Predefined-Time Adaptive Sliding Mode Speed Control of PMSM Drives Using a Switchable Power Exponent and Dual-Gain Surface

1  Introduction

The permanent magnet synchronous motor (PMSM) has been widely used in many applications to achieve high performance, such as industrial servo drives, precision manufacturing, robotic actuators and aircraft actuators, etc., because of its high-power density value, compact dimensions and low maintenance requirements [1–6]. This is further enhanced by the proliferation of EVs in the last decade, which has resulted in the emergence of tight speed control, low torque ripple and low power loss at extreme speeds and load requirements for modern traction drives [1,2]. In such applications, the performance of the speed control has direct impacts on the system performance, such as positioning accuracy in servo systems, production quality in manufacturing, and driving range in traction systems. Therefore, high-performance PMSM speed control has been a research focus that has been continuously pursued and is still an active area of research [3–8].

There are many more challenges to speed control of a PMSM than just nonlinearity. The voltage equations are cross-coupled in a speed-dependent manner in the synchronously rotating d–q reference frame. Therefore, whenever the rotor speed changes, both axes are affected at the same time. But there is another layer of complexity due to the mechanical dynamics; the torque of the electromagnetics, the load torque, the inertia, and viscous friction play back and forth in a nonlinear fashion. In addition to the nominal dynamics, the parameter uncertainties include: temperature-dependent stator resistances, wear and aging of flux linkages, load torque variations (including sudden load torque changes), and cogging torque. Convergence time predictability is another practically important, but seldom mentioned, problem: many nonlinear controllers ensure that the speed error will ultimately be driven to zero, but the time needed will vary with the magnitude of the initial error and/or with the system parameters, in a manner that is not directly specifiable by the designer. The fundamental challenge of high-performance PMSM drive design is the simultaneous resolution of cross-axis coupling and parameter uncertainty, external disturbance, and assignable convergence time.

Based on the field-oriented control (FOC) technique, the conventional proportional integral (PI) controllers have been well applied to the industrial standard of motor speed and current regulations [9,10]. They are widely used in commercial drives because they are easy to integrate into the decoupled model and because of the well-known gain selection rules. They have a basic structural restriction, though: consistency of optimization for disturbance rejection bandwidth and transient stability margin for all loading conditions in a fixed-gain pair. An increase in the proportional gain improves load rejection but risks closed-loop instability, whereas a reduction to restore stability margin sacrifices response speed [11]. Under large disturbances and significant parameter variation, tracking performance degrades in ways that retuning alone cannot correct.

This structural limitation has motivated substantial research into the advanced nonlinear control strategies for PMSM drives. Several approaches have been proposed, each one focusing on a certain aspect in relation to the bigger picture of robustness, and all reintroducing some additional constraints into the problem formulation. MPC is capable of dealing with multi-input multi-output interaction effects and constrained control input actions through one optimization problem. However, its requirement for real-time computations makes it unsuitable for high-speed drives [12]. More tractable in implementation, active disturbance rejection control (ADRC) lumps all model uncertainty into a single term estimated by an extended state observer. However, the observer bandwidth tuning remains a critical sensitivity, particularly under abrupt load torque steps [13–15]. It has been shown that adaptive controllers that adjust gains online typically cannot converge quickly enough when the disturbance varies at a rate faster than their adaptation bandwidth [16]. Passivity-based control (PBC) preserves the physical energy structure of the plant within the control design, a theoretically principled approach. However, its synthesis procedure is computationally intensive for high-order motor models [17,18]. Fuzzy and neural network controllers extend representational capacity at the cost of reduced closed-loop transparency and substantial offline training effort [19,20]. Each of these approaches addresses one or two of the three core difficulties: cross-axis coupling, parameter uncertainty, and fast external disturbance, but none resolves all three comprehensively.

The sliding mode control (SMC) technique is well known as a robust control approach whose basis lies in structural aspects, not just the parameters. This technique is based on variable structure control ideas, which were initially developed by Emelyanov and then enhanced by Utkin in his study [21,22]. SMC will drive the dynamic system’s state to move towards a well-defined sliding surface in the error space. On reaching this surface, the behavior of the closed-loop system becomes fully dependent upon the definition of the sliding surface, which does not depend at all on matched perturbations or changes in the parameters within the bounded set. Thus, to realize a PMSM speed controller using SMC, the sliding surface should be designed according to the error of motor speed and the error derivative of motor speed. Research over the subsequent decades extended this foundation in several directions, thereby producing a family of variants each targeting a specific performance gap that the basic first-order formulation leaves unresolved. Other classical sliding mode architectures have evolved robustness paradigms [23,24]. Integral SMC incorporates the tracking error integral into the surface definition, thus eliminating the reaching phase and ensuring that system robustness holds from the first instant of operation [25]. Terminal SMC introduces a fractional-power term into the surface that drives the tracking error to zero in finite time rather than asymptotically, which is significant when rapid convergence is required [26,27]. A known difficulty with fractional-power surfaces is a singularity near the origin, where the required control input grows without bound as the system state approaches equilibrium. Non-singular terminal SMC formulations resolve terminal SMC limitations by substituting a linear surface term in the neighborhood of the equilibrium, maintaining finite-time convergence without the unbounded control action [28,29]. Composite SMC strategies superimpose a switching component onto a nominal feedback law, confining the role of the switching term to compensating for the residual uncertainty that the nominal controller does not handle [30].

A further dimension of the SMC design space concerns the rate and predictability of convergence. Finite-time SMC guarantees that the tracking errors reach zero in a finite duration, an improvement over asymptotic stability; however, the settling time depends mainly on the system’s initial conditions, so the accurate convergence deadline cannot be specified in advance [8]. Fixed-time SMC eliminates this dependency by establishing an upper bound in the settling time that holds for all initial conditions; the bound is nonetheless an implicit function of system parameters rather than a freely assignable design variable, limiting the designer’s ability to impose an explicit deadline [6]. Predefined-time adaptive control is able to resolve both limitations: the convergence time upper bound is set as an explicit, freely chosen parameter independent of initial conditions and system parameters alike. Predefined-time adaptive SMC has been used for chaotic synchronization, fractional-order systems, and nonlinear systems in general, confirming its theoretical effectiveness.

The principal limitation of the SMC is chattering problems, which arise because the nominal discontinuous switching law, when implemented on digital hardware at a finite sampling rate which produces high-frequency oscillations rather than exact sliding [31,32]. In PMSM drives, chattering appears as torque and current ripple, acoustic noise, and faster wear of mechanical components, which reduces the tracking precision that the control aims to achieve. However, using the discontinuous sign function instead of a smooth approximation such as a hyperbolic tangent function reduces chattering substantially; the residual error within the approximation region is small but non-zero, representing a practical trade-off between chattering elimination and exact sliding [33]. Higher-order sliding mode control algorithms, especially the super-twisting algorithms, act on higher derivatives of the sliding variable and thus produce a continuous control signal, which reduces chattering significantly without loss of robustness [34,35]. Adaptive and state-dependent reaching laws modulate the switching gain according to the distance of the state from the surface, so that the gain is large when robustness demands it and small when the state is near the surface [36,37]. Despite these developments, a key trade-off still exists in most of the current designs.

The switching gain chattering tension has a logical resolution: if the disturbance were known, the switching gain would only need to account for the residual estimation error, rather than the full disturbance amplitude, a much smaller quantity. Disturbance observers do essentially this: they estimate the lumped disturbance from available state measurements and feed the estimate forward as a compensation signal, so the switching law only needs to handle the residual estimation error [38,39]. Across numerous PMSM studies, this composite architecture observer-based feedforward paired with an SMC switching law has been shown to simultaneously improve tracking accuracy and reduce chattering [40–43]. Despite these improvements, the convergence-time literature for PMSM SMC reveals three unresolved gaps. First, existing predefined-time stability inequalities assign fixed values conventionally expressed as −1 and +1 to the power terms of the Lyapunov condition, leaving no freedom to adjust the convergence dynamics beyond selecting the predefined time itself. Second, the systematic application of predefined-time SMC to PMSM speed control within an FOC architecture, with rigorous treatment of parametric uncertainty and bounded load disturbance, has not been reported. Third, no existing predefined-time SMC for PMSM incorporates an adaptive dual-gain function that simultaneously amplifies control action for large tracking errors and for small ones, achieving fast convergence throughout the full error magnitude range. This paper addresses all three gaps.

This work presents a PTASMSC of surface-mounted PMSM drives, developed within an FOC structure. The PI controller in the outer speed loop is replaced with the proposed PTASMSC. The inner current loops are retained in order to ensure accurate and fast tracking of current references, and this results in the commanded current reaching the motor without delay. The approach relies on a flexible predefined-time stability concept that introduces a tunable parameter to shape convergence behavior, which extends existing methods and provides better control over system response without affecting the guaranteed convergence time. An adaptive sliding surface with a dual-gain structure delivers strong correction, significant speed error and precise adjustment for minor error, which improves performance across the full operating range without extra control effort. The system’s overall response converges within the predefined time irrespective of the initial operating conditions. A smooth approximation substitutes the discontinuous switching action to minimize chattering and ensure practical implementation that preserves the stability. A comparative summary of the proposed method and alternative SMC approaches reported in the literature is shown in Table 1 to clearly differentiate the proposed method from other alternative approaches.

The rest of this paper is arranged as follows. Section 2 presents the PMSM plant model and the derivation of the lumped disturbance. Section 3 presents the theoretical foundation and realization of the proposed PTASMSC. Section 4 presents numerical simulation and HIL results based on benchmark comparisons. Section 5 provides concluding remarks and future work.

2  System Formulation and Preliminaries

2.1 PMSM Mathematical Model

The dynamic behavior of the surface-mounted PMSM is described in the synchronously rotating d–q reference frame. To derive a tractable model, motor iron-core saturation, eddy-current losses, and hysteresis effects are neglected. Additionally, for the surface-mounted topology, rotor saliency is absent, so the d–q axis stator inductances are equal: Lq=Ld=Ls. When representing the PMSM in the d–q reference frame, the motor stator voltages are described by the following relations [44]:

Vd=Rid−Lqiqωe+LddiddtVq=Riq+Ldidωe+ωeφf+Lqdiqdt(1)

where Vd, Vq are the stator voltages; id, iq represent the stator current components; R is the resistance of the stator; φf is the motor permanent magnet flux linkage; and ωe represents the rotor electrical angular speed. Under the FOC strategy with id=0, Thus, electromagnetic torque simplifies to:

Te=32pnφfiq(2)

where pn denotes the number of motor pole pairs. The following dynamic equation of the PMSM gives the rotational behaviour of the rotor shaft.

Te−TL=Jdωmdt+Bωm(3)

where B is the viscous friction coefficient, TL is the load torque, J is the rotor moment of inertia, ωm represents the mechanical angular speed.

2.2 Uncertainty and Disturbance Formulation

In real motor operating conditions, the parameters J and B do not remain constant and may change with time. Decomposing each into a nominal value and an uncertain increment, (3) becomes:

Te−TL=(J+ΔJ)dωmdt+(B+ΔB)ωm(4)

Rearranging (4) and collecting all uncertain and load terms on the right-hand side yields:

Te=Jdωmdt+Bωm+ΔJdωmdt+ΔBωm+TL(5)

Let x=ωm denote the state variable, (5) is recast as the compact state-space equation:

x˙=TeJ−BxJ+D(6)

The term D aggregates all disturbance contributions into a single lumped quantity, defined as:

D=−(ΔJJdωmdt+ΔBJωm+TLJ)(7)

The derivative of D is assumed to remain within known bounds, i.e., |D˙|≤β, where β is a positive constant. This assumption is standard in disturbance-observer design [26] and is readily satisfied under realistic operating conditions. To facilitate controller synthesis, the speed tracking error and its rate of change are considered as follows:

e1=ωm∗−ωme2=e˙1(8)

where ωm∗ is the desired reference speed. Differentiating e1 and substituting (6) gives:

e˙1=ω˙m∗−TeJ+BJx−D(9)

Eqs. (6)–(9) constitute the complete plant model on which the proposed PTASMSC controller and its disturbance observer are designed in the following sections. The lumped disturbance D, defined in (7) as D=−(ΔJJdωmdt+ΔBJωm+TLJ), satisfies |D|≤β. For PI speed regulator drives, the error dynamics of (9) are expressed as e˙1=−kpe1−ki∫e1dτ+D; for any non-zero D, this yields a persistent steady-state error ess=D/ki≠0, which fixed gains cannot eliminate, kp and ki alone. Sliding mode control resolves this structurally: a surface s(e1,e2)=0 is designed so that, once reached, the closed-loop dynamics are independent of all matched disturbances satisfying |D|≤Dmax, and (e1,e2)→0 is guaranteed. The classical linear surface s=ce1+e2 confines convergence to (e1,e2)→0ast→∞; finite-time convergence requires a fractional-power or nonlinear surface term. Furthermore, the nominal discontinuous switching control us =−εsign(s) implemented at the sampling rate fs produces chattering of amplitude O(ε/fs), exciting current ripple and mechanical resonances. Both limitations, asymptotic-only convergence and chattering, are addressed by the predefined-time nonlinear surface and tanh() approximation.

3  Proposed Controller Design

3.1 Predefined-Time Stability and Lyapunov Foundations

This section presents the stability definitions and the predefined-time Lyapunov theorem that underpin the controller design in this section. Consider a nonlinear system x˙=φ(x,t),x(0)=x0∈Rn, φ(0,t)=0.

Definition 1: The origin of the finite-time stability requires Lyapunov stability and the existence of a settling-time function T:Rn→(0,∞) such that the state x(t) becomes zero for all t≥T(x0) [29,45]. In general, T(xo) grows with ‖x0‖ and cannot be bounded independently of xo.

Definition 2: The origin of the fixed-time stability is ensured when convergence occurs within finite time, and the settling time admits a uniform upper bound Tmax>0, independent of the initial state, such that T(xo)≤Tmax for all x0∈Rn [45]. However, Tmax is an implicit function of the controller parameters and cannot be specified freely.

Definition 3: Predefined-time stability holds for the origin when fixed-time stability is satisfied, and a constant settling-time bound Tp>0 is specified independently of the initial condition and system and controller parameters [46]. The designer specifies Tp a priori; the remaining controller gains are then derived from it.

The following theorem, proved in full in [47], provides a flexible Lyapunov sufficient condition for predefined-time stability with a switchable power exponent. It is the theoretical foundation for the sliding surface (8) and control law (10).

Theorem 1: Flexible predefined-time stability: establishes the generalized Lyapunov foundation for predefined-time stability. Assume V:Rn→(0,∞) is a positive definite Lyapunov function with continuous differentiability. Therefore, the core differential inequality is formulated as:

V˙≤−ln32−2k+δ+ln32k+δ−2Tp(2V+Vk+(δ/2)−sign(V−1))(10)

where Tp is the predefined time which is greater than one, 1≤k<2, 0<δ<1, and 2k<2+δ; under these conditions, the system achieves predefined-time stability with T(x0)≤Tp for all x0∈Rn. The power term k+(δ/2)−sign(V−1) represent switching exponent.

Remark 1: Setting k=1 in (10) recovers the symmetric predefined-time condition of [15,18–23] based on fixed exponents (1−δ/2) and (1+δ/2). The parameter k∈(1,2) is therefore a strict generalisation: every prior result with symmetric 1 ± exponents is a special case k=1 of Theorem 1.

3.2 Predefined-Time Adaptive Sliding Mode Speed Control

In the FOC architecture, the d-axis motor reference current is set to in id∗=0, decoupling magnetic flux and torque control so that the motor electromagnetic torque is regulated solely through the q-axis motor current. The inner current control loops are assumed to operate with sufficiently high bandwidth such that in id=0 and iq≈iq∗ at all times, where iq∗ is the q-axis reference current, which is produced by the outer speed controller. The speed control objective is to drive the rotor’s mechanical speed ω to track a smooth reference ωr∗ within a user-specified predefined time Tp=Tp1+Tp2, despite disturbances and parameter variations.

Define the tracking error of the motor speed as:

e=ωr∗−ω(11)

Differentiating (11) and substituting the motor mechanical dynamics (3) yields the error dynamics:

e˙=ω˙r∗−Ktiq∗−TL−Bω+Δ(t)J(12)

The nominal value of the load torque TL is set to zero when unknown for practical robust nature consideration.

To facilitate controller design, define the computable feedforward term F(ω,t) and the bounded normalised disturbance d(t) as:

F(ω,t)=ωr∗+TL+BωJ(13a)

d(t)=−Δ(t)J,|d(t)|<=dmax(13b)

Substituting iq≈iq∗ (nominal inner-loop assumption), the error dynamics simplify to:

e˙=F(ω,t)−u+d(t)(14a)

u=KtJiq∗(14b)

The predefined-time control problem reduces to designing u (equivalently iq∗ via (14b)) such that the error system (14a) achieves predefined-time stability with a settling time bounded by Tp=Tp1+Tp2.

Assumption 1: The lumped disturbance d(t) remains within a known bound, such that, |d(t)|=dmax for all t≥0. where dmax considered as a known constant.

Now, from Theorem 1 and the error dynamics (7a), a novel predefined-time adaptive sliding mode surface is established for the speed loop of PMSM. The sliding surface is formulated as:

s=e+∫0t(c1e+c2|e|2k−1+δsign(|e|−1)(β1|e|b+β2|e|b)⋅tanh⁡(e))dτ(15)

where c1 and c2 are defined consistently with Theorem 1 as:

c1=2.ln3(2−2k+δ)+ln32k+δ−2Tp1(16a)

c2=ln3(2−2k+δ)+ln32k+δ−22Tp1(16b)

where constraints 1≤k<2, 0<δ<1, 2k<2+δ, β1>β2>1, 0<b<1, and Tp1>0 is the designer-specified convergence time for the sliding phase. The developed PTASMSC law will generate motor reference q-axis current as:

iq∗=JKt(F(ω,t)+c1e+c2|e|2k−1+δsign(|e|−1)⋅(β1|e|b+β2|e|b)⋅tanh(e)+c4s+c5|s|2k−1+δsign(|s|−1)⋅(β1|s|b+β2|s|b)⋅tanh(s)+ε⋅tanh(s))(17)

The gains c4 and c5 are defined analogously to c1 and c2 but with Tp1 replaced by Tp2:

c4=2.ln3(2−2k+δ)+ln32k+δ−2Tp2(18a)

c5=ln3(2−2k+δ)+ln32k+δ−22Tp2(18b)

The disturbance compensation gain ε must satisfy:

ε≥dmax(19)

where Tp2 > 0 is the predefined reaching time. The total predefined convergence time for the speed tracking error is Tp=Tp1+Tp2, which depends solely on the designer-chosen constants Tp1 and Tp2, and they are independent of all system initial conditions. The structure of the proposed FOC-based PMSM drive with the proposed PTASMSC in the outer motor speed loop is illustrated in Fig. 1. The proposed PTASMSC is responsible for generating the motor reference current iq∗, which is tracked by inner PI current controllers through Park/Clarke transformations and an SVPWM inverter. The detailed structure and the flowchart of the proposed PTASMSC controller are depicted in Figs. 2 and 3, respectively.

Figure 1: Detailed block diagram of the FOC of PMSM drives using proposed PTASMSC for speed loop and PI controllers for current loop.

Figure 2: Algorithmic block configuration of the proposed PTSMC for speed control using nonlinear error mapping and adaptive dual-gain surface feedback path.

Figure 3: Operational flowchart of the proposed PTASMSC for PMSM drives showing the execution procedure from parameter reading to final generation of reference quadrature current (iq∗).

3.3 Controller Stability Analysis

This subsection provides a rigorous proof that the proposed control schemes (15)–(19) ensures that the tracking speed error of the motor converges to zero according to the predefined time Tp=Tp1+Tp2.

Theorem 2: Consider the PMSM speed error dynamics (8) under Assumption 1. With the sliding surface defined by (15) and the control law defined by (17)–(19), the following holds:

(i)   When the error trajectory lies on the sliding surface (s=0), the speed tracking error e(t) converges to zero within the predefined time Tp1.

(ii)   Starting from any initial condition, the tracking speed error of the motor reaches the sliding surface within the predefined time Tp2. Consequently, e(t)→0 within the total predefined time Tp=Tp1+Tp2, independent of all initial conditions.

(A) Proof (Phase 1—convergence of the sliding surface)

When s = 0, the Lyapunov function candidate is considered:

V1=12e2(20)

Differentiating (20):

V1˙= ee˙(21)

Setting s=0 and differentiating (15) with respect to time gives s˙=0, from which the reduced error dynamics on the sliding surface are:

e˙=−(c1e+c2|e|2k−1+δsign(|e|−1)⋅(β1|e|b+β2|e|b)⋅tanh(e))(22)

Substituting (22) into (21):

V˙1=−c1e2−c2|e|2k+δsign(|e|−1)⋅(β1|e|b+β2|e|b)⋅tanh(e)(23)

The given adaptive function h(e)=(β1|e|b+β2/|e|b) attains its global minimum hmin(e)=2β1β2≤2 by using the arithmetic mean and geometric mean (AM–GM) inequality, since β1>β2>1, tanh(e)⋅e≥0and, c2|e|(2k+δ)sign(|e|−1)≥0, then Eq. (16) satisfies:

V1˙≤−c1e2−2c2|e|2k+δsign(|e|−1)(24)

Expressing the right-hand side in terms of V1 using e2=2V1 and |e|∧(2k+δ)=(2V1)∧(k+δ/2), and substituting definitions (16a) and (16b):

V˙1≤−ln32−2k+δ+ln32k+δ−2Tp1(2V1+V1k+(δ/2)−sign(|e|−1))(25)

Condition (25) exactly matches the predefined-time stability criterion of Theorem 1. Therefore, V1(t)→0 and e(t)→0 within the specified predefined time Tp1. This completes the proof of Phase 1.

(B) Proof (Phase 2—convergence to the sliding surface)

The following Lyapunov function candidate is used for the reaching phase.

V2˙=12s2(26)

Its time derivative is:

V2˙=ss˙(27)

Differentiating sliding surface of (15) with respect to time and using the error dynamics of Eq. (14a):

s˙=F(ω,t)−u+d(t)+c1e+c2|e|2k−1+δsign(|e|−1)⋅(β1|e|b+β2|e|b)⋅tanh⁡(e)(28)

Substituting the control law (17) into (28), the feedforward term F(ω, t) and the error-feedback component cancel exactly, yielding:

s˙=d(t)−(c4s+c5|s|2k−1+δsign(|s|−1)⋅(β1|s|b+β2|s|b)⋅tanh⁡(s))−ε⋅tanh⁡(s)(29)

Therefore, the derivative of the V2˙ becomes:

V˙2=s⋅s˙<=s⋅d(t)−(c4s2+c5|s|2k+δsign(|s|−1)⋅(β1|s|b+β2|s|b))−ε|s|⋅tanh⁡(s)(30)

Since |d(t)|≤dmax≤ε| from condition (19), and |s|.tanh(s)>0 for s≠0, the disturbance term is dominated by compensation term: s.d(t)≤ε|s|.hanh(s). Applying the same adaptive function bound hmin≥2, Eq. (23) reduces to:

V˙2≤−(c4s2+2c5|s|2k+δsign(|s|−1))(31)

Expressing in terms of V2˙ using s2=2V˙2 and substituting definitions (18a) and (18b):

V˙2≤−ln3(2−2k+δ)+ln32k+δ−2Tp2(2V2+V2k+δ2sign(|s|−1))(32)

By Theorem 2, condition (32) guarantees V2(t)→0, hence s(t)→0, within the predefined time Tp2. Once the sliding surface is reached at t=t1≤Tp2, Phase 1 guarantees e(t)→0 within an additional interval Tp1. Therefore, the total speed tracking error converges to zero within Tp=Tp1, for all initial conditions.

Remark 2: The settling time T(x0) for the closed-loop control system satisfies T(x0)≤Tp=Tp1≤Tp=Tp1+Tp2 for all xo∈Rn, where Tp1,Tp2>0 are freely assigned scalars independent of all system parameters. This contrasts with finite-time stability, where T≈c1|xo|α grows with |xo|, and with fixed-time stability, where Tmax=f(k,δ) is implicit in the controller gains. The designer selects Tp1 and Tp2, and this leads to a shift in the convergence time without modifying any other controller parameters.

Remark 3: The parameter k∈(1,2) modulates the power exponents k=±δ/2 in the Lyapunov inequality (18). Lemma 2.2 shows that the baseline predefined-time stability is ensured by choosing the symmetric power exponents 1±δ/2 as the baseline. Setting k=1 scales the flexible power exponents (k±δ/2) back to this standard symmetric configuration. For k>1, the upper exponent k+δ/2>1, which steepens the decay rate V1 for V1>1, while constraint 2k<2+δ keeps k−δ/2<1, preserving finite-gain behavior for V1<1. Because the sign (|e|−1) switching couples both branches non-linearly, the net reduction in settling time is not monotone in k; the recommended tuning sequence is: fix Tp1, Tp2, then sweep k∈(1,2) and δ∈(0,1) subject to 2k<2+δ.

Remark 4: Define h(e)=β1|e|b+β2|e|b. By the AM–GM inequality, h(e)≥2β1β2=hmin for all e≠0, with equality at |e|=(β2/β1)1/b. As |e|→∞, h(e)∼β1|e|b→∞; as |e|→0, h(e)∼β2/|e|b→∞. Consequently, the effective nonlinear gain c2h(e)|e|n on the surface (8) satisfies c2h(e)|e|n≥c2hmin|e|n≥2c2|e|n, which underpins the bound in (17), while growth at both extremes accelerates convergence beyond what a fixed-gain term c2|e|n alone achieves.

Remark 5: The substitution sign(⋅)→tanh(μ) with μ=100 introduces the pointwise approximation error |sign(s)−tanh(μs)|=1−tanh(μ|s|)≤2e−2Tx. This modifies inequality (24) to V˙2≤−(c4s2+2c5|s|nsign(|s|−1))+ρ(s), where ρ(s)=ε|s|(1−tanh⁡(μ|s|))≤2ε|s|e−2Tx. Since ρ(s)→0 exponentially as μ→∞, trajectories converge to a residual set of radius O(e−2Tx/μ); for μ=100, this is numerically negligible. Simultaneously, the continuous signal tanh(100s) eliminates the discontinuous switching that causes current chattering in the q-axis reference iq∗.

4  Results and Discussion

This section of the article investigates the validity and robustness of the proposed PTASMSC of SMPMSM drive through six case studies that have been simulated in MATLAB/Simulink and experiments in hardware-in-the-loop (HIL) based on an OPAL-RT platform. The two inner PI current controllers of the FOC are implemented with a bandwidth equal to ten times the outer speed loop. The parameters of the SMPMSM that were used for verification purposes are given in Table 2.

Case 1: Controller’s parameters tuning: The design parameters govern the proposed PTASMSC controller, Eqs. (15)–(19), and the evolution of three key parameters across the two simulation versions is summarized in Table 3. The key tuning parameters of the proposed PTASMSC V2 based on the sliding gain coefficient (c1), hyperbolic tangent smoothing factor (μ), and adaptive disturbance compensation gain (ε) are thoroughly analyzed and compared to the standard baseline configuration V1 as shown in Fig. 4.

Figure 4: Quantitative comparison of critical controller tuning parameters for V˙1 and V˙2 version: (a) gain coefficient c1 reduced 7.8× by dual-gain surface adjustment (b) dynamic boundary layer variation through tanh smoothing factor µ and (c) disturbance compensation gain ε=50.

Moving from v1 to v2 attenuates the required switching gain c1 from 464 s−1 to a conservative 60 s−1. This achieves identical tracking convergence speed while relaxing unnecessary high-gain control stress on the PMSM drive system by an absolute reduction of the primary switching gain magnitude by a factor of 7.8 through selecting A1=0.6 instead of 0.1, suppressing chattering to approximately 0.55 A peak-to-peak as illustrated in Fig. 4a. The tuning factor μ is moderated from 100 down to 50. This smoothly doubles the functional boundary layer width (δb=1/μ) from 0.010 to 0.020(rad/s), providing exceptional high-frequency chattering mitigation without compromising tracking precision as shown in Fig. 4b. From Fig. 4c, it can be observed that V1 violates the sliding reaching condition because ε=5, which is much less than dmax=1000 rad/s2, leaving it vulnerable to sliding phase breakdown. Conversely, the proposed V2 configuration safely boosts the adaptive disturbance compensation gain threshold to ε=50, ensuring robust, invariant tracking performance against severe off-scale load variations. Finally, the transition process from the controller V1 to V2 are summarized in Table 4.

Case 2: Performance of step response at no load torque: The comparison of the motor speed step response between the proposed PTASMSC controller, conventional PI controller, and the PTSMC at 200 (rad/s) reference speed and no-load torque is visualized in Fig. 5a. The stator current in the q-axis has been verified using the aforementioned controller, and the results are illustrated in Fig. 5b.

Figure 5: Step response simulation results under 200 (rad/s) step speed command at zero load torque (a) rotor mechanical speed track highlighting 0% overshoot and a 0.03 s settling time for the proposed PTASMSC, (b) Iq stator current (A).

It is clear from the comparison results that the dynamic performance of the proposed PTASMSC strategy is better than that of the conventional methods. The conventional PI controller has a substantial overshoot of 20% and a settling time of 0.25 s. The PTSMC is able to eliminate this overshoot and reduce settling time to 0.05 s while introducing a longer rise time. In contrast, the proposed PTASMSC achieves the most efficient response, reaching the 200 rad/s reference in 0.03 s in a completely stable manner with no overshoot or oscillation. The convergence is very fast, which validates the efficiency of the adaptive predefined-time mechanism to improve the dynamic performance of motor drive systems.

Case 3: Performance of reversal speed at no load torque: As shown in Fig. 6a, the performance of the developed PTASMSC controller is tested by abruptly changing the motor speed from 250 (rad/s) to −250 (rad/s) at 0.4 s after motor running with no load and comparing the results with both PI and PTSMC controllers. Among the other two controllers, the proposed PTASMSC successfully combines the benefits of high-speed tracking capability with high stability. It mitigates the overshoot of the linear PI control and overcomes the sluggish response in speed characteristic of standard PTSMC. This makes it the most effective and powerful approach for motor speed control under these specific test conditions. The performance of the motor current in the q-axis at 0.4 s speed reversal is illustrated in Fig. 6b. The result confirms the superiority of the proposed PTASMSC in disturbance rejection, minimizing the current stress (peak magnitude), and providing the fastest stabilization.

Figure 6: Dynamic response evaluation during a motor high-speed four-quadrant step from +250 (rad/s) to −250 (rad/s) executed at t = 0.4 s (a) motor speed tracking comparison presenting linear overshoot mitigation, (b) quadrature current transient showing optimized peak current containment.

Case 4: Performance of step response at 5 N·m load torque: Fig. 7a shows a comprehensive comparative analysis of the motor speed response when a reference speed of 250 (rad/s) and a 5 N·m load torque are applied at 0.4 s. The results confirm that the proposed PTASMSC successfully enhances the anti-disturbance capability. By integrating a switchable power exponent and dual-gain surface, the controller provides an optimized balance of high-speed tracking and robust disturbance rejection, making it idealized for high-performance industrial applications. The transient response of the quadrature current Iq for the compared control strategies is illustrated in Fig. 7b. It can be observed that the proposed PTASMSC achieves the best balance between high-speed transient response and robust disturbance rejection. By introducing a high-amplitude, fast-converging torque current, it eliminates the oscillatory tendencies of classical PI control and the sluggishness of standard sliding mode techniques.

Figure 7: Anti-disturbance verification at 250 (rad/s) motor speed with a sudden 5 N.m load torque applied at 0.4 s (a) motor speed droop profile shows minimized 0.04% speed drop, (b) Iq current adjustment at 0.01 s recovery time.

Case 5: Speed step change response performance: Fig. 8a shows the speed response performance of the motor drive under different step speed commands, and Fig. 8b shows the stator q-axis current response of the three control strategies. The reference speed is 50 (rad/s) at the starting instant, 200 (rad/s) after 0.26 s, and 100 (rad/s) at 0.52 s respectively. The results show that the PTASMC achieves better performance compared to the other two methods in motor speed control of the three. It minimizes the compromise of fast response time and stability, and allows fast tracking without overshoots as found with conventional PI control. It is also well adapted for high-performance applications where exactness and rapid recovery from disturbances are important. Both the PTSMC and the PTASMC exhibit the “predefined-time” property; the error tends to zero after a finite amount of time from the step change, no matter what the initial conditions are or how large the step change is. The PTASMC is very stable at the speed drop at 0.52 s. The PI controller takes nearly 0.2 s to converge, and the PTASMC achieves convergence very quickly. After the predefined time, all three controllers converge to the reference; the steady state signal of the sliding mode variants PTSMC and PTASMC is very clean, without chattering as is generally seen with standard SMC.

Figure 8: Multi-step motor speed tracking profile under varying reference sequences 50 (rad/s) to 200 (rad/s) to 100 (rad/s) (a) motor speed response showing condition-independent predefined-time tracking (b) transient Iq current response validating seamless stabilization.

To provide a rigorous benchmark evaluation, Table 5 outlines the quantitative performance metrics of the conventional PI, standard PTSMC, and the proposed PTASMSC strategies under both transient startup and sudden load disturbance conditions. The proposed PTASMSC has a settling time of 0.03 s during the no-load startup transient, which is 75% and 41.67% faster than conventional PI (0.14 s) and conventional PTSMC (0.06 s), respectively. The fast tracking is believed to be due to the proposed switchable power exponent mechanism which dynamically changes the contraction trajectory according to the size of the state-space error margin.

In addition, the proposed control framework has greater anti-external load disturbance stiffness when subjected to abrupt external load disturbance of 5 N.m. Furthermore, the proposed control framework has better anti-external load disturbance stiffness when it is subjected to 5 N.m abrupt external load disturbance. The maximum dynamic speed drop is precisely kept to a minimal value of only 0.05 rad/s, corresponding to a speed drop ratio of almost 0.025%. By contrast, the conventional PI controller suffers a severe transient deceleration of 12.5 rad/s (6.25% speed drop ratio) while the base line of the conventional PTSMC has a drop of 0.60 rad/s (0.30% speed drop ratio). This is a 95.6% decrease in speed deviation along the PI loop and an 81.67% increase compared to the baseline PTSMC. At the same time, the speed recovery time (trec) is reduced to an ultra-fast value of 0.002 s, which proves that the adaptive dual-gain surface very quickly estimates and nullifies the lumped disturbance vector. Lastly, the steady-state tracking error is shown to be very tightly bounded to a very small tracking error band of ±0.01 rad/s. These comparative results clearly show the proposed PTASMSC method is able to overcome the classic engineering trade-off between chattering suppression at high frequencies and aggressive disturbance rejection in high performance PMSM drive applications.

Case 6: Proposed controller evaluation using four types of disturbance profiles: In this case, a number of comprehensive simulation tests were then performed at a fixed reference speed of 200 rad/s for the aim assessing of the tracking resilience and disturbance rejection performance of the control loops under four challenging disturbance types. Under these disturbances, the comparative study of the conventional PI controller, PTSMC, and the proposed PTASMSC strategy is shown in Fig. 9. For an abrupt step load change as shown in Fig. 9a, the linear integrator propagation delays result in a significant speed droop in the PI loop, despite this, the Improved PTSMC reduces the speed droop recovery time, but it still exhibits an obvious droop; on the other hand, the Proposed PTASMSC shows an outstanding dynamic stiffness, with small speed deviation and quickly brings back the tracking errors to zero within its pre-defined time window. For continuous sinusoidal disturbances as illustrated in Fig. 9b, time-varying periodic loads cause permanent phase lag in the PI loop, and bounded oscillations in the Improved PTSMC, while the Proposed PTASMSC completely decouples the ripple by means of the real-time dual-gain adaptive mechanism. In case of high-frequency cogging torque pulsation as depicted in Fig. 9c, the trajectory of the PI loop is distorted, and fixed-gain interactions result in severe chattering phenomenon in the Improved PTSMC, however, the proposed PTASMSC uses an adaptive layer of singularity free design to eliminate the disturbance with high fidelity and smooth regulation. Lastly, when the PI loop is subject to heavy jitter while under concurrent stochastic noise and parameter variation in rotor inertia J and flux linkage φf as demonstrated in Fig. 9d, the proposed PTASMC shows a high sensitivity to noise, while the Proposed PTASMSC still maintains very good tracking integrity demonstrating that the switchable power exponent and dual-gain surface provide excellent parameter insensitivity and robustness to the PI loop under heavy jitter and concurrent stochastic noise.

Figure 9: Motor speed tracking under 200 (rad/s) reference speed (a) abrupt step load change (b) continuous sinusoidal disturbance (c) cogging torque pulsations (d) stochastic noise and parameter shifts.

Table 6 is a summary of the quantitative performance comparison of the three control strategies for different challenging disturbance profiles. The metrics are based on tracking accuracy, transient response, quality of control effort (chattering), and robustness to parameter uncertainties.

Case 7: Experimental performance response using HIL: To verify the practical applicability of the developed PTASMSC controller, HIL simulations are conducted using OPAL-RT. The Fig. 10 shows the reference and the measured motor rotor speeds with the reference motor speed set to 100 (rad/s). The actual motor speed tracks the reference within a short time and achieves zero steady-state error. Fig. 11 presents the speed response for a reference speed of 200 (rad/s), which demonstrates the fast dynamic response of the controller. Fig. 12 illustrates the PMSM response under a sudden change in reference speed from 100 (rad/s) to 200 (rad/s).

Figure 10: Real-time HIL experimental result capture on the OPAL-RT platform validating actual motor speed tracking and zero steady-state error under a steady 100 (rad/s) reference motor speed.

Figure 11: Real-time HIL experimental result capture on the OPAL-RT platform displaying high-speed tracking convergence and boundary layer stability for a steady 200 (rad/s) reference motor speed.

Figure 12: Real-time HIL experimental waveform capture on the OPAL-RT platform displaying the transient physical response and tracking agility of the drive system during a sharp, step-wise speed alteration.

5  Conclusions

A new predefined-time adaptive sliding mode speed control (PTASMSC) strategy based on switchable power exponent and a dual-gain surface is proposed and validated in this paper for dynamic response enhancement of SMPMSM drives. The adaptive sliding surface embedded with a dual-gain function addresses a persistent limitation of fixed-gain sliding mode controllers: the corrective gain adjusts to the instantaneous magnitude of the speed error, so the controller delivers high corrective action during large transients and precise fine correction near the equilibrium without any compromise to the stability proof. The guaranteed convergence time is determined entirely by two designer-specified constants and remains independent of all initial conditions and motor parameters. Extensive simulation using MATLAB and experimentation in hardware-in-the-loop on the OPAL-RT real-time platform under different motor and load torque conditions have been conducted. Comparative results show that the proposed PTASMSC performs much better than the conventional PI and PTSMC methods under various step-speed references and sudden load torque disturbances. The quantitative analysis confirms that the proposed controller can indeed provide an extremely fast tracking within a desired time window; in this case, a transient settling time of only 0.035 s, which is 75% less than the conventional PI baseline, without any maximum overshoot (0%). In addition, the adaptive dual gain allows for an even better robustness to external disturbances with a maximum speed drop ratio of only 0.04% and a recovery time of almost zero, 0.01 s. Most significantly, the switchable power exponent and dual-gain surface make the error converge quickly for large deviations and prevent high-frequency chattering in the quadrature current (Iq) loop when the error is small in the steady state. Therefore, the proposed PTASMSC provides deterministic predefined-time convergence and chattering attenuation for the high-performance industrial PMSM speed regulation application.

Future work will pursue two directions. The first is an investigation of the nonlinear relationship between the switchable power exponent and the actual settling time, intending to derive a closed-form expression that replaces the current simulation-based tuning step. The second is an extension of the predefined-time technique to more complex drive configurations, including multi-motor systems and drives subject to demagnetization and inter-turn fault conditions, where the structural robustness of the predefined-time approach may offer a particular advantage.

Acknowledgement: None.

Funding Statement: The authors received no specific funding for this study.

Author Contributions: Ali H. Numan conceived and conducted the experiments using HIL, analyzed the results, interpreted the findings, and wrote the draft manuscript. also contributed to the selection of the evaluation procedures. Ashwaq Q. Hameed performed supervision, simulation, and analysis. All authors reviewed and approved the final version of the manuscript.

Availability of Data and Materials: The authors confirm that the data used in this study are available on request.

Ethics Approval: Not applicable.

Conflicts of Interest: The authors declare no conflicts of interest.

References

1. Busarello TDC, Bubshait A, Varaprasad OVSR, Alsaleem A, Simões MG. A comprehensive methodology of field-oriented control design with parameter variation analysis for interior permanent magnet synchronous machine drives. IEEE Access. 2025;13(4):89524–41. doi:10.1109/ACCESS.2025.3571610. [Google Scholar] [CrossRef]

2. Pilla R, Tummala AS, Chintala MR. Tuning of extended Kalman filter using self-adaptive differential evolution algorithm for sensorless permanent magnet synchronous motor drive. IJE Trans B Appl. 2016;29(11):1565–73. doi:10.5829/idosi.ije.2016.29.11b.10. [Google Scholar] [CrossRef]

3. Zhang B, Zhang W, Luo H. Two-dimensional control of a fractional-order permanent magnet synchronous motor model for industrial robots. Electr Power Syst Res. 2025;249(9):111963. doi:10.1016/j.epsr.2025.111963. [Google Scholar] [CrossRef]

4. Belkhier Y, Fredj S, Rashid H, Benbouzid M. Robust nonlinear control of permanent magnet synchronous motor drives: an evolutionary algorithm optimized passivity-based control approach with a high-order sliding mode observer. Eng Appl Artif Intell. 2025;145(12):110256. doi:10.1016/j.engappai.2025.110256. [Google Scholar] [CrossRef]

5. Ping Z, Li Y, Song Y, Huang Y, Wang H, Lu JG. Nonlinear speed tracking control of PMSM servo system: a global robust output regulation approach. Control Eng Pract. 2021;112(3):104832. doi:10.1016/j.conengprac.2021.104832. [Google Scholar] [CrossRef]

6. Numan AH, Hameed AQ. Optimum model predictive torque control of PMSM drives for PV water pumping systems. J Eur Des Systèmes Autom. 2025;58(9):1975–86. doi:10.18280/jesa.580919. [Google Scholar] [CrossRef]

7. Quyet Than D, Thai Nguyen D, Dich Nguyen Q, Nam Giap V. Fixed time homogeneous stability-based sliding mode control and disturbance observer for surface mounted permanent magnet synchronous motors. Control Eng Pract. 2026;172(4):106895. doi:10.1016/j.conengprac.2026.106895. [Google Scholar] [CrossRef]

8. Rafaq MS, Jung JW. A comprehensive review of state-of-the-art parameter estimation techniques for permanent magnet synchronous motors in wide speed range. IEEE Trans Ind Inform. 2020;16(7):4747–58. doi:10.1109/TII.2019.2944413. [Google Scholar] [CrossRef]

9. Gao Z, Yuan H, Lv G, Dai R, Wang H, Liu Y, et al. A composite finite-time control scheme with disturbance observer for robust speed tracking of permanent magnet synchronous motor. Electr Power Syst Res. 2026;254:112620. doi:10.1016/j.epsr.2025.112620. [Google Scholar] [CrossRef]

10. Song W, Li J, Ma C, Xia Y, Yu B. A simple tuning method of PI regulators in FOC for PMSM drives based on deadbeat predictive conception. IEEE Trans Transp Electrif. 2024;10(4):9852–63. doi:10.1109/TTE.2024.3385114. [Google Scholar] [CrossRef]

11. Zwerger T, Mercorelli P. Inner loop model predictive control and outer loop PI reference governor for PMSMs with input and state saturation for torque control. Math Comput Simul. 2025;228:178–201. doi:10.1016/j.matcom.2024.08.006. [Google Scholar] [CrossRef]

12. Yu R, Liu M, Wang M, Chen YH, Li C. Robust control design and experimental validation of permanent magnet synchronous motors with cyber interference and physical uncertainty. Mech Syst Signal Process. 2026;244(1):113792. doi:10.1016/j.ymssp.2025.113792. [Google Scholar] [CrossRef]

13. Velarde-Gomez S, Giraldo-Suarez E, Garcés-Ruiz A. Horizon-one model-predictive control with stability guarantee for a permanent magnet synchronous motor. Control Eng Pract. 2026;173(6):106948. doi:10.1016/j.conengprac.2026.106948. [Google Scholar] [CrossRef]

14. Xu W, Junejo AK, Liu Y, Hussien MG, Zhu J. An efficient antidisturbance sliding-mode speed control method for PMSM drive systems. IEEE Trans Power Electron. 2021;36(6):6879–91. doi:10.1109/TPEL.2020.3039474. [Google Scholar] [CrossRef]

15. Tian Y, Du K, Qu J, Feng L, Chai Y. A learning observer-based control strategy for PMSM with position sensor fault in railway. Control Eng Pract. 2024;142(7):105705. doi:10.1016/j.conengprac.2023.105705. [Google Scholar] [CrossRef]

16. Fan Y, Chen J, Zhang Q, Cheng M. An improved inertia disturbance suppression method for PMSM based on disturbance observer and two-degree-of-freedom PI controller. IEEE Trans Power Electron. 2023;38(3):3590–9. doi:10.1109/TPEL.2022.3218842. [Google Scholar] [CrossRef]

17. Zhang J, Ren W, Sun XM. Current-constrained adaptive robust control for uncertain PMSM drive systems: theory and experimentation. IEEE Trans Transp Electrif. 2023;9(3):4158–69. doi:10.1109/TTE.2023.3236431. [Google Scholar] [CrossRef]

18. Khanchoul M, Hilairet M, Normand-Cyrot D. A passivity-based controller under low sampling for speed control of PMSM. Control Eng Pract. 2014;26:20–7. doi:10.1016/j.conengprac.2013.12.013. [Google Scholar] [CrossRef]

19. Belkhier Y, Oubelaid A. Passivity-based control of PMSM servo system with load torque adaptation: theoretical and experimental validation. IEEE Trans Transp Electrif. 2025;11(1):4494–503. doi:10.1109/TTE.2024.3463540. [Google Scholar] [CrossRef]

20. Zheng Y, Zhao H, Zhen S, Sun H. Fuzzy-set theory based optimal robust constraint-following control for permanent magnet synchronous motor with uncertainties. Control Eng Pract. 2021;115(3):104911. doi:10.1016/j.conengprac.2021.104911. [Google Scholar] [CrossRef]

21. Emelyanov SV. Variable structure control systems. Moscow, Russia: Nauka; 1967. (In Russian). [Google Scholar]

22. Utkin VI. Variable structure systems with sliding modes. IEEE Trans Autom Control. 1977;22(2):212–22. doi:10.1109/TAC.1977.1101446. [Google Scholar] [CrossRef]

23. Hung JY, Gao W, Hung JC. Variable structure control: a survey. IEEE Trans Ind Electron. 1993;40(1):2–22. doi:10.1109/41.184817. [Google Scholar] [CrossRef]

24. Wu L, Liu J, Vazquez S, Mazumder SK. Sliding mode control in power converters and drives: a review. IEEE. 2022;9(3):392–406. doi:10.1109/jas.2021.1004380. [Google Scholar] [CrossRef]

25. Li K, Ding J, Sun X, Tian X. Overview of sliding mode control technology for permanent magnet synchronous motor system. IEEE Access. 2024;12(12):71685–704. doi:10.1109/ACCESS.2024.3402983. [Google Scholar] [CrossRef]

26. Hou Q, Ding S, Yu X. Composite super-twisting sliding mode control design for PMSM speed regulation problem based on a novel disturbance observer. IEEE Trans Energy Convers. 2021;36(4):2591–9. doi:10.1109/TEC.2020.2985054. [Google Scholar] [CrossRef]

27. Zhang K, Wang L. Adaptive dynamic sliding mode control of permanent magnet linear synchronous motor based on feedback linearization. Trans China Electrotech Soc. 2021;36(19):4016–24. doi:10.1109/iciea.2007.4318554. [Google Scholar] [CrossRef]

28. Wu L, Lei B, Sun M. Steady-state error band improvement using dead-beat terminal sliding mode control. IEEE Trans Circuits Syst II Express Briefs. 2022;69(8):3590–4. doi:10.1109/TCSII.2022.3169834. [Google Scholar] [CrossRef]

29. Shi S, Guo X, Xu S, Min H, Dai L. Recursive higher order non-singular terminal sliding mode control with prescribed convergence time: application to PMSM servo systems. IEEE Trans Autom Sci Eng. 2025;22:17955–66. doi:10.1109/TASE.2025.3586612. [Google Scholar] [CrossRef]

30. Xu B, Zhang L, Ji W. Improved non-singular fast terminal sliding mode control with disturbance observer for PMSM drives. IEEE Trans Transp Electrif. 2021;7(4):2753–62. doi:10.1109/TTE.2021.3083925. [Google Scholar] [CrossRef]

31. Hong J, Lin X, Zhang J, Huang W, Yan B, Li X. A composite sliding mode control with the first-order differentiator and sliding mode observer for permanent magnet synchronous machine. ISA Trans. 2024;147(3):489–500. doi:10.1016/j.isatra.2024.02.006. [Google Scholar] [PubMed] [CrossRef]

32. Wang Y, Feng Y, Zhang X, Liang J. A new reaching law for antidisturbance sliding-mode control of PMSM speed regulation system. IEEE Trans Power Electron. 2020;35(4):4117–26. doi:10.1109/TPEL.2019.2933613. [Google Scholar] [CrossRef]

33. Yim J, You S, Lee Y, Kim W. Chattering attenuation disturbance observer for sliding mode control: application to permanent magnet synchronous motors. IEEE Trans Ind Electron. 2023;70(5):5161–70. doi:10.1109/tie.2022.3189074. [Google Scholar] [CrossRef]

34. Nguyen AT, Basit BA, Choi HH, Jung JW. Disturbance attenuation for surface-mounted PMSM drives using nonlinear disturbance observer-based sliding mode control. IEEE Access. 2020;8:86345–56. doi:10.1109/ACCESS.2020.2992635. [Google Scholar] [CrossRef]

35. Ullah A, Ullah S, Zhang Z, Pan J. Smooth super-twisting sliding mode control design with high order super-twisting observer for speed tracking control of permanent magnet synchronous motor drive system. ISA Trans. 2025;162(4):227–42. doi:10.1016/j.isatra.2025.04.007. [Google Scholar] [CrossRef]

36. Hou S, Han J, Chu Y, Zhou C, Fei J. Practical terminal sliding mode control of active power filters by self-organizing Hermite fuzzy neural network. IEEE Trans Power Electron. 2026;41(1):626–39. doi:10.1109/TPEL.2025.3603892. [Google Scholar] [CrossRef]

37. Ayyarao T, Ganesh M. Enhanced direct torque control of PMSM drives using adaptive sliding mode strategy. AIP Conf Proc. 2025;3298:040039. doi:10.1063/5.0279223. [Google Scholar] [CrossRef]

38. Xu X, Wang L, Zhang Z, Wu Z. A novel robust control for permanent magnet synchronous motor integrating deadbeat predictive current control and sliding mode observer. Alex Eng J. 2025;130(4):57–67. doi:10.1016/j.aej.2025.08.057. [Google Scholar] [CrossRef]

39. Chen WH, Ballance DJ, Gawthrop PJ, O’Reilly J. A nonlinear disturbance observer for robotic manipulators. IEEE Trans Ind Electron. 2000;47(4):932–8. doi:10.1109/41.857974. [Google Scholar] [CrossRef]

40. Yang J, Li S, Yu X. Sliding-mode control for systems with mismatched uncertainties via a disturbance observer. IEEE Trans Ind Electron. 2013;60(1):160–9. doi:10.1109/TIE.2012.2183841. [Google Scholar] [CrossRef]

41. Yang T, Deng Y, Li H, Sun Z, Cao H, Wei Z. Fast integral terminal sliding mode control with a novel disturbance observer based on iterative learning for speed control of PMSM. ISA Trans. 2023;134(4):460–71. doi:10.1016/j.isatra.2022.07.029. [Google Scholar] [CrossRef]

42. Zhao K, Jia N, She J, Dai W, Zhou R, Liu W, et al. Robust model-free super-twisting sliding-mode control method based on extended sliding-mode disturbance observer for PMSM drive system. Control Eng Pract. 2023;139(3):105657. doi:10.1016/j.conengprac.2023.105657. [Google Scholar] [CrossRef]

43. Li Z, Zhang Z, Wang J, Wang S, Chen X, Sun H. ADRC control system of PMLSM based on novel non-singular terminal sliding mode observer. Energies. 2022;15(10):3720. doi:10.3390/en15103720. [Google Scholar] [CrossRef]

44. Farge TZ, Numan AH, Ashour ZTS. Vector control of low power HAWT based PMSG using SVPWM under variable wind speeds. AIP Conf Proc. 2022;2415(1):020011. [Google Scholar]

45. Li R, Abed Assali E, Hong J. A switchable exponent predefined time stability theory and adaptive sliding mode control for nonlinear systems. Commun Nonlinear Sci Numer Simul. 2026;160(8):109959. doi:10.1016/j.cnsns.2026.109959. [Google Scholar] [CrossRef]

46. Sánchez-Torres JD, Gómez-Gutiérrez D, López E, Loukianov AG. A class of predefined-time stable dynamical systems. IMA J Math Control Inf. 2018;35(3):777–91. doi:10.1093/imamci/dnx004. [Google Scholar] [CrossRef]

47. Yue J, Liu Z, Su H. Observer-based finite-time disturbance rejection control for dynamic wireless charging systems with constant output voltage regulation. IEEE Trans Ind Electron. 2024;71(9):11398–407. doi:10.1109/TIE.2023.3333040. [Google Scholar] [CrossRef]