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Robust Control Design Using a Hybrid-Gain Finite-Time Sliding-Mode Controller

Amit Shivam, Kiran Kumari, Fernando A. C. C. Fontes

TL;DR

This work introduces a hybrid-gain finite-/fixed-time sliding-mode control (HG-FTSMC) that combines an outer bounded finite-time gain with an inner fixed-time gain to achieve rapid, robust convergence while respecting actuator limits. By avoiding norm-based normalization and using a two-region gain structure, the method attains fast transient performance with bounded control effort. Lyapunov analysis confirms finite-time reaching into a predefined boundary layer and fixed-time convergence within the layer, and simulations on a perturbed first-order integrator and a two-link Euler–Lagrange system demonstrate reduced control effort with comparable settling times to SATO. The experimental-significant outcome is a practically feasible, smoother, and robust SMC framework applicable to both simple and multi-DOF robotic systems.

Abstract

This paper proposes a hybrid-gain finite-time sliding-mode control (HG-FTSMC) strategy for a class of perturbed nonlinear systems. The controller combines a finite-time reaching law that drives the sliding variable to a predefined boundary layer with an inner mixed-power or exponential law that guarantees rapid convergence within the layer while maintaining smooth and bounded control action. The resulting control design achieves finite-time convergence and robustness to matched disturbances, while explicitly limits the control effort. The control framework is first analyzed on a perturbed first-order integrator model, and then extended to Euler-Lagrange (EL) systems, representing a broad class of robotic and mechanical systems. Comparative simulations demonstrate that the proposed controller achieves settling times comparable to recent finite-time approaches [1], while substantially reducing the control effort. Finally, trajectory-tracking simulations on a two-link manipulator further validate the robustness and practical feasibility of the proposed HG-FTSMC approach.

Robust Control Design Using a Hybrid-Gain Finite-Time Sliding-Mode Controller

TL;DR

This work introduces a hybrid-gain finite-/fixed-time sliding-mode control (HG-FTSMC) that combines an outer bounded finite-time gain with an inner fixed-time gain to achieve rapid, robust convergence while respecting actuator limits. By avoiding norm-based normalization and using a two-region gain structure, the method attains fast transient performance with bounded control effort. Lyapunov analysis confirms finite-time reaching into a predefined boundary layer and fixed-time convergence within the layer, and simulations on a perturbed first-order integrator and a two-link Euler–Lagrange system demonstrate reduced control effort with comparable settling times to SATO. The experimental-significant outcome is a practically feasible, smoother, and robust SMC framework applicable to both simple and multi-DOF robotic systems.

Abstract

This paper proposes a hybrid-gain finite-time sliding-mode control (HG-FTSMC) strategy for a class of perturbed nonlinear systems. The controller combines a finite-time reaching law that drives the sliding variable to a predefined boundary layer with an inner mixed-power or exponential law that guarantees rapid convergence within the layer while maintaining smooth and bounded control action. The resulting control design achieves finite-time convergence and robustness to matched disturbances, while explicitly limits the control effort. The control framework is first analyzed on a perturbed first-order integrator model, and then extended to Euler-Lagrange (EL) systems, representing a broad class of robotic and mechanical systems. Comparative simulations demonstrate that the proposed controller achieves settling times comparable to recent finite-time approaches [1], while substantially reducing the control effort. Finally, trajectory-tracking simulations on a two-link manipulator further validate the robustness and practical feasibility of the proposed HG-FTSMC approach.

Paper Structure

This paper contains 10 sections, 4 theorems, 43 equations, 4 figures, 3 tables.

Table of Contents

  1. Introduction
  2. Problem Formulation
  3. Proposed Controller Design
  4. Intuitive Settling-Time Estimate for the SATO Controller
  5. First-Order System
  6. EL Systems
  7. Simulation Results
  8. First-order system
  9. EL System
  10. Conclusion

Key Result

Theorem 1

Given the system dynamics eq:first-order, the proposed control law and, hybrid gain drives the system into equilibrium (at origin). For any $\varepsilon>0$ such that $\varepsilon \in (0,\varepsilon_o]$, the variable $x(t)$ enters the boundary layer $|x|\le\varepsilon$ in finite time, employing the hybrid gain proposed as The convergence time follows the bound

Figures (4)

  • Figure 1: Results for comparative controller design
  • Figure 2: Results for HG-FTSMC with polynomial inner law
  • Figure 3: Results for HG-FTSMC with exponential inner law
  • Figure 4: Results for two link manipulator

Theorems & Definitions (11)

  • Remark 1: Interpretation of the SATO settling time
  • Remark 2: Limitations and motivation for proposed design
  • Theorem 1
  • proof
  • Corollary 1: Exponential inner (erf) variant
  • Remark 3: Bounded effort and tuning
  • Theorem 2
  • proof
  • Corollary 2: Exponential Inner Law
  • Remark 4: Boundedness and Tuning Guidelines
  • ...and 1 more