Modified global finite-time quasi-continuous second-order robust feedback control

TL;DR

This paper introduces a modified finite-time robust control law for second-order systems that ensures parameterizable amplitude limits, full state-space control, and exact convergence time estimation, with stability proven under perturbations.

Contribution

It presents a novel modification to existing sliding mode control that enhances control amplitude limitations, expands the controllable state space, and provides analytic solutions for unperturbed cases.

Findings

Control amplitude can be effectively limited.

Full state-space control is achieved.

Finite convergence time can be exactly estimated.

Abstract

A non-overshooting quasi-continuous sliding mode control with sub-optimal damping was recently introduced in Ruderman and Efimov (2025) for perturbed second-order systems. The present work proposes an essential modification of the nonlinear control law which (i) allows for a parameterizable control amplitude limitation in a large subset of the initial values, (ii) admits an entire state-space R2 (that was not given in Ruderman and Efimov (2025)) for the finite-time control, and finally (iii) enables for the found analytic solution of the state trajectories in the unperturbed case. The latter allows also for an exact estimation of the finite convergence time, and open an avenue for other potentially interesting analysis of the control properties in the future. For a perturbed case, the solution-based and Lyapunov function-based approaches are developed to show the uniform global asymptotic stability. The proposed robustness and convergence analysis are accompanied by several illustrative numerical examples.