Fractional Positive Continuous-Time Linear Systems and Their Reachability

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 behaving as positive linear systems can be found in engineering, management science, economics, social sciences, biology, 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 is given in the monographs (Farina and Rinaldi, 2000; Kaczorek, 2002). An extension of positive systems are cone systems (Kaczorek, 2006; Kaczorek, 2007b). The notion of cone systems was introduced in (Kaczorek, 2006). Roughly speaking, a cone system is a system obtained from a positive one by substitution of the positive orthants of states, inputs and outputs by suitable arbitrary cones. The realization problem for cone systems was addressed in (Kaczorek, 2006; Kaczorek, 2007a). The positive controllability of dynamical systems was investigated in (Klamka, 2002) and the approximate constrained controllability of mechanical systems in (Klamka, 2005). The first definition of the fractional derivative was introduced by Liouville and Riemann at the end of the 19-th century (Nishimoto, 1984; Miller and Ross, 1993; Podlubny, 1999). This idea was used by engineers for modelling various processes in the late 1960s (Vinagre et al., 2002; Vinagre and Feliu, 2002; Zaborowsky and Meylaov, 2001). Mathematical fundamentals of fractional calculus are given in the monographs (Miller and Ross, 1993; Nishimoto, 1984; Podlubny, 1999; Oldham and Spanier, 1974; Oustalup, 1993). Fractional-order controllers were developed in (Oustalup, 1993; Podlubny et al. 1997). A generalization of the Kalman filter for fractional-order systems was proposed in (Sierociuk and Dzieliński, 2006). Some others applications of fractionalorder systems can be found in (Engheta, 1997; Ostalczyk, 2000; Ostalczyk, 2004a; Ostalczyk, 2004b; Ferreira and Machado, 2003; Moshrefi-Torbati and Hammond, 1998; Reyes-Melo et al.,2004; Riu et al., 2001; Sjöberg and Kari, 2002; Vinagre et al., 2002; Samko et al., 1993). In (Ortigueira, 1997), a method for computation of the impulse responses from the frequency responses for fractional standard (nonpositive) discrete-time linear systems was given. Fractional polynomials and nD systems were investigated in (Gałkowski and Kummert, 2005).