Stability properties of a class of positive switched systems with rank one difference

Abstract Given a single-input continuous-time positive system, described by a pair ( A , b ) , with A a diagonal matrix, we investigate under what conditions there exists a state-feedback law u ( t ) = c ⊤ x ( t ) that makes the resulting controlled system positive and asymptotically stable, by this meaning that A + b c ⊤ is Metzler and Hurwitz. In the second part of this note we assume that the state-space model switches among different state-feedback laws ( c i ⊤ , i = 1 , 2 , … , p ) each of them ensuring the positivity, and show that the asymptotic stability of this type of switched system is equivalent to the asymptotic stability of all its subsystems, while its stabilizability is equivalent to the existence of an asymptotically stable subsystem.

[1]  Robert Shorten,et al.  Quadratic Stability and Singular SISO Switching Systems , 2009, IEEE Transactions on Automatic Control.

[2]  Peng Shi,et al.  Stability of switched positive linear systems with average dwell time switching , 2012, Autom..

[3]  Thomas A. Henzinger,et al.  Hybrid Systems: Computation and Control , 1998, Lecture Notes in Computer Science.

[4]  Patrizio Colaneri,et al.  Stabilization of continuous-time switched linear positive systems , 2010, Proceedings of the 2010 American Control Conference.

[5]  Franco Blanchini,et al.  Co-Positive Lyapunov Functions for the Stabilization of Positive Switched Systems , 2012, IEEE Transactions on Automatic Control.

[6]  Valery Opoitsev,et al.  Absolute stability of positive systems with differential constraints , 2000 .

[7]  Ettore Fornasini,et al.  On the stability and stabilizability of a class of continuous-time positive switched systems with rank one difference , 2013, 2013 European Control Conference (ECC).

[8]  Ettore Fornasini,et al.  On the stability of continuous-time positive switched systems , 2010, Proceedings of the 2010 American Control Conference.

[9]  Franco Blanchini,et al.  Continuous-time optimal control for switched positive systems with application to mitigating viral escape , 2010 .

[10]  Ettore Fornasini,et al.  Stability and Stabilizability Criteria for Discrete-Time Positive Switched Systems , 2012, IEEE Transactions on Automatic Control.

[11]  M. Yu. Churilova,et al.  On absolute stability of positive systems , 2010 .

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

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

[14]  Christopher King,et al.  On the existence of a common quadratic Lyapunov function for a rank one difference , 2004 .

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

[16]  P. Curran,et al.  A unifying framework for the circle criterion and other quadratic stability criteria , 2003, 2003 European Control Conference (ECC).

[17]  Franco Blanchini,et al.  Is stabilization of switched positive linear systems equivalent to the existence of an Hurwitz convex combination of the system matrices? , 2011, IEEE Conference on Decision and Control and European Control Conference.

[18]  Raymond A. DeCarlo,et al.  Switched Controller Synthesis for the Quadratic Stabilisation of a Pair of Unstable Linear Systems , 1998, Eur. J. Control.

[19]  Wassim M. Haddad,et al.  Hybrid nonnegative and compartmental dynamical systems , 2002, Proceedings of the 2002 American Control Conference (IEEE Cat. No.CH37301).

[20]  Michael Margaliot,et al.  On the Stability of Positive Linear Switched Systems Under Arbitrary Switching Laws , 2009, IEEE Transactions on Automatic Control.

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

[22]  Wassim M. Haddad,et al.  Hybrid nonnegative and computational dynamical systems , 2002 .

[23]  Michael Margaliot,et al.  A Maximum Principle for the Stability Analysis of Positive Bilinear Control Systems with Applications to Positive Linear Switched Systems , 2012, SIAM J. Control. Optim..

[24]  Robert Shorten,et al.  Stability Criteria for Switched and Hybrid Systems , 2007, SIAM Rev..

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

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

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