Stability analysis of networked systems containing damped and undamped nodes

This paper answers the question if a qualitatively heterogeneous passive networked system containing damped and undamped nodes shows consensus in the output of the nodes in the long run. While a standard Lyapunov analysis shows that the damped nodes will always converge to a steady-state value, the convergence of the undamped nodes is much more delicate and depends on the parameter values of the network as well as on the topology of the graph. A complete stability analysis is presented based on an eigenvector analysis involving the system parameters and the topology of both the original graph and the reduced graph obtained by a Kron reduction that eliminates the damped nodes.

[1]  Murat Arcak,et al.  Passivity as a Design Tool for Group Coordination , 2007, IEEE Transactions on Automatic Control.

[2]  B. Siswoyo.  Introduction to the Mathematical Theory of Systems and Control , 2012 .

[3]  Karl Henrik Johansson,et al.  On robust synchronization of heterogeneous linear multi-agent systems with static couplings , 2015, Autom..

[4]  Arjan van der Schaft,et al.  Port-Hamiltonian Systems Theory: An Introductory Overview , 2014, Found. Trends Syst. Control..

[5]  David J. Hill,et al.  Transient Stability Analysis of Multimachine Power Networks with Linear Frequency Dependent Loads , 1982 .

[6]  Magnus Egerstedt,et al.  Controllability of Multi-Agent Systems from a Graph-Theoretic Perspective , 2009, SIAM J. Control. Optim..

[7]  Jun Zhao,et al.  Synchronization of Dynamical Networks With Nonidentical Nodes: Criteria and Control , 2011, IEEE Transactions on Circuits and Systems I: Regular Papers.

[8]  Claudio De Persis,et al.  An internal model approach to frequency regulation in inverter-based microgrids with time-varying voltages , 2014, 53rd IEEE Conference on Decision and Control.

[9]  Frank Allgöwer,et al.  Duality and network theory in passivity-based cooperative control , 2013, Autom..

[10]  D. Helbing.  Traffic and related self-driven many-particle systems , 2000, cond-mat/0012229.

[11]  Guanrong Chen,et al.  Pinning control and synchronization on complex dynamical networks , 2014, International Journal of Control, Automation and Systems.

[12]  Giuseppe Notarstefano,et al.  On the Reachability and Observability of Path and Cycle Graphs , 2011, IEEE Transactions on Automatic Control.

[13]  Kathryn Fraughnaugh,et al.  Introduction to graph theory , 1973, Mathematical Gazette.

[14]  Frank Allgöwer,et al.  Hierarchical Clustering of Dynamical Networks Using a Saddle-Point Analysis , 2013, IEEE Transactions on Automatic Control.

[15]  Reza Olfati-Saber,et al.  Consensus and Cooperation in Networked Multi-Agent Systems , 2007, Proceedings of the IEEE.

[16]  Henk Nijmeijer,et al.  Partial Network Synchronization Using Diffusive Dynamic Couplings , 2014 .

[17]  Florian Dörfler,et al.  Kron Reduction of Graphs With Applications to Electrical Networks , 2011, IEEE Transactions on Circuits and Systems I: Regular Papers.

[18]  Arjan van der Schaft,et al.  Port-Hamiltonian Systems on Graphs , 2011, SIAM J. Control. Optim..

[19]  Ali Saberi,et al.  Output synchronization for heterogeneous networks of introspective right‐invertible agents , 2014 .