State-feedback stabilization of multi-input compartmental systems

Abstract In this paper we address the positive (state–feedback) stabilization of multi-input compartmental systems, i.e. the design of a state–feedback matrix that preserves the compartmental property of the resulting feedback system, while achieving stability. We first provide necessary and sufficient conditions for the positive stabilizability of compartmental systems whose state matrix is irreducible. Then we address the case when the state matrix is reducible, identify two sufficient conditions for the problem solution, and then extend them to a general algorithm that allows to verify when the problem is solvable and to produce a solution.

[1]  Georges Bastin,et al.  Congestion control in compartmental network systems , 2006, Syst. Control. Lett..

[2]  M. E. Valcher,et al.  Continuous-Time Compartmental Switched Systems , 2016 .

[3]  L. Mirsky,et al.  The Theory of Matrices , 1961, The Mathematical Gazette.

[4]  Dirk Aeyels,et al.  Stabilization of positive linear systems , 2001, Syst. Control. Lett..

[5]  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.

[6]  Alessandro Astolfi,et al.  Design of Positive Linear Observers for Positive Linear Systems via Coordinate Transformations and Positive Realizations , 2008, SIAM J. Control. Optim..

[7]  Edward J. Davison,et al.  Necessary and sufficient conditions for stabilizability of positive LTI systems , 2009, Syst. Control. Lett..

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

[9]  Shengyuan Xu,et al.  Control for stability and positivity: equivalent conditions and computation , 2005, IEEE Transactions on Circuits and Systems II: Express Briefs.

[10]  Richard A. Brualdi,et al.  Combinatorial matrix theory , 1991 .

[11]  D. Hinrichsen,et al.  Robust Stability of positive continuous time systems , 1996 .

[12]  John A. Jacquez,et al.  Qualitative Theory of Compartmental Systems , 1993, SIAM Rev..

[13]  J. M. V. den.  Positive Linear Observers for Linear Compartmental Systems , 1998 .

[14]  J. Jacquez.  Compartmental analysis in biology and medicine: Kinetics of distribution of tracer-labeled materials , 1972 .

[15]  Hans Schneider,et al.  The influence of the marked reduced graph of a nonnegative matrix on the Jordan form and on related properties: A survey , 1986 .

[16]  G. P. Barker.  Theory of cones , 1981 .

[17]  O. Taussky.  A Recurring Theorem on Determinants , 1949 .

[18]  W. Haddad,et al.  Nonnegative and Compartmental Dynamical Systems , 2010 .

[19]  G. Bastin,et al.  Positive linear observers for positive linear systems , 1999, 1999 European Control Conference (ECC).

[20]  Robert J. Plemmons,et al.  Nonnegative Matrices in the Mathematical Sciences , 1979, Classics in Applied Mathematics.