Sliding mode control of the speed loop for PMSM based on variable parameter reaching law and adaptive observer

Abstract

To improve the speed control performance of the permanent magnet synchronous motor (PMSM) drive system, a sliding mode control strategy based on variable parameter reaching law (VPRL) and adaptive disturbance observer is proposed. The VPRL introduces state variables into the conventional exponential reaching law (CERL) and designs a power term function related to the arctangent function. In response to the issue of poor performance in traditional Luenberger disturbance observers (LDOB), by introducing an adaptive function that represents the relationship between system state variables and parameters into the traditional LDOB, an adaptive Luenberger disturbance observer (ALDOB) is proposed to estimate and compensate for the total disturbances in the system. Simulations and experiments demonstrate that the proposed sliding mode control strategy significantly improves both the dynamic performance and robustness of the PMSM.

Keywords

Permanent magnet synchronous motor (PMSM)new reaching law sliding mode disturbance observer robustness

Introduction

Permanent magnet synchronous motors (PMSMs) have gained widespread application in high-precision servo control systems, including robotics, CNC machine tools, and large-aperture telescopes, owing to their superior characteristics of high dynamic performance, high power density, high torque-to-current ratio, and high efficiency (Junejo et al., 2018; Lee et al., 2020). Consequently, research on high-precision speed control strategies for PMSMs holds significant importance and value. At present, the PMSM usually adopts the proportional integral PI (proportional integral) control to regulate the speed, which is a simple algorithm and easy to adjust the parameters. However, PMSM represents a complex system characterized by multiple variables, strong coupling, and nonlinear dynamics. While traditional PI controllers are sufficient for basic control needs within a limited scope, they struggle to deliver high-performance control when confronted with variations in the motor’s internal parameters or external disturbances to the system. This limitation has driven extensive investigations into modern control theory-based speed control strategies (Deng et al., 2020; Liu et al., 2024a; Zou et al., 2023). Notably, sliding mode control (SMC) has been widely implemented in PMSM drive systems due to its insensitivity to system model accuracy and strong robustness against external disturbances, effectively enhancing system anti-interference capabilities (Komurcugil et al., 2022; Mu and He, 2017).

In practical applications, conventional SMC methods exhibit two critical limitations: slow convergence rate of the reaching law and high-frequency chattering induced by time-delayed switching control laws (Zuo et al., 2023), both of which degrade system dynamic response performance. To address these shortcomings, numerous scholars have proposed alternative approaches to improve traditional reaching laws. Feng and Zhang improved upon the exponential reaching law, effectively suppressing the inherent chattering of sliding modes and accelerating the convergence rate toward the sliding manifold (Feng et al., 2024; Zhang et al., 2011). Similarly, Mishra et al. enhanced the traditional reaching law through the addition of terminal attractors to ensure rapid convergence near the selected sliding surface. However, the introduction of terminal attractors paradoxically reduces the approaching speed prior to reaching the sliding surface compared with linear reaching laws (Mishra et al., 2016). Sun implemented discrete-time fractional-order terminal sliding mode variable structure in speed controllers, demonstrating improved dynamic performance of controlled systems, albeit with increased theoretical complexity (Sun et al., 2017). Bodur proposed a hybrid reaching law combining piecewise functions with super-twisting algorithms to mitigate chattering while maintaining fast convergence, though at the expense of increased algorithmic complexity (Bodur and Kaplan, 2023). Bartoszewicz developed a novel composite reaching law through exponential and power-law piecewise functions, proving its capability to achieve sliding surface arrival time independence from initial system states while shortening convergence time and smoothing output signals (Bartoszewicz and Latosiński, 2018).

Traditional sliding mode variable structure control typically employs sufficiently large switching gains to ensure system stability and suppress disturbances (Liu et al., 2024b). However, such large switching gains inevitably exacerbate high-frequency chattering, and the inherent difficulty in determining disturbance upper bounds significantly compromises control performance. Observer-based approaches offer promising solutions to this challenge. Yim et al. (2022) proposed a disturbance observer that reduces unnecessary high-frequency components in system estimation, effectively mitigating sliding mode chattering. The Luenberger disturbance observer (LDOB), widely adopted in disturbance observation, demonstrates particular advantages in this domain. Guezmil et al. (2015) implemented a combined SMC-LDOB approach under various disturbance and uncertainty conditions, achieving enhanced online estimation of motor speed and position accuracy. Wang addressed external disturbances and parameter mismatches in surface-mounted PMSMs by developing an extended model that employs LDOB for estimating lumped disturbances in both speed and current loops, subsequently compensating these disturbances through speed controllers (He et al., 2019). Nevertheless, the fixed bandwidth constraints of LDOBs in these studies result in limited environmental adaptability and suboptimal observation performance.

This paper designs a novel reaching law (variable parameter reaching law, VPRL). It adjusts the switching gain adaptively based on the distance from the system to the sliding mode surface, addressing the conflict between system response time and chattering. To further enhance the disturbance rejection capability, an adaptive disturbance observer (adaptive Luenberger disturbance observer, ALDOB) is also designed. This observer introduces an adaptive function that represents the relationship between the system state variable and the parameter into the traditional LDOB. The bandwidth of ALDOB can change adaptively with the system speed error. While estimating and compensating for the system’s lumped disturbances, it does not affect the system’s steady-state performance, further enhancing the system’s anti-interference capability and speed tracking performance. Simulation and experimental validation have confirmed the effectiveness of the VPRLSMC + ALDOB control method, which significantly enhances the robustness and dynamic response performance of the PMSM servo system.

Sliding mode reaching law

Conventional exponential reaching law and its analysis

The proposal for conventional exponential reaching law (CERL), as presented by the distinguished authority Gao Weibing, is shown in equation (1)

\frac{ds}{dt} = - ε sign (s) - ks, s > 0, k > 0

(1)

where s is a sliding mode surface function; $ε sign (s)$ stands for isokinetic reach term; ks stands for index reach term.

In equation (1), when $s > 0$

\frac{ds}{dt} = - ε - ks

(2)

The time to reach the sliding mode surface is

t = \frac{1}{k} (\ln (s (0) + \frac{ε}{k}) - \ln \frac{ε}{k})

(3)

The CERL is characterized by fast convergence, a simple structure, and ease of implementation. However, as the system nears the sliding surface, this method tends to introduce high-frequency chattering. Leading to frequent switching of the control input and degrading system performance. Moreover, the exponential reaching law is highly sensitive to parameter selection, requiring careful tuning of the switching gain and coefficient of variation; otherwise, it may result in performance degradation or system instability. Consequently, the selection of the $k, ε$ value, along with the chattering issue, has driven researchers to further explore improved approaches.

According to the CERL, when the value of $| s |$ is relatively large, a larger ε value and a smaller k value can accelerate the reaching speed. As $| s | \to 0$ , selecting a smaller ε and a larger k can still maintain a certain reaching speed while reducing chattering as the system reaches the sliding surface. This also provides us with a new train of thought.

Proposal of a variable parameters reaching law

The novel VPRL

{\begin{matrix} \frac{ds}{dt} = k_{1} H (x) {| s |}^{s \cdot \arctan (\frac{π}{2} \cdot s)} sign (s) - k_{2} {| x |}^{α} \cdot s \\ H (x) = \frac{| x |}{| x | + ε} \\ \lim_{t \to \infty} | x | = 0 \end{matrix}

(4)

where x is the system state variables; $k_{1} > 0, k_{2} > 0, ε > 0, 0 < α < 1$ .

The VPRL introduces functions $H (x)$ and ${| x |}^{α}$ related to the state variables $| x |$ in both isokinetic reach term and index reach term of CERL. As the state variables $| x |$ approaches zero, function $H (x)$ gradually decreases and ultimately tends toward zero, ensuring that throughout the entire process, $k_{1} H (x)$ remains less than $k_{1}$ . This implies that the variable gain is consistently smaller than the original switching gain, thus effectively suppressing sliding mode chatter. In addition, The introduction of function ${| x |}^{α}$ in the index reach term relates the approach speed of the reaching law to the state variables $| x |$ , significantly accelerating the convergence speed. In addition, the convergence speed can be adjusted based on the value of α. Ultimately, as the state variable $| x |$ decreases to zero, the convergence speed also approaches zero.

For the power function: $s \cdot \arctan (\frac{π}{2} \cdot s)$

Since the domain of definition of arctan functions are all real numbers and monotonically increasing in the interval $(- \infty, + \infty)$ . So the power function $\arctan (\frac{π}{2} \cdot s)$ is monotonically increasing in the $(- \infty, + \infty)$ interval.

When $s > 0$ , $\arctan (\frac{π}{2} \cdot s) > 0$ , and when $s < 0$ , $\arctan (\frac{π}{2} \cdot s) < 0$ , so the power function $s \cdot \arctan (\frac{π}{2} \cdot s) \geq 0$ .

The power function can adaptively adjust based on the different sliding mode state.

Performance analysis of new reaching law control

The system shown in equation (5) is established, and the performance of CERL and the VPRL are compared and analyzed

\overset{\cdot\cdot}{θ} (t) = - \frac{k}{m} θ^{3} + \frac{1}{m} u

(5)

where $θ (t)$ denotes the position signal; $u (t)$ denotes the control input.

The tracking errors $e (t)$ and $\overset{\cdot}{e} (t)$ are

{\begin{matrix} e (t) = θ_{d} (t) - θ (t) \\ \overset{\cdot}{e} (t) = {\overset{\cdot}{θ}}_{d} (t) - \overset{\cdot}{θ} (t) \end{matrix}

(6)

where $θ_{d} (t)$ denotes the target signal.

Define the sliding mode function as

s (t) = ce (t) + \overset{\cdot}{e} (t)

(7)

where the sliding mode surface coefficient c is a positive constant.

This can be obtained from equations (5)–(7)

\begin{matrix} \overset{\cdot}{s} (t) = c \overset{\cdot}{e} (t) + \overset{\cdot\cdot}{e} (t) \\ = c ({\overset{\cdot}{θ}}_{d} (t) - \overset{\cdot}{θ} (t)) + ({\overset{\cdot\cdot}{θ}}_{d} (t) + \frac{k}{m} θ^{3} - \frac{1}{m} u) \end{matrix}

(8)

Using VPRL, the improved SMC law is obtained from equations (3) and (8) as

\begin{matrix} u (t) = m [k_{1} H (x) {| s |}^{s \cdot \arctan (\frac{π}{2} \cdot s)} sign (s) + k_{2} {| x |}^{α} \cdot s + \\ c ({\overset{\cdot}{θ}}_{d} (t) - \overset{\cdot}{θ} (t)) + ({\overset{\cdot\cdot}{θ}}_{d} (t) + \frac{k}{m} θ^{3})] \end{matrix}

(9)

Simulation comparison of VPRL and CERL control is compared under the optimal parameter conditions, with the simulation parameters of $c = 15, ε = 2000, k_{1} = 1, k_{2} = 100, α = 0.5$ , the target position signal of the system $θ_{d} (t) = 5 \sin (\frac{1}{10} π t)$ , and the original state variables $x (0)$ as $[x_{1}, x_{2}] = [2, 0]$ . Figures 1 –3 show the comparison of the control performance between CERL and VPRL. From Figures 1 –3, it can be seen that the VPRL is excellently than the CERL in tracking the performance of a target signal, convergence speed and chatter in controller output. The controller output curve shows that the output signal u(t) chatter amplitude of VPRLSMC + ALDOB is reduced by about 60% compared with that of CERL, and the high-frequency component is significantly reduced. This improvement is attributed to the state-variable-based power function design in VPRL, which allows the reaching law to dynamically reduce the switching gain when approaching the sliding mode surface, effectively suppressing the mechanical vibration and noise caused by high-frequency switching in the traditional SMC, thus extending the life of the actuator.

In order to comprehensively assess the adaptability and feasibility of the sliding mode reaching law designed in this paper, its key parameters are analyzed in conjunction with the data collected during the optimal parameter selection process discussed earlier and the theoretical analysis underlying the sliding mode reaching law design, including $k_{1}, k_{2}, α, ε$ . The switching gain $k_{1}$ controls the strength of the isokinetic reach term in the reaching law, and the speed of convergence is improved by about 20% when $k_{1}$ is increased from 0.5 to 2, but the chatter amplitude at $| x | > 1$ increases by about 25%. Conversely, a smaller $k_{1}$ ( $k_{1}$ = 0.2) reduces the chatter by about 30%, but extends the convergence phase by 0.15 seconds. The linear gain $k_{2}$ regulates the weight of the index reach term in the reaching law, and increasing $k_{2}$ significantly accelerates the convergence process when $| x |$ is large in the initial phase, but may lead to chatter due to excessive speed, we deem that $k_{1} \in [0.5, 2]$ and $k_{2} \in [50, 200]$ can strike a balance between fast convergence and suppression of chatter. The power exponent function α serves to regulate the sensitivity of the adaptive function to the state variables. When $| x |$ larger, it provides stronger convergence power and accelerates convergence, and when $| x |$ is small, ${| x |}^{α}$ tends to 0, reduces the convergence speed, and smooths the transition to the sliding mode surface. Figure 4 illustrates the relationship between the speed of convergence of the system to the sliding mode surface and the sliding modulus for different values of α. The variable coefficient ε is used to regulate the decay rate of H(x), and increasing ε can keep the high gain for a long time and accelerate the convergence. The isokinetic reach term composed of H(x) with the designed inverse tangent function can make up for the shortcomings of a single sign function and a high gain, Figure 4 shows the performance of the isokinetic reach term at the origin under different values of ε. Compared with the sign function, H(x) with the inverse tangent function shows smoother characteristics at the origin, and this smoothing effectively suppresses the chattering phenomenon. $k_{1}, k_{2}, α, ε$ regulates the convergence speed and chattering through the nonlinear coupling mechanism, and the reasonable design of these coefficients can ensure the dynamics performance while effectively suppressing the chattering.

SMC strategy based on VPRL

Mathematical model of PMSM

Ignoring the effects of core saturation, eddy currents, and hysteresis losses on the PMSM, in the synchronous rotating (d-q) coordinate system, the voltage of the surface type PMSM can be obtained as

{\begin{matrix} u_{d} = R i_{d} + L \frac{d i_{d}}{dt} - p_{n} ω_{m} L i_{q} \\ u_{q} = R i_{q} + L \frac{d i_{q}}{dt} + p_{n} ω_{m} L_{d} i_{d} + p_{n} ω_{m} ϕ_{f} \end{matrix}

(10)

where $u_{d}, i_{d}$ and $u_{q}, i_{q}$ denote the voltage, current along the d and q-axis respectively;L denote the stator inductance;R denote the stator resistance; $ω_{m}$ is the electrical angular velocity; $P_{n}$ denote the pole pairs; $ϕ_{f}$ denote the flux linkage.

The dynamic model of PMSM can be described as

\frac{d ω_{m}}{dt} = \frac{K_{t}}{J} i_{q} - \frac{T_{L}}{J} - \frac{B}{J} ω_{m}

(11)

where: B is the coefficient of viscous friction; J is the moment of inertia; $T_{L}$ is the load torque; $K_{t} = \frac{3 P_{n} ϕ_{f}}{2}$ is the torque constant.

Considering the system parameters and torque variations, we get

(J + Δ τ) \frac{d ω_{m}}{dt} = (K_{t} + Δ κ) i_{q} - (B + Δ ρ) ω_{m} - T_{L}

(12)

where: $Δ τ, Δ κ, Δ ρ$ are the parameter variations of moment of inertia, damping coefficient, and torque constant, respectively.

Furthermore, the dynamic equation of PMSM can be described as

\frac{d ω_{m}}{dt} = \frac{K_{t}}{J} i_{q}^{ref} + D_{ω}

(13)

where: $i_{q}^{ref}$ is the q-axis reference current; $D_{ω}$ represents the lumped disturbance, specifically expressed as

D_{ω} = (\frac{K_{t} + Δ κ}{J + Δ τ}) i_{q} - \frac{K_{t}}{J} i_{q}^{ref} - (\frac{B + Δ ρ}{J + Δ τ}) ω_{m} - \frac{T_{L}}{J + Δ τ}

(14)

Anti-disturbance sliding mode speed controller based on VPRL

First, the velocity error $x_{1}$ is defined as

x_{1} = ω_{m}^{ref} - ω_{m}

(15)

where $ω_{m}^{ref}$ refers to the given mechanical angular velocity of the rotor. Combining equation (13) and taking the derivative of $x_{1}$ , we get

{\overset{\cdot}{x}}_{1} = - {\overset{\cdot}{ω}}_{m} = - \frac{K_{t}}{J} i_{q}^{ref} - D_{ω}

(16)

Introducing the velocity error micro-component can easily lead to high-frequency noise. Therefore, in this paper, the integral of the velocity error denoted as $x_{2}$ is introduced in the design of the sliding mode surface as follows

x_{2} = \int_{- \infty}^{t} x_{1} dt = \int_{- \infty}^{t} (ω_{m}^{ref} - ω_{m}) dt

(17)

Then, the sliding mode surface s is designed as follows

s = x_{1} + c x_{2}

(18)

Setting $s = 0$ in equation (18) and taking the derivative, we get

{\overset{\cdot}{x}}_{1} = - c x_{1}

(19)

From equation (19), we can solve for $x_{1} (t) = x_{1} (0) e^{- ct}$ , which indicates that the velocity error exponentially approaches zero with a time constant c. Therefore, the dynamic characteristics of the system during the sliding phase can be preset through the coefficient c.

Combining equations (4), (16), and (18), we get sliding mode controller control equation

- \frac{K_{t}}{J} i_{q}^{ref} - D_{ω} + c x_{1} = - k_{1} H (x_{1}) {| s |}^{s \cdot \arctan (\frac{π}{2} \cdot s)} sign (s) - k_{2} {| x_{1} |}^{α} \cdot s

(20)

Select the speed tracking error $x_{1}$ as the state variable.

Furthermore, from equation (20), we can derive

i_{q}^{ref} = \frac{J}{K_{t}} [c x_{1} + k_{1} H (x_{1}) {| s |}^{s \cdot \arctan (\frac{π}{2} \cdot s)} sign (s) + k_{2} {| x_{1} |}^{α} s - D_{ω}]

(21)

Selection of Lyapunov function

V = \frac{1}{2} s^{2}

(22)

Then we can obtain from equations (18) and (22)

\overset{\cdot}{V} = s (- k_{1} H (x_{1}) {| s |}^{s \cdot \arctan (\frac{π}{2} \cdot s)} sign (s) - k_{2} {| x_{1} |}^{α} \cdot s)

(23)

In equation (23), $k_{1}$ and $k_{2}$ are both positive constants, $| s |$ and $x_{1}$ are positive values. Thus, $\overset{\cdot}{V} \leq 0$ can be easily obtained and the Lyapunov stability conditions are satisfied, confirming that the proposed reaching law guarantees the stability of the system state.

ALDOB design

To further enhance the system’s disturbance rejection performance, an ALDOB is designed. Since the sampling period of the velocity loop is very small, $D_{ω}$ can be considered a constant, thus $\frac{d D_{ω}}{d t}$ . Therefore, the state–space equation of the PMSM velocity loop can be described as

{\begin{matrix} \frac{dx}{dt} = Ax + Bu \\ y = Cx \end{matrix}

(24)

where $x = {[\begin{matrix} ω_{m} & D_{ω} \end{matrix}]}^{T}$ represents the state vector; $y = ω_{m}$ represents the output; $u = i_{q}^{ref}$ represents the control input

A = [\begin{matrix} 0 & 1 \\ 0 & 0 \end{matrix}], B = [\begin{matrix} \frac{K_{t}}{J} \\ 0 \end{matrix}], C = [\begin{matrix} 1 & 0 \end{matrix}]

(25)

In the LDOB, state variables can be corrected through state estimation errors. Specifically, the LDOB can be designed as

{\begin{matrix} \frac{d \hat{x}}{dt} = A \hat{x} + Bu + H (y - \hat{y}) \\ \hat{y} = C \hat{x} \end{matrix}

(26)

In which $H = {[\begin{matrix} h_{1} & h_{2} \end{matrix}]}^{T}$ is the observer gain matrix, $\hat{x}$ is the observation of vector x, and $\hat{y}$ is the observation of system output y. From equations (25) and (26), the derivative of the observation error $\tilde{x} = \hat{x} - x$ can be obtained as

\frac{d \tilde{x}}{dt} = (A - HC) \tilde{x}

(27)

Choose an appropriate observer system matrix $(A - HC)$ such that the real parts of its eigenvalues are negative, that is

Re {λ_{i} (A - HC)} < 0, i = 1 < 1, 2 .

(28)

Assuming $λ_{1} = λ_{2} = - l (l > 0)$ , we get $H = {[\begin{matrix} 2 l & l \end{matrix}]}^{T}$ , then from equation (26) we can obtain

{\overset{\cdot}{\hat{D}}}_{ω} = l^{2} (ω_{m} - {\hat{ω}}_{m})

(29)

where l is an adjustable parameter related to the observer bandwidth.

It is worth noting that choosing a large value of l can accelerate the convergence speed of the observer, but an excessively large l can lead to overshoot in the system’s step response and poor steady-state performance. Therefore, this paper introduces an adaptive function representing the relationship between the system state variable $x_{1}$ and the parameter l in the LDOB, and designs an ALDOB, allowing the parameter l to vary adaptively at different stages of the system response, effectively solving the aforementioned problems. Specifically, the adaptive function is as follows

l_{A} (x_{1}) = l {\tan h [μ (| x_{1} | - ω_{a})] + 1}

(30)

where $μ = 1, ω_{a} = 0.8$ , l is the total gain, $ω_{a}$ is the switching threshold, μ is a constant related to the slope of the hyperbolic arctangent function at the switching threshold.

When the rotational speed approaches the given speed, $| x_{1} | < ω_{a}$ , the gain $l_{A}$ of the ALDOB is small, so the observer will not overcompensate for the lumped disturbance, thus not affecting the system’s steady-state performance. When a sudden load is applied to the system, $| x_{1} | > ω_{a}$ , the gain $l_{A}$ of the ALDOB is larger, the observer has a faster response speed, and the system has strong disturbance rejection capability.

Disturbance compensation and stability proof

The disturbance observation value ${\hat{D}}_{ω}$ is used for feedforward compensation at the output end of the velocity sliding mode controller. According to equation (21), the compensated velocity SMC law is

i_{q}^{ref} = \frac{J}{K_{t}} [c x_{1} + k_{1} H (x_{1}) {| s |}^{s \cdot \arctan (\frac{π}{2} \cdot s)} sign (s) + k_{2} {| x_{1} |}^{α} s - {\hat{D}}_{ω}]

(31)

To prove the stability of the overall closed-loop control system, Lyapunov stability theory is used for analysis. The Lyapunov function for the overall closed-loop control system is selected as $V = \frac{1}{2} s^{2}$ , Taking the derivative of V, we get

\overset{\cdot}{V} = s \overset{\cdot}{s} = s (- {\overset{\cdot}{ω}}_{m} + c x_{1}) = s (- \frac{K_{t}}{J} i_{q}^{ref} + c x_{1} - D_{ω})

(32)

Substituting equation (31) into equation (32), we get

\overset{\cdot}{V} = s (- k_{1} H (x_{1}) {| s |}^{s \cdot \arctan (\frac{π}{2} \cdot s)} sign (s) - k_{2} {| x_{1} |}^{α} s + {\hat{D}}_{ω} - D_{ω})

(33)

In the text, the observer’s estimation error ${\hat{D}}_{ω} - D_{ω}$ is bounded. When $k_{1}$ and $k_{2}$ are sufficiently large, it can satisfy $\overset{\cdot}{V} \leq 0$ , thereby proving that the system using this control strategy is asymptotically stable.

Simulation and experimental analysis

Simulation analysis

To prove the validity of the VPRLSMC + ALDOB control, simulation and experimental studies were conducted. In terms of controllers, a PI control is used for the current loop, and for the speed loop, conventional PI controllers, SMC controllers, and VPRLSMC + ALDOB control are used for comparative simulation and experimental validation. The PMSM employs field-oriented control (FOC) vector control strategy. The PMSM system control diagram is shown in Figure 5.

To compare and verify the dynamic response performance of the motor under the control of various controllers, the target speed is configured at 1000 r/min. The initial load is set to 1 N m. The control parameters of each controller are as follows: PI: P = 0.14, I = 7. SMC: c = 15, k = 10, ε = 1.5. DTSMC: α = 0.001, γ = 0.001, β = 0.01. VPRLSMC: c = 15, k₁ = 1, k₂ = 100, α = 0.5, ε = 2000. Figure 6 shows the relationship curve of motor speed, torque and time under the conventional PI control, conventional SMC, VPRLSMC + ALDOB, and DTSMC (Samantaray and Chakrabarty, 2024). From Figure 6, the system under the PI control has an overshoot of about 185 r/min at startup, and the startup time is 0.12 seconds; under conventional SMC, there is an overshoot of about 155 r/min at startup, and the startup time is 0.19 seconds; under the VPRLSMC + ALDOB control method, the overshoot can be realized without any overshoot and the startup time is only 0.07 seconds; under the DTSMC, while overshoot-free startup is possible, the startup time is slightly slower than the VPRLSMC + ALDOB control method.

To verify the disturbance rejection performance of the VPRLSMC + ALDOB control under sudden load conditions, a sudden load of 10 N m is applied to the system at 0.6 seconds. Figure 6 shows that compared to the other three control methods, the VPRLSMC + ALDOB control method exhibits smaller maximum speed fluctuations and shorter settling times when the system is subjected to load disturbances.

The fact that DTSMC employs a reaching law based on the inverse hyperbolic sine function makes it dependent on the inherent nonlinear properties of the inverse hyperbolic sine function (smooth near the origin, approximately linear when away) to suppress chatter. The function shape is fixed, the performance is dependent on parameter tuning, and if the chatter suppression is improved, it sacrifices the initial convergence speed when moving away from the sliding mode surface. In contrast, the VPRL proposed in this paper does not need to manually weigh “convergence speed” and “chatter suppression” due to the introduction of the state-variable-dependent function, and the system automatically adjusts the switching gain according to the state. At the same time, it realizes the fast convergence when moving away and the smoothness when approaching. In addition, the pursuit of extremely low quasi-sliding bandwidth of DTSMC may require sacrificing certain robustness margins or further reducing the convergence speed, which may be the reason why DTSMC performs poorly when facing load variations in simulation experiments.

The simulation indicate that the PMSM speed loop controller based on the VPRLSMC + ALDOB control has good speed control performance. Compared with conventional PI, conventional SMC, and DTSMC, it shows significant improvements in terms of system dynamic response performance, vibration suppression, and disturbance rejection capabilities.

Experimental analysis

To verify the validity of the VPRLSMC + ALDOB control regarding dynamic response performance, chatter suppression, and disturbance rejection in the PMSM dual closed-loop control system, experiments were conducted on a Surface-Mounted Permanent Magnet Synchronous Motor (SPMSM) experimental platform, which involved speed variation and sudden load addition tests. The experimental platform includes a host machine, a real-time simulator, a synchronous motor counter-drag platform, a drive-side driver, and a load-side driver. The control system and drivers use the Links-Box real-time simulator and Links-HS050A driver from LingSiChuangQi Company. A dual-motor counter-drag loading mode is employed, the observer bandwidth is set to 20, and the constant μ related to the slope of the hyperbolic arctangent function at the switching threshold is set to 1. The switching threshold $ω_{a}$ is 0.8, the speed loop sampling frequency is 2 kHz, and the inverter sampling frequency is 10 kHz. The experimental platform is shown in Figure 7.

The speed response performance of several control strategies was experimentally validated. The target speed was set to 500 and 750 r/min, and the speed response curves of the three methods are compared in Figure 8. When the initial speed was 500 r/min, it shown that the adjustment times for PI control, SMC control, and VPRLSMC + ALDOB control are 1.6, 0.91, and 0.57 seconds, respectively. When the initial speed was changed to 750 r/min, the adjustment times for PI control, SMC control, and VPRLSMC + ALDOB control are 2.51, 2.01, and 0.55 seconds, respectively. It can be seen that PI control has the longest adjustment time and exhibits overshoot, SMC control has a relatively longer adjustment time, VPRLSMC + ALDOB control has the shortest adjustment time. Compared with PI control and SMC, when the initial speed is 500 r/min, its convergence speed is improved by about 40%–63%. When the initial speed is 750 r/min, its convergence speed is improved by about 70%–76%. This improvement stems from the dynamic gain adjustment mechanism of VPRL, which adaptively adjusts the reaching law parameter through the state variables to avoid the overshooting and response lag caused by the fixed switching gain in conventional SMC. In addition, the introduction of ALDOB further reduces the accumulation of dynamic errors by compensating the aggregate perturbation in real time.

To demonstrate the superiority of the VPRLSMC + ALDOB control system under sudden load conditions, an initial reference speed set at 500 r/min, a sudden load of 5 and 10 N m was added at 4 seconds, respectively. The experimental results under the three control actions are shown in Figure 8. In the experiment with sudden load, PI control has the longest adjustment time and the largest speed fluctuation value. SMC control has a relatively long adjustment time and a larger maximum speed fluctuation value. Compared to the other two control strategies, when a sudden 5 N·m load is applied, the VPRLSMC + ALDOB control suppresses the fluctuation to 47.62 r/min (47.6% lower than PI and 22.6% lower than SMC), and reduces the adjustment time to 1.42 seconds (59.5% improvement over PI and 41.1% improvement over SMC). When a 10 N m load is applied suddenly, the VPRLSMC + ALDOB still maintains optimal performance, with a maximum speed fluctuation of 83.33 r/min and an adjustment time of 1.85 seconds, which is 54.5% and 36.2% lower than that of the PI and SMC, respectively. VPRLSMC + ALDOB control exhibits better disturbance rejection performance in the PMSM dual closed-loop control system during sudden load conditions. This shows that ALDOB is able to quickly boost the observer gain in the early stage of perturbation to enhance the real-time compensation capability for sudden load changes through adaptive bandwidth design, while reducing the gain in the steady-state stage to avoid the impact of overcompensation on the steady-state accuracy, and through feedforward compensation to weaken the impact of the disturbance on the speed loop, forming a synergistic control effect with VPRL. Table 1 summarizes the specific data on the speed response of the three control methods for different conditions of startup and sudden load increase.

Table 1.

Description of the dataset for Figures 8 and 9.

	Control methods	Adjustment time, t/s	Overshooting
500 r/min	PI	1.60	Yes
	SMC	0.91	Yes
	VPRLSMC + ALDOB	0.57	No
750 r/min	PI	2.51	Yes
	SMC	2.01	Yes
	VPRLSMC + ALDOB	0.60	No
	Control methods	Adjustment time, t/s	Maximum speed fluctuation, r/min
5 N m	PI	3.51	90.90
	SMC	2.41	61.54
	VPRLSMC + ALDOB	1.42	47.62
10 N m	PI	4.16	183.33
	SMC	2.61	130.77
	VPRLSMC + ALDOB	1.85	83.33
	Control methods	Adjustment time, t/s	Maximum speed fluctuation, r/min
5 N m	VPRLSMC + ALDOB	1.42	47.62
	VPRLSMC	1.59	55.23
10 N m	VPRLSMC + ALDOB	1.85	83.33
	VPRLSMC	1.99	107.69
20 N m	VPRLSMC + ALDOB	2.08	133.33
	VPRLSMC	2.20	176.92

To highlight the performance of the VPRLSMC + ALDOB control regarding observing system disturbances, comparative experiments were conducted on the system’s disturbance rejection capabilities with and without an adaptive observer. The initial reference speed was set to 500 r/min, and at 4 seconds, loads of 5, 10, and 20 N m were suddenly added, respectively. The experimental group without the adaptive observer was named VPRLSMC. The comparative results of the experiments with and without the adaptive observer are shown in Figure 9. VPRLSMC + ALDOB control has shorter adjustment times and smaller speed fluctuations when the adaptive observer is added. Therefore, it can be concluded that under the condition of an adaptive observer, when a sudden load is added, the VPRLSMC + ALDOB control method has a strong superiority in disturbance rejection performance in the PMSM control system. Table 1 summarizes the data from the experiments comparing the perturbation rejection capabilities of the systems with and without the adaptive observer.

Conclusion

In this paper, based on the CERL, a new type of reaching law is proposed, which improves the convergence speed compared with the CERL and effectively suppresses the chatter of the sliding mode motion. For the external disturbance problem, an ALDOB is designed, and the anti-interference capability of the system is further improved by VPRLSMC + ALDOB control method. The superiority of the control method is verified through simulation and experiment, and the results show that the VPRLSMC + ALDOB control method can successfully enhance the dynamic performance and robustness of the PMSM. However, there is still some room for improvement in this study, and future research directions can be centered on the following points:

This study mainly focuses on the compensation of a single external load disturbance, but in real industrial scenarios, motors may face multiple sources of coupled disturbances such as parametric uptake, nonlinear friction, and harmonic disturbances at the same time. Subsequent research can explore the design of a multidimensional disturbance observer to enhance the robustness of the system against composite disturbances.

Combining VPRL with model predictive control (MPC), fuzzy logic control, or fault-tolerant control is expected to further improve the dynamic accuracy while guaranteeing the robustness and enhance the reliability of the system under abnormal operating conditions such as sensor failure.

Footnotes

Author contributions

Lin Yang, Yu Zhang: Formal analysis, Conceptualization. Priya Singh: Data curation, Writing- Original draft preparation. Minli Zheng, Hui Jiang: Resources, Funding acquisition. Yinfeng Liu, Tengyu Ma: Writing- Reviewing and Editing.

Declaration of conflicting interests

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

Funding

The authors disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This research was funded by the National Nature Science Foundation of China, grant number 52205462.

ORCID iD

Lin Yang

Data availability statement

All data generated or analyzed during this study are included in this manuscript.

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