Finding the exact delay bound for consensus of linear multi-agent systems

Abstract

This paper focuses on consensus problems for high-order, linear multi-agent systems. Undirected communication topologies along with a fixed and uniform communication time delay are taken into account. This class of problems has been widely studied in the literature, but there are still gaps concerning the exact stability bounds in the domain of the delays. The novelty of this paper lies in the determination of an exact and explicit delay bound for consensus. This is done in a very efficient manner by using the cluster treatment of characteristic roots (CTCR) paradigm. Before the stability analysis, a state transformation is performed to decouple the system and simplify the problem. CTCR is then deployed to the individual subsystems to obtain the stability margin in the domain of the delays without the conservatism introduced by other approaches more frequently found in the literature. Simulation results are presented to support the analytical claims.

Keywords:

• consensus
• cooperative systems
• time-delay systems
• CTCR

Notes

1. See the appendix A and the works of Brewer (1978) and Bernstein (2005) for a definition and properties of the Kronecker summation operation.

2. We think there is a typo somewhere in the paper by Zhang and Tian (2014) and a minus is missing in front of the feedback gain. We added it to the presentation of K in (18(18) $$\mathbf{K}=-\left[\begin{array}{ccc}\hfill -0.2694& \hfill 0.0402& \hfill -0.0899\\ \hfill 0.0386& \hfill -0.2857& \hfill -0.1238\end{array}\right].$$(18) ) in order to reproduce the results of the original paper with fidelity. Without that minus sign the system is unstable for any value of the delay.

Additional information

Notes on contributors

Rudy Cepeda-Gomez

Rudy Cepeda-Gomez received the BSc degree from Universidad Industrial de Santander (UIS), Bucaramanga, Colombia, in 2006; the MSc degree from Pontificia Universidad Javeriana, Bogotá, Colombia, in 2010, both in electronic engineering; and the PhD degree in mechanical engineering from the University of Connecticut, Storrs, CT, USA, in 2012. He was a full-time lecturer and leader of the research group with the Mechatronic Engineering Department at Santo Tomás University, Bucaramanga, Colombia, and a part-time lecturer at UIS (2012–2013). In January 2014, he joined the Institute of Automation at the University of Rostock, Germany. He was a visiting scholar at the Electrical Engineering Department of the University of Delaware (Summer 2008) and at the Computer Science Department of KU Leuven (Spring 2014). His research interests lie in the general area of analysis and control of time-delay systems. Dr Cepeda-Gomez is a member of ASME, IEEE, and the honour society of Phi-Kappa-Phi.