsrespresents the leakage inductane;ψfis the flux amplitude of the permanent magnet in the motor winding; andωeis the electric angular velocity.

2) Flux vector equation

Ψdq=LdqIdq+ΨfT(θe)Λ,

(2)

T(θe)=

(3)

3) Electromagnetic torque equation

(4)

whereNpis the number of motor pole pairs.

1.2 FP-PMSM discrete mathematical model

According to Eqs.(1), (2) and (4), the motor state equation discretized by the forward Euler approximation method is given by

(5)

(6)

In addition,A,BandCare the coefficient matrices. For convenience, we define

(7)

thenA,BandCare expressed as

(8)

2 Design of FCS-MPTC

FCS-MPTC is based on the system model and its constraints to achieve the optimization of the objective function within a predetermined time. The corresponding optimal action sequence is obtained by using the latest measurement data. The process of selecting the optimal voltage vector of the FCS-MPTC strategy is shown in Fig.1.

Fig.1 Optimal voltage vector selection of FCS-MPTC

2.1 Inverter voltage vector

The main circuit topology diagram of the FP-PMSM drive system powered by a five-phase voltage source inverter is shown in Fig.2.

Fig.2 Drive system for FP-PMSM

According to the theory of five-phase coordinate transformation, the set of voltage vectors in the Eq.(5) can be transformed into

(9)

whereTis the five-phase inverter switching matrix as

(10)

Let the inverter phase voltage and switch status be

Sk=[SaSbScSdSe]T,

(11)

the inverter space voltage vector can be expressed as

(12)

All switching states of the five-phase inverter are combined into Eq.(12). Then 32 basic voltage vectors are obtained, including 30 non-zero voltage vectors and 2 zero vectors. The distribution of the space voltage vectors in thed1-q1andd2-q2coordinates is shown in Fig.3.

(a) d1-q1 coordinates

(b) d2-q2 coordinates

2.2 Design of objective function and selection of voltage vector

The FCS-MPTC calculates the predicted values for each possible execution by the objective function. To predict the future of stator flux and torque, the design objective function is given as

s.t.V(i)∈{V1,V2,…,V31},

(13)

wherek1,k2are the weight coefficients. The given electromagnetic torque is obtained from the output of the speed regulator. The given flux linkage is calculated according to the maximum torque current ratio (MTPA)[35], that is,

(14)

2.3 Delay compensation and improvement strategy

To improve the effect of delay on the performance of FCS-MPTC, the variable at timet(k+1) is used as the starting value for prediction, and the variable at timet(k+2) is predicted. Considering a beat delay, the flux linkage and torque at timet(k+2) are predicted according to Eq.(5), and the FCS-MPTC objective function with the maximum allowable stator current limit is re-designed as

s.t.V(i)∈{V1,V2,…,V31}.

(15)

The last term in Eq.(15) is a nonlinear function that limits the amplitude of the FP-PMSM stator current, which is expressed as

(16)

whereImaxis the maximum allowable stator current amplitude. If the predicted current amplitude generated by the specified voltage vector is greater thanImax, the voltage vector is not selected by the objective function. If the predicted current amplitude is less thanImax, the objective function contains only the first two terms and the optimal voltage vector is selected.

3 Design of FOSM-MRAS observer

In the MRAS algorithm, the equations that do not contain unknowns are used as reference models, and the equations that include parameter variables are used as adjustable models, and the output difference is used to adjust the parameters to be estimated of the adjustable model in real time according to the adaptive law.

3.1 Design of reference model and adjustable model

Assuming that the five-phase windings are distributed symmetrically, according to Eq.(1), the current model of FP-PMSM with stator current as the variable in the dual rotating coordinate system can be obtained as

(17)

It can be seen that the harmonic subspace current model does not contain rotor speed information. This means that the harmonic current does not participate in the energy conversion of the motor, and the existence of harmonic subspace (including zero-sequence subspace) current has no effect on the construction of the speed observer. The current fundamental current model contains both rotor speed information and flux linkage information. In order to obtain an adjustable model, the fundamental current model needs to be transformed as

(18)

For surface mounted FP-PMSM, the inductance of motor underd-qcoordinate axis is uniformly expressed byLs, i.e.Ld=Lq=Ls. Then Eq.(18) can be expressed as

(19)

Let the parameters be

(20)

Eq.(19) can be further expressed as

(21)

(22)

In Eq.(22), the state matrixAcontains the speed information of FP-PMSM, which can be used as an adjustable model. FP-PMSM itself is chosen as the reference model, andωeis the parameter to be identified. Thus, we have

(23)

3.2 Adaptive law of MRAS observer

Since the motor speed is unknown, the state equation represented by the estimated value of the stator current is shown as where

(24)

(25)

The error dynamic equation of stator current vector can be obtained by subtracting the phase of Eqs.(22) and (24), and then we get

(26)

Eq.(26) is equivalent to a standard feedback system, where

Ae=A,

andJis the coefficient matrix as

(27)

3.3 Stability law of MRAS observer

According to Popov’s super stability theory, to make the feed back system shown in Eq.(26) asymptotically stable, the following constraints must be satisfied, namely

(28)

wherer0is any finite positive number.

To obtain the MRAS adaptive law, it is necessary to solve Popov’s integral inequality inversely. SubstitutingeandWinto the constraint conditions, we can get

(29)

(30)

According to Eqs.(29) and (30), we can get

η(0,t1)=

η1(0,t1)+η2(0,t1),

(31)

where

(32)

Choosing

(33)

and taking the derivatives on both sides of the first expression of Eq.(33), we can get

(34)

It is proved that the dynamic equation of stator voltage vector error expressed by Eq.(26) can ensure the stabilization of the feedback system. Then the speed estimator is obtained as

(35)

(36)

Eq.(36) can be expressed as

(37)

3.4 Design of FOSM-MRAS observer

For integer-order sliding mode observers, when the initial error of the observer is large or the given signal is abrupt, the cumulative effect of the integral term on the deviation will lead to integral saturation and deteriorate the dynamic performance of motor speed. This problem can be solved by designing fractional integral sliding surface.

In order to satisfy the condition of fractional integral type symbolic function, the sliding surface is designed as a global sliding mode as

(38)

The design of reaching law can guarantee the dynamic quality of sliding mode motion. The isokinetic approaching law is selected as

(39)

where sgn(s) is a symbolic function. In order to reduce the chattering phenomenon of the system, the nonlinear exponential function fal(s,α,γ) is used instead of sgn(s) as

(40)

where fal(s,α,γ) is linearl continuousγandαare constant. Whenγ>0 and 0<α<1, small error and large gain can be realized. For convenince, we write fal(s,α,γ) as fal(s). The observer with fal(s) function is designed as

(41)

whereksis a constant greater than zero.

3.5 Stability of FOSM-MRAS observer

The derivation of the stator current error is shown as

(42)

where

(43)

In order to prove the stability of the designed observer, the Lyapunov function is selected as

(44)

(45)

(45)

It can be seen that a sufficiently large sliding mode gainkshould be selected to ensure the observer to be asymptotically stable in a large range.

Based on the above analysis, the designed FOSM-MRAS observer structure diagram is shown in Fig.4.

Fig.4 Structure block diagram of FOSM-MRAS observer

4 Simulation

The frame diagram of the sensorless MPTC system of FP-PMSM is shown in Fig.5.

Fig.5 Block diagram of PMSM-MPTC based on FOSM-MRAS observer

The parameters of FP-PMSM are shown in Table 1.

Table 1 Parameters of FP-PMSM

4.1 Comparison of SM-MRAS and FOSM-MRAS

The given speed is 1 000 r/min and the simulation time is set to 0.4 s. After no-load starting of the motor, the load of 3 N·m is added suddenly at 0.2 s. Fig.6 is the speed response of sensorless MPTC system of FP-PMSM.

Fig.6 Speed response of sudden load

In order to compare objectively and fairly, the overshoot of the speed response curves of both systems is close to zero by adjusting the parameters. It can be seen that the control system based on two kinds of observers can respond to the system load quickly when running without load, but FOMS-MRAS has a shorter rise time. After 0.2 s sudden load, both of them can quickly respond to the load change. However, the dynamic descent and response time of FOSM-MRAS are significantly shorter. It means that the control system based on FOSM-MRAS has good dynamic and static performance and strong anti-load disturbance.

(a) Observation error of SM-MRAS angular velocity

(b) Observation error of FOSM-MRAS angular velocityFig.7 Angular velocity error

(a) SM-MRAS Rotor position>

(b) FOSM-MRAS rotor positionFig.8 Curve of rotor position estimation and actual value change

Fig.9 shows the load response of the control system based on two observers. The torque ripple of FOSM-MRAS is smaller. After the sudden load is applied, the SM-MRAS has obvious torque fluctuations, while the FOSM-MRAS torque ripple remains stable.

(a) Torque response of SM-MRAS under sudden load

(b) Torque response of FOSM-MRAS under sudden loadFig.9 Torque response to sudden load

Fig.10 shows the five-phase current response of the two control systems. Total harmonic distortion (THD) is shown in Table 2, and it is defined as

(a) Five-phase current of SM-MRAS

(b) Five-phase current of FOSM-MRAS

Table 2 Current harmonic distortion

(47)

whereX1is the fundamental wave,Xnis the high-order harmonic wave, andDTHD,xis the distortion of each phase harmonic wave.

It can be seen that the FOSM-MRAS based control system has smaller THD value and better control performance.

4.2 Comparison of FCS-MPTC and DTC

Based on FOSM-MRAS observer, the sensorless PMSM drive systems using FCS-MPTC and DTC control strategies are presented, respectively. The five-phase current response during no-load operation is shown in Fig.11.

(a) Five-phase current response of DTC control

(b) Five-phase current response of FCS-MPTC controlFig.11 Five-phase current response

It can be seen that the current fluctuation of FCS-MPTC is significantly smaller than that of the DTC.

5 Conclusions

In this study, the FCS-MPTC system based on FP-PMSM discrete mathematical model is realized. Firstly, in order to optimize the selection of the voltage vector of the control system and improve the delay of one cycle when the digital control system is realized, the objective function of the FCS-MPTC system is improved and designed. Secondly, a new type of FOSM-MRAS speed observer is designed, and the stability of the designed observer is proved. Finally, the improved FP-PMSM speed sensorless FCS-MPTC control system is realized through simulation. Simulation results show that the FOSM-MRAS speed observer has high observation accuracy and stability. Under different operating conditions of the motor, the improved FCS-MPTC system based on FOSM-MRAS speed observation is stable in operation, more robust to load disturbance, and has good dynamics. Compared with the DTC system, the designed FCS-MPTC system significantly reduces the pulsation of the motor’s five-phase current.