1 Introduction

In recent years, with the development of power electronic technology, AC servo system has been paid more and more attention and research. The PMSM has the advantages of small size, high efficiency, and high power density, which plays an important role in AC servo system and has been widely used in high-performance drive system (Yu et al. 2015; Du et al. 2014; Yang and Wang 2014). At present, FOC or direct torque control (DTC) is usually used in the drives of PMSM for control strategy. However, either for the control strategy, the speed and the rotor position angle are required. At present, there are two schemes for obtaining the two parameters, namely the sensors and sensorless. In sensor, this can directly obtain the position information through installing encoder and hall sensor. But this scheme will undoubtedly increase the cost of the design of the system, and the adaptability is also relatively weak. In sensorless, first proposed and most simple method is back electromotive force, but at the low speed, the back electromotive force is small and the accuracy is not high. Another more mature method is high-frequency signal injection. This approach relies on external incentives and also does not apply at high speed (Yu et al. 2015). With the development of modern control theory, sliding mode observer, MRAS, Kalman filter, and other sensorless scheme are also developed (Guo 2013). These methods are more and more concerned by researchers for their good robustness.

On the other hand, with the increase in temperature, the resistance and inductance of the motor windings will change (Chen et al. 2015; Rashed et al. 2007). For sensorless control, these parameters are very important. So, it is necessary to identify these parameters at the same time as estimating speed.

MRAS has the advantages of simple algorithm, easy to implement in the digital control system, and has the advantages of faster adaptive speed. This has been proposed and applied to the PMSM sensorless control (Qi et al. 2007). There are two different models. One is the reference model and the other is the adjustable model including the identified parameters. The deviation signal of the output of the two models send to the adaptive mechanism and then the output of the adaptive mechanism are the identified parameters (Wang et al. 2014). From the current research, the emphases of MRAS method parameter identification are the establishment of adjustable model and the construction of adaptive law in adaptive mechanism. They are related to the accuracy of the identification and the stability of the system.

For speed estimation, there are two equations as adjustable model. One is stator flux linkage equation Qi (2007) and another is stator current equation. However, for parameters identification, there is only stator current equation. So, this paper chooses stator current equation as adjustable model. For the adaptive law, there are also two method, Lyapunov stability and the Popov stability, respectively. There is a PI regulator in adaptive law using Popov stability. But using Lyapunov stability, there is only a integrator (Liu 2011; Liu et al. 2010).

In Lindita and Aida (2013), Qu and Ye (2011), Zhang and Ruan (2015), and Fan et al. (2012), the basic theoretical knowledge of MRAS theory for estimating PMSM speed was given, and the q-axis and d-axis equations of stator current using for adjustable model was described. The adaptive law was constructed by Popov stability theory. These establish the theoretical foundation for this paper. In An et al. (2008), it identified the stator resistance, d-axis and q-axis inductance, and rotor flux linkage, but speed is acquired by sensor. In Pradeep et al. (2015), it used MRAS for sensorless control in the case of load changes. This fully demonstrates the good robustness of the MRAS. In Zhang et al. (2014), it used fuzzy controller to adjust the parameters of the PI regulator in the speed observer based on MRAS. The whole system had a good dynamic and steady performance in a wide speed range. In Liu (2011); Liu et al. (2010), they constructed adaptive law by Popov stability and Lyapunov stability analysis method for parameter identification and made a contrast for the two kinds of stability criterion. But the speed was also acquired by sensor. In Maiti et al. (2008), reactive power equation as reference model was used, and speed adaptive law by Popov stability was established. In Yang et al. (2011), Zaltni and Abdelkrim (2010), and Hamida et al. (2013), they established different adaptive laws according to different stability. Different adaptive law leads to different system performance. In Khlaief et al. (2013), the stability of MRAS system was analyzed, and the transfer function of the system was constructed with the modern control theory, and analyzed the transfer function. It also identified stator resistance based on these.

In summary, the problem of using MRAS to estimate the speed or identify parameters is the construction of adaptive law and the selection of the adjustable model.

First of all, this paper uses q-axis and d-axis equations of stator current as adjustable model. Then, for speed estimation, the adaptive law is constructed by Popov stability. The stability is analyzed through modern control theory, and the transfer function of the speed observer is constructed. Then, for parameters identification, if continue to use Popov stability to construct adaptive laws, it must consider the problem of losing rank. In addition, it also needs to regulate PI parameters of adaptive laws. So, this has brought inconvenience. Thus, this paper constructs adaptive law through Lyapunov stability, and this cannot consider the problem of losing rank and need not regulate PI parameters. This method is simple and effective and also can estimate speed and identifies stator resistance, d-axis and q-axis inductance simultaneously. Finally, this paper uses FOC as a control strategy and gives the result verification through simulation and experiment. It can be seen that the methods proposed in this paper are feasible and effective.

2 Speed Estimation

2.1 Mathematical Model of PMSM

The stator voltage equation of the surface mounting PMSM in d–q axis is:

$$\begin{aligned} \left[ \begin{array}{l} u_d\\ u_q\\ \end{array}\right] =\left[ \begin{array}{ll} R_s +DL_d&{}\quad -\omega _r L_q\\ \omega _r L_d&{}\quad R_s +DL_q\\ \end{array}\right] \left[ \begin{array}{l} i_d\\ i_q\\ \end{array}\right] +\left[ \begin{array}{l} 0\\ \psi _r \omega _r\\ \end{array}\right] \quad \quad \end{aligned}$$
(1)

where \(u_{d}\), \(u_{q}\), \(i_{d}\), \(i_{q}\) are the stator voltage and current of the motor in the d–q axis; \(R_{s}\), \(L_{d}\), \(L_{q}\) are the stator resistance and the inductance of the d–q axis; D is differential operator; \(\omega _{r}\) and \(\psi _{r}\) are rotor electric angular speed and flux. According to the Eq. (1), the stator current state equation can be obtained.

$$\begin{aligned} D\left[ \begin{array}{l} i_d\\ i_q\\ \end{array}\right] =\left[ \begin{array}{ll} -\frac{R_s}{L_s} &{}\quad \omega _r\\ -\omega _r &{}\quad -\frac{R_s}{L_s}\\ \end{array}\right] \left[ \begin{array}{l} i_d\\ i_q\\ \end{array}\right] +\left[ \begin{array}{l} \frac{u_d}{L_s}\\ \frac{u_q -\psi _r \omega _r}{L_s}\\ \end{array}\right] \end{aligned}$$
(2)

For surface mounting PMSM, \(L_{d} =L_{q} =L_{s}\) Motor use (Eq. (2)) for reference model and the Eq. (3) use for adjustable model. For speed estimation, \(R_{s}\) and \(L_{s}\) can be regarded as fixed values. So, the adjustable model for speed estimation can be given as:

$$\begin{aligned} D\left[ \begin{array}{l} \hat{i}_{d}\\ \hat{i}_{q}\\ \end{array}\right] =\left[ \begin{array}{ll} -\frac{R_s}{L_s}&{}\quad \hat{\omega }_{r}\\ -\hat{\omega }_{r}&{}\quad -\frac{R_s }{L_s }\\ \end{array}\right] \left[ \begin{array}{l} \hat{i}_{d}\\ \hat{i}_{q}\\ \end{array}\right] +\left[ \begin{array}{l} \frac{u_d}{L_s}\\ \frac{u_q -\psi _r \hat{\omega }_{r}}{L_s}\\ \end{array}\right] \end{aligned}$$
(3)

Defined state error:

$$\begin{aligned} \varepsilon _d =i_d -\hat{i}_{d} ,\varepsilon _q =i_q -\hat{i}_{q} \end{aligned}$$
(4)

The state error equation of Eq. (2) subtracting Eq. (3) can be given as:

$$\begin{aligned} D\left[ \begin{array}{l} \varepsilon _d\\ \varepsilon _q\\ \end{array}\right] =\left[ \begin{array}{ll} -\frac{R_s}{L_s}&{}\quad \hat{\omega }_{r}\\ -\hat{\omega }_{r}&{}\quad -\frac{R_s}{L_s}\\ \end{array}\right] \left[ \begin{array}{l} \varepsilon _d\\ \varepsilon _q\\ \end{array}\right] +\left[ \begin{array}{l} i_q\\ -i_d -\frac{\psi _r}{L_s}\\ \end{array}\right] \left( \omega _r -\hat{\omega }_{r} \right) \end{aligned}$$
(5)

Written state space expression:

$$\begin{aligned} D\varepsilon =A\varepsilon +Bu \end{aligned}$$
(6)

2.2 Adaptive of Speed Observer Design

MRAS basic block diagram of speed estimation is shown in Fig. 1.

Fig. 1
figure 1

MRAS basic block diagram of speed estimation

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Obviously, the stability and precision of the system are related to the construction of the adaptive mechanism. From Fig. 1, it can be seen that the adaptive mechanism is related to the state error Eq. (5). The structural diagram of the Eq. (5) is shown in Fig. 2:

Fig. 2
figure 2

Equation (5) structure diagram

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For speed estimation, the adaptive law is constructed by Popov stability. So, the conditions for the stability of Fig. 2 are about two sides. One is the zero pole of the transfer function of the forward channel in the left half of the s domain. Another is feedback channel to satisfy Popov stability.

For first condition, the transfer function of forward channel can be deduced according to modern control theory. Its state space expression is:

$$\begin{aligned} \dot{\varepsilon }= & {} A\varepsilon +u \nonumber \\ y= & {} \varepsilon \end{aligned}$$
(7)

transfer function:

$$\begin{aligned} H(s)=\frac{s+\frac{R_{s} }{L_{s} }}{s^{2}+2\frac{R_s }{L_s }s+\left( {\frac{R_s }{L_s }} \right) ^{2}+\hat{\omega }_{r}^{2}} \end{aligned}$$
(8)

The pole-zero loci of Eq. (8) is shown in Fig. 3 for a range of \(\hat{\omega }_{r}\) starting at \(-200\) up to 200 rad/s. From the graph, we can see that with the increase in the speed, the poles also have negative real parts. So this condition is confirmed.

Fig. 3
figure 3

Pole-zero loci of H(s) about range of \(\hat{\omega }_{r}\) starting at \(-200\) up to 200 rad/s

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For second condition, the derivation of Popov stability is introduced in the above literature, and it is not described in this article. The adaptive law can be constructed by Popov stability.

$$\begin{aligned} \hat{\omega }= & {} K_i \int \left( \varepsilon _d i_q -\varepsilon _q i_d -\varepsilon _q \frac{\psi _r}{L_s} \right) \hbox {d}t\nonumber \\&+\,K_p \left( {\varepsilon _d i_q -\varepsilon _q i_d -\varepsilon _q \frac{\psi _r }{L_s }} \right) + \hat{\omega }_{r} \left( 0\right) \end{aligned}$$
(9)

The adaptive law of the observer is constructed, as shown in Eq. (9). So, the MRAS structure diagram can be obtained in Fig. 4.

Fig. 4
figure 4

MRAS structure diagram

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The dotted line in Fig. 4 is represented by the state space expression.

$$\begin{aligned} \dot{\varepsilon }= & {} A\varepsilon +Bu \nonumber \\ y= & {} B^{T}\varepsilon \end{aligned}$$
(10)

So, its transfer function is:

$$\begin{aligned} G\left( s \right) =\frac{\left( {s+\frac{R_s }{L_s }} \right) \left( {i_q ^{2}+\left( {i_d +\frac{\psi _r }{L_s }} \right) ^{2}} \right) }{s^{2}+2\frac{R_s }{L_s }s+\left( {\frac{R_s }{L_s }} \right) ^{2}+\hat{\omega }_{r}^{2}} \end{aligned}$$
(11)

Because G(s) and H(s) have the same characteristic root, so the stability is not changed even add PI regulator. Thus, the speed observer is stable.

3 Parameters Identification

Because speed estimation is based on Popov stability, parameters identification does not need to consider the problem of rank based on Lyapunov stability .

For parameters identification, \(\omega _{r}\) can be regarded as fixed values. So, the adjustable model for adjustable parameters can be given.

$$\begin{aligned} D\left[ \begin{array}{l} \hat{i}_{d}\\ \hat{i}_{q}\\ \end{array}\right] =\left[ \begin{array}{ll} -\hat{a}&{}\quad \omega _r\\ -\omega _r&{}\quad -\hat{a}\\ \end{array}\right] \left[ \begin{array}{l} \hat{i}_{d}\\ \hat{i}_{q}\\ \end{array}\right] +\left[ \begin{array}{l} u_d \hat{b}\\ \left( u_q -\psi _r \omega _r\right) \hat{b}\\ \end{array}\right] \end{aligned}$$
(12)

where \(a=\frac{R_s }{L_s },\hat{a} =\frac{\hat{R}_{s} }{\hat{L}_{s}}\), \(b=\frac{1}{L_s }, \hat{b} =\frac{1}{\hat{L}_{s}} \).

The state error equation of Eq. (2) subtracting Eq. (12) can be given. The derivation process of the Eq. (13) is given in the “Appendix.”

$$\begin{aligned} D\left[ \begin{array}{l} \varepsilon _d\\ \varepsilon _q\\ \end{array}\right]= & {} \left[ \begin{array}{ll} - \hat{a} &{}\quad \omega _r\\ -\omega _r &{}\quad - \hat{a}\\ \end{array}\right] \left[ \begin{array}{l} \varepsilon _d\\ \varepsilon _q\\ \end{array}\right] +\left[ \begin{array}{l} -i_d\\ -i_q\\ \end{array}\right] \left( a-\hat{a}\right) \nonumber \\&+\,\left[ \begin{array}{l} u_d\\ u_q -\psi _r \omega _r\\ \end{array}\right] \left( b- \hat{b} \right) \end{aligned}$$
(13)

Written state space expression:

$$\begin{aligned} D\varepsilon =A_1 \varepsilon +B_1 u_1 +Cu_2 \end{aligned}$$
(14)

So, MRAS basic block diagram of parameters identification is shown in Fig. 5.

Fig. 5
figure 5

MRAS basic block diagram of parameters identification

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For parameters identification, it uses Lyapunov stability to construct the adaptive law.

Definition \(\phi ^{T}=(u_{1} \, u_{2})\,s=(B_{1} \, C)^{T}\)

Equation (14) becomes:

$$\begin{aligned} D\varepsilon =A_1 \varepsilon +\phi ^{T}s \end{aligned}$$
(15)

The design of the Lyapunov equation is:

$$\begin{aligned} V\left( X \right) =\frac{1}{2}\left( {\varepsilon ^{T}P\varepsilon +\phi ^{T}\Gamma \phi } \right) \end{aligned}$$
(16)

where \(P=\left( \begin{array}{ll} 1&{} 0\\ 0&{} 1\\ \end{array}\right) \), \(\Gamma =\left( \begin{array}{ll} 1&{} 0\\ 0&{} 1\\ \end{array}\right) \).

Fig. 6
figure 6

Control block diagram of speed estimation and parameters identification of PMSM

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According to Lyapunov stability second method, if the system is stable, two conditions should be met. One is \(V\left( X \right) \)-positive definite and the other is \(\dot{V} (X)\)-negative definite. Because the P and \(\Gamma \) are positive definite, \(V\left( X \right) \) is positive definite.

$$\begin{aligned} \dot{V}\left( X\right)= & {} \frac{1}{2} \left( \dot{\varepsilon }^{T}P\varepsilon +\varepsilon ^{T}P \dot{\varepsilon } \right) \nonumber \\&+\,\frac{1}{2}\left( \dot{\phi }^{T}\Gamma \phi +\phi ^{T}\Gamma \dot{\phi } \right) \end{aligned}$$
(17)

where

$$\begin{aligned}&\dot{\varepsilon }^{T}P\varepsilon =\varepsilon ^{T}A_1 ^{T}P\varepsilon +s^{T}\phi P\varepsilon \\&\varepsilon ^{T}P\dot{\varepsilon }=\varepsilon ^{T}P\left( {A_1 \varepsilon } \right) +\varepsilon ^{T}P\left( {\phi ^{T}s} \right) \\&\dot{\phi }^{T}\Gamma \phi +\phi ^{T}\Gamma \dot{\phi }=2\left( u_1\dot{u}_1+u_2 \dot{u}_2\right) \\ \end{aligned}$$

So, Eq. (17) becomes

$$\begin{aligned} \dot{V}\left( X \right)= & {} \frac{1}{2}\varepsilon ^{T}\left( {PA_1 +A_1 ^{T}P} \right) \varepsilon \nonumber \\&+\,\frac{1}{2}\left( {s^{T}\phi P\varepsilon +\varepsilon ^{T}P\left( {\phi ^{T}s} \right) } \right) +\left( u_1 \dot{u}_1+u_2 \dot{u}_2\right) \nonumber \\ \end{aligned}$$
(18)

Because \(PA_1 +A_1 ^{T}P=\left[ {{\begin{array}{ll} {-a}&{} 0 \\ 0&{} {-a} \\ \end{array} }} \right] \) is negative definite, \(\frac{1}{2}\varepsilon ^{T}\left( {PA_1 +A_1 ^{T}P} \right) \varepsilon \) is negative definite.

To meet the condition about \(\dot{V}\left( X\right) \) is negative definite, we can let

$$\begin{aligned} \frac{1}{2}\left( {s^{T}\phi P\varepsilon +\varepsilon ^{T}P\left( {\phi ^{T}s} \right) } \right) +\left( u_1 \dot{u}_1+u_2 \dot{u}_2\right) =0 \end{aligned}$$

By calculation, we can get:

$$\begin{aligned}&u_1 \left( \dot{u}_1-\varepsilon _d i_d -\varepsilon _q i_q \right) \\&\quad +u_2 \left( \dot{u}_2+\varepsilon _d u_d +\varepsilon _q u_q -\psi _r \omega _r\right) =0 \\&\dot{u}_1=\varepsilon _d i_d +\varepsilon _q i_q, \dot{u}_2=-\varepsilon _d u_d -\varepsilon _q u_q +\psi _r \omega _r \end{aligned}$$

So, the adaptive law of a and b can be given as

$$\begin{aligned} \hat{a}= & {} a(0)-k_a \int {\left( {\varepsilon _d i_d +\varepsilon _q i_q } \right) }\hbox {d}t \end{aligned}$$
(19)
$$\begin{aligned} \hat{b}= & {} b(0)+k_b \int {\left( {\varepsilon _d u_d +\varepsilon _q u_q -\psi _r \omega _r } \right) }\hbox {d}t \end{aligned}$$
(20)

where a(0) and b(0) are the initial values. \(K_{a}\) and \(K_{b}\) are gains. These parameters cannot break away from the initial value suddenly. They must be changed based on the initial value. So, the Eqs. (19) and (20) express these transformations and earn the estimation value.

4 Simulation Results

In this paper, the field-oriented control (FOC) is built firstly in MATLAB/Simulink. Then, the adjustable model and adaptive law were built, as shown in Fig. 6. The PMSM parameters involved in this paper are shown in Table 1. The simulation parameters are shown in Table 2.

The initial rotor given speed of the PMSM is 50 rad/s, and the rotor given speed is 30 rad/s at 0.1 s. The initial load torque of the motor is 0.1 N m and the load torque is 0.2 N m at 0.15 s.

Table 1 PMSM parameters
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Table 2 Simulation parameters
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where \(U_{dc}\) is DC bus voltage, P is pole pairs of PMSM, \(K_{ip} \) and \(K_{id}\) are current loop PI parameters of FOC, \(K_{\omega p} \) and \(K_{\omega i}\) are speed loop PI parameters of FOC, \(K_{i}\) and \(K_{p}\) are speed observer PI parameters of MRAS.

Fig. 7
figure 7

Real and observed speed without identification parameters

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Fig. 8
figure 8

Real and observed speed with identification parameters

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As shown in Fig. 7, speed observer based on MRAS has a very good dynamic response. Figure 8 shows that the speed observer also has a good dynamic response with identification parameters. By comparing speed estimation with identification and without identification, we can see that the speed estimation with identification has a slight oscillations. But we also can find that the error between the estimated speed and real speed is small. So, the method of speed estimation with parameters identification is useful. Also, at 0.15 s, when load transient from 0.1 to 0.2 N m, the speed observer still has a good dynamic response. The estimated speed can reach about 29 rad/s fast. And this value has a little error with the given value. Therefore, this method can cope with the disturbance of load transient.

As shown in Figs. 9 and 10, the stator resistance, d-axis and q-axis inductance calculated by Eqs. (19) and (20) are stable around the given value. Also, they have a little error with the given value. So, this can solve the problem of parameters variation due to external factors in PMSM running. From the simulation results, we can see the effectiveness and feasibility of the proposed method in this paper.

Fig. 9
figure 9

Real and identified resistance

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Fig. 10
figure 10

Real and identified inductance

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5 Experiment Results

In order to further verify the feasibility and effectiveness of the proposed method in this paper, a experimental platform was built according to Fig. 6. This paper uses TMS320F28035 and DRV8301 as controller and driver. The PMSM is 24 V, 62 W, and some basic parameters are shown in Table 1.

The initial rotor given speed of the PMSM is 50 rad/s. The initial load torque of the motor is 0 N m.

The experimental platform is shown in Fig. 11.

Fig. 11
figure 11

Picture of the experimental platform

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Fig. 12
figure 12

Real and observed speed with identification parameters under experimental platform

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Fig. 13
figure 13

Identified resistance under experimental platform

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Fig. 14
figure 14

Identified inductance under experimental platform

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As shown in Fig. 12, the speed observer with identification parameters has a good dynamic response. So, the method of speed estimation with parameters identification is useful. Because the PMSM needs a starting process at experimental process, the response time is about 11 s in Fig. 12. Figures 13 and 14 show that the method of this paper can identify parameters usefully. In fact, we can find that these parameters are related to current and voltage through Eqs. (9), (19), and (20). It means that their stability is related to the stability of the whole system. So, we can find that they also have similar stability through simulation and experimental results.

6 Conclusion

In this paper, the adjustable model is built according to the stator current equations in d-q axis, and the adaptive law of speed estimation is constructed according to the Popov stability. Then, for the problems of speed estimation and parameters identification simultaneously, the adaptive laws of stator resistance, d-axis and q-axis inductance are constructed through Lyapunov stability.

So, this paper solves the change of parameters due to external factors in PMSM running. In sensorless control, some parameters are important, and we need to identify the parameters at the same time as speed estimation. This paper proposes a solution to estimate speed and identify stator resistance, d-axis and q-axis inductance simultaneously. The feasibility and effectiveness of this method are proved by simulation and experimental results.