Trajectory Convergence From Coordinate-Wise Decrease of Quadratic Energy Functions, and Applications to Platoons

Abstract

We consider trajectories where the sign of the derivative of each entry is opposite to that of the corresponding entry in the gradient of an energy function. We show that this condition guarantees convergence when the energy function is quadratic and positive definite and partly extend that result to some classes of positive semi-definite quadratic functions including those defined using a graph Laplacian. We show how this condition allows establishing the convergence of a platoon application in which it naturally appears, due to deadzones in the control laws designed to avoid instabilities caused by inconsistent measurements of the same distance by different agents.

On the benefits of saturating information in consensus networks with noise

“…The following result shows that if at certain time node i enters S i2 (η), it will indefinitely remain in this set. (20), which are interconnected over the graph G, let each local control input be generated in accordance with (21) and (22) with ε(t) and α(t) satisfying (23). For each η ∈ (0, ε(0)], define T (η) as in the proof of Theorem 3.…”

“…In the noiseless case, the node disagreement normalized by the norm of the initial state can be arbitrarily reduced. See [23] for another recent paper on the topics.…”

Erratum: Consensus for Networks with Unknown but Bounded Disturbances

“…As pointed out by the authors of [2], the fault in the proof cannot be fixed without changing the proof approach. In particular, convergence results based on condition (1) in [2] must be used.…”

“…As pointed out by the authors of [2], the fault in the proof cannot be fixed without changing the proof approach. In particular, convergence results based on condition (1) in [2] must be used. In this case, Theorem 1 in [1] follows from Theorems 5 and 4 in [2], as Lemma 3 in [1] implies that the main assumptions in [2] are satisfied.…”

“…In particular, convergence results based on condition (1) in [2] must be used. In this case, Theorem 1 in [1] follows from Theorems 5 and 4 in [2], as Lemma 3 in [1] implies that the main assumptions in [2] are satisfied.…”