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(1.Inverter Technologies Engineering Research Center of Beijing,North China University of Technology,Beijing 100144,China;2.School of Electrical and Electroric Engineering,North China Electric Power University,Beijing 102206,China;3.College of Engineering,Universidad Andres Bello,Santiago 8370146,Chile)

Abstract:Model predictive current control can achieve fast dynamic response and satisfactory steady-state performance for induction motor (IM) drives.However,many motor parameters are required to implement the control algorithm.Consequently,if the motor parameters used in the controller are not accurate,the performance may deteriorate.In this paper,a new robust predictive current control is proposed to improve robustness against parameter mismatches.The proposed method employs an ultra-local model to replace the mathematical model of the IM.Additionally,to improve the control performance,a linear extended state observer is developed for disturbance estimation.Experimental tests confirm that satisfactory tracking performance can still be obtained although the motor parameters may not be accurately set in the controller.

Keywords:Induction motor drives,current control,predictive control,observers,robustness

1 Introduction1

Induction motors (IMs) are widely used in industrial fields due to their advantages,such as simple structure,high reliability,easy maintenance,and ability to adapt to various complex environments.At present,the high-performance control algorithms for IMs used in industrial fields include field-oriented control (FOC)[1]and direct torque control (DTC)[2].The FOC can achieve acceptable steady-state performance,but proper proportional-integral parameters should be designed.The DTC can achieve fast dynamic performance with a simple structure;however,classic switching table-based DTC presents relatively high torque ripples[3].In recent years,model predictive control (MPC)[4-5]has received increasing attention owing to its simple principle,fast transient response,and flexibility in handling nonlinear constraints and multiple variable control[6-7].

Many researchers have studied MPC and proposed new methods to improve its performance or reduce the control complexity.In Ref.[8],a model predictive flux control is proposed to eliminate weighting,and the amount of calculations is significantly reduced.In Ref.[9],the design of the weighting factor for stator flux is optimized to reduce torque ripples.Although the MPC exhibits satisfactory steady-state performance and fast dynamic response,it is highly dependent on the precision of motor parameters because these are required for predicting the evolution of the control variable.

In practical application,the motor parameters can change under different working conditions.For example,the stator and rotor resistances can change with temperature variations[10].When the motor parameters used in the predictive controller do not conform with their actual values,the derived voltage vector may not be the optimal;consequently,the overall control performance deteriorates.To resolve these problems,several methods for increasing robustness have been investigated in the existing literature,including online parameter adaption[11],disturbance observer[12],extended state observer (ESO)[13]and model-free control (MFC)[14].

An online adaption method can be applied to track the actual model parameters.In Ref.[15],an online parameter estimation scheme based on a discrete-time dynamic model was developed to estimate the stator inductance,stator resistance,rotor flux linkage,and load torque.In Ref.[16],an online identification method based on a reference model,which estimates the motor parameters by nullifying the current estimation error,is presented.Compared with online identification based on Kalman filter,this method improves convergence dynamics and overall system stability.However,during steady state,the simultaneous estimation of multiple model parameters based on the fundamental motor model is infeasible[16].

The disturbance observer and ESO have also received considerable attention in the field of motor control.These methods can improve the robustness of the control system by observing and compensating for the lumped disturbance.In Ref.[13],a disturbance observer is investigated for the predictive torque control of IM drives subject to load disturbance,parameter errors,and digital delay.In Ref.[17],a robust speed and flux estimation is developed to reduce the impact of external disturbance and internal estimation error.Using the observer-based method,the motor can still maintain satisfactory control performance despite motor parameter variations and external disturbances.However,most of the above techniques are developed based on the machine mathematical model,and motor parameters are still required[18].

Recently,various MFC schemes have been introduced to improve parameter robustness for motor drives.Lin et al.[19]proposed a model-free predictive current control (MFPCC) based on the information of the previous current difference.Although this method does not require any motor parameters,it necessitates current sampling twice in one control period.Therefore,proper timing to trigger analog-to-digital conversion for current measurement is necessary to avoid the switching harmonics,which can increase implementation complexity in the sampled current.Fliess et al.[20]combined the principles of MFC and intelligent proportional-integral-derivative controllers.The method exhibits satisfactory robustness against unknown disturbances and parameter mismatches.The basic idea of this strategy is that the system is defined as an ultra-local model,which is estimated using differential algebra.In Ref.[14],an ultra-local model-based model-free current control (MFCC),which only relies on the input and output data of the controlled system and achieves satisfactory control performance for a surface-mounted permanent-magnet synchronous machine drive,is proposed.However,up to five parameters require tuning.

In this paper,a new robust predictive current control (RPCC) method based on the ultra-local model and a linear extended state observer (LESO) is investigated for IM drives.The work contribution can be highlighted by the following points.

(1) In the proposed strategy,the unknown part in the ultra-local model is estimated using the LESO.Compared with the conventional MFCC,the proposed RPCC reduces the amount of calculations.Further,it only uses two control parameters,thus simplifying the tuning effort.

(2) In existing literature,observers are used to estimate the influence of mismatched parameters based on the mathematical motor model.These methods still require knowing the motor parameters.The proposed method introduces an ultra-local model;it can estimate the total unknown part of the system without the motor model.Hence,applying the control scheme to other motors is convenient.

The effectiveness of the proposed RPCC is validated by experimental tests performed on a 2.2 kW IM drive.

2 IM drive system model

The voltage and flux equations of IM described by space vectors in a stationary frame can be expressed as[21]

3 Principle of proposed method

The proposed RPCC combines the MFC,LESO,and predictive current control.The control diagram of the proposed method is shown in Fig.1.The details of the proposed RPCC and LESO are explained in the following sections.

Fig.1 Control diagram of RPCC

3.1 Ultra-local model of IM

In Ref.[20],an MFC theory that defines a single-input single-output system is proposed in the form of a first-order ultra-local model

whereuandyare the input and output variables,respectively;αis the weighting factor of the input variable;Fcontains uncertain parameters and disturbances.The hatdenotes variable differentiation.

By comparing Eqs.(5) and (7),the stator current dynamics can be rewritten as follows

Althoughαcontains motor parameters,the desired control performance can still be achieved by adjustingFwhen the motor parameters change.Thus,Fshould be accurately estimated to ensure satisfactory control performance.

In Ref.[22],differential algebra[23]is applied to estimateFusingthe inputand output data overashort t ime.Theestimated value()can be calculatedby Eq.(9)[22,24]

3.2 Introduction of LESO

The ESO only requires a relative degree of the system[25];hence,it can be used without the motor model.

By using the ESO to estimate the unknown part of the system in RPCC rather than using differential algebraic parameter identification techniques,the performance of RPCC has been improved,and the implementation has been simplified.

The ESO can be divided into nonlinear extended state observer (NLESO) and LESO,which has better parameter robustness[26]and simpler structure.Thus,the LESO is used in this study.

The control objects of a first-order system can be defined as[27]

3.3 Parameter design of LESO

The structural diagram of the RPCC is shown in Fig.2.

To select the proper observer gain,assume that inductance error does not occur.The relationship between the real current and the set value can be expressed as follows

Fig.2 Structural diagram of RPCC in z domain

3.4 Prediction of current and voltage

4 Experimental results

To verify the advantages of the proposed RPCC,it is compared with the deadbeat predictive current control (DBPCC) and conventional MFPCC on a two-level inverter-fed IM drive platform.The conventional MFPCC represents an MFPCC that uses differential algebra (Eq.(9)) to estimate the ultra-local model;Fig.3 shows the experimental setup.The machine and observer parameters applied to the test are listed in Tab.1.The control algorithm is implemented on a 32-bit floating DSP TMS320F28335,and the sampling frequency is 10 kHz.In the experimental results,the stator current is sampled by a current probe.The motor speed andq-axis andd-axis currents are obtained via an on-board digital-to-analog converter.All data are acquired by a scopecorder(YOKOGAWA DL850) and plotted on a personal computer using Matlab.

Fig.3 Experimental setup of two-level inverter-fed IM drive

Tab.1 Machine and control parameters

4.1 Dynamic performance

The performance during the transient process is tested for the DBPCC,conventional MFPCC,and proposed RPCC.The results when the motor starts from standstill to 1 500 r/min with no load and with inaccurate parameters set in the controller are shown in Figs.4 and 5.From top to bottom,the curves in these figures represent the motor speed,q-axis current (iq),d-axis current (id),and one-phase stator current (ia).

Fig.4 Starting response from standstill to rated speed when all parameters increase to three times normal values

The responses when all parameters (stator resistanceRs,rotor resistanceRr,stator inductanceLs,rotor inductanceLr,and mutual inductanceLm) used in the controller become three times their normal values are shown in Fig.4.The results when all parameters are half of their nominal values are presented in Fig.5.For the DBPCC,significant current ripples or tracking errors are observed when the parameters are not accurate.As shown in Figs.4b and 5b,although the conventional MFPCC can achieve a fast starting response when the motor parameters are not accurate,some ripples are observed in thedq-axis current during motor acceleration.The dynamic responses of the proposed RPCC with different parameter mismatches are shown in Figs.4c and 5c.The figures clearly show that the proposed RPCC can achieve the best performance without the influence of parameter mismatches.

Fig.5 Starting response from standstill to the rated speed when all parameters become half of normal values

4.2 Steady-state response

Fig.6 Responses at 1 500 r/min with rated load and accurate parameters

The steady-state responses of the DBPCC,conventional MFPCC,and proposed RPCC with accurate motor parameters are shown in Fig.6.Based on the figure,all methods can achieve a stable operation.However,the conventional MFPCC exhibits slightly larger current ripples.The total harmonic distortions (THDs) of the current at different speeds with the rated load are shown in Fig.7.The steady-state performance of the proposed RPCC is comparable with that of the DBPCC when the model parameters are accurate.The RPCC clearly has a lower stator current THD than the conventional MFPCC at all testing points.

Fig.7 THD of stator current at different speeds without parameter mismatches for DBPCC,conventional MFPCC and the proposed RPCC

Performance comparisons with parameter errors are also conducted.The steady-state responses of the DBPCC,conventional MFPCC,and proposed RPCC with inaccurate motor parameters are shown in Figs.8-11.In these figures,All=npu denotes that all model parameters used in the control algorithm arentimes their actual values.A similar description is applied to resistance (RsandRr) and inductance (Ls,Lr,andLm).

The results at 150 r/min under the rated load with smaller estimated motor parameters for the DBPCC are shown in Fig.8a.The results show that the performance under low-speed operation is not significantly affected when the motor parameters set in the control algorithm are smaller.

The results at 150 r/min under the rated load with larger estimated motor parameters for the DBPCC are shown in Fig.9a.Larger stator resistance results are observed in certain deviations of thedq-axis current.If the rotor resistance is large,the DBPCC can still achieve satisfactory control performance.However,if the inductance is large,significant ripples in thedq-axis current are observed.From the foregoing tests,the DBPCC is found to be sensitive to overestimated inductance at a low-speed operation.

Fig.8 Responses at 150 r/min with rated load with smaller parameters

In contrast,Figs.8b and 9b indicate that the conventional MFPCC can achieve stable operation with different motor parameter errors.

The results for the proposed RPCC are shown in Figs.8c and 9c.Compared with the differential algebra method,the LESO can more accurately estimate;hence,the proposed RPCC can achieve a better steady-state performance.

Fig.9 Responses at 150 r/min with rated load with larger parameters

The results at 1 500 r/min under the rated load for the DBPCC are illustrated in Figs.10a and 11a.If the inductance is less than the accurate value,the system finally loses stability.If the inductance is larger than the accurate value,distinct stator current harmonics are observed.

The results at 1 500 r/min under the rated load for the conventional MFPCC are shown in Figs.10b and 11b.The conventional MFPCC exhibits acceptable parameter robustness,and satisfactory performance can still be achieved despite parameter mismatches.

Fig.10 Responses at 1 500 r/min with rated torque and smaller parameters

The results shown in Figs.10c and 11c are consistent with those shown in Figs.8c and 9c.The proposed RPCC can achieve better steady-state performance under parameter mismatches.

Fig.11 Responses at 1 500 r/min with rated torque with larger parameters

5 Conclusions

In practical implementation,motor parameters may change under different working conditions and environments.The conventional predictive current control is sensitive against parameter mismatches because many motor parameters are used in the algorithm.If large parameter errors exist,both the dynamic and steady-state performance levels deteriorate.To resolve this problem,the RPCC method for IM drives based on the ultra-local model and LESO is proposed in this paper.Compared with the DBPCC,the proposed method uses the input and output data of the system without requiring an accurate system model.Hence,it is robust against the estimation errors of motor parameters.The experimental results validate that the proposed RPCC achieves satisfactory control performance over a wide speed range even with parameter mismatches.