Finite-time fuzzy dynamic surface control for permanent magnet synchronous motor stochastic systems with input constraint and load disturbance

Abstract

In this paper, a finite-time adaptive fuzzy dynamic surface control (DSC) method is proposed for the position tracking control of permanent magnet synchronous motors (PMSMs) stochastic nonlinear system with input constraint and load disturbance. First, the stochastic disturbance of PMSMs is considered in operation, and the fuzzy control method is applied to cope with the stochastic nonlinear function in the motor model. Second, the DSC technique is applied to avoid the “explosion of complexity” in the backstepping design. Moreover, the finite-time control is applied to the stochastic nonlinear system of PMSMs to improve the convergence speed of the system, tracking accuracy, and anti-interference ability. Conclusive, simulation results are given to verify the method that can achieve fast tracking of the desired signal.

Keywords

Adaptive fuzzy control dynamic surface control finite-time control stochastic systems

Introduction

Permanent magnet synchronous motors (PMSMs) have been widely used in agriculture and industry because of its low energy, small size, high efficiency, and long service life. However, the PMSMs are typically strong coupling nonlinear system, which is easy to be affected by uncertain factors such as the change of motor parameters and external load disturbance in practical application. To achieve effective control of the nonlinear systems, many control strategies have been studied, for instance, sliding mode control (Barambones and Alkorta, 2014; Song et al., 2018; Yang et al., 2018; Zhu et al., 2020), direct torque control (Lin et al., 2020; Yip et al., 2020), adaptive control (Yu et al., 2020), backstepping control (Tong et al., 2011b; Yu et al., 2017), robust control (Liu et al., 2020; Zhang et al., 2020a), and so on.

The backstepping method is to design the controller step-by-step by introducing the virtual control variable, which has been successfully applied to the PMSMs (Yu et al., 2011; Zhou and Wang, 2002). However, there are still some problems in the traditional backstepping method.

On one hand, what is noteworthy is that these control strategies hardly think about the impact of stochastic disturbances. Stochastic nonlinear system is an important part of nonlinear system and has a wider range of applications. Due to the existence of stochastic interference, the design and stability analysis of the controller are more complex. In the actual application, the system will be subject to stochastic interference, such as stochastic surge of power supply voltage and stochastic fluctuation of motor load. Damping torque and magnetic saturation may cause changes in motor torque, self-inductance, and other related parameters, and these parameter changes will also produce stochastic disturbances (Li et al., 2017a; Liu et al., 2018; Yu et al., 2018). On the other hand, the “explosion of complexity” is caused by the repeated differentiations of virtual input in the process (Cui et al., 2017; Li et al., 2017a). To overcome this issue, the dynamic surface control (DSC) method was proposed by introducing a first-order filtering in every step of the backstepping design process of the actual input (Peng et al., 2016; Wang et al., 2012; Yu et al., 2015). However, the above methods are all asymptotically stable control laws; none of the methods have certain limitations in tracking speed.

For the past few years, more and more attention has been paid to the application of finite-time method. Bhat and Bernstein (1998) and Yu et al. (2020) first proposed the stability theory of nonlinear system based on Lyapunov for finite time. In comparison with the traditional control method, the finite-time method has many advantages, such as high tracking accuracy, fast response, and good anti-interference ability (Li et al., 2019; Song et al., 2020). Yin et al. (2011) applied the finite-time control method to stochastic nonlinear systems for the sake of achieving semi-global finite-time stability practical (SGFSP). Sui et al. (2019) combined the adaptive fuzzy control method with the finite-time control strategy to the stochastic nonlinear systems. It is generally known that the input constraint is a common issue in many practical systems, which seriously affects the control performance of the systems. The saturation of input voltage will make the iron core tend to magnetic saturation, increase the excitation current and no-load current, and increase the loss, resulting in the decrease of motor efficiency, serious heating, and even burning, which seriously affects the normal use of the motor. So, it is necessary to consider the problem of input constraint in the stochastic nonlinear systems of PMSMs.

In view of the above problems, a finite-time DSC technology for the stochastic nonlinear systems of PMSMs with load disturbance and input constraint is proposed. In comparison with the previous methods, the main advantages of the finite-time fuzzy backstepping position tracking control proposed are as follows:

Our method primarily proposes a finite-time DSC control method for the stochastic nonlinear systems of PMSMs with input constraint and load disturbance. By combining adaptive fuzzy control with DSC strategy, the unknown stochastic nonlinear functions are solved, and that the “explosion of complexity” issue is coped with.

In comparison with Sui et al. (2021), our control method introduces the finite-time control method, which is applied to the PMSM stochastic system to improve the response speed, tracking accuracy, and anti-interference ability of the system.

In comparison with Yu et al. (2015), our control method considers the stochastic interference and input constraint during the operation of PMSM system, making the designed control system more accord with the actual engineering requirements.

The remaining of the paper is organized as follows. In the “Mathematical model of PMSMs” section, the mathematical model of PMSM is proposed. In the “Stochastic finite-time adaptive control based on DSC technique for PMSMs” section, stochastic adaptive fuzzy DSC control technique for PMSMs is devised. In the “Stability analysis” section, the stability analysis is given. In the “Simulation experiment” section, a simulation experiment is furnished. Ultimately, the conclusions are presented in the “Conclusion” section.

Mathematical model of PMSMs

Consider the dynamic mathematical model of PMSMs (Qu et al., 2015)

{\begin{matrix} \frac{d θ}{dt} = ω, \\ \frac{d ω}{dt} = \frac{3 n_{p}}{2} [(\frac{L_{d}}{J} - \frac{L_{q}}{J}) i_{d} i_{q} + \frac{Φ i_{q}}{J}] - \frac{B ω}{J} - \frac{T_{L}}{J} \\ L_{d} \frac{d i_{d}}{dt} = - R_{s} i_{d} + n_{p} ω L_{q} i_{q} + u_{d}, \\ L_{q} \frac{d i_{q}}{dt} = - R_{s} i_{q} + n_{p} ω L_{d} i_{d} + u_{q} . \end{matrix}

(1)

where $n_{p}$ is the pole pairs, $T_{L}$ denotes the load torque, $J$ stands for inertia, $ω$ is rotor angular velocity, $θ$ represents rotor position, and $Φ$ denotes rotor flux linkage. $i_{d}$ , $i_{q}$ , $u_{d}$ , and $u_{q}$ denote the $d - q$ axis currents and voltages. $L_{d}$ and $L_{q}$ mean the $d - q$ axis stator inductance, and $R_{s}$ means the stator resistance.

In order to simplify the design process and calculation steps of the above model, the notations are introduced as follows

\begin{matrix} x_{1} = θ, x_{2} = ω, x_{3} = i_{q}, x_{4} = i_{d}, a_{1} = \frac{3 n_{p} Φ}{2}, \\ a_{2} = \frac{3 n_{p} (L_{d} - L_{q})}{2}, b_{1} = - \frac{R_{s}}{L_{q}}, b_{2} = - \frac{n_{p} L_{d}}{L_{q}} \end{matrix}

b_{3} = - \frac{n_{p} Φ}{L_{q}}, b_{4} = \frac{1}{L_{q}}, c_{1} = - \frac{R_{s}}{L_{d}}, c_{2} = \frac{n_{p} L_{q}}{L_{q}}, c_{3} = - \frac{1}{L_{q}}

Then, the model of the PMSM systems can be expressed as the following form

\begin{matrix} {\overset{\cdot}{x}}_{1} = x_{2} \\ {\overset{\cdot}{x}}_{2} = \frac{a_{1}}{J} x_{3} + \frac{a_{2}}{J} x_{3} x_{4} - \frac{B}{J} x_{2} - \frac{T_{L}}{J} \\ {\overset{\cdot}{x}}_{3} = b_{1} x_{3} + b_{2} x_{2} x_{4} + b_{3} x_{2} + b_{4} u_{q} \\ {\overset{\cdot}{x}}_{4} = c_{1} x_{4} + c_{2} x_{2} x_{3} + c_{3} u_{d} \end{matrix}

(2)

Stochastic system block diagram for PMSMs is illustrated in Figure 1.

Lemma 1. As one of the common uncertain systems, the randomness of stochastic systems can cause uncertainties (Chen et al., 2014; Cui et al., 2020; Sui et al., 2020). Consider stochastic systems as

dx = f (x) dt + h (x) dw

(3)

where $w$ is r-dimensional standard Brownian motion, $f (0) = 0$ and $h (0) = 0$ .

Considering the influence of stochastic factors, the stochastic system model of PMSMs is presented as

\begin{matrix} d x_{1} = x_{2} dt + ψ_{1}^{T} dw \\ d x_{2} = (\frac{a_{1}}{J} x_{3} + \frac{a_{2}}{J} x_{3} x_{4} - \frac{B}{J} x_{2} - \frac{T_{L}}{J}) dt + ψ_{2}^{T} dw \\ d x_{3} = (b_{1} x_{3} + b_{2} x_{2} x_{4} + b_{3} x_{2} + b_{4} u_{q}) dt + ψ_{3}^{T} dw \\ d x_{4} = (c_{1} x_{4} + c_{2} x_{2} x_{3} + c_{3} u_{d}) dt + ψ_{4}^{T} dw \end{matrix}

(4)

where $ψ_{1}$ , $ψ_{2}$ , $ψ_{3}$ , and $ψ_{4}$ denote unknown smooth nonlinear functions.

In this paper, the constraint nonlinearity $u$ is described as

u = sat (v) {\begin{matrix} sign (v) u_{\max}, | v (t) | \geq u_{\max} \\ v, | v (t) | \geq u_{\max} \end{matrix}

(5)

where $u_{\max}$ is an unknown parameter, and $v$ denotes the input signal of constraint nonlinearity.

Remark 1. Note that for $| v | = u_{\max}$ , there exists a sharp corner which restricts the application of the backstepping method. To overcome this drawback, a smooth function is introduced to approximate the constraint function and define it as

s (v) = u_{\max} \times \tanh (v / u_{\max}) = u_{\max} \times \frac{e^{v / u_{\max}} - e^{- v / u_{\max}}}{e^{v / u_{\max}} + e^{- v / u_{\max}}}

(6)

Then, $sat (v)$ in equation (5) can be rewritten as

sat (v) = s (v) + d (v)

(7)

where $d (v) = sat (v) - s (v)$ has a bound and

| d (v) | = | sat (v) - s (v) | \leq u_{\max} (1 - \tanh (1)) = D

(8)

From the mean-value theorem, there exists a constant $η$ with $0 < η < 1$ , which satisfies

s (v) = s (v_{0}) + s_{v_{η}} (v - v_{0})

(9)

where $s_{v_{η}} = \frac{\partial s (v)}{\partial v} |_{v = v_{η}}, v_{η} = η v + (1 - η) v_{0}$ . By choosing $v_{0} = 0$ , we have

s (v) = s_{v_{η}} v, u = s_{v_{η}} v + d (v)

(10)

where there exists an unknown positive constant $s_{m}$ , and it satisfies $0 < s_{m} < s_{v_{η}} \leq 1$ . The following definition and useful lemma are needed to facilitate the design.

Definition 1. For any given $V (x) ϵ C^{2}$ , define the differential operator $L$ as (Li et al., 2017)

L V = \frac{\partial V}{\partial x} f + \frac{1}{2} Tr {h^{T} \frac{\partial^{2} V}{\partial x^{2}} h}

(11)

Lemma 2. Suppose that there is a function $V : R^{n} \to R_{+}$ and constants satisfying $a_{0} > 0$ , $b_{0} \geq 0$ , $0 < β < 1$ , class $K_{\infty}$ functions $\underline{k}$ , and $\bar{k}$ , such that (Tong et al., 2011b)

\begin{array}{l} - k (| x |) \leq V (x) \leq \bar{k} (| x |) \\ ℒ V (x) \leq - a_{0} V^{β} (x) + b_{0} \end{array}

(12)

for all $x \in R^{n}$ and $t \geq t_{0}$ .

Design $Ω_{x} = {x | E [V^{β} (x)] \leq (b_{0} / (1 - ϕ_{3}) a_{0})}$ where $\forall 0 < ϕ_{3} < 1$ .

When the time approaches $T_{r} = (1 / ϕ_{3} a_{0} (1 - β)) [E [V^{1 - β} (x_{0})] - (b_{0} / (1 - ϕ_{3}) a_{0})^{1 - β / β}]$ . $Ω_{x}$ is bounded and then $dx = f (x) dt + h (x) dw$ satisfies SGFSP.

Stochastic finite-time adaptive control based on DSC technique for PMSMs

A stochastic finite-time adaptive controller based on DSC technique for PMSM will be proposed based on backstepping.

The tracking error variables are defined as

{\begin{matrix} z_{1} = x_{1} - x_{1 d} \\ z_{2} = x_{2} - α_{1 d} \\ z_{3} = x_{3} - α_{2 d} \\ z_{4} = x_{4} \end{matrix}

(13)

where $x_{1 d}$ is the reference signals, $α_{id} (i = 1, 2)$ are the output signals of the first-order filter based on DSC technique, $α_{i}$ are the input signals of the first-order filter, then $ζ_{i} {\overset{\cdot}{α}}_{id} + α_{id} = α_{i}$ and $α_{id} (0) = α_{i} (0)$ . $ζ_{i}$ are the time constants, and $α_{id} (0)$ and $α_{i} (0)$ are the initial values of $α_{id}$ and $α_{i}$ . Denote $ϑ_{i} = ‖ W_{i} ‖^{2}, (i = 1, 2, 3, 4)$ , ${\hat{ϑ}}_{i}$ is the estimate of $ϑ_{i}$ , ${\tilde{ϑ}}_{i} = ϑ_{i} - {\hat{ϑ}}_{i}$ . $l_{i}, h_{i}, ε_{i}$ are known to be positive design parameters (i = 1,2,3,4).

Denote $y_{1} = α_{1 d} - α_{1}$ , $y_{2} = α_{2 d} - α_{2}$ , then ${\overset{\cdot}{y}}_{i} = {\overset{\cdot}{α}}_{id} - {\overset{\cdot}{α}}_{i} = - \frac{{\overset{\cdot}{α}}_{id} - {\overset{\cdot}{α}}_{i}}{ζ_{i}} - {\overset{\cdot}{α}}_{i} = - \frac{y_{i}}{ζ_{i}} + B_{i} (i = 1, 2)$

According to Wang et al. (2012), $| B_{i} |$ has a maximum $B_{iMax}$ on compact set $| Ω_{i} | (i = 1, 2)$ and $B_{i} \leq B_{iMax}$ ; therefore, $y_{i}^{3} {\overset{\cdot}{y}}_{i} \leq - \frac{y_{i}^{4}}{ζ_{i}} + | B_{iMax} | | y_{i}^{3} | \leq - \frac{y_{i}^{4}}{ζ_{i}} + \frac{3 τ}{4} y_{i}^{4} + \frac{B_{iMax}^{4}}{4 τ} (i = 1, 2)$ , where $τ$ is a constant, $τ > 0$ .

According to universal approximation theorem (Qu et al., 2015), for a given $ε_{i} > 0$ , there exists a fuzzy system $W_{i}^{T} S_{i} (Z)$ such that $f_{i} (Z) = W_{i}^{T} S_{i} (Z) + δ_{i} (Z)$ , $δ_{i} (Z)$ is the approximation error satisfying $| δ_{i} (Z) | \leq ε_{i}$ , and the following inequality can be obtained as

z_{i}^{3} f_{i} (Z) \leq \frac{1}{2 h_{i}^{2}} z_{i}^{6} ‖ W_{i} ‖^{2} S_{i}^{T} (Z) S_{i} (Z) + \frac{1}{2} h_{i}^{2} + \frac{1}{4} ε_{i}^{4} + \frac{3}{4} z_{i}^{4}

(14)

Step 1. Choose Lyapunov function candidate as $V_{1} = \frac{1}{4} z_{1}^{4} + \frac{1}{4} y_{1}^{4} + \frac{1}{2 r_{1}} {\tilde{ϑ}}_{1}^{2}$ . Then

\begin{matrix} L V_{1} = z_{1}^{3} ({\overset{\cdot}{x}}_{1} - {\overset{\cdot}{x}}_{1 d}) + \frac{3}{2} z_{1}^{2} ψ_{1}^{T} ψ_{1} + \frac{1}{r_{1}} {\tilde{ϑ}}_{1} {\tilde{ϑ}}_{1}^{.} + \dot{y_{1}} y_{1}^{3} \\ = z_{1}^{3} (z_{2} + α_{1 d} - {\overset{\cdot}{x}}_{1 d}) + \frac{3}{2} z_{1}^{2} ψ_{1}^{T} ψ_{1} + \frac{1}{r_{1}} {\tilde{ϑ}}_{1} {\tilde{ϑ}}_{1}^{.} + \dot{y_{1}} y_{1}^{3} \end{matrix}

(15)

By using Young’s inequality (Wang et al., 2014), we have

\begin{matrix} z_{1}^{3} z_{2} \leq \frac{3}{4} z_{1}^{4} + \frac{1}{4} z_{2}^{4} \\ \frac{3}{2} z_{1}^{2} ψ_{1}^{T} ψ_{1} \leq \frac{3}{4} l_{1}^{- 2} z_{1}^{4} ‖ ψ_{1} ‖^{4} + \frac{3}{4} l_{1}^{2} \\ y_{1}^{3} {\overset{\cdot}{y}}_{1} \leq - \frac{y_{1}^{4}}{ζ_{1}} + \frac{3 τ}{4} y_{1}^{4} + \frac{B_{1 M}^{4}}{4 τ} \end{matrix}

(16)

Let $f_{1} (Z) = \frac{3}{4} l_{1}^{- 2} z_{1}^{4} ‖ ψ_{1} ‖^{4}$ , then substituting equations (14) and (16) into equation (15) results in

\begin{matrix} L V_{1} \leq z_{1}^{3} (\frac{3}{2} z_{1} + α_{1} - {\overset{\cdot}{x}}_{1 d} + \frac{1}{2 h_{1}^{2}} z_{1}^{3} ‖ W_{1} ‖^{2} S_{1}^{T} (Z) S_{1} (Z)) \\ + z_{1}^{3} (α_{1 d} - α_{1}) \\ + \frac{3}{4} l_{1}^{2} + \frac{1}{2} h_{1}^{2} + \frac{1}{4} ε_{1}^{4} + \frac{1}{4} z_{2}^{4} - \frac{1}{r_{1}} {\tilde{ϑ}}_{1} {\hat{ϑ}}_{1}^{.} - \frac{y_{1}^{4}}{ζ_{1}} + \frac{3 τ}{4} y_{1}^{4} + \frac{B_{1 M}^{4}}{4 τ} \end{matrix}

(17)

where $z_{1}^{3} (α_{1 d} - α_{1}) = z_{1}^{3} y_{1} \leq \frac{3}{4} z_{1}^{4} + \frac{1}{4} y_{1}^{4}$ .

Select virtual control law and adaptive law as

\begin{matrix} α_{1} = - k_{1} z_{1}^{4 β - 3} - \frac{9}{4} z_{1} + {\overset{\cdot}{x}}_{1 d} - \frac{1}{2 h_{1}^{2}} z_{1}^{3} {\hat{ϑ}}_{1} S_{1}^{T} (Z) S_{1} (Z) \\ {\hat{\overset{\cdot}{ϑ}}}_{1} = \frac{r_{1}}{2 h_{1}^{2}} z_{1}^{6} S_{1}^{T} (Z) S_{1} (Z) - m_{1} {\hat{ϑ}}_{1} \end{matrix}

(18)

where $r_{1}$ and $m_{1}$ are positive numbers. Substituting equation (18) into equation (17) gets

\begin{matrix} L V_{1} \leq - k_{1} z_{1}^{4 β} + \frac{1}{4} z_{2}^{4} + \frac{3}{4} l_{1}^{2} + \frac{1}{2} h_{1}^{2} + \frac{1}{4} ε_{1}^{4} + \frac{1}{4} y_{1}^{4} \\ + \frac{m_{1}}{r_{1}} {\tilde{ϑ}}_{1} {\hat{ϑ}}_{1} - \frac{y_{1}^{4}}{ζ_{1}} + \frac{3 τ}{4} y_{1}^{4} + \frac{B_{1 M}^{4}}{4 τ} \end{matrix}

(19)

Step 2. Premeditate the following candidate as $V_{2} = V_{1} + \frac{J}{4} z_{2}^{4} + \frac{1}{4} y_{2}^{4} + \frac{1}{2 r_{2}} {\tilde{ϑ}}_{2}^{2}$ . Then,

\begin{matrix} L V_{2} = L V_{1} + {Jz}_{2}^{3} {\overset{\cdot}{z}}_{2} + y_{2}^{3} {\overset{\cdot}{y}}_{2} + \frac{1}{r_{2}} {\tilde{ϑ}}_{2} {\tilde{ϑ}}_{2}^{.} + \frac{3}{2} z_{2}^{2} ψ_{2}^{T} ψ_{2} \\ = L V_{1} + z_{2}^{3} [a_{1} (z_{3} + α_{2 d}) + a_{2} x_{3} x_{4} - B x_{2} - T_{L} - J {\overset{\cdot}{α}}_{1 d}] \\ + \frac{3}{2} z_{2}^{2} ψ_{2}^{T} ψ_{2} + y_{2}^{3} {\overset{\cdot}{y}}_{2} + \frac{1}{r_{2}} {\tilde{ϑ}}_{2} {\tilde{ϑ}}_{2}^{.} \end{matrix}

(20)

The parameter $T_{L}$ is bounded in practical systems; there exists a constant $d > 0$ that satisfies $| T_{L} | \leq d$ , by invoking Young’s inequality

\begin{matrix} a_{1} z_{2}^{3} z_{3} \leq \frac{3 a_{1}}{4} z_{2}^{4} + \frac{a_{1}}{4} z_{3}^{4} \\ \frac{3}{2} z_{2}^{2} ψ_{2}^{T} ψ_{2} \leq \frac{3}{4} l_{2}^{- 2} z_{2}^{4} ‖ ψ_{2} ‖^{4} + \frac{3}{4} l_{2}^{2} \\ - z_{2}^{3} T_{L} \leq \frac{3}{4} z_{2}^{4} + \frac{1}{4} d^{4} \\ y_{2}^{3} {\overset{\cdot}{y}}_{2} \leq - \frac{y_{2}^{4}}{ζ_{2}} + \frac{3 τ}{4} y_{2}^{4} + \frac{B_{2 M}^{4}}{4 τ} \end{matrix}

(21)

Let $f_{2} (Z) = \frac{3}{4} l_{2}^{- 2} z_{2}^{4} ‖ ψ_{2} ‖^{4} + a_{2} x_{3} x_{4} - J {\overset{\cdot}{α}}_{1 d} - B x_{2}$ , then substituting equations (14) and (21) into equation (20) results in

\begin{matrix} L V_{2} \leq - k_{1} z_{1}^{4 β} + a_{1} z_{2}^{3} (α_{2 d} - α_{2}) \\ + \sum_{i = 1}^{2} (\frac{3}{4} l_{i}^{2} + \frac{1}{2} h_{i}^{2} + \frac{1}{4} ε_{i}^{4} - \frac{y_{i}^{4}}{ζ_{i}} + \frac{3 τ}{4} y_{i}^{4} + \frac{B_{iM}^{4}}{4 τ}) \\ - \frac{1}{r_{2}} {\tilde{ϑ}}_{2} {\hat{ϑ}}_{2}^{.} + z_{2}^{3} (\frac{7}{4} + \frac{3 a_{1}}{4}) z_{2} + \frac{1}{2 h_{2}^{2}} z_{2}^{3} ‖ W_{2} ‖^{2} S_{2}^{T} (Z) S_{2} (Z) + a_{1} α_{2} \\ + \frac{a_{1}}{4} z_{3}^{4} + \frac{1}{4} d^{4} + \frac{m_{1}}{r_{1}} {\tilde{ϑ}}_{1} {\hat{ϑ}}_{1} \end{matrix}

(22)

where $a_{1} z_{2}^{3} (α_{2 d} - α_{2}) = a_{1} z_{2}^{3} y_{2} \leq \frac{3 a_{1}}{4} z_{2}^{4} + \frac{a_{1}}{4} y_{2}^{4}$ .

Construct virtual control law and adaptive law as

\begin{matrix} α_{2} = \frac{1}{a_{1}} [- k_{2} z_{2}^{4 β - 3} - \frac{6 a_{1} + 7}{4} z_{2} - \frac{1}{2 h_{2}^{2}} z_{2}^{3} {\hat{ϑ}}_{2} S_{2}^{T} (Z) S_{2} (Z)] \\ {\hat{\overset{\cdot}{ϑ}}}_{2} = \frac{r_{2}}{2 h_{2}^{2}} z_{2}^{6} S_{2}^{T} (Z) S_{2} (Z) - m_{2} {\hat{ϑ}}_{2} \end{matrix}

(23)

where $r_{2}$ and $m_{2}$ are positive numbers. Combining equations (22) and (23) gives

\begin{matrix} L V_{2} \leq \sum_{i = 1}^{2} (- k_{i} z_{i}^{4 β} + \frac{m_{i}}{r_{i}} {\tilde{ϑ}}_{i} {\hat{ϑ}}_{i} + \frac{3}{4} l_{i}^{2} + \frac{1}{2} h_{i}^{2} + \frac{1}{4} ε_{i}^{4} - \frac{y_{i}^{4}}{ζ_{i}} + \frac{3 τ}{4} y_{i}^{4} + \frac{B_{iM}^{4}}{4 τ}) \\ + \frac{a_{1}}{4} z_{3}^{4} + \frac{1}{4} y_{1}^{4} + \frac{a_{1}}{4} y_{2}^{4} + \frac{1}{4} d^{4} \end{matrix}

(24)

Step 3. Choose Lyapunov function candidate as $V_{3} = V_{2} + \frac{1}{4} z_{3}^{4} + \frac{1}{2 r_{3}} {\tilde{ϑ}}_{3}^{2}$ . Then

\begin{matrix} L V_{3} = L V_{2} + z_{3}^{3} {\overset{\cdot}{z}}_{3} + \frac{1}{r_{3}} {\tilde{ϑ}}_{3} {\tilde{ϑ}}_{3}^{.} + \frac{3}{2} z_{3}^{2} ψ_{3}^{T} ψ_{3} \\ = L V_{2} + z_{3}^{3} (b_{1} x_{3} + b_{2} x_{2} x_{4} + b_{3} x_{2} + b_{4} (s_{v_{q η}} v_{q} + d (v_{q})) - {\overset{\cdot}{α}}_{2 d}) \\ + \frac{3}{2} z_{3}^{2} ψ_{3}^{T} ψ_{3} + \frac{1}{r_{3}} {\tilde{ϑ}}_{3} {\tilde{ϑ}}_{3}^{.} \end{matrix}

(25)

By invoking Young’s inequality, we can obtain

\begin{matrix} z_{3}^{3} b_{4} d (v_{q}) \leq \frac{3 b_{4}}{4} z_{3}^{4} + \frac{b_{4}}{4} D_{q}^{4} \\ \frac{3}{2} z_{3}^{2} ψ_{3}^{T} ψ_{3} \leq \frac{3}{4} l_{3}^{- 2} z_{3}^{4} ‖ ψ_{3} ‖^{4} + \frac{3}{4} l_{3}^{2} \end{matrix}

(26)

Let $f_{3} (Z) = b_{1} x_{3} + b_{2} x_{2} x_{4} + b_{3} x_{2} + \frac{3}{4} l_{3}^{- 2} z_{3} ‖ ψ_{3} ‖^{4} - {\overset{\cdot}{α}}_{2 d}$ , then substituting equations (14) and (26) into equation (25) results in

\begin{matrix} L V_{3} \leq \sum_{i = 1}^{2} (- k_{i} z_{i}^{4 β} - \frac{y_{i}^{4}}{ζ_{i}} + \frac{3 τ}{4} y_{i}^{4} + \frac{B_{iM}^{4}}{4 τ} + \frac{m_{i}}{r_{i}} {\tilde{ϑ}}_{i} {\hat{ϑ}}_{i}) \\ + \sum_{i = 1}^{3} (\frac{3}{4} l_{i}^{2} + \frac{1}{2} h_{i}^{2} + \frac{1}{4} ε_{i}^{4}) \\ + z_{3}^{3} (\frac{a_{1} + 3 + 3 b_{4}}{4} z_{3} + \frac{1}{2 h_{3}^{2}} z_{3}^{3} ϑ_{3}^{2} S_{3}^{T} (Z) S_{3} (Z) + b_{4} s_{v_{q η}} v_{q}) \\ + \frac{1}{4} y_{1}^{4} + \frac{a_{1}}{4} y_{2}^{4} - \frac{1}{r_{3}} {\tilde{ϑ}}_{3} {\hat{ϑ}}_{3}^{.} + \frac{1}{4} d^{4} + \frac{1}{4} D_{q}^{4} \end{matrix}

(27)

There exists a constant $b_{5} > 0$ that satisfies $b_{5} \leq b_{q} \leq b_{4} s_{v_{q η}}$ ; the control law $v_{q}$ is designed as

v_{q} = \frac{1}{b_{5}} (- k_{3} z_{3}^{4 β - 3} - \frac{1}{2 h_{3}^{2}} z_{3}^{3} ϑ_{3}^{2} S_{3}^{T} (Z) S_{3} (Z) - \frac{a_{1} + 3 + 3 b_{4}}{4} z_{3})

(28)

Then, we have

\begin{matrix} b_{4} s_{v_{d η}} v_{q} \leq b_{q} v_{q} \leq b_{5} v_{q} = - k_{3} z_{3}^{4 β - 3} - \frac{1}{2 h_{3}^{2}} z_{3}^{3} ϑ_{3}^{2} S_{3}^{T} (Z) S_{3} (Z) \\ - \frac{a_{1} + 3 + 3 b_{4}}{4} z_{3} \end{matrix}

(29)

Selecting adaptive law as

{\hat{\overset{\cdot}{ϑ}}}_{3} = \frac{r_{3}}{2 h_{3}^{2}} z_{3}^{6} S_{3}^{T} (Z) S_{3} (Z) - m_{3} {\hat{ϑ}}_{3}

(30)

where $r_{3}$ and $m_{3}$ are positive numbers. Substituting equations (29) and (30) into equation (27) gives

\begin{matrix} L V_{3} \leq \sum_{i = 1}^{3} (- k_{i} z_{i}^{4 β} + \frac{m_{i}}{r_{i}} {\tilde{ϑ}}_{i} {\hat{ϑ}}_{i} + \frac{3}{4} l_{i}^{2} + \frac{1}{2} h_{i}^{2} + \frac{1}{4} ε_{i}^{4}) \\ + \sum_{i = 1}^{2} (- \frac{y_{i}^{4}}{ζ_{i}} + \frac{3 τ}{4} y_{i}^{4} + \frac{B_{iM}^{4}}{4 τ}) + \frac{1}{4} d^{4} + \frac{1}{4} D_{q}^{4} + \frac{1}{4} y_{1}^{4} + \frac{a_{1}}{4} y_{2}^{4} \end{matrix}

(31)

Step 4. Choose a Lyapunov function candidate as $V_{4} = V_{3} + \frac{1}{4} z_{4}^{4} + \frac{1}{2 r_{4}} {\tilde{ϑ}}_{4}^{2}$ . Then

\begin{matrix} L V_{4} = L V_{3} + z_{4}^{3} {\overset{\cdot}{z}}_{4} + \frac{1}{r_{4}} {\tilde{ϑ}}_{4} {\tilde{ϑ}}_{4}^{.} + \frac{3}{2} z_{4}^{2} ψ_{4}^{T} ψ_{4} \\ = L V_{3} + z_{4}^{3} (c_{1} x_{4} + c_{2} x_{2} x_{3} + c_{3} (s_{v_{d η}} v_{d} + d (v_{d}))) \\ + \frac{3}{2} z_{4}^{2} ψ_{4}^{T} ψ_{4} + \frac{1}{r_{4}} {\tilde{ϑ}}_{4} {\tilde{ϑ}}_{4}^{.} \end{matrix}

(32)

According to Young’s inequality

\begin{matrix} z_{4}^{3} c_{3} d (v_{d}) \leq \frac{3 c_{3}}{4} z_{4}^{4} + \frac{c_{3}}{4} D_{d}^{4} \\ \frac{3}{2} z_{4}^{2} ψ_{4}^{T} ψ_{4} \leq \frac{3}{4} l_{4}^{- 2} z_{4}^{4} ‖ ψ_{4} ‖^{4} + \frac{3}{4} l_{4}^{2} \end{matrix}

(33)

Let $f_{4} (Z) = c_{1} x_{4} + c_{2} x_{2} x_{3} + \frac{3}{4} l_{4}^{- 2} z_{4} ‖ ψ_{4} ‖^{4}$ , then substituting equations (14) and (33) into equation (32) results in

\begin{matrix} L V_{4} \leq - \sum_{i = 1}^{3} (k_{i} z_{i}^{4 β} + \frac{m_{i}}{r_{i}} {\tilde{ϑ}}_{i} {\hat{ϑ}}_{i}) + \sum_{i = 1}^{2} (- \frac{y_{i}^{4}}{ζ_{i}} + \frac{3 τ}{4} y_{i}^{4} + \frac{B_{iM}^{4}}{4 τ}) \\ - \frac{1}{r_{4}} {\tilde{ϑ}}_{4} {\hat{ϑ}}_{4}^{.} \\ + \sum_{i = 1}^{4} (\frac{3}{4} l_{i}^{2} + \frac{1}{2} h_{i}^{2} + \frac{1}{4} ε_{i}^{4}) + \frac{1}{4} d^{4} + \frac{1}{4} D_{q}^{4} + \frac{1}{4} y_{1}^{4} + \frac{a_{1}}{4} y_{2}^{4} \\ + z_{4}^{3} (\frac{1}{2 h_{4}^{2}} z_{4}^{3} ‖ W_{4} ‖^{2} S_{4}^{T} (Z) S_{4} (Z) + \frac{3 b_{4} + 3}{4} z_{4} + c_{3} s_{v_{d η}} v_{d}) \end{matrix}

(34)

There exists a constant $b_{6} > 0$ that satisfies $b_{6} \leq b_{d} \leq c_{3} s_{v_{d η}}$ ; the control law $v_{d}$ is designed as

v_{d} = \frac{1}{b_{6}} (- k_{4} z_{4}^{4 β - 3} - \frac{1}{2 h_{4}^{2}} z_{4}^{3} ϑ_{4}^{2} S_{4}^{T} (Z) S_{4} (Z) - \frac{3 + 3 b_{4}}{4} z_{4})

(35)

Then we have

\begin{matrix} c_{3} s_{v_{d η}} v_{d} \leq b_{d} v_{d} \leq b_{6} v_{d} = - k_{4} z_{4}^{4 β - 3} - \frac{1}{2 h_{4}^{2}} z_{4}^{3} ϑ_{4}^{2} S_{4}^{T} (Z) S_{4} (Z) \\ - \frac{(3 + 3 b_{4})}{4} z_{4} \end{matrix}

(36)

Selecting adaptive law as

{\hat{\overset{\cdot}{ϑ}}}_{4} = \frac{r_{4}}{2 h_{4}^{2}} z_{4}^{6} S_{4}^{T} (Z) S_{4} (Z) - m_{4} {\hat{ϑ}}_{4}

(37)

where $r_{4}$ and $m_{4}$ are positive numbers. Substituting equations (36) and (37) into equation (34) gives

\begin{matrix} L V_{4} \leq \sum_{i = 1}^{4} (- k_{i} z_{i}^{4 β} + \frac{3}{4} l_{i}^{2} + \frac{1}{2} h_{i}^{2} + \frac{1}{4} ε_{i}^{4} + \frac{m_{i}}{r_{i}} {\tilde{ϑ}}_{i} {\hat{ϑ}}_{i}) \\ - (\frac{1}{ζ_{1}} - \frac{3 τ}{4} - \frac{1}{4}) y_{1}^{4} \\ - (\frac{1}{ζ_{2}} - \frac{3 τ}{4} - \frac{a_{1}}{4}) y_{2}^{4} + \frac{B_{1 M}^{4}}{4 τ} + \frac{B_{2 M}^{4}}{4 τ} + \frac{d^{4}}{4} + \frac{b_{4}}{4} D^{4} + \frac{c_{3}}{4} D^{4} \end{matrix}

(38)

Stability analysis

Premeditate the following candidate as $V = V_{4}$ . Then

\begin{matrix} L V \leq \sum_{i = 1}^{4} (- k_{i} z_{i}^{4 β} + \frac{3}{4} l_{i}^{2} + \frac{1}{2} h_{i}^{2} + \frac{1}{4} ε_{i}^{4} + \frac{m_{i}}{r_{i}} {\tilde{ϑ}}_{i} {\hat{ϑ}}_{i}) \\ - (\frac{1}{ζ_{1}} - \frac{3 τ}{4} - \frac{1}{4}) y_{1}^{4} \\ - (\frac{1}{ζ_{2}} - \frac{3 τ}{4} - \frac{a_{1}}{4}) y_{2}^{4} + \frac{B_{1 M}^{4}}{4 τ} + \frac{B_{2 M}^{4}}{4 τ} + \frac{d^{4}}{4} + \frac{b_{4}}{4} D^{4} + \frac{c_{3}}{4} D^{4} \end{matrix}

(39)

Similarly, by Young’s inequality

- {\tilde{ϑ}}_{i} {\hat{ϑ}}_{i} \leq - \frac{1}{2} {\tilde{ϑ}}_{i}^{2} + \frac{1}{2} ϑ_{i}^{2}

(40)

Substituting equation (40) into equation (39) holds

\begin{matrix} L V \leq \sum_{i = 1}^{4} (- k_{i} z_{i}^{4 β} + \frac{3}{4} l_{i}^{2} + \frac{1}{2} h_{i}^{2} + \frac{1}{4} ε_{i}^{4} + \frac{m_{i}}{r_{i}} {\tilde{ϑ}}_{i} {\hat{ϑ}}_{i}) - (\frac{1}{ζ_{1}} - \frac{3 τ}{4} - \frac{1}{4}) y_{1}^{4} \\ - (\frac{1}{ζ_{2}} - \frac{3 τ}{4} - \frac{a_{1}}{4}) y_{2}^{4} + \frac{B_{1 M}^{4}}{4 τ} + \frac{B_{2 M}^{4}}{4 τ} + \frac{d^{4}}{4} + \frac{b_{4}}{4} D^{4} + \frac{c_{3}}{4} D^{4} \\ \leq \sum_{i = 1}^{4} (- 4^{β} k_{i} {(\frac{z_{i}^{4}}{4})}^{β} + \frac{3}{4} l_{i}^{2} + \frac{1}{2} h_{i}^{2} + \frac{1}{4} ε_{i}^{4} - \frac{m_{i}}{2 r_{i}} {\tilde{ϑ}}_{i}^{2} + \frac{m_{i}}{2 r_{i}} ϑ_{i}^{2}) \\ - (\frac{1}{ζ_{1}} - \frac{3 τ}{4} - \frac{1}{4}) y_{1}^{4} \\ - (\frac{1}{ζ_{2}} - \frac{3 τ}{4} - \frac{a_{1}}{4}) y_{2}^{4} + \frac{B_{1 M}^{4}}{4 τ} + \frac{B_{2 M}^{4}}{4 τ} + \frac{d^{4}}{4} + \frac{b_{4}}{4} D^{4} + \frac{c_{3}}{4} D^{4} \\ \leq - a_{0} V^{β} + b_{0} \end{matrix}

(41)

where $a_{0} = \min {4^{β} k_{1}, \frac{4^{β}}{J} k_{2}, 4^{β} k_{3}, 4^{β} k_{4}, m_{i}, 4 (\frac{1}{ζ_{2}} - \frac{3 τ}{4} - \frac{1}{4}), 4 (\frac{1}{ζ_{1}} - \frac{3 τ}{4} - \frac{a_{1}}{4})}$ , $i = 1, 2, 3, 4, k_{1} = k_{1}, k_{2} = k_{2}, k_{3} = k_{3} b_{q}, k_{4} = k_{4} b_{d}$

\begin{matrix} b_{0} = \sum_{i = 1}^{2} (\frac{B_{i M}^{4}}{4 τ}) + \sum_{i = 1}^{4} (\frac{3}{4} l_{i}^{2} + \frac{1}{2} h_{i}^{2} + \frac{1}{4} ε_{i}^{4}) \\ + \sum_{i = 1}^{4} (\frac{m_{i}}{2 r_{i}} ϑ_{i}^{2}) + \frac{d^{4}}{4} + \frac{b_{4}}{4} D^{4} + \frac{c_{3}}{4} D^{4} \end{matrix}

By Lemma 1, $E [V^{β} (x)] \leq \frac{b_{0}}{(1 - ϕ_{3}) a_{0}}$ ; when $\forall t > T_{r}$ , all signals are bounded in closed-loop systems and there exists

E [{(\frac{1}{4} | z_{i} |^{4})}^{β}] \leq \frac{b_{0}}{(1 - ϕ_{3}) a_{0}}

(42)

From equations (41) and (12), it follows that

L V \leq - a_{0} V^{β} + b_{0}

(43)

Remark 2. From the above analysis, the tracking error can converge to a small enough neighborhood of the origin under the control law designed in this paper by choosing the appropriate $a_{0}$ and $b_{0}$ . It can be concluded from equations (41) and (42) that $b_{0} / ((1 - ϕ_{3}) a_{0})$ will be sufficiently small by selecting large enough $r_{i}$ and sufficiently small $ε_{i}, h_{i}, l_{i}$ .

Simulation experiment

To test the effectiveness and feasibility of the proposed method, we have compared the finite-time control method with the traditional backstepping method; simulation and comparison experiments were carried out. Our parameters of the PMSMs are selected as

\begin{matrix} R_{s} = 0.68 Ω, L_{d} = L_{q} = 0.00977 H, J = 0.002 kg \cdot m^{2}, \\ B = 0.001158 N \cdot m / (rad / s), Φ = 0.1245 H, n_{p} = 3 \end{matrix}

The design parameters of two control methods in the controllers are selected as

\begin{matrix} k_{1} = 50, k_{2} = 15, k_{3} = 6000, k_{4} = 700 \\ r_{1} = r_{2} = r_{3} = r_{4} = 0.02 \\ m_{1} = m_{2} = m_{3} = m_{4} = 0.05 \\ h_{1} = h_{2} = h_{3} = h_{4} = 2 \end{matrix}

Choosing the reference signals as

x_{1 d} = 0.5 \sin (t) + 0.3 \sin (0.5 t) . T_{L} = {\begin{matrix} 1.0, 0 \leq t \leq 15 \\ 1.5, t > 15 \end{matrix}

Figures 2 –5 reveal the simulation results. Figures 2(a)–5(a) display the proposed finite-time control method, and Figures 2(b)–5(b) show the traditional backstepping control method. Figure 2(a) and (b) shows the angular position tracking signal $x_{1}$ and the reference signal $x_{1 d}$ . The tracking errors are displayed in Figure 3(a) and (b). Figure 4(a) and 4(b) display the q-axis voltage $u_{q}$ , and the d-axis voltage $u_{d}$ is shown in Figure 5(a) and (b).

Figures 2 and 3 show that the controller designed in this paper can make the system track the desired signal quickly and ensure that the tracking error is small enough. When $t = 15 s$ , the external load torque $T_{L}$ changes from 1.0 to 1.5. From Figures 4 and 5, it can be seen that the inputs $u_{q}$ and $u_{d}$ of the controller are stable in a bounded area. In Figure 4, the small fluctuation of $u_{q}$ is caused by the change of external load.

Remark 3. From Figure 2(a) and (b), it turns out that the two approaches can track the given reference signal. In addition, the proposed method avoids the repeated derivation of the virtual control signal in the process and reduces the calculation of the control law.

Conclusion

The position tracking controller based on the finite-time control and adaptive fuzzy backstepping of DSC has been designed for the stochastic nonlinear system of PMSMs with load disturbance and input constraint. The matter of “explosion of complexity” is solved and the stochastic nonlinear control of PMSMs is realized. The simulation results that testify the control strategy described in this paper can overcome the effect of input constraint and stochastic disturbance, and make sure that the control strategy can follow the track of expected signal more effectively. In the future work, we will add the application of command filtering technology to stochastic systems of PMSMs, which will reduce the filtering error caused by the DSC.

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 in part by the National Natural Science Foundation of China under Grant Number 61973179, in part by the Taishan Scholar Special Project Found under Grant Number TSQN20161026 and Qingdao Key Research and Development Special Project (21-1-2-6-nsh).

ORCID iDs

Sijia Song

Jinpeng Yu

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