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A hierarchical consistency framework for real-time supervisory control

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Abstract

The control framework of hierarchical consistency of timed discrete-event systems (TDES’s) is investigated in a standard two-level hierarchy. Real-time concepts and the associated theoretical results supporting consistent TDES hierarchies are developed. Where the given low-level system model of the hierarchy possesses time fidelity, a consistency version that assures time fidelity of the high-level system model is also developed. Importantly, this version furnishes a sound real-time high-level specification design foundation for hierarchical control. An example illustrates the new time-fidelity control foundation. Given that in general, a given two-level TDES hierarchy is not hierarchically consistent between the levels, the structural existence and synthesis of the sufficiency structure for hierarchical consistency is investigated. Both the timed versions of hierarchical consistency - without and with output-time fidelity guarantee - are successively treated. The abstraction or output-system refinement procedures for the version without output-time fidelity guarantee are first developed for a class of TDES hierarchies under mild output-system design restrictions. The abstraction methods for the version with output-time fidelity are then developed for a subclass ‘linearly’ structured under further output-system design restrictions. A detailed example explains and illustrates the use of an overarching method developed.

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Notes

  1. A timeout event can be used in timed models as explained in Cassandras and Lafortune (2008b). It is a marker that specifies the maximal duration that a system can stay in a system state, within which an activity event it is defined for at the state is expected to occur. A timeout occurrence indicates that the activity event has failed to occur within the specified duration. Because the timeout occurrence may model disablement of the activity and other events upon entering a new state, it is different from time ticks that simply model time progression.

  2. Note that, since TDES G is a (finite-state) TTG, the language of interest for control synthesis, K ∩ Lm(G), is a regular language and can thus be modeled by a TTG. In the algorithmic computation (Wonham 2016) of Supcon(G, K), K can be practically expressed as a regular language by a specification TTG.

  3. Although the same 5-tuple notation is used as in Section 2.4, it should be clear in the context that the structure of Glo is in general not the same as that of a given TDES G.

  4. In this example, the high-level tickthis eligible at every state in both the system abstractions, hence their satisfying Property 4.

  5. Examining Glo for the curious reader, this high-level tick models the minimum time required by the underlying system for auto-adjusting the setting of the camera shutter speed and aperture to anticipate a clear photo finish.

  6. In principle, in place of reachability trees, more efficient and compact representations are available for practical implementation that can be stored (Wonham 2016; Zhong and Wonham 1989).

  7. An event denoted by symbol \(\overset {\mathrm {x}}{\gamma }\) is simply called a force-don’t-care event, and is either \(\overline {\gamma }\) or γ.

  8. By original, we refer to the low-level TDES prior to undergoing Procedure OCC-SR.

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Authors and Affiliations

  1. School of Computer Science and Engineering, Nanyang Technological University, Singapore, 639798, Republic of Singapore

    Quang Ha Ngo

  2. Robot Intelligence Technology Laboratory, School of Electrical Engineering, KAIST, Daejeon, 305-701, South Korea

    Kiam Tian Seow

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  1. Quang Ha Ngo
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  2. Kiam Tian Seow
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Correspondence to Quang Ha Ngo.

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Ngo, Q.H., Seow, K.T. A hierarchical consistency framework for real-time supervisory control. Discrete Event Dyn Syst 28, 375–426 (2018). https://doi.org/10.1007/s10626-018-0267-z

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