On the Properties of Reachability, Observability, Controllability, and Constructibility of Discrete-Time Positive Time-Invariant Linear Systems with Aperiodic Choice of the Sampling Instants

Abstract

This paper investigates the properties of reachability, observability, controllability, and constructibility of positive discrete-time linear time-invariant dynamic systems when the sampling instants are chosen aperiodically. Reachability and observability hold if and only if a relevant matrix defining each of those properties is monomial for the set of chosen sampling instants provided that the continuous-time system is positive. Controllability and constructibility hold globally only asymptotically under close conditions to the above ones guaranteeing reachability/observability provided that the matrix of dynamics of the continuous-time system, required to be a Metzler matrix for the system's positivity, is furthermore a stability matrix while they hold in finite time only for regions excluding the zero vector of the first orthant of the state space or output space, respectively. Some related properties can be deduced for continuous-time systems and for piecewise constant discrete-time ones from the above general framework.