The fundamental problem in hierarchical supervisory control under partial observation is to find conditions preserving observability between the original (low-level) and the abstracted (high-level) plants. Two conditions for observable specifications were identified in the literature -- observation consistency (OC) and local observation consistency (LOC). However, the decidability of OC and LOC were left open. We show that both OC and LOC are decidable for regular systems. We further show that these conditions do not guarantee that supremal (normal or relatively observable) sublanguages computed on the low level and on the high level always coincide. To solve the issue, we suggest a new condition -- modified observation consistency -- and show that under this condition, the supremal normal sublanguages are preserved between the levels, while the supremal relatively observable high-level sublanguage is at least as good as the supremal relatively observable low-level sublanguage, i.e., the high-level solution may be even better than the low-level solution.

翻译：分层监督控制在部分观测下的基本问题是寻找保持原始（低层）与被抽象（高层）被控对象之间可观性的条件。文献中确定了两种适用于可观规范的条件——观测一致性（OC）与局部观测一致性（LOC）。然而，OC与LOC的可判定性问题仍未解决。我们证明了对正则系统而言，OC与LOC均是可判定的。进一步研究表明，这些条件无法保证在低层与高层计算出的上确界（正规或相对可观）子语言总是保持一致。为解决此问题，我们提出新条件——修正观测一致性——并证明在该条件下，上确界正规子语言在层级间得以保持，而高层上确界相对可观子语言至少不劣于低层上确界相对可观子语言（即高层解可能优于低层解）。