Prescribed-Time Disturbance Observer-Based Sliding Mode Control for Speed Regulation of Permanent Magnet Synchronous Motor Drives

Permanent magnet synchronous motors (PMSMs) are widely used in high-performance electric drive applications, including electric vehicles, industrial robotics, aerospace actuators, and renewable energy systems [1-3]. Their widespread adoption is attributed to high power density, a wide torque-speed range, and compatibility with the field-oriented control (FOC) framework, which decomposes the nonlinear PMSM dynamics into independently regulated flux and torque channels [4]. Within the FOC architecture, the inner current loops are typically governed by high-bandwidth proportional-integral (PI) controllers or finite-control-set model predictive schemes, while the outer speed loop remains the principal locus of dynamic performance and disturbance rejection [5-7]. It is the behavior of this speed loop under realistic, non-vanishing load disturbances that motivates the present work.

Under the assumption that the inner current loop tracks its reference ideally, a standard and well-validated approximation for drives with fast current dynamics, the PMSM mechanical subsystem reduces to a first-order equation relating rotor speed ω to the q-axis current reference $\left(i_q^*\right)$ [8]. The lumped disturbance in this equation is the normalized load torque $d(t)=-\frac{T_l}{J}$, which is genuinely non-vanishing: it persists indefinitely and exhibits time-varying amplitude arising from cutting forces in machine tools, road-gradient variations in traction drives, or wind-gust torques in generator systems [9-11]. Conventional PI speed controllers rely on error accumulation to drive a corrective response, which introduces a fundamental trade-off between bandwidth and disturbance attenuation a trade-off that cannot be resolved by gain tuning alone [12].

Sliding mode control (SMC) is widely employed for disturbance rejection due to its robustness, stemming from the invariance property of the sliding surface and its straightforward implementation within the FOC speed loop [13]. Numerous SMC variants have been developed for PMSM drives. Terminal SMC achieves finite-time convergence; the super-twisting algorithm (STA) reduces chattering while preserving second-order sliding behavior [14], adaptive SMC adjusts the switching gain based on disturbance estimates, and fixed-time SMC-SMO schemes guarantee convergence within a bounded time independent of initial conditions [15]. Despite these advances, a fundamental challenge remains: robust rejection of non-vanishing disturbances requires the switching gain to exceed the disturbance bound, which inevitably increases chattering when disturbances are large. Reducing the gain alleviates chattering but compromises robustness, revealing an inherent limitation of conventional SMC in the absence of disturbance information.

Disturbance observer-based (DOB) control is a commonly adopted approach for addressing this trade-off [16]. The inclusion of a disturbance estimate in the control law as a feedforward compensation term reduces the required switching gain from the disturbance bound to the estimation-error bound, which is typically one to two orders of magnitude smaller. Extended state observers (ESOs), nonlinear disturbance observers (NDOs), and sliding mode observers (SMOs) have all been successfully integrated with SMC for PMSM speed control. A representative recent contribution is, which combines a fixed-time-convergent SMO with a fixed-time SMC, which demonstrates that the switching gain need only bound the observer residual rather than the full disturbance magnitude. A predefined-time DOB combined with adaptive SMC for PMSM was proposed, and a two-stage predefined-time exact SMC framework for generic nonlinear systems appeared in the study [17]. Despite these advances, the following limitation remains: the disturbance estimation error is bounded or converges to zero in finite time, but the convergence instant is either initial-condition dependent or represents only an upper bound, not a user-specified exact deadline.

The prescribed-time control paradigm introduced to overcome precisely this limitation guarantees that all closed-loop errors converge to zero at an exact, user-specified settling time $\left(T^i\right)$ regardless of initial conditions and system parameters [18, 19]. This is a strictly stronger guarantee than finite-time control (settling time depends on initial conditions), fixed-time control (settling time is an upper bound), or predefined-time control (actual convergence may precede the predefined instant). Prescribed-time methods achieve this through time-varying feedback gains that grow to infinity as $\left(t \rightarrow T^{i-}\right)$. Significant progress has been made for linear systems [20, 21], nonlinear systems in strict-feedback form [22-24], multi-agent coordination [25-27], and fault-tolerant control [28, 29]. A PTO with time-varying high gain was proposed in the study [30, 31] to achieve zero-error state estimation within a prescribed observation time $\left(T_o\right)$, and this observer was used as the foundation of a two-stage backstepping controller for nonlinear systems with non-vanishing disturbances. However, the combination of a prescribed-time zero-error observer with SMC and its application to PMSM speed regulation within the FOC framework remains relatively unexplored.

When the disturbance estimation error converges to zero by $\left(T_o\right)$, the Stage-2 scheme performs disturbance estimation and prescribed-time tracking separately at different stages, and the two stages are prescribed with different time limits. This separation leads to a two-timescale architecture: The observer is used first, before the high-gain tracking phase kicks on, so the prescribed-time SMC gain acts on a disturbance-free plant, and chattering elimination is a structural property of the design, and not just an approximation.

The novelty of the proposed prescribed-time disturbance observer-based sliding mode controller (PT-DOBSMC) method can be summarized as follows:

1. The existing "fixed-time" and "predefined-time" observer-based controllers usually can ensure that the estimation error satisfies boundedness or convergence to an arbitrarily small residual at the end of the period. Unlike the PTO in existing literature, however, the proposed one is designed with a time-varying high-gain injection mechanism that can make the estimation error converge to exactly zero at a user-specified observation deadline $\left(T_o\right)$.

2. The traditional SMC for chattering elimination (e.g., adaptive SMC or observer-based SMC) usually needs a switching gain to be large enough to cover the residual error of the observer. Our two-stage controller is able to achieve exactly the cancellation of disturbance by reducing the switching gain in the second stage ($\left(t \geq T_o\right)$) to a negligible level.

3. The framework of the proposed method is based on two independent prescribed deadlines: observation deadline $\left(T_o\right)$ and tracking deadline $\left(T_f\right)$. This enables the designer to independently adjust the disturbance rejection speed and tracking performance. This is not the case in most single-stage prescribed time controllers, i.e., observer and controller convergence are often linked.

4. The suggested approach explicitly solved the problem of having $C^p$ continuity of the control input from stage to stage $\left(T_o\right.$ and $\left.T_f\right)$. Impulsive current transients at switching stages have often been a problem in this field in the past. Still, the proposed gain structuring method provides a smooth and continuous transition, avoiding the loading or stress on the power electronics, which is crucial for practical applications of PMSM drives.

The DOB and ESOs can only ensure asymptotic (fixed-time) convergence, which entails some estimation error and high, persistent switching gains for the controller and chattering phenomenon. The PTO, on the other hand, forces the disturbance estimation error to converge to zero at a user-defined time $\left(T_o\right)$ for any initial condition and disturbance magnitude. This makes the switching gain of the controller much smaller for eliminating the chattering phenomenon in the structure. Furthermore, it offers two timescale tunings, where one can independently tune the observation deadline $\left(T_o\right)$ and tracking deadline $\left(T_f\right)$ to adjust the speed of disturbance rejection and the transient tracking performance without risking impulsive current transients.

Motivated by the above, this paper proposes a PT-DOBSMC strategy for PMSM speed regulation within the FOC framework. The speed tracking error serves as the sliding variable, and the q-axis current reference is the control input. A PTO with a time-varying high gain is designed to achieve zero-error disturbance estimation at exactly the prescribed observation time $\left(T_o\right)$.

1. A prescribed-time DOB is developed for PMSM speed regulation within the FOC framework. Unlike predefined-time and fixed-time observers, which achieve only a bounded estimation error at the observation deadline, the proposed observer drives the estimation error to zero at a user-specified observation instant.
2. A two-stage sliding mode controller is proposed. The first stage maintains bounded speed tracking while the observer converges to the disturbance. The second stage activates after the observation deadline and drives the speed error to zero at a user-specified tracking deadline. The switching gain in the second stage is substantially reduced compared to the first stage, resulting in reduced chattering.
3. The transition between the two control stages is proven to be continuous. No impulsive current commands arise at either stage boundary, which protects the power electronics from stress associated with abrupt current changes. The two prescribed deadlines, one for observation and one for tracking, are independently selectable by the designer and do not depend on initial conditions or load magnitude.
4. The proposed strategy is evaluated using MATLAB/Simulink simulations under four different operating conditions. The results demonstrate the efficacy of the developed strategy with reduced chattering.

The limitations of the fixed-time, predefined-time and extended state observer-based approaches are overcome with the proposed PT-DOBSMC framework, which has a convergence time that is independent of initial system conditions and/or conservative upper bounds, or chattering due to high switching gains. The framework addresses these challenges by separating the process of observing and tracking in user-specific $\left(T_o\right.$ and $\left.T_f\right)$ deadlines. In particular, the time-varying high-gain injection mechanism, which has a decay-stabilization term, forces the disturbance estimation error to go to zero exactly at $\left(T_o\right)$. This exact alignment allows the structural reduction of the Stage-2 sliding mode switching gain $\left(\rho_{s w 2}\right)$ to an arbitrarily small value, which fundamentally makes chattering disappear.

The remainder of this paper is organized as follows. Section 2 formulates the PMSM control problem and defines the disturbance exosystem model. Section 3 describes the PTO design and proves zero-error convergence. Section 4 develops the proposed two-stage PT-DOBSMC framework and analyses its stability. Section 5 reports the simulation results, and Section 6 concludes the paper.

To achieve objective Eq. (5), the disturbance estimation problem is reformulated through a coordinate transformation. The standard approach to designing an asymptotic observer and accepting a bounded residual is incompatible with the objective of exact zero-error at a user-specified deadline. Instead, we exploit the finite-dimensional exosystem structure of (3) through a coordinate transformation that converts the disturbance estimation problem into a linear observer canonical form, and then inject a time-varying gain that grows to infinity at exactly $\left(T_o\right)$ to enforce exact convergence. An additional decaying term in the gain ensures that the error remains at zero after $\left(T_o\right)$, without any persistent high gain - a property absent from fixed-time and predefined-time observers.

3.1 Known composite driving term

Before constructing the observer, it is useful to isolate the portion of the plant dynamics that is fully known at every instant. Substituting Eqs. (1) and (2), the composite term is:

$\varphi(t)=K_t, i_q^*-b$     (7)

This equation is computable in real time from the measured speed $\omega$ and the commanded current reference $\left(i_y^*\right)$. The term $(\varphi(t))$ plays the role of a known input to the observer, and the entire observer design proceeds without requiring knowledge of the exosystem parameters $\left(b_r\right)$.

3.2 Coordinate transformation and transformed dynamics

To convert the coupled plant–exosystem into an observable canonical form, we introduce the following state transformation that absorbs the coupling between the mechanical state ω and the exosystem outputs:

$\begin{gathered}\xi_1=\omega \\ \xi_2=d-b_m \omega \\ \xi_k=d^{(k-2)}-b_m d^{(k-3)}-\cdots-b_{m-k+2} \omega, \\ k=3, \ldots, m+1\end{gathered}$     (8)

Transformation Eq. (8) avoids repeated differentiation of exosystem states, which eliminates the complexity explosion that afflicts backstepping-based observer schemes [34]. Differentiating Eq. (8) along the trajectories of Eqs. (1) and (3), and using Eq. (7), the transformed states $\left(\xi_k\right)$ evolve as

$\begin{gathered}\dot{\xi}_1=\xi_2+b_m \omega+\varphi \\ \dot{\xi}_k=\xi_{k+1}+b_{m-k+1} \omega-b_{m-k+2} \varphi, k=2, \ldots, m \\ \dot{\xi}_{m+1}=-b_1 \varphi\end{gathered}$      (9)

System Eq. (9) is driven entirely by the known term ($\varphi(t)$) and the measurable speed $\omega$; the unknown exosystem parameters $\left(b_r\right)$ appear only as coefficients of known quantities. This property results in linear estimation-error dynamics, which are amenable to prescribed-time design.

3.3 Prescribed-Time Observer equations and linear error dynamics

The PTO mirrors the transformed dynamics Eq. (9) with output-injection terms driven by the measurable error in the first transformed state ($\xi_1=\omega$, so $\xi_1-\xi_1=\omega-\xi_1$ is directly observable):

$\dot{\hat{\xi}}_1=\widehat{\xi}_2+b_m \omega+\varphi+\rho_1\left(\omega-\widehat{\xi}_1\right)$

$\dot{\hat{\xi_k}}=\widehat{\xi_{k+1}}+b_{m-k+1} \omega-b_{m-k+2} \varphi+\rho_k\left(\omega-\widehat{\xi_1}\right)$,

$k=2, \ldots, m$     (10)

$\dot{\hat{\xi_{m+1}}}=-b_1 \varphi+\rho_{m+1}\left(\omega-\widehat{\xi}_1\right)$

Subtracting the observer dynamics Eq. (10) from the plant dynamics Eq. (9) causes the unknown $d(t)$ and the exosystem parameters $\left(b_r\right)$ to be explicitly canceled and leaves the error system Eq. (11) linear and independent of the disturbance. The observation errors ($e_k=\xi_k-\hat{\xi}_k$) satisfy a system that takes the linear form:

$\begin{gathered}e_k=\xi_k-\widehat{\xi_k}, k=1, \ldots, m+1 \\ \dot{e_1}=e_2-\rho_1 e_1 \\ \dot{e_k}=e_{k+1}-\rho_k e_1, k=2, \ldots, m \\ \dot{e}_{m+1}= - \rho_{m+1}  e_1\end{gathered}$     (11)

The error system Eq. (11) is linear in e and completely free of the unknown parameters $\left(b_r\right)$ - they canceled exactly in the subtraction. This linearity results from the coordinate transformation Eq. (8), which allows ($\rho_k(\mathrm{t})$) to be designed purely from the desired error polynomial, independent of the disturbance. The equations in Eq. (11) are written in matrix form $(\dot{e}=\Lambda(t) e)$, with the companion matrix given as:

$\bar{\Lambda}=\left[\begin{array}{ccccc}-p_1 & 1 & 0 & \cdots & 0 \\ -p_2 & 0 & 1 & \cdots & 0 \\ \vdots & \vdots & \vdots & \ddots & \vdots \\ -p_m & 0 & 0 & \cdots & 1 \\ 0 & 0 & 0 & \cdots & 0\end{array}\right] \in R^{(m+1) \times(m+1)}$     (12)

The designer chooses the coefficients $\left(p_k\right)$ such that the polynomial $\left(s^{m}+p_1 s^{m-1}+\cdots+p^{\mathrm{m}}\right)$ is Hurwitz. For the PMSM case with $(m=1)$, the stability condition simplifies to $\left(p_1>0\right)$, which requires only a positive observer pole.

3.4 Time-varying prescribed-time injection gains

Zero-error convergence is achieved through the timevarying scaling function ($\mu(t)$), which grows without bound as $\left(t \rightarrow T_o{ }^{-}\right)$:

$\begin{gathered}\mu(t)=\frac{T_o}{T_o-t}, 0 \leq t<T_o \\ \mu(t)=0, t \geq T_o\end{gathered}$     (13)

As $\left(t \rightarrow T_o^{-}\right)$, $(\mu(t) \rightarrow+\infty)$, and this causes the injection gains to increase and enforce convergence at the prescribed instant. After $\left(T_o\right)$ and $(\mu)$ are set to zero, and the gains vanish, which releases the observer from the high-gain regime. For the two-state case (m = 1), the gains take the explicit form:  

$\begin{gathered}\left.\rho_1(t)= (2 \alpha \mu+T_o-t\right) \alpha^2, t<T_o \\ \rho_2(t)=\alpha^2 \mu^2, t<T_o \\ \rho_k(t)=0, t \geq T_o, k=1,2\end{gathered}$     (14)

Here $(\alpha>0)$ is the single tuning parameter that sets the error convergence rate: the error characteristic polynomial becomes $(s+\alpha \mu)^2$, whose roots track $(-\alpha \mu$ to $-\infty)$ as $(t \rightarrow$ $\left.T_o^{-}\right)$. The term $\left(T_o-t\right) \alpha^2$ in $\left(\rho_1\right)$ is the "v-term" introduced in the study [26]: This is a decay-stabilization contribution that vanishes precisely at ($T_o$), thereby ensuring a smooth transition from the regime $\left(t<T_o\right)$ to $\left(t \geq T_o\right)$, as well as perfect continued tracking by the observer even after the injection gains have reduced to zero. For numerical robustness, a cap $\left(\rho_{\mathrm{k}} \leq \rho^{\text {max}}\right)$ is imposed in the final moments before ($T_o$); since the error is already vanishingly small at that point, the cap does not affect the theoretical guarantee.

3.5 Disturbance estimate recovery

The disturbance is estimated by inverting the coordinate transformation given in Eq. (8) after the computation of the observer states $\left(\hat{\xi}_{\mathrm{k}}\right)$. From the definition $\left(\xi_2=d-b^{\mathrm{m}} \omega\right)$, solving for d gives:

$\begin{gathered}\hat{d}(t)=\widehat{\xi}_2+b_m \omega(t) \\ \hat{d}(t)=\widehat{\xi}_2[m=1]\end{gathered}$     (15)

For the practically important case $(m=1)$ with a constant load torque model ($b_1=0$), the recovery simplifies to $(\widehat{d}=$ $\hat{\xi}_2)$ - the observer's second state directly carries the disturbance estimate, without any additional computation. Higher-order exosystems ($m=2$ for sinusoidal loads, $m=3$ for the simulation example in Section V) require the full recovery formula but add no structural complexity; only the observer dimension increases.

3.6 Main convergence result

This section is intended to state the main result formally.

Theorem 1 (Prescribed-Time Zero-Error Convergence): Let Assumption 2 hold and let the observer gains ($\rho_k(t)$) be given by Eq. (14) [or its general-m counterpart], with $(\alpha>0)$ chosen such that the polynomial $\left(s^m+p_1 s^{m-1}+\cdots+p^m\right)$ is Hurwitz. Then the observation errors ($e_k(t)=\xi_k-\xi_k^{\wedge}$) satisfy: (i) $\left|e_k(t)\right|$ is bounded for all $t \in\left(0, T_o\right)$; (ii) $\left(e_k(t) \rightarrow\right.$ 0 ) as ($t \rightarrow T_o^{-}$); and (iii) ($e_k(t) \equiv 0$) for all ($t \geq T_o$), provided the disturbance trajectory continues to be generated by the same exosystem realization that was active during $\left[0, T_o\right]$. Consequently, $(\hat{d}(t)=d(t))$ for all $\left(t \geq T_o\right)$ in the ideal continuous-time setting, subject to the same proviso.Proof: Consider the Lyapunov candidate for the error system Eq. (11):

$V_e=\frac{1}{2} e^T e=\frac{1}{2} \sum_{k=1}^{m+1} e_k^2$     (16)

where, $(P>0)$ is the unique solution to the matrix Lyapunov equation:

$\bar{\Lambda}^T P+P \bar{\Lambda}=-2 I$     (17)

which exists since $(\bar{\Lambda})$ is Hurwitz. Define $\left(\kappa=\frac{1}{\lambda_{\max }(p)}>0\right)$, so that $\left(e^T e \geq 2 \kappa V_e\right)$ by the Rayleigh quotient. The central step is to show that the gain structure Eq. (14) produces the decomposition:

$\Lambda(t)=\mu(t) \cdot \bar{\Lambda}+R(t)$     (18)

where, $\|R(t)\|=O\left(T_{\mathrm{o}}-t\right) \rightarrow 0$ as $\left(t \rightarrow T_{\mathrm{o}}\right)$. This holds because each gain $\left(\rho_k(t)\right)$ in Eq. (14) splits as $\left(\rho_k(t)=\right.$ $\mu .(\bar{\Lambda}$ entery$\left.)+\left(T_o-t\right). \rho_k\right)$; the first term produces the $(\mu \bar{\Lambda})$ factor, the second is the v-term and forms $R(t)$. The Lyapunov derivative is dominated by the ($-2 \mu$) term as ($t$) approaches ($t \rightarrow T_o^{-}$), from below, so the residual matrix $R(t)$ is negative definite for all $(t)$ close to $\left(t \rightarrow T_o^{-}\right)$and for all ($t \rightarrow T_o^{-}$), because the deviation of the time-varying gain from the prescribed-time structure is dominant. Differentiating $\left(V_e\right)$ along Eq. (11) and substituting:

$\begin{gathered}\dot{V}_e=\frac{1}{2} e^T\left(\Lambda(t)^T P+P \Lambda(t)\right) e \\ =\frac{1}{2} e^T\left[\mu\left(\bar{\Lambda}^T P+P \bar{\Lambda}\right)+R(t)^T P+P R(t)\right] e \\ =\frac{1}{2} e^T\left(-2 \mu I+R(t)^T P+P R(t)\right) e \\ \leq-2 \kappa \mu(1-c\|R\| / \mu) V_e \leq-2 \kappa \mu V_e\end{gathered}$     (19)

The last inequality uses ($e^{\mathrm{T}} e \geq 2 \kappa V_{\mathrm{e}}$) and the bound $\left(c\|R\| / \mu=O\left(\left(T_{\mathrm{o}}-t\right)^2 / T_{\mathrm{o}}\right) \rightarrow 0\right)$, confirming Eq. (19) is a strict inequality throughout $\left(0, T_{\mathrm{o}}\right)$. where $(\kappa)$ Integrating Eq. (19) using $\left(\mu=T_{\mathrm{o}} /\left(T_{\mathrm{o}}-t\right)\right)$ gives $\left(V_{\mathrm{e}}(t) \leq V_{\mathrm{e}}(0) \cdot\left(T_{\mathrm{o}}-\right.\right.$ $\left.t)^2 \kappa / T_{\mathrm{o}}{ }^2 \kappa \rightarrow 0\right)$ as $\left(t \rightarrow T_{\mathrm{o}}^{-}\right)$. For $\left(t \geq T_{\mathrm{o}}\right):(\mu=0)$ gives $(\dot{\mathrm{e}}=0) ;\left(e\left(T_{\mathrm{o}}\right)=0\right)$ gives $(e \equiv 0)$ (claim (iii)), provided no new exosystem realization is introduced after ($T_{\mathrm{o}}$). Using (15), to obtain ($\hat{d}(t)=d(t)$), subject to the condition: $(\hat{d}(t)=d(t))$ for $\left(t \geq T_{\mathrm{o}}\right)$.

Remark 1 (Ideal-Model Qualifier and Sampling Effects): The convergence guarantee of Theorem 1 holds exactly under ideal continuous-time dynamics. In a sampled-data implementation, the observation error at $\left(T_o\right)$ is not exactly zero but remains of order $(O(h))$, where h is the sampling period. The small but nonzero Stage-2 switching gain compensates this residual ($\rho_{s w 2}$) and this provides robustness against both sampling-induced estimation errors and disturbance variations that occur after $\left(T_o\right)$. The gain cap $\left(\rho_k \leq 5000\right)$ becomes active only during the final few milliseconds before $\left(T_o\right)$, when the observation error is already negligible, it does not affect the theoretical guarantee.

Remark 2 (Observer Implementation): Although the disturbance parameters $\left(b_r\right)$ are treated as unknown, the recovery Eq. (15) appears to involve $b_m$. This apparent contradiction is resolved as follows. The error dynamics in Eq. (11) are obtained by subtracting the observer Eq. (10) from the transformed plant model Eq. (9). This makes all terms that contain $\left(b_r\right)$ to cancel identically. Consequently, the observer gains ($\rho_k(t)$) can be tuned without any knowledge of the disturbance parameters. Moreover, in the PMSM application studied here, $(m=1)$ and $\left(b_1=0\right)$. Thus, the observer can be fully implemented without any prior knowledge of the disturbance parameters.

Remark 3 (Observer Gain Discontinuity vs. Error Continuity): The scaling function $\mu(t)$ transitions abruptly from ($+\infty$ to 0) at ($t=T_o$), which renders the injection gains ($\rho_k(t)$) discontinuous at ($T_o$) by design. This gain discontinuity does not contradict the smooth behavior of the observation errors: ($e_k(t)$) decays to zero continuously throughout $\left(0, T_o\right)$ and remains identically zero afterward. It is the gains that are discontinuous, not the errors, and the paper asserts smoothness only for the latter at ($T_o$), not for the observer's gains.

The effectiveness of the proposed PT-DOBSMC is validated through simulation studies on a PMSM drive under FOC. Four different operating scenarios are considered: (S1) nominal step-speed tracking under constant load; (S2) sensitivity to prescribed tracking time $T^i$; (S3) sensitivity to prescribed observation time $T_o$; (S4) step disturbance rejection and estimation performance. In each case, observer convergence, speed-tracking performance, control-signal continuity, and chattering characteristics are examined. All simulations are performed in MATLAB/Simulink (R2024b) using the ode4 solver with a fixed step size as 10^-5.

5.1 Permanent Magnet Synchronous Motor drive parameters and controller setup

The base observer and controller parameters are given in Table 1. Only the parameters being varied are changed in each scenario, with all others held at their base values. The selected parameters are parameters of a typical intermediate interior PMSM commonly found in industrial applications and EV systems, which allow the validity of the simulation results to be verified by comparison with the actual driving situation. The gains were chosen to meet the theoretical stability conditions and to remain within the hardware limit for the maximum current demand of 20 A, thus showing a practical application of the PT-DOBSMC strategy. The gains are calculated depending on the mechanical time constant of the system and the specified performance requirements, while simultaneously trying to minimize chattering in Stage-2 and ensure robustness in Stage-1.

Table 1. Base observer and controller parameters

The simulation platform is a representative interior PMSM with the following parameters: rotor inertia $(J=$ $0.003 \mathrm{Kg} . \mathrm{m}^2$) , viscous friction coefficient ($B=$ 0.008 (N.m.s/rad)), pole pairs ($P_p=3$), and permanentmagnet flux linkage. ($\psi_{\mathrm{f}}=0.175 \mathrm{~Wb}$), torque constant $\left(K_t=1.5 P_p \psi^i=1.575(\mathrm{~N} . \mathrm{m} / \mathrm{A})\right)$, and a DC bus voltage of 310 V . These values yield the effective gain $\left(K_t / J=\right.$ $525 \mathrm{rad} / \mathrm{s}^2)$ per ampere and friction ratio $\left(B / J=2.667 \mathrm{~s}^{-1}\right)$. These fixed parameters represent a mid-range interior PMSM for EV auxiliary and industrial servo applications.

The performance is quantified using three metrics computed over the full simulation window. The integral of time-weighted absolute error (ITAE), which penalizes late-persisting errors:

ITAE $=\int_0^{T_f} t|e(t)| d t$     (34)

The integral of squared error (ISE), which penalizes large transient deviations:

$\mathrm{ISE}=\int_0^{T_f} e(t)^2 d t$     (35)

And the post-$T^i$ steady-state absolute error, which directly tests the prescribed-time claim:

$e_{s s}=\lim _{t \rightarrow T_f}+\left|\omega_{r e f}-\omega(t)\right|=0$     (36)

According to Theorem 2, the steady-state error is zero regardless of load conditions or initial states. Any nonzero value in the simulations would indicate a stability or numerical issue.

5.2 Proposed prescribed-time disturbance observer-based sliding mode controller validation

5.2.1 Scenario 1: Nominal speed step with constant load torque

In the first scenario, the reference speed is set to ($\omega_{\text {ref}}=$ $100 \mathrm{rad} / \mathrm{s}$) and the motor starts from rest ($\omega(0)=0 \mathrm{rad} / \mathrm{s}$) at $(\mathrm{t}=0 \mathrm{~s})$. A constant load torque $\left(T_L=5 N \cdot m\right)$ is applied to the motor throughout the simulation and thus the normalized disturbance value is $(\mathrm{d}(\mathrm{t})=-5 / 0.003 \approx$ $-1667 \mathrm{rad} / \mathrm{s}^2)$ . The disturbance is non-vanishing and unknown to the controller; only its order $(m=1)$ is assumed (constant model). The set values of the prescribed times are $\left(T_o=0.10 \mathrm{~s}\right)$ and $\left(T^i=0.40 \mathrm{~s}\right)$.

Figure 2 shows a comparison of the speed tracking performance and speed tracking error $\left(\mathrm{s}(t)=\omega_{\text {Ref}}-\right. \omega(t))$ for a motor control system using the conventional SMC, adaptive SMC, and proposed PT-DOBSMC.

(a) Motor speed response

(b) Speed tracking error $s(t)$

Figure 2. Comparative performance of the proposed prescribed-time disturbance observer-based sliding mode controller (PT-DOBSMC), conventional SMC, and adaptive SMC controllers

In the Stage-1 period (0, 0.10) s, the proposed PT-DOBSMC has the best transient performance compared to the conventional SMC and the adaptive SMC. The PT-DOBSMC has a convergence speed that is faster than that of the conventional SMC and the adaptive SMC, and the chattering phenomenon of the adaptive SMC is reduced to a bounded speed error within the compact set ($\omega$), defined in Eq. (24). At $\left(T_{\mathrm{o}}=0.10 \mathrm{~s}\right)$, the observer's estimate of the disturbance is accurate, and the Stage-2 controller is activated. As per the steady-state error being zero ($e_{s s}=0$), Theorem 2, claim ii), there is an error that reaches zero at the exact time ($T^i=$ 0.4 s). Unlike the conventional SMC where the steady-state chattering is persistent, the PT-DOBSMC is stable with minimal oscillation in the steady state. Moreover, the proposed control law is also robust and smooth with respect to the inherent switching artifacts of the comparison controllers, since no step discontinuity can be seen at the transition time of the stage (Theorem 2, claim 3). Figure 3 shows the disturbance true value $d(t)$, PTO estimate. $\hat{d}(t)$, and the estimation error $\tilde{d}(t)=d(t)-\hat{d}(t)$. The estimation error decays within (0, $\mathrm{T}_{\mathrm{o}}$) and converges to zero at $\mathrm{t}=0.30 \mathrm{~s}$ and remains at zero in the ideal continuous-time model. This matches the analytically predicted behavior from Theorem 1 and verifies objective Eq. (5):

$\tilde{d}(t)=d(t)-\hat{d}(t) \equiv 0, \quad \forall t \geq T_o$     (37)

(a) $d(t)$ and $\hat{d}(t)$

(b) Estimation error $\tilde{d}(t)$

Figure 3. Scenario 1—Disturbance performance

(a) q-axis reference current $i_q^*(t)$

(b) Prescribed-time sliding mode control (SMC) gain $\psi_0(t)$

Figure 4. Scenario 1—Control signals

Figure 4 plots $\left(i_y^*(t)\right)$ and the prescribed-time gain $\left(\psi_0(t)\right)$. The gain structure is given by Eq. (23):

$\psi_0(t)=\varpi_0 \frac{\left(t-T_0\right)^p}{T_f-t}+k_0$     (38)

Note that $\left(\psi_0(t)\right)$ rises smoothly from $\left(k_0=5\right)$ at $\left(T_o\right)$ and diverges as ($t \rightarrow T^{i-}$), while simultaneously ($s(t) \rightarrow 0$). The product $\left(\psi_0(t) \cdot s(t)\right)$ remains finite and continuous (verified by the smooth $\left(i_q^*\right)$ trace), confirming the L'Hôpital continuity argument. Stage-2 chattering is negligible: the tanh switching term amplitude is $\left(\rho_{s w 2}=1.0 \mathrm{rad} / \mathrm{s}^2\right)$ vs. $\left(\rho_{s w 1}=\right.$ $500 \mathrm{rad} / \mathrm{s}^2)$ in Stage-1.

5.2.2 Scenario 2: Sensitivity to prescribed tracking time $T^i$

A useful property of prescribed-time control is that the settling deadline is designer-specified. To validate this, three values of ($T^i$) are tested: $0.4 \mathrm{~s}, 0.5 \mathrm{~s}$ (base), and 0.7 s, with all other parameters held fixed ($T_o=0.10 \mathrm{~s}$ and load $T_L=5 \mathrm{~N} \cdot \mathrm{~m}$). The stability condition $\varpi_{o}\left(T^i-T_o\right)^{p} \geq 0.5$ is verified for each case in Table 2. Note that $\left(T_o=0.10 \mathrm{~s}\right)$ is held fixed: The observation deadline remains unchanged when the tracking deadline is varied, illustrating the independent design of the two prescribed times.

Table 2. Stability condition verification for scenario 2

Figure 5 shows the speed responses for the three values of $\left(T^i\right)$. The speed error reaches zero exactly at the prescribed deadline. Smaller values of $\left(T^i\right)$. Require faster growth of the Stage-2 gain $\left(\psi_0(t)\right)$ and this result in higher peak current demands near the deadline. This illustrates the fundamental trade-off in prescribed-time control: faster convergence requires greater control effort. For $\left(T^i=0.4 s\right)$, the peak current is approximately 15 A, which remains below the inverter limit of 20 A. For longer deadlines, the current demand is lower and can be reduced further by decreasing $\varpi_0$. Observer performance remains unchanged in all cases, with $\tilde{d}(t)$ converging to zero at ($T_o=$ 0.30 s). This demonstrates that the observation and tracking deadlines can be selected independently, unlike in single-stage prescribed-time controllers [9-11].

Figure 6 illustrates the transient response of the estimated metric command q-axis current $\left(I_q^*\right)$ at three different event-triggering time stamps, which are 0.4 s, 0.5 s, and 0.7 s, respectively. The results show that a severe time-dependent attenuation characteristic occurs in the system's sensitivity. When the disturbance is presented early at (t = 0.4 s), the system exhibits a pronounced response with a sharp peak at approximately 15 A. However, when the injection time is applied, the peak magnitude reduces significantly to 11.5A at (t = 0.5 s), and becomes almost totally suppressed by (t = 0.7 s), where it barely deviates from the baseline noise. This proves that the system develops enhanced damping and robustness as time progresses, effectively mitigating late-stage transient disturbances that occur.

Figure 5. Scenario 2—speed response for $T^i \in$ [0.4, 0.5, 0.7] s

Figure 6. Scenario 2—q-axis current $i_y^*(t)$ for $T^i \in$ (1.0, 1.5, 2.5) s. Larger ($T^i$) reduces peak current demand near the deadline

5.2.3 Scenario 3: Sensitivity to prescribed observation time $T_o$

The prescribed observation time $\left(T_o\right)$ determines how quickly the PTO converges to the disturbance estimate. A smaller $\left(T_o\right)$ enables earlier disturbance cancellation and shortens the Stage-1 tracking period, but requires higher observer gains and greater numerical stiffness. Three values are compared: $T_o \in$ (0.20, 0.30, 0.70) s, with the tracking time fixed at ($T^i$ = 0.70 s).

Figure 7. Scenario 3—speed tracking error s(t) for $T_o \in$ (0.10, 0.20, 0.30) s with $T^i$ = 0.70 s) fixed

Figure 7 depicts the speed error $s(t)$ for each case. For ($T_o$ $=0.20 \mathrm{~s}$), the Stage-1 tracking error shown in red color is present for only (0.20 s) before Stage-2 activates, which results in a slightly larger error peak during Stage-1 due to the shorter window for the controller to establish boundedness. For ($T_o=$ 0.30 s), the bounded-tracking phase lasts longer; the error remains larger for a longer time, but the Stage-2 prescribedtime convergence to zero at ($T^i=0.70 \mathrm{~s}$) is unaffected. In all three cases, $\left(e_{S S}=0\right)$ exactly at $T^i$, confirming Theorem 2 claim Eq. (2) holds regardless of $\left(T_o\right)$ provided $\left(T_o<T^i\right)$.

The disturbance estimation errors are shown in Figure 8. It is clear from the result that the disturbance-estimation trajectories are consistent with the prescribed observation times, which confirms the theoretically predicted behavior.

Figure 8. Scenario 3—disturbance estimation error $\tilde{d}(t)$ for three $\left(T_0\right)$ values

5.2.4 Scenario 4: Step disturbance rejection and estimation performance

It is important to note that the sudden step loads that are applied in this scenario intentionally violate Assumption 2: impulsive load-torque step changes fall outside the disturbance class generated by the autonomous linear exosystem Eq. (3). These disturbances are therefore incorporated deliberately to examine the robustness of the PT-DOBSMC under conditions that extend beyond its theoretical assumptions. This scenario is used to assess the robustness of the developed controller against disturbance changes that violate Assumption 2. This scenario tests robustness against sudden load changes. The motor runs at $\omega_{\text {ref}}=200 \mathrm{rad} / \mathrm{s}$ with an initial constant load $T_l=5$ N·m at t = 1.0 s, a step load of +5 N·m is applied. A second step of −5 N·m is applied at t = 2.0 s. These steps model sudden mechanical loading events — for example, a tool engaged in a CNC spindle drive or a traction motor that hit an incline.

The two load steps applied at $t=1.0 \mathrm{~s}$ and $t=2.0 \mathrm{~s}$ both occur after $T_o$, which exposes an important practical limitation: beyond $T_o$, the observer injection gains $\rho_k=0$, which means the PTO ceases to track changes in the disturbance after this point. The exact cancellation guaranteed by Theorem 1 covers only the disturbance profile observed during the period $\left(0, T_o\right)$; any subsequent variation in disturbance $d(t)$ beyond $T_o$ leaves an uncompensated residual. The Stage-2 controller manages this residual via its switching term. $\left(\rho_{s w 2} \cdot \tanh \left(\frac{s}{\varphi_{s w}}\right)\right)$ and this provides input-to-state stability (ISS)-type robustness against bounded post $T_o$ disturbance changes. The peak speed deviation and recovery time in Figure 9 therefore characterize the ISS robustness margin rather than the prescribed-time convergence performance, the latter applying strictly to the initial tracking phase. This limitation is inherent to the current design. Extending the observer to continue updating with reduced gains beyond $T_o$ remains an important topic for future investigation.

Figure 9 shows speed response, speed error, disturbance estimate, and $i_y^*(t)$ across the full simulation. At each step event, the speed error rises or falls briefly before being rejected. The rejection time and peak speed drop are presented in Table 3.

(a) Speed response

(b) Tracking error

(c) $d(t)$ and $\hat{d}(t)$

(d) q-axis command current

Figure 9. Scenario 4 controller performance

Table 3. Step disturbance rejection metrics (scenario 4)

Four consistent properties of the PT-DOBSMC emerge across all five operating scenarios. First, ($e_{S S}$ = 0) is attained in all cases, and this confirms that prescribed-time convergence is a structural property of the design rather than a probabilistic outcome. Second, the observation and tracking deadlines ($T_o$ and $T^i$) are genuinely decoupled: adjusting one leaves the other unchanged, which provides the drive engineer with two independent handles for separately tuning disturbance rejection bandwidth and speed-response settling time. Third, chattering in Stage-2 is negligible by construction: once $\hat{d}(t)=d(t)$, the switching gain $\left(\rho_{s w 2}\right)$ is only a numerical regularisation term, not a true switching driver. This is qualitatively different from reducing chattering via boundary-layer approximation or gain adaptation, both of which leave a nonzero residual.

Figure 10. Speed response comparison of the proposed strategy with [6]

(a) Variable mechanical load step $T_L(t)=5 \mathrm{~N} . \mathrm{m}$

(b) Periodic cogging and harmonic torque ripple $T_{\text {ripple }}(t, \omega)$ [6^th and 12^th]

(c) Total lumped mechanical disturbance $T_d(t, \omega)$

(d) Speed response and disturbance rejection comparison at ($\omega_{\text {ref}}$ = 140 rad/s)

Figure 11. Comparison of speed response of the permanent magnet synchronous motor (PMSM) drive under variable mechanical loads and periodic torque ripple disturbances changes using conventional sliding mode controller (SMC), adaptive SMC and proposed prescribed-time disturbance observer-based sliding mode controller (PT-DOBSMC)

The main limitation observed in the simulation is the sensitivity of the control current near ($T^i$) to the choice of $\varpi_o$ and (p). For aggressive deadlines (e.g. $T^i=1.0 \mathrm{~s}$), the prescribed-time gain grows rapidly, and the current saturation limit is approached. Practitioners are therefore advised to verify the peak $\left(i_y^*(t)\right)$ in simulation before commissioning, and to choose $\left(T^i\right)$ conservatively relative to the mechanical time constant of the load.

The proposed strategy is compared with the predefined sliding mode control (PSMC) approach presented in the study [6]. Unlike the proposed method, which employs a two-stage control structure, the controller in the study [6] uses a single-stage design. The control parameters are selected such that both controllers achieve settling within 0.6 s, irrespective of the load condition. The comparison, shown in Figure 10, demonstrates the superiority of the proposed two-stage approach. Specifically, the proposed controller achieves an ISE value of (8352), whereas the controller in the study [6] yields an ISE value of (9768), indicating improved tracking performance.

The sharp slope of the speed tracking curve in the initial prescribed time interval $\in\left(0, T_f\right)$, physically represents a transient increase in the electromagnetic torque output ($T_e \propto$ $i_q^{\wedge}$) generated by the fast temporal scaling function $\mu(t)$ imparting kinetic energy to the rotor mass $J$, thus forcing the rotor speed to follow the reference command in time before urgently $\left(T_f\right)$. The next maximum in the disturbance estimation curve is PTO rapidly rebuilding and compensating the external load torque ($T_L$) and internal reactive voltage drops (back-EMF and inductive dynamics) instantaneously.

The performance of the PMSM drive is tested as described in Figure 11 for different operating conditions, such as a variable mechanical load step $\left(T_L(t)=5 N . m\right)$, periodic cogging torque ripples $\left(T_{\text {ripple}}(t, \omega)\right)$ and total lumped mechanical disturbance ($T_d(t, \omega)$). Specifically, the step changes of the lumped load torque, the high-frequency periodic disturbances and the lumped disturbance profile are shown in the plot of Figure 11(a) through Figure 11(c), respectively, over time. Moreover, Figure 11(d) presents a comparison of the speed response and disturbance rejection performance for the conventional SMC, adaptive SMC and the proposed PT-DOBSMC methods at a reference speed ($\omega_{\text {ref}}=$ $140 \mathrm{rad} / \mathrm{s})$ of the vehicle. The comparative results in terms of speed tracking show that the proposed controller exhibits good disturbance rejection and minimal overshoot with a comparatively faster settling time than the conventional and adaptive SMC approaches.

Table 4. Performance comparison of sliding mode controller (SMC), adaptive SMC, and proposed prescribed-time disturbance observer-based sliding mode controller (PT-DOBSMC) schemes

The performance of the PMSM drive is tested as described in Figure 11 for different operating conditions, such as a variable mechanical load step $\left(T_L(t)=5 N . m\right)$, periodic cogging torque ripples $T_{\text {ripple}}(t, \omega)$ and total lumped mechanical disturbance $T_d(t, \omega)$. Specifically, the step changes of the lumped load torque, the high-frequency periodic disturbances and the lumped disturbance profile are shown in the plot of Figure 11(a) through Figure 11(c), respectively, over time. Moreover, Figure 11(d) presents a comparison of the speed response and disturbance rejection performance for the conventional SMC, adaptive SMC, and the proposed PT-DOBSMC methods at a reference speed ($ω_{ref}$ = 140rad/s) of the vehicle. The comparative results in terms of speed tracking show that the proposed controller exhibits good disturbance rejection and minimal overshoot with a comparatively faster settling time than the conventional SMC and adaptive SMC approaches.

The effectiveness of the proposed control framework is tested by comparing its performance with that of the conventional SMC and adaptive SMC strategies in order to establish a rigorous evaluation of the efficacy of the proposed control framework. Based on the results shown in Table 4, the proposed method provides substantial improvements in terms of tracking accuracy, transient response, and input chattering reduction.