Synchronization of coupled nonlinear oscillators with shifted gamma-distributed delays

Systems & Control Letters

Niculescu

2015

International audienceThe paper considers the problem of stabilization of systems possessing a multiple zero eigenvalue at the origin. The controller that we propose, uses multiple delayed measurements instead of derivative terms. Doing so, we increase the performances of the closed loop in presence of system uncertainties and/or noisy measurements. The problem formulation and the analysis is presented through a classical engineering problem which is the stabilization of an inverted pendulum on a cart moving horizontally. On one hand, we perform a nonlinear analysis of the center dynamics described by a three dimensional system of ordinary differential equations with a codimension-three triple zero bifurcation. On the other hand, we present the complementary stability analysis of the corresponding linear time invariant system with two delays describing the behavior around the equilibrium. The aim of this analysis is to characterize the possible local bifurcations. Finally, the proposed control scheme is numerically illustrated and discussed

Section: Concluding Remarkmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Inverted pendulum stabilization: Characterization of codimension-three triple zero bifurcation via multiple delayed proportional gains

Boussaada

Systems & Control Letters

Niculescu

2015

“…The aim is to characterize the occurrence of synchronous behavior as a function of the parameters of the distributed delay. Preliminary results have been reported in . Here, we are going further by emphasizing the delay distribution effect on stability of the linearized system as well as the information we can get for the nonlinear one.…”

Section: Introductionmentioning

confidence: 94%

“…I.-C. MORȂRESCU, W. MICHIELS AND M. JUNGERS in [14]. Here, we are going further by emphasizing the delay distribution effect on stability of the linearized system as well as the information we can get for the nonlinear one.…”

mentioning

confidence: 96%

Effect of a distributed delay on relative stability of diffusely coupled systems, with application to synchronized equilibria

Int. J. Robust. Nonlinear Control

Michiels

Jungers

2015

Self Cite

SUMMARYWe perform a linear stability analysis of synchronized equilibria in networks of diffusely coupled oscillators, affected by distributed delays in the coupling, and we characterize the structure of the emanating solutions in the bifurcations, under the assumption that the delay kernels are equal to each other. The motivation comes from the fact that valuable quantitative and qualitative information about the occurrence and type of synchronous or partially synchronous solutions can be obtained from this linear analysis. We analyze stability of synchronized equilibria as a function of the parameters of a so-called shifted gamma-distributed delay, which allows to represent or approximate a large class of distributed delay kernels. We also present an asymptotic analysis method, which is particularly suitable for studying the effect of a distribution of delays on stability. We apply the methods to networks of coupled Lorenz systems, where we highlight that stability is favored by a distributed delay compared with a discrete delay with the same average value. Among others, we show that if the coupling is diffusive, the synchronized equilibria become asymptotically stable for large values of the coupling gain, that is, the systems locally asymptotically synchronize at an equilibrium, independently of the network topology.

“…Consensus and synchronization are challenging problems widely studied in the context of linear agents interacting through a directed or undirected graph with a fixed or dynamically changing topology. There are also contributions related to nonlinear agents such as oscillators dynamics [4], nonholonomic robots [1] or general nonlinear systems [5].…”

Section: Introductionmentioning

confidence: 99%

Guaranteed cost control design for synchronization in networks of linear singularly perturbed systems

Rejeb

2017 IEEE 56th Annual Conference on Decision and Control (CDC)

Daafouz

2017

Self Cite

Abstract-This work presents the design of a decentralized control strategy that allows singularly perturbed multi-agent systems to achieve synchronization with global performance guarantees. The study is mainly motivated by the presence of two features that characterize many physical systems. The first is the complexity in terms of interconnected subsystems and the second is that each subsystem involves processes evolving on different time-scales. The main difficulty that we have to overcome is that we have to avoid the use of centralized information related to the interconnection network structure. This problem is solved by rewriting the synchronization problem in terms of stabilization of a singularly perturbed uncertain linear system. The singularly perturbed dynamics of subsystems generates theoretical challenges related to the stabilizing controller design but also numerical issues related to the computation of the controller gains. We show that these problems can be solved by decoupling the slow and fast dynamics. Our theoretical developments are illustrated by some numerical examples.