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Finite abstractions for hybrid systems with stable continuous dynamics

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Abstract

This paper outlines an abstraction process in which a particular class of hybrid automata with continuous dynamics that have parameterized positive limit sets, are being abstracted into finite transition systems. The limit sets with their corresponding attraction regions define pre- and post-conditions for the continuous dynamics, and determine the transitions in the discrete abstraction. An observable (weak) bisimulation equivalence is established between the two models. The abstraction process described can find application in verification, as well as in planning and symbolic control.

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Notes

  1. The region of attraction of a particular component \(L^+_b\) of the positive limit set L  +  of the flow of vector field f in X can be defined as (Athanasopoulos 2003) \( \mathcal A^+_b( X) \triangleq \{ x\in X \mid \lim_{t\to\infty}\) \(\ensuremath{\mathrm{dist} \big(\Phi_t(x),L^+_b\big)} = 0\}\).

  2. This is in fact a semi-automaton, since there is no specification for initial and final states.

  3. Alternatively, we can define \(\mathfrak R\) as an equivalence relation in \(\mathcal{H}\) by identifying it with the map \(V_{{\kern-1.75pt}M}^{-1} \circ V_{{\kern-1.75pt}M} (\cdot, s(\cdot))\).

  4. These type of transitions in H correspond to an on-line re-parameterization of the controller already activated.

  5. The set S n denotes the surface of an n-dimensional sphere.

  6. We assume that the user is not right next to the printer and so \(q_u \notin \mathcal{W}(p_n)\), because then there would be no need to send the robot to bring the printout.

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Authors and Affiliations

  1. Mechanical Engineering, University of Delaware, Newark, DE, 19716, USA

    Herbert G. Tanner, Jie Fu & Chetan Rawal

  2. Electrical and Computer Engineering, University of New Mexico, Albuquerque, NM, 87131-0001, USA

    Jorge L. Piovesan & Chaouki T. Abdallah

Authors
  1. Herbert G. Tanner
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  2. Jie Fu
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  3. Chetan Rawal
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  4. Jorge L. Piovesan
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  5. Chaouki T. Abdallah
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Corresponding author

Correspondence to Herbert G. Tanner.

Additional information

The work of the first author is supported by the National Science Foundation under grant # 0447898. The work of the other authors was also supported by the National Science Foundation (CNS0626380) under the FIND initiative.

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Tanner, H.G., Fu, J., Rawal, C. et al. Finite abstractions for hybrid systems with stable continuous dynamics. Discrete Event Dyn Syst 22, 83–99 (2012). https://doi.org/10.1007/s10626-011-0119-6

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