## Compositional abstractions of hybrid control systems (2001)

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Venue: | In Proceedings of the 40th IEEE Conference on Decision and Control |

Citations: | 7 - 1 self |

### BibTeX

@INPROCEEDINGS{Tabuada01compositionalabstractions,

author = {Paulo Tabuada and George J. Pappas and Pedro Lima},

title = {Compositional abstractions of hybrid control systems},

booktitle = {In Proceedings of the 40th IEEE Conference on Decision and Control},

year = {2001},

pages = {203--238}

}

### OpenURL

### Abstract

Abstract. Abstraction is a natural way to hierarchically decompose the analysis and design of hybrid systems. Given a hybrid control system and some desired properties, one extracts an abstracted system while preserving the properties of interest. Abstractions of purely discrete systems is a mature area, whereas abstractions of continuous systems is a recent activity. In this paper we present a framework for abstraction that applies to discrete, continuous, and hybrid systems. We introduce a composition operator that allows to build complex hybrid systems from simpler ones and show compatibility between abstractions and this compositional operator. Besides unifying the existing methodologies we also propose constructions to obtain abstractions of hybrid control systems.

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Citation Context ...operty that needed to be preserved in this abstraction was the detectability of the discrete switching conditions. Related but orthogonal work considers purely discrete abstractions of hybrid systems =-=[2, 10, 12, 36-=-]. The similarities between notions of abstraction for discrete, continuous, and hybrid systems immediately raise the question of a more unied theory of abstraction. In this paper, we begin addressing... |

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Citation Context ... the subsystems and is guaranteed that the interconnection of those abstractions is an abstraction of the original large-scale system. Our composition operator diers from the approaches described in [=-=23, 13-=-], in that we model synchronization by restricting the behavior of the systems without a priori dening inputs and outputs. We specialize the developed results for hybrid systems, presenting a construc... |

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Citation Context ...operty that needed to be preserved in this abstraction was the detectability of the discrete switching conditions. Related but orthogonal work considers purely discrete abstractions of hybrid systems =-=[2, 10, 12, 36-=-]. The similarities between notions of abstraction for discrete, continuous, and hybrid systems immediately raise the question of a more unied theory of abstraction. In this paper, we begin addressing... |

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Citation Context ... in various temporal logics [25]. For purely continuous systems, the notions of simulation, and bisimulation had no counterparts. Recently, similar notions were introduced in the collection of papers =-=[32, 31, 33, 30, 42, 43]-=-. This research resulted in automatic constructions of abstractions for linear control systems [31], while characterizing abstracting maps that preserve properties of interest such as controllability.... |

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Citation Context ...id to the modeling, analysis and control of large-scale, multi-agent, complex, hybrid systems. The impulse from the applications side has been tremendous and includes among others: automotive engines =-=[7, 6]-=-, air-trac management [45], chemical batch plants [29], manufacturing systems [14], TCP congestion control [16] and biomolecular networks [1] among many others. One approach to deal with the inherent ... |

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Citation Context ...operty that needed to be preserved in this abstraction was the detectability of the discrete switching conditions. Related but orthogonal work considers purely discrete abstractions of hybrid systems =-=[2, 10, 12, 36-=-]. The similarities between notions of abstraction for discrete, continuous, and hybrid systems immediately raise the question of a more unied theory of abstraction. In this paper, we begin addressing... |

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Citation Context ... each layer of abstraction. The notion of abstraction is quite mature in theoretical computer science, and, in particular, in the areas of concurrency theory [27] [46], and computer aided verication [=-=26]-=-. This has resulted in formal and very meaningful notions of abstraction which are used to tackle exponential explosion of purely discrete systems. Given a discrete system, an abstraction is simply a ... |

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Citation Context ...he applications side has been tremendous and includes among others: automotive engines [7, 6], air-trac management [45], chemical batch plants [29], manufacturing systems [14], TCP congestion control =-=[16-=-] and biomolecular networks [1] among many others. One approach to deal with the inherent complexity of hybrid control systems is to organize them in a hierarchical framework where dierent layers of a... |

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Citation Context ...make this fact precise. Instead of trying to dene trajectories of abstract control systems (which would be as dicult as dening trajectories of hybrid control systems, see the dierent approaches in [19=-=, 28, -=-35]) we will restrict our attention to the orbits of abstract control systems. Denition 2.6. Let X be an abstract control system over X with monoid MX . The set O x is an orbit of X through the point ... |

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Citation Context ...s and model matching problems is claried. Furthermore, it is shown how recasting supervisory control problems for discrete event systems as a bisimulation problem leads to more ecient algorithms. In [=-=24-=-], bisimulation is fundamentally used at the level of the problem formulation by requiring the closed loop system to simulate or bisimulate the specication. Based on these results, in [41], we took th... |

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Citation Context ...portant issue. We start bysrst considering a more unied and abstract model for control systems. Our abstract control systems model is inspired by categorical denitions of systems that are as old as [4=-=, 37]-=- and as recent as [39]. Although categorically inspired, the paper is accessible to readers that are not familiar with category theory, except for some proofs that rely on simple category theory notio... |

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Citation Context ... multi-agent, complex, hybrid systems. The impulse from the applications side has been tremendous and includes among others: automotive engines [7, 6], air-trac management [45], chemical batch plants =-=[29]-=-, manufacturing systems [14], TCP congestion control [16] and biomolecular networks [1] among many others. One approach to deal with the inherent complexity of hybrid control systems is to organize th... |

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Citation Context ...d systems. The impulse from the applications side has been tremendous and includes among others: automotive engines [7, 6], air-trac management [45], chemical batch plants [29], manufacturing systems =-=[14]-=-, TCP congestion control [16] and biomolecular networks [1] among many others. One approach to deal with the inherent complexity of hybrid control systems is to organize them in a hierarchical framewo... |

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Citation Context ...r dening morphisms as relations, in which case a bisimulation would simply be a symmetric simulation relation, that is, a relation R such that both R and R 1 are simulations. We direct the reader to [44] for an account of such an approach and proceed with the denition: Denition 2.5 (Bisimulations of Abstract Control Systems). Let X and Y be abstract control systems over X and Y with monoids MX a... |

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Citation Context ... in various temporal logics [25]. For purely continuous systems, the notions of simulation, and bisimulation had no counterparts. Recently, similar notions were introduced in the collection of papers =-=[32, 31, 33, 30, 42, 43]-=-. This research resulted in automatic constructions of abstractions for linear control systems [31], while characterizing abstracting maps that preserve properties of interest such as controllability.... |

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Citation Context ...orithms. In [24], bisimulation is fundamentally used at the level of the problem formulation by requiring the closed loop system to simulate or bisimulate the specication. Based on these results, in [=-=41]-=-, we took thesrst steps towards constructing abstractions of hybrid systems while preserving timed languages. Even though only the continuous part of the system was abstracted, the important property ... |

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Citation Context ... in various temporal logics [25]. For purely continuous systems, the notions of simulation, and bisimulation had no counterparts. Recently, similar notions were introduced in the collection of papers =-=[32, 31, 33, 30, 42, 43]-=-. This research resulted in automatic constructions of abstractions for linear control systems [31], while characterizing abstracting maps that preserve properties of interest such as controllability.... |

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Citation Context ...make this fact precise. Instead of trying to dene trajectories of abstract control systems (which would be as dicult as dening trajectories of hybrid control systems, see the dierent approaches in [19=-=, 28, -=-35]) we will restrict our attention to the orbits of abstract control systems. Denition 2.6. Let X be an abstract control system over X with monoid MX . The set O x is an orbit of X through the point ... |

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Citation Context ...make this fact precise. Instead of trying to dene trajectories of abstract control systems (which would be as dicult as dening trajectories of hybrid control systems, see the dierent approaches in [19=-=, 28, -=-35]) we will restrict our attention to the orbits of abstract control systems. Denition 2.6. Let X be an abstract control system over X with monoid MX . The set O x is an orbit of X through the point ... |

2 |
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Citation Context ...otion of abstraction is quite mature in theoretical computer science, and, in particular, in the areas of concurrency theory (Milner, 1989; Winskel and Nielsen, 1994), and computer aided veri®cation (=-=Manna and Pnueli, 1995-=-). This has resulted in formal and very meaningful notions of abstraction which are used to tackle exponential explosion of purely discrete systems. Given a discrete system, an abstraction is simply a... |