Perturbed Integrators Chain Control via Barrier Function Adaptation and Lyapunov Redesign

TL;DR

This paper addresses robust stabilization of a perturbed chain of integrators with an unknown control coefficient by blending Lyapunov redesign with barrier‑function adaptation. It introduces a two‑phase control: a predefined‑time reaching phase using a time‑varying PNF‑type gain, followed by a barrier‑function phase that enforces confinement within a prescribed neighborhood using an adaptive gain, all governed by a single quadratic Lyapunov function $V(x)=x^T P x$ derived from the ARE. The main result guarantees arrival at $\mathcal{S}_{\epsilon/2}$ in finite time and perpetual confinement in $\mathcal{S}_{\epsilon}$ despite disturbances, with a single switching in the control gain. The approach is validated via simulations on a torsional spring–damper model and experimentally on Furuta’s pendulum, demonstrating practical effectiveness and tunable neighborhood and settling‑time properties.

Abstract

Lyapunov redesign is a classical technique that uses a nominal control and its corresponding nominal Lyapunov function to design a discontinuous control, such that it compensates the uncertainties and disturbances. In this paper, the idea of Lyapunov redesign is used to propose an adaptive time-varying gain controller to stabilize a class of perturbed chain of integrators with an unknown control coefficient. It is assumed that the upper bound of the perturbation exists but is unknown. A proportional navigation feedback type gain is used to drive the system's trajectories into a prescribed vicinity of the origin in a predefined time, measured using a quadratic Lyapunov function. Once this neighborhood is reached, a barrier function-based gain is used, ensuring that the system's trajectories never leave this neighborhood despite uncertainties and perturbations. Experimental validation of the proposed controller in Furuta's pendulum is presented.

Figures (9)

• Figure 1: Tracking with $\epsilon = 1$.
• Figure 2: Tracking with $\epsilon = 1\times 10^{-2}$.
• Figure 3: Tracking with $\epsilon = 1\times 10^{-4}$.
• Figure 4: Angles for different values of $T$ starting from $\theta_p(0) = 0.5$.
• Figure 5: control signal and gain $\Lambda(t)$ for different values of $T$ starting from $\theta_p(0) = 0.5$.