Approximation of fractional positive stable continuous-time linear systems by fractional positive stable discrete-time systems

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, or 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 is more complicated and less advanced. An overview of state of the art in positive systems theory is given by Farina and Rinaldi (2000) as well as Kaczorek (2002). The stability of positive linear systems was investigated by Farina and Rinaldi (2000) as well as Kaczorek (2002), and that of fractional linear systems by Busłowicz (2008; 2012), Busłowicz and Kaczorek (2009), as well as Kaczorek (2011). The problem of preservation of positivity by approximating the continuous-time linear systems by the corresponding discrete-time linear systems was addressed by Kaczorek (1999). A linear Padé-type approximation of the exponential matrix of positive asymptotically stable continuous-time linear systems was applied by Kaczorek (2013). It was shown that the approximation preserves the positivity and asymptotic stability of the systems. In this paper the linear Padé-type approximation will be applied to fractional positive continuous-time linear systems. The paper is organized as follows. In Section 2, basic definitions and theorems concerning positive standard and fractional continuous-time and discrete-time linear systems are recalled. In Section 3, the main result of the paper is presented. It is shown that the approximation preserves the positivity and asymptotic stability of the systems. Concluding remarks are given in Section 4. The following notation will be used: