Temporal logic control for stochastic linear systems using abstraction refinement of probabilistic games

Abstract

We consider the problem of computing the set of initial states of a dynamical system such that there exists a control strategy to ensure that the trajectories satisfy a temporal logic specification with probability 1 (almost-surely). We focus on discrete-time, stochastic linear dynamics and specifications given as formulas of the Generalized Reactivity(1) fragment of Linear Temporal Logic over linear predicates in the states of the system. We propose a solution based on iterative abstraction-refinement, and turn-based 2-player probabilistic games. While the theoretical guarantee of our algorithm after any finite number of iterations is only a partial solution, we show that if our algorithm terminates, then the result is the set of satisfying initial states. Moreover, for any (partial) solution our algorithm synthesizes witness control strategies to ensure almost-sure satisfaction of the temporal logic specification. We demonstrate our approach on an illustrative case study.

Temporal logic control for stochastic linear systems using abstraction refinement of probabilistic games

“…The almost-sure winning set Almost G ((E, F )) for Büchi implication acceptance condition can be solved in quadratic time [4,7]. In this work, we use more intuitive, cubic time algorithm described in detail in [25]. Moreover, in the states of the set Almost G ((E, F )), there always exist witness strategies, called almost-sure winning strategies, that are memoryless and indeed pure, i.e., not randomized, as defined above.…”

“…We analyze the game Table 1: Definitions of polytopic operators Post (posterior), Pre (predecessor), PreR (robust predecessor), PreP (precise predecessor), Attr (attractor) and AttrR (robust attractor), where X ⊆ X , U ⊆ U are polytopes, and {X j } j∈J is a set of polytopes in X . The algorithms to compute all the operators are listed in [25].…”

“…All polytopic operators that are used in this section are formally defined in Tab. 1 and their computation is described in detail in [25].…”

“…The set U J i can be computed using only polytopic computations as described in [25]. For a state Xi ∈ {Xi} ⊂ SN and action U J i ∈ ActN , we let…”

“…6. Due to space constraints, detailed descriptions of involved technical computations together with additional simulations can be found in [25].…”