Persistence analysis of interconnected positive systems with communication delays

This is a continuation of our preceding studies on the analysis of interconnected positive systems. Under mild conditions on positive subsystems and a nonnegative interconnection matrix, we showed that the state of the interconnected positive system converges to a positive scalar multiple of a prescribed positive vector. As a byproduct of this property, called persistence, it turned out that the output converges to the positive right eigenvector of the interconnection matrix. This result is effectively used in the formation control of multi-agent positive systems. The goal of this paper is to prove that the essential property of persistence is still preserved under arbitrary (time-invariant) communication delays. In the context of formation control, this preservation indicates that the desired formation is achieved robustly against communication delays, even though the resulting formation is scaled depending upon initial conditions for the state. From a mathematical point of view, the key issue is to prove that the delay interconnected positive system has stable poles only except for a pole of degree one at the origin, even though it has infinitely many poles in general. To this end, we develop frequency-domain (s-domain) analysis for delay interconnected positive systems.

[1]  J. Lam,et al.  Positive state‐bounding observer for positive interval continuous‐time systems with time delay , 2012 .

[2]  F. Tadeo,et al.  Positive observation problem for linear time-delay positive systems , 2007, 2007 Mediterranean Conference on Control & Automation.

[3]  R. Bellman,et al.  Differential-Difference Equations , 1967 .

[4]  Takashi Tanaka,et al.  The Bounded Real Lemma for Internally Positive Systems and H-Infinity Structured Static State Feedback , 2011, IEEE Transactions on Automatic Control.

[5]  George J. Pappas,et al.  Stable flocking of mobile agents, part I: fixed topology , 2003, 42nd IEEE International Conference on Decision and Control (IEEE Cat. No.03CH37475).

[6]  Charles R. Johnson,et al.  Matrix analysis , 1985, Statistical Inference for Engineers and Data Scientists.

[7]  Wassim M. Haddad,et al.  Stability theory for nonnegative and compartmental dynamical systems with time delay , 2004, Proceedings of the 2004 American Control Conference.

[8]  Dimitri Peaucelle,et al.  Stability and persistence analysis of large scale interconnected positive systems , 2013, 2013 European Control Conference (ECC).

[9]  J. Hale.  Theory of Functional Differential Equations , 1977 .

[10]  Dimitri Peaucelle,et al.  Dominant pole analysis of stable time-delay positive systems , 2014 .

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

[12]  Jon Rigelsford,et al.  Positive 1D and 2D Systems , 2002 .

[13]  George J. Pappas,et al.  Stable flocking of mobile agents part I: dynamic topology , 2003, 42nd IEEE International Conference on Decision and Control (IEEE Cat. No.03CH37475).

[14]  S. Rinaldi,et al.  Positive Linear Systems: Theory and Applications , 2000 .

[15]  Anders Rantzer,et al.  On the Kalman-Yakubovich-Popov Lemma for Positive Systems , 2012, IEEE Transactions on Automatic Control.

[16]  Corentin Briat,et al.  Robust stability and stabilization of uncertain linear positive systems via integral linear constraints: L1‐gain and L∞‐gain characterization , 2012, ArXiv.

[17]  Anders Rantzer,et al.  Distributed control of positive systems , 2011, IEEE Conference on Decision and Control and European Control Conference.