Synthesis of Supervisory Control With Partial Observation on Normal State-Tree Structures

Abstract

Supervisory control theory (SCT) of a discrete-event system (DES) is well developed to find its maximally permissive supervisor. As an extension of SCT, a new framework, state-tree structures (STS), has been deployed to manage the state explosion problem of SCT. The supervisory control with <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">partial observation</i> is investigated in SCT, of which the state explosion issue remains to be explored. This paper dwells upon an approach to the synthesis of supervisory control with partial observation in the <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">normal</i> STS framework, requiring that the conjunction of any two projected subpredicates should be false. First, following the definition of the observability based on STS, it is proven that there does not exist the supremal observable subpredicate. Second, we construct a subclass of observable subpredicates of a given predicate (obtained as a specification) on the basis of <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">normal</i> STS. Third, a largest element is proven to exist in the subclass, and thus a <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">weakly controllable, coreachable, and observable</i> subpredicate is computed to solve the supervisory control problem of the <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">normal</i> STS by an iterative algorithm. An illustrative example with state size over 10 <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">9</sup> is given to show that the proposed algorithm based on STS is superior to the approach based on SCT, which leads to the program crashes in SCT. <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">Note to Practitioners</i> —DES is a collection of many contemporary computer-integrated industrial and social service systems. For the sake of technology or cost considerations, the occurrences of some events are not fully observed, rendering the supervisory control algorithmically complicated and exacerbating the originally notorious state explosion problem. This paper aims to ameliorate the state explosion problem using a novel formalism, namely, state-tree structures, that can be applied to real-world systems thanks to the computational efficiency that is demonstrated by an industrial system. Compared with the traditional SCT of DES, the significance of the proposed methodology will intrigue academic researchers and industrial practitioners.