Harmonic-Separation-Based Direct Extraction and Compensation of Inverter Nonlinearity for State Observation Control of PMSM

Nonideal switching characteristics of voltage source inverter (VSI) will result in distortions in both phase currents and reference voltages of a permanent magnet synchronous machine (PMSM) drive system, which will also consequently introduce ripples into the output speed and shaft torque, and result in a performance deterioration in the state observation control. To solve this issue, a method of directly extracting and compensating the nonlinear voltage disturbance due to VSI nonlinearity employed in PMSM drives is proposed in this paper. Firstly, the distorted voltage due to VSI nonlinearity is directly extracted via a harmonic separation scheme, which can be easily compensated by a proportional integral controller afterwards. Since both 6th harmonic and DC terms of dq-axis distorted voltages due to VSI nonlinearity are well compensated, the proposed method shows sufficiently good performance under both $i_{d}=0$ and maximum torque per ampere controls ( $i_{d} < 0$ ), as verified on two prototype PMSMs being salient pole and non-salient pole, respectively, both developed for a low-speed rotary actuator. In addition, the application of proposed method to state observation controls such as the model-based sensorless control and the deadbeat predictive current control, is finally investigated. Test results manifest that the proposed method can effectively improve the accuracy of observed rotor position/speed, and the ability of current tracking.


V dead
Distorted voltage term due to voltage source inverter nonlinearity (V). T s , T dead Switching period and dead time (µs). T on , T off Turn-on/turn-off delay of switching tubes (µs). V dc DC link voltage (V). V sat , V d Forward voltage drops of switching tubes and freewheeling diodes (V). L d , L q dq-axis inductances (H).
The associate editor coordinating the review of this manuscript and approving it for publication was Jinquan Xu . ω, ω e Mechanical and electrical speed (rad/s). θ, θ e Rotor electrical position, and θ = θ e + π/2 (rad).

∧
Denoting the estimated value.

I. INTRODUCTION
The conventional three-phase half-bridge inverter and space vector pulse width modulation (SVPWM) techniques have been extensively employed in drive control of permanent magnet synchronous machines (PMSMs), which usually work on the assumption that the switching characteristics of power devices are ideal. However, in reality, nonlinearities of voltage source inverters (VSIs), such as dead time, switch-on/switch-off delay, and forward voltage drops due to on-state insulated gate bipolar translators (IGBTs) and freewheeling diodes, would result in a mismatch between reference voltages and real ones of PMSMs, and ultimately lead to non-negligible 6th order harmonic in the synchronous reference frame [1]- [3]. The voltage distortion due to 6th order harmonic would eventually generate disturbances in three-phase currents, output torque and rotor speed, especially in low-speed applications. Furthermore, stator observation control such as sensorless control and deadbeat predictive current control (DPCC) are now widely used in the high-performance drive control of PMSMs. In reality, the VSI nonlinearity can cause 5th and 7th harmonics of αβ-axis estimated back electromotive force (EMF) in the model-based sensorless technique [4], which degrades the accuracy of rotor position/speed estimation. And on account of the voltage distortion resulted from VSI nonlinearity, the tracking performance of stator current gets worse in the DPCC. Overall, accurate machine voltages and low-harmonic stator current are usually essential for the model-based sensorless control and DPCC [5], which need a precise compensation on the VSI nonlinearity. Thus, the compensation of VSI nonlinearity has been extensively studied recently in both academia and industry, whose mainstream methods can be classified into two main types being reviewed below.
The first type can be regarded as a pulse-based correction scheme, which adjusts gating signals of power devices to compensate those nonlinear effects and needs accurate detection of current polarities [6]- [14]. A simple solution is to directly determine polarities of three-phase currents according to current sensors [6]- [12] or terminal voltages [13], whose accuracy depends on the analog-to-digital converter (ADC). And a circuit for the detection of current polarities is proposed in [14], which can significantly improve the accuracy of detection [14]. However, it would result in higher cost and increase the complexity of hardware system. Besides, it would be simpler if only the influence of dead time is taken into account [6]- [9] whereas the accuracy of compensation is still unsatisfactory in low DC link voltage applications, in which on-state voltage drops of power devices and turn-on/off delay play quite important roles in three-phase voltages [10]- [13]. Although the voltage drop is taken into consideration in [11], it is regarded as a constant and the accuracy of compensation cannot be guaranteed under changeable phase current conditions. Furthermore, the curve between the conduction voltage drops of power devices and phase currents can be measured from experimental tests [15], whose precision really depends on the stator resistance of the measurement model.
The second type, which takes into account the estimation and compensation of VSI nonlinearities including on-state voltage drops of power devices, dead time, switch-on/off delay etc. and does not require offline tests, has now become the mainstream solution in recent studies [16]- [30]. This solution is based on the average value theory in which the volt-second error in a switching cycle is compensated to the reference voltage, then the reference voltage acts on the SVPWM to generate the drive signal of IGBTs. In this solution, a disturbance voltage term is usually defined for representing the voltage distortion due to VSI nonlinearities, and will be estimated and compensated by way of harmonic analysis [16]- [23] or disturbance voltage observer [24]- [30].
The former one, being based on harmonic analysis, can extract the disturbance voltage term from reference voltages or measured currents, which can be employed for the online compensation of VSI nonlinearity afterwards. For example, it is reported in [16]- [23] that the VSI nonlinearity can be approximately compensated via the minimization of harmonics. Specifically, the disturbance voltage term can be identified by waveform analysis [17], self-commissioning algorithm [18], iterative learning control (ILC) [19], total least square algorithm [20], current injection [21], neural network [22], [23] and so on. The strategy in [17] is insensitive to the variation of machine parameters, whereas it only works well at steady state, and the estimation process of disturbance voltage is also relatively slow at low-speed region due to the calculation of average slope. In [18], the proposed algorithm enhances the accuracy of flux identification, and has a good application in induction motor drives. According to eliminating harmonics in the estimated d-axis back EMF, the method in [19] achieves a better performance in sensorless control. Note that the update law gain of ILC needs to be carefully selected, otherwise the algorithm cannot converge. Similarly, it is proposed in [20] to estimate the disturbance voltage by eliminating the harmonics of estimated back EMF. In [21], current injection is used to estimate VSI nonlinearity, but its application is quite limited due to the ripple generated during injection. As detained in [22] and [23], adjusting the weight factor of neural network can estimate VSI nonlinearity online, but the design of algorithms is relatively complex. In addition, the PMSM model can be simplified by using i d = 0 to be less parameter-dependent and achieve the nonlinearity compensation [16], [22]. However, it would not be effective for i d < 0 control, for example, the maximum torque per ampere (MTPA) control of an interior permanent magnet synchronous machine (IPMSM). It is noted that the disturbance voltage varies with the angle between current vector and q-axis [26], that is, the disturbance voltage is related to the control mode. Research on the compensation of VSI nonlinearity under MTPA control is presented in [31], which utilizes the resistance of inverter and machine to calculate the voltage error.
The observer based method usually has good performance in the identification of the disturbance voltage term, whereas it inherently relies on the accuracy of electrical machine parameters [24]- [30]. In this scheme, the observed disturbance voltage term is compensated online, and the observation error due to parameter variation will act directly on the control system. In short, the observer based method is usually quite sensitive to machine parameters, and its accuracy would deteriorate in applications with uncertain or VOLUME 9, 2021 changeable machine parameters. In particular, it is proposed in [26] to indirectly observe an intermediate parameter to calculate the voltage distortion, which avoids the observation delay caused by filtering. In fact, the intermediate parameter is the distorted voltage term V dead in this paper, whereas the proposed method in this paper does not require any machine parameters. Furthermore, a simplified model of PMSM is needed to establish a disturbance observer [25], [29], for example, i d = 0.
As analyzed hereinbefore, the pulse-based correction scheme usually only considers the nonlinearity due to switching time delay and dead time, or requires offline tests in low DC link voltage applications. As for the scheme based on estimation and compensation of VSI nonlinearities, the harmonic analysis based method usually has relatively complex design of algorithms or can only work well at steady state and i d = 0 control, while the observer based method is quite parameter-dependent. In order to achieve simplicity, multimode controls and parameter-independence in the compensation of VSI nonlinearities, a method of directly extracting and compensating the nonlinear voltage disturbance is proposed in this paper. It extracts the disturbance voltage term according to a harmonic separation scheme, which is parameterindependent and needs quite few calculations. Afterwards, the determined distorted voltage term is eventually online compensated by a close-loop proportional integral (PI) control. Since both 6th harmonic and DC terms in dq-axis distorted voltages are considered in the extraction model, it can be employed for both i d = 0 and i d = 0 controls. The effectiveness of proposed method is finally evaluated on a non-salient pole surface-mounted PMSM (SPMSM) and a salient pole IPMSM, respectively, which shows quite good performance under both i d = 0 and MTPA controls (i d = 0). Besides, the application of proposed method to state observation control is also verified. Experimental results show that the proposed method can significantly improve the performance of state observation controls such as the sensorless control and the DPCC. The whole paper is organized as follows: Section II is about the theory fundamentals of extraction of disturbance voltage due to VSI nonlinearity; Section III introduces the design of a feedback PI control to suppress the VSI nonlinearity; Section IV shows the experimental verifications and analyses; Section V details the application of proposed method to state observation controls.

II. MODELING OF PMSM DRIVE SYSTEM AND EXTRACTION OF DISTORTED VOLTAGE DUE TO VSI NONLINEARITY A. MODELING OF PMSM DRIVE SYSTEM
Neglecting the influence of cross-coupling effect and the measurement error of DC link voltage, the dq-axis voltage equations of PMSM considering VSI nonlinearity can be expressed as follows: [25]- [29] where u * d and u * q are dq-axis reference voltages, i d and i q are dq-axis currents, V dead Dd and V dead Dq are dq-axis distorted voltage terms due to VSI nonlinearity. V dead can be considered as a constant at steady state, and it can be written as [25], [27] (2) In addition, Dd and Dq are functions of electrical angular position and current polarities, and can be expressed below: where i a,b,c represents one of three-phase currents. As shown in Table 1, Dd and Dq will vary under different current polarities and electrical angular position [28] while λ is the current angle between current vector and q-axis, as indicated in Appendix A. The current angle λ determines the average values of V dead Dd and V dead Dq in theory. For example, in the first case of Table 1, Dd = 4sin(θ e ) and Dq = 4cos(θ e ) while in the second case, Dd = 4sin(θ e -π/3) and Dq = 4cos(θ e -π/3), etc. Thus, Dd and Dq can also be re-written as follows: Dd where α could be 0, -π /3, -2π/3, -π , -4π /3, or -5π /3, as shown in Table 1. As a result, the frequency of distorted voltage due to VSI nonlinearity will be 6 times of fundamental frequency in theory, which can also be validated by experiments as shown in Fig. 1. It can be seen in Fig. 1(a) that during the test, calculated Dd and Dq are oscillating at 6 times of fundamental frequency (10Hz). Consequently, resulting 6th harmonic will be introduced into dq-axis currents, as shown in Fig. 1(b), and generate ripples in both the output torque and speed, and eventually result in deterioration of control performance.

B. EXTRACTION OF DISTORTED VOLTAGE DUE TO VSI NONLINEARITY
According to the theory of Fourier series expansion, a periodic signal can be expressed as the synthesis of DC and harmonic components. Thus, the dq-axis reference voltages of a PMSM drive system can be expressed as follows: where u * dd and u * qd are DC components of dq-axis reference voltages; u * dh and u * qh are harmonic components. Similarly, the dq-axis distorted voltages caused by VSI nonlinearity can be represented as follows: can be extracted by low-pass filters (LPFs), as shown in Fig. 2.
In order to extract the distorted voltage term V dead generated by VSI nonlinearity, a harmonic separation scheme is proposed in this paper and introduced below: Firstly, u * dh and u * qh in (6) can be re-written as follows: where u dh and u qh are dq-axis harmonic components except VSI nonlinearity, which are mainly related to back EMF harmonics, cogging torque, etc. It should be noted that for those low voltage/low speed PMSMs, the harmonics generated by the VSI nonlinearity are usually dominant compared with other harmonic sources. Through (6)- (8), dq-axis reference voltages can be eventually re-written as follows: Substituting (5) into (9) yields: Since V dead can be regarded as a constant at steady state, it can be extracted by the harmonic separation scheme introduced as follows: Firstly, (10a) and (10b) multiplied by sin(θ e + α) and cos(θ e + α), respectively, will yield (11): Finally, substituting (5) into (12) yields: Equation (13) is the key model for the extraction of V dead , and the whole harmonic separation scheme can be illustrated by the control block diagram shown in Fig. 2. Its working principle is explained as follows: In (13), Dd and Dq can be directly calculated by (3) and (4), while u * dd , u * qd , (V dead Dd) d and (V dead Dq) d can be obtained by low-pass filters, as shown in Fig. 2. The 4V dead in (13) can be considered as a DC term. And for those low voltage/low speed PMSMs designed for servo control or high-accuracy actuators, the harmonics in u dh and u qh are quite negligible, especially in unsaturated region. Based on this hypothesis, it can be known that the two terms, u dh Dd/4 and u qh Dq/4, are both high-frequency harmonic terms. In short, as can be seen in (13), the expression on the left side consists of variables that are measurable, while the expression on the right side has a constant DC term 4V dead and two high-order harmonic terms u dh Dd/4 and u qh Dq/4. Thus, the DC component V dead in (13) can be directly extracted by the control block diagram illustrated in Fig. 2. Besides, V dead Dd, V dead Dq, u * d , u * q and the left side of (13) in Fig. 2 mainly contain 6th harmonic denoted by 6f e , and f e is the fundamental frequency of machine operation. The cut-off frequency f c of LPFs is set to f c = 0.1 × 6f e . And a detailed description of LPFs and the extraction of V dead at different cut-off frequencies are given in Appendix B.

III. FEEDBACK COMPENSATION SCHEME OF VSI NONLINEARITY
Once V dead can be real-time extracted, disturbance voltage terms V dead Dd and V dead Dq can be online compensated afterwards [26], [28]. Since the disturbance voltage term V dead can be considered as a constant at steady state, a PI control scheme of close-loop feedback is proposed for its compensation, as illustrated in Fig. 3. It can be seen in the control block diagram that the reference value of V dead is set to 0V, which means that it will be eventually compensated to 0V. In this case, V dead Dd and V dead Dq will be consequently minimized to 0V, as a consequence, reference voltages and actual ones can be equivalent. As shown in Fig. 3, the PI output is multiplied by Dd and Dq, respectively, and their products u dc and u qc will be employed for the compensation of dq-axis disturbance voltages. Both u * d2 and u * q2 are the dq-axis reference voltages after compensation, and will be sent to the SVPWM control. In order to prevent over compensation under transient conditions, a saturation function is applied to limit the PI output. The limit can be determined according to the output peak of PI regulator at the rated current.
As can be seen from Fig. 1(a), it is note that the average value of Dd is zero under i d = 0 control, which means that the DC component (V dead Dd) d in (13) is also be zero. However, (13) is not be zero and needs to be considered. The above mentioned variation in (V dead Dd) d is a key factor in the harmonic analysis based compensation of VSI nonlinearity [22], whereas most existing researches cannot adapt to working under both i d = 0 and i d = 0 controls. Thus, two PMSMs designed for low-speed actuators, being salient pole and nonsalient pole, respectively, will be employed for evaluating the performance of proposed method under i d = 0 and i d = 0 (selecting MTPA as an example), respectively. The main design parameters of two prototype PMSMs are listed in Table 2. For the SPMSM, it will operate under conventional i d = 0 control while the IPMSM will operate under the conventional MTPA control mode being introduced by following equations: [32] where I s is the stator current from the speed loop output.

IV. EXPERIMENTAL VERIFICATION A. TEST RIG AND HARDWARE CONFIGURATION OF DRIVE SYSTEM
The proposed method is experimentally validated on the PMSM test rig shown in Fig. 4, in which two prototype PMSMs are employed for investigation, respectively. The DC link voltage of test rig is set to 60V. A magnetic powder brake is firstly employed as the external load of PMSMs, which has an accuracy in torque production and will be replaced by a DC load machine for the step test afterwards. The product model of employed power inverter module is FSBB30CH60F,   of which typical electrical characteristic parameters are listed in Table 3. Besides, the control board and the power board are integrated together, and the product model of employed microprocessor is MKV31F256VLL12 being of ARM architecture whose ADC resolution is 16bits. According to the technical information of power inverter module, the switching frequency of inverter is set to 12kHz and the dead time is set to 3µs.
In order to test the performance of proposed method under i d = 0 control, it is firstly employed on the drive control of the non-salient pole SPMSM. Fig. 5 shows the extraction and compensation of V dead under i d = 0 control when load torque is 1.5N·m. It can be seen that the proposed extraction scheme has a good performance in convergence, and afterwards, the extracted disturbance voltage term V dead can be minimized to 0V. After the settling down of compensation, disturbance voltages in u * d and u * q would be eventually minimized to zero, which implies a good application potential in advanced drive controls such as sensorless control [18]- [20], predictive cur-  rent control [5], [38], and parameter identification [21], [22], thanks to its parameter-independent features. It is worth noting that the extracted V dead increases with the increase of load torque, which indicates that the proposed method recognizes the variation of voltage drops of power devices without offline experiments.
In Fig. 6, u * α and u * β are αβ-axis reference voltages obtained by inverse Park-transformation of u * d and u * q under i d = 0 control, while u * α2 and u * β2 are αβ-axis voltages obtained by inverse Park-transformation of u * d2 and u * q2 , as detailed in Fig. 3. Obviously, the synthesized voltage vector of u * α and u * β becomes more circular with the proposed compensation, which indicates that the real stator winding voltages would have less harmonics after compensation [2]. Similarly, as can be seen in Fig. 7, the current vector in αβ-axis coordinate becomes more circular after the compensation and the waveforms of dq-axis currents become much smoother, which is quite important in the reduction of torque ripples. Besides, as shown in Fig. 8, the clamping phenomenon during the zerocrossing becomes negligible, and the current of phase A is almost sinusoidal after the proposed compensation. The FFT results shown in Figs. 8(b) and (d) confirms that the 5th and 7th harmonics are almost zero after the compensation. Fig. 9 shows the waveforms of rotor speed with and without the proposed compensation, together with their FFT results, respectively. In Fig. 9, 10Hz is the fundamental frequency of three-phase currents. The 20Hz component is mainly related to the scaling error in current measurement [33], while the 60Hz component is mainly caused by the VSI nonlinearity. It can be seen in Fig. 9 that after the proposed compensation, the 6th harmonic in rotor speed can be significantly suppressed. As a consequence, torque ripples would be therefore suppressed [34].
Furthermore, as can be clearly seen in Fig. 6(b), the disturbance voltage due to VSI nonlinearity will result in a significant distortion in the reference voltage, especially at relatively low-speed region, for example, 150r/min. There-  fore, the compensation of disturbance voltages is quite essential in reality, especially in a low-speed actuator system which works at relatively low voltage and is sensitive to current and voltage harmonics. Besides, in high speed applications such as the flux weakening control, the compensation of VSI nonlinearity is also important for achieving a higher speed expansion and a larger constant output power region [35].
In Figs. 10-12, the dynamic performance of proposed method is further evaluated on a non-salient pole SPMSM, by way of speed ramp/step response and load step response, respectively. Fig. 10 shows measured dq-axis currents and rotor speeds under variable speed control. The ramp response of rotor speed is at the rate of 100r/s. It can be seen in Fig. 10(a) that the 6th harmonic oscillations in d-axis current is quite obvious during the whole test, while those oscillations could be significantly suppressed within ± 0.05A after the proposed compensation. It is also noteworthy that small 6th harmonic oscillations in q-axis current and rotor speed can be also minimized after the proposed compensation. Fig. 11 shows measured waveforms during the step response of rotor speed. It can be seen that with the proposed compensation, the measured currents have quite negligible 6th harmonic during the whole step response test. Fig. 12 shows measured waveforms during the load step VOLUME 9, 2021   response. As a result, the q-axis current will immediately increase from 1A to 2.7A after the addition of external load from 0.5N·m to 1.4N·m. In Fig. 12(a), 6th harmonic in dq-axis currents and clamping effect during the zero-crossing transients are quite obvious without compensation, whereas those phenomena will be eliminated with the proposed compensation. Based the above tests and analysis, it can be concluded that the proposed method can achieve a continuous accurate compensation under both dynamic and steady state tests.

D. TEST RESULTS OF IPMSM UNDER MTPA CONTROL
In order to verify the effectiveness of proposed method under i d = 0 control, a prototype IPMSM operating under MTPA control (14) is employed for performance evaluation, and relevant experimental results and FFT analysis are depicted in Figs. 13-15. In Fig. 13, it is obvious that the rotation of voltage vector in αβ-axis coordinate becomes more circular with the proposed     Fig. 14, with the proposed compensation, the rotation of current vector in αβ-axis coordinate becomes more circular, and the dq-axis currents become quite smooth. It is worth noting that compared with i d = 0 control, the 6th harmonic in q-axis current will become more significant, which can be explained that the current angle λ has been changed to non-zero under i d = 0 control, as described in Table 1. Besides, as can be seen in Fig. 15, the phase A current is almost sinusoidal and its total harmonic distortion (THD) is significantly reduced from 5.75% to 1.44% after the compensation. From the above test results and analyses, it manifests that the proposed scheme also has excellent performance under i d = 0 control, which confirms the theory proposed in section II and Fig. 3.

V. APPLICATION OF PROPOSED METHOD TO STATE OBSERVATION CONTROL OF PMSM
As mentioned hereinbefore, conventional state observation controls, such as model-based sensorless control and DPCC, needs accurate reference voltages and/or less-harmonic stator currents to observe rotor position/speed, stator current and so on. However, due to the VSI nonlinearity, the reference voltages and measured stator currents usually have non-negligible harmonics. Since the proposed method is parameter-independent and needs few CPU calculation and no additional hardware, it is inherently proper for improving the performance of state observation controls. Thus, the application of proposed method will be studied in this section, and the SPMSM introduced in Table 2 will be employed for tests. Block diagrams of application of proposed method including PMSM-model-based sensorless control and DPCC are shown in Fig. 16. In Fig. 16(a), the sensorless strategy is based on the conventional sliding mode observer (SMO) and phase-locked loop (PLL), whose detailed design process is introduced in Appendix C. In Fig. 16(b), the prediction model of investigated DPCC is cited from [38], which is introduced in Appendix D.

A. EXAMPLE 1: APPLICATION OF PROPOSED METHOD TO SENSORLESS CONTROL
In traditional SMO-based sensorless control, the αβ-axis voltage equations can be expressed as follows: [36], [37] where u α and u β are actual αβ-axis voltages, i α and i β are αβ-axis currents, L s is the stator inductance (L s = L d = L q ), p is the differential operator, E α and E β are the extended back EMF. Usually, the real machine voltages u α and u β are not measurable due to the high frequency PWM, and αβ-axis reference voltages u * α and u * β are employed for replacing u α and u β in control algorithms. However, u * α and u * β contain obvious harmonic components resulted from VSI nonlinearity, and so do i α and i β , which are employed as the inputs of SMO. Thus, the performance of sensorless control will deteriorate owing to the harmonics in input signals.
Considering the distorted voltage due to VSI nonlinearity, and neglecting the influence of cross-coupling effect, the αβ-axis voltage equations can be written as follows: As shown in (16), it is essential to compensate the VSI nonlinearity to ensure the accuracy of u * α and u * β . Otherwise,  the performance of sensorless control will deteriorate due to those nolinearities. Figs. 17-19 are test results using sensorless control without VSI nonlinearity compensation under T dead = 2µs, 3µs and 4µs, respectively. As the dead time increases, the 5th and 7th harmonics of u * α , u * β , i α , i β , and the value of extracted V dead increase. As a result, the estimated back EMF will contain more 5th and 7th harmonics with the increase of dead time. As a consequence, the back EMF harmonics will increase the position-error in the PLL based position observation, as shown in Figs. 17-19(f).
On condition that the rotor speed and dead time are set to 150r/min and 3µs, respectively, Figs. 20-23 are test results without compensation of VSI nonlinearity, while Figs. 24-27 are waveforms with the proposed compensation. In Fig. 20, the estimated E α has obvious 5th and 7th harmonics due to the influence of VSI nonlinearity. Comparing Fig. 20(d) with Fig. 24(d), it is evident that the proposed method can effectively reduce both 5th and 7th harmonics in estimated back EMF. Besides, the extracted V dead will quickly converge to 0V after the proposed compensation, as shown in Fig. 25. Comparing Fig. 22 with Fig. 26,   FIGURE 22. Error between measured and estimated rotor positions without compensation of VSI nonlinearity (T dead = 3µs, 150r/min, 1N·m).

FIGURE 23.
Error between measured and estimated rotor speeds without compensation of VSI nonlinearity (T dead = 3µs, 150r/min, 1N·m). it can be seen that the maximum position-error will distinctly decreases with the proposed compensation, which decreases from 2.52 • to 1.12 • . In addition, as shown in Figs. 23 and 27, the oscillation of observed rotor speed is also significantly reduced with the proposed compensation, whose peak-topeak error decreases from 12.5r/min to 10.1r/min.

B. EXAMPLE 2: APPLICATION OF PROPOSED METHOD TO DPCC
The conventional DPCC can directly obtain the voltage vector via. PMSM model, which guarantees the current loop can VOLUME 9, 2021   achieve good dynamic and steady characteristics at the same time. As detailed in Appendix D, the conventional DPCC model does not consider the influence of VSI nonlinearity in predicted dq-axis voltages: [38] where k denotes the kth sampling period. The predicted dq-axis voltages u pre d (k) and u pre q (k) are used as reference voltages for the SVPWM control. However, as discussed in Section II, the reference and output voltages are usually not exactly the same due to the existence of VSI nonlinearity. u pre d (k) and u pre q (k) considering the VSI nonlinearity can be re-written as follows: According to the Table 1, assuming that the value of V dead is 0.5V and i d = 0, the term V dead Dd(k) mainly contains 6th harmonic and its average value will be 0V while the value of V dead Dq(k) is approximately 2V. Since the back EMF is relatively small at low-speed, the term V dead Dq(k) will have great influence on the current loop control, in which the tracking performance of feedback current will deteriorate. Thus, it is necessary to take into account the compensation of VSI nonlinearity for DPCC. Fig. 16(b) is the block diagram of DPCC using the proposed method. It should be noteworthy that with the proposed method, the predicted u pre d (k) and  u pre q (k) shown in Fig. 16(b) are equivalent to the reference voltages u * d and u * q shown in Fig. 3. Fig. 28 shows the experimental results of DPCC without compensation of VSI nonlinearity under T dead = 2µs, 3µs and 4µs, respectively. In Fig. 28, the measured i q cannot accurately track the corresponding reference current i * q . When the dead time is set to 2µs, 3µs and 4µs, respectively, the steadystate error will be 0.38A, 0.43A and 0.47A, respectively. Obviously, the tracking ability of stator current deteriorates with the increase of dead time. This can be explained that the term V dead Dq(k) takes a non-negligible portion of q-axis reference voltage, and i * q cannot accurately matches i q . Since V dead Dd(k) is of zero mean under i d = 0 control, i d will fluctuate around i * d and contains obvious 6th harmonic component, as shown in Fig. 28. Fig. 29 shows the application of proposed method to the DPCC. It can be seen that the DPCC with the proposed   compensation can accurately track the reference currents i * q and i * d , and significantly reduces the oscillation of i d and i q . Fig. 30 shows the test results of DPCC without compensation of VSI nonlinearity under T dead = 2µs, 3µs and 4µs, respectively, while Fig. 31 shows the DPCC with the proposed compensation under T dead = 3µs. It can be seen that when T dead = 2µs, 3µs and 4µs, respectively, the steadystate error becomes 0.41A, 0.45A and 0.48A, respectively.
It can be concluded that the steady-state error between i * q and i q increases with the dead time. In addition, since the overall distorted voltage due to VSI nonlinearity will increase with the current, it can be seen in Figs. 28 and 30 that the increase of load torque will increase the steady-state error. It also can be seen in Fig. 31 that the proposed compensation can significantly reduce the steady-state error and improve the tracking performance of dq-axis currents.

VI. CONCLUSION
A simple and parameter-independent compensation of VSI nonlinearity has been proposed in this paper, which is based on the cooperation of a harmonic separation scheme and a PI feedback. Compared with existing methods, the proposed scheme does not need extra hardware, system parameters, and can be used in both i d = 0 and i d = 0 controls. Its dynamic and steady state performances are evaluated on a non-salient pole SPMSM and a salient pole IPMSM, respectively, which shows good performances during the speed ramp/step response and load step response. Besides, the experiments show that the proposed method has a good application prospect in state observation controls such as the conventional sensorless control and DPCC. It is worth noting that, for electrical machines usually working under saturated regions or with quite significant cogging torque, an over compensation on the 6th harmonics will probably occur. In this case, an offline test for the 6th harmonics in back EMF and cogging torque will be needed for improving the accuracy of proposed method. It will be our future work to investigate the above challenge.

APPENDIX A
In this study, the relationships among abc, αβ and dq coordinates are shown in Fig. 32.

APPENDIX B. SETUP OF CUT-OFF FREQUENCIES OF LOW-PASS FILTERS
The digital LPFs used in Fig. 2 are discretized from the firstorder RC LPFs. The expression after discretization is: where u o is the signal output; u i is the signal input; f c is the cut-off frequency; T is the sampling period (T = 83.3µs). Fig. 33 shows the extraction of V dead at different cut-off frequencies when the f e is 10Hz. With the increasing of f c , the convergence speed can be improved while the steadystate fluctuation will synchronously rise up. The f c has no influence on the average value of extracted V dead , However, it will influence on the dynamic performance of extraction of V dead , for example, the convergence speed and peak to peak oscillation at steady state. After a tradeoff between filtering effect and convergence speed, f c = 0.1 × 6f e is set as the employed f c in principle during the whole speed range.

APPENDIX C. DESIGN PRINCIPLE OF SMO AND PLL
The αβ-axis state equations of SPMSM including the extended back EMF can be expressed as follows: [4], [36] d dt In practical applications, the values of u α and u β are acquired from u * α and u * β in Fig. 16(a), respectively. It is noted that, after the compensation of VSI nonlinearity, u * α and u * β basically have no harmonic components. The EMF-based SMO is insensitive to parameter variation and converges rapidly, which is usually constructed as whereî α andî β are the observed values of stator current, k s is the sliding mode gain and its value is designed by Lyapunov theory. Then (C2) subtract (C1) becomes: In order to estimate E α and E β , the sliding mode surface When the state variables of SMO reach the sliding mode surface, the back EMF can be expressed as follows: The rotor position can be extracted from the estimated back EMF. A simple method is to use an arctangent function, while it sensitive to system noise. In this paper, the PLL is used to extract the position information, as shown in Fig. 34.
Besides, the estimated position error is normalized to improve the stability of PLL. Due to the back EMF related to the rotor speed, the poles of the transfer function of PLL-based estimator may vary with different speeds, which will transform the response performance of PLL. The normalized position error is expressed as The estimated position can be obtained as follows: where k p and k i are the proportion and integration coefficients of PI controller, respectively.

APPENDIX D. BASIC PRINCIPLE OF DPCC
According to Euler's method, the dq-axis voltage equations of PMSM can be discretized as follows: where u d and u q are actual dq-axis voltages.
In (D1), replacing i d (k +1) and i q (k +1) with the reference currents i * d (k) and i * q (k), the predicted voltages u pre d (k) and u pre q (k) can be obtained as follows: −ω e L q i q (k) In a microprocessor, i d (k + 1) and i q (k + 1) are generated by the previous predicted voltages [38]. Thus, it is necessary to predict i d (k + 1) and i q (k + 1) to improve the accuracy of voltage prediction, as shown below: where the superscript com represents the delay compensation. Finally, substituting (D3) into (D2) yields: Equations (D3) and (D4) are the core models of DPCC, and their specific applications are shown in Fig. 16 Research Centre. He is currently an Associate Professor with the School of Automation, Wuhan University of Technology. His research interests include the power electronics system design and control, such as dc-dc converter sensorless control strategies, electrical machine parameters estimation by control theory, system nonlinearity compensation for dc-dc converters, and voltage source inverters. VOLUME 9, 2021