Positivity of a class of fractional descriptor continuous-time nonlinear 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. An overview of state of art in positive systems theory is given in [1–3]. Descriptor (singular) linear systems have been considered in many papers and books [4–13]. The eigenvalues and invariants assignment by state and output feedbacks have been investigated in [8, 9, 12, 14]. The positive linear systems with different fractional orders have been addressed in [15–17]. Selected problems in theory of fractional linear systems has been given in monograph [16]. Descriptor standard positive linear systems by the use of Drazin inverse has been addressed in [4–6, 17, 18]. The checking of the positivity of descriptor linear systems is addressed in [5, 11, 15]. The stability of positive descriptor systems has been investigated in [14, 19] and the stability of fractional linear systems with delays of the retarded type has been investigated in [20] and of linear systems consisting of n subsystems with different fractional orders in [21]. The practical stability of positive fractional discrete-time systems has been analyzed in [22]. The descriptor standard and positive discrete-time nonlinear systems have been considered in [23] and the stability of a class of positive nonlinear systems in [24]. In this paper sufficient conditions for the positivity of the fractional descriptor continuous-time nonlinear systems will be presented. The paper is organized as follows. In Sec. 2 sufficient conditions for the positivity of the fractional descriptor nonlinear systems are established by the use of the WeierstrassKronecker decomposition of the pencil of linear part of nonlinear system. An illustrating example of positive fractional descriptor nonlinear system is presented in Sec. 3. Concluding remarks are given in Sec. 4. The following notation will be used: R – the set of real numbers, R – the set of n ×m real matrices, Z+ – the set of nonnegative integers, R + – the set of n × m matrices with nonnegative entries and R+ = R n×1 + , Mn – the set of n×n Metzler matrices (real matrices with nonnegative off-diagonal entries), In – the n× n identity matrix.