1 Introduction

Recently, permanent magnet synchronous motors are widely used in industrial applications for high efficiency and high output density. As shown in Fig. 1, there are mainly three kinds of synchronous motors. PMSM drivers generally have closed-loop control systems, based on vector control method. Vector control is one of the most popular controls for PMSM, known as decoupling or field orientated control (FOC). It decouples three phase stator currents into two phase d- and q-axis currents which produce the flux and torque, respectively such that it allows direct control of flux and torque.

Fig. 1
figure 1

Classification according to rotor shape of PMSM. a Spoke-type permeant magnet synchronous motor. b Interior buried permanent magnet synchronous motor. c Surface mounted permanent magnet synchronous motor

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Therefore, the exact information of the d- and q-axis inductances is indispensable for successful controller design. If the exact inductances of PMSM cannot be measured or estimated, it may occur as a result of low-efficient operation, output power reduction and even out-of-synchronization. In addition, PMSM has a severe local self-saturation and cross-saturation effects, so PMSM parameters such as resistance and d- and q-axis inductances are changed nonlinearly by conditions such as electric current, phase angle, etc. In particular, the d- and q-axis inductances change irregularly depending on the mechanical power, shape, and operating characteristics of motor, so the incorrect estimation of parameters cause the performance degradation of the control systems. Thus, accurate estimation of the inductances that change in real-time is a necessary element for designing the controller and ensuring the improved control performance.

The estimation methods are categorized into two methods: the offline and online parameter estimations according to the estimation time in system operation.

The offline parameter estimation techniques are classified according to the implementation point at standstill or operating conditions. The estimation techniques at the standstill conditions include the DC current decay test and AC standstill methods. On the other hand, the estimation techniques for the operating conditions include the vector-controlled method and generator test. The offline parameter estimation methods have the merit to be easily understood by its simple algorithm. However, there still exist disadvantages to require additional equipment and measurement error caused by the estimation at the single operating point.

Meanwhile, the online parameter estimation techniques include the model reference adaptive control techniques, recursive least square based techniques, extended Kalman filter based techniques, and artificial neural network techniques. The online parameter estimation techniques are appropriate for applications with various operation ranges, because they are performed during the system operation. However, the high-efficient microprocessor is required for dealing with the relatively complex procedure. In this paper, the state-of-the-art offline and online parameter identification techniques are entirely reviewed and summarized by researching the conventional and recent advancement of parameter identification methods for PMSM.

2 IPMSM Equivalent Circuit

For vector control of IPMSM, the three-phase voltage equation of IPMSM is converted into the d-q axes voltage equation as in (1) through Clarke transformation and Park transformation.

$$\left[ {\begin{array}{*{20}c} {v_{ds}^{r} } \\ {v_{qs}^{r} } \\ \end{array} } \right] = \left[ {\begin{array}{*{20}c} {R_{a} } & { - \omega L_{q} } \\ {\omega L_{d} } & {R_{a} } \\ \end{array} } \right]\left[ {\begin{array}{*{20}c} {i_{ds}^{r} } \\ {i_{qs}^{r} } \\ \end{array} } \right] + \left[ {\begin{array}{*{20}c} 0 \\ {\omega_{e} \Psi_{a} } \\ \end{array} } \right]$$
(1)

where

p: differential operator.

Ra: stator resistance.

ψa: permanent magnet flux linkage.

Ld, Lq: d-axis and q-axis stator inductance.

vds, vqs: d-axis and q-axis stator voltages in rotor frame.

ids, iqs: d-axis and q-axis stator currents in rotor frame.

ωe: electrical stator angular velocity.

The stator resistance is easily measured through DC test [1,2,3,4,5,6], and permanent magnetic flux linkage is obtained by the residual flux density of permanent magnet. However, as shown in Fig. 2, Ld and Lq are nonlinearly changed according to the d- and q-axis current, and their cross-coupling effect [7,8,9]. Therefore, a technique for accurately estimating Ld and Lq is certainly required for designing various control algorithms such as predicting torque and flux-weakening capabilities. Various estimation methods to obtain the exact values of Ld and Lq have been increasingly studied. That is the reason why this paper focuses on analyzing and classifying these techniques in order to provide a complete guideline.

Fig. 2
figure 2

d- and q-axis Inductances following Stator Current and Current Phase Angle. a d-axis inductance b q-axis inductance

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3 Offline Parameter Estimation Techniques

3.1 DC Current Decay Test

In the DC current decay test maintains the rotor on the d-axis or q-axis of the stator in order to measure the inductance of the u-phase DC current which decreases from rated current to zero [10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25]. The test configuration is depicted in Fig. 3a. The test consists of the two stages to apply a step voltage onto the stator winding of the machine at a standstill and to measure the corresponding voltage and current. The procedures to fix the rotor in d- and q-axis direction by applying stator voltage are illustrated in Fig. The inductances are obtained by

$$L_{d} = \frac{{\left[ {\int_{0}^{\infty } {\frac{2}{3}V_{dD} \left( t \right)dt} + R_{s} \int_{0}^{\infty } {i_{da} \left( t \right)dt} } \right]}}{{i_{da0} }}$$
(2)
$$L_{q} = \frac{{\left[ {\int_{0}^{\infty } {\frac{2}{3}V_{qD} \left( t \right)dt} + R_{s} \int_{0}^{\infty } {i_{qa} \left( t \right)dt} } \right]}}{{i_{qa0} }}$$
(3)
Fig. 3
figure 3

a Connection diagram for measuring the Ld, Lq. b Connecting to align the direct axis. c Connecting to align the quadrature axis

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The advantages of DC current decay test are that the test equipment is simple and easy to measure [22]. But, because the test is executed only at the stationary state, it inevitably has inductance errors and iron loss not considered during operating state [26].

In [16] dealing with these drawbacks, the armature of synchronous machine is supplied at standstill by DC-Chopper, pseudo random binary sequences (PRBS) voltages, PWM voltages and DC decay. The methods described in [23] are performed to determine the optimum set of the measured samples in order to gain the machine’s parameters. An extended DC decay test technique for arbitrary positions of the rotor is proposed in [24]. Another DC current decay test is described in [14], in which the method is proposed in order to measure it with regard to saturation and cross saturation effects.

3.2 AC Standstill Method

In the ac standstill method for the IPMSM, the currents and voltages of this phase and another phase are measured under the standstill condition to supply the single phase sinusoidal voltage [27,28,29,30,31,32,33,34,35,36]. So, d- and q-axis inductances are calculated from the self and mutual inductances of the stator winding. The a-phase self and a and c-phase mutual inductance are expressed as a function of the electrical angle

$$L = L_{ls} + L_{0} - L_{1} \cos 2\theta_{r}$$
(4)
$$M = - M_{0} - M_{1} \cos 2\left( {\theta_{r} - \frac{\pi }{3}} \right)$$
(5)

where Lls is leakage inductance, L0, M0 are dc term of the self and mutual inductances, L1, M1 represent second-harmonic components of the self and mutual inductances. The self and mutual inductances of the others can be expressed in the same way.

The d- and q-axis inductances are obtained by using Park’s transformation as follows:

$$L_{d} = L_{ls} + \left( {L_{0} - M_{0} } \right) - \left( {{{L_{1} } \mathord{\left/ {\vphantom {{L_{1} } 2}} \right. \kern-\nulldelimiterspace} 2} + M_{1} } \right)$$
(6)
$$L_{q} = L_{ls} + \left( {L_{0} - M_{0} } \right) + \left( {{{L_{1} } \mathord{\left/ {\vphantom {{L_{1} } 2}} \right. \kern-\nulldelimiterspace} 2} + M_{1} } \right)$$
(7)

The connection diagram for testing is reported in Fig. 4. One of the phase windings is excited by AC supply such that the line current and induced phase voltages in one of the other two windings are measured at different rotor positions. At each rotor position, the self and mutual inductances are calculated by Eqs. (8) and (9).

$$L_{a} = \frac{{\sqrt {\left( {{{V_{a} } \mathord{\left/ {\vphantom {{V_{a} } {I_{a} }}} \right. \kern-\nulldelimiterspace} {I_{a} }}} \right)^{2} - R_{s}^{2} } }}{2\pi f}$$
(8)
$$M_{ac} = \frac{{V_{c} }}{{2\pi fI_{a} }}$$
(9)
Fig. 4
figure 4

Circuit connection of the ac standstill test

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The parameters Lls, L0, M0, L1 and M1 are determined by (8) and (9) such that d- and q-axis inductances are calculated by (6) and (7).

Even though the AC standstill test gives the simplicity and relatively better accuracy on the estimation of the inductances, it still has the problem of not reflecting the actual operating conditions because the inductance of IPMSM was measured only in the condition of standstill. Also, there exists the time-consuming factor because of the high number of measurements and long measurement times.

In [34], the multi-sine AC standstill test is introduced for the swift identification, by which the effect of saturation, cross saturation, and frequency on the d- and q-axis parameters are rapidly evaluated with a VSI for signal generation. Another improved AC standstill test is described in [27], which the d- and q-axis currents that are produced by the two single phase currents passing through the proposed circuit are identical with that produced by the three phase currents at running condition. In [35], a 3-phase AC voltage source is applied such that the vector control drive is not required. Hence, it is very suitable for normal laboratory experiments since the d- and q-axis inductances are estimated, simultaneously considering the saturation and cross-magnetizing effect. The method in [32] focuses on the new PMSM model with the stator iron loss. Using this model, the d- and q-axis inductances and the equivalent iron loss resistance on the stator are measured by AC standstill test.

3.3 Vector Control Method

The vector control method is based on terminal measurements of the fundamental voltage and current peaks and their phases with respect to the rotor position [33, 37,38,39,40,41,42]. The a-phase current, voltage, and rotor position are measured by a current probe, a differential probe, and a position sensor, respectively.

Shown in Fig. 5, the d- and q-axis current are calculated from the a-phase current waveform and the rotor position. Similarly, from the measured amplitude and the phase of the fundamental of the voltage, the d- and q-axis voltages are obtained.

Fig. 5
figure 5

Calculation of the individual d and q components from the measured current and the phase relationship [37]

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The d- and q-axis flux linkages can be calculated by

$$v_{d} = i_{d} R_{s} - \omega_{e} \phi_{q}$$
(10)
$$v_{q} = i_{q} R_{s} + \omega_{e} \phi_{d}$$
(11)

If the magnet flux is constant and the cross-coupling inductances are zero, the d- and q-axis inductance can be directly estimated from the flux linkages.

The d- and q-axis inductance as follows

$$L_{d} \left( {i_{d} ,\,\,i_{q} } \right) = \frac{{\phi_{d} - \phi_{f} }}{{i_{d} }}$$
(12)
$$L_{q} \left( {i_{d} ,\,\,i_{q} } \right) = \frac{{\phi_{q} }}{{i_{q} }}$$
(13)

Since vector control method are based on the measured voltages and currents, estimated parameters include the effect of saturation and cross-coupling effects. Vector control method is also used to estimate the inductance of the motor even with space harmonics. However, the vector controlled method requires the extra equipment such as the dynamometer, the oscilloscope and the position sensor.

3.4 Other Methods

The other existing approaches to off-line inductance estimation are based on finite element method, described in [7, 43,44,45,46,47]. The methods of [43, 46] calculate motor parameters considering magnetic nonlinearity by using equivalent magnetic circuits, whereas the procedures of [44 ,7] consider the saturation and cross-coupling effects.

The method to obtain the parameters of PMSM without torque measurement has been proposed in [48], in which there are advantages with taking into account iron losses, avoiding uncertainties due to the value of copper resistance. In [49], the inductances are identified by injecting high-frequency signal to the estimated d- and q-axes. The method considers the detailed identification error caused by the inverter nonlinearity influence at different rotor positions.

Reference [50] provides the well summarized review on the off-line synchronous inductance estimation methods.

4 Online Parameter Estimation Techniques

4.1 Recursive Least Square

This group of methods uses known parameters, such as voltages and currents, to identify unknown parameters through mathematical models [51,52,53,54,55,56,57,58,59,60,61,62,63,64,65]. Unknown parameters can be found to minimize the error between observation and estimation.

Mathematical model of least squares is expressed as

$$Y\left( k \right) = \Theta^{T} \left( k \right)Z\left( k \right)$$
(14)

where Y(k) is the output, Θ(k) is the unknown parameter vector of the model and Z(k) is the input vector.

The unknown parameter vector Θ(k) can be obtained from the known vectors Y(k) and Z(k) by a general recursive least square method. The unknown parameters of the mathematical model can be obtained by

$$\varepsilon_{i} \left( k \right) = \left( {Y\left( k \right) - \hat{\Theta }\left( k \right)Z\left( k \right)} \right)^{2}$$
(15)

where εi(k) is the square value of the prediction error and \(\widehat{\Theta }\)(k) is the estimated parameter matrix.

By minimizing the least square function from (14), the estimated parameters are determined by the discrete time approach with respect to the parameter matrix Θ(k), as follows:

$$\hat{\Theta }\left( k \right) = \hat{\Theta }\left( {k - 1} \right) + K\left( k \right)\left( {Y\left( k \right) - Z^{T} \left( k \right)\hat{\Theta }\left( {k - 1} \right)} \right)$$
(16)
$$K\left( k \right) = P\left( {k - 1} \right)Z\left( k \right)\left( {\lambda I + Z^{T} \left( k \right)P\left( {k - 1} \right)Z\left( k \right)} \right)^{ - 1}$$
(17)
$$P\left( k \right) = {{\left( {I - K\left( k \right)Z^{T} \left( k \right)} \right)P\left( {k - 1} \right)} \mathord{\left/ {\vphantom {{\left( {I - K\left( k \right)Z^{T} \left( k \right)} \right)P\left( {k - 1} \right)} \lambda }} \right. \kern-\nulldelimiterspace} \lambda }$$
(18)

where K(k) is the gain matrix which updates parameters proportional to the error, P(k) is covariance matrix which must be definite and λ is forgetting factor given by 0 < λ < 1. The mathematical model for RLS is obtained from voltage equations of IPMSM. The observation consists of the indirect reference stator voltage and the measured stator current.

Generally, in the RLS method, the influence of the noise on the parameter identification is trivial, as shown in [56]. However, when the four parameters are simultaneously estimated by using the RLS algorithm, it imposes a heavy burden on the controller and could not be converged on the solution due to poorness of available data. Thus, in [60], the method for estimating the d- and q-axis inductances at the sampling rate and estimating the resistance and torque constant in a separate program executed at a lower frequency is proposed. In [54, 57], the model of the estimated rotating reference frame is used for the parameter identification method.

Since the RLS parameter estimator uses the fixed gain, the accuracy of the estimation is not guaranteed without the parameter variation, and the disturbance observer could provide the unstable transient response characteristic. Parameter fluctuation and inaccuracy in real-time estimation lead to degrading system performances.

4.2 Model Reference Adaptive System based Techniques

The model reference adaptive system (MRAS) is the very popular control method tuned by control factors that can be updated according to the change of system responses. The output of the system is compared to the desired response from the reference model, which is independent of d- and q-axis inductances but the adjustable model dependents on these parameters. The error signal is input to the adaptation mechanism. The output of the adaptation mechanism is determined in order to apply to tuning the adjustable model and also for feedback. The d- and q-axis inductances are corrected based on this error [66,67,68,69,70,71,72,73,74,75,76,77,78]. The stability of the closed loop estimator is ensured using Popov's hyper-stability theory. Figure 6 shows the structure of the MRAS.

Fig. 6
figure 6

Structure of MRAS scheme for inductance estimation

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Based on MEAS, the identified algorithm of d- and q-axis inductances can be written as

$$\hat{L}_{d} \left( t \right) = \hat{L}_{d} \left( 0 \right) + k_{1} \int_{0}^{t} {\left( {i_{q} - \hat{i}_{q} } \right)d\tau } + k_{2} \left( {i_{q} - \hat{i}_{q} } \right)$$
(19)
$$\hat{L}_{q} \left( t \right) = \hat{L}_{q} \left( 0 \right) + k_{1} \int_{0}^{t} {\left( {i_{d} - \hat{i}_{d} } \right)d\tau } + k_{2} \left( {i_{d} - \hat{i}_{d} } \right)$$
(20)

The MRAS method is relatively easy to estimate the motor parameters, so it is applied to solve the voltage imbalance of inverter or converter [73]. Also, the Ld and Lq estimation methods of the motor using this technique can accurately obtain the estimated values under various conditions.

However, the error of the d- and q- axis inductance directly affecting the stator current command has a vital influence on the motor efficiency, so the stator resistance must be estimated to compensate the voltage drop component. This method also has drawbacks in that it is difficult to design the adaptive mechanism and synchronize the PI gain at various operating points.

4.3 Artificial Neural Network (ANN) Techniques

The artificial neural networks have single or multi-layers consisting of input and output, which requires less computation and gives faster convergence time compared to other algorithms such as MRAS and extended Kalman filter (EKF) [79,80,81,82,83,84,85]. In [80,81,82,83], The adaptive linear neuron (adaline) networks with only inputs and outputs are used to estimate the parameters. The ANN structure for the PMSM is shown in Fig. 7.

Fig. 7
figure 7

ANN structure for PMSM

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The mathematical model of adaline neural networks is as

$$O\left( {W_{i} ,X_{i} } \right) = \sum\limits_{i = 0}^{n} {W_{i} X_{i} }$$
(21)

where Xi is the inputs, Wi is the weights and O(Wi,Xi) is the activation function.

A learning control mechanism samples the inputs, the output, and the desired output and uses these to adjust the weights. The weighting adjustment is obtained through the least mean square (LMS) algorithm as follows:

$$W_{i} \left( {t + 1} \right) = W_{i} \left( t \right) + 2\eta \left[ {d\left( t \right) - O\left( {W_{i} ,X_{i} } \right)} \right]X_{i} \left( t \right)$$
(22)

where 0 < i < 1 and η is the learning rate and usually is a small number between 0 and 1 (typically η < 1/n).

In [85], the harmonics in the rotor flux linkage is considered and the corresponding torque ripple is minimized by on-line torque constant estimation. A novel feature of adaptive on-line weights and biases up-dating of the ANN has also been included in [84]. The method of [79] consists of four layered feed-forward neural networks with an input layer, an output layer and two hidden layers.

Using a neural network, it is possible to calculate parameter variations such as inductance, armature resistance and back emf constant for motor drive in real-time and to enable high performance and robustness control. The controller of the neural network can reduce the computational complexity and simplify the control method, thus constituting a more practical and efficient control system for the IPMSM drive system by vector control. In addition, field weakening control is implemented by using neural network in order to achieve fast response by minimum loss in operating range. However, there is a large inductance error when the initial value is inaccurate.

4.4 Extended Kalman Filter (EKF) Based Techniques

The extended Kalman filter is an optimal recursive estimator for nonlinear systems [70, 75, 86,87,88,89,90]. It provides a solution that considers the effects of the disturbance noises including system and measurement noises. The EKF is developed in the discrete-time state by taking into account the nonlinear model of the IPMSM as follows

$$\left\{ {\begin{array}{*{20}l} {x_{k} = f\left( {x_{k - 1} ,u_{k - 1} } \right) + w_{k - 1} } \\ {y_{k} = h\left( {x_{k} } \right) + v_{k} \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,} \\ \end{array} } \right.$$
(23)

where f is the system transition function, h is the measurement function and uk-1 is the control input. wk−1 and vk are process and output noise respectively. They are assumed to be zero mean Gaussian white noises with covariance Qk and Rk respectively.

For the EKF, mathematically, the predictor step is given by

$$\left\{ {\begin{array}{*{20}l} {\hat{x}_{k|k - 1} = f\left( {\hat{x}_{k - 1|k - 1} ,u_{k - 1} } \right)\,\,\,\,\,\,\,\,\,\,\,} \\ {P_{k|k - 1} = F_{k - 1} P_{k - 1|k - 1} F_{k - 1}^{T} + Q_{k - 1} } \\ \end{array} } \right.$$
(24)

And the corrector step is given by

$$\left\{ {\begin{array}{*{20}c} {K_{k} = P_{k|k - 1} H_{k}^{T} \left( {H_{k} P_{k|k - 1} H_{k}^{T} + R} \right)^{ - 1} } \\ {\hat{x}_{k|k} = \hat{x}_{k|k - 1} + K_{k} \left( {y_{k} - H_{k} \hat{x}_{k|k - 1} } \right)\,\,\,\,} \\ {P_{k|k} = \left( {I - K_{k} H_{k} } \right)P_{k|k - 1} \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,} \\ \end{array} } \right.$$
(25)

In (23) and (24), Pk is the covariance matrix corresponding to the state estimation error and Kk is called the Kalman filter gain. After both the prediction and correction steps have been performed then \({\widehat{x}}_{k}\) is the current estimate of the states and \({y}_{k}\) can be calculated directly from it. Both \({\widehat{x}}_{k}\) and Pk are stored and used in the predictor step of the next time period. The state transition and observation matrices are defined to be the following Jacobians

$$F_{k - 1} = \left. {\frac{\partial f}{{\partial x}}} \right|_{{\hat{x}_{k - 1|k - 1} ,\,u_{k - 1} }} ,\,\,\,\,H_{k} = \left. {\frac{\partial h}{{\partial x}}} \right|_{{\hat{x}_{k|k - 1} }}$$
(26)

The EKF is a recursive optimal filter with the ability to estimate the state variables of a nonlinear system with the minimized estimation error variance so that the state variables and parameters of the system in a noise environment can be estimated appropriately. In addition, the EKF can estimate the parameters from the mathematical model of the device through the process of measurement noise covariance matrices from operating input–output data.

However, the EKF may fail to converge on the appropriate state values if the system model is inaccurate. It also processes input data with noise repeatedly which may lead to a high computational burden.

4.5 Other Methods

As another on-line parameter estimation technique, affine projection algorithm is reported in [91, 92]. The d- and q-axis stator inductances are estimated at high convergence rate, while the stator resistance, the magnetic flux linkage, and the motor torque are estimated at slow convergence rate. Therefore, the estimated parameters and motor torque from the affine projection algorithms can be used to continuously update the control gains for the adaptive controllers.

A direct method of calculating the phase inductance is proposed in [93], which calculates the phase inductance from the phase voltage equations. In [94], the motor parameters are estimated by tuning the controller gains that cancel the pole of the motor transfer function. The real-time estimation algorithms proposed by [95,96,97] identify the inductance matrix including the rotor position from current harmonics generated by inverter.

5 Conclusion

The inductance is an essential parameter of PMSM to design a controller to obtain high-performances with respect to torque, speed, or sensorless controls. The various research outputs on inductance identification techniques for PMSM are introduced and summarized in this paper.

The inductance estimation techniques of PMSM are classified into off-line and on-line parameter identification methods according to the operating time. The pros and cons of off-line parameter estimation, DC decay, AC standstill, vector controlled and other methods, and on-line parameter identification methods, RLS, MRAC, ANN, EKF and introduced other methods, are analyzed with the viewpoint of the applicability, responding performance, robustness against the disturbance and noise, and calculation complexity. PMSM parameter identification techniques introduced in this paper are summarized in Table 1.

Table 1 Comparison of PMSM Parameter Identification Techniques
Full size table

Most of the significant identification methods popular in an industry are categorized into the criteria systems and dealt with in this paper, even if there exist the untouched techniques for lack of space. We attempted to provide the sufficient references for a better understanding of readers with various research backgrounds.