Finite-Time Stability in Nonlinear Control Systems

Finite-time stability in nonlinear control systems denotes the ability of a closed-loop system to reach an equilibrium point within a predetermined finite interval, rather than merely approaching asymptotically. This property is highly desirable in applications requiring rapid convergence and guaranteed performance within strict time bounds, such as robotic manipulators, autonomous aerial vehicles and precision hydraulic actuators. Achieving finite-time stability typically relies on specialised Lyapunov functions whose time derivatives remain negative definite outside the equilibrium, coupled with homogeneity and sliding-mode techniques to enforce strong convergence. Recent advances have extended classical homogeneity concepts to systems with delays, perturbations and uncertain dynamics, while continuous higher-order sliding-mode controllers have mitigated chattering and measurement noise. The resulting frameworks offer explicit upper bounds on settling time, robustness against bounded disturbances and broad applicability to multi-input multi-output configurations. Overall, finite-time stability theory has matured into a flexible toolkit for designers seeking both rapid response and rigorous performance guarantees in complex nonlinear settings.

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Recent studies have addressed robustness and generality of finite-time stability in systems with distributed delays. One work has analysed homogeneous nonlinear systems whose dynamics depend on a continuum of past states through a distributed delay kernel. By comparing Lyapunov–Krasovskii and Razumikhin approaches and applying averaging methods for time-varying perturbations, the authors derived conditions ensuring that the zero solution retains practical finite-time stability, illustrated on mechanical oscillators with Liénard-type dynamics.

Another line of development has focused on geometric tracking for multirotor aerial platforms. A continuous nonsingular terminal sliding-mode controller was formulated to guarantee finite-time convergence of both attitude and disturbance-compensation errors without incurring high-frequency chattering. The controller relies on a novel disturbance observer and a homogeneous design, and simulations confirm its superior settling time compared with traditional sliding-mode schemes under aerodynamic perturbations.

In the domain of hydraulic actuation, robust trajectory-tracking has been achieved using continuous higher-order sliding-mode observers and controllers. By estimating unmeasured velocity derivatives in finite time and embedding them within a third-order sliding-mode control law, the scheme attains guaranteed settling-time bounds while handling antagonistic load disturbances. Experimental results on a dual-cylinder setup demonstrate precise position control and disturbance rejection, highlighting the practical viability of finite-time methods in industrial systems.

Finite-Time Stability in Nonlinear Control Systems publication trend

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

Technical terms

Finite-time stability: The property that all system trajectories converge to an equilibrium in a uniformly bounded time.

Homogeneity: A scaling characteristic of system dynamics that enables construction of Lyapunov functions yielding finite-time convergence.

Sliding-mode control: A robust control methodology employing discontinuous or high-order corrective actions to drive system states onto a predesigned manifold, ensuring rapid convergence.

Lyapunov function: A positive-definite scalar function whose negative-definite derivative along trajectories certifies stability properties.

Distributed delay: A modelling framework in which current system behaviour depends on an integral of past states weighted by a delay kernel, capturing memory effects.

References

1. Stability of homogeneous systems with distributed delay and time-varying perturbations. Automatica (2023).
2. Hydraulic actuator control based on continuous higher order sliding modes. Control Engineering Practice (2025).
3. Finite-Time Geometric Tracking Controller for Multirotor Systems. IEEE Access (2022).