Practical stability of positive fractional discrete-time 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 in more complicated and less advanced. An overview of state of the art in positive systems theory is given in monographs [1, 2]. Mathematical fundamentals of fractional calculus are given in the monographs [3–6]. The fractional positive linear continuous-time and discrete-time systems have been addressed in [7–11]. The first monograph on analysis and synthesis of control systems with delays was the monograph published by Gorecki in 1989 [12]. Stability of positive 1D and 2D systems has addressed in [13–17] and the stability of positive fractional linear systems has been investigated in [18, 19]. The reachability and controllability to zero of positive fractional linear systems have been considered in [20–22]. The fractional order controllers have been developed in [23]. A generalization of the Kalman filter for fractional order systems has been proposed in [24]. Fractional polynomials and nD systems have been investigated in [25]. The notion of standard and positive 2D fractional linear systems has been introduced in [26, 27]. In this paper a new concept of the practical stability of positive fractional discrete-time linear systems will be introduced and necessary and sufficient conditions for the practical stability will be established. The paper is organized as follows. In Section 2 the basic definition and necessary and sufficient conditions for positivity and asymptotic stability of the linear discrete-time systems are introduced. In Section 3 the positive fractional linear discrete-time systems are introduced. The main results of the paper are given in Section 4, where the concept of practical stability of the positive fractional systems is proposed and necessary and sufficient conditions for the practical stability are established. Concluding remarks are given in Section 5. To the best author’s knowledge the practical stability of the positive fractional systems has not been considered yet. The following notation will be used in the paper. The set of real n × m matrices with nonnegative entries will be denoted by R + and R n + = R n×1 + . A matrix A = [aij ] ∈ R nxm + (a vector) will be called strictly positive and denoted by A > 0 if aij > 0 for i = 1, . . . , n, j = 1, . . . ,m. The set of nonnegative integers will be denoted by Z+.