Stability Analysis and Constrained Control of Positive Systems with Time-Varying Delays: A Linear Copositive Lyapunov-Krasovskii Functional Method

This paper considers the analysis and constrained control problems of positive systems with time-varying delays. A system is a called positive system if its states keep nonnegative whenever its initial conditions and inputs are nonnegative, and the control is under positivity constraint means that the resulting closed-loop system is not only stable, but also positive. The main contribution lies in the following aspects. Firstly, based on a linear copositive Lyapunov-Krasovskii functional, some stability criteria are established for positive systems with time-varying delays. Secondly, constrained controller for a class of systems with time-varying delays is designed, which guarantees the corresponding closed-loop system is positive and asymptotically stable.

[1]  J. Kurek.  Stability of positive 2-D system described by the Roesser model , 2002 .

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

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

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

[5]  Long Wang,et al.  Constrained Control of Positive Discrete-Time Systems With Delays , 2008, IEEE Transactions on Circuits and Systems II: Express Briefs.

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

[7]  T. Kaczorek.  Positive 1D and 2D Systems , 2001 .

[8]  M. Busłowicz.  Robust stability of positive discrete-time linear systems with multiple delays with linear unity rank uncertainty structure or non-negative perturbation matrices , 2007 .

[9]  Uwe Helmke,et al.  Positive observation problem for linear time-delay positive systems , 2007, MED 2007.

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

[11]  Long Wang,et al.  Stability Analysis for Continuous-Time Positive Systems With Time-Varying Delays , 2010, IEEE Transactions on Automatic Control.

[12]  F. Tadeo,et al.  Control of constrained positive discrete systems , 2007, 2007 American Control Conference.

[13]  F. Tadeo,et al.  Positive observation problem for linear discrete positive systems , 2006, Proceedings of the 45th IEEE Conference on Decision and Control.

[14]  Guo-Ping Liu,et al.  Parameter-dependent Lyapunov functional for stability of time-delay systems with polytopic-type uncertainties , 2004, IEEE Transactions on Automatic Control.

[15]  Yuanqing Xia,et al.  New stability and stabilization conditions for systems with time-delay , 2007, Int. J. Syst. Sci..

[16]  T. Kaczorek.  Stabilization of positive linear systems by state-feedbacks , 1999 .

[17]  A. Berman,et al.  Nonnegative matrices in dynamic systems , 1979 .