Further results on synergistic Lyapunov functions and hybrid feedback design through backstepping

TL;DR

This paper advances hybrid feedback design by utilizing generalized synergistic Lyapunov function and feedback pairs, enabling stabilization without additional dynamic variables, demonstrated on a 3D pendulum.

Contribution

It introduces a generalized form of SLFF pairs that do not require extra variables and constructs SLFF pairs for extended systems with integrator control inputs.

Findings

SLFF pairs can be directly used for backstepping without extra variables.

Hybrid feedback asymptotically stabilizes extended systems with positive synergy gap.

Existence of SLFF pairs is guaranteed when a hybrid stabilizer exists.

Abstract

We extend results on backstepping hybrid feedbacks by exploiting synergistic Lyapunov function and feedback (SLFF) pairs in a generalized form. Compared to existing results, we delineate SLFF pairs that are ready-made and do not require extra dynamic variables for backstepping. From an (weak) SLFF pair for an affine control system, we construct an SLFF pair for an extended system where the control input is produced through an integrator. The resulting hybrid feedback asymptotically stabilizes the extended system when the synergy gap for the original system is strictly positive. To highlight the versatility of SLFF pairs, we provide a result on the existence of a SLFF pair whenever a hybrid feedback stabilizer exists. The results are illustrated on the 3D pendulum.