A Contraction Approach to the Hierarchical Analysis and Design of Networked Systems

This brief is concerned with the stability of continuous-time networked systems. Using contraction theory, a result is established on the network structure and the properties of the individual component subsystems and their couplings to ensure the overall contractivity of the entire network. Specifically, it is shown that a contraction property on a reduced-order matrix that quantifies the interconnection structure, coupled with contractivity/expansion estimates on the individual component subsystems, suffices to ensure that trajectories of the overall system converge towards each other.

[1]  Eduardo D. Sontag,et al.  Monotone and near-monotone biochemical networks , 2007, Systems and Synthetic Biology.

[2]  David Angeli,et al.  Monotone control systems , 2003, IEEE Trans. Autom. Control..

[3]  A. Michel,et al.  Stability of Dynamical Systems — Continuous , Discontinuous , and Discrete Systems , 2008 .

[4]  Mathukumalli Vidyasagar,et al.  Input-Output Analysis of Large-Scale Interconnected Systems , 1981 .

[5]  Fabian R. Wirth,et al.  Small gain theorems for large scale systems and construction of ISS Lyapunov functions , 2009, 2012 IEEE 51st IEEE Conference on Decision and Control (CDC).

[6]  Nathan van de Wouw,et al.  Convergent dynamics, a tribute to Boris Pavlovich Demidovich , 2004, Syst. Control. Lett..

[7]  David Angeli,et al.  A Lyapunov approach to incremental stability properties , 2002, IEEE Trans. Autom. Control..

[8]  Fabian R. Wirth,et al.  A Small-Gain Condition for Interconnections of ISS Systems With Mixed ISS Characterizations , 2010, IEEE Transactions on Automatic Control.

[9]  Eduardo Sontag,et al.  Untangling the wires: A strategy to trace functional interactions in signaling and gene networks , 2002, Proceedings of the National Academy of Sciences of the United States of America.

[10]  Hongtao Lu.  Global exponential stability analysis of Cohen-Grossberg neural networks , 2005, IEEE Transactions on Circuits and Systems II: Express Briefs.

[11]  Mario di Bernardo,et al.  Contraction Theory and Master Stability Function: Linking Two Approaches to Study Synchronization of Complex Networks , 2009, IEEE Transactions on Circuits and Systems II: Express Briefs.

[12]  Eduardo D. Sontag,et al.  A passivity-based stability criterion for a class of interconnected systems and applications to biochemical reaction networks , 2007, 2007 46th IEEE Conference on Decision and Control.

[13]  Mario di Bernardo,et al.  Stability of networked systems: A multi-scale approach using contraction , 2010, 49th IEEE Conference on Decision and Control (CDC).

[14]  Eduardo Sontag,et al.  A passivity-based stability criterion for a class of biochemical reaction networks. , 2008, Mathematical biosciences and engineering : MBE.

[15]  G. Dahlquist.  Stability and error bounds in the numerical integration of ordinary differential equations , 1961 .

[16]  P. Hartman.  On Stability in the Large for Systems of Ordinary Differential Equations , 1961, Canadian Journal of Mathematics.

[17]  Roy D. Williams,et al.  Error estimation for numerical differential equations , 1996 .

[18]  Hong Qiao,et al.  A critical analysis on global convergence of Hopfield-type neural networks , 2005, IEEE Transactions on Circuits and Systems I: Regular Papers.

[19]  R. FitzHugh.  Mathematical models of threshold phenomena in the nerve membrane , 1955 .

[20]  D. C. Lewis.  Metric Properties of Differential Equations , 1949 .

[21]  Fabian R. Wirth,et al.  An ISS small gain theorem for general networks , 2007, Math. Control. Signals Syst..

[22]  Mario di Bernardo,et al.  How to Synchronize Biological Clocks , 2009, J. Comput. Biol..

[23]  A. Winfree.  Biological rhythms and the behavior of populations of coupled oscillators. , 1967, Journal of theoretical biology.

[24]  Mario di Bernardo,et al.  Global Entrainment of Transcriptional Systems to Periodic Inputs , 2009, PLoS Comput. Biol..

[25]  Eduardo Sontag.  Input to State Stability: Basic Concepts and Results , 2008 .

[26]  W. Lohmiller,et al.  Contraction analysis of non-linear distributed systems , 2005 .

[27]  Jean-Jacques E. Slotine,et al.  On Contraction Analysis for Nonlinear Systems Analyzing stability differentially leads to a new perspective on nonlinear dynamic systems , 1999 .

[28]  P. Moylan,et al.  Stability criteria for large-scale systems , 1978 .

[29]  T. Ström.  On Logarithmic Norms , 1975 .

[30]  J. Jouffroy.  Some ancestors of contraction analysis , 2005, Proceedings of the 44th IEEE Conference on Decision and Control.

[31]  Jean-Jacques E. Slotine,et al.  On Contraction Analysis for Non-linear Systems , 1998, Autom..