Realization Problem for Positive Linear Systems with Time Delay

In positive systems inputs, state variables and outputs take only nonnegative 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 systems behavior can be found in engineering, management science, economics, social sciences, biology, and medicine, and so forth. Positive linear systems are defined on cones and not on linear spaces. Therefore, the theory of positive systems is more complicated and less advanced. An overview of state of the art in positive systems theory is given in the monographs [4, 5]. Recent developments in positive systems theory and some new results are given in [5]. Realizations problem of positive linear systems without time delays has been considered in many papers and books [1, 4, 5]. Explicit solution of equations describing the discrete-time systems with time delay has been given in [2]. Recently, the reachability, controllability, and minimum energy control of positive linear discrete-time systems with time delays have been considered in [3, 6]. In this paper, the realization problem for positive single-input single-output discretetime systems with time delay will be formulated and solved. Necessary and sufficient conditions for the solvability of the realization problem will be established and a procedure for computation of a minimal positive realization of a proper rational function will be presented. To the best knowledge of the author, the realization problem for positive linear systems with time delays has not been considered yet.