On positively invariant sets for discrete-time linear systems with disturbance: an application of maximal disturbance sets

This note concerns the investigation of maximal sets, with respect to which a set is positively invariant, and, given a set a, some properties of Δ-p.i. sets are analyzed. The approach uses one step of the backward recursion introduced in d´Alessandro and De Santis (1991), where feasibility and optimality conditions for linear dynamical input and/or state constrained systems were analyzed. In Section II, the author gives the main definitions and some basic results. In the subsequent Section III some properties of Δ-p.i. sets are investigated. The particular case of polyhedral sets is analyzed in Section IV. In Section V, the author shows an application to linear systems with parametric uncertainties in the dynamic matrix A. The author develops a technique to determine a set in the parameters space such that, for any value of the parameter in the set, a given set in the state space is positively invariant. In the last section a numerical example is developed