Geometrization of Higher-Order Linear Control Laws for Attitude Control on $\mathsf{SO(3)}$

TL;DR

This paper develops a geometric framework for analyzing the stability of higher-order nonlinear attitude control laws on SO(3), extending PID control results and using LMI-based conditions for practical controller design.

Contribution

It extends stability analysis of geometric control laws on SO(3) to higher-order systems using Lyapunov functions and LMIs, with practical application to multicopter attitude control.

Findings

Almost globally asymptotically stable equilibrium achieved.

LMI conditions provide less conservative stability criteria.

Practical 21-state control law successfully designed for multicopter.

Abstract

This paper presents a theoretical framework for analyzing the stability of higher-order geometric nonlinear control laws for attitude control on the Special Orthogonal Group $\mathrm{SO(3)}$. In particular, the paper extends existing results on the analysis of PID-type geometric nonlinear control laws to more general higher-order dynamic state-feedback compensators on $\mathrm{SO(3)}$. The candidate Lyapunov function is motivated by quadratic Lyapunov functions of the form $V(x)=x^{\top}Px$ typically considered in the analysis of linear time-invariant (LTI) systems. The stability analysis is carried out in two steps. In the first step, a sufficient condition is obtained for the positive definiteness of the candidate Lyapunov function, and a necessary and sufficient condition for the negative definiteness of the corresponding Lyapunov rate. These conditions ensure that the desired equilibrium is almost globally asymptotically stable (AGAS). In the second step, a convex relaxation of the proposed conditions is used to obtain sufficient conditions in the form of linear matrix inequalities (LMIs). Overall, the approach is motivated by the widespread use of LMI-based analysis and design tools for LTI systems. To reduce conservatism, matrix gains are considered for the controller gains as well as the Lyapunov function coefficients. The applicability of the approach to practical problems is illustrated by designing and analyzing a 21-state geometric nonlinear attitude control law for a multicopter.