Sign properties of Metzler matrices with applications

Several results about sign properties of Metzler matrices are obtained. It is first established that checking the sign-stability of a Metzler sign-matrix can be either characterized in terms of the Hurwitz stability of the unit sign-matrix in the corresponding qualitative class, or in terms the negativity of the diagonal elements of the Metzler sign-matrix and the acyclicity of the associated directed graph. Similar results are obtained for the case of Metzler block-matrices and Metzler mixed-matrices, the latter being a class of Metzler matrices containing both sign- and real-type entries. The problem of assessing the sign-stability of the convex hull of a finite and summable family of Metzler matrices is also solved, and a necessary and sufficient condition for the existence of common Lyapunov functions for all the matrices in the convex hull is obtained. The concept of sign-stability is then generalized to the concept of Ker$_+(B)$-sign-stability, a problem that arises in the analysis of certain jump Markov processes. A sufficient condition for the Ker$_+(B)$-sign-stability of Metzler sign-matrices is obtained and formulated using inverses of sign-matrices and the concept of $L^+$-matrices. Several applications of the results are discussed in the last section.

[1]  Pierre Apkarian,et al.  Self-scheduled H∞ control of linear parameter-varying systems: a design example , 1995, Autom..

[2]  John N. Tsitsiklis,et al.  Parallel and distributed computation , 1989 .

[3]  F. Tadeo,et al.  Controller Synthesis for Positive Linear Systems With Bounded Controls , 2007, IEEE Transactions on Circuits and Systems II: Express Briefs.

[4]  Francesca Parise,et al.  On the reachable set of the controlled gene expression system , 2014, 53rd IEEE Conference on Decision and Control.

[5]  Corentin Briat,et al.  Positive systems analysis via integral linear constraints , 2015, 2015 54th IEEE Conference on Decision and Control (CDC).

[6]  James Lam,et al.  $\ell_{\infty}/L_{\infty} $-Gain Analysis for Positive Linear Systems With Unbounded Time-Varying Delays , 2015, IEEE Transactions on Automatic Control.

[7]  Corentin Briat.  L∞-to-L∞ interval-observers for continuous-time systems with persistent inputs , 2015, ArXiv.

[8]  D.L. Elliott,et al.  Feedback systems: Input-output properties , 1976, Proceedings of the IEEE.

[9]  M. Shubik,et al.  Convex structures and economic theory , 1968 .

[10]  Maria Elena Valcher,et al.  On the zero pattern properties and asymptotic behavior of continuous-time positive system trajectories , 2007 .

[11]  Juan R. Torregrosa,et al.  Sign pattern matrices that admit M-, N-, P- or inverse M-matrices , 2009 .

[12]  Germán A. Enciso,et al.  Stochastic analysis of biochemical reaction networks with absolute concentration robustness , 2013, Journal of The Royal Society Interface.

[13]  Donald Ervin Knuth,et al.  The Art of Computer Programming , 1968 .

[14]  E. Verriest,et al.  Robust stability of delay-difference equations , 1995, Proceedings of 1995 34th IEEE Conference on Decision and Control.

[15]  Gerard J. Foschini,et al.  A simple distributed autonomous power control algorithm and its convergence , 1993 .

[16]  Annalisa Zappavigna,et al.  Unconditional stability of the Foschini-Miljanic algorithm , 2012, Autom..

[17]  D. Handelman.  Representing polynomials by positive linear functions on compact convex polyhedra. , 1988 .

[18]  Victor M. Preciado,et al.  Stability of Spreading Processes over Time-Varying Large-Scale Networks , 2015, IEEE Transactions on Network Science and Engineering.

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

[20]  Stephen P. Boyd,et al.  Distributed Optimization and Statistical Learning via the Alternating Direction Method of Multipliers , 2011, Found. Trends Mach. Learn..

[21]  V. Klee,et al.  When is a Matrix Sign Stable? , 1977, Canadian Journal of Mathematics.

[22]  James Quirk,et al.  Qualitative Problems in Matrix Theory , 1969 .

[23]  Donald E. Knuth,et al.  The art of computer programming: V.1.: Fundamental algorithms , 1997 .

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

[25]  W. Ebeling.  Stochastic Processes in Physics and Chemistry , 1995 .

[26]  João Pedro Hespanha,et al.  Lyapunov conditions for input-to-state stability of impulsive systems , 2008, Autom..

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

[28]  Mustafa Khammash,et al.  Determining the long-term behavior of cell populations: A new procedure for detecting ergodicity in large stochastic reaction networks , 2013, 1312.2879.

[29]  Dimitri Peaucelle,et al.  L1 gain analysis of linear positive systems and its application , 2011, IEEE Conference on Decision and Control and European Control Conference.

[30]  M. Feinberg.  Complex balancing in general kinetic systems , 1972 .

[31]  David F. Anderson,et al.  Continuous Time Markov Chain Models for Chemical Reaction Networks , 2011 .

[32]  Ho-Chan Kim,et al.  Hinfinity control of discrete-time linear systems with time-varying delays in state , 1999, Autom..

[33]  François Le Gall,et al.  Powers of tensors and fast matrix multiplication , 2014, ISSAC.

[34]  Marko Wagner,et al.  Convex Structures And Economic Theory , 2016 .

[35]  Mustapha Ait Rami,et al.  Stability Analysis and Synthesis for Linear Positive Systems with Time-Varying Delays , 2009 .

[36]  Jacques L. Willems,et al.  Structural controllability and observability , 1986 .

[37]  Richard A. Brualdi,et al.  Matrices of Sign-Solvable Linear Systems , 1995 .

[38]  S. Niculescu.  Delay Effects on Stability: A Robust Control Approach , 2001 .

[39]  R. Jackson,et al.  General mass action kinetics , 1972 .

[40]  J. Goutsias,et al.  Markovian dynamics on complex reaction networks , 2012, 1205.5524.

[41]  John J. Tyson,et al.  Classification of instabilities in chemical reaction systems , 1975 .

[42]  Franco Blanchini,et al.  Switched Positive Linear Systems , 2015, Found. Trends Syst. Control..

[43]  Carsten Thomassen.  When the sign pattern of a square matrix determines uniquely the sign pattern of its inverse , 1989 .

[44]  Hai Lin,et al.  Stability and Stabilizability of Switched Linear Systems: A Survey of Recent Results , 2009, IEEE Transactions on Automatic Control.

[45]  Takashi Tanaka,et al.  DC-dominant property of cone-preserving transfer functions , 2013, Syst. Control. Lett..

[46]  Robert Shorten,et al.  On linear co-positive Lyapunov functions for sets of linear positive systems , 2009, Autom..

[47]  Vitaly Katsnelson,et al.  Sign patterns for chemical reaction networks , 2009, 0904.2960.

[48]  Victor M. Preciado,et al.  Dynamic analysis of bet-hedging strategies as a protection mechanism against environmental fluctuations , 2016, 2016 IEEE 55th Conference on Decision and Control (CDC).

[49]  Tadeusz Kaczorek,et al.  Robust stability of positive discrete-time interval systems with time-delays , 2004 .

[50]  Corentin Briat.  Linear Parameter-Varying and Time-Delay Systems: Analysis, Observation, Filtering & Control , 2014 .

[51]  Mustafa Khammash,et al.  Antithetic Integral Feedback: A new motif for robust perfect adaptation in noisy biomolecular networks , 2014, bioRxiv.

[52]  Thomas G. Kurtz,et al.  Stochastic Analysis of Biochemical Systems , 2015 .

[53]  Corentin Briat,et al.  Dwell-time stability and stabilization conditions for linear positive impulsive and switched systems , 2016, Nonlinear Analysis: Hybrid Systems.

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

[55]  Mi-Ching Tsai,et al.  Robust and Optimal Control , 2014 .

[56]  B. L. Clarke.  Theorems on chemical network stability , 1975 .

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

[58]  Marcello Colombino,et al.  Convex characterization of robust stability analysis and control synthesis for positive linear systems , 2014, 53rd IEEE Conference on Decision and Control.

[59]  Marcello Colombino,et al.  A Convex Characterization of Robust Stability for Positive and Positively Dominated Linear Systems , 2015, IEEE Transactions on Automatic Control.

[60]  R. May,et al.  Stability and Complexity in Model Ecosystems , 1976, IEEE Transactions on Systems, Man, and Cybernetics.

[61]  Corentin Briat,et al.  Robust ergodicity and tracking in antithetic integral control of stochastic biochemical reaction networks , 2016, 2016 IEEE 55th Conference on Decision and Control (CDC).

[62]  Corentin Briat,et al.  Computer control of gene expression: Robust setpoint tracking of protein mean and variance using integral feedback , 2012, 2012 IEEE 51st IEEE Conference on Decision and Control (CDC).

[63]  Daniel Liberzon,et al.  Switching in Systems and Control , 2003, Systems & Control: Foundations & Applications.

[64]  Victor Klee,et al.  Linear algorithms for testing the sign stability of a matrix and for findingZ-maximum matchings in acyclic graphs , 1977 .

[65]  Corentin Briat,et al.  Interval peak-to-peak observers for continuous- and discrete-time systems with persistent inputs and delays , 2015, Autom..

[66]  ROBERT M. MAY,et al.  Will a Large Complex System be Stable? , 1972, Nature.

[67]  Robert Shorten,et al.  On the Stability of Switched Positive Linear Systems , 2007, IEEE Transactions on Automatic Control.

[68]  Vladimir L. Kharitonov,et al.  Stability of Time-Delay Systems , 2003, Control Engineering.

[69]  J. Doyle,et al.  Robust and optimal control , 1995, Proceedings of 35th IEEE Conference on Decision and Control.

[70]  Abdellah Benzaouia,et al.  Properties of power control algorithms in wireless networks with delays: A Positive Systems Approach , 2010 .

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

[72]  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.

[73]  M. Buslowicz.  Simple stability conditions for linear positive discrete-time systems with delays , 2008 .

[74]  Olivier Bernard,et al.  Interval observers for linear time-invariant systems with disturbances , 2011, Autom..

[75]  Takashi Tanaka,et al.  Symmetric Formulation of the S-Procedure, Kalman–Yakubovich–Popov Lemma and Their Exact Losslessness Conditions , 2013, IEEE Transactions on Automatic Control.

[76]  J. Daafouz,et al.  Equivalence between the Lyapunov-Krasovskii functionals approach for discrete delay systems and that of the stability conditions for switched systems , 2008 .

[77]  Ching-tai Lin.  Structural controllability , 1974 .

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

[79]  Frank J. Hall,et al.  When does the inverse have the same sign pattern as the transpose? , 1999 .

[80]  S. Meyn,et al.  Stability of Markovian processes III: Foster–Lyapunov criteria for continuous-time processes , 1993, Advances in Applied Probability.

[81]  A. Morse,et al.  Stability of switched systems: a Lie-algebraic condition ( , 1999 .

[82]  Alexander Felzenbaum,et al.  Sign patterns of matrices and their inverses , 1987 .

[83]  Charles R. Johnson.  Sign patterns of inverse nonnegative matrices , 1983 .

[84]  Jiang-Wen Xiao,et al.  Impulsive positive observers and dynamic output feedback stabilization of positive linear continuous systems , 2017 .

[85]  Donald E. Knuth.  The art of computer programming: fundamental algorithms , 1969 .

[86]  Hal L. Smith,et al.  Monotone Dynamical Systems: An Introduction To The Theory Of Competitive And Cooperative Systems (Mathematical Surveys And Monographs) By Hal L. Smith , 1995 .

[87]  Bryan L. Shader,et al.  Sign-consistency and solvability of constrained linear systems , 1998 .

[88]  P. van den Driessche,et al.  Allow problems concerning spectral properties of sign pattern matrices: A survey , 2009 .

[89]  Stephen P. Boyd,et al.  Convex Optimization , 2004, Algorithms and Theory of Computation Handbook.

[90]  E. Kaszkurewicz,et al.  Matrix diagonal stability in systems and computation , 1999 .

[91]  Charles R. Johnson,et al.  Self-Inverse Sign Patterns , 1993 .

[92]  M. Khammash.  Necessary and sufficient conditions for the robustness of time-varying systems with applications to sampled-data systems , 1993, IEEE Trans. Autom. Control..

[93]  C. Jeffries.  Qualitative Stability and Digraphs in Model Ecosystems , 1974 .

[94]  Olga Pryporova.  Qualitative convergence of matrices , 2009 .

[95]  J. William Helton,et al.  Determinant Expansions of Signed Matrices and of Certain Jacobians , 2008, SIAM J. Matrix Anal. Appl..

[96]  U. Jönsson.  A Lecture on the S-Procedure , 2006 .

[97]  Robert Shorten,et al.  On Linear Copositive Lyapunov Functions and the Stability of Switched Positive Linear Systems , 2007, IEEE Transactions on Automatic Control.

[98]  M. Khammash,et al.  Antithetic Integral Feedback Ensures Robust Perfect Adaptation in Noisy Biomolecular Networks. , 2016, Cell systems.

[99]  James P. Quirk,et al.  Qualitative Economics and the Stability of Equilibrium , 1965 .

[100]  R. Shorten,et al.  Quadratic and Copositive Lyapunov Functions and the Stability of Positive Switched Linear Systems , 2007, 2007 American Control Conference.

[101]  ALAN ROBERTS,et al.  The stability of a feasible random ecosystem , 1974, Nature.

[102]  Ricardo G. Sanfelice,et al.  Hybrid Dynamical Systems: Modeling, Stability, and Robustness , 2012 .

[103]  Ettore Fornasini,et al.  Linear Copositive Lyapunov Functions for Continuous-Time Positive Switched Systems , 2010, IEEE Transactions on Automatic Control.

[104]  Jiang-Wen Xiao,et al.  Stability Analysis of Impulsive Positive Systems , 2014 .

[105]  Ankit Gupta,et al.  A Scalable Computational Framework for Establishing Long-Term Behavior of Stochastic Reaction Networks , 2013, PLoS Comput. Biol..