Reachability Of Fractional Continuous-Time Linear Systems Using The Caputo-Fabrizio Derivative

The Caputo-Fabrizio definition of the fractional derivative is applied to analysis of the positivity and reachability of continuous-time linear systems. Necessary and sufficient conditions for the reachability of standard and positive fractional continuous-time linear systems are established. INTRODUCTION A dynamical system is called positive if its trajectory starting from any nonnegative initial condition state remains forever in the positive orthant for all nonnegative inputs. An overview of state of the art in positive system theory is given in the monographs (Farina and Rinaldi 2000; Kaczorek 2001) and in the papers (Kaczorek 1997, 1998, 2011b, 2014a, 2014b, 2015b). Models having positive behavior can be found in engineering, economics, social sciences, biology and medicine, etc. The positive standard and descriptor systems and their stability have been analyzed in (Kaczorek 1997, 1998, 2001, 2011b, 2014b, 2015b). The positive linear systems with different fractional orders have been addressed in (Kaczorek 2011b, 2012) and the descriptor discrete-time linear systems in (Kaczorek 1998). Descriptor positive discrete-time and continuous-time nonlinear systems have been analyzed in (Kaczorek 2014a) and the positivity and linearization of nonlinear discrete-time systems by state-feedbacks in (Kaczorek 2014b). New stability tests of positive standard and fractional linear systems have been investigated in (Kaczorek 2011a). The stability and robust stabilization of discrete-time switched systems have been analyzed in (Zhang, Xie, Zhang and Wang 2014; Zhang, Han, Wu and Hung 2014). Minimum energy control of 2D systems in Hilbert spaces has been analyzed in (Klamka 1983). Controllability of dynamical systems has been investigated in (Kalman 1960; Klamka 1991, 1997, 1998). Recently a new definition of the fractional derivative without singular kernel has been proposed in (Caputo and Fabrizio 2015; Losada and Nieto 2015). In this paper the Caputo-Fabrizio definition of the fractional derivative will be applied to analysis of the reachability of the standard and positive linear systems. The paper is organized as follows. In section 2 necessary and sufficient conditions for the reachability of fractional standard continuous-time linear systems are established. Necessary and sufficient conditions for the positivity of the fractional systems and sufficient conditions for the reachability of the positive systems are proposed in section 3. Concluding remarks are given in section 4. The following notation will be used: R the set of real numbers, m n× R the set of m n × real matrices, m n× + R the set of m n × matrices with nonnegative entries and 1 × + + R = R n n , n M the set of n n× Metzler matrices, n I the n n× identity matrix. REACHABILITY OF STANDARD FRACTIONAL SYSTEMS The Caputo-Fabrizio definition of fractional derivative of order α of the function ) (t f for 1 0 < < α has the form (Caputo and Fabrizio 2015; Losada and Nieto 2015) . 0 , ) ( ) ( , ) ( ) ( 1 exp 1 1 ) (