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Abstract

This study proposes a novel hierarchical nonlinear proportional-integral fast terminal sliding mode (HNLPIFTSM) control for permanent magnet synchronous motor (PMSM) speed regulation system. A new type of sliding surface called HNLPIFTSM surface, which combines the benefits of a nonlinear proportional-integral (PI) sliding mode surface and a fast terminal sliding mode (FTSM) surface, is proposed to enhance the robustness and improved the dynamic response, whilst preserving the great property of the conventional hierarchical fast terminal sliding mode (HFTSM) control strategy. The proposed HNLPIFTSM surface uses the novel nonlinear PI sliding mode surface as its inner loop and uses the FTSM surface as its outer loop. Meanwhile, an extended state observer (ESO) is used to estimate the uncertain terms of the PMSM speed regulation system. Furthermore, the stability of the closed-loop control system under the ESO and the HNLPIFTSM control strategy is proved by the Lyapunov stability theorem. Finally, the simulations and experimental demonstrations verify the effectiveness and superiorities of our proposed HNLPIFTSM control strategy over the conventional HFTSM control strategy.

Keywords

Hierarchical nonlinear proportional-integral fast terminal sliding mode control permanent magnet synchronous motor nonlinear PI sliding mode surface fast terminal sliding mode surface extended state observer

Introduction

Sliding mode (SM) control strategy is a common robust control method, which was proposed by former Soviet scholars in the 1950s. The sliding mode variable structure control system based on the SM control has strong robustness and anti-interference to uncertain system, and the control strategy is simple and easy to realize, which has been widely in practical applications.^1–3

The inherent chattering problem of sliding mode variable structure seriously limits the application of SM control strategy in practical engineering. In the meantime, in the face of complex control problems, it is difficult to achieve efficient control effect by using traditional SM control strategy. Hence, compound SM control methods are effective measures for such requirements. In recent years, in order to improve the control effect of uncertain systems, many scholars have studied various composite SM control strategies, such as fractional order SM control,^4,5 adaptive SM control,^6–9 passivity-based SM control,¹⁰ mode-free SM control,^11,12 fuzzy SM control,^13,14 event-triggered SM control,^15–17 backstepping SM control,¹⁸ and hierarchical SM control. Since the hierarchical SM control strategy not only eliminates the chattering issue but also increases the control precision of the system. To the best of our knowledge, as a high-order SM control algorithm, the hierarchical SM control strategy has been widely studied by many scholars. Qian et al.¹⁹ proposed a hierarchical SM control for a class of SIMO under-actuated systems. Shao et al.²⁰ developed a hierarchical SM control strategy with an adaptive disturbance observer was designed for a linear motor positioner. Wang et al.²¹ presented a hierarchical SM control strategy with a proportional integral derivative fractional order sliding mode control for cable driven manipulators.

A fast terminal sliding mode (FTSM) control strategy was first proposed by Yu and Man²² Because of its superior performance, the FTSM control strategy has been widely used in permanent magnet in-wheel motor,²³ nonlinear uncertain mass-spring system,²⁴ gantry cranes,²⁵ second-order underactuated systems,²⁶ and so on. In the conventional approach, the hierarchical SM control strategy was motivated to eliminate the chattering of the conventional SM control strategy. However, since the great properties of both hierarchical SM control strategy and FTSM control strategy, there is a desire to combine the benefits of both to obtain defined finite time convergence, chattering elimination and strong robust simultaneously. Abolvafaei and Ganjefar²⁷ and Liu et al.²⁸ combined the FTSM surface with a linear sliding surface is used to design the hierarchical FTSM control strategy. By using the strategy, the advantages of two sliding mode surfaces can be obtained simultaneously. However, the conventional linear sliding mode surface also has the disadvantages of simple and rough signal processing. To overcome the issue, a novel nonlinear PI sliding mode surface is designed in this article.

Gao²⁹ proposed a linear Extended State Observer (ESO). Because of its simple structure and high estimation efficiency, the linear ESO has been widely used in many applications such as networked multi-axis servo system,³⁰ networked interconnected motion system,³¹ unmanned powered parafoil,³² three-dimensional elastic wing,³³ and so on.

Permanent magnet synchronous motor (PMSM) speed regulation system is a typical uncertain system due to its own parameter uncertainty and external load disturbance. Recently, PMSM has been widely used in many industrial applications. Meanwhile, the control of PMSM speed regulation system is a challenging task because of the uncertain system. Many scholars have applied many advanced control strategies to the control of the uncertain systems.^34–41

Motivated by the above analysis and previous studies, a novel hierarchical nonlinear proportional-integral fast terminal sliding mode (HNLPIFTSM) control strategy is proposed for the PMSM speed regulation system in the present work. A novel type of sliding mode surface called HNLPIFTSM surface, which combines the benefits of the nonlinear proportional-integral (PI) sliding mode surface and the FTSM surface, is proposed to enhance the robustness and improved the dynamic response. The proposed HNLPIFTSM surface uses the nonlinear PI sliding mode surface as its inner loop and uses the FTSM surface as its outer loop. Meanwhile, a linear ESO is used to estimate the uncertain terms of the PMSM speed regulation system. The stability of the closed-loop control system under the linear ESO and HNLPIFTSM control strategy is proved using the Lyapunov stability theory.

Through comparisons with the existing studies, our contributions of this article are further highlighted as follows.

In order to improve the dynamic response speed and anti-disturbance ability, a nonlinear PI sliding mode surface is designed.

The combination of the FTSM surface with the nonlinear PI sliding mode surface is used to design a novel HNLPIFTSM control strategy.

The effectiveness and superiorities of our ESO based the HNLPIFTSM control strategy are proved by the comparative simulations and experimental demonstrations.

The remaining parts of this paper are arranged as:

In Section 2, some important knowledge of the mathematical model of PMSM are briefly presented. Control strategies and stability analysis are given in Section 3. the simulations and experimental demonstrations are shown in Section 4. Finally, Section 5 concludes the study.

Problem description

The torque equation of PMSM is first described as follows:⁴²

T_{e} = \frac{3}{2} p_{n} (ϕ_{f} i_{q} (t) + (L_{d} - L_{q}) i_{d} (t) i_{q} (t)),

(1)

where $T_{e}$ is the electromagnetic torque; $ϕ_{f}$ presents the flux of permanent magnet; $L_{d}$ , $L_{q}$ , $i_{d} (t)$ and $i_{q} (t)$ present inductance components and current components of the $dq -$ axes.

When $L_{d} = L_{q}$ is satisfied, the motion equation of PMSM is given as:⁴²

{\begin{matrix} T_{e} - T_{L} = J \frac{d ω (t)}{dt} + B ω (t), \\ T_{e} = \frac{3}{2} p_{n} ϕ_{f} i_{q} (t), \end{matrix}

(2)

\frac{d ω (t)}{dt} = \frac{3 p_{n} ϕ_{f}}{2 J} i_{q} (t) - \frac{B}{J} ω (t) - \frac{1}{J} T_{L},

(3)

where $ω (t)$ is actual mechanical speed of PMSM; $J$ implies rotational inertia; $T_{L}$ is load torque; $B$ presents friction coefficient of the rotor and load. To simplify analysis, the motion equation of PMSM can be rewritten as:

\frac{d ω (t)}{dt} = \frac{3 p_{n} ϕ_{f}}{2 J} i_{q}^{*} (t) + Δ d,

(4)

where $Δ d$ is the total disturbance of the system, $i_{q}^{*} (t)$ is the control input.

The error equation of the PMSM speed regulation system is given as:

e (t) = ω_{r} (t) - ω (t),

(5)

where $ω_{r} (t)$ is the reference mechanical speed; $e (t)$ is the error of the mechanical speed.

Control strategies and stability analysis

Control strategies designed

The traditional linear sliding mode surface as the inner loop is selected as:

s_{1} (t) = K_{p} e (t) + K_{i} \int e (t),

(6)

where $K_{p} > 0$ and $K_{i} > 0$ .

Based on the FTSM control strategy,²² the following FTSM surface as the outer loop is proposed as:

s_{2} (t) = a_{1} s_{1}^{\frac{p}{q}} (t) + a_{2} s_{1} (t) + \frac{d s_{1} (t)}{dt},

(7)

where $a_{1} > 0$ and $a_{2} > 0$ , $p$ and $q$ are positive odd integers, $\frac{1}{2} < \frac{p}{q} < 1$ .

In order to resist the occurrence of uncertain interference, the following reaching law is applied:

\frac{d s_{2} (t)}{dt} = - η sign (s_{2} (t)),

(8)

where $η$ is positive constant, the symbolic function $sign (s_{2} (t))$ is as follows.

sign (s_{2} (t)) = {\begin{matrix} 1 & if s_{2} (t) > 0 \\ 0 & if s_{2} (t) = 0 \\ - 1 & if s_{2} (t) < 0 \end{matrix}

(9)

According to (7), (8), and (9), one can obtain:

a_{1} s_{1}^{\frac{p}{q}} (t) + a_{2} s_{1} (t) + \frac{d s_{1} (t)}{dt} = - \int η sign (s_{2} (t))

(10)

By substituting (6) into (10), one has:

a_{1} s_{1}^{\frac{p}{q}} (t) + a_{2} s_{1} (t) + K_{p} \frac{de (t)}{dt} + K_{i} e (t) = - \int η sign (s_{2} (t))

(11)

By substituting (4) and (5) into (11) gives:

\begin{matrix} a_{1} s_{1}^{\frac{p}{q}} (t) + a_{2} s_{1} (t) + K_{i} e (t) + \int η sign (s_{2} (t)) \\ + K_{p} (\frac{d ω_{r} (t)}{dt} - \frac{3 p_{n} ϕ_{f}}{2 J} i_{q}^{*} (t) - Δ d) = 0 \end{matrix}

(12)

According to (12), the traditional HFTSM control input is defined as:

\begin{matrix} i_{q}^{*} (t) = \frac{2 J}{3 p_{n} ϕ_{f} K_{p}} \\ (a_{1} s_{1}^{\frac{p}{q}} (t) + a_{2} s_{1} (t) + K_{i} e (t) + \int η sign (s_{2} (t))) \\ + \frac{2 J}{3 p_{n} ϕ_{f}} (\frac{d ω_{r} (t)}{dt} - Δ d) \end{matrix}

(13)

In this paper, a nonlinear PI sliding mode surface as the inner loop is defined as:⁴³

s_{1} (t) = K_{p} sig {(e (t))}^{\tilde{α}} + K_{i} \int sig {(e (t))}^{\tilde{α}},

(14)

where the nonlinear function $sig {(e (t))}^{\tilde{α}}$ is defined as:²¹

sig {(e (t))}^{\tilde{α}} = {| e (t) |}^{\tilde{α}} sign (e (t)),

(15)

where $0 < \tilde{a} < 1$ .

Remark 1. Compare the traditional linear sliding surface (6), the proposed nonlinear PI sliding mode surface (14) can effectively overcome the disadvantages of simple and rough signal processing of the traditional linear sliding mode surface.

Combining (7), (8), and (14), we have:

\begin{matrix} a_{1} s_{1}^{\frac{p}{q}} (t) + a_{2} s_{1} (t) + K_{p} \frac{dsig {(e (t))}^{\tilde{α}}}{dt} + K_{i} sig (e (t))^{\tilde{α}} \\ = - \int η sign (s_{2} (t)) \end{matrix}

(16)

Substituting (4) and (5) into (16) yields:

\begin{matrix} a_{1} s_{1}^{\frac{p}{q}} (t) + a_{2} s_{1} (t) + K_{i} sig (e (t))^{\tilde{α}} + \int η sign (s_{2} (t)) \\ + K_{p} \tilde{α} {| e (t) |}^{\tilde{α} - 1} (\frac{d ω_{r} (t)}{dt} - \frac{3 p_{n} ϕ_{f}}{2 J} i_{q}^{*} (t) - Δ d) = 0 \end{matrix}

(17)

According to (17), the proposed HNLPIFTSM control input is designed as:

\begin{matrix} i_{q}^{*} (t) = \frac{2 J}{3 p_{n} ϕ_{f} K_{p} \tilde{α} {| e (t) |}^{\tilde{α} - 1}} \\ (\begin{matrix} a_{1} s_{1}^{\frac{p}{q}} (t) + a_{2} s_{1} (t) + K_{i} sig (e (t))^{\tilde{α}} \\ + \int η sign (s_{2} (t)) \end{matrix}) \\ + \frac{2 J}{3 p_{n} ϕ_{f}} (\frac{d ω_{r} (t)}{dt} - Δ d) \end{matrix}

(18)

In this subsection, the stability of the proposed HNLPIFTSM control strategy will be discussed.

The Lyapunov function for the proposed HNLPIFTSM control strategy is selected:

V = \frac{1}{2} s_{2}^{2} (t) > 0

(19)

By substituting (14) into (7), we can get:

\begin{matrix} s_{2} (t) = a_{1} s_{1}^{\frac{p}{q}} (t) + a_{2} s_{1} (t) + K_{i} sig (e (t))^{\tilde{α}} \\ + K_{p} \tilde{α} {| e (t) |}^{\tilde{α} - 1} (\frac{d ω_{r} (t)}{dt} - \frac{3 p_{n} ϕ_{f}}{2 J} i_{q}^{*} (t) - Δ d) \end{matrix}

(20)

Then, substituting the proposed HNLPIFTSM control input (18) into (20), the following equation can be obtained.

s_{2} (t) = - η \int sign (s_{2} (t))

(21)

According to (19) and (21), we can get:

\begin{matrix} \frac{dV}{dt} = s_{2} (t) \frac{d s_{2} (t)}{dt} \\ = s_{2} (t) (- η sign (s_{2} (t))) \\ = - η | s_{2} (t) | \end{matrix}

(22)

According to the Lyapunov criterion, the FTSM surface as the outer loop can reach zero. Then, the FTSM surface satisfies the following condition.

a_{1} s_{1}^{\frac{p}{q}} (t) + a_{2} s_{1} (t) + \frac{d s_{1} (t)}{dt} = 0

(23)

When chosen $p$ and $q$ properly, we can get that the nonlinear PI sliding mode surface $s_{1} (t)$ as the inner loop reach zero in finite time, which can be derived from the following expression.²²

t = \frac{q}{a_{2} (q - p)} (\ln (a_{2} s_{1} {(0)}^{\frac{(q - p)}{q}} + a_{1}) - \ln a_{1}),

(24)

where $s_{1} (0) \neq 0$ is an initial state.

When the nonlinear PI sliding mode surface reach zero in finite time, the novel nonlinear PI sliding mode surface can be written as:

K_{p} sig {(e (t))}^{\tilde{α}} + K_{i} \int sig {(e (t))}^{\tilde{α}} = 0

(25)

Let $ℜ = {| e (t) |}^{\tilde{α}} sign (e (t))$ , equation (25) can be written as:

ℜ + \frac{K_{i}}{K_{p}} \int ℜ = 0

(26)

Then

ℜ = ℜ_{0} e^{- \frac{K_{i}}{K_{p}} t},

(27)

where $ℜ_{0}$ is the coefficient determined by the initial condition. According to (15) and (27), we know $e (t) \to 0$ when $t \to \infty$ , the stability of the system (25) can be guaranteed. Consequently, the stability of the proposed HNLPIFTSM control strategy is proved.

Linear ESO designed

The motion equation of PMSM can be expressed as:

\frac{d x_{1}}{dt} = x_{2} + b_{0} u,

(28)

where $x_{1} = ω (t)$ ; $x_{2} = Δ d$ ; $u = i_{q}^{*} (t)$ and $b_{0} = \frac{3 p_{n} ϕ_{f}}{2 J}$ .

Then, the linear ESO was designed as:²⁹

{\begin{matrix} \hat{e} (t) = Z_{21} - y (t) \\ \frac{d Z_{21}}{dt} = Z_{22} - β_{1} \hat{e} (t) + b_{0} u (t) \\ \frac{d Z_{22}}{dt} = - β_{2} \hat{e} (t) \end{matrix},

(29)

where $β_{1}$ and $β_{2}$ are positive constants; $Z_{21}$ and $Z_{22}$ are the estimated values of $x_{1}$ and $x_{2}$ , respectively.

According to the previous studies,⁴⁴ the stability of the linear ESO will be discussed.

The linear ESO (29) can be rewritten as:

{\begin{matrix} \frac{d Z_{21}}{dt} = Z_{22} + β_{1} (x_{1} - Z_{21}) + b_{0} u (t) \\ \frac{d Z_{22}}{dt} = β_{2} (x_{1} - Z_{21}) \end{matrix}

(30)

Equation (30) can be written as:

[\begin{matrix} \frac{d Z_{21}}{dt} \\ \frac{d Z_{22}}{dt} \end{matrix}] = A [\begin{matrix} Z_{21} \\ Z_{22} \end{matrix}] + B [\begin{matrix} x_{1} \\ u (t) \end{matrix}],

(31)

where

A = [\begin{matrix} - β_{1} & 1 \\ - β_{2} & 0 \end{matrix}]

(32)

B = [\begin{matrix} β_{1} & b_{0} \\ β_{2} & 0 \end{matrix}]

(33)

According to (31), (32), and (33), the stability of system can be guaranteed.

So, we can get:

{\begin{matrix} Z_{21} \to x_{1} \\ Z_{22} \to x_{2} \end{matrix}

(34)

Therefore, the linear ESO exhibits stability.

Theorem 1: By substituting (27) into (18), the proposed control input with the linear ESO is designed as:

\begin{matrix} i_{q}^{*} (t) = \frac{2 J}{3 p_{n} ϕ_{f} K_{p} \tilde{α} {| e (t) |}^{\tilde{α} - 1}} \\ (\begin{matrix} a_{1} s_{1}^{\frac{p}{q}} (t) + a_{2} s_{1} (t) + K_{i} sig (e (t))^{\tilde{α}} \\ + \int η sign (s_{2} (t)) \end{matrix}) \\ + \frac{2 J}{3 p_{n} ϕ_{f}} (\frac{d ω_{r} (t)}{dt} - Z_{22}) \end{matrix}

(35)

By substituting (35) into (20), we can obtain:

s_{2} (t) = - η \int sign (s_{2} (t)) + K Δ D

(36)

where $K = K_{p} \tilde{α} {| e (t) |}^{\tilde{α} - 1}$ and $Δ D$ is the estimation error of the linear ESO.

Inspired by previous articles,^45,46 the stability of the proposed HNLPIFTSM control strategy with the linear ESO will be discussed. The Lyapunov function for the proposed HNLPIFTSM control with the linear ESO is selected:

V = \frac{1}{2} s_{2}^{2} (t) > 0

(37)

Taking derivative of both sides of the equation (37), it can be shown as:

\begin{matrix} \frac{dV}{dt} = \frac{d (s_{2} (t))}{dt} s_{2} (t) \\ = s_{2} (t) (- η sign (s_{2} (t)) + \frac{d (K Δ D)}{dt}) \\ = - η | s_{2} (t) | + s_{2} (t) \frac{d (K Δ D)}{dt} \\ \leq - η | s_{2} (t) | + | s_{2} (t) | | \frac{d (K Δ D)}{dt} | \end{matrix}

(38)

Assumption 1: The following condition is assumed for analysis in the paper.

| \frac{d (K Δ D)}{dt} | \leq Θ

(39)

where $Θ$ is a constant parameter.

According to (39), we know that $\frac{dV}{dt} < 0$ can be guaranteed as long as $η > Θ$ . Consequently, the stability of the proposed HNLPIFTSM control strategy with the linear ESO is proved.

A novel field oriented control (FOC) of the PMSM speed regulation system based on the proposed HNLPIFTSM control strategy (18) with the linear ESO (29) is given in Figure 1.

Figure 1.

The block diagram of the PMSM speed regulation system based on the proposed HNLPIFTSM control strategy with the ESO.

Comparative simulations and experimental demonstrations

In order to verify the effectiveness and superiorities of our proposed HNLPIFTSM control strategy over the conventional HFTSM control strategy, the FOC of the PMSM speed regulation system based on different control strategies are constructed in this paper.

Table 1 shows the key parameters of the PMSM in the simulation. The selected parameters of the ESO are given as: $β_{1} = 2000$ and $β_{2} = 1000000$ . The selected parameters of the traditional HFTSM control strategy and the proposed HNLPIFTSM control strategy in the simulation are given in Tables 2 and 3, respectively. In particular, to better eliminate the effect of parameter setting on the comparison results, we set some same parameters in the traditional HFTSM control strategy and the proposed HNLPIFTSM control strategy.

Table 1.

Parameters of the PMSM.

$L$	$ϕ_{f}$	$J$	$p_{n}$	$B$
$8.5 mH$	$0.175 Wb$	$0.003 kg \cdot m^{2}$	4	$0.008 N \cdot m \cdot s$

Table 2.

Parameters selected of the traditional HFTSM control strategy.

$K_{p}$	$K_{i}$	$p$	$q$	$η$	$a_{1}$	$a_{2}$
3	15	7	9	10	1	1

Table 3.

Parameters selected of the proposed HNLPFTSM control strategy.

$K_{p}$	$K_{i}$	$p$	$q$	$η$	$\tilde{α}$	$a_{1}$	$a_{2}$
3	15	7	9	10	0.25	1	1

The dynamic response of rotational speed is first used for comparison in the simulation. Figures 2 to 5 show the rotational speed response curves under the different control strategies, respectively. To better compare the control performance of various controllers at different speeds, the reference speed is set to 30 and 60 rpm, respectively. Based on different control schemes, the actual speed response of the PMSM without load starting are compared in Figures 2 and 3, respectively. Based on different control schemes, the actual speed response of the PMSM with $1 N \cdot m$ load starting are compared in Figures 4 and 5, respectively. Taking Figures 2 to 5 to analyse, the reaching time under the proposed algorithm HNLPIFTSM control strategy is shorter than the reaching times under the conventional existing controller. The Figures show that the speed response curves based on the proposed HNLPIFTSM control strategy are faster and better than the conventional existing controller. As a result, the proposed HNLPIFTSM control strategy has better dynamic characteristics than the traditional control strategy.

Figure 2.

(a) Actual speed response under the traditional HFTSM control strategy without load when the reference speed is set at 30 rpm and (b) actual speed response under the traditional HFTSM control strategy without load when the reference speed is set at 60 rpm.

Figure 3.

(a) Actual speed response under the proposed HNLPIFTSM control strategy without load when the reference speed is set at 30 rpm and (b) actual speed response under the proposed HNLPIFTSM control strategy without load when the reference speed is set at 60 rpm.

Figure 4.

(a) Actual speed response under the traditional HFTSM control strategy with $1 N \cdot m$ load when the reference speed is set at 30 rpm and (b) actual speed response under the traditional HFTSM control strategy with $1 N \cdot m$ load when the reference speed is set at 60 rpm.

Figure 5.

(a) Actual speed response under the proposed HNLPIFTSM control strategy with $1 N \cdot m$ load when the reference speed is set at 30 rpm and (b) speed response curves under the proposed HNLPIFTSM control strategy with $1 N \cdot m$ load when the reference speed is set at 60 rpm.

In the following article, an external load with step change is used to demonstrate the anti-disturbance capability of the proposed HNLPIFTSM control strategy. The reference speed is set to 30 rpm. In Figures 6 and 8, the external load suddenly becomes 1 $N \cdot m$ from 0 $N \cdot m$ at 4.5 s under the traditional HFTSM control strategy and the proposed HNLPIFTSM control strategy, respectively. In Figures 7 and 9, the external load suddenly becomes 0 $N \cdot m$ from 1 $N \cdot m$ at 4.5 s under the traditional HFTSM control strategy and the proposed HNLPIFTSM control strategy, respectively. Table 4 gives the speed recovery time under the traditional HFTSM control strategy and the proposed HNLPIFTSM control strategy with the load changed suddenly. It can be seen from Figures 6 to 9 and Table 4 that the recovery time under the proposed HNLPIFTSM control strategy is smaller than that of the traditional HFTSM control strategy, which reveal the proposed HNLPIFTSM control strategy has better robustness than that of the conventional existing controller.

Figure 6.

(a) Actual speed response curve under the traditional HFTSM control strategy when sudden increases 1 $N \cdot m$ load disturbance at 4.5 s, (b) detailed view of the diagram (a), and (c) iq current response curve under the traditional HFTSM control strategy when sudden increases 1 $N \cdot m$ load disturbance at 4.5 s.

Figure 7.

(a) Actual speed response curve under the proposed HNLPIFTSM control strategy when sudden increases 1 $N \cdot m$ load disturbance at 4.5 s, (b) detailed view of the diagram (a), and (c) iq current response curve under the proposed HNLPIFTSM control strategy when sudden increases 1 $N \cdot m$ load disturbance at 4.5 s.

Table 4.

The comparison of robustness under the traditional HFTSM control strategy and the proposed HNLPIFTSM control strategy.

Control strategies	The external load suddenly increase at 4.5 s	The external load suddenly decrease at 4.5 s
Control strategies	Speed recovery time (s)
The traditional HFTSM control strategy	0.31	0.25
The proposed HNLPIFTSM control strategy	0.08	0.07

Figure 8.

(a) Actual speed response curve under the traditional HFTSM control strategy when sudden decreases of 1 $N \cdot m$ load disturbance at 4.5 s, (b) detailed view of the diagram (a), and (c) iq current response curve under the traditional HFTSM control strategy when sudden decreases of 1 $N \cdot m$ load disturbance at 4.5 s.

Figure 9.

(a) Actual speed response curve under the proposed HNLPIFTSM control strategy when sudden decreases of 1 $N \cdot m$ load disturbance at 4.5 s, (b) detailed view of the diagram (a), and (c) iq current response curve under the proposed HNLPIFTSM control strategy when sudden decreases of 1 $N \cdot m$ load disturbance at 4.5 s.

In this subsection, we will further validate the effectiveness and superiority of our proposed HNLPIFTSM control strategy. A PMSM speed control experimental platform based a cSPACE (Control signal process and control engineering) is used in this paper, which is shown in Figure 10. The cSPACE PMSM speed control experimental platform consists of a SM060R20B30M0AD PMSM, a MY1016 DC Generator, a diver experiment box with TI TMS320F28335 DSP and a Matlab/Simulink software. The key parameters of the proposed HNLPIFTSM control strategy in the experiment are given as: $\tilde{α} = 0.25$ , $p = 7$ , $q = 9$ , $η = 1$ . The selected parameters of the ESO in the experiment are given as: $β_{1} = 200$ and $β_{2} = 10000$ .

Figure 10.

The PMSM speed control experimental platform based cSPACE.

When the reference speed is set to the same value, the performances of the traditional HFTSM control strategy and the proposed HNLPIFTSM control strategy are showed in Figure 11. It can be seen from Figure 11 that the experimental demonstrations verify the excellent speed tracking performance of the proposed HNLPIFTSM control strategy.

Figure 11.

(a) The experimental speed response under the traditional HFTSM control strategy and (b) the experimental speed response under the proposed HNLPIFTSM control strategy.

When the reference speed is set to 100 rpm and the external load is changed from no load to load suddenly, the performances of the traditional HFTSM control strategy and the proposed HNLPIFTSM control strategy are showed in Figures 12(a) and 13(a), respectively. When the reference speed is set to 100 rpm and the external load is changed from load to no load suddenly, the performances of the traditional HFTSM control strategy and the proposed HNLPIFTSM control strategy are showed in Figures 12(b) and 13(b), respectively. Such Figures show that the proposed control strategy has the superior kinematics and strong anti-disturbance ability of load disturbance. In summary, the experimental demonstrations verify the excellent speed tracking performance and robustness of the proposed HNLPIFTSM control strategy.

Figure 12.

(a) The experimental speed response under the traditional HFTSM control strategy when the external load is changed from no load to load suddenly and (b) the experimental speed response under the traditional HFTSM control strategy when the external load is changed from load to no load suddenly.

Figure 13.

(a) The experimental speed response under the proposed HNLPIFTSM control strategy when the external load is changed from no load to load suddenly and (b) the experimental speed response under the proposed HNLPIFTSM control strategy when the external load is changed from load to no load suddenly.

Conclusions and future work

In this paper, we propose a novel ESO based the HNLPIFTSM control strategy to obtain high performance of the PMSM speed regulation system. Firstly, a novel HNLPIFTSM surface is designed, which combines the advantages of the nonlinear PI sliding mode surface and the FTSM surface, is proposed to improved the robustness and the dynamic response, while maintaining the good performance of traditional HFTSM control strategy. The proposed HNLPIFTSM surface uses the nonlinear PI sliding mode surface as its inner loop and uses the FTSM surface as its outer loop. Meanwhile, an ESO is used to estimate the uncertain terms of the speed regulation system. Comparative simulations and experimental demonstrations verify the excellent speed tracking performance and robustness of the proposed strategy.

At present, we only use the control strategy in the middle and low speed control of PMSM. In the future work, we will further complete the application of the proposed HNLPIFTSM control strategy in the control of high speed motor.

Footnotes

Handling Editor: James Baldwin

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 research was supported by the National Natural Science Foundation of China under Grant (no. 61703202), by the Natural Science Foundation of Universities of Anhui Province (no. KJ2020A0692) and by the Building Electrical and Intelligent Professional Excellence Engineer Training Innovation Project (no. 2020zyrc153).

ORCID iD

Peng Gao

References

Cao

Jing

CN.

Improved discrete sliding mode control strategy for pulse-width modulation rectier. Turk J Electr Eng Comput Sci 2012; 26: 184–193.

Santiesteban

. Time convergence estimation of a perturbed double integrator: family of continuous sliding mode based output feedback synthesis. In: Proceedings of the 2013 European control conference, Zurich, Switzerland, 17–19 July 2013, pp.3764–3769. New York: IEEE.

Wang

, et al. Adaptive time-delay control for cable-driven manipulators with enhanced nonsingular fast terminal sliding mode. IEEE Trans Ind Electron 2020; 68: 2356–2367.

Wang

, et al. Practical tracking control of robot manipulators with continuous fractional-order nonsingular terminal sliding mode. IEEE Trans Ind Electron 2016; 63: 6194–6204.

Gao

Zhang

Ouyang

, et al. A sliding mode control with nonlinear fractional order PID sliding surface for the speed operation of surface-mounted PMSM drives based on an extended state observer. Math Probl Eng 2019; 2019: 1–13.

Liu

Sun

Zhang

JH.

Adaptive sliding mode control for 4-wheel SBW system with Ackerman geometry. ISA Trans 2020; 96: 103–115.

Oucheriah

Guo

LP.

PWM-based adaptive sliding-mode control for boost DC-DC converters. IEEE Trans Ind Electron 2013; 60: 3291–3294.

Baek

Jin

Han

A new adaptive sliding-mode control scheme for application to robot manipulators. IEEE Trans Ind Electron 2016; 63: 3628–3637.

Jiang

Karimi

Yang

, et al. Observer-based adaptive sliding mode control for nonlinear stochastic Markov jump systems via T-S fuzzy modeling: applications to robot arm model. IEEE Trans Ind Electron 2021; 68: 466–477.

10.

Liu

Karimi

JP.

Passivity-based robust sliding mode synthesis for uncertain delayed stochastic systems via state observer. Automatica 2020; 111: 108596.

11.

Gao

Ouyang

, et al. A novel model-free intelligent proportional-integral super twisting nonlinear fractional-order sliding mode control of PMSM speed regulation system. Complexity 2020; 2020: 1–15.

12.

Gao

Zhang

Model-free hybrid control with intelligent proportional integral and super-twisting sliding mode control of PMSM drives. Electronics 2020; 9: 1427.

13.

Hou

Shen

, et al. Fuzzy sliding mode control for systems with matched and mismatched uncertainties/disturbances based on ENDOB. IEEE Access 2019; 7: 666–673.

14.

Song

Smith

. A comparison of sliding mode fuzzy controller and fuzzy sliding mode controller. In: Proceedings of the 19th international conference of the North American fuzzy information processing society, Atlanta, GA, 13–15 July 2020, pp.480–484. New York: IEEE.

15.

Wang

Rong

Reduced-order extended state observer based event-triggered sliding mode control for DC-DC buck converter system with parameter perturbation. Asian J Control. Epub ahead of print 14 January 2019. DOI: 10.1002/asjc.2301.

16.

Fan

Wang

ZS.

Event-triggered integral sliding mode control for linear systems with disturbance. Syst Control Lett 2020; 138: 104669.

17.

Zhang

Liu

Zhang

, et al. Speed regulation of permanent magnet synchronous motor using event triggered sliding mode control. Math Probl Eng 2018; 2018: 1–11.

18.

Shen

Huang

Wang

Robust backstepping sliding mode controller investigation for a port plate position servo system based on an extended states observer. Asian J Control 2019; 21: 302–311.

19.

Qian

Zhao

Hierarchical sliding mode control for a class of SIMO underactuated system. Control Cybern 2008; 37: 160–175.

20.

Shao

Zheng

Wang

, et al. Recursive sliding mode control with adaptive disturbance observer for a linear motor positioner. Mech Syst Signal Process 2021; 146: 107014.

21.

Wang

Peng

Zhu

, et al. Adaptive PID-fractional-order nonsingular terminal sliding mode control for cable-driven manipulators using time-delay estimation. Int J Syst Sci 2020; 51: 3118–3133.

22.

Man

ZH.

Fast terminal sliding-mode control design for nonlinear dynamical systems. IEEE Trans Circuits Syst I Fundam Theory Appl 2002; 49: 261–264.

23.

Huang

Pan

, et al. Fast terminal sliding mode control of permanent magnet in-wheel motor based on a fuzzy controller. Energies 2020; 13: 188.

24.

Amirkhani

Mobayen

Iliaee

, et al. Fast terminal sliding mode tracking control of nonlinear uncertain mass-spring system with experimental verifications. Int J Adv Robot Syst 2019; 16: 172988141982817.

25.

Singh

Hoang

. Fast terminal sliding mode control for gantry cranes. In: Proceedings of 33rd international symposium on automation and robotics in construction, Auburn, Alabama, 18–21 July 2016, pp.437–443. IAARC.

26.

Singh

QP.

Fast terminal sliding control application for second-order underactuated systems. Int J Control Autom Syst 2019; 17: 1884–1898.

27.

Abolvafaei

Ganjefar

Maximum power extraction from a wind turbine using second-order fast terminal sliding mode control. Renew Energy 2019; 139: 1437–1446.

28.

Liu

Chen

Huang

HX.

Second-order hierarchical fast terminal sliding model control for a class of underactuated systems using disturbance observer. Autom Control Intell Syst 2019; 7: 65–78.

29.

Gao

. Active disturbance rejection control: a paradigm shift in feedback control system design. In: Proceedings of the 2006 American control conference, Minneapolis, MN, USA, 14–16 June 2006, pp.2399–2405. New York: IEEE.

30.

Wang

, et al. LESO-based position synchronization control for networked multi-axis servo systems with time-varying delay. IEEE-CAA J Automatica Sinica 2020; 7: 1116–1123.

31.

Wang

Zhang

A linear active disturbance rejection control approach to position synchronization control for networked interconnected motion system. IEEE Trans Control Netw Syst 2020; 7: 1746–1756.

32.

Luo

Sun

, et al. On decoupling trajectory tracking control of unmanned powered parafoil using ADRC-based coupling analysis and dynamic feedforward compensation. Nonlinear Dyn 2018; 92: 1619–1635.

33.

Yang

Huang

Zhao

, et al. Transonic flutter suppression for a three-dimensional elastic wing via active disturbance rejection control. J Sound Vib 2019; 445: 168–187.

34.

Pillay

Krishnan

Control characteristics and speed controller design for a high performance permanent magnet synchronous motor drive. IEEE Trans Power Electron 1989; 5: 151–159.

35.

Wai

RJ.

Total sliding-mode controller for PM synchronous servo motor drive using recurrent fuzzy neural network. IEEE Trans Ind Electron 2001; 48: 926–944.

36.

Dominguez

Navarrete

Meza

, et al. Digital sliding-mode sensorless control for surface-mounted PMSM. IEEE Trans Ind Electron 2014; 10: 137–151.

37.

Xiao

Jun

HL.

Load disturbance observer-based control method for sensorless PMSM drive. IET Electr Power Appl 2016; 10: 735–743.

38.

Ramırez-Villalobos

Aguilar

Coria

LN.

Sensorless H∞ speed-tracking synthesis for surface-mount permanent magnet synchronous motor. ISA Trans 2017; 67: 140–150.

39.

Gao

Zhou

, et al. Reinforcement learning control of a flexible two-link manipulator: an experimental investigation. IEEE Trans Syst Man Cybern Syst. Epub ahead of print 5 March 2020. DOI: 10.1109/TSMC.2020.2975232.

40.

, et al. Adaptive fuzzy full-state and output-feedback control for uncertain robots with output constraint. IEEE Trans Syst Man Cybern Syst. Epub ahead of print 3 February 2020. DOI: 10.1109/TSMC.2019.2963072.

41.

Fang

Yuan

Liu

Active disturbance rejection control and fractional order proportional integral derivative hybrid control for hydroturbine speed governor system. Meas Control 2018; 51: 192–201.

42.

Liu

A novel sliding mode single-loop speed control method based on disturbance observer for permanent magnet synchronous motor drives. Adv Mech Eng 2018; 10: 168781401881527.

43.

Gao

Zhang

Model-free control using improved smoothing extended state observer and super-twisting nonlinear sliding mode control for PMSM drives. Energies 2021; 14: 922.

44.

Zhang

Wang

Yuan

, et al. Active disturbance rejection control strategy for signal injectionbased sensorless IPMSM drives. IEEE Trans Transp Electrif 2018; 4: 330–339.

45.

Van

Mavrovouniotis

Self-tuning fuzzy PID-nonsingular fast terminal sliding mode control for robust fault tolerant control of robot manipulators. ISA Trans 2020; 96: 60–68.

46.

Van

An enhanced robust fault tolerant control based on an adaptive fuzzy PID-nonsingular fast terminal sliding mode control for uncertain nonlinear systems. IEEE ASME Trans Mechatron 2018; 23: 1362–1371.

A novel hierarchical nonlinear proportional-integral fast terminal sliding mode control for PMSM drives

Abstract

Keywords

Introduction

Problem description

The torque equation of PMSM is first described as follows: 42

Control strategies and stability analysis

Control strategies designed

Linear ESO designed

Comparative simulations and experimental demonstrations

Conclusions and future work

Footnotes

Declaration of conflicting interests

Funding

ORCID iD

References

The torque equation of PMSM is first described as follows:⁴²