On the Semi-Global Stability of an EK-Like Filter

Abstract

This paper proposes to apply the Kalmanlike observer paradigm to general nonlinear systems by linearization along the estimated trajectory, similarly to an Extended Kalman Filter. The main difference is that the quadratic Riccati equation is replaced by a linear Lyapunov equation which can be solved and explicitly related to a determinability Gramian. This allows to show by Lyapunov analysis and without any ad-hoc assumption on the Riccati solution, that the resulting observer, called Extended Kalman-like Filter, can be made semi-globally convergent if the input is actively used to a) stabilize the (unknown) true trajectory, b) sufficiently excite the determinability of the linearized systems along the (known) estimated trajectory. A class of systems where this compromise can be reached is provided.

Observer design for continuous-time dynamical systems

“…-in a Kalman-like design, Q = 0, λ > 0 modelling a (stabilizing) forgetting factor, Π(0) −1 = P(0) (resp R −1 ) describing the weight of the initial error and the output error in the optimized cost (48), and thus the confidence in our initial guess and the output comparatively.…”

“…Recent contributions though, suggest that for controlled systems, the input could be actively chosen online to maximize the Gramian associated to (A( x, t), C( x, t)), namely sufficiently excite the observability of the linearization along the known estimate trajectory x, in order to guarantee (52) holds and obtain semi-global convergence [206,48].…”

Online Decentralized Perception-Aware Path Planning for Multi-Robot Systems

“…, where 1 ψi (t) is the current estimated orientation of robot i provided by the EKF (and, thus, locally available to robot i at the current time t). The block decomposition detailed in (17)(18)(19) then becomes fully decentralized since each block can be evaluated by robot i by only resorting to local and 1-hop information.…”

“…It is worth noting that formation rigidity is only a sufficient condition for solving the localization problem which can also be solved without the rigidity condition but at the cost of introducing additional requirements on the group motion. When not rigid, the group must typically satisfy some persistent excitation condition [16], [17] for ensuring a converging localization (whereas no special robot motion is required in the rigid case). Clearly, the possibility of relaxing the rigidity requirement can be important during a mission for allowing better maneuverability and flexibility in the group shape (especially when coping with sensing/communication constraints) even if this imposes some constraints on the robot motion.…”

Information-Aware Lyapunov-Based MPC in a Feedback-Feedforward Control Strategy for Autonomous Robots

“…We will show here that control inputs computed on the state estimates generate an uncertainty on the time derivative of the candidate Lyapunov function proportional to the state estimation uncertainty, which is directly related to the active sensing choices. To quantify the amount of the collected information (and hence of the uncertainty) along the planned trajectories, the Constructability Gramian (CG), quantifying the level of constructibility of the current/future state, is used as the guiding metric [11], [15].…”