Stable linear systems possess invariant sets which have hyperellipsoidal regions associated with their Lyapunov function. In real systems, however, state and control variables are often confined in bounded polyhedral regions (polytopes) so that a set of linear inequalities has to be satisfied. In this paper, necessary and sufficient conditions for the existence of positively invariant polytopes for both discrete-time and continuous-time linear systems are given in terms of their spectral properties.
Invariant polytopes of linear systems / Farina, Lorenzo; Benvenuti, Luca. - In: IMA JOURNAL OF MATHEMATICAL CONTROL AND INFORMATION. - ISSN 0265-0754. - 15:3(1998), pp. 233-240. [10.1093/imamci/15.3.233]