Toward Minimal Data Rate Enforcing Regular Safety Properties: An Invariance Entropy Approach

The study of minimal data rate for control using some notions of entropy has been so far limited to classical control tasks such as set invariance, state-estimation, or stabilization. In this letter, for the first time, we present a study on sufficient data rates to enforce regular safety properties over uncertain systems with dynamics described by set valued maps. Every regular safety property has a set of bad prefixes which can be modelled by a deterministic finite automaton (DFA). The main idea is to construct a hybrid system by taking the product of the deterministic finite automata with the given system and studying the invariance feedback entropy (IFE) of controlled invariant sets of the hybrid system. If there exists a nonempty controlled invariant set for the hybrid system satisfying a certain property then there exists a coder-controller with a data rate not less than the IFE that can enforce the regular safety property over the original control system. We demonstrate the effectiveness of our results by designing a coder-controller enforcing a regular safety property over a linear control system.