Realization Problem for Positive Multivariable Discrete–time Linear Systems with Delays in the State Vector and Inputs

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 systems behaviour 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 is more complicated and less advanced. An overview of the state of the art in positive systems theory is given in the monographs (Farina and Rinaldi, 2000; Kaczorek, 2002). Recent developments and some new results are given in (Kaczorek, 2003). Realization problems of positive linear systems without time delays were considered in many papers and books (Benvenuti and Farina, 2004; Farina and Rinaldi, 2000; Kaczorek, 2002). Explicit solution of equations describing discrete-time systems with time delay was discussed in (Busłowicz, 1982). Recently, reachability, controllability and minimum energy control of positive linear discrete-time systems with delay were considered in (Busłowicz and Kaczorek, 2004; Xie and Wang, 2003). In this paper, the realization problem for positive multivariable discrete-time systems with delays in the state and inputs will be formulated and solved. Conditions for the solvability of the realization problem and the existence of a minimal positive realization will be established. A procedure for the computation of a positive realization of a proper rational matrix will be presented. To the best of the author’s knowledge, the realization problem for positive linear systems with delays in the state vector and inputs has not been considered yet.