Discrete-time systems properties defined on state-space regions

In this paper various properties of discrete-time dynamic control systems trajectories with respect to state-space corner regions are considered. The known notion of state-space invariance serves as a basis for derivation of the whole family of dynamics behaviors for which both necessary and sufficient conditions are derived in a general nonlinear case as well as in the linear time-invariant case, shortly LTI-case. Specific examples are given for every case considered, and the proposed notions are analyzed with both theoretical and practical usefulness in mind. For the general nonlinear case a geometric approach is used, which provides a direct insight into the nature of trajectories' behavior. In the LTI-case a geometric approach is used as well, but it is also translated into the purely algebraic set of conditions allowing for a direct analysis of system matrices. The presented family of control systems expands on the classical state-space invariance (and positive systems) analysis, thus potentially opening new research venues in this branch of control theory and system dynamics in general.