Reach and Robust Positively Invariant Sets Revisited

Abstract: This paper considers linear dynamical systems subject to additive and bounded disturbances, and studies properties of their forward reach, robust positively invariant (RPI) and the minimal RPI sets. The analysis is carried out for discrete-time (DT), continuous-time (CT), and sampled-data (SD) systems from a unified perspective. In the DT and CT cases, we review key existing results, while for the SD case novel results that reveal substantial structural differences to the DT and CT cases are presented. In particular, the main topological and computational properties associated with the DT and CT forward reach and RPI sets fail to be directly applicable to SD systems. In light of this, we introduce and develop topologically compatible notions for the SD forward reach, RPI and mRPI sets. We address and enhance computational aspects associated with these sets by complementing them with approximate, but guaranteed, and numerically more plausible notions.Keywords: Forward Reachability, Forward Reach Sets, Robust Positive Invariance, Robust Positively Invariant Sets, Minimal Robust Positively Invariant Sets, Bounded Disturbances, Discrete-Time, Continuous-Time, Sampled-data