Linear Constraints and Guarded Predicates as a Modeling Language for Discrete Time Hybrid Systems

Model based design is particularly appealing in software based control systems (e.g., embedded software) design, since in such a case system level specifications are much easier to define than the control software behavior itself. In turn, model based design of embedded systems requires modeling both continuous subsystems (typically, the plant) as well as discrete subsystems (the controller). This is typically done using hybrid systems. Mixed Integer Linear Programming (MILP) based abstraction techniques have been successfully applied to automatically synthesize correct-by-construction control software for discrete time linear hybrid systems, where plant dynamics is modeled as a linear predicate over state, input, and next state variables. Unfortunately, MILP solvers require such linear predicates to be conjunctions of linear constraints, which is not a natural way of modeling hybrid systems. In this paper we show that, under the hypothesis that each variable ranges over a bounded interval, any linear predicate built upon conjunction and disjunction of linear constraints can be automatically translated into an equivalent conjunctive predicate. Since variable bounds play a key role in this translation, our algorithm includes a procedure to compute all implicit variable bounds of the given linear predicate. Furthermore, we show that a particular form of linear predicates, namely guarded predicates, are a natural and powerful language to succinctly model discrete time linear hybrid systems dynamics. Finally, we experimentally show the feasibility of our approach on an important and challenging case study taken from the literature, namely the multi-input Buck DC-DC Converter. As an example, the guarded predicate that models (with 57 constraints) a 6-inputs Buck DC-DC Converter is translated in a conjunctive predicate (with 102 linear constraints) in about 40 minutes.