Directed Control of Discrete Event Systems with $\mathbf{LTL} [\mathcal{F}]$ Specifications

In a formal approach to controller synthesis, temporal logic is used as a rigorous description of a control specification. In this paper, we describe control specifications by extended linear temporal logic formulas with propositional quality operators, which are evaluated with real numbers between 0 and 1. Such quantitatively extended logic enables us to consider how better the specification is satisfied. Our control objective is to design a controller such that the satisfaction value of the specification formula on the controlled system is larger than or equal to a given threshold. We propose a game-based controller synthesis algorithm. First, the specification formula and the threshold are translated into a generalized nondeterministic Büchi automaton, which is composed with the plant model. Next, we extend events of the product automaton in order to make its transitions deterministic. The extended product automaton is then transformed into a generalized Büchi game, from whose solution we obtain a desirable controller.