Predefined-Time Super-Twisting Disturbance-Observer-Based Speed Control for PI-Based FOC Induction Motor Drives

Quang Huy Le, Ngoc Thuy Pham

Faculty of Electrical Engineering Technology, Industrial University of Ho Chi Minh City (IUH), Ho Chi Minh, Viet Nam

1. INTRODUCTION

Induction motors (IMs) remain the most widely used electrical machines in industrial drive systems owing to their rugged structure, low manufacturing cost, high reliability, and minimal maintenance requirements. With the rapid development of energy-efficient industrial processes and electric drive technologies, high-performance control of induction motor drives has become a critical research and application topic [1]–[3]. Among various control strategies, FOC has been established as the industrial standard for IM drives, as it enables decoupled control of torque and flux, achieving dynamic performance comparable to that of separately excited DC motors [4]–[7].

In practical FOC-based IM drive systems, proportional–integral (PI) controllers are predominantly employed in the speed and current control loops due to their simple structure, ease of implementation, and low computational and hardware costs. These advantages make PI controllers particularly attractive for industrial applications where robustness, reliability, and cost-effectiveness are primary concerns [8]–[10]. Consequently, PI-based control architectures continue to dominate commercial IM drive products despite the availability of more advanced nonlinear control techniques.

However, it is well recognized that conventional PI speed controllers exhibit inherent limitations when operating under real-world conditions. Specifically, their performance degrades significantly in the presence of load torque disturbances, parameter variations (e.g., rotor resistance and rotor time constant changes), and unmodeled nonlinear dynamics [11]-[14]. These issues typically manifest as large speed overshoot, prolonged settling time, and weak disturbance rejection capability, particularly during abrupt load changes or low-speed operation. Extensive studies have shown that fixed-gain PI controllers tuned under nominal conditions cannot simultaneously guarantee fast transient response and robust disturbance attenuation over a wide operating range [15]-[16].

To mitigate these drawbacks, numerous PI enhancement strategies have been reported in the literature. Intelligent tuning approaches based on fuzzy logic, neural networks, and reinforcement learning have been investigated to improve adaptability and robustness [17]–[27]. Although such methods can enhance performance under specific conditions, they generally introduce increased structural complexity, require substantial training or heuristic design procedures, and often lack rigorous stability guarantees. Moreover, their reliance on additional computational resources and tuning parameters may limit applicability in cost-sensitive industrial drive systems.

Sliding mode control (SMC) has emerged as an effective alternative for robust electric drive control due to its inherent insensitivity to matched disturbances and parameter uncertainties [28]-[32]. In particular, higher-order sliding mode techniques, such as the super-twisting algorithm (STA), achieve finite-time convergence while significantly reducing chattering compared with classical first-order SMC [33]–[37]. STA-based control strategies have demonstrated strong disturbance rejection and robustness in motor drive applications [38][39]. Nevertheless, most existing SMC- or STA-based approaches replace conventional PI controllers entirely, leading to more complex control architectures and reduced compatibility with existing industrial FOC implementations.

To preserve the simplicity and industrial feasibility of PI-based FOC systems while enhancing robustness, hybrid PI–SMC and PI–STA schemes have been proposed [40]–[42]. Despite these efforts, two important limitations remain. First, many hybrid designs rely on fixed-gain STA structures, whose convergence properties depend on gain selection and may require prior knowledge of disturbance bounds. Second, most existing approaches focus on finite-time convergence, where the settling time depends on initial conditions and cannot be explicitly prescribed, which limits performance predictability in time-critical industrial applications.

Motivated by these challenges, this paper proposes a predefined-time super-twisting disturbance-observer-based speed control strategy for FOC-based induction motor drives. The proposed method retains the conventional PI controllers in the current and flux regulation loops, thereby preserving the original industrial control architecture, while introducing a predefined-time super-twisting disturbance observer exclusively in the mechanical speed loop. Unlike traditional SMC-based controllers that replace the PI structure, the proposed design operates as an auxiliary disturbance compensation mechanism, enhancing robustness without altering the fundamental PI framework.

Furthermore, a nonlinear predefined-time sliding manifold combined with time-varying adaptive gains is constructed to explicitly shape the convergence profile of the speed tracking error. This design guarantees convergence within a user-defined time independent of initial conditions and without requiring prior knowledge of disturbance bounds, thereby overcoming key limitations of conventional finite-time STA-based methods [43]–[48]. As a result, the proposed approach achieves predictable transient performance, improved disturbance rejection capability, and strong robustness, while maintaining low implementation complexity and full compatibility with industrial PI-based FOC systems.

 The main contributions of this paper can be summarized as follows:

1. A predefined-time super-twisting disturbance-observer-based speed control framework is developed for PI-based FOC induction motor drives (PI-PT STA DOB), enhancing robustness while preserving the conventional PI architecture. In this framework, a nonlinear predefined-time sliding manifold with time-varying adaptive gains is designed to guarantee convergence within a prescribed time, independent of initial conditions and disturbance bounds.
2. Comparative simulation studies demonstrate that the proposed method achieves superior transient performance and disturbance rejection capability compared with the conventional PI, while maintaining structural simplicity suitable for industrial applications.

The remainder of this paper is organized as follows. Section II presents the mathematical model of the induction motor. Section III describes the proposed predefined-time disturbance-observer-based speed control structure. Section IV provides the stability analysis of the closed-loop system. Section V presents the simulation results, and Section VI concludes the paper.

2. METHODS

1. Mathematical model of IMDs

In this section, the dynamic model of the induction motor is formulated in the synchronous rotating reference frame. This representation enables the decoupling of flux and torque components and provides the basis for implementing FOC [49][50]. For improved readability and to facilitate understanding of the mathematical development, the principal symbols and parameters employed in this study are listed in Table 1.

Table 1. Notation List

The state-space model representation can be mathematically written as the following equation:

Where:  and  can be defined as:

With:

The stator flux can be estimated such as:

The stator and rotor flux linkage phase is given by:

The electromagnetic torque of the induction motor can be written as:

The mechanical dynamics of the induction motor are governed by:

2. Hybrid PI–PTSTADOB Speed Controller Design

Substituting the electromagnetic torque equation (4) into the mechanical dynamics (5) and grouping parameter uncertainties and load torque variations into a lumped disturbance term, the speed subsystem can be rewritten as:

where  denotes the lumped disturbance term including load torque variation, friction uncertainty and parameter perturbations.

Let the speed tracking error be defined as:

To guarantee convergence within a user-defined time independent of initial conditions, a nonlinear predefined-time sliding surface is constructed as:

Where:

This structure combines fast far-from-origin convergence ( -term) and strong near-origin contraction ( -term), forming a predefined-time stable manifold. Taking the derivative:

Using (6):

The proposed hybrid structure retains the conventional PI controller:

An auxiliary predefined-time super-twisting disturbance observer (PTSTADOB) is introduced:

The total control input becomes:

Unlike classical STA with constant gains, predefined-time convergence is enforced via time-varying gains:

where is a user-defined convergence time. These gains grow unbounded as , enforcing convergence before .

Consider the Lyapunov function:

Taking derivative:

Substituting (12):

Choosing  and assuming bounded disturbance  we obtain:

Substituting (14):

Since  there exists such that:

Integrating both sides yields:

Therefore: which implies . The settling time is explicitly determined by the design parameter: . The closed-loop PI–PTSTADOB speed control system guarantees global predefined-time stability, and the speed tracking error converges to zero exactly within the user-specified time .

3. RESULT AND DISCUSSION

The proposed hybrid PI–PTSTADOB control strategy was validated through MATLAB/Simulink 2023b simulations using a squirrel-cage induction motor with the following ratings: . The simulation model is presented in Figure 1.

1. Dynamic Performance Evaluation under Rated Load Torque

Figure 2 and Table 2 provide a comprehensive comparative evaluation of the dynamic performance of the conventional PI, PI–STA, and the proposed PI–PTSTADOB controllers under rated load conditions. As observed from the speed responses in Figure 2, all three strategies are capable of tracking the reference command; however, their transient characteristics, robustness to load disturbances, and steady-state accuracy differ significantly.

During acceleration and deceleration intervals, the conventional PI controller exhibits pronounced overshoot and oscillatory behavior. Although the settling time is relatively short (0.0223s), the transient response is poorly damped, indicating limited robustness against load torque disturbances and parameter uncertainties. This is further reflected by the relatively high overshoot and large error indices in Table 2.

The PI–STA controller improves disturbance rejection and reduces oscillatory behavior compared with the PI scheme. The overshoot is significantly reduced to 0.1209, and the integral error indices (ISE and IAE) decrease accordingly. However, residual oscillations remain observable in the zoomed-in responses, which can be attributed to the fixed-gain super-twisting mechanism and its sensitivity to gain selection. Although the settling time (0.0227 s) is comparable to that of the PI controller, the overall transient quality remains limited by the trade-off between robustness and chattering attenuation.

In contrast, the proposed PI–PTSTADOB controller achieves a substantially improved dynamic response. The predefined-time sliding manifold combined with time-varying super-twisting gains ensures rapid convergence with enhanced damping characteristics. As shown in Table 2, the overshoot is reduced to only 0.0131, corresponding to approximately 98.4% and 89.2% reductions compared with PI and PI–STA, respectively. Unlike conventional STA approaches, the predefined-time gain scheduling enforces strong convergence near the sliding surface while avoiding excessive oscillatory behavior.

Moreover, the error performance indices demonstrate the superiority of the proposed method. The ISE decreases from 0.0213 (PI) and 0.0083 (PI–STA) to 0.00082, representing reductions of approximately 96% and 90%, respectively. Similarly, the IAE is reduced to 0.0124, confirming improved disturbance rejection and faster attenuation of tracking errors. Most notably, the speed RMSE drops dramatically to 0.0641, which corresponds to an 80.4% reduction relative to PI and a 68.6% reduction relative to PI–STA. This significant reduction in steady-state fluctuation highlights the effectiveness of the predefined-time disturbance observer in compensating load variations and modelling uncertainties.

Although the settling times of PI–STA and PI–PTSTADOB are numerically similar, the qualitative transient behavior differs markedly. The proposed controller achieves near-zero overshoot, minimal oscillations, and smoother torque and current responses, as illustrated in Figure 2. This indicates that the improvement is not merely in convergence speed but in overall closed-loop damping, robustness, and control smoothness.

These results confirm that integrating a predefined-time super-twisting disturbance observer into the conventional PI–FOC framework preserves the simplicity and industrial feasibility of PI control while significantly enhancing transient stability, disturbance rejection capability, and steady-state precision. The predefined-time design further guarantees convergence within a user-specified time horizon independent of initial conditions, which distinguishes the proposed approach from conventional finite-time or fixed-gain STA-based methods.

Figure 1. Simulation Model of Hybrid PI-PTSTADOB Based FOC

Figure 2. Dynamic Performance of PI, PI-STA and PI-PTSTADOB under Rated Load Moment

Table 2. Quantitative Dynamic Performance Comparison under Rated Load Torque

2. Robustness Evaluation under Load Disturbance

Figure 3 presents a comprehensive robustness assessment of the conventional PI, PI–STA, and the proposed PI–PTSTADOB controllers under stepwise load torque variations conditions. The comparison focuses on speed regulation accuracy, torque response smoothness, and current behavior, thereby providing a systematic evaluation of disturbance rejection capability.

In the left column of Figure 3, the reference speed is subjected to multiple step changes while the load torque varies abruptly. Under these disturbance intervals, the conventional PI controller exhibits noticeable speed deviations and prolonged recovery periods, reflecting limited robustness against sudden load perturbations. Although the PI–STA controller improves disturbance attenuation due to the sliding-mode action, residual oscillations and ripple remain observable, particularly during successive load transitions. These oscillations are mainly caused by the fixed-gain super-twisting mechanism, which cannot adapt its convergence intensity to varying disturbance levels.

By contrast, the proposed PI–PTSTADOB controller maintains close tracking of the reference speed with negligible deviation throughout all disturbance intervals. The enlarged views demonstrate rapid recovery immediately after each load step, with significantly suppressed oscillations. This enhanced behavior results from the predefined-time sliding manifold combined with time-varying super-twisting gains, which enforce strong convergence within a user-defined time horizon independent of initial conditions and disturbance magnitude. Unlike conventional finite-time or fixed-gain STA schemes, the predefined-time design ensures uniform convergence speed and consistent disturbance compensation performance under varying load conditions.

The torque responses further validate these observations. The PI controller generates large torque spikes during load transitions, which induce mechanical stress and amplify speed oscillations. The PI–STA controller reduces peak torque magnitude but still exhibits oscillatory behavior. In contrast, the PI–PTSTADOB strategy produces smoother torque transitions with reduced ripple, indicating effective lumped disturbance estimation and compensation. The coordinated torque–speed response demonstrates improved damping characteristics and enhanced closed-loop stability.

The stator current responses further corroborate the superiority of the proposed approach. The PI controller exhibits pronounced current spikes and increased ripple in the current components, indicating higher electrical stress and reduced noise immunity. The PI–STA controller partially mitigates these effects but still introduces oscillatory current components. In contrast, the PI–PTSTADOB controller maintains smooth and bounded current trajectories with reduced peak amplitudes, demonstrating improved current stress management and enhanced robustness against disturbance-induced fluctuations.

Overall, the results confirm that integrating the predefined-time super-twisting disturbance observer into the conventional PI–FOC framework significantly enhances robustness without increasing structural complexity. The proposed method achieves superior disturbance rejection, smoother torque and current responses, and consistent predefined-time convergence, thereby preserving industrial feasibility while substantially improving closed-loop performance under both deterministic and stochastic disturbance conditions.

Figure 3. Robustness Evaluation under Load Disturbance

4. CONCLUSIONS

This paper has proposed a hybrid predefined-time super-twisting disturbance-observer-based speed control strategy for PI-based FOC induction motor drives. Unlike conventional sliding-mode-based approaches that replace the PI structure, the proposed method preserves the simplicity and industrial compatibility of the classical PI–FOC framework while enhancing robustness through an auxiliary predefined-time super-twisting disturbance observer integrated into the speed loop. By constructing a nonlinear predefined-time sliding manifold and employing time-varying super-twisting gains, the proposed controller guarantees convergence of the speed tracking error within a user-defined time independent of initial conditions and without requiring prior knowledge of disturbance bounds. Lyapunov-based analysis has rigorously established global predefined-time stability of the closed-loop system. Comprehensive simulation studies under rated load and stepwise load variations demonstrate that the proposed PI–PTSTADOB strategy achieves significantly reduced overshoot, substantially lower error indices (ISE, IAE, and RMSE), smoother torque transitions, and improved current behavior compared with conventional PI and PI–STA controllers. In particular, the results confirm that the improvement is not limited to faster convergence but extends to enhanced damping characteristics, stronger disturbance rejection capability, and reduced oscillatory behavior. Importantly, these performance gains are obtained without increasing structural complexity or computational burden, thereby maintaining practical feasibility for industrial implementation. Overall, the proposed approach provides an effective and theoretically guaranteed solution for high-performance and robust speed regulation of induction motor drives operating under uncertain and disturbance-prone conditions.

Future research will focus on extending the proposed PI–PTSTADOB strategy toward experimental validation on real-time hardware platforms to further confirm its practical applicability under industrial operating conditions. Furthermore, he development of a sensorless control scheme incorporating the predefined-time disturbance observer is also a promising direction to reduce system cost and improve reliability.

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