Pointwise Completeness and Pointwise Degeneracy of Positive Fractional Descriptor Continuous-Time Linear Systems with Regular Pencils

Descriptor (singular) linear systems have been considered in many papers and books (Bru et al., 2003; 2002; Choundhury, 1972; Campbell et al., 1976; Dai, 1989; Guang-Ren Duan, 2010; Kaczorek, 2013; 2011a; 2014b; 2004; 1992; 2014c; 2011c; 2011d; 2011e; Virnik, 2008). The eigenvalue and invariant assignment by state and output feedbacks was investigated by Kaczorek (2004; 2011d), along with minimum energy control of descriptor linear systems (Kaczorek, 2014b). In positive systems inputs, state variables and outputs take only nonnegative values (Farina and Rinaldi, 2000; Kaczorek, 2002). 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 fractional linear systems and selected problems in the theory of fractional systems were addressed by Kaczorek (2011d). Descriptor standard positive linear systems by the use of the Drazin inverse were discussed by Bru et al. (2003; 2002), Campbell et al. (1976) and Kaczorek (2013; 2011a; 2014a; 2011c; 2011e), who also applied the shuffle algorithm to check the positivity of descriptor linear systems (Kaczorek, 2011a). The stability of positive descriptor systems was investigated by Virnik (2008). Reduction and decomposition of descriptor fractional discrete-time linear systems were considered by Kaczorek (2011c), who also introduced a new class of descriptor fractional linear discrete-time systems (Kaczorek, 2011e). Pointwise completeness and pointwise degeneracy for standard and fractional linear systems were investigated by Busłowicz (2008), Busłowicz et al. (2006), Choundhury (1972), Kaczorek and Busłowicz (2009), Kaczorek (2011b; 2009; 2010), Olbrot (1972), Popov (1972), Trzasko et al. (2007) and Weiss (1970). The Drazin inverse of matrices was applied to find solutions of the state equations of fractional descriptor continuous-time linear systems with regular pencils by Kaczorek (2014a).