Predefined-Time Control Strategies for Nonlinear Dynamical Systems

Predefined-time control refers to a class of feedback design techniques that guarantee convergence of a nonlinear system’s state to a desired equilibrium within a user-specified time bound, irrespective of initial conditions or bounded disturbances. This approach extends classical finite-time and fixed-time stability by allowing the settling time to be chosen as an explicit design parameter. Central to these strategies is the construction of Lyapunov-based stability theorems and associated control laws—often leveraging sliding-mode surfaces, polynomial or vector-function formulations, adaptive elements and event-triggered mechanisms—to impose a hard deadline on convergence. Such methods have been applied to a breadth of systems, from robotic manipulators and permanent-magnet motors to secure synchronisation of chaotic oscillators and power electronics. The global significance of predefined-time control lies in its capacity to offer guaranteed performance in safety-critical and time-sensitive systems, including multi-agent coordination, aerospace trajectory tracking and industrial servo systems. Recent efforts have prioritised robustness against model uncertainties, external disturbances and measurement constraints, while enhancing flexibility through unified theoretical frameworks and novel approximators such as neural networks and fuzzy logic.

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A unified Lyapunov-based theorem has been proposed to consolidate existing predefined-time stability results into three sufficient conditions. By framing these within a single analytical method, designers can derive standard non-singular sliding-mode controllers for Lagrangian systems, ensuring that the convergence time is an explicit parameter that degenerates to finite-time stability under relaxed conditions. Another line of work focuses on high-precision industrial applications: a robust sliding-mode controller for permanent-magnet linear motors achieves position tracking with convergence guaranteed within a preset interval and independent of initial states or system parameters, significantly reducing sensitivity to friction and external disturbances compared with traditional PID or linear sliding-mode schemes. In the domain of powertrain control, a predefined-time algorithm for a four-degree-of-freedom synchronous motor demonstrates resilience to both deterministic disturbances and stochastic noise, with simulations confirming that all state variables reach equilibrium within the designer-specified time. Collectively, these studies illustrate the interplay between unified theoretical advances and tailored implementations, highlighting practical gains in accuracy, robustness and resource efficiency across diverse nonlinear platforms.

Predefined-Time Control Strategies for Nonlinear Dynamical Systems publication trend

The graph below shows the total number of articles in predefined-time control strategies for nonlinear dynamical systems across all publications each year (not limited to Nature Index journals).

Technical terms

Predefined-time stability: The property by which a controlled system converges to equilibrium within a user-defined time bound, irrespective of initial conditions.

Lyapunov function: A scalar, positive-definite function whose time derivative along system trajectories is used to certify stability and convergence rates.

Sliding mode control: A robust control methodology that drives system trajectories onto a predetermined sliding surface and maintains them there despite perturbations.

Fractional-order system: A dynamical system described by derivatives of non-integer order, capturing memory and hereditary effects in the model.

Settling time: The elapsed time required for the system state to enter and remain within a specified neighbourhood of the equilibrium point.

References

1. Predefined-Time Control of Full-Scale 4D Model of Permanent-Magnet Synchronous Motor with Deterministic Disturbances and Stochastic Noises. Actuators (2021).
2. Predefined-Time Polynomial-Function-Based Synchronization of Chaotic Systems via a Novel Sliding Mode Control. IEEE Access (2020).
3. Predefined-time vector-polynomial-based synchronization among a group of chaotic systems and its application in secure information transmission. AIMS Mathematics (2021).
4. Unified Sufficient Conditions for Predefined-Time Stability of Non-Linear Systems and Its Standard Controller Design. Actuators (2024).
5. Adaptive Fuzzy Event-Triggered Cooperative Control for Multi-Robot Systems: A Predefined-Time Strategy. Sensors (2023).
6. Design of Predefined Time Convergent Sliding Mode Control for a Nonlinear PMLM Position System. Electronics (2023).