On Structured Lyapunov Functions and Dissipativity in Interconnected LTI Systems

In this paper, we study connections between structured storage or Lyapunov functions of a class of interconnected systems (dynamical networks) and dissipativity properties of the individual systems. We prove that if a dynamical network, composed as a set of linear time invariant systems interconnected over an acyclic graph, admits an additive quadratic Lyapunov function, then the individual systems in the network are dissipative with respect to a (nonempty) set of interconnection neutral supply functions. Each supply function from this set is defined on a single interconnection link in the network. Specific characterizations of neutral supply functions are presented, which imply robustness of network stability/dissipativity to removal of interconnection links.

[1]  D. Siljak,et al.  Control of Complex Systems: Structural Constraints and Uncertainty , 2010 .

[2]  Geir E. Dullerud,et al.  Distributed control design for spatially interconnected systems , 2003, IEEE Trans. Autom. Control..

[3]  Dimitri Peaucelle,et al.  Analysis and Synthesis of Interconnected Positive Systems , 2017, IEEE Transactions on Automatic Control.

[4]  A. Papachristodoulou,et al.  Convex Design of Structured Controllers using Block-Diagonal Lyapunov Functions , 2017, 1709.00695.

[5]  J. Willems.  Dissipative dynamical systems part I: General theory , 1972 .

[6]  On additive Lyapunov functions and existence of neutral supply rates in acyclic LTI dynamical networks , 2016 .

[7]  G. Dullerud,et al.  A Course in Robust Control Theory: A Convex Approach , 2005 .

[8]  Robin Wilson,et al.  Modern Graph Theory , 2013 .

[9]  J. Willems.  Dissipative dynamical systems Part II: Linear systems with quadratic supply rates , 1972 .

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

[11]  Harry L. Trentelman,et al.  On quadratic differential forms , 1994, Proceedings of 1994 33rd IEEE Conference on Decision and Control.

[12]  Carsten W. Scherer,et al.  LPV control and full block multipliers , 2001, Autom..

[13]  I. Postlethwaite,et al.  Linear Matrix Inequalities in Control , 2007 .

[14]  Cédric Langbort,et al.  Distributed control design for systems interconnected over an arbitrary graph , 2004, IEEE Transactions on Automatic Control.