A Problem in Positive Systems Stability Arising in Topology Control

Abstract

We present a problem in the stability of switched positive systems that arises in network topology control. Preliminary results are given that guarantee stability of a network topology control problem under certain assumptions. Roughly speaking, these assumptions reduce the underlying stability problem to a nonlinear consensus problem with a driving term, that eventually becomes a Lur'e problem. Simulation results are given to illustrate our algorithm. While these results indicate that our assumptions can be removed, a proof of the general stability problem remains open. IntroductionRecent years have witnessed a growing interest in the control community in problems that arise when dynamic systems evolve over graphs. While the most high profile of these applications are in consensus applications such as formation flying and synchronisation problems, [4,8,11], many other applications have arisen where the manner in which network topologies change affect the performance of algorithms that are run over these networks. In such applications, an essential requirement is that the topology of the graph be such that some properties required to support communication and control are satisfied, the most basic of these being that the network is connected. Considerations of this kind have given rise to the emerging field of network topology control. Clearly, graph connectivity is an essential component in

A framework for decentralised feedback connectivity control with application to sensor networks

“…This would correspond to weighting the nodes' radii in the averaging scheme. It is possible to include such weighting in our framework, and all the proofs directly hold with but a small modification, Knorn et al (2009). An example of this is given in Figure 18, where by design we wish one node to have twice the radius as the others, and one node half the radius.…”