Design of an adaptive nonsingular terminal sliding model control method for a bearingless permanent magnet synchronous motor

Abstract

This paper presents an adaptive nonsingular terminal sliding mode controller for a bearingless permanent magnet synchronous motor. In order to rapidly converge state variables associated with terminal sliding mode control, an adaptive variable-rated exponential reaching law, in which the L1 norm of state variables is introduced, is proposed for the second-order uncertain nonlinear dynamical system. Exponential and constant reaching speed can adaptively adjust according to the distance between state variable and equilibrium point, which can shorten the reaching time and weaken system chattering. The mathematical models for rotating speed and the rotor radial displacement of the bearingless permanent magnet synchronous motor system are set up. The proposed method is then applied to the speed and radial displacement control. Simulation results are provided to validate the effectiveness of the proposed method.

Keywords

Nonsingular terminal sliding mode control bearingless permanent magnet synchronous motor reaching law robust control speed control radial displacement control

Introduction

The bearingless permanent magnet synchronous motor (BPMSM) is being actively researched and developed around the world due to its advantages of simple structure, reliable operation, high efficiency, high torque density, power density, etc. They have demonstrated the potential applications in high-speed precision mechanical processing, aeronautics and astronautics, centrifugal machines, turbomolecular pumps, flywheel energy storage, etc. (Asama et al., 2011; Schneider et al., 2008; Sun et al., 2013). However, the BPMSM control system is a nonlinear, time-varying and complex system with unavoidable and unmeasured disturbances, as well as parameter variations (Chiba et al., 1994). It is very difficult to achieve a satisfactory performance in the entire operating range by only using linear control algorithms (Ooshima et al., 1996, 2004; Zhu et al., 2009).

In recent years, with the development of modern control theory and motion control, in order to obtain high performance of the BPMSM control system with the characteristics of rapid response, small overshoot, high tracking precision and strong anti-disturbance ability, various methods of nonlinear control theory have been proposed, such as independent control (Qiu et al., 2006), direct radial displacement control (Zhang and Luo, 2009), nonlinear feedback control (Grabner et al., 2008), sliding mode control (Sun et al., 2013), fuzzy self-adaptive proportional–integral–derivative (PID) control (Meng et al., 2014), artificial neural network inverse intelligent control (Sun et al., 2009), etc. These control methods have improved the performance of the BPMSM system from different aspects.

Among these methods, the sliding mode control methods are regarded to be efficient methods to improve the disturbance rejection and robustness properties of BPMSM systems (Sun et al., 2013). However, the sliding mode control method has a problem that system states converge to the equilibrium point at an infinite time. Accordingly, a nonlinear sliding mode called terminal sliding mode control (TSMC) has been proposed (Zak, 1988). By introducing the nonlinear term in the sliding mode, TSMC can converge to the specified trajectory in finite time. After that, in order to avoid the singularity problem associated with conventional TSMC, nonsingular terminal sliding mode control (NTSMC) was developed (Feng et al., 2002; Yang et al., 2013). TSMC, NTSMC and their improved algorithms have been widely used in high-speed and high-precision motor control systems due to their limited time convergence, fast dynamic response and high tracking accuracy (Chen et al., 2011; Feng et al., 2009; Jin et al., 2009; Li et al., 2013; Tong et al., 2008; Zhang et al., 2011). In order to further improve the convergence rate of the state variables in NTSMC, Chen et al. (2011) introduced a decay time variant into the sliding mode and designed an integral time variable dynamics sliding mode, which increased the complexity of the sliding mode surface design. Tong et al. (2008) presented a variable-rated exponential reaching law, but did not consider the reaching law selected when the state variables become zero. Based on the variable-rated exponential reaching law, Zhang et al. (2011) introduced a power function of the terminal attractor and the system state variable. This method has improved the convergence rate of the state variable, and suppressed the chattering of sliding mode control. However, the complexity of the system design and calculation was increased due to the introduction of a power function.

In this paper, an adaptive variable-rated exponential reaching law is presented, where the L1 norm of state variables is introduced. Exponential and constant approach speed can adaptively adjust according to the state variables’ distance to the equilibrium point, which can shorten the reaching time and weaken system chattering. The reaching law is applied to design a nonsingular terminal sliding mode controller, which is simulated and verified based on a second-order uncertain nonlinear system. By constructing the vector control platform of the BPMSM, the BPMSM speed and radial displacement adaptive NTSM controller are designed, and the effectiveness of the proposed method is verified.

Adaptive nonsingular terminal sliding mode control

Consider a second-order uncertain nonlinear dynamical system (Feng et al., 2002)

{\begin{matrix} {\overset{\cdot}{x}}_{1} = x_{2} \\ {\overset{\cdot}{x}}_{2} = f (x) + b (x) u + g (t) \end{matrix}

(1)

where $x = {[x_{1}, x_{2}]}^{T}$ is the system state vector, f(x) and $b (x) \neq 0 are smooth nonlinear of x, and g (t)$ represents the uncertainties and disturbances satisfying $| g (t) | \leq l_{g}, where l_{g} > 0$ , and $u$ is the scalar control input. For system (1), the conventional index reaching law used to design the sliding mode controller is (Gao, 1989)

\overset{\cdot}{s} = - ε sgn (s) - ks, k > 0, ε > 0

(2)

The reaching law constant has two parts: $\overset{\cdot}{s} = - ks$ is the exponential reaching item $and \overset{\cdot}{s} = - ε sgn (s)$ is the constant speed reaching item. Coefficients $k$ and $ε$ defined in equation (2) cannot adjust themselves. Then, the convergence cannot achieve the best performance for state variables in a different position. In this paper, an adaptive variable speed exponential reaching law is given by

\overset{\cdot}{s} = - ε \frac{1}{1 + c x_{1}} sgn (s) - (k + c x_{1}) s

(3)

where $‖ x ‖_{1} = \sum_{i = 1}^{n} | x_{i} |$ is the first-order norm of system state variables, $k > 0$ , $ε > 0$ , $c > 0$ and $n > 0$ .

The first-order norm of state variables introduced in the adaptive variable speed exponential reaching law (3) can adaptively adjust its exponential reaching speed and constant reaching speed. For equation (3), the instant answer of exponential reaching is $s = s (0) e^{- (k + c x_{1}) t}$ ; when $x_{1}$ is large and the exponential decay rate is far greater than in equation (2), the reaching time will be greatly shorten. In addition the reaching rate $ε \frac{1}{1 + c x_{1}}$ of constant reaching is far less than $ε$ of equation (2). When $x_{1}$ is small, the reaching time of the sliding mode can be shortened and the system chattering can be weakened by increasing adjustment coefficient $c$ . When the selected state variable $x$ converges to zero in the course of system stability, the adaptive variable speed exponential reaching law will transform into constant exponential reaching law.

This reaching law is used to design a nonsingular terminal sliding model controller as adaptive NTSMC. The theorem is as follows.

Theorem 1. For system (1) with the adaptive NTSM described by the following terminal sliding variable

s = x_{1} + \frac{1}{β} x_{2}^{p / q}

(4)

If the control is designed as

u = - b^{- 1} (x) [f (x) + β \frac{q}{p} x_{2}^{2 - \frac{p}{q}} + l_{g} sgn (s) + ε \frac{1}{1 + c x_{1}} sgn (s) + (k + c x_{1}) s]

(5)

where $β > 0 is a design constant, l_{g} > 0, ε > 0, c > 0$ , and $p$ , $q$ are positive odd integers, $1 < p / q < 2,$ then the system (1) will converge to zero in finite time.

Proof. For (4), its derivative along the dynamics system (1) is

\overset{\cdot}{s} = {\overset{\cdot}{x}}_{1} + \frac{1}{β} \frac{p}{q} x_{2}^{\frac{p}{q} - 1} {\overset{\cdot}{x}}_{2} = x_{2} + \frac{1}{β} \frac{p}{q} x_{2}^{\frac{p}{q} - 1} {\overset{\cdot}{x}}_{2}

Then

s \overset{\cdot}{s} = s (x_{2} + \frac{1}{β} \frac{p}{q} x_{2}^{\frac{p}{q} - 1} {\overset{\cdot}{x}}_{2})

= \frac{1}{β} \frac{p}{q} x_{2}^{\frac{p}{q} - 1} {s [g (t) - l_{g} sgn (s)] - ε \frac{1}{1 + c x_{1}} | s | - (k + c x_{1}) s^{2}}

(6)

Two different cases will be discussed as follows.

(1) When $x_{2} \neq 0$ , the following is obtained:

s \overset{\cdot}{s} \leq \frac{1}{β} \frac{p}{q} x_{2}^{\frac{p}{q} - 1} [- ε \frac{1}{1 + c x_{1}} | s | - (k + c x_{1}) s^{2}] \leq 0

(7)

Therefore, for the case $x_{2} \neq 0$ , the condition for reaching sliding mode $s = 0$ with finite time is satisfied.

(2) When $x_{2} = 0$ , substituting (5) into system (1) yields

{\overset{\cdot}{x}}_{2} = g (t) - l_{g} sgn (s) - ε \frac{1}{1 + c x_{1}} | s | - (k + c x_{1}) s

(8)

{\overset{\cdot}{x}}_{2} \leq - ε \frac{1}{1 + c x_{1}}, s > 0

(9)

{\overset{\cdot}{x}}_{2} \geq ε \frac{1}{1 + c x_{1}}, s < 0

(10)

For $s > 0$ and $x_{2} = 0$ , it is obtained from (9) that $x_{2}$ is not in steady state. $x_{2}$ will change into $x_{2} < 0$ in finite time. It is obtained from (7) that the sliding mode $s = 0$ can be reached in finite time. Similarly, for $s < 0$ and $x_{2} = 0$ , it is obtained from (10) and (7) that the system reaches the sliding mode $s = 0$ from initial state in finite time.

It is concluded that the sliding mode can be reached from anywhere in the phase plane in finite time. It is obtained that the system will converge to zero in finite time after reaching the sliding mode from Yang et al. (2013). This completes the proof.

Mathematical model of the bearingless permanent magnet synchronous motor

There have two sets of windings nested in BPMSM stator. One is a torque winding, the other is a suspension winding. The pole pairs of these two windings is different about 1. A surface-mounted BPMSM usually uses rotor field oriented control to decouple the torque and suspension force; the vector control strategy of $i_{d}^{*} = 0$ is used in the torque section of the BPMSM. In $d - q$ coordinates the mathematical model of rotating speed is expressed as follows (Xu et al., 2012)

\frac{d ω}{dt} = \frac{1.5 P_{1} ψ_{f}}{J} i_{q} - \frac{F}{J} ω - \frac{1}{J} T_{L}

(11)

where $ω$ is the rotor angular velocity, $i_{d}$ and $i_{q}$ are the stator current components, respectively, at the d-axis and q-axis, $P_{1}$ is the number of pole pairs of the torque winding, $ψ_{f}$ is the flux linkage of permanent magnet poles and stator windings linkage, F is the friction coefficient of rotor and load, $T_{L}$ is the mechanical torque and J is the moment of inertia.

The radial suspension force of the BPMSM is obtained by the virtual displacement method of the electromagnetic field (Chiba et al., 1994). The radial suspension forces $F_{x}$ and $F_{y}$ at x (horizontal) direction and y (vertical) directions are expressed as follows:

[\begin{matrix} F_{x} \\ F_{y} \end{matrix}] = M' I [\begin{matrix} - \cos Ø & \sin Ø \\ \sin Ø & \cos Ø \end{matrix}] [\begin{matrix} i_{x} \\ i_{y} \end{matrix}]

(12)

where $M'$ is the derivative of x and y with the mutual inductance of two sets of windings, ignoring the motor nonlinear magnetic saturation. $Ø = ω t + θ_{0}, θ_{0} = \arctan (I_{q} / I_{p})$ , $I_{p}$ is the field current component of the permanent magnet, where $I_{q}$ is the torque current component.

Consider the influence of the BPMSM rotor eccentricity and radial force perturbation; the rotor radial displacement mathematical model is established as follows by the dynamic equations of the BPMSM

{\begin{matrix} \overset{\cdot\cdot}{x} = \frac{1}{m} f_{x} + \frac{1}{m} F_{x} - \frac{1}{m} F_{xr} \\ \overset{\cdot\cdot}{y} = \frac{1}{m} f_{y} + \frac{1}{m} F_{y} - \frac{1}{m} F_{yr} \end{matrix}

(13)

where m is the rotor mass, $F_{xr}$ and $F_{yr}$ are the radial disturbance forces at the horizontal and vertical directions, $f_{x}$ and $f_{y}$ are the Maxwell radial forces, which are proportional to displacement at rotor eccentricity. This can be expressed as

f_{x} = k_{s} x, f_{y} = k_{s} y

(14)

where $k_{s}$ is the radial force displacement stiffness, only related to the structure of the motor.

Adaptive nonsingular terminal sliding mode control design for the bearingless permanent magnet synchronous motor

Design of speed adaptive NTSMC

A speed controller is used to accurately track the given speed $ω^{*}$ , and should have robustness to parameter variations and load disturbances. If $ω^{*}$ is smooth and second-order continuously differentiable, the state variables can be defined as follows

{\begin{matrix} e_{ω 1} = ω^{*} - ω \\ e_{ω 2} = \frac{d e_{ω 1}}{dt} = {\overset{\cdot}{ω}}^{*} - \overset{\cdot}{ω} \end{matrix}

(15)

It is obtained from (11) and (15) that

{\overset{\cdot}{e}}_{ω 2} = {\overset{\cdot\cdot}{ω}}^{*} - \overset{\cdot\cdot}{ω} = - \frac{F}{J} e_{ω 2} - \frac{1.5 P_{1} ψ_{f}}{J} i_{q} + \frac{1}{J} {\overset{\cdot}{T}}_{L} + {\overset{\cdot\cdot}{ω}}^{*} + \frac{F}{J} {\overset{\cdot}{ω}}^{*}

(16)

Considering the motor parameter uncertainty, ${\overset{\cdot}{e}}_{ω 2}$ can be shown as

{\overset{\cdot}{e}}_{ω 2} = (- \frac{F}{J} + Δ a_{1}) e_{ω 2} + (- \frac{1.5 P_{1} ψ_{f}}{J} + Δ b_{1}) {\overset{\cdot\cdot}{i}}_{q} + (\frac{1}{J} {\overset{\cdot}{T}}_{L} + {\overset{\cdot\cdot}{ω}}^{*} + \frac{F}{J} {\overset{\cdot}{ω}}^{*} + Δ d_{1})

(17)

where $Δ a_{1}$ , $Δ b_{1}$ and $Δ d_{1}$ are corresponding uncertainty items that are value bounded. It is assumed that $g_{1} (t)$ is the total uncertainty, therefore

g_{1} (t) = Δ a_{1} e_{ω 2} + Δ b_{1} {\overset{\cdot\cdot}{i}}_{q} + \frac{1}{J} {\overset{\cdot}{T}}_{L} + {\overset{\cdot\cdot}{ω}}^{*} + \frac{F}{J} {\overset{\cdot}{ω}}^{*} + Δ d_{1}

(18)

where $Δ a_{1} e_{ω 2}$ , ${\overset{\cdot\cdot}{ω}}^{*}$ and $\frac{F}{J} {\overset{\cdot}{ω}}^{*}$ are all bounded due to the assumptions of $ω^{*}$ . The motor current is a sine function, so $Δ b_{1} {\overset{\cdot\cdot}{i}}_{q}$ is bounded. $\frac{1}{J} {\overset{\cdot}{T}}_{L}$ is the external disturbance, setting its value as bounded (Tong et al., 2008). Therefore, $g_{1} (t)$ is bounded. It is assumed that $| g_{1} (t) | < l_{g 1}$ is the uncertain part and $l_{g 1}$ is the upper bound of disturbance.

The state equation of speed error system can be obtained from (15), (17) and (18)

{\begin{matrix} {\overset{\cdot}{e}}_{ω 1} = e_{ω 2} \\ {\overset{\cdot}{e}}_{ω 2} = - \frac{F}{J} e_{ω 2} - \frac{1.5 P_{1} ψ_{f}}{J} {\overset{\cdot\cdot}{i}}_{q} + g_{1} (t) \end{matrix}

(19)

Theorem 2. For the speed error control system (19) of the BPMSM, choose sliding mode

s_{1} = e_{ω 1} + \frac{1}{β_{1}} e_{ω 2}^{p_{1} / q_{1}}

(20)

If $i_{q}$ can be designed to satisfy the control law below

i_{q} = \frac{J}{1.5 P_{1} ψ_{f}} \int_{0}^{t} [- \frac{F}{J} e_{ω 2} + β_{1} \frac{q_{1}}{p_{1}} e_{ω 2}^{2 - \frac{p_{1}}{q_{1}}} + l_{g 1} sgn (s_{1}) + ε_{1} \frac{1}{1 + c_{1} e_{ω 1}} sgn (s_{1}) + (k_{1} + c_{1} e_{ω 1}) s_{1}] d τ

(21)

then the speed error will converge in finite time. Here $e_{ω} = {[e_{ω 1}, e_{ω 2}]}^{T}; β_{1}, q_{1}, p_{1}, ε_{1}, k_{1}, c_{1}$ are system parameters.

Design of rotor radial displacement adaptive NTSMC

The radial displacement controller is used to accurately track the given rotor radial displacements $x^{*}$ and $y^{*}$ , and its robustness to parameter variations and load disturbances is also required. This paper takes the x direction as an example to design the adaptive NTSMC. The y direction is the same as x. Assuming that $x^{*}$ is smooth and second-order continuously differentiable, the state variables can be defined as follows

{\begin{matrix} e_{x 1} = x^{*} - x \\ e_{x 2} = \frac{d e_{x 1}}{dt} = {\overset{\cdot}{x}}^{*} - \overset{\cdot}{x} \end{matrix}

(22)

It is obtained from (13) and (22) that

{\overset{\cdot}{e}}_{x 2} = \frac{k_{s}}{m} e_{x 1} - \frac{1}{m} F_{x} + {\overset{\cdot\cdot}{x}}^{*} - \frac{k_{s}}{m} x^{*} + \frac{1}{m} F_{xr}

(23)

Considering the motor parameter uncertainty, ${\overset{\cdot}{e}}_{x 2}$ can be shown as

{\overset{\cdot}{e}}_{x 2} = (\frac{k_{s}}{m} + Δ a_{2}) e_{x 1} + (- \frac{1}{m} + Δ b_{2}) F_{x} + ({\overset{\cdot\cdot}{x}}^{*} - - \frac{k_{s}}{m} x^{*} + \frac{1}{m} F_{xr} + Δ d_{2})

(24)

where $Δ a_{2}$ , $Δ b_{2}$ and $Δ d_{2}$ are corresponding uncertainty items and are value bounded. It is assumed that $g_{2} (t)$ is the total uncertainty, therefore

g_{2} (t) = Δ a_{2} e_{x 1} + Δ b_{2} F_{x} + ({\overset{\cdot\cdot}{x}}^{*} - - \frac{k_{s}}{m} x^{*} + \frac{1}{m} F_{xr} + Δ d_{2})

(25)

The state equation of radial displacement error control system based on the x direction can be obtained from (22), (24) and (25):

{\begin{matrix} {\overset{\cdot}{e}}_{x 1} = e_{x 2}, \\ {\overset{\cdot}{e}}_{x 2} = \frac{k_{s}}{m} e_{x 1} - \frac{1}{m} F_{x} + g_{2} (t) \end{matrix}

(26)

Theorem 3. For the radial displacement error control system (26) based on the x direction, choose sliding mode

s_{2} = e_{x 1} + \frac{1}{β_{2}} e_{x 2}^{p_{2} / q_{2}}

(27)

Radial force $F_{x}$ can be designed to satisfy the control law below

F_{x} = m [\frac{k_{s}}{m} e_{x 1} + β_{2} \frac{q_{2}}{p_{2}} e_{x 2}^{2 - \frac{p_{2}}{q_{2}}} + l_{g 2} sgn (s_{2}) + ε_{2} \frac{1}{1 + c_{2} e_{x 1}} sgn (s_{2}) + (k_{2} + c_{2} e_{x 1}) s_{2}]

(28)

Then radial displacement error on the x direction will converge in finite time. Here $e_{x} = {[e_{x 1}, e_{x 2}]}^{T}; β_{2}, q_{2}, p_{2}, ε_{2}, k_{2}, c_{2}$ are the system parameters; $l_{g 2} > | g_{2} (t) |$ is the uncertain part and $l_{g 2}$ is the upper bound of disturbance.

Simulation studies

Performance analysis of adaptive NTSMC and NTSMC

Take the below second-order nonlinear system as an example to design the adaptive nonsingular terminal sliding mode controller and verify the effectiveness of the proposed method

{\begin{matrix} {\overset{\cdot}{x}}_{1} = x_{2} \\ {\overset{\cdot}{x}}_{2} = - 16 x_{2} + 125 u + 0.1 \sin (20 t) \end{matrix}

(29)

With nonsingular terminal sliding mode (4) and reaching law (3), the control law is designed as follows

u = - \frac{1}{125} [- 16 x_{2} + β \frac{q}{p} x_{2}^{2 - \frac{p}{q}} + l_{g} sgn (s) + ε \frac{1}{1 + c x_{1}} sgn (s) + (k + c x_{1}) s]

(30)

Therefore, system (29) will converge in finite time.

Two cases are chosen to analyse and compare the proposed method with NTSMC with reaching law (2). One is that the initial state variable is close to the equilibrium point, namely $x (0) = [0.1, 0.1]^{T}$ . The other is that the initial state variable is far from the equilibrium point, namely $x (0) = [30, 30]^{T}$ . According to Feng et al. (2009) and Zheng et al. (2009), the controller parameters designed are as shown in Table 1.

Table 1.

Controller parameters.

Initial state	Controller parameters
[0.1 0.1]^T	β = 1, q = 3, p = 5, l_g = 0.15, ε = 0.20, k = 2, c = 500
[30 30]^T	β = 6, q = 3, p = 5, l_g = 0.15, ε = 0.20, k = 2, c = 50

The simulation results are shown in Figures 1 and 2, which indicate that the proposed method can make state variables converge in finite time and the convergence rate is faster than the NTSM controller with exponential reaching law. In Figure 2(b), the initial state of x₂ is far from the equilibrium point and x₂ converges rapidly from 30 to −20 in a very short period of time under the proposed reaching law control. Because t = 5 s in Figure 2(b), the initial state of x₂ is coincident with the coordinate axes, so it looks as if it is starting to converge from −20. However, in fact the initial state of x₂ converges from 30. The phase plane response of the control system is also provided, as shown in Figures 1(c) and 2(c).

Figure 1.

Performance comparison of the proposed method and nonsingular terminal sliding mode control $(NTSMC) at x (0) = [0.1, 0.1]^{T} .$

Figure 2.

Performance comparison of the proposed method and nonsingular terminal sliding mode control $(NTSMC) at x (0) = [30, 30]^{T} .$

Simulation of adaptive NTSMC for the BPMSM

Figure 3 shows the structure diagram of the BPMSM based on vector control. In this diagram, $2 / 3$ is the $α β / abc$ transform, CRPWM is the current tracking inverter, $θ$ is the rotor angle position and $θ_{1}$ and $θ_{2}$ are the initial phase angles between the d direction and A phase winding centre line of torque winding and levitation force winding, respectively. The BPMSM parameters are shown in Table 2.

Figure 3.

Schematic diagram of the bearingless permanent magnet synchronous motor (BPMSM) system based on vector control.

Table 2.

Data of bearingless permanent magnet synchronous motor parameters.

Paper size parameter	Symbol	Value
Number of poles pairs in torque winding	P ₁	4
Number of poles pairs in suspension force winding	P ₂	3
Stator resistance	R_s	2.875 Ω
D-axis inductance	L_d	8.5 mH
Q-axis inductance	L_q	8.5 mH
Magnet flux linkage	ψ_f	0.175 Wb
Rotor inertia	J	0.8×10⁻⁴ kg·m²
Rotor mass	m	1 kg
Specified voltage of torque winding	V	240 V
Specified current	I	4.87 A
The maximum electromagnetic torque	T	13.44 N·m
Mechanical time constant		0.21 s
Specified load torque	T_L	6.72 N·m

The simulation results of the BPMSM speed regulation system with different controllers is shown in Figure 4, where the ordinary exponential reaching law is used by the NTSMC. The parameters of adaptive NTSMC are as follows: $β_{1} = 1000, p_{1} = 9, q_{1} = 7, l_{g 1} = 80, ε_{1} = 75, k_{1} = 50, c_{1} = 50$ . The parameters of the PI controller are as follows: proportional coefficient $k_{p} = 2$ ; integral coefficient $k_{i} = 6$ ; resistant saturated coefficient $k_{t} = 0.02$ . The step response of rotating speed ( $ω = 600 rad / s$ ) is shown in Figure 4(a). The response of rotating speed sudden change (speed change from 500 to 100 rad/s) is shown in Figure 4(b). It can be seen that the proposed method can make the rotating speed quickly track the given value and shows small overshoots and short settling times. Compared with ordinary NTSMC, the response of the proposed method is faster. Compared with the PI controller, the proposed method has smaller overshoot and static error and higher robustness of the system. Figure 4(c) shows the curve of rotating speed control given by different control methods with 800 rad/s rotating speed and torque disturbance. When load torque disturbance is added, the speed can return to the steady state in a short time and the maximum speed drop of the proposed method is much smaller than for the other two control methods.

Figure 4.

Comparison of responses under the proportional–integral (PI), nonsingular terminal sliding mode (NTSM) and proposed method.

A comparison of performance indexes of the three methods is shown in Table 3. Please note that the tolerance band of settling time and recovery time in this paper is always selected as 2%. It can be observed that the proposed method is still effective under different operating conditions. radial displacement control curve on the $x and y$ directions under the rotating speed mutated from 500 to 100 rad/s is shown in Figure 5. The parameters of adaptive NTSMC are as follows: $β_{2} = 1000, p_{2} = 9, q_{2} = 7, l_{g 2} = 10, ε_{2} = 10, k_{2} = 2, c_{2} = 5$ . The parameters of the PI controller are as follows: proportional coefficient $k_{p} = 20$ ; integral coefficient $k_{i} = 65$ ; resistant saturated coefficient $k_{t} = 0.01$ . It can be seen that the radial displacement with the adaptive NTSMC can reach the equilibrium faster than the other methods. From these results, we can conclude that the proposed method has a better performance at most conditions.

Table 3.

Comparison of performance indices on the three methods.

Index	w = 600 rad/s		w change from 500 to 100 rad/s		Torque disturbance at w = 800 rad/s
Control scheme	OS (%)	ts (s)	OS (%)	ts (s)	Speed decrease (rad/s)	tv (s)
PI	5.91	0.0130	16.7	0.0123	10.96	0.0062
NTSMC	0.18	0.0092	0.31	0.0083	9.20	0.0051
Proposed method	0.16	0.0074	0.21	0.0061	8.40	0.0049

OS: overshoot; ts: settling time; speed decrease: maximum decrease of speed caused by a load torque disturbance; tv: recovery time; PI: proportional–integral; NTSM: nonsingular terminal sliding mode.

Figure 5.

Radial displacement control curve under the proportional–integral (PI), nonsingular terminal sliding mode (NTSMC) and proposed method.

Conclusion

In this paper, an adaptive nonsingular TSMC method has been proposed for the BPMSM. An adaptive variable-rated exponential reaching law is presented to obtain faster convergence of state variables during the whole process in TSMC, where the L1 norm of state variables is introduced. Simulation results show that the proposed method can rapidly track the given values, overshoot and static error are small, and higher robustness can be obtained. It is effective to design a nonsingular terminal sliding mode controller by using the adaptive variable-rated exponential reaching law. In the future, how to apply the proposed method on an actual system based on Digital Signal Processor (DSP) will be the next work.

Footnotes

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This work was supported by National Natural Science Foundation of China (grant number 61573170), Natural Science Foundation of Jiangsu Province (grant number BK20150530), Jiangsu Province Colleges and Universities Natural Science Research Project (grant number 12KJB470002), a project funded by the Priority Academic Program Development of Jiangsu Higher Education Institutions (PAPD) and the Professional Research Foundation for Advanced Talents of Jiangsu University (grant number 14JDG077).

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