Stability Analysis of Trajectories on Manifolds with Applications to Observer and Controller Design

Abstract: This paper examines the local exponential stability (LES) of trajectories for nonlinear systems on Riemannian manifolds. We present necessary and sufficient conditions for LES of a trajectory on a Riemannian manifold by analyzing the complete lift of the system along the given trajectory. These conditions are coordinate-free which reveal fundamental relationships between exponential stability and incremental stability in a local sense. We then apply these results to design tracking controllers and observers for Euler-Lagrangian systems on manifolds; a notable advantage of our design is that it visibly reveals the effect of curvature on system dynamics and hence suggests compensation terms in the controller and observer. Additionally, we revisit some well-known intrinsic observer problems using our proposed method, which largely simplifies the analysis compared to existing results.