Reachability and observability of positive discrete-timelinear systems with integer positive and negative powers of the state Frobenius matrices

A dynamical system is called positive if its state variables and outputs take nonnegative values for all nonnegative inputs and nonnegative initial conditions. The positive linear and nonlinear continuous-time and discrete-time systems have been addressed in many papers and books [1–23]. Positive descriptor systems have been analyzed in [1–3, 6–11, 15, 17, 22, 23] and positive nonlinear systems in [18, 19]. The minimum energy control of positive systems has been investigated in [9–12, 14] and the stability of positive systems in [4, 14, 21, 23]. The positive systems consisting of n subsystems with different fractional orders have been introduced in [13, 16]. In this paper the reachability and observability of positive discrete-time linear systems with integer positive and negative powers of state monomial generalized Frobenius matrices will be addressed. The paper is organized as follows. In Section 2 the basic definitions and theorems concerning positive linear systems are recalled. The notion of monomial generalized Frobenius matrices has been introduced and the reachability of positive linear systems with these state matrices has been analyzed in Section 3.