Jonathan Rogge

Existence of Partial Entrainment and Stability of Phase Locking Behavior of Coupled Oscillators

We study a network of all-to-all interconnected phase oscillators as modeled by the Kuramoto model. For coupling strengths larger than a critical value, we show the existence of a collective behavior called phase locking: the phase differences between all oscillators are constant in time. As the coupling strength increases, the distance between each pair of phases decreases. Stability of each phase locking solution is proven for general frequency distributions. There exist one unique asymptotically stable phase locking solution. Furthermore a description is given of partial entrainment, which can be regarded as the finite number analogon of partial synchronization in the infinite number case. When the network is partially entraining some phase differences possess an upper and lower bound. Partial entrainment of the three-cell network is analyzed: an estimate of the onset of partial entrainment is given and the existence of partial entrainment is proven. Furthermore, local stability of partial entrainment is proven for the three-cell network with two identical oscillators.

Stability of phase locking in a ring of unidirectionally coupled oscillators

In this paper we discuss the behavior of a finite group of phase oscillators unidirectionally coupled in a ring. The dynamics is described by the Kuramoto model. In the case of identical oscillators, all phase locking solutions are obtained explicitly, together with their stability properties. For nonidentical oscillators it is proven that there exist phase locking solutions for sufficiently strong coupling. An algorithm for obtaining all phase locking solutions is given. These solutions can be subdivided into classes, the stability properties of each of which are established separately. The stability results are extended to interconnections belonging to a class of odd functions. Finally, a connection with the field of active antenna arrays is made, generalizing some results earlier obtained in this field.

Vehicle Platoons Through Ring Coupling

In this paper, a novel strategy for the control of a string of vehicles is designed. The vehicles are coupled in a unidirectional ring at the interaction level: each vehicle is influenced by the position of its immediate forward neighbor; the first vehicle in the platoon is influenced by the position of the last vehicle. Through these interactions a cooperative behavior emerges and a platoon of vehicles moving at a constant velocity with constant inter-vehicle spacings is formed. This contrasts with more traditional control schemes where an independent leader vehicle is followed by the remaining vehicles.

Consensus over ring networks as a quadratic optimal control problem

This paper presents the consensus problem in the framework of optimal control. Our aim is to synchronize a set of identical linear systems. We propose a cost which penalizes mutual differences between the states of these systems. The feedback matrix resulting from this linear quadratic control problem represents the interconnection network which synchronizes the systems. In general the interconnection structure is of the all-to-all type. We show that it is possible to devise an LQR problem in which the cost results in an interconnection structure representing ring coupling. Care has to be taken that the effect of the feedback control is restricted to synchronizing the systems, i.e. when the systems are synchronized, the feedback control signal is required to be equal to zero.

Distributed collision avoidance for autonomous vehicles: world automata representation

The automatic control of interacting autonomous vehicles (AVs) is one of the problems that engineers are currently trying to solve. The present paper deals with the design of local control laws governing the movement and collision avoidance of such groups of AVs, enforcing the safeness of the operations as well as task completions. This problem is inspired by the automation of a container terminal where each AV executes tasks assigned by a supervisor. A task involves moving an AV from an assigned origin to an assigned destination by a given deadline. The constraints imposed by the bounded workspace (a long and narrow quay in the container terminal example), the deadlines assigned to each task, and the uncertainty in the detection and communication make the problem difficult to solve in a centralized way. Therefore a distributed control approach is preferred with a local control agent in each AV adjusting its trajectory, so that its task is completed without collisions. By applying a fixed set of priority rules the computational complexity for each agent is reduced compared to the centralized case. Whenever an AV detects a possible conflict, i.e. the estimated position of another AV within the detection range, it must adjust its own speed and trajectory in order to avoid a future collision, reducing the number of cases where a supervisor has to intervene in order to resolve conflicts that degenerate in dead-locks. The modelling and validation of the system is performed by using the world automata theory.

Decentralized control of vehicle platoons with interconnection possessing ring topology

In this paper we design a control strategy for platoons of identical vehicles. It is assumed that each vehicle measures the distance with its immediate forward neighbor. The lead vehicle in the platoon only receives information on the position of the last vehicle in the platoon. We prove that the resulting behavior of the system is a platoon of vehicles moving at a constant velocity with constant distance between each pair of consecutive vehicles and that for a class of identical controllers this solution is asymptotically stable for sufficiently small coupling strength. An upper limit of this coupling strength is calculated, below which the solution is asymptotically stable, independent of the number of vehicles in the platoon. Moreover, simulations indicate that the platoon is string stable. To improve the behavior, integral action is added between the first and last vehicle of the platoon. The resulting behavior is determined and its stability properties are discussed.

Stability of phase locking and existence of entrainment in networks of globally coupled oscillators

Abstract We study a network of a finite number of all-to-all interconnected phase oscillators as modeled by the Kuramoto model. For coupling strengths larger than a critical value, we show the existence of a collective behavior called phase locking: the phase differences between all oscillators are constant in time. Stability of each phase locking solution is proven for general frequency distributions. Furthermore a description is given of partial entrainment: some but not all phase differences remain bounded. For the three cell network an estimation of the coupling strength at the onset of partial entrainment is computed.

A Strategy for Exploration with a Multi-robot System

The present paper develops a novel strategy for the exploration of an unknown environment with a multi-robot system. Communication between the robots is restricted to line-of-sight and to a maximum inter-robot distance. The algorithm we propose is related to methods used for complete coverage of an area, where all free space is physically covered. In the present paper it is required that the entire free space is covered by the sensors of the robots, enabling us to scan more space in less time, compared to complete coverage algorithms. The area to be scanned contains disjoint convex obstacles of unknown size and shape. The geometry of the robot group has a zigzag shape, which is stretched or compressed to adapt to the environment. The robot group is allowed to split and rejoin when passing obstacles. A direct application of the algorithm is mine field clearance.