LMI-based Stability Analysis of a Geometric PID-type Attitude Control Law

This paper presents an analytical framework for analyzing the stability of a geometric PID controller, with a prescribed structure, for attitude control on the Special Orthogonal Group
$SO(3)$
. A key feature of the proposed approach is the use of linear matrix inequalities (LMIs) to formulate sufficient conditions which ensure that the closed-loop tracking error system is almost globally asymptotically stable (AGAS). To this end, a candidate Lyapunov function is considered which is slightly more general than those traditionally employed in the
$SO(3)$
literature. In particular, the Lyapunov function contains terms which couple the attitude and velocity errors with the integrator state, as well as matrix gains for four of the six Lyapunov function coefficients. The LMI-based stabilization conditions are then cast as a feasibility problem which can be used to search for Lyapunov function coefficients that confirm AGAS for a given PID controller with matrix gains. Using the proposed approach, control designers can use linearized models of the attitude kinematics and dynamics to tune the PID gains, and then solve a semidefinite programming problem to obtain AGAS guarantees for the corresponding geometric nonlinear PID controller. The effectiveness of this method is demonstrated on a practical problem involving the design and analysis of a geometric PID controller for a hexacopter UAV with local performance requirements specified in terms of rise time, settling time, gain and phase margins, and closed-loop bandwidth.