Higher-order attitude control using rotation matrices

This paper presents a theoretical framework for analyzing the stability of higher-order attitude control laws on the Special Orthogonal Group SO(3). In particular, it considers a class of dynamic state-feedback compensators in which the controller state xK∈Rn has arbitrary finite dimension. This class of compensators includes, as special cases, several existing geometric nonlinear PID-type attitude controllers in which the dynamic component is an integrator state having dimension 3. As a result, the proposed analytical framework extends existing results on geometric nonlinear PID-type attitude control to more general dynamic state-feedback compensators on SO(3). The stability analysis of the higher-order attitude controller is carried out in two main steps. First, a quadratic candidate Lyapunov function is used to obtain sufficient conditions, in the form of bilinear matrix inequalities (BMIs), which ensure that the desired equilibrium of the closed-loop tracking error system is almost globally asymptotically stable (AGAS). Then, a convex relaxation of the proposed conditions is used to obtain sufficient conditions in the form of linear matrix inequalities (LMIs). To reduce conservatism, matrix gains are used for the controller gains as well as the Lyapunov function coefficients. The use of quadratic Lyapunov functions and LMIs is motivated by their ubiquitous use in the design and analysis of dynamic compensators for linear time-invariant (LTI) systems. The applicability of the proposed analytical framework to practical problems on SO(3) is illustrated by designing a 9-state almost globally asymptotically stabilising attitude controller for an agile multicopter.