Geometrization of Linearization-based Control Laws on the Unit Circle

Although the single-axis attitude control of a rigid body is an inherently nonlinear control problem wherein the angular position evolves on the unit circle, controller design is often carried out by linearizing the dynamics and applying linear control theory. Since the resulting approximation is valid only for small angular errors, linear control design methods typically provide only local stability and performance guarantees. Moreover, for large angular errors, linear control laws can cause unwinding. In contrast, nonlinear controllers which respect the geometry of the unit circle can achieve almost global asymptotic stability. Given this trade-off between the stability and performance guarantees possible with linear and geometric nonlinear control laws, there is a need for design and analysis methods which can bridge the gap between the two approaches. In this paper, we propose one such method for geometrizing linear control laws on the unit circle. More precisely, we consider a linearization-based controller for single-axis attitude control, and introduce a nonlinear rotational error function into the closed loop. Then, we use Lyapunov stability analysis to show that the proposed nonlinear control architecture achieves almost global asymptotic stability on the unit circle. Moreover, the Lyapunov analysis yields sufficient conditions in the form of linear matrix inequalities (LMIs) which can be used to analyze the stability properties of a given linearization-based control law. Thus, our approach allows the control designer to use standard methods for designing linear controllers with local stability and performance guarantees, and to obtain almost global asymptotic stability guarantees by checking the feasibility of the proposed LMIs. After detailing the theoretical developments, we demonstrate the applicability of our method on a practical problem.