Reachability of Cone Fractional Continuous-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 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 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. The notion of cone systems was introduced in (Kaczorek, 2006). Roughly speaking, a cone system is a system obtained from a positive one by the 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, 2007c). The first definition of the fractional derivative was introduced by Liouville and Rieman at the end of the 19th century (Miller and Ross, 1993; Nishimoto, 1984; Podlubny, 1999). This idea was used by engineers for modelling various processes in the late 1960s (Vinagre et al., 1960; Vinagre and Feliu, 2002; Zaborowsky and Meylaov, 2001). Mathematical fundamentals of fractional calculus are given in the monographs (Miller and Ross, 1993; Nishimoto, 1984; Oldham and Spanier, 1974; Oustaloup, 1995; Podlubny, 1999). Fractional order controllers were developed in (Oustaloup, 1993; Podlubny et al., 1997). A generalization of the Kalman filter for fractional order systems was proposed in (Sierociuk and Dzieliński, 2007). Some other applications of fractional order systems can be found in (Engheta, 1997; Ferreira and Machado, 2003; Ostalczyk, 2000; Ostalczyk, 2004; Ostalczyk, 2004; Reyes-Melo et al., 2004; Riu et al., 2001; Samko et al., 1993; Sjöberg and Kari, 2002; Vinagre and Feliu, 2002). In (Ortigueira, 1997), a method was set forth for the computation of the impulse responses from the frequency responses for fractional standard (non-positive) discrete-time linear systems. Fractional polynomials and nD systems were investigated in (Gałkowski, 2005). Necessary and sufficient conditions for the reachability and controllability to zero of cone fractional discrete-time linear systems were established in (Kaczorek, 2007a; Kaczorek, 2007b; Kaczorek, 2007d). In this paper, the notion of cone fractional linear continuous-time systems will be introduced. Sufficient conditions for the reachability of cone fractional linear systems will be established. To the best of the author’s knowledge, cone fractional continuous-time linear systems have not been considered yet.