##
**Formulae relating controllability, observability, and co-observability.**
*(English)*
Zbl 0911.93020

In the context of supervisory control of discrete-event systems, the notion of controllability can be used to establish necessary and sufficient conditions for the existence of a supervisor that achieves a desired controlled behavior of a given discrete-event plant under complete observation of events. If the events are not completely observed by the supervisor, but rather their observations are filtered by an observation mask, then an additional condition of observability is needed to guarantee its existence. In the more general case of decentralized supervision, more than one supervisor control the plant; the uncontrollable event set for the \(i\)-th supervisor is \(\Sigma_{ui}\), and its observation is filtered through the mask \(M_i\). In this setting, the existence condition for co-observability has been established for the case when the controlled behaviour is given as a prefix-closed language; it can also be generalized to the non-prefix-closed case.

In the paper under review, the authors first obtain a relationship between controllability, observability and co-observability. Given a plant \(G\) with event set \(\Sigma\), uncontrollable event sets \(\Sigma_{ui}\), and observation masks \(M_i\) for the \(i\)-th supervisor, they prove that the infimal, prefix-closed, \((L(G),\bigcap_i \Sigma_{ui})\)-controllable and \((L(G),\Sigma_{ui},M_i)\)-co-observable superlanguage of the specification language equals the intersection (taken over all \(i\)’s) of infimal prefix-closed \((L(G),\Sigma_{ui})\)-controllable and \((L(G),M_i)\)-observable superlanguages. This result can be used to obtain the individual supervisors for decentralized control, where the \(i\)-th supervisor generates the infimal prefix-closed, \((L(G),\Sigma_{ui})\)-controllable and \((L(G),M_i)\)-observable superlanguage of the given specification.

In the paper under review, the authors first obtain a relationship between controllability, observability and co-observability. Given a plant \(G\) with event set \(\Sigma\), uncontrollable event sets \(\Sigma_{ui}\), and observation masks \(M_i\) for the \(i\)-th supervisor, they prove that the infimal, prefix-closed, \((L(G),\bigcap_i \Sigma_{ui})\)-controllable and \((L(G),\Sigma_{ui},M_i)\)-co-observable superlanguage of the specification language equals the intersection (taken over all \(i\)’s) of infimal prefix-closed \((L(G),\Sigma_{ui})\)-controllable and \((L(G),M_i)\)-observable superlanguages. This result can be used to obtain the individual supervisors for decentralized control, where the \(i\)-th supervisor generates the infimal prefix-closed, \((L(G),\Sigma_{ui})\)-controllable and \((L(G),M_i)\)-observable superlanguage of the given specification.

Reviewer: Miroslav Lovrić (Hamilton)

### MSC:

93B07 | Observability |

93C30 | Control/observation systems governed by functional relations other than differential equations (such as hybrid and switching systems) |

93B05 | Controllability |

93A14 | Decentralized systems |

93A13 | Hierarchical systems |

### Keywords:

discrete-event system; supervisory control; controllability; observability; centralized control; decentralized control; co-observability
PDFBibTeX
XMLCite

\textit{R. Kumar} and \textit{M. A. Shayman}, Automatica 34, No. 2, 211--215 (1998; Zbl 0911.93020)

Full Text:
DOI

### References:

[1] | Cieslak, R.; Desclaux, C.; Fawaz, A.; Varaiya, P., Supervisory control of discrete event processes with partial observation, IEEE Trans. Automat. Control, 33, 3, 249-260 (1988) · Zbl 0639.93041 |

[2] | Hopcroft, J. E.; Ullman, J. D., (Introduction to Automata Theory, Languages and Computation (1979), Addison-Wesley: Addison-Wesley Reading, MA) · Zbl 0426.68001 |

[3] | Kumar, R., Formulas for observability of discrete event dynamical systems, (Proc. 1993 Conf. on Information Sciences and Systems (1993), Johns Hopkins University: Johns Hopkins University Baltimore, MD), 581-586 |

[4] | Kumar, R.; Garg, V. K., (Modeling and Control of Logical Discrete Event Systems (1995), Kluwer Academic Publishers: Kluwer Academic Publishers Boston, MA) · Zbl 0875.68980 |

[5] | Lafortune, S.; Chen, E., On the infimal closed and controllable superlanguage of a given language, IEEE Trans. Automat. Control, 35, 4, 398-404 (1990) · Zbl 0709.68028 |

[6] | Lin, F.; Wonham, W. M., On observability of discreteevent systems, Information Sci., 44, 3, 173-198 (1988) · Zbl 0644.93008 |

[7] | Ramadge, P. J.; Wonham, W. M., Supervisory control of a class of discrete event processes, SIAM J. Control Optim., 25, 1, 206-230 (1987) · Zbl 0618.93033 |

[8] | Rudie, K.; Willems, J. C., The computational complexity of decentralized-event control problems, IEEE Trans. Automat. Control, 40, 7, 1313-1319 (1995) · Zbl 0833.93007 |

[9] | Rudie, K.; Wonham, W. M., The infimal prefix closed and observable superlanguage of a given language, Systems Control Lett., 15, 5, 361-371 (1990) · Zbl 0746.93061 |

[10] | Rudie, K.; Wonham, W. M., Think globally, act locally: decentralized supervisory control, IEEE Trans. Automat. Control, 37, 11, 1692-1708 (1992) · Zbl 0778.93002 |

This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.