Permanent magnet synchronous motor active disturbance rejection control based on ultra-local modeling and improved extended state observer

Abstract

In this paper, we propose an active disturbance rejection control (ADRC) method—termed ADRC-IESO-ULM—that integrates an improved extended state observer (IESO) with an ultra-local modeling (ULM) adaptive quadratic sliding mode control strategy. This approach is designed to accelerate system response and effectively counter both external and internal disturbances in a permanent magnet synchronous motor (PMSM). First, we enhance the conventional extended state observer to better capture time-varying perturbations affecting the control system. Moreover, we employ a polynomial approximation to approximate a new fal function, addressing the derivative discontinuity issues at the segmentation points of the traditional fal function and thereby improving the overall control performance of the PMSM system. Second, the ULM technique eliminates the need for explicit state observer construction, offering robust performance while avoiding the instability problems associated with improperly designed observers. Finally, simulation and experimental results demonstrate that the IESO can more accurately track system state variables and errors and that the ADRC-IESO-ULM method achieves faster response and superior disturbance rejection compared to both the proportional-integral (PI) method and the traditional ADRC approach.

Keywords

Permanent magnet synchronous motor active disturbance rejection control newfal function ultra-local modeling improved extended state observer sliding mode control

Introduction

With the rapid development of electric motor technology, PMSMs are increasingly used in industrial automation, rail transport, aerospace, and new energy vehicles. As a result, control systems based on motor drive technology have become a prominent focus of research (Guo et al., 2017; Kuang et al., 2017; Yang et al., 2020). Due to their high efficiency and simple structure, PMSMs have become key components in modern electric transmission systems. However, in practical applications, PMSM drive systems encounter numerous, complex external perturbations as well as internal uncertainties, particularly under high-speed operation and load fluctuations. Traditional methods, such as PID and linear control, often struggle to achieve satisfactory performance (Chen et al., 2020; Li et al., 2021).

In recent years, research aimed at enhancing the robustness and stability of PMSM drive systems has increasingly turned to robust control (Ahmed et al., 2025; Kogan and Stepanov, 2025), adaptive control (Denny and Kumar, 2024; Rahimi et al., 2016), and intelligent control (Abdalla et al., 2025; AL-Wesabi et al., 2025). Notably, the ADRC technique (Liu et al., 2020, 2024) has emerged as a promising solution to address PMSM control challenges. The core principle of ADRC is to manage external disturbances, parameter variations, and nonlinearities by estimating and compensating for system disturbances in real time (Han, 2009). Unlike traditional PID control, which relies on an exact mathematical model, ADRC utilizes real-time feedback from the system output, making it more robust in the presence of uncertainties and nonlinear dynamics (Garrido and Luna, 2021).

However, the modeling of PMSM control systems is subject to significant uncertainties and complex nonlinear phenomena. Traditional ADRC nonlinear state feedback methods encounter considerable modeling errors and computational burdens when the system’s dynamic characteristics change rapidly or when strong nonlinearities are present (Yan et al., 2024; Zhu and et al., 2022). To address these limitations in PMSM control systems, recent studies have proposed integrating sliding mode control (SMC) theory into the ADRC framework (Chen et al., 2023; Li et al., 2025; Yao et al., 2021). SMC is inherently robust, particularly in rejecting external perturbations and handling system uncertainties, and it naturally excels in managing nonlinear systems. Therefore, SMC serves as an effective alternative to traditional nonlinear state feedback in ADRC when dealing with pronounced nonlinear behavior. To more accurately address complex nonlinearities, this paper replaces the traditional nonlinear feedback of ADRC with a combination of ULM and SMC. ULM models the system’s nonlinear characteristics through local modeling techniques, thereby avoiding the high complexity associated with global modeling (Zhang et al., 2024). This approach allows for flexible adjustment of the control strategy based on the system’s local dynamic characteristics, enhancing both control accuracy and robustness (Ghanipoor et al., 2025). By substituting traditional nonlinear state feedback with the ULM-SMC combination, the ADRC control system can more efficiently manage perturbations and uncertainties in complex systems, particularly under strong nonlinear and time-varying dynamics, ultimately yielding superior control performance (Yang et al., 2024).

Meanwhile, the ESO remains a core component of ADRC, and its enhancement has naturally become a research trend. Researchers have proposed designing a continuous fal function to overcome the discontinuities present in the original fal function (Jiang and Liu, 2024; Yan et al., 2024). In addition, the switching extended state observer (SESO) has been introduced to improve the accuracy and robustness of state and perturbation estimation by dynamically switching observer parameters or structures (Hao et al., 2021). Similarly, the cascaded extended state observer (CESO) has been developed to not only enhance estimation accuracy but also to bolster the system’s anti-interference capabilities (Deng et al., 2022; Wang et al., 2024; Xu et al., 2023). Building on these pioneering studies and addressing the limitations observed in conventional ESO approaches, this paper proposes an IESO. The IESO refines the traditional fal function using a polynomial approximation and introduces a sign function to heighten the observer’s sensitivity to the direction of perturbations, thereby enhancing its anti-interference performance. Compared to the SESO and CESO, the proposed IESO boasts a simpler structure and easier implementation while being more effective at suppressing time-varying disturbances. The ADRC-IESO-ULM method presented in this paper combines ULM state feedback with IESO. In this configuration, the IESO estimates system state variables and perturbations, providing real-time feedback and compensation. Meanwhile, the ULM state feedback further detects and compensates for residual perturbations through an adaptive sliding mode disturbance observer, and then processes the state variables using a quadratic sliding mode controller. This integrated design not only reduces the computational workload of the IESO but also improves the accuracy of disturbance estimation. Therefore, this paper not only enhances the stability and response speed of the control system through structural improvements to the ADRC but also significantly boosts the system’s anti-interference capability when dealing with time-varying disturbances. The main contributions of this paper are:

By employing ULM-based quadratic sliding mode control, the PMSM speed loop error converges rapidly to zero, significantly reducing system vibrations.

Using a polynomial approximation, a new fal function with continuous and smooth segment transitions is developed. This enhancement compensates for the deficiencies of the original fal function—whose segment points were not smooth—and helps eliminate the potential for control system vibrations.

Conventional ESOs are unable to reduce the error between observed and desired trajectories to zero under time-varying disturbances, typically maintaining a constant residual error near the origin. To overcome this limitation, this paper enhances the observer’s performance by refining its internal structure and parameter adaptation.

The structure of this paper is as follows: section I describes the dynamics model of permanent magnet synchronous motor. Section II designs the ADRC-IESO-ULM controller and analyses it. Simulation analysis and experimental verification simulates and analyses the designed controller and compares the PI and ADRC, obtains the practicability of ADRC-IESO-ULM controller by physical verification and compares it with PI control. Finally, concludes the paper.

Mathematical modeling of PMSM

In this paper, a surface-mounted PMSM is chosen as the plant, and a control strategy with I_d = 0 is adopted. The mathematical model of the PMSM is expressed in the d–q reference frame. Neglecting core saturation, iron losses, and motor parameter variations, the rotor voltage equations for the PMSM is given by

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

(1)

where u_d, u_q, i_d, and i_q denote the d–q axis components of the rotor voltage and current, respectively; L_s is the stator inductance; φ_f is the permanent magnet flux; R represents the rotor resistance; P_n is the number of pole pairs; and ω_m is the rotor’s mechanical angular velocity. The mechanical equation of motion of the motor is given by

J \frac{d ω_{m}}{dt} = T_{e} - T_{L} - B ω_{m}

(2)

Where, B is the damping factor, J is the rotational inertia, T_L is the load torque and T_e is the electromagnetic torque. The electromagnetic torque equation is

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

(3)

Where L_d and L_q are the d-q axis components of the rotor inductance, respectively. For surface-mounted permanent magnet synchronous motor, satisfying L_d = L_q = L_s, the following mathematical model is derived

{\begin{matrix} \frac{d i_{q}}{dt} = \frac{1}{L_{s}} (- R i_{q} - p_{n} ω_{m} ϕ_{f} + u_{q}) \\ \frac{d ω_{m}}{dt} = \frac{1}{J} (- T_{L} + \frac{3 p_{n} ϕ_{f}}{2} i_{q}) \end{matrix}

(4)

The parameters of the PMSM can vary due to temperature fluctuations, magnetic saturation, aging, and other factors, which in turn affect control accuracy and system stability. In this paper, ADRC is employed to control the PMSM’s rotational speed as it can automatically adapt to such parameter variations without relying on an exact motor model, thereby enhancing the overall system robustness.

Figure 1 shows the motor’s speed control loop and current control loop. ADRC-IESO-ULM is employed in the speed control loop, while a PI controller governs the current loop. The ADRC-IESO-ULM scheme comprises two main parts: a quadratic sliding-mode controller with an adaptive sliding-mode perturbation observer, and a modified dilation state observer.

Figure 1.

Structure of ADRC-IESO-ULM control strategy for PMSM.

Design improved active disturbance rejection control

This section presents an active disturbance rejection control strategy that integrates ultra-local modeling, nonlinear quadratic sliding-mode control, and an improved extended state observer. The enhanced observer precisely estimates the states and disturbances of the PMSM system. These estimates are then fed back into the adaptive controller to compensate for system uncertainties and thereby enhance control accuracy.

Improved nonlinear state error feedback

Ultra-local modeling

According to model free control (MFC), the ultra-local modeling (ULM) of a nonlinear single-input single-output (SISO) system is represented as

y^{(v)} = F + λ u

(5)

where y represents the output and v indicates the differential order; F encompasses unknown perturbations, including both modeled and unmodeled disturbances; u is the controller output; and λ is a constant. To enhance control accuracy for the PMSM system while also simplifying its structure, it is crucial to develop a framework that is well suited for PMSM control. The design of model-free control (MFC) is achieved by setting v = 1 to construct a first-order ULM, from which the above equation can be rewritten as

\overset{\cdot}{y} = F + λ u

(6)

From equation (6), it is clear that the ULM consists of a known component, λu, and an unknown component, F, which represents the system’s nonlinear and time-varying perturbations. Therefore, a two-fold approach is necessary: a controller is designed to handle the known input, and an observer is developed to estimate and compensate for the unknown disturbances. The following section details the design for each part of the ULM.

Design an adaptive quadratic sliding mode controller

Combining equation (6) the ULM of the velocity loop of the PMSM system yields

{\overset{\cdot}{z}}_{1} = F + λ u_{0}

(7)

Based on ULM, the controller for state feedback is designed as

u_{0} = \frac{- F + {\overset{\cdot}{ω}}_{m}^{*} + u_{c}}{λ}

(8)

Defining state variables

{\begin{matrix} e = ω_{m}^{*} - z_{1} \\ \overset{\cdot}{e} = - u_{c} \end{matrix}

(9)

where $ω_{m}^{*}$ is the reference speed of the PMSM, z₁ is the observation of the system state variable $ω_{m}$ by the expanded state observer observation. As the secondary slip mode surface increases the control force of the system when the error is large, the system converges to the slip mold surface quickly. When the error is small, the system control force is relatively reduced to avoid large vibration. Therefore, the design of the secondary type sliding mode surface is

s_{1} = \overset{\cdot}{e} + c_{1} e + c_{2} e^{2}

(10)

where c₁ and c₂ are design parameters greater than 0 used to adjust the convergence rate and stability. Derivation of equation (10) gives

{\overset{\cdot}{s}}_{1} = \overset{\cdot\cdot}{e} + c_{1} e + 2 c_{2} e \overset{\cdot}{e}

(11)

Traditional exponential convergence laws fail to significantly accelerate convergence when the state is far from the sliding-mode surface. They also cannot adaptively reduce the state’s velocity as it approaches the surface. As a result, system vibrations are intensified. To address these issues, this paper proposes an adaptive exponential convergence law that adapts based on the system’s state variables, as expressed below

{\overset{\cdot}{s}}_{1} = - (1 + | \tanh (s_{1}) |) l_{2} sign (s_{1}) - (l_{1} + {| s_{1} |}^{l}) s_{1}

(12)

where 0 < l < 1, l₁ > 0, l₂ > 0. When the state variables of the system are farther away from the sliding mode surface ${| s_{1} |}^{l}$ and tanh (s₁) is larger, The system can approach the sliding mold surface faster, further reducing the time it takes for the system to converge on the sliding mold surface. When the system state variable nears the sliding mode surface, the adaptive coefficients adjust the convergence law so that it approaches zero. This ensures that the state variable enters the sliding mode surface and converges toward the origin, thereby reducing motion inertia and suppressing vibrations. This result is obtained by combining equations (11) and (12)

\overset{\cdot}{e} = - u_{c} = \frac{- (1 + | \tanh (s_{1}) |) l_{2} sign (s_{1}) - (l_{1} + {| s_{1} |}^{l}) s_{1} - \overset{\cdot}{e}}{c_{1} + 2 c_{2} e}

(13)

Combined with equation (8), this gives

u_{0} = \frac{- (1 + | \tanh (s_{1}) |) l_{2} sign (s_{1})}{(c_{1} + 2 c_{2} e) λ} + \frac{- (l_{1} + {| s_{1} |}^{l}) s_{1} - \overset{\cdot}{e}}{(c_{1} + 2 c_{2} e) λ} + \frac{- F + {\overset{\cdot}{ω}}_{m}^{*}}{λ}

(14)

To analyze the stability of the adaptive quadratic sliding mode controller designed in this paper, the following positive definite function is established as the Lyapunov function

V_{1} = \frac{1}{2} {s_{1}}^{2}

(15)

Deriving equation (15) and bringing equation (12) into it yields

\begin{matrix} {\overset{\cdot}{V}}_{1} & = s_{1} {\overset{\cdot}{s}}_{1} \\ = s_{1} (- (1 + | \tanh (s_{1}) |) l_{2} sign (s_{1}) \\ - (l_{1} + {| s_{1} |}^{l}) s_{1}) \\ = - (1 + | \tanh (s_{1}) |) l_{2} | s_{1} | - (l_{1} + {| s_{1} |}^{l}) {s_{1}}^{2} \end{matrix}

(16)

As $l_{1} > 0$ and $l_{2} > 0$ can be obtained

{\overset{\cdot}{V}}_{1} \leq 0

(17)

Since the Lyapunov function is positive definite and its derivative is negative definite, it is feasible to design adaptive quadratic sliding mode controllers that stabilize all state variables, ensuring that the system converges to the sliding mode surface.

Design of ULM observer

Real-time observation of F, as defined in equation (14), is performed using an adaptive sliding mode perturbation observer (ASMPO) based on model-free theory, which accurately estimates the unknown component of F. Building on the design of the adaptive quadratic sliding mode controller

{\begin{matrix} {\overset{\cdot}{z}}_{1} = F + λ u_{0} \\ \overset{\cdot}{F} = f (t) \end{matrix}

(18)

where f(t) is the rate of change of the unknown part of the system. For the controller shown in equation (18), the adaptive terminal sliding mode disturbance observer is designed as follows

{\begin{matrix} {\overset{\cdot}{\hat{z}}}_{1} = \hat{F} + λ u_{0} + u_{smo} \\ \overset{\cdot}{\hat{F}} = g u_{smo} \end{matrix}

(19)

where ${\overset{\cdot}{\hat{z}}}_{1}$ is the observed rotational speed of the system, $\hat{F}$ is an unknown perturbation of the systematic observation, $\overset{\cdot}{\hat{F}}$ is the rate of change of the unknown perturbation of the systematic observation, $u_{smo}$ is the sliding mode function of the system design and g is the sliding mode function gain. Combining equations (18) and (19), the error equation is obtained as

{\begin{matrix} {\overset{\cdot}{e}}_{z} = u_{smo} + e_{f} \\ {\overset{\cdot}{e}}_{f} = g u_{smo} - f (t) \end{matrix}

(20)

where $e_{z} = {\hat{z}}_{1} - z_{1}$ is the rotational speed observation error, $e_{f} = \hat{F} - F$ is the observation error of the unknown perturbation. Combined with the error equation the linear sliding mode surface is selected as follows

s_{2} = {\overset{\cdot}{e}}_{z} + k_{1} e_{z}

(21)

where k₁ > 0, derivation of equation (21) yields equation (22) as follows

{\overset{\cdot}{s}}_{2} = {\overset{\cdot\cdot}{e}}_{z} + k_{1} {\overset{\cdot}{e}}_{z}

(22)

In order to effectively suppress chatter and shorten the convergence time, the adaptive exponential convergence rate is designed to be

{\overset{\cdot}{s}}_{2} = - k_{2} s_{2} - k_{3} (1 - {e_{z}}^{- k_{4} | s_{2} |}) sign (s_{2})

(23)

Where k₂ > 0, k₃ > 0 and k₄ > 0. Combining equations (22) and (23), the proposed adaptive sliding mode perturbation control law is obtained as

\begin{matrix} {\overset{\cdot}{e}}_{z} = (- k_{2} s_{2} - {\overset{\cdot\cdot}{e}}_{z}) \frac{1}{k_{1}} - k_{3} (1 - {e_{z}}^{- k_{4} | s_{2} |}) sign (s_{2}) \frac{1}{k_{1}} \\ ⇓ \\ u_{smo} = (- k_{2} s_{2} - {\overset{\cdot\cdot}{e}}_{z}) \frac{1}{k_{1}} - e_{f} - k_{3} (1 - {e_{z}}^{- k_{4} | s_{2} |}) sign (s_{2}) \frac{1}{k_{1}} \end{matrix}

(24)

To demonstrate the stability of the designed adaptive sliding mode disturbance observer, the Lyapunov function is designed as follows

V_{2} = \frac{1}{2} {s_{2}}^{2}

(25)

Deriving equation (25) and bringing equation (23) into it yields

\begin{matrix} {\overset{\cdot}{V}}_{2} & = s_{2} {\overset{\cdot}{s}}_{2} \\ = s_{2} (- k_{2} s_{2} - k_{3} (1 - {e_{z}}^{- k_{4} | s_{2} |}) sign (s_{2})) \\ = - k_{2} {s_{2}}^{2} - k_{3} (1 - {e_{z}}^{- k_{4} | s_{2} |}) | s_{2} | \end{matrix}

(26)

Since k₂ > 0, k₃ > 0, k₄ > 0, it follows that $\overset{\cdot}{V} < 0$ . So by Lyapunov stability analysis it can be obtained that the designed adaptive sliding mode perturbation observer is able to converge the observation error and keep it on the sliding mode surface. Therefore, the adaptive sliding mode perturbation observer designed in this section is stable.

Improved extended state observer

The nonlinear ESO relies on the fal function as its core element. However, the conventional fal function is nondifferentiable at its segmentation points, which can induce chattering in the control system. Moreover, traditional ESO struggle to track time-varying perturbations and cannot drive perturbation errors to zero in finite time. To address these issues, we develop a new fal function via polynomial approximation, ensuring smooth differentiability at the segmentation points and thus preventing potential instability. This improved observer also incorporates a sign function to enhance directional sensitivity, enabling the observer to eliminate perturbation errors in finite time-even under time-varying disturbances.

Improvement and analysis of the newfal function

The nonlinear function is critical for ADRC because it must converge to zero near the origin, be smooth, and continuously differentiable. The fal function was evaluated against these criteria by varying the control parameters δ (0.01, 0.05, 0.1, 0.15) and α (0, 0.25, 0.5, 1). As shown in Figure 2(a) and (b), a larger α makes the fal function behave more linearly, while a larger δ extends its linear region. However, regardless of the parameter adjustments, the fal function consistently exhibits a distinct inflection point. This inflection point can introduce high-frequency perturbations in the state observer output, which may compromise system stability.

To enhance both the smoothness and control accuracy of the fal function, segmented polynomials are employed to approximate its linear and nonlinear components

new fal = {\begin{matrix} h_{1} e + h_{2} e^{2} + h_{3} e^{3}, | e | \leq δ \\ {| e |}^{α} sign (e), | e | > δ \end{matrix}

(27)

where h₁, h₂, h₃ are the gains of the function. Since the soft-sign function $softsign (x) = \frac{x}{1 + | x |}$ is nearly linear near the origin and smoother compared to $y = x$ , so $e \Rightarrow softsign (e)$ , Similarly the hyperbolic tangent function tanh(e) replaces e³. Combining the properties of nonlinear functions out at the origin leads to the following conditions

{\begin{matrix} t_{1} = softsign (e), t_{2} = \tanh (e) \\ h_{1} t_{1} + h_{2} e^{2} + h_{3} t_{2} = δ^{α}, e = δ \\ - h_{1} t_{1} + h_{2} e^{2} - h_{3} {t_{2}}^{3} = - δ^{α}, e = - δ \\ h_{1} {\overset{\cdot}{t}}_{1} + 2 h_{2} e \overset{\cdot}{e} + h_{3} {\overset{\cdot}{t}}_{2} = α δ^{α - 1}, e = \pm δ \end{matrix}

(28)

Solving equation (28) gives

{\begin{matrix} \begin{matrix} t_{1} = softsign (δ) \\ t_{2} = \tanh (δ) \end{matrix} \\ h_{1} = \frac{σ^{α}}{t_{1}} - \frac{t_{2}}{{\overset{\cdot}{t}}_{2} t_{1} - t_{2} {\overset{\cdot}{t}}_{1}} (α σ^{α - 1} - \frac{{\overset{\cdot}{t}}_{1}}{t_{1}} σ^{α}) \\ h_{2} = 0 \\ h_{3} = \frac{t_{1}}{{\overset{\cdot}{t}}_{2} t_{1} - t_{2} {\overset{\cdot}{t}}_{1}} (α σ^{α - 1} - \frac{{\overset{\cdot}{t}}_{1}}{t_{1}} σ^{α}) \end{matrix}

(29)

Summing up leads to the expression for the newfal function

newfal = {\begin{matrix} h_{1} softsign (e) + h_{3} \tanh (e), | e | \leq δ \\ {| e |}^{α} sign (e), | e | > δ \end{matrix}

(30)

Again δ = 0. 01, α = 0, 0.25, 0.5, 1; the newfal function response image is shown in Figure 2(c);then α = 0.25, δ = 0.01, 0.05, 0.1, 0.15 were chosen; the newfal function image is shown in Figure 2(d). α = 0.25 and δ = 0.01 are selected to compare the newfal function and fal function. Figure 3(a) shows the graphs of both functions, while Figure 3(b) compares the system chattering vibration of ESO using the fal function and the newfal function. From Figure 3(b), it can be concluded that the improved newfal function can effectively reduce the system chattering vibration to a limited extent.

Figure 2.

Analysis of the parameters of the traditional fal function and newfal function: (a) delta = 0.01, (b) alpha = 0.25, (c) delta = 0.01, and (d) alpha = 0.25.

Figure 3.

Fal and newfal: (a) comparison between fal and newfal with the same parameters and (b) comparison of system chattering.

Design and analysis of improved extended state observer

The structure of the extended state observer can be obtained by combining the newfal function

{\begin{matrix} e_{2} = z_{1} - y \\ {\overset{\cdot}{z}}_{1} = z_{2} - β_{01} newfal (e_{2}, γ_{2}, ϕ_{2}) + bu (t) \\ {\overset{\cdot}{z}}_{2} = - β_{02} newfal (e_{2}, γ_{2}, ϕ_{2}) \end{matrix}

(31)

Existing ESO techniques guarantee that, in the absence of time-varying perturbations, estimation errors asymptotically converge to zero globally. However, in regenerative systems subject to time-varying perturbations, errors settle near zero rather than converge exactly to it. This behavior hampers accurate state-trajectory estimation and disturbance identification. As a result, designing control laws to maintain system stability becomes more challenging. To address this issue, we enhance the ESO structure by incorporating a sign(e₂) feedback term, thereby enabling global asymptotic convergence of observation errors despite unknown bounded time-varying perturbations. The improved ESO structure is shown in Figure 4

{\begin{matrix} e_{2} = z_{1} - y \\ {\overset{\cdot}{z}}_{1} = z_{2} - β_{01} newfal (e_{2}, γ_{2}, ϕ_{2}) + bu (t) \\ {\overset{\cdot}{z}}_{2} = - β_{02} newfal (e_{2}, γ_{2}, ϕ_{2}) + Gsign (e_{2}) \end{matrix}

(32)

Figure 4.

Structure of traditional ESO and improved ESO.

Stability analysis of the improved ESO

Define the error dynamic equation

{\begin{matrix} e_{2} = z_{1} - y \\ {\overset{\cdot}{e}}_{2} = {\overset{\cdot}{z}}_{1} - \overset{\cdot}{y} \end{matrix}

(33)

Carrying forward equation (32) into equation (33)

{\begin{matrix} {\overset{\cdot}{e}}_{2} = z_{2} - β_{01} newfal (e_{2}, γ_{2}, ϕ_{2}) - \overset{\cdot}{y} + bu \\ {\overset{\cdot}{z}}_{2} = - β_{02} newfal (e_{2}, γ_{2}, ϕ_{2}) + Gsign (e_{2}) \end{matrix}

(34)

To analyze the stability of the system, the following Lyapunov function is constructed

V = \frac{1}{2} e_{2}^{2} + \frac{1}{2} z_{2}^{2}

(35)

where V > 0 and V = 0 if and only if e₂ = 0, z₂ = 0. Derivation of equation (35)

\overset{\cdot}{V} = e_{2} {\overset{\cdot}{e}}_{2} + z_{2} {\overset{\cdot}{z}}_{2}

(36)

Carrying equation (34) into equation (36)

\begin{matrix} \overset{\cdot}{V} & = e_{2} [z_{2} - β_{01} newfal (e_{2}, γ_{2}, ϕ_{2}) - \overset{\cdot}{y} + bu] \\ + z_{2} [- β_{02} newfal (e_{2}, γ_{2}, ϕ_{2}) + Gsign (e_{2})] \\ = e_{2} z_{2} - β_{01} e_{2} newfal (e_{2}, γ_{2}, ϕ_{2}) + e_{2} (bu - \overset{\cdot}{y}) \\ - β_{02} z_{2} newfal (e_{2}, γ_{2}, ϕ_{2}) + z_{2} Gsign (e_{2}) \end{matrix}

(37)

Analyzing each of the terms in equation (37), $- β_{01} e_{2} newfal (e_{2}, γ_{2}, ϕ_{2})$ , Since the design of the nonlinear function $newfal (e_{2}, γ_{2}, ϕ_{2}) > 0$ ensures that $e_{2} newfal (e_{2}, γ_{2}, ϕ_{2}) > 0$ , therefore this term is always negative and provides convergence; $- β_{02} z_{2} newfal (e_{2}, γ_{2}, ϕ_{2})$ , Since z₂ is the higher-order dynamic response to e₂, z₂ is consistent with the sign of e₂, Therefore $z_{2} newfal (e_{2}, γ_{2}, ϕ_{2}) > 0$ , Therefore this term is also always negative and serves to stabilize the extended state z₂; $e_{2} z_{2} + z_{2} Gsign (e_{2})$ , e₂ is the error, z₂ is that the extended state is driven by unmodeled dynamics or perturbations, directionally related to the trend of e₂, Thus the dynamics of the system makes e₂ and z₂ have the same sign, as a result $e_{2} z_{2} > 0$ ; due to $e_{2} z_{2} + z_{2} Gsign (e_{2}) = z_{2} e_{2} + Gsign (e_{2})$ , From the expression it can be seen that the key to the transformation of the symbols is $e_{2} + Gsign (e_{2})$ , e₂ is the direction of the error, either positive or negative, $Gsign (e_{2})$ is the sign function compensation term, whose magnitude and direction are determined by the gain G, G is designed to provide enough feedback to make $Gsign (e_{2}) > | e_{2} |$ , In order for the compensation term of the sign function to cover the error effects. When $G > | e_{2} |$ , the opposite sign of $e_{2} + Gsign (e_{2})$ , making the sum $e_{2} z_{2} + z_{2} Gsign (e_{2}) < 0$ ; The gain G of $Gsign (e_{2})$ is chosen to be large enough, So that $G > | e_{2} |$ , which ensures that the sign of $e_{2} + Gsign (e_{2})$ is ultimately dominated by $Gsign (e_{2})$ , $e_{2} z_{2} + z_{2} Gsign (e_{2}) < 0$ and of opposite sign to z₂, thus being able to ensure that, This enables effective control of the extended state and error, ensuring system stability and rapid decay of the error. $e_{2} (bu - \overset{\cdot}{y})$ , When $e_{2} > 0 (z_{1} > y)$ , observer state z₁ is higher than the actual output y, need to control the input u to decrease the output y by an increasing law, as a result $- \overset{\cdot}{y} + bu < 0 \Rightarrow bu < \overset{\cdot}{y}$ ; When $e_{2} < 0 (z_{1} < y)$ , the same reasoning follows, so that by designing a reasonable control input u, It is possible to ensure that the sign of $bu - \overset{\cdot}{y}$ is the opposite of the sign of e₂, thus making $e_{2} (bu - \overset{\cdot}{y}) < 0$ . By the above analysis carried out on each term of the Lyapunov function, it can be obtained that taking a value at $β_{01}, β_{02}, G$ that satisfies the above analysis guarantees $\overset{\cdot}{V} < 0$ . Therefore, by the Lyapunov stability proof analysis, it can be obtained that the improved ESO is asymptotically stable.

Observational performance analysis of improved ESO

In order to further understand the tracking effect of IESO on the control system z₁, By comparing the experiments to understand the control system of the same PMSM, the observation of the system state using ESO as well as IESO is obtained in Figure 5. variable z₁ faster. The comparison of Figure 5(b) clearly yields that IESO has a smaller tracking error for the state variable z₁. The performance of IESO is superior to that of ESO as can be obtained from the two figures.

Figure 5.

Effect of ESO and IESO on state z₁ observation: (a) is the tracking speed of the two observers for the observation target, and (b) is the observation accuracy of the two observers for the observation target.

Simulation analysis and experimental verification

Simulation analysis

Simulation analysis 1

Set U_dc = 311 V, f_pwm = 6000 Hz, Parameters of the simulated motor pole pair number P_n = 4, J = 0.003, magnetic chain flux = 0.175, using a period T_s = 1 × 10⁻⁶ seconds, the simulation time is 1 seconds. Motor starts with no load, Set the speed to 1000 r/min, in 0.2 seconds is to increase the load 30 N m, increase the speed to 1500 r/min when the simulation time reaches 0.5 seconds, At 0.7 seconds of simulation, the previously added load is removed until the end of the run. In order to more effectively verify the sophistication of the control scheme proposed in this paper, it is compared with the model-free adaptive terminal sliding mode control based on a sliding mode observer (MFC-ULM-ASMC) (Zhao et al., 2023).

Figure 6(e) shows the PMSM speed simulation, and Figure 6(a)–(d) shows the local zoom-in of different stages of the PMSM under the four control strategies. where the controller parameters for ADRC-IESO-ULM are c₁ = 25, c₂ = 25, l₁ = 10,000, l₂ = 100, l = 0.1, k₁ = 10, k₂ = 20, k₃ = 15, k₄ = 0.002, β₀₁ = 10,000, β₀₂ = 160,000, γ₂ = 0.5, φ₂ = 0.25, G = 20,000, b = 2000.

Figure 6.

Rotation speed waveforms of the four controllers; (a) No-load startup; (b) Load application; (c) Sudden speed surge; (d) Load removal; (e) Overall motor speed variation diagram.

Figure 6 compares the performance of four control strategies under various operating scenarios:

Figure 6(a) presents a local zoom-in of the motor’s startup. The ADRC-IESO-ULM exhibits an overshoot of 3.4 r/min, which is reduced by 94.8%, 61.4%, and 52.1% compared to the PI, ADRC, and MFC-ULM-ASMC methods, respectively. Its stabilization time is 0.01 seconds—a reduction of 99.2%, 89.5%, and 23.1% relative to the other strategies.

Figure 6(b), when a 30 N m load is applied at 0.2 seconds, the ADRC-IESO-ULM results in a speed fluctuation of 27.7 r/min. This fluctuation is 72.9%, 45.1%, and 37.5% lower than that of the other three strategies. In addition, the system regains stability in 0.004 seconds, corresponding to a reduction in stabilization time of 90.2%, 88.6%, and 20% compared to the alternatives.

Figure 6(c) shows the motor speed increasing from 1000 to 1500 r/min over 0.5 seconds. Here, the ADRC-IESO-ULM incurs a speed overshoot of 2 r/min, which is 94.6%, 50%, and 44.7% less than the overshoots observed with the other strategies. The system recovers in 0.006 seconds, reflecting reductions of 99.4%, 72.7%, and 33.3% in recovery time relative to the other methods.

Figure 6(d), following the application of a load at 0.2 seconds that is removed at 0.7 seconds, the ADRC-IESO-ULM produces a speed fluctuation of 9.1 r/min. This is 91.1%, 82.3%, and 68.3% lower than the fluctuations recorded by the other control strategies. The restoration of stability takes 0.005 seconds, which is significantly faster than the alternatives.

Simulation analysis 2

In order to verify that the improved controller suppresses the output fluctuations of the control system, the following experimental comparisons are carried out.

The results in Figure 7 show that ADRC-IESO-ULM is significantly better than ADRC and PI control in suppressing output fluctuations, and is superior to the other two control strategies.

Figure 7.

Comparison of control system output fluctuation: (a) comparison of speed fluctuation, (b) comparison of id fluctuation, and (c) comparison of iq fluctuation.

Experimental validation

The implementation steps of the experimental platform shown in Figure 8 are as follows: the simulink model of the control strategy is transferred to the DSPACE micro lab box via Ethernet, the PWM driver signals generated by the DSPACE micro lab box are transferred to the DC motor as a load and to the PMSM as a control object, and when the motor is running, the status signals generated by the Torque Transducer are returned to the computer’s display.

Figure 8.

Comprehensive experimental platform for multi-motor drive control.

In order to further verify the reliability of ADRC-IESO-ULM for PMSM control, this section presents an experimental validation comparison of PMSM using ADRC-IESO-ULM and PI control strategies. The experimental equipment uses a permanent magnet synchronous motor and a DC motor pair drag. By having the DC motor reverse with the PMSM, it allows the PMSM to generate a load. The experiments were conducted by starting the PMSM at no load and letting the speed increase to 500 r/min, and running at 500 r/min for 5 seconds, adding a 3000 mA current to a trailing DC motor allows a permanent magnet synchronous motor to apply a sudden load, remove the applied load after 10 seconds; after 30 seconds let the speed surge to 1000 r/min, After 10 seconds a current of 3000 mA is again given to the DC motor with the pair of drags, The load was removed after 10 seconds and run in this state until the end of the experiment.

Figures 9 and 10 represent the results of the experiments performed on the experimental platform with the PI control strategy, the MFC-ULM-ASMC control strategy, and the ADRC-IESO-ULM control strategy, respectively. Figure 9(a) represents the output speed of the PMSM, and through the experimental comparison, it can be obtained that the ADRC-IESO-ULM control strategy improves the disturbance immunity of the motor by 37.5% and 63.4% compared to the MFC-ULM-ASMC control strategy and the PI control strategy when the load increases or decreases abruptly, respectively.

Figure 9.

(a) Speed and (b) electromagnetic torque of motors with three control strategies.

Figure 10.

(a) id and (b) iq of motors with three control strategies.

Figure 9(b) represents the output results of the electromagnetic torque of the PMSM, when the rotational speed is increased suddenly, the electromagnetic torque of the PI control mutates to 690 N·m, the MFC-ULM-ASMC control mutates to 440 N·m, and the ADRC-IESO-ULM control mutates to 420 N·m, and comparing with the rest two control strategies, the ADRC-IESO-ULM control strategy reduces 36.2% and 4.5%.

Figure 10(a) shows the d-axis current output results of the PMSM, when the speed is increased suddenly, the current fluctuation of PI control is 2.2 A, the current fluctuation of MFC-ULM-ASMC control is 1.2 A, and the current fluctuation of ADRC-IESO-ULM control is 0.9 A, comparing with the remaining two control strategies, the ADRC-IESO-ULM control strategy reduces the current fluctuation by 59.1% and 25%.

Figure 10(b) shows the results of the q-axis current output of the PMSM, when the speed is increased suddenly, the current fluctuation of the PI control is 11 A, the current fluctuation of the MFC-ULM-ASMC control is 6.7 A, and the current fluctuation of the ADRC-IESO-ULM control is 5.8 A, and comparing with the rest of the two control strategies, the ADRC-IESO-ULM control strategy reduces the current fluctuation by 47.3% and 13.4%, respectively.

Through the above experiments and data analysis, it can be obtained that the ADRC-IESO-ULM control strategy has an effective application prospect in the field of PMSM application, and the motor speed, the fluctuation of electromagnetic torque, and the fluctuation of d-q-axis current have been effectively improved.

Conclusion

To enhance the control of permanent magnet synchronous motor under time-varying perturbations, this paper proposes an IESO that addresses key limitations of conventional designs. First, it resolves the non-differentiability issue of the traditional fal function by employing a polynomial approximation, yielding a continuous fal function that significantly reduces system chattering. Furthermore, the proposed method integrates a quadratic sliding mode control based on the ULM with an adaptive exponential convergence law, accelerating system response and improving overall stability. Comparative simulations demonstrate that the ADRC-IESO-ULM control system outperforms both PI controllers and conventional ADRC controllers in terms of tracking speed, accuracy, and anti-interference capability. Experimental results further validate these advantages: the ADRC-IESO-ULM strategy reduces q-axis current volatility by 47.3% and 13.4%, d-axis current volatility by 59.1% and 25%, and electromagnetic torque volatility by 36.2% and 4.5%, compared to PI and MFC-ULM-ASMC, respectively. In addition, it exhibits faster dynamic response, lower startup overshoot, stronger disturbance rejection, and quicker stabilization. These findings collectively confirm the robustness and efficacy of the ADRC-IESO-ULM approach for PMSM control. However, existing IESO designs face a trade-off between immunity and responsiveness. Future research should focus on decoupling the IESO to optimize these two aspects independently.

Footnotes

Author contributions

All authors contributed to the study conception and design. The first draft of the manuscript was written by J.W., and all authors commented on previous versions of the manuscript. All authors read and approved the final manuscript.

Declaration of conflicting interests

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

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This work was supported by Wuhu Core Technology Research Project (2022hg11 and 2023yf012).

ORCID iD

Jiqiang Wu

Data availability statement

Data sharing is not applicable to this article as no datasets were generated or analyzed during the current study.

References

Abdalla

Abdullah

Kassem

(2025) An adaptive frame and intelligent control approach for an autonomous hybrid renewable energy technology consisting of PV, wind, and fuel cell innovation. Alexandria Engineering Journal 114: 279–291.

Ahmed

del Rio-Chanona

Mercangöz

(2025) ARRTOC: Adversarially robust real-time optimization and control. Computers & Chemical Engineering 194: 108930.

AL-Wesabi

Al-Shamma’a

, et al. (2025) Intelligent adaptive PSO and linear active disturbance rejection control: A novel reinitialization strategy for partially shaded photovoltaic-powered battery charging. Computers and Electrical Engineering 123: 110037.

Chen

Jiang

Tang

, et al. (2023) Revised adaptive active disturbance rejection sliding mode control strategy for vertical stability of active hydro-pneumatic suspension. ISA Transactions 132: 490–507.

Chen

, et al. (2020) A variable-order fractional proportional-integral controller and its application to a permanent magnet synchronous motor. Alexandria Engineering Journal 59(5): 3247–3254.

Deng

Yao

Jiao

, et al. (2022) High dynamic output feedback robust control of hydraulic flight motion simulator using a novel cascaded extended state observer. Chinese Journal of Aeronautics 35(7): 300–309.

Denny

Kumar

(2024) An adaptive continuous nonsingular fast terminal SMC for permanent magnet synchronous motor fed electric vehicle. European Journal of Control 75: 100931.

Garrido

Luna

(2021) Robust ultra-precision motion control of linear ultrasonic motors: A combined ADRC-Luenberger observer approach. Control Engineering Practice 111: 104812.

Ghanipoor

Murguia

Mohajerin Esfahani

, et al. (2025) Robust fault estimators for nonlinear systems: An ultra-local model design. Automatica 171: 111920.

10.

Guo

Zhang

, et al. (2017) Design and implementation of a loss optimization control for electric vehicle in-wheel permanent-magnet synchronous motor direct drive system. Energy Procedia 105: 2253–2259.

11.

Han

(2009) From PID to active disturbance rejection control. IEEE Transactions on Industrial Electronics 56(3): 900–906.

12.

Hao

Yang

Gong

, et al. (2021) Linear/nonlinear active disturbance rejection switching control for permanent magnet synchronous motors. IEEE Transactions on Power Electronics 36(8): 9334–9347.

13.

Jiang

Liu

(2024) Active disturbance rejection control of permanent magnet synchronous motor speed control system based on improved fal function. In: 2024 5th international conference on mechatronics technology and intelligent manufacturing (ICMTIM), Nanjing, China, 26–28 April, pp. 219–223. New York: IEEE.

14.

Kogan

Stepanov

(2025) Robust H∞ control with transients for LTV systems based on prior knowledge and data. Automatica 173: 112083.

15.

Kuang

Guo

, et al. (2017) Research on a six-phase permanent magnet synchronous motor system at dual-redundant and fault tolerant modes in aviation application. Chinese Journal of Aeronautics 30(4): 1548–1560.

16.

Wang

Liu

, et al. (2025) A sliding mode active disturbance rejection control of DC active power filter for enhancing DC microgrid voltage robustness. Electric Power Systems Research 241: 111388.

17.

Song

Chen

, et al. (2021) Adaptive fuzzy PI controller for permanent magnet synchronous motor drive based on predictive functional control. Journal of the Franklin Institute 358(15): 7333–7364.

18.

Liu

Luo

Chen

, et al. (2020) Measurement delay compensated LADRC based current controller design for PMSM drives with a simple parameter tuning method. ISA Transactions 101: 482–492.

19.

Liu

Zhang

Shi

(2024) Improved current control for PMSM via an active disturbance rejection controller. European Journal of Control 78: 101005.

20.

Rahimi

Bavafa

Aghababaei

, et al. (2016) The online parameter identification of chaotic behaviour in permanent magnet synchronous motor by Self-Adaptive Learning Bat-inspired algorithm. International Journal of Electrical Power & Energy Systems 78: 285–291.

21.

Wang

Deng

Zhang

, et al. (2024) Enhanced disturbance observer-based hybrid cascade active disturbance rejection control design for high-precise tracking system in application to aerospace satellite. Aerospace Science and Technology 146: 108939.

22.

Wang

Yin

(2023) Practical fixed-time observer-based generalized cascade predictor design for synchronization of integrator systems with delays and complex unknowns. ISA Transactions 135: 159–172.

23.

Yan

Deng

Yang

, et al. (2024) Improved nonlinear active disturbance rejection control for a continuous-wave pulse generator with cascaded extended state observer. Control Engineering Practice 146: 105897.

24.

Yang

Huang

, et al. (2024) Approximate optimal condition model-free predictive velocity control of a direct-drive wave energy converter based on ultra-local model. Ocean Engineering 307: 118214.

25.

Yang

, et al. (2020) Efficiency improvement of permanent magnet synchronous motor for electric vehicles. Energy 213: 118859.

26.

Yao

Gao

(2021) On multi-axis motion synchronization: The cascade control structure and integrated SMC–ADRC design. ISA Transactions 109: 259–268.

27.

Zhang

Chen

, et al. (2024) A novel ultra-local model-based finite time synergetic robustness control for uncertain robotic manipulator. Journal of the Franklin Institute 361(18): 107362.

28.

Zhao

Liu

Zhou

, et al. (2023) Model-free fast integral terminal sliding-mode control method based on improved fast terminal sliding-mode observer for PMSM with unknown disturbances. ISA Transactions 143: 572–581.

29.

Zhu

Zhang

Jing

, et al. (2022) Nonlinear active disturbance rejection control strategy for permanent magnet synchronous motor drives. IEEE Transactions on Energy Conversion 37(3): 2119–2129.