Nonblocking Supervisory Control with Finite-Step Constraints in Agent-Task Systems

In this paper, we investigate the problem of bounded nonblocking supervisory control in Agent-Task systems, with a particular emphasis on quantitative control during task execution. We consider a specific scenario in which an agent is assigned a task within an environment, aiming to ensure the completion of the task without encountering deadlocks, while also limiting the number of actions within predefined bounds. The agent automaton model <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$A$</tex> and the task automaton model <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$T$</tex> are synchronized in parallel to construct the combined model <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$G$</tex>, where the marked states of <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$G$</tex> indicate the completion of the task. For the purpose of achieving finite-step supervisory control, we first employ a model compression algorithm to iteratively compress the model <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$G$</tex> from its initial state, yielding the model <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$K$</tex>-bounded <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$G$</tex>. Subsequently, we perform a reverse compression process on the model <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$K$</tex>-bounded <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$G$</tex>, starting from the marked states and other relevant states, resulting in the inverse model <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$K$</tex>-bounded <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$R G$</tex>. We then apply an estimation procedure to account for unobservable events, leading to the estimated models <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$\text{OBS}(K$</tex>-bounded <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$G)$</tex> and OBS (<tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$K$</tex>-bounded <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$R G$</tex>). By combining the products of these two estimated models, we derive a bidirectional estimation model, which forms the basis for seeking the existence of a finite-step supervisory controller <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$Sp$</tex>.