Fuzzy adaptive sliding mode control of PMSM based on extended state observer

Abstract

An extended state observer (ESO) based fuzzy adaptive sliding mode control (FASMC) for permanent magnet synchronous motor (PMSM) drive system is proposed in this paper. In order to improve the disturbance rejection property, a dynamic disturbance compensation method is developed based on extended state observer. A fuzzy adaptive sliding mode control method with a novel sliding mode reaching law is presented to enhance the tracking performance. Then, the proposed method called FASMC-ESO, is applied to regulate the speed of PMSM. The global asymptotic stability analysis of the system is proved by Lyapunov functions. Comparative simulation and experiment results verify the effectiveness of the proposed method.

Keywords

Disturbance observer (DO)permanent magnet synchronous motor (PMSM)fuzzy adaptive (FA)sliding mode control (SMC)extended state observer (ESO)

1. Introduction

Permanent magnet synchronous motors (PMSMs) are widely used in wind power generation, electric vehicles and industrial control due to its high-power density and reliability [1–4]. The traditional PI controller cannot meet the high-performance control requirements of PMSM system because of the nonlinearities, uncertainties and interferences of the system. Sliding mode control (SMC) has received great attention due to its good robustness, strong anti-interference performance, low sensitivity to parameter variations, external disturbance, and fast response [5–8]. In [9], a nonlinear speed-control method based on sliding mode control and disturbance compensation techniques is proposed for PMSM servo systems. In [10], a new exponential reaching law is proposed to improve the dynamic performance of PMSM control system, and it is verified by comparing with the traditional PI controller. In order to suppress the disturbance and decrease the chattering of a terminal sliding mode controller, a speed regulation method based on the disturbance observer and non-singular terminal sliding mode is proposed in [11]. In [12], combined speed and direct thrust force control with sensorless speed estimation using a SMC with integral action is proposed, which shows an excellent transient and steady-state control performance in linear PMSM. To reduce the switching frequency variations of the direct torque control method [13], presents a fixed switching period SMC method, whose performance is demonstrated by the experimental results. In [14], a non-linear H _∞ and SMC method is proposed for the position control of the PMSM, which can minimize the effects of motor parameters variations and load disturbances. The speed control performance of the PMSM system under the SMC has been greatly improved compared with the traditional PI controller, however, the robustness of SMC control system needs to be further enhanced in the case of parameters variations and unknown load torque disturbances. A lot of work has been done to improve the disturbance rejection performance of the SMC system. A load torque observer based on a fuzzy sliding mode speed controller is proposed in [15], which can improve the dynamic performance and robustness of the PMSM speed control system. A modified SMC based on the nonlinear disturbance estimation is proposed in [16], which shows a better control performance than the traditional SMC and the integral SMC methods. An extended nonlinear disturbance observer (ENDOB)-based fuzzy sliding mode control approach is proposed for single-input and single-output systems with matched and mismatched uncertainties/disturbances in [17], and the effectiveness are verified by simulations.

This paper is organized as follows. In Section 2, the mathematical model of PMSM is introduced. To compensate the lumped disturbance, the extended state observer is studied in Section 3. A novel sliding surface based on fuzzy adaptive control is constructed in Section 4. In Section 5, simulations and experiments are performed to verify the effectiveness of the proposed method. Finally, conclusion is drawn in Section 6.

2. Mathematical model of PMSM

Considering a surface-mounted PMSM, and for the simplify analysis, an ideal mathematical model of PMSM is adopted and the assumptions are as follows [18,19].

1. The magnetic saturation of the PMSM iron core is negligible;

2. The influences of the magnetic hysteresis and the eddy currents are ignored;

3. The distribution of the magnetomotive forces is sinusoidal;

4. The magnetic circuit of the machine is not saturated;

5. The gap irregularities owing to the stator slots are neglected.

Define (i _d, i _q, ω) as the state variables of the PMSM system, and the voltage equations can be described as follows [20]: $\begin{eqnarray}\displaystyle \left\{\begin{array}{@{}l@{}}u_{d}=R_{s}i_{d}-{\omega}L_{q}i_{q}+L_{d}{\displaystyle \frac{di_{d}}{dt}}\\[5.0pt] u_{q}=R_{s}i_{q}+{\omega}{\psi}_{f}+{\omega}L_{d}i_{d}+L_{q}{\displaystyle \frac{di_{q}}{dt}}.\end{array}\right. & & \displaystyle\end{eqnarray}$ (1)

The magnetic chain equations can be given by [21] $\begin{eqnarray}\displaystyle \left\{\begin{array}{@{}l@{}}{\psi}_{d}=L_{d}i_{d}+{\psi}_{f}\\ {\psi}_{q}=L_{q}i_{q}.\end{array}\right. & & \displaystyle\end{eqnarray}$ (2)

The torque and motion equations can be expressed as follows [22]: $\begin{eqnarray}\displaystyle \left\{\begin{array}{@{}l@{}}T_{e}={\displaystyle \frac{3}{2}}[P_{n}{\psi}_{f}i_{q}+(L_{d}-L_{q})i_{d}i_{q}]\\[8.0pt] {\displaystyle \frac{d{\omega}}{dt}}={\displaystyle \frac{1}{J}}(T_{e}-T_{L}-B{\omega})\\[8.0pt] {\omega}=P_{n}{\omega}_{r}\end{array}\right. & & \displaystyle\end{eqnarray}$ (3) where u _d and u _q are the d- and q- axis stator voltages; i _d and i _q are the d- and q- axis currents; L _d and L _q are the d- and q- axis stator inductances; 𝜓_d and 𝜓_q are the d- and q- axis flux linkages; R _s is the stator resistance; 𝜓_f is the flux linkage of permanent magnets; J is the moment of inertia of the motor; P _n is the pole pairs; B is the viscous friction coefficient; ω_r is the mechanical angular velocity; ω is the electrical angular velocity; T _L and T _e represent the mechanical load torque and electromagnetic torque respectively; The PMSM studied in this paper is a surface mounted permanent motor [23–25], such that L _d = L _q = L.

3. Mathematical model of extended state observer

3.1. The design of extended state observer

Nonlinear dynamics, parameter uncertainties and external disturbances in the PMSM system are regarded as the lumped disturbance, which can be observed by an extended state observer (ESO). The extended state observer can be described as $\begin{eqnarray}\displaystyle \left\{\begin{array}{@{}l@{}}{\dot{x}}_{1}=x_{2}\\ {\dot{x}}_{2}=f(x_{1},x_{2},t)+bu\\ y=x_{1}\end{array}\right. & & \displaystyle\end{eqnarray}$ (4) where x _j(j = 1,2) is the system state, u is the control input, and f (x _j, t) is the unknown disturbance of the system, b is a gain coefficient. The observer can be designed as $\begin{eqnarray}\displaystyle \left\{\begin{array}{@{}l@{}}e_{1}=z_{1}-y\\ {\dot{z}}_{1}=z_{2}-{\beta}_{1}e_{1}\\ {\dot{z}}_{2}=f(z_{1},z_{2},t)-{\beta}_{2}e_{1}+bu.\end{array}\right. & & \displaystyle\end{eqnarray}$ (5)

3.2. The disturbance observer of PMSM based on ESO

For purpose of improving the disturbance rejection performance, a lumped disturbance observer is introduced in the PMSM speed control system. Based on the differential equations of PMSM, the disturbance of the PMSM drive system can be expressed as [26–28] $\begin{eqnarray}\displaystyle \dot{{\omega}} & = & \displaystyle \frac{3P_{n}{\psi}_{f}i_{q}}{2J}-\frac{B{\omega}}{J}-\frac{T_{L}}{J}\nonumber\\ \displaystyle & = & \displaystyle \frac{3P_{n}{\psi}_{f}i_{q}^{\ast }}{2J}-\frac{B{\omega}}{J}-\frac{T_{L}}{J}-\frac{3P_{n}{\psi}_{f}}{2J}(i_{q}^{\ast }-i_{q})\nonumber\\ \displaystyle & = & \displaystyle \frac{3P_{n}{\psi}_{f}i_{q}^{\ast }}{2J}+d(t).\end{eqnarray}$ (6)

Set $d(t)=-(B{\omega}/J)-(T_{L}/J)-b_{0}(i_{q}^{\ast }-i_{q})$ , b ₀ = 3P _n𝜓_f∕2J. It can be seen that friction, external load disturbances, q-axis current tracking errors are contained in d(t), which considered as the lumped disturbances.

Define x ₂ = d(t), x ₁ = ω. The equation ((6)) can be rewritten as $\begin{eqnarray}\displaystyle \left\{\begin{array}{@{}l@{}}{\dot{x}}_{1}=x_{2}+{\displaystyle \frac{3P_{n}{\psi}_{f}}{2J}}i_{q}^{\ast }\\[5.0pt] {\dot{x}}_{2}=c(t)\end{array}\right. & & \displaystyle\end{eqnarray}$ (7) where $c(t)={\dot{d}}(t)$ . According to literatures [29] and [30], a second-order linear ESO is constructed for system ((7)) as follows: $\begin{eqnarray}\displaystyle \left\{\begin{array}{@{}l@{}}{\dot{z}}_{1}=z_{2}-2p(z_{1}-x_{1})+b_{0}u\\ {\dot{z}}_{2}=-p^{2}(z_{1}-x_{1})\end{array}\right. & & \displaystyle\end{eqnarray}$ (8) where p is the expected pole of ESO, z ₁ is the estimated value of angular velocity ω, and z ₂ is the estimated value of d(t). That is to say, as t →∞, z ₁(t) →ω, z ₂(t) → d(t). Based on the mentioned method, the system disturbance compensation can be realized.

3.3. Stability analysis and pole assignment

Equations (7) and (8) can be rewritten as $\begin{eqnarray}\displaystyle \text{} & \text{} & \displaystyle \left[\begin{array}{@{}c@{}}{\dot{x}}_{1}\\ {\dot{x}}_{2}\end{array}\right]=\left[\begin{array}{@{}cc@{}}0 & 1\\ 0 & 0\end{array}\right]\left[\begin{array}{@{}c@{}}x_{1}\\ x_{2}\end{array}\right]+\left[\begin{array}{@{}c@{}}{\displaystyle \frac{3P_{n}{\psi}_{f}}{2J}}i_{q}^{\ast }\\[7.0pt] c(t)\end{array}\right]\end{eqnarray}$ (9) $\begin{eqnarray}\displaystyle \text{} & \text{} & \displaystyle \left[\begin{array}{@{}c@{}}{\dot{z}}_{1}\\ {\dot{z}}_{2}\end{array}\right]=\left[\begin{array}{@{}cc@{}}-2p & 1\\ -p^{2} & 0\end{array}\right]\left[\begin{array}{@{}c@{}}z_{1}\\ z_{2}\end{array}\right]+\left[\begin{array}{@{}cc@{}}2p & 0\\ p^{2} & 0\end{array}\right]\left[\begin{array}{@{}c@{}}x_{1}\\ x_{2}\end{array}\right]+\left[\begin{array}{@{}c@{}}b_{0}i_{q}^{\ast }\\ 0\end{array}\right].\end{eqnarray}$ (10)

Then the following system can be obtained $\begin{eqnarray}\displaystyle \left[\begin{array}{@{}c@{}}{\dot{z}}_{1}-{\dot{x}}_{1}\\ {\dot{z}}_{2}-{\dot{x}}_{2}\end{array}\right]=\left[\begin{array}{@{}cc@{}}-2p & 1\\ -p^{2} & 0\end{array}\right]\left[\begin{array}{@{}c@{}}z_{1}-x_{1}\\ z_{2}-x_{2}\end{array}\right]+\left[\begin{array}{@{}c@{}}\left(b_{0}-{\displaystyle \frac{3P_{n}{\psi}_{f}}{2J}}i_{q}^{\ast }\right)\\[5.0pt] -c(t)\end{array}\right]. & & \displaystyle\end{eqnarray}$ (11)

Equation (11) can be expressed as $\begin{eqnarray}\displaystyle \left[\begin{array}{@{}c@{}}{\dot{e}}_{1}\\ {\dot{e}}_{2}\end{array}\right]=\left[\begin{array}{@{}cc@{}}-2p & 1\\ -p^{2} & 0\end{array}\right]\left[\begin{array}{@{}c@{}}e_{1}\\ e_{2}\end{array}\right]+\left[\begin{array}{@{}c@{}}\left(b_{0}-{\displaystyle \frac{3P_{n}{\psi}_{f}}{2J}}\right)i_{q}^{\ast }\\[5.0pt] -c(t)\end{array}\right]. & & \displaystyle\end{eqnarray}$ (12)

Set $A=\left[\begin{array}{@{}ll@{}}-2p & 1\\ -p^{2} & 0\end{array}\right]$ , $S=\left[\begin{array}{@{}c@{}}\left(b_{0}-\frac{3P_{n}{\psi}_{f}}{2J}\right)i_{q}^{\ast }\\ -c(t)\end{array}\right]$ . The asymptotic stability of the system (11) can be guaranteed when all the characteristic roots of A are located in the left of the complex plane. The characteristic equation of A can be expressed as $\begin{eqnarray}\displaystyle |sI-A|=\left|\begin{array}{@{}cc@{}}s+2p & -1\\ p^{2} & s\end{array}\right|=s^{2}+2ps+p^{2}=0. & & \displaystyle\end{eqnarray}$ (13)

According to Routh criterion, when p > 0, ESO is stable, then e ₁(t) → 0, e ₂(t) → 0.

4. Design of speed controller

4.1. Fuzzy adaptive sliding mode speed controller

For the sake of enhancing the dynamic response speed, a fuzzy adaptive sliding mode control (FASMC) method is proposed.

4.1.1. Design of a new sliding mode controller

The velocity tracking error is defined as $\begin{eqnarray}\displaystyle e={\omega}^{\ast }-{\omega} & & \displaystyle\end{eqnarray}$ (14) where e is the angular velocity tracking error, ω^∗ is the reference speed and ω is the actual angular velocity. Taking the derivative of (14) and combining with (6), the following can be obtained $\begin{eqnarray}\displaystyle {\dot{e}}=\dot{{\omega}}^{\ast }-\dot{{\omega}}=\dot{{\omega}}^{\ast }-\frac{3P_{n}{\psi}_{f}}{2J}i_{q}^{\ast }-x_{2}. & & \displaystyle\end{eqnarray}$ (15)

The sliding surface of the PMSM system can be designed as $\begin{eqnarray}\displaystyle s=e+c\int _{0}^{t}ed{\tau} & & \displaystyle\end{eqnarray}$ (16) where c is the gain coefficient of the sliding mode surface, c > 0 and the derivative of s can be described as $\begin{eqnarray}\displaystyle {\dot{s}}={\dot{e}}+ce. & & \displaystyle\end{eqnarray}$ (17)

An equivalent sliding mode control based on the fuzzy adaptive method is constructed to improve the robustness of the system. The sliding mode controller based on ESO can be designed as $\begin{eqnarray}\displaystyle i_{q}^{\ast }=\frac{2J}{3P_{n}{\psi}_{f}}[\dot{{\omega}}^{\ast }-z_{2}+ce+{\varepsilon}sign(s)+ks]. & & \displaystyle\end{eqnarray}$ (18)

The robustness of the PMSM drive system will be reduced if the values of ϵ and k are not appropriate chosen. Therefore, for improving the response speed and disturbance rejection performance of the PMSM drive system, the values of ϵ and k must be determined properly.

In this paper, the traditional exponential reaching law is chosen as the sliding mode reaching law, which can be described as ${\dot{s}}=-{\varepsilon}\mathrm{sgn}(s)-ks$ , where, ϵ, k > 0. In order to improve the adaptive performance of SMC, a new reaching law is designed as follows: $\begin{eqnarray} \displaystyle {\dot{s}}=-{\varepsilon}\mathrm{sgn}(s)-\frac{k}{{\varepsilon}}s,\quad k>0,{\varepsilon}>0. & & \displaystyle \end{eqnarray}$ (19)

Then the following equations can be obtained $\begin{eqnarray} \displaystyle \begin{array}{@{} l@{}}ce+{\dot{e}}=-{\varepsilon}\mathrm{sgn}(s)-{\displaystyle \frac{k}{{\varepsilon}}}s\\ [5.0pt] ce+{\omega}^{\ast }-{\displaystyle \frac{3P_{n}{\psi}_{f}}{2J}}i_{q}^{\ast }-x_{2}=-{\varepsilon}\mathrm{sgn}(s)-{\displaystyle \frac{k}{{\varepsilon}}}s\\ [5.0pt] {\displaystyle \frac{3P_{n}{\psi}_{f}}{2J}}i_{q}^{\ast }=ce+{\omega}^{\ast }-x_{2}+{\varepsilon}\mathrm{sgn}(s)+{\displaystyle \frac{k}{{\varepsilon}}}s\\ [5.0pt] i_{q}^{\ast }={\displaystyle \frac{2J}{3P_{n}{\psi}_{f}}}\left[ce+{\omega}^{\ast }-x_{2}+{\varepsilon}\mathrm{sgn}(s)+{\displaystyle \frac{k}{{\varepsilon}}}s\right]. \end{array} & & \displaystyle \end{eqnarray}$ (20)

After estimated by ESO, z ₂ will converge to x ₂ asymptotically. And the following formulation can be obtained $\begin{eqnarray}\displaystyle i_{q}^{\ast }=\frac{2J}{3P_{n}{\psi}_{f}}\left[\dot{{\omega}}^{\ast }-z_{2}+ce+{\varepsilon}sign(s)+\frac{k}{{\varepsilon}}s\right]. & & \displaystyle\end{eqnarray}$ (21) The reaching speed of the system can be guaranteed adjusting ϵ and k.

4.1.2. Design of a fuzzy controller

The equivalent control item of the sliding mode controller is processed by a fuzzy controller. The existence condition of the sliding mode is $s{\dot{s}}<0$ [20].

The Lyapunov function is chosen as $\dot{V}=s{\dot{s}}=s(-{\varepsilon}\mathrm{sgn}(s)-\frac{k}{{\varepsilon}}s)$ , $\dot{V}=-{\varepsilon}|s|-\frac{k}{{\varepsilon}}s^{2}$ , where ϵ, k > 0, then $\dot{V}\leq 0$ is always true. $s{\dot{s}}\lt 0$ is the condition for the existence of sliding mode.

According to the fuzzy control principle, s and ${\dot{s}}$ are defined as the input of the fuzzy controller and $\hat{k}$ is the output of the fuzzy controller [28–30]. $\begin{eqnarray}\displaystyle \begin{array}{@{}l@{}}s=\{\text{NB},\text{NM},\text{NS},\text{ZO},\text{PS},\text{PM},\text{PB}\}\\ {\dot{s}}=\{\text{NB},\text{NM},\text{NS},\text{ZO},\text{PS},\text{PM},\text{PB}\}\\ \hat{k}=\{\text{NB},\text{NM},\text{NS},\text{ZO},\text{PS},\text{PM},\text{PB}\}.\end{array} & & \displaystyle\end{eqnarray}$ (22)

The output of the fuzzy controller is designed under the condition $s{\dot{s}}<0$ . The control rules are listed in Table 1.

Table 1

Fuzzy sliding mode control rules table

	${\dot{s}}$
s	NB	NM	NS	ZO	PS	PM	PB
PB	ZO	PS	PM	PB	PB	PB	PB
PM	NS	ZO	PS	PM	PB	PB	PB
PS	NM	NS	ZO	PS	PM	PB	PB
ZO	NB	NM	NS	ZO	PS	PM	PB
NS	NB	NB	NM	NS	ZO	PS	PM
NM	NB	NB	NB	NM	NS	ZO	PS
NB	NB	NB	NB	NB	NM	NS	ZO

The triangle functions are selected as the membership function in this paper. The input central values of the triangle functions are $\{-6,-4,-2,0,2,4,6\}$ corresponded to $\{\text{PB},\text{PM},\text{PS},\text{ZO},\text{NS},\text{NM},\text{NB}\}$ .

The output values of the fuzzy controller corresponded to $\{\text{PB},\text{PM},\text{PS},\text{ZO},\text{NS},\text{NM},\text{NB}\}$ are designed as $\{3,2,1,0,1,2,3\}$ . The output of the fuzzy controller is obtained by the center of gravity method. $\begin{eqnarray}\displaystyle \text{} & \text{} & \displaystyle u^{\ast }=\frac{\displaystyle \mathop{\sum }_{i}U_{i}\cdot {\mu}_{i}(U_{i})}{\displaystyle \mathop{\sum }_{i}{\mu}_{i}(U_{i})}\end{eqnarray}$ (23) $\begin{eqnarray}\displaystyle \text{} & \text{} & \displaystyle u^{\ast }=\hat{k}\end{eqnarray}$ (24) where u ^∗ is the output of the fuzzy control system, and μ_i(U _i) is the proportion in the ith membership, and U _i is the center value of the ith membership, and $\hat{k}$ is the input parameter of the sliding mode controller.

Replacing k of the equation (18) by $\hat{k}$ , the control law can be described as $\begin{eqnarray}\displaystyle i_{q}^{\ast }=\frac{2J}{3P_{n}{\psi}_{f}}\left[\dot{{\omega}}^{\ast }-z_{2}+ce+{\varepsilon}sign(s)+\frac{\hat{k}}{{\varepsilon}}s\right]. & & \displaystyle\end{eqnarray}$ (25)

The whole control block diagram of the fuzzy adaptive SMC system is shown in Fig. 1.

Fig. 1.

Fuzzy adaptive SMC structure diagram.

The whole control block diagram of the fuzzy adaptive SMC system based on ESO is shown in Fig. 2.

Fig. 2.

Control system block diagram.

Table 2

PMSM nominal parameters

Parameters	Value	Unit
Line resistance	0.33	Ω
Line inductance	9 × 10⁻⁴	H
Magnetic poles	4	pairs
Torque constant	0.087	N.m/A
Nominal voltage	36	VAC
Nominal current	7.5	A
Inertia	1.89 × 10⁻⁵	J/kg ⋅ m²
Rated speed	3000	r/min

Fig. 3.

Simulation responses under PI controller. (a) speed without load disturbance; (b) torque without load disturbance; (c) i _d, i _q without load disturbance; (d) i _a, i _b, i _c without load disturbance; (e) speed with load disturbance; (f) torque with load disturbance; (g) i _d, i _q with load disturbance; (h) i _a, i _b, i _c with load disturbance.

Fig. 3 (Continued).

4.2. Stability analysis

TheoremA .
A fuzzy adaptive sliding mode speed controller is designed to control PMSM. Equation (16) is selected as the sliding mode surface and (25) is designed as the control law. The system state arrives and moves along the sliding mode surface to achieve stability, which enhances the disturbance rejection performance and robustness of the system.

Proof of Theorem A .

The Lyapunov function is defined as $\begin{eqnarray}\displaystyle V=\frac{1}{2}s^{2}. & & \displaystyle\end{eqnarray}$ TheoremB .

Derivative of V , the following can be obtained $\begin{eqnarray}\displaystyle \dot{V}=s{\dot{s}}=s\left(\dot{{\omega}}^{\ast }-\frac{3P_{n}{\psi}_{f}}{2J}i_{q}^{\ast }-z_{2}+ce\right). & & \displaystyle\end{eqnarray}$ (27)

Then $\dot{V}$ can be rewritten as $\begin{eqnarray}\displaystyle \dot{V} & = & \displaystyle s\left[\dot{{\omega}}^{\ast }-\frac{3P_{n}{\psi}_{f}}{2J}\cdot \frac{2J}{3P_{n}{\psi}_{f}}\left(\dot{{\omega}}^{\ast }-z_{2}+ce+{\varepsilon}sign(s)+\frac{\hat{k}}{{\varepsilon}}s\right)-x_{2}+ce\right]\nonumber\\ \displaystyle & = & \displaystyle s\left[z_{2}-x_{2}-{\varepsilon}sign(s)-\frac{\hat{k}}{{\varepsilon}}s\right]\nonumber\\ \displaystyle & = & \displaystyle s(z_{2}-x_{2})-{\varepsilon}sign(s)\cdot s-\frac{\hat{k}}{{\varepsilon}}s^{2}\nonumber\\ \displaystyle & \leq & \displaystyle |e_{2}||s|-{\varepsilon}|s|-\frac{\hat{k}}{{\varepsilon}}s^{2}.\end{eqnarray}$ (28)

Because ϵ ≥ |e ₂|, so $\dot{V}\leq 0$ always holds. That is to say the system is stable, namely s will be reaching to zero, i.e. $c\int _{0}^{t}ed{\tau}+e=0$ . The stability of the fuzzy adaptive sliding mode speed control system based on ESO is proved.

5. Simulation experimental analysis

5.1. Simulation verification

To demonstrate the effectiveness of the proposed method, simulations and experiments among the conventional PI method, the SMC method, and the FASMC-ESO method in PMSM system are carried out. Simulations are built in MATLAB/Simulink, and the experimental platform is implemented by a TMS320F28335 controller. Parameters of PMSM are listed in Table 2.

Fig. 4.

Simulation responses under SMC controller. (a) speed without load disturbance; (b) torque without load disturbance; (c) i _d, i _q without load disturbance; (d) i _a, i _b, i _c without load disturbance; (e) speed with load disturbance; (f) torque with load disturbance; (g) i _d, i _q with load disturbance; (h) i _a, i _b, i _c with load disturbance.

Fig. 4 (Continued).

Fig. 5.

Simulation responses under FASMC-ESO controller. (a) speed without load disturbance; (b) torque without load disturbance; (c) i _d, i _q without load disturbance; (d) i _a, i _b, i _c without load disturbance; (e) speed with load disturbance; (f) torque with load disturbance; (g) i _d, i _q with load disturbance; (h) i _a, i _b, i _c with load disturbance.

Fig. 5 (Continued).

Loading and unloading simulation results of the PI controller, the SMC controller and the FASMC-ESO controller are shown in Figs 3, 4 and 5, PMSM starts without load torque and the reference speed is given as 1000 r/min. The responses of the PMSM system are shown in Fig. 3(a)–(d), Fig. 4(a)–(d) and Fig. 5(a)–(d), which represent the speed-response, torque, i _d, i _q and i _a, i _b, i _c, respectively. And the responses of the PMSM with load torque increased from 0 to 0.1 N ⋅ m at t = 0.1 s and decreased to 0 N ⋅ m at t = 0.2 s are shown in Fig. 3(e)–(h), Fig. 4(e)–(h) and Fig. 5(e)–(h). The FASMC-ESO method has smaller overshoot and shorter response time compared with the PI and SMC control methods during the PMSM starts. When the load torque is changed suddenly at the same time, the FASMC-ESO gives less speed fluctuation and shorter recovering time than the other two control methods. The results show that the FASMC-ESO method have smaller current and torque fluctuations than the PI and SMC methods. In a summary, the FASMC-ESO method has many advantages over the conventional PI and SMC methods, then the good robustness in PMSM control system is verified.

5.2. Experimental verification

Firstly, the SMC controller is constructed according to the new reaching law, and the fuzzy adaptive controller for the proposed reaching law is designed. Secondly, by designing the gain parameters, ESO is established to solve the disturbance variation problem. Then, the estimation of disturbance is used to compensate the system. The current sampling period of the PMSM control system is 200 us, and the parameters of the SMC and FASMC-ESO are set as c = 10, p = 50, ϵ = 0.1. $s,{\dot{s}}$ and $\hat{k}$ of the fuzzy controller are designed as $\{-600,-400,-200,0,200,400,600\}$ , $\{-30000,-20000,-10000,0,10000,20000,30000\}$ and $\{0.8,0.5,0.2,0.1,0.2,0.5,0.8\}$ respectively.

Fig. 6.

Experimental platform.

Fig. 7.

Experiment result under PI controller without load disturbance.

Fig. 8.

Experiment result under SMC controller without load disturbance.

Fig. 9.

Experiment result under FASMC-ESO controller without load disturbance.

Fig. 10.

Experiment result under PI controller with load disturbance.

Fig. 11.

Experiment result under SMC controller with load disturbance.

Fig. 12.

Experiment result under FASMC-ESO controller with load disturbance.

The experimental platform is shown in Fig. 6, and TMS320F28335 digital controller is used. The experimental results without load disturbance are shown in Figs 7–9. It can be observed that the FASMC-ESO controller has faster response speed and smaller overshoot when the target speed is set to 1000 r/min. The load disturbance experimental results of the PI controller, the SMC controller and the FASMC-ESO controller are shown in Figs 10–12. The speed and the current responses of the FASMC-ESO scheme, the PI control and the SMC control are compared under the same conditions. When the steady state is 1000 r/min and the load torque changed suddenly in PMSM, it can be observed that FASMC-ESO has a better disturbance rejection ability than PI and SMC methods. From a practical point of view, the method proposed in this paper has the ability to operate in various environments.

6. Conclusions

In this paper, for improving the robustness of PMSM drive system, a fuzzy adaptive sliding mode control system based on extended state observer has been proposed. The major contributions of this paper include: (1) in order to improve the response speed of PMSM system, a new fuzzy adaptive sliding mode controller is proposed; (2) for the sake of estimating the load disturbance, an extended state observer is presented; (3) the fuzzy adaptive sliding mode controller based on extended state observer is used to improve the speed-control performance of the PMSM system. Simulation and experimental results have illustrated the proposed method. In the future, for further improving the control performance of the PMSM system, we will appropriately determine the parameters of the FASMC-ESO approach for PMSM system.

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