Positively Invariant Families of Sets for Interconnected and Time-Delay Discrete-Time Systems

Abstract

Methods for the construction of positively invariant (PI) sets, both for interconnected and time-delay systems, either suffer from computational intractability or come with considerable conservatism, with respect to their ability to provide a nontrivial PI set as well as their ability to recover or even approximate the maximal positively invariant (mPI) set. Therefore, we apply the notion of PI families of sets to interconnected and time-delay systems. We show that for such systems, this notion enjoys computational practicability and, at the same time, is nonconservative, both in terms of the type of sets it produces and its ability to approximate and recover the mPI set. Moreover, this technique also provides a tractable stability analysis tool. In this paper, for both interconnected and time-delay systems, the properties of PI families of sets are analyzed, their use is illustrated via several examples, and their construction via convex optimization algorithms is discussed.