Practical stability and asymptotic stability of positive fractional 2D linear 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, 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 in the monographs [1, 2]. The most popular models of two-dimensional (2D) linear systems are the models introduced by Roesser [3], FornasiniMarchesini [4, 5] and Kurek [6]. These models have been extended for positive systems in [2, 7, 8]. An overview of positive 2D systems theory has been given in the monograph [2]. Stability of positive 1D and 2D linear systems has been considered in [1, 8–10] and the robust stability in [11, 12]. The positive fractional linear systems have been addressed in [13–16] and their stability has been investigated in [12, 17–19]. LMI approaches to checking the stability of positive 2D systems have been proposed in [10, 20]. In this paper new necessary and sufficient conditions for the asymptotic stability of the positive fractional 2D linear systems will be established. It will be shown that the checking of the asymptotic stability of positive fractional 2D linear systems can be reduced to testing the stability of corresponding 1D positive linear systems.