Positive linear Lyapunov systems

Tadeusz Kaczorek, Przemysław Przyborowsky

Warsaw University of Technology

Institute of Control and Industrial Electronics

Koszykowa 75, 00-662, Warsaw, Poland

 

In positive systems inputs, state variables and outputs take only non-negative values. Examples of positive systems are industrial processes involving chemical reactors, heat exchangers and distillation columns, storage systems, compartmental systems, water and atmospheric pollution models. A variety of models having positive linear behavior can be found in engineering, management science, economics, social sciences, biology and medicine, etc.

Positive linear systems are defined on cones and not on linear spaces. Therefore, the theory of positive systems in more complicated and less advanced. An overview of state of the art in positive systems theory is given in some monographs. The realization problem for positive linear systems without and with time delays has been considered in different works.

The reachability, controllability to zero and observability of dynamical systems have been considered (J.Klamka). The reachability and minimum energy control of positive linear discrete-time systems have been investigated (M.Busłowicz, T.Kaczorek). The positive discrete-time systems with delays have been considered in some another works. The controllability and observability of Lyapunov systems have been investigated by Murty Apparao. The positive discrete-time and continuous-time Lyapunov systems have been considered also. The positive linear time-varying Lyapunov systems have been investigated in early works (T.Kaczorek, P.Przyborowsky), and the Lyapunov cone systems have been considered. The positive discrete-time Lyapunov systems with delays have been investigated by some authors.

In this paper, the positive linear continuous and discrete-time Lyapunov systems will be addressed. For the continuous-time Lyapunov systems sufficient and necessary conditions for the positivity and asymptotic stability and the sufficient conditions for reachability and observability will be established. For the discrete-time Lyapunov systems the sufficient and necessary conditions for positivity, asymptotic stability, reachability, controllability to zero and observability will be given. Extensions of the Cayley-Hamilton theorem for the Lyapunov systems will be presented. The considerations will be illustrated on the numerical examples.

kaczorek@isep.pw.edu.plљљљљљ przyborp@isep.pw.edu.pl