Autonomous Error and Constructive Observer Design for Group Affine Systems

Abstract

Inertial Navigation Systems (INS) are algorithms that fuse inertial measurements of angular velocity and specific acceleration with supplementary sensors including GNSS and magnetometers to estimate the position, velocity and attitude, or extended pose, of a vehicle. The industry-standard extended Kalman filter (EKF) does not come with strong stability or robustness guarantees and can be subject to catastrophic failure. This paper exploits a Lie group symmetry of the INS dynamics to propose the first nonlinear observer for INS with error dynamics that are almost-globally asymptotically and locally exponentially stable, independently of the chosen gains. The observer is aided only by a GNSS measurement of position. As expected, the convergence guarantee depends on persistence of excitation of the vehicle's specific acceleration in the inertial frame. Simulation results demonstrate the observer's performance and its ability to converge from extreme errors in the initial state estimates.

Data-Driven Control of Nonlinear Systems: Beyond Polynomial Dynamics

“…We emphasize again that the closed loop under our proposed certified control law µ operates reliably in the sense that it exponentially converges to the origin. On the contrary, the established approach of designing an LQR controller µ LQR based on the linear lifted system (19) learned via linear EDMD (20) leads to an unstable and, hence, unsafe behavior of the nonlinear system.…”

“…In this paper, we use [a : b] := Z ∩ [a, b] and [b] := [1 : b].2 Here, we tacitly assumed invariance of X under the flow such that the observable functions are defined on x(t; x, u) for all x ∈ X. This assumption can be relaxed by considering initial values contained in a tightened version of X, see, e.g.,[20] for details.…”